# Spin and Polarization in High-Energy Hadron-Hadron and Lepton-Hadron
Scattering

László Jenkovszky  Bogolyubov ITP, Kiev, 03143 Ukraine; e-mail: jenk at
bitp.kiev.ua

###### Abstract

The role of spin degrees of freedom in high-energy hadron-hadron and lepton-
hadron scattering is reviewed with emphasis on the dominant role of soft,
diffractive, non-perturbative effects. Explicit models based on analyticity
and Regge-pole theory, including the pomeron trajectory (gluon exchange in the
$t$ channel) are discussed. We argue that there is a single, universal pomeron
in Nature, manifest as relatively “soft” or “hard”, depending on the
kinematics considered. Both the pomeron and the non-leading (secondary) Regge
trajectories, made of quarks are non-linear, complex functions. They are
populated by a finite number of resonances: known baryons and mesons in case
of the reggeons and hypothetical glueballs in case of the pomeron (“oddballs”
on the odderon trajectory). Explicit models and fits are presented that may be
used in recovering generalized parton distributions from deeply virtual
Compton scattering and electoproduction of vector mesons.

spin; polarization; diffraction; pomeron; gluon; confinement

##  I Introduction

Interest in spin physics at high-energies was varying with time. In general,
there is a prejudice that the role of spin decreases as energy is increasing.
This may be true in general, especially in hadronic reaction, with some
exceptions. First, one has to understand the origin of polarization in the
scattering of unpolarized hadrons, as for example in $\Lambda$ production.
Another issue is the still purely understood delicate mechanism of the
diffraction minimum in high-energy proton-proton scattering. This point will
be discussed in Section II. Here one inevitably encounters related problems
such as: diffraction and the nature of the pomeron, that could be utilized as
made of gluons; reconciling soft (non-perturabative, Regge) and hard
(perturbative QCD) aspects of the theory, nucleon structure, revealed in
inclusive deep-inelastic scattering (DIS) and exclusive deeply virtual Compton
scattering (DVCS) as means to reveal the nucleon structure, closely related to
spin and polarization, thus connecting structure and dynamics.

The present paper is a recapitulation of the earlier results by the author,
revised, updated and extended with account for the recent developments in the
field, especially those connected to “nucleon holography”, based on
generalized parton distributions (GPDs). Parallel to these practical
developments, the still open basic problem of quark confinement is always
present in our discussions.

Due to recent progress in understanding the internal structure of the nucleon,
especially connected with deeply virtual Compton scattering (DVCS) and
generalized parton distributions (GPD) as well as the prospects of building
new experimental facilities, such as the planned Electron Ion Collider (EIC),
the interest definitely shifted to lepton induced inelastic reactions.

Recent interest in spin physics was triggered two decades ago by the
publication of the EMC data (EMC, ) (so-called “spin crisis”). We briefly
mention it in Section VII recommending for further reading the excellent
presentation of this issue in Ref. Rev2 .

Since all high-energy processes are dominated by pomeron exchange in the $t$
channel, we pay special attention to the derivation and properties of the
pomeron and its relation to gluon exchange, by which we mean the
identity/diversity of/between the BFKL pomeron derived from QCD and that based
on the analytic $S$ matrix theory. In particular, we discuss the following
issues:

  * •

Is the input pomeron a simple Regge pole? What are the alternatives, if any?

  * •

Do resonances terminate, abruptly replaced by a continuum or they gradually
fade, their peaks becoming progressively wider and lower? This issue is
connected with the Hagedorn spectrum and possible phase transition between
hadronic and quark-gluon matter.

  * •

Origin of the diffraction (dip-bump) pattern in elastic hadron scattering;

  * •

Role of unitarization in producing the dip-bump structure;

  * •

Are there two (or more) pomerons—a “soft” and a “QCD-inspired”, “hard” one?

  * •

Can the Regge-pomeron pole be $Q^{2}-$dependent?

The paper is organized as follows. In Section II we discuss elastic $NN$
scattering with spin. Recall that the dominant point of view is that the role
of spin decreases as energy increases, and it can be ignored at multi-GeV
energies, i.e., beyond those of the ISR, and even more so at the LHC. Spin
effect may be important in the region of the diffraction minimum, i.e., around
$t=-1$ GeV2. The point is that the origin and mechanism of the dip-bump
phenomenon is still disputable. There are many models but no theoretical
understanding of the origin of this important phenomenon. Predictions, mostly
based on variants of the Regge-eikonal approach, failed in predicting the
position and depth of the dip at the LHC. Spin effects are not the favourite
but still a viable possibility, alternative to the dominant ones, based on
unitarity (Section II.3).

The importance and properties of the Regge trajectories are repeatedly
stressed throughout this paper, with a dedicated Section II.2. Dual models
have shown that Regge trajectories are sort of dynamical variables. The
trajectories are non-linear, complex function. Their thresholds and asymptotic
behaviour should satisfy known constraints. For example, the lowest,
$4m_{\pi}$ threshold affects the behaviour of the differential cross section
in the Coulomb interference region, competing with possible spin effects.

In Section III we focus on the procedure of recovering Generalized Parton
Distributions (GPDs) from electron–proton scattering with spin, measured at
the JLab Grav . Here again the pomeron exchange is a basic element of the
theory. Different from hadron-hadron scattering, dominated by a “soft” pomeron
exchange, here the pomeron is “hard”, i.e., its intercept is much larger
($\alpha(0)\approx 1.3)$ than that considered in Section II. In Section IV we
introduce a model for Deep Inelastic Scattering (DIS) amplitude with
$Q^{2}$-dependent pomeron intercept. That model will be generalized to deeply
virtual Compton scattering (DVCS) in Section V, applicable both to “soft”,
e.g., $NN$ scattering Section II, and “hard” DVCS.

Various aspects related to the origin of nucleon’s spin are discussed in
Section VII. We share the point of view that there is no “crisis”, over-
dramatized after the EMS measurements. The proton spin $1/2$ can be collected
from that of its constituents and their motion.

Finally, let us mention recent findings in spin physics revealed in ultra-
relativistic heavy-ion collisions; STAR collaboration STAR  discovered a
significantly nonzero global polarization of Lambda hyperons produced in non-
central Au–Au collisions in the RHIC Beam Energy Scan (BES) Program STAR .
Different hydrodynamic models Karp ; Karp1 ; Karp2  generally reproduce the
magnitude of the measured polarization. In the hydrodynamic models, the Lambda
hyperons produced at “particlization” (fluid to particle transition)
hypersurface acquire polarization via a thermodynamic spin-vorticity coupling
mechanism. This effect is interesting as a possible manifestation of the most
vortic fluid ever made.

Interestingly, this is not the first case that $\Lambda$ polarization raises
interest in the high energy community. In the nineteens of the past century,
much discussion was related Lambda  to the origin of polarization in the
scattering of unpolarized particles. According to Ref. Strum  polarization of
inclusively produced lambdas arises due to the spin–orbit interaction in a
scalar field in which quarks recombine into hadrons.

The present paper to a large extent reflects the author’s vision of the spin
effects in the “soft” region and his personal contribution in the field. For a
much wider panorama we recommend to the reader the excellent papers Leader0 ;
Rev1 ; Rev2 .

##  II Hadron-Hadron Scattering

In this section we collect the basic notions and rules indispensable in
describing hadronic reactions. We restrict our discussion of this vast field
to elastic nucleon scattering in the nearly forward region, dominated by the
diffraction cone, and nominated “soft” or “non-perturbative”, implying that
perturbative QCD methods here are not applicable. Instead, the basic tools are
those based on the analytic $S$ matrix, namely, dispersion relations,
unitarity and Regge poles. We relate these means in the context of the subject
of the present paper—spin and polarization effects at high energies. Below is
a short overview of the relevant tools.

Soft events are described by Regge pole models, sometimes appended by “QCD-
inspirations”. In most of the papers on the subject, a spin-independent
invariant scattering amplitude is used at high-energies. Since spin effects
cannot be excluded a priori, below we briefly present an introduction to the
relevant formalism.

In the Regge pole model, the trajectories contain most of the basic
information on the dynamics. Their form is constrained by unitarity and
analyticity, and is intimately connected by the Chew–Frautchi plot to the
spectrum of hadrons. This is a central issue in high energy physics, since it
is related to the problem of confinement: how do heavy resonances melt
producing a quark–gluon soup? In our approach, the real part of Regge
trajectories is limited, implying a finite number of resonances in Nature.
This issue is related to the Hagedorn spectrum and “ultimate temperature”, now
interpreted as a the temperature of the phase transition between hadrons and
quark–gluon plasma Biro .

Unitarity is both important and technically difficult to be satisfied. Various
approaches and approximations to unitarity are presented and compared in this
Section. Let us mention that the final result (construction of a viable
scattering amplitude) depends both on the input and subsequent unitarization
procedure. The better the input (“born term”), the better are chances for its
subsequent successful unitarization: unitarity correction to a reasonable
input (for example, a dipole pomeron, reproducing itself under unitarization)
are small DP .

###  II.1 Elastic Proton-Proton Scattering

Elastic proton–proton scattering is described by five helicity amplitudes
Lehar ; Lehar1 , that in the Regge limit, $t$ fixed and $s\rightarrow\infty$,
can be expressed in terms of the Regge-pole contributions as Capella ; Troshin
; Troshin1 ; Selyugin1 ; Selyugin2

$\displaystyle\Phi_{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}}(s,t)\approx\sum_{i}g^{i}_{\lambda_{1},\lambda_{2}}(t)g^{i}_{\lambda_{3},\lambda_{4}}(t)[\sqrt{|t|}]^{|\lambda_{1}-\lambda_{2}|+|\lambda_{3}-\lambda_{4}|}\left(\frac{s}{s_{0}}\right)^{\alpha_{i}}(1\pm
e^{-i\pi\alpha_{i}}),$ (1)

where $\alpha_{i}\equiv\alpha_{i}(t)$ are Regge trajectories. In high-energy
elastic scattering the dominant trajectory is that of the pomeron. In the
simplest case, it is linear, parametrized, for simplicity as
$\alpha(t)=\alpha(0)+\alpha^{\prime}t$. We will use also advanced models of
Regge trajectories, to be introduced in the next Section II.2. Recently
Glueballs  it was used to predict glue and oddballs—resonances made of two or
three gluons.

The above helicity amplitudes differ only by powers of $\sqrt{t}$ leaving the
Regge propagator (i.e., nature of the Regge pole) intact. This means that the
addition of spin-flip amplitudes by themselves will not provide the dip
mechanism, although they will slightly modify it. The (single) diffraction
minimum followed by maximum was shown DP  to arise even at the Born level by
choosing the pomeron to be a dipole (double Regge pole or dipole pomeron (DP)
DP —an interesting and unique alternative to a simple Regge pole). Fits to the
the data on $pp$ scattering, including those from the LHC (TOTEM
Collaboration) using DP was performed recently DP1 . The global fit to all
energies, including the dip-bump is good, although it leaves room for further
perfection. Inclusion of the spin-flip component may improve the situation.

In most of the relevant papers the dip-bump structure is generated by
unitarity corrections, see Section II.3, although there is no consensus on the
details.

The differential cross sections is given by

$\displaystyle\frac{d\sigma}{dt}=\frac{2\pi}{s^{2}}\left(|\Phi_{1}|^{2}+|\Phi_{2}|^{2}+|\Phi_{3}|^{2}+|\Phi_{4}|^{2}+4|\Phi_{5}|^{2}\right)\,.$
(2)

The helicity amplitudes can be written as
$\Phi_{i}(s,t)=\Phi^{h}_{i}(s,t)+\Phi^{\rm em}_{i}(s,t)e^{\varphi(s,t)}$ ,
where $\Phi^{h}_{i}(s,t)$ comes from the strong interactions, $\Phi^{\rm
em}_{i}(s,t)$ from the electromagnetic interactions and $\varphi(s,t)$ is
their interference.

The spin correlation parameters, the analyzing power—$A_{N}$ and the and
double-spin parameter $A_{NN}$, can be extracted from experimental
measurements:

$\displaystyle A_{N}$ $\displaystyle=$
$\displaystyle\frac{\sigma(\uparrow)-\sigma(\downarrow)}{\sigma(\uparrow)+\sigma(\downarrow)}=\frac{\Delta\sigma^{s}}{\sigma_{0}}\,,$
(3) $\displaystyle A_{NN}$ $\displaystyle=$
$\displaystyle\frac{\sigma(\uparrow\uparrow)-\sigma(\uparrow\downarrow)}{\sigma(\uparrow\uparrow)+\sigma(\uparrow\downarrow)}=\frac{\Delta\sigma^{d}}{\sigma_{0}}\,,$
(4)

where $\Delta\sigma^{s}$ and $\Delta\sigma^{d}$ refer to the difference of
single- and double-spin-flip cross sections. The expressions for these
parameters are

$\displaystyle A_{N}\frac{d\sigma}{dt}$ $\displaystyle=$
$\displaystyle-\frac{4\pi}{s^{2}}\left[Im(\Phi_{1}+\Phi_{2}+\Phi_{3}-\Phi_{4})\Phi^{*}_{5}\right]\,;$
(5) $\displaystyle A_{NN}\frac{d\sigma}{dt}$ $\displaystyle=$
$\displaystyle\frac{4\pi}{s^{2}}\left[Re(\Phi_{1}\Phi^{*}_{2}-\Phi_{3}\Phi^{*}_{4})+|\Phi_{5}|^{2}\right]\,.$
(6)

###  II.2 Regge Trajectories

Unitarity imposes a severe constraint on the threshold behaviour of the
trajectories, as shown on the diagram, Figure 1:

$\Im m\alpha(t)_{t\rightarrow t_{0}}\sim(t-t_{0})^{\Re e\alpha(t_{0})+1/2},$
(7)

while asymptotically the trajectories are constrained by

$\Bigg{|}\frac{\alpha(t)}{\sqrt{t}\ln{t}}\Bigg{|}_{t\rightarrow\infty}\leq{\rm
const}.$ (8)

The above asymptotic constraint can be still lowered to a logarithm by
imposing wide-angle power behaviour for the amplitude.

Figure 1: Diagram of elastic scattering with $t$-channel exchange containing a
branch point at $t=4m_{\pi}^{2}$.

The trajectory satisfying the above constraints is:

$\alpha(t)=\frac{1+\delta+\alpha_{1}t}{1+\alpha_{2}\big{(}\sqrt{t_{0}-t}-\sqrt{t_{0}})},$
(9)

where $t_{0}=4m_{\pi}^{2}$ for the pomeron and $t_{0}=9m_{\pi}^{2}$ for the
odderon and $\delta,\alpha_{1},\alpha_{2}$ are parameters fitted to the data
with the obvious constraints: $\alpha(0)\approx 1.08$ and
$\alpha^{\prime}(0)\approx 0.3$ (in the case of the pomeron trajectory).
Trajectory (9) has square-root asymptotic behaviour, in accordance with the
requirements of the analytic $S$-matrix theory.

The contribution of the lowest threshold in the $t$ channel of the trajectory
and the scattering amplitude is shown on the diagram, Figure 1. The presence
of this threshold produces a “break” in the diffraction cone and generates the
pion cloud of the nucleon, as shown in Figure 2.

For $t>>t_{0},\ \
|\alpha(t)|\rightarrow\frac{\alpha_{1}}{\alpha_{2}}\sqrt{|t|}$. For $t>t_{0}$
(on the upper edge of the cut), $\Im m\alpha>0.$

The intercept is $\alpha(0)=1+\delta$ and the slope at $t=0$ is

$\alpha^{\prime}(0)=\alpha_{1}+\alpha_{2}\frac{1+\delta}{2\sqrt{t_{0}}}.$ (10)

Resonances terminate, i.e., they do not appear beyond the maximum value or the
finite real part of a given Regge trajectory.

For a recent application of this trajectory to glueball spectroscopy see Ref.
Glueballs .

###  II.3 Unitarity

The best known unitarization procedure is that of Regge-eikonal, where the
input (“Born term”) is a Regge-pole amplitude.

It implies two Fourier–Bessel transforms (integrals), one from the Mandelstam
variable $t$ (momentum space) to the impact parameter $b$ (coordinate space).
While direct integration is usually performed analytically (trivial for linear
Regge trajectories and still simple for a single square-root threshold, see
Ref. NC , the inverse integral is much more cumbersome, and can be done only
numerically. The technicalities become even more complicated when spin degrees
are included (see below).

A simple unitarization procedures is by the use Selyugin1 ; Selyugin2  of the
equation:

$\displaystyle\frac{dN}{dy}=\Delta N\ [\ 1-N\ ]\,,$ (11)

where $y=\log(s/s_{0})$ and $\Delta=1-\alpha(0)$, ($\alpha(0)$ is the
intercept of the leading Regge pole trajectory.

Its solution is

$\displaystyle N=\frac{\chi(s,b)}{1\ +\ \chi(s,b)}\,,$ (12)

where $\chi(s,b)\approx s^{\Delta}$ is connected with the Born term of the
scattering amplitude.

The scattering amplitude is

$\displaystyle\Phi^{h}(s,t)\ =\ \frac{i}{2\pi}\ \int\ d^{2}b\
e^{i\vec{b}\vec{q}}\frac{\chi(s,b)}{1+\chi(s,b)}\,.$ (13)

The phase $\chi(s,b)$ is connected to the interaction potential:

$\displaystyle\chi(s,b)\ =\ F_{\rm Born}(s,b)\ \approx\frac{1}{k}\
\int\hat{V}\left(\sqrt{b^{2}+z^{2}}\right)dz.$ (14)

If the potential contains a non-spin-flip part and, for example, spin-orbital
and spin-spin interactions, the phase is:

$\displaystyle\chi(s,b)=\chi_{0}(s,b)-i\
\vec{n}\cdot(\vec{\sigma}_{1}+\vec{\sigma}_{2})\chi_{\rm
LS}(s,b)-i(\vec{\sigma}_{1}\cdot\vec{\sigma}_{2})\ \chi_{\rm SS}(s,b).$ (15)

If however only the spin-flip and spin-non-flip parts are accounted for, the
overlap function is

$\displaystyle\Gamma(s,b)=\frac{\chi_{0}(s,b)+\sigma\chi_{sf}(s,b)}{1+\chi_{0}(s,b)+\sigma\chi_{sf}(s,b)}=1-\frac{\left(1+\chi_{0}(s,b)\right)-\sigma\chi_{sf}(s,b)}{\left(1+\chi_{0}(s,b)\right)^{2}-\left(\sigma\chi_{sf}(s,b)\right)^{2}}\,.$
(16)

Using the representation for the Bessel functions

$\displaystyle J_{0}(x)\ =\frac{1}{2\pi}\ \int_{0}^{2\pi}\ e^{ix\cos\phi}\
d\phi\ \ \ \ \ J_{1}(x)\ =-\frac{1}{2\pi}\ \int_{0}^{2\pi}\ e^{ix\cos\phi}\
\sin{\phi}\ d\phi\,,$ (17)

the representation of spin-non-flip and spin flip amplitude becomes

$\displaystyle\Phi^{h}_{1}(s,t)\ =\ i\ \int_{0}^{\infty}\
bJ_{0}(bq)\frac{\chi_{0}(s,b)}{1+\chi_{0}(s,b)}db\,;$ (18)
$\displaystyle\Phi^{h}_{5}(s,t)\ =\ i\ \int_{0}^{\infty}\
b^{2}J_{1}(bq)\frac{\chi_{sf}(s,b)}{\left(1+\chi_{0}(s,b\right))^{2}}db\,.$
(19)

####  II.3.1 “$U$-matrix” Unitarization

With an extra coefficient $n$

$\displaystyle\frac{dN}{dy}=\Delta N\ [\ 1-N/n\ ]\,.$ (20)

one gets the so-called $U$-matrix unitarization form, see Troshin  and
references therein.

This way of unitarization, developed mainly in Serpukhov, Dubna and Kiev, is
less familiar than the eikonal one, nevertheless it results in a number of
interesting predictions, among which is the so-called reflective scattering
Troshin2 , resulting in an increasing ratio $\sigma_{el}(s)/\sigma_{tot}(s)$
at the LHC. The unorthodox predictions of the $U$ matrix approach may bringing
new ideas in the complicated field of “soft physics”.

In the impact parameter representation, the properties of the $U$-matrix were
explored in Ref. Troshin . In that approach, the hadronic amplitude is given
by

$\displaystyle\Phi^{h}(s,t)\ =\ \frac{i}{2\pi}\ \int\ d^{2}b\
e^{i\vec{b}\vec{q}}\frac{\chi(s,b)}{1+\chi(s,b)/2}\,,$ (21)

where $\chi(s,b)$ is the same Born amplitude as before.

Comparing Equation (21) with Equation (13), we see that they are of similar
rational form, just they differ by the additional coefficient in the
denominator. This additional coefficient leads to different analytic
properties: the upper bound at which the overlapping function saturates will
be twice that compared with the eikonal or the $U_{e}$ representations, and
the inelastic overlap function at $b=0$ tends to zero at high energies.
Furthermore, $\sigma_{\rm el}/\sigma_{\rm tot}\rightarrow 1$.

For the $pp$ helicity amplitudes the solution of the unitarity equations is
Troshin :

$\displaystyle\Phi_{\lambda_{3},\lambda_{4},\lambda_{1},\lambda_{2}}({\bf
p},{\bf q})$ $\displaystyle=$ $\displaystyle
U_{\lambda_{3},\lambda_{4},\lambda_{1},\lambda_{2}}({\bf p},{\bf q})+$
$\displaystyle
i\frac{\pi}{8}\sum_{\lambda^{\prime},\lambda^{\prime\prime}}\int
d\Omega_{{\bf\hat{k}}}U_{\lambda_{3},\lambda_{4},\lambda^{\prime},\lambda^{\prime\prime}}({\bf
p},{\bf
k})\Phi_{\lambda^{\prime},\lambda^{\prime\prime},\lambda_{1},\lambda_{2}}({\bf
k},{\bf q})\,,$

####  II.3.2 Eikonal

To get the standard eikonal representation of the elastic scattering amplitude
in the impact parameter representation one uses the non-linear equation

$\displaystyle\frac{dN_{e}}{dy}=-\Delta\log(1-N_{e})[1-N_{e}]\,.$ (23)

where $y=\log(s/s_{0})$ and the subscript “e“ implies that the solution
$N_{e}$ is of the standard eikonal form

$\displaystyle N_{e}=\Gamma(s,b)=[1-e^{-\chi(s,b)}].$ (24)

The eikonal representation is then

$\displaystyle\Phi^{h}(s,t)\ =\frac{i}{2\pi}\ \int\ e^{i\vec{b}\vec{q}}\
\left[1-e^{-\chi(s,b)}\right]\ d^{2}b,$ (25)

where:

$\displaystyle\chi(s,b)=\chi_{0}(s,b)-i\
\vec{n}\cdot(\vec{\sigma}_{1}+\vec{\sigma}_{2})\chi_{LS}(s,b)-i(\vec{\sigma}_{1}\cdot\vec{\sigma}_{2})\
\chi_{SS}(s,b).$ (26)

With account for Equations (17) and (27), one gets respectively for spin non-
flip and spin-flip

$\displaystyle\Phi^{h}_{1}(s,t)\ =\ i\ \int_{0}^{\infty}\
bJ_{0}(bq)\left[1-e^{\chi_{0}(s,b)}\right]\ \left[1-b^{2}\chi^{2}_{\rm
LS}(s,b)-3/2\chi^{2}_{\rm SS}(s,b)\right]\ db,$ (27)
$\displaystyle\Phi^{h}_{5}(s,t)\ =\ i\ \int_{0}^{\infty}\
J_{1}(bq)\chi_{1}e^{\chi_{0}(s,b)}\ b\ \left[\chi_{\rm LS}(s,b)+i\ \chi_{\rm
LS}(s,b)\ \chi_{\rm SS}(s,b)\right]\ db,$ (28)

where

$\displaystyle\chi(s,b)_{0}$ $\displaystyle\approx$ $\displaystyle\
\int_{-\infty}^{\infty}V_{0}(s,b,z)dz\,;$ (29) $\displaystyle\chi(s,b)_{1}$
$\displaystyle\approx$ $\displaystyle\frac{b}{2}\
\int_{-\infty}^{\infty}V_{1}(s,b,z)dz\,.$ (30)

For Gaussian potentials $V_{0}$ and $V_{1}$

$\displaystyle V(s,b)_{0,1}\ \approx\ \int_{-\infty}^{\infty}e^{-Br^{2}}\
dz=\frac{\sqrt{\pi}}{\sqrt{B}}e^{-Bb^{2}}\,,$ (31)

in the first Born approximation, $\Phi_{0}^{h}$ and $\Phi_{1}^{h}$ take the
forms

$\displaystyle\Phi^{h}_{1}(s,t)\ \approx\ \int_{0}^{\infty}\
bJ_{0}(bq)e^{-Bb^{2}}db=e^{-Bq^{2}};$ (32) $\displaystyle\Phi^{h}_{5}(s,t)\
\approx\ \int_{0}^{\infty}\ b^{2}J_{1}(bq)e^{-Bb^{2}}db=qBe^{-Bq^{2}}.$ (33)

Figure 2: (a) Typical shape of the high-energy differential (diffractive)
cross section, and (b) Its impact-parameter image. A “dictionary” relating the
shape of the diffraction cone (in $t$), left icon to its impact-paramter image
(rignt), and highlighting their two silent features: the “break”, origin of
nucleon’s atmosphere and the dip-bump, resulting from absorptions (unitarity!)
at small values of the impact parameter, see DP .

In view of these technical complications, of interest is the unique case of a
dipole pomeron (DP), reproducing itself under eikonal corrections, as noticed
in Ref. Phillips , see also the Appendix in Fortschritte . In other words, DP
in the impact parameters representation remains stable (intact) under
eikonalization. In addition, a DP produces logarithmically rising cross
sections at unit pomeron intercept and obeys geometrical scaling, making it
attractive as an efficient model of diffraction. Note that higher order Regge
poles violate unitarity.

##  III Nuclear Structure: From Deeply Virtual Compton Scattering (DVCS) to
Generalized Parton Distributions (GPDs)

An important part of spin physics is lepton-hadron scattering. Deeply virtual
exclusive $ep$ processes provide an important tool in accessing the
generalized parton distributions (GPDs) Mueller:1998fv ; Radyushkin:1996nd ;
Ji:1996nm . The underlying mechanism is Regge-pole exchange in the
$t$-channel. Based on factorization theorems Collins:1996fb ; Collins:1998be ,
GPDs offer a partonic interpretation of these processes, where unobserved
transverse degrees of freedom are integrated out. Thereby, these universal
functions, defined in terms of matrix elements of quark and gluon operators
or, alternatively, as a non-diagonal overlap of light-cone wave functions
Diehl:1998kh ; Diehl:2000xz , encode the non-perturbative aspects of the
nucleon Diehl:2003ny ; Belitsky:2005qn . In particular, GPDs provide access to
the transverse spatial distribution of patrons Burkardt:2000za ; Die02 ;
KogSop70 , and to the decomposition of the nucleon spin in terms of quark and
gluon degrees of freedom Ji:1996ek .

Phenomenologically, exclusive electroproduction of a real photon, DVCS,
depicted in Figure 3 (left), is the golden channel to constrain GPDs as it is
theoretically clean and the phase of its amplitude can be measured using the
interference with the Bethe–Heitler (BH) amplitude (see Figure 3; cf. Figures
4 and 5).

More details, especially those related to experiments can be found in
Accardi:2012hwp ; Fazio .

###  III.1 Deeply Virtual Compton Scattering

The differential photon electroproduction cross section is the sum of the BH
amplitude squared, DVCS amplitude squared, and the interference (INT) terms
(see Figure 4)

$\frac{d\sigma^{ep\to ep\gamma}}{dx_{\rm B}dtd{\cal Q}^{2}d\phi
d\varphi}=\frac{d\sigma^{ep\to ep\gamma,\mbox{\tiny BH}}(F_{1},F_{2})}{dx_{\rm
B}dtd{\cal Q}^{2}d\phi d\varphi}\pm\frac{d\sigma^{ep\to ep\gamma,\mbox{\tiny
INT}}(F_{1},F_{2},{\cal F})}{dx_{\rm B}dtd{\cal Q}^{2}d\phi
d\varphi}+\frac{d\sigma^{ep\to ep\gamma,\mbox{\tiny(D)VCS}}({\cal F},{\cal
F}^{\ast})}{dx_{\rm B}dtd{\cal Q}^{2}d\phi d\varphi}\,.$ (34)

Figure 3:  Deeply virtual Compton scattering (DVCS) and Bethe–Heitler (BH)
amplitudes contributing to the photon leptoproduction cross section in leading
order approximation of QED.

Here the $+(-)$ signs correspond to the electron (positron) beam, $x_{\rm B}$
is the common Bjorken scaling variable, $\phi$ is the azimuthal angle between
lepton and hadron scattering planes, and $\varphi=\Phi-\phi$, where
$\Phi(\equiv\phi_{S})$ is the angle between the lepton scattering plane and a
possible transverse spin component of the incoming proton at rest. To the
leading order (LO) in the electromagnetic fine structure constant $\alpha_{\rm
em}=\frac{e^{2}}{4\pi}\approx\frac{1}{137},$ and neglecting the electron mass,
the three terms on the r.h.s. of (34) are known in terms of the
electromagnetic Pauli form factor $F_{1}(t)$ and the Dirac form factor
$F_{2}(t)$, parameterizing the BH amplitude, and a set of 12 photon helicity
dependent CFFs ${\cal F}_{ab}(x_{\rm B},t,{\cal Q}^{2})$, parameterizing the
DVCS amplitude, see Figure 3. These CFFs are labelled by the helicities of the
incoming $a\in\\{+,0,-\\}$ and outgoing photon $b\in\\{+,-\\}$ called

${\cal F}_{ab}\in\\{{\cal H}_{ab},{\cal E}_{ab},\widetilde{\cal
H}_{ab},\widetilde{\cal E}_{ab}\\}\quad\mbox{with}\quad{\cal F}_{0-}={\cal
F}_{0+}\,,\quad{\cal F}_{+-}={\cal F}_{-+}\,,$ (35)

Figure 4: Interference in DVCS (Equation (34)) provides a holographic picture
of the proton similar to the classical one. Figure 5: Generalized parton
distributions (GPD) (right) unify in a non-trivial way the the impact
parameter image of a nucleon (left) with the $x$ dependence of DIS structure
functions (middle).

###  III.2 Relating DVCS Observables to GPDs

GPDs, denoted as

$F(x,\eta=\xi,t,\mu^{2})\quad\mbox{with}\quad
F\in\\{H,E,\widetilde{H},\widetilde{E}\\}\,,$

besides depending on the partonic momentum fraction $x$ and the momentum
transfer squared $t$, depend also on the $t$-channel longitudinal momentum
fraction $\eta$, called skewness (often denoted by $\xi$ in the literature),
and on the factorization scale $\mu^{2}$. The unpolarized parton GPDs are
called $H$ and $E$ Ji:1998pc , where the former (latter) GPD can be loosely
associated with a proton helicity (non)conserved distribution.

DVCS observables can be evaluated in terms of the helicity CFFs (35). To
express them in terms of GPDs, it is appropriate to utilize a conventionally
defined GPD-inspired CFF basis, such as the one introduced in Belitsky:2001ns
:

${\cal F}\in\\{{\cal H},{\cal E},\widetilde{\cal H},\widetilde{\cal E},{\cal
H}_{3},{\cal E}_{3},\widetilde{\cal H}_{3},\widetilde{\cal E}_{3},{\cal
H}_{\rm T},{\cal E}_{\rm T},\widetilde{\cal H}_{\rm T},\widetilde{\cal E}_{\rm
T}\\}\,.$ (36)

Here, the CFFs ${\cal H},{\cal E},\widetilde{\cal H}$, and $\widetilde{\cal
E}$ are associated with twist-two GPDs
$F\in\\{H,E,\widetilde{H},\widetilde{E}\\}$ and govern the photon helicity
non-flip DVCS amplitude, i.e., at leading twist-two accuracy one has

$\displaystyle{\cal F}_{++}(x_{\rm B},t,{\cal Q}^{2})={\cal F}(x_{\rm
B},t,{\cal Q}^{2})+{\cal O}(1/{\cal Q}^{2})\quad\mbox{for}\quad{\cal
F}\in\\{{\cal H},{\cal E},\widetilde{\cal H},\widetilde{\cal E}\\}.$ (37)

To LO they are calculated from the handbag diagram, depicted in Figure 6,
yielding the convolution formula

${\cal F}(x_{\rm B},t,{\cal Q}^{2})\stackrel{{\scriptstyle\rm
LO}}{{=}}\sum\limits_{i}\int_{-1}^{1}\\!dx\left[\frac{e^{2}_{i}}{\xi-
x-i\epsilon}\mp\\{x\rightarrow-x\\}\right]F_{i}(x,\xi,t,\mu^{2})\;\;\mbox{for}\;\;{\cal
F}\in\left\\{{{\cal H},{\cal E}\atop\widetilde{\cal H},\widetilde{\cal
E}}\right\\},$ (38)

where $e_{i}$ are the fractional quark charges. The variable $\xi\sim x_{\rm
B}/(2-x_{\rm B})$ is a conventionally defined Bjorken-like scaling variable,
equated to the longitudinal momentum fraction in the $t$-channel, and
$\mu^{2}\sim{\cal Q}^{2}$ being the factorization scale.

For twist-two and LO accuracy, and light quarks one adopts the conventions

$\displaystyle\xi=\frac{x_{\rm B}}{2-x_{\rm
B}}\quad\mbox{and}\quad\mu^{2}={\cal Q}^{2}\,.$ (39)

From Regge-pole models, consistent with the data, see Section IV below, the
real part of the dominant CFF ${\cal H}$ in the small- and even
moderate-$x_{\rm B}$ region is much smaller than its imaginary part (at least
for smaller values of $-t$). One concludes that the interference term is
negligible and one can simplify the $t$-differential cross section to

$\displaystyle\frac{d\sigma^{\mbox{\tiny
TOT}}}{dt}\approx\frac{y^{2}\left[\frac{d\sigma_{\rm T}^{\mbox{\tiny
BH,red}}}{dt}+\varepsilon(y)\frac{d\sigma_{\rm L}^{\mbox{\tiny
BH,red}}}{dt}\right]}{\left(1-y\frac{(1-x_{\rm B})t}{{\cal
Q}^{2}+t}\right)\left(\frac{{\cal Q}^{2}+t}{{\cal Q}^{2}+x_{\rm
B}t}-y\right)}+\frac{d\sigma^{\mbox{\tiny\rm DVCS}}(y)}{dt}$ (40)

with
$\frac{d\sigma^{DVCS}(y)}{dt}=\frac{d\sigma_{T}^{DVCS}}{dt}+\varepsilon(y)\frac{d\sigma_{L}^{DVCS}}{dt}.$

Figure 6:  Factorization of the DVCS amplitude to leading order in
perturbative QCD and to leading twist-two accuracy.

##  IV Modelling DVCS

Below we present models of DVCS and vector meson production (VMP), potentially
useful in calculating GPDs.

Regge pole models provide an adequate framework to describe high-energy, low
$t$ scattering phenomena. Being part of the $S$ matrix theory, however,
strictly speaking, they are valid only for the scattering of on-mass-shall
particles. Still, the successful application of the Regge pole models in
describing the off-mass-shall HERA data opened the way to their use in deep-
inelastic scattering, DVCS and vector meson production at HERA. Off-shell
extension was realized e.g., by calling the $Q^{2}$-dependent Regge
trajectories “effective” ones. A particularly simple and efficient Regge pole
model Capua  with $Q^{2}$-dependent residues (vertices) is presented below, in
Section IV.

In a more advanced Regge-pole model, Section V the Regge trajectories and the
residues do not depend on virtuality. Instead, the amplitude contains two (or
more) Regge-pole terms, whose relative weight depends on $Q^{2}$, mimicking
the multi-pole nature of the so-called QCD pomeron.

### Simple Model of DVCS

By Regge factorization, Figure 7, the DVCS amplitude can be written as

$A(s,t,Q^{2})_{\gamma^{*}p\rightarrow\gamma
p}=-A_{0}V_{1}(t,Q^{2})V_{2}(t)(-is/s_{0})^{\alpha(t)},$ (41)

where $A_{0}$ is a normalization factor, $V_{1}(t,Q^{2})$ is the
$\gamma^{*}P\gamma$ vertex, $V_{2}(t)$ is the $pPp$ vertex and $\alpha(t)$ is
the exchanged pomeron trajectory, which we assume to be logarithmic:

$\alpha(t)=\alpha(0)-\alpha_{1}\ln(1-\alpha_{2}t).$ (42)

Such a trajectory is nearly linear for small $|t|$, thus reproducing the
forward cone of the differential cross section, while its logarithmic
asymptotic provides for the large-angle scaling behaviour, typical of hard
collisions at small distances, with power-law fall-off in $|t|$, compatible
with the quark counting rules. Here we are referring to the dominant pomeron
contribution eventually appended by a secondary trajectory, e.g., the
$f$-Reggeon.

Figure 7:  Diagrams of DVCS (a) and VMP (b); (c) DVCS (VMP) amplitude in a
Regge-factorized form.

For convenience, and following the arguments based on duality, the $t$
dependence of the $pPp$ vertex is introduced via the $\alpha(t)$ trajectory:
$V_{2}(t)=e^{b\alpha(t)}$ where $b$ is a parameter. A generalization of this
concept will be applied also to the upper, $\gamma^{*}P\gamma$ vertex by
introducing the trajectory

$\beta(z)=\alpha(0)-\alpha_{1}\ln(1-\alpha_{2}z),$ (43)

where the value of the parameter $\alpha_{2}$ may be different in $\alpha(t)$
and $\beta(z)$ (a relevant check will be possible when more data are
available).

Hence the scattering amplitude, with correct signature, becomes

$A(s,t,Q^{2})_{\gamma^{*}p\rightarrow\gamma
p}=-A_{0}e^{b\alpha(t)}e^{b\beta(z)}(-is/s_{0})^{\alpha(t)}=-A_{0}e^{(b+L)\alpha(t)+b\beta(z)},$
(44)

where $L\equiv\ln(-is/s_{0})$.

The model contains a limited number of parameters. Moreover, most of them can
be estimated a priori. The product $\alpha_{1}\alpha_{2}$ is just the forward
slope $\alpha^{\prime}$ of the Reggeon ($\approx$0.2 GeV-2 for the pomeron,
but much higher for $f$ and/or for an effective Reggeon). The value of
$\alpha_{1}$ can be estimated from the wide-angle quark counting rules. For
large $t$ ($|t|>>$1 GeV2) the amplitude goes roughly as $\sim
e^{-\alpha_{1}\ln(-t)}=(-t)^{\alpha_{1}},$ where the power $\alpha_{1}$ is
related to the number of quarks in a collision.

From Equation (44) the slope of the forward cone is

$B(s,Q^{2},t)=\frac{d}{dt}\ln|A|^{2}=2\left[b+\ln\left({s\over{s_{0}}}\right)\right]{\alpha^{\prime}\over{1-\alpha_{2}t}}+2b\,{\alpha^{\prime}\over{1-\alpha_{2}z}},$
(45)

which, in the forward limit, $t=0$ reduces to

$B(s,Q^{2})=2\left[b+\ln\left({s\over{s_{0}}}\right)\right]\alpha^{\prime}+2b\,{\alpha^{\prime}\over{1+\alpha_{2}Q^{2}}}.$
(46)

Thus, the slope shows shrinkage in $s$ and antishrinkage in $Q^{2}.$

In the $Q^{2}\rightarrow 0$ limit the Equation (44) becomes

$A(s,t)=-A_{0}e^{2b\alpha(t)}(-is/s_{0})^{\alpha(t)}$ (47)

where we recognize a typical Regge-behaved photoproduction (or, for
$Q^{2}\rightarrow m_{H}^{2},$ on-shell hadronic ($H$)) amplitude. The related
deep inelastic scattering structure function is recovered by setting
$Q_{2}^{2}=Q^{2}_{1}=Q^{2}$ and $t=0$, to get a typical elastic virtual
forward Compton scattering amplitude:

$A(s,Q^{2})=-A_{0}e^{b(\alpha(0)-\alpha_{1}\ln(1+\alpha_{2}Q^{2}))}e^{(b+\ln(-is/s_{0}))\alpha(0)}\propto-(1+\alpha_{2}Q^{2})^{-\alpha_{1}}(-is/s_{0})^{\alpha(0)}.$
(48)

In the Bjorken limit, when both $s$ and $Q^{2}$ are large and $t=0$ (with
$x\approx Q^{2}/s$ valid for large $s$), the structure function is given by:

$F_{2}(s,Q^{2})\approx{(1-x)Q^{2}\over{\pi\alpha_{e}}}\Im A(s,Q^{2})/s,$ (49)

where $\alpha_{e}$ is the electromagnetic coupling constant and the
normalization is $\sigma_{t}(s)=\frac{4\pi}{s}\Im A(s,Q^{2}).$ The resulting
structure function has correct (required by gauge invariance)
$Q^{2}\rightarrow 0$ limit and approximate scaling (in $x$) behavior for large
enough $s$ and $Q^{2}$.

##  V Reggeometry

In a series of papers (see Refs. Salii1 ; Salii2 ), photon virtuality was
incorporated in a “geometrical” way, reflecting the observed trend in the
decrease of the forward slope as a function of $\widetilde{Q^{2}}$. This
geometrical approach, combined with the Regge-pole model, was named “Reg-ge-
ometry” (a wordplay, pun). A Reggeometric amplitude dominated by a single
pomeron shows reasonable agreement with the HERA data on VMP and DVCS, when
fitted separately to each reaction.

As further step, to reproduce the observed trend of hardening as
$\widetilde{Q^{2}}$ increases, and following Donnachie and Landshoff DL1 ; DL2
, a two-term amplitude, characterized by a two-component—“soft” +
“hard”—pomeron, was suggested Salii1 . We stress that the pomeron is unique,
but we construct it as a sum of two terms. Then, the amplitude is defined as

$A(\widetilde{Q^{2}},s,t)=A_{s}(\widetilde{Q^{2}},s,t)+A_{h}(\widetilde{Q^{2}},s,t)$
(50)

($s=W^{2}$ is the square of the c.m.s. energy), such that the relative weight
of the two terms changes with $\widetilde{Q^{2}}$ in the right way, i.e., the
ratio $r=A_{h}/A_{s}$ increases as the reaction becomes “harder” and v.v. It
is interesting to note that this trend is not guaranteed “automatically”: both
the “scaling” model Capua  or the Reggeometric one Salii1 ; Salii2  show the
opposite tendency, that may not be merely an accident and whose reason should
be better understood. This “wrong” trend can and should be corrected, and in
fact it was corrected DL1 ; DL2  by means of additional
$\widetilde{Q^{2}}$-dependent factors $H_{i}(\widetilde{Q^{2}}),\ i=s,h$
modifying the $\widetilde{Q^{2}}$ dependence of the amplitude, in a such way
as to provide increasing of the weight of the hard component with increasing
$\widetilde{Q^{2}}$. To avoid conflict with unitarity, the rise with
$\widetilde{Q^{2}}$ of the hard component is finite (or moderate), and it
terminates at some saturation scale, whose value is determined
phenomenologically. In other words, the “hard” component, invisible at small
$\widetilde{Q^{2}}$, gradually takes over as $\widetilde{Q^{2}}$ increases. An
explicit example of these functions is presented below.

Recall that the invariant scattering amplitude is defined as

$A(Q^{2},s,t)=\widetilde{H}e^{-\frac{i\pi\alpha(t)}{2}}\left(\frac{s}{s_{0}}\right)^{\alpha(t)}e^{2\left(\frac{a}{\widetilde{Q^{2}}}+\frac{b}{2m_{N}^{2}}\right)t},$
(51)

where

$\alpha(t)=\alpha_{0}+\alpha^{\prime}t$ (52)

is the linear pomeron trajectory (the use of more advanced, non-linear
trajectories of Section II.2 is straightforward), $a$ and $b$ are two
parameters to be determined from the fitting procedure and $m_{N}$ is the
nucleon mass. The coefficient $\widetilde{H}$ is a function providing the
right behavior of elastic cross sections in $\widetilde{Q^{2}}$:

$\widetilde{H}\equiv\widetilde{H}(\widetilde{Q^{2}})=\frac{\widetilde{A_{0}}}{\left(1+\frac{\widetilde{Q^{2}}}{{Q_{0}^{2}}}\right)^{n_{s}}},$

where $\widetilde{A_{0}}$ is a normalization factor, $Q_{0}^{2}$ is a scale
for the virtuality and $n_{s}$ is a real positive number.

In this model one uses an effective pomeron, which can be “soft” or “hard”,
depending on the reaction and/or kinematic region defining its “hardness”. In
other words, the values of the parameters $\alpha_{0}$ and $\alpha^{\prime}$
must be fitted to each set of the data. Apart from $\alpha_{0}$ and
$\alpha^{\prime}$, the model contains five more sets of free parameters,
different in each reaction, as shown in Table 1. The exponent in the
exponential factor in Equation (51) reflects the geometrical nature of the
model: $a/\widetilde{Q^{2}}$ and $b/2m_{N}^{2}$ correspond to the “sizes” of
upper and lower vertices in Figure 7c.

With the norm

$\frac{d\sigma_{el}}{dt}=\frac{\pi}{s^{2}}|A(Q^{2},s,t)|^{2},$ (53)

the differential and integrated elastic cross sections become,

$\frac{d\sigma_{el}}{dt}=\frac{A_{0}^{2}}{\left(1+\frac{\widetilde{Q^{2}}}{{Q_{0}^{2}}}\right)^{2n}}\left(\frac{s}{s_{0}}\right)^{2(\alpha(t)-1)}e^{4\left(\frac{a}{\widetilde{Q^{2}}}+\frac{b}{2m_{N}^{2}}\right)t}$
(54)

and

$\sigma_{el}=\frac{A_{0}^{2}}{\left(1+\frac{\widetilde{Q^{2}}}{{Q_{0}^{2}}}\right)^{2n}}\frac{\left(\frac{s}{s_{0}}\right)^{2(\alpha_{0}-1)}}{4\left(\frac{a}{\widetilde{Q^{2}}}+\frac{b}{2m_{N}^{2}}\right)+2\alpha^{\prime}\ln\left(\frac{s}{s_{0}}\right)},$
(55)

where

$A_{0}=-\frac{\sqrt{\pi}}{s_{0}}\widetilde{A_{0}}.$

The results of the fits are presented in Table 1. The “mass parameter” for
DVCS was set to $M=0$ GeV, therefore in this case $\widetilde{Q^{2}}=Q^{2}$.
Each type of reaction was fitted separately. As it can be seen from the right
plot of Figure 8, the single-term model fails to fit both the high- and
low-$|t|$ regions properly, especially when soft (photoproduction or low
$Q^{2}$) and hard (electroproduction or high $Q^{2}$) regions are considered.
One of the problems of the single-term Reggeometric pomeron model, Equation
(51), is that the fitted parameters in this model acquire particular values
for each reaction,

Figure 8:  Representative fits of Equations (54) and (55) to the data on
$J/\psi$ production. The values of the fitted parameters are compiled in Table
1. Table 1: Values of the parameters in Equations (54) and (55) fitted to data
on vector meson production (VMP) and DVCS at HERA. The parameters with
indefinite error bars were fixed at the fitting stage.

|  $A_{0}$ $\left[\frac{\sqrt{\text{{nb}}}}{\text{{GeV}}}\right]$ |  $\widetilde{Q^{2}_{0}}$ $\left[\text{{GeV}}^{\textbf{2}}\right]$ |  $n$ | $\alpha_{0}$ |  $\alpha^{\prime}$ $\left[\frac{1}{\text{GeV}^{\textbf{2}}}\right]$ | $a$ | $b$ | ${\tilde{\chi}}^{2}$
---|---|---|---|---|---|---|---|---
$\rho^{0}$ |  344 $\pm$ 376 |  0.29 $\pm$ 0.14 |  1.24 $\pm$ 0.07 |  1.16 $\pm$ 0.14 |  0.21 $\pm$ 0.53 |  0.60 $\pm$ 0.33 |  0.9 $\pm$ 4.3 | 2.74
$\phi$ |  58 $\pm$ 112 |  0.89 $\pm$ 1.40 |  1.30 $\pm$ 0.28 |  1.14 $\pm$ 0.19 |  0.17 $\pm$ 0.78 |  0.0 $\pm$ 19.8 |  1.34 $\pm$ 5.09 | 1.22
$J/\psi$ |  30 $\pm$ 31 |  2.3 $\pm$ 2.2 |  1.45 $\pm$ 0.32 |  1.21 $\pm$ 0.09 |  0.077 $\pm$ 0.072 | 1.72 | 1.16 | 0.27
$\varUpsilon$(1S) |  37 $\pm$ 100 |  0.93 $\pm$ 1.75 |  1.45 $\pm$ 0.53 |  1.29 $\pm$ 0.25 |  0.006 $\pm$ 0.6 | 1.90 | 1.03 | 0.4
$DVCS$ |  14.5 $\pm$ 41.3 |  0.28 $\pm$ 0.98 |  0.90 $\pm$ 0.18 |  1.23 $\pm$ 0.14 |  0.04 $\pm$ 0.71 | 1.6 |  1.9 $\pm$ 2.5 | 1.05

The VMP results clearly show hardening of the pomeron in the change of
$\alpha_{0}$ and $\alpha^{\prime}$ when going from light to heavy vector
mesons.

It is also interesting to note that the effective pomeron trajectory for DVCS
($\alpha_{0}=1.23,$ $\alpha^{\prime}=0.04$, see Table 1), contrary to
expectations is rather “hard”.

### Two-Component Reggeometric Pomeron

Now we introduce the universal, “soft” and “hard”, pomeron model. Using the
Reggeometric ansatz of Equation (51), we write the amplitude as a sum of two
parts, corresponding to the “soft” and “hard” components of a universal,
unique pomeron:

$A(Q^{2},s,t)=\widetilde{H_{s}}\,e^{-i\frac{\pi}{2}\alpha_{s}(t)}\left(\frac{s}{s_{0s}}\right)^{\alpha_{s}(t)}e^{2\left(\frac{a_{s}}{\widetilde{Q^{2}}}+\frac{b_{s}}{2m_{N}^{2}}\right)t}+\widetilde{H_{h}}\,e^{-i\frac{\pi}{2}\alpha_{h}(t)}\left(\frac{s}{s_{0h}}\right)^{\alpha_{h}(t)}e^{2\left(\frac{a_{h}}{\widetilde{Q^{2}}}+\frac{b_{h}}{2m_{N}^{2}}\right)t}.$
(56)

Here $s_{0s}$ and $s_{0h}$ are squared energy scales, and $a_{i}$ and $b_{i}$,
with $i=s,h$, are parameters to be determined with the fitting procedure. The
two coefficients $\widetilde{H_{s}}$ and $\widetilde{H_{h}}$ are functions
similar to those defined in Refs. DL1 ; DL2 :

$\widetilde{H_{s}}\equiv\widetilde{H_{s}}(\widetilde{Q^{2}})=\frac{\widetilde{A_{s}}}{{\Bigl{(}1+\frac{\widetilde{Q^{2}}}{{Q_{s}^{2}}}\Bigr{)}}^{n_{s}}},~{}~{}~{}~{}~{}~{}\quad\widetilde{H_{h}}\equiv\widetilde{H_{h}}(\widetilde{Q^{2}})=\frac{\widetilde{A_{h}}\Bigl{(}\frac{\widetilde{Q^{2}}}{Q_{h}^{2}}\Bigr{)}}{{\Bigl{(}1+\frac{\widetilde{Q^{2}}}{{Q_{h}^{2}}}\Bigr{)}}^{n_{h}+1}},$
(57)

where $\widetilde{A_{s}}$ and $\widetilde{A_{h}}$ are normalization factors,
$Q_{s}^{2}$ and $Q_{h}^{2}$ are scales for the virtuality, $n_{s}$ and $n_{h}$
are real positive numbers. Each component of Equation (56) has its own, “soft”
or “hard”, Regge (here pomeron) trajectory:

$\alpha_{s}(t)=\alpha_{0s}+\alpha_{s}^{\prime}t,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\quad\alpha_{h}(t)=\alpha_{0h}+\alpha_{h}^{\prime}t.$

The terms “soft” and “hard” may be misleading, alluding to the wide-spead
nomenclature by which the “soft” pomeron is associated with the traditional
Regge-pole theory, while the “hard” one is the BFKL pomeron derived from QCD,
having little to do with the former, by meaning that they are different
object. Instead, we insist that there is only one pomeron, although it may be
complicated, consisting of more terms etc. In our case the two terms refer to
a single object, THE pomeron, while its (two) parts are weighted with
virtuality $Q^{2}$.

As an input we use the parameters suggested by Donnachie and Landshoff DL1 ;
DL2 , so that

$\alpha_{s}(t)=1.08+0.25t,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\quad\alpha_{h}(t)=1.40+0.1t.$

The “pomeron” amplitude (56) is unique, valid for all diffractive reactions,
its “softness” or “hardness” depending on the relative
$\widetilde{Q^{2}}$-dependent weight of the two components, governed by the
relevant factors $\widetilde{H}_{s}(\widetilde{Q}^{2})$ and
$\widetilde{H_{h}}(\widetilde{Q^{2})}$.

To reduce the number of free parameters, we simplified the model, by fixing
$a_{s,h}=0$ and substituting the exponent
$2\left(\frac{a_{s,h}}{\widetilde{Q^{2}}}+\frac{b_{s,h}}{2m_{N}^{2}}\right)$
with $b_{s,h}$ in Equation (56). The proper variation with $\widetilde{Q^{2}}$
will be provided by the factors $\widetilde{H_{s}}(\widetilde{Q^{2}})$ and
$\widetilde{H_{h}}(\widetilde{Q^{2}})$.

Consequently, the scattering amplitude assumes the form

$A(s,t,Q^{2},M_{V}^{2})=\widetilde{H_{s}}\,e^{-i\frac{\pi}{2}\alpha_{s}(t)}\left(\frac{s}{s_{0s}}\right)^{\alpha_{s}(t)}e^{b_{s}t}+\widetilde{H_{h}}\,e^{-i\frac{\pi}{2}\alpha_{h}(t)}\left(\frac{s}{s_{0h}}\right)^{\alpha_{h}(t)}e^{b_{h}t}.$
(58)

The “Reggeometric” combination
$2\left(\frac{a_{s,h}}{\widetilde{Q^{2}}}+\frac{b_{s,h}}{2m_{N}^{2}}\right)$
was important for the description of the slope $B(Q^{2})$ within the single-
term pomeron model (see previous Section), but in the case of two terms the
$Q^{2}$-dependence of $B$ can be reproduced without this extra combination,
since each term in the amplitude (58) has its own $Q^{2}$-dependent factor
$\widetilde{H_{\ }}_{\\!\\!s,h}(Q^{2})$.

By using the amplitude (58) and Equation (53), we calculate the differential
and elastic cross sections, by setting for simplicity $s_{0s}=s_{0h}=s_{0}$,
to obtain

$\frac{d\sigma_{el}}{{dt}}=H_{s}^{2}e^{2\\{L(\alpha_{s}(t)-1)+{b_{s}}t\\}}+H_{h}^{2}e^{2\\{L(\alpha_{h}(t)-1)+{b_{h}}t\\}}$
(59)

$+2H_{s}H_{h}e^{\\{L(\alpha_{s}(t)-1)+L(\alpha_{h}(t)-1)+({b_{s}}+{b_{h}})t\\}}\cos\Bigl{(}\frac{\pi}{2}(\alpha_{s}(t)-\alpha_{h}(t))\Bigr{)},$
$\sigma_{el}=\frac{H_{s}^{2}e^{2\\{L(\alpha_{0s}-1)\\}}}{2(\alpha_{s}^{\prime}L+{b_{s}})}+\frac{H_{h}^{2}e^{2\\{L(\alpha_{0h}-1)\\}}}{2(\alpha_{h}^{\prime}L+{b_{h}})}+2H_{s}H_{h}e^{L(\alpha_{0s}-1)+L(\alpha_{0h}-1)}\frac{\mathfrak{B}\cos\phi_{0}+\mathfrak{L}\sin\phi_{0}}{\mathfrak{B}^{2}+\mathfrak{L}^{2}}.$
(60)

In these two equations we use the notations

$\begin{array}[]{l}L=\ln\left({s/s_{0}}\right),\\\
\phi_{0}=\frac{\pi}{2}(\alpha_{0s}-\alpha_{0h}),\end{array}\qquad\begin{array}[]{l}\mathfrak{B}=L\alpha_{s}^{\prime}+L\alpha_{h}^{\prime}+({b_{s}}+{b_{h}}),\\\
\mathfrak{L}=\frac{\pi}{2}(\alpha_{s}^{\prime}-\alpha_{h}^{\prime}),\end{array}$
$H_{s}(\widetilde{Q^{2}})=\frac{A_{s}}{{\Bigl{(}1+\frac{\widetilde{Q^{2}}}{{Q_{s}^{2}}}\Bigr{)}}^{n_{s}}},\quad
H_{h}(\widetilde{Q^{2}})=\frac{A_{h}\Bigl{(}\frac{\widetilde{Q^{2}}}{{Q_{h}^{2}}}\Bigr{)}}{{\Bigl{(}1+\frac{\widetilde{Q^{2}}}{{Q_{h}^{2}}}\Bigr{)}}^{n_{h}+1}},$

with

$A_{s,h}=-\frac{\sqrt{\pi}}{s_{0}}\widetilde{A_{\ }}_{\\!\\!s,h}.$

Notice that amplitude (58) can be rewritten in the form

$A(s,t,Q^{2},{M_{v}}^{2})=\widetilde{A_{s}}e^{-i\frac{\pi}{2}\alpha_{s}(t)}\left(\frac{s}{s_{0}}\right)^{\alpha_{s}(t)}e^{b_{s}t-n_{s}\ln{\left(1+\frac{\widetilde{Q^{2}}}{\widetilde{Q_{s}^{2}}}\right)}}$
$+\widetilde{A_{h}}e^{-i\frac{\pi}{2}\alpha_{h}(t)}\left(\frac{s}{s_{0}}\right)^{\alpha_{h}(t)}e^{b_{h}t-(n_{h}+1)\ln{\left(1+\frac{\widetilde{Q^{2}}}{\widetilde{Q_{h}^{2}}}\right)}+\ln{\left(\frac{\widetilde{Q^{2}}}{\widetilde{Q_{h}^{2}}}\right)}},$
(61)

where the two exponential factors
$e^{b_{s}t-n_{s}\ln{\left(1+\frac{\widetilde{Q^{2}}}{\widetilde{Q_{s}^{2}}}\right)}}$
and
$e^{b_{h}t-(n_{h}+1)\ln{\left(1+\frac{\widetilde{Q^{2}}}{\widetilde{Q_{h}^{2}}}\right)}+\ln{\left(\frac{\widetilde{Q^{2}}}{\widetilde{Q_{h}^{2}}}\right)}}$
can be interpreted as the product of the form factors of upper and lower
vertices (see Figure 7c). Interestingly, the amplitude (61) resembles the
scattering amplitude of Ref. Capua .

#### Fitting the Two-Component Pomeron to VMP and DVCS HERA Data

The fitting strategy is based on the minimization of the quantity
$\tilde{\chi}^{2}=\frac{1}{N}\sum_{i=1}^{N}{\tilde{\chi}_{i}^{2}}$, where $N$
is the number of all reactions involved (i.e., $\rho$, $\phi$, $\omega$,
$J/\psi$, $\varUpsilon$ and $\gamma$ production); $\tilde{\chi}_{i}$ is the
mean value of $\chi^{2}$ for different types of data for selected class of
reactions, defined as
$\tilde{\chi}_{i}=\frac{1}{N_{i}}\sum_{k=1}^{N_{i}}{\tilde{\chi}_{k,i}^{2}}$,
where $\tilde{\chi}_{k,i}^{2}$ is $\chi_{k,i}^{2}/d.o.f.$ for i-th class of
reactions and k-th type of data, i.e., those relative to $\sigma_{el}(Q^{2})$,
$\sigma_{el}(W)$ and ${d\sigma_{el}(t)}/{dt}$; $N_{i}$ is number of different
type of data for i-th class of reactions.

The normalization parameters were fixed at

$f_{\rho}=0.680,\ f_{\phi}=0.155,\ f_{\omega}=0.068,\ f_{J/\psi}=1,\
f_{\varUpsilon}=0.750$ (62)

and $s_{0}$ was set $1$ GeV2.

DVCS and VMP are similar in the sense that in both reactions a vector particle
is produced. However there are differences between the two because of the
vanishing rest mass of the produced real photon. The unified description of
these two types of related reactions does not work by simply setting
$M_{\gamma}=0.$ From fits we found $M_{DVCS}^{eff}=1.8$ GeV and a
normalization factor $f_{DVCS}=0.091$ follows.

Some results of the fits are shown in Figures 9–12 (for $\sigma_{el}(W)$) and
Figure 13 (for $d\sigma_{el}(t)/dt$) for vector meson production, with the
values of the fitted parameters given in Table LABEL:tab:fit1(s+h). The mean
value of the total $\tilde{\chi}^{2}$ (see above its definition) is equal to
0.986. The mean values of $\tilde{\chi}^{2}$ of the fit for different
observables (i.e., $\sigma_{el}(Q^{2})$, $\sigma_{el}(W)$ or
$d\sigma_{el}(t)/{dt})$ and different reactions (VMP or DVCS), together with
the numbers of degrees of freedom (number of data points) and the global mean
value $\tilde{\chi}_{i}^{2}$, are shown in Table LABEL:tab:fit1_chi.

In Table 2 the parameters of the two-component pomeron model (Equations (59)
and (60)) fitted to the combined VMP and DVCS data are quoted, when the
pomeron trajectories are fixed to $\alpha_{s}(t)=1.08+0.25t$ and
$\alpha_{h}(t)=1.20+0.01t$.

Next, by using Equation (59) with the values of the parameters from Table
LABEL:tab:fit1(s+h) and the formula

$B(Q^{2},W,t)=\frac{d}{dt}\ln{\frac{d\sigma_{el}}{dt}},$ (63)

one calculates the forward slopes and compares them with the experimental data
on VMP, including those for the $\Psi$(2S) production. A compilation of all
results is presented in Figure 14.

The number of fitted parameters of the two-component pomeron model (Equation
(62)) is 12 (Table LABEL:tab:fit1(s+h)), with five additional normalization
factorsfor six vector particle productions ($\rho^{0}$, $\phi$, $\omega$,
$J/\psi$, $\varUpsilon$ and $\gamma$). By fixing the pomeron trajectories
$\alpha_{s}(t)$ and $\alpha_{h}(t)$ (see Table 2), the number of free
parameters reduces to $8$.

Figure 9:  Fit of Equation (60) to data on elastic cross section $\sigma_{el}(W)$ for $\rho^{0}$, for different values of $Q^{2}$. Figure 10:  Fit of Equation (60) to the data on elastic cross section $\sigma_{el}(W)$ for $\phi$, for different values of $Q^{2}$. Figure 11:  Fit of Equation (60) to the data on elastic cross section $\sigma_{el}(W)$ for $J/\psi$, for different values of $Q^{2}$. Figure 12:  Fit of Equation (60) to the data on elastic cross section $\sigma_{el}(W)$ for $\varUpsilon$, for different values of $Q^{2}$. Figure 13:  Fit of Equation (59) to the data on the differential elastic cross section $d\sigma_{el}/dt$ for $\rho^{0}$, for different values of $Q^{2}$ and $W$. Figure 14:  Experimental data on the slope $B$ as function of $\widetilde{Q^{2}}$ for $\rho^{0},\phi,J/\psi$, $\varUpsilon$ and $\Psi$(2S), and our theoretical predictions from Equation (63). Table 2:  Parameters of the two-component pomeron model (Equations (59) and (60)) fitted to the combined VMP and DVCS data, with fixed parameters of the pomeron trajectories $\alpha_{s}(t)=1.08+0.25t$ and $\alpha_{h}(t)=1.20+0.01t$. |  $A_{0s,h}$ $\left[\frac{\sqrt{\text{{nb}}}}{\text{{GeV}}}\right]$ |  $\widetilde{Q^{2}_{s,h}}$ $\left[\text{{GeV}}^{\textbf{2}}\right]$ | $n_{s,h}$ | $\alpha_{0\,s,h}$ |  $\alpha^{\prime}_{s,h}$ $\left[\frac{\textbf{1}}{\text{{GeV}}^{\textbf{2}}}\right]$ |  $b_{s,h}$ $\left[\frac{\textbf{1}}{\text{{GeV}}^{\textbf{2}}}\right]$
---|---|---|---|---|---|---
soft | 807 $\pm$ 1107 | 0.46 $\pm$ 0.70 | 1.79 $\pm$ 0.79 | 1.08 | 0.25 | 3.41 $\pm$ 2.48
hard | 47.9 $\pm$ 46.9 | 1.30 $\pm$ 1.12 | 1.33 $\pm$ 0.26 | 1.20 | 0.01 | 2.15 $\pm$ 1.14

More results and discussion on the fitting details can be found in Refs.
Salii1 ; Salii2 .

##  VI Balancing between “Soft” and “Hard” Dynamics

In this section we illustrate the important and delicate interplay between the
“soft” and “hard” components of our unique pomeron. Since the amplitude
consists of two parts, according to the definition (50), it can be written as

$A(Q^{2},s,t)=A_{s}(Q^{2},s,t)+A_{h}(Q^{2},s,t).$ (64)

As a consequence, the differential and elastic cross sections contain also an
interference term between “soft” and “hard” parts, so that they read

$\frac{d\sigma_{el}}{dt}=\frac{d\sigma_{s,el}}{dt}+\frac{d\sigma_{h,el}}{dt}+\frac{d\sigma_{interf,el}}{dt}$
(65)

and

$\sigma_{el}=\sigma_{s,el}+\sigma_{h,el}+\sigma_{interf,el},$ (66)

according to Equations (59) and (60), respectively.

Given Equations (65) and (66), we can define the following ratios for each
component:

$R_{i}(\widetilde{Q^{2}},W,t)=\frac{\frac{d\sigma_{i,el}}{dt}}{\frac{d\sigma_{el}}{dt}}$
(67)

and

$R_{i}(\widetilde{Q^{2}},W)=\frac{\sigma_{i,el}}{\sigma_{el}},$ (68)

where $i$ stands for $\\{s,h,interf\\}$.

Figure 15 shows the interplay between the components for both $\sigma_{i,el}$
and $R_{i}(\widetilde{Q^{2}},t)$, as functions of $\widetilde{Q^{2}}$, for $W$
= 70 GeV. In Figure 16 both plots show that not only $\widetilde{Q^{2}}$ is
the parameter defining softness or hardness of the process, but such is also
the combination of $\widetilde{Q^{2}}$ and $t$, similar to the variable
$z=t-Q^{2}$ introduced in Ref. Capua . On the whole, it can be seen from the
plots that the soft component dominates in the region of low
$\widetilde{Q^{2}}$ and low $|t|$, while the hard component dominates high
$\widetilde{Q^{2}}$ and high $|t|$.

In other words, the variables $t$ and $Q^{2}$ have much in common: both refer
to squared momentum transfer and have the meaning of “hardness”. The present
model provides a valuable laboratory to study this important issue.

Hadron-induced reactions, discussed in Section II differ from those induced by
photons at least in two aspects. First, hadrons are on the mass shell and
hence the relevant processes are typically “soft”. Secondly, the mass of
incoming hadrons is positive, while the virtual photon has negative squared
“mass”. Our attempt to include hadron-hadron scattering into the analysis with
our model has the following motivations: (a) by vector meson dominance (VMD)
the photon behaves partly as a meson, therefore meson–baryon (and more
generally, hadron-hadron) scattering has much in common with photon-induced
reactions. Deviations from VMD may be accounted for the proper $Q^{2}$
dependence of the amplitude (as we do hope is in our case!); (b) connection
between space- and time-like reactions; (c) according to some claims the
highest-energy (LHC) proton–proton scattering data indicate the need for a
“hard” component in the pomeron (to anticipate, our fits do not found support
the need of any noticeable “hard” component in $pp$ scattering).

Figure 15:  Interplay between soft (green line), hard (blue line) and
interference (yellow line) components of the cross section $\sigma_{i,el}$
(left plot) and $R_{i}(\widetilde{Q^{2}},t)$ (right plot) as functions of
$\widetilde{Q^{2}}$, for $W=70$ GeV.

We do not intend to perform here a high-quality fit to the $pp$ data; that
would be impossible without the inclusion of subleading contributions and/or
the odderon.

The $pp$ scattering amplitude is written in the form similar to the amplitude
(58) for VMP or DVCS, the only difference being that the normalization factor
is constant since the $pp$ scattering amplitude does not depend on $Q^{2}$:

$A^{pp}(s,t)=A^{pp}_{s}\,e^{-i\frac{\pi}{2}\alpha_{s}(t)}\left(\frac{s}{s_{0}}\right)^{\alpha_{s}(t)}e^{b_{s}t}+A^{pp}_{h}\,e^{-i\frac{\pi}{2}\alpha_{h}(t)}\left(\frac{s}{s_{0}}\right)^{\alpha_{h}(t)}e^{b_{h}t}.$
(69)

We fix the parameters of pomeron trajectory at

$\alpha_{s}(t)=1.084+0.35t,\qquad\alpha_{h}(t)=1.30+0.10t.$

With these trajectories the total cross section

$\sigma_{tot}=\frac{4\pi}{s}Im\;A(s,{t=0})$ (70)

was found compatible with the LHC data. From the comparison of Equation (70)
to the LHC data we get

$A^{pp}_{s}=-1.73\text{\,mb}\cdot\text{GeV}^{2},\,\,\quad\quad
A^{pp}_{h}=-0.0012\text{\,mb}\cdot\text{GeV}^{2}.$

We conclude that, while the data on total cross section are compatible with a
small “hard” admixture in the amplitude, the slope parameter with a hard
component included seems to manifest a wrong tendency, by slowing down with
increasing energy, while the TOTEM measurements show the opposite.

Figure 16:  Left column: soft (upper surface), hard (middle surface) and
interference (bottom surface) components of the ratio
$R_{i}(\widetilde{Q^{2}},W,t)$ are shown as functions of $\widetilde{Q^{2}}$
and $t$, for $W=70$ GeV. Right column: some representative curves of the
surfaces projected on the ($t,\widetilde{Q^{2}}$ ) plane.

The resulting fits are reasonable, despite the following open problems:

  * •

sub-leading Regge contributions must be included in any extension of the model
to lower energies (below $30$ GeV);

  * •

the $\widetilde{Q^{2}}$ dependence of the scattering amplitude, introduced
empirically has to be compared with the results of unitarization and/or QCD
evolution.

  * •

as seen from Section VI, the “soft” component of the pomeron dominates in the
region of small $t$ and small $\widetilde{Q^{2}}$. Hence, any parameter
responsible for the “softness” and/or “hardness” of processes, should be a
combination of $t$ and $Q^{2}$. A simple solution was suggested in Ref. Capua
with the introduction of the variable $z=t-Q^{2}$. The interplay of these two
variables remains an important open problem that requires further
investigation.

The extension of our formalism to hadronic reactions ($pp$ scattering) shows
that the available data can be will described by a single—soft—component.

##  VII Spin of the Proton in Terms of Its Constituents

The origin of spin in the nucleon is a rich subject deserving a dedicated
presentation. Here I only briefly sketch some recent developments in the
field. A complete and comprehensive treatment of the subject can be found
e.g., in Refs. Leader0 ; Rev2 .

In the simplest case, if only valence quarks contributed, the nucleon spin
would be simple a sum

$J=\frac{\Delta\Sigma}{2}+L_{q}=\frac{1}{2},$ (71)

where $\Delta\Sigma$ is the quark spin contribution and $L_{q}$ is the orbital
angular momentum (OAM) contribution. Partition between the quark spin and the
OAM for a long time were subject of debates generally called “spin crisis”.
Estimates of shares varied in a wide range. Among the reasons of disagreement
and contradictions was the role of gluons, and sea as compared to that of the
valence quarks. The phenomenon known as “spin crisis” arose from an EMC
experiment at CERN EMC  quoting $\Sigma\approx 0$ (with large uncertainty),
thus allegedly contradicting the naive quark model. Recent COMPAS, HERMES and
Jlab measurements are consistent with a value of $\Sigma\approx 0.3.$
Actually, $J$ must obey conservation of the total angular momentum known as
Jaffe-Manohar sum rule Jaffe_Manohar ,

$J=\Delta\Sigma(Q^{2})+L_{q}(Q^{2})+\Delta(Q^{2})+L_{q}(Q^{2})=\frac{1}{2}.$
(72)

An important task in the context of the above some rule is the separation of
sea and valence quarks, indistinguishable in DIS observables. In addition, as
noted in Ref. BMa , strange and anti-strange sea quarks can contribute
differently in to the nucleon spin.

Paper Rev2  discusses also how DIS data on proton, neutron and deuteron
targets can be used to separate the contributions from different quark
polarizations assuming validity of $SU(3)_{f}$.

That paper contains many more ideas, among which is the use of parton–hadron
duality of Bloom and Gilman B-G —another effective tool to be used in studies
of spin effects and a possible key to understand better problems related to
quark confinement.

Before the advent of QCD, a nucleon (N) was visualized as a bound state of 3
massive quarks (Q) ( $M_{Q}\approx M_{N}/3$) lying in some kind of potential.
In the simplest non-relativistic case, for an s-state the constituent quarks
have no orbital angular momentum (OAM) and one has, for the nucleon at rest,
say polarized in the positive Z-direction

$1/2=S_{z}^{N}=\sum_{Q}S_{z}^{Q}.$ (73)

With relativistic corrections, the bottom components of the quark Dirac
spinors contain OAM and Equation (73) is modified to

$\sum_{Q}S_{z}^{Q}\approx 0.3$ (74)

In the naive parton model, for a fast moving proton with helicity $+1/2$,

$a_{0}=\Delta\Sigma\equiv(\Delta u+\Delta\overline{u})+(\Delta
d+\Delta\overline{d})+(\Delta s+\Delta\overline{s})$ (75)

where the $\Delta q$, $\Delta\bar{q}$ are the first moments of the polarized
quark–parton helicity densities. Hence, bearing in mind that $\Delta
q=q_{+}-q_{-}$, where $\pm$ corresponds to the number densities with spin
along or opposite to the proton’s momentum, one obtains in the naive parton
model

$a_{0}=\Delta\Sigma=2\left[\sum_{q}\langle
S_{z}^{q}\rangle+\sum_{\bar{q}}\langle S_{z}^{\bar{q}}\rangle\right]$ (76)

and if there is no other source of angular momentum one expects

$\left[\sum_{q}\langle S_{z}^{q}\rangle+\sum_{\bar{q}}\langle
S_{z}^{\bar{q}}\rangle\right]=S_{z}^{\textrm{proton}}=1/2$ (77)

implying, naively,

$a_{0}=1.$

After the famous EMC experiments revealed that only a small fraction of the
nucleon spin is due to quark spins, there has been great interest in ‘solving
the spin puzzle’, i.e., in decomposing the nucleon spin into contributions
from quark/gluon spin and orbital degrees of freedom.

The EMC experiment gave $a_{0}\approx 0$ and later experiments confirmed that
$a_{0}\ll 1$, giving rise to the spin crisis in the (_naive_) parton model.
However, Equation (VII) cannot possibly be true because the right hand side is
a fixed number, whereas the left hand side is, beyond the naive level, equal
to $a_{0}(Q^{2})$, i.e., a function of $Q^{2}$! Thus failure of Equation (VII)
to hold cannot be used to infer that there is crisis. It is obvious that a
correct relation between the spin of a nucleon and the angular momentum of its
constituents should include their orbital angular momentum and should also
include a contribution from the gluons.

### Ji’s and J-M’s Decompositions

Finally, we briefly mention two sum rules visualising in different albeit
complementary ways the spin decomposition in the nucleon.

The Ji decomposition Ji:1996ek

$\frac{1}{2}=\frac{1}{2}\sum_{q}\Delta q+\sum_{q}{L}_{q}^{z}+J_{g}^{z}$ (78)

appears to be very useful, as not only the quark spin contributions $\Delta q$
but also the quark total angular momenta $J_{q}\equiv\frac{1}{2}\Delta
q+{L}_{q}^{z}$ (and by subtracting the spin piece also the the quark orbital
angular momenta $L_{q}^{z}$) entering this decomposition can be accessed
experimentally, through generalized parton distributions (GPDs). The terms in
(78) are defined as expectation values of the corresponding terms in the
angular momentum tensor

$M^{0xy}=\sum_{q}\frac{1}{2}q^{\dagger}\Sigma^{z}q+\sum_{q}q^{\dagger}\left({\vec{r}}\times
i{\vec{D}}\right)^{z}q+\left[{\vec{r}}\times\left({\vec{E}}\times{\vec{B}}\right)\right]^{z}$
(79)

in a nucleon state with zero momentum. Here
$i{\vec{D}}=i{\vec{\partial}}-g{\vec{A}}$ is the gauge-covariant derivative.

Jaffe and Manohar proposed Jaffe_Manohar  an alternative decomposition of the
nucleon spin, which does have a partonic interpretation

$\frac{1}{2}=\frac{1}{2}\sum_{q}\Delta q+\sum_{q}{\cal
L}_{q}^{z}+\frac{1}{2}\Delta G+{\cal L}_{g}^{z},$ (80)

whose terms are defined as matrix elements of the corresponding terms in the
$+12$ component of the angular momentum tensor

$M^{+12}=\frac{1}{2}\sum_{q}q^{\dagger}_{+}\gamma_{5}q_{+}+\sum_{q}q^{\dagger}_{+}\left({\vec{r}}\times
i{\vec{\partial}}\right)^{z}q_{+}+\varepsilon^{+-ij}\mbox{Tr}F^{+i}A^{j}+2\mbox{Tr}F^{+j}\left({\vec{r}}\times
i{\vec{\partial}}\right)^{z}A^{j}.$ (81)

In Figure 17 the share of various components to the nucleon spin, according to
the two (Ji and Jaffe-Manohar) distribtutions is shown.

Figure 17: Origin of the proton spin according to sum rules of Jaffe, and
Monahor and Ji.

What appeared to be spin crisis in the parton model, over three decades years
ago, was a consequence of a misinterpretation of the results of the famous
European Muon Collaboration experiment on polarized deep inelastic scattering.

##  VIII Conclusions

Interest in spin physics and polarization has moved from hadron-hadron to
lepton-hadron processes. This is mainly due to the fact that in proton–proton
and proton–antiproton scattering spectacular spin effects were/are not
expected at highest energies (ISR, Fermilab, BNL or LHC), except maybe for the
dip-bump structure at the diffraction cone. Various aspects of polarization in
inelastic hadronic reactions reactions were treated in simple intuitive models
in Refs. Strum .

Instead, deeply virtual Compton scattering (DVCS), with spin degrees of
freedom is now at the focus of interest as a source of information on
generalized parton distributions (GPDs).

In both cases (hadron-hadron and lepton-hadron) diffraction and the pomeron
play a crucial role. Therefore much space was dedicated to the theoretical
problems of diffractive scattering, properties of the vacuum Regge trajectory
(pomeron), and confinement of quarks and gluons.

In the near future, research programs at the Electron-Ion Colliders (EICs)
will play an important role in understanding gluon GPDs. Apart from providing
constraints on the total quark/gluon contributions to the proton spin, the
GPD’s will provide important information on the nucleon tomography, for
example, the 3D imaging of partons inside the proton. Together with the
gravitational form factors extracted from the DVCS, this will deepen our
understanding of the nucleon spin structure in return. The EIC may shed light
on the quark/gluon orbital angular momentum (OAM) directly through various
hard diffractive processes. Pioneer experimental effort to constrain the
gravitation form factor from DVCS experiment at JLab has been carried out in
Ref. Grav . The quark and gluon helicity contributions to the proton spin with
the unique coverage in both $x$ and $Q^{2}$ at EIC will provide the most
stringent constraints on $\Delta\Sigma$ and $\Delta G$ EIC1 ; EIC2 .

Last but not least, Regge trajectories Section II.2, relating uniquely the
spin and mass of particles, are building blocks of the theory, containing the
basic information on the dynamics. By crossing symmetry, unitarity and duality
they relate various aspects of high-energy strong interaction dynamics.

This research received no external funding

###### Acknowledgements.

During many years of my professional activity I enjoyed and profited from the
collaboration with Salvatore Fazio, Roberto Fiore, Francesco Paccanoni,
Alessandro Papa, Enrico Predazzi, Rainer Schicker, István Szanyi and Andrii
Salii as well from discussions with Victor Fadin, Sergey Troshin and Oleg
Selyugin, whom I thank very much. I acknowledge the intelligent and useful
remarks by the Referee that helped me to improve this presentation. The work
was supported by the National Academy of Sciences of Ukraine, grant N
012r100935, “Fundamental properties of matter in relativistic collisions of
nuclei and in the early Universe”. The author declares no conflict of
interest.  References

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