# T-odd Parton Distribution Functions and Azimuthal Anisotropy at High
Transverse Momentum in $p$-$p$ and $p$-$A$ Collisions

Ismail Soudi<EMAIL_ADDRESS>Department of Physics and Astronomy,
Wayne State University, Detroit, MI 48201.  University of Jyväskylä,
Department of Physics, P.O. Box 35, FI-40014 University of Jyväskylä, Finland.
Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki,
Finland.  Abhijit Majumder<EMAIL_ADDRESS>Department of Physics and
Astronomy, Wayne State University, Detroit, MI 48201.

###### Abstract

Various azimuthal anisotropies ($v_{1},v_{2},v_{3},v_{4}$), at high transverse
momentum (high-$p_{T}$), are shown to arise from the asymmetric scattering of
transverse polarized quarks and gluons, arising from unpolarized nucleons (the
Boer-Mulders’ effect) and resulting in unpolarized hadrons (the Collins
effect). Combined with the asymmetric scattering of partons from polarization
independent but transverse momentum dependent (TMD) distributions, we obtain a
possible mechanism to understand the azimuthal anisotropy of hadrons at large
transverse momentum observed in $p$-$p$ collisions. Constraining the ratio of
polarization dependent TMD distributions to polarization independent
distributions by comparing with the data from $p$-$p$ collisions, we find that
a simple $A^{1/3}$ enhancement of the intrinsic transverse momentum
$(k_{\perp}^{2})$ distribution, of the initial state partons from the proton,
straightforwardly yields the azimuthal anisotropy at high $p_{T}$ in $p$-$A$
collisions.

##  I Introduction

One of the foremost signals of the formation of a quark-gluon plasma (QGP)
during a heavy ion collision is the elliptic flow of the bulk medium [1, 2, 3,
4, 5], in semi-central collisions. 111Centrality of the collision indicates
the degree of transverse spatial overlap of the two incoming nuclei (with
$0\%$ indicating most central or maximal overlap, and $100\%$ indicating most
peripheral or a grazing collision). Due to the geometry of the collision, the
overlap region of the two nuclei is not circular, and contains azimuthal
anisotropies in density. Pressure gradients created by these spatial
inhomogeneities lead to azimuthal anisotropies in the momentum distribution
[6], quantified by the Fourier coefficients of the differential hadronic
yield,

$\displaystyle\frac{dN}{d\phi}=\frac{dN}{d\phi}\left(1+\sum_{n}2v_{n}\cos[n(\phi-\Psi_{n})]\right)\;.$
(1)

In the above equation, $\phi$ is the azimuthal angle and $\Psi_{n}$ is the
$n^{\rm th}$ order event plane angle, and $dN$ is the differential yield of
hadrons, in some range of transverse momentum and rapidity.

Due to the azimuthally anisotropic spatial density distribution in semi-
central heavy-ion collisions, hard partons produced in rare collisions, tend
to experience azimuthally asymmetric energy loss [7, 8, 9]. This leads to an
azimuthal anisotropy in both high transverse momentum (high $p_{T}$) hadron
[10] and jet distributions [11, 12]. This high $p_{T}$ anisotropy, goes hand-
in-hand with the angle-integrated, centrality-dependent suppression of the
high $p_{T}$ jet and hadron spectrum in heavy-ion collisions (with greater
suppression appearing in the more central collisions) [13, 14, 15, 16, 17, 18,
19, 20].

Simulations of jet energy loss in a fluid medium [21, 22, 23] compute and
obtain both the angle integrated and angle dependent suppression by changing
the underlying space-time fluid profile used to compute the local jet
transport coefficients [24, 25, 26, 27]. The underlying fluid profile is
obtained by rigorous comparison of bulk evolution simulations with
experimental data [28, 29, 29, 30, 31].

Recently, measurements of $p$-$p$ and $p$-$Pb$ collision events (referred to
as small systems), with high multiplicity ($N\gtrsim 90$), yield a variety of
azimuthal anisotropies [32, 33, 34] in the soft bulk sector
($p_{T}\\!\lesssim\\!3$ GeV). This indicates the possible formation of a QGP
in these small systems. The corroborating evidence from high $p_{T}$ probes
is, however, inconclusive. Experimental data on hadrons at large $p_{T}$ show
a significant azimuthal anisotropy [35, 36], in both $p$-$p$ (measured up to
10 GeV), and $p$-$Pb$ (measured up to 50 GeV) collisions. However, there is no
evidence of suppression in the angle integrated spectrum in $p$-$Pb$
collisions (at high-$p_{T}\gtrsim 5$ GeV)[37, 38, 39].

The origin of an azimuthal anisotropy at high-$p_{T}$, in the absence of any
suppression of the angle integrated spectrum in $p$-$A$, can be found in the
presence of the same anisotropy in $p$-$p$ collisions themselves. In the
current effort, this anisotropy will arise due to the existence of a small
transverse momentum in the initial state: Partons in the proton carry a small
transverse momentum $k_{\perp}$, in addition to the large longitudinal
momentum, which is a fraction ($x$) of the proton’s momentum, i.e.
$p^{+}=xP^{+}$.

This small transverse momentum will lead to azimuthal anisotropies in both the
polarization dependent and polarization averaged cross sections. In this
Paper, we study how transverse momentum dependent parton distribution
functions (TMDPDFs) and fragmentation functions (TMDFFs) [40], can generate
azimuthal anisotropy at high $p_{T}$ in $p$-$p$ and $p$-$Pb$ collisions. In
particular, we find that the combination of the Boer-Mulders’ process of
polarized partons from unpolarized hadrons [41] and the Collins’ process [42]
of polarized parton fragmentation into unpolarized hadrons lead to a large
azimuthal anisotropy. The small transverse momentum $k_{\perp}$ in $p$-$p$
will be enhanced by a factor of $A^{1/3}$ (with $A$ the mass number of the
nucleus) in $p$-$A$ collisions, to account for the possibility of soft pre-
scattering of the hard parton from the proton within the nucleus, prior to the
hard scattering that leads to jet formation.

Due to the small system size, momentum conservation in the initial state
becomes more apparent in the particle distributions from the final state in
$p$-$p$ and $p$-$A$: The transverse momentum $k_{\perp}$ of the hard parton,
that will participate in hard scattering is balanced by the remainder of the
proton. As a result, the net transverse momentum of the produced dijet system
is partially balanced by the net transverse momentum of the bulk system.
Simulations using the PYTHIA generator [43] will be used to show that an
intrinsic transverse momentum will yield both greater particle production over
a range of rapidities, as well as the correlation between the dijet pair
momentum ($\bm{q}_{T}=\bm{p}_{jet\,1}+\bm{p}_{jet\,2}$) and the event plane
angle(s). Momentum conservation will correlate the $\cos(n\phi)$ asymmetries
of the hard sector with the event plane of the bulk system, leading to
corresponding $v_{n}$’s in the hard sector. PYTHIA simulations, currently do
not include polarization dependent matrix elements. Thus, our final results
will be semi-analytic, assuming that the dijet pair angle is completely
correlated with the event plane angle(s).

The paper is organized as follows: In Sec. II, we present the theoretical
framework of polarization dependent and polarization averaged scattering with
TMDPDFs and TMDFFs. We also present the matrix elements for the processes that
will be used. In Sec. III, we present a phenomenological description of TMDPDF
and TMDFFs that will be used to numerically evaluate these quantities. In Sec.
IV, we present the overarching phenomenological picture of how an elliptic
anisotropy can originate in a small system, in the presence of hard
scattering. We also carry out a modified PYTHIA simulation to demonstrate this
picture and correlate the event plane angle(s) with the direction of the net
momentum of the dijets ($\bm{q}_{T}=\bm{p}_{jet\,1}+\bm{p}_{jet\,2}$). In Sec.
V, we present our results for the cross section and the azimuthal anisotropy,
and compare with data from the ALICE and ATLAS collaborations. A summary and
outlook is presented in Sec. VI.

##  II Theoretical Framework

In the standard collinear factorization of hard processes [44], the outgoing
hadron distributions are azimuthally symmetric. The two incoming partons that
participate in the hard scattering, approach head-to-head, perfectly
collinearly, and leave the hard scattering back-to-back. There is no preferred
direction in the azimuthal plane of the initial state. Instead, if one were to
include the more realistic transverse momentum dependent (TMD-) PDFs [45, 46,
47, 48], and FFs [49, 50, 51, 52], the two incoming partons ($a,b$) are no
longer perfectly collinear. Each has a transverse momentum
($\bm{k}^{a}_{\perp},\bm{k}^{b}_{\perp}$). As a result, the direction
$\bm{k}^{a}_{\perp}+\bm{k}^{b}_{\perp}$ breaks the azimuthal symmetry of the
process, resulting in momentum correlations in the transverse plane of the
hadronic scattering.

Following [50, 51], we assume that at high energies the momentum hierarchy
between the longitudinal momenta and the transverse momenta allows the pion
production cross section to factorize into a soft parton correlator and a hard
scattering matrix element

$\displaystyle\frac{d\sigma}{dyd^{2}P_{T}}=\sum_{a,b,c,d}\int\frac{dx_{a}dx_{b}dz}{2\pi^{2}z^{3}s}d^{2}k_{\bot
a}d^{2}k_{\bot b}d^{3}k_{\bot C}$ $\displaystyle\delta(\bm{k}_{\bot
C}\cdot\hat{p}_{c})J(\bm{k}_{\bot C})\Gamma^{\sigma\mu}(x_{a},k_{\bot
a})\Gamma^{\alpha\nu}(x_{b},k_{\bot b})$
$\displaystyle\hat{M}_{\mu\nu\rho}\left(\hat{M}_{\sigma\alpha\beta}\right)^{*}\Delta^{\rho\beta}(z,k_{\bot
C})\delta(\hat{s}+\hat{t}+\hat{u})\;,$ (2)

where $J(\bm{k}_{\bot C})=\frac{(E_{C}^{2}+\sqrt{\bm{p}_{C}^{2}-\bm{k}_{\bot
C}^{2}})^{2}}{4(\bm{p}_{C}^{2}-\bm{k}_{\bot C}^{2})}$ and the Mandelstamn
variables of the partonic system are given by $(\hat{s},\hat{t},\hat{u})$. In
the equation above,
$\hat{M}_{\mu\nu\rho}\left(\hat{M}_{\sigma\alpha\beta}\right)^{*}$ is the hard
partonic squared amplitude, $\Gamma$ are the initial state correlators and
$\Delta$ are the final state fragmentation correlator.

The leading twist gluon generalized PDF reads [53]

$\displaystyle\Gamma^{\mu\nu}_{P}(x,k_{\bot})=$
$\displaystyle\int\frac{d(\xi\cdot
P)d^{2}\xi_{T}}{(2\pi)^{3}}~{}e^{i(p\cdot\xi)}\frac{\langle P|{\rm
Tr}\left[F^{\mu\rho}(0)F^{\nu\sigma}(\xi)\right]|P\rangle
n_{\rho}n_{\sigma}}{(p\cdot n)^{2}}$
$\displaystyle=\frac{-g_{T}^{\mu\nu}f_{g}(x,k_{\bot})+\left(\frac{k_{\bot}^{\mu}k_{\bot}^{\nu}}{M_{p}^{2}}+g_{T}^{\mu\nu}\frac{k_{\bot}^{2}}{2M_{p}^{2}}\right)h_{g}^{\bot}(x,k_{\bot}^{2})}{2x}\;,$
(3)

where $f(x,k_{\bot})$ is the unpolarized gluon distribution. The gluon Boer-
Mulders’ function $h^{\bot}(x,k_{\bot}^{2})$ can be interpreted as the
distribution of linearly polarized gluons in the nucleon. Using the following
gluon polarization vectors,

$\displaystyle\epsilon^{\mu}_{\lambda_{1}}=$
$\displaystyle\left(0,-\lambda_{1}e^{-i\phi},-i\lambda_{1}e^{i\phi},0\right)\;,\;,$
(4) $\displaystyle\epsilon^{\mu*}_{\lambda_{2}}=$
$\displaystyle\left(0,-\lambda_{2}e^{-i\phi},-i\lambda_{2}e^{i\phi},0\right)\;,$
(5)

we can project the gluon generalized PDF onto gluon helicity states

$\displaystyle\Gamma^{\lambda_{1},\lambda_{2}}_{P}(x,k_{\bot})=$
$\displaystyle\frac{-\delta^{\lambda_{1},\lambda_{2}}f_{g}(x,k_{\bot})+\frac{k_{T}^{2}}{2M_{p}^{2}}\delta^{\lambda_{1},-\lambda_{2}}h_{g}^{\bot}(x,k_{\bot}^{2})}{2x}\;.$
(6)

The quark generalized PDF is given by

$\displaystyle\Gamma^{\mu\nu}_{P}(x,k_{\bot})=$
$\displaystyle\int\frac{d(\xi\cdot
P)d^{2}\xi_{T}}{(2\pi)^{3}}~{}e^{i(p\cdot\xi)}\frac{\langle
P|\bar{\psi}(0)\psi(\xi)|P\rangle n^{\mu}n^{\nu}}{(p\cdot n)^{2}}$
$\displaystyle=\frac{1}{2}\left[\not{P}f_{q}(x,k_{\bot})+h_{q}^{\bot}(x,k_{\bot})\frac{\sigma_{\mu\nu}k_{\bot}^{\nu}P^{\nu}}{2M_{p}}\right]\;,$
(7)

where $\sigma_{\mu\nu}=[\gamma_{\mu},\gamma_{\nu}]$. Similar to the gluon, the
distribution of unpolarized quarks is given by $f_{q}(x,k_{\bot})$. The time-
reversal odd function $h_{q}^{\bot}(x,k_{\bot})$ is the quark Boer-Mulders’
function, which can be interpreted as the distribution of transversely
polarized quarks in an unpolarized proton.

Analogously, the fragmentation correlators are written

$\displaystyle\Delta_{g}^{\lambda_{1},\lambda_{2}}(z,k_{\bot})=$
$\displaystyle\frac{-\delta^{\lambda_{1},\lambda_{2}}D_{g}(z,k_{\bot}^{2})+\delta^{\lambda_{1},-\lambda_{2}}\frac{k_{\bot}^{2}}{2M_{\pi}^{2}}H_{g}^{\bot}(z,k_{\bot}^{2})}{2/z},$
(8)
$\displaystyle\Delta_{q}(z,k_{\bot})=\frac{\not{p}D_{q}(z,k_{\bot}^{2})+\frac{\sigma_{\mu\nu}k_{\bot}^{\nu}P^{\nu}}{2M_{p}}H^{\bot}_{q}(z,k_{\bot}^{2})}{2/z}.$
(9)

The TMD fragmentation of unpolarized gluons or quarks to unpolarized pions
with momentum $z\bm{p}_{c}+\bm{k}_{\bot C}$ is given by
$D_{g/q}(z,k_{\bot}^{2})$. The equivalent to the BM function is the Collins’
function $H_{g/q}^{\bot}(z,k_{\bot}^{2})$, which describes the fragmentation
of a transversely polarized gluon or transversely polarized quarks to an
unpolarized pion [42].

###  II.1 Unpolarized partons cross section

For the case of unpolarized partons, the cross section from Eq. (2) can be
simplified further by setting the same helicities between the matrix element
and its complex conjugate, leading to

$\displaystyle\frac{d\sigma}{dyd^{2}P_{T}}=\sum_{a,b,c,d}\int\frac{dx_{a}dx_{b}dz}{16\pi^{2}x_{a}x_{b}z^{2}s}d^{2}k_{\bot
a}d^{2}k_{\bot b}d^{3}k_{\bot C}$ $\displaystyle\delta(\bm{k}_{\bot
C}\cdot\hat{p}_{c})J(\bm{k}_{\bot C})f^{a/A}\left(x_{a},{k}_{\bot
a}\right)f^{b/B}\left(x_{b},{k}_{\bot b}\right)$
$\displaystyle|\hat{M}_{ab}^{cd}|^{2}D^{C/c}\left(z,{k}_{\bot
C}\right)\delta(\hat{s}+\hat{t}+\hat{u})\;,$ (10)

where $|\hat{M}_{ab}^{cd}|^{2}$ is now the usual unpolarized partonic squared
matrix element that can be found in the literature [54].

Due to the transverse momenta of the partons, the partonic center of mass
frame $(\bm{q}=\bm{k}_{a}+\bm{k}_{b})$ is off-center in the transverse plane
of the hadronic scattering. This leads to transverse momentum correlations
between the final state partons. Taking the limit of small transverse momentum
$k_{\bot a}\ll p_{a}$, the incoming partonic momenta, with a small component
in the $x$-direction, can be written
$\bm{p}_{a}=p_{a}(1,\sin\theta_{a},0,\cos\theta_{a})$, where $\theta_{a}$ is
the polar angle in the hadronic center of mass frame. Consider a typical
partonic matrix element squared, where the outgoing parton $c$ is produced at
mid-rapidity ($\theta_{c}=\pi/2$), along an azimuthal angle $\phi_{c}$,

$\displaystyle\mathcal{M}\propto\frac{\hat{s}}{\hat{t}}\simeq\frac{-\hat{s}}{2p_{a}p_{c}(1-\theta_{a}\cos(\phi_{c}))}\;.$
(11)

By expanding the denominator in the small angle limit $\theta_{a}\ll 1$, we
obtain

$\displaystyle\mathcal{M}$
$\displaystyle\simeq\frac{-\hat{s}}{4p_{a}p_{c}}\left[2+\theta_{a}^{2}+2\theta_{a}\cos(\phi_{c})+\theta_{a}^{2}\cos(2\phi_{c})\right]\;.$
(12)

These azimuthal correlations generated by the intrinsic transverse momentum
are suppressed by powers of the large transverse momentum, as
$\theta_{a}=\nicefrac{{k_{\bot a}}}{{p_{a}}}$. Thus, while azimuthal
anisotropies do arise in unpolarized cross sections, they are suppressed by
powers of the hard scale, and have a diminishing contribution at high-$p_{T}$.

###  II.2 Polarized partonic cross section

Since polarized distributions change the helicity between the matrix elements
and its complex conjugate (c.f. Eqs. (6-II)), inclusive unpolarized pion
production requires two polarized distributions of the same parton species.
Either both initial partons are polarized or one initial parton and one final
parton are polarized, which we will refer to as $\rm BM\otimes BM$ or $\rm
BM\otimes C$ effects, respectively. In the following four sections we will
present the relevant contributions for gluon initiated processes and quark
initiated processes.

####  II.2.1 Gluon initiated: Boer-Mulders’ Collins’ effect

We start with the $\rm BM\otimes C$ effect, where a polarized gluon, stemming
from a Boer-Mulders’ function, scatters with an unpolarized gluon, producing a
polarized gluon that fragments to an unpolarized pion, with a Collins’
function. Here, the matrix element times the PDFs and FF is written as,

$\displaystyle\Sigma^{\rm BM\otimes C}={\rm Re}\sum_{\lambda=\pm
1}H^{\bot(1)}_{C/g}(z,k_{\bot C})$
$\displaystyle\times\left[h^{\bot(1)}_{g/A}(x_{a},k_{\bot
a}^{2})f_{g/B}(x_{b},k_{\bot
b}^{2})\hat{M}_{\lambda,\lambda}^{\lambda,\lambda}\left(\hat{M}_{-\lambda,\lambda}^{-\lambda,\lambda}\right)^{*}\right.$
$\displaystyle\left.+f_{g/A}(x_{a},k_{\bot
a}^{2})h^{\bot(1)}_{g/B}(x_{b},k_{\bot
b}^{2})\hat{M}_{\lambda,\lambda}^{\lambda,\lambda}\left(\hat{M}_{\lambda,-\lambda}^{-\lambda,\lambda}\right)^{*}\right]\;.$
(13)

We list the Matrix element times the PDFs and FF for gluon-gluon

$\displaystyle\mathcal{T}^{\rm BM\otimes C}_{gg\to gg}=$ $\displaystyle
g_{s}^{4}\frac{N^{2}}{N^{2}-1}\frac{t^{2}+tu+u^{2}}{t^{2}}e^{-4\lambda
i(\phi_{12}-\phi_{23})}\;,$ (14)

where using the momentum angle $\bm{p}=(p,\theta,\phi)$, we write

$\displaystyle\tan\phi_{ij}=\tan\frac{\phi_{j}-\phi_{i}}{2}\frac{\sin\frac{\theta_{j}-\theta_{i}}{2}}{\sin\frac{\theta_{j}+\theta_{i}}{2}}\;.$
(15)

After performing the helicity sums, the contribution to the cross section
becomes

$\displaystyle\Sigma^{\rm BM\otimes C}_{gg\to gg}=2\hat{D}^{C/g}(z,k_{\bot
C})$ $\displaystyle\Bigl{[}h^{\bot(1)}_{g/A}(x_{a},k_{\bot
a}^{2})f_{g/B}(x_{b},k_{\bot
b}^{2})\hat{M}^{0}_{1}\hat{M}^{0}_{2}\cos\left(4(\phi_{12}-\phi_{23})\right)$
$\displaystyle+f_{g/A}(x_{a},k_{\bot a}^{2})h^{\bot(1)}_{g/B}(x_{b},k_{\bot
b}^{2})\hat{M}^{0}_{1}\hat{M}^{0}_{3}\cos\left(4(\phi_{12}-\phi_{13})\right)\Bigr{]}.$
(16)

We define matrix elements stripped of their angular correlations,

$\displaystyle\hat{M}^{0}_{1}\hat{M}^{0}_{2}=\frac{9}{2}g_{s}^{4}\left[1+\frac{u}{t}+\frac{u^{2}}{t^{2}}\right]\;,$
(17)
$\displaystyle\hat{M}^{0}_{1}\hat{M}^{0}_{3}=\frac{9}{2}g_{s}^{4}\left[1+\frac{t}{u}+\frac{t^{2}}{u^{2}}\right]\;.$
(18)

####  II.2.2 Gluon initiated: Double Boer-Mulders’ effect

Another contribution to the cross section comes from the scattering of two
polarized gluons leading to unpolarized gluons in the final state. The matrix
element times the PDFs and FF is now written

$\displaystyle\Sigma^{\rm BM\otimes BM}_{gg\to gg}=$
$\displaystyle\sum_{\lambda=\pm 1}h^{\bot(1)}_{g/A}(x_{a},k_{\bot
a}^{2})h^{\bot(1)}_{g/B}(x_{b},k_{\bot b}^{2})$
$\displaystyle\hat{M}_{\lambda,\lambda}^{\lambda,\lambda}\left(\hat{M}_{-\lambda,-\lambda}^{\lambda,\lambda}\right)^{*}D_{C/g}(z,k_{\bot
C})\;,$ (19)

with the following matrix element

$\displaystyle\mathcal{T}^{\rm BM\otimes BM}_{gg\to gg}=$ $\displaystyle
g_{s}^{4}\frac{N^{2}}{N^{2}-1}\frac{t^{2}+tu+u^{2}}{s^{2}}e^{-4\lambda
i(\phi_{23}-\phi_{13})}\;.$ (20)

Performing the helicity sums, yields

$\displaystyle\Sigma^{\rm BM\otimes BM}_{gg\to
gg}=4h^{\bot(1)}_{g/A}(x_{a},k_{\bot a}^{2})h^{\bot(1)}_{g/B}(x_{b},k_{\bot
b}^{2})D_{C/g}(z,k_{\bot C})$
$\displaystyle\hat{M}^{0}_{2}\hat{M}^{0}_{3}\cos\left(4(\phi_{13}-\phi_{23})\right)\;,$
(21)

with

$\displaystyle\hat{M}^{0}_{2}\hat{M}^{0}_{3}=\frac{9}{2}g_{s}^{4}\left[\frac{t^{2}+tu+u^{2}}{s^{2}}\right]\;.$
(22)

####  II.2.3 Quark initiated: Boer-Mulders’ Collins’ effect

For the quark-quark scattering, the square matrix element of the $BM\otimes C$
effect lead to odd number of Dirac matrices and only the interference of the
$t$\- and $u$-channel diagrams contribute to the cross section. The matrix
element times the PDFs and FF is written

$\displaystyle\Sigma^{\rm BM\otimes C}_{qq\to qq}=2H^{\bot(1)}_{C/q}(z,k_{\bot
C})\biggl{[}\hat{M}^{0}_{1}\hat{M}^{0}_{2}h^{\bot(1)}_{q/A}(x_{a},k_{\bot a})$
$\displaystyle\times f_{q/B}(x_{b},k_{\bot
b})+\hat{M}^{0}_{1}\hat{M}^{0}_{3}f_{q/A}(x_{a},k_{\bot
a})h^{\bot(1)}_{q/B}(x_{b},k_{\bot b})\biggr{]}.$ (23)

The matrix elements are given by the trace of ten Dirac matrices which
evaluates to

$\displaystyle\hat{M}^{0}_{1}\hat{M}^{0}_{2}=$
$\displaystyle-\frac{8}{27}\frac{1}{tu}\biggl{[}t\left(\bm{k}_{\bot
a}\cdot\bm{p}_{2}\right)\left(\bm{k}_{\bot C}\cdot\bm{p}_{4}\right)$
$\displaystyle+t\left(\bm{k}_{\bot a}\cdot\bm{p}_{2}\right)\left(\bm{k}_{\bot
C}\cdot\bm{p}_{2}\right)-su(\bm{k}_{\bot a}\cdot\bm{k}_{\bot C})\biggr{]}\;,$
(24) $\displaystyle\hat{M}^{0}_{1}\hat{M}^{0}_{3}=$
$\displaystyle-\frac{8}{27}\frac{1}{tu}\biggl{[}u\left(\bm{k}_{\bot
b}\cdot\bm{p}_{1}\right)\left(\bm{k}_{\bot C}\cdot\bm{p}_{4}\right)$
$\displaystyle+u\left(\bm{k}_{\bot b}\cdot\bm{p}_{1}\right)\left(\bm{k}_{\bot
C}\cdot\bm{p}_{1}\right)-st(\bm{k}_{\bot b}\cdot\bm{k}_{\bot C})\biggr{]}\;.$
(25)

The scalar products above lead to an azimuthal correlation of the form
$\cos(\phi_{\bm{k}_{\bot a}})$ and $\cos(\phi_{\bm{k}_{\bot b}})$, which
produces a contribution to the $v_{1}$ in the azimuthal distribution (c.f.
Appendix A).

####  II.2.4 Quark initiated: Double Boer-Mulders’ effect

Similarly, the quark $BM\otimes BM$ effect is given by the interference of the
$t$\- and $u$-channel diagrams which leads to the following Dirac trace

$\displaystyle\Sigma^{\rm BM\otimes C}_{qq\to qq}=$ $\displaystyle
2D^{C/q}(z,k_{\bot C})h^{\bot(1)}_{q/A}(x_{a},k_{\bot
a})h^{\bot(1)}_{q/B}(x_{b},k_{\bot b})$
$\displaystyle\times\hat{M}^{0}_{2}\hat{M}^{0}_{3}\;,$ (26)

with the matrix element

$\displaystyle\hat{M}^{0}_{2}\hat{M}^{0}_{3}=$
$\displaystyle-\frac{8}{27}\frac{1}{tu}\biggl{[}s\left(\bm{k}_{\bot
a}\cdot\bm{p}_{3}\right)\left(\bm{k}_{\bot b}\cdot\bm{p}_{4}\right)$
$\displaystyle+s\left(\bm{k}_{\bot a}\cdot\bm{p}_{4}\right)\left(\bm{k}_{\bot
b}\cdot\bm{p}_{3}\right)+tu(\bm{k}_{\bot a}\cdot\bm{k}_{\bot b})\biggr{]}\;.$
(27)

Here the scalar products lead to azimuthal correlation of the form
$\cos(\phi_{\bm{k}_{\bot a}})\times\cos(\phi_{\bm{k}_{\bot b}})$ which
produces a contribution to the $v_{2}$ in the azimuthal distribution (c.f. A).

All the matrix elements listed above for quark and gluon scattering will have
to be convoluted with the appropriate TMDPDF and TMDFF, according to Eqs. (2).
These non-perturbative functions, typically obtained from fits to data from
other experiments, are not well constrained in many cases. In the subsequent
section, we will describe how these functions are parameterized and how we
modify these for use in $p$-$A$ collisions. Final numerical results are
presented in Sec. V.

##  III Phenomenology of TMDPDF and TMDFF

In the preceding section, we outlined the matrix elements of all the processes
that could contribute to high-$p_{T}$ hadron production in $p$-$p$ (and
$p$-$A$) collisions. These include both polarization averaged and polarization
dependent contributions. To complete the process, these matrix elements have
to be convoluted with the appropriate TMDPDF and TMDFF.

In this section, we will discuss our chosen parameterizations for both the
polarization averaged and polarization dependent PDF and FF. It will come as
no surprise that some of these functions are not well constrained by
experimental data. Our choice of parameterization is further constrained by
the fact that our parameterization also has to be easily modified to the case
of $p$-$A$. As implemented below, this will force us to use factorized ($x$
and $k_{\perp}$) distributions.

###  III.1 Mean transverse momentum

Figure 1: Different models of the mean transverse momentum $\langle
k_{\perp}^{2}\rangle$ for $p$-$p$ and $p$-$A$ collisions where one side is
enhanced by $A^{1/3}$. The dot-dashed lines indicate the case of
$x$-independent $\langle k_{\perp}^{2}\rangle=A^{1/3}$ GeV. Solid lines
indicate the $x$-dependent model of Eq. (III.1). The dotted lines indicate
what would happen to this model if the model of $x<x_{0}$ is allowed to apply
for the entire range of $x$. The solid blue line is the hard kinematic bound
in Eq. (30).  Figure 2: Mean transverse momentum as a function of the hard
scale $Q$, obtained by carrying out the averaging of $\langle
k_{\perp}^{2}\rangle(x,Q^{2})$, over the allowed range of $x$ as described in
Eq. (33). Due to the shift of parton distribution functions to smaller $x$ at
larger $Q$, the mean transverse momentum gradually reduces with increasing
$Q$.

There have been several experimental and Lattice QCD results that have enabled
multi-dimensional parameterizations of TMDPDFs [$f(x,k_{\perp}^{2},Q^{2})$,
where the $x$ and $k_{\perp}$ dependence is not factorized] [55, 56]. However,
there are no parameterizations of un-factorized TMDPDFs including nuclear
effects, that can be used for $p$-$A$ collisions. There are two specific
effects that concern us, the modification of the $x$ distribution of partons
in the nucleus $A$, referred to as shadowing [57], and the modification of the
transverse momentum of a parton in the $p$ undergoing pre-scattering in the
nucleus prior to the hard exchange that leads to dijets, referred to as the
Cronin effect [58]. In the absence of a multiple scattering formalism that can
simultaneously describe both these effects, we are left with a factorized
parameterization of both these effects.

In the literature, for the case of factorized distributions, the transverse
momentum dependence is typically approximated by a Gaussian _Ansatz_ [59, 52]

$\displaystyle f(x,k_{\bot},Q^{2})=$ $\displaystyle\frac{4\pi}{\langle
k_{\bot}^{2}\rangle}e^{-\frac{k_{\bot}^{2}}{\langle
k_{\bot}^{2}\rangle}}f(x,Q^{2})\;.$ (28)

In the presence of multiple nucleons, partons may undergo multiple scattering
before the hard exchange that leads to the energetic dijets. For the case of
polarization averaged processes, it has been demonstrated that the mean
transverse momentum in $p$-$A$ collisions is enhanced by a factor $A^{1/3}$
[60, 61, 62], i.e.,

$\displaystyle\langle k_{\perp}^{2}\rangle_{pA}\simeq A^{1/3}\langle
k_{\perp}^{2}\rangle_{pp}\;.$ (29)

In order to make sure the partons are generally collinear with the proton, we
impose a kinematic constraint on the transverse momentum of the partons [51]

$\displaystyle k_{\bot}\leq\sqrt{s}\times{\rm
min}\left(x,\sqrt{x(1-x)}\right)\;.$ (30)

While the factorization of the transverse momentum simplifies the calculation
significantly, due to kinematic constraints the low-$x$ partons will not
contribute to the cross section.

In an attempt to capture the complex relation between the transverse momentum
and the momentum fraction, we use an $x$-dependent $\langle
k_{\bot}^{2}\rangle$ _ansatz_. A parameterization of the mean transverse
momentum in $p$-$p$ collisions was presented in Ref. [52] as follows

$\displaystyle\langle k_{\bot}^{2}\rangle^{1/2}(x)=\langle
k_{\bot}^{2}\rangle_{0}^{1/2}x^{0.8}(1-x)^{0.15}\;.$ (31)

However, this _ansatz_ can become large at intermediate $x$ as shown by the
doted lines in Fig. 1.

Multiple scattering in deconfined matter leads to transverse momentum
broadening governed by the transport coefficient $\hat{q}$ [63]. The mean
transverse momentum is proportional to $\sqrt{\hat{q}L}$, where $L$ is the
length of the medium. Perturbatively, $\hat{q}$ is governed by a logarithmic
dependence $(\log E/E_{0})$ on the energy of the hard parton $E$ and a soft
screening scale $E_{0}$ [63]. Guided by this result, we modify the
$x$-dependent _ansatz_ of Ref. [52] as follows222For momentum fractions in the
range $x_{0}\leq x\leq 10x_{0}$, we use a hyperbolic tangent
$\left(\tanh\frac{x-x_{0}}{x_{0}}\right)$ to smoothly match the logarithmic
behavior to the power-law.,

$\displaystyle\langle k_{\bot}^{2}\rangle^{1/2}(x)=\langle
k_{\bot}^{2}\rangle_{0}^{1/2}\left((1-x)\sqrt{\frac{s}{s_{0}}}\right)^{0.15}A^{1/6}$
$\displaystyle\times\begin{cases}\left(\frac{x}{x_{0}}\sqrt{\frac{s}{s_{0}}}\right)^{0.8}\;,&\text{
if }x\leq x_{0}\\\
\left[1+b_{0}\log\left(\frac{x}{x_{0}}\sqrt{\frac{s}{s_{0}}}\right)\right]^{1/2}\;,&\text{
if }x>x_{0}\end{cases}$ (32)

where we take $\langle k_{\bot}^{2}\rangle_{0}=5$ GeV2, $x_{0}=10^{-2}$,
$\sqrt{s_{0}}=8.16$ TeV and $b_{0}=2$. This is indicated by the solid red line
in Fig. 1. Using these constants one can reproduce the distribution of Ref.
[52] at $x\leq x_{0}$.

Since Eq. (III.1) involves non-trivial $x$ dependence, the true mean
transverse momentum of the PDF is not directly the value of $\langle
k_{\bot}^{2}\rangle^{1/2}(x)$. To obtain what will refer to as the effective
mean transverse momentum, one needs to integrate the full PDF as follows

$\displaystyle\langle k_{T}^{2}\rangle(Q)=\frac{\int dx_{a}\langle
k_{\bot}^{2}\rangle(x_{a})f(x_{a},Q)}{\int dx_{a}f(x_{a},Q)}\;,$ (33)

where $Q$ is the hard scale of the PDF. In contrast to Fig. 1, the curve we
obtain in Fig. 2 leads to a mean transverse momentum lower than $2$ GeV for
all scales. This is because the support of the PDF becomes more and more
concentrated at low $x$ as the scale increases.

###  III.2 Fragmentation Functions

The theoretical description of the production of pions from the hadronic
scatterings, requires fragmentation functions as described in Eq. (II).
Collinear and TMD unpolarized fragmentation functions have been studied
extensively and are well constrained by data [64, 65, 66]. In order to
simplify the analysis, we will employ the same factorized Gaussian _Ansatz_
for the TMD fragmentation functions, as we have done for PDFs, i.e.

$\displaystyle D_{C/c}(z,k_{\bot C}^{2})=\frac{4\pi}{\langle k_{\bot
C}^{2}\rangle}e^{-\frac{k_{\bot C}^{2}}{\langle k_{\bot
C}^{2}\rangle}}D_{C/c}(z)\;.$ (34)

One can also consider a $z$-dependent $\langle k_{\bot C}^{2}\rangle$ _ansatz_
for the fragmentation functions [52]. However, this plays a very minor role
for the processes that are being studied. Since the pion transverse momentum
$\bm{k}_{\bot C}$ is transverse to the momentum of the parton $c$, the
fragmentation functions will not contribute to the azimuthal anisotropies in
the plane transverse to the beam axis. For the $z$-dependence of the
fragmentation functions, we will use the KKP parameterization [64].

###  III.3 Polarized distributions

While the unpolarized collinear and TMD distributions can be extracted from
global fits to data, the polarized Boer-Mulders’ and Collins’ functions arise
in sub-leading effects making them harder to extract. However, constraints on
the positivity of the density matrix lead to a bound on the polarized
distributions known as the Soffer bound [67, 50]

$\displaystyle h^{\bot(1)}_{q}(x,k_{\bot}^{2})\equiv$
$\displaystyle\frac{k_{\bot}}{2M_{p}}h^{\bot}_{q}(x,k_{\bot}^{2})\leq
f_{q}(x,k_{\bot}^{2})\;,$ (35) $\displaystyle
h^{\bot(1)}_{g}(x,k_{\bot}^{2})\equiv$
$\displaystyle\frac{k_{\bot}^{2}}{2M_{p}^{2}}h^{\bot}_{g}(x,k_{\bot}^{2})\leq
f_{g}(x,k_{\bot}^{2})\;,$ (36) $\displaystyle
H^{\bot(1)}_{q}(x,k_{\bot}^{2})\equiv$
$\displaystyle\frac{k_{\bot}}{2M_{p}}H^{\bot}_{q}(x,k_{\bot}^{2})\leq
D_{q}(x,k_{\bot}^{2})\;,$ (37) $\displaystyle
H^{\bot(1)}_{g}(x,k_{\bot}^{2})\equiv$
$\displaystyle\frac{k_{\bot}^{2}}{2M_{p}^{2}}H^{\bot}_{g}(x,k_{\bot}^{2})\leq
D_{g}(x,k_{\bot}^{2})\;.$ (38)

For the present work, we will consider the polarized distributions to be
proportional to the unpolarized distributions, i.e.
$h^{\bot(1)}(x,k_{\bot}^{2})=bf_{q}(x,k_{\bot}^{2})$ and
$H^{\bot(1)}(x,k_{\bot}^{2})=BD_{q}(x,k_{\bot}^{2})$. Since all the processes
that we consider are proportional to the product of two polarized
distributions, we will only modulate the overall factor of333For the process
with two Boer-Mulders functions we will consider that $B\cdot B=B\cdot b$.
$B\cdot b$. Once may assume that $B\simeq b$. We present results for different
values ($0.2\leq B\cdot b\leq 0.4$) in Sec. V.

##  IV Bulk simulations and elliptic flow

In the preceding section, we have described the 2-to-2 scattering of polarized
and unpolarized partons ($a+b\rightarrow c+d$), with transverse momentum in
the initial state. If the two out-going hard partons have transverse momenta
$\bm{p}_{T,c}$ and $\bm{p}_{T,d}$, then there _will_ be an asymmetry in the
azimuthal angle $\phi$ between $\bm{q}_{T}=\bm{p}_{T,c}+\bm{p}_{T,d}$ and
$\bm{Q}_{T}=\bm{p}_{T,c}-\bm{p}_{T,d}\approx 2\bm{p}_{T,c}$ [45, 46].

However, the angle $\phi_{q_{T}}$, which indicates the direction of the vector
$\bm{q}_{T}$, varies from event to event. The azimuthal asymmetry in the angle
between $\bm{q}_{T}$ and $\bm{Q}_{T}$ will yield anisotropy coefficients
($v_{n}$) if there is a correlation between $\phi_{q_{T}}$ and the event plane
angles $\Psi_{n}$. That is, if the system can be reoriented in each event,
based on the detected particles in that event. In this section, we will
demonstrate that $\phi_{q_{T}}$ is strongly correlated with the event plane
angles of the colliding system.

###  IV.1 Elliptic flow in small systems

Figure 3:  Left: A cartoon of a $p$-$p$ collision with relative transverse
momentum in the initial state. The blue proton indicated by the solid blue
circle comes out of the page, and contains a hard parton (blue dot) with
transverse momentum, recoiling off the rest of the proton represented by the
blue blob moving in the opposite direction. The net transverse momentum of the
proton is zero, and so the hard parton and the remainder of the proton are
balanced in transverse momentum. The red proton going into the page has a
similar situation with its hard parton and remainder also balanced in
transverse momentum. Right: the final state after the collision can be
separated in a part containing the dijet pair resulting from the scattering of
the two hard partons recoiling off the remainder of the colliding system.

Azimuthal angular distributions of particles are expressed in terms of the
Fourier series

$\displaystyle\frac{dN}{d\phi
dp_{T}dy}=\frac{dN}{dp_{T}dy}\left(1+2\sum_{n}v_{n}\cos[n(\phi-\Psi_{n})]\right)\;,$
(39)

where the $\Psi_{n}$ are event plane angles, with $dN$ the differential yield
of particles with momentum given as $p_{z}=\sqrt{p_{T}^{2}+m^{2}}\sinh(y)$ and
$p_{x}=p_{T}\cos(\phi),p_{y}=p_{T}\sin(\phi)$.

In the case of a heavy-ion collision, the origin of azimuthal anisotropies in
the momentum distribution of particles is now well understood: An initial
geometric asymmetry, i.e. a spatially asymmetric density distribution leads to
pressure gradients, which in turn lead to an asymmetric momentum distribution.
In the case of small systems, there is no geometrical asymmetry in the initial
state. In events without hard scattering, any asymmetries are generated
entirely from fluctuations in the initial state [68]. However, the presence of
a hard scattering leads to a much larger multiplicity, that overwhelms the
yield of particles produced in entirely soft interactions [69]. In the case of
collisions with hard partonic scattering, multiplicity or transverse energy
production at almost all rapidities is dominated by hadrons that originate in
the hard scattering, or are connected, in some way, to the hard scattering,
e.g., in the breaking of strings that stretch from the hard high $p_{T}$
parton to the remnants that go down the beam line [70].

Particle production, and the eventual momentum asymmetries, that are effected
by the hard sub-process, may be described as follows: Let us consider a hard
parton carrying momentum fraction $x_{a}$ of the proton $A$. If the hard
parton possesses, in addition, a transverse momentum $\bm{k}_{\bot,a}$,
orthogonal to the proton’s momentum, the remaining partons (and any other
constituents) inside the proton must have the same magnitude of transverse
momentum, but in the opposite direction, i.e., $-\bm{k}_{\bot,a}$.

On the left side of Fig. 3 we present the initial state of a proton-proton
scattering where the blue proton with a solid boundary is coming out of the
page, the red proton with the dotted boundary goes into the page. The hard
parton is indicated by the solid blue/red dot and the remainder of the proton
is indicated by the blue/red blob. The balanced transverse momenta between the
hard parton and the remainder, in the blue proton, are indicated by the
opposing blue arrows. The opposing red arrows indicate a similar situation in
the red proton. In the illustration, we show the hard parton as physically
separated from the remainder of the photon. This is just for illustration, the
hard parton may indeed be within the vicinity of the remainder.

After the scattering, hard partons produce 2 back-to-back jets in the right
panel of Fig. 3. The center of mass of the final state partons is shifted by
the transverse momentum
$\bm{q}_{T}=\bm{k}_{\bot,a}+\bm{k}_{\bot,b}=\bm{p}_{\bot,c}+\bm{p}_{\bot,d}$.
While the entire remainder does not participate in the collision, one expects
that the total transverse momentum of the bulk is generally opposite to the
hard scattering $\sim-\bm{q}_{T}$.

In the illustration described above (and in Fig. 3), we have considered the
case of a single 2-to-2 hard scattering, with the remainder of the colliding
system being soft. This does not have to be the case, the remainder of the
participating system may engender multiple hard scatterings as well. In this
case, the dots in our illustration indicate the two partons that undergo the
hardest exchange. The blob will represent the remainder of hard and soft
scatterings. None of our considerations above are affected by this. We will
investigate these assumptions further by considering a simple PYTHIA [43]
simulation in the subsequent subsec. IV.2.

###  IV.2 PYTHIA simulations

In this paper, we have studied the azimuthal distribution of high-$p_{T}$
hadrons produced in the scattering of hard partons that possess transverse
momentum in the initial state. To experimentally determine this azimuthal
distribution, one needs a reference direction in the azimuthal plane. Since
the incoming state consists of unpolarized protons, no reference angle is
provided by the initial state; it has to be determined from the distribution
of hadrons in the final state.

In the preceding subsection, we argued that the vector
$\bm{q}_{T}=\bm{k}_{\bot,a}+\bm{k}_{\bot,b}=\bm{p}_{\bot,c}+\bm{p}_{\bot,d}$
would be correlated with the event plane angle(s). In this subsection, we
demonstrate this with a PYTHIA simulation [43]. As will be demonstrated in the
next section, the largest component of our calculated azimuthal asymmetry is
for the case of polarization dependent scattering. Such simulations are not
possible using PYTHIA, or any other event generator that correlates hard and
soft particle production. Our simulations will thus demonstrate the angular
correlation only for the case of polarization averaged scattering, which will
constitute a small component of the calculated azimuthal anisotropy. Also, in
PYTHIA simulations with multiple interactions turned on, the initial
transverse momentum of each parton that eventually encounters a hard
scattering, is balanced of its sole remnant and not distributed among other
partons and remnants in the same proton.

To overcome the above limitations of typical PYTHIA simulations, we set up
events where initial state radiation, final state radiation and multiple
parton interactions are turned off. Each proton now consists of a hard parton
that will participate in the hard scattering (small dots in Fig. 3), other
partons that do not participate in the hard scatterings, and diquarks, that
along with the non-participating partons, constitute the remnant or reminder
of the proton (blobs in Fig. 3).

The entire events thus consist of two hard partons which undergo a hard
scattering and remnants that proceed down the beam line. To simulate TMD
effects, we generate a transverse momentum $k_{Ti}$ for each remnant, from a
Gaussian distribution with a width of $\langle k_{\bot}^{2}\rangle^{1/2}$. The
total transverse momentum of all remnants $\bm{q}_{T}=\sum_{i}\bm{k}_{Ti}$ is
compensated by applying a Lorentz boost to the center of mass frame of the
hard scattering with the velocity

$\displaystyle\bm{v}=\frac{\bm{q}_{T}}{\sqrt{(E_{a}+E_{b})^{2}+\bm{q}_{T}^{2}}}\;.$
(40)

This leads to initial state partons with a transverse momentum following a
Gaussian distribution with a width comparable to the remnants’ width. By using
a boost to generate the transverse momentum of the initial state partons, we
ensure that the kinematics of the scattering is unmodified, such that we
ensure energy-momentum conservation in the hard-scattering while balancing the
transverse momentum between the hard scattering and the remnants partons.

The above equation also does not work in the case there is initial state
radiation, as the hard partons are then not directly balanced off the
remnants. The remnants and final state partons are hadronized in PYTHIA using
the Lund string model [71], where strings are connected between the hard
outgoing partons and the beam remnants.

###  IV.3 Event plane angle

As described in the previous subsection, we have generated events where the
hard scattering system of two partons has a net transverse momentum,
$\bm{q}_{T}=\bm{p}_{\bot,c}+\bm{p}_{\bot,d}$, where $\bm{p}_{\bot,c}$ and
$\bm{p}_{\bot,d}$ are the large momenta of the outgoing partons, which are now
no longer back-to-back. The remainder of the event, which consists of beam
remnants that are connected via strings with the hard partons and remnants
which are not connected to a hard scattering, will recoil with a net
transverse momentum $-\bm{q}_{T}$.

Figure 4:  Histogram of the correlation between the event plane angle and the
azimuthal angle of the total transverse momentum for $\langle
k_{\bot}^{2}\rangle^{1/2}=1$ GeV (left) and $\langle
k_{\bot}^{2}\rangle^{1/2}=5$ GeV (right) using the event plane angle
$\Psi_{1}$ (top) and $\Psi_{2}$ (bottom).

Once the entire event is hadronized, the event plane angle $\Psi_{n}$ in Eq.
(39) can be obtained using the flow vector $\bm{Q}_{n}$ [72] defined as
follows,

$\displaystyle\bm{Q}_{n}=\frac{1}{\sqrt{M}}\sum_{j}^{M}e^{in\phi_{j}}\;,$ (41)

where the sum runs over all hadrons in the event (typically over some rapidity
range) and $\phi_{j}$ is the azimuthal angle of the hadron $j$. The event
plane angle is then defined as the Euler angle of the flow vector $\bm{Q}_{n}$
as follows

$\displaystyle\Psi_{n}=\frac{1}{n}\arg[\bm{Q}_{n}]\;.$ (42)

In Fig. 4, we compute correlation histograms of the values of the event plane
angle $\Psi_{n}$ and the azimuthal angle of the total transverse momentum
$\bm{q}_{T}$. For both $\Psi_{1}$ and $\Psi_{2}$, we observe a clear
correlation between the two angles which becomes more pronounced as the mean
transverse momentum is increased $\langle k_{\bot}^{2}\rangle^{1/2}=5$ GeV.
This implies that in the presence of hard scattering, leading to two high
$p_{T}$ jets, the net transverse momentum of the jets
$\bm{q}_{T}=\bm{p}_{\bot,c}+\bm{p}_{\bot,d}$, is tightly correlated with the
$1^{\rm st}$ and $2^{\rm nd}$ order event plane (it is also correlated with
higher order event planes, though more weakly). Thus, in events with a hard
scattering, the direction of the $\bm{q}_{T}$ vector can be determined, by
analyzing all the hadrons in an event, and thus the azimuthal distribution of
$\bm{p}_{\bot c/d}$ with respect of $\bm{q}_{T}$ can indeed be determined.

###  IV.4 Centrality selection

While not a central issue in our calculations, our PYTHIA simulations reveal
another interesting feature of these small system collisions: The correlation
between the transverse momentum of the incoming partons and the transverse
energy produced in the rapidity of forward calorimeters. There are several
methods used by experiments to select the centrality of the collision. One
such method is based on the transverse energy in the forward calorimeter
(FoCal) which measures hadrons in the pseudo-rapidity range $3.9\leq|\eta|\leq
5.0$. Transverse momentum of the initial hard partons can lead to additional
transverse energy in the FoCal.

Figure 5: Average transverse energy $\langle E_{T}\rangle=E\sin\theta$ in the
forward direction $3\leq|\eta|\leq 10$ as a function of the average transverse
momentum of the initial hard parton $\langle k_{T}^{2}\rangle^{1/2}$.

Using the results of the simulations in the previous subsection, we compute
the transverse energy in each event as,

$\displaystyle E_{T}=\sum_{i}E_{i}\sin\theta_{i}\;,$ (43)

where $E_{i}$ is the energy of the final state hadron and $\theta_{i}$ is the
angle between the hadron momentum and the $z$-axis. The sum runs over all
hadrons in the event within a prescribed range in (pseudo-)rapidity. The
average transverse energy $\langle E_{T}\rangle$ is shown in Fig. 5 as a
function of the average transverse momentum of the initial hard parton
$\langle k_{T}^{2}\rangle^{1/2}$. We find a clear linear relation between the
initial transverse momentum of the hard parton and the transverse energy in
the forward direction.

Experimental data on the elliptic anisotropy, from the ATLAS detector [36],
that will be compared against, are extracted from the most central (0-5%)
events in $p$-$Pb$ collisions. In these experiments, centrality is determined
from the amount of transverse energy deposited in the FoCal. The results of
our simulations above indicate that larger transverse energy deposited in the
FoCal isolates events where the mean intrinsic transverse momentum of the
initial state partons is larger. These are events where the correlation
between the direction of $\bm{q}_{T}$ and the event plane angles is more
enhanced (See Fig. 4). This further enhances our argument that the angle of
$\bm{q}_{T}$ can be determined in experiment.

##  V Results

In the previous section, we argued that events with hard scattering that
contain a dijet pair recoiling off the remainder of the colliding protons will
have a well defined event plane, with several event plane angles coinciding
with the angle of the dijet momentum
$\bm{q}_{T}=\bm{p}_{\bot,c}+\bm{p}_{\bot,d}$ (where $\bm{p}_{\bot,c}$ and
$\bm{p}_{\bot,d}$ are the large transverse momenta of the out-going jets, see
Fig. 3). We further argued that more central events isolated via larger
transverse energy deposition in the forward calorimeter, enhances events with
larger intrinsic transverse momentum in the initial state, leading to events
with a larger $\bm{q}_{T}$.

In the calculations that follow, we will assume that $\bm{q}_{T}$ _is
coincident_ with the event plane angle(s). In reality, the event plane angle
will be strongly correlated with the direction of $\bm{q}_{T}$. To compensate
for this assumed coincidence, we will keep the mean $\langle
k_{\bot}^{2}\rangle=1$ GeV2 for $p$-$p$ collisions, for the case of
$x$-independent $\langle k_{\bot}^{2}\rangle$, and a comparable value for the
case of $x$-dependent $\langle k_{\bot}^{2}\rangle$. Thus, we are using values
of $\langle k_{\bot}^{2}\rangle$ that are closer to the lower limit considered
in our PYTHIA simulations.

Beyond this, there will be no further reference to the PYTHIA simulations. No
further simulations will be carried out. Our results will be from analytical
calculations. The anisotropy coefficients will be calculated assuming that
$\bm{q}_{T}$ coincides with the direction of the event plane. We remind the
reader that a majority of the calculated anisotropy will result from
polarization dependent scattering; and currently, there is no event generator
that can simulate polarization dependent processes. Thus, there is no
alternative other than semi-analytical calculations.

Figure 6:  Pion production cross section in $p$-$p$ at $5.02$ TeV. The top
panel displays unpolarized contributions from $gg\to gg$ (blue), $qg\to qg$
(red), $qq\to qq$ (green) and $gq\to gq$ (orange) partonic scatterings. The
bottom panel displays the ratio to the $gg\to gg$ scattering which dominates
the cross section for $p_{T}\lesssim 90$ GeV. Compared with ALICE data [73].
Figure 7:  Pion production cross section using gluon-gluon scattering only in
$p$-$p$ at $5.02$ TeV (top) and $p$-$Pb$ at $8.16$ TeV (bottom). Using a fixed
$\langle k_{\bot}^{2}\rangle$, the unpolarized contribution (blue full lines)
is dominant over the $BM\otimes C$ (purple lines) and the $BM\otimes BM$
contributions (green lines). The cross section using unpolarized $gg\to gg$
partonic scattering and an $x$-dependent $\langle k_{\bot}^{2}\rangle$ (red
dashed line squares) reproduces the collinear results at high $p_{T}\leq 5$
GeV. Experiment results from ALICE [73] and ATLAS [74].

For our calculations, we employ the nCTEQ parameterization [75] for integrated
PDFs [$f(x,Q^{2})$] and leading order KKP [64] for FFs. While we will use
either a constant or $x$-dependent mean transverse momentum for the TMD-PDFs,
we will only employ a constant $\langle k_{\bot}^{2}\rangle^{1/2}=0.8$ GeV for
the pion’s TMD-FFs, following the work of Ref. [52, 51].

When using a constant transverse momentum $\langle k_{\bot}^{2}\rangle$ the
Gaussian $k_{\bot}$ dependence integrates out, recovering the results from
collinear factorization. We display the angular integrated cross section,
using full lines in Fig. 6, for different partonic scatterings $ab\to cd$,
with the fragmenting parton $c$. As expected, the gluon PDFs are dominant at
lower values of the $Q^{2}$ scales; this is partially responsible for the
$gg\to gg$ partonic scattering contribution remaining dominant up to
$p_{T}\lesssim 80$ GeV, beyond which quark initiated processes start to
contribute. We also include the experimental data from the ALICE collaboration
[73], which, given the limited range in $p_{T}$, is almost completely
described by the $gg\to gg$ process.

In Fig. 7, we further decompose the $gg\to gg$ cross section with fixed
$\langle k_{\perp}^{2}\rangle$ into the polarization averaged (solid blue
line), and polarization dependent contributions which include a Boer-Mulders
distribution with a Collins fragmentation function ($BM\otimes C$ in green
solid) and a double Boer-Mulders distribution ($BM\otimes BM$ in maroon
solid). The upper panel represents $p$-$p$ at $\sqrt{s_{\rm NN}}=5.02$ TeV,
while the lower panel is for $p$-$Pb$ collisions, at $\sqrt{s_{\rm NN}}=8.16$
TeV.

Since polarization phases do not cancel for the polarized cross sections, the
angular integrated spectrum is suppressed compared to the unpolarized cross
section. The red dotted line with squares represents the case of $gg\to gg$,
with an $x$-dependent mean transverse momentum. The $x$-dependence leads to a
deviation from the collinear cross section at low $p_{T}$, visibly for the
case of $p$-$Pb$ (and barely for the case of $p$-$p$). However, at
$p_{T}\gtrsim 5$ GeV, the effect of the transverse momentum on the cross
section becomes negligible. Above this $p_{T}\;(\gtrsim 5$ GeV$)$, the cross
section is indistinguishable from the collinear result as shown in Fig. 7. As
mentioned in Sec. III, for the case of $p$-$Pb$, shown in the bottom panel of
Fig. 7, the $\langle k_{\perp}^{2}\rangle$ for the proton is enhanced by
$A^{1/3}$ to account for multiple soft interaction in the $Pb$ prior to the
hard scattering.

Figure 8:  Elliptic flow coefficient $v_{2}$ for pion production in $p$-$p$ at
$5.02$ TeV (top) and $p$-$Pb$ at $8.16$ TeV (bottom). The cyan and red filled
areas are the uncertainty over the functional form of the mean transverse
momentum between the $x$-dependent and fixed $\langle
k_{\perp}^{2}\rangle^{1/2}=1$ GeV using $B\cdot b=0.4$ for the polarized
contributions. The cyan and red hatched areas are the uncertainty on the
polarized distribution using the $x$-dependent $\langle
k_{\perp}^{2}\rangle(x)$ and varying the polarized contributions between
$0.2\leq B\cdot b\leq 0.4$.

Figure 9:  The different contributions to the elliptic flow coefficient
$v_{2}$ in $p$-$Pb$ at $8.16$ TeV using a fixed $\langle k^{2}\rangle^{1/2}=1$
GeV and $B\cdot b=0.4$ for the unpolarized distribution in $p$-$p$ (top) and
$p$-$Pb$ (bottom). The hatched blue area is the unpolarized contribution, the
green area is the $BM\otimes C$ terms.

Figure 10:  The different contributions to the elliptic flow coefficient
$v_{2}$ in $p$-$Pb$ at $8.16$ TeV using the $x$-dependent $\langle
k^{2}\rangle^{1/2}(x)$ GeV and $B\cdot b=0.4$ for the unpolarized distribution
in $p$-$p$ (top) and $p$-$Pb$ (bottom). The hatched blue area is the
unpolarized contribution, the green area is the $BM\otimes C$ terms.

The cross section can be written as an expansion in the azimuthal angle
$\phi_{\pi}$ as follows

$\displaystyle\frac{d\sigma}{dp_{T}dyd\Delta\phi}=$
$\displaystyle\frac{d\sigma}{dp_{T}dy}\left(1+2\sum_{n=1}v_{n}\cos(n\Delta\phi)\right)\;,$
(44)

where $\Delta\phi\equiv\phi_{\pi}-\phi_{\bm{q}_{T}}$ is the azimuthal angle of
the pion $\phi_{\pi}$ with respect to the total transverse momentum of the
hard scattering $\bm{q}_{T}$. The flow coefficients $v_{n}$, can be obtained
by integrating over the azimuthal angle $\Delta\phi$ as follows

$\displaystyle v_{n}=\langle\cos(n\Delta\phi)\rangle=$
$\displaystyle\frac{\int
d\Delta\phi~{}cos(n\Delta\phi)\frac{d\sigma}{dp_{T}dyd\Delta\phi}}{\int
d\Delta\phi~{}\frac{d\sigma}{dp_{T}dyd\Delta\phi}}\;.$ (45)

The elliptic flow coefficient $v_{2}$ obtained is displayed in Fig. 8 ($p$-$p$
in the top panel and $p$-$Pb$ on the bottom panel). The uncertainty over the
functional form of the mean transverse momentum is reflected in the red
($p$-$p$) and cyan ($p$-$Pb$) filled areas which is the bound between the
fixed $\langle k_{\perp}^{2}\rangle$ and $x$-dependent one using $B\cdot
b=0.4$ for the polarized contribution. The upper limit of the filled red and
cyan band is the $v_{2}$ for the case of fixed $\langle k_{\perp}^{2}\rangle$
(=1 GeV2 for $p$-$p$ and $A^{1/3}$ GeV2 for $p$-$Pb$). The lower limit of the
filled region is the upper bound from the case of $x$-dependent $\langle
k_{\perp}^{2}\rangle$ with $B\cdot b=0.4$ for the polarized portion of the
contribution.

We also present the uncertainty on the polarized distribution by varying the
$B\cdot b$ parameter in the range $0.2\leq B\cdot b\leq 0.4$ and using
$x$-dependent mean transverse momentum in the hatched red and cyan areas.
Within uncertainties, the elliptic flow coefficient $v_{2}$ is consistent with
the experimental data from ATLAS [35, 36] for $p_{T}\gtrsim 6$ GeV.

To investigate the importance of the polarization dependent terms versus
polarization averaged terms, we present the different contributions to the
elliptic flow using a fixed mean transverse momentum in Fig. 9 and the
$x$-dependent mean transverse momentum in Fig. 10. The unpolarized
contribution is shown in blue hatched area, and the $BM\otimes C$ terms are
shown in green area. Although we use $B\cdot b=0.4$ for the polarized
contribution, for both the fixed and $x$-dependent mean transverse momentum,
the $BM\otimes C$ terms are dominant compared to the unpolarized contribution
at high-$p_{T}$. For a fixed $\langle k_{\perp}^{2}\rangle$, we observe that
while the unpolarized contribution becomes negligible when $p_{T}\gg
k_{\perp}$, the $BM\otimes C$ terms continue to contribute, and lead to a
large azimuthal anisotropy coefficient, even at large transverse momentum.
This because the unpolarized contribution is power suppressed. The
$x$-dependence of $\langle k_{\perp}^{2}\rangle$ tends to reduce both the
polarization dependent and polarization averaged contributions at low $p_{T}$,
but enhances them at higher $p_{T}$. In this case as well, the polarization
averaged contribution tends to decrease at high $p_{T}$, though it depreciates
at a slower rate than in the case of constant $\langle k_{\perp}^{2}\rangle$.

Figure 11:  Same as Fig. 8 but for $v_{3}$ (top) and $v_{4}$ (bottom). The red
filled area is for $BM\otimes BM$ contribution.

To further explore the azimuthal anisotropies arising from TMDPDFs, we compute
higher order harmonics $v_{3}$ and $v_{4}$ which are displayed in Fig. 11. The
experimental results of $v_{3}$ and $v_{4}$ display a different behavior than
$v_{2}$. While $v_{2}$ starts decreasing with $p_{T}$ already at $p_{T}\gtrsim
5$ GeV, $v_{3}$ and $v_{4}$ are increasing up to the highest data points
($p_{T}\simeq 10$ and $p_{T}\simeq 4$, respectively). However, the data
displays large uncertainties at high-$p_{T}$ and is limited to $p_{T}\leq 10$
GeV. While experimental data is only available for $p_{T}\leq 10$ GeV, the
polarized contributions lead to non-zero coefficients at high-$p_{T}$.
Markedly, $v_{4}$ is entirely driven by the $BM\otimes BM$ terms at
high-$p_{T}$, which does not contribute to any $v_{n}$ coefficient with $n<4$.

Figure 12:  Azimuthal angle distribution $\frac{d\sigma}{d\phi}$ for $gg\to
gg$ scattering pion production in $p$-$p$ at $5.02$ TeV (top) and $p$-$Pb$ at
$8.16$ TeV (bottom) at a fixed $p_{T}=6$ GeV. The panels from left to right
represents the unpolarized, $BM\otimes C$ and $BM\otimes BM$ contributions.

To obtain further insight into the origin of different $v_{n}$’s from
different terms, we compute the azimuthal angular distribution of different
contributions to the cross section, at a fixed $p_{T}$. In Fig. 12, we present
the azimuthal angle distribution $\frac{d\sigma}{d\phi}$ for $gg\to gg$
scattering in $p$-$p$ and $p$-$Pb$ collisions. We find that for the $p$-$p$
scattering the unpolarized contribution has a large constant term and a small
$v_{1}$ contribution. While the $BM\otimes C$ and $BM\otimes BM$ terms
contribute predominantly to $v_{2}$ and $v_{4}$, respectively. Conversely, for
$p$-$Pb$ collisions, while different terms display similar symmetries, they
gain an additional elliptic shape due to the asymmetry of the transverse
momentum distribution, leading to a larger $v_{2}$. We remind the reader that
all terms in Fig. 12 are for the process $gg\to gg$ (quark scattering
contributions are included in the appendix).

##  VI Summary and Outlook

In hard scattering processes calculated within the ambit of collinear
factorization, the incoming partons possess no transverse momentum with
respect to the direction of the incoming nucleons (in $p$-$p$ or $p$-$Pb$
collisions). In such processes, the outgoing partons and hadrons demonstrate
no azimuthal anisotropy. However, in the event the incoming partons have a
small amount of transverse momentum, anisotropies in the azimuthal
distribution arise. The presence of transverse momentum in the initial state
parton distribution, both with and without the presence of transverse
polarization (linear polarization for gluons), leads to a scattering cross
section that is not isotropic in azimuthal angle.

In this paper, we have presented detailed results on the first attempt to
calculate these anisotropies and compare them to experimentally measured
Fourier coefficients (of these anisotropies) in $p$-$p$ and $p$-$Pb$
collisions. This attempt presents complete details of our calculations, which
were briefly presented in Ref. [76]. In the range of $p_{T}$ explored by
experimental data, the dominant channel is $gg\to gg$ (Fig. 6). Within this
channel, the dominant contribution to the angle integrated cross section is
the unpolarized process (Fig. 7). However, $gg\to gg$, with one gluon linearly
polarized (Boer-Mulders’ distribution), followed by the fragmentation of the
linearly polarized gluon to an unpolarized hadron (Collins’ function),
provides the largest contribution to the elliptic coefficient $v_{2}$ (Fig.
8), followed by the polarization averaged process (Fig. 9,10).

We calculate using two rather different models of the TMDPDF: A
straightforward model with a constant mean $\langle k_{\perp}^{2}\rangle$, and
a modified version of the $x$-dependent model of Ref. [52]. The mean $\langle
k_{\perp}^{2}\rangle$ is set to be 1 GeV2, for the case of constant $\langle
k_{\perp}^{2}\rangle$ in $p$-$p$, with a similar value for the $x$-averaged
$\langle k_{\perp}^{2}\rangle$ for the x-dependent _ansatz_ (Fig. 2). These
values are enhanced by $A^{1/3}$ of the $p$ in $p$-$A$ to account for multiple
scattering prior to the hard exchange that produces the dijet.

The $A^{1/3}$ enhancement required a factorized form for the TMDPDF (separate
functions for $x$ and $k_{\perp}$ dependence), as multiple scattering based
calculations for polarization dependent TMDs do not exist at this time. We
further vary the ratio between the polarization dependent and independent
distributions ($0.2\leq b\cdot B\leq 0.4$) to obtain a range of results for
$v_{2}$. This is done as the exact magnitudes of the Boer-Mulders and Collins
functions are not well known, at this time. In the absence of a bulk
simulation which determines the event plane, we have argued in Sec. III that
there is a strong correlation between the event plane angle and that of the
dijet momentum $\bm{q}_{T}$. Calculations of $v_{n}$ were then done using the
direction of $\bm{q}_{T}$ as the event plane angle.

In spite of these overlapping aspects of ignorance, our modest range of values
for the Boer-Mulders’ and Collins’ functions seems to straightforwardly
encapsulate the $v_{2}$ in $p$-$p$. Even more encouraging is the $A^{1/3}$
enhancement for $p$-$A$ collisions reproducing the larger $v_{2}$ in $p$-$Pb$.
Our calculations show a mild tension with the measured $v_{3}$, and the
comparisons with $v_{4}$ are inconclusive, as these have not been measured to
a sufficiently high $p_{T}$.

We remind the reader that this is the first calculation of the azimuthal
anisotropy at high $p_{T}$, arising from transverse momentum of the incoming
partons in the initial state. As such, no solace should be drawn from the
agreement with the measured $v_{2}$, and no alarm should be perceived in the
tension with the measured $v_{3}$. The results of this paper, and the
accompanying letter [76], should be taken as an argument for further
exploration of TMDPDFs as a source of azimuthal anisotropy in small system
collisions. Without having done a full study, our naïve expectation is that
these effects will be diminished in systems with an extended final state, such
as in heavy-ion collisions, where energy loss may completely wash out these
initial state effects. In fact, it may even be interesting to study the
interference of this effect with the typical anisotropy from energy loss in
semi-central collisions of intermediate size ions, e.g., $O$+$O$ collisions
[77].

Restricting our attention to $p$-$p$ and $p$-$A$ collisions, we point out that
all parton distribution functions used were leading twist and there could be
sizable contributions at lower $p_{T}$ from higher twist contributions [78].
These contributions could also be important for $v_{3}$ (and higher
harmonics). As one moves towards a description of the remainder of the proton
participating in the collision [79], yet another contribution could originate
in generalized transverse momentum dependent distributions [80].

The $A^{1/3}$ enhancement in the case of $p$-$A$ is only known for the case of
polarization averaged scattering, and has to be carried out for polarization
dependent scattering. Also, one should carefully check for the possibility of
energy loss in the case of multiple scattering of partons in a large nucleus
prior to the hard scattering that leads to high $p_{T}$ jets [81, 60]. These
calculations will rigorously test the hypothesised broadening without energy
loss, required for the existance of an elliptic anisotropy with no
modification to the angle integrated spectrum.

Most prior calculations of multiple scattering of partons in nuclear matter
[81, 60, 82, 61, 62] have not considered gluons as the hard parton. For the
currently measured $p_{T}$ range, gluon scattering is the most dominant
effect, and thus, many of these calculations will have to be repeated for
gluons. While gluon-gluon scattering is by far the largest contribution in the
considered $p_{T}$ range, quark initiated processes will begin to become
comparable at higher $p_{T}$ (see the Appendix). Quark mass effects (not
necessarily for heavy-flavors) may allow for additional enhancement [46].

With further improvement in semi-analytical calculations, these effects will
eventually have to be studied via extensive simulations, where both energy-
momentum and angular momentum correlations between the hard and soft sectors
can be thoroughly investigated. Any simulation, similar to the semi-analytic
calculations in this paper, will have to assume factorization between hard and
soft sectors in transverse momentum dependent high-$p_{T}$ hadron production.
Such studies will of course suffer from the lack of a formal proof of
factorization in traverse momentum dependent processes in $p$-$p$ (and by
extension $p$-$A$) collisions [83].

##  VII Acknowledgments:

A. M. is indebted to S. Pratt and J. Putschke for discussions and for
suggesting hard scattering as a source of the elliptic anisotropy. Both
authors are members of the JETSCAPE collaboration and have benefited from
internal discussions. This work is supported in part by the U.S. Department of
Energy under Grant No. DE-SC0013460. I. S. was funded as a part of the
European Research Council project ERC-2018-ADG-835105 YoctoLHC , and as a part
of the Center of Excellence in Quark Matter of the Academy of Finland (project
346325).

##  Appendix A Quark Scattering

Figure 13:  Azimuthal angle distribution $\frac{d\sigma}{d\phi}$ for $qq\to
qq$ scattering pion production in $p$-$p$ at $5.02$ TeV (top) and $p$-$Pb$ at
$8.16$ TeV (bottom) at a fixed $p_{T}=100$ GeV. The panels from left to right
represents the unpolarized, $BM\otimes C$ and $BM\otimes BM$ contributions.
Figure 14:  Azimuthal anisotropy $v_{2}$ for pion production in $p$-$Pb$ at
$8.16$ TeV using a fixed $\langle k^{2}\rangle^{1/2}=1$ GeV and $B\cdot b=0.4$
for the unpolarized distribution. The blue lines are the $gg\to gg$
contribution, the green lines are the $qq\to qq$ contribution. Full lines
represent the unpolarized contribution, dashed lines represent $BM\otimes C$
terms and dashed lines circles represent $BM\otimes BM$ terms.

Since the gluon PDFs and FFs are dominant at small scales, we have found that
gluon-gluon scattering is the only relevant partonic scattering at $p_{T}$
range of the experimental results. For completeness, we present the results
obtained for quark scattering at high-$p_{T}$. In Fig. 13, we present the
azimuthal angle distribution $\frac{d\sigma}{d\phi}$ for $qq\to qq$ scattering
in $p$-$p$ and $p$-$Pb$ collisions. While the unpolarized contribution
displays a similar shape as the gluon scattering, the polarized contributions
are dominated by lower order coefficients than the gluon scattering. For the
quark the $BM\otimes C$ terms are dominated by $v_{1}$, while the $BM\otimes
BM$ terms are dominated by $v_{2}$.

Computing the azimuthal anisotropy $v_{2}$, we find that the $qq\to qq$
scattering is subdominant compared to the gluon scattering as shown in Fig.
14. However, the $BM\otimes BM$ term is increasing with $p_{T}$ only
surpassing the gluon contribution at $p_{T}\gg 200$ GeV.

##  Appendix B Kinematics of the cross section integration

While we use a different approach to obtain the matrix element, the kinematics
we use are based on [52, 84, 51]. We reproduce the kinematics here.

We use the hadronic center of mass reference frame with Mandelstamn variables
$(s,t,u)$, where the hadronic momenta are given by

$\displaystyle p_{A}^{+}=$ $\displaystyle-p_{B}^{-}=\sqrt{\frac{s}{2}}\;,\quad
p_{A}^{-}=p_{B}^{+}=0\;,$ (46) $\displaystyle p_{C}=$
$\displaystyle\left(\sqrt{p_{T}^{2}+p_{L}^{2}},p_{T},0,p_{L}\right)\;.$ (47)

The partonic scattering $a+b\to c+d$ is

$\displaystyle p_{a}=$ $\displaystyle p_{A}+\left(\frac{k_{\perp
a}^{2}}{2x_{a}\sqrt{s}},k_{\perp a}\cos\phi_{a},k_{\perp
a}\sin\phi_{a},-\frac{k_{\perp a}^{2}}{2x_{a}\sqrt{s}}\right)\;,$ (48)
$\displaystyle p_{b}=$ $\displaystyle p_{B}+\left(\frac{k_{\perp
b}^{2}}{2x_{b}\sqrt{s}},k_{\perp b}\cos\phi_{b},k_{\perp
b}\sin\phi_{b},\frac{k_{\perp b}^{2}}{2x_{b}\sqrt{s}}\right)\;,$ (49)

The detected hadron $C$ has light-cone momentum fraction
$z=\frac{p_{C}^{+}}{p_{c}^{+}}$ and transverse momentum $\bm{k}_{\perp C}$
with respect to the parton $c$. Using the delta function $\delta(\bm{k}_{\bot
C}\cdot\hat{p}_{c})$ from Eq. (2), we find that the partonic momentum is given
by

$\displaystyle E_{c}=$ $\displaystyle\frac{E_{C}+\sqrt{E_{C}^{2}-k_{\bot
C}^{2}}}{2z}\;,$ (50) $\displaystyle\bm{p}_{c}=$
$\displaystyle\frac{E_{c}}{\sqrt{E_{C}^{2}-k^{2}_{\bot
C}}}(\bm{p}_{C}-\bm{k}_{\bot C})\;.$ (51)

The integration phase space can be written as

$\displaystyle\int
d\Omega=\int\frac{dx_{a}dx_{b}dz}{2\pi^{2}z^{3}s}d^{2}k_{\bot a}d^{2}k_{\bot
b}d^{3}k_{\bot C}$ $\displaystyle\delta(\bm{k}_{\bot
C}\cdot\hat{p}_{c})J(\bm{k}_{\bot C})\delta(\hat{s}+\hat{t}+\hat{u})\;.$ (52)

We define arbitrary transverse momentum $\bm{k}_{\bot C}$ in the hadronic
center of mass frame as follows

$\displaystyle\bm{k}_{\bot C}$ (53) $\displaystyle=k_{\bot
C}\left(\sin\theta_{k_{\bot C}}\cos\phi_{k_{\bot C}},\sin\theta_{k_{\bot
C}}\sin\phi_{k_{\bot C}},\cos\theta_{k_{\bot C}}\right)\;.$

The momentum of the outgoing parton $c$ is related to the pion’s momentum
$\bm{p}_{\pi}=p_{T}(1,0,\sinh y)$ as follows

$\displaystyle\bm{p}_{c}=$ $\displaystyle\bm{p}_{\pi}-\bm{k}_{\bot C}\;.$ (54)

The transverse momentum delta function can be performed to define the
azimuthal angle of $\bm{k}_{\bot C}$

$\displaystyle\cos\phi_{k_{\bot C}}=$ $\displaystyle\frac{k_{\bot
C}-p_{T}\sinh y\cos\theta_{k_{\bot C}}}{pT\sin\theta_{k_{\bot C}}}\;,$ (55)
$\displaystyle\sin\phi_{k_{\bot C}}^{\pm}=$
$\displaystyle\pm\sqrt{1-\cos^{2}\phi_{k_{\bot C}}}\;,$ (56)

leading to the Jacobian

$\displaystyle\int d^{3}\bm{k}_{\perp C}\delta(\bm{k}_{\bot
C}\cdot\hat{p}_{c})f(\bm{k}_{\bot C})$ $\displaystyle=\int
k_{C}dk_{C}d\theta_{k_{C}}\frac{\sqrt{p_{C}^{2}-k_{C}^{2}}}{p_{T}|\sin\phi_{k_{C}}|}\left[f(\bm{k}_{\bot
C}^{+})+f(\bm{k}_{\bot C}^{-})\right]\;,$ (57)

where one needs to sum over the two solutions $\sin\phi_{k_{C}}^{\pm}$.

Using the definition of the hadronic momentum, we obtain the Jacobian
$J(\bm{k}_{\bot C})$ for the transformation from the parton $c$ to the hadron
$C$ as follows

$\displaystyle\frac{d^{3}p_{c}}{E_{c}}=\frac{1}{z^{2}}J(\bm{k}_{\bot
C})\frac{d^{3}p_{C}}{E_{C}}\;,$ (58)

obtaining

$\displaystyle J(\bm{k}_{\bot
C})=\frac{(E_{C}^{2}+\sqrt{\bm{p}_{C}^{2}-\bm{k}_{\bot
C}^{2}})^{2}}{4(\bm{p}_{C}^{2}-\bm{k}_{\bot C}^{2})}\;.$ (59)

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