# The CSS Hamiltonian: high energy evolution of rapidity dependent observables

Haowu Duana, Alex Kovnera and Michael Lublinskyb a Physics Department,
University of Connecticut, Storrs, CT 06269, USA
b Physics Department, Ben-Gurion University of the Negev, Beer Sheva 84105,
Israel

###### Abstract

We consider evolution of observables which depend on a small but fixed value
of longitudinal momentum fraction $x$, to high rapidity, such that $\eta>\ln
1/x$. We show that this evolution is not given by the JIMWLK (or BK) equation.
We derive the evolution Hamiltonian - $H_{CSS-JIMWLK}$ which generates this
evolution in the cases of dilute and dense projectile wave function. The two
limits yield identical results for $H_{CSS-JIMWLK}$. We show that the
resulting evolution for the gluon TMD is identical to the (double logarithmic)
perturbative Collins-Soper-Sterman evolution equation in the longitudinal
resolution parameter at a fixed and very large transverse resolution.

##  I Introduction

The study of high energy evolution of hadronic observables has been a very
lively and active research area in the QCD community. The evolution of
hadronic cross sections with energy has been formulated in several different
ways, which are fundamentally equivalent to each other: the Balitsky hierarchy
(or BK equation in its mean field incarnation) for correlators of Wilson
lines, or the JIMWLK functional evolution equation for the color field quasi
probability density functionalBalitsky:1995ub ; JIMWLK . At leading order in
$\alpha_{s}$ these have been known for over 25 years, and their NLO form has
also been derived and studied more recently NLOJIMWLK .

In all of these approaches the object of interest is a set of hadronic
observables dominated by gluons (or quarks) with lowest allowed longitudinal
momenta present in the hadronic state boosted to high rapidity. However these
do not exhaust all interesting observables.

In particular lately there has been a lot of interest of understanding the
interplay of the Sudakov suppression and high energy evolution in processes
with two very different transverse scales at high energy. One example of such
process considered recently by several groups is production of high
$p_{\perp}$ dijets with small momentum imbalance in photon (real or virtual)
hadron scattering at high energy dijet . The Sudakov suppression in such a
process is expected. Usually such a suppression can be understood as either
gluon radiation attributable to the ”hard part” of the process, in this case
emi=ssion from the dijet, or alternatively as a result of the Collins, Soper,
Sterman (CSS) evolution of the target transverse momentum distribution (TMD)
CSS . In order to understand it as latter, one needs to be able to calculate
gluon TMD at fixed longitudinal momentum fraction $x$ in the target wave
function which is evolved well past this value of $x$. This is not an
observable of the type described by JIMWLK evolution, since the gluons one is
tagging are not the softest gluons in the evolved wave function. We will call
such observables, observables of the CSS type.

Another example of an observable where not only the softest gluons are
important is multi gluon production cross section where the rapidity
differences between the tagged gluons are large Jalilian-Marian:2004vhw ;
Kovner:2006wr . In this case one has to evolve the hadronic wave function in
rapidity well past the rapidities of some of the gluons, and this part of the
evolution is not going to be JIWMLK like.

In the present paper we study the evolution of observables of this type. We
derive the analog of the JIMWLK Hamiltonian, which has to be used for evolving
observables of non-JIMWLK type, in the sense that they measure properties of
gluons at fixed, small value of $x$ which is nevertheless greater than $x$ of
the softest gluons in the evolved wave function. We call this Hamiltonian -
the CSS-JIMWLK Hamiltonian, $H_{CSS-JIMWLK}$.

We start in Sec. 2 by reviewing the derivation of the standard JIMWLK
Hamiltonian. Several such derivations exist in the literature using different
perspectives Balitsky:1995ub ; JIMWLK ; Mueller:2001uk ; ming . Here we work
with the derivation presented in a way convenient for the
generalization/modification we are seeking ming . In Sec. 3 we explain how
this derivation has to be modified if observables of the CSS-type are to be
considered and derive $H_{CSS-JIMWLK}$ both in the dense and dilute projectile
limits. Perhaps surprisingly, in these two limits the Hamiltonian turns out to
be the same when written in terms of the color charge currents of the
projectile. In Sec. 4 we show that when applied to the gluon (and quark) TMD
the evolution generated by $H_{CSS-JIMWLK}$ yields the CSS equation with
respect to longitudinal resolution with transverse resolution equal to UV
scale. Finally Sec.5 contains a short discussion of our results.

##  II The JIMWLK evolution: a recap

We start with a recap of the derivation of the JIMWLK evolution equation. In
this we mostly follow the exposition in ming . Our aim here is not to repeat
the derivation of JIMWLK for the n-th time, but rather to set the stage for
the subsequent discussion of the CSS-JIMWLK Hamiltonian.

Consider a wave function of a hadronic system, which we will refer to as ”the
projectile”. We separate the degrees of freedom of this system into ”valence”
and ”soft”, see Fig. 1. The valence degrees of freedom are the gluon fields
with longitudinal momentum greater than some cutoff value $\Lambda^{+}$, or
equivalently longitudinal momentum fraction greater than
$\Lambda^{+}/P^{+}\equiv e^{-\eta}$, where $P^{+}$ is the total longitudinal
momentum of the hadron and $\eta$ is the maximal rapidity of the gluons in the
hadron. The soft degrees of freedom are gluons with smaller longitudinal
momentum fraction. They occupy a window of rapidity $\Delta\eta$ below the
cutoff $\Lambda^{+}$.

At initial energy the valence gluons are assumed to scatter eikonally on the
target, while the soft gluons do not contribute to scattering. If the target
is dense, one takes into account multiple ”soft” scatterings, if it is dilute
the scattering is considered to leading order only. To increase the energy of
the collision we boost the hadronic projectile. As a result, part of the soft
gluons become energetic enough to scatter, and this leads to the evolution of
the scattering amplitude.

The valence system evolved to rapidity $\eta$ is described by the valence
state denoted as $|\eta\rangle$. The wave function of the soft gluons in the
presence of the valence charges has the form of a unitary operator $\Omega$
acting on the Fock space vacuum $|0\rangle_{s}$ of the soft modes, so that the
complete wave function is

$\Psi=\Omega[\hat{j},a^{\dagger},a]|0\rangle_{s}\otimes|\eta\rangle\,;~{}~{}~{}~{}~{}~{}~{}~{}~{}\hat{j}^{a}(\mathbf{x}_{\perp})\,=\,\int_{\Lambda^{+}}^{\infty}{dk^{+}\over
2\pi}\,a_{v}^{\dagger}(k^{+},\mathbf{x}_{\perp})\,T^{a}\,a_{v}(k^{+},\mathbf{x}_{\perp})$
(1)

where $a^{\dagger}$ and $a$ are creation and annihilation of the soft gluon
modes, while $a_{v}^{\dagger}$ and $a_{v}$ refer to the valence gluon creation
and annihilation operators. The unitary operator $\Omega$ has been calculated
for dilute projectile, where the color current parametrically is considered to
be $j\sim 1$, and for dense case where $j\sim 1/\alpha_{s}$. For dilute
projectile, $\Omega$ is a simple coherent operator

$\Omega=\mathrm{Exp}\left\\{i\int
d^{2}\mathbf{x}_{\perp}\,\hat{b}_{i}^{a}(\mathbf{x}_{\perp})\int_{\Lambda^{+}e^{-\Delta\eta}}^{\Lambda^{+}}\frac{dk^{+}}{\pi|k^{+}|^{1/2}}\left({a}_{i}^{\dagger
a}(k^{+},\mathbf{x}_{\perp})+{a}_{i}^{a}(k^{+},\mathbf{x}_{\perp})\right)\right\\}$
(2)

where the longitudinal integral is over the ”window” of rapidity width
$\Delta\eta$, and $\hat{b}^{a}_{i}(x_{\perp})$ is the ”classical” field
created by the color charge $\hat{j}$.

Figure 1: The mode structure of the wave function and the two types of
operators/evolutions.

$\hat{b}_{i}^{a}(\mathbf{x}_{\perp})=g\int
d^{2}\mathbf{y_{\perp}}\frac{\partial_{i}}{\partial^{2}}(\mathbf{x_{\perp},y_{\perp}})\,\hat{j}^{a}(\mathbf{y}_{\perp})$
(3)

For dense projectile the field is given by the solution of classical Yang-
Mills equations

$\begin{split}&\partial_{i}\hat{b}_{i}^{a}(\mathbf{x}_{\perp})=g\hat{j}^{a}(\mathbf{x}_{\perp})\,,\\\
&\partial_{i}\hat{b}_{j}^{a}(\mathbf{x}_{\perp})-\partial_{j}\hat{b}_{i}^{a}(\mathbf{x}_{\perp})-gf^{abc}\hat{b}_{i}^{b}(\mathbf{x}_{\perp})\hat{b}_{j}^{c}(\mathbf{x}_{\perp})=0\,.\\\
\end{split}$ (4)

In the dense case, $\Omega$ is not a simple coherent operator, but in addition
contains a factor Gaussian in the creation and annihilation operators
Kovner:2007zu ; tolga . The explicit form of $\Omega$ is not important here,
but an interested reader can find the derivation in tolga .

Prior to the evolution by a rapidity $\Delta\eta$ the ”valence” system is
described by a density matrix $\hat{\rho}_{v}=|\eta\rangle\langle\eta|$ (the
derivation is equally valid for a valence state which is a mixed state). For
observables $\hat{O}$ that depend on the total valence charge density
$\hat{j}$,

${\rm Tr}[\hat{O}(\hat{j})\hat{\rho}_{v}]\,=\,\int
dj\,\tilde{O}(j)\,W_{\eta}[j]$ (5)

Here the expectation value of $\hat{O}$ has been represented using the Wigner-
Weyl formulation of quantum mechanics, introducing the phase space (quasi)
probability distribution $W_{\eta}[j]$ of the classical phase space variable
$j$. The function $\tilde{O}(j)$ is the Wigner-Weyl representation of the the
operator $\hat{O}$ (see ming ,Kovner:2005uw ).

After one step of the evolution the density matrix is

$\hat{\rho}(\Delta\eta)\,=\,{\Omega}|0\rangle\hat{\rho}_{v}\langle
0|\,{\Omega}^{\dagger}\,.$ (6)

The JIMWLK equation is derived for a set of observables that depend on the
total valence charge density, i.e. charge density integrated over rapidity
(see eq.(1)). As such at one step of the evolution not only the wave function
of the system evolves, but also the operator of the observable gets an
additional contribution due to the color charge density in the new rapidity
interval that opens up due to the evolution. The evolution of such an
observable follows from the calculation of the expectation value

${\rm
Tr}[\hat{O}(\hat{j}+\hat{j}_{\mathrm{soft}})\,\hat{\rho}(\Delta\eta)]={\rm
Tr}[\hat{R}^{\dagger}\,\hat{O}(\hat{j})\,\hat{R}\,\hat{\rho}(\Delta\eta)]={\rm
Tr}[\hat{O}(\hat{j})\,\hat{R}\,\hat{\rho}(\Delta\eta)\,\hat{R}^{\dagger}]$ (7)

where $\hat{\rho}(\Delta\eta)$ is given in (6), and the operator $\hat{R}$
shifts the valence color charge density according to

$\hat{R}^{\dagger}\hat{j}^{a}(\mathbf{x}_{\perp})\hat{R}\,=\,\hat{j}^{a}(\mathbf{x}_{\perp})+\hat{j}^{a}_{\mathrm{soft}}(\mathbf{x}_{\perp})\,;~{}~{}~{}~{}~{}~{}~{}\hat{j}^{a}_{soft}(\mathbf{x}_{\perp})\,=\,\int^{\Lambda^{+}}_{\Lambda^{+}e^{-\Delta\eta}}{dk^{+}\over
2\pi}\,a^{\dagger}(k^{+},\mathbf{x}_{\perp})\,T^{a}\,a(k^{+},\mathbf{x}_{\perp}).$
(8)

That such an operator can be formally mathematically defined was shown in ming
. The evolution can be conveniently represented as the evolution of the
probability distribution $W_{\eta}[j]$.

The evolution equation follows from performing the average in (7) over the
soft gluon Hilbert space:

$\frac{dW_{\eta}[j^{a}]}{d\eta}\,=\,-\,H_{RFT}\,W_{\eta}[j^{a}]$ (9)

with the ”master Hamiltonian”

$H_{RFT}=\frac{1}{4\pi}\Bigg{[}(\tilde{b}_{L}^{a}-b_{R}^{a})^{\dagger}(N^{\dagger}_{\perp,R}N_{\perp,R})^{ab}(\tilde{b}_{L}^{b}-b_{R}^{b})+h.c.\Bigg{]}$
(10)

where the summation over color indices and integration over transverse
coordinates is implicit.

The Hamiltonian in (10) is written in terms of classical fields $b$ defined as
solutions of classical Yang-Mills equations with the source $j$ which formally
look identical to (4), except that the fields and charges are not operators in
the Hilbert space, but functions on the classical phase space. The factor
$N_{\perp}$ is given by

$N_{\perp}[b]=\left[\delta_{ij}-\partial_{i}\frac{1}{\partial^{2}}\partial_{j}-D_{i}\frac{1}{D^{2}}D_{j}\right]^{ab}(\mathbf{x}_{\perp},\mathbf{y}_{\perp})\,;~{}~{}~{}~{}~{}~{}~{}~{}~{}D_{i}^{ab}=\delta^{ab}\partial_{i}-igT^{e}_{ba}b_{i}^{e}$
(11)

In (10),

$b_{L}^{b}\equiv b^{b}[j_{L}];~{}~{}~{}~{}~{}b_{R}^{b}\equiv
b^{b}[j_{R}];~{}~{}~{}~{}~{}~{}N_{\perp,R}=N_{\perp}[b_{R}].\,;~{}~{}~{}~{}~{}~{}\tilde{b}_{L}^{a}=b_{L}^{b}\,{\cal
R}_{p}^{ab}\,;~{}~{}~{}~{}~{}~{}~{}~{}~{}{\cal
R}_{p}\,=\,e^{\tau}\,;~{}~{}~{}~{}~{}\tau=gT^{e}\frac{\delta}{\delta j^{e}}$
(12)

The left and right currents $j_{L(R)}$ are expressed in terms of the classical
phase space variable $j$ as Kovner:2005uw ; Altinoluk:2013rua

$\begin{split}j_{L}^{a}=&j^{b}\left[\frac{\tau}{2}\coth{\frac{\tau}{2}}+\frac{\tau}{2}\right]^{ba}\\\
=&j^{a}+\frac{1}{2}j^{b}\left(gT^{e}\frac{\delta}{\delta
j^{e}}\right)_{ba}+\frac{1}{12}j^{b}\left(gT^{e}\frac{\delta}{\delta
j^{e}}\right)^{2}_{ba}-\frac{1}{720}j^{b}\left(gT^{e}\frac{\delta}{\delta
j^{e}}\right)^{4}_{ba}+\ldots\\\
j_{R}^{a}=&j^{b}\left[\frac{\tau}{2}\coth{\frac{\tau}{2}}-\frac{\tau}{2}\right]^{ba}\\\
=&j^{a}-\frac{1}{2}j^{b}\left(gT^{e}\frac{\delta}{\delta
j^{e}}\right)_{ba}+\frac{1}{12}j^{b}\left(gT^{e}\frac{\delta}{\delta
j^{e}}\right)^{2}_{ba}-\frac{1}{720}j^{b}\left(gT^{e}\frac{\delta}{\delta
j^{e}}\right)^{4}_{ba}+\ldots\\\ \end{split}$ (13)

The introduction of distinct left and right operators is an integral part of
representing quantum mechanics as a theory in the classical phase space
Agarwal:1971wc . Although $\tau$ is a differential operator, in the above
expressions it is treated as a number. This is because at the end of the day
it acts only on $W_{\eta}[j]$ and does not act on any of the factors $j$ in
the evolution Hamiltonian (10) itself. Also ${\cal R}_{p}$ is treated as a
matrix in color indices and not a differential operator.

A physically intuitive way of thinking about the classical derivative operator
$\tau$ is that it represents the color fields of the target on which the
projectile is scattering. Formally the derivation does not assume scattering
at all, but deals only with the projectile wave function. However if we
consider the scattering matrix of the projectile on a target described by the
color fields $\alpha_{T}$, in the eikonal approximation, every factor of
$\tau$ turns into a factor of $\alpha_{T}$, since the calculation of the
S-matrix is equivalent to functional Fourier transform Kovner:2005en .

Although Eq.(10) is written in the form which seemingly can be applied to the
situation where the target and projectile can be both dense, this is not quite
the case. The derivation of Eq.(10) is fully under control only in the
situation where at least one of the scattering objects (target and projectile)
is dilute. The general form given in Eq.(10) should be understood as an
interpolating function between these limiting situations. The two physical
limits of Eq.(10) are the so called JIMWLK and KLWMIJ Hamiltonians obtained in
the limit of dilute target and dilute projectile respectively.

The JIMWLK Hamiltonian applies to a dense projectile scattering on a dilute
target. To obtain the JIMWLK Hamiltonian one expands Eq.(10) to leading
(second) order in $\tau$. At this order we have $N_{\perp,R(L)}=N_{\perp}$ and

$\begin{split}&b_{L}{\cal R}_{p}-b_{R}=b_{i}^{a}[j_{L}]{\cal
R}_{p}^{ab}-b_{i}^{b}[j_{R}]\\\ \simeq&\left(b_{i}^{a}[j]+\frac{\delta
b_{i}^{a}}{\delta
j^{e}}(j_{L}^{e}-j^{e})\right)\left(\delta_{ab}+gT^{d}_{ab}\frac{\delta}{\delta
j^{d}}\right)-\left(b_{i}^{b}[j]+\frac{\delta b_{i}^{b}}{\delta
j^{e}}(j_{R}^{e}-j^{e})\right)\\\
\simeq&gb_{i}^{a}T^{d}_{ab}\frac{\delta}{\delta j^{d}}+\frac{\delta
b_{i}^{b}}{\delta j^{e}}gj^{c}T^{d}_{ce}\frac{\delta}{\delta j^{d}}\\\
=&gb_{i}^{a}(\mathbf{x}_{\perp})T^{d}_{ab}\frac{\delta}{\delta
j^{d}(\mathbf{x}_{\perp})}+\int d^{2}\mathbf{z}_{\perp}\frac{\delta
b_{i}^{b}(\mathbf{x}_{\perp})}{\delta
j^{e}(\mathbf{z}_{\perp})}gj^{c}(\mathbf{z}_{\perp})T^{d}_{ce}\frac{\delta}{\delta
j^{d}(\mathbf{x}_{\perp})}\\\ =&i\left[\partial_{i}-D_{i}\frac{1}{\partial
D}D\partial\right]^{bd}(\mathbf{x},\mathbf{z})\,\frac{\delta}{\delta
j^{d}(\mathbf{z}_{\perp})}\end{split}$ (14)

where the covariant derivative is defined in (12). In the last line we have
used $igT^{e}_{ba}b_{i}^{e}=\delta^{ab}\partial_{i}-D_{i}^{ab}$ and
$igT^{e}_{ba}j^{e}=-(\partial D-D\partial)^{ab}=igj^{e}T^{a}_{eb}$ as well as

$\frac{\delta b_{i}^{b}(\mathbf{x}_{\perp})}{\delta
j^{e}(\mathbf{z}_{\perp})}=\left[D_{i}\frac{1}{\partial
D}\right]^{be}(\mathbf{x}_{\perp},\mathbf{z}_{\perp})\,$ (15)

which follows by differentiating the equations (4).

Additionally using eq. (11) we calculate

$Q_{i}^{a}({\mathbf{x}}_{\perp};U)\equiv[UN_{\perp}(b_{L}\mathcal{R}_{p}-b_{R})]^{a}_{i}({\mathbf{x}}_{\perp})=-i\left[U({\mathbf{x}})\left(D_{i}\frac{1}{D^{2}}-\partial_{i}\frac{1}{\partial^{2}}\right)D\partial\right]^{ab}(\mathbf{x}_{\perp},\mathbf{z}_{\perp})\frac{\delta}{\delta
j^{b}(\mathbf{z}_{\perp})}\,.$ (16)

where the standard eikonal scattering matrix $U$ is related to the classical
field $b$ via

$U(\mathbf{x}_{\perp})=\mathcal{P}\mathrm{exp}\left[ig\int_{\mathcal{C}}d\mathbf{y}_{\perp}\cdot
b^{c}(\mathbf{y}_{\perp})T^{c}\right]$ (17)

with the path $\mathcal{C}$ starting at a fixed reference point (which can be
taken at infinity) on the transverse plane and ending at the point
$\mathbf{x}_{\perp}$.

It is customary to represent the JIMWLK evolution in terms of the eikonal
matrix $U$ as well as the left and right rotation operators that act on it. To
do that one has to change variables from $j$ to $U$ ming ,

$\frac{\delta}{\delta j^{a}(\mathbf{x}_{\perp})}=\int
d^{2}\mathbf{z}_{\perp}\frac{\delta U^{cd}(\mathbf{z}_{\perp})}{\delta
j^{a}(\mathbf{x}_{\perp})}\frac{\delta}{\delta U^{cd}(\mathbf{z}_{\perp})}$
(18)

with

$\begin{split}\frac{\delta U^{cd}(\mathbf{z}_{\perp})}{\delta
j^{a}(\mathbf{x}_{\perp})}&=\int d^{2}\mathbf{y}_{\perp}\frac{\delta
U^{cd}(\mathbf{z}_{\perp})}{\delta b_{l}^{b}(\mathbf{y}_{\perp})}\frac{\delta
b_{l}^{b}(\mathbf{y}_{\perp})}{\delta
j^{a}(\mathbf{x}_{\perp})}\,=\,-\,ig\left[U(\mathbf{z}_{\perp})T^{d}\frac{1}{\partial
D}\right]^{ca}(\mathbf{z}_{\perp},\mathbf{x}_{\perp})\end{split}$ (19)

This yields

$\frac{\delta}{\delta j^{a}(\mathbf{x}_{\perp})}=\int
d^{2}\mathbf{z}_{\perp}\frac{\delta U^{cd}(\mathbf{z}_{\perp})}{\delta
j^{a}(\mathbf{x}_{\perp})}\frac{\delta}{\delta U^{cd}(\mathbf{z}_{\perp})}\
=-ig\int
d^{2}\mathbf{z}_{\perp}\left[\frac{1}{D\partial}\right]^{ae}(\mathbf{x}_{\perp},\mathbf{z}_{\perp})\mathcal{J}^{e}_{R}(\mathbf{z}_{\perp})\,$
(20)

with the right rotation operator

$\mathcal{J}^{b}_{R}(\mathbf{z}_{\perp})=-\mathrm{Tr}\left(U(\mathbf{z}_{\perp})T^{b}\frac{\delta}{\delta
U^{\dagger}(\mathbf{z}_{\perp})}\right)\,$ (21)

This operator when acting on $U$ multiplies it on the right by $T^{b}$. Also
recall that

$\partial_{i}\frac{1}{\partial^{2}}(\mathbf{x}_{\perp},\mathbf{y}_{\perp})=\frac{1}{2\pi}\frac{(\mathbf{x}_{\perp}-\mathbf{y}_{\perp})_{i}}{(\mathbf{x}_{\perp}-\mathbf{y}_{\perp})^{2}}\,,\quad
D_{i}\frac{1}{D^{2}}(\mathbf{x}_{\perp},\mathbf{y}_{\perp})=\frac{1}{2\pi}U^{\dagger}(\mathbf{x}_{\perp})\frac{(\mathbf{x}_{\perp}-\mathbf{y}_{\perp})_{i}}{(\mathbf{x}_{\perp}-\mathbf{y}_{\perp})^{2}}U(\mathbf{y}_{\perp})\,.$
(22)

Finally, eq. (16) becomes

$Q_{i}^{a}[\mathbf{x}_{\perp};U]=-\frac{g}{2\pi}\int
d^{2}\mathbf{y}_{\perp}\frac{(\mathbf{x}_{\perp}-\mathbf{y}_{\perp})_{i}}{(\mathbf{x}_{\perp}-\mathbf{y}_{\perp})^{2}}\left[U(\mathbf{y}_{\perp})-U(\mathbf{x}_{\perp})\right]^{ab}\mathcal{J}_{R}^{b}(\mathbf{y}_{\perp})$
(23)

The standard JIMWLK kernel is reproduced as

$\begin{split}H_{JIMWLK}\,=&-\frac{1}{2\pi}\int
d^{2}\mathbf{z}_{\perp}Q_{i}^{a\dagger}[\mathbf{z}_{\perp};U]Q_{i}^{a}[\mathbf{z}_{\perp};U]\\\
=&-\frac{\alpha_{s}}{2\pi^{2}}\int_{\mathbf{z}_{\perp},\mathbf{y}_{\perp},\mathbf{x}_{\perp}}\frac{(\mathbf{z}_{\perp}-\mathbf{x}_{\perp})_{i}}{(\mathbf{z}_{\perp}-\mathbf{x}_{\perp})^{2}}\frac{(\mathbf{z}_{\perp}-\mathbf{y}_{\perp})_{i}}{(\mathbf{z}_{\perp}-\mathbf{y}_{\perp})^{2}}\mathcal{J}^{e}_{R}(\mathbf{y}_{\perp})\\\
&\times[1+U^{\dagger}(\mathbf{y}_{\perp})U(\mathbf{x}_{\perp})-U^{\dagger}(\mathbf{y}_{\perp})U(\mathbf{z}_{\perp})-U^{\dagger}(\mathbf{z}_{\perp})U(\mathbf{x}_{\perp})]^{ed}\mathcal{J}^{d}_{R}(\mathbf{x}_{\perp})\,.\end{split}$
(24)

If (10) is expanded to leading order in $j$ rather than $\tau$, one obtains
the so called KLWMIJ equation, which describes the scattering of a dilute
projectile on a dense target. It is dual to the JIMWLK equation (24) and is
obtained from it by the Dense-Dilute Duality transformation
$U(\mathbf{x}_{\perp})\leftrightarrow{\cal R}_{p}(\mathbf{x}_{\perp})$
Kovner:2005en .

This completes our recap of the JIMWLK equation.

##  III The ”CSS-JIMWLK” Hamiltonian

Our goal now is to derive the equation that governs the evolution of another
set of variables, i.e. those that involves gluons at fixed value of rapidity
$\ln 1/x$ even as the rapidity ”resolution” is finer, or in other words the
wave function has been evolved past the rapidity of these gluons to
$\eta\gg\ln 1/x$. The total evolution interval $\eta$ clearly provides the
longitudinal resolution in the same sense as usually discussed in the context
of longitudinal resolution of TMD, as the softest gluons present in a wave
function evolved to $\eta$, have rapidity $\eta$. We note that technically the
”factorization scheme” inherently provided by the evolution discussed here is
a little different than in the standard approach. However to leading order in
$\alpha_{s}$ to which we are working here this does not matter.

In the context of JIMWLK equation, the evolution in $\eta$ physically
represents evolution in the total energy of the process. In the context of the
present work however one can think about it alternatively as an evolution in
the longitudinal factorization scale, which for some observables does not have
to be identical with the total energy. It is useful to keep this distinction
in mind in the following. In both cases however, the evolution is generated by
boosting the hadronic wave function and probing its gluonic structure.
Formally it is not difficult to derive the equation we are after.

The starting point of the derivation is again (6) - the choice of observable
does not affect the evolution of the wave function, or density matrix. Now
however (7) is modified. We are not interested in an ”JIMWLK-like” observable
$\hat{O}$ that depends only on the valence charge, and certainly not on the
charge integrated over the whole range of rapidity all the way to $\eta$.
Instead the observable $\hat{\cal O}$, whatever it is, stays the same
throughout the evolution and its expectation value changes solely due to the
evolution of the soft gluon wave function. In other words, if for the JIMWLK
we integrated over soft gluons only partially, keeping track of their
contribution to $j$ which modified the observable, now we should integrate out
the soft gluons completely, in the Wilsonian sense, since the observable does
not depend on these degrees of freedom. We note that this is a typical
situation considered in the physics of open quantum systems, which generally
(if the system is Markovian) leads to the Kossakowski-Lindblad equation for
time evolution of reduced density matrix Gorini:1975nb ; Lindblad:1975ef .

In practical terms this means that the operator $\hat{R}$ in (7) should be set
equal to unity. Retracing our steps in the derivation of JIMWLK we see that
this leads to one ”minor” modification in the evolution kernel in (10). Namely
$\tilde{b}_{L}$ should be replaced by $b_{L}$, since $\mathcal{R}_{p}$ in the
definition of $\tilde{b}_{L}$ originates precisely from the operator $\hat{R}$
in (7). Note that we still have two ”classical fields” $b_{L}$ and $b_{R}$,
since the index here differentiates between the fields in the wave function
and the conjugate wave function that are ordered in a specific way relative to
the operator $\hat{\cal O}$footnote .

With this modification the new ”master Hamiltonian” is:

$H^{\prime}_{RFT}=\frac{1}{4\pi}\Bigg{[}({b}_{L}^{a}-b_{R}^{a})^{\dagger}(N^{\dagger}_{\perp,R}N_{\perp,R})^{ab}({b}_{L}^{b}-b_{R}^{b})+h.c.\Bigg{]}$
(25)

As before, this Hamiltonian is applicable for either dilute target or dilute
projectile. We first explore the limit of dense projectile and dilute target,
i.e. expand the Hamiltonian (25) to second order in $\tau$.

$\begin{split}&b_{L}^{a}-b_{R}^{a}=b_{i}^{a}[j_{L}]-b_{i}^{a}[j_{R}]\\\
\simeq&\left(b_{i}^{a}[j]+\frac{\delta b_{i}^{a}}{\delta
j^{e}}(j_{L}^{e}-j^{e})\right)-\left(b_{i}^{a}[j]+\frac{\delta
b_{i}^{a}}{\delta j^{e}}(j_{R}^{e}-j^{e})\right)\\\ \simeq&\frac{\delta
b_{i}^{a}}{\delta j^{e}}gj^{c}T^{d}_{ce}\frac{\delta}{\delta j^{d}}=\int
d^{2}\mathbf{z}_{\perp}\frac{\delta b_{i}^{a}(\mathbf{x}_{\perp})}{\delta
j^{e}(\mathbf{z}_{\perp})}gj^{c}(\mathbf{z}_{\perp})T^{d}_{ce}\frac{\delta}{\delta
j^{d}(\mathbf{x}_{\perp})}\\\ =&i\left[D_{i}-D_{i}\frac{1}{\partial
D}D\partial\right]^{ad}(\mathbf{x}_{\perp},\mathbf{z}_{\perp})\,\frac{\delta}{\delta
j^{d}(\mathbf{z}_{\perp})}\end{split}$ (26)

A little bit more algebra and

$\displaystyle[N_{\perp}(b_{L}-b_{R})]^{a}\simeq\partial_{i}\frac{1}{\partial^{2}}(D\partial-\partial
D)^{ab}(\mathbf{x}_{\perp},\mathbf{z}_{\perp})\frac{\delta}{\delta
j^{b}(\mathbf{z}_{\perp})}=ig\partial_{i}\frac{1}{\partial^{2}}(\mathbf{x}_{\perp},\mathbf{z}_{\perp})\left[j(\mathbf{z}_{\perp})T^{a}\frac{\delta}{\delta
j(\mathbf{z}_{\perp})}\right]$

Thus the CSS -JIMWLK Hamiltonian for a dense projectile can be conveniently
written as

$\displaystyle H_{CSS-
JIMWLK}=\frac{g^{2}}{2\pi}\int_{\mathbf{x_{\perp},y_{\perp}}}\left[j(\mathbf{x}_{\perp})T^{a}\frac{\delta}{\delta
j(\mathbf{x}_{\perp})}\right]\left[\frac{1}{\partial^{2}}\right](\mathbf{x_{\perp},y_{\perp}})\left[j(\mathbf{y_{\perp}})T^{a}\frac{\delta}{\delta
j(\mathbf{y}_{\perp})}\right]$ (28)

We can now perform the same exercise for a dilute projectile. This time we
expand (25) to lowest order in $j$ but keep all orders in $\tau$.
Interestingly, the result we obtain is identical to (28). We thus discover
that the CSS-JIMWLK Hamiltonian we have derived is universal and is applicable
equally to dilute and dense hadronic system.

Another way of writing $H_{CSS-JIMWLK}$ is in terms of the dual charge
rotation operators $\mathcal{I}_{R(L)}$. These are analogous to
$\mathcal{J}_{R(L)}$ but act as rotation operators on the matrix ${\cal
R}_{p}$ rather than $U$:

$\mathcal{I}^{b}_{R}(\mathbf{z}_{\perp})=-\mathrm{Tr}\left({\cal
R}_{p}(\mathbf{z}_{\perp})T^{b}\frac{\delta}{\delta{\cal
R}_{p}^{\dagger}(\mathbf{z}_{\perp})}\right)\,;\ \ \ \ \ \ \
\mathcal{I}^{a}_{L}(\mathbf{z}_{\perp})={\cal
R}_{p}^{ab}(\mathbf{z}_{\perp})\mathcal{I}^{b}_{R}(\mathbf{z}_{\perp});\ \ \ \
\ \mathcal{I}^{V}\equiv\mathcal{I}_{L}-\mathcal{I}_{R}$ (29) $\displaystyle
H_{CSS-
JIMWLK}=\frac{g^{2}}{2\pi}\int_{\mathbf{x}_{\perp},\mathbf{y}_{\perp}}\mathcal{I}^{V}(\mathbf{x}_{\perp})\frac{1}{\partial^{2}}(\mathbf{x}_{\perp},\mathbf{y}_{\perp})\mathcal{I}^{V}(\mathbf{y}_{\perp})$
(30)

Eqs. (28) and (30) have a very simple form and its physics is easy to
understand. The operator $jT^{a}\frac{\delta}{\delta j}$ is an emission vertex
of a gluon from the charge density $j$. Recall that the charge density $j$ is
integrated over rapidity up to the longitudinal resolution scale $\eta$, thus
$jT^{a}\frac{\delta}{\delta j}$ is a gluon emission vertex from any gluon with
rapidity $\eta_{g}<\eta$ present in the wave function, since all such gluons
contribute to $j$. Thus the action of $H_{CSS-JIMWLK}$ amounts to emission of
a soft gluon in the wave function and its reabsorption (either in the wave
function or its conjugate) from any of the ”valence” gluons in the wave
function.

In the Wigner-Weyl representation

$\langle\eta|\hat{\cal O}|\eta\rangle\,=\,\int dj\,W_{\eta}[j]\,\tilde{\cal
O}(j)\,\equiv\,\langle\,\tilde{\cal O}\,\rangle_{\eta}\ $ (31)

Algebraically, the action of $jT^{a}\frac{\delta}{\delta j}$ on the operator
$\tilde{\cal O}(j)$ is nothing but a color rotation of any of the valence
gluon creation and annihilation operators. The evolution equation, generated
by $H_{CSS-JIMWLK}$ is

$\frac{d}{d\eta}\langle\eta|\hat{\cal O}|\eta\rangle=-\langle\,H_{CSS-
JIMWLK}\,\tilde{\cal O}\,\rangle_{\eta}$ (32)

The action of $H_{CSS-JIMWLK}$ on $\tilde{\cal O}$ follows from the action of
$jT^{a}\frac{\delta}{\delta j}$. As we have discussed above,
$j(\mathbf{x}_{\perp})T^{a}\frac{\delta}{\delta j(\mathbf{x}_{\perp})}$ is the
Wigner-Weyl representation of the color rotation operator Agarwal:1971wc ,
therefore

$\langle\,j(\mathbf{x}_{\perp})T^{a}\frac{\delta}{\delta
j(\mathbf{x}_{\perp})}\tilde{\cal
O}\,\rangle_{\eta}\equiv\langle\eta|[\hat{j}^{a}(\mathbf{x}_{\perp}),\hat{\cal
O}]|\eta\rangle\,.$ (33)

To be mathematically precise we would have to find a Wigner-Weyl
representation $\tilde{\cal O}$ of $\hat{\cal O}$, but we are not going to do
it here since the exact form depends on the operator $\hat{\cal O}$ while (33)
holds for all operators of the type we are discussing.

The evolution equation then is

$\frac{d}{d\eta}\langle\eta\,|\hat{\cal
O}|\,\eta\rangle=-\frac{g^{2}}{2\pi}\langle\eta|\,\int
d^{2}\mathbf{u}_{\perp}d^{2}\mathbf{v}_{\perp}\frac{1}{\partial^{2}}(\mathbf{u}_{\perp},\mathbf{v}_{\perp})[\hat{j}^{a}(\mathbf{u}_{\perp}),[\hat{j}^{a}(\mathbf{v}_{\perp}),\hat{\cal
O}]]\,|\eta\rangle$ (34)

This form of the evolution is the most straightforward to use for generic
operators $\hat{\cal O}$. This equation is of the type of Kossakowski-Lindblad
equation Gorini:1975nb ; Lindblad:1975ef  ubiquitous in discussions of
dynamics of open quantum systems, even though in our case the dynamics is in
rapidity rather than in time.

One might wonder why did we go into trouble of working with the Wigner-Weyl
representation for the evolution in the first place, if we are going to use
the operator representation (34) at the end of the day. In fact, in the dilute
case, (34) can be easily derived directly using the wave function (1) with
(2). However for dense case this derivation albeit possible, is considerably
more cumbersome. The advantage of our approach here is that it yields equally
easily $H_{CSS-JIMWLK}$ in both dilute and dense cases.

##  IV The evolution of the gluon TMD and the CSS equation

Let us see in more detail how $H_{CSS-JIMWLK}$ evolves some basic observables.
As the simplest observable let us consider the unpolarized gluon TMD. The
unpolarized gluon TMD in the most naive way is defined as the correlator of
the gluon fields at transverse momentum $\mathbf{k}$. In addition we have to
be clear about the longitudinal momentum fraction and the longitudinal
resolution dependence of the TMD. We are interested in
$T(\mathbf{k},x,\zeta)$, where $\mathbf{k}$ denotes the transverse momentum of
the TMD, $x=k^{+}/P^{+}$ denotes the longitudinal momentum fraction and
$\zeta$ denotes the longitudinal resolution parameter. As noted earlier we use
a dimensionless longitudinal resolution parameter rather than a dimensional
one usually introduced in CSS. The relation between the two is logarithmic,
and to leading order in $\alpha_{s}$ the two are equivalent. We stress that
here we define TMD as the number of gluons at a given transverse and
longitudinal momentum at a fixed resolution. In any calculation of physical
process the most convenient choice of the resolution scale is dictated by the
kinematics of the process, and we are not considering this question here.

We do not introduce in this discussion the transverse resolution parameter
$\mu^{2}$, since as we shall see in the standard CGC approach this parameter
is naturally associated with the UV cutoff.

###  IV.1 The CSS equation

Given our previous discussion, it is clear how $x$ and $\zeta$ are
accommodated in the high energy evolution framework. From the point of view of
the high energy evolution, the hadronic wave function evolved to a particular
value of rapidity $\eta$ contains only gluons with longitudinal momentum
$k^{+}\leq P^{+}e^{-\eta}$. Thus the parameter of the evolution $\eta$
naturally provides the longitudinal resolution in the same sense as
longitudinal resolution in the perturbative TMD physics. Therefore the
longitudinal resolution of a TMD calculated in the hadronic wave function
evolved to rapidity $\eta$, is in fact $\zeta=\eta$. The operator definition
of the TMD is therefore

$T(\mathbf{k},x,\zeta)=\langle\zeta|\,\hat{T}(\mathbf{k},x)\,|\zeta\rangle\,;~{}~{}~{}~{}~{}~{}~{}~{}~{}\hat{T}(\mathbf{k},x)=\int
d^{2}\mathbf{x}_{\perp}d^{2}\mathbf{y}_{\perp}\,e^{i\mathbf{k\cdot(x_{\perp}-y_{\perp})}}\
a^{\dagger a}_{i}(x,\mathbf{x}_{\perp})\,a^{a}_{i}(x,\mathbf{y}_{\perp})$ (35)

where $|\zeta\rangle$ is the hadronic state evolved to rapidity $\eta=\zeta$.

It is obvious that gluon TMD at longitudinal momentum fraction $x$ vanishes as
long as $\eta\leq\ln 1/x$. When the wave function is evolved up to $\eta=\ln
1/x$, the gluon TMD is generated for the first time by splittings of the
valence gluons. At this point in the evolution, the TMD is given by the
correlator of the Weizsacker-Williams field feng

$\begin{split}T(\mathbf{k},x,\eta=\ln 1/x)&=\,\int
d^{2}\mathbf{x}_{\perp}d^{2}\mathbf{y}_{\perp}\,e^{i\mathbf{k\cdot(x_{\perp}-y_{\perp})}}\langle\eta|\hat{b}^{a}_{i}(\mathbf{x}_{\perp})\hat{b}^{a}_{i}(\mathbf{y}_{\perp})|\eta\rangle\\\
&=\,\int
d^{2}\mathbf{x}_{\perp}d^{2}\mathbf{y}_{\perp}\,e^{i\mathbf{k\cdot(x_{\perp}-y_{\perp})}}\,\int
dj\,W_{\eta}[j]\,b^{a}_{i}(\mathbf{x}_{\perp})\,b^{a}_{i}(\mathbf{y}_{\perp})\end{split}$

where the gluon field $b$ is determined through (4) by the color charge
density $j$ integrated over rapidity up to $\eta=\ln 1/x$footnote2 .

Now consider further evolution in $\eta$. Naively, one could think that one
has to evolve the product of Weizsacker - Williams fields (or equivalently
$W_{\eta}$) with $H_{JIMWLK}$, but this is incorrect. The $\eta$ evolution
increases the longitudinal resolution in the wave function, but does not
change the longitudinal momentum fraction $x$ in the TMD. On the other hand
the evolution with $H_{JIMWLK}$ increases the resolution but also decreases
the value of the momentum fraction $x$. The TMD at fixed $x$ and $\zeta>\ln
1/x$ has to be calculated by evolving with the Hamiltonian $H_{CSS-JIMWLK}$
(28) in $\eta$ from $\eta=\ln 1/x$ to $\eta=\zeta$ with the initial condition
(IV.1).

Acting with the Hamiltonian (28) on the operator $\hat{T}(\mathbf{k},x)$ leads
to a simple homogeneous equation for the TMD.

$\begin{split}\frac{\partial
T(\mathbf{k},x,\eta)}{\partial\eta}=-\frac{g^{2}}{2\pi}\int
d^{2}\mathbf{u}_{\perp}d^{2}\mathbf{v}_{\perp}\frac{1}{\partial^{2}}(\mathbf{u}_{\perp},\mathbf{v}_{\perp})\,\langle\eta|[\hat{j}^{a}(\mathbf{u}_{\perp}),[\hat{j}^{a}(\mathbf{v}_{\perp}),\hat{T}(\mathbf{k},x)]]|\eta\rangle\end{split}$
(36)

To evaluate the right-hand side, we use

$\begin{split}[\hat{j}^{a}(\mathbf{v}_{\perp}),a^{\dagger\,b}_{i}(x,\mathbf{x}_{\perp})]&=\Big{[}a_{i}^{\dagger}(x,\mathbf{v}_{\perp})T^{a}\Big{]}^{b}\delta^{(2)}(\mathbf{x}_{\perp}-\mathbf{v}_{\perp})\end{split}$
(37)

and similarly for the commutator with $a$. This of course follows from the
canonical commutation relation

$\begin{split}[a_{i}^{a}(p^{+},\mathbf{x}_{\perp}),a_{j}^{\dagger
b}(k^{+},\mathbf{y}_{\perp})]=(2\pi)\delta(p^{+}-k^{+})\delta_{ij}\delta_{ab}\delta^{(2)}(\mathbf{x}_{\perp}-\mathbf{y}_{\perp})\end{split}$
(38)

Two consecutive color rotations yield

$\begin{split}[\hat{j}^{a}(\mathbf{u}_{\perp}),[\hat{j}^{a}(\mathbf{v}_{\perp}),a_{i}^{\dagger\,b}(x,\mathbf{x}_{\perp})]]&=N_{c}a_{i}^{\dagger\,b}(x,\mathbf{u}_{\perp})\delta^{(2)}(\mathbf{x}_{\perp}-\mathbf{v}_{\perp})\end{split}$
(39)

The final result is

$\displaystyle\frac{\partial
T(\mathbf{k},x,\eta)}{\partial\eta}=-\frac{2g^{2}N_{c}}{2\pi}\int
d^{2}\mathbf{u}_{\perp}d^{2}\mathbf{v}_{\perp}\int
d^{2}\mathbf{x}_{\perp}d^{2}\mathbf{y}_{\perp}\
e^{i\mathbf{k\cdot(x_{\perp}-y_{\perp})}}\frac{1}{\partial^{2}}(\mathbf{u}_{\perp},\mathbf{v}_{\perp})$
$\displaystyle\times\Big{[}\delta^{(2)}(\mathbf{u}_{\perp}-\mathbf{v}_{\perp})\delta^{(2)}(\mathbf{v}_{\perp}-\mathbf{x}_{\perp})-\delta^{(2)}(\mathbf{u}_{\perp}-\mathbf{x}_{\perp})\delta^{(2)}(\mathbf{v}_{\perp}-\mathbf{y}_{\perp})\Big{]}\langle\eta|a^{\dagger
a}_{i}(x,\mathbf{x}_{\perp})a^{a}_{i}(x,\mathbf{y}_{\perp})|\eta\rangle$
$\displaystyle=-\frac{2\alpha_{s}N_{c}}{2\pi}\int
d^{2}\mathbf{x}_{\perp}d^{2}\mathbf{y}_{\perp}\
e^{i\mathbf{k\cdot(x_{\perp}-y_{\perp})}}\,\langle\eta|a^{\dagger
a}_{i}(x,\mathbf{x}_{\perp})a^{a}_{i}(x,\mathbf{y}_{\perp})|\eta\rangle$
$\displaystyle\times\Big{[}\int
d^{2}\mathbf{u}_{\perp}\ln\frac{1}{(\mathbf{u}_{\perp}-\mathbf{x}_{\perp})^{2}\lambda^{2}}\delta^{(2)}(\mathbf{u}_{\perp}-\mathbf{x}_{\perp})-\ln\frac{1}{(\mathbf{x}_{\perp}-\mathbf{y}_{\perp})^{2}\lambda^{2}}\Big{]}$
(40)

where we have used,

$\frac{1}{\partial^{2}}(\mathbf{u}_{\perp},\mathbf{v}_{\perp})=\frac{1}{4\pi}\ln\frac{1}{(\mathbf{u}_{\perp}-\mathbf{v}_{\perp})^{2}\lambda^{2}}$
(41)

with $\lambda$ \- an IR cut-off. The result does not depend on the choice of
$\lambda$ due to the cancellation between real and virtual terms. The
integration over $\mathbf{u}_{\perp}$ in the virtual contribution (second term
in (IV.1) simply introduces a UV cut-off $\Lambda$.

$\int
d^{2}\mathbf{u}_{\perp}\ln\frac{1}{(\mathbf{u}_{\perp}-\mathbf{x}_{\perp})^{2}\lambda^{2}}\delta^{(2)}(\mathbf{u}_{\perp}-\mathbf{x}_{\perp})=\ln\frac{\Lambda^{2}}{\lambda^{2}}\,.$
(42)

Since the TMD is a decreasing function of the transverse momentum and
logarithm is a slowly varying function, the real term can be approximated as

$\begin{split}&\int d^{2}\mathbf{x}_{\perp}d^{2}\mathbf{y}_{\perp}\
e^{i\mathbf{k\cdot(x_{\perp}-y_{\perp})}}\ln\frac{1}{(\mathbf{x}_{\perp}-\mathbf{y}_{\perp})^{2}\lambda^{2}}\langle\eta|a^{\dagger
a}_{i}(x,\mathbf{x}_{\perp})a^{a}_{i}(x,\mathbf{y}_{\perp})|\eta\rangle\\\
&\approx\ln\frac{\mathbf{k}^{2}}{\lambda^{2}}\int
d^{2}\mathbf{x}_{\perp}d^{2}\mathbf{y}_{\perp}\
e^{i\mathbf{k\cdot(x_{\perp}-y_{\perp})}}\langle\eta|a^{\dagger
a}_{i}(x,\mathbf{x}_{\perp})a^{a}_{i}(x,\mathbf{y}_{\perp})|\eta\rangle\end{split}$
(43)

In summary, (IV.1) reduced to the simple momentum space form of the CSS
equation CSS ,

$\frac{\partial
T(\mathbf{k},x,\eta)}{\partial\eta}=\frac{-2\alpha_{s}N_{c}}{2\pi}\ln\frac{\Lambda^{2}}{\mathbf{k}^{2}}\
T(\mathbf{k},x,\eta)$ (44)

with the transverse resolution scale $\mu^{2}=\Lambda^{2}$.

Without the approximation (43), Eq. (IV.1), when Fourier transformed into
coordinate space yields the standard perturbative form of the CSS equation in
impact parameter space as is commonly used in the TMD literature TMDbook .

The solution for this simple homogeneous equation is of course given by the
doubly logarithmic Sudakov suppression factor

$T(\mathbf{k};x,\zeta)=e^{-\frac{\alpha_{s}N_{c}}{\pi}\ln\frac{\Lambda^{2}}{\mathbf{k}^{2}}[\zeta-\ln
1/x]}T(\mathbf{k};x,\ln 1/x)$ (45)

with the initial condition (IV.1).

###  IV.2 The transverse resolution

Eq.(44) is the CSS equation for the evolution of the gluon TMD in the
longitudinal resolution parameter with the transverse resolution scale equal
to the UV cutoff. It is quite natural that the UV cutoff arises as the
transverse resolution scale in the present framework. The reason is that in
the JIMWLK approach, where the evolution is with respect to the logarithm of
the longitudinal momentum, all transverse momentum modes, up to the UV cutoff
populate the wave function at a fixed $\eta$. Therefore the transverse
resolution for TMD defined in (35) is $\Lambda$.

One could try to rectify this by introducing the resolution scale directly in
the definition of the TMD. This seems to make perfect sense, as at the end of
the day it is the physical process that determines which observable is being
probed in the wave function. However this fails to modify the resolution scale
in the evolution equation. Let us try to do it nevertheless to understand the
problem.

Let us modify the definition of the TMD by hand to introduce a transverse
resolution scale. To do that we introduce smeared creation and annihilation
operators

$a^{a\mu}_{i}(x,\mathbf{x}_{\perp})\equiv\int_{{z}_{\perp}}a_{i}^{a}(x,\mathbf{x_{\perp}-z_{\perp}})f^{\mu}(\mathbf{z}_{\perp})$
(46)

where the smearing function $f$ has the properties

$\int d^{2}\mathbf{z}_{\perp}f^{\mu}(\mathbf{z}_{\perp})=1;\ \ {\rm and}\ \
f^{\mu}(\mathbf{z}_{\perp})\rightarrow 0;\ \ \
\mathbf{z}_{\perp}^{2}>1/\mu^{2}$ (47)

The exact shape of the function $f$ does not matter and amounts to the choice
of the ”transverse factorization scheme”. In particular a simple Gaussian form
$f(\mathbf{z}_{\perp})=\frac{\mu^{2}}{\pi}e^{-\mu^{2}\mathbf{z}_{\perp}^{2}}$
is a convenient choice. We further define the transverse momentum resolution
dependent TMD

$T(\mathbf{k},\mu^{2};x,\zeta)=\int
d^{2}\mathbf{x}_{\perp}d^{2}\mathbf{y}_{\perp}\
e^{i\mathbf{k\cdot(x_{\perp}-y_{\perp})}}\ \langle\zeta|a^{\dagger
a\mu}_{i}(x,\mathbf{x}_{\perp})a^{a\mu}_{i}(x,\mathbf{y}_{\perp})|\zeta\rangle$
(48)

This definition is eminently reasonable. Now let us consider its evolution
with $H_{CSS-JIMWLK}$. The UV divergence in the calculation above arose from
the term where both charge densities $j$ in $H_{CSS-JIMWLK}$ hit the same
creation or annihilation operators, setting the coordinates of the two charge
densities to be equal, thereby bringing the UV cutoff into the game. With the
smeared gluon field the problem unfortunately is not going away. When acting
with $H_{CSS-JIMWLK}$ on a single smeared field we still get the contribution
only when the coordinates of $\hat{j}(\mathbf{u}_{\perp})$ and
$\hat{j}(\mathbf{v}_{\perp})$ are equal, and thus this contribution is
proportional to $\lim_{|\mathbf{u}_{\perp}-\mathbf{v}_{\perp}|\rightarrow
0}\ln|\mathbf{u}_{\perp}-\mathbf{v}_{\perp}|$. This yields the same UV
divergence as before which is not regulated by $\mu$.

Physically it is quite clear what is going on. The evolution in $k^{+}$ allows
eikonal emissions with arbitrary transverse momenta of ”soft” gluons. Thus in
one step of the evolution the transverse phase space for emissions is infinite
and the probability of a given gluon at longitudinal momentum $x$ to decay
diverges. Kinematic restriction in the definition (48) does not change this
fact. The only way to regulate this UV divergence would be to smear the charge
density $j$ that enters $H_{CSS-JIMWLK}$. However this is not in our power
since $H_{CSS-JIMWLK}$ is determined unambiguously by the evolution of the
wave function. On the other hand, were we to change the evolution parameter
from $k^{+}$ to $k^{-}$ the situation would be quite different. This would
limit the transverse space for emissions and indeed regulate the UV divergence
at a transverse scale determined by $x$ and the resolution $\zeta$ inprep .
The evolution parameter would then simultaneously determine the longitudinal
and transverse resolution as was observed in balitsky-tarasov .

To conclude this section we make the following comment. Since at $\eta=\ln
1/x$ the TMD is given by (IV.1), one might be naturally tempted to simply
evolve this expression by acting on it directly with $H_{CSS-JIMWLK}$ (28),

${d\tilde{O}\over d\eta}\,=\,H_{CSS-
JIMWLK}\,\tilde{O}\,;~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\tilde{O}\,\sim\,b^{a}b^{a}$
(49)

This is however incorrect. Such a procedure would amount to CSS-evolving the
charge density integrated up to $\ln 1/x$, let us call it $j_{x}$, to rapidity
higher than $\ln 1/x$ without adding to $j_{x}$ the charge of softer gluons,
and then calculating the classical field $b_{x}$ determined by the evolved
$j_{x}$ via classical Yang-Mills equations. This is not the correct evolution
of TMD given in Eq. (IV.1). Such an evolution would take into account
correctly emission of the softest gluons from higher $x$ gluons that comprise
$j_{x}$ and thus decrease in the number of such gluons footnote3 . These
however are not the gluons with the longitudinal momentum fraction $x$, but
rather those with the momentum fraction greater than $x$! The gluons at $x$
themselves also split, and it is this splitting that is first and foremost
responsible for the decrease in their number via the Sudakov form factor. This
splitting is not accounted for by simply CSS - evolving $j_{x}$. To reiterate,
the correlator of classical fields equals to the gluon TMD only at rapidity
$\eta=\ln 1/x$. At $\eta<\ln 1/x$, the TMD vanishes, while at $\eta>\ln 1/x$
it is not given by the correlator of classical fields created by the charge
density $j_{x}$, but is suppressed by the Sudakov form factor (see (45)).

###  IV.3 The quark TMD

The previous discussion can be extended easily to other similar quantum
observables. As another example consider the quark TMD. Although in the high
energy framework the quark TMD itself arises only at higher order in
$\alpha_{s}$ relative to the gluon TMD, its evolution is a leading order
effect. The unpolarized quark TMD is defined similarly to the unpolarized
gluon TMD

$T_{q}(\mathbf{k},x,\zeta)=\langle\zeta|\,\hat{T}_{q}(\mathbf{k},x)\,|\zeta\rangle\,;~{}~{}~{}~{}~{}~{}\hat{T}_{q}(\mathbf{k},x)=\int
d^{2}\mathbf{x}_{\perp}d^{2}\mathbf{y}_{\perp}\,e^{i\mathbf{k\cdot(x_{\perp}-y_{\perp})}}\
d^{\dagger\alpha}_{\sigma}(x,\mathbf{x}_{\perp})\,d_{\sigma}^{\alpha}(x,\mathbf{y}_{\perp})$
(50)

with $d^{\dagger}$ and $d$ \- quark creation/annihilation operators ($\alpha$
is a fundamental color index and $\sigma$ is a spinor index).

In the presence of quarks $H_{CSS-JIMWLK}$ is a functional of valence charge
density, that also includes quarks.

$\hat{j}^{a}\,=\,\hat{j}_{g}^{a}\,+\,\hat{j}^{a}_{q}\,;~{}~{}~{}~{}~{}~{}~{}~{}\hat{j}^{a}_{q}(\mathbf{x}_{\perp})\,=\,\int_{\Lambda^{+}}^{\infty}{dk^{+}\over
2\pi}\,d^{\dagger}(k^{+},\mathbf{x}_{\perp})\,t^{a}\,d(k^{+},\mathbf{x}_{\perp})$
(51)

Here $j_{g}$ is defined by (1) and $t^{a}$ is a generator in fundamental
representation of the color group.

The charge density $j_{q}^{a}$ acts on the quark operators as a color
rotation:

$[\hat{j}^{a}_{q}(\mathbf{v}_{\perp}),d_{\sigma}^{\alpha}(x,\mathbf{x}_{\perp})]\,=\,d_{\sigma}^{\beta}(x,\mathbf{v}_{\perp})\,t^{a}_{\beta\alpha}\,\delta^{(2)}(\mathbf{x}_{\perp}-\mathbf{v}_{\perp})$
(52)

and similarly on $d^{\dagger}$. Hence, the CSS-JIMWLK evolution for quark TMD,
up to the change $N_{c}\rightarrow C_{F}=(N_{c}^{2}-1)/2N_{c}$, is identical
to the gluon one:

$\frac{\partial
T_{q}(\mathbf{k},x,\eta)}{\partial\eta}\,\simeq\,-\frac{\alpha_{s}C_{F}}{\pi}\ln\frac{\Lambda^{2}}{\mathbf{k}^{2}}\
T_{q}(\mathbf{k},x,\eta)$ (53)

which is again the longitudinal CSS evolution equation.

##  V Discussion

In this note we have discussed the evolution with energy of a set of
observables that are defined at a fixed, albeit small value of the
longitudinal momentum fraction $x$. These observables are distinct from the
ones described by the JIMWLK equation. The latter governs the evolution of
observables that formally are functions of the color charge integrated over
the whole range of rapidity of gluons present in the wave function. These
observables are dominated by the softest gluons present in the wave function
at high energy. We have derived the Hamiltonian- $H_{CSS-JIMWLK}$ that is
appropriate for evolving the ”non-JIMWLK” observables with energy.

The Hamiltonian $H_{CSS-JIMWLK}$ was obtained in the dilute and dense
projectile limits. Interestingly, the form of the Hamiltonian in both limits
is identical, and we believe on physical grounds that this form should be
universal beyond these two limits. This may sound surprising at first sight.
In both limits the Hamiltonian has a simple form of the leading order emission
probability of a single soft gluon. In the dilute case this is very natural,
since this is the essence of the CSS physics - the gluon TMD depletes due to
perturbative emissions of gluons into the new phase space opened up by
evolution. In the dense case the basic physics is the same, but one could have
expected the emission probability to be affected by nonlinearities inherent in
the dense wave function. However the fact that comes into play here, is that
the wave function is ”dense” only due to high density of the softest gluons -
the gluons with the smallest possible longitudinal momentum fraction. Here on
the other hand we are interested in the emission off higher $x$ gluons, and
those are not plentiful even in the highly evolved wave function. Although the
soft gluon emitted from the TMD can interact with other soft gluons, this
interaction does not affect the change of the state of the primary emitter.
Thus the nonlinear corrections are irrelevant for the evolution of the TMD,
and TMD-like observables. We believe this is the reason why $H_{CSS-JIMWLK}$
is unaffected by the finite gluon density effects included in JIMWLK.

Interestingly, the evolution equation generated by $H_{CSS-JIMWLK}$ is of the
Kossakowski-Lindblad type Gorini:1975nb ; Lindblad:1975ef . As explained in
Armesto:2019mna ; ming , the JIMWLK Hamiltonian also corresponds to the
Kossakowski-Lindblad evolution of the reduced density matrix, albeit the form
of the evolution is different. In the case of JIMWLK this is not obvious since
one does not integrate the soft gluons from the density matrix fully, but
keeps their contribution to the color charge. In the CSS case the situation is
much cleaner - one fully integrates over the soft gluons in the evolution and
thus the Kossakowski-Lindblad form of the evolution has exactly the same
origin as in open quantum systems.

We have shown that when applied to gluon TMD, the evolution generated by
$H_{CSS-JIMWLK}$ yields the CSS evolution equation. To be more precise, we
recover one of the two CSS equations - the evolution with respect to the
longitudinal resolution scale. This is of course the best we can hope for
given that we only ever evolve with respect to one parameter. The dynamics of
the transverse evolution is not formally included in the evolution of the wave
function. The major drawback of the high energy evolution paradigm that gives
rise to the standard JIMWLK equation is that the transverse resolution scale
is necessarily given by the UV cutoff. The reason is that the high energy
evolution as formulated in the original BFKL and JIMWLK approach is the
evolution in the longitudinal momentum. The transverse phase space is
unrestricted and emissions of gluons with arbitrarily large transverse momenta
are allowed. Therefore having finite evolution interval introduces a finite
resolution in longitudinal momentum, but without restricting transverse
momenta.

This is not a problem fundamentally, but has to pose a significant
inconvenience in practical terms. Any physical process resolves the TMD at a
finite transverse resolution scale, usually determined by the hard physical
scale of the process. This means that using TMD defined with this particular
resolution scale in TMD factorization formulae simplifies calculations
significantly. It makes the hard part of the process simple and perturbative
with all relevant resummations of large logarithms contained in the
calculation of TMD. One could in principle instead use TMD at a very high
scale, i.e. the UV cutoff. But in this case the hard part will also contain
this high scale, therefore it will contain large logarithms that will have to
be resummed. There should be large cancellations between the TMD and the hard
part in the final result. This clearly is far from the optimal way to organize
any calculation. Additionally we note that the leading order JIMWLK evolution
contains only the double logarithmic part of the DGLAP splittings. In order to
fully recover the DGLAP evolution, and therefore also the single Sudakov
logarithm in CSS, one needs to resum higher order corrections in the JIMWLK
Hamiltonian dglapbfkl . Both of these problems are rectified if one considers
the evolution in $k^{-}$ rather than $k^{+}$ balitsky-tarasov ; inprep .

To summarize, in order to properly understand effects related to the evolution
of gluon TMD it is important to realize that its evolution is not given by the
JIMWLK equation. Instead TMD evolves with $H_{CSS-JIMWLK}$, and only the
initial condition for this evolution at $\zeta=\ln 1/x$ is generated by the
standard JIMWLK evolution. This has to be consistently taken into account in
processes where the relevant value of $x$ is set by the kinematics while one
considers rapidity evolution beyond this value. We believe this must be
important for the correct treatment of Sudakov emissions in processes like
back to back dijet production in DIS dijet .

## Acknowledgement

We thank Tolga Altinoluk, Nestor Armesto, Guillaume Beuf and Alina Czajka for
many useful discussions related to the subject of this work. We are also
grateful to Nestor Armesto and Florian Cougulic for comments on the draft. The
work of H.D. and A.K. is supported by the NSF Nuclear Theory grant 2208387.
This research was supported by Binational Science Foundation grant #2022132.
ML’s work was funded by Binational Science Foundation grant #2021789 and by
the ISF grant #910/23. This research was also supported by MSCA RISE 823947
“Heavy ion collisions: collectivity and precision in saturation physics”
(HIEIC).

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