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\begin{document}
\selectlanguage{english}
	
\title{Covariant reggeization framework and cross-sections of diffractive processes}

\author{\firstname{R.~A.}~\surname{Ryutin}}
\affiliation{%
	NRC \flqq Kurchatov Institute\frqq -- IHEP, Protvino, Moscow region, 142281, Russia
}%

%\date{\today}
%\today

\begin{abstract}
	A version of the Regge approach is considered, in which the Reggeon is initially treated as a quantum field with arbitrary spin in Minkowski space, followed by an analytical continuation of the diffractive amplitudes into the complex angular momentum domain. Results are presented for irreducible tensors and their contractions, from which cross sections of all basic diffraction processes are calculated. The three-Reggeon approximation and the limit at small momentum transfer is also considered.
\end{abstract}

\maketitle

%\keywords{Covariant Regge formalism, irreducible representations, Rarita-Schwinger conditions, Poincar\'e group, hadronic diffraction}

%\pacs{
%	{11.55.Jy}{ Regge formalism}   \and
%	{12.40.Nn}{ Regge theory, duality, absorptive/optical models} \and
%	{13.85.Ni}{ Inclusive production with identified hadrons}\and
%	{13.85.Lg}{ Total cross sections}
%} % end of PACS codes

%%\pacs[JEL Classification]{D8, H51}

%\pacs[MSC Classification]{81V05,81-01,81-08,81Q99,81T99,81U99,81V25,81V99}

%\maketitle	
	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	

%----------------Introduction---------------------------------------------

\section*{Introduction}

% english

In this paper, we consider Reggeons as objects
 arising from irreducible tensors of the 
Poincaré group in Minkowski space 
$\mathbb{M}_D=\mathbb{R}^{1,D-1}$ 
for arbitrary dimension $D$ and spin $J$ after contracting 
them over all indices and then analytically continuing 
the resulting cross-sections to the domain of complex 
angular momentum. This sentence contains the 
entire description of the approach.

This work was started quite a long time ago~\cite{spinparity}-\cite{SD} (see 
also old papers~\cite{CovariantReggeIn}-\cite{CovariantReggeOut}) and was 
motivated by several facts:
\begin{itemize}
\item
First, study of the spin-parity analysis in the exclusive 
central diffraction~\cite{spinparity},\cite{KMRspinparity}, where 
the covariant approach yields strictly defined cross-section 
behavior for fixed spins. However, since 
calculations for arbitrary spins, as they 
should be in principle, are quite cumbersome, many 
authors (see~\cite{Nachtmann} for example) mix 
integer spins and trajectories in their approach, this is 
incorrect. For high energies, this is
 less critical, and may give similar results, but if the 
model is used for low (like in U-70, for example) and 
intermediate energies, significant discrepancies are possible.

\item
Secondly, the direct application of this approach
 leads to zeros in the cross sections in the limit 
of small  momentum transfer~\cite{SD}, which
 was demonstrated quite a long time ago for 
conserved currents~\cite{Close}. This is
 particularly visible in single dissociation
and central production. However, experimental 
data do not show such a behaviour. Of course, reggeon 
is a rather complex object, and the requirement
 for current conservation is likely unnecessarily
 strong. However, the idea arose to try to 
eliminate this behavior near zero momentum 
transfer squared in various ways, so that the model 
could be used for data analysis. It was shown 
 that unitarization can smooth out the zeros 
effect\cite{SD}. Another solution is to assume that the 
current is conserved in more 
dimensions; then, in four dimensions, it will 
be effectively non-conserved.
 
\item
Third, we would like to have a clearer 
understanding of the reggeon from a 
quantum field theory perspective, and 
what the obtained cross sections for 
its scattering by hadrons mean. We can 
conditionally consider it analogous to an 
atom with its energy levels and scattering, or
 a string, or simply a mathematical 
object corresponding to a mixture of 
various processes. Or we can consider 
it, among other things, as a reggeized 
quantum field of arbitrary spin. In the last 
case it leads to rigorous, rather beautiful 
mathematical consequences, which can 
be tested experimentally to determine 
the extent to which they are fulfilled or violated.

\item
The next question to be solved is whether 
we can extract reggeon–hadron and 
reggeon–reggeon cross sections 
from data, which was made first by 
Kaidalov~\cite{Kaidalov}, and what is 
the physical meaning of these cross 
sections, which can be of the order 
of hadronic cross sections, as was
 shown in the previous paper~\cite{SD}.

\end{itemize}

\section{Basical scheme and tensors for diffractive processes}

We start with the field
$\Phi^{\mu_1...\mu_J}(x)$
and its current tensor 
${\mathcal I}^{\mu_1\dots\mu_J}$
 corresponding to the spin
$J$, which are an irreducible 
tensors of the Poincaré group in 
Minkowski space 
$\mathbb{M}_D=\mathbb{R}^{1,D-1}$ of
 arbitrary dimension $D$, satisfying, together 
with its current, the standard Rarita-Schwinger
 conditions: transverseness, symmetry,
tracelessness (TST).
 
Then we obtain the scalar–scalar–tensor vertex
${\mathcal V}$ and the propagator $ {\mathcal P}$
\begin{eqnarray}
 \label{eq:TvertexDef}
{\mathcal V}_{\mu_1\dots\mu_J}(p,q)&=&<p-q|{\mathcal I}_{(\mu_1\dots\mu_J)}|p>,
\\
 {\mathcal P}_{(\mu_1...\mu_J),(\nu_1...\nu_J)}(q)&=&
\int d^4x\; e^{iq(x-y)}
\left< 0|\;  
\Phi_{\mu_1...\mu_J}(x) 
\Phi_{\nu_1...\nu_J}(y)\; 
|0 \right>=\nonumber\\
&=& \Pi_{(\mu_1...\mu_J), (\nu_1...\nu_J)}(q)/(m^2(J)-q^2),
\label{eq:WtensorDef}
\end{eqnarray}
which have the poles at
$
m^2(J)-q^2=0,\; \mbox{i.e.}\; 
J=\alpha_{{\mathbb R}}(q^2)\equiv 
\alpha_{{\mathbb R}} \;,
\label{eq:polesJ}
$
after an appropriate analytic continuation 
of the signatured amplitudes
in $J$, where $\alpha_{\mathbb R}$ is the 
reggeon trajectory (reggeization prescription):
\begin{equation}
\label{reggeizationpr}
\sum_JF^J/(q^2-m^2)\to
\alpha_{{\mathbb R}}^{\prime}
\eta_{{\mathbb R}}(q^2)
\Gamma(-\alpha_{{\mathbb R}})
F^{\alpha_{{\mathbb R}}}
/2.
\end{equation}

TST conditions for them are
\begin{equation}
 q_{\lambda} {\mathcal T}_{\mu_1...\lambda ...\mu_J}=0,
\quad
 {\mathcal T}_{...\mu_i...\mu_j ...}=
{\mathcal T}_{...\mu_j...\mu_i ...},\quad
g_{\mu_i\mu_j} {\mathcal T}_{\mu_1...\mu_i...\mu_j...\mu_J}=0.
\label{eq:Vcond}
\end{equation}
for any irreducible 
${\mathcal T}_{...}={\mathcal T}_{...}(\{q_i\},\{p_i\})$,
\begin{equation}
\begin{array}{cccccc}
\mathcal{T}\;= & \mathcal{V} & \mathcal{F} & \mathcal{W}(\mathrm{or}\;\mathcal{P})
& \mathcal{Y} & \mathcal{H}
 \\
\Omega_T= & \{i\} & \{i,i^{\star}\} &\{i,i'\} 
& \{1,2,3\} & \{1,1',2,2'\}
 \\
\end{array}
\end{equation}
$$
1^{\star}=2, 2^{\star}=1; \qquad
(i^{\star})'=(i')^{\star};\qquad
i\in\{1,1',2,2'\},
$$
\begin{equation}
(r) \equiv \alpha^{(r)}_s,\; s\in [1,J_r],
\qquad
r\in\{1,1',2,2',3\} \leftrightarrow 
\alpha^{(r)} \in\{\mu,\mu',\nu,\nu',\rho\}
\label{def:Tindex}
\end{equation}

Then we can construct
 vertexes with multiple tensor legs 
with different spins and also different symmetric 
groups of Lorentz indexes at each leg. And we
 have the same conditions, but for each 
symmetric group of indexes and 
corresponding momentum 
transfers (see Fig.~\ref{fig}).

\begin{figure}[htb]
	\centering
	%  \fbox{
	\includegraphics[width=0.6\linewidth, %bb=140 120 390 290
	]{fig.pdf}
	%  }
	\caption{Amplitudes of diffractive 
		processes represented as contractions 
		of basic Lorentz tensors. Abbreviations: EL (elastic), SD (single 
		dissociation), DD (double dissociation), CEDP (central exclusive 
		diffractive production), CIDP (central inclusive 
		diffractive production).\label{fig}}
\end{figure}

In our approach we consider only totally 
symmetric tensors (in each symmetric 
group of Lorentz indices) without mixed 
symmetry for $D>4$. This is an 
effective, fairly simple model that 
requires further refinement and 
generalization, at least to curved 
spaces of the anti-de Sitter type, but 
the fundamental influence of additional 
dimensions can be traced even in this way.

Let us consider the "harmonical" basis, where 
we have basic transverse symmetric tensors 
$S_{\vec{k}',\vec{n}}^{T\; \vec{J}}$
which are made of all possible 
combinations (marked by vector indexes) of 
simple transverse tensors
\begin{eqnarray}
\mathcal{G}_{\alpha\beta} (v_1,v_2)&=&
 g_{\alpha\beta} + 
\frac{
v_2^2 v_{1\alpha} v_{1\beta} +
v_1^2 v_{2\alpha} v_{2\beta} -
(v_1v_2) (
v_{1\alpha} v_{2\beta} + v_{2\alpha} v_{1\beta}
)	
}{
(v_1v_2)^2-v_1^2 v_2^2
}
\label{def:GHarmrs}\\
P_{\alpha}(v_1,v_2)&=&
(v_{1\alpha}-\frac{v_1v_2}{v_2^2}v_{2\alpha})/
\sqrt{|v_1^2-\frac{(v_1v_2)^2}{v_2^2}|}
\end{eqnarray}
for different $v_2=q,q_i$ (momentum transfer) 
and $v_1=p,p_i$ (incoming/outcoming momenta 
of a vertex) (see~\cite{covreggeI} for 
details). Powers of tensors
$\mathcal{G}$, $P$ denote symmetrized 
products (sum of unique terms without any multiplier) 
of these tensors in each group 
of Lorentz indices:
\begin{equation}
S_{\vec{k}\,\vec{n}}^{T\; \vec{J}} \equiv 
\left(\!
\prod_{i\in\Omega_T}\!\!
P_{i(i)}^{\otimes J^*_i}
\mathcal{G}_{(ii)}^{\otimes n_i}
\!\!\!\prod_{\forall r\neq s \atop r,s\in\Omega_T}
 \!\!\!\mathcal{G}_{(rs)}^{\otimes k_{rs}}
\!\right),\quad
J^*_i=J_i-2n_i-\kappa_i,\quad
 \kappa_i=\sum_{r\neq i} k_{ir}.
\label{def:ST}
\end{equation}
Each irreducible tensor is decomposed over 
a basis of irreducible representations, the 
choice of which is quite arbitrary, and contains
 a certain number of independent components 
corresponding to the degrees of freedom for
 the chosen spin values. In the 
work~\cite{covreggeI} two
different basises were considered. In every 
basis tracelessness is ensured by the tensor
$
G_{\alpha\beta}=g_{\alpha\beta}-q_{i\alpha}q_{i\beta}/q_i^2
$,
which is essentially a metric orthogonal 
only to the momentum transfer and "lives" 
in Minkowski space $\mathbb{M}_{D-1}$.

The formulas and equations for the coefficients
 in the "harmonical" basis are much simpler, and
 the basic tensors are finite in the limit of small
momenta transfers, which is very convenient for 
calculations. Any irreducible tensor can be 
represented as follows
\setlength{\fboxrule}{1pt}
\begin{empheq}[box=\fcolorbox{red}{white}]{align}
\mathcal{T}^{\vec{J}}=
\sum_{\bar{\Omega}^T_{\vec{k}}} 
\hat{\tau}^{\vec{J}}_{\vec{k}} 
\mathcal{T}^{*\,\vec{J}}_{\vec{k}},\qquad
\mathcal{T}^{*\,\vec{J}}_{\vec{k}} =
\sum_{\tilde{\Omega}^T_{\vec{k}'\,\vec{n}}}
\tau^{\vec{k}'(\vec{k};\vec{J})}_{\vec{n}} 
S^{T\,\vec{J}}_{\vec{k}'\,\vec{n}},
\label{def:anyirrep}
\end{empheq}
$\hat{\tau}^{\vec{J}}_{\vec{k}}$ are form-factors,
 and regions of summation are
\begin{equation}
\tilde{\Omega}^T_{\vec{k}'\,\vec{n}}\!\!:\!
J^*_r\ge 0,\; \kappa_r\ge\kappa'_r,\; 
n_r\ge 0,\; k'_{rs}\ge 0,\quad
 r,s\in\Omega_T,
\; r\neq s; 
\qquad \bar{\Omega}^T_{\vec{k}} = 
\tilde{\Omega}^T_{\vec{k}\,\vec{0}}.
\label{def:regionsirrep}
\end{equation}
The basis of generating functions is
%\setlength{\fboxrule}{1pt}
\begin{empheq}[box=\fcolorbox{red}{white}]{align}
{\varPhi^{\star}}^{\mathcal{T}}_{\vec{k}}
(\vec{x},\vec{u},\dvec{u}) 
&= 
\sum_{\tilde{\Omega}^T_{\vec{k}',\vec{n}}}  
\tau^{\vec{k}'(\vec{k};\vec{J})}_{\vec{n}} 
\mathcal{N}^{\vec{J}}_{\vec{k}'\vec{n}} \,
\vec{x}^{\,\otimes\vec{J}^*}\!\!
\vec{u}^{\,\otimes\vec{n}}\,
\dvec{u}^{\,\otimes\vec{k}'}=
\frac{\mathcal{N}^{\vec{J}}_{\vec{k}\vec{0}}}{\Pi_0^{\vec{J}\,\vec{k}}} \,
\prod_{s\in\Omega_T} 
\hat{\Pi}(\square_{\omega_s})\;
\vec{x}^{\,\otimes\vec{J}-\vec{k}}
\dvec{u}^{\,\otimes\vec{k}}
\label{eq:irrepHarmProjT}
\end{empheq}
\begin{eqnarray}
\Pi_0^{\vec{J}\,\vec{k}}&=&
\prod_{i\in\Omega_T}
{}_2F_1 \left(-\frac{J_i-\kappa_i}{2},\frac{1-J_i+\kappa_i}{2};
c^{J_i}; 1\right)
,
\\
 \square_{\omega_s} &=&
g_{\alpha\beta}\; 
\partial_{\omega_{s\,\alpha}}\! 
\partial_{\omega_{s\,\beta}},
 \quad
\partial_{\omega_{s\,\alpha}} = 
\frac{\partial x_{s}}{\partial_{\omega_{s\,\alpha}}}
\partial_{x_{s}}
+
\frac{\partial u_{s}}{\partial_{\omega_{s\,\alpha}}} 
\partial_{u_{s}}
+
\sum_{r\neq s}
\frac{\partial u_{rs}}{\partial_{\omega_{s\,\alpha}}} 
\partial_{u_{rs}}
\label{eq:irrepHarmProj}
\end{eqnarray}
%\setlength{\fboxrule}{1pt}
\begin{empheq}[box=\fcolorbox{red}{white}]{align}
 \hat{\Pi}(\square_{\omega_s}) 
\vec{x}^{\,\otimes\vec{J}-\vec{k}}
\dvec{u}^{\,\otimes\vec{k}}
=
\sum_{m_s=0}^{[J_s/2]} 
\frac{(u_s+x_s^2)^{m_s}}{4^{m_s} m_s! (c^{J_s})_{m_s}} 
\left(\square_{\omega_s}\right)^{m_s} 
\vec{x}^{\,\otimes\,\vec{J}-\vec{k}}
\dvec{u}^{\,\otimes\,\vec{k}} 
|_{\omega_s\to 0}
\label{eq:irrepHarmProjTsingle}
\end{empheq}
%\setlength{\fboxrule}{1pt}
\begin{empheq}[box=\fcolorbox{red}{white}]{align}
 \mathcal{T}^{\vec{J}} =
\sum_{\bar{\Omega}^T_{\vec{k}}} 
\hat{\tau}_{\vec{k}}\;
\frac{1}{\vec{J}!}\; 
\prod_{s\in\Omega_T}
\partial^{\otimes J_s}_{\omega_s}\;
 {\varPhi^{\star}}^{\mathcal{T}}_{\vec{k}}
(\vec{x},\vec{u},\dvec{u})
|_{\{\omega_s\}\to 0}
\label{eq:irrepTensorTHarm}
\end{empheq}
The notations are $\vec{x}=\{x_s\}, \vec{u}=\{u_s\}, 
\dvec{u}=\{u_{rs}\}$,
\begin{eqnarray}
x_s&=&P_{s(s)}\otimes\omega_s,\quad
u_s=\mathcal{G}_{(ss)}\otimes
\omega_s
\otimes
\omega_s,\quad
u_{rs}=\mathcal{G}_{(rs)}\otimes
\omega_r
\otimes
\omega_s,
\label{eq:irrepHarmProjs}
\\
 c^{J} &=&-\left( J+(D-5)/2\right),\qquad
 \mathcal{N}^{\vec{J}}_{\vec{k}\vec{n}} =
 \vec{J}!/(2^{|\vec{n}|}\vec{n}!\vec{k}!\vec{J}^*).
\label{eq:irrepcJ}\\
A^{\,\otimes b} &=&
A_{\alpha_1}...A_{\alpha_b},\quad
\vec{a}^{\;\vec{b}} =
\prod_i a_i^{b_i},\quad 
\vec{a}\;! = \prod_i a_i!,\quad 
|\vec{a}| =\sum_i a_i 
\label{vecfuns}
\end{eqnarray}

\section{General formula for cross-sections}

The algorithm for cross sections (convolutions of tensors) is presented below 
for $W$ tensors (DD cross section). It easily generalizes to other 
contractions~\cite{covreggeII}. Then cross-sections can be expressed in
terms of hypergeometric like functions and continued to the complex
$J$.
\begin{eqnarray}
\frac{d\sigma_{DD}}{d\Phi}&=&\sum_{k_1,k_2=0}^{\min(J,J')}
\hat{w}_{k_1}\hat{w}_{k_2}
\frac{d\sigma^{k_1k_2}_{DD}}{d\Phi}\nonumber\\
\frac{d\sigma^{k_1k_2}_{DD}}{d\Phi}&\sim& 
\mathcal{W}^{*\vec{J}}_{k_2}(p_2,q) 
\otimes
\mathcal{W}^{*\vec{J}}_{k_1}(p_1,q) =
S_{k_2,\vec{0}}^{W\; \vec{J}}(p_2,q) \otimes \mathcal{W}^{*\vec{J}}_{k_1}(p_1,q) =\nonumber\\
&=&  \frac{(P_2\partial_{\omega_1})^{J-k_2}
(P_2\partial_{\omega'_1})^{J'-k_2}
\left[ 
\mathcal{G}_{\alpha\beta}\otimes
\partial_{\omega_{1\alpha}}
\partial_{\omega_{1'\beta}}
\right] ^{k_2}}{k_2!(J-k_2)!(J'-k_2)!} 
{\varPhi^{\star}}^{\mathcal{W}}_{k_1}
(\vec{x},\vec{u},\dvec{u}) 
 |_{\Omega_{ch}}
\nonumber\\
\Omega_{ch}: 
x_i&=&P_1\omega_i\to Z=P_1P_2,\quad 
\xi_i =P_2\omega_i\to P_2^2,\quad
u_i,u_{11'}\to 1-\frac{Z^2}{P_1^2}
\label{csDD}
\end{eqnarray}
SD and DD cross-sections for $D=4$ are presented in~\cite{SD} 
in the forward limit.

Nevertheless, the number of form factors sometimes remains
large enough, and all they need to be fitted. It makes 
difficult to extract any physical information from 
the data. Therefore, one can, for example, require the finiteness of
tensors in the forward limit and get linear equations for 
form-factors. Then, for tensors $W$, $F$, everything 
is expressed in terms of a single form factor, which can
be extracted from the data. For $H$ there still remains a 
fairly large number of form factors, for even $|\vec{J}|$
\begin{equation}
N_H=\mathbb{C}_{J_0}^2,\qquad
J_0=\min\left(J_1,(J_1+J_2+J_3-J_4)/2\right)
\end{equation}
But we can use, for example, their cross-channels symmetry to reduce their number to a minimum.

\begin{figure}[htb]
	\centering
	%  \fbox{
	\includegraphics[width=0.35\linewidth, %bb=140 120 390 290
	]{fig2.pdf}
	%  }
	\caption{Contractions of 3-Reggeon vertexes to obtain
		$W_{3\mathbb{R}}$ and $H_{3\mathbb{R}}$.
		\label{fig2}}
\end{figure}
A three-Reggeon approximation can be used in $W$ and $H$. We compute the forward limit of $Y$ vertices, contract them and average over an arbitrary space-like direction that appears in this limit (see Fig.~\ref{fig2}):
\begin{eqnarray}
\mathcal{W}^{\vec{J}}_{3\mathbb{R}}(p,q)&=&
<
\mathcal{Y}_{fwd}^{\{J,J',J_3\}}(q,n)\otimes
\mathcal{V}^{J_3}(n)
>_{p,q},
\label{W3R}
\\
\mathcal{H}^{\vec{J}}_{3\mathbb{R}}(q_{1,2})&=&
<
\mathcal{Y}_{fwd}^{\{J_1,J_{1'},J_3\}}(q_1,n)\otimes
\mathcal{Y}_{fwd}^{\{J_2,J_{2'},J_3\}}(q_2,n)
>_{q_1,q_2},
\label{H3R}\\
\mathcal{Y}_{fwd}^{\vec{J}}(q,n) &=&
\lim_{\epsilon\to 0\atop qn=0}
\mathcal{Y}^{\vec{J}}
(q_1=q+\frac{\epsilon}{2}n,q_2=-q+\frac{\epsilon}{2}n),
\label{Yfwd}\\
 <n_{\alpha_1}...n_{\alpha_{2N}}>_{p,q}&=&
 \frac{
 \int d^D n \;\delta(n^2+1)\;\delta(pn)\;\delta(qn)\;
  n_{\alpha_1}...n_{\alpha_{2N}}
  }{
  \int d^D n \;\delta(n^2+1)\;\delta(pn)\;\delta(qn)
  }=
\nonumber\\
 &=& 
 \frac{
 (-1)^{N}
 }{
 2^{N} 
 \left(\frac{D-2}{2}\right)_N
 } 
 \left(
 \mathcal{G}_{(rs)}(p,q)
^{\otimes N} \right)_{all\;sym.},\qquad 
 N=|\vec{J}|/2,
\end{eqnarray}
where "$all\;sym.$" means the symmetrization of
all indices (even from different groups), and $r$, $s$ can be from 
any group of indices defined by $\Omega_{W,H}$.

\section{Conclusions}

In this work we shortly presented the algorithm to calculate irreducible
tensors in $\mathbb{M}_D$ and express all the basic diffractive 
cross-sections in terms of their contractions. The general form of irreducible tensor 
form factors can be used not only in diffraction processes. All the details can be found 
in~\cite{covreggeI},\cite{covreggeII}.

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R.~Ryutin, 
%Covariant reggeization framework for diffraction. Part I: Hadronic tensors in Minkovsky space-time of any dimension, 
arXiv:2507.16019 [hep-ph];
%https://doi.org/10.48550/arXiv.2507.16019

\bibitem{covreggeII}
%\refitem{report}
R.~Ryutin, Covariant reggeization framework for diffraction. 
Part II: Hadronic diffractive cross-sections, \textit{in preparation}.

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