PHYSICAL REVIEW D VOLUME 33, NUMBER 12

Chiral symmetry at finite baryonic density

A. Trzupek Institute of Physics, Jagellonian University, Reymonta 4, Krakow, Poland

J. Wosiek Institute of Computer Science, Jagellonian University, Reymonta 4, Krakow, Poland

(Received 13 January 1986)

Variational description of the chiral-symmetry breaking in QCD is generalized to the case of fi- nite baryonic density, at zero temperature. A modified trial state is used which allows us to perform the whole analysis directly in the sector of Hilbert space with a finite baryonic background. Our re- sults indicate that chiral symmetry is restored for a finite density of baryons, presumably in the first-order phase transition.

I. INTRODUCTION

Spontaneous breakdown of chiral symmetry is an im- portant property of quantum chromodynamics. Equally interesting is the transition from the broken, low-energy phase, to the chirally symmetric one-the quark-gluon plasma. Such a transition can be driven by two parame- ters: the temperature and/or the matter density. The behavior of QCD at finite temperatures was intensively studied by lattice simulations1~' and in the c ~ n t i n u u m . ~ ' ~ On the other hand, the effect of a finite baryonic back- ground is less Suitable lattice methods have been developed only recently.' In this paper we use the variational approach to study the latter question in the framework of the nonrelativistic model of QCD in the continuum.

It was shown in Refs. 8 and 9 that the model gives spontaneous breaking of chiral symmetry at zero tempera- ture and at vanishing baryonic background. We prove that a chirally nonsymmetric vacuum becomes unstable for some finite value of baryonic density. This is accom- plished by explicit construction of the trial state with a nonzero baryonic background. The energy of the pertur- bative vacuum turns out to be lower than that of the non- perturbative one. Since this is the result of the variational approach, one cannot definitely claim that chiral symme- try is restored; however, such a conclusion is plausible.

The paper is organized as follows. In Sec. I1 we recall the variational techniaue of Ref. 8. Section I11 contains a generalization of this approach to the finite density of baryons. Our results are presented in Sec. IV. Finally Sec. V contains conclusions and a list of further problems. Some technical details are shifted to the appendixes.

11. VARIATIONAL METHOD AND BOGOLIUBOV-VALATIN VACUUM

Let us briefly recall the variational approach of Amer et al.' The model QCD Hamiltonian reads

(see Ref. 10 for a careful discussion of the choice between the normally ordered and nonordered Hamiltonians, cf. also Refs. 8 and 91, where V(x)= - vo2 I x / and the tem- porary lattice regulator is introduced. a ,kB,b= 1, . . . , 8 are Dirac and Gell-Mann matrices, respectively. $(x) is the colored massless quark field, which can be expanded in the chirally invariant basis of free massless spinors:

where b z ( k ) [d:+(k)] is the annihilation [creation] opera- tor of a quark [antiquark] with the helicity s and color a.

The trial state / In) is defined as9

where N = nl, [1+7D2(k)] is a normalization factor, P(k) is a trial wave function of a Cooper pair, and T

denotes the volume element in the momentum space. / 0) stands for the perturbative vacuum. Equivalently, one can use the Bogoliubov-Valatin (BV) transformation to define new creation and annihilation operators

The state I R) can be considered as a quasiparticle vacu- um provided

3753 @ 1986 The American Physical Society

3754 A. TRZUPEK AND J. WOSIEK - 3 3

and (2.5)

COS &- - 1

2 [1+P2(k)] ' /2 '

Quark fields $(XI can be expanded in the new basis (2.41,

Now the spinors us( k ) and us( k ) need not satisfy the massless Dirac equation.

In the variational approach one looks for the stationary point of the average energy

o [ c p l = ( n l ~ 1 0 ) . (2.7)

The resulting gap equation 68/6q?=0 cannot be solved analytically for a linear potential. Nevertheless, it is pos- sible to investigate chiral-symmetry breaking without solving this equation. In order to show the instability of the perturbative vacuum, it is sufficient to find an unsta- ble direction in the functional space of all p's. Hence, the negative value of the difference

for some p( k ) would signal breakdown of the chiral sym- metry. Condition (2.8) was the starting point of the analysis of Ref. 8 and will be used in the next chapter after suitable modifications.

The average energy (2.7) can be easily calculated using Wick's theorem. The latter implies that H may be rewrit- ten in the form

where

+4--!-- 2 j V ( k - p ) ~ r [ i \ + t k ) ~ - ( p ) ] , (an )3 k , p

is the energy of the trial BV state (2.3). A+ denote the projectors

and v ( k ) is the Fourier transform of the potential

T o simplify the analysis further, one linearizes Eq. (2.12) for small p (k ) . This amounts to searching for an instabil-

ity in the small neighborhood of the chirally invariant vacuum. Further analytical calculations are possible with the aid of the normalized Gaussian ansatz for

111. INCLUDING BARYONIC BACKGROUND

Introduction of the chemical potential to the formalism is not necessary since the baryonic number B is a constant of motion, and we work at zero temperature. Instead, the trial state (2.3) will be generalized to contain a baryonic background with finite density. This allows for perform- ing instability analysis directly in the sector of Hilbert space with definite value of B.

In constructing a trial state we included simplest phenomenological properties of the low-energy phase, namely, that the nuclear matter is built from color-singlet baryons containing three valence and/or constituent quarks. The whole construction is done in the quasiparti- cle basis. This provides a state with a well-defined B since the Bogoliubov-Valatin transformation preserves baryonic number.

Our state with a finite baryonic background is

with / 0) and bt defined by (2.3) and (2.4), respectively. The color-singlet character of / B ) is evident from (3.1).

We prefer to think of the state (3.1) in terms of ideal- ized baryons created by the operators

where s labels various spin and flavor states of a baryon built from three quasiparticles with spins s,, s2, and s3. C,SIs2,, are the appropriate Clebsch-Gordan (CG) coeffi- cients. All dynamical calculations are done for one flavor f= 1 only. The f = 2 case will be discussed shortly. To carry on with the analytical calculations, we used a sim- plified momentum wave function of a baryon. Our as- sumption, which is apparent from Eq. (3.2), may be called a crude valence approximation: all three constituent auarks have the same momentum. The wave function in other variables is determined by a suitable nonrelativistic internal symmetry: SU(2),,,, for one flavor, and SU(2) , , i ,~ SU(2 for two flavors. We believe that violation of the chiral symmetry is not very sensitive to the details of the baryonic wave function. At present, we consider Eq. (3.2) as a zeroth-order approximation and plan to refine it in the future. In spite of this simplifica- tion a rather interesting physical picture of the chiral properties of the model emerges.

The density of baryons in the state (3.1) is given by

where

33 - CHIRAL SYMMETRY AT FINITE BARYONIC DENSITY 3755

The number of baryonic states of, which can be con- structed from three quarks, is reduced by the Pauli princi- ple and crude valence approximation. Antisymmetry in color, together with a space-symmetric wave function, im- plies symmetrized spin and flavor variables. As a result, only four (20) baryonic states can be built from one (two) flavor(s), for a given momentum mode.

Using Wick's theorem repeatedly and the normalization of the CG coefficients, one obtains, from (3.3) and (3.4),

Let us now discuss the role of Fermi momentum KF as introduced in Eq. (3.1). [One can simply show that the operators (3.2) satisfy usual fermionic anticommutation relations when restricted to the states of the type (3.1).] One might doubt the usefulness of the definition (3.1) since the momentum of a single quasiparticle is not con- served and consequently the momentum distribution in (3.1) is not realistic. But the main purpose of introducing KF in (3.1) was to confine ourselves to the sector of Hil- bert space with definite B. KF directly controls [via Eq. (3.5)] baryonic density. As such, KF is a constant of motion. The Fermi sphere does not have sharp boundaries, in this case, and KF should be treated as its average radius.

A straightforward relation between KF and the baryon- ic density was the main advantage of our definition of the state (3.1). In all subsequent formulas, various energies will directly depend on KF. This allows for easy monitor- ing of the influence of p~ on the stability of various vacua.

We now repeat the instability analysis for the baryonic state (3.1). ( B / H 1 B ) depends, in this case, on two pa- rameters: R and KF. For a given KF (i.e., p ~ ) R will be adjusted in order to minimize the total difference A of average energies in I B ) and in the perturbative vacuum. The calculation of ( B / H / B ) is a little bit more involved than in the previous case. Again we use the Wick decom- position of the Hamiltonian (2.1). However, for KF#O, : H z : and : H 4 : also contribute to the average correspond- ing to the interactions with "baryonic quarks."

The energy difference is the sum

with A 8 given by (2.8) and

coming from the mutual interactions among baryonic constituents and the sea. Accordingly,

AEself = A E 3 f + rnf:,f and (3.8)

A E,,, = AE :{, + AE~,", .

Sandwiching : H z : and : H 4 : with / B ) and collecting non- vanishing contractions allows us to identify all contribu- tions listed in Eqs. (3.7) and (3.8). Linearizing for small p ( k ) one obtains, after simple Dirac and color algebra,

With the Gaussian ansatz (2.13) for p(p) most of the in- tegrals can be calculated analytically. A 8 was obtained by Le Yaouanc et Their result reads

For AEk,,, Eqs. (3.9) and (2.13) give (P=KFR)

It is convenient to calculate AEi<lf by rewriting the in- tegral (3.10) in the configuration space (cf. Appendix A). One obtains

where

AEf& is more complicated to derive. Careful treatment of the product of three distributions P(k-p) , O(KF-p), and 0 ( K ~ - k ) which appear under the integral in (3.11) is required. After lengthy calculations (cf. Appendix C) one arrives at the following result:

3756 A. TRZUPEK AND J. WOSIEK

where

with

and

The addition of the principal-value prescription to the naive (divergent) result is the final outcome of the regular- ization discussed in Appendix C .

The calculation of AE;$ is done in Appendix B. The result reads

with

The calculation of AEQ; is analogous to that for AEf& and is given in Appendix D. The answer is

2 2 - l p Z e -P 12 -2.-

I - L d - ? I 7 Collecting together all the formulas one obtains the fol-

lowing structure for the total energy difference:

where the dimensionless functions A (KFR ) and B (KFR ) can be easily read off from Eqs. (3 .14)-(3 .19) and are dis- cussed in the next section.

IV. RESULTS

To begin with, we discuss our results in two limiting cases, i.e., for zero and infinite baryonic densities. It fol- lows from Eqs. ( 3 . 1 3 - 4 3 . 2 0 ) that AE-+O when KF-+O, as it should; without baryonic background the result of Ref. 7 is recovered, i.e.,

There exists a (infinite) range of R in which A is negative. Hence, chiral symmetry is spontaneously broken.

FIG. 1. The energy differences AE:{,f, AEfSf, A E ~ , , and AE:ft, as obtained from Eqs. (3.16)-(3.19).

CHIRAL SYMMETRY AT FINITE BARYONIC DENSITY

FIG. 2. Contributions to the kinetic ( A ) and potential ( B ) energy differences as functions of RKF [cf. Eq. (3.20)J.

FIG. 4. Dependence of VoR,,, on K F / V o .

On the other hand, when KF+m (infinite p ~ ) it is readily seen from Eqs. (3.9)-(3.13) that AE+-A%'. Hence A-0. This is expected from Pauli blocking (cf. below). Gradually filling all levels with bare fermions leaves no room for the qij pairs; therefore, the difference between chirally symmetric and asymmetric states van- ishes. The behavior of A for intermediate densities was studied numerically.

Figure 1 presents separate contributions from various interactions, while Fig. 2 shows our results for A (RKF) and B (RKF ). The complete energy difference A is plotted versus VoR in Fig. 3. For small KF (and R ) the situation is similar to that for At3 [Eq. (4.1)]. However, for large R, the R dependence of A is stabilized due to the ex- ponential vanishing of B (RKF ), as discussed above. As a result A is negative in a finite range of R, for small KF. Increasing KF we reach the critical value KE above which A is positive for all R; hence, the chiral symmetry is re- stored in agreement with the intuitive arguments5 and the mean-field result^.^ More precisely, the instability analysis we have done justifies a slightly weaker state- ment, namely, that the instability of the perturbative vac- uum towards the Bogoliubov-Valatin vacuum vanishes for large enough baryonic densities. The critical value of Fer- mi momentum is Kf- =O. 12V0. Recent measurements of the string tension1' give Vo=400 MeV, which leads to Kk=48 MeV. This in turn gives p~ =(33 M ~ v ) ~ for two flavors. This number is much smaller than the generally accepted value of (250 M ~ v ) ~ . Similar large discrepancies were also found by Adler and ~ a v i s . ~ They have attribut- ed them to the lack of the Coulomb part in the interac- tion. Such an explanation seems natural since, as is seen from Fig. 2, the main portion of the chiral-symmetry breaking comes from small Cooper pairs. Also ~ o c i c ' ~ reached a similar conclusion. We point out that our value of Kg is consistent with that obtained in Ref. 12 in spite

FIG. 3. Total energy difference A plotted vs VoR for three values of KF /Vo.

of the different approach. The influence of the baryonic background is artificially

enhanced, in our model, by the effect of Pauli blocking. Filling up baryonic levels according to (3.1) and (3.2) de- stroys all Cooper pairs with constituent momentum K <KF. This leaves open the question of the optimal choice of the wave function p(k) . It might turn out that the commonly accepted ansatz (2.13) is inadequate to the present situation. Other choices of p ( k ) are currently in- vestigated. On the other hand, a more realistic selection of the baryonic wave function in (3.2) should allow for soft Cooper pairs. Then the conventional choice of p ( k ) will again be suitable. This point was brought to our at- tention by Zalewski.

It turns out that A has one minimum in R. We may in- terpret Rm,,(KF) (cf. Fig. 4) as the optimal size of the qq pair for a given density. The best radius depends weakly on KF showing instability around the critical point-a behavior similar to that found in Ref. 12. We see from Figs. 3 and 4 that the restoration of the chiral symmetry occurs through the first-order phase transition, at least in the linearized approximation. This is in agreement with earlier predictions.s~6

For K i < KF < EF =O. 3 V , the system may remain in the metastable state. When KI. > Kb, a local minimum vanishes and the chirally symmetric state is preferred. This structure may be an artifact of the linear approxima- tion. It would be very interesting to see if this scenario is supported by the solution of the exact gap equation.

ACKNOWLEDGMENT

We would like to thank Professor K. Zalewski for many stimulating discussions and for a critical reading of the manuscript.

APPENDIX A

AE& is given by Eq. (3.10). sinceX

we see immediately that

In order to obtain the second term, containing p2(p), we use the configuration space representation of v ( k ) . In- tegrating over p we define

3758 A. TRZUPEK AND J. WOSIEK - 33

Using the identity a / 2 m = Som d q / ( q 2 + m ' ) we rewrite (B6) as

One can easily show that

I may be found by setting k=r. Then

Now the q=O singularity cancels manifestly and we can take the m -0 limit under the integral. This gives

After some manipulations we obtain

The Dawson function D ( x ) is defined by

~ ( x ) = e - ' ~ Joxe dt .

Integrating now over k we finally arrive at Eq. (3.16) .

After changing orders of integrations one obtains r l

where APPENDIX B

In this appendix we calculate AEf:, as defined by Eq. (3.12). We use the same definition of the Fourier transform of the linear potential as Amer et al. :' Integration by parts gives

v ( k ) = lim 8 a v o 2 I 1 2 7 ~ ~ m -0 k 2 ( k 2 + r n 2 ) m

AE;;, can be written as

A E g t = - R 3 v O 2 ~ , , 7T3G

with which can be rewritten in the form

J1 can be rewritten in the form

Finally, integration over x gives

1 / 2

- p e - ~ ~ " + [i] # J ( p ) ] , 0 1 3 )

with the Dawson and error functions defined as usual:

where

Cancellation of the 9 - 0 singularity inside the integral (B4) can be readily seen using the following trick. After the angular integration one obtains

APPENDIX C

Self-energy contribution AE{sf is given by Eq. (3.11) . After integrating over angles of p one obtains [with the Gaussian ansatz for p ( p ) ]

CHIRAL SYMMETRY AT FINITE BARYONIC DENSITY

FIG. 5. Various ranges of q integrations used in Appendix C.

(C1)

with f ( KF/p) defined by

Singularity of the potential at q=O can be treated as in Appendix B. The angular integration of (C2) gives

with QF(p,q)=min( KF,p +q). Inserting (C5) into (C4) and performing an elementary integration over q one ob- tains ( u = KF /p) expression (3.17) without the principal- value prescription. In order to understand the latter, we must take a closer look at the origin of the p -KF singu- larity in (3.20). Because of the nonanalyticity of QF(p,q) [cf. Eq. (CS)], the q integration in (C4) splits into various ranges depending on p (see Fig. 5). In case (a) ( 0 < p < KF/2) we are far from the singular region and can calculate f ( u) naively as above. For KF/2 < p < KF [case (b)], however, the q=O singularity of the potential starts to show up. Namely, cancellation of the q=O singularity in (C3) is effective only if the upper limit of the A1 or BI interval is not equal to 0. In case (b), howev- er, this is not true for p +KF; hence f ( u ) is singular at u - 1. More formally, a Fourier transform of the linear potential can be defined only in the space of generalized functions. As such it should be always integrated with regular test functions. Sharp momentum cutoffs in (C1) do not satisfy this requirement.

This last remark suggests proper solutions to this prob- lem. We shall replace

= -p JOm [ ~ ( q , p ) - ~ ( ~ , p ) ] * (C3) 4' '

/.I-0 7T where

1 I - qt in (C2), and the p-+O limit will be taken after q and p in- H ( q , p ) = i 1-1 (p2+$-2pqt )I/' tegrations. It is necessary to do this only in the BI and

BII cases. In other situations the naive calculation suf- ~ 6 ( ~ ~ - - ( ~ ' + ~ ' - - 2 ~ ~ t ) " ~ ) d t . (c4) fices.

Inserting (C6) into (C4), and integrating by parts one Integration of (C4) is elementary; we obtain obtains after some algebra

~ ~ ~ ' A H " ' = [ ~ ( K F ~ - ~ ~ ) + ~ ~ ( K ~ - - ~ ) ] arctan I KF-P +q - a r c t a n - - +(q+-q) .

P P P ~ + ( K F - ~ ) ~

Singular contributions come only from the A H " and AH"' terms. Integrating AH"/q2 by parts, over BI and BII inter- vals, one obtains, up to 0 (p ) ( a = KF -p ),

3760 A. TRZUPEK AND J. WOSIEK

Again the q integration is elementary. Dropping further nonleading (with p ) terms one obtains

The second singular contribution comes from AH1''. Similar steps lead to

In deriving (C9) and ((210) we used the following represen- tation for the Dirac delta and the principal-value distribu- tions:

with p-0. The remaining integrations can be done naively, finally

giving (3.17).

APPENDIX D

Calculation of AE;$ is similar to that for AE{zf. From (3.13) one has

16 AEPP int - - ---RvO2 Jo* ~ ( ~ ~ - ~ j ~ ~ e - ~ ~ p ~ / ~ ~ ~ d ~ ,

a& ( D 1)

J 3 = JOm % $ [ G ( ~ ) - G ( o ) ] , 4

with 1

G ( q ) = R dr e - R ~ P - ~ " " B ( K - 1

F-(q2+P2-2pqt)'/2)

As in Appendix C, the p integration is divergent at p =KF. We shall separate the singular contribution expli- citly. The remaining nonelementary integrals will be reg- ular, and, hence, can be treated naively.

The q integration splits into three domains:

AI: O<q <KF ,

AII: KF -p < q <K, +p , AIII: KF+p < q < cc .

Accordingly,

11 ~ 1 1 1 J" - JSlng + J y g J ~ = J : + J ~ + 3 , 3 - 3

J\ and J!" are regular at p =KF:

KF-P J ) = Jo ~ ( ~ - ~ - ~ , ~ ~ ~ ~ - ~ - p + q ~ ~ ~ / 2 ~ P‘?

Whole singularity comes from the J!' term

KF+P 1 I -ie - R 2 ( p -qI2/2 -R2KF2/2 J;' = J K F - p pq -e )

In order to single out the most divergent contribution we add and subtract ( l/pq)e-R2p2/2 under the integral (D4). J!' splits into

and

which has only a logarithmic singularity at p =KF. Linear divergence of J y n g is now explicit:

According to Appendix C, the naive result should be sup- plemented with the principal-value prescription for the in- tegration of (D7). This gives (3.19).

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