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Aspects of chiral anomalies at finite density Metlitski, Max A. 2006

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Aspects of Chiral Anomalies at Finite Density b y • Max A. Metlitski B.Sc, The University of British Columbia, 2003 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in The Faculty of Graduate Studies (Physics) The University Of British Columbia December 13, 2005 © Max A. Metlitski 2005 i i Abstract This thesis is devoted to the study of various aspects of chiral anomaly in an environment with a non-vanishing fermion density. It is comprised of 3 semi-autonomous projects. In the first part, we examine the problem of Witten's superconducting strings at finite fermion chemical potential. We demonstrate how various symmetries of the hamiltonian can be used to exactly compute the fermion electric current in the string background. We show that the current along the string is not sensitive to the profiles of the string fields, and at fixed chemical potential and temperature depends only on the string winding number, the total gauge flux through the vortex and, possibly, the fermion mass at infinity. In the second part, we study a complementary problem of appearance of axial current on magnetic flux tubes in dense matter. We demonstrate the appearance of the axial current by an explicit microscopic calculation. We further show how the results of both parts I and II can be derived by thinking of the fermion chemical potential as a background gauge field, which induces an anomaly of the axial current. In the third part, we use 2-color QCD as a model to study the effects of simultaneous presence of the so-called 9 parameter, chemical potentials for baryon number, ps and for isospin charge, [ij. We pay special attention to 9, HB, HJ dependence of different vacuum condensates, including chiral and diquark condensates, as well as the gluon condensate and the topological susceptibility. We find that two phase transitions of the second order will occur when 9 relaxes from 9 = 2n to 9 = 0, if ii is of order of the pion mass, m„. We demonstrate that the transition to the superfluid phase at 9 = IT occurs at a much lower chemical potential than at 9 — 0. We also show that the strong 9 dependence present near 9 = n in vacuum (Dashen's phenomenon), becomes smoothed out in the superfluid phase. Finally, we note that all results of this study easily generalize to Nc = 3 QCD at finite isospin chemical potential. Contents Abstract ii Contents iii List of Tables v List of Figures vi Acknowledgements vii Co-Authorship Statement viii 1 Introduction 1 1.1 Chiral Anomalies - A Beautiful Nuisance . 1 1.2 Review of Chiral Anomalies 3 1.2.1 Classical Symmetries. Noethef's Theorem 3 1.2.2 Currents in the Quantum World 6 1.2.3 Chiral Anomalies 7 1.2.4 Global Aspects of the Anomaly. Index Theorem 9 2 Superconducting Strings at Finite Chemical Potential and Temperature 15 2.1 Introduction 15 2.2 Motivation . 16 2.3 Currents on Strings 19 2.3.1 The Model 19 2.3.2 The Spectrum 21 2.3.3 Current - Naive Approach 23 2.3.4 Current - Corrections from Polarized Continuum . . . . 25 2.4 Conclusion 29 Contents iv 3 Axia l Current on Magnetic Flux Tubes 31 3.1 Introduction 31 3.2 Fictitious Anomaly Argument 31 3.3 Zero Modes on Flux Tubes 35 3.4 Evading the Fermion Mass Gap 40 3.5 Conclusion 43 4 6- Parameter in 2 Color Q C D at Finite Baryon and Isospin Density 44 4.1 Motivation . . . '. 44 4.2 The Effective Theory at Finite /J,B and /// 46 4.3 The mass term and 6 parameter 48 4.4 Phase Diagram 50 4.4.1 Vacuum Alignment and Spectrum . 50 4.4.2 Chiral Condensates and Densities 53 4.4.3 6 Dependence 59 4.5 Ward Identities 66 4.6 Gluon Condensate 67 4.7 Conclusion. Speculations 71 5 Summary and Outlook 74 A Fermion Current on a Domain Wall 76 B Fermion Number on a Thick Domain Wall with Flux . . . . 80 C Parametrization of the Vacuum Manifold 83 Bibliography 87 V L i s t of T a b l e s 4.1 Chiral condensates in Nc = Nf = 2 Q C D at finite 9 55 4.2 Densities in Nc = Nf = 2 QCD at finite 9 56 vi List of Figures 4.1 9 dependence in Nc = 2, Nf = 2 QCD without chemical potential 61 4.2 Phase diagram of Nc = 2, Nf = 2 QCD with non-degenerate quarks '. . . 62 4.3 9 dependence of the topological susceptibility in iV c = 2, Nf = 2 QCD at various values of chemical potential 64 4.4 Conjectured form, of the Phase diagram of Nc = 2, Nf — 2 QCD with degenerate quarks 65 V l l Acknowledgements I would like to thank A. R. Zhitnitsky - it has been my great pleasure to work with him. I am grateful to M. Forbes, D. T. Son, G. E. Volovik, P. B. Wiegmann, D. Kutasov, P. van Baal, M. Stephanov, M. Stone and V. Liu for useful comments and conversation. I would also like to thank the organizers of the workshop, "QCD and Dense Matter: From Lattices to Stars" at the Institute for Nuclear Theory, Seattle where a part of this work was initiated. Vll l Co-Authorship Statement The work that appears in Chapter 2 is based on Ref. [1]. It has been pre-sented by the author at the Lake Louise Winter Institute (Lake Louise), UBC String Theory Seminar (Vancouver), APS Northwestern Meeting (Victoria), Theory Canada I Conference (Vancouver) and the Workshop on Embedded Defects (Montreal). The work presented in Chapters 3 and 4 is based on Refs. [2, 3], works on which the author collaborated. 1 Chapter 1 Introduction 1.1 C h i r a l A n o m a l i e s - A B e a u t i f u l N u i s a n c e Symmetry considerations are extremely powerful in Quantum Field Theory and form a basis for our understanding of particle physics. Usually, sym-metries of the classical theory remain true in the quantum world. However, under some circumstances quantum effects ruin the classical symmetry, lead-ing to a phenomenon of quantum anomalies. Although a nuisance from a first sight, quantum anomalies have many beautiful properties. On one hand, the appearance of anomalies has to do with the fact that QFT has infinitely many degrees of freedom. To make the theory well de-fined, one needs to introduce an ultra-violet cut-off in the calculations. The presence of such a dimensional cut-off often leads to conformal anomalies -breakdown of scale invariance of the theory. The particular fashion in which the regularization is introduced can also break other symmetries - most no-tably chiral symmetries in theories, which involve fermions. Thus, naively, anomalies have to do with UV properties of the theory. However, as can be explicitly seen from 2d theories, one can often introduce the high-energy regularization in a very implicit way. For instance, the cen-tral charge in the Virassoro algebra can be understood as a consequence of the mere fact that the quantum theory is defined in a Hilbert space. Like-wise, the chiral anomaly in 2d is a consequence of a filled Dirac sea about which the theory is quantized. Thus, although formally derived in the UV, anomalies greatly affect the low-energy physics. Conformal anomaly in QCD leads to an appearance of a scale in the theory: AQCD- Perhaps even more dramatically, the axial anom-aly is the only dynamical element in the Schwinger model - in its absence, mass gap generation and appearance of a chiral condensate would not oc-cur. Similarly, the axial anomaly is believed to give the solution of the U(l) Chapter 1. Introduction 2 problem in QCD, providing the rf boson with a large mass. Chiral anomalies have deep connections to topology. Besides violating local chiral current conservation, the anomaly may result in global non-conservation of axial charge. In path integral formalism this is connected to appearance of fermion zero modes of definite chirality in the gauge field background. The so called index of the Dirac operator (the difference between the numbers of zero modes of opposite chirality) is strictly controlled by the anomaly and depends only on the topology of the background gauge field. The index of the Dirac operator is clearly a manifestly infrared property, but it can conveniently be evaluated in the ultraviolet. One can sometimes embed the topologically non-trivial background field configuration of Euclidean QFT as a time independent object in a Minkowski space of one dimension higher. For instance, in this fashion instantons of Euclidean QED2, become magnetic flux-tubes of QED2+i- The original zero modes of 2d then lead to a degeneracy of the flux-tube ground state, induc-tion of a Chern-Simons term in the effective bosonic action and appearance of fermion number on the flux-tube of QED in 2 + 1 dimensions. Similar observations can be made if the magnetic flux is confined into vortices of Nielsen-Olisen type. Indeed, the use of chiral anomaly type equations is very frequent in the field devoted to evaluation of fermion number on topological defects. What happens if one adds another spatial dimension to the system? The flux-tubes (vortices) now become line-like string defects in 3 + 1 dimensions. The zero modes in the string core become free to move along the string. As these zero modes move, they carry currents along the string. As shown by Witten [6] in certain models such currents can be induced on the string by an application of an electric field along the string. The string then behaves like a superconducting wire. We will return to discuss Witten's superconducting strings in detail in the first part of this thesis. This thesis is concerned with the effects of chiral anomaly in environments with a non-vanishing fermion density. As already stressed, the anomaly itself can be derived in the U V and hence cannot be directly affected by infra-red properties of the theory, such as finite fermion chemical potential and temperature. Nevertheless, as we will see, chiral anomalies can manifest themselves at finite fermion density in rather interesting ways. Chapter 1. Introduction 3 1.2 Review of Chi ra l Anomalies In this section we present a very brief review of chiral anomalies. This review will by no means be exhaustive - we will only touch on results which will be crucial for the discussion in further chapters. 1.2.1 Classical Symmetries. Noether's Theorem. In the world of classical field theory, there is an important result, known as Noether's theorem, which states that for each symmetry of the action, there exists a corresponding conserved current. By a symmetry of the action, S = J dxC{4,d^(x),x) (1.1) we will understand a transformation, <f>{x) -»• 4>\x) = <f>{x) + e8<p{x) (1.2) which leaves the Lagrangian density invariant up to a total derivative, C{(t>'{x),d^{x), x) = C(</>(x), d„cf>(x), x) + edMT" + 0(e 2) (1.3) Here T M is some current. In most relevant examples that we shall consider will be zero. Differentiating, eq. (1.3) with respect to e, one obtains, So far, our field 4> has been an arbitrary function of space-time coordinates. Now, we take (j> to be the classical minimum of the action, which satisfies the Euler-Lagrange (EL) equation of motion, Substituting the E L equation into (1.4), % ( 5 ^ * * - . T ' ) - 0 ( 1 ' 6 ) Chapter 1. Introduction 4 Defining, r - M t f * - T ' ( L 7 ) we see that the current is conserved, = 0 (1.8) A more illuminating understanding of the above proof is provided by the following argument. In most cases of interest, the transformation (1.2) will be the infinitesimal version of a global symmetry, (f> —> cf>', such that S[(f>'} = S[<p]. We implicitly assume that the global symmetry preserves the boundary conditions of the theory. It is now clear why we have allowed for a total derivative d^T^ in the variation (1.3) - such a total derivative does not change the action, provided that the flux of T M through the boundary of our space-time is zero (this can often be ensured by suitable boundary conditions). Let us now extend the global symmetry to a local transformation, whose infinitesimal version is still given by equation (1.2), but with e being a space-time dependent function. We assume that e(x) is chosen so that the boundary conditions are preserved under the local transformation. As the theory is local and the action is preserved under transformations, where e is constant, for non-constant e we should obtain, S[<f/] = Sty] + J dxfd^e + 0(e2) (1.9) A short calculation demonstrates that j M in the variation above is given precisely by (1.7). But the variation of the action should vanish on classical extrema, yielding conservation of the current j11 (1.8). We finally note that given a locally conserved current, one may construct a conserved charge, ,Q= [ dxj°(x) (1.10) where the integral in (1.10) is over the spatial coordinates only. Indeed, jtQ = J dxd0f(x) = - J dxdJix) = - JdSlf(x) = 0 (1.11) provided that there is no out-flux of the current through the spatial boundary (which can again be arranged by suitable boundary conditions). Chapter 1. Introduction 5 Let us discuss some examples of classical currents. The most simple one arises in a theory of a complex scalar field cf), £ = fyW-F(^) (1-12) This theory possesses a global C/(l) symmetry, U(l) : <j> -* eiacp (1.13) The infinitesimal variations are, 54> = i<j>, 8qb* = —id)*. Applying the equa-tion (1.7), we construct a conserved current, jfl = i(dli<f>*<f>-&*</> <F) (1.14) In the following chapters, the most important example of currents will arise from the Dirac Lagrangian for a fermion field ip — (ipL,ipR) in even-dimensional space-time, £ = tpi-y^id^ - ieA^ip - rmptp (1.15) This Lagrangian possesses a global symmetry, U(l) : i\> -» e i Q V (1:16) to which corresponds the conserved vector current, f = T / Y Y (1.17) In case the fermion mass is zero, the left and right components of the field ip are decoupled in (1.15), so that we can perform independent phase rotations of ipi, and IJJR. Thus, we obtain another classical global symmetry, U(1)A : V -> (1.18) with a corresponding conserved axial current, f5 = ijjj^ip (1.19) Chapter 1. Introduction 6 1.2.2 Currents in the Quantum World Let us discuss what happens to our conserved currents once we quantize our theory. We know that quantizing the theory in the Hamiltonian formalism is equivalent to replacing Poisson brackets by quantum commutators. In par-ticular, all the classical equations of motion still hold as operator equations in the quantum world. Thus, we expect that classical currents will remain conserved in the quantum theory. The most straight-forward way to make contact between quantum and classical field theory is through the path integral formalism. Consider the generating functional for our theory of a complex scalar field (1.12), Z[rf, 77*] = J VcpVtp* exp(z J dx (d^P - V{<p*4>) + r)<p* + rf<p)) (1.20) Performing a local version of the transformation (1.13), we obtain, Z[r),T]*} = J V<pV(f>* exp(i J dx {£+j»d^a + ne-ia(p*+ri*eia(p+(f)*4)dlladlda) (1.21) Performing a variation of (1.21) with respect to a, one obtains, V(PV<p*{-d^jli{x)-i(r](x)(p*(x)-T]*{x)<p{x)))exp(i j d x ( £ + # * + r f </>)) = 0 (1.22) The equation (1.22) is a generating functional for an infinite set of Ward Iden-tities associated with conservation of j^. For instance, taking two variational derivatives with respect to 77,77*, (d,f(x)<t>(Xl)(P*(x2)) = (4>(Xl)<p*(x2))(5(x - Xl) - 5(x - x2)) (1.23) Thus, correlation functions involving d^j^ vanish, up to contact terms ap-pearing on the righthand side of eq. (1.23). Actually, such contact terms should not be interpreted as leading to current non-conservation, rather they are related to the order of the <9P operator and T ordering sign, which is implicit in (1.23). In path-integral formalism, the automatically appears to the left of the correlator, so that switching to the Hamiltonian form, dfi(Tf(x)(l>(x1)<p*(x2)) = (Td^fix^ix.Wix,)) + +*(x-xi)(T[j 0(x) )^(xi)]^(x 2))H-<J(x-x 2)(r^(x 1)b- 0(x) )^( a;2)]> (1.24) Chapter 1. Introduction 7 Hence, <9Mj'M itself vanishes as a Heisenberg operator, where's the contact terms in (1.23) come from commutators of j°(x) with insertions of 4>, <j>*. 1.2.3 Chiral Anomalies In the last section we argued that classical currents are generally conserved on the quantum level. The path integral formalism has provided us with a derivation of this fact that exactly mirrors the proof of the classical Noether's theorem. In this section, we will expose a weakness in this argument. Let us consider the theory of fermions coupled to a gauge field that we discussed earlier (1.15). We shall temporarily work in Euclidean space-time, such that the partition function of the theory is, Z = J VipVtp exp(-J dx-piD^ + m)^) (1.25) Here, = (d^ - ieA^^D = D^j^ and {7^, 7„} = 2<5M„. The vector field is an arbitrary Hermitian matrix (we will be separately interested in cases where A^ is a gauge field associated with a U(l) or SU(N) symmetry). The matrix 7 5 is defined as, 7 5 = — ( — i) d / / 27o7i--7d - i and satisfies { 7 ^ , 7 5 } = 0. Let us perform a local chiral rotation (1.18), ip = e-a(-xh5ip\ ip = $ e i a { x ) l b (1.26) The action of the theory transforms as, S$, tp] = J dx{4>'{D^ + me2ia^)iP' + i d ^ ' w ^ ' ) (1.27) However, when performing a change of variables (1.26), we should also com-pute the Jacobian of the transformation J, so that, VipVip = J-2V4>'V-ip' (1.28) where1, J = det{e2iai") = exp(Trloge i Q 7 5 ) (1.29) Working to leading order in a, log J = Trlog(l + ialb) = i T r ( a 7 5 ) (1.30) 1 Here tr denotes matrix trace and Tr denotes a general operator trace. Chapter 1. Introduction 8 The Jacobian appears to vanish as to"(75) = 0. However, we should first regularize the expression (1.30). A convenient (and gauge invariant) way of doing this is, l o g J = lim iTr{al5e-DD'/M2) = lim i [ dx a{x)tr{^{x\e~DD]'M2\x)) M—>oo M—>oo J • (1-31) One may use the heat kernel expansion to represent (1.31) as a series in M 2. We rewrite the operator, Drf = - D l i D l i - E , E = ^FIU,<TIU, (1.32) where [D^Dy] — —ieF^, crM„ — — 1 [ 7 ^ , 7 » / ] - Inserting a complete basis of states, (x\e~DDl/M2\x) = [ ^ c-ir>xc(DuDu+E)/M^px = f c((Du+JPu)2+E)/M* N 1 . J ( 2 7 r ) d J (2^)d = f -P2/M* (2ipuDu+D?+E)/M* ( 1 J (2vr)d V ' Expanding the second exponential in a' series in M ~ 2 and performing the integral over p, we obtain, (x\e-W\x) = (^ )d/V + ^ M - 2 + ( ^ 2 + l [ ^ , J D , ] [ J D , , J D , ] + 1 + ^[ J D M , [ J D M , J B]])M- 4 + 0 ( M - 6 ) ) (1.34) In the limit M —> 0 0 , the terms in brackets in eq. (1-34) survive only up to order M~2 (M~4) in dimension d — 2 (d = 4). Moreover, when (1.34) is traced with 7 5 , the result is finite, with only EM~2 contributing for d = 2 and \E2M~A for d = 4. Thus, T ( i ) t 2 -^e„„(rF„„ (1.35) 47T = 4 e2 T(x) = —e^trF^F^ (1.36) Chapter 1. Introduction 9 The traces in (1.35), (1.36) are over the internal (colour, flavour) indices, and £oi2..d-i = 1- Thus, we see that the Jacobian does not vanish - this is the source of chiral anomaly. Having computed the Jacobian we find, Z = J V$V*Pexp(- J dx(4>(D^ + me2^5W + id^ajl + 21ogj) (1.37) Performing a variation of (1-37) with respect to a, we obtain, V^V^(-id^{x)+2imi>'y5'iJj(x)-2iT(x)) exp(- J dx^^D^+m)^) '= 0 (1.38) The above equation is a generating functional for an infinite set of anomalous Ward Identities. In particular, switching to Hamiltonian formalism, we see that as a Heisenberg operator, satisfies, Wx) = -1T{x) + 2m4>l5TP{x) (1.39) Thus, gauge-invariant regularization of the theory implies that even in the chiral limit m —> 0, the axial current is not conserved. 1.2.4 Global Aspects of the Anomaly. Index Theorem. In the previous section, we saw that local conservation of axial current is lost in the quantum world. A related question is whether the overall axial charge is conserved. Let us consider the variation of the generating functional of a gauge theory (1.37) under a global axial transformation, a(x) = a. The anomalous contribution to the variation of the action is, 5Sanom = -2ia IdxT(x) (1:40) A crucial observation is that T(x) is a total derivative, T(x) — d^K^. Indeed, in case of a U(l) theory in d = 2, one obtains, K» = T ^ W ^ (1-41) For a U(l) theory in d — 4, we have, e2 K» = TT^e^A^ (1.42) 167T^ Chapter 1. Introduction 10 For an SU(N) non-abelian theory in d = 4, the expression is slightly more complicated, e2 2 K» = {tr{A„FXa) + -ie tr(AuAxAa)) (1.43) Thus, for a global variation, the anomalous contribution is a surface term, dxT(x) = J dS^Kp (1.44) It appears that if we work on a compact space, the surface term must vanish. This, however, is not necessarily so. When we place gauge fields on a compact manifold, we may need to resort to the use of several coordinate patches. Notice, that is not gauge invariant and, thus, doesn't -transform as a 3-form under change of coordinate patch. Consider for instance the theory on a sphere Sd (d = 2 or d = 4). Then we can use two patches: M i = Sd — {PN} ~ R d and M2 = Sd — {ps} ~ Md,where and ps are the north and south pole respectively. Now, on the intersection of the two patches, the gauge fields defined separately in each patch, must agree up to a gauge transformation. Such gauge transformations are just maps from Sd — {PN,PS} to the gauge group G. But Sd — {PN,PS} deformation retracts to the equatorial "circle" Sd~~. Thus, we have to classify all the maps from to G, i.e. compute 7Td_i(G). For a U(l) theory in d = 2, we obtain, ni(U(l)) — Z. It turns out that the integer number of the topological class is exactly equal to q in this case, g = [ F = n, neZ (1.45) 2?r J S 2 where we've switched to a notation more appropriate to curved space: F — iF^dx^ A dx". On the other hand, in d = 4, 7r 3(£/(l)) = 0. Thus, a U(l) theory on a sphere in d = 2 contains topologically non-trivial fields, while such fields are absent in d = 4 (however, they appear if we consider the U(l) theory on a 4-torus). The non-abelian SU(N) gauge theories in d = 4 also possess non-trivial configurations, as 7~3(SU(N)) = Z (this is most obvious for N = 2, as SU(2) ~ S 3 ) , which again satisfy, [ tr(FAF)=n, n e Z (1.46) e 8TT2 We note that by stereographic projection, the study of gauge fields on Sd is equivalent to the study of gauge fields on M d , which tend to a pure gauge at Chapter 1. Introduction 11 infinity, VX|=°°-;W (L47) The topological classification is then in terms of the maps of the spherical boundary at \x\ — oo, S^1, to G. We also note that the topological number q is actually invariant under continuous deformations of on any compact manifold. As we have been discussing only anomalies on flat manifolds in the previous sections, we note that for U(l) theory over the torus T2 and for U(l), or SU(N) theory over T 4 , again any integer value of q can be realized. It is known that on any manifold q is actually an integer. Thus, the anomalous contribution (1.40) does not generally vanish and the global axial symmetry of U(l) theory in d = 2 and SU(N) theory in d = 4 is lost. This observation is believed to provide a solution to the so-called [/(l)-problem of QCD - the absence of a 9 t h pseudo-Goldstone rj'. The so-called, Schwinger model (a U(l) gauge theory in d = 2) provides, perhaps, an even more striking manifestation of the loss of axial symmetry. This model exhibits the appearance of a chiral condensate {"p"p) ~£ 0, which would be impossible in the absence of axial anomaly, as continuous symmetries cannot be violated in d — 2. The reverse side of the axial anomaly in the afore mentioned theories is the loss of 9 dependence in gauge theories involving massless fermions. Indeed, as we saw, the parameter q naturally appears in gauge theory, and thus, may enter the gauge field action as, •Se = i0 J dxT(x) (1.48) where the variable 9 is 27r periodic due to the integer nature of q. Pure SU(N) Yang-Mills theory in d = 4 is expected to have a strong ^-dependence, with E(9) ~ N2f(9/N), where E is the vacuum energy of the theory. Yet, if we couple massless fermions into our theory, the 9 dependence disappears. Indeed, the 9 parameter can be completely removed by a global axial rotation (1.26), with a choice of a = 9/{2Nf), where Nf is the number of massless quarks in our theory. This fact is shocking from the point of view of the 1 /N expansion, as the contributions of a light quark loop to correlation functions of T(x) are expected to be suppressed relative to gluon contributions. A way out of this paradox has been suggested by E . Witten[4], who noticed that the apparent contradiction can be resolved by assuming that the mass of the Chapter 1. Introduction 12 rj' meson, is of order m2, ~ ^ . Thus, rj' actually becomes light in the large TV limit. If the quarks are not exactly massless, the 9 parameter can be incorpo-rated into complex valued quark masses, see (1.37). We will extensively use this fact in chapter 4 of this thesis. We may express the fact that the par-tition function of the theory loses its 9 dependence in the limit of massless quarks, by deriving the following Ward Identities for the correlation functions of T(x). The identities are obtained by differentiating (1.37) with respect to a. The first derivative yields, j m = m J d M l ) ) (1.49) where we have averaged both over fermions and gauge-fields. If we assume translational invariance, this implies, (T) = m(fy 5^> (1.50) Taking the second derivative of (1.37) with respect to a, J dx(T(x)T(0))conn = -m(4>ip) + m2 J dx{tpj5tp(x)tp'j5ip(Q))conn (1.51) Identities for n-point correlators of T(x) can be obtained in a similar fashion. We see that all the correlators are proportional to powers of m. So why do we lose the 9 dependence in a theory with massless fermions? The reason is that the partition function vanishes in every sector with q ^ 0 due to the appearance of fermion zero modes. Let us demonstrate this fact explicitly. We have presented eq. (1.50) as already averaged over all gauge configurations. However, it follows from (1.37) that this equation also holds for the fermion correlator computed separately in the background of a field A M 2 . This implies, q — J dxT(x) = —m J dxtr(x\j5j^-^—\x) = . ' (1.52) - - ^ ( T O ) - - ^ 7 T O ) < - ) 2We may also obtain (1.50) by integrating the expectation value of eq. (1.39) over the whole compact manifold. The resulting surface term on the left-handside of (1.39) vanishes, as is a gauge-invariant current. Chapter 1. Introduction 13 where in the last step we have used the fact that {75, D} — 0 struction, D is antihermitian and hence, in a basis where 75 may be written as D — iH where, On a compact space, all the eigenvalues of the hermitian operator H will be discrete. Moreover, as {75, i f} = 0, the non-zero eigenvalues of H will come in pairs A, —A with | — A) = 7s |A). Correspondingly, the non-zero eigenvalues of H2, A 2 , come in pairs, which have opposite sign under 75: i i ^ | A ) , i^ps |A). Thus, only eigenstates of H with A = 0 contribute to the trace (1.53). Yet, all zero modes of H are simultaneously eigenstates of 75. If there are A L zero modes with 75 = 1 and A L zero modes with 75 = —1, we observe, A L = dim(ker(V])), A L = dim(ker(V)) (1.55) The difference, dim(ker(V^)) — dim(ker(D)) is known as the index of V. The above consideration of the spectrum of H implies that the index of an operator on a compact manifold is invariant under smooth variations of the background fields. Indeed, as the operator V is perturbed continuously, the eigenstates of H begin to move. It is possible that at a certain time, some eigenstate of H2 descends onto 0. However, all eigenstates of H2 except the zero modes have partners of opposite chirality. Thus, if an eigenstate with 75 = 1 descends to 0, its partner with 75 = —1 must follow suit. Hence, the index does not change. Now, eq. (1.53) implies, N+-N- = -q . (1.56) Thus, for any gauge-field, such that the topological number q ^ 0, we are assured that there exists at least one zero mode. If fermions are massless, the presence of fermion zero modes, makes the determinant of the Dirac operator vanish, and, thus, non-trivial topological sectors do not contribute to the partition function. We will use the index theorem (1.56) and its close counterparts in chapters 2 and 3 of this thesis. Incidentally, by eq. (1.56), the fact that the index of an operator is invariant under small deformations of the background fields, . Now, by con-•(i-°0-Chapter 1. Introduction 14 implies that the topological number q also has this property. Moreover, we see from (1.56) that q must be an integer. Actually, the effects of non-integer q on an open space have been examined[5]. For the case of a U(l) theory in d = 2, it was found that eq. (1.56) is still satisfied. The fractional part of q in eq. (1.56) is saturated by'a continuum of scattering states, which descend onto A = 0. 15 Chapter 2 Superconducting Strings at Finite Chemical Potential and Temperature 2.1 I n t r o d u c t i o n Ever since Witten's pioneering paper[6], it has been known that cosmic strings can posses fermion zero modes concentrated in the string core. One remarkable feature of this system, is that an application of a constant elec-tric field in the string direction induces an electric current along the string carried by the zero modes. This current will grow linearly with time, while the electric field is turned on and will persist even after the field is turned off. The string, thus, becomes superconducting. It must be noted, however, that the behaviour of the system is known precisely only for induced currents smaller than a certain critical current - once the current exceeds this critical value, the energies of the zero modes become larger than the fermion mass at infinity m, and it becomes possible for the charge carriers to move off the string, quenching the superconductivity. The question of build up of charge and current on the superconducting string in an external electric field has been analyzed extensively in [6, 7, 8, 9, 10, 11]. In this section, we investigate a very different mechanism for inducing a current on the string. Namely, we compute the current on a superconducting string in the presence of a non-zero fermion chemical potential \i and tem-perature T. It is a rather trivial exercise to calculate the current J along the string for \i < m, T <C m, when then the low energy dynamics of the fermion-string system are governed by an effective 1 + 1 dimensional theory of zero modes moving along the string. In this case it is straightforward to show that the current for each fermion species is J — ^p, where e is the fermion charge and n is the winding number of the string. However, we make a much stronger statement: it is possible to calculate exactly the total Chapter 2. Superconducting Strings at Finite Chemical Potential and Temperature 16 electric current in the string direction for any value of the fermion chemical potential \i and temperature T. We shall show that the result is topological in nature, and is independent of the particular profiles of the background string fields. The result will depend crucially on whether the string is local (as considered by Witten) or global (as, for instance, in the case of axion strings). In particular, if the string is local, the naive prediction for J of the effective 1 + 1 dimensional theory, remains valid for any value of /J,, T. The appearance of quantum numbers (particularly of fermion number) on topological defects is a very well-developed subject with known compu-tational methods[12], such as trace identities and adiabatic expansion. At zero chemical potential, the fermion charge induced on defects is usually a topological quantity and, frequently, can be evaluated exactly. However, at arbitrary finite chemical potential, the fermion charge induced is generally not topological[13], and difficult to compute exactly. In view of this, our result is particularly interesting, since we show that quantum numbers such as total current can remain topological and exactly calculable at arbitrary fermion chemical potential. Mathematically, our analysis can be easily gener-alized to a large class of Hamiltonians involving fermions in d + 1 dimensions in the background of a d dimensional defect, which is uniform in the (d+l)'st direction. As far as we know, the problem of computation of electric current on a superconducting string in the background of an arbitrary fermion chemical potential and temperature has not been considered before, although some of the techniques we use have been previously discussed in conjunction to index theorem[14] for string zero modes, and charge induced on the string by an electric field[9, 10]. 2.2 Motivat ion The main physical motivation for analyzing the superconducting string in the background with a non-zero fermion chemical potential originally came from studying the so-called "Condensed Matter Physics of QCD." [15, 16]. It is well-known that at large baryon chemical potential, the ground state of QCD develops a diquark condensate, which spontaneously breaks the SU(3) color symmetry, as well as a number of global chiral symmetries. Such a state of matter may be realized in.our universe in the dense interiors of neutron stars. From a theoretical standpoint the dense quark matter with its large Chapter 2. Superconducting Strings at Finite Chemical Potential and Temperature array of spontaneously broken global symmetries offers a wonderful thought laboratory for the study of topological structures (some of which may be relevant for neutron star physics). In particular, in the case of both two and three light quark flavors, it is believed that at sufficiently large baryon chemical potential, the axial U(1)A symmetry of QCD becomes spontaneously broken.1 Correspondingly, strings associated with spontaneous breaking of the U(1)A symmetry may exist[18]. Recently, it was shown [19] that such strings will carry an electric current given by, -ye^n2_ • a Here, the sum is over all the quark species, ea are the quark electric charges, txa are the quark chemical potentials, n is the winding number of the string and q denotes the axial charge of the chiral condensate (q = 4 in dense QCD). This result was obtained[19] using a technical trick of fictitious chiral anomalies. Here, I review the anomaly based arguments of [19]. Consider a general neutral Goldstone boson rj,2 with the following trans-formation properties under one of the spontaneously broken diagonal axial symmetries of QCD: A - ^ e ^ ^ i P a , TJ^TJ + O (2.2) Here Qa denotes the flavor content of the Goldstone boson and 7  is created out of the vacuum by the current, J ^ ^ Q a A - f l 5 ^ (2-3) a For a Goldstone associated with a pure singlet U(1)A symmetry, Qa — K As is well known, it is useful to represent quark chemical potentials as the zeroth components of a fictitious vector gauge field = (1,0) (this trick will again appear in chapter 4 of this thesis). Then the coupling of quarks to and to the usual electromagnetic gauge field takes the form: £ = ^ ( / i a ^ - eaA^a7^a (2.4) x O f course, the U(1)A symmetry is also explicitly broken by instantons. However, at large chemical potential one can show[17] that all instanton effects are suppressed. 2 T h i s need not be the "77" boson of QCD. Chapter 2. Superconducting Strings at Finite Chemical Potential and Temperature where \ia and ea are quark chemical potentials and electromagnetic charges respectively. The anomaly equation for the current j1* in the background of fields and A M takes the form: d„f = e^iC^A F^FXa + CvAv V^FXa + C „ v v V^VXa) (2.5) where the field tensors FX<J, VXa are defined as, FXa = dxAa — daAx, VXa = dxVa — daVx, and the coefficients, a a a The anomalous current non-conservation (2.5) must be reproduced in the effective Lagrangian for the neutral Goldstone boson n. Thus, as was shown in[19], the effective Lagrangian for 7  pics up the following anomalous term describing its interaction with the fields A M and V^: L V = L ° + 2diJj]^°(CT]AAA„FXa + CAVVVFX,, + C^vV^) (2.7) Here L° is the standard, non-anomalous part of the effective Lagrangian for 77, which transforms as L° —> L° — j^d^O under (2.2). We now restore the fictitious field to its true value V M = (1,0). Then the last term in (2.7) vanishes3, and we are left with, Lr, = L ° v - CvAA 7  ^°F^FXa + ±CvAV T? • B (2.8) where B is the magnetic field. The first anomalous term in eq. (2.8) describes the usual decay of a Goldstone boson to two photons, and is absent on the classical level, if there is no background electric field present. We now concentrate on the second anomalous term in (2.8), which does not occur in vacuum (at /1 = 0): Lr, = L l + 4CvAvVri • B (2.9) Retaining only the anomalous contribution and integrating by parts, • = 4 C V u / ( V x V77) • i (2.10) Thus, the anomalous contribution (2.10) vanishes in a topologically trivial background. However, in the presence of a U(1)A string, the field 7  varies 3However, as shown in [19] this term can become important if the quark matter is rotating and/or superfluid vortices appear. Chapter 2. Superconducting Strings at Finite Chemical Potential and Temperature 19 by 2irn as one goes in a circle around the core of the string, and one has to replace V x Vry —> 2im52(xA-)z. Hence, the action for the interaction of the axial string with the electromagnetic field takes the form, S = J Jdtjdl-A (2.11) where the integral is over the string core, and J is given by eq. (2.1). Using the field A as a probe of electric current, one observes that the current J is flowing along the string. Notice that the above derivation is completely insensitive to the micro-scopic details of QCD and depends solely on the pattern of spontaneous symmetry breaking and anomaly structure of the theory. Thus, an applica-tion of this method to a model of Witten's superconducting strings would yield the same result, J = ^ per each fermion species (the charge of the chiral condensate q = 2 in the model that we will consider). As shown below, this result is, indeed, microscopically confirmed in the studied model in the regime when p < m. This is precisely the regime where the Goldstone modes are the only relevant degrees of freedom in the bulk and the analysis of [19] is applicable. Outside of this regime, the fictitious axial anomaly approach based on the effective chiral Lagrangian treatment is not justified. Never-theless, in the model of Witten's strings, in some cases this result remains correct for all p, while in other cases it receives corrections of order 1. 2.3 Currents on Strings 2.3.1 The Model Consider the following model of a Dirac fermion ip coupled to a string: C = # y " ( 0 „ - ieA, - | ^ 7 5 ) ^ - + ^ - ^ ^ (2.12) Here A^ and are gauge fields and <f> is a complex scalar field. The model has the following classical gauge symmetries: U(l) : i> - e^V, Aft —> An + c > , <j> -> <j> (2.13) U(l) : i> -> e^(sh5/V, R„ -* + d^6, <f> -+ e^xU (2.14) Chapter 2. Superconducting Strings at Finite Chemical Potential and Temperature 20 This model is exactly equivalent to Witten's model of superconducting cos-mic strings with a particular choice of gauge charges4. By convention, we associate the vector field with electromagnetism. We note that the above model suffers from gauge anomalies, which can be removed, for example, by adding another fermion ip to the model with the opposite R charge q = —q and the electric charge e, such that e2 = e2. The Lagrangian for the ip fermion is then: t = tir(d» - ieA, + | i? , 7 5 )^ - 4(0^^ + < / > * ^ ^ (2.15) Notice that ip now couples to (p* rather than (p. We could also consider the situation when the U(l) symmetry is global, such that the gauge field R^ is absent, the remaining U(l) gauge symmetry of the Lagrangian (2.12) is by itself anomaly free, and the addition of the ip field is unnecessary. In our calculations, we will recover this case by taking q = 0..-We assume that the U(l) symmetry is spontaneously broken, the <p field acquires a non-zero expectation value and strings of the cp field are possible. We wish to consider the fermion ip in the background of an infinitely long static string uniform in the z direction. The string is characterized by a non-zero winding number n of the scalar field: n = f^_^M (216) J 2™ \<p\2 [ Z A b ) where the integral is over a contour in the xy plane at infinity, and the absolute value of the scalar field \<p\ tends to some constant cp0 as r —> 00 in the xy plane. If the U(l) symmetry is local, then in most models (such as the Abelian Higgs model), the condition that the string energy is finite, implies that D^tp = {d/j, — iqRfj,)(p —> 0 fast enough as r —> co in the xy plane. This in turn implies the quantization of the string flux: $ = -T- [ d2xeabRab = n (2.17) 4 W e could have easily considered the completely general version of Witten's model, however, to simplify the algebra slightly we concentrate on the above choice of gauge charges, which makes the field couple to the vector current and the field couple to the axial current. Chapter 2. Superconducting Strings at Finite Chemical Potential and Temperature 21 From here on a, b = 1,2 and i?M„ is the usual field strength tensor. It must be noted that the condition (2.17) is not present in the case of global strings, so we will throughout our calculations keep the flux $ arbitrary and at the end set $ = n for local strings and <J> = 0 for global strings. Our objective is to calculate the expectation value of the electromagnetic fermion current in the string direction, J3 — e J d2x$^ip) (2.18) at finite fermion chemical potential /J. and temperature T. Note that if the U(l) symmetry is local, there is an additional contribution to the electro-magnetic current from the ijj fermions. This, however, can be obtained from the result for the ip fermions by setting e —> e, q —> q = —q, qb —* (j>*, which translates into <& —> — <&, n —> —n. 2.3.2 The Spectrum Let's start by analyzing the spectrum of our fermions in the string back-ground. The one-particle Hamiltonian is: H = -ia-idt - ieAi - | i ? t 7 5 ) + ^ V ^ T ^ + « ^ ^ ^ ) (2-19) where a% = 7 ° 7 * and i = 1,2,3. For a static background string uniform in the third direction, Ai = 0, R3 = 0, and hence, H = -idsa^ + H1 (2.20) EL = -iaa(da-lJRal5) + hj°(^^^ + ^]~^) (2.21) Since all the fields are assumed uniform in the third direction, we can choose —id^ip = kip and work at fixed k.5 In each k sector, Hk = ko? + H1 (2.22) and the operator Hk now acts solely in the transverse xy plane. At this point, we make our choice of the 7 matrices to be: - • - ( V O - r ) - ^ = ( ! s ) ^ = ( o -"0 (2.23) 5We take the third direction z to be compact of length L so that the eigenvalues k are discrete. As usual, we will take L —* 00 at the end of the calculation. Chapter 2. Superconducting Strings at Finite Chemical Potential and Temperature The operator HL then takes the form: " L = U o) <2-24> where, V = aa a + | i t ! a e a V + / i ( i ^ 0 + i ^ ^ ) (2.25) Let's discuss the properties of the operator H1. Since \4>\ —> (po as r —* oo, the continuum spectrum of i f - 1 starts at eigenvalues |A|' — rn — \h\cpo. H1-may also have bound states. We let TO;, be the smallest positive eigenvalue of H-1. By dimensional reasons, ~ rn. Now, observe, {a3,H±} = 0 (2.26) Thus, a3 maps a properly normalized eigenstate |A) of Hx with eigenvalue A into a properly normalized eigenstate of H1- with- eigenvalue —A. More-over, since a3 maps zero modes of H1- into zero modes of H1-, all the zero-modes of Hx can be classified by their eigenvalue under a3. Writing, A(x) = (u(x), v(x)), we note that the zero modes of H1- with a3 = 1 satisfy v = 0, T>^u = 0, while the zero modes of H1- with a3 = — 1 satisfy it = 0, P i ; = 0. So letting /V+'be the number of a 3 = L zero modes, and the number of a 3 = — 1 zero modes, we have, N = N+-N_ = dim(ker(V^)) - dim{ker(V)) = Index(H^) • (2.27) Hence, N is the index of an elliptic operator, which is usually a strongly topological quantity. ./V has been first computed explicitly for a particular background string configuration in [20] to be: N = n (2.28) This result was later generalized [9],[14] to arbitrary background string fields. We now return to the operator Hk- Observe, [Hk,Hx2] = 0. So, we can obtain the spectrum of Hk from the spectrum of H1 in the following way. Let, H1\(x) = \\{x) (2.29) First, suppose, A > 0. Then, the state, ip(x) = C i A(x) + c 2 a 3 A(x) (2.30) Chapter 2. Superconducting Strings at Finite Chemical Potential and Temperature is going to be an eigenstate of Hk with eigenvalue E, provided that, The eigenvalues of the above equation are, E = ± V A 2 + k2 (2.32) and the eigenvectors, c i \ 1 ( ±sgn{k){{\2 + k2y±X)' , ( 2 3 3 ) °2 J ± (2{\2 + k2)"f V ( ( A 2 + A 2 ) ' T A ) i Thus, each eigenstate of H1 with positive eigenvalue, generates one positive energy and one negative energy eigenstate of Hk- However, this correspon-dence has to be taken with a grain of salt, since most eigenstates of H1- are continuum states, and the "1 to 2" map discussed above between eigenstates of H1- and eigenstates of Hk need not preserve the density of states. The zero modes of Hx are also simultaneously eigenstates of Hk. These have the dispersion, E = ka3 (2.34) So the zero modes of H1 become chiral fermions moving up or down the string depending on the sign of their eigenvalue under a3. 2.3.3 Current - Naive Approach We now proceed to the computation of electric current at finite /J, T. This is given by: J3 — e J d2x(xpj3i)) = e J d2xtr{x\a3n(H)\x) (2.35) where, - e^T,(Jl+l <2-36> is the usual Fermi-Dirac distribution. Summing over each momentum sector k, we obtain, J3 = e^J2 J d2x tr{x\a3n(Hk)\x) = e^ Tr{a3n{Hk)) (2.37) Chapter 2. Superconducting Strings at Finite Chemical Potential and Temperature 24 Using the correspondence between spectra of Hk and H1, we may schemat-ically write the operator trace (2.37) as: j 3 = ezH E^ I^ I^H^) k E(Hk) = e\Y<( E (^\kA^x,k,s)n(Es(X,k))+ ] T (X\a3\\)n(E(\,k)) k \A ( / / - J - ) > 0 , s = ± A(if->-)=0 Here E(Hk) denote eigenstates of Hk, A ( i f ± ) denote eigenstates of H1, and ip\,k,± denote eigenstates of Hk generated by an eigenstate |A) of H-1, with energies - E ^ A , k) = ± \ / A 2 + k2. Again, we stress that the above representa-tion would have been absolutely correct if all the states contributing to the operator trace were discreet, and normalizable (for instance if T = 0 and fj. < m). In our case, this is not generally so, but we choose for now to ignore this problem, in order to illustrate the general idea behind the computation. We will later return to take the continuum states into consideration more carefully. For the moment suppose, T = 0, 0 < p, < mj. Then n(E) = 6(E)6(/j,—E). Hence, only states generated by zero modes contribute to the sum in (2.38), as all the other states have energies \E\ = \/X2 + k2 > |A| > > p. The zero modes are eigenstates of a 3 , and thus, satisfy, E — a3k and (A|cv3|A) = a 3 . Thus, J 3 = ej ^{N+6(fi - k)0(k) - N_8(u. + k)0{-k)) = k where we've used the fact that index N is equal to the winding number of the vortex n. Now, let's relax our assumption and work at arbitrary T, p. We first need to evaluate the matrix element {ip\,k,s\®3\'*P\,k,s) (in what follows we suppress the indices A, k, s). Using eq.(2.30) and (a 3 ) 2 = 1,-we see (tp\a3\ip) = (|ci|2 + |c 2 | 2)(A|a: 3 |A) + (c*c2 + c*2ci). Recalling a 3 |A) = | - A), we obtain, (2.38) Chapter 2. Superconducting Strings at Finite Chemical Potential and Temperature (ip\a3\?p) — c\ci + c*2C\ — -|, where we've used eq. (2.33). Hence, j 3 = 4 z M E ^4^n(^.(A,A:)) + JV+n(fc)-JV_n(-fc) (2.40) Now, observe that Es(X,k) = Es(X,—k) for A > 0. Hence, the first sum in the brackets in eq. (2.40) is odd in k, and, thus, the contribution to J3 from non-zero modes of Hx cancels out exactly, leading to: J3 = eN-Y^n(k) = e n / ^n(k) = enn0{p,T) (2.41) k Here no(p,T) is the number density of a free massless chiral fermion in 1 dimension, at finite chemical potential \i and temperature T. It is a peculiar fact that no(p,T) is temperature independent and equals ^ , so that, J*=e-f ' (2.42) 2.3.4 Current - Corrections from Polarized Continuum Although, the result (2.42) is very attractive, it is actually generally incor-rect. We know that this result is exact for T = 0, p < m, when J 3 receives contributions only from normalizable eigenstates of Hk- We will now show, that the presence of long range vortex fields polarizes the continuum eigen-states of Hk in a way, which might significantly modify the result (2.42) for Ii > m. Let's return to the trace (2.37). We can rewrite this expression in terms of spectral current density as: J 3 = J dEn(E)j3(E) (2.43) j3(E) = e^Tr(a36(Hk-E)) (2.44) k We use the following representation of the delta function, 6{x) = ± lim e_ 0+ - J=ii)> t o rewrite, ' j3(E) = ^- lim e~YTr(a3(—^-— - )) (2.45) v ' 27re-o+ V Hk + z+ Hk + z-JJ v ; Chapter 2. Superconducting Strings at Finite Chemical Potential and Temperature where z+ = —E + ie, z = —E — ie. From here on, the limit e —> 0 + is implied. Simplifying (2.45), l r ^ i e / ka3 + Hx - z+ ka3 + H1- - z L ^ 2 T T V Hx + k2 - (z+)2 H±2 + k2 - {z-)2' (2.46) where we've used Hi = H1 + k2. The terms in (2.46), which are odd in k cancel out, and we obtain, i , ^ 1 „ / Hx — z+ HL — z~ A . j (E) = - > —Tr a3( 5 5 ) (2.47) v ; L ^ 2 7 r V KH±2 + k2 ~{z+)2 H±2 + k2 - {z-)2') K ' Now, {a3,HL} = 0. Hence, for any function / , Tr{azHLf(H±2)) = -Tr( t f - L / ( f / ' - L 2 )a 3 ) = -Tr(a3H±f(H±2)) = 0, and, f(E) = ±Y ^ T r (a3( 2 ~ Z + )) (2.48) J K ' L ^ 2 n \ KH±2 + k2 - (z+)2 H±2+ k2 - (z-)2JJ V ' We now introduce the function g, (2.49) This function is very well known [12],[14] as it satisfies, N = Index{HL) = Yxm^g(M2) (2.50) More generally, g(M2) is related to the spectral asymmetry <Jk(E), of the Hamiltonian ii/fc[12], and hence to its ry-invariant as, o-k(E) = ±-k{G(k2-(z+)2)-G(k2-(z-)2)) (2.51) / ' O O % = 2 / dEak{E) (2.52) Jo G(z) = ^ (2.53) Chapter 2. Superconducting Strings at Finite Chemical Potential and Temperature 27 Here, g(z) is understood as the analytic continuation of g from R to C. From eq. (2.48), we can express j3(E) in terms of G as, ?w = L E ^ (z+G(k2 - (*+)2) - z ~ G ^ - (^ )2)) = i E e f k k (2.54) Following the technique of trace identities described in detail by [12],[14], one can explicitly calculate g{M2) to be: M2 ^ ) = n - ( n - * ) ^ T 7 ? 5 (2.55) Hence, the index N = \imM2_tQ g(M2) = n, in agreement with previous calculations [9],[14],[20]. Continuing g to the complex plane, we obtain, c<*>-1 -<-•> j h < 2 - 5 6 ) Hence, generically, G has a pole at z = 0, and a pole at z — — m2, i.e. at the continuum threshold. Notice, however, that the pole at z = m2 disappears when n — We can now substitute the result (2.56) into (2.51) and take the limit e —> 0 + to calculate the spectral asymmetry, ak(E) = k sgn(E)(nS(E2 - k2) - (n - $)6{E2 - k2 - m2)) (2.57) which yields the 77-invariant, k r,k = n sgn(k) - (n - $ ) - — = = (2.58) V /C ~r" Tfl We note that eq. (2.58) is in agreement with previous calculation of the 77-invariant [9]. Returning to the evaluation of current, we substitute the result (2.57) into eq. (2.54) to obtain, j\E) = ^nJ2(S(E-k) + 6(E + k))-k " i f ^ ~ E ^ -Vk2 + m2) + 5{E + Vk2 + m 2 )) k (2.59) Chapter 2. Superconducting Strings at Finite Chemical Potential and Temperature 28 and the total current in the string direction (2.43) becomes, J 3 = e J ^ (nn(k) - ^ ( n ( v / F + ^ ) + n ( - V k 2 + m2))^ (2.60) This can be conveniently rewritten as, J 3 = e(nn 0(/j,T) - ^ ^ n m ( M , T ) ) (2.61) where no( jU, T) — ^  is the familiar number density of one-dimensional chiral massless fermions, and, — {n{Vk2 + m2) + n ( -v / / c 2 + m 2)) (2.62) is the number density of one-dimensional 2-component (Dirac) fermions of mass m. Several comments are in order here. First of all, we see from eq. (2.61) that the naive result (2.41) is generally modified by a contribution from modes located at the continuum threshold. Observe, that for / i = 0, the current J 3 vanishes for all temperatures. At non-zero chemical potential, two cases are of particular interest. The first case is that of a local string, satisfying the finite energy condition, D(p —> 0 faster than 1/r, which implies <3? = n. In this case, the contribution from continuum modes vanishes, and we recover our initial result (2.42), which is due solely to the zero modes, 3 _ ejm 2?r v ; This "coincidence" can be explained as follows. If Dcp —> 0 fast enough, the fields in the problem are, essentially, short range, and hence we can easily put the system in a box, making the spectrum discrete, so that the argument in section II C is correct. Let's briefly discuss what happens when we add the second fermion ip to the problem. Recall, we used this fermion to cancel gauge anomalies of our model. As noted in section 2.3.1, the contribution of ip to J 3 can be obtained by taking e —» e, fi —> /}, n —> —n, $ —> — <£. In particular, the continuum modes at threshold again cancel out, and J 3 = — ^ p. In particular, if the chemical potentials of tp and ip fermions are the same, we can obtain a non-vanishing total electromagnetic current along the string, by letting 6 e — — e, This choice certainly respects the anomaly cancellation condition e Chapter 2. Superconducting Strings at Finite Chemical Potential and Temperature 29 so that, 4M = ^ (2-64) The second practically interesting case is that of a global string. This case can be recovered by taking <& —» 0. Then, J 3 = en(n0{p, T) - \nm(p, T)) (2.65) In this case, the field 4> is long range, and there is a significant modification of the result (2.41). Note that nm(p,T) is no longer temperature independent, so for simplicity, we choose to work at T = 0, p > 0. Then, n(E) = 6{E)6{ii - E) and, J 3 = %L ffj, - efr _ m ) ^ 2 _ m 2 ) (2.66) 27T Thus, for /J, < m, the current is governed by our original result (2.42), while for / i > m, we also get a counterflow current from the states at continuum threshold. Thus, J3{p) has a cusp at p. = rn, and for p 3> m falls off to 0 as emn m 4TV U We would like to note that the computation of the current presented in this section has been somewhat formal. In order to confirm the validity of the present computation, we have also analyzed a 2 + 1 dimensional prob-lem of current on a domain wall. The advantage of this problem is that it also allows for an independent evaluation of current by the use of scatter-ing techniques. In appendix A , we show that the scattering techniques and the trace-identities method developed in the present section yield the same result, when applied to the domain wall. Note that the problem of current computation on the domain wall is not purely academic and may be relevant for px + ipy superconductors[21]. 2.4 Conclusion In this chapter, we have found an exact expression for the electric current on superconducting strings as a function of fermion chemical potential and temperature. We've analyzed the case of both local and global strings, and our analysis has not been limited to a low energy theory of zero modes in the string core. Our ability to obtain such an exact result has been due to Chapter 2. Superconducting Strings at Finite Chemical Potential and Temperature a cancelation (or partial cancelation) of contributions of all, but the zero fermion modes to the current. For local strings, we've seen that for all values of T and / i , the current is due to zero modes in the string core. On the other hand, for global strings, the current receives contributions both from the zero modes and from certain states at continuum threshold. The latter contribution tends to cancels out the contribution from the zero modes as the fermion chemical potential becomes much larger than the fermion mass m. The results of this chapter confirm the validity of the "fictitious chiral anomaly" approach, which was originally developed in the context of dense quark matter [19]. It is interesting whether an exact microscopic calculation similar to the one presented in this section can be performed to directly confirm the results of [19] in the dense matter setting. We would like to note that the study of persistent topological currents and spin currents in conjunction with problems, such as, for example, Quan-tum Hall Effect [22] and Spin-Hall Effect [23], has over the past years become an active subject of research in condensed matter physics. It would be inter-esting to investigate the relation of the phenomenon discussed in this paper to problems in condensed matter systems. For instance, persistent supercur-rents, are known to appear on vortices in superfluid 3 He — A and, somewhat similarly to currents considered in this paper, are due to chiral anomalies[24]. 31 Chapter 3 A x i a l Current on Magnetic F lux Tubes 3.1 Introduction This chapter is devoted to a study of a phenomenon complementary to that discussed in chapter 2, namely, the appearance of axial current on'magnetic flux tubes in dense matter. Similarly to the existence of vector current on axial vortices, the phenomenon examined in this chapter can be derived, as shown below, from the "fictitious axial anomaly" trick, originally developed in [19]. The appearance of a macroscopic current in the ground state of the system seems to directly contradict Baym's theorem[25]. We show that, in fact, there is no true contradiction once the current is properly defined. We also gain a physical understanding of the fictitious anomaly: it represents the fact that the fermion number is not invariant under large chiral transformations in a system pierced by a finite magnetic flux. In this chapter we microscopically compute the axial current on flux tubes in a dense environment in two simple examples. The first example is just QED3+i and, thus, does not exhibit spontaneous axial symmetry breaking. The second example is a cr-model where the axial symmetry is broken. Both examples are shown to support an axial current predicted by the fictitious anomaly technique. However, the nature in which the axial current is carried, is very different in the two models. 3.2 Fictitious Anomaly Argument Let's return to the discussion of the anomalous effective Lagrangian (3.1). (3.1) Chapter 3. Axial Current on Magnetic Flux Tubes 32 5S One effect of this Lagrangian has been investigated in detail in the previous chapter. Here we discuss a different consequence of this term. Let's vary the action obtained from Lagrangian (3.1), with respect to 7  —> 7  + 6, to derive the classical equations of motion. By construction, L° —> L° — j^d^O, hence, J dAx{-fd^e + ACnAVBidie)= (3.2) = / dixdlxf9 + I dt I dSii-f + 4CflAvBi)6 (3.3) J J JdR Here the surface integral is over the boundary of the region R where our dense matter is realized. So, as V • B — 0, the anomalous term does not contribute to the equation of motion = 0. However, if we do not restrict 9 to vanish on the boundary, we also obtain a boundary condition, j • dS = ACvAVB • dS (3.4) Now, in the steady state situation, there is no build up of axial charge, and we have V • j = 0. Hence for any cross-section S of the region R let Sb be the part of dR such that dS — dSb- Then, f dS-j= f dS-j=4C7lAv f dS-B = ACriAv I dS-B = J2^^ (3.5) where $ is the total magnetic flux through the cross-section S. So we see that the anomalous term in eq. (3.1) implies the existence of an axial current flowing through the dense matter which is proportional to the magnetic flux. At this point we make the following important remark regarding the for-mula (3.5): the final expression for the current does not depend on the specific properties of the pseudo-Goldstone boson 77, such as its coupling constant /,,. This is not due to our choice of units, and this is not a typo, so it leads us to weaken our assumption of spontaneous chiral symmetry breaking and existence of the 7  Goldstone. Let us formalize the above argument in the following way. We consider our microscopic theory, coupled to electromagnetism and chemical potentials, £ = . ^ ( / i a ^ - e a ^ ) 7 / ; 7 ^ (3.6) a We take A^ to be a background static magnetic field. Consider the generating functional of the microscopic theory in the background of an axial-like gauge Chapter 3. Axial Current on Magnetic Flux Tubes 33 field Rfj,, Z\R^\ = J D[fields}exp{iS + i j d^xfR^) = exp{iW {R^}) (3.7) Note that in section 3.3, we will consider a case where the microscopic the-ory involves fields other than fermions charged under R^. In that case the coupling to i? M in (3.7) will be modified, this, however, will not alter our con-clusions below. We take R^ to be a constant vector potential R^{x) = (0, R). Observe that eq. (3.2) is the correct form of the variation of the action of the theory under an axial transformation, irrespective of whether the symmetry is spontaneously broken or not. Note that the field R^ does not contribute to the anomaly, as it is chosen to be constant. Thus, W[R„ - 3^9} = W[Rp} - J d4x 4 C^vB-dJ (3.8) Clearly, if the variation parameter 9 satisfies periodic boundary conditions, the anomalous piece on the right-handside of (3.8) vanishes. On the other hand, we are not aware of existing computations of anomaly in the case when 9 is not periodic. We shall assume that the anomalous contribution in this case is'still given by expression (3.8). Thus, let us choose the magnetic field to be uniform in the 3d direction. We take 9 = qx3, so that, ^W[Ra-q] = ^W[Ra]-4CVAv^qL (3.9) where T is the time interval in the path integral and L is the length of our system in the third direction. Now, 1 dW T dR3 R=0 = L f {j3{x))d2x (3.10) If the parameter q is arbitrarily small, by differentiating (3.9) with respect to q we arrive at the desired result (3.5). However, if our theory is quantized in a box of length L , we are allowed to perform only those transformations (2.2), which preserve the boundary conditions of all the fields in the path integral. For instance, if Qa are relatively prime integers, we must have q = 27rn/L, where n & Z. Perhaps, this periodicity condition can be lifted in some phys-ical situations (e.g. when our system is embedded in a theory defined in a Chapter 3. Axial Current on Magnetic Flux Tubes 34 much bigger, box). However, if we adhere to the periodicity conditions, then the corrections associated with the fact that q is not a continuous variable, generally preclude us from obtaining the result (3.5). For instance, expand-ing (3.9) in a series about q = 0, we see that the q2 term will generally be multiplied by a coefficient of order of the volume of the system V = LA. If <P is of order of several elementary flux quanta, and q ~ 1/L, then the anomalous term on the right hand side of (3.9) is thermodynamically neg-ligible compared to the q2 term on the left hand side. Yet, if the magnetic flux through the system is of a thermodynamic magnitude, for instance if B is uniform, or, at least, lim /i_» 0 0 <&/A = (B) ^ 0, then the anomalous term is significant (it is of order V) and we recover the relation (3.5). Below, we shall perform microscopic calculations of the current on flux tubes in two examples. The first example will exhibit a current for any number of flux-quanta through the system. In the second example, only in the limit of microscopic flux we are able to show evidence that (3.5) is satisfied. Incidentally, the above calculation of the axial current is very similar to the proof of Baym's theorem[25], which states that all the conserved currents vanish in the ground state of the system. The only difference between our discussion and that of Baym is the appearance of an anomalous piece in eq. (3.8). The presence of the anomalous piece can be understood as follows. In the Hamiltonian formalism, the grand-canonical potential operator is = H — X^ aMa-^ a- Under a symmetry transformation U (2.2), UHU^ = H + / d3xf did + 0{\di9y2). On the other hand, Na transforms as, Thus, Na is invariant under topologically trivial chiral transformations, but transforms non-trivially when 9 has a finite winding. Our "fictitious axial anomaly" is a simple reflection of this fact. The factor of chemical potential in the anomaly simply keeps track of the fermion number. Observe that we could define a new current j1 = J1—4CVAVB1. Both j1 and f are conserved. Nevertheless, the grand-canonical potential fi transforms as UtlW = fi + / d3xjidi9 + 0((di9)2). Thus, it is the current f to which Baym's theorem is applicable and the flux of this current will be 0. So there is no true discrepancy between our result and Baym's theorem. Chapter 3. Axial Current on Magnetic Flux Tubes 35 3.3 Zero Modes on F lux Tubes We will show in this section that the appearance of current on magnetic flux tubes at finite chemical potential derived in the previous section using anomaly arguments can be understood very simply microscopically within the following model. We consider the following Lagrangian, £ = ipi^ + ieA^-p - rmp-p + / J ^ / V . (3.12) which describes the interactions of a single light quark ip of mass m with a background electromagnetic field A^, at finite baryon chemical potential \i. Hence, the discussion in this section actually applies to any QED-like system at finite chemical potential. We are interested in the case of magnetic flux tubes, i.e. A^ is static and the magnetic field B = V x A — B(x, y)z is uniform in the third direction z. Our goal is to compute the total axial current J | = J d 2x(V'7 37 5V') along the flux tube. The Dirac Hamiltonian is, H = -i(di + i e A i h V + m 7 ° (3.13) and the Dirac equation becomes, -HRipL + mipR = EipL (3.14) m-pL + HRiljR = ETJJR (3.15) where we use the conventions of Peskin and Schroeder and, HR = (-idi + eAi)ai (3.16) So, I/JL = ±(E- HR)ifjR, where • (H2R + m2)1>R = E2^R (3.17) Hence every eigenstate ipR of HR with eigenvalue e generates two solutions of the Dirac equation with energies E — ±\Je2 + m2 and, *±=(1L) = ( 4 ( - 2 + e 2 ) ) - ( ^ + 2? ffR ) (3-18) \4>RJ± V ((m 2 + e 2 ) 2 ±e)>il>R J Now we concentrate on the right sector HRipR — eipR. Due to invariance with respect to translation in z direction, we go to momentum eigenstates Chapter 3. Axial Current on Magnetic Flux Tubes 36 —id^ipR = p^R (we take the third direction to be periodic of length L, and take the limit L —> co at the end of the calculation). In each momentum sector, the operator HR takes form, HR = p3a3 + HX (3.19) HX = (-ida + eAa)aa, a = 1,2. (3.20) We note that {cr3, HX} = 0. Hence, if |A) is a properly normalized eigen-state of H1 with eigenvalue A then cr31 A) is a properly normalized eigenstate of H 1 - with eigenvalue —A. So, all eigenstates of H 1 - with non-zero eigen-values are of form |A), | — A) = cr31A), where A > 0. Also, a3 maps zero eigenstates of H1 to zero eigenstates of H 1 - and hence we can classify all zero modes of H1 by their eigenvalue under cr3. The eigenstates of HR can now be expressed in terms of eigenstates of H - 1 . Clearly, [HR, H3-2} = 0, so HR only mixes states |A), | - A). For A > 0, we write, ^ = c i | A ) + c 2 a 3 |A) (3.21) where C i , c 2 satisfy: A P 3 \ ( C i \ = ^ / C l P3 - A M c 2 J I c 2 (3.22) Hence e = ± \ J \ 2 + p2 and, U J ± - ( 4 ( A + f t ) ) I ((A 2 + P ! ) ^ A ) ^ J ( 3 - 2 3 ) So each eigenstate of H 1 - with an eigenvalue A > 0 generates two eigenstates Of HR. 1 The zero modes of H1 are simultaneously eigenstates of HR with eigen-value, e = pza3 (3.24) xThere are known examples[12], such as fermion number appearing on domain walls, when this is not strictly speaking true. Indeed, some of the energy levels of HR are contin-uous rather than discreet and the correspondence discussed above between the eigenstates of H 1 - and HR need not preserve the density of states. However, for the particular Hamil-tonian H R , it can be shown that if B{x) —> 0 as x —* oo sufficiently fast, this problem does not arise. Chapter 3.. Axial Current on Magnetic Flux Tubes 37 Hence, when the mass m —» 0, zero modes of Hx become gapless modes of H capable of travelling up or down the flux tube depending on the sign of <T3 and on the chirality. We will shortly see, that at finite chemical potential, precisely these modes carry an axial current along the flux tube. The following quantity will be of particular importance to us: N — A L — A L , where A L and A L are the numbers of zero modes of HL with cr3 = 1 and a 3 = — 1 respectively. Observe, that if |A) is a zero mode of Hx with |A) = (u, v) then, = 0, P f i i = 0 • . (3.25) where, V = -idl -d2 + e{Ax - iA2) (3.26) Hence A L = dim(kerCD^)), A L = dim{ker(T>)), and, N = IndexiH-1) = A L - A L = dim(ker{V^)) - dim{ker(V)) (3.27) The index of the elliptic operator H1 has been computed in numerous works[12, 26, 27] using two types of methods: i) complex analysis methods, ii) trace identities and axial Euclidean anomaly in 2 dimensions (this is par-ticularly interesting in the light of our using 4 dimensional anomalies above to derive axial currents on flux tube at finite u). The zero modes have also been computed exactly for some simple configurations of the gauge field [27]. In general the index is given by: IndexiH1) = — (3.28) ZTT $ = J d2xB3{x) (3.29) Hence the index measures the number of flux quanta through the xy plane, which is in essence a topological quantity. Now let's proceed to compute the axial fermion current induced at finite chemical potential /i. For further generality, we also include the effects of non-zero temperature T. The axial current density in the third direction is given by, jl(x) = ^(x) 7 3 7 5 i / ; (x) = iPla34>L(x) + ^Ra3^R(x) (3.30) Chapter 3. Axial Current on Magnetic Flux Tubes 38 We wish to compute the expectation value of the total current along the flux tube, J 3 = J d2x(j3(x)). At finite chemical potential and temperature we have, 053(*)> = ^ n ( J B ) ^ ( x ) 7 0 7 3 7 5 ^ ( 2 : ) E = ^ ( n ( ( e 2 + m 2 ) 5 ) + n(-(e 2 + m 2 ) > ) ) 4 ( i ) f f V f t W (3.31) Here, n(E) = ^{E1^^\E)+1 is the usual Fermi-Dirac distribution, ipE are eigenstates of H with energy E, ipRe are eigenstates of HR with eigenvalue e, and we've used eq. (3.18). The explicit form of ipRe in terms of eigenstates of H1 implies, < J 5 > = l E E K ( A 2 + ^ + m 2 ) i ) + n ( - ( A 2 + p 2 + m 2 )5 ) ) P3 A > 0 , s = ± < ^ ( A , p 3 ) | a 3 | ^ ( A , p 3 ) ) + + zEEK(P3 + ^ 2 ) " ) + M - ( p l + ^ 2 ) " ) ) ( A k 3 | A ) (3.32) p 3 A=0 Here A > 0 label eigenstates of H L , which generate eigenstates ip)\(X,p^) of HR with momentum p 3 and eigenvalue e± = ±-y/A 2 + P3 , while A = 0 label the zero modes of H X . Now, let's evaluate the matrix element ( ^ ( A , P 3 ) | C 3 | V , R ( A , P 3 ) ) for A > 0. Using eq. (3.21) and dropping the sub-scripts A, p3, s, we obtain, (ipR\a3\ipR) = (|c i| 2 + |C 2 | 2)(A|CT 3 |A) + {c\c2 +cic*2). Noting, (A|CT3|A) = (A| — A) = 0 for A > 0, and using the explicit formula (3.23) for C i , c 2, we obtain, (i/jR|cr3|V>ij) = SP3(A 2 + P 3 2 ) - 5 . This matrix el-ement is odd in both p 3 and s, hence the sum over all A > 0 in eq. (3.32) vanishes , and only the zero modes of H1 contribute to J | . The zero modes carry a definite value of 0 3 , so that (A|cr3|A) = cr3. Thus, we are left with, J 3 = ( jv + - iv_)i2W(rf + ^ a)*) + «(-(^+ P3 = 7rnrn{T,n) (3.33) Z 7 T nm{T,fj) = j^(n((pl + m2)^+n(-(pl + m2)^) (3.34) Chapter 3. Axial Current on Magnetic Flux Tubes 39 Here, nm(T, fj,) is just the number density of one-dimensional two-component (Dirac) fermions of mass m at finite temperature T and chemical potential / i . Hence, our final result (3.33) is topological in nature, since for each value of T and / i , it is sensitive only to the total magnetic flux and is independent of the particular distribution of the magnetic field. Several limits of the result (3.33) are noteworthy. First of all, in the massless limit m —> 0, one has n(T, fi) = ^ and, ( ' • * ) If there are several species of quarks present, we can sum eq. (3.35) over quark flavours and colours to obtain the current J 3 of eq. (2.3), which in the true dense QCD creates the 77 boson, J 3 = Y * (3.36) a This agrees with our result (3.5) of the previous subsection, where we explic-itly used the fact m = 0 (and, hence, chiral symmetry) in assuming that the axial current conservation is violated only by anomalies. So, we see that the appearance of axial current on flux tubes, which was derived somewhat mys-teriously in the previous section using the trick of fictitious chiral anomalies, is microscopically due to fermion zero modes. Our microscopic approach supports the validity of the fictitious chiral anomaly trick and serves as a check of the anomalous effective Lagrangian derived in [19]. Let us note that the result (3.35) is also independent of temperature for m — 0, which is a quite natural feature of a truly topological phenomenon. More explicitly, this fact is due to the special property of massless one-dimensional fermions, namely, their density at finite chemical potential is temperature independent. For arbitrary mass the density of one-dimension fermions n(T, u.) is generally temperature dependent, so for simplicity we consider the limit T — 0: Then, n(0,/J) = \J \x2 — m2/n and, JJ.fyZE^. (3.37) It is instructive to take the non-relativistic limit of eq. (3.37). Writing, fi = m + /j,nr, where the non-relativistic chemical potential \inr <C m, J 3 * £ v ^ $ ( 3 3 g ) 27T Chapter 3. Axial Current on Magnetic Flux Tubes 40 In the non-relativistic setting, J | is just the spin <S3, and for the case of uniform magnetic field, our result stems from the familiar fact that all Landau levels are doubly degenerate with respect to spin, except the lowest Landau level. It is amazing that this simple fact has such deep connections to chiral anomalies in 2 and 4 dimensions. 3.4 Evading the Fermion Mass Gap In the previous section we analyzed the microscopic origin of the result (3.5) in a model of free fermions coupled to a static magnetic field. The axial current in this model arose from the appearance of fermion zero modes on magnetic flux tubes. We. also saw that if one introduces a mass term into the Lagrangian, which explicitly violates the axial symmetry, the result (3.5) is lost. However, one may also consider theories where the axial symmetry is present, but the fermions have a finite mass gap, m. In this case it is apparently impossible to realize eq. (3.5) as we expect that no physical quantity depends on fi, for u. < m. In this section, we show that in such a situation, at least formally, there may still be a way to recover the result (3.5). We consider a U(l) cr-model, £ = ^ ( ^ + ze^)1/»-^(r^^ + 0 ^ I ^ ) ^ + >C0 (3.39) The Lagrangian for the <j> field is constrained only by the U(1)A symmetry, U{l)A • 4> -> e ^ f y , <f> -> e*V (3-40) For definiteness, we consider, = aM W - V(\<j>\2) (3.41) One may generalize (3.39) to include several fermion flavors (in particular, if we wish, we can ensure that the global U(1)A symmetry is non-anomalous). We assume that the U(1)A symmetry is spontaneously broken and (0 ) ^  0. The fermions then acquire a mass gap, m = h{<j>). This mass gap precludes us from exciting any "elementary" fermions for m < \i. However, as shall be shown below, fermion number can also be induced in other ways. Chapter 3. Axial Current on Magnetic Flux Tubes 41 Let us consider the fermions in the background of a field (/>(x), which has the spatial dependence, cp{x) = e^3 {<p) (3.42) This configuration can be thought of as a very thick axial domain wall (in fact, as thick as our physical system). It is well known[28] that an axial domain wall in the background of a magnetic field carries a finite fermion number! Assuming that the 4> field is slowly varying, the adiabatic expansion (con-sisting essentially of evaluating the same triangle graph, which contributes to the 4d axial anomaly) yields [7, 28], N = J d 3 x ( ^ 7 V ) = ^ J dx3^ arg{<p{z)) (3.43) where <3? is the magnetic flux and 4>(x) is assumed uniform in the x,y plane. Applying the expression (3.43) to our thick domain wall, We are not aware of a microscopic derivation of expression (3.44) in the literature 2 , so we will present one in Appendix B. Microscopically, fermion number on the wall is due to polarization of the whole Dirac sea. We will see that the expression (3.44) is exactly correct at least for q < m, \i < m. We will discuss only the regime \i < m in this work, as it presents the most serious problems from the point of view of evading the fermion mass gap. Now let us consider the thermodynamic potential Q of the system in the background of our large soliton, 0, = E(q) — fj,N(q). Here E(q) includes the contributions of both fermions and the boson field (j). In the limit q < m, B < m 2 , the largest contribution to the energy comes from the bosons, as their dynamics are governed by the scale (0) >^ m. Classically, the boson energy is, ^ = ?W (3-45) Using eq. (3.44) and minimizing Q with respect to q, 2 The microscopic calculations that have been carried out[28] considered <p(x) — <j>\ + i4>2(x), where <f>i is a constant Chapter 3. Axial Current on Magnetic Flux Tubes 42 where {B) = <P/A We see that in the regime that we are considering ftCl, (B) < m 2 , [i < m, our original assumption q < m is self-consistent. The axial current associated with the U(1)A symmetry is, f = ^ y y t y + 2i((f)*dfl(p - qbd^cp*) • (3.47) The largest contribution to again comes from bosons, J 3 = 4g|W|2 (3.48) so that from (3.46), Jd>xj\x) = ^ (3.49) Thus, we have recovered the formula (3.5). Unlike the case of a QED^+i where the current is carried by fermion zero modes, in the present scenario, the current is due to bosons, associated with spontaneously broken U(1)A symmetry. This has been expected as the chiral anomaly, which is the basis of the argument in section 3.2, is independent of whether the symmetry is spontaneously broken or not. Indeed, it is well-known, that the triangle diagram (correlator of axial and two vector currents), which is the source of anomaly, has a pole [29, 30]. In the case when the axial symmetry is not broken, the pole is saturated by a fermion-antifermion pair with a very specific momentum-helicity structure. In the case when the axial symmetry is broken, the pole is saturated by a Goldstone boson. We note that the calculation of the current presented in this section is variational in nature, and hence, cannot be considered rigorous. Our goal has been simply to illustrate a possibility via which one can recover the result (3.5) in theories, where fermions possess a mass gap. In fact, the present calculation has some serious difficulties. A glance at the formula (3.46) shows that if A —> co with $ fixed, the minimum q ~ 1/L 2 . Thus, the minimum q ceases to be a sensible quantity, unless $ is scaled in the same fashion as the area of the system A. The above problem is a natural consequence of our variational approach. In fact, we have set up our variational ansatz in such a way that we have a flow of current associated with the 4> field through our whole system, so that our "domain wall" stretches across the whole xy plane. But the flux <3?, may be localized over some macroscopic, but finite, area of the xy plane. In this case, a more energetically favorable ansatz needs to be considered. It is possible that there is in-fact no axial current for <E> of order of a few flux Chapter 3. Axial Current on Magnetic Flux Tubes 43 quanta (recall that we have been able to derive the result (3.5) in 3.2 only in the limit of thermodynamically large flux). On the other hand, perhaps the result (3.5) is cleverly saturated for any <&. This problem is subject to further investigation. 3.5 Conclusion In this chapter we have discussed the appearance of axial current on magnetic flux tubes at finite fermion chemical potential using several approaches. A l l of these approaches are weaved together by chiral anomalies. In a system where chiral symmetry is not spontaneously broken, the current can be understood in terms of fermion zero modes on the flux tube. These fermion zero modes are in a certain sense themselves due to anomaly in 2 dimensional Euclidean field theory and have implications for the 2 + 1 dimensional QED. Thus, we see that our trick with fictitious anomalies at finite chemical potential in 3 + 1 dimensions, in a sense continues the propagation of anomaly from 2 to 3 to 4 dimensions. This is a common pattern in the study of anomalies. 44 C h a p t e r 4 6— Pa rame te r i n 2 C o l o r Q C D at F i n i t e B a r y p n and Isospin D e n s i t y 4.1 Mot iva t ion In this chapter we investigate the behavior of 2-color QCD under the influence of three parameters: 9, \XB and / j / . The main motivation for such a study is, of course, the attempt to understand the cosmological phase transition when 9, being non-zero and large at the very beginning of the phase transition, slowly relaxes to zero, as the axion resolution of the strong CP problem suggests. Therefore, the universe may undergo many QCD phase transitions when 9 relaxes to zero. Another motivation is the attempt to understand the complicated phase diagram of QCD as a function of external parameters 9, HB and Finally, our study may be of interests for the lattice community - the determinant of the Dirac operator for 7VC = 2 is real when 9 = n in the presence of nonzero \i. As we show, in this case the superfluid phase is realized at a much lower chemical potential than at 9 — 0 . This gives a unique chance to study the superfluid phase on the lattice at a much smaller /j, than would normally be required. To study all these problems in real 3 color Q C D at finite HB is, of course, a very difficult task. To get some insight into what might happen we shall use a controlled analytical method to study these questions in the non-physical (but nevertheless, very suggestive) Nc = 2 theory. We use the chiral effective Lagrangian approach to attack the problem. We shall determine the phase diagram in the HB, HI, 9 planes, various condensates and lowest lying excita-tions. We expect that our approach is valid as long as all external parameters 1^3,1^1 and the quark mass, mq, are much smaller than AQCD- We perform most of our calculations for the case of two flavors Nf = 2 where the algebra simplifies considerably. Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 45 One exciting effect that we find is that 9 dependence of the theory at fixed (j, may become non-analytic. This is due to the fact that the critical chemical potential for transition to the superfluid phase varies with 9. Therefore, a change of 9 might trigger a second-order phase transition, accompanied by a discontinuity in the topological susceptibility x- We also find that the strong 9 dependence, present near 9 = ir in vacuum, is washed out in the superfluid phase. We expect that for equal quark masses a first order phase transition (Dashen's phenomenon) will occur in the Nc = 2, Nf = 2 theory at 9 = TT, in the normal phase, but will disappear in the superfluid phase. We also find some interesting results, which appear even at 9 = 0. Most importantly we compute the dependence of the gluon condensate, {^^G^l/G^l'a), on the chemical potential. The gluon condensate decreases with density near the normal to superfluid phase transition, but, counter-intuitively, increases for <C p, <C AQCD-We also evaluate novel vacuum expectation values which appear in the superfluid phase: {iuTy0C,y5T2d) in the baryon breaking phase and (1127075 d) in the isospin breaking phase. .These densities, being nonzero even at 9 = 0, nonetheless have never been discussed in the literature previously. These densities, themselves, break the baryon and isospin symmetries respectively, and so may be considered as additional order parameters. The presentation of our results is organized as follows. In section 4.2, we introduce our notations for the low energy effective Lagrangian. In section 4.3, we introduce the 9 parameter into the effective Lagrangian description. In section 4.4, we discuss the phase diagram of our theory in detail, computing the spectrum of lowest lying excitations, characterizing the phases in terms of chiral condensates and densities and paying special attention to physics near the point 9 = n. In section 4.5, we check that our results satisfy known Ward Identities supporting the self consistency of our approach. In section 4.6, we study the gluon condensate {^^G^G^11"1) as a function of pi and 9. In Conclusion, we discuss the relevance of the obtained results for 3 color QCD and make some speculative remarks on evolution of the early universe during the QCD phase transition. In appendix C, we clarify some technical issues associated with global aspects of the goldstone manifold. Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 46 4.2 The Effective Theory at Finite fig and LLI Two color QCD at zero chemical potential is invariant under SU(2iv%-) ro-tations in the chiral limit. This enhanced symmetry (as compared to the SU(A^)xSU( iV/ )xU( l ) of three color QCD) is manifest in the Lagrangian if we choose to represent it in a basis of quarks ip a n d conjugate quarks ip [31, 32]. For Nf = 2 we use, d u \ d j \ 0-2T2(uR) \ 0-2T2(dR)* ) (4.1) where the Pauli matrices r 2 and <r2 act in colour and spin space respectively. We work in Euclidean space and use the definitions, 7„ = 0. t 0 7 5 = - 1 0 0 1 ov = (—i,o~k)- The microscopic Lagrangian then reads, L = i¥auD^ (4.2) and possesses a symmetry, U G SU{4) (4.3) The enhanced symmetry manifests itself in the low energy effective the-ory through the manifold of goldstone modes associated with spontaneous breaking of chiral symmetry, SU(2Nf) —> Sp(2Nf). In our case, Nf — 2, and the goldstone manifold is SU(4)/Sp(4), corresponding to the condensation of — SU(4) flavor sextet. The fields on this manifold can be represented by a 4x4 antisymmetric unitary matrix 2, with det T, = 1, that transforms under (4.3) as, E -> UEUT We parameterize the vacuum manifold as, j T U e SU(4), E c S - UEcU1 (4.4) (4.5) In what follows we use notations suggested in [33, 34, 35] for the descrip-tion of baryonic as well as isospin chemical potentials. In these notations Chapter 4. 6— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 47 the baryon charge of the quark is 1/2, which comes from 1/NC, so that the baryon (diquark in Nc = 2) has baryon charge 1. Thus, chemical potentials enter the microscopic Lagrangian as, L = %hvDvip - ^/WTV - ^ / V h ^ V (4.6) In the basis of SU(4) spinors (4.1) the baryon and isospin (third compo-nent) charge matrices in block-diagonal form are[31, 32, 33, 34, 35], so that the Lagrangian reads, L = i¥auDuV -¥(fiBB + iiiI)V (4.8) The effective Lagrangian for the field 2 of goldstone modes is determined by the symmetries inherited from the microscopic two-color Q C D Lagrangian. To lowest order in derivatives and at zero quark mass the effective Lagrangian is [35], F2 ' £ = — T r V j / E W E 1 " (4.9) The /x-dependence enters the effective Lagrangian through the covariant ex-tension of the derivative, <90£ -» V 0 £ = d 0 £ - [(HBB + u.II)-E + l2(pJBB + HiI)T} , V , S = doEt -» V 0 E t = SoE.t + [(HBB + Ai//)S + Z(HBB + i^lf] \ V , E = d{Y) (4.10) required by an extended local gauge symmetry[31]. Therefore, to this order in chiral perturbation theory, the Lagrangian at finite \i does not require any extra phenomenological parameters beyond the pion decay constant, F. This fact gives predictive power to chiral perturbation theory at finite / i . In using the effective Lagrangian constructed above we must, of course, assume that chiral symmetry for Nc = Nf = 2 QCD is spontaneously broken. Since we have regarded the hadronic modes as heavy, the theory is expected to be valid only up to the mass of the lightest non-goldstone hadron. Chapter 4. 9^ Parameter in 2 Color QCD at Finite Baryon and Isospin Density 48 4.3 The mass term and 6 parameter The mass term in the fundamental Lagrangian is defined as, LM = muuu + m^dd (4-11) while the 9 term in the fundamental Lagrangian is, . < 4 ' 1 2 ) We keep mu ^ md on purpose: as is known mu — is a very singular limit when one discusses 9 dependence, see below. We would like to incorporate the 8 dependence directly into the mass matrix. This can be achieved by performing a chiral rotation, ^ _> e i * w / 2JV /^ (4. 1 3 ) With this field redefinition, the topological 9 term in the Lagrangian disap-pears, due to the axial anomaly, and the mass term becomes, LM = ^ ^ M H + ^—^-M^ (4.14) where the mass matrix M is, M = e-ie'N' ( m n u ° ) (4.15) In the basis of SU(4) spinors (4.1), the mass term becomes, LM = ^ T M O 2 T ^ + h.c. (4.16) where, in block-diagonal form,' M = \ - M 0 ) ^ The transformation properties of LM under (4.3) imply that to lowest order, M. enters the effective Lagrangian as, £m = -gReTr(MX), (4.18) Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 49 where the coefficient g is determined by the chiral condensate in the limit m —> 0~^ , 9 = 0, \IQ — fij — 0 [32], 9 = (4-19) as will be confirmed below. In our notations the chiral condensate includes the sum over all flavors, (ip'ip) = 2~2f CV'/V'/)-The chiral effective Lagrangian incorporating the effects of a n d non-zero quark masses, thus becomes, F2 C = — T r V ^ E V ^ - gReTr (MT) (4.20) We shall use the Lagrangian (4.20) for the rest of this work. So far our discussion easily generalizes to arbitrary Nf. However, signifi-cant algebraic simplification can be obtained by considering = 2. Indeed, for Nf = 2, the effective Lagrangian (4.20) with mu ^ md, 9 ^ 0, can be reduced to the same Lagrangian but with m'u = m'd and 9' — 0. This is achieved, by performing an SU(4) (more specifically SU(2)A) rotation, E = U0TU0T (4.21)' with the particular choice of, cos a / 0 y (mu + md) cos(f?/2) ^ n = 0 R* * = e i a a 3 ' 2 yj{mu + md)2 cos2(9/2) + (mu - md)2 sin2(6/2) s i n a = ( T ^ - m d ) s i n ( f l / 2 ) V ( m u + md)2 cos2(0/2) + (mu - md)2 sin2 (6/2) Our parameter a is related to the commonly used Witten's variables <f>u, </></. [37], via, (j)u = 9/2 - a , </>d = B/2 + a (4.23) <t>u + <f>d = 9, mu sin c6„ = md sin <pd (4.24) After such a transformation, the Lagrangian (4.20) takes the form, £ = ^ T r V ^ E V ^ E 1 - gm(9)ReTv (M0EJ (4.25) Chapter 4. 0— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 50 with, M0 = ( _°x m{6) = i ( K + mdf cos2(0/2) + (m* - md)2 sin 2(0/2))* (4.26) The detailed explanation of this reduction, which along the way clarifies certain global properties of the vacuum manifold, is presented in apppendix C. Technically, the simplification is due to pseudo-reality of SU(Nj = 2) (see also section 4.4.3 for a more quantitative discussion). 4.4 Phase Diagram 4.4.1 V a c u u m Al ignment and Spec t rum Our next step is to find the classical minimum of the effective Lagrangian (4.20) to determine the phase diagram, pattern of spontaneous symmetry breaking and, subsequently, the spectrum of excitations. For arbitrary Nf, quark masses, 6 and chemical potentials this is a non-trivial algebraic prob-lem. However, as was shown in the previous section, for Nf = 2, the effective Lagrangian reduces to the form (4.25), which was already analyzed in [35]. Thus, we may immediately read off all quantities of interest. First, let's study the phase diagram for fixed mu, md, 6. To get acquainted with our theory, let's begin with the trivial environment /jg = pi — 0. The effective Lagrangian possesses an 5p(4) symmetry at this point. The classical minimum is given by, <£} = S c (4.27) The Sp(4) symmetry is unbroken. The low-lying excitations are a. quintet of pseudo-goldstones (3 pions and 2 diquarks), with dispersions, E = v/p 2 + ml{6) (4.28) mliO) = ^ = (4.29) The pseudo-Goldstone mass m , acquires a dependence on 9 through the effective quark mass parameter m{6) (4.26) (this 6 dependence is implicitly implied in all formulas below, unless otherwise stated). As we shall see, the whole phase diagram turns out to be determined by the parameter mn(9). Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 51 We note that mn(9) reaches its maximum at 9 = 0 and minimum at 9 = ir. Moreover, for mu = m^, 6 = TT, m , vanishes to first order in M. We note that strong P and CP symmetries are explicitly broken in the system with 9 ^ 0. So at 9 ^ 0, the pions (diquarks) are no longer pure pseudoscalars (scalars). This will become particularly clear when we discuss Bose-condensates of our goldstones in the superfluid phase. Now let's turn on chemical potentials. For \IB ^ 0, u-i ^ 0, the symmetry of the problem is broken to U(1)B X U(l)r-1 We introduce the following notations to describe vacuum alignment of E at finite chemical potentials, 7:). ^i-i o 1 ) • <«»> As is known[35], there are 3 distinct phases in the ( / / B , / J / ) plane,2 I. Normal Phase (N): \HB\ < mTr{9), \/J>i\ < m^O) (£} = E C (4.31) Symmetry breaking: U{l)B x [/(1)7 - • U{l)B x t / ( l ) 7 Spectrum: E = A / P 2 + m 2 ± HB TT 0 E = v V + m 2 7 T ± E = V P 2 + ml ± ll! (4.32) II. Baryon Phase (B): \HB\ > "v(#), \fJ-i\ < IMSI ml „ / m 4 (E) = - f E c + l - - f ) E B (4.33) M B V M B . : I f only one of the chemical potentials is turned on, say HJ = 0, /ig ^ 0, then the symmetry is actually, SU(2)y x U(1)B-2In the original paper[35], a certain physically reasonable ansatz was taken for the classical static minimum (E) of (4.20). It was shown that this ansatz is, indeed, a local minimum, and the authors assumed that this minimum is also global. We note, that using the explicit parametrization of the vacuum manifold presented in Appendix C, it is possible to prove that the ansatz is, indeed, a global minimum. Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 52 Symmetry breaking: U(l)B x U(l)j —> U(l)j Spectrum: E> = p a ^ ,» ( 1 + 3 ^ ) ± M B ( 4 p ^ 4 ^ ( l + 3 | ) y -E = ^Jtf + nl (4.34) III. Isospin Phase (I): \HI\ > m„(9), \HB\ < \HI\ (E) = ^ S c + f l - ^ ) 5 E / (4.35) A4/ \ Hi / Symmetry breaking: U(1)B X U(1)I —> U(1)B Spectrum: Same as for B Phase, but with q± <-> 7 r ± and <-> Hi-The phase transition between N phase and B phase, as well as N phase and / phase is second order, whereas the phase transition between B phase ' and / phase is first order. As noted in [35], the symmetry of the phase diagram/spectrum, with respect to U-B ^ A*/ 'is a direct consequence of the symmetry of the microscopic theory, di <-> — dR, dR <-> di, HB Hi-Thus, the phase diagram in the (HB, A*/) plane looks the same at 9 ^ 0 as at 9 = 0, with the important replacement, m? —> ml(9). This is a very natural conclusion. Indeed, at 9 ^ 0 diquarks (pions) still carry baryon (isospin) number. Hence, their energy is lowered at finite baryon (isospin) chemical potential. As soon as U-B (HI) reaches the vacuum diquark (pion) mass mW(9), Bose-condensation occurs leading to spontaneous breaking of U{1)B (U(1)I) symmetry. Quantitatively, the 9 dependence of the Goldstone mass m„(9) implies that the transition to superfluid phase is shifted to a smaller chemical po-tential HB, Hii compared to 9 = 0. In the limiting case, when mu = and 9 = n, the transition occurs in the vicinity of / i = 0 (see Section 4.4.3 for Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 53 a more precise discussion). For physical values, = IMeV, mu = AMeV, the transition at 9 = ir occurs at /j = ^t+ml) * m *-(°) ~ 70MeV. 4.4.2 Chiral Condensates and Densities In section 4.4.1, we have established the phase diagram of Nc — 2, Nf = 2 QCD at finite \IB, p-i and 9. In this section, we wish to characterize this phase diagram in terms of chiral condensates and densities. Similar computations have been performed[32, 35] at 9 = 0. However, we evaluate a wider range of expectation values and find some condensates that are non-zero even at 9 = 0, which have not been discussed in the original papers. As expected, we also find new condensates at 9 ^ 0. We follow the standard procedure for computing microscopic condensates from the effective Lagrangian. We start from a slightly generalized version of the microscopic Lagrangian (4.8) together with the mass term (4.14), Here the hermitian, traceless matrix T incorporates the chemical poten-tials for all 15 charges associated with the SU{4) symmetry, and the chiral condensate source J is an arbitrary, antisymmetric matrix (12 real compo-nents). We may express T and J in terms of a basis, T = tAXA, J = jaXa, XA = XA, Tr(XA) = 0, XaT = -Xa, tA,ja e M Differentiating the vacuum free energy density T, we obtain our conden-sates and charge densities: (4.37) dT dtA dT (4.38) The relations (4.38) hold for any T, J, however, we will apply them when the derivatives and expectation values are evaluated at physical parameters, T = T 0 = U-BB + ml and J = M. Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 54 In the effective theory, the sources T and J are incorporated by replacing To —> T in the covariant derivative (4.10), and M —> J in the mass term (4.18). The condensates (4.38), thus, become, = F 2 ( T r ( E t A A V 0 S - V 0 E U A E ) ) or, A dT — = -g(ReTr(XaE)) (4.39) OJa It remains to evaluate the expressions (4.39), with E given by the time-independent, classical minimum of the effective Lagrangian (4.20). We must remember that to simplify algebra we expressed, E = UQT,U0T', with UQ given by (4.22). We should also remember that we incorporated 6 dependence into the mass-matrix by a chiral rotation (4.13) of the quark fields. The condensates and densities, expressed in terms of the original quark fields, are listed in Tables 1,2. We define the charge conjugation matrix C = 707275-We also introduce the parameter A(#) in Tables 1,2, 1 Normal Phase A (0) = { f^p Baryon Phase ( 4 . 4 0 ) Isospin Phase which obtains its 9 dependence through m 2(0). We can now see, how our phase diagram is described in terms of conden-sates and charge densities. First, let's gauge our intuition by considering the Normal phase. At 9 = 0, the parameter a of eq. (4.22) is 0, and the only con-densates (Table 1) are {uu) = (dd). At non-zero 9, we also get condensates {iu-fu), {id*y5d), while {uu), {dd) get depleted. The appearance of P and CP odd condensates {iu^u), {idq^d) is a direct consequence of explicit P and CP breaking by the 9 term. Finally, for 9^0, mu ^ m&, the parameter a y£ 0, and we see explicit effects of isospin symmetry breaking: (uu) ^ {dd) (correspondingly for P odd condensates). Such effects are absent to lowest order in M at 9 = 0. As expected, all charge densities (Table 2) in the Normal phase vanish. Let's now see what happens in superfluid phases. At 9 = 0, the Baryon phase is characterized by a scalar diquark condensate {i^Cf^d), which breaks the U(1)B symmetry. The Isospin phase is characterized by a pseudo-scalar pion condensate {iu^d), which breaks the U(l)i symmetry. These Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 55 Table 4.1: Chiral condensates in Nc = Nf = 2 Q C D at finite 9 Condensate/{ipip)Q N Phase (A = 1) B Phase (A = ^ ) I Phase (A - ^ ) iuT C^r2d 0 - i c o s ( i ) ( l - A 2 P 0 uTCT2d 0 - i s i n ( f ) ( l - A 2 ) ^ 0 iu-y5d 0 0 - i c o s ( f ) ( l - A 2 P ud 0 0 - i s i n ( f ) ( l - A 2 P uu |Acos(^ - a) — |As in ( f — a) dd | A c o s ( | + a) id-y5d - i A s i n ( f + a ) condensates appear at the expense of depleting ('tpip). As expected, at finite U(1)B and U(l)i violating condensates of opposite parity also appear: {uTCr2d) in B phase and (ud) in I phase. The Baryon and Isospin phases also carry non-vanishing SU(4) charge densities. The I phase, is characterized by the isospin density, nj = \ $7o<^> = 4 F 2 M / ( l - (4.41) This is precisely the density, which one expects to induce by applying an isospin chemical potential ui- At 9 = 0 it coincides-with the previous results [34], [35]. In addition, we also obtain the following axial charge density, nA = (iul0l5d) = 4 F 2 / i 7 ^ P (l - 2 cosQ(c?) (4.42) which has not been discussed previously in the literature even at 9 = 0. This is the axial charge density, corresponding to off-diagonal generators of the SU(2)A group, which is both spontaneously and explicitly broken. Note that the axial charge density (4.42) does not vanish already at 9 — 0. Thus, we for Chapter 4. 8— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 56 Table 4.2: Densities in iV c = Nf = 2 Q C D at finite 8 Density N Phase (A = 1) B Phase (A = ^ ) . I Phase (A - ^ ) 0 4FVB (1 - A 2) 0 iuT-y0Cj5r2d 0 - 4 F 2 M i 3 A ( l - A 2 ) ^ c o s ( a ) 0 uTj0CT2d 0 - 4 F 2 M B A ( 1 - A 2 )5 s in(a) •0 0 0 4 F 2 M / ( 1 - A 2 ) iujo-y5d 0 0 4 F V / A ( l - A 2 ) = c o s ( a ) l uiod 0 0 4 F 2 / z / A ( l - A 2 ps in (a ) U T^0CT2U 0 0 0 dTj0CT2d 0 0 0 now concentrate on 8 = 0, and hence a(9 = 0) = 0, to better understand the physical nature of this new density (4.42). For simplicity, we take \UB\ < m r The density nA spontaneously breaks the U(l)i symmetry and, hence, may be considered as an order parameter alongside the pion condensate, < O = («7hd> = - ^ > 0 ( 1 " ^ ) 8 (4.43) There was no explicit chemical potential conjugate to nA in the Lagrangian -once U(l)j is already spontaneously broken by (TT), nA is induced automati-cally. The quantitative behaviour of these two order parameters is somewhat different. The pion condensate monotonically increases with ui after the Normal to Isospin phase transition, and (TT) —> —|(T/5I/>)0 for jir 2> m^. On the other hand, the new charge density nA first increases after the'phase tran-sition, reaches a peak at = S ^ m ^ , and then decreases to 0 for u S> mn. Of course, we always consider only /zr, \IQ <C AQCD-One can understand the appearance of a new condensate nA = (ra7o75ci) in the following simple way. We are in the phase where the isospin density, rii ~ (ujou) — (d'jod), as well as the condensate, (ir") ~ (iu^d), do not vanish. This implies that our ground state can be understood as a coher-ent superposition of an infinitely large number of TT~ mesons. We expect Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 57 that we do not disturb the ground state of the system by adding one of these 7r~ mesons. On the other hand, we can relate the matrix element with an extra ir~ meson to the matrix element without the 7r~ using the standard P C A C technique, (A\0\Bn) ~ i(A\[0,Q5}\B).. The coefficient of proportionality would not be precisely 1/F in the present case because our pions are not in a trivial vacuum, but rather in the (TT~) condensed phase. However, we expect that the general algebraic structure of the vacuum expec-tation value will be obtained correctly using this approach. Indeed, taking O = ujou — d^od, as the isospin density and calculating the commutator [0 ,Q 5 ] , where Q5 = J d^xu^Q^d = J d?xv)^d is the axial charge, one ob-tains the structure (iu"fo^y5d) entering the eq. (4.42). Therefore, if we expect the vacuum expectation value of (O) = (U^QU — d"f0d) to be nonzero, we should also expect a nonzero value for the axial density (iu-yo^d). This logic is definitely supported by the explicit calculations (4.42). One can also test formula (4.42) at small isospin density nj. In this case our system may be understood as a dilute Bose-Condensate of non-relativistic 7T _ particles[32]. How is nA manifested in this terminology? We shall work at fixed isospin number density (instead of at fixed /jj). Moreover, we will temporarily work in Minkowski space. In the Isospin phase, the diquarks are not important as we saw, so we parameterize £ as, The field U transforms as U -» LUR) under SU(2)L x SU(2)R and the effective Lagrangian for U reads, Thus, we see that the Lagrangian describing the pion sector oi Nc — Nf = 2 QCD is exactly the same as the one describing Nc = 3, Nf — 2 QCD. We express, U = exp (I7r2°p°)• Similarly to eq. (4.38),(4.39), we identify, U € SU{2) (4.44) £ = F2 (Trd^Ud^ + 2mlReTrU) (4.45) - e a 6 c < 9 V 7 r c 7T F .a 2F&1- .a (4.47) (4.48) (4.46) Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 58 where we have expanded the corresponding currents to leading order in TT fields. We also expand the Lagrangian to fourth order in TT fields, (4.49) We can ignore the TT° particles as they are irrelevant for TT- condensation. It is useful to combine TT = ^ ( v 1 + wr2). To describe non-relativistic physics involving TT~ particles, we can replace 8QTT —> imnir, diir —> 0, in the quartic terms of Lagrangian (4.49). Finally, we adopt a non-relativistic normalization of our TT~ field, by introducing, a canonical, non-relativistic Bose field (of dimension 3/2), 0 f = V2m~^TT (4.50) The Hamiltonian density in terms of the <p field reads, H = + + 3 ^ ( < ^ ) 2 ( 4 5 1 ) while the condensates and densities become, nj = ^4>a3ip = $<p (4.52) TIA = iujo^d = — 2 F y / m 7 r $ (4.53) iul5d = -—^—<t>] (4.54) F^Jmn In this language, we see that both of our U{\)j order parameters, nA and iujsd are expressed in terms of the same non-relativistic Bose field 0L The energy density of a spatially uniform Bose-Condensate as a function of isospin density is, e = r rvn, + 3 2 ^ / (4.55) Therefore, the isospin chemical potential, V = ^ = mv + ^ n j (4.56) One can check that (4.56) agrees to first order in n; with the result (4.41) obtained in the grand-canonical ensemble treatment. Re-expressing the order parameters in terms of isospin density, we obtain (up to U(l)i phase), nA = 2F(m7rnI)1*, (iuj5d) = ~(H>)0 (jj^A ' (457) Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 59 in agreement to leading order with previous result (4.42), (4.43). Thus, the appearance of the second order parameter n& is quite natural. Finally, we remark that the situation in the B phase is the mirror image of the above discussion. The new U(1)B breaking density is, {iuTi0Ci5T2d) = -4F2nBX ( l - A 2 ) * cos(a) (4.58) 4.4.3 6 Dependence So far we have been mostly investigating the phase diagram in the (HB^I) plane at fixed 9. In this section we would like to focus more on the 9 depen-dence, drawing the phase diagram in the (9, /i) plane. This trivial exercise leads to rather interesting consequences, namely, the 9 dependence at fixed /a becomes non-analytic. We further characterize the phase diagram in terms of the (GG) correlator and the topological susceptibility x- Finally, we confirm our calculations by checking the validity of low energy theorems. To simplify the discussion we shall take fij = 0 and focus on 9 dependence in the Normal and Baryon phases. The situation in the Isospin phase is again just the mirror image, as can be explicitly checked. We begin by considering 9 dependence at ji = 0. The story is exactly the same as in the well-studied case Nc = 3, Nf = 2. The vacuum energy density T{9) is, jT(6),/j = 0) = -4F2ml(9) (4.59) where, < { e ) = TmM (4.60) m(0) = l- ((mu + md)2 cos2(#/2) + (m u - md)2 sin2(#/2)) > (4.61) By differentiating F{9) we can compute correlation functions of GG, dT ,.g2GG, , A n n . d2F .. [ ^ g ' G G . ^ G G 892 = X = ~ Jd4x(T9-^(x)9-^(0))conn (4.63) Chapter 4'. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 60 At p, = 0 we find, ,.g2GG lmumd . -( * 3 ^ = ° = - 4 H 9 ) S m { d ) { ^ ) o *<" = °> = I ^ H + ^ S i n 2 ^ ) ^ o (4-64) The expressions (4.64) reflect the well-known strong 9 dependence in the region mu « md = mq, 9 « TT. Let's introduce the asymmetry parameter, \mu — md\ e = - (4.65) mu + md and assume e « 1. The CP odd order parameter (iGG) (see Fig. 4.1,a) starts out at 0 when 9 = 0 and increases smoothly with 9, reaching its maximum just before 9 = TT at, .92GG _ _ m , 32TT2 ) e = x " ~ 2 Afterwards, the order parameter (iGG) experiences a steep crossover, drop-ping to its minimum of, < ^ ) e = x - « - ^ ( ^ > 0 (4.66) < ^ W « + ^ W > o (4-67) The crossover occurs in the region \9 — ir\ ~ e and hence, the topological susceptibility x has a sharp peak around 9 = 7r of width A9 ~ e and height, x(7r) / |x(0) | = l/e(seeFig.4.1,b). Such behaviour of the CP odd order parameter (iGG) strongly suggests that for mu = md, spontaneous breaking of CP symmetry occurs at 9 = ir. This situation, known as Dashen's phenomenon, has been extensively studied in Nc = 3 QCD with Nf = 3 and Nf = 2[36, 37, 38, 39, 40, 41]. For Nf = 3 with ms 3> mu,md it is believed that spontaneous CP breaking occurs at 9 = TT for \mu — md\ms < mumd. For = 2, CP violation occurs at 9 = TT, mu = md and possibly in a small window of \mu — md\ ^ 0[41]. However, it is important to note that Dashen's phenomenon is not un-der complete theoretical control in our effective Lagrangian(4.20). Indeed, for a moment, we fix mu — md. Then, for general 9, the mass term explic-itly breaks the symmetry of the effective Lagrangian (4.20) from SU(4) to Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 61 0.5 / / / / 100 80 60 JC/|X{0)| 0 -0.5 & 40 I I H 2n Figure 4.1: 9 dependence in iV c = 2, Nf = 2 Q C D at // = 0, e = 0.01. a) The C P odd order parameter ( i ^ r ) - See eq. (4.64) for precise normalization, b) Topological susceptibility x-Sp(4). However, for 9 = n, the mass term in the effective Lagrangian van-ishes, restoring the symmetry to SU (4) and giving rise to apparently massless goldstones: rn\{9 = TT) = 0. Yet, no such symmetry restoration occurs in the fundamental microscopic QCD Lagrangian at 9 = TX. This contradiction is resolved by including higher order (quadratic) mass terms in the effective Lagrangian, which would explicitly break SU(4) even at 9 = 7T[40]. It is precisely these terms, which control the physics of Dashen's phenomenon, and which are not included in the present work. We do not wish to consider such higher order mass terms in this paper. For any fixed ^ 0 these terms can be neglected by considering suffi-ciently small mq. If the higher order mass terms are largely saturated by a third quark of mass mUtd -C ms <C AQCD, then we require, \mu - md\ mUid m\(9 = 0) » — ( 4 - 6 8 ) mu + md ms M 2 This condition is, indeed, realized in the true physical world. If, on the other hand, the higher order terms are controlled by a light rf (as motivated by Nc —> co), we consider, \mu - md\ ^ m ( ^ ) 0 m\(9 = 0) •"irr/|| ————_ ZL (4 69) mu + md * F 2 M 2 M 2 Of course, by imposing restrictions (4.68), (4.69) we automatically exclude the regions of parameter space where Dashen's transition is realized, and Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 62 1.2-1-l V m „ ( 0 ) 0.8-0.6 \ B / 0.4 N \ I I 0.2 \ / 0 n In Figure 4.2: Phase diagram of JVc = 2, Nf = 2 Q C D as a function of UB and 9. Here, e = = 0.01 A rapid crossover occurs in the Normal phase at 9 = TT, which is conjectured to become a first order phase transition, when mu = md. we may discuss only the quantitatively steep crossover in the Normal phase. However, we shall see in a moment that by considering the system at finite p, the rapid changes in the vicinity of 9 ~ TT observed in the Normal phase will be washed out. Let us now turn on finite p,B- Once conditions (4.68), (4.69) are met, all the results of previous sections hold for any 9. In particular, the transition to the Baryon phase occurs at p = mn(9) (see Fig. 4.2). As explained above, we can consider arbitrarily small ratio vn\(9 = Tr)/m2v(9 = 0) = e as long as mq —> 0. Thus, for ps < m^(9 = TT) the Normal phase is realized for all 9, while for ps > rnn(9 = 0) we are entirely in the Baryon phase. Finally, if we fix PB with m„(9 = TT) < ps < rnn(9 = 0) and vary 9 from 0 to 2TT we encounter two phase transitions: from Normal to Baryon phase and then back to Normal. Thus, the 9 dependence becomes non-analytic in this region! Since the N to B phase transition is second order, we expect the topological susceptibility, x to be discontinuous across the phase boundary. The transitions between B and N phases occur at-9 = 9C and 8 =' 27r — 9C, with the critical 9C given by m f f(0 c) = p. Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 63 In the Baryon'phase, the free energy density reads, F(9) = - 2 F V I ( l + (4.70) Clearly, the 9 dependence in the superfluid phase is different from that in the Normal phase (4.59). This is most clearly seen by computing, :92GG _ mumd 3271-2 / - i6F 2/4 Cfer> = 7 ^ O - W > > 0 sin(0), rnumd - , 2 We have to remember that expressions (4.71) hold for all 9 only once we are entirely in the Baryon phase: /J,B > .m„(9 = 0). On the other hand, if mv(9 = TT) < \IB < mn(9 = 0), then we use expression (4.64), for 9 where the Normal phase is realized, and expression (4.71), for 9 where the Baryon phase exists. Focusing for a moment on U.B > m1T(9 = 0), we see that the 9 dependence is very smooth: there is no sign of rapid crossover in (iGG) near 9 = TT and the large peak in the susceptibility x disappears. Moreover, as \IB increases, the 9 dependence is suppressed, as expected. This smooth 9 dependence at 9 ~ TT in the superfluid phase should be contrasted with sharp behavior in the Normal phase discussed above, see Figs 4.1,a,b. Now we' would like to understand, how the strong 9 dependence at p = 0 gets smoothed out as the chemical potential us increases. For, 0 < \IB < mn(9 = TT), 9 dependence is the same as at \i = 0. The key region is m7r(9 = TT) < HB < mv{9 — 0), where at fixed JJ,B, the Baryon phase is realized for 9C < 9 < 2TT — 9C and the Normal phase is realized otherwise. Let us investigate the behaviour of the topological susceptibility x i n this region. As we expected, x is discontinuous across the phase transition, ( 0 ) W ) o „ , „ 2 |X(0)| 64FVI sm\9c) (4.72) As the chemical potential increases slightly past m^{9 = TT), a narrow region of Baryon phase appears around 9 = TT, deep inside the crossover region shown on Fig. 4.1. In terms of susceptibility x, this affects only the very top of the peak of x(#) by' introducing small discontinuities at 9 = 9C and 9 = 2TT — 9c (see Fig. 4.3,a). As u, further increases, the range of 9 where the B Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 64 A 100-/ \ / \ • " / \ x/ixi")! A / \ / \ / \ 3.06 3 08 3.1 a) X/\W)\ 3.12 3.14 3.16 , C ' " \ / . \ / .18 3.2 3.22 3.24 0 j 2.5 2 1.5 0.5 306 3.08 3.1 3.12 3.14 b) X/|X(«)[ 1 3.16 3.16 3.2 322 3.24 y " y 27t 0 e " \ -0-5 y v •« y \ Figure 4.3: a) 9 dependence of the topological susceptibility x, m Nc = 2, Nf = 2 Q C D at \i = 1.02771^(0 — n) (broken curve) together with x at [x = 0 (unbroken curve). Here, e = 0.01 b) the same with p, = 1 .3m„(S = TT); C) the same with u = 0.7mn(9 = 0); d) the same with \i — m„(9 — 0) phase exists starts growing. This is accompanied by growth in discontinuities of x a t the transition points. Eventually, for m„(6 = ix) < \i <C mv(9 = 0), the original peak in x associated with the CP crossover, is entirely replaced by discontinuities associated with the second order Normal to Baryon phase transition (see Fig. 4.3,b). Once this occurs, the magnitude of the jump x(#+) — x(@c) starts to decrease (Fig.4.3,c), until finally at u = m^(9 = 0), x becomes continuous again and we are entirely in the Baryon phase (Fig. 4.3,d). The washout of the sharp 9 dependence near 9 = TI has been realized! The most exciting result of this section is that for mu ^ the "Dashen's crossover" first splits into two second order Normal to superfluid phase tran-sitions and for p > m f f (0) gets entirely washed out. We would like now to pro-vide some speculations regarding the degenerate case mu = m<j, 9 = TT. This point might be of importance for lattice fermions[39, 44], as it is equivalent Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 65 to a theory where one quark mass is negative and 9 parameter is not explic-itly present. In principle, it is possible to analyze this situation rigourously by going to higher order Chiral Perturbation Theory. However, since the algebra becomes rather involved even at /J. = 0, we confine ourselves to a conjecture based on the above results and common 9 = TT lore. For Nc = 2, Nf = 2, at JJ, = 0, we expect that Dashen's phenomenon will occur along the same lines[40] as for Nc = 3. Spontaneous CP violation will happen at 0 = TT, however, no continuous symmetries will be broken and Goldstones will have a small, but finite mass mn(9 — TT) > 0. At finite [x, we expect a line of first order phase transitions at 9 = TT to extend to / i = m^iO = TT), where it splits into two second order phase transition lines (see Fig. 4.4). We note that we have recently been able to confirm this conjecture in Nc = 3, Nf = 2 QCD [45], where the higher order terms in the effective Lagrangian are slightly better understood than for Nc = 2. We remark that in the Baryon phase, P-parity is still spontaneously bro-ken at 0 = TT, while in the Isospin phase, P-parity is broken at 9 = 0, but not a 9 = TT. ' <iGG> > 0 <iOO> < 0 Figure 4.4: Conjectured form of the Phase diagram of Nc = 2, Nf = 2 QCD for mu = md. Solid line indicates a first order phase transition, while dashed lines indicate second order phase transitions. The region near 9 = TT is not to scale. Chapter 4. 6— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 66 4.5 Ward Identities In this section we check the validity of Ward Identities (WI) [4, 38, 42, 43] at nonzero / i , 8. We anticipate that WI must remain untouched when external parameters such as /j, 8 or temperature T are introduced. Indeed, the anom-aly (chiral and/or conformal) is a short distance (UV) phenomenon, which is not affected by medium effects (density \x ^ 0, 8 and/or temperature T). This fact was implicitly used when we constructed the effective Lagrangian (4.20). However, we are in a position to calculate each term entering the WI explicitly. Therefore, the check of the WI is a nontrivial test of self consistency of our results. The first identity that we consider, relates the two CP odd order para-meters, ^l^) = W^m) (4-73) The identity (4.73) reflects the well known fact that there is no 6 dependence when m —> 0. By consulting Table 1 and eqs. (4.64), (4.71), we can ex-plicitly check that our results satisfy the identity (4.73) both in Normal and superfluid phases. The next WI we would like to discuss is, /n2CC n2CC 1 d 4 x { T 3 ^ { x ) | ^ ( 0 ) ) C O N " = N2 + 0 { M 2 ) ( 4 ' 7 4 ) The 0(M2) term in (4.74) is usually dropped in the chiral limit, SU(Nf)v symmetric limit at 8 = 0, assuming the resolution of the U(l) problem when flavor singlet v( is a heavy state. Indeed, in this case Table 1 and eqs. (4.64),(4.71) imply that the WI (4.74) holds both in Normal and super-fluid phases. An important remark is that both sides of (4.73) and (4.74) depend on /J, in a very nontrivial way. Nevertheless, the identities are pre-served as expected. Now, we would like to see what happens with (4.74) when we relax the requirement of the SU(N/)v symmetric limit and also consider 9 ^ 0. In this case it is important to keep the 0(M2) term, 0(M2) = f d4x(Tijl5M,p(x)iPl5MiP(0))conn (4.75) Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 67 We begin in the Normal phase and evaluate, The above result implies that m~l singularities develop in the 0(M2) term of eq. (4.74), due to ^'/goldstone mixing, which occurs for mu ^ m^. However, the singularities disappear when mu = md, so that the 0(M2) term can be neglected in the chiral, SU(Nf)v limit, as long as we are sufficiently far from 9 = TT. This is the physically expected result. What happens with (4.74) in the superfluid phase, when 9 ^  0? We can immediately see that the 0(M2) term can no longer be neglected even when mu = md- As has been discussed above, the Normal to superfluid transition is second order, so that the topological susceptibility x generically experiences a jump across the phase boundary (4.72), while the chiral condensate (ipMtp) (Table 1) is continuous. Thus, for the WI (4.74) to hold,'the 0(M2) term must jump across the phase boundary, accounting for the discontinuity in x and, thus, contributing on the same footing as (TpMijj) to the righthand side of (4.74). It is not surprising that the 0(M2) term becomes important. Indeed, from Table 1 and (4.71) we see that both x and (ipMip) are of order M2 jy?B in the superfluid phase. The fact that x becomes 0(M2) rather than O(M) is part of smoothing out of 9 dependence in the superfluid phase. The contact between 0(M) dependence in the Normal phase and 0(M2) dependence in the superfluid phase is provided by the fact that u2 ~ O(M) at the Normal to superfluid phase transition. Thus, all correlators in eq. (4.74) develop ii~2 singularities in the superfluid phase, which are due to the modification of the goldstone spectrum (4.34). Finally, from (4.74), to leading order in M, in the superfluid phase, J d4x(T^5Mi;(x) $<y5M4>(0))conn = - (ml+m2d-2mumdcos{9))(4>ii)l (4.77) which vanishes only if mu = rrid, 9 = 0. 4.6 G luon Condensate Having determined the 9 and \x dependence of different condensates and densities containing the quark degrees of freedom (Tables 1, 2), one can Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 68 wonder if similar results can be derived for the gluon condensate (G2^), which describes the gluon degrees of freedom. As is known, the gluon condensate-represents the vacuum energy of the ground state in the limit mq = 0, p = 0 and plays a crucial role in such models as the MIT Bag model, where a phenomenological "bag constant" B describes the non- perturbative vacuum energy of the system. The question we want to answer: how will the gluon condensate (G 2^) (bag constant B) depend on p, 9 if the system is placed into dense matter? This question is relevant for a number of different studies such as the equation of state in the interior of neutron stars, see e.g. [46], or stability of dense strangelets[47]. Of course, it is difficult to answer this question in full 3 color QCD at finite however, the answer can be easily obtained in 2 color QCD for p <C A.QCD, which is the subject of the present work. We limit ourselves to considering only the Normal and Baryon phases, the results in the Isospin phase, as always, are obtained by replacing /IB —• /-*/• We work in Minkowski space in this section. We start from the equation for the conformal anomaly, % = -^G%,Ga>» + ipMip (4.78) where we have taken the standard 1 loop expression for the (3 function and b = yiV,. — ^Nf = 6 for Nc = Nf = 2. As ususal, a perturbative constant is subtracted in expression (4.78). For massless quarks and in the absence of chemical potential, eq. (4.78) implies that the QCD vacuum carries a negative non-perturbative vacuum energy due to the gluon condensate. Now, we can use the effective Lagrangian (4.20) to calculate the change in the trace of the energy-momentum tensor (9£) due to a finite chemical potential ps *C AQCD- The energy density e and pressure p are obtained from the free energy density F, e = F + pBnB (4.79) p = -T (4.80) Therefore, the conformal anomaly implies, ba2 ba2 ^32^2 }'l'm'0 ~ ^32^2 ^ ' ° ~ - 4 (F(p, m, 9) - Fo) - M W M , m, 8) + {tM^^e (4.81) Here, the subscript 0 on an expectation value means that it is evaluated at / i = m = 0, 9 = 0. The good news is that we have already calculated all Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 69 quantities on the righthand side of eq. (4.81) - see expressions (4.59), (4.70) and Tables 1, 2. Thus, in the Normal phase we obtain, (•^G^Gr*)^ - (J^G%G>"a)0 = -3m{6){H>)0 (4.82) When 9 = 0, (4.82) reduces to the standard result[43], which was derived in a different manner. As expected, (G 2 „) does not depend on \i in the Normal phase. The Baryon phase is more exciting, < | ^ G ^ % , m , e - ( ^ G ^ G ^ ) o = 4 F V 2 , ( l + 2 ^ ) . (4.83) It is instructive to represent the same formula in a somewhat different way, (^2G%Gn,,rn:e-(^2G^Gn,=o,m,e = 4F2(p2B-ml(9)) ( l - , . (4.84) which makes contact with the fact that in the Normal phase, when \IB < mn(9), the gluon condensate does not vary with \xB. However, for /J,B > mn(9), the dependence of the gluon condensate (G 2 „) on jj,B in the Baryon phase becomes rather interesting. The condensate decreases with \1Q for " V < HB < 2 1 / ' 4 m 7 r and increases afterwards. The qualitative difference in the behaviour of the gluon condensate for [IQ K, mn and for mv <C u,B <C AQCD can be explained as follows. Right after the Normal to Baryon phase transition occurs, the baryon density ns is small and our system can be un-derstood as a weakly interacting gas of diquarks. The pressure of such a gas is negligible compared to the energy density, which comes mostly from diquark rest mass. Thus, (0^) increases with ns and, according to the anomaly equa-tion (4.78), (G 2^) decreases. A similar decrease in (G 2^) with baryon density is expected to occur in "dilute" nuclear matter (see [48] and review [49]). On the other hand, for \IB S> mn, energy density is approximately equal to pressure, and both are mostly due to self-interactions of the diquark con-densate. Luckily, the effective Chiral Lagrangian (4.20) gives us control over these self-interactions as long as \IB <S AQCD- Such control is largely absent in corresponding calculations of (G 2 „) in nuclear matter. As Ae ~ Ap, the trace (0JJ) decreases and the gluon condensate increases with baryon density. Such behaviour of (G 2^) is quite unusual, as finite baryon density, on general grounds, is expected to suppress the gluons. Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 70 At small baryon density, we can also use a slight variant of the above method for calculation of the gluon condensate, originally developed in the context of nuclear matter. As long as our Bose-condensate of diquarks is dilute, we may neglect interactions between diquarks, and approximate the change in (G 2 „) as just the expectation value of G 2 ^ in each diquark times their number, (4.85) Here q denotes a diquark state relativistically normalized to (q (p)\q {p')} = 2Ep(2ir)353(p —p'), giving rise to the factor in (4.85). It remains to calcu-late the matrix element, (c /~ |G 2 Jg _ ) . This can be done by sandwiching the anomaly equation (4.78) between two diquark states. As (q~\Q£\q~) = 2m 2 , 2m 2 = -(q-\J-2G%G^a\q-) + (q-\W^\q-) (4.86) 1 3 2 7 T 2 We are used to the fact that the goldstone mass comes entirely from the symmetry breaking term TpMip, so we might, naively, expect from eq. (4.86) that G 2 ^ vanishes in a diquark. However, the diquark mass, to first order in M is given by, m2 = (q-$Mi/>\q-) (4.87) therefore, bn2 (q-\^-2G^Gnq-) = -ml (4.88) so that G 2 ^ and TpMip contribute equally to the goldstone mass in eq. (4.86). Now from eq. (4.85), / ^9 y~<a riuva\ i ^9 r~tu.va\ ^ ^27r2G%Gn,,m,e - (~-2G%G^)0<m,e = -^nBmn (4.89) This is in agreement with our full result (4.83),(4.84) to leading order in nB. Finally, we note that by differentiating (4.82), (4.83) with respect to 9 we can obtain correlation functions of G 2 ^ with GG, in Normal and superfluid phases. Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 71 4.7 Conclusion. Speculations. We conclude this paper with the following speculative remarks: 1. Naively, one could say that we studied in the present paper a purely academic question by considering Nc = 2 rather than the realized in nature QCD with Nc = 3. We should comment on this as follows. First, for Nc = 2, the so-called, diquarks become well-defined gauge invariant objects. How-ever, diquarks, as has been argued in a number of papers, (see, e.g. recent papers on the subject[50, 51, 52, 53, 54, 55]) may play an important role in 3-color Q C D dynamics. If the passage from SU(2)coior to SU(3)coior does not lead to dramatic disturbances of these diquarks, these predictions based on SU(2)coior remain qualitatively valid in real QCD! Arguments supporting the conjecture of smoothness of the transition SU(2)coior to SU(3)coior are presented in [54]. We also note that there is some similarity between the proposal of [55] and thepresent work to study the diquark dynamics. In the proposal [55] the idea is to introduce a color source to study the diquark dy-namics, while in our paper with Nc = 2 the diquarks automatically become gauge invariant objects, and no source is required to study them. 2. It has been suggested that the ^-induced CP odd state can be formed in heavy ion collisions at RHIC, see original papers[56] and a recent review[57]. Our analysis could be quite relevant for the study of the decays of a CP odd configuration, if it is formed. 3. It has been known for quite some time that violation of parity and CP parity (which is the case when 9^0) may completely change the phase structure of a theory. Some lattice calculations, for example, suggest that the behavior of the system could be very nontrivial when 9^0, and some singular behavior and even phase transitions are expected, see e.g. [58]. In an environment where C, P and CP are strongly violated, the interaction of quarks and anti-quarks is not identical, as it is usually assumed, but rather, could be very different. Under such conditions the QCD phase transition in the early universe could have a much more complicated history than it is typically assumed. In particular, one can imagine that some very nontrivial objects, such as Witten's nuggets[59], which behave as dark matter particles, can be formed. Moreover, due to the differences in interactions between quarks and anti-quarks in the presence of 9, local separation of baryon charges may take place, and chunks of quarks or anti-quarks in condensed color superconducting phase may form during the QCD phase transition, serving Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 72 as dark matter [60]. This scenario is based on the idea that while the universe is globally symmetric, the anti-baryon charge can be stored in chunks of dense color superconducting antimatter. In this case, instead of baryogenesis, one should discuss the separation of baryon charges. Such a global picture of our universe, is definitely, not in contradiction with the present observational constraints[60]. Rather, it may give a natural explanation for some global parameters such as ^IOM/^BI^O], or even can naturally explain the famous 511KeV line from the bulge of our galaxy [61]. Typical relaxation time for 9 is much larger than AQCD, therefore, one can neglect the dynamics of 9 for studying the possible phases for each given 9. This was the main reason for us to study 9 dependence of the QCD phase diagram. We find in the present analysis that, indeed, two phase transitions of the second order will take place when 9 relaxes from 9 = 2TT to 9 = 0. These phase transitions will occur under very generic conditions when \ii is smaller than mn, but of the same order of magnitude as m , . The physical consequences of these phase transitions are still to be explored. 4. Aside from these far reaching speculations, we would like to mention here a few much more terrestrial consequences of the present study, which may have some impact on the lattice simulations. "First of all, 2 color Q C D is a nice laboratory to study a variety of different very deep problems of gauge theories at nonzero temperature and density, see e.g. [62, 63, 64, 65]. New elements, which were not studied previously and which are the subject of the present work, are related to 9 dependence of different condensates. There are a few interesting observations which deserve to be mentioned here: a) At 9 = TT, when the determinant of the Dirac operator is real, the super-fluid phase is realized at much lower critical chemical potential / i c than at 9 = 0. In the limit mu = md, we expect, ^ C ^ ~ Q | ~ ( A ( ^ , D ) 2 • ^ & V E S A unique .chance to study the superfluid phase on the lattice at a much smaller \i than would normally be required. We hope that our conjecture for the dis-appearance of Dashen's transition in the superfluid phase can be explicitly tested on the lattice. b) Knowledge of 9 dependence of different condensates allows one to calcu-late the topological susceptibility and other interesting correlation functions as a function of \i. Corresponding Ward Identities at nonzero /_i can be tested on the lattice. c) Physics of gluon degrees of freedom and \i dependence of the gluon conden-sate can also be tested on the lattice. The behavior of the gluon condensate Chapter 4. 9— Parameter in 2 Color QCD at Finite Baryon and Isospin Density 73 as a function of ii is very nontrivial, as has been explained in the text. Nev-ertheless, our prediction is robust in a sense that it is based exclusively on the chiral dynamics and no additional assumptions have been made to derive the corresponding expression. Finally, we should emphasize that all results presented above are valid only for very small chemical potentials HB, Hi AQCD when the chiral effec-tive theory is justified. For larger chemical potentials we expect a transition to a deconfined phase at ^ 7^-QCD [66]. We should also add that all results presented above can be easily generalized to Nc = 3 QCD with MB = 0[45]. 74 C h a p t e r 5 S u m m a r y a n d O u t l o o k In this thesis we have investigated the role of chiral anomalies in dense matter. The first two parts of this work were concerned with the interplay between chiral anomalies and currents on topological defects in dense matter. This research was motivated by the recent derivation of the anomalous effective Lagrangian[19] in the context of dense QCD, which predicts the appear-ance of quantum numbers on certain topological defects. We have indirectly confirmed the validity of the effective Lagrangian approach of [19] by micro-scopically checking its predictions in a number of simple models. Thus, we have realized that the appearance of electric current on axial strings in dense QCD[19] is closely connected to the old problem of Witten's superconduct-ing strings. It is amazing that 20 years after the introduction of Witten's strings in the context of cosmology, it appears that similar objects might be physically realized in our universe in the dense interiors of neutron stars. We have microscopically shown that once put into an environment with a non-vanishing fermion chemical potential, Witten's strings automatically carry an electric current. Utilizing the machinery originally developed to compute fermion number on defects, we have demonstrated that this cur-rent is topological in nature, being independent of the particular profiles of background fields. We have also discussed in detail certain intricacies related to the computation of current on local and global strings. Our analysis of Witten's strings at finite density easily generalizes to other defects which are uniform in one directions. For instance, we have explicitly discussed in this thesis currents on domain walls in 2 + 1 dimensions. It would be extremely interesting to try to extend the microscopic com-putation of currents on Witten's strings considered in this paper to the "real-istic" case of axial strings in dense QCD. Although based on the anomalous chiral Lagrangian arguments we expect that the result will be the same, it is bound to have a slightly different microscopic origin. In particular, for the case of QCD, the fermion gap is developed above the fermi surface, as opposed to a mass gap of the Witten's model. The chemical potential in the Chapter 5. Summary and Outlook 75 case of dense QCD is the most important energy scale - all the physics of spontaneous symmetry breaking, gap formation and axial string appearance is secondary to the existence of the fermi sea. This is certainly in sharp con-trast to the toy model of Witten's strings, making the relevant computation in dense Q C D more complex (and, possibly, more rewarding). The automatic appearance of the current on strings may hold promise for various mechanisms "designed" to stabilize string loops against shrink-ing, producing defects, known as vortons[67, 68, 69]. The possibility of ax-ial string loop stabilization in the context of dense QCD is under current consider ation [70]. We have also analyzed in detail another consequence of the anomalous Lagrangian[19] - the appearance of axial current on magnetic flux tubes. We have been able to microscopically confirm this phenomenon for a simple model of free fermions in the background of magnetic field. In this case, the current is due to fermion zero modes, which are localized on flux-tubes. We have also considered a model, where the axial symmetry is actually spon-taneously broken and a fermion mass gap is present. From first sight, it seems that the mass gap prevents the appearance of any axial current for sufficiently small values of chemical potential. However, we have indicated a way, in which it is possible to evade the mass gap and reproduce the re-sult predicted by the anomalous Lagrangian[19]. The resulting axial current is due to a "flow" of bosons associated with spontaneous axial symmetry breaking. The final part of this thesis has been concerned with 9 dependence in Nc = 2 QCD at finite baryon and isospin chemical potential. The crucial fact that permitted us to undertake this analysis is that, due to the chiral anomaly, the 9 dependence disappears if the quark masses are precisely 0. If the quark masses are small, but finite, the 9 parameter may be incorporated into the effective chiral Lagrangian as a phase of the complex quark masses. Following this approach and also introducing baryon and isospin chemical potentials, we have been able to derive a number of results regarding the interplay of the parameters, #,/JB,/J/ in Nc = 2,Nj = 2 QCD. The most striking of our results is the disappearance of Dashen's phenomenon in the superfluid phase. We have recently generalized our findings to the case of Nc = 3 QCD[45]. We hope that some of our results on Nc = 2 and Nc = 3 QCD will be confirmed by lattice simulations in the future. 7 6 Appendix A Fermion Current on a Domain Wall In this appendix, we consider a domain wall in 2 + 1 dimensions. We show that at finite chemical potential there appears a fermion current along the boundary of the wall. The current is computed using the method of trace identities developed in section 2.3.4. The result is independently confirmed by the use of scattering techniques. We point out that a similar check has been performed for the original computation of fermion number fractional-ization on domain walls in 1 + 1 dimensions [12]. The domain wall is an excellent setting for studying the intricacies associated with "hidden" spec-tral asymmetries as it involves long-range fields. We begin by considering the following Lagrangian describing fermions in the background of the domain wall, L = $ii»d^ - (jnH* (A. l ) We take c/> to be a domain wall, which varies along the xl direction interpo-lating between (j)(xl = —oo) = — <f>o, (/){xl = °°) = </>o- We assume that the domain wall is straight along the x2 direction, i.e. c/> is independent of x%. Our goal is to calculate the current along the domain wall at finite fermion chemical potential, . J = j cfcc^V)*. (A.2) The single particle Hamiltonian is, H = -rffdi + <f>j° (A.3) For definiteness, we use the convention 7 0 = cr1, 7 1 = ia3, 7 2 = — ia2. Working in sectors of constant momentum k along the second direction, H = ka3 + H (A.4) Appendix A. Fermion Current on a Domain Wall 77 where H1 acts only in the direction perpendicular to the surface of the domain wall, H1 = - i c V 2 + eV 1 (A.5) We observe, {<r3, HL} = 0. Since the current in the second direction is given by tp^ip = ip^cr^ip, we see that'the problem under consideration exactly par-allels that considered in chapter 2. In particular, all the formal manipulations of section 2.3.4 still apply, and we may express the current as, J = J dE j(E)n(E) (A.6) where, k with o~k(E) being the odd spectral density of H/.. The odd spectral density has been computed [12], ak(E) = k sgn(E) (sgn((P0)5(E2 - k2) - ^ ~ ^ ~ ^ 0 0 ^ Substituting this into (A.7) yields 1 n(E2-k2) (E2 - k2 - (P2)~2 J (A.S) j(E) = ^sgn((P0)(l - 6(E2 - $ ) ) (A.9) Hence, at zero temperature, j _ | „ | < { A M ) J = ^ , \u\xPo ( A . l l ) We now present an alternative derivation of the result (A. 10) using scat-tering theory. We closely follow the analysis of [12], which was concerned with fermion number fractionalization on domain walls in 1 + 1 dimensions. Using the familiar form, Appendix A. Fermion Current on a Domain Wall 78 V = -b\ + <p{x), £>f = d1 + <p{x) (A.13) the eigenstates ip = of Hk satisfy, Vv = (E-k)u (A.14) V^u = (E + k)v (A.15) •For E ^ —k, we may write, 232?^ - (E2-k2)u (A.17) We see that for every eigenstate u of "DD* with eigenvalue A 2 , we obtain two eigenstates of Hk with energies E = ±(A 2 + k2) 2 of the following form, (If)5 V (2£(£;+fc))i / The remaining eigenstates of H^, have \E\ = \k\ and are of the form, X>tu = o, u = 0, £ = fc (A.19) Vv = 0, it = 0, E = -fc (A.20) Letting 7V+ = dim{ker(V^), A L = dim(ker(V^)), one can show[12] that, Index{HL) = N+- N-= sgn((t>0) (A.21) We now proceed to compute the current along the domain wall, J = ±£ E n(^ ) [dxl^Ea3iPE = jJ2(N+n(k)-NM-k)) + k E(Hk) J k + i £ £ « ( £ ) / ^ ' ( ^ » ^ - 2 ^ j ( I , , " £ r i ' w ) k E(Hk),\E\^\k\ V 7 (A.22) Appendix A. Fermion Current on a Domain Wall Integrating the last term in (A.22) by parts, 79 J = - ]T sgn(4>o)n{k) + k k E(Hk),\E\?\k\  W V 1 7 (A.23) Since, the functions ug are even in k (they are eigenfunctions of (A.17) with eigenvalue E2 — k2), the second term in (A.23) is 0. Thus, it remains to analyze the surface term. Notice that only the continuum eigenstates have a chance to contribute to the surface term. We write the continuum states in the following form, The corresponding energies of the continuum states are E — ±EP. with FP = (p2 + $ + k2)^. Substituting the asymptotics (A.24) into (A.23) and using unitarity, \R\2 + | T | 2 = 1, we obtain, J = z E - i E / 1 E Ejtwy (A'25) k k E — i Ep Symmetrizing (A.25) with respect to k —> —A; and changing the integration variable from p to E one obtains, J=TT f d E < E ) \E\ (s9n(<Po)5(E2 - k2) - ° ^ t Z f * r) L^J V J l \y ^ 7 r ( £ 2 - / c 2 ) ( £ 2 - ^ - ^ ) ^ (A.26) Comparing (A.26) with eqs. (A.6), (A.7) and (A.8), we see that the scattering analysis yields the same results as the trace identities. We end our discussion by noting that it would be very interesting to apply the methods developed in this thesis to compute the fermion current on a domain wall in &px+ipy superconductor[21]. In order to do so one first needs to generalize the present discussion to treat currents on topological defects, which have 2 + 1 (as opposed to 1 + 1) world-volume dimensions. 80 Appendix B Fermion Number on a Thick Domain Wall with Flux The purpose of this appendix is to compute the fermion number on a "thick" domain wall immersed in a finite magnetic flux. We consider the model (3.39) with the background field (f> having a uniform phase variation in the third direction 4> = e%qx We take the magnetic field to be static and uniform in the third direction. The one particle Hamiltonian is, H = - i 7 V ( d i + ieAi) ~ /i7o'(0*^4^ + (R1) We let tp be an eigenstate of H with energy E. One may get rid of the phase variation of 0 by a redefinition of fermion wave-functions, ip — & i q x 7 ^2ip'> s.t. H'ip' = Eip' and, H' = -i7°f{di + ^ 7 5 + ieAi) + m 7 ° (B.2) provided that q = n € Z , so that the transformation (B.2) does not affect the boundary conditions. In the limit when q is an intensive quantity, we can neglect the quantization of q (that's why we have insisted that lim' J4_ 0 O ^ 0 in eq. (3.46)). The transformation that we have just performed, is exactly the "large" chiral rotation that we were discussing in section 3.2. It is straightforward to diagonalize H'. We rewrite, ( -H1 - (p3 - q/2)^ m ' m H1- + (p3 + q/2)a3 (B.3) where we have utilized the invariance of H' in the third direction. We are us-ing the same notations as in section 3.3. We expand ip' in terms of eigenstates | A ) o f i f x , Appendix B. Fermion Number on a Thick Domain Wall with Flux 81 where c's are coefficients depending on A and p3. We observe that H' does not mix states with different A > 0. So, in sectors with A > 0, / - A - ( P 3 - 5 / 2 ) m . 0 - ( P 3 - 9 / 2 ) A 0 m m 0 X p3 + q/2 \ 0 m P3 + Q/2 - A / (B.5) For each A > 0, we obtain 4 eigenvalues, E = (X2 + ((pl + m2)li±q/2)2)K E = - ( A 2 + ((pj + m 2)5 ± g / 2 ) 2 ) 2 (B.6) Thus, gaped modes of H1 generate eigenstates of H, which come in pairs of opposite energy. Now we turn our attention to zero modes of H-1, -(p3-q/2)s m m •(p3 + g/2)s CL CR = E CL CR (B.7) Here, s denotes the sign of the corresponding zero mode of H1- under a 3 . The energies of the corresponding eigenstates of H, therefore, are, E = qs/2±{p23 + m2Y- (B.8) We are now ready to compute the fermion number of the configuration. We see that for \q\ < 2m, the spectrum of H has a gap. Thus, for \q\ < 2m and p <§C m, no "elementary" fermions are excited. The fermion number of the configuration comes from the entire Dirac sea polarized by the domain wall. As usual, we define the fermion number TV as, N = - y i - ~ y -9 0 (B.9) E<0 E>0 This is the conventional definition, which preserves charge conjugation [12]. The expression (B.9) is formal: it is understood to be regularized in a way, which is symmetric with respect to interchange E —> — E. Now, we see that the eigenstates of H generated by gapped modes of H1-do not contribute to the fermion number as their dispersion is symmetric with respect to E —> — E (B.6) 1. Thus, we can concentrate on the contribution of : We do not have to worry about intricacies related to hidden asymmetry of the fermion density of states, as all the fields in the problem are short range. Appendix B. Fermion Number on a Thick Domain Wall with Flux 82 the fermion zero modes. We calculate the odd density of states, a(E) = \ Yl(S(E - E') - 5{E + E')) (B.IO) 1 E> where the sum is over the eigenvalues E' of H. Using eq. (B.8), o{E) = ^Lj2(Pm(E-qs/2)+pm(-E+qs/2)-pm(-E-qs/2)-pm(E+qs/2)) A =0 ( B . l l ) where pm(E) is given by, Pm(E) = f pS(E - (pj + m 2)^) = E 9(E - m) (B.12) The fermion number as defined by (B.9) is just the integral of the odd spectral density, poo N = - / dEo-(E) (B.13) Jo Evaluating the integral (B.13), one obtains, N = ]—L V q s ( l - 0(\q\ - 2m)(1 - 4m2/q2f*) (B.14) 2TT ^-^ A=0 Now using the index theorem (3.28), J2\=o s ~ e$/2ir, N = ^Lq(l - 9(\q\ - 2m)(l - 4m2/q2)^) (B.15) Thus, for \q\ < 2m, we reproduce the result of the adiabatic approximation[7, 28], N = ^-2Lq (B.16) On the other hand, for \q\ > 2m the full result (B.15) disagrees with the adiabatic expansion. This is not surprising as the adiabatic approximation generally breaks down due to spectral flow, which occurs precisely for | g | > 2m, see (B.8). Thus, we see that the thick domain wall immersed in a magnetic flux carries a non-vanishing fermion number. 83 Appendix C Parametrization of the Vacuum Manifold The purpose of this appendix is to clarify some global aspects of the manifold of goldstone fields in i V c = 2, Nf = 2 QCD. Once the vacuum manifold is parameterized, we show that to first order in quark mass, all effects of the 8 parameter can be incorporated into a common real quark mass. We begin with the assumption that the chiral symmetry breaking pattern of Nc = 2, Nf = 2 QCD is, exactly, SU{4) -> Sp(4), so that the vacuum manifold X = SU(4)/Sp(4). We represent the manifold as, X = {UECUT, U e SU(4)} ( C l ) with E c given by eq. (4.5). Note that X C A, where A is the set of all 4 x 4 antisymmetric, unitary matrices of determinant 1. The original work[32] had implicitly assumed that X = A. As we shall show, this is almost, but not quite true. In fact, A = XUiX, i.e A is a disjoint union of two pieces, both of which are homeomorphic to SU{4)/Sp(4)\ Even though such technical details do not affect the analysis of [32], they become important, once the 8—parameter is introduced. In particular, if we minimized the effective Lagrangian (4.20) over £ 6 A, we would obtain very different 8 dependence. In fact, the theory (4.20) with E G iX corresponds to the theory with £ G X with the redefinition 8 —> 8 + TT. As long as we represent our vacuum manifold by any one of the two pieces X or iX, we obtain the same physics, if we define 8 appropriately. However, if we enlarge the vacuum manifold to contain both pieces, the physics changes: we obtain cusps in 8 dependence at 8 = ± | , instead of Dashen's phenomenon at 8 = TT. Since we find no evidence of an additional spontaneously broken discrete symmetry that would connect the two disjoint pieces of A, we insist on our original assumption that the vacuum manifold is SU(4)/Sp(4) = X. , Now, let us demonstrate the above claims. We begin by writing any Appendix C. Parametrization of the Vacuum Manifold 84 £ G A as, Here, a, b are complex numbers and we have used the fact that £ is anti-symmetric. The requirement, d e t £ = 1, implies, (dei(C) + ab)2 = 1 (C.3) We call, A± = { £ G A, det(C) + ab = ± 1 } . Then A is the disjoint union, A = A+UA-. We shall show, A+ = X, A- = iX. We begin with the observation that X C A+. Indeed, SU(A) is connected, so X is connected. But, £ c G X, and det(C) + ab = 1 for £ c . Therefore, det(C) + ab= 1 for any £ G X, so X C ^4+. Now, take £ G .4+. The condition ££^ = 1 implies, CC^ + \a\2 = 1 (C.4) &C + \b\2 = 1 (C.5) a*a2CTa2 = bC] (C.6) The remaining condition for £ G A+ is, det(C) + ab=l (C.7) Equation (C.4) implies \a\ < 1, and C can be written (non-uniquely) in the form, C = y/l - \a\2e^S, S G SU{2) (C.8) Substituting (C.8) into (C.5), produces, \a\ = \b\. We write, a = |a|eI<i6a, b = |6|e^\ If a = 0, the condition (C.6) is satisfied automatically, and (C.7) implies, e2l*c = 1. If \a\ = 1, the condition (C.6) is again satisfied, and (C.7) gives, ei^a+^ = 1. Finally, if 0 < a < 1, (C.6) becomes, e2i^o2STo2 = e ^ + ^ S 1 (C.9) But SU(2) is pseudo-real: a2STa2 = S\ so e2i^ = e^+^l Substitution into (C.7) gives We note that if eirt>c = —1, we can always reabsorb the negative sign into the definition of 5 G 5(7(2). Thus, £ G A+ if and only if it may be written as, Appendix C. Parametrization of the Vacuum Manifold 85 We now show that any £ of form ( C . l l ) is in X. We let, ex(- = yj\ — |a | 2 + i\a\. Define matrices, Ui, £/2, U3,U 6 SU{4) as, One can now check that UTicUT = 2. This concludes the proof of the fact that X = A+. It is now trivial to show that A- = iA+ = iX. The most practically useful'result of the above discussion is the explicit parametrization ( C . l l ) of the vacuum manifold X. For instance, this para-metrization allows one to prove that the local minimum of the effective La-grangian (4.20) at finite /J,B and /zy, originally constructed in [35], is, actually, global. Moreover, we can now use the form ( C . l l ) to study the topology of the vacuum manifold X. Indeed, the matrix 5 £ SU(2) can be uniquely written as S = X0 + iX^i, X0X0 + XiXi = 1. Also, ei<t>a = X4 + iX5, Xl + Xl = 1. Clearly, X is in one to one correspondence with the set of vec-tors i n R 6 , y/1 - \a\2(X0,X1,X2,X3,0,0) + \a\(0,0,0,0,X4,X5), 0 < |a| < 1. But this is just a parametrization of S 5 . Hence, X = SU{A)/Sp(A) = S5. Finally, let us discuss the SU(4) transformation (4.21). 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