Quark screening lengths in finite temperature QCD

284

NucJear Physics B (Proc_

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QUARK SCREENING LENGTHS IN FINITE TEMPERATURE QCD

Andreas GOCKSCH I n s t i t u t e for T h e o r e t i c a l P h y s i c s , U n i v e r s i t y o f C a l i f o r n i a , S a n t a B a r b a r a ; C A 93106 and P h y s i c s D e p a r t m e n t , B r o o k h a v e n N a t i o n a l L a b o r a t o r y , U p t o n ~ N e w Y o r k , 119731

We have computed Landau gauge quark propagators in both the confined and deconfined phase o f QCD. ! discuss the magnitude o f the resulting sueening lengths as well as aspec~ o f chiral symmetry relevant to the quark propagator. Here I would like to report on some recent work that l have done in collaboration with A. Soni at Bmokhaven. The idea o f this investigation is to extend previous work 2 on the quark propagetor in lattice QCD to the case o f non-zero temperature. Let me summarize quickly what was clone earlier:. The quark propagator, being a gauge variant quantity, vanishes when averaged over all ganges (which of course is what we are doing in the compact formulation o f lattice gauge theory). If we fix the gauge however a nice w..xponentia] decay is observed in < z~,(0)t~(t) > indicating a partide pole. The resuiting mass is a typical "constituent' ma~ o f roughly 300 MeV. in particular this mass does not vanish in the limit of vanishing current quark mass - it has two pieces, a bare and a dynamicaly generated one. Later on we will see that there is an argument due to Mandula and Ogilvie3 which makes the interpretation of Landau gauge quark masses as constituent masses appear quite plausible. The extension to finite temperature is trivial. Again, we fix the gauge configurations to lattice Landau gauge by maximizing the gauge fixing functional

SgI = Re ~-~TrU~(x)

(1)

x,p

over all gauge transformations

g(x).

We then mea-

sure the quanta1 w,y.~

on the gauge foted configurations and extract the *quark ~Dreening length" from the exponential decay in the z-direcfiou. (note that in pnndple there should be a factor e~-~ indde the sum to project onto the zero frequency mode with antipe~o~c boundary conditions, but on our L: = 4 lattice this does not matter). The screening length is defined as the inverse of the finite temperature quark "mass'. Why is the finite temperature quark propagator an interesting object to look at? Our hope is that it ~.~l! help us understand the nature of the high temperature phase of QCD. To appreciate the last statement let me hack up a little. The standard folklore of the high T phase of QCD is o f course that of a plasma o f deconfined, "almost free' quarks and gluons. This simple picture even has some evidence backing it up: The values of the e n e r ~ density o f quarks and gluons are close to their ideal gas values just above the transition. However as was pointed out by DeTar4 this does not neccessarily imply the existence o f colored modes in the plasma at long distances. As a matter of fact OeTar introduced the concept of 'dynamical confinement' in which the long distance modes of the plasma are actually color singlet. This idea was in some

0920--5632/91/$3.50 ~) Elsevier Sdence Publishers B.V. (North-Holland)

.4.

5 5.~ 5.68 5.72 5.72

285

(~ck.~h /Quark ~.~ning leugths in finite temperature QCD

z~z) z.eo(2) z.s~(2) -0.~(z) ' 0.~(z) 1.20(2) 1.26(2) 1.19(2) -oo2(5) 0.5(z) o~8z(2) o.~(2) o.~e(2) 'i -0.2(I) 0~89(8) z.7o(z) x.72(~) z.~2) i °~z) i ~.~(z) 0-~0) z.4~O) L~o(z) 1.4~(2) 0.16 i 0.~(z) I ~.2(z) o.ez(O 1.26(2) L37(1) 1.27(2)

[ 0.14 I 0.15 I z.~(~) I 1 0.16 i0.~(~) i i 0 14 0.4(2) i

Table 1: A compitat~ o~ sun-~ ofo~r d~ta.

sen~ °~o~emed" when DeT~ and Kogut6 aad ~tmr also G o ~ b ~/ =/9 ~ r e d ha~ro.;c ~ e e ~ g ie~gd~ in the deco~l;ned ~ of ( ~ D ~ dynamical femmas. They fou=~l ~.zt ~ si~Oe~ modes propa~te ~ masses several times d ~ temp~atree, that the modes ==repa~ty do=bled ( i ~ = g in pardo=lar tl=e u d e o m ) aod that eke ~ -- # ~ is less than t w ~ the lowest Matse_.~"-a ftequeacy z T , iediQd,~ bound propzga~o=. L~ter ka~t-ve~ it (o~ed8 that the same ~_-~__ _" -_re also eme~es in the

quenched appemdmat~. This preses~Ls for the inte~etatiem of the above r e m ~ as being p m p e ~ e s of t~%edyaamical modes of the Idasm=. In the quenched a l q x = d m d o o t h e ~ =re m~ cal modes. In addition it was also foum]10 that the

quark number " ~ _ ~ T ~ b ~ is large, iadkatiag the presence of lf~lbt excitatim~ Hence the ~ m a t ~ is ~ m ~ m that more information is dearly needed (Ideally, ~ would rd~ to be able to

~a~

mas~" ~

t h m = ~ eke a ~ c~re~ ==rd

Le~ zke c m ~ r z q m ~ mma~ ~ ! bad n m m ~ l

be~e

~ z~ro_ fl~

qamrk z~ a o0msl~m~lt q ~ d ~

~

~ a

~-

l ~ e ~eac~ at

the argmm~t is as k~m~_ Imge~ d mmg ga~l~ r=ed ¢ m r ~ m 6 m s =e ca= me d ~ ~ ==~=~ ga~-e f~ds to ~ t l ~ pmpal~m~ a=d ~ c r p = g e traasfmm =be pml=gat~- $ ~ ) = < o v ~ the ~ 6dds m d t J ~ ~J(3) valoed ~ w ~ =re r e a h a~q~,~,.g the w a p a g a ~ d a m l = glet b==~l star= d q = ~ a~l a ( q = ~ l ~ ) ga=ge as w~ do. ~orrespoe~ to a "a~oe" for d ~ 5 ~ 3

study the real time response of the Idasma wbkh is of coupe very d ~ u l t ) .

this ~ndt c m r e ~ l s

Back to the quark propagator. Our study was done in the quenched approximation, which in the light of the above remarks should su~ce. We have generated 10 statL~;,~lly independent gauge con~g-

so that ~e rare deal[eegm~lk a ~

,,rations for each value of fl = 5.60,5~68,5.72 and 5.865 on an 8~ - 20 - 4 lattice. The gauge configurations were then brought into Landau gauge and the quark propagator measured at ~ = 0.]],0.]~ and 0.16 (Wilson ferm~ons). The last value of ~ is close to nc~ at the largest value of ft. As this point is in the symmetric high T phase, one might ask what the meaning of~;o is when there is no massless pion. The answer was given by Iwasaki c~ aL7 who

fmma~ to a = = = ~

~cala~

~

o f a m a ~ e femiom and a m~ak~s scalar, Yhis cerr~nly a restorable d e ~ m of a T I ~ argument alse m ~ g e s ~ ~ a t a propagating quark ~n a ~ gauge does ~mt ~ m ~ c ~ p r ~ a gation of colored objects. Of course this is in alpre~meat ~ h the fact that we do measure ~onst~m~t quark propagation at zero T. Convemely, I do not think that t h e fact that we measure hadrooic propagation in the decon~ned phase ~mpliesthe edstence of color ~n~et ~X~tl~mS only. Oeady, we v*~uld not have expected the ¢~Ior dn~let modes not to propagate in the deconfined phase (although s~ng|et

286

A, Gocksch ,"Quark screening ]eng~hs in finite to~,,orm,~:::. . . . . .

masses significa ntly lighter that twice the quark mass might indicate that bound propagation is favored). Before I present some of our data ra more detailed account of this work will be pu,b[/shecl elsewhere11) I will shortly discuss the expectations one might have about the behavior of the quark propagator based on symmetry arguments. The most general form of a fermion propagator consistent with our boundary conditions (antiperiodic in time, periodic in space) is ss(~) = (1 + r ) ( c + ~ - ~ + +0 - r)(C+e -~-~)

C_e-~(~-~) + c-~-~%

(3)

I).2

f

,

~-~n'~n

!

;

~ = 5 ~ 8 6 5 ,~s=O ".6

i 2;-

f

,

_/

-i 7

/

l

-0.2 ~-

/ /

/

Ja

-o~,-

/

L

where r is a linear combination of 7 matrices, in view of the above interpretation of the constituent quark as a bound state it is possible to have heavier states (possibly of oppomte parity). We did not find any evidence for a higher spectral component. The accuracy of our calculation is however not good enough to rule out such a state completely. The propagator does not transform with a delinite sign under "z-reversal'. It does howev~ s a t i ~ the well known relation 7sSj,(O,z)7s = S / ( z , O ) . If we now assume that the quark propagator is invariant under chiral rotations in the deconfined phase we easily conclude that it must satisfy 7sSl7 s = - S S. This implies that S~ ~ 7~, i.e_

s , ( z ) = -~A(~ -~-" - ~-~r-=~)

(4)

where A = (7+ -- C - . All our fits were done according to Eq.(3). A typical fit is shown in Fig. 1. A compilation of some data from both the deconfined and the confined phase can be found in Table 1. The phase transition takes place at ~ = 5.7 on our lattice so the data are just below and above T~. Note that in the deconfined phase our data are certainly consistent with a symmetric propagator. Furthermore recall that due to the antiperiodic boundary conditions we expect the mass of the quark to be bounded from below by roughly sin(TrT) ~ 0.71 in the deconfined phase where it makes sense to model the propagator by the free (or perturbative) fermion propagator. The values in the table are at least consistent with this expectation. Note that the pion mass in the deconfined phase is close to twice the quark mass, a result which is different from what on finds 8 with

-0.~ O

5

~D

Z~

28

C

Figure 1: The quark propagator including a fit with the sum of two exponentia~ The subscript on P~ refers to the spine* components. staggered fermions in the chiral limit. Also shown is the color octet pion which is degenerate with the ordinar/singtet pion. in the confined phase we find a very fight quark mass at the highest •. We also see that the the pion begins to get light, at ~ = 5-60 and = 0.16 the quark mass is m a = 0.9(1) whereas the pion only weighs rr~a = 1.04(2). Let me summarize: We have measured constituent quark screening lengths in finite temperature QCD. In the deconfined phase we find evidence for a (nearly) symmetric but massive propagator. This is consistent with perturbation theory12 which also predicts a thermally generated mass of order gT where 9 is the gauge coupling. Our data are too crude to be able to say anything about this latter prediction. In the deconfined phase the pion is heavy, whereas it is light in the confined phase.

ACKNOWLEDGEM ENTS This manuscript has been authored under Contract No. DE-AC02-76-CH00016 with the U.S. Department of Energy and NSF Grant No. PHYS904035 (supplemented by funds from NASA). The

.&. Goch~ch / Q~arl~"screening ien~hs in ~nite tempera¢ure QCE~ computat~ns reported hef,~ were pe~ormed at ~tEL~5C a~d SDSC

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