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Primordial galactic magnetic fields from the QCD phase transition Forbes, Michael McNeil 2001

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Primordial Galactic Magnetic Fields from the QCD Phase Transition by Michael McNeil Forbes B.Sc, The University of British Columbia, 1999 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Physics and Astronomy) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 2001 © Michael McNeil Forbes, 2001 In presenting this thesis in partial fulfilment of the requirements for an ad-vanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics and Astronomy The University Of British Columbia Vancouver, Canada Abstract ii Abstract In this thesis I describe in detail a mechanism which we proposed to gener-ate large-scale primordial magnetic fields with correlation lengths of 100 kpc today. Domain walls with QCD scale internal structure form, coalesce and attain Hubble scale correlations. These domain walls subsequently align nucleon spins. Due to strong CP violation, nucleons in these walls have anomalous electric and magnetic dipole moments and the walls are ferro-magnetic. This induces electromagnetic fields with Hubble size correlations. The same CP violation also induces a maximal helicity (Chern-Simons) cor-related through the Hubble volume which may support an "inverse cas-cade" allowing the initial correlations to grow to 100 kpc today. Details of the physics and estimation methods are presented as well as necessary background and a discussion of the numerical methods used to obtain the classical domain wall solutions. In particular, a nice method for estimating properties of flat domain walls is presented. In addition, possible flaws with the argument are examined and other applications of QCD domain walls to astrophysical problems are discussed. Contents iii Contents Abstract i i List of Figures vi Forward v i i i Acknowledgements ix Part I Thesis 1 1. Introduction 2 2. Overview of the Mechanism 4 3. Evolution of Magnetic Fields 8 3.1 Helicity Conservation 8 3.2 Scaling Laws 9 3.3 Phase Transitions 10 3.4 Applicability of Scaling Laws 11 4. Generation of Magnetic Fields by Domain Walls 12 4.1 Hubble Size Correlations 12 4.1.1 Q C D Phase Transition -. . . 13 4.2 Essential Domain Wall Properties 15 4.3 Strong C P Violations in Domain Walls 17 5. QCD Domain Walls 20 5.1 Domain Wall Solutions 22 5.2 Domain Wall Decay 25 6. Alignment of Spins in the Domain Wall 28 6.1 Fermion Degeneracy in the Domain Wall 33 Contents iv 7. Generation of Electromagnetic Field 34 7.1 Helicity 35 8. Conclusion 37 8.1 Summary 37 8.2 Speculations and Future Directions 39 Bibliography. 42 Part II Appendices 48 A. Observations 49 A . l Techniques 49 A.1.1 Zeeman Splitting 49 A.1.2 Synchrotron Emissions 50 A.1.3 Faraday Rotation Measures (RMs) 50 A.2 Results 51 A.2.1 Local Fields 51 A.2.2 Clusters 52 A. 2.3 Intergalactic Media 52 A. 3 Summary 53 B. QCD Domain Wall Structure 54 B. l Introduction 54 B.2 Effective Lagrangian and 9 dependence in QCD 55 B. 2.1 The (7(1) Problem 56 B.2.2 Chiral Effective Lagrangian 57 B.3 Topological Stability and Instabilities 64 B.3.1 Higher Degrees of Freedom 66 B.4 Domain Walls 69 B.4.1 Domain Wall Equations 69 B.4.2 QCD Domain Walls 71 B.4.3 Axion Dominated Domain Walls 74 B.4.4 Axion-Pion Domain Wall 77 B.4.5 Axion-Eta' Domain Wall 77 B. 5 Decay of the QCD domain walls 79 C. Numerical Solution of Domain Wall 90 C. l Introduction 90 C.2 COLNEW 91 C.3 Purpose 91 Contents v C.4 Usage 92 C.5 Change of Variables 99 C.6 QCD Domain Walls 100 C.6.1 Example System 101 C.7 Original Documentation 102 C.8 Code Listings 102 C.8.1 Example Problem 102 C.8.2 QCD Domain Wall Problem 103 C.8.3 System Files 103 C.8.4 Automatic Differentiation 104 C.8.5 COLNEW 105 List of Figures vi List of Figures 5.1 Basic form of the QCD domain wall (soliton). The analytic approximation 5.10 is plotted as a dotted line to show the good agreement. We have taken Nc = 3 here. Notice that the wall thickness is set by the parameter \i 24 B . l Three examples of transitions that are topologically different. Paths A and B wind around the hold once (n = 1) and twice (n = 2) respectively whereas path C does not wind (n = 0). Path C can be shrunk to a point whereas the others cannot. Each path is said to belong to a different homotopy class. Only paths A and B are affected by the topology of the space. Path C might imagine that it is living in a space with no hole. 65 B.2 Here we show the same picture as in Figure B . l except that we show the third dimension. Here we can see that all the paths are now homotopically equivalent. We can deform the paths by "lifting" them over the obstacle so that we can un-wind them. If the paths were strings with some weight, then it would require some energy to "lift" the strings over the obsta-cle. If this energy was not available, then we would say that, classically, the configurations that wind around the "peg" are stable. Quantum mechanically, however, the strings could still tunnel through the "peg", and so the configurations are unstable quantum mechanically. The probability that one string could tunnel into another configuration would depend on the height of the "peg" 67 B.3 Basic form of the QCD domain walls. Notice that the scale for the pion transition is larger than for the eta' transition. Notice that the width of the rf wall is set by the scale m^. . 75 B.4 Same form as in Figure B.3 except in terms of the variables <t>u and <j)d 76 List of Figures vii B.5 Basic form of the an domain wall. Notice that all the fields have a structure on the scale of m~l and that the isotopical singlet fields plays a very small role 78 B.6 Basic form of the avi domain walls with a closeup where the axion field a « ir/2. Notice that the large scale structure is similar to that of the an wall, but that there is also a small scale structure on the scale of m^. Near the center of the wall, the pion regains its mass and undergoes a transition on the scale of 79 B.7 Zoom in on Figure B.6 showing detailed structure of n' core. Near the center of the wall, the pion regains its mass and undergoes a transition on the scale of m% 80 B.8 Profile of the "Mexican-hat" potential (B.50). The slice is made along the axis through (j>g = n to the left and (f>s = 0 to the right. The trough of the potential lowers from the cusp at p = 1, <j> = TT where V = —ECOS(TT/NC) down to the vacuum state VM-M — —E. The hump where V = 0 is at the origin is where h = pexp(i(ps/NC) = 0 and hence the singlet field 4>s can have any value at this point. It is by passing across this point that a QCD domain wall can tunnel and a hole can form 86 Forward viii Forward The focus of this thesis is the discussion of a mechanism for forming large scale galactic magnetic fields from domain walls at the QCD phase transition in the early universe. As such, the purpose of the main body is to present this mechanism in as clear a manner as possible. The mechanism, however, includes concepts from several branches of physics. In order to elucidate the main presentation, several of the details, especially concerning the domain wall structure and QCD background, have been placed in appendices. The result is that the main body of this thesis serves as a presentation and tour of the mechanism for generating magnetic fields, referring to the (rather lengthy) appendices for technical details. By organising the thesis in this manner I hope to make it accessible to audiences who do not have the specific expertise, while providing the details for those readers who do. In particular, this thesis bridges both astrophysics and particle physics and it is my hope that the main body is accessible to members of both communities. The main body was written almost entirely by me and parallels the paper [1], though the ideas were developed jointly with my supervisor Ariel Zhitnitsky: the use of "we" generally refers to the two of us. Appendix B parallels the paper [2] which was written by both of us: I have removed or modified most of the content that was written by my supervisor, however, the arguments and estimates of the domain wall lifetime must be credited to him as must the detailed justification of the chiral effective Lagrangian. The rest of the thesis is exclusively written by me. The main ideas presented in this thesis were first communicated in the letter [3], and later in the papers [1, 4]. Details about QCD domain walls were first presented in.[2]. This thesis should be regarded as a collection and elucidation of these papers in a cohesive unit, with added explanations and background so as to be accessible to advanced undergraduate and graduate students working in the field. A cknowledgem ents ix Acknowledgements I am greatly indebted to my supervisor Dr. Ariel Zhitnitsky for the idea and much of the drive behind the ideas presented in this thesis. Without his support and contributions none of this would have been possible. I would like to thank him for exposing me to high energy physics and for showing me the possibilities in bridging high energy physics together with astrophysics and cosmology. I would also like to thank Robert H . Brandenberger and M . B . Voloshin for many useful discussions. My supervisor would like to thank: M . Sha-poshnikov and I. Tkachev for discussions which motivated this study; Larry McLerran and D. Son for discussions on Silk damping; and A . Vainshtein for discussions on the magnetic properties of domain walls. I also wish to thank F. Wilczek, K . Rajagopal, for useful discussions about high density matter and astrophysics, and A . Vilenkin and T. Kibble for discussions about phase transitions in the early universe. This work was supported by the N S E R C of Canada. Part I Thesis 1. Introduction 2 1. Introduction M a n y different observations suggest that there exist substantial (microgauss) magnetic fields i n the universe today [5, 6], however, there has yet to emerge a theory which adequately explains the origins of these fields. Most of the data on large scale astrophysical magnetic fields comes from the observa-t ion of synchrotron radiat ion emitted i n galaxies. T h i s radiat ion is plane polarized, and as it passes through magnetic fields, the plane of polar izat ion rotates due to the Faraday effect: an effect which depends on the frequency of the radiat ion, electron density, and the strength and orientation of the magnetic fields. B y comparing several sources or radia t ion w i t h different fre-quencies, one can extrapolate to determine the or iginal plane of polar izat ion and then estimate the magnetic field strengths. 1 W h a t is s t r ik ing, is not just the existence of magnetic fields, but that there appear to be microgauss fields which have correlations and field re-versals as large as 500 kpc in clusters. To put this into perspective, the luminous cores of galaxies have typical scales of up to 10 kpc while it is estimated that the galactic dark matter halos extend to 50 kpc. The current models for producing these fields involve two ma in processes: 1) dynamica l amplif icat ion and/or generation of magnetic fields by galactic processes (galactic mechanisms) and 2) p r imord ia l mechanisms which take place prior to gravitat ional structure formation. The galactic mechanisms are pr imar i ly based on gravitat ional dynamos, al though there are sugges-tions that supernovae or other stellar phenomena may play a role. W h i l e it is l ikely that galactic dynamos amplify fields, it seems difficult to account for the large scale correlations of the magnetic fields when only galactic mechanisms are considered. It is also not certain that galactic mechanisms can generate magnetic fields: instead they may serve only as an amplifier, requiring seed fields to be present for the dynamo to work. The inadequacies of the galactic mechanisms have lead to many proposals that the magnetic fields may have a pr imordia l origin. In this case, some process i n the early universe (typically at a cosmic phase transi t ion or dur ing 1 See Appendix A for details about the observational methods and results. 1. Introduction 3 inflation) is thought to generate magnetic turbulence. This turbulence then sustains itself as the universe expands and the fields that we observe are the remnants of this turbulence. Most primordial sources, however, also produce fields which end up with very small correlations today or which are very weak. Most likely, a complete picture of the history of astrophysical magnetic fields requires some primordial inputs as well as an amplification mechanism provided by gravitational dynamics. In this thesis, however, we discuss an-other primordial mechanism which may be able to produce fields of 100 kpc correlations today. In combination with dynamic amplification mechanisms, this mechanism might provide a solid foundation for the theory of large-scale astrophysical magnetic fields. At this point we would like to refer the reader to the several reviews and sample papers in this field. The primary discussions of observations, which contain reviews of the theory, are presented in [5, 6]. Good current reviews are given in [7-9]. Many different types of primordial mechanisms are dis-cussed, for example: Inflationary mechanisms [10], cosmic strings [11, 12], charge asymmetries [13, 14], and phase transitions [15, 16]. The evolution of primordial magnetic fields is discussed in [9, 17, 18]. In particular, the inverse cascades discussed will be important for our mechanism. 2. Overview of the Mechanism 4 2. Overview of the Mechanism The mechanism that we propose has the following core components: 1. Sometime near the Q C D phase transition, T Q C D ~ 1 GeV, domain walls form which can interact with Q C D scale physics. 2. These domain walls rapidly coalesce until there remains, on average, one domain wall per Hubble volume with Hubble scale correlations. 3. Baryons interact with the domain walls and align their spins along the walls. The Hubble scale correlation of the domain walls thus induces a Hubble scale correlation in the spin density. 4. The anomalous magnetic and induced electric dipole moments of the baryons generate helical electromagnetic fields also correlated on the Hubble scale. 5. The domain walls move rapidly and vibrate, effectively filling the Hub-ble volume with helical magnetic turbulence with a Hubble scale cor-relation. 6. The domain walls decay and the electric fields are screened leaving magnetic turbulence with Hubble scale correlations. 7. As the universe expands, an "inverse cascade" mechanism transfers energy from small to large scale modes, effectively increasing the re-sulting correlation lengths but diluting the field strengths. 8. Galactic dynamos amplify the fields in galaxies, but the fields should also persist in the extra-galactic media. The idea that domain walls might generate magnetic fields is not original. It was suggested that standard axion domain walls could be ferromagnetic in [19] , however mechanism discussed their seems to be flawed: The scale of the standard axion walls is of the order m~l which is at least some twelve orders of magnitude larger than the Q C D scale A Q £ D (ma is the axion 2. Overview of the Mechanism 5 mass.) It is hard to see how these walls can efficiently affect QCD physics at the temperatures that were present in the early universe where thermal fluctuations will destroy all coherence.1 Another problem with proposals including standard axion domain walls is that these walls must decay to prevent cosmological problems [22]. There are still questions about how the standard so-called N ^ 1 axion domain walls can decay. The N — 1 axion model, which has a known decay mode [23], may be relevant if it an be endowed with a QCD scale substructure. This thesis outlines the properties that domain walls must have to gen-erate sufficient magnetic seed fields via the proposed mechanism. The exact source could be one of several types of walls, including modified axion do-main walls, or something entirely different. To be concrete, we present our model in terms of a recently conjectured quasi-stable QCD domain wall [2] which may exist independently of axion physics or which can add additional QCD scale structure to the standard axion domain walls. These domain walls are characterized by a transition in the singlet rj field which has a size and energy scale set by A Q C D - Hence, QCD domain walls can directly couple to QCD physics. In addition, the singlet field transition at the center of the wall induces an effective non-zero CP violating 6 background which in turn will induce an electric dipole moment and alter the magnetic dipole moment in the fermions [24]. Thus, both the electric and magnetic dipole moments of all the particles are on the same order. In the presence of these anomalous dipole moments, the potential cancellations discussed in [25, 26] are no longer a problem and the walls are ferromagnetic. Another crucial aspect of our mechanism is some sort of "inverse cas-cade" mechanism which governs the evolution of the magnetic fields after they are formed. This mechanism was suggested by Cornwall [27], discussed by Son [28] and confirmed by Field and Carroll [29]. It is based on the idea that magnetic helicity (Abelian Chern-Simons number) H = J A • Bd 3 £ is approximately conserved in the universe where temperatures are higher than To ~ 100 eV. This conservation of helicity causes energy to cascade up the turbulent modes, increasing the energy in large scale modes, and in-creasing the effective correlation length of the turbulence. The importance of helicity was originally demonstrated by Pouquet and collaborators [30]. Without this helical inverse cascade, there is no known way to generate large correlations fields today from sub-Hubble scale fields formed in early phase 1 For some other discussions about the magnetic properties of the domain walls, see [20, 21] and reference therein. We should note, however, that in all these discussions, the most difficult problem of generating large scale correlations has not been addressed. 2. Overview of the Mechanism 6 transitions, and one must consider super-Hubble scale fields generated by inflationary mechanisms. It turns out, however, that if the inverse cascade functions, then Hubble scale correlations at the Q C D phase transition (the last major phase transition) result in ~ 1 0 0 kpc correlations today. Thus it is natural to consider Q C D physics as the source of primordial fields (earlier physics can only produce even smaller correlations). We must point out, however, that the mechanism of this inverse cascade is not well understood. Indeed, most of the results are based on simple scaling arguments with somewhat restrictive assumptions. We shall discuss this point later. In this thesis (and in our previous papers [1, 3 , 4 ] ) we have assumed what appears to be rather maximal efficiency for the inverse cas-cade. This mechanism relies on such a cascade to produce the correlations. More wil l be said about this later. Another little understood aspect of this mechanism concerns the dynam-ics of the domain walls and the interactions of the domain walls, nucleons and electromagnetic fields. As we shall show, all of these components inter-act on the same scale A Q C D and hence there are complicated back-reactions and nonlinear dynamics. Presently, we do not have the tools to fully analyze these features, but we show here in detail, quantitative calculations which we believe are good estimates of the scale of the effects. If there is an inverse cascade mechanism which functions at the maximal efficiency estimated in [ 2 8 , 2 9 ] , then the mechanism described here may produces magnetic fields today with correlations up to I ~ 1 0 0 kpc and with strengths of i Q-9Q Brms ~ 77 , 1 ~ 1 0 0 kpc ( 2 . 1 ) s A Q C D where the parameter 1 < A Q C D ^ 1 0 1 9 depends on the dynamics of of the domain walls as discussed in Chapter ( 4 ) . If the correlation £ turns out to be small, then this mechanism might generate detectable extra-galactic fields, otherwise we still require a galactic dynamo to amplify the fields. In any case, however, if the inverse cascade is sufficient to maintain the large scale correlations, then it seems that, even if £ is large, the resulting fields may be strong enough to seed the galactic dynamos [ 3 1 ] . We shall begin by discussing the inverse cascade mechanism in Chap-ter ( 3 ) and then give estimates of the field strengths in an idealized case of static, flat walls. Finally, We shall discuss the dynamics of the domain walls and describe the whole process, justifying the mechanism. The impor-tant details about the domain walls will be summarised in this body: A n 2. Overview of the Mechanism 7 extended discussion about the relevant physics and numerical methods is presented later in the appendices. 3. Evolution of Magnetic Fields 8 3. Evolut ion of Magnetic Fields Given a stable magnetic field configuration in the universe, one might naively expect the size of the correlations of the field to expand with space as gov-erned by the universe's scale parameter I oc R(T) and the field strength to be correspondingly diluted B oc R(T)~2. It was discovered by Pouquet and collaborators [30], however, that, if the magnetic fields have a non-zero he-licity (Abelian Chern-Simons number) H = f A • Bd 3 a; , then the fields will scale differently. Cornwall [27] suggested that helical fields might undergo an inverse cascade. The magnetohydrodynamic (MHD) equations were studied by Son [28] who derived the scaling relations (3.1) presented below. These have subsequently been confirmed by Field and Carroll [29]. We assume throughout this section, that helicity can be generated which supports the inverse cascade. 3.1 Helicity Conservation The basic idea behind the inverse cascade is that the magnetic helicity H is an approximately conserved quantity in the early universe. It is also known that the small scale turbulent modes decay more rapidly than the higher scale modes. In order to conserve the helicity, as the small scale modes decay, the helicity must be transfered to the larger modes. Wi th this transfer of helicity is a corresponding transfer of energy. This is the source of the inverse cascade. To understand the origins of the conservation of helicity, note that it is a topological quantity that describes the Gaussian linking number of the vector potential lines of flux (see for example [32]). In a perfectly conducting medium, these lines of flux cannot cross, and there is no way to unlink the flux lines: helicity is thus perfectly conserved. Even when the conductivity is finite, the helicity is also well conserved. There is a direct analogy between magnetohydrodynamics and fluid me-chanics. In fluid mechanics, the equivalent of helicity is vorticity ( = f v • (V x v)d 3a; which is the Gaussian linking number of the fluid flow lines. If the fluid has no viscosity, then ( is perfectly conserved because 3. Evolution of Magnetic Fields 9 the fluid flow lines are not allowed to cross.1 Even viscous fluids, however, approximately conserve vorticity. This is why, for example, smoke rings and tornados are quite stable. These relate the initial field strength Bvms(Ti) with initial correlation l(Ti) to the present fields today (T n o w % 2 x 10 - 4 eV) BTma(Tnow) with correla-tion Z(T n o w). During the period when the universe supports turbulence (as indicated by a large Reynolds number Re), the inverse cascade mechanism functions and we have the scalings B oc T 7 / 3 and I oc T ~ 5 / 3 as indicated by the second factors in (3.1). In the early universe, Re is very large and the turbulence is well supported. As the universe cools, eventually, for tempera-tures below some To, the turbulence is no longer well supported. Exactly at what effective temperature To the turbulence ceases is not clear: Son points out that, at To w 100 eV, the Reynolds number drops to unity and thus turbulence is not well supported because of the viscosity of the plasma [28]. We take this as a conservative estimate. Field and Carroll argue that the turbulence is force-free and hence unaffected by the viscosity. Thus, they take To ~ 1 eV, the epoch when the matter and radiation energy densities are in equilibrium, and argue that the cascade may even continue into the matter dominated phase of the universe. If this is true, then it might be possible to increase the correlation lengths of the fields by one or two orders of magnitude beyond the estimate (2.1). In any case, for temperatures lower than To, the turbulence and inverse cascade are not supported and so we assume that the fields are "frozen in" and experience only the naive scaling I oc T - 1 and B oc T 2 indicated by the first factors in (3.1). 1 If the flow lines could cross, consider the behaviour of a test particle sitting at the crossing. The trajectory would depend on the direction from which the particle came, but as the particle has no inertia (m = 0 ) , its behaviour is indeterminate. This is represented by a singularity in the fluid flow equations: if no singularity is present to start with, then one cannot develop. Once there is viscosity, one cannot discuss such test particles because the fluid now experiences shear forces and the argument breaks down. 3.2 Scaling Laws The main results of [28, 29] are: (3.1b) (3.1a) 3. Evolution of Magnetic Fields 10 3.3 Phase Transitions As pointed out by Son [28], the only way to generate turbulence is either through a phase transition T2 or through gravitational instabilities. Thus, until gravitational dynamos are active, the scalings (3.1) should be valid. In any case, galactic dynamos will amplify the fields, but will not affect the maximum correlation length. In particular, (3.1b) should be a good estimate, regardless of the role of galactic dynamics (the uncertainty coming from the transition period T ~ To when the scaling laws change). Now we consider, as the source of the magnetic turbulence, a phase transition in the early universe. As we shall show, our mechanism generates Hubble size correlations l{ at a phase transition Ti. In the radiation domi-nated epoch, the Hubble size scales as T^2. Combining this with (3.1b), we — 1/3 see that lnov/ oc T{ , thus, the earlier the phase transition, the smaller the resultant correlations. The last phase transition is the QCD transition, Ti = T Q C D ~ 0.2 GeV with Hubble size / ( T Q C D ) ~ 30 km. With our estimates (7.3) of the initial magnetic field strength BTms(Ti) « e A ^ C j D / ( £ A Q C D ) ~ ( 1 0 1 7 G)/(£A Q C D ) we use Equations (3.1) to arrive at the estimate (2.1). The meaning of the correlation length £ will be discussed in detail later in Chapter 4. The most important result here is that, as long as one has a mechanism to generate Hubble scale correlations and a maximally helical magnetic field at the QCD phase transition such that the inverse cascade proceeds via (3.1), magnetic turbulence of 100 kpc correlations is naturally produced. The questions: 'How can helical magnetic fields with Hubble-scale correlations be produced at the QCD phase transition' and, 'Are these fields strong enough to account for the observed microgauss fields?' will be addressed in the rest of this thesis. The estimate (2.1) suggests, however, that even in the worst case of almost maximal suppression £ A Q C D ~ 10 1 9, an efficient galaxy dynamo may be able to amplify the fields to the microgauss level. In the best case, the mechanism would produce measurable extra-galactic fields. In either case, the important result is the generation of the 100 kpc cor-relations: if observations show that the fields have much larger correlations, then the proposed mechanism can only be salvaged if a more efficient "in-verse cascade" mechanism is shown to work between T Q C D and now. Having said this, one might consider the electroweak or earlier phase transitions. As we mentioned, the earlier the phase transition, the smaller the resulting cor-— 1/3 relations / n o w oc T{ For the electroweak transition, the scaling (3.1b) suggests that Hubble scale helical fields could generate 100 pc correlations 3. Evolution of Magnetic Fields 11 today. Thus it might be possible that electroweak phenomena could act as the primordial source, but this presupposes a mechanism for generating fields with Hubble scale correlations. Such a mechanism does not appear to be possible in the Standard Model. Instead, the fields produced are corre-lated at the scale T~l which can produce only ~ 1 km correlations today which are of little interest. Thus, the previous analysis seems to suggest that, in order to obtain magnetic fields with 100 kpc correlation lengths, helical fields must be gen-erated with Hubble scale correlations near or slightly after the QCD phase transition T Q C D - The same conclusion regarding the relevance of the QCD scale for this problem was also reached by Son, Field and Carroll [28, 29]. 3.4 Applicability of Scaling Laws The applicability of the scaling laws (3.1) rests on several assumptions. In particular, that there be maximal helicity on all scales at which the inverse cascade functions. In our previous papers [3, 4] we assumed that Hubble scale correlations in the helicity sufficed to power the inverse cascade, but, as pointed out by A. Vilenkin [private communication] and confirmed by D. T. Son [private communication], it seems that this may not be sufficient and that the helicity must be correlated on all scales where the cascade is to occur. The mechanism as we present in this thesis naturally produces maximal helicity only on the Hubble scale near the QCD transition. Thus, as it stands, it may not produce the desired correlations. We still feel that the mechanism may have merit because: 1) The inverse cascade mechanism is far from well understood and thus, perhaps a similar mechanism will function with helicity generated on the same Hubble scale (this might be unlikely) and 2) Other CP violating physics may arrange for a net generation of helicity on larger scales which could power the cascade. This second point will be discussed in more detail in Section 7.1. Without further ado, we now present our picture of the mechanism and justify the the estimate (2.1) of the magnetic field strength. 4. Generation of Magnetic Fields by Domain Walls 12 4. Generation of Magnetic Fields by Domain Walls The key players in this mechanism are domain walls, which form shortly after the Q C D phase transition. Details of the walls were first presented in [2], wil l be described in Appendix B , and summarized in Chapter (5). In Chapters (6) and (7) we shall show that these walls tend to align nuclear magnetic and electric dipole moments along the plane of the wall. A n im-portant feature of the walls is that across the wall there is maximal strong C P violation due to an induced nonzero 6. Because of this, the electric and magnetic dipole moments of the nucleons are of the same order. Thus, both neutrons and protons will have non-zero electric and magnetic dipole moments and play a role in generating the electromagnetic fields. Because of the correlation between the electric and magnetic fields along the domain wall, the generated fields have an induced helicity as we shall examine in Chapter (7). This helicity has the same sign along the entire domain wall and we expect that the domain wall will fill the entire Hubble volume, thus the helicity will be correlated on the Hubble scale. C P asym-metry in the universe prior to domain wall formation may preferentially form walls of a particular CP. This would be transferred to the magnetic fields, extending the helicity to scales beyond the Hubble scale. Finally, the domain walls wil l decay so that the universe is not domi-nated by domain walls today. By this point, however, the helical magnetic turbulence has been generated. 4.1 Hubble Size Correlations The reason that we feel that domain walls may hold the key to explaining primordial magnetic seeds is that, in a short time, they can generate Hubble scale correlations. The initial fields must have a Hubble scale correlation or else there is no known way—even with an inverse cascade—to generate the huge correlations today. Let us briefly summarize the behaviour we expect of domain walls at the Q C D phase transition. For a nice description of general domain wall dynamics see [22] from which most of these results were derived. 4. Generation of Magnetic Fields by Domain Walls 13 1. Prior to the phase transition Ti = A Q C D I the fields are in random fluctuations on the scale Ti and domain walls are not present. 2. After the phase transition, the fields settle into their vacuum states. Domains are formed where the fields are settling into different1 vacuum states. These domains are separated by domain walls and have a scale set by A Q C D -3. Numerical studies suggest that these small-scale domain walls rapidly merge increasing the correlation length of the walls. This coarsening occurs simultaneously throughout space and the correlation length of the domain walls can increase faster than the speed of light. 4. The coarsening stops once the domain walls attain a Hubble scale. On average, one ends up with one domain wall per Hubble volume, but which curls and moves through space, essentially filling the volume. It is these Hubble sized domain walls that can generate magnetic tur-bulence with Hubble size correlations. As we shall see below, there are two types of domain walls corresponding to opposite field transitions. One we call a "soliton", and the other we call an "anti-soliton". Together a soliton and an anti-soliton can annihilate, but the coarsening essentially separates regions of solitons from anti-soliton regions by a distance of the Hubble scale so that they do not annihilate. In Chapter (7.1) we shall show that the soli-tons and anti-solitons are associated with helicity of the opposite sign. Thus, the domain walls effectively separate the helicity generating a Hubble scale correlation length in the fields and in the helicity. At this point, it may be possible to bring in some unknown C P violating physics which preferentially forms solitons over anti-solitons. Such an effect would convert this C P vi-olation into a net helicity in the magnetic fields on a scale larger that the Hubble scale. This would power the inverse cascade discussed in Section 3.2. 4.1.1 QCD Phase Transition There is some question about what conditions must be like at the Q C D phase transition in order for domain walls to form. Q C D lattice simulations suggest that at low densities (such as those present in the early universe), the 1 In the case of QCD domain walls, the vacuum states are actually the same but the field configuration, going from one domain to the next undergoes a classically stable transition. This behaviour is qualitatively similar to the sine-Gordon model Csa = (dn<f>)2 — cosc/> where c/> is interpreted as a phase so that the vacuum states <j> = 2irn are actually identical. 4. Generation of Magnetic Fields by Domain Walls 14 transition between the quark-gluon plasma and the normal hadronic phase is a smooth crossover, and that the critical point sits at some finite density (see the recent review [33]). If the rate at which the universe cools is sufficiently slow, then it is possible that no domain walls form. In the preface to the paperback edition [22], the authors discuss this scenario as the Kibble-Zurek picture: to estimate the size of the correlations produced, one must consider the relaxation timescale r (T) : the time it takes to establish correlations on the scale £ ( r ) . The freezout temperature Tf is determined by the condition T(TJ) ~ tr) = \Tj — T C | / | T / | , i.e. when the relaxation time is on the same order as the dynamical timescale of temperature variations. Since the true critical point is at a somewhat higher densities than the universe, to my be bounded from below and, if the cooling is sufficiently slow, it is possible that r <C to and domain walls will not form. Relative to Q C D scale physics, the cooling rate of the universe is very slow suggesting that a smooth crossover might imply no quasi-stable topolog-ical defects can form. One must, however, estimate the relaxation timescale, which diverges as one approaches the second-order tri-critical point. If the second-order point is at a low enough density, then it is conceivable that the timescale may be large compared to—or at least on the same order as—the cooling rate. Perhaps a better way to look at this is to say that: If pure Q C D domain walls do form (as opposed to axion domain walls which will be discussed in a moment), then the second order point must be at a sufficiently low density to provide a large enough relaxation timescale. Domain walls do not necessarily require a phase transition or sharp crossover to form (although purely Q C D domain walls probably do require a rapid transition). Take for example the so-called N = 1 axion domain walls [34, 35]: In this model, axion strings are first formed at a higher energy scale where the Peccei-Quinn symmetry is broken. After this transition, the axion field (essentially, a complex field) sits in a Mexican-hat potential with an approximately degenerate ground state about the rim. Above this scale, the axion field is in thermal fluctuations, but below this scale, the thermal fluctuations relax and the field relaxes to the "brim" of the potential. It is likely that regions of space in thermal fluctuation will have formed vortices. Upon cooling these vortices contract to minimize their energy, but the cen-tral region cannot unwind due to the topological configuration of the vortex. If the potential is truly degenerate, then the resulting objects that form are called cosmic strings and are topologically stable. As the universe cools toward the Q C D scale, this potential starts to tip. As the potential tips, the strings "stretch" as the field attempts to relax to the minimum state. Because of the topology of the string vortex, however, 4. Generation of Magnetic Fields by Domain Walls 15 the field cannot fully relax. Bounded by the string and extending in the "direction" away from which the potential tilts, a domain wall forms. This domain wall has the same type of structure described later in this thesis (Section 5 and Appendix B) . These domain walls become unstable due to a tunneling process similar to that discussed in Sections 5.2 and B.5. Notice, however, that the wall tension slowly increases as the potential tilts, and so a slow transition at the Q C D scale does not forbid their formation. Rather, as can be seen from Equation (5.15), this means that the walls become slowly unstable and so will tend to resist decay until after the transition. In fact, with axions, the problem is not in forming the walls, but in making sure that they decay to avoid the cosmological problems associated with persistent domain walls of large energy density [23]). Thus, even if the transition is a smooth crossover, it is possible for do-main walls to form. In this thesis we are primarily concerned with Q C D domain walls, and so we assume that the dynamics of the crossover are such that domain walls do form, coalesce and, attain Hubble-scale correlations as described in [22]. To estimate these dynamical effects, unfortunately, re-quires a better understanding of the dynamics of the domain walls and of the phase transition than we have presently. 4.2 Essential Domain Wall Properties We can now formulate a set of properties that must be satisfied by domain walls if they are to be considered as sources for the primordial magnetic fields described in this thesis: 1. The walls must attain Hubble scale correlations near the Q C D phase transition to generate the observed correlations. 2. The walls must have structure on the scale of A Q Q D in order to interact effectively with nucleons. 3. There must be some way to avoid the cancellations discussed in [25, 26] so that the walls are ferromagnetic rather than diamagnetic. 4. The walls must somehow induce a definite helicity in the magnetic field throughout the Hubble volume and possible extending beyond the Hubble volume to a comoving scale similar to that attained by the fields when the cascade stops. 5. The domain walls must be unstable or have other features so that the problems of domain wall domination in the universe are avoided, 4. Generation of Magnetic Fields by Domain Walls 16 but they must be sufficiently stable that they can generate the appro-priate fields. They must also leave nucleosynthesis production ratios relatively unaffected (though it may actually be desirable that the the walls effect a slightly inhomogeneous nucleosynthesis. See Section 8.2.) It seems to be a rather general property of cosmological domain wall networks that they rapidly coarsen through the Kibble mechanism until the walls have a Hubble-scale correlation length [22]. Thus, criterion 1 should be easily satisfied by almost all types of domain walls (see Section 4.1.1). Criterion 2 rules out the standard axion domain walls discussed in [19, 23, 36], however, there may be features of these walls that have QCD scale which were previously neglected. In particular, Shifman and Gabadadze discuss a gluonic core sandwiched at the center of axion domain walls [37]. This structure has a QCD scale and may be able to align nuclear matter. In our paper [2] we discuss another possibility: that the rf field may provide domain wall structure with QCD scale. This structure, which we shall refer to as a QCD domain wall, can exist within the standard axion domain wall providing the required QCD scale structure, but can also exist, even if there is no axion (unlike the walls of [37] which require an axion field). A further property of QCD domain walls and axion domain walls is that, at their center, there is strong CP violation. This CP violation has several effects as described in Chapter (4.3). In particular, the CP violation induces an anomalous electric dipole moments in nucleons. Thus, CP violating do-main walls can satisfy criterion 3. Helicity is also associated with CP violation as it is a CP odd quantity. As discussed above, the soliton and anti-soliton domain walls solutions have opposite CP. Thus, each is associated with opposite helicity. Typically, the solitons and anti-solitons separate spatially so that Hubble-sized regions are filled with one type or another. The helicity is generated through the corre-lation of both electric and magnetic fields along the walls. Thus, the Hubble scale spatial separation of soliton and anti-soliton domain walls also sepa-rates the helicity and thus generates helical turbulence with Hubble scale correlations. If the inverse cascade requires helicity on larger regions as discussed in Section 3.4, then some CP violating mechanism which prefer-entially favours the formation of one type of domain wall (soliton) over the other type (anti-soliton) could have the desired effect. In this case, criterion 4 would be satisfied. Whether or not such an effect is required depends on details of the inverse cascade which are not presently well understood. 4. Generation of Magnetic Fields by Domain Walls 17 Another major problem with axion domain walls is that most varieties appear to be absolutely stable. The N = 1 axion model discussed recently by Chang, Hagmann and Sikivie [23] has a decay mode that satisfies the criterion 5. However, this is not an aesthetically satisfying model and, to date, the other axion models are plagued by this problem. If axion domain walls are to be considered, then a satisfactory solution to this problem must be found. Q C D domain walls without an axion exhibit a decay mode similar to the the N — 1 axion model. The scales, however, are set by A Q C D rather than ma and so criterion 5 must be address from the point of view: Do the walls live long enough to generate the turbulence? As addressed in Section B.5 of Appendix B , the answer may be yes. The only decay mode is through a nucleation process suppressed by quantum mechanical tunneling. Consequently, these walls may have a macroscopic lifetime long enough to generate the fields. In any case, however, they decay fast enough to avoid affecting nucleosynthesis and other cosmological effects.2 Thus, there may be several types of domain walls that could act as sources for primordial magnetic seed fields. In this thesis, we now specialize to discuss the Q C D domain walls presented in Appendix B showing that they may be able to generate sufficiently large magnetic fields to seed galactic dynamos and possibly to observe in the extra-galactic medium. Here we briefly summarize the essence of strong C P violation to show how axion and Q C D domain walls may satisfy the criteria discussed above. The most general form for the fundamental Q C D Lagrangian is known to contain the following term related to the anomaly: where G£„ is the gluon field tensor and Ga^ = \£^p,TGapa is its dual. This term is odd under the discrete symmetry CP, thus, iff? is non-zero, then the strong interaction should violate CP. Experiments, however, have placed tight limits \9\ < 10~ 9. The contributions to the final 9 arise from several sources, and it remarkable that these seem to exactly cancel. The origin of this cancellation is known as the strong C P problem. 2 It is possible that, as nucleons are attracted to the domain walls, the back-reactions which we have neglected in this analysis might stabalize the walls. If this is the case, then the walls might affect nucleosynthesis as described in Section 8.2. . 4.3 Strong CP Violations in Domain Walls (4.1) 4. Generation of Magnetic Fields by Domain Walls 18 One solution is to promote 9 from a parameter to a dynamical field called the axion. The idea is that, prior to the QCD phase transition, the axion field is massless and 9 can take on any value. After the transition, the axion acquires a mass and sits in a potential with a minimum energy where 9 = 0.3 The axion field thus relaxes to the minimum, restoring CP conservation today. To date, axions have not been detected, however, there is an allowed region consistent with experimental, astrophysical and cosmological constraints: A very light axion with a mass of ma ~ 1 0 - 5 -1 0 - 3 eV may still resolve the strong CP problem. In addition, axions of this mass are a strong cold dark candidate [38]. As mentioned, axions provided a nice mechanism for generating domain walls, but because the axion must be extremely light, there is no way for such structures to efficiently interact with nucleons. For a good reviews of the strong CP problem and the role of axions, see [39-41]. In any case, we assume that some method exists to solve the strong CP problem. What is important about the 9 parameter is that, in the low-energy limit, it only appears in the combinations (9 + (j)) (Equation 5.4) and (9 + (j> — a), where (p is the dynamical field related to the rj' meson and a is the axion field. Thus, even when 9 = 0, CP will be violated in strong interactions in a domain wall background where </> or </> — a is non-zero over a macroscopically large region. Hence, QCD and axion domain walls induce strong CP violations over their central regions. One of the consequences of this strong CP violation is that nucleons have an induced electric dipole moment as well as a magnetic dipole moment4 [24]. We summarize these results here. In the chiral limit mq —> 0, and for small 9, the electric dipole moment is where gVNN is the strong nNN coupling constant and g^NN is the CP odd 3 In the language of Section 3.3, prior to some high-energy scale (Peccie-Quinn), the axion field is free. After this transition, the flat Mexican hat potential is formed restricting the degrees of freedom to a phase angle 9 except at the center of strings (This defect energy is associated with the string tension.) Through this spontaneous symmetry breaking, the massless phase excitation (the axion) acts as a Goldstone boson. As the temperature cools, this potential tilts and 9 = 0 is favoured. The tilt explicitly breaks the original symmetry and gives the axion a mass (thus it is usually referred to as a pseudo-Goldstone boson). This is when domain walls form. 4 Normally the electric dipole moment is suppressed to the same order as 9 in that C P is conserved. (4.2) 4. Generation of Magnetic Fields by Domain Walls 19 TTNN coupling constant which was estimated to be g^NN ~ O.O4|0|. In these formulae the 9 parameter should be treated as the singlet (f> domain wall solution (j){z) with nontrivial z dependence. From these formulae one can compute the following relation Thus, for all nucleons, including the neutron, both the electric and magnetic dipole moments are non-zero and of the same order in the domain wall background when 9(z) = (j>(z) ~ 1. 5. QCD Domain Walls 20 5. QCD Domain Walls We saw in Chapter (4.2) that several types of domain walls might act as sources for seed fields. To be concrete, we shall now restrict our attention to QCD domain walls to show how domain walls might produce magnetic seed fields. In this chapter, we shall present a short review of the results presented in Appendix B, simplifying the model for presentation. To describe these walls, we consider the low-energy effective theory of QCD, including the pions, and the rf singlet field. The rf field is not as light, but it represents the essential physics associated with the anomaly that is responsible for the QCD domain walls. The pions and rf enter the Lagrangian through the matrix representation U = exp r 7r aA a 2 rf tV2— Vi- (5.1) where 7ra are the Nj — 1 pseudo-Goldstone fields, A a are the Gell-Mann matrices for SU(Nf), and rj' is the singlet field. From now on, we limit ourselves to the simplest case of one flavour, Nf = 1, which contains only the rf field but which captures all of the relevant physics. The Nf = 2 case is described in [2] and presented in Appendix B. Although the models are quantitatively different, the phenomena described by both is the same. In this model, we see that (5.1) reduces to a single complex phase U = e^, <f>=% (5.2) frf The effective Lagrangian density then reduces to _ frf with the effective potential C=J-^(d^)2-V{(j>) (5.3) V{(j)) = -min<jMcoscA + £cos fd + ^ + 2 n l ) i ( 5 4 ) 5. QCD Domain Walls 21 which was first introduced in [42]. The minimization on the right comes from choosing the lowest energy branch of the multi-valued potential. Details about this potential are discussed in the original paper [42] but several points wil l be made here. A l l dimensionful parameters are expressed in terms of the Q C D chiral and gluon vacuum condensates: M = diag(mg|(<f ql)\) is the mass matrix and E = (bas/(32ir)G2) is the vacuum energy. These are well known numerically: mq ~ 5 MeV (see for example [43]), (qq) ~ -(240 M e V ) 3 [44], (as/nG2) ~ 0.012 G e V 4 [45, 46] and b = UNc/3-2Nf/3 is the first term in the beta function. This potential correctly reproduces the D i Vecchia-Veneziano-Witten ef-fective chiral Lagrangian in the large JVC limit [47, 48], it reproduces the anomalous conformal and chiral Ward identities of Q C D , and it reproduces the known dependence in 9 for small angles [47, 48]. It also exhibits the correct 27r periodicity in 9. This periodicity is the most important property of the potential and is the reason that Q C D domain walls form: The qual-itative results do not depend on the exact form (cosine) of the potential. Rather, the domain walls form because of the 2n periodicity (9 -» 9 + 2n) which represents the discrete nature of the ground state symmetries. It is exactly these symmetries that lead to the existence of axion domain walls when 9 is promoted to a dynamical axion field [23, 34-36, 49]. As described in Chapter (4.3), we see that the singlet U(l) field cp occurs in the same place as the C P violating 9 parameter. Thus, even though to a high degree of precision, 9 = 0, in the macroscopic regions where (rf) ^ 0 there wil l be C P violating physics. We see that the potential(5.4) has ground states characterized by 0o = 2-7m, (5.5) where n is an integer, with vacuum energy Vm-m = —M — E. Expanding about the minimum (j> = 4>Q + 5$ we find the mass of the field1 The most important point to realize is that all of the ground states (5.5) in fact represent the same physical state U = 1. Thus, it is possible for the 0 field to make a transition 2irn -> 2irm for different integers n and m. Within this model (5.3), where all heavy degrees of freedom have been integrated 1 In the more general case of Nf quarks with equal masses, the right-hand side of Equation (5.6) should be multiplied by a factor Nf (Compare with (B.18).) 5. QCD Domain Walls 22 out, these transitions are absolutely stable and represent the domain walls. When one includes the effects of the heavier degrees of freedom, however, we find that the walls are unstable on the quantum level. This is described in detail in Appendix B and briefly reviewed in Chapter 5.2. To study the structure of the domain walls, we look at a simplified model where one half of the universe is in one ground state and the other half is in another. The fields wil l orient themselves in such a way as to minimize the energy density in space, forming a domain wall between the two regions. In this model, the domain walls are planar and we shall neglect the x and y dimensions: A complete description of this wall is given by specifying the boundary conditions and by specifying how the fields vary along z. We present here the two basic domain wall solutions. These are charac-terized by interpolations from the state (j> = 0 to: Soliton: <j> — 2TT, Anti-soliton: (j) = —2ir. It is possible to consider transitions between further states (i.e. 0 —> 2rm) but these can be thought of as multiple domain walls. They also have higher energies, are less stable, and are thus less important for our discussion. To gain an understanding of the structure of the domain walls we look for the solution which minimizes the energy density. The energy density (wall tension) per unit area is given by the following expression where the first term is the kinetic contribution to the energy and the last term is the potential. Here, a dot signifies differentiation with respect to z: a = — To minimize the wall tension (5.7), we can use the standard variational principle to arrive at the following equation of motion for the domain wall solutions: 5.1 Domain Wall Solutions (5.7) hi = sine/) + E (5.8) AM MNC 5. QCD Domain Walls 23 Again, the last term of Equation (5.8) should be understood as the lowest branch of a multi-valued function as described by Equation (5.3). The general analytical solution of Equations (5.8) is not enlightening and we present the numerical solution in Fig. 5.1. In order to gain an intuitive understanding of this wall, we examine the solution in the chiral limit M <C E/N% (physically, when Nf > 1, this is the limit <C mvi). In this case, the last term of (5.8) dominates. Thus, the structure of the cj> field is governed by the differential equation: 72""'" 1^ * = *3I.si"w- (5'9) Now, there is the issue of the cusp singularity when (j> = n which results from switching branches of the potential (see Equation (5.3).) By defini-tion, we keep the lowest energy branch, such that the right-hand side of Equation (5.9) is understood to be the function sin(c/>/Arc) for 0 < </> < TT and sin([t/> — 2TT]/NC) for n < <f> < 2n. However, since the equations of mo-tion are symmetric with respect to the center of the wall (which we take as z = 0), 4> = TT only at the center of the wall and not before, so we can simply look at half of the domain, z £ (—oo,0], with boundary conditions 4>{—oo) = 0 at z = —oo and 0(0) = TT at z = 0. The rest of the solution will be symmetric with <j) = 2n at z = +oo. Equation (5.9) with the boundary conditions above has the solution 4 ^ t a n _ 1 2TT - 4iV r t a n - 1 ^ z tan j^- \, z<0, 4>{z)={ - L r C J - (5.10) ^ ' - 1 e-" 2 tan ^ - | , z > 0. which is a good approximation of the solution to Equation (5.8) when M <g E/N%. Here, the scale of the wall is set by the parameter fj,: 2VE H=——, l im M = n y , (5.11) which is the inverse width of the wall and which is equal to the field mass mvi in the chiral limit m 2, —> 0 (see Equation (5.6)). Thus, we see that, indeed, the Q C D domain walls have a Q C D scale. Solution (5.10) describes the soliton. The anti-soliton can be found by taking z —> —z: thus, we have the transition soliton —> anti-soliton under the discrete C P symmetry. The numerical solution for the cj) field is shown in Fig. 5.1. It turns out that the approximation is reasonable even in the physical case where N^M/E ~ 1 0 _ 1 . 5. QCD Domain Walls 24 The wall surface tension defined by Equation (5.7) and can be easily calculated analytically in the chiral limit when the analytical solution is known and is given by Equation (5.10). Simple calculations leads to the following result:2 In the case when mq ^ 0, a is numerically close to the estimate (5.12). r>r-— , , -5 0 5 Z/J, Fig. 5.1: Basic form of the QCD domain wall (soliton). The analytic approximation 5.10 is plotted as a dotted line to show the good agreement. We have taken Nc = 3 here. Notice that the wall thickness is set by the parameter fi. 2 In the general case of Nf quarks of equal mass, the right-hand side of Equation (5.12) should be multiplied by the factor \/yJWf. In this case, we can compare (5.12) with (B.41) where Nf = 2. 5. QCD Domain Walls 25 5.2 Domain Wall Decay-Finally, we note that these domain walls are not stable: as mentioned ear-lier, the vacuum states (5.5) represent the same physical state. When one includes the heavier gluonic degrees of freedom, it becomes possible for the fields to "unwind" through this extra degree of freedom. Classically this is not allowed because the heavy degrees of freedom are constrained by a large potential barrier. It is still possible, however, for the field to tunnel through this barrier forming a hole in the domain wall. Once a large enough hole is formed, it will expand and consume the domain wall. This process is called "nucleation" and is similar to the mechanism consuming N = 1 axion domain walls [22, 23, 34, 50]. In Appendix B, we estimate the lifetime of these domain walls borrowing the same methods used to estimate the lifetime of axion domain walls in the N = 1 axion models [22, 23, 34, 50]. We should point out one major difference between the N = 1 axion model and our model. In the axion model, strings form first, and then the domain walls form, primarily bounded by strings [23] as discussed in Section 4.1.1. Q C D domain walls, on the other hand, would form simultaneously with strings at the Q C D transition. Some walls my form which are bounded by strings: these would probably disappear quite rapidly. Closed domain walls, however, could only decay through the nucleation process and would this would greatly enhance their lifetime. We do not fully understand the dynamics at the phase transition, but the picture we have is that many small closed domain walls may form and them merge to reduce the wall tension. This is the coarsening phenomenon discussed in Section 4.1 (see also [22] for a nice discussion). The tunneling probability can be estimated by computing the action So of an instanton solution of the Euclidean (imaginary time, t = ir) field equations, which approaches the unperturbed wall solution at T —>• ±oo. In this case, the probability P of creating a hole is proportional to the factor where So is the classical instanton action. If the radius Rc of the nucleating hole is much greater than the wall thickness, we can use the thin-string and thin-wall approximation. In this case, the action for the string and for the wall are proportional to the cor-responding world-sheet areas [50], P ~ e -So (5.13) S0 = 4irR2a (5.14) 5. QCD Domain Walls 26 Here o is the wall tension (5.12), and a ~ y2E is the string tension which we estimate based on dimensional arguments. The string tension a tries to close the hole while the wall tension a tries to widen the hole. Minimizing (5.14) with respect to R we find that p 2 a c 1 6 7 r c * 3 / M ^ Rc = —, o0= » o • (5.15) If a hole forms with radius R > Rc then the hole will expands with time as x2 + y2 = R2 + t2, rapidly approaching the speed of light and consuming the domain wall. Inserting numerical values for the phenomenological relevant case Nf = 2 we find that a ~ (0.28 GeV) 2 , a = (200 MeV) 3 , S0 ~ 120. (5.16) What is important is that So is numerically large, and hence the lifetime is much larger than the Q C D scale because of the huge tunneling suppression e~s° ~ I O - 5 2 . A more complete analysis is presented in Appendix B where we estimate the lifetime (Equation (B.56)) of the walls to be on the order •T ~ I O - 5 s (5.17) even though the walls are governed by the microscopic Q C D scale. This result should be interpreted with some caution: in the low energy regime, we do not have very good control over the quantitative physics.3 Arguments presented in Appendix B show, however, that it is at least possible for domain walls of purely Q C D origin to live for macroscopically large lifetimes. To summarize, we have a Q C D domain wall with all of the properties required to generate magnetic fields: 1. The walls form shortly after the Q C D phase transition and attain Hubble-scale correlations through the Kibble mechanism. 2. The Q C D domain walls have a structure on the scale of m~} ~ A Q C D and thus they can efficiently interact with nucleons and other Q C D matter. 3. The transition in the singlet n' field produces an environment near the wall where the effective 6 parameter is non-zero. Thus, across the wall, 3 It is possible to regain control of the calculation in the high density limit. See the discussion in Chapter (8.1). 5. QCD Domain Walls 27 there is maximal C P violation. In such an environment, it is known that the electric and magnetic dipole moments of the nucleons are of the same order [24]. Because of the induced electric dipole moments and anomalous magnetic moments, the cancellations discussed in [25, 26] do not qualitatively affect the physics and the domain walls are truly ferromagnetic. 4. The strong C P violation also provides a mechanism for generating helicity on a Hubble scale4 by aligning both the electric and magnetic dipole moments along the domain wall. 5. The decay mechanism renders the Q C D domain walls unstable such that the walls themselves do not pose a cosmological problem. How-ever, the suppression in the decay mechanism due to quantum tun-neling might extend the lifetime of the walls to a macroscopic scale (5.17) which is long enough to generate the required electromagnetic turbulence as we shall show. 4 To be precise, the domain walls separate the helicity into Hubble size regions to that globally the total helicity is zero, but within Hubble scale regions, the helicity is maximal and correlated with the same sign. 6. Alignment of Spins in the Domain Wall 28 6. Alignment of Spins in the Domain W a l l Now, by present a simplified method for estimating the magnitudes of bulk properties on the domain wall, we shall show that, indeed, the domain walls acquire a magnetization. This method makes the approximation that the domain wall is flat, and that translational and rotational symmetries are preserved in the plane of the wall (which we take to be the x-y plane). These approximations are valid in the case of domain walls whose curvature is large in comparison to the length scale of the pertinent physics. Once this approximation is made, we can reformulate the problem in 1 + 1 dimensions (z and t) and calculate the density of the desired bulk properties along the domain wall. To regain the full four-dimensional bulk properties, we must estimate the density of the particles in the x-y plane to obtain the appropriate density and degeneracy factors for the bulk density. Thus, the final results are not independent of physics in the x-y plane, but rather, these effects are only accounted for through degeneracy factors. We proceed to demonstrate this technique by calculating the alignment of fermionic spins along the wall. We take the standard form for the inter-action between the pseudo-scalar n' field and the nucleons which respects all relevant symmetries: Here 0 = cj>(z) characterizes our domain wall solution as expressed in Equa-tion (5.10) and is the nucleon mass. For our approximations, we assume that fluctuations in the nucleon field ^ do not affect the domain walls and treat the domain walls as a background field. 1 The strategy is to break (6.1) into two 1 + 1 dimensional components by setting dx = dy — 0 (this is the 1 A full account would take into account the effects of this back-reaction. We expect that such back-reactions would affect the potential (5.4) by altering the form of the last term and possibly adding higher order corrections. This may affect the magnitudes of some of the estimates, but would certainly not alter the topology of the fields and thus the domain walls would still form with a similar structure. Quantitatively this would alter the results, but not the order of magnitude. What may be substantially affected is the lifetime of the domain walls. In particular, the result [51] that domain walls are very (6.1) 6. Alignment of Spins in the Domain Wall 29 approximation that the physics in the z direction decouples from the physics in the x-y plane) and then by manipulating the system of equations that result. First, we introduce the following chiral components of the Dirac spinors:2 • • - ( £ ) • *-=(!)• ( 6 - 2 ) * = V2 V%2 - 6/ (6.3) Now we re-express (6.1) by noting that 7 2 = / : d0 -o-jdj The associated Dirac equation is do mN -do mN cos(0) 2sin(c/>) is\n(4>) cos(c/>) cos(c/>) isin(c/>) * = 0. s^in((/>) cos(c/>) This is equivalent to the coupled system: 2i(d0 + <Jidi)V- = 2mNe^+, (6.4a) 2i{do - o-idi)y+ = 2mJV-e_i*tf_. (6.4b) Now, we decouple the z coordinates from x and y by setting dx = dy = 0: 'dt + dz 0 0 dt-d.. dt-dz 0 0 dt + d, (6.5a) (6.5b) stable at high densities might suggest that an accumulation of matter on these walls could stabailize them. See the further discussion in Section 8.2. We are using the standard representation here: I 0 7o = -I 1\ Tl = 1 o 7 j ff2 75 = 0 J 1 0 C3 = - c ' ) 6. Alignment of Spins in the Domain Wall 30 Both these equations are diagonal. Thus, we see that the top components and bottom components of ^ ± mix independently. 0, (6.6a) 0. (6.6b) Remember that we are looking for a 2-dimensional Dirac equation, thus we want the kinetic terms to look the same. For this reason we should flip the rows and columns of the second equation. Doing this and defining the two 2-dimensional spinors ' •<•>-(&)• *<2> = (S)- (6-7) the equations have the following structure: ( i f f y - mNe+i^)y{1) = 0 (6.8a) ( i f f y - m i v e - ^ 5 ) ^ ) = o (6.8b) where the index \i 6 {t,z}, the Lorentz signature is (1,-1) and we define the following 2-dimensional version of the gamma matrices: It = cri , 7z = -io~2, 75 = 0-3. These satisfy the proper 2-dimensional relationships 75 = jtlz and 7^ 7,/ = 9[LV + C/ii/75- We can reproduce equation (6.8) from the following effective 2-dimensional Lagrangian density, £ 2 = * { 1 ) ( i f f y - m N e ^ ) + + * ( 2 ) ( i f f y - mne-^) * ( 2 ) , (6.9) where two different species of fermion with opposite chiral charge interact with the domain wall background 4>{z). We have thus successfully reduced our problem to a two-dimensional fermionic system. It is known that for several systems in 1 + 1 dimensions, the fermionic representation is is equivalent to a 1 + 1 dimensional bosonic -mjve«* i(dt + dz)\ [xi i{dt-dz) -mtfe-^J V£i -mNe^ i(dt - dz)\ (X2 i(dt + dz) -mNe-^) U 2 6. Alignment of Spins in the Domain Wall 31 system through the following equivalences3 [54, 55]: * U ) W * < j ) \(W2, (6.10a) - 7 = ^ 5 " ^ , (6.10b) -> -ncos{2^0j), (6.10c) * ( j ) i 7 5 * ( j ) -> -/isin(2v/7r6'j). (6.10d) After making these replacements, we are left with the following 2-dimensional bosonic effective Lagrangian density describing the two fields 9\ and 02 in the domain wall background <j>(z) £ = \(9A)2 + - u(9ue2) (6.ii) where the effective potential is U(61,62) = -mNri [cos(2v^6>i - <f>) + COS(2VTF6>2 + </>)] . (6.12) The next approximation that we make is to neglect the dynamics of the 6% fields: we assume that they relax slowly in the domain wall background such that their dynamics do not contribute appreciably to the final state4 which minimizes the potential (6.12). The classical minimizing solution is thus w = 57? < 6 ' 1 3 ) We are now ready to show that the domain walls align the spins of the fermions. The relevant spin operator is 5 = § 7 7 5 * = ^+a^>+ + tfLtr#_ (6.14) 3 The constant /i in the last two equations is a scale parameter of order TUN • The exact coefficient of this term depends on the model an is only known for exactly solvable systems but in all cases, is of order unity. This technique is well-known to the condensed matter and particles physics communities. See for example [52, 53]. 4 This is the same adiabatic approximation used by Goldstone and Wilczek [52]. 5 Here we use the convention that atj = § [ 7 i , 7 j ] , thus the spin operator = ^eijkaij-In matrix form with the standard representation, this becomes: In terms of the gamma matrices, this is Sj, — 707 /175. 6. Alignment of Spins in the Domain Wall 32 Let us consider the z component of the spin. We then have = x i x i - xlx2+eki - & 2 = * ( i)7t* (l) - *(2)7t*(2) (6-15) and so we see that the four-dimensional spin operator S z = 707*75 is ex-pressed in terms of a pair of two-dimensional fermion charge operators. We can calculate the expectation value of the spin operator in the domain wall background using this two-dimensional correspondence (6.15) and the bosonic representation of the fermions M/tS.vj, = * ( 1 ) 7 t * ( 1 ) - * ( 2 ) 7 t * ( 2 ) = -±dz(02 - 6X). (6.16) Finally, we use our minimizing bosonic solution (6.13) to obtain the following 4-dimensional average spin aligned along the domain wall: (tfts,*) = - I W (6.17) 7T OZ We will also need the following matrix elements later on: = < x k i - x k 2 + e l x i - d x 2 > = <*(1)*(1)> - <*(2)*(2)> = 0 (6.18) and = * ( - £ i X i + £2X2 + x k i - x k s ) = -<*(1)*75*(1)> + (*(2)*75*(2)> = 2^sin(0) (6.19) Remember that we have restricted ourselves to a 1 + 1 dimensional theory. We must now estimate the density and degeneracy of the nucleons along the wall so we can obtain a true 1 + 3 dimensional estimate of the spin density. 6. Alignment of Spins in the Domain Wall 33 6.1 Fermion Degeneracy in the Domain Wall We have assumed that, locally, the domain walls have only a spatial z depen-dence. There is still a 2-dimensional translational and rotational symmetry in the x-y plane. These translational degrees of freedom imply that momen-tum in the plane is conserved. Hence, we can treat the neglected degrees of freedom for the fermions as free. The degeneracy in a region of area S will simply be a sum over these degrees of freedom with a discrete factor g — 4 = 2 x 2 for spin and isospin degeneracy \\P\\<PF Here we estimate the Fermi momentum pp « A Q C D by the thermal scale of the fermions.6 and assume that the Fermi sea is filled 7 This completes our estimate of the induced spin along the domain wall in a small region 5. Combining our estimate of the spin (6.17) from the bosonisation scheme with the fermion degeneracy (6.20) we obtain the spin density along the wall: &W)AD = (6-21) As a check, note that the dimension here is 3 as it should be. We are treating the fermions along the wall as a two-dimensional massless Fermi-gas. 7 Voloshin obtained a similar estimate [26]. Furthermore, he estimates that there are sufficiently many fermions in the Hubble volume to diffuse into the domain wall potential justifying our assumption about the filled Fermi sea. 7. Generation of Electromagnetic Field 34 7. Generation of Electromagnetic F ie ld Once the spins are aligned, the nucleon electric and dipole moments interact with the electromagnetic fields through the interaction ^ ( d * * < V 7 5 * + + V^D^D")*. (7.1) Here the nucleons have both electric and magnetic dipole moments ~ [iy respectively (4.3). Now, we make the approximation that the nucleons align independently of the electromagnetic field, and we treat the nucleon field $ as a back-ground. The situation is analogous to having a field of dipoles aligned along the domain wall. The net fields generated by surface of area £ 2 willed with a constant density of aligned dipoles is proportional to £ _ 1 since the dipoles tend to cancel. For a perfectly flat domain wall of infinite extent, £ —>• oo and thus no net field remains, as pointed out in [26]. The Q C D domain walls, however, are far from flat: the walls have many wiggles and high frequency dynamic excitations. Thus, the fields generated by the dipoles wil l not can-cel on the domain wall, but will be suppressed by a factor1 of ( £ A Q C D ) _ 1 where £ is an effective correlation length that depends on the dynamics of the domain walls. As an upper bound, the extent of the domain walls is limited by the Hubble scale. Typically, domain walls remain space filling, thus we expect A Q C D < £ <§; Hubble scale. Unfortunately, we presently cannot make a tighter bound on iota. We shall see, however, that even in the worst case, this mechanism can at least generate feasible seed fields for galactic dynamos to amplify. The result of Chapter (6) is a method for estimating the strengths of various sources in the domain walls. We now need to couple these to the generation of electromagnetic turbulence. To do this properly requires the solution to Maxwell's equations as coupled to the sources in (7.1). This is difficult, though no doubt important for accurate numerical estimates, and so for an order of magnitude estimate we make a dimensional estimate 1 The density of the dipoles is governed by the Q C D scale A Q Q D . 7. Generation of Electromagnetic Field 35 considering the sources as a set of dipoles sitting in the domain walls. The spacing between the dipoles is set by the Q C D scale A Q C D , and the strengths of the field can be estimated from (7.1) using dimensional arguments: (Fp,) ~ - r ^ — (<M*<V75*) + . (7.2) ? A Q C D This includes the dipole suppression discussed above. From (6.18), (6.19) and (7.2) we arrive at the following estimates for the average electric and magnetic fields (correlated on the Hubble scale) which includes the degen-eracy factors (remember that u. ~ mjv): 0 5 e A n r r ) i o 1 7 G \'EZ)\ ~\{BZ)\ ~ ^ I U U ( 7 3 ) £ A Q C D ? A Q C D This method of estimating the electric and magnetic fields produced is extremely crude: we have not solved Maxwell's equations, we have not taken back reactions into account and we have not fully accounted for the motion and geometry of the domain walls. Nevertheless, we expect that the es-timates (7.3) to be valid as an order of magnitude estimate for the field strengths. The approximations we have made and effects that we have ne-glected will be discussed in Chapter (8.1). Thus, we have the approximations (7.3) which, along with (3.1), justify the estimate (2.1). 7.1 Helicity Finally, we note that the turbulence discussed in Chapter (3) should be highly helical. This helicity arises from the fact that both electric and mag-netic fields are correlated together along the entire domain wall, ( E ) ~ ( A ) / T where ( A ) is the vector potential and r is a relevant timescale for the electrical field to be screened (we expect T ~ A Q C D as we discuss below). The magnetic helicity density is thus: h ~ A • B ~ T(EZ)(BZ) ~ r4%^- (7.4) It can be seen from (7.2) that both the electric field and magnetic fields have the same structure in the domain wall. This implies that domain walls (solitons) have the same sign of helicity everywhere. Under C P the wall becomes an "anti-wall" (anti-soliton) and the orientation of the magnetic field B changes direction: thus anti-walls have opposite helicity. 7. , Generation of Electromagnetic Field 36 Note what happens here: If the total helicity was zero in the quark-gluon-plasma phase and remains zero in the whole universe, then the helicity is separated so that in one Hubble volume where one domain wall dominates, the helicity has the same sign. The reason for this is that, as the domain walls coalesce, initial perturbations cause either a soliton or an anti-soliton to dominate and fill the Hubble volume. In the neighboring space, there will be other solitons and anti-solitons so that there is an equal number of both, but they are separated and this spatial separation prevents them from annihilating. This is similar to how a particle and anti-particle may be created and then separated so they do not annihilate. In any case, the helicity is a pseudo-scalar and thus maintains a constant sign everywhere along the domain wall: thus, the entire Hubble volume is filled with helicity of the same sign. This is the origin of the Hubble scale correlations in the helicity and in B2. The correlation parameter £ which affects the magnitude of the fields plays no role in disturbing this correlation. If, however, there is hidden C P violating physics which somehow favours the formation of soliton domain walls rather than the anti-soliton, then there will be a net helicity throughout the universe. If this is the case, then this helicity would support the type of inverse cascade discussed in Chapter 3. Such a situation would be very similar to the corresponding asymmetry in baryon number (matter verses anti-matter) and likely associated with the same physics. As we mentioned, eventually, the electric field will be screened. The timescale for this is set by the plasma frequency for the electrons (protons will screen much more slowly) uip which turns out to be numerically close to A Q C D near the Q C D phase transition. The nucleons, however, also align on a similar timescale A Q ^ , and the helicity is generated on this scale too, so the electric screening will not qualitatively affect the mechanism.2 Finally, we note that the turbulence requires a seed which remains in a local region for a timescale set by the conductivity [56] a ~ cT /e 2 ~ A Q C D where for T = 100 MeV, c « 0.07 and is smaller for higher T. Thus, even if the domain walls move at close to the speed of light (due to vibrations), there is still enough time to generate turbulence throughout the Hubble volume. 2 If the screening were more efficient, then one might worry that the electric field would not last long enough to generate the helicity. 8. Conclusion 37 8. Conclusion 8.1 Summary Here is a brief summary of what we have done and the approximations that were made: In Section (4.2) we outlined the generic features that domain wall models should posses if they are to successfully generate primordial seeds by the method described in this thesis. We proceeded with the QCD domain wall model to estimate the magnitude of the effects.1 Making the approximation of thin, fiat, domain walls, we proceeded to calculate the averages of quantities like (^io^^) by reducing the interaction (6.1) to a 1 + 1 dimensional system. In this approximation we discarded the momenta in the x-y plane to get (6.5). We recapture the effects of these momenta by the degeneracy factors in Section (6.1). This approximation is valid only if the physics in the z direction is independent of motion in the x-y plane, and breaks down when thermal (or other) fluctuations are large enough that the physics in the z-t directions no longer decouples from the physics in the x-y plane. In making estimates like (6.17), another approximation we have made is to ignore back reactions. We have treated the domain wall as a static back-ground: in reality, the presence of fermions in the domain wall would affect the structure. What we say is that such back reactions will not change the 1 Further support for the existence of QCD domain walls comes from calculations pre-sented in [51] where it is shown that these walls almost certainly exist in the high density regime of QCD. Thus, domain walls seem to be important features at high density. Fur-thermore, if one accepts a conjecture on quark-hadron continuity at low temperature with respect to variations in the chemical potential fi [57], then one can make the following argument: If for large fi domain walls exist, but for low ji they do not, then there should be some sort of phase transition as one lowers fi. Thus, the continuity conjecture supports the existence of quasi-stable QCD domain walls at lower densities, at least down to the densities of hyper-nuclear matter. Coupled with the fact that gluon and quark conden-sates do not vary much as one moves from the domain of hyper-nuclear matter to the low density limit, one suspects that the qualitative picture holds even for zero density. Differ-ent arguments based on large Nc counting, also support the existence of the meta-stable QCD domain walls at zero chemicalpotential /x (See Appendix B.) 8. Conclusion 38 overall structure or scale of the phenomenon, however, it wil l definitely alter the quantitative results. Thus, estimates like (6.17) should only be taken as qualitative approximations to the structure in the domain wall. A com-prehensive analysis would take into account the effects of fermions on the domain walls through additional interactions to (5.3). These interactions, however, would not alter the £7(1) nature of the rj' field, and thus the basic domain wall structure would still be present. Also, it is unlikely that the back-reaction of the electromagnetic fields can substantially affect the do-main wall structure or the alignment of the spins. Indeed, the domain walls and the spin alignment are due to Q C D interactions on the scale A Q C D -Any back-reaction, would be suppressed at least by a factor of a ~ 1/137, thus, the quantitative results might be altered, but we expect the qualitative behaviour and orders of magnitudes to be preserved. The next step was to estimate the strengths of the generated fields by using dimensional arguments and considering a collection of dipoles aligned in the domain wall background arriving at the estimates (7.3). The actual fields generated will be sensitive to the geometry and dynamics of the domain walls: this is something that we do not understand very well. We have captured these effects in the unknown scale length £ but there is much more that could be said about this. To study these effects we will need to solve Maxwell's equations (7.1), however, in the non-trivial geometry of the domain walls, this will probably have to be simulated. Of particular importance is the question: Can larger fields be generated by motion of the domain walls? The interaction of the fields with the plasma is also important because the electric fields will be screened. Simple estimates, however, show that the screening timescale is at least as long as the other timescales. We have estimated the scaling of helical the magnetic turbulence (3.1) and showed that, if an inverse cascade can be supported, then the fields generated by domain walls at the Q C D phase transition would be of astro-physical interest. The behaviour of the inverse cascade has the possibility of eliminating this method of generating primordial seed fields: If no cascade action occurs beyond the Hubble scale, then there is no way to generate astrophysically interesting seeds from Hubble scale correlations at the Q C D (or earlier) phase transition (s). It is somewhat encouraging, though, that if the cascade does work as described by (3.1), then Hubble scale correlations at the Q C D transition will be increased to required order of magnitude to explain the correlations in clusters. The point we make here is that fields are generated and correlated on the 8. Conclusion 39 same length as the domain walls. Thus, the domain walls provide a natural mechanism for generating Hubble scale correlations. Without a better understanding of the dynamics of the domain walls, we cannot better estimate the field strengths, but it is possible that the gen-erated fields are quite large (nanogauss to microgauss scale) even without any amplification. It seems that the fields will still require some amplifi-cation by galactic dynamos, but these primordial seeds provide the large scale correlation lengths that have been difficult to achieve through other mechanisms. We have seen that domain walls at the Q C D phase transition may pro-vide a nice way to resolve some of the problems with generating large scale magnetic fields. It is important to note that the mechanism described here is seated in well-established physics and makes definite predictions. The only free "parameters" are the correlation length £ which affects only the strengths of the fields generated and the degree to which the inverse cas-cade works. These parameter are not really free, but represent a lack of understanding of the formation and dynamics of domain walls and of the inverse cascade mechanism. As our understanding is improved, the scale of this parameter will be fixed, and the method will make a definite prediction about the scale and strengths of the fields. In particular, the mechanism can be ruled out in several ways: First, if no inverse cascade functions, then Hubble scale correlations at any phase-transition are uninteresting. Second, if the cascade does work as described, then the method makes a definite prediction about the maximum possible size of the correlations produced. In particular, if new observations show that correlations exist on scales much larger than 1 Mpc, then other pri-mordial methods (probably rooted in inflation) must be considered. In any case, however, domain walls may affect other astrophysical phenomena as we shall discuss in the next section. 8.2 Speculations and Future Directions This mechanism has been a first attempt to describe astrophysical conse-quences of recently conjectured Q C D domain walls [2]. These walls may affect several areas of astrophysics, both through the field generation mech-anism described here and through other effects. We discuss several of these applications below. The most promising possibility to test the idea of primordial seeds is to measure extra-galactic and extra-cluster magnetic fields. This may be possible in the near future through measurements of anisotropies in the 8. Conclusion 40 cosmic microwave background (CMB) radiation spectrum [58, 59] or through non-thermal radiation from Compton scattering of relativistic electrons off of the C M B [60]. Current C M B observations [61] place upper bounds on the strengths of primordial fields, which the fields described in this thesis respect. There are other limits imposed on the strengths of primordial magnetic fields from nucleosynthesis production rates, all of which this mechanism respect. For a thorough review, see [7]. There are many possible consequences of primordial magnetic fields dis-cussed in [7], but we point out a particularly interesting possibility that might support domain wall generated fields. It seems that certain types of field structures might explain the apparent violations of the G Z K cutoff [62]. It is likely that primordial fields from domain walls may posses a planar structure that could assist in explaining these violations. Independent of the magnetic fields, Q C D domain walls might affect early cosmology in another way. As we discussed in the text, baryons are con-centrated on the wall causing inhomogeneities in the baryon density. While it seems that Q C D domain walls wil l decay before T = 1 MeV, the inho-mogeneity in baryon density must be redistributed by diffusion and other dissipative processes and thus inhomogeneities might persist that affect nu-cleosynthesis. Recent measurements of the C M B by B O O M E R A N G [63] and M A X I M A [64] lead to a value of baryon density $7^ is larger than the value allowed by the conventional model. The agreement can be achieved [65] through inhomogeneous big bang nucleosynthesis (IBBN) if regions of baryon inho-mogeneity are separated by a distance scale of about 70 km (at T = 1 MeV), and if these regions have a planar structure with high surface to volume ra-tio. Models have been proposed whereby such a planar structure can occur [66], however, we suggest that Q C D domain walls might automatically cre-ate this kind of structure with the appropriate scale if the baryon diffusion and other dissipative processes are sufficiently slow (or if something stabi-lizes the Q C D domain walls). Conversely, nucleosynthesis provides another constraint to check the validity of Q C D domain walls. Additional work is required [67] before any definite statements can be made. At this point, we would like to comment on a speculative application of Q C D domain walls that may be of significant astrophysical interest. Given that Q C D domain walls are very stable at high densities [51], it is not in-conceivable that the accumulation of baryons along the domain wall by a mechanism similar to that described in Chapter (6) may stabilize the do-main walls. If this is the case, then Q C D domain walls might form the seeds for primordial baryonic compact objects. Such objects would certainly affect 8. Conclusion 41 nucleosynthesis, the cosmic microwave background radiation spectrum and possible structure formation. The result, however, would be stable, cold, "invisible" baryonic matter that would contribute to the dark matter. In addition, this matter would have Q C D scale cross-section recently suggested for dark matter [68] to explain several discrepancies with the standard cold dark matter picture. Without further calculations, it is not possible to make even qualitative predictions about these speculations, but, as our under-standing improves, such a model would be able to make definite predictions. Furthermore, the physics of Q C D domain walls may lend itself to signatures that can be tested in relativistic heavy ion collisions. This might provide some concrete foundations for Q C D phenomena affecting cosmology and astrophysics. Likewise, the accuracy of nuclear abundance measurements and C M B anisotropy measurements could provide serious constraints on the behaviour of Q C D phenomena related to domain walls. Thus, while we cannot yet seriously advocate the idea of compact primordial baryonic ob-jects forming from this mechanism, one should at least keep them in mind when studying these and similar processes. We would like to close by emphasizing the relationship that is developing between astrophysics and particle physics at the Q C D scale. Particle physics provides concrete models for astrophysical phenomena that, as our under-standing of fundamental physics increases, has definite predictive power. Thus, there is the potential to ground many astrophysical phenomena in well founded, testable physics. In return, astrophysical observations provide a means for testing particle physics theories under conditions not possible on earth. 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D15, 2929"(1977). [103] A. A. Migdal and M . A. Shifman, Phys. Lett. B114, 445 (1982). [104] R. Gomm, P. Jain, R. Johnson, and J. Schechter, Phys. Rev. D33, 801 (1986). [105] B. D. Wandelt, R. Dave, G. R. Farrar, P. C. McGuire, D. N. Spergel, and P. J. Steinhardt (2000), arXive:astro-ph/0006344. [106] U. Ascher, J. Christiansen, and R. Russell, A C M Trans. Math Software 7, 209 (1981), this paper contains examples where use of the code is demonstrated. 48 Part II Appendices A. Observations 49 A . Observations The current status of observations of cosmic magnetic fields is reviewed in [7] and described in detail in [5, 6]. In this appendix we shall briefly describe the methods and results. The results quoted are from the review [7] unless otherwise noted. A.l Techniques There are three primary means through which magnetic fields are observed: 1. Zeeman splitting of spectral lines, 2. Synchrotron radiation of free relativistic electrons and, 3. Faraday rotation measures of polarized light. A.1.1 Zeeman Splitting The basic idea behind the method of measuring Zeeman is to observe the spectral absorption patterns in the light received from distance stars and galaxies. The light from distant sources passes through regions filled pri-marily with hydrogen and helium gas. In doing so, light is absorbed at the frequencies where these molecules exhibit resonances, or bound states.1 In the absence of a magnetic field, the spectral lines corresponding to states of spin up and spin down with the other quantum numbers in common are practically degenerate in energy. A n external magnetic field, however, couples to the spin through an interaction H • S splitting the energy of these two states. This effect, the Zeeman splitting, is proportional to the strength 1 This technique is, for example, commonly used to determine how quickly distant sources are receding from us: for example, the light undergoes a redshift (similar to the Doppler shift in sound from a receding ambulance) which lowers the frequency of these spectral lines. B y comparing the measured lines with the absorption lines one can estimate the relative velocity of a distant galaxy through this "red-shift", and thus, given a model for the universe expansion rate, one can estimate the distance of the object. A. Observations 50 of the magnetic field and is measurable. It gives.a direct measurement of the magnetic field strength in the region of the absorbing gas. Unfortunately, the effect is quite small and is of little use for sources that lie outside of our own galaxy. Thus the two other techniques are used for extra-galactic sources. A.1.2 Synchrotron Emissions Many of the electrons in galaxies are moving very rapidly with relativistic velocities. In the presence of a magnetic field, these electrons are acceler-ated perpendicular to the field and thus radiate energy called synchrotron radiation. The characteristics of this type of radiation are well known (see for example [70]). In particular, the polarization of the radiation is pre-dominantly in the plane of the orbit: i.e. perpendicular to the magnetic field. The intensity of the radiation, however, depends on the density of the electrons emitting the radiation, and so one must assume some model for this electron density n e . Once this is assumed, by measuring the intensity of the synchrotron radiation from a source on can estimate the strengths of the magnetic fields at the source which is proportional to neB2. This method works for very distance sources, but requires an indepen-dent estimate the electron density. For very hot electron gases, ne may be estimated by measuring, for example, X-ray emissions. Such an approach sometimes works in galaxy clusters, however, it it often difficult to directly measure ne directly, especially where the electron gas is cool or rarefied as it is in the inter-galactic medium (IGM). In these cases, assumptions about the equipartition of magnetic and plasma densities, or other models must be used to obtain estimates of ne. A.1.3 Faraday Rotation Measures (RMs) The third techniques used to measure distant magnetic fields is to observe the Faraday rotation of the plane of polarization of synchrotron emissions. The idea here is based on the Faraday effect (see for example [70]). Linearly polarized light can be decomposed as a linear combination of left handed and right handed circular polarizations. These two circularly polarized com-ponents propagate differently in certain dielectric media when immersed in a magnetic field. This causes the relative phase between to two polariza-tions to shift polarization resulting in the rotation of the linear plane of polarization. The synchrotron radiation is highly linearly polarized and so A. Observations 51 provides a good source. The result is that we measure a different plane of polarization than that emitted from the source:2 0(w) ~ £ j ne(z)B(z) • dz. (A. l ) Here, 9(ui) is the angle of rotation of the the plane of polarization between the source and the observer; OJ is the frequency of the light; ne(z) is the electron density along the line of sight (parameterized by z) and B(z) is the magnetic field along the line of sight. Thus, if one can estimate the electron density along the line of sight and initial angle of polarization, then one can measure the average strength B \ \ of the component of the magnetic field along the line of sight. The problem of determining the total field strength is also exacerbated by the possibility of field reversals which wil l cancel some of the effects. The first problem is solved by comparing several different frequencies and triangulating back to the source. Again, determining ne is typically difficult as is counting the field reversals, though pulsar frequency and decay rates can also help determine these for nearby sources. The advantage of rotation measures is that, apart from the preceding limitations, they are effective up to very far distances and can give some clues about the structure of the magnetic fields along the line of sight. In practice, all three methods are compared for near sources and the first two are used to calibrate and validate the results from the rotation measures. Rotation measures are then used to estimate distant fields by making various models of the electron density n e . The latter results for distant sources, however, are still very uncertain and represent, at best, model dependent upper bounds on the field strengths, especially in the inter-galactic medium. A.2 Results A.2.1 Local Fields Locally, within our galaxy, the average field strength appears to be 3-4 /xG. The field orientation is the same to the order of several kiloparsecs which is comparable to the size of the galaxy, and two reversals have been observed between the arms of the galaxy. These two reversals may suggest that the field is symmetrical. The average energy density B2/8TT is approximately 2 For distant sources, the cosmological redshift must also be accounted for in the fre-quency dependence and the integration measure. See [7] for a review. A. Observations 52 equivalent to an equipartition of magnetic energy, radiation density from the cosmic microwave background (CMB) and small-scale turbulent motion. In several other galaxies, similar field strengths have been observed, though there are galaxies where the filed strength appears to be stronger than equipartition would suggest. The structure of fields seems to vary: In some galaxies there is a definite axially symmetric structure while in others there appears to be no structure. Structure on the same scale and symmetric with the galaxy structure provides evidence that gravitational mechanisms are responsible for either amplifying, generating or, at least, aligning the fields. A.2.2 Clusters Galaxy clusters provide a means of studying the inter-cluster medium (ICM). As well, some clusters have well observed detailed magnetic field structures. Typically, in the I C M , fields with strengths 1-10 / J G exists with reversal lengths of 10-100 kpc. The largest observed coherence lengths in clusters is on the order of 1 Mpc. In the Hydra A cluster, for example, rotation measures imply a ~ 6 / J G field coherent on scales over 100 kpc. Superimposed onto this background are fields of 30 / i G tangled on a smaller scale. In some clusters, fields as large as 70 / J G have been observed. Detailed observations of these strong central fields suggest that they may be organized into filamentary structures [71]. This may suggest that the fields are associated with plasma flows. A.2.3 Intergalactic Media Fields outside of galaxies and clusters cannot be measured as easily, but distant radio sources can constrain the field strengths. In these regions, the electron gas is very rarefied and the coherence length of the fields is unknown. Cosmological models, however, provide some estimates. For fields coherent on the 1 Mpc scale, current rotation measure to a redshift of 2.5 one estimates limits of . B I G M ^ 1 0 - 9 G [5]. A n interesting measurement that may be possible in the future is to check for distortions in the cosmic microwave background due to magnetic fields present in the extragalactic media now and primordially [8, 59, 72]. Present constraints are consistent with the 10~ 9 G estimate above [58, 61]. A. Observations 53 A.3 Summary While it appears that the structure of astrophysical magnetic fields is some-how correlated with galactic structure, the mechanism of these correlations is not well understood.. In particular, the large correlation lengths of fields in clusters of up to 1 Mpc are apparently difficult to explain using conven-tional dynamo mechanism. While the current debate about the origin of these fields is far from resolved [7], it seems likely that a combination of primordial sources with galactic dynamics conspire to form the currently observed fields. One would really hope for some positive measurement of intergalactic fields or much tighter bounds. Future C M B anisotropy mea-surements might reach a sensitivity to provide such constraints. B. QCD Domain Wall Structure 54 B . Q C D Domain W a l l Structure B.l Introduction This appendix contains a discussion about the Q C D domain wall structure. It represents a distillation of the paper [2] with added background material. The starting point of the analysis is the low energy effective theory of Q C D described in [73]. Experience with supersymmetric (SUSY) models demon-strated that the effective Lagrangian approach is a very effective tool for the analysis of large distance dynamics in a strong coupling regime. There are two different definitions of an effective Lagrangian in quantum field theory: One of them is the Wilsonian effective Lagrangian describing the low energy dynamics of the lightest particles in the theory. The idea here is that one integrates over high energy modes so that only low energy degrees of freedom remain. In Q C D , this is implemented by an effective chiral Lagrangian for pseudoscalar mesons. Another type of effective Lagrangian is defined by taking the Legendre transform of the generating functional for connected Green's functions to obtain an effective potential. This object is useful in addressing questions about the vacuum structure of the theory in terms of vacuum expectation values (VEVs) of composite operators—these V E V s minimize the effective action. This approach is well suited for studying the dependence of the vacuum state on external parameters (such as the light quark masses or the vacuum angle 9 in QCD) . However, there is no way of recovering the kinetic term through this approach: Hence, it cannot be used to study any dynamic features of the theory such as scattering S-matrix elements. The utility of the second approach for gauge theories was recognized long ago for supersymmetric models, where the anomalous effective potential was found for both the pure supersymmetric gluodynamics [74] and supersym-metric Q C D (SQCD) [75] models. Properties of the vacuum structure in the SUSY models were correctly understood only after analyzing this kind of effective potential. The structure of this appendix is given as follows: B. QCD Domain Wall Structure 55 Section B.2: Here we review the properties of the Q C D effective Lagrangian [42, 76] which is a generalization of the D i Vecchia-Veneziano-Witten ( W W ) Lagrangian [47, 48] to include terms subleading in 1/NC as well as to account for a constraint due to the quantization topologi-cal charge.1 One should emphasize from the very beginning that the specific form of the effective potential used in this paper is not critical for the present analysis: only the topological structure and winding 9 —> 9+2im—a consequence of the topological charge quantization—is essential. Section BA: This is devoted to the analysis of the structure of two different types of the domain walls for the specific case of two flavours Nf = 2. The results presented in the main body of the thesis are a simplification of those presented here. In this section, the role of the axion field in these domain walls is also discussed: in particular, we show that these walls can exists with or without a dynamical axion. Section B.5: Here we argue the Q C D domain walls are not stable on the quantum level due to a tunneling phenomena. We estimate their life-time which is macroscopic, but short on a cosmological scale. Thus, these domain walls are not a cosmological problem and may be quite useful in explaining certain astrophysical phenomena as discussed in the main body of this thesis. B.2 Effective Lagrangian and 9 dependence in QCD Our analysis begins with the effective low energy Q C D action derived in [42, 76], which allows the ^-dependence of the ground state to be analyzed. The essential physics for the effective theory follows from the symmetries of Q C D which has the following Lagrangian density: £ Q C D = - ^ 2 GG + *(* p - M)tt + GG (B.l) This is a gauge field theory with Nf x Nc fermions (quarks) in the vector The whole theory has an SU(NC) colour symmetry which remains exact (quarks of different colour but the same flavour have the same mass and charges for example) and so I shall not refer to colour except where it appears 1 Such a generalization was also motivated by S U S Y considerations [77], see also [78] for a review. B. QCD Domain Wall Structure 56 numerically. We find that nature is well described if one takes Nf = 9 and Nc = 3. It turns out in nature that the masses of the quarks satisfy the following relationship: mu,m,i <C ms <C mt,m(,,m c . Thus, it is often a good ap-proximation to treat the lightest two or three quarks as massless and the other quarks as having infinite mass. In these limits (sometimes referred to as "QCD Lite"), one neglects the effects of the heavy quarks and considers Nf = 2 or Nf = 3. B.2.1 The U(l) Problem If the lightest quarks were truly massless, then the matrix M would be zero and one could separate the left handed and right handed helicities which would transform independently under the group G = U(Nf)L <g> U(Nf)R. In this vacuum state, however, this symmetry is broken to yield a non-zero quark condensate ( ^ L ^ R ) = const ^ 0. The remaining symmetry is H = U(NF)L=R and hence one expects, from Goldstone's theorem, dim(G !) — dim(iz") = 2NJ - (Nf) = Nj massless Goldstone bosons.2 This symmetry, however, is not exact, and so we expect that these par-ticles should in fact not be massless, but should acquire a mass perturba-tively, proportional to the amount by which the symmetry is broken. In any case, due to the large difference in masses between the lightest and heaviest quarks, one expects that there should be 9 = 3 3 (i.e. Nf = 3) particles which behave almost like Goldstone bosons (aka pseudo-Goldstone bosons). In nature, however, only eight such particles are found: the pions . These particles are light and well approximated by Goldstone bosons but there is no ninth pseudo-Goldstone boson. This is often referred to as the U(l) problem. Later, with the discovery of instantons, it was found that Q C D had an anomaly: the U(1)L=R classical symmetry of the theory was explicitly broken by quantum effects. Effecting this symmetry causes a shift in the 9 parameter which can affect the physics of the system. Thus, the full Q C D Lagrangian must include the C P violating 9 term proportional to G G . In any case, the proper description of the symmetries of the theory is reduced because of this explicit breaking of the symmetry to the fol-2 Common notation is to call this remaining symmetry the "vector" symmetry and the broken symmetry the "Axial" symmetry, thus one often sees SU(Nf)v = SU(Nf)v and SU(Nf)A- One should also note that this is a special case in that G = H®H is a product of the residual symmetry group. The Goldstone bosons correspond to generators of the coset space G/H which in general is not a group. B. QCD Domain Wall Structure 57 lowing: G = SU{Nf)L <g> SU{Nf)R -> H = SU{Nf)L=R, which yields Nj — 1 = 8 pions resolving the £7(1) problem. Further explicit breaking of this SU(Nf)i=ji symmetry by unequal quark masses breaks the mass degeneracies of the pions giving rise to the kaons. B.2.2 Chiral Effective Lagrangian The idea behind the formulation of a chiral effective Lagrangian is to con-struct an effective theory of the lightest particles, in this case the pions, that has the desired symmetry properties. The symmetry giving rise to the pions is spontaneously broken by the chiral condensate (^r^r) which acquires a non-zero magnitude v = \(^L^R)\- The residual symmetry associated with the pions are the phases of this condensate. Thus, we can parameterize the fluctuations about the ground state as a matrix ( ^ L ^ R ) = — K ^ ^ I U j j where i,j are flavour indices. The symmetries L € U(Nj)i and R € U(NJ)R are effected by U —>• L U R J . In keeping with the goal of describing the low energy spectrum of the theory, we wish to keep only terms with the lowest number of derivatives. In the massless theory, we see that the Lagrangian £eff = T r ^ U ^ U ) (B.2) respects these symmetries. Now, consider adding a mass term. When the masses are non-zero but degenerate, the symmetries should break down to SU(Nf)L=R- Furthermore, when the masses are not degenerate, this symmetry should also break. The following Yukawa interaction mass term satisfies these symmetry constraints: CY = T r ( M U - t - U t M t ) . (B.3) The effective theory formed by these two terms is the starting point for many analyses. We would also like to include the effects of the 6 parameter on this effective theory. The effective description of this provided a nice explanation of the formal resolution to the £7(1) problem and was first dis-cussed in [47, 48] where, in the large Nc limit, the effects of the 9 parameter are incorporated through a term like: E / i log det U - 0 C * = 2 \ ~N~C These terms combined form the Di Vecchia-Veneziano-Witten ( W W ) effec-tive Chiral Lagrangian which describes the low energy effective physics and dependence of this physics on small angles 6 (or correspondingly large Nc.) B. QCD Domain Wall Structure 58 In addition, this term explicitly violates the U(1)L=R symmetry and gives rise to the mass of the rj' particle. Thus, the U(l) problem was solved by associating the mass of the rf particle with the vacuum energy E instead of the pion mass scale M ~ (^r^r) mq. The connection with the bare physical fields and the respective decay constants is made through the definition of U in terms of the chiral fields: U = exp / * A / A 7 / fa U+U = 1. (B.5) Here we have the bare pseudoscalar fields 7r ,^ the bare eta' field rj and the respective decay constants ~ /„• = 130 MeV. The matrices generate the fundamental representation of SU(Nf) (i.e. the Gell-Mann matrices of for 5/7(3)). (The normalizations are chosen to put the kinetic term in canonical form: —1/4 Tr fylWU.) The W W effective Lagrangian was later modified to include higher order effects [42, 76]. This Lagrangian is the starting point for our discussion. In terms of U the low-energy effective potential is given by: N c ~ 1 ' , n Z + i l o g D e t U - 0 — 1 J V c _ 1 r /2JT WQCD{0,U) = l im — l o g V exp \VE cos — NK + ^ y T r ( M U + U t M t ) | . (B.6) A l l dimensional parameters in this potential are expressed in terms of the Q C D vacuum condensates, and are well known: M = diag(m* |(3P\1/J)|); and the constant E is related to the Q C D gluon condensate E = (If^G 2 ), where numerically 3b = IINC - 2Nf; quark condensate (***') ~ -(240 M e V ) 3 , gluon condensate (^-G2) ~ 1.2 x 1 0 - 2 G e V 4 . It is possible to argue that equation (B.6) represents the anomalous effective Lagrangian realizing broken conformal and chiral symmetries of Q C D . The arguments, as presented in [79] are that Equation (B.6): 1. correctly reproduces the W W effective chiral Lagrangian, [47, 48] in the large Nc limit [ For small values of (9 — i logDet U), the term with I = 0 dominates the infinite volume limit. Expanding the cosine (this corresponds to the expansion in 1/NC), we recover exactly the W W effective poten-tial [47, 48] together with the constant term -E = - ( 6a s / ( 327 r )G 2 „ ) B. QCD Domain Wall Structure 59 required by the confermal anomaly: Wvvw(0,V,U^) = -E - (fl _ ilogdetU)2 Li - ^ T r ( M U + U t M t ) + . . . (B.7) here we used the fact that at large Nc, E/N2 = — {V2)YM is the topological susceptibility in pure Yang-Mills theory. Corrections in 1/NC stemming from Equation (B.6) constitute a new result of [42, 76].] 2. reproduces the anomalous conformal and chiral Ward identities of Q C D , [ Let us check that the anomalous Ward Identities (WI's) in Q C D are reproduced from Equation (B.6). The anomalous chiral WI's are automatically satisfied with the substitution 9 —> (9 — i log det U) for any Nc, in accord with [47, 48]. Furthermore, it can be seen that the anomalous conformal WI's of [80] for zero momentum correlation functions of the operator G 2 „ in the chiral limit mq —> 0 are also satisfied when E is chosen as above. As another important example of WI's, the topological susceptibility in Q C D near the chiral limit will be calculated from Equation (B.6). For simplicity, the limit of SU(Nf) isospin symmetry with Nf light quarks, mq -C A Q C D will be considered. For the vacuum energy for small 9 one obtains [42, 76] Evac(9) = -E + mq{**)Nf cos (J^j + 0 (m 2 ) . (B.8) Differentiating this expression twice with respect to 9 reproduces the chiral Ward identities [81, 82]:. lim i J dx*** (0\T{^GG{x) ^ G G ( O ) } |0> = Other known anomalous Ward Identities of Q C D can be reproduced from Equation (B.6) in a similar fashion. Consequently, Equation (B.6) reproduces the anomalous conformal and chiral Ward identities of Q C D , and in this sense passes the test for it to be the effective anomalous potential for QCD.] B. QCD Domain Wall Structure 60 3. reproduces the known results for the 9 dependence [47, 48]. [ As mentioned earlier, our results are similar to those found in [47, 48]. A new element which was not discussed in the 1980's is the procedure of summation over I in (B.6). As we shall discuss in a moment, this leads to the cusp structure of the effective potential which seems to be an unavoidable consequence of the topological charge quantization. 3 These singularities are analogous to the ones arising in SUSY models and show the non-analyticity of phases at certain values of 9. The origin of this non-analyticity is clear, it appears when the topological charge quantization is imposed explicitly at the effective Lagrangian level.] A n interesting note is that, in general, the 9 dependence appears in the combination 9/Nc, (see equation (B.6)) which naively does not provide the desired 2ir periodicity for the physical observables; equation (B.6), however, explicitly demonstrates the 2ir periodicity of the partition function. This seeming contradiction is resolved by noting that in the thermodynamic limit, V —> oo, only the term of lowest energy in the summation over I is retained for a particular value of 9, creating the illusion of 9/Nc periodicity in the observables. Of course, the values 9 and 9 + 2ir are physically equivalent for the entire set of states, but relative transitions—switching branches— between different 9 states have physical significance. Exactly this important property will be used in what follows for the construction of the domain walls which are classically stable, non-trivial interpolations between physically equivalent vacuum states (see below). Consequently, the 9 dependence in the infinite volume limit appearing in the combination 9/Nc is a result of being stuck in a particular state: The local geometry gives the impression of 9/Nc periodicity but the topology gives the true 9 periodicity. The reader is referred to the original papers [42, 76] for more detailed discussions of the properties of the effective potential (B.6). In the next section we shall discuss different types of domain walls which interpolate between various vacuum states, but first we should study the classification of vacuum states themselves. In order to do so, it is convenient 3 This element was not explicitly imposed in the approach of [47, 48]; the procedure was suggested much later to cure some problems in SUSY models [77] and references therein; an analogous construction was discussed for gluodynamics and QCD in [42, 76]. B. QCD Domain Wall Structure 61 to parameterize the fields U as 4 0 U = o \ 0 0 \ o o o such that the potential (B.6) takes the form 1 (B.10) V(4>u9) = -Ecos (J^O - - ^ M i C o s & . ( B . l l ) The minimum of this potential is determined by the following equation: M i ( B . 1 2 ) At lowest order in 1/NC this equation coincides with that of [47, 48]. For general values of Mi/E, it is not possible to solve Equation (B.12) analyti-cally, however, in the realistic case eu, ed <C 1, es ~ 1 where E\ = NcMi/E, the approximate solution can be found: sin 4>u sin (fid sin 6* md sin# yjm2 + md + 2mumd cos 6 mu sin 9 yjm\ + m2d + 2mumd cos 9 0(eu,ed) . + 0(eu,ed) , + 0(eu,ed) , (B.13) This solution coincides with the one of [47, 48] to leading order in eu,ed. In what follows, for numerical estimates and for simplicity, we shall use the SU(2) limit mu = md <C ms where the solution (B.13) can be approximated as: 2 ' <t>u -6 ~ <fis - 0 , 0 <9 < TT, TT <9 < 2TT, (B.14) etc. 4 In classifying the ground states of the theory, it is sufficient to consider a maximally commuting subset of operators. In this case, where the operator valued fields are operators in U(Nf), such a set of operators is spanned by a Cartan subalgebra of U(Nj). This can always be chosen to be diagonal. For U(3) with the Gel l -Mann basis, for example, the Cartan subalgebra is obtained from A 3 , As and the identity (U(l)) matrices. Together, these span the space of 3 x 3 diagonal unitary matrices. B. QCD Domain Wall Structure 62 Once solution (B.14) is known, one can calculate the vacuum energy and topological charge density Q = (0|§£GG?|0) as a function of 9. In the limit mu = m-d = m, and (dd) = (uu) = one has: Vvac(9) ~ V6=0 + 2m|(tttt)|(l - |cos § |) (0\%GG\0) = -^^1 = - m | ( * * ) | 8 i n f . (B.15) As expected, the 6 dependence appears only in combination with the quark mass m and goes away in the chiral limit. One can also calculate the chiral condensate (^L^fR) in the 0 vacua using solution (B.14) for vacuum phases: <e|**|0> = cos( | )<O|**|O) f l = O ) <0hT»75*|0> = - sin(f) <0|*tt|0>,= 0 (B.16) A remark is in order. As is well known, in thermal equilibrium and in the limit of infinite volume, the \9) vacuum state is a stable state for all values of 6. Thus, it is possible to conceive of a world with ground state \6) where 6 =fi 0. The physics of this world would be quite different from that of our own: In particular, P and C P symmetries would be strongly violated due to the non-zero value of the P and C P violating condensates (B.15,B.16). Despite the fact that the state \9) has a higher energy than |0) (B.15), it is stable because of a superselection rule: There exists no gauge invariant observable A in Q C D that can communicate between different \9) states, i.e. (#'|.4|0) ~ 8(9 — 9') for all gauge invariant observables A. Therefore, there are no possible transitions between these states [83-85] and any such state \9) may happen to be the ground state for our world. On account of this superselection rule, one might ask why 9 = 0 is so finely tuned in our universe. Indeed, within standard Q C D , there is no reason to prefer any particular value of 9. This is known as the strong C P problem. One of the best solutions to this problem has been known for many years: introduce a spontaneously broken symmetry (Peccei-Quinn symmetry [86]). The corresponding pseudo-Goldstone particle—the axion [39, 40, 82, 87-92]—behaves exactly like the parameter 9 but is now a dynamical field, thus we can absorb the parameter 9 by redefining the axion field a(x) and set 9 = 0. The axion is now dynamical and so the corresponding states \9) ~ \a(x)) are no longer stable: the axion field relaxes to the true minimum 9 ~ a(x) = 0 (B.15). The axion is included in the potential (B.6) through the Yukawa interaction (B.3) T r ( M U + l ^ M t ) T r ( M U e i a + U t M t e _ i a ) . (B.17) B. QCD Domain Wall Structure 63 and kinetic term / 2 ( f y a ) 2 / 4 . Examining the potential (B.6) we see that the parameter 8 can be absorbed into the U(l) phase of U which in turn can be removed by a redefinition of the axion field a —t a + 9/Nj. Although axions have not been detected and experiments have ruled out the possibility of the original electroweak scale axion [87, 88] there is still an allowed window with very small coupling constant /„•/'fa <C 1 emphasizing that the axion arises from a very different scale than the electroweak or Q C D . Axions with this scale are also good dark matter candidates (see for example [38-40, 92]). The axion thus provides a way for the vacuum state to relax to the lowest energy state \9 = 0) (B.15). In the following we shall consider two types of domain walls: Axion domain walls where 9 is the dynamical axion field a and Q C D domain walls where 9 is the fundamental parameter of the theory. In the latter case, we assume that the strong C P problem has been solved by some means and take 9 = 0 to be fixed. The effective potential (B.6) can be used to study the vacuum ground state (B.15,B.16) as well as the pseudo-Goldstone bosons as its lowest en-ergy excitations. In particular, one could study the spectrum as well as mixing angles of the pseudo-Goldstone bosons by analyzing the quadratic fluctuations in the background field (B.15,B.16). We refer to the original papers [42, 73, 76] on the subject for details, but here we want to quote the following mass relationships for the rf meson to be used in the following discussions: f ^ = ^ f E + W E ^ I ( 0 | ^ * i 0 ) | + O(m 2 ) . (B.18) c -{u,d,s} This relation is in a good agreement (on the level of 20%) with phenomenol-ogy. In the chiral limit this formula takes especially simple form < = J§T2E> (B.19) which demonstrates that, in the chiral limit, the rf' mass is proportional to the gluon condensate, and is therefore related to the conformal anomaly. What is more important for us in this paper is that the combination on the right hand side of this equation exactly coincides with a combination describing the width of the Q C D domain wall (see Equation (B.38)). For this reason, the properties of the Q C D domain walls are dominated by the rf field. B. QCD Domain Wall Structure 64 B.3 Topological Stability and Instabilities The domain walls that we will discuss are examples of quasi-stable topolog-ical defects: classical solutions to the equations of motion which are stable due to the topological configurations of the fields. Examples of topological configurations abound in the literature, for example, Instantons, Skyrmions, Strings, Domain walls etc. [69, 93]. The basic idea is that the theory con-tains some conserved charge density that is a total derivative. In this case, the total charge Q = f qQdzx is quantized and represents some topo-logical property of the fields such as winding or linking. Magnetic charge in the Georgi-Glashow model is an example (see [93] for details). The essential point is that the charges Q are exactly conserved quantities: the subspace of configurations with Q = Q\ is orthogonal to the subspace with Q = Q2 7^  Q\- In particular, Q2 — Q\ = n so there is no continuous way to vary a configuration from one subspace to the other and thus there is no overlap. Thus, objects with non-zero charges are absolutely stable: even if have an energy higher than the true vacuum state where Q = 0. Referring back to our effective theory, we shall show that there is a conserved topological charge associated with domain wall configurations. Physically, the states 9 = 0 and 9 = 2im are identical: they represent the same state. However, as one can see from Equation (B.14), the solutions for the ground states corresponding to 9 — 0 and 9 = 2-K are not described in the same way: 9 = 0 corresponds to (f>u = (fid = 0 while 9 = 2n corresponds to 4>u = 27T, (fid = 0. It clear that the physics in both these states is exactly the same: If we lived in one of these state and ignored the others, then we could assign an arbitrary phases ~ 2nn for each cfiu or (fid separately and independently. However, if we want to interpolate between these states to get feeling about both of them, the difference in phases between these states can no longer be a matter of choice, but rather is specified by Equation (B.14). This classification arises because the singlet combination (fis = ^2<fii really lives on a U(l) manifold which has the topology of a circle. The integer n is only important if we are discussing transitions around the U(l) circle: in this case, it is important to keep track of how many times the field winds around the center. Thus, n is a topological winding number which plays an important role when the physics can interpolate around the entire U(l) manifold. We illustrate this idea in Figure B . l . Here, we show three topologically distinct paths in a two-dimensional space with an impenetrable barrier in the center. The paths that wind around the barrier cannot be deformed into the other paths. Each path is characterized by a winding number n. B. QCD Domain Wall Structure 65 Fig. B.l: Three examples of transit ions that are topologica l ly different. Pa ths A and B w i n d around the hold once (n = 1) and twice (n = 2) respectively whereas pa th C does not w i n d (n = 0). P a t h C can be shrunk to a point whereas the others cannot. E a c h pa th is said to belong to a different homotopy class. O n l y paths A and B are affected by the topology of the space. P a t h C might imagine that i t is l i v i n g in a space w i t h no hole. A n example of the situation described above is the well-known 1 + 1 dimensional sine-Gordon model defined by the Lagrangian £ S G = ^ ( fy^) 2 — Acos(c/>). Here, the topological current = ^e^d^cf) and is related to the (f> -> (f) + 2-KU symmetry of the ground state. The well known soliton and anti-soliton (kink) solutions are absolutely stable objects with topological charges Q = ± 1 . These cannot be recovered by the standard methods of quantum field theory when one starts from the vacuum state (<p) = 0 and ignores the topology. In our case, an analogous ground state symmetry is realized by the procedure of summation over I in equation (B.6), which makes symmetry 9 —y 6 + 27rn explicit. Thus, in 1 + 1 spatial dimensions we have the anal-ogous stable objects. The fact that we actually consider 3 + 1 dimensions means that the objects are not point-like solitons as they are in the sine-Gordon model, but rather, are two-dimensional domain walls with finite surface tension. B. QCD Domain Wall Structure 66 B.3.1 Higher Degrees of Freedom What we have said up to this point is well known. In the sine-Gordon model, the solitons are absolutely stable objects as can be seen by the fact that they are associated with the conserved current = ^^d^cp (here, the indices run over the 1 + 1 dimensions z and t). The conservation is trivial: d^q^ = ^e^d^d^cp = 0. The corresponding topological charge is Q = fqtdz = ± J -dz(j)(z)dz = i ( 0 2 = _ o o - <j>z=+00) = n- - n+ which is described by the winding number n = n + — n _ . Here, the field is in a vacuum state <t>z=±oo — 2rm± at infinity. Thus, we see that the charge is absolutely conserved and is integral. In our effective theory (B.6), we consider an analogous conserved current <fo = -^-e^Tr tfd»U = -U^d" T = jjU^ d^ s (B.20) i where we have introduced the notation (f>s = J2i 4>i ~ v' * n a t w e shall use later for the isotopical singlet (rf) field. By the same argument, we see that the two dimensional domain walls of the theory (B.6) are absolutely stable.5 In the sine-Gordon model, this is the end of the story: the solitons are absolutely stable. In our effective Lagrangian (B.6), however, we have neglected the gluon degrees of freedom. In reality, however, the gluon degrees of freedom are not very heavy. Thus, we must consider these extra degrees and look at how they affect the charge conservation. What we find is that, when we account for the extra gluon degrees of freedom, the topology of the fields is no longer restricted to the U(l) manifold. These extra degrees of freedom allow the domain walls to continuously deform and to decay so that the ground state exists everywhere. To see how an extra degree of freedom can change the topology, consider Figure B.2. Here we have added a third dimension to show that the barrier was actually a "peg" of finite height. Now that we can move in the extra dimension, we can use this degree of freedom to "lift" the paths over the "peg": thus, they are no longer topological!/ stable. In the Q C D analogue, paths A and B represents domain walls (path C would is a trivial closed loop 5 One might think that, since the domain walls directly involve the rf field, that the stability of the 77' particle might affect the stability of the domain walls. This is not so. Even in the effective theory (B .6), the rj particle can be considered as unstable decaying rj —• 27 for instance. This instability is related to the fact that the rj number charge is not conserved. . Irrespective of this non-conservation, the current (B.20) is still perfectly conserved. The only way for these domain walls to decay is by violating the conservation of this current. This is what we consider next. B. QCD Domain Wall Structure 67 which would relax to a point representing the same vacuum state everywhere with no domain wall.). There is an energy cost to "lift" the path over the barrier and at low temperatures T >C A Q C D there is not enough energy to do this, so classically, the domain walls are stable. It is still possible, however, for the walls to overcome the barrier by tunneling through the barrier. The tunneling probability, however, could be low due to the height of the obstacle and hence the lifetime of the walls could be much larger than the A Q C D scale which one might naively expected for standard Q C D fluctuations. See Section B.5 for details of the dynamics of the gluon fields. We should also remark that with each winding, the domain walls become more energetic. Walls with a large number of windings either have enough energy to rapidly unwind, or else separate spatially forming several domain walls of winding number ~ ± 1 . For this reason, we shall discuss in this, paper only the simplest walls which wind once. Fig. B.2: Here we show the same picture as in Figure B . l except that we show the third dimension. Here we can see that all the paths are now homotopically equivalent. We can deform the paths by "lifting" them over the obstacle so that we can unwind them. If the paths were strings with some weight, then it would require some energy to "lift" the strings over the obstacle. If this energy was not available, then we would say that, classically, the configurations that wind around the "peg" are stable. Quantum mechan-ically, however, the strings could still tunnel through the "peg", and so the configurations are unstable quantum mechanically. The probability that one string could tunnel into another configuration would depend on the height of the "peg". To tie this picture together, consider the formerly conserved current (B.20) and effective Lagrangian (B.6) and ask: from where do the phase field fa come? These phases arise from some sort of complex field $ = pe%<^s. B. QCD Domain Wall Structure 68 In general, we must consider the dynamics, not only of the phase <f>s, but of the radial component p = |$ | . We assume that this field lives in some sort of Mexican-hat potential with approximate symmetry </>s —>• c/>s + a and minimum valley where (p) = 1. This approximate symmetry is sponta-neously broken by the vacuum expectation values of the condensate ($) = exp(i(</>s)). To recover the effective Lagrangian: we "integrate" over the "heavy" p degree of freedom by setting p = (p) = 1 equal to its classical ex-pectation value and ignoring the quantum fluctuations about this minimum. This must reproduce the effective potential (B.6) with the pseudo-Goldstone field 4>s (rf a n d in the real theory). The appropriate Mexican-hat poten-tial reproducing (B.6) is given in (B.47). One can now consider an appropriate generalization of the current (B.20) ^ = - ^ e ^ T r $ t ^ $ (B.21) = --^e^pe-^dr-pe** = -±elivp(ff/p + ipff/<l>s) (B.22) = ^eiu,{-ip9'p + p2dr<l>s) (B.23) which reduces to (B.20) when we set p = (p) = 1. Now we have 0 % = -^t^dOpVp- ^vpd»dvp+ ^pe^pd^s + ^p2^vd»dv4>s = -peilvd»pd»<S>s (B.24) which is no longer conserved as expected. This was only conserved in the effective theory (B.6) because we integrated out the heavy degrees of freedom by setting p — 1 so that d^p — 0. Thus, the decay of the domain walls is directly related to the dynamics of the heavy degrees of freedom. We consider this effect in Section B.5. The physical object responsible for this behaviour in (B.6) is the gluon condensate E ~ (G2), which is the most essential contribution to the mass mv> of the rf particle. 6 6 If one assumes that all of the rf physics comes exclusively from the phase of the chiral condensate rather than from gluon condensate, then one might argue (using the linear sigma model) that the walls are classically unstable. Were this the case, then mvi ~ mq —» 0 in the chiral l imit and the U(l) problem would remain unresolved [47]. Thus, we see the importance of the gluon condensate E ~ (G2) 2> mq which ensures the classical stability of the domain walls for Nc > 3 as explained in Section B.5. B. QCD Domain Wall Structure 69 B.4 Domain Walls In the rest of this paper we limit ourselves with the simplest case Nf = 2 and neglect the difference between /„• and /,,/ which numerically are very close to each other. To describe the basic structure of the Q C D domain walls as well as that of axion domain walls we replace the parameter 9 in Equation (B.6) by a dynamical axion field 9 —> Nfa = 2a (this corresponds to the so-called N = 2 axion model). We also introduce here the follow-ing dimensionless phases, (j>s describing the isotopical "singlet" field, and 4>T describing the isotopical "triplet" field. These fields correspond to the dynamical rj' (singlet) and pion 7r° (triplet) fields defined in (B.5). <t>s = <f>u + <t>d, 4>T = 4>u- <i>d, v' = T^TI^5' N ° = (B-25) In what follows we also need to know the masses of the relevant fields in terms of parameters of the effective potential (B.6): ml = -^; m 2 = - 2 - ( l - e 2 ) ; m 2 = - ^ ^ + — , (B.26) where mass relation for 7/ follows from (B.18) with Nf = 2. Here we have neglected all possible mixing terms, and have introduced the following no-tations _ Mu + Md m M | ( 0 | ^ ^ | 0 ) | + m r f K O | i ^ d l O ) | M = = , (B.27) * = £ r ^ r * ° - 3 - (B-28) Md + Mu v ' B.4.1 Domain Wall Equations To study the structure of the domain wall we look at a simplified model where one half of the universe is in one ground state and the other half is in another. The fields will orient themselves in such a way as to minimize the energy density in space, forming a domain wall between the two regions. In this model, the domain walls are planar and we shall neglect the x and y dimensions for now. Thus, a complete description of the wall is given by specifying the boundary conditions and by specifying how the fields vary along z. The energy density of a domain wall is given by the following expression ° = / " [jth2 + l l ^ 2 + i f ^ + F ( < ^ 5 ' a ) _ F m i n ) d z ( R 2 9 ) B. QCD Domain Wall Structure 70 where the first three terms are the kinetic contribution to the energy and the last term is the potential. The kinetic term is actually a four divergence, but we have assumed the wall to be a stationary solution—hence the time derivatives vanish—and symmetric in the x-y plane. The only dependence remaining is the z dependence. Here, a dot signifies differentiation with respect to z: a = g | . Now, to find the form of the domain walls, it is convenient to use a form of the potential which follows from ( B . l l ) : V{<f>s,(fr,a) = -2M ( c o s ( ^ ) c o s ( ^ + a) + £ s i n ( ^ ) s i n ( ^ - + a)) - S c o s ( ^ ) . (B.30) Here we have redefined the fields (f>u —> <pu + a and (fid —> (fid + a m order to remove the axion field from the last term ~ E and to insert it into the terms ~ M. We have assumed here that only the ir° field varies along the wall. This is valid if sin(7r°) sin(a + rf) > 0 along the entire wall. In this case, it is clear from (B.30) that this term will be minimized if 7r-z = ± 1 and the kinetic term is minimized by rotating away the -K\ and iT2 contributions. This assumption holds in all of the cases we consider. To minimize these equations, we can apply a standard variational prin-ciple and arrive at the following equations of motion for the domain wall solutions: (B.31) (B.32) (B.33) (B.34) where the last term of equation (B.33) should be understood as the lowest branch of the multivalued function described by equation (B.6). Namely, for (fis, [0, TT], this should be interpreted as sin ({(/)S~2TTI)/NC) with the integer I chosen to minimize the potential term cos(((fis — 2rrl)/NC). For example, with 7r < 4>s < 27r, the last term should be of the form sin((05 — 2TT)/NC). Notice the following features: first, the trigonometric terms on the right hand side are of, at most, order 1; thus the scale for the curvature (or rather, the second derivative) of the domain wall solutions is limited by j2jM and ^ = cos ( & ) sin + a) - £sin ( ^ ) cos (f + a) , ^ = sin ( & ) cos (f + a) - £ cos (f) sin (f + a) , jjjjS = cos (f) sin (f + a) - £sin (f) cos (f + a) + - V - S i n (iA ~ MNC ° l n \ Nc J ' B. QCD Domain Wall Structure 71 f%/M etc. In particular, the axion domain wall must have a characteristic scale larger than m~l/(l — £ 2) and the pion domain wall must have a scale larger than m~l. The last term in equation governing the (ps field can potentially be somewhat larger than 1, hence the smallest scale for the (ps field is related to the r/ mass. We see immediately that an axion domain wall must have a structure some thirteen orders of magnitude larger than the natural QCD scale and that the rj field can have structure one order of magnitude smaller than that of the pion field. B.4.2 QCD Domain Walls Here we consider the most important case of the QCD domain wall solution which exists with or without an axion field. So we now set a = 0. The equations of motion become: £ c o s ( ^ ) s i n ( ^ ) , (B.35a) e s i n ( f ) c o s ( £ ) + ^ s i n ( £ ) . (B.35b) For convenience, we shall label the vacuum states using the notation (<pu, (pa). Thus, we have only one physical ground state {(pui^d) = (0,0), however, because of the conserved topological current (B.20), classically stable domain walls can form and interpolate from the ground state (<pu, (pa) = (0,0) along a path which is not homotopic to the null path. To classify the paths we use the redundant notation where (0,0) and (27r, 0) etc. are considered as different states and we talk about the field interpolating between these states. Keep in mind that this is only a way of classifying the homotopy classes and that in fact all the states represented by (2im, 2irm) for integers m and n are one in the same vacuum state. The simplest domain wall is described by a continuous transition from the ground state ((pu,(pd) = (0,0) to the state labeled ((pu,<fid) — (27r, 0) as described by the vacuum solution Equation (B.14) with Q = 2ir (or, equivalently, I = -1 in Equation (B.6)). This wall corresponds to a single winding around the U(l) manifold. It is also possible to wind in the opposite sense. To summarize, the two topologically stable domain walls of minimal energy correspond to one winding in each direction and are classified by the transitions from (<pu,(pd) = (0,0) to: Soliton {(puAd) = (27r,0). 8 M M\ ^ = s in ( f ) cos( f ) -% - Cos ( f ) sin ( £ ) -B. QCD Domain Wall Structure 72 Anti-soliton (<pu,(f>d) = ( — 27r,0). Note that, in the chiral limit (mu = rrid), £ = 0 and the transitions to {4>u,<frd) = (0, ±2TT) have the same energy, thus there is a degeneracy. If mu > md, then these transition in 4>d are the minimal energy solutions and the <j>u solutions above become unstable. In reality > m u , and the transitions to (c/>u,c/>d) = (±27r, 0) are the only stable transition. 7 The general case of Equations (B.35) cannot be solved analytically and we present the numerical solution of Equation (B.35) in Figures B.3 and B.4. In order to gain an intuitive understanding of this wall, we examine the solution in the limit m , <C . In this case, the last term of (B.35) dominates unless <f>s is very close to the vacuum states, (j>s — 27m. Thus, the central structure of the (f>s field is governed by the differential equation: Now, there is the issue of the cusp singularity when c/>s = n because we change from one branch of the potential to another as expressed in Equation (B.6). By definition, we keep the lowest energy branch, such that the right hand side of Equation (B.36) is understood to be the function sm(<f>s/Nc) for 0 < (f>s < TT and sin((c/>5 — 2TT)/NC) for TT < ifis < 2TT. However, we notice that the equations of motion are symmetric with respect to the center of the wall (which we take as z = ZQ), hence 4>s = TT only at the center of the wall and not before, so we can simply look at half of the domain, z 6 (—oo,0], with boundary conditions (fis = 0 at z = — oo and 4>s = TT at z = 0. The rest of the solution will be symmetric with c/>s = 2ir at z — +oo. Equation 7 There appear to be four singlet transitions, but since the fields live on the SU(2) xU(l) manifold, and there should only be two—one corresponding to a single winding around the U(l) manifold, and the other corresponding to winding the opposite way. The problem here is that we are only considering the vacuum parameters <f>u and cpd- Together, these live on the J7(l) x U(l) manifold—a torus—which admits four topologically stable transitions, but we have neglected the other massless pseudo-Goldstone fields n^. When we bring these into the game, we see that these walls are really paired off with the transitions to [TT, ±7T] being homomorphic and the transitions to [—IT, ±7r] being homomorphic. Thus, we are left with only two topologically stable transitions with the singlet field tps going to TT (which we call the soliton) and with cps going to — TT (which we call the anti-soliton). The other apparent solutions are actually saddle points rather than minima and thus the solution is classically unstable. B. QCD Domain Wall Structure 73 (B.36) with the boundary conditions above has the solution <t>s{z) = [<t>u + 4>d) = { ANr tan" t a n m: exp[/x(2 - zo)] 27T - 4Nr tan - l tan ^ - exp [-fj,(z - z0)] where ZQ is the position of the center of the domain wall and fi = l im fi = rrirf z <z0 z > z0 (B.37) (B.38) is the inverse width of the wall, which is equal to the mvt mass in the chiral limit (see Equation (B.26)). Thus, we see that the dynamics of the central portion of Q C D domain walls is governed by the rf' field. We shall also refer to the 05 transition (B.37) which occurs in several places (see for example Section B.4.5) as the rf domain wall. The first derivative of the solution is continuous at z = zo, but the second derivative exhibits a finite jump. Before we continue our discussions regarding the structure of Q C D do-main walls, a short remark is in order: The sine-Gordon equation which is similar to Equation (B.36) with a cusp singularity, was first considered in [73] where a solution similar to (B.37) was presented. There is a funda-mental difference, however, between domain walls discussed in [73] and the domain walls we consider here. In [73], the domain walls were constructed as auxiliary objects in order to describe a decay of metastable vacuum states which may exist under the certain circumstances. The walls we discuss here classically stable physical objects where the solutions interpolate between the same vacuum state; their existence is a consequence of the topology of the U(l) singlet rf' field. This topology, represented by the exact symmetry 9 = 9 + 2-irn in Equation (B.6) is a very general property of Q C D and does not depend on the specific choice of parameters or functional form of the effective potential. Similar equations and solutions with application to the axion physics were also discussed in [37]. The solution described above dominates on scales where \z\ < how-ever, the isotopical triplet pion transition can only have a structure on scales larger than m " 1 3> fi~l and so the central structure of the rf wall can have little effect on the pion field. Indeed, we can see that, for \z\ 3> (f>s is ap-proximately constant with the vacuum values. Making this approximation, we see that the isotopical triplet field is governed by the equation ^ = 27/4 sin ( ^ (B.39) B. QCD Domain Wall Structure 74 This has the same form as (B.36) and hence the solution is 0 ~ I ^ t a n _ 1 [tan (I) e x P [ m 7 r ( - z - 20)]] ; - 0 0 < z - z0 <C ^27r — 8 t a n - 1 [tan ( | ) exp[-m7T(z - zo)]] , fi~l <C z — ZQ < +00 (B.40) which is a reasonable approximation for all z. Numerical solutions for the (j>s and (f>T fields are shown along with the same solution in terms of the (j)u and <pd in Figures B.3 and B.4. As we can see from the explicit form of the presented solution, the 77' transition is sandwiched in the pion transition. This is a key feature for some applications of this type of the domain wall as discussed in the main body of this thesis. The wall surface tension defined by Equation (B.29) can be easily calcu-lated analytically in the chiral limit when the analytical solution is known and is given by Equations (B.37) and (B.40). Simple calculations leads to the following result " = W/V(^G2}(1~C0S^)+0K/' )- (R41) In case when mq 7^  0, an analytical solution is not known, but numerically, cr is close to the estimate (B.41). B.4.3 Axion Dominated Domain Walls In the previous subsection when the Q C D domain walls were discussed, the axion was not introduced as a dynamical field. In this subsection we assume that the axions exist. In this case there are domain walls in which the axion is the dominant player. The introduction of axions, in most cases, makes the domain wall an absolutely stable object. Our case is no exception and the axion model under discussion, (which is the N = 2 axion models according to the classification [39, 40, 92]) is an absolutely stable object. At the same time it is well-known [69, 94], that stable domain walls can be a cosmological disaster. We do not address in this paper the problem of avoiding a domain wall dominated universe. Rather, we would like to describe some new elements in the structure of axion domain walls, which were not previously discussed. The first and most natural type of the axion domain wall was discussed by Huang and Sikivie [36] who neglected the rj field in their construction. We shall refer to this wall as the Axion-Pion domain wall (o^). As shown B. QCD Domain Wall Structure 75 zrriw Fig. B.3: Basic form of the QCD domain walls. Notice that the scale for the pion transition is larger than for the eta' transition. Notice that the width of the rj' wall is set by the scale mn'. in [ 3 6 ] , it has a width of the scale ~ m~l 3> A Q C D for both the axion and 7T meson components. As Huang and Sikivie expected, the rj' plays a very small role in this wall. In what follows we include a discussion of this type of domain wall for the completeness. Our original result is to description a new type of the axion domain wall in which the 77' field is a dominant player. We shall call this new solution the Axion-Eta ' domain wall (a^). This new type of the domain wall was considered for the first time in [3] as a possible source for galactic magnetic fields in early universe. In what follows we give a detail description of the solution for the wall. Here we want to mention the fundamental difference between the wall discussed in [36] and the av> wall introduced in [3]. Unlike the a% wall which has structure only on the huge scale of m~l, the avi wall has nontrivial structure at both the axion scale m~l as B. QCD Domain Wall Structure 76 i i i i i — i i i i i -20 -15 -10 -5 0 5 10 15 20 Fig. BA: Same form as in Figure B.3 except in terms of the variables <pu and ind-well as at the Q C D scale m~} ~ A Q C d . The reason for this is that, in the presence of the non-zero axion field (which is equivalent to a non-zero 9 parameter), the pion mass is efficiently suppressed due to its Goldstone nature, thus the pion field follows the axion field and has a structure on the same m~l scale. The / / , however, is not very sensitive to 6 and so it remains massive. Again, the solution has a singlet transition occurring at the center of the wall. One can adopt the viewpoint that the cty domain wall is an axion domain wall with a Q C D domain wall in the center. This phenomenon is critical for applications involving the interaction of domain walls with strongly interacting particles. Indeed, there is no way for the 0,^ wall to trap any strongly interacting particles, like nucleons, because of the huge difference in scales; the cjy wall, however, has a Q C D structure and can therefore efficiently interact with nucleons on the Q C D scale A Q C D . Regarding the wall tension (B.29), it is dominated by the axion physics and B. QCD Domain Wall Structure 77 thus has the same order of magnitude for both types of the axion domain walls which is proportional to cr ~ —- ~ fafnmn as found in [36]. BAA Axion-Pion Domain Wall The solution discussed by Huang and Sikivie corresponds to the transition (a,(fiu,(fid) : (0,0,0) —>• (TT, TT, — TT), i.e., by a transition in the axion and pion fields only. This transition describes the aw wall. Indeed, in terms of (a, (f>s, (fir) this transition corresponds to a nontrivial behavior of the axion and triplet pion component (fir of the (fiu, (fid fields: (a, (fis, (fir) '• (0,0,0) —> (TT, 0, 2TT). The other transition (the av' wall) corresponds to the transition (a,(fiu,(fid) • (0,0,0) —> (—TT,TT,TT) in which the singlet n' field is dominant: (a, (fis, (fir) '• (0,0,0) —> (—TT, 2TT, 0). It will be considered later on. Huang and Sikivie discussed the solution to this wall in the limit where the rf field is extremely massive and hence they neglected its role. It can be integrated out which effectively corresponds to fixing it rf(z) = 0. Indeed, if we simply fix r)'(z) = 0 in our equations, then we reproduce their solution. When E is large, then <C m'^ as Huang and Sikivie assumed, the effects of the r/ particle can be neglected and the solution for the the axion and pion fields presented in [36] is valid for the boundary conditions described above. We plot this numerical solution which includes the rf effects in Figure B.5. As announced above, the an solution has the only scale of ~ m~x for both components, axion as well as the pion field ~ (fix- The rf field remains close to the its vacuum value and only slightly corrects the solution. BA.5 Axion-Eta' Domain Wall Having considered the solution of the an wall, we now investigate the struc-ture of the a^i domain wall which is a new solution. This solution corre-sponds to the transition (a,(fiu,(fid) : (0,0,0) —y (TT,TT,TT). N O W the singlet rf field undergoes a transition instead of the triplet pion field: (a, (fis, (fir) • (0,0,0) -» (—TT,2TT, 0). As we discussed above, the singlet field never be-comes massless, and therefore, a new structure at the Q C D scale ~ A Q C D emerges in sharp contrast to the well-studied an domain wall [36] where no such structure appears. We should note that the potential (B.30) has the same vacuum energy at (fis = 0 as well as at (fis = 2TT due to the change of (fis field to the lowest energy branch at (fis = TT as described by Equation (B.6). Therefore, this domain wall interpolates between two degenerate states, and thus, like the an domain wall [36], the solution under these considerations is absolutely stable. B. QCD Domain Wall Structure 78 0 - 5 0 zma Fig. B.5: Basic form of the a„ domain wall. Notice that all the fields have a struc-ture on the scale of m " 1 and that the isotopical singlet fields plays a very small role. For \z\ the last term in (B.33) is negligible and so the solution behaves like the aw wall. What happens is that, away from the wall, the axion field dominates and shapes the wall as it does in with the an solution. Again, the pion mass is suppressed and (ps ~ 0. As z ~ / J - 1 however, the last term of (B.35) starts to dominate the behaviour. At this point, the cty wall undergoes a sharp transition similar to the Q C D domain wall described by (B.37). We plot this solution Figure B.6 along with a blowup in Figure B.7. Notice also, that the singlet field cancels the effects of the axion near the center of the wall, and so the pion field becomes massive again as it undergoes its transition. B. QCD Domain Wall Structure 79 Fig. B.6: Bas ic form of the an> domain walls w i t h a closeup where the ax ion field a « n/2. Not ice that the large scale s tructure is s imi lar to that of the an wa l l , but that there is also a smal l scale structure on the scale of mni. Near the center of the wa l l , the p ion regains its mass and undergoes a t rans i t ion on the scale of B.5 Decay of the QCD domain walls Up to this moment, we have treated the domain walls as topologically stable objects. We do not have much new to say regarding the stability for general types of the domain walls, nor do we have a resolution for the general prob-lem of avoiding a domain wall dominated universe. We refer to the textbook [69] for the references on the subject. Instead, this section is devoted specif-ically to the Q C D domain walls discussed in Section B.4.2. Here we argue that Q C D domain walls (not the axion walls) are unstable on the quantum level due to tunneling. We estimate the relevant life-time of this quantum transition which turns out to be quite short on the cosmological scale but B. QCD Domain Wall Structure 80 -40 -30 -20 -10 10 20 30 40 Fig. B.7: Zoom in on Figure B.6 showing detailed structure of rj' core. Near the center of the wall, the pion regains its mass and undergoes a transition on the scale of . quite long with respect to the Q C D scale A Q C D albeit exponentially sup-pressed as it should be. Therefore, these walls do not pose a cosmological problem as one might naively suspect.8 We have nothing new to say re-garding the stability or evolution of axion domain walls [34, 35, 49, 95-98] (which were also discussed in the previous sections), see [23] and references therein for a recent review on the subject. Having said that the Q C D domain walls are unstable, a very natural question occurs: "What sense does it make to study an object which does not 8 We should remind the reader once again that the existence of the Q C D domain walls described above is the reflection of the well-understood symmetry 6 —> 0 + 2irn and, is the consequence of the topological charge quantization; their existence is not based on any model-dependent assumptions we have made to support the specific calculations in the previous sections. B. QCD Domain Wall Structure 81 presently exist?" The answer is: "It does not make much sense if we study Q C D at the present cosmological epoch. However, if we want to understand what happened shortly after the Q C D phase transition in the evolution of the early universe, or what may happen at R H I C after the transition from a quark-gluon plasma to the hadronic phase when system cools, then this is perfectly reasonable question to ask (and hopefully answer!)" Indeed, according to the standard theory of the cosmological phase transitions [69, 99], if, below a critical temperature Tc, the potential develops a number of degenerate minima, then the choice of minima will depend on random fluctuations in fields. The minima that the fields settle to can be expected to differ in various regions space. If neighboring volumes fall into different minima, then a kink (domain wall) will form as a boundary between them. For the present estimate of the decay rate for the Q C D domain wall, the only information we need from the previous sections are: the domain wall tension o and the wall thickness. In the chiral limit for two light flavors Nf = 2, the tension a is given by Equation (B.41) which we would like to present in the following form: a = -= 1 — cos —— , (B.42) V2 V 2 N c J In this formula, the \/2 in denominator should be replaced by ^Nf for an arbitrary Nf, however in all numerical estimates we shall use Nf = 2 which we believe is very good approximation for our problem in the limit mu ~ m<2 <C ms as Equations (B.13) and (B.14) suggest. Besides that, for iV c > 3, one can approximate 1—cos[7r/(2Afc)] ~ n2 f(8N2) such that Equation (B.42) takes especially simple form which will be used in the numerical estimates which follow, a = *2Jj^ - (200 M e V ) 3 . (B.43) What to use as the wall thickness is a more subtle question due to the structure of the Q C D domain walls described above. As we discussed in the previous sections, the core of the Q C D domain wall is determined by the flavour isotopical singlet rj field with thickness ti~l ~ , while the 7T meson halo has a much larger size ~ m~l. Nevertheless, in what follows we treat the Q C D domain wall as the object with wall thickness of order ~ m", 1 due to the fact that the most important contribution to a comes from this small region of Az ~ mtfl • Indeed, as Equation (B.29) suggests, the contribution to a from the rj field is proportional to EAz ~ E/m^i. At the B. QCD Domain Wall Structure 82 same time, the contribution to a from the TT field, which was ignored in (B.42) and (B.43), is proportional to mq Az ~ mq (tyty) /m^ ~ —» 0 which vanishes in the chiral limit. Therefore, despite of the fact that the TT meson cloud is of a much larger in size, its contribution to a is much smaller, as announced earlier. Numerically, the TT contribution consists only 10% of a and we ignore this contribution in what follows. We are now prepared to discuss the decay of the Q C D domain walls. The decay mechanism is due to a tunneling process which creates a hole in the domain wall. This hole connects the configurations (<fiui<fid) = (0,0) to the ((j>ui4>d) = (27T,0) domain (see Equation (B.14)). This decay mechanism is similar to the decay of a metastable wall bounded by strings, and we . use a similar technique to estimate the tunneling probability. In this case, the walls can decay by quantum nucleation of holes which are bounded by strings. The idea of the calculation was suggested by Kibble [50], and has been used many times since then (see [69] for a review). A well known exam-ple is the calculation of the decay rate in the so-called NPQ = 1 axion model where the axion domain wall become unstable for a similar reason due to the presence of axion strings [23, 95-98]. However, as was emphasized in [100], the existence of strings as the solutions to the classical equations of motion is not essential for this decay mechanism (see below). Some configurations, not necessarily the solutions of classical equations of motion, which satisfy appropriate boundary conditions, may play the role played by strings in the NPQ = 1 axion model. To be more specific, let us consider a closed path starting in the first do-main with (</>„, (fid) = (0,0), which goes through the hole and finally returns back to the starting point by crossing the wall somewhere far away from the hole. The phase change along the path is clearly equals to <j>u + <fid = 27T. Therefore, the absolute value of a field which gives the mass to the r/ field (the dominant part of the domain wall) has to vanish at some point in-side the region encircled by the path. By moving the path around the hole continuously, one can convince oneself that there is a loop of a string-like configuration (where the absolute value of a relevant field vanishes such that the rf singlet phase is a well defined) enclosing the hole somewhere. In this consideration we did not assume that a hole, or string enclosing the hole, are solutions of the equations of motion. 9 They do not have to be solutions. 9 It is quite obvious that such a configuration cannot be described within our non-linear a model given by Equation (B.6) where it was assumed that the gluon as well as the chiral condensates are non-zero constants. In this case, the singlet phase is not well defined everywhere. However, in the case of a triplet IT meson string, such a configuration can easily be constructed within a linear a model by allowing the absolute value of the B. QCD Domain Wall Structure 83 However, if we want to describe the hole nucleation semi-classically [50, 69], then we should look for a corresponding instanton which is a solu-tion of Euclidean (imaginary time, t = ir) field equations, approaching the unperturbed wall solution at r —> ±oo. In this case the probability P of creating a hole with radius R per area S per time T can be estimated as follows 1 0 [50, 102]: 3 e " 5 ° x Det, (B.44) where So is the classical instanton action; Det can be calculated by analyzing small perturbations (non-zero modes contribution) about the instanton, (see [102] for an explanation of the meaning of this term) and will be estimated using dimensional arguments; and (y/So/{2n))3 is the contribution due to three zero modes describing the instanton position. 1 1 If the radius (a critical radius Rc to be estimated later) of the nucleating hole is much greater than the wall thickness, we can use the thin-string and thin-wall approximation, in this case, the action for the string and for the wall are proportional to the corresponding worldsheet areas [50], A'K S0 = AirR2a - —R3a. (B.45) o Here a is the wall tension (B.43), and a is the string tension, to be es-timated. The world sheet of a static wall lying in the x-y plane is the three-dimensional hyperplane z = 0. In the instanton solution, this hyper-plane has a "hole" which is bounded by the closed worldsheet of the string. chiral condensate to fluctuate along with the Goldstone phase (TT meson field). The a term in the linear a model essentially describes the rigidity of the potential. Indeed, the corresponding calculations within a linear a model were carried out in [101] where it was demonstrated that the solution describing the n meson string exists, albeit unstable as expected from the topological arguments. To carry out a similar calculations in our case for the singlet 7 / phase, one should allow fluctuations of the gluon fields: the fields that give mass to rj meson and that describe the rigidity of the relevant potential. (See (B.20) and the surrounding discussion.) 1 0 The estimate given below is designed for illustrative purposes only, and should be considered as a very rough estimation of the effect to an accuracy not better than the order of magnitude. 1 1 The three zero modes in our case should be compared with the four zero modes from the calculations of [102]. This difference is due to the fact that in [102] the decay of three dimensional metastable vacuum state was discussed. In our case, we discuss a decay of a two-dimensional object. B. QCD Domain Wall Structure 84 Minimizing (B.45) with respect to R we find that n 2a „ 167ra3 Rc = —, <So = ^ ~ . B.46 CT OCT The Lorentzian evolution of the hole after nucleation can be found by making the inverse replacement r —> —it from Euclidean to Minkowski space-time. The hole expands with time as x2 + y2 = R2 +12, rapidly approaching the speed of light. Now we want to estimate the string tension a. As discussed in Sec-tion B.3.1, we cannot make this estimate with Equation (B.6) where the gluon degrees of freedom are integrated out and replaced by their vacuum expectation values. Instead, we have to take one step back and describe the dynamics of the gluon condensate by considering the original Lagrangian [42, 76] which includes the complex gluon degrees of freedom h: V(0, h, U) = N c Det U 2eE) e~ie This satisfies all the conformal and chiral anomalous Ward identities, has the correct large behaviour etc. (see [42, 73, 76] for details). Integrating out the heavy h field wil l bring us back to Equation (B.6). It is interesting to note that the structure of Equation (B.47) is quite similar in structure to the effective potential for SQCD [75] and gluodynamics [103]. Let us stop for a moment to consider the form of this potential. When we integrate out the heavy gluon degrees of freedom, we replace h with its vacuum expectation value (h). The vacuum expectation value 1 2 of the h is given by (h) = 2Eexp(-iY2&/Nc) (B.48) such that (B.47) becomes "( i"L,H 1) = " £ ™ ( ^ * ) - E « - « * • (B-49) in agreement with ( B . l l ) . Wi th h fixed at its vacuum expectation value, the singlet combination cf>g = Y^4>i exhibits the U(l) topology described 1 2 For more accurate treatment of the minimization procedure which carefully accounts for the different branches, see the original paper [73]. B. QCD Domain Wall Structure 85 above and our domain walls are stable. Now, however, we allow the gluon condensate to fluctuate, which can be parameterized as h = p (h) where p is a real physical fields related to glueballs. In this case, the potential V is given by V(p, <f>s) = +Ep (log p-1) cos ^ (B.50) where we have neglected the terms proportional to M as they are negligible since they only contribute a constant offset due to the cpr field. For an early phenomenological discussion of the potential (B.50) without the cps fields, see [104]. Now the combined degrees of freedom p and (j>s are no longer restricted to the circle p = 1 as they were in the effective theory (B.6). The U(l) topology is no longer a constraint of the fields and thus, the walls are not topologically stable. Instead, the restriction of p « 1 is dynamical, and made by a barrier at h = 0. Thus, with these degrees of freedom, the fields parameterize the plane, however, the potential is that of a tilted Mexican-hat with a barrier at h = 0. The barrier is high enough that a domain wall interpolating around the trough of the hat is classically stable. If the barrier were infinitely high as we assumed when we fixed p = 1 as we did in (B.6), then the cps field could wind around the barrier and would be topologically stable. Wi th a finite barrier, however, the field can tunnel through the barrier as described above. This situation is analogous to the case of the string and "peg" shown in Figure B.2. We show a more accurate picture of the barrier (B.50) in Figure B . 8 . 1 3 The most important property of the potential is the following: The ab-solute minimum of the potential in the chiral limit corresponds to the value Vmin = —E which is the ground state of our world with p = 1 and <f>s — 0. At the same time, the maximum of the potential (B.6), where one branch changes to another one is V = -ECOS(TT/NC) where cj>s = TT (we are still taking 0 = 0). This corresponds to the point p = 1 and cps = n in the po-tential (B.50). Thus, the trough of the Mexican-hat is given where h = (h), i.e. at radius p = 1 and the maximum V = —Ecos(n/NC) of the potential (B.6) is exactly the barrier through which the rf field interpolate to form the Q C D domain wall. It is important to note that the height of the barrier for the potential (B.6) is numerically is quite high ~ E(l — cos j^-), but that vanishes in large NC limit. Indeed, in this limit, the peak of the barrier is 1 3 We are indebted to Misha Stephanov for suggesting this nice intuitive picture for explaining the domain wall decay mechanism. B. QCD Domain Wall Structure 86 AV (j)S = TT ^ i <f>S = 0 Fig. B.8: Prof i le of the "Mexican-ha t" potent ia l (B.50) . The slice is made along the axis through (f>s = -K to the left and <f>s — 0 to the r ight. T h e t rough of the potent ia l lowers from the cusp at p = 1, cf> = n where V = —ECOS(IT/NC) down to the vacuum state V m i n = —E. T h e hump where V = 0 is at the or ig in is where h = pexp(i<f>S/NC) = 0 and hence the singlet field 4>s can have any value at this point . It is by passing across this point that a Q C D domain wal l can tunnel and a hole can form. degenerate with the absolute minimum Vm\n = — E of the potential as it should be. 1 4 The height of this barrier describes how much the Mexican-hat it tilted. The other important property of the potential (B.50) is its value where the singlet phase ^ is not well defined. From (B.50) it is clear that this occurs when h = 0. This is exactly the height of the peak of the Mexican-hat (or the "peg" in Figure B.2) which classically prevents p —> 0 and which makes the Q C D domain wall classically stable. The energetic barrier at h = 0 that tells us the energetic cost to create the "string" around the edge of the hole which must accompany the hole, and which includes the region with h = 0. Potential (B.50) vanishes at this point, V(h = 0) = 0, which implies that the barrier at h = 0 is quite high: AV = E. As expected, the barrier at h = 0 should be order of E ~ N2 in contrast with the barrier to 77' where one should expect a suppression by some power of Nc. We also 1 4 Remember, the rj direction becomes flat in the large Nc l imit as mv> —¥ 0. B. QCD Domain Wall Structure 87 note that the total number of classically stable solutions can be estimated from the condition cos(7rk/Nc) > 0 where the barrier for the rj field is still lower than the peak h = 0. Thus, —Nc/2 < k < Nc/2 where k labels the solutions. As a consequence, we estimate the string tension a using pure dimen-sional arguments: a ~ y/2E where 2E is the only relevant dimensional pa-rameter of the problem in the chiral limit. This parameter enters Equation (B.47) such that (h) = 2E. Therefore,1 5 a ~ V2E ~ (0.28GeV) 2, Rc = — ~ ~ ^ , (B.51) K2fn / T T ' and * = ^ ~ 2 ^ P ~ ( n o - u o ) , (B.52) where the uncertainty in 5o reflects a 10% variation in a from the central value given in (5.16). Of course, a more realistic error in estimate (5.16) would be much higher. It is quite remarkable: in spite of the fact that all parameters in our problem are Q C D parameters of order A Q C D , the classical action So could be numerically quite large, and thus the corresponding prob-ability (B.44) may be quite small. We do not see any simple explanation for this phenomenon except for the fact that expressions (B.45) and (5.16) for So contains a huge numerical factor 167r/3 of purely geometrical origin. In addition, since So ~ N2, one should expect an additional enhancement in S0. Here we mention an interesting note: As we mentioned earlier, in Q C D there is only one dimensional parameter, A Q C D , and it is thus generally be-lieved that the semi-classical approximation in Q C D cannot be parametri-cally justified. Nevertheless, numerically Rc is quite large, much larger than the width of the domain wall (B.38) which is essentially set by m^,1. Therefore, the semi-classical approximation (B.44) is somewhat justified a posteriori. Now, we are prepared to make our last step and estimate the probability 1 5 The magnitude for a ~ \fl~E ~ (0.28GeV) 2 should not be considered as a strong over-estimation; rather it should be considered as a lower bound. Indeed, if one considers the ix meson string[101] which should be much softer (and therefore, would possess much smaller a,r) one finds, nevertheless, that a„ ~ 7 r / 2 is very close numerically to our estimation for the rj string tension. B. QCD Domain Wall Structure 88 of creating a hole with radius Rc ~ 8A 7 c / (7r 2 / 7 r ) : 3 e ~ 5 ° . (B.53) We have estimated Det ~ E3/4 through dimensional analysis. Using the numerical values given above for Rc and So — 130 we arrive to the following final result: ^ ~ 1 0 3 e - 1 3 0 G e V ~ l O ^ V 1 . (B.54) The most amazing result of the estimate (B.54) is the astonishingly small probability for the decay P ~ 10~ 5 0 GeV which one might naively expect to be on the GeV level. This small number leads to a very large life time for the domain walls, and consequently, makes them relevant to cosmology at the Q C D scale. Of course, our estimation of So is not robust, and even small variation of parameters may drastically change our estimate (B.54), making it much larger or much smaller. In what follows we stick with estimate (B.54) for cosmological applications. First of all, let us estimate an average size I of a domain wall before it collapses with an average lifetime of T; ~ l/c. This corresponds the situation when the probability of the decay is close to one. If L is the Hubble size scale, L ~ 30 km, then P J 2 _ P L2 L f l \ 3 ~ TirR2/1 ~ TirR2c \Lj ~ ( l O - ^ X l O ^ X l O - S ) ~ 102 ^y ~ 1. (B.55) Formula (B.55) implies that the average size / of the domain wall before it collapses could be as large as the Hubble size / ~ 1 0 " 2 / 3 L ~ 0.2L. It also implies that on a Hubble scale domain wall L, the average number of holes which will be formed is approximately (n) ~ {L/l)2 ~ 25. Finally, the lifetime of a Hubble size domain wall is expected to be on the scale of n ^= ~ 2 x 10 _ 5 s , (B.56) cy/(n) which is macroscopically large! It is clear that all these phenomena are due to the astonishingly small number (B.54) which makes the link between Q C D and cosmology feasible. B. QCD Domain Wall Structure 89 This small number is due to the tunneling process rather than some spe-cial fine-tuning arrangements. As we mentioned above, we have not made any adjustments to the phenomenological parameters used in the estimates. Rather, we have used the standard set of parameters introduced in Equa-tion (B.6). To conclude the discussion of (B.54) we would like to remind the reader that similar miracles related to tunneling processes happens in physics quite often. For example, the difference in life time for U 2 3 8 and P o 2 1 2 under a decay is on the order of 10 2 0 in spite of the fact that the "internal" physics of these nuclei, and all internal scales are very similar. We should be careful to qualify this result. First, it assumes that the effective Lagrangian (B.6) is a good description of the low energy physics of Q C D including the physics of the rf field. This assumption is well justified in the large Nc limit. The theory is not quantitatively justified as Nc —> 3 but the qualitative features of the potential responsible for producing the domain walls persist for all Nc > 3 as shown in Figure B.8 so we expect the qualitative features to remain. Second, it assumes that the semi-classical calculation leading to (B.53) is justified. This is certainly not justified a-priori, but is somewhat justified a-posteriori. Third, we have assumed T = 0 for this analysis. Thermal fluctuations may considerably reduce the lifetime of these walls. Finally, the interaction of the domain wall with nucleons can drastically change the properties of the domain wall due to the strong interaction of all relevant fields. In particular, the domain walls may become much more stable in the presence of nucleons, and even account for the strong self-interacting dark matter discussed in [68, 105]. We hope to return to these considerations in a separate publication. C. Numerical Solution of Domain Wall 90 C. Numerical Solution of Domain Wall Cl Introduction This appendix describes one method useful for obtaining numerical solutions to domain wall (or other similar types of boundary value) problems. The difficulty is that the problem is specified with boundary conditions at two points (in this case ±00) so there is no direct method for computing the solution (like Euler's method, or variants like Rung-Kutta). One could try to shoot—guess the initial slope to produce a solution that "hits" the other boundary—but the system is highly sensitive to the initial conditions. In addition, the boundary points are at infinity, so several limits must be taken. In any case, it is not straightforward to implement such a scheme accurately. A better solution is to discretize the solution, provide a guess that inter-polates between the boundary points, then "massage" the solution (some-times called relaxation) until it is a good approximate solution. In the case of a domain wall, this makes particular sense because the wall is the solution that minimizes the surface tension (energy density) and so one could sim-ply view the problem as constrained optimization problem with the energy being a measure of the "goodness" of the fit. A third problem encountered is that some parts of the solution are very smooth (far from the transition) requiring few sample points, while other parts are very sharp and require many sample points. A good solver will dynamically change the solution lattice to sample the solution more near rapid changes. Several such programs exists: the one we used, C O L N E W , is described here. In addition to the base C O L N E W code (which was writ-ten in F O R T R A N ) we describe a C++ interface as well as an automatic differentiation class to simplify the solving of similar problems. This appendix discusses the following topics: 1. Inputs and outputs from C O L N E W routine. 2. C++ interface for C O L N E W and automatic differentiation scheme. 3. Transforming infinite domain boundary conditions to finite boundary C. Numerical Solution of Domain Wall 91 points. 4. Explicit example of numerical domain wall solution setup. C.2 COLNEW The C O L N E W package is available from: http://www.netlib.org/ode/colnew.f This section is simply an encapsulation of the documentation provided in the comments of the source file and a description of the C++ interface. The C++ interface shall be referred to as Au toCOLSYS. C O L N E W is a modification of the package C O L S Y S by Ascher, Chris-tiansen and Russell [106]. It incorporates a new basis representation re-placing b-splines, and improvements for the linear and nonlinear algebraic equation solvers. The package can be referenced as either C O L N E W or C O L S Y S . The C + + interface supplied here contains a full implementation of C O L -SYS as well as the following added features: 1. C++ interface: A l l code can be written in C++ rather than FOR-T R A N . 2. Automatic differentiation is used to compute the Jacobian and other desired derivatives. Thus, the programmer only need enter the dif-ferential equations and the boundary conditions. This is quite a bit slower than if you hardcode the Jacobian etc. For high performance, one should hard code the Jacobian. This feature is provided as a trade-off between performance and easy of use. For modest problems, more time wil l be spent computing the Jacobian by hand (and ensuring no mistakes are made) than are lost because of performance issues. 3. Data output and macros to load solutions into M A T L A B for analysis. C.3 Purpose This package solves a multi-point boundary value problem for a mixed order system of ODE-s. The differential equations are specified by i € { 1 , . . . , n} (n < 20) differential equations fi() of the solution vector u = (m,U2, • • •, un) C. Numerical Solution of Domain Wall 92 denned on the range x € [aieft, aright]: dx' l-(ui(x)) =fi(x,z(x)). ( C l ) Each equation fa is of order mj. The solution consist of the n functions ui(x), however, all the derivatives must also be computed. The vector z(x) consists of all of the solutions and their derivatives up to the (m-i — l ) t h derivative, _ / d m d V d ^ - V dun d ™ - - 1 ^ z= r 1 ' d 7 ' d ^ " - ' d^^'U2"-- '"*» d ^ ' - ' d ^ r j ' ( a 2 ) thus the vector is of length m* = m-i + 7712 + • • • + mn. To fully specify the solution, m* boundary conditions are required. These are specified at ra* boundary points Q where j € { 1 , . . . ,m*} by a system of constraint functions gj() which are 0 when the boundary condition is met: 0i(G,*(Ci)) =0 . (C.3) These equations are specified in terms of the full vector z and so can fix the solutions or any of their derivatives at the boundary points. The boundary points must also satisfy: Oleft < Cl < C2 < • • • < Cm* < Aright (C.4) The user must define the functions fi and the boundary conditions gj as well as specify the number and order of the equations n and m-i and the boundary points (j. C.4 Usage To use the C++ code as specified, you must create a class which is a subclass of the Sys tem class ("system" of equations) defined in S y s t e m , hpp. The example directory contains the files described here. Your class file should contain the following: TySfndef __Example_hpp__ ^ d e f i n e Examp le_hpp__ // Provides the Example class which specifies the // differential equations described in the documentation. : #end i f / / ExampleJipp— C. Numerical Solution of Domain Wall 93 These prevent the file from being included more than once. It is good coding practice to put this in all header files. e ^include "omstream.hpp" 7 ^include " System, hpp" The file omstream.hpp contains the M A T L A B output code. It is needed for the outputParamsFcn () function. The file System.hpp contains the System class definition. 9 class Example : public System 1 0 { Here the class Example inherits from System. The public ensures that System's variables can be read by Example. 11 public: 12 / / Optional Parameters 1 3 double A; This section consists of optional parameters you may want to use in your code. The test system has one parameter A. 15 / / Required functions 16 virtual void diffeqFen (double x_, 1 7 const Param &vars_, is ValueArray &eqs_); This and the following three functions are required. Here the types Param and ValueArray are arrays used to implement automatic differentiation. Their •use will be described below. 2 0 virtual void bFcn(int i_ , 21 const Param &vars_, 2 2 Value &g-); This function computes the boundary conditions. 24 virtual void outputParamsFcn (omstream &mos_); This outputs the parameters. 26 / / Hie constructor must call the System:: Initialize () 2 7 / / after it as initialized all the required parameters in 28 / / the System class. 2 9 ExampleQ; 30 31 / / Optional functions 3 2 virtual void guessFcn (double x_, 3 3 double * va rs_ , 3 4 double *eqs_){}; 3 5 } ; The Example class does not provide a guess. If you want to provide a guess, then you should change line 34 to read: i double * eqs_); C. Numerical Solution of Domain Wall 94 and then include the definition in E x a m p l e . c c . The section between lines 12 and 13 is used to define parameters that you wil l use for your system. They can be anything you want: the rest of the code wil l not use them. (The only thing to beware of is name conflicts with members of the base System class). You must define the following functions: v o i d d i f feqFcn ( d o u b l e x _ , const Pa ram &vars_ , Va lueAr ray & e q s _ j This function evaluates the differential equations (fi) at the point x . (x) using the current solution array vars_ (z) and places the results in the array pointed to by e q s . (/). To refer to the parameters, use the index notation with the text string of the parameter name as defined in the varnames array defined below. A n example implementation might be: 5 v o i d 6 E x a m p l e : : d i f f e q F c n (double x_ , 7 c o n s t P a r a m & v a r s - , 8 V a l u e A r r a y & e q s _ ) ^ { 1 0 / / Here we make references to the parameters for coding n / / convienience . The names in the parentheses must 12 / / correspond to the names put in the "varnames" array. 1 3 c o n s t V a l u e & y l = va rs_ [ " y l " ] ; 14 c o n s t V a l u e & d y l = v a r s . [ " d y l " ] ; 15 c o n s t V a l u e & y 2 = v a r s . [ " y2 " ] ; 16 c o n s t V a l u e &dy2 = v a r s . [ " dy2 " ] ; 17 is / / These are the results so they will be modified and 1 9 / / should not be declared const. The name in the 2 0 / / parentheses must correspond to the names put in the 21 / / "eqnames" array. 22 V a l u e fcddyl = e q s - [ " d d y l " ] ; 2 3 V a l u e &ddy2 = e q s . [ " d d y 2 " ] ; 24 2 5 d d y l = y l ; 26 ddy2 = y l * ( y l * y 2 - e x p ( — A * y l ) ) ; 2 7 } ; Lines 13-16 and 22-23 are cosmetic, but they show you how, to use the arrays. The result is that eqs_ contains the differential equations as well as the derivatives. Notice that the parameters have been used as well. If you need a temporary variable that contains any of the Va lue variables on the right hand side, then , declare it as type Va lue rather than double. v o i d b F c n f i n t i . , const Param &vars_ , Va lue &cg-) C. Numerical Solution of Domain Wall 95 This function evaluates the L boundary condition (gi) at the point zeta [ L ] (Q) using the current solution array va rs . (z) and places the results in the referenced variable g_. Note that due to the way C O L S Y S works, only the index L is specified. The function need not use the zeta 0 array at all. This function can be implemented using a s w i t c h -c a s e construct. 2 9 v o i d 3 0 E x a m p l e : : b F c n ( i n t i _ , 31 c o n s t Param & v a r s _ , 3 2 V a l u e & g - ) 33 { 3 4 c o n s t V a l u e & y l = va r s_ [ " y l " ] ; 3 5 c o n s t V a l u e & y 2 = v a r s . | " y 2 " ] ; 36 37 / / This is a bit of a funny function because of the use 38 / / of the variable i to specify which component function 3 9 //to compute. 40 41 s w i t c h ( i - ) { 42 c a s e 0 : 4 3 g . = y l - e x p ( - 2 . 0 ) ; 4 4 b r e a k ; 4 5 c a s e 1: 46 g_ = y2-exp(—exp( —2)) ; 4 7 b r e a k ; 48 c a s e 2 : 4 9 g_ = y l - e x p ( l . O ) ; so b r e a k ; 51 c a s e 3: 52 g_ = y 2 - e x p ( - e x p ( 1 . 0 ) ) ; 53 b r e a k ; 54 d e f a u l t : 5 5 / / This should never happen. 56 e x i t ( — 1) ; 57 } 58 }; v o i d ou tpu tPa ramsFcn (omstream &mos_j This outputs the variables for M A T L A B to read. It only needs to output the internal parameters that you defined: the solution will be output automatically. 60 v o i d 61 E x a m p l e : : o u t p u t P a r a m s F c n (omstream &mos_) 62 { 63 mos_. n e w v a r ( " A " ); 64 mos_ < < A ; C. Numerical Solution of Domain Wall 96 65 } ; E x a m p l e ^ This is the constructor. It must set all of the parameters. It is a bit long in that it sets all of the relevant parameters for C O L S Y S . It is probably a good idea to just copy the example file and make the required changes. In particular, the size of the arrays nd imf and ndimi is set statically to a rather large value here. If you run out of space while running the program, then you may want to increase these values here. If you want, you could also prompt the user for values in the constructor. This might be very useful for filenames etc. To facilitate scripting, I usually have all the parameters set as in lines 71-73 in the constructor. Then, I can run the program and use a script to supply the parameter values by redirecting an input file. 67 / / The constructor must perform all initializations , especially the 68 / / required parameters, and then call System:: Initialize (). 69 E x a m p l e : : E x a m p l e Q 70 { 71 / / Prompt the user for the parameter. ' 72 cou t « "Parameter„A?„>"; 7 3 c i n > > A ; 74 7 5 / / The following are required parameters: 76 ncomp = 2; 77 78 / / mstar = 4 is computed automatically . 79 so n fxpn t = 0; 81 82 m. p u s h _ b a c k ( 2 ) ; / / Order of each diff eq. 83 m. p u s h _ b a c k ( 2 ) ; 84 85 a l e f t = —2.0; • // Endpoints 86 a r i g h t = 1.0; 87 88 z e t a . p u s h . b a c k ( —2.0 ) ; / / Location of boundary points. 89 z e t a . p u s h . b a c k ( —2.0 ) ; 9 0 ze t a . p u s h _ b a c k ( 1 . 0 ) ; 91 z e t a . p u s h . b a c k ( l . O ) ; 92 9 3 / / Fixedpoints other than boundaries. 9 4 / / fixpnt. push-back() 95 96 / / The following are optional parameters. These are already defined 97 //by default by System(). C. Numerical Solution of Domain Wall 97 98 usermesh = f a l s e ; / / User defined mesh? 99 a d a p t i v e m e s h = t r u e ; / / Should mesh be adapted by program? 100 / / ipar(8). 101 use rguess = f a l s e ; / / User provide guess? 1 0 2 u s e r j a c o b i a n = f a l s e ; / / user provide jacobian? 1 0 3 u s e r b d i f f = f a l s e ; / / User provide boundary diff? 1 0 4 n o n l i n = t r u e ; / / Is the problem nonlinear? ipar(l) 1 0 5 n c o l l o c p t s = 0; / / Default. # of collocation points 106 //per subinterval ipar(2). 1 0 7 nmesh = 0; / / Default. # of initial mesh 108 / / subintervals . ipar(3) 1 0 9 r e g u l a r = 0; / / Regular of sensitive n o / / problem. ipar(lO). i n n t o l = 4 ; / / Number of tolerances. The default 1 1 2 / / is one per variable. 113 d e f t o l = l e — 5 ; / / The default tolerance. The 114 / / function sysinit will set Itol and u s / / tol if ntol < 1. 116 nd imf=10000000 ; / / Size of float workspace. ipar(5) 117 nd im i=500000 ; / / Size of integer workspace. ipar(6) u s i p r i n t = — 1 ; // Diagonostie output, ipar(l) 119 120 121 1 2 2 I t o l . p u s h . b a c k ( 0 ) ; / / Which variables are affected by 1 2 3 I t o l . p u s h _ b a c k ( l ) ; / / tolerances (C indexing)? 1 2 4 I t o l . p u s h . b a c k ( 2 ) ; 1 2 5 I t o l . p u s h _ b a c k ( 3 ) ; 126 1 2 7 d e f t o l = l e — 5 ; 1 2 8 t o l . p u s h _ b a c k ( d e f t o l ); / / Tolerances 1 2 9 t o l . p u s h _ b a c k ( d e f t o l ); 1 3 0 t o l . push_back( d e f t o l ); 131 t o l . p u s h _ b a c k ( d e f t o l ); 132 1 3 3 d a t a f i l e n a m e = "exampledata.dat"; 134 1 3 5 g u e s s f i l e n a m e = "exampleguess. dat" ; 136 1 3 7 / / These are the names that you use in diffeqFcn () 138 xname = " z " ; 139 1 4 0 va rnames . p u s h _ b a c k ( " y l " ) ; 141 va rnames . p u s h _ b a c k ( " d y l " ) ; 1 4 2 va rnames . p u s h _ b a c k ( " y 2 " ) ; 1 4 3 va rnames . p u s h . b a c k ( " d y 2 " ); 144 1 4 5 eqnames. p u s h _ b a c k ( " d d y l " ); 146 eqnames. p u s h - b a c k ( " d d y 2 " ); C. Numerical Solution of Domain Wall 98 147 148 //Must call this at the end of the constructor. 149 I n i t i a l i z e (); 150 } ; The Fortran C O L S Y S code has been given the following C++ interface in the file colsys .hpp: 1 ^ S f n d e f - _ c o l s y s . h p p _ _ 2 ^ d e f i n e _ _ c o l s y s _ h p p _ _ 3 4 e x t e r n " C v o i d c o l s y s _ ( c o n s t i n t *ncomp, 5 c o n s t i n t * m , 6 c o n s t d o u b l e * a l e f t , 7 c o n s t d o u b l e * a r i g h t , 8 c o n s t d o u b l e z e t a [ ] , 9 i n t i p a r [ ] , 1 0 i n t I t o l [ ] , n d o u b l e t o l [], 12 d o u b l e f i x p n t [] , 1 3 i n t i s p a c e [ ] , 14 d o u b l e f s p a c e [], 1 5 i n t * i f l a g , 16 v o i d ( * f s u b ) ( d o u b l e * x , 1 7 d o u b l e * z , 18 d o u b l e * f ) , 1 9 v o i d ( * d f s u b ) ( d o u b l e * x , 2 0 d o u b l e * z , 21 d o u b l e * d f ) , /'/ 2d array 22 v o i d ( * g s u b ) ( i n t * i , 2 3 d o u b l e * z , 24 d o u b l e * g ) , 2 5 v o i d ( * d g s u b ) ( i n t * i , 26 d o u b l e * z , 2 7 d o u b l e * d g ) , 28 v o i d ( * g u e s s ) ( d o u b l e * x , 2 9 d o u b l e * z , 3 0 d o u b l e * d m v a l ) 31 ) ; 32 3 3 e x t e r n " C v o i d a p p s l n . ( d o u b l e * x , 34 d o u b l e z [], 3 5 d o u b l e f s p a c e [], 36 i n t i s p a c e [ ] ) ; 3 7 3 8 : #end i f / / —colsys.hpp--C. Numerical Solution of Domain Wall 99 C.5 Change of Variables Sometimes the domain of the problem is not good for computations. This is the case with domain walls the domain is infinite. In this case, it is useful to transform the domain. This works as follows: the original range is R = [aieft, aright] a n d the transformed range is R = [a\e{t, bright] • You specify the transformation t : R i - » R. To be clear, you have specified the problem on the range R but wish C O L S Y S to work with the problem on the range R for numerical reasons. The independent variables are x € R and x € R and are related by t through: x = t{x) (C.5) The program must address the following issue: Given a function f(x) defined on R with derivatives Jrn ~ d x m , (C.6) how does one compute the function f(x) = f(x) on the desired computa-tional range R and its derivatives , def d " / ( £ ) We define fQ = f = f0(x) = f0 = f = f(x). To start, we note that we know (or can compute) either fm or fm for all required m and we know tm ^ ^ (C.8) dxm - v ' For any function / , we compute the derivatives as follows: fo = fo ^ = d/p _ d / 0 dx dx dx dx = hh dx dx dx dx dx h = ht\ + Zhht2 + hh k = ht\ + 6 / 3 i 2 t 2 + / 2 ( 3 t | + 4*1*3) + hU C. Numerical Solution of Domain Wall 100 Presently such changes of variables must be computed manually. Future version will use automatic differentiation to do this. C.6 QCD Domain Walls As a presentation of these numerical methods, we shall show how to find a numerical solution to the Q C D domain walls for one flavour Nf = 1 without axions as discussed in Chapter 5. The source for the solution to these equations is provided in the files q c d l . hpp and q c d l . cc . To compile these files instead of the example system, one should change the variable N A M E in the makefile from Example to q c d l . The system of equations representing the domain wall are given by Equa-tion (5.8): AM MNC Nc For numerical purposes, it is better to combine all parameters into as few parameters as possible. Thus we introduce the two parameters c\ and AM AE s ci = 1 r C2 = WW (ai0) Notice that the parameter fj, as defined in (5.11) is related to c 2 by \i = \fc2JNc. Wi th these parameters the equation of motion is: d24> <t> , = ci sin0 + c 2 sin — . ( C . l l ) The boundary conditions are at 1 z G {—oo,0} so we need to transform the problem to a finite domain. From the approximate solution (5.10) we see that the largest relevant scale in the problem is set by /J so we define the scale length s ~ 1//J and transform z according to: dz z = tan _ 1(,z7s) — = s(l + tan 2(z)) (C.12a) dz d2z dz z = stan(z) — r = 2stan(S)(l + tan2(5)) = 2tan(5)— (C.12b) dzz dz 1 Recall that the solution is symmetric about z = 0 so we only consider half of the wall to avoid the cusp singularity at z = 0. C. Numerical Solution of Domain Wall 101 d2f(~z) _ d2f(z) (dz\2 d2zdf(z)dz • 77 + J=2 J : 77 ( U . 1 2 C J dz 2 dz2 \dz J dz2 dz dz d 2 ^ d ^ % 2 t a n ( ^ (C.12d) dz2 \dzj ' ™ ^ dz = ^ A ^ 1 + tan 2(5)) 2 + 2tan(5) d4P (C.12e) dz' dz Now, including ^ as a variable we have the following equivalent equation of motion in the transformed variable z: ^ = - s 2 ( l + t a n 2 5 ) 2 ^ c 1 s i n ^ + c 2 s i n - ^ - ^ + 2 t a n ( z ) d ^ (C.13) and the boundary conditions are: 0 ( - f ) = O , 0(0) = 71. (C.14) The choice of the tan() function as the scaling was motivated by the approximate solution (5.10). This scaling makes the approximate solution linear. In general, it is easiest to numerically find solutions that are linear: these are well approximated on a discrete lattice. Thus, one chooses the transformation which makes the solution as nice as possible. In the case of the axion domain wall, for example, such a transformation is not simple because there are two scales, l / m a and 1/m^ separated by many orders of magnitude. In this case, even a numerical solution runs into problems because of the dynamic range. On the other hand, the huge dynamic range allows one to separate the components of the problem and solve for each on the appropriate scale. C.Q.I Example System The following system is used to test the input parameters and driver. It is the system presented in the files Example.cc and E x a m p l e . h p p : n ) (C.15) (C.16) (C.17) This has the solution: Vi = ex y2 = e-eX (C.18) when A = 1. dV dz2 = V l d2y2 dz2 = yi(ym - e yi(-2) = e-2 y2(-2) 3/i(l)'= e 2/2(1) = e~e C. Numerical Solution of Domain Wall 102 C.7 Original Documentation These are relevant snipits from the original documentation2 found in the file c o l n e w . f: Input Variables — Listing of src/colnew.f has been skipped. — User Supplied Routines — Listing of src/colnew.f has been skipped. — Output Variables — Listing of src/colnew.f has been skipped. — Simple Continuation — Listing of src/colnew.f has been skipped. — Package Subroutines — Listing of src/colnew.f has been skipped. — C.8 Code Listings The full source code will not be listed here as the usefulness of hundreds of pages of printed code is unclear. It wil l be included with the source to this thesis at the Los Almos National Laboratories pre-print archives located at http://www.arXive.org where it can be downloaded electronically and used. The code is not con-sidered ready for distribution: if and when it is, especially the automatic differentiation code, it will most likely be distributed through the N E T L I B repository http://www.netlib.org. C.8.1 Example Problem This is the code for the example problem described in Section C.6.1. The Makefile is configured for a Unix system for use with the G N U make utility (though others might work). 2 These and similar code listings are suppressed in the printed version because they can be found easily on the web. They may be useful in an electronic version of this document. C. Numerical Solution of Domain Wall 103 Example.hpp — Listing of src/Example.hpp has been skipped. — Examp7e.cc — Listing of src/Example.cc has been skipped. — Makefile — Listing of src/Makefile has been skipped. — C.8.2 QCD Domain Wall Problem This is the code for the Q C D domain wall problem described in the body of the thesis and in Section C.6. The same Makefile listed above with work with the modification that in line 7, Example should be changed to qcd l . Also included is a parameter file qcdl.params. This can be redirected as input to the compiled qcdl program to produce the datafile qcd l .da t which can be read into M A T L A B using the script myload. m also listed below. This is how Figure 5.1 was produced. qcdl .hpp — Listing of src/qcdl .hpp has been skipped. — qcdl.cc — Listing of src/qcdl.cc has been skipped. — qcdl .params — Listing of src/qcdl.params has been skipped. — C.8.3 System Files The following files are the core of the system. They should not need to be modified (except to make improvements etc.) System.cc — Listing of src/System.cc has been skipped. — C. Numerical Solution of Domain Wall 104 System.hpp — Listing of src/System.hpp has been skipped. — main.cc This is a template file that the Makefile uses to generate testmain . cc by-replacing the text N A M E with the appropriate file name. You may want to modify this file to allow different interface functionality. — List ing of src/main.cc has been skipped. — omstream.cc The two omstream files provide methods for storing data that can be read by the myload.m script into M A T L A B . — Listing of src/omstream.cc has been skipped. — omstream.hpp — Listing of src/omstream.hpp has been skipped. — myload.m This is the M A T L A B script used to load the data output by the omstream methods. — Listing of src/myload.m has been skipped. — C.8.4 Automatic Differentiation These are a first stab at an automatic differentiation class. They are not fully tested but work for the cases tested above. They make heavy use of templates, so some compilers may have problems. It should be pretty easy to extricate the dependencies from the other code if you want to provide the derivatives by hand. Also, there is a huge performance increase if you turn on optimizations with templates (some 20x!), however, it does take substantially longer to compile. ADMath.hpp — Listing of s rc /ADMath .hpp has been skipped. — C. Numerical Solution of Domain Wall 105 AutoDifF.hpp — Listing of src/AutoDiff.hpp has been skipped. — C.8.5 COLNEW This is the original C O L N E W program and the C++ encapsulation used above. Following that is a simple "Hello World" program in Fortran that can be used with the '-v compiler option to locate the relevant libraries on a given system. colsys.hpp — Listing of src/colsys.hpp has been skipped. — colnew.f — Listing of src/colnew.f has been skipped. — hello.f — Listing of src/hello.f has been skipped. — 

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