Bethe-Salpeter amplitudes of Upsilons
Rasmus Larsen, Stefan Meinel, Swagato Mukherjee, Peter Petreczky
BBethe-Salpeter amplitudes of Upsilons
Rasmus Larsen, Stefan Meinel,
2, 3
Swagato Mukherjee, and Peter Petreczky Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA. ∗ Department of Physics, University of Arizona, Tucson, Arizona, USA. RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973, USA. (Dated: August 7, 2020)Based on lattice non-relativistic QCD (NRQCD) studies we present results for Bethe-Salpeteramplitudes for Υ(1 S ), Υ(2 S ) and Υ(3 S ) in vacuum as well as in quark-gluon plasma. Our studyis based on 2 + 1 flavor 48 ×
12 lattices generated using the Highly Improved Staggered Quark(HISQ) action and with a pion mass of 161 MeV. At zero temperature the Bethe-Salpeter amplitudesfollow the expectations based on non-relativistic potential models. At non-zero temperatures theinterpretation of Bethe-Salpeter amplitudes turns out to be more nuanced, but consistent with ourprevious lattice QCD study of excited Upsilons in quark-gluon plasma.
I. INTRODUCTION
Potential models give a good description of the quarko-nium spectrum below the open charm and bottom thresh-olds, see e.g. Refs. [1, 2] for reviews. Even some ofthe states above the threshold are also reproduced wellwithin this model. Potential models can be justified us-ing an effective field theory approach [3, 4]. This ap-proach is based on the idea that for a heavy quark withmass m , there is a separation of energy scales relatedto the quark mass, inverse size of the bound state, andbinding energy, m (cid:29) mv (cid:29) mv , with v being the ve-locity of the heavy quark inside the quarkonium boundstate. The effective field theory at scale mv is the non-relativistic QCD (NRQCD), where the heavy quark andanti-quark are described by non-relativistic Pauli spinorsand pair creation is not allowed in this theory [5]. Theeffective theory at scale mv is potential NRQCD (pN-RQCD) and the quark anti-quark potential appears as aparameter of the pNRQCD Lagrangian. Potential modelappears as the tree level approximation of pNRQCD [4].Non-potential effects are manifest in the higher order cor-rections. For very large quark mass, v ∼ α s (cid:28)
1. There-fore, the large energy scales can be integrated out per-turbatively [3, 4]. However, for most of the quarkoniumstates realized in nature this condition is not fulfilled. IfΛ
QCD (cid:29) mv , all the energy scales can be integrated outnon-perturvatively and the potential is given in terms ofWilson loops calculated on the lattice [3, 4]. So in thislimit too the potential description is justified. However,for many quarkonia, Λ QCD (cid:39) mv , and it is not clearhow to justify the potential models.In potential models one can also calculate the quarko-nium wave function. On the other hand, in lattice QCDwe can calculate the Bethe-Salpeter amplitude, which inthe non-relativistic limit reduces to the wave function.Thus, one can use the Bethe-Salpeter amplitude for fur-ther tests of the potential models. In particular, one canalso reconstruct the potential from the Bethe-Salpeter ∗ [email protected] amplitude [6–10]. Most of these studies focused on quarkmasses close to or below the charm quark mass, thoughin Ref. [9] quark masses around the bottom quark havealso been considered. The resulting potential turned outto be similar to the static potential calculated on thelattice, but some differences have been found. The po-tential description is expected to work better for largerquark masses and therefore bottomonium is best suitedfor testing this approach. Studying bottomonium on thelattice using a fully relativistic action is more difficultbecause of the large cutoff effects and the rapid fall-off ofthe correlators. One of our aims is to test the potentialmodel by calculating the bottomonium Bethe-Salpeteramplitude using lattice NRQCD [11, 12], which is verywell suited for studying bottomonium [13–20].The existence and the properties of quarkonia in thehot medium attracted a lot of attention in the last 30years. It has been proposed a long time ago that quarko-nium production in heavy-ion collisions can be used toprobe quark-gluon plasma (QGP) formation [21]. Thestudy of in-medium properties of quarkonia and theirproduction in heavy ion collisions is an extensive re-search program, see e.g. Refs. [22–24] for reviews. Thein-medium properties of quarkonia as well as their dis-solution (melting) are encoded in the finite tempera-ture spectral functions. Quarkonium states show up aspeaks in the spectral function that become broader as thetemperature increases and eventually disappear abovesome temperature ( T ). The temperature above which nopeaks in the spectral function can be identified is oftencalled the melting temperature. Reconstructing quarko-nium spectral functions from lattice calculations at non-zero temperature appeared to be very challenging (seee.g. discussions in Refs. [25–28]). The study of Bethe-Salpeter amplitudes has been proposed as an alternativemethod to address this problem. The idea behind thisapproach is to compare the Bethe-Salpeter amplitude cal-culated on the lattice with the expectations of the freefield theory that would indicate an unbound heavy quarkanti-quark pair. Bethe-Salpeter amplitudes at non-zerotemperature for charmonium have been calculated in pre-vious lattice QCD studies [29–34], but presently our un- a r X i v : . [ h e p - l a t ] A ug derstanding regarding the interpretations of quarkoniaBethe-Salpeter amplitudes at T > S ), Υ(2 S )and Υ(3 S ) states at T >
0. By comparing with the cor-responding T = 0 results, where the interpretations ofBethe-Salpeter amplitudes are more straightforward, wepoint out and discuss subtleties associated with interpre-tations of Bethe-Salpeter amplitudes at T > II. BETHE-SALPETER AMPLITUDES AT T = To define the Bethe-Salpeter amplitude for bottomo-nium we consider the correlation function˜ C rα ( τ ) = (cid:68) O rqq ( τ ) ˜ O α (0) (cid:69) , (1)where ˜ O α is the meson operator that has a good overlapwith a given quarkonium state α and O rqq is a point-split meson operator with the quark and antiquark fieldsseparated by distance r , O rqq ( τ ) = (cid:88) x ¯ q ( x , τ )Γ q ( x + r , τ ) . (2)Here, Γ fixes the quantum number of the meson. Further-more, in the present work we use Coulomb gauge fixedensembles to define the expectation value. Inserting acomplete set of states we obtain the following spectraldecomposition of the correlator:˜ C rα ( τ ) = (cid:88) n (cid:104) | O rqq (0) | n (cid:105) (cid:104) n | ˜ O α (0) | (cid:105) e − E n τ . (3)Assuming that only one state | α (cid:105) contributes at large τ ,which is correct for an appropriately chosen ˜ O α , at largeEuclidean time we have˜ C rα ( τ ) = A ∗ α (cid:104) | O rqq (0) | α (cid:105) e − E α τ , (4)where A ∗ α = (cid:104) α | ˜ O α (0) | (cid:105) . The matrix element φ α ( r ) = (cid:104) | O rqq (0) | α (cid:105) (5)is called the Bethe-Salpeter (BS) amplitude and describesthe overlap of the quarkonium state | α (cid:105) with the statethat is obtained by letting the two field operators atdistance r act on the vacuum. In the non-relativisticlimit it reduces to the wave function of the given quarko-nium state. Thus, up to normalization factor the Bethe-Salpeter amplitude is given by the large τ behavior ofexp( E α τ ) C α ( τ ), with E α being the energy of quarkoniumstate | α (cid:105) , which is also calculated on the lattice. In the following we will use the terms BS amplitude and wavefunction interchangeably.As mentioned in the introduction, we aim to calcu-late the bottomonium BS amplitudes using NRQCD. Weperformed calculations using 2+1 flavor gauge configura-tions generated by HotQCD with the highly improvedstaggered quark (HISQ) action [37, 38]. The strangequark mass was fixed to its physical value, while thelight quark masses correspond to the pion mass of 161MeV in the continuum limit [37, 38]. We use the sameNRQCD formulation as in our previous study [39, 40].For the calculations at zero temperature we use 48 lat-tices and β = 10 /g = 6 .
74 corresponding to lattice spac-ing a = 0 . O i ( τ, x ) = (cid:88) r ψ i ( r )¯ q ( τ, x )Γ q ( τ, x + r ) . (6)Here, ψ i ( r ) is the trial wave function of the i th bottomo-nium state obtained by solving the Schr¨odinger equa-tion with the Cornell potential modified by discretiza-tion effects [15]. Since G ij ( τ ) = (cid:104) O i ( τ ) O j (0) (cid:105) is non-zero(though small) also for i (cid:54) = j we have to solve the gener-alized eigenvalue problem G ij ( τ )Ω jα = λ α ( τ, τ ) G ij ( τ )Ω jα (7)to obtain the optimized operator for bottomonium state α ˜ O α = (cid:88) j Ω jα O j . (8)Thus, to obtain the BS amplitude we consider the large τ behavior of the following combination: e E α τ ˜ C rα ( τ ) = e E α τ (cid:88) j Ω jα (cid:10) O rqq ( τ ) O j (0) (cid:11) . (9)In practice, the value of τ does not have to be very large.We find that τ > . τ = 0the BS amplitude will be equal to the trial wave function ψ i ( r ). To obtain the proper normalization of the BS am-plitude we require that (cid:82) ∞ drr | φ α ( r ) | = 1. In Fig. 1we show the BS amplitude φ α ( r ) for Υ(1 S ), Υ(2 S ) andΥ(3 S ) states compared to the corresponding trial wavefunctions ψ α ( r ) used to construct the optimized mesonoperators. We see that the r -dependence of the BS ampli-tudes is in qualitative agreement with the expectations ofnon-relativistic potential model. However, the details ofthe r dependence are different from the input trial wavefunction. We also note that the orthogonalization proce-dure is important for getting the correct r dependence ofthe BS amplitudes.If the potential picture is valid the BS amplitudeshould satisfy the Schr¨odinger equation (cid:18) −∇ m b + V ( r ) (cid:19) φ α = E α φ α , (10) -2-1.5-1-0.5 0 0.5 1 1.5 2 2.5 0.2 0.4 0.6 0.8 1 1.2 1.4r [fm]r φ α (r) Υ (1S) Υ (2S) Υ (3S) FIG. 1. The BS amplitudes for Υ(1 S ), Υ(2 S ) and Υ(3 S )states at T = 0 as function of r (filled symbols) comparedwith the corresponding trial wave functions (open symbols). with m b being the b-quark mass of the potential model.Note that the reduced mass in the b ¯ b system is m b / m b and the potential V ( r ). Wedetermine the b -quark mass using Υ(1 S ) and Υ(2 S )states as follows m b = ∇ φ Υ(1 S ) φ Υ(1 S ) − ∇ φ Υ(2 S ) φ Υ(2 S ) E Υ(2 S ) − E Υ(1 S ) (11)To evaluate ∇ φ α we use the simplest difference scheme.The value of m b determined from the above equation fordifferent values of quark antiquark separation r is shownin Fig. 2. The r -range was chosen such that it doesnot include the node of Υ(2 S ) and large distances, wherethe statistical errors are large. We see some modula-tion of the extracted m b in r , which may indicate thatthe BS amplitude cannot be completely captured by theSchr¨odiner equation, but there is no clear tendency of m b as function of r . Therefore we fitted the values of m b obtained for different r to a constant. This resulted in m b = 5 . ± .
33 GeV . (12)This value of the effective bottom quark mass obtained byus is not very different from the one used by the originalCornell model, m b = 5 .
17 GeV [41] but is significantlylarger than the b -quark mass used in most of the potentialmodels (see e.g. Ref. [42]). We also determined the valueof m b using the BS amplitudes and the energy levels ofΥ(1 S ) and Υ(3 S ) and obtained m b = 5 . .
51) GeV.This agrees with the above result within the errors.Having determined m b , we can also calculate the po-tential, V ( r ), using the BS amplitudes and the bottomo- m b [ G e V ] r [fm] FIG. 2. The effective bottom quark mass, m b , in the poten-tial approach determined for different quark antiquark sep-arations r (see text). The horizontal solid line is the fittedvalue of m b , while the dashed lines indicate the correspond-ing uncertainty. nium energy levels as V ( r ) = E α + ∇ φ α m b φ α . (13)The results are shown in Fig. 3. Given our findings for m b it is not surprising that the values of the potential ob-tained using Υ(1 S ), Υ(2 S ) and Υ(3 S ) states agree withinerrors. In the figure we also compare the value of V ( r )determined from the different states to the phenomeno-logical potential of the original Cornell model [41] andthe energy of static quark antiquark pair obtained fromWilson loops at lattice spacing a = 0 .
06 fm [38]. It isquite non-trivial that all these potentials agree with eachother. A similar conclusion has been reached in Refs. [7–9] when the limit of quark mass going to infinity wastaken. We note that the relativistic corrections to thespin-dependent part of the potential are quite small forthe b quark mass [43] and, thus, are not visible given ourstatistical errors.The discussion above ignored spin-dependent effects.To address the spin-dependent part of the potential wealso calculated the BS amplitude for η b ( nS ) states, n =1 , ,
3. We have found that the corresponding BS am-plitudes agree with the ones of the Υ( nS ) states withinerrors. Therefore, with the present statistics we cannotresolve the spin-dependent part of the potential.As discussed above, the r -dependence of the BS am-plitudes qualitatively follow the r -dependence of the trialwave function ψ i ( r ) obtained from the potential model.But at qualitative level significant differences can be seen,c.f. Fig. 1. This potential model used m b = 4 . nS ) bottomonium states using the staticquark anti-quark energy [38] as a potential and m b = 6GeV. The results are shown in Fig. 4 and we see thatthe agreement between the BS amplitude and the wave -1-0.5 0 0.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 V (r) [ G e V ] r [fm] Cornell modelWloops, a=0.06 fm Υ (1S) Υ (2S) Υ (3S) FIG. 3. The potential, V ( r ) obtained from the BS ampli-tude of Υ(1 S ), Υ(2 S ) and Υ(3 S ) states compared to the phe-nomenological Cornell potential [41] shown as solid line aswell as to the the energy of the static quark antiquark pairobtained from Wilson loops using a = 0 .
06 fm lattice [38].All the lattice results were normalized to coincide with theCornell potential at r = 0 . -2-1.5-1-0.5 0 0.5 1 1.5 2 2.5 0.2 0.4 0.6 0.8 1 1.2 1.4r [fm]r φ α (r) Υ (1S) Υ (2S) Υ (3S) FIG. 4. The BS amplitude for Υ( nS ) states as function of r (filled symbols) compared with the non-relativistic wave func-tions obtained from potential model with m b = 6 GeV (opensymbols). functions is significantly improved. We also note thatthe dependence of the energy levels on m b is rather mild,e.g. changing m b from 4 GeV to 6 GeV only reduces thespin-averaged 2S-1S splitting by 3 . m b in the potential model is a viable option. III. BETHE-SALPETER AMPLITUDES AT T > We can also consider the mixed correlator ˜ C rα ( τ, T )defined in Eq. (1) for T > T = 1 /β ˜ C rα ( τ, T ) = 1 Z ( β ) Tr (cid:104) O rqq ( τ ) ˜ O α (0) e − βH (cid:105) , (14)with the thermal partition function Z ( β ) = Tr (cid:2) e − βH (cid:3) .Using energy eigenstates to evaluate the trace and in-serting a complete set of states we obtain the followingexpression for the correlator ˜ C rα ( τ, T )˜ C rα ( τ, T ) =1 Z ( β ) (cid:88) n,m e − ( E n − E m ) τ (cid:104) m | O rqq | n (cid:105) (cid:104) n | ˜ O α | m (cid:105) e − βE m . (15)Since we perform calculations in NRQCD the sum over m should be restricted to states that do no contain theheavy quark anti-quark pair; heavy quark pair creationis not allowed in NRQCD. We denote those states as | m (cid:48) (cid:105) . If we write the states | n (cid:105) as | n (cid:48) γ (cid:105) , where index n (cid:48) labels the light degrees of freedom and γ labels thequarkonium states, the above expression for ˜ C rα ( τ, T ) canbe re-written as˜ C rα ( τ, T ) = 1 Z ( β ) (cid:88) γ,n (cid:48) ,m (cid:48) (cid:104) e − ( E n (cid:48) ,γ − E (cid:48) m ) τ e − βE m (cid:48) (cid:104) m (cid:48) | O rqq | n (cid:48) γ (cid:105) (cid:104) n (cid:48) γ | ˜ O α | m (cid:48) (cid:105) (cid:105) . (16)If we write E m (cid:48) γ = E γ + E m (cid:48) + ∆ E m (cid:48) γ and assume thatthe operator ˜ O α mostly projects on to quarkonium state | α (cid:105) we can obtain a simplified form˜ C rα ( τ, T ) = e − E α τ (cid:20) φ α A ∗ (cid:48) α + 1 Z ( β ) × (cid:88) m (cid:48) (cid:104) m (cid:48) | O rqq | m (cid:48) α (cid:105) (cid:104) m (cid:48) α | ˜ O α | m (cid:48) (cid:105) e − βE m (cid:48) − ∆ E m (cid:48) α τ (cid:35) (17)with A ∗ (cid:48) α = A ∗ α /Z ( β ). In the above equation we separatedout the the m (cid:48) = 0 vacuum contribution in the sum cor-responding to the thermal trace. At small temperaturethe first term in the above equation is the dominant oneand the correlator is approximately given by the T = 0BS amplitude. In general, however, there is no simple in-terpretation of the correlator ˜ C rα ( τ, T ) in terms of somefinite temperature quarkonium wave function. The tem-perature dependence of this correlator crucially dependson the value of the matrix elements (cid:104) m (cid:48) | O rqq | m (cid:48) α (cid:105) and (cid:104) m (cid:48) α | ˜ O α | m (cid:48) (cid:105) . The size of (cid:104) m (cid:48) | O rqq | m (cid:48) α (cid:105) depends on theseparation r and, therefore, also the size of the thermaleffect will be r dependent. For values of r that are aboutthe size of the bottomonium state of interest the ma-trix elements (cid:104) m (cid:48) | O rqq | m (cid:48) α (cid:105) and (cid:104) m (cid:48) α | ˜ O α | m (cid:48) (cid:105) should beof similar size and, thus, the temperature dependence of˜ C rα ( τ, T ) is expected to be comparable to the correlatorof ˜ O α explored in Ref. [40].We performed calculations of ˜ C rα at six different tem-peratures using 48 ×
12 lattices from HotQCD collabo-ration. The parameters of the calculations including the
FIG. 5. The effective masses M r eff ( τ, T ) in GeV of the Υ(1 S ) correlator at T = 151 MeV (left) and T = 334 MeV (right) asfunction of τ and r . β T [MeV] T > ×
12 lattices. gauge coupling β = 10 /g and number of configurationsare summarized in Table I. As at zero temperature, weused 8 sources per gauge configuration.We could use the same approach as in Ref. [40] toexplore the temperature dependence of the correlator˜ C rα ( τ, T ) and define the effective mass for a fixed raM r eff ( τ, T ) = ln (cid:32) ˜ C rα ( τ, T )˜ C rα ( τ + a, T )) (cid:33) . (18)Now the effective mass also depends on the distance r between the quark and antiquark in the point-split cur-rent. In Fig. 5 we show the effective mass of Υ(1 S )correlator as function of r and τ at the lowest and thehighest temperature. The errors of the effective massesare not shown to improve the visibility. Since the energylevels in NRQCD are only defined up to a lattice spacingdependent constant, as in Ref. [39] we calibrate the ef-fective masses with respect to the energy level of η b (1 S )state at zero temperature. At large τ and r the errorsare quite large and within these errors we do no see anymedium effects in the effective mass at the lowest tem-perature. For small r the effective mass quickly reaches aplateau with increasing τ . For large r the effective massat 151 MeV reaches the plateau from below. At the high-est temperature, T = 335 MeV the r and τ dependenceof the effective masses looks similar for not too large val-ues of r . However, the behavior of the effective mass isqualitatively different for large r . In particular the ef-fective mass does not reach a plateau with increasing τ .For excited states the results for M r eff ( τ, T ) look similar, except that the errors are very large for r > .
65 fm.As an example we show the effective mass for Υ(3 S ) inFig. 6 at two values of r , r = 0 .
25 fm and r = 0 .
65 fmfor different temperatures. For the smaller r we see notemperature dependence of the Υ(3 S ) effective mass at T = 172 MeV and T = 251 MeV. This is likely due tothe fact that the matrix elements (cid:104) m (cid:48) | O rqq | m (cid:48) Υ(3 S ) (cid:105) aresmall for r = 0 .
25 fm and the first term in Eq. (17) dom-inates. Note, however, that the errors are large. For thehighest temperature, T = 334 MeV we start to see signif-icant temperature dependence. For the larger distance, r = 0 .
65 fm the medium effects are more pronounced.While the modifications of M r eff are small for T = 172MeV, thermal effects are significant for T = 251 MeVand 334 MeV, comparable in size to the thermal effectsin the effective masses of correlators of optimized opera-tors [40].Since the correlator ˜ C rα does not correspond to a pos-itive definite spectral function it is difficult to infer in-medium properties of bottomonia from M r eff . The largestatistical errors make this even more complicated. An-other way to analyze the temperature dependence of ˜ C rα is to consider the integral N α ( τ, T ) = (cid:90) ∞ drr (cid:16) ˜ C rα (cid:17) . (19)At zero temperature this quantity should be proportionalto exp( − E α τ ) for sufficiently large τ . This is also ex-pected to be true below the crossover temperature. Thecombination N norm ( τ, T ) = exp(2 E α τ ) N α ( τ, T ) (20)should be independent of τ and can be interpreted as thenormalization of the BS amplitude. In Fig. 7 we show N norm ( τ, T ) as function of τ for different temperaturesnormalized to one at t = τ /a = 3. For the lowest tem-perature as well as for T = 0 we see that N norm ( τ, T ) isapproximately constant as expected. Here we note thatthe τ range in Fig. 7 is different for Υ(1 S ), Υ(2 S ) andΥ(3 S ) states. This is due to the fact that the correlators C r Υ(2 S ) and C r Υ(3 S ) will be contaminated by the lowest -1-0.5 0 0.5 1 1.5 0.1 0.2 0.3 0.4 0.5 τ [fm]M [GeV] T=172 MeVT=251 MeVT=334 MeV -1-0.5 0 0.5 1 1.5 0.1 0.2 0.3 0.4 0.5 τ [fm]M [GeV] T=172 MeVT=251 MeVT=334 MeV FIG. 6. The effective masses M r eff ( τ, T ) in GeV of the Υ(3 S ) correlator for r = 0 .
25 fm (left) and r = 0 .
65 fm (right) at differenttemperatures as function of τ . Υ(1 S ) state at large τ as the projection is not perfectdue to the small operator basis of only three operatorsused in this study. As the temperature increases we seethat N norm ( τ, T ) no longer approaches a constant but in-creases at large τ . This implies that the correlator ˜ C rα isno longer dominated by the first term in Eq. (17). The τ -dependence of N norm ( τ, T ) is larger for high tempera-tures and is also more pronounced for excited states, asexpected.We could also analyze the τ -dependence of N α ( τ, T ) interms of the corresponding effective masses aM N α eff ( τ, T ) = ln (cid:18) N α ( τ, T ) N α ( τ + a, T ) (cid:19) . (21)At large τ these effective masses should reach a plateauequal to 2 E α . Our results for M N α eff for the differentΥ( nS ) states are shown in Fig. 8. As before the ef-fective masses have been calibrated with respect to theenergy of η b state at T = 0. We see that at T = 0 as wellas at the lowest temperature the effective masses reach aplateau corresponding to the physical mass (energy), butat higher temperatures this is not the case, in general.For the ground state the errors are large enough so thatno clear medium shift can be seen, except at the highesttemperature, T = 334 MeV. For the Υ(2 S ) the corre-sponding effective masses decrease with increasing τ for T ≥
251 MeV. For the Υ(3 S ) we see a significant shiftin M N α eff ( τ, T ) already for T >
191 MeV. The behavior of M N α eff ( τ, T ) is qualitatively similar to the behavior of theeffective masses of the correlator of optimized operatorsstudied in Ref. [40]. This corroborates the findings ofRef. [40] on the in-medium modifications of the bottomo-nium spectral functions. For the Υ(1 S ) state our findingsare also consistent with other studies of bottomonium atnon-zero temperature using NRQCD [44–49].Before concluding this section we mention that so farwe only discussed Υ( nS ) states but very similar resultshave been obtained for η b ( nS ) states as well. IV. COMPARISONS BETWEEN T > = If we insist on the interpretation of the correlator˜ C rα ( τ, T ) in terms of the wave function we could sim-ply divide it by N α ( τ, T ) and study the r -dependenceof the corresponding ratio for sufficiently large τ . Atsmall temperatures this ratio will have an r -dependencethat closely follows the r -dependence of the BS ampli-tude at T = 0. In Fig. 9 we compare φ α ( τ, T ) =˜ C rα ( τ, T ) /N α ( τ, T ) for the lowest temperature, T = 151MeV and τ = 0 .
65 fm with the corresponding zero tem-perature BS amplitudes. For the Υ(1 S ) and the Υ(2 S )we do not see any difference between the zero tempera-ture BS amplitude and φ α ( τ, T ). For the Υ(3 S ) some dif-ference between the zero temperature and finite tempera-ture result for φ α ( τ, T ) can be seen at large r , though it isnot statistically significant. In any case the r -dependenceof φ α at T = 0 and T = 151 MeV is quite similar evenfor the Υ(3 S ). The lack of medium effects in the BS am-plitude for T = 151 MeV is not surprising since at thistemperature all bottomonia should exist as well definedstates. Next, we compare φ α ( τ, T ) at the lowest andthe highest temperature at τ = 0 . τ value the contribution of the second term inEq. (17) is too small. Therefore, in Fig. 11 we show ourresults for φ α ( τ, T ) at T = 251 MeV and several val-ues of τ . As one can see from the figure for the Υ(2 S )and Υ(3 S ) there is a significant τ -dependence of φ α forlarge r . This suggests that the normalized BS amplitudecannot be interpreted simply as the wave function of in-medium Υ in potential model picture. Yet, the r depen-dence of φ α ( τ, T ) does not change much from one τ valueto another. In summary, the correlation ˜ C rα ( τ, T ) showssignificant temperature dependence as one would expectbased on the previous studies. However, the r depen- τ [fm]N norm T=0T=151 MeVT=172 MeVT=199 MeVT=251 MeVT=334 MeV 0.5 1 1.5 2 2.5 3 3.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 τ [fm]N norm T=0T=151 MeVT=172 MeVT=199 MeVT=251 MeVT=334 MeV τ [fm]N norm T=0T=151 MeVT=172 MeVT=199 MeVT=251 MeVT=334 MeV
FIG. 7. Norm of the squared BS wave function at differenttemperatures for the Υ(1 S ) (Top), Υ(2 S ) (Middle) and Υ(3 S )(Bottom) states. dence of this correlator does not change significantly asthe temperature and τ is varied. Thus, focusing only onthe r dependence of ˜ C rα ( τ, T ) without a detailed study ofits τ dependence may result in wrong conclusions aboutthe fate of Υ(2 S ) and Υ(3 S ) states at high temperature.For the Υ(1 S ) there is only little dependence of φ α on -0.1 0 0.1 0.2 0.3 0.4 0.2 0.4 0.6 0.8 1 1.2 τ [fm]M N Υ (1S)eff T=0T=151 MeVT=172 MeVT=199 MeVT=251 MeVT=334 MeV-0.5 0 0.5 1 1.5 2 2.5 3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 τ [fm]M N Υ (2S)eff T=0T=151 MeVT=172 MeVT=199 MeVT=251 MeVT=334 MeV -1 0 1 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 τ [fm]M N Υ (3S)eff T=0T=151 MeVT=172 MeVT=199 MeVT=251 MeVT=334 MeV
FIG. 8. Effective mass M N α eff in GeV at different tempera-tures for Υ(1 S ) (Top), Υ(2 S ) (Middle) and Υ(3 S ) (Bottom)correlators. τ and therefore, in Fig. 11 we only show the numericalresults for τ = 0 . τ -dependence indi-cates that Υ(1 S ) can exist in the deconfined medium at T = 251 MeV as a well defined state with little mediummodification, in agreement with the previous studies ofbottomonium at non-zero temperature based on NRQCD[44–49].The lack of temperature dependence of the normalizedBS amplitude at T > τ = 0 . φ α ( r, T ) can be used as a proxy for the T = 0 BS am- -2-1.5-1-0.5 0 0.5 1 1.5 2 2.5 0.2 0.4 0.6 0.8 1 1.2 1.4r [fm]r φ α (r) Υ (1S) Υ (2S) Υ (3S) FIG. 9. The BS amplitudes times r for the Υ(1 S ) , Υ(2 S )and Υ(3 S ) at T = 0 MeV (filled symbols) and T = 151 MeV(open symbols) for τ = 0 .
65 fm. -2-1.5-1-0.5 0 0.5 1 1.5 2 2.5 0.2 0.4 0.6 0.8 1 1.2 1.4r [fm]r φ α (r) Υ (1S) Υ (2S) Υ (3S) FIG. 10. BS amplitude times r for the Υ(1 S ) , Υ(2 S ) andΥ(3 S ) at T = 334 MeV (filled symbols) and T = 151 MeV(open symbols) at τ = 0 . plitude at zero temperature. Since the two temperaturesshown in Fig. 10 correspond to two different lattice spac-ings this result also implies that the lattice spacing de-pendence of the BS amplitude is small. Therefore, thecomparison of the wave function obtained from potentialmodel and BS amplitude obtained on the lattice with a = 0 . V. CONCLUSIONS
Using lattice NRQCD in this paper we studied thecorrelation functions, ˜ C rα , between operators optimizedto have good overlaps with the of Υ(1 S ), Υ(2 S ) andΥ(3 S ) vacuum wave functions and simple spatially non-local bottomonium operators, where the bottom quark -2-1.5-1-0.5 0 0.5 1 1.5 2 2.5 0.2 0.4 0.6 0.8 1 1.2 1.4r [fm]r φ α (r) τ =0.2 fm τ =0.4 fm τ =0.6 fm FIG. 11. The BS amplitude times r for the Υ(2 S ) (filledsymbols) and Υ(3 S ) (open symbols) at T = 251 MeV for τ = 0 . , .
40 and 0 .
65 fm. Also shown as crosses is theresult for the Υ(1 S ). and anti-quark are separated by distance r . This cor-relator has been calculated at zero as well as at non-zero temperature. At zero temperature ˜ C rα can be in-terpreted in terms of the Bethe-Salpeter amplitude. Wehave found that the r -dependence of the Bethe-Salpeteramplitude closely resembles the corresponding potentialmodel based bottomonium wave function. Moreover, bychoosing the bottom quark mass used in the Schr¨odingerequation to be ∼ . C rα in terms of effective masses. For Υ(1 S )we see only very small temperature and Euclidean timedependence of the corresponding effective masses, ex-cept at the highest temperature of 334 MeV. For Υ(2 S )and especially for Υ(3 S ) significant dependence on theEuclidean time were observed, making it difficult todraw parallels between Bethe-Salpeter amplitudes andpotential model based in-medium wave functions. Sincethe r -dependence changes very little with varying Eu-clidean time and temperature, focusing solely on the r -dependence of ˜ C rα at a fixed τ might lead to misleadingconclusions regarding existence of well-defined Υ(2 S ) andΥ(3 S ) in medium. On the other hand, we found thatthe behavior of the effective masses is similar to the onepreviously studied by us using correlators of optimizedbottomonium operators [40], supporting the picture ofthermal broadening of bottomonium states. ACKNOWLEDGMENTS
This material is based upon work supported by: (i)The U.S. Department of Energy, Office of Science, Officeof Nuclear Physics and High Energy Physics through theContract No. DE-SC0012704; (ii) The U.S. Departmentof Energy, Office of Science, Office of Nuclear Physics andOffice of Advanced Scientific Computing Research withinthe framework of Scientific Discovery through AdvanceComputing (ScIDAC) award Computing the Propertiesof Matter with Leadership Computing Resources. (iii) S. Meinel acknowledges support by the U.S. Department ofEnergy, Office of Science, Office of High Energy Physicsunder Award Number DE-SC0009913. (iv) Computa-tions for this work were carried out in part on facilities ofthe USQCD Collaboration, which are funded by the Of-fice of Science of the U.S. Department of Energy. (v) Thisresearch used awards of computer time provided by theINCITE and ALCC programs at Oak Ridge LeadershipComputing Facility, a DOE Office of Science User Facilityoperated under Contract No. DE-AC05- 00OR22725. [1] N. Brambilla et al. (Quarkonium Working Group),(2004), arXiv:hep-ph/0412158 [hep-ph].[2] N. Brambilla et al. , Eur. Phys. J.
C71 , 1534 (2011),arXiv:1010.5827 [hep-ph].[3] N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Nucl.Phys.
B566 , 275 (2000), arXiv:hep-ph/9907240 [hep-ph].[4] N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Rev.Mod. Phys. , 1423 (2005), arXiv:hep-ph/0410047 [hep-ph].[5] W. Caswell and G. Lepage, Phys. Lett. B , 437(1986).[6] Y. Ikeda and H. Iida, Prog. Theor. Phys. , 941 (2012),arXiv:1102.2097 [hep-lat].[7] T. Kawanai and S. Sasaki, Phys. Rev. Lett. , 091601(2011), arXiv:1102.3246 [hep-lat].[8] T. Kawanai and S. Sasaki, Phys. Rev. D85 , 091503(2012), arXiv:1110.0888 [hep-lat].[9] T. Kawanai and S. Sasaki, Phys. Rev.
D89 , 054507(2014), arXiv:1311.1253 [hep-lat].[10] K. Nochi, T. Kawanai, and S. Sasaki, Phys. Rev.
D94 ,114514 (2016), arXiv:1608.02340 [hep-lat].[11] G. P. Lepage, L. Magnea, C. Nakhleh, U. Magnea, andK. Hornbostel, Phys. Rev.
D46 , 4052 (1992), arXiv:hep-lat/9205007 [hep-lat].[12] B. A. Thacker and G. P. Lepage, Phys. Rev.
D43 , 196(1991).[13] C. T. H. Davies, K. Hornbostel, A. Langnau, G. P. Lep-age, A. Lidsey, J. Shigemitsu, and J. H. Sloan, Phys.Rev.
D50 , 6963 (1994), arXiv:hep-lat/9406017 [hep-lat].[14] S. Meinel, Phys. Rev.
D79 , 094501 (2009),arXiv:0903.3224 [hep-lat].[15] S. Meinel, Phys. Rev.
D82 , 114502 (2010),arXiv:1007.3966 [hep-lat].[16] T. C. Hammant, A. G. Hart, G. M. von Hip-pel, R. R. Horgan, and C. J. Monahan, Phys.Rev. Lett. , 112002 (2011), [Erratum: Phys. Rev.Lett.115,039901(2015)], arXiv:1105.5309 [hep-lat].[17] R. J. Dowdall et al. (HPQCD), Phys. Rev.
D85 , 054509(2012), arXiv:1110.6887 [hep-lat].[18] J. O. Daldrop, C. T. H. Davies, and R. J. Dow-dall (HPQCD), Phys. Rev. Lett. , 102003 (2012),arXiv:1112.2590 [hep-lat].[19] R. Lewis and R. M. Woloshyn, Phys. Rev.
D85 , 114509(2012), arXiv:1204.4675 [hep-lat].[20] M. Wurtz, R. Lewis, and R. M. Woloshyn, Phys. Rev.
D92 , 054504 (2015), arXiv:1505.04410 [hep-lat].[21] T. Matsui and H. Satz, Phys. Lett.
B178 , 416 (1986). [22] G. Aarts et al. , Lorentz workshop: Tomography of theQuark-Gluon Plasma with Heavy Quarks Leiden, Nether-lands, October 10-14, 2016 , Eur. Phys. J.
A53 , 93 (2017),arXiv:1612.08032 [nucl-th].[23] A. Mocsy, P. Petreczky, and M. Strickland, Int. J. Mod.Phys.
A28 , 1340012 (2013), arXiv:1302.2180 [hep-ph].[24] A. Bazavov, P. Petreczky, and A. Velytsky, “Quarko-nium at Finite Temperature,” in
Quark-gluon plasma 4 ,edited by R. C. Hwa and X.-N. Wang (2010) pp. 61–110,arXiv:0904.1748 [hep-ph].[25] I. Wetzorke, F. Karsch, E. Laermann, P. Petreczky, andS. Stickan, Nucl. Phys. Proc. Suppl. , 510 (2002),arXiv:hep-lat/0110132 [hep-lat].[26] S. Datta, F. Karsch, P. Petreczky, and I. Wetzorke, Phys.Rev.
D69 , 094507 (2004), arXiv:hep-lat/0312037 [hep-lat].[27] A. Jakovac, P. Petreczky, K. Petrov, and A. Velytsky,Phys. Rev.
D75 , 014506 (2007), arXiv:hep-lat/0611017[hep-lat].[28] A. Mocsy and P. Petreczky, Phys. Rev.
D77 , 014501(2008), arXiv:0705.2559 [hep-ph].[29] T. Umeda, R. Katayama, O. Miyamura, and H. Mat-sufuru, Int. J. Mod. Phys.
A16 , 2215 (2001), arXiv:hep-lat/0011085 [hep-lat].[30] H. Ohno, T. Umeda, and K. Kanaya (WHOT-QCD),
Proceedings, 26th International Symposium onLattice field theory (Lattice 2008): Williamsburg, USA,July 14-19, 2008 , PoS
LATTICE2008 , 203 (2008),arXiv:0810.3066 [hep-lat].[31] T. Umeda, S. Ejiri, S. Aoki, T. Hatsuda, K. Kanaya,Y. Maezawa, and H. Ohno,
Proceedings, 26th In-ternational Symposium on Lattice field theory (Lattice2008): Williamsburg, USA, July 14-19, 2008 , PoS
LAT-TICE2008 , 174 (2008), arXiv:0810.1570 [hep-lat].[32] H. Ohno, S. Aoki, S. Ejiri, K. Kanaya, Y. Maezawa,H. Saito, and T. Umeda (WHOT-QCD), Phys. Rev.
D84 , 094504 (2011), arXiv:1104.3384 [hep-lat].[33] P. W. M. Evans, C. R. Allton, and J. I. Skullerud, Phys.Rev.
D89 , 071502 (2014), arXiv:1303.5331 [hep-lat].[34] C. Allton, W. Evans, P. Giudice, and J.-I. Skullerud,(2015), arXiv:1505.06616 [hep-lat].[35] M. Laine, O. Philipsen, P. Romatschke, and M. Tassler,JHEP , 054 (2007), arXiv:hep-ph/0611300 [hep-ph].[36] N. Brambilla, J. Ghiglieri, A. Vairo, and P. Petreczky,Phys. Rev. D78 , 014017 (2008), arXiv:0804.0993 [hep-ph]. [37] A. Bazavov et al. , Phys. Rev. D85 , 054503 (2012),arXiv:1111.1710 [hep-lat].[38] A. Bazavov et al. (HotQCD), Phys. Rev.
D90 , 094503(2014), arXiv:1407.6387 [hep-lat].[39] R. Larsen, S. Meinel, S. Mukherjee, and P. Petreczky,Phys. Rev.
D100 , 074506 (2019), arXiv:1908.08437 [hep-lat].[40] R. Larsen, S. Meinel, S. Mukherjee, and P. Petreczky,Phys. Lett.
B800 , 135119 (2020), arXiv:1910.07374 [hep-lat].[41] E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, andT.-M. Yan, Phys. Rev.
D21 , 203 (1980).[42] S. Jacobs, M. G. Olsson, and C. Suchyta, III,Phys. Rev.
D33 , 3338 (1986), [Erratum: Phys.Rev.D34,3536(1986)]. [43] G. S. Bali, K. Schilling, and A. Wachter, Phys. Rev. D , 2566 (1997), arXiv:hep-lat/9703019.[44] G. Aarts, S. Kim, M. P. Lombardo, M. B. Oktay, S. M.Ryan, D. K. Sinclair, and J. I. Skullerud, Phys. Rev.Lett. , 061602 (2011), arXiv:1010.3725 [hep-lat].[45] G. Aarts, C. Allton, S. Kim, M. P. Lombardo, M. B.Oktay, S. M. Ryan, D. K. Sinclair, and J. I. Skullerud,JHEP , 103 (2011), arXiv:1109.4496 [hep-lat].[46] G. Aarts, C. Allton, S. Kim, M. P. Lombardo, M. B.Oktay, S. M. Ryan, D. K. Sinclair, and J.-I. Skullerud,JHEP , 084 (2013), arXiv:1210.2903 [hep-lat].[47] G. Aarts, C. Allton, T. Harris, S. Kim, M. P. Lombardo,S. M. Ryan, and J.-I. Skullerud, JHEP , 097 (2014),arXiv:1402.6210 [hep-lat].[48] S. Kim, P. Petreczky, and A. Rothkopf, Phys. Rev. D91 ,054511 (2015), arXiv:1409.3630 [hep-lat].[49] S. Kim, P. Petreczky, and A. Rothkopf, JHEP11