NEW ALGEBRAIC-NUMERIC METHODS
FOR LOOP INTEGRALS.
SOME 1-LOOP EXPERIENCE


D .Y u. Bardina, L.V. Kalinovskayaa and F. V. Tkachovb

a Joint Institute for Nuclear Research, Dubna, Moscow Region, 141980 Russia

b Institute for Nuclear Research of Russian Academy of Sciences
Moscow, 117312 Russia



We report our experiences with the generalized integration-by-parts algorithm  in the
context of calculations of a realistic one-loop subset of diagrams.




This project combined the experience of the 1 (x + A)
x  1
+ 
ZFITTER1 team [1] with the experience of designing 1 - V ( x) = V (x)
. 0.1
2(
 +1)
algorithms for multiloop calculations2 [4], [5] in order to
explore the potential of the "algebraic" scenario of loop Here x is the vector of Feynman parameters,
calculations suggested in [6].
T T
V (x) = x Vx + 2R x + Z is a quadratic polynomial,
The scenario generalizes the idea of [4], which is to T 1 T 1
-
= -
exploit the rich algebraic structure of integrands in order A R V -
=
and (Z R V R).
to reduce them to a maximally simple form, depending Any one-loop integral can be represented as
on the specific application. The original algorithm of [4] -n
-

was about reduction of massless self-energies to recur- dxQ(x)V ( x)
S . 0.2

sively one-loop integrals easily calculable analytically; An iterative application of 0.1 followed by integration by
it demonstrated that such a reduction may be an extre- parts reduces 0.2 to a sum of the form
mely powerful calculational tool. The scenario of [6] is
N -
about reduction of integrands to a form maximally dx Q (x )V (x ),
S 0.3
suitable for subsequent numerical integration. It is based where the sum involves integrals over simplices with no
on a subtle algebraic fact (a generalization of the so- higher dimensionality than in the original integral 0.2.
called Bernstein theorem [7]) and is applicable to arbitra- The Laurent expansion in yields integrals of the form
ry loop integrals (arbitrary topologies and patterns of
masses and external momenta). This is in theory.
N V (x )
dx Q(x )
V (x )ln ,
0.4
In practice, the complexity of the problem of finding S 2

the differential identities required by the scenario of [6]
arXiv: 17 Dec 2000 where  is the unit of mass. (In a general one loop
for integrands with 2 or more loops now appears to be so integral second powers of log V are possible.)
great (and the resulting identities so hugely cumbersome) By choosing N large enough (which is achieved by
that the feasibility of the scenario of [6] beyond one loop suitably many applications of the identity 0.1) one can
level is rather doubtful. achieve an arbitrarily large degree of smoothness of the
However, for one-loop integrals, all algebraic difficul- integrands in 0.4, ensuring their better amenability to
ties reduce to an iterative application of a simple dif- numerical integrations. With N =0, one has a simple
ferential identity: absolute convergence (logarithmic singularities in the
integrand). With N =1, the integrand is continuous; with
N =2, it has continuous first derivatives, etc. Simple 1-
1 ZFITTER [1], along with TOPAZ0 [2], is a program which dimensional examples show that the optimal rate of
incorporates all the available theoretical knowledge about the Z- numerical convergence (with integration algorithms that
resonance in the process of e+e- annihilation [3]. They have been used make use of continuity of derivatives) is achieved for
to extract Standard Model parameters from the precision data
obtained at LEP1 by the experimental collaborations at LEP. values of N =4,5; by rate of convergence we understand
2 These algorithms are behind the flourishing industry of NNLO QCD the number of integrand evaluations required to attain a
calculations (sum rules and moments of DIS structure functions; mass precision of 8-10 digits.
expansions; (
tot e +e - hadrons)).





1


On the downside, the quantity which appears in the  1
(k ) P
B ( 0.9
 )  k
dx P ln x ,
denominator may contain small factors near thresholds, 0 x 2
0 
leading to large numerical cancellations between differ-
ent terms in the sum 0.3, which offset the benefits of where
smoother integrands.3 The scenario of [6] offers no
 2   2  2
P Q x(1 x) M x M (1  x) . 0.10
options for that. Note, however, that a complete set of x 1 2
rules for obtaining systematic asymptotic expansions of Conventional expressions for the formfactors
arbitrary Feynman integrals in arbitrary asymptotic Z
F , # = ,
L ,
Q D are in terms of the standard C 0 and B0.
regimes was obtained in [8], so the handling of cases #
with a small is in theory not an insurmountable One application of the identity 0.1 (in n ) together
 4
problem (although, of course, one would prefer a uniform with integration by parts yields expressions in terms of
algorithm to treat the various cases). (0)
C and (0)
B . The total length of the three expressions
0 0
Our aim was to explore the behavior of this algorithm is one LaTeX page. Two applications yield expressions
in a realistic case of a subset of diagrams which in terms of (1)
C and (1)
B , with the total length of two
0 0
contribute to the ZFITTER [1]. The subset (which we pages.
call Z cluster; for details see [9]) is gauge invariant and After the algebraic simplifications, numerical integra-
UV finite. It is shown in the following figure: tion was performed using a simple Simpson integrator
f which is sensitive to smoothness of integrands.
The numerical results are presented in the following
Z 0
table:
f

k Re Im
We will consider and evaluate three standard scalar
Z
F
formfactors Z
F , # = ,
L ,
Q D which are present in the Z L 1.62989336 2.32844153
#
cluster with incoming photon in the Z vertex. These 0 < 10-9 1.5E-3
can be expressed in terms of the standard Passarino- 1 < 10-9 1.9E-5
Veltman functions C 0 and B0 (see e.g. [3]). For instance: Z
FQ -0.366725194 0.808473633

2 2 2 2 n 4 1
( , , ; , , ) n
i C p p Q m m m 
 d q , 0.5
0 1 2 1 2 d d d 0 < 10-9 2.0E-4
0 1 2
1 < 10-9 2.2E-6
where  2  2
d q m  i ,   2  2
d (q p ) m  i and
0 1 1 1 Z
FD -0.00234117123 -0.0063346893
  2  2
d (q Q) m  i . Also Q  p  p . In terms
2 2 1 2 0 < 10-9 1.9E-3
of Feynman parameters (in n = 4):
1 < 10-9 2.3E-5
 1  1 x
2 2 2
 1
C ( p , p ,Q ; m , ,
m m ) dx dy P , 0.6
0 1 2 1 2 x, y
0 0
For each formfactor, the first line lists exact results
where (obtained in the usual way via dilogarithms). The second
and third lines give absolute relative errors for the results
 2 2  2 2  2  2  2
P x p y p xy Q p p
x, y 1 2 ( 1 2 ) of numerical calculation of the same quantities. The lines
 2  2  2  2  2  2  2
x( p m m ) y( p m m ) m . 0.7
1 1 2 2 k = 0|1 correspond to the cases when the identity 0.1 is
used once
We introduce the following generalizations: | twice so that all resulting integrands have the
singularity ln P |PlnP (absolute integrability with loga-
 1  1 x P
(k ) rithmic singularities [step function in the imaginary
 k x, y
C ( 0.8
 ) dx dy P ln .
0 x, y 2
0 0  parts] in the first case; continuous integrands in the
second case). The simplest Simpson integrator was used
Similarly: which likes continuous second order derivatives. But we
felt the complexity of the formulas resulting from three
application of the differential identity was too great to
3 Examples of this kind were provided to us by K. Kato. Note that justify the effort involved in the interfacing of the
maximal benefits from the algebraic transformations being described algebraic and numerical parts of this project for more
are obtained with integration routines that make use of the increased
regularity of resulting integrands. But since the method effectively than two applications of the identity 0.1 at the current
subtracts all types of divergencies/singularities simultaneously, it may exploratory stage of the project.
be beneficial even with MC integrators.




2


A conclusion that can be drawn from our experiment References
is that the scenario based on the use of the algebraic [1] D. Bardin et al: ZFITTER v.6.21 - A semi-analytical
identity 0.1 may, at least in some one loop applications, + -
program for fermion pair production in e e
be a viable alternative to the standard procedures based annihilation,  [DESY 99-070].
on the use of analytical expressions in terms of diloga- [2] G. Montagna, O. Nicrosini, F. Piccinini, and
rithms (cf. [3]). Moreover, we have observed a number of G.Passarino [Comp. Phys. Comm. 93 (1996)120; 117
attractive properties: (1999) 278]
 Gauge cancellations within gauge invariant subsets of [3] D. Bardin and G. Passarino: The Standard Model in
diagrams occur prior to numerical integrations. This the Making. Precision Study of the Electroweak
means that numerical cancellations due to at least this Interactions [Clarendon, Oxford, 1999].
source are not encountered. [4] F.V. Tkachov: A theorem on analytical calculability
 of 4-loop renormalization group functions
Similarly cancel UV divergences. [Phys. Lett. 100B (1981) 65].
 In our example, we have observed little (if any) [5] F.V. Tkachov: Advanced methods for studying
evidence of numerical instabilities. It is essential here radiative corrections. Theory of asymptotic
that a deterministic numerical integration algorithm was operation ,  [QFTHEP'92;
used which took into account the increased smoothness Phys. Part. Nucl. 25 (1994) 649].
of the integrands; Monte Carlo integration algorithms are [6] F.V. Tkachov: Algebraic algorithms for multiloop
inappropriate here. calculations. The first 15 years. What's next?,
 The method offers an interesting alternative to the  [NUIMA A389 (1996) 309].
conventional Passarino-Veltman reduction to scalar inte- [7] I.N. Bernstein: Analytical continuation of generalized
functions in a parameter [Functional Analysis And
grals. Note that the new reduction (i.e. the representation Its Applications, 6 (1972) 26].
in terms of the basic scalar functions 0.8, 0.9 with all x'
excluded from 0.4) was implemented in n dimensions. [8] F.V. Tkachov: Towards systematic near-threshold
expansions in perturbative QFT , 
The greatest technical difficulty for implementation [Phys. Lett. B412 (1997) 350].
of the described scenario -- both in one loop and to a [9] D. Bardin, L. Kalinovskaya, and G. Nanava:
very much greater extent beyond one loop -- appears to An electroweak library for the calculation of EWRC
be the dominant software engineering platform in the to e+e- f f within the topfit project,
HEP community (conventional inefficient symbolic ma- .
nipulation programs + numerical calculations with the
archaic Fortran/C or the clay-feet monster of C++, with [10] F.V. Tkachov: Theoretical physicist's personal
software engineering in the post-Java world
no easy connection between the two). Another problem is [to be posted].
that the computer resources needed even in simpler cases
are huge (although Moore's Law is slowly taking care of
that; perhaps too slowly).

Since there are physical problems in which further
progress via conventional approaches seems to be all but
excluded, experimentation with the scenario for loop
calculations of [6] will continue. On a limited scale and
in some special problems one may even expect realistic
calculations to occur within a few years -- provided a
sufficient high-quality software engineering effort could
be mustered. But the main problem is the inadequacy of
the dominant software engineering platform in HEP.4


Acknowledgments

This work was supported in part by the Russian
Foundation for Basic Research (quantum field theory
section) under grant 99-02-18365. F.T. thanks K.Kato
for providing some numerical examples of calculations of
one-loop integrals.


4 It cannot be emphasized too strongly: C++ cannot be a foundation
on which to build future. Some related comments together with a
discussion of alternatives are collected in [10].




3



