Modeling the nonperturbative contributions to the complex heavy-quark potential

Modeling the nonperturbative contributions to the complex heavy-quark potential Yun Guo,1,2 Lihua Dong,1,2 Jisi Pan,1,2 and Manoel R. Moldes3 1Department of Physics, Guangxi Normal University, Guilin, 541004, China 2Guangxi Key Laboratory of Nuclear Physics and Technology, Guilin, 541004, China 3Departamento de Fisica de Particulas, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Galicia, Spain (Received 7 May 2019; published 27 August 2019) In this paper, we construct a simple model for the complex heavy quark potential which is defined through the Fourier transform of the static gluon propagator. Besides the hard thermal loop resummed contribution, the gluon propagator also includes a nonperturbative term induced by the dimension two gluon condensate. Within the framework of thermal field theory, the real and imaginary parts of the heavy quark potential are determined in a consistent way without resorting to any extra assumption as long as the exact form of the retarded/advanced gluon propagator is specified. The resulting potential model has the desired asymptotic behaviors and reproduces the data from lattice simulation reasonably well. By presenting a direct comparison with other complex potential models on the market, we find the one proposed in this work shows a significant improvement on the description of the lattice results, especially for the imaginary part of the potential, in a temperature region relevant to quarkonium studies. DOI: 10.1103/PhysRevD.100.036011 I. INTRODUCTION The heavy-ion experiments at RHIC and the LHC have shown very rich and interesting physics that cannot be interpreted by simple extrapolation from protonproton collisions, which indicates the formation of a new form of matter—the quark-gluon plasma (QGP) during the ultrarelativistic heavy-ion collisions. Heavy quarkonium dissociation has been proposed long time ago as a very sensitive probe to study the hot and dense medium [1]. Bound states of heavy quarks could survive inside the plasma where the temperature Tis higher than the deconfining temperature. However, color screening produced by the light quarks and gluons weakens the interaction between the quark-antiquark pair and leads to the dissociation of quarkonia. Since excited states are more weakly bound than the lower ones, the successive dissociations can possibly serve as a thermometer of QGP [2]. The studies on quarkonia can be carried out in the nonrelativistic limit due to their large masses, where a quantum mechanical description becomes available. As the basic input in the Schrödinger equation, the heavy-quark (HQ) potential turns to be very crucial to understand the physical properties of the bound states. At zero temperature, the well-known Cornell potential successfully describes the experimentally observed quarkonium spectroscopy and agrees with the lattice simulations very well. Within the framework of effective field theory (EFT) of QCD, i.e., potential nonrelativistic QCD (pNRQCD), substantial development has been achieved in the heavy quarkonium physics [3,4]. The EFT was also generalized to finite temperature QCD which justified the description of heavy quarkonia in terms of an in-medium potential. However, the EFTat finite temperature involves much more complications due to the appearance of some extra T-dependent scales [5]. As a result, constructing phenomenological potential models has been widely considered over the past decades which provides an alternative way to analyze the in-medium behaviors of the bound states. In previous studies, the color singlet free energy or internal energy of a static quark pair obtained from lattice simulations was identified with the HQ potential. In addition, based on these lattice results, various proposals for the potential models have also been extensively discussed, see Refs. [6–9] for examples. However, the realvalued potential models cannot really represent the HQ potential in the hot medium because it must include an imaginary part induced by the color singlet-octet transition Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 100, 036011 (2019) 2470-0010=2019=100(3)=036011(14) 036011-1 Published by the American Physical Society as well as the Landau damping of the low-frequency gauge fields [10]. A first step toward a QCD derivation of the HQ potential at finite temperature was carried out in Ref. [11]. In hard thermal loop (HTL) resummed perturbation theory, the static Wilson loops were computed in the imaginary-time formalism. After analytical continuation to Minkowski space, it was found that besides a Debye screened potential as its real part, the potential also contains an imaginary part which determines the decay width of a quarkonium state. Such a perturbative calculation in the weak-coupling limit, however, is only valid when the distance rbetween the quark and antiquark is small. In the past long period of time, the large distance behavior of the complex potential is not clear due to the lack of the corresponding lattice data. Fortunately, progress has been made in recent years [12–15]. Burnier et al. have measured the complex-valued static potential by first principle simulations in quenched QCD. In a latest publication [16], the improved results with reduced finite volume artifacts have been provided. To sufficiently describe the interaction between the quark pair at finite temperature, there have already been some attempts to develop complex HQ potential models. In Ref. [17], Thakur et al. defined the complex HQ potential by Fourier transforming the product of the Cornell potential in momentum space and the inverse dielectric function ϵ−1ðpÞ. Therefore, medium effects are entirely encoded in the complex dielectric function which has been calculated in HTL perturbation theory. Solving the Schrödinger equation with such a complex HQ potential, the binding energies and decay widths of quarkonia have been obtained. However, they did not make a comparison between their potential model and the corresponding lattice results. As we will show later, predictions from this model cannot reproduce the data very well and some asymptotic behaviors are also found to be unphysical. In Ref. [18], Burnier et al. constructed a complex potential model based on the generalized Gauss law [19,20] and similarly as Ref. [17], medium effects are incorporated by using the same dielectric function. The predicted imaginary part of the potential based on the model is only satisfactory when T is large and ris small. Therefore, for better understanding the in-medium properties of quarkonia, a more accurate HQ potential model is required which is expected to be in agreement with the lattice data at a quantitative level. For the above mentioned purpose, the current paper aims to construct a complex HQ potential model which can be used for other phenomenological studies on the heavy quarkonia. The rest of the paper is organized as follows. In Sec. II, we briefly review the calculation of the complex potential in perturbation theory which provides the Coulombic contribution in our potential model. In Sec. III, we adopt a phenomenological gluon propagator whose nonperturbative term is induced by the dimension two gluon condensate. Performing Fourier transform of such a gluon propagator in Keldysh representation, the obtained HQ potential has a real part which is identical to the Karsch–Mehr–Satz (KMS) potential model. On the other hand, the imaginary part presents some unexpected features and does not agree with the lattice simulation. Improvements are discussed in Sec. IV where, by inspecting the asymptotic behaviors of the model proposed in Sec. III, an additional string contribution is introduced in the gluon propagator. The resulting HQ potential model has been compared to other available models in Refs. [17,18] as well as the lattice results in Ref. [16]. In a temperature region relevant to quarkonium physics, a significant improvement on the imaginary part of the HQ potential is observed. Finally, we give a short summary in Sec. V. II. PERTURBATIVE HEAVY QUARK POTENTIAL AT FINITE TEMPERATURE At zero temperature, the interaction between a static quark pair can be successfully described by the Cornell potential. It takes a form of a Coulomb plus a linear part, VCornell ¼−αs rþσr; ð1Þ where αs¼g2CF=ð4πÞis the strong coupling constant, σis the so-called string tension which has the dimension of energy square. At finite temperature, the potential at short distances can be computed in thermal field theory with perturbation expansion. In the real time formalism, the propagator is given by a 2×2matrix. It is more convenient to use the Keldysh representation where we have three independent components named retarded (DR), advanced (DA) and symmetrical (DF) propagators. Their relation to the physical “11”component is given by D11 ¼ðDRþDAþDFÞ=2. Within hard-thermal-loop approximation, one can compute the self-energy contributions which are used to determine the resummed gluon propagators through the Dyson-Schwinger equation. The perturbative HQ potential Vpcan be obtained from the following Fourier transform1 Vpðˆ rÞ¼−g2CFZd3p ð2πÞ3ðeip·r−1ÞDp 11ðp0¼0;pÞ;ð2Þ where Dp 11 refers to the physical component of the resummed gluon propagator and ˆ r¼rmDwith the Debye mass given by m2 D¼ðNfþ2NcÞg2T2 6. At leading order, the static gluon propagator in the above Fourier transform reads 1From here on, Donly denotes the temporal component of the gluon propagator which is relevant to the HQ potential. We introduce a supper script “p”to indicate perturbative quantities, accordingly a supper script “np”stands for the nonperturbative quantities. GUO, DONG, PAN, and MOLDES PHYS. REV. D 100, 036011 (2019) 036011-2 ReDp 11ðp0¼0;pÞ¼Dp Rðp0¼0;pÞ ¼Dp Aðp0¼0;pÞ ¼1 p2þm2 D ;ð3Þ ImDp 11ðp0¼0;pÞ¼1 2Dp Fðp0¼0;pÞ¼ −πTm2 D pðp2þm2 DÞ2: ð4Þ The real part of the potential is obtained by the Fourier transform of the retarded/advanced propagator while the imaginary part comes from the symmetric propagator in Keldysh representation. Explicitly, we have [11,21] ReVpðˆ rÞ¼−g2CFZd3p ð2πÞ3ðeip·r−1Þ1 p2þm2 D ¼−αsmDþe−ˆ r r;ð5Þ ImVpðˆ rÞ¼−g2CFZd3p ð2πÞ3ðeip·r−1Þ−πTm2 D pðp2þm2 DÞ2 ¼−αsTϕ2ðˆ rÞ;ð6Þ where ϕnðˆ rÞ¼2Z∞ 0 dz z ðz2þ1Þn1−sinðzˆ rÞ zˆ r:ð7Þ Notice that for the real part, the r-independent term is divergent and we have subtracted a vacuum contribution 1=p2in the integrand to get a finite result. As compared to the vacuum case, the Coulombic behavior at small distances gets screened and a nonzero imaginary contribution appears. However, the above perturbation theory is not capable of dealing with the medium corrections to the string contribution in the Cornell potential which will be considered by constructing phenomenological models and discussed in the next section. III. AN EXTENDED KARSCH-MEHR-SATZ HEAVY-QUARK POTENTIAL MODEL To study the in-medium properties of the heavy bound states, such as charmonia and bottomonia, in the nonrelativistic limit, a proper potential that needs to be specified in the Schrödinger equation contains nonperturbative physics due to the typical size of the charm and bottom quark bound states. Therefore, we cannot directly use the above perturbative potential to describe the interactions. In Ref. [22], a new phenomenological term has been added to the perturbative (retarded) gluon propagator Dp Rin order to account for the effects coming from the low frequency modes incorporated in the dimension two gluon condensates. As a result, the full retarded propagator DRat static limit takes the following form DRðp0¼0;pÞ≡Dp Rðp0¼0;pÞþDnp Rðp0¼0;pÞ ¼1 p2þm2 Dþm2 G ðp2þm2 DÞ2:ð8Þ The above equation can be considered as an analogy to the condensates at zero temperature [23] which implies a term m2 G=p4to be added to the vacuum perturbative gluon propagator 1=p2. Here, m2 Gis a dimensional constant. Several applications based on the propagator given in Eq. (8) have been carried out, see Refs. [22,24,25] for examples. Here, we are interested in the Fourier transform of Dnp Rat static limit which leads to the following nonperturbative string contribution to the real part of the potential, ReVnp Iðˆ rÞ¼αsm2 G 2mD½1−exp ð−ˆ rÞ:ð9Þ We use VI¼VpþVnp Ito denote the complex HQ potential model discussed in this section. Improvements on Vnp Iwill be discussed in Sec. IV and the resulting potential model is then denoted as VII ¼VpþVnp II . The real part of VIis the sum of Eqs. (5) and (9). By matching it onto the Cornell potential at small distances, we find the dimension two constant m2 Gcan be related to the string tension through σ¼αsm2 G=2. Therefore, ReVIis actually identical to the famous KMS potential model [26] in which the large distance interaction is described as a QCD string screened at the same scale as the perturbative contribution. Explicitly, we have ReVIðˆ rÞ¼−αsmDþe−ˆ r rþσ mD½1−exp ð−ˆ rÞ:ð10Þ Inspired by the above analysis on the real part of the potential, we will also consider adding a string contribution, which describes the large distance behavior of ImV,to the perturbative symmetric propagator Dp F. In equilibrium, the symmetric propagator can be related to the retarded and advanced ones through the following identity DFðPÞ¼ð1þ2nBðp0ÞÞsgnðp0Þ½DRðPÞ−DAðPÞ;ð11Þ which is valid for full propagators as a consequence of the KMS condition [27]. In the above equation, nBis the BoseEinstein distribution function and the four-momentum P≡ðp0;pÞ. Although only the static forms of the propagators are required in the Fourier transform, one still need to know the p0-dependent propagators DRðPÞand DAðPÞ in order to compute DFðp0¼0;pÞthrough Eq. (11). MODELING THE NONPERTURBATIVE CONTRIBUTIONS TO …PHYS. REV. D 100, 036011 (2019) 036011-3 To make it more clear, we consider the distribution function nBof on-shell thermal gluons in small p0limit ð1þ2nBðp0ÞÞsgnðp0Þ¼2T p0þOðp0 0Þ;ð12Þ which indicates that the leading contribution from DRðPÞ−DAðPÞshould be linear in p0in order to have a nonzero and finite symmetric propagator at static limit. In fact, for the perturbative terms, we have [21] Dp R=AðPÞ¼ðp2−ΠR=AðPÞÞ−1 ¼p2−m2 Dp0 2pln p0þpiϵ p0−piϵ−1−1 ; ð13Þ where ΠR=AðPÞis (the temporal component of) the retarded/advanced gluon self-energy at leading order. Performing a Taylor expansion assuming p0→0,itis straightforward to show Dp RðPÞ−Dp AðPÞ¼m2 D 2p −2πi ðp2þm2 DÞ2p0þOðp2 0Þ:ð14Þ Together with Eqs. (11) and (12), one can get the symmetric propagator Dp Fðp0¼0;pÞwhose Fourier transform determines ImVpas already calculated in Eq. (6). However, the p0-dependence of the nonperturbative propagators Dnp R=AðPÞis not known. As a “minimal” extension of the corresponding perturbative result, it is also introduced in a similar way by replacing m2 Dwith −ΠR=AðPÞand we assume Dnp R=AðPÞ¼m2 Gðp2−ΠR=AðPÞÞ−2 ¼m2 Gp2−m2 Dp0 2pln p0þpiϵ p0−piϵ−1−2 ; ð15Þ which has the desired static limit and leads to the following result Dnp RðPÞ−Dnp AðPÞ¼m2 Gm2 D p −2πi ðp2þm2 DÞ3p0þOðp2 0Þ: ð16Þ Accordingly, we can obtain the nonperturbative symmetric propagator through Eq. (11) as Dnp Fðp0¼0;pÞ¼m2 Gm2 D p −4πTi ðp2þm2 DÞ3:ð17Þ After Fourier transforming Eq. (17), the string contribution to the imaginary part of the HQ potential is found to be ImVnp Iðˆ rÞ¼−g2CFZd3p ð2πÞ3ðeip·r−1Þ−2πTm2 Gm2 D pðp2þm2 DÞ3 ¼− 4σT m2 D ϕ3ðˆ rÞ:ð18Þ Summing up Eqs. (6) and (18), the full imaginary part in the potential model VIreads ImVIðˆ rÞ¼−αsTϕ2ðˆ rÞ− 4σT m2 D ϕ3ðˆ rÞ:ð19Þ As an extension of the real-valued KMS model, the complex version VI¼ReVIþiImVIis also called the extended KMS potential model. It is interesting to see if the above simple model could reproduce the lattice data. To do so, we use the lattice data in quenched QCD from Ref. [16]. The two parameters αsand σwere assumed to be unchanged in a hot medium, once determined at zero temperature. Due to the absence of a T¼0lattice measurement, αs¼0.272 and σ¼0.215 GeV2are determined by using the data at 113 MeV [16]. Applying these values of αsand σto the relation σ¼αsm2 G=2leads to a dimension two condensate which coincides with the corresponding lattice simulations [28–30]. Notice that at short distances, the running of αsis controlled by the scale 1=r. Given the shortest quark-pair separation available in the data from Ref. [16], which is about 0.1 fm, it turns out that ReVat short distances can be well described by a naive Cornell potential with fixed coupling constant. On the other hand, with the upcoming high resolution lattice simulations at shorter resolved distances, it is certainly important to take into account the running of αsðrÞ. Furthermore, although the model study on the Polyakov loop in Ref. [22] suggests that the nonperturbative finite temperature condensate is consistent with that at zero temperature, a possible medium dependence of the string tension σcannot be ruled out in principle. However, the exact T-dependent form of such a nonperturbative quantity has not been fully clear yet. In this work, we will employ a constant σfor simplicity and assume all the medium effects on the HQ potential are encoded in the only free parameter mDin the model. It is worthwhile to mention that such an assumption can effectively avoid any double counting of the medium effects. Since the extraction of the imaginary part from lattice simulations gets much more challenging than the real part, the lattice data of ReVis used to determine the Debye mass. In addition, we only consider the lattice data of ReVup to 1 fm in our fit because in this region of the quark pair separations, the lattice reconstruction is most reliable and the error bars are actually very small. As a crosscheck, the values of mDfrom the fit to ReVwill be adopted to evaluate GUO, DONG, PAN, and MOLDES PHYS. REV. D 100, 036011 (2019) 036011-4 the imaginary part of the potential. The optimized values we obtain for mDat different temperatures are given in Table I. A critical behavior is found by inspection of the data and the deconfining temperature Tcis around 290 MeV. This is actually consistent with the T-dependence of mD as given in the above table. For temperatures below Tc, the values of mDfrom the fit turn to be extremely small which are at the order of 10−6or even smaller. On the other hand, once the temperature exceeds Tc,mDgets a nonzero value which increases with temperature Tas expected. Since we are more interested in the behavior of the HQ potential in the deconfined phase, when T<T c, the values of mDare simply taken to be zero in Table I.As a result, ˆ rvanishes for finite quark pair separation and ReVIis exactly identical to the vacuum Cornell potential in the confined phase. As we can see from Table I,mDdoes not have a simple linear dependence on Twhich clearly indicates the nonperturbative effects in the temperature region relevant to the quarkonium studies. Interestingly, we find that the extracted Debye mass can be simply parametrized as mDðTÞ¼aT þb=T. Besides the usual leading order result, a new term inversely proportional to Thas been included which accounts for the nonperturbative contributions and becomes important when the temperature is decreasing to Tc. As shown in Fig. 1, in the deconfined phase, the values of mDcan be well reproduced when taking the parameters as a¼1.719 and b¼−0.123 GeV2. Notice that this simple parametrization of mDdoes not apply in the asymptotically high temperature limit because the parameter ais considered as a constant. Furthermore, the negative value of bindicates that the ratio mD=T decreases as Tapproaches to Tcfrom above. The same has also been observed in a massive quasiparticle model when fitting to the equation of state [31]. In addition, lattice measurements of the gauge invariant correlation function between Polyakov loops shows that the associated screening mass behaves similarly as that presented in Fig. 1 [32,33]. However, based on the two point function of gluons computed on lattice, a contradictory conclusion was obtained where the corresponding gauge dependent mass increases as Tapproaches to Tc[34,35]. Such an upward trend of the ratio mD=T actually coincides with the result from perturbation calculation [36,37]. Therefore, as discussed in Ref. [38], it would be important to reanalyze the lattice data by using a Higgsed propagator where different modes with both increasing and decreasing masses are combined. The comparisons between the extended KMS model and the lattice data are given in Fig. 2for ReVand in Fig. 32for ImV. We also plot the pure perturbative results Vpwhich clearly indicate the necessity to include string contributions even for relatively small distances. As shown in Fig. 2, ReVIhas a good agreement with the lattice data. At very small distances, the Coulombic interaction is dominated while at large distances, it exhibits a screened behavior as suggested by the data. In addition, the lattice data at T¼271 MeV is nicely reproduced by the vacuum Cornell potential, therefore, our assumption of vanishing Debye mass in the confined phase is justified. On the other hand, ImVIgets a rapid increase with the quark pair separation which obviously overshoots the lattice data as shown in Fig. 3. Besides the quantitative deviations in the deconfined phase, a qualitative difference appears when T<T c. Neglecting the cold nuclear effects, in the confined phase, one would expect ImVis approximately zero which is actually supported by lattice data despite the huge uncertainties. Unfortunately, the model prediction at T¼271 MeV is apparently contradictory to the lattice results. The lack of success of Eq. (19) requires improvements on the current potential model, especially for the imaginary part. Given the model above, it is also important to discuss its asymptotic behaviors which hints at some possible modifications on the extended KMS potential model. In the small distance limit where ˆ r≪1, the real part of the potential ReVIreduces to the vacuum Cornell potential and the Coulombic interaction dominates over the string contribution. We can define a distance scale rsðTÞwhere the nonperturbative effects start to matter. It is determined by requiring jReVpðrsÞj ¼ jReVnpðrsÞj and we find that rsðTÞ¼ ffiffiffiffiffiffiffiffiffiffi αs=σ p. This result is actually T-independent because medium effect appears as higher order correction to the Cornell potential when we expand ReVIwith respect FIG. 1. Comparison between the parametrization of Debye mass (red solid curve) and its values extracted from the lattice data (blue dots) in Ref. [16]. TABLE I. Debye mass extracted from the extended KMS model VIfit to the lattice result for ReVin Ref. [16]. T[MeV] 406 369 338 312 290 271 254 226 113 VI:mD[MeV] 423 258 231 134 87.8 0 0 0 0 2Notice that for the imaginary part of the potential, we actually plot its absolute values in all the figures. MODELING THE NONPERTURBATIVE CONTRIBUTIONS TO …PHYS. REV. D 100, 036011 (2019) 036011-5 to ˆ r. Since this perturbative expansion is valid for ˆ r≪1, the above result is applicable when mD≪ffiffiffiffiffiffiffiffiffiffi σ=αs pwhich can be satisfied for not very high temperatures. To study the asymptotic behavior of ImVIin small ˆ rlimit, we need to expand the following functions,3 ϕ2ðˆ rÞ≈− 1 9 ˆ r2ð3ln ˆ r−4þ3γEÞ;ð20Þ ϕ3ðˆ rÞ≈ 1 12 ˆ r2þ1 900 ˆ r4ð15 ln ˆ r−23 þ15γEÞ;ð21Þ ϕ4ðˆ rÞ≈ 1 36 ˆ r2− 1 360 ˆ r4;ð22Þ FIG. 2. Comparison of ReVbetween the lattice data in quenched QCD (blue dots) from Ref. [16] and the extended KMS potential model VIas discussed in Sec. III. The red solid curve denotes the model prediction based on ReVIwhile the black dashed curve denotes the results from pure perturbative contribution ReVp. The critical temperature Tc¼290 MeV. FIG. 3. Comparison of ImVbetween the lattice data in quenched QCD (blue dots) from Ref. [16] and the extended KMS potential model VIas discussed in Sec. III. The red solid curve denotes the model prediction based on ImVIwhile the black dashed curve denotes the results from pure perturbative contribution ImVp. The critical temperature Tc¼290 MeV. 3The expansion of ϕ4ðˆ rÞwill be used later. GUO, DONG, PAN, and MOLDES PHYS. REV. D 100, 036011 (2019) 036011-6 where γEis the Euler-Gamma constant. The imaginary part of the potential develops a nonzero value at finite temperature and quark pair separation. In general, one can expect that the imaginary part of the potential behaves similarly as the real part, namely, ImVpis dominant at very short distances, when starting to separate the quark pair, the contribution from ImVnp gets increased and eventually becomes comparable to ImVpat the same distance scale rsðTÞ∼ffiffiffiffiffiffiffiffiffiffi αs=σ p. However, this desired feature does not show up in the analysis based on the above potential model ImVI. In fact, the distance scale rsðTÞdetermined through jImVpðrsÞj ¼ jImVnpðrsÞj is found to be rsðTÞ≈ 1 mD e −σ αsm2 D;ð23Þ which is exponentially suppressed when mD≪ffiffiffiffiffiffiffiffiffiffi σ=αs p. Therefore, for the imaginary part of the potential, the string contribution becomes important at much smaller distances as compared to the real part. For example, taking αs¼0.272 and σ¼0.215 GeV2, we find that rsðTÞ≈ 0.2fm for ReVIwhich differs the distance scale for ImVI by orders of magnitude. For some typical value of the Debye mass, mD∼0.3GeV, rsðTÞis about 10−4fm for the imaginary part. As already mentioned before, ImVIhas finite values in the confined phase which increase quickly with the distance r. The origin of such an incorrect behavior actually comes from ImVnp Iin the small ˆ rlimit. One can easily check that for vanishing Debye mass, jImVIjreduces to σTr2=3which perfectly reproduces the solid curve in the last plot of Fig. 3. On the other hand, the above discussed problems can be solved if the leading order contribution in ImVnp is proportional to ˆ r4ln ˆ rinstead of ˆ r2. As a result, the same distance scale rsðTÞ∼ffiffiffiffiffiffiffiffiffiffi αs=σ pis found for both real and imaginary part of the HQ potential and in the confined phase, ImValso vanishes if mDis assumed to be zero. When ˆ r→∞, the asymptotic value of ReVIequals σ=mD−αsmD. In general, the Debye screening mass increases with T, therefore, ReVIðˆ r→∞Þdecreases as Tis getting larger. This is qualitatively in agreement with that suggested by lattice data. In addition, for the imaginary part, we have ImVIðˆ r→∞Þ¼−αsT− 2σT m2 D :ð24Þ According to this equation, the asymptotic value of ImVI could have a nontrivial dependence on the temperature T. Only at very high temperatures where the Debye mass mD∼T, we can expect jImVIðˆ r→∞Þj increase with increasing Tprovided mD>ffiffiffiffiffiffiffiffiffiffiffiffi 2σ=αs p. However, the current lattice simulations on the complex HQ potential cannot provide us sufficient information about the asymptotic values of ImVat large ˆ r. IV. AN IMPROVED KARSCH-MEHR-SATZ HEAVY-QUARK POTENTIAL MODEL For the purpose of quantitatively describing the lattice data, in this section, we will discuss the improvements on the extended KMS potential model VIas proposed in Sec. III. In fact, the analysis on the asymptotic behavior of ImVIsuggests the leading order contribution from ImVnp should behave like ∼ˆ r4ln ˆ rwhen ˆ r→0. Therefore, an extra nonperturbative term could be introduced in the symmetric propagator and the resulting contribution to ImVis expected to cancel the ∼ˆ r2term in ImVnp Iin the small ˆ rlimit. This can be achieved in a consistent way through Eq. (11) and the key point is to find a proper string contribution which needs to be added to the retarded/ advanced propagator Dnp R=A in Eq. (8). At finite temperature, a nonperturbative term m2 G=ðp2þ m2 DÞ2in the retarded propagator was introduced based on the extension of the vacuum dimension two gluon condensates. From a phenomenological point of view, however, we cannot rule out some other possible forms, for example, adding a term ∼m2 Gm2 D=ðp2þm2 DÞ3in Eq. (8) does not ruin the vacuum limit m2 G=p4since this term vanishes as mD→0. Furthermore, both terms produce the same kind of condensate related to m2 Gand in the Gaussianlike approximation, lead to the same nonperturbative contribution ∼1=T2to the (logarithm of the) Polyakov loop in deconfined phase[24]. Therefore, we formally write the improved retarded propagator as ˜ DR≡DRþδDR and (the static limit of) the newly added string contribution δDRcan be written as δDRðp0¼0;pÞ≡am2 Gm2 D ðp2þm2 DÞ3;ð25Þ where ais a dimensionless constant. Fourier transforming Eq. (25), an extra contribution to the real part of the potential reads ReδVðˆ rÞ¼a 4 σ mDð1−e−ˆr−ˆ re−ˆrÞ:ð26Þ Qualitatively, the asymptotic behavior of ReVis not affected by the above contribution. Since Eq. (26) vanishes when taking ˆ r→0, one still gets the Cornell potential in this limit. On the other hand, the asymptotic value at infinitely large ˆ ris changed into ð1þa=4Þσ=mD−αsmD. In vacuum, the main contribution to ReVis dominated by σrfor large quark pair separation. For 0<T<T c,ReVis very close to the Cornell potential as observed by the lattice simulation. Therefore, the medium effects in the confined phase, which strictly speaking are not exactly zero, can be treated perturbatively by assuming ˆ r≪1. For large but finite quark pair separation r, we can always assume ˆ r≪1in the MODELING THE NONPERTURBATIVE CONTRIBUTIONS TO …PHYS. REV. D 100, 036011 (2019) 036011-7 confined phase due to mD→0. Since the potential ReV at finite Tcannot overshoot the vacuum potential, such medium effects should set in as a negative correction to σr. According to Eq. (9), the leading order correction equals −σˆ r2=ð2mDÞwhich is negative as expected. After including the extra contribution in Eq. (26),thiscorrection becomes −σˆ r2=ð2mDÞþaσˆ r2=ð8mDÞ. To keep it nonpositive, we choose the maximum value of a,i.e.,a¼4and it is expected to give the “most confining”potential. Accordingly, the leading order correction appears at higher order in ˆ rwhich is still negative and given by −σˆ r3=ð6mDÞ. In the above discussion, we ignore the medium correction coming from the perturbative terms because it is small as compared to the corresponding nonperturbative correction when ris large. With the above choice of the constant a,the asymptotic value ReVnpðˆ r→∞Þis changed from σ=mD to 2σ=mDwhich is identical to other potential models discussed in Refs. [17,39]. With the improved propagator ˜ DR=A, we will also study the corresponding changes in the imaginary part of the potential. To do so, the p0-dependence should be properly introduced in δDR=A. Here, we adopt the following assumption for the p0-dependent δDR=A δDR=AðPÞ¼bm2 Gm2 D ðp2−ΠR=AðPÞÞ3 þb0m2 Gð−m2 D−ΠR=AðPÞÞ ðp2−ΠR=AðPÞÞ3:ð27Þ There is a subtlety in the above equation due to the fact that the term with the dimensionless constant b0vanishes in the static limit. Therefore, the recovery to Eq. (25) when p0¼0which requires the dimensionless constant bto be equal to a, however, does not impose any constraint on the value of b0. In fact, Eq. (27) can be considered as a generalized expression of δDR=AðPÞas compared to its simplest form one could imagine ∼m2 GΠR=AðPÞ=ðp2− ΠR=AðPÞÞ3. The latter is obtained from Eq. (25) by replacing mDwith the gluon self-energy −ΠR=AðPÞand identical to our assumption Eq. (27) only when b¼b0. The necessity of considering a more general form of δDR=A,as we will see later, is based on the fact that b0has to take some value different from bin order to meet the crucial requirement on ImVnp, namely, its leading order contribution should be proportional to ∼ˆ r4ln ˆ rin small ˆ rlimit. Using Eq. (11), we can calculate the extra contribution to the symmetric propagator induced by Eq. (27), δDFðp0¼0;pÞ¼−2πTim2 Gm2 D pb3m2 D ðp2þm2 DÞ4 −b01 ðp2þm2 DÞ3;ð28Þ which after performing Fourier transform, gives rise to the following correction to the imaginary part of the HQ potential ImVnp I, ImδVðˆ rÞ¼−b6σT m2 D ϕ4ðˆ rÞþb02σT m2 D ϕ3ðˆ rÞ; ≈−bσT m2 Dˆ r2 6− ˆ r4 60 þb0σT m2 Dˆ r2 6− 23 −15γE−15 ln ˆ r 450 ˆ r4; for ˆ r≪1:ð29Þ Here, the small ˆ rexpansion is obtained by using Eqs. (21) and (22). As we can see the leading order contribution from the ϕ3ðˆ rÞterm is ∼ˆ r2and the same holds for the ϕ4ðˆ rÞterm but with opposite sign. If the values of band b0were chosen to be the same, the leading order contribution from Eq. (29) would be proportional to ∼ˆ r4ln ˆ rand there is no way to cancel the ∼ˆ r2term in ImVnp I. On the other hand, the desired result can be obtained when the relation between the two dimensionless constants b0−b¼2is satisfied. After including the correction in Eq. (29), the nonperturbative contribution to ImVnow takes the form ImVnp II ðˆ rÞ¼ImVnp Iðˆ rÞþImδVðˆ rÞ ¼b2σT m2 D½ϕ3ðˆ rÞ−3ϕ4ðˆ rÞ; ≈bσT m2 D 30 ln ˆ r−31 þ30γE 900 ˆ r4;for ˆ r≪1: ð30Þ In the above equation, b0has already been replaced by bþ2. Furthermore, in order to have the “most confining” potential, the constant bis uniquely determined as b¼a¼4. The above discussion clearly demonstrates the rationality of using the more general assumption Eq. (27) for the p0-dependence of δDR=AðPÞ. However, one may wonder that what would happen if we added more than one possible term like ∼m2 GðΠR=AÞn=ðp2−ΠR=AÞnþ2(with n¼1;2;…) to Dnp R=AðPÞ. By inspecting the asymptotic behavior of the real part of the potential, no qualitative change was found when two or more terms are added simultaneously. However, in small ˆ rlimit, the correction to the imaginary part induced by each individual term is subleading with respect to the term ∼ˆ r2in ImVI, as a result, no cancellation could happen. Unavoidably, we need to use the same trick as Eq. (27) to split at least one added term into two parts with different coefficients. Therefore, adding more terms turns to be not helpful which on the other hand, makes the model complicated. GUO, DONG, PAN, and MOLDES PHYS. REV. D 100, 036011 (2019) 036011-8 Now, we are ready to write down the improved KMS potential model VII which, as compared to the extended KMS model VIin Sec. III, contains the corrections from Eqs. (26) and (29). Explicitly, the results are listed below ReVIIðˆ rÞ¼−αsmDþe−ˆ r rþ2σ mD½1−exp ð−ˆ rÞ −σ mD ˆ rexp ð−ˆ rÞ; ImVIIðˆ rÞ¼−αsTϕ2ðˆ rÞþ8σT m2 D ϕ3ðˆ rÞ− 24σT m2 D ϕ4ðˆ rÞ: ð31Þ The asymptotic values of the above potential model at ˆ r→0are found to be ReVIIðˆ r→0Þ¼−αs=r þσrand ImVIIðˆ r→0Þ¼αsT 3 ˆ r2ln ˆ rþ2σT 15m2 D ˆ r4ln ˆ r: ð32Þ Correspondingly, for ˆ r→∞we have ReVIIðˆ r→∞Þ¼−αsmDþ2σ mD and ImVIIðˆ r→∞Þ¼−αsT− 4σT m2 D :ð33Þ According to Eq. (32), now the distance scale rsðTÞis at the order of ∼ffiffiffiffiffiffiffiffiffiffi αs=σ pfor both real and imaginary part of the HQ potential. Furthermore, at infinitely large ˆ r, the nonperturbative contribution equals ð1þa=4Þσ=mDfor the real part and −bσT=m2 Dfor the imaginary part. Therefore, Vnp II ðˆ r→∞Þis determined solely by the dimensionless constant a(a¼bis required). Before we show the comparison between the improved KMS potential model VII ¼ReVII þiImVII and lattice data, it is also worthwhile to mention other phenomenological models which have been studied in Refs. [17,18]. We refer to the one in Ref. [17] as Thakur–Kakade–Patra (TKP) model. Accordingly, the model in Ref. [18] is referred to as Burnier-Rothkopf (BR) model. Explicit forms of these two potential models can be found in the above mentioned references. Although the basic ideas of model construction are very different from each other, the perturbative terms in these models are all expressed by the leading order HTL result, i.e., Eqs. (5) and (6). On the other hand, despite owning different nonperturbative forms, their asymptotic behaviors of ReVnp are actually very similar. Taking ˆ r→0,ReVnp reduces to the linear rising Cornell potential. At infinitely large r, the asymptotic value ReVnpðˆ r→∞Þobtained from TKP model coincides with the improved KMS model. While for the BR model, the corresponding value becomes ∼σ3=4=ffiffiffiffiffiffiffi mD pwhich may indicate a different T-dependence as compared to the other two models.4 We would like to also mention that there exist some differences among these potential models when we consider the medium effect as a perturbation to the Cornell potential. In the deconfined phase where mDis very small, the potential can be expanded in term of ˆ r. As just discussed before, in the improved KMS model, the medium correction from ReVnp is −σˆ r3=ð6mDÞ, comparing with that from ReVpwhich is −αsmD ˆ r=2, we find a critical distance ∼ffiffiffiffiffiffiffiffiffiffi αs=σ pabove which the leading order medium correction is from ReVnp, while in the region where ris smaller than the critical distance, the correction from ReVp is dominated. On the contrary, the nonperturbative correction due to medium effect is ∼ˆ r2in TKP model and ∼ðrμÞ4 in BR model with μ∼ðm2 Dσ=αsÞ1=4. As a result, for mD→0, the leading order correction comes from the nonperturbative terms even at very small distances. Despite such a qualitative difference, all these medium corrections have negative contributions and none of them overshoots the vacuum potential. As for the nonperturbative terms in ImV, the improved KMS model and BR model share some common features, namely, the Coulombic HTL part dominates at small distances while at asymptotically large distances, the string contribution saturates to some constant as required. As pointed in Ref. [18], when ris small, the string term in the BR model rises according to r3which is subleading with respect to the Coulombic contribution and the corresponding distance scale rsðTÞis comparable to that in ImVII. The TKP model, on the other hand, shows some unexpected differences. In small ˆ rlimit, the leading order contribution from the nonperturbative term is proportional to ˆ r2, so the string part would contribute equally as the Coulombic term even at very small r. This is exactly the same as the previous discussed potential model in Eq. (18). As a result, ImVin this model gets an unwanted increase proportional to r2when the temperature is below Tc. Finally, we find that ImVnpðˆ r→∞Þin TKP model does not converge to some constant, instead a logarithmic divergence ∼ln ˆ rexists. Similar as what we did in Sec. III, the strong coupling constant αsand the string tension σare assumed to be T-independent and the lattice data of the real part of the potential is used to extract the only free parameter mDin the above models. The corresponding results can be found in Table II. We point out that the value of mDat a given temperature varies according to the different forms of ReV under consideration, however, the T-dependence of mDin these models looks very similar and all of them can be well described by using the previous parametrization aT þb=T. For the improved KMS model, the set of parameters are 4At relatively large distances (ˆ r≫1), the real parts of the improved KMS model and the BR model decay exponentially, however, the TKP model retains an ∼1=r behavior. MODELING THE NONPERTURBATIVE CONTRIBUTIONS TO …PHYS. REV. D 100, 036011 (2019) 036011-9