Vector and axialvector mesons at nonzero temperature within a gauged linear sigma model

# Vector and axialvector mesons at nonzero temperature within a gauged linear sigma model

Stefan Strüber Institut für Theoretische Physik, Johann Wolfgang Goethe-Universität,
Max von Laue-Str. 1, D-60438 Frankfurt/Main, Germany
Dirk H. Rischke Institut für Theoretische Physik and Frankfurt Institute for Advanced Studies,
Johann Wolfgang Goethe-Universität, Max von Laue-Str. 1, D-60438 Frankfurt/Main, Germany
July 12, 2019
###### Abstract

We consider vector and axialvector mesons in the framework of a gauged linear sigma model with chiral symmetry. For , we investigate the behavior of the chiral condensate and the meson masses as a function of temperature by solving a system of coupled Dyson-Schwinger equations derived via the 2PI formalism in double-bubble approximation. We find that the inclusion of vector and axialvector mesons tends to sharpen the chiral transition. Within our approximation scheme, the mass of the meson increases by about 100 MeV towards the chiral transition.

###### pacs:
11.10.Wx, 12.38.Lg, 12.40.Yx
preprint:

## I Introduction

The fundamental theory of the strong interaction is quantum chromodynamics (QCD). QCD has a local gauge symmetry which determines the interaction between matter constituents, the quarks, and gauge fields, the gluons. For massless quarks, the quark sector of the QCD Lagrangian also has a global chiral symmetry, where is the number of quark flavors. The anomaly induced by instantons 'tHooft:1986nc () breaks this symmetry explicitly to , where vector and axial vector symmetries are introduced via . The discrete symmetry plays no role for the dynamics and will be omitted in the following. Nonzero, degenerate quark masses explicitly break , such that the remaining symmetry is . Non-degenerate quark masses further break this symmetry to , corresponding to baryon number conservation.

At low momenta , where MeV is the QCD scale parameter, quarks and gluons are confined inside hadrons. Therefore, on a typical hadronic length scale fm, the gauge symmetry of QCD is (at best) of minor importance, and the interactions between hadrons are predominantly determined by the global chiral symmetry of QCD. In the QCD vacuum, the axial part of the latter symmetry is spontaneously broken by a non-vanishing expectation value of the quark condensate Vafa:1983tf (). According to Goldstone’s theorem, this would lead to Goldstone bosons. However, since the explicit symmetry breaking induced by the anomaly reduces the axial symmetry to , spontaneous symmetry breaking of the latter gives rise to only Goldstone bosons. These Goldstone bosons acquire a mass due to the explicit chiral symmetry breaking by nonzero quark masses.

At temperatures of the order of , the thermal excitation energy is large enough to restore the symmetry of QCD. If instantons are sufficiently screened Gross:1980br (), this could additionally lead to a restoration of the explicitly broken . For vanishing quark masses, the high- and the low-temperature phases of QCD have different symmetries, and therefore must be separated by a phase transition. The order of this chiral phase transition is determined by the global symmetry of the QCD Lagrangian; for , the transition is of first order if , for , the transition can be of second order if Pisarski:1983ms (). If the quark masses are nonzero, the second-order phase transition becomes cross-over.

Lattice QCD calculations predict the chiral phase transition to happen at a temperature MeV Aoki:2006br (); Karsch:2007vw () for zero quark chemical potential . The phase transition temperature is expected to decrease when increases and vanishes at some value corresponding to quark number densities of the order of a few times nuclear matter density. Therefore, the chiral transition of QCD is the only phase transition in a theory of one of the fundamental forces of nature which can be studied under laboratory conditions: heavy-ion collision experiments performed at the accelerator facilities CERN-SPS, BNL-RHIC and, in the near future, CERN-LHC and GSI-FAIR create matter which is sufficiently hot and/or dense, such that chiral symmetry is restored. Indeed, the primary goal of these experiments is to find evidence for the restoration of chiral symmetry by the creation of the so-called quark-gluon plasma, i.e., the phase of QCD where quarks and gluons are liberated from confinement.

When chiral symmetry is restored, the masses of hadrons with the same quantum numbers except for parity and G-parity, so-called chiral partners, become degenerate. Chiral partners are, for instance, the sigma and the pion in the (pseudo-)scalar sector, or the and the in the (axial) vector sector. A promising signal for chiral symmetry restoration in heavy-ion collisions is therefore to study changes of the spectral properties of hadrons in the hot and dense environment Pisarski:1994yp (); Brown:2005kb (); Rapp:1999ej (). One of the prime candidates is the meson. The meson decays sufficiently fast (and with – for experimental purposes – sufficiently large branching ratio) into a pair of dileptons which, due to their small (since electromagnetic) cross section, are able to carry information from the hot and dense stages of a heavy-ion collision to the detector. The CERES and NA60 experiments at the CERN-SPS have found convincing evidence for a modification of the meson spectral function in Pb+Pb and In+In collisions, respectively Agakichiev:2005ai (); Damjanovic:2007qm ().

It is important to clarify whether the modification of the meson spectral function observed by the CERN-SPS experiments is in any way related to chiral symmetry restoration or is merely due to many-body interactions in the hot and dense medium. This question can be decided by calculating the dilepton production rate from QCD and then using this rate in a dynamical model for heavy-ion collisions in order to compute the dilepton spectrum. The low-invariant mass region of the dilepton spectrum is dominated by the decay of hadronic states. Therefore, for the calculation of the dilepton rate it is more convenient to apply a low-energy effective theory for QCD, featuring hadronic states as elementary degrees of freedom and respecting the chiral symmetries of QCD, rather than using QCD itself. Since we are interested in the restoration of chiral symmetry at nonzero temperature, the low-energy effective theory of choice is a linear sigma model which treats hadrons and their chiral partners on the same footing.

Linear sigma models have been used for quite some time in order to study chiral symmetry restoration in hot and dense strongly interacting matter. For instance, Pisarski and Wilczek have applied renormalization group arguments to a symmetric linear sigma model with scalar degrees of freedom and have drawn important qualitative conclusions regarding the order of the chiral phase transition for different numbers of quark flavors Pisarski:1983ms (). The calculation of hadronic properties at nonzero temperature (and density) faces serious technical difficulties. For instance, for the following reasons it is impossible to apply standard perturbation theory. First, it turns out that the coupling constants of effective low-energy models of QCD are of the order of one, rendering a perturbative series in the coupling constant unreliable. Second, nonzero temperature (or density) introduces an additional scale which invalidates the usual power counting in terms of the coupling constant Dolan:1973qd (). A consistent calculation to a given order in the coupling constant then may require a resummation of whole classes of diagrams Braaten:1989mz ().

A convenient technique to perform such a resummation and thus arrive at a particular many-body approximation scheme is the so-called two-particle irreducible (2PI) or Cornwall-Jackiw-Tomboulis (CJT) formalism Cornwall:1974vz (), which is a relativistic generalisation of the functional formalism Luttinger:1960ua (); Baym:1962sx (). The 2PI formalism extends the concept of the generating functional for one-particle irreducible (1PI) Green’s functions to that of one for 2PI Green’s functions , where and are the expectation values of the one- and two-point functions. The central quantity in this formalism is the sum of all 2PI vacuum diagrams, . Any many-body approximation scheme can be derived as a particular truncation of .

An advantage of the 2PI formalism is that it avoids double counting and fulfills detailed balance relations and thus is thermodynamically consistent. Another advantage is that the Noether currents are conserved for an arbitrary truncation of as long as the one- and two-point functions transform as rank-1 and -2 tensors. A disadvantage is that Ward-Takahashi identities for higher-order vertex functions are no longer fulfilled vanHees:2002bv (). As a consequence, Goldstone’s theorem is violated Petropoulos:1998gt (); Lenaghan:1999si (). Another consequence is that sum rules of the Weinberg type at zero and nonzero temperature Kapusta:1993hq () are not necessarily fulfilled. A strategy to restore Goldstone’s theorem is to perform a so-called “external” resummation of random-phase-approximation diagrams with internal lines given by the full propagators of the underlying approximation in the 2PI formalism vanHees:2002bv (). In this work, however, this problem is less severe since we shall only focus on the case of explicit chiral symmetry breaking by (small) non-vanishing quark masses.

Effective theories with chiral and symmetry have already been studied within the 2PI formalism in the so-called double-bubble approximation which gives rise to Hartree-Fock-type self-energies for the individual particles Baym:1977qb (); Bochkarev:1995gi (); Polchinski:1996na (); Amelino-Camelia:1997dd (); Petropoulos:1998gt (); Lenaghan:1999si (); Lenaghan:2000ey (); Roder:2003uz (). The model has also been studied within the 2PI formalism beyond the double-bubble approximation including sunset-type diagrams Roder:2005vt (). The chiral models in these works, however, only include scalar and pseudoscalar particles. Here, we extend these investigations to vector and axial vector mesons.

Our ultimate goal is a calculation of the dilepton rate for given temperature (and density). In this work, we perform the first step in this direction studying chiral symmetry restoration in the mesonic mass spectrum at nonzero temperature. We apply the so-called gauged linear sigma model as outlined in Ref. Gasiorowicz:1969kn (). This model has been used by Pisarski Pisarski:1994yp () who varied the tree-level mass parameters in order to derive qualitative statements about the behavior of meson masses as a function of temperature. Here, we extend these studies to a full-fledged self-consistent many-body calculation of meson masses at nonzero temperature using the 2PI formalism in double-bubble approximation.

This paper is organized as follows. The following section is dedicated to a discussion of the gauged linear sigma model with chiral symmetry, with being the number of quark flavors. In Sec. III we consider the case in more detail and rewrite the Lagrangian explicitly in terms of scalar, pseudoscalar, vector, and axial vector degrees of freedom. We also discuss issues related to the breaking of chiral symmetry such as the mixing of pseudoscalar and axial vector mesons. In Sec. IV we use this model to calculate the behavior of the meson masses and the condensate as a function of temperature. Section V concludes this paper with a summary of our results and an outlook for further studies.

We use the imaginary-time formalism to compute quantities at nonzero temperature. Our notation is

 ∫kf(k)=T∞∑n=−∞∫d3k(2π)3f(2πinT,→k). (1)

We use units . The metric tensor is .

## Ii The gauged linear sigma model

In this section we present the gauged linear sigma model with symmetry. The Lagrangian reads Gasiorowicz:1969kn ()

 L = Tr[(DμΦ)†DμΦ]−m20Tr% [Φ†Φ]−λ1(Tr[Φ†Φ])2−λ2Tr[(Φ†Φ)2]−14Tr[(Lμν)2+(Rμν)2] (2)

Here, the field stands for scalar and pseudoscalar hadronic degrees. These are chosen to be organized into representations under transformations, such that

 Φ→U†RΦUL, (3)

where are unitary matrices acting on the fundamental representation of . This allows for an explicit representation of in terms of a complex matrix,

 Φ=(σa+iπa)Ta, (4)

where , are the generators of in the fundamental representation, and , , parametrize scalar and pseudoscalar fields, respectively.

In order to introduce vector and axialvector degrees of freedom, one defines right- and left-handed vector fields,

 Rμ=(Vμa+Aμa)Ta,Lμ=(Vμa−Aμa)Ta, (5)

where , are vector and axialvector fields, respectively. These are treated as massive Yang-Mills fields, with field-strength tensor

 Rμν=∂μRν−∂νRμ−ig[Rμ,Rν],Lμν=∂μLν−∂νLμ−ig[Lμ,Lν]. (6)

Right- and left-handed fields transform as gauge fields under ,

 Rμ→U†R(Rμ+ig∂μ)UR,Lμ→U†L(Lμ+ig∂μ)UL, (7)

while the field-strength tensors transform covariantly under ,

 Rμν→U†RRμνUR,Lμν→U†LLμνUL. (8)

The covariant derivative in Eq. (2) couples scalar and pseudoscalar degrees of freedom to right- and left-handed vector fields,

 DμΦ=∂μΦ+ig(ΦLμ−RμΦ). (9)

The first line in Eq. (2) contains terms which are invariant under local transformations. The vector meson mass term renders this symmetry a global one. The determinant terms represent the anomaly of QCD and break the symmetry explicitly to , where stands for baryon number conservation. The last term in Eq. (2) corresponds to the quark mass term in the QCD Lagrangian. Since this term is flavor-diagonal, we have . For degenerate nonzero quark masses, , while all other vanish. In this case, is explicitly broken to . For non-degenerate quark masses, also the other have to be nonzero. Exact isospin symmetry requires .

Local invariance implies universality of the coupling, i.e., the coupling between right- and left-handed vector fields to scalar and pseudoscalar fields is the same as the coupling of the vector fields among themselves. Note that the Lagrangian (2) would also be locally invariant under , had we introduced separate coupling constants for right- and left-handed vector fields and modified the covariant derivative, . However, if , one obtains non-vanishing parity-violating terms in the Lagrangian, which must not occur in viable theories of the strong interaction.

The assumption of local invariance further restricts possible couplings between scalar and vector mesons. Under global transformations, terms like and would transform as itself. Taking discrete symmetries into account possible coupling terms would be , and Pisarski:1994yp ().

According to Noether’s theorem, continuous symmetries lead to conserved currents. Explicit symmetry breaking induces source terms in the conservation laws. For global symmetries, there is a simple way to derive the currents and the conservation laws Gell:1960 (). Consider the symmetry transformation to be of the form . Promoting the global symmetry to a local one, , and computing the variation of the Lagrangian, one can then read off the Noether currents and the conservation laws from the identity

 δL=−∂μJμaθa−Jμa∂μθa. (10)

For QCD, the symmetry leads to the following vector and axial-vector currents and the conservation laws:

 JaVμ = −∂δLQCD∂(∂μθaV)=¯qγμTaq, (11a) JaAμ = −∂δLQCD∂(∂μθaA)=¯qγμγ5Taq, (11b) ∂μJaμV = −∂δLQCD∂θaV=i¯q[M,Ta]q, (11c) ∂μJaμA = −∂δLQCD∂θaA=i¯q{Ta,M}γ5q+δa0g2Nf64π2ϵμναβGμνbGαβb, (11d)

where is the quark mass matrix and the gluon field-strength tensor. The last term on the right-hand side of the conservation law for the axial current cannot be obtained from the variation of the classical QCD Lagrangian. It represents the anomaly and arises from instantons 'tHooft:1986nc ().

In the gauged linear sigma model, the vector and axial-vector currents and the corresponding conservation laws can be obtained analogously from the variation of the Lagrangian (2) under local transformations. The result is

 JaVμ = −∂δL∂(∂μθaV)=m21gVaμ,JaAμ=−∂δL∂(∂μθaA)=m21gAaμ, (12a) ∂μJaμV = −∂δL∂θaV=−fabcσbhc,∂μJaμA=−∂δL∂θaA=dabcπbhc+4δa0cIm[det(Φ)], (12b)

where are the totally antisymmetric and symmetric structure constants of , respectively. Since the first line in Eq. (2) is locally invariant under , it cannot contribute to the currents or the conservation laws. Then, the vector meson mass term which violates local invariance gives rise to the celebrated current-field proportionality (12a) Gasiorowicz:1969kn ().

One may wonder why the vector current, , and axial vector current, , arising from scalar and pseudoscalar particles do not appear in the expressions (12a) for the vector and axial vector currents. They read explicitly

 ~JaVμ = −12fabc[ϕ∗b(Dμϕ)c−(Dμϕ)∗bϕc], (13a) ~JaAμ = −i2dabc[ϕ∗b(Dμϕ)c−(Dμϕ)∗bϕc], (13b)

where

 (Dμϕ)a≡[∂μδac+g(fabcVbμ−idabcAbμ)]ϕc (14)

is the covariant derivative. However, employing the equations of motion, these currents are recognized as simply being parts of the total vector and axial vector currents (12a),

 JaVμ = ~JaVμ−1g∂νVaνμ−fabc(VνbVcνμ+AνbAcνμ), (15a) JaAμ = ~JaAμ−1g∂νAaνμ−fabc(VνbAcνμ+AνbVcνμ), (15b)

where

 Vaμν ≡ (16a) Aaμν ≡ (16b)

are the field strength tensors for vector and axial vector fields, respectively.

In the following, we briefly discuss consequences of Eqs. (12b). In the chiral limit, , and the isosinglet vector current and the vector currents , as well as the axial vector currents , are exactly conserved. The isosinglet axial vector current receives a contribution from the anomaly. In the case of explicit chiral symmetry breaking, we have . However, since , the isosinglet vector current is still exactly conserved. If only , but , for the same reason also the other vector currents are exactly conserved. For instance, this is the case for assuming isospin invariance which requires for . In contrast, since , the axial vector currents are only partially conserved; the famous partial conservation of axial currents (PCAC).

## Iii The case Nf=2

So far, the discussion of the Lagrangian (2) was valid for an arbitrary number of quark flavours. In the following, we restrict ourselves to the case of mass-degenerate up and down quarks. For this case, the fields , , and can be written in terms of the physical scalar (), pseudoscalar (), as well as vector () and axial vector fields () as:

 Φ = (σa+iπa)ta=(σ+iη)t0+(→a0+i→π)⋅→t, (17a) Rμ = (Vμa+Aμa)ta=(ωμ+fμ1)t0+(→ρμ+→aμ1)⋅→t, (17b) Lμ = (Vμa−Aμa)ta=(ωμ−fμ1)t0+(→ρμ−→aμ1)⋅→t. (17c)

The vector and axial vector mesons enter the Lagrangian through the following terms:

 Tr[(DμΦ)†DμΦ] = 12[∂μσ+g(ηf1μ+→π⋅→a1μ)]2 (18a) +12[∂μη−g(σf1μ+→a0⋅→a1μ)]2 +12[∂μ→a0+g(→ρμ×→a0+η→a1μ+→πf1μ)]2 +12[∂μ→π−g(→π×→ρμ+σ→a1μ+→a0f1μ)]2, −14Tr[(Lμν)2+(Rμν)2] = −14(∂μων−∂νωμ)2 (18b) −14(∂μf1ν−∂νf1μ)2 −14[∂μ→ρν−∂ν→ρμ+g(→ρμ×→ρν+→a1μ×→a1ν)]2 −14[∂μ→a1ν−∂ν→a1μ+g(→ρμ×→a1ν+→a1μ×→ρν)]2, m212Tr[(Lμ)2+(Rμ)2] = m212(ω2μ+→ρ2μ+f21μ+→a21μ). (18c)

Note that the -meson completely decouples from the dynamics. In order to have non-vanishing coupling of the to the other fields, we would have to include the Wess-Zumino-Witten term Wess:1971yu (); Kaymakcalan:1983qq (). Note that, had we included globally -invariant terms such as e.g. , the -meson would not decouple.

Since the -meson is protected by the symmetry, it does not mix with the . Consequently, we do not expect the to become degenerate in mass with the when chiral symmetry is restored. This is in contrast to the and the , which mix under transformations: these mesons are expected to become degenerate in mass when chiral symmetry is restored.

In the phase where chiral symmetry is broken, the -field assumes a non-vanishing expectation value, . In order to examine the fluctuations around the ground state, we shift the -field by its vacuum expectation value, . After this shift the covariant terms (18a) read

 Tr[(DμΦ)†DμΦ] = 12[∂μσ+g(ηf1μ+→π⋅→a1μ)]2 (19) +12[∂μη−g(σf1μ+ϕf1μ+→a0⋅→a1μ)]2 +12[∂μ→a0+g(→ρμ×→a0+η→a1μ+→πf1μ)]2 +12[∂μ→π−g(→π×→ρμ+σ→a1μ+ϕ→a1μ+→a0f1μ)]2.

The shift of the -field leads (after an integration by parts) to the bilinear terms and . Physically, these terms correspond to a mixing between the longitudinal component of the axial vector mesons and the Goldstone modes arising from chiral symmetry breaking. If the mesons were true gauge fields, i.e., , and if , the theory would be invariant under gauge transformations. In this case, the mixing could be removed by an ’t Hooft gauge-fixing term

 LGF=−12ξ(∂μωμ)2−12ξ(∂μ→ρμ)2−12ξ(∂μfμ1+ξgϕη)2−12ξ(∂μ→aμ1+ξgϕ→π)2. (20)

In unitary gauge, , the pseudoscalar particles (the Goldstone modes from spontaneously breaking to ) would become infinitely heavy, i.e, they are no longer dynamical degrees of freedom.

Since , the ’t Hooft gauge-fixing procedure is not at our disposal. We follow a method commonly used in the literature Gasiorowicz:1969kn (); Ko:1994en (); Pisarski:1994yp (), which is a redefinition of the axial fields,

 fμ1 = f′1μ+w∂μη, (21a) →aμ1 = →a′μ1+w∂μ→π, (21b)

where the new fields ared denoted by a prime (which is later omitted) and is defined such that the bilinear terms mixing the axial vector and pseudoscalar fields are eliminated,

 w=gϕm21+(gϕ)2. (22)

One could also perform a redefinition of the axial vector fields using covariant derivatives Gasiorowicz:1969kn (); Ko:1994en (),

 fμ1 = f′1μ+w(∂μη−g→aμ1⋅→a0), (23a) →aμ1 = (23b)

Note, however, that the old fields appear in the covariant derivates and couple the set of equations. Solving for the old fields in terms of the new one obtains

 fμ1 = (f′1μ+w{∂μη−g→a0⋅[→a′μ1+w(∂μ→π+g→ρμ×→π)]})(1−g2w2→a20)−1, (24a) →aμ1 = (→a′μ1+w{∂μ→π−g[→π×→ρμ+→a0(f′μ1+w∂μη)]})⋅(1−g2w2→a0→a0)−1. (24b)

Note that the term on the right-hand side in Eq. (24b) is a dyadic product, and consequently the corresponding term in parentheses a matrix in isospin space, which has to be multiplied from the left with the preceding isospin vector. The ensuing set of equations couples different isospin components of the old and new fields. All this can be avoided by simply performing the redefinition with the partial derivative, Eq. (21). It should be noted that such complications did not arise in previous treatments Gasiorowicz:1969kn (); Ko:1994en (), because they did not consider the and mesons. In that case, the redefinition of the field simply reads It has been noted in Ref. Ko:1994en () that this gives the correct seagull term when coupling the vector mesons to the photon. However, in this work we are not concerned with coupling the vector mesons to leptonic currents, and thus we restrict the consideration to the simpler version (21) of the redefinition. A different method to cope with the problem is to define a non-diagonal propagator Urban:2001ru ().

After the redefinition (21), the kinetic terms of the pseudoscalar mesons acquire a wave-function renormalization,

 12(∂μπa)2⟶12Z−2(∂μπa)2, (25)

where

 Z2=m21+(gϕ)2m21. (26)

In order to have correctly normalized asymptotic states, we have to redefine the pseudoscalar fields,

 πa⟶Zπa. (27)

After the redefinition of the pseudoscalar particles the Lagrangian reads

 L = +12[∂μσ+gZ(ηf1μ+wZη∂μη+→π⋅→a1μ+wZ→π⋅∂μ→π)]2 (28) +12[(∂μη)2+g2(σf1μ+wZσ∂μη+→a0⋅→a1μ+wZ→a0⋅∂μ→π)2 +12(−2g(Z∂μη−gϕfμ1−gϕwZ∂μη)(σf1μ+wZσ∂μη+→a0⋅→a1μ+wZ→a0⋅∂μ→π)] +12[∂μ→a0+g(→ρμ×→a0+Zη→a1μ+wZ2η∂μ→π+Z→πf1μ+wZ2→π∂μη)]2 +12[(∂μ→π)2+g2(Z→π×→ρμ+σ→a1μ+wZσ∂μ→π+→a0f1μ+wZ→a0∂μη)2 +12(−2g(Z∂μ→π−gϕ→aμ1−gϕwZ∂μ→π)⋅(Z→π×→ρμ+σ→a1μ+wZσ∂μ→π+→a0f1μ+wZ→a0∂μη)] −14(∂μων−∂νωμ)2−14(∂μf1ν−∂νf1μ)2 −14{∂μ→ρν−∂ν→ρμ+g[→ρμ×→ρν+→a1μ×→a1ν+wZ(→a1μ×∂ν→π+∂μ→π×→a1ν)+w2Z2∂μ→π×∂ν→π]}2 −14{∂μ→a1ν−∂ν→a1ν+g[→ρμ×→a1ν+→a1μ×→ρν+wZ(→ρμ×∂ν→π+∂μ→π×→ρν)]}2 −12[m20+c+(λ1+3λ22)ϕ2]→a20−Z22[m20−c+(λ1+λ22)ϕ2]→π2 +m212(ω2μ+→ρ2μ)+m21+(gϕ)22(f21μ+→a21μ) −14(λ1+λ22)(σ2+Z2η2+→a20+Z2→π2)2−(λ1+λ22)ϕσ(σ2+Z2η2+→a20+Z2→π2) −λ22[(σ→a0+Z2η→π)2+Z2→a20→π2−Z2(→a0⋅→π)2]−λ2ϕ→a0⋅(σ→a0+Z2η→π) −V(ϕ),

where

 V(ϕ)=12(m20−c)ϕ2+14(λ1+λ22)ϕ4−h0ϕ (29)

is the classical potential energy density. In the derivation of Eq. (28), we have exploited the fact that is the minimum of the potential energy density, i.e.,

 0=dVdϕ=[m20−c+(λ1+λ22)ϕ2]ϕ−h0. (30)

From the Lagrangian (28), one reads off the tree-level masses for the scalar, pseudoscalar, vector, and axial vector mesons,

 m2σ = m20−c+3(λ1+λ22)ϕ2, (31a) m2η = Z2[m20+c+(λ1+λ22)ϕ2], (31b) m2a0 = m20+c+(λ1+3λ22)ϕ2, (31c) m2π = Z2[m20−c+(λ1+λ22)ϕ2], (31d) m2ω = m2ρ=m21, (31e)