Distribution amplitudes of light-quark mesons from lattice QCD
Jorge Segovia, Lei Chang, Ian C. Cloet, Craig D. Roberts, Sebastian M. Schmidt, Hong-shi Zong
aa r X i v : . [ nu c l - t h ] N ov Distribution amplitudes of light-quark mesons from lattice QCD
Jorge Segovia a , Lei Chang b , Ian C. Clo¨et a , Craig D. Roberts a , Sebastian M. Schmidt c , Hong-shi Zong d,e,f a Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA b CSSM, School of Chemistry and Physics University of Adelaide, Adelaide SA 5005, Australia c Institute for Advanced Simulation, Forschungszentrum J¨ulich and JARA, D-52425 J¨ulich, Germany d Department of Physics, Nanjing University, Nanjing 210093, China e State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, CAS, Beijing 100190, China f Joint Center for Particle, Nuclear Physics and Cosmology, Nanjing 210093, China
Abstract
We exploit a method introduced recently to determine parton distribution amplitudes (PDAs) from minimal information in order toobtain light-quark pseudoscalar and vector meson PDAs from the limited number of moments produced by numerical simulationsof lattice-regularised QCD. Within errors, the PDAs of pseudoscalar and vector mesons constituted from the same valence quarksare identical; they are concave functions, whose dilation expresses the strength of dynamical chiral symmetry breaking; and SU(3)-flavour symmetry is broken nonperturbatively at the level of 10%. Notably, the appearance of precision in the lattice moments ismisleading. The moments also exhibit material dependence on lattice volume, especially for the pion. Improvements need thereforebe made before an accurate, unified picture of the light-front structure of light-quark pseudoscalar and vector mesons is revealed.
Keywords: quantum chromodynamics, dynamical chiral symmetry breaking, Dyson-Schwinger equations, lattice-regularisedQCD, light-front quantum field theory, light mesons, parton distribution amplitudes, strange quarks
Preprint no . ADP-13-21 / T841
1. Introduction . A valence parton distribution amplitude(PDA) is a light-front wave-function of an interacting quantumsystem. It provides a connection between dynamical propertiesof the underlying relativistic quantum field theory and notionsfamiliar from nonrelativistic quantum mechanics. In particular,although particle number conservation is generally lost in rela-tivistic quantum field theory, ϕ ( x ) has a probability interpreta-tion. It can therefore translate features that arise purely throughthe infinitely-many-body nature of relativistic quantum fieldtheory into images whose interpretation seems more straightfor-ward [1–6]. For a meson, the argument of the PDA, x , expressesthe light-front fraction of the bound-state total-momentum car-ried by the meson’s valence quark, which is equivalent to themomentum fraction carried by the valence-quark in the infinite-momentum frame; and momentum conservation entails that thevalence antiquark carries the fraction ¯ x = (1 − x ).In the theory of strong interactions, the cross-sections formany hard exclusive hadronic reactions can be expressed ac-curately in terms of the PDAs of the hadrons involved [7–13].For example, in the case of the electromagnetic form factor oflight pseudoscalar mesons [7–10]: ∃ Q > Λ QCD | Q F P ( Q ) Q > Q ≈ πα s ( Q ) f P w ϕ , (1) w ϕ = Z dx x ϕ P ( x ) , (2)where α s ( Q ) is the strong running coupling, f P is the meson’sleptonic decay constant and ϕ P ( x ) is its PDA. Such formulae areexact. However, the PDAs are not determined by the analysis framework; and the value of Q is not predicted. (N.B. Dy-namical generation of the mass-scale Λ QCD ∼ . ff ective synergy [4], which is highlightedby a prediction for a pion’s elastic electromagnetic form factor[26]. These advances can be summarised succinctly: it is nowpossible to compute bound-state PDAs from Poincar´e-covarianthadron BSAs and thereby place useful and empirically verifi-able constraints on both. Preprint submitted to Physics Letters B 19 October 2013 wo significant features emerged in developing the connec-tion between BSAs and PDAs. The first is an appreciation thatthe so-called asymptotic PDA; i.e., ϕ asy ( x ) = x ¯ x , (3)provides an unacceptable description of meson internal struc-ture at all scales which are either currently accessible or fore-seeable in experiments [4, 5, 25, 26]. This should not be surpris-ing because evolution with energy scale in QCD is logarithmic .The second important point is that at all energy scales, theleading twist PDAs for light-quark mesons are concave func-tions. This last is a powerful observation because it elimi-nates the possibility of “humped” distributions [27] and enablesone to obtain a pointwise accurate approximation to a meson’svalence-quark PDA from a very limited number of moments[5], which is all that is available, e.g., from numerical simula-tions of lattice-regularised QCD [28–30].An e ff ort has recently begun, focused on computation of me-son PDAs directly from BSAs obtained using QCD’s Dyson-Schwinger equations (DSEs) [23–25]. With the well con-strained kernels for bound-state equations that are now avail-able [31–34], these studies have a direct connection to QCD;and hence comparison of their results with experiment willserve as meaningful tests of this theory, as were previous com-putations of parton distribution functions [20, 35–37].Independently, it is worth capitalising on the second obser-vation reported above; namely, to use extant results from nu-merical simulations of QCD in order to obtain insights into thepointwise behaviour of meson PDAs, as has already been triedfor parton distribution functions (PDFs) [38–40]. The resultsshould be valuable in the analysis and planning of contempo-rary and future experiments. They will also serve as a bench-mark by which to gauge outcomes of attempts at the computa-tion of meson PDAs, including the DSE studies already men-tioned but also results from QCD sum-rules (e.g., Refs. [41–47]) and models (e.g., Refs. [48–52]).
2. Computing PDAs from moments . One should properly de-note a meson PDA by ϕ ( x ; τ ). It is a function of two arguments: x , the parton light-front momentum fraction; and τ = /ζ ,where ζ is the momentum-scale that characterises the exclu-sive process in which the meson is involved. On the domainwithin which QCD perturbation theory is valid, the equationdescribing the τ -evolution of ϕ ( x ; τ ) is known and has the solu-tion [9, 10] ϕ ( x ; τ ) = ϕ asy ( x ) (cid:20) + ∞ X j = , ,... a / j ( τ ) C (3 / j ( x − ¯ x ) (cid:21) , (4)where { C (3 / j , j = , , . . . , ∞} are Gegenbauer polynomialsof order α = / ffi cients { a / j , j = , , . . . , ∞} evolve logarithmically with τ , vanishing as τ →
0. This result expresses the fact that in the neighbourhood τ Λ QCD ≃
0, QCD is invariant under the collinear conformalgroup SL(2; R ) [53, 54]. Gegenbauer- α = / O (3)-invariant systems in quantum mechanics is plain. Nonperturbative methods in QCD typically provide access tomoments of the PDA; viz., the quantities h ( x − ¯ x ) m i τϕ = Z dx ( x − ¯ x ) m ϕ ( x ; τ ) . (5)Until recently it was commonly assumed that at any length-scale τ , an accurate approximation to ϕ ( x ; τ ) is obtained by us-ing just the first few terms of the expansion in Eq. (4); and hencethat the best use of a limited number of moments was to deter-mine the first few Gegenbauer coe ffi cients, a / j ( τ ), in Eq. (4).We will call this Assumption A . It leads to models for ϕ ( x )whose pointwise behaviour is not concave on x ∈ [0 , ϕ ( x ) lies within theclass of distributions produced by a meson BSA which may becharacterised as vanishing at zero relative momentum, insteadof peaking thereat. No ground-state pseudoscalar or vector me-son solution exhibits such behaviour [34, 55, 56]. Assumption A is certainly valid on τ Λ QCD ≃
0. However,it is grossly incorrect at any energy scale accessible in con-temporary or foreseeable experiments. This was highlightedin Ref. [5] and in Sec. 5.3 of Ref. [25]. The latter used the fact[57–59] that ϕ asy ( x ) can only be a good approximation to a me-son’s PDA when it is accurate to write u v ( x ) ≈ δ ( x ), where u v ( x )is the meson’s valence-quark PDF, and showed that this is notvalid even at energy scales characteristic of the large hadroncollider (LHC). Hence, realistic meson PDAs are necessarilymuch broader than ϕ asy ( x ). It follows that an insistence on us-ing just a few terms in Eq. (4) to represent a hadron’s PDA mustlead to unphysical oscillations; i.e., humps, just as any attemptto represent a box-like curve via a Fourier series will inevitablylead to slow convergence and spurious oscillations.An alternative to Assumption A , advocated and explained inRefs. [4–6, 25], is to accept that at all accessible scales, thepointwise profile of PDAs is determined by nonperturbativedynamics; and hence PDAs should be reconstructed from mo-ments by using Gegenbauer polynomials of order α , with thisorder – the value of α – determined by the moments themselves,not fixed beforehand. In illustrating this procedure, Ref. [4]considered DSE results for the pion’s BSA, wrote ϕ ( x ; τ ) = N α [ x ¯ x ] α − (cid:20) + j s X j = , ,... a α j ( τ ) C ( α ) j ( x − ¯ x ) (cid:21) , (6)where α − = α − / N α = / B ( α + / , α + /
2) and obtaineda converged, concave result for the PDA with j s =
2. (N.B. Inthe case of mesons in a multiplet that contains an eigenstate ofcharge-conjugation, ϕ ( x ) = ϕ ( ¯ x ); and hence only even termscontribute to the sum in Eq. (4).) Naturally, once obtained inthis way, one may project ϕ ( x ; τ ) onto the form in Eq. (4); viz.,for j = , , . . . , a / j ( τ ) =
23 2 j + j +
2) ( j + Z dx C (3 / j ( x − ¯ x ) ϕ ( x ; τ ) , (7)therewith obtaining all coe ffi cients necessary to represent anycomputed distribution in the conformal form without ambiguity2 able 1: Meson PDA moments obtained using numerical simulations of lattice-regularised QCD with N f = + ζ = s -quark mass, discretisation and renormalisation. meson h ( x − ¯ x ) n i ×
32 24 × π n = ρ k n = φ n = K n = K ∗k n = K n = K ∗k n = ffi culty. In this form, too, one may determine the distribu-tion at any τ ′ < τ using the ERBL evolution equations for thecoe ffi cients { a / j ( τ ) , i = , , . . . } [9, 10].In connection with the challenge of reconstructing a distri-bution from moments, consider that since discretised spacetimedoes not possess the full rotational symmetries of the Euclideancontinuum, then, with current algorithms, at most two nontriv-ial moments of ϕ ( x ) can be computed using numerical simu-lations of lattice-regularised QCD. In the case of mesons in amultiplet that contains an eigenstate of charge-conjugation onehas h x − ¯ x i ≡
0, which means that, on average, the valence-quark and -antiquark share equally in the light-front momen-tum of the bound-state; and hence only one nontrivial momentis accessible. Herein we propose to follow Ref. [5] and use thislimited information to reconstruct PDAs from lattice-QCD mo-ments using an analogue of Eq. (6) that is also valid for mesonscomprised from valence-quarks with nondegenerate masses: ϕ ( x ) = x α (1 − x ) β / B ( α, β ) . (8)The moments listed in Table 1 are su ffi cient to determine α , β inall instances; and, as mentioned above, if one wishes to evolvethe distribution obtained to another momentum scale, τ ′ < τ ,then this may be achieved by projecting Eq. (8) onto the formin Eq. (4) using Eq. (7), and subsequently employing the ERBLevolution equations [9, 10].
3. Light pseudoscalar and vector mesons with equal-massvalence-quarks . Consider now that a vector meson has twoPDAs, one associated with light-front longitudinal polarisation, ϕ V k , and the other with light-front transverse polarisation, ϕ V ⊥ .Simulations of lattice-QCD performed thus far have produced τ = /ζ , ζ = ϕ V k and ϕ V ⊥ which areequal within errors [29]. Similarly, it is apparent in Table 1that contemporary lattice-QCD cannot distinguish between ϕ V k and ϕ P , where the latter is the PDA associated with the vectormeson’s pseudoscalar analogue. We expect, however, that inreality these PDAs are di ff erent. Indeed, since a vector meson’selectric radius is greater than its magnetic radius, and the lat-ter, in turn, is greater than the charge radius of the pseudoscalarmeson analogue [60, 61], we anticipate the following ordering Table 2: Selection of computed quantities associated with the meson PDAs inEqs. (10), (11), (14), (15). x max is the location of the PDA’s maximum, whichlies at x = for the du case ( ϕ asy ( x max ) = . w is defined in Eq. (2).The fact that the n = , &
60% of their kin-dred lower moments highlights the statements made in connection with Eq. (6);i.e., that any attempt to reconstruct the PDA using Eq. (4) must converge veryslowly. ×
32 24 × ϕ du ( x max ) 1 . + . − . . + . − . ϕ su ( x max ) 1 . + . − . . + . − . h ( x − ¯ x ) i su .
019 0 . h ( x − ¯ x ) i su . ± .
02 0 . ± . h ( x − ¯ x ) i du . + . − . . ± . w du . + . − . . + . − . w su . + . − . . + . − . at accessible energy scales: ϕ V k narrower-than ϕ V ⊥ narrower-than ϕ P , (9)where “narrower” means pointwise closer to ϕ asy π ( x ). This ex-pectation requires confirmation via explicit calculations withinthe same DSE framework that delivered the stated ordering ofradii.The need for such a study is highlighted by the followingobservations. The pattern of Eq. (9) is seen in Refs. [42, 51],which report ϕ V k a little narrower than ϕ V ⊥ , and both narrowerthan ϕ P . In contrast, combining Refs. [49, 52] one finds ϕ V k ( x ) ≈ ϕ P ( x ) but ϕ P much narrower than ϕ V ⊥ , whereas Ref. [50] pro-duces ϕ V k ( x ) ≈ ϕ P ( x ) but ϕ V ⊥ much narrower than ϕ V k .The inconsistency just described is plainly unsatisfactory.So, absent a well-constrained DSE study, herein we simplywork with the contemporary lattice-QCD result: ϕ V k ≈ ϕ V ⊥ ≈ ϕ P = : ϕ du , and report PDAs obtained from the pseudoscalarmoments in Table 1. Using Eq. (8), the two rightmost columnsof this Table yield:16 × α du = β du = . + . − . , (10)24 × α du = β du = . + . − . . (11)The PDAs in Eqs. (10), (11) precisely reproduce the valuesof the moments in Table 1 and predict the quantities listed inTable 2.It is worth remarking here that there are two extremes for thePDA: ϕ du = ϕ point = constant, which describes a point-particle;and ϕ du = ϕ asy , which is the result in conformal QCD. Thismeans that the second moment is bounded as follows:12 = h ( x − ¯ x ) i ϕ asy ≤ h ( x − ¯ x ) i ϕ ≤ h ( x − ¯ x ) i ϕ point = . (12)Therefore, instead of using an absolute scale, the accuracy ofand deviations between the moments in Table 1 should be mea-3 .0 0.25 0.50 0.75 1.00.00.51.0 x Φ du H x L H ´ L A B Φ du H x L H ´ L AC Figure 1: PDA for pseudoscalar and vector mesons constituted from equal massvalence-quarks, reconstructed using Eq. (8).
Upper panel – solid curve and as-sociated error band (shaded region labelled “B”): Eq. (10), obtained from the16 × lower panel – solid curve and as-sociated error band (shaded region labelled “C”): Eq. (11), obtained from the24 × sured against these bounds. Consequently, the di ff erence be-tween the central values of the π – n = ff erences between Eqs. (10) and (11).The analogous bounds on the fourth moment are given by3 / = . < h ( x − ¯ x ) i < / α du = . α du = .
29. Thedashed curve labelled “A” in both panels is the DSE predictionfor the chiral-limit pion: ϕ π ( x ; τ ) = . x (1 − x )] a [1 + ˜ a C a + / (2 x − , (13) a = .
31, ˜ a = − .
12, which was obtained elsewhere [4]using the most sophisticated symmetry-preserving kernels forthe gap and Bethe-Salpeter equations that are currently avail-able [32]. These kernels incorporate essentially nonperturba-tive e ff ects associated with DCSB, which are omitted in theleading-order (rainbow-ladder) truncation and any stepwise im- Φ s u H x L H ´ L A D
Figure 2: Solid curve and associated error band (shaded region labelled “D”):PDA in Eq. (15), describing s ¯ u pseudoscalar and vector mesons, reconstructedusing Eq. (8) and obtained from the 24 × × ff erent.The dashed curve “A” is the DSE prediction for the pion’s PDA in Eq. (13). provement thereof [31]. They have exposed a key role playedby the dressed-quark anomalous chromomagnetic moment [62]in determining observable quantities; e.g., clarifying a causalconnection between DCSB and the splitting between vectorand axial-vector mesons [32]. If one chooses to approximateEq. (13) via Eq. (8), then it corresponds to α = β = . ff erent lattice spacings have overlap-ping error bands. Notwithstanding this, the di ff erences arematerial, something which may be illustrated by consideringthe “1 / x ” moment of the PDAs that, according to Eqs. (1),(2), sets the large- Q magnitude in the perturbative QCD for-mulae for a pseudoscalar meson’s elastic form factor. Thesemoments are presented in Table 2. The DSE prediction is w du = (1 / h x − i = .
53, a result compatible with that ob-tained using the PDAs of Refs. [41, 49]; and a QCD sum rulesanalysis produces [43] w du = . ± .
1. These continuum-QCDresults are compatible with experiment. It appears, therefore,that the 24 ×
64 lattice configurations produce a form of ϕ du ( x )that is too broad.The preceding analysis emphasises anew that information isgained using the procedure advocated in Refs. [4–6, 25] but notlost. It has enabled an informed analysis of the lattice results,providing context and highlighting possible shortcomings. s ¯ u pseudoscalar and vector mesons . When reconstructinga PDA for s ¯ u mesons, we choose to focus on the pseudoscalarmeson moments in Table 1 because they show the least sensi-tivity to lattice volume and possess the smallest errors. Usingthe procedure described in association with Eq. (6), they yield16 × α su = . + . − . , β su = . + . − . , (14)24 × α su = . + . − . , β su = . + . − . . (15)These PDAs precisely reproduce the values of the moments inTable 1; and the positive value of the first moment indicates4 .0 0.25 0.50 0.75 1.00.00.51.01.5 x Φ s u H x L H ´ L F E D
Figure 3: Solid curve (labelled “E”) is ERBL evolution to ζ =
10 GeV ofkaon PDA defined by Eq. (15) (dashed curve, labelled “D” to match the samePDA in Fig. 2). Dotted curve (labelled “F”) is ϕ asy ( x ) in Eq. (3). that, on average, the s -quark carries more of the bound-state’smomentum than the ¯ u -quark. In addition, the PDAs predict thequantities listed in Table 2.The positive value of h ( x − ¯ x ) i su is responsible for the shiftin position, relative to the peak in the pion’s PDA, of the max-imum in ϕ su ( x ); viz., from x = . x = .
55, which is ap-parent in Fig. 2. This 10% increase is a measure of nonpertur-bative SU(3)-flavour-symmetry breaking. It is comparable withthe 15% shift in the peak of the kaon’s valence s -quark PDF, s Kv ( x ), relative to u Kv ( x ) [37]. By way of context, it is notablethat the ratio of s -to- u current-quark masses is approximately28 [63], whereas the ratio of nonperturbatively generated Eu-clidean constituent-quark masses is typically [64] 1 . f K / f π ≈ . Q = via Eqs. (1),(2). Let us first, however, provide some background. Owing tocharge conservation, F K ( Q = / F π ( Q = =
1; and in theconformal limit, F K / F π = f K / f π = .
50. Moreover, given that r π / r K >
1, we anticipate that the ratio F K ( Q ) / F π ( Q ) growsmonotonically toward its conformal limit because anything elsewould indicate the presence of a new, dynamically generatedmass-scale. This expectation is supported by DSE form factorpredictions [65], which produce F K ( ζ ) / F π ( ζ ) = .
13. Now,using Eqs. (1), (2) and the results in Table 2, one has16 ×
32 24 × F K ( ζ ) / F π ( ζ ) 1 . + . − . . + . − . . (16)The central value of this ratio obtained from the 16 ×
32 latticeis consistent with expectations and the DSE prediction; but thelarge errors on w P ; i.e., the (1 / x )-moment, diminish the signif-icance of this outcome. Regarding the result obtained from the @ Ζ (cid:144) Ζ D < x - x > s u H ´ L Figure 4:
Solid curve – Leading-order evolution of h x − ¯ x i su with scale, ζ ,computed from the PDA defined by the central values in Eq. (15). The vertical dashed line marks ζ = ζ . The horizontal dotted line marks 50% of thismoment’s ζ = ζ value. It is not reached until the energy scale ζ = e . ζ = . ×
64 lattice, the central value suggests that this larger lat-tice produces a pion PDA which is too broad, consistent withthe discussion in the penultimate paragraph of Sec. 3. However,given the even larger errors in this case, little can safely be con-cluded.
5. ERBL evolution . As noted above, with decreasing τ = /ζ ,all meson PDAs shift pointwise toward ϕ asy in Eq. (3). Thisevolution was canvassed elsewhere for the symmetric pion PDA[5, 25]. Herein, it is therefore interesting to elucidate the e ff ectof evolution on the skewed kaon distribution associated with themoments produced by lattice-QCD.The solid curve (labelled “E”) in Fig. 3 is the 24 × α, β in Eq. (15),evolved to τ = /ζ , ζ =
10 GeV, using the leading-orderERBL equations. The evolved distribution is described by24 × ( τ → τ ) : α su = . + . − . , β su = . + . − . , (17)and has a central-value peak-location shifted just 2.4% closerto x = . It is apparent in Figure 3 that PDA evolution is slow.The slow pace of evolution can be quantified as follows.Consider the moment h x − ¯ x i su , which measures the average ex-cess of momentum carried by the valence s -quark in the meson.As indicated above, this moment is a measure of the magnitudeand flavour-dependence of DCSB. It is zero in the conformallimit. The ζ -evolution of h x − ¯ x i su is depicted in Fig. 4. Plainly,the s -quark momentum-excess remains more than 50% of its ζ -value until energy scales exceeding those generated at theLHC. Hence, consistent with similar analyses of ϕ du , nonper-turbative phenomena govern the pointwise behavior of ϕ su atall energy scales that are currently conceivable in connectionwith terrestrial facilities. (Higher-order evolution [66, 67] doesnot materially a ff ect these results or conclusions.)Finally, we return to the ratio in Eq. (16) and consider theimpact of ERBL evolution. Working with the 16 ×
32 lattice,which produces the more reasonable value, the ratio in Eq. (16)becomes F K ( ζ ) / F π ( ζ ) = . + . − . . (18)5ubtracting the unit Q = ζ =
100 GeV; i.e., Q =
10 000 GeV ,the central value of the ratio is 1 .
34, which is a further increaseof 17%. However, one still remains at only 68% of the pertinentconformal limit value.
6. Epilogue . Light-front parton distribution amplitudes (PDAs)have numerous applications in the analysis of hard exclusiveprocesses in the Standard Model but predictive power is lack-ing unless they can be calculated. Many nonperturbative meth-ods for the estimation of nonperturbative matrix elements inQCD produce moments of the PDAs, instead of the pointwisebehaviour directly. Therefore, in order to make progress, oneneeds an e ff ective means by which to reconstruct the PDA fromits moments.The method introduced in Refs. [4–6] enables one to obtaina pointwise accurate approximation to meson PDAs from lim-ited information. We employed it to extract the PDAs of light-quark pseudoscalar and vector mesons from the restricted num-ber of moments made available by numerical simulations oflattice-regularised QCD. Our analysis shows that, at all energyscales currently accessible to terrestrial experiments, the PDAsare concave functions whose dilation and asymmetry, when thelatter is present, express the strength of dynamical chiral sym-metry breaking.Notably, within errors, the lattice moments indicate thatwhen constituted from the same valence quarks, the PDAs ofpseudoscalar and vector mesons are identical. Some studiesin continuum QCD support an approximate equality betweenthese amplitudes, however, there is significant disagreement be-tween methods and models. In addition, the lattice momentsappear precise. However, our analysis showed this appearanceto be misleading because the errors on the moments admit anerror band on a given PDA which is e ff ectively large. More-over, especially for the pion, the lattice moments exhibit mate-rial dependence on lattice volume. It is plain, therefore, that im-provements must be made in both continuum- and lattice-QCDbefore we arrive at an accurate, unified picture of the light-frontstructure of pseudoscalar and vector mesons constituted fromlight-quarks. Acknowledgments . We are grateful for insightful commentsfrom G. Gao, Y.-x. Liu and P. C. Tandy; and for the enthu-siasm and hospitality of students, especially Z.-f. Cui, in thePhysics Department at Nanjing University where this work wasconceived and partially completed. Work supported by: Uni-versity of Adelaide and Australian Research Council throughgrant no. FL0992247; Forschungszentrum J¨ulich GmbH; De-partment of Energy, O ffi ce of Nuclear Physics, contract no. DE-AC02-06CH11357; the National Natural Science Foundation ofChina (Grant nos. 11275097, 10935001 and 11075075); and theResearch Fund for the Doctoral Program of Higher Education(Grant no. 2012009111002). References [1] B. D. Keister, W. N. Polyzou, Adv. Nucl. Phys. 20 (1991) 225–479.[2] F. Coester, Prog. Part. Nucl. Phys. 29 (1992) 1–32.[3] S. J. Brodsky, H.-C. Pauli, S. S. Pinsky, Phys. Rept. 301 (1998) 299–486.[4] L. Chang, I. C. Clo¨et, J. J. Cobos-Martinez, C. D. Roberts, S. M. Schmidt,et al., Phys. Rev. Lett. 110 (2013) 132001.[5] I. C. Clo¨et, L. Chang, C. D. Roberts, S. M. Schmidt, P. C. Tandy, Phys.Rev. Lett. 111 (2013) 092001.[6] L. Chang, C. D. Roberts, S. M. Schmidt (arXiv:1308.4708 [nucl-th]).
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