%Dear Organizers of SPIN96,
%
% I am sending the manuscript entitled
% "$Q^2$-evolution of chiral-odd twist-3 distributions
% $h_L(x,Q^2)$ and $e(x, Q^2)$ in the large $N_c$ limit"
% (Session IV.2)
% by I.I. Balitsky, V.M. Braun, Y. Koike and K. Tanaka,
% which I would like to submit for the proceedings of
% SPIN96 Symposium.
% It is prepared by using LaTeX.
%
% Yours sincerely,
% Kazuhiro Tanaka
%  Department of Physics, Juntendo University
%  Inba-gun, Chiba 270-16, Japan
%
%  TEL: +81-476-98-1001 (ext.367)
%  FAX: +81-476-98-1011
%  Email: tanakak@rikaxp.riken.go.jp
%
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\begin{document}
\begin{center}
{\large \bf
$Q^2$-evolution of 
chiral-odd twist-3 distributions\\
$h_L(x,Q^2)$
and $e(x, Q^2)$ in the large $N_c$ limit
\\ }
\vspace{5mm}
I.I. Balitsky$^1$, V.M. Braun$^{2}$, Y. Koike$^3$ 
and K. Tanaka$^4$
\\
\vspace{5mm}
{\small\it
(1) Center for Theoretical Physics, MIT,
Cambridge, MA 02139, U.S.A\\
(2) NORDITA, Blegdamsvej 17, DK-2100 Copenhagen, Denmark\\
(3) Dept. of Physics, Niigata University, Niigata 950-21, Japan\\
(4) Dept. of Physics, Juntendo University, Inba-gun, 
Chiba 270-16, Japan
\\ }
\end{center}

\begin{center}
ABSTRACT

\vspace{5mm}
\begin{minipage}{130 mm}
\small
We prove that the twist-3 chiral-odd parton distributions 
obey simple GLAP 
evolution equations in the limit $N_c\to\infty$ 
and give analytic
formulae for the corresponding anomalous dimensions. 
The results are valid to $O(1/N_{c}^2)$ accuracy and
will be useful in confronting
with future experiments.
\end{minipage}
\end{center}

The nucleon has three independent twist-3 parton
distributions $g_{2}(x, Q^{2})$, $h_{L}(x, Q^{2})$ 
and $e(x, Q^{2})$[1].
$g_{2}$ is chiral-even while $h_{L}$ and $e$ are chiral-odd.
The increasing precision of experiment 
%data 
%from LEP, HERA and the TEVATRON
requires understanding of these higher twist effects induced by 
correlations of partons.
% in the colliding (produced) hadrons. 
In particular, $g_{2}$ and $h_{L}$ 
play a distinguished role in spin physics,
since they can be measured as leading effects for certain 
asymmetries[2,1]. 
%The E143 collaboration has reported the 
%measurement of 
%$g_2$ via the polarized DIS[2], providing with 
%the first experimental test of quark-gluon 
%correlations in the nucleon. $h_L$ 
%is measurable in the Drell-Yan production by the polarized beam[1].
    
The $Q^2$-evolution of twist-3 distributions is generally
quite sophisticated due to mixing with quark-antiquark-gluon operators,
the number of which increases with spin (moment of the distributions).
For the flavor-nonsinglet part of
$g_2$, a crucial simplification has been pointed out
%the operators involving gluons effectively decouple from the 
%evolution equation 
in the large $N_c$ limit, that is neglecting $O(1/N_c^2)$
corrections[3]:
$g_2(x,Q^2)$ obeys a simple GLAP evolution equation
and the corresponding anomalous dimension is known in analytic form.
%The statement holds true with full account for effects
%subleading in $N_c$ but for large moments $n$. 
%Thus the claimed accuracy is 
%in fact $O(1/N_c^2 \cdot \ln(n)/n)$. The ABH evolution equation gives a 
%guide to the expected small x and large x behavior 
%which is important for experimental extrapolations, and is
%used\cite{bag} to rescale the model predictions to high values 
%of $Q^2$ of the actual experiments.

%Physics of this decoupling is so far not understood, and it is 
%probably not related to usual simplifications of the $N_c\to\infty$ limit.
%The observed phenomenon appears, however, to be quite general.
In this work[4] we demonstrate that the same pattern is
obeyed by chiral-odd distributions $h_L(x, Q^{2})$ 
and $e(x, Q^{2})$ and that the simplification 
is of universal nature.
% as well, albeit with different 
%anomalous dimensions. 
{}For all practical purposes this solves the problem of  
the $Q^2$-evolution of twist-3 nonsinglet parton distributions,
since the corrections $1/N_c^2$ are small.

{}Following[5,1]
we define the parton distributions 
as nucleon matrix elements of nonlocal light-cone operators
(we do not show the gauge phase factors)
\begin{equation}
\int\frac{dz}{2 \pi} e^{-i (P\cdot z)x}
\langle PS|\bar\psi(z/2)\Gamma \psi(-z/2)|PS\rangle.
\end{equation}
Here $x$ is the Bjorken variable,
$z$ is a light-like vector $z^2=0$, 
and $|PS\rangle$ is the nucleon state with its momentum $P$ and 
the spin $S$.
The substitution $\Gamma = 1$ and $\sigma_{\mu \nu}i \gamma_{5}$
generates $e(x)$ and $h_{L}(x)$ respectively 
as the twist-3 contributions[1].

By using the equations of motion, 
the twist-3 
quark-antiquark distributions can be 
expressed in terms of quark-gluon
correlations. The relevant operator identities are[6] 
%(see Eqs.(27),(28) in \cite{BF2}). 
\begin{equation}
 \bar\psi(z)\psi(-z) = \bar\psi(0)\psi(0)
 +\int_0^1 du\int_{-u}^u dt\, S_{1}(u,t,-u)\,,
\end{equation}
\vspace{-0.5cm}
\begin{equation}
 \bar\psi(z)\sigma_{\mu\nu}z_\nu i\gamma_5\psi(-z) =        
\big[\bar\psi(z)\sigma_{\mu\nu}z_\nu i\gamma_5\psi(-z)]_{\rm twist 2}
+i z_\mu \int_0^1 \!\!\! udu\int_{-u}^u \!\!\!
tdt\,S_{i \gamma_{5}}(u,t,-u),
\end{equation}
where 
$S_{\Lambda}(u,t,v;\mu^2) = \bar \psi(u z)\sigma_{\mu\xi}\Lambda 
   g G_{\nu\xi}(tz)z_\mu z_\nu \psi(vz)$
with $\Lambda = 1, i\gamma_{5}$,
and we neglect operators containing total 
derivatives which are irrelevant for our purposes.

The $Q^2$-dependence of the twist-3 distributions is governed by the
renormalization group (RG) equation for the corresponding nonlocal operators
$S_{\Lambda}$. 
To leading logarithmic accuracy the 
evolution for $\Lambda=1$ and $i \gamma_{5}$ is the same; hence we
drop the subscript in what follows.
%We find that an approach of \cite{BB} to treat
%RG equations directly for the nonlocal operators
%is convenient for searching the analytic solutions.
We introduce the Mellin transformed operators[7,3]
\begin{equation}
     S(u,t,v)=
     \frac{1}{2\pi i}\int^{1/2+i\infty}_{1/2-i\infty}\!\!dj\,
      (u-v)^{j-2}S(j,\xi), \hspace{0.5cm}
      \xi=\frac{u+v-2t}{u-v},
\end{equation}
with $j$ the complex angular momentum; operators with different $j$ 
do not mix with each other. Neglecting contributions 
down by $1/N_c^2$, we obtain 
the RG equation[4]
\begin{equation}
     \left(\mu \frac{\partial}{\partial\mu} 
           + \beta(g)\frac{\partial}{\partial g}\right)
      S(j,\xi;\mu)= -\frac{\alpha_{s}}{2\pi}
   \int_{-1}^{1}\!\!d\eta \,K_{j}(\xi,\eta)
 S(j,\eta;\mu).
\end{equation}
{}For the explicit form of the kernel $K_{j}(\xi,\eta)$,
we refer the readers to Ref.[4].

To solve (5) we consider the  {\em conjugate} 
homogeneous equation
\begin{equation}
   \int_{-1}^{1}\!\!d\eta \,K_{j}(\eta,\xi) \phi_{j}(\eta)
=     \gamma_{j}\phi_{j}(\xi).
\end{equation}
Then it is easy to see that 
$\int^{1}_{-1} d\xi \phi_{j}(\xi) S(j, \xi; \mu)$
gives a multiplicatively
renormalizable {\em nonlocal} operator corresponding to 
the anomalous dimension 
$\gamma_j$.

One can prove that Eq.(6) has two solutions 
analytic 
at the points $\xi = \pm 1$:
%\begin{equation}
$\phi^{+}(\xi)=1$ and 
%\hspace{0.5cm}
$\phi^{-}(\xi)=\xi$
%\end{equation}
(all other solutions have logarithmic branching points).
The corresponding eigenvalues (anomalous dimensions) 
respectively equal
%\begin{eqnarray}
\begin{equation}
%     \gamma_{j}^{+} &=&
     \gamma_{j}^{\pm} =
2 N_c\left\{\psi(j+1)+\gamma_{E}-\frac{1}{4}-\frac{1}{j+1}
\left(- \frac{1}{2} \pm 1 \right) \right\},
%; \;
%\label{gamma1}\\
%     \gamma_{j}^{-} &=&
%     \gamma_{j}^{-} =
%2 N_c\left\{\psi(j+1)+\gamma_{E}-\frac{1}{4}+\frac{3}{2(j+1)}\right\},
%                             \label{gamma2}
%\end{eqnarray}
\end{equation}
where $\psi(z)=\frac{d}{dz}\ln\Gamma(z)$
and $\gamma_{E}$ is the Euler constant.

The superscript $\pm$ corresponds to the ``parity''
under $\xi \rightarrow - \xi$: due to the 
symmetry of the kernel $K_{j} (-\eta, -\xi)= K_{j} (\eta, \xi)$[4], 
one can look for separate solutions 
which are even (odd) under $\xi\to -\xi$. 
{}From Eqs.(2) and (3), 
the relevant quantities for $e(x)$ and $h_L(x)$ are
%, respectively,
%$S(u,t,-u)+S(u,-t,-u)$ and ${\tilde S}(u,t,-u) - {\tilde S}(u,-t,-u)$,
%which correspond to 
even and odd ``$\xi$-parity'' 
pieces of the nonlocal operators.

Substituting the definition (4) into (2) 
and (3), 
we observe that our solutions
in the large $N_{c}$ limit give the 
$Q^{2}$-evolution for the moments:
\begin{equation}
  {\cal M}_n[e](Q) = L^{\gamma^+_n/b}{\cal M}_n[e](\mu)\,;
\;\;
  {\cal M}_n[\widetilde{h}_L](Q) = L^{\gamma^-_n/b}
{\cal M}_n[\widetilde{h}_L](\mu),
\end{equation}
where 
%${\cal M}_n[\widetilde{h}_L]\equiv 
%\int_{-1}^1 dx x^n \widetilde{h}_L(x)$,
${\cal M}_n[e]\equiv \int_{-1}^1 dx x^n e(x)$, 
$b = (11N_c-2N_f)/3$,
and 
$L\equiv \alpha_s(Q)/\alpha_s(\mu)$.
$\widetilde{h}_L$ is the genuine twist-3 contribution to $h_{L}$,
after subtracting out the twist-2 piece[1].
These results show simple GLAP evolutions
without any complicated operator mixing.

Expansion of nonlocal operators at small quark-antiquark separations
generates the series of local operators of increasing dimension.
The anomalous dimension matrix for these local operators
has been obtained for $h_{L}$[8] as well as for $e$[9].
By comparing our solutions (7) with the spectrum of the anomalous 
dimensions in the $N_{c} \rightarrow \infty$ limit,
which is obtained by the numerical diagonalization
of the mixing matrix in [8,9], we conclude that our 
solutions always correspond to operators with the {\em lowest}
anomalous dimension in the spectrum (for the detail, see Refs.[4,9]). 

%The complete spectrum of anomalous dimensions in the $N_c\to\infty$ limit
%obtained by the numerical diagonalization of the mixing
%matrix in \cite{KT} is shown in Fig.~1, together with our analytic 
%solution for the lowest eigenvalue.
%We conclude that our 
%solutions always correspond to operators with the {\em lowest}
%anomalous dimension in the spectrum. 

%
%
% the following 8 cm wide figure will be placed in 
% a 8 cm wide box on the right side of the page 
%
%\begin{wrapfigure}{r}{8cm}
%\epsfig{figure=figure.eps,width=8cm}
%{\small Figure 1: Spectrum of anomalous dimensions of twist 3 operators 
%for $h_L(x)$ in the limit $N_c\to\infty$. The solid line shows the analytic 
%solution ().}
%\end{wrapfigure}

To illustrate numerical accuracy of the leading-$N_c$ approximation,
consider the result[8] including $1/N_c^2$ corrections 
for the evolution of the $n=5$ moment of $h_L$:
%which is the lowest moment
%in which mixing appears: 
\begin{eqnarray}
{\cal M}_{5}[\widetilde{h}_{L}](Q)
&=& \left[ 0.416 b_{5,2}(\mu) + 0.193 b_{5,3}(\mu) \right]
L^{12.91/b}
\nonumber \\
&+& \left[ 0.013 b_{5,2}(\mu) - 0.050 b_{5,3}(\mu) \right]
L^{18.05/b},
\end{eqnarray}
where $b_{n, k}(\mu)$ 
%($k = 2, \cdots, [(n+1)/2]$) 
are reduced matrix 
elements of the independent 
quark-antiquark-gluon local operators in the notation of [1,8].
This is reduced in the large $N_c$ limit to
\begin{equation}
 {\cal M}_{5}[\widetilde{h}_{L}](Q) = \left[
\frac{3}{7}b_{5,2}(\mu) + \frac{1}{7} b_{5,3}(\mu) \right]
L^{13.7/b}.
\end{equation}
One observes: the contribution of the 
operator with the higher anomalous dimension in (9) is small
($\sim 1/N_{c}^{2}$), while the one with
the lowest anomalous dimension is close to the large $N_{c}$ limit.
This observation is crucial for phenomenology, 
since description of each 
moment of the twist-3 distribution now 
requires one single nonperturbative
parameter. Similar phenomenon is observed also for $e(x)$[9].

%We can make this point even stronger, by observing that admixture of 
%operators with higher anomalous dimensions is suppressed at large $n$
%for arbitrary values of $N_c$.
%The full evolution equation with account of all $1/N_c^2$ terms is complicated
%and will be given elsewhere \cite{BBKT2}. However, it
% simplifies drastically in the limit $j\to\infty$ and
%coincides with the large-$j$ evolution equation considered in
%To avoid a singularity in the last term at $j=0$ it is better to substitute
%$\ln j \to \psi(j+1)$ which is within the accuracy.
%With this modification of the anomalous dimensions,
%the results in (\ref{moments}) are valid to the $O(1/N_c^2\cdot \ln(n)/n)$
%accuracy.

To summarize,  our solutions
provide a powerful framework both in confronting with 
experimental data and for the model-building. From a general
point of view, they are interesting as providing with 
an example of an interacting
three-particle system in which one can find an exact energy 
of the lowest state. For phenomenology, main lesson is that inclusive
measurements of twist-3 distributions are complete (to our accuracy)
in the sense that knowledge of the distribution at one value of $Q_0^2$ 
is enough to predict its value at arbitrary $Q^2$, in the spirit
of GLAP evolution equation. 
%This allows to  
%relate results of different 
%experiments to each other, and to compare with model calculations which 
%typically are given at a very low scale. 

\vspace{0.2cm}
\vfill
{\small\begin{description}
\item{}
This work is financially supported in part by RIKEN.
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Phys. Lett. {\bf B266} (1991) 117.
\item{[4]} 
I.I. Balitsky, V.M. Braun, Y. Koike and K. Tanaka,
% 
Phys. Rev. Lett. {\bf 77} (1996) 3078.
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\end{description}}

\end{document}

