                           





! " # $ % & ' (





) 0 1 2 3 2 4 2 5 6 7 8 9 4 @ A 5 B 8 C 5 1 5 B 8 @ D 7 6 2 D 5 C 4 1 1 3 B 0 E @ B F 5 G H 7 6 I @ 3 5 0 @ 5 1 P





Q 7 1 @ 7 R S S T U S V P C W I I ) E





Abstract The pinched/nonpinched classification of intersections of causal singularities of
propagators in Minkowski space is reconsidered in the context of the theory of asymptotic
operation as a first step towards extension of the latter to non-Euclidean asymptotic re-
gimes. A highly visual distribution-theoretic technique of singular wave fronts is taylored
to the needs of the theory of Feynman diagrams. Besides a simple derivation of the usual
Landau equations in the case of the conventional singularities, the technique naturally ex-
tends to other types of singularities e.g. due to linear denominators in non-covariant
gauges etc. As another application, the results of Euclidean asymptotic operation are ex-
tended to a class of quasi-Euclidean asymptotic regimes in Minkowski space.





X Y ` a b c d e ` f b Y g proofs within the method of As-operation and established the
latter as a superior alternative to the traditional (BPHZ-type)
The problem of constructing asymptotic expansions of mul- methods in the theory of Feynman diagrams (for a systematic
tiloop Feynman diagrams in masses and momenta for various dicussion of the differences between the two paradigms see
asymptotic regimes in a phenomenologically meaningful and [1]).
calculationally useful form is the key analytical problem of ap- It became clear quite early that the method of asymptotic
plied Quantum Field Theory. This is because some sort of as- operation easily extends to the most general Euclidean asymp-
ymptotic expansion seems to always be involved in an impor- totic regimes [4] and that simplicity of the resulting formulas
 26 Mar 1997 tant way -- explicitly or implicitly -- whenever one deals with remains its characteristic feature [5]. Moreover, the general
Feynman diagrams in applications (see [1] for a systematic dis- philosophy of the method of As-operation [4], [1] is by no
cussion). means specific to Euclidean problems (first examples of non-
The method of asymptotic operation (As-operation for trivial application of the method in essentially non-Euclidean
short; for a review see [1]) has proved to be the most powerful situations can be found in [14], [15]).
one for solving the asymptotic expansion problem within ap- The purpose of the present paper is to begin a systematic
plied QFT. It originated [2][5] in the context of studies of extension of the theory of asymptotic operation to non-
analytical calculational methods for the coefficient functions of Euclidean asymptotic regimes.
short-distance operator product expansion. The resulting algo- h i p q r s t s u v t r w x y s u t y v y i y i u y x u s u w i





rithms [3], [6] (extended to arbitrary Euclidean expansions in
[5]) allow one to obtain contributions to coefficient functions A necessary preliminary step is to reinterprete the well-
from individual diagrams in a maximally simple form, which known pinch/non-pinch classification of intersections of light-
comes about as a natural consequence of a property of perfect cone and mass-shell singularities of causal propagators in
factorization of the expansions obtained with As-operation [3], purely distribution-theoretic terms as is necessary within the
[4]. Due to the perfect factorization, calculations of coefficient framework of As-operation. Indeed, the first question one is
functions are much simplified and are directly reduced to inte- confronted with is as follows.
grals for which there exist powerful algebraic algorithms [7]. The formulas of As-operation for Euclidean asymptotic re-
As a result, a class of next-next-to-leading order calculations gimes were derived (cf. [4]) using the Wick rotation so that one
had been reduced to feeding the corresponding diagrams into a dealt with Euclidean momenta throughout the analytical part of
computer (e.g. [8][11]). A regularization-independent analysis the derivation. However, the final formulas are valid irrespec-
[12], [13] demonstrated a remarkable compactness of formal tive of the Wick rotation used at intermediate steps. But then it


                





should be possible to derive the Euclidean As-operation di- of a point which one studies. This, however, cannot be
rectly in Minkowski space. achieved with analytic functions, so that analytic continuation
The technical problem one encounters here is due to the is no longer allowed.
more involved structure of singularities of Feynman propaga- Lastly, as was stressed e.g. in [17] (Chapter 2 and refs.
tors in Minkowski space as compared with the Euclidean therein), an accurate derivation of the Landau equations via
situation. Indeed, the Euclidean scalar propagator has the form contour deformations in a general case requires a use of the
( p2 m2)! 1 where p2 " 0 for all p 0 . When formally ex- notoriously cumbersome methods of algebraic topology. This is
#





unsatisfactory because a calculationist would always prefer a
panded in powers of m , this results in singularities at the iso- direct analytical argument at a "microlocal" level -- and argu-
$ %





lated point p 0 . If there are other integration momenta, ment that could be directly connected with how one calculates,
then the singularity of a single propagator is localized on a lin- say, asymptotic expansions -- to an indirect one such as the
ear manifold. On the other hand, in a Minkowskian situation one based on contour deformations.
the standard scalar propagator becomes ( !
2 2 1
& &
p m i )
0
) 0 1 2 3 4 5 6 7 8 0 6 9 @ 2 A 6 B 0 6 9 C 0 D 4 0 B 8 E F G





where p2 can be both positive and negative so that there is a
singularity localized on the non-linear manifold described by An alternative technique that would allow a direct study of
overlapping causal singularities is based on the notion of sin-
%





the second-order algebraic equation p2 m2 . If m is one of gular wave front of a distribution (or simply wave front) intro-
%





the expansion parameters ( m O(' ) ), then the expanded ex- duced by Hrmander [18] and Sato [19] in the context of stud-
%
pression has singularities on the light cone p2 0 . Moreover, ies of propagation of singularities of generalized solutions of
%
the point p 0 is a singular point of the light cone manifold partial differential equationsiii. Whereas Hrmander worked
with distributions proper [20], Sato considered a related notion
in the sense of differential geometry and should be treated of hyperfunctions [21]. From a practical point of view, how-
%





separately. The singularity at p 0 is traditionally referred to ever, the two definitions are equivalent (cf. the review [22]).
as soft singularityi. Other singular points have the property that The wave front describes singularities of a given function
the propagator near each such point becomes a well-known So- (their positions and "orientation", cf. Sec. 4.23 below; for a
hotsky distribution, namely, (x !
& i )
0 1 where x is a properly general definition see [22], [23]). The key technical result of
chosen one-dimensional local coordinate. To emphasize their interest to us here is a criterion of existence of products of dis-
analytical nature, such singularities can be referred to as So- tributions in terms of their wave fronts (described e.g. in [23],
hotsky-type singularities. To emphasize their physical origin, Theorem IX.45 essentially due Hrmander). However, we will
the term causal singularities can be empoyed. While the causal not need the criterion in full generality, and in Sec. 4 its sim-
plified version will be taylored to the needs of the theory of
( i0 prescription ensures that the propagator itself is a well-
Feynman diagrams.
defined distribution everywhere in the space of p , this is not Sato et al. [24] were first to apply the wave-front-related
necessarily true for products of such distributions. The causal techniques to Feynman diagrams. They did so from the point of
Sohotsky-type singularities may overlap and make the product view of the theory of hyperfunctions [21], which was rather
non-integrable by power counting. natural because hyperfunctions can be roughly thought of as
In general, singularities that are non-integrable by power boundary values of analytic functions.iv The main result of [24]
counting result in non-trivial contributions to asymptotic ex- is a description of non-analyticities of Feynman diagrams in
pansions. Therefore, to obtain a Minkowski-space derivation of terms of certain closed formulas (to be compared with the im-
the Euclidean As-operation one has to understand why the in- plicit description via the Landau equations). Despite being in a
tersections of causal singularities do not contribute in the case formal sense explicit, such formulas nevertheless seem to be
of Euclidean asymptotic regimes. less useful in physical applications than, say, the transparent
The usual explanation of non-contribution of causal singu- interpretation of pinched singularities due to Coleman and
larities in Euclidean case is, because the energy integration Norton [26].v, vi, vii Yet the notion of wave front -- if used prop-
contours are not pinched and one can deform them away from erly, i.e. without excessive generality -- is a handy tool in the
such singularities (cf. the derivation of the Landau equations theory of Feynman diagrams due to its visual simplicity.
that give a criterion of occurence of pinches [16], [17]). But
this is the same as to say that the Wick rotation can be per-
formed, and cannot satisfy us. Indeed, eventually one will have
to answer the question of what happens near the points that
iii
separate pinched and non-pinched singularities.ii Then one Cf. the light cone singularities of causal propagators that are special so-
lutions of the wave equation in coordinate representation.
would have to cope with the fact that the latter are disposed of iv The relevant physically significant results include e.g. the edge of the
via contour deformations while no such trick is allowed for the wedge theorem in the context of studies of dispersion relations (see e.g.
former. To avoid such a situation one has to reinterprete the [25]).
mechanism of non-contribution of non-pinched singularities in v I than J. Collins for a discussion of the Coleman and Norton interpreta-
a manner that avoids contour deformations. tion.
vi
Moreover, the technique of As-operation is, strictly speak- The work of Sato et al. should not be judged too narrowly: they studied a
new "mechanism of proofs" in an applications-motivated problem in ex-
ing, not naturally compatible with contour deformations. In- actly the same way as physicists would design an experiment to test, say, a
deed, whenever distributions are involved one has to deal with particle production mechanism -- irrespective of its immediate usefulness.
test functions which in the theory of As-operation should be lo- vii The results of [24] were later reproduced in a series of publications by
calizable -- i.e. zero everywhere except a small neighbourhood Smirnov (see his compilation [27] and refs. therein) who claimed to have
used a "different" technique of "singular wave front" of Hrmander rather
than the "analytical wave front" of Sato et al. Such claims cannot be sus-
i By association with soft (zero-momentum) photons in QED; analytically, tained in view of the exact equivalence of the two versions of the definition
this is the hardest singularity generated by a propagator. of wave front in the context of Feynman diagrams, as was emphasized in
ii I first heard this question explicitly stated from G. Sterman. connection with Smirnov's "results" in [28].





2


                





! " # $ % & ' ( )





Because the method of As-operation iteratively reduces
studying any however complex geometrical patterns of singu-
It was already emphasized elsewhere [1] that the mathe- larities to the case of singularities localized at isolated points,
matical apparatus of asymptotic operation includes a collection Sec. 3 contains an exhaustive treatment of one-dimensional So-
of techniques that cannot be found in any single source on dis- hotsky distributions, the one-dimensional prototypes of causal
tribution theory.i This is why it would be useful to present an singularities. We consider in detail the issue of when products
account of the corresponding techniques in a form taylored to of such singularities exist despite divergence by power count-
the specific needs of theorists that have to deal with Feynman ing. Then this fact is reinterpreted in terms of Fourier trans-
diagrams in applied problems. The goal of the present paper form thus preparing ground for an extension to many dimen-
from a technical point of view is to do that with respect to the sions. Finally, a simple extension to asymptotic expansions of
technique of wave fronts. products of Sohotsky distributions is presented.
In this respect note, first, that applications to Feynman dia- Sec. 4 treats the multidimensional case. First, general inter-
grams do not require the use of wave fronts in full generality. sections of Sohotsky-type singularities are considered, and a
Second, whereas the usual treatments consider "static" prod- special version of the highly visual notion of singular wave
ucts of singular functions, we wish to study asymptotic expan- front is deduced in a few easy steps, along with the necessary
sions so that not only existence of products has to be estab- heuristic motivations. A criterion of local existence of such
lished but also the fact that they possess the required asymp- products (a special case of the more abstract one due to Hr-
totic approximation properties. All the necessary mathematical mander) is derived as an obvious generalization of the one-
apparatus will be explained in detail. dimensional results, both in an analytical and geometrical
From the point of view of the theory of multiloop diagrams, form. Then it is shown that the same criterion holds for the
we are going to: rederive the Landau equations [16], [17] in a case of osculating singularities; at this point the heuristic
purely distribution-theoretic fashion, i.e. avoiding the use of power of the notion of wave front becomes apparent.
contour deformations as is appropriate within the theory of As- Finally, in Sec. 5 we turn to multiloop diagrams and present
operation; demonstrate how the technique can be applied to a one-step derivation of the Landau equations. Then we briefly
analysis of singular functions of a more general form than the discuss how the developed formalism helps to understand the
standard causal propagators (cf. the problem of singularities in structure of singularities in the theory of non-covariant gauges,
non-covariant gauges [31]); extend the Euclidean theory of as- and how the results of Euclidean theory of asymptotic opera-
ymptotic operation [4], [5] to the simplest class of non- tion are extended to the class of quasi-Euclidean asymptotic
Euclidean asymptotic regimes (the so-called quasi-Euclidean regimes.
ones).
It should be emphasized that the motivations of this work go
4 5 6 7 5 8 7 9 @ A 6 8 5 B 9 C 9 D 7 5 @ E 5 7 F G H C I C 9 P


somewhat beyond providing a "better derivation" of the Lan-
dau equations. The problem of asymptotic expansions of mul- I H D B 5 C 9 I @ Q





tiloop diagrams in non-Euclidean regimes is a notoriously
complex one, and complex problems are solved by building an R $ S 1 S T $ 2 % U $ " T 2 S & V " 1 2 W % X ( '





adequate intuition based on -- and together with -- and ade-
quate formalism. So our aim is to present a more flexible tech- Our notations will generally agree with [4]. Let us briefly
nique than the one based on contour deformations and to re- summarize them together with whatever extensions are needed
consider the classification of singularities in a manner suitable in the non-Euclidean case. This will also introduce one into the
for a subsequent construction of non-Euclidean infrared R- and context of the problem.
As-operations. The fact that the Landau equations reemerge as The object one deals with is G( p) , the integrand of a mul-
a result is in principle of little consequence. tiloop diagram G , whereas p Y ( p , pl )
1 is the collection of
`





0 1 2 ' ( 3





all its integration (loop) momenta: dp a dDp dDp
1 2 . The
b





The plan of the paper is as follows. In the descriptive Sec. 2 space of p is denoted as P. The concrete value of D (the num-
we establish the context in which to study non-Euclidean As- ber of scalar components of each pi ) is not important. It is of-
operation. First the systematic notations are summarized to de- ten convenient to ignore the structure of P altogether and to
scribe the products of singular functions we wish to deal with. consider it as an abstract vector space of finite dimensionality.
Then a summary of As-operation is presented emphasizing the The diagrammatic interpretation of G will not be important
point that its structure is independent of whether one works in either and we will treat G( p) simply as a product of singular
Euclidean or Minkowski space. Further, we identify the non- factors. Its structure is described below.
linear causal singularities and briefly discuss the distinctively A general scheme is that one starts with G( p) and con-
non-Euclidean geometrical phenomenon of osculating singu-
larities. The conventional treatments avoid discussing osculat- structs a distribution from it. The conditions governing such a
ing singularities at a microlocal level altogether, but the issue construction are always local. This means one should first per-
cannot be circumvented if one aims (as we do in the theory of form such a construction for all sufficiently small c
As-operation) at a complete solution of the asymptotic expan- (neighbourhoods of singular points). After that the transition to
sion problem. a distribution defined on the entire P is achieved via the stan-
dard trick of the decomposition of unit [29], [30]. For this rea-
son, we focus attention only on what happens in a small open
region .
c

i The original source on the theory of distributions is [20]. Good textbooks Furthermore, we will ignore the polynomial factors in the
are [29] (with many examples to different physical problems) and [30]
(with proofs of the general results of the theory of distributions that avoid numerator of G( p) that are due to interaction vertices or non-
the abstract theory of topological spaces). Wave fronts are briefly dis- scalar particles. This is because the As-operation, by definition,
cussed in [23].





3


                





$ % `





U A E A 8 H V I 7 8 9 @ 7 9 8 A B C 7 W A H I X F Y 7 B 7 G @ B Y A 8 H 7 G B E




constructs a complete asymptotic expansion in powers and
logarithms of the expansion parameter. Such expansions are
unique and, therefore, commute with multiplications by poly- It is important to understand that the general structure of the
nomials [4]. This is a general property that remains valid in the As-operation is the same irrespective of whether one deals with
non-Euclidean case. Therefore, the presence of polynomial Euclidean or non-Euclidean problems. This is because neither
factors is of no significance as far as the analytical mechanism the motivations for the distribution-theoretic point of view pre-
of As-operation is concerned. They play no role in the study of sented in Sec. 4 of [4], nor the extension principle of Sec. 6 of
the geometry of causal singularities and will be ignored. [4], nor even the general structure of the arguments of Sec. 11
So, the products of singular factors that we consider in this of [4] which established the structure of As-operation, depend
paper have the following form on that assumption. Let us summarize the prescriptions of As-
operation emphasizing this fact.
" # $ % $





G( p) ! g( p) The asymptotic operation (with respect to a concrete
g G a b c





small parameter T which implies a concrete choice of the as-
where
ymptotic regime) is meant to be applied to a product of singu-
% $ % 2



0 1
) n
T
& & ' ( g
g( p) ( p) D ( p) i lar factors G( p, ) regarded as a distribution in the integration
g g 0
momenta p . It is then supposed to yield an asymptotic expan-
D p T
g ( ) is whatever expression happens to occur in the de- sion of G( p, ) in powers and logarithmsi of T in the sense of
nominator of the g -th factor. Eq. 2.3 fixes the relative sign of distributions. At an operational level, the construction involves
D p the following steps:
g ( ) with respect to the infinitesimal imaginary part. In this

work we consider only such regions P that the change of (i) Applying the formal Taylor expansion in powers of T to
d c





3 4





)
sign does not occur within in all factors in the product. each factor T
g ( p, ) in the product,
5





$ % P





f





6 7 8 9 @ 7 9 8 A B C D A E B F G E H 7 B 8 I





G( p, &
!
T ) g( p,T )
g G
In the case of the standard scalar propagator, # $ % i
f f





g g





&

e



!
h g( p,T ) !
h g( p,T
c c )
# $ % R g G g G
D ( p) 2 2
' Q
l ( p) m
g g g which results in a formal series in powers of T , identification of
where mg may or may not be zero. The quantity l p
g ( ) is the potentially non-integrable singularities in the resulting expres-
sion and their geometrical classification. Here one determines
momentum flowing through the g -th line of G ; it is a linear the number and geometric structure of the manifolds in P on
combination of pi and the external momenta. The concrete which those singularities are localized. One here deals with an
form of lg is determined by the topology of the diagram and hierarchy of intersecting manifolds of various dimensionalities.
the pattern of its external momenta. When studying the general Note that all such manifolds occur as intersections of singular
T
analytical mechanisms, we make assumptions neither about the manifolds for individual factors g( p, ) .
topology, nor about the pattern of external momenta (i.e. (ii) Determining which factors from the product contribute at
whether they are put on mass shell, taken at zero values etc.). each singular point. Here one establishes a correspondence
The construction of As-operation requires to perform vari- between the singular manifolds , the subproducts G
ous expansions of the factors [1], [4]. This explains why we p q r s





allow the powers n consisting of factors that are singular there, and the diagram-
g to differ from 1. Moreover, in the non-
matic interpretation of such subproducts as some kind of sub-
Euclidean case, such expansions may change the form of de- graphs of the initial Feynman graph.ii
nominators. In particular, there can emerge denominators with
a linear dependence on the momentum, (iii) Studying the analytical structure of the singularities (some
# $ % S form of power counting etc.) and constructing a subtraction
D ( p) 2
Q
n p m
g g g procedure that transforms the singular and, in general, non-
integrable formal expansion into a correctly defined distribu-
where ng is a parameter vector, etc. The examples where such ~
t

tion on the entire P.iii As in [13], we denote it as :
factors occur are: the string operators in the context of the Su-
dakov problem etc. [32]; the propagator of the heavy quark in
i The powers need not be integer, e.g. square roots are allowed etc. We are
the effective heavy quark theory (cf. e.g. the effective QCD La- talking about "powers and logarithms" although the analytical fact that the
grangian in [33]). The non-covariant gauges [34] also involve expansions we deal with run in powers and logarithms of the expansion pa-
factors with denominators similar to 2.6 but with the sign of rameter and do not contain, say, log log terms, cannot be decided a
u





the imaginary part different in different regions of P. priori. However, such an assumption is not used in the actual construction
In view of the variety of possible denominators, it makes of As-operation and its correctness is established along the way. Note also
sense to keep irrelevant details out of the way and simply re- that proving the power-and-log structure of expansions is a much simpler
gard D p
g ( ) as a smooth function of p . problem than establishing their precise diagrammatic form. References to
some such studies can be found e.g. in [4].
The expansion parameter is denoted as T . The dependence ii Note that the resulting diagrammatic images are determined by the un-
on T of various quantities is indicated, as usual [4], with an ad- derlying analytical structures (in particular, the asymptotic regime) and
ditional argument, e.g. G( p, need not fit exactly into the standard categories of the theory of graphs. In
T ) , D p T
g ( , ) etc. Its particular fact, a graph-theoretic characterization, even if possible, may be too cum-
form is determined by the specific problem and the asymptotic bersome to be useful, or simply irrelevant (except in a very special context,
regime chosen. e.g. when rewriting known results in a "rigorous" form; cf. [27]).
iii In certain cases one may do without such a subtraction procedure if a
convenient regularization is available (the dimensional regularization is





4


                





! !


The singularities of the g -th factor are localized on the mani-
& ~
# $ #





"



& ' ( ' )

% % %
( , ) ( , )
g p
g p
g G g G fold V described by the equation D p 0 :
3 g P g ( ) W
~
0

The subtraction procedure should be minimal in that it { | ( ) }
0 ' ( ' @ (
W p D p
should not modify the leading power behaviour near singular 3 g g W





points. It is analytically somewhat similar to the Bogoliubov R- The singularity corresponding to the g -th factor is referred to
operation in coordinate representation. as elementary singularity.
(iv) Constructing the counterterms Note that
&
( p, ) localized at the 3 g may be a smooth manifold everywhere in P
1 2





singular manifolds (thus fixing the finite arbitrariness in (e.g. the cases 2.5 with mg 0 and 2.6). On the other hand,
3 2

X





~ the points where l p 0 for 2.5 with m 0 (the so-called
0
the definition of ): g ( ) W g W
~ 6 soft singularity) correspond to the apex of the light cone where
$ #





&

% %





9
g( p, )
g G the light cone is not a smooth manifold. More generally, such
special singularities occur whenever the gradient nullifies, i.e.
" ~ 6 6
$ #





7 8



& & & ( ' @ A

% %





Y
9 g( p, ) %
9
2 p g p
1 ( , ) 4 5 ( , ) . D p W
g ( ) 0 , where the gradient operator is
g G \2 g G

The inclusion of
&
( ,
p ) is necessary in order to transform b c
Y





8 8 `
1 2 , , ' ( ' @ h
g




f f




a d e

the expression into a correct expansion in the sense of distri- p p p
1 2

butions. &
( ,
p ) contain all the non-analytic (logarithmic, Y
2
1 If D p
g ( )
0 but D p
W 0 g ( )
0 0 , then there is a small neigh-
X





square root etc.) dependence on & . They are localized on the
bourhood of p
singular manifolds and, therefore, are linear combinations i 0 in which the local coordinates
3 2





p ( , , may be chosen so that D p x y .
p
W x
p )
W p x y
B B
of -functions. The number of derivatives of the -functions is g ( ( , , ))
determined by the power counting at the previous step, Then the factor becomes
whereas the non-trivial coefficients are found using the con- u v n
q


p x y 0 ' ( ' @ w
r s x t i g
( ( , , ))
sistency conditions [4]. The coefficients of the counterterms of g
the As-operation are detemined uniquely [4]. We call the distributions that can be locally reduced to such
Note that one needs some form of explicit parametrization form Sohotsky-type distributions. Their integrals with test
B



of to write down the structure of -functions (their argu-
3 2 functions are well-defined despite the fact that they are diver-
ments etc.). gent by power counting (see also Sec. 3).
In this paper we consider only intersections of causal sin-
C D E F G H I P Q I E D R S H T U





( ' @ @ gularities. Formally, this means that all 3 g are smooth mani-

Strictly speaking, in the context of asymptotic expansions folds within . A sufficient condition to ensure that is to re-
i





one deals with the singularities of the formal expansions of the quire that
product G( p,& ) in & . Note, however, that the structure of Y
D p 0 p ' ( ' @
g
g ( ) , ,
X x y x





those singularities is the same (up to the strength of singulari-
ties) for all terms of the formal expansion. This is because the This means that we exclude from consideration the soft singu-
singularities are due to denominators, and the terms resulting larities -- their study requires additional techniques that would
from expansion of a factor have the same expression raised to lead us outside the scope of the present paper. The restriction
different powers in the denominator. Therefore, it is conven- will remain valid throughout this paper.
ient first to ignore the "dynamic" aspect of the problem (i.e.
S E E D S E U R S P F P I T D E
the fact that one deals with expansions) and consider the p ( ' @
s





problem as "static" (i.e. existence of products only). After that
Geometric representation of the gradient vector D( p) in
we will see what modifications are needed to take into account
approximation properties etc. the space P of independent variable p depends on how one
The singularities of the product G( p) are due to the fact chooses the scalar product in P. This is because the gradient is,
that each of the denominators may have zeros, and the singula- strictly speaking, a vector from the co-space P whose embed-






rities of individual factors in the space P may overlap in a non- ding into P is a matter of convention. However, the properties
trivial fashion. One wishes to study such overlapping singula- we are going to consider are expressed via relations that are


Y
rities based on the information about the individual factors. linear in (e.g. D( p) 0 ) and are, therefore, unaffected by
the conventions adopted. This allows some flexibility of nota-
tion depending on specific purposes: To build insight into the
nature of intersections of Sohotski-type distributions, we will
the preferred choice). Then the task of recasting the final answers into an consider simple examples, in which it is convenient to regard
explicitly convergent form is postponed till the end of the construction --
P is a Euclidean space. Then the gradient vector D( p )
0 is
cf. [4], [5] where such an approach was motivated by calculational con-
venience of final formulas in a situation where the subtraction procedure is normal to the singular manifold D( p) 0 . But after the gen-
W





understood well enough from analogy with the UV renormalization in co- eral theorems are understood and we turn to concrete examples
ordinate representation (cf. [36], [12]). Such an approach, however, may in the context of Feynman diagrams and Minkowski space, it is
not be always desirable, especially in the non-Euclidean problems where convenient to proceed in a formal algebraic manner. In par-
the multi-level recursion structure of the problem with secondary expan-
sions obscures the connection between various types of divergences that ticular, p is then an array of Minkowskian vectors (the collec-
occur at intermediate steps. Moreover, the dimensional regularization may tion of loop momenta) and is of a mixed Euclidean/Minkowski
not always work in non-Euclidean situations [14].





5


                





s t 4 6 B 3 g u @ g 3 7 v g 0 g 5 1 w D E D F





$





type, e.g. p p! " # p p !
i i i . Then the gradient D( p )
$
% 0 may
Consider the integrand of a one-loop scalar self-energy in-
happen to be tangential to the corresponding manifold sertion:
D( p) 0 (for instance, if p is a Minkowskian 4-vector and
&





1 1
$ D E D D
D( p) 2 then D( p ) 2 p ). But that is a purely formal
& p ' &
0 0 x 2 y 2 x
x x 2 y 2 x

p 1
m i ( p q) 2
m i
fact of no analytical consequence.
where m a
i 0 , p is a 4-dimensional integration momentum
( ) 0 1 2 3 4 5 6 4 0 ) 7 8 3 9 @ 4 0 9 @ ) 0 1 2 3 4 5 @ 2 A 1 5 4 B C @ D E F G





while q is the external momentum. The integral of 2.22 over p
Consider an arbitrary subset of lines, G , and define:
H I (after appropriate UV renormalization) is a function of q2 and

U
E D E F V the masses. The interesting asymptotic regimes are as follows:
R S T



P P





Q g
Q
g G (i) q2 x (the Euclidean regime),

In other words, is the collection of points through which
Q

W (ii) q2 y (the quasi-Euclidean regime; cf. Sec. 5.9);
pass the singularities of each factor g . It is called singular
X H





(iii) q2 m y
( m 2
) (the threshold regime).
manifold corresponding to the subproduct . 1 2
H





As in the Euclidean case, one may have even To study (i), one chooses q to be purely space-like and then
Q Q





W & W Y





Wick-rotates p into the Euclidean region. Then the formalism
when a
! . However, the analytical nature of the singularity
` ` of [4] becomes applicable.
of the entire product G( p) on is determined by all the
Q
W The case (ii) is more interesting because from the point of
elementary singularities and by all factors g( p) that view of momentum space, the picture here is much different
Q




W W
g b from the Euclidean case. Indeed, choose q to be purely time-
are singular on the entire , i.e. . So, whenever one
Q Q





W W W
g b
like, q 2
"
q

0
, 0
, 0
, . An immediate Wick rotation is prohib-
deals with a singular manifold , the corresponding G
Q





W H I





2 1 2
contains all g such that . The subproduct ited. Rescale the dimensional parameters, p p(q ) / and
Q





W W
g b
2 p
2 2
T
U m m q , so that the regime (ii) is replaced by m 0 for
D E F c
( p) i i i
" g( p)
`





Q
g q fixed, with both masses going to zero at the same asymptotic
that contains all the factors from G( p) that are singular on rate. Within the framework of the theory of As-operation, one
constructs the corresponding expansion of 2.22 in the sense of
is described as complete. The singular manifold to-
Q Q




W W distributions. The first step of the construction is to identify the
gether with the associated complete subproduct will be called singularities of the formal Taylor expansion, whose terms have
singular subgraph although at this stage we do not associate the following general form:
any graphical image with it. 1 1 E D E D
The collection of all singular subgraphs of G is denoted as N N

x 2 x
x x 2
1 x 2
S[G] . ( p i ) ( ( p q) i )

The singularities are localized on two light cones, one shifted
d @ e 2 3 4 f ) 0 1 @ ) 0 1 2 3 4 5 ) f ) g @ D E D h





with respect to the other by q as shown in 2.24:

In the Euclidean case, there were two basic geometric pat- ( p - q)2 = 0
terns of intersections of elementary singularities: factorizable
and non-factorizable. Because now singular manifolds are non- p = q






linear, an additional pattern has to be dealt with, namely, os- A
culatingi singularities.
Let g( p) be an intersection point of several non-linear p = 0
p2 = 0
manifolds , , and let be its small neighbourhood.
X
W H
g g i
D E D





One can always find a smooth deformation of coordinates
p p p! within that transforms any one of the singular mani- Two counterterms will be needed for the singularities at the
i





folds apices of the light cones. Recall that in the Euclidean regime
W g into a linear manifold so that the corresponding fac-
(i) one would have to deal with these two points only. In the
n
tor in the product takes the form (v p! q i )r g
g 0 where vg is a present case, however, there are also singularities localized on
the light cones away from their apices. Thus, in a small neigh-
constant vector. There are two cases possible:
 bourhood of a general point A of any of the light cones the
i





A general case is when all the factors g( p) that are
X H local coordinates can be chosen so that the corresponding factor
singular at p0 can be simultaneously linearized. becomes (x x i ) N
0 , which is non-integrable by power
r i

 When a simultaneous linearization is not possible, one deals counting for Ni large enough. The prescriptions of As-
with osculating singularities. operation would require counterterms to be introduced for such
singularities. However, it turns out that such light-cone singu-
larities do not contribute to the final answer. The usual argu-
ments explaining this rely on deformations of the contour of
integration over p0 into the complex plane. But for reasons
i i.e. "kissing" (from Latin).




6


                





discussed in the Introduction we would like to find an expla- But according to the known analyticity properties, the expan-
nation for the absence of the counterterms in such cases di- sion in near 2.25 should not contain non-anaytic terms. This
rectly in real Minkowskian momentum space. means that, unlike the preceding case, the osculation point of
The threshold regime (iii) is the most difficult one. Indeed, the two hyperboloids does not require a non-trivial coun-
the small expansion parameter can be introduced by repre- terterm.
senting q as follows:
& % G f





I P Q R S T U V W X T T Y ` U Q a b Y ` c d T Q a Y e Y U b





# $ % & % & '





q ! m " m "
1 2 , 0 0
, 0
,
Another rather typical geometrical pattern of intersection of
Then the expansion in ( 0 is equivalent to the expansion in causal singularities occurs in the following product of two
q2 massless propagators:
( m "
( m 2
1 2 ) .
A typical term of the formal expansion of 2.22 in has the 1 1 & % G &





following structure of denominators that determine the struc- N N
C 2 C 0 g C C 2
1 C
p i p q i0 2
[ ] [ ( ) ] 3





ture of singularities:
where q2 ! 0 (both p and q are 4-dimensional Minkowskian
1
vectors). Such a product may have resulted from a formal ex-
[) 2 0 ) 0 2 0 2 )
1





1
m r (r ) r i ] pansion in one of the high-energy regimes (cf. [35]). The sin-
& % & 4 gularities of the two factors are localized on the two light cones
2 1

[ as shown in 2.33.
0 2 0 ) 0 2 0 2 )
1
m r r r i N
2 ( ) ] 3





p - q 2 =
where we have introduced the variable r : ( ) 0
A A
% & % & 6
p 0 0
5 p !
p m "
( , ) ( r , r)
1 p = q
The singularities of the two factors are localized on two hyper-
boloids which touch (osculate) at the point r p2 = 0 p = 0
! 0
& % G G





( p 5 7
! q q ).
8 0

!
p = q The first factor is singular on the light cone p2 0 while the
p2 = m2 , p0 > 0
1 second, on a similar light cone whose apex is shifted to the
point p ! q . The two light cones intersect over the line
9 @ 2
p - q = m22
& % G p




p = 0 h i
! !
p2 = 0 L p| p zq

p2 = m2 , p0 < 0 & % & A
1 (i.e. p should be collinear to the light-like momentum q which
is the reason why this is called collinear singularity). Note
By analogy with the preceding example one has to conclude that, in general, an intersection of two 3-dimensional manifolds
that the general points of the hyperboloids need not be supplied is 2-dimensional, while in the present example it is 1-
with counterterms when the expansion in the sense of distribu- dimensional. To see what happens here, consider a cross sec-
tions is constructed. But the argument does not work for the tion of the space of q with a hyperplane p0 ! const > q0
point r B p B
0 ( q ) . On the other hand, it is well-known from
(represented by the line AA in 2.33) as in 2.35.
studies of analyticity properties of Feynman diagrams that the
asymptotic regime under consideration corresponds to the point p2 = 0
of non-analyticity; therefore, the expansion must contain a non-
analytic term. In view of the fact that non-analytic dependences
are contained in the coefficients of counterterms of the As-
E F 2
operation (cf. [4]), one concludes that it is exactly the coun- p - q = 0 & % G '





terterm for the singularity at r B p B
0 ( q ) that should contain
the non-analytic threshold behaviour. The intersections of the two light cones with this hyperplane
Consider, however, the expansion in are spheres that osculate at a point. When the cross section
defined as follows:
shifts along p0 , the spheres inflate/deflate and shift, but there
# $ % & % & D





q ! m C m "
1 2 , 0 0
, 0
,
is always one osculation point. If 0 q 0 q 0
p q then the two
Then there will also be a point where the two hyperboloids spheres are side-by-side, otherwise one is inside the other; the
touch (cf. 2.30; only the osculating upper hyperboloids are two cases correspond to pinched and non-pinched singularity.
shown; the dotted line shows the position of the unshifted hy-
perboloid ).

E F 2
p - q = m22
p = q

p2 = m2 , p0 > 0
1




& % G H





7


                





! " # $ % & " ! ' % ( ! ) 0 1 ) ' " 2 that are singular at p 0 is determined by the behaviour of
y





Fourier transforms at infinity. Such a behaviour may be differ-
Within the formalism of asymptotic operation, all problems ent in different directions, and under certain conditions this
are recursively reduced to investigation of one-dimensional may result in a well-defined product even if the factors are sin-
singularities. Therefore, it is convenient to review the proper- gular.
ties of the relevant one-dimensional distributions and remind First recall that Fourier transforms of test functions are
ourselves of some basic properties of Fourier transforms that smooth functions that exhibit rapid decrease in any direction
will be needed in the discussion of the multidimensional case. (i.e. decrease faster than any inverse power of their argument).
3 4 5 6 7 4 7 8 9 @ A B C 9 8 D C E F 8 C 7 G H I P





However, Fourier transforms of distributions need not be de-
creasing. On the other hand, they may behave badly in some
Let p be a one-dimensional integration variable running directions, and exhibit a rapid decrease in others. For instance,
from and . Recall the familiar one-dimensional So-
Q R S T by straightforward integration one sees that
hotsky distribution e
d





1 I H I
c ikx c 0x
1 1d ( )
y i x e e x 0
H I a
W 1
lim p Q i0
p c d
X V Y `
U i0 0 p U i

which is well-defined despite the fact that the singularity is The Fourier transform ( x 0 remains constant as x ,
)
R





logarithmically divergent by power counting. As usual, the but decreases rapidly as x .
Q R





limit should be understood in the sense of the distribution the- For the complex-conjugate distribution the pattern of dan-
ory, i.e. the above expression is not expected to be interpreted gerous and non-dangerous directions is mirrored.
otherwise than in integrals with test functions ( )
p (that are Note that raising the l.h.s. of 3.9 to an arbitrary power
b





smooth everywhere and non-zero only in a compact region): n 0 results in a power of x on the r.h.s.:






e e



e






d d n d
1 1 c
d
1
I H I H

I H I P e
c ipx n 1
g 1
dp ( ) lim ( ) c 0
1d ( )

y i x e Q ix e x
x 0
b p
X 1dp b p f
p 0 0 p Q i0
Q i V p Q i
c d c d

c d





Note that both the integral and the limit on the r.h.s. exist for The important point is that the growth of the integrand corres-
any such ( )
p .
b ponding to a power singularity is no more than polynomial.
The well-known Sohotsky formula reads: Further, consider the integral of 3.7 against a test function
1 1 ( )
p . Using 3.9, we find:
H I q



PV ( ) b
S ii p p
p `
U i0 h p
e





d





where the first term is defined with the usual principal value 1 1 1
dp g ( p)
prescription. p U i U
v
0 p i0
c d





e e





d d





r D 7 B F s 8 9 7 t 6 7 4 7 8 9 @ A B C 9 8 D C E F 8 C 7 G 9 H I u





I H I P P





g
const ( ) ( ) ~( )
0 0 S

h 1dx 1dx x x x x
f f f



The next point is existence and non-existence of products of c d c d





Sohotsky distributions. For instance, the product of the Sohot- Thus, the convergence of the integral on the l.h.s. around
sky distribution and its complex conjugate, p 0 has transformed into convergence of the integral on the
1 1 y
H I w





r.h.s. at infinity in various directions. (Since the Fourier trans-
p `
U i0 v p S i0 form ~
g ( x) is a smooth function, integration over finite regions
does not exist as a distribution defined on all test functions is not dangerous.) One sees that the -functions ensure conver-





unlike the square gence in all directions other than those in the positive quadrant
1 1 x 0, 0 . However, convergence in those directions is en-
x
H I x











f





p v `
U i0 p U i0 sured by the properties of ~
g ( x) , as explained above.

which does. One concludes that the integral 3.11 exists for any test
In the context of analytical properties of Feynman diagrams function, therefore, the product 3.7 is a well-defined distribu-
(cf. e.g. [17]), this phenomenon is well-known and described in tion.
terms of pinched (Eq. 3.6) vs non-pinched (Eq. 3.7) integration On the other hand, a similar calculation for the product 3.6
contours. Indeed, if one integrates Eq. 3.7 with a function that gives:
is analytical in a neighbourhood of p 0 , then the Cauchy e
d





y



1 1
theorem allows one to simply deform the contour of integration 1dp g ( p)
U v
into the upper complex halfplane near p 0 because the poles p i0 p 0
S i
c d





y





e e
of the two factors are on the same side of the contour; then the d d





I H I P a
integrand will have no singularities at all on the integration h 1dx 1dx g
( ) ( ) ~( )
x 0 0 S
x x x
f f i f





contour. In the case of 3.6 this is not possible -- the contour is c d c d





"pinched" by the two poles. One can see that there are directions in the plane ( x, x ) --
f





j j
G 8 5 D D 5 8 8 C 7 G C G 8 5 D 9 7 t 7 F D C 5 D 8 D G 9 t 7 D 9 H I e.g. x U
x x -functions
S x0 , S 0 -- in which the
h h

f f





are both equal to 1 while ~
g ( x x ) ~
g
U U x x
S ( S )
0 0 const as
Following, essentially, Hrmander (see [23] and refs. there- h h
f f





k

. The integral on the r.h.s., therefore, cannot be con-
in), consider the products 3.6 and 3.7 from the point of view of R





Fourier transformation. Then existence of products of functions




8


                





vergent for all test functions, and the product 3.6 is not a well- e n
1 1 ( m) 9 8 9 @
defined distribution. 0 0 ` a
b c nd 2
p i p m i m 0 ( p i0)
As a further excercise, replace the second factor by, say, a a n 0
a









( p) . Then the second -function on the r.h.s. of 3.11 is re-
! It can be proved in a way similar to the preceding example by
placed by a constant, and again, the integral does not exist for using the procedure described in sec. 7 of [4]. Another way is
some test functions: the product of a -function and a Sohot- to use partial fractioning,
"





sky-type distribution is ill-defined.
1 1 1
8 9
1 V 1
On the other hand, if one of the distributions is a function
p V i0 p U m V i0 V U V
# m p i0 p m i0 W
that has any number of derivatives in a small neighbourhood
around p 0 , then there are no dangerous directions associ- and reduce this example to the case of one factor already con-
#





sidered.
ated with it at that point, so that its product with any distribu- Note that if one changes the sign of the imaginary parts of
tion exists around p 0 .
# one of the factors, then non-trivial m -dependent counterterms
$ % & ' ( ) & 0 1 2 3 4 5 ' 2 6 7 2 8 9 @ 8 appear in the expansion. For example:

It is important to realize that any definition of an integral in e n
1 1 ( m) 2 i 9 8 9 8
( )
` a p
infinite bounds contains -- explicitly or implicitly -- some sort 0 0 b c nd 2
p i p m i m 0 ( p i0 a m
)
a n 0
D D
of a cutoff, say, | x | . Such a cutoff plays the role of a
A , B C





a natural regularization. Then the absolute convergence of the The simplest way to obtain this result is via a trick -- switch-
integral 3.11 (established e.g. by power counting) implies that ing the sign of imaginary part of the first factor using the So-
whatever (correct) regularization method one uses to give hotsky formula 3.4. However, one can also derive it in a sys-
meaning to the product of the two Sohotsky distributions on the tematic way as in the preceding examples.
l.h.s. of 3.11, the limiting value for the integral on the l.h.s. The conclusion is that the expansion 3.21 looks as if obtain-
will exist for any test function and it will be the same for all ed by simply taking a product of the first factor (whose expan-
regularizations. This gives a precise meaning to the phrase that sion in m is trivial) and the expansion of the second factor,
the product in such a case is "naturally defined" in the sense of Eq. 3.17. Each term of such a product is a well-defined distri-
distributions. bution. However, strictly speaking, existence of such a product
does not guarantee that it will possess the necessary approxi-
E 5 F G H ' I ' 4 7 2 3 H & 6 5 4 I 6 5 8 9 @ P





mation properties to be the asymptotic expansion of the initial
m -dependent product (the l.h.s. of 3.21). But it turns out that it
E 5 4 6 Q 0 2 R 4 5 ' ) 4 S ( ' 4 I 6 8 9 @ T





does. Let us understand why this is so using the techniques of
As a warm-up excercise, consider the distribution Sec. 3.8: in a general multidimensional situation, explicit cal-
1 culations similar to those above may not be possible.
8 9 @ X





p U m V i0 W
6 ' 2 ) H ) 2 ' & ' 4 I 6 4 6 ' 2 ) G 5 I I ( ) 4 2 ) ' ) & 6 5 I ) G 5 8 9 P





where m is a real parameter. Its expansion in the sense of dis-
tributions as m 0 is: Similarly to 3.11, one has:
Y





n
e ( m) u
9 8 9 @ f


Eq. 3.16 d
` a n 1 1 1






b c d
m 0 ( p i0) 1 dp (p)
n 0 p i0 p m i0
a





t u







a a





P R O O F . It is sufficient to consider only the case n 0 . As u u
#



d d





usual, the expansion is sought in the form 9 8 9 T
d d


t
imx
1 dx 1
dx ( x )
0 [ ( x 0)e
] ~( x x )






1 1 t u t u
8 9 @ g





U U
( ) " ( ) ( )
U V # V c m p o 1
p m i0 p i0 W
If one performs the Taylor expansion in m on the r.h.s., the


and the coefficient c(m) is determined from a consistency factor exp( imx ) (which is bounded by 1 everywhere) trans-
a





condition (in a manner similar to the Euclidean case; see forms into a polynomial of x . The remainder term is then
e





sec. 7.4 of [4]): bounded by something like O mN x N
( which exhibits a poly-
e )

w f





u
h nomial growth as x -- but not faster than polynomial.
d





e Y





c 9 8 9 @ v
(m) 1
V U U





p q
1 d 1 1 0 1 Therefore, the pattern of behaviour at infinity (polynomial
i r s
# p #
U V V o( ) o( )
p m i0 p i0
t u growth vs rapid decrease) is the same both for non-expanded
The integration can be performed using the explicit represen- factors, the expanded factors, and the remainder term. This al-
D
tation 3.4 and symmetric cutoffs, | p | . lows termwise integration of the product of expansions with
A





preservation of approximation properties.
The expansion 3.17 can be regarded as the Taylor expan- One sees that the "natural" existence (in the sense of
sion of a translation along the p -axis. This explains the ab- Sec. 3.13) of the product of expansions of individual factors
sence of non-trivial counterterms with logarithmic dependence implies that such a product is indeed a correct asymptotic ex-
on m : an infinitesimal translation involves differentiation, and pansion for the initial non-expanded product; in other words,
differentiations in p of the distribution ( p t
w i ) n
0 are per- taking product and performing expansion are two commuting
formed in a straightforward manner. operations in such a case.

x y I & 7 ' I ) 5 8 9 The above examples provide enough motivation for analog-
Consider a product of two factors like 3.16 with their sin- ous multidimensional constructions to which we now turn.
gularities separated by a distance o(m) . Then one has the fol-

lowing expansion:





9


                





! " # $ % & ' " ( ) " 0 " & ' 1 2 # 3 ' & ! 3 4 $ & 3 " 5 '





t u u
+
1 v w w
F G y
c 0
3 5 i 1 d e i A
D 1 x
A W i0
6 7





0

We are going to generalize the notion of dangerous direc- one can rewrite 4.5 as follows:
tions to the multidimensional case. We first consider the case
t t t





v v v v v v





F G
of linear denominators (Sec. 4.1) and derive a simple geometric Eq. 4.5 ~
c i3 1 d d d ( )
1 1 2 1 x
U Y Y
3 1 1 2 2 3 3
criterion of existence of products of Sohotsky-type dis- 0 0 0
tributions. We then consider smooth deformations of coordi- where ~
( x) is the Fourier transform of ( )
p :
nates (Sec. 4.14) and see what form the criterion takes. The U
u





notion of singular wave front emerges naturally (Sec. 4.23) and ~ G F G H
ipx
( )
x 1dpe ( p)
allows one to easily obtain a criterion (essentially due to Hr-
mander; cf. Theorem IX.45 in [23]) for the general non-linear One can see that th
v e rapid decrease of ~
makes the integral
case including the so-called osculating singularities (Sec. 4.34). convergent at large
Finally, the extension of the criterion to the case of asymptotic g in any directions if the following con-
expansions of Sohotsky-type distributions becomes straight- dition is satisfied
v v v v





forward (Sec. 4.41). G F G H H
v , for all
Y v Y v
1 1 2 2 3 3 0 g 0

8 9 @ A B C D B E A F G H





Then the product in the integrand of 4.5 exists in the sense of
distributions.
I P Q R B D S Q C E F G T





Note that for, say, , the first inequality in 4.11 de-
3 0
Consider a product of two Sohotsky-type distributions inte- generates into the condition of existence of the product of the
grated against a test function ( )
p localized in the region : first two factors. Therefore, instead of making the assumption
U V





of existence of subproducts ovne can simp
v ly require that the
1 1 G F G X
dp ( ) inequality in 4.11 hold for all
0, 0 .
U p g g g
W W
(v p i )(
0 v p i )
0
1 2

A Q A S C 9 D D C 9 S A C 9 Q @ F G H T



For simplicity of notation, we do not consider general powers
explicitly. The distributions with i0 can be taken into ac-
Y Consider the set of
v all possible linear combinations
count by changing signs of (some of) vi . v with 0 . We call it proper cone if it
v v
1 1 2 2 3 3 g
One can see that the product is a well-defined distribution
unless the two vectors v lies strictly within one half of the linear space of vi :
1 and v2 look in opposite directions.

Indeed, if they are linearly dependent then the problem effec-
tively degenerates into a 1-dimensional one: if they look in the v1 v2
same direction then the product is a well-defined distribution; v3
if they look in opposite directions then the product is non-
integrable. If the two vectors are linearly independent then one
deals with a direct product of two distributions that are well- F G H X





defined separately, and such a product is a well-defined distri-
bution too. In such a case the integral 4.9 and the product 4.5 are well-
defined.
I ` C A A R B D S Q C E F G F





The generalization to more than three factors is straightfor-
ward.
Under the same assumptions, consider the product of three Note that the property of a set of vectors to span a proper
Sohotsky-type distributions: cone is invariant under arbitrary linear transformations.
1 1 G F G a
dp ( ) Q @ q 9 @ A B C h A R Q C B S 9 Q @ E Q R D Q Q C h 9 @ B S A E F G H F
U p
W W W
(v p i )(
0 v p i )(
0 v p i )
0
1 2 3

The singularities of the g-th factor are localized on the linear As a next step towards the general case, notice that exis-
manifold tence of a distribution is a fact that survives any non-linear
G F G d smooth mapping of coordinates. To rewrite the above criterion
b

c c
g { p | vg p }
0
in a form that is "covariant" under such transformations, con-
First, without loss of generality one assumes that the origin is sider a neighbourhood of the origin and perform a smooth
V





the only point of intersection of the three manifolds. Second, t
( C ) deformation of the coordinates so that
one has to assume that the product of each pair of the three 2 G F G H a

factors is well-defined as a distribution everywhere. A standard v p v p ( ) ( )
Y O p D p
g g g
reasoning [4] shows that the product 4.5 is then integrable with
any test function that is equal to zero in any small neighbour- Fig. 4.13 is modified as follows:
hood of the origin of coordinates. It remains to study the be-
haviour of the product near that isolated point. v1 v2
v3
e f 9 E S A @ D A Q R S ` A g C Q h i D S p B @ B q r S 9 D B q D C 9 S A C 9 Q @ F G s





Using the representation

F G H d





10


                





Then Eq. 4.5 becomes: singular in the sense of differential geometry) within , which
2





1 means that
!
1 dp ( p) % & % ' (
# # #
D p i0 D p i0 D p i0
$ D( p) 0 p
" 1 2 3 % & % T &
( ( ) )( ( ) )( ( ) )
supp c
d x





y
D( p) 0

The singularities of the three factors are now localized on non- Construct the wave front of the distribution (D( )
p e i )f n
0 by
linear manifolds described by non-linear equations: attaching the vector D( p )
d 0 (describing the "dangerous" di-
)

0 0
g { p| g
D ( p) }
0 % & % ' 1 rection; cf.also Sec. 4.27 below) to each singular point p v
0 :






The integration measure is transformed with a Jacobian that is D( p )
0
smooth within and it is absorbed into the redefined test
2





function. The region is deformed accordingly.
2





Moreover, the choice of the origin of coordinates cannot af- p0
fect existence of the product, so the origin can be shifted arbi- & % T





trarily. Denote the point of intersection of the three manifolds Now the criterion 4.22 can be reformulated in geometric terms:
4.18 as p 0 3
0 . Since v D ( p
g g )
0 , the criterion 4.11 is rewrit- C D P C B @ A R F D @ P D F B p C R @ A B C @ A D E





& % T





ten as follows:
For each point p
4 4 4 4 0 where some of the factors of the product
3 5 3 5 3 6 7





1 1
D ( 0
p ) 2 2
D ( 0
p ) 3 3
D ( 0
p ) 0 for all 0
g % & % ' 8
4.21 are singular, select all those and only those factors that
9 @ A B C @ A D E F D @ G E H E I P Q C @ D F F G R B D @ S are singular at that point. If the corresponding dangerous di-
& % T U





rections 3 D p
Consider now a product containing an arbitrary number of g ( )
0 span a proper cone -- and if this holds true

Sohotsky-type factors: for any such point p0 -- then the product is integrable with
W 1 arbitrary test functions localized within .
2





n % & % T '
#
g V ( ( ) 0) g p H A S G t G E C @ D I S t A @ C R B A D E
X g
D p i g
D( p ) & % T (
d 0

Without loss of generality one can assume that there is only
one point p The above analysis was based on a deformation of coordi-
0 (not a higher-dimensional manifold) where the nates that made all singular manifolds flat simultaneously.
singular manifolds ) g intersect all at once; otherwise by a This is not always possible (recall osculating singularities,
smooth change of coordinates one can deform the intersection Sec. 2.20). Therefore, it is useful to consider the microlocal
`





Y
) ) ), and then mechanism that ensures existence of the products in terms of
V





X X
g g into a linear subspace (within 2
Fourier transforms without invoking global deformations. It is
only the coordinates that are transverse to ) will matter.
X at this point that the notion of wave front becomes genuinely
By analogy with the preceding examples, one writes down non-trivial.
the following criterion of existence of the product 4.21 at p
f
0 a
0 : Consider again the distribution ( )
p D( )
p e i near






a a





w
4 4 4 a point p ( ) but D( p )
3 6 7 0 where D p0 0 d x
0 0 . It turns out
g g
D ( p )
0 0 for all g 0 0 % & % T T
c g
b





g V V
X g X that any direction n other than the one described by D( p )
d 0

If all subproducts are exist, then it is sufficient to consider only is not dangerous at p0 in the following sense: one can find a
4
7 test function ( p) that is localized sufficiently closely around
g 0 .





The heuristic meaning of 4.22 is the same as in the example p0 and such that the Fourier transform
above: as is discussed in 4.27 the vector D( p ) ~

w ixp
1
d 0 describes the x dp e p p decreases rapidly (i.e. faster than
( ) ( ) ( )

dangerous direction for the distribution (D( )
p ~
e i )f n
0 at its 4
any power) in the direction of n , i.e. n 0 faster than







( )
singular point p0 , i.e. the only direction in which its Fourier
4
any power N 5
f as .
transform does not decrease rapidly, thus jeopardizing conver-
The proposition is quite intuitive: behaviour at infinity of
gence in that direction of Fourier-transformed integrals with Fourier transforms is determined by the behaviour of the origi-
test functions. nal in an arbitrarily small neighbourhood of the origin. The de-
Recall that the above reasoning allows one to prove the cri- nominators of the distributions we deal with are linear func-
terion only in situations where all the non-linear singularities 2
can be "flattened" simultaneously by a smooth transformation tions up to O( p ) terms which are -- as the preceding exam-
to the form of 4.13. Nevertheless, it is useful to summarize the ples suggest -- negligible if one focuses on a sufficiently small
experience gained in a intuitive geometrical form. neighbourhood, so that the distributions resemble the standard
Sohotsky distribution that depends on the coordinate directed
g G h C F @ D E B D F G i D p D B S q H r B H s C t A S B @ A Q I B A D E





& % T u along D( p )
d 0 . One may also note that multiplication by a

Consider a distribution (D( )
p smooth function does not change this property.
e i )f n
0 defined within a re-
The formal proof which is a straightforward technical im-
gion . Its singularities are localized at the points where
2 plementation of the above heuristic idea is given in Sec. 4.28
D( p) 0 0 . Such points constitute a manifold, below.
v w w
{ p | D( p) 0} . Assume that ) is well-behaved (i.e. is non- A somewhat stronger version of the above property will be
~ 4 4
useful: the rapid decrease n 0 in is uniform with re-







( )





11


                





spect to all directions n outside some (arbitrarily narrow) cone Similarly to case A, change the coordinates in 4.29 as fol-
Y h




C containing D( p ) lows: X G X G G
H H
L p p . Denote N n
! 0 . This is established as follows. First, || ( ) , || 0 . Then using
the proof of the above proposition allows one to obtain bounds 4.8 (with the integration variable changed to ), rewrite the
i





that are, in fact, uniform with respect to all directions from a integral as:
sufficiently narrow cone Cn around n. Then one notices that c p





directions can be represented as points of the unit sphere which U Y U
6 5 a 7





X X X X q X X

H
d ( ) exp ( ) r
i d d i i N nH H
1 1 D
@
|| ||





is compact; it follows that for any cone C one can select a fi- E
N
nite number of directions ni whose cones Cn contain all di-
i Y
where q (X ) G p (X ) X . If the neighbourhood was chosen
A
|| ||
rections outside C . Choosing the largest bound among those
s t


corresponding to C small enough, one can ensure that q (X ) u 1 . This allows
n , one obtains the required uniform one.
i
one to integrate by parts:

" # $ $ % $ % & ' ( ) # $ ) $ 0 1 & 1 $ 2 $ % 3 ( 4 5 6 5 7 8 6 5 7 9





e E e







c
1 i N ( )
d ||
e { }
Let the test functions ( p) be localized within a small ||
@





v





neighbourhood of the point p e E e






c
i N ( ) 1 6 5 a a
A 0 . By a shift of coordinates x y
||
B 1 B
d e { }
etc.
w





one ensures that p W
B N ( )
0 0 . Consider the integral

1 E 1 5 6 5 7 F
dp e i np ( ) The new factor does not cause problems because it is smooth
@ p a
C D
( L( p) i )
0 and the denominator does not turn to zero. In this way one ob-
tains bounds for the integral over X which, after integration
Split p G pH
( , p )
|| where the first component is orthogonal to over , make manifest the rapid decrease of the integral. This
i





the manifold L( p) G 0 while the second, tangential to it at 0. completes the proof.

The Taylor theorem guarantees that L( p) 2
D
G p I P L( )
0 O( p ) 0 4 & 1 2 0 1 2 # 1 & 1 ( 0 6 5 a 6





for p 0 . Therefore, choosing small enough allows one to
Q A




This is the situation when the singular manifolds
find a smooth interpolating function ( p) such that
R D p
g ( ) G 0 touch rather than intersect in a general fashion. We
( p) B p S 0 for all p sufficiently large, and such that
! L( )
R





are going to show that the criterion obtained above works here
( p) B L( p) within . Without loss of generality, the fol-
R A too.
lowing reasoning assumes L( p) T .
R ( p) It is sufficient to consider the simplest case of 2-
There are two cases to be considered: dimensional Euclidean p B ( p , p )
1 2 and the product of two
A. n is directed at an angle to L(0) ;
! factors:
B. The directions of n and L(0) are exactly opposite.
!





Y



G 1
( )
p ( )
p , ( p) (D ( )
p i0) ,
C A S E A . Let us exhibit the fact that the oscillations of 1 2 g g C





the exponential along the singular manifold L( p) Y Y Y
6 5 a
G 0 for G G 2
1
D ( p) 1
p i0, 2
D ( )
p 1
p 2
p i0.
U



D V suppress the integral as in the case of a smooth func-
Q


The dangerous directions are shown in 4.36:
tion -- irrespective of the singular behaviour in the transverse
directions. To this end, split pH B pW pWW
( , ) where p is directed p1
2
along the projection of n onto the subspace of p =
H , so that WW
p p p
1 2 p2
is orthogonal to both n and L(0) . In the coordinates
!







( p , p , p ) G =
|| , one has n (n , ,
0 n )
|| . Finally, choose new coordi- p 0
1 p = p = 0
1 2 6 5 a






nates (X , X ,X )
|| so that:
It is natural to suppose that since the quadratic terms played no
Y



5 6 5 a b

`





X G X G X X G
|| ( p), n n p n|| || , p role in Secs. 4.144.20 they should not be important here, too.
Because the singular manifolds do not intersect away from





One can see that the closer n to L(0) , the smaller n , and
! p G 0 , it is sufficient to consider an arbitrarily small neigh-
the closer ( p) should be to p S ( )
0 , which requires a
! L
@ bourhood of that point.
smaller . The dangerous direction at p G 0 , the same for both fac-
A





In the new coordinates, the integral becomes tors, is

5 6 5 a 8




E e e





c d d T
1 i (n n ) 1 B B
D (0) D (0) 1
( 0
, ) V
5 6 5 a f


d ! !
X X
e || || ( ) 1 2
@
C a
D



X
( i
|| )
0 Take a cone Csing of directions around V. Choose it narrow
The Jacobian of the coordinate transformation was absorbed enough so that C D C
sing sing -- which is the cone spanned by




into the test function X
( ) . The integrand is smooth in X so
@


all possible pairs of vectors from C
that one can perform the standard integration-by-parts trick sing -- is a proper cone.
(Sec. 3.8) to establish the rapid decrease of the entire integral. Furthermore, according to the last paragraph of Sec. 4.27,
C A S E B . Here one makes use of the fact that one deals given Csing , one can find a pair of test functions p and
1( )
with a "slightly deformed" Sohotsky distribution whose Fourier p localized around k G 0 such that Fourier transforms

transform is a "slightly deformed" -function with essentially 2 ( )
g ~







similar nullification properties. ( )
g x of the distributions ( p) ( p) are rapidly and
@ g g





12


                





uniformly decreasing in all directions except those in the cone i p p q r s t u r v w x y





Csing .
P F P E P D H I F G U





Because the functions ( ) are smooth and non-zero in a
g p $





small neighbourhood of p 0 , it is sufficient to prove exis-
! The original Landau equations [16], [17] provided an an-
tence of the product swer to this question: at what values of external parameters --
masses and momenta -- does a given multiloop diagram cease
( p) ( p)

1 2 % & '
to be an analytic function? The problem we are considering
D p " i # D p " i $ $
1( ) 0 2 ( ) 0 now is phrased differently: given an integrand of a multiloop
Consider the integral diagram for some fixed values of external momenta and
masses, where in the space of integration momenta does the
( ) ( )
p p
1 ( % & )
dp dp 1 2 ( p integrand (the product of propagators) cease to be a well-
1 2 )
D ( p "
) i0 # D ( p "
) i # $
1 2 0 defined distribution? The close relation of the two questions is
clearly seen from the point of view of the theory of asymptotic
(


for an arbitrary test function ( )
p (whose localization proper- operation which establishes a connection between the follow-
ties are not important because the distributions themselves are ing two effects:
localized as needed). Its representation in terms of Fourier (i) non-analyticity of the integrated diagram as a function of
transforms is as follows: external parameters at some values of the latter (i.e. a presence
1 ~ of, say, logarithmic terms in the corresponding asymptotic ex-
% % 3
1 ~
1 2

d2x d2 y ( x) ( y) ~( x y
1 2 )
0 pansion near those values) and
$ $





One sees that the only directions in the 4-dimensional space of (ii) non-integrability of the Taylor-expanded integrand (for
the aggregate integration variable (x, y) where the product a discussion see [1], [4]).
~
4 ~
4
(x) ( y) E H P D H I F
1 2 does not exhibit a rapid decrease are the ones
$





that lie within the direct product C C
sing , i.e. the direc- Consider a product of propagators of the form 2.2. It need
# sing
tions of the form ( x, y) not be a complete integrand -- just a subproduct whose fac-
5 6 (V ,V )
1 2 with V C
i sing . However, d
7 tors's causal singularities pass over a region P . Consider
the construction of Csing ensures that the integrand in such di- the points where the causal singularities of all the factors from
rections is suppressed by the rapid decrease of the test func- G overlap. Such points are solutions of the system of equations
tion. This proves absolute convergence of the integral 4.40 and,






&

therefore, existence of the products 4.38 and 4.35. L ( p
g ) 0
!





g G $ $
The proof immediately generalizes to the case of any num-
ber of factors. However, at some of those points the product exists, and at
others, it does not -- as described by the criterion of Sec. 4.20.
To summarize, the criterion of Sec. 4.20 is valid irrespective Recalling that p is the collection of all loop momenta,
of the pattern of intersection of the singular manifolds 8 g -- p 5 ( p , p , pl )
1 2 , and that the gradient operator is then repre-






whether they intersect in a transverse fashion or osculate -- as sented as 2.13, one rewrites 4.22 as follows:
long as the other conditions are satisfied (most notably,





smoothness of all manifolds D p ). 2( )
l p d
8 g , or 9 @
g ( ) 0 within A g

h





d

j 1
p


d


B C D E F G H I F D I P G Q R S D I D H T E C S P F G H I F G % % U





i i i

%





e f
g 0 for all 0, 0
d g g f
$





$ $






2
g
g g
lg ( p d
) g
One can immediately extend the above results to obtain a d
h






criterion of existence of asymptotic expansions (cf. Sec. 3.14). p
l
It will be sufficient to consider the case of two factors.
5
Let The points p ( p , p , p ) that satisfy 5.3 and 5.4 are the
V be a parameter that goes to zero, 0 . Consider a 1 2 l
W X





0





n points where the product exists in a natural sense (as explained
product of Sohotsky-type distributions ( D ( p, 0 . As-
W ) Y i ) g
g ` in Sec. 3.13) -- such singularities are traditionally called non-
d
sume that for all ( p, b c 0 ), pinched, while p satisfying 5.3 but not 5.4 constitute a mani-
W ) (Y a , W )
0 (with some e f
D p fold in the integration space where the product does not exist
V V V
g ( , ) are smooth in ( p, ) and D p 0 . Then if
9 @
g ( , ) -- the manifold of pinched singularities.
the product of (D ( p, )
0 " i ) N
g 0 exists for arbitrarily large N The system of Landau equations is obtained if one replaces
`





inequalities by equalities in 5.4, and reinterpretes it as a de-
in the "natural" sense (i.e. in virtue of the Hrmander crite- scription of pinched singularities rather than conditions for ex-
rion)i, then the asymptotic expansion in V g 0 in the sense of istence of the product.
distributions of the original product is equal to the product of
the asymptotic expansions of factors (recall that the latter ex- k I F l T I P H P F D m P m E G





pansions are simply the Taylor expansions). $





The proof is a straightforward modification of the reasoning In the theory of non-covariant light-cone gauges ( n A 5 0
in Sec.4.28 in the spirit of Sec. 3.14.
with n2 5 0 ) one encounters propagators of the following
form:

i which means that the product of Taylor expansions of factors is well-
defined to all orders in .
h





13


                





#






1 n theless result in the same formulas as in the Euclidean case. A
) p0 n0 p) 1 2 3 4


% &




!

) 0 simple example of such a regime was given in Sec. 2.21.
$ g "
2 ' (
p i0 n p It is not difficult to understand (cf. Fig. 2.24) that no modi-
fications (no new counterterms) to the formulas of Euclidean
where n2 5 n0
0 6
, 0 . An important technical problem (cf. asymptotic operation are necessary as long as the following
e.g. [31]) is the singular structure of Feynman diagrams in- holds true in the asymptotic limit corresponding to a particular
volving such propagators in view of the additional singularity regime: Apices of light cones corresponding to various propa-
in 5.6. The so-called Mandelstam-Leibbrandt prescription (for gators should either merge in the asymptotic limit, or stay away
a review see [34]) consists in using from each other at a non-light-like separation; they are also not
3 2 3 @
n p 0 allowed to approach mass shell surfaces in the asymptotic
7 n p 8 i 9
0 sign p
limit. More precisely, if, as in [4], m , q and M , Q are small
Using the formalism developed above, it is not difficult to see and large masses and external momenta of the problem, re-
why this prescription works: spectively (for definiteness, all the momenta enter the dia-
Indeed, the singular manifolds corresponding to the two grams), and if one formulates the asymptotic regime in terms

7
factors in the denominator of 5.6 are described by p2 5 0 and of m,q 0 with M ,Q fixed, then q should vanish compo-
n p 0 , and intersect over the line p B n . Consider any point nentwise whereas Q are allowed to be arbitrary non-lightlike
A





p on that line and evaluate the gradients: provided the asymptotic configuration does not correspond to
any of the thresholds. More precisely, in the context of a con-
3 2 3 F


D E





C C
2 0 0
! !
p 2 p , sign p n p sign p n crete 1PI diagram G , let C be a cut of the diagram with Q
) ) ) )




A A g g





being the momentum entering any one of the two resulting
(The overall sign is switched so as to make the imaginary in- components of G , and let M be the sum of masses of the
finitesimal parts the same sign, as required by the formalism of g




the preceding sections.) lines cut by C ; further suppose that Q2 h 2 for any cut i
g M
g g





One sees that the two gradients are the same up to a posi- in the asymptotic limit ( m 5 q 5 0 ). Then the asymptotic ex-
tive coefficient -- irrespective of whether p is on the upper or pansion is given by the same formulas as in the case of purely
lower half-cone. The criterion 4.26 then tells us that the prod- Euclidean asymptotic regimes [4], [5].
uct 5.6 with the prescription 5.7 is a well-defined distribution.
This also means that the wave front of the propagator 5.6 on Finally, it should be emphasized that, strictly speaking, the
the light cone is not different from that of the simple scalar pinch/non-pinched classification of the singularities can be in-
propagator. From the point of view of studying intersections of terpreted in the context of asymptotic operation as equality to
causal singularities of several such propagators, this means that zero of counterterms at the corresponding points of singular
there will be no new pinches etc. as compared with the covari- manifolds. Therefore, one might proceed with the construction
ant case. Of course, the propagator 5.6 has an additional sin- of non-Euclidean asymptotic operation ignoring such issues,
gularity n p 0 outside the light cone. But since such addi- and then rediscover nullification of counterterms for non-
A





tional singularities are localized on a linear manifold, studying pinched singularities by an inspection of the resulting integrals.
their intersections presents no difficulty. As a matter of fact, analytical study of the most complex types
A reasoning of this sort allows one to largely avoid explicit of non-Euclidean singularities (boundary points of pinched
calculations when studying the singular structure of diagrams manifolds) may be psychologically easier if one ignores the
(cf. examples of such calculations in [31]). pinch problem altogether. But the existing treatments start
with the pinch/non-pinched classification (cf. [35]), so it would
2 3 f





G H I P Q R S H T U Q V W I X I P Y ` a b c b Q T d W e Q ` W P have been inconvenient to postpone establishing a connection
with what is already known until the new theory is completely
As a last application, we consider the class of asymptotic developed.
regimes which, although not allowing Wick rotation, never-





p T q X c r U W V e ` W X b P





I thank A. V. Radyushkin and G. Sterman for discussions of the problem of non-Euclidean asymptotic expansions.
Parts of this work were done during two visits to the Physics Department of Penn State University.
At different stages, this work was supported in parts by the CTEQ collaboration, the Russian Fund for Fundamental Research
95-02-05794, the International Science Foundation under grants MP9000/9300, and the U.S. Department of Energy under grant
DE-FG02-90ER-40577.
I am indebted to P. Cherzor for pointing out several misprints.





14


                





! " ! # ! $ % ! &





[1] F.V.Tkachov: Advanced methods for studying radiative [17] R. J. Eden, P.V. Landshoff, D. I. Olive, and
corrections. Theory of asymptotic operation, in "New J. C. Polkinghorne: The Analytic S-Matrix. Cambridge:
Techniques for Calculating Higher Order QCD Correc- Cambridge University Press, 1966.
tions" (1992, 16-18 December 1992, ETH, Zrich). ETH: [18] L. Hrmander: Acta Math. 121 (1971) 79.
Zrich, 1992; Sov. J. Nucl. Phys. 56 (1993) 180; Sov. J. [19] M. Sato: in Actes Congr. Intern. Math., Nice, 1970.
Part. Nuclei 25 (1994) 649. [20] L. Schwartz: Thorie des Distributions. Paris: Hermann,
[2] F.V.Tkachov: in "Quarks-82" (1982, May 5-7, Sukhumi, 1966.
USSR). INR, USSR Acad. Sci.: Moscow, 1983. [21] M. Sato, T. Kawai, and M. Kashiwara: Lect. Notes in
[3] F.V.Tkachov: Phys. Lett. B124 (1983) 212. Math. 287 (1973) 265.
[4] F.V.Tkachov: preprints INR, Russ. Acad. Sci. P-332, P- [22] Y.V. Egorov and M. A. Shubin: Linear differential equa-
358, Moscow, 1984; tions with partial derivatives. Elements of the modern
Int. J. Mod. Phys. A8 (1993) 2047. theory. (Modern problems of mathematics. Fundamental
[5] G.B.Pivovarov and F.V.Tkachov: preprints INR, Russ. directions, Vol. 31.) Moscow: VINITI, 1988.
Acad. Sci. P-370, Moscow, 1984, P-459, 1986; Int. J. [23] M. Reed and B. Simon: Methods of Modern Mathematical
Mod. Phys. A8 (1993) 2241. Physics. II: Fourier Analysis, Self-Adjointness. New
[6] S. G. Gorishny, S.A.Larin, and F.V.Tkachov: Phys. Lett. York: Academic Press, 1975.
B124 (1983) 217. [24] M. Sato, T. Miwa, M. Jimbo, and T. Oshima: Publ. RIMS
[7] F.V. Tkachov: Phys. Lett. B100 (1981) 65; Teor. Mat. 12, Suppl. (1977) 387.
Fiz. 56 (1983) 350; Theses INR, USSR Acad. Sci., Mos- [25] N. N. Bogoliubov, A. A. Logunov, A. I. Oksak, and
cow 1984. I. T. Todorov: General Principles of Quantum Field The-
[8] G. T. Loladze, L. R. Surguladze, and F.V.Tkachov: Phys. ory. Dordrecht: Kluwer, 1990.
Lett. B162 (1985) 363. [26] S. Coleman and R. E. Norton: Nuovo Cim. 38 (1965) 438.
[9] L. R. Surguladze and F.V.Tkachov: Nucl. Phys. B331 [27] V. A. Smirnov: Renormalization and asymptotic expan-
(1990) 35. sions. Progress in Physics, Vol. 14. Basel: Birkhauser,
[10] S. A. Larin, F.V.Tkachov, and J. A. M. Vermaseren: Phys. 1991.
Rev. Lett. 66 (1991) 862. [28] A. I. Zaslavsky: Theor. Math. Phys. 80 (1989) 372.
[11] S. A. Larin, F.V.Tkachov, and J. A. M. Vermaseren: Phys. [29] L. Schwartz: Mthodes mathmatique pour les sciences
Lett. B272 (1991) 121. physiques. Paris: Hermann, 1961.
[12] A. N. Kuznetsov, F.V.Tkachov, and V.V.Vlasov: Tech- [30] V. S.Vladimirov: Generalized Functions in Mathematical
niques of distributions in applied quantum field theory Physics. Moscow: Mir Publishers, 1979.
(I) Euclidean As-operation for products of singular [31] A. Bassetto: Renormalization of Yang-Mills theories in
functions. Preprint Penn State Univ. PSU-92/108-T, light-cone gauge: recent results. (Invited talk at the
1992. Workshop on Light-Cone Quantization, 1992, Dallas,
[13] A. N. Kuznetsov and F.V.Tkachov: Techniques of distri- Texas). Preprint DFDP 92/TH/32, Padova.
butions in perturbative quantum field theory. (II) Appli- [32] G. P. Korchemsky and A.V. Radyushkin: Phys. Lett. B279
cations to theory of Feynman diagrams. Preprint INR, (1992) 359.
Russ. Acad. Sci. INR-809/93, Moscow, 1992. [33] E. Eichten and B. Hill: Phys. Lett B234 (1990) 511.
[14] J. C. Collins and F.V.Tkachov: Phys. Lett. B294 (1992) [34] G. Leibbrandt: Rev. Mod. Phys. 59 (1987) 1067.
403. [35] S. B. Libby and G. Sterman: Phys. Rev. D18 (1978) 4737.
[15] J. C. Collins and F.V.Tkachov: The dimensional regulari- [36] H. Epstein and V. Glaser: Ann. Inst. H. Poincare A XIX
zation in the Sudakov problem (in preparation). (1973) 211.
[16] L. D. Landau: Nucl. Phys. 13 (1959) 181.





15



