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Instantaneous chiral quark model for relativistic mesons in a hot and dense medium
NUCLEAR PHYSICS A EISEVIER
Instantaneous
Nuclear Physics A586 (1995) 711-733
chiral quark model for relativistic mesons in a hot and dense medium *
D. Blaschke a, Yu.L. Kalinovsky a,1g2,L. Miinchow b, V.N. Pervushin b,3, G. Rijpke a, S. Schmidt a a MPG Arbeitsgruppe “Theoretische VXelteilchenphysik”, Uniuersitiit Restock, D-18051 Restock, Germany b Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia Received
18 August
1994; revised
18 November
1994
Abstract A chiral quark model with covariant instantaneous interactions is formulated using relativistic thermodynamic Green functions. The approach is applied to the description of mesons as relativistic bound states in hot and dense quark matter. The Schwinger-Dyson equation for the quark-mass operator and the Salpeter equations for quark-antiquark bound states are obtained for a nonlocal covariant four-point interaction kernel. Special attention is paid to the medium modification of meson properties as a function of their total momentum. Numerical results for the pion mass and the pion decay constant at finite temperature are presented for the special case of a separable interaction. We obtain a “protection” of the pseudoscalar bound state against medium influence due to the finite (thermal) velocity relative to the surrounding matter.
1. Introduction
One of the most interesting fields of current research in nuclear and particle physics is the search for effects occurring in hot and dense hadronic matter [l]. This research is becoming more topical in the light of the construction of new
* Work supported in part by the Federal Minister for Research and Technology (BMFT) within the Heisenberg-Landau Programme. ’ Permanent address: Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141 980 Dubna, Russia. * Supported by DFG Grant No. Ro 905/7-l. 3 Supported by Russian Fund of Fundamental Investigations No. 94/02/14411. 0375-9474/95/$09.50 0 1995 Elsevier SSDI 0375-9474(94)00809-4
Science
B.V. All rights reserved
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accelerators such as the LHC at CERN and the RHIC at Brookhaven, which are planned to provide temperatures and particle densities where the quark (and gluon) substructure of the hadrons are to become evident not only by changes in the hadronic properties but, even more interestingly, by the emergence of a new phase of matter characterized by chiral symmetry and quark deconfinement. This, therefore, makes an accurate simultaneous description of all the modifications of the properties of hadrons and their interactions at finite temperatures and densities necessary. An appropriate theoretical description of hot and dense hadronic matter should be based on quark and gluon degrees of freedom and should obey the symmetries of quantum chromodynamics. However, since up to now a QCD-based description of a many-hadron system is far from being tractable, most of the investigations at finite temperature and density have been performed within the framework of QCD-motivated effective models. These approaches consider the gluonic degrees of freedom as “frozen” and implicitely accounted for by an effective quark-quark interaction. The Nambu-Jona-Lasinio (NJL) type models [2-81 approximate the interaction by contact forces. They describe well the phenomenon of spontaneous chiral symmetry breaking and reproduce the basic low-energy theorems of hadronic physics. These chiral models, however, are not appropriate for the description of heavy quarkonia and excited meson-mass spectra. A rather successful description of heavy-quark bound states can be given by the nonrelativistic Schriidinger equation with phenomenological confining potentials [9] of finite range. In particular, the spectra of charmonium and bottomonium states have been described to high accuracy [lO,ll]. However, both the nonrelativistic potential models and the relativistic NJL model have only a limited region of validity, and a unified approach would be desirable. Such a generalized approach would be of interest, e.g., for the description of mesons consisting of heavy- as well as light-quark flavours, see [12]. The first steps in this direction have been done for the case of zero temperature and density. A QCD-inspired approach to the description of the spectrum of mesons and their interactions has recently been formulated in [13-161 as a covariant generalization of nonrelativistic potential models [9]. This instantaneous approach to mesons contains, as limiting cases, the above-mentioned descriptions of both heavy as well as light quarkonia. It is based on the following three assumptions: (i) The Markov-Yukawa definition of bilocal fields as irreducible representation of the Poincare group [17,181. (ii) The dominant role of instantaneous interactions in the formation of these bound states [13-161. (iii) The method of the reduced phase-space quantization of chromodynamics, which leads to an unambiguous separation of the instantaneous and the retardation parts in gluon-exchange interactions with respect to the eigenvector of the total momentum operator of the bound state [19,20]. In the present paper we present a generalization of this nonlocal chiral quark model with covariant instantaneous interactions to the case of finite temperature
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and density. Particular interest is devoted to the consequences of this description for the meson dispersion relations. As a result we will show that the temperature dependence of the pion mass and the pion decay constant is overestimated in approaches which neglect the finite velocity of the bound state relative to the medium. The paper is organized as follows: In Section 2 we shortly review the path-integral bosonization for an instantaneous four-point interaction kernel which allows for the introduction of covariant bilocal meson fields. We derive the SchwingerDyson as well as the Bethe-Salpeter equation from the effective chiral-quark-model lagrangian by functional-integral techniques. In Section 3 we generalize the covariant description of the bilocal mesons to the case of finite temperature and density by applying the Matsubara technique of finite-temperature Green functions. We examine the particular case of pions which are Lorentz-boosted relative to the medium. In Section 4, we apply the present approach to the case of a separable quark-antiquark interaction and discuss results of a model calculation. Finally, Section 5 contains a summary of the obtained results and the conclusions.
2. Covariant bound states in effective QCD A systematic field-theoretic approach to the description of hadronic matter on the basis of an effective-action functional in the quark sector of QCD can be formulated within the path-integral approach [21,22]. In this section we present the principal notations and results of this approach. The quark-action functional can be represented in the form
- +
/
dx, dx, dy, dy, [&(x&,(y~)l
X[X”(X,,
y,;
x2,
Here, the first term is the free-quark
Y2)]AB;CD[~~C(X2)qD(Y2)1.
(1)
action, with
(2) and &, = diag(m$, a = 1,. . . , N,, is the current quark mass matrix. The indices A, B, C, D are a compact notation for Dirac as well as flavour and colour indices. The channel decomposition of the four-point interaction kernel Z reads [22,23] [-+qq,
Y,; x2, YZ)]AB;CD= (4o’@Bp(~~
Y 1x3 Y),
where
(4)
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is the spin-flavour-colour part of the quark-antiquark correlation in the hadronic channel H, which is given as a direct product of the respective matrices in Dirac, flavour and colour spaces,
a = 1,2,3
C’= ($lC, kAi),
i=1,2
for SU(2)r,
(6)
forSU(3),.
(7)
,..., 8
The orbital part of the interaction, %7(x,
yIX,Y)=w(x*,
Y l)a4(X-
W(X7)6(Y
‘77),
(8)
has a relativistic covariant form [15,16], with x =x1 -x2, X= (x1 +x,)/2 being the four-vectors of the relative and the center-of-mass coordinates of the incoming quark-antiquark pair; y = y1 - yZ, Y = (yl + y2)/2 are those of the outgoing one, respectively. The four-dimensional a-function guarantees the condition of the center-of-mass conservation, which is a consequence of the homogeneity of the space-time continuum. The instantaneous interaction kernel is further assumed to neglect retardation effects in the s-channel so that it depends via W(x I, y ‘) only on the transverse components of the four-distances xP and y, with respect to the (conserved) four-vector P, of the center-of-mass momentum:
(9) and yj are introduced accordingly. The neglect of retardation effects in the s-channel is motivated by the analogy with quantum electrodynamics where the assumption of the dominance of instantaneous interactions in the formation of bound states can be justified [24]. The projection on the subspace of equal-time processes is represented by 6(x * 71). The potential of the effective quark interaction (8) can be constructed in a gauge-invariant way within the reduced phasespace-quantization scheme [19,20]. For the application of the general nonlocal theory, there are two important classes of potentials which are contained as special cases in the general form of W(xL, y l>, i.e. (i) bilocal potentials:
Yt
W(xL 7 yL)=63(xL
-yL)V(xL),
(10)
(ii) separable potentials: W(xl,
Y ‘)
= bg(x’)g(Y
‘).
(11)
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Within the standard functional-integral approach to bilocal field theory [3,211 the action (1) can be transformed by introducing bilocal bosonic fields J =A”&,, x2). After integration over the quark fields, the effective bosonized action takes the form ,S,,[A]=
-N,[+.N(zJ)-‘A+~T~~~(-G;‘+_.N)].
(12)
In this equation the trace Tr acts on the four-momenta as well as on the flavour and Dirac indices. We confine ourselves to the case of colour-singlet states in such a way that the trace over colour indices is performed yielding the factor NC. Due to the particular choice of the instantaneous interaction kernel (8), the fields .&&(x1, xz> = AzB_@(x 1X) can be introduced in such a way that they satisfy the Markov-Yukawa condition [17,18] for instantaneous bilocal meson fields:
x,-&_lH(x IX) = 0.
(13)
P
These fields are irreducible representations of the Poincare group with definite mass, P2 = it4$, spin and orbital parts which can be expressed as d”(x(X)
=I
dP (21r)3’21/2w,o +ei(p.xTH(xL
[e-icr~x)~H(xL (P)a$(P) I -P)a,(P)]G(x*qH),
(14)
where u&(P), a;(P) are creation and annihilation operators of a bound state. The index H denotes the set of hadron quantum numbers and PO is fixed as PO= wH =JFGg. T&x ’ I P) and I,(x L I -P> are the quark-meson-vertex amplitudes which are functions of the transverse component of the four-distance X: with respect to the four-vector P, of the center-of-mass momentum. The extremum condition for action (12),
~&fL4 = s”H L =2-h, 0, coincides with the Schwinger-Dyson
(15) equation for the quark-mass operator
2,
-X4&1~ x2) = &4,4B +K
/ dy, dy, [z”(x,,
~2;
Y,, Y~)]~~;~J&,(Y~,
Y*), (16)
where [G-‘(+
%)lAB = i&4B+1-4
-LA%
4
(17)
is the quark Green function in the mean-field approximation. This equation defines the dynamical quark-mass spectrum which is spontaneously generated by
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the mechanism of chiral symmetry breaking. The expansion of the action (12) with respect to fluctuations, .4’ =A- (2 - &,), around the classical solution (161, gives S,,[.M’]=S~~[~-&)I
+S&P]+s&4V],
(18)
where (19)
(20) The equation of motion for the fluctuating part X(x1, x2) of the bilocal field can be found from the stationarity condition for the effective action Seff[X],
Wff[ A’1 SA?
1:_r=
(21)
0.
Neglecting the interaction part (20) of the effective action (18), thus restricting ourselves to the treatment of gaussian fluctuations, we obtain the homogeneous Bethe-Salpeter equation in the ladder approximation for the vertex function r of the (#j) state, C&u
~2)
=i
/
dy,
xG,,(x,,
dy,
[X’Yxu
Y&F(YD
~2;
~1, Y,)],,;,, Y,)%(Y,~
~2).
(22)
Eq. (22) has to be solved together with Eq. (16) and describes two-particle states of the quark-antiquark system. It has been shown in [12-161 for bilocal potentials and in [23,25,26] for separable potentials that Eqs. (16) and (22) describe the spontaneous chiral symmetry breaking in accordance with the Goldstone theorem, so that the action (19) can be considered as the generalization of chiral lagrangians. In the nonrelativistic limit, Eqs. (16) and (22) reduce to the Schrodinger equation [16].
3. Pair correlations in the medium at finite temperature A short summary of the covariant treatment of pair correlations in the vacuum can be found in Appendix A. We want to generalize the approach by considering a many-particle system at finite temperature and density. In this case, we have to define the thermodynamical ensemble in a covariant way. This results in the introduction of covariant distribution functions. Equivalently, we can choose a preferable system of description, which is the center-of-mass system of the manyparticle system, where temperature and chemical potential will be defined. Then the description of pair correlations in this system contains an additional parameter, the velocity of the pair relative to the surrounding thermal medium.
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3.1. Gap equation for quark pairs moving in the medium We will start from the Schwinger-Dyson space has the form
equation (16) which in the momentum
Here the index a denotes the quark flavour. The quark four-momentum is decomposed into components parallel, pl = ~(p * ~1, and perpendicular, p L =pp -p$ to the center-of-mass four-momentum P, of the bilocal field &p, P>, dp = dp’ dpll. The interaction potential in momentum space W(p I, q ‘) is obtained from the one introduced in Eq. (8) in coordinate representation by Fourier transformation (A.2). Let us consider a Lorentz-boosted quark pair in the rest system of the medium. The Lorentz boost is defined by the velocity
IPI
(24)
US--
PO’
which corresponds to the center-of-mass momentum P,, = (PO, 1P I,0,0) of the quark-antiquark pair, where we have chosen the direction of the three-momentum vector P parallel to the PI-axis. The four-vectors in the boosted system are ~2’ = h,,pv and A$‘)= h,,yy, with ’ l/h=7 A,”
=
-l&=7
-V/AT
\
l/h=2
0 0
0 0
0
0
0
0
(25)
10 0 1
Accordingly, the separation of parallel and perpendicular components relative momentum p:) is realized using the 4-vector qlL in the form
of the
(26) such that
(27)
The projected gamma-matrices
y I(“) and yll(“) are defined accordingly.
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For the solution of the Schwinger-Dyson equation, we consider the decomposition of the self-energy into the complete set of gamma-matrices (1, r$“, y *@)), ~Ql(U)
I u) = y$“AF( p I(“)) + E”( p -y
b
+qpy
cos[26”( p +‘)]
J-(u)
;S:,“;‘)
{sin[26a(p1(“))]
- I pL(‘:)I}.
IP
In contrast to the representation of the mass operator in the vacuum (A.3), we have taken into account that particle and antiparticle energies are modified in a different way in a medium with finite baryochemical potential such that an energy splitting AE”(p ‘@‘)) between quark and antiquark states may arise. Now, the Green function (A.61 can be represented as G”( p(oo), p l(u))
A?( p I(“))
n”+(p*‘“‘)
=
p($-Ea,(pL(“))
+ie +
-iI5
p’o”‘+p(pl(“))
where
n=*(pl’“‘)
= ;(1 fy~‘[SQ~(‘:))]2}
are the projectors on positive- and negative-energy Wouthuysen matrix Y(p L@‘)),
(30) states, defined by the Foldy-
[ S”( p L(U))] *I2= cos[26”( pL(“))] T &&sin(2fiY(p
‘@))],
(31)
and E$(pl@))
=Ea(pL@))
+AE”(pL’“‘).
(32)
In order to introduce the temperature and the chemical potential of the surrounding medium, we have to replace the zero-component of the relative four-momentum in the rest frame of the medium (p,) by Matsubara frequencies of the boosted system, i.e. pp po=
dg
+ vpl”)
(33)
+pon+lf*
Inserting (33) in the expression in (291, we obtain
k!_(pL(“)) +po”+~.+Ea(p~(u)Iv)+i~
YP
i
43’
(34)
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where
En+(pl@)I u) =
E”,(p*‘“‘)
+upp
67
.
(35)
The spectral function is found to be 2 Im G”(p,,
p’(‘))
=2~[~~(pi’o’)~(p~+~a-~~(~~‘rr’~~))
Inserting the spectral function (36) in the Landau spectral representation [28] for the Green function, we obtain the finite-temperature propagator of a moving quark, ga(Pa,
PA(“))
+ n~(Pl(U))[l-fl:(Pl(u)lU)]
p~;P~yP”“l”’
=
i +
a
po+/_La-Ea+(pL(U)IU) +i.c
L(U)1u) - ie
a
iqpyi1 -f”(p*‘“‘I
u)] -ie
p,+pa+EqpLqU)
(37) where we have used the notation fq(PL@)
i
E’;(p-wl
(
exp
Iu)=
u)
T/f
-l
1I +1
T
(38)
for the Fermi distribution function of the boosted argument. The Schwinger-Dyson equation for the self-energy takes the form l(U)
.SCa(pL(“)I
u)
=mG + 2N,N,
/
dq --~(pw,
qw)
(2r13
(39) After performing the go-integration and taking into account the properties of the projection operator -yg’rg’ = 1, w e compare the result with the representation (28) of the self-energy 2Yp I(‘) I u>. We can then rewrite (39) as a coupled system of equations, E”(
p I(‘))
cos[26”( p +‘))I
xcos[26”(q~‘“‘)][l
-f:(q-yu)
-fa(qqU)],
(40)
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X6 sin[26”(p*(“))][l-f~(q*(“)Iv)-f~(q””’l~)],
(41)
A,??“( p L(“)) l(u)
=N,N,/ -W(p~‘ dq “‘,
qL(U))[f~(qL(U)Iv)
-fa(qL(U)Iv)].
(2r13
(42)
In Eq. (411, the abbreviation 5 = (p I(‘) * q L(“))/ ( p I(‘) I( q I(“) I has been introduced. The system of Eqs. (40)-(42) is the new result of the presented approach. It generalizes the result obtained in the zero-temperature case [8,13,29] in a twofold way: 6) Medium effects are included, and (ii) a Lorentz boost with respect to the medium is performed. The effect of the boost relative to the rest system of the medium can be reformulated as a change in the distribution functions of the surrounding matter. The result (40) we would have obtained using the covariant distribution function f’;(Pl(“)
[
Iv)+fz(p)=
(P2yW) + l]-’
exp
(43)
in performing the Matsubara summation. Let us comment on the case where the quark pair is at rest in the medium. This case corresponds to the choice 77= (1, 0, 0,0) for the projection operator. The four-momentum vector p,, is decomposed into pp =pk +pi with p! = (pO, 0) and p,‘= (0, p) and d p - d p0 dp. In the Schwinger-Dyson equations (40)-(42) we have to replace f:(pI(') Iu>by the usual Fermi function f",(plO)= [exp(
E’(3Tp’+)l]-l-iq(p),
and obtain the result which is form-equivalent
to the system of Eqs. (A.7), (A.@.
3.2. Salpeter equation for moving bound states in the medium The Salpeter equation (22) with the kernel (8) in the medium has the form l(u)
rab(qL(u)1 P)
= 2iN,N,
/
dp
dpli
w(q w,
p W)
(W4
XP( p” + ;P, p -+))I+(
p l(O) I P)I9(
p” - +P, p I@)). (45)
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After integrating (4.5) with temperature Green functions (37) over $1 we obtain the the Bethe-Salpeter equation in the form r=b(q
L(“)l P)
x
A0:(p~(“))y~)T~b(p~(U)IP)y~)/lb(-p~(~))[l-f:(p~(“)(u)-f6(p~(~)Iu)] E’:(pl(“‘)+Eb(pI(U))-Mab(~JU)
( +
AT (p ~(“))y&wb(
p ~‘“‘IP)y6”‘Ab+(-pp(“))[l-fl(p~(“)IU)-f~(p*(U)IU)] En(pI(“))+EbC(pI(u))+M.b(~Iv)
+
A~(pI(“))y6U)T”b(pI(“)IP)y6u)lib,(-pI(”))[f~(pI(U)Iu)-ff;(pI(U)Iu)] EL;(pI(“))-Eb+(pI(u))+M.b(~l”)
+
A”(P
UU))y&G~ab(p E’i(pL’“))-
Eb_(p I(“))-
M”b(@l”)
(46) where
kfob(,h=Iu) = @ + (/‘, -,.‘b)m.
(47)
This expression shows that in the vacuum (T = 0, pa = kb = 0) the solution of the Salpeter equation depends on the 4-momentum of the bound state in an invariant way. Thus, the medium effects on the moving mesons in Eqs. (46) and (47) can be studied starting from manifestly covariant equations of motion in the vacuum. We observe by inspecting Eqs. (46) and (47) that the influence of a finite-temperature and chemical potential on mesons propagating with a finite velocity u is less than the medium influence on mesons at rest. This effect occurs because at a finite velocity parameter u the overlap of that region in momentum space where the interaction is nonzero with the occupation of quark states by the medium particles becomes smaller and consequently the meson gets “stabilized”. In the present work, we focus on the investigation of the temperature and velocity dependences in a flavour-symmetric quark plasma with r(~,= pb. In addition to the quark- and meson-mass spectra, we calculate also the pion decay constant. For the derivation of the normalization condition, see also [25].
4. Model calculations
for a separable model
In order to give an example for the solution of the Eqs. (40)-(42) we have to specify the interaction potential in momentum space. If the potential depends only on the difference of the transverse momentum vectors,
W(P’, 4’) = qPL+),
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one arrives at the bilocal relativistic theory of mesons [8,14] which has recently been applied to the case of heavy mesons at zero temperature [12]. Here we give an evaluation for the case of a separable representation of the interaction [23,29], w(P~,q~)=~,g(IP~I)g(lq~l),
(49)
where the integrals can be immediately performed, see also [23,29]. A particular simple interaction formfactor arises in the Nambu-Jona-Lasinio model, where
g(wI)=e(~-IPLI).
(50)
It corresponds to a constant interaction in momentum space characterized by the coupling constant V, and the cutoff parameter A which regularizes the occurring integrals. Note that in the framework of a separable approach it is possible to use more realistic formfactors such as gaussian or lorenzian shapes. A systematic study of this issue has been published recently [23]. For the sake of simplicity, we confine ourselves here to the simple NJL formfactor. Furthermore, we consider the degenerate two-flavour case, where m,=mz=md,. We have fixed the parameters of the model (A = 900 MeV, V, = 1.92 X 10e5 MeV-*, m, = 5.1 MeV, rni = 140 MeV) in such a way that the pion mass (M, = 140 MeV), the kaon mass (Mk = 490 MeV), the pion decay constant (f, = 93 MeV) and the constituent quark mass (m = 330 MeV) are recovered at vanishing temperature (T = 0) and baryon density (p = 0) for mesons at rest (rJ= ]PI/P,=O). The results are shown for the masses of the constituent quarks, the pion and kaon masses as well as the pion decay constant at finite values for temperature and boost velocity. 4.1. Solution of the gap equation In the case of the separable interaction (49) with the cutoff formfactor (501, the integral in Eq. (41) vanishes, such that the set of Eqs. (40)-(42) reduces to
(51) AE=
I@)1u) -f-(p?
-___
u)],
(52)
where E,(P~‘“‘)
= {I p’(‘)[*+m*(T,
p,
u)
+AE.
The only difference with the corresponding expressions within the NJL model with 3-d cutoff is that the occupation numbers f,(p *(O)I u) which are defined according to Eq. (38) depend on the velocity parameter u via E +(p I(‘) I u).
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m”+mS
800 mu+md
z z
600
2 2
400 200 0
I
0
100
300
200
Temperature T [MeV] Fig. 1. The temperature dependence of the pseudoscalar meson masses M,, M, and the corresponding continua given by m, + md and m, + m,, respectively, at rest in the medium.
For further numerical evaluation, we will replace E + with E, thus neglecting the contribution of the energy shift AE, which is small in the region of interest. The solution of Eq. (51) for the quark mass is shown in Fig. 1 both for the m, + md and for the m, + m, thresholds as a function of the temperature for zero chemical potential and zero velocity relative to the medium. The restoration of the chiral symmetry which would occur at temperature T, - 220 MeV in the chiral limit m, = 0 is washed out due to the explicit chiral symmetry breaking by the finite current quark masses. Of special interest is the dependence of the quark mass on the boost velocity with respect to the medium. In solving the gap equation (51) for finite values of the boost velocity relative to the medium, we have to perform the integration over the angle 13 between the boost velocity (chosen in the direction of pr) and the momentum p ’(“) which occurs in the quark distribution functions (38) due to + up I(“) cos O]/ m, see Eq. (35). The integraEdP L(u)l v> = [E,(p-‘)) tion can be performed analytically and leads to the angular-averaged Pauli-blocking factor
Q(PL(“)l v)
= &/
=
dL’+,[l
-~+(J+‘)I
2Tmlncosh[E+(p P
L(U$)
u)
-f-(P?
I(‘) I v)/T]
cosh[i%(plc”)Iv)/T]
u)]
+ cosh( p/T) +cosh(,u/T)
’
(53)
D. Blaschke et al./Nuclear PhysicsA586 (1995) 71l-733
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a) m,=O 0.8 T=150 MeV
s 8 Y
0.6
j
0.4
b) m,=5.1 C
0.8
$
0.6
P
0.4
MeV
-
0.2 0
0
0.2
0.6 0.4 Velocity v [c]
0.8
1.0
Fig. 2. The dependence of the constituent quark mass on the boost velocity for different temperatures T = 150,220 and 300 MeV is displayed (a) in the chiral limit m,, = 0 and (b) for a finite bare quark mass m, = 5.1 MeV. The chiral restoration temperature for u = 0 is T, = 220 MeV. The mechanism of the restoration of the chiral symmetry is inhibited for finite velocities.
where E,(p I(“) lu)=E,(pLo’)lv)l~=a. Th e results for the dependence of the constituent quark mass m, as a function of the boost velocity are shown in Fig. 2 for zero chemical potential and different values of the medium temperature, both in the chiral limit (Fig. 2a) and with an explicit chiral symmetry breaking (Fig. 2b). For vanishing boost velocity the standard result of a strong temperature dependence of the quark mass (chiral symmetry restoration) around T - 220 MeV is recovered. As an important result we see in Fig. 2 that for finite boost velocity (u 2 0.9c) the temperature dependence is smoothed out and chiral symmetry restoration is inhibited. This behaviour is caused by the ineffectiveness of the Pauli-blocking for quarks with large momenta relative to the medium. 4.2.
Solution of the Salpeter equation for pseudoscalar mesons
In order to calculate the meson-mass spectrum, we have to perform the decomposition of the Salpeter equation into the different Dirac channels. In the
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Physics A586 (1995) 711-733
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framework of a separable approach, this decomposition leads to a matrix equation for the meson vertex functions, see [291. In particular, the pion mass A4, is defined as a solution of the equation l-Re.P(@=M,Iu)=O,
(54)
if one neglects pseudoscalar-axial-vector J”(P I u) is given as J”(PIu)
=
2V,N,N, T2
A /
dp*“’
mixing. The
(P~@‘))~
polarization
Jqpy E2(pl’“‘)_
function
(p,2)2Q(pL'""u). (55)
We notice that the Goldstone theorem is also valid for T,p # 0 and arbitrary boost velocity u. The proof is straightforward: In the chiral limit m, = 0 the equation for the pion mass (54) and the gap equation (51) for the constituent mass coincide only if the mass of the pseudoscalar meson (the pion) vanishes, P2 = A42= 0,and it therefore appears as a massless Goldstone boson. The finite pion mass A4, = 140 MeV is a result of the explicit chiral symmetry breaking due to the finite current quark mass ma. We will now discuss the behaviour of the mass of pseudoscalar mesons in dependence on the temperature and on the boost velocity. In Fig. 1, we show the result of the numerical solution of Eq. (54) for the pion mass as a function of the
F iii
‘0°) 100
-
GF
fn
--____
-a__
_
2”
--._
-___---
”
-
4bT
=
-.-\ -.*\
‘\
0
‘\
\
\ I , I I
I I
O0
100 Temperature
200 T [MeV]
Fig. 3. The temperature dependence of the mass MT and the decay constant f, for a pion at rest (dotted line) and for a pion moving with the mean thermal velocity (u>r (solid line). A comparison of both cases shows that a fast-moving pion is “protected” against modification by the medium.
126
D. Blaschke et al. /Nuclear
Physics AS86 (1995) 71 I-733
temperature if the pion is at rest in the medium for vanishing baryon-number density (F = 0). For comparison, the behaviour of the sum of the constituent quark masses is also shown in that figure. Note the fact that the mass of the pion remains fairly constant up to temperatures of about T - 100 MeV where it starts to increase slightly. The continuum of unbound quark-antiquark states (mg + m,), however, decreases rapidly and falls below the pion mass at the Mott transition temperature T, N 220 MeV. This effect is well known in many-particle physics, and it occurs since self-energy effects in the one-particle sector are partially compensated by Pauli-blocking for the two-particle states, see [30]. Let us now study the effects of a finite boost velocity relative to the medium on the behaviour of the pion mass. In Fig. 3 the solution of Eq. (55) is compared for a pion at rest and for a pion moving with a mean thermal velocity u = ( u >r, where
(56) Simultaneously, we show in Fig. 3 the results for the electroweak decay constant of the pion,
where the quark-meson dp
coupling constant g,,, A(U)
pl@)2
is defined by
E( p l(U)) [E2(p”u’)
_
(MT,2)2]2Q(pL”“u)*
(58)
In accordance with the Goldberger-Treiman relation, the decay constant is proportional to the quark mass and drops to zero when the pion enters the quark-antiquark continuum [26]. One can see from Fig. 3, that the medium influence on the pion mass and decay constant is overestimated when pions are considered at rest in the medium. At temperatures T- 200 MeV the mean thermal velocity of the pion is already (v>r N 0.95c, and the corresponding shift of the Pauli-blocking operator in phase space leads to a considerable “protection” of the pion against temperature effects. This is the main new result obtained within the present approach. 4.3. Inclusion of the strangeness degree of freedom In the flavour SU(3) case we have to consider two different bare quark masses E”, , Eb, in the
m, = mt; = rn8 and mS, with the result that the quark energies
D. Blaschke et al. /Nuclear
Physics A586 (1995) 71 I-733
727
Salpeter equation (46) are no longer degenerate and the contributions of all four terms have to be considered in the numerical solution. In order to obtain the kaon mass M, = 490 MeV at zero temperature we choose rni = 140 MeV. The temperature dependence of M, for a zero chemical potential and vanishing boost velocity is shown in Fig. 1. The behaviour of the kaon as a pseudoscalar meson in the flavour SU(3) multiplet is similar to that of the pion. Note that the continuum for the kaon does not coincide with that of the pion since the constituent mass of the strange quark differs from that of the light flavours.
5. Conclusions We have considered a covariant description of relativistic mesons as quark-antiquark bound states in hot and dense quark matter. The starting point was the relativistic theory of mesons (inspired by the Heisenberg-Pauli-type minimal quantization of QCD [13]) which unifies the chiral lagrangian for light quarkonia and the nonrelativistic potential model for heavy ones [12]. At zero density, a covariant formulation of two-particle bound states can be given by discarding retardation effects in the co-moving frame. A main result is the derivation of relativistic covariant equations for the quark self-energy and the meson vertex functions describing the behaviour of the quark and meson spectra in a hot and dense medium. As a qualitatively new aspect, we included in our considerations the influence of a nonvanishing three-momentum of mesons P = uwH. Some general conclusions may be obtained just from the structure of the equations. One interesting property with possible experimental relevance concerns the dependence of the spectrum on the total momentum of the bound states. Looking at the asymptotics of fast particles with P2 x-if,'(O), M&, one finds a characteristic reduction of Pauli-blocking leading to a stabilization of the bound states against break-up into continuum states. This behaviour is similar to that observed for clusters in nuclear matter 130,311. We emphasize that part of the peculiarities observed in the low-momentum part of the pion and kaon spectra from violent heavy-ion collisions could be caused by the momentum dependence of the quark-pair correlations in hot and dense matter. However, the investigation of the parametric velocity dependence of meson properties is only an intermediate step. The self-consistent determination of the velocity distribution of pair correlations in quark matter has to be given within a thermodynamical approach which is beyond the scope of the present work and will be given elsewhere [32].
Acknowledgment The authors would like to thank J. Hiifner, D. Ebert and M. Volkov for their interest and critical comments on this work. One of us (V.N.P.) thanks I. Puzynin,
728
D. Blaschke et al. /Nuclear Physics A586 (1995) 71l-733
P. Akishin, I. Amirkhanov and T. Strizh for discussions. The work of Yu.K. was supported by DFG grant No. Ro 905/7-l.
Appendix A. Covariant treatment of quark-pair correlations at zero temperature In this appendix, we summarize the instantaneous interaction model at zero temperature. Previous attempts to describe mesonic fluctuations at finite total momentum within a systematic many-particle approach, as, e.g., the NJL model with three-dimensional cutoff, explicitely break Lorentz covariance already in the vacuum. We will show that a manifestly covariant description of two-particle bound states can be given even for nonlocal interactions as long as retardation effects can be neglected. A.1. Gap equation
In momentum space, the gap equation (16) for the self-energy (mass operator) with the covariant notation (8) for the interaction kernel is given by [33-401 2:“( q ‘) = rnz + 2iN,N,
=mt + 2N,N,
dp
/
---+(qL, (2a)
/
dpl yW(ql, (ET)
P~)G’(P) dp” P’)
i/~G”(p”,
P’)
(A-1)
where the index a denotes the quark flavour. The quark four-momentum is decomposed into components parallel, pj = q&p - q), and perpendicular, p * = pp -pi, to the center-of-mass four-momentum P, of the bilocal field J(!p, P), dp = dp * dp”. The interaction potential in momentum space W(p' , q *) is obtained from the one introduced in Eq. (8) in coordinate representation by Fourier transformation,
W( p L , q ‘) = j dx dy ei(p’*+4’y)W( x ’ , y ‘)S( x. ~)8( y * 77). In order to solve Eq. (A.11, we introduce self-energy: -V(p’)
=Ea(pL)[Sa(P~)]2+d~~
The matrix Y(p
= [Y(p+
matrix,
d’ T -, p I , sin[26”(
that allows to define the projection states
of the
(A-3)
‘1 is the Foldy-Wouthuysen
[Sa(pL)]*2=cos[26~(pL)]
k*(p’)
the following representation
(A.9
operators
+(I +~)S’(P’)
p ‘)I,
(A-4)
on positive- and negative-energy
= +(I
~I#~(P’)]*).
(A.5)
D. Blaschke et al. /Nuclear Physics A586 (1995) 71l-733
Using Eq. (A.31, the propagator G’(P”> P’)
1 = ti _za(P~)
of a massive quark can be represented A”+(+) pll- EU( p ‘) + iE
=
729
as [13,161
’
in such a way that the dependence on the parallel component is separated. The poles of the propagator (A.6) at positive and negative energies correspond to quarks and antiquarks, respectively. After integration over $1 in (A.0 with the representation of Green function (A.61 the gap equation (A.l) can be written as a set of two coupled integral equations:
=m;S-NcNf E”(p’)
/
dq’ yJqqL
cw
7
P’) cos[2qq1)]7
(A.71
sin[26”(pL)]
for the two unknown functions E‘Yp ‘1 and WYp ‘). The system (A.7), (A.@ describes parametrically the quark dispersion relation for a nonlocal interaction in a relativistic covariant form. A.2. Salpeter equation For the instantaneous interaction kernel (81, the Salpeter equation (22) for the vertex functions in momentum space has the form [25] TUb(qL I q = 2iNcNf/
dr,
--$V(qL,pL)Ga(p++P)Tnb(pLIP)Gb(p-+P) (24
=2NcNfj~W(qL,
p’)
(2p)
/
dp"
~G”(~“+$Pl~~)~‘b(piI~)Gb(p”-~~Ip~)
In this formula, P is the total 4-momentum of the bound state. Due to the choice of the interaction kernel (8), the vertex function rnb,(p L 1PI has no dependence on the parallel component pll. The factor 4 appearing in Eq. (A.91 for the effective interaction in the case of the moving bound states, P # 0, was evaluated by taking q in the direction of P so that 7, = PJ @.
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D. Blaschke et al. /Nuclear Physics A586 (1995) 711-733
Integrating (A.9) over pII, we obtain the Salpeter equation in the form
+ Aa(P*)[wb(PL IP)+q-PL) Ea(pL)+Eb(pL)+*+iE
I
(A.lO)
*
Eq. (A.lO) has the relativistic covariant form since it depends on the center-of-mass momentum of the quark pair in an invariant way via F.P Using the explicit form of the projectors A *(p I), it is convenient to introduce the “undressed” vertex functions Fab(p I): fab(p*)
(A.ll)
=Sa(pL)rab(p+sb(pL).
The vertex functions (A.ll) P(qL
satisfy the equation
I P)
x i
A0_
A0
E”(p’)
Xf;ab(pL
+Eb(dL) - \lp2-k
+ Ea(pL)+Eb(pL)+dF+ie
i
(A.12)
Ip)[s,(pL)s,(&)]~
where Aok = (1 f rj)/2. We can see that the vertex function fab can be represented as the sum of two functions Ffb and f;b expanding over the full system of Dirac matrices [34-361 (see also Eq. (5)), Fab(pL
I P) =r;pb(pL
I P) + @fb(pL
(A.13)
I P),
with i’p” = sk( rk)qb
(A.14)
and (P)fb
= (L7;“, LP;b, (Vb)Y,
(JPb$),
I= 1, 2.
The quantities YFb, Pyb, (Yl;“b>P, (JZ’~“Y‘are the vertex functions of pseudoscalar, vector, axial-vector mesons, respectively. The investigation (A.12) in the rest frame for the zero-temperature case was recently carried [8,12]. In the main text of the present work we generalize these results finite-temperature case, see Section 3.
(A.15) scalar, of Eq. out in to the
D. Blaschke et al. /Nuclear
A.3. Pseudoscalar meson decay constant
Physics AS86 (I 995) 711- 733
731
f,
In order to calculate the pseudoscalar-meson decay constant f,, we consider the inclusion of weak interactions in the effective action (1). The weak interactions change the quark flavour indices. We must therefore restore the flavour indices in the effective action. The quadratic part of the effective action (19) in momentum space has the form
jj$)=iNc tr /
dp’ (2T)4
2
dp” _
rob(P
L
I P)Gb( P” + +J’ I P ‘)
xTb”(pllP)Ga(p”-+PJpl).
(A.16)
Now we consider the inclusion of weak interactions lagrangian of the weak interactions has the form
in the action (A.16). The
(A.17) with the elements V,, of the Kobayashi-Maskawa matrix, and the Fermi constant Q denotes the column of (u, c, t) quarks and 4 stands for the G = 10-5/m$ column of (d, s, b) quark fields. The leptons I = e, II, r and their neutrinos vI = v,, v~, V~ couple to the quark-antiquark pairs as a pair field 1, = jOPvI, where 0, = ~~(1 + y5). Thus the weak-interaction lagrangian (A.17) is very similar in its structure to the quark-antiquark effective action (A.16). For the definition of the meson decay constants we Awill introduce in, the nonlocal effective action the local leptonic weak current L,, = (G/ fi)V,,I(P,) by the substitution r
ab ~
yab
+
jab,
(A.18)
where i= O,l, and P, is the total momentum of the leptonic pair. This results in the following additional contribution to the effective (A. 161, which corresponds to weak interactions: 2 Tr(G”PQbGbib,)
= 2cI&OP(GPG)“b/P. fi
action
(A.19)
The matrix element for the decay of a meson ~3 into a leptonic pair (IV) is then (IvIs~~lI)=(2~)4s(Pyp.-PpL)
/&
+
Ii, I”)v,bf;bpg~
(A.20) where fiibPp
=
ix
j
dpL(2T)4
dp’itr[O~G~(pl~-_PIp~)~~bGb~p~~+fPlpi)]
(A.21)
D. Blaschke et al. /Nuclear Physics A586 (1995) 711-733
732
defines the electroweak decay constant of the meson 9. After integration over pll and performing the trace operation, we obtain for the decay constant f$ of pseudoscalar mesons
(A.22) where M9ab - @
is the mass of the pseudoscalar
meson, see Eq. (A.12), and
1 &!*(P’>
=
E”(p’)
+Eb(pL)
1
-M$’
* E”(pl)
+Eb(pL)
+M$ib
(A.23)
denotes the symmetric ( + > and antisymmetric ( - > modes of the particle-antiparticle propagator.
References
PI Quark Matter ‘93: Proc. Tenth Int. Conf. on Ultra-relativistic nucleus-nucleus collisions, Borliinge, Sweden, June 1993, E. Stenlund, H.-A. Gustavson, A. Oskarsson and I. Otterlund, eds. (NorthHolland, Amsterdam, 1994). Dl M.K. Volkov, Ann. Phys. 157 (1984) 282. [31 D. Ebert and H. Reinhardt, Nucl. Phys. B 271 (1986) 188. 141 D. Ebert and M.K. Volkov, Phys. Lett. B 272 (1991) 86. t51U. Vogl and W. Weise, Prog. Part. Nucl. Phys. 27 (1991) 195. [61S.P. Klevansky, Rev. Mod. Phys. 64 (1992) 649. 171T. Hatsuda and T. Kunihiro, Phys. Rev. Lett. 55 (1985) 185; Phys. Lett. B 185 (1987) 304; Prog. Theor. Phys. Suppl. 91 (1987) 284; Nucl. Phys. B 351 (1991) 593. h31Yu.L. Kalinovsky, L. Kaschluhn and V.N. Pervushin, Phys. Lett. B 231 (1989) 288; Fortschr. Phys. 38 (1990) 353. [91 W. Lucha, F. Schiiberl and D. Gromes, Phys. Reports 200 (1991) 127. [lOI E. Eichten et al., Phys. Rev. D 17 (1978) 3090; D 21 (1980) 203. t111W. Buchmiiller and S.-H.H. Tye, Phys. Rev. D 24 (1991) 132. [121 Yu.L. Kalinovsky and C. Weiss, Z. Phys. C 63 (1994) 275. 1131V.N. Pervushin, Yu.L. Kalinovsky, W. Kallies and N.A. Sarikov, Fortschr. Phys. 38 (1989) 4, 333. 1141Yu.L. Kalinovsky, W. Kalhes, B.N. Kuranov, V.N. Pervushin and N.A. Sarikov, Sov. J. Nucl. Phys. 49 (1989) 1059. 1151V.N. Pervushin, Nucl. Phys. B (Proc. Suppl.) 15 (1990) 197. 1161Yu.L. Kalinovsky, W. Kallies, L. Kaschluhn, L. Miinchow, V.N. Pervushin and N.A. Sarikov, Few-Body Systems 10 (1991) 87. t171 M.A. Markov, J. Phys. (Moscow) 3 (1940) 452. [I81 H. Yukawa, Phys. Rev. 77 (1949) 219. [191 V.N. Pervushin, Riv. Nuovo Cimento 8 (1985) 1. @I N.S. Han and V.N. Pervushin, Mod. Phys. L&t. A2 (1987) 367; Fortschr. Phys. 37 (1989) 611. [211 H. Kleinert, Understanding the fundamental constituents of matter, ed. A. Zichichi (Plenum, New York, 1978); V. Pervushin, H. Reinhardt and D. Ebert, Sov. J. Part. Nucl. 10 (1979) 444. [221 D. Ebert, H. Reinhardt and M.K. Volkov, Prog. Part. Nucl. Phys. 33 (1994) 1. 1231S. Schmidt, D. Blaschke and Yu. L. Kalinovsky, Phys. Rev. C 50 (1994) 435. 1241 S. Love, Ann. of Phys. 113 (1978) 135.
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[25] D. Blaschke et al., Dubna preprint JINR E2-94-307 (19941. [26] S. Schmidt, D. Blaschke and Yu. L. Kalinovsky, University Restock report MPG-VT-UR 38/94 (1994), Z. Phys. C, in press. [27] T. Matsubara, Prog. Theor. Phys. 14 (1957) 570; A.A. Abrikosov, L.P. Gor’kov and I.E. Dzyaloshinskii, Methods of quantum field theory in statistical physics (Prentice-Hall, Englewood Cliffs, NJ, 1963). [28] J.I. Kapusta, Finite-temperature field theory (Cambridge Univ. Press, Cambridge, 1989); A.L. Fetter and J.D. Walecka, Quantum theory of many-particle systems (McGraw-Hill, New York, 1971). [29] D. Blaschke, Yu.L. Kalinovsky, V.N. Pervushin, G. Rijpke and S. Schmidt, Z. Phys. A 346 (1993) 85. [30] W.-D. Kraeft, D. Kremp, W. Ebeling and G. Ropke, Quantum statistics of charged particle systems, (Akademie Verlag, Berlin, 1986). [31] G. Ropke, L. Miinchow and H. Schulz, Nucl. Phys. A 379 (1982) 536. [32] D. Blaschke, Yu.L. Kalinovsky and S. Schmidt, in preparation. [33] S.L. Adler and AS. Davis, Nucl. Phys. B 244 (1984) 277. [34] A. Le Yaouanc, L. Oliver, S. Ono, 0. Pene and J.-C. Raynal, Phys. Rev. D 31 (1985) 137. [35] F. Gross and J. Milana, Phys. Rev D 43 (1991) 2401. [36] J.-F. Lagae, Phys. Rev. D 45 (1992) 305. [37] A. KociE, Phys. Rev. D 33 (1986) 1785. [38] A. Trzupek and J. Wosiek, Phys. Rev. D 33 (1986) 3753. [39] H. Hirata, Prog. Theor. Phys. 77 (19871 939. [40] A.S. Davis and A.M. Matheson, Nucl. Phys. B 246 (1984) 203.
Instantaneous
Nuclear Physics A586 (1995) 711-733
chiral quark model for relativistic mesons in a hot and dense medium *
D. Blaschke a, Yu.L. Kalinovsky a,1g2,L. Miinchow b, V.N. Pervushin b,3, G. Rijpke a, S. Schmidt a a MPG Arbeitsgruppe “Theoretische VXelteilchenphysik”, Uniuersitiit Restock, D-18051 Restock, Germany b Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia Received
18 August
1994; revised
18 November
1994
Abstract A chiral quark model with covariant instantaneous interactions is formulated using relativistic thermodynamic Green functions. The approach is applied to the description of mesons as relativistic bound states in hot and dense quark matter. The Schwinger-Dyson equation for the quark-mass operator and the Salpeter equations for quark-antiquark bound states are obtained for a nonlocal covariant four-point interaction kernel. Special attention is paid to the medium modification of meson properties as a function of their total momentum. Numerical results for the pion mass and the pion decay constant at finite temperature are presented for the special case of a separable interaction. We obtain a “protection” of the pseudoscalar bound state against medium influence due to the finite (thermal) velocity relative to the surrounding matter.
1. Introduction
One of the most interesting fields of current research in nuclear and particle physics is the search for effects occurring in hot and dense hadronic matter [l]. This research is becoming more topical in the light of the construction of new
* Work supported in part by the Federal Minister for Research and Technology (BMFT) within the Heisenberg-Landau Programme. ’ Permanent address: Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141 980 Dubna, Russia. * Supported by DFG Grant No. Ro 905/7-l. 3 Supported by Russian Fund of Fundamental Investigations No. 94/02/14411. 0375-9474/95/$09.50 0 1995 Elsevier SSDI 0375-9474(94)00809-4
Science
B.V. All rights reserved
712
D. Blaschke et al. /Nuclear
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accelerators such as the LHC at CERN and the RHIC at Brookhaven, which are planned to provide temperatures and particle densities where the quark (and gluon) substructure of the hadrons are to become evident not only by changes in the hadronic properties but, even more interestingly, by the emergence of a new phase of matter characterized by chiral symmetry and quark deconfinement. This, therefore, makes an accurate simultaneous description of all the modifications of the properties of hadrons and their interactions at finite temperatures and densities necessary. An appropriate theoretical description of hot and dense hadronic matter should be based on quark and gluon degrees of freedom and should obey the symmetries of quantum chromodynamics. However, since up to now a QCD-based description of a many-hadron system is far from being tractable, most of the investigations at finite temperature and density have been performed within the framework of QCD-motivated effective models. These approaches consider the gluonic degrees of freedom as “frozen” and implicitely accounted for by an effective quark-quark interaction. The Nambu-Jona-Lasinio (NJL) type models [2-81 approximate the interaction by contact forces. They describe well the phenomenon of spontaneous chiral symmetry breaking and reproduce the basic low-energy theorems of hadronic physics. These chiral models, however, are not appropriate for the description of heavy quarkonia and excited meson-mass spectra. A rather successful description of heavy-quark bound states can be given by the nonrelativistic Schriidinger equation with phenomenological confining potentials [9] of finite range. In particular, the spectra of charmonium and bottomonium states have been described to high accuracy [lO,ll]. However, both the nonrelativistic potential models and the relativistic NJL model have only a limited region of validity, and a unified approach would be desirable. Such a generalized approach would be of interest, e.g., for the description of mesons consisting of heavy- as well as light-quark flavours, see [12]. The first steps in this direction have been done for the case of zero temperature and density. A QCD-inspired approach to the description of the spectrum of mesons and their interactions has recently been formulated in [13-161 as a covariant generalization of nonrelativistic potential models [9]. This instantaneous approach to mesons contains, as limiting cases, the above-mentioned descriptions of both heavy as well as light quarkonia. It is based on the following three assumptions: (i) The Markov-Yukawa definition of bilocal fields as irreducible representation of the Poincare group [17,181. (ii) The dominant role of instantaneous interactions in the formation of these bound states [13-161. (iii) The method of the reduced phase-space quantization of chromodynamics, which leads to an unambiguous separation of the instantaneous and the retardation parts in gluon-exchange interactions with respect to the eigenvector of the total momentum operator of the bound state [19,20]. In the present paper we present a generalization of this nonlocal chiral quark model with covariant instantaneous interactions to the case of finite temperature
D. Biaschke et al. /Nuclear
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and density. Particular interest is devoted to the consequences of this description for the meson dispersion relations. As a result we will show that the temperature dependence of the pion mass and the pion decay constant is overestimated in approaches which neglect the finite velocity of the bound state relative to the medium. The paper is organized as follows: In Section 2 we shortly review the path-integral bosonization for an instantaneous four-point interaction kernel which allows for the introduction of covariant bilocal meson fields. We derive the SchwingerDyson as well as the Bethe-Salpeter equation from the effective chiral-quark-model lagrangian by functional-integral techniques. In Section 3 we generalize the covariant description of the bilocal mesons to the case of finite temperature and density by applying the Matsubara technique of finite-temperature Green functions. We examine the particular case of pions which are Lorentz-boosted relative to the medium. In Section 4, we apply the present approach to the case of a separable quark-antiquark interaction and discuss results of a model calculation. Finally, Section 5 contains a summary of the obtained results and the conclusions.
2. Covariant bound states in effective QCD A systematic field-theoretic approach to the description of hadronic matter on the basis of an effective-action functional in the quark sector of QCD can be formulated within the path-integral approach [21,22]. In this section we present the principal notations and results of this approach. The quark-action functional can be represented in the form
- +
/
dx, dx, dy, dy, [&(x&,(y~)l
X[X”(X,,
y,;
x2,
Here, the first term is the free-quark
Y2)]AB;CD[~~C(X2)qD(Y2)1.
(1)
action, with
(2) and &, = diag(m$, a = 1,. . . , N,, is the current quark mass matrix. The indices A, B, C, D are a compact notation for Dirac as well as flavour and colour indices. The channel decomposition of the four-point interaction kernel Z reads [22,23] [-+qq,
Y,; x2, YZ)]AB;CD= (4o’@Bp(~~
Y 1x3 Y),
where
(4)
D. Blaschke et al. /Nuclear
714
Physics A586 (1995) 71 l-733
is the spin-flavour-colour part of the quark-antiquark correlation in the hadronic channel H, which is given as a direct product of the respective matrices in Dirac, flavour and colour spaces,
a = 1,2,3
C’= ($lC, kAi),
i=1,2
for SU(2)r,
(6)
forSU(3),.
(7)
,..., 8
The orbital part of the interaction, %7(x,
yIX,Y)=w(x*,
Y l)a4(X-
W(X7)6(Y
‘77),
(8)
has a relativistic covariant form [15,16], with x =x1 -x2, X= (x1 +x,)/2 being the four-vectors of the relative and the center-of-mass coordinates of the incoming quark-antiquark pair; y = y1 - yZ, Y = (yl + y2)/2 are those of the outgoing one, respectively. The four-dimensional a-function guarantees the condition of the center-of-mass conservation, which is a consequence of the homogeneity of the space-time continuum. The instantaneous interaction kernel is further assumed to neglect retardation effects in the s-channel so that it depends via W(x I, y ‘) only on the transverse components of the four-distances xP and y, with respect to the (conserved) four-vector P, of the center-of-mass momentum:
(9) and yj are introduced accordingly. The neglect of retardation effects in the s-channel is motivated by the analogy with quantum electrodynamics where the assumption of the dominance of instantaneous interactions in the formation of bound states can be justified [24]. The projection on the subspace of equal-time processes is represented by 6(x * 71). The potential of the effective quark interaction (8) can be constructed in a gauge-invariant way within the reduced phasespace-quantization scheme [19,20]. For the application of the general nonlocal theory, there are two important classes of potentials which are contained as special cases in the general form of W(xL, y l>, i.e. (i) bilocal potentials:
Yt
W(xL 7 yL)=63(xL
-yL)V(xL),
(10)
(ii) separable potentials: W(xl,
Y ‘)
= bg(x’)g(Y
‘).
(11)
D. Blaschke et al. /Nuclear
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715
Within the standard functional-integral approach to bilocal field theory [3,211 the action (1) can be transformed by introducing bilocal bosonic fields J =A”&,, x2). After integration over the quark fields, the effective bosonized action takes the form ,S,,[A]=
-N,[+.N(zJ)-‘A+~T~~~(-G;‘+_.N)].
(12)
In this equation the trace Tr acts on the four-momenta as well as on the flavour and Dirac indices. We confine ourselves to the case of colour-singlet states in such a way that the trace over colour indices is performed yielding the factor NC. Due to the particular choice of the instantaneous interaction kernel (8), the fields .&&(x1, xz> = AzB_@(x 1X) can be introduced in such a way that they satisfy the Markov-Yukawa condition [17,18] for instantaneous bilocal meson fields:
x,-&_lH(x IX) = 0.
(13)
P
These fields are irreducible representations of the Poincare group with definite mass, P2 = it4$, spin and orbital parts which can be expressed as d”(x(X)
=I
dP (21r)3’21/2w,o +ei(p.xTH(xL
[e-icr~x)~H(xL (P)a$(P) I -P)a,(P)]G(x*qH),
(14)
where u&(P), a;(P) are creation and annihilation operators of a bound state. The index H denotes the set of hadron quantum numbers and PO is fixed as PO= wH =JFGg. T&x ’ I P) and I,(x L I -P> are the quark-meson-vertex amplitudes which are functions of the transverse component of the four-distance X: with respect to the four-vector P, of the center-of-mass momentum. The extremum condition for action (12),
~&fL4 = s”H L =2-h, 0, coincides with the Schwinger-Dyson
(15) equation for the quark-mass operator
2,
-X4&1~ x2) = &4,4B +K
/ dy, dy, [z”(x,,
~2;
Y,, Y~)]~~;~J&,(Y~,
Y*), (16)
where [G-‘(+
%)lAB = i&4B+1-4
-LA%
4
(17)
is the quark Green function in the mean-field approximation. This equation defines the dynamical quark-mass spectrum which is spontaneously generated by
716
D, Blaschke et al. /Nuclear Physics A586 (1995) 711- 733
the mechanism of chiral symmetry breaking. The expansion of the action (12) with respect to fluctuations, .4’ =A- (2 - &,), around the classical solution (161, gives S,,[.M’]=S~~[~-&)I
+S&P]+s&4V],
(18)
where (19)
(20) The equation of motion for the fluctuating part X(x1, x2) of the bilocal field can be found from the stationarity condition for the effective action Seff[X],
Wff[ A’1 SA?
1:_r=
(21)
0.
Neglecting the interaction part (20) of the effective action (18), thus restricting ourselves to the treatment of gaussian fluctuations, we obtain the homogeneous Bethe-Salpeter equation in the ladder approximation for the vertex function r of the (#j) state, C&u
~2)
=i
/
dy,
xG,,(x,,
dy,
[X’Yxu
Y&F(YD
~2;
~1, Y,)],,;,, Y,)%(Y,~
~2).
(22)
Eq. (22) has to be solved together with Eq. (16) and describes two-particle states of the quark-antiquark system. It has been shown in [12-161 for bilocal potentials and in [23,25,26] for separable potentials that Eqs. (16) and (22) describe the spontaneous chiral symmetry breaking in accordance with the Goldstone theorem, so that the action (19) can be considered as the generalization of chiral lagrangians. In the nonrelativistic limit, Eqs. (16) and (22) reduce to the Schrodinger equation [16].
3. Pair correlations in the medium at finite temperature A short summary of the covariant treatment of pair correlations in the vacuum can be found in Appendix A. We want to generalize the approach by considering a many-particle system at finite temperature and density. In this case, we have to define the thermodynamical ensemble in a covariant way. This results in the introduction of covariant distribution functions. Equivalently, we can choose a preferable system of description, which is the center-of-mass system of the manyparticle system, where temperature and chemical potential will be defined. Then the description of pair correlations in this system contains an additional parameter, the velocity of the pair relative to the surrounding thermal medium.
D. Blaschke et al. /Nuclear Physics A586 (I 995) 711- 733
717
3.1. Gap equation for quark pairs moving in the medium We will start from the Schwinger-Dyson space has the form
equation (16) which in the momentum
Here the index a denotes the quark flavour. The quark four-momentum is decomposed into components parallel, pl = ~(p * ~1, and perpendicular, p L =pp -p$ to the center-of-mass four-momentum P, of the bilocal field &p, P>, dp = dp’ dpll. The interaction potential in momentum space W(p I, q ‘) is obtained from the one introduced in Eq. (8) in coordinate representation by Fourier transformation (A.2). Let us consider a Lorentz-boosted quark pair in the rest system of the medium. The Lorentz boost is defined by the velocity
IPI
(24)
US--
PO’
which corresponds to the center-of-mass momentum P,, = (PO, 1P I,0,0) of the quark-antiquark pair, where we have chosen the direction of the three-momentum vector P parallel to the PI-axis. The four-vectors in the boosted system are ~2’ = h,,pv and A$‘)= h,,yy, with ’ l/h=7 A,”
=
-l&=7
-V/AT
\
l/h=2
0 0
0 0
0
0
0
0
(25)
10 0 1
Accordingly, the separation of parallel and perpendicular components relative momentum p:) is realized using the 4-vector qlL in the form
of the
(26) such that
(27)
The projected gamma-matrices
y I(“) and yll(“) are defined accordingly.
D. Blaschke et al. /Nuclear Physics AS86 (1995) 71l-733
718
For the solution of the Schwinger-Dyson equation, we consider the decomposition of the self-energy into the complete set of gamma-matrices (1, r$“, y *@)), ~Ql(U)
I u) = y$“AF( p I(“)) + E”( p -y
b
+qpy
cos[26”( p +‘)]
J-(u)
;S:,“;‘)
{sin[26a(p1(“))]
- I pL(‘:)I}.
IP
In contrast to the representation of the mass operator in the vacuum (A.3), we have taken into account that particle and antiparticle energies are modified in a different way in a medium with finite baryochemical potential such that an energy splitting AE”(p ‘@‘)) between quark and antiquark states may arise. Now, the Green function (A.61 can be represented as G”( p(oo), p l(u))
A?( p I(“))
n”+(p*‘“‘)
=
p($-Ea,(pL(“))
+ie +
-iI5
p’o”‘+p(pl(“))
where
n=*(pl’“‘)
= ;(1 fy~‘[SQ~(‘:))]2}
are the projectors on positive- and negative-energy Wouthuysen matrix Y(p L@‘)),
(30) states, defined by the Foldy-
[ S”( p L(U))] *I2= cos[26”( pL(“))] T &&sin(2fiY(p
‘@))],
(31)
and E$(pl@))
=Ea(pL@))
+AE”(pL’“‘).
(32)
In order to introduce the temperature and the chemical potential of the surrounding medium, we have to replace the zero-component of the relative four-momentum in the rest frame of the medium (p,) by Matsubara frequencies
dg
+ vpl”)
(33)
+pon+lf*
Inserting (33) in the expression in (291, we obtain
k!_(pL(“)) +po”+~.+Ea(p~(u)Iv)+i~
YP
i
43’
(34)
D. Blaschke et al. /Nuclear
Physics A586 (1995) 711-733
719
where
En+(pl@)I u) =
E”,(p*‘“‘)
+upp
67
.
(35)
The spectral function is found to be 2 Im G”(p,,
p’(‘))
=2~[~~(pi’o’)~(p~+~a-~~(~~‘rr’~~))
Inserting the spectral function (36) in the Landau spectral representation [28] for the Green function, we obtain the finite-temperature propagator of a moving quark, ga(Pa,
PA(“))
+ n~(Pl(U))[l-fl:(Pl(u)lU)]
p~;P~yP”“l”’
=
i +
a
po+/_La-Ea+(pL(U)IU) +i.c
L(U)1u) - ie
a
iqpyi1 -f”(p*‘“‘I
u)] -ie
p,+pa+EqpLqU)
(37) where we have used the notation fq(PL@)
i
E’;(p-wl
(
exp
Iu)=
u)
T/f
-l
1I +1
T
(38)
for the Fermi distribution function of the boosted argument. The Schwinger-Dyson equation for the self-energy takes the form l(U)
.SCa(pL(“)I
u)
=mG + 2N,N,
/
dq --~(pw,
qw)
(2r13
(39) After performing the go-integration and taking into account the properties of the projection operator -yg’rg’ = 1, w e compare the result with the representation (28) of the self-energy 2Yp I(‘) I u>. We can then rewrite (39) as a coupled system of equations, E”(
p I(‘))
cos[26”( p +‘))I
xcos[26”(q~‘“‘)][l
-f:(q-yu)
-fa(qqU)],
(40)
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D. Blaschke et al. /Nuclear Physics A586 (1995) 71l-733
X6 sin[26”(p*(“))][l-f~(q*(“)Iv)-f~(q””’l~)],
(41)
A,??“( p L(“)) l(u)
=N,N,/ -W(p~‘ dq “‘,
qL(U))[f~(qL(U)Iv)
-fa(qL(U)Iv)].
(2r13
(42)
In Eq. (411, the abbreviation 5 = (p I(‘) * q L(“))/ ( p I(‘) I( q I(“) I has been introduced. The system of Eqs. (40)-(42) is the new result of the presented approach. It generalizes the result obtained in the zero-temperature case [8,13,29] in a twofold way: 6) Medium effects are included, and (ii) a Lorentz boost with respect to the medium is performed. The effect of the boost relative to the rest system of the medium can be reformulated as a change in the distribution functions of the surrounding matter. The result (40) we would have obtained using the covariant distribution function f’;(Pl(“)
[
Iv)+fz(p)=
(P2yW) + l]-’
exp
(43)
in performing the Matsubara summation. Let us comment on the case where the quark pair is at rest in the medium. This case corresponds to the choice 77= (1, 0, 0,0) for the projection operator. The four-momentum vector p,, is decomposed into pp =pk +pi with p! = (pO, 0) and p,‘= (0, p) and d p - d p0 dp. In the Schwinger-Dyson equations (40)-(42) we have to replace f:(pI(') Iu>by the usual Fermi function f",(plO)= [exp(
E’(3Tp’+)l]-l-iq(p),
and obtain the result which is form-equivalent
to the system of Eqs. (A.7), (A.@.
3.2. Salpeter equation for moving bound states in the medium The Salpeter equation (22) with the kernel (8) in the medium has the form l(u)
rab(qL(u)1 P)
= 2iN,N,
/
dp
dpli
w(q w,
p W)
(W4
XP( p” + ;P, p -+))I+(
p l(O) I P)I9(
p” - +P, p I@)). (45)
D. Blaschke et al. /Nuclear
Physics A586 (1995) 711-733
721
After integrating (4.5) with temperature Green functions (37) over $1 we obtain the the Bethe-Salpeter equation in the form r=b(q
L(“)l P)
x
A0:(p~(“))y~)T~b(p~(U)IP)y~)/lb(-p~(~))[l-f:(p~(“)(u)-f6(p~(~)Iu)] E’:(pl(“‘)+Eb(pI(U))-Mab(~JU)
( +
AT (p ~(“))y&wb(
p ~‘“‘IP)y6”‘Ab+(-pp(“))[l-fl(p~(“)IU)-f~(p*(U)IU)] En(pI(“))+EbC(pI(u))+M.b(~Iv)
+
A~(pI(“))y6U)T”b(pI(“)IP)y6u)lib,(-pI(”))[f~(pI(U)Iu)-ff;(pI(U)Iu)] EL;(pI(“))-Eb+(pI(u))+M.b(~l”)
+
A”(P
UU))y&G~ab(p E’i(pL’“))-
Eb_(p I(“))-
M”b(@l”)
(46) where
kfob(,h=Iu) = @ + (/‘, -,.‘b)m.
(47)
This expression shows that in the vacuum (T = 0, pa = kb = 0) the solution of the Salpeter equation depends on the 4-momentum of the bound state in an invariant way. Thus, the medium effects on the moving mesons in Eqs. (46) and (47) can be studied starting from manifestly covariant equations of motion in the vacuum. We observe by inspecting Eqs. (46) and (47) that the influence of a finite-temperature and chemical potential on mesons propagating with a finite velocity u is less than the medium influence on mesons at rest. This effect occurs because at a finite velocity parameter u the overlap of that region in momentum space where the interaction is nonzero with the occupation of quark states by the medium particles becomes smaller and consequently the meson gets “stabilized”. In the present work, we focus on the investigation of the temperature and velocity dependences in a flavour-symmetric quark plasma with r(~,= pb. In addition to the quark- and meson-mass spectra, we calculate also the pion decay constant. For the derivation of the normalization condition, see also [25].
4. Model calculations
for a separable model
In order to give an example for the solution of the Eqs. (40)-(42) we have to specify the interaction potential in momentum space. If the potential depends only on the difference of the transverse momentum vectors,
W(P’, 4’) = qPL+),
D. Blaschke et al. /Nuclear
722
Physics A586 (1995) 711-733
one arrives at the bilocal relativistic theory of mesons [8,14] which has recently been applied to the case of heavy mesons at zero temperature [12]. Here we give an evaluation for the case of a separable representation of the interaction [23,29], w(P~,q~)=~,g(IP~I)g(lq~l),
(49)
where the integrals can be immediately performed, see also [23,29]. A particular simple interaction formfactor arises in the Nambu-Jona-Lasinio model, where
g(wI)=e(~-IPLI).
(50)
It corresponds to a constant interaction in momentum space characterized by the coupling constant V, and the cutoff parameter A which regularizes the occurring integrals. Note that in the framework of a separable approach it is possible to use more realistic formfactors such as gaussian or lorenzian shapes. A systematic study of this issue has been published recently [23]. For the sake of simplicity, we confine ourselves here to the simple NJL formfactor. Furthermore, we consider the degenerate two-flavour case, where m,=mz=md,. We have fixed the parameters of the model (A = 900 MeV, V, = 1.92 X 10e5 MeV-*, m, = 5.1 MeV, rni = 140 MeV) in such a way that the pion mass (M, = 140 MeV), the kaon mass (Mk = 490 MeV), the pion decay constant (f, = 93 MeV) and the constituent quark mass (m = 330 MeV) are recovered at vanishing temperature (T = 0) and baryon density (p = 0) for mesons at rest (rJ= ]PI/P,=O). The results are shown for the masses of the constituent quarks, the pion and kaon masses as well as the pion decay constant at finite values for temperature and boost velocity. 4.1. Solution of the gap equation In the case of the separable interaction (49) with the cutoff formfactor (501, the integral in Eq. (41) vanishes, such that the set of Eqs. (40)-(42) reduces to
(51) AE=
I@)1u) -f-(p?
-___
u)],
(52)
where E,(P~‘“‘)
= {I p’(‘)[*+m*(T,
p,
u)
+AE.
The only difference with the corresponding expressions within the NJL model with 3-d cutoff is that the occupation numbers f,(p *(O)I u) which are defined according to Eq. (38) depend on the velocity parameter u via E +(p I(‘) I u).
D. Blaschke et al. /Nuclear
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Physics A586 (I 99.5) 71 I-733
723
m”+mS
800 mu+md
z z
600
2 2
400 200 0
I
0
100
300
200
Temperature T [MeV] Fig. 1. The temperature dependence of the pseudoscalar meson masses M,, M, and the corresponding continua given by m, + md and m, + m,, respectively, at rest in the medium.
For further numerical evaluation, we will replace E + with E, thus neglecting the contribution of the energy shift AE, which is small in the region of interest. The solution of Eq. (51) for the quark mass is shown in Fig. 1 both for the m, + md and for the m, + m, thresholds as a function of the temperature for zero chemical potential and zero velocity relative to the medium. The restoration of the chiral symmetry which would occur at temperature T, - 220 MeV in the chiral limit m, = 0 is washed out due to the explicit chiral symmetry breaking by the finite current quark masses. Of special interest is the dependence of the quark mass on the boost velocity with respect to the medium. In solving the gap equation (51) for finite values of the boost velocity relative to the medium, we have to perform the integration over the angle 13 between the boost velocity (chosen in the direction of pr) and the momentum p ’(“) which occurs in the quark distribution functions (38) due to + up I(“) cos O]/ m, see Eq. (35). The integraEdP L(u)l v> = [E,(p-‘)) tion can be performed analytically and leads to the angular-averaged Pauli-blocking factor
Q(PL(“)l v)
= &/
=
dL’+,[l
-~+(J+‘)I
2Tmlncosh[E+(p P
L(U$)
u)
-f-(P?
I(‘) I v)/T]
cosh[i%(plc”)Iv)/T]
u)]
+ cosh( p/T) +cosh(,u/T)
’
(53)
D. Blaschke et al./Nuclear PhysicsA586 (1995) 71l-733
724
a) m,=O 0.8 T=150 MeV
s 8 Y
0.6
j
0.4
b) m,=5.1 C
0.8
$
0.6
P
0.4
MeV
-
0.2 0
0
0.2
0.6 0.4 Velocity v [c]
0.8
1.0
Fig. 2. The dependence of the constituent quark mass on the boost velocity for different temperatures T = 150,220 and 300 MeV is displayed (a) in the chiral limit m,, = 0 and (b) for a finite bare quark mass m, = 5.1 MeV. The chiral restoration temperature for u = 0 is T, = 220 MeV. The mechanism of the restoration of the chiral symmetry is inhibited for finite velocities.
where E,(p I(“) lu)=E,(pLo’)lv)l~=a. Th e results for the dependence of the constituent quark mass m, as a function of the boost velocity are shown in Fig. 2 for zero chemical potential and different values of the medium temperature, both in the chiral limit (Fig. 2a) and with an explicit chiral symmetry breaking (Fig. 2b). For vanishing boost velocity the standard result of a strong temperature dependence of the quark mass (chiral symmetry restoration) around T - 220 MeV is recovered. As an important result we see in Fig. 2 that for finite boost velocity (u 2 0.9c) the temperature dependence is smoothed out and chiral symmetry restoration is inhibited. This behaviour is caused by the ineffectiveness of the Pauli-blocking for quarks with large momenta relative to the medium. 4.2.
Solution of the Salpeter equation for pseudoscalar mesons
In order to calculate the meson-mass spectrum, we have to perform the decomposition of the Salpeter equation into the different Dirac channels. In the
D. Blaschke et al. /Nuclear
Physics A586 (1995) 711-733
725
framework of a separable approach, this decomposition leads to a matrix equation for the meson vertex functions, see [291. In particular, the pion mass A4, is defined as a solution of the equation l-Re.P(@=M,Iu)=O,
(54)
if one neglects pseudoscalar-axial-vector J”(P I u) is given as J”(PIu)
=
2V,N,N, T2
A /
dp*“’
mixing. The
(P~@‘))~
polarization
Jqpy E2(pl’“‘)_
function
(p,2)2Q(pL'""u). (55)
We notice that the Goldstone theorem is also valid for T,p # 0 and arbitrary boost velocity u. The proof is straightforward: In the chiral limit m, = 0 the equation for the pion mass (54) and the gap equation (51) for the constituent mass coincide only if the mass of the pseudoscalar meson (the pion) vanishes, P2 = A42= 0,and it therefore appears as a massless Goldstone boson. The finite pion mass A4, = 140 MeV is a result of the explicit chiral symmetry breaking due to the finite current quark mass ma. We will now discuss the behaviour of the mass of pseudoscalar mesons in dependence on the temperature and on the boost velocity. In Fig. 1, we show the result of the numerical solution of Eq. (54) for the pion mass as a function of the
F iii
‘0°) 100
-
GF
fn
--____
-a__
_
2”
--._
-___---
”
-
4bT
=
-.-\ -.*\
‘\
0
‘\
\
\ I , I I
I I
O0
100 Temperature
200 T [MeV]
Fig. 3. The temperature dependence of the mass MT and the decay constant f, for a pion at rest (dotted line) and for a pion moving with the mean thermal velocity (u>r (solid line). A comparison of both cases shows that a fast-moving pion is “protected” against modification by the medium.
126
D. Blaschke et al. /Nuclear
Physics AS86 (1995) 71 I-733
temperature if the pion is at rest in the medium for vanishing baryon-number density (F = 0). For comparison, the behaviour of the sum of the constituent quark masses is also shown in that figure. Note the fact that the mass of the pion remains fairly constant up to temperatures of about T - 100 MeV where it starts to increase slightly. The continuum of unbound quark-antiquark states (mg + m,), however, decreases rapidly and falls below the pion mass at the Mott transition temperature T, N 220 MeV. This effect is well known in many-particle physics, and it occurs since self-energy effects in the one-particle sector are partially compensated by Pauli-blocking for the two-particle states, see [30]. Let us now study the effects of a finite boost velocity relative to the medium on the behaviour of the pion mass. In Fig. 3 the solution of Eq. (55) is compared for a pion at rest and for a pion moving with a mean thermal velocity u = ( u >r, where
(56) Simultaneously, we show in Fig. 3 the results for the electroweak decay constant of the pion,
where the quark-meson dp
coupling constant g,,, A(U)
pl@)2
is defined by
E( p l(U)) [E2(p”u’)
_
(MT,2)2]2Q(pL”“u)*
(58)
In accordance with the Goldberger-Treiman relation, the decay constant is proportional to the quark mass and drops to zero when the pion enters the quark-antiquark continuum [26]. One can see from Fig. 3, that the medium influence on the pion mass and decay constant is overestimated when pions are considered at rest in the medium. At temperatures T- 200 MeV the mean thermal velocity of the pion is already (v>r N 0.95c, and the corresponding shift of the Pauli-blocking operator in phase space leads to a considerable “protection” of the pion against temperature effects. This is the main new result obtained within the present approach. 4.3. Inclusion of the strangeness degree of freedom In the flavour SU(3) case we have to consider two different bare quark masses E”, , Eb, in the
m, = mt; = rn8 and mS, with the result that the quark energies
D. Blaschke et al. /Nuclear
Physics A586 (1995) 71 I-733
727
Salpeter equation (46) are no longer degenerate and the contributions of all four terms have to be considered in the numerical solution. In order to obtain the kaon mass M, = 490 MeV at zero temperature we choose rni = 140 MeV. The temperature dependence of M, for a zero chemical potential and vanishing boost velocity is shown in Fig. 1. The behaviour of the kaon as a pseudoscalar meson in the flavour SU(3) multiplet is similar to that of the pion. Note that the continuum for the kaon does not coincide with that of the pion since the constituent mass of the strange quark differs from that of the light flavours.
5. Conclusions We have considered a covariant description of relativistic mesons as quark-antiquark bound states in hot and dense quark matter. The starting point was the relativistic theory of mesons (inspired by the Heisenberg-Pauli-type minimal quantization of QCD [13]) which unifies the chiral lagrangian for light quarkonia and the nonrelativistic potential model for heavy ones [12]. At zero density, a covariant formulation of two-particle bound states can be given by discarding retardation effects in the co-moving frame. A main result is the derivation of relativistic covariant equations for the quark self-energy and the meson vertex functions describing the behaviour of the quark and meson spectra in a hot and dense medium. As a qualitatively new aspect, we included in our considerations the influence of a nonvanishing three-momentum of mesons P = uwH. Some general conclusions may be obtained just from the structure of the equations. One interesting property with possible experimental relevance concerns the dependence of the spectrum on the total momentum of the bound states. Looking at the asymptotics of fast particles with P2 x-if,'(O), M&, one finds a characteristic reduction of Pauli-blocking leading to a stabilization of the bound states against break-up into continuum states. This behaviour is similar to that observed for clusters in nuclear matter 130,311. We emphasize that part of the peculiarities observed in the low-momentum part of the pion and kaon spectra from violent heavy-ion collisions could be caused by the momentum dependence of the quark-pair correlations in hot and dense matter. However, the investigation of the parametric velocity dependence of meson properties is only an intermediate step. The self-consistent determination of the velocity distribution of pair correlations in quark matter has to be given within a thermodynamical approach which is beyond the scope of the present work and will be given elsewhere [32].
Acknowledgment The authors would like to thank J. Hiifner, D. Ebert and M. Volkov for their interest and critical comments on this work. One of us (V.N.P.) thanks I. Puzynin,
728
D. Blaschke et al. /Nuclear Physics A586 (1995) 71l-733
P. Akishin, I. Amirkhanov and T. Strizh for discussions. The work of Yu.K. was supported by DFG grant No. Ro 905/7-l.
Appendix A. Covariant treatment of quark-pair correlations at zero temperature In this appendix, we summarize the instantaneous interaction model at zero temperature. Previous attempts to describe mesonic fluctuations at finite total momentum within a systematic many-particle approach, as, e.g., the NJL model with three-dimensional cutoff, explicitely break Lorentz covariance already in the vacuum. We will show that a manifestly covariant description of two-particle bound states can be given even for nonlocal interactions as long as retardation effects can be neglected. A.1. Gap equation
In momentum space, the gap equation (16) for the self-energy (mass operator) with the covariant notation (8) for the interaction kernel is given by [33-401 2:“( q ‘) = rnz + 2iN,N,
=mt + 2N,N,
dp
/
---+(qL, (2a)
/
dpl yW(ql, (ET)
P~)G’(P) dp” P’)
i/~G”(p”,
P’)
(A-1)
where the index a denotes the quark flavour. The quark four-momentum is decomposed into components parallel, pj = q&p - q), and perpendicular, p * = pp -pi, to the center-of-mass four-momentum P, of the bilocal field J(!p, P), dp = dp * dp”. The interaction potential in momentum space W(p' , q *) is obtained from the one introduced in Eq. (8) in coordinate representation by Fourier transformation,
W( p L , q ‘) = j dx dy ei(p’*+4’y)W( x ’ , y ‘)S( x. ~)8( y * 77). In order to solve Eq. (A.11, we introduce self-energy: -V(p’)
=Ea(pL)[Sa(P~)]2+d~~
The matrix Y(p
= [Y(p+
matrix,
d’ T -, p I , sin[26”(
that allows to define the projection states
of the
(A-3)
‘1 is the Foldy-Wouthuysen
[Sa(pL)]*2=cos[26~(pL)]
k*(p’)
the following representation
(A.9
operators
+(I +~)S’(P’)
p ‘)I,
(A-4)
on positive- and negative-energy
= +(I
~I#~(P’)]*).
(A.5)
D. Blaschke et al. /Nuclear Physics A586 (1995) 71l-733
Using Eq. (A.31, the propagator G’(P”> P’)
1 = ti _za(P~)
of a massive quark can be represented A”+(+) pll- EU( p ‘) + iE
=
729
as [13,161
’
in such a way that the dependence on the parallel component is separated. The poles of the propagator (A.6) at positive and negative energies correspond to quarks and antiquarks, respectively. After integration over $1 in (A.0 with the representation of Green function (A.61 the gap equation (A.l) can be written as a set of two coupled integral equations:
=m;S-NcNf E”(p’)
/
dq’ yJqqL
cw
7
P’) cos[2qq1)]7
(A.71
sin[26”(pL)]
for the two unknown functions E‘Yp ‘1 and WYp ‘). The system (A.7), (A.@ describes parametrically the quark dispersion relation for a nonlocal interaction in a relativistic covariant form. A.2. Salpeter equation For the instantaneous interaction kernel (81, the Salpeter equation (22) for the vertex functions in momentum space has the form [25] TUb(qL I q = 2iNcNf/
dr,
--$V(qL,pL)Ga(p++P)Tnb(pLIP)Gb(p-+P) (24
=2NcNfj~W(qL,
p’)
(2p)
/
dp"
~G”(~“+$Pl~~)~‘b(piI~)Gb(p”-~~Ip~)
In this formula, P is the total 4-momentum of the bound state. Due to the choice of the interaction kernel (8), the vertex function rnb,(p L 1PI has no dependence on the parallel component pll. The factor 4 appearing in Eq. (A.91 for the effective interaction in the case of the moving bound states, P # 0, was evaluated by taking q in the direction of P so that 7, = PJ @.
730
D. Blaschke et al. /Nuclear Physics A586 (1995) 711-733
Integrating (A.9) over pII, we obtain the Salpeter equation in the form
+ Aa(P*)[wb(PL IP)+q-PL) Ea(pL)+Eb(pL)+*+iE
I
(A.lO)
*
Eq. (A.lO) has the relativistic covariant form since it depends on the center-of-mass momentum of the quark pair in an invariant way via F.P Using the explicit form of the projectors A *(p I), it is convenient to introduce the “undressed” vertex functions Fab(p I): fab(p*)
(A.ll)
=Sa(pL)rab(p+sb(pL).
The vertex functions (A.ll) P(qL
satisfy the equation
I P)
x i
A0_
A0
E”(p’)
Xf;ab(pL
+Eb(dL) - \lp2-k
+ Ea(pL)+Eb(pL)+dF+ie
i
(A.12)
Ip)[s,(pL)s,(&)]~
where Aok = (1 f rj)/2. We can see that the vertex function fab can be represented as the sum of two functions Ffb and f;b expanding over the full system of Dirac matrices [34-361 (see also Eq. (5)), Fab(pL
I P) =r;pb(pL
I P) + @fb(pL
(A.13)
I P),
with i’p” = sk( rk)qb
(A.14)
and (P)fb
= (L7;“, LP;b, (Vb)Y,
(JPb$),
I= 1, 2.
The quantities YFb, Pyb, (Yl;“b>P, (JZ’~“Y‘are the vertex functions of pseudoscalar, vector, axial-vector mesons, respectively. The investigation (A.12) in the rest frame for the zero-temperature case was recently carried [8,12]. In the main text of the present work we generalize these results finite-temperature case, see Section 3.
(A.15) scalar, of Eq. out in to the
D. Blaschke et al. /Nuclear
A.3. Pseudoscalar meson decay constant
Physics AS86 (I 995) 711- 733
731
f,
In order to calculate the pseudoscalar-meson decay constant f,, we consider the inclusion of weak interactions in the effective action (1). The weak interactions change the quark flavour indices. We must therefore restore the flavour indices in the effective action. The quadratic part of the effective action (19) in momentum space has the form
jj$)=iNc tr /
dp’ (2T)4
2
dp” _
rob(P
L
I P)Gb( P” + +J’ I P ‘)
xTb”(pllP)Ga(p”-+PJpl).
(A.16)
Now we consider the inclusion of weak interactions lagrangian of the weak interactions has the form
in the action (A.16). The
(A.17) with the elements V,, of the Kobayashi-Maskawa matrix, and the Fermi constant Q denotes the column of (u, c, t) quarks and 4 stands for the G = 10-5/m$ column of (d, s, b) quark fields. The leptons I = e, II, r and their neutrinos vI = v,, v~, V~ couple to the quark-antiquark pairs as a pair field 1, = jOPvI, where 0, = ~~(1 + y5). Thus the weak-interaction lagrangian (A.17) is very similar in its structure to the quark-antiquark effective action (A.16). For the definition of the meson decay constants we Awill introduce in, the nonlocal effective action the local leptonic weak current L,, = (G/ fi)V,,I(P,) by the substitution r
ab ~
yab
+
jab,
(A.18)
where i= O,l, and P, is the total momentum of the leptonic pair. This results in the following additional contribution to the effective (A. 161, which corresponds to weak interactions: 2 Tr(G”PQbGbib,)
= 2cI&OP(GPG)“b/P. fi
action
(A.19)
The matrix element for the decay of a meson ~3 into a leptonic pair (IV) is then (IvIs~~lI)=(2~)4s(Pyp.-PpL)
/&
+
Ii, I”)v,bf;bpg~
(A.20) where fiibPp
=
ix
j
dpL(2T)4
dp’itr[O~G~(pl~-_PIp~)~~bGb~p~~+fPlpi)]
(A.21)
D. Blaschke et al. /Nuclear Physics A586 (1995) 711-733
732
defines the electroweak decay constant of the meson 9. After integration over pll and performing the trace operation, we obtain for the decay constant f$ of pseudoscalar mesons
(A.22) where M9ab - @
is the mass of the pseudoscalar
meson, see Eq. (A.12), and
1 &!*(P’>
=
E”(p’)
+Eb(pL)
1
-M$’
* E”(pl)
+Eb(pL)
+M$ib
(A.23)
denotes the symmetric ( + > and antisymmetric ( - > modes of the particle-antiparticle propagator.
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