 in: QFTHEP'97 (XII Int. Workshop, Samara, Russia, 4-10 September 1997) 1




Algebraic algorithm for next-to-leading order calculations
in the large-s/small-x regime


A.V. Stepanovaa and F.V. Tkachovb

aPhysics Department, Moscow State University, Moscow 119899
bInstitute for Nuclear Research of Russian Academy of Sciences, Moscow 117312



An algebraic algorithm is presented for analytical calculation of arbitrary dimensionally regulated massless
two-loop forward-scattering diagrams that constitute the most cumbersome part of next-to-leading order calcu-
lations in the large-s /small-x regime.





Introduction 1 point for us here is that those counterterms always






contain -functions instead of some propagators of
There is much interest in studying processes that the original diagram, so they are easier to take into
can be described as almost-forward scattering: e.g. account than the formally expanded part, although
the small-x problem in the context of deep-inelastic from theoretical point of view it is the obtaining of
scattering in QCD [1] (via the optical theorem), the counterterms that is the most interesting part of the
small-angle Bhabha scattering in QED [2], etc. In problem. From calculational point of view, however,
such problems one deals with a kinematical situation the most cumbersome part is always the calculation of
when the scattering of two particles k k  k k
1 2 3 4 the formally expanded diagrams of the original proc-
(e.g. virtual photon and proton in the case of DIS; ess (this is also the case with the NNLO calculations
electron and positron in the case of Bhabha scatter- of coefficient functions of OPE -- e.g. [5] -- accord-
ing) occurs at the values of the Mandelstam variable ing to the prescriptions of the Euclidean theory of
s 2
 k 
( k ) (corresponding to
1 2 W 2 in the DIS case) asymptotic operation [6], [7]). It is this problem that
much larger than any other Lorentz-invariant dimen- we are going to address in this work.
sional parameter in the problem (masses and virtuali-
ties of participating and virtual particles as well as the Topologies and algorithm 2
Mandelstam variables t and u ). A systematic way to The loop integrals we are going to consider have
treat such processes in perturbative Quantum Field the following kinematics:
Theory is via an asymptotic expansion corresponding
to the asymptotic regime s 2
  t ,u, k k k
i . This asymptotic 1 1

regime belongs to the so-called class of non-
 v2 7 Nov 1998 Euclidean asymptotic regimes for which, unlike the k2 k2 2.1
Euclidean class, systematic prescriptions for obtain-
ing asymptotic expansions to all orders of the small One-loop diagrams of this type are elementarily
expansion parameter (in our case, s 1 ) did not exist integrated in terms of Euler's -functions. So we are












(see e.g. [3]). The first systematic theory of non- going to consider two-loop diagrams. The method we
Euclidean asymptotic regimes based on the distribu- will use is an extension to this kinematics of the well-
tion-theoretic method of asymptotic operation [3] has known algebraic (integration-by-parts) algorithm of
been reported in another talk at this workshop [4]. [8] that was originally developed for 3-loop massless
According to the theory of [3], [4], systematic as- self-energy diagrams.
ymptotic expansions are obtained by first formally There are eighteen non-equivalent two-loop to-
Taylor-expanding the corresponding diagrams in the pologies. (For instance, topologies that differ e.g. by
 
small parameter, and then adding certain coun- k k are equivalent.) Here is the list:
1 2
terterms whose structure is dictated by the prescrip-
tions of the theory of asymptotic operation. The key


 in: QFTHEP'97 (XII Int. Workshop, Samara, Russia, 4-10 September 1997) 2





k1 k1


k2 k2





2.2


Four of them (enclosed in the dashed box) can be whenever a k 2
i occurs in the identity, the corre-
calculated in -functions by direct integration.
sponding contribution vanishes whereas in the origi-
Twelve topologies can be treated using integration- nal algorithm it caused a line to shrink to a point. In
by-parts recurrencies. The remaining two (in solid particular, the term that corresponded to shrunk top
boxes) are irreducible in the sense that they contain right line in 2.4 is nullified in the case of 2.3.
some integrals that can neither be reduced to other The resulting recurrence relation has a standard
topologies via integration by parts nor easily ex- form, and we only point out that the integrals result-
pressed in terms of -functions.
ing from repeated application of the rule of triangle
The integration-by-parts method [8] is well known according to 2.3 all belong to the same topology:
and we do not explain it in details. We just present
one example of a recursion relation for one of the
topologies: 2.5


2.3 Irreducible topologies 3

This in fact is similar to one of the topologies en- One class of irreducible diagrams corresponds to
countered in the original integration-by-parts algo- the following topology:
rithm for three-loop self energies:


3.1

2.4 Note that for this topology, there is an irreducible
It is sufficient to apply the rule of triangle [8] to the scalar product which can be chosen as the scalar
triangle indicated on Fig. 2.3. But now we have product of the momenta flowing through the lines
massless external momenta ( shown with hollow arrows. In the original integra-
k 2  k 2  ). Therefore
1 2 0
tion-by-parts algorithm it was possible to get rid of


 in: QFTHEP'97 (XII Int. Workshop, Samara, Russia, 4-10 September 1997) 3



the powers of the irreducible scalar product by differ- Acknowledgments
entiations with respect to the external momentum. In We thank N.I. Ussyukina for discussion of irre-
the present case the two external momenta are re- ducible diagrams, the organizing committee and
stricted to mass shell, so the differentiation does not sponsors of the QFTHEP'97 workshop for financial
work. Using symmetry with respect to reflections, one support, and the participants of the workshop for
can get rid of odd powers of the irreducible invariant. comments. This work was supported in part by the
In practice we expect only zeroth and second powers Russian Foundation for Basic Research under grant
to occur. 95-02-05794.
The other irreducible topology contains only one
irreducible diagram: References


[1] A. Ali and J. Bartels (ed.), Small-x Behaviour of
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Detailed discussion of calculation of the irreducible [2] A. Arbuzov and et al., Phys. Lett. B383 (1996) 238
topologies goes beyond the scope of the present paper. .
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extracted as a special case from the double box result .
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the topologies 3.1 can be done analytically using .
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In conclusion we compare our algorithm with the [9] N.I. Ussyukina, in QFTHEP'97, Samara, September
original algorithm for massless 3-loop self-energies. 4-10, 1997 (to be published).
[10] S.G. Gorishny, S.A. Larin, L.R. Surguladze, and F.V.
First, in the case of 2-loop forward scattering we have Tkachov, Comp. Phys. Comm. 55 (1989) 381.
more topologies than in the case of 3-loop self- [11] S.A. Larin, F.V. Tkachov, and J.A.M. Vermaseren,
energies. Second, now there are more types of dia- preprint NIKHEF-H/91-18 (1991).
grams to be computed using special formulas. Third, [12] M. Veltman, in New Computing Techniques in Phys-
there are more irreducible diagrams. Forth, there are ics Research III (AIHENP'93). Oberammergau, Ger-
more poles here -- fourth order poles (two poles per many. Oct. 4-8, 1993. World Scientific (1993).
loop as is typical for Minkowski-space integrals) in-
stead of third order poles in the case of the original
algorithm. So, on the whole the algorithm is more
cumbersome. However, this is not expected to be a
problem because we are going to use a specialized
software package BEARa which is designed to be
much faster and much more economical with disk
space than analogous software currently available
such as the well-known MINCER program [10], [11]
for systems such as Veltman's SCHOONSCHIP [12].






a http://www.inr.ac.ru/~ftkachov/projects/bear/index.htm



