Symmetries of QCD

partition function is extended by an external source J, providing a generating functional for finite temperature expectation values.

2.1 quantum chromodynamics 27

SU_R(2) : ψ_L7→ψ_L, ψ_R 7→e^iθRiT(f)i ψ_R, ψ¯_L7→ψ¯_L, ψ¯_R 7→ψ¯_Re^{−iθRiT(f)i} , U_V(1) : ψ_L7→e^iθVψ_L, ψ_R 7→e^iθVψ_R, ψ¯_L7→ψ¯_Le^−iθV, ψ¯_R 7→ψ¯_Re^−iθV,

U_A(1) : ψ_L7→e^−iθAψ_L, ψ_R 7→e^iθAψ_R, ψ¯_L7→ψ¯_Le^iθA, ψ¯_R 7→ψ¯_Re^−iθA, (2.29) where in the fundamental representation the generators T_(fⁱ ₎ = τⁱ/2 of the SU_f(2) flavor group are given by thePauli matricesτⁱ. On a classical level, continuous symmetries of a field theory can be related to corresponding conserved currents byNoether’s theorem [217,218].

In a simple version, cf. Ref. [215], the theorem states that the invariance of theLagrangian L under an infinitesimal change δφi of the field multiplet φi implies the conservation of a currentj^µ given by

j^µ= ∂L

∂(∂_µφ_i)δφ_i, (2.30)

i.e.,∂_µj^µ= 0. For a Lie group with generatorsT^a and an invariance under the infinitesimal transformation φ_i 7→ φ_i−i^aT_ij^aφ_j ≡ φ_i +^aδφ^a_i with parameters ^a, we find a conserved currentj_µ^a for each generator. The corresponding conserved charges are obtained by taking the space integral of the zero component of the conserved currents as the associated charge densities, i.e., Q^a =^R d³x j₀^a. We note that in a quantized theory the charge operators of a conservedNoether current are the generators of the related symmetry, i.e., the field operator φ_i transforms according to

φ_i7→φ⁰_i= e^iaQaφ_ie^−iaQa ≈φ_i+ i^a[Q^a, φ_i] =φ_i−i^aT_ij^aφ_j, (2.31) where the matricesT^a form a representation of the generatorsQ^a.

Noether’s theorem applied to the U_V(1) symmetry in Eq. (2.29) leads to baryon number conservation. The conserved charge is given by

QV =^Z d³xψ¯Lγ0ψL+ ¯ψRγ0ψR

=^Z d³x ψ^†(x)ψ(x), (2.32) with the quark number density N_q(x) =ψ^†(x)ψ(x). In fact, the QCDLagrangian is invariant under separate UV(1) phase transformations of the single quark flavors and the currents ψ¯uγ^µψu, ¯ψdγ^µψd, . . . of the individual quark flavors are also conserved. The quark number Eq. (2.32) describes the sum of these individually conserved charges. In order to study strong-interacting matter at finite density, the conserved quark number densityN_q(x) can be incorporated into the path integral representation of the partition function according to Eq. (2.24) and gives rise to the additional contribution ¯ψ(−iµγ₀)ψto the kinetic part of the quark fields ˆTψψ¯ which then reads⁹

Tˆψψ¯ := ¯ψ i∂/−iµγ₀ψ . (2.33)

9 The imaginary unit i originates from the replacement ¯ψ→i ¯ψaccording to our chosen conventions for the imaginary-time formalism withEuclideanspace-time, see AppendixA.2.

The quark chemical potentialµ is a Lagrange multiplier and introduces, somewhat loosely speaking, an imbalance between quarks and antiquarks.

At low energy the chiral symmetry SU_L(2) ⊗SU_R(2) is not manifest, i.e., hidden in nature. Only the subgroup of the isospin rotations SU_V(2) that describes the simultaneous transformation of left- and right-handed quark fields with θL =θR is a realized symmetry in the chiral limit¹⁰ and gives rise to isospin conservation. Indeed, the ground state can be shown to be necessarily invariant under SU_V(2)⊗U_V(1) in the chiral limit [219].

The invariance under the orthogonal axial transformations generated by the linear combi-nation

Qⁱ_A=Qⁱ_R−Qⁱ_L= 1 2

Z d³xψγ¯ ₀γ₅τⁱψ (2.34) of the right- and left-handed charge operators of the chiral symmetry Qⁱ_R andQⁱ_L, respectively, would imply degenerate states of opposite parity in the hadron spectrum. However, this so-called parity doubling is not observed, e.g. a low-energy baryon octet of negative parity corresponding to the existing octet of positive parity is missing in the particle spectrum.

The symmetry of the ground state is strongly connected to the symmetry of the spectrum by Coleman’s theorem [47]. Therefore, the symmetry pattern of the hadron spectrum shows strong evidence that the ground state is not invariant under axial transformations.

Based on an analogy with superconductivity [45,46], this led to the hypothesis that the invariance of the QCD action under chiral transformations is an accurate symmetry which is spontaneously broken down toSU_L(2)⊗SU_R(2)→SU_V(2) by the ground state, i.e., the generators Qⁱ_A|0i 6= 0 do not annihilate the vacuum. This non-perturbative phenomenon of the theory of the strong interaction is called chiral symmetry breaking (χSB). According toGoldstone’s theorem [48,49], the spontaneous breaking of a continuous global symmetry gives rise to the appearance of massless Nambu-Goldstone bosons in the channels of each broken symmetry, i.e., the number of massless bosons equals the number of generators of transformations that do not leave the ground state invariant. The properties of the Nambu-Goldstone bosons are closely related to the associated “broken” generators, cf. Appendix Cfor

more details. The hadron spectrum contains indeed particles of unusually low masses, i.e., in the case of 2-flavor QCD the pion triplet, which are considered to be pseudoNambu-Goldstone bosons. The reason for the addition pseudo is the small but non-zero masses of the pions as the QCD Lagrangian is only approximately invariant under chiral transformations due to the small but finite current quark masses at the physical point.

The pions have negative parity which matches the transformation behavior of the generators QAof the axial transformations. Indeed, the QCD vacuum can be expected to be not invariant under axial transformations. The strong attractive interactions give rise to the appearance of condensates of quark-antiquark pairs as the energy cost to produce a pair of at least approximately massless quarks is small. Such pairs, due to the requirement of vanishing momentum and angular momentum, possess a net chiral charge [202]. The spontaneous

10 Regarding the two lightest flavors up and down, this symmetry is approximately realized for physical current quark masses as well, since both masses are almost equal and negligibly small compared to, e.g., the QCD scale ΛQCD.

2.1 quantum chromodynamics 29 breakdown of the chiral symmetry is therefore associated with the formation of a corresponding chiral condensatehψψi¯ which is according to

h0|[iQⁱ_A,ψ¯iγ₅τⁱψ]|0i=h0|ψψ|¯ 0i=h0|ψ¯_Lψ_R+ ¯ψ_Rψ_L|0i, i∈ {1,2,3}, (2.35) a sufficient criterion for χSB. The formation of the chiral condensate renders the quarks massive and provides in this way a mechanism for the dynamical generation of the constituent quark masses.

The restoration of chiral symmetry does not necessarily imply the axialU_A(1) symmetry to be restored. In fact, this symmetry is not realized in nature but is broken by quantum corrections. Global symmetries of a classical action that are not realized in a quantum theory are referred to as anomalous symmetries [220–222].¹¹The so-called axial or chiral anomaly is caused by topologically non-trivial gauge configurations [223,224] that lead to a divergence of the classicalNoether current j_A^µ = ¯ψγ^µγ₅ψ in the form of

∂_µj_A^µ ∼g²_sN_f^µνρσF_µν^a F_a,ρσ, with⁰¹²³= 1, (2.36) c.f. Refs. [202,203]. For the phenomenologically more relevant three-flavor case, the explicit breaking of theU_A(1) symmetry has been put forward even earlier to explain theη-η⁰ mixing in the mesonic particle spectrum [225,226], resulting in the mass splitting of the respective particles. The absence of theU_A(1) symmetry may be deduced from other observations as well, e.g. from the missing parity doubling of hadronic states [227] again, at least at low energies.¹² Silver-Blaze property

In order to study strong-interaction matter at finite density, the kinetic term of the quark fields is modified to include the quark chemical potential, see Eq. (2.33). It appears that any finite chemical potential changes the eigenspectrum of the propagator and consequently the partition functionZ and derived thermodynamic quantities such as the free energy or the quark number density as well. Atzero temperature, however, the so-calledSilver-Blaze property, or “problem”, refers to the fact that the partition function of, e.g., a fermionic system does not exhibit a dependence on the chemical potential, i.e., it remains as that of the vacuum, provided that the chemical potential is less than some critical value [228], see also, e.g., Refs. [229–231]. The critical value is determined by the (pole) mass of the lightest particle carrying a finite charge associated with the chemical potential, i.e., in case of the quark chemical potential a finite baryon number, see below. TheSilver-Blaze property of the partition function or free energy carries over to the correlation functions, see, e.g., Ref. [229]. Phenomenologically speaking, this property simply states that the fermion density (corresponding to the difference in the numbers of fermions and antifermions) remains zero at zero temperature as long as the chemical potential is less than the (pole) mass of the lightest charged particle. For the sake of simplicity, let us consider in the following a theory where

11 In the path integral formalism the anomalous breaking of a classical symmetry can be traced back to an integral measure that is not invariant.

12 The spontaneous breaking of chiral symmetry dynamically generates the constituent quark mass which breaks theUA(1) symmetry anyway.

only the fermions carry a charge. Mathematically speaking, the Silver-Blazeproperty is then a consequence of the fact that the theory is invariant under the following transformation:¹³

ψ¯7→ψ¯e^−iατ, ψ7→e^iατψ , µ7→µ+ iα , (2.37) where α parametrizes the transformation and τ is the imaginary time. Setting α = q0, Eq. (2.37) immediately implies the following invariance of the partition functionZ:

Z

µ→µ+iq0

=Z . (2.38)

Thus, the partition function is invariant under a shift of the chemical potentialµalong the imaginary axis. Assuming that Z is analytic, it follows thatZ does not depend onµat all. In particular, we deduce from an analytic continuation of Eq. (2.38) from q₀ to iq₀ thatZ does not depend on the actual value of the real-valued chemical potentialµ.

For the effective action Γ, it follows in the same way from Eq. (2.37) that Γ[ ¯ψe^−iq0τ,e^iq0τψ]

µ→µ+iq₀ = Γ[ ¯ψ, ψ], (2.39)

and similarly for higher n-point functions, since the latter are obtained from Γ by taking functional derivatives with respect to the fields and setting them to zero subsequently.

On the level of correlation functions, we recall that, for example, the two-point function has a pole at p²₀ = −m²_f at ~p = 0, where m_f is the (pole) mass of the fermion. Thus, an analytic continuation of the two-point function in the complex p₀-plane is restricted to the domain |p₀| ≤ mf, as the pole mass is the singularity closest to the origin of the complex p₀-plane. From Eq. (2.39), on the other hand, we find the following relation for the two-point function:

Γ^(1,1)(p0−q0, ~p;p⁰₀−q0, ~p⁰)_µ→µ+iq

0 = Γ^(1,1)(p0, ~p;p⁰₀, ~p⁰), (2.40) where the four-momenta (p⁽⁰⁾₀ , ~p⁽⁰⁾) are associated with the ingoing and outgoing fermion lines, respectively. Note that Γ^(1,1) is diagonal in momentum space, i.e.,

Γ^(1,1)(p₀, ~p;p⁰₀, ~p⁰) = ˜Γ^(1,1)(p₀, ~p)(2π)δ(p₀−p⁰₀)(2π)³δ⁽³⁾(p~−~p⁰). (2.41) For q₀ = iµ, Eq. (2.40) implies

lim

µ→0Γ^(1,1)(p0, ~p;p⁰₀, ~p⁰)

p₀⁽⁰⁾→p₀⁽⁰⁾−iµ

= Γ^(1,1)(p0, ~p;p⁰₀, ~p⁰) (2.42) for µ < m_f. The latter constraint follows from the definition of the pole mass: Γ^(1,1) = 0 for (p0 −iµ)² = −m²_f, i.e., p0 = i(µ±mf), and ~p = 0. Note that mf refers to the pole mass at µ= 0. This line of argument can be generalized straightforwardly to highern-point functions.

13 Here, we effectively treat the chemical potential as an external constant background field.

2.2 aspects of color superconductivity 31