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Evidence for meson clouds around hadrons in full lattice QCD
[ILI[II I ;t~ :11~: t'b~][Ik~J
KI.,qKVIKR
Nuclear PhysicsB (Proc. Suppl.) 39B,C (1995) 228-230
PROCEEDINGS SUPPLEMENTS
Evidence for Meson Clouds a r o u n d H a d r o n s in Full Lattice QCD M. Faber, M. Muller, M. Schaler ~ and H. Gausterer b aInstitut ffir Kernphysik, Technische Universit~t Wien, Wiedner Hauptstr. 8-10, A-1040 Vienna, Austria bInstitut fdr Theoretische Physik, Universit/at Graz, Universit~itsplatz 5, A-8010 Graz, Austria We investigate chiral symmetry breaking locally around a static quark in the framework of lattice QCD by calculating the correlation of the Polyakov loop with the chiral order parameter. We observe a restoration of chiral symmetry in the vicinity of the color source with a different mechnism in both phases of QCD. In the hadronic phase the chiral condensate is screened by a pion cloud, and the effect is more pronounced for lighter dynamical quarks. Above the deconfinement phase transition we find that the Polyakov loop is surrounded by a cloud of scalar a-mesons.
L THEORY
pseudoscalar Goldstone boson [2].
To investigate the breaking ofchiral symmetry locally around static quarks we calculate the chiral condensate in a system containing a static source at finite temperature. The Polyakov loop L(r-) describes the time propagation of a static quark [1]. We obtain insight into the local chiral structure in the vicinity of the static quark by computing the correlations between the Polyakov loop L and the order parameter of chiral symmetry ~¢ in the form < L(0)¢¢(r-) >. This observable can be written as a path-integral
f v[u]v[6, ¢] f ¢]
(1)
The functional integration extends over all degrees of freedom of the gauge fields U and the fermionic fields ¢ and ¢. The integral (1) is calculated numerically in a Monte Carlo simulation. For the lattice regularized action of the gauge fields we use Wilson's plaquette action. As far as the fermionic degrees of freedom are concerned we employ the Kogut-Susskind formulation. This method retains for m = 0 a chiral U(1)v ® (f(1)A symmetry of the chiral flavor group S U ( n ] ) v @ S U ( n / ) A in continuum space-time. The spontaneous b r e a k i n g of the (}(1)a symmetry leads to the appearance of a 0920-5632/95/$09.50 © 1995 ElsevierScienceB,V. All rights reserved. SSD1 0920-5632(95)00074-7
2. N U M E R I C A L S I M U L A T I O N S The system was simulated on an 8 s x 4 lattice with inverse gluonic coupling constants # = 4.9 below the deconfinement phase transition and # = 5.2 for the system in the quarkgluon plasma phase. We used the Hybrid Monte Carlo algorithm. The quark mass was decreased from m a = 0.1 to m a = 0.0125 in both phases, and the number of flavors in the Kogut-Susskind action was n/ -- 4. The lattice constant a corresponds to approximately 0.3 fm giving for the lowest quark mass m ~ 8 MeV. As a result, we observe that the chiral condensate is lowered in the vicinity of a static source. This suppression decreases with increasing quark mass in the hadronic phase, whereas the effect is nearly mass independent and less pronounced above the deconfinement phase transition [3]. Taking ¢¢ as the order p a r a m e t e r ofchiral symmetry breaking we find t h a t chiral symmetry is locally restored around static quarks, which is in agreement with the basic ideas of bag models. The d a t a has been fitted with an exponential function taking lattice anisotropies into ac-
229
M. Faber et al./Nuclear Physics B (Proc. Suppl.) 39B, C (1995) 228-230
Table 1 Amplitude a, screening m a s s / / a n d asymptotic constant 7 for the chiral condensate around a static quark according to Eq. (2) for various dynamical quark masses m
rn 0.1 0.05 0.025 0.0125
t3 = 4.9 // 1.806 1.322 0.947 0.863
~ 0.117 0.108 0.102 0.118
7 0.999 0.999 0.997 0.998
count < < L ><
>
= 7 - ~re-Ur
(2)
>
and the systematics extracted from our analysis is summarized in Table 1. Since clustering occurs in both phases the constant 7 is in agreem e n t with one. In the chirally broken phase we obtain an increase of the screening m a s s / / with higher quark mass, and the amplitude a appears to be quark mass independent. On the other hand, a should depend on the charge of the static quark. In this respect, it was shown in Ref. [4] t h a t ~ increases with the eigenvalue of the quadratic Casimir operator of the representation ot the Polyakov loop. Together with the slight reverse quark mass dependence of the screening m a s s / / w e obtain a suppression of the chiral condensate, which is roughly independent of the quark mass in the quark-gluon plasma. In Fig. 1 we plot the square of the screening mass/12 against the quark mass m in the confinement phase 03 = 4.9). We see a linear behaviour #2 ~ m for m _> 0.025, and therefore the G e l l - M a n n - O a k e s - R e n n e r relation rrz2 ~ m < ¢ ¢ > with the pion mass mr tells us t h a t the chiral condensate around a static quark source is screened by pions. For the lightest dynamical quarks (m = 0.0125) we are faced with the finite-size effect of the pion as the Goldstone boson of chiral s y m m e t r y breaking [5]. The data in Fig. 1 can be fitted by a hyperbola 3 #4 _ 3100 m 2 = 1,
(3)
/3 = 5.2 # 0.924 1.087 1.323 1.321
a 0.047 0.052 0.067 0.066
7 0.998 0.998 0.998 0.998
with the asymptote p2 = k ra, k = 32.2. Its slope k turns out to be about a factor 4 larger than the corresponding value in the relation m2 ~ 8.5 m obtained in a recent lattice calculation [6]. These observations sugget that the Polyakov loop L is surrounded by a pion cloud, and it interacts with the scalar density 5 ¢ via two-pion exchange. The appearance of a two-pion exchange reflects the fact t h a t 2¢ is a scalar operator, which can only be saturated by two pseudoscalar pions in order to fulfill parity conservation. The correlation < L(0)~¢(r) > measures the
4
t"q
'~,2
¢-q
0
0.00
I
0.02
I
....
0.04
I
I
0.06 0.08 m[a a ]
I
0.10
0.12
Figure 1. The square of the screening mass /12 as a function of the quark mass m in the confinement phase.
230
M. Faber et al./Nuclear Physics B (Proc. Suppl.) 39B, C (1995) 228-230
screening of the Polyakov loop by a pion cloud because a Polyakov loop represents an infinitely heavy quark Q in a color triplet state. Due to the Z(3)-symmetry of the QCD vacuum in the confined phase a single color source has infinite energy and therefore cannot exist. The fermionic determinant provides the possibilities to screen the color field of the infinitely heavy quark Q by means of a light antiquark or two light quarks qq [7]. The corresponding heavy-light bound states Q~ and Qqq have finite energy. The sum of its Boltzmann factors gives rise to a finite value of the Polyakov loop even in the confined phase when dynamical fermions are included in the lagrangian. On the other hand, ¢ ¢ represents a colorless operator, which can create or destroy a particle with the quantum numbers of the a-meson. The energetically most favorable interaction between the two colorless states (the heavy-light system and the chiral condensate) is via the exchange of almost massless Goldstone bosons. Above the deconfinement phase transition (/~ = 5.2) chiral symmetry is realized through parity doubling, and the pion and its scalar parity partner, the a - m e s o n , are degenerate. The data for the screening masses in Table 1 can be extrapolated linearly to the chiral limit, with the result p (rn = 0) ~ 1.4. Using a critical gluonic coupling of/?c = 4.9 in the chiral limit/~ = 5.2 in our simulation corresponds roughly to T/Tc "~ 1.6. In this temperature range Gocksch [8] obtained for the mass of the a - m e s o n in the chiral limit ma -- 1.2, which is in good agreement with our screening mass given above. Therefore, we can say that in the plasma phase of full QCD the scalar density ¢ ¢ interacts with the static source L via the exchange of a a - m e s o n . Since a is a scalar particle this screening mechanism includes parity conservation trivially. 3. C O N C L U S I O N S We have shown that for light dynamical quarks the chiral condensate is suppressed in the vicinity of a static quark, whereas the screening mechanism is different in the con-
finement and the deconfinement phase. In the hadronic phase the static quark is surrounded by a pion cloud, and the chiral condensate is screened due to two-pion exchange. In the quark-gluon plasma phase chiral symmetry is realized through parity doubling of the hadron spectrum. The chiral condensate and the Polyakovloop interact via a-exchange, and the local restoration of chiral symmetry is only weakly mass dependent. REFERENCES
1. L. D. McLerran and B. Svetitsky, Phys. Rev., D 24 (1981) 450. 2. J. B. Kogut, M. Stone, H. W. Wyld, S. H. Shenker, J. Shigemitsu and D. I~ Sinclair, Nucl. Phys., B 225 (1983) 326. 3. M. Faber, H. Gausterer and M. Schaler, Phys. Lett., B 317 (1993) 1021. 4. W. B~rger, M. Faber, H. Markum and M. M~ller, Phys. Rev., D 47 (1993) 3034. 5. T. Jolicoeur and A_ Morel, Nucl. Phys., B 262 (1985) 627. 6. R. Altmeyer etal., Nucl. Phys., B 389 (1993) 445. 7. M. Faber, 0. Borisenko, S. Mashkevich, G. Zinovjev, Fresh look on triality, IK-TUW-Preprint 9308401, 1993; Triality in QCD, Proceedings of the International School-Seminar'93 "Hadrons and Nuclei from QCD" , ed.by tCFyjii, Y.Akaishi, B.L.Reznik; World Scientific Singapore-New-Jersey-London-HongKong, p.333-340. 8. A. Gocksch, Phys. Rev. Lett., 67 (1991) 1701.
KI.,qKVIKR
Nuclear PhysicsB (Proc. Suppl.) 39B,C (1995) 228-230
PROCEEDINGS SUPPLEMENTS
Evidence for Meson Clouds a r o u n d H a d r o n s in Full Lattice QCD M. Faber, M. Muller, M. Schaler ~ and H. Gausterer b aInstitut ffir Kernphysik, Technische Universit~t Wien, Wiedner Hauptstr. 8-10, A-1040 Vienna, Austria bInstitut fdr Theoretische Physik, Universit/at Graz, Universit~itsplatz 5, A-8010 Graz, Austria We investigate chiral symmetry breaking locally around a static quark in the framework of lattice QCD by calculating the correlation of the Polyakov loop with the chiral order parameter. We observe a restoration of chiral symmetry in the vicinity of the color source with a different mechnism in both phases of QCD. In the hadronic phase the chiral condensate is screened by a pion cloud, and the effect is more pronounced for lighter dynamical quarks. Above the deconfinement phase transition we find that the Polyakov loop is surrounded by a cloud of scalar a-mesons.
L THEORY
pseudoscalar Goldstone boson [2].
To investigate the breaking ofchiral symmetry locally around static quarks we calculate the chiral condensate in a system containing a static source at finite temperature. The Polyakov loop L(r-) describes the time propagation of a static quark [1]. We obtain insight into the local chiral structure in the vicinity of the static quark by computing the correlations between the Polyakov loop L and the order parameter of chiral symmetry ~¢ in the form < L(0)¢¢(r-) >. This observable can be written as a path-integral
f v[u]v[6, ¢] f ¢]
(1)
The functional integration extends over all degrees of freedom of the gauge fields U and the fermionic fields ¢ and ¢. The integral (1) is calculated numerically in a Monte Carlo simulation. For the lattice regularized action of the gauge fields we use Wilson's plaquette action. As far as the fermionic degrees of freedom are concerned we employ the Kogut-Susskind formulation. This method retains for m = 0 a chiral U(1)v ® (f(1)A symmetry of the chiral flavor group S U ( n ] ) v @ S U ( n / ) A in continuum space-time. The spontaneous b r e a k i n g of the (}(1)a symmetry leads to the appearance of a 0920-5632/95/$09.50 © 1995 ElsevierScienceB,V. All rights reserved. SSD1 0920-5632(95)00074-7
2. N U M E R I C A L S I M U L A T I O N S The system was simulated on an 8 s x 4 lattice with inverse gluonic coupling constants # = 4.9 below the deconfinement phase transition and # = 5.2 for the system in the quarkgluon plasma phase. We used the Hybrid Monte Carlo algorithm. The quark mass was decreased from m a = 0.1 to m a = 0.0125 in both phases, and the number of flavors in the Kogut-Susskind action was n/ -- 4. The lattice constant a corresponds to approximately 0.3 fm giving for the lowest quark mass m ~ 8 MeV. As a result, we observe that the chiral condensate is lowered in the vicinity of a static source. This suppression decreases with increasing quark mass in the hadronic phase, whereas the effect is nearly mass independent and less pronounced above the deconfinement phase transition [3]. Taking ¢¢ as the order p a r a m e t e r ofchiral symmetry breaking we find t h a t chiral symmetry is locally restored around static quarks, which is in agreement with the basic ideas of bag models. The d a t a has been fitted with an exponential function taking lattice anisotropies into ac-
229
M. Faber et al./Nuclear Physics B (Proc. Suppl.) 39B, C (1995) 228-230
Table 1 Amplitude a, screening m a s s / / a n d asymptotic constant 7 for the chiral condensate around a static quark according to Eq. (2) for various dynamical quark masses m
rn 0.1 0.05 0.025 0.0125
t3 = 4.9 // 1.806 1.322 0.947 0.863
~ 0.117 0.108 0.102 0.118
7 0.999 0.999 0.997 0.998
count < < L ><
>
= 7 - ~re-Ur
(2)
>
and the systematics extracted from our analysis is summarized in Table 1. Since clustering occurs in both phases the constant 7 is in agreem e n t with one. In the chirally broken phase we obtain an increase of the screening m a s s / / with higher quark mass, and the amplitude a appears to be quark mass independent. On the other hand, a should depend on the charge of the static quark. In this respect, it was shown in Ref. [4] t h a t ~ increases with the eigenvalue of the quadratic Casimir operator of the representation ot the Polyakov loop. Together with the slight reverse quark mass dependence of the screening m a s s / / w e obtain a suppression of the chiral condensate, which is roughly independent of the quark mass in the quark-gluon plasma. In Fig. 1 we plot the square of the screening mass/12 against the quark mass m in the confinement phase 03 = 4.9). We see a linear behaviour #2 ~ m for m _> 0.025, and therefore the G e l l - M a n n - O a k e s - R e n n e r relation rrz2 ~ m < ¢ ¢ > with the pion mass mr tells us t h a t the chiral condensate around a static quark source is screened by pions. For the lightest dynamical quarks (m = 0.0125) we are faced with the finite-size effect of the pion as the Goldstone boson of chiral s y m m e t r y breaking [5]. The data in Fig. 1 can be fitted by a hyperbola 3 #4 _ 3100 m 2 = 1,
(3)
/3 = 5.2 # 0.924 1.087 1.323 1.321
a 0.047 0.052 0.067 0.066
7 0.998 0.998 0.998 0.998
with the asymptote p2 = k ra, k = 32.2. Its slope k turns out to be about a factor 4 larger than the corresponding value in the relation m2 ~ 8.5 m obtained in a recent lattice calculation [6]. These observations sugget that the Polyakov loop L is surrounded by a pion cloud, and it interacts with the scalar density 5 ¢ via two-pion exchange. The appearance of a two-pion exchange reflects the fact t h a t 2¢ is a scalar operator, which can only be saturated by two pseudoscalar pions in order to fulfill parity conservation. The correlation < L(0)~¢(r) > measures the
4
t"q
'~,2
¢-q
0
0.00
I
0.02
I
....
0.04
I
I
0.06 0.08 m[a a ]
I
0.10
0.12
Figure 1. The square of the screening mass /12 as a function of the quark mass m in the confinement phase.
230
M. Faber et al./Nuclear Physics B (Proc. Suppl.) 39B, C (1995) 228-230
screening of the Polyakov loop by a pion cloud because a Polyakov loop represents an infinitely heavy quark Q in a color triplet state. Due to the Z(3)-symmetry of the QCD vacuum in the confined phase a single color source has infinite energy and therefore cannot exist. The fermionic determinant provides the possibilities to screen the color field of the infinitely heavy quark Q by means of a light antiquark or two light quarks qq [7]. The corresponding heavy-light bound states Q~ and Qqq have finite energy. The sum of its Boltzmann factors gives rise to a finite value of the Polyakov loop even in the confined phase when dynamical fermions are included in the lagrangian. On the other hand, ¢ ¢ represents a colorless operator, which can create or destroy a particle with the quantum numbers of the a-meson. The energetically most favorable interaction between the two colorless states (the heavy-light system and the chiral condensate) is via the exchange of almost massless Goldstone bosons. Above the deconfinement phase transition (/~ = 5.2) chiral symmetry is realized through parity doubling, and the pion and its scalar parity partner, the a - m e s o n , are degenerate. The data for the screening masses in Table 1 can be extrapolated linearly to the chiral limit, with the result p (rn = 0) ~ 1.4. Using a critical gluonic coupling of/?c = 4.9 in the chiral limit/~ = 5.2 in our simulation corresponds roughly to T/Tc "~ 1.6. In this temperature range Gocksch [8] obtained for the mass of the a - m e s o n in the chiral limit ma -- 1.2, which is in good agreement with our screening mass given above. Therefore, we can say that in the plasma phase of full QCD the scalar density ¢ ¢ interacts with the static source L via the exchange of a a - m e s o n . Since a is a scalar particle this screening mechanism includes parity conservation trivially. 3. C O N C L U S I O N S We have shown that for light dynamical quarks the chiral condensate is suppressed in the vicinity of a static quark, whereas the screening mechanism is different in the con-
finement and the deconfinement phase. In the hadronic phase the static quark is surrounded by a pion cloud, and the chiral condensate is screened due to two-pion exchange. In the quark-gluon plasma phase chiral symmetry is realized through parity doubling of the hadron spectrum. The chiral condensate and the Polyakovloop interact via a-exchange, and the local restoration of chiral symmetry is only weakly mass dependent. REFERENCES
1. L. D. McLerran and B. Svetitsky, Phys. Rev., D 24 (1981) 450. 2. J. B. Kogut, M. Stone, H. W. Wyld, S. H. Shenker, J. Shigemitsu and D. I~ Sinclair, Nucl. Phys., B 225 (1983) 326. 3. M. Faber, H. Gausterer and M. Schaler, Phys. Lett., B 317 (1993) 1021. 4. W. B~rger, M. Faber, H. Markum and M. M~ller, Phys. Rev., D 47 (1993) 3034. 5. T. Jolicoeur and A_ Morel, Nucl. Phys., B 262 (1985) 627. 6. R. Altmeyer etal., Nucl. Phys., B 389 (1993) 445. 7. M. Faber, 0. Borisenko, S. Mashkevich, G. Zinovjev, Fresh look on triality, IK-TUW-Preprint 9308401, 1993; Triality in QCD, Proceedings of the International School-Seminar'93 "Hadrons and Nuclei from QCD" , ed.by tCFyjii, Y.Akaishi, B.L.Reznik; World Scientific Singapore-New-Jersey-London-HongKong, p.333-340. 8. A. Gocksch, Phys. Rev. Lett., 67 (1991) 1701.