Gravitational catalysis of chiral and color symmetry breaking of quark matter in hyperbolic space

Gravitational catalysis of chiral and color symmetry breaking of quark matterin hyperbolic space

D. Ebert,1,2,* A.V. Tyukov,3 and V. Ch. Zhukovsky3,†

1Joint Institute for Nuclear Research, Dubna, R-141980, Russia2Institut fur Physik, Humboldt-Universitat zu Berlin, 12489 Berlin, Germany

3Faculty of Physics, Department of Theoretical Physics, Moscow State University, 119991, Moscow, Russia(Received 26 November 2008; revised manuscript received 5 August 2009; published 15 October 2009)

We study the dynamical breaking of chiral and color symmetries of dense quark matter in the ultrastatic

hyperbolic spacetime R �H3 in the framework of an extended Nambu–Jona-Lasinio model. On the basis

of analytical expressions for chiral and color condensates as functions of curvature and temperature, the

phenomenon of dimensional reduction and gravitational catalysis of symmetry breaking in strong

gravitational field is demonstrated in the regime of weak coupling constants. In the case of strong

couplings it is shown that curvature leads to small corrections to the flat-space values of condensate and

thus enhances the symmetry breaking effects. Finally, using numerical calculations phase transitions

under the influence of chemical potential and negative curvature are considered and the phase portrait of

the system is constructed.

DOI: 10.1103/PhysRevD.80.085019 PACS numbers: 11.30.Qc, 04.62.+v, 12.39.�x

I. INTRODUCTION

Dynamical breaking of chiral and color symmetries hasbeen successfully studied within field theories of theNambu–Jona-Lasinio (NJL) type with four-fermion inter-actions [1]. They are also quite useful in describing thephysics of light mesons (see e.g. [2–4] and referencestherein) and diquarks [5,6].

The possibility for the existence of the color supercon-ductive (CSC) phase with a nonzero colored diquark con-densate was proposed both in the region of high baryondensities [7–9] and moderate densities [10–14]. In theframework of NJL models the CSC phase formation hasgenerally been considered as a dynamical competitionbetween diquark hqqi and usual quark-antiquark conden-sation h �qqi. Special attention has been paid to the catalyz-ing influence of external fields on chiral symmetrybreaking (�SB) in the regime of weak coupling [15–17](constant magnetic field), [18,19] (chromomagnetic fields)and on the condensation of diquarks [20,21] (chromomag-netic fields). In particular, it was demonstrated that in astrong field the considered symmetry is dynamically bro-ken for an arbitrary weak attraction between quarks. Thephysical explanation for this is that the effect of dynamicalsymmetry breaking is accompanied by an effective low-ering of dimensionality in strong fields, where the numberof reduced units of dimensions depends on the concretetype of the background field. Dynamical symmetry break-ing in a magnetic field in spacetimes of dimension higherthan four was also considered in [22].

For cosmological and astrophysical applications, it isalso interesting to study the influence of spacetime curva-

ture on symmetry breaking. One of the ways to account forthe effects of gravity is to use the adiabatic expansion ofGreen’s functions in the vicinity of a fixed point in powersof small curvature (see, for example, the review [23]).However, since second-order phase transitions take placein the infrared region, the considered processes may be-come sensitive to the global structure of spacetime, andthen one needs exact expressions for the propagators in thecurved spacetime.In particular, an exact solution can be found for space-

time with high symmetry. A variety of examples of �SB indifferent symmetric spaces both at zero and finite tempera-ture has been considered in the literature (see e.g. [23]).One of the well-known examples of spaces with constantpositive curvature is the Einstein universe of the form R �S3. �SB at finite temperature and chemical potential in thestatic Einstein universe was recently considered in [24].Further investigations of quark matter in this gravitationalbackground concerning, in particular, diquark and pioncondensation were performed in [25,26]. There, the posi-tive curvature was shown to lead to the restoration ofbroken symmetries, thus acting in a similar way as thetemperature. Another symmetric space, where exact solu-tion may be found, is the ultrastatic hyperbolic spacetimeR �H3. It was noted that in spaces with negative curva-ture, the chiral symmetry is broken even at weak couplingconstant (see e.g. [23]). The detailed analysis of the heatkernel in hyperbolic space showed that the physical reasonfor this phenomenon is the effective dimensional reductionfor fermions in the infrared region [27]. Recently, thecombined influence of gravitational and magnetic fieldson �SB in the special case of the 2D space of negativecurvature, i.e., on the Lobachevsky plane, was consideredin [28]. As discussed by these authors, the study of theeffects of surface curvature may be important for some

*[email protected]†[email protected]

PHYSICAL REVIEW D 80, 085019 (2009)

1550-7998=2009=80(8)=085019(12) 085019-1 � 2009 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.80.085019

condensed matter systems concerning, in particular, thequantum Hall effect in graphene [29,30].

The aim of this paper is to study the effects of dynamicalbreaking of chiral and/or color symmetries in dense quarkmatter under the influence of negative curvature of ultra-static hyperbolic spacetime R �H3. In the framework ofan extended Nambu–Jona-Lasinio model, including ð �qqÞ-and ðqqÞ-interactions, an exact in curvature expression forthe thermodynamic potential is derived, which contains allthe necessary information about the condensates. Basingupon the analytical solutions of gap equations for quarkand diquark condensates we show that in strong gravita-tional field there arises a gravitational catalysis of dynami-cal symmetry breaking and chiral and color symmetriesmay be simultaneously broken even for weakly interactingquarks. This situation resembles the influence of magneticor chromomagnetic fields on symmetry breaking in flatcase. We also consider the role of finite temperature andfind that for any fixed value of curvature there exists acritical temperature at which chiral and color symmetriesbecome restored. Moreover, using numerical calculations,phase transitions under the influence of a chemical poten-tial are investigated and the phase portrait of the system atzero temperature is constructed. Finally, it is shown that inthe strong coupling regime negative curvature enhances thevalues of condensates as compared to the flat case. Ouranalysis demonstrates that negative curvature acts on chiraland color condensates in a way similar to that of a magneticfield in flat spacetime.

II. THE EXTENDED NJL MODEL IN CURVEDSPACETIME

Let us briefly remind the reader of the basic definitionsnecessary for the description of fermions in curved space-time. In 4-dimensional curved spacetime with signatureðþ;�;�;�Þ, the line element is written as

ds2 ¼ �a bea�e

b�dx

�dx�:

The gamma-matrices ��, metric g�� and the vierbein e�a ,

as well as the definitions of the spinor covariant derivative

r� and spin connection !a b� are given by the following

relations [31,32]:

f��ðxÞ; ��ðxÞg ¼ 2g��ðxÞ; f�a; �bg ¼ 2�a b;

�a b ¼ diagð1;�1;�1;�1Þ; g��g�� ¼ �

��;

g��ðxÞ ¼ e�a ðxÞe�aðxÞ; ��ðxÞ ¼ ea�ðxÞ�a:

(1)

r� ¼ @� þ��; �� ¼ 12!

ab� �a b; �a b ¼ 1

4½�a;�b�;!a b

� ¼ 12e

aeb�½C�� C�� C��;C�� ¼ ea@½�e��a: (2)

Here, the index a refers to the flat tangent space defined bythe vierbein at the spacetime point x, and the �aða ¼

0; 1; 2; 3Þ are the usual Dirac gamma-matrices ofMinkowski spacetime. Moreover, �5 is defined as usual(see, e.g., [31,33,34]), i.e. to be the same as in flat space-time and thus independent of spacetime variables.The extended NJL model which includes the ð �qqÞ- and

ðqqÞ-interactions of colored up- and down-quarks can beused to describe the formation of the color superconduct-ing phase. For the color group SUcð3Þ, its Lagrangian takesthe form

L ¼ �q½i��r� þ��0�qþ G1

2Nc

½ð �qqÞ2 þ ð �qi�5 ~qÞ2�

þG2

Nc

½i �qc"�b�5q�½i �q"�b�5qc�: (3)

Here, Nc ¼ 3 is the number of colors, G1 and G2 arecoupling constants (their particular values will be chosenin what follows), � is the quark chemical potential, qc ¼C �qt, �qc ¼ qtC are charge-conjugated bispinors (t standsfor the transposition operation). The charge conjugation

operation is defined with the help of the operator C ¼i�2�0, where the flat-space matrices �2 and �0 are used(see, e.g., [31]). The quark field q � qi� is a doublet offlavors and triplet of colors with indices i ¼ 1, 2; � ¼ 1, 1,2, 3. Moreover, ~ � ð1; 2; 3Þ denote Pauli matrices inthe flavor space; ð"Þik � "ik, ð�bÞ� � �� b are the totallyantisymmetric tensors in the flavor and color spaces,respectively.Next, by applying the usual bosonization procedure, we

obtain the linearized version of the Lagrangian (3) withcollective boson fields �, ~� and �,

~L ¼ �q½i��r� þ��0�q� �qð�þ i�5 ~ ~�Þq� 3

2G1

ð�2 þ ~�2Þ � 3

G2

��b�b

� ��b½iqtC"�b�5q� � �b½i �q"�b�5C �qt�: (4)

The Lagrangians (3) and (4) are equivalent, as can beseen by using the Euler-Lagrange equations for the bosonicfields, from which it follows that

�b ¼ �G2

3iqtC"�b�5q; � ¼ �G1

3�qq;

~� ¼ �G1

3�qi�5 ~q:

(5)

The fields � and ~� are color singlets, and �b is a colorantitriplet and flavor singlet.In what follows, it is convenient to consider the effective

action for boson fields, which is expressed through theintegral over quark fields

expfiSeffð�; ~�;�b;��bÞg ¼ NZ½dq�½d �q�

� exp

�iZ

d4xffiffiffiffiffiffiffi�g

p ~L�; (6)

D. EBERT, A. V. TYUKOV, AND V.CH. ZHUKOVSKY PHYSICAL REVIEW D 80, 085019 (2009)

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where

Seffð�; ~�;�b;��bÞ ¼ �Z


p �3ð�2 þ ~�2Þ

2G1

þ 3�b��b

G2

�þ Sq; (7)

with Sq standing for the quark contribution to the effective

action.In the mean field approximation, the fields�, ~�,�b,��b

can be replaced by their ground state averages: h�i, h ~�i,h�bi, and h��bi, respectively. Let us choose the followingground state of our model:

h�1i ¼ h�2i ¼ h ~�i ¼ 0:

If h�i � 0, the chiral symmetry is broken dynamically, andif h�3i � 0, the color symmetry is broken. Evidently, thischoice breaks the color symmetry down to the residualgroup SUcð2Þ. (In the following we denote h�i, h�3i � 0by letters �, �.)

The quark contribution has the following form (for moredetails see [21])

Sqð�;�Þ ¼ �i lnDet½ir � �þ��0� � i

2lnDet½4j�j2

þ ð�ir � �þ��0Þðir � �þ��0Þ�: (8)

Here, the first determinant is over spinor, flavor, and coor-dinate spaces, and the second one is over the two-

dimensional color space as well, and r ¼ ��r�.

Note that the effective potential of the model, the globalminimum point of which will determine the quantities �and �, is given by Seff ¼ �Veff

Rd4x

ffiffiffiffiffiffiffi�gp

, where

Veff ¼ 3�2

2G1

þ 3��

G2

þ ~Veff ; ~Veff ¼ �Sqv;

v ¼Z


p:

(9)

III. THERMODYNAMIC POTENTIAL

In this work we will consider the ultrastatic spacetimeR �H3 with constant negative curvature. The metric isgiven by

ds2 ¼ dt2 � a2ðd�2 þ sinh2�d�2Þ; (10)

where a is the radius of the hyperboloid which is related tothe scalar curvature by the relation R ¼ � 6

a2, and d�2 is

the metric on the two-dimensional unit sphere.

Let us next introduce the one-particle Hamiltonian H ¼�i ~� ~rþ��0, where ~� ¼ �0 ~�. Using this operator thequark contribution to the effective action can be writtenin the following form (for more details see [25]):

Sq ¼ � i

2flnDet½H2 � ðp0 þ�Þ2� þ 2 lnDet½4j�j2

þ ðH ��Þ2 � p20�g; (11)

where p0 ¼ i@0, and we have summed over colors (theDet-operator now does not include color space).In order to calculate the effective action one needs to

solve the equation for eigenfunctions of the Hamiltonian Hin hyperbolic space

ð�i ~� ~rþ��0Þ� ¼ �: (12)

Decomposing � into upper c ;1 and lower c ;2 com-

ponents and further separating variables by setting c 1;2 ¼f1;2ð�Þ�1;2, the solutions can be found in terms of hyper-

geometric functions (for details see e.g. [27,34–36]).Let us now consider the diagonal element:

trh ~xj ln½H2 � ðp0 þ�Þ2�j ~xi ¼Z 1

0d�

X�¼�

Xslm

ln½ð�ð�ÞÞ2

� ðp0 þ�Þ2�c ðsÞ��lmð ~xÞy

� c ðsÞ��lmð ~xÞ; (13)

where l, m, � ¼ affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 � �2

pand s are the quantum num-

bers which characterize the spinor �. Here � is the sign ofthe energy and we will now take ð�Þ> 0. Notice that thelast sum does not depend on the point ~x due to the homo-geneity of the space HN . Therefore, it can be evaluated atthe origin � ¼ 0, when only the l ¼ 0 term survives [34](for generality, we quote below the formulas referring tothe N-dimensional space):

Xslm

c ðsÞ��lmð0Þyc ðsÞ

��lmð0Þ ¼ 2ðN�1Þ=2�Nð�Þ;

�Nð�Þ � 1

aN�ðN=2Þ2N�3

�N=2þ1jCl¼0ð�Þj�2;

(14)

where

Clð�Þ ¼ 2N�2ffiffiffiffi�

p �ðlþ N=2Þ�ði�þ 1=2Þ�ði�þ lþ N=2Þ : (15)

Notice that the definition of the measure �Nð�Þ (the den-sity of states) coincides with that used in [36].Thus we can calculate the contribution from the first

term in (11):

GRAVITATIONAL CATALYSIS OF CHIRAL AND COLOR . . . PHYSICAL REVIEW D 80, 085019 (2009)

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lnDet½H2 � ðp0 þ�Þ2�¼ Tr ln½H2 � ðp0 þ�Þ2�¼ Nf

ZdNxdt trh ~x; tj ln½H2 � ðp0 þ�Þ2�j ~x; ti

¼ vNf2ðN�1Þ=2 Z dp0

2�

X�¼�

Z 1

0d��Nð�Þ

� ln½ð�ð�ÞÞ2 � ðp0 þ�Þ2�; (16)

where v is the spacetime volume of R �HN , andNf ¼ 2 is

the number of flavors.The second term gives

lnDet½4j�j2 þ ðH ��Þ2 � p20�

¼ vNf2ðN�1Þ=2 Z dp0

2�

X�¼�

Z 1

0d��Nð�Þ ln½4j�j2

þ ðð�Þ þ ��Þ2 � p20�: (17)

Thus the quark contribution to the effective potential reads

~V eff ¼�Sqv

¼ i

2Nf2

ðN�1Þ=2Z dp0

2�

X�¼�

Z 1

0d��Nð�Þfln½ðð�Þ

þ��Þ2�p20�þ2ln½4j�j2þðð�Þþ��Þ2�p2

0�g:(18)

In the case of finite temperature T ¼ 1= > 0, the fol-lowing substitutions should be made:Z dp0

2�ð� � �Þ ! i

Xn

ð� � �Þ; p0 ! i!n;

!n ¼ 2�

�nþ 1

2

�; n ¼ 0;�1� 2; . . . ;

where !n is the Matsubara frequency. Then the quarkcontribution to the effective potential (18) becomes thethermodynamic potential �q:

�q ¼�2ðN�1Þ=2Nf

2

Xþ1

n¼�1

X�¼�

Z 1

0d��Nð�Þfln½ðð�Þ

þ��Þ2þ!2n�þ 2 ln½4j�j2þðð�Þþ��Þ2þ!2

n�g:(19)

Summing over the Matsubara frequencies, we obtain thethermodynamic potential:

�ð�;�Þ ¼ Nc

��2

2G1

þ j�j2G2

�� 2ðN�1Þ=2NfðNc � 2Þ

Z 1

0d��Nð�Þfð�Þ þ T lnð1þ e� ðð�Þ��ÞÞ

þ T lnð1þ e� ðð�Þþ�ÞÞg � 2ðN�1Þ=2Nf

Z 1

0d��Nð�Þf

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðð�Þ ��Þ2 þ 4j�j2

qþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðð�Þ þ�Þ2 þ 4j�j2

qþ 2T lnð1þ e�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðð�Þ��Þ2þ4j�j2

pÞ þ 2T lnð1þ e�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðð�Þþ�Þ2þ4j�j2

pÞg: (20)

The spectrum of the Dirac operator depends only on one dimensionless parameter �. Instead of �, one can introduce thequantity with the dimension of momentum p ¼ �=a. Then the spectrum of the Dirac operator may be written as

ðpÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2 þ �2

q� Ep: (21)

In the case of the 3D space, N ¼ 3, the density of states is

�3ð�Þ ¼�2 þ 1

4

2�2a3: (22)

Thus the thermodynamic potential in the R �H3 spacetime becomes1:

�ð�;�Þ¼ 3

��2

2G1

þj�j2G2

�� 2

�2

Z 1

0dp

�p2þ 1

4a2

�fEpþT lnð1þe� ðEp��ÞÞþT lnð1þe� ðEpþ�ÞÞþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðEp��Þ2þ4j�j2

q

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðEpþ�Þ2þ4j�j2

qþ2T lnð1þe�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðEp��Þ2þ4j�j2

pÞþ2T lnð1þe�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðEpþ�Þ2þ4j�j2

pÞg: (23)

1Note that after introducing the quantity with dimension of momentum p ¼ �=a, the spectrum of the Dirac operator (21) formallycoincides with the usual dispersion relation in flat Minkowski spacetime and does not depend on curvature. At the same time, thedensity of states (22) in the hyperbolic space differs from the usual measure of integration over momentum in Minkowski spacetime byan additional term depending on the curvature (in fact, this is the only curvature dependent part of the thermodynamic potential). Toobtain the correctmeasure of integration over the continuum spectrum, one needs to use the properly normalized eigenfunctions in (14).


085019-4

It should be noted that the thermodynamic potential is divergent at large p. Therefore, we must use some regularizationprocedure. Since we interpret p as the module of the momentum, the easiest way to regularize the divergent integral is tointroduce the momentum cutoff, p �. Then the regularized potential looks as follows:

�regð�;�Þ ¼ 3

��2

2G1

þ j�j2G2

�� 2

�2

Z �

0dp

�p2 þ jRj

24

�fEp þ T lnð1þ e� ðEp��ÞÞ þ T lnð1þ e� ðEpþ�ÞÞ

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðEp ��Þ2 þ 4j�j2

qþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðEp þ�Þ2 þ 4j�j2

qþ 2T lnð1þ e�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðEp��Þ2þ4j�j2

pÞ

þ 2T lnð1þ e� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðEpþ�Þ2þ4j�j2

pÞg; (24)

where we used the expression for the scalar curvature R ¼� 6

a2. This formula consists of two parts arising from the

two terms in the first round bracket of the integrand: thefirst part is the same as in (3þ 1)D Minkowski spacetime,while the second one, which is linear in curvature, corre-sponds to the contribution from (1þ 1)D spacetime.Hence, when the contribution of the second term domi-nates over the first one, dimensional reduction by two unitstakes place.

The values of condensates � and j�j correspond to thepoint of global minimum of the regularized thermody-namic potential and are determined as solutions of thegap equations

@�reg

@�¼ 0;

@�reg

@j�j ¼ 0: (25)

In the following sections we will consider the behaviorof the condensates as functions of curvature, temperature,and chemical potential.

IV. ANALYTICAL SOLUTIONS

A. Chiral condensate

Let us first consider the case when chiral symmetry isbroken while the color symmetry remains unbroken (� �0 and � ¼ 0). Since the quark condensate appears even inthe vacuum, we put for simplicity � ¼ 0 and T ¼ 0.

Then one can obtain the following expression for theeffective potential

Veffð�Þ ¼ �4

�2v0ðxÞ;

v0ðxÞ ¼ 3x2

2g� 3

4FðxÞ � jrj

8GðxÞ;

x ¼ �

�; g ¼ �2

�2G1; r ¼ R

�2;

(26)

where

FðxÞ ¼ ð2þ x2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ x2

p� x4 ln

1þffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ x2

p

x;

GðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ x2

pþ x2 ln


p

x:

(27)

The gap equation for the condensate � reads

1

g¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ x2

pþ

�jrj12

� x2�ln1þ


p

x: (28)

An analytical solution of this equation can be obtained onlyfor small x 1 (� �). Expanding the right-hand sideof (28) in x we have�

1

g� 1

�¼

�jrj12

� x2�ln2

x: (29)

We will consider three different cases: (a) subcritical g,g < gc ¼ 1, where gc is the critical constant in flat four-dimensional spacetime; (b) near critical g, when g ! gc �0, and (c) overcritical g, g > 1.In the case of subcritical g, a nontrivial solution of the

gap equation (29) exists only if x2 < jrj12 . In the strong

curvature limit �2 jRj12 , the chiral condensate is given by

�0 ¼ 2� exp

�� 12

jrj�1

g� 1

��

¼ 2� exp

�� 12�2ð1� gÞ

jRjG1

�: (30)

One can distinguish two subcases in which the last expres-sion is consistent with the above made assumptions:(i) g < 1, r 1 or (ii) g 1, r� 1. The case (ii) showsthat the gap equation has a nontrivial solution even atarbitrary weak coupling constant.The expression (30) looks very similar to the chiral

condensate in the two-dimensional Gross-Neveu model.After excluding the coupling constant from the effectivepotential by using the gap equation (29), we obtain

Veffð�Þ ¼ jRj�2

16�2

�ln�2

�20

� 1

�; (31)

and this is indeed (up to a dimensional factor) the effectivepotential of the Gross-Neveu model. Hence, we concludethat in the case of subcritical coupling the strong gravita-tional field of hyperbolic space leads to the effective di-mensional reduction from (3þ 1) to (1þ 1) (see also[27]).One should also note that the nonanalytical dependence

of the chiral condensate on curvature in the exponent of(30) looks quite similar to that in a magnetic field


085019-5

�0 ¼ffiffiffiffiffiffiffiffiffijeBj�

sexp

�� 2�2ð1� gÞ

jeBjG1

�; (32)

but the preexponential factors differ (see, for example, thereview [37]). This fact demonstrates that in the case ofhyperbolic space the negative curvature plays an analogouscatalyzing role as the magnetic field does in flat space,where it gives rise to the catalysis of �SB even at arbitraryweak attraction between quarks.

In the near critical regime g ! gc � 0 the chiral con-densate is just the constant

�0 ¼ffiffiffiffiffiffiffijRj12

s; (33)

where the curvature must be small jRj12 �2.

In the overcritical regime g > 1, a nontrivial solution of

(29) exists only if x2 > jrj12 . In the weak curvature limit

jRj12 m2 we obtain

�0 ¼ m

�1þ jRj

24m2þO

�R2

m4

��; (34)

where m is the solution of the gap equation at R ¼ 0. It isseen that in this case the curvature leads to small analyticalcorrections to the flat-space value of chiral condensate.

Next, let us consider the influence of finite temperatureon the behavior of the chiral condensate. The temperature-dependent contribution to the thermodynamic potentiallooks as follows

�Tð�Þ ¼ � 12

�2TZ 1

0dp

�p2 þ jRj

24

�lnð1þ e� EpÞ:

Expanding the logarithm into a series and performing theintegration over momentum, we obtain

�Tð�Þ ¼ 12

�2T�

�jRj24

X1n¼1

ð�1Þnn

K1ðn �Þ

þ �2X1n¼1

ð�1Þnn

K2ðn �Þn �

�;

where K�ðxÞ is the Macdonald function (modified Besselfunction). Then the gap equation at finite temperature reads

1

g¼


pþ

�jrj12

� x2�ln1þ


p

x� jrj

12I1

� x2ðI3 � I1Þ; (35)

where

I1ð �Þ ¼ �2X1n¼1

ð�1ÞnK0ðn �Þ;

I3ð �Þ ¼ �2X1n¼1

ð�1ÞnK2ðn �Þ:

Here we consider only the most interesting case of sub-critical coupling and strong gravitational field. Thus, as

previously at g < 1 and �2 jRj12 , we obtain for the chiral

condensate

�0ðTÞ ¼ 2� exp

�� 12�2ð1� gÞ

jRjG1

� I1ð �0ðTÞÞ�

¼ �0ð0Þ exp½�I1ð �0ðTÞÞ�: (36)

The function I1ðxÞ is positive and monotonically decreases,which means that temperature leads to the restoration ofbroken symmetry. The critical temperature Tc is defined bythe condition �0ðTcÞ ¼ 0 which gives the BCS-like rela-tion

Tc ¼ ��1eC�0ð0Þ ’ 0; 57�0ð0Þ; (37)

where �0ð0Þ is given by (30). It should be mentioned thatthe temperature dependence of the chiral condensate (36)is the same as in a constant chromomagnetic field [20].This brief analysis shows that in the case of subcritical

coupling the negative curvature leads to the catalysis of�SB while in the overcritical regime it leads to the en-hancement of the chiral condensate. One might expect thesame behavior in the case of a color condensate. The caseof overcritical coupling will also be considered in the nextsection using numerical methods.

B. Mixed phase

Let us now consider a more general situation, when bothcondensates can take nonzero values. What concerns colorsymmetry breaking in flat spacetime, it is well known thatdiquark pairing and color superconductivity arise (for largequark number densities and, correspondingly, at largechemical potential) due to an instability of the Fermisurface so that the color superconducting state is energeti-cally more preferable. For studying the influence of gravityon the formation of a color condensate in its most ‘‘pureform,’’ we find it convenient, contrary to the flat case, in thefollowing to take � ¼ 0. Then the effective potential atzero temperature can be written in the form

Veffð�;�Þ ¼ �4

�2v0ðx; yÞ;

v0ðxÞ ¼ 3A

2x2 þ By2 � 1

4ðFðxÞ þ 2FðzÞÞ

� jrj24

ðGðxÞ þ 2GðzÞÞ; (38)

where

x ¼ �

�; y ¼ 2j�j

�; z ¼ m�

�¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

q;

A ¼ 1

g¼ �2

�2G1

; B ¼ 3�2

4�2G2

; r ¼ R

�2;

(39)

and the functions F and G are the same as in (27). The


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nontrivial solutions for condensates satisfy the followinggap equations obtained from (38)

3A ¼ HðxÞ þ 2HðzÞ þ jrj12

ðKðxÞ þ 2KðzÞÞ; (40)

B ¼ HðzÞ þ jrj12

KðzÞ; (41)

where

HðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ x2

p� x2 ln


p

x;

KðxÞ ¼ ln1þ


p

x:

(42)

Substituting (41) in (40), we obtain a separate equation forthe chiral condensate

3A� 2B ¼ HðxÞ þ jrj12

KðxÞ; (43)

which formally coincides with (28) in the previoussubsection IVA, when 1=g becomes replaced by (3A�2B). As in the previous case, we will solve Eqs. (41) and(43) in the limit of small condensates x 1 and y 1and consider only the most interesting case of strong

curvature x2 jrj12 , y

2 jrj12 and subcritical couplings A >

1, B> 1, where the effect of gravitational catalysis is moreclearly seen. In this limit we simply have HðxÞ ¼ 1 andKðxÞ ¼ lnð2=xÞ. The solutions of the gap equations are asfollows

�20 ¼ 4�2 exp

�� 24

jrj ð3A� 2B� 1Þ�; (44)

m2�0 ¼ �20 þ 4j�0j2 ¼ 4�2 exp

�� 24

jrj ðB� 1Þ�; (45)

where the coupling constants must satisfy the inequalities:3A� 2B> 1 and B> 1. From this, the color condensatefollows

4j�0j2 ¼ m2�0�1� exp

�� 24

jrj ð3A� 3BÞ��; (46)

which exists only if A> B (i.e. G2 >34G1). Hence, both

condensates may exist simultaneously in the region A >B> 1 (in this region the inequality 3A� 2B ¼ Aþ 2ðA�BÞ> 1 holds automatically).

Excluding the coupling constants from the effectivepotential one obtains

Veffð�;�Þ ¼ jRj48�2

��2

�ln�2

�20

� 1

�þ 2m2�

�lnm2�m2�0

� 1

��(47)

The other possible stationary point of the effective po-tential is � ¼ 0 and � � 0. This type of solution may beobtained from the previous one by letting�2

0 ¼ 0 in (45) so

that the combined condensate m2�0 reduces to the tilded

expression 4j~�0j2.The phase structure of the model is defined by the global

minimum of the effective potential. There are four types ofstationary points of the effective potential: ð0; 0Þ, ð~�0; 0Þ,ð0; ~�0Þ, and ð�0;�0Þ. Let us denote the correspondingvalues of the effective potential at these points by v1, v2,v3, and v4. The effective potential is normalized in such away that v1 ¼ 0. The other values are

v4 ¼ �jRj48

½�20 þ 2m2�0�; v3 ¼ �jRj

6j~�0j2;

v2 ¼ �jRj16

~�20;

(48)

where ~�0 is given by (30). First of all, we see that theminimum v1 ¼ 0 of the symmetry case is higher than theother minima. This means in contrast to the flat case thatfor subcritical couplings the symmetric phase in hyperbolicspace is unstable under the formation of different conden-

sates and symmetry breaking. Since m2�0 ¼ 4j~�0j2, it iseasily seen that v4 < v3 in the region A > B> 1. Themixed phase is the true vacuum of our model if v4 < v2

or �20 þ 2m2�0 > 3~�2

0. Dividing both sides of the last in-

equality by ~�20 and using (30), (44), and (45), we obtain the

following condition

exp

�24

jrj 2ðB� AÞ�þ 2 exp

�24

jrj ðA� BÞ�> 3:

Introducing a new variable u ¼ exp½24jrj ðB� AÞ�, 0< u<

1 for A > B> 1, we see that the above condition is just thesimple cubic inequality u3 þ 2> 3u, which is automati-cally satisfied for 0< u< 1. Thus, the condition A > B>1 is sufficient for the mixed phase to be the true vacuumstate of our model.In the opposite case B � A > 1 the chiral and color

condensates cannot exist simultaneously and we need to

compare v2 and v3. The inequality v3 < v2 leads to A�B> jrj

24 ln32 which contradicts our previous assumption.

Therefore, in the region B � A > 1 only the phase withbroken chiral symmetry occurs, and the chiral condensateis described by (30).Let us also briefly consider the influence of finite tem-

perature on the mixed phase. Proceeding in the samemanner as in subsection IVA, we obtain the followingtemperature dependence of condensates

�20ðTÞ ¼ �2

0ð0Þ exp½�2I1ð �0ðTÞÞ�; (49)

m2�0ðTÞ ¼ m2�0ð0Þ exp½�2I1ð m�0ðTÞÞ�; (50)

where �20ð0Þ and m2

�0ð0Þ are given by (44) and (45), re-

spectively. In analogy with (37), the critical temperaturesfor chiral and color condensates are

T�c ¼ ��1eC�0ð0Þ; T�

c ¼ ��1eCm�0ð0Þ: (51)


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Since the inclusion of temperature results in a smoothdiminution of condensates, the relation between couplingswhich determines the phase structure of the model shouldnot change. Again both condensates may exist simulta-neously if A > B which implies the relation T�

c > T�c .

Therefore at T < T�c the mixed phase is realized with

chiral and color condensate of the form

j�0ðTÞj2 ¼ 14fm2�0ðTÞ � �2

0ðTÞg: (52)

Obviously, at T�c < T < T�

c , chiral symmetry is restored,while the color symmetry remains broken

j�0ðTÞj2 ¼ j~�0ðTÞj2 ¼ m2�0ðTÞ4

: (53)

Finally, at T > T�c both symmetries become restored.

The phase portraits of the system are presented in Fig. 1for fixed values of coupling constants A and B, for A B(left picture) and for A > B (right picture). The criticalcurves which separate different phases are described bycorresponding formulas (37) and (51) for criticaltemperatures.

In fact the solutions (49) and (50) of the gap equationsgive only an implicit dependence of the condensates �0ðTÞand m�0ðTÞ on temperature. Because of the complexity ofthe function I1ðxÞ, no analytical solutions for condensatescan be found. However, Eqs. (49) and (50) can be easilysolved by using numerical methods, for example, an iter-ative procedure. When the temperature is fixed, the resultof such procedure will depend only on the first step ofiteration. The most obvious choice is to take the conden-sates at zero temperature as a first approximation. Theschematic picture of condensates inside the mixed phaseis shown in Fig. 2 for fixed values of �0ð0Þ and �0ð0Þ. It isinteresting to note that the condensates �0 and m�0 havethe usual form of the BCS theory of superconductivity anddecrease with temperature, while the color condensate �0

slightly grows at T & T�c and has a maximum around T�

c .

We have not specified the values of condensates on thesecurves since we have no phenomenologically motivatedchoice of coupling constants and curvature. Figure 2 dem-onstrates only qualitatively the behavior of condensates asfunctions of the temperature at fixed curvature jrj andcouplings A and B. Moreover, since the condensates atnonzero temperature depend on the curvature only throughthe condensates at zero temperature, the increasing curva-ture will produce the appropriate stretching of curves inFig. 2.

V. PHASE TRANSITIONS

In the previous section the phase structure of the modelwas extensively studied in the regime of subcritical cou-plings by formally considering arbitrary ratios of couplingconstants. In particular, in order to investigate the influenceof curvature on condensates in ‘‘pure form,’’ the chemicalpotential was taken to be zero. Now, in this section, we turnto the more general case of phase transitions at finitechemical potential (quark number density).In the following we want to compare our results with the

well-established case of flat spacetime. Since the role of a

r

T

Tc

0, 0

0, 0

0 r

T

Tc

Tc

0, 0

0, 0

0, 0

0

FIG. 1. The phase portraits of the system showing critical curves as functions of jrj for fixed values of couplings A and B: for A B(left) and for A > B (right).

TTcTc

m T

T

T

0

0

0

m 0

FIG. 2. Condensates �0, �0 and m�0 as functions of tempera-ture inside the mixed phase. (Henceforth we will omit the sub-script 0 of condensates on the plots.)


085019-8

growing temperature in the restoration of broken symme-tries was already demonstrated, our considerations will berestricted here to the case of zero temperature. The inves-tigation of the phase transitions at finite chemical potentialbeyond the limit of small condensates is difficult to per-form analytically, and hence numerical methods will nowbe used.

In order to be able to make comparison with the flat case,we consider the following ‘‘more physical’’ relation ofcouplings taken from the instanton-motivated NJL-model[11] (see also [21]):

G2 ¼ 38G1: (54)

Let us now fix g ¼ G1�2=�2 ¼ 1:4 in such a way that

the chiral condensate at zero curvature, R ¼ 0, in thevacuum is equal to the usual value of the constituent quarkmass in flat space (�0 ¼ 350 MeV). Here the cutoff pa-rameter is taken to be � ¼ 600 MeV.

In terms of dimensionless couplings A and B [see (39)],relation (54) corresponds to B ¼ 2A, and one might thinkthat it excludes the existence of a mixed phase in accor-dance with our previous discussion. However, we shouldstress that the inequality B � A, which guarantees theabsence of a mixed phase, was obtained only for subcriticalcouplings. Here we have fixed g ¼ 1:4 and thus A ¼1=g < 1, and the overcritical regime is realized. The dif-ference between these cases is that in the subcritical regimethe main contribution to the values of condensates is givenby the two-dimensional part of the effective potential,while in the overcritical regime, the condensates receivetheir contribution mainly from the flat four-dimensionalpart. Therefore, our previous arguments are not applicablein this case.

It was observed earlier that in a flat four-dimensionalspacetime the relation (54) between coupling constantsleads to the absence of a mixed phase in a wide range ofparameters [11]. In the previous section, we have foundthat in the overcritical regime finite curvature gives onlysmall corrections to the flat-space value of chiral conden-

sate [see (34)]. Using numerical calculations, we will nowanalogously show that in a wide range of the values ofcurvature its contribution to condensates is small in com-parison with their values in flat case. It is clear that suchcorrections cannot change the phase structure of the modelobtained in flat case in [11], and thus cannot lead toformation of new phases. Therefore, in what follows, weassume that there is no mixed phase, where both conden-sates simultaneously take nonzero values.As is well known, for increasing chemical potential

there arises diquark pairing, whereas the chiral condensatebecomes suppressed.The corresponding behavior of the chiral and color

condensates as functions of the chemical potential at r ¼0 and jrj ¼ 1 is shown in Fig. 3.As is seen from Fig. 3, the critical chemical potential, at

which the phase transition takes place, in flat space, r ¼ 0,is�c 330 MeV, while at jrj ¼ 1�c 366 MeV. Fromthis we can conclude that the critical line �cðjrjÞ whichseparates the two phases is a growing function of thecurvature. The numerical study of the number density,which is the first derivative of the thermodynamical poten-tial with respect to the chemical potential, shows that it isdiscontinuous at �c. Thus, an increasing chemical poten-tial leads to a first order phase transition.It should also be mentioned that for the limiting case of

zero curvature our results for the value of the criticalchemical potential �c 330 MeV and the maximumvalue of the color condensate � 70 MeV (see Fig. 3)are in agreement with the results obtained in [38] for thesame values of �0 and � (note that our value of the colorcondensate � is by definition 2 times less than condensate� in [38]).We can also examine the condensates � and � as

functions of the absolute value of curvature jrj. The be-havior of chiral and color condensates inside both phases ispresented in Fig. 4.It is clear that with growing curvature the values of

condensates slowly increase, which results in the enhance-

0.1 0.2 0.3 0.4 0.5 0.6

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.1 0.2 0.3 0.4 0.5 0.6

0.1

0.2

0.3

0.4

0.5

FIG. 3. Condensates � and � as functions of � (all quantities are given in units of GeV) at r ¼ 0 (left) and at jrj ¼ 1 (right).


085019-9

ment of the symmetry breaking effects. As we have alreadymentioned above, in a wide range of curvature values theincrement of the condensates is small with respect to theirvalues at r ¼ 0. Moreover, it is seen from Fig. 4 thatcondensates grow with curvature almost linearly which isin agreement with our previous analytical consideration[see (34)].

Finally, the r��—phase portrait of our system isshown in Fig. 5. As it was already mentioned, the criticalchemical potential, at which the first order phase transitiontakes place, slightly grows with increasing curvature.

IV. SUMMARY

In the framework of an extended NJL model we haveinvestigated chiral and/or color symmetry breaking indense quark matter under the influence of negative curva-ture of hyperbolic space by employing the thermodynamic(effective) potential of the system as a function of chiraland diquark condensates, temperature, curvature andchemical potential.

Two different regimes of dynamical symmetry breakinghave been studied. First, the regime of subcritical cou-plings, where the values of coupling constants are lowerthan their critical values in flat space, was considered. As is

well known, in this situation in four dimensions, no sym-metry breaking takes place, and gap equations have onlytrivial solutions. Unlike the flat case, in hyperbolic space,symmetry may, however, be broken even for an arbitrarysmall coupling constant. In the case of subcritical cou-plings we obtained expressions for the chiral and colorcondensates that depend nonanalytically on curvature.Second, the regime of overcritical coupling constants wasinvestigated, when the symmetry may be broken even inflat space. In this case it was shown that curvature leads tosmall analytical corrections which increase the flat-spacevalues of condensates and thus enhances the symmetrybreaking effects.It is interesting to note that in the subcritical regime of

coupling constants the strong gravitational field of hyper-bolic space serves as a catalyzing factor similar to the roleof the magnetic field [15–17] or chromomagnetic fields[18–21] in the effects of dynamical symmetry breaking. Aswe have explicitly demonstrated, the gravitational catalysistakes place for chiral and color condensates. It is also worthmentioning that the effect of gravitational catalysis isaccompanied by a lowering of dimensions of the system.In the regime of subcritical couplings the solutions of gapequations look quite similar to those for the 2D Gross-Neveu model. Therefore, we concluded that the strongcurvature of hyperbolic space leads to an effective dimen-sional reduction by two units (see also [27]).As we have already mentioned, in the subcritical regime

the negative curvature essentially changes the phase struc-ture of the NJL model found in [39] making the symmetricphase unstable under the formation of condensates, whilein the overcritical case one expects only minor modifica-tions to the phase structure obtained in flat space.Therefore, we have extensively studied the phase structurein the subcritical regime. It is interesting to note that theoverall phase structure depends only on the ratio of theinverse (dimensionless) coupling constants A and B, butnot on the curvature. For subcritical couplings the mixed

0.0 0.2 0.4 0.6 0.8 1.0r0.0

0.1

0.2

0.3

0.40

0

FIG. 5. Phase portrait for the coupling relation G2 ¼ 38G1.

0.2 0.4 0.6 0.8 1.0r

0.1

0.2

0.3

0.4

0.2 0.4 0.6 0.8 1.0r

0.02

0.04

0.06

0.08

FIG. 4. Condensates (in units of GeV) � at � ¼ 200 MeV (left) and � at � ¼ 400 MeV (right) as functions of jrj.


085019-10

phase is realized only if A=B > 1, while if A=B 1 onlythe chiral symmetry may be broken. The same critical ratioA=B ¼ 1 was found in the flat space in the framework ofthe random matrix model [40]. It was argued by theseauthors that this relation is a consequence of global sym-metries of the model.

Moreover, the influence of finite temperature on phasestructure was investigated. The phase portraits of the sys-tem for different relations between couplings were con-structed. In particular, we have shown that for any fixedvalue of curvature there exists a critical temperature atwhich the phase transition takes place and symmetry be-comes restored.

Finally, using numerical calculations, we have investi-gated the phase transitions between �SB and CSC phasesunder the influence of chemical potential and curvature inthe regime of overcritical couplings. It was demonstratedthat similar to the flat case there arises a diquark pairing forincreasing chemical potential, while the chiral condensatebecomes suppressed. The phase portrait of the system atzero temperature was also constructed, and it was shownthat the critical line �cðjrjÞ, separating the two differentphases, is a growing function of curvature. The chiral anddiquark condensates, � and �, acquire only small correc-

tions due to curvature increasing the flat-space values ofcondensates and this leads to an enhancement of the sym-metry breaking effects.The results of this paper, although describing a model

situation with symmetry breaking in a hyperbolic space,may hopefully find further development in more realisticsituations with phase transitions in quark matter under theinfluence of strong gravitational fields.

ACKNOWLEDGMENTS

We are grateful to A. E. Dorokhov, M.K. Volkov andK.G. Klimenko for useful discussions. We also appreciatethe remarks and helpful suggestions made by the referee inhis report. Two of us (A. V. T. and V. Ch. Zh.) thank M.Mueller-Preussker for hospitality during their stay at theInstitute of Physics of Humboldt-University, where part ofthis work has been done, and also DAAD for financialsupport. D. E. thanks the colleagues of the BogolyubovLaboratory for Theoretical Physics of JINR Dubna forkind hospitality and the Bundesministerium fur Bildungund Forschung for financial support. This work has alsobeen supported in part by the Deutsche Forschungs-gemeinschaft under Grant No. 436 RUS 113/477.

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Documents

Gravitational catalysis of chiral and color symmetry breaking of quark matter in hyperbolic space