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
Article funded by SCOAP3
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, $a_k,\, k
4m_\pi^{2}$), 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}
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} where $I_{V}$ 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 ($\rho,\omega,\ldots$). ]]>