The ρ-meson light-cone distribution amplitudes from lattice QCD

Journal of High Energy Physics, Apr 2017

We present the results of a lattice study of the normalization constants and second moments of the light-cone distribution amplitudes of longitudinally and transversely polarized ρ mesons. The calculation is performed using two flavors of dynamical clover fermions at lattice spacings between 0.060 fm and 0.081 fm, different lattice volumes up to m π L = 6.7 and pion masses down to m π = 150 MeV. Bare lattice results are renormalized non-perturbatively using a variant of the RI′-MOM scheme and converted to the \( \overline{\mathrm{MS}} \) scheme. The necessary conversion coefficients, which are not available in the literature, are calculated. The chiral extrapolation for the relevant decay constants is worked out in detail. We obtain for the ratio of the tensor and vector coupling constants f ρ T /f ρ  = 0.629(8) and the values of the second Gegenbauer moments a 2 ‖  = 0.132(27) and a 2 ⊥  = 0.101(22) at the scale μ = 2 GeV for the longitudinally and transversely polarized ρ mesons, respectively. The errors include the statistical uncertainty and estimates of the systematics arising from renormalization. Discretization errors cannot be estimated reliably and are not included. In this calculation the possibility of ρ → ππ decay at the smaller pion masses is not taken into account.

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The ρ-meson light-cone distribution amplitudes from lattice QCD

Received: December -meson light-cone distribution amplitudes from lattice QCD Vladimir M. Braun 0 1 2 4 5 6 7 Peter C. Bruns 0 1 2 4 5 6 7 Sara Collins 0 1 2 4 5 6 7 John A. Gracey 0 1 2 3 5 6 7 Michael Gruber 0 1 2 4 5 6 7 Meinulf Gockeler 0 1 2 4 5 6 7 Fabian Hutzler 0 1 2 4 5 6 7 Paula Perez-Rubio 0 1 2 4 5 6 7 Andreas Schafer 0 1 2 4 5 6 7 Wolfgang Soldner 0 1 2 4 5 6 7 Andre Sternbeck 0 1 2 5 6 7 Philipp Wein 0 1 2 4 5 6 7 E-mail: 0 1 2 5 6 7 Open Access 0 1 2 5 6 7 c The Authors. 0 1 2 5 6 7 0 avors of dynamical clover 1 P. O. Box 147, Liverpool, L69 3BX , United Kingdom 2 Universitatsstra e 31 , 93040 Regensburg , Germany 3 Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool 4 Institut fur Theoretische Physik, Universitat Regensburg 5 decay at the smaller pion masses is not taken 6 In this calculation the possibility of 7 mesons , respectively We present the results of a lattice study of the normalization constants and second moments of the light-cone distribution amplitudes of longitudinally and transversely polarized mesons. The calculation is performed using two fermions at lattice spacings between 0:060 fm and 0:081 fm, di erent lattice volumes up to m L = 6:7 and pion masses down to m = 150 MeV. Bare lattice results are renormalized non-perturbatively using a variant of the RI0-MOM scheme and converted to the MS scheme. The necessary conversion coe cients, which are not available in the literature, are calculated. The chiral extrapolation for the relevant decay constants is worked out in detail. We obtain for the ratio of the tensor and vector coupling constants f T =f = 0:629(8) and the values of the second Gegenbauer moments a2 = 0:132(27) and a2 = 0:101(22) at the scale = 2 GeV for the longitudinally and transversely polarized The errors include the statistical uncertainty and estimates of the systematics arising from renormalization. Discretization errors cannot be estimated reliably and are not included. cTheoretisch-Physikalisches Institut; Friedrich-Schiller-Universitat Jena - into account. 1 Introduction 2 3 4 General formalism Continuum formulation Lattice formulation Lattice correlation functions Decay constants Second moments | the longitudinal case Second moments | the transverse case Details of the lattice simulations Renormalization Data analysis Results and conclusion A Transversity operators in the continuum B Chiral extrapolation B.1 E ective eld theory framework B.2 Chiral Lagrangians for resonances B.3 Extrapolation formulae B.4 Contributions to matrix elements B.5 Loop functions Introduction V = nal state are attracting increasing attention. Prominent examples are provided by B-meson weak decays, B ! V , B ! V ` `, B ! V , B ! V + Among these processes, the decays B ! K are of particular relethat are sensitive to new physics, see, e.g., ref. [3] for a recent review. Another example high energy, eN ! e N , that, besides deeply-virtual Compton scattering (DVCS), allows one to resolve the transverse distribution of partons inside the nucleon. The corresponding cross sections were measured by the HERA collider experiments H1 and ZEUS and xed target experiments HERMES (DESY), CLAS (JLAB), and Hall A (JLAB) at small and moderate values of the Bjorken momentum fraction xBj, respectively. In the the electron-ion collider (EIC) [4]. The standard framework for the theoretical description of such processes is based on the transverse degrees of freedom are integrated out. In general, meson and baryon DAs the momentum fractions) are given by matrix elements of local operators. From the pheat the origin) and the rst non-trivial Gegenbauer moment that characterizes the width of the DA are the most relevant quantities. For example, knowledge of the second moment of the DA of the longitudinally polarized meson is crucial for global ts of generalized parton distributions from the DVMP and DVCS data [5]. The -meson coupling to the vector current is known experimentally and the other parameters were estimated in the past using QCD sum rules [6], see also ref. [7] for an update. Lattice calculations of the tensor coupling have been reported in refs. [8{12] and the second moments in ref. [13]. In this work we present new results using two avors of dynamical clover fermions at lattice spacings between 0:060 fm and 0:081 fm, di erent lattice volumes and pion masses down to m = 150 MeV. Our approach is similar to the strategy used in our paper on the pion DA [14]. In addition to a much larger set of lattices as compared to the previous studies, a new element of our analysis is a consistent use of non-perturbative coe cients for the conversion between our non-perturbative renormalization scheme on the lattice and the MS scheme are not available in the literature for tensor operators, we accuracy. The chiral extrapolation for the relevant quantities is worked out in detail. Although our calculation presents a considerable improvement as compared to earlier we only consider mesons and leave the e ects of the SU(3) avor breaking for a future study. Likewise, we do not consider ! mesons that would require the calculation of dise ects due to the decay that becomes possible at the smaller pion masses used seems unlikely that such e ects are of principal importance. Last but not least, discretizabe an important problem in such calculations. We expect to be able to improve on some framework of the CLS initiative [15]. This work, aiming in the long run at smaller lattice spacings with the help of open boundary conditions, is in progress. The presentation is organized as follows. Section 2 is introductory, we collect the necessary de nitions and specify the quantities that will be considered in this work. Section 3 contains a list of the correlation functions that we compute on the lattice. The lattice ensembles at our disposal and the procedure used to extract the signal are described in section 4. A non-perturbative calculation of the necessary renormalization factors is deof the same operators in the continuum and sketch a two loop calculation of the correcoe cients between our RI0-SMOM scheme (de ned as in ref. [14]) and the MS scheme are presented in the auxiliary le attached to the electronic version of this paper. Section 6 is devoted to the data analysis and the extrapolation to the physical pion mass using, where available, chiral e ective eld theory expressions derived in appendix B. The nal section 7 contains a summary of our results and a discussion. meson has two independent leading twist (twist two) DAs, [16], corbreaking and electromagnetic e ects, the DAs of charged and neutral 0 mesons are related so that it is su cient to consider one of them, for example, + . The DAs are de ned h0jd(z1n)n=[z1n; z2n]u(z2n)j +(p; ) [z1n; z2n]u(z2n)j +(p; ) = m f (e( ) n) General formalism Continuum formulation e( ) p = 0 ; X e( )e( ) = e( ; ) = e( ) and we use the notation = if T e( 0) e( ) (p n) tion . The straight-line path-ordered Wilson line connecting the quark elds, [z1n; z2n], is inserted to ensure gauge invariance. The -meson polarization vector e( ) has the following The variable x has the meaning of the fraction of the meson's light-cone momentum p n which is carried by the u-quark, whereas 1 x is the momentum fraction carried by the antiquark d, and is the renormalization scale (we assume the MS scheme). The scale dependence will often be suppressed in what follows. The couplings f and f T appearing in (2.1) are de ned as matrix elements of local h0jd(0) u(0)j +(p; )i = f m e( ) ; u(0)j +(p; )i = if T e( )p in ref. [17] for a detailed discussion. One obtains [17] f + = (210 4) MeV ; f (0u) = (221:5 3) MeV ; f (0d) = (209:7 3) MeV ; where for the neutral meson we quote separate values for the uu and dd currents. The difwhich will be neglected throughout this study. The tensor coupling f T is scale dependent and is not directly accessible from experiment. To leading order one obtains The symmetry property (2.8) implies and, neglecting isospin breaking e ects, are symmetric under the interchange of the momentum fractions of the quark and the antiquark, For convenience we introduce a generic notation h as weighted integrals of the type ; for the moments of the DAs de ned f T ( ) = f T ( 0) CF = 0 where CF = (Nc2 Nf the number of active avors. The DAs are normalized to unity, 1)=(2Nc), 0 = (11Nc dx ; (x) = 1 ; ; (x) = x)li ; = x)kxli ; = h(1 and in addition we have the (momentum conservation) constraint x)kxli ; = h(1 = 2x corresponding to the di erence of the momentum fraction between the quark and the antiquark and consider or, alternatively, Gegenbauer moments n ; = h(2x n = 2; 4; 6; : : : ; n; = 2(2n + 3) 3(n + 1)(n + 2) h Cn3=2(2x relations, e.g., 2; = dence at the one loop level The anomalous dimensions are given by (0) = 8CF 2(n + 1)(n + 2) As Gegenbauer polynomials form a complete set of functions, the DAs can be written as an expansion Gegenbauer coe cients, e.g., an( ) = an( 0) an( ) = an( 0) ; (x; ) = 6x(1 x) 41 + X an; ( )Cn3=2(2x ; (x; ) = 3 1 + X an; ( ) : n(0)=(2 0) (0) =(2 0) (0) = 8CF Since the anomalous dimensions increase with n, the higher-order contributions in the term survives, usually referred to as the asymptotic DA: ! 1) = as(x) = 6x(1 Beyond the leading order, higher Gegenbauer coe cients an mix with the lower ones, ak; k < n [18, 19]. This implies, in particular, that Gegenbauer coe cients with higher values of n are generated by the evolution even if they vanish at a low reference scale. This e ect is numerically small, however, so that it is usually reasonable to employ the Gegenbauer expansion to some xed order. Lattice formulation From now on we work in Euclidean space, using the same conventions as in ref. [14]. The renormalized light-ray operators entering the de nition of the DAs are de ned as the means that moments of the DAs, by construction, are given by matrix elements of local operators and can be evaluated on the lattice using the Euclidean version of QCD. Our aim in this work is to calculate the couplings f , f T and the second DA moments. To this end we de ne bare operators V (x) = d(x) u(x) ; T (x) = d(x) (x) = d(x) (x) = d(x) D~D~ + D~ D~ D~D~ + D~ D~ On the lattice the covariant derivatives will be replaced by their discretized versions. Projection onto the leading twist corresponds to symmetrization over the maximal possible set of Lorentz indices and subtraction of traces. The operation of symmetrization and trace subtraction will be indicated by enclosing the involved Lorentz indices in parentheses, for instance, O( ) = 12 (O O . Note that for the operators involving the -matrix also those traces have to be subtracted which correspond to index pairs where one of the indices equals Using the shorthand D~~ = D~ )(x) can be rewritten as )(x) = d(x) ( D~~D~~)u(x) and its matrix element between the vacuum and the state is proportional to the bare value of the second moment h 2i : +(p; )i = N( where N( ) is a kinematical prefactor. The operator V(+ to the second derivative of the vector current, )(x) in the continuum reduces )(x) = @( @ d(x) )u(x) ; +(p; )i = N( with the same prefactor. While in the continuum h 12ibare = 1 by construction, this is no longer true on the lattice because the Leibniz rule holds for discretized derivatives only up to lattice artefacts and hence (2.25) is violated. As we will see below, the deviation from unity for the renormalized h 12i is small. Nevertheless, it still has to be taken into account and a ects the relation between h 2i and the Gegenbauer moment at nite lattice spacing [14]: a2 = The situation with the tensor operators T ( ) and the corresponding matrix elements The operators V( ) mix under renormalization even in the continuum, as . Additional mixing could result from the fact that the continuous Fortunately, in the case at hand it is possible to avoid additional mixing by using suitably chosen operators, which will be detailed below. Lattice correlation functions The basic objects from which moments of the DAs can be extracted on the lattice are 2-point correlation functions. In order to \create" the meson we use the interpolating current V (x), which is de ned as V (x) with smeared quark elds. For details of our smearing algorithm see section 4. Let O be a local (unsmeared) operator, e.g., one of the operators de ned in eq. (2.22) above. One then obtains for the 2-point function in the region where the ground state dominates 3 X e ip x hO(t; x)V y(0)i = A(O; V j p) = Here T is the time extent of the lattice, a is the lattice spacing, and E denotes the energy state. The sign factors are determined by the Dirac matrices in the creation operator (which is in our case always table 1), and nt is the number of time derivatives in O. O are the analogous factors for O (see For the decay constants and the second DA moments of the meson we have to evaluate the following set of correlation functions: Decay constants Cjj (t; 0) = Z Cjj (t; 0) = m f pZ C4jj (t; 0) = In the actual ts we average over the forward and backward running states. As in in the time direction is reached (see gure 1 for an example), the \mixing" of these two contributions is completely negligible. Therefore we work with simple exponential ts, where the averaging operator t^ is de ned as 1 X3 t^+Cjj (t; 0) = 1 X3 t^+Cjj (t; 0) = f 1 X3 t^ C4jj (t; 0) = t^ C(t; p) = The decay constants f and f T can be obtained by simultaneously tting the correlation functions (3.5a){(3.5c). The result for the mass is then dominated by the two from excited states so that the isolation of the ground state is less reliable. Therefore we rst t the correlator with a smeared operator at the sink, (3.5a), to extract Z and m . These values are then inserted in eqs. (3.5b) and (3.5c) in order to obtain f and f T as well an estimation of the statistical error. Second moments | the longitudinal case ments consist of the operators Here and in the following f g denotes symmetrization of the enclosed n indices with an form according to the irreducible representation (4) of the hypercubic group H(4) [21]. other, but mixing with additional operators of the same or lower dimension is forbidden. The amplitudes (3.2) of the 2-point functions (3.1) where O is one member of these multiplets are related to the amplitudes where O is a component of the vector current V by A(O1 ; V j p) = A(O2 ; V j p) = A(O3 ; V j p) = A(O4 ; V j p) = p2p3A(V4; V j p)+ip2EA(V3; V j p)+ip3EA(V2; V j p) ; p1p3A(V4; V j p)+ip1EA(V3; V j p)+ip3EA(V1; V j p) ; p1p2A(V4; V j p)+ip2EA(V1; V j p)+ip1EA(V2; V j p) ; p1p2A(V3; V j p)+p1p3A(V2; V j p)+p2p3A(V1; V j p) : In order to be able to write these and some of the following formulae in a compact form we have introduced the notation R , where R+ (R ) is the bare value of h12i (h 2i ). We will try to increase the signal-to-noise ratio by considering only correlation functions lattice of spatial extent L. Therefore we exclude O4 from our calculation. After averaging over all suitable combinations as well as over forward and backward running states, the O1 = Vf234g ; O2 = Vf134g ; O3 = Vf124g ; O4 = Vf123g : second longitudinal moments can be obtained from the ratio j=1 k=1 k6=j 1 X3 p^+t^+Ckk t; 2L ej where momentum averaging is accounted for by the operator p^ : Second moments | the transverse case In the transverse case we consider the following multiplets: p^ C(t; p) = C(t; p) : O1;T = T13f32g + T23f31g O2;T = T12f23g + T32f21g O3;T = T12f24g + T42f21g O4;T = T21f13g + T31f12g O5;T = T21f14g + T41f12g O6;T = T31f14g + T41f13g The two multiplets O1+;T ; : : : ; O6;T and O1;T ; : : : ; O6;T both transform according to the + irreducible representation 2 (6) of the hypercubic group H(4). As in the case of the multiplets (3.7), mixing with additional operators of the same or lower dimension is forbidden by symmetry. The amplitudes (3.2) of the 2-point functions (3.1) where O is one member of the multiplets (3.11) are related to the amplitudes where O is a component of the tensor A(O1;T ; V j p) = p2p3 A(T13; V j p) + p1p3 A(T23; V j p) A(O2;T ; V j p) = p2p3 A(T12; V j p) + p1p2 A(T32; V j p) + ip2E A(T41; V j p) + ip1E A(T42; V j p) ; A(O3;T ; V j p) = p1p2 A(T42; V j p) p1p3 A(T43; V j p) + ip3E A(T41; V j p) + ip1E A(T43; V j p) ; + ip2E A(T12; V j p) + ip3E A(T31; V j p) ; A(O4;T ; V j p) = p1p3 A(T21; V j p) + p1p2 A(T31; V j p) A(O5;T ; V j p) = p1p2 A(T41; V j p) p2p3 A(T43; V j p) + ip3E A(T42; V j p) + ip2E A(T43; V j p) ; + ip1E A(T21; V j p) + ip3E A(T32; V j p) ; A(O6;T ; V j p) = p1p3 A(T41; V j p) p2p3 A(T42; V j p) + ip1E A(T31; V j p) + ip2E A(T23; V j p) : Here R+ (R ) is the bare value of h12i (h 2i ). 1999( 4) 1998( 2) 1999( 1) 2028( 2) 1237( 2) 1599( 3) 982( 2) 1999( 2) 2178( 2) m [MeV] Nconf( Nsrc) = 5:20, a = 0:081 fm, a 1 = 2400 MeV = 5:29, a = 0:071 fm, a 1 = 2800 MeV = 5:40, a = 0:060 fm, a 1 = 3300 MeV hopping parameter , the pion mass m , the nite volume corrected pion mass m1 determined in ref. [22], the lattice size, the value of m L, where L is the spatial lattice extent, the number of con gurations Nconf and the number of sources Nsrc used on each con guration. Note that the pion masses have been slightly updated compared to the numbers in ref. [14]. The ensembles marked earlier within the QCDSF collaboration. As in the longitudinal case, we only consider correlation functions with the smallest non-zero momentum in one spatial direction and perform averages similar to those in eq. (3.9). This leads to the following ratio for the second transverse moments: 1 X3 j=1 l=1 k=1 l6=j k6=j k6=l Details of the lattice simulations For this work we used gauge con gurations which have been generated using the Wilson fermions. A list of the ensembles used is shown in table 2. We used lattices with three different inverse couplings 0:06 fm and 0:081 fm. The pion masses vary between 150 MeV and 500 MeV, with spatial volumes between (1:9 fm)3 and (4:5 fm)3. In order to increase the overall statistics we performed multiple measurements per conguration. The source positions of these measurements were selected randomly to reduce the autocorrelations. To obtain a better overlap with the ground state we applied Wuppertal smearing [23] in the interpolating current V using APE smeared gauge links [24]. calculated from the time- and momentumaveraged correlation functions according to eq. (3.9) on the = 5:29, = 0:13632, L = 32a, T = 64a ensemble. The cyan-colored bar indicates the tted value of R , the error and the tting range. For the statistical analysis we generate 1000 bootstrap samples per ensemble using a binsize of 4 to further eliminate autocorrelations. For the purpose of maximizing the statistics of the second moments, we average for each bootstrap sample over all suitable and backward running states as pointed out in eqs. (3.9) and (3.13). In order to reduce contributions from excited states the choice of the starting point of the t range is important. As an example, gure 1 demonstrates that, with increasing source-sink distance, the excited states fall below the noise and plateaus of the correlation functions for R The starting time tstart is then chosen in such a way that ts with even larger starting times no longer show any systematic trend in the tted values. Multi-state ts (over larger t ranges) yield consistent results. Renormalization Having computed the bare values of the second DA moments, we are left with the task of renormalizing these bare quantities to obtain results in the standard continuum MS there is only mixing between the respective + and operator multiplets, so we have to determine 2 2 mixing matrices such that One then obtains for the second moments of the DAs in the MS scheme = Z11O = Z22O+ : = 1;1 R = 2;2 R +; ; ij = ij = with the renormalization factors ZV and ZT of the vector and the tensor currents, respectively. Note that one cannot expect 2;2 to be equal to one, since the Leibniz rule holds on the lattice only up to discretization artefacts. We want to evaluate the renormalization and mixing coe cients non-perturbatively on the lattice employing a variant of the RI0-MOM scheme, because lattice perturbation theory is not su ciently reliable. Since forward matrix elements of the + operators vanish in the continuum limit, we cannot work with the momentum geometry of the original RI0-MOM scheme but must use a kind of RI0-SMOM scheme [25]. We follow exactly the the MS vertex functions of our operators in order to convert the results from our SMOM scheme to the MS scheme. While these are known to two loops in the longitudinal case, i.e., for the operators (2.22a), see ref. [26], as well as for the currents (2.21), see ref. [27], the corresponding results for operators with derivatives involving the matrix , e.g., the operators (2.22b), are not yet available in the literature. Therefore we discuss the latter case, the so-called transversity operators, in appendix A. In the end, we determine the matrix Z(a; 0) (and analogously (a; 0)) at the reference the expression Z(a; )MC = W ( ; 0)Z(a; 0) + A1a 2 2 + A2(a2 2)2 + A3(a2 2)3 ; where the three matrices Ai parametrize the lattice artefacts and W ( ; 0) describes the running of Z in the three loop approximation of continuum perturbation theory. Ignoring the very small statistical errors, we estimate the much more important systematic uncertainties of Z(a; 0) by performing a number of ts, where exactly one element of the analysis is varied at a time. More precisely, we choose as representative examples for t intervals 4 GeV2 < 2 < 100 GeV2 and 2 GeV2 < 2 < 30 GeV2, and we use the Finally, we consider values for r0 and r0 MS corresponding to the results given in ref. [28]. The various t possibilities are compiled in table 3. As in the case of the pion DA, the largest e ect comes from the variation of nloops. In order to obtain our nal numbers for the second moments of the DAs we extract them from the bare data R ; using each of these sets of values for 11, 12 and 22. So we have six results for each of our gauge eld ensembles. As our central values we take the results from Fit 1. De ning i as the di erence between the result obtained with the s from Fit i and the result determined with the s from Fit 1, we estimate the systematic uncertainties due to the renormalization factors as p 22 + (0:5 one loop to two loops. This should amount to a rather conservative error estimate. The (in GeV2) artefacts (in fm) A3 6= 0 A3 6= 0 A3 6= 0 A3 = 0 A3 6= 0 A3 6= 0 renormalization factors ZV and ZT needed for the evaluation of f and f T , respectively, are calculated in the same way. Data analysis From the bare values of f etc. we obtain renormalized results in the MS scheme with the help of our renormalization (and mixing) coe cients on each of our gauge eld ensembles. With the range of ensembles available (see table 2) we are able to study the pion mass dependence and, to only a limited extent, volume and discretization e ects. Since our for our nal numbers. decay constants Considering the pion mass dependence, we make use of Chiral Perturbation Theory (ChPT) for vector mesons [29{31] to obtain the one loop extrapolation formulae for the Re f = f (0) 1 Re f T = f T (0) 1 = f (0) 1 + + f (2)m2 + f (3)m3 + O(m4 ) ; + f T (2)m2 + f T (3)m3 + O(m4 ) ; + f (2)m2 + f (3)m3 + O(m4 ) : Details on the ChPT calculation are given in appendix B. For 2m < m , i.e., below the neglect instability e ects in our lattice computation, which is necessarily done on volumes, we use only the real part to t the mass dependence of our data. The pion decay constant F is kept xed at its physical value 92:4 MeV, and the chiral renormalization is chosen to be 775 MeV. Estimates within ChPT suggest that the third-order term / m3 is not negligible for most of our masses (see appendix B). Our data con rm this expectation | the third order term is required in order to t over the full range of pion masses. Consistent ts are obtained including only second order terms for m < 300 MeV, however, we have, essentially, only L = 40a L = 48a L = 64a L = 40a L = 64a L = 40a L = 64a L = 40a L = 48a L = 64a L = 40a L = 64a L = 40a L = 64a ts using eqs. (6.1) for the decay constants f , f T and their ratio, including (left) and excluding (right) the data point at m position of the physical pion mass. The band indicates the one sigma statistical error. two pion masses in this range. Alternatively, one can ignore the information from ChPT and perform polynomial ts, i.e., drop the logarithmic term in the t functions (6.1). This yields very similar results. We expect that a t including the larger pion masses will yield more reliable numbers than simply taking the values at m = 150 MeV as our nal results because, in particular, the lattice used at this pion mass is relatively small. In order to get at least some idea of the in uence of the instability of the two kinds of ts, including all masses or excluding the results at m = 150 MeV, which constants are shown in gure 2. Note that the extrapolated values at the physical point are reasonably consistent with the data at the lowest pion mass. L = 40a L = 64a L = 40a L = 64a L = 40a L = 64a L = 40a L = 64a the data point at m mass. The band indicates the one sigma statistical error. For the second moments of vector meson distribution amplitudes (see gure 3) no ChPT and the nucleon [34] do not contain chiral logarithms in leading one-loop order. The reasons are rather generic and may apply to vector mesons as well. Therefore we stick to simple linear ts in m2 depicted in gure 3. There is no discernible dependence on the gures 2{3. We perform an extrapolation for every choice given in table 3 and compute renormalization factors from the di erences of the extrapolated numbers as indicated at the end of section 5. Although our data do not allow us to study nite-size and discretization e ects systematically, we can make some observations. Considering volume e ects, for = 5:29, The e ects for the decay constants are sizable, see gures 2 and 4. Unlike the well-known forward to compute the leading nite-volume corrections for this ratio (see appendix B). = 5:29, L = 40a L = 64a L = 40a L = 64a indicates the position of the physical pion mass. It turns out that the corrections are numerically tiny so that from the ChPT analysis one expects that themselves. This expectation is in agreement with our data, as shown in gure 4: the vector coupling f itself (left panel). Since in phenomenological studies of hard reactions f accessible, is a much more interesting quantity. So we do not perform an in nite-volume extrapolation for f and use this measurement mostly for normalization purposes (e.g. computing the second moments). On the other hand, the observed volume dependence would not have any signi cant e ect. One can see from gure 3 that the second moments tend to increase with the spatial no leading chiral logarithms and a very mild nite-volume dependence. We have checked noticeable in uence on our results. f T [MeV] the main text. The numbers in parentheses denote the statistical error and our estimate of the uncertainty introduced by the renormalization procedure. Discretization errors are notoriously di cult to control. A certain insight can be obtained looking at the quantities h12iMS and h12iMS, which indicate the violation of the Leibniz rule at nite lattice spacing. In the continuum limit they should equal one for all pion masses. Results for all ensembles are plotted in gure 5. Again only statistical smaller. While h12iMS deviations are noticeably smaller than what we found in the case of the pion [14]. of about 1%, we observe deviations from one of up to 2% for h12iMS. Note that these equals one within the statistical errors with a maximal deviation Results and conclusion In table 4 we compare the results of the two kinds of nal ts that we have performed. The values in the row labelled \analysis 1" have been obtained by ts to all data points, while the row labelled \analysis 2" contains the results from ts where the data with the t functions (6.1), whereas the second Gegenbauer moments have been tted with linear functions of m2 . One sees that the results of the two ts are in very good agreement, which may be an indication that -meson decay, , is not of major importance for the shortmight be more important, but, unfortunately, cannot be estimated reliably using the set of lattices at our disposal. We expect to be able to quantify the discretization errors using the CLS initiative [15]. Comparing to the pion case we observe that for the meson we are able to access nents in order to compute a2 in the pion. This helps to reduce the statistical noise and the corresponding error. As our nal results we adopt the numbers from analysis 1. Although the systematic in the case of the decay constants f T and f , the agreement with the experimental value Sum rule calculations [6, 7] yield f numbers given in ref. [7] at the renormalization scale = 1 GeV have been evolved to lattice [10] lattice [12] lattice [13] sum rules [6, 7] 0:74(5) renormalization scale is = 2 GeV. the systematics are not yet fully controlled, the discrepancies do not look worrying. of our results have been added in quadrature. Again, the sum rule numbers at = 2 GeV have been obtained from the original results at with Nf = 2. Some of these quantities have already been investigated on the lattice. The BGR = 0:742(14) All existing results are, generally, in good agreement, apart from the ratio of decay investigations. This ratio depends strongly on the pion mass, cf. gure 2, and the extrapolation could be a ected by the decay at this level of accuracy. Clari cation of this at large recoil (see, e.g., ref. [17]), where, in some cases, there is a tension with predictions of the Standard Model. Our value for the second Gegenbauer coe cient a2 is signi cantly more precise compared to previous results. At this level of accuracy, we start to be sensitive to the di erence between the longitudinally and transversely polarized mesons. Our results suggest that a2 may be slightly larger than a2, although the di erence is not yet statistically signi cant. The 20% accuracy for a2 achieved in our work is interesting for GPD formalism [5]. Such processes will be investigated with high priority at the JLAB 12 GeV upgrade and, in the future, at the EIC. tions [15]. Apart from the study of discretization errors our goal is to consider DAs of the whole SU(3)f meson octet, with emphasis on properties of the K meson, which is of prime importance for avor physics. This work is in progress. Acknowledgments IRG 256594. The ensembles were generated primarily on the QPACE systems of the thank Benjamin Gla le for software support. Part of the analysis was also performed on of various institutions which we acknowledge below. One of the authors (JAG) thanks Dr. P.E.L. Rakow for useful discussions. Helpful conversations with G.S. Bali are gratefully acknowledged. The work of JAG was carried out with the support of STFC through the Consolidated Grant ST/L000431/1. The authors gratefully acknowledge the Gauss Centre for Supercomputing (GCS) for providing computing time for a GCS Large-Scale Project on the GCS share of the suand Nordrhein-Westfalen (MIWF). Transversity operators in the continuum we refer the interested reader for more background. First, the two classes of operators we are interested in are the avor non-singlet operators, where the operators with a single derivative have been included for completeness. We = 12 [ ; ] which is related to OT2 = S & = S & @T2 = S@ @T3 = S@ @@T3 = S@ @ = i& : Our use of & is to retain the same conventions with earlier renormalization of similar operators [26, 27, 40] and our use of generalized -matrices which we discuss later. To de ne the action of the symbol S, which imposes certain symmetrization and tracelessness conditions, it is best to consider the generalized transversity operators OT 1::: i::: n from O 1::: i::: n = 0 (i O 1::: i::: j::: n = 0 for the rst operator of the T2 sector with again a parallel de nition for the total derivative operator [42]. In our construction for the T3 operators we have taken the convention to as well as for the non-total derivative operator of each set. It will be clear from the context which is meant. The labelling for each derivative of a total derivative operator is one @ symbol applied to the sector label. In de ning the operators we have omitted the explicit avor indices and note that our perturbative renormalization will be for massless quarks; in other words we are in the chiral limit. The total derivative operators are required since there is operator mixing within each separate sector. It would not usually be necessary to include these but since the Green's functions they are needed for are non-forward matrix elements, then a momentum will ow through the operator insertion and the mixing will OiTol = ZiTjl OjTl ZiTj2 = ZiTj3 = B 0 We use 1 and 2 to label the elements of the T2 matrix where 1 is the operator T2. Similarly 1, 2 and 3 label the T3 matrix elements which respectively correspond to T3, @T3 and @@T3. The explicit mixing matrix for the T2 system has been determined in ref. [42] to three loops in the MS scheme. Prior to the results we present here, the T3 matrix was known only operator T3 itself and the remaining two diagonal entries are the same as the operator T2 and the tensor current [42{44]. In other words the operators of the T2 system without because the non-zero entries of the nal two rows of ZiTj3 are the non-zero entries of the ZiTj2 matrix. We have determined the nal two o -diagonal elements of ZiTj3 by renormalizing legs is nulli ed. In other words there is a non-zero momentum owing through the inserted operator. This was the method used to determine a similar mixing matrix for the third moment of the usual twist-2 Wilson operators in deep inelastic scattering [26]. However, in ref. [26] it was noted that such a computational setup was not su cient to determine piece of information was required. This is achieved here for T3 by the identity Z1T23 = Z2T23 which is straightforward to establish by integration by parts. Thus to deduce these reloop renormalization of the operator T3. As the resulting anomalous dimensions for T2 are given in ref. [42], we record the rst row of the three loop anomalous dimension mixing matrix for T3 which is 1T13(a) = CF a + CF [1195CA + CF 10368 3CA2 + 126557CA2 31104 3CACF 30197CACF 67392 3CANf TF 38900CANf TF + 20736 3CF2 17434CF2 + 67392 3CF Nf TF 50552CF Nf TF 18557CACF + 10368 3CANf TF 10368 3CF Nf TF + 10696CF Nf TF 1T23(a) = 1T33(a) = 3 CF a + CF [ 125CA + 34CF + 64Nf TF ] 27 + 694CANf TF 6790CA2 + 15552 3CACF 10368 3CF2 + 16736CF2 + O(a4) ; + O(a4) ; 3 CF a + CF [11CA 109CF + 20Nf TF ] 54 47952 3CA2 + 32969CA2 + 132192 3CACF 138749CACF + 25920 3CANf TF 3200CANf TF 72576 3CF2 + 27332CF2 25920 3CF Nf TF + 39040CF Nf TF + 2000Nf2TF2 4860 + O(a4) ; gauge parameter cancels as it ought to for gauge invariant operators in the MS scheme. Having summarized the renormalization of the operators of interest the next stage is to provide the perturbative corrections to the Green's function where the operator is Green's function the fully symmetric point given by from which we have inserted in a quark 2-point function. As we are considering non-forward matrix elements there is a momentum owing through the operator. More speci cally we consider the two independent external momenta p and q and we will evaluate the Green's function at is a mass scale. For this section we will take this scale to be the same mass scale 2 dimensions to ensure the coupling will not have any logarithms of mass parameter ratios. As each Green's function has free Lorentz indices we choose to decompose them into a basis of Lorentz tensors denoted by (p; q) and P(k) (p; q). Here T2 and T3 indicate the sector as the basis will be the same for the Green's function with the total derivative operators of each sector too. The choice of tensors in each basis is not unique. However, each basis is large due to the number of objects available to build the tensors. These include the momenta p and q as . In addition there are Lorentz tensors built from the -matrices. As in previous perturbative evaluations [26, 40] we use the generalized -matrices of [46{48] denoted by for integers n span the spinor space when dimensional regularization is used. As an aside we note that it is in this context that our choice of & in the operator de nition ts naturally. The algebra and properties of these matrices is well-established [49, 50]. We note one speci c property which is important here which is p2 = q2 = (p + q)2 = p q = where there is no sum over repeated m or n and I 1::: m 1::: n is the generalized unit matrix. The key point is that this trace partitions the space spanned by the tensors in the basis into distinct sectors. As we consider the operators in massless QCD, only will be needed. For T3 it might be expected that the symmetrization conditions exclude this -matrix from the basis. Finally with these objects we have constructed the tensor basis for each sector. For T2 that involves 30 tensors consistent with the symmetry properties of the inserted operator. A sample set is presented below. For T3 there are 42 tensors and for space reasons these as well as the full T2 set are given in the attached data le. The next step is to compute the coe cients in the decomposition of each Green's func1::: 6 would be required but p2=q2= = Cl for the di ering dimensionalities of the tensor basis and Green's functions for each sector. Thus we have C2 = i and C3 = 2. The algorithm to determine these coe cients has been given in refs. [26, 40] for instance. Brie y, to apply the multiloop perturbative integration techniques to nd these amplitudes we have to extract scalar Feynman integrals which is achieved by a projection method. The projection matrix, MiTjl , required for each sector is constructed from the respective tensor basis [26, 40] as it is the inverse of the matrix NiTjl = tr (p; q)P(Tjl) 1::: l+1 (p; q) p2=q2= Due to the size of the matrices, their explicit form is given in the auxiliary data le provided. shortcut. Hence we have p2=q2= Next we brie y note the practical details of actually carrying out the two loop evaluanotation after all the Lorentz and color indices have been included. There are 3 graphs at one loop. At two loops there are 32 graphs for OT2 and 37 for OT3 with fewer graphs for total derivative operators. After this the Feynman rules for either operator together out to produce a large number of scalar Feynman integrals that need to be calculated. To products of the momenta in the numerators of the integrals are written in terms of the propagators. In addition there may be propagator forms which are not present which are referred to as irreducible. In this format the Laporta algorithm [54] is then applied which uses integration by parts to systematically construct all the algebraic relations between expansion is known from direct computation [55{58]. Therefore, we are able to and two loops. Whilst this is in essence the Laporta method [54] one has to construct the relations in a practical way. We have chosen to use the Reduze package [59]. Moreover, topology at one loop and two at two loops. The latter are the ladder and non-planar three cases. The nal stage is to carry out the overall renormalization. This is achieved by and gauge parameter, following the procedure introduced in ref. [60] for automatic symbeing extracted at the end to leave the nite expressions for each scalar amplitude. To allow orientation to the full data available in the attached data le we give a selection each (n)-matrix partition for both operators of the T2 sector. For instance, we have (T22)(p; q) = 1:000000 + [0:271008 + 2:395659]a (T223)(p; q) = [0:472269 + 1:416806]a (T229)(p; q) = [ 0:222222 + [1:329626 2 + 2:430759 6:178403Nf + 55:151461]a2 + O(a3) ; + [1:795895 2 + 3:195370 2:817413Nf + 36:018151]a2 + O(a3) ; + [ 0:808446 2 4:040708 + 0:886539Nf 14:783322]a2 + O(a3) ; (@2T)2 (p; q) = 1:000000 + [ 0:062325 + 0:062325]a + [0:054445 2 + 0:640942 1:600114Nf + 17:009954]a2 + O(a3) ; (@2T32)(p; q) = [0:347245 + 1:041736]a (@2T92)(p; q) = [ 1:041737 + [1:302171 2 + 3:618039 1:851976Nf + 25:400736]a2 + O(a3) ; + [ 3:906512 2 10:854117 + 5:555928Nf 76:202209]a2 + O(a3) ; is the gauge parameter and the restriction stands for evaluation at (A.9) and (A.10). Although we are only interested in the values in the Landau gauge, de ned in the MS scheme. Next we summarize some aspects of the tensor basis and projection matrix for the T2 sector. Indeed one purpose of this summary is to provide an aid to the understanding of the full information given in the attached data le for both T2 and T3. Due to the size of the bases and matrices we used, a useable electronic format is more appropriate for their representation. First, we present a selection of tensors in the T2 basis choosing several representatives from each (n)-matrix partition. When one of the external momenta is (p; q) = & q + & q + [2& qq q (p; q) = & p + [2& pq q + d& pq q + & pq q + d& qp q + 3& qp q + d& qp q +& qp q + d& qq q + 2& qq q + d& pq q + & pq q + d& qp q (p; q) = & pp p (p; q) = MT2 = B where the subscript on the block matrices corresponds to the label of the analogous (n)matrix appearing in the projection tensor. Each of these partitions is of di erent size being respectively 22, 6 and 2 dimensional. Given the size of the rst two submatrices it is again not feasible to display all entries. Instead we choose to give a few reference entries to facilitate the extraction of the full matrices from the data le. We have M(T22) 6 20 = M(T02) 3 6 = M(T22) 15 10 = M(T02) 4 2 = where indices of M(0) i j range from 1 to 6 and these can be mapped to the labels of the tensor basis by adding 22. Finally, the remaining sector is compact enough to record it (p; q) = [p p p (p; q) = We have only shown one tensor from the nal partition as the other is given by replacing the uncontracted vector p by q. For each of the bases we have explicitly constructed the projection matrix coe cients. For T2 as there are 30 projectors this would correspond to a 30 30 matrix where the entries are rational polynomials in d. However, as we are using the generalized basis of -matrices in d-dimensions the projector matrix is block diagonal due to the property of (A.12). In other words completely as M(T42) = Overall the matrix MT2 is symmetric as is MT3 . Finally, this information should be su cient to connect with the full electronic representation for both sectors. Chiral extrapolation E ective eld theory framework In the speci c framework of Chiral Perturbation Theory (ChPT, see, e.g., refs. [61{63]) applied here, the generating functional of all QCD correlators is evaluated by means of a path integral involving an e ective low-energy Lagrangian Le (U; v; a; s; p; : : :) (compare with ref. [61], and eqs. (1) and (2) of ref. [30]), eiZ[v;a;s;p;t] = h0j T exp i [dU ] exp i Formally, all QCD Green's functions can be obtained by taking functional derivatives axial-vector and antisymmetric tensor source elds s; p; v ; a ; t . It should be noted that the tensor structure with an additional 5 is not independent due to the identity 5 = 2i . The dots stand for other possible source elds (for example, the coupling to symmetric tensor elds has been considered in refs. [64, 65]). The tensor source has been incorporated in ref. [66]. The matrix eld U collects the pion (Goldstone elds in a convenient way (see below). The e ective Lagrangian has to be invariant under local chiral transformations of the Goldstone boson and source elds, and shares all other symmetries of LQCD. A formal proof that low-energy QCD can indeed be analyzed in this way has been given by Leutwyler [62]. Under chiral transformations (L; R) 2 SU(2)L SU(2)R, the quark and external source elds transform as qL := l := v s + ip ! R(s + ip)Ly ; qR := ! Ll Ly + iL@ Ly ; r := v + a ! Rr Ry + iR@ Ry ; := P = P The e ective Lagrangian and the perturbative series are ordered by a low-energy power (or quark masses). For details and further references, we refer to refs. [61{63]. At leading chiral order, the e ective Lagrangian describing the interaction of the pseudo-Goldstone bosons (pions) with the external source elds and each other is given by (see ref. [61]) L(M2) = part of s. The brackets h constant in the chiral limit (F Here U = exp(ip2 =F ) with i denote the avor (or isospin) trace, F is the pion decay 86 MeV), and r U = @ U i(v + a )U + iU (v labels the speci c pion, and are the pertaining channel matrices. We write out + = = p where the a are the Pauli matrices. The matrix eld U transforms as U ! RU Ly under chiral transformations. K(L; R; U ) (which is also unitary). Below we shall set the external elds p; a to zero, = ta 2 . At fourth order, we have L(M4) = i `6 F + [u ; u ]i where we only show the terms needed for our present work (see refs. [61, 66, 67] for the complete Lagrangian at that order, and eq. (B.7) for the de nition of the operators u , and T ). Chiral Lagrangians for resonances of ChPT already in refs. [29, 30]. In the following, a \heavy vector meson" framework was set up [31, 68{71] to deal with problems related to the modi ed power counting in the mass in the chiral limit). Today, it is better understood how to deal with such problems in scheme for the e ective eld theory [72{74]. Such methods have been applied to the case of heavy meson resonances in refs. [75{82]. We refer to these references for details on the vector meson e ective eld theory outlined below. Keeping in mind the transformation behavior of the external source elds v , a and given above, we can write down the following terms describing the interaction of the vector mesons with the external source elds and the pions (compare also the previous LiVnVt = = uF L uy = uyt uy LiVnt = F R;L = @ (v r U = @ U i(v + a )U + iU (v u = iuy (r U ) uy ; u = p = D V D V := @ V i(v + a )]u + u[@ Extrapolation formulae For the sake of completeness, we rst discuss the pion matrix elements see eq. (B.3) for the channel matrices . We have used a large-Nc argument here to cast and ! elds in the matrix form of the last line in eq. (B.7), compare also with eq. (27) of ref. [31]. The dots indicate terms of higher chiral order, terms involving external source elds s; p (which are not needed here), or terms involving more derivatives, which result in contributions of the same form as those resulting from the terms given above, when using the equations of motion or eld transformations [84]. The vector eld propagator in momentum space is (q) = ( i) f v (k2) = 1 f t (k2) = + rmM 2 + rkk2 + O(p4) ; The standard framework of ChPT yields p)i = i abc(2p p)i = abc(k p I (k2) + O(p4) ; (B.11) where the loop function I (k2) is given at the end of this appendix, in eq. (B.25) (it vanishes for k2 ! 0, and is complex for k2 > 4m2 ), and `r6; rm; rk are renormalized lowenergy constants, which depend on the scale . M is the leading term in the quark-mass expansion of the pion mass m , derived from the Lagrangian (B.2) (at the order we are working, it can be set equal to the pion mass). We note that, up to corrections of two elastic unitarity, Im f v (k2) = f v (k2) (k2)t11 (k2) ; Im f t (k2) = f t (k2) (k2)t11 (k2) ; 4m2 < k2 < 16m2 ; where (s) := t11(s) = for 4m2 < s < 16m2 : It easily follows that the form factors f v;t must have the phase 11(s) in the elastic region. Contributions to matrix elements Here we use the de nitions Here we have to set k2 s equal to the rho pole, k2 function renormalization factor is derived from the ! spole = m2 [85]. The wave where the contribution of the one loop graphs to the self-energy is given by (compare loop(s) := qj b(k; )i = i ab(e( )k e( )k )f T =p2 ; V M 2 + A IA!(k2) : and nd at the one loop level up to O(p4) The \tadpole" terms can be taken to be energy independent at the order we are working to. The integral IA can be deduced from eqs. (B.23){(B.26) below, and IA! is given by eq. (B.29) (with mV ! m! associated with local operators hF + m ). The local terms proportional to cV , cTV can be i etc., and can be used to absorb (real) terms of O(M 2) from the loop integrals. The loop functions are given at the end of this appendix (IA! = IAV (mV ! m! graphs). The leading non-analytic term in IA! is given by m )). In the one loop approximation, we evaluate m2!) = and the terms of order M 0 and M 2 are absorbed in the corresponding LECs. The chiral logarithm of this integral is of O(M 4). One nds where c is the following combination of (renormalized) LECs, c := The coe cient of the leading chiral logarithm is in agreement with ref. [67]. With gV (see ref. [80], and references therein), the coe cient of the third-order term should be . Inserting this estimate, and = 770 MeV, the third-order term becomes comparable to the leading chiral logarithm for M & 200 MeV, so it might give a non-negligible contribution for most data points. motivates the extrapolation formula (6.1c), while the formulae (6.1a) and (6.1b) result from (B.16) and (B.17), respectively, upon inserting the explicit expressions for the loop functions given below. The cusp e ects and imaginary parts in the chiral behavior of the on Euclidean lattices with a nite volume. A more thorough analysis is needed to deal with given above are not a icted by this de ciency. This can be deduced from the fact that they agree with the corresponding results in the heavy vector meson framework [67, 71], where the unitarity e ects due to the loops are either dropped or derived from contact terms of a non-Hermitean part of the e ective Lagrangian (see, e.g., ref. [68]). In the expression for the ratio given in eq. (B.21), the factors of pZ and the nonanalytic terms in the loop function IA containing the imaginary part cancel at one-loop order. Due to this simpli cation, it is straightforward to compute the nite-volume corrections for this ratio. Here, we attempt only an estimate of the leading correction, related to the O(M 2) `chiral-log' term contained in the tadpole loop integral I (compare eq. (B.27) below). According to the standard formalism of ChPT in a nite I(L) = I + 06=k2Z3 Here K1(z) is the modi ed Bessel function of the second kind, which decays exponentially for large positive z, K1(z) ! 2z e z . Inserting (B.22) in (B.16) and (B.17) yields the nite-volume correction to the ratio of eq. (B.21) upon a straightforward chiral Loop functions in the formulae above. The loop integral with two pion propagators is given by I (s) = k2 =: s : It diverges when the space-time dimension d approaches 4, I (0) = 2 + however the di erence I (s) := I (s) I (0) is nite, I (s) = where it is understood that real values of s are approached from the upper complex plane for s 2 [4M 2; 1). Explicitly, In the chiral limit (M ! 0), I (s) = 1 + 0(s) artanh 0(s) := s > 4M 2. We have also employed the abbreviation 0(s) (s 4M 2)=(16 ) for real A := I := M 2 = 2M 2 for d ! 4. The scalar integral including two di erent propagators can be written as s) = = I V (m2V ) and we refer to appendix B of ref. [80] for details on the chiral expansion. We also use where IV is given by the formula for I with M ! mV . Here, the letter V stands for the vector meson running in the loop ( ; !; : : :). Open Access. This article is distributed under the terms of the Creative Commons any medium, provided the original author(s) and source are credited. 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Vladimir M. Braun, Peter C. Bruns, Sara Collins, John A. Gracey, Michael Gruber, Meinulf Göckeler, Fabian Hutzler, Paula Pérez-Rubio, Andreas Schäfer, Wolfgang Söldner, André Sternbeck, Philipp Wein. The ρ-meson light-cone distribution amplitudes from lattice QCD, Journal of High Energy Physics, 2017, 82, DOI: 10.1007/JHEP04(2017)082