Gluons in Two-Color QCD at High Baryon Density
GGluons in two-color QCD at high baryondensity
V. G. Bornyakov a,b,c and R. N. Rogalyov a a) Institute for High Energy Physics NRC ”Kurchatov Institute”, 142281 Protvino, Russiab) School of Biomedicine, Far East Federal University, 690950 Vladivostok, Russiac) Institute of Theoretical and Experimental Physics NRC ”Kurchatov Institute”,117259 Moscow, Russia January 7, 2021
Abstract
Landau gauge longitudinal and transverse gluon propagators are studiedin lattice QCD with gauge group SU (2) at varying temperature and quarkdensity. In particular, it is found that the longitudinal propagator decreaseswith increasing quark chemical potential at all temperatures under study,whereas the transverse propagator increases with increasing quark chemicalpotential at T <
MeV and does not depend on it at higher temperatures.The relative strength of chromoelectric and chromomagnetic interactions isalso discussed.
We study Landau gauge gluon propagators in N f = 2 lattice QCD withgauge group SU (2) at nonzero temperature and quark density. The detailsof the lattice action used to generate the gauge field configurations are de-scribed in recent papers[1, 2].It is well known that lattice QCD approach is very successful at zerobaryon density but is inapplicable so far at high baryon density due to the socalled sign problem [3]. It is then useful to study in the lattice regularizationthe theories similar to QCD (QCD-like) but without sign problem. In partic-ular, QCD with SU (2) gauge group [4] (to be called below QC D) is one ofsuch theories. a r X i v : . [ h e p - l a t ] J a n C D was studied using various approaches which are also applicable toQCD at high baryon density. It is thus possible to check their predictions inthe case of QC D confronting respective results with first principles latticeresults.The phase diagram of QC D in the T − µ q plane is still not fixed solidly.In the studies of the N f = 2 lattice QC D with staggered fermionic actionat high quark density and T = 0 it was demonstrated[1] that the stringtension σ decreases with increasing µ q and becomes compatible with zerofor µ q above 850 MeV. In a more recent paper [5], where N f = 2 latticeQC D with Wilson fermionic action was studied, the authors did not findthe confinement-deconfinement transition at low temperature. The reasonfor this discrepancy might be the lattice artifacts at large lattice values of µ q a .In this paper we make a step toward the study of QC D phase diagramin the T − µ q plane. We concentrate on the study of the Landau gauge gluonpropagators at nonzero temperature and varying quark chemical potentialand compare the results with our earlier findings obtained at zero tempera-ture. We try to find the signs of the confinement-deconfinement transition inthe gluon propagators dependence on the temperature and the quark chemi-cal potential.The gluon propagators are among important quantities to study, e.g. theyplay crucial role in the Dyson-Schwinger equations approach. The µ q and/ortemperature dependence of the gluon propagators in Landau gauge in latticeQC D were studied before in Refs. [6, 7, 8, 9].One of the conclusions of Ref. [9] was that, beyond the hadronic phasein the µ q − T phase diagram of QC D, the chromoelectric interactions aresuppressed at low momenta, whereas the chromomagnetic - practically donot change.The gluon propagators in QC D at nonzero µ q were also studied inRef. [10] with help of the Dyson-Schwinger equations approach and in Ref. [11]using the massive Yang-Mills theory approach at one-loop. The authors em-phasize that after the agreement with the lattice results for the gluon propa-gators will be reached their methods could be applied to real QCD at nonzerobaryon density. Thus to provide unbiased lattice results is very important.Our computations are completed on lattices , · , · , · with respective temperature values T = 0 , , , MeV and quarkchemical potential ≤ µ q < . GeV. The lattice spacing for parametersused in this study was determined in Ref.[1] as a = 0 . fm. In this work Strictly speaking, for our symmetric lattice, N t a = 140 MeV. We consider this value as agood approximation for zero temperature in accordance with common practice. e do not determine the line of confinement-deconfinement transition in T − µ q plane leaving this important task to future studies. It is pretty clearthat T = 560 MeV is in the deconfinement phase for any value of µ q . Sameis true for T = 280 MeV, most probably. We expect that T = 188 MeV is inthe confinement phase at µ q = 0 and transition to the deconfinement phasehappens at high µ q . The analysis of our data indicates that both the longitudinal and the trans-verse propagators are decreasing functions of the momentum; our data canbe described using the fit function D L,T ( p ) = Z L,T δ L,T p p + 2 R L,T p + M L,T , (1)which, in particular, stems from the Refined Gribov-Zwanziger approach[12]. It has received considerable attention in the literature [13, 14, 16, 17,15]. The respective fitting procedure was considered in detail in Refs. [18,19] at T = 0 , here we describe the results for higher temperatures. We usethe usual normalization condition for the propagators, D renL,T ( κ ) = 1 κ , (2)at κ = 6 GeV. Below we consider renormalized quantities omitting super-script ’ren’. We consider only the soft modes p = 0 . The gluon dressing functions J L,T ( p ) = p D L,T ( p ) (3)as well as the curves derived from the Gribov-Stingl fit (1) are shown inFig.1. It should be emphasized that the zero-momentum propagator valueswere taken into account in the fit; the respective information is encoded inthe behavior of the dressing functions at p → . J L p, GeV T=0 MeVT=188 MeVT=280 MeVT=560 MeV µ q = 0.00 GeV J T p, GeV T=0 MeV T=188 MeVT=280 MeVT=560 MeV µ q = 0.00 GeV J L p, GeV T=0 MeVT=188 MeVT=280 MeVT=560 MeV µ q = 0.90 GeV J T p, GeV T=0 MeV T=188 MeVT=280 MeVT=560 MeV µ q = 0.90 GeV J L p, GeV T=0 MeVT=188 MeVT=280 MeVT=560 MeV µ q = 1.57 GeV J T p, GeV T=0 MeV T=188 MeVT=280 MeVT=560 MeV µ q = 1.57 GeV Figure 1: Longitudinal (left column) and transverse (right column) gluondressing functions at µ q = 0 . (first row), µ q = 0 . GeV (second row),and µ q = 1 . GeV (third row) for four temperatures.4 n general, a shape of the dressing functions is similar to that in puregluodynamics at finite temperature; specific features of their behavior are asfollows.When we increase either T or µ q (or both) the value of the momentum atwhich the longitudinal dressing function J L has a maximum increases, thewidth of the peak also increases, while the maximal value of the dressingfunction decreases. At high T and/or µ q , J L tends to that of a free massiveparticle; at T = 560 MeV and µ q = 1 . GeV it comes close to the dressingfunction of a free particle of mass m ≈ GeV.A completely different situation is observed in the transverse case. At T = 560 MeV the dressing function is practically independent of µ q ; atlower temperatures the peak position does not depend on µ q , whereas thepeak value moderately increases with increasing µ q , the width of the peakdepends only weakly on µ q . Therewith, an increase of the temperature ata given chemical potential gives (i) an increase of the peak position, (ii) adecrease of the peak value, (iii) a moderate increase of the peak width. Thetransverse gluon dressing functions at all temperatures under study coincideat momenta above ≈ GeV for µ q = 0 and at momenta above ≈ . GeVfor high µ q values. This is an indication of smallness of the magnetic screen-ing mass at all values of T and µ q . Both propagators change only a little when the temperature varies below280 MeV, whereas they show a significant decrease as the temperature in-creases from T = 280 MeV to T = 560 MeV.This being so, temperature dependence of the longitudinal propagator is substantial at p (cid:46) GeV at small µ q and it is seen up to the normalizationpoint p = 6 GeV at high µ q . In the infrared, the longitudinal propagatordecreases significantly with the temperature at µ q (cid:46) . GeV, whereas atgreater values of µ q it shows a less pronounced decrease. This can be seenalso in Fig. 2 (left, upper row) where dependence of the propagators at zeromomentum on temperature and quark chemical potential are depicted. Thevalue of D L,T (0) is of special importance since it is used in one of the defi-nitions of the screening mass, see, e.g. review Ref. [20].Temperature dependence of the transverse propagator is seen only atrelatively low momenta ( p (cid:46) ÷ . GeV) as was already said above. Sim-ilar to the longitudinal case, it decreases with temperature. However, incontrast to the longitudinal propagator, its variation with the temperature ncreases with increasing µ q (see also Fig.2 (right)). D L ( ) , G e V - µ q , GeV T=0 MeV T=188 MeVT=280 MeVT=560 MeV 0 5 10 15 20 25 0 0.5 1 1.5 D T ( ) , G e V - µ q , GeV T=0 MeV T=188 MeVT=280 MeVT=560 MeV 0.1 1 10 0 0.1 0.2 0.3 0.4 0.5 0.6 D L ( ) , G e V - T, GeV µ q =0.00 GeV µ q =0.45 GeV µ q =0.90 GeV µ q =1.35 GeV µ q =1.80 GeV 10 0 0.1 0.2 0.3 0.4 0.5 0.6 D T ( ) , G e V - T, GeV µ q =0.00 GeV µ q =0.45 GeV µ q =0.90 GeV µ q =1.35 GeV µ q =1.80 GeV Figure 2: Zero-momentum longitudinal (left column) and transverse (rightcolumn) gluon propagators as functions of µ q at various temperatures (up-per row) and as functions of T at various µ q (lower row). The longitudinal propagator gradually decreases with the chemical poten-tial at all momenta ( p ≤ GeV) and temperatures under consideration. Thedecreasing is more pronounced at low momenta and temperatures. he transverse propagator at low momenta and temperatures ( T <
MeV, p < ∼ ÷ . GeV) and µ q > ∼ GeV increases with increasing µ q . At T = 280 MeV its dependence on µ q becomes less pronounced andit disappears completely at T = 560 MeV. At high momenta the transversepropagator is independent of µ q at all temperatures. In the general case, a widely used definition of the screening mass stemsfrom the on-mass-shell renormalization of the propagator: the inverse prop-agator is considered as a regular function in some neighborhood of p = 0 and thus represented in the form D − L,T ( p ) = Z − ( m E,M + ˜Π( p ) + p ) , (4)where the Taylor expansion of the function ˜Π( p ) starts from the order O (( p ) ) terms. The propagator of renormalized fields A R = Z − / A hasthe form D renL,T ( p ) = 1 m E,M + p + ˜Π( p ) (5)and, if ˜Π( p ) is small in the infrared, it has a pole at p ≈ − m E,M . Thusthe parameter m E,M can be associated both with the mass of the particle andwith the asymptotic behavior of the propagator at spatial infinity D L,T (0 , (cid:126)x ) (cid:39) C E,M e − m E,M | (cid:126)x | . (6)In Refs. [21, 22, 23] the chromoelectric and chromomagnetic screeningmasses were determined using the Yukawa-type fit function D − L,T ( p ) = Z − E,M ( m E,M + p ) (7)at zero and finite temperatures. It was shown [23] that the Yukawa-type fitfunction (7) provides a good quality of this fit over rather wide range ofmomenta giving evidence for smallness of ˜Π( p ) in the infrared.The above definition of m E,M can be related to the correlation length: m E,M = ξ − E,M , (8) here the correlation length ξ E,M is conventionally defined in terms of thecorrelation function (propagator in our case) by the expression [24] ξ = 12 (cid:82) V dx d(cid:126)x ˜ D ( x , (cid:126)x ) | (cid:126)x | (cid:82) V dx d(cid:126)xD ( x , (cid:126)x ) = − D (0 ,(cid:126) (cid:88) i =1 (cid:18) ddp i (cid:19) (cid:12)(cid:12)(cid:12) (cid:126)p =0 D (0 , (cid:126)p ) . (9)We consider definition of the screening masses based on the fit of D − L,T ( p ) at low momenta to a polynomial in p : D − L,T ( p ) = Z − E,M (cid:0) m E,M + p + c · ( p ) + ... (cid:1) . (10)This method was used in Refs. [25, 18]. However, we use the function(10) rather than function (7) because we have no enough data points in theinfrared region where the propagator can be described by the function (7).Thus, to obtain a reasonable fit results we had to use terms up to ( p ) for D L ( p ) and terms up to ( p ) for D T ( p ) . Still, we hope that making use ofthe fit function (10) provides reasonably good estimates of the parameters ineq. (7).When the screening masses are large, it is natural to assume that the one-gluon exchange dominates. In the approximation of one-gluon exchange,the potential of chromoelectrostatic interaction between static external colorsources is given by the Fourier transform of the longitudinal propagator (thechromomagnetostatic potential between currents - by the Fourier transformof the transverse propagator). Both chromoelectric and chromomagnetic po-tentials are short-range; that is, they can be roughly described by a potentialwell of certain width and depth. The parameter characterizing the width isprovided by ξ E,M = 1 m E,M , whereas the parameter V E,M characterizingthe depth can be defined as follows: (cid:90) d(cid:126)x V E,M ( (cid:126)x ) = D L,T ( p = 0 , (cid:126)p = 0) = V E,M ξ E,M (11) = ⇒ V E,M = D L,T (0) m E,M , (12)where V E,M ( (cid:126)x ) is the chromoelectric (chromomagnetic) potential describ-ing interaction between static color charges (currents).Before proceeding further, we recollect again another definition of thescreening mass [20]: M E = 1 D L (0) , M M = 1 D T (0) . (13) learly, it depends on renormalization and is rather sensitive to the finitevolume effects. Loosely speaking, eq. (13) characterizes “the total amount”of the interaction since M E,M = (cid:90) dx d(cid:126)xD L,T ( x , (cid:126)x ) = (cid:90) d(cid:126)x V E,M ( (cid:126)x ) , (14)where D L,T ( x , (cid:126)x ) are the propagators in the coordinate representation.It follows from (12) and (10) that V E,M = Z E,M m E,M , (15)where Z E,M is the respective renormalization factor. The numerical resultsfor V E,M will be presented below. µ q and T m E , G e V µ q , GeV T=0 T=188 MeVT=280 MeVT=560 MeV 0 0.5 1 1.5 2 0 0.5 1 1.5 m M , G e V µ q , GeV T=0 T=188 MeVT=280 MeVT=560 MeV
Figure 3: Chromoelectric (left) and chromomagnetic (right) screeningmasses as functions of µ q at various temperatures. In Fig.3 we show the electric (left panel) and magnetic (right panel)masses as functions of the quark chemical potential at various temperatures.Our value for m E / √ σ at µ q = 0 and T = 0 is 1.50(4) [18]. This valueshould be compared with the value 1.47(2) obtained in SU(3) gluodynamics t zero temperature [22] by fitting the inverse propagator to the form (7) atsmall momenta . We also quote a value 1.48(5) obtained for a mass domi-nating the small momentum behavior of a gluon propagator in SU (2) latticegluodynamics in [26].At temperatures T ≤ MeV, m E does not significantly change atsmall µ q corresponding to the hadron phase. Above µ q ≈ MeV itstarts to increase and continues to increase up to 1.8 GeV. This behavioris analogous to that of the electric screening mass in QCD at
T > T c as wasdemonstrated by lattice simulations [27, 28, 23, 21]. No such increasing wasreported in the earlier studies[9] of QC D. µ q < MeV µ q > MeV
T <
MeV m E (cid:39) . GeV m E (cid:39) . ÷ . GeV m M (cid:39) . GeV m M (cid:39) . ÷ . GeV T = 560 MeV m E (cid:39) . ÷ . GeV m E (cid:39) . ÷ . GeV m M (cid:39) . ÷ . GeV m M (cid:39) . ÷ . GeVTable 1: Dependence of the screening masses on the quark chemical po-tential and temperature. It should be emphasized that, at T = 0 and µ q > MeV, the magnetic mass decreases from m M (cid:39) MeV to m M (cid:39) MeV.
Information on the behavior of the screening masses is summarized inthe table. Chromoelectric forces feature the longest range at T = 0 and µ q = 0 ; their screening increases with an increase of both T and µ q .Chromomagnetic forces have the same radius as chromoelectric at T =0 and µ q = 0 ; however, their screening increases with T , whereas µ q -dependence of their screening depends on the temperature: m M decreaseswith µ q at low temperatures, remains constant at T = 280 MeV and tendsto increase at T = 560 MeV.It should be emphasized that an incerase of m M is clearly seen pro-vided that high temereatures are taken into consideration. No such increasewas observed in Ref. [9] where the temperature range T < . T c was in-vestigated; in this temperature range our data tend to confirm the previousresults: a very limited growth of m M is observed at T <
MeV. We obtained this value taking mass value 647(7) MeV, obtained in [22] and dividing it by √ σ = 440 MeV used in [22] to set the scale. t should also be noticed that the Z factors obtained here differ from theratio η E,M ( µ q ) = m E,M ( µ q ) M E,M ( µ q ) = m E,M ( µ q ) D L,T (0) (16)considered in our previous study [18]. The reason is as follows: we obtain Z from the fitting procedure, which is in fact equivalent to making use of theformula (16), however, with D L,T (0) evaluated as zero-momentum value ofthe fit function (10) rather than the value extracted from data immediately.In our previous study [18] devoted to zero-temperature case, Z E wasconsidered as independent of µ q apart from variation at small µ q values.However, in the present study we found that both Z E and Z M depend eitheron µ q or on T . V E , G e V µ q , GeV T=0 MeV T=188 MeVT=280 MeVT=560 MeV 0 2 4 6 8 10 12 0 0.5 1 1.5 V M , G e V µ q , GeV T=0 MeV T=188 MeVT=280 MeVT=560 MeV
Figure 4: The parameter V E,M characterizing interaction between colorstatic sources as functions of the quark chemical potential at various tem-peratures. Huge errors make only qualitative estimates possible.
The depth of the potential well can be roughly estimated from the for-mula (15), the results are shown in Fig.4. It varies over the range (cid:46) V E,M (cid:46) GeV. Huge errors do not allow to perform quantitative compari-son of V E with V M , as seen by eye, V M tends to be a little greater. • We investigated the dependence of D T and D L on the temperature andisospin chemical potential. Both propagators decrease with increas- ng temperature; however, they behave differently as the functions of µ q . D L decreases with increasing µ q at all temperatures under study,whereas D T increases with increasing µ q at T <
MeV and isindependent of µ q at higher temperatures.• In the model under study, the radius of action of the chromomagneticforces is greater than that of chromoelectric forces at all temperaturesand chemical potentials excepting a neighborhood of the point µ q =0 , T = 0 . Within the range of action, the strength of chromomagneticforces is also approximately the same or a little greater than that ofchromoelectric ones (at µ q > , T > ). For this reason, the stronginteracting matter described by QC D at nonzero temperatures andquark chemical potentials can be named a chromomagnetic medium. Acknowledgments
The work was completed due to support of the Russian Foundation for BasicResearch via grant 18-02-40130 mega. The authors are thankful to VictorBraguta, Andrey Kotov and Alexander Nikolaev for providing gauge fieldconfigurations and useful discussions. The research is carried out usingthe Central Linux Cluster of the NRC “Kurchatov Institute” - IHEP, theequipment of the shared research facilities of HPC computing resourcesat Lomonosov Moscow State University, the Linux Cluster of the NRC“Kurchatov Institute” - ITEP (Moscow). In addition, we used computerresources of the federal collective usage center Complex for Simulation andData Processing for Mega-science Facilities at NRC Kurchatov Institute,http://ckp.nrcki.ru/.
References [1] V. Bornyakov, V. Braguta, E. M. Ilgenfritz, A. Y. Kotov, A. Molochkovand A. Nikolaev, JHEP , 161 (2018) [arXiv:1711.01869 [hep-lat]].[2] N. Astrakhantsev, V. Bornyakov, V. Braguta, E. M. Ilgenfritz,A. Kotov, A. Nikolaev and A. Rothkopf, JHEP , 171 (2019)[arXiv:1808.06466 [hep-lat]].[3] S. Muroya, A. Nakamura, C. Nonaka and T. Takaishi, Prog. Theor.Phys. , 615-668 (2003) [arXiv:hep-lat/0306031 [hep-lat]].
4] J. Kogut, M. A. Stephanov, D. Toublan, J. Verbaarschot and A. Zhitnit-sky, Nucl. Phys. B , 477-513 (2000) [arXiv:hep-ph/0001171 [hep-ph]].[5] T. Boz, P. Giudice, S. Hands and J. I. Skullerud, Phys. Rev. D ,no.7, 074506 (2020) [arXiv:1912.10975 [hep-lat]].[6] S. Hands, S. Kim and J. I. Skullerud, Eur. Phys. J. C , 193 (2006)[arXiv:hep-lat/0604004 [hep-lat]].[7] T. Boz, S. Cotter, L. Fister, D. Mehta and J. I. Skullerud, Eur. Phys. J.A , 87 (2013) [arXiv:1303.3223 [hep-lat]].[8] O. Hajizadeh, T. Boz, A. Maas and J. I. Skullerud, EPJ Web Conf. ,07012 (2018) [arXiv:1710.06013 [hep-lat]].[9] T. Boz, O. Hajizadeh, A. Maas and J. I. Skullerud, Phys. Rev. D ,no.7, 074514 (2019) [arXiv:1812.08517 [hep-lat]].[10] R. Contant and M. Q. Huber, Phys. Rev. D , no.1, 014016 (2020)doi:10.1103/PhysRevD.101.014016 [arXiv:1909.12796 [hep-ph]].[11] D. Suenaga and T. Kojo, Phys. Rev. D , no.7, 076017 (2019)doi:10.1103/PhysRevD.100.076017 [arXiv:1905.08751 [hep-ph]].[12] D. Dudal, J. A. Gracey, S. P. Sorella, N. Vandersickel and H. Ver-schelde, Phys. Rev. D , 065047 (2008) [arXiv:0806.4348 [hep-th]].[13] D. Dudal, O. Oliveira and N. Vandersickel, Phys. Rev. D , 074505(2010) [arXiv:1002.2374 [hep-lat]].[14] A. Cucchieri, D. Dudal, T. Mendes and N. Vandersickel, Phys. Rev. D , 094513 (2012) [arXiv:1111.2327 [hep-lat]].[15] R. Aouane, V. Bornyakov, E. Ilgenfritz, V. Mitrjushkin, M. Muller-Preussker and A. Sternbeck, Phys. Rev. D , 034501 (2012)[arXiv:1108.1735 [hep-lat]].[16] O. Oliveira and P. J. Silva, Phys. Rev. D , 114513 (2012)[arXiv:1207.3029 [hep-lat]].[17] D. Dudal, O. Oliveira and P. J. Silva, Annals Phys. , 351-364 (2018)[arXiv:1803.02281 [hep-lat]].[18] V. Bornyakov, A. Kotov, A. Nikolaev and R. Rogalyov, Particles ,no.2, 308-319 (2020) [arXiv:1912.08529 [hep-lat]].[19] V. Bornyakov, V. Braguta, A. Nikolaev and R. Rogalyov,[arXiv:2003.00232 [hep-lat]].
20] A. Maas, Phys. Rept. (2013), 203-300 [arXiv:1106.3942 [hep-ph]].[21] V. Bornyakov and V. Mitrjushkin, Phys. Rev. D , 094503 (2011)[arXiv:1011.4790 [hep-lat]].[22] O. Oliveira and P. Bicudo, J. Phys. G , 045003 (2011)[arXiv:1002.4151 [hep-lat]].[23] P. Silva, O. Oliveira, P. Bicudo and N. Cardoso, Phys. Rev. D , no.7,074503 (2014) [arXiv:1310.5629 [hep-lat]].[24] S. Ma, Modern Theory of critical phenomena (W. A. Benjamin, Ad-vanced Book Program, Minnesota University, 1976).[25] V. Bornyakov and V. Mitrjushkin, Int. J. Mod. Phys. A , 1250050(2012) [arXiv:1103.0442 [hep-lat]].[26] K. Langfeld, H. Reinhardt and J. Gattnar, Nucl. Phys. B , 131-156(2002) [arXiv:hep-ph/0107141 [hep-ph]].[27] C. S. Fischer, A. Maas and J. A. Muller, Eur. Phys. J. C , 165-181(2010) [arXiv:1003.1960 [hep-ph]].[28] A. Maas, J. M. Pawlowski, L. von Smekal and D. Spielmann, Phys.Rev. D , 034037 (2012) [arXiv:1110.6340 [hep-lat]]., 034037 (2012) [arXiv:1110.6340 [hep-lat]].