We investigate correlation functions of the Polyakov loop and static
meson/diquark systems with the chiral condensate and the quark number density
at finite temperature. In particular the latter observable can give insight in
the mechanism of screening and string breaking at finite temperature. We use
for our analysis gauge field configurations generated in 2+1 flavor QCD with an
improved staggered fermion action with almost physical light quark masses and a
physical value of the strange quark mass on lattices with temporal extent Nt=4
and 6.Comment: 7 pages, 6 figures, presented at the XXV International Symposium on
  Lattice Field Theory, July 30 - August 4 2007, Regensburg, German

Huebner, Kay

English

arXiv

We investigate correlation functions of the Polyakov loop and static mesodiquark systems with the chiral condensate and the quark number density at finite temperature. In particular the latter observable can give insight into the mechanism of screening and string breaking at finite temperature. We use for our analysis gauge field configurations generated in 2+1 flavor QCD with an improved staggered fermion action with almost physical light quark masses and a physical value of the strange quark mass on lattices with temporal extent N{sub {tau}} = 4 and 6

HUEBNER,K.

UNT (University of North Texas) Digital Library

BRO@K@M&WEN N A T I O N  A L LAB 0 R A T O R Y  BNL-7 928 5-2007-CP Correlations of chiral condensates and quark number densities with static quark sources Kay Hubner Presented at the XXV International Symposium on Lattice Field Theory (Lattice 2007) Regensburg, Germany July 30 to August 4,2007 Physics Department ' Nuclear Theory Group Brookhaven National Laboratory P.O. Box 5000 Upton, NY 1 1973-5000 www.bnl.gov Notice: This manuscript has been authored by employees of Brookhaven Science Associates, LLC under Contract No. DE-ACOZ 98CH10886 with the U.S. Department of Energy. The publisher by accepting the manuscript for publication acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. . This preprint is intended for publication in a journal or proceedings. Since changes may be made before publication, it may not be cited or reproduced without the author's permission. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility .for the accuracy, completeness, or any third party’s use or the results of such use of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof or its contractors or subcontractors. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. Correlations of chiral condensates and quark number densities with static quark sources Kay Hubner" Brookhaven National Lab E-mail: huebner@bnl . gov We investigate correlation functions of the Polyakov loop and static mesoddiquark systems with the chiral condensate and the quark number density at finite temperature. In particular the latter observable can give insight into the mechanism of screening and string breaking at finite temper- ature. We use for our analysis gauge field configurations generated in 2+1 flavor QCD with an improved staggered fermion action with almost physical light quark masses and a physical value of the strange quark mass on lattices with temporal extent N ,  = 4 and 6. The XXV International Symposium on Lattice Field Theory Regensburg, Germany July 30-4 August 2007 *Speaker. 'for the RBC-Bielefeld Collaboration, together with C. Schroeder, UC San Diego @ Copyright o w n 4  by the author(s) under the terms of the Cnativc Commons Attribution-NonCommcrcial-ShanAlikc Licence. http:llpos.sissa.itl Correlations of chiral condensates and auark number densities with static auark sources Kav Hiibner 1. Introduction Global expectation values of the chiral condensate and quark number density give information about the structure of the thermal medium at finite temperature and non-zero chemical potential p.  Calculating these observables locally, i. e. depending on their spatial position, allows for a spatially resolved analysis of dynamical quark clouds and the chiral condensate around static quark, diquark or Meson sources. We investigate correlation functions of the chiral condensate/quark number density with the Polyakov loop or products of Polyakov loops, which allow to probe the screening and string breaking mechanism of systems composed of static color charges at finite temperature. Lattice calculations of these operators have been performed in the past with staggered and Wilson quarks [ 1,2,3,4] for rather large quark masses and on small lattices. The investigations presented here form the basis for future studies of the spatial correlators of the quark number density, which has been proposed as a means to decide whether the QGP has a liquid phase [5 ] .  2. Operators We have calculated the chiral condensate and the quark number density depending on their spatial position x by using where x = (%,x), f = 1,. . . ,nf labels the flavor, ccf is the corresponding chemical potential and Tr, denotes the trace in color space, with TG 1 = 1. Derivatives with respect to pf are evaluated at pf = 0. We can now calculate the correlator of kxf (x) and n{(x) with the Polyakov loop L(x) or products of Polyakov loops, for which we use where L, and therefore also CQ0 and CQQ, is properly renormalised by the multiplicative renor- malisation constant Z(&Nr, which is determined at T = 0 from a matching of the static quark potential to the string potential. The C& and CQQ describe static meson and diquark systems, properly renormalised as well. The renormalisation for kxf involves both multiplicative and ad- ditive renormalisation constants. In order to eliminate the additive contributions, we calculate the connected correlator respectively. The correlation functions rt{(x)L(y)) and (n{(x)C.(r)) for s = QQ,QQ are then ( kx f  (XMY)), = (kXf(X)L(Y)) - ( i ix f )  (L)  7 (2.3) which approaches the corresponding continuum expression(@Vf(x)L(y)), in the limit a --+ 0. In this work, we analyze gauge field configurations generated in 2+1 flavor QCD with the p4fat3 staggered fermion action, where the strange quark mass 4 is fixed to its physical value and the light quark mass is choosen as iq = O.lm,. Lattice sizes are 163 x 4,6 [6, 71. For the calculation of expectation values we employed the random noise vector method with 100 random noise vectors. 2 Correlations of chiral condensates and auurk number densities with static auark sources Kay Hubner -0.04 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 Trr,  - 1.029 I-*-- - 1.190 1.507 - Ix-YP 2.050 * I 0.937 - lxrP , 0,959 , a  0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 . .. 1 ' +b 1 . i Tlr, . 1.029 - m. - 1.190 I = . 1.507 - I Ix-YP 2.050 4 0.2 0.4 0.6 0.8 1 1 2  1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.01 0 -0.01 -0.02 1. -I -0.03 -0.04 . *  -0.05 1 -0.07 0.959 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -0.02 -0.03 m *  f . +  0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Figure 1: (j&y"L), (upper row) and (niL) (lower row) for the strange quark species. 3. Results for (XxL), and (n&) In fig. 1 we show the results for @xsL), (upper row) and (r@) (lower row) for the strange quark species obtained from Nt = 4 lattices. We observe that all correlators vanish rapidly at distances rT21 ,  i. e. when kxf and 4 approach their thermal expectation value undisturbed by the presence of the static quark. For small separations (jixsL), assumes smaller values, signalling a gradual restoration of chiral symmetry in the vincinity of the static quark. The correlator(n:L) becomes smaller than zero for smaller separations as well, as the static quark source is screened by an excess of dynamical anti-quarks arranged in a cloud around the static quark. The effect becomes smaller above T,, as thermal gluons are available in the deconfined phase to provide screening. All these findings are in qualitative agreement with those of earlier studies [ 1,2,3,4].  For (r$L) we observe a peculiar dip at rT x 0.5, which we investigate in more detail in fig. 2. Varying the lattice spacing, we see in fig. 2 (left), that the position of the dip varies with a as well, pointing at a lattice artifact that vanishes in the continuum limit. In fig. 2 (right) we computed (nGL) at T/Tc = 0.905,Nt = 4 for p4, Naik and naive staggered discretisations of the fermion matrix M. The artifact is present in all three discretisation schemes, thus its appearance is not dependent on the particular action used. Rather the mixing of different quantum channels in the staggered formulation is probably responsible for this effect. In order to quantify further the behavior of these correlators, we have performed simple expo- 3 I Correlations of chiral condensates and quark number densities with static Quark sources Kav Hiibner -0.oocn - -0.oo04 -0.oo06 -0.O008 -0.001 -0.0012 ::::I: , , , , , " a k , *  1 -0.004 naive I 4 w- 1 x - M  -0005 1 2 3 4 5 6 7 8  Figure 2: (.$I!,) for NT = 4 and 6 (left) and for different staggered discretisations of the fermion matrix M at TITc = 0.905,N, = 4 (right). nential fits. The resulting screening masses dJ of the correlator involving the chiral condensate are, of course, not affected by the multiplicative renormalisation constants. In fig. 3 we show ( zxsL) ,  (upper row) and (n;TIL) (lower row) for the strange quark species on a log-scale together with the results of the X2-fits of exponential functions. We obtain similar results for the light quark species. The fit ranges employed were 0.3 5 rT 5 1 for n f L  , where we left out the values due to the artifact at rT x 0.5, and 0.1 5 rT 5 1.2 for ( X x f L ) , .  The results for the screening masses obtained from the fits with exponential functions are shown in fig. 4(left). As expected' the screening masses obtained from nqL are larger than those from ( z ~ f L ) ,  and we generally find mT is larger than m'. We observe a drop of all screening masses close to T,, as all correlation functions become large in the cross over region. For large temperatures we have d J / T  << 1 and we therefore expect td = ms to hold, which is not yet the case at T / &  = 2. Moreover, the deviation of the data from the lattice with smaller aspect ratio points to volume effects which are still present in our calculations of dJ. The Debye masses extracted from the Nz = 6 lattices are roughly of the same size as the screening masses [9]. In order to avoid contamination from excited states, we plan to study p = 0 projections of the correlators in the future. The total induced quark number of flavor f around a static anti-quark can be computed by b )  V )  where we show the results of our calculation in fig. 4(right). As was already observed in 2-flavor QCD [lo], vanishes rapidly for T > T,, where screening is predominantly provided by ther- mal gluons. For T 4 0 the induced quark number approaches one, where this limit is approached faster for light quarks than the strange flavor, since the heavier sea quarks are more difficult to induce. 'For heavy sea quarks the correlation functions (5~fL.)~ and n f L  can be related to hopping parameter expansions of real and imaginary parts of Polyakov loops, respectively[2]. In this limit one finds at high temperatures tn+/mxxL = 1.5 [8]. L) 4 Correlations of chiral condensates and quark number densities with static quark sources Kay Hubner f 1 0.1 0.01 02 04 06 08 1 12 14 1 0937 - 0959 = 01 0 01 ow1 * .  0.2 0.3 0.4 0.5 0.6 0.7 0.8 '0.9 1 1 0.1 0.01 0.001 0.1 0.01 0.w1 o.ooo1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 3: (2xsL),  (upper row) and (n",) (lower row) for the strange quark species on a log-scale. Lines are the results of the  fits with simple exponential functions. 4. Results for ( n q C ~ ~ )  and (nqCQ0) We now turn to the computation of three point correlation functions (nqcQQ) and (nqcQQ). In fig. 5 we show (nSq(x1 ,xz )cQQ(r) )  for T / G  = 0.905, calculated on lattices with temporal extent Nr = 4 for separations rT = 1.25 (left) and rT = 0.25 (right) of the Polyakov loops, which are placed at X I  = f rT /2 ,x2  = 0. We see that for large separations r the correlator assumes negative values centered around the individual positions of the static quarks. This signals an excess of dynamical anti-quarks and the screening mechanism is mesonic, i. e. QQ t Qq+ Qtj, where Q stands for a static quark and q for a dynamical quark. For small separations we observe positive values of the correlator. Thus the screening mechanism has changed to baryonic, i. e. QQ t QQq. This behavior has been observed in the induced quark number of the diquark system in 2-flavor QCD as well [ 101. In fig. 6 we show the values of ( ~ ; C Q Q )  (left) and ( $ C Q ~ )  (right) along the line connecting the static quarks for several temperatures and two separations, where q = 4 x 2  = 0. The dotted lines mark the position of the Polyakov loops, where the anti-quark source in the @-system is located at larger values of x. We see mesonic screening in the diquark system for large separations r at all T calculated here, whereas at the smallest distance only the lowest temperature turns to baryonic screening. In the @-system the screening mechanism is mesonic for all T and rT, i. e. QQ -+ Qij+ Qq. 5 Correlations of chiral condensates'and quark number densities with static quark sources Kay Hiibner 1.0 0.8 0'9 0.7 0.6 0.4 0.3 :.: 0.0 ' light'*- *- - strange --a - ; i 0 5 ' t (  * * - c : ' 5  - m m . .  Trr,, Figure 4: Left: Screening masses obtained from an exponential fit. Closed symbols: N ,  = 4, open symbols: N,  = 6. Debye masses are taken from [9]. Right: Induced quark number around a static anti-quark. <n:(x,,x2)CaQ(rT=1 25)>/? 0 005 0 0 3  0 -0 005 0.01 -0 015 0.00 -0.02 -0.01 -0 02 -0 025 OM 1 -0 01 -0.02 Figure 5: ( ~ ~ C Q Q )  for T/Tc = 0.9O5,NT = 4 at separations rT = 1.25 (left) and rT = 0.25 (right) of the Polyakov loops. 5. Conclusion and Outlook We have studied correlators of the local chiral condensate and the local quark number density with Polyakov loops in 2+1 flavor QCD with almost realistic quark masses at finite temperature on 163 x 4,6 lattices. We found that the correlator of the Polyakov loop with the quark number density and the connected correlator with the chiral condensate are well described by an exponential fit- Ansatz. We calculated the screening masses from these correlators and the induced quark number in the static quark system. Moreover, we investigated the quark number density in static diquark and mesonic systems, where we found the screening mechanism in the diquark system to change from mesonic to baryonic at small separations for small temperatures. In the future we want to study the p = 0 projection of the correlators with the Polyakov loop to avoid contamination from excited states in the screening masses. Furthermore we like to examine lattices with a larger physical volume in order to get better control over finite volume effects. This work forms the basis for future computations of spatial correlators of the quark number density (nq(0)nq(r)), which might shed light on whether the QGP has a liquid phase [5 ] .  6 t Correlations of chiral condensates and quark number densities with static quark sources Kay Hubner 0.005 O.Oo0 -0.005 -0.010 -0.015 -0.020 -2 -1 0 1 2 0.010 0.005 0 .W -0.005 -0.010 -0.015 -0.020 -2 -1 0 1 2 0.020 0.015 0.010 0.005 0.000 -0.W5 -0.010 -0.015 -0.020 -2 -1 0 1 2 0.010 0.005 0.000 -0.005 -0.010 -2 -1 0 1 2 Figure 6: Cut along the connecting axis of the static quarks, (nicQQ) (left) and (n",Qg) (right) for several temperatures. The dotted lines give the position of the Polyakov loops. 6. Acknowledgement This manuscript has been authored under contract number DE-AC02-98CH10886 with the U. S .  Department of Energy. References [ 13 W. Feilmair, M. Faber and H. Markum, Phys. Lett. B 221 (1989) 363. [2] W. Feilmair, M. Faber and H. Markum, Phys. Rev. D 39, 1409 (1989). [3] W. Sakuler et al., Phys. Lett. B 276, 155 (1992). [4] W. Buerger, M. Faber, H. Markum, M. Muller and M. Schaler, Nucl. Phys. Proc. Suppl. 34,269 [5] M. H. Thoma, Nucl. Phys. A 774,307 (2006) [6] Jan van der Heide, "Thermodynamics of 2+1 flavour QCD", this Proceedings [7] M. Cheng et al., Phys. Rev. D 74,054507 (2006) [8] S. Nadkami, Phys. Rev. D 33 (1986) 3738. (1994). . [9] K. Petrov, "Entropy and Internal energy of Heavy quark-anti-quark pair", this Proceedings [ 101 M. Doring, K. Hubner, 0. Kaczmarek and E Karsch, Phys. Rev. D 75,054504 (2007) 7 

CORRELATIONS OF CHIRAL CONDENSATES AND QUARK NUMBER DENSITIES WITH STATIC QUARK SOURCES.

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Correlations of chiral condensates and quark number densities with static quark sources

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