Light-cone Distribution Amplitudes of the Nucleon and Negative Parity Nucleon Resonances from Lattice QCD
V.M. Braun, S. Collins, B. Gläßle, M. Göckeler, A. Schäfer, R. W. Schiel, W. Söldner, A. Sternbeck, P. Wein
LLight-cone Distribution Amplitudes of the Nucleon and Negative Parity Nucleon Resonances fromLattice QCD
V.M. Braun, S. Collins, B. Gläßle, M. Göckeler, A. Schäfer, R. W. Schiel, ∗ W. Söldner, A. Sternbeck, and P. Wein
Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany (Dated: October 1, 2018)We present the results of a lattice study of light-cone distribution amplitudes (DAs) of the nucleon and nega-tive parity nucleon resonances using two flavors of dynamical (clover) fermions on lattices of different volumesand pion masses down to m π (cid:39) MeV. We find that the three valence quarks in the proton share their mo-mentum in the proportion
37% : 31% : 31% , where the larger fraction corresponds to the u -quark that carriesproton helicity, and determine the value of the wave function at the origin in position space, which turns out tobe small compared to the existing estimates based on QCD sum rules. Higher-order moments are constrained byour data and are all compatible with zero within our uncertainties. We also calculate the normalization constantsof the higher-twist DAs that are related to the distribution of quark angular momentum. Furthermore, we usethe variational method and customized parity projection operators to study the states with negative parity. Inthis way we are able to separate the contributions of the two lowest states that, as we argue, possibly correspondto N ∗ (1535) and a mixture of N ∗ (1650) and the pion-nucleon continuum, respectively. It turns out that thestate that we identify with N ∗ (1535) has a very different DA as compared to both the second observed negativeparity state and the nucleon, which may explain the difference in the decay patterns of N ∗ (1535) and N ∗ (1650) observed in experiment. PACS numbers: 12.38Gc, 12.38Lg, 13.40Gp, 14.20 GkKeywords: Lattice QCD, Nucleon Wave Function, Nucleon Resonances
I. INTRODUCTION
Understanding nucleon structure in terms of quarks andgluons is an important goal of Quantum Chromodynamics(QCD). The full nucleon wave function is very complicatedand remains elusive but substantial progress was made for ob-servables which require only specific limited nonperturbativeinput. In particular, hard exclusive reactions involving largemomentum transfer from the initial to the final state baryonare dominated by the overlap of the light-cone wave functionsat small transverse separations [1–3] that are usually referredto as light-cone distribution amplitudes (DAs).The DAs are fundamental nonperturbative functions thatare complementary to conventional parton distributions, butare much less well-known because their relation to experi-mental observables is less direct as compared to quark par-ton densities and on a more subtle theoretical footing. Theyare scale-dependent and for very large scales approach sim-ple asymptotic expressions called asymptotic DAs [1, 3] thatare widely believed, however, to provide one with a ratherpoor approximation for the momentum transfers accessible inmodern experiments.The theoretical description of DAs is based on the relationof their moments, i.e., integrals with powers of the momen-tum fractions, to matrix elements of local operators. Suchmatrix elements can be estimated using nonperturbative tech-niques, at least in principle, and the DAs reconstructed asan expansion in a suitable basis of polynomials in the mo-mentum fractions. Historically, the first and the second mo-ments of the nucleon DA have first been estimated using QCD ∗ Electronic address: [email protected] sum rules [4–7] and the results indicated a very large devia-tion from the asymptotic expressions. They were used ex-tensively for model building of the DAs [4–8] and allowedone to get a reasonable description of the experimental dataon nucleon electromagnetic form factors and several other re-actions within a purely perturbative framework, see, e.g., thereview [3].Despite a certain phenomenological success, this approachhas remained controversial for many years. First, the QCDsum rules used to calculate the moments have been criticizedas unreliable, see, e.g., [9]. Second, it is commonly acceptednowadays that perturbative contributions to hard exclusive re-actions at accessible energy scales must be complemented bythe so-called soft or end-point corrections that correspond to adifferent (Feynman) mechanism to transfer the large momen-tum to a loosely bound system. Estimates of the soft contri-butions using QCD sum rules, e.g., [10], quark models [11]and, more recently, light-cone sum rules [12–14] favor nu-cleon DAs that deviate from the asymptotic expressions onlymildly.With the advent of lattice QCD it has become possibleto calculate moments of the DAs starting from first princi-ples [15], however, this task appears to be technically com-plicated so that detailed calculations are just beginning. Thefirst quantitative results of lattice calculations of the momentsof nucleon DAs have been obtained by the QCDSF collabora-tion [16, 17] using two flavors of dynamical (clover) fermions.The same group also made an exploratory study of the DAs ofnucleon resonances with negative parity [18].In this work we extend the analysis in [16–18] by makinguse of a much larger set of lattices with different volumes,lattice spacings and pion masses down to m π (cid:39) MeV, andmaking various refinements in the procedure how the requiredmatrix elements are extracted from lattice data. Our data allow a r X i v : . [ h e p - l a t ] M a r one to perform, for the first time, a reliable chiral and infinitevolume extrapolation of the results to the physical limit, andalso a continuum extrapolation (to a lesser extent).Our main results can be summarized as follows: • We have calculated the nucleon coupling f N that cor-responds to the probability amplitude to find the threevalence quarks at one space point, f N = 2 . − GeV . (1)Here and below all numbers refer to the scale µ =4 GeV , the first error is statistical, including the chi-ral extrapolation, and the second is due to the contin-uum extrapolation. This number appears to be ∼ smaller than the existing estimates, which further de-creases the perturbative contribution to nucleon formfactors. • We have also calculated the nucleon couplings λ and λ that are related to the normalization of the P -wavethree-quark wave functions that involve orbital angularmomentum λ = − . − GeV ,λ = 8 . − GeV . (2) • We have determined the momentum fractions carried bythe three valence quarks in the proton (cid:104) x (cid:105) = 0 . , (cid:104) x (cid:105) = 0 . , (cid:104) x (cid:105) = 0 . , (3)where the first number corresponds to the u -quark thatcarries the proton helicity and the other two to the u, d quarks with helicities opposite to one another that aresometimes thought of as coupled in a scalar “diquark”.The approximate equality (cid:104) x (cid:105) (cid:39) (cid:104) x (cid:105) was not ex-pected and can be viewed as being consistent with the“diquark” picture. • We use the variational method and customized parityprojection operators to study the states with negativeparity. In this way we are able to separate the contribu-tions of the two lowest states that, as we argue, possiblycorrespond to N ∗ (1535) and a mixture of N ∗ (1650) with the pion-nucleon continuum, respectively. It turnsout that the state that we identify with N ∗ (1535) hasa qualitatively different DA compared to both the sec-ond observed negative parity state and the nucleon: Ithas a very small value at the origin and is almost anti-symmetric with respect to the interchange of the quarksin the scalar “diquark”. This result is important for theforthcoming studies of the electroproduction of nucleonresonances at large momentum transfers at the 12 GeVupgrade of the Jefferson Lab accelerator facility [19]and may explain the difference in the decay patterns of N ∗ (1535) and N ∗ (1650) observed in experiment. The presentation is organized as follows. Section II is in-troductory. We explain the relation between DAs and light-cone wave functions and introduce the required definitionsand notations. The necessary steps to compute moments ofthe DAs from lattice QCD are detailed in Section III. The nu-merical analysis of our lattice data and their extrapolation tothe physical point is presented in Section IV. The final resultsare collected in Section V, while Section VI is reserved forthe summary and conclusions. The paper also contains sev-eral Appendices with a discussion of more technical issues. II. NUCLEON WAVE FUNCTIONS AND DISTRIBUTIONAMPLITUDES
The quantum-mechanical picture of a nucleon as a super-position of states with different numbers of partons is for-mulated in the infinite momentum frame or using light-conequantization. Although a priori there is no reason to expectthat nucleon wave function components with, say, 100 partons(quarks and gluons) are suppressed as compared to those withonly the three valence quarks, the phenomenological successof naive quark models suggests that only the first few Fockcomponents are relevant. At least in hard exclusive reactionswhich involve a large momentum transfer to the nucleon, thedominance of valence states is widely expected and can beproven within QCD perturbation theory [1, 3].The most general parametrization of the three-quark sectorinvolves six scalar light-cone wave functions [20, 21] whichcorrespond to different possibilities to couple the quark helic-ities and the total orbital angular momentum to produce thehelicity- / nucleon state: λ + λ + λ + L z = 1 / . Inparticular, zero angular momentum is allowed, L = 0 , if thequark helicities λ i sum up to / . The corresponding contri-bution can be written as [1, 3, 20]: | N ( p ) ↑ (cid:105) L =0 = (cid:15) abc √ (cid:90) [ dx ][ d (cid:126)k ]6 √ x x x Ψ N ( x i , (cid:126)k i ) | u ↑ a ( x , (cid:126)k ) (cid:105)× (cid:104)(cid:12)(cid:12) u ↓ b ( x , (cid:126)k ) (cid:105)| d ↑ c ( x , (cid:126)k ) (cid:105)− (cid:12)(cid:12) d ↓ b ( x , (cid:126)k ) (cid:105)| u ↑ c ( x , (cid:126)k ) (cid:105) (cid:105) . (4)Here Ψ N ( x i , (cid:126)k i ) is the light-cone wave function that dependson the momentum fractions x i and the transverse momenta (cid:126)k i of the quarks. The integration measure is defined by (cid:90) [ dx ] = (cid:90) dx dx dx δ (cid:0) (cid:88) x i − (cid:1) , (cid:90) [ d (cid:126)k ] = (16 π ) − (cid:90) d (cid:126)k d (cid:126)k d (cid:126)k δ (cid:0) (cid:88) (cid:126)k i (cid:1) . (5)In hard processes the contribution of Ψ N ( x i , (cid:126)k i ) is dominantwhereas the other existing three-quark wave functions giverise to a power-suppressed correction, i.e., a correction ofhigher twist.The light-front description of a nucleon is very attractivefor model building, but faces conceptual difficulties that donot allow the calculation of light-cone wave functions fromfirst principles, at least at present. In particular there are sub-tle issues related to renormalization and gauge dependence.An alternative approach describes nucleon structure in termsof distribution amplitudes corresponding to matrix elementsof nonlocal gauge-invariant light-ray operators. The classifi-cation of DAs is based on twist rather than the number of con-stituents as for the Fock state wave functions. For example theleading-twist-three nucleon (proton) DA ϕ N ( x i ) is defined bythe matrix element [22] (cid:104) | (cid:15) ijk (cid:16) u ↑ i ( a n ) C (cid:54) nu ↓ j ( a n ) (cid:17) (cid:54) nd ↑ k ( a n ) | N ( p ) (cid:105) = − f N p · n (cid:54) n u ↑ N ( p ) (cid:90) [ dx ] e − ip · n (cid:80) x i a i ϕ N ( x i ) , (6)where q ↑ ( ↓ ) = (1 / ± γ ) q are quark fields of given he-licity, p µ is the proton momentum, p = m N , u N ( p ) theusual Dirac spinor in relativistic normalization, n µ an aux-iliary light-like vector n = 0 and C is the charge-conjuga-tion matrix. The relativistic normalization is tacitly assumedalso for the state vector, | N ( p ) (cid:105) . The Wilson lines that ensuregauge invariance are inserted between the quarks; they are notshown for brevity. The normalization constant f N is definedin such a way that (cid:90) [ dx ] ϕ N ( x i ) = 1 . (7)In principle, only the complete set of nucleon DAs carries thefull information on the nucleon structure, in the same manneras the complete basis of light-cone wave functions. In prac-tice, however, both expansions have to be truncated and theusefulness of a truncated version, taking into account eitherthe first few Fock states or a few lowest twist contributions,may depend on the concrete physics application.Using the wave function in Eq. (4) to calculate the matrixelement in Eq. (6) it is easy to show that the DA ϕ N ( x i ) isrelated to the integral of the wave function Ψ N ( x i , (cid:126)k i ) overtransverse momenta, which corresponds to the limit of zerotransverse separation between the quarks in position space [1]: f N ( µ ) ϕ N ( x i , µ ) ∼ (cid:90) | (cid:126)k | <µ [ d (cid:126)k ] Ψ N ( x i , (cid:126)k i ) , (8)where we have now explicitly stated the dependence on thescale µ . Thus, the normalization constant f N can be inter-preted as the nucleon wave function at the origin (in positionspace).As always in a field theory, extraction of the asymptoticbehavior produces divergences that have to be regulated. Asa result, the DAs become scheme- and scale-dependent. Inthe calculation of physical observables this dependence iscancelled by the corresponding dependence of the coefficientfunctions. The DA ϕ N ( x i , µ ) can be expanded in orthogonalpolynomials P nk ( x i ) defined as eigenfunctions of the corre-sponding one-loop evolution equation: ϕ N ( x i , µ ) = 120 x x x ∞ (cid:88) n =0 n (cid:88) k =0 ϕ Nnk ( µ ) P nk ( x i ) , (9) where f N ( µ ) = f N ( µ ) (cid:18) α s ( µ ) α s ( µ ) (cid:19) / (3 β ) ,ϕ Nnk ( µ ) = ϕ Nnk ( µ ) (cid:18) α s ( µ ) α s ( µ ) (cid:19) γ nk /β (10)and (cid:90) [ dx ] x x x P nk ( x i ) P n (cid:48) k (cid:48) ( x i ) ∝ δ nn (cid:48) δ kk (cid:48) . (11)Here β = 11 − n f is the first coefficient of the QCD beta-function and γ nk are the respective anomalous dimensions.The first few polynomials are P = 1 , P = 21( x − x ) , P = 7( x − x + x ) , P = 6310 [3( x − x ) − x ( x + x ) + 2 x ] , P = 632 ( x − x + x )( x − x ) , P = 95 [ x +9 x ( x + x ) − x x − x + x ] . (12)The corresponding anomalous dimensions are γ =0 , γ = 209 , γ = 83 ,γ = 329 , γ = 409 , γ = 143 . (13)The normalization condition (7) implies that ϕ N = 1 . In whatfollows we will refer to the coefficients ϕ Nnk ( µ ) as shape pa-rameters. For a given order of the polynomials n , the coeffi-cients ϕ Nnk , k = 0 , , . . . , n are ordered according to increas-ing anomalous dimension, cf. Eq. (13). They are related to theexpansion coefficients used in Refs. [17, 18] by ϕ N = 121 c , ϕ N = 17 c ,ϕ N = 1063 c , ϕ N = 263 c , ϕ N = 59 c . (14)The set of ϕ Nnk together with the normalization constant f N ( µ ) at a certain reference scale µ specifies the momen-tum fraction distribution of the valence quarks in the nucleon.They are nonperturbative parameters that can be related to ma-trix elements of local gauge-invariant three-quark operators(see below).In the last twenty years evidence has mounted that thesimple-minded picture of a proton with the three valencequarks in an S-wave is incomplete, so that for example theproton spin is definitely not constructed from the quark spinsalone and also the electromagnetic Pauli form factor F ( Q ) cannot be explained without quark orbital angular momentumcontributions. The general classification of three-quark light-cone wave functions with nonvanishing angular momentumhas been worked out in Refs. [20, 21]. As shown in Ref. [23],the light-cone wave functions with L z = ± reduce, in thelimit of small transverse separation, to the twist-four nucleonDAs introduced in Ref. [22]: (cid:104) | (cid:15) ijk (cid:16) u ↑ i ( a n ) C /nu ↓ j ( a n ) (cid:17) /pd ↑ k ( a n ) | N ( p ) (cid:105) = − p · n /p u ↑ N ( p ) (cid:90) [ dx ] e − ip · n (cid:80) x i a i × (cid:104) f N Φ N,W W ( x i ) + λ N Φ N ( x i ) (cid:105) , (cid:104) | (cid:15) ijk (cid:16) u ↑ i ( a n ) C /nγ ⊥ /pu ↓ j ( a n ) (cid:17) γ ⊥ /nd ↑ k ( a n ) | N ( p ) (cid:105) = − p · n (cid:54) n m N u ↑ N ( p ) (cid:90) [ dx ] e − ip · n (cid:80) x i a i × (cid:104) f N Ψ N,W W ( x i ) − λ N Ψ N ( x i ) (cid:105) , (cid:104) | (cid:15) ijk (cid:16) u ↑ i ( a n ) C/p /nu ↑ j ( a n ) (cid:17) (cid:54) nd ↑ k ( a n ) | N ( p ) (cid:105) = λ N p · n (cid:54) n m N u ↑ N ( p ) (cid:90) [ dx ] e − ip · n (cid:80) x i a i Ξ N ( x i ) , (15)where Φ N,W W ( x i ) and Ψ N,W W ( x i ) are the so-calledWandzura-Wilczek contributions that can be expressed interms of the leading-twist DA ϕ N ( x i ) [24]: Φ N,W W ( x i ) = − (cid:88) n,k ϕ Nnk ( n + 2)( n + 3) (cid:18) n + 2 − ∂∂x (cid:19) × x x x P nk ( x , x , x ) , Ψ N,W W ( x i ) = − (cid:88) n,k ϕ Nnk ( n + 2)( n + 3) (cid:18) n + 2 − ∂∂x (cid:19) × x x x P nk ( x , x , x ) . (16)The two new constants λ N and λ N are defined in such a waythat the integrals of the “genuine” twist-4 DAs Φ , Ψ , Ξ arenormalized to unity, similar to Eq. (7). They have the samescale dependence to one-loop accuracy: λ N , ( µ ) = λ N , ( µ ) (cid:18) α s ( µ ) α s ( µ ) (cid:19) − /β . (17)The nonlocal operators entering the definitions of nucleonDAs do not have a definite parity. Thus the same operatorscouple also to the negative parity spin-1/2 nucleon resonances N ∗ (1535) , N ∗ (1650) , etc. One can define the leading-twistDA of these resonances from (cid:104) | (cid:15) ijk (cid:16) u ↑ i ( a n ) C (cid:54) nu ↓ j ( a n ) (cid:17) (cid:54) nd ↑ k ( a n ) | N ∗ ( p ) (cid:105) = 12 f N ∗ p · n (cid:54) n u ↑ N ∗ ( p ) (cid:90) [ dx ] e − ip · n (cid:80) x i a i ϕ N ∗ ( x i ) , where, of course, p = m N ∗ . The constant f N ∗ has the phys-ical meaning of the wave function of N ∗ at the origin. TheDA ϕ N ∗ ( x i ) is normalized to unity (7) and has the expansion ϕ N ∗ ( x i , µ ) = 120 x x x ∞ (cid:88) n =0 n (cid:88) k =0 ϕ N ∗ nk ( µ ) P nk ( x i ) , (18) with the shape parameters ϕ N ∗ nk .Similarly, there exist three independent subleading twist-4distribution amplitudes Φ N ∗ , Ψ N ∗ , Ξ N ∗ (as for the nucleon).They can be defined as [18] (cid:104) | (cid:15) ijk (cid:16) u ↑ i ( a n ) C /nu ↓ j ( a n ) (cid:17) /pd ↑ k ( a n ) | N ∗ ( p ) (cid:105) = 14 p · n /p u ↑ N ∗ ( p ) (cid:90) [ dx ] e − ip · n (cid:80) x i a i × (cid:104) f N ∗ Φ N ∗ ,W W ( x i ) + λ ∗ Φ N ∗ ( x i ) (cid:105) , (cid:104) | (cid:15) ijk (cid:16) u ↑ i ( a n ) C /nγ ⊥ /pu ↓ j ( a n ) (cid:17) γ ⊥ /nd ↑ k ( a n ) | N ∗ ( p ) (cid:105) = − p · n (cid:54) n m N ∗ u ↑ N ∗ ( p ) (cid:90) [ dx ] e − ip · n (cid:80) x i a i × (cid:104) f N ∗ Ψ N ∗ ,W W ( x i ) − λ ∗ Ψ N ∗ ( x i ) (cid:105) , (cid:104) | (cid:15) ijk (cid:16) u ↑ i ( a n ) C/p/nu ↑ j ( a n ) (cid:17) (cid:54) nd ↑ k ( a n ) | N ∗ ( p ) (cid:105) = λ ∗ p · n (cid:54) n m N ∗ u ↑ N ∗ ( p ) (cid:90) [ dx ] e − ip · n (cid:80) x i a i Ξ N ∗ ( x i ) , (19)where Φ N ∗ ,W W ( x i ) and Ψ N ∗ ,W W ( x i ) are given by the sameexpressions (16) in terms of the expansion of the leading-twistDA ϕ N ∗ ( x i ) as for the nucleon.The asymptotic distribution amplitudes (at very largescales) for the nucleon and the resonances are the same: ϕ as ( x i ) = 120 x x x , Φ as4 ( x i ) = 24 x x , Φ W W, as4 ( x i ) = 24 x x (1 + 23 (1 − x )) , Ψ W W, as4 ( x i ) = 24 x x (1 + 23 (1 − x )) , Ξ ( x i ) = 24 x x , Ψ as4 ( x i ) = 24 x x . (20)For the sake of completeness we also give the definitions ofthe normalization constants in terms of matrix elements of lo-cal three-quark operators. For the nucleon (cid:104) | (cid:15) ijk ( u i C /nu j )(0) γ /nd k (0) | N ( p ) (cid:105) = f N p · n /n u N ( p ) , (cid:104) | (cid:15) ijk ( u i Cγ µ u j )(0) γ γ µ d k (0) | N ( p ) (cid:105) = λ N m N u N ( p ) , (cid:104) | (cid:15) ijk ( u i Cσ µν u j )(0) γ σ µν d k (0) | N ( p ) (cid:105) = λ N m N u N ( p ) , (21)and similarly for N ∗ (cid:104) | (cid:15) ijk ( u i C /nu j )(0) γ /nd k (0) | N ∗ ( p ) (cid:105) = f N ∗ p · nγ /n u N ∗ ( p ) , (cid:104) | (cid:15) ijk ( u i Cγ µ u j )(0) γ γ µ d k (0) | N ∗ ( p ) (cid:105) = λ N ∗ m N ∗ γ u N ∗ ( p ) , (cid:104) | (cid:15) ijk ( u i Cσ µν u j )(0) γ σ µν d k (0) | N ∗ ( p ) (cid:105) = λ N ∗ m N ∗ γ u N ∗ ( p ) . (22) III. DISTRIBUTION AMPLITUDES AND LATTICE QCD
On the lattice one can calculate moments of the DAs, e.g., Φ lmn = (cid:90) [ dx ] x l x m x n ϕ ( x i ) , (23)which are related to matrix elements of local three-quark oper-ators with covariant derivatives, as explained below. The nor-malization is such that Φ = 1 . Starting from this Sectionspacetime is Euclidian and we use the Weyl representation forthe γ –matrices; our conventions follow [17].A traditional classification of leading-twist three-quark op-erators (in continuum theory) corresponds to a vector, axialand tensor Lorentz structure of the u -quark pair: V ρ ¯ l ¯ m ¯ nτ (0) = (cid:15) abc (cid:104) i l D ¯ l u (0) (cid:105) aα ( Cγ ρ ) αβ × (cid:2) i m D ¯ m u (0) (cid:3) bβ (cid:2) i n D ¯ n ( γ d (0)) (cid:3) cτ , A ρ ¯ l ¯ m ¯ nτ (0) = (cid:15) abc (cid:104) i l D ¯ l u (0) (cid:105) aα ( Cγ ρ γ ) αβ × (cid:2) i m D ¯ m u (0) (cid:3) bβ (cid:2) i n D ¯ n d (0) (cid:3) cτ , T ρ ¯ l ¯ m ¯ nτ (0) = (cid:15) abc (cid:104) i l D ¯ l u (0) (cid:105) aα (cid:0) C ( − iσ ξρ ) (cid:1) αβ × (cid:2) i m D ¯ m u (0) (cid:3) bβ (cid:2) i n D ¯ n ( γ ξ γ d (0)) (cid:3) cτ , (24)where we tacitly assume taking the leading-twist part, i.e.,symmetrization and subtraction of traces. The multi-index ¯ l ≡ λ · · · λ l , D ¯ l ≡ D λ . . . D λ l (and similarly for ¯ m and ¯ n ) denotes the Lorentz structure associated with the covari-ant derivatives D µ = ∂ µ − igA µ , whereas the indices l, m, n (without bars) stand for the total number of covariant deriva-tives acting on the first, second and third quark, respectively.Matrix elements of these operators define a set of couplings V lmn , A lmn , T lmn , (cid:104) |V ρ ¯ l ¯ m ¯ nτ | N ( p ) (cid:105) = − f N V lmn p ρ p ¯ l p ¯ m p ¯ n u N,τ ( p ) , (cid:104) |A ρ ¯ l ¯ m ¯ nτ | N ( p ) (cid:105) = − f N A lmn p ρ p ¯ l p ¯ m p ¯ n u N,τ ( p ) , (cid:104) |T ρ ¯ l ¯ m ¯ nτ | N ( p ) (cid:105) = 2 f N T lmn p ρ p ¯ l p ¯ m p ¯ n u N,τ ( p ) , (25)which can be viewed as moments of auxiliary nucleon DAs V ( x , x , x ) , A ( x , x , x ) , T ( x , x , x ) . These DAs areoften used in practical calculations.Identity of the two u –quarks implies the symmetry relations V lmn = V mln , A lmn = − A mln , T lmn = T mln . (26)In addition, the requirement that the nucleon has isospin 1/2allows one to express all T –moments in terms of V − A : T lmn = ( V − A ) lnm + ( V − A ) mnl . (27)The nucleon DA moments (23) are recovered as Φ lmn = ( V − A ) lmn . (28)Note that the operators defined in Eqs. (24) and (28) do nothave definite isospin themselves. We define F ρ ¯ l ¯ m ¯ nτ = 13 (cid:104) V ρ ¯ l ¯ m ¯ nτ − A ρ ¯ l ¯ m ¯ nτ − T ρ ¯ l ¯ n ¯ mτ (cid:105) (29) which is an isospin-1/2 operator: It is annihilated by theisospin raising operator which is easy to verify using Fierzidentities.Knowing the matrix elements of F ρ ¯ l ¯ m ¯ nτ is sufficient. With (cid:104) |F ρ ¯ l ¯ m ¯ nτ | N ( p ) (cid:105) = − f N φ lmn p ρ p ¯ l p ¯ m p ¯ n u N,τ ( p ) (30)one gets φ lmn = 13 (cid:104) ( V − A ) lmn + 2 T lnm (cid:105) (31)and Φ lmn = 2 φ lmn − φ nml . (32)The shape parameters of the nucleon DA (9) can be obtainedfrom the set of moments φ lmn as follows: ϕ = 32 (cid:0) φ − φ (cid:1) ,ϕ = 12 (cid:0) φ − φ + φ (cid:1) ,ϕ = 3 (cid:0) φ + φ − φ − φ (cid:1) + 2 φ − φ ,ϕ = 3 (cid:0) φ − φ (cid:1) + 9 (cid:0) φ − φ (cid:1) ,ϕ = φ − φ + φ + 9 (cid:0) φ + φ (cid:1) − φ . (33)Momentum conservation ( x + x + x = 1 ) implies the fol-lowing constraints: φ lmn = φ ( l +1) mn + φ l ( m +1) n + φ lm ( n +1) . (34)These relations can be used to rewrite (33) in equivalent al-ternative representations. This possibility should, however,be used with caution, as the momentum conservation (in thisform) is a consequence of the Leibniz rule for derivatives thatis only fulfilled to O ( a ) accuracy in lattice simulations, cf.Subsection IV E.For the next-to-leading twist DAs we only consider the op-erators without derivatives L τ (0) = (cid:15) abc u (0) aα ( Cγ ρ ) αβ u (0) bβ [ γ γ ρ d (0)] cτ , M τ (0) = (cid:15) abc u (0) aα ( Cσ µν ) αβ u (0) bβ [ γ σ µν d (0)] cτ , (35)which yield the next-to-leading twist normalization constants λ and λ defined in Eqs. (21) and (22). A. Lattice operators
Discretization of space and time reduces the Lorentz sym-metry of the continuum theory to the discrete hypercubic sym-metry of a four-dimensional lattice. Thus, additional mix-ing between discretized versions of continuum operators be-comes allowed, and this mixing has to be reduced as muchas possible by choosing a suitable operator basis. To thisend, the three-quark operators that appear in the calculation
TABLE I: Irreducibly transforming multiplets.dimension 9/2 dimension 11/2 dimension 13/2(0 derivatives) (1 derivative) (2 derivatives) τ B (0)1 ,i , B (0)2 ,i , B (0)3 ,i , B (0)4 ,i , B (0)5 ,i B (2)1 ,i , B (2)2 ,i , B (2)3 ,i τ B (2)4 ,i , B (2)5 ,i , B (2)6 ,i τ B (0)6 ,i B (1)1 ,i B (2)7 ,i , B (2)8 ,i , B (2)9 ,i τ B (0)7 ,i , B (0)8 ,i , B (0)9 ,i B (1)2 ,i , B (1)3 ,i , B (1)4 ,i B (2)10 ,i , B (2)11 ,i , B (2)12 ,i , B (2)13 ,i τ B (1)5 ,i , B (1)6 ,i , B (1)7 ,i , B (1)8 ,i B (2)14 ,i , B (2)15 ,i , B (2)16 ,i , B (2)17 ,i , B (2)18 ,i of DAs have to be classified according to their transforma-tion properties under the spinorial hypercubic group. Theirreducibly transforming multiplets of three-quark operatorshave been found in Refs. [25, 26] and their structure is shownschematically in Table I. The left column contains the listof the five irreducible spinorial representations. Each entryin the table corresponds to a multiplet of baryon operators;e.g., B (0)7 ,i , B (0)8 ,i , B (0)9 ,i correspond to the three independent do-decuplets ( i = 1 , , . . . , ) of three-quark operators withoutderivatives which transform according to the τ representa-tion. Explicit expressions for all operators with up to twoderivatives are given in Refs. [25, 26]. We refer to them asKGS operators in what follows.For example B (0)7 , = u u d , B (0)7 , = 1 √ u u d + u u d ) , B (0)7 , = u u d , B (0)7 , = u u d , B (0)7 , = 1 √ u u d + u u d ) , B (0)7 , = u u d , B (0)7 , = u u d , B (0)7 , = 1 √ u u d + u u d ) , B (0)7 , = u u d , B (0)7 , = u u d , B (0)7 , = 1 √ u u d + u u d ) , B (0)7 , = u u d , (36)where, e.g., u stands for the third component of the u -quarkbispinor (in the Weyl representation). The operators B (0)8 ,i ( B (0)9 ,i ) can be obtained from B (0)7 ,i by exchanging the spinorindices of quark one and two (one and three).The KGS operators can be mapped to certain componentsof the three-quark operators in the V , A , T basis. In our ex-ample, B (0)7 ,i , B (0)8 ,i and B (0)9 ,i correspond to the combinations V + A , V − A and T , respectively. A little algebra yields −B (0)8 , B (0)8 , −B (0)8 , B (0)8 , τ = 14 (cid:0) − γ ( V τ − A τ ) + γ ( V τ − A τ ) (cid:1) , −B (0)9 , B (0)9 , −B (0)9 , B (0)9 , τ = 14 (cid:0) − γ T τ + γ T τ (cid:1) , (37)and similar representations can be worked out for all othercases.The relations of this type reveal that the particular combi-nations of V , A , T operators that appear on the r.h.s. trans-form according to a particular irreducible representation ofthe spinorial hypercubic group (so that they are “good” latticeoperators, in principle), but they do not have definite isospinyet. Isospin-1/2 operators can easily be constructed, however,from suitable combinations of the KGS operators belongingto the same representation, and they can be expressed in termsof the F operators defined in Eq. (29). For the above example,e.g., taking the difference between the two given operators oneobtains O B, ≡ − γ F + γ F . (38)Another suitable combination is [17] O C, ≡ − γ F − γ F + γ F + γ F . (39)Both operators, O B, and O C, , transform according to the τ representation.The lattice operators with one and two derivatives areconstructed in a similar fashion. In the following, curlybraces indicate symmetrization over indices, e.g., F { } = (cid:0) F + F (cid:1) . We use in our calculations three operatorswith one derivative ( l + m + n = 1 ) from the τ representa-tion, O lmnA, = − γ γ F { } + γ γ F { } + γ γ F { } − γ γ F { } − γ γ F { } , O lmnB, =2 γ γ F { } + γ γ F { } − γ γ F { } + γ γ F { } − γ γ F { } , O lmnC, = − γ γ F { } + γ γ F { } + γ γ F { } − γ γ F { } , (40)and the only existing isospin-1/2 operator with two derivatives( l + m + n = 2 ) from the τ representation, O lmn = − γ γ γ F { } + γ γ γ F { } − γ γ γ F { } + γ γ γ F { } . (41) It turns out that the twist-four operators L and M whichwere defined in Eq. (35) are already good lattice operators andtransform according to the τ representation. B. Correlation functions
On the lattice we measure correlation functions of these op-erators with a smeared nucleon source N τ , which will be dis-cussed in detail in Subsection IV B.For O B, as an example, the contributions of the lowestpositive and negative parity states to such a correlation func-tion read (cid:104)O B, ( t, (cid:126)p ) τ ¯ N (0 , (cid:126)p ) τ (cid:48) (cid:105) == √ Zf N ( ip γ + E N γ )( E N γ − i(cid:126)p · (cid:126)γ + m N ) e − E N t E N + (cid:112) Z ∗ f ∗ ( ip γ + E ∗ γ )( − E ∗ γ + i(cid:126)p · (cid:126)γ + m ∗ ) e − E ∗ t E ∗ −√ Zf N ( ip γ − E N γ )( − E N γ − i(cid:126)p · (cid:126)γ + m N ) e − E N ( T − t ) E N − (cid:112) Z ∗ f ∗ ( ip γ − E ∗ γ )( E ∗ γ + i(cid:126)p · (cid:126)γ + m ∗ ) e − E ∗ ( T − t ) E ∗ . (42)Here (cid:126)p = { p , p , p } is the momentum and we use the short-hand notations f ∗ = f N ∗ , m ∗ = m N ∗ etc., for the quantitiesrelated to the negative parity state, N ∗ . √ Z is a – usuallymomentum-dependent – factor that indicates the overlap ofthe smeared nucleon source with the “physical” nucleon onthe lattice. We explain how to eliminate this unknown factorat the end of this Subsection.In this work we are specifically interested in a clean sep-aration of states with different parity. Note that the correla-tion function in (42) is a matrix with respect to the spinor in-dices τ and τ (cid:48) . For convenience we multiply this expressionby γ and try to find a parity projection operator in the form γ ± ≡ (1 + k ± γ ) , cf. [27], with k ± to be determined fromthe condition that positive and negative parity states are distin-guished by propagating forwards and backwards in time. Fordefiniteness let us consider the forward movers. We get (cid:104) (cid:0) γ O B, ( t, (cid:126)p ) (cid:1) τ ¯ N (0 , (cid:126)p ) τ (cid:48) (1 + k ± γ ) τ (cid:48) τ (cid:105) = √ Zf N ( k ± p + k ± E N + m N E N ) e − E N t E N + (cid:112) Z ∗ f ∗ ( − k ± p − k ± E ∗ + m ∗ E ∗ ) e − E ∗ t E ∗ . (43)One sees that if p = 0 (but p and p arbitrary) choos-ing k + = m ∗ /E ∗ annihilates the negative parity contribu-tion and thus extracts the positive parity (nucleon) state, and,vice versa, k − = − m N /E N projects onto the negative parity state. It turns out that, under certain restrictions for the mo-menta (cid:126)p , the same choice yields the correct parity projectionfor all correlation functions we are interested in. In the fol-lowing expressions we show the positive parity contributionsonly and abbreviate k ≡ k + and E ≡ E N ( (cid:126)p ) : C B, = (cid:104) ( γ O B, ( t, (cid:126)p )) τ ( N (0 , (cid:126)p )) τ (cid:48) ( γ + ) τ (cid:48) τ (cid:105) = f N (cid:112) Z N E ( m N + kE ) + kp E e − Et ,C C, = (cid:104) ( γ O C, ( t, (cid:126)p )) τ ( N (0 , (cid:126)p )) τ (cid:48) ( γ + ) τ (cid:48) τ (cid:105) = f N (cid:112) Z N E ( m N + kE ) + k ( p + p − p ) E e − Et , (44) C lmnA, = (cid:104) ( γ γ O lmnA, ( t, (cid:126)p )) τ ( N (0 , (cid:126)p )) τ (cid:48) ( γ + ) τ (cid:48) τ (cid:105) = − f N φ lmn (cid:112) Z N p E ( m N + kE )+ k (2 p − p ) E e − Et ,C lmnB, = (cid:104) ( γ γ O lmnB, ( t, (cid:126)p )) τ ( N (0 , (cid:126)p )) τ (cid:48) ( γ + ) τ (cid:48) τ (cid:105) = f N φ lmn (cid:112) Z N p E ( m N + kE ) + kp E e − Et ,C lmnC, = (cid:104) ( γ γ O lmnC, ( t, (cid:126)p )) τ ( N (0 , (cid:126)p )) τ (cid:48) ( γ + ) τ (cid:48) τ (cid:105) = − f N φ lmn (cid:112) Z N p E ( m N + kE ) + kp E e − Et , (45)and C lmn = (cid:104) ( γ γ γ O lmn ( t, (cid:126)p )) τ ( N (0 , (cid:126)p )) τ (cid:48) ( γ + ) τ (cid:48) τ (cid:105) = − f N φ lmn (cid:112) Z N p p E ( m N + kE ) + kp E e − Et . (46)For example, in order to have a nonzero overlap with theground states we must keep p and p nonvanishing but set p = 0 for the case of C lmn . With this choice the contri-bution from negative parity states with mass-to-energy ratio m ∗ /E ∗ is completely eliminated. Contributions from excitednegative parity states, which have a different ratio m/E , arenot completely eliminated but strongly suppressed. Togetherwith the suppression due to smearing and the suppression dueto the exponential decay with a higher mass, the positive par-ity states will dominate the signal, as desired.For the twist-four correlation functions, we find: C L = (cid:104) ( L ( t, (cid:126)p )) τ ( N (0 , (cid:126)p )) τ (cid:48) ( γ + ) τ (cid:48) τ (cid:105) = λ m N (cid:112) Z N m N + kEE e − Et ,C M = (cid:104) ( M ( t, (cid:126)p )) τ ( N (0 , (cid:126)p )) τ (cid:48) ( γ + ) τ (cid:48) τ (cid:105) = λ m N (cid:112) Z N m N + kEE e − Et . (47)In order to determine the coupling constants, we have toeliminate the √ Z N from the above equations. We do thisby considering yet another correlation function, that of thesmeared nucleon interpolator with itself: C N = (cid:104) ( N ( t, (cid:126)p )) τ ( N (0 , (cid:126)p )) τ (cid:48) ( γ + ) τ (cid:48) τ (cid:105) = Z N m N + kEE e − Et . (48)Taking the following ratio will then yield the desired result: C B, √ C N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126)p =0 = C C, √ C N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126)p =0 = f N m N e − m N t/ , (49)and similarly for λ and λ . Finally, the moments φ lmn arebest determined by taking the following ratios: C lmnA, C B, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = p =0 = − C lmnB, C B, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = p =0 = C lmnC, C B, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = p =0 = − φ lmn p ,C lmn C C, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p =0 ,p = p = − φ lmn p p . (50) C. Renormalization
The set of operators belonging to a given representation isclosed under renormalization. For f N , we have f rN = Z f N f lat N , (51) where the renormalization constant Z f N should not be con-fused with the √ Z -factor from the previous Subsection, r de-notes the renormalized value and “lat” the lattice value. For λ , , λ ri = Z λij λ lat j , (52)where a sum over repeated indices is implied, and for the mo-ments of the distribution amplitude, φ (1) i = ( φ , φ , φ ) and φ (2) i = ( φ , φ , φ , φ , φ , φ ) , φ (1) ,ri = Z (1) ij φ (1) , lat j , φ (2) ,ri = Z (2) ij φ (2) , lat j . (53)The renormalization factor Z f N and the renormalizationmatrices Z λij , Z (1) ij and Z (2) ij have been calculated in [26, 28].There, the matching of the lattice data to a kind of RI-MOM scheme has been performed non-perturbatively, and thematching of the RI-MOM scheme to the MS scheme has beencalculated in one-loop perturbation theory with the help of“naive” dimensional regularization that has certain shortcom-ings, cf. [29]. We use these results for our present study.In [26, 28], the renormalization matrices are only given forlattices of size up to × . But since there seems to be nosignificant volume dependence (the values for the × and × lattices agree within error bars), we felt comfortableto use their renormalization matrices for the × latticealso for our larger lattices. IV. DATA ANALYSISA. Ensembles used
The calculations in this paper have been done using the Wil-son gauge action and n f = 2 non-perturbatively improvedWilson (Clover) fermions. A list of the ensembles used isgiven in Table II. We would like to highlight that we have nowanalyzed ensembles with pion masses of MeV, very closeto the physical value. Hence the older ensembles used in [16–18], with large pion masses m π (cid:38) MeV can be neglectedaltogether. Another important improvement is that we havegenerated data for different lattice volumes (three volumes for β = 5 . , κ = 0 . ) and lattice spacings (three spacingsfor m π ≈ MeV) which allows us to quantify finite vol-ume and discretization effects. To set the scale, we use theSommer parameter r = 0 . fm [30, 31]. B. Isolating physical states
A major task in any lattice data analysis is the isolation andidentification of physical states. To suppress excited states,we have smeared the source using Wuppertal smearing [32]with APE smoothed [33] links. We have adjusted the numberof smearing steps to optimize the plateau for the proton.In our previous work ([17, 18] and the data points with m π > MeV in [34, 35]), we used a different smearing(Jacobi smearing [36, 37]) with a less-optimized number of
TABLE II: Ensembles used for this work. κ m π / MeV Size m π L Number ofconfigs. a β = 5 . , a = 0 . fm , a − = 2427 MeV0.13596 † × × β = 5 . , a = 0 . fm , a − = 2764 MeV0.13620 † × × † × × ∗ × × × × † × × ∗ × × † × × β = 5 . , a = 0 . fm , a − = 3270 MeV0.13647 † × × × × a The number of measurements per configuration is shown in parentheses. † These ensembles were generated on the QPACE systems, financed primarilyby the SFB/TR 55, while the others were generated earlier within the QCDSFcollaboration. ∗ For these ensembles, we have computed only the N interpolator and thuswe do not use them for the analysis of the negative parity states. smearing steps. The “jump” seen in the coupling constants at m π ≈ MeV in [35] disappeared when we re-computedthem with the improved smearing. It was, therefore, an arti-fact of our analysis rather than a physics effect.The difference to these older results is of the order of for the proton and up to for the negative parity statesin the case of the couplings; the shape parameters are lessaffected. The lesson is that source optimization proves to bevery important for calculations of this kind, i.e., for matrixelements of local operators.To separate the positive and negative parity states, we usethe parity projectors γ ± , as described above.For positive parity, the state that we are interested in is thenucleon. It has a large overlap with the (smeared) interpolatorof the form N = N ≡ ( uCγ d ) u , the “standard” nucleoninterpolator. Since the mass of the nucleon is significantlylower than that of excited states, it is relatively easy to isolate.To identify a suitable time range for the fit, its start and endcan be considered separately. The end can be determined bydemanding that the influence of the backward-in-time runningparity partner is negligible, i.e., much less than the statisti-cal error for the state under consideration. The starting timeshould be large enough that higher mass excitations are suf-ficiently suppressed but as small as possible to optimize thesignal-to-noise ratio for the observables. In order to find opti-mal starting times we have generated plots for all observableslike the mass plot shown in Fig. 1 and made fits with fixed endpoint and varying starting point. We further plot the fit resultswith error bars and demand that, for a good starting point, onedoes not observe any obvious systematic trend compared tothe points with larger starting times. Using this starting point,the χ / d.o.f. of the fit turned out to be on the order of one or t start m N ( l a tti ce un it s ) FIG. 1: Procedure to find a good fit range, illustrated by the exampleof the β = 5 . , κ = 0 . , × lattice. The end point ofthe fit range has been fixed to t end = 29 and the starting point t start has been varied. Based on this plot, we have chosen t start = 9 , sincevariations for data points with larger starting times appear to be ofstatistical nature. Note the highly stretched scale, indicating the highstatistical accuracy of our data. smaller, indicating a good fit.Identification of negative parity baryons on the lattice isconsiderably more difficult than that of the nucleon: In addi-tion to the two lowest-lying J P = 1 / − states N ∗ (1535) and N ∗ (1650) , which only have a small mass difference, there arealso contributions of pion-nucleon scattering states.The study [38] suggested that the two negative parity statescan be separated using the variational method with the three-quark interpolating operators N and N ≡ ( uCd )( γ d ) . In amore recent investigation using the same interpolating opera-tors [39] it was found that the mass of the lower state comesout to be very close to the sum of the nucleon and pion massesfor the same lattice, suggesting it is an (S-wave) N π scatter-ing state. The higher mass state in this study has — due to thelarge error bars — a mass consistent with both N ∗ (1535) and N ∗ (1650) so that they could not be distinguished.Yet another study [40] uses the same interpolating operators N and N and includes in addition a third, five-quark inter-polator to represent the nucleon-pion continuum. In a two-state analysis, using only the three-quark operators, their re-sults agree with the results from [39], yielding one state closeto the nucleon-pion threshold and one heavier state. The fullthree-state analysis produces one state slightly below the nu-cleon pion threshold (indicating attractive interaction) and twoheavier states that may be identified with the N ∗ (1535) and N ∗ (1650) . Comparing the eigenvectors of the variational ba-sis for the two- and three-state analyses, the authors suggestthat the lower mass state of the two-state analysis splits intothe N π state and the N ∗ (1650) , while the higher mass state ofthe two-state analysis becomes the N ∗ (1535) , see Fig. 2. Thisis also phenomenologically plausible, since the N ∗ (1535) isnot expected to mix strongly with the N π continuum as theobserved N ∗ (1535) → N π decay width is rather small [41].Due to the high cost of five-quark interpolators we haveused only the three-quark interpolators, N and N , for ouranalysis. Following the identification suggested in Ref. [40],cf. Fig. 2, we will label the lower mass state of our two-state variational analysis N ∗ (1650?) and the higher mass state0 E / G e V ? N − N − , Nπ FIG. 2: Negative parity energy levels from experiment (left), thetwo-state lattice variational analysis using N and N (middle) andthe three-state analysis including a five-quark operator (right). Thedashed lines show the sum of the nucleon and pion masses. This fig-ure is taken from [40], with arrows added to indicate the conjecturedsplitting of the lower state. N ∗ (1535?) , where the question marks indicate that this iden-tification is still uncertain and requires further study. In thecase of N ∗ (1650?) we expect that there is also considerablecontamination by nucleon-pion scattering states.The masses that we find for the nucleon and the negativeparity states are shown in Fig. 3. The nucleon mass has beenstudied in more detail in [30] and is – when extrapolated tothe physical point – consistent with experiment.The mass of the higher negative parity state (labeled N ∗ (1535?) , as explained above) changes rather smoothlywith the pion mass and is compatible with both known res-onances N ∗ (1535) and N ∗ (1650) within the error bars.For the mass of the lower state N ∗ (1650?) , [39] and [40]obtain a value close to the sum of the nucleon and pionmasses. Our ensembles with m π (cid:39) MeV confirm thisbehavior, but at smaller pion masses, the fitted mass appearsto be significantly higher than the
N π threshold. Whether thisis due to a smaller admixture of
N π scattering states at lowerpion masses or due to larger relative momentum of the nu-cleon and pion within the scattering state is unclear. To solvethis puzzle and to separate the N ∗ (1650?) from N π scatteringstates, studies with a larger variational basis, preferably withfive-quark interpolators, are required, but they are too expen-sive at present. Meanwhile, the identification of the negativeparity states should be regarded with caution.
C. Autocorrelations
Since lattice QCD data are based on configurations whichhave been generated by a Markov process, they are subject m π / GeV m / G e V FIG. 3: Masses of the nucleon (black), N ∗ (1650?) (blue) and N ∗ (1535?) (red, double line) as a function of the pion mass. Thesum of the nucleon and pion masses (green, dotted error bars) isshown for comparison. The crosses, circles and stars designate β = 5 . , β = 5 . and β = 5 . data points, respectively. Theexperimental values at the physical point (vertical dotted red line) arehighlighted by an arrow. to autocorrelations between consecutive trajectories. A pow-erful method to reduce these autocorrelations is to move thesource when going from one configuration to the next: Usinga different part of the lattice volume reduces the correlations.To determine the remaining autocorrelations and the re-sulting increase in the errors, we have applied the binningmethod. For most of our observables, the binned error wasonly slightly, if at all, greater than the error from the “naive”error analysis. Therefore, autocorrelations were only mini-mal. Merely a few observables on some ensembles showedgreater autocorrelation effects and in the worst case, everyother configuration was still statistically independent. D. Chiral and infinite volume extrapolations
The chiral extrapolations to the physical point and to infi-nite volume for the couplings f N and λ , are shown in Figs. 4and 5 and for the shape parameters ϕ nk in Appendix B. Theyhave been handled differently for the nucleon and the negativeparity states.For the nucleon, extrapolation formulae for both the leadingand next-to-leading twist normalization constants based onchiral perturbation theory ( χ PT) are available from Ref. [42].For the next-to-leading twist parameters we used the combi-nations m N λ and m N λ in the fits, which are more naturalfrom a χ PT point of view as compared to the couplings them-selves.Our extrapolation formulae for the moments of the leading1 m π / GeV f N / G e V m π / GeV f N ∗ / G e V FIG. 4: Chiral extrapolations of the wave functions at the origin f N , f N ∗ for the nucleon [left panel] and the negative parity resonances N ∗ (1535?) (red, double line) and N ∗ (1650?) (blue) [right panel]. Circles correspond to the lattice data for β = 5 . , crosses to β = 5 . and stars to β = 5 . . The dotted lines on the left panel show the central value of the lowest order fit scaled to the three lattice spacings (seeSubsection IV E), where the lowest line is for β = 5 . , the middle one for β = 5 . and the highest one for β = 5 . . On the right panel, the σ and σ error bands of the fit are shown in red for N ∗ (1535?) and in blue with dashed lines for N ∗ (1650?) . The physical point is indicatedby the vertical dotted red lines. m π / GeV − λ m N / G e V m π / GeV λ N ∗ m N ∗ / G e V m π / GeV λ m N / G e V m π / GeV λ N ∗ m N ∗ / G e V FIG. 5: Chiral extrapolations of the normalization constants of the twist-4 DAs λ , , λ N ∗ , for the nucleon [left panel] and negative parityresonances [right panel]. The identification of the curves and the data points is the same as in Fig. 4. twist distribution amplitude are new results. Details of theircalculation can be found in Appendix B. All expressions wereobtained in leading one-loop covariant baryon χ PT and in-clude correction terms for finite volume effects.We have fit our data with these extrapolation formulae andquote our final results for m π → m phys π and V → ∞ . We havealso checked the β = 5 . , κ = 0 . ensembles (where we have three different volumes) for residual finite volume ef-fects, but have concluded that the remaining small discrepan-cies between the three data points must be of statistical nature.For the negative parity states, extrapolation formulae basedon chiral perturbation theory do not exist yet. Therefore, wehave used naive (linear) extrapolations to the physical point.Given that our smallest pion mass is already very close to the2 a / fm f N / G e V a / fm − λ m N / G e V a / fm λ m N / G e V FIG. 6: Continuum extrapolation of the couplings f N [top], λ [bottom left] and λ [bottom right] using the largest volume data for m π (cid:39) MeV. The shaded areas correspond to 1 σ statistical error bars for the linear extrapolation (green, f N only), quadratic extrapolation(orange, dashed lines) and cubic extrapolation (blue, λ and λ only). physical value, the deviation of the linear extrapolation frommore sophisticated approaches should be marginal.Since we have analyzed the negative parity states for atmost two volumes per β and κ , a consistent study of finitevolume effects for N ∗ (1535?) and N ∗ (1650?) is not possible.However, the relatively small finite volume effects that wereobserved for the nucleon suggest that the finite volume effectsfor the negative parity states should be reasonably small aswell, i.e., at most of the order of the statistical error. E. Continuum extrapolation
We have analyzed ensembles with three lattice spacings, a = 0 . fm (corresponding to β = 5 . ), a = 0 . fm( β = 5 . ) and a = 0 . fm ( β = 5 . ).For f N , λ and λ of the nucleon, the statistical accuracy isso high that discretization effects can be observed, see Fig. 6.Since the exact form of the finite a corrections is unknown,we have treated the continuum extrapolation as follows: For f N , we have tried two extrapolations, one with a linear de-pendence and one with a quadratic dependence on a , fittingthe constants c (1) N and c (2) N simultaneously with the low-energyconstants in f (1) N ( m π , a ) = f N ( m π )(1 + c (1) N a ) ,f (2) N ( m π , a ) = f N ( m π )(1 + c (2) N a ) , where f N ( m π ) is the χ PT formula for f N and the volume dependence is suppressed for brevity. Both fits were almostequally good, which can be attributed to the small leverageof our three lattice spacings. Therefore, it is not possible todecide which fit is more accurate. As the central value ofour final result, we quote the average of f (1) N ( m phys π , and f (2) N ( m phys π , and as uncertainty in the continuum extrapola-tion one half of the difference between the two fit results.For λ and λ , we know that there are no O ( a ) effects, sincethere are no dimension / operators in the τ representationwhich could give rise to corrections linear in a , cf. Table I.Therefore, we have tried extrapolations with a quadratic and acubic dependence on a , λ (2)1 , ( m π , a ) = λ , ( m π )(1 + c (2)1 , a ) ,λ (3)1 , ( m π , a ) = λ , ( m π )(1 + c (3)1 , a ) . Again, both fits were almost equally good and we quote theaverage and one half of the difference of the two fits as ourcentral value and uncertainty of the continuum extrapolation,respectively.Of course, also a combination of linear and quadratic cor-rections for f N (quadratic and cubic for λ , ) is possible and,with only three lattice spacings available, will yield resultswith enormous uncertainties for a → . Therefore, additionalfiner lattices will be required for a more reliable analysis ofthe discretization effects.In turn, the statistical errors for the shape parameters ϕ nk are so large that no clear discretization effects could be ob-served. This does not imply, however, that there are no signif-3 a / fm P φ ( ) a / fm P φ ( ) FIG. 7: Check of the momentum conservation constraints Eq. (55) for the nucleon as a function of lattice spacing a . For each a we have usedthe largest volume at m π (cid:39) MeV. icant effects for these quantities and an uncertainty due to thecontinuum extrapolation of at least the order of the statisticalerror should be assumed.An indirect argument for the consistency of the continuumextrapolation for the relevant matrix elements of the operatorsincluding derivatives acting on the quark field can be obtainedby the verification of the energy conservation relations (34)for the sums of first and second moments: (cid:88) φ (1) ≡ φ + φ + φ , (cid:88) φ (2) ≡ φ + φ + φ + 2 (cid:0) φ + φ + φ (cid:1) . (54)It follows from Eq. (34) that these sums should be equal toone in the continuum limit, (cid:88) φ (1) = 1 , (cid:88) φ (2) = 1 , (55)and the deviations (due to discretization errors in the Leibnizrule for derivatives) are a good measure for the discretizationartifacts.Since the shape parameters ϕ nk are extracted from dif-ferences of matrix elements corresponding to the moments φ lmn , they have much larger statistical errors than the mo-ments themselves and especially the sums of the momentsin Eqs. (54), which can be determined with high precision.These sums are plotted for the three available lattice spacingsusing the largest volume data for m π (cid:39) MeV in Fig. 7.It is seen that the deviations are not large and the continuumextrapolated values fulfill the energy conservation constraintswithin the statistical accuracy, at the percent level for the firstand 2-3% for the second moments. These results are veryencouraging and suggest that the continuum extrapolation isunder control.
V. FINAL RESULTS
The final results for the normalization constants and shapeparameters of the nucleon and the two lowest negative par-ity states, N ∗ (1535?) and N ∗ (1650?) , are shown in Table III.The question marks are a reminder that the identification of TABLE III: The final results of the normalization constants andshape parameters of the nucleon and negative parity resonances, N ∗ (1535?) and N ∗ (1650?) , at the scale µ = 4 GeV . The firsterror is the combined statistical error and the one due to chiral andinfinite volume (for the nucleon only) extrapolation. The second er-ror (for the nucleon couplings) is the uncertainty of the continuumextrapolation. Nucleon N ∗ (1535?) N ∗ (1650?)10 f N / GeV . . . λ m/ GeV − . . . λ m/ GeV . . − . ϕ . . . ϕ . − . . ϕ − . . − . ϕ − . − . − . ϕ − . . . the results with physical negative parity resonances needs fur-ther study and in particular we expect that the numbers for N ∗ (1650?) include significant contributions from the pion-nucleon continuum. For each state, the normalization con-stants and the moments φ lmn were fit simultaneously. Theshape parameters were then determined from the φ lmn usingEqs. (33).The following extrapolations have been performed: chiralextrapolation to the physical pion mass (for all quantities),infinite volume extrapolation (only for the nucleon) and thecontinuum extrapolation (only for the nucleon normalizationconstants). It is seen that the continuum extrapolation is thesingle largest source of uncertainties for the nucleon normal-ization constants. For the negative parity states, on the otherhand, the results for the different lattice spacings agree withinthe errors. The uncertainty in their normalization constants re-lated to the continuum extrapolation can be expected on gen-eral grounds to be of the same order of magnitude as for thenucleon. For the shape parameters, we expect the error due tothe continuum extrapolation to be of the same order or smallerthan the shown statistical error.Our result for λ N appears to be in a very good agree-4 TABLE IV: Comparison of our results for the nucleon shape parameters to the existing models. The values are given at a renormalizationscale µ = 2 GeV .this work KS CZ COZ SB BK BLW ABO1 ABO2 ϕ . .
144 0 .
191 0 .
163 0 .
152 0 . . .
05 0 . ϕ . .
169 0 .
252 0 .
194 0 .
205 0 . . .
05 0 . ϕ − . .
56 0 .
32 0 .
41 0 .
65 0 .
000 0 .
000 0 . . ϕ − . − .
01 0 .
03 0 . − .
27 0 .
000 0 . − . − . ϕ − . − . − . − .
163 0 .
020 0 .
000 0 .
000 0 . − . ϕ ϕ ϕ ϕ ϕ -0.3-0.2-0.100.10.20.30.40.50.6 LatticeKSCZCOZSBBKBLWABO1ABO2
FIG. 8: Comparison of our results for the nucleon shape parameters(black circles) to QCD sum rule predictions (red symbols), light-conesum rules (blue symbols) and the BK model (orange crosses). ment with the next-to-leading order QCD sum rule calculation m N λ N ( QCD-SR ) = − (3 . ± . · − GeV [43], but thewave function at the origin, f N , comes out to be significantlybelow QCD sum rule estimates which give f N ( QCD-SR ) =(4 . ± . · − GeV [43], where in both cases we haverescaled the QCD sum rule results from µ = 1 GeV to µ = 4 GeV using two-loop anomalous dimensions, see Ap-pendix A. This result deals a further blow to all attempts to de-scribe hard exclusive reactions involving nucleons at realisticenergies in the classical perturbative QCD framework [1–3].The main achievement of this study is the determination ofthe first order shape parameters of the DAs with significantprecision. These parameters are responsible for the globalstructure of the DAs in the momentum fraction space and, inparticular, determine the average momentum fractions carriedby the valence quarks: (cid:104) x (cid:105) = 13 + ϕ + 13 ϕ , (cid:104) x (cid:105) = 13 − ϕ , (cid:104) x (cid:105) = 13 − ϕ + 13 ϕ . (56)The corresponding numbers are given in Eq. (3).The approximate equality ϕ (cid:39) ϕ for the nucleon andas a consequence (cid:104) x (cid:105) (cid:39) (cid:104) x (cid:105) attracts attention. This equality cannot be exact at all scales since ϕ and ϕ have differentanomalous dimensions. However, it is very interesting andsuggests that the nucleon wave function (at low virtualities)is symmetric under the interchange of the two quarks coupledin the scalar “diquark”. The diquark symmetry for the secondorder shape parameters would imply the constraint ϕ − ϕ + 2 ϕ = 0 . This relation cannot be checked with our data due to insuffi-cient precision and should be addressed in future lattice cal-culations.A comparison of our results for the nucleon shape pa-rameters to the existing estimates is shown in Table IVand Fig. 8. These are due to QCD sum rule calculationsof Chernyak and Zhitnitsky (CZ)[4], King and Sachrajda(KS)[5], Chernyak, Ogloblin and Zhitnitsky (COZ)[6], andStefanis and Bergmann (SB)[8], light-cone sum rule calcu-lations of nucleon electromagnetic form factors by Braun,Lenz and Wittmann (BLW)[13], and Anikin, Braun and Of-fen (ABO1 and ABO2)[14], and the QCD-inspired model byBolz and Kroll (BK)[11]. For this table (and plot) we used therenormalization scale µ = 2 GeV . Our results clearly ruleout the old QCD sum rule calculations of the first-order shapeparameters (alias the momentum fractions), but agree withinerrors with the parameters extracted from the light-cone sumrules and the BK model. For the second-order parameters, ourresults rule out a large value of ϕ found in [4–6, 8] but areotherwise consistent with zero (and with different models).For the negative parity states, we observe that the lead-ing twist DA of N ∗ (1650?) is similar to that of the nucleon,whereas N ∗ (1535?) is qualitatively different: with a verysmall value at the origin f N ∗ (cid:28) f N and large first-ordershape parameters ϕ N ∗ , ϕ N ∗ that have opposite sign to eachother. This striking difference is illustrated by the barycen-tric plots of the DAs in Fig. 9. It can be seen that the DA of N ∗ (1650?) (in reality, probably a mixture of N ∗ (1650) andthe pion-nucleon background) is similar to the nucleon, butwith larger deviations from the asymptotic form. The DA of N ∗ (1535?) appears to be completely different: It is approxi-mately antisymmetric under the exchange of the quarks in thediquark. This feature can be related to the observed small de-cay width of the N ∗ (1535) to a pion-nucleon final state. It isalso interesting that the next-to-leading twist couplings λ , for the nucleon and both negative parity states are compara-ble, which is an indication that the quark angular momentumplays a similar role. The consequences of this structure for5 x x x x x x x x x FIG. 9: Barycentric plots of the nucleon [left], N ∗ (1650?) [center] and N ∗ (1535?) [right] wave functions. Only the first moments of thedistribution amplitude have been used to create these plots. the electroproduction cross section of the negative parity res-onances at large momentum transfer [18, 19] will be studiedelsewhere. VI. CONCLUSIONS AND OUTLOOK
We have presented the results of a lattice study of light-conedistribution amplitudes of the nucleon and negative paritynucleon resonances using two flavors of dynamical (clover)fermions on lattices of different volumes and pion massesdown to m π (cid:39) MeV. Our data allow us to perform, for thefirst time, a reliable chiral and finite volume extrapolation ofthe results to the physical limit, and also a continuum extrap-olation for some observables. These are, to our knowledge,the first baryon structure calculations from first principles thatgo beyond the studies of the mass spectrum for the nucleonresonances. Our results are shown in Table III and Fig. 9, andsummarized in the Introduction so that we do not need to re-peat this discussion here.The present study can be continued and improved in sev-eral directions. Moving to lattices with N f = 2 + 1 dynami-cal quarks is an obvious step. In this way one can investigateDAs for the full baryon octet, Λ , Σ and Ξ . The decay patternof the N ∗ (1535) (its decay fraction to N η is (42 ± [41])implies that the addition of the strange quark is important forstudies of negative parity states. Further work is needed to im-prove the identification of the two lowest-lying negative parityresonances, N ∗ (1535) and N ∗ (1650) . The continuum extrap-olation remains the largest source of errors and will be of con-cern as well. There are also several other technical issues tobe addressed, e.g., the matching of the RI-MOM scheme tothe MS scheme has to be calculated to two-loop accuracy. Acknowledgments
This work has been supported in part by the DeutscheForschungsgemeinschaft (SFB/TR 55) and the EuropeanUnion under the Grant Agreement numbers 238353 (ITN STRONGnet) and 256594 (FP7-PEOPLE-2009-RG). Thecomputations were performed on the QPACE systems of theSFB/TR 55, Regensburg’s Athene HPC cluster, the Super-MUC system at the LRZ/Germany and Jülich’s JUGENE us-ing the Chroma software system [44] and the BQCD software[45] including improved inverters [46, 47].
Appendix A: Two-loop renormalization of the normalizationconstants f N , λ , For a generic nucleon coupling f = f N , λ , λ the scaledependence is given by f ( µ ) = E f ( µ, µ ) f ( µ ) (A1)where E NLO f ( µ, µ ) = (cid:20) α s ( µ ) α s ( µ ) (cid:21) γ (0) f /β (A2) × (cid:26) α s ( µ ) − α s ( µ )2 πβ (cid:18) γ (1) f − β β γ (0) f (cid:19)(cid:27) . The first two coefficients of the beta-function are β = 11 − n f , β = 102 − n f . (A3)Anomalous dimensions are defined such that (cid:20) µ ∂∂µ + β ( α s ) ∂∂α s + 12 γ f ( α s ) (cid:21) f = 0 ,γ f ( α s ) = γ (0) f α s π + γ (1) f (cid:16) α s π (cid:17) + . . . . (A4)The leading order anomalous dimensions are given by γ (0) f N = 23 , γ (0) λ = − , γ (0) λ = − . (A5)6The next-to-leading order (NLO) anomalous dimensions inthe KM scheme [29] are γ (1) f N = 239 + 149 β ,γ (1) λ = −
193 + 43 β ,γ (1) λ = − β . (A6)We stress that the NLO anomalous dimensions are scheme-dependent. Two of them, γ (1) λ and γ (1) λ , have been calculatedalso in a different scheme in Ref. [48]. Appendix B: Chiral extrapolation
We employ two-flavor baryon χ PT in order to obtain a sys-tematic framework for the extrapolation of the nucleon distri-bution amplitudes to physical quark masses and infinite vol-ume. The necessary extrapolation formulae for the leadingand next-to-leading twist normalization constants have beenderived in [42]. For completeness we quote here the relevantexpressions: ( λ m N ) ( m π ) = α (0)1 (cid:18) − m π πF π ) (cid:18) g A + (3 + 9 g A ) ln m π µ (cid:19)(cid:19) + 16 α (2)1 ( µ ) m π + O ( m π ) , ( λ m N ) ( m π ) = β (0)1 (cid:18) − m π (4 πF π ) (cid:18) g A + (3 + 9 g A ) ln m π µ (cid:19)(cid:19) + 32 β (2)1 ( µ ) m π + O ( m π ) ,f N ( m π ) = κ (0)1 (cid:18) − m π πF π ) (cid:18) g A + (19 + 9 g A ) ln m π µ (cid:19)(cid:19) + 4 κ (2)1 ( µ ) m π + O ( m π ) , (B1)where α (0 , , β (0 , and κ (0 , are low-energy constants(LECs). The dependence of the renormalized LECs α (2)1 , β (2)1 and κ (2)1 on the χ PT-scale µ cancels the µ -dependence of thelogarithm ln( m π /µ ) . Finite volume corrections, which donot introduce additional low-energy constants, have also beencomputed. Explicit expressions can be found in Ref. [42].To obtain the quark mass dependence of the higher mo-ments of the leading twist DA we follow the same proce-dure. Let us briefly describe the main steps. To begin with,we define three-quark operators with mixed antisymmetric (MA) and mixed symmetric (MS) flavor structure (MA ∝ uud − udu , MS ∝ − uud + udu + duu ) η MA lmn = q Tp (cid:0) γ L U Rlmn − γ R U Llmn (cid:1) ,η MS lmn = 43 q Tp τ a (cid:16) γ L U R,almn − γ R U L,almn (cid:17) , (B2)where q p ≡ (1 , T projects onto the quark content of theproton and we use the notation U L/Rlmn ≡ (cid:15) ijk n µ n ν (cid:16)(cid:16) ( in · D ) n q iTL/R (cid:17) Cσ µρ ( iτ ) (cid:16) ( in · D ) l q jL/R (cid:17)(cid:17) σ νρ (cid:0) ( in · D ) m q k (cid:1) , U L/R,almn ≡ (cid:15) ijk n µ n ν (cid:16)(cid:16) ( in · D ) n q iTL/R (cid:17) Cσ µρ ( iτ ) τ a (cid:16) ( in · D ) l q jL/R (cid:17)(cid:17) σ νρ (cid:0) ( in · D ) m q k (cid:1) , (B3)with the quark doublet field q ≡ ( u, d ) T . In the case l = m = n = 0 , the MS operator reduces to the isospin-improvedChernyak-Zhitnitsky current given in [42]. Since the trans-formation properties under chiral rotations are not affectedby additional derivatives, γ L U R,almn and γ R U L,almn transform as (2 L , R ) and (3 L , R ) , while γ L U Rlmn and γ R U Llmn transformas (2 L , R ) and (1 L , R ) , respectively. Utilizing the standardDA decomposition [22], one finds that these operators project onto the moments defined in Eq. (23): (cid:104) | η MA lmn | N ( p ) (cid:105) = f N (cid:0) Φ lmn − Φ nml (cid:1) × ( n · p ) l + m + n +1 /nN ( p ) , (cid:104) | η MS lmn | N ( p ) (cid:105) = f N (cid:0) Φ lmn + Φ nml (cid:1) × ( n · p ) l + m + n +1 /nN ( p ) . (B4)7 TABLE V: Low-energy operators for the antisymmetric (MA) and symmetric (MS) moments of the leading twist DA grouped according totheir chiral dimension d . We have only listed terms that contribute to the proton-to-vacuum matrix element of the operators at leading one-looplevel in the limit of exact isospin symmetry and have used the shorthand D n ≡ n · D . d k O MA , ( d ) k,r,LR O MA , ( d ) k,r,RL O MS , ( d ) ,ak,r,LR O MS , ( d ) ,ak,r,RL u † γ L /n ( iD n ) r +1 Ψ − uγ R /n ( iD n ) r +1 Ψ u † τ a uγ L /n ( iD n ) r +1 Ψ − u τ a u † γ R /n ( iD n ) r +1 Ψ2 1 tr { χ + } u † γ L /n ( iD n ) r +1 Ψ − tr { χ + } uγ R /n ( iD n ) r +1 Ψ tr { χ + } u † τ a uγ L /n ( iD n ) r +1 Ψ − tr { χ + } u τ a u † γ R /n ( iD n ) r +1 Ψ m π / GeV ϕ (a) m π / GeV ϕ (b) m π / GeV ϕ (c) m π / GeV ϕ (d) FIG. 10: Chiral extrapolations of the shape parameters of the first order ϕ , ϕ (9) for the nucleon [left panels] and the negative parityresonances N ∗ (1535?) (red, double line) and N ∗ (1650?) (blue) [right panels]. Circles correspond to the lattice data for β = 5 . , crosses to β = 5 . and stars to β = 5 . . The darker and lighter bands correspond to the σ and σ error bands of the chiral perturbation theory fit,respectively. On the right panels, the data and error bands are shown in red for N ∗ (1535?) and in blue with dashed lines (much more narrow)for N ∗ (1650?) . The physical point is indicated by the vertical dotted red lines. The low-energy form of the operators reads η MA lmn = q Tp ∞ (cid:88) d =0 i d (cid:88) k =1 κ MA , ( d ) k lmn (cid:16) O MA , ( d ) k, ( l + m + n ) ,LR − O MA , ( d ) k, ( l + m + n ) ,RL (cid:17) ,η MS lmn = 43 q Tp τ a ∞ (cid:88) d =0 i d (cid:88) k =1 κ MS , ( d ) k lmn (cid:16) O MS , ( d ) ,ak, ( l + m + n ) ,LR − O MS , ( d ) ,ak, ( l + m + n ) ,RL (cid:17) , (B5)i.e., all operators of the same symmetry class containing thesame number of derivatives only differ in the LECs, since theoperators built of chiral fields cannot be sensitive to the ac- tual position of the derivatives. By construction, the occuringLECs obey the following constraints: κ MS , ( d ) k lmn = κ MS , ( d ) k nml , κ MA , ( d ) k lmn = − κ MA , ( d ) k nml . (B6)The LECs are further constrained by Eq. (34) which en-sures energy-momentum conservation in plus direction (thisreduces the number of parameters for a simultaneous fit of the0th, 1st and 2nd moments from to ). The terms con-tributing to the leading one-loop calculation of the respectivematrix elements in the limit of exact isospin symmetry can betaken from Table V.One finds that the additional Lorentz indices can only comefrom derivatives acting on the nucleon field (all other possibil-ities are either of higher order, contain too many pion fields or8 m π / GeV ϕ (a) m π / GeV ϕ (b) m π / GeV ϕ (c) m π / GeV ϕ (d) m π / GeV ϕ (e) m π / GeV ϕ (f) FIG. 11: Chiral extrapolations of the shape parameters of second order ϕ , ϕ , ϕ (9). 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