# The Nambu-Goto string in QCD:
Dipole interactions, scattering and entanglement

Yizhuang Liu<EMAIL_ADDRESS>Institute of Theoretical Physics,
Jagiellonian University, 30-348 Kraków, Poland  Maciej A. Nowak
<EMAIL_ADDRESS>Institute of Theoretical Physics and Mark Kac Center
for Complex Systems Research, Jagiellonian University, 30-348 Kraków, Poland
Ismail Zahed<EMAIL_ADDRESS>Center for Nuclear Theory,
Department of Physics and Astronomy, Stony Brook University, Stony Brook, New
York 11794–3800, USA

###### Abstract

We revisit some aspects of the stringy approach to dipole-dipole interactions,
scattering and entanglement in QCD, using the Nambu-Goto (NG) string, without
recourse to holography. We first show that the potential between two static
dipoles exchanging closed NG strings is attractive at all separations.
Underlining the exchange there is an emergent entropy, that is dominated by
the tachyon at large separation, and vanishes at short separation, a measure
of the confinement-deconfinement transition. The same tachyon is dominant in
the scattering amplitude, as a correlator of two Wilson loops for two fixed
dipole-like hadrons separated at large rapidity gap, where the contribution of
the worldsheet fermions is included. While the tachyon causes the mean string
bit density to grow exponentially with the rapidity, the total scattering
cross section still satisfies the Froissart bound by quantum shadowing. The
stringy scattering exchange also carries an entanglement entropy, that
saturates when the bound is reached. For hadrons with varying dipole sizes,
the tachyon exchange takes place in hyperbolic space in the conformal limit.
The result for the full S-matrix is reminiscent of the one from Mueller′s
evolved dipole wavefunction, for the total dipole-dipole cross section in
perturbative QCD.

##  I Introduction

String theory has been hailed as a possible consistent theory of quantum
gravity. By pertinent compactifications, it has the potential to lead to
various extensions of the standard model at higher energies. But perhaps the
most compelling application of string theory may still be in strong
interactions, where it originated from. It is plausible that QCD in the large
number of colors $N_{c}$ limit, may be dual to an effective string theory with
a weak string coupling $g_{s}\sim\frac{1}{N_{c}}$, as suggested by holography
in higher dimensions Ammon and Erdmenger (2015) (and references therein).

In so far, the compelling arguments for a QCD string stem from lattice QCD
simulations Kuti (2006) (and references therein). Indeed, detailed QCD lattice
simulations in 1+2 dimensions have shown that the closed flux tube in gauge
theories with various $N_{c}$, are well described by a Nambu-Goto (NG) string
in flat space dimensions Athenodorou  _et al._ (2007). Detailed studies of the
string mass on its length for various $N_{c}$, yield results that are in good
agreement with the NG string even for short lengths, well beyond the
contribution of the Luscher term. Although fuzzy, the string is well
approximated by a fundamental NG string. The extension of the lattice analysis
to 1+3-dimensions for $N_{c}=3,5$ for the closed string spectrum, has also
shown convincing agreement with the NG string Athenodorou  _et al._ (2009).

The description of the heavy quark-antiquark potential for fixed separation
$R$, using a fully quantized NG string in 2+D⟂-dimensions, was carried by
Arvis with the result Arvis (1983)

$\displaystyle
V(R)=\bigg{(}\sigma_{T}^{2}R^{2}-\frac{D_{\perp}}{24\alpha^{\prime}}\bigg{)}^{\frac{1}{2}}\
,$

with the string tension $\sigma_{T}=1/2\pi\alpha^{\prime}$ and
$\alpha^{\prime}=l_{s}^{2}$. The large distance expansion of the NG potential,
yields

$\displaystyle V(R)\approx\sigma_{T}R-\frac{\pi
D_{\perp}}{24R}-\frac{\pi^{2}}{2\sigma_{T}R^{3}}\bigg{(}\frac{D_{\perp}}{24}\bigg{)}^{2}\
,$ (1)

the linear confining potential, plus the universal Luscher correction Luscher
_et al._ (1980) and the Luscher-Weisz correction Luscher and Weisz (2004).
These three contributions are well reproduced by the high precision QCD
lattice analysis of the inter-quark potential in Brandt (2017). Yet, the full
NG potential becomes imaginary below a critical distance

$\displaystyle
R_{c}=\pi\sqrt{\frac{D_{\perp}\alpha^{\prime}}{6}}=\frac{1}{2}\frac{|M_{0}|}{\sigma_{T}}\rightarrow\frac{1}{3}\,{\rm
fm}\ .$

The rightmost numerical estimate following from $D_{\perp}=2$, with
$\alpha^{\prime}=1/2m_{\rho}^{2}$ the Regge rho meson slope. This behavior is
tied to the tachyon with negative squared mass $M_{0}^{2}<0$ in the dual
spectrum, and doomed the NG string as a fundamental string in 4-dimensions.
Yet the tachyon mass is also at the origin of the measured large distance
$1/R$ and $1/R^{3}$ corrections exhibited by the QCD string, which is fuzzy
and not fundamental. The inter-quark potential below the critical $R_{c}$ in
QCD, is non-confining. The lattice results Brandt (2017, 2021) (see Figs. 1-3
therein) set the deviations at small separations $r<r_{0}$, with the Sommer
scale $r_{0}\sim 0.5$ fm. Hence, the use of the NG string to model QCD
interactions, should prove useful away from criticality, for separations
$r>{r_{0}}>R_{c}$. The additional corrections to the NG string stemming from
the constraints of Lorentz symmetry Aharony and Field (2011) are observed to
be small in this range. In a way, this is consistent with the empirical
success of the Cornell potential for charmonia Eichten  _et al._ (1978). In
what follows, we will make use of these observations.

More specifically, we will use the NG string to analyse the interaction
potential and scattering amplitude in the Regge limit ($s\gg-t\sim\Lambda_{\rm
QCD}^{2}$), between two QCD dipoles using the exchange of a NG string in flat
4-dimensional space, without recourse to holography. When formulated in
Euclidean signature, the two calculations parallel each other with the
scattering following from the potential with an Euclidean-angle valued
rotation, followed by a pertinent analytical continuation to the hyperbolic
angle or rapidity. This construction was initially suggested for a pair of
quark scattering in Meggiolaro (1996). For completeness, we recall that this
stringy approach to scattering in QCD, was initially proposed in higher
dimensions using the gauge-gravity duality Rho  _et al._ (1999); Janik and
Peschanski (2000); Polchinski and Strassler (2002), and since then discussed
by many e.g. Polchinski and Strassler (2003); Brower  _et al._ (2007); Brower
(2003); Amorim  _et al._ (2021) (and references therein).

The present work relies on preceding constructions Basar  _et al._ (2012);
Stoffers and Zahed (2013), to derive a number of new results: 1/ The static
dipole-dipole potential is dominated by the exchange of NG closed strings,
with the tachyon dominant at large distances; 2/ The NG exchanges are
characterized by an emergent entropy, that undergoes a phase-like transition
with varying separation; 3/ The scattering of two dipoles at large rapidity,
is dominated by the two-particle irreducible (2PI) NG surface of genus 2, with
a total cross section that is in good agreement with the recently reported
data at the LHC; 4/ The NG estimate of the rapidity and parton-$x$ at
saturation; 5/ The contribution of the NG worldsheet fermions to both the
Pomeron intercept, and quantum entanglement at low parton-$x$; 6/ A new
entanglement entropy for multiple NG exchanges in the process of shadowing,
that asymptotes a single qubit at the Froissart bound; 7/ The generalization
of the NG tachyon diffusion from flat to curved hyperbolic and confining
space, to include evolution in the size of the probing dipoles.

The organization of the paper is as follows: In section II we detail the
construction of the potential between two static dipoles of fixed and equal
size, via the exchange of closed NG strings. The potential is found to be
attractive at all separations, with the NG tachyon dominant at large
separations. In section III we extend the potential analysis to the scattering
amplitude, through a simple rotation in Euclidean signature, followed by an
analytical continuation to rapidity in Minkowski signature. The NG tachyon is
shown to dominate the scattering amplitude at large rapidities. Both the
potential and scattering amplitude exponentiate through 2PI “webs” that are
identified with the exchange of a NG string, in leading order in $1/N_{c}$.
This resummation in the scattering channel, yields to saturation of the total
cross section by quantum shadowing, even though the string bits density keeps
increasing exponentially at large rapidities. In section IV we review and
extend the string results for quantum entanglement for hadron-hadron in the
Regge limit, and DIS scattering in the low-x regime. We suggest that the
quantum entanglement entropy saturates at the Froissart bound. In section V we
show how to extend the NG tachyon diffusion in curved transverse space, by
including the size of the probe dipoles and enforcing conformal symmetry. The
result is reminiscent of the BFKL evolution result following from Mueller′s
wavefunction evolution for the total cross section in perturbative QCD. Our
conclusions are in section VI. A short parallel between the NG tachyon results
and pQCD in the Regge limit, is discussed in the Appendix.

##  II Stringy dipole-dipole interaction

There are many indications from lattice simulations that flux tubes in
quenched QCD, can be described by an effective theory of strings of which the
Nambu-Goto (NG) action is the leading contribution Kuti (2006). Remarkably,
the NG string appears to describe remarkably well the fuzzy QCD string, even
for relatively short distances. We now use this lattice observation, to
analyze the potential between a pair of static dipoles, in quenched QCD. This
potential is amenable to a measurement on the lattice.

This construction parallels closely the analysis of the scattering of a pair
of light-like dipoles discussed in Basar  _et al._ (2012) (and references
therein), using the holographic construction. To our knowledge, the potential
between two parallel dipoles was never analyzed using the holographic string.
It cannot be deduced from the twisted dipoles in Basar  _et al._ (2012) by
taking the twisting angle to zero, since the twisted exchange presents an
essential singularity in this limit. The singularity reflects on the existence
of a pair creation process through a string-like Schwinger mechanism in the
twisted configuration, that is absent in the untwisted configuration for the
potential analysis.

###  II.1 Dipole-dipole correlator

Consider the static correlators of two identical Wilson loops

$\displaystyle{\bf WW}=\frac{\langle{\bf W}(a,{\bf b}_{\perp}){\bf W}(a,{\bf
0}_{\perp})\rangle}{\langle{\bf W}\rangle\langle{\bf W}\rangle}\ ,$ (2)

with

$\displaystyle{\bf W}(a,{\bf x}_{\perp})=\frac{1}{N_{c}}Tr\bigg{(}{\bf P}{\rm
exp}\bigg{(}ig_{Y}\int_{{\cal C}_{a}}d\tau{\bf A}.{\bf v}\bigg{)}\bigg{)}\ .$
(3)

The contour ${\cal C}_{a}$ runs along the rectangular loop of side $a$,
located at ${\bf x}_{\perp}$, with infinite extent along the temporal
direction in the v-direction. As usual, the Wilson-loop correlator
exponentiates through 2PI “webs” Gatheral (1983); Frenkel and Taylor (1984) ,
which we identify to leading order as a closed NG string with genus 2. More
specifically,

$\displaystyle\ln{\bf WW}\equiv{\bf WW}_{\rm
2PI}=g_{s}^{2}\int\frac{dT}{2T}\,{\bf K}(T)\ ,$ (4)

with $g_{s}\sim\frac{1}{N_{c}}$ the string coupling, and

$\displaystyle{\bf K}(T)=\int_{T}D[x]\,e^{-S[x]+{\rm ghost}}\ ,$ (5)

the NG string partition function with cylindrical topology and modulus $T$.
The NG action in conformal gauge is Polyakov (1986)

$\displaystyle S[x]$ $\displaystyle=$
$\displaystyle\frac{\sigma_{T}}{2}\int_{0}^{T}d\tau\int_{0}^{1}d\sigma~{}(\dot{x}^{\mu}\dot{x}_{\mu}+{x^{\prime}}^{\mu}{x^{\prime}}_{\mu})\
.$ (6)

Here $\dot{x}=\partial_{\tau}x$ and $x^{\prime}=\partial_{\sigma}x$. The
string tension is $\sigma_{T}=1/(2\pi\alpha^{\prime})$.

The evaluation of (4) in string theory, is in general difficult due to the
finite dipole sizes, and we need to make reasonable approximations. For small
size dipoles, we will assume that the cylindrical boundaries are highly
pinched, and approximate them by straight lines. The exchanged closed string
forms a funnel, with linear end-points, much like the exchange between static
D0 branes, as detailed for the twisted dipoles in Basar  _et al._ (2012).
However, funnels with higher windings or N-ality are not suppressed between D0
branes, but are suppressed between dipoles of finite transverse size. A
physical interpretation of the final result, will allow for a simple
extraction of the dipole-dipole potential from this approximation below.

###  II.2 Static dipole-dipole potential

With this in mind, we now decompose the string embedding coordinates using the
worldsheet normal modes in $2+D_{\perp}$ flat space, with linear and periodic
boundary conditions in the affine time with period $T$ Qian and Zahed (2015)

$\displaystyle x^{0}(\tau,\sigma)$ $\displaystyle=$ $\displaystyle
X+\frac{cW}{\sigma_{T}}\tau+\sum_{m=-\infty}^{\infty}\sum_{n=1}^{\infty}x_{m,n}^{0}\exp\bigg{(}{i2\pi
m\frac{\tau}{T}}\bigg{)}\cos(\pi n\sigma)\ ,$ $\displaystyle
x^{1}(\tau,\sigma)$ $\displaystyle=$
$\displaystyle\sum_{m=-\infty}^{\infty}\sum_{n=1}^{\infty}x_{m,n}^{1}\exp\bigg{(}{i2\pi
m\frac{\tau}{T}}\bigg{)}\sin(\pi n\sigma)\ ,$ $\displaystyle
x_{\perp}(\tau,\sigma)$ $\displaystyle=$
$\displaystyle\bigg{(}\sigma-\frac{1}{2}\bigg{)}b_{\perp}+\sum_{m=-\infty}^{\infty}\sum_{n=1}^{\infty}x_{m,n}^{\perp}\exp\bigg{(}{i2\pi
m\frac{\tau}{T}}\bigg{)}\sin(\pi n\sigma)\ .$ (7)

with $c=1/{l_{s}}$. Since

$x^{0}(\tau+T,\sigma)-x^{0}(\tau,\sigma)=W\bigg{(}\frac{T}{\sigma_{T}l_{s}}\bigg{)}\
,$ (8)

we interpret $W=0,\pm 1,\pm 2,...$ as a winding number, with $2\pi l_{s}T$ the
circumference of the cylindrical funnel. Below we will show that this is at
the origin of an effective temperature for the exchanged closed strings, in
the dipole-dipole potential. $X,W$ plays the role of collective coordinates.

In terms of (II.2), all integrals in (5) are Gaussian with the result

$\int
dX\sum_{W}\mathbb{K}(T,W)=\frac{a^{2}X}{l_{s}^{3}}\sum_{W}\exp\left(-\frac{T}{2}\bigg{[}\sigma_{T}b^{2}+\frac{c^{2}W^{2}}{\sigma_{T}T^{2}}\bigg{]}\right)\left[\prod_{n=1}^{\infty}2\sinh\left(\frac{n\pi
T}{2}\right)\right]^{-D_{\perp}}\ .$ (9)

The diverging products can be regularized by standard zeta function
regularization, using the representation

$\sinh(\pi x)=\pi x\prod_{m=1}^{\infty}\left(1+\frac{x^{2}}{m^{2}}\right)\ .$
(10)

With this in mind, and trading the re-summation over the windings using the
Poisson summation formula, we obtain

$\displaystyle\int
dX\sum_{W}\mathbb{K}(T,W)=\sqrt{\frac{1}{c^{2}\alpha^{\prime}T}}\frac{a^{2}X}{l_{s}^{3}}\sum_{k}\exp\left(-\frac{T}{2}\bigg{[}\sigma_{T}b^{2}+\frac{2\pi
k^{2}}{c^{2}\alpha^{\prime}T^{2}}\bigg{]}\right)\eta^{-D_{\perp}}\left(i\frac{T}{2}\right)\
,$

where $\eta(x)$ is Dedekind eta function

$\displaystyle\eta(\tau)=q^{\frac{1}{24}}\prod_{n=1}^{\infty}(1-q^{n})\qquad
q=e^{2i\pi\tau}$ (12)

Note that (12) satisfies $\eta(ix)=\eta(i/x)/\sqrt{x}$, and relates to the
string density of modes Fubini  _et al._ (1969)

$\displaystyle\eta^{-D_{\perp}}\big{(}ix\big{)}=x^{\frac{D_{\perp}}{2}}e^{\frac{\pi
D_{\perp}}{12x}}\sum_{n=0}^{\infty}d(n)e^{-n\frac{2\pi}{x}}\ ,$ (13)

with $d(n)$ being the string density of states normalized to $d(0)=1$, with
asymptotically

$\displaystyle d(n)\approx
C\,n^{-\frac{D_{\perp}+3}{4}}\,{e^{2\pi\sqrt{D_{\perp}n/6}}}\ .$ (14)

The Poisson re-summation trades the sum over the windings $W$ of the closed
string exchanges, with the dual sum over the N-alities or $k$ fluxes Basar
_et al._ (2012). For source dipoles as Wilson loops in the fundamental
representation, $k=1,..,[\frac{N_{c}}{2}]$ which runs to infinity in the large
$N_{c}$ limit. For QCD with $[\frac{3}{2}]=1$, only the $k=1$ N-ality is to be
retained.

The leading contribution to the static potential for two parallel dipoles of
size $a$, is given by the 2-particle irreducible (2PI) string exchange

$\displaystyle V_{DD}(b)=-\frac{{\rm
ln}\mathbb{WW}}{X}=-g_{0}^{2}\sum_{n=0}^{\infty}d(n)\Delta(m_{n},b)$ (15)

with the exchanged scalar propagator

$\displaystyle\Delta(m_{n},b)=\int\frac{dk^{D_{\perp}+1}}{(2\pi)^{D_{\perp}+1}}\frac{e^{ik\cdot
b}}{k^{2}+m_{n}^{2}}=\frac{1}{2\pi}\bigg{(}\frac{m_{n}}{2\pi
b}\bigg{)}^{\frac{D_{\perp}-1}{2}}\,K_{\frac{D_{\perp}-1}{2}}(m_{n}b)$ (16)

and coupling to the dipole
$g_{0}=\frac{g_{s}a}{\alpha^{\prime}}(\sqrt{2\pi}\alpha^{\prime})^{\frac{D_{\perp}}{4}}$.
Here $\mathbb{K}_{\alpha}(x)$ is the modified Bessel function, and $d(n)$ is
the canonical string density of states with $d(0)=1$, and $g_{s}$ the string
coupling. (15) amounts to a tower of closed string exchanges or glueballs,
with radial masses

$\displaystyle m_{n}=\frac{1}{c\alpha^{\prime}}\
\left(1-\frac{D_{\perp}c^{2}}{12\pi\sigma_{T}}+\frac{2n\pi
c^{2}}{\pi^{2}\sigma_{T}}\right)^{\frac{1}{2}}=\sigma_{T}\beta\left(1-\frac{\beta_{H}^{2}}{\beta^{2}}+\frac{8\pi
n}{\sigma_{T}\beta^{2}}\right)^{\frac{1}{2}}\ ,$ (17)

and with the inverse Hagedorn temperature $\beta_{H}=\sqrt{\pi
D_{\perp}/3\sigma_{T}}$. $\beta=2\pi/c=2\pi l_{s}$ is the circumference of the
exchanged cylindrical worldsheet.

For large $b$, the exchange in (15) is dominated by the tachyon mode with
$n=0$

$\displaystyle
m_{0}=\sigma_{T}\beta\left(1-\frac{\beta_{H}^{2}}{\beta^{2}}\right)^{\frac{1}{2}}=\frac{1}{l_{s}}\left(1-\frac{D_{\perp}}{6}\right)^{\frac{1}{2}}\
,$ (18)

which is still real positive for $D_{\perp}<6$. For $D_{\perp}=2$ and using
half the rho meson Regge slope $\alpha^{\prime}=1/4m_{\rho}^{2}$ for a closed
string, we have $m_{0}\approx 1257$ MeV, which is close to the
$m_{0^{++}}=1475$ MeV glueball reported on the lattice Meyer (2004). The
attractive and static dipole-dipole potential follows as

$\displaystyle
V_{DD}(b)\approx-\frac{g_{s}^{2}a^{2}\sqrt{\pi\sigma_{T}}}{\alpha^{\prime}}\left(\frac{\pi\alpha^{\prime}m_{0}}{b}\right)^{\frac{D_{\perp}-1}{2}}\mathbb{K}_{\frac{D_{\perp}-1}{2}}\left(m_{0}b\right)\
.$ (19)

For short separations, the exchange is also attractive

$\displaystyle
V_{DD}(b)\approx-\bigg{(}\frac{g_{s}^{2}a^{2}\sqrt{\pi\sigma_{T}}}{2\alpha^{\prime}}\sum_{n}d(n)\bigg{)}\Gamma\bigg{(}\frac{D_{\perp}-1}{2}\bigg{)}\,\bigg{(}\frac{1}{\sigma_{T}b^{2}}\bigg{)}^{\frac{D_{\perp}-1}{2}}\
,$ (20)

and Coulombic for $D_{\perp}=2$, i.e. $V_{DD}(b)\sim-g_{s}^{2}/b$. However,
this contribution signals the onset of a critical NG string, with a diverging
mode sum for the overall coefficient. At short separations, the exchange is
not confining. It is dominated by 2-gluon Coulomb exchange in the $0^{++}$
channel. The Casimir-Polder contribution characterizes the fully non-confining
potential at large separations Shuryak and Zahed (2000) (and references
therein).

Recall that the string coupling is $g_{s}=f(\lambda)/N_{c}$, with $f(\lambda)$
a non-universal function of the large ′t Hooft coupling
$\lambda=g_{Y}^{2}N_{c}$. For instance, in holographic models,
$f(\lambda)=\lambda/4\pi$ (${\cal N}$=4 SUSY) and
$f(\lambda)=(\lambda/3)^{\frac{3}{2}}/\pi$ (Witten model). This observation
shows that the 2-PI contribution (15) is dominant in large $N_{c}$, as the
higher #-PI contributions are suppressed by $(1/N_{c}^{2})^{1+\\#}$.

###  II.3 Emergent entropy in dipole-dipole interaction

The circumference $\beta=2\pi l_{s}$ in (15), plays the role of an inverse
effective temperature, associated with the spatial exchange of closed strings
or glueballs, in the transverse b-direction. Remarkably, $T_{R}=1/\beta=1/2\pi
l_{s}$ is identical to the Rindler temperature of falling matter on the
stretched horizon, a membrane a string length away from the event horizon of a
stationary black-hole. With this in mind, we may interpret the potential in
(15) as a free energy. As a result, the stringy exchange in the dipole-dipole
potential, carries an emergent entropy,

$\displaystyle S_{EV}(b)=\beta^{2}\bigg{(}\frac{\partial
V_{DD}(b)}{\partial\beta}\bigg{)}$ (21)

To understand the nature of this entropy and how it may relate to the spatial
entanglement entropy in QCD4 at large $N_{c}$, we note that (15) at large
separation is given by

$\displaystyle
V_{DD}(b)\sim\sqrt{\frac{\pi}{2}}\bigg{(}\frac{g_{s}^{2}a^{2}}{{\alpha^{\prime}}^{\frac{3}{2}}}\bigg{)}\bigg{(}\frac{2\pi\alpha^{\prime}}{b^{2}}\bigg{)}^{\frac{D_{\perp}-1}{2}}\sum_{n=0}^{\infty}d(n)\,x_{n}^{\frac{D_{\perp}}{2}-1}\,e^{-x_{n}}$
(22)

with $\alpha^{\prime}=l_{s}^{2}$ and $x_{n}=m_{n}b$. Inserting (22) in (21)
gives

$\displaystyle
S_{EV}(b)\sim(2\pi)^{\frac{D_{\perp}}{2}+4}\bigg{(}\frac{g_{s}^{2}a^{2}}{2b^{2}}\bigg{)}\bigg{(}\frac{l_{s}}{b}\bigg{)}^{D_{\perp}-1}\,\sum_{n=0}^{\infty}d(n)\,x_{n}^{\frac{D_{\perp}}{2}-2}\,e^{-x_{n}}$
(23)

(23) amounts to the entropic function at large $x_{n}\gg 1$

$\displaystyle C_{DD}(b)=\frac{\partial S_{EV}(b)}{\partial{\rm
ln}b}\sim-(2\pi)^{\frac{D_{\perp}}{2}+4}\bigg{(}\frac{g_{s}^{2}a^{2}}{2b^{2}}\bigg{)}\bigg{(}\frac{l_{s}}{b}\bigg{)}^{D_{\perp}-1}\,\sum_{n=0}^{\infty}d(n)\,x_{n}^{\frac{D_{\perp}}{2}-1}\,e^{-x_{n}}$
(24)

Using the radial mass spectrum (17) and the string density of states (14), the
sum is dominated by the large-n contribution

$\displaystyle\sum_{n=0}^{\infty}d(n)\,x_{n}^{\frac{D_{\perp}}{2}-1}\,e^{-x_{n}}\sim\int_{n_{\rm
low}}^{\infty}dn\,n^{-\frac{5}{4}}\,e^{-m_{n}b+2\pi\sqrt{\frac{D_{\perp}n}{6}}}$
(25)

Using the change of variable $t=\sqrt{8\pi n/\sigma_{T}}/\beta$ we can rewrite
(25) as

$\displaystyle\int_{t_{\rm
low}}^{\infty}\frac{dt}{t^{\frac{3}{2}}}\,e^{\frac{1}{2}\sigma_{T}\beta\beta_{H}t-\sigma_{T}\beta
b\sqrt{1+t^{2}}}$ (26)

which can be undone by saddle point approximation for $\beta/b\ll 1$, with the
result

$\displaystyle\sum_{n=0}^{\infty}d(n)\,x_{n}^{\frac{D_{\perp}}{2}-1}\,e^{-x_{n}}\sim
e^{-\sigma_{T}b\sqrt{b^{2}-(\beta_{H}/2)^{2}}}$ (27)

to exponential accuracy. The branch point singularity
$b=\frac{1}{2}\beta_{H}=R_{c}$ reflects on the diverging sum for small
distances in agreement with (I).

We now recall that the entropic function for spatial entanglement in QCD4 at
large $N_{c}$, was argued to be of the form Klebanov  _et al._ (2008)

$\displaystyle
C_{1+3}(b)\sim\frac{1}{32\sqrt{\pi}}\,\frac{V_{2}}{b^{2}}\,\sum_{n=0}^{\infty}d(n)(M_{n}b)^{\frac{3}{2}}e^{-2M_{n}b}\sim\int^{\infty}dM\,M^{\alpha}e^{(\beta_{H}-2b)M}$
(28)

where $M_{n}\sim\sqrt{n}/\alpha^{\prime}$ was used for the asymptotic of the
glueball spectrum. If instead, the exact radial glueball spectrum (17) with
$M_{n}=\frac{1}{2}m_{n}$ which carries the same asymptotics is used in (25),
then to exponential accuracy in the entropic function for spatial entanglement
in QCD4

$\displaystyle\sum_{n=0}^{\infty}d(n)(M_{n}b)^{\frac{3}{2}}e^{-2M_{n}b}\sim
e^{-\sigma_{T}b\sqrt{b^{2}-(\beta_{H}/2)^{2}}}$ (29)

in agreement with (27).

We conclude that the entropic function constructed from the emergent entropy
(21) associated to two interacting dipoles, shares the same asymptotic
behavior as the entropic function for spatial entanglement in QCD4 at large
$N_{c}$. It also diverges at the same location $b=\frac{1}{2}\beta_{H}$. The
divergence reflects on the transition from a confining to a deconfining phase
as probed by (21) and similarly by (29). Amusingly, the mode sums in both
cases are identical for $D_{\perp}=5$ or a NG in 2+5-dimension, somehow
reminiscent of the holographic proposal.

##  III Stringy dipole-dipole scattering

At large center of mass energy with a large rapidity gap, the hadron-hadron
scattering amplitude is universal. The amplitude is that of two fixed size
dipoles scattering elastically. For small angle scattering, the amplitude is
dominated by gluon exchanges with vacuum quantum numbers. In perturbative QCD,
the exchange is captured by the BFKL resummation of rapidity ordered gluons,
the so-called hard Pomeron. In non-perturbative QCD, the resummation is
captured by Reggeized gluons. In the planar approximation, the exchange is
string-like with the topology of a cylinder, the so-called soft Pomeron. The
existence of a hard and soft Pomeron at large rapidity, was initially pointed
out in Donnachie and Landshoff (1998). It finds a natural description in the
gravity dual approach to QCD Polchinski and Strassler (2003); Brower  _et al._
(2007), using a critical string in 10 dimensions in the conformal limit (hard
Pomeron), followed by conformal symmetry breaking (soft Pomeron).

The purpose of this section is to show that the NG string which is non-
critical, allows for the description of the elastic dipole-dipole scattering
amplitude using $1/N_{c}$ counting rules, already in 4 dimensions. The result
is a soft Pomeron with parameters that are distinct from the holographic
results. The scattering construction parallels that of the dipole-dipole
interaction, showing the inter-connecteness of the potential and scattering
problems. The hard Pomeron can also be retrieved, by allowing the tachyon mode
in the NG string to diffuse both in transverse and longitudinal size, a point
inspired by holography Stoffers and Zahed (2013). Since the NG string provides
the closest description of the QCD string potential, it should prove relevant
for the QCD scattering amplitude in the eikonal limit. In particular, a more
transparent approach to the unitarization of the cross section, as well as the
partonic string bit content and entanglement, are seen to emerge.

The present construction using a NG string in $D=4$ dimensions is the closest
to the holographic construction using $D>4$ presented in Stoffers and Zahed
(2013); Basar  _et al._ (2012), with an essential difference in the Pomeron
intercept. The twisted dipole-dipole correlator follows from the reduction of
the eikonal QCD scattering amplitude, and then evaluated using the closed and
non-critical NG string in flat $D=4$ dimensions. The exchange is solely in the
confining regime of QCD. It differs from the holographic Pomeron discussed in
Polchinski and Strassler (2003), by the value of its intercept, the coupling
to the sourcing dipoles, and the absence of a conformal limit. This can
remedied as we discuss below.

###  III.1 Scattering amplitude

In Euclidean signature, the scattering between two dipoles follows the same
analysis as that for the potential between two static dipoles presented above,
with one difference: the dipoles are not palarell but slated at an angle
$\theta$. This angle maps by analytical continuation to the rapidity or boost
angle $\chi$, thanks to Minkowski historical observation. With this in mind, a
rerun of the preceding arguments gives for the twisted worldsheet propagator
Basar  _et al._ (2012)

$\mathbb{K}(T,\theta)=\frac{a^{2}}{\alpha^{\prime}}\frac{e^{-\frac{\sigma_{T}}{2}Tb^{2}}}{2\sinh\left(\frac{\theta
T}{2}\right)}\prod_{n=1}^{\infty}\prod_{s=\pm}\frac{\sinh\left(\frac{n\pi
T}{2}\right)}{\sinh\left[\frac{T(n\pi+s\theta)}{2}\right]}\left[\prod_{n=1}^{\infty}2\sinh\left(\frac{n\pi
T}{2}\right)\right]^{-D_{\perp}}\ .$ (30)

The details regarding the twisted string worldsheet mode decomposition
analogous to (II.2), followed by the detailed mode integration leading to (30)
are given in Basar  _et al._ (2012). The double analytical continuation
$T\rightarrow iT$ and $\theta\rightarrow-i\chi$, maps this twisted dipole-
dipole correlator onto the scattering amplitude of two light-like Wilson
loops. With this in mind, inserting (30) in the corresponding ${\bf WW}$ 2PI
correlator and using (12,13), yield Basar  _et al._ (2012)

$\displaystyle\mathbb{WW}_{\rm 2PI}(\chi,a,b)$ $\displaystyle=$
$\displaystyle\frac{g_{s}^{2}a^{2}}{4\alpha^{\prime}}\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}e^{-k\frac{\pi\sigma_{T}\mathbb{b}^{2}}{\chi}}\eta^{-D_{\perp}}\bigg{(}\frac{ik\pi}{\chi}\bigg{)}$
$\displaystyle=$
$\displaystyle\frac{g_{s}^{2}a^{2}}{4\alpha^{\prime}}\sum_{k=1}^{\infty}\sum_{n=0}^{\infty}\,d(n)\,\frac{(-1)^{k}}{k}\bigg{(}\frac{k\pi}{\chi}\bigg{)}^{\frac{D_{\perp}}{2}}{\rm
exp}\bigg{(}-\frac{2\chi}{k}\bigg{[}n+\frac{\mathbb{b}^{2}}{\alpha^{\prime}(2\chi/k)^{2}}-\frac{D_{\perp}}{24}\bigg{]}\bigg{)}\
,$

after analytical continuation, with $\chi\approx\ln(\alpha^{\prime}s)$
identified as the rapidity, for large invariant mass $\sqrt{s}$. The
scattering amplitude in momentum space is

$\displaystyle\frac{1}{-2is}\mathcal{T}_{DD}(\chi,q)$ $\displaystyle\approx$
$\displaystyle\int
d^{2}\mathbb{b}~{}e^{i\mathbb{q}_{\perp}\cdot\mathbb{b}}\,\mathbb{WW}_{\rm
2PI}(\chi,a,b)$ $\displaystyle\approx$
$\displaystyle\frac{\pi^{2}g_{s}^{2}a^{2}}{2}\sum_{n=0}^{\infty}\sum_{k=1}^{\infty}d(n)\frac{(-1)^{k}}{k}\left(\frac{k\pi}{\chi}\right)^{\frac{D_{\perp}-2}{2}}\exp\left(-\frac{2\chi}{k}\left[n+\frac{\alpha^{\prime}}{4}{\bf
q}_{\perp}^{2}-\frac{D_{\perp}}{24}\right]\right)\ .$

Again, $k$ sums over the N-ality with $k=1,..,[\frac{N_{c}}{2}]$ all the way
to infinity at large $N_{c}$. In our case, only the $k=1$ term contributes to
the scattering of two dipoles as twisted Wilson loops, in the fundamental
representation of SU$(3_{c})$. With this in mind, and in the large rapidity
limit, (III.1) simplifies

$\displaystyle\mathcal{T}_{DD}(\chi,q)\approx is\,(\pi
g_{s}a)^{2}\,\left(\frac{\pi}{\chi}\right)^{\frac{D_{\perp}}{2}-1}\exp\left(-\chi\left[\frac{\alpha^{\prime}}{2}\bigg{(}{\bf
q}_{\perp}^{2}+M_{0}^{2}\bigg{)}\right]\right)\ ,$ (33)

with the tachyon squared mass $M_{0}^{2}=-\frac{D_{\perp}}{6\alpha^{\prime}}$.

The closed string exchange amounts to a Pomeron exchange, with a Regge
trajectory

$\displaystyle\alpha_{\mathbb{P}}(t)=\frac{D_{\perp}}{12}+\frac{\alpha^{\prime}}{2}t\
,$ (34)

hence a dipole-dipole (hadron-hadron) scattering amplitude that rises as
$\sigma_{\mathbb{P}}(s)\sim s^{\alpha_{\mathbb{P}}(t)}$. In the Regge limit
with $-t\ll s$, this amplitude is dominated by a single NG string exchange,
given by

$\displaystyle{\cal A}(s,t)\sim-2is\int d^{2}be^{iq\cdot b}\,{\bf WW}_{\rm
2PI}(s,a,b)\sim is^{1+\alpha_{\mathbb{P}}(t)}\ ,$ (35)

with $t=-q^{2}$.

###  III.2 Cross section and Froissart bound

The elastic scattering amplitude (36) yields the total cross section

$\displaystyle\sigma(s)=\frac{1}{s}{\rm Im}\,{\cal A}(s,0)\sim-2\int
d^{2}b\,{\bf WW}_{\rm 2PI}(s,a,b)\sim s^{\alpha_{\mathbb{P}}(0)}\ ,$ (36)

by the optical theorem. (36) increases with the squared invariant mass $s$, in
violation of unitarity. This shortcoming can be addressed by noting that
$\langle{\bf WW}\rangle$ as a correlator of two Wilson loops, requires the
exponentiation of all the 2PI contributions in leading order in
$\frac{1}{N_{c}}$, much like in the potential between the two static dipoles
discussed earlier

$\displaystyle{\bf WW}(\chi,a,b)=\frac{\langle{\bf W}_{\frac{\chi}{2}}(a,{\bf
b}_{\perp}){\bf W}_{-\frac{\chi}{2}}(a,{\bf 0}_{\perp})\rangle}{\langle{\bf
W}\rangle\langle{\bf W}\rangle}=\exp\bigg{[}{\bf WW}_{\rm
2PI}(\chi,a,b)\bigg{]}\ ,$ (37)

${\bf WW}_{\rm 2PI}$ is the 2PI “web” contributions Gatheral (1983); Frenkel
and Taylor (1984), which is dominated by a string exchange with genus 2 as
detailed above, with higher geni contributions suppressed by powers of
$g_{s}^{2}\sim 1/N^{2}_{c}$. Since ${\bf WW}\equiv{\cal S}$ identifies with
the full $S$-matrix, and using ${\cal S}=1+i{\cal T}$ as detailed in Appendix
A, we obtain

$\displaystyle\sigma(s)=2\int d^{2}b\ {\rm Re}\big{(}1-{\cal
S}(\chi,a,b)\big{)}=2\int d^{2}b\,\big{(}1-e^{{\bf WW}_{\rm
2PI}(\chi,a,b)}\big{)}\ ,$ (38)

since ${\bf WW}_{\rm 2PI}(\chi,a,b)$ is real. To proceed, it is useful to
recast the tachyon contribution in (III.1) in impact parameter space, as
follows Basar  _et al._ (2012)

$\displaystyle\mathbb{WW}_{\rm
2PI}(\chi,a,b)\approx-\frac{g_{s}^{2}a^{2}}{4\alpha^{\prime}}\bigg{(}\frac{\pi}{\chi}\bigg{)}^{\frac{D_{\perp}}{2}}\,e^{-S_{\rm
cl}-S_{\rm 1loop}}\ ,$ (39)

The first contribution in the exponent of (39)

$\displaystyle S_{\rm cl}=\sigma_{T}\int_{0}^{T_{P}}{\rm
cos}^{2}(\chi\tau)\,bd\tau\int_{0}^{1}bd\sigma=\frac{1}{2}\sigma_{T}\beta
b=\frac{b^{2}}{2\alpha^{\prime}\chi}\ ,$ (40)

is identified with a semi-classical worldsheet instanton, with a tunneling
time $T_{P}=1/\beta=2\pi b/\chi$. The second contribution in the exponent of
(39)

$\displaystyle S_{\rm 1loop}=\frac{D_{\perp}}{2}{\rm
lndet}(-\partial_{\perp}^{2})=-\frac{\pi
D_{\perp}}{6}\frac{b}{\beta}=-\frac{D_{\perp}}{12}\chi\ ,$ (41)

is the 1-loop zeta regulated corrective action, around the worldsheet
instanton.

Figure 1: Total cross section in mb versus $\sqrt{s}$ in GeV: solid-red-lower
curve is the empirical $pp$ cross section as parametrized by the COMPETE
collaboration in Cudell  _et al._ (2002) and quoted in (45). It is in
agreement with the cross sections currently measured by the LHC Antchev  _et
al._ (2019). The solid-blue-lower curve is the NG result (44) with $a=l_{s}$
and $g_{s}=1$ and ${\cal O}$=30, with $\alpha^{\prime}=1/4m_{\rho}^{2}$, while
the solid-green-upper curve is for $\alpha^{\prime}=1/2m_{\rho}^{2}$.

Using (39-41), we now note that the integrand in (38) is controlled by the
exponent in (III.1), which is a tradeoff between the minimal worldsheet
instanton action $S_{\rm cl}$ and its 1-loop quantum correction $S_{\rm
1loop}$. The black disc radius is reached when

$\displaystyle S_{\rm cl}=|S_{\rm 1loop}|\rightarrow b_{\rm
max}^{2}=\frac{D_{\perp}\alpha^{\prime}}{6}\chi^{2}\ ,$ (42)

so that $e^{{\bf WW}_{\rm 2PI}}\rightarrow\theta(b-b_{\rm max})$. As a result
(38) yields the total cross section

$\displaystyle\sigma(s)\sim 2\int d^{2}b\,\theta(b_{\rm max}-b)\sim
2\pi\,b_{\rm max}^{2}=2\pi\alpha^{\prime}\frac{D_{\perp}\,\chi^{2}}{6}\ .$
(43)

At large rapidity, the 2PI NG contribution saturates the Froissart bound, with
a scale fixed by the string tension $\sigma_{T}=1/2\pi\alpha^{\prime}$, and
not the pion mass as suggested in Nussinov (2008); Basar  _et al._ (2012);
Kharzeev  _et al._ (2018).

More specifically, (38) evaluates exactly to

$\displaystyle\sigma(s)=2\pi\alpha^{\prime}\bigg{(}\frac{D_{\perp}\chi^{2}}{6}-\chi\ln(D_{\perp}\chi)+\left(\frac{a^{2}g_{s}^{2}\pi}{4\alpha^{\prime}}\chi+\gamma_{E}\right)+{\cal
O}\bigg{(}e^{-\frac{a^{2}g_{s}^{2}\pi}{4\alpha^{\prime}\chi}e^{\frac{D_{\perp}\chi}{6}}}\bigg{)}\bigg{)}\
,$ (44)

with $\chi={\rm ln}(\alpha^{\prime}s)$, in agreement with the estimate (43).
The new result (44) stemming from the NG exchange, is to be compared with the
empirical parametrization of the $pp$ data by the COMPETE collaboration Cudell
_et al._ (2002)

$\displaystyle\sigma^{pp}(s)\sim\bigg{(}35.5+0.307\,{\rm
ln}^{2}\bigg{(}\frac{s}{29.1\,{\rm GeV}^{2}}\bigg{)}\bigg{)}\,{\rm mb}\ ,$
(45)

after dropping the Reggeon contributions at large $\sqrt{s}$. In Fig. 1 we
show the NG result for the total cross section (44) for $a=l_{s}$ and
$g_{s}=1$ with ${\cal O}$=30: green-solid-upper curve with
$\alpha^{\prime}=l_{s}^{2}=1/2m_{\rho}^{2}$ (the rho meson trajectory slope),
and red-solid-curve with $\alpha^{\prime}=l_{s}^{2}=1/4m_{\rho}^{2}$ (half the
rho meson trajectory slope). The empirical parametrization (45) blue-solid-
lower curve, has been used by the COMPETE collaboration to reproduce the
compiled $pp$ and $p\bar{p}$ data, two decades ago. It is in good agreement
with the recently reported TOTEM measurements for $pp$ at the highest
$\sqrt{s}=13$ TeV at the LHC Antchev  _et al._ (2019). The NG result is mostly
sensitive to the string length, and is undistinguishable from the COMPETE
parametrization for $\alpha^{\prime}=l_{s}^{2}=1/4m_{\rho}^{2}$. Note that the
value of this parameter is tied to glueball spectrum in (18). This shows the
duality between the scattering amplitude of two dipoles, and the static
potential between two dipoles.

Figure 2: Differential cross section (46) for fixed impact parameter $b$
between two dipoles of fixed size $a=1$ and $g_{s}=1$, as it unitarizes by
shadowing, at large rapidity $\chi$. The green-solid upper curve, red-solid
middle curve and the blue-solid lower curve are for impact parameters
$b=2,4,6$, in units of the string length. The dashed line follows from the
saturation condition (55).

###  III.3 Shadowing of wee string bits

We now note that the total cross section (38) amounts to

$\displaystyle\sigma(s)\sim 2\int d^{2}b\,N(s,a,b)\ ,$ (46)

with

$\displaystyle\frac{1}{2}\frac{d\sigma}{d^{2}b}=N(s,a,b)=1-S(s,a,b)=1-e^{{\bf
WW}_{\rm 2PI}(s,a,b)}\ .$ (47)

$\frac{1}{2}$ the effective number of wee-string-bits flowing through the
cylindrical annulus $2\pi bdb$. The effective number $N$, obeys a non-linear
diffusion-like equation

$\displaystyle\bigg{(}\partial_{\xi}+M_{0}^{2}-\nabla_{\perp}^{2}\bigg{)}\ln(1-N)=0\
,$ (48)

with $\xi=\frac{1}{2}\alpha^{\prime}\chi$ playing the role of an effective
(Gribov) time, or

$\displaystyle\partial_{\xi}N-M_{0}^{2}(1-N)\ln(1-N)-\nabla_{\perp}^{2}N-\frac{(\nabla_{\perp}N)^{2}}{1-N}=0\
.$ (49)

In the small number limit $\ln(1-N)\sim-N$, one recovers the linear diffusion
equation, with non-linear corrections for larger $N$, that cause saturation
asymptotically. For instance, in the quadratic approximation, the non-linear
evolution is given by

$\displaystyle\bigg{(}\partial_{\xi}+M_{0}^{2}-\nabla_{\perp}^{2}\bigg{)}N-\frac{M_{0}^{2}}{2}N^{2}-(\nabla_{\perp}N)^{2}+{\cal
O}(N^{3})=0\ ,$ (50)

which is reminiscent of the non-linear Gribov-Levin-Ryskin equation, for the
unintegrated gluon distribution Gribov  _et al._ (1983).

In Fig. 2 we show the behavior of the integrand in the total cross section
(46) versus the rapidity, for three different values of the impact parameter
$b$. The green-solid upper curve, red-solid middle curve and the blue-solid
lower curve are for impact parameters $b=2,4,6$, in units of the string
length. We have set the string coupling $g_{s}=1$, and the static dipole sizes
$a=1$ in units of the string length. The dependence on the impact parameter is
mild.

###  III.4 DIS view of wee string bits

In the stringy approach to the Pomeron and unitarization, the picture of a
hadron at large rapidities or small-x, is different from that following from
pQCD, where a hadron at large rapidity $\chi$ preserves its transverse size,
and shrinks its longitudinal size by the gamma factor
$\gamma=e^{\frac{1}{2}\chi}$. In contrast, when a string is exchanged, the
hadron transverse size grows logarithmically as $|\Delta
x_{\perp}|\sim\sqrt{\chi\alpha^{\prime}}$, while its light front longitudinal
size grows parametrically as $|\Delta
x^{-}|\sim\chi^{0}\alpha^{\prime}/0^{+}$, with $0^{+}$ the time resolution in
the light front coordinate $x^{+}$ Susskind (1994) (note that $\Delta
x^{-}\Delta x^{+}\sim\alpha^{\prime}$ by the uncertainty principle). Parton as
wee string bits do not behave as normal matter under Lorentz boost.

The number of wee-string-bits grows exponentially with ${\bf WW}_{\rm 2PI}\sim
e^{\alpha_{\mathbb{P}}\chi}$. This growth is similar to the growth of the
longitudinal light front momentum $P^{+}\sim\gamma\sim e^{\frac{1}{2}\chi}$,
of the boosted hadron. The string growth persists, even though the total cross
section saturates by quantum shadowing, with $\rho(\chi)$ the number of string
bits per light front volume $|\Delta x_{\perp}||\Delta x^{-}|$,

$\displaystyle{\bf
diffusive\,\,regime:}\,\,b_{\perp}\sim\sqrt{\alpha^{\prime}\chi}$
$\displaystyle\rho(\chi)\sim\frac{e^{\alpha_{\mathbb{P}}\chi}}{\chi\alpha^{\prime
2}/0^{+}}\ ,$ $\displaystyle{\bf
ballistic\,\,regime:}\,\,b_{\perp}\sim\sqrt{\alpha^{\prime}}\chi$
$\displaystyle\rho(\chi)\sim\frac{1}{\chi^{2}\alpha^{\prime 2}/0^{+}}\ .$ (51)

An illustration of this spatial growth under boosting of the nucleon is shown
in Fig. 3. The ballistic regime dominates the total cross section.

Figure 3: Boosted nucleon with large longitudinal momentum $P^{+}\sim
e^{\frac{1}{2}\chi}$. The confined quark-diquark pair is highly contracted,
with a vanishingly small longitudinal size $1/P^{+}$, and fixed transverse
size $\chi^{0}$. It is surrounded by a halo of partons as string bits, which
extends transversely as $\sqrt{\chi}$ (diffusive regime) and up to $\chi$
(ballistic regime). The halo remains parametrically large longitudinally as
$\chi^{0}/0^{+}$, with $0^{+}$ the time resolution along $x^{+}$ Susskind
(1994). Throughout, it is described by a continuous NG string with
longitudinal momentum $P^{+}$. All dimensions are in string units.

These features are accessible to DIS scattering at large $Q^{2}/m_{H}^{2}\gg
1$ and small parton faction $x\ll 1$, where the virtual photon can be viewed
as a small projectile dipole of size $a_{P}\sim 1/\sqrt{Q^{2}}$, scattering
off a target hadron also as a dipole of a larger size $a_{T}$, as illustrated
in Fig. 4. We recall the DIS kinematics

$s-m_{H}^{2}={Q^{2}}\left(\frac{1}{x}-1\right)$

with the identification $\chi\rightarrow{\rm ln}{\frac{1}{x}}$. For a fixed
target size, (39) translates to

$\displaystyle{\bf WW}_{\rm 2PI}(Q^{2},x,b)\sim
g_{s}^{2}\frac{1}{\sqrt{\alpha^{\prime}Q^{2}}}\frac{1}{x^{{\alpha_{\mathbb{P}}(0)}}}e^{-\frac{{\bf
b}_{\perp}^{2}}{2\alpha^{\prime}{\rm
ln}\frac{1}{x}}}\sim\left(\frac{Q^{2}(x)}{Q^{2}}\right)^{\frac{1}{2}}e^{-\frac{{\bf
b}_{\perp}^{2}}{2\alpha^{\prime}{\rm ln}\frac{1}{x}}}\ ,$ (52)

after re-insertion of the pre-exponent, with

$\displaystyle
Q(x)=\frac{g^{2}_{s}}{\sqrt{\alpha^{\prime}}}\frac{1}{x^{{\alpha_{\mathbb{P}}(0)}}}\
.$ (53)

The arguments presented in Appendix A show that the $F_{2}$ structure function
is

$\displaystyle F_{2}(x,Q^{2})\sim
xG_{\mathbb{P}}(x,Q^{2})\sim\left(\frac{Q^{2}(x)}{Q^{2}}\right)^{\frac{1}{2}}\
,$ (54)

at low-x, where $Q(x)$ maybe regarded as the stringy analogue of the so-called
saturation momentum, with differences with the original proposal by Golec-
Biernat-Wusthoff (GBF) Golec-Biernat and Wusthoff (1999). The standard
condition for saturation is set by the requirement that
${S}(s,a,b)|_{S}=e^{-\frac{1}{2}}$ in (47) (a drop by one standard deviation
in the GBF Gaussian proposal for $S$), or equivalently

$\displaystyle\frac{d\sigma}{d^{2}b}\bigg{|}_{S}=2(1-e^{-\frac{1}{2}})=0.79\qquad\longrightarrow\qquad
14<\chi_{S}={\rm ln}\frac{1}{x_{S}}<20\ .$ (55)

The rightmost result follows numerically from the black-dashed curve in Fig.
2, for the impact parameter $b$ in the range $2<b/l_{s}<6$. This stringy
estimate puts a lower bound on parton-x at saturation $x_{S}>10^{-6}$, which
falls outside the reach of current colliders, including the future EIC. This
conclusion is specific to the Pomeron as the exchange of non-critical NG
string in $D=4$ dimensions.

Figure 4: DIS scattering as two Wilson loops ${\bf W}_{D}$ exchanging a closed
NG string, in the Regge limit.

Finally, (52) may be viewed as the number of wee string bits with parton-x, at
a distance $b_{\perp}=\sqrt{{\bf b}_{\perp}^{2}}$ in the transverse plane,
surrounding a fast moving hadron sourced by a fixed size dipole. The number is
small for $Q^{2}\gg Q^{2}(x)$, whatever $b_{\perp}$. It is large for $Q^{2}\ll
Q^{2}(x)$, only in the disc ${{b}_{\perp}}\sim\sqrt{\alpha^{\prime}{\rm
ln}\frac{1}{x}}$, which is seen to grow diffusively in the immediate
surrounding of the target dipole. It drops substantially in the much wider
corona ${{b}_{\perp}}\sim\sqrt{\alpha^{\prime}}{\rm ln}\frac{1}{x}$, where the
growth is ballistic.

##  IV Entanglement in scattering

In the large rapidity limit, (III.1) is dominated by the tachyon contribution
in the closed string exchange described by the Nambu-Goto string. The tachyon
as a mode encodes the quantum entanglement between the projectile and the
target, carried geometrically by the worldsheet. A way to quantify this, is to
recast (III.1) in the form (41). This is readily identified as the free energy
of $D_{\perp}$ massless bosons trapped in a box of size $b$ at temperature
$1/\beta$ as illustrated in Fig. 5, with Liu and Zahed (2019)

$\displaystyle S_{\rm 1loop}=\beta F_{B}=D_{\perp}\int\frac{bdp}{2\pi}\,{\rm
ln}\bigg{(}1-e^{-\beta|p|}\bigg{)}\ .$ (56)

As a result, the exchange in Fig. 5 carries a quantum or entanglement bosonic
entropy

$\displaystyle S_{EB}=\beta^{2}\frac{\partial
F_{B}}{\partial\beta}=D_{\perp}\int\frac{bdp}{2\pi}\,\frac{2\beta|p|}{e^{\beta|p|}-1}=\frac{D_{\perp}}{6}\chi=2\alpha_{\mathbb{P}}(0)\chi\
.$ (57)

Eq. (57) clearly captures the entropy of fluctuating string bits (gluonic
dipoles) on the instanton worldsheet of size $\beta\times b$.

###  IV.1 Fermionic contribution to the Pomeron intercept and DIS

The fermionic correction to (57) follows immediately from this physical
observation, as $n_{f}$ massless worldsheet fermions also trapped in
$\beta\times b$ as illustrated in Fig. 5, with the result

$\displaystyle\beta F_{F}=n_{f}\int\frac{bdp}{2\pi}\,{\rm
ln}\bigg{(}1+e^{-\beta|p|}\bigg{)}\ ,$ (58)

in total analogy with (56). The corresponding quantum or entanglement
fermionic entropy is then

$\displaystyle S_{EF}=\beta^{2}\frac{\partial
F_{F}}{\partial\beta}=n_{f}\int\frac{bdp}{2\pi}\,\frac{2\beta|p|}{e^{\beta|p|}+1}=\frac{1}{2}\frac{n_{f}}{6}\chi\
,$ (59)

with a net entanglement entropy

$\displaystyle
S_{EE}=S_{EB}+S_{EF}=\bigg{(}1+\frac{1}{2}\frac{n_{f}}{{D_{\perp}}}\bigg{)}\frac{D_{\perp}}{6}\chi\
,$ (60)

Figure 5: String worldsheet exchange $\beta\times b$ in the Regge limit. The
transverse fluctuations $x_{\perp}^{i}$ with $i=1,..,D_{\perp}$, and the
$n_{f}$ massless fermions $q$, are subject to periodic boundary conditions in
$\beta$.

The result (60) implies that the stringy Pomeron intercept is affected by the
fermionic corrections on the worldsheet. More specifically, the Pomeron
contribution in hadron-hadron scattering is modified, with a shifted intercept

$\displaystyle\alpha_{\mathbb{P}}(t)=\frac{D_{\perp}}{12}+\frac{\alpha^{\prime}}{2}t\rightarrow\tilde{\alpha}_{\mathbb{P}}(t)=\bigg{(}1+\frac{1}{2}\frac{n_{f}}{{D_{\perp}}}\bigg{)}\frac{D_{\perp}}{12}+\frac{\alpha^{\prime}}{2}t\
.$ (61)

due to the fermionic contribution.

Also in DIS, we can re-interpret the gluonic $F_{2}$ structure function (54)
at low-x as

$\displaystyle F_{2}(x,Q^{2})\sim
xG_{\mathbb{P}}(x,Q^{2})\sim\frac{1}{x^{\tilde{\alpha}_{\mathbb{P}(0)}}}\ ,$
(62)

at the resolution fixed by the probing dipole size $Q\sim 1/a$. Note that (60)
is still solely given by the gluon density (62) at low-x

$\displaystyle S_{EE}(x,Q^{2})\sim{\rm ln}(xG_{\mathbb{P}}(x,Q^{2})\ ,$ (63)

albeit with a fermion corrected gluonic intercept. At low-x, the partonic
evolution does not follow from DGLAP, but rather BFKL (weak coupling) or
surfaces (strong coupling). An alternative proposal to account for the
fermionic contribution to the entanglement entropy at low-x was suggested in
Kharzeev and Levin (2021); Hentschinski and Kutak (2022); Hentschinski  _et
al._ (2022).

Figure 6: The entanglement entropy between two light-light scattering dipoles
in the Regge limit, versus the rapidity $\chi$, following from the 2PI NG
string contribution to the total cross section shown in Fig. 6. The green-
solid lower curve, red-solid middle curve and blue-solid upper are for impact
parameters $b=2,4,6$, for a pair of dipoles of fixed size $a=1$, in units of
the string length. The rapid and linear rise in the entanglement entropy with
rapidity, is stopped and reversed by quantum shadowing. It levels off
asymptotically when the Froissart bound is reached.

###  IV.2 Entanglement and Froissart bound

Finally, we suggest that in Reggeized hadron-hadron scattering at the
Froissart bound, the entanglement entropy saturates by quantum shadowing, even
though the entanglement entropy as measured by (63) in DIS does not. For that,
we interpret the 2PI stringy exchanges in the shadowing process in (38), as a
net quantum free energy

$\displaystyle F_{\rm 2PI}=-\frac{1}{\beta}\,{\rm ln}\bigg{(}2\big{(}1-e^{{\bf
WW}_{\rm 2PI}}\big{)}\bigg{)}\ .$ (64)

Note that it reduces to the stringy free energy for small ${\bf WW}_{\rm 2PI}$
and large rapidity. Hence the 2PI quantum or entanglement entropy

$\displaystyle S_{\rm 2PI}=\beta^{2}\frac{\partial F_{\rm
2PI}}{\partial\beta}={\rm ln}\big{(}1-e^{{\bf WW}_{\rm
2PI}}\big{)}+\beta\frac{\partial_{\beta}{\bf WW}_{\rm 2PI}}{1-e^{-{\bf
WW}_{\rm 2PI}}}+{\rm ln}2\rightarrow{\rm ln}2\ ,$ (65)

which is seen to asymptote a constant for fixed $b$ and large rapidity
$\chi\gg 1$. In the unitarity limit, the entanglement is that of a single
qubit! Recall that in the black disc limit, the scattering choice appears to
be binary, as the elastic and inelastic cross sections are equal to the
classical cross section (Babinet theorem).

In Fig. 6 we show the entanglement entropy versus rapidity, using our proposal
(64) for the 2PI contribution, for different values of the impact parameter
$b$. The green-solid lower curve, red-solid middle curve and blue-solid upper
curve, are for impact parameters $b=2,4,6$ and fixed dipole size $a=1$, all
units of the string length. We have fixed the string coupling $g_{s}=1$. The
rapid initial rise with rapidity, is caused by the NG tachyon in the single
string exchange (60). This rise overshoots the unitarity line, before it is
overtaken by quantum shadowing in (65), to level off at the Froissart bound.
This levelling off is generic of chaotic systems in their approach to
equilibrium Latora and Baranger (1999); Liu and Zahed (2019), although here
from above and not below.

##  V NG tachyon diffusion in a confining warped space

The scattering amplitude of two fixed dipoles of size $a$ in (33), is
dominated by the exchange of the NG tachyon

$\displaystyle\exp\left(-\chi\left[\frac{\alpha^{\prime}}{2}\bigg{(}{\bf
q}_{\perp}^{2}+M_{0}^{2}\bigg{)}\right]\right)\ ,$ (66)

in the transverse $D_{\perp}$-dimensions, where the rapidity $\chi$ emerges as
a proper time. This exchange is diffusive, and can be recast (66) using the
contour integral

$\displaystyle\int_{-i\infty}^{+i\infty}\frac{dj}{2i\pi}\frac{e^{\chi
j}}{j+\frac{\alpha^{\prime}}{2}(M_{0}^{2}+{\bf
q}_{\perp}^{2})}=\frac{2}{\alpha^{\prime}}\int_{-i\infty}^{+i\infty}\frac{dj}{2i\pi}{e^{\chi
j}}G(j,{\bf q}_{\perp})\ .$ (67)

The propagator in the complex j-plane satisfies

$\displaystyle({\bf q}_{\perp}^{2}+m_{j}^{2})\,G(j,{\bf q}_{\perp})=1\ .$ (68)

with the j-dependent mass $m_{j}^{2}=\frac{2}{\alpha^{\prime}}(j-j_{0})$ and
$j_{0}=\frac{D_{\perp}}{12}$. This stringy result captures the exchange of an
emergent spin-j in the Regge limit, for fixed momentum transfer ${\bf
q}_{\perp}$, sourced by a projectile and a target dipole of fixed sizes (both
set to $a$).

###  V.1 Warped diffusion

In QCD a boosted hadron is a collection of wee dipole of various sizes as we
discussed earlier. The scattering between these dipoles is mediated by a non-
critical NG string. Large size wee dipoles scattering mostly in the IR,
through the exchange of a closed NG string in the confining regime. Small size
dipoles scatter mostly in the conformal regime, albeit at strong coupling in
our case. The chief results are different transverse growth size of the
string, at the origin of the transverse size dependence of the saturation
scale: 1/ in the confining regime the growth is linear in the root of the
rapidity as in (52); 2/ in the conformal regime, the growth is exponential in
the rapidity as we show below.

With this in mind, the sourcing dipole sizes vary as well, with the combined
change in $a,b_{\perp}$, expected to be conformal in the UV, and stringy in
the IR. To realize this, we rewrite (68) in a general coordinate space

$\displaystyle-\frac{1}{\sqrt{|g|}}\partial_{\mu}(\sqrt{|g|}g^{\mu\nu}\partial_{\nu}G)+m_{j}^{2}G=\frac{1}{\sqrt{|g|}}\delta(z-z^{\prime})\delta^{D-1}(\vec{x}-\vec{x}^{\prime})\
,$ (69)

by combining $(a,{\bf b}_{\perp})\rightarrow(z,\vec{x})=x^{\mu}$ in
$D=1+D_{\perp}$ space. But what is the metric $g_{\mu\nu}$ when $z$ is added
as a coordinate? (69) can be viewed as an evolution equation in our QCD
analysis of the scattering amplitude, with the evolution taking place in
$z\sim 1/\sqrt{Q^{2}}$ and rapidity $\chi$. Hence, the metric should exhibit
conformal symmetry for small $z$. Inspired by holography, we fix $g_{\mu\nu}$
through the line element

$\displaystyle
ds^{2}=\frac{R^{2}}{z^{2}}e^{\mp\kappa^{2}z^{2}}(dz^{2}+d^{2}{x}_{\perp})\ .$
(70)

For small size dipoles $z\rightarrow 0$, (70) reduces to that of AdS3 which is
conformal. For large size dipoles $z\rightarrow 1/\kappa$ as expected from
confinement for both warping signs. Although (70) is reminiscent of the
holographic analysis of the Pomeron in AdS${}_{5}\times$S5 Polchinski and
Strassler (2003, 2003), we emphasise that the present construction is not
holographic. The starting point is the NG string in flat space with a tachyon
for $D_{\perp}=2$, with no reference to type IIB string theory in
10-dimensions with no tachyon Ammon and Erdmenger (2015) (and references
therein). As we noted earlier, the NG string in 4-dimension is the only
effective string model currently supported by QCD lattice simulations. We
define

$\displaystyle\frac{R^{2}}{\alpha^{\prime}}\equiv\sqrt{\lambda}\ ,$ (71)

with $\kappa,R$ to be tied below. Since our approach is not holographic, the
identification $\lambda=g_{Y}^{2}N_{c}$ does not follow. However, it is
natural to expect that $R/l_{s}\gg 1$, since $R$ is the radius of the
hyperbolic space, where the warped evolution of the NG tachyon is justified
for large transverse separations.

With this in mind, and using (70) in (69), we obtain

$\displaystyle-\partial_{z}^{2}G(z,z^{\prime},t)+(D-2)\bigg{(}\frac{1}{z}\pm\kappa^{2}z\bigg{)}\partial_{z}G(z,z^{\prime},t)$
$\displaystyle+\bigg{(}t+\frac{S}{z^{2}}e^{\mp\kappa^{2}z^{2}}\bigg{)}G(z,z^{\prime},t)=z^{D-2}e^{\pm(\frac{D}{2}-1)\kappa^{2}z^{2}}\,\delta(z-z^{\prime})\
.$ (72)

with $\partial_{\perp}^{2}\rightarrow t$ and

$\displaystyle S\equiv S_{j}=2\sqrt{\lambda}(j-j_{0})\ .$ (73)

To remove the first order derivative, we redefine

$\displaystyle G(z,z^{\prime},t)\rightarrow
z^{\frac{D-2}{2}}e^{\pm\frac{D-2}{4}\kappa^{2}z^{2}}z^{{}^{\prime}\frac{D-2}{2}}e^{\pm\frac{D-2}{4}\kappa^{2}z^{{}^{\prime}2}}G(z,z^{\prime},t)\
,$ (74)

set $u=\kappa z$, expand $e^{\mp u^{2}}=1\mp u^{2}+{\cal O}(u^{4})$ (moderatly
small size dipoles) to obtain

$\displaystyle-\frac{d^{2}}{du^{2}}G(u,u^{\prime},t)+$
$\displaystyle\bigg{(}\frac{S_{j}+\frac{D(D-2)}{4}}{u^{2}}+\frac{u^{2}}{4}(D-2)^{2}+\frac{t}{\kappa^{2}}\mp
S_{j}\pm\frac{1}{2}(D-2)(D-3)\bigg{)}\ G(u,u^{\prime},t)=\delta(u-u^{\prime})\
.$

For $D=1+D_{\perp}=3$, we have explicitly

$\displaystyle-\frac{d^{2}}{du^{2}}G(u,u^{\prime},t)+\bigg{(}\frac{S_{j}+\frac{3}{4}}{u^{2}}+\frac{u^{2}}{4}+\frac{t}{\kappa^{2}}\mp
S_{j}\bigg{)}G(u,u^{\prime},t)=\delta(u-u^{\prime})\ .$ (76)

###  V.2 Repulsive warping

The repulsive warping with $e^{+\kappa^{2}z^{2}}$ acts as absolute confinement
for the dipole sizes in hyperbolic space, characterizing the evolution. (In
holography, it is a regulated hard wall in bulk AdS). In this case the linear
and homogeneous equation (76) becomes

$\displaystyle-\frac{d^{2}}{du^{2}}G(u)+\bigg{(}\frac{S+\frac{D(D-2)}{4}}{u^{2}}+\frac{u^{2}}{4}(D-2)^{2}+\frac{t}{\kappa^{2}}+S-\frac{1}{2}(D-2)(D-3)\bigg{)}G(u)=0\
.$ (77)

To simplify the equation, we consider $u\rightarrow
u^{\prime}=\sqrt{D-2}u=\sqrt{D-2}\kappa z$ , for which (77) reads

$\displaystyle-\frac{d^{2}}{du^{2}}G(u)+\bigg{(}\frac{S+\frac{D(D-2)}{4}}{u^{2}}+\frac{u^{2}}{4}+\tilde{t}+\tilde{S}-\frac{1}{2}(D-3)\bigg{)}G(u)=0\
.$ (78)

With

$\displaystyle D_{\perp}=D-1\ ,\ \tilde{t}=\frac{t}{(D-2)\kappa^{2}}\ ,\
\tilde{S}=\frac{S}{D-2}\ .$ (79)

the general solutions are of the form

$\displaystyle G_{1}(u)$ $\displaystyle=$ $\displaystyle
e^{-\frac{u^{2}}{4}}u^{1+\sqrt{\frac{D_{\perp}^{2}}{4}+(D_{\perp}-1)\tilde{S}}}\,{\mathbb{M}}\left(\frac{2-\frac{D_{\perp}}{2}+\tilde{S}+\tilde{t}+\sqrt{\frac{D_{\perp}^{2}}{4}+(D_{\perp}-1)\tilde{S}}}{2},1+\sqrt{\frac{D_{\perp}^{2}}{4}+(D_{\perp}-1)\tilde{S}},\frac{u^{2}}{2}\right)\
,$ $\displaystyle G_{2}(u)$ $\displaystyle=$ $\displaystyle
e^{-\frac{u^{2}}{4}}u^{1+\sqrt{\frac{D_{\perp}^{2}}{4}+(D_{\perp}-1)\tilde{S}}}\,{\mathbb{U}}\left(\frac{2-\frac{D_{\perp}}{2}+S+\tilde{t}+\sqrt{\frac{D_{\perp}^{2}}{4}+(D_{\perp}-1)\tilde{S}}}{2},1+\sqrt{\frac{D_{\perp}^{2}}{4}+(D_{\perp}-1)\tilde{S}},\frac{u^{2}}{2}\right)\
,$

where ${\mathbb{M}}(a,b,z)$ and ${\mathbb{U}}(a,b,z)$ are the Kummer
${\mathbb{M}}$ function and the Tricomi ${\mathbb{U}}$ function, respectively.
${\mathbb{M}}$ is regular at $u=0$, while ${\mathbb{U}}$ has a branch cut at
$u=0$. The solution to (76) reads

$\displaystyle G(u,u^{\prime})$ $\displaystyle=$ $\displaystyle{\cal
A}\,G_{2}(u)G_{1}(u^{\prime})\qquad\qquad u>u^{\prime}\ ,$ $\displaystyle
G(u,u^{\prime})$ $\displaystyle=$ $\displaystyle{\cal
A}\,G_{1}(u)G_{2}(u^{\prime})\qquad\qquad u<u^{\prime}\ .$ (81)

with ${\cal A}$ given by the Wronskian

$\displaystyle{\cal A}=\frac{1}{{{\cal
W}(G_{1},G_{2})}}\sim-{\Gamma\left(\frac{2-\frac{D_{\perp}}{2}+\tilde{t}+\tilde{S}+\sqrt{(D_{\perp}-1)\tilde{S}+\frac{D_{\perp}^{2}}{4}}}{2}\right)}\
.$ (82)

###  V.3 Reggeized trajectories

${\cal A}$ has poles when

$\displaystyle\frac{\tilde{t}+\tilde{S}+2-\frac{D_{\perp}}{2}+\sqrt{(D_{\perp}-1)\tilde{S}+\frac{D_{\perp}^{2}}{4}}}{2}=-n\
.$ (83)

or squared masses $|t(j,n)|$ given by

$\displaystyle|\tilde{t}(j,n)|=2n+\tilde{S}+2-\frac{D_{\perp}}{2}+\sqrt{(D_{\perp}-1)\tilde{S}+\frac{D_{\perp}^{2}}{4}}\
.$ (84)

By extending the tachyon diffusion to AdS3 space, the original tachyon pole
morphs into a multitude of Regge poles,

$\displaystyle\tilde{S}=\frac{2\sqrt{\lambda}(j-j_{0})}{D_{\perp}-1}=\frac{-5+2D_{\perp}-4n+\frac{2|t|}{(D_{\perp}-1)\kappa^{2}}\pm\sqrt{(2D_{\perp}-3)^{2}-8(D_{\perp}-1)n+\frac{4|t|}{\kappa^{2}}}}{2}\
.$ (85)

For $n=0$ and small $t$, the Regge trajectories for the $\pm$ signs are

$\displaystyle j_{+}$ $\displaystyle=$ $\displaystyle
j_{0}+\frac{(D_{\perp}-1)(D_{\perp}-2)}{\sqrt{\lambda}}+\bigg{(}\frac{3D_{\perp}-4}{D_{\perp}-2}\bigg{)}\frac{\alpha^{\prime}}{2}|t|+{\cal
O}(t^{2})$ $\displaystyle j_{-}$ $\displaystyle=$ $\displaystyle
j_{0}-\frac{D_{\perp}-1}{2\sqrt{\lambda}}+\frac{\alpha^{\prime}}{2}|t|+{\cal
O}(t^{2})\ .$ (86)

provided that

$\kappa^{2}R^{2}=\frac{D_{\perp}-2}{2D_{\perp}-3}$

with $\sqrt{\lambda}=R^{2}/\alpha^{\prime}$. The dominant contribution to the
Pomeron stems from the $j_{-}$ trajectory, which is the closest to the origin
in the j-plane. The warping shifts down the flat space intercept $j_{0}$.

###  V.4 Confining regime

For large $t$ and large $n$ the poles in the j-plane become imaginary, but
with negative real parts. To proceed with the contour integration in (67) for
the tachyon propagator, we select the branch cut of $\sqrt{S+1}$ from
$-1-i\infty$ to $-1$. With this in mind, the dominant contribution stems from
the first pole $n=0$ with $j=j_{-}$. At the pole, the hypergeometric functions
in (V.2) simplify

$\displaystyle{\mathbb{M}}\bigg{(}0,1+\sqrt{S+\frac{D_{\perp}^{2}}{4}},\frac{u^{2}}{2}\bigg{)}=1$
$\displaystyle{\mathbb{U}}\bigg{(}0,1+\sqrt{S+\frac{D_{\perp}^{2}}{4}},\frac{u^{2}}{2}\bigg{)}=1$
(87)

Using (V.2,V.2,V.4) in (67) and carrying the j-integral gives the warped
tachyon propagator

$\displaystyle
G(z,z^{\prime},b_{\perp})\sim(zz^{\prime})^{D_{\perp}-\frac{1}{2}}\int\frac{d^{2}q}{(2\pi)^{2}}\,e^{\chi(j_{0}-\frac{D_{\perp}-1}{2\sqrt{\lambda}}-\frac{\alpha^{\prime}q^{2}}{2})+iq\cdot
b}$ (88)

after re-winding the re-definition (74), for large $\chi$. The result for the
warped tachyon propagator is

$\displaystyle
G(z,z^{\prime},b_{\perp})\sim(zz^{\prime})^{D_{\perp}-\frac{1}{2}}e^{\chi(j_{0}-\frac{D_{\perp}-1}{2\sqrt{\lambda}})-\frac{b^{2}_{\perp}}{2\chi\alpha^{\prime}}}$
(89)

(89) reduces to the unwarped result with a shifted down intercept.

###  V.5 Conformal regime

The effects of the warping is mostly in action away from the confining regime.
Indeed, in the conformal regime, the hypergeometric functions are limited to
small $u=\kappa z$ and large $t\gg\kappa^{2}$, which we will refer to as the
conformal limit. More specifically, this amounts to the limits

$\displaystyle\lim_{\kappa\rightarrow 0}G_{1}(u),\ G_{2}(u)\ ,$ (90)

which are not simply the $u\rightarrow 0$ limits, because
$\frac{t}{\kappa^{2}}$ in the second argument has to go to infinity. To obtain
these limits, the simplest way is to note that the differential equation (78)
reduces to

$\displaystyle-\frac{d^{2}}{dz^{2}}G(z)+\bigg{(}\frac{S+\frac{D(D-2)}{4}}{z^{2}}+t\bigg{)}G(z)=0\
,$ (91)

with two solutions

$\displaystyle\tilde{G}_{1}(z)=\sqrt{z}J_{-\sqrt{S+\frac{D_{\perp}^{2}}{4}}}(-i\sqrt{t}z)\
,$ (92)
$\displaystyle\tilde{G}_{2}(z)=\sqrt{z}Y_{-\sqrt{S+\frac{D_{\perp}^{2}}{4}}}(-i\sqrt{t}z)\
.$ (93)

Here $-\sqrt{S+\frac{D_{\perp}^{2}}{4}}$ is chosen to have a negative real
part. With this in mind, the warped tachyon propagator in the conformal limit
is of the form

$\displaystyle G(z,z^{\prime},b_{\perp})=\frac{\pi
t}{2}\sqrt{zz^{\prime}}J_{-\sqrt{S+\frac{D_{\perp}^{2}}{4}}}(-i\sqrt{t}z_{<})Y_{-\sqrt{S+\frac{D_{\perp}^{2}}{4}}}(-i\sqrt{t}z_{>})\
.$ (94)

For $b_{\perp}\gg z,z^{\prime}$, the Bessel functions reduce to

$\displaystyle J_{-\sqrt{S+\frac{D_{\perp}^{2}}{4}}}(-i\sqrt{t}z)$
$\displaystyle\rightarrow$
$\displaystyle\frac{1}{\Gamma\bigg{(}1-\sqrt{S+\frac{D_{\perp}^{2}}{4}}\bigg{)}}\bigg{(}\frac{\sqrt{t}z}{2}\bigg{)}^{-\sqrt{S+\frac{D_{\perp}^{2}}{4}}}\
,$ (95) $\displaystyle Y_{-\sqrt{S+\frac{D_{\perp}^{2}}{4}}}(-i\sqrt{t}z_{>})$
$\displaystyle\rightarrow$
$\displaystyle\cos\bigg{(}\pi\sqrt{S+\frac{D_{\perp}^{2}}{4}}\bigg{)}\Gamma\bigg{(}\sqrt{S+\frac{D_{\perp}^{2}}{4}}\bigg{)}\bigg{(}\frac{\sqrt{t}z}{2}\bigg{)}^{-\sqrt{S+\frac{D_{\perp}^{2}}{4}}}\
.$ (96)

As a result, the warped tachyon propagator in the conformal regime is
dominated by the following S-integral

$\displaystyle
G(z,z^{\prime},b_{\perp})\sim\int\frac{d^{2}q}{(2\pi)^{2}}e^{i{q}\cdot
n_{\perp}}\int
dS\exp\bigg{(}\chi\bigg{(}j_{0}+\frac{S}{2\sqrt{\lambda}}\bigg{)}+\bigg{(}\frac{1}{2}-\sqrt{\frac{D_{\perp}^{2}}{4}+S}\bigg{)}\ln\bigg{(}\frac{zz^{\prime}q^{2}}{2b_{\perp}^{2}}\bigg{)}\bigg{)}\
,$ (97)

with $n_{\perp}$ a unit vector along $q_{\perp}$ after rescaling away the
transverse momentum, and trading $j\rightarrow\tilde{S}$ using (79). The
result is

$\displaystyle G(z,z^{\prime},b_{\perp})\sim\frac{\xi}{(4\pi{\cal
D}\chi)^{\frac{3}{2}}}\exp\bigg{(}\chi\bigg{(}j_{0}-\frac{{\cal
D}D_{\perp}^{2}}{4}\bigg{)}-\frac{\xi^{2}}{4\chi\cal D}\bigg{)}\ ,$ (98)

with the conformal variable $\xi=\ln\frac{b_{\perp}^{2}}{zz^{\prime}}$ and
diffusion like constant ${\cal D}=\frac{1}{2\sqrt{\lambda}}$.

###  V.6 Cross section in conformal regime

In terms of (98), the dipole-dipole scattering amplitude in the conformal
regime is now

$\displaystyle{\bf WW}_{\rm
2PI}=-\frac{g_{s}^{2}}{4}(2\pi)^{\frac{3}{2}}G=-\frac{g_{s}^{2}\xi}{4(2{\cal
D}\chi)^{\frac{2}{3}}}e^{\tilde{j}_{0}\chi-\frac{\xi^{2}}{4\chi{\cal D}}}\ ,$
(99)

with $\tilde{j}_{0}=j_{0}-\frac{1}{4}{\cal D}D_{\perp}^{2}$. Inserting (99) in
the 2PI contribution to the total cross section (38) gives

$\displaystyle\sigma(s)=2\int d^{2}b\bigg{(}1-e^{{\bf WW}_{\rm
2PI}}\bigg{)}=2\pi zz^{\prime}\int_{0}^{\infty}d\xi
e^{\xi}\bigg{(}1-\exp\left(-\frac{g_{s}^{2}\xi}{4(2{\cal
D}\chi)^{\frac{2}{3}}}e^{\tilde{j}_{0}\chi-\frac{\xi^{2}}{4\chi{\cal
D}}}\right)\bigg{)}\ .$ (100)

For $\tilde{j}_{0}<4{\cal D}$, the integral in (100) can be undone for large
$\chi$, with the result for the total cross section in the conformal regime

$\displaystyle\sigma(s)\rightarrow 2\pi
zz^{\prime}\frac{g_{s}^{2}\sqrt{\pi}}{2\sqrt{2}}e^{\chi\bigg{(}j_{0}-{\cal
D}\bigg{(}\frac{D_{\perp}^{2}}{4}-1\bigg{)}\bigg{)}}\ .$ (101)

It grows exponentially, i.e. $\sigma(s)\sim zz^{\prime}s^{j_{0}}$ with
$j_{0}=\frac{D_{\perp}}{12}=\frac{1}{6}$ for $D_{\perp}=2$, much like the BFKL
result in pQCD, which is also conformal at weak coupling.

Modulo the NG string assignments for $j_{0}$ and ${\cal D}$, the contribution
(98) to the Reggeized scattering amplitude, and the total dipole-dipole cross
section, is analogous to the result following from Mueller′s dipole
wavefunction evolution in pQCD Mueller (1995). More importantly, the results
(89) (confining regime) and (98) (conformal regime), show that the general and
warped NG result (V.2) interpolates continuously between these two regimes in
Reggeized scattering. This point was originally made in the context of the
gravity dual construction Polchinski and Strassler (2003); Brower  _et al._
(2007), with no tachyon in bulk.

##  VI Conclusions

One of the most striking features of the detailed lattice studies of the fuzzy
QCD string, is its description as a fundamental NG string for large and even
relatively small lengths. This observation has been numerically checked in
both 3- and 4-dimensions, and for different SU(Nc) realizations. We have used
this observation, to analyze the stringy potential between a pair of static
and scattering dipoles, as well as their quantum entanglement.

The derivation of the static potential of two fixed size dipoles, follows from
the exchange of closed strings or glueballs in the quenched approximation
(large $N_{c}$ limit). Using the NG string, we have shown that this exchange
is dominated by the tachyonic mode and attractive at large distances. The
attraction persists at short distances, albeit in a power law form following
from the resummation over the string states. This change in the static
potential is captured by an emergent quantum entropy.

The derivation of the scattering amplitude for two fixed size dipoles can be
obtained using similar arguments, by noting that it follows from a potential
between two dipoles set at an angle $\theta$, which is then analytically
continued to the rapidity $i\chi$. In the Regge limit, the scattering
amplitude is totally fixed by the tachyonic mode of the NG string, as all the
excited modes are suppressed at large rapidity. There is a total parallel
between the potential channel and the scattering channel, where the 2PI
contribution is retained in both, in leading order in $1/N_{c}$. This
contribution yields unitarization by saturation of the Froissart bound.

We have extended some results regarding the quantum entanglement as captured
by the NG tachyon in dipole-dipole scattering. The inclusion of the worldsheet
fermions, modify the intercept of the stringy Pomeron, thereby changing the
quantum entanglement as measured by DIS through solely a modification of the
gluonic density. In the presence of shadowing, the quantum entanglement
entropy is depleted, and saturates at the Froissart bound.

The NG tachyon contribution to the Reggeized scattering amplitude of two
dipoles, captures two key aspects of the exchanged string: 1/ an exponential
growth in the number of string bits at the origin of the growth of the total
cross section at large rapidity prior to saturation; 2/ a diffusive spread of
the string bits in the transverse plane. It is the balance between these two
phenomena that yields saturation by quantum shadowing, as captured by the 2PI
contribution.

Perturbative QCD arguments using BFKL evolution of gluons as dipoles, have
shown that the gluon sizes evolve and that the evolution is conformal in the
UV. This aspect of QCD in the Regge limit, can be extended to the exchanged NG
string, by considering the diffusive spreading of the string bits in the
transverse plane together with the changes in the source and target dipole
sizes. In other words, the NG tachyonic mode should diffuse in conformal
3-dimensional space (transverse space plus dipole size) as opposed to simply
2-dimensional space (transverse space).

We have shown how to explicitly extend the diffusion of the NG tachyon mode in
flat transverse space, to curved AdS3 plus a repulsive wall. We have
emphasised that this approach is not holographic, since no string-gauge
duality is used. In the conformal limit, the modified NG tachyon diffusion in
proper time and curved space, yields results for the scattering amplitude of
two dipoles with evolving sizes, similar to those following from the BFKL
evolution of Mueller′s wavefunction in QCD. In the confining regime, the NG
tachyon diffusion is preserved, albeit with a shifted down intercept.

Acknowledgements

We thank Krzysztof Kutak for discussions. This work is supported by the Office
of Science, U.S. Department of Energy under Contract No. DE-FG-88ER40388 and
by the Priority Research Areas SciMat and DigiWorld under program Excellence
Initiative - Research University at the Jagiellonian University in Kraków.

##  Appendix A Relation to pQCD dipoles

There are few parallels between the stringy results we have developed, and
those established using pQCD. In particular, the 2PI string amplitude (38) at
fixed $b$, can be recast in the form

$\displaystyle 1-e^{{\bf WW}(s,a,b)}=1-{\rm
exp}\bigg{(}-\frac{g_{s}^{2}}{4}\frac{a^{2}}{\alpha^{\prime}}\,xG_{\mathbb{P}}(x,1/a^{2})S(b)\bigg{)}\
,$ (102)

with $g_{s}=f(\lambda)/N_{c}$, the Gaussian profile

$\displaystyle
S(b)=\bigg{(}\frac{\pi}{\chi}\bigg{)}^{\frac{D_{\perp}}{2}}\,e^{-S_{\rm
min}}=\bigg{(}\frac{\pi}{\chi}\bigg{)}^{\frac{D_{\perp}}{2}}\,e^{-\frac{b^{2}}{2\chi\alpha^{\prime}}}\
,$ (103)

and $\chi={\rm ln}\frac{1}{x}$ in DIS. The diffusion of the string bits in the
transverse plane is manifest in (103). (102) is very similar to the Glauber-
Mueller formula for multiple dipole-target interactions Mueller (1990)

$\displaystyle N_{\rm GM}(x_{01},x,b)=1-{\rm
exp}\bigg{(}-\frac{\lambda}{N^{2}_{c}}\frac{x_{01}^{2}}{8R^{2}}\,xG_{\rm
DGLAP}(x,4/{x_{01}^{2}})S(b)\bigg{)}\ ,$ (104)

where the dipole size is $x_{01}$ ($a$ in our case), $R$ is the radius of the
target ($\\#\sqrt{\alpha^{\prime}}$ in our case), and $S(b)$ is identified
with the dipole profile function inside the target ((103) in our case). (104)
is an extension of the original Golec-Biernat-Wusthoff formula for the
analysis of saturation in DIS at HERA Golec-Biernat and Wusthoff (1999),
through the addition of the profile function.

In general, the forward amplitude ${\cal T}(b_{\perp},\chi)$ of dipole-nucleus
scattering at impact parameter $b_{\perp}$ and rapidity $\chi$, relates to the
full S-matrix ${\cal S}(b_{\perp},\chi)$ as

$\displaystyle 1-{\cal S}({b}_{\perp},\chi)=-i{\cal T}({b}_{\perp},\chi)\ ,$
(105)

and the total cross section is given by

$\displaystyle\sigma(\chi)=2\int d^{2}b{\rm Re}\left(1-{\cal
S}({b}_{\perp},\chi)\right)\ .$ (106)

In our case or in the color glass condensate model (CGC), the full S-matrix is
approximated by the Wilson-loop average

$\displaystyle{\cal S}=\frac{1}{N_{c}}\langle{\rm
Tr}V_{\vec{x}_{0}}V^{\dagger}_{\vec{x}_{1}}\rangle_{\rm target}\ ,$ (107)

with $V_{\vec{x}}$ a Wilson line located at $\vec{x}$. If the target is
another Wilson-loop, it simply reduces to the ${\bf WW}$ corrector. In the CGC
model, this Wilson-loop average exponentiates Kovchegov and Levin (2022) (and
references therein)

$\displaystyle{\cal
S}=\exp\bigg{(}-\frac{1}{4}{x}^{2}_{\perp}Q_{s}^{2}({b}_{\perp},Y)\ln\frac{1}{|{x}_{\perp}|\Lambda}\bigg{)}\
.$ (108)

Our large ${b}_{\perp}$ result cuts off the growth by a delicate balance
between the factors $e^{\frac{D_{\perp}\chi}{12}}$ (growth of the string bits)
and $e^{-\frac{b^{2}}{2\chi\alpha^{\prime}}}$ (diffusion penalty).

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