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SUMMARY:S-Matrix unitarity and Pomeron shadowing corrections
DTSTART;VALUE=DATE-TIME:20201222T133000Z
DTEND;VALUE=DATE-TIME:20201222T135000Z
DTSTAMP;VALUE=DATE-TIME:20230926T122601Z
UID:indico-contribution-178@indico.bitp.kiev.ua
DESCRIPTION:Speakers: Georgy Tersimonov (Bogolyubov Institute for Theoreti
cal Physics)\nRegge theory is the only valid framework to describe soft sc
attering processes where the perturbative QCD is not applicable. In Regge
theory\, the particle diffraction is treated as an exchange of some 'objec
t' called Pomeron (which in some way generalizes a particle — in particu
lar\, it is described by variable complex angular momentum which generaliz
es a spin). That approach was found surprisingly useful to phenomenologica
lly calculating cross sections.\nIn 1960s\, it was shown that multi-Pomero
n shower production reactions $pp \\rightarrow p + X_1 + X_2 + ... + p$\,
where showers $(X_i)$ are separated by large rapidity gaps\, are breaking
the S-matrix unitarity because corresponding cross-sections $\\sigma_{tot}
$ grow with the rapidity ($\\xi$) faster than allowed by unitarity (the u
pper bound is $\\sigma_{tot} \\leq \\xi^2$). This issue is known as Finkel
stein-Kajantie problem. In 1974\, a possible solution was proposed [1] in
multi-channel Eikonal model. It considered the gap survival probability $S
^2$ — the probability to observe the pure process where the gap is not p
opulated by secondaries produced in the additional inelastic interaction.
In the impact parameter representation the probability is given by $S^{2}(
b) = |e^{-\\Omega(b)}|$\, where $b$ is the impact parameter and $\\Omega$
is the proton opacity. In the black disc limit $Re(\\Omega) \\rightarrow \
\infty$\, so $S^2(b) \\rightarrow 0$. So the additional rescatterings sho
uld close the rapidity gaps. The work [2] shows that decreasing of the sur
vival probability should overcompensate the original cross-sesction growth
so\, as a result\, the cross-sections should also vanish with energy: $\\
frac{d\\sigma}{d\\xi_1} \\sim e^{-{\\Delta}\\xi_1} \\rightarrow 0$\, where
$\\xi_1$ is the shower width on the rapidity scale. If the result is corr
ect then the unitarity is restored. Over the past decades\, it has been co
nsidered a cure for the FK problem [3].\nThe work [4] had discovered that
such an approach still fails to unitarize the Pomeron contribution to the
single diffraction dissociation amplitude due to an error in the calculati
ons. The suspicion had arised: is the cure really effective in terms of al
l the processes it is purposed for? Recent TOTEM soft scattering data rene
wed the interest to these questions.\nIn the work [5] we investigate the s
urvival probability method for all the diffractive processes. The main pro
cesses are next. The first is single diffraction dissociation where one of
the two incoming protons transforms into a shower: $pp \\rightarrow X +
p$. The second is double diffraction dissocastion where both protons trans
forms: $pp \\rightarrow X_1 + X_2$. The third is central production: $pp
\\rightarrow p + X + p$. Integrated cross-sections of all these processes
behave similar to each other\, so only the simplest\, the single dissocia
tion\, will be considered in this talk. Its cross-section contains a mult
iplier $e^{\\Delta(\\xi_1 + 2\\xi_2 - a\\xi)}$\, where $a \\rightarrow 2\\
frac{\\xi}{\\xi + \\xi_1}$ as $\\xi \\rightarrow \\infty.$ Here $\\xi_2$ i
s the rapidity gap between the produced shower and the initial proton\; $\
\xi_1 + \\xi_2 = \\xi$ — the overall rapidity difference between interac
ting protons. While investigating the high energy asymptotics ($\\xi \\rig
htarrow \\infty$)\, the authors of [2] considered $a$ as $2$ and $e^{\\Del
ta(\\xi_1 + 2\\xi_2 - a\\xi)}$ simply became $e^{-\\Delta{\\xi_1}}$. Howev
er\, if the calculations are done in an explicit way\, one can see that $a
= 2(1 -\\frac{\\xi_1}{\\xi} + O(\\frac{\\xi^2_1}{\\xi^2}))$ and so $e^{\\
Delta(\\xi_1 + 2\\xi_2 - a\\xi)} = e^{\\Delta(\\xi_1 + 2\\xi_2 - 2\\xi (1
-\\frac{\\xi_1}{\\xi}))}$ = $e^{+\\Delta{\\xi_1}}$\, thus the fast cross-
section growth is in fact maintained.\nThereby the existing survival proba
bility methods are unable to keep the cross-section growth within the unit
arity bound. We develop a different approach based on the Pomeron and trip
le-Pomeron vertex renormalization via Schwinger-Dyson equations. We take t
he Pomeron in it's maximal form providing the maximal strong interactions
strength allowed by unitarity. The triple-Pomeron vertex is chosen to cont
ain zeroes at some transferred momenta and complex angular momenta. The pa
rameters of developing model can be chosen in such a way that the unitarit
y bounds are not violated.\n\n[1] J. L. Cardy. General Features of the Reg
geon Calculus with $\\alpha > 1$. *Nucl. Phys. B*\, 75 (1974)\n[2] E. Gots
man\, E.M. Levin\, U. Maor. Diffractive Dissociation and Eikonalization in
High Energy $pp$ and $p\\bar{p}$ Collisions\, *Phys. Rev. D*\, 49 (1994)\
n[3] V.A. Khoze\, A.D. Martin\, M.G. Ryskin. Black disc\, maximal Odderon
and unitarity. *Phys.Lett. B*\, 780 (2018)\n[4] E. Martynov\, B. Strumins
ky. Unitarized model of hadronic diffractive dissociation. *Phys.Rev.
D*\, 53 (1996)\n[5] E. Martynov\, G. Tersimonov. Multigap diffraction cr
oss sections: Problems in eikonal methods for the Pomeron unitarization. *
Phys. Rev. D*\, 101 (2020)\n\nhttps://indico.bitp.kiev.ua/event/7/contribu
tions/178/
LOCATION:Online meeting
URL:https://indico.bitp.kiev.ua/event/7/contributions/178/
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