F.V. Tkachov, Nucl. Instr. Meth. Phys. Res. A389 (1996) 309 





Algebraic algorithms for multiloop calculations
The first 15 years. What's next?

Fyodor V. Tkachov

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




The ideas behind the concept of algebraic ("integration-by-parts") algorithms for multiloop calculations are reviewed. For any
topology and mass pattern, there exists a finite iterative algebraic procedure which transforms the corresponding Feynman-
parameterized integrands into a form that is optimal for numerical integration, with all the poles in D  4 explicitly extracted.




The Concept (3-loop finite parts of massless self-energy diagrams)
A victory has a hundred fathers seemed within grasp. The appetites whetted, it was sug-
The concept of algebraic algorithms for multiloop in- gested that "we" tackle the many multiple series that
tegrals [1][4] has brought about an industry of analyti- emerged there within the technique that I had shown to
cal multiloop calculationsa, so it would not be inappro- be successful with 2-loop finite parts. However, I balked:
priate to reminisce upon the anguish of its birth [1], if one were to appropriate the fruits of my meditations,
given a non-negligible amount of research funds being one could at least leave me alone in the joy of the proc-
ess. (That tacit arrangement was adhered to by the other
misallocated through misinformationb.
side with much gusto all the way through the theory of
Back in 1979, when perturbative QCD, IBM, and the asymptotic expansions, which is another story [7].)
USSR were in full bloom, the first project which I took
part in was successfully completed. Our analytical cal- Being thus vigorously motivated, I conceived of a
culation was receiving a considerable attention, and my method that would, ideally, meet the following criteria:
V





contribution to it was such that there was no doubt in my My method would reduce human intervention to an
mind that it would be my Ph.D. thesis. However, the absolute minimum. First, because my eyesight was at
senior member of our team of three intimated to me that stake. Second, although I remember being able (then) to
the result would go instead into another thesis that would expand -functions in 2-loops pretty reliably, I was less
W





not be able to get composed "otherwise" (which was true sure about 3 loops.
X





beyond any shade of doubt). The deal (in which I had no That meant computer algebra and SCHOONSCHIP [8],
negotiation powers) was accompanied by a swift promo- so that my ideal method would be basically simple to
tion of the senior member.c A concrete reward for me make a program impleme
X ntation feasible.
was a deterioration of my eyesight by two diopters as a
Third, the usual approach was (and largely remains)
result of my innocent calculational enthusiasm. Moreo-
to do all basic integrals "analytically" (i.e. by hand in the
ver, due to the rigidity of Soviet system and my then in-
usual human fashion) and leave to computer only traces,
significance, I was stuck in that vulnerable position for
scalar products, and substitutions of answers for the basic
what seemed to be forever, although I was too light-
 v2 7 Nov 1998 integrals. Since there were so many different integrals in
hearted and inexperienced at the time to foresee all the
3 loops, I decided that such an approach was unaccept-
arXiv: v2 7 Nov 1998 consequences (which may have been just as well). able: My method would have to do something about all
But research was fun and the next level of complexity
those int
X egrals too.

a A number of papers in the proceedings of all AIHENP work- Lastly, my ideal method -- my baby -- had to be
shops discuss calculations that make an essential use of identi- beautiful . That was the only kind of consolation I could
ties obtained via differentiations in momentum space. expect to obtain with any degree of certainty.
b often undocumented; but cf. an unsuspecting citation in [5]. (NB The list is by no means obsolete after 15 years.)
Of course, Germany (say) is rich enough to afford some over- Where did I have to look for such a method? When-
head, especially if one takes into account the grant-fetching ever one has to start from scratch (which one has to do



value of the Russian multiloop expertise. However, " surprisingly often [7], [9]), a little philosophizing at the
  #  '()0 234)5  98@3  A





 "! %$ & !1 67





level of first principles is appropriate (because nothing
 S 


CB#DEFHG IP&!QR T U " [6]
c works like a sound first principle implemented with due
Human nature is, of course, much the same everywhere in the simplicity and relentless logic):
world. But one ought to remember about behavioral and social One general consideration was not unknown (see e.g.
renormalization factors -- in the present case, roughly,
Mr. James Carter over tovarishch Leonid Ilyich [Brezhnev]. [10]) but rarely heeded, namely, that best algorithms for


F.V. Tkachov, Nucl. Instr. Meth. Phys. Res. A389 (1996) 309  2




computer algebra applications should avoid imitating sible variants and trying to combine them in all sorts of
human ways of doing formulas. A rule of thumb is, one ways. It would have been easier to deal with 2-loop dia-
has to trade the true complexity of the amazing pattern grams, but I did not know the answer and chose the 3-
matching that human beings are capable of, for a dull loop ladder as my playing ground, so it took several
hugeness of more or less uniform data structures (e.g. months to stumble upon the celebrated triangle relation
polynomials) that computers are so good at. A difficulty [1]. That took care of planar diagrams. A little more ef-
here is that one has to understand quite well what's go- fort yielded me non-planars in terms of one basic non-
ing on inside computer algebra systems. Another diffi- planar diagram (as well as a rebuke for having failed to
culty is in imagining an "analytical" algorithm that one explain sooner the entire significance of differential
may not even be able to perform oneself. identities). I had a complete algorithm but could not re-
Furthermore, "calculation" is nothing but a "trans - sist the temptation of wasting some more time discover-
formation". In order to "transform" an object one has to ing (once a week or so) ways to reduce the one remaining
understand its "structure". What is "structure" of 3-loop scalar non-planar integral to planar ones before the algo-
massless self-energy integrals? (Exercise Answer this rithm was submitted for publication.
before reading on.) First of all, it is the set of identities But that was not the whole story. When actual pro-
expressing momentum conservation at vertices. That is gramming of the algorithm based on pure recursions [1],
already used in simplifications of scalar products in nu- [2] was attempted, it proved a failure (the huge program
merators. Next, "masslessness" means that Euler's ide n- simply would not work as a whole). The project remained
tity Q f (Q) = f (Q) holds for massless self-energies at a standstill until an explicit solution of the triangle
with Q the external momentum. (At that point I already recursion was found [3]. It resulted in a great speedup in
knew better to be bothered by comments about how it was both debugging and performance of the program which
silly to attempt to do integrals by differentiations.) Doing finally began to turn diagrams into numbers in a way that
differentiation on the integrand and simplifying scalar caused me to baptize it "MINCER". I emphasize a central
products one obtains an identity connecting several dif- role of that solution in the actual functioning of
ferent integrals. The identity depends on how Q is passed MINCER [4] because misleadingly few references have
through the propagators, so I started writing out all pos- been made to the fact in the literature.

--------------------


What's Next? Lastly, the true purpose of algebraic identities is not so
...For the duration of 40 hours he was expounding to much to "calculate" the integrals as to make them
Messrs. Gentlemen a great project, the execution of which simpler OR reduce their number.d
subsequently brought a great glory to England and showed
how far the human mind can reach out sometimes ! NB The meaning of "simpler" is, again, strictly co n-
The Drilling Through The Moon With A Colossal Auger text-dependent: in the case of numerical integration it
-- such was the subject of Mr. Lund's speech !... [11] need not mean "fewer terms" in the integrand but e.g.
"more continuous derivatives". For instance, the new
The range of applications of algebraic algorithms of multidimensional integration package MILX [15]
the conventional type (differential identities in momen- provides an option of using higher-order quadratures
tum space [1]) continues to expand (cf. the recent appli- along with Monte Carlo, thus greatly improving per-
cations to the electron magnetic moment [12] and to the formance when integrands are sufficiently smooth.
effective heavy quark theories [13]). What I'd like to
discuss now is the case of integrals with several external So, let x = (x , , xK )
1 K be the vector of Feynman pa-
parameters (s, t , masses etc.), which is vital in modern rameters; S is the integration region, a simplex described
high energy physics applications. by x > 0; x < 1; j = 1,K, K ( dim );
j j j K S Q(x)
First of all, with several parameters in the problem,
the variety of analytical details is so enormous that one is and V x
i ( ) are arbitrary polynomials of x with symbolic
driven by desperation and logic to ignore them all and to coefficients; i are symbolic (complex, transcendental)
consider calculation of an arbitrary multiloop integral -- parameters. Consider integrals of the form
however preposterous such an idea may seem!
Second, one has to be clear about the meaning of
"calculation". The enormity of the problem forces one to
interpret it in a strictly pragmatic fashion as obtaining d Contrast this with the "algebraic calculability" of [14] which
a family of "satisfactorily fast" algorithms to produce attempts to normalize the spectacular success achieved in the
required curves. Such algorithms need not be based on case of massless self-energies which was merely a matter of
"analytical" formulas. In fact, with many parameters in luck. A situation much more like what I envisaged in the be-
the problem, numerics at the final stage can hardly ever ginning occured in the case of anomalous magnetic moment
be avoided. where everything reduces to 18 independent integrals [12].


F.V. Tkachov, Nucl. Instr. Meth. Phys. Res. A389 (1996) 309  3




 -1 +1 
x
d Q(x) V i ( x
z ) . (1) b P (x, )V ( x) = V (x) , (4)
i
S i
where both b and P depend polynomially on and on






Dimensionally/analytically regulated diagrams are ob- the coefficients of V. Take V ( x) = V (x) , perform the
tained by appropriate choices of Q(x) , V x i i
i ( ) and i . differentiations in (4), cancel the symbolic powers and
i are equal to a fixed integer (denoted ni ; it can be equate coefficients of monomials of x on both sides; one
positive, negative or zero) plus a symbolic part (denoted obtains a linear system of equations for the unknown
i ): i are regarded as infinitesimal because an expan- coefficients of P . Bernstein's result is equivalent to ex-
sion in istence of a non-zero solution of the system, i.e. some
i has to be performed ultimately. The point is
that, whereas the integral (1) may be well-defined as a determinant is non-zero. Perform a similar reduction for
meromorphic function of  (3), and compare the resulting linear system with that
i , the expansion in i may obtained for (4). The former differs from the latter by
not be performed directly in the integrand because some replacements of different entries of by different 

n i , so
i may be too large and negative. that the determinant mentioned above cannot be zero.
The structure I propose to exploit can be described as Therefore, the system has a non-zero solution, which
"polynomials to symbolic powers" and is fully specified yields the identity (3) with the coefficients of P that are
by (1): no further information about V x
i ( ) will be used. rational functions of both i and the coefficients of
The key result I would like to present is as follows: V x
i ( ) . Taking out all the denominators and combining
Theorem A ("algorithm of algorithms of algebraic them into B yields (3). 
multiloop calculations"). Any integral (1) can be re- P r o o f o f T h eo r em A One iteratively performs re-
duced -- via a finite number of purely algebraic steps -- placements of the integrand with the corresponding l.h.s.
to the following form with any given integer 0 : of (3) and performs integrations by parts to get rid of the
1 differentiations. One does this (with appropriate P and
x
d ' Q' (x') +
V i (x' )
, (2)
z i
B B ) also for all integrals corresponding to the boundary
S ' S i
' terms resulting from integration by parts. After a finite
where the sum contains a finite number of terms; S' can number of iterations the integer parts of all complex
be either S or any of its boundary simplices with vari- powers become . 
ous dim S' S ( dim S' = 0 is allowed); x' parameterize In dimensionally regulated diagrams, all i are linear
S' in a natural fashion (e.g. some sets x' are obtained combinations of one symbolic parameter . This may






by nullifying certain components of x ); Q'( x' ) are cause problems due to nullification of some B . That
polynomials of x' ; B' are independent of x' ; B' and all would mean a breakdown of dimensional regularization
coefficients of Q'( x' ) are polynomials of all i and of so that an additional analytic regularization would have
the coefficients of Q(x) and V x to be used to make all
i ( ) ; V x
i ( ' ) are restric- i independent. But since the
identities (3) have to be found for arbitrary
tions of V x i anyway (to
i ( ) to the corresponding boundary simplices handle integrals with different integer parts of
S' ; all numeric coefficients involved are integer. i ), there
is no practical problem here. 
The proof uses the following result:
Discussion
Theorem B ("generalized Bernstein functional
equation"). For any finite set of polynomials V x  The operators P for the same topology but different
i ( ) mass patterns need not be connected in any simple way.
there exists an identity of the following form:  There is an infinite family of non-trivially different
1
B-1 +
P (x,  
) V i (x) = V i (x) , (3)
i i operators P satisfying (3) for the same V x
i ( ) .
i i  The only way, known to me at the time of this writ-
where P ( x, ) is a polynomial of x and =
i / ix ; B ing, of finding the polynomials P is via a direct study of
and all coefficients of P are polynomials of i and of the linear system of equations for the unknown coeffi-
the coefficients of V x
i ( ) . cients of P (cf. the proof of Theorem B). This is cumber-
some (the form of the system depends on the degree of P
P r o o f o f T h eo r em B Using methods of abstract
algebra ("finitely generated ideals" of "graded left mo d- w.r.t. derivatives), but one has to find such an identity
ules", etc.), one can establish (see e.g. [16]) existence of only once for each topology/mass pattern, and only poly-
identities (3) for the case of exactly one (arbitrary) poly- nomial algebra with rational coefficients is involved.
nomial -- a result known as the Bernstein functional  Multiplying both sides of (3) from the left by some of
equation: V x
i ( ) 's and absorbing the latter into P on the l.h.s., one


F.V. Tkachov, Nucl. Instr. Meth. Phys. Res. A389 (1996) 309  4



proves existence of identities with only a subset of expo- F bx Agx I
1 +  1
+ 
nents raised by +1, with others intact. In general, the 1- ( ) ( ) , (5)
HG V x V x
2( + 1) KJ =
family of operators of this latter form contains ones that
cannot be obtained from (3) as just described. These ad- where = - T ~- T ~ -
Z R V 1
e Rj and A = R V 1 . (Special
ditional identities may result in more economical algo-
rithms (iteratively raising each symbolic power to exactly cases of this relation in integral form have been known
the minimal value required for achieving the desired for some time; cf. [18].) (Exercise Write this out for
~
smoothness of the integrand). your favorite one-loop integral.) (NB If V is singular
 Similarly, one can directly seek an identity with +1's then one should determine its null space and treat trans-
on the l.h.s. replaced with + N
i 's with required Ni 0 verse and longitudinal coordinates separately.) Usually,
(or even with some N < one first reduces non-scalar one-loop integrals to scalar
i 0 if the integer part of the i -th
original power was already positive), and with Q on the ones and then uses formulas for the latter in terms of
r.h.s. This is similar to explicit solution of recurrent re- dilogarithms etc. -- a typical example of a procedure
lations in the conventional algorithms [3], [12]. motivated by how human beings perform calculations.
 More iterations make integrands smoother, but the The computer-oriented approach based on (5) is to apply
original integrals did have singularities in external pa- it iteratively to the integrand (with all the numerators)
rameters! The paradox is resolved by noticing that the until required regularity is achieved; then one can take
denominators B have zeros. Those zeros, therefore, must advantage of the numerical integration schemes that em-
correspond to both threshold singularities (hence B must ploy higher-order quadratures such as MILX [15].e
be connected to determinants of the system of Landau  For 2-loop diagrams, the situation is as follows. For
equations) and to the poles in D - 4 (in the case of di- the simplest sunset topology, if P is second order w.r.t.
mensional regularization). 's (a minimal possible number of derivatives that
 It follows that the algorithm automatically extracts yields a non-trivial identity (3)) then P has 5-th powers
the poles in D - 4 : just expand integrands in the end in of x 's. ( NB The maximal powers of 's and x 's are ant i-
D - 4 ; complex powers give rise to integer 0 powers correlated.) This is quite cumbersome, but it is a dull
of V x complexity of a polynomial.
i ( ) and logV ( x)
i ; all the poles come from B 's.
With well-defined integrals for the coefficients of the  Being a monotonous repetition of the same cumber-
poles, their cancellations may be checked numerically. some algebraic identity, the algorithm is as unfitted for
 Due to zeros of B's, the numerical advantage of hand calculations as it is perfect for programming.
smoother integrands is lost near thresholds (where nu-
merical convergence is poor anyway) due to cancellations Conclusions
in the sum (2). However, the regions near thresholds may It took about 10 years for MINCER [4], an implementa-
be systematically treated via asymptotic expansions ob- tion of the first algebraic multiloop algorithm [1][3], to
tained by the method of asymptotic operation [7]. deliver reliable results. The amount of algebra involved
 The identities of the original "integration-by-parts" in the construction and iterative application of (3) is even
algorithm [1][2] can be regarded as special cases of (3). greater, and may prove huge enough to prevent the new
In that sense, Eq. (3) represents the most general basic "algorithm of algorithms" from being actually used b e-
type of algebraic identities for multiloop integrals, which yond one loop to any greater extent than the SONG OF
justifies the name of Theorem A. Concerning the identi- SONGS is sung.
ties connecting integrals with different values of the But consider this: If an identity (3) is explicitly con-
space-time dimension D (cf. [18], [20], [19]), one notices structed for a given topology/mass pattern, then it simul-
that D enters exponents of different factors with different taneously does the following things -- each of which
signs, whereas in (3) all exponents change in the same would normally be an achievement per se:
direction. So one cannot naturally interpret the new 
It reduces integrands to a form that is optimal for
identities as recurrences w.r.t. D (beyond one loop).
 numerical integration.
As a simple application, consider dimensionally

regulated one-loop integrals which contain just one com- It allows straightforward program implementation.
plex exponent, z -n-
x
d Q( x)V (x) , where n is a non- 
S It explicitly extracts all (!) the poles in D - 4 .
negative integer, is the complex regularization pa-






Taking into account the unrestricted generality of the
rameter, and V ( x) is a quadratic polynomial of x .
~ new approach (it is applicable -- at least in theory --
Without loss of generality V ( x) x V
T x 2RT
= + x + Z to any multiloop integral, including ones with, say, non-
~
where V is a K  K matrix and R is a K-vector. Then relativistic propagators etc. etc.) and the pace of progress


e A specialized version of MILX is to be made available.


F.V. Tkachov, Nucl. Instr. Meth. Phys. Res. A389 (1996) 309  5




of computer industry, it looks like it might possess a po- constructing an identity (3) for one's favorite two-loop
tential to relieve us from some grueling chores in future topology.
and to give a boost to the industry of loop calculations A c k n o wl ed g m en t s The trigger for this work was a
(which needs it [21]; remember also the funds that went conversation with K. Kato at the Xth Zvenigorod work-
into the lattice QCD). The operators P could be found by shop (INP MSU, September, 1995) concerning numerical
solving a system of linear algebraic equations for their computation of loop integrals within the framework of
coefficients -- a highly cumbersome procedure given all the GRACE project [22]. I thank N. Sveshnikov for a dis-
the symbolic parameters involved (D, masses etc.) but cussion, and J. Vermaseren, D. Perret-Gallix and V. Ilyin
requiring nothing that would go much beyond the stan- for helping a signal to get through. Financial support
dard arsenal of polynomial computer algebra. So, anyone came from AIHENP' 96, ISF-Logovaz and the Russian
equipped with a sufficiently fast and reasonably flexible Foundation for Basic Research (grant 95-02-05794).
computer algebra system may already try one's hand at

--------------------


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