%Title: Vacuum polarization in pQCD: first complete O(\alpha_s^4) result
%Author: P.A. Baikov, K.G. Chetyrkin and J.H. K\"uhn
%Published: <EM> Nucl. Phys. Proc. Suppl. </EM> <B> 135 </B> (2004) 243-246. 
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Vacuum polarization in pQCD: first complete $O(\alpha_s^4)$ result\thanks{
Talk presented at 7th DESY Workshop on Elementary Particle Theory: 
Loops and Legs in Quantum Field Theory, Zinnowitz, Germany, 25-30 April  2004.
}}

\author{P.~A.~Baikov\address{Institute of Nuclear Physics,
        Moscow State University, \\
        Moscow~~119992, Russia}%
       ,
        K.G.Chetyrkin\address[KUNI]{Institut f\"ur Theoretische Teilchenphysik, \\
        Universit\"at Karlsruhe, D-76128 Karlsruhe, Germany
        }%
\thanks{On leave from Institute for Nuclear Research
of the Russian Academy of Sciences, Moscow, 117312, Russia.}
        and
        J.H.~K\"uhn\addressmark[KUNI]}



\begin{document}

\begin{abstract}
We report  the results of the   first complete QCD five-loop
calculation, viz. analytical contribution of order $m_q^2 \alpha_s^4$, 
to the (absorptive part of) the vacuum polarization function. The
contribution directly leads to the knowledge of the the full 
${\cal O}( \als^4  m_q^2/ s)$ 
correction to the total cross-section
of $e^+ e^-$ annihilation into hadrons. We  compare
our results with predictions following from various optimization
schemes. 
\end{abstract}
% typeset front matter (including abstract)
\maketitle

\section{Introduction}

Important physical observables like the famous ratio,
$
  R(s) = {\sigma(e^+e^-\to {\rm hadrons})\over \sigma(e^+e^-\to
    \mu^+\mu^-)}\,
$,
and the decay rate of the $Z$ boson are
related to the vector and axial-vector current correlators (see,
e.g.~\cite{ChKK:Report:1996}). Perturbative QCD (pQCD) provides reliable
predictions for $R(s)$ in the continuum, i.e. sufficiently  above the respective
quark threshold and the respective resonance region.




The current status of the pQCD predictions for $R(s)$  is as follows:
\vspace{3mm}

\noindent{
\boldmath \bf 
${\cal O}(\als^2)$ order.}
The full knowledge of $R(s)$ with massive quarks at
${\cal O}(\als^2)$ is  available\cite{Chetyrkin:1996ii,Chetyrkin:1996cf}.
The polarization function in order $\alpha_s^2$ for massive
quarks is therefore under full control.
\vspace{3mm}



\noindent{\boldmath \bf  ${ \cal O}(\als^3)$ order.}
The massless result is known since long \cite{GorKatLar91:R(s):4l,SurSam91,gvvq}. 
For high energy quark mass effects  can be taken into account via the
expansion in quark masses.  This has been done for the
quadratic and quartic terms  in 
\cite{ChetKuhn90,ChK:mq4as2,ChKH:mq4as3}.
\vspace{3mm}

\noindent{\boldmath \bf  ${ \cal O}(\als^4)$ order.}
%As is well-known the 
At two previous workshops  we have discussed the possibility to 
compute at least some  contributions to $R(s)$ at this order  
\cite{Chetyrkin:2000uw,Baikov:2003zg} 
(which   amounts  to the knowledge of the 
polarization function at the record-breaking 
{\em five-loop}  level).
The first genuine QCD calculations of the polarization function at 
five  loops have been reported in 
\cite{ChBK:vv:as4nf2,ChBK:tau:as4nf2,ChBK:vv:mq2as4nf2,Baikov:2003zg}; 
they have taken into account 
the gauge invariant subset of diagrams proportional to $\as^4 n_f^3$ and 
$\as^4 n_f^2$.
\vspace{3mm}



This talk presents  complete results for the (diagonal) vector correlator
at   order $m_q^2 \als^4$. We also compare  our
results with some of the  available predictions following from  various
``optimization'' schemes.

\section{Calculation and results}
Consider  the two-point correlator
of vector   quark currents and the corresponding vacuum polarization
function  ($j_\mu^v= \ovl{Q}\gamma_\mu Q$;
$Q$ is a  quark field with  mass $m$, all other $n_f-1$ quarks  are
assumed to be massless)
\begin{eqnarray}
\nnb
\Pi_{\mu\nu}(q)  &=&
 i \int {\rm d} x e^{iqx}
\langle 0|T[ \;
\;j_{\mu}^{v}(x)j_{\nu}^{v}(0)\;]|0 \rangle
\\
&=&
\displaystyle
(-g_{\mu\nu}q^2  + q_{\mu}q_{\nu} )\Pi(q^2)
{}.
\label{PiV}
\end{eqnarray}
The  physical observable $R(s)$ is related to $\Pi(q^2)$
by
\beq
R(s) = 12 \pi \Im \, \Pi(q^2 + i\epsilon)
{}.
\label{R(S):def}
\eeq
It is convenient to decompose $R(s)$ into the  massless contribution  and those 
quadratic in the quark mass  as follows ($ a_s = \als(\mu^2)/\pi$):
\bea
\nnb
R(s) &=& 3 \left\{r^V_0 +  \frac{m^2}{s} r^{V}_2 \right\}+\dots
\\
 &=& 3\left\{\sum_{i \ge 0} a_s^i\left(
 r^{V,i}_0 + \frac{m^2}{s} r^{V,i}_2 \right)
                                    \right\} +\dots
\nnb
{}.
\eea

For the calculation of $r^{V,4}_0$ one should compute the divergent
parts of five-loop diagrams and finite parts of the four-loop ones
\cite{ChKT:vv:as2}.  Fortunately, as was first shown in Ref.~\cite{ChetKuhn90},
the quadratic mass correction $r^{V,4}_2$ can be obtained with the
help of renormalization group methods {\em exclusively} from the
four-loop function $\Pi_2$ defined as
\[
\nnb
\Pi   = \Pi_0  + \frac{m^2}{-q^2} \Pi_2,
\ \ \
\Pi_2 = \frac{3}{16 \pi^2}\sum_{i \ge 0}\   a_s^i k^{V,i}_2
{}.
\]
This is because the renormalization group equation for $\Pi_2$ can 
be presented in the form ($L = \ln\frac{\mu^2}{-q^2}$):
 \begin{equation} 
\frac{ \partial  }{ \partial  L} \Pi_2 =
\left(
-2 \gamma _m( a_s )
-
 \beta( a_s) a_s\frac{ \partial  }{ \partial  a_s}
\right)
\Pi_2
\label{rg:Pi2}
 \end{equation} 
The rhs of eq.~\re{rg:Pi2} depends on the  function $\Pi_2$ 
multiplied by {\em at least} one factor of $\alpha_s$ coming either from
$\gamma_m$ or from $\beta$,  both of which  are known to $\alpha_s^4$.
Thus, one needs only  $\Pi_2$ to $\alpha_s^3$ in order to find 
the derivative on the lhs  of \re{rg:Pi2} to  $\alpha_s^4$.
Once  $\frac{ \partial  }{ \partial  L} \Pi_2$ is  known, one can  find  the function 
$\Pi_2$ up to a constant contribution which  has no effect   on $R(s)$.  

Using the new technique described in 
\cite{Baikov:tadpoles:96,Baikov:explit_solutions:97,%
BkvSthr:tadples:98,Baikov:criterion:00,Baikov:2003zq}.
and the parallel version of FORM \cite{Fliegner:2000uy,Vermaseren:2000nd},
we have computed the the function  $\Pi_2$ at four loops.

The coefficients  $k^{V,i}_2$ for $i=1,2$ are known from \cite{Gorishnii:1986pz}
while our result for  $k^{V,3}_2$ reads
\begin{eqnarray}
\lefteqn{k^{V,3}_{2} =  } 
\nonumber\\
&{+}& \, n_f^2
\left[
-\frac{5161}{1458} 
-\frac{8}{27}  \,\zeta_{3}
%zero == 0
\right]
\nonumber\\
&{+}& \,n_f 
\left[
\frac{62893}{162} 
-\frac{4150}{243}  \,\zeta_{3}
+\frac{424}{27}  \,\zeta_3^2
+\frac{20}{3}  \,\zeta_{4}
\BreakI
\phantom{+ \,n_f }
-\frac{28880}{243}  \,\zeta_{5}
%zero == 0
\right]
+ k^{V,3}_{2,0}
{},
\label{kV4_2:qcd}
\end{eqnarray}
The last term 
\bea
k^{V,3}_{2,0} &=& 
-\frac{10499303}{1944} + \frac{66820}{81} \zeta_3 - \frac{7225}{27} \zeta_3^2
\nonumber
\\
&+&
 \frac{281390}{81} \zeta_5 - \frac{1027019}{648} \zeta_7
{}
\label{k^{V,3}_{2,0}}
\eea
and allows the first evaluation of a five loop contribution to $R(s)$.
Numerically,
\bea
\lefteqn{\Pi_2 = 
-  8 - 21.333 \, a_s
+  a_s^2 \, ( 10.56 \, n_f -224.80 )}
\nonumber
\\
&+& a_s^3\, (- 3.896 \, n_f^2 +  274.37\, n_f  - 2791.81 )
\label{Pi_2:numer}
\eea

Now the use of eqs.~(\ref{R(S):def},\ref{rg:Pi2}) yields the following 
results for the corresponding contribution to $R(s)$ ($a_s =  a_s (s), \mu^2 = s$):
 \begin{eqnarray} 
\lefteqn{ r^{V}_{2} = 12\   a_s  + a_s^2 \left\{
-\frac{13}{3} \,n_f +\frac{253}{2}\right\}
}
\nonumber
 \\ &+& a_s^3 \left\{
  \left[  \frac{125}{54}-\frac{1}{9} \pi^2  \right]   \, n_f^2 + \left[ 
 -\frac{4846}{27}+
\frac{17}{3} \pi^2
  \right. \right.  
\nonumber 
\\ &{}& 
\left.\left.  
 \phantom{ a_s }
 -\frac{466}{27} \,\zeta_{3}
+\frac{1045}{27} \,\zeta_{5}  \right]   \,n_f +2442
\right.
\nonumber
\\
 &{}&
\left.
\phantom{ a_s }
-\frac{285}{4} 
\pi^2+\frac{490}
{3} \,\zeta_{3}-\frac{5225}{6} \,\zeta_{5}\right\}
\nonumber
 \\ &+&a_s^4\left\{
  \left[  -\frac{2705}{1944}+\frac{13}{108} \pi^2  \right]   
\, n_f^3 + \left[  \frac{91943}{486}
 \right. \right.  
\nonumber 
\\ &{-}&
\left.
\frac{1121}{108} \pi^2+\frac{53}{9} \,\zeta_3^2
+\frac{13}{324} \,\zeta_{3}-\frac{3610}{81} \,\zeta_{5}  
\right]   \, n_f^2 
\nonumber
\\
&+& 
\left.
r_{2,1}^{V,4} \,n_f +r_{2,0}^{V,4} 
\right\} 
{},
\label{rV4_2V:nf} 
\end{eqnarray}

\ice{
I
In[39]:= Coef[ Coef[ Ex[rf],as^4*ma^2] ,nf,1]//InputForm

Out[39]//InputForm= 
-5129327/864 + (19301*Pi^2)/72 - Pi^4/9 + 
 (126593*Zeta[3])/648 - (15917*Zeta[3]^2)/72 + 
 (409585*Zeta[5])/162 - (1027019*Zeta[7])/1728
}

\bea
r_{2,1}^{V,4} =   
&-&\frac{5129327}{864} + \frac{19301}{72} \pi^2- \frac{\pi^4}{9}
\nonumber
\\
&+& \frac{126593}{648} \, \zeta_3 - \frac{15917}{72}\, \zeta_3^2 
\nonumber
\\
&+&   \frac{409585}{162} \, \zeta_5  - \frac{1027019}{1728}\, \zeta_7
\label{r_{2,1}^{V,4}}
\eea

\ice{
Coef[ Coef[ Ex[rf],as^4*ma^2] ,nf,0]//InputForm

Out[38]//InputForm= 
128033845/2592 - (68003*Pi^2)/32 - (203227*\zeta[3])/36 + 
 (296225*\zeta[3]^2)/144 - (2129485*\zeta[5])/72 + 
(42107779*\zeta[7])/3456
}
\bea
r_{2,0}^{V,4} =
&{}& \frac{128033845}{2592} - \frac{68003}{32}\pi^2 - \frac{203227}{36} \, \zeta_3
\nonumber
\\
&+&
\frac{296225}{144}\, \zeta_3^2 - \frac{2129485}{72} \, \zeta_5 
\nonumber
\\
&+&
\frac{42107779 }{3456} \, \zeta_7
\label{r_{2,0}^{V,4}}
\eea
Finally, eqs.~(\ref{rV4_2V:nf},\ref{r_{2,0}^{V,4}}) in numerical form read
\begin{eqnarray} 
\lefteqn{r^V_2 = {}
12\,  a_s +   a_s^2 \left(-4.3333 \ n_f+  126.5  \right)
}
\nonumber
\\
&+&
a_s^3
\left(
 1.2182 \, n_f^2- 104.167  \, n_f  + 1032.14 
\right)
  \nonumber  
\\
&+& a_s^4
\left(
- 0.20345 \,  n_f^3
+ 49.0839  \,n_f^2  
  \nonumber  
\right.
\\
&{}&
\phantom{||||||}
-
\left.
\  1364.04 \, n_f \ + 6225.84
\right)
{}.
\label{rV_2:num}
\end{eqnarray} 

%6225.84 - 1364.04 nf

\section{Discussion}

It is interesting to compare eqs.~(\ref{k^{V,3}_{2,0}}-\ref{rV_2:num})
with predictions made in \cite{ChBK:vv:mq2as4nf2} and based on the optimization
methods FAC (Fastest Apparent Convergence) and PMS (Principle of
Minimal Sensitivity) 
\cite{stevenson:1981vj,Stevenson:1981du,Stevenson:1982wn,Stevenson:1984qw,Kataev:1995vh}. 
As in many cases FAC and PMS
predictions are quite close \cite{Kataev:1995wd} we will also use
so-called NNA (Naive NonAbelianization) approach of \cite{Broadhurst:1995se}.

We start from purely {\em Euclidian} quantity $k^{V,3}_2$. The predictions read
(should be compared to the second line of eq.~(\ref{Pi_2:numer}))
 \bea 
k^{V,3}_{2}(\mathrm{FAC})
 &=&   -4.2 \, n_f^2 + 277 \, n_f -2886    
\label{FAC:nf:3}
\\
k^{V,3}_{2}(\mathrm{PMS}) 
&=& -4.2 \, n_f^2   + 282 \, n_f   -2916
{},
\label{PMS:nf:3}
\\  
{k^{V,3}_{2}(\mathrm{NNA})}
&=& - 3.896 \, n_f^2     + 129 \, n_f  
-1060
%\\
%\nonumber
%&{}& \phantom{|||||||||}-1060
{}.
\label{NNA}
\eea 
We observe a remarkably good agreement between FAC/PMS predictions and
our result. In fact, the relative accuracy for all terms in
eqs.~(\ref{FAC:nf:3},\ref{PMS:nf:3}) is better than 10\%. Even more,
for the calculationally most demanding contribution, namely,
$k^{V,3}_{2,0}$, the relative error decreases up to a level of astonishingly small
5\%! However,  the observation can not be automatically  translated to
the essentially {\em Minkowskian} case of  $r^V_2$.
Indeed, for, say, $n_f=4$ we get
\bea
r^V_2/12\ (\mbox{exact})
 &=&  \as + 9.09722 \as^2 
\nnb
\\
 &+& 52.913 \as^3 
 + 128.499 \, \as^4
{},
\label{exact}
\eea
\bea
 r^V_2/12\ (\mbox{FAC}) 
&=& \as + 9.09722 \as^2 
\nnb
\\
 &+& 52.913 \as^3
+ {177} \, \as^4
{},
\label{FAC:NF:4}
\eea
\bea
r^V_2/12\ (\mbox{PMS}) 
&=&  \as + 9.09722 \as^2
\nnb
\\  
&+& 52.913 \,  \as^3  
+ {193}\, \as^4
{},
\label{PMS:nf:4}
\eea
\bea
r^V_2/12\ (\mbox{NNA}) 
&=&  \as + 9.09722 \as^2
\nnb
\\  
&+& 52.913 \,  \as^3  
- {754}\, \as^4
{}.
\label{NNA:nf:4}
\eea


\noindent
Now the error in the predicted $\as^4$ term increases to 
about  40\%! 
The reason is sitting in  the peculiar structure of the
$\as^4$ term in eqs.~(\ref{exact}-\ref{NNA:nf:4}):
\beq
-1311.03  - 0.515625 \  k^{[V]3}_{2,0}
{}.
\label{cancellation}
\eeq
Here the first term describes the ``kinematical'' effects due to the
analytic continuation as well as the contributions coming from the
interference between (of order $\as^3$ and less) terms in the beta
function, the quark anomalous dimension and $\Pi_2$ in
eq.~\re{rg:Pi2}.  Numerically $k^{[V]3}_{2,0}$ is about $3 \cdot
10^3$, hence there are strong cancellations in
eq.~(\ref{cancellation}) which lead to a significant loss in accuracy
if the predicted value of $k^{[V]3}_{2,0}$  even slightly
deviates from the exact result.







As for the NNA approach the corresponding prediction is in obvious
conflict\footnote{The quadratic in $n_f$ term in $k^{V,3}_2$ was used
as the input.} with the result of the direct calculation already for
the Euclidian quantity $k^{V,3}_2$.






\section{Conclusions}


We have presented the analytical calculation of contributions of order
$m_q^2 \alpha_s^4 $ to $R(s)$.  This constitutes the first full five
loop contribution to the fundamental quantity $R(s)$, albeit  only to
its  mass correction.


Predictions of FAC, PMS and NNA
optimization methods are tested against the exact results.  Good
quantitative agreement is found for the the FAC/PMS predictions
in the Euclidean region. The agreement deteriorates  after
analytical continuation to the physical Minkowskian region. 
NNA fails to predict the result.


This work was supported by Sonder\-forschungs\-bereich-Transregio {\it
``Computational Particle Physics''} (SFB-TR 9), by INTAS (grant
00-00313), by RFBR (grants 03-02-17177), by Volkswagen Foundation and
by the European Union under contract HPRN-CT-2000-00149.


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