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 flavors of dynamical clover fermions at lattice spacings between fm and fm, different lattice volumes up to and pion masses down to MeV. Bare lattice results are renormalized non-perturbatively using a variant of the RI-MOM scheme and converted to the 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 and the values of the second Gegenbauer moments and at the scale GeV for the longitudinally and transversely polarized 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 decay at the smaller pion masses is not taken into account.

Lattice field theory simulation Non-perturbative renormalization

Article funded by SCOAP3

Introduction

General formalism

Continuum formulation

1-x} \end{align} For convenience we introduce a generic notation $\langle \cdots \rangle^{\goodparallel,\goodperp}$ for the moments of the DAs defined as weighted integrals of the type \begin{align} \langle x^k (1-x)^l \rangle^{\goodparallel,\goodperp} & = \int_0^1 \!dx\, x^k (1-x)^l\, \phi^{\goodparallel,\goodperp}_\rho(x)\,. \label{phi:moments} \end{align} The symmetry property~\eqref{phi:x->1-x} implies \begin{align} \langle (1-x)^k x^l \rangle^{\goodparallel , \goodperp} = \langle (1-x)^l x^k \rangle^{\goodparallel , \goodperp}\,, \end{align} and in addition we have the (momentum conservation) constraint \begin{align} \langle (1-x)^k x^l \rangle^{\goodparallel , \goodperp} = \langle (1-x)^{k+1} x^l \rangle^{\goodparallel , \goodperp} + \langle (1-x)^k x^{l+1} \rangle^{\goodparallel , \goodperp}\,. \end{align} Hence the set of moments~\eqref{phi:moments} for positive integers $k,l$ is overcomplete. We introduce the variable \begin{align} \xi = 2x-1 \,, \end{align} corresponding to the \emph{difference} of the momentum fraction between the quark and the antiquark and consider $\xi$ moments \begin{align} \langle \xi^n \rangle^{\goodparallel , \goodperp} = \langle (2x-1)^n \rangle^{\goodparallel , \goodperp}\,,\qquad n=2,4,6,\ldots\,, \end{align} or, alternatively, Gegenbauer moments \begin{align} a_n^{\goodparallel , \goodperp} = \frac{2(2n+3)}{3(n+1)(n+2)} \langle\, C_n^{3/2}(2x-1)\, \rangle^{\goodparallel , \goodperp} \end{align} as independent non-perturbative parameters. The two sets are related by simple algebraic relations, e.g., \begin{align} \label{eq:a2xi2} a_2^{\goodparallel,\goodperp} &= \frac{7}{12}\bigl(5\langle\xi^2\rangle^{\goodparallel,\goodperp}-1\bigr)\,. \end{align} The rationale for using Gegenbauer moments is that they have autonomous scale dependence at the one loop level \begin{subequations} \begin{align} a^\goodparallel_n(\mu) &= a^\goodparallel_n(\mu_0) \biggl(\frac{\alpha_s(\mu)}{\alpha_s(\mu_0)}\biggr)^{\gamma^{\goodparallel(0)}_n/(2\beta_0)}\,, \\ a^\goodperp_n(\mu) &= a^\goodperp_n(\mu_0) \biggl(\frac{\alpha_s(\mu)}{\alpha_s(\mu_0)}\biggr)^{\bigl(\gamma^{\goodperp(0)}_n-\gamma^{\goodperp(0)}_0\bigr)/(2\beta_0)}\,. \end{align} \end{subequations} The anomalous dimensions are given by \begin{align} \gamma^{\goodparallel(0)}_n = 8C_F \left( \sum_{k=1}^{n+1} \frac{1}{k} - \frac{3}{4} - \frac{1}{2(n+1)(n+2)} \right), && \gamma^{\goodperp(0)}_n = 8C_F \left( \sum_{k=1}^{n+1} \frac{1}{k} - \frac{3}{4} \right). \end{align} As Gegenbauer polynomials form a complete set of functions, the DAs can be written as an expansion \begin{align} \phi_\rho^{\goodparallel,\goodperp}(x,\mu) &= 6x(1-x) \left[ 1 + \smashoperator{\sum_{n=2,4,\ldots}^\infty} a^{\goodparallel,\goodperp}_n(\mu) C_n^{3/2}(2x-1)\right]. \end{align} Typical integrals that one encounters in applications can also be expressed in terms of the Gegenbauer coefficients, e.g., \begin{align} \int_0^1\!\frac{dx}{1-x}\phi^{\goodparallel,\goodperp}_\rho(x,\mu)&= 3\, \Biggl[1 + \smashoperator{\sum_{n=2,4,\ldots}^\infty} a^{\goodparallel,\goodperp}_n(\mu)\Biggr]\,. \end{align} Since the anomalous dimensions increase with $n$, the higher-order contributions in the Gegenbauer expansion are suppressed at large scales so that asymptotically only the leading term survives, usually referred to as the asymptotic DA: \begin{align} \label{def:phias} \phi^{\goodparallel,\goodperp}_\rho(x,\mu\to\infty) &= \phi_{\text{as}}(x) = 6x(1-x)\,. \end{align} Beyond the leading order, higher Gegenbauer coefficients $a_n$ mix with the lower ones, $a_k,\, k Lattice formulation Lattice correlation functions Decay constants Second moments --- the longitudinal case Second moments --- the transverse case Details of the lattice simulations Renormalization Data analysis Results and conclusion Acknowledgments Transversity operators in the continuum Chiral extrapolation Effective field theory framework Chiral Lagrangians for resonances Extrapolation formulae 4m_\pi^{2}$), and $\ell_{6}^{r},\lambda^{r}_{m},\lambda^{r}_{k}$ are renormalized low-energy constants, which depend on the scale $\mu_\chi$. $M$ is the leading term in the quark-mass expansion of the pion mass $m_\pi$, derived from the Lagrangian~(\ref{eq:L2M}) (at the order we are working, it can be set equal to the pion mass). We note that, up to corrections of two loop order, these expressions for the form factors are consistent with the constraints from elastic unitarity, \begin{align} \operatorname{Im}f_{\pi\pi}^{v}(k^2) &= f_{\pi\pi}^{v}(k^2)\sigma(k^2)t_{1}^{1\ast}(k^2)\,, \nonumber \\ \operatorname{Im}f_{\pi\pi}^{t}(k^2) &= f_{\pi\pi}^{t}(k^2)\sigma(k^{2})t_{1}^{1\ast}(k^{2})\,,\quad 4m_\pi^{2}

Contributions to \texorpdfstring{$\rho$}{rho} matrix elements

Loop functions

4M^2$. We have also employed the abbreviation \begin{equation}\label{eq:IpipiA} I_{\pi\pi}^{A}:=\frac{1}{4(d-1)}\left(2I_{\pi}-(s-4M^2)I_{\pi\pi}\right)\,, \end{equation} where \begin{equation}\label{eq:tadpi} I_{\pi} := \int\frac{d^{d}l}{(2\pi)^{d}}\frac{i}{l^{2}-M^{2}}=2M^{2}\bar{\lambda}+\frac{M^{2}}{16\pi^{2}}\log\left(\frac{M^{2}}{\mu_\chi^{2}}\right)\,, \end{equation} for$d\rightarrow 4. The scalar integral including two different propagators can be written as \begin{align}\label{eq:IpiV} I_{\pi V}(k^{2}\equiv s) = \int\frac{d^{d}l}{(2\pi)^{d}}\frac{i}{((k-l)^{2}-m_V^{2})(l^{2}-M^{2})} = I_{\pi V}(m_V^{2})-\frac{(s-m_V^{2})}{16\pi^{2}}J^{\pi V}(s)\,, \end{align} and we refer to appendix~B of ref.~\cite{Bruns-ml-2013tja} for details on the chiral expansion. We also use \begin{equation}\label{eq:IMVA} I_{\pi V}^{A}=\frac{1}{4s(d-1)}\big((4sM^2-(s+M^2-m_V^2)^2)I_{\pi V}+(s+M^2-m_V^2)I_{\pi} +(s-M^2+m_V^2)I_{V}\big), \end{equation} whereI_{V}$is given by the formula for$I_{\pi}$with$M\rightarrow m_V$. Here, the letter$V$stands for the vector meson running in the loop ($\rho,\omega,\ldots\$). ]]>

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