%Title: Using differential equations to compute two-loop box integrals
%Author: T.Gehrmann, E. Remiddi
%Published: * Nuclear Physics B (Proc. Suppl.) * ** 89 ** (2000) 251-255. * Proceedings of the Workshop ``Loops and Legs in Quantum Field Theory''*, Koenigstein-Weissig, April 2000
%hep-ph/0005232
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\title{
Using differential equations to compute
two-loop box integrals\thanks{Presented at {\it Loops and Legs
in Quantum Field Theory},
April 2000, Bastei, Germany}}
\author{\underline{T.\ Gehrmann}\address{Institut
f\"ur Theoretische Teilchenphysik,
Universit\"at Karlsruhe, D-76128 Karlsruhe,
Germany}
and
E.\ Remiddi\address{Dipartimento di Fisica,
Universit\`{a} di Bologna, and INFN, Sezione di Bologna,
I-40126 Bologna, Italy}}
\begin{document}
\unitlength 1cm
\begin{abstract}
The calculation of exclusive observables beyond the one-loop level
requires elaborate techniques for the computation of
multi-leg two-loop integrals. We
discuss how the large number of different integrals
appearing in actual two-loop calculations can be reduced to a small
number of master integrals. An efficient method to
compute these master integrals is to derive and solve
differential equations in the external
invariants for them.
As an application of the differential equation method, we compute
the ${\cal O}(\epsilon)$-term of a particular combination of
on-shell massless planar double
box integrals, which appears in the tensor reduction of $2 \to 2$
scattering amplitudes at two loops.
\end{abstract}
% typeset front matter (including abstract)
\maketitle
\thispagestyle{myheadings}
\markright{TTP00-10 -- hep-ph/0005232}
\section{Introduction}
Perturbative corrections to many inclusive quantities have been computed
to the two- and three-loop level in past years. From the technical
point of view, these inclusive calculations correspond to the computation of
multi-loop two-point functions, for which many elaborate
calculational tools have been developed. In contrast,
corrections to exclusive
observables, such as jet production rates, could up to now only be
computed at the one-loop level. These calculations demand the
computation of multi-leg amplitudes to the required number of loops,
which beyond the one-loop level
turn out to be a calculational challenge obstructing further progress.
Despite considerable progress made in recent times, many of the
two-loop integrals relevant for the calculation of jet observables
beyond next-to-leading order are still unknown. One particular class of
yet unknown
integrals appearing in the two-loop corrections to three jet production
in electron-positron collisions, to two-plus-one jet production in
electron-proton collisions and to vector boson plus jet production in
proton-proton collisions are two-loop four-point functions with massless
internal propagators and one external leg off-shell.
We elaborate on in this talk several
techniques to compute multi-leg amplitudes beyond one loop. We
demonstrate how integration-by-parts identities (already known to be a very
valuable tool in inclusive calculations) and identities following from
Lorentz-invariance (which are non-trivial only for integrals depending
on at least
two independent external momenta) can be used to reduce the large number
of different integrals appearing in an actual calculation to a small
number of master integrals. This reduction can be carried out
mechanically (by means of a small chain of
computer programs), without
explicit reference to the actual structure of the integrals under
consideration and can also be used for the reduction of
tensor integrals beyond one loop.
The master integrals themselves, however, can not be computed from
these identities. We derive differential equations in the external
momenta for them. Solving these
differential equations, it is possible to compute the master integrals
without explicitly carrying out any loop integration, so that this
technique appears to be a valuable alternative to conventional
approaches for the computation of multi-loop integrals.
We demonstrate the application of these tools on
several examples.
\section{Reduction to master integrals}
Any scalar massless two-loop integral can be brought into the form
\begin{eqnarray}
\lefteqn{I(p_1,\ldots,p_n) = }\nonumber \\
& & \int \frac{\d^d k}{(2\pi)^d}\frac{\d^d l}{(2\pi)^d}
\frac{1}{D_1^{m_1}\ldots D_{t}^{m_t}} S_1^{n_1}
\ldots S_q^{n_q} \; ,
\label{eq:generic}
\end{eqnarray}
where the $D_i$ are massless scalar propagators, depending on $k$, $l$ and the
external momenta $p_1,\ldots,p_n$ while $S_i$ are scalar products
of a loop momentum with an external momentum or of the two loop
momenta. The topology (interconnection of
propagators and external momenta) of the integral is uniquely
determined by specifying the set $(D_1,\ldots,D_t)$
of $t$ different propagators in the graph. The integral itself is then
specified by the powers $m_i$ of all propagators and by the selection
$(S_1,\ldots,S_q)$ of scalar products and their powers $(n_1,\ldots,n_q)$
(all the $m_i$ are positive integers greater or equal to 1, while the
$n_i$ are greater or equal to 0).
Integrals of the same topology with the same dimension $r=\sum_i m_i$
of the denominator and same total number $s=\sum_i n_i$ of scalar products
are denoted as a class of integrals $I_{t,r,s}$. The integration measure and
scalar products appearing the above expression are in Minkowskian space,
with the usual causal prescription for all propagators. The loop
integrations are carried out for arbitrary space-time dimension $d$,
which acts as a regulator for divergencies appearing due to the
ultraviolet or infrared behaviour of the integrand (dimensional
regularisation, \cite{dreg,hv}).
The number $N(I_{t,r,s})$ of the integrals grows quickly as $r, s$
increase, but the integrals are related among themselves
by various identities.
One class of identities follows from the fact that the integral over the
total derivative with respect to any loop momentum vanishes in
dimensional regularisation
\begin{equation}
\int \frac{\d^d k}{(2\pi)^d} \frac{\partial}{\partial k^{\mu}}
J(k,\ldots) = 0,
\end{equation}
where $J$ is any combination of propagators, scalar products
and loop momentum vectors. $J$ can be a vector or tensor of any rank.
The resulting identities~\cite{hv,chet} are called integration-by-parts (IBP)
identities.
In addition to the IBP identities, one can also exploit the fact that
all integrals under consideration are Lorentz scalars (or, perhaps
more precisely, ``$d$-rotational'' scalars) , which are
invariant under a Lorentz (or $d$-rotational) transformation of the
external momenta~\cite{gr}. These Lorentz invariance (LI) identities
are obtained from:
\begin{eqnarray}
\lefteqn{\left(p_1^{\nu}\frac{\partial}{\partial
p_{1\mu}} - p_1^{\mu}\frac{\partial}{\partial
p_{1\nu}} + \ldots\right. }\nonumber \\
&&
\left. \ldots + p_n^{\nu}\frac{\partial}{\partial
p_{n\mu}} - p_n^{\mu}\frac{\partial}{\partial
p_{n\nu}}\right) I(p_1,\ldots,p_n) = 0 \;. \nonumber \\ &&
\label{eq:li}
\end{eqnarray}
\markright{}
In the case of two-loop four-point functions, one has a total of 13
equations (10 IBP + 3 LI) for each integrand corresponding to an
integral of class $I_{t,r,s}$, relating integrals of the same topology
with up to $s+1$ scalar products and $r+1$ denominators, plus integrals
of simpler topologies ({\it i.e.}~with a smaller number of different
denominators).
The 13 identities obtained starting from an integral $I_{t,r,s}$ do contain
integrals of the following types:
\begin{itemize}
\item $I_{t,r,s}$: the integral itself.
\item $I_{t-1,r,s}$: simpler topology.
\item $I_{t,r+1,s}, I_{t,r+1,s+1}$ : same topology, more complicated than
$I_{t,r,s}$.
\item $I_{t,r-1,s}, I_{t,r-1,s-1}$: same topology, simpler than
$I_{t,r,s}$.
\end{itemize}
Quite in general, single identities of the above kind can be used
to obtain the reduction of $I_{t,r+1,s+1}$ or $I_{t,r+1,s}$ integrals
in terms of $I_{t,r,s}$ and simpler integrals - rather than to
get information on the $I_{t,r,s}$ themselves.
If one considers the set of all the identities obtained starting from
the integrand of all the $N(I_{t,r,s})$ integrals of class $I_{t,r,s}$,
one obtains
$(N_{{\rm IBP}}+ N_{{\rm LI}}) N(I_{t,r,s})$ identities
which contain $N(I_{t,r+1,s+1})+N(I_{t,r+1,s})$
integrals of more complicated structure. It was first noticed by S.\
Laporta~\cite{laporta}
that with increasing $r$ and $s$
the number of identities grows faster than the number
of new unknown integrals.
As a consequence, if for a given $t$-topology one considers the set of
all the possible equations obtained by considering all the integrands up to
certain values $r^*, s^*$ of $r, s$, for large enough $r^*, s^*$
the resulting system of equations, apparently overconstrained,
can be used for
expressing the more complicated integrals, with greater values of $r, s$
in terms of simpler ones, with smaller values of $r, s$.
An automatic procedure to preform
this reduction by means of computer algebra
is discussed in more detail in~\cite{eproc}.
For any given four-point two-loop topology,
this procedure can result either in a reduction
towards a small number (typically one or two) of integrals of the
topology under consideration and integrals of simpler
topology (less different denominators), or even in a complete
reduction of all integrals of the topology under consideration
towards integrals with simpler topology.
Left-over integrals of the topology under consideration are called
irreducible master integrals or just
master integrals.
\section{Differential equations for master integrals}
\label{sec:de}
The IBP and LI identities allow to
express integrals of the form (\ref{eq:generic}) as a linear
combination of a few master integrals, i.e.\ integrals which are
not further reducible, but have
to be computed by some different method.
At present, the complete set of master integrals for massless on-shell
two-loop box integrals is known
analytically~\cite{smirnov,tausk,smir2,dur} up to
finite terms in $\epsilon=(4-d)/2$. For massless two-loop box integrals
with one off-shell leg, several topologies are yet to be computed
analytically. A purely numerical approach for computing
these integrals order by order in $\epsilon$ has recently been
proposed by Binoth and Heinrich in~\cite{numbox}.
A method for the analytic
computation of master integrals avoiding the explicit
integration over the loop momenta is to derive differential equations in
internal propagator masses or in external momenta for the master integral,
and to solve these with appropriate boundary conditions.
This method has first been suggested by Kotikov~\cite{kotikov} to relate
loop integrals with internal masses to massless loop integrals.
It has been elaborated in detail and generalised to differential
equations in external momenta in~\cite{remiddi}; first
applications were presented in~\cite{appl}.
In the case of four-point functions with one external off-shell leg
and no internal masses, one has three independent
invariants, resulting in three differential equations.
The derivatives in the invariants $s_{ij}=(p_i+p_j)^2$
can be expressed by derivatives in the external momenta:
\begin{eqnarray}
\sab \frac{\partial}{\partial \sab} & = & \frac{1}{2} \left( +
p_1^{\mu} \frac{\partial}{\partial p_1^{\mu}} +
p_2^{\mu} \frac{\partial}{\partial p_2^{\mu}} -
p_3^{\mu} \frac{\partial}{\partial p_3^{\mu}}\right)\;, \nonumber \\
\sac \frac{\partial}{\partial \sac} & = & \frac{1}{2} \left( +
p_1^{\mu} \frac{\partial}{\partial p_1^{\mu}} -
p_2^{\mu} \frac{\partial}{\partial p_2^{\mu}} +
p_3^{\mu} \frac{\partial}{\partial p_3^{\mu}}\right)\;,\nonumber \\
\sbc \frac{\partial}{\partial \sbc} & = & \frac{1}{2} \left( -
p_1^{\mu} \frac{\partial}{\partial p_1^{\mu}} +
p_2^{\mu} \frac{\partial}{\partial p_2^{\mu}} +
p_3^{\mu} \frac{\partial}{\partial p_3^{\mu}}\right)\;. \nonumber\\
\label{eq:derivatives}
\end{eqnarray}
It is evident that acting with the right hand sides of (\ref{eq:derivatives})
on a master integral $I_{t,t,0}$
will, after interchange of
derivative and integration, yield
a combination of
integrals of the same type as appearing in the IBP and LI identities for
$I_{t,t,0}$, including integrals of type $I_{t,t+1,1}$ and
$I_{t,t+1,0}$. Consequently, the scalar derivatives (on left hand side of
(\ref{eq:derivatives}))
of $I_{t,t,0}$ can be expressed by a linear combination of
integrals up to $I_{t,t+1,1}$ and
$I_{t,t+1,0}$.
These can all be reduced (for topologies containing only one
master integral) to $I_{t,t,0}$
and to integrals of simpler topology
by applying the IBP and LI identities. As a result, we obtain
for the master integral $I_{t,t,0}$
an inhomogeneous linear
first order differential equation in each invariant. For topologies with
more than one master integral, one finds a coupled system of
first order differential equations.
The inhomogeneous term in these differential equations contains only
topologies simpler than $I_{t,t,0}$, which are considered to be known
if working in a bottom-up approach.
The master integral $I_{t,t,0}$ is obtained by matching the general
solution of its differential equation to an appropriate boundary
condition. Quite in general, finding a boundary condition is
a simpler problem than evaluating the whole
integral, since it depends on a smaller number of kinematical
variables. In some cases, the boundary condition can even be
determined from the differential equation itself.
In writing down the differential equations for all master integrals
appearing in the reduction of two-loop four-point functions with up to
one off-shell leg, one finds that all boundary conditions can be
related to the following four one-scale two-loop integrals:
\begin{eqnarray}
\bubbleNLO & = & A_3
\left(-s\right)^{d-3}\nonumber \\[3mm]
\doublebubbleNLO
& = & A_{2,LO}^2 \left(-s\right)^{d-4}\nonumber\\[3mm]
\triangleNLO
& = & A_4 \left(-s\right)^{d-4}\nonumber\\[3mm]
\trianglecrossNLO
& = & A_6 \left(-s\right)^{d-6}\nonumber\\
\end{eqnarray}
These integrals fulfill homogeneous differential equations in $s$; their
normalisation can therefore not be calculated in the differential
equation method. They can however be evaluated explicitly
using Feynman parameters~\cite{kl}.
\section{Applications}
Using the differential equation method, we computed in~\cite{gr} all
master integrals with up to $t=5$ different denominators that can appear
in the reduction of two-loop four-point functions with one off-shell
leg.
Differential equations were also used in the computation of two-loop
on-shell master integrals with planar~\cite{smir2} and
crossed~\cite{dur} topologies. These topologies contain two master
integrals each. The differential equations were applied
only to obtain a
relation among both master integrals, while one of the master integrals
was calculated by Smirnov for the planar case~\cite{smirnov} and
Tausk for the non-planar case~\cite{tausk} using a different method.
At this workshop, Nigel Glover reported first progress towards the
computation of massless on-shell $2\to 2$ scattering amplitudes at two
loops~\cite{glover}. In the reduction of planar amplitudes, it turned
out that it is not sufficient to know the planar double box integrals
up to their finite terms in $\epsilon$.
In the tensor reduction procedure,
the following combination of master integrals arises:
\begin{eqnarray}
\lefteqn{\frac{1}{\epsilon}\left[
\doubleboxNLO - t \doubleboxdotNLO \right]}\nonumber \\
& \sim & \frac{1}{\epsilon}\left[
K_1(x,\epsilon) + K_2(x,\epsilon)\right]\;,
\end{eqnarray}
where $x=t/s$. The functions $K_1(x,\epsilon)$ and $ K_2(x,\epsilon)$
were computed only up to finite terms in $\epsilon$ by Smirnov and
Veretin in~\cite{smir2}. The structure of the up to now unknown
${\cal O}(\epsilon)$-term was subject of debate at this workshop. We
did therefore decide after the workshop
to attempt its computation using the
differential equation method.
Starting from the differential equations for the massless double box
integrals with one off-shell leg, which were obtained using the
algorithms described in Section~\ref{sec:de}, we can derive the
differential equations for the above on-shell integrals
by constraining $\sab+\sac+\sbc =0$ (see also~\cite{dur}). Using $s$ and
$x$ as independent variables, we obtain two homogeneous differential
equations in $s$ (corresponding to a rescaling relation,~\cite{appl}) and
two coupled
inhomogeneous equations in $x$, which can be employed to compute the
two master integrals. Using the fact that the master integrals and their
derivatives are regular in $t$ at
$t=-s$, the boundary condition at $x=-1$
is inferred from the differential equations. This condition is
however not yet sufficient to
determine the boundary conditions for both master
integrals, since it only fixes one combination of them.
It does however turn out that it is sufficient to match the
non-logarithmic terms in $K_1(x,\epsilon)$, obtained by Smirnov
in~\cite{smirnov}, up to finite order in
$\epsilon$ to find the boundary condition
for $K_1(x,\epsilon)+K_2(x,\epsilon)$ up to ${\cal O}(\epsilon)$.
The differential equations are then solved order by order in $\epsilon$
by expressing the unknown as
sum of harmonic polylogarithms~\cite{hpl}.
Requiring the differential equation to be fulfilled and the
boundary conditions to be matched, the coefficients of the
harmonic polylogarithms in the ansatz can all be determined.
Using the same normalisation factor as~\cite{smir2},
the coefficient of the ${\cal O}(\epsilon)$-term of the above
combination of double box integrals reads:
\begin{eqnarray}
\lefteqn{\left[
K_1(x,\epsilon) + K_2(x,\epsilon)\right] \Big|_{{\cal
O}(\epsilon)} }\nonumber \\
& = & -\frac{4}{3} \ln^4 x + \frac{4}{3} \ln^3 x - 4(4+3\pi^2) \ln^2 x
\nonumber \\
& & -\Big(46 + 21 \pi^2 - \frac{16}{3} \zeta (3)\Big) \ln x \nonumber \\
& & - 50 - 8\pi^2 - \frac{139}{60}\pi^4 + \frac{698}{3}\zeta(3)
\nonumber\\
& & -32\, S_{2,2}(-x) + 32 \ln x\, S_{1,2}(-x) \nonumber \\
& & - 128\, \Li_4(-x)
+32 \Big( 3\ln x - \ln (1+x) \nonumber \\
& & -2\Big)\, \Li_3(-x)+32\Big( -\ln^2 x + 2 \ln x \nonumber \\
& & + \ln x \ln (1+x) - \frac{5}{6} \pi^2 \Big)
\Li_2(-x) \nonumber \\
& & + 8 \left(\ln^2 x + \pi^2 \right)\ln^2(1+x) \nonumber\\
& & + \frac{16}{3} \Big( -\ln^3 x + 6 \ln^2 x + 6 \pi^2 - 2 \pi^2 \ln x
\nonumber \\
&& + 6 \zeta(3) \Big) \ln(1+x) \nonumber \\
&& + x \bigg[
+\frac{16}{3} \ln^4 x - \frac{52}{3} \ln^3 x +\frac{46}{3}\pi^2 \ln^2 x
\nonumber \\
&& + \Big(2 -\frac{169}{3} \pi^2 - \frac{496}{3} \zeta (3)\Big) \ln x
\nonumber \\
& & - 84 - \frac{46}{3}\pi^2 + \frac{823}{360}\pi^4 + 536\,\zeta(3)
\nonumber\\
& & -56\, S_{2,2}(-x) + 56 \ln x\, S_{1,2}(-x) \nonumber \\
&& + 40 \, \Li_4(-x)
+ \Big( 16 \ln x - 56 \ln (1+x) \nonumber \\
&& - 104\Big)\, \Li_3(-x) - \Big( 36 \ln^2 x +\frac{8}{3}\pi^2 \nonumber \\
&&
- 56 \ln x \ln(1+x) - 104 \ln x
\Big) \Li_2(-x) \nonumber \\
&& + 14 (\ln^2 x + \pi^2) \ln^2 (1+x) \nonumber \\
&& + \frac{4}{3} \Big(-16 \ln^3 x + 39 \ln^2 x - 23\pi^2 \ln x \nonumber \\
&&
+ 39 \pi^2 +
42 \zeta(3) \Big) \ln(1+x) \bigg] \;.
\end{eqnarray}
\section{Outlook}
We have demonstrated how techniques developed for
multi-loop calculation of two-point functions can be extended towards
integrals with a larger number of external legs. In particular, we
have shown that the use of differential equations in external invariants
can be used as a powerful method to compute master integrals without
carrying out explicit loop integrations.
As a first example of
the application of these tools in practice, we computed some
up to now unknown
two-loop four-point functions, relevant for jet calculus beyond the
next-to-leading order.
The most important potential application of these
tools is the yet outstanding derivation of two-loop virtual corrections to
exclusive quantities, such as jet observables.
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\end{document}