%Title: Four-Loop Tadpoles and Precision Calculations in Perturbative QCD
%Author: Johann H. Kühn
%Published: Contibution to the International Conference on High Energy Physics, Moscow, July 2006, to be published in the proceedings.
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\begin{document}
\title{Four-Loop Tadpoles and Precision Calculations\\ in Perturbative QCD}
\author{Johann H. K\"uhn$^*$}
\address{Institut f\"ur Theoretische Teilchenphysik, Universit\"at Karlsruhe,
D-76128 Karlsruhe, Germany}
\twocolumn[\maketitle\abstract{Recent results on four-loop tadpoles
amplitudes and their implications for precision calculations are
presented.}
\keywords{perturbative QCD; decoupling, quark masses}]
\section{Introduction}
%
%\subsection{Producing the Hard Copy}\label{subsec:prod}
%
During the first and partly even the second decade after QCD had been
proposed as the theory of strong interactions, a number of tests
were suggested to probe the qualitative features of this novel theory.
In recent years the emphasis has shifted to different topics:
on one hand the precise
determination of the fundamental parameters of the theory, in particular
the strong coupling constant $\alpha_s$ and the quark masses,
on the other hand to
the prediction of quantities that enter observables of
interest, with the top quark contribution to the $\rho$ parameter or
lepton spectra and rates for $B$-meson decays as characterisic
examples. This
second application then allows to determine the parameters of the
electroweak theory, e.g. the elements of the CKM matrix or, indirectly,
the mass of the
Higgs bosons, or to even find indications for a theory beyond the
Standard Model. Most of these investigations are based on the
perturbative treatment of QCD, although interesting and successful cross
checks and predictions have been made within lattice gauge theory.
As a consequence of the relatively large value of the strong coupling,
and to match the increasing experimental precision, the inclusion of
quantum corrections is a must for any
meaningful evaluation of the forementioned fundamental parameters. The
evaluation of Feynman amplitudes, corresponding to diagrams with several
external legs, with several mass- and energy-scales and more than one
loop is difficult, often impossible. In practice, however, many physical
problems exhibit a drastic hierarchy of scales, with the quark masses
$m_t^2\gg m_b^2\gg m_c^2\gg m_s^2\gg m_{u,d}^2$ as most prominent
examples. Frequently only one of them can be considered as
non-vanishing, the effects of the other ones can be considered through
appropriate expansions. Furthermore, the characteristic energy scale may
often be assumed to be significantly smaller or larger than the quark
mass, an assumption leading to further drastic simplifications.
In the high energy limit quark masses are neglected or again taken into
account through expansions. In the low energy limit, in contrast,
the external momentum flow may be ignored, and one arrives at
vacuum, also denoted tadpole diagrams. As long, as
only one quark mass $m_Q$ is kept non-vanishing, the scale of the
corresponding amplitude can be factored out and the calculation is
reduced to the evaluation of a dimensionless integral.
In many cases the approximation of small external momentum,
$q^2\ll m_Q^2$,
can be ameliorated by considering a Taylor series in $q^2/m_Q^2$.
The Taylor coefficients can again be expressed by tadpole diagrams, and
often an excellent approximation to the complete result is
constructed\cite{Chetyrkin:1995ii}
from the low- and the high-energy behaviour. For the
correlator of two vector currents, the vacuum polarization, this has been
acchieved in three-loop approximation about ten years ago and
similar results are available for axial vector and scalar correlators.
Important applications along these lines were the evaluation of the
$R$-ratio (with $R\equiv \sigma(e^+e^-\to had) / \sigma_{pt}$) for the
production of massive quarks,
the Higgs boson decay rate into massive
quarks, the coupling of the Higgs boson to gluons and other observables
of interest\cite{Steinhauser:2002rq}.
The Taylor coefficients
of the vacuum polarization are closely related to moments of the
$R$-ratio for the production of heavy quarks, and this correspondence has
lead to a highly precise evaluation of the charm and bottom quark
mass\cite{Kuhn:2001dm}.
In view of the importance and the success of this approach a vigorous
research program has been initiated during the last years towards the
evaluation of tadpole amplitudes in four-loop approximation. A variety of
results with important physical applications has been presented during
the past months which will be described below.
%
\section{Calculational Techniques}
%
From the technical side all these calculations are based on the
the traditional Integration-by-Parts method in combination with the Laporta
algorithm\cite{Laporta:1996mq},
which allows to reduce the huge number of Feynman integrals
algebraically to a few master integrals through an
automated FORM3 program\cite{STURMAN}. The master
integrals are then evaluated either analytically or numerically with
high precision. Difference equations\cite{Laporta:2002pg}
and, alternatively, numerical methods
based on a Pad\'e approximation for three-loop propagator
integrals\cite{faisst} lead to a precision of thirty or seven
digits, respectively.(For a detailled discussion and a list of references
see\cite{Chetyrkin:2006dh}.)
In this context it is useful to select a set of master integrals whose
coefficient functions are finite in the limit $\epsilon \to 0$.
This choice of an ``epsilon-finite basis'' \cite{Chetyrkin:2006dh} gives
important insights into the structure of the master integrals.
In particular it leads directly to
analytic expressions for those terms which cancel algebraically through
renormalization and is particularly suited for the
numerical evaluation through the formentioned Pad\'e
method. Let us now describe the results in more detail.
%
\section{Heavy quark decoupling and matching conditions}
%
According to the Appelquist-Carazzone theorem,
effects due to heavy particles should eventually
``decouple'' from the low-energy physics.
However, a peculiarity of mass-independent renormalization schemes is that the
decoupling theorem does not hold in its naive form in theories
renormalized in this framework: In general potentially large mass
logarithms appear, when one calculates a physical observable.
The well-known way to bring these logarithms under control is to
construct an effective field theory by ``integrating out'' the heavy
quark field $h$.
For QCD with $n_l=n_f-1$ light and one heavy quark $h$
the coupling constants in both theories, the full $n_f$-flavor QCD
and the effective $n_l$-flavor one,
$\alsnf$ and $\alsnl$ are then related by the so-called
matching condition
\begin{equation}
\alsnl(\mu)=
\zeta_g^2 (\mu,\alsnf(\mu),m)\,\,
\alsnf(\mu)
{}.
\label{als_matching_cond}
\end{equation}
Here ${m} = m(\mu)$ is the (running) mass of the heavy quark and $\zeta_g$
the decoupling function is now available in four-loop
approximation\cite{Schroder:2005hy,Chetyrkin:2005ia}.
The bare decoupling constants can all be expressed in terms of
massive Feynman integrals without any external
momenta\cite{Chetyrkin:1997un},
namely the gluon and ghost vacuum polarization functions and the
ghost-gluon vertex function, all at vanishing external momenta.
Inclusion of the additional term leads to a minute shift of
$\alpha_s(M_Z)$ by 0.001 when running from $m_\tau$ to $M_Z$ and a
slight reduction of the theory error.
%
\section{Coupling of the Higgs boson to gluons}
%
With the SM Higgs boson mass below 200~GeV, its $ggH$ coupling is
essentially generated by the top quark only and
becomes independent of the top quark mass $M_t$ in the limit
$M_H\ll 2M_t$.
%
Then the situation may be greatly simplified\cite{Chetyrkin:1997un} by
constructing a heavy-top-quark effective Lagrangian, which
is a linear combination of certain dimension-four
operators acting in QCD with five quark flavors and an
effective coupling $C_1$, directly related to the logarithmic
derivative of $\zeta_g$ w.r.t.\ $m$.
Now, armed with the newly computed four-loop term in $\zeta_g$,
the old result\cite{Chetyrkin:1997un} for $C_1$
(earlier obtained indirectly) was checked.
In fact, one could even construct $C_1$ at five loops in
terms of the known four-loop QCD decoupling function, the quark anomalous
dimension $\g_m$ and the $\beta$-function and the only yet unknown
parameter: the $n_f$-dependent piece of the five-loop contribution to
the $\beta$-function.
\section{Taylor coefficients of the vacuum polarization
in ${\cal O}(\alpha_s^3)$}
The vacuum polarization function $\Pi(q^2)$
can be linked to moments $\mathcal{M}_n$
with the help of dispersion relations:
\begin{equation}
\mathcal{M}_n^{\mbox{\scriptsize{exp}}}=\!\!\int\!ds \frac{R(s)}{s^{n+1}}=
\frac{12\pi^2}{n!}\left(\frac{d}{dq^2}\right)^n \left.\Pi(q^2)\right|_{q^2=0}\!,
\label{moments}
\end{equation}
$ \mathcal{M}_n^{\mbox{\scriptsize{exp}}}$
depends on the experimental measurable $R$-ratio,
the derivatives of the vacuum
polarization function can be calculated in perturbative QCD.
This method can be used to determine charm- and
bottom-quark masses. Taking into
account moments up to three-loop
order\cite{Chetyrkin:1995ii}, precise results for
both quark masses have been obtained\cite{Kuhn:2001dm}.
In view of the present and
foreseeable precision the
four-loop contributions are of phenomenological importance and have been
evaluated\cite{Chetyrkin:2006xg,Boughezal:2006px} for the two lowest terms of the
expansion. The uncertainties from
the variation of the renormalization scale are thus significantly reduced.
For the lowest moment, however, the experimental uncertainties are still dominant.
A preliminary analysis\cite{kss}
based on the lowest three moments which includes
the latest experimental information and an estimate of the still unknown
four-loop terms of $\mathcal{M}_2$ and $\mathcal{M}_3$ leads to
\begin{eqnarray}
\overline{m}_c(3 \GeV) = 1007\pm16 \MeV {},\\
\overline{m}_b(10\GeV) = 3630\pm35 \MeV {}.
\end{eqnarray}
and thus to a significant reduction of the error.
\section{Four-loop QCD corrections to the $\boldsymbol{\rho}$ parameter}
A group of dominant radiative corrections can be absorbed in the
shift of the $\rho$ parameter from its lowest order value
$\rho_{\text{Born}}=1$. The result for the one-loop approximation
$\delta \rho = 3 \, x_t = 3 \, G_F m_t^2/(8\sqrt{2}\, \pi^2)$,
hence quadratic in $m_t$, was first obtained by Veltman
and used to establish a limit on the mass splitting within one fermion
doublet. In order to make full use of the present experimental
precision, this one-loop calculation was improved by two-loop
and even three-loop QCD corrections.
Also important are two-loop and three-loop
electroweak effects proportional
to $x_t^2$ and $x_t^3$, respectively, and the three-loop mixed
corrections\cite{Faisst:2003px} of order $\alpha_s x_t^2$.
For fixed pole mass of the top quark, the three-loop result leads to a
shift of about 10 MeV in the mass of the $W$-boson.
(This applies both to the pure QCD corrections\cite{Chetyrkin:1995js}
and the mixed QCD-electroweak ones\cite{Faisst:2003px}.)
Conversely, the corresponding shift of the top
quark pole mass amounts to 1.5 GeV.
These values are comparable to the experimental precision anticipated
for top- and $W$-mass measurements at the International Linear
Collider. An improvement of the theoretical accuracy seems,
therefore, desirable.
The $W$-self-energy receives contributions from the correlator of the
``non-diagonal'' $t$-$b$-current only. Contributions to the
$Z$ self-energy originate only from the ``diagonal'' axial current
correlator induced by top quark loops.
The nonvanishing parts of $\Pi_T^Z(0)$ are conveniently
decomposed into non-singlet and singlet pieces characterized by
Feynman diagrams where the external current couples to the same and to
two different closed fermion lines, respectively. In three-loop
approximation the singlet-piece is larger than the non-singlet piece
by nearly a factor twenty. This has motivated the authors
of\cite{Schroder:2005db} to evaluate, in a first step, the four-loop
singlet piece.
To complete the evaluation of the four-loop QCD corrections to
the $\rho$ parameter, $\Pi^{W}_T(0)$ and the non-singlet parts of
$\Pi^{Z}_T(0)$ are required.
The result for the shift in the $\rho$ parameter can be cast into the
following form\cite{Chetyrkin:2006bj,Boughezal:2006xk}:
\begin{equation}
\label{eq:rho-define}
\delta \rho^{\overline{\mathrm{MS}}} =
3 x_t \sum_{i=0}^{3} \left(\frac{\alpha_s}{\pi}\right)^i
\delta \rho^{\overline{\mathrm{MS}}}_i
\end{equation}
For $m_t$ defined in the $\overline{\mathrm{MS}}$-scheme one
finds\cite{Chetyrkin:2006dh,Boughezal:2006xk}
\begin{eqnarray}
\label{eq:rho-number}
\delta \rho^{\overline{\mathrm{MS}}}_3 & =
\nonumber
\delta \rho^{\overline{\mathrm{MS}}}_3(\mbox{singlet})
+\delta \rho^{\overline{\mathrm{MS}}}_3(\mbox{non-singlet})\\
& = -3.2866 + 1.6067 = -1.6799
{}.
\end{eqnarray}
If expressed in terms of the pole mass, a major shift originates from the
large correction in the pole-$\overline{\mathrm{MS}}$ relation:
$\delta \rho^{pole}_3 = - 93.1501$.
For fixed top mass, this corresponds to shift of around 2~MeV in the
$W$-boson mass.
The result based on the three-loop calculation is thus stabilized.
\section*{Acknowledgments} I would like to thank K. Chetyrkin, M. Faisst,
P. Maierh\"ofer and C. Sturm for a fruitful and exciting collaboration.
Supported by DFG through
SFB/TR~9 and by BMBF, Grant No. 05HT4VKA/3.
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\end{document}