Infrared Evolution Equations: Method and Applications
Infrared Evolution Equations: Method and Applications
B.I. Ermolaev
Ioffe Physico-Technical Institute, 194021 St.Petersburg, Russia
M. Greco
Department of Physics and INFN, University Rome III, Rome, Italy
S.I. Troyan
St.Petersburg Institute of Nuclear Physics, 188300 Gatchina, Russia
It is a brief review on composing and solving Infrared Evolution Equations. They can be used in
order to calculate amplitudes of high-energy reactions in different kinematic regions in the double-
logarithmic approximation.
PACS numbers: 12.38.Cy
I. INTRODUCTION
Double-logarithmic (DL) contributions are of a special interest among radiative corrections. They are interesting
in two aspects: first, in every fixed order of the perturbation theories they are the largest terms among the radiative
corrections depending on the total energy and second, they are easiest kind of the corrections to sum up. DL corrections
were discovered by V.V. Sudakov in Ref. [1] in the QED context. He showed that DL terms appear from integrations
over soft, infrared (IR) -divergent momenta of virtual photons. All-order resummation of such contributions led to
their exponentiations.
Next important step was done in Refs. [2] where calculation and summation of DL contributions was considered
in a systematic way. They found a complementary source of DL terms: soft virtual fermions. This situation appears
in the Regge kinematics. The all-order resummations of DL contributions in the Regge kinematic are quite involved
and yield more complicated expressions than the Sudakov exponentials. Nonetheless important was the proof of the
factorization of bremsstrahlung photons with small k⊥ in the high-energy hadronic reactions found in Ref. [3] and
often addressed as the Gribov’s bremsstrahlung theorem. This statement, suggested originally in the framework of
the phenomenological QED of hadrons was extended to QCD in Refs. [4].
Calculation in the double-logarithmic approximation (DLA) amplitudes of the fermion-antifermion annihilation in
the Regge forward and backward kinematics involves accounting for DL contributions from soft quarks and soft gluons.
These reactions in QED and QCD have many common features. The e+e− -annihilation was studied in Refs. [2].
The quark-aniquark annihilation DLA was investigated in Ref. [5]. The method of calculation here was based on
factorization of virtual quarks and gluons with minimal k⊥. Generally speaking, the results obtained in Ref. [5] could
be obtained with the method of Ref. [2], however the technique of calculations suggested in Ref. [5] was much more
elegant and efficient. Although Ref. [5] is about quark scattering only, it contains almost all technical ingredients
necessary to compose Infrared Evolution Equations for any of elastic scattering amplitudes. Nevertheless it could not
directly be applied to inelastic processes involving emission of soft particles. Such a generalization was obtained in
Refs. [4, 6]. The basic idea of the above-mentioned method was suggested by L.N. Lipatov: to investigate evolution
with respect to the infrared cut-off. The present, sounding naturally term ”Infrared Evolution Equations” (IREE) for
this method was suggested by M. Krawczyk in Ref. [7] where amplitudes for the backward Compton scattering were
calculated in DLA.
The aim of the present brief review is to show how to compose and solve IREE for scattering amplitudes in different
field theories and kinematic regions. The paper is organized as follows: in Sect. II we consider composing IREE in the
technically simplest hard kinematics. In Sect. III we consider composing IREE in the forward kinematics and apply
it to studying the structure function g1 of the polarized Deep-Inelastic scattering (DIS) at small x. The point is that
the commonly used theoretical instrument to study g1 is DGLAP [11]. It collects logarithms of Q
2 to all orders in
αs but does not include the total resummation of logarithms of 1/x, though it is important at small x. Accounting
for such a resummaton leads to the steep rise of g1 at the small-x region. As is shown in Sect. IV, DGLAP lacks the
resummaion but mimics it inexplicitly, through the special choice of fits for the initial parton densities. Invoking such
peculiar fits together with DGLAP to describe g1 at x ≪ 1 led to various misconceptions in the literature. They are
enlisted and corrected in Sect. V. The total resummaion of the leading logarithms is essential in the region of small x.
In the opposite region of large x, DGLAP is quite efficient. It is attractive to combine the resummation with DGLAP.
http://arxiv.org/abs/0704.0341v1
The manual for doing it is given in Sect. VI. Finally, Sect. VII is for concluding remarks.
II. IREE FOR SCATTERING AMPLITUDES IN THE HARD KINEMATICS
From the technical point of view, the hard kinematics, where all invariants are of the same order, is the easiest for
analysis. For the simplest, 2 → 2 -processes, the hard kinematics means that the Mandelstamm variables s, t, u obey
s ∼ −t ∼ −u . (1)
In other words, the cmf scattering angles θ ∼ 1 in the hard kinematics. This kinematics is the easiest because the
ladder Feynman graphs do not yield DL contributions here and usually the total resummation of DL contributions
leads to multiplying the Born amplitude by exponentials decreasing with the total energy. Let us begin with composing
and solving an IREE for the well-known object: electromagnetic vertex Γµ of an elementary fermion (lepton or quark).
As is known,
Γµ = ū(p2)
γµf(q
2)− σµνqν
g(q2)
u(p1) (2)
where p1,2 are the initial and final momenta of the fermion, m stands for the fermion mass and the transfer momentum
q = p2−p1. Scalar functions f and g in Eq. (2) are called form factors. Historically, DL contributions were discovered
by V. Sudakov when he studied the QED radiative corrections to the form factor f at |q2| ≫ |p21,2|. Following him,
let us consider vertex Vµ at
|q2| ≫ p21 = p22 = m2 (3)
i.e. we assume the fermion to be on–shell and account for DL electromagnetic contributions. We will drop m for the
sake of simplicity.
A. IREE for the form factor f(q2) in QED
Step 1 is to introduce the infrared cut-off µ in the transverse (with respect to the plane formed by momenta p1,2)
momentum space for all virtual momenta ki:
ki ⊥ > µ (4)
where i = 1, 2, ...
Step 2 is to look for the softest virtual particle among soft external and virtual particles. The only option we have
is the softest virtual photon. Let denote its transverse momenta ≡ k⊥. By definition,
k⊥ = min ki ⊥ . (5)
Step 3: According to the Gribov theorem, the propagator of the softest photon can be factorized (i.e. it is attached
to the external lines in all possible ways) whereas k⊥ acts as a new cut-off for other integrations. Adding the Born
contribution fBorn = 1 we arrive at the IREE for f in the diagrammatic form. It is depicted in Fig. 1. IREE in the
analytic form are written in the gauge-invariant way, but their diagrammatical writing depends on the gauge. In the
present paper we use the Feynman gauge.
Applying to it the standard Feynman rules, we write it in the analytic form:
f(q2, µ2) = fBorn − e
dαdβdk2
− µ2) f(q2, k2
(sαβ − k2
+ ıǫ)(−sα+ sαβ − k2
+ ıǫ)(sβ + sαβ − k2
+ ıǫ)
where we have used the Sudakov parametrization k = αp2 + βp1 + k⊥ and denoted s = −q2 ≈ 2p1p2. As f(q2, k2⊥)
does not depend on α and β, the DL integration over them can be done with the standard way, so we are left with a
simple integral equation to solve:
f(q2, µ2) = fBorn − e
ln(s/k2
)f(q2, k2
) . (7)
FIG. 1: The IREE for the Sudakov form factor. The letters in the blobs stand for IR cut-off.
Differentiation of Eq. (7) over µ2 (more exactly, applying −µ2∂/∂µ2) reduces it to a differential equation
∂f/∂(ln(s/µ2)) = −(e2/8π2) ln(s/µ2)f (8)
with the obvious solution
f = fBorn exp[−(α/4π) ln2(q2/m2)] (9)
where we have replaced µ by m and used α = e2/4π. Eq. (9) is the famous Sudakov exponential obtained in Ref. [1].
B. IREE for the form factor g(q2) in QED
Repeating the same steps (see Ref. [8] for detail) leads to a similar IREE for the form factor g:
g(q2,m2, µ2) = gBorn(s,m2)− e
ln(s/k2
)g(q2,m2, k2
) (10)
where gBorn(s,m2) = −(m2/s)(α/π) ln(s/m2). Solving this equation and putting µ = m in the answer leads to the
following relation between form factors f and g:
g(s) = −2
, (11)
with ρ = s/m2. Combining Eqs. (9,11) allows to write a simple expression for the DL asymptotics of the vertex Γµ:
Γµ = ū(p2)
σµνqν
u(p1) exp[−(α/4π) ln2 ρ] . (12)
C. e+e− -annihilation into a quark-antiquark pair
Let us consider the e+e− -annihilation into a quark q(p1) and q̄(p2) at high energy when 2p1p2 ≫ p21,2. We consider
the channel where the e+e− -pair annihilates into one heavy photon which decays into the q(p1) q̄(p2) -pair:
e+e− → γ∗ → q(p1) q̄(p2) . (13)
We call this process elastic. In this case the most sizable radiative corrections arise from the graphs where the quark
and antiquark exchange with gluons and these graphs look absolutely similar to the graphs for the electromagnetic
vertex Γµ considered in the previous subsection. As a result, accounting for the QCD radiative corrections in DLA
to the elastic form factors fq, gq of quarks can be obtained directly from Eqs. (9,11) by replacement
α → αsCF , (14)
with CF = (N
2 − 1)/2N = 4/3.
D. e+e− -annihilation into a quark-antiquark pair and gluons
In addition to the elastic annihilation (13), the final state can include gluons:
e+e− → γ∗ → q(p1) q̄(p2) + g(k1), ..g(kn) . (15)
We call this process the inelastic annihilation. The QED radiative corrections to the inelastic annihilation (15) in
DLA are absolutely the same as the corrections to the elastic annihilation. On the contrary, the QCD corrections
account for gluon exchanges between all final particles. This makes composing the IREE for the inelastic annihilation
be more involved (see Ref. [4]). The difference to the considered elastic case appears at Step 2: look for the softest
virtual particle among soft external and virtual particles. Indeed, now the softest particle can be both a virtual gluon
and an emitted gluon. For the sake of simplicity let us discuss the 3-particle final state, i.e. the process
e+e− → γ∗ → q(p1) q̄(p2) + g(k1) . (16)
The main ingredient of the scattering amplitude of this process is the new electromagnetic vertex Γ
µ of the quark.
In DLA, it is parameterized by new form factors F (1) and G(1)
Γµ = B1(k1)ū(p2)
(1)(q, k1)−
σµνqν
G(1)(q, k1)
u(p1) (17)
where (1) corresponds to the number of emitted gluons, q = p1 + p2 and l is the polarization vector of the emitted
gluon. The bremsstrahlung factor B1 in Eq. (17) at high energies is expressed through k1 ⊥:
( p2l
− p1l
. (18)
We call F (n), G(n) inelastic form factors. Let us start composing the IREE for F (1). Step 1 is the same like in
the previous case. Step 2 opens more options. Let us first choose the softest gluon among virtual gluons and denote
its transverse momentum k⊥ The integration over k⊥ runs from µ to s. As µ < k1 ⊥ < s, we have two regions to
consider: Region D1 were
µ < k1⊥ < k⊥ <
s (19)
and Region D2 were
µ < k⊥ < k1⊥ <
s (20)
Obviously, the softest particle in Region D1 is the emitted gluon, so it can be factorized as depicted in graphs (b,b’)
of Fig. 2.
On the contrary, the virtual gluon is the softest in Region D2 were its propagator is factorized as shown in graphs
(c,d,d’) of Fig. 2. Adding the Born contribution (graphs (a,a’) in Fig. 2) completes the IREE for F (1) depicted in
Fig. 2. Graphs (a-b’) do not depend on µ and vanish when differentiated with respect to µ. Blobs in graphs (c-d’) do
not depend on the longitudinal Sudakov variables, so integrations over α, β can be done like in the first loop. After
that the differential IREE for F (1) is
∂F (1)
CF ln
(2p2k1
(2p1k1
F (1) . (21)
Solving Eq. (21) and using that (2p1k1)(2p2k1) = sk
1⊥ leads to the expression
F (1) = exp
CF ln
(k21⊥
suggested in Ref. [9] and proved in Ref. [4] for any n. The IREE for the form factor G(n) was obtained and solved in
Ref. [8]. It was shown that
G(n) = −2∂F (n)/∂ρ . (23)
FIG. 2: The IREE for the inelastic quark form factor.
E. Exponentiation of Sudakov electroweak double-logarithmic contributions
The IREE -method was applied in Ref. [10] to prove exponentiation of DL correction to the electroweak (EW)
reactions in the hard kinematics. There is an essential technical difference between the theories with the exact gauge
symmetry (QED and QCD) and the EW interactions theory with the broken SU(2) ⊗ U(1) gauge symmetry: only
DL contributions from virtual photons yield IR singularities needed to be regulated with the cut-off µ whereas DL
contributions involving W and Z -bosons are IR stable because the boson masses MW and MZ act as IR regulators.
In Ref. [10] the difference between MW and MZ was neglected and the parameter
M & MW ≈ MZ (24)
was introduced, in addition to µ, as the second IR cut-off. It allowed to drop masses MW,Z . The IREE with two IR
cut-offs was composed quite similarly to Eq. (6), with factorizing one by one the softest virtual photon, Z-boson and
W -boson. As a result the EW Sudakov form factor FEW is
FEW = exp
− α(Q
ln2(s/µ2)−
SU(2)
(Y 21 + Y
− α(Q
ln2(s/M2)
where Q1,2 are the electric charges of the initial and final fermion (with W -exchanges accounted, they may be
different), Y1,2 are their hyper-charges and C
SU(2)
F = (N
2 − 1)/2N , with N = 2. We have used in Eq. (25) the
standard notations g and g′ for the SU(2) and U(1) -EW couplings. The structure of the exponent in Eq. (25) is quite
clear: the first, µ -dependent term comes from the factorization of soft photons like the exponent in Eq. (9) while
other terms correspond to the W and Z -factorization; the factor in the squared brackets is the sum of the SU(2)
and U(1) Casimirs, with the photon Casimir being subtracted to avoid the double counting. In the limit µ = M the
group factor in the exponent is just the Casimir of SU(2)⊗ U(1).
III. APPLICATION OF IREE TO THE POLARIZED DEEP-INELASTIC SCATTERING
Cross-sections of the polarized DIS are described by the structure functions g1,2. They appear from the standard
parametrization of the spin-dependent part Wµν of the hadronic tensor:
Wµν = ıǫµνλρqλ
Sρg1(x,Q
Sρ − pρ
g2(x,Q
where p, m and S are the momentum, mass and spin of the incoming hadron; q is the virtual photon momentum;
Q2 = −q2; x = Q2/2pq. Obviously, Q2 > 0 and 0 6 x 6 1.
Unfortunately, g1,2 cannot be calculated in a straightforward model-independent way because it would involve QCD
at long distances. To avoid this problem, Wµν is regarded as a convolution of Φq,g - probabilities to find a polarized
quark or gluon and the partonic tensors W̃
(q,g)
µν parameterized identically to Eq. (26). In this approach W̃
(q,g)
µν involve
only QCD at short distances, i.e. the Perturbative QCD while long-distance effects are accumulated in Φq,g. As Φq,g
are unknown, they are mimicked by the initial quark and gluon densities δq, δg. They are fixed aposteriori from
phenomenological considerations. So, the standard description of DIS is:
Wµν ≈ W (q)µν ⊗ δq +W (g)µν ⊗ δg . (27)
The standard theoretical instrument to calculate g1 is DGLAP[11] complemented with standard fits[12] for δq, δg.
We call it Standard Approach (SA). In this approach
g1(x,Q
2) = Cq(x/z)⊗∆q(z,Q2) + Cg(x/z)⊗∆g(z,Q2) (28)
where Cq, g are coefficient functions and ∆q(z,Q2), ∆g(z,Q2) are called the evolved (with respect to Q2)quark and
gluon distributions. They are found as solutions to DGLAP evolution equations
d lnQ2
Pqq∆q + Pqg∆g
d lnQ2
Pgq∆q + Pgg∆g
where Pab are the splitting functions. The Mellin transforms γab of Pab are called the DGLAP anomalous dimensions.
They are known in the leading order (LO) where they are ∼ αs and in the next-to-leading order (NLO), i.e. ∼ α2s.
Similarly, Cq,g are known in LO and NLO. Details on this topic can be found in the literature (e.g. see a review
[13]). Structure function g1 has the flavor singlet and non-singlet components, g
1 and g
1 . Expressions for g
1 are
simpler, so we will use mostly them in the present paper when possible. It is convenient to write g1 in the form of the
Mellin integral. In particular,
gNS DGLAP1 (x,Q
2) = (e2q/2)
CNS(ω)δq(ω) exp
γNS(ω, αs(k
where µ2 is the starting point of the Q2 -evolution; CNS and γNS are the non-singlet coefficient function and anomalous
dimension. In LO
γNS(ω,Q
ω(1 + ω)
+ S2(ω)
, (31)
CLONS(ω) = 1 +
2ω + 1
ω(1 + ω)
S1(ω) + S
1(ω)− S2(ω)
FIG. 3: The IREE for the non-singlet component of the spin structure function g1.
with Sr(ω) =
j=1 1/j
r . The initial quark and gluon densities in Eq. (30) are defined through fitting experimental
data. For example, the fit for δq taken from the first paper in Ref. [12] is
δq(x) = Nx−α
(1− x)β(1 + γxδ)
, (32)
with N being the normalization, α = 0.576, β = 2.67, γ = 34.36 and δ = 0.75.
DGLAP equations were suggested for describing DIS in the region
x . 1, Q2 ≫ µ2 (33)
(µ stands for a mass scale, µ ≫ ΛQCD) and there is absolutely no theoretical grounds to apply them in the small-x
region, however being complemented with the standard fits they are commonly used at small x. It is known that SA
provide a good agreement with available experimental data but the price is invoking a good deal of phenomenological
parameters. The point is that DGLAP, summing up leading lnk Q2 to all orders in αs, cannot do the same with
leading lnk(1/x). The later is not important in the region (33) where lnk(1/x) ≪ 1 but becomes a serious drawback
of the method at small x. The total resummation of DL contributions to g1 in the region
x ≪ 1, Q2 ≫ µ2 (34)
was done in Refs. [14]. The weakest point in those papers was keeping αs as a parameter, i.e. fixed at an unknown
scale. Accounting for the most important part of single-logarithmic contributions, including the running coupling
effects were done in Refs. [15]. In these papers µ2 was treated as the starting point of the Q2 -evolution and as the IR
cut-off at the same time. The structure function g1 was calculated with composing and solving IREE in the following
It is convenient to compose IREE not for g1 but for forward (with |t| . µ2) Compton amplitude M related to g1
as follows:
ℑM . (35)
It is also convenient to use for amplitude M the asymptotic form of the Sommerfeld-Watson transform:
ξ(−)(ω)F (ω,Q2/µ2) (36)
where ξ(−)(ω) = [e−ıπω − 1]/2 ≈ −ıπω/2 is the signature factor. The transform of Eq. (36) and is often addressed
as the Mellin transform but one should remember that it coincides with the Mellin transform only partly. IREE for
Mellin amplitudes F (ω,Q2) look quite simple.
For example, the IREE for the non-singlet Mellin amplitude FNS related to gNS1 by Eqs. (35,36) is depicted in
Fig. 3. In the Mellin space it takes the simple form:
[ω + ∂/∂y]FNS = (1 + ω/2)HNSF
NS (37)
where y = ln(Q2/µ2). Eq. (37) involves a new object (the lowest blob in the last term in Fig. 3): the non-singlet
anomalous dimension HNS accounting for the total resummaton of leading logarithms of 1/x. Like in DGLAP, the
anomalous dimension does not depend on Q2 but, in contrast to DGLAP, HNS can be found with the same method.
The IREE for it is algebraic:
ωHNS = A(ω)CF /8π
2 + (1 + ω/2)H2NS +D(ω)/8π
2 . (38)
The system of Eqs. (37,38) can be easily solved but before doing it let us comment on them. The left-hand sides
of Eqs. (37,38) are obtained with applying the operator −µ2∂/∂µ2 to Eq. (36). The Born contribution in Fig. 3
does not depend on µ and therefore vanishes. The last term in Fig. 3 (the rhs of Eq. (37)) is the result of a new, t
-channel factorization which does not exist in the hard kinematics defined in Eq. (1). In order to compose the IREE
for the Compton amplitude M , in accordance with the prescription in the previous section we should first introduce
the cut-off µ. Then Step 2 is to tag the softest particles. In the case under discussion we do not have soft external
particles. Had the softest particle been a gluon, it could be factorized in the same way like in Sect. II. However, the
only option now is to attach the softest propagator to the external quark lines and get ln(t/µ2) = 0 from integration
over β (cf Eq. (7)). So, the softest gluon does not yield DL contributions. The other option is to find a softest quark.
The softest t -channel quark pair factorizes amplitude M into two amplitudes (the last term in Fig. 3) and yield DL
contributions. The IREE for HNS is different:
(i) HNS does not depend on Q
2, so there is not a derivative in the lhs of Eq. (37).
(ii) The Born term depends on µ and contributes to the IREE (term A in Eq. (37))).
(iii) As all external particles now are quarks, the softest virtual particle can be both a quark and gluon. The case
when it is the t -channel quark pair, corresponds to the quadratic term in the rhs of Eq. (37). The case of the softest
gluon yields the term D, with
D(ω) =
dηe−ωη ln
(ρ+ η
[ ρ+ η
(ρ+ η)2 + π2
where b = (33− 2nf )/12π and η = ln(µ2/Λ2QCD).
The term A in Eq. (37) stands instead of αs. The point is that the standard parametrization αs = αs(Q
2) cannot
be used at x ≪ 1 and should be changed (see Ref. [16] for detail). It leads to the replacement αs by
A(ω) =
η2 + π2
dρe−ωρ
(ρ+ η)2 + π2
. (40)
Having solved Eqs. (37,38), we arrive at the following expression for gNS1 in the region (34):
gNS1 (x,Q
2) = (e2q/2)
(1/x)ωCNS(ω)δq(ω) exp
HNS(ω)y
where the coefficient function CNS(ω) is expressed through HNS(ω):
CNS(ω) =
ω −HNS(ω)
and HNS(ω) is the solution of algebraic equation (43):
HNS = (1/2)
ω2 −B(ω)
where
B(ω) = (4πCF (1 + ω/2)A(ω) +D(ω))/(2π
2) . (44)
It is shown in Ref. [17] that the expression for g1 in the region
x ≪ 1, Q2 . µ2 (45)
can be obtained from the expressions obtained in Refs. [15] for g1 in region (34) by the shift
Q2 → Q2 + µ20 (46)
where µ0 = 1 GeV for the non-singlet g1 and µ0 = 5.5 GeV for the singlet.
IV. COMPARISON OF EXPRESSIONS (30) AND (41) FOR gNS1
Eqs. (30) and (41) read that the non-singlet g1 is obtained from δq with evolving it with respect to x (using the
coefficient function) and with respect to Q2 (using the anomalous dimension). Numerical comparison of Eqs. (30)
and (41) can be done when δq is specified.
A. Comparison of small-x asymptotics, neglecting the impact of δq
In the first place let us compare the small-x asymptotics of for gNS DGLAP1 and g
1 , assuming that δq does not
affect them. In other words, we compare the differencee in the x-evolution at x → 0. Applying the saddle-point
method to Eqs. (30) and (41) leads to the following expressions:
gNS DGLAP1 ∼ exp
ln(1/x) ln ln(Q2/Λ2QCD)
gNS1 ∼ (1/x)∆NS(Q2/µ2)∆NS/2 (48)
where ∆NS = 0.42 is the non-singlet intercept
1. Expression (47) is the well-known DGLAP asymptotics. Obviously,
the asymptotics (48) is much steeper than the DGLAP asymptotics (30).
B. Numerical comparison between Eqs. (30) and (41), neglecting the impact of δq
A comparison between Eqs. (30) and (41) strongly depends on the choice of δq but also depends on the difference
between the coefficient functions and anomalous dimensions. To clarify the latter we choose the simplest form of δq:
δq(ω) = Nq . (49)
It corresponds to the evolution from the bare quark where δq(x) = Nqδ(1 − µ2/s). Numerical results for R =
[gNS1 − gNS DGLAP1 ]/gNS DGLAP1 with δq chosen by Eq. (49) manifest (see Ref. [19] for detail) that R increases when
x is decreases. In particular, R > 0.3 at x . 0.05. It means that the total resummation of leading ln
(1/x) cannot
be neglected at x . 0.05 and DGLAP cannot be used beyond x ≈ 0.05. On the other hand, it is well–known that
Standard Approach based on DGLAP works well at x ≪ 0.05. To solve this puzzle, we have to consider the standard
fit for δq in more detail.
C. Analysis of the standard fits for δq
There are known different fits for δq. We consider the fit of Eq. (32). Obviously, in the ω -space Eq. (32) is a sum
of pole contributions:
δq(ω) = Nη
(ω − α)−1 +
mk(ω + λk)
, (50)
with λk > 0, so that the first term in Eq. (50) corresponds to the singular term x
−α of Eq. (32) and therefore the
small-x asymptotics of fDGLAP is given by the leading singularity ω = α = 0.57 of the integrand in Eq. (50) so that
the asymptotics of gNS DGLAP1 (x,Q
2) is not given by the classic exponential of Eq. (47) but actually is the Regge-like:
gNS DGLAP1 ∼ C(α)(1/x)α
ln(Q2/Λ2)/ ln(µ2/Λ2)
)γ(α)/b
, (51)
with b = (33 − 2nf)/12π. Comparison of Eq. (48) and Eq. (51) demonstrates that both DGLAP and our approach
lead to the Regge behavior of g1, though the DGLAP prediction is more singular than ours. Then, they predict
1 The singlet intercept is much greater: ∆S = 0.86.
different Q2 -behavior. However, it is important that our intercept ∆NS is obtained by the total resummation of the
leading logarithmic contributions and without assuming singular fits for δq whereas the SA intercept α in Eq. (47)
is generated by the phenomenological factor x−0.57 of Eq. (32) which makes the structure functions grow when x
decreases and mimics in fact the total resummation2. In other words, the role of the higher-loop radiative corrections
on the small-x behavior of the non-singlets is, actually, incorporated into SA phenomenologically, through the initial
parton densities fits. It means that the singular factors can be dropped from such fits when the coefficient functions
account for the total resummation of the leading logarithms and therefore fits for δq become regular in x in this
case. They also can be simplified. Indeed, if x in the regular part N
(1 − x)β(1 + γxδ)
of the fit (32) is not large,
all x -dependent terms can be neglected. So, instead of the rather complicated expression of Eq. (32), δq can be
approximated by a constant or a linear form
δq(x) = N(1 + ax) . (52)
with 2 phenomenological parameters instead of 5 in Eq. (32).
V. CORRECTING MISCONCEPTIONS
The total resummation of lnk(1/x) allows to correct several misconceptions popular in the literature. We list and
correct them below.
Misconception 1: Impact of non-leading perturbative and non-perturbative contributions on the intercepts of g1
is large.
Actually: Confronting our results and the estimates of the intercepts in Refs. [18] obtained from fitting
available experimental data manifests that the total contribution of non-leading perturbaive and non-perturbative
contributions to the intercepts is very small, so the main impact on the intercepts is brought by the leading logarithms.
Misconception 2: Intercepts of g1 should depend on Q
2 through the parametrization of the QCD coupling
αs = α(Q
Actually: This is groundless from the theoretical point of view and appears only if the the parametrization of the
QCD coupling αs = α(k
) is kept in all ladder rungs. It is shown in Ref. [16] that this parametrization cannot be
used at small x and should be replaced by the parametrization of Eq. (40).
Misconception 3: Initial densities δq(x) and δg(x) are singular but they are defined at x not too small. Later,
being convoluted with the coefficient functions, they become less singular.
Actually: It is absolutely wrong: Eq. (50) proves that the pole singularity x−α in the fits does not become weaker
with the x-evolution.
Misconception 4: Fits for the initial parton densities are complicated because they mimic unknown non-
perturbative contributions.
Actually: Our results demonstrate that the singular factors in the fits mimic the total resummation of lnk(1/x)
and can be dropped when the resummation is accounted for. In the regular part of the fits the x -dependence is
essential for large x only, so impact of non-perturbative contributions is weak at the small-x region.
Misconception 5: Total resummations of lnk(1/x) may become of some importance at extremely small x but not
for x available presently and in a forthcoming future.
Actually: The efficiency of SA in the available small-x range is based on exploiting the singular factors in the
standard fits to mimic the resummations. So, the resummations have always been used in SA at small x in an
inexplicit way, through the fits, but without being aware of it.
2 We remind that our estimates for the intercepts ∆NS ,∆S were confirmed (see Refs. [18]) by analysis of the experimental data
VI. COMBINING THE TOTAL RESUMMATION AND DGLAP
The total resummaton of leading logarithms of x considered in Sect. IV is essential at small-x. When x ∼ 1, all
terms ∼ lnk(1/x) in the coefficient functions and anomalous dimensions cannot have a big impact compared to other
terms. DGLAP accounts for those terms. It makes DGLAP be more precise at large x than our approach. So,
there appears an obvious appeal to combine the DGLAP coefficient functions and anomalous dimensions with our
expressions in order to obtain an approach equally good in the whole range of x : 0 < x < 1. The prescription
for such combining was suggested in Ref. [19]. Let us, for the sake of simplicity, consider here combining the total
resummation and LO DGLAP. The generalization to NLO DGLAP can be done quite similarly. The prescription
consists of the following points:
Step A: Take Eqs. (31) and replace αs by A of Eq. (40), converting γNS into γ̃NS and C
NS into C̃
Step B: Sum up the obtained expressions and Eqs. (42,43):
c̃NS = C̃
NS +HS , h̃NS = γ̃NS +HNS . (53)
New expressions c̃NS , h̃NS combine the total resummation and DGLAP but they obviously contain the double count-
ing: some of the first–loop contributions are present both in Eqs. (31) and in Eqs. (42,43). To avoid the double
counting, let us expend Eqs. (42,43) into series and retain in the series only the first loop contributions3:
A(ωCF )
NS = 1 +
A(ωCF )
. (54)
Finally, there is Step C: Subtract the first-loop expressions (54) from Eq. (53)) to get the combined, or ”synthetic”
as we called them in Ref. [19], coefficient function cNS and anomalous dimension hNS :
cNS = c̃NS − C(1)NS , hNS = h̃NS −H
NS . (55)
Substituting Eqs. (55) in Eq. (41) leads to the expression for gNS1 equally good at large and small x. This description
does not require singular factors in the fits for the initial parton densities. An alternative approach for combining
DLA expression for g1 was suggested in Ref. [20]. However, the parametrization of αs in this approach was simply
borrowed from DGLAP, which makes this approach be unreliable at small x.
VII. CONCLUSION
We have briefly considered the essence of the IREE method together with examples of its application to different
processes. They demonstrate that IREE is indeed the efficient and reliable instrument for all-orders calculations in
QED, QCD and the Standard Model of EW interactions. As an example in favor of this point, let us just remind that
there exist wrong expressions for the singlet g1 in DLA obtained with an alternative technique and the exponentiation
of EW double logarithms obtained in Ref. [10] had previously been denied in several papers where other methods of
all-order summations were used.
VIII. ACKNOWLEDGEMENT
B.I. Ermolaev is grateful to the Organizing Committee of the Epiphany Conference for financial support of his
participation in the conference.
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http://arxiv.org/abs/hep-ph/9609243
http://arxiv.org/abs/hep-ph/0605133
http://arxiv.org/abs/hep-ph/0108224
http://arxiv.org/abs/hep-ph/0110229
http://arxiv.org/abs/hep-ph/9802386
	Introduction
	IREE for scattering amplitudes in the hard kinematics
	IREE for the form factor f(q2) in QED
	IREE for the form factor g(q2) in QED
	e+e- -annihilation into a quark-antiquark pair
	e+e- -annihilation into a quark-antiquark pair and gluons
	Exponentiation of Sudakov electroweak double-logarithmic contributions
	Application of IREE to the polarized Deep-Inelastic Scattering
	Comparison of expressions (30) and (41) for g1NS
	Comparison of small-x asymptotics, neglecting the impact of q 
	Numerical comparison between Eqs. (30) and (41), neglecting the impact of q
	Analysis of the standard fits for q
	Correcting misconceptions
	Combining the total resummation and DGLAP
	Conclusion
	Acknowledgement
	References
