# Transition from the shadow to reflective scattering

S.M<EMAIL_ADDRESS>N.E. Tyurin
NRC “Kurchatov Institute”–IHEP
Protvino, 142281, Russian Federation

We consider transition from shadow to relective scattering mode and behavior
of the amplitude and cross–sections of hadron interactions in the both modes.

Keywords: elastic scattering, shadow scattering, reflective scattering,
unitarity saturation.

## Introduction

The elastic and inelastic overlap functions introduced by Van Hove [1] are
related through unitarity with the elastic scattering amplitude (its imaginary
part) and are strongly interdependent. Transition from a shadow to reflective
scattering is determined by an energy evolution of the overlap functions. It
has been suggested earlier that this evolution will result in reaching the
black disc limit and then its exceeding by the elastic scattering amplitude
[2]. Such evolution is associated with the formation of the peripheral impact
parameter profile of the inelastic overlap function (black ring formation).
Similar ideas have been discussed in [3] and relevant references can be also
found in [4]. Indication on the existence of this effect has been found under
the LHC experimental data analysis at $\sqrt{s}=13$ TeV (see [5]).

We discuss a transition from the shadow to reflective scattering mode and use
$U$-matrix unitarization scheme (see e.g. [2] and references therein) for that
purpose. This scheme provides a continuious transition between the two
scattering modes and is therefore convinient for a model construction. It does
not require extra assumption on existence of the discontinuity in energy
dependence of the scattering phase under the standard exponential unitarizaton
[6]. Indeed, this latter option naturally covers only half of the range
allowed by unitarity for the scattering amplitude values.

##  1 Standard relations

A partial elastic scattering matrix element can be represented as the complex
function:

$S_{l}(s)=\kappa_{l}(s)\exp[2i\delta_{l}(s)]$ (1)

with the two real functions $\kappa_{l}$ and $\delta_{l}$. Herewith
$\kappa_{l}$ can vary in the interval $0\leq\kappa_{l}\leq 1$ and is known as
an absorption factor connected with the inelastic overlap function
$h_{l,inel}(s)$ (see below). The value $\kappa_{l}=0$ means a complete
absorption of the respective initial state. The functions $h_{l,el}(s)$ and
$h_{l,inel}(s)$, the contributions of the elastic and inelastic intermediate
states into the unitarity relation, are interrelated:

$\mbox{Im}f_{l}(s)=h_{l,el}(s)+h_{l,inel}(s).$ (2)

The inelastic overlap function can be expressed through the function
$\kappa_{l}(s)$ by the relation

$\kappa_{l}^{2}(s)=1-4h_{l,inel}(s).$ (3)

The normalization is $S_{l}=1+2if_{l}$ and saturation of unitarity relation,
$\mbox{Im}f_{l}\to 1$, leads to $\mbox{Re}f_{l}\to 0$ and $h_{l,inel}(s)\to 0$
at fixed value of $l$.

Representation Eq. (1) is able, in principle, to provide negative values of
the real part of the function $S_{l}$, but an extra assumption on the presence
of discontinuity in the $s$ –dependence of the phase $\delta_{l}(s)$ is
required. Without this assumption, representation of the partial amplitude
$f_{l}(s)$ in the exponential form with the phase $\delta_{l}(s)$ and factor
$\kappa_{l}(s)$ as the input quantities inevitably limits the consideration by
the shadow mode when $|f_{l}(s)|\leq 1/2$. It has just took place in ref. [1]
where the second solution of the unitarity equation, relevant for the
reflective scattering mode $|f_{l}(s)|>1/2$, was neglected because of apparent
absence of a smooth transition between the shadow and reflective mode. Indeed,
as it was already noted, transition to the reflective mode requires an
additional artificial assumption on the critical phase behavior in the range
of energies far beyond the domain populated by resonances. It would be
difficult to reconcile such a behavior with a physics interpretation of a
model based on Eq. (1).

Contrary to that, the use of the amplitude representation in the rational form
($U$–matrix) and the function $U$ as an input for the model construction of
the hadron scattering amplitude allows one to continuously examine the both
scattering modes. The rational form of unitarization does not require the
singularity presence in the input function, that is an obvious obstacle under
model construction. In this sense we use the term “smooth transition”, and
connection of the two scattering modes due to rational form of the amplitude
unitarization naturally serves in favor of the conclusion on the both shadow
and reflective modes presence.

Thus, in the framework of $U$-matrix unitarization scheme, the partial elastic
scattering amplitude $f_{l}(s)$ is represented as the ratio

$f_{l}(s)=u_{l}(s)/[1-iu_{l}(s)].$ (4)

In the impact parameter representation, Eq. (4) has the form

$f(s,b)=u(s,b)/[1-iu(s,b)].$ (5)

The amplitude $f(s,b)$ is the Fourier–Bessel transform of the scattering
amplitude $F(s,t)$:

$F(s,t)=\frac{s}{\pi^{2}}\int_{0}^{\infty}bdbf(s,b)J_{0}(b\sqrt{-t}).$ (6)

This unitarization scheme is a solution of unitarity provided that
Im$u(s,b)\geq 0$.

Our aim is to discuss a continuous transition from the shadow to reflective
scattering mode in the unitarization approach based on Eqs. (4) and (5) and to
provide respective angular distributions relevant for the both modes. Such a
transition, if we trace analogy with optics, can be interpreted as the
continuous energy increase of the imaginary part of a matter refraction index
in the transient state.

As it was noted above, the exponential form, Eq. (1), implies a scattering
phase discontinuity and therefore is not fully appropriate for model
constructions. However, it should also be noted, that the exponential form
might be modified in order to reproduce a smooth transition to the negative
values of $S$ at the price of specific reduction of allowed scattering phase
variation domain and introduction of an extra free parameter into a scheme [7]
(see also [6] and [8] for discussion).

##  2  Angular dependence of the amplitude in the reflective and shadow
scattering modes

Eqs. (6) and (5) determine the scattering amplitude in the whole region of of
transferred momentum variation. The Froissart–Gribov projection formula [9,
10] assumes an exponential decrease of the amplitude at large values of $b$
and fixed value of the variable $s$. We consider the pure imaginary case and
use the relation

$u(s,b)=f(s,b)/[1-f(s,b)].$ (7)

The amplitude $f(s,b)$ is allowed to variate from zero to unity by the
unitarity. This is the only restriction imposed on the input function $u$.
Actually, the representation of the amplitude in the form of (4) or (5)
provides the unitarity as well as continuous transition between the both
scattering modes. The variation of $u$ from zero to infinity covers the both
possible intervals of the amplitude variation: the shadow one (0,1/2] and the
region (1/2,1) which can be interpreted as the reflective scattering range by
analogy with the optics [11]. The reflective mode appears in the range of
variables where the amplitude values are in the range $1/2<f(s,b)<1$. It means
that the elastic scattering $S$-matrix element is in the region $-1<S(s,b)<0$.
The inelastic overlap function is invariant under sign–changing of $S$ [12].

Negative values of $S(s,b)$ correspond to the reflective scattering. The
values of amplitude $f$ in the range $1/2<f<1$ correspond to $u>1$ and
therefore the amplitude cannnot be expanded over the $u$.

The method of the scattering amplitude calculation has been developed long
time ago and is based on the analysis of singularities of the scattering
amplitude in the complex $\beta=b^{2}$ plane [13]. The amplitude expressed in
the form of Eq. (5) is particularly relevant for such an analysis. The
singularities responsible for the region of small and moderate momentum
transfers are the poles in the complex $\beta$–plane. The residues in the
poles $\beta=\beta_{n}(s)$ allow to obtain the following expression for the
scattering amplitude $F(s,t)$ in this region of the $t$–variation:

$F(s,t)\propto\sum_{n}\sqrt{\beta_{n}(s)}K_{0}(\sqrt{t\beta_{n}(s)}),$ (8)

where $K_{0}(z)$ is the Macdonald function. Locations of the poles
$\beta_{n}(s)$ are determined by the roots of equation

$1+u(s,\beta)=0.$ (9)

Those are determined by an explicit form of the function $u$ which should take
into account general properties of the scattering matrix under model
construction of $u$ encoded into the representation

$u(s,\beta)=\frac{\pi^{2}}{s}\int_{t_{0}}^{\infty}\rho(s,x)K_{0}(\sqrt{x\beta})dx,$
(10)

where $\rho(s,x)$ is the spectral density which can be used for the model
construction of $u$ and $t_{0}=4m^{2}$. In the potential scattering, the
function $u$ is proportianal to the integral from the respective potential
[14]:

$u\propto\int_{-\infty}^{\infty}V[\sqrt{(z^{2}+\beta)}]dz.$ (11)

On the base of the Mandelstam representation for elastic scattering amplitude,
definition of generalized potential which is complex and energy–dependent was
given in [15].

The poles $\beta_{n}(s)$ are the direct–channel analogues of the Regge poles.
These singularities are generated by implementation of the unitarity, i.e. by
unitarization of the input amplitude in a rational form while the function
$u(s,\beta)$ itself does not have poles at high energies. It has also the
branching point at $\beta=0$. [13, 16].

The knowledge of the poles origin is important for solution of the problem
related to separation of the unitarity effects from effects related to the
adopted input amplitude form and its analytical properties.

There is also separation of contributions related to the different kinds of
singularities regarding the specific momentum transfer ranges. In fact, this
separation corresponds to the contributions from different interaction scales.
We do not consider here the fixed angle scattering region where the amplitude
is governed by the singularity other than the poles, namely by a cut of
$u(s,\beta)$ along the negative real axis in the $\beta$-plane [16].

There is no need to perform continuation into the complex $b$ or $\beta$ plane
for calculation of the amplitude $F(s,t)$ when $u\leq 1$ since the amplitude
$f(s,\beta)$ can be obtained by iteration over $u$. The expansion series over
$u$ is convergent and has the form:

$f(s,\beta)=\sum_{n=1}^{\infty}(-1)^{n-1}[u(s,\beta)]^{n}.$ (12)

The respective representation for the scattering amplitude $F(s,t)$ is:

$F(s,t)=\sum_{n=1}^{\infty}(-1)^{n-1}U_{n}(s,t),$ (13)

where

$U_{n}(s,t)=\frac{s}{2\pi^{2}}\int_{0}^{\infty}d\beta[u(s,\beta)]^{n}J_{0}(\sqrt{-t\beta}).$
(14)

Eqs. (12)-(14) correspond to the shadow scattering regime.

The particular case of $u=1$ (at $s=s_{r}$ and $\beta=0$) is treated by a
limiting transition $\beta\to 0$. In the impact parameter space the
expansions, Eqs. (8) and (12) correspond to the two scattering modes and
reflect a continuous transition between them.

Regarding the Eq. (13) it should be noted, that there is no clear
understanding what the term “rescattering” means in the framework of
$S$–matrix formalism operating the asymptotic states. It might happen that the
so called successive terms of expansion series correspond, in fact, to the
simultaneous interactions [18].

##  3 Transition from the shadow to reflective scattering

Both the iteration series Eq.(13) and expansion of the scattering amplitude
Eq. (8) result from the chosen unitarization in the rational form. Unitarity
plays a decisive role in the hadron interaction dynamics.

Another, but interrelated component of this dynamics is associated with a form
of the input function $u$. The particular dependencies of the $u$ on energy
and impact parameter are important for a transition from the shadow to
reflective scattering (the value of energy) and, of course, for other aspects
of hadron interactions. Since the discussion here is qualitative, we do not
make particular model assumptions on explicit form of the function $u$. We
just suppose that this function decreases monotonically with the impact
parameter growth in consistence with the Froissart–Gribov formula for the
scattering amplitude [9, 10]. This formula is valid at large values of the
impact parameter.

We also suppose that $u(s,\beta)$ monotonically increases with the energy in
order to cover the whole range of its allowed values and to provide saturation
of the unitarity limit at $s\to\infty$, (see Eq. (7)). This saturation is a
necessary prerequisite for reproduction of the principle of maximal strength
of strong interactions introduced by Chew and Frautchi [19].

The above assumptions imply existence of the two modes and continuous
transition to the reflective scattering starting at $\beta=0$ when $s=s_{r}$:

$u(s_{r},0)=1\implies f(s_{r},0)=1/2\;\mbox{and}\;\partial f/\partial
u|_{s=s_{r},\beta=0}=1/4.$ (15)

The function $u$ is a regular one at $s=s_{r}$ contrary to the singular
scattering phase $\delta$. Transition to reflective scattering mode generates
singularity in the scattering phase [11]. It makes the $u$ a preferable
quantity for model constructions in the energy region far beyond the resonance
region and the use $u$ leads to a continuous dependence of the scattering
amplitude $f$ as $s=s_{r}$, Eq. (15).

Genesis of hadron scattering interaction region can be described as a
transition from a gray to black disk and then to a black ring with reflective
region in the center of this ring. The reflective ability [6] becomes positive
at the energies $s>s_{r},\beta=0$. Scattering at larger impact parameters
remains to be of the shadow nature moving to the periphery of the interaction
region with the energy growth.

The current estimation for the value of $\sqrt{s_{r}}$ in $pp$–scattering
where a transition from the shadow to reflective scattering starts has been
obtained under quantitative analysis of the LHC data [5] and has a magnitude
$\sqrt{s_{r}}\simeq 13$ TeV.

##  4 Conclusion

The $U$–matrix representation of the amplitude provides a regular transition
between the two scattering modes. This is due to the obvious fact that the
amplitude, Eq. (5), is not limited beforehand by the range of $|f|\leq 1/2$,
or the black disk bound.

Transition to the reflective scattering mode would affect the multiparticle
production dynamics [20]. The related effects, being observed experimentally,
would indicate on its existence additionally.

The mean multiplicity and average transverse momentum of the produced
particles would slow down their energy increases at the LHC energies [21, 22].
Expected change of the multiplicity distribution shape has been described on
base of dipole Pomeron parameterization of the function $u$ in [23].

It should be noted again that asymptotically the total and elastic
cross–sections are rising as $\ln^{2}s$ while the inelastic cross-section
growth is $\ln s$ at $s\gg s_{r}$, due to transition to the reflective
scattering mode. The difference in asymptotics of the elastic and inelastic
interactions is due to the unitarity limit saturation by the scattering
amplitude and should not depend on a particular way of the amplitude
unitarization [24].

To perform a quantitative analysis of the experimental data, the explicit
models for an input amplitude have to be used. The most recent illustrations
of a quantitative model-based analysis in the framework of $U$–matrix approach
can be found in refs. [25, 26]222We are grateful to V.A. Petrov for draw our
attention to ref. [26]..

Appearance of the reflective scattering mode may be interpreted as a formation
of a color conducting state of QCD matter in the transient state at hadron
collision [27]. Proceeding along this line, one can consider a smooth
transition to the reflective scattering mode with energy growth as a
cross–over between the two phases of matter: the one with hadron degrees of
freedom (shadow regime) when the scattering is determined by the multiparticle
intermidiate states and the color conducting phase with constituents as the
main degrees of freedom (reflective regime) which may be associated with high
energy head-on collision.

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