Static quark-antiquark pair free energy and screening masses: continuum results at the QCD physical point
Szabolcs Borsanyi, Zoltan Fodor, Sandor D. Katz, Kalman K. Szabo, Attila Pasztor, Csaba Torok
SStatic quark-antiquark pair free energy andscreening masses: continuum results at theQCD physical point
Szabolcs Borsányi
University of Wuppertal
Zoltán Fodor
University of Wuppertal
Sándor D. Katz
Eötvös University and MTA-ELTE Lendület Lattice Gauge Theory Research Group
Attila Pásztor ∗ University of WuppertalE-mail: [email protected]
Kálmán K. Szabó
University of Wuppertal and Jülich Supercomputing Center
Csaba Török
Eötvös University and MTA-ELTE Lendület Lattice Gauge Theory Research Group
We study the correlators of Polyakov loops, and the corresponding gauge invariant free energyof a static quark-antiquark pair in 2+1 flavor QCD at finite temperature. Our simulations werecarried out on N t = 6, 8, 10, 12, 16 lattices using a Symanzik improved gauge action and a stoutimproved staggered action with physical quark masses. The free energies calculated from thePolyakov loop correlators are extrapolated to the continuum limit. For the free energies we usea two step renormalization procedure that only uses data at finite temperature. We also measurecorrelators with definite Euclidean time reversal and charge conjugation symmetry to extract twodifferent screening masses, one in the magnetic, and one in the electric sector, to distinguish twodifferent correlation lengths in the full Polyakov loop correlator. This conference contribution isbased on the paper: JHEP 1504 (2015) 138 The 33rd International Symposium on Lattice Field Theory14 -18 July 2015Kobe International Conference Center, Kobe, Japan ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ a r X i v : . [ h e p - l a t ] O c t tatic Q ¯ Q pair free energy and screening masses: continuum results at the physical point
Attila Pásztor
1. Introduction
At high temperatures strongly interacting matter undergoes a transition where colorless hadronsturn into a phase dominated by colored quarks and gluons, the quark gluon plasma (QGP). Decon-finement properties of the transition can be studied by infinitely heavy, static test charges. Here,we calculate the gauge invariant static quark-antiquark pair free energy.The free energy of a static quark-antiquark pair as a function of their distance at various tem-peratures is determined by the Polyakov loop correlator [1], which gives the gauge invariant ¯ QQ free energy as: F ¯ QQ ( r ) = − T ln C ( r , T ) = − T ln (cid:28) ∑ x Tr L ( x ) Tr L + ( x + r ) (cid:29) . (1.1)In the above formula, x runs over all the lattice spatial sites, and the Polyakov loop, L ( x ) , is definedas the product of temporal link variables U ( x , x ) ∈ SU ( ) : L ( x ) = N t − ∏ x = U ( x , x ) , (1.2)A related problem is distinguishing correlation lengths in the correlator of Polyakov loops,which give inverse screening masses in the plasma. In the full Polyakov loop correlator the electricand magnetic sectors both contribute. In order to investigate the effect of electric and magneticgluons separately, one can use the symmetry of Euclidean time reflection [2], that we will call R .The crucial property of magnetic versus electric gluon fields A and A i is that under this symmetry,one is intrinsically odd, while the other is even: A ( τ , x ) R −→ − A ( − τ , x ) , A i ( τ , x ) R −→ A i ( − τ , x ) (1.3)Under this symmetry the Polyakov loop transforms as L R −→ L † . One can easily define correlatorsthat are even or odd under this symmetry, and thus receive contributions only from the magnetic orelectric sector, respectively [2, 3]: L M ≡ ( L + L † ) / L E ≡ ( L − L † ) / . (1.4)We can further decompose the Polyakov loop into C even and odd states, using A C −→ A ∗ and L C −→ L ∗ as: L M ± = ( L M ± L ∗ M ) / L E ± = ( L E ± L ∗ E ) / . (1.5)Next, we note that Tr L E + = = Tr L M − , so the decomposition of the Polyakov loop correlator todefinite R and C symmetric operators contains two parts . We define the magnetic correlation More precisely, the excess free energy that we get when inserting two static test charges in the medium. In the literature, a factor of N c is often included in the definition. Including this factor leads to a term in the staticquark free energy that is linear in temperature. Note that the Polyakov loop correlator does not overlap with the R ( C ) = +( − ) and R ( C ) = − (+) sectors. Toaccess these sectors, other operators are needed. tatic Q ¯ Q pair free energy and screening masses: continuum results at the physical point
Attila Pásztor function as: C M + ( r , T ) ≡ (cid:28) ∑ x Tr L M + ( x ) Tr L M + ( x + r ) (cid:29) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:28) ∑ x Tr L ( x ) (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) , (1.6)and the electric correlator as : C E − ( r , T ) ≡ − (cid:28) ∑ x Tr L E − ( x ) Tr L E − ( x + r ) (cid:29) . (1.7)Then, from the exponential decay of these correlators, we can define the magnetic and electricscreening masses. Note that with our definition Tr L M + = Re Tr L and Tr L E − = i Im Tr L , and: C ( r , T ) − C ( r → ∞ , T ) = C M + ( r , T ) + C E − ( r , T ) , (1.8)from which it trivially follows that if the magnetic mass screening mass is lower than the electricmass, we will have C ( r , T ) − C ( r → ∞ , T ) asymptotic to C M + ( r , T ) as r → ∞ , or equivalently, thehighest correlation length in C equal to that of C M + . We will determine the correlation lengths byfitting a Yukawa ansatz to these correlators.
2. Simulation details
The simulations were performed by using the tree level Symanzik improved gauge, and stout-improved staggered fermion action, that was used in [4]. We worked with physical quark masses,and fixed them by reproducing the physical ratios m π / f K and m K / f K .Compared to our previous investigations of Polyakov loop correlators, reported in the confer-ence proceedings [5], here we used finer lattices, namely we carried out simulations on N t = N t = , ,
10 lattices. Our results were obtained in the temperaturerange 150 MeV ≤ T ≤
450 MeV. We use the same configurations as in [6] and [7].
3. The gauge invariant free energy
We use a renormalization procedure based entirely on our T > F ren ¯ QQ ( r , β , T ; T ) = F ¯ QQ ( r , β , T ) − F ¯ QQ ( r → ∞ , β , T ) , (3.1)with a fixed T . This renormalization prescription corresponds to the choice that the free energyat large distances goes to zero at T . We choose T = Here our definition differs from that used in [3] in a sign. tatic Q ¯ Q pair free energy and screening masses: continuum results at the physical point
Attila Pásztor -500-400-300-200-100 0a=0.081fm 2 a 3 a 2a 5 a 3a 4a F (r , N t , β )- F (r = ∞ , N t , β ) [ M e V ] r[fm]N t =8 β =3.94 HYP smearingNo smearing Figure 1:
The smeared and unsmeared free energies at a given β and N t , after the first step of the renormal-ization procedure. In the first step we renormalize the single static quark free energy which satisfies:2 F Q ( β , T ) = F ¯ QQ ( r → ∞ , β , T ) = − T log |(cid:104) Tr L (cid:105)| . (3.2)We define its renormalized counterpart as: F renQ ( β , T ; T ) = F Q ( β , T ) − F Q ( β , T ) . (3.3)In the second step the full renormalized ¯ QQ free energy can be written as: F ren ¯ QQ ( r , β , T ; T ) = ˜ F ¯ QQ ( r , β , T ) + F renQ ( β , T ; T ) , (3.4)where ˜ F ¯ QQ ( r , β , T ) = F ¯ QQ ( r , β , T ) − F ¯ QQ ( r → ∞ , β , T ) = F ¯ QQ ( r , β , T ) − F Q ( β , T ) . (3.5)Note, that this second step of the renormalization procedure is completely straightfoward to imple-ment, at each simulation point in N t and β we just subtract the asymptotic value of the correlator.The main advatage of this 2 step procedure is that it allows us to extend the temperature range wecan do a continuum limit in without performing T = β values. Formore details see [11]. The Polyakov loop correlator behaves similarly to baryon correlators in imaginary time: atlarge values of r we can get negative values of C at some configurations . For this reason, it ishighly desirable to use gauge field smearing which makes for a much better behavior at large r , atthe expense of unphysical behavior at small r . For this reason, we measured the correlators bothwithout and with HYP smearing. We expect that outside the smearing range (i.e. r ≥ a ) the twocorrelators coincide. This is supported by Figure 1. Therefore we use the smeared correlators for r ≥ a and the unsmeared ones for r < a . Of course, the ensemble average should in principle be positive definite. tatic Q ¯ Q pair free energy and screening masses: continuum results at the physical point
Attila Pásztor -1000-800-600-400-200 0 200 400 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 F r en ( T , r) [ M e V ] r[fm] T=150MeVT=175MeVT=200MeVT=225MeVT=250MeVT=275MeVT=300MeVT=325MeVT=350MeV Figure 2:
Continuum values of the static ¯ QQ free energy at different temperatures. The continuum extrapolations were done with N t = , ,
12 and when available N t =
16 lat-tices. For details on the systematic error estimation see [11]. The final results are in Figure 2.Note that the curves seem to tend to the same curve as r →
0, corresponding to the expectation thatUV physics is temperature independent. Also note that the error bars get smaller as we approach T = MeV , which was chosen as a renormalization point. This is a natural consequence of theimplementation of our renormalization prescription. It is also the reason why the correlator tendsto zero at that point. A different renormalization would correspond to a constant shift in this graph.We note, that on this conference an other determination of the same free energy was reported[12] using the HISQ action, and there is a slight difference in the asymptotic value (or the singlequark free energy), at the higher temperature values of T > − . σ . In the lower temperature range, where published continuum data on the Polyakov loopis also available, we have an agreement, see e.g. [13].
4. The screening masses
We continue with the discussion of the electric and magnetic screening masses obtained fromthe correlators (1.6) and (1.7). For this analysis we only use lattices above the (pseudo)criticaltemperature, since that is the physically interesting range for screening. Next, we mention that forthis analysis, we only use the data with HYP smearing, since we are especially interested in thelarge r behavior. To chose the correct fit interval for the Yukawa ansatz, we use the Kolmogorov-Smirnov test to check whether the χ -s are properly distributed. Note that the determination of thescreening masses does not need additional renormalization. We fit linear functions to all screeningmasses at all values of N t , and use these to do a continuum extrapolation from the N t = , , tatic Q ¯ Q pair free energy and screening masses: continuum results at the physical point
Attila Pásztor m / T T[MeV]m M /Tm E /T 1 1.5 2 2.5 150 200 250 300 350 400 450 m E / m M T[MeV] This work3D EFT, N f =3AdS/CFTWHOT QCD, N f =2, N t =4, m π /m ρ =0.65 Figure 3:
The continuum extrapolations of the screening masses and the ratio of the screening masses.For the ratio m E / m M we also included different estimates from the literature: Lattice results from Ref. [3],dimensionally reduced 3D effective field theory results at T = T c from Ref. [14], and results from N = We finish this section by comparing our results to those from earlier approximations in theliterature. For comparison let us use our results at T = ≈ c . Here we have: • This work: 2+1 flavour lattice QCD at the physical point after continuum extrapolation: m E / T = . ( ) m M / T = . ( ) m E / m M = . ( ) • Ref. [3]: 2 flavour lattice QCD with Wilson quarks, a somewhat heavy pion m π / m ρ = . m E / T = . ( ) m M / T = . ( ) m E / m M = . ( ) • From Table 1 of Ref. [15]: N = N c limit, AdS/CFT m E / T = . m M / T = . m E / m M = . • From Figure 3 of Ref. [14]: dimensionally reduced 3D effective theory, N f = m E / T = . ( ) m M / T = . ( ) m E / m M = . ( ) • From Figure 3 of Ref. [14]: dimensionally reduced 3D effective theory, N f = m E / T = . ( ) m M / T = . ( ) m E / m M = . ( ) tatic Q ¯ Q pair free energy and screening masses: continuum results at the physical point
Attila Pásztor
We note, that our results are closest to the results from dimensionally reduced effective fieldtheory.
5. Summary
In this paper we have determined the renormalized static quark-antiquark free energies in thecontinuum limit. We introduced a two step renormalization procedure using only the finite temper-ature results. The low radius part of the free energies tended to the same curve, corresponding tothe expectation that at small distances, the physics is temperature independent. We also calculatedthe magnetic and electric screening masses, from the real and imaginary parts of the Polyakov looprespectively. As expected, both of these masses approximately scale with the temperature as m ∝ T ,with m M < m E , therefore, magnetic contributions dominating at high distances. The values we gotfor the screening masses are close to the values from dimensionally reduced effective field theory. Acknowledgment
Computations were carried out on GPU clusters at the Universities of Wuppertal and Budapestas well as on supercomputers in Forschungszentrum Juelich.This work was supported by the DFG Grant SFB/TRR 55, ERC no. 208740. and the Lenduletprogram of HAS (LP2012-44/2012).
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