aainstitutetext: Physics Department, University of Connecticut, 2152 Hillside
Road, Storrs, CT 06269, USAbbinstitutetext: Department of Particle Physics,
Tel Aviv University, Tel Aviv 69978, Israelccinstitutetext: Physics
Department, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel

#  High energy scattering in the Unitary Toy Model

Alex Kovner, b Eugene Levin c and Michael Lublinsky

###### Abstract

We continue exploring the Unitary Toy Model (UTM) as a playground for high
energy collisions in QCD. Our new approach is based on the diagonalization of
the evolution Hamiltonian. Part of the spectrum can be identified with
intercepts of dressed Pomerons. Analogously to QCD, a multi-Pomeron expansion
of the $S$-matrix is badly divergent asymptotic series. Yet we succeeded to
establish resummation procedures resulting in a well behaved $S$-matrix. In
addition the Hamiltonian possesses negative eigenvalues, which dominate the
approach of the $S$-matrix to saturation. We are hopeful that important
lessons about BFKL-based Pomeron calculus could be taken from the toy world to
real QCD.

††dedication:

##  1 Introduction

Zero transverse dimension toy models ACJ ; AAJ ; JEN ; ABMC ; CLR ; CIAF ;
MUSA ; RS ; BIT ; SHXI ; KOLEV ; KLremark2 ; nestor ; LEPRI ; utm ; utmm  can
be viewed as a realization of the Pomeron calculus or more generally of the
Reggeon Field Theory (RFT). Over the years they have been intensively used to
model high energy collisions in QCD. These models encode various fundamental
features of QCD such as unitarity, but are much simpler than the latter and
frequently solvable analytically. Hence they provide a valuable playground to
gain intuition about the dynamics of real QCD.

Our discussion focuses on two toy models. The first one is the BFKL cascade
model which mimics Mueller’s dipole model in QCD MUDI  (for dipoles of fixed
sizes). We refer to this model as the BFKL cascade BFKL , although it is a
closer analog to the Balitsky-Kovchegov (BK) model in QCD BK  ***The gluon
emissions in the projectile wave function in BFKL and BK models are the same.
The physical difference between the two is that BK equation BK  accounts for
multiple scatterings on the target, while the original BFKL equation BFKL
does not.. The BFKL cascade lacks both $s$ and $t$-channel unitarity KLremark2
; utm  and hence has to be corrected. A simple modification of the BFKL
cascade wich restores the proper unitarity properties is the Unitary Toy Model
(UTM) first introduced in MUSA  and later explored in KLremark2 ; BIT ; utm .
Additional improvements to the model have been proposed (see utmm ), but they
will not be addressed here.

All of the models share the following simple probabilistic expression for the
total $S$-matrix for scattering of $n$ dipoles of the projectile on $m$
dipoles of the target

$S(Y)\,\,=\,\,\sum_{n,m}e^{-m\,n\,\gamma}\,P_{n}^{P}(Y_{0})\,P_{m}^{T}(Y-Y_{0})$
(1.1)

Here $\gamma\sim\alpha_{s}^{2}$ is the Born approximation to the low energy
dipole-dipole scattering amplitude. $P_{n}^{P}(Y_{0})$ is the probability to
find $n$ dipoles in the projectile boosted to rapidity $Y_{0}$, and similarly
for the target. The probabilities are found by solving the evolution equations
in rapidity, and it is in the form of these equations that the models differ
from each other. The important consistency condition on the evolution is
$t$-channel unitarity, which is essentially the requirement that the
$S$-matrix does not depend on the reference frame, that is the choice of
$Y_{0}$ at fixed $Y$MUSA ; BIT ; utmm . UTM is defined to satisfy this
condition.

Initial conditions for the evolution determine the dipole content of the
colliding particles. In this paper, we will be largely concerned with dipole-
dipole scattering, that is with initial condition $P_{1}^{P,T}(0)=1$,
$P^{P,T}_{n>1}(0)=0$. We will also briefly discuss DIS-like processes off a
nucleus, in which the target is assumed to consist of many dipoles at rest.

While both the BFKL and UTM models have been extensively explored in the past,
in this paper we address a somewhat different set of questions and use a new
approach. Our formulation follows algebraic approach based on diagonalization
of the respective evolution Hamiltonians, for which we find spectrum and
eigenfunctions. Part of the spectrum can be identified with Pomeron
intercepts, which differ between the models. Pomeron calculus emerges as a
representation of the $S$-matrix.

The multi-Pomeron expansion in BFKL is an asymptotic, Borel summable series,
while in UTM it is badly divergent. The UTM asymptotic series cannot be summed
via Borel resummation. Nevertheless the $S$-matrix in UTM is physical and well
defined, and we are able to establish resummation methods which circumvent the
apparent divergence of the Pomeron calculus and yield the finite, physical
result for the $S$-matrix.

A similar and a very old problem also exists in QCD: due to $1/N_{c}$
corrections, the $2n$-reggeon (reggeized gluon) BKP states BKP  have
intercepts which grow faster than $n$ times the single BFKL Pomeron intercept.
As a consequence, for many years, this was believed to be the major challenge
even to very existence of the BFKL-based Pomeron calculus in real QCD. We are
hopeful that our explorations in the toy models provide an important insight
into RFT in QCD, suggesting that the $1/N_{c}$ corrections can be re-summed.

Beyond discussing the $S$-matrix we touch upon the probability distributions.
The latter contain all the information about the dipole cascade and could be
instrumental in understanding particle production in these toy models. This
question is postponed for a separate study.

We review the BFKL cascade model in Sec. 2. UTM is introduced in Sec. 3.
$S$-matrix in UTM is explored in Sec 4. Sec 5 is the summary section. Several
Appendices complement our calculations.

##  2 The BFKL cascade

###  2.1 The dipole-dipole scattering: probabilities and Pomeron exchanges

The simplest model describes a single BFKL-like cascade. It is specified by
the following evolution equation for the dipole probabilities $P^{\mbox{\tiny
BFKL}}_{n}(Y)$:

$\frac{d\,P^{\mbox{\tiny
BFKL}}_{n}(Y)}{d\,Y}\,\,=\,\,-\Delta\,n\,P^{\mbox{\tiny
BFKL}}_{n}(Y)\,\,+\,\,\Delta\,(n-1)\,P^{\mbox{\tiny BFKL}}_{n-1}(Y)$ (2.1)

Here we regard the ”BFKL intercept” parameter $\Delta$ as parametrically of
order $\Delta\sim\alpha_{s}N_{c}$ inspired by the QCD BFKL equation.

The solution for this equation for a single dipole initial condition is:

$P_{n}^{BFKL}(Y)=\frac{1}{N(Y)}\left(1-\frac{1}{N(Y)}\right)^{n-1}\,\,\,\xrightarrow{Y\,\gg\,1/\Delta}\,\,\,\frac{1}{N(Y)}\exp\left(-\frac{n\,-\,1}{N(Y)}\right)$
(2.2)

where the average dipole multiplicity in the BFKL cascade is

$N(Y)\,=\,e^{\Delta\,Y}$ (2.3)

The $S$-matrix calculated via eq.(1.1) with $P^{\mbox{\tiny BFKL}}_{n}(Y)$
does not depend on $Y_{0}$ only if we expand the exponential factor
$e^{-m\,n\,\gamma}$ to leading order in $\gamma$, $e^{-m\,n\,\gamma}\approx
1-mn\gamma$:

$S^{\mbox{\tiny BFKL}}_{\mbox{\tiny LO}}=1-\gamma N(Y)$ (2.4)

Keeping all terms in the exponents leads to a nontrivial dependence of $S$ on
$Y_{0}$. However, it turns out that if both $Y-Y_{0}$ and $Y_{0}$ are large,
the dependence on $Y_{0}$ disappears in the term that leads to the largest
contribution. Indeed,

$\displaystyle S^{\mbox{\tiny BFKL}}(Y)$ $\displaystyle=$
$\displaystyle\sum_{n,m}e^{-m\,n\,\gamma}\,P^{\mbox{\tiny
BFKL}}_{n}(Y_{0})\,P^{\mbox{\tiny BFKL}}_{m}(Y-Y_{0})$ (2.5a) $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\frac{\left(-\gamma\right)^{k}}{k!}\underbrace{\Bigg{(}\sum_{n=0}^{\infty}n^{k}\,P_{n}(Y_{0})\Bigg{)}}_{c_{k}\left(Y_{0}\right)}\underbrace{\Bigg{(}\sum_{m=0}^{\infty}m^{k}\,P_{m}(Y-Y_{0})\Bigg{)}}_{c_{k}\left(Y-Y_{0}\right)}\,\,$
(2.5b)

Eq. (2.5b) gives the expression for the scattering amplitude which can be
viewed as the sum of multi Pomeron exchanges. Each of these exchanges is
proportional to $\left(e^{\Delta Y}\right)^{n}$. Eq. (2.4) is the contribution
of the first two terms in Eq. (2.5a). Lets define short notations
$N_{1}=N\left(Y_{0}\right)$, $N_{2}=N\left(Y-Y_{0}\right)$, and assume both to
be large. We start with Eq. (2.5a) which takes the form

$\displaystyle S^{\mbox{\tiny BFKL}}(Y)$ $\displaystyle=$
$\displaystyle\sum_{n,m}e^{-m\,n\,\gamma}\,P^{\mbox{\tiny
BFKL}}_{n}(Y_{0})\,P^{\mbox{\tiny
BFKL}}_{m}(Y-Y_{0})\,\simeq\,\sum^{\infty}_{n=1,m=1}e^{-m\,n\,\gamma}\frac{e^{-\frac{n}{N_{1}}}}{N_{1}}\,\,\frac{e^{-\frac{n}{N_{2}}}}{N_{2}}$
(2.6a) $\displaystyle\simeq\,(\gamma\,\ll\,1)$ $\displaystyle\simeq$
$\displaystyle\frac{1}{N_{1}\,N_{2}}\int\limits^{\infty}_{0}d\bar{m}\int\limits^{\infty}_{0}d\bar{n}\exp\left(-\gamma\bar{n}\bar{m}-\gamma\bar{n}-\gamma\bar{m}-\gamma\,-\bar{n}/N_{1}-\bar{m}/N_{2}\right)$
(2.6b) $\displaystyle=$ $\displaystyle\frac{1}{\gamma
N_{1}N_{2}}e^{\frac{1}{\gamma\text{N}_{1}\text{N}_{2}}+\frac{1}{\text{N}_{1}}+\frac{1}{\text{N}_{2}}}\Gamma\left(0,\frac{(\gamma\,\text{N}_{1}\,+\,1)(\gamma\,\text{N}_{2}+1)}{\gamma\,\text{N}_{1}\text{N}_{2}}\right)$
(2.6c) $\displaystyle\xrightarrow{N_{2}\gg 1,N_{1}\gg 1}$
$\displaystyle\,\frac{1}{\gamma\,N_{1}\,N_{2}}\Bigg{(}-\ln\gamma+\mathcal{O}(1/N_{1})\,+\mathcal{O}(1/N_{2})\Bigg{)}$
(2.6d)

where $\bar{n}=n-1$, $\bar{m}=m-1$, and $\Gamma\left(0,z\right)$ is the
incomplete Gamma function (see formulae 8.35 of Ref.RY ). The leading term
does not depend on $Y_{0}$. However, all other terms do depend on the value of
$Y_{0}$.

It is instructive to see how the result Eq. (2.6d) arises from the ”Pomeron
calculus”, i.e. Eq. (2.5b) which represents the scattering amplitude as a sum
of Pomeron exchanges. The power moments $c_{k}$ are given by

$c_{k}\left(N\right)=\frac{1}{N}\sum_{n=1}^{\infty}n^{k}\exp\left(-\frac{n}{N}\right)\,=\,\,\text{Li}_{-k}\left(e^{-1/N}\right)\,\,\xrightarrow{N\gg
1}k!N^{k}\,\,+\,\,\frac{\zeta\left(-k\right)}{N}\,\,+\,\,\mathcal{O}\left(1/N^{2}\right)$
(2.7)

where $\zeta(z)$ is the Riemann zeta function. For $\zeta\left(-k\right)$ we
use the following equation (Abel–Plana formula):

$\zeta(-k)=-\frac{1}{k+1}+\tilde{\zeta}\left(-k\right);~{}~{}\mbox{with}~{}~{}\tilde{\zeta}\left(-k\right)\,\,=\,\,\frac{1}{2}\,+\,2\int\limits^{\infty}_{0}dt\frac{\sin\left(k\arctan(t)\right)\left(1-t^{2}\right)^{k/2}}{e^{2\pi
t}\,-\,1}$ (2.8)

$\tilde{\zeta}\left(k\right)$ is equal $\frac{1}{2}+$ function that decreases
faster than $1/k$. The first term in Eq. (2.8) has a simple meaning if in Eq.
(2.7) summation is replaced by integration

$c_{k}\left(N\right)=\frac{1}{N}\int_{n=1}^{\infty}dn\,n^{k}\exp\left(-\frac{n}{N}\right)\,=k!\,N^{k}\,\,-\,\,\frac{1}{k+1}\frac{1}{N}\,\,+\,\,\mathcal{O}\left(\frac{1}{N^{2}}\right)$
(2.9)

. Substituting $\zeta(-k)=-\frac{1}{k+1}$ into Eq. (2.7), for the $S$-matrix
we obtain

$\displaystyle S^{\mbox{\tiny
BFKL}}(Y)=\sum_{k=0}^{\infty}\frac{\left(-\gamma\right)^{k}}{k!}\,c_{k}\left(Y_{0}\right)\,c_{k}\left(Y-Y_{0}\right)\,\simeq\,\sum_{k=0}^{\infty}\frac{\left(-\gamma\right)^{k}}{k!}\left(k!\,N_{1}^{k}\
\,-\frac{1}{k+1}\frac{1}{N_{1}}\right)\,\left(k!\,N^{k}_{2}\,-\frac{1}{k+1}\frac{1}{N_{2}}\right)$
$\displaystyle\simeq\,\,\frac{1}{\gamma\,N_{1}\,N_{2}}e^{\frac{1}{\gamma
N_{1}\,N_{2}}}E_{1}\left(\frac{1}{\gamma
N_{1}\,N_{2}}\right)\,\,-\,\,\frac{\ln\left(1\,\,+\,\,\gamma
N_{1}\right)}{\gamma\,N_{1}\,N_{2}}\,\,-\,\,\frac{\ln\left(1\,\,+\,\,\gamma
N_{2}\right)}{\gamma\,N_{1}\,N_{2}}$ $\displaystyle\xrightarrow{\gamma
N_{1};\,\gamma\,N_{2}\gg\,1}\,\frac{1}{\gamma\,N_{1}\,N_{2}}\Bigg{\\{}\ln\left(\gamma
N\left(Y\right)\right)\,-\,\ln\left(\gamma N_{1}\right)\,-\,\ln\left(\gamma
N_{2}\right)\Bigg{\\}}\,=\,\frac{\ln\left(1/\gamma\right)}{\gamma\,N_{1}\,N_{2}}\,\,=\,\frac{\ln\left(1/\gamma\right)}{\gamma}\,e^{-\Delta\,Y}$
(2.10)

where $E_{1}\left(z\right)=-Ei\left(-z\right)$ is the Exponential integral
(Ref.RY , formulae 8.21). Eq. (2.1) gives the same asymptotics as Eq. (2.6d).

In the context of this calculation we note that in order to obtain the correct
asymptotics using the moment expansion (or ”Pomeron calculus”) we need to keep
the subleading term in (2.9) for large $N$. If we were to drop this subleading
term we would get the factor $\ln(\gamma N(Y))$ instead of $\ln 1/\gamma$ in
front of the exponential in (2.1), i.e. a significantly different asymptotics
at large rapidity. This may sound surprising, since this term is a very small
correction to high moments at any $Y$, and to all moments at large $Y$.
However while the series representation in terms of probabilities is
convergent, as probabilities for large number of dipoles $n$ always decrease
as $n\rightarrow\infty$, the expansion in moments (or in Pomeron exchanges) is
a very different animal. The individual terms in this expansion grow
factorially with $k$ and exponentially in $Y$. Thus even a small correction to
a fixed moment can generate a leading correction to the whole sum. This should
caution us against using naive Pomeron calculus (i.e. approximating
$n$-Pomeron exchange by a simple product of $n$ single Pomerons) in a frame
where both $Y_{0}$ and $Y-Y_{0}$ are large, and thus a single term in the
Pomeron series is given by a product of two large numbers,
$c_{k}(Y_{0})c_{k}(Y-Y_{0})$ as in (2.5b). Thankfully, terms suppressed by
additional powers of $1/N$ in (2.9) do not contribute to the asymtotic result.

We emphasize that in the BFKL cascade model, the $Y_{0}$ independence is
approximate only and is seen in a frame when both projectile and target are
boosted to sufficiently high energies. The resulting expressions for the
$S$-matrix also differ significantly from the result computed in the target
rest frame. The latter we quote below in Eq. (2.25). While the exponential
fall with the total rapidity is the same, the leading coefficient differs by
$-\ln\gamma$, which is quite dramatic. This signifies the problem of the BFKL
cascade model and its frame dependence.

###  2.2 The $S$-matrix in the target rest frame

Let us now consider the calculation of the scattering matrix in the rest frame
of the target. This means that we evolve only the projectile probability
distribution, while the target remains fixed. In most of this paper we are
interested in the target comprised of a single dipole.

To calculate the S-matrix it is convenient to introduce the notion of the
generating functional.

####  2.2.1 Generating functional and eigenfunctions

The standard definition of the generating functionalMUDI ; LELU  is

$Z^{BFKL}(u,\tilde{Y})\equiv\sum_{n}u^{n}\,P_{n}^{BFKL}(\tilde{Y})$ (2.11)

Beyond the formal meaning of generating the probabilities $P_{n}$ in the
projectile, the functional $Z(u)$ for $u=e^{-\gamma m}$ also has a direct
physical meaning of a scattering matrix of the projectile on a target
consisting of $m$ dipoles, calculated in the target rest frame. The BFKL
equation for the generating function can be written as

$\frac{\partial}{\partial\tilde{Y}}Z^{BFKL}(u)\,=\,{\cal
H}_{BFKL}\,Z^{BFKL}(u)$ (2.12)

where $\tilde{Y}=\frac{\Delta}{\gamma}Y$ and the BFKL Hamiltonian operator is

${\cal H}_{BFKL}=\gamma u(u-1)\frac{d}{du}$ (2.13)

The standard way of solving an evolution equation of the type (2.12) is first
to find the eigenfunctions of the evolution operator. The operator ${\cal
H}_{BFKL}$ is not Hermitian and it therefore has distinct right and left
eigenfunctions. It is straightforward to verify that the following are its
right eigenfunctions:

$\Phi_{n}^{BFKL}\left(u\right)\,=\,\left({u-1\over
u}\right)^{n}=\phi^{n}\,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Psi_{n}^{BFKL}\left(u\right)\,=\,\left({u-1\over
u}\right)^{-n}=\phi^{-n}\,$ (2.14)

with the corresponding eigenvalues $\Delta^{BFKL}_{n}=n\gamma$ and
$\tilde{\Delta}^{BFKL}_{n}=-n\gamma$. Here we have defined for convenience

$\phi\equiv\frac{u-1}{u}$ (2.15)

Note, that in principle $n$ in the above expressions is not quantized, and
therefore in this sense the spectrum is continuous. However our goal will be
to expand the generating function in these eigenfunctions. By definition
$Z(u)$ should be expandable in powers of $u$, which would suggest that the
relevant eigenfunctions for the expansion are $\Psi_{BFKL}^{n}$ for integer
$n$. On the other hand for $u$ close to unity we expect $Z(u)$ to have a good
Taylor expansion in powers of $1-u$. This singles out the set
$\Phi_{BFKL}^{n}$ for integer $n$. We will therefore only need the
eigenfunctions $\Phi_{BFKL}^{n}$ and $\Psi_{BFKL}^{n}$ for integer values of
$n$.

The left eigenfunctions with the same eigenvalues are

$\displaystyle\tilde{\Phi}_{n}^{BFKL}\left(u\right)\,=\,{1\over(1-u)}\,\left({u-1\over
u}\right)^{n}\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\tilde{\Psi}_{n}^{BFKL}\left(u\right)\,=\,{1\over(1-u)}\,\left({u\over
u-1}\right)^{n}$ (2.16)

Since ${\cal H}_{BFKL}$ is not Hermitian, it’s left and right eigenfunctions
are different. In general for finite dimensional spaces there is a very useful
theorem, that states that the left and right eigenfunctions $\phi^{L}_{n}$ and
$\phi^{R}_{m}$ that correspond to different eigenvalues are orthogonal to each
other. Therefore with appropriate normalization one can use the set of all
right eigenfunctions to ”invert” the left eigenfunctions

$\langle\phi^{L}_{n}|\phi^{R}_{m}\rangle=\delta_{nm}$ (2.17)

This property extends to infinite dimensional spaces if the eigenfunctions in
question are well behaved (normalizable). Our eigenfunctions on the other hand
clearly are not normalizable as they have nonintegrable singularity either at
$u=1$ or $u=0$. Nevertheless, one can show by direct calculation that with an
appropriate definition of scalar product (2.17) still holds. In particular the
two sets of eigenfunctions are orthonormal with the scalar product defined as
a contour integral in the complex plane:

$\frac{1}{2\pi i}\oint_{\Gamma_{1}}{du\over
u}\,\,\tilde{\Psi}_{m}^{BFKL}\left(u\right)\,\Phi_{n}^{BFKL}\left(u\right)\,=\,\frac{1}{2\pi
i}\oint_{-\Gamma_{2}}{du\over
u}\,\,\tilde{\Psi}_{m}^{BFKL}\left(u\right)\,\Phi_{n}^{BFKL}\left(u\right)\,=\,\delta_{mn}$
(2.18)

where the contour $\Gamma_{1}$ in the complex plane is taken to encircle only
the point $u=1$ while $\Gamma_{2}$ is the contour that only encircles the
point $u=0$ (see Fig. 1). Equivalently,

$\frac{1}{2\pi i}\oint_{\Gamma_{1}}{du\over
u}\,\,\Psi_{m}^{BFKL}\left(u\right)\,\tilde{\Phi}_{n}^{BFKL}\left(u\right)\,=\,\frac{1}{2\pi
i}\oint_{-\Gamma_{2}}{du\over
u}\,\,\Psi_{m}^{BFKL}\left(u\right)\,\tilde{\Phi}_{n}^{BFKL}\left(u\right)\,=\,\delta_{mn}$
(2.19)

In the following sections we will see that a similar scalar product that
preserves the property (2.17) can also be defined in the UTM.

As the previous equation suggests we can choose either $\Gamma_{1}$ or
$\Gamma_{2}$ as the contour for the definition of the scalar product.

Figure 1:  Contours of integrations in the complex $u$ plane. The red dots
mark the poles in the BFKL cascade model while the black dotes are those in
the UTM.

When expanding a function $Z^{BFKL}(u)$ in the basis of right eigenfunctions,
in order to extract the coefficients of the expansion we can use the
”operator” representation of the left eigenfunctions:

$\displaystyle\tilde{\Phi}_{n}^{BFKL}\left(u\right)\,=\,\,\delta(u)\,\,\frac{1}{n!}\partial_{\phi^{-1}}^{n}=(-1)^{n}\delta(u)\frac{1}{n!}\left((u-1)^{2}\frac{d}{du}\right)^{n},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\tilde{\Psi}_{n}^{BFKL}\left(u\right)\,=\,\delta(1-u)\,\,\frac{1}{n!}\partial_{\phi}^{n}=\delta(1-u)\frac{1}{n!}\left(u^{2}\frac{d}{du}\right)^{n}$
(2.20)

This corresponds to the choice of $\Gamma_{1}$ in eq.(2.18) and $\Gamma_{2}$
in eq.(2.19).

The problem of solving the BFKL equation now becomes similar to a Schroedinger
equation, except with the Hamiltonian which is not Hermitian. Nevertheless, we
can expand the generating function and therefore the scattering matrix in the
set of right eigenfunctions. Here we can choose either $\Phi_{n}$ or
$\Psi_{n}$ as the basis for expansion, since each one of them forms a complete
basis (a peculiarity of a non Hermitian Hamiltonian). Which set to choose is
dictated by convenience.

Suppose first we are in the genuine BFKL situation, where the target is
dilute. This means that the single dipole scattering amplitude is small, or
$\gamma\ll 1$, or $1-u\ll 1$. In this situation it is clear that the
generating functional should be expanded in Taylor series in $1-u$. Since in
this regime $\Phi_{n}^{BFKL}\approx(u-1)^{n}$, it means that we should be
expanding the generating function in the basis of $\Phi_{n}^{BFKL}$:

$Z^{BFKL}\left(u,\tilde{Y}\right)\,\,=\,\,\sum^{\infty}_{n=0}C_{n}\,\,\Phi_{n}^{BFKL}\left(u\right)\,e^{n\,\gamma\,\tilde{Y}}\,.$
(2.21)

We can use $\tilde{\Psi}_{n}^{BFKL}$ to find the expansion coefficients
$C_{n}$. For this we could either use directly the representation (2.2.1), or
use (2.16) noting that

$\frac{1}{2\pi i}\oint_{\Gamma_{1}}{du\over
u}\,\,\tilde{\Psi}_{m}^{BFKL}\left(u\right)\,Z^{BFKL}\left(u,\tilde{Y}\right)\,=\,C_{m}\,e^{m\,\gamma\,\tilde{Y}}$
(2.22)

Applying to a single dipole initial condition,
$Z^{BFKL}_{(1)}\left(u,\tilde{Y}\,=\,0\right)\,=\,u$ we find†††Interestingly,
if we were to try to use the contour $\Gamma_{2}$ to calculate the
coefficients $C_{m}$ we would find that they all vanish. This is the
consequence of the fact that the expansion (2.21) converges at $u=1$ with the
unit radius of converges, while the contour $\Gamma_{2}$ includes a region
$u<0$ and is therefore outside of the radius of convergence. ,

$C_{m}^{(1)}\,=\frac{1}{2\pi i}\,\oint_{\Gamma_{1}}{du\over
u}\,\,\tilde{\Psi}_{m}^{BFKL}\left(u\right)\,u\,=\,1$ (2.23)

and

$Z^{BFKL}_{(1)}\left(u,\tilde{Y}\right)\,\,=\,\,\sum^{\infty}_{n=0}\left({u-1\over
u}\,e^{\gamma\,\tilde{Y}}\right)^{n}$ (2.24)

The $S$-matrix of a single dipole in the target rest frame is then

$S\left(\tilde{Y}\right)\,\,=\,\,Z^{BFKL}_{(1)}\left(e^{-\gamma},\tilde{Y}\right)\,\,=\,\,\sum^{\infty}_{n=0}\left({(1-e^{\gamma})}\,e^{\gamma\,\tilde{Y}}\right)^{n}=\frac{1}{1-(1-e^{\gamma})e^{\gamma\,\tilde{Y}}}$
(2.25)

which reproduces the well known solution of the BK equation.

More generally for an $n$ dipole initial state,
$Z^{BFKL}_{(n)}\left(u,\tilde{Y}\,=\,0\right)\,=\,u^{n}$. we have

$C^{(n)}_{m}\,=\frac{1}{2\pi i}\,\oint_{\Gamma_{1}}{du\over
u}\,\,\tilde{\Psi}_{m}^{BFKL}\left(u\right)\,u^{n}\,=\,\frac{(m+n-1)!}{m!\,(n-1)!}$
(2.26)
$Z^{BFKL}_{(n)}\left(u,\tilde{Y}\right)\,\,=\,\,\sum^{\infty}_{m=0}\frac{(m+n-1)!}{m!(n-1)!}\left({u-1\over
u}\,e^{\gamma\,\tilde{Y}}\right)^{m}$ (2.27)

And

$S^{(n)}\left(\tilde{Y}\right)\,\,=\,\,Z^{BFKL}_{(n)}\left(e^{-\gamma},\tilde{Y}\right)\,\,=\,\,\sum^{\infty}_{m=0}\frac{(m+n-1)!}{m!(n-1)!}\left({(1-e^{\gamma})}\,e^{\tilde{Y}}\right)^{m}=\left[\sum^{\infty}_{m=0}\left({(1-e^{\gamma})}\,e^{\gamma\,\tilde{Y}}\right)^{m}\right]^{n}$
(2.28)

Now consider the opposite situation, i.e. dense target. This means that
$\gamma\gg 1$, and $u\ll 1$. This is not what we normally call BFKL, but this
is certainly in the region of applicability of the BK equation which applies
to scattering of a small number of dipoles on a dense target. Now one would
want to expand $Z(u)$ in powers of $u$ rather than in powers of $1-u$, and
therefore use the $\Psi^{BFKL}_{n}$ basis.

$Z^{BFKL}\left(u,\tilde{Y}\right)\,\,=\,\,\sum^{\infty}_{n=0}B_{n}\,\,\Psi_{n}^{BFKL}\left(u\right)\,e^{-n\,\gamma\,\tilde{Y}}\,.$
(2.29)

The coefficients can be found in the analogous way‡‡‡Here the situation is the
mirror image of (2.23). If we try to use the contour $\Gamma_{1}$ to extract
the expansion coefficients we will fail. This is because the expansion (2.29)
converges around $u=0$ and not around $u=1$, and the contour $\Gamma_{1}$ is
partially outside the radius of convergence.:

$\frac{1}{2\pi
i}\oint_{-\Gamma_{2}}{du\over{u}}\,\,\tilde{\Phi}_{m}^{BFKL}\left(u\right)\,Z^{BFKL}\left(u,\tilde{Y}\right)\,=\,B_{m}\,e^{-m\,\gamma\,\tilde{Y}}$
(2.30)

For a single dipole initial condition we find

$B_{m}^{(1)}=-1,\ \ m>0$ (2.31)

Then

$Z^{BFKL}_{(1)}\left(u,\tilde{Y}\right)\,=-\sum^{\infty}_{m=1}\left(\frac{u}{u-1}e^{-\gamma\,\tilde{Y}}\right)^{m}=1-\frac{1}{1+\frac{u}{1-u}e^{-\gamma\,\tilde{Y}}}=\frac{1}{1-(1-\frac{1}{u})e^{\gamma\,\tilde{Y}}}$
(2.32)

which unsurprisingly coincides with (2.25).

For an $n$ dipole initial condition, $B^{(n)}_{m<n}=0$, while

$B^{(n)}_{m\geq n}=\frac{1}{2\pi
i}\,\oint_{-\Gamma_{2}}{du\over{u}}\,\tilde{\Phi}_{m}^{BFKL}\left(u\right)\,u^{n}=\sum_{k=1}^{n}(-1)^{k}\frac{n!(k+m-1)!}{k!(n-k)!(k-1)!m!}\,=\,(-1)^{n}\frac{(m-1)!}{(m-n)!\,(n-1)!}\,=\,-\,C^{(n)}_{-m}$
(2.33)

It is instructive to compare the two expansions (2.25) and (2.32), now both
considered for arbitrary $\gamma$. For small $\gamma\ll 1$, (2.25) is a good
expansion, i.e. has decreasing terms for initial rapidity, and up to
”critical” rapidity $\tilde{Y}_{C}\sim\frac{1}{\gamma}\ln\frac{1}{\gamma}$.
For higher rapidities the individual terms are growing and to make sense out
of it, the whole series has to be resummed. For large $\gamma\gg 1$ the
expansion is poor right from the very beginning, and has to be resummed even
at initial rapidity. As for (2.32), this for large $\gamma\gg 1$ is good at
any rapidity, since for high rapidities the individual terms only get smaller.
For small $\gamma\ll 1$ on the other hand, the expansion is poor at initial
rapidity and all the way up to $\tilde{Y}_{C}$. For higher rapidities,
$\tilde{Y}>\tilde{Y}_{C}$ however the expansion becomes good and gives
reliable results only keeping a small number of terms.

Again we stress for future reference, that the expansion eq.(2.25) for
$Y<Y_{c}$ is convergent and not asymptotic - all the terms in the series are
decreasing.

A comment is due here about the origin of the two sets of eigenvalues and the
corresponding eigenfunctions. The positive eigenvalues $n\gamma$ are the zero
dimensional analogs of the familiar n-Pomeron intercepts of QCD. These lead to
the exponential growth of the scattering amplitudes at weak coupling and small
rapidity. We will discuss the Pomeron calculus in more detail in the next
section. On the other hand we are not so familiar with the negative
eigenvalues $-n\gamma$ in QCD. The eigenfunctions corresponding to those are
singular at $u=1$ and therefore, as discussed above is not a good basis for
expansion at small rapidity. However for large rapidity where the scattering
matrix is small, these eigenfunctions dominate the $s$-matrix, as is obvious
from (2.32). Thus this set of eigenvalues governs the approach of the
scattering matrix to saturation. In the context of QCD the approach to
saturation in BK evolution is given by the Levin-Tuchin law LETU , which
indeed yields the $s$-matrix exponentially decreasing with rapidity. We thus
identify the largest negative eigenvalue with the zero dimensional incarnation
of the Levin-Tuchin decay. The rest of the negative eigenvalues provide
correction to this leading exponential decay, and presumably should exist in
QCD as well as subleading corrections close to saturation.

##  3 The UTM

The UTM was proposed in Ref.MUSA  (see also Refs.KLremark2 ; BIT ; utm ) as
the model that satisfies the $t$ channel unitarity condition. It was shown in
utm  that it is also $s$-channel unitary. Requiring $Y_{0}$ independence of
the $S$-matrix in Eq. (1.1) one is lead to the evolution equation for
$P^{\mbox{\tiny UTM}}_{n}\left(Y\right)$ MUSA ; KLremark2 ; BIT :

$\frac{dP^{\mbox{\tiny
UTM}}_{n}(Y)}{dY}\,=\,-\frac{\Delta}{\tilde{\Delta}_{1}}\left(1\,-\,e^{-\gamma
n}\right)P^{\mbox{\tiny
UTM}}_{n}(Y)\,\,+\,\,\frac{\Delta}{\tilde{\Delta}_{1}}\left(1\,-\,e^{-\gamma(n-1)}\right)\,P^{\mbox{\tiny
UTM}}_{n-1}(Y)$ (3.1)

with $\tilde{\Delta}_{1}=1-e^{-\gamma}\approx\gamma$ at weak coupling. While
$\tilde{\Delta}_{1}$ is a free parameter of the UTM model, the specific choice
is made so to exactly match the single Pomeron intercept of the BFKL cascade
model (see below).

The crucial difference between (Eq. (3.1)) and the BFKL cascade (Eq. (2.1)) is
that in the UTM the dipole emission probability saturates when the number of
dipoles is large $n>1/\gamma$ §§§in Ref. utmm  we have proposed an even more
realistic model in which multiple dipole emissions are allowed in a single
step of the evolution, but we are not considering it in this paper..

###  3.1 The generating functional

Again we define the generating functional as

$Z(u,\tilde{Y})\equiv\sum_{n}u^{n}\,P_{n}(\tilde{Y}).$ (3.2)

where $\tilde{Y}=\frac{\Delta}{\tilde{\Delta}_{1}}\,Y$. Notice that
$\tilde{Y}$ in the UTM is slightly different from that in the BFKL cascade. In
the UTM, the generating functional obeys the equation (see Eq. (2.16) of
Ref.utmm ):

$\frac{\partial}{\partial\tilde{Y}}Z(u,\tilde{Y})=-(1-u)\left(1-e^{-\gamma
u\frac{\partial}{\partial u}}\right)Z(u,\
Y)=\,(u-1)\Bigg{(}Z(u,\tilde{Y})-\,Z\left(e^{-\gamma}\,u,\,\tilde{Y}\right)\Bigg{)}$
(3.3)

with the initial and boundary conditions for a single dipole projectile:

$\displaystyle\mbox{Initial
condition:}~{}~{}Z\left(u,\tilde{Y}=0\right)\,=\,u$ (3.4)
$\displaystyle\mbox{Boundary
condition:}~{}~{}Z\left(u=1,\tilde{Y}\right)\,\,=\,\,1.$ (3.5)

The second condition (3.5) can be also imposed as the initial condition at
$\tilde{Y}=0$, since it is preserved by the evolution (3.3) at any
$\tilde{Y}$. As in the BFKL case, the evolution of the generating function is
generated by a non-Hermitian operator, which for the UTM is

${\cal H}_{UTM}\,=\,-\,(1-u)\,\left(1-e^{-\gamma u\frac{\partial}{\partial
u}}\right)$ (3.6)

###  3.2 The UTM Pomeron calculus

As is obvious from the previous discussion, the UTM is a well defined, t and s
- channel unitary model that yields a finite $S$-matrix which at large
rapidity falls to zero. It also exhibits saturation, as the dipole emission
probability (per unit rapidity) for large dipole number $n$ approaches a
constant as $n\rightarrow\infty$. Nonetheless, if we were to approach the
model from the point of view of Pomeron calculus based on the BFKL
approximation, we would have to face a serious problem.

The BFKL Pomeron calculus is understood as perturbation theory around the BFKL
limit of the Hamiltonian. To facilitate this we rewrite the UTM Hamiltonian as

$\mathcal{H}_{\rm
UTM}\,\,=\,\,-\,\,t\,\left(1-e^{\gamma(1-t)\frac{\partial}{\partial
t}}\right)\,\,=\,\,\underbrace{\tilde{\Delta}_{1}\,t\frac{\partial}{\partial
t}}_{\mathcal{H}_{0}}\,-\,\underbrace{t\,\left(1-e^{\gamma(1-t)\,\frac{\partial}{\partial
t}}\,\,+\,\,\tilde{\Delta}_{1}\,\frac{\partial}{\partial\,t}\right)}_{\mathcal{H}_{I}}$
(3.7)

where $t=1-u$. Using the variable $t$ rather than $u$ is natural at weak
coupling/small rapidities as it has the meaning of the dipole-dipole
scattering amplitude and is small in this regime.

The choice of $\mathcal{H}_{0}$ dictated by the interpretation of the zero
dimension models as the QCD with fixed dipole size BFKL ; KOLEB . At zeroth
order in the perturbation, the eigenfunctions of $\mathcal{H}_{UTM}$ are pure
powers

$\mathcal{H}_{0}\,t^{n}\,=\,\Delta^{0}_{n}\,t^{n};\ \ ~{}~{}~{}\ \
\Delta^{0}_{n}\,=\,\tilde{\Delta}_{1}\,n\,.$ (3.8)

Although (3.8) is valid for arbitrary $n$ in the context of the Pomeron
calculus we choose to consider positive integer values of $n$ only. This
choice corresponds to summation of BFKL ladders in QCD, and ensures that
expansion of the scattering matrix in Taylor series in $t$ is interpretable as
expansion in multi Pomeron exchanges. In the same spirit we identify the
perturbative BFKL Pomeron with the first eigenfunction
${I\\!\\!P}(\tilde{Y})\equiv
t\,e^{{\tilde{\Delta}_{1}}\tilde{Y}}\,=\,t\,e^{{\Delta}Y}$.

We can write the Hamiltonian in terms of the ”Schroedinger picture” Pomeron
field ${I\\!\\!P}(\tilde{Y}=0)\equiv t$. In addition to $\mathcal{H}_{0}$, the
Hamiltonian contains $n\rightarrow m$ Pomeron vertices

$\displaystyle\mathcal{H}_{\rm UTM}$ $\displaystyle=$
$\displaystyle\tilde{\Delta}_{1}\,t\frac{\partial}{\partial
t}-\sum_{m=1}^{\infty}\sum_{n=m-1}^{\infty}(-1)^{m-1}\left[1-\delta_{m=1,n=0}-\delta_{m=1,n=1}\right]\frac{1}{(m-1)!(n-m+1)!}\tilde{\Delta}_{1}^{n}t^{m}\left(\frac{\partial}{\partial
t}\right)^{n}$ (3.9) $\displaystyle=$
$\displaystyle\tilde{\Delta}_{1}\,t\frac{\partial}{\partial
t}-\sum_{m,n}^{\infty}V_{m}^{n}t^{m}\left(\frac{\partial}{\partial
t}\right)^{n}$

with

$V_{m}^{n}=(-1)^{m-1}\frac{1}{(m-1)!(n-m+1)!}\tilde{\Delta}_{1}^{n};\ \ \ \ \
n\geq m-1;\ \ \ \ m\geq 1$ (3.10)

except $(m=1,n=0)$ and $(m=1,n=1)$.

The interaction contains vertices ${I\\!\\!P}^{n}\to{I\\!\\!P}^{m}$ for an
infinite number of values of $n$ and $m$ even though in terms of the number of
dipoles, the UTM Hamiltonian only allows emission of a single dipole in one
step of the evolution (see Eq. (3.1)). The UTM is therefore an example of a
simple Pomeron calculus with the infinite number of vertices, which describes
a fully unitary system with saturation.

There are three distinct types of Pomeron vertices in (3.9). The first type is
$V_{m}^{n}$ with $n=m-1$. These are the Pomeron splittings. All the splitting
vertices increase the number of bare Pomerons by exactly one. The second type
is $V_{m}^{n}$ with $n>m$. Those are the Pomeron merging vertices. In UTM any
number of Pomerons can disappear as a result of a merging via a single merging
vertex. Finally the third type are vertices with $n=m$. Those preserve the
number of Pomerons, but give corrections directly to the intercept of the
$n$-Pomeron state (see Fig. 2).

Imagine now a perturbative calculation of an $n$-Pomeron intercept in this
Pomeron calculus. In the language of the previous section that corresponds to
calculating perturbatively eigenvalues of $\mathcal{H}_{UTM}$. It is the
common lore that the first most important contribution in such a calculation
comes from the so called ”enhanced diagrams”, i.e. diagrams with small Pomeron
loops. There are many types of such diagrams in the UTM, since there exist
merging vertices where any number of Pomerons disappear. Thus, for example one
contribution to two Pomeron propagator would be a diagram where each one of
the two Pomerons split into two, and subsequently the resulting four Pomerons
merge into two via $V_{2}^{4}$. At any rate the enhanced diagrams are those
where both the splitting and the merging vertices are close to each other in
rapidity. The size in rapidity of a fluctuation from $n$ to $m$ Pomerons is
given by the usual ”time-energy” uncertainty relation $\Delta
Y\sim\frac{1}{\Delta^{0}_{m}-\Delta^{0}_{n}}\sim\frac{1}{\bar{\gamma}}$.
”Integrating out” this time scale leads to ”renormalization” of the
$n\rightarrow n$ vertices, generating effective vertices $\tilde{V}_{n}^{n}$.
In the second step we would have to resum all diagrams with insertions of
these effective $\tilde{V}_{n}^{n}$ vertices. Those diagrams yield corrections
to the $n$-Pomeron intercepts.

The resulting values of $n$-Pomeron intercepts in principle can be either
smaller or larger than their bare values. The latter option is problematic in
view of the fact that in order to calculate the $S$ matrix one has to sum over
all dressed Pomeron exchanges. If the $n$-Pomeron intercept grows faster than
$n$, such a series is not Borel summable BORSUM , and making sense of such an
expansion presents a serious challenge. In fact one could be sufficiently
discouraged to believe that the model itself is not well defined. In QCD, for
example the simple $1/N_{c}^{2}$ corrections ($N_{c}$ is the number of colors)
lead to the intercepts which are larger than the intercepts of n-BFKL
Pomerons¶¶¶ The very brief review of the problem, references and the most
pessimistic point of view can be found in Ref.LENC  . . Many experts therefore
believe that the Pomeron calculus in QCD has a deadly problem in the order
$1/N^{2}_{c}$ (including one of us LENC ).

In this context UTM provides an argument against such pessimism. Although we
are not going to perform explicitly perturbative calculation of the type
described above in UTM, from the results presented later in this section it
will become clear that the perturbative Pomeron calculation does lead to
positive corrections to the $n$-Pomeron intercepts. In fact, as we will show
these intercepts behave as $\Delta_{n}=e^{\gamma n}-1$ and grow exponentially
at large $n$. Nevertheless as we have stressed, this does not signify a
problem with the model, as the $S$-matrix in UTM is well defined. We will show
below, that even though standard Borel transform is not applicable to
summation of the series of multi Pomeron exchanges, this series can be indeed
summed with a finite and well defined result. We hope this may teach us
something about dealing with a similar apparent problem of the QCD Pomeron
calculus.

Figure 2:  The typical Pomeron diagrams in the UTM. Fig. 2-a: the diagrams for
two Pomeron Green’s function. Fig. 2-b: the diagrams for the three Pomeron
Green’s function. $\tilde{V}^{2}_{2}$ is effective vertex for
${I\\!\\!P}^{2}\to{I\\!\\!P}^{2}$ interaction, while $\tilde{V}^{3}_{3}$ is
the effective vertex for ${I\\!\\!P}^{3}\to{I\\!\\!P}^{3}$ interaction .

In the rest of this section we discuss the solution of the ”Shroedinger
equation” (3.3) and the calculation of the physical scattering matrix.

###  3.3 The Eigenfunctions

We now proceed to solve UTM along the same lines as the BFKL cascade model in
the previous section. The first step in this approach is to find the
eigenfunctions of the UTM Hamiltonian.

####  3.3.1 Right eigenfunctions

It has distinct sets of right and left eigenfunctions. The right
eigenfunctions are relevant for solving the equation for the generating
function, or $S$-matrix. The left eigenfunctions are also interesting in their
own right, since as we will show below, they are relevant for expanding the
probabilities.

We consider the right eigenfunctions first. Let
$\Phi_{n}\left(u,\gamma\right)$ denote the set of eigenfunctions with positive
eigenvalues $\Delta_{n}$:

${\cal
H}_{UTM}\,\Phi_{n}(u)\,=\,\Delta_{n}\,\Phi_{n}(u)\,~{}~{}\rightarrow\,~{}~{}\Delta_{n}\,\,\Phi_{n}\left(u\right)\,\,=\,(u-1)\Bigg{(}\Phi_{n}\left(u\right)\,\,-\,\Phi_{n}\left(e^{-\gamma}\,u\right)\Bigg{)};$
(3.11)

By inspection we find a set of positive eigenvalues
$\Delta_{n}=e^{\gamma\,n}\,-\,1$ with the eigenfunctions

$\Phi_{n\geq
1}\left(u\right)\,\,=\,\,\prod^{n-1}_{l=0}\left(e^{-\gamma\,l}\,\,-\,\,\frac{1}{u}\right)\,,~{}~{}~{}~{}~{}~{}~{}~{}\Phi_{0}\left(u\right)\equiv
1\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Delta_{n}=e^{\gamma\,n}\,-\,1\,.$ (3.12)

Notice that $\Phi_{n}\left(u\right)$ are well behaved near $u=1$. In
particular $\Phi_{n>1}\left(u=1\right)=0$. On the other hand,
$\Phi_{n}\left(u\right)$ diverge at $u\rightarrow 0$. The functions $\Phi_{n}$
share these properties with $\Phi_{n}^{BFKL}$. However as opposed to
$\Phi_{n}^{BFKL}$ which has a multiple pole at $u=0$ for $n>1$, $\Phi_{n}$ has
a single pole for any $n$. The functions $\Phi_{n}\left(u\right)$ could be
expressed in terms of the q-Pochhammer symbols (see formula 6.1.22 of Ref.AS
and Ref.QP ).

In the formal limit $\gamma\rightarrow 0$ one has
$\Phi_{n}(u)\rightarrow\Phi_{n}^{BFKL}(u)$, and also
$\Delta_{n}\rightarrow\Delta^{0}_{n}$. Since $\Delta_{n}$ is perturbatively
expandable in $\gamma$, this is the multi-Pomeron intercept one would obtain
from perturbative approach described in the previous subsection. We thus
indeed see that the resummed perturbation theory leads to n-Pomeron intercept
that grows with $n$ exponentially, as advertised in the previous subsection.
Our goal in calculating the $S$-matrix is then to sum the series in
$n$-Pomeron exchanges.

Formally representation of the $S$-matrix in terms of these exchanges is
equivalent to expansion of the generating function that solves (3.3) in the
basis of eigenfunctions $\Phi_{n}$:

$Z\left(u,\tilde{Y}\right)\,\,=\,\,\sum^{\infty}_{n=0}C_{n}\,\,\Phi_{n}\left(u\right)e^{\Delta_{n}\,\tilde{Y}}\,.$
(3.13)

In this expansion the boundary conditions at $u=1$ are trivially satisfied by
choosing $C_{0}=1$.

To find the coefficients in the expansion eq.(3.13) we need to find the set of
left eigenfunctions $\tilde{\Psi}_{n}$ and (with proper normalization) use

$\langle\tilde{\Psi}_{n}|\Phi_{m}\rangle\,=\,\delta_{mn}$ (3.14)

with appropriate definition of scalar product. The result, that we derive
below, is that for a single dipole projectile

$C_{n}\rightarrow C_{n}^{(1)}=e^{\frac{\gamma}{2}n(n+1)}.$ (3.15)

We can understand a little better the nature of the series in eq.(3.13) by
scrutinizing the small $\gamma$ limit, and considering the physical point
$u=e^{-\gamma}$ (which corresponds to the single dipole target). In this limit
we have

$\Phi_{n}(e^{-\gamma})\approx(-1)^{n}\gamma^{n}n!;\ \ ~{}~{}~{}~{}~{}~{}\ \
n\ll 1/\gamma$ (3.16)

and

$\Phi_{n}(e^{-\gamma})\approx(-1)^{n}e^{\gamma\left(n-\frac{1}{\gamma}\right)};\
\ ~{}~{}~{}~{}~{}\ \ \ \ \ n\gg 1/\gamma$ (3.17)

The $n$-th term in the expansion (3.13) is
$(-1)^{n}C_{n}\Phi_{n}e^{\Delta_{n}\tilde{Y}}$. As opposed to the BFKL case,
where the analogous series was convergent for small values of $\tilde{Y}$, in
the UTM this is an asymptotic series even for vanishing $\tilde{Y}$. The
magnitudes of the successive terms decrease until about $n\sim 1/\gamma$, and
thereafter increase very fast due to the Gaussian growth of $C_{n}$. This
behavior persists at small $\tilde{Y}$ until the rapidity parametrically
reaches the critical value
$\tilde{Y}_{c}\approx\frac{1}{\gamma}\ln\frac{1}{\gamma}$. For larger
rapidities the terms in the series grow starting at $n=1$ and the series is
divergent. So quantitatively this is similar to BFKL in the sense that at
small $Y$ the series can be truncated, although this time it is not convergent
but asymptotic. At large $\tilde{Y}$ the series is very problematic, as the
eigenvalues $\Delta_{n}$ grow exponentially, and thus a clever resummation is
necessary to make sense of the result.

Just like in the BFKL case, the operator ${\cal H}_{UTM}$ also has another set
of eigenfunctions, which we denote as $\Psi_{n}\left(u\right)$, with negative
eigenvalues $-\tilde{\Delta}_{n}$.

${\cal
H}_{UTM}\,\Psi_{n}(u)\,=\,-\,\tilde{\Delta}_{n}\,\Psi_{n}(u)\,~{}~{}~{}\rightarrow\,~{}~{}~{}-\tilde{\Delta}_{n}\,\,\Psi_{n}\left(u\right)\,\,=\,(u-1)\Bigg{(}\Psi_{n}\left(u\right)\,\,-\,\Psi_{n}\left(e^{-\gamma}\,u\right)\Bigg{)}$
(3.18)

The eigenvectors corresponding to the negative eigenvalues are

$\Psi_{n\geq
1}\left(u\right)\,\,=\,\prod^{n}_{l=1}\left(e^{\gamma\,l}\,\,-\,\,{1\over
u}\right)^{-1}\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\tilde{\Delta}_{n}\,=\,1\,-\,e^{-\gamma\,n}\,.$
(3.19)

The functions $\Psi_{n}(u)$ are well behaved both at $u=0$ and $u=1$, however
they have poles at the physical points $u=e^{-\gamma k}$: in fact
$\Psi_{n}(u)$ has $n$ poles at $u_{k}=e^{-\gamma\,k}$ for all integer $k$ in
the interval $(1\leq k\leq n)$. This is significant as we will see in a short
while.

The most interesting point for us that corresponds to a single dipole
scattering is $u=e^{-\gamma}$. At this point all the functions are singular
with the behavior at the pole

$\Psi_{n}(u\rightarrow e^{-\gamma})=\frac{1}{u-e^{-\gamma}}\
e^{-\frac{\gamma}{2}(n+2)(n-1)}\prod_{l=2}^{n}\frac{1}{1-e^{-\gamma(l-1)}}$
(3.20)

Decomposing the general solution of Eq. (3.3) in this basis, we write

$Z\left(u,\tilde{Y}\right)\,\,=\,\,\sum^{\infty}_{n=1}B_{n}\,\,\Psi_{n}\left(u\right)e^{-\tilde{\Delta}_{n}\,\tilde{Y}}\,.$
(3.21)

We note that the fact that $\Psi_{n}(u)$ have poles at physical values of $u$
does not by itself mean that the expansion of the generating function in this
basis is nonsensical. As long as all the poles are of the same order (and in
our case all are indeed simple poles) a simple rescaling by a common
(divergent) factor will make the values of $\Psi_{n}$ at all the physical
points finite, and everywhere else vanishing. We will not pursue this strategy
here and instead continue working with the functions $\Psi_{n}(u)$ defined
above. The price we will have to pay however, is that we will not be able to
determine the coefficients $B_{n}$ by themselves, but only their contribution
to the physical $S$-matrix.

####  3.3.2 Left eigenfunctions

Let us now consider left eigenfunctions of ${\cal H}_{UTM}$. These are the
same as right eigenfunctions of the operator ${\cal H}_{UTM}^{\dagger}$
(notice the anti-Hermitian property of the dilatation operator):

${\cal H}^{\dagger}_{UTM}=-\left(1-e^{\gamma u\frac{\partial}{\partial
u}}\right)(1-u)\,,$ (3.22)

Explicitly the equations satisfied by the left eigenfunctions are

$\lambda_{n}g_{n}(u)\,=\,(u-1)\,g_{n}(u)\,-\,(ue^{\gamma}-1)\,g_{n}(ue^{\gamma})$
(3.23)

We again find two sets of solutions

$\displaystyle\tilde{\Phi}_{n}\left(u\right)\,={1\over(1-u)}\,\prod^{n-1}_{l=0}\left(e^{\gamma\,l}\,\,-\,\,\frac{1}{u}\right)\,,~{}~{}~{}~{}~{}\tilde{\Phi}_{0}\left(u\right)\,=\,(1-u)^{-1}\,,~{}~{}~{}~{}~{}\lambda_{n}=-\,\tilde{\Delta}_{n}\,=\,-\,(1\,-\,e^{-\gamma\,n}).$
(3.24)

and

$\tilde{\Psi}_{n}\left(u\right)\,\,=-\,{1\over
u}\,\prod^{n}_{l=0}\left(e^{-\gamma\,l}\,-\,{1\over
u}\right)^{-1}\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\tilde{\lambda}_{n}\,=\,\Delta_{n}\,=\,e^{\gamma\,n}\,-\,1.$
(3.25)

The left eigenfunctions are regular for physical values of $u$, i.e. $0\leq
u\leq 1$, with $\tilde{\Phi}_{n}(e^{-\gamma k})=0$ for $k<n$. On the other
hand for $u>1$ the functions $\tilde{\Psi}_{n}(u)$ have poles at $u=e^{\gamma
k}$ for $k\leq n$.

For the record we note that $\tilde{\Phi}_{n}$ obey two recurrent relations:

$\tilde{\Phi}_{k}\left(e^{-\gamma
n}\right)\,=\,\left[e^{\gamma\,(k-1)}\,-\,e^{\gamma\,n}\right]\,\tilde{\Phi}_{k-1}\left(e^{-\gamma
n}\right)\,,~{}~{}~{}~{}~{}~{}~{}\tilde{\Phi}_{k}\left(e^{-\gamma
n}\right)\,=\,{1-e^{-\gamma\,(n-1)}\over
e^{-\gamma\,k}-e^{-\gamma\,n}}\,\tilde{\Phi}_{k}\left(e^{-\gamma(n-1)}\right)\,$
(3.26)

####  3.3.3 Expansion Coefficients $C_{n}$ and Exponential Moments

To calculate the expansion coefficients $C_{n}$ in (3.13) we use a somewhat
roundabout procedure. Although the generating function $Z(u)$ in the physical
region (where it represents the scattering matrix) is defined for $0\leq u\leq
1$, we can consider it as a function of all real $u$. We assume that the
expansion (3.13) holds for all $u$ with the same coefficients $C_{n}$.

Define an auxiliary variable $\xi_{n}\equiv e^{\gamma n}$. The value of $Z$ at
the points $u=e^{\gamma k}$ is equal to the $k$-th exponential moment:

$\,Z(e^{\gamma\,k},Y)=M_{k}\left(Y\right)\,\equiv\,\left\langle(\xi_{n})^{k}\right\rangle\,\,=\,\,\sum^{\infty}_{n=1}\xi_{n}^{k}\,P_{n}\left(Y\right)\,$
(3.27)

The moments are expanded in terms of the right eigenfunctions as

$M_{k}\left(Y\right)\,=\,\sum^{\infty}_{n=0}C_{n}\,\,\Phi_{n}\left(\xi_{k}\right)\,e^{\Delta_{n}\,\tilde{Y}}\,=\,\sum^{k}_{n=0}C_{n}\,\,\Phi_{n}\left(\xi_{k}\right)\,e^{\Delta_{n}\,\tilde{Y}}$
(3.28)

where the last equality holds since $\Phi_{n}(\xi_{k})=0$ for $n>k$ as is
obvious from (3.12). To find the coefficients $C_{n}$ we introduce the inverse
matrix $Q_{m}^{k}$ that satisfies

$\sum_{k=0}^{\infty}Q_{m}^{k}\,\Phi_{n}\left(\xi_{k}\right)\,=\,\sum_{k=n+1}^{\infty}Q_{m}^{k}\,\Phi_{n}\left(\xi_{k}\right)\,=\,\delta_{nm}$
(3.29)

Hence

$\sum_{k}Q_{m}^{k}\,M_{k}\,=\,C_{m}\,e^{\Delta_{m}\,\tilde{Y}}$ (3.30)

We now utilize the theorem about the orthogonality of left and right
eigenfunctions. It is straightforward to check that the appropriate measure
under which this orthogonality holds is

$\frac{1}{2\pi i}\oint_{\,\Gamma_{3}}{du\over
u}\,\tilde{\Psi}_{m}(u)\,\Phi_{n}\left(u\right)\,=\,\delta_{mn}$ (3.31)

Here the contour $\Gamma_{3}$ encloses the real axis to the right (and
including) the point $u=1$, and thus encircles all the poles of
$\tilde{\Psi}_{m}$, (see Fig. 1). Using this relation we find

$C_{n}\,=\frac{1}{2\pi i}\,\oint_{\,\Gamma_{3}}{du\over
u}\,\tilde{\Psi}_{n}(u)\,Z\left(u,Y=0\right)$ (3.32)

For the $k$-dipole target initial condition, $Z\left(u,Y=0\right)\,=\,u^{k}$
this is:

$C_{n}^{(k)}\,=\,\prod_{l=1}^{n}{e^{\,l\,\gamma}\,(e^{\,\gamma\,(l+k-1)}-1)\over
e^{\,l\,\gamma}-1}$ (3.33)

For $k=1$,

$C_{n}^{(1)}\,=\,e^{\gamma\,n(n+1)/2}\,.$ (3.34)

which is the result quoted in the beginning of this section.

###  3.4 The Probabilities

One interesting property of the operator ${\cal H}_{UTM}$ is that it’s action
on a function of continuous variable, $\phi(u)$ is such that it only relates
the values of the function at discrete values of its argument $\phi(e^{-\gamma
k}u)$. This is very different from an action of a differential operator, such
as ${\cal H}_{BFKL}$ (2.13). Thus, although we have been treating the
eigenfunctions as continuous functions of continuous variable, strictly
speaking the eigenvalue equation only determines eigenfunctions at discrete
points $e^{-\gamma k}u_{0}$ for an arbitrary value of $u_{0}$. Since the point
$u=1$ is a physical point for scattering amplitude, our calculations in the
previous sections corresponded to choosing $u_{0}=1$ and determining the
generating function at points $u_{k}=e^{-\gamma k}$. The value of the
generating function at these particular values is indeed physically
significant, since it is equal to the scattering matrix for a $k$-dipole
target. The values of any eigenfunction at other points, excluding this
discrete set, is to some extent arbitrary and were chosen to make functions
continuous.

We can use this property of ${\cal H}_{UTM}$ to cast the equation for
probabilities in the form similar to that for the generating function. Let us
formally extend the probabilities to a ”probability function” - a function of
continuous variable such that

$P_{n}(\tilde{Y})\rightarrow P(u,\tilde{Y});\ \ \ \ ~{}~{}~{}~{}~{}~{}~{}~{}\
\ \ P(u=e^{-\gamma n},\tilde{Y})=P_{n}(\tilde{Y})$ (3.35)

The equation (3.1) for $P^{UTM}_{n}$ can be rewritten as an equation for the
function $P(u)$

$\frac{dP(u,\tilde{Y})}{d\tilde{Y}}\,=\,{\cal
H}_{UTM}^{\dagger}\,P(u,\tilde{Y})~{}~{}~{}~{}~{}$ (3.36)

The function $P(u)$ can be expanded in the set of left eigenfunctions of
${\cal H}_{UTM}$,

$P\left(u,\tilde{Y}\right)\,=\,\sum^{\infty}_{k=0}\tilde{C}_{k}\,\tilde{\Phi}_{k}\left(u\right)\,e^{-\tilde{\Delta}_{k}\,\tilde{Y}}=\,\sum^{\infty}_{k=1}\tilde{B}_{k}\,\tilde{\Psi}_{k}\left(u\right)\,e^{\Delta_{k}\,\tilde{Y}}\,.$
(3.37)

The coefficients $\tilde{C}_{k}$ are found by using the orthogonality
condition

$\frac{1}{2\pi i}\oint_{\Gamma}{du\over
u}\,\Psi_{m}(u)\,\tilde{\Phi}_{n}\left(u\right)\,=\,\delta_{mn}$ (3.38)

where the contour $\Gamma$ encircles the physical interval $0<u\leq 1$,
excluding the origin $u=0$ (see Fig. 1). This relation can be verified
directly using the explicit form of the functions $\Psi_{n}$ and
$\tilde{\Phi}_{n}$.

For the single dipole initial condition, at initial rapidity the function
$P(u)$ should be chosen such that its value is equal to unity at
$u=e^{-\gamma}$, and the function vanishes quickly away from this point along
the real axis. For example
$P(u,\tilde{Y}=0)=e^{-\left[\Lambda(u-e^{-\gamma})^{2}\right]}$ with
$\Lambda\gg\frac{1}{\gamma}$ will do. For such functions, in the integral
$\frac{1}{2\pi i}\oint_{\Gamma}{du\over u}\,\Psi_{k}(u)P(u,\tilde{Y}=0)$ only
the pole of $\Psi_{k}(u)$ at $u=e^{-\gamma}$ contributes, and we find

$\displaystyle\tilde{C}_{0}$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\tilde{C}_{1}$ $\displaystyle=$ $\displaystyle-e^{-\gamma}$
$\displaystyle\tilde{C}_{m}\,$ $\displaystyle=$
$\displaystyle\,\bar{Q}_{m>1}^{1}\,=\,-\,2^{(1-m)}\,e^{-\gamma\,m(m+3)/4}\,\prod_{l=1}^{m-1}{1\over\sinh[\gamma\,l/2]};~{}~{}~{}~{}~{}~{}\
\ \ m>1;$ (3.39)

Note that since $\tilde{\Phi}_{k}\left(e^{-\gamma n}\right)\,=\,0$ for all
$k>n>0$, the probability function at physical points where it is equal to the
dipole probability in the cascade, is in fact a finite sum

$P_{n}^{UTM}\left(\tilde{Y}\right)\,=\,\sum^{n}_{k=1}\tilde{C}_{k}\,\tilde{\Phi}_{k}\left(e^{-\gamma
n}\right)\,e^{-\tilde{\Delta}_{k}\,\tilde{Y}}$ (3.40)

In principle we can find similarly the coefficients $\tilde{B}$ in the second
expansion in (3.37). However it is not clear to us that expansion of the
probability function in positive exponents of rapidity can ever be useful, and
we will not do it here.

##  4 Dipole scattering in the UTM

Our goal now is to understand the behavior of the $S$-matrix in the UTM and to
show that notwithstanding apparent problems with Pomeron calculus it can be
calculated. In particular we will explicitly calculate the asymptotic behavior
at large $\tilde{Y}$.

Since UTM is frame independent by construction, below we carry calculations in
the frame which is the most convenient for us, and that is the target rest
frame.

###  4.1 Scattering on a target dipole

From Eq. (1.1), the $S$-matrix on a single target dipole reads

$S\left(\tilde{Y}\right)\,\,=\,\,Z\left(e^{-\gamma},\tilde{Y}\right)\,\,=\,\,\sum^{\infty}_{n=1}e^{-\gamma\,n}\,P_{n}^{UTM}\left(\tilde{Y}\right)$
(4.1)

Using (3.40) and expanding in powers of $\tilde{Y}$ we can write it as

$S(\tilde{Y})=e^{-\,\tilde{Y}}\sum^{\infty}_{j=0}\sum_{n=1}^{j+1}\frac{\tilde{Y}^{j}}{j!}e^{-\gamma}\,{e^{-\gamma
nj}}\prod^{n-2}_{l=0}\left(1\,\,-\,\,e^{\gamma n(l-j)}\right)\ $ (4.2)

We will use this expression for numerical evaluation in the following.

In the previous section we have provided two expansions for $S(\tilde{Y})$ \-
expansion in multi Pomeron amplitudes and expansion in decreasing exponents of
rapidity. We will show in this section that both lead to the same asymptotic
result for the $S$-matrix. We start with the expansion in Pomeron exchanges.

####  4.1.1 Pomeron calculus

We start with the representation of $S$-matrix in terms of ”dressed” multi-
Pomeron exchanges

$S\left(\tilde{Y}\right)\,\,=\,\,\sum^{\infty}_{n=0}C_{n}^{(1)}\,\Phi_{n}\left(e^{-\gamma}\right)\,e^{\Delta_{n}\,\tilde{Y}}\
$ (4.3)

Here $\Delta_{n}=e^{\gamma n}-1$ is the $n$-Pomeron intercept (in $\tilde{Y}$
variable) ∥∥∥Naively one might want to identify $C_{n}\Phi_{n}$ as the product
of the residues (impact factors), with $C_{n}$ being related to the projectile
while $\Phi_{n}$ to the target. However, residues factorize for bare single
Pomeron exchange only. Beyond one Pomeron exchange the factorizations breaks
down. .

Eq. (4.3) is an asymptotic series, which, at first glance, cannot be summed
because the coefficients grow too fast with $n$: $\Delta_{n}\,>\,\gamma\,n$,
while
$C_{n}^{(1)}\left(\gamma\right)=\exp\left(\frac{n(n+1)}{2}\,\gamma\right)$
grows as Gaussian, which is badly divergent too ******In Appendix A we suggest
how to sum the asymptotic series with the Gaussian growth of the
coefficients.. In addition $\Phi_{n}$ also grows with $n$ exponentially. The
only redeeming feature of this expression is that the terms in the series are
alternating sign since which is clear from (3.12).

Summing series of the type Eq. (4.3) is a long standing problem in the Pomeron
calculus (see Ref.LENC  for a short review). Summation of asymptotic series
entails finding an analytical function with identical series expansion. Borel
summation BORSUM  is one of the standard tools of the trade in this business.
The series Eq. (4.3) however cannot be summed via Borel procedure due to the
aforementioned very fast growth of the coefficients. We will therefore
rearrange the series in the following way. We first expand the exponential
factors using the explicit form of $\Delta_{n}$ as

$\exp\left(\Delta_{n}\,\tilde{Y}\right)\,\,=\,\,e^{-\,\tilde{Y}}\sum^{\infty}_{j=0}\frac{e^{\gamma\,n\,j}\,\tilde{Y}^{j}}{j!}.$
(4.4)

Substituting Eq. (4.4) in Eq. (4.3) we obtain:

$\displaystyle
S\left(\tilde{Y}\right)\,\,=\,\,Z\left(e^{-\gamma},\tilde{Y}\right)$
$\displaystyle=$ $\displaystyle
e^{-\,\tilde{Y}}\sum^{\infty}_{j=0}\sum^{\infty}_{n=0}\frac{e^{\gamma\,n\,j}\,\tilde{Y}^{j}}{j!}C_{n}^{(1)}\Phi_{n}\left(e^{-\gamma}\right)$
(4.5)

In Eq. (4.5) we have two series: an absolutely convergent sum over $j$ and an
asymptotic series over $n$.

|
---|---
Figure 3:  $S\left(\tilde{Y}\right)$ versus $\tilde{Y}$ for (i) exact solution
of Eq. (4.2), and (ii) small $\gamma$ approximation of Eq. (4.8). For
$\gamma=0.025$ (left plot), the curves coincide with $0.1\%-1\%$ accuracy. For
$\gamma=0.25$ (right plot), the accuracy ranges from $5\%$ to $20\%$ in the
displayed interval, worsening at larger rapidities.

#### Small $\gamma$ limit

We first explore the limit $\gamma\ll 1$. For small $\gamma$ we have
$C_{n}^{(1)}\simeq 1$ and
$\Phi_{n}\left(1-\gamma\right)\,\,=\,\,\left(-\gamma\right)^{n}\,\,n!\,\,+\,\,O\left(\gamma^{n+1}\right)$.
Keeping only leading terms we can write

$S\left(\tilde{Y}\right)\,\,=\,\,\,\,\,\,\sum^{\infty}_{n=0}\left(-\gamma\right)^{n}\,\,n!\,\,e^{\Delta_{n}\,\tilde{Y}}\,\,\Bigg{\\{}1\,\,\,+\,\,{\cal
O}\left(\gamma\right)\Bigg{\\}}\,,\,\quad\quad\,\gamma\ll 1\,.$ (4.6)

Neglecting subleading terms may seem risky in view of our discussion in
Section 2. In particular keeping only the leading term for $C_{n}$ we omit the
Gaussian enhancement at large $n$. However it is the presence of exponentially
growing $\Delta_{n}$ that poses the main problem for resummation, and we keep
$\Delta_{n}$ in full glory in (4.6).

If we were to expand $\Delta_{n}$ for formally small $\gamma$, that is
$\Delta_{n}\simeq n\gamma$, we would reproduce the large $Y$ limit of the BFKL
cascade model in the center of mass frame (2.6d).

Applying the expansion (4.4) to (4.6) we have,

$S\left(\tilde{Y}\right)\,\,=\,\,\,e^{-\,\tilde{Y}}\sum^{\infty}_{j=0}\sum^{\infty}_{n=0}\frac{e^{\gamma\,n\,j}\,\tilde{Y}^{j}}{j!}\,\left(-\gamma\right)^{n}\,\,n!$
(4.7)

The asymptotic series in $n$ is Borel summable and gives

$S\left(\tilde{Y}\right)\,\,=\,\,\,\,e^{-\,\tilde{Y}}\sum^{\infty}_{j=0}\frac{\tilde{Y}^{j}}{j!}\frac{1}{\gamma}e^{-(j+1)\gamma}\exp\left(\frac{1}{\gamma}e^{-(j+1)\gamma}\right)\,\Gamma\left(0,\frac{1}{\gamma}e^{-(j+1)\gamma}\right)$
(4.8)

For large $\tilde{Y}\gg 1$, the sum is dominated by terms with large $j$, and
$\frac{1}{\gamma}e^{-(j+1)\gamma}\ll 1$. Taking the asymptotic form of the
incomplete $\Gamma$-function at small argument, we get

$\displaystyle S\left(\tilde{Y}\right)\,\,$ $\displaystyle=$
$\displaystyle\,\,\,\,e^{-\,\tilde{Y}}\sum^{\infty}_{j=0}\frac{\tilde{Y}^{j}}{j!}\left(\frac{1}{\gamma\xi_{j}}\left(\ln\left(\gamma\xi_{j}\right)\
\,-\gamma_{E}-\frac{1}{4}\gamma\right)\right)$ (4.9) $\displaystyle\,=\,$
$\displaystyle
e^{-(1-e^{-\gamma})\tilde{Y}}\left(e^{-2\gamma}\,\tilde{Y}+e^{-\gamma}\left(\frac{1}{\gamma}\left(\ln\gamma-\gamma_{E}\right)+\frac{3}{4}\right)\right)$

$\gamma_{E}=0.577$ is the Euler constant. Eq. (4.9) gives the leading
asymptotic behavior of the $S$-matrix at small $\gamma$. The validity of this
approximation is demonstrated in Fig. 3 alongside the exact result. We observe
that even for $\gamma=0.25$ the small $\gamma$ approximation is very good
unless extremely large rapidities are considered.

Note that even though we started with expansion of $S$ in multi-Pomeron
exchanges which grow exponentially, the final result (4.9) is given by the
first decreasing exponent $-\tilde{\Delta}_{1}$. This suggests that, as in the
BFKL limit at large rapidity the appropriate expansion should be that in the
negative exponents $e^{-\tilde{\Delta}_{n}\tilde{Y}}$ rather than in the
multi-Pomeron exchanges. However, the presence of the prefactor $\tilde{Y}$ in
(4.9) indicates that a straightforward expansion of $S$-matrix in this basis
is not possible, as the asymptotics is not given by a pure exponential. Indeed
let us try to find coefficients $B_{n}$ in (3.21). For this we should use the
following orthogonality condition

$\frac{1}{2\pi
i}\oint_{\Gamma}\frac{1}{u}\Psi_{m}(u)\tilde{\Phi}_{m}(u)=\delta_{mn}$ (4.10)

However, naively calculating the overlap with the initial condition
$Z(u,\tilde{Y}=0)=u$ we find

$B_{m}=\frac{1}{2\pi i}\oint_{\Gamma}\frac{1}{u}u\ \tilde{\Phi}_{m}(u)=0$
(4.11)

as the integrand has no poles to the right of $u=0$. This result is consistent
with our observation below (3.21), but is not helpful for finding the
$S$-matrix. Thus to find a better approximation for the $S$-matrix we need to
look for a somewhat more refined method than just formal expansion in the
basis $\tilde{\Psi}(u)$. In the rest of this section we develop such a method
based on the analytic continuation.

####  4.1.2 The analytic continuation

We start with the expansion of the generating function in multi-Pomeron
exchanges, but represent it formally as an integral in the complex $n$-plane:

$Z\left(u,\tilde{Y}\right)\,\,=\,{1\over 2\pi i}\,\oint_{C}{(-1)^{n}\pi
dn\over\sin(\pi\,n)}\,C_{n}^{(1)}\,\,\Phi_{n}\left(u\right)e^{\Delta_{n}\,\tilde{Y}}$
(4.12)

where the contour $C$ runs along the real positive semi-axes (see Fig. 4). Let
us assume that all the functions of $n$ can be analytically continued to the
complex plane in such a way that the integrand vanishes fast enough at
infinity. In this case there is no contribution from the circle at infinity
and we can deform the integration contour and put it along the real negative
semi-axes, the contour $C^{\prime}$.

Figure 4:  Contours of integration in the complex $n$ plane.

To integrate along $C^{\prime}$ we only need to know the analytic continuation
of the integrand in (4.12) close to the negative real axis. To find this
analytic continuation we first note that eigenfunctions $\Phi_{n}$ satisfy the
recurrence relation

$\Phi_{n+1}\left(u\right)\,=\,\Phi_{n}\left(u\right)\,\left(e^{-\gamma\,n}-{1\over
u}\right)$ (4.13)

We can use this relation to continue the function $\Phi$ to negative integer
$n$:

$\Phi_{-n+1}\left(u\right)\,=\,\Phi_{-n}\left(u\right)\,\left(e^{\gamma\,n}-{1\over
u}\right)$ (4.14)

This can be iterated to give

$\Phi_{-n}\left(u\right)\,=\,\Phi_{-(n-1)}\left(u\right)\,\left(e^{\gamma\,n}-{1\over
u}\right)^{-1}\,=\,\,\prod^{n}_{l=1}\left(e^{\gamma\,l}\,\,-\,\,{1\over
u}\right)^{-1}\,=\,\Psi_{n}\left(u\right)$ (4.15)

We have already noticed that $\Psi_{n}(u)$ have simple poles in $n$ at
$u=e^{-\gamma\,k}$, for all integer $k\leq n$. Thus in the vicinity of integer
$n$ we can write

$\Phi_{-n}\left(u=e^{-\gamma\,k}\right)\,=\,\Psi_{n}\left(u=e^{-\gamma
k}\right)\,=\,-\,{(-1)^{n}\pi\,\tilde{Q}_{n}^{k}\over\gamma\,\sin(\pi\,n)}\,$
(4.16)

where $\tilde{Q}_{n}^{k}$ are introduced as residues of the poles in $n$. Our
goal now is to find these residues.

For a single dipole target we need $k=1$:

$\displaystyle\Phi_{n}\left(e^{-\gamma}\right)$ $\displaystyle=$
$\displaystyle(-1)^{n}\,e^{\gamma\,n}\prod^{n}_{l=1}\left(1\,-\,e^{-\gamma\,l}\right)=(-1)^{n}\,e^{\gamma\,n-\gamma\frac{n(n+1)}{4}}\prod^{n}_{l=1}\left(2\sinh\left(\frac{1}{2}\gamma
l\right)\right)$ $\displaystyle=$
$\displaystyle(-\gamma)^{n}\,n!\,e^{\gamma\,n-\gamma\frac{n(n+1)}{4}}\exp\left(\sum^{n}_{l=1}\ln\left(\frac{2\sinh\left(\frac{1}{2}\gamma
l\right)}{\gamma l}\right)\right)$ $\displaystyle=$ $\displaystyle
e^{\gamma\,n-\gamma\frac{n(n+1)}{4}}(-\gamma)^{n}n!\exp\left(\sum^{\infty}_{k=1}\sum^{n}_{l=1}\mu_{k}(\gamma\,l)^{2k}\right)=e^{\gamma\,n-\gamma\frac{n(n+1)}{4}}(-\gamma)^{n}n!\exp\left(\sum^{\infty}_{k=1}\mu_{k}\,\gamma^{2k}H_{n}^{(-2k)}\right)$

Here $H_{n}^{k}$ is the generalized harmonic number. Coefficient $\mu_{k}$ can
be easily calculated (see Ref.RY , 1.518(1)):

$\displaystyle\mu_{k}\,=\,\frac{2^{k+1}\,{\cal
B}_{2k}}{2k\,(2k)!}\,=\,-\,(-1)^{k}\,\frac{\zeta\left(2k\right)}{k\,(2\pi)^{2k}}$
(4.18)

where ${\cal B}_{2k}$ are Bernoulli numbers. Thus we get

$\Phi_{n}\left(e^{-\gamma}\right)=e^{\gamma\,n-\gamma\frac{n(n+1)}{4}}(-\gamma)^{n}\Gamma\left(n+1\right)\exp\left(\sum^{\infty}_{k=1}(-1)^{k+1}\frac{1}{k}\zeta\left(2k\right)\left(\frac{\gamma}{2\,\pi}\right)^{2k}H_{n}^{(-2k)}\right)$
(4.19)

and

$\Psi_{n}\left(e^{-\gamma}\right)\,=\,\Phi_{-n}\left(e^{-\gamma}\right)=e^{-\gamma\,n-\gamma\frac{n(n-1)}{4}}(-\gamma)^{-n}\Gamma\left(-n+1\right)\exp\left(\sum^{\infty}_{k=1}(-1)^{k+1}\frac{1}{k}\zeta\left(2k\right)\left(\frac{\gamma}{2\,\pi}\right)^{2k}H_{-n}^{(-2k)}\right)$
(4.20)

The function $\Gamma\left(-n+1\right)$ has poles at integer values of $n$.
Using $-\,H^{(-2k)}_{-n-1}=H_{n}^{(-2k)}$ and
$\Gamma(1-z)\,\Gamma(z)\,=\,{\pi\over\sin(\pi\,z)}\,$ we can finally write

$\displaystyle\tilde{Q}_{n}^{1}\,=\,-\,{1\over\gamma^{n-1}\,(n-1)!}\,e^{\gamma\frac{n(n-1)}{4}-n\gamma}\exp\left(\sum_{k=1}^{\infty}\gamma^{2k}\,\mu_{k}\,H^{(-2k)}_{n-1}\right)\,.$
(4.21)

This expression with $(n-1)!\rightarrow\Gamma(n)$ provides the analytic
continuation of $\tilde{Q}^{1}_{n}$ to the complex $n$ plane. Notice that
$\tilde{Q}^{1}_{n}\,=\,\bar{Q}^{1}_{n}$ computed in (3.4).

The $S$-matrix then is

$\displaystyle S\left(\tilde{Y}\right)\,$ $\displaystyle=$
$\displaystyle{1\over 2\pi i}\
\oint_{C}{\pi\,(-1)^{n}dn\over\sin(\pi\,n)}\,C_{n}^{(1)}\,\Phi_{n}\left(u=e^{-\gamma}\right)e^{\Delta_{n}\,\tilde{Y}}\,=\,{1\over
2\pi
i}\oint_{C^{\prime}}{\pi^{2}dn\over\gamma\,\sin^{2}(\pi\,n)}\,C_{-n}^{(1)}\,\tilde{Q}_{n}^{1}\,e^{-\tilde{\Delta}_{n}\,\tilde{Y}}$
(4.22) $\displaystyle=$
$\displaystyle\,{1\over\gamma}\,\sum^{\infty}_{n=1}{d\over
dn}\left[C_{-n}^{(1)}\,\tilde{Q}_{n}^{1}\,e^{-\tilde{\Delta}_{n}\,\tilde{Y}}\right]\,=\,\tilde{Y}\sum_{n=1}\bar{\beta}_{n}\,e^{-\tilde{\Delta}_{n}\,\tilde{Y}}\,+\,\sum_{n=1}\beta_{n}\,e^{-\tilde{\Delta}_{n}\,\tilde{Y}}$

with

$\beta_{n}\,\equiv\,{1\over\gamma}{d\over
dn}\left[C_{-n}^{(1)}\,\tilde{Q}_{n}^{1}\right]\,;~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\bar{\beta}_{n}\,\equiv\,-\,C_{-n}^{(1)}\,\tilde{Q}_{n}^{1}\,e^{-\gamma\,n}$
(4.23)

In the limit $n\rightarrow\infty$,

$\lim_{n\rightarrow\infty}\left[C_{-n}^{(1)}\,\tilde{Q}_{n}^{1}\,e^{-\tilde{\Delta}_{n}\,\tilde{Y}}\right]\,=\,-\,e^{-\gamma\,n}\,e^{-\,\tilde{Y}}$
(4.24)

thus the series in $n$ is convergent even though the coefficients $C_{n}$
individually have a Gaussian growth.

For large enough $\tilde{Y}$ the dominant contribution comes from $n=1$.
Keeping also terms that are large for small $\gamma$ we have

${1\over\gamma}{d\over
dn}\left[C_{-n}^{(1)}\,\tilde{Q}_{n}^{1}\,e^{-\tilde{\Delta}_{n}\,\tilde{Y}}\right]_{n=1}\,\simeq\,-\,{d\over
dn}\left[e^{-\gamma}\gamma^{-n}\,e^{-\tilde{\Delta}_{n}\,\tilde{Y}}\right]_{n=1}\,=\,e^{-2\gamma}\,\left[\tilde{Y}\,+\,e^{\gamma}\,{1\over\gamma}\,\ln\gamma\right]\,e^{-\tilde{\Delta}_{1}\,\tilde{Y}}$
(4.25)

which provides both the asymptotic and pre-asymptotic behavior consistent with
Eq. (4.9). At finite $\gamma$, the leading large $Y$ asymptotics is

$S\left(\tilde{Y}\right)\,=\,{1\over\gamma}\,C_{-1}^{(1)}\,\tilde{Q}_{1}^{1}\,{d\over
dn}\,\left[e^{-\tilde{\Delta}_{n}\,\tilde{Y}}\right]_{n=1}\,=\,e^{-2\gamma}\,\tilde{Y}\,e^{-\tilde{\Delta}_{1}\,\tilde{Y}}$
(4.26)

After exploring the asymptotics, we will now keep all the subleading terms.
For convenience, let us define new variable $\alpha_{n}$:

${1\over\gamma}\,C_{-n}^{(1)}\left(\gamma\right)\,\tilde{Q}_{n}^{1}\,=\,-\,{e^{\gamma\,n^{2}/4\,-\,5\gamma\,n/4}\over\gamma^{n}}\,\alpha_{n}\,;~{}~{}~{}~{}~{}~{}~{}~{}~{}\alpha_{n}\,\equiv\,\prod_{l=1}^{n-1}{\gamma\over
2\sinh[\gamma\,l/2]}$ (4.27)

These equations define $\alpha_{n}$ at integer values of $n$, but we
understand it in the sense of analytic continuation. Hence

$\displaystyle\bar{\beta}_{n}$ $\displaystyle=$
$\displaystyle{e^{\gamma\,n^{2}/4\,-\,9\,\gamma\,n/4}\over\gamma^{n-1}}\,\alpha_{n}\,$
$\displaystyle\beta_{n}$ $\displaystyle=$
$\displaystyle{e^{\gamma\,n^{2}/4\,-\,5\,\gamma\,n/4}\over\gamma^{n}}\,\left[{\ln(\gamma)}\,\alpha_{n}\,-(n/2\,-\,5/4)\,\gamma\,\alpha_{n}-\alpha^{\prime}_{n}\right]$
(4.28)

Formally Eq. (4.22) is an infinite series. However as is clear from (4.24) the
series is convergent at any value of rapidity $\tilde{Y}$. Let us introduce a
separation between ”large” and ”small” terms in the expansion, $n_{0}$. At the
initial rapidity $Y=0$,

$S\left(\tilde{Y}=0\right)\,=\,e^{-\gamma}\,=\,\sum_{n=1}^{n_{0}}\beta_{n}\,+\,\sum_{n=n_{0}+1}^{\infty}\beta_{n}\,$
(4.29)

Hence

$\sum_{n=n_{0}+1}^{\infty}\beta_{n}\,=\,e^{-\gamma}\,-\,\sum_{n=1}^{n_{0}}\beta_{n}$
(4.30)

We choose $n_{0}(\tilde{Y})$ such that for $n>n_{0}$,

$e^{-\tilde{\Delta}_{n}\,\tilde{Y}}\,\simeq\,e^{-\tilde{Y}}\,(1\,{+}\,e^{-\gamma\,n}\,\tilde{Y})\,;\,~{}~{}~{}~{}~{}~{}~{}~{}\,\tilde{Y}\,\ll\,e^{\gamma\,n_{0}}.$
(4.31)

Using this relation, we can rewrite the $S$-matrix at arbitrary rapidity
$\tilde{Y}$

$\displaystyle S\left(\tilde{Y}\right)$ $\displaystyle\simeq$ $\displaystyle
e^{-\gamma}\,e^{-\tilde{Y}}+\sum_{n=1}^{n_{0}}\beta_{n}\,\left[e^{-\tilde{\Delta}_{n}\,\tilde{Y}}\,-\,e^{-\tilde{Y}}\right]+\tilde{Y}\sum_{n=1}^{n_{0}}\bar{\beta}_{n}\,e^{-\tilde{\Delta}_{n}\,\tilde{Y}}+\tilde{Y}\,e^{-\tilde{Y}}\sum_{n=n_{0}+1}^{\infty}[\bar{\beta}_{n}\,-\,\beta_{n}\,e^{-\gamma\,n}]$
(4.32) $\displaystyle\simeq$ $\displaystyle
e^{-\gamma}\,e^{-\tilde{Y}}+\sum_{n=1}^{n_{0}}\beta_{n}\,\left[e^{-\tilde{\Delta}_{n}\,\tilde{Y}}\,-\,e^{-\tilde{Y}}\right]+\tilde{Y}\sum_{n=1}^{n_{0}}\bar{\beta}_{n}\,e^{-\tilde{\Delta}_{n}\,\tilde{Y}}$

The last term in the first line can be dropped since for very large $n$,
$\bar{\beta}_{n}\,=\,\beta_{n}\,e^{-\gamma\,n}$ as follows from the
asymptotics (4.24). The very same result could be arrived at in a somewhat
shorter way:

$\displaystyle S\left(\tilde{Y}\right)\,$ $\displaystyle=$
$\displaystyle\tilde{Y}\sum_{n=1}^{n_{0}}\bar{\beta}_{n}\,e^{-\tilde{\Delta}_{n}\,\tilde{Y}}\,+\,\sum_{n=1}^{n_{0}}\beta_{n}\,e^{-\tilde{\Delta}_{n}\,\tilde{Y}}\,+\,{1\over\gamma}\,\sum^{\infty}_{n=n_{0}+1}{d\over
dn}\left[C_{-n}\left(\gamma\right)\,\tilde{Q}_{n}^{1}\,e^{-\tilde{\Delta}_{n}\,\tilde{Y}}\right]$
(4.33) $\displaystyle\simeq$
$\displaystyle\tilde{Y}\sum_{n=1}^{n_{0}}\bar{\beta}_{n}\,e^{-\tilde{\Delta}_{n}\,\tilde{Y}}\,+\,\sum_{n=1}^{n_{0}}\beta_{n}\,e^{-\tilde{\Delta}_{n}\,\tilde{Y}}\,+\,e^{-Y}\,\sum^{\infty}_{n=n_{0}+1}e^{-\gamma\,n}$
$\displaystyle=$
$\displaystyle\tilde{Y}\sum_{n=1}^{n_{0}}\bar{\beta}_{n}\,e^{-\tilde{\Delta}_{n}\,\tilde{Y}}\,+\,\sum_{n=1}^{n_{0}}\beta_{n}\,e^{-\tilde{\Delta}_{n}\,\tilde{Y}}\,+\,e^{-\tilde{Y}}\,{e^{-\gamma\,n_{0}}\over
1-e^{-\gamma}}$

The expression above simultaneously incorporates both the initial condition
and large $Y$ asymptotics. So far, the only undetermined entry in Eq. (4.1.2)
and the $S$-matrix is $\alpha^{\prime}_{n}$. It is not trivial to compute
$\alpha^{\prime}_{n}$ because it implies differentiation of $\alpha_{n}$ over
discrete index $n$, requiring careful analytical continuation in $n$. The
expression for $\alpha^{\prime}_{n}$ is derived in the Appendix B: we first
relate all $\alpha^{\prime}_{n}$ to $\alpha^{\prime}_{1}$ (see (B.2)) and then
provide expression for $\alpha^{\prime}_{1}$ (see (B.5) for representation in
the form of an infinite series and (B.6) for an alternative, integral
representation).

In the limit of small $\gamma$, i.e. $\gamma\,n\,<1$, the expressions are

$\alpha_{n}\simeq{1\over(n-1)!}~{}~{}~{}~{}~{}~{}~{}~{}~{}\rightarrow~{}~{}~{}~{}~{}~{}~{}\alpha^{\prime}_{n}\simeq{\gamma_{E}\,-\,H_{n-1}\over(n-1)!};~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\alpha^{\prime}_{1}\simeq\gamma_{E};$
(4.34)
$\displaystyle\bar{\beta}_{n}={e^{\gamma\,n^{2}/4\,-\,9\,\gamma\,n/4}\over\gamma^{n-1}\,(n-1)!}\,;~{}~{}~{}~{}~{}~{}~{}~{}\beta_{n}={e^{\gamma\,n^{2}/4\,-\,5\,\gamma\,n/4}\over\gamma^{n}\,(n-1)!}\,\left[{\ln(\gamma)}\,\,-(n/2\,-\,5/4)\,\gamma\,-\gamma_{E}\,+\,H_{n-1}\right]$
(4.35)

As mentioned earlier, the leading large $\tilde{Y}$ asymptotics comes from
$n=1$, reproducing Eq. (4.9). Furthermore,
$S\left(\tilde{Y}=0\right)\,=\,\sum_{n}\beta_{n}$.

Although we have not proven rigorously that the contribution of the circle at
infinity is negligible, and the two contours on Fig. 4 can be deformed into
each other, the fact that we found an expression for the $S$-matrix that
yields correct both low and high rapidity limits is a strong indication that
this is indeed the case. A more thorough investigation of this point is given
in Appendix A.

###  4.2 Scattering on a target nucleus

We now consider a large nucleus target, modeling nucleus in its rest frame as
$A$ dipoles. This is typical dilute-on-dense process, in QCD frequently
described by the BK equation BK . In the BFKL cascade model of Section 2, the
$S$-matrix computed in the nucleus rest frame is

$S^{BK}\left(Y\right)=\sum_{n=1}^{\infty}e^{-\gamma\,n\,A}P^{\mbox{\tiny
BFKL}}_{n}\left(\tilde{Y}\right)=\sum_{n=1}^{\infty}e^{-\gamma\,n\,A}\frac{1}{N(Y)\,-\,1}\Bigg{(}1\,-\,\frac{1}{N(Y)}\Bigg{)}^{n}\,\,=\,\,\frac{1}{1\,\,-\,\,\Big{(}1\,-\,e^{\gamma\,A}\Big{)}N(Y)}$
(4.36)

which indeeds satisfies the ”BK equation” BK

$\frac{d\,S^{BK}\left(Y\right)}{d\,Y}\,\,=\,\,\Delta\Bigg{(}(S^{BK})^{2}\,\,-\,\,S^{BK}\Bigg{)}$
(4.37)

The large $Y$ asymptotics

$S^{BK}\left(Y\rightarrow\infty\right)\simeq\frac{e^{-\Delta\,Y}}{e^{\,\gamma\,A}\,-\,1}\xrightarrow{A\rightarrow\infty}e^{-\gamma\,A}\,e^{-\Delta\,Y}$
(4.38)

while at small rapidities we have multi-Pomeron expansion:

$S^{BK}\left(Y\rightarrow
0\right)=1\,+\,\Big{(}1\,-\,e^{\gamma\,A}\Big{)}\,e^{\Delta\,Y}\,+\,\Big{(}1\,-\,e^{\gamma\,A}\Big{)}^{2}\,e^{2\,\Delta\,Y}\,+\,\cdots\xrightarrow{A\rightarrow\infty}1\,-\,e^{\gamma\,A}\,e^{\Delta\,Y}\,+\,e^{2\gamma\,A}\,e^{2\,\Delta\,Y}\,+\,\cdots$
(4.39)

As we have extensively discussed above, the BFKL cascade does not satisfy the
$t$-channel unitarity (independence of the calculational frame). Therefore,
the $S$-matrix computed in the UTM will differ from Eq. (4.36).

In the UTM, the $S$-matrix

$S^{A}\left(\tilde{Y}\right)\,=\,\sum_{n=1}^{\infty}e^{-\gamma\,n\,A}\,P^{\mbox{\tiny
UTM}}_{n}\left(\tilde{Y}\right)$ (4.40)

Equivalently,

$\displaystyle S^{A}\left(\tilde{Y}\right)\,$ $\displaystyle=$
$\displaystyle{1\over 2\pi i}\
\oint_{C}{\pi\,{(-1)}^{n}dn\over\sin(\pi\,n)}\,C_{n}^{(1)}\,\Phi_{n}\left(u=e^{-\gamma\,A}\right)e^{\Delta_{n}\,\tilde{Y}}$
(4.41) $\displaystyle=$
$\displaystyle-\,\sum^{A}_{n=1}\,C_{-n}^{(1)}\,\,\Psi_{n}\left(u=e^{-\gamma\,A}\right)e^{-\tilde{\Delta}_{n}\,\tilde{Y}}+{1\over\gamma}\,\sum^{\infty}_{n=A+1}{d\over
dn}\left[C^{(1)}_{-n}\,\tilde{Q}_{n}^{A}\,e^{-\tilde{\Delta}_{n}\,\tilde{Y}}\right]\
$

We notice that for $A>n$, $\Psi_{n}$ is not singular. Hence the leading large
$Y$ asymptotics given by $n=1$ is very easy to write down:

$S^{A}\left(\tilde{Y}\rightarrow\infty\right)\,\simeq\,{e^{-\tilde{\Delta}_{1}\,\tilde{Y}}\over
e^{\gamma\,A}\,-\,e^{\gamma}}\,\xrightarrow{A\rightarrow\infty}e^{-\gamma\,A}\,e^{-\tilde{\Delta}_{1}\,\tilde{Y}}$
(4.42)

Pomeron expansion (the first line in Eq. (4.41)) is more suitable for small
rapidities:

$\displaystyle S^{A}\left(\tilde{Y}\rightarrow 0\right)$ $\displaystyle=$
$\displaystyle
1+e^{\gamma}\,(1-e^{\gamma\,A})\,e^{\Delta_{1}\,\tilde{Y}}\,+\,e^{2\,\gamma}(1-e^{\gamma\,A})(1-e^{\gamma\,(A+1)})\,e^{\Delta_{2}\,\tilde{Y}}+\cdots$
(4.43)
$\displaystyle\xrightarrow{A\rightarrow\infty}1-e^{\gamma}e^{\gamma\,A}\,e^{\Delta_{1}\,\tilde{Y}}\,+\,e^{2\,\gamma}e^{2\gamma\,A}\,e^{\Delta_{2}\,\tilde{Y}}\,+\cdots$

Comparing the asymptotics of the BFKL and UTM models, we notice that for small
$\gamma$ and moderate rapidities they are practically identical. In particular
the $A$ dependence in both models is the same for $A>1/\gamma$.

##  5 Summary

In this paper we have further explored the toy world focussing on the BFKL
cascade model and the UTM that has been devised to satisfied the $t$-channel
unitarity, that is independence of the $S$-matrix on the reference frame.

We have solved for the eigenfunctions and spectrum of both models. The
$S$-matrix for high energy scattering is given in terms of the probabilistic
formula Eq. (1.1). Both model Hamiltonians possess negative and positive
eigenvalues, each set separately corresponding to a complete basis. The
probabilities $P_{n}$ entering Eq. (1.1) are naturally expanded in the basis
of negative eigenvalues with the expansion truncated to a finite ($n$) number
of terms, resulting in a well defined calculational procedure both for the
probabilities and for the $S$-matrix via Eq. (1.1).

The Pomeron calculus emerges as an expansion of the $S$-matrix in terms of the
eigenfunctions with the positive eigenvalues, the latter being identified with
the dressed Pomeron intercepts. In the BFKL cascade, the intercept of
$n$-Pomeron exchange scales linearly with $n$ as $n\,\Delta$, where $\Delta$
is the intercept of a single ”BFKL Pomeron”. The expansion for the $S$-matrix
is asymptotic, yet Borel summable.

In the UTM (see Eq. (4.3) for dipole-dipole scattering), the $n$-Pomeron
intercept grows much faster with $n$, so that $\Delta_{n}>n\,\Delta_{1}$. In
fact, at large $n$, the intercept grows exponentially
$\Delta_{n}\simeq\exp\left(\gamma\,n\right)$. Thus multi-Pomeron expansion in
the UTM cannot be summed a la Borel. Similar though somewhat less severe
problem has been known in QCD for a long time: the $1/N_{c}$ corrections to
the intercepts also lead to $\Delta_{n}^{QCD}>n\,\Delta_{1}^{BFKL}$ ††††††A
brief review of the problem, references and the most pessimistic point of view
on this problem can be found in Ref.LENC .. This lead many experts (including
one of us LENC ) to believe that the Pomeron calculus has a deadly problem
beyond strict leading order in $1/N_{c}$.

In the present paper we have managed to establish a resummation procedure that
provides a meaningful $S$-matrix in the UTM, starting from the Pomeron
calculus (see Eq. (4.3)). The main idea is based on analytical continuation
which maps the positive spectrum into negative one. The approach is
particularly suitable for computing asymptotic behaviour of the $S$-matrix at
large rapidities. One of the interesting new observations is that the approach
to unitarity features double poles in the complex angular momentum plane, with
the leading asymptotics $S(Y)\sim Y\,e^{-\Delta\,Y}$.

The UTM can be interpreted as a theory of interacting bare (”BFKL”) Pomerons.
To make this representation explicit we split the UTM Hamiltonian into that of
the BFKL and self-interactions, ${\cal H}_{UTM}\,=\,{\cal H}_{0}\,+\,{\cal
H}_{I}$. The ”BFKL” Pomeron has bare intercept $\Delta$. Self-interactions,
which involve infinitely many vertices both of the splitting and merging type,
”dress” the bare Pomeron resulting in the UTM spectrum. Since the bare Pomeron
is supercritical ($\Delta>0$), the dominant contributions to the ”dressing” of
the intercept are due to enhanced diagrams (small bare Pomeron loops). The
solution of the UTM is a means of summing such diagrams, the lesson which can
hopefully be taken to QCD.

We have seen in Section 2 that in the BFKL cascade model, due to its frame
dependence, the results computed in the central mass frame differ from those
obtained in the target rest frame. For dipole-dipole collisions, the target
rest frame result (BK solution) is quoted in Eq. (2.25). Compared with the UTM
(see Eq. (4.43) for $A=1$), even at small $\gamma$, we observe a significant
difference: e.g. the residues of two-Pomeron exchanges differ by a factor two.
This is not surprising. Theoretically it is of course well understood that the
BFKL cascade model (a.k.a. the BK equation) is not applicable for dipole-
dipole, or somewhat more general dilute-on-dilute scattering, beyond the
leading single Pomeron exchange at small rapidities. It is interesting however
to see explicitly how the unitarization corrections not included in the BK,
affect the scattering amplitude. The BK model is devised rather to describe
dilute-on-dense scattering. Modeling the dense system as $A$ target dipoles,
we see indeed that the BK solution (4.36) and the UTM result coincide as long
as $\gamma$ is small, and $\gamma A>1$ as long as the rapidity is not too
large. This is the standard condition for absence of Pomeron loops and
dominance of semi-enhanced (fan) diagrams. For large enough rapidities
$\tilde{Y}>\frac{1}{\gamma^{2}}$ the two results diverge as the rate of the
exponential decrease of the scattering matrix in the two models is slightly
different even at small $\gamma$.

The existence of the set of negative eigenvalues and the corresponding
eigenfunctions in either the BFKL cascade or UTM is not a widely recognized
feature. While positive eigenvalues correspond to intercepts of multi-Pomeron
exchanges and is a convenient basis for expanding the $S$-matrix at low
rapidities, the negative eigenvalues have a very different meaning. They
dominate the approach of the $S$-matrix to. saturation at $Y\rightarrow\infty$
‡‡‡‡‡‡Some discussion along these lines could be found in Kovner:2006vb . In
Kovner:2006vb  though the spectrum was deemed to be double degenerate, which
is not what we have discovered in the toy models. . The analogous exponential
approach to saturation has been observed in QCD, the so called Levin-Tuchin
law LETU . Although the BFKL cascade model (a.k.a. the BK equation) in QCD is
certainly more complicated due to nontrivial transverse dynamics, we believe
that one leading Levin-Tuchin exponent is the direct analog of the largest
negative eigenvalue $-\tilde{\Delta}_{1}$ in our toy model. Additional
negative eigenvalues (and corresponding eigenfunctions) should also exist in
QCD and their study is an interesting open question. We also note that a model
analogous of the UTM has been also formulated in two transverse dimensions.
This is the so called ’Diamond action” model diamond , which has been shown to
be $t$-channel unitary, although its $s$-channel unitarity has not been
formally established.

We conclude by repeating our hope that the lessons learnt from the toy world
can be helpful in further development of the BFKL Pomeron calculus and better
understanding of the saturation dynamics in real QCD.

Acknowledgements

We thank our colleagues at Tel Aviv University for encouraging discussions. We
also thank Carlos Contreras for useful discussions related to the subject of
this work. This research was supported by Binational Science Foundation grant
#2022132. AK is supported by the NSF Nuclear Theory grant 2208387. ML’s work
was funded by Binational Science Foundation grant #2021789 and by the ISF
grant #910/23. This research was also supported by MSCA RISE 823947 “Heavy ion
collisions: collectivity and precision in saturation physics” (HIEIC).

##  Appendix A  Large $n$ behavior of the Pomeron exchanges

As has been discussed, the scattering matrix can be written as the sum over
different numbers of the Pomeron exchanges, viz.:

$S\left(\tilde{Y}\right)\,\,=\,\,Z\left(e^{-\gamma},\tilde{Y}\right)\,\,\,\,\,=\,\,\,\sum^{\infty}_{n=0}C_{n}^{(1)}\,\Phi_{n}\left(e^{-\gamma},\gamma\right)\,e^{\Delta_{n}\,\tilde{Y}}\
$ (A.1)

where $e^{\Delta_{n}\,\tilde{Y}}$ is the Green’s function of the exchanges of
$n$ Pomerons. The asymptotic series of Eq. (A.1) has two problems: the severe
problem of large $n$ behaviour for $\Delta_{n}\xrightarrow{n\gg
1}\,e^{\gamma\,n}$ and the Gaussian behaviour of
$C_{n}^{(1)}=\exp\left(\frac{1}{2}\gamma n(n+1)\right)$. The first problem has
been solved, introducing $j$ summation in Eq. (4.4). It turns out that we can
solve the second problem by replacinng

$C_{n}^{(1)}=\exp\left(\frac{1}{2}\gamma
n(n+1)\right)\,\,=\,\,\int\limits_{-\infty}^{\infty}d\lambda\frac{e^{-\frac{(\gamma-2\lambda)^{2}}{8\gamma}}}{\sqrt{2\pi\gamma}}e^{\lambda\,n}$
(A.2)

Integral over $\lambda$ is well convergent. Using Eq. (4.4) and Eq. (A.2) we
can rewrite $S$ \- matrix as

$S\left(\tilde{Y}\right)\,\,=\,\,Z\left(e^{-\gamma},\tilde{Y}\right)\,\,\,\,\,=\,\,\,e^{-\tilde{Y}}\sum_{j=0}\frac{\tilde{Y}^{j}}{j!}\sum^{\infty}_{n=0}\underbrace{\int\limits_{-\infty}^{\infty}d\lambda\frac{e^{-\frac{(\gamma-2\lambda)^{2}}{8\gamma}}}{\sqrt{2\pi\gamma}}e^{\lambda\,n}}_{C_{n}^{(1)}}e^{\gamma\,n\,j}\Phi_{n}\left(e^{-\gamma},\gamma\right)$
(A.3)

At large $n$,
$\Phi_{n}\left(e^{-\gamma},\gamma\right)\,\to\,\left(-e^{\gamma}\right)^{n}$
and therefore, Eq. (A.3) can be summed using the Borel transform:

$\displaystyle S\left(\tilde{Y}\right)\,\,\,$ $\displaystyle=$
$\displaystyle\,\,\,e^{-\tilde{Y}}\sum_{j=0}\frac{\tilde{Y}^{j}}{j!}\int\limits_{-\infty}^{\infty}d\lambda\frac{e^{-\frac{(\gamma-2\lambda)^{2}}{8\gamma}}}{\sqrt{2\pi\gamma}}\int^{\infty}_{0}d\tau
e^{-\tau}\sum^{\infty}_{n=1}e^{\left(\lambda+\gamma\,j\right)\,n}\tau^{n}\frac{\Phi_{n}\left(e^{-\gamma},\gamma\right)}{n!}$
, $\displaystyle=$
$\displaystyle\,\,\,e^{-\tilde{Y}}\sum_{j=0}\frac{\tilde{Y}^{j}}{j!}\int\limits_{-\infty}^{\infty}d\lambda\frac{e^{-\frac{(\gamma-2\lambda)^{2}}{8\gamma}}}{\sqrt{2\pi\gamma}}\int^{\infty}_{0}d\tau\,e^{-\tau}\mathcal{B}\left(\lambda,j,\tau\right)$
(A.4)

where Borel function $\mathcal{B}\left(\lambda,j,\tau\right)$ is analytical
function of all its variables, since the series in $n$ is the absolute
converged series as well as sum over $j$ and integrals over $\lambda$ and
$\tau$.

##  Appendix B Computing $\alpha^{\prime}_{n}$

In this Appendix we compute $\alpha^{\prime}_{n}$ defined in Eq. (4.27).
Notice the recurrent relation:

$\displaystyle\alpha_{n+1}$ $\displaystyle=$
$\displaystyle\alpha_{n}\,{\gamma\over 2\sinh[\gamma n/2]}$ (B.1a)
$\displaystyle\alpha^{\prime}_{n+1}$ $\displaystyle=$
$\displaystyle\alpha^{\prime}_{n}\,{\gamma\over 2\sinh[\gamma
n/2]}\,-\,\alpha_{n}\,{\gamma^{2}\cosh[\gamma n/2]\over 4\sinh^{2}[\gamma
n/2]}\,=\,\alpha^{\prime}_{n}\,{\gamma\over 2\sinh[\gamma
n/2]}\,-\,\alpha_{n+1}\,{\gamma\cosh[\gamma n/2]\over 2\sinh[\gamma n/2]}$
(B.1b)

Thus we can relate all the derivatives of $\alpha_{n}$ to the one at $n=1$:

$\alpha^{\prime}_{n}=\alpha^{\prime}_{1}\,\alpha_{n}\,-\,\gamma\,\alpha_{n}\,{1\over
2}\sum_{k=1}^{n-1}\coth[\gamma
k/2];~{}~{}~{}\frac{\alpha^{\prime}_{n}}{\alpha_{n}}=\alpha^{\prime}_{1}\,-\,\gamma\,{1\over
2}\sum_{k=1}^{n-1}\coth[\gamma k/2]$ (B.2)

Our next goal is to analytically continue $\alpha_{n}$ from integer $n$ to
continuous. From Eq. (4.27),

$\ln\alpha_{n}\,=\,-\ln[(n-1)!]-\sum_{n=1}^{n-1}\ln\left[\frac{2\sinh\left(\frac{1}{2}l\gamma\right)}{\gamma
l}\right]\,=\,-\ln[(n-1)!]-\sum_{k=1}^{\infty}\mu_{k}\,\gamma^{2k}\,\sum^{n-1}_{l=1}l^{2k}$
(B.3)

The last expression is an analytic function of $n$:

$s_{k}(n)\equiv\sum^{n-1}_{l=1}l^{2k}=H_{n-1}^{(-2k)};~{}~{}~{}\frac{ds_{k}(n)}{dn}=-2k\left(\zeta(1-2k)-H_{n-1}^{(1-2k)}\right);~{}~{}\frac{ds_{k}(n)}{dn}\Bigg{|}_{n=1}=-2k\zeta(1-2k);$
(B.4)

$\alpha^{\prime}_{1}$ then has several equivalent representations:

$\displaystyle\alpha^{\prime}_{1}$ $\displaystyle=$
$\displaystyle\gamma_{E}-\sum^{\infty}_{k=1}\mu_{k}\gamma^{2k}\left(-2k\zeta(1-2k)\right)$
(B.5) $\displaystyle=$
$\displaystyle\gamma_{E}-2\sum^{\infty}_{k=1}(-1)^{k}\left({\gamma\over
2\pi}\right)^{2k}\,\zeta(2k)\,\zeta(1-2k){=\gamma_{E}-4\sum^{\infty}_{k=1}\left({\gamma\over
4\pi^{2}}\right)^{2k}\Gamma\left(2k\right)\zeta^{2}(2k)}$ $\displaystyle=$
$\displaystyle\gamma_{E}-4\sum^{\infty}_{k=1}\left({\gamma\over
4\pi^{2}}\right)^{2k}\zeta(2k)\int^{\infty}_{0}dt\frac{t^{2k-1}}{e^{t}-1}={\color[rgb]{0,0,0}\gamma_{E}-2\int^{\infty}_{0}\frac{dt}{t(e^{t}-1)}\Bigg{\\{}1-\frac{\gamma
t}{4\pi}\cot\left(\frac{\gamma t}{4\pi}\right)\Bigg{\\}}}$ (B.6)

In the integral representation for $\alpha^{\prime}_{1}$, one notices that
$\cot$ has poles along the integration counter, so the integral should be
understood in the principle value sense. For small $\gamma$,
$\alpha^{\prime}_{1}=\gamma_{E}$, while the first correction is quadratic in
$\gamma$ as is obvious from Eq. (B.6).

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