# Neutrino-tagged jets at the Electron-Ion Collider

Miguel Arratia  Department of Physics and Astronomy, University of California,
Riverside, CA 92521, USA  Thomas Jefferson National Accelerator Facility,
Newport News, VA 23606, USA  Zhong-Bo Kang  Department of Physics and
Astronomy, University of California, Los Angeles, CA 90095, USA  Mani L.
Bhaumik Institute for Theoretical Physics, University of California, Los
Angeles, CA 90095, USA  Center for Frontiers in Nuclear Science, Stony Brook
University, Stony Brook, NY 11794, USA  Sebouh J. Paul  Department of Physics
and Astronomy, University of California, Riverside, CA 92521, USA  Alexei
Prokudin  Division of Science, Penn State University Berks, Reading, PA 19610,
USA  Thomas Jefferson National Accelerator Facility, Newport News, VA 23606,
USA  Felix Ringer  Thomas Jefferson National Accelerator Facility, Newport
News, VA 23606, USA  Department of Physics, Old Dominion University, Norfolk,
VA 23529, USA  C.N. Yang Institute for Theoretical Physics, Stony Brook
University, Stony Brook, NY 11794,USA  Department of Physics and Astronomy,
Stony Brook University, Stony Brook, NY 11794, USA  Fanyi Zhao  Department of
Physics and Astronomy, University of California, Los Angeles, CA 90095, USA
Mani L. Bhaumik Institute for Theoretical Physics, University of California,
Los Angeles, CA 90095, USA  Center for Frontiers in Nuclear Science, Stony
Brook University, Stony Brook, NY 11794, USA

###### Abstract

We explore the potential of jet observables in charged-current deep inelastic
scattering (CC DIS) events at the future Electron-Ion Collider (EIC). Tagging
jets with a recoiling neutrino, which can be identified by the event’s missing
transverse momentum, will allow for flavor-sensitive measurements of
Transverse Momentum Dependent parton distribution functions (TMDs). We present
the first predictions for transverse-spin asymmetries in azimuthal neutrino-
jet correlations and hadron-in-jet measurements. We study the kinematic reach
and the precision of these measurements and explore their feasibility using
parameterized detector simulations. We conclude that jet production in CC DIS,
while challenging in terms of luminosity requirements, will complement the EIC
experimental program to study the three-dimensional structure of the nucleon
encoded in TMDs.

††preprint: JLAB-THY-22-3758, YITP-SB-2022-38

##  I Introduction

The Electron-Ion Collider (EIC) will usher in a new era for the study of the
3D structure of the nucleon Accardi:2012qut ; AbdulKhalek:2021gbh . Its high
luminosity and polarization of both electron and hadron beams will enable
precise measurements of observables related to Transverse Momentum Dependent
parton distribution and fragmentation functions (TMDs).

Jets are energetic sprays of particles observed at high-energy collider
experiments that are closely related to the underlying quark and gluon
dynamics of hard-scattering events. Jets at the EIC will have transverse
momenta up to $\sim 40$ GeV Arratia:2019vju ; Page:2019gbf . The EIC will
produce the first jets in deep-inelastic scattering off transversely-polarized
nucleons. The potential of jets produced in neutral-current deep-inelastic
scattering (NC DIS) has recently been explored, see e.g. Refs.
Hinderer:2015hra ; Boughezal:2018azh ; Zheng:2018ssm ; Gutierrez-Reyes:2018qez
; Gutierrez-Reyes:2019vbx ; Liu:2018trl ; Kang:2020xyq ; Borsa:2020yxh ;
Arratia:2020azl ; Arratia:2020nxw ; Liu:2020dct ; Arratia:2020ssx ;
Kang:2020fka ; Guzey:2020gkk ; Kang:2021ffh ; delCastillo:2021znl ;
Makris:2021drz ; Zhang:2021tcc ; AbdulKhalek:2022hcn ; Tong:2022zwp ;
Liu:2022wop ; Lee:2022kdn . In this work, we will focus on jets produced in
charged-current deep-inelastic scattering (CC DIS).

The CC DIS process, which involves the exchange of a virtual $W^{\pm}$ boson,
enables jet measurements that are sensitive to the flavor of the scattered
quark. The leading-order process, $W^{\pm}q\to q^{\prime}$, is illustrated in
Fig. 1. Due to the conservation of electric charge, electrons can only scatter
via the exchange of a $W^{-}$ off positively charged partons, which are
predominantly $u$-quarks, especially at large $x$. Likewise, with a positron
beam, scattering processes occur predominantly with $d$-quarks through the
exchange of a $W^{+}$ boson. Moreover, tagging either charm or strange jets
can further enhance the flavor sensitivity of jet measurements Zheng:2018ssm ;
Arratia:2020azl ; Kang:2020fka ; Arrington:2021yeb ; Lee:2022kdn .

Figure 1: Charged-current deep-inelastic scattering where the produced jet
recoils against a neutrino.

The H1 and ZEUS collaborations measured inclusive CC DIS off unpolarized
protons with longitudinally polarized electron and positron beams Aaron:2012qi
; Collaboration:2010xc ; Chekanov:2008aa ; Chekanov:2002zs ; Breitweg:1999aa ;
Derrick:1996sw . These measurements allowed for constraining the flavor
dependence of collinear parton distribution functions (PDFs)
Abramowicz:2015mha . In addition, CC DIS jet production measurements by the
ZEUS collaboration Chekanov:2003jd ; Chekanov:2008af , were compared to
precise next-to-next-to-next-to-leading order QCD calculations
Gehrmann:2018odt .

One of the main challenges in measuring CC DIS is the measurement of the
events’ kinematic variables, Bjorken $x$ and $Q^{2}$, in the presence of an
undetected final-state neutrino. Several methods exist to address this
challenge Blumlein:2012bf ; Collaboration:2010xc . The feasibility studies of
CC DIS at the EIC have been performed in Refs. Aschenauer:2013iia ;
AbdulKhalek:2021gbh  for DIS off longitudinally polarized protons with the
goal to access helicity PDFs. In this study we will focus on the CC DIS off
transversely polarized protons that will lead to measurements of the
transverse-spin effects related to TMDs.

In semi-inclusive DIS (SIDIS), transverse-spin asymmetries can be extracted
from modulations of the azimuthal angles with respect to the virtual-boson
direction, typically in the Breit frame Avakian:2019drf ; Anselmino:1993tc ;
Chen:2020ugq ; PhysRevD.103.016011 ; Yang:2022sbz . In CC DIS, this approach
requires a measurement of the three-momentum of the scattered neutrino to
define the azimuthal angle, which is challenging due to acceptance losses at
forward rapidities.

Jet-based measurements of spin asymmetries can reduce these difficulties.
Following Liu et al. Liu:2018trl , TMDs can be accessed in lepton-jet
azimuthal correlation measurements in the laboratory frame instead of the
conventional Breit frame. Liu et al. Liu:2018trl  considered NC DIS, but the
formalism can be extended to CC DIS as well. The advantage of this approach is
that the measurement of the azimuthal angle only requires the neutrino’s
transverse momentum in the lab frame, which, in general, can be measured more
precisely than the full three-momentum Gao:2022bzi . In addition, hadron-in-
jet asymmetry measurements can be performed by defining an azimuthal angle of
the hadron with respect to the jet axis Yuan:2007nd ; Procura:2009vm ;
Kang:2017glf ; Arratia:2020nxw ; Kang:2021ffh .

The jet-based TMD measurements have the additional advantage of decoupling
initial- and final-state TMD effects (at leading power in the jet radius)
Kang:2021ffh . That is, they do not involve a convolution of TMD PDFs and
fragmentation functions which can introduce strong correlations in global fits
of SIDIS data Bacchetta:2017gcc ; Scimemi:2019cmh .

In this paper, we present the first study of neutrino-jet and hadron-in-jet
spin asymmetries in CC DIS. We determine all possible spin asymmetries within
the TMD factorization formalism. We present numerical estimates for transverse
single-spin asymmetries in CC DIS. We also perform feasibility studies of
these measurements using fast detector simulations and we quantify the
expected kinematic reach and the statistical uncertainties.

The remainder of this paper is organized as follows. We describe the proposed
measurements in Section II and the theoretical framework in Section III. We
describe the fast detector simulation in Section IV and the expected
experimental performance in Section V. We estimate the background in Section
VI, and show projections for the transverse-spin asymmetries in Section VII.
We conclude in Section VIII.

##  II Proposed measurements

Following Liu et al. Liu:2018trl , we propose the measurement of the
distribution of the azimuthal separation between the outgoing neutrino (as
determined from the missing transverse momentum), and the jet. Due to the
momentum conservation, the jet and neutrino are expected to be predominantly
produced back-to-back. Therefore, the azimuthal distribution is expected to be
centered around $\phi_{\rm jet}-\phi_{\nu}-\pi=0$, with some finite width due
to out-of-cone QCD radiation and the non-zero initial momentum of the
scattered quark. In the next section, we determine all spin asymmetries that
can be measured in neutrino-jet and hadron-in-jet production within TMD
factorization.

Moreover, we propose to measure the transverse single-spin asymmetry in
neutrino-jet correlations, also known as the left-right asymmetry

$A_{\mathrm{UT}}=\frac{{\rm d}\sigma^{\uparrow}-{\rm
d}\sigma^{\downarrow}}{{\rm d}\sigma^{\uparrow}+{\rm
d}\sigma^{\downarrow}}\,,$ (1)

where, ${\rm d}\sigma^{\uparrow,\downarrow}$ refers to the differential cross
section measured with transverse polarization of the initial proton pointing
up or down. This asymmetry is expected to exhibit a modulation with respect to
the angular separation between the incoming proton spin, $\phi_{S}$, and the
momentum imbalance, $\phi_{q}$, i.e.,

$A_{\mathrm{UT}}=A^{\sin(\phi_{S}-\phi_{q})}_{\mathrm{UT}}\sin(\phi_{S}-\phi_{q}).$
(2)

Here, the momentum imbalance between the jet and the neutrino is defined by
$\vec{q}_{T}=\vec{p}_{T}^{\,\rm jet}+\vec{p}_{T}^{\,\nu}$. This asymmetry is
sensitive to the Sivers function Liu:2018trl ; Liu:2020dct , which describes
the anisotropy of unpolarized partons in a transversely polarized proton.

We also propose to perform a hadron-in-jet measurement in CC DIS of the
asymmetry $A^{\sin(\phi_{S}-\phi_{\rm h})}_{\mathrm{UT}}$ defined for the
azimuthal angle of the hadron in jet $\phi_{\rm h}$. In NC DIS, the hadron-in-
jet asymmetry is sensitive to both the Collins and the transversity functions
Arratia:2020nxw .

##  III Theoretical framework

In this section, we discuss inclusive jet production in CC DIS, neutrino-jet
correlations, and hadron-in-jet observables.

###  III.1 Inclusive jet production

We follow the theoretical framework developed in Refs. Liu:2018trl ;
Arratia:2020nxw ; Liu:2020dct ; Kang:2021ffh  for the NC DIS. At the parton
level, we consider the leading-order process $eq\rightarrow\nu q^{\prime}$
mediated via the exchange of a virtual $W$ boson. We consider the cross
section differential in Bjorken $x$ and the transverse momentum of the
produced neutrino, $p_{T}^{\nu}$, which is defined relative to the beam
direction in the laboratory frame. The leading-order cross section can be
written as

$\displaystyle\frac{{\rm d}\sigma^{ep\rightarrow\nu\mathrm{jet}X}}{{\rm
d}x\,{\rm d}^{2}\vec{p}^{\;\nu}_{T}}=\,\sum_{q}\sigma_{0}^{eq\rightarrow\nu
q^{\prime}}f_{q}(x,\mu)\,,$ (3)

where the renormalization scale $\mu$ of the PDF $f_{q}$ is chosen at the
order of the hard scale of the process $\mu\sim p_{T}^{\nu}$. The prefactor
$\sigma_{0}$ for initial quarks $u$ and $\bar{d}$ are given by

$\displaystyle\sigma_{0}^{eu\rightarrow\nu
d}=\frac{|\overline{\mathcal{M}}_{eu\rightarrow\nu
d}|^{2}}{16\pi^{2}\hat{s}^{2}}\frac{\hat{t}}{x(\hat{t}-\hat{u})}$
$\displaystyle=8(G_{F}m_{W}^{2})^{2}|V_{ud}|^{2}\frac{\hat{s}^{2}}{(\hat{t}-m_{W}^{2})^{2}+m_{W}^{2}\Gamma_{W}^{2}}\frac{\hat{t}}{x(\hat{t}-\hat{u})}\,,$
(4)
$\displaystyle\sigma_{0}^{e\bar{d}\rightarrow\nu\bar{u}}=\frac{|\overline{\mathcal{M}}_{e\bar{d}\rightarrow\nu\bar{u}}|^{2}}{16\pi^{2}\hat{s}^{2}}\frac{\hat{t}}{x(\hat{t}-\hat{u})}$
$\displaystyle=8(G_{F}m_{W}^{2})^{2}|V_{ud}|^{2}\frac{\hat{u}^{2}}{(\hat{t}-m_{W}^{2})^{2}+m_{W}^{2}\Gamma_{W}^{2}}\frac{\hat{t}}{x(\hat{t}-\hat{u})}\,,$
(5)

where $G_{F}$ is the Fermi constant, $m_{W}$ and $\Gamma_{W}$ are the $W$
boson mass and decay width, and $V_{ud}$ is the standard CKM matrix element.
Here ${\hat{t}}/(x(\hat{t}-\hat{u}))$ is the Jacobian factor, which is
obtained by transforming the cross section to be differential in Bjorken-$x$
instead of the neutrino rapidity $y_{\nu}$. These two variables are related by

$\displaystyle
x=\frac{p_{T}^{\nu}e^{y_{\nu}}}{\sqrt{s}-p_{T}^{\nu}e^{-y_{\nu}}}.$ (6)

The Mandelstam variables in Eqs. (4) and (5) can be written in terms of the
kinematic variables of the produced neutrino and the center-of-mass energy,
namely

$\displaystyle\hat{s}=xs\,,$ (7)
$\displaystyle\hat{t}=-Q^{2}=-\sqrt{s}p_{T}^{\nu}e^{y_{\nu}}=-x\sqrt{s}p_{T}^{\mathrm{jet}}e^{-y_{\mathrm{jet}}},$
(8)
$\displaystyle\hat{u}=-x\sqrt{s}p_{T}^{\nu}e^{-y_{\nu}}=-\sqrt{s}p_{T}^{\mathrm{jet}}e^{y_{\mathrm{jet}}}\,.$
(9)

Here $p_{T}^{\text{jet }}$ and $y_{\text{jet }}$ denote the jet transverse
momentum and rapidity, respectively.

###  III.2 Neutrino-jet correlations

Next, we discuss neutrino-jet correlations via the exchange of a $W^{-}$ boson
in polarized electron-proton scattering

$\displaystyle
p(P_{A},\lambda_{p},\vec{S}_{T})+e(P_{B},\lambda_{e})\rightarrow\mathrm{jet}(P_{J})+\nu(P_{D})+X\,.$
(10)

Here $\lambda$ indicates the longitudinal polarization and $\vec{S}_{T}$
denotes the transverse spin vector of the proton. In order to access TMDs, we
study back-to-back neutrino-jet production in the $ep$ collision frame. By
defining light-cone vectors $n_{+}^{\mu}=\frac{1}{\sqrt{2}}(1,0,0,1)$ and
$n_{-}^{\mu}=\frac{1}{\sqrt{2}}(1,0,0,-1)$, we write the momentum of the
incoming proton $P_{A}$ and the electron $P_{B}$ as

$\displaystyle P_{A}^{\mu}$
$\displaystyle=P^{+}n_{+}^{\mu}+\frac{M^{2}}{2P^{+}}n_{-}^{\mu}\approx
P^{+}n_{+}^{\mu}\ =\sqrt{\frac{s}{2}}n_{+}^{\mu}\,,$ (11) $\displaystyle
P_{B}^{\mu}$ $\displaystyle=\sqrt{\frac{s}{2}}n_{-}^{\mu}\,.$ (12)

Here $s=(P_{A}+P_{B})^{2}$ is the center-of-mass energy. We set the final
observed jet to be produced in the $xz$-plane, with the following momentum
$P_{J}^{\mu}=E_{J}\left(1,\sin\theta_{J},0,\cos\theta_{J}\right)$. Here
$E_{J}$ is the jet energy and the angle $\theta_{J}$ is measured with respect
to the beam direction. We find that the differential cross section can be
written in terms of the structure functions as follows 111Notice that unlike
the usual practice for SIDIS or DY cross sections, we do not factor out the
elementary cross-sections from the structure functions. Our structure
functions therefore become dimension-full quantities, see for instance Ref.
Kang:2022dpx .

$\displaystyle\frac{{\rm d}\sigma^{ep\rightarrow\nu\mathrm{jet}X}}{{\rm
d}x\,{\rm d}^{2}\vec{p}^{\,\nu}_{T}\,{\rm
d}^{2}q_{T}}=F_{UU}+\lambda_{p}F_{UL}$ $\displaystyle\hskip
28.45274pt+|S_{T}|\left[\sin(\phi_{q}-\phi_{S_{A}})F_{UT}^{\sin(\phi_{q}-\phi_{S_{A}})}\right.$
$\displaystyle\hskip
62.59596pt\left.+\cos(\phi_{q}-\phi_{S_{A}})F_{UT}^{\cos(\phi_{q}-\phi_{S_{A}})}\right]$
$\displaystyle\hskip
28.45274pt+\lambda_{e}\left[F_{LU}+\lambda_{p}F_{LL}\right.$
$\displaystyle\hskip
54.06006pt+|S_{T}|\sin(\phi_{q}-\phi_{S_{A}})F_{LT}^{\sin(\phi_{q}-\phi_{S_{A}})}$
$\displaystyle\hskip
54.06006pt\left.+|S_{T}|\cos(\phi_{q}-\phi_{S_{A}})F_{LT}^{\cos(\phi_{q}-\phi_{S_{A}})}\right]\,.$
(13)

The subscripts $E$ and $P$ of a structure function $F_{EP}$ indicate the
polarization of the incoming electron and incoming proton respectively: $U$
for unpolarized, $L$ for longitudinally polarized or $T$ for transversely
polarized. For example, for unpolarized scattering, the structure function is
denoted by $F_{UU}$. In the limit of small values of the transverse-momentum
imbalance $|\vec{q}_{T}|\ll p_{T}^{\rm jet}\sim p_{T}^{\nu}$, one can write
this structure function in the following form using the TMD factorization
formalism

$\displaystyle F_{UU}=$
$\displaystyle\sum_{q}\frac{|\overline{\mathcal{M}}_{eq\rightarrow\nu
q^{\prime}}|^{2}}{16\pi^{2}\hat{s}^{2}}H(Q,\mu)\,\mathcal{J}_{q}(p_{T}^{\mathrm{jet}}R,\mu)$
$\displaystyle\times\int\frac{{\rm
d}b_{T}b_{T}}{2\pi}J_{0}(q_{T}b_{T})f_{1}^{\mathrm{TMD}}(x,b_{T},\mu,\zeta)$
$\displaystyle\qquad\times S_{q}(b_{T},y_{\mathrm{jet}},R,\mu)\,.$ (14)

Here $H(Q,\mu)$ is the hard function, which accounts for virtual corrections
at the hard scale $Q$. The jet function $\mathcal{J}_{q}$ is associated with
collinear dynamics of the jet with characteristic scale $\mu_{J}\sim
p_{T}^{\rm jet}R$ Ellis:2010rwa . For our numerical results presented below we
use the anti-$k_{T}$ algorithm Cacciari:2008gp . The quark TMD PDF including
the appropriate soft factor for a generic TMD in $b_{T}$-space is defined by
Boer:2011xd ; Kang:2021ffh

$\displaystyle{f}_{q}^{(n),{\rm TMD}}(x,b_{T},\mu,\zeta)=$
$\displaystyle\frac{2\pi n!}{\left({M^{2}}\right)^{n}}\int{\rm
d}k_{T}\,k_{T}\left(\frac{k_{T}}{b_{T}}\right)^{n}J_{n}\left(k_{T}b\right)$
$\displaystyle\times\tilde{f}_{q}^{\,\rm
TMD}\\!\left(x,k_{T}^{2},\mu,\zeta\right)\,,$ (15)

where $M$ is the mass of the nucleon and $J_{n}$ is the $n$-th order Bessel
function. Here $\mu$ is the renormalization scale, while $\zeta$ is the so-
called Collins-Soper scale Collins:2011zzd . Notice that for the unpolarized
TMD $f_{1}^{\rm TMD}$, $n=0$. The remaining soft function $S_{q}$ in Eq. (14)
includes a contribution from the global soft function which depends on the
Wilson lines in the beam and jet directions, and the collinear-soft function
associated with the soft jet dynamics. Since $S_{q}$ accounts for different
soft contributions, it depends on both the jet rapidity $y_{\rm jet}$ and the
jet radius $R$ and the expression of $S_{q}$ is given in Arratia:2020nxw .

The other cross sections or structure functions in Eq. (13) depend on the
polarization state of the nucleon (longitudinal or transverse) and the
longitudinal lepton polarization. As an example, we consider the case where
the initial proton is transversely polarized. The cross section or structure
function $F_{UT}^{\sin(\phi_{q}-\phi_{S_{A}})}$ sensitive to correlations of
an unpolarized quark in the transversely polarized proton is given by

$\displaystyle F_{UT}^{\sin(\phi_{q}-\phi_{S_{A}})}=$
$\displaystyle\sum_{q}\frac{|\overline{\mathcal{M}}_{eq\rightarrow\nu
q^{\prime}}|^{2}}{16\pi^{2}\hat{s}^{2}}H(Q,\mu)\,\mathcal{J}_{q}(p_{T}^{\mathrm{jet}}R,\mu)$
$\displaystyle\times\int\frac{{\rm d}b_{T}b_{T}^{2}}{4\pi
M}J_{1}(q_{T}b_{T})f_{1T}^{\perp(1),\mathrm{TMD}}(x,b_{T},\mu,\zeta)$
$\displaystyle\qquad\times S_{q}(b_{T},y_{\mathrm{jet}},R,\mu)\,.$ (16)

In this case, the structure function is related to the Bessel function of the
1st order $J_{1}(q_{T}b_{T})$ and the Sivers function
$f_{1T}^{\perp(1),\mathrm{TMD}}(x,b_{T},\mu,\zeta)$ in $b_{T}$-space as
defined in Eq. (III.2).

We provide more details about the other structure functions in Eq. (13) in the
Appendix A. For example, for an incoming electron with helicity $\lambda_{e}$,
one replaces $\sigma_{0}^{eq\rightarrow\nu q^{\prime}}$ with
$\sigma_{0}^{e_{L}q\rightarrow\nu q^{\prime}}$ as given in Eq. (29).

###  III.3 Hadron distributions inside jets

Figure 2: Normalized distribution of the neutrino-jet imbalance momentum
$q_{T}/p^{\nu}_{T}$ in unpolarized electron-proton scattering via the exchange
of a $W^{-}$ boson. We show our theoretical results at NLL accuracy with QCD
scale uncertainties (orange) compared to Monte-Carlo event-generator
simulations obtained with Pythia8 Sjostrand:2007gs  (blue).

Figure 3: Distributions of the $\pi^{\pm}$-in-jet longitudinal-momentum
fraction $z_{h}$ (left) and the transverse momentum $j_{T}$ (right) in CC DIS
events. We show our theoretical results at NLL accuracy with QCD scale
uncertainties (orange) compared to Monte-Carlo event-generator simulations
obtained with Pythia8 Sjostrand:2007gs  (blue).

In this subsection, we study the longitudinal- and transverse-momentum
distributions of hadrons in the identified jet for electron-proton scattering
via the exchange of a $W^{-}$ boson:

$\displaystyle p(P_{A},\lambda_{p},\vec{S}_{T})+e(P_{B},\lambda_{e})$
$\displaystyle\hskip
34.14322pt\rightarrow\big{(}\mathrm{jet}(P_{J})h(z_{h},\vec{j}_{T})\big{)}+\nu(P_{D})+X.$
(17)

The production of unpolarized final-state hadrons at leading twist is encoded
in two TMD jet fragmentation functions (JFFs), $\mathcal{D}_{1}$ and
$\mathcal{H}_{1}^{\perp}$ Kang:2021ffh

$\displaystyle\Delta(z_{h},\vec{j}_{T})=\,$
$\displaystyle\mathcal{D}_{1}^{h/q}(z_{h},{j}_{T})\frac{\not{n}_{-}}{2}$
$\displaystyle-i\mathcal{H}^{\perp,h/q}_{1}(z_{h},{j}_{T})\frac{\not{j_{T}}}{z_{h}M_{h}}\frac{\not{n}_{-}}{2}\,.$
(18)

Thus, we find the following differential cross section expressed in terms of
structure functions

$\displaystyle\frac{{\rm d}\sigma^{ep\rightarrow\nu+\mathrm{jet}X}}{{\rm
d}x\,{\rm d}^{2}\vec{p}^{\,\nu}_{T}\,{\rm d}^{2}q_{T}\,{\rm d}z_{h}\,{\rm
d}^{2}{j_{T}}}=F^{h}_{UU}+\lambda_{p}F^{h}_{UL}$
$\displaystyle\qquad+|S_{T}|\left[\cos(\phi_{q}-\phi_{S_{A}})F_{UT}^{h,\cos(\phi_{q}-\phi_{S_{A}})}\right.$
$\displaystyle\quad\hskip
38.41139pt\left.+\sin(\phi_{q}-\phi_{S_{A}})F_{UT}^{h,\sin(\phi_{q}-\phi_{S_{A}})}\right]$
$\displaystyle\qquad+\lambda_{e}\left[F^{h}_{LU}+\lambda_{p}F^{h}_{LL}\right.$
$\displaystyle\qquad\hskip
31.2982pt+|S_{T}|\sin(\phi_{q}-\phi_{S_{A}})F_{LT}^{h,\sin(\phi_{q}-\phi_{S_{A}})}$
$\displaystyle\qquad\hskip
31.2982pt\left.+|S_{T}|\cos(\phi_{q}-\phi_{S_{A}})F_{LT}^{\cos(\phi_{q}-\phi_{S_{A}})}\right]\,.$
(19)

In total, we find 8 structure functions and the full expression is provided in
Appendix B. Following the same convention we used in Eq. (13), the subscripts
$E$ and $P$ of a structure function $F^{h}_{EP}$ here also indicate the
polarization of the incoming electron and the incoming proton, respectively.
Notably, we found that none of the Collins-type jet fragmentation functions
contribute in Eq. (19). The reason is that chiral-odd functions have to be
coupled with another chiral-odd function, the chirality between two factors of
$(1-\gamma_{5})$ resulting from the weak charged-current vertices must be odd.
As a result, see Eq. (B), we always have $(1-\gamma_{5})(1+\gamma_{5})=0$,
which implies that all terms involving chiral-odd functions vanish. This
conclusion is robust at leading power of the TMD factorization formalism we
are using. However, whether this still holds going beyond the leading power
TMD factorization, see e.g. within TMD factorization at sub-leading power
Ebert:2021jhy ; Rodini:2022wki ; Gamberg:2022lju  or including higher loops as
discussed in Refs. Benic:2019zvg ; Benic:2021gya , needs further
investigation. We leave such a study to a future publication.

Using TMD factorization at leading power, see Refs. Yuan:2007nd ;
Procura:2009vm ; Ellis:2010rwa ; Jain:2011xz ; Kaufmann:2015hma ; Kang:2016ehg
; Bain:2016rrv ; Kang:2017glf ; Kang:2017btw ; Kang:2019ahe , we can write the
unpolarized structure function where a hadron is measured inside the jet as
follows:

$\displaystyle F^{h}_{UU}$
$\displaystyle=\,H(Q,\mu)\sum_{q}\sigma_{0}^{eq\rightarrow\nu
q^{\prime}}\,{\cal D}_{1}^{h/q}(z_{h},j_{T},p_{T}^{\rm jet}R,\mu)$ (20)
$\displaystyle\times\,\int\frac{{\rm
d}^{2}\vec{b}_{T}}{(2\pi)^{2}}\,e^{i\vec{q}_{T}\cdot\vec{b}_{T}}\,f_{1}^{\rm
TMD}(x,b_{T},\mu,\zeta)\,S_{q}(b_{T},y_{\rm jet},R,\mu)\,.$

Here the variables $z_{h}=\vec{p}_{h}\cdot\vec{p}_{\rm jet}/|\vec{p}_{\rm
jet}|^{2}$ and ${j}_{T}=\left|\vec{p}_{h}\times\vec{p}_{\rm
jet}\right|/|\vec{p}_{\rm jet}|^{2}$ denote the longitudinal momentum fraction
and the transverse momentum relative to the (standard) jet axis of the hadron
inside the jet, respectively. In the factorized cross section in Eq. (20),
${\cal D}_{1}^{h/q}$ is a TMD fragmenting jet function. It describes the
hadron-in-jet measurement and replaces the jet function $\mathcal{J}_{q}$ in
Eq. (14). At next-to-leading logarithmic (NLL) accuracy, we can write ${\cal
D}_{1}^{h/q}$ as

$\displaystyle{\cal D}_{1}^{h/q}(z_{h},j_{T},p_{T}^{\rm jet}R,\mu)$
$\displaystyle\hskip 28.45274pt=\int\frac{{\rm
d}^{2}\vec{b}_{T}^{\,\prime}}{(2\pi)^{2}}e^{{i\vec{j}_{T}\cdot\vec{b}_{T}^{\,\prime}}/{z_{h}}}D_{1}^{q/h}(z_{h},\vec{b}_{T}^{\,\prime},p_{T}^{\rm
jet}R)\,.$ (21)

Here we work in Fourier conjugate space and $D_{1}^{q/h}$ is a TMD
fragmentation function (TMDFF) evaluated at the jet scale. We use the Fourier
variable $\vec{b}_{T}^{\,\prime}$ here to indicate that there is no
convolution of the TMD fragmentation function with the TMD PDF in Eq. (20).
Also note that the TMDFFs can be matched to the collinear FFs Aybat:2011zv ;
Collins:2011zzd  and in this work, we apply the extraction of collinear
fragmentation functions in deFlorian:2014xna  for constructing the TMDFFs. See
Ref. Kang:2017btw  for more details.

##  IV Simulation

In this section, we discuss Monte-Carlo event-generator results for neutrino-
jet correlations as well as detector-response simulations. We show comparisons
between theoretical calculations discussed in the previous section and the
Monte-Carlo simulations for unpolarized cross sections in CC DIS events.

###  IV.1 Event-Generation with Pythia8

We used Pythia8 Sjostrand:2007gs  to simulate CC DIS events in unpolarized
electron-proton and positron-proton collisions. We choose the energies of the
incoming electron and proton as 10 GeV and 275 GeV, respectively. These beam-
energy values, which yield a center-of-mass energy of $\sqrt{s}=105$ GeV,
correspond to the operation point that maximizes the luminosity of the EIC
design EICdesign . Following Ref. Aschenauer:2013iia , we selected events with
$Q^{2}>100$ $\mathrm{GeV}^{2}$. QED radiative effects Badelek:1994uq ;
Liu:2021jfp  are not included in the simulation to match the calculations in
Sec. III 222Based on similar measurements in NC DIS H1:2021wkz , the QED
corrections are expected to be small for the observables considered in this
work. Therefore, we do not expect that our conclusions are affected by these
effects.. We used the Fastjet3.3 package Cacciari:2011ma  to reconstruct jets
with the anti-$k_{T}$ algorithm Cacciari:2008gp  and jet radius parameter
$R=1$. The input particles for the generator-level jet-finding algorithm are
all stable particles ($c\tau>$ 10 mm), except for neutrinos.

Figure 2 shows our theoretical results at NLL accuracy for the transverse
momentum imbalance of the neutrino and jet $q_{T}/p^{\nu}_{T}$. In addition,
we show the Pythia8 simulations for unpolarized CC DIS events. The theoretical
uncertainties are obtained by varying the scales renormalization scale
$\mu\sim p_{T}^{\rm jet}$ and the jet scale $\mu_{J}\sim p_{T}^{\rm jet}R$ by
a factor of 2 around their central values and taking the envelope. We observe
good agreement between the resummed TMD calculation at NLL and the Pythia8
results. However, the tail of the $q_{T}/p_{T}^{\nu}$ distribution falls
slower at high $q_{T}/p_{T}^{\nu}$ for the Pythia8 simulations compared to the
resummed TMD result. This is likely due to multi-jet events, which are not
included as a matching contribution in the TMD result at large $q_{T}$.

Figure 3 shows our theoretical results including QCD scale uncertainties (we
again take the envelope of the results when varying the scales $\mu\sim
p_{T}^{\rm jet}$ and $\mu_{J}\sim p_{T}^{\rm jet}R$ by a factor of 2 around
their central values) for the longitudinal $z_{h}$ and transverse momentum
$j_{T}$ distributions for $\pi^{\pm}$ compared to the Pythia8 results. We use
the same simulated event sample as described above, and we observe reasonable
agreement between the two results.

Lastly, Figure 4 shows the neutrino yields expected for 100 fb-1, which can be
collected in about a year of running at 1034 cm-2s-1, as a function of the
neutrino’s transverse momentum. We also show the mean of the parton momentum
fraction $x$ as a function of transverse momentum (red dots). Values up to
$x=0.8$ can be probed with jet/neutrino transverse momenta of $p_{T}=45$ GeV,
which corresponds to the kinematic limit. With 100 fb-1, the statistical
uncertainty of the cross-section measurement is expected to be negligible over
the entire kinematic range. However, high luminosity is needed to measure the
corresponding spin asymmetries, as will be further discussed in Sec. VII
below.

Figure 4: Expected yield of neutrinos and jets in CC DIS events with an
electron beam and 100 fb-1 integrated luminosity. In addition, we show the
average parton momentum fraction $x$, which is probed as a function of the
neutrino transverse momentum in the laboratory frame. The cross sections
generated in Pythia8 have been scaled to match the total cross section
calculated at NLO in Ref. Aschenauer:2013iia .

###  IV.2 Detector-response simulations

We use the Delphes package deFavereau:2013fsa  to perform fast detector
simulations with parameters specified in Ref. Arratia:2021uqr . We consider a
general-purpose detector geometry including tracking, electromagnetic and
hadronic calorimeters with coverage up to $|\eta|=4.0$ and full azimuthal
coverage, as described in the EIC Yellow Report AbdulKhalek:2021gbh . This is
in line with proposed EIC detector design Adkins:2022jfp ; ATHENA:2022hxb ;
CORE:2022rso  that considered a high degree of hermeticity, which can be
ensured with dedicated detectors at forward angles Arratia:2022quz . We show a
representative charged-current event in Fig. 5.

Figure 5: Display of a simulated CC DIS event using Delphes deFavereau:2013fsa
. Top: 3D view. Bottom: Transverse view.

To reconstruct jets in the detector-response simulation, we use again the
Fastjet3.3 package Cacciari:2011ma  with the anti-$k_{T}$ algorithm
Cacciari:2008gp  and $R=1$ Newman:2013ada . As input to the jet algorithm, we
use the set of particle-flow objects reconstructed with Delphes.

In Fig. 6, we show the hadron-in-jet momenta for reconstructed $\pi^{\pm}$, as
well as the average $z_{h}$ in each momentum bin. We find that the charged
pions in jets are mostly in $-$0.5$<\eta<$3.5, and have momenta up to about 45
GeV, which can be identified with high purity using gas-based Cherenkov
detectors AbdulKhalek:2021gbh .

Figure 6: Pseudorapidity and momentum distribution of charged pions in jets
with $p_{T}>5$ GeV in $ep$ CC DIS. The average longitudinal-momentum fraction
of the hadron with respect to the jet axis is shown by the red dots.

##  V Event reconstruction and kinematic resolution

As typically done at particle colliders, neutrinos can be identified by
measuring the missing transverse momentum, $\vec{p}_{T}^{\rm\,miss}$, which is
defined as the vector sum of the transverse momenta of all measured particles
(identified using the particle-flow algorithm to avoid double-counting).

In this section we estimate the performance of this reconstruction method for
EIC. We expect this estimate to be reasonable given that the Delphes fast
smearing was shown to reproduce reasonably well the performance obtained from
a comprehensive detector simulation of the CMS experiment down to about
$|p_{T}^{\mathrm{miss}}|=20$ GeV deFavereau:2013fsa .

Figure 7: Performance of the reconstruction of $\phi_{\nu}$ in CC DIS events.
The red error bars indicate the means and standard deviations for given slices
of $p_{T}^{\nu}$

We define $\phi_{\nu}$ as the azimuthal angle of $-\vec{p}_{T}^{\rm\,miss}$.
We show the reconstruction performance of $\phi_{\nu}$ in Fig. 7. The standard
deviation is less than 0.06 radians, which is of similar order as the di-jet
azimuthal-angle resolution of the measurement presented in Ref. Abelev:2007ii
.

We employ the Jacquet-Blondel (JB) method of Ref. Amaldi:1979yh  to
reconstruct the lepton kinematics. The event inelasticity is given by
$y_{JB}=\sum(E_{i}-p_{z,i})/(2E_{e})$, where the sum is over all the
reconstructed particles. The four-momentum transfer is given by
$Q^{2}_{JB}=(p^{\rm miss}_{T})^{2}/(1-y_{JB})$ and the Bjorken scaling
variable is $x_{JB}=Q^{2}_{JB}/(sy_{JB})$, where $s=4E_{e}E_{p}$ and $E_{e}$
($E_{p}$) is the energy of the electron (proton) beam. The resolution of
reconstructing these variables for inclusive DIS was investigated in Ref.
Aschenauer:2013iia , and was found to be reasonable for all three of these
variables. The performance of the Jacquet-Blondel method might be improved
with Machine-Learning methods such as those proposed in Refs. Arratia:2021tsq
; Diefenthaler:2021rdj ; Arratia:2022wny .

In Fig. 8, we compare the reconstructed values of $q_{T}/p_{T}^{\nu}$ with the
value obtained at generator level. In the bottom panel of this figure, we show
that the “bin purity”, or the fraction of events generated in a given bin that
are reconstructed to be in the same bin. The purity is more than 50%, which is
a level amenable to standard unfolding methods.

Figure 8: Top: 2D histogram of the generated $q_{T}/p_{T}^{\nu}$ ($x$ axis)
vs. the reconstructed value ($y$ axis). Middle: Spectra of reconstructed and
generated $q_{T}/p_{T}^{\nu}$. Bottom: Purity as a function of
$q_{T}/p^{\nu}_{T}$.

##  VI Suppression of the background from NC DIS and photoproduction

Given the relatively low rate of charged-current DIS events relative to
neutral-current DIS and photo-production, the background suppression generally
represents a significant challenge. If the scattered electron is missed, the
event topologies of neutral- and charged-current DIS can become identical. We
expect that this scenario will be significantly suppressed at the EIC compared
to the HERA experiments due to improved low-angle taggers for low-$Q^{2}$
events AbdulKhalek:2021gbh , although the performance of these systems is hard
to estimate at this point.

Rather than using a low-angle scattering veto to suppress photoproduction, we
follow the approach used by the CC DIS analyses at HERA Collaboration:2010xc
which relied on two kinematic variables: $\delta=\sum_{i}E_{i}-p_{z,i}$ (where
$E_{i}$ and $p_{z,i}$ are the reconstructed energy and longitudinal momentum
of detected particles, and the sum runs over all reconstructed particles) and
the ratio of the anti-parallel component $V_{AP}$ and the parallel component,
$V_{P}$, of the hadronic final state. The two components are defined as

$V_{AP}=-\sum_{i}\vec{p}_{T,i}\cdot\hat{n},\;\;\;\;{\rm
for}\;\;\vec{p}_{T,i}\cdot\hat{n}<0\,,$ (22)

and

$V_{P}=\sum_{i}\vec{p}_{T,i}\cdot\hat{n},\;\;\;\;{\rm
for}\;\;\vec{p}_{T,i}\cdot\hat{n}>0\,.$ (23)

Here $\vec{p}_{T,i}$ are the transverse parts of the individual particles’
momenta, $\hat{n}=-\vec{p}^{\,\nu}_{T}/|\vec{p}^{\,\nu}_{T}|$, and we sum over
all reconstructed particles in the event. The purpose of the cuts on this
variable is to ensure an azimuthally collimated energy flow. For charged-
current events, the ratio $V_{AP}/V_{P}$ is small – in particular for the
events that we are interested in for TMD studies.

In order to test the efficacy of these variables for background reduction, we
ran simulations of photoproduction reactions in the same manner as our CC DIS
simulations, see Sec. IV above. We focus on photoproduction because it is
expected to be the dominant background, based on experience from HERA
Collaboration:2010xc .

We used the following cuts: $p^{\,\nu}_{T}>15$ GeV, $V_{AP}/V_{P}<0.35$, and
$\delta<30$ GeV, which are similar to the values used in Ref.
Collaboration:2010xc . We found that $\approx 30\%$ of the generated CC DIS
events passed the cuts, whereas only 0.0005$\pm$0.0002% of photoproduction
events passed the cuts. However, the photoproduction cross section is three
orders of magnitude larger compared to CC DIS (58 nb, compared to 14 pb,
estimated using Pythia8). Therefore, we estimate that about $8\pm 3\%$ of the
identified event sample would be background from photoproduction when using
only cuts on kinematic variables and no additional low-angle electron tagger.

Given that our estimate suggests that the background will be reduced to
manageable levels, we neglect it from the projections we show in Sec. VII.

##  VII Statistical precision of spin asymmetry measurements

In Fig. 9, we show the statistical uncertainty projected for the transverse
single-spin asymmetry $A^{\sin(\phi_{q}-\phi_{S_{A}})}_{UT}$ as a function of
Bjorken $x$. Here we assume a luminosity of 100 fb-1 and the absolute
uncertainty of the asymmetry measurement is estimated to be
$\sqrt{2}/(p\sqrt{N})$, where $p$ is the polarization of the proton beam,
which we take to be 70$\%$, and $N$ is the number of events in a given bin
that pass our cuts, scaled to match the NLO total inclusive cross section of
Ref. Aschenauer:2013iia  and an integrated luminosity of 100 fb-1. Following
Ref. Anselmino:2011ay , we include a factor of $\sqrt{2}$ to account for the
fitting of the azimuthal modulations.

Figure 9: Projected statistical precision for the neutrino-jet asymmetry,
which is sensitive to the Sivers distribution, for $e^{+}p$ collisions (open
circles) and $e^{-}p$ collision (closed circles), for 100 fb-1. The yellow and
blue curves show theoretical results and the corresponding bands show the
uncertainty of the extracted Sivers function in Ref. Echevarria:2020hpy .

We compare these results to the numerical results of our calculations, see
Sec. III, which are integrated over the transverse-momentum imbalance
$0<q_{T}<5$ GeV and inelasticity $0.1<y<0.9$. The uncertainty bands of the
calculations show the uncertainty of current extractions of the Sivers
function, see Ref. Echevarria:2020hpy . The projected statistical error bars
are smaller than the predicted asymmetry for the first three bins, allowing
the proposed measurement to provide a decent comparison to theoretical
calculations.

Figure 10: Projected statistical precision for the $z_{h}$ dependence of the
$\pi^{\pm}$-in-jet Collins asymmetries (open circles for $\pi^{+}$, closed
circles for $\pi^{-}$) in electron-proton collisions.

While these measurements would provide weaker constraints on TMDs than
analogous ones in the neutral-current channel, they offer an independent check
with different flavor sensitivity. Moreover, they could test the consistency
and universality of the theoretical predictions.

The measurement that is projected in Fig. 9 would require electron and
positron beams. The capability to operate with positron beam is not included
in the EIC baseline design although it might be a possible upgrade. Figure 9
shows that the positron data would yield an opposite asymmetry compared to the
electron data, and their comparison could help constrain in particular the
TMDs associated with the $d$ quark.

We likewise show in Fig. 10 the projected statistical uncertainties for
hadron-in-jet Collins asymmetries as a function of $z_{h}$ and Bjorken $x$ for
charged pions with $j_{T}<1.5$ GeV. As explained in Sec. III, this asymmetry
is expected to vanish in CC DIS within the TMD factorization formalism at
leading power due to the chiral-odd nature of the transversity and Collins
functions. The projected statistical uncertainties are at the level of $1\%$
or smaller for small $z_{h}$. As we have mentioned earlier, performing these
measurements will enable precise tests of the theory and the assumptions of
factorization and chirality of the functions involved. If the asymmetry is not
observed to be exactly zero, it might indicate sensitivity to sub-leading
contributions that we have neglected in Sec. III or some other non-standard
effect.

##  VIII Summary and conclusions

We have proposed a novel channel to study the 3D structure of the nucleon at
the EIC that offers unique sensitivity to different quark flavors: neutrino-
jet correlations in charge-current deep-inelastic scattering.

We have presented first calculations of unpolarized cross-sections and
transverse-spin asymmetries for this channel. In addition, we performed
calculations of longitudinal and transverse momentum distributions of
identified hadrons inside jets and we compared our results to Monte-Carlo
event-generator simulations. We find that the hadron-in-jet Collins asymmetry
is exactly zero as a consequence of the chiral-odd nature of the effect.

We used the expected EIC machine parameters in terms of luminosity and energy
to estimate the kinematic reach of the proposed measurement. We also used fast
detector simulations to estimate the performance of the neutrino (missing-
momentum) reconstruction, and the neutrino-jet momentum imbalance. We found
that these measurements should be feasible with a general-purpose detector at
the EIC running at nominal luminosity.

Jet-based TMD measurements in charged-current DIS will provide important
cross-checks and complement analogous measurements in the neutral-current
channel. As such, we conclude that this channel represents an interesting
addition to the growing science program that can be carried out with jet
measurements at the future EIC.

###### Acknowledgements.

This work was supported by the MRPI program of the University of California
Office of the President, award number 00010100. M.A. was supported through DOE
Contract No. DE-AC05-06OR23177 under which JSA operates the Thomas Jefferson
National Accelerator Facility. Z.K. and F.Z. are supported by the National
Science Foundation under grant No. PHY-1945471. F.R. was supported by the
Simons Foundation under the Simons Bridge program for Postdoctoral Fellowships
at SCGP and YITP award number 815892; the NSF, award number 1915093; the DOE
Contract No. DE-AC05-06OR23177, under which Jefferson Science Associates, LLC
operates Jefferson Lab; and Old Dominion University. A.P. is supported by the
National Science Foundation Grant No. PHY-2012002 and by the DOE Contract No.
DE-AC05-06OR23177, under which Jefferson Science Associates, LLC operates
Jefferson Lab.

##  Appendix A Inclusive jet production

In this appendix, we provide the expressions for all the structure functions
that appear in Eq. (13). We start with the squared matrix element of the
process $e+p\rightarrow\nu+\mathrm{jet}+X$, where the incoming proton and
electron are unpolarized, which is given by

$\displaystyle|\mathcal{M}|^{2}=\left(\frac{e^{2}}{{2}\sin^{2}\theta_{w}}\right)^{2}|V_{ud}|^{2}\frac{(g^{\mu\nu}-\frac{q^{\mu}q^{\nu}}{m_{W}^{2}})(g^{\mu^{\prime}\nu^{\prime}}-\frac{q^{\mu^{\prime}}q^{\nu^{\prime}}}{m_{W}^{2}})}{(q^{2}-m_{W}^{2})^{2}+(m_{W}\Gamma_{W})^{2}}$
$\displaystyle\times\mathrm{Tr}\left[{\not{P}_{D}}\gamma^{\mu}\left(\frac{1-\gamma_{5}}{2}\right)\not{P}_{B}\gamma^{\mu^{\prime}}\left(\frac{1-\gamma_{5}}{2}\right)\right]$
$\displaystyle\times\mathrm{Tr}\left[\hat{\not{P}}_{C}\gamma^{\nu}\left(\frac{1-\gamma_{5}}{2}\right)\hat{\not{P}}_{A}\gamma^{\nu^{\prime}}\left(\frac{1-\gamma_{5}}{2}\right)\right]$
$\displaystyle=$
$\displaystyle\,8(G_{F}m_{W}^{2})^{2}|V_{ud}|^{2}\frac{1}{(\hat{t}-m_{W}^{2})^{2}+(m_{W}\Gamma_{W})^{2}}$
$\displaystyle\times\mathrm{Tr}\left[\not{P}_{D}\gamma^{\mu}\left(\frac{1-\gamma_{5}}{2}\right)\not{P}_{B}\gamma^{\nu}\left(\frac{1-\gamma_{5}}{2}\right)\right]$
$\displaystyle\times\mathrm{Tr}\left[\hat{\not{P}}_{C}\gamma_{\mu}\left(\frac{1-\gamma_{5}}{2}\right)\hat{\not{P}}_{A}\gamma_{\nu}\left(\frac{1-\gamma_{5}}{2}\right)\right]\,.$
(24)

Here $\hat{{P}}_{A}=xP_{A}$ and $\hat{{P}}_{C}={P}_{J}$ and we used
$4G_{F}/\sqrt{2}=e^{2}/(2m_{W}^{2}\sin^{2}\theta_{w})$. For a longitudinally
polarized proton with helicity $\lambda_{p}$, we substitute
$\hat{\not{P}}_{A}\rightarrow\gamma_{5}\hat{\not{P}}_{A}$. For a transversely
polarized proton with transverse spin $S_{T}^{i}$, we have
$\hat{\not{P}}_{A}\rightarrow\gamma_{5}\gamma_{i}\hat{\not{P}}_{A}$. However,
note that the trace of the hadronic tensor vanishes for a transversely
polarized proton. For a longitudinally polarized electron with helicity
$\lambda_{e}$, one substitutes
$\not{P}_{B}\rightarrow\not{P}_{B}+\lambda_{e}\gamma_{5}\not{P}_{B}$. The
leptonic tensor is given in terms of the momenta of the electron and the left-
handed neutrino:

$\displaystyle L^{\mu\nu}$
$\displaystyle=\mathrm{Tr}\left[\not{P}_{D}\gamma^{\mu}(1+\lambda_{e}\gamma_{5})\not{P}_{B}\gamma^{\nu}\left(\frac{1-\gamma_{5}}{2}\right)\right]$
$\displaystyle=(1-\lambda_{e})\left(P_{B}^{\mu}P_{D}^{\nu}+P_{B}^{\nu}P_{D}^{\mu}-g^{\mu\nu}P_{B}\cdot
P_{D}+i\epsilon^{\mu\nu P_{B}P_{D}}\right)$
$\displaystyle=L_{u}^{\mu\nu}+L_{p}^{\mu\nu}\,,$ (25)

where

$\displaystyle L_{u}^{\mu\nu}$
$\displaystyle=\left(P_{B}^{\mu}P_{D}^{\nu}+P_{B}^{\nu}P_{D}^{\mu}-g^{\mu\nu}P_{B}\cdot
P_{D}+i\epsilon^{\mu\nu P_{B}P_{D}}\right)\,,$ (26) $\displaystyle
L_{p}^{\mu\nu}$
$\displaystyle=-\lambda_{e}\left(P_{B}^{\mu}P_{D}^{\nu}+P_{B}^{\nu}P_{D}^{\mu}-g^{\mu\nu}P_{B}\cdot
P_{D}+i\epsilon^{\mu\nu P_{B}P_{D}}\right),$

represent the polarized and unpolarized components of the leptonic tensor. We
can then obtain the differential cross section given in Eq. (13) with the
following structure functions

$\displaystyle F_{UU}=$
$\displaystyle\sum_{q}\frac{|\overline{\mathcal{M}}_{eq\rightarrow\nu
q^{\prime}}|^{2}}{16\pi^{2}\hat{s}^{2}}H(Q,\mu)\,\mathcal{J}_{q}(p_{T}^{\mathrm{jet}}R,\mu)$
$\displaystyle\times\int\frac{{\rm
d}b_{T}b_{T}}{2\pi}J_{0}(q_{T}b_{T})\,f_{1}^{\mathrm{TMD}}(x,b_{T},\mu,\zeta)$
$\displaystyle\qquad\times S_{q}(b_{T},y_{\mathrm{jet}},R,\mu)\,,$
$\displaystyle=$
$\displaystyle\,\mathcal{C}\left[f_{1}\right]_{eq\rightarrow\nu
q^{\prime}}\,,$ (28) $\displaystyle F_{LU}=$
$\displaystyle\,\mathcal{C}\left[f_{1}\right]_{e_{L}q\rightarrow\nu
q^{\prime}}\,,$ (29) $\displaystyle F_{UL}=$
$\displaystyle\,\mathcal{C}\left[g_{1L}\right]_{eq_{L}\rightarrow\nu
q^{\prime}}\,,$ (30) $\displaystyle F_{LL}=$
$\displaystyle\,\mathcal{C}\left[g_{1L}\right]_{e_{L}q_{L}\rightarrow\nu
q^{\prime}}\,,$ (31) $\displaystyle F_{UT}^{\cos(\phi_{q}-\phi_{S_{A}})}=$
$\displaystyle\sum_{q}\frac{|\overline{\mathcal{M}}_{eq_{L}\rightarrow\nu
q^{\prime}}|^{2}}{16\pi^{2}\hat{s}^{2}}H(Q,\mu)\,\mathcal{J}_{q}(p_{T}^{\mathrm{jet}}R,\mu)$
$\displaystyle\times\int\frac{{\rm d}b_{T}b_{T}^{2}}{4\pi
M}J_{1}(q_{T}b_{T})\,g_{1T}^{(1),\mathrm{TMD}}(x,b_{T},\mu,\zeta)$
$\displaystyle\qquad\times S_{q}(b_{T},y_{\mathrm{jet}},R,\mu)\,,$
$\displaystyle=$
$\displaystyle\tilde{\mathcal{C}}\left[g_{1T}\right]_{eq_{L}\rightarrow\nu
q^{\prime}}\,,$ (32) $\displaystyle F_{LT}^{\cos(\phi_{q}-\phi_{S_{A}})}=$
$\displaystyle\,\tilde{\mathcal{C}}\left[g_{1T}\right]_{e_{L}q_{L}\rightarrow\nu
q^{\prime}}\,,$ (33) $\displaystyle F_{UT}^{\sin(\phi_{q}-\phi_{S_{A}})}=$
$\displaystyle\,\tilde{\mathcal{C}}\left[f^{\perp}_{1T}\right]_{eq\rightarrow\nu
q^{\prime}}\,,$ (34) $\displaystyle F_{LT}^{\sin(\phi_{q}-\phi_{S_{A}})}=$
$\displaystyle\,\tilde{\mathcal{C}}\left[f^{\perp}_{1T}\right]_{e_{L}q\rightarrow\nu
q^{\prime}}\,.$ (35)

The relevant leading-order matrix elements squared are given by

$\displaystyle|\overline{\mathcal{M}}_{eu\rightarrow\nu d}|^{2}=$
$\displaystyle\,8(G_{F}m_{W}^{2})^{2}|V_{ud}|^{2}\frac{\hat{s}^{2}}{(\hat{t}-m_{W}^{2})^{2}+m_{W}^{2}\Gamma_{W}^{2}}\,,$
(36)
$\displaystyle|\overline{\mathcal{M}}_{e\bar{d}\rightarrow\nu\bar{u}}|^{2}=$
$\displaystyle\,8(G_{F}m_{W}^{2})^{2}|V_{ud}|^{2}\frac{\hat{u}^{2}}{(\hat{t}-m_{W}^{2})^{2}+m_{W}^{2}\Gamma_{W}^{2}}\,,$
(37) $\displaystyle|\overline{\mathcal{M}}_{e_{L}q\rightarrow\nu
q^{\prime}}|^{2}=$ $\displaystyle\,-|\overline{\mathcal{M}}_{eq\rightarrow\nu
q^{\prime}}|^{2}\,,$ (38)
$\displaystyle|\overline{\mathcal{M}}_{eq_{L}\rightarrow\nu q^{\prime}}|^{2}=$
$\displaystyle\,-|\overline{\mathcal{M}}_{eq\rightarrow\nu
q^{\prime}}|^{2}\,,$ (39)
$\displaystyle|\overline{\mathcal{M}}_{e_{L}q_{L}\rightarrow\nu
q^{\prime}}|^{2}=$ $\displaystyle\,|\overline{\mathcal{M}}_{eq\rightarrow\nu
q^{\prime}}|^{2}\,,$ (40)

The matrix elements for an unpolarized and polarized electron are related to
each other, see Eqs. (39), (40) due to the factor $(1-\lambda_{e})$ in the
expression of the leptonic tensor. As a result, we obtain the following
relations between the different structure functions:

$\displaystyle F_{LU}$ $\displaystyle=-F_{UU}\,,$ (41) $\displaystyle F_{LL}$
$\displaystyle=-F_{UL}\,,$ (42) $\displaystyle
F_{LT}^{\cos(\phi_{q}-\phi_{S_{A}})}$
$\displaystyle=-F_{UT}^{\cos(\phi_{q}-\phi_{S_{A}})}\,,$ (43) $\displaystyle
F_{LT}^{\sin(\phi_{q}-\phi_{S_{A}})}$
$\displaystyle=-F_{UT}^{\sin(\phi_{q}-\phi_{S_{A}})}\,.$ (44)

##  Appendix B Hadron distributions inside the jet

For an unpolarized final-state hadron, we obtain TMD JFFs $\mathcal{D}_{1},\
\mathcal{H}_{1}^{\perp}$ at leading-twist Kang:2021ffh . The corresponding
correlator can be written as follows

$\displaystyle\Delta(z_{h},\vec{j}_{T})=$
$\displaystyle\mathcal{D}_{1}^{h/q}(z_{h},{j}_{T})\frac{\not{n}_{-}}{2}$
$\displaystyle-i\mathcal{H}^{\perp,h/q}_{1}(z_{h},{j}_{T})\frac{\not{j_{T}}}{z_{h}M_{h}}\frac{\not{n}_{-}}{2}\,,$
(45)

where we suppress the depenence on the renormalization scale $\mu$ and the
Collins-Soper scale $\zeta$ Collins:2011zzd . The different traces of the
correlator are given by

$\displaystyle\Delta^{h/q[\gamma^{-}]}=$
$\displaystyle\mathcal{D}_{1}^{h/q}(z_{h},{j}_{T})\,,$ (46)
$\displaystyle\Delta^{h/q[i\sigma^{i-}\gamma_{5}]}=$
$\displaystyle\frac{\epsilon_{T}^{ij}{j_{T}}^{j}}{z_{h}M_{h}}\mathcal{H}^{\perp,h/q}_{1}(z_{h},{j}_{T})\,,$
(47)

For an electron colliding with an unpolarized or a longitudinally polarized
initial proton, we obtain the same partonic scattering amplitudes as shown in
Appendix A. However, if the electron collides with a transversely polarized
quark from the initial proton, the corresponding term in the hadronic tensor
is given by

$\displaystyle H^{\mu\nu}_{\mathcal{H}_{1}^{\perp h/q}}={\rm
Tr}\left[\frac{\not{j_{T}}}{z_{h}M_{h}}\hat{\not{P}}_{C}\gamma^{\mu}\left(\frac{1-\gamma_{5}}{2}\right)\not{v}\hat{\not{P}}_{A}\gamma^{\nu}\left(\frac{1-\gamma_{5}}{2}\right)\right]\,$
(48)

where $\hat{{P}}_{A}=xP_{A}$ and $\hat{{P}}_{C}={P}_{J}$. For different TMD
PDFs, the vector $\not{v}$ is given by $\big{(}-\not{S}_{T}\gamma_{5}\big{)}$
for $h_{1}$, $\big{(}-\lambda_{p}\not{k}_{T}\gamma_{5}/M\big{)}$ for
$h_{1L}^{\perp}$, $\big{(}-i\not{k}_{T}/M\big{)}$ for $h_{1}^{\perp}$ and
$\big{(}(\vec{k}_{T}\cdot\vec{S}_{T}\not{k}_{T}-\vec{k}_{T}^{2}\not{S}_{T}/2)\gamma_{5}/M^{2}\big{)}$
for $h_{1T}^{\perp}$. Note that there are always three $\gamma$ matrices
between the $(1-\gamma_{5})/2$ factors in the expression of the hadronic
tensor in Eq. (48). Thus, we find

$\displaystyle\left(\frac{1-\gamma_{5}}{2}\right)\gamma_{\alpha}\gamma_{\beta}\gamma_{\rho}\left(\frac{1-\gamma_{5}}{2}\right)$
$\displaystyle=\gamma_{\alpha}\gamma_{\beta}\gamma_{\rho}\left(\frac{1+\gamma_{5}}{2}\right)\left(\frac{1-\gamma_{5}}{2}\right)=0\,.$
(49)

Therefore, the expression in Eq. (48) vanishes and spin asymmetries involving
transversely polarized quarks in CC DIS are zero. As a result, all
contributions related to chiral-odd Collins jet fragmentation function do not
appear in the differential cross section for hadron-in-jet production. Here,
we show the differential cross section in terms of the remaining non-zero
structure functions

$\displaystyle\frac{{\rm d}\sigma^{ep\rightarrow\nu+\mathrm{jet}X}}{{\rm
d}y_{J}\,{\rm d}^{2}p_{JT}\,{\rm d}^{2}q_{T}\,{\rm d}z_{h}\,{\rm
d}^{2}{j_{T}}}=F^{h}_{UU}+\lambda_{p}F^{h}_{UL}$
$\displaystyle+|S_{T}|\left[\cos(\phi_{q}-\phi_{S_{A}})F_{UT}^{h,\cos(\phi_{q}-\phi_{S_{A}})}\right.$
$\displaystyle\hskip
34.14322pt\left.+\sin(\phi_{q}-\phi_{S_{A}})F_{UT}^{h,\sin(\phi_{q}-\phi_{S_{A}})}\right]$
$\displaystyle+\lambda_{e}\left[F^{h}_{LU}+|S_{T}|\sin(\phi_{q}-\phi_{S_{A}})F_{LT}^{h,\sin(\phi_{q}-\phi_{S_{A}})}\right.$
$\displaystyle\quad+\left.\lambda_{p}F^{h}_{LL}+|S_{T}|\cos(\phi_{q}-\phi_{S_{A}})F_{LT}^{h,\cos(\phi_{q}-\phi_{S_{A}})})\right]\,.$
(50)

In total there are 8 structure functions, which are given by

$\displaystyle F^{h}_{UU}=$
$\displaystyle\sum_{q}\frac{|\overline{\mathcal{M}}_{eq\rightarrow\nu
q^{\prime}}|^{2}}{16\pi^{2}\hat{s}^{2}}H(Q,\mu)\,\mathcal{D}_{1}(p_{T}^{\mathrm{jet}}R,\mu)$
$\displaystyle\times\int\frac{{\rm
d}b_{T}b_{T}}{2\pi}J_{0}(q_{T}b_{T})\,f_{1}^{\mathrm{TMD}}(x,b_{T},\mu,\zeta)$
$\displaystyle\times S_{q}(b_{T},y_{\mathrm{jet}},R,\mu)\,,$ $\displaystyle=$
$\displaystyle\,\mathcal{C}^{h}\left[f_{1}\mathcal{D}_{1}\right]_{eq\rightarrow\nu
q^{\prime}}\,,$ (51) $\displaystyle F^{h}_{LU}=$
$\displaystyle\,\mathcal{C}^{h}\left[f_{1}\mathcal{D}_{1}\right]_{e_{L}q\rightarrow\nu
q^{\prime}}\,,$ (52) $\displaystyle F^{h}_{UL}=$
$\displaystyle\,\mathcal{C}^{h}\left[g_{1L}\mathcal{D}_{1}\right]_{eq_{L}\rightarrow\nu
q^{\prime}}\,,$ (53) $\displaystyle F^{h}_{LL}=$
$\displaystyle\,\mathcal{C}^{h}\left[g_{1L}\mathcal{D}_{1}\right]_{e_{L}q_{L}\rightarrow\nu
q^{\prime}}\,,$ (54) $\displaystyle F_{UT}^{h,\cos(\phi_{q}-\phi_{S_{A}})}=$
$\displaystyle\sum_{q}\frac{|\overline{\mathcal{M}}_{eq_{L}\rightarrow\nu
q^{\prime}}|^{2}}{16\pi^{2}\hat{s}^{2}}H(Q,\mu)\,\mathcal{D}_{1}(p_{T}^{\mathrm{jet}}R,\mu)$
$\displaystyle\times\int\frac{{\rm d}b_{T}b_{T}^{2}}{4\pi
M}J_{1}(q_{T}b_{T})\,g_{1T}^{(1),\mathrm{TMD}}(x,b_{T},\mu,\zeta)$
$\displaystyle\times S_{q}(b_{T},y_{\mathrm{jet}},R,\mu)\,,$ $\displaystyle=$
$\displaystyle\,\tilde{\mathcal{C}}^{h}\left[g_{1T}\mathcal{D}_{1}/M\right]_{eq_{L}\rightarrow\nu
q^{\prime}}\,,$ (55) $\displaystyle F_{LT}^{h,\cos(\phi_{q}-\phi_{S_{A}})}=$
$\displaystyle\,\tilde{\mathcal{C}}^{h}\left[{g_{1T}\mathcal{D}_{1}}/{M}\right]_{e_{L}q_{L}\rightarrow\nu
q^{\prime}}\,,$ (56) $\displaystyle F_{UT}^{h,\sin(\phi_{q}-\phi_{S_{A}})}=$
$\displaystyle\,\tilde{\mathcal{C}}^{h}\left[{f^{\perp}_{1T}\mathcal{D}_{1}}/{M}\right]_{eq\rightarrow\nu
q^{\prime}}\,,$ (57) $\displaystyle F_{LT}^{h,\sin(\phi_{q}-\phi_{S_{A}})}=$
$\displaystyle\,\tilde{\mathcal{C}}^{h}\left[{f^{\perp}_{1T}\mathcal{D}_{1}}/{M}\right]_{e_{L}q\rightarrow\nu
q^{\prime}}\,.$ (58)

We can obtain relations between the different hadron-in-jet structure
functions analogous to inclusive jets production (see Appendix A):

$\displaystyle F^{h}_{LU}$ $\displaystyle=-F^{h}_{UU}\,,$ (59) $\displaystyle
F^{h}_{LL}$ $\displaystyle=-F^{h}_{UL}\,,$ (60) $\displaystyle
F_{LT}^{h,\cos(\phi_{q}-\phi_{S_{A}})}$
$\displaystyle=-F_{UT}^{h,\cos(\phi_{q}-\phi_{S_{A}})}\,,$ (61) $\displaystyle
F_{LT}^{h,\sin(\phi_{q}-\phi_{S_{A}})}$
$\displaystyle=-F_{UT}^{h,\sin(\phi_{q}-\phi_{S_{A}})}\,.$ (62)

## Bibliography

  * (1) A. Accardi et al., “Electron Ion Collider: The Next QCD Frontier,” Eur. Phys. J. A52 no. 9, (2016) 268, arXiv:1212.1701 [nucl-ex].
  * (2) R. Abdul Khalek et al., “Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report,” Nucl. Phys. A 1026 (2022) 122447, arXiv:2103.05419 [physics.ins-det].
  * (3) M. Arratia, Y. Song, F. Ringer, and B. Jacak, “Jets as Precision Probes in Electron-Nucleus Collisions at the Electron-Ion Collider,” arXiv:1912.05931 [nucl-ex].
  * (4) B. S. Page, X. Chu, and E. C. Aschenauer, “Experimental Aspects of Jet Physics at a Future EIC,” Phys. Rev. D 101 no. 7, (2020) 072003, arXiv:1911.00657 [hep-ph].
  * (5) P. Hinderer, M. Schlegel, and W. Vogelsang, “Single-Inclusive Production of Hadrons and Jets in Lepton-Nucleon Scattering at NLO,” Phys. Rev. D 92 no. 1, (2015) 014001, arXiv:1505.06415 [hep-ph]. [Erratum: Phys.Rev.D 93, 119903 (2016)].
  * (6) R. Boughezal, F. Petriello, and H. Xing, “Inclusive jet production as a probe of polarized parton distribution functions at a future EIC,” Phys. Rev. D 98 no. 5, (2018) 054031, arXiv:1806.07311 [hep-ph].
  * (7) L. Zheng, E. C. Aschenauer, J. H. Lee, B.-W. Xiao, and Z.-B. Yin, “Accessing the gluon Sivers function at a future electron-ion collider,” Phys. Rev. D 98 no. 3, (2018) 034011, arXiv:1805.05290 [hep-ph].
  * (8) D. Gutierrez-Reyes, I. Scimemi, W. J. Waalewijn, and L. Zoppi, “Transverse momentum dependent distributions with jets,” Phys. Rev. Lett. 121 no. 16, (2018) 162001, arXiv:1807.07573 [hep-ph].
  * (9) D. Gutierrez-Reyes, I. Scimemi, W. J. Waalewijn, and L. Zoppi, “Transverse momentum dependent distributions in $e^{+}e^{-}$ and semi-inclusive deep-inelastic scattering using jets,” JHEP 10 (2019) 031, arXiv:1904.04259 [hep-ph].
  * (10) X. Liu, F. Ringer, W. Vogelsang, and F. Yuan, “Lepton-jet Correlations in Deep Inelastic Scattering at the Electron-Ion Collider,” Phys. Rev. Lett. 122 no. 19, (2019) 192003, arXiv:1812.08077 [hep-ph].
  * (11) Z.-B. Kang, K. Lee, and F. Zhao, “Polarized jet fragmentation functions,” Phys. Lett. B 809 (2020) 135756, arXiv:2005.02398 [hep-ph].
  * (12) I. Borsa, D. de Florian, and I. Pedron, “Inclusive-jet and dijet production in polarized deep inelastic scattering,” Phys. Rev. D 103 no. 1, (2021) 014008, arXiv:2010.07354 [hep-ph].
  * (13) M. Arratia, Y. Furletova, T. J. Hobbs, F. Olness, and S. J. Sekula, “Charm jets as a probe for strangeness at the future Electron-Ion Collider,” Phys. Rev. D 103 no. 7, (2021) 074023, arXiv:2006.12520 [hep-ph].
  * (14) M. Arratia, Z.-B. Kang, A. Prokudin, and F. Ringer, “Jet-based measurements of Sivers and Collins asymmetries at the future electron-ion collider,” Phys. Rev. D 102 no. 7, (2020) 074015, arXiv:2007.07281 [hep-ph].
  * (15) X. Liu, F. Ringer, W. Vogelsang, and F. Yuan, “Lepton-jet Correlation in Deep Inelastic Scattering,” Phys. Rev. D 102 no. 9, (2020) 094022, arXiv:2007.12866 [hep-ph].
  * (16) M. Arratia, Y. Makris, D. Neill, F. Ringer, and N. Sato, “Asymmetric jet clustering in deep-inelastic scattering,” Phys. Rev. D 104 no. 3, (2021) 034005, arXiv:2006.10751 [hep-ph].
  * (17) Z.-B. Kang, X. Liu, S. Mantry, and D. Y. Shao, “Jet Charge: A Flavor Prism for Spin Asymmetries at the EIC,” Phys. Rev. Lett. 125 (2020) 242003, arXiv:2008.00655 [hep-ph].
  * (18) V. Guzey and M. Klasen, “Diffractive dijet photoproduction at the EIC,” JHEP 05 (2020) 074, arXiv:2004.06972 [hep-ph].
  * (19) Z.-B. Kang, K. Lee, D. Y. Shao, and F. Zhao, “Spin asymmetries in electron-jet production at the future electron ion collider,” JHEP 11 (2021) 005, arXiv:2106.15624 [hep-ph].
  * (20) R. F. del Castillo, M. G. Echevarria, Y. Makris, and I. Scimemi, “Transverse momentum dependent distributions in dijet and heavy hadron pair production at EIC,” JHEP 03 (2022) 047, arXiv:2111.03703 [hep-ph].
  * (21) Y. Makris, “Revisiting the role of grooming in DIS,” Phys. Rev. D 103 no. 5, (2021) 054005, arXiv:2101.02708 [hep-ph].
  * (22) Y.-Y. Zhang and X.-N. Wang, “Parton rescattering and gluon saturation in dijet production at EIC,” Phys. Rev. D 105 no. 3, (2022) 034015, arXiv:2104.04520 [hep-ph].
  * (23) R. Abdul Khalek et al., “Snowmass 2021 White Paper: Electron Ion Collider for High Energy Physics,” in 2022 Snowmass Summer Study.  3, 2022.  arXiv:2203.13199 [hep-ph].
  * (24) X.-B. Tong, B.-W. Xiao, and Y.-Y. Zhang, “Harmonics of Parton Saturation in Lepton-Jet Correlations at the EIC,” arXiv:2211.01647 [hep-ph].
  * (25) X. Liu and H. X. Zhu, “The Nucleon Energy Correlators,” arXiv:2209.02080 [hep-ph].
  * (26) K. Lee, J. Mulligan, M. Płoskoń, F. Ringer, and F. Yuan, “Machine learning-based jet and event classification at the Electron-Ion Collider with applications to hadron structure and spin physics,” arXiv:2210.06450 [hep-ph].
  * (27) J. Arrington et al., “EIC Physics from An All-Silicon Tracking Detector,” arXiv:2102.08337 [nucl-ex].
  * (28) H1 Collaboration, F. Aaron et al., “Inclusive Deep Inelastic Scattering at High $Q^{2}$ with Longitudinally Polarised Lepton Beams at HERA,” JHEP 09 (2012) 061, arXiv:1206.7007 [hep-ex].
  * (29) ZEUS Collaboration, H. Abramowicz et al., “Measurement of high-$Q^{2}$ charged current deep inelastic scattering cross sections with a longitudinally polarised positron beam at HERA,” Eur. Phys. J. C 70 (2010) 945–963, arXiv:1008.3493 [hep-ex].
  * (30) ZEUS Collaboration, S. Chekanov et al., “Measurement of charged current deep inelastic scattering cross sections with a longitudinally polarised electron beam at HERA,” Eur. Phys. J. C 61 (2009) 223–235, arXiv:0812.4620 [hep-ex].
  * (31) ZEUS Collaboration, S. Chekanov et al., “Measurement of high Q2 charged current cross-sections in e-p deep inelastic scattering at HERA,” Phys. Lett. B 539 (2002) 197–217, arXiv:hep-ex/0205091. [Erratum: Phys.Lett.B 552, 308–308 (2003)].
  * (32) ZEUS Collaboration, J. Breitweg et al., “Measurement of high Q**2 charged current e+ p deep inelastic scattering cross-sections at HERA,” Eur. Phys. J. C 12 (2000) 411–428, arXiv:hep-ex/9907010. [Erratum: Eur.Phys.J.C 27, 305–309 (2003)].
  * (33) ZEUS Collaboration, M. Derrick et al., “Study of charged current $ep$ interactions at $Q^{2}$ ¿ $200-Gev^{2}$ with the ZEUS detector at HERA,” Z. Phys. C 72 (1996) 47–64, arXiv:hep-ex/9606014.
  * (34) H1, ZEUS Collaboration, H. Abramowicz et al., “Combination of measurements of inclusive deep inelastic ${e^{\pm}p}$ scattering cross sections and QCD analysis of HERA data,” Eur. Phys. J. C 75 no. 12, (2015) 580, arXiv:1506.06042 [hep-ex].
  * (35) ZEUS Collaboration, S. Chekanov et al., “Jet production in charged current deep inelastic e+ p scattering at HERA,” Eur. Phys. J. C31 (2003) 149–164, arXiv:hep-ex/0306018 [hep-ex].
  * (36) ZEUS Collaboration, S. Chekanov et al., “Multi-jet cross-sections in charged current e+- p scattering at HERA,” Phys. Rev. D78 (2008) 032004, arXiv:0802.3955 [hep-ex].
  * (37) T. Gehrmann, A. Huss, J. Niehues, A. Vogt, and D. M. Walker, “Jet production in charged-current deep-inelastic scattering to third order in QCD,” Phys. Lett. B792 (2019) 182–186, arXiv:1812.06104 [hep-ph].
  * (38) J. Blumlein, “The Theory of Deeply Inelastic Scattering,” Prog. Part. Nucl. Phys. 69 (2013) 28–84, arXiv:1208.6087 [hep-ph].
  * (39) E. C. Aschenauer, T. Burton, T. Martini, H. Spiesberger, and M. Stratmann, “Prospects for Charged Current Deep-Inelastic Scattering off Polarized Nucleons at a Future Electron-Ion Collider,” Phys. Rev. D88 (2013) 114025, arXiv:1309.5327 [hep-ph].
  * (40) H. Avakian, B. Parsamyan, and A. Prokudin, “Spin orbit correlations and the structure of the nucleon,” Riv. Nuovo Cim. 42 no. 1, (2019) 1–48, arXiv:1909.13664 [hep-ex].
  * (41) M. Anselmino, P. Gambino, and J. Kalinowski, “Polarized deep inelastic scattering at high-energies and parity violating structure functions,” Z. Phys. C 64 (1994) 267–274, arXiv:hep-ph/9401264.
  * (42) K.-b. Chen and W.-h. Yang, “Parity violating semi-inclusive deeply inelastic scattering at the Electron-Ion Collider,” arXiv:2004.01359 [hep-ph].
  * (43) W.-h. Yang, “Charged current semi-inclusive deeply inelastic scattering at the electron-ion collider,” Phys. Rev. D 103 (Jan, 2021) 016011. https://link.aps.org/doi/10.1103/PhysRevD.103.016011.
  * (44) X. Yang and W. Yang, “Semi-inclusive deeply inelastic (anti)neutrino nucleus scattering,” arXiv:2211.10899 [hep-ph].
  * (45) A. Gao, J. K. L. Michel, I. W. Stewart, and Z. Sun, “A Better Angle on Hadron Transverse Momentum Distributions at the EIC,” arXiv:2209.11211 [hep-ph].
  * (46) F. Yuan, “Azimuthal asymmetric distribution of hadrons inside a jet at hadron collider,” Phys. Rev. Lett. 100 (2008) 032003, arXiv:0709.3272 [hep-ph].
  * (47) M. Procura and I. W. Stewart, “Quark Fragmentation within an Identified Jet,” Phys. Rev. D 81 (2010) 074009, arXiv:0911.4980 [hep-ph]. [Erratum: Phys.Rev.D 83, 039902 (2011)].
  * (48) Z.-B. Kang, X. Liu, F. Ringer, and H. Xing, “The transverse momentum distribution of hadrons within jets,” JHEP 11 (2017) 068, arXiv:1705.08443 [hep-ph].
  * (49) A. Bacchetta, F. Delcarro, C. Pisano, M. Radici, and A. Signori, “Extraction of partonic transverse momentum distributions from semi-inclusive deep-inelastic scattering, Drell-Yan and Z-boson production,” JHEP 06 (2017) 081, arXiv:1703.10157 [hep-ph]. [Erratum: JHEP 06, 051 (2019)].
  * (50) I. Scimemi and A. Vladimirov, “Non-perturbative structure of semi-inclusive deep-inelastic and Drell-Yan scattering at small transverse momentum,” JHEP 06 (2020) 137, arXiv:1912.06532 [hep-ph].
  * (51) Z.-B. Kang, K. Lee, D. Y. Shao, and F. Zhao, “Spin Asymmetries in Electron-jet Production at the EIC,” arXiv:2201.04582 [hep-ph].
  * (52) S. D. Ellis, C. K. Vermilion, J. R. Walsh, A. Hornig, and C. Lee, “Jet Shapes and Jet Algorithms in SCET,” JHEP 11 (2010) 101, arXiv:1001.0014 [hep-ph].
  * (53) M. Cacciari, G. P. Salam, and G. Soyez, “The anti-$k_{t}$ jet clustering algorithm,” JHEP 04 (2008) 063, arXiv:0802.1189 [hep-ph].
  * (54) D. Boer, L. Gamberg, B. Musch, and A. Prokudin, “Bessel-Weighted Asymmetries in Semi Inclusive Deep Inelastic Scattering,” JHEP 10 (2011) 021, arXiv:1107.5294 [hep-ph].
  * (55) J. Collins, Foundations of perturbative QCD, vol. 32.  Cambridge University Press, 11, 2013.
  * (56) T. Sjostrand, S. Mrenna, and P. Z. Skands, “A Brief Introduction to PYTHIA 8.1,” Comput. Phys. Commun. 178 (2008) 852–867, arXiv:0710.3820 [hep-ph].
  * (57) M. A. Ebert, A. Gao, and I. W. Stewart, “Factorization for azimuthal asymmetries in SIDIS at next-to-leading power,” JHEP 06 (2022) 007, arXiv:2112.07680 [hep-ph].
  * (58) S. Rodini and A. Vladimirov, “Definition and evolution of transverse momentum dependent distribution of twist-three,” JHEP 08 (2022) 031, arXiv:2204.03856 [hep-ph].
  * (59) L. Gamberg, Z.-B. Kang, D. Y. Shao, J. Terry, and F. Zhao, “Transverse-momentum-dependent factorization at next-to-leading power,” arXiv:2211.13209 [hep-ph].
  * (60) S. Benic, Y. Hatta, H.-n. Li, and D.-J. Yang, “Single-spin asymmetries at two loops,” Phys. Rev. D 100 no. 9, (2019) 094027, arXiv:1909.10684 [hep-ph].
  * (61) S. Benić, Y. Hatta, A. Kaushik, and H.-n. Li, “gT(x) contribution to single spin asymmetries in SIDIS,” Phys. Rev. D 104 no. 9, (2021) 094027, arXiv:2109.05440 [hep-ph].
  * (62) A. Jain, M. Procura, and W. J. Waalewijn, “Parton Fragmentation within an Identified Jet at NNLL,” JHEP 05 (2011) 035, arXiv:1101.4953 [hep-ph].
  * (63) T. Kaufmann, A. Mukherjee, and W. Vogelsang, “Hadron Fragmentation Inside Jets in Hadronic Collisions,” Phys. Rev. D 92 no. 5, (2015) 054015, arXiv:1506.01415 [hep-ph]. [Erratum: Phys.Rev.D 101, 079901 (2020)].
  * (64) Z.-B. Kang, F. Ringer, and I. Vitev, “Jet substructure using semi-inclusive jet functions in SCET,” JHEP 11 (2016) 155, arXiv:1606.07063 [hep-ph].
  * (65) R. Bain, Y. Makris, and T. Mehen, “Transverse Momentum Dependent Fragmenting Jet Functions with Applications to Quarkonium Production,” JHEP 11 (2016) 144, arXiv:1610.06508 [hep-ph].
  * (66) Z.-B. Kang, A. Prokudin, F. Ringer, and F. Yuan, “Collins azimuthal asymmetries of hadron production inside jets,” Phys. Lett. B774 (2017) 635–642, arXiv:1707.00913 [hep-ph].
  * (67) Z.-B. Kang, K. Lee, J. Terry, and H. Xing, “Jet fragmentation functions for $Z$-tagged jets,” Phys. Lett. B 798 (2019) 134978, arXiv:1906.07187 [hep-ph].
  * (68) S. M. Aybat and T. C. Rogers, “TMD Parton Distribution and Fragmentation Functions with QCD Evolution,” Phys. Rev. D 83 (2011) 114042, arXiv:1101.5057 [hep-ph].
  * (69) D. de Florian, R. Sassot, M. Epele, R. J. Hernández-Pinto, and M. Stratmann, “Parton-to-Pion Fragmentation Reloaded,” Phys. Rev. D 91 no. 1, (2015) 014035, arXiv:1410.6027 [hep-ph].
  * (70) BNL, “An electron-ion collider study.” https://wiki.bnl.gov/eic/upload/EIC.Design.Study.pdf, 2020.
  * (71) B. Badelek, D. Y. Bardin, K. Kurek, and C. Scholz, “Radiative correction schemes in deep inelastic muon scattering,” Z. Phys. C 66 (1995) 591–600, arXiv:hep-ph/9403238.
  * (72) T. Liu, W. Melnitchouk, J.-W. Qiu, and N. Sato, “A new approach to semi-inclusive deep-inelastic scattering with QED and QCD factorization,” JHEP 11 (2021) 157, arXiv:2108.13371 [hep-ph].
  * (73) H1 Collaboration, V. Andreev et al., “Measurement of Lepton-Jet Correlation in Deep-Inelastic Scattering with the H1 Detector Using Machine Learning for Unfolding,” Phys. Rev. Lett. 128 no. 13, (2022) 132002, arXiv:2108.12376 [hep-ex].
  * (74) M. Cacciari, G. P. Salam, and G. Soyez, “FastJet User Manual,” Eur. Phys. J. C72 (2012) 1896, arXiv:1111.6097 [hep-ph].
  * (75) DELPHES 3 Collaboration, J. de Favereau, C. Delaere, P. Demin, A. Giammanco, V. Lemaître, A. Mertens, and M. Selvaggi, “DELPHES 3, A modular framework for fast simulation of a generic collider experiment,” JHEP 02 (2014) 057, arXiv:1307.6346 [hep-ex].
  * (76) M. Arratia and S. Sekula, “A Delphes card for the EIC yellow-report detector,” arXiv:2103.06886 [physics.ins-det].
  * (77) J. K. Adkins et al., “Design of the ECCE Detector for the Electron Ion Collider,” arXiv:2209.02580 [physics.ins-det].
  * (78) ATHENA Collaboration, J. Adam et al., “ATHENA detector proposal — a totally hermetic electron nucleus apparatus proposed for IP6 at the Electron-Ion Collider,” JINST 17 no. 10, (2022) P10019, arXiv:2210.09048 [physics.ins-det].
  * (79) CORE Collaboration, R. Alarcon et al., “CORE – a COmpact detectoR for the EIC,” arXiv:2209.00496 [physics.ins-det].
  * (80) M. Arratia et al., “A high-granularity calorimeter insert based on SiPM-on-tile technology at the future Electron-Ion Collider,” arXiv:2208.05472 [physics.ins-det].
  * (81) P. Newman and M. Wing, “The Hadronic Final State at HERA,” Rev. Mod. Phys. 86 no. 3, (2014) 1037, arXiv:1308.3368 [hep-ex].
  * (82) STAR Collaboration, B. I. Abelev et al., “Measurement of transverse single-spin asymmetries for di-jet production in proton-proton collisions at s**(1/2) = 200-GeV,” Phys. Rev. Lett. 99 (2007) 142003, arXiv:0705.4629 [hep-ex].
  * (83) U. Amaldi et al., “REPORT FROM THE STUDY GROUP ON DETECTORS FOR CHARGED CURRENT EVENTS,” in ECFA Study of an ep Facility for Europe.  1979\.
  * (84) M. Arratia, D. Britzger, O. Long, and B. Nachman, “Reconstructing the kinematics of deep inelastic scattering with deep learning,” Nucl. Instrum. Meth. A 1025 (2022) 166164, arXiv:2110.05505 [hep-ex].
  * (85) M. Diefenthaler, A. Farhat, A. Verbytskyi, and Y. Xu, “Deeply Learning Deep Inelastic Scattering Kinematics,” arXiv:2108.11638 [hep-ph].
  * (86) M. Arratia, D. Britzger, O. Long, and B. Nachman, “Optimizing observables with machine learning for better unfolding,” JINST 17 no. 07, (2022) P07009, arXiv:2203.16722 [hep-ex].
  * (87) M. Anselmino et al., “Transverse Momentum Dependent Parton Distribution/Fragmentation Functions at an Electron-Ion Collider,” Eur. Phys. J. A47 (2011) 35, arXiv:1101.4199 [hep-ex].
  * (88) M. G. Echevarria, Z.-B. Kang, and J. Terry, “Global analysis of the Sivers functions at NLO+NNLL in QCD,” JHEP 01 (2021) 126, arXiv:2009.10710 [hep-ph].


Therefore, the function $\mathcal{S}(k_{1},k_{2},k_{3})$ oscillates as
$\sin\left(\mu_{c}\log\left(\frac{k_{1}+k_{2}+k_{3}}{k_{0}}\right)\right)$
with an envelope given by
$((k_{1}+k_{2}+k_{3})/k_{0})^{-n}\times(k_{3}/k_{1})$ in the squeezed limit.
This functional form explains all the orange curves in Fig. 12. On top of this
resonant signal, NG induced by oscillatory feature also exhibits sinusoidal
running, $\mathcal{S}\propto\sin(K/k_{0})$, due to the abrupt onset of the
feature at $t=t_{0}$ (see [85] for more explanation), which is manifest in
Fig. 12 orange curves.

Figure 12: Contribution to the shape function $\mathcal{S}(k_{1},k_{2},k_{3})$
for the oscillatory feature scenario from Fig. 2b (blue) and Fig. 2c (orange).
The vertical scale corresponding to the blue (orange) curve is shown on the
left (right) axis for each panel. The left and the right panels correspond to
$n=0$ and $n=3/2$, respectively. The fist row shows the dependence of
$\mathcal{S}$ on $k_{3}/k_{0}$ with $(k_{1}+k_{2})/k_{3}$ fixed; the second
row shows the dependence of $\mathcal{S}$ on $(k_{1}+k_{2})/k_{3}$ with
$k_{3}/k_{0}$ fixed. Note that the orange curves are power-law suppressed in
$\mu$ and do not contain the clock signal with mass $\mu H$. Their scale-
dependent oscillation patterns are determined by the background frequency
$\mu_{c}$ as $\mathcal{S}\propto\sin[\mu_{c}\log(K/k_{0})]$ for fixed
$(k_{1}+k_{2})/k_{3}$ (upper) and $k_{3}/k_{0}$ (lower). Here $K\equiv
k_{1}+k_{2}+k_{3}$. In addition, we observe sinusoidal running in orange
curves due to the sharp feature, given by $\mathcal{S}\propto\sin(K/k_{0})$
for fixed $(k_{1}+k_{2})/k_{3}$ (upper) and $k_{3}/k_{0}$ (lower).

Now we move on to Fig. 13 where we show results for the sharp feature case
given in Eq. (3.8). As before, Fig. 2c again reduces to a contact interaction
which we can roughly parametrize as a sharp feature acting on the contact
vertex,

$\displaystyle\propto
B_{0}\theta(t-t_{0})\dot{\delta\phi}\left[(\dot{\delta\phi})^{2}-\frac{1}{a^{2}}(\partial_{i}\delta\phi)^{2}\right].$
(D.5)

Given this structure, we can immediately notice that the oscillation of
$\mathcal{S}$ will be given by
$\propto\sin\left((k_{1}+k_{2}+k_{3})/k_{0}\right)$, which explains both
oscillation patterns of the orange curves in Fig. 13. We notice from the right
panel that the Fig. 2c contribution can actually be comparable or bigger than
the Fig. 2b contribution in some kinematic configurations. This is likely a
consequence of an infinitely sharp feature and in realistic models this
contribution is expected to become smaller. Fig. 14 contains some further
comparison between the oscillatory feature and sharp feature case.

Figure 13: Contribution to the shape function $\mathcal{S}(k_{1},k_{2},k_{3})$
for the sharp feature scenario from Fig. 2b (blue) and Fig. 2c (orange). The
vertical scale corresponding to the blue (orange) curve is shown on the left
(right) axis for each panel. The sinusoidal running in orange curves is
determined by $\mathcal{S}\propto\sin((k_{1}+k_{2}+k_{3})/k_{0})$ which
reduces to the observed frequencies for fixed $(k_{1}+k_{2})/k_{3}$ (left) and
$k_{3}/k_{0}$ (right). We note that, for the right panel, the Fig. 2c (orange)
contribution is larger than Fig. 2b (blue) contribution due to the infinite
sharpness of the feature in the setup.  Figure 14: Amplitudes of the feature-
induced NG in the contact interaction limit. First row: NG induced by the
positive frequency component of the oscillatory feature. This component has
sinusoidal running due to the step function as well as resonant signal due to
the oscillatory feature. Second row: NG induced by negative frequency
component of the oscillatory feature which only contains sinusoidal running as
this component has no resonance point. Third row: Subtraction of the second
and first rows, corresponding to an oscillatory feature
$\propto\sin(t-t_{0})\theta(t-t_{0})$. The amplitude of sinusoidal running is
reduced compared to each of the positive and negative frequency components as
the sharp feature is less sharp, i.e., the background feature $B(t)$ is
continuous but with an abrupt onset. Fourth row: Step sharp feature exhibits
similar behavior as negative frequency component of the oscillatory feature
since in both of these cases only sharp feature contributes to NG.

##  Appendix E Useful formulae

##### Asymptotic expansions:

  * •

Gamma function at $y\gg 1$:

$\displaystyle\Gamma(x\pm iy)\simeq\sqrt{2\pi}e^{\mp i(\pi/4-\pi
x/2+y)}y^{x-1/2\pm iy}e^{-\pi y/2}\,,$ (E.1)

where $x$ and $y$ are positive real numbers.

  * •

Hypergeometric function at $p\gg 1$:

$\displaystyle_{2}F_{1}(a,b;c;p)\simeq\dfrac{\Gamma(c)\Gamma(b-a)}{\Gamma(b)\Gamma(c-a)}(-p)^{-a}+\dfrac{\Gamma(c)\Gamma(a-b)}{\Gamma(a)\Gamma(c-b)}(-p)^{-b}\,.$
(E.2)

  * •

Hankel function at $z\lesssim\sqrt{\mu}$ (late-time expansion):

$\displaystyle H^{(1)}_{i\mu}(z)$
$\displaystyle\simeq~{}{1+\coth(\pi\mu)\over\Gamma(1+i\mu)}\left({z\over
2}\right)^{i\mu}-i{\Gamma(i\mu)\over\pi}\left({z\over 2}\right)^{-i\mu}$
$\displaystyle\xrightarrow[]{\mu\gg
1}e^{-i\pi/4}\sqrt{{2\over\pi\mu}}e^{\pi\mu/2}e^{i\mu(1-\log\mu)}\left({z\over
2}\right)^{i\mu},$ (E.3) $\displaystyle H^{(2)}_{i\mu}(z)$
$\displaystyle\simeq~{}{1-\coth(\pi\mu)\over\Gamma(1+i\mu)}\left({z\over
2}\right)^{i\mu}+i{\Gamma(i\mu)\over\pi}\left({z\over 2}\right)^{-i\mu}$
$\displaystyle\xrightarrow[]{\mu\gg
1}e^{i\pi/4}\sqrt{{2\over\pi\mu}}e^{-\pi\mu/2}e^{-i\mu(1-\log\mu)}\left({z\over
2}\right)^{-i\mu}\,.$ (E.4)

  * •

Hankel function at $z\gtrsim\mu^{2}$ (early-time expansion):

$\displaystyle H^{(1)}_{i\mu}(z)\simeq$
$\displaystyle~{}e^{-i\pi/4}\sqrt{{2\over\pi z}}e^{\pi\mu/2}\,e^{iz}~{},$
(E.5) $\displaystyle H^{(2)}_{i\mu}(z)\simeq$
$\displaystyle~{}e^{i\pi/4}\sqrt{{2\over\pi z}}e^{-\pi\mu/2}\,e^{-iz}\,.$
(E.6)

##### Hypergeometric integrals:

$\displaystyle e^{-\pi\mu/2}\int_{0}^{\infty}{\rm
d}z\,z^{n}H^{(1)}_{i\mu}(z)\,e^{+ipz}=$
$\displaystyle\frac{(i/2)^{n}}{\sqrt{\pi}\,\Gamma(n+3/2)}\Gamma(n+1+i\mu)\Gamma(n+1-i\mu)_{2}F_{1}(n+1-i\mu,n+1+i\mu,n+3/2,(1-p)/2)\;,$
(E.7) $\displaystyle e^{\pi\mu/2}\int_{0}^{\infty}{\rm
d}z\,z^{n}H^{(2)}_{i\mu}(z)\,e^{-ipz}=$
$\displaystyle\frac{(-i/2)^{n}}{\sqrt{\pi}\,\Gamma(n+3/2)}\Gamma(n+1+i\mu)\Gamma(n+1-i\mu)_{2}F_{1}(n+1+i\mu,n+1-i\mu,n+3/2,(1-p)/2)\;.$
(E.8)

## References

  * [1] Planck Collaboration, Y. Akrami et al., “Planck 2018 results. X. Constraints on inflation,” Astron. Astrophys. 641 (2020) A10, arXiv:1807.06211 [astro-ph.CO].
  * [2] N. Bartolo, E. Komatsu, S. Matarrese, and A. Riotto, “Non-Gaussianity from inflation: Theory and observations,” Phys. Rept. 402 (2004) 103–266, arXiv:astro-ph/0406398.
  * [3] M. Liguori, E. Sefusatti, J. R. Fergusson, and E. P. S. Shellard, “Primordial non-Gaussianity and Bispectrum Measurements in the Cosmic Microwave Background and Large-Scale Structure,” Adv. Astron. 2010 (2010) 980523, arXiv:1001.4707 [astro-ph.CO].
  * [4] X. Chen, “Primordial Non-Gaussianities from Inflation Models,” Adv. Astron. 2010 (2010) 638979, arXiv:1002.1416 [astro-ph.CO].
  * [5] Y. Wang, “Inflation, Cosmic Perturbations and Non-Gaussianities,” Commun. Theor. Phys. 62 (2014) 109–166, arXiv:1303.1523 [hep-th].
  * [6] X. Chen and Y. Wang, “Quasi-Single Field Inflation and Non-Gaussianities,” JCAP 04 (2010) 027, arXiv:0911.3380 [hep-th].
  * [7] N. Arkani-Hamed and J. Maldacena, “Cosmological Collider Physics,” arXiv:1503.08043 [hep-th].
  * [8] X. Chen and Y. Wang, “Large non-Gaussianities with Intermediate Shapes from Quasi-Single Field Inflation,” Phys. Rev. D 81 (2010) 063511, arXiv:0909.0496 [astro-ph.CO].
  * [9] D. Baumann and D. Green, “Signatures of Supersymmetry from the Early Universe,” Phys. Rev. D 85 (2012) 103520, arXiv:1109.0292 [hep-th].
  * [10] V. Assassi, D. Baumann, and D. Green, “On Soft Limits of Inflationary Correlation Functions,” JCAP 11 (2012) 047, arXiv:1204.4207 [hep-th].
  * [11] X. Chen and Y. Wang, “Quasi-Single Field Inflation with Large Mass,” JCAP 09 (2012) 021, arXiv:1205.0160 [hep-th].
  * [12] E. Sefusatti, J. R. Fergusson, X. Chen, and E. P. S. Shellard, “Effects and Detectability of Quasi-Single Field Inflation in the Large-Scale Structure and Cosmic Microwave Background,” JCAP 08 (2012) 033, arXiv:1204.6318 [astro-ph.CO].
  * [13] J. Norena, L. Verde, G. Barenboim, and C. Bosch, “Prospects for constraining the shape of non-Gaussianity with the scale-dependent bias,” JCAP 08 (2012) 019, arXiv:1204.6324 [astro-ph.CO].
  * [14] S. Pi and M. Sasaki, “Curvature Perturbation Spectrum in Two-field Inflation with a Turning Trajectory,” JCAP 10 (2012) 051, arXiv:1205.0161 [hep-th].
  * [15] T. Noumi, M. Yamaguchi, and D. Yokoyama, “Effective field theory approach to quasi-single field inflation and effects of heavy fields,” JHEP 06 (2013) 051, arXiv:1211.1624 [hep-th].
  * [16] J.-O. Gong, S. Pi, and M. Sasaki, “Equilateral non-Gaussianity from heavy fields,” JCAP 11 (2013) 043, arXiv:1306.3691 [hep-th].
  * [17] R. Emami, “Spectroscopy of Masses and Couplings during Inflation,” JCAP 04 (2014) 031, arXiv:1311.0184 [hep-th].
  * [18] X. Chen, M. H. Namjoo, and Y. Wang, “Quantum Primordial Standard Clocks,” JCAP 02 (2016) 013, arXiv:1509.03930 [astro-ph.CO].
  * [19] E. Dimastrogiovanni, M. Fasiello, and M. Kamionkowski, “Imprints of Massive Primordial Fields on Large-Scale Structure,” JCAP 02 (2016) 017, arXiv:1504.05993 [astro-ph.CO].
  * [20] A. Kehagias and A. Riotto, “High Energy Physics Signatures from Inflation and Conformal Symmetry of de Sitter,” Fortsch. Phys. 63 (2015) 531–542, arXiv:1501.03515 [hep-th].
  * [21] X. Chen, Y. Wang, and Z.-Z. Xianyu, “Loop Corrections to Standard Model Fields in Inflation,” JHEP 08 (2016) 051, arXiv:1604.07841 [hep-th].
  * [22] H. Lee, D. Baumann, and G. L. Pimentel, “Non-Gaussianity as a Particle Detector,” JHEP 12 (2016) 040, arXiv:1607.03735 [hep-th].
  * [23] P. D. Meerburg, M. Münchmeyer, J. B. Muñoz, and X. Chen, “Prospects for Cosmological Collider Physics,” JCAP 03 (2017) 050, arXiv:1610.06559 [astro-ph.CO].
  * [24] X. Chen, Y. Wang, and Z.-Z. Xianyu, “Standard Model Background of the Cosmological Collider,” Phys. Rev. Lett. 118 no. 26, (2017) 261302, arXiv:1610.06597 [hep-th].
  * [25] X. Chen, Y. Wang, and Z.-Z. Xianyu, “Standard Model Mass Spectrum in Inflationary Universe,” JHEP 04 (2017) 058, arXiv:1612.08122 [hep-th].
  * [26] X. Chen, Y. Wang, and Z.-Z. Xianyu, “Schwinger-Keldysh Diagrammatics for Primordial Perturbations,” JCAP 12 (2017) 006, arXiv:1703.10166 [hep-th].
  * [27] A. Kehagias and A. Riotto, “On the Inflationary Perturbations of Massive Higher-Spin Fields,” JCAP 07 (2017) 046, arXiv:1705.05834 [hep-th].
  * [28] H. An, M. McAneny, A. K. Ridgway, and M. B. Wise, “Quasi Single Field Inflation in the non-perturbative regime,” JHEP 06 (2018) 105, arXiv:1706.09971 [hep-ph].
  * [29] H. An, M. McAneny, A. K. Ridgway, and M. B. Wise, “Non-Gaussian Enhancements of Galactic Halo Correlations in Quasi-Single Field Inflation,” Phys. Rev. D 97 no. 12, (2018) 123528, arXiv:1711.02667 [hep-ph].
  * [30] A. V. Iyer, S. Pi, Y. Wang, Z. Wang, and S. Zhou, “Strongly Coupled Quasi-Single Field Inflation,” JCAP 01 (2018) 041, arXiv:1710.03054 [hep-th].
  * [31] S. Kumar and R. Sundrum, “Heavy-Lifting of Gauge Theories By Cosmic Inflation,” JHEP 05 (2018) 011, arXiv:1711.03988 [hep-ph].
  * [32] X. Chen, W. Z. Chua, Y. Guo, Y. Wang, Z.-Z. Xianyu, and T. Xie, “Quantum Standard Clocks in the Primordial Trispectrum,” JCAP 05 (2018) 049, arXiv:1803.04412 [hep-th].
  * [33] X. Chen, Y. Wang, and Z.-Z. Xianyu, “Neutrino Signatures in Primordial Non-Gaussianities,” JHEP 09 (2018) 022, arXiv:1805.02656 [hep-ph].
  * [34] W. Z. Chua, Q. Ding, Y. Wang, and S. Zhou, “Imprints of Schwinger Effect on Primordial Spectra,” JHEP 04 (2019) 066, arXiv:1810.09815 [hep-th].
  * [35] S. Kumar and R. Sundrum, “Seeing Higher-Dimensional Grand Unification In Primordial Non-Gaussianities,” JHEP 04 (2019) 120, arXiv:1811.11200 [hep-ph].
  * [36] Y.-P. Wu, “Higgs as heavy-lifted physics during inflation,” JHEP 04 (2019) 125, arXiv:1812.10654 [hep-ph].
  * [37] A. Moradinezhad Dizgah, H. Lee, J. B. Muñoz, and C. Dvorkin, “Galaxy Bispectrum from Massive Spinning Particles,” JCAP 05 (2018) 013, arXiv:1801.07265 [astro-ph.CO].
  * [38] R. Saito and T. Kubota, “Heavy Particle Signatures in Cosmological Correlation Functions with Tensor Modes,” JCAP 06 (2018) 009, arXiv:1804.06974 [hep-th].
  * [39] X. Tong, Y. Wang, and S. Zhou, “Unsuppressed primordial standard clocks in warm quasi-single field inflation,” JCAP 06 (2018) 013, arXiv:1801.05688 [hep-th].
  * [40] S. Alexander, S. J. Gates, L. Jenks, K. Koutrolikos, and E. McDonough, “Higher Spin Supersymmetry at the Cosmological Collider: Sculpting SUSY Rilles in the CMB,” JHEP 10 (2019) 156, arXiv:1907.05829 [hep-th].
  * [41] S. Lu, Y. Wang, and Z.-Z. Xianyu, “A Cosmological Higgs Collider,” JHEP 02 (2020) 011, arXiv:1907.07390 [hep-th].
  * [42] A. Hook, J. Huang, and D. Racco, “Searches for other vacua. Part II. A new Higgstory at the cosmological collider,” JHEP 01 (2020) 105, arXiv:1907.10624 [hep-ph].
  * [43] A. Hook, J. Huang, and D. Racco, “Minimal signatures of the Standard Model in non-Gaussianities,” Phys. Rev. D 101 no. 2, (2020) 023519, arXiv:1908.00019 [hep-ph].
  * [44] S. Kumar and R. Sundrum, “Cosmological Collider Physics and the Curvaton,” JHEP 04 (2020) 077, arXiv:1908.11378 [hep-ph].
  * [45] L.-T. Wang and Z.-Z. Xianyu, “In Search of Large Signals at the Cosmological Collider,” JHEP 02 (2020) 044, arXiv:1910.12876 [hep-ph].
  * [46] T. Liu, X. Tong, Y. Wang, and Z.-Z. Xianyu, “Probing P and CP Violations on the Cosmological Collider,” JHEP 04 (2020) 189, arXiv:1909.01819 [hep-ph].
  * [47] D.-G. Wang, “On the inflationary massive field with a curved field manifold,” JCAP 01 (2020) 046, arXiv:1911.04459 [astro-ph.CO].
  * [48] Y. Wang and Y. Zhu, “Cosmological Collider Signatures of Massive Vectors from Non-Gaussian Gravitational Waves,” JCAP 04 (2020) 049, arXiv:2001.03879 [astro-ph.CO].
  * [49] L. Li, S. Lu, Y. Wang, and S. Zhou, “Cosmological Signatures of Superheavy Dark Matter,” JHEP 07 (2020) 231, arXiv:2002.01131 [hep-ph].
  * [50] L.-T. Wang and Z.-Z. Xianyu, “Gauge Boson Signals at the Cosmological Collider,” JHEP 11 (2020) 082, arXiv:2004.02887 [hep-ph].
  * [51] J. Fan and Z.-Z. Xianyu, “A Cosmic Microscope for the Preheating Era,” JHEP 01 (2021) 021, arXiv:2005.12278 [hep-ph].
  * [52] K. Kogai, K. Akitsu, F. Schmidt, and Y. Urakawa, “Galaxy imaging surveys as spin-sensitive detector for cosmological colliders,” JCAP 03 (2021) 060, arXiv:2009.05517 [astro-ph.CO].
  * [53] A. Bodas, S. Kumar, and R. Sundrum, “The Scalar Chemical Potential in Cosmological Collider Physics,” JHEP 02 (2021) 079, arXiv:2010.04727 [hep-ph].
  * [54] S. Aoki and M. Yamaguchi, “Disentangling mass spectra of multiple fields in cosmological collider,” JHEP 04 (2021) 127, arXiv:2012.13667 [hep-th].
  * [55] N. Arkani-Hamed, D. Baumann, H. Lee, and G. L. Pimentel, “The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities,” JHEP 04 (2020) 105, arXiv:1811.00024 [hep-th].
  * [56] D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee, and G. L. Pimentel, “The cosmological bootstrap: weight-shifting operators and scalar seeds,” JHEP 12 (2020) 204, arXiv:1910.14051 [hep-th].
  * [57] D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee, and G. L. Pimentel, “The Cosmological Bootstrap: Spinning Correlators from Symmetries and Factorization,” SciPost Phys. 11 (2021) 071, arXiv:2005.04234 [hep-th].
  * [58] N. Maru and A. Okawa, “Non-Gaussianity from $X,Y$ gauge bosons in Cosmological Collider Physics,” arXiv:2101.10634 [hep-ph].
  * [59] S. Lu, “Axion isocurvature collider,” JHEP 04 (2022) 157, arXiv:2103.05958 [hep-th].
  * [60] Q. Lu, M. Reece, and Z.-Z. Xianyu, “Missing scalars at the cosmological collider,” JHEP 12 (2021) 098, arXiv:2108.11385 [hep-ph].
  * [61] L.-T. Wang, Z.-Z. Xianyu, and Y.-M. Zhong, “Precision calculation of inflation correlators at one loop,” JHEP 02 (2022) 085, arXiv:2109.14635 [hep-ph].
  * [62] X. Tong, Y. Wang, and Y. Zhu, “Cutting rule for cosmological collider signals: a bulk evolution perspective,” JHEP 03 (2022) 181, arXiv:2112.03448 [hep-th].
  * [63] L. Pinol, S. Aoki, S. Renaux-Petel, and M. Yamaguchi, “Inflationary flavor oscillations and the cosmic spectroscopy,” arXiv:2112.05710 [hep-th].
  * [64] Y. Cui and Z.-Z. Xianyu, “Probing Leptogenesis with the Cosmological Collider,” arXiv:2112.10793 [hep-ph].
  * [65] X. Tong and Z.-Z. Xianyu, “Large Spin-2 Signals at the Cosmological Collider,” arXiv:2203.06349 [hep-ph].
  * [66] M. Reece, L.-T. Wang, and Z.-Z. Xianyu, “Large-Field Inflation and the Cosmological Collider,” arXiv:2204.11869 [hep-ph].
  * [67] G. L. Pimentel and D.-G. Wang, “Boostless Cosmological Collider Bootstrap,” arXiv:2205.00013 [hep-th].
  * [68] Z. Qin and Z.-Z. Xianyu, “Phase Information in Cosmological Collider Signals,” arXiv:2205.01692 [hep-th].
  * [69] N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space.  Cambridge Monographs on Mathematical Physics. Cambridge Univ. Press, Cambridge, UK, 2, 1984.
  * [70] Particle Data Group Collaboration, P. A. Zyla et al., “Review of Particle Physics,” PTEP 2020 no. 8, (2020) 083C01.
  * [71] C. M. Sou, X. Tong, and Y. Wang, “Chemical-potential-assisted particle production in FRW spacetimes,” JHEP 06 (2021) 129, arXiv:2104.08772 [hep-th].
  * [72] P. Creminelli, “On non-Gaussianities in single-field inflation,” JCAP 10 (2003) 003, arXiv:astro-ph/0306122.
  * [73] J. Chluba, J. Hamann, and S. P. Patil, “Features and New Physical Scales in Primordial Observables: Theory and Observation,” Int. J. Mod. Phys. D 24 no. 10, (2015) 1530023, arXiv:1505.01834 [astro-ph.CO].
  * [74] A. Slosar et al., “Scratches from the Past: Inflationary Archaeology through Features in the Power Spectrum of Primordial Fluctuations,” Bull. Am. Astron. Soc. 51 no. 3, (2019) 98, arXiv:1903.09883 [astro-ph.CO].
  * [75] WMAP Collaboration, H. V. Peiris et al., “First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Implications for inflation,” Astrophys. J. Suppl. 148 (2003) 213–231, arXiv:astro-ph/0302225.
  * [76] X. Chen, “Primordial Features as Evidence for Inflation,” JCAP 01 (2012) 038, arXiv:1104.1323 [hep-th].
  * [77] X. Chen, “Fingerprints of Primordial Universe Paradigms as Features in Density Perturbations,” Phys. Lett. B 706 (2011) 111–115, arXiv:1106.1635 [astro-ph.CO].
  * [78] X. Chen and C. Ringeval, “Searching for Standard Clocks in the Primordial Universe,” JCAP 08 (2012) 014, arXiv:1205.6085 [astro-ph.CO].
  * [79] T. Battefeld, J. C. Niemeyer, and D. Vlaykov, “Probing Two-Field Open Inflation by Resonant Signals in Correlation Functions,” JCAP 05 (2013) 006, arXiv:1302.3877 [astro-ph.CO].
  * [80] X. Gao, D. Langlois, and S. Mizuno, “Oscillatory features in the curvature power spectrum after a sudden turn of the inflationary trajectory,” JCAP 10 (2013) 023, arXiv:1306.5680 [hep-th].
  * [81] T. Noumi and M. Yamaguchi, “Primordial spectra from sudden turning trajectory,” JCAP 12 (2013) 038, arXiv:1307.7110 [hep-th].
  * [82] R. Saito, M. Nakashima, Y.-i. Takamizu, and J. Yokoyama, “Resonant Signatures of Heavy Scalar Fields in the Cosmic Microwave Background,” JCAP 11 (2012) 036, arXiv:1206.2164 [astro-ph.CO].
  * [83] R. Saito and Y.-i. Takamizu, “Localized Features in Non-Gaussianity from Heavy Physics,” JCAP 06 (2013) 031, arXiv:1303.3839 [astro-ph.CO].
  * [84] X. Chen and M. H. Namjoo, “Standard Clock in Primordial Density Perturbations and Cosmic Microwave Background,” Phys. Lett. B 739 (2014) 285–292, arXiv:1404.1536 [astro-ph.CO].
  * [85] X. Chen, M. H. Namjoo, and Y. Wang, “Models of the Primordial Standard Clock,” JCAP 02 (2015) 027, arXiv:1411.2349 [astro-ph.CO].
  * [86] Q.-G. Huang and S. Pi, “Power-law modulation of the scalar power spectrum from a heavy field with a monomial potential,” JCAP 04 (2018) 001, arXiv:1610.00115 [hep-th].
  * [87] G. Domènech, J. Rubio, and J. Wons, “Mimicking features in alternatives to inflation with interacting spectator fields,” Phys. Lett. B 790 (2019) 263–269, arXiv:1811.08224 [astro-ph.CO].
  * [88] M. Braglia, X. Chen, and D. K. Hazra, “Comparing multi-field primordial feature models with the Planck data,” JCAP 06 (2021) 005, arXiv:2103.03025 [astro-ph.CO].
  * [89] M. Braglia, X. Chen, and D. K. Hazra, “Uncovering the History of Cosmic Inflation from Anomalies in Cosmic Microwave Background Spectra,” arXiv:2106.07546 [astro-ph.CO].
  * [90] M. Braglia, X. Chen, and D. K. Hazra, “Primordial Standard Clock Models and CMB Residual Anomalies,” arXiv:2108.10110 [astro-ph.CO].
  * [91] X. Chen, R. Easther, and E. A. Lim, “Generation and Characterization of Large Non-Gaussianities in Single Field Inflation,” JCAP 04 (2008) 010, arXiv:0801.3295 [astro-ph].
  * [92] R. Flauger, L. McAllister, E. Pajer, A. Westphal, and G. Xu, “Oscillations in the CMB from Axion Monodromy Inflation,” JCAP 06 (2010) 009, arXiv:0907.2916 [hep-th].
  * [93] R. Flauger and E. Pajer, “Resonant Non-Gaussianity,” JCAP 01 (2011) 017, arXiv:1002.0833 [hep-th].
  * [94] X. Chen, “Folded Resonant Non-Gaussianity in General Single Field Inflation,” JCAP 12 (2010) 003, arXiv:1008.2485 [hep-th].
  * [95] A. A. Starobinsky, “Spectrum of adiabatic perturbations in the universe when there are singularities in the inflation potential,” JETP Lett. 55 (1992) 489–494.
  * [96] J. A. Adams, B. Cresswell, and R. Easther, “Inflationary perturbations from a potential with a step,” Phys. Rev. D 64 (2001) 123514, arXiv:astro-ph/0102236.
  * [97] R. Bean, X. Chen, G. Hailu, S. H. H. Tye, and J. Xu, “Duality Cascade in Brane Inflation,” JCAP 03 (2008) 026, arXiv:0802.0491 [hep-th].
  * [98] A. Achucarro, J.-O. Gong, S. Hardeman, G. A. Palma, and S. P. Patil, “Features of heavy physics in the CMB power spectrum,” JCAP 01 (2011) 030, arXiv:1010.3693 [hep-ph].
  * [99] N. Bartolo, D. Cannone, and S. Matarrese, “The Effective Field Theory of Inflation Models with Sharp Features,” JCAP 10 (2013) 038, arXiv:1307.3483 [astro-ph.CO].
  * [100] L. Kofman, A. D. Linde, and A. A. Starobinsky, “Towards the theory of reheating after inflation,” Phys. Rev. D 56 (1997) 3258–3295, arXiv:hep-ph/9704452.
  * [101] D. J. H. Chung, E. W. Kolb, A. Riotto, and I. I. Tkachev, “Probing Planckian physics: Resonant production of particles during inflation and features in the primordial power spectrum,” Phys. Rev. D 62 (2000) 043508, arXiv:hep-ph/9910437.
  * [102] L. Kofman, A. D. Linde, X. Liu, A. Maloney, L. McAllister, and E. Silverstein, “Beauty is attractive: Moduli trapping at enhanced symmetry points,” JHEP 05 (2004) 030, arXiv:hep-th/0403001.
  * [103] N. Barnaby, Z. Huang, L. Kofman, and D. Pogosyan, “Cosmological Fluctuations from Infra-Red Cascading During Inflation,” Phys. Rev. D 80 (2009) 043501, arXiv:0902.0615 [hep-th].
  * [104] R. Flauger, M. Mirbabayi, L. Senatore, and E. Silverstein, “Productive Interactions: heavy particles and non-Gaussianity,” JCAP 10 (2017) 058, arXiv:1606.00513 [hep-th].
  * [105] J. H. Kim, S. Kumar, A. Martin, and Y. Tsai, “Cosmological particle production and pairwise hotspots on the CMB,” JHEP 11 (2021) 158, arXiv:2107.09061 [hep-ph].
  * [106] J. M. Maldacena, “Non-Gaussian features of primordial fluctuations in single field inflationary models,” JHEP 05 (2003) 013, arXiv:astro-ph/0210603.
  * [107] S. Weinberg, “Quantum contributions to cosmological correlations,” Phys. Rev. D 72 (2005) 043514, arXiv:hep-th/0506236.
  * [108] A. Achúcarro et al., “Inflation: Theory and Observations,” arXiv:2203.08128 [astro-ph.CO].