# TMD factorization bridging large and small x

Swagato Mukherjee  Vladimir Skokov  Andrey Tarasov  Shaswat Tiwari

###### Abstract

QCD factorization takes different forms in the large-x and small-x regimes. At
large-x, collinear factorization leads to the DGLAP evolution equation, while
at small-x, rapidity factorization results in the BFKL equation. To unify
these different regimes, a new TMD factorization based on the background field
method is proposed. This factorization not only reduces to CSS and DGLAP in
the large-x limit and BFKL in the small-x limit, but also defines a general
evolution away from these regimes.

##  1 Introduction

In these proceedings, we shall summarize the results of [1]. In the TMD
factorization approach [2, 3, 4, 5], the production of a color-singlet state
(Drell-Yan pair, Higgs and $W$ boson productions etc.) in an unpolarized
hadron-hadron scattering reads

$\displaystyle\frac{d\sigma}{dQdyd^{2}q_{\perp}}=\sum_{ij}H_{ij}(Q,\mu)\int
d^{2}b_{\perp}e^{iq_{\perp}b_{\perp}}f_{i}(x_{a},b_{\perp},\mu,\zeta_{a})f_{j}(x_{b},b_{\perp},\mu,\zeta_{b})+O\left(\frac{q^{2}_{\perp}}{Q^{2}}\right)\,$
(1)

where $Q$ is an invariant mass of the final state, $y$ is its rapidity, and
$q_{\perp}$ is the measured transverse momentum111We use the following
notation $q_{\perp}b_{\perp}\equiv q_{m}b_{m}$. and $q_{\perp}^{2}<<Q^{2}$.
The sum goes over partons participating in the hard scattering which is
defined by a hard function $H_{ij}$. The transverse-momentum dependent parton
distribution functions (TMDPDFs) depend on an impact parameter variable
$b_{\perp}$. The $x_{a}$ and $x_{b}$ are the longitudinal-momentum fractions
of the colliding hadrons carried by the partons.

The functions in Eq. (1) depend on unphysical factorization scales, $\zeta$
and $\mu$, that separate the dynamical modes in the TMD factorization scheme.
Dependence on these parameters can be studied by perturbative methods and is
governed by the anomalous dimensions $\gamma^{i}_{\mu}$ and
$\gamma^{i}_{\zeta}$ which form a part of the so called Collins-Soper equation
[2, 6]. One can then construct the solutions to these equations, and evolve
the TMDPDF from initial scales, $(\mu_{0},\zeta_{0})$, to final scales,
$(\mu,\zeta)$, using

$\displaystyle
f_{i}(x,b_{\perp},\mu,\zeta)=\exp\Big{(}\int^{\mu}_{\mu_{0}}\frac{d\tilde{\mu}}{\tilde{\mu}}\gamma^{i}_{\mu}(\tilde{\mu},\zeta_{0})\Big{)}\exp\Big{(}\frac{1}{2}\gamma^{i}_{\zeta}(\mu,b_{\perp})\ln\frac{\zeta}{\zeta_{0}}\Big{)}f_{i}(x,b_{\perp},\mu_{0},\zeta_{0})\,.$
(2)

The above evolution resums large logarithms $\ln(\mu^{2}b^{2}_{\perp})$ and
$\ln(\zeta b^{2}_{\perp})$. These logarithms are of the ultraviolet (UV)
origin, and the infrared (IR) structure of the TMD factorization is encoded in
the boundary term $f_{i}(x,b_{\perp},\mu_{0},\zeta_{0})$.

The TMDPDF, $f_{i}(x,b_{\perp},\mu_{0},\zeta_{0})$, is intrinsically
nonperturbative. However, for small values of $b_{\perp}\ll\Lambda^{-1}_{\rm
QCD}$ the TMDPDF can be expanded around $b_{\perp}=0$ and matched onto
collinear parton distribution functions (PDFs) [3, 7, 8, 9] and for small
values of x it can be expanded around $x=0$ and matched onto dipole amplitudes
[10, 11]. While both these methods are well developed, one needs to sum over
the series in $b_{\perp}$ (higher twist terms) and the series in x (sub-
eikonal terms) to get the full IR structure of TMDs. This does not seem
feasible.

Hence we propose a new approach, where we study the IR structure of TMDPDF’s
at next to leading order (NLO) in the background field method. We work in the
dilute limit, in the background of 2 gluons.

##  2 Background Field method

The concept of factorization can be elegantly introduced within the background
field method. Starting with a matrix element of an arbitrary operator
$\mathcal{O}(\hat{C})$ which depends on quark and gluon fields,222Here we
understand $C$ as a general notation for the quark and gluon fields. one can
write it as a functional integral over those fields

$\displaystyle\langle
P_{1}|\mathcal{O}(\hat{C})|P_{2}\rangle=\int\mathcal{D}C~{}\Psi^{\ast}_{P_{1}}(\vec{C}(t_{f}))\mathcal{O}(C)\Psi_{P_{2}}(\vec{C}(t_{i}))e^{iS_{\rm
QCD}(C)}\,$ (3)

where $\Psi_{P_{2}}$ and $\Psi_{P_{1}}$ are initial and final state wave
functions at $t_{i}\to-\infty$ and $t_{f}\to\infty$.

The main logic of the background field approach is to split the field into
different modes (using some factorization scale $\sigma$),

$\displaystyle C_{\mu}\to A_{\mu}+B_{\mu}\,.$ (4)

We integrate over the “quantum" modes A while keeping the “background" B
fixed. One can think about the “background" fields as fields associated with
the target, and the “quantum" fields as fields of the hard scattering. One can
then integrate over the “quantum" modes in the functional integral, to get

$\displaystyle\langle
P_{1}|\mathcal{O}(\hat{C})|P_{2}\rangle=\sum_{i}H_{i}(\sigma)\otimes\langle
P_{1}|\mathcal{V}_{i}(\sigma)|P_{2}\rangle\,$ (5)

where $H_{i}(\sigma)$ is obtained by integration over the “quantum" modes and
the matrix element depends only on the “background" modes. This is the
statement of factorization in the language of background fields.

In order to study the matrix element, we further split the background modes
$B=B^{q}+B^{bg}$ using a scale $\sigma^{\prime}$ and then integrate over
$B^{q}$ modes to get

$\displaystyle\langle
P_{1}|\mathcal{V}_{i}(\sigma)|P_{2}\rangle=\sum_{j}C_{ij}(\sigma,\sigma^{\prime})\otimes\langle
P_{1}|\mathcal{V}_{j}(\sigma^{\prime})|P_{2}\rangle\,$ (6)

where $C_{ij}(\sigma,\sigma^{\prime})$ was obtained by integration over the
dynamical modes $B^{q}$ between scales $\sigma$ and $\sigma^{\prime}$. This is
the statement of evolution in the backgound field language. Here $\sigma$ acts
as the UV scale and $\sigma^{\prime}$ acts as the IR scale. We shall now apply
this technique to TMD factorization.

##  3 TMD factorization

In the background field approach, the hard function is separated from the TMD
modes by the scale $\mu_{UV}^{2}$ and the are collinear and anti-collinear
modes (for the two colliding hadrons in Drell-Yan) of the TMD matrix element
are given by the beam function,

$\begin{split}&\mathcal{B}_{ij}(x_{B},b_{\perp})\\\
&=\int^{\infty}_{-\infty}dz^{-}e^{-ix_{B}P^{+}z^{-}}\langle
P,S|\bar{T}\\{F^{m}_{-i}(z^{-},b_{\perp})[z^{-},\infty]^{ma}_{b}\\}T\\{[\infty,0^{-}]^{an}_{0}F^{n}_{-j}(0^{-},0_{\perp})\\}|P,S\rangle\,.\end{split}$
(7)

In Eq. (7), $i$ and $j$ are Lorentz indices. The adjoint Wilson lines are
along the light-cone direction

$\displaystyle[x^{-},y^{-}]_{z_{\perp}}=\mathcal{P}\exp\Big{[}ig\int^{x^{-}}_{y^{-}}dz^{-}A_{-}(z^{-},z_{\perp})\Big{]}\,.$
(8)

The collinear and anti-collinear modes are separated by their rapidity,
$y\>=\>\ln{k^{+}\over k^{-}}$. Depending on how the modes are separated, there
might be an intersection region that is populated by the so-called soft modes,
which have the matrix element

$\displaystyle\mathcal{S}(b_{\perp})=\frac{1}{N^{2}_{c}-1}\langle 0|{\rm
Tr}[S^{\dagger}_{\bar{n}}(b_{\perp})S_{n}(b_{\perp})S^{\dagger}_{n}(0_{\perp})S_{\bar{n}}(0_{\perp})]|0\rangle\,$
(9)

where the Wilson line

$\displaystyle
S_{n}(b_{\perp})=P\exp\Big{[}ig\int^{\infty}_{0}d(x\cdot\bar{n})n\cdot
A(x\cdot\bar{n},b_{\perp})\Big{]}\,.$ (10)

The full matrix element of the background modes is defined as

$\displaystyle
f_{ij}(x_{B},b_{\perp})=\sqrt{\mathcal{S}(b_{\perp})}\mathcal{B}_{ij}(x_{B},b_{\perp})\,.$
(11)

##  4 NLO calculation

Figure 1: Illustration of our factorization scheme. Green is soft, blue is
collinear. $\mu_{IR}$ and $\rho$ are IR cut-offs. $\nu$ defines the region of
intersection between two collinear modes. The double counting of this region
is removed by the inclusion of the soft factor $\mathcal{S}(b_{\perp})$ in the
factorization formula. A mass-shell hyperbolas are defined by a scale
$\mu^{2}$ via $2k^{+}k^{-}=k^{2}_{\perp}=\mu^{2}$.

We calculate the one loop real and virtual corrections to the beam function
and the soft function using the background field approach. The UV and IR
divergences are regulated using dimensional regularization. This gives us the
scales $\mu_{UV}$ and $\mu_{IR}$. The rapidity divergences are regulated using
the $\eta$-regularization, which for the beam function takes the form

$\displaystyle\int^{\infty}_{0}\frac{dk^{-}}{k^{-}}\to\nu^{\eta}\int^{\infty}_{0}\frac{dk^{-}}{k^{-}}|k^{+}|^{-\eta}\,,$
(12)

with $k^{+}\>=\>{k_{\perp}^{2}\over 2k^{-}}$ and for the soft function,

$\displaystyle\int\frac{dk^{+}dk^{-}}{k^{+}k^{-}}=\nu^{\eta}2^{-\eta/2}\int\frac{dk^{+}dk^{-}}{k^{+}k^{-}}|k^{+}-k^{-}|^{-\eta}\,.$
(13)

One encounters transverse UV and IR divergences in both real and virtual
corrections. However the real corrections only diverge in the UV of rapidity
($k^{-}\rightarrow\infty$) while the virtual corrections diverge in the IR of
rapidity ($k^{-}\rightarrow 0$). Regularization of these rapidity divergences
introduces scales $\nu$ for the rapidity UV divergence and the scale $\rho$
for rapidity IR divergence. The soft factor has a rapidity UV divergence and
hence depends on the scale $\nu$ (see figure 1 for scale structure). The UV
poles are then removed by multiplying the UV renormalization factor $Z_{UV}$
and the 1-loop soft factor $\mathcal{S}^{(1)}(b_{\perp})$ to the beam function
$\mathcal{B}_{ij}(x,b_{\perp})$. The IR poles are absorbed into the
nonpertubative $f_{ij}$ constructed from the background fields. Carrying out
this procedure, one then gets the final result

$\displaystyle f_{ij}(x_{B},b_{\perp},\mu^{2}_{\rm
UV},\zeta)=f_{ij}(x_{B},b_{\perp},\mu^{2}_{\rm
IR},\rho)-4\alpha_{s}N_{c}\int{\textstyle d}\lower
0.12915pt\hbox{\kern-3.80005pt${}^{\scriptstyle-}$}\kern-0.50003pt{}^{2}p_{\perp}e^{ip_{\perp}b_{\perp}}\int^{1}_{0}\frac{dz}{z(1-z)}\int{\textstyle
d}\lower
0.12915pt\hbox{\kern-3.80005pt${}^{\scriptstyle-}$}\kern-0.50003pt{}^{2}k_{\perp}$
$\displaystyle\Big{[}\mathcal{R}^{a}_{ij;lm}(z,p_{\perp},k_{\perp})+\mathcal{R}^{b}_{ij;lm}(z,p_{\perp},k_{\perp})\Big{]}\int
d^{2}z_{\perp}e^{-i(p_{\perp}-k_{\perp})z_{\perp}}f_{lm}(\frac{x_{B}}{z},z_{\perp},\mu^{2}_{\rm
IR},\rho)$
$\displaystyle+\frac{\alpha_{s}N_{c}}{2\pi}\Big{(}-\frac{1}{2}(L^{\mu_{\rm
UV}}_{b})^{2}+L^{\mu_{\rm UV}}_{b}\ln\frac{\mu^{2}_{\rm
UV}}{\zeta^{2}}-\frac{\pi^{2}}{12}\Big{)}f_{ij}(x_{B},b_{\perp},\mu^{2}_{\rm
IR},\rho)-\frac{\alpha_{s}N_{c}}{\pi}L^{\mu_{\rm IR}}_{b}$
$\displaystyle\times\int^{1}_{0}dz\Big{[}\frac{1}{(1-z)_{+}}+\frac{1}{z}\Big{]}f_{ij}(\frac{x_{B}}{z},b_{\perp},\mu^{2}_{\rm
IR},\rho)-\frac{\alpha_{s}N_{c}}{2\pi}\int d^{2}z_{\perp}\int{\textstyle
d}\lower
0.12915pt\hbox{\kern-3.80005pt${}^{\scriptstyle-}$}\kern-0.50003pt{}^{2}p_{\perp}e^{ip_{\perp}(b-z)_{\perp}}$
$\displaystyle\times\Big{(}\frac{1}{2}\ln^{2}\frac{\mu^{2}_{\rm
IR}}{p^{2}_{\perp}}+\ln\frac{\mu^{2}_{\rm
IR}}{p^{2}_{\perp}}\ln\frac{\rho}{\zeta}-\frac{\pi^{2}}{12}\Big{)}\frac{g_{il}p_{j}p_{m}+p_{i}p_{l}g_{mj}}{p^{2}_{\perp}}f_{lm}(x_{B},z_{\perp},\mu^{2}_{\rm
IR},\rho)+\frac{\alpha_{s}N_{c}}{2\pi}\int d^{2}z_{\perp}$
$\displaystyle\times\int{\textstyle d}\lower
0.12915pt\hbox{\kern-3.80005pt${}^{\scriptstyle-}$}\kern-0.50003pt{}^{2}p_{\perp}e^{ip_{\perp}(b-z)_{\perp}}\Big{(}\frac{\beta_{0}}{2N_{c}}\ln\frac{\mu^{2}_{\rm
UV}}{p^{2}_{\perp}}+\frac{67}{18}-\frac{5N_{f}}{9N_{c}}\Big{)}f_{ij}(x_{B},z_{\perp},\mu^{2}_{\rm
IR},\rho)+O(\alpha^{2}_{s})\,$ (14)

with
$L^{\mu}_{b}\equiv\ln\Big{(}\frac{b^{2}_{\perp}\mu^{2}}{4e^{-2\gamma_{E}}}\Big{)}$,
$\mu$ is the $\overline{\rm{MS}}$ scale and $\zeta\>=\>x_{B}P^{+}$. It can be
further shown that the equation reduces to the well-known collinear matching
formula in the limit of small $b_{\perp}$

$\displaystyle f_{1}(x_{B},b_{\perp},\mu^{2}_{\rm
UV},\zeta)=f_{1}(x_{B},0_{\perp},\mu^{2}_{\rm
IR})-\frac{\alpha_{s}N_{c}}{\pi}L^{\mu_{\rm
IR}}_{b}\int^{1}_{0}\frac{dz}{z}P_{gg}(z)f_{1}(\frac{x_{B}}{z},0_{\perp},\mu^{2}_{\rm
IR})$ (15)
$\displaystyle+\frac{\alpha_{s}N_{c}}{2\pi}\Big{(}-\frac{1}{2}(L^{\mu_{\rm
UV}}_{b})^{2}+L^{\mu_{\rm UV}}_{b}\ln\frac{\mu^{2}_{\rm
UV}}{\zeta^{2}}-\frac{\pi^{2}}{12}\Big{)}f_{1}(x_{B},0_{\perp},\mu^{2}_{\rm
IR})+\dots\,$

where $f_{1}(x_{B},b_{\perp})$ is the unpolarized TMDPDF and $P_{gg}(z)$ is
the DGLAP splitting function. Similarly, in the small-x limit it can be shown
that the equation reduces to

$\displaystyle f_{1}(x_{B},p_{\perp},\mu^{2}_{\rm
UV},\zeta)\simeq\mathcal{H}_{1}(p_{\perp},\rho)+\ln\frac{\rho}{\zeta}\int{\textstyle
d}\lower
0.12915pt\hbox{\kern-3.80005pt${}^{\scriptstyle-}$}\kern-0.50003pt{}^{2}k_{\perp}K_{\rm
BFKL}(p_{\perp},k_{\perp})\mathcal{H}_{1}(p_{\perp}-k_{\perp},\rho)$
$\displaystyle+\frac{\alpha_{s}N_{c}}{2\pi}\int
d^{2}b_{\perp}\Big{(}-\frac{1}{2}(L^{\mu_{\rm UV}}_{b})^{2}+L^{\mu_{\rm
UV}}_{b}\ln\frac{\mu^{2}_{\rm
UV}}{\zeta^{2}}-\frac{\pi^{2}}{12}\Big{)}\int{\textstyle d}\lower
0.12915pt\hbox{\kern-3.80005pt${}^{\scriptstyle-}$}\kern-0.50003pt{}^{2}k_{\perp}e^{ik_{\perp}b_{\perp}}\mathcal{H}_{1}(p_{\perp}-k_{\perp},\rho)$
$\displaystyle+\frac{\alpha_{s}N_{c}}{2\pi}\Big{(}\frac{\beta_{0}}{2N_{c}}\ln\frac{\mu^{2}_{\rm
UV}}{p^{2}_{\perp}}+\frac{67}{18}-\frac{5N_{f}}{9N_{c}}\Big{)}\mathcal{H}_{1}(p_{\perp},\rho)\,$
(16)

where $\mathcal{H}$ is the dipole amplitude and $K_{BFKL}$ is the well-known
BFKL kernel.

##  5 Conclusions

In this paper, we study the IR structure of the gluon TMDPDFs. We go beyond
the standard collinear matching procedure, which is performed in the
$b_{\perp}\ll\Lambda^{-1}_{\rm QCD}$ approximation, and develop a new approach
that is valid in a wide region of $x_{B}$ and
$b_{\perp}\lesssim\Lambda^{-1}_{\rm QCD}$. To do that, we employ the
background field method and perform the calculation of the TMDPDFs in the
dilute limit at the NLO order. This gives us a general matching kernel, which
reduces to the known collinear matching kernel and eikonal expansion results
in the appropriate limits.

##  6 Acknowledgnents

We thank Ian Balitsky, Paul Caucal, Florian Cougoulic, Yuri V. Kovchegov,
Farid Salazar, Björn Schenke, Tomasz Stebel, Yossathorn Tawabutr, and Raju
Venugopalan for the discussions.

This material is based upon work supported by The U.S. Department of Energy,
Office of Science, Office of Nuclear Physics through Contract Nos. DE-
SC0012704 and DE-SC0020081, and within the frameworks of Saturated Glue
(SURGE) Topical Collaboration in Nuclear Theory. V.S. thanks the Binational
Science Foundation grant #2021789 for support.

## References

  * [1] Swagato Mukherjee, Vladimir V. Skokov, Andrey Tarasov, and Shaswat Tiwari.  Unified description of DGLAP, CSS, and BFKL evolution: TMD factorization bridging large and small x.  Phys. Rev. D, 109(3):034035, 2024.
  * [2] John C. Collins and Davison E. Soper.  Back-To-Back Jets in QCD.  Nucl. Phys. B, 193:381, 1981.  [Erratum: Nucl.Phys.B 213, 545 (1983)].
  * [3] John C. Collins, Davison E. Soper, and George F. Sterman.  Transverse Momentum Distribution in Drell-Yan Pair and W and Z Boson Production.  Nucl. Phys. B, 250:199–224, 1985.
  * [4] John C. Collins and Davison E. Soper.  The Theorems of Perturbative QCD.  Ann. Rev. Nucl. Part. Sci., 37:383–409, 1987.
  * [5] John C. Collins, Davison E. Soper, and George F. Sterman.  Factorization of Hard Processes in QCD.  Adv. Ser. Direct. High Energy Phys., 5:1–91, 1989.
  * [6] John C. Collins and Davison E. Soper.  Back-To-Back Jets: Fourier Transform from B to K-Transverse.  Nucl. Phys. B, 197:446–476, 1982.
  * [7] John C. Collins and Davison E. Soper.  Parton Distribution and Decay Functions.  Nucl. Phys. B, 194:445–492, 1982.
  * [8] Zhong-Bo Kang and Jian-Wei Qiu.  QCD evolution of naive-time-reversal-odd parton distribution functions.  Phys. Lett. B, 713:273–276, 2012.
  * [9] V. M. Braun, A. N. Manashov, and B. Pirnay.  Scale dependence of twist-three contributions to single spin asymmetries.  Phys. Rev. D, 80:114002, 2009.  [Erratum: Phys.Rev.D 86, 119902 (2012)].
  * [10] Fabio Dominguez, Bo-Wen Xiao, and Feng Yuan.  $k_{t}$-factorization for Hard Processes in Nuclei.  Phys. Rev. Lett., 106:022301, 2011.
  * [11] Fabio Dominguez, Cyrille Marquet, Bo-Wen Xiao, and Feng Yuan.  Universality of Unintegrated Gluon Distributions at small x.  Phys. Rev. D, 83:105005, 2011.