%Title: Relating the Quark Mass defined in the ``Regularization Invariant'' Scheme to the MS bar Mass at O(\alpha_s^3) 
%Author: K.G. Chetyrkin and A. Retey
%Published: <EM> Act. Phys. Pol. </EM> <B> 30 </B> (1999) 3373-3377. 
\documentclass{appolb}
\usepackage{epsfig}

\newcommand{\RI}{\mathrm{RI}}

\catcode`\@=11
\def\slash{\mathpalette\make@slash}
\def\make@slash#1#2{\setbox\z@\hbox{$#1#2$}
  \hbox to 0pt{\hss$#1/$\hss\kern-\wd0}\box0}
\catcode`\@=12  
 
\begin{document}

\title{ Relating the Quark Mass defined in the ``Regularization
Invariant'' Scheme to the $\overline{\mbox{MS}}$ Mass at ${\cal
O}(\alpha_s^3)$}

\author{ K. G. Chetyrkin\thanks{ Permanent address:
          Institute for Nuclear Research, Russian Academy of Sciences,
          60th October Anniversary Prospect 7a, Moscow 117312, Russia.}
\and 
         A. R\'etey
\address{ Institut f\"ur Theoretische Teilchenphysik,
          Universit\"at Karlsruhe, % Kaiserstr. 12,    Postfach 6980,
          D-76128 Karlsruhe, Germany
        }
}

\maketitle

\begin{abstract} 
We report on the analytical calculation of the ${\cal O}(\alpha_s^3)$
conversion factor between the $\overline{\mbox{MS}}$ quark mass and
the one defined in the so-called ``Regularization Invariant'' scheme.
\end{abstract}

\PACS{12.38.B, C; 12.38.G; 12.15.F}

\section{Introduction}\label{sec:intro}

Although the quark masses are fundamental parameters of the QCD
Lagrangian, their relation to measurable physical quantities is not
direct. They depend on the renormalization scheme and, within a given
one, on the renormalization scale $\mu$.

In the realm of perturbative QCD the most frequently used mass
definition is the so-called short distance $\overline{\mbox{MS}}$
mass, based on the $\overline{\mbox{MS}}$-scheme
\cite{ms,MSbar}. Unfortunately it is difficult to obtain precise
information on the quark masses from pQCD, as the mass dependence of
it's predictions is relatively weak.

One possibility to obtain such information are lattice QCD
calculations, which provide a direct way to determine quark masses
from first principles (for recent discussions see
\cite{Dominguez:1997eu, Narison:1997fm, Chetyrkin:1998ej,
Kenway:1998ew, Lubicz:1998kc} ). The resulting quark mass is the
(short distance) bare lattice quark mass. A scheme that is directly
accessible in lattice calculations is the ``Regularization
Independent'' scheme, which has been used in some recent lattice
calculations and was proposed in \cite{NMPmethod}.

Relating lattice quark masses to those defined in a continuum
perturbative scheme as the $\overline{\mbox{MS}}$ one requires the
calculation of the corresponding renormalization constants.  These
constants can be defined and computed only {\em perturbatively}. The
conversion factor for the mass defined in the $\RI$ scheme and the
$\overline{\mbox{MS}}$ scheme is now known at next-to-next-to-leading
order (NNLO) from ~\cite{Franco:1998bm} and happens to be
numerically significant. This makes mandatory to know the NNNLO
$O(\alpha_s^3)$ term in the conversion factor.

In this work we report on the calculation of this term. It turns out
that the size of this term is comparable to the previous one at a
renormalization scale of 2 GeV --- the typical scale currently used in
lattice calculations of the light quark masses. 

\section{Scheme Dependence of the Quark Mass}
\label{sec:massfield}

To calculate the conversion factors, we start with the bare quark
propagator (for simplicity we stick to the Landau gauge and do not
explicitly display the gauge dependence)
\begin{equation}
\label{def:fermi}
S_0(q,\alpha_s^0,m_0) = 
\mathrm{i} \int \mathrm{d}x \mathrm{e}^{\mathrm{i}qx} 
\langle \mathrm{T}[\psi_0 (x) \bar\psi_0 (0)] \rangle = 
(m_0  - \slash{q} - \Sigma_0)^{-1}
\end{equation}
with the quark mass operator $\Sigma_0$ being conveniently decomposed
into Lorentz invariant structures according to $\Sigma_0 = \slash{q}
\Sigma^0_V + m_0 \Sigma^0_S$.  Here $m_0$ and $\psi_0$ are the bare
quark mass and field respectively and $\displaystyle a_s^0 \equiv
\alpha_s^0/\pi = g^2/(4\pi^2)$, where $g$ is the bare QCD gauge
coupling.

Higher order corrections to physical quantities in pQCD will give
finite results only after regularization and reparameterization of all
parameters and fields of the theory. Moreover, Greens functions as
(\ref{def:fermi}) also need to be renormalized. To be precise we
assume that (\ref{def:fermi}) is dimensionally regulated by going to
non-integer values of the space-time dimension $D=4-2\epsilon$
\cite{dim.reg-a,dim.reg-b}.  Then the $\overline{\mbox{MS}}$
renormalized Green function (\ref{def:fermi}) reads:
\begin{equation}
\label{def:fermi-ren}
S(q,\alpha_s,m,\mu) 
=  (m  - \slash{q} - \Sigma)^{-1} 
\\
= Z_2^{-1} S_0(q,\alpha_s^0,m_0)|_{{}_{
\scriptstyle
m_0=Z_m m,\, \alpha_s^0= \mu^\epsilon Z_\alpha \alpha_s } }
\end{equation}
where $\psi= Z_2^{-1/2} \psi_0 $ is the renormalized quark field and
the {}'t Hooft mass parameter $\mu$ is a scale at which the renormalized
quark mass is defined.  The renormalization constants $Z_2,Z_\alpha$
and $Z_m$ are series of the generic form
\begin{equation}
\label{def:Z-expansions}
Z_? = 1 + \sum_{i>0} Z_?^{(i)} \frac{1}{\epsilon^i}, \, \, \, \,
\, Z_?^{(i)} = \sum_{ j \ge i } Z_?^{(i,j)} \left(\frac{\alpha_s}{\pi} 
\right )^j {}, \quad ? = 2,\alpha,m{}.
\end{equation}
The quark propagator renormalized according to a different subtraction
procedure reads (parameters marked with a prime belong to the other
scheme)
\begin{equation}
\label{def:fermi-ren-prime}
\begin{array}{l}
\displaystyle
S(q,\alpha'_s,m',\mu) = 
 \frac{1}{m'  - \slash{q} - \Sigma'}
=(Z'_2)^{-1} S_0(q,\alpha_s^0,m_0)|_{{}_{
\scriptstyle
m_0=Z'_m m, \,  \alpha_s^0= \mu^\epsilon Z_\alpha' \alpha_s } }
{},
\end{array}
\end{equation}
where without essential loss of generality we have set $\mu' =
\mu$. The finiteness of the renormalized fields and parameters in both
schemes implies that, within perturbation theory, the relation between
them is uniquely:
\begin{equation} 
m = \frac{Z_m'}{Z_m} \cdot m' = C_m
\cdot m' , \, \,\,
\psi = \sqrt{\frac{Z_2'}{Z_2}} \cdot \psi' = \sqrt{ C_2} \cdot \psi'  
\label{def:C2}
{},
\end{equation}
with the ``conversion functions'' being themselves {\em finite} series
in $\alpha_s'$, i.e.
\begin{equation}
C_?  \equiv        
1 + \sum_{i>0} C_?^{(i)} \left( \frac{\alpha_s'}{\pi}\right)^i 
{}\qquad,\qquad ? = m,2.
\label{def:C-expansions-ap}
\end{equation}

In general the coefficients $C_?^i$ may depend on the ratio $m'/\mu$.
If such a dependence is absent the corresponding scheme is called a
``mass independent'' one.  In what follows, we will assume that the
function $C_\alpha$ is known and, thus, will deal with series of the
type (\ref{def:C-expansions-ap}) in terms of the
$\overline{\mbox{MS}}$ $\alpha_s$

From eqs.~(\ref{def:fermi}) it is easy to see that 
\begin{equation}
C_2 \cdot ( 1 + \Sigma_V ) = 1 + \Sigma'_V  
\quad,\quad
C_2 \cdot C_m \cdot ( 1 - \Sigma_S ) = 1 - \Sigma'_S 
\label{form:C2Cm}
\end{equation}
These equations together with renormalization conditions for the
non-$\overline{\mbox{MS}}$ scheme then provide the necessary
information to determine the conversion factors $C_m$ and $C_2$, given
the $\overline{\mbox{MS}}$ renormalized $\Sigma_V$ and $\Sigma_S$.

A mass independent $\mathrm{MOM}$\footnote{The so-called momentum
subtraction schemes require the values of Green Functions with fixed
$\mu$ dependent external momentum configurations to be fixed} scheme
has recently been suggested in \cite{NMPmethod} under the name of
$\RI$ (``Regularization Invariant'') and is defined by\footnote{Traces
are to be taken over color, Lorentz and Dirac indices}:
\begin{equation}
\scriptstyle
\lim \limits_{m\rightarrow 0} \frac{1}{48} Tr \left[ {\scriptstyle 
\gamma_\mu \frac{\scriptstyle \partial\left( \slash{q}
(1+\Sigma^{\RI}_V)\right)}{\scriptstyle \partial 
q_\mu} } \right]_{\scriptscriptstyle q^2=-\mu^2} {\scriptstyle = \, 1}
\,,\,
\lim \limits_{m\rightarrow 0} \frac{1}{12} Tr \left[{\scriptstyle 1 -
\Sigma^{\RI}_S } \right]_{\scriptscriptstyle 
q^2=-\mu^2} { 
\scriptstyle = \, 1 } 
\label{def:RI-rencond}
\end{equation}
From these we get the conversion constants ($\ell =
log(-\frac{q^2}{\mu^2})$):
\begin{equation}
C^{\RI}_2 = \left[{\scriptstyle 1 + \Sigma_V + 
\frac{1}{2}\cdot
\frac{\scriptstyle \partial\Sigma_V(\ell)}{\scriptstyle \partial \ell}}
\right]^{-1}_{ {\scriptscriptstyle q^2=-\mu^2}\atop{m=0} } 
,\,
C^{\RI}_m = \left[ 
\frac{\scriptstyle 1 + \Sigma_V + \frac{1}{2} \cdot \frac{\scriptstyle
\partial\Sigma_V(\ell)}{\scriptstyle \partial \ell}}
     {\scriptstyle  1 - \Sigma_S } \right]_{ {\scriptscriptstyle 
q^2=-\mu^2}\atop{m=0}}
\label{form:CRI}
\end{equation}

\section{Three Loop $\overline{\mbox{MS}}$ Quark Propagator}

We have analytically computed the functions $\Sigma_V$ and $\Sigma_S$
in the massless limit to order $\alpha_s^3$. In addition, we have
calculated the first 6 (3 for the 3-loop case) terms in a small mass
expansion of these functions. The calculation was making intensive use
of computer algebra programs.  In particular, we have used {\tt
QGRAF}~\cite{qgraf} for the generation of diagrams and {\tt
LMP}~\cite{rhdiss} for the asymptotic expansions (for an introduction
see \cite{Harlander:ustron1999}). The resulting massless propagator
diagrams and massive tadpole diagrams have been evaluated with the
form packages {\tt MINCER}~\cite{mincer2} and {\tt
MATAD}~\cite{Stediss}.  Up to 2 loops the analytical mass dependence
of the functions $\Sigma_V$ and $\Sigma_S$ are known
\cite{Fleischer:1998dw}. These results have been used as cross checks
for our results and we found full agreement in numerical evaluation
and small and large mass expansions of their result. The full results
of these functions are published in \cite{preparation}.

\section{Results}
\label{results}

Our result for the conversion function for the $\RI$ mass to the
$\overline{\mbox{MS}}$ mass reads\footnote{$\zeta_x$ are the values of
Riemann's Zeta function, note that $C_m^{\RI}$ is gauge independent}
as functions of $n_f$:

\begin{eqnarray}C_m^{\RI} &=& 
1  \quad + \frac{\alpha_s}{4\pi}
\left[ 
-\frac{16}{3}\right] 
\quad + \left(\frac{\alpha_s}{4\pi}\right)^{2}
\left[
-\frac{1990}{9} 
+\frac{152}{3}  \,\zeta_{3}
+\frac{89}{9}  \,n_f 
\right]
\nonumber\\
 &{+}& \left(\frac{\alpha_s}{4\pi}\right)^{3}
\left[
-\frac{6663911}{648} 
+\frac{408007}{108}  \,\zeta_{3}
-\frac{2960}{9}  \,\zeta_{5}
+\frac{236650}{243}  \,n_f 
\right. \nonumber \\ &{}& \left.
\nonumber
-\frac{4936}{27}  \,\zeta_{3} \,n_f 
+\frac{80}{3}  \,\zeta_{4} \,n_f 
-\frac{8918}{729}  \, n_f^{2}
-\frac{32}{27}  \,\zeta_{3} \, n_f^{2}
\right]
{}.
\label{CmRIqcd}
\end{eqnarray}
At a scale $\mu = 2$ GeV and $n_f=4$, the numerical contributions of
the leading order to NNNLO terms are as follows (with $\alpha_s(2 \,
\mbox{GeV}) /\pi = 0.1$)
\[
C_m^{\RI} = 1.-0.133333-0.0754071-0.0495357
{}
\]
One observes that the sizes of the NNLO and NNNLO contributions to
$C_m^{\RI}$ at this scale amount to about 7.5\% and 5\%
respectively. This shows that the NNNLO term is numerically
significant and should be taken into account when transforming the
$\RI$ quark masses to the $\overline{\mbox{MS}}$ ones. Indeed, the
size of the NNNLO term makes the applicability of pQCD at this scale
doubtful. For a more elaborate discussion on the result see
\cite{preparation}.

% \section{Conclusions}

% We have computed the small mass expansion of the quark propagator to
% ${\cal O}(\alpha_s^3)$. From that result we calculated the conversion
% factor for the $\RI$ to the $\overline{\mbox{MS}}$ quark mass to
% the same order and found the last term to give a quite sizable
% contribution at a scale of $\mu=$ 2 GeV.

\section{Acknowledgments}

A.R. would like to thank R. Harlander and T. Seidensticker for technical
help and advice. This work was supported by the {\it
DFG-Forschergruppe ``Quantenfeldtheorie, Computeralgebra und
Monte-Carlo-Simulationen''}, and the {\it Graduiertenkolleg
``Elementarteilchenphysik''} at the University of Karlsruhe.

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\end{document}

