Charmonium-nucleon potential from lattice QCD
TTKYNT-10-14
Charmonium-nucleon potential from lattice QCD
Taichi Kawanai ∗ and Shoichi Sasaki † Department of Physics, The University of Tokyo,Hongo 7-3-1, Tokyo 113-0033, Japan (Dated: November 5, 2018)We present results for the charmonium-nucleon potential V c ¯ c - N ( r ) from quenched lattice QCD,which is calculated from the equal-time Bethe-Salpeter amplitude through the effective Schr¨odingerequation. Detailed information of the low-energy interaction between the nucleon and charmonia( η c and J/ψ ) is indispensable for exploring the formation of charmonium bound to nuclei. Oursimulations are performed at a lattice cutoff of 1 /a ≈ . withthe nonperturbatively O ( a ) improved Wilson action for the light quarks and a relativistic heavyquark action for the charm quark. We have found that the potential V c ¯ c - N ( r ) for either the η c and J/ψ states is weakly attractive at short distances and exponentially screened at large distances. Thespin averaged
J/ψ - N potential is slightly more attractive than that of the η c - N case. Heavy quarkonium states such as charmonium ( c ¯ c )states do not share the same quark flavor with the nu-cleon ( N ). This suggests that the heavy quarkonium-nucleon interaction is mainly induced by the genuineQCD effect of multi-gluon exchange [1–3]. Therefore the c ¯ c - N system is ideal to study the effect of multi-gluonexchange between hadrons. As an analog of the van derWaals force, the simple two-gluon exchange contributiongives a weakly attractive, but long-ranged interaction be-tween the heavy quarkonium state and the nucleon [4, 5].However, the validity of calculations based on a pertur-bative theory is questionable for QCD where the stronginteraction influences the long distance region.The c ¯ c - N scattering at low energy has been studiedfrom first principles of QCD. The s -wave J/ψ - N scatter-ing length is about 0.1 fm by using QCD sum rules [6]and 0 . ± .
48 fm (0 . ± .
66 fm for η c - N ) by latticeQCD [7], while it is estimated as large as 0.25 fm from thegluonic van der Waals interaction [2]. All studies suggestthat the c ¯ c - N interaction is weakly attractive. This indi-cates that the formation of charmonium bound to nucleiis enhanced. In 1991, Brodsky et al. had argued thatthe c ¯ c -nucleus ( A ) bound system may be realized for themass number A ≥ c ¯ c - N potential V c ¯ c - N ( r )is indispensable for exploring nuclear-bound charmoniumstates like the η c - He or
J/ψ - He bound state in few bodycalculations [9].We recall the recent great success of the N - N potentialfrom lattice QCD [10]. In this new approach, the poten-tial between hadrons can be calculated from the equal-time Bethe-Salpeter (BS) amplitude through an effectiveSchr¨odinger equation. Thus, direct measurement of the c ¯ c - N potential is now feasible by using lattice QCD. Itshould be important to give a firm theoretical predictionabout the nuclear-bound charmonium, which is possiblyinvestigated by experiments at J-PARC and FAIR/GSI.The method utilized here to calculate the hadron-hadron potential in lattice QCD is based on the same idea originally applied for the N - N potential [10, 11].We first calculate the equal-time BS amplitude of twolocal operators (hadrons h and h ) separated by givenspatial distances ( r = | r | ) from the following four-pointcorrelation function: G h - h ( r , t ; t , t )= (cid:88) x (cid:88) x (cid:48) , y (cid:48) (cid:104) h ( x , t ) h ( x + r , t ) ( h ( x (cid:48) , t ) h ( y (cid:48) , t )) † (cid:105) , where r is the relative coordinate of two hadrons at sinkposition ( t ). Each hadron operator at source positions( t and t ) is separately projected onto a zero-momentumstate by a summation over all spatial coordinates x (cid:48) and y (cid:48) . To avoid the Fierz rearrangement of two-hadron op-erators, it is better to set t (cid:54) = t . Without loss of gen-erality, we choose t = t + 1 = t src hereafter. Supposethat | t − t src | (cid:29) G h - h ( r , t ; t src ) ∝ φ h - h ( r ) e − E h h ( t − t src ) , (1)where the r -dependent amplitude, which is defined as φ h - h ( r ) = (cid:88) x (cid:104) | h ( x ) h ( x + r ) | h h ; E h - h (cid:105) (2)with the total energy E h - h for the ground state of thetwo-particle h - h state, corresponds to a part of theequal-time BS amplitude and is called the BS wave func-tion [12, 13]. After an appropriate projection with re-spect to discrete rotation φ A +1 h - h ( r ) = 124 (cid:88) R∈ O h φ h - h ( R − r ) , (3)where R represents 24 elements of the cubic group O h ,one can get the BS wave function projected in the A +1 representation, which corresponds to the s -wave in con-tinuum theory at low energy. Once the BS wave func-tions φ A +1 h - h ( r ) are calculated in lattice simulations, the a r X i v : . [ h e p - l a t ] O c t TABLE I: Simulation parameters in this study. The Sommerparameter r = 0 . a a − Lattice size ∼ Laβ [fm] [GeV] ( L × T ) [fm] Statistics6.0 0.0931 2.12 16 ×
48 1.5 20032 ×
48 3.0 602 hadron-hadron “effective” central potential with the en-ergy eigenvalue E of the stationary Schr¨odinger equation,can be obtained by V h - h ( r ) − E = 12 µ ∇ φ A +1 h - h ( r ) φ A +1 h - h ( r ) , (4)where µ is the reduced mass of the h - h system and ∇ isdefined by the discrete Laplacian with nearest-neighborpoints. Although the energy eigenvalue E is supposedto be the energy difference between the total energy oftwo hadrons ( E h - h ) and the sum of the rest mass ofindividual hadrons ( M h + M h ), we instead determine E with the condition of lim r →∞ { V h - h ( r ) − E } = 0 [13].More details of this method can be found in Ref. [11].Let us first consider the low energy η c - N interac-tion, which does not possess a spin-dependent part. Weuse the conventional interpolating operators, h ( x ) = (cid:15) abc ( u a ( x ) Cγ d b ( x )) u c ( x ) for the nucleon and h ( y ) =¯ c a ( y ) γ c a ( y ) for the η c state, where a , b and c are colorindices, and C = γ γ is the charge conjugation matrix.We have performed quenched lattice QCD simulations ontwo different lattice sizes, L × T = 32 ×
48 and 16 × β = 6 /g = 6 . a − ≈ . La ≈ La ≈ . O (600) for L = 32 and O (200) for L = 16, respectively. The simulation parame-ters and the number of sampled gauge configurations aresummarized in Table I.We use non-perturbatively O ( a ) improved Wilsonfermions for light quarks ( q ) [16] and a relativistic heavyquark (RHQ) action for the charm quark ( Q ) [17].The RHQ action is a variant of the Fermilab approach[18], which can remove large discretization errors forheavy quarks. The hopping parameter is chosen to be κ q = 0 . , . , . M π =0 . , . , .
87 GeV ( M N = 1 . , . , .
70 GeV), and κ Q = 0 . M η c = 2 .
92 GeV and M J/ψ = 3 .
00 GeV) [19]. Eachhadron mass is obtained by fitting the correspondingtwo-point correlation function with a single exponentialform. We calculate quark propagators with wall sources,which are located at t src = 5 for the light quarks and at t src = 4 for the charm quark, with Coulomb gaugefixing. It is worth mentioning that Dirichlet boundaryconditions are imposed for quarks in the time directionin order to avoid wrap-round effects, which are very cum-bersome in systems of more than two hadrons [20]. Inaddition, the ground state dominance in four-point func-tions is checked by an effective mass plot of total energiesof the c ¯ c - N system.The left panel of Fig. 1 shows a typical result of theprojected BS wave function at the smallest quark mass,which is evaluated by a weighted average of data inthe time-slice range of 16 ≤ t − t src ≤
35. The wavefunctions are normalized to unity at a reference point r = (16 , , η c - N interaction is certainly attractive. Thisattractive interaction, however, is not strong enough toform a bound state as is evident from this figure, wherethe wave function is not localized, but extends to largedistances.In the right panel of Fig. 1, we show the effective cen-tral η c - N potential, which is evaluated by the wave func-tion through Eq. (4) with measured values of E and µ .As is expected, the η c - N potential clearly exhibits an en-tirely attractive interaction between the charmonium andthe nucleon without any repulsion at either short or largedistances. The short range attraction is deemed to be aresult of the absence of Pauli blocking, that is a relevantfeature in this particular system of the heavy quarkoniumand the light hadron. The interaction is exponentiallyscreened in the long distance region r (cid:38) − exp( − r m ) /r n , which in-clude the Yukawa form ( m = 1 and n = 1), and ii) in-verse power law functions − /r n , where n and m are notrestricted to be integers. The former case can easily ac-commodate a good fit with a small χ /ndf value, whilein the latter case we cannot get any reasonable fit. Forexample, the functional forms − exp( − r ) /r and − /r give χ / ndf (cid:39) . . − γe − αr /r to fit our dataof V c ¯ c - N ( r ), we obtain γ ∼ . α ∼ . c ¯ c - N potential adopted in Refs. [1], where the parameters( γ = 0 . α = 0 . W a v e f un c t i on (cid:113) (r) r [fm] -60-50-40-30-20-10 0 10 0 0.5 1 1.5 2 2.5 V (r) [ M e V ] r [fm] lattice dataYukawa fitModel potential FIG. 1: The wave function (left) and the effective central potential (right) in the s -wave η c - N system for M π = 0 .
64 GeV asa typical example. In the right panel we fit a Yukawa potential (solid line) and compare with the phenomenological potential(dashed) adopted in Ref. [1]. exchange model. The strength of the Yukawa potential γ is six times smaller than the phenomenological value,while the Yukawa screening parameter α obtained fromour data is comparable. The c ¯ c - N potential obtainedfrom lattice QCD is rather weak.We next show both finite-size and quark-mass depen-dence of the η c - N potential in Fig. 2. Firstly, as shownin the left panel of Fig. 2, there is no significant differ-ence between potentials computed from lattices with twodifferent spatial sizes ( La ≈ . η c - N potentialis quickly screened to zero and turns out to be some-how short ranged. In principle, the short range part ofthe potential, which is represented by ultraviolet physics,should be insensitive to the spatial extent associated withan infrared cutoff. As a result, it is assured that the largerlattice size is large enough to study the η c - N system.No large quark-mass dependence is also observed inthe right panel of Fig. 2. This is a non-trivial featuresince there is an explicit dependence on the reduced mass µ in the definition of the effective central potential (4).However, if one recalls that the c ¯ c - N interaction is mainlygoverned by multi-gluon exchange, the resulting potentialis expected to be less sensitive to the reduced mass of theconsidered system ignoring the internal structures of the η c and nucleon states.If one takes a closer look at the inset of Figure 2, itis found that the nature of the attractive interaction inthe η c - N system tends to get slightly weaker as the lightquark mass decreases. Does this mean that the strengthof the η c - N potential at the physical point becomes muchweaker than what we measured at the quark mass sim-ulated in this study? The answer to this question is notsimple. It is worth to remember that the ordinary van der Waals interaction is sensitive to the size of the charge dis-tribution, which is associated to the dipole size. Largerdipole size yields stronger interaction. We may expectthat the size of the nucleon becomes large as the lightquark mass decreases. However, the very mild but oppo-site quark-mass dependence observed here does not ac-commodate this expectation properly.Recent detailed studies of nucleon form factors tell usthat the root mean-square (rms) radius of the nucleon,which is a typical size of the nucleon, shows rather mildquark-mass dependence and its value is much smallerthan the experimental value up to at M π ∼ . c ¯ c - N potential from dynamical simulations would becomemore strongly attractive in the vicinity of the physicalpoint, where the size of the nucleon is much larger thanat the quark mass simulated in this study.We also have calculated the J/ψ - N potential. It shouldbe noted here that different total spin states are al-lowed as spin-1/2 and spin-3/2 states in the J/ψ - N sys-tem. This fact introduces a little complexity regard-ing the spin-dependence on the J/ψ - N potential. Theinterpolating operator of the J/ψ state is defined as h ,i ( y ) = ¯ c a ( y ) γ i c a ( y ), which carries the specific spinpolarization direction. Therefore, the four-point correla-tion function for the J/ψ - N state becomes a matrix formwith spatial Lorentz indices, G h - h ij = (cid:104) h h ,i ( h h ,j ) † (cid:105) . -60-50-40-30-20-10 0 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 V (r) [ M e V ] r [fm] L xT=32 x48L xT=16 x48 -60-50-40-30-20-10 0 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 V (r) [ M e V ] r [fm] M (cid:47) =640 MeVM (cid:47) =730 MeVM (cid:47) =870 MeV -35-30-25-20-15 0.2 0.3 0.4 FIG. 2: The volume dependence (left) and the quark-mass dependence (right) of the η c - N potential. -60-50-40-30-20-10 0 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 V (r) [ M e V ] r [fm] (cid:100) c -NJ/ (cid:115) -N FIG. 3: The central and spin-independent part of the
J/ψ - N potential at M π = 640 MeV. The η c - N potential is alsoincluded for comparison. It can be expressed by an orthogonal sum of spin-1/2 andspin-3/2 components [7]: G h - h ij = G / P / ij + G / P / ij where proper spin projection operators for the spin-1 / / P / ij = γ i γ j and P / ij = δ ij − γ i γ j in the center of mass frame [22].Then, each spin part can be projected out as G / = (cid:80) i,j P / ij G h - h ji and G / = (cid:80) i,j P / ij G h - h ji wherethe indices i and j are also summed over all spatial di-rections.As a result, we can obtain the BS wave function andthe resulting J/ψ - N potential for each spin channel. Al-though the lower-spin state (spin-1/2) is not free from thecontamination of the η c - N state through channel mix-ings [7], we simply consider the spin averaged four-point correlation function for the J/ψ - N system as G J/ψ - N ave = 13 G / + 23 G / = 13 (cid:88) i G J/ψ - Nii . (5)This procedure may provide only a spin-independent partof the J/ψ - N potential through the same analysis appliedto the η c - N system.We show our result of the J/ψ - N potential in Fig. 3where the η c - N potential is included for comparison. The J/ψ - N potential shows short-range attraction. Similarto what we discussed in the η c - N case it does not havea normal “van der Waals type” − /r n behavior. Thereis no qualitative difference between the η c - N and J/ψ - N potentials. Quantitatively, the attractive interactionobserved in the J/ψ - N potential is rather stronger thanthat of the η c - N system as shown in Fig. 3, though itis still not strong enough to form a bound state in the J/ψ - N system.What is a possible origin of the stronger attraction ap-pearing in the J/ψ - N system? As was discussed previ-ously, the attractive interaction in the c ¯ c - N system tendsto be slightly stronger as the light quark mass increases.It should be recalled that the reduced mass of the c ¯ c - N system is also changed through a variation of the lightquark mass. Supposed that the strength of the c ¯ c - N po-tential depends simply on the reduced mass of the c ¯ c - N system, a mass difference of the η c and J/ψ state mayaccount for the difference between the η c - N and J/ψ - N potentials. However, this is not the case. As shownin the inset of Figure 2, the η c - N potential gets deeper,when the reduced mass of the η c - N system is increased byabout 10% through a variation of the light quark mass.On the other hand, although the reduced mass receivesonly about 1% gains in the J/ψ - N system relative to the η c - N system, the difference between the η c - N and J/ψ - N potentials shown in Fig. 3 is much bigger than thereduced mass dependence observed in Fig. 2. This mayindicate that the stronger interaction in the J/ψ - N sys-tem than in the η c - N system is caused by some dynamicsassociated with the structure of quarkonia.We again recall that the ordinary van der Waals inter-action is sensitive to the size of the charge distribution.Therefore, what we observed here is intuitively accountedfor by the simple speculation that the size of the J/ψ state is larger than that of the η c state. Nevertheless,similar to the η c - N system in the J/ψ - N no appreciablefinite-size effects are observed.We have studied the charmonium-nucleon potential V c ¯ c - N ( r ) in quenched lattice QCD. Potentials betweencharmonia ( η c and J/ψ ) and the nucleon are calculatedfrom the equal-time BS amplitude through the effectiveSchr¨odinger equation. We have found that the centraland spin-independent potential V c ¯ c - N ( r ) in both the η c - N and J/ψ - N systems is weakly attractive at short dis-tances and exponentially screened at large distances. It isobserved that both potentials have no appreciable finite-size dependence and no significantly large quark-mass de-pendence within the pion mass region 640MeV ≤ M π ≤ J/ψ - N system is slightly stronger attrac-tive than the η c - N system. This should be accounted forby the dynamics associated with the structure of quarko-nia.In order to make a reliable prediction about thenuclear-bound charmonium, an important step in the fu-ture is clearly an extension to dynamical lattice QCDsimulations in the lighter quark mass region. Such plan-ning is now underway. Once we obtain a realistic poten-tial, we will proceed exploring the nuclear-bound charmo-nium state with theoretical inputs of the charmonium-nucleon potential by using the exact few-body calcula-tion.We would like to thank T. Hatsuda for helpful sug-gestions and fruitful discussions. We also thank to T.Misumi for his careful reading of the manuscript. T.K.is supported by Grant-in-Aid for the Japan Society forPromotion of Science (JSPS) Research Fellows (No. 22-7653). S.S. is supported by the JSPS Grant-in-Aids forScientific Research (C) (No. 19540265) and Scientific Re-search on Innovative Areas (No. 21105504). Numericalcalculations reported here were carried out on the PACS-CS supercomputer at CCS, University of Tsukuba andalso on the T2K supercomputer at ITC, University ofTokyo. ∗ Electronic address: [email protected] † Electronic address: [email protected][1] S. J. Brodsky, I. A. Schmidt and G. F. de Teramond,Phys. Rev. Lett. , 1011 (1990).[2] S. J. Brodsky and G. A. Miller, Phys. Lett. B , 125(1997).[3] M. E. Luke, A. V. Manohar and M. J. Savage, Phys. Lett.B , 355 (1992).[4] T. Appelquist and W. Fischler, Phys. Lett. B , 405(1978).[5] G. Feinberg and J. Sucher, Phys. Rev. D , 1717 (1979).[6] A. Hayashigaki, Prog. Theor. Phys. , 923 (1999).[7] K. Yokokawa, S. Sasaki, T. Hatsuda and A. Hayashigaki,Phys. Rev. D , 034504 (2006).[8] D. A. Wasson, Phys. Rev. Lett. , 2237 (1991).[9] V. B. Belyaev et al., Nucl. Phys. A , 100 (2006).[10] N. Ishii, S. Aoki and T. Hatsuda, Phys. Rev. Lett. ,022001 (2007).[11] S. Aoki, T. Hatsuda and N. Ishii, Prog. Theor. Phys. (2010) 89.[12] M. L¨uscher, Nucl. Phys. B , 531 (1991).[13] S. Aoki et al. [CP-PACS Collaboration], Phys. Rev. D , 094504 (2005).[14] R. Sommer, Nucl. Phys. B , 839 (1994) [arXiv:hep-lat/9310022].[15] M. Guagnelli, R. Sommer and H. Wittig [ALPHA col-laboration], Nucl. Phys. B , 389 (1998); S. Necco andR. Sommer, Nucl. Phys. B , 328 (2002)[16] M. L¨uscher, S. Sint, R. Sommer, P. Weisz and U. Wolff,Nucl. Phys. B , 323 (1997).[17] S. Aoki, Y. Kuramashi and S. I. Tominaga, Prog. Theor.Phys. , 383 (2003).[18] A. X. El-Khadra, A. S. Kronfeld and P. B. Mackenzie,Phys. Rev. D , 3933 (1997).[19] Y. Kayaba et al. [CP-PACS Collaboration], JHEP ,019 (2007).[20] T. T. Takahashi, T. Umeda, T. Onogi and T. Kunihiro,Phys. Rev. D , 114509 (2005) [arXiv:hep-lat/0503019].[21] S. Matsuyama and H. Miyazawa, Prog. Theor. Phys. ,942 (1979).[22] M. Benmerrouche, R. M. Davidson and N. C. Mukhopad-hyay, Phys. Rev. C (1989) 2339.[23] T. Yamazaki et al. , Phys. Rev. D , 114505 (2009)[arXiv:0904.2039 [hep-lat]].[24] M. A. B. Beg and A. Zepeda, Phys. Rev. D , 2912(1972).[25] The potential for r (cid:38) . . It is worth noting that the r -dependence of the potential in the region of r (cid:38) ..