B. Ducloué et al. / Physics Letters B 803 (2020) 135305 3
The elastic S
-matrix
S
xy
also depends on the rapidity difference between the dipole and the proton through the high-energy evolution.
The physical picture of this evolution and its analytical description depend on how the total energy is divided between the (dipole)
projectile and the (proton) target i.e. upon the choice of the “dipole frame” in which one is working. It is useful to consider the two
extreme situations: the “target frame”, in which most of the total energy (and hence the whole high-energy evolution) is carried by the
proton, and the “projectile frame”, where the energy is mostly carried by the dipole (and the high-energy evolution is encoded in the
dipole wavefunction). Importantly the rapidity variable which represents the “evolution time” for the high-energy evolution, is different in
these two situations:
target rapidity: η ≡ ln
P
−
|q
−
|
=
ln
2q
+
P
−
Q
2
= ln
1
x
(2)
dipole rapidity: Y
≡ ln
q
+
q
+
0
= ln
2q
+
P
−
Q
2
0
= ln
1
x
+ ln
Q
2
Q
2
0
= η + ρ (3)
For the target rapidity, the typical gluon from the proton which interacts with the dipole has a longitudinal momentum k
−
= Q
2
/2
q
+
=
|
q
−
|, and hence a longitudinal extent ∼ 1/k
−
of the order of the lifetime x
+
of the q
¯
q
pair.
For the projectile rapidity, the softest dipole
to participate in the collision has a longitudinal momentum q
+
0
= Q
2
0
/2
P
−
— i.e. a lifetime 2
q
+
0
/Q
2
0
equal to the longitudinal extent 1/P
−
of the proton —, where the scale Q
0
Q
is
the transverse momentum scale for the onset of unitarity (multiple scattering) effects in the
(unevolved) proton. The two rapidities differ by ρ ≡ ln(Q
2
/Q
2
0
) which is large when Q Q
0
.
The non-linear effects associated with the high gluon density are described differently in the two frames. In the target frame, soft
gluon emissions occur in the proton wavefunction which is a dense environment. Accordingly, these emissions are modified by non-linear
effects like gluon recombinations. The non-linear evolution of the dense hadron wavefunction has been computed only to leading order,
yielding the (functional) JIMWLK equation [21–26]. Conversely, in the dipole frame one views the evolution as successive, soft, gluon
emissions within the dipole wavefunction, a dilute hadronic system. Gluon emissions from the dipole occur like in the vacuum and non-
linear
effects exclusively refer to multiple scattering. This leads to the Balitsky hierarchy (and the BK equation), currently known to NLO
accuracy [8–12]. Since our purpose in this work is to go beyond LO accuracy, we systematically use the dipole frame in what follows.
Time ordering and collinear improvements in Y (dipole frame) Besides being less suited for applications to DIS, the evolution with Y
has
a
more severe conceptual drawback: the typical emissions contributing to this evolution at leading order can violate proper time ordering,
that is, the condition that the formation time of a daughter gluon be smaller than the lifetime of its parent.
2
To understand this, we
first recall that, when Q
2
Q
2
0
, the typical emissions associated with the high-energy evolution of the dipole wavefunction are strongly
ordered both in longitudinal momenta (
k
+
) and in transverse momenta (
k
⊥
):
q
+
k
+
1
k
+
2
···q
+
0
, Q
2
k
2
1
⊥
k
2
2
⊥
··· Q
2
0
. (4)
This corresponds to soft and collinear emissions which yield the dominant, double-logarithmic, contributions proportional to powers of
¯
α
s
Y ρ. However, an explicit calculation of the relevant Feynman graphs shows [41] that this double-logarithmic enhancement only holds
so long as the gluon lifetimes are ordered as well:
2q
+
Q
2
2k
+
1
k
2
1
⊥
2k
+
2
k
2
2
⊥
···
2q
+
0
Q
2
0
. (5)
This condition reduces the rapidity phase-space available for the evolution from Y to Y − ρ ≡ η. The condition Eq. (5)is already violated
(due to the emission of daughter gluons with sufficiently soft k
⊥
) in the LO BK evolution which resums an infinite series in
¯
α
s
Y ρ,
instead of the correct series in powers of
¯
α
s
(Y − ρ)ρ. The difference between the two corresponds to an alternating series of double
“anti-collinear” logarithms proportional to
¯
α
s
ρ
2
which spoil the convergence of the perturbative expansion in Y . In particular, the NLO BK
equation includes the first (negative) contribution proportional to
¯
α
s
ρ
2
[8] which makes the evolution unstable [29] and hence unsuitable
for physical studies.
3
To overcome this difficulty, it was originally proposed [31,41]to enforce time-ordering directly in the dipole frame evolution. Two
“collinearly improved” BK equations have been proposed: in the first [31]the evolution is non-local in rapidity and has the same kernel
as at LO, while in the second [41]the evolution is local in Y , but both the kernel and the initial condition receive corrections to all orders
in
¯
α
s
ρ
2
. Both methods allow for a faithful resummation of the dominant series in powers of
¯
α
s
ρ
2
, but the subleading terms (proportional
to powers of
¯
α
k
s
ρ
2
with k ≥ 2) are not under control. At a first sight, both strategies appear to be successful: the respective equations are
stable [41,42], they can be extended to full NLO accuracy [42], and moreover they allow for good fits to the HERA data for DIS at small
x [43,44].
Recasting dipole evolution in terms of η A more recent study has revealed that these apparent successes were in fact deceiving [45]. The
numerical studies in [42–44]have been presented in terms of Y instead of the physical rapidity η = Y − ρ = ln(1/x), and in the DIS fits
in [43,44], the variable Y has been abusively interpreted as ln(1/x). The correct procedure would require to first transform the results
from Y to η by a simple change of variables, before attempting a physical interpretation or a fit to the data. When following this correct
procedure, one finds [45]an unacceptably large resummation-scheme dependence. For example when solved with the same initial condition
2
Notice that formation times and lifetimes are parametrically the same for this space-like evolution.
3
More generally there is a tower of series of such spurious terms: series appears to correct the time-ordering violation in the previous one. The dominant series includes
all powers of
¯
α
s
ρ
2
, the subdominant one, those of
¯
α
2
s
ρ
2
, etc.
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