VIII Int. Workshop on High Energy Physics. Zvenigorod, Russia. 15-21 Sept. 1993. INP MSU: Moscow, 1994
 corrected 7 Nov 1998


ON DIMENSIONAL REGULARIZATION AND MATHEMATICAL RIGOUR


F.V. Tkachov#1

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


The controversy concerning the phenomenon of breakdown of dimensional regularization in the problems involving asymptotic
expansions of Feynman diagrams in non-Euclidean regimes is discussed with some pertinent bibliographic comments.




I would like to clarify some points in the discus- one encounters the singularity at = 0 generated
sion of whether or not the dimensional regulari- after the denominator is Taylor-expanded in , and
zation [1] (see also [2]) breaks down in the explicit that singularity is not regulated by 0 . The simp-
and very simple example presented in [3]. The lest manifestation of this is that one will not be able
example is a one-loop scalar triangle diagram with to perform the integration in the expanded expres-
only one non-zero internal mass m in the horizon- sion termwise--even at = 1 (4 - D) 0 -- without
2
tal "gluon" propagator in the figure below. One running into infinities. It is hard to imagine how the
considers the case when Q2 ( p p 2
= - - ) 
1 2 standpoint of ref. [4] could be defended in view of
with m fixed and p2 = 0 , and the explicit evidence provided by eq. (1).
i
q
one attempts to construct a To reiterate: if one attempts to expand the above
perfectly factorized asymptotic triangle graph in powers and logarithms of (i.e.
2 expansion (a one-loop Sudakov to perform a factorization), then the first subtraction
1
k analogue of OPE). Then the one has to make corresponds to the collinear singu-
soft singularity corresponding larities, and after collinear subtractions, the resul-
p p
1 3 2 to the gluon can not be regula- ting expression--as was found in [3]--contains a
ted by dimensional continu- singularity that is not regulated by dimensional
ation after the first subtraction (corresponding to continuation. This pathology persists (as it should)
collinear singularities) is done. irrespective of the representation one uses (e.g. it is
Surprisingly, our finding was disputed in [4] manifest in the above integral representation).
which concluded that "one should not be afraid of It should be stressed that the errors in [4] were
using dimensional regularization in practical calcu- committed at the level of interpretation rather than
lations connected with the Sudakov limit". calculations. Indeed, ref. [4] failed to point out any
Referring the reader for a detailed critique of [4] concrete mistakes in [3], and the simple calculation
to [5], I'd like to discuss some additional points. performed in [4] (which correctly found the pre-
 v4 23 Nov 1998 Without further ado, let me present an exact sence of the second logarithm in the expansion--a
expression (pointed out by N. Ussyukina) for the logarithm that is clearly not associated with poles in
arXiv: v4 23 Nov 1998 triangle graph in terms of Feynman parameters in ) confirms rather than disproves the conclusions
which integrations over two of the three Feynman of [3].
parameters have been done (an interested reader Ref. [4] contains statements that are breathtaking
will easily reproduce it): to an expert--e.g. that some UV divergences may
cancel IR ones (recall the proverbial apples and
1 - -
1 d [ ]
 oranges)--a revelation declared in [4] to constitute
1
( - -

) -
z , (1)
2 1
Q ( + ) - a "principal new feature" of the expansion problem
0 in non-Euclidean asymptotic regimes (see [5] for an
explanation of the origin of this fallacy).
where = m2 / Q2 > 0 , and a trivial overall coeffi- The misinterpretations of [4] are not accidental.
cient is dropped. The representation of the diagram They are fortified by references to the book [6] that
in terms of hypergeometric functions quoted in [4] purports to provide an uncompromisingly
can be directly obtained from this. "rigorous" account of the state of the art in the
If one attempts to extract the powers and loga- theory of multiloop calculations and asymptotic
rithms of 0 prior to integration over , then properties of Feynman diagrams. Because all our


#1 E-mail: ftkachov@ms2.inr.ac.ru


knowledge comes at second hand in some way, a a "rigorous" lower level formalism is only a matter
number of theorists took for granted the awesome of having spare time and nothing better (sic!) to do.
rigor of [6] and were misled by [4]. Further, consider the method of defining dimensi-
Therefore, I would like to take the bull by the onally regulated diagrams via the -parametric
horns and examine the book [6] itself. A book representation described in [6]. It implements, in a
review being a perfectly legitimate scientific genre, most straightforward fashion, the elementary idea of
it is also clear that (i) to trace the origins of a splitting the integration domain into as many
misconception is no less instructive than those of a sectors  la Sudakov-Hepp as needed to make the
discovery; (ii) the apparent discrepancy between the integral in each subdomain absolutely convergent
grandiose formalism of [6] and the elementary for some values of D , and then using trivial tricks
nature of the errors of [4] may be the most puzzling (integration by parts) to perform analytical continu-
aspect of the current dispute; (iii) references to the ation. (Apart from minor modifications, the forma-
"rigor" of [6] constitute to many (let us face it) the lism of this kind was thoroughly worked through in
single most convincing argument of [4]. [11].) This is extremely cumbersome (like program-
The proof of the pudding is in the eating. Why ming in machine codes)--and it is also obvious that
mathematical research into the nature of multiloop an intermediate layer of abstraction is missing to
Feynman diagrams should be judged otherwise? hide the ugly details (cf. high level programming
Indeed, it is useful inasmuch as one can learn some- languages). The method of [6] is also totally impra-
thing from it about how to deal with those noto- ctical--the resul ting formalism is far too cumberso-
riously complex beasts. (One could argue that such me to induce any calculationist to use it to check a
a research could be interesting from the point of new trick. The reader is left in disappointment to
view of pure mathematics, but there is nothing decide whether or not to trust the outlines of proofs
in [6] worth mentioning in that respect.) presented in [6].
The design of ref. [6] is simple. It outlines a An examination of the journal publications on
method of purportedly rigorous definition of dimen- which ref. [6] is based reveals that the actual proofs
sionally regulated Feynman diagrams, and then it are as a rule omitted there as being "technical"
sketches how the method could be used to while most of the text consists of lengthy formu-
"rigorously prove" many well-known results in lations of "theorems" that simply rephrase other
applied QFT. An unwary reader will be impressed people's results in an obscure formal manner. But
by the formidable formalism, while the biblio- what is then the claim of the author of [6]? If the
graphic comments are phrased in such a way as to proofs are not worth publishing in full, why are they
leave no doubt that the formulae calculationists had as important as ref. [6] attempts to convince us?
been using for years in highly non-trivial calcu- And the crucial question: if those proofs are so
lations, did not, really, exist before they were cumbersome that cannot be presented in full, who
declared "rigorously proved" by the author of [6]. and how should prove that they are correct?
A closer examination, however, reveals the fol- An experienced theorist would of course never
lowing interesting circumstance. All more or less rely on such an evidence and would always prefer
significant results described in [6] (starting from the concrete checks of his/her findings (calculations,
results on singular structure of Feynman diagrams formulas etc.) by another method. Meaningful--
[8]) were found by other authors by other methods even if not completely formalized--heuristics are
(which the author of [6] did not bother to describe), incomparably more useful in that respect than the
in some cases confirmed by as many as three groups sterile rigor of obscure "proofs".
(e.g. [7], [9], [10]); have been checked for self- Ref. [6] also contains a compilation of formulas
consistency in dozens of sophisticated calculations for multiloop calculations. The compilation is, un-
(e.g. [12]); the complexity of those results should derstandably, rather indiscriminate--it takes a first-
leave no doubt that their authors certainly knew hand experience of real-life calculations to make
what they were doing. To translate their heuristics such a compilation useful. Instead, the emphasis of
(if one may thus qualify their exact algorithms) into [6] is on justification of the tricks within dimen-
sional regularization. However, the "justi fication" is


F.V.Tkachov  156



mostly reduced to formal references to the cumber- Unfortunately, a student of Applied Quantum
some constructions of dimensionally regularized Field Theory seeking insight into the nature of
diagrams described previously. multiloop Feynman diagrams, will not profit
The last chapters of [6] present a version of from [6]. Motivation, heuristics, and understanding
"proof" of general Eucli dean asymptotic expansions are the concepts banished from the compilation [6]
of Feynman diagrams in the MS scheme discovered with the utmost "rigor" of an empty formalism.
and verified in different ways by three groups of Acknowledgments. I thank E. Kazes for lending
authors [7], [9], [10], where all the formulas that me a copy of ref. [6], J. Collins for discussions, and
are needed in practical calculations were found and N. Ussyukina for pointing out the representation (1).
subsequently verified in several record-setting 3- to
5-loop calculations (for a review see [7]). Ref. [6]  !#"$%&'()01"'325476398@AB"
CADE !#"$%3FDE(G)0(1"D'3254'6398@"





treats the pioneer publications--the publi cations E6IHIA)0P"#"DQQ5BRS(T
DE6IHI3)0PF"AD#"QUQ5RS(T





that contributed to defining the state of the art in '"6V!'"DT!'"D2)XW(Y"D'3254'63
'"6V!'"DT!'"D2)XW1"D'3254'6C3





multiloop calculations--with dismis sive (sometimes `3a0bdce(fhgpi&bqhgXrGias@``tu`v





grossly misleading) small-print comments. Appa-
rently, those papers are not up to the standards of References
"rigor" of [6]. But a more likely reason is that the [1] G. ' t Hooft and M. Veltman:
Nucl. Phys. B44
book [6] contains hardly any genuinely new ideas or (1972) 189.
results as compared with the original publications of [2] J. C. Collins: Renormalization. Cambridge:
the other authors. In fact, the version of proofs Cambridge University Press, 1984.
described in [6] is extremely cumbersome and [3] J. C. Collins and F. V. Tkachov: Phys. Lett.
exhibits the worst features of the BPHZ method (cf. B294 (1992) 403.
the discussion in [7])--for lack of space I cannot [4] V. A. Smirnov: Phys. Lett. B309 (1993) 397.
quote e.g. the incomprehensible definition of IR [5] J. C. Collins and F. V. Tkachov: The dimen-
subgraphs given in [6], to be compared with the sional regularization cannot be salvaged in
simple criterion of the original publications (see the Sudakov problem (to be published)
[7]). And again, neither ref. [6] nor the journal [6] V. A. Smirnov: Renormalization and asympto-
publications it is based upon contain complete tic expansions. Basel: Birkhuser, 1991.
proofs. [7] F. V. Tkachov: Advanced methods for study-
The nature of the book [6] explains the fiasco of ing radiative corrections. Theory of asymp-
[4]: all of the meaningful problems considered and totic operation. Sov. J. Nucl. Phys. 56 (1993)
"proved" in [6] had already been posed, thought out 180 , and refs. therein.
and solved by others. Ref. [4] marked the first time [8] M. Sato, T. Miwa, M. Jimbo, and T. Oshima:
the author of [6] ventured to consider a genuinely Publ. RIMS 12, Suppl. (1977) 387.
novel and delicate problem. [9] C. H. Llewellyn Smith and J. P. de Vries:
Mathematics is not about rigor--or, at least, not Nucl. Phys. B296 (1988) 991.
about that bad kind of rigor that hides a lack of [10] S. G. Gorishny: Nucl. Phys. B319 (1989) 633.
genuine ideas. Like any natural science, a good [11] O. I. Zav' yalov: Renormalized Feynman diag
-
mathematics is first and foremost about understand- rams. Moscow: NAUKA, 1979 (in Russian).
ing, about solving difficult problems, and about [12] S. A. Larin, F. V. Tkachov, and J. A. M.
making discoveries. Vermaseren: Phys. Rev. Lett. 66 (1991) 862.



