F.V. Tkachov: On the large-s behavior ... 2001-01-08 06:29 Page 1 of 12





ON THE LARGE-S BEHAVIOR OF TOTAL CROSS SECTIONS
IN PERTURBATIVE QCD

[Talk at QFTHEP-2000, Tver, 14-20 September 2000]


Fyodor V. Tkachov

INR RAS, Moscow, 117312, Russia
ftkachov@ms2.inr.ac.ru


Reported is a factorization theorem for large-s behavior of total cross sections obtained in a straightforward
fashion from ordinary perturbation theory. The theorem extends the standard factorization results for the
Bjorken and Drell-Yan processes. The obtaining of useful evolution equations is found to be problematic.
A by-product is a direct method for systematic calculations of the parton evolution kernels.



Disclaimer.
The literature on the Regge limit is vast and hard to be familiar with for a newcomer to the field,
so this text is likely to have missed some relevant publications, for which I apologize.
I would appreciate to be notified of a relevant result.


Introduction 1 direct extension of the standard results for the Bjorken
and Drell-Yan processes. As to the corresponding evolu-
I am going to report first results of a project in which I tion equation, I have not (yet) found one as straightfor-
attempt to reexamine the problem of large-s behavior of ward to use in phenomenological applications as the
total cross sections in perturbative QCD in a simple- renormalization group equations or the GLAPD
minded but systematic fashion, i.e. starting from ordinary equation.
Feynman diagrams and carefully expanding them in the It is far from possible to present complete details in
large-s limit.1 The specific motivations for this project this text due to space-time limitations. However, there
were: seem to be many interesting things one can do based on
 importance of the subject for the cosmic neutrino the result 5.9 (see Sec. 5.13) even without entering into
physics [2]; details of its derivation.
 the TOTEM experiment being planned for LHC [3]; Technical foundation. Asymptotic operation 1.1
 my personal frustration with the non-transparency of
The technical foundation which made this project
arXiv: 8 Jan 2001 the BFKL theory [4] (especially with the concept of
reggeized gluons) and many other results of the theore- possible is the theory of asymptotic operation (AO; see
tical school of logarithmic approximations where it is the review [8]) -- a systematic modern theory of asymp-
customary to stick a running coupling into a formula by totic expansions of Feynman diagrams.3
hand. Asymptotic expansions in parameters is a key piece of
Since the theory of large-s behavior is closely connec- any physical formalism. In perturbative quantum field
ted to the theory of deeply inelastic scattering (DIS; see
e.g. [5]), an ancillary motivation was 3 The theory of AO originated in the dark pre-Gorbachev and pre-
 the problem of calculation of NNLO corrections to the Internet age behind the Iron Curtain at the fringes of the established
parton evolution (GLAPD [6]) kernels.2 theoretical communities [9], which circumstances provided fertile soil
for a desinformation of the theoretical community in regard of author-
Briefly, my conclusion is that an accurate pQCD deri- ship, originality and theoretical underpinnings of some of the most
vation of the large-s behavior leads to formulae that are a important theoretical results in perturbative quantum field theory ob-
tained since the early 80s -- a desinformation which has been affect-
ing a sizeable fraction of theoretical HEP community and causing a
1 This talk is an updated version of [1] where a technical difficulty considerable waste (bound to continue into the future) of both
was overlooked at the final step of derivation (see below Eqs. 5.6, 5.7 research funds and human effort. The brief review of the original
and the following discussion). This is corrected in the present version, papers given below is of course incomplete without a parallel review
leading to somewhat less optimistic conclusions about predictability
of "secondary" publications -- but ( sorry! no fun this time ) space
of the large-s behavior within perturbative QCD. limitations and the focus of this talk do not allow it to branch out in
2 The current state of the art is summarized in [7]. that direction.





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theory, the difficulty of obtaining asymptotic expansions calculations (e.g. [19]; a recent example is [20]).
is that formal expansions result in non-integrable singu- After the mathematical spadework performed in [10]6
larities in integrands. The theory of AO is based on an and [22] the extension of AO to non-Euclidean7 asymp-
efficient algorithmic treatment of the problem of singula- totic regimes required only a simple additional trick of
rities via a powerful mathematical method derived from secondary expansion (the so-called homogenization [8],
the theory of distributions [10], [11]. Distributions are [23]8). The non-Euclidean AO yielded a complete fully
singular limits of ordinary functions, and it would have algorithmic solution for the problem of asymptotic
been surprising if they did not emerge in studies of expansions of Feynman diagrams summarized in [23]:
integrands with small parameters. Distributions in this For any diagram and for any asymptotic regime, the pre-
context play an algorithmic role similar to complex or scriptions of [23] yield the corresponding asymptotic
irrational numbers in the theory of algebraic equations. expansion in a maximally simple form which is suitable
This simple analogy should give one an idea of the for obtaining factorization theorems.
inefficiencies implied in the use of the old-fashioned The first application of non-Euclidean AO was to the
Sudakov-Zimmermann-Lipatov-... methods. construction of a systematic gauge-invariant perturbation
The theory of AO extends and generalizes [9] the theory for models with unstable fundamental fields [25].9
pattern of reasoning pioneered by Bogolyubov in the dis- This is because the results for amplitudes with only
covery [12] of a correct UV renormalization procedure. virtual loops can always be rewritten so as to eliminate
The variant of AO which treated the class of so-called all traces of distributions10 and thus pretended to have
Euclidean asymptotic regimes was developed in 1982 been derived via ordinary methods (the so-called method
1988 [9], [13][17], [10]. It was a major theoretical of regions), so I had to focus on applications to problems
breakthrough in several respects (see [8] for more with real phase space11. The present project addresses
details): another real phase space problem.
(i) a key concept of perfect factorization (including the The method of AO can be viewed as a systematic
requirement of purely power-and-log dependence on the reorganization of the conventional Bogolyubov-Para-
expansion parameter) with far-reaching consequences4 syuk-Sudakov-Hepp-Zimmermann-Lipatov-... tech-
for both theory and applications of OPE, etc. [13], [15];5 niques in such a way as to hide all the complexity of
(ii) putting Wilson's operator-product expansion on a splitting the integration domain into subregions by
firm foundation by obtaining it in a form valid for models providing a layer of abstraction based on the notion of
with zero-mass particles such as QED and QCD [13]; singular distribution. AO offers a finite set of rules to
generate a power-and-log asymptotic expansion of any
(iii) discovery [13] and simple mnemonic rules [14] for Feynman diagram for a given asymptotic regime, with
efficient formulae for calculations of Wilson's coeffici- the emphasis not just on splitting the integration domain
ents in the MS scheme; but also on appropriate modifications of the integrand,
(iv) an elegant general algorithmic scheme of derivation depending on the subregion (such modifications may
of factorization theorems [16], [17] (see below Sec. 3.2); take the form of secondary expansions [8], [23]).
(v) extension of the results for OPE (including efficient
calculational formulae) to arbitrary Euclidean asymptotic
regimes, including mass expansions [16], [17];

(vi) a rigorous regularization-independent distribution-
theoretic formalism which systematized the method of
AO and prepared ground for its extensions (see [10] and 6 Construction of a regularization-independent treatment was an im-
refs. therein). portant step e.g. in view of the failure of dimensional regularization in
The discovered formulae in combination with the so- some Minkowski-space situations ([21] and Sec. 5 below).
called integration-by-parts algorithms [18] formed a 7 I do not use the term Minkowski-space regimes because Minkowski
theoretical foundation for a flourishing large-scale space allows Euclidean (and quasi-Euclidean) regimes.
calculational industry, resulting in an array of NNLO 8 The time gap is explained by my involvement with the (then)
burning problem of finding the best jet algorithm [24], which
prevented me from doing a simple calculation in order to see that the
4 Only such expansions attain the ultimate goal of any expansion, paradox I was puzzled by in 1992 (different results from different
which is maximal simplification of calculations; only perfectly facto- scalings) does not exist (because the secondary expansion is done in
rized results allow a correct phenomenological interpretation and the sense of distributions [23]).
treatment of power corrections; uniqueness of such expansions result- 9
ing in an automatic inheritance of gauge identities as well as a drastic The work was torpedoed by anonymous referees similarly to [17].
simplification of the corresponding analyses (sums and products of The details are at http://www.inr.ac.ru/~ftkachov/projects/unstable.
such expansions again possess this property). 10 This is a technical foundation for the effects mentioned in ftn. 3.
5 Earlier versions of OPE -- however rigorously proved -- were 11 where I'd be more safe from plagiarism if not from anonymous
flawed in this respect. See [8] for a discussion. referees.





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AO effectively algebraizes the problem of obtaining and 2
m . Componentwise, this is described by
asymptotic expansions by offering a higher level of abs- 2 1/ 2
p - p = O( )
m , p = 0, p = O(s )
+ + + - . The quantities
traction over -- and thus hiding the complexities of -- pertaining to the asymptotic limit will carry tildes.
the splittings of integration region which old-fashioned The Regge (large-s ) regime corresponds to the case
techniques must tackle directly.

AO, therefore, stands in the same relation to old- when only s is large: 2
p - p = O( )
m , p = 0.
  
fashioned techniques as does integral calculus with
respect to ancient methods based on explicit summations. ... 2.1

Mathematically, the method of AO is rooted in the
theory of distributions (for a discussion see [11]). The It will be convenient to regard the large parameters as
distribution-theoretic viewpoint on singularities of per- O (1), and expand in the small parameters.
turbation theory was pioneered by Bogolyubov in his It is also convenient to work with cross sections
studies of UV divergences [12]. This idea (largely (p+,p-,m) normalized so as to be dimensionless.
ignored by the experts for a quarter of a century) was In perturbation theory, this is a sum of unitarity diagrams
extended to the problem of asymptotic expansion in the such as shown in Fig. 2.1. The first one is identically
theory of AO [9], with great practical benefits as zero in the Regge limit (it does however play a role at
discussed above. intermediate steps of the analysis).
But the distribution-theoretic power of AO is best My analysis will also involve the quantity
seen in the fact that for it, there is no fundamental diffe-
( p , p ) = ( p , p ,0) -- which is usually referred to
rence between, say, an ordinary singular function such as + - + -
-
x 1 and the corresponding singular distribution (x). as the hard parton cross section -- with the kinematical
parameters formally set to their asymptotic values. Such
A remarkable consequence of the distribution- a quantity is replete with collinear and IR singularities12
theoretic nature of AO is that it works equally well for which are manifest in the form of poles in 1
= (4 - D)
2
singular functions such as propagators (m 2 - p 2 - i0)-1 if dimensional regularization is used. Such singularities
and for the related distributions such as (m 2 - p 2 - i0). correspond to large logarithms in the asymptotic
This means that the prescriptions of AO [23] are behavior of (p+ ,p- ,m ). The method of AO establishes
applicable equally well to Feynman diagrams corres- a direct algorithmic connection between such singula-
ponding to amplitudes (virtual loops) and to unitarity rities and the large logarithms.
diagrams corresponding to matrix elements squared A correct expansion of (p+ ,p- ,m ) in the asymptotic
(phase space loops). regime (see below) yields








( p , p )
+ - with all the
The treatment of the two kinds of diagrams is com- singularities/poles appropriately subtracted via special
pletely uniform with AO, allowing an efficient analysis counterterms, so that only logs of s survive.
of real phase-space problems similar to the case of purely
virtual loop diagrams. One-loop example 2.2

Consider the simplest box. p-
The old-fashioned methods cannot be applied in such In the asymptotic limit, its
cases without first integrating out -functions, resulting singularities are generated by s k
in considerable complications due to distortions of the the two denominators and the
original fundamental product-of-propagators structure of two -functions, i.e. by the p+
diagrams, which seems to be at least partially respon- product
sible for the complexities of the large-s theory. 2
- 2
k k - ( 2
k + p+ ) ( 2
( ) (k - p )
- ). There is also a nume-
Setup 2 rator (a polynomial of k ). In the space of k  the
singularities can be shown as in the following picture.
Consider the total cross p-
section of the process shown
in the picture. We will be s
having in view standard
renormalizable gauge models p+
such as QED and QCD. Masses of the participating
fields (electrons or quarks) are collectively denoted as m .
The Bjorken (DIS) regime corresponds to the case
12 UV singularities play no role and present no complications, and I
when s and 2 2
Q - p- are both much larger than 2
p+ simply ignore them in the present discussion.





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Prescriptions of AO 3







p p






Prescriptions of AO consist of the following steps:
0 0  An accurate formulation of the asymptotic regime in
terms of kinematical parameters such as masses and
p components of external momenta (this we've done
 p






2.3 already, see Sec. 2).

The left and right figures correspond to the Bjorken and Formal (Taylor) expansion of the integrands in the
Regge limits, respectively. Each contains three light small parameter.
cones. Their intersections correspond to more complex  Enumeration, description, and analysis (power count-
singularities. The patterns of intersections are related but ing) of the singularities of the formal expansion, and con-
different: in the Regge case there are more overlaps. nection of the non-integrable ones to the diagrammatic
Non-integrable singularities are shown with fat lines: images ("singular subgraphs"). This amounts to con-





green and blue segments correspond to the collinear struction of the so-called operation R which eliminates
 
singularities related to p+ and p- , and described by non-integrable singularities via additive counterterms
2
that are local in the space of integration momenta. This I
k
= 0,
2.4 will call renormalization by analogy with the subtraction
 
k =  xp , x (0,1) .

2
(k
 p ) = 0
 of UV divergences. In this case it is usually sufficient to
determine a minimal form of the coefficients; if dimen-
The red dot on the right figure 2.3 corresponds to the soft sional regularization is applicable, it is sufficient to
singularity: require that the coefficients contain only pure poles in .
2
k = 0, 
R generalizes the regularization-independent prescrip-
 2
(k + p ) = 0, k = 0. tions of [10] which in turn are analytically similar to
+ 2.5
Bogolyubov's R-operation in coordinate representation.
 2
(k
- p ) = 0
-  Construction of the so-called bare asymptotic opera-






Diagrammatically, the factors contributing to the three tion A s which differs from R only by finite contributions
singularities are as follows: to counterterms. Such finite contributions accumulate the
non-analytical dependencies of the final answer (a.k.a.
large logarithms of the small parameter) and ensure that
the result is a correct asymptotic expansion of the origi-
   nal integral.
The green and blue lines correspond to configurations  A combinatorial rearrangement (usually called
 

with internal momentum collinear to p and
p
+ - , res- "factorization" although it may actually lead to exponen-
pectively. Red lines (the vertical lines in the third figure) tiation or additive structures) of the contributions from





are soft (zero momentum). Purely green and blue confi- different diagrams for both operations ( R and As ). This
gurations correspond to GLAPD-type contributions; the can be followed by a derivation of the corresponding
third configuration corresponds to a BFKL-type contribu- evolution equation.
tion. In the Bjorken limit, only the green configuration
contributes, giving rise to the standard result. In the AO establishes a direct correspondence between:
Regge regime, however, the leading logarithmic (L) con- -- singular manifolds in the space of integration
tribution comes from the third configuration, whereas the momenta;
collinear contributions are (a) next-to-leading logarithmic
-- subsets of singular factors contributing to the
(NL) level and (b) proportional to the first diagram in 2.1
corresponding singularities;
which is zero in the Regge limit. So the GLAPD-type
contributions actually emerge only at the NNL level. -- the corresponding diagrammatic images
Neighborhoods of the singularities generate large ("subgraphs");
logarithms in the corresponding limits. The technical -- the countereterms to be added to "bare" expressions.
problem is to extract those contributions without explicit Such counterterms are localized exactly at the singula-
evaluation of the original diagram. The Sudakov-Lipatov rities of the bare expressions, and are proportional to
techniques is one way. Asymptotic operation is another. (derivatives of) -functions.





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an expansion always follows the renormalization 3.3
The beauty of the technique is that the writing of - except that the renormalization factor/kernel now
functions is mechanical once the singular manifold is contains, in addition to poles, finite parts with non-trivial
parametrized: e.g. for 2.4 the counterterms are13 (large-logarithmic) dependencies on the small
( )
dx c (x)
(k  xp ), parameters:

 3.1

(s, m) = As (s, m) = K (m)  (s) . 3.4
where c is a coefficient to be determined. as

The theory of AO guarantees that (s, m)
 (s, m) to
The least trivial part of the problem reduces to deter- as
mining the coefficients of counterterms. The systematic the desired asymptotic precision, so that the quantity
prescriptions were summarized in ref. [23]. I only note (s,m) is what one seeks except for the fact that the
as
that an essential element here is the so-called homogeni- two bare quantities on the r.h.s. of 3.4 contain divergen-
zation, or secondary expansion, and that the formalism is ces and cannot be directly interpreted in a phenomenolo-
kind of fool-proof in the sense that the problem of choos- gical context.
ing a correct scaling to do the power counting for
complicated singularities does not arise (it is resolved (vi) To this end one constructs an inversion of the renor-
automatically if the prescriptions of AO are followed malization operator Z:
unwaveringly). Fortunately, sometimes important results  1
Z Z - 1. 3.5
can be obtained without writing out explicit expressions
for such coefficients. This important trick was introduced as a tool for deriving
renormalization group equations, asymptotic expansions
Standard Factorization Scenario 3.2 and studies of evolution equations in [28]. The inversion
The general all-logs, all-powers algorithmic scheme is possible as a formal power series because Z differs
of systematic derivation of factorization theorems was from unit operator only by higher-order corrections.
pioneered in the theory of AO for OPE and mass (vii) From 3.5 and 3.4 one obtains
expansions [16], [17], and is as follows.
 1
(s, m) K (m) Z -
=  Z  (s)
as
(i) One starts with a quantity to be expanded, say

( = K m s
s ( ) ( ) .
, m ), and formulates the asymptotic regime, say,   3.6
 2
s m (here s and m represent the scales of large and Alternatively, one can construct a renormalization for K
small parameters). and use its inversion. The result is the same up to nota-
tions.
(ii) One writes down the so-called "bare" expression
(s) -- which is (s,m) formally (Taylor-)expanded in (viii) The last step is to obtain evolution equations. To
the asymptotic limit. this end one notes that both quantities on the r.h.s. of 3.4
are renormalization group invariant. One simply applies
(iii) One examines (enumerates and classifies) singulari- the standard differential renormalization group operator
ties of the integrand of






(s) in the space of integration D = d d ln  = ln  + (g) g
to 3.3:
momenta.
  1
-
= =  
(iv) One constructs the corresponding minimal renorma- D (s) DZ (s) DZ Z Z (s)








lization procedure R . Represent this symbolically as =  (s) ,
 3.7

(s) = R(s) = Z  (s) .
 3.3 where the operator is given by


 -
R is applied to integrands, and the operator Z is a 1
= DZ Z . 3.8






"factorized" version of R obtained for the entire PT Note that, essentially by construction,
series. The exact form of Z depends on details of the






problem. It may be a multiplicative or additive operation, D (s, m) 0 .
as 3.9
an integral operator, or any mix of these. The subscript 
indicates that the usual arbitrariness is involved here, I called the above Standard Factorization Scenario
parameterized by , as usual. because: it is the simplest and cleanest way to derive
OPE in MS scheme; it worked fine in the discovery of
(v) One constructs the correct asymptotic expansion in mass expansions and, more generally, expansions for
the bare form. The point here is that the structure of such arbitrary Euclidean regimes (see [8] and refs. therein); it
also works fine for the Bjorken regime in DIS (as well as
the Drell-Yan process away from the threshold).
13 The default lower and upper integration limits are 0 and 1
throughout this text.





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Application to the Bjorken regime 4  For clarity's sake: objects like K(x,m) incorporate,
within PT, large logarithms of the small parameter,
A straightforward application of the above prescrip- ln(m/ ). Numerical values for such objects cannot be
tions gets one the following results. The key point is to obtained within pQCD and must therefore be treated as
write down the explicit form of renormalization 3.3. phenomenological input determined from experimental
In this regime (in the leading power approximation but data. This was discussed in [15] in connection with the
taking into account all logarithms) non-integrable (there- so-called vacuum condensates of local operators. Parton
fore requiring non-zero counterterms) are only collinear distributions K(x,m ) are another example.
singularities, which are all logarithmic, and one obtains  An interesting related problem is whether the small-x
-

1
( p ) = dx x Z (x) (
xp ) . behavior of K
 +  ( x, m ) can be reliably predicted from PT.
col + 4.1
Any equation obtained from PT and claimed to predict
such a behavior must be based on a strong hypothesis


Only the dependence on p+ is shown. The form of this ("behavior at small x obtained from PT persists outside
integral operator is predetermined by the structure of the PT even if K(x,m ) is uncalculable from PT at any non-
simplest collinear singularity 2.4. The factorization zero x " or something essentially equivalent) and must
which leads to Eq. 4.1 is depicted as follows: assume a necessarily non-trivial quark-and-gluon model
for hadrons.

On calculability of P(x ) 4.8
+ + The explicit expression for Z col has the following
structure:

+ + + 4.9
 ^
Z (x) = dx dx (1- x - x ) Z (x ,...x ) ,
col 1 l i col 1 l
4.2
where ^
Z (x , is directly connected with the colli-
 , x )
col 1 l
Comments related to internal loops within subgraphs are near singularities of diagrams contributing to the bare
given in Sec. 4.8. ("hard") cross section. The variables xi are the fractions
From 4.1, one obtains the evolution equation (cf. 3.7): of longitudinal momenta of unobserved final state
partons (these variables correspond to the cut phase
1 -
1
D ( p ) = dx x P( x) (xp ) ,
 + 4.3
2  +
Q / S space loops in 4.2). One has

^
where the lower limit results from the restriction Z (x , 4.10
 , x ) = - P R G ,

col 1 l col
G

2 2
(xp + p ) = xs - Q > 0 .
+ - 4.4 where summation runs over all contributing diagrams,






R is the specific variant
P(x ) is nothing but the GLAPD kernel, and its P is the pole part operator, col






expression in terms of Z is this (cf. 3.8): of R for collinear singularities, and R is R with-
col col

out the main counterterm. This formula is related to those
1 - -
1 1
P(x) = dz z Z (z /x) DZ ( z) .
encountered in the theory of UV renormalization; in the
col col 4.5
x latter case such formulae are written and used in integral
The kernel is inverted in the operator sense. sense; in our case Eq. 4.10 directly generalizes the for-
mulae for Euclidean infrared divergences in momentum
One immediately obtains the AO in bare form (cf. 3.4) space (without momentum integrations) given in [10].
1
1
dx x-
= K (x,m) (xs) , In regard of the outstanding problem of NNLO calcu-
as 4.6
xBjorken lation of parton evolution kernels,14 I can say the follow-
ing. The calculation of 4.10 is not so much difficult ana-
and its analog in terms of finite quantities (cf. 3.6):
lytically as it is cumbersome due to many parameters (3
1
1
dx x-
= K (x, m) (xs) . in the case of NNLO). In the NLO case, the quantities
as   4.7
xBjorken 4.10 are expressed in terms of simple integrals. In the
NNLO case, it is not difficult to obtain simple low-dim-
This agrees with the standard DIS factorization. K is ensional integrals. Then there remain the integrals in 4.9
directly seen to be the parton distribution at the scale  and the one in 4.5. In the NLO case these integrals yield
(trivial summations over parton flavors etc. are omitted). results15 that in my examples are clearly reminiscent of
Exercise. Obtain the standard (GLAPD) evolution
equation for K 14
 (x) from 4.7, 4.3 and 3.9. See [7] for a review of the state of the art.
15 I am indebted to A.Kotikov for a help with this.





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the available answers [26]. In the NNLO case, it is single additive contribution:
possible to arrive at answers in the form of low-dimen-







sional absolutely convergent integrals (all the poles are ( p , p ) = + ( p , p ) .
 + - col + - 5.3
extracted at the level of 4.10, so afterwards one only
deals with the finite coefficients of such poles) over The new additive term has no dependence on the external
standard compact regions. Such integrals seem to be momenta (this is not quite trivial but is the only piece of
easily amenable to numerical integration (I have not information from the analysis of singularities that will be
studied their analytical calculability). Although rather used below) and contains only poles in .
cumbersome, this scenario does not rely on matching One immediately writes down the AO in bare form as
conditions nor on inversion of Mellin transform (as in it must differ from 5.3 only by dependence of renormali-
a comparably cumbersome scenario of [20]). In short, zation kernels and on the small parameters (cf. 3.4):
this seems to be a realistic scenario for a direct calcu-

lation of parton evolution kernels. ( , ) = ( )
as p p m
+ -
dx dy

Application to the Regge regime 5 + K (x, m) K ( y, m) (xp , yp ).
+ - 5.4
x y

The construction of AO for this case has many paral- Recall that






is the regularized hard parton cross sec-
lels with the Bjorken case. I first show the analog of 4.2: tion with all IR and collinear singularities in place.

The form of 5.4 with two convolutions corresponding
to the two partons' momenta is meant to bring out the
similarity with the standard factorization results for DIS
+ + and Drell-Yan processes. From now on I will be using

the fact that






are scalar functions of s =
2 p p
+ - only.
+ + + The next step is to reexpress






in 5.4 in terms of the
finite quantity 5.3. First use inversion of the collinear
+ renormalization similarly to the Bjorken case:
+
 dz

(s) = (m) + Q (z) (zs),
as  col 5.5
z
+ + 1 dx
where Q ( z) K (x, m) K (z /x, m) .
  
z x

+ + + + Q(z ) is a well-defined function with no stronger than
5.1 logarithmic singularity near z = 0 within PT.
(In the case of the standard Drell-Yan process, there
There is a full complement of terms corresponding to are no soft singularities, = 0 and Eq. 5.5 is equivalent to

configurations collinear with respect to p+ (purely the standard factorization theorem; see e.g. [5]. Non-zero
green; they are the same as in 4.2) as well as a similar set lower cutoffs for integrals in 5.5 then follow from kine-


of terms corresponding to p- (purely blue; they mirror matical restrictions in






.)
col






the green ones). Their contribution to R can be imme-
diately written down in the factorized form as follows The importance of Eq. 5.5 is that it follows from an
accurate analysis of ordinary PT diagrams in the same

( p , p ) R ( p , p )
col + - col + - manner as all the standard factorization theorems.
dx dy
=


Z (x) Z ( y) (xp , yp ) .
In this respect, Eq. 5.5 is unambiguous.
col col + - 5.2
x y

To complete the transformation, one is tempted to add
There are also qualitatively new terms (the last row; and subtract from






-- in order to make it finite
red lines correspond to soft [zero-momentum] singula- col
rities). These are a major headache to accurately analyze (recall 5.3) -- the quantity  and rewrite the result as
and compute.16 Fortunately, the cumulative contribution follows:
of such terms after momentum integrations reduce to a ? dz

(s) = (m) + Q (z) (xs) ,
as   r 5.6
z
16 The subtleties of mixed soft-collinear singularities could be a source where
of grave mistakes if not handled with care.





7


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? Bjorken regime 4.7. A restriction observed within PT is
dz

( )
m = ( )
m -  Q (z) .
 
5.7 = =
z that the singularity at x y 0 is logarithmic, so that

If these two formulae made sense then the result would -
1
K (x) < x , < . 5.8
have been extremely attractive:17 in order to describe the 
~ 2

Regge limit within pQCD one would only have to intro-
duce a single new phenomenological constant (m); This is the weakest simple hypothesis that could (per-
given such a constant and the parton distributions, it haps ought to, unless its validity is explicitly disproved)

would have been sufficient to evolve
be made about parton distributions based on PT.
 in 5.6.18  The behavior of 5.5 near lower integration limit
Alas, there is no reason for the square-bracketed implicates the non-perturbative parton distributions K.
integral to be convergent at z = 0; in fact, within PT it is This is a novel feature compared with the conventional
logarithmically divergent. So the two expressions 5.6 and situations (Euclidean problems, the Bjorken regime, etc.)
5.7 are ill-defined within PT. where subtractions never require knowledge of non-
A naive attempt to obtain an evolution equation for perturbative dependencies.
 with respect to  similarly to 5.6 also fails: one
Therefore, an important decision one has to make at
runs into divergent expressions of the form 5.7 with Z col this point is whether or not one is allowed to make hypo-
in place of K, and Z (x) ~ O(1) near zero in one loop
col theses about small-x behavior of parton distributions.
with powers of ln x in higher orders.
In fact, it is apparent how to make such hypotheses in
At this point we've run into a major problem which a satisfactory fashion given that the -dependence of
is the source of all subsequent complications in the (perturbative) parton distributions is essential for a cor-
study of the large-s regime. rect cancellation of poles in 5.5.
The value of the above formalism is that it offers If one chooses not to make such hypotheses then there
a rather simple explicit formal framework to discuss is essentially no choice but to introduce a cut at (low) z
(if not solve) the problem. in order to split the integration region, and then perform
Let us summarize the relevant properties of 5.5: the described identical transformation in the subregion
 which is not adjacent to zero in order to render






There is no natural kinematic cut as in the Bjorken col
and Drell-Yan cases, so the integrations over x and y finite. Obtain:
extend all the way down to zero.
  1 dz 
The quantity



(s) is the hard parton cross section = +
col (s) , (s) Q (z) (zs),
as c   5.9
c z
with (only) collinear singularities removed by the colli-
near renormalization of 5.2, and since the tree level con- where
tribution is zero, the singularity at z = 0 in 5.5 is dimensi- (s) = lnn s
 c  
c n 5.10
n ( 2 .
, , , )
onally regulated just fine. However, any constant term

along with






causes the integral over z to diverge.19
col Eq. 5.10 is obtained within PT after a complete expan-
 K are perfectly well-defined finite parton distribu- sion in of -- and canceling the poles (which must
tions -- exactly the same objects already occurred in the cancel by construction) in -- the exact expression

17 c 1
It was reported in [1]. The analysis of [1] was correct up to Eq. 5.5. dz dz
( )
m + Q ( z) ( zs) -  Q ( z) .
5.11
18  
After I had reported it in [1], B. Ermolaev notified me of the posting col
z z
0 c
[27] which apparently builds upon an earlier work of the same
authors. I am not prepared to comment on [27] due to my unfamili- The peculiar features of the factorization result 5.9
arity with their formalism which it is a matter of taste to characterize 5.10 are these:
as either more or less cumbersome than mine (it is certainly indirect,
inlike mine) -- it is just cumbersome enough that I cannot afford to (1) a mixed integral-multiplicative-additive form of the
study it in detail immediately or even soon enough. Ref. [27] also factorization;
claims that a large-s (small-x in the notations of [27]) pQCD predic-
tion involving only one (if I am not mistaken) phenomenological con- (2) an infinite sum in 5.10 which is a function of s ;
stant. In view of my findings (see e.g. Eq. 5.10) I am not sure whether (3) the cutoff c at the lower integration limit.
it is possible to obtain such a result from pQCD without some hidden
hypothesis (or an oversight, as was the case with [1]). I can only ex- All this introduces substantial complications in the
press my regret that B. Ermolaev did not offer any comments at the study of evolution equations.
time of [1].

19 A similar failure of dimensional regularization was found in [21].





8


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Discussion 5.12 On the subject of evolution equations 5.13

(i) The derivation of 5.95.10 (including the transition There seems to be too many options in regard of deri-
from 5.11 to 5.10) rests solely on an accurate study of vation of evolution equations, so my purpose here is not
ordinary PT diagrams. No physical arguments were used to provide a specific algorithmic procedure but to demon-
(whatever the adjective physical might mean). The non- strate the most obvious options.
PT interpretations are made solely after all the expan- As discussed above, keeping  and c independent
sions, cancellations, etc., leading to 5.95.10 are means one should study evolution with respect to both
completed. parameters. Evolution in c is useful if one chooses
(ii) All finite quantities depend on  as indicated; the c  / s .
expressions are sums of integer powers of ln  , as is First of all, one could write down two formal evolu-
usual with minimal subtractions. In the case of and Q , tion equations from the fact that the l.h.s. of 5.9 is inde-
the argument of the logarithm actually contains the ratio pendent of  and c. The differentiation in c yields a
 /m within PT. simple result:

(iii) The role of the cutoff c is, in the final respect,
similar to that of  -- and one may even be tempted to d
 =
, (s) Q (c) (cs).
c   5.14
choose c = / s , perhaps with a simple numerical coef- d ln c
ficient. I prefer to keep  and c separate to emphasize The differentiation in  yields
the fact that whereas  corresponds to ordinary PT sub-
tractions, the subtraction associated with c involves a D (s) = - (s; K ) ,
,c ,c  5.15
non-perturbative function and is performed in a techni-
cally different manner. 1 dz
where s K = D Q z zs

. 5.16
(iv) The latter fact implies that along with the evolution , ( ; ) ( ) ( )
c   
c z
in  , one should study evolution in c . I will not discuss
the technical problem of how to combine both in a The latter is a functional of the parton distributions
convenient numerical procedure. which enter via Q. The functional is quadratic in the
parton distributions and does not involve their deriva-
(v) To obtain cancellation of poles within PT, one tives. It is finite and formally obtainable from PT because
should take into account the -dependence of Q(z ). As the derivative in the integrand is easily calculable: the
a result, the coefficients ,c, n receive contributions from evolution of parton distributions is known (Sec. 4),
O (k) corrections to perturbative Q(z ), in addition to 
some weighted integrals of Q whereas D (s) is found similarly to 5.9:
 ( z ) at = 0 . So the objects 
,c, n have to be regarded as new non-perturbative
1 dx
constants in phenomenological interpretations. The latter D (s) = 2 P(x) (xs)
  + , (s) ,
c 5.17
fact means (if one follows the usual logic) that the c x
expression (s)
 given by 5.10 is a non-perturbative n 2
,c where (s) = ln s  .
,c c, 5.18
1 n
n ( )
function to be extracted from experimental data.

The GLAPD evolution kernels P(x ) are given by 4.5 and
Since there is an infinite number of diagrams in the all c,n calculable within PT as power series in the cou-
PT series one starts with, there is no a priori principle to pling (normalized at ) with numeric coefficients.
prevent appearance of an infinite number of independent Eq. 5.18 is obtained by expansion in of, and canceling
objects non-controllable by PT -- and this seems to be poles in, the following expression which is similar to
exactly the situation with the large-s regime. 5.11:

However, the factorization theorem for the Bjorken 1 c
  dx dx
= - + 
regime also involves unknown functions (parton distribu- s D P x P x xs
, ( ) 2 ( ) 2 ( ) ( ) .
c col 5.19
x x
tions) yet it proves possible to obtain meaningful results c 0
from it. So it would be premature to draw pessimistic Contributions to c,n for larger n start from higher
conclusions from the above (although the picture is not orders of PT but this may not be enough to offset the
rosy either). For instance, it may be important for an double-logarithmic nature of the series (log squared per
efficient phenomenological use of the above factorization each power of the coupling). So there is a potential pro-
to study the (in)dependence of the coefficients , blem with convergence here (or a missing exponentiation
c, n on
the process. of the Sudakov type).

Assuming the representation 5.10, we can in principle
extract coefficients of ln n (s /2) and obtain evolution




9


F.V. Tkachov: On the large-s behavior ... 2001-01-08 06:29 Page 10 of 12





equations for the coefficients ,c, n : 5.10) dominate asymptotically, but this requires further
analysis.
d
Remember, however, that all
 = , c, n are in principle
,c, Q (c) (c) ,
n  n 5.20
d ln c independent constants -- independent between them-
selves and independent from the parton distributions (at
D = (n +1) - (K ) .
,c,n ,c,n 1
+ c,n  5.21 least within PT).

Just how useful all the above equations could be, is a
(
n z ) is a well-defined power series in the coupling question which goes beyond the scope of this text.
which is calculable to any perturbative order. (
c, n K ) is
a functional of the parton distribution whose form and Conclusions 6
coefficients can be found from PT.
One could in principle fix  (say, at 10 GeV), take A systematic diagram-by-diagram all-logarithms ana-
externally provided parton distributions and the constants lysis based on the method of asymptotic operation [23],
[8] allows one to reproduce the standard factorization
, c, n , both normalized at the chosen value of . Then it
theorems and evolution equations for the Bjorken asymp-
would be sufficient to evolve






 with respect to s . totic regime in DIS and for the Drell-Yan process. The
Since






 is a function of the ratio s /2, the evolution same method was earlier used to find powerful calcula-
tional formulae for OPE and mass expansions [13], [16],
equation in  (Eq. 5.17) is in principle (I am not saying [17], now in a continuous large-scale use (cf. [20]).
"in practice") sufficient.
For the large-s behavior of total cross sections, the
Let us approach the problem from a different direc- same systematic method yields a factorization theorem
tion. Normalize couplings and parton distributions at (Eq. 5.9) in gauge theories such as QCD and QED. The
 = s so that Eqs.5.95.10 become theorem directly extends the standard results for the
1 Bjorken regime in DIS and for the Drell-Yan processes.
dz

(s) = + Q ( z) (zs).
as 5.22 The obtained factorization theorem involves an infini-
s ,c,0 s s
z
c te number of new independent phenomenological coef-

Recall that the perturbative






(s)
 is a sum of ficients in addition to parton distributions (equivalently,
a function which is uncalculable within pQCD). This
k 2
ln (s / ) with coefficients power series in , so
S , seems to give a literal interpretation to the characteriza-
tion20 of the Regge regime as infinitely more complicated
that






(zs) is a sum of lnk z with coefficients power
s than the Bjorken one. I cannot say at this point whether
series in or not this infinite arbitrariness can be reduced in a
S , = ; the coefficients of the latter power
s
meaningful fashion (see comments after 5.24).
series are just numbers. Then the only quantity that needs
to be evolved is . If a closed evolution equation Whereas I am rather confident that the obtained facto-
s ,c,0 rization theorem 5.9 is correct and that no hidden hypo-
for this quantity were available, one would be able to thesss was used in its derivation from ordinary PT dia-
predict the asymptotic cross section 5.22 with just one grams, the situation with the corresponding evolution
new phenomenological number. However, Eq. 5.21 equations is less clear. I have explored some straight-
couples ,c, n with different n. forward options for deriving evolution equations, and the
The form of 5.21 indicates that a diagonalization may conclusions are not encouraging so far: the resulting
be possible. To this end assume the following Ansatz: equations (5.17, 5.21 and 5.20) appear to be rather more
involved than the usual renormalization group equations
(s) = lnn s s .
,c 5.23
n ,c,n ( 0) or the GLAPD equations, and it is not clear how to make
Then Eq. 5.21 becomes an efficient and systematic practical use of them.
I am not claiming that the presented formulae consti-

D
= -
(K ) . tute a complete formalism to study evolution at large s
,c,n ,c,n  5.24
-- only the surface has been scratched. There are clearly
many options, and determining an optimal way to evolve
The functional on the r.h.s. (which is formally calculable cross sections to large s is a subject for a separate
from PT as a coefficient of lnn (s s ) in 5.16) contains
0 project.
powers of logarithm of the ratio s /2 On the up side, the analysis of the Bjorken regime
0 . Unfortunately,
this diagonalized form cannot be used with 5.22. uncovers an algorithmic (although necessarily cumber-
 some) scenario for direct (i.e. not relying on calculations
It is not inconceivable that the dynamics of 5.24 (or
5.21) is such that only terms with lowest n in 5.23 (or 20 which I heard in exactly this form from A. White.





10


F.V. Tkachov: On the large-s behavior ... 2001-01-08 06:29 Page 11 of 12





of moments of structure functions) and systematic (i.e. radiative corrections. Theory of asymptotic
suitable for next-next-to-leading order) calculations of operation ,  [QFTHEP'92;
parton evolution kernels. Phys. Part. Nucl. 25 (1994) 649].
Anyhow, the presented formalism offers what seems [9] F.V. Tkachov: Factorization via R-operation: the
to be an explicit formal framework to discuss the large-s Extension Principle [in: Quarks-82. Proc. Int.
Seminar. Sukhumi, May 5-7, 1982. INR, Moscow,
behavior of total cross sections, and some options are 1982].
likely to have not been explored yet. [10] A.N. Kuznetsov, F.V. Tkachov, and V.V. Vlasov:
Lastly, it is not necessary to enter into details of deri- Techniques of distributions in applied quantum
vation of the theorems 5.5 and 5.9 in order to play field theory. (I) Euclidean As-operation for
theoretical games with their implications, for which a products of singular functions, ; (II)
quite rich field is available (see Sec. 5.13). Applications to theory of Feynman diagrams, hep-
.
Acknowledgments [11] F.V. Tkachov: Distribution-theoretic methods in
A.Butkevich [2] has been pressing me to help with perturbative quantum field theory, 
clarifying the subject of large-s behavior of total cross [talk at The Bogolyubov Conf. on Problems of Theor.
sections for over two years. M. Kienzle-Focacci and Math. Physics, Moscow-Dubna-Kyiv, 27 Sept.  6
W. Kienzle kindly provided information about the Oct. 1999, JINR].
TOTEM experiment [3]. Many conversations with [12] N.N. Bogoliubov: Variational equations of quantum
L. Lipatov were useful and encouraging, and the inspira- field theory, Doklady USSR Acad. Sci. 82 (1952)
217;
tional invitation to give a talk at [1] prompted me to sit N.N. Bogoliubov and D.V. Shirkov: Introduction to
down to formulae. I thank K. Christos, B. Ermolaev, the Theory of Quantized Fields. Interscience, 1959.
V. Fadin, M. Grazzini, K. Kato, A. Kotikov, E. Kuraev, [13] F.V. Tkachov: On the operator product expansion in
A.Vasiljev and A.White for their interest and comments. the MS scheme [Phys. Lett. 124B (1983) 212].
This work was supported in part by the Russian Foun- [14] S.G. Gorishny, S.A. Larin and F.V. Tkachov: The
dation for Basic Research (quantum field theory section) algorithm for OPE coefficient functions in the MS
under grant 99-02-18365. scheme [Phys. Lett. 124B (1983) 217].
[15] F.V. Tkachov: The limit m
q 0 of perturbative
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As-operation, 
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[8] F.V. Tkachov: Advanced methods for studying





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3
S corrections to the Bjorken sum rule

[Phys. Rev. Lett. 66 (1991) 862];
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[Theor. Math. Phys. 77 (1988) 51].





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