Inhomogeneous chiral condensate in the quark-meson model

(1)

Inhomogeneous chiral condensate in the quark-meson model

Prabal Adhikari,^1,* Jens O. Andersen,^2,† and Patrick Kneschke^3,‡

1St. Olaf College, Physics Department, 1520 St. Olaf Avenue, Northfield, Minnesota 55057, USA

2Department of Physics, Faculty of Natural Sciences, NTNU, Norwegian University of Science and Technology,

Høgskoleringen 5, N-7491 Trondheim, Norway

3Faculty of Science and Technology, University of Stavanger, N-4036 Stavanger, Norway (Received 15 February 2017; published 24 July 2017)

The two-flavor quark-meson model is used as a low-energy effective model for QCD to study inhomogeneous chiral condensates at finite baryon chemical potentialμB. The parameters of the model are determined by matching the meson and quark masses, and the pion decay constant to their physical values using the on-shell and modified minimal subtraction schemes. Using a chiral-density wave ansatz for the inhomogeneity, we calculate the effective potential in the mean-field approximation and the result is completely analytic. The size of the inhomogeneous phase depends sensitively on the pion mass and whether one includes the vacuum fluctuations or not. Finally, we briefly discuss the mean-field phase diagram.

DOI:10.1103/PhysRevD.96.016013

I. INTRODUCTION

The phase structure of QCD has been subject of interest since its phase diagram was first conjectured in the 1970s.

Today, we have a relatively good understanding of the phase transition at zero baryon chemical potential μB. At μB ¼0 there is no sign problem and one can use lattice simulations. For2þ1flavors and physical quark masses, the transition is a crossover at a temperature of around 155 MeV[1–4]. Above the transition temperature QCD is in the quark-gluon plasma phase. At temperatures up to a few times the transition temperature, this is a strongly interacting liquid [5]. For higher temperatures, resummed perturbation theory yields results for the thermodynamic functions that are in good agreement with lattice data[6,7].

The situation is less clear at finite density and low temperature. Due to the sign problem, this part of the phase diagram is not accessible to standard Monte Carlo tech- niques based on importance sampling. Only at asymptoti- cally high densities are we confident about the phase and the properties of QCD. In this limit, the ground state of QCD is the color-flavor locked phase which is a color-superconducting phase [8]. The color symmetry is completely broken and all the gluons are screened. The low-energy excitations of this phase are Goldstone modes which can be described by a chiral effective Lagrangian. At medium densities, information about the phase diagram has been obtained mainly by using low-energy effective models that share some features with QCD such as chiral symmetry breaking in the vacuum. Examples of low-energy models

are the Nambu-Jona-Lasinio (NJL) model and the quark- meson (QM) model as well as their Polyakov-loop extended versions PNJL and PQM models.

Details and further motivation of the QM model can be found in[9] and[10], although historically the fermionic degrees of freedom were nucleons instead of quarks. One may object to having both quark and mesonic degrees of freedom present at the same time in the QM model, since quarks are confined at low temperatures. The Polyakov loop is introduced in order to mimic confinement in QCD in a statistical sense by coupling the chiral models to a constantSUðNcÞbackground gauge fieldA^a_μ[11], which is expressed in terms of the complex-valued Polyakov loop variableΦ. Consequently the effective potential becomes a function of the expectation value of the chiral condensate and the expectation value of the Polykov loop, where the latter then serves as an approximate order parameter for confinement. Finally, one adds the contribution to the free energy density from the gluons via a phenomenological Polyakov loop potential[11].

At these lower densities, QCD is still in a color- superconducting phase, but the symmetry-breaking pattern is different[8,12]. The ground state for a given value of the baryon chemical potential is very sensitive to the values of the parameters of the effective models. It turns out that some of the color-superconducting phases are inhomogeneous [8,12,13]. Inhomogeneous phases do not exist only in dense QCD, but also for example in ordinary superconductors and in imbalanced Fermi gases. In the present paper, we reconsider the problem of inhomogeneous chiral-symmetry breaking phases in dense QCD [14,15]within the QM model. To be specific, we focus on a chiral-density wave (CDW). The problem of inhomogeneous phases has been addressed before in the context

*[email protected]

†[email protected]

‡[email protected]

(2)

of the Ginzburg-Landau approach[16–19], the NJL[20–25]

and PNJL models[26,27], the QM model[22,28,29], and the nonlocal chiral quark model[30]. Numerical methods for the calculation of the phase diagram for a general inhomogeneous condensate are available[31,32], but we resort to a chiral-density wave ansatz in order to present analytical results.

Most of the work has been done in the mean-field approximation; however, the properties of the Goldstone modes that are associated with the spontaneous symmetry breaking of space-time symmetries are important as they may destabilize the inhomogeneous phase [18,19]. The destabilization is caused by long-wavelength fluctuations at finite temperature, where long-range order is replaced by algebraic decay of the order parameter. This does not apply at T ¼0 since the long-wavelength fluctuations are sup- pressed in this case.

In the next section, we briefly discuss the QM model and explain how we calculate the one-loop effective potential in the large-N_c limit using the on-shell (OS) and modified minimal subtraction (MS) schemes together with dimensional regularization. We also calculate analytically the medium-dependent part of the effective potential and the quark density at zero temperature. In Sec.III, we present and discuss our results for the different phases. We also discuss the mean-field phase diagram as a function ofTand μ. In Appendix A we calculate some integrals and sum integrals that we need, and in AppendixB, we calculate the parameters of the Lagrangian as functions of physical observables to leading order in the large-N_c expansion.

Finally, in AppendixC, we calculate the effective potential to the same order.

II. QUARK-MESON MODEL AND EFFECTIVE POTENTIAL

The Euclidean Lagrangian of the two-flavor quark- meson model is

L¼1

2½ð∂_μσÞ²þ ð∂_μπÞ² þ1

2m²ðσ²þπ²Þ þ λ

24ðσ²þπ²Þ²−hσ

þψ¯f½∂ −γ⁰μfþgðσþiγ⁵τ·πÞψf; ð1Þ where f¼u, d is the flavor index and μf is the corre- sponding chemical potential. Forμu ¼μd, in addition to a global SUðN_cÞsymmetry, the Lagrangian has a Uð1Þ_B× SUð2Þ_L×SUð2Þ_Rsymmetry in the chiral limit, while away from it, the symmetry is reduced toUð1Þ_B×SUð2Þ_V. For μu≠μd, the symmetry is reduced to Uð1Þ_B×Uð1Þ_I₃_L× Uð1ÞI₃R for h¼0 and Uð1ÞB×Uð1ÞI₃ for h≠0. In the remainder of this paper we choose μu¼μd¼μ¼¹₃μB, whereμis the quark chemical potential andμBis the baryon chemical potential.

In the vacuum, the σ field acquires a nonzero vacuum expectation value, which we denote byϕ0. We next make an ansatz for the inhomogeneity. In the literature, mainly one-dimensional modulations have been considered, for example CDW and soliton lattices. Since the results seem fairly independent of the modulation[28], we opt for the simplest, namely a one-dimensional chiral-density wave.

The ansatz is

σðzÞ ¼ϕ0cosðqzÞ; π3ðzÞ ¼ϕ0sinðqzÞ; ð2Þ where ϕ0 is the magnitude of the wave and qis a wave vector. The mean fields can be combined into a complex order parameter MðzÞ ¼g½σðzÞ þiπ3ðzÞ ¼Δe^iqz, where Δ¼gϕ₀. The dispersion relation of the quarks in the background(2)is known [33]

E² ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p²_∥þΔ²

q q

2 ₂

þp²_⊥; ð3Þ wherep_∥¼p₃andp²_⊥ ¼p²₁þp²₂. In the QCD vacuum, the chiral symmetry is broken by forming pairs of left-handed quarks and right-handed antiquarks (and vice versa). These quark-antiquark pairs have zero net momentum and so the chiral condensate is homogeneous withq¼0. An inhomogeneous chiral condensate in the vacuum would imply the spontaneous breakdown of rotational symmetry. At finite density, it is possible to form an inhomogeneous condensate by pairing a left-handed quark with a right-handed quark with the same momentum. The net momentum of the pair is nonzero, resulting in an inhomogeneous chiral condensate.

A nonzero wave vectorqlowers the energy of the negative branch in(3)and as a result only this branch is occupied by the quarks in this phase[14].

At tree level, the parameters of the Lagrangian(1)m²,λ, g², andhare related to the physical quantitiesm²_σ,m²_π,m_q, andf_π by

m²¼−1

2ðm²_σ−3m²_πÞ; λ¼3ðm²_σ−m²_πÞ f²_π ; ð4Þ g²¼m²_q

f²_π; h¼m²_πf_π: ð5Þ Expressed in terms of physical quantities, the tree-level potential is

V_tree¼1

2f²_πq²Δ² m²_q−1

4f²_πðm²_σ −3m²_πÞΔ² m²_q þ1

8f²_πðm²_σ−m²_πÞΔ⁴

m⁴_q−m²_πf²_π Δ

m_q: ð6Þ The relations in Eqs.(4)–(5)are the parameters determined at tree level and are often used in practical calculations.

However, this is inconsistent in calculations that involve loop corrections unless one uses the OS renormalization

ADHIKARI, ANDERSEN, and KNESCHKE PHYSICAL REVIEW D 96,016013 (2017)

(3)

scheme. In the on-shell scheme, the divergent loop integrals are regularized using dimensional regularization, but the counterterms are chosen differently from the (MS) scheme.

The counterterms in the on-shell scheme are chosen so that they exactly cancel the loop corrections to the self-energies and couplings evaluated on shell, and as a result the renormalized parameters are independent of the renormalization scale and satisfy the tree-level relations(4)–(5). In the MS scheme, the relations (4)–(5) receive radiative corrections and the parameters depend on the renormalization scale. The divergent part of a counterterm in the OS scheme is necessarily the same as the divergent part of the counterterm in the MS scheme. Since the bare parameters

are independent of the renormalization scheme, one can write down relations between the renormalized parameters in the MS and the OS scheme. The latter are expressed in terms of the physical masses and couplings in Eqs.(4)–(5) and we can therefore express the renormalized running parametersm²

MS,λ_MS,g²

MS, andh_MS in the MS scheme in terms of the massesm²_σ,m²_π, andm_q, and the pion decay constantf_π. In Ref.[34], we calculated the parameters in the chiral limit. In this paper we generalize these relations to the physical point, which are derived in AppendixB. The result for the renormalized one-loop effective potential in the large-N_c limit is derived in AppendixC and reads

V_1-loop¼1 2f²_πq²

1− 4m²_qN_c ð4πÞ²f²_π

logΔ²

m²_qþFðm²_πÞ þm²_πF⁰ðm²_πÞ

Δ²

m²_qþ3 4m²_πf²_π

1− 4m²_qN_c

ð4πÞ²f²_πm²_πF⁰ðm²_πÞ Δ²

m²_q

−1 4m²_σf²_π

1þ 4m²qN_c ð4πÞ²f²_π

1−4m²q

m²_σ

Fðm²_σÞ þ4m²q

m²_σ −Fðm²_πÞ−m²_πF⁰ðm²_πÞ

Δ²

m²_q

þ1 8m²_σf²_π

1− 4m²_qN_c ð4πÞ²f²_π

4m²_q m²_σ

logΔ²

m²_q−3 2

−

1−4m²_q m²_σ

Fðm²_σÞ þFðm²_πÞ þm²_πF⁰ðm²_πÞ

Δ⁴

m⁴_q

−1 8m²_πf²_π

1− 4m²_qN_c

ð4πÞ²f²_πm²_πF⁰ðm²_πÞ Δ⁴

m⁴_q−m²_πf²_π

1− 4m²_qN_c

ð4πÞ²f²_πm²_πF⁰ðm²_πÞ

Δ

m_q− N_cq⁴ 6ð4πÞ² þ N_c

3ð4πÞ²

"

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q²

4 −Δ² r

ð26Δ²þq²Þ−12Δ²ðΔ²þq²Þlog

q

2þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q² 4 −Δ² q

Δ

# θ

q 2−Δ

−2N_cT Z

p

flog½1þe^−βðE^−μÞ þlog½1þe^−βðE^þμg; ð7Þ

whereE is given by Eq.(3)and a sum overis implied.

Moreover, Fðp²Þ andF⁰ðp²Þ are defined in AppendixA.

We note that the vacuum part of the effective potential (obtained by setting q¼μ¼T¼0) of Eq. (7) has its minimum atΔ¼m_qby construction, as does the tree-level potential equation(6). The result for the vacuum part of the effective potential is completely analytic and obtained using dimensional regularization. At this point, a few remarks on the regularization of the effective potential are appropriate. A physically meaningful effective potential cannot depend on the wave vectorqwhen the amplitudeΔ vanishes. It is straightforward to show that theT¼μ¼0 part of Eq. (7) satisfies this. The finite T=μ part of the effective potential, i.e. the last line of Eq.(7)also satisfies this, but at finiteTone must show it numerically. AtT¼0, it can be show analytically, see below. If one regularizes the effective potential with a sharp momentum cutoffΛ[35], it is not independent of q for Δ¼0. The residual q dependence in the limit Δ→0 is then an artifact of the regulator which can be dealt with by introducing extra subtraction terms. Different regularization methods are discussed in some detail in [35,36].

In the limit T¼0, we can calculate the medium contribution to the effective potentialV_1-loop analytically.

Since this contribution is finite, the calculation can be done directly in three dimensions. This contribution is given by the zero-temperature limit of the last line in Eq.(7)and is denoted byV^med₁ . We first consider the contribution fromE_þ in Eq.(7), which we denote byV^med_1þ. AtT¼0, this reads

V^med_1þ ¼−2Nc

Z

p

ðμ−E_þÞθðμ−E_þÞ

¼−16Nc

ð4πÞ² Z _∞

0 dp_∥

× Z _∞

0 ðμ−E_þÞθðμ−E_þÞp_⊥dp_⊥: ð8Þ The integral overp_⊥is straightforward to do, but we have to be careful with the upper limit due to the step function. The upper limit, denoted byp^f_⊥, is a function ofp_∥and is given by

ðp^f_⊥Þ²¼μ²− ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p²_∥þΔ²

q þq

2 ₂

: ð9Þ

(4)

Integrating overp_⊥ fromp_⊥ ¼0 top_⊥ ¼p^f_⊥ yields

V^med_1þ ¼−16N_c ð4πÞ²

Z _pf

∥

0

1 6μ³þ1

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p²_∥þΔ²

q þq

2 ₃

−1

2μ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p²_∥þΔ²

q þq

2 ₂

dp_∥; ð10Þ

where the upper limit of integration is denoted byp^f_∥. The upper limit can be found by settingp_⊥ ¼0in the dispersion relation orp^f_⊥ ¼0 in(9) and is therefore given by

p^f_∥¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

μ−q 2

₂

−Δ² s

: ð11Þ

Changing variables tou¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p²_∥þΔ² q

, we obtain

V^med_1þ ¼−16N_c ð4πÞ²

Z _uf þ

Δ

1 6μ³þ1

3

uþq 2

₃

−1 2μ

uþq

2 ₂

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu du u²−Δ²

p ; ð12Þ

where the upper limit isu^f_þ ¼μ−^q₂. In order to get a nonzero contribution, we must haveμ≥Δþ^q₂. Integrating overu, we find

V^med_1þ ¼− 2N_c ð4πÞ²

"

2 3

μ−q 2

₂

−Δ²

s

μþq 2

μ−q

2 ₂

þ1

4Δ²ð13q−10μÞ

þΔ²ðΔ²−2μqþq²Þlogμ−^q₂þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðμ−^q₂Þ²−Δ² q

Δ

# θ

μ−q

2−Δ

: ð13Þ

The second contribution is forE₋in Eq.(7). It is denoted byV^med₁₋ and is found from Eq.(12)by the substitutionq→−q.

This gives

V^med₁₋ ¼−16N_c ð4πÞ²

Z u^f−

ulow

1 6μ³þ1

3 u−q

2 ³−1

2μ

u−q 2

₂

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu du u²−Δ²

p ; ð14Þ

where the upper limit isu^f₋ ¼μþ^q₂and the lower limit isu_low. The lower limit depends on the relative magnitude ofμ,Δ, and^q₂. The different cases are discussed below.

(1) Δ>^q₂: The dispersion relation is shown in the left panel of Fig.1.u_low ¼Δand there is a nonzero contribution if μ−Δþ^q₂>0. This contribution is obtained from(13) by the substitution q→−q. This yields

V^med₁₋ ¼− 2Nc

ð4πÞ²

"

2 3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μþq

2 ₂

−Δ²

r h

μ−q 2

μþq 2

₂

−1

4Δ²ð13qþ10μÞi

þΔ²ðΔ²þ2μqþq²Þlogμþ^q₂þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðμþ^q₂Þ²−Δ² q

Δ

# θ

μþq 2−Δ

: ð15Þ

(2) Δ<^q₂: The dispersion relation is shown in the right panel of Fig.1 (blue curve) and the minimum of jE₋j is at p¼p₀¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q² 4 −Δ² q

and is zero. For p < p₀, we havejE₋j ¼^q₂−u and for p > p₀, we havejE₋j ¼u−^q₂. For Δ<^q₂, we also have to distinguish between the casesμ>^q₂−Δ andμ<^q₂−Δ.

(5)

(a) Ifμ>^q₂−Δ, we have to integrate fromp¼0top¼p_f ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðμþ^q₂Þ²−Δ² q

, or fromu_low ¼Δto u¼μþ^q₂. The green horizontal line indicates the value of the chemical potential and the intersection with the dispersion relation gives the upper limit of integration. This yields

V^med₁₋ ¼− 2N_c ð4πÞ²

"

2 3

μþq 2

₂

−Δ²

s "

μ−q 2

μþq

2 ₂

−1

4Δ²ð13qþ10μÞ

#

þΔ²ðΔ²þ2μqþq²Þlnμþ^q₂þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðμþ^q₂Þ²−Δ² q

Δ þ1

6q

4 −Δ² r

ð26Δ²þq²Þ−2Δ²ðΔ²þq²Þln

q2þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q² 4 −Δ² q

Δ

# θ

μ−q

2þΔ

: ð16Þ

(b) Ifμ<^q₂−Δ, we must integrate fromp¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðμ−^q₂Þ²−Δ² q

top¼p_f¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðμþ^q₂Þ²−Δ² q

, or fromu_low ¼^q₂−μ tou^f₋ ¼μþ^q₂. The value of the chemical potential is indicated by the orange line and the intersection with the dispersion relation gives the upper and lower limits of integration. This yields

V^med₁₋ ¼− 2Nc

ð4πÞ² (

−2 3

μ−q 2

₂

−Δ²

s "

μþq 2

μ−q

2 ₂

þ1

4Δ²ð13q−10μÞ

#

þ2 3

μþq 2

₂

−Δ²

s "

μ−q 2

μþq

2 ₂

−1

4Δ²ð13qþ10μÞ

#

þΔ²ðΔ²−2μqþq²Þln

q

2−μþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðμ−^q₂Þ²−Δ² q

Δ

þΔ²ðΔ²þ2μqþq²Þlnμþ^q₂þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðμþ^q₂Þ²−Δ² q

Δ þ1

6q

4 −Δ² r

ð26Δ²þq²Þ−2Δ²ðΔ²þq²Þln

q

2þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q² 4 −Δ² q

Δ

) θ

q

2−μ−Δ

: ð17Þ

The expression for V^med_1þ and the different expressions forV^med₁₋ can be combined to give our final result for the matter- dependent part of the Eq.(7)

FIG. 1. Left: Dispersion relationE₋for^q₂<Δ. The horizontal orange line is forμ>Δ−^q₂. Right: Dispersion relationE₋for^q₂<Δ. The horizontal green line is for the caseμ>^q₂−Δand the horizontal orange line is for the caseμ<^q₂−Δ. See main text for discussion of the regions of integration in the different cases.

(6)

V^medþ_1þ þV^med₁₋ ¼− 2N_c ð4πÞ²

(2 3

μ−q 2

₂

−Δ²

s "

μþq 2

μ−q

2 ₂

þ1

4Δ²ð13q−10μÞ

# sign

μ−q

2

þΔ²ðΔ²−2μqþq²Þlogjμ−^q₂j þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðμ−^q₂Þ²−Δ² q

Δ

) θ

μ−q 2

−Δ

− 2Nc

ð4πÞ² (2

3

μþq 2

₂

−Δ²

s

μ−q 2

μþq

2 ₂

−1

4Δ²ð13qþ10μÞ

þΔ²ðΔ²þ2μqþq²Þlogμþ^q₂þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðμþ^q₂Þ²−Δ² q

Δ

) θ

μþq

2−Δ

− N_c 3ð4πÞ²

"

q

4 −Δ² r

ð26Δ²þq²Þ−12Δ²ðΔ²þq²Þlog

q2þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q² 4 −Δ² q

Δ

# θ

q 2−Δ

: ð18Þ SettingΔ¼0in Eq.(18), it is straightforward to verify that the matter part of the effective potential is independent ofq, as discussed after Eq.(7). Moreover, we note that the last line of Eq.(18)cancels against the penultimate line in Eq.(7)in the complete thermodynamic potential.

In the limitT ¼0, we can obtain an analytic result for the quark densityn_q as well. It is given by n_q¼−∂V^med_1þ

∂μ −∂V^med₁₋

∂μ ¼2N_c Z

p½θðμ−E_þÞ þθðμ−E₋Þ

¼ 4N_c ð4πÞ²

"

2 3

μþq 2

₂

−Δ²

s

μ²−Δ²þ1

4μq−q² 8

þΔ²qlogμþ^q₂þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðμþ^q₂Þ²−Δ² q

Δ

# θ

μþq

2−Δ

þ 4N_c ð4πÞ²

"

2 3

μ−q 2

₂

−Δ²

s

μ²−Δ²−1

4μq−q² 8

sign

μ−q

2

−Δ²qlogjμ−^q₂j þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðμ−^q₂Þ²−Δ² q

Δ

#

×θ μ−q 2 −Δ

: ð19Þ

The quark density (19) is also independent of the wave vector q when the amplitudeΔ is set to zero, as can be verified by inspection.

III. RESULTS AND DISCUSSION

In the numerical work, we set N_c¼3 everywhere. We use a constituent quark mass m_q¼300MeV. Since the sigma mass is not very well known experimentally [37], one typically allows it to vary betweenm_σ ¼400MeV and m_σ ¼800MeV. We choosem_σ ¼600MeV. At the physical point we takem_π¼140MeV and for the pion decay constant we usef_π¼93MeV. In the chiral limit the pion mass is zero.

It is known from earlier studies in the homogeneous case that vacuum fluctuations play an important role. If we omit the quantum fluctuations, the phase transition in the chiral limit is first order in the entireμ-Tplane. If they are included the transition is first order forT¼0and second order forμ¼0. The first-order line starting on theμ axis ends at a tricritical point. In the inhomogeneous case, we therefore examine the importance of these fluctuations as

well. In Fig. 2, we show the phase diagram in the μ-T plane in the chiral limit without vacuum fluctuations.

The solid lines indicate a first-order transition while the dashed line indicates a second-order transition. The region between the two red lines is the inhomogeneous phase.

The black line is the first-order transition line in the homogeneous case.

In Fig.3, we show the phase diagram in theμ-Tplane in the chiral limit where vacuum fluctuations are included.

The inhomogeneous phase in the entireμ-Tplane has now been replaced by a small region at low temperatures. The second-order line starting at μ¼0 ends at the Lifschitz point indicated by the full red circle. Sincem_σ ¼2mq this is also the position of the tricrital point[16]. The region between the two red lines is the inhomogeneous phase.

Comparing Figs.2 and 3, we see the dramatic effects of including the fermionic vacuum fluctuations.

Although we find an inhomogeneous phase for finite temperature, it has to be mentioned that this phase might not survive if effects beyond the mean-field approximation are included. There is evidence that in the chiral limit the existence of the Lifshitz point is simply an artifact of the

(7)

mean-field approximation as pointed out in Ref. [19]

and Ref. [38].

In Ref.[19], it is shown that the Goldstone bosons that arise from the breaking of the translational and rotational

symmetry have a quadratic dispersion relation in some directions and a linear dispersion relation in other directions. At finite temperature, the former leads to strong long- wavelength fluctuations (phase fluctuations) that destroy off-diagonal long-range order altogether. Long-range order is replaced by quasi-long-range order where the order parameter is decaying algebraically. AtT ¼0, the phase fluctuations are not strong enough to destroy this order and there is a true condensate.

In Fig. 4, we show the modulus Δ (solid blue line) and the wave vectorq(dashed red line) as functions ofμat T¼0in the chiral limit withm_σ ¼2m_q¼600MeV. The left panel shows the results without quantum fluctuations and the right panel with. The transition from a phase with homogeneous condensate to a phase with a chiral-density wave is first order, while the transition to a chirally symmetric phase is second order. In the case with no vacuum fluctuations, the vacuum state, i.e. with zero quark density extends all the way to the transition to the CDW phase which extents fromμ¼291MeV toμ¼384MeV.

This is not the case if we include quantum fluctuations. The vacuum state extends from μ¼0 up to μ¼291MeV, where there is a transition to a homogeneous phase with a nonzero quark density andΔdecreases. This phase extends up toμ≈322.7MeV. In both cases, the vanishing quark density forμ<μc, where μc is the critical density for the transition to either the CDW phase (left panel) or another homogeneous phase (right panel) with decreasingΔ is an example of the silver-blaze property. In this phase, all physical quantities are independent of the quark chemical potential[39].

In Fig.5, we show the modulusΔ(solid blue line) and the wave vector q (dashed red line) as functions of μ at T¼0 at the physical point with m_σ ¼2mq¼600MeV andm_π¼140MeV. In the left panel, we have omitted the quantum fluctuations and in the right panel, they have been included. Without quantum corrections, there is a transition from a phase with a homogeneous chiral condensate to a phase with a chiral-density wave. This transition is first order. Again, this is in contrast to the case where we include the vacuum fluctuations; the vacuum phase extends from FIG. 2. The phase diagram in theμ-Tplane form_q¼300MeV

andm_σ¼600MeV in the chiral limit without quantum fluctuations. A dashed line indicates a second-order transition, while a solid line indicates a first-order transition. The region between the red lines is the inhomogeneous phase.

FIG. 3. The phase diagram in theμ-Tplane form_q¼300MeV and m_σ¼600MeV in the chiral limit including vacuum fluctuations. A dashed line indicates a second-order transition, while a solid line indicates a first-order transition. The region between the red lines is the inhomogeneous phase.

FIG. 4. GapΔ(solid blue line) and wave vectorq(dashed red line) as functions of the quark chemical potentialμin the chiral limit, at T¼0, and for m_σ¼2m_q¼600MeV. Left panel is without vacuum fluctuations and right panel with vacuum fluctuations.

(8)

μ¼0toμ¼300MeV and then a second order transition occurs to a phase with a homogeneous quark chiral condensate and a nonzero quark density. In this phase, the chiral condensate decreases. There are two more transitions, one from the phase with a homogeneous chiral condensate (and a nonzero quark density) to a phase with an inhomogeneous phase and a transition to a chirally symmetric phase. Both transitions are first order.

The present work can be extended in different directions.

For example, it would be of interest to study inhomogeneous phases in a constant magnetic background.

ACKNOWLEDGMENTS

The authors would like to thank Stefano Carignano for useful discussions. P. A. would like to acknowledge the research travel support provided through the Professional Development Grant and would like to thank the Faculty Life Committee and the Dean’s Office at St. Olaf College. P. A. would also like to acknowledge the computational support provided through the Computer Science Department at St. Olaf College and thank Richard Brown, Tony Skalski and Jacob Caswell. P. A. and P. K.

would like to thank the Department of Physics at Norwegian University for Science and Technology (NTNU) for kind hospitality during the latter stages of this work.

APPENDIX A: INTEGRALS AND SUM INTEGRALS

In the imaginary-time formalism for thermal field theory, a fermion has Euclidean 4-momentum P¼ ðP₀;pÞ with P²¼P²₀þp². The Euclidean energy P₀ has discrete values: P₀¼ ð2nþ1ÞπTþiμ, where n is an integer.

Loop diagrams involve a sum over P₀ and an integral over spatial momenta p. We define the dimensionally regularized sum integral by

XZ

fPg¼TX

P₀

Z

p

; ðA1Þ

where the integral is ind¼3−2ϵ dimensions Z

p

¼

e^γ^EΛ² 4π

_ϵZ d^dp ð2πÞ^d

¼

e^γ^EΛ² 4π

_ϵZ

d^d−1p_⊥ ð2πÞ^d−1

Z dp_∥ 2π

¼

e^γ^EΛ² 4π

_ϵZ

d^d⁻¹p_⊥ ð2πÞ^d−1 1 π

Z _∞

Δ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidu u u²−Δ²

p : ðA2Þ

HereΛis the renormalization scale in the modified minimal subtraction scheme MS, p_∥¼p₃, p²_⊥¼p²₁þp²₂ and u¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p²_∥þΔ² q

. We need

I₀¼−XZ

fPglog½P²₀þE²: ðA3Þ Summing over the Matsubara frequenciesP₀, we obtain

I₀¼− Z

p

fEþTlog½1þe^{−βðE−μÞ}

þTlog½1þe^{−βðEþμÞ}g: ðA4Þ The integral in Eq. (A4) is needed for E¼E and is calculated by expanding it in powers ofq, see AppendixC.

The integrals that appear are Z

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u²þp²_⊥

q ¼− Δ⁴

ð4πÞ²

e^γ^EΛ² Δ²

_ϵ

Γð−2þϵÞ;

¼− Δ⁴ 2ð4πÞ²

Λ² Δ²

_ϵ 1 ϵþ3

2þOðϵÞ

; ðA5Þ Z

p

p²_⊥

ðu²þp²_⊥Þ³²¼− 4Δ² ð4πÞ²

e^γ^EΛ² Δ²

_ϵ ΓðϵÞ

¼− 4Δ² ð4πÞ²

Λ² Δ²

_ϵ 1 ϵþOðϵÞ

; ðA6Þ FIG. 5. GapΔ(solid blue line) and wave vectorq(dashed red line) as functions of the quark chemical potentialμat the physical point forT¼0and m_σ¼2mq¼600MeV. Left panel is without vacuum fluctuations and right panel with vacuum fluctuations.

(9)

Z

p

p²_⊥ð4u²−p²_⊥Þ ðu²þp²_⊥Þ⁷²

¼− 16 3ð4πÞ²

e^γ^EΛ² Δ²

_ϵ

ð−1þϵÞΓð1þϵÞ ¼ 1

3π²þOðϵÞ:

ðA7Þ We also need some integrals in D¼4−2ϵ dimensions.

Specifically, we need the integrals

Aðm²Þ ¼ Z

p

1 p²−m²

¼ im² ð4πÞ²

Λ² m²

_ϵ 1

ϵþ1þOðϵÞ

; ðA8Þ

Bðp²Þ ¼ Z

k

1

ðk²−m²_qÞ½ðkþpÞ²−m²_q

¼ i ð4πÞ²

Λ² m²_q

_ϵ 1

ϵþFðp²Þ þOðϵÞ

; ðA9Þ

B⁰ðp²Þ ¼ i

ð4πÞ²F⁰ðp²Þ; ðA10Þ where the functions Fðp²Þ andF⁰ðp²Þ are

Fðp²Þ ¼− Z ₁

0 dxlog p²

m²_qxðx−1Þ þ1

¼2−2rarctan 1

r

; ðA11Þ

F⁰ðp²Þ ¼ 4m²_qr

p²ð4m²_q−p²Þarctan 1

r

− 1

p²; ðA12Þ were we defined r¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4m²q

p² −1 q

.

APPENDIX B: PARAMETER FIXING In this appendix, we find the relation between the parameters in the Lagrangian(1)and the physical observables using the MS and OS renormalization schemes.

The sigma and pion self-energies are given by

Σ_σðp²Þ ¼−8g²N_c

Aðm²_qÞ−1

2ðp²−4m²_qÞBðp²Þ

þ4λgϕ₀N_cm_q

m²_σ Aðm²qÞ; ðB1Þ Σ_πðp²Þ ¼−8g²N_c

Aðm²_qÞ−1

2p²Bðp²Þ

þ4λgϕ₀N_cm_q

3m²_σ Aðm²qÞ; ðB2Þ

where the last term of Eqs.(B1) and(B2) is the tadpole contribution to the self-energies, and where the integrals Aðm²ÞandBðp²Þare defined in Eqs.(A8)and(A9). We do not need the quark self-energy since it is of orderN⁰_c. Thus Z_ψ ¼1andδm_q¼0at this order. The inverse propagator for the sigma or pion can be written as

p²−m²_σ;π−iΣ_σ;πðp²Þ þcounterterms: ðB3Þ In the on-shell scheme, the physical mass is equal to the renormalized mass in the Lagrangian.¹Thus we can write

Σ_σ;πðp²¼m²_σ;πÞ þcounterterms¼0: ðB4Þ The residue of the propagator on shell equals unity, which implies

∂

∂p²Σ_σ;πðp²Þj_p²_¼m²_σ;πþcounterterms¼0: ðB5Þ The large-N_c contribution to the one-point function is

δΓ^ð1Þ ¼−8g²N_cϕ₀Aðm²_qÞ þiδt; ðB6Þ where δt is the tadpole counterterm. The equation of motion is equivalent to the vanishing one-point function, which yields on tree levelt¼h−m²_πϕ₀¼0. This has to hold also on one-loop level, which gives the renormalization condition

δΓ^ð1Þ ¼0: ðB7Þ

The counterterms are given by

Σ^ct1_σ ðp²Þ ¼i½δZ_σðp²−m²_σÞ−δm²_σ; ðB8Þ Σ^ct1_π ðp²Þ ¼i½δZ_πðp²−m²_πÞ−δm²_π; ðB9Þ

Σ^ct2_σ ¼3Σ^ct2_π ¼−iλϕ₀

m²_σ δt; ðB10Þ δt¼δh−f_πδm²_π−m²_πδf_π; ðB11Þ where the counterterm in Eq. (B10) cancels the tadpole contribution to the self-energies. The on-shell renormalization constants are given by the self-energies and their derivatives evaluated at the physical mass. This yields

δm²_σ ¼−iΣσðm²_σÞ; ðB12Þ δm²_π ¼−iΣπðm²_πÞ; ðB13Þ

1In defining the mass, we ignore the imaginary parts of the self-energy.

(10)

δZ_σ ¼i ∂

∂p²Σ_σðp²Þj_p2¼m²_σ; ðB14Þ δZ_π¼i ∂

∂p²Σ_πðp²Þj_p2¼m²_π: ðB15Þ From Eqs.(B1)–(B6), we find²

δm²_σ ¼8ig²N_c

Aðm²_qÞ−1

2ðm²_σ−4m²_qÞBðm²_σÞ

; ðB16Þ

δm²_π ¼8ig²N_c

Aðm²_qÞ−1

2m²_πBðm²_πÞ

; ðB17Þ δZ_σ ¼4ig²N_c½Bðm²_σÞ þ ðm²_σ −4m²_qÞB⁰ðm²_σÞ; ðB18Þ δZ_π¼4ig²N_c½Bðm²_πÞ þm²_πB⁰ðm²_πÞ; ðB19Þ δt¼−8ig²N_cf_πAðm²_qÞ: ðB20Þ The countertermsδm²,δλ,δg², andδhcan be expressed in terms of the counterterms δm²_σ, δm²_π, δZ_π, and δt. Since

there is no correction to the quark-pion vertex in the large- N_c limit, we find

δg²¼−g²δZ_π: ðB21Þ Since there is no correction to the quark mass in the large- N_c limit, we find δm_q¼0or

δg²¼−g²δf²_π

f²_π : ðB22Þ This yieldsδZ_π¼^δf_f2²^π

π. From this relation, Eq.(B11), and h¼tþm²_πf_π, one finds

δm²¼−1

2ðδm²_σ−3δm²_πÞ; ðB23Þ δλ¼3δm²_σ−δm²_π

f²_π −λδZ_π; ðB24Þ δh¼δtþf_πδm²_πþ1

2m²_πf_πδZ_π: ðB25Þ The expressions for the counterterms are

δm²_OS¼8ig²N_c

Aðm²_qÞ þ1

4ðm²_σ−4m²qÞBðm²_σÞ−3

4m²_πBðm²_πÞ

¼δm²_divþ4g²N_c ð4πÞ²

m²logΛ²

m²_q−2m²q−1

2ðm²_σ−4m²qÞFðm²_σÞ þ3

2m²_πFðm²_πÞ

; ðB26Þ

δλOS¼−12ig²N_c

f²_π ðm²_σ−4m²_qÞBðm²_σÞ þ12ig²N_c

f²_π m²_πBðm²_πÞ−4iλg²N_c½Bðm²_πÞ þm²_πB⁰ðm²_πÞ

¼δλdivþ12g²N_cm²_σ ð4πÞ²f²_π

1−4m²_q m²_σ

logΛ²

m²_qþFðm²_σÞ

þlogΛ²

−12g²N_cm²_π ð4πÞ²f²_π

2logΛ²

m²_qþ2Fðm²_πÞ þm²_πF⁰ðm²_πÞ

; ðB27Þ

δg²_OS¼−4ig⁴N_c½Bðm²_πÞ þm²_πB⁰ðm²_πÞ ¼δg²_divþ4g⁴N_c ð4πÞ²

logΛ²

; ðB28Þ

δh^OS¼−2ig²N_cm²_πf_π½Bðm²_πÞ−m²_πB⁰ðm²_πÞ

¼δh_divþ2g²N_cm²_πf_π ð4πÞ²

logΛ²

m²_qþFðm²_πÞ−m²_πF⁰ðm²_πÞ

; ðB29Þ δZ^OS_σ ¼δZ_σ;div−4g²N_c

ð4πÞ²

logΛ²

m²_qþFðm²_σÞ þ ðm²_σ−4m²_qÞF⁰ðm²_σÞ

; ðB30Þ

δZ^OS_π ¼δZ_π;div−4g²N_c ð4πÞ²

logΛ²

; ðB31Þ whereFðm²ÞandF⁰ðm²Þare defined in AppendixA, and the divergent quantities are

δm²_div¼4m²g²N_c

ð4πÞ²ϵ ; δλdiv¼ 8N_c

ð4πÞ²ϵðλg²−6g⁴Þ;

δg²_div¼ 4g⁴N_c

ð4πÞ²ϵ; ðB32Þ

2The self-energies are without the tadpole contributions.