Perturbative QCD Core of Hadrons and Color Transparency Phenomena

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Citation: Frankfurt, L.; Strikman, M.

Perturbative QCD Core of Hadrons

and Color Transparency Phenomena.

Preprints 2022,4, 0. https://doi.org/

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Article

Perturbative QCD Core of Hadrons and Color Transparency

Phenomena

Leonid Frankfurt 1,† and Mark Strikman 2,*

1Tel Aviv University; leonidfrankfurt@gmail.com

2Penn State University

*Correspondence: mxs43@psu.edu

† Current address: Tel Aviv University

Abstract:

In the current paper, we argue that the ground state of a hadron contains a signiﬁcant

perturbative quantum chromodynamics (pQCD) core as the result of color gauge invariance and of the

values of chiral and gluon vacuum condensates. The evaluation within the method of dispersion sum

rules (DSR) of the vacuum matrix elements of the correlator of local currents with the proper quantum

numbers leads to the value of the radius of the pQCD core of a nucleon of about 0.4–0.5 fm. The selection

of the initial and ﬁnal states allows to select processes in which the pQCD core of the projectile gives

the dominant contribution to the process. It is explained that the transparency of nuclear matter for the

propagation of a spatially small and color-neutral wave packet of quarks and gluons—a color transparency

(CT) phenomenon—for a group of hard processes off nuclear targets can be derived in the form of the

QCD factorization theorem accounting for the color screening phenomenon Based on the success of

the method of DSR, we argue that a pQCD core in a hadron wave function is surrounded by the layer

consisting of quarks interacting with quark and gluon condensates. As a result, in the quasi-elastic

processes

e+A→e0+N+ (A−

)∗

, the quasi-Feynman mechanism could be dominating in a wide

range of the momentum transfer squared,

. In this scenario, a virtual photon is absorbed by a single

quark, which carries a large fraction of the momentum of the nucleon and dominates in a wide range of

. CT should reveal itself in these processes at an extremely large

as the consequence of the presence

of the Sudakov form factors, which squeeze a nucleon.

Keywords:

quantum chromodynamics (QCD); exclusive processes; color transparency; nucleon structure

1. Introduction

The color transparency (CT) phenomenon is the suppression of the ﬁnal and/or initial

state interaction for the small size wave packet of quarks and gluons produced in the hard

processes and propagated through a nucleus. The quantitative approach is to calculate the

cross-section of the interaction of small size wave packet scattering off a target, as well as to

describe some properties of the bound state hadrons speciﬁc for quantum chromodynamics

(QCD).

CT has been derived (i) for the deep inelastic scattering (DIS) processes initiated by highly

virtual photons; (ii) for the processes of the diffractive electroproduction of vector mesons such

γ∗+A→V+A

for

V=ρ

J/ψ

etc.

; (iii) for the processes:

π+A→two jets +A

p+A→three jets +A

; (iv) in the case of small parton momentum fraction, small-

processes,

factorization theorems can be derived for the diffractive photoproduction of the bound states

of heavy quarks such as

J/ψ

and

, produced in the ultraperipheral collisions of heavy ions.

and (v) CT is generalized to include effects of the leading twist gluon shadowing for small

processes.

Within QCD, a hadron consists of the three overlapping layers, corresponding to two

distinctive phases of the QCD matter. The outer layer is formed by the pion cloud of a hadron.

Just this layer produces internucleon attraction in low-energy nuclear phenomena. The next

arXiv:2210.11569v1 [hep-ph] 20 Oct 2022

2 of 13

layer is formed by quarks interacting with chiral and gluon condensates. The existence of this

layer is the basis of the success of the method of dispersion sum rules in the calculation of

parameters of the ground states of hadrons. The important role of the vacuum condensate of

chiral quark pairs is implied by the phenomenon of spontaneously broken chiral symmetry in

QCD. The boundary between both layers can be evaluated as the boundary of the region where

two-pion exchange between nucleons dominates. This value of the boundary was estimated

in Ref. [

] long ago. In any case, the position of this boundary is not well deﬁned since it

ﬂuctuates.

The second layer is relevant for the competition between soft and hard processes for the

large momentum transfer behavior of hadron form factors. This competition results from the

speciﬁc of the Lorentz transformation within light cone quantum mechanics. It was found

that for a two-body system that large momentum transfer

is multiplied by the factor, 1

−α

in the argument of the wave function of a ﬁnal state of a two-body system [

]. A similar

pattern holds for many body systems. Thus, the relative contribution into form factors of

conﬁgurations where one constituent carries most of the light cone (LC) fraction—

—of the

hadron is enhanced. Feynman has pointed out that this may lead to a dominance of the

conﬁgurations where one parton carries practically all the momentum of the hadron. Hence,

this contribution is referred to as the Feynman mechanism. In practice, there is a wide pre-

asymptotic region, e.g.,

α≥

0.8, which is strongly enhanced in a wide range of

. In what

follows, this kinematics is referred to as a quasi-Feynman mechanism.

Together, these two layers describe the QCD phase of the spontaneously broken chiral

symmetry. The signiﬁcant probability of the perturbative QCD (pQCD) core within the wave

function of a hadron follows from the analysis of the vacuum correlator of local currents in

the coordinate space, i.e., effectively from the color gauge invariance and the values of the

chiral and gluon condensates. Thus, the evaluation of the radius of the pQCD core of a hadron

requires using a model accounting for the condensates.

The discovery of heavy quarks such as

etc.

allows to expand CT to the number of the

processes, where CT has already been observed and can be further investigated.

The understanding of the QCD structure of the wave function of a hadron and of a nucleus

allows to separate a group of hard processes, where the hard interaction produces spatially

small wave packets of quarks and gluons. The strength of the interaction of such wave packets

with hadrons is unambiguously calculable within QCD for high-energy processes. Selecting

special initial and ﬁnal states is necessary for ensuring the dominance of the contribution of

the pQCD core of the hadrons. For this group of phenomena, CT is one of the elements of

the QCD factorization theorem. In these processes, CT was already unambiguously observed.

The second group is formed by the processes, where there is no constraint for the pQCD core

to dominate in the wave function of a hadron. In this case, the quasi-Feynman mechanism

dominates in a wide range of momentum transfer. At an extremely large momentum transfer,

the contribution of the pQCD core should dominate since the Sudakov form factors gradually

squeeze the wave packet.

2. Three-Layer Structure of the Nucleon Wave Function in QCD

2.1. The Spatial Distribution of Valence Quarks in a Nucleon

The valence quark and momentum sum rules for the parton distributions within a hadron

unambiguously follow from the Wilson operator expansion. For certainty, the discussion in

this paper is restricted to the case of a nucleon target. The sum rules for the generalized valence

quark distributions, VN(x,Q2,t), at a non-zero momentum transfer,

ZVN(x,Q2,t)dx =FV(t), (1)

3 of 13

is of a prime interest to us. Here,

N(t)

is the isotopic (SU(3)) vector form factor of a nucleon

calculable in terms of the combination of the electromagnetic form factors of a proton and a

neutron. The generalized valence quark sum rules follow for any

from the combination of

the Ward identities for electroweak currents and the energy dependence of the high-energy

amplitudes in QCD in non-vacuum quantum numbers in the crossed channel [

]. To investigate

the pQCD core of a nucleon at moderate

, the method of dispersion sum rules (DSR),

developed in [4], is used here.

The radius of valence quark distribution follows from the sum rules and from the data on

the electromagnetic form factors of a nucleon:

(r2

V)1/2 =0.65 fm. (2)

2.2. The pQCD Core of the Wave Function of a Nucleon

The Dirac sea is an important property of relativistic quantum ﬁeld theories. Ignoring the

Dirac sea in the non-relativistic approximation leads to the non-conservation of baryon and

electric charges of the energy—momentum tensor as deﬁned in QCD and a related violation

of probability conservation for the high-energy processes. For example, these violations

lead to the so-called West correction for the structure functions of the deuteron,

σtot(eD)<

σtot(ep) + σtot(eN)

in the impulse approximation [

]. This calculation used the non-relativistic

wave function of the deuteron. Such a correction results in the violation of the exact QCD sum

rules such as the baryon charge sum rule, in the violation of the Glauber decomposition over

rescatterings, etc.

The only approach known so far to account for the Dirac sea in a way consistent with

the exact QCD sum rules is to use the LC. mechanics of nuclei. It resembles the parton model

approximation for a quantum ﬁeld theory, suggested by Feynman. Imposing angular mo-

mentum conservation and the requirement of Lorentz symmetry for on-mass-shell amplitude,

LC mechanics can be transformed into the instant time form for a wide range of nucleon

momenta [

]. In the LC mechanics of a nucleus, the West correction disappears, as can be found

in Ref. [

]. This phenomenon is important in nuclear theory, in the kinematics, where

k2/m2

not negligible; here, kdenotes nucleon momentum and mNdenotes nucleon mass.

The phenomenon is also present in quantum electrodynamics (QED) in the calculation of

high-order corrections to the wave functions of molecules.

To investigate the properties of an LC wave function of a nucleon, the vacuum element

of the retarded commutator of local color-neutral currents,

, with the quantum numbers of a

nucleon, Omitting Lorentz indices we can write

K(y0,~

y) = h0|θ(y0)[J(y0,~

y),J(0)]|0i. (3)

are analyzed here.

h0|

and

|0i

denote in and out states,

is the difference of 4 D coordinates of

the in and out currents, and y0is the zero component of y.θis the Heaviside function,

The retarded commutator is equivalent to the

- product but allows the analytic contin-

uation of the Fourier transform into the complex plane of energies and derives dispersion

relations. This correlator in the momentum space was analyzed in Refs. [

] within the DSR

approximation. For the aims of the study, given here, it is convenient to analyze this correlator

in the coordinate space.

The intermediate states in the correlator with the quantum numbers of a nucleon are

accounted for as the full system of eigen-states of the QCD Hamiltonian:

K(y0,~

y)) = ∑

h0|J(0)|ni2exp(i(En−E0)y0−i(pn·~

y))θ(y0)dτn. (4)

Here, Enand E0are ..., respectively; pnis ..., and τnis ...

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Citation:Frankfurt,L.;Strikman,M.PerturbativeQCDCoreofHadronsandColorTransparencyPhenomena.Preprints2022,4,0.https://doi.org/Publisher'sNote:MDPIstaysneutralwithregardtojurisdictionalclaimsinpublishedmapsandinstitutionalafl-iations.Copyright:©2022bytheauthors.LicenseeMDPI,Basel,Switzerland.Thisarti...

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