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Calculations in modified gauge theory : testing some ideas from QCD in a toy model Thomas, Evan Cameron 2017

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Calculations in Modified GaugeTheoryTesting Some Ideas From QCD in a Toy ModelbyEvan Cameron ThomasB.Sc., The University of Washington, 2007M.Sc., The University of British Columbia, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2017c© Evan Cameron Thomas 2017AbstractWe use a deformed “center-stablised” gauge theory, which can be broughtinto a weak coupling regime while remaining confined and gapped, as a toymodel to study some ideas from real QCD. The deformed model has thecorrect nontrivial θ-dependence and degeneracy of topological sectors con-jectured for QCD, and is, apparently, smoothly connected to the stronglycoupled undeformed Yang-Mills, so that we can perhaps expect to get somequalitative insights into QCD. We demonstrate the presence of a nondisper-sive contact term in the topological susceptibility, which contributes withthe opposite sign to normal dispersive contributions coming from physicalpropagating degrees of freedom. We further show that, despite the systembeing completely gapped with no massless physical degrees of freedom, thesystem has a Casimir-like, power scaling, dependence on boundaries, in con-trast with the naive expectation that a system with only massive degrees offreedom should have a weak (exponentially small) dependence on long dis-tance effects. This behaviour suggests the possibility for a solution for thecosmological dark energy problem coming from the strongly coupled QCDsector on a manifold with a boundary, which would have the correct sign andbe of the correct order of magnitude. Next, we investigate the interactionbetween point-like topological charges (monopoles) and extended sheet-liketopological defects (domain walls) in attempt to explain some recent latticeQCD results suggesting that extended topological objects are more impor-tant to understanding the relevant field configurations in QCD than theinstantons traditionally expected. Finally, we derive the existence of ex-cited metastable vacuum states and calculate their decay rate to the trueground state of the theory, comparing with the expected results discussedyears ago in proper QCD. The presence of metastable vacuum states witha nonzero effective θ parameter, like those present in the deformed model,could explain P and CP violation in heavy ion collisions observed on anevent by event basis, which seem to average away over many events.iiLay SummaryWe investigate a toy model related to the true theory for the strong nuclearinteraction which binds nuclear matter. The toy model is much easier towork with, but preserves many important aspects of the true theory, espe-cially related to the vacuum structure. We use this simplified model to studysome ideas in the true theory, in which there are no obvious ways to performthe relevant calculations. Our computations provide some insight into a par-ticular dependence of the bulk energy density on the size of the system, thestructure of the type of configurations relevant in the theory, and some oldquestions about semi-stable vacuum states. These results could help explainthe origins for cosmological dark energy, some recent results in a differentlattice approximation which differ from conventional wisdom, and some oddobservations in heavy ion collisions.iiiPrefaceChapter 2 is primarily a review of the deformed “center-stabilised” gaugetheory model discussed by Lawrence Yaffe and Mithat U¨nsal in [85], andas such, aside from a few corrections and expanded discussions, does notrepresent any original work attributable to myself. Chapter 3 is adaptedfrom work titled “Topological Susceptibility and Contact Term in QCD: AToy Model” published in Physical Review D 85:044039 [80]. The content inChapter 4 is published as “Casimir Scaling in Gauge Theories with a Gap:Deformed QCD As a Toy Model” in Physical Review D 86:065029 [79].Chapter 5 is published as “Long Range Order in Gauge Theories: DeformedQCD As a Toy Model” in Physical Review D 87:085027 [81]. Chapter 6was the result of a project in collaboration with another student, AmitBhoonah, with whom I worked on the numerical simulations presented, andis published as “Metastable vacuum decay and θ dependence in gauge theory.Deformed QCD as a toy model.” in Nuclear Physics B 890:30 [7]. I did thebulk of the writing for that paper, and actually removed the short sectionAmit wrote in the text of Chapter 6, since it was not particularly relevant forthis presentation, such that the text presented here represents my writing.In all four of these works my supervisor, Ariel Zhitnitsky, suggested thefundamental ideas and wrote many of the parts about historical contextand relation to other theories and models scattered throughout.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Description of the Model . . . . . . . . . . . . . . . . . . . . . 42.1 Formulation of the Theory . . . . . . . . . . . . . . . . . . . 42.2 Infrared Description . . . . . . . . . . . . . . . . . . . . . . . 62.3 Monopole Sine-Gordon Equivalence . . . . . . . . . . . . . . 102.4 Mass Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Topological Susceptibility and Contact Term . . . . . . . . 153.1 The Contact Term and Degeneracy of Topological Sectors . . 153.1.1 The Contact Term . . . . . . . . . . . . . . . . . . . . 153.1.2 Topological Susceptibility and Contact Term in 2DQED . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1.3 The Contact Term from Summation Over TopologicalSectors . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Topological Susceptibility in the Deformed QCD . . . . . . . 223.2.1 Topological Susceptibility in the Monopole Picture . 223.2.2 Topological Susceptibility in the Presence of the LightQuarks . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.3 Interpretation . . . . . . . . . . . . . . . . . . . . . . 283.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29vTable of Contents4 Casimir Scaling and Dark Energy . . . . . . . . . . . . . . . 314.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Casimir-Type Behaviour in Deformed QCD . . . . . . . . . . 344.2.1 Casimir-Type Corrections for 4D Instantons . . . . . 364.2.2 Casimir-Type Corrections for 3D Monopoles . . . . . 394.2.3 Non-Zero Mode Contributions . . . . . . . . . . . . . 414.3 Topological Sectors and the Casimir Correction in QCD . . . 434.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 Long Range Order and Domain Walls . . . . . . . . . . . . . 485.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.2 Domain Walls in Deformed Gauge Theory . . . . . . . . . . 515.2.1 Domain Wall Solution . . . . . . . . . . . . . . . . . . 525.3 Domain Wall - Monopole Interaction . . . . . . . . . . . . . 585.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 Metastable Vacuum Decay . . . . . . . . . . . . . . . . . . . . 666.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.2 Metastable Vacuum States . . . . . . . . . . . . . . . . . . . 696.3 Metastable Vacuum Decay . . . . . . . . . . . . . . . . . . . 726.4 Computations . . . . . . . . . . . . . . . . . . . . . . . . . . 756.4.1 Numerical Technique . . . . . . . . . . . . . . . . . . 756.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 786.4.3 Improved Results . . . . . . . . . . . . . . . . . . . . 816.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89AppendicesA Domain Wall Decay . . . . . . . . . . . . . . . . . . . . . . . . 98B Metastable Vacuum Decay . . . . . . . . . . . . . . . . . . . . 101C Asymptotic Vacuum Decay . . . . . . . . . . . . . . . . . . . 105viList of Figures2.1 Diagram depicting the deformation removing the phase tran-sition to a deconfined phase at weak coupling. It is based ona similar diagram from [85]. . . . . . . . . . . . . . . . . . . 75.1 Picture depicting the transition between paths correspondingto the decay of some domain wall state to a domain wall freeground state. Inspired by a similar picture in [28]. . . . . . . 555.2 Graph showing the two layer structure of the topologicalcharge density plotted against one direction across the Do-main Wall and the other one of the two dimensions alongit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.3 Plot of the numerical result for the binding energy at variousseparation distances between domain wall and monopole. No-tice that for z0 < 0, the monopole to the right of the domainwall, there is an “attractive” potential with a minimum nearz0 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.4 Close up plot of the points near the minimum in Figure 5.3showing that the minimum is to the z0 < 0 side of the center. 625.5 Close up plot of the points to the right in Figure 5.3 showingthe small barrier present on the z0 > 0 side. Notice the muchfiner vertical scale. . . . . . . . . . . . . . . . . . . . . . . . . 636.1 Plot of some simulation data for the one dimensional action(6.16) as a function of the angle ϕ between the boundaryconditions done for N = 7. . . . . . . . . . . . . . . . . . . . 786.2 Plot of some simulation data for the σ field configurationplotted across the domain wall done for N = 7 and φ =−8pi/7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.3 Plot of some simulation data for the decay exponent F (N)plotted for N in the range 15 to 75. . . . . . . . . . . . . . . 806.4 Plot of the improved simulation data for the decay exponentF (N) plotted for N in the range 7 to 75. . . . . . . . . . . . 83viiList of FiguresB.1 Qualitative picture for the potential of a general system witha global ground state, φ(−), and a higher energy metastablestate, φ(+), with an energy splitting between the two givenby . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102viiiChapter 1IntroductionGauge theories, the class of Quantum Field Theories (QFTs) exhibiting gen-eralised versions of the gauge symmetry from electrodynamics, have been ex-tremely successful in describing the fundamental forces (excepting gravity)via vector boson mediated particle interactions. Quantum Electrodynam-ics (QED) has been relatively easy to work with, being weakly coupled andamenable to perturbation theory at all accessible scales, and has matched ex-periment extremely well. By contrast, Quantum Chromodynamics (QCD),the theory for the strong nuclear force, is strongly coupled (highly nonlin-ear) at low temperature. The coupling runs down at higher temperature,but unfortunately the system undergoes a phase transition (crossover) from“confined” hadronic matter to a “deconfined” quark gluon plasma state be-fore the coupling is small in a perturbative sense. This means that, whilethere is a weak coupling regime, the perturbative calculations we can do atweak coupling cannot tell us much about the confined phase. As a result,people have looked for other ways to approach QCD, including: phenomeno-logical models that attempt to write down informed guesses for an effectivetheory; lattice models that discretise space and attempt to calculate cor-relators by brute force Monte-Carlo simulation of the path integral; andAdS/CFT calculations that produce weakly coupled dual gravity theoriesto QCD-like strongly coupled field theories.Each of the approaches mentioned above has its shortcomings. Phe-nomenological approaches can be difficult to get any predictive power from,since they are not necessarily describing any behaviour accurately outsidethe observed phenomena they are based on. Lattice QCD is extremely nu-merically expensive, requiring significant time on expensive supercomputersto do realistic calculations. It can also be quite difficult to get any kind ofphysical intuition about the reasons for effects seen on the lattice. AdS/CFTdoes not have a known direct dual to QCD, so instead makes some argu-ments in supersymmetric models and/or gauge models with different gaugegroups than the SU(3) describing real QCD.Hopefully, despite not having a perfect calculational tool, using multipledesperate approaches will allow us to form an increasingly good picture of1Chapter 1. Introductionthe behaviour of real QCD as we see similar effects in different approxi-mations. As such, we investigate a different approximation to those listedabove, the so called “center-stablised” or “deformed” gauge theory devel-oped by Lawrence Yaffe and Mithat U¨nsal [85], and apply it to carry outsome calculations. The model is built by taking a normal gauge theory,with Yang-Mills Lagrangian, and adding an extra term (deformation) to theLagrangian acting as a potential penalty for the order parameter for the de-confinement phase transition. This means, by suitable choice of parameters,we can enforce a confined phase to arbitrarily large temperature (and soarbitrarily small coupling). Thus, we can work in a model which is weaklycoupled, so amenable to perturbation theory and semiclassical treatment,but describes a confined system which is, apparently, smoothly connectedto the real strongly coupled system without a phase transition between thetwo.We begin, in Chapter 2, by reviewing the relevant aspects of the model,showing the two dual descriptions, as a Coulomb gas of topological monopolesand as a coupled sine-Gordon model, for the low energy effective theory atweak coupling. In Chapter 3, we discuss the topological susceptibility, akey element in the resolution of the U(1)A problem in QCD, and demon-strate the presence of a nondispersive “contact” term in both dual descrip-tions of the low energy effective theory. The contact term has the oppositesign to the contribution from any physical propagating degrees of freedomas is necessary to satisfy the Ward Identity, which requires a cancellationwith the contribution coming from the physical fields. Previously, there wasno consistent method for deriving the contact term in a four dimensionalgauge theory. Instead, Witten inserted this term directly by hand [93] andVeneziano added an extra “ghost” field that leads to a contact term whenintegrated out [90, 91], but in this model the contact term arises naturally.[80]Next, we show, in Chapter 4, that a zero mode analysis of the monopoleconfigurations describing the Coulomb gas description gives a Casimir-likepower scaling for the bulk energy density, as described by the topologicalsusceptibility, rather than the naive expectation for a gapped system withno massless physical. If only physical degrees of freedom contribute, theirdispersion relations dictate an exponential suppression of any bulk depen-dence on the boundary. In contrast, the deformed model has a nondispersiveWe further argue that, if it persists in undeformed QCD at strong coupling,such a Casimir scaling could lead to a solution for the cosmological darkenergy naturally following from QCD on a bounded manifold, without theneed for new fields or new physics. [79]2Chapter 1. IntroductionIn Chapter 5, we address some recent lattice QCD results suggestingthat the topological defects, relevant in gauge configurations that saturatethe path integrals, are extended objects looking like interleaved sheets ofopposite topological charge, rather than the point-like instantons peoplehave traditionally discussed. A class of domain wall objects that appear inthe deformed model have similar properties, being (classically) topologicallystable with sheets of opposite topological charge interleaved. We considertheir interaction with point-like monopoles and explain a possible dynamicalreason for the absence of point-like objects in relevant configurations. [81]Finally, in Chapter 6, we demonstrate the presence of metastable vacuumstates (with energy higher than the true vacuum) in the deformed model andcalculate the decay rate from the metastable states to the true vacuum. Wealso discuss how the presence of such metastable states can, if similar statesexist in undeformed strongly coupled QCD, lead to P and CP effects thathave apparently been observed in some heavy ion collisions. [7]3Chapter 2Description of the ModelIn this chapter we discuss the “center-stabilised” deformed Yang-Mills de-veloped in [85] and references therein, before moving on to a discussion ofthe topological properties of this theory in Chapter 3, and some applicationsin Chapters 4, 5, and 6. In the deformed theory an extra term is put into theLagrangian in order to prevent the center symmetry breaking that charac-terises the QCD phase transition between “confined” hadronic matter and“deconfined” quark-gluon plasma. The extra term is a penalty for statesin which the order parameter for such a transition develops an expectationvalue. Thus, we have a theory which remains confined at high temperaturein a weak coupling regime, and for which it is claimed [85] that there does notexist an order parameter to differentiate the low temperature (non-Abelian)confined regime from the high temperature (Abelian) confined regime. Thismeans we can do some simple semiclassical calculations in a confined the-ory that, as we shall discuss, retains some interesting properties of, and issmoothly connected to, undeformed Yang-Mills. For some other extensionsof this model related to inclusion of adjoint fermions, extensions to generalgauge groups, and so on, see[60–62] and references therein. We follow [85]in deriving the relevant parts of the theory.2.1 Formulation of the TheoryWe start with pure Yang-Mills (gluodynamics) with gauge group SU(N) onthe manifold R3 × S1 with the standard actionSYM =∫R3×S1d4x12g2tr[F 2µν(x)], (2.1)and add to it a deformation action,∆S ≡∫R3d3x1L3P [Ω(x)] , (2.2)42.1. Formulation of the Theorybuilt out of the Wilson loop (Polyakov loop) wrapping the compact dimen-sion,Ω(x) ≡ P[ei∮dx4 A4(x,x4)]. (2.3)The “double-trace” deformation potential P [Ω] respects the symmetries ofthe original theory and is built to stabilise the phase with unbroken centersymmetry. It is defined byP [Ω] ≡bN/2c∑n=1an |tr [Ωn]|2 . (2.4)Here bN/2c denotes the integer part of N/2 and {an} is a set of suitablylarge positive coefficients.The centre of the gauge group is the subgroup of elements which com-mute with all elements of the full group, is isomorphic to ZN for the gaugegroup SU(N), and is the symmetry corresponding to the confinement decon-finement phase transition. The first term of P [Ω], proportional to |tr [Ω]|2(with a sufficiently large positive coefficient), will prevent breaking of thecenter symmetry from ZN to Z1 with order parameter 〈tr [Ω]〉, but will notprevent tr[Ω2]from developing a vacuum expectation value so that it willnot prevent the center symmetry breaking from ZN to Z2 with order pa-rameter 〈tr [Ω2]〉. The term proportional to ∣∣tr [Ω2]∣∣2 however does preventsuch a symmetry breaking. Likewise, for each other subset Zp of ZN (withN mod p = 0), there needs to be a corresponding term, proportional to|tr [Ωp]|2, in the deformation potential. This is the reason for includingterms up to∣∣tr [ΩbN/2c]∣∣2. Note that for real life QCD the gauge group isSU(3) and so in that case only one term would be necessary,P [Ω] = a |tr [Ω]|2 . (2.5)In undeformed pure gluodynamics the effective potential for the Wilsonloop, whose expectation value acts as an order parameter, is minimised forΩ an element of ZN , which corresponds to a deconfined phase. The deforma-tion potential (2.4) with sufficiently large {an} however changes the effectivepotential for the Wilson line so that it is minimised instead by configura-tions in which tr [Ωn] = 0, which in turn implies that the eigenvalues of Ωare uniformly distributed around the unit circle. Thus, the set of eigenvaluesis invariant under the ZN transformations, which multiply each eigenvalueby e2piik/N (rotate the unit circle by k/N). The center symmetry is thenunbroken by construction. The coefficients, {an}, can be suitably chosen52.2. Infrared Descriptionsuch that the deformation potential, P [Ω], forces unbroken symmetry atany compactification scales [85], but for our purposes we are only interestedin small compactifications (L Λ−1 where L is the length of the compact-ified dimension, interpreted as an inverse temperature, and Λ is the QCDscale). The idea here is to go to weak coupling for mathematical controlbut to look at low-energy behaviour at that scale since we are interestedin vacuum behaviour. At small compactification, the gauge coupling at thecompactification scale is small so that we can work in a perturbative regimeand explicitly evaluate the potential for the Wilson loop due to Quantumfluctuations as in [32, 85]. The one-loop potential isV [Ω] =∫R3d3x1L4V [Ω(x)] , (2.6)withV [Ω] = − 2pi2∞∑n=11n4|tr [Ωn]|2 . (2.7)The undeformed potential for the Wilson loop (2.7) is minimised when Ω isan element of the center, ZN , so that Ω = e2piik/N . The deformation poten-tial (2.4) must therefore overcome this one-loop potential and force Ω to notchoose one particular element of ZN . We must choose the coefficients {an}to be larger than 2/(pi2n4). A simple choice is an = 4/(pi2n4). With thischoice, the full one-loop effective potential for the Wilson loop is minimisedfor tr [Ωn] = 0 for all n 6= 0 mod N , indicating unbroken center symmetry.2.2 Infrared DescriptionAs mentioned in the previous section, we are interested in the regime inwhich the compactification size is small, L Λ−1, and so the gauge couplingis small at the compactification scale, g2 (1/L)  1. So, in our deformedtheory, the combined effective potential for the Wilson loop is the sum of(2.7) and (2.4), which is minimised by field configurations withΩ = Diag(1, e2pii/N , e4pii/N , . . . , e2pii(N−1)/N), (2.8)up to conjugation by an arbitrary element of SU(N). The configuration(2.8) can be thought of as braking the gauge symmetry down to its max-imal Abelian subgroup, SU(N) → U(1)N−1. In the gauge in which Ω isdiagonal, the modes of the diagonal components of the gauge field with zero62.2. Infrared Description0L∞0Lc∞deformationcouplingStandardYang-MillsDeformedYang-MillsdeconfinedphaseFigure 2.1: Diagram depicting the deformation removing the phase transi-tion to a deconfined phase at weak coupling. It is based on a similar diagramfrom [85].momentum along the compactified dimension describe the U(1)N−1 pho-tons. Modes of the diagonal gauge field with non-zero momentum in thecompactified dimension form a Kaluza-Klein tower and receive masses thatare integer multiples of 2pi/L and become large for small L. The remain-ing off-diagonal components of the gauge field form a Kaluza-Klein tower ofcharged W -bosons which receive masses that are integer multiples of 2pi/NL.Then the lightest W -boson mass, mW ≡ 2pi/NL, describes the scale belowwhich the dynamics are effectively Abelian.As described in [85], the proper infrared description of the theory is a di-lute gas of N types of monopoles, characterised by their magnetic charges,which are proportional to the simple roots and affine root of the Lie al-gebra for the gauge group U(1)N . Although the symmetry breaking isSU(N) → U(1)N−1, it is simpler to work with U(1)N and, as we will see,the extra degree of freedom will completely decouple from the dynamics.72.2. Infrared DescriptionThe extended root system is given by the simple roots,α1 = (1,−1, 0, . . . , 0) = eˆ1 − eˆ2,α2 = (0, 1,−1, . . . , 0) = eˆ2 − eˆ3,...αN−1 = (0, . . . , 0, 1,−1) = eˆN−1 − eˆN ,(2.9)and the affine root,αN = (−1, 0, . . . , 0, 1) = eˆN − eˆ1.We denote this root system by ∆aff and note that the roots obey the innerproduct relationαa · αb = 2δa,b − δa,b+1 − δa,b−1. (2.10)For a fundamental monopole with magnetic charge αa ∈ ∆aff , the topo-logical charge is given byQ =∫R3×S1d4x116pi2tr[FµνF˜µν]= ± 1N, (2.11)and the Yang-Mills action is given bySYM =∫R3×S1d4x12g2tr[F 2µν]=∣∣∣∣∫R3×S1d4x12g2tr[FµνF˜µν]∣∣∣∣ = 8pi2g2 |Q| . (2.12)The second equivalence hold because the classical monopole solutions areself dual [32],Fµν = F˜µν .For an antimonopole with magnetic charge −αa, the Yang-Mills action isthe same (2.12) and the topological charge changes sign, Q = −1/N .So the infrared description, at distances larger than the compactificationlength L, is given by a three dimensional dilute monopole gas with N typesof monopoles (and so N types of anti-monopoles) interacting by a speciesdependent Coulomb potential with interactions defined by the inner product(2.10),Va,b(r) = L(2pig)2 (±αa) · (±αb)4pi |r|= ±L(2pig)2 2δa,b − δa,b−1 − δa,b+14pi |r| , (2.13)82.2. Infrared Descriptionwhere the overall sign is plus for a monopole-monopole or antimonopole-antimonopole interaction and minus for a monopole-antimonopole interac-tion. For a given monopole configuration with n(a) monopoles and n¯(a)antimonopoles of types a = 1, . . . , N , at positions r(a)k , k = 1, . . . , na andr¯(a)k , k = 1, . . . , n¯a respectively, the three dimensional U(1)N magnetic fieldis given byB(x) =N∑a=12pigαan(a)∑k=1x− r(a)k4pi∣∣∣x− r(a)k ∣∣∣3 −n¯(a)∑l=1x− r¯(a)l4pi∣∣∣x− r¯(a)l ∣∣∣3 . (2.14)LettingM (a) = n(a) + n¯(a),r(a)k ={r(a)k for k ≤ n(a)r¯(a)k−n(a) for k > n(a),Q(a)k ={+1 for k ≤ n(a)−1 for k > n(a) ,(2.15)we can write (2.14) in a more compact form,B(x) =N∑a=12pigαaM(a)∑k=1Q(a)kx− r(a)k4pi∣∣∣x− r(a)k ∣∣∣3 . (2.16)The action for such a monopole configuration is a combination of the monopoleself-energies and the Coulomb interaction potential energies for each pair ofmonopoles,SMG = SselfN∑a=1M (a) + Sint, (2.17)whereSint =2pi2Lg2N∑a,b=1αa · αbM(a)∑k=1M(b)∑l=1Q(a)k Q(b)l G(r(a)k − r(b)l) (2.18)andG(r) ≡ 14pi |r| . (2.19)The canonical partition function is then given, as usual, by a sum over allpossible monopole configurations with a statistical weight e−S ,Z =∫ N∏a=1dµ(a) e−Sint , (2.20)92.3. Monopole Sine-Gordon Equivalencewith measuredµ(a) =∞∑n(a)=0(ζ/2)n(a)n(a)!∞∑n¯(a)=0(ζ/2)n¯(a)n¯(a)!∫R3n(a)∏k=1dr(a)k∫R3n¯(a)∏l=1dr¯(a)k . (2.21)The monopole fugacity, ζ, describes the density of monopoles and is givenby,ζ ≡ C e−Sself = Am3W(g2N)−2e−∆Se−8pi2/Ng2(mW ), (2.22)where the C factor is the one-loop functional determinant in the monopolebackground as described in the appendix of [85].Next we show explicitly that the above monopole partition function(2.20) is equivalent to a sine-Gordon partition function which describes theproper θ-dependence for the QCD vacuum.2.3 Monopole Sine-Gordon EquivalenceThe sine-Gordon partition function for this model describes a three dimen-sional N -component real scalar field theory, given byZ =∫ N∏a=1Dσa e−Sdual[σ], (2.23)withSdual =∫R3d3x[12L( g2pi)2(∇σ)2 − ζN∑a=1cos(αa · σ)]. (2.24)Considering the cosine term,exp[ζ∫R3d3xN∑a=0cos(αa · σ)]=N∏a=1exp[ζ∫R3d3x cos(αa · σ)]=N∏a=1exp[ζ2∫R3d3x(eiαa·σ + e−iαa·σ)],(2.25)we can apply the power series representation for the exponential, ex =∑xn/n!, and getN∏a=1∞∑M(a)=0(ζ/2)M(a)M (a)!M(a)∏m=0[∫R3d3xm(eiαa·σ(xm) + e−iαa·σ(xm))] .(2.26)102.3. Monopole Sine-Gordon EquivalenceWe then make use of the binomial theorem,(x+ y)n =n∑k=0(nk)xn−kyk with(nk)=n!(n− k)!k! , (2.27)and arrive at,N∏a=1∞∑M(a)=0M(a)∑m=0(ζ/2)M(a)m!(M (a) −m)!M(a)∏k=1∫R3d3xk expiM(a)∑k=1Q(a)k αa · σk=[N∏a=1∫dµ(a)]expi N∑a=0M(a)∑k=0Q(a)k αa · σ(xk) , (2.28)where dµ(a) is given in (2.21). Thus, inserting (2.28) into the sine-Gordonpartition function (2.23), we haveZ =∫ N∏a=1[Dσa dµ(a)]exp[−β∫R3d3x L]L = 12(∇σ)2 − iβN∑a=1M(a)∑k=1Q(a)k δ(x(a)k − x)αa · σ(x), (2.29)whereβ ≡ 1L( g2pi)2. (2.30)Treating the last term in the exponent as a source term,J(x) ≡ −iβN∑a=1M(a)∑k=1Q(a)k δ(x(a)k − x)αa, (2.31)and completing the square with the shift σ(x)→ σ(x)+∫ d3y G(x−y)J(y),we have,Z = Z0∫ [ N∏a=0dµ(a)]exp{β2∫R3d3x∫R3d3y [J(x)G(x− y)J(y)]},(2.32)in which Z0 is the functional determinantZ0 ≡∫ [ N∏a=0Dσa]exp[−β2∫R3d3x (∇σ)2]. (2.33)112.3. Monopole Sine-Gordon EquivalenceThe above determinant does not contain any of the relevant physics and isjust a constant prefactor that will drop out of any calculation of operator ex-pectation values in the monopole ensemble. Finally, inserting the expressionfor the source (2.31), the partition function becomes,Z = Z0∫ [ N∏a=0dµ(a)]×exp−2pi2Lg2N∑a,b=1M(a)∑k=1M(b)∑l=0αa · αb Q(a)k Q(b)l G(x(a)k − x(b)l ) , (2.34)which is the partition function for the monopole gas from (2.20).Next, including a θ-parameter in the Yang-Mills action,SYM → SYM + iθ∫R3×S1116pi2tr[FµνF˜µν], (2.35)with F˜µν ≡ µνρσFρσ, multiplies each monopole fugacity by eiθ/N and anti-monopole fugacity by e−iθ/N . In the dual sine-Gordon theory this inclusionis equivalent to shifting the cosine term so that 1Sdual →∫R3[12L( g2pi)2(∇σ)2 − ζN∑a=1cos(αa · σ + θN)]. (2.36)The θ parameter enters the effective Lagrangian (2.36) as θ/N which is thedirect consequence of the fractional topological charges of the monopoles(2.11). Nevertheless, the theory is still 2pi periodic. This 2pi periodicity ofthe theory is restored not due to the 2pi periodicity of Lagrangian (2.36).Rather, it is restored as a result of summation over all branches of thetheory when the levels cross at θ = pi(mod 2pi) and one branch replacesanother and becomes the lowest energy state. Indeed, the ground stateenergy density is determined by minimisation of the effective potential (2.36)when summation∑N−1l=0 over all branches is assumed in the definition of thecanonical partition function (2.20). It is given byEmin(θ) = − limV→∞1V Lln{N−1∑l=0exp[V ζN cos(θ + 2pi lN)]}, (2.37)1We note in passing that there is a typo in [85] in sine-Gordon representation whichis corrected here. Also, it has been stated (incorrectly) in [85] that the sine-GordonLagrangian is 2pi periodic as a result of a symmetry. This statement is incorrect, as theclaimed symmetry is not in fact a symmetry of the theory, such that θ parameter entersthe Lagrangian as θ/N as it should. To check this insert σ = 0 and notice that theθ-dependence is explicitly different after the transformation suggested in [85].122.3. Monopole Sine-Gordon Equivalencewhere V is 3d volume of the system. (2.37) shows that in the limit V →∞cusp singularities occur at the values at θ = pi (mod 2pi) where the lowestenergy vacuum state switches from one analytic branch to another. Thefirst derivative of the vacuum energy, which is proportional to the topologicaldensity condensate, is two-valued at these points. This means that wheneverθ = pi (mod 2pi) we stay with two degenerate vacua in the thermodynamiclimit. If, on the other hand, the thermodynamic limit is performed fora fixed value of θ, any information on other states is completely lost in(2.37). Correspondingly, the 2pi periodicity in θ is also lost in infinite volumeformulae. We miss the chance to know about additional states when wework in the infinite volume limit from the very beginning. As a result, usualV =∞ formulae become blind to the very existence of a whole set of differentvacua, which is in fact responsible for restoration of the 2pi periodicity in θ.The model under consideration explicitly supports this pattern in deformedQCD where all computations are under complete theoretical control.Such a pattern is known to emerge in many four dimensional supersym-metric models, and also gluodynamics in the limit N = ∞. It has beenfurther argued [34] that the same pattern also emerges in four dimensionalgluodynamics at any finite N . We follow, in fact, the technique from [34]to arrive at (2.37) in analysing the θ periodicity of the theory. The samepattern emerges in holographic description of QCD [96] at N =∞ as well.Finally, considering the expectation value〈e±iαb·σ(y)〉 = 1Z∫ [ N∏a=1Dσa dµ(a)]×exp{−β∫R3d3x[12(∇σ)2 + J · σ ∓ iβδ(x− y)αb · σ]}=Z0Z∫ [ N∏a=0dµ(a)]e−SMG exp±4pi2Lg2N∑a=1M(a)∑k=1Q(a)k αa · αbG(x(a)k − y) ,(2.38)we note that the operator eiαa·σ(x) is the creation operator for a monopoleof type a at x, i.e.Ma(x) = eiαa·σ(x). (2.39)Likewise, the operator for an antimonopole is M¯a(x) = e−iαa·σ(x). Theexpectation values of these operators in fact determine the ground state ofthe theory.132.4. Mass Gap2.4 Mass GapThe cosine potential in the sine-Gordon action (2.24) gives rise to a massterm for the dual scalar fields. Expanding the potential around the minimumσ = 0 up to quadratic order and rescaling σ → L(2pi)2/g2σ to put thekinetic term into canonical form, and gives (up to a constant term)V (σ) ∼= 12m2σN∑a=1(σa+1 − σi)2, (2.40)withm2σ ≡ Lζ(2pig)2. (2.41)The above mass term is diagonalised by the discrete Fourier transformσ˜b ≡ 1√NN∑a=0e−2piiabN σa, (2.42)becomingV (σ) ∼= 12N∑a=1m2a |σ˜a|2 , (2.43)where ma = mσ sin(pia/N). So the only scalar field which remains masslessis the Nth field, which is the field associated with the affine root. Insertingthe discrete Fourier transform (2.42) into the full cosine potential howevershows that the Nth field drops out of the cosine potential completely, soalthough it remains massless, it completely decouples from the theory anddoes not interact with the other components at all.14Chapter 3Topological Susceptibilityand Contact TermThis chapter reproduces the work presented in [80]. We explicitly demon-strate the presence of a contact term in the topological susceptibility for thedeformed gauge theory. It contributes with a sign opposite the contributionsfrom physical propagating degrees of freedom such that the relevant Wardidentities are satisfied.3.1 The Contact Term and Degeneracy ofTopological SectorsIn this section we present an overview of the nature of the contact termwhich is not related to any physical degrees of freedom . We explain howwe know about its mere existence because of requirements imposed by theanomalous Ward Identities, which require its presence. We also give a sim-ple two dimensional example explaining how this term emerges in gaugetheories. The nature of this “weird” contribution is entirely determined bythe topological properties of the model rather than the physical propagatingdegrees of freedom of the system. Thus, we will find similar behaviour intheories with similar topological properties irrespective of the particularitiesof the theories. Such calculations cannot be carried out at present in un-deformed QCD, but can in the simplified deformed model we consider. Assuch, because this model exhibits a similar topological structure to what weexpect for proper undeformed QCD, we have some hope that calculations,which can be performed here, can provide some useful insight.3.1.1 The Contact TermWe start with definition of the topological susceptibility χ which is themain ingredient of the resolution of the U(1)A problem in QCD [90, 91,93], see also [42, 54, 64]. The necessity for the contact term in topological153.1. The Contact Term and Degeneracy of Topological Sectorssusceptibility χ can be explained in few lines as follows. We define thetopological susceptibility χ in the standard way:χ(θ = 0) =∂2Evac(θ)∂θ2∣∣∣∣θ=0= limk→0∫d4xeikx〈T{q(x), q(0)}〉, (3.1)where θ is the conventional θ parameter which enters the Lagrangian alongwith topological density operator q(x), see precise definitions below. Themost important feature of the topological susceptibility χ, for our presentdiscussion, is that it does not vanish in spite of the fact that q = ∂µKµ istotal divergence. Furthermore, any physical state of mass mG, momentumk → 0 and coupling 〈0|q|G〉 = cG contributes to the dispersive portion ofthe topological susceptibility with negative sign 2χdispersive ∼ limk→0∫d4xeikx〈T{q(x), q(0)}〉∼ limk→0〈0|q|G〉〈G|q|0〉−k2 −m2G' −|cG|2m2G≤ 0, (3.2)while the resolution of the U(1)A problem, which would provide a physicalmass for the η′ meson, requires a positive sign for the topological suscepti-bility (3.3), see the original reference [90] for a thorough discussion,χnon−dispersive = limk→0∫d4xeikx〈T{q(x), q(0)}〉 > 0 . (3.3)Therefore, there must be a contact contribution to χ, which is not related toany propagating physical degrees of freedom, and it must have the “wrong”sign, by which we mean opposite to any term originating from physicalpropagators, in order to saturate the topological susceptibility (3.3). In theframework [93] the contact term with the “wrong” sign has been simplypostulated, while in refs.[90, 91] the Veneziano ghost had been introducedto saturate the required property (3.3). This Veneziano ghost field is simplyan unphysical degree of freedom with the “wrong” sign in the kinetic termsuch that it generates the same contact term when integrated out. It shouldbe emphasised that these two descriptions are equivalent and simply twoseparate ways of describing the same physics and that in these two pictures,the claim that the “contact” term does not come from physical propagatingdegrees of freedom is manifest.2We use the Euclidean notations where path integral computations are normally per-formed.163.1. The Contact Term and Degeneracy of Topological SectorsIt should be mentioned here that the “wrong” sign in topological sus-ceptibility (3.3) is not the only manifestation of this “weird” unphysicaldegree of freedom. In fact, one can argue that the well known mismatchbetween Bekenstein-Hawking entropy and the entropy of entanglement forgauge fields is due to the same gauge configurations which saturate the con-tact term in the topological susceptibility in QCD as discussed in [101]. Inboth cases the extra term with a “wrong” sign is due to distinct topologi-cal sectors in gauge theories. This extra term is non-dispersive in nature,can not restored from the conventional spectral function through dispersionrelations, and is not associated with any physical propagating degrees offreedom.3.1.2 Topological Susceptibility and Contact Term in 2DQEDThe goal here is to give some insights on the nature of the contact term usinga simple exactly solvable two dimensional QED2 [48]. We follow [100, 101]to discuss all essential elements related to the contact term.We start by considering two dimensional photodynamics (QED formu-lated without fermions) which is naively a trivial theory as it does not haveany physical propagating degrees of freedom. However, we shall argue thatthis (naively trivial) two dimensional photodynamics nevertheless has a con-tact term which is related to the existence of different topological sectorsin the theory. Thus, the presence of degenerate topological sectors in thesystem, which we call the “degeneracy” for short3 , is the source for thiscontact term which is not related to any physical propagating degrees offreedom.The topological susceptibility χ in this model is defined as followsχ ≡ e24pi2limk→0∫d2xeikx 〈TE(x)E(0)〉 , (3.4)3Not to be confused with conventional term “degeneracy” when two or more physicallydistinct states are present in the system. In the context of this paper, the “degeneracy”references the existence of winding states |n〉 constructed as follows: T |n〉 = |n + 1〉. Inthis formula the operator T is the large gauge transformation operator which commuteswith the Hamiltonian [T , H] = 0, implying the “degeneracy” of the winding states |n〉.The physical vacuum state is unique and constructed as a superposition of |n〉 states. Inpath integral approach the presence of N different sectors in the system is reflected bysummation over k ∈ Z in (3.12), (3.13), and (3.14).173.1. The Contact Term and Degeneracy of Topological Sectorswhere q = e2piE is the topological charge density operator and∫d2x q(x) =e2pi∫d2x E(x) = k (3.5)is the integer valued topological charge in the 2d U(1) gauge theory, E(x) =∂1A2 − ∂2A1 is the field strength. The expression for the topological sus-ceptibility in 2d Schwinger QED model when the fermions are included intothe system is known exactly [67, 68]χQED =e24pi2∫d2x[δ2(x)− e22pi2K0(µ|x|)], (3.6)where µ2 = e2/pi is the mass of the single physical state in this model, andK0(µ|x|) is the modified Bessel function of order 0, which is the Green’s func-tion of this massive particle. The expression for χ for pure photodynamicsis given by (3.6) with coupling e = 0 in the brackets 4 which corresponds tothe de-coupling from matter field ψ, i.e.χE&M =e24pi2∫d2x[δ2(x)]=e24pi2. (3.7)The crucial observation here is as follows: any physical state contributes toχ with negative signχdispersive ∼ limk→0∑n〈0| e2piE|n〉〈n| e2piE|0〉−k2 −m2n< 0, (3.8)in accordance with the general formula (3.2) in four dimensions discussedpreviously. In particular, the term proportional −K0(µ|x|) with negativesign in equation (3.6) results from the only physical field of mass µ. How-ever, there is also a contact term∫d2x[δ2(x)]in (3.6) and (3.7) whichcontributes to the topological susceptibility χ with the opposite sign, andwhich can not be identified according to (3.8) with any contribution fromany physical asymptotic state. In the two-dimensional theory without afermion (photodynamics), there are no asymptotic states since there are nopossible polarisation states, and so it is clear that the contact term (3.7) isnot related to any physical propagating degree of freedom. Likewise, witha fermion included, there is one physical degree of freedom, yet we see alsothe additional “contact” contribution in (3.6).4the factor e24pi2in front of (3.6) does not vanish in this limit as it is due to our definition(3.4) rather than result of dynamics183.1. The Contact Term and Degeneracy of Topological SectorsThis term has fundamentally different, non-dispersive nature. In fact itis ultimately related to different topological sectors of the theory and thedegeneracy of the ground state as we shortly review below. Without thiscontribution it would be impossible to satisfy the Ward Identity becausethe physical propagating degrees of freedom can only contribute with sign(−) to the correlation function as (3.8) suggests, while the Ward Identityrequires χQED(m = 0) = 0 in the chiral limit m = 0. One can explicitlycheck that Ward Identity is indeed automatically satisfied only as a result ofexact cancellation between conventional dispersive term with sign (−) andnon-dispersive term (3.7) with sign (+),χQED =e24pi2∫d2x[δ2(x)− e22pi2K0(µ|x|)]=e24pi2[1− e2pi1µ2]=e24pi2[1− 1] = 0. (3.9)Therefore, contact term actually plays a crucial role in maintaining theconsistency of the theory, because the Ward Identity can not be satisfiedwithout it. While the exact formula (3.6) is known, it does not hint atthe kind of physics responsible for the contact term with the “wrong sign”,mainly what sort of field configurations should saturate the contact term.Below, we provide some insights on this matter.3.1.3 The Contact Term from Summation Over TopologicalSectorsThe goal here is to demonstrate that the contact term in the exact formulae(3.6) and (3.7) is a result of the summation over different topological ksectors in the 2d pure U(1) gauge theory. The relevant “instanton-like”configurations are defined on a two dimensional Euclidean torus with totalarea V as follows [67, 68],A(k)µ = −pikeVµνxν , eE(k) =2pikV, (3.10)such that the action of this classical configuration is12∫d2xE2 =2pi2k2e2V. (3.11)This configuration corresponds to the topological charge k as defined by(3.5). The next step is to compute the topological susceptibility for the193.1. The Contact Term and Degeneracy of Topological Sectorstheory defined by the following partition functionZ =∑k∈Z∫DAe− 12∫d2xE2 . (3.12)All integrals in this partition function are Gaussian and can be easily eval-uated. The result is determined essentially by the classical configurations(3.10) and (3.11) since real propagating degrees of freedom are not presentin the system of pure U(1) gauge field theory in two dimensions. We areinterested in computing χ defined by equation (3.4). In the path integralapproach it can be represented as follows,χE&M =e24pi2Z∑k∈Z∫DA∫d2xE(x)E(0)e−12∫d2xE2 . (3.13)This Gaussian integral can be easily evaluated and the result is as follows[100, 101],χE&M =e24pi2· V ·∑k∈Z4pi2k2e2V 2exp(−2pi2k2e2V)∑k∈Zexp(−2pi2k2e2V). (3.14)In the large volume limit V →∞ one can evaluate the sums entering (3.14)by replacing∑k∈Z →∫dk, and the leading term in equation (3.14) takesthe formχE&M =e24pi2· V · 4pi2e2V 2· e2V4pi2=e24pi2. (3.15)A few comments are in order. First, the obtained expression for thetopological susceptibility (3.15) is finite in the limit V →∞, coincides withthe contact term from exact computations (3.6), (3.7) performed for the2d Schwinger model, and has the “wrong” sign in comparison with anyphysical contributions (3.8). Second, the topological sectors with very largek ∼√e2V saturate the series (3.14). As we can see from the computationspresented above, the final result (3.15) is sensitive to the boundaries, in-frared regularisation, and many other aspects which are normally ignoredwhen a theory from the very beginning is formulated in infinite space withconventional assumption about trivial behaviour at infinity. Lastly, the con-tribution (3.15) does not vanish in a trivial model with no propagatingdegrees of freedom present in the system. This term is entirely determinedby the behaviour at the boundary, which is conveniently represented by203.1. The Contact Term and Degeneracy of Topological Sectorsthe classical topological configurations (3.10) describing different topologi-cal sectors (3.5), and accounts for the degeneracy of the ground state. In thisway, large distance physics enter despite the lack of physical long distancedegrees of freedom. Furthermore, we know that this term must be presentin the theory when the dynamical quarks are introduced into the system.Indeed, it plays a crucial role in this case as it saturates the Ward Identityas (3.9) shows.We conclude this section by noting that the contact term in the frame-work of [48] can be computed in terms of the Kogut-Susskind ghost byreplacing the standard path integral procedure of summation over differenttopological sectors above as follows. The topological density q = e2piE in2d QED is given by e2piE = (e2pi )√pie(2φˆ−2φ1)where φˆ is the physicalmassive field of the model and φ1 is the ghost [48]. The relevant correlationfunction in coordinate space which enters the expression for the topologicalsusceptibility (3.4) can be explicitly computed using the ghost as followsχQED(x) ≡〈Te2piE(x),e2piE(0)〉=( e2pi)2 pie2∫d2p(2pi)2p4e−ipx[− 1p2 + µ2+1p2]=( e2pi)2 [δ2(x)− e22pi2K0(µ|x|)](3.16)where we used the known expressions for the Green’s functions. The ob-tained expression precisely reproduces the exact result (3.6) as claimed. Inthe limit e → 0 when the fermion matter field decouples from gauge de-grees of freedom we reproduce the contact term (3.7), (3.15) which waspreviously derived as a result of summation over different topological sec-tors of the theory. The non-dispersive contribution manifests itself in thisdescription in terms of the unphysical ghost scalar field which provides therequired “wrong” sign for the contact term. These two different descriptionsare analogous to the same two computations in four dimensions mentionedin Section 3.1.1, with and without the Veneziano ghost, and again we em-phasise the equivalence of the two. In the picture wherein the contact termis saturated by a ghost field we see again how the contact term is not relatedto physical propagating degrees of freedom.213.2. Topological Susceptibility in the Deformed QCD3.2 Topological Susceptibility in the DeformedQCDNext we consider the topological susceptibility in the deformed theory dis-cussed in the previous chapter in both the monopole and sine-Gordon for-malisms. We define the topological density q(x) and topological charge QbyQ ≡ 116pi2∫R3×S1d4x tr[FµνF˜µν]= L∫R3d3x [q(x)] (3.17)and as in (3.1) the topological susceptibility χ is given byχ = L limk→0∫d3x eik·x 〈q(x)q(0)〉 . (3.18)First, in next subsection we compute the topological susceptibility directly,using the monopole gas representation. As the next step, we reproduce ourresults using sine Gordon representation of the theory. Finally, we computethe topological susceptibility with a single massless quark introduced intothe system. Essentially, the goal here is to discuss the same physics relatedto the nondispersive contact term and topological sectors in the deformedmodel for QCD, in close analogy to our discussions in 2d QED in Section3.1.3.2.1 Topological Susceptibility in the Monopole PictureIn order to compute the functional form of the topological susceptibility inthe monopole theory we consider the topological density,q(x) =116pi2tr[FµνF˜µν]=−18pi2ijk4N∑a=1F(a)jk F(a)i4=g4pi2N∑a=1〈A(a)4〉 [∇ ·B(a)(x)],(3.19)where the U(1)N magnetic field, Bi = ijk4Fjk/2g is given byB(a)(x) =2pigαan(a)∑k=1x− r(a)k4pi∣∣∣x− r(a)k ∣∣∣3 −n(a)∑k=1x− r(a)k4pi∣∣∣x− r(a)k ∣∣∣3 , (3.20)223.2. Topological Susceptibility in the Deformed QCDand〈A(a)4〉is just the expectation value of the diagonal gauge fields in thecompact direction, 〈A(a)4〉=2piNLµa. (3.21)The above µa are the fundamental weights for the SU(N) algebra and aredefined byµa · αb ≡ 12δabα2b = δab . (3.22)Inserting the magnetic field for the monopole ensemble into the topologicaldensity expression (3.19) and applying Gauss’s theorem to the result, wearrive atq(x) =N∑a=11LNn(a)∑k=1δ(r(a)k − x)−n¯(a)∑l=1δ(r¯(a)l − x)=1LNN∑a=1M(a)∑k=1Q(a)k δ(r(a)k − x),(3.23)which obviously gives the proper topological charge for a single monopoleor antimonopole, Q = ±1/N . The topological density operator q(x) hasdimension four as it should.The expectation value 〈q(x)q(0)〉 is the topological density operator(3.23) evaluated at each point inserted in the partition function (2.20),〈qq〉 = 1Z∫ N∏a=1dµ(a) [q(x)q(0)] e−Sint=1ZN2L2∫ N∏a=1dµ(a)N∑a,b=1M(a)∑k=1M(b)∑l=1[Q(a)k Q(b)l δ(r(a)k − x)δ(r(b)l )]e−Sint=1ZN2L2∫dµ∑m∑n[QmQn δ(rm − x)δ(rn)] e−Sint=1ZN2L2∫dµ{δ(x)∑mδ(rm)e−Sint +∑m∑n6=m[QmQn δ(rm − x)δ(rn)] e−Sint=ζNL2{δ(x)−O(ζ)} , (3.24)233.2. Topological Susceptibility in the Deformed QCDwhere we have condensed the indices to just m and n which run over eachmonopole in the ensemble. In the above expression, the double sum of deltafunctions gives a set of terms in which each pair of monopoles in the ensembleare moved to the points x and 0 and computes the partition function giventhat arrangement. The monopole gas experiences Debye screening so thatthe field due to any static charge falls off exponentially with characteristiclength m−1σ . The number density N of monopoles is given by the monopolefugacity, ∼ ζ, so that the average number of monopoles in a “Debye volume”is given byN ≡ m−3σ ζ =( g2pi)3 1√L3ζ 1. (3.25)The last inequality holds since the monopole fugacity is exponentially sup-pressed, ζ ∼ e−1/g2 , and in fact we can view (3.25) as a constraint onthe validity of our approximation. The statement here is that inserting orremoving a particular monopole will not drastically affect the monopole en-semble as a result of condition (3.25), so that we can compute expectationsof operators in the original ensemble without considering the back-reactionon the ensemble itself. Then, because removing any given monopole doesnot significantly change the ensemble, we can treat the delta functions in thethird line of (3.24) as simply creation operators. The second term in (3.24),which is a dispersive term, reduces to the form 〈M †M〉 since it is only non-zero for monopole-antimonopole pairs of the same type. The factor overallfactor of ζ and additional factor in O(ζ) in formula (3.24) appears becauseeach monopole we remove from the ensemble leaves a factor of ζ/N (a) in themonopole measure, and there are N (a) such terms for each type of monopoleso that we are left with just a factor of ζ.The computed non-dispersive contribution (3.24) to the topological sus-ceptibility in the deformed gauge model has exactly the same structure weobserved in two dimensional QED discussed in Section 3.1. In particular, itis expressed in terms of a δ(x) function, and it has the “wrong sign” sim-ilar to (3.7). Furthermore, this contribution is not related to any physicalpropagating degrees of freedom, but rather, it is determined by degener-ate topological sectors of the theory. The corresponding “degeneracy” isformulated in terms of monopoles which essentially describe the tunnellingtransitions between those “degenerate” sectors, see Section 3.2.3 for morecomments on the physical meaning of the formula. If we neglect a smallterm O(ζ) in formula (3.24) we arrive to the following final expression for243.2. Topological Susceptibility in the Deformed QCDthe topological susceptibility in deformed gauge theory without quarksχYM =ζNL2∫d4x [δ(x)] =ζNL. (3.26)It has dimension four as it should. This expression is a direct analog ofequation (3.7) derived for two dimensional QED. The same formula (3.26)can be computed also in the dual sine-Gordon theory by differentiation ofthe ground state energy density (2.37) with respect to the θ as generalexpression (3.1) statesχYM (θ = 0) =∂2Emin(θ)∂θ2∣∣∣∣θ=0=ζNL, Emin(θ = 0) = −NζL. (3.27)Agreement between the two computations can be considered as a consistencycheck of our approach in the weakly coupled regime. One can explicitly seethat the general relation χYM ∼ Emin(θ = 0)/N2 holds for the deformedmodel as a result of θ/N dependence in equation (2.36). Real stronglycoupled QCD is that the vacuum energy scales as N2 in QCD rather than,apparently, ∼ N in (3.27). To rectify these two, note that the domain ofvalidity for this model is defined by LNΛ 1, as discussed in [85], so thatL ∼ 1/N .3.2.2 Topological Susceptibility in the Presence of theLight QuarksOur goal here is to introduce a single massless quark ψ into the system tosee how the topological susceptibility changes in this case. We anticipatethat the emerging structure should be very similar to (3.6) as the topo-logical susceptibility must vanish in the presence of massless quark in thesystem: χQCD(mq = 0) = 0 as the direct consequence of the Ward Identitiesdiscussed in Section 3.1.The low energy description of the system in confined phase with a singlequark is accomplished by introducing the η′ meson. As usual, the η′ would beconventional massless Goldstone boson if the chiral anomaly is ignored. Inthe dual sine-Gordon theory the η′ field appears exclusively in combinationwith the θ parameter as θ → θ − η′. As it is well known, this is the directresult of the transformation properties of the path integral measure under253.2. Topological Susceptibility in the Deformed QCDthe chiral transformations ψ → exp(iγ5 η′2 )ψ. Therefore we have,Z =∫ N∏a=0DσaDη′ exp{−Sσ − Sη′ − Sint} (3.28)Sσ =∫R3d3x · 12L( g2pi)2(∇σ)2Sη′ =∫R3d3x · c2(∇η′)2Sint = −∫R3d3x · ζN∑a=1cos(αa · σ + θ − η′N),where coefficient c determines the normalisation of the η′ field and has di-mension one. This coefficient, in principle, can be computed in this model,but such a computation is beyond the scope of the present work and notparticularly illuminating. In four dimensional QCD the coefficient c is ex-pressed in terms of standard notations as (c/L) → f2η′ . In terms of theseparameters the η′ mass is given bym2η′ =ζcN. (3.29)Since η′ shows up in the Yang-Mills Lagrangian as η′FF˜ , we can com-pute our requisite expectation, 〈FF˜ , F F˜ 〉, by functional differentiation withrespect to η′,〈q(x)q(y)〉 = 1Zi δδη′(x)i δδη′(y)Z∣∣∣∣θ=0. (3.30)Thus we have,〈q(x)q(y)〉 = 1Zδδη′(y)∫DσDη′[−ζNL2∫R3d3r1 δ(r1 − x)N∑a=1η′(r1)N]e−S=1Z∫DσDη′[ζNL2∫d3r1 δ(r1 − x)δ(r1 − y)]e−S− 1Z∫DσDη′[ζ2NL2∫d3r1∫d3r2 δ(r1 − x)δ(r2 − y)×N∑a,b=1η′(r1)Nη′(r2)N e−S=ζNL2[δ(x− y)− ζN〈η′(x)η′(y)〉]. (3.31)263.2. Topological Susceptibility in the Deformed QCDThe first term in (3.31) is precisely non-dispersive contact term with the“wrong sign” that we computed previously in pure gauge theory using twodifferent methods, see (3.26) and (3.27). The second term represents theconventional dispersive contribution of the physical η′ state 5. One cancompute it by redefining η′ → η′/√c field to bring its kinetic term Sη′ tothe canonical form. In the lowest order approximation it is reduced to theconventional Green’s function of the free massive η′ scalar field with massdetermined by (3.29), such thatχQCD =∫d4x〈q(x)q(y)〉 = ζNL∫d3x[δ(x)−m2η′e−mη′r4pir]= 0, (3.32)where we represented the canonical η′ propagator in terms of its free Green’sfunction in three dimensions.The structure of this equation follows precisely the same pattern we ob-served in analysis of two dimensional QED, see (3.6). Indeed, it contains thenon-dispersive term due to the degeneracy of the topological sectors of thetheory. This contact term (which is not related to any physical propagatingdegrees of freedom) has been computed using monopoles describing the tran-sitions between these topological sectors (3.26). The second term emergesas a result of insertion of the massless quark into the system. It entersχQCD precisely in such a way that the Ward Identity χQCD(mq = 0) = 0 isautomatically satisfied as a result of cancellation between the two terms inclose analogy with the two dimensional case (3.9). We should also mentionthat very similar structure emerges in real strongly coupled QCD in theframework wherein the contact term is saturated by the Veneziano ghost.This structure has been confirmed by QCD lattice studies, see [102, 104] forsome details and references to original lattice results.5This additional interactions due to the η′ exchange may in fact be used as a probeto study the relevant topological charges present in the system. It was precisely the ideabehind the proposal, see relatively recent papers [56, 99] and earlier references therein,that θ/N behaviour unambiguously implies that the relevant vacuum fluctuations musthave fractional topological charges 1/N . In the present weakly coupled regime these ideashave a precise realisation as the basic vacuum fluctuations are indeed the fractionallycharged monopoles (3.23). The results of [56, 99] are in fact much more generic as theyare not based on a weakness of the interaction or semiclassical expansion, but rather, ongeneric features of the η′ system which are unambiguously fixed by the Ward Identity.In the approach advocated in [56, 99] one can not study the dynamics of fractionallycharged constituents in contrast with the present paper where the dynamics is completelyfixed and governed by (3.28). However, the fact that the constituents carry fractionaltopological charge 1/N can be recovered in the approach [56, 99] because the color-singletη′ field enters the effective Lagrangian in a unique way and serves as a perfect probe ofthe relevant topological charges of the constituents in the system.273.2. Topological Susceptibility in the Deformed QCD3.2.3 InterpretationThe results derived in previous sections were formulated in Euclidean spaceusing conventional Euclidean path integral approach. Our goal here is togive a physical interpretation of these results in physical terms formulatedin Minkowski space time. First of all, the δ(x) function which appear in theexpression for topological susceptibility (3.26) should, in fact, be understoodas total divergence,χ ∼∫δ(x) d3x =∫d3x ∂µ(xµ4pix3)=∮S2dΣµ(xµ4pix3). (3.33)Indeed, the starting point to derive χ was the topological density operator(3.23) which is expressed in terms of δ(x − xi) functions, but in fact rep-resents the topologically nontrivial boundary conditions determined by thebehaviour at a distant surface S2 as (3.33) states. The representation (3.33)explicitly shows that we are not dealing with ultraviolet (UV) propertiesof the problem wherein our approximation breaks down. Our treatment ofthe problem is perfectly justified as δ(x − xi) functions actually representthe far infrared (IR) part of physics rather than UV physics. This explainswhy our description is valid for x L in spite of presence of the apparentlyUV singular elements such as the δ(x − xi) functions which appear in theequations in sections 3.2.1 and 3.2.2.Our next comment is about the interpretation of the classical monopolegas from chapter 2. The monopoles in our framework are not real particles,they are pseudo-particles which live in Euclidean space and describe thephysical tunnelling processes between different winding states |n〉 and |n+1〉. The grand canonical partition function written in terms of the classicalCoulomb gas (2.34) is simply a convenient way to describe this physics oftunnelling. In particular, the monopole fugacity ζ together with factor L−1should be understood as number of tunnelling events per unit time per unitvolume (N of tunnelling eventsV L)=NζL, (3.34)where extra factor N in (3.34) accounts for N different types of monopolespresent in the system. The expression (3.34) is precisely the contact term,up to factor 1/N2 computed in (3.26). It is not a coincidence that numberof tunnelling events per unit time per unit volume precisely concurs withthe absolute value of the energy density of the system (3.27), since the en-ergy density (3.27) in our model is saturated by the topological fluctuationswhich are not related to any physical propagating degrees of freedom. We283.3. Commentsemphasise that while this energy density is not related to any fluctuationsof real physical particles, this energy, nevertheless, is still real physical ob-servable parameter, though it can not be defined in terms of conventionalDayson T-product. Instead, it is defined in terms of the Wick’s T-product,see Appendix of [101] on a number of subtleties with definition of the energy.Finally, the characteristic Debye screening length which appears in theCoulomb gas representation in chapter 2rD ≡ m−1σ =g2pi√Lζ L (3.35)should be interpreted as a typical distance in physical 3d space in which thetunnelling event is felt by other fields present in the system. The tunnellinginterpretation also explains the “wrong” sign in residues of the correlationfunction (3.26) as we describe the tunnelling in terms of the Euclideanobjects interpolating between physically equivalent topological sectors |n〉rather than the tunnelling of conventional physical degree of freedom be-tween distinct vacuum states in condensed matter physics.3.3 CommentsThe main results of this chapter can be formulated as follows. We studied anumber of different ingredients related to θ dependence, the non-dispersivecontribution in topological susceptibility with the “wrong sign”, topologicalsectors in gauge theories, and related subjects using a simple “deformedQCD”. This model is a weakly coupled gauge theory, which however has allthe relevant essential elements allowing us to study difficult and nontrivialquestions which are known to be present in real strongly coupled QCD.Essentially we tested the ideas related to the U(1)A problem formulatedlong ago in [42, 54, 64, 90, 91, 93] in a theoretically controllable mannerusing the deformed gauge theory as a toy model. One can explicitly seehow all the crucial elements work. Here we compute the contact term in thetopological susceptibility, (3.24) or (3.31), directly, reproducing conjecturedresults by Witten [93], who put in the contact term by hand, and Veneziano[90, 91], who put in some auxiliary ghost fields that saturate the contact termwhen integrated out. See [106] for more details about the use of auxiliaryfields in this context.As this model is a weakly coupled gauge theory, one can try to formu-late (and answer) many other questions which are normally the preroga-tive of numerical Monte Carlo simulations. One such question is the study293.3. Commentsof scaling properties of the contact term. We can address what happensto the contact term when the Minkowski space-time R3,1 gets slightly de-formed. For example, what happens when infinite Minkowski space-timeR3,1 is replaced by a large, but finite size torus? Or, what happens whenthe Minkowski space-time R3,1 is replaced by FRW metric characterised bythe dimensional parameter R ∼ H−1 describing the size of horizon (H beingthe Hubble constant)? A naive expectation based on common sense suggeststhat any physical observable in QCD must not be sensitive to very large dis-tances ∼ exp(−ΛQCDR) as QCD has a mass gap ∼ ΛQCD. Such a naiveexpectation seems to formally follows from the dispersion relations similarto (3.2), which dictate that a sensitivity to very large distances must be ex-ponentially suppressed with a mass gap present in the system, and there arenot any physical massless states in the spectrum. However, as we discussed,along with conventional dispersive contribution (3.2) in the system, there isalso the non-dispersive contribution (3.3) which emerges as a result of topo-logically nontrivial sectors in four dimensional QCD. This contact term maylead to a power-like corrections R−1 +O(R−2) rather than exponential-like∼ exp(−ΛQCDR) because the dispersion relations do not dictate the scalingproperties of this term. In fact, this term in the deformed model in infiniteMinkowski space has been explicitly computed in this chapter and it is givenby (3.26), (3.32). As our model is a weakly coupled gauge theory, one can,in principle, compute the correction to the formulae (3.26), (3.32) due to thefinite size of the system [79]. In other words, one can then try to computethe corrections to the monopole fugacity ζ when the model is formulatedin a finite manifold determined by size R. This is the content of the nextchapter.30Chapter 4Casimir Scaling and DarkEnergyThis chapter reproduces the work presented in [79]. We show that thedeformed model exhibits a power-like Casimir scaling with the size of themanifold it is placed on, rather than the exponential scaling one might expectfor a gapped theory.4.1 MotivationThe main motivation for the study presented in this chapter is a suggestionon the dynamical Dark Energy (DE) model which is entirely rooted in thestrongly coupled QCD, without any new fields and/or coupling constants[88, 89, 100, 101]. The key element of the proposal [88, 89, 100, 101] is basedon paradigm that the relevant energy which enters the Einstein equations isin fact the difference ∆E ≡ E−EMink between the energies of a system in anon-trivial background and Minkowski space-time geometry, similar to thewell known Casimir effect when the observed energy is a difference betweenthe energy computed for a system with conducting boundaries (positionedat finite distance L) and infinite Minkowski space6. This paradigm is basedon the conjecture that gravity, as described by the Einstein equations, is alow-energy effective interaction which, as such, should not be sensitive tothe microscopic degrees of freedom in the system but to some effective scale.Thus, the energy density that enters the semiclassical Einstein equationsshould not be the “bare” energy as computed in QFT, and indeed we knowit cannot be, but rather a “renormalised” energy density. We propose therenormalisation scheme given above which sets the vacuum energy to zero in6Here and in what follows we use term “Casimir effect” to emphasise the power likesensitivity to large distances irrespective of their nature. A crucial distinct feature whichcharacterises the system we are interested in is the presence of dimensional parameterL ∼ H−1 (where H is a Hubble constant) in the system which discriminates it frominfinitely large and flat Minkowski space-time.314.1. MotivationMinkowski space wherein the Einstein equations are automatically satisfiedas the Ricci tensor identically vanishes.The above prescription is in fact the standard subtraction procedurethat is normally used for the description of horizon thermodynamics [6, 35]as well as in a course of computations of different Green’s function in acurved background by subtracting infinities originating in the flat space[8]. In the present context such a definition ∆E ≡ (E − EMink) for thevacuum energy was first advocated in 1967 by Zeldovich [97] who arguedthat ρvac ∼ Gm6p with mp being the proton’s mass. Subsequently, such adefinition for the relevant effective energy ∆E ≡ (E − EMink) which entersthe Einstein equations has been advocated from different perspectives in anumber of papers, see, for example, the relatively recent works [45, 46, 50,52, 59, 69, 78, 98] and references therein.We study the scaling behavior of ∆E when the background deviatesslightly from Minkowski space. The difference ∆E must obviously vanishwhen any deviations (parametrised by Hubble constant or inverse size of thevisible universe, H ∼ L−1) go to zero as this corresponds to the transition toinfinite flat Minkowski space. A naive expectation based on common sensesuggests that ∆E ∼ exp(−ΛQCD/H) ∼ exp(−1041) as QCD has a massgap ∼ ΛQCD ∼ 100 MeV , and therefore, ∆E must not be very sensitive tosize of our universe L ∼ H−1. Such a naive expectation formally followsfrom the dispersion relations which dictate that a sensitivity to very largedistances must be exponentially suppressed when the mass gap is present inthe system7.However, as emphasised in [100, 101] in strongly coupled gauge theoriesalong with conventional dispersive contribution there exists a non-dispersivecontribution, not related to any physical propagating degrees of freedom.This non-dispersive (contact) term generally emerges as a result of topolog-ically nontrivial sectors in four dimensional QCD. The variation of this con-tact term with variation of the background may lead to a power like scaling∆E ∼ H+O(H)2 rather than to an exponential like ∆E ∼ exp(−ΛQCD/H)since its contribution is not determined by some gapped dispersion relations.If true, the difference between two metrics (FLRW and Minkowski) wouldlead to an estimate∆E ∼ Λ3QCDL∼ (10−3eV )4, 1/L ∼ H ∼ 10−33eV (4.1)7The Casimir effect due to the massless E&M field obvious shows such power depen-dence ∆E = − pi2720L4 . Similar computations for a massive scalar particle with mass mleads to an exponentially suppressed result ∆E ∼ exp(−mL) as expected, see e.g.[57].324.1. Motivationwhich is amazingly close to the observed dark energy value today. It isinteresting to note that expression (4.1) reduces to Zeldovich’s formulaρvac ∼ Gm6p if one replaces ΛQCD → mp and H → GΛ3QCD. The laststep follows from the solution of the Friedman equationH2 =8piG3(ρDE + ρM ) , ρDE ∼ HΛ3QCD (4.2)when the dark energy component dominates the matter component, ρDE ρM . In this case the evolution of the universe approaches a de-Sitter statewith constant expansion rate H ∼ GΛ3QCD as follows from (4.2).A comprehensive phenomenological analysis of this model has been re-cently performed in [12], see also [65, 71] where comparisons with currentobservational data including SnIa, BAO, CMB, BBN have been presented.The conclusion was that this model is consistent with all presently availabledata. The main goal here is not comparison of this model with observations;we refer the reader to [12] on this matter. Rather, the purpose of this chap-ter is to attempt to get some deep theoretical insights behind the Casimirtype behaviour (4.1) in a gapped theory such as QCD.Another motivation to study the Casimir like behaviour in QCD is aproposal [102, 104] that the P odd correlations observed at RHIC and LHCis in fact another manifestation of long range order advocated in this work.Furthermore, an apparently universal thermal spectrum observed in all highenergy collisions when the statistical thermalisation could never be reachedin the systems, might be also related to the same contact term, not relatedto any physical propagating degrees of freedom, see [102, 104] and referencestherein for the details.There are a number of arguments supporting the power like behaviour∆E ∼ H +O(H)2 in gauge theories, see Section 4.3 where we present somegeneral arguments suggesting the Casimir like corrections in gauge theorieswith nontrivial topological structure. However, it is always desirable andvery instructive to see how the general arguments work in some simplifiedsettings.First, one can examine the exactly solvable two dimensional QED. De-spite this model containing only a single physical massive field, still one canexplicitly compute ∆E ∼ L−1 which is in drastic contrast with the naivelyexpected exponential suppression, ∆E ∼ e−L [86].Another piece of support for this power-like behaviour is an explicitcomputation in a simple case of a Rindler space-time in four dimensionalQCD [55, 100, 102]. These computations explicitly show that the powerlike behaviour emerges in four dimensional gauge systems in spite of the334.2. Casimir-Type Behaviour in Deformed QCDfact that the physical spectrum is gapped. Thus, a power-like behaviouris not a specific feature of two dimensional physics. Accounting for thenon-trivial topological sectors in QCD in Rindler space was accomplishedin [55, 100, 102] using unphysical auxiliary field, the so-called Venezianoghost, which encodes the same “contact term” described in the previouschapter. As discussed, the inclusion of different topological sectors wasinstead introduced by an unphysical ghost field which saturates the “contactterm”.Finally, power like behaviour ∆E ∼ L−1 is also supported by recentlattice results [36]. The approach advocated in [36] is based on the physicalCoulomb gauge wherein nontrivial topological structure of the gauge fieldsis represented by the so-called Gribov copies. The power like correction∼ L−1 had been also noticed, though in quite different context, in [74]where numerical computations were performed using the so-called instantonliquid model.While a number of supporting arguments presented above suggest theCasimir-type power law scaling ∆E ∼ H+O(H)2 in strongly coupled QCD,a simple explanation for this behaviour is still lacking. Indeed, skepticswould argue that two dimensional example [86] is a special case, while infour dimensions everything could be very different. A similar skepticism canalso be expressed with the ghost based computations [55, 100, 102] as theentire treatment of the problem is based on an auxiliary ghost field whichdoes not belong to the physical Hilbert space and has been inserted byhand. Finally, the numerical computations [36, 74] can not provide a simplephysical picture explaining the nature of the phenomenon as the entire effectis hidden in numerics.This is precisely the goal for the present study: to consider the energydependence on the boundary conditions this simplified (“deformed”) ver-sion of QCD which, on one hand, is a weakly coupled gauge theory whereincomputations can be performed in theoretically controllable manner. Onother hand, this deformation preserves all the relevant elements of stronglycoupled QCD such as confinement, degeneracy of topological sectors, non-trivial θ dependence, presence of non-dispersive contribution to topologicalsusceptibility, and other crucial aspects, for this phenomenon to emerge.4.2 Casimir-Type Behaviour in Deformed QCDUp to this point the theory was formulated on R3×S1 with small compact-ification size L for compact time coordinate S1 and infinitely large space344.2. Casimir-Type Behaviour in Deformed QCDR3 describing three other dimensions. Here however, we are actually in-terested in behaviour of the system when a space with large dimensions R3receives some small modifications, for example the theory is defined in a ballR3 → B3 with L being a very large size of the compact dimension of thesphere S2 which is a boundary of the ball B3. Such a modification can bethought as a simplest way to model and test the sensitivity of our theory toarbitrary large distances such as size of our visible universe determined bythe Hubble constant H/ΛQCD ∼ 10−41. We want to know how the topolog-ical susceptibility of the system which describes the θ dependent portion ofthe vacuum energy Evac(θ = 0) changes with slight variation of that largesize of the system. We assume that L ∼ H−1 ∼ 10 Gpc is much larger thanany other scales of the problem. Essentially we want to see whether ourdeformed model with a mass gap mσ predicts an exponential scaling typicalfor a free massive particle∆E(L) ≡ [E(B3)− E(R3)] ∼ exp(−mσL) (4.3)or demonstrates a Casimir type behaviour∆E(L) ≡ [E(B3)− E(R3)] ∼ 1L+O(1L)2. (4.4)If we did not have a non-dispersive contribution in our system, we wouldimmediately predict the behaviour (4.3) as the only available option for agapped theory in close analogy with conventional Casimir computations fora massive particle ∆E(L) ∼ exp(−mL), see for example the review paper[57]. However, our system is more interesting as it exhibits a non-dispersiveterm resulting from degeneracy of topological sectors in gauge theory asdiscussed in the previous chapter. This contact term, being unrelated to anyphysical degrees of freedom, may provide different scaling properties sinceconventional dispersion relations do not dictate its behaviour at very largedistances. As we shall argue, the deformed gauge model indeed exhibits theCasimir type behaviour (4.4) in a drastic departure from the conventionalviewpoint represented by equation (4.3). As we reviewed in the previoussection we interpret a tiny deviation of the θ-dependent vacuum energyEvac in expanding universe (in comparison with Minkowski space-time) as amain source of the observed dark energy. The Casimir type behaviour (4.4)plays a key role in possibility of such an identification.We start our discussion in Section 4.2.1 with conventional 4d instantoncomputations [77] in which infrared regularisation for some gauge modes isrequired and achieved by putting the system into a sphere with finite radius354.2. Casimir-Type Behaviour in Deformed QCDL. It allows us to compute power like corrections to the standard instan-ton density [77]. However, the corresponding corrections being computedfor a fixed instanton size ρ can not be interpreted as a physically observ-able quantity because the integral (∫dρ) over large size instantons divergesfor this system when semiclassical approximation for large ρ breaks down.Nevertheless, this example explicitly shows when and why a Casimir typecorrection (to conventional formula computed in infinite R4 space) emerges.Next, we compute a similar correction for the deformed model in Section4.2.2 wherein a Casimir type correction also appears, resulting from the samephysics related to topological sectors of the theory. In contrast with the pre-vious case, the correction computed in this system is physically “observable”quantity as it represents the vacuum energy of the system. Indeed, the tun-nelling transitions in this case are described by weakly coupled monopoles,such that semiclassical computations of the vacuum energy (3.34),(6.1) ex-pressed in terms of the density ζ of pseudo-particles are fully justified. Thesize of pseudo-particles (fractionally charged monopoles) which describe thetunnelling events in this model is fixed by construction [80, 85] so there isno divergence as seen in the instanton case.We conclude in Section 4.4 with a few final comments.4.2.1 Casimir-Type Corrections for 4D InstantonsOur goal here is to study a power like correction to the instanton densitydescribed in the classic paper [77]. As such, we adopt ’t Hooft’s notation,and in particular, use the same background-dependent gauge C4 = DµAa quµ ,which drastically simplifies all computations. Essentially, the problem is re-duced to analysis of the normalisation factors for finite number of zero modes(8 for SU(2) gauge group) in this gauge wherein the system is defined in asphere with large but finite radius radius L. Essentially we follow the con-struction described in section XI of [77]. The corresponding normalisationfactor explicitly enters the expression for the instanton density as it ac-companies the integration over collective variables. The contribution fromnon-zero modes does not exhibit such corrections as we argue in Section4.2.3. We now concentrate on the zero modes and power like correctionswhich accompany the normalisation factors if the system is defined on alarge but finite space B4L (four dimensional interior of a ball of radius L)rather than an infinite space R4.We start with four translational zero modes which have the formAa quµ (ν) ∼ ηaµν(1 + r2)−2, ν = 1, ..., 4 (4.5)364.2. Casimir-Type Behaviour in Deformed QCDwhere we use ’t Hooft’s notations for ηaµν symbols and dimensionless coor-dinate r2 = x2µ measured in units of ρ = 1. Computing the correspondingcorrection factor due to the translation zero modes κtr., we haveκtr. ≡∫ L0 d4x[Aa quµ (ν)]2∫∞0 d4x[Aa quµ (ν)]2'[1− 3L4+O( 1L6)]. (4.6)The corresponding correction factor to the instanton density has power likecorrection as anticipated. As a result of additional rotational symmetry oneshould expect, in general, L−2 corrections, while translation zero modes leadto a much smaller correction ∼ L−4 as equation (4.6) shows. As such, itwill be neglected in what follows. Dilation and global gauge rotations leadto ∼ L−2 as we discuss below.For the dilation zero modeAa quµ ∼ ηaµνxν(1 + r2)−2 (4.7)a similar formula readsκdil. ≡∫ L0 d4x[Aa quµ (ν)]2∫∞0 d4x[Aa quµ (ν)]2'[1− 3L2+O( 1L4)], (4.8)such that the correction to the instanton density is proportional to√κdil. '(1− 32L2 ).Computing the corresponding contribution due to three zero modes re-lated to global gauge rotations requires much more refined analysis as ex-plained in [77]. This is due to the specific features of the background de-pendent gauge C4 = DµAa quµ when the corresponding three modes are puregauge artifact. As shown in [77] the corresponding contribution is finite,but very sensitive to the infrared regularisation determined by the size Rof large sphere. The corresponding contribution to the instanton density is∼ (λ4V )3/2 where V is the four volume, while λ4 ∼ V −1 is defined as followsλ4 =∫V d4x[ψaµ(b)]2∫V d4x[ψa(b)]2, b = 1, 2, 3, (4.9)ψa(b) = ηaµν η¯bµλxνxλ(1 + x2),ψaµ(b) = Dµψa(b) = ηaλµη¯bλνxν(1 + x2)2.The corresponding power like corrections can be computed in a similar man-ner to the other zero modes, except that we must retain the regularisation374.2. Casimir-Type Behaviour in Deformed QCDsince the denominator above diverges as ∼ V . So we have the two correctionfactorsκnum. ≡∫ L0 d4x[ψaµ(b)]2∫∞0 d4x[ψaµ(b)]2'[1− 3L2+O( 1L4)],andκden. ≡ V (R)V (L)∫ L0 d4x[ψa(b)]2∫ R0 d4x[ψa(b)]2'[1− 4L2+O( 1L4)].The fraction, V (R)/V (L), is the correction to V in the instanton densityfactor, and is included here so that we can take the regularisation R →∞.The combined gauge rotation correction factor is thenκrot. ≡ κnum.κden.'[1 +1L2+O( 1L4)], (4.10)such that the correction to the instanton density is proportional to (κrot.)3/2 '(1 + 32L2 ). Accidentally, for SU(2) gauge group the leading L−2 correctionfrom the dilation (4.8) and global gauge rotations (4.10) exactly cancel eachother. This accidental cancellation does not hold for general SU(N) gaugegroup when power of κrot. enters the instanton density with a different power.We remark here that the technique used in [77] is essentially a varia-tional approach wherein the boundary conditions are implemented implicitlyrather than explicitly. It allows us to use all the zero modes (4.5),(4.7),(4.9)as well as standard classical instanton solution in the original form definedon R4 in which the conformal invariance is a symmetry of the system. So inthis approach, neither the instanton itself, nor its zero modes (4.5),(4.7),(4.9)are solutions of the equation of motions which vanish at the boundary. Thisapproach has been tested in many follow up papers, and we adopt it in thepresent work using the same technique in the next section. We also pointout that the conformal invariance is explicitly broken in the one instantonsector by the size of the instanton ρ, such that corrections take the form( ρ2L2 )n. It is restored by the integration∫dρ. However, in this paper we areinterested in by the computation in one instanton sector only when dimen-sional parameter ρ is explicitly present in the system, and is small and fixed,as it is in the deformed model discussed in chapters 2 and 3.The important message here is that such kind of power correction do ap-pear in general. The source of these corrections is a long range tail of zeromodes. We can not derive a definite conclusion from these computationsbecause the integral over large size instantons (∫dρ) diverges and the semi-classical approximation breaks down. However, the same problem studied384.2. Casimir-Type Behaviour in Deformed QCDin the deformed gauge theory model considered in Section 4.2.2 does notsuffer from such deficiencies as semiclassical computations are under com-plete theoretical control. Thus, a Casimir like correction to the monopolefugacity ζ in this model is explicitly translated to the correction to the vac-uum energy density and topological susceptibility (4.17), supporting (4.4)and in huge contrast with naive expectation (4.3). It is important to notethat the source of the corrections in the deformed model is the same as inthe undeformed QCD considered here, and that source is the long rangetails of the zero modes, which lead to large distance sensitivity. The onlydifference is that the role of the instanton size ρ in computations above inthe one instanton sector is played by the inverse monopole mass m−1W in thenext section. Because it is a true scale of the problem however, m−1W is notintegrated over as ρ is.4.2.2 Casimir-Type Corrections for 3D MonopolesWe now turn to the deformed gauge theory described in chapters 2 and3 wherein the low-energy behaviour is given by a U(1)N Coulomb gas ofmonopoles in Euclidean R3. Basically, we want to understand the depen-dence of the monopole fugacity, ζ, which comes out of the measure trans-formation to collective coordinates, on the size of the system, L. In thiscase, as in the previous section, we consider the interior of a sphere of largebut finite radius L. There are four zero-modes present in this system: threetranslations since the monopoles are in R3, no dilations since the monopolesize is fixed by the symmetry-breaking scale in this model mW , and onegauge rotation since the gauge group for a given monopole is U(1). As in[77], we work in a regular gauge to remain sensitive to the large distancephysics. The monopole solution in the “hedgehog” regular gauge is given byvaµ(x) = µνaxν|x|2[1− mW |x|sinh (mW |x|)],φa(x) =xa|x|2 [mW |x| coth (mW |x|)− 1] , (4.11)where we adapted notations from [19, 21] treating the monopole measure insupersymmetric Yang-Mills theory. In formula (4.11) vaµ denotes the threespacial gauge fields for the classical solution, and φa the gauge field in thecompact time direction (the “Higgs” field in this model) when all fields canbe combined in a single 4d field vm.We then want to compute the correction factors for the collective coordi-nate measure coming from these four zero modes when the system is defined394.2. Casimir-Type Behaviour in Deformed QCDin a large but finite sphere. We closely follow ’t Hooft’s treatment [77] pre-sented in the previous section 4.2.1. We start by considering the translationmodes defined by the spacial derivative of the classical monopole solution(4.11) with respect to the collective coordinate positionZam(ν) = −∂νvam(x− z) +Dmvaν = vamν (4.12)where the minus sign is because ∂/∂z = −∂/∂x since z only enters as x− z,and vamν is the field strength since the covariant derivative is Dm = ∂m −i [vm, ∗]. The second term on the right hand side of (4.12) is necessary to keepZam(ν) in the background gauge, see [19, 21] for more details.8 This leads usto the following expression for correction factor due to the translation zeromodesκtr. ≡∫ L0 d4x[Zam(ν)]2∫∞0 d4x[Zam(ν)]2'[1− 1mWL+O( 1L2)](4.13)Next we consider the gauge rotation zero-mode. As in the previoussection, the contribution to the collective coordinate measure, and so themonopole fugacity, is ∼ (λV ) 12 where V is the three-volume and λ is givenbyλ =∫V d3x[Baµ]2∫V d3x[φa]2Baµ =12µνρ∂νvaρ = Dµφa. (4.14)Again, the denominator diverges as ∼ V and we look at the two correctionfactorsκnum. ≡∫ L0 d3x[Baµ]2∫∞0 d3x[Baµ]2'[1− 1mWL+O( 1L2)],andκden. ≡ V (R)V (L)∫ L0 d3x[φa]2∫ R0 d3x[φa]2'[1− 3mWL+O( 1L2)].The total correction factor for the gauge rotation mode is thenκrot. ≡ κnum.κden.'[1 +2mWL+O( 1L2)], (4.15)8There is also a more extended (and careful) discussion of both the derivation of the“hedgehog” solution, (4.11), and this gauge transformation that must added to the simplederivatives with respect to the zero mode collective coordinates in order to satisfy thegauge condition, presented in Chapter 4 of [72].404.2. Casimir-Type Behaviour in Deformed QCDand therefore the total correction to the monopole fugacity from the (4.14)is√κrot. ' (1 + 1L). Assembling the total correction to the fugacity,κ3/2tr. κ1/2rot. '[1− 12mWL+O( 1L2)]. (4.16)Thus, the deformed gauge theory, when put on a manifold with a bound-ary, receives some corrections to the monopole fugacity compared to Minkowskispace that are power-like in the manifold size. The correction (4.16) to themonopole fugacity leads immediately to the same correction to the topolog-ical susceptibility and so the background energy density since, as we saw inthe previous chapter,EYM(θ) = −NcζLcos(θNc),χYM (θ = 0) =∂2EYM(θ)∂θ2∣∣∣∣θ=0=ζNcL. (4.17)Here we considered only the lowest branch from (2.37) at θ = 0 for simplicity.To be more precise,ζ(L) = ζ ·[1− 12mWL+O( 1L2)], (4.18)where ζ is the monopole fugacity which enters the relation (4.17) computedin infinite Minkowski space. We emphasise that the energy density changesin the bulk of space-time, not only in the vicinity of the boundaries, similarto the Casimir effect when the bulk energy density changes as a result ofmerely presence of the boundary. To reiterate, the deformed model, despitethe presence of a mass gap, displays a surprising Casimir-like sensitivityto large distance boundaries, such that the energy density differs from theMinkowski space value by ∆E ∼ 1mWL . Again, this is in contrast to the naiveexpectation based on analysing the physical degrees of freedom, ∆E ∼ e−mLwith m ∼ mσ being the lowest mass scale of the problem (4.3).4.2.3 Non-Zero Mode ContributionsComputations of the Casimir corrections presented in the previous sectionwere based on an analysis of the zero modes when the corresponding nor-malisation factor explicitly enters the instanton/monopole density. Now, wewant to present some arguments suggesting that corrections due to the non-zero mode contributions can be neglected, and, therefore, cannot cancel the414.2. Casimir-Type Behaviour in Deformed QCDzero modes contribution. Indeed, the computation of non-zero mode contri-bution is reduced to an analysis of the phase shifts in the scattering matrixwhich can not change the normalisation of the wave function itself. The onlychanges that occur are phase shifts. Furthermore, an absolute normalisationis dropped from the final formula for the instanton/monopole density whenthe ratio of the eigenvalues is considered. This argument is consistent withobservation that non-zero mode contribution depends on matter context ofthe theory as it varies when massive scalar of spinor fields in different rep-resentations are part of the consideration. At the same time, the Casimirtype corrections computed above are exclusively due to the gauge portion ofthe theory, not its matter context. Indeed, these Casimir corrections werederived in pure gluodynamics. So, it is difficult to imagine how a Casimircorrection to a non-zero mode contribution (even if it is nonzero) mightcancel a Casimir type correction originating from an analysis of gauge zeromodes.We also comment that the correction L−1 occurs as a manifestation of aslow power like decay of the zero modes in the background of a topologicallynontrivial gauge configuration. It should be contrasted with conventionalbehaviour of zero modes with a mass gap present in the system from the verybeginning (for example, the well studied problem of a double well potential).In former case, the zero modes decay according to a power law leading tothe Casimir type correction, while in the later case, the zero modes are welllocalised configurations which decay exponentially fast at large distances andcan not be sensitive to large distance physics. The mass gap is present for allphysical degrees of freedom in both models. However, in the former case themass gap emerges as a result of the same instanton/monopole dynamics,while in the later a mass gap was present in the system from the verybeginning and it was not associated with any instanton/monopole dynamics.QCD obviously belongs to the former case, and we therefore expect this effectwill persist in real strongly coupled QCD.Next, our computations of the Casimir correction to the instanton/monopole density are based on assumption of the dilute gas approximation.This is enforced in Section 4.2.1 by a finite instanton size ρ which is keptfixed and small. On other hand, the semiclassical approximation in Section4.2.2 is automatically justified due to the parametrically small fugacity ζ,and total neutrality in this system is automatically achieved as long as thesize of the system L is much larger than the Debye screening length m−1σ , see(3.25). In other words, we assume L m−1σ such that neutrality of the sys-tem is automatically satisfied with exponential accuracy. The finite size ofthe manifold does not spoil this neutrality if condition L m−1σ is satisfied.424.3. Topological Sectors and the Casimir Correction in QCDFurthermore, the computation of the monopole’s fugacity ζ and correspond-ing corrections (4.18) can be performed without taking into account of theinteraction of a monopole with other particles from the system as it wouldcorrespond to higher order corrections in density expansion ∼ ζ2. This isprecisely the procedure which was followed in the original computations byPolyakov in [58] and in the deformed model in [85] at weak coupling.Also, we emphasise that in the variational approach developed in [77]neither the classical solution nor the corresponding zero modes vanish atthe boundary of a finite size manifold. The constraints related to the finitesize L of the manifolds are accounted for implicitly rather than explicitly inthis approach. In particular, one should not explicitly cut off the classicalaction of the configuration as a result of finite size L in which the instan-ton/monopole is defined as this contribution is implicitly taken into accountby the variational approach. However, even if we use an explicit cutoff forclassical solution it still cannot cancel the zero mode corrections as theseterms have different behaviour in N . The correction to the classical solutionwould be one and the same for any N , while corrections due to zero modesdepend on N as the correction (4.15) counts number of gauge rotations forSU(N) gauge theory.Finally, it is quite possible that we overlooked some other possible cor-rections (for example, some corrections due to the boundaries which mayoccur in the vicinity of these boundaries). We emphasise that our mainresult is not the computation of a specific coefficient in front of the correc-tion to fugacity in equation (4.18). Rather, our main point is that thesetypes of corrections do occur in a system with a gap, and it is very difficultto imagine that some boundary corrections might mysteriously cancel thesecomputed bulk corrections. Therefore, we next present some arguments andexamples suggesting that a Casimir type behaviour in gauge theories is infact quite generic, rather than a peculiar feature of our choice of system.4.3 Topological Sectors and the CasimirCorrection in QCDIn this section we want to present few generic arguments suggesting thatthe emergence of a Casimir-like behaviour is not an accident, and not acomputational peculiarity. Rather, the effect has a deep theoretical rootsas argued in [103]. We review these arguments starting with analogy withthe well known Aharonov-Casher effect as formulated in [63]. The relevantpart of that work can be stated as follows. If one inserts an external charge434.3. Topological Sectors and the Casimir Correction in QCDinto superconductor wherein the electric field is exponentially suppressed∼ exp(−r/λ) with λ being the penetration depth, a neutral magnetic fluxonwill be still sensitive to an inserted external charge at arbitrary large dis-tance. The effect is purely topological and non-local in nature. The crucialpoint is that this phenomenon occurs, in spite of the fact that the systemis gapped, due to the presence of different topological states in the system.We do not have a luxury of solving a similar problem in strongly coupledfour dimensional QCD analytically. However, one can argue that the roleof the “modular operator” of [63], which is the key element in the demon-stration of long range order, is played by the large gauge transformationoperator T in QCD, which also commutes with the Hamiltonian [T , H] = 0,such that our system must be transparent to topologically nontrivial puregauge configurations, similar to the transparency of the superconductor tothe “modular electric field”, see [103] for the details.We interpret the computational results in a number of systems whereCasimir like corrections have been established as a manifestation of thesame physics which can be described in terms of the operator T . We shouldmention that there are a few other systems, such as topological insulators,where a topological long range order emerges in spite of the presence of agap in the system.There are a number of simple systems in which the Casimir type be-haviour ∆E ∼ L−1 + O(L)−2 has been explicitly computed. In all knowncases this behaviour emerges from non-dispersive contributions such thatthe dispersion relations do not dictate the scaling properties of this term.The first example is an explicit computation [86] in exactly solvabletwo-dimensional QED defined in a box size L. The model has all elementscrucial for present work: non-dispersive contact term which emerges due tothe topological sectors of the theory. This model is known to be a theory ofa single physical massive field. Still, one can explicitly compute ∆E ∼ L−1in contrast with naively expected exponential suppression, ∆E ∼ e−L. An-other piece of support for a power like behaviour is an explicit computationin a simple case of Rindler space-time in four dimensional QCD [55, 100, 102]where Casimir like correction have been computed using the unphysicalVeneziano ghost which effectively describes the dynamics of the topolog-ical sectors and the contact term when the background is slightly modi-fied. Thus, power-like behaviour is not a specific feature of two dimensionalphysics.Our next example is 2d CPN−1 model formulated on finite interval withsize L [53]. In this case one can explicitly see emergence of ∆E ∼ L−1in large N limit in close analogy to our case (4.18) where a theory has a444.4. Commentsgap, but nevertheless, exhibits the power like corrections. The correctioncomputed in [53] also comes from a non-dispersive contribution which cannot be associated with any physical propagating degrees of freedom, similarto our case (4.18).Power like behaviour ∆E ∼ L−1 is also supported by recent lattice re-sults [36]. The approach advocated in [36] is based on physical Coulombgauge, in which nontrivial topological structure of the gauge fields is repre-sented by the so-called Gribov copies leading to a strong infrared singularity.Thus, the same Casimir-like scaling emerges in a different framework wherethe unphysical Veneziano ghost (used in [55, 100, 102]) is not even men-tioned.The very same conclusion also follows from the holographic descriptionof the contact term presented in [103]. The key element for this conclusionfollows from the fact that the contact term in holographic description isdetermined by massless Ramond-Ramond (RR) gauge field defined in thebulk of 5-dimensional space. Therefore, it is quite natural to expect thatmassless R-R field in holographic description leads to power like correctionswhen the background is slightly modified.To avoid any confusion with terminology we follow [103] and call thiseffect as “Topological Casimir Effect” where no massless degrees of freedomare present in the system, but nevertheless, the system itself is sensitive toarbitrary large distances. It is very different from conventional Casimir effectwhere physical massless physical photons are responsible for power like be-haviour. From the holographic viewpoint discussed in [103] the “TopologicalCasimir Effect” in our physical space-time can be thought as conventionalCasimir effect in multidimensional space when massless propagating R-Rfield in the bulk is responsible for this type of behaviour, although this fieldis not a physical asymptotic state in our four dimensional world.4.4 CommentsWe tested a sensitivity of the deformed gauge theory model with non-trivialtopological features to arbitrary large distances. A naive expectation basedon dispersion relations dictates that a sensitivity to very large distancesmust be exponentially suppressed (4.3) when a mass gap is present in thesystem. However, we argued that along with conventional dispersive contri-bution there exists a non-dispersive contribution, not related to any physicalpropagating degrees of freedom. This non-dispersive (contact) term with the“wrong sign” emerges as a result of topologically nontrivial sectors, and can454.4. Commentsbe explicitly computed in our model. The variation of this contact termwith variation of the background leads to a power like “Topological CasimirEffect” (4.4) in accordance with the arguments presented in Section 4.3 andin contrast with the naively expected exponential suppression (4.3).The Topological Casimir Effect in QCD, if confirmed by future analyticaland numerical studies, may have profound consequences for understandingof the expanding FLRW universe we live in. We already mentioned in Sec-tion 4.1 that the observed cosmological dark energy (4.1) may is fact be justa manifestation of this Topological Casimir Effect without adjusting anyparameters. In the adiabatic approximation the universe expansion can bemodeled as a slow process in which the size of the system adiabatically de-pends on time L(t) which leads to extra energy as equations (4.4) and (4.18)suggest. Such a model is obviously consistent with observations if L(t) issufficiently large [87]. We do not insist that this is the model of our uni-verse. Rather, we claim that if the effect persists in strongly coupled QCD,the energy density which can not be identified with any physical propagat-ing degrees of freedom, is sensitive to arbitrary large distances as a resultof nontrivial topological features of QCD. Different geometries (such as anFLRW universe) obviously would lead to different coefficients. Nonetheless,the important message from these computations in our simplified model isthat the energy density in the bulk is sensitive to arbitrary large distancescomparable with the visible size of the universe, and that this sensitivitycomes not from any new physics but simply from the proper treatment ofthe topological structure of QCD.We should mention, also, that with regard to extending from imposedboundaries in flat space, as we considered here, to effective boundary con-ditions due to curvature, the nontrivial holonomy along the compact (S1)dimension is an important aspect of this model that warrants consideration.Mainly, there is no contradiction with the conventional argument that onlya curvature R ∼ H2 should enter the bulk energy density in such an analysison a curved (FLRW) manifold. This is because∮Aµdxµ around the com-pact dimension is an invariant characteristic of the system which cannot bereduced to the curvature, similar to the Aharonov-Bohm effect, where therelevant phenomenon is expressed in terms of the potential Aµ rather thanthe field strength Fµν . Essentially, this is just the statement that the effectis due to topological properties not local field configurations. For more onthe topic of nontrivial holonomy and calculations in curved space see [107].Finally, we add that a comprehensive phenomenological analysis basedon this idea has been recently performed in [12] where comparison with464.4. Commentscurrent observational data including SnIa, BAO, CMB, BBN has been pre-sented, see also [13, 27, 55, 65, 66, 71] with related discussions. The conclu-sion was that the model (4.1) is consistent with all presently available data,and we refer to these papers on analysis of the observational data.47Chapter 5Long Range Order andDomain WallsThis chapter reproduces the work presented in [81]. We consider the interac-tion between extended two dimensional domain walls and localised point-liketopological monopoles. The domain walls considered here are topological de-fects that interpolate between the vacuum state and itself, essentially just awinding.5.1 MotivationThe main motivation for the work presented in this chapter is the recentMonte Carlo studies in pure glue gauge theory which have revealed somevery unusual features. To be more specific, the relevant gauge configura-tions display a laminar structure in the vacuum consisting of extended, thin,coherent, locally low-dimensional sheets of topological charge embedded in4d space, with opposite sign sheets interleaved, see the original lattice QCDresults [3, 37–39]. A similar structure has been also observed in lattice QCDby different groups [10, 11, 40, 41, 49] and also in a two dimensional CPN−1model [1]. Furthermore, the studies of localisation properties of Dirac eigen-modes have also shown evidence for the delocalisation of low-lying modeson effectively low-dimensional surfaces. The following is a list of the keyproperties of these gauge configurations which we wish to study:1) The tension of the “low dimensional objects” vanishes below the crit-ical temperature and these objects percolate through the vacuum, forminga kind of a vacuum condensate;2) These “objects” do not percolate through the whole 4d volume, butrather, lie on low dimensional surfaces 1 ≤ d < 4 which organise a coherentdouble layer structure;3) The total area of the surfaces is dominated by a single percolatingcluster of “low dimensional object”;485.1. Motivation4) The contribution of the percolating objects to the topological suscep-tibility has the same sign compared to its total value;5) The width of the percolating objects apparently vanishes in the con-tinuum limit;6) The density of well localised 4d objects (such as small size instantons)apparently vanishes in the continuum limit.It is very difficult to understand the above properties using conventionalquantum field theory analysis. Indeed, the QCD lattice results [3, 10, 37–41, 49] imply that the topological density distribution is not localised in anyfinite size configurations such as instantons; rather the topological densityis spread out on the surface of low-dimensional sheets. Such a structure cannot be immediately seen in gluodynamics, at least not at the semiclassicallevel. At the same time, these Monte Carlo results could be interpreted verynicely with a conjecture that the observed structure is identified with theextended D2 branes in a holographic description[30, 31, 103].One of the key elements of this conjecture is assumption that the ten-sion of the D2 branes vanishes below the QCD phase transition T < Tcsuch that an arbitrarily large number of these objects can be formed. Thesecond key element in identification of the structure observed on the lat-tice [3, 10, 37–41, 49] with the holographic description in terms of the Dbranes is the assumption that the topological density distribution which isoriginally localised in well defined D0 branes (instantons), somehow spreadsout along extended D2 branes as a result of the interaction between D0-D2branes, leading to their binding. Such a picture was basically motivated, asmentioned in [31, 103], by the structure which emerges in supersymmetricfield theories [20] where the relevant dynamics can be indeed formulated interms of the strongly bound D0-D2 configurations.In this chapter, we investigate precisely the second idea above in theframework of the “deformed gauge theory” developed in [85] and discussedin Chapter 2. The deformation allows us to bring the gauge theory into aweakly coupled regime wherein calculations can be performed in theoreti-cally controllable manner. In spite of the great deal of analytic control pro-vided, the deformed theory preserves many of the relevant structures presentin strongly coupled QCD including confinement, degeneracy of topologicalsectors, and the correct nontrivial θ dependence. Furthermore, it seems,there is no order parameter differentiating the weakly coupled deformedregime from the strongly coupled regime, which reproduces undeformedQCD [85], so that the qualitative behaviour of the two theories may bequite similar.In particular, the deformed theory exhibits two important structures of495.1. Motivationnote: first, the topological charge in this model is carried by the fractionallycharged monopoles with topological charges Q = ±1/N ; and second, thereare domain walls present in the system as a result of a generic 2pi period-icity of the effective low energy Lagrangian governing the dynamics. Giventhese ingredients, we would like to test the following two ideas which areapparently related to the configurations observed in the lattice simulations[3, 10, 37–41, 49]:1) the domain walls form precisely a double layer structure with oppositesign sheets of the topological charge density interleaved;2) the monopoles and domain walls attract each other and the topologicalcharge originally localised on monopoles spreads out along the domain walls.If the second occurs, there will be few well-localised finite sized sourcescarrying the topological charge. Instead, the topological charge density willbe spread over extended domain walls, which is precisely the pattern thathas been observed in simulations [3, 10, 37–41, 49]. For other discussionsrelated to long range order in this model see [4, 106].We note that a similar picture of attraction between monopoles anddomain walls was originally discussed in a cosmological context [23], see alsothe related papers [2, 22, 24] and references therein. The basic idea there isthat if physical monopoles and domain walls are present in the system, therewill be an attractive force between them. Then, if these objects collide, themonopole’s winding number (monopole charge) spreads out on the surfaceof the domain wall, and will be eventually pushed to the boundaries atinfinity. This effect was suggested as a solution of the so-called “cosmologicalmonopole problem”. In our context we do not have real physical monopolesand real physical domain walls in Minkowski space, but rather Euclideanmonopoles and domain walls which must be interpreted as configurationsdescribing the tunnelling processes in physical Minkowski space, see [103]for a detailed discussion of this point. Nevertheless, the formal structure ofthe problem and relevant features (such as attraction between the objectsand spreading the magnetic charge over the surface) are very much the same.The structure of our presentation is as follows. In Section 5.2, we con-struct the domain walls and explicitly demonstrate the double layer struc-ture apparently observed on the lattices. Then, in Section 5.3 we study theinteraction of the domain walls and monopoles. And, finally, in Section 5.4we comment on the important aspects of these results.505.2. Domain Walls in Deformed Gauge Theory5.2 Domain Walls in Deformed Gauge TheoryIn the deformed theory there is a discrete set of degenerate vacuum statesas a result of the 2pi periodicity of the effective Lagrangian (2.36) for the σfields, and thus there exist domain wall configurations interpolating betweenthese states. The corresponding configurations are not however conventionaldomain walls similar to the well known ferromagnetic domain walls in con-densed matter physics which interpolate between physically distinct vacuumstates. Here, instead, the corresponding configuration interpolates betweentopologically different but physically equivalent winding states |n〉, whichare connected to each other by a large gauge transformation. Therefore,the corresponding domain wall configurations in Euclidean space are inter-preted as configurations describing tunnelling processes in Minkowski space,similar to Euclidean monopoles which also interpolate between topologicallydifferent, but physically identical states. This interpretation should be con-trasted with the conventional interpretation of static domain walls definedin Minkowski space when the corresponding solution interpolates betweenphysically distinct states.In fact, a similar domain wall which has an analogous interpretation isknown to exist in QCD at high temperature (in the weak coupling regime)where it can be described in terms of classical equations of motion. Theseare the so-called ZN domain walls which separate domains characterised bya different value for the Polyakov loop at high temperature. As is known,see the review papers [29, 75] and references therein, these ZN domain wallsinterpolate between topologically different but physically identical statesconnected by large gauge transformations similar to our case. These objectscan be described in terms of classical equation of motion and have finitetension ∼ T 3 such that their contribution to path integral is strongly sup-pressed. While the corresponding topological sectors are still present in thesystem at low temperature (though they are realised in a different way) it isnot known how to describe the fate of these ZN walls in QCD in the strongcoupling regime where the semiclassical approximation breaks down.The domain walls to be discussed below in the deformed model are verymuch the same as ZN domain walls at high temperature and their contri-bution to path integral is also strongly suppressed as their tension is finitein the weak coupling regime. Nevertheless, one can study the structure ofthese domain walls, as well as their interaction with dynamical magneticmonopoles. Furthermore, as we discussed in Section 5.1 the domain wallstructure is apparently observed in the lattice simulations, which imply that515.2. Domain Walls in Deformed Gauge Theorythey may have effectively vanishing tension at low temperature. We conjec-ture that the domain walls we describe below in the weak coupling regimein the deformed model slowly become the objects (with effectively vanish-ing tension) which are observed in lattice simulations [3, 10, 37–41, 49] inthe strong coupling regime, as we adiabatically increase the coupling con-stant without hitting the phase transition as argued in [85]. This portionof the theory can not be tested in the deformed model in the semiclassi-cal approximation, but hopefully this portion of strongly coupled dynamicscan be understood in the future using different techniques, such as the dualholographic description as advocated in the present context in [103].5.2.1 Domain Wall SolutionThere are a few different types of domain walls supported by the system(2.36) which have different physical meanings. Here we focus on the discretesymmetry of the effective Lagrangian (2.36)Sdual →∫R3[12L( g2pi)2(∇σ)2 − ζN∑a=1cos(αa · σ + θN)].given by the 2pi shift, σa → σa + 2pi, where any component of σ field canbe shifted by 2pi independently. To simplify analysis, we consider a singlespecific non-vanishing component for the σ field sitting at a−th position,σ =(0, 0, σ(a), 0, . . . , 0)a = 1, ...N. (5.1)This component describes a specific diagonal element of the original non-Abelian field strength. For example, χ(1) corresponds to the following struc-ture in conventional matrix notationsB(1) =g2piL∇σ(1) ·1 0 ... 00 −1 ... 0... ... ... 00 0 0 0 . (5.2)There are N different domain wall types similar to the monopole case sinceclassification of our system is based on αi ∈ ∆aff . We emphasise that thereare only (N−1) physical propagating photons in the system as one scalar sin-glet field, though it remains massless, completely decouples from the system,and does not interact with other components at all, as we saw in Chapter2. As a result of this structure, a configuration with N different types of525.2. Domain Walls in Deformed Gauge Theorymagnetic monopoles will carry zero magnetic charge and one unit of thetopological charge Q = 1 as each monopole carries Q = 1/N topologicalcharge. The corresponding configuration can be identified with a conven-tional instanton with Q = 1 which is made of N constituents. A similarcomment also applies to the domain wall structure: a configuration with Ndifferent types of domain walls on top of each other will produce a trivialvacuum configuration as the Abelian components of the magnetic field willcancel each other, similar to the magnetic monopole construction. Thus, al-though there are N different types of the domain walls in our construction,only (N − 1) of them are independent exactly as with the monopoles.In what follows, without loss of generality, we consider the N = 2 case.In this case there is only one physical field χ = (σ1− σ2) which correspondsto a single diagonal component from the original SU(2) gauge group. Theorthogonal combination (σ1 +σ2) decouples from the system as explained inthe original paper [85] and can be seen immediately in this situation. Theaction (2.36) becomes,Sχ =∫R3d3x14L( g2pi)2(∇χ)2− ζ∫R3d3x[cos(χ+θ2)+ cos(−χ+ θ2)]. (5.3)In terms of the χ field, the classical equation of motion which follows from(5.3) and which determines the profile of the domain wall has the form∇2χ−m2χ sinχ = 0, (5.4)where we take θ = 0 for simplicity, and the mass of χ field mχ = 2mσ isrelated to the Debye correlation length (2.41). The solution of this sine-Gordon equation which interpolates in one direction centered at the originbetween χ(z = −∞) = 0 and χ(z = +∞) = 2pi, and which is centered atz0 = 0 being independent of x, y coordinates is well knownχ(z) = 4 arctan [exp(mχz)] . (5.5)We are now in position to explain the physical meaning of this solution. Aswe mentioned before, this domain wall solution (5.5) does not describe aphysical domain wall which interpolates between physically distinct vacuumstates, but rather interpolates between topologically different but physicallyidentical states. We remark that a similar construction has been consideredpreviously in relation to the so-called N = 1 axion model [33, 92], more535.2. Domain Walls in Deformed Gauge Theoryrecently in the QCD context in [28], and in high density QCD in [76]. In thepreviously considered cases [28, 33, 76, 92] as well as in present case (5.5)there is a single physical unique vacuum state, and interpolation (5.5) cor-responds to the transition from one to the same physical state. Therefore,such domain walls are not stable objects, but will decay quantum mechani-cally, see Appendix A for corresponding estimates. Nevertheless, if life timeof the configuration (5.5) is sufficiently large, it can be treated as stableclassical background, and it can be used to study the interaction of domainwalls with monopoles, which is one of the main objectives of present work,see Figure 5.1 depicting the transition between two topologically differentpaths corresponding to the decay of some domain wall state to a domain wallfree ground state. The path wrapping the peg represents a state with somedomain walls, while the path that does not denotes a state with no domainwalls. We can deform the domain wall path by lifting it over the obstacleso that we can unwind it and deform it into the domain wall free path. Ifthe path describes domain walls with some weight, then it would requiresome energy to lift over the obstacle. If this energy is not available, thenclassically, the configurations that wind around the peg are stable. Quan-tum mechanically, however, the domain wall could still tunnel through thepeg, and so the configurations are unstable quantum mechanically, see theestimate for this probability in Appendix A.One can view these “additional” vacuum states, which are physicallyidentical states and which have extra 2pi phase in operator (2.39), as ananalog to the Aharonov-Bohm effect with integer magnetic fluxes whereelectrons do not distinguish integer fluxes from identically zero flux. Ourdomain wall solution (5.5) describes interpolation between these two phys-ically identical states. Finally, we should also comment that, formally, asimilar soliton-like solution which follows from the action (5.3) appears inthe computation of the string tension in Polyakov’s 3d model [58, 85]. Thesolution considered there emerges as a result of the insertion of externalsources in a course of computing the vacuum expectation of the Wilsonloop. In contrast, in our case, the solution (5.5) is an internal part of thesystem without any external sources. Furthermore, the physical meaningof these solutions are fundamentally different. In our case the interpreta-tion of the solution (5.5) is similar to an instanton describing the tunnellingprocesses in Minkowski space, while in the computations [58, 85] it was anauxiliary object which appears in the computation of the string tension.The width of the domain wall is determined by m−1χ , while the domain545.2. Domain Walls in Deformed Gauge TheoryNon-Trivial Path(Domain Wall State)Trivial Path(Ground State)Figure 5.1: Picture depicting the transition between paths correspondingto the decay of some domain wall state to a domain wall free ground state.Inspired by a similar picture in [28].wall tension σ for the profile (5.5) can be computed and is given byσ = 2 ·∫ +∞−∞dz14L2( g2pi)2(∇χ)2=mχL2( g2pi)2 ∼√ ζL3. (5.6)In the deformed model, the topological charge density distribution (2.11)555.2. Domain Walls in Deformed Gauge Theorycan be written asq(x) =116pi2tr[FµνF˜µν]=−18pi2ijk4N∑a=1F(a)jk F(a)i4=g4pi2N∑a=1〈A(a)4〉 [∇ ·B(a)(x)],(5.7)where the U(1)N magnetic field, Bi = ijk4Fjk/2g is expressed in terms ofthe scalar magnetic potential asF(a)ij =g22piLijk∂kσ(a), B(a) =g2piL∇σ(a). (5.8)In the last step of (5.7) we have replaced the field in the compact directionby it’s vacuum expectation value since we are considering a semiclassicalapproximation.With the explicit solution at hand (5.5), the magnetic field distribution(5.8) for the domain wall is given byBz =( g4piL) 4mχ(emχz + e−mχz), (5.9)and the topological density can then be computed using formula (5.7) withthe following resultq(z) =ζLsinχ =4ζL(emχz − e−mχz)(emχz + e−mχz)2. (5.10)From equation (5.10), we see that the net topological chargeQ ∼ ∫ dz q(z)on the domain wall vanishes. However, the charge density has an interestingdistribution; it is organised in a double layer structure, which is preciselywhat apparently has been measured in the lattice simulations mentionedearlier [3, 37–39]. For a graphical depiction see Figure 5.2 which is a 3d ren-dering of the domain wall solution (5.5). The same double layer structurecan be seen by computing the magnetic charge density ρM which is definedasρ(a)M ≡[∇ ·B(a)(x)]=( g4piL) ∂2χ∂z2= 4ζ ·(4pig)(emσz − e−mσz)(emσz + e−mσz)2. (5.11)565.2. Domain Walls in Deformed Gauge TheoryThus, the relation between the topological charge density (5.10) and mag-netic charge density (5.11) holds for the domain wallq(z) =( g2pi)·(1LN)· ρM (z) (5.12)in agreement with the general expression (5.7).From eqs. (5.10), (5.11), we see that an average density of magneticmonopoles filling the interior of domain wall is expressed in terms of thesame parameter ζ which characterises the average monopole’s density in thesystem (2.36). One can interpret this relation as a hint that the topologicalcharge sources have a tendency to reside in vicinity of the domain wallsrather than being uniformly distributed. We further elaborate on this matterin Section 5.3.It is interesting to note that the domain walls in the deformed model arevery similar (algebraically) to well known domain walls studied previouslyin some SUSY models, see the review article [82]. Of course, there are fun-damental differences between the two: in SUSY models the domain wallsinterpolate between physically distinct vacuum states, in contrast with ourdomain walls which correspond to interpolation between topologically differ-ent but physically identical states. Therefore, the interpretation in these twocases is fundamentally different: in SUSY models the domain walls are realphysical objects, while in our deformed model they should be interpretedsimilar to instantons, objects which describe the tunnelling processes, see[103] for more comments on this interpretation. Furthermore, the classifi-cation of the domain walls in SUSY models is based on the flavour groupsymmetry breaking SU(NF )→ U(1)NF−1, in contrast with colour symmetrybreaking we consider here. However, the formal classification of the domainwalls in SUSY models based on simple roots from the flavour group is verymuch the same as classification in our case based on SU(N) → U(1)N−1breaking pattern, see equations (2.9) and (2.10). These similarities include,in particular, highly nontrivial properties such as ordering of the domainwalls or their passing through each other. However, these questions will notbe elaborated on in the present work.The most important lesson from this analysis is that the double layerstructure naturally emerges in the construction of the domain walls in theweak coupling regime in deformed gauge theory. As claimed in [85] thetransition from the high temperature weak coupling regime to the low tem-perature strong coupling regime should be smooth without encountering anyphase transitions on the way. Therefore, it seems reasonable to identify the575.3. Domain Wall - Monopole InteractionFigure 5.2: Graph showing the two layer structure of the topological chargedensity plotted against one direction across the Domain Wall and the otherone of the two dimensions along it.double layer structure found in this work (5.10) with the double layer struc-ture from lattice measurements [3, 37–39] when one slowly moves along asmooth path from the weak coupling to the strong coupling regime.5.3 Domain Wall - Monopole InteractionFrom expressions (2.39) and (5.1) one can infer that the interaction, or to bemore precise, the algebraic structure, of the domain wall with monopoles isvery similar to monopole - monopole/antimonopole interactions. Since ourdomain walls are not dynamical configurations of the system, but rather,should be treated as a background classical fields, we can not address somehard fundamentally quantum issues such as: what the density of domain585.3. Domain Wall - Monopole Interactionwall configurations looks like, or why the topological charge density is mostlyspread out in the domain walls rather than in localised objects, similar tothe pattern lattice simulations suggest [3, 37–39]. Rather we can formulatea different question which can be addressed in the weak coupling regime.What happens to the monopoles if they are formed in the presence of thedomain walls? As the number of different types of domain walls, N , is equalto the number of different types of monopoles, one could assume that aspecific monopole type “a” will find a corresponding most attractive domainwall. Therefore, we concentrate below on analysis of a specific configurationcontaining two relevant elements: a domain wall of type (5.2) and a nearbyanti-monopole with magnetic charge −α1 and topological charge Q = −1/N .We now consider the domain wall configurations discussed in the previoussection interacting with monopole configurations. In these computationsthe domain walls are treated as classical background fields, and as suchwe do not consider fundamentally quantum questions, such as the density ofdomain walls. Instead, we consider some questions which can be answered inthe semiclassical context. We focus on the interaction between a monopoleand domain wall, each acting as magnetic sources, and compute the energyof the configuration as a function of separation distance between the two.Therefore, the question we are addressing is where would a point chargeprefer to sit in the presence of our domain wall? As stated in Section 5.1,this question is motivated by lattice simulations [3, 37–39] which suggestthat the density of well localised 4d objects (such as small size 4d instantons)apparently vanishes. We should emphasise that our domain walls should notbe thought of as empty objects, but should instead be thought of as alreadyfilled by magnetic monopoles with density determined by (5.10). Indeed, asdiscussed back in Section 2.3, the sigma fields are a dual description for amonopole gas model, so any configuration of sigma fields can be thought ofas some distribution of monopoles.Again, we consider the simplified scenario of SU(2), which correspondsto considering the interaction between a single type of domain wall, a, anda monopole of the same type, or to be more precise an antimonopole so thatthe magnetic charge is −αa. The domain wall is defined as previously, (5.5),but centered at a distance z0 from the origin, so that the magnetic scalarpotential is given by (letting m = mχ)χz0 (x) = 4 arctan[em(z−z0)]. (5.13)595.3. Domain Wall - Monopole Interaction-0.04-0.03-0.02-0.010.000.01-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0Bi nd in g En er gy  ( ∆E ) [ m2 ]Separation Distance (z0) [1/m]Binding energy as a function of separation distance∆E(z0)Figure 5.3: Plot of the numerical result for the binding energy at variousseparation distances between domain wall and monopole. Notice that forz0 < 0, the monopole to the right of the domain wall, there is an “attractive”potential with a minimum near z0 = 0.The monopole is defined such that it is a point source solution to the Klein-Gordon equation,∇2xϕ (x)−m2ϕ (x) = δ (x) , (5.14)centered at the origin (x0 = 0), and is thus an approximate solution to thesine-Gordon equation, (5.4), away from the origin. The magnetic potentialof the monopole is then given by the well known Yukawa potential,ϕ (x) = − e−m|x|4pim|x| . (5.15)We then consider the configuration of monopole and domain wall sepa-rated by a distance z0 and would like to compute the magnetostatic energy605.3. Domain Wall - Monopole Interaction(Euclidean action) as a function of z0. The energy associated with just adomain wall alone is proportional to the area of the domain wall, which isinfinite in this case, so we compute instead the difference between the energyof the two together and the energy of the two independently,∆E (z0) = S [χz0 + ϕ]− S [χz0 ]− S [ϕ] , (5.16)where S is given by (axes have been rescaled relative to (5.3))S [χ] =∫R3d3x[12(∇χ)2 −m2 cosχ]. (5.17)The quantity ∆E defines a “binding energy” and is finite. We cannot how-ever compute it analytically, and so we compute above integrals numericallyinstead, for z0 varying near the domain wall. Some technical details of thecomputation are as follows. We work in a cylindrical volume oriented acrossthe domain wall such that it respects the symmetries of the physical geome-try. The cylinder is defined around the origin with radius 10/m and length30/m, so that we neglect the space outside of this region. It is valid todo so since the monopole potential is exponentially suppressed with lengthconstant m and we are considering a binding energy. We were forced toremove a small volume around the origin when computing the potential en-ergy term because the structure is that of the cosine of a divergent quantity,which is highly oscillatory. The potential energy due to the removed pieceis bounded by the volume removed since it is a cosine so that we can makeit arbitrarily small. These two approximations make up the bulk of thenumerical uncertainty, which is ∼ m2/106.Performing the numerical integration results in the plot given in Figure5.3. There is an attractive potential between the monopole and domainwall with the monopole on one side (z0 < 0), and a slightly repulsive onefor the other side (z0 > 0). The small barrier for z0 > 0 is difficult to seein Figure 5.3 but obvious in Figure 5.5 which is just a plot of only pointsbeyond z0 > 3 with a much finer vertical scale. Also, there is a minimumat z0 ∼ 1/10m (see Figure 5.4), while the peak of the domain wall chargedistribution is ∼ 1/m. Thus the monopole would prefer to sit “inside” thedomain wall, between the center and the peak of the sheet with the samecharge density. It is interesting that the monopole (with charge −α) isattracted to the domain wall sheet with the same charge (−α) rather thanthe sheet of opposing charge (α), but the theory is non-linear so it is notaltogether unexpected. It also suggests a dynamical stability (at least at theclassical level) of the domain wall in addition to the topological stability.615.3. Domain Wall - Monopole Interaction-0.035-0.034-0.033-0.032-0.031-0.030-0.4 -0.2 0.0 0.2 0.4Bi nd in g En er gy  ( ∆E ) [ m2 ]Separation Distance (z0) [1/m]Binding energy as a function of separation distance (close up)∆E(z0)Figure 5.4: Close up plot of the points near the minimum in Figure 5.3showing that the minimum is to the z0 < 0 side of the center.Figure 5.3 is not the complete story since we have not considered pos-sible changes in the magnetic flux distribution coming from the monopole.Basically, the monopole shape could deform in response to the interactionwith the domain wall, so as to become less spherically symmetric. In orderto properly treat this problem, we should allow the spherical distribution ofthe monopole to vary to some superposition of solutions to the Klein-Gordonequation (5.14). This described further calculation is beyond the scope ofthis work, but we conjecture that the magnetic field will prefer to orientitself along the domain wall, so that the magnetic flux will be pushed outto the edge of the domain wall at the boundary of space, similar to argu-ments presented in refs. [2, 22, 24] in cosmological context. In this way, wehave a picture in which any point-like magnetic monopoles become boundto extended domain walls with any magnetic flux being pushed along thedomain walls to infinity. Apparently, this is precisely the picture discovered625.3. Domain Wall - Monopole Interaction-0.00015-0.00010-0.000050.000000.000050.000100.000153.0 3.5 4.0 4.5 5.0 5.5 6.0Bi nd in g En er gy  ( ∆E ) [ m2 ]Separation Distance (z0) [1/m]Binding energy as a function of separation distance (close up)∆E(z0)Figure 5.5: Close up plot of the points to the right in Figure 5.3 showingthe small barrier present on the z0 > 0 side. Notice the much finer verticalscale.in lattice simulations [3, 37–39] wherein very few localised 4d objects areobserved in the system.As a preliminary toward calculating the angular dependence if we allowthe angular distribution to vary, we write a more general expression for amonopole-like solution to the Klein-Gordon equation (5.14), which dependson the angular coordinates:ϕmn (x) ∼ H(1)n (imr)Y mn (θ, φ), (5.18)where H(1)n are the spherical Hankel functions of the first kind and the Y mnare the spherical harmonics. Assuming the azimuthal axis is oriented acrossthe domain wall, the problem is azimuthally symmetric and the sphericalharmonics reduce to Legendre polynomials of cos(θ).When we attempt to calculate the binding energy as defined above for635.4. Commentsϕn it is negative and divergent for n ≥ 2. It thus appears that the sys-tem is very sensitive to angular changes. Furthermore, the divergence in∆E seems to come from the core (near the divergence in ϕ) since it ishighly sensitive to the amount of the core we remove when performing thenumerical calculations. This however is also the region in which this ap-proximation by Klein-Gordon monopoles is not really justified, and in factthe whole low-energy effective theory is suspect. We therefore conclude thatsome other more sophisticated techniques will be required to address thisproblem of angular distribution, and as such it is well beyond the scopeof this work. Nevertheless, we do conjecture that the flux will have a ten-dency to spread along domain wall, but unfortunately cannot make a morequantitative claim.5.4 CommentsThere are two important results of this work. Firstly, a double layer struc-ture similar to that which is observed in lattice simulations [3, 37–39] natu-rally emerges in the construction of the domain walls in the weak couplingregime in the deformed gauge model. Secondly, monopole configurationscharacterised by well localised topological (and magnetic) charge interactwith domain walls in such a way that there is an attraction between the two,and the monopole favors a position inside the domain wall. We introducedthese domain walls as external background fields, while they are expected tobe dynamical configurations with effectively vanishing tension in the strongcoupling regime. We further suggest a tendancy that the magnetic field duea monopole in the presence of a domain will tend to align with the domainwall, such that the flux is pushed to the boundary of the domain wall. Ifthis effect persists in strongly coupled regime, it could be an explanationfor the observation in lattice simulations [3, 37–39] that there are no welllocalised objects with finite size which would carry the topological charge.In weak coupling the domain wall solution is a nicely behaved smoothfunction, but what happens when we transition slowly to the strong cou-pling regime? The holographic picture suggests that the effective domainwall tension vanishes and so they can be formed easily in vacuum. It ispossible that the domain walls become “clumpy” with a large number offolders. Such fluctuations would then increase the entropy of the domainwall, which eventually could overcome the intrinsic tension. If this happens,the domain walls would look like very crumpled and wrinkled objects withlarge number of folds, and as such, the domain walls may loose their natural645.4. Commentsdimensionality, and become characterised by a Hausdorff dimension as somerecent lattice simulations suggest [11]. Nevertheless, the topological chargedistribution on larger scales after averaging over a large number of thesefoldings should be sufficiently smooth so that the double layer structurewould not disappear because the transition from weak to strong couplingshould be sufficiently smooth as argued in [85]. Therefore, we identify thedouble layer structure found in this work (5.10) with the double layer struc-ture from the lattice measurements [3, 37–39]. These particularities of thetransition from weak to strong coupling are also interesting future questions,which will require an analysis beyond the semi-classical level.65Chapter 6Metastable Vacuum DecayThis chapter reproduces the work presented in [7]. We demonstrate thepresence of metastable vacuum states and calculate the decay rate fromsuch states to the true ground state in the context of the deformed modeldiscussed in Chapter 2.6.1 MotivationA study of the QCD vacuum state in the strong coupling regime is the pre-rogative of numerical Monte Carlo lattice computations. However, a numberof very deep and fundamental questions about the QCD vacuum structurecan be addressed and, more importantly, answered using some simplifiedversions of QCD. Here, we study a set of questions related to metastablevacuum states and their decay to the true vacuum state using the deformedgauge theory model wherein we can work analytically. This model describesa weakly coupled gauge theory, which however preserves many essential ele-ments expected for true QCD, such as confinement, degenerate topologicalsectors, proper θ dependence, and so on, as we have seen in previous chap-ters. This allows us to study difficult and nontrivial features, particularlyrelated to vacuum structure, in an analytically tractable manner.The fact that some high energy metastable vacuum states must bepresent in a gauge theory system in the large N limit has been known forquite some time [94]. A similar conclusion also follows from the holographicdescription of QCD as originally discussed in [96]. Furthermore, the decayrate of these excited vacua in the large N limit in strongly coupled puregauge theory can be estimated as Γ ∼ exp(−N4) [73].The fundamental observation on the emergence of these excited vacuumstates was made in a course of studies related to the resolution of the U(1)Aproblem in QCD in the large N limit [90, 91, 93]. In the present work wedo not introduce quarks (which play an essential role in the formulation ofthe U(1)A problem) into the system, but rather, study pure gluodynam-ics, and the metastable vacuum states which occur there. Nevertheless, thekey object relevant for the resolution of the U(1)A problem, the so-called666.1. Motivationtopological susceptibility χ, still emerges in our discussions in pure gluody-namics because it plays an important role in understanding the spectrumof the ground state and multiple metastable states. Indeed, the topologi-cal susceptibility is defined as χ(θ) = ∂2Evac(θ)∂θ2. Therefore, the informationabout the ground (or in general metastable) states Evac(θ) is related to the θbehaviour of the system formulated in terms of the topological susceptibilityχ(θ).When some deep questions are studied in a simplified version of a theory,there is always a risk that some effects which emerge in the simplified ver-sion of the theory could be just artifacts of the approximation, rather thangenuine consequences of the original underlying theory. Our study usingthis deformed theory as a toy model is not free from this potential difficultywith potential misinterpretation of artifacts as inherent features underlyingQCD. Nevertheless, there are few strong arguments suggesting that we in-deed study some intrinsic features of the system rather than some artificialeffects. The first argument is discussed in the original paper on “centre-stablised Yang-Mills” [85], which we have been calling the “deformed gaugetheory model”, where it has been claimed that this model describes a smoothinterpolation between a strongly coupled gauge theory and the weakly cou-pled deformed model without any phase transition by combining a smoothdeformation and an apparently smooth transition to small compactification.In addition, there are a few more arguments based on our previous expe-rience with the this model, which also suggest that we indeed study someintrinsic features of QCD rather than some artifact of the deformation.Our arguments are based on the computations presented in Chapter 3(published in [80]) of the contact term in the deformed theory, see also [84]for some related discussions. The key point is that this contact term with apositive sign (in the Euclidean formulation) in the topological susceptibilityχ is required for the resolution of the U(1)A problem [90, 91, 93]. At thesame time, any physical propagating degrees of freedom must contributewith a negative sign. In [93] this positive contact term has been simplypostulated while in [90, 91] an unphysical Veneziano ghost was introducedinto the system to saturate this term with the “wrong” sign in the topologicalsusceptibility. This entire non-trivial picture has been successfully confirmedby numerical lattice computations. More importantly for the present studies,this picture has been supported by our analytical computations in whichall the nontrivial crucial elements for the resolution of the U(1)A problememerge.Indeed, the non-dispersive contact term in the topological susceptibility676.1. Motivationcan be explicitly computed in this model and is given by [80]χcontact =∫d4x〈q(x), q(0)〉 ∼∫d3x [δ(x)] , (6.1)where q(x) is the topological density operator. It has the required “wrong”sign as this contribution is not related to any physical propagating degrees offreedom, but is rather related to the topological structure of the theory, andhas a δ(x) function structure as it should. In this model χ is saturated byfractionally charged weakly interacting monopoles describing the tunnellingtransitions between topologically distinct, but physically equivalent topolog-ical winding sectors as discussed previously. Furthermore, the δ(x) functionin (6.1) should be understood as a total divergence related to the infrared(IR) physics, rather than to ultraviolet (UV) behaviour as explained in [80]χcontact ∼∫δ(x) d3x =∫d3x ∂µ(xµ4pix3). (6.2)The singular behaviour of the contact term has been confirmed by latticecomputations where it has been found that the singular behaviour at x→ 0is an inherent IR feature of the underlying QCD rather than some latticesize effect [10, 38, 40, 41].In addition, one can explicitly see how the Veneziano ghost postulated in[90, 91] is explicitly expressed in terms of auxiliary topological fields whichsaturate the contact term (6.1) in this model as was shown in [106]. In otherwords, the η′ field in this model generates its mass (which is precisely theformulation of the U(1)A problem) as a result of a mixture of the Goldstonefield with the topological auxiliary field governed by a Chern-Simons likeaction, see [106] for the details.All these features related to the θ dependence which are known to bepresent in the strongly coupled regime also emerge in the weakly coupleddeformed toy model. Therefore, we interpret such behaviour as a strongargument supporting our assumption that the deformed model properly de-scribes, at least qualitatively, the features related to the θ dependence andvacuum structure of QCD, including the presence of metastable states whichis main subject of the present work.The specific computations we perform related to the metastable vacuumstates have never been performed using numerical lattice (or any other)methods. Therefore, we do not have the same luxury present in our previousstudies of the contact term [80] in which our results were supported bynumerous lattice computations. Nevertheless, as the specific questions aboutthe metastable states are closely related to much more generic studies of the686.2. Metastable Vacuum Statesθ dependence in the system, as reviewed above, we are still confident thatour results presented below, based on the deformed model, are inherentqualitative properties of QCD rather than some artificial effects which mayoccur due to the deformation.Our presentation is organised as follows. In Section 6.2 we explic-itly demonstrate the presence of metastable states in the deformed gaugemodel. In Section 6.3 we review the general strategy to compute a decay ofmetastable vacuum states to the true vacuum in the path integral formula-tion. In Section 6.4 we present our numerical analysis on the life time of themetastable states as a function of a “semi-classicality” which is a parameterdetermining the region of validity of our semiclassical computations. Fi-nally, we conclude in Section 6.5 with speculations on possible consequencesand manifestations of our results for physics of heavy ion collisions where ametastable state might be formed as a result of collision, and the system,which is order the size of a nuclei, might be locked in this state for sufficientlylong period of time ∼ 10 fm/c.6.2 Metastable Vacuum StatesHere we concentrate on the Euclidean potential density for the σ fields atθ = 0,U(σ) =N∑n=1[1− cos (σn − σn+1)] , (6.3)where again σN+1 is identified with σ1. To simplify notations we skip a largecommon factor N in our discussions which follow. We restore this factor inour final formula. Also, we have added a constant (N) so that the potentialis positive semi-definite. The lowest energy state, denoted by σ(−), is thestate with all σ fields sitting at the same value (σn = σn+1) and has zeroenergy. This is clearly the true ground state of the system, but there arealso potentially some higher energy metastable states. For an extremal statewe must have∂U∂σn= 0 (6.4)for all n, which immediately givessin (σn − σn+1) = sin (σn−1 − σn) . (6.5)A necessary condition for a higher energy minimum of the potential is thusthat the σ fields are evenly spaced around the unit circle or (up to a total696.2. Metastable Vacuum Statesrotation),σn = m2pinN, (6.6)where m is an integer. A sufficient condition is then∂2U∂σ2n> 0, (6.7)again for all n. This gives uscos (σn − σn+1) + cos (σn−1 − σn) > 0, (6.8)which using (6.6) givescos(m2piN)> 0. (6.9)So, we get a constraint on m in the form of (6.9), and also on N . From (6.9)it is quite obvious that metastable states always exist for sufficiently largeN , which is definitely consistent with old generic arguments [94]. In oursimplified version of the theory one can explicitly see how these metastablestates emerge in the system, and how they are classified in terms of thescalar magnetic potential fields σ(x).We should also remark here that a non-trivial solution with m 6= 0 in(6.9) does not exist9 in this simplified model for the lowest N = 2, 3, 4.Therefore, in our study we always assume N ≥ 5.Looking back at the potential (6.3), the lowest energy of the possibilitiesare given by m = ±1, so that the lowest energy metastable states, denotedby σ(+), are given by (again up to a constant rotation)σ(+)n = ±2pinN. (6.10)To understand the physical meaning of the solutions describing the non-trivial metastable vacuum states, we recall that the operator eiαa·σ(x) is thecreation operator for a monopole of type a at point x, as it was explicitlydemonstrated in [80],Ma(x) = eiαa·σ(x). (6.11)Therefore, the vacuum expectation value 〈Ma(x)〉 describes the magneti-sation of a given metastable ground state classified by the parameter m.9N = 4 deserves a special consideration as at m = ±1 the second derivative (6.7)vanishes. It may imply a presence of the massless particles in the spectrum for theseexcited vacuum states. It may also correspond to a saddle point in configuration space.We shall not elaborate on this matter in the present work.706.2. Metastable Vacuum StatesAs one can see from (6.6), the corresponding vacuum expectation value〈Ma(x)〉 always assumes the element from the centre of the SU(N) gaugegroup. Specifically, for the lowest metastable vacuum states given by (6.10),the magnetisation is given by〈Ma(x)〉 = exp[±i2piN]. (6.12)The fact that the confinement in this model is due to the condensationof fractionally charged monopoles has been known since the original paper[85]. Now we understand the structure of the excited metastable states also;mainly, these metastable vacuum states can be also thought of as a con-densate of the monopoles. However, the condensates of different monopoletypes, n from (6.10), are now shifted by a phase such that the correspondingmagnetisation receives a non-trivial phase (6.12).A different, but equivalent way to classify all these new metastable vac-uum states is to compute the expectation values for the topological densityoperator for those states. By definition〈 116pi2tr[FµνF˜µν]〉m ≡ −i∂Sdual(θ)∂θ|θ=0 (6.13)= −i ζL〈sin (αa · σ)〉m = −i ζLsin(2pimN).The imaginary i in this expression should not confuse the readers as we workin Euclidean space-time. In Minkowski space-time this expectation value isobviously a real number. A similar phenomenon is known to occur in theexactly solvable two dimensional Schwinger model wherein the expectationvalue for the electric field in the Euclidean space-time has an i, see [105]for discussions in present context. The expectation value (6.13) is the orderparameter of a given metastable state.A crucial point we want to make here is that a metastable vacuum statewith m 6= 0 in general violates P and CP invariance since the topologicaldensity operator itself is not invariant under these symmetries. Preciselythis observation inspires our suggestion, discussed in Section 6.5, that suchmetastable states could be the major source of the local P and CP viola-tion observed in heavy ion collisions at the Relativistic Heavy Ion Collider(RHIC) at Brookhaven, and the Large Hadron Collider (LHC).Now we come back to our discussions of the lowest metastable states(6.10). Putting the metastable configuration back into the potential (6.3)we find that the energy density separation between the true ground state716.3. Metastable Vacuum Decayand the lowest metastable states (6.10) is given by10 ≡(E(+) − E(−))= N[1− cos(2piN)]. (6.14)The choice of sign in (6.10) is irrelevant for the purposes of calculating thevacuum decay since the two states m = ±1 are degenerate in terms of en-ergy and so have the same energy splitting with respect to the ground state.These states, however, are physically distinct as the expectation value of thegauge invariant operator (6.13) has opposite signs for these two metastablevacuum states. This implies that all P and CP effects will have the op-posite signs for these two states, while the probability to form these twometastable states is identical, as is the decay rate. So, while our fundamen-tal Lagrangian is invariant under these symmetries, the metastable vacuumstates, if formed, may spontaneously break that symmetry.6.3 Metastable Vacuum DecayIn this section we briefly review the general theory and framework for calcu-lating metastable vacuum decay rates in Quantum Field Theory, restatingthe important results for the three dimensional model discussed above. For amore thorough discussion see Appendix B or the original papers [14, 15, 47].There is also a great, and fairly extensive, discussion on this topic in Chap-ter 7 of [72]. The process for the decay of a metastable vacuum state to thetrue vacuum state is analogous to a bubble nucleation process in statisticalphysics. Considering a fluid phase around the vaporisation point, thermalfluctuations will cause bubbles of vapor to form. If the system is heatedbeyond the vaporisation point, the vapor phase becomes the true groundstate for the system. Then, the energy gained by the bulk of a bubble tran-sitioning to the vapor phase goes like a volume while the energy cost forforming a surface (basically a domain wall) goes like an area. Thus, there10We should comment here that the vacuum energy of the ground state E(±) ∼ N inthis model scales as N in contrast with conventional N2 scaling in strongly coupled QCD.However, the ratio /E(±) ∼ N−2 shows the same scaling as in strongly coupled QCD.The difference in behaviour in large N limit between weakly coupled “deformed QCD” andstrongly coupled QCD obviously implies that we should anticipate a different asymptoticscaling for the decay rate in the large N limit in our simplified model in comparison withresult [73]. As we discuss in Sections 6.4.2 and 6.4.3 this is indeed the case. Furthermore,the region of validity in this model shrinks to a point in the limit N → ∞ as discussedin [85]. Therefore, the asymptotic behaviour at N →∞ should be considered with greatcaution.726.3. Metastable Vacuum Decayis some critical size such that smaller bubbles represent a net cost in energyand will collapse while larger bubbles represent a net gain in energy. Once abubble forms which is larger than the critical size it will grow to consume theentire volume and transition the whole of the sample to the vapor phase. Tounderstand the lifetime of such a ’superheated’ liquid state, the importantcalculation is, therefore, the rate of nucleation of critical bubbles per unittime per unit volume (Γ/V ). Similarly, we aim to calculate this decay ratefor our system from the metastable state σ(+) to the ground state σ(−),though through quantum rather than thermal fluctuations. Classically, asystem in the configuration σ(+) is stable, but quantum mechanically thesystem is rendered unstable through barrier penetration (tunneling).The semiclassical expression for the tunneling rate per unit volume isgiven by [14, 15, 47]ΓV= Ae−SE(σb)/~ [1 +O (~)] (6.15)where SE is the Euclidean action (3.11) and is evaluated in the field config-uration called the “Euclidean bounce” which we have denoted σb. The Eu-clidean bounce is a finite action, spherically symmetric configuration whichsolves the classical equations of motion and interpolates from the metastablestate to a configuration “near” the ground state and back.In the limit of small separation energy  the bounce approaches σ(−)more closely and spends longer in the region nearby, so that the bounceconfiguration resembles a bubble with the interior at σ(−), the exterior atσ(+), and a domain wall surface interpolating between the two11. If thebubble is very large, corresponding to very small , then the curvature atthe interpolating surface is small and the surface appears flat.Therefore, if the separation energy, , between the two states is small, weneed only solve for the one dimensional soliton interpolating between σ(+)and σ(−) which solvesS1 =∫dxN∑n=1[12(dσndx)2+ 1− cos (σn − σn+1)]. (6.16)This is called the thin-wall approximation, and is the framework in whichwe will work. In the deformed model, as discussed in the previous section,11One should comment here that this model also exhibits very different types of thedomain walls, mainly those considered in Chapter 5 (and [81]). The objects discussedthere are fundamentally different from solutions in the present work as they essentiallydescribe the tunnelling events between different topological sectors, while in here thedomain wall-like objects play the auxiliary role in order to evaluate the life time of ametastable state.736.3. Metastable Vacuum Decaythe separation  ∼ 1/N , so that the thin-wall approximation coincides withthe large N approximation.For the thin wall approximation the full action reduces toS3 = 4piR2S1 − 43piR3 =163piS312, (6.17)where the last step is computed by using variational analysis to get R =2S1/. Notice again the similarity to a bubble nucleation problem. Thisextremal action with respect to the bubble size is in fact a maximum, andas such the action increases with R for smaller size and decreases with Rfor larger. Hence, the bounce configuration which saturates the decay rateis essentially a bubble of critical size as discussed when making this analogyto bubble nucleation.The condition for the validity of the thin wall approximation is essentiallythat the interior of the bubble is very near the true ground state σ(−) sothat it is nearly stable and stays near σ(−) for large ρ. We want Rµ  1,where µ2 = ∂2U/∂σ2n(σ(−)) is the curvature of the potential at the groundstate; here µ =√2. Thus, we need2√2S1  , (6.18)where  is given by (6.14).We now have everything required to calculate the exponent for the vac-uum decay (B.1) assuming we can solve for a classical path associated withthe one dimensional action (6.16) interpolating between the two states σ(+)and σ(−). We have not discussed the coefficient A, and indeed it is a muchmore complicated problem related to the functional determinant of the fulldifferential operator, δ2S/δσ2. This calculation is beyond the scope of thiswork for the deformed model as it does not change the basic physical pic-ture advocating in this work. We want to see that the leading factor in thedecay rate is indeed exponentially small, and our computations are justifiedas long as our semiclassical parameter (3.25) is sufficiently large, N  1.Furthermore, we anticipate that the dependence on N in the exponent ismuch more important than the N dependence in pre exponential factor∼ δ2S/δσ2. The only power -like corrections which may emerge from thedeterminant is through a factor of√SE/2pi for each zero mode [15], and wecan safely neglect these corrections in comparison with much more profoundexponential behaviour in N , see Sections 6.4.2 and 6.4.3.746.4. Computations6.4 ComputationsWe now proceed to solve the equations of motiond2σndx2= sin (σn − σn+1)− sin (σn−1 − σn) , (6.19)with σN+1 identified with σ1, derived from the action (6.16) subject to theboundary conditionsσn (x→ −∞) = 0, (6.20)andσn (x→ +∞) = 2piNn+ ϕ. (6.21)The ϕ in (6.21) is a relative rotation angle between the two boundaries,since each of the two states are only defined up to a rotation. The angle isdetermined by demanding a minimal action interpolation. That is, we shouldminimise the action with respect to the interpolating field configuration andalso with respect to this angle ϕ. The final solution thus obtained will thenbe defined only up to an arbitrary total rotation which will be importantlater. Additionally, we expect the solution to be a soliton (instanton-like) inthe sense that it should be well contained with only exponential tails awayfrom the center so that the interpolation occurs in an exponentially smallregion. The characteristic size of this region, we expect, is given by m−1σ inthe original model, or just 1 in dimensionless notations used here. Then, wecan calculate the vacuum decay rate asΓV∼(S32pi)3e−S3 , (6.22)where we have put in the part of the coefficient that we can calculate relatedto the zero modes in the system. There are six zero modes: three transla-tions, two spacial rotations, and the one global σ-rotation discussed earlier.We now discuss in Section 6.4.1 the numerical technique employed to solvethis problem, and in Section 6.4.2 the results of these numerical calculations.6.4.1 Numerical TechniqueThe sine-Gordon equation for a single field, u′′ = sin(u), has a soliton solu-tion given byu (x) = 4 arctan (ex) , (6.23)756.4. Computationsinterpolating between 0 and 2pi, which seems like good starting point. Assuch, we choose a similar form for the initial guess at the solution for thecoupled equations, and hence define our initial guess to be of the formσn =( nN+ϕ2pi)4 arctan (ex) . (6.24)This initial guess has two important properties; it satisfies the boundaryconditions (6.20) and (6.21), and tails off toward those boundaries as decay-ing exponentials for x → ±∞, which is the type of behaviour expected, asdiscussed previously.The equations of motion (6.19) are on an infinite domain and must betruncated to be solved numerically. We want to truncate the domain to a re-gion beyond which changes in the σi (r) are numerically insignificant. Giventhat the tails of the σn (r) (and we expect the final solution) are decayingexponentials, choosing the domain [−16, 16] means that the boundary valuesare within ∼ 10−7 of their final values and is suitable for our purpose.In order to promote numerical stability particularly around the boundaryvalues we employ Chebyshev spectral methods for integrals and derivatives,as described in [9][83], using an unevenly spaced Chebyshev grid given byxi = cos(piiNp), ∀ 0 ≤ i ≤ Np (6.25)where Np is the number of grid points. Notice that the Chebyshev griddefines a domain [−1, 1] and so we scale x→ x16 in order to express functionswith the chosen domain on the spectral grid.From now on, we will use the following notation: σin denotes the ith gridpoint of the nth field, where N is the total number of fields while Np is thenumber of grid points. The differentiation matrix (Np ×Np) is given by [9](page 570) asDij =2N2p+16 i = j = 0−2N2p+16 i = j = Np− xj2(1−x2j)0 < i = j < Np(−1)i+jpipj(xi−xj) i 6= j(6.26)where,pj ={2 j = 0 or N1 otherwise.766.4. ComputationsAny higher derivative is then given by repeated multiplication by D.This differentiation matrix (6.26) is basically just the linear operator describ-ing interpolating a function on the grid points by an N thp order polynomialand differentiating that polynomial. Since it uses knowledge of the entirefunction rather than just the few nearby points like a finite difference, theaccuracy of the derivative is generally much better than any small order fi-nite difference. Furthermore, using a grid spaced in this way provides muchmore numerical stability, counteracting the so called Runge phenomenon[83], which leads to large oscillations near boundaries of uniform grids.The algorithm we employ to minimise the action with respect to thefield configuration is a gradient descent method, which is to treat the actionas a potential over the configuration space formed by each σ field at eachgrid point, then to take steps in the negative gradient direction. Essentially,iterating the expressionσin → σin − δdS1dσinσin → σin + δ(D2σn)i − δ sin (σin − σin+1)+ δ sin(σin−1 − σin)(6.27)where δ is a chosen step size, which we start as δ = 1. At each iteration,we enforce the boundary conditions and check if the action (6.16) applied tothe new configuration is in fact smaller than the old configuration. If so, wemove to the new configuration. If not, the step was too large and we haveoverstepped the section of the potential with a downward slope, so we goback to the old configuration and reduce the step size by δ → δ/2 and iteratethis procedure until we find a good step or reduce the step size below ourdesired precision. We then reset δ = 1 and continue until we cannot find agood step within our desired precision. Once reaching a position from whichno step reduces the action, we are within the defined precision of a minimumof the action. In order to probe more of the configuration space we took aMonte-Carlo-like approach wherein we adjusted our initial guess by addingsome Gaussian noise in an envelope, (1−|x|)2. We chose an envelope of thisform because we expect that the solution has the sort of exponential tailsof our initial guess (6.24), while we do not know the form of the core of thedomain wall. Thus, it is sensible to probe more of the configuration spacerelated to the specific details of the core.776.4. Computations3.544.55-6pi/7 -4pi/7 -2pi/7 0 2pi/7 4pi/7 6pi/71 -D Ac ti onφ-Angle (-8pi/7)Simulation Data for S1(φ) in Nc = 7Figure 6.1: Plot of some simulation data for the one dimensional action(6.16) as a function of the angle ϕ between the boundary conditions donefor N = 7.6.4.2 ResultsThe first issue we address is the question about the favoured angle, ϕ, be-tween the two boundaries, (6.20) and (6.21). In order to find the angle wechose (arbitrarily) N = 7 and varied the angle in the range [−pi, pi]− 8pi/7,and look at the action S1 as a function of ϕ. The results of that simulationare plotted in figure 6.1. The center point on the plot, −8pi/7, may seemodd, but it is the value for ϕ which leads to a maximally symmetric solution,so it is very believable minimum for the potential, and indeed this is whatwe see. The solution for the minimal σ field configuration corresponding toϕ = −8pi/7 is shown in figure 6.2 across the domain wall. Extending these786.4. Computations-6pi/7-4pi/7-2pi/702pi/74pi/76pi/7-0.6 -0.4 -0.2  0  0.2  0.4  0.6σ-F ie ld sChebyshev Grid PointsSimulation Data for σ-Fields in Nc=7Figure 6.2: Plot of some simulation data for the σ field configuration plottedacross the domain wall done for N = 7 and φ = −8pi/7.results to arbitrary N we setϕ = −pi(N + 1N), (6.28)which just ensures that the solution we look for is maximally symmetric inthe same sense as the fields in figure 6.2, basically that σn = −σN+1−n. Wehave checked that this choice of ϕ, (6.28), does in fact lead to the lowestaction configuration for N = 20 and N = 35 with results much like thoseshown in figure 6.1 for N = 7, so we are comfortable with our assumption.Next, we are expecting a non-perturbative function of the form Γ/V ∼exp [−F (N)] [73, 94], so running simulations for Γ/V (N) we plot the resultsin the form F (N). This plot is given in figure 6.3 where the points and errorbars given are the mean and standard deviation for 25 trials of our simulationat each N between 15 and 75 using 312 Chebyshev grid points; it is shown796.4. Computations 0.001 0.01 0.1 1 10 20  30  40  50  60  70F (N c)  = - Lo g[ Γ/ V]  (1 06 )NcSimulation Data for Log[Γ/V] (Nc)  Simulation Data  F(Nc) = a(Nc+c)b  a = 3.906e-3  +/- 6.18e-4  b = 4.83304   +/- 0.03145  c = 4.26324   +/- 0.3429Figure 6.3: Plot of some simulation data for the decay exponent F (N)plotted for N in the range 15 to 75.806.4. Computationson a log-log scale to emphasise the power law behaviour of F (N).The particular fit parameters are given for completeness but should notbe regarded as terribly important. In fact, the most important result for thepresent analysis is that the computations are performed in a theoreticallycontrollable manner where every single step is justified in the semi-classicallimit (3.25) governed by the parameter N  1. We are after all working ina toy model, and as such should expect a good qualitative picture but nottake the numerical details too seriously. It is however interesting that thefinal form for the decay rate is given as, putting the parameter N back in,ΓV∼ exp{−N(aN b)}(6.29)with both a and b positive. Thus, the decay rate does drop off exponentiallyin N and our other semiclassical parameter N , and indeed faster than anyperturbation term would describe as previously conjectured. It is a semi-classical calculation, but the behaviour is fundamentally non-perturbative,and it is only parametrically justified when N  1.A few comments are in order. First, our numerical estimates (6.29) canbe only trusted for finite N > 5, but not for parametrically large N → ∞where the region of validity of the model shrinks to a point. Furthermore,if the external parameter N were allowed to vary in a very large region itmay lead (and, in fact, it does) to a systematic error in our numerical sim-ulations. This is because in our numerical simulations we assume that allour variables are order of unity, rather than having some functional depen-dence on N , which may not be the case when the external parameter N isallowed to vary in wide region of parameter space. Finally, one should notexpect that our formula (6.29) would reproduce the asymptotic behaviour[73] due to the differences in large N scaling between the deformed modeland strongly coupled QCD. As mentioned previously, the main goal of ourcomputations is to support the qualitative, rather than quantitative pictureof metastable vacua and their decay, conjectured in [94, 96] in a simplifiedmodel where calculations are parametrically justified at N  1 and finiteN . Nevertheless, there is room to improve our numerical simulations in amuch wider range of N as a result of a recent analysis [51] in which theasymptotic expression at N → ∞ has been analytically computed. Theseimprovements are discussed in the next section.6.4.3 Improved ResultsRecently, an analytical analysis of the asymptotic behaviour of this calcula-tion, inspired by the above numerical results, has been carried out [51] with816.4. Computationsthe asymptotic expression for the decay rate per unit volume given byΓV∼ exp[−N 256N7/29√3pi (pi − θ)2]. (6.30)We briefly reproduce these results in Appendix C. The asymptotic expression(6.30) gives us a hint about how to produce a better estimate for the decayrate for very large N  1 in comparison with our naive numerical resultspresented in Section 6.4.2 above, wherein we assumed that parameter N isnot allowed to vary in an extended region of parameter space.Indeed, the analysis in [51] suggests the specific reason for the disparitybetween the asymptotic expression and the numerical results shown in figure6.3 for large N . Mainly, the asymptotic guess solution is given byσn (x) =(4nN− 2)arctan[exp(−2√3Nx)], (6.31)which has a size ∼√N changing with the parameter N . Our guess solution(6.24) does not scale with N and so becomes an increasingly bad guess atasymptotically largerN , such that our numerical solver becomes increasinglylikely to find some other local minimum of the action. Furthermore, ourintegration domain was fixed for all N as we did not even attempt to considerany large variations with N in our analysis in Section 6.4.2. When weallow the external parameter N to become large, the true minimal actioninterpolating trajectory eventually will not fit in the finite size numericalgrid which we fixed for all N . Essentially, forcing boundary conditions ontoo small a domain also forces a higher action local minimum as N increases.This is precisely the mechanism by which a systematic error is introducedinto the numerical simulations as a result of large variation in the externalparameter N , as suggested in the previous section.Fortunately, the analytical expression (6.31) which is valid for asymp-totically large N suggests a simple fix to improve our numerical solution athigher N by explicitly taking into account the variation of the trajectorysize with this parameter. Technically, we can allow the integration domainto scale ∼√N , and start with the asymptotic guess (6.31) in which the largeparameter N explicitly enters, as the initial guess for the “improved” numer-ical algorithm. Again, we added some Gaussian noise to get an ensemble of25 initial guesses for each N and relaxed them as described in Section 6.4.1to arrive at a minimum of the action. These improved results are shown infigure 6.4 plotted along with the asymptotic expression for the decay rate826.5. Comments 0.0001 0.001 0.01 0.1 1 10 10  20  30  40  50  60  70F (N c)  = - Lo g[ Γ/ V]  (1 06 )NcSimulation Data for Log[Γ/V] (Nc)  Previous Simulation  Improved Simulation  F(Nc) = a(Nc+c)b  G(Nc) = (256/9√3 pi3) Nc7/2  a = 0.815267  +/- 0.007364  b = 3.3837      +/- 0.00207  c = -1.53268   +/- 0.01316Figure 6.4: Plot of the improved simulation data for the decay exponentF (N) plotted for N in the range 7 to 75.and the first few points from the original simulations in Section 6.4.2. Theyreproduce the previous results for finite N ≤ 15 and approach the asymp-totic result (6.30) given in [51] from below for large N . Numerically, theasymptotic expression (6.30), which is formally valid at N → ∞, describesour improved simulation data sufficiently well (with accuracy better than10%) only at large N ≥ 35.6.5 CommentsOur comments here can be separated into two different parts: solid theo-retical results within the deformed model; and some speculations related tostrongly coupled QCD realised in nature.We start with the first part of the conclusion in which our basic result836.5. Commentsis as follows. We have demonstrated that the deformed model shows (onceagain) that some qualitative features expected to occur in the strongly cou-pled regime in the large N limit as argued in [94] do emerge in the simplifiedversion of the theory as well. We demonstrated the existence of metastablevacuum states with energy density higher than the ground state by  ∼ 1/N ,and have shown that the lifetime of the metastable states is exponentiallysuppressed in this model with respect to the semi-classicality parameter N .The suppression increases even further with increasing number of colours Nfor a fixed N , and it is given by (6.29).In this simplified system one can explicitly see these metastable states,how they are classified, and the microscopic dynamics which govern thecorresponding physics. The P and CP invariance is generally violated inthese metastable vacuum states as the expectation value for the topologicaldensity (6.13) explicitly shows. We believe that this feature of spontaneousbreaking of the P and CP invariance in metastable states is quite a genericfeature which is shared by strongly coupled pure gauge theories (for suf-ficiently large N). Precisely this feature of the metastable states plays acrucial role in our speculative portion of the conclusion.Therefore, we now speculate that precisely this spontaneous symmetrybreaking effect is responsible for the asymmetries in event by event studiesobserved at the RHIC (Relativistic Heavy Ion Collider) and the LHC (LargeHadron Collider). To be more specific, the violation of local P and CPsymmetries has been the subject of intense studies for the last couple ofyears as a result of very interesting ongoing experiments at RHIC [16, 26]and, more recently, at the LHC [17, 18, 25, 70], see [44] for a recent reviewand introduction to the subject with a large number of references to originalpapers.The main idea for explaining the observed asymmetries is to assume[43, 44] that an effective θ(~x, t)ind 6= 0 is induced in the process of coolingof the system representing the high temperature quark-gluon plasma. Inother words, the system in the process of cooling may spontaneously chooseone or another state which is not the absolute minimum of the system cor-responding to the θ = 0, but rather, some excited state, similar to the oldidea when the disoriented chiral condensate can be formed as a result ofheavy ion collisions. The key assumption is that this induced θ(~x, t)ind 6= 0is coherent on a relatively large scale, of order the size of nuclei ∼ 10 fm. Ifa state with 〈θ(~x, t)ind〉 6= 0 is indeed induced, it implies a violation of thelocal P and CP symmetries on the same scales where θ(~x, t)ind 6= 0 is corre-lated. It may then generate a number of P and CP violating effects, such asCharge/Chiral Separation (CSE) and Chiral Magnetic (CME) Effects, see846.5. Comments[44] for a recent review.One of the critical questions for the applications of the CME to heavyion collisions is a correlation length of the induced 〈θ(~x, t)ind〉 6= 0. Why arethese P odd domains large?We suggest that the system being originally formed at high temperaturemight be locked in one of these metastable states during the cooling stage12.If this happens one should obviously expect a number of P and CP effectsto occur coherently in the entire system characterised by a large scale oforder the size of nuclei L  Λ−1QCD. We therefore identify θ(~x, t)ind 6= 0from [43] with the effective theta parameter 2pi/N which enters (6.13) andwhich manifests a spontaneous violation of the P and CP symmetries in thesystem.The presence of such long range order (which itself is a consequence ofa spontaneous selection of a metastable vacuum state in the entire systemduring the cooling process) may explain why the CME is operational inthis system and how the asymmetry can be coherently accumulated. Thisidentification would justify the effective Lagrangian approach advocatedin [43, 104] wherein θ(~x, t)ind is treated as a slow background field withcorrelation length much larger than any conventional QCD fluctuations,L  Λ−1QCD. It is important to emphasise that the P and CP symmetriesare good symmetries of the fundamental QCD. As mentioned in footnote12, the asymmetries can only be observed in heavy ion collisions in eventby event analyses when the system might be locked, for sufficiently longperiod of time τ ∼ L/c Λ−1QCD, in a metastable state in one collision withone specific sign for the topological density (6.13). Because the metastablestates with opposite signs for the topological density operator (6.13) havethe same energy, which state is chosen for a particular event is random andevenly distributed. Thus, it is clear that if one averages over a large num-ber of events, the asymmetry will be washed out as the probability to formthese metastable states is identical and the lifetime for the two is the sameas we mentioned in Section 6.2. However, in the event by event studies theasymmetry will be evident in the system. Apparently, this is precisely whathas been observed. The P and CP violation is seen in collider events onlyon an event by event basis but averages to zero over many events, see the12 The P and CP symmetries, of course, are good symmetries in QCD. The probabilityto produce the m = +1 state from equation (6.9) is identical to that to produce them = −1 state. Therefore, there will not be any P and CP violating effects if one averagesover a large number of events. However, one should expect some asymmetries if oneanalyses the system on an event by event basis, which is precisely the procedure used atRHIC and the LHC, see [44] for a recent review.856.5. Commentsrecent review paper [44] for details.86Chapter 7ConclusionWe examined some interesting aspects of the modified deformed gauge the-ory developed by Lawrence Yaffe and Mithat U¨nsal [85] in a confined phasein an theoretically controllable manner, which is impossible with currentmethods in real strongly coupled QCD. This model, as discussed in Chapter2, is constructed by taking a standard Yang-Mills Lagrangian and puttingin an extra potential by hand that penalises an expectation value for theWilson line in the compact direction. The Wilson line acts as an orderparameter for a center symmetry breaking that characterises the deconfine-ment phase transition, so that the extra potential, if chosen to be strongenough, forces a confined phase. Thus, we have a system at weak coupling(small compactification scales) that is nonetheless confined and gapped, forwhich the low energy effective dynamics is given by two dual descriptions:a multi-species Coulomb gas; or a coupled sine-Gordon model.In Chapter 3 we calculated the topological susceptibility analytically inboth dual descriptions, demonstrating the presence of a nondispersive con-tact term with the sign opposite that of the contribution from any physicalpropagating degrees of freedom. We discussed the necessity of such a termin the resolution of the U(1)A problem in QCD, which provides the physicalmass for the η′ meson, and explained how such a term has previously beenpostulated either directly or via an extra unphysical ghost field. In the de-formed model however, the contact term emerges naturally and can be seenin both descriptions.Next, in Chapter 4, we considered the Coulomb gas description for thedeformed model, and performed an analysis of the zero modes for the col-lective coordinates of a monopole, following a similar analysis performed byGerard ’t Hooft [77] in his classic paper on four dimensional instantons. Wecalculated the corrections to these zero mode contributions to the measuredue to a finite size of the manifold. The results of this analysis are thatthe monopole fugacity, and so also the bulk energy density, receives someCasimir-like power law corrections based on the size of the manifold. This isin contrast to the naive expectation that in a gapped system with only mas-sive degrees of freedom should only have a weaker exponentially suppressed87Chapter 7. Conclusiondependence on the boundary. We further argued that, if such an effect per-sists in strongly coupled undeformed QCD, it provides a natural solutionto the cosmological dark energy problem, with a rough prediction for themagnitude, HΛ3QCD ∼ (10−3eV )4, that is of the correct order of magnitude.Furthermore, this explanation requires no new physics, coming merely fromlong distance “nondispersive” effects in the QCD sector interacting with afinite sized manifold.Then, in Chapter 5, performed a numerical analysis of the interactionbetween a point-like topological monopole and an extended topological do-main wall in the sine-Gordon description. The domain wall solution in thesine-Gordon picture is qualitatively similar to relevant gauge configurationsdiscussed in the context of some Lattice QCD simulations [3, 10, 37–41, 49],which suggest that extended topological objects are more relevant than thepoint-like instantons that have been discussed traditionally. We found thatthe lowest energy configuration involves the monopole sitting within the do-main wall toward the side with the same topological charge as the monopole.This result suggests a dynamical reason for the absence of instanton-like con-figurations in lattice simulations, and perhaps a dynamical stability for thedomain walls also, above and beyond the classical topological stability.Finally, in Chapter 6, we demonstrated the presence of metastable vac-uum states with energy greater than the true ground state in the deformedgauge model, and calculated the decay rate from the lowest energy of suchstates to the ground state following the procedure developed by Sidney Cole-man [14, 15]. We solved for configurations interpolating between the truevacuum state and the higher energy metastable state numerically, then usedthese to find the decay rate as a function of Nc, the number of colours de-fined by the gauge group SU(N) for the model, confirming the predictedbehaviour predicted by Edward Witten for undeformed gauge theory atlarge Nc [95]. As expected, the result is given as Γ/V ∼ exp[aN b], for somecoefficients a and b which we computed, confirming a nonperturbative ori-gin for such behaviour since this dependence cannot arise at any order ofperturbation expansion.The deformed gauge theory model is an extremely useful toy model forstudying ideas in true undeformed gauge theory, especially topological prop-erties, in a confined phase. It allows for semiclassical analysis of a topologi-cally nontrivial, confined theory with a mass gap at weak coupling, which issmoothly connected parametrically to undeformed strongly coupled Yang-Mills gauge theory. 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This lowers the energy of the configuration over that where the holewas filled by the domain wall transition by an amount proportional to R2where R is the radius of the hole. The hole, however, must be surroundedby a string-like field configuration which interpolates between an unwoundconfiguration and a wound one. This string represents an excitation in theheavy degrees of freedom and thus costs energy, however, this energy scaleslinearly as R. Thus, if a large enough hole can form, it will be stable andthe hole will expand and consume the wall. This process is commonly calledquantum nucleation and is similar to the decay of a metastable wall boundedby strings; therefore, we use a similar technique to estimate the tunnellingprobability. The idea of the calculation was suggested in [92] to estimate thedecay rate in the so-called N = 1 axion model. In a QCD context similarestimates have been discussed for the η′ domain wall in large N QCD in [28]and for the η′ domain wall in high density QCD in [76].If the radius of the nucleating hole is much greater than the wall thick-ness, we can use the thin-string and thin-wall approximation. This approxi-mation is justified, as we shall see, when we calculate the critical radius Rc.In this case, the action for the string and for the wall are proportional tothe corresponding worldsheet areasS0(R3 × S1) = 2piRLα− piR2Lσ. (A.1)The first term is the energy cost of forming a string, where α is the stringtension and 2piRL is its worldsheet area. The second term is the energy gainby forming the hole over keeping the domain wall, in which σ is the walltension and piR2L is its worldsheet volume. We should note that formula98Appendix A. Domain Wall Decay(A.1) replaces the following more familiar expression for the classical action,which was used in previous similar computations [28, 76]S0(R4) = 4piR2α− 4pi3R3σ. (A.2)Minimizing (A.1) with respect to R we find the critical radius Rc and theaction S0,Rc =ασ, S0(R3 × S1) = piα2Lσ, (A.3)which replace the more familiar expressions for the critical radius Rc =2ασand classical action S0(R4) = 16piα33σ2from [28, 76].Therefore, the semiclassical probability of this process is proportional toΓ ∼ exp(−piα2Lσ)(A.4)where σ is the DW tension determined by (5.6), while α is the tension of thevortex line in the limit when the interaction term ∼ ζ due to the monopole’sinteraction in the low energy description (2.36) is neglected and the U(1)symmetry is restored. In this case the vortex line is a global string withlogarithmically divergent tensionα ∼ 2pi 14L2( g2pi)2lnRRcore(A.5)where R ∼ m−1χ is a long-distance cutoff which is determined by the widthof the domain wall, while Rcore ∼ L where the low energy description breaksdown. The vortex tension is dominated by the region outside the core, soour estimates for computing α to the logarithmic accuracy are justified.Furthermore, the critical radius can be estimated asRc =ασ∼ pi2mχln(1mχL), (A.6)which shows that the nucleating hole ∼ Rc is marginally greater than thewall thickness ∼ m−1χ as the logarithmic factor ln( 1mχL) ∼ lnN  1 whenN  1 is the large parameter of the model, see (3.25). Therefore, ourthin-string and thin-wall approximation is marginally justified.As a result of our estimates (A.4), (5.6), (A.5) the final expression forthe decay rate of the domain wall is proportional toΓ ∼ exp(−piα2Lσ)∼ exp(−pi3 ( g4pi)3 ln2( 1mχL )√L3ζ)∼ exp (−γ · N ln2N ) 1, (A.7)99Appendix A. Domain Wall Decaywith γ being some numerical coefficient. The estimate (A.7) supports ourclaim that in the deformed gauge theory model, with a weak coupling regimeenforced and N  1, our treatment of the domain walls as stable objects isjustified.100Appendix BMetastable Vacuum DecayIn this appendix we briefly review the general theory and framework forcalculating metastable vacuum decay rates in Quantum Field Theory. Fora more thorough discussion see [14, 15]. The process for the decay of ametastable vacuum state to the true vacuum state is analogous to a bubblenucleation process in statistical physics. Considering a fluid phase aroundthe vaporisation point, thermal fluctuations will cause bubbles of vapor toform. If the system is heated beyond the vaporisation point, the vaporphase becomes the true ground state for the system. Then, the energygained by the bulk of a bubble transitioning to the vapor phase goes likea volume while the energy cost for forming a surface (basically a domainwall) goes like an area. Thus, there is some critical size such that smallerbubbles represent a net cost in energy and will collapse while larger bubblesrepresent a net gain in energy. Once a bubble forms which is larger thanthe critical size it will grow to consume the entire volume and transitionthe whole of the sample to the vapor phase. To understand the lifetime ofsuch a ’superheated’ liquid state, the important calculation is, therefore, therate of nucleation of critical bubbles per unit time per unit volume (Γ/V ).Similarly, we aim to calculate this decay rate for our system with from themetastable state σ(+) to the ground state σ(−), though through quantumrather than thermal fluctuations.Consider a general system with a ground state field configuration, φ(−),and metastable field configuration, φ(+), with an energy density differencebetween the two given by . Qualitatively the potential for the systemshould be understood as something like Figure B.1. Classically, a systemin the configuration φ(−) is stable, but quantum mechanically the system isrendered unstable through barrier penetration (tunneling).The semiclassical expression for the tunneling rate per unit volume isgiven by [15]ΓV= Ae−SE(φb)/~ [1 +O (~)] (B.1)where the Euclidean action, SE , is the action upon analytically continuing101Appendix B. Metastable Vacuum DecayE ne rg y de ns i tyφ-field configurationQualitative diagram for potentialεφ(-)φ(+)Figure B.1: Qualitative picture for the potential of a general system with aglobal ground state, φ(−), and a higher energy metastable state, φ(+), withan energy splitting between the two given by .to imaginary time and is given bySE =∫d4x[12(∂tφ)2 +12(∇φ)2 + U (φ)]. (B.2)We have explicitly left ~ in (B.1) to emphasise the semiclassical expansion.The action, (B.2), in the exponent of (B.1) is evaluated in the field con-figuration called the “Euclidean bounce” which we have denoted φb. TheEuclidean bounce is a finite action configuration which solves the classicalequations of motion and interpolates, in Euclidean time, from the metastablestate to a configuration “near” the ground state and back. Making referenceto the potential depicted in Figure B.1, continuing to Euclidean time essen-tially describes a system with the sign of the potential flipped. As such, thebounce describes a path starting at the now local maximum φ(+) at t→ −∞rolling down into the valley and up to the classical tuning point near thehigher peak φ(−), then reversing and traveling back to φ(+) at t→ +∞. Inorder for the action to be finite the bounce must also tend to φ(+) as thespacial coordinates go to infinity in any direction.Additionally, we have glossed over one technicality by representing φ asa single dimension while in fact it is not. In principle there is a classicalturning surface, call it Σ, rather than a single point and so there may bemany paths from the peak at φ(+) to the surface Σ. The resolution howeveris straightforward. Each such path contributes as (B.1) and so the pathof minimal action is the dominant path. For details see [5]. Furthermore,102Appendix B. Metastable Vacuum Decaythe minimal action path is spherically symmetric so that the action can bewritten in terms on a single radial dimension,SE =∫ ∞0(2pi2ρ3)dρ[12(dφdρ)2+ U (φ)]. (B.3)Thus, the bounce we should find is the minimal action configuration whichsolves the equation of motion,d2φdρ2+3ρdφdρ= U ′ (φ) , (B.4)subject to the boundary conditions φ → φ(+) as ρ → ∞ and dφ/dρ → 0 ast→ 0.In the limit of small separation energy  the bounce approaches the peakφ(−) more closely and spends longer in the region around the peak, so thatthe bounce configuration resembles a bubble with the interior at φ(−), theexterior at φ(+), and a domain wall surface interpolating between the two. Ifthe bubble is very large, corresponding to very small , then the curvature atthe interpolating surface is small and the surface appears flat. Alternately,simply note that the second term in (B.4) goes like 1/ρ, so if the fields onlychange appreciably around a thin surface at large ρ, the second term can beneglected and the equation of motion reduces further to the much simplerformd2φdρ2= U ′ (φ) . (B.5)Therefore, if the separation energy, , between the two states is small, weneed only solve for the one dimensional soliton interpolating between φ(+)and φ(−) which solves (B.5). This is called the thin-wall approximation, andis the framework in which we work in Chapter 6. In the deformed modeldiscussed in Chapter 2, the separation  ∼ 1/N2, so that the thin-wallapproximation coincides with the large N approximation.For the thin wall approximation the full action reduces toSE ≈ −12pi2R4+ 2pi2R3S1 (B.6)where S1 is the one dimensional action across the domain wall given byS1 =∫dx[12(dφdx)2+ U (φ)]. (B.7)103Appendix B. Metastable Vacuum DecayWhat remains then is to determine the size of the bubble, R. The stipula-tion that the bounce configuration describes a classical path implies that itextremises the action (B.6). Thus, by variation,dSEdR= 0 = −2pi2R3+ 6pi2R2S1, (B.8)which yields R = 3S1/. Notice again the similarity to a bubble nucleationproblem. This extremal action with respect to the bubble size is in facta maximum, and as such the action increases with R for smaller size anddecreases with R for larger. Hence, the bounce configuration which saturatesthe decay rate is essentially a bubble of critical size as discussed when makingthis analogy to bubble nucleation.104Appendix CAsymptotic Vacuum DecayHere we briefly reproduce the largeNc asymptotic calculation for the metastablevacuum decay rate performed in [51]. Starting with the action (2.24),S =1L( g2pi)2 ∫R3d3xN∑n=1[12(∇σn)2 −m2σ cos(σn − σn+1 + θN)], (C.1)with mσ = ζL (2pi/g)2, adding the constant shift and rescaling x → x/mσwe have the action used for our calculations in Chapter 6,S = N∫R3d3xN∑n=1[12(∇σn)2 + 1− cos(σn − σn+1 + θN)], (C.2)whereN = 1mσL( g2pi)2is our semi-classicality parameter. Then, considering the same two vacuumstates (σn = 0 and σn = 2pin/N) with the energy difference  = 2pi(pi−θ)/N ,we introduce the ”centre of gravity” (Σ) and ”distance” (σ) asΣ =1NN∑n=1σn , σ = σN − σ1 , (C.3)in terms of which the vacuums are given byσn = Σ +(n− 1N − 1 −12)σ −−−→N1(nN− 12)σ , (C.4)for σ = 0 and σ = 2pi, with Σ = 0 being equivalent to our previouslydiscussed choice of φ, (6.28). In terms of those redefinitions, we have theactionS → N∫d3x[N24(∇σ)2 + 1− cosσ + σ22N], (C.5)105Appendix C. Asymptotic Vacuum Decaywhich immediately gives (ignoring the last term which is small for large N)the kink solution,σ (x) = 4 arctan[exp(−2√3Nx)], (C.6)and the 1d actionS1 = 4√N3(C.7)as claimed.106

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