# Predictions for the sPHENIX physics program

Ron Belmont  Jasmine Brewer  Quinn Brodsky  Paul Caucal  Megan
Connors111Workshop organizers Magdalena Djordjevic  Raymond
Ehlers222Contributions on behalf of the JETSCAPE Collaboration Miguel A.
Escobedo  Elena G. Ferreiro  Giuliano Giacalone  Yoshitaka Hatta  Jack Holguin
Weiyao Ke  Zhong-Bo Kang  Amit Kumar333Contributions on behalf of the JETSCAPE
Collaboration Aleksas Mazeliauskas  Yacine Mehtar-Tani444Workshop organizers
Genki Nukazuka555Workshop organizers Daniel Pablos  Dennis V.
Perepelitsa666Workshop organizers Krishna Rajagopal  Anne M.
Sickles777Workshop organizers Michael Strickland  Konrad Tywoniuk  Ivan Vitev
Xin-Nian Wang  Zhong Yang  Fanyi Zhao

###### Abstract

sPHENIX is a next-generation detector experiment at the Relativistic Heavy Ion
Collider, designed for a broad set of jet and heavy-flavor probes of the
Quark-Gluon Plasma created in heavy ion collisions. In anticipation of the
commissioning and first data-taking of the detector in 2023, a RIKEN-BNL
Research Center (RBRC) workshop was organized to collect theoretical input and
identify compelling aspects of the physics program. This paper compiles
theoretical predictions from the workshop participants for jet quenching,
heavy flavor and quarkonia, cold QCD, and bulk physics measurements at
sPHENIX.

###### keywords:

Relativistic Heavy Ion Collider , heavy-ion collisions , quark-gluon plasma ,
jet quenching , heavy flavor , thermalization

††journal: Nuclear Physics A

[uncg]organization=University of North Carolina, Greensboro
[cern]organization=CERN [mit]organization=Massachusetts Institute of
Technology [subatech]organization=SUBATECH (IMT Atlantique, Université de
Nantes, IN2P3/CNRS) [gsu]organization=Georgia State University
[belgrade]organization=Institute of Physics Belgrade
[berkeley]organization=University of California Berkeley
[santiago]organization=Universidade de Santiago de Compostela
[barcelona]organization=Universitat de Barcelona
[heidelberg]organization=Institut für Theoretische Physik, Universität
Heidelberg [bnl]organization=Brookhaven National Laboratory
[rbrc]organization=RIKEN BNL Research Center [cnrs]organization=Ecole
Polytechnique [lanl]organization=Los Alamos National Laboratory
[ucla]organization=University of California Los Angeles
[wayne]organization=Wayne State University [infn]organization=INFN, Sezione di
Torino [oviedo]organization=Departamento de Física, Universidad de Oviedo
[ictea]organization=Instituto Universitario de Ciencias y Tecnologías
Espaciales de Asturias (ICTEA) [cu]organization=University of Colorado Boulder
[uiuc]organization=University of Illinois Urbana Champaign
[ksu]organization=Kent State University [bergen]organization=Bergen University
[ccnu]organization=Central China Normal University [lbnl]organization=Lawrence
Berkeley National Laboratory

###### Contents

  1. 1 Introduction
  2. 2 sPHENIX Experiment at RHIC
  3. 3 Jet probes of the QGP
    1. 3.1 Baseline for jet and hadron nuclear modification factors in sPHENIX
    2. 3.2 JetMed predictions for sPHENIX
    3. 3.3 JETSCAPE predictions for sPHENIX
    4. 3.4 $R_{\rm AA}$ and $v_{2}$ as jet substructure observables with sPHENIX
    5. 3.5 CoLBT-hydro predictions for sPHENIX
    6. 3.6 Disentangling jet modification in sPHENIX
    7. 3.7 Comprehensive SCETG predictions for sPHENIX
    8. 3.8 Energy correlations in jets with sPHENIX
    9. 3.9 Bayesian inference with JETSCAPE using sPHENIX data
  4. 4 Heavy flavor quark probes of the QGP
    1. 4.1 DREENA-A predictions for open heavy flavor in sPHENIX
    2. 4.2 Heavy flavor jets in sPHENIX with LIDO
    3. 4.3 KSU and QTraj predictions for Upsilon suppression in sPHENIX
    4. 4.4 Modeling quarkonium suppression in sPHENIX
  5. 5 Bulk probes of the QGP
    1. 5.1 Non-Gaussian flow fluctuations at sPHENIX
  6. 6 QCD with polarized protons and cold nuclei
    1. 6.1 Azimuthal angular correlation in dijet events at sPHENIX
    2. 6.2 Hadron-in-jet production with sPHENIX
  7. 7 Conclusion

##  1 Introduction

Relativistic collisions of heavy ion nuclear beams produce Quark-Gluon Plasma
(QGP), a high-density and high-temperature phase of matter comprised of
deconfined and strongly-interacting quarks and gluons [1, 2]. The QGP is the
primordial substance which dominated the observable universe in the
microseconds after its creation in the Big Bang, and has been the subject of
intensive experimental studies for decades at facilities around the world. In
the modern era of collider-based experiments, QGP has been produced and
studied at the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National
Laboratory (BNL) since 2000 and at the Large Hadron Collider (LHC) at CERN
since 2010.

The QGP exhibits several remarkable many-body phenomena, such as a collective
expansion which is describable by relativistic hydrodynamics with a near-
perfect fluidity [3], first observed by experiments at RHIC [4, 5, 6, 7] and
subsequently confirmed by experiments at the LHC [8, 9, 10]. The QGP is
composed of quarks and gluons whose interactions are described by quantum
chromo-dynamics (QCD), the theory of the strong nuclear interaction. Despite
this, it is not understood how the properties and observed long-wavelength
behavior of the QGP emerge from these fundamental degrees of freedom - a key
open question in nuclear physics [11]. To address this question, high
transverse momentum ($p_{\mathrm{T}}$) jets or heavy-flavor hadrons that are
formed from the fragmentation of hard partons produced in the early stages of
the nucleus–nucleus collision, have been recognized as unique probes of the
QGP over a wide range of scales. As they propagate through the expanding,
cooling medium, their interactions with the QGP probe its properties over
momentum scales ranging from the deeply perturbative to those comparable to
fluid scales. Experimental measurements of the “quenching” of
high-$p_{\mathrm{T}}$ jets and the modification of heavy-flavor hadrons have
been used for this purpose at RHIC, and then greatly expanded under the later-
generation experiments at the LHC [12, 13].

To definitively address the above open questions and complete the scientific
mission of RHIC, the sPHENIX experiment was designed as a new, next-generation
collider detector to measure jet and heavy-flavor observables with a level of
precision not previously achievable at RHIC. The particular timeliness and
necessity of the sPHENIX physics program has been widely recognized, for
example in white papers contributed by the community to the U.S. Long-Range
Plan for Nuclear Science process starting in 2022 [14, 15]. As this paper is
being written, sPHENIX is undergoing final installation in the experimental
hall of the former PHENIX detector, with first collisions for detector
commissioning expected in May 2023.

In order to continue to develop the theoretical context for the first sPHENIX
data and strengthen the scientific relationships between the sPHENIX
Collaboration and the theoretical community, a three-day workshop sponsored by
the RIKEN BNL Research Center (RBRC) was held at BNL in July 2022 with over
one hundred registered participants, entitled “Predictions for sPHENIX” [16].
The workshop was held in a plenary style, with heavy-ion theorists asked to
highlight interesting physics which is potentially accessible with sPHENIX
and, when possible, to give concrete predictions before the arrival of first
data. This paper represents a summary of the predictions and physics
discussion at the workshop, with written contributions from the theory
speakers and their close collaborators collected in the sub-sections that
follow. The intention of the authors is that the summaries below help motivate
further theoretical work which is focused on the quantitative extraction of
physics information from the future sPHENIX data.

##  2 sPHENIX Experiment at RHIC

Figure 1: 3-D rendering of the sPHENIX detector. An azimuthal slice has been
removed to better show the component sub-detectors. Not shown are the ZDC and
SMD, which are situated out of frame on either side of the collision point.

The sPHENIX experiment is a new collider detector at RHIC, situated at the 8
o’clock position of the ring, in the interaction region of the predecessor
PHENIX experiment. Some principal highlights of sPHENIX relevant to the
physics program include its high data rate, large acceptance, precision
tracking close to the vertex, high momentum resolution, hadronic calorimetry,
unbiased triggering in $p$+$p$ collisions, and streaming readout capability of
the tracking detectors. The detector and its major subsystems are rendered in
Fig. 1.

The sPHENIX detector is constructed around and within a superconducting
solenoid magnet, originally used in the BaBar experiment [17], which produces
an axial magnetic field that reaches 1.4 Tesla in its center. The tracking
system consists of four detectors, listed here in order of their proximity to
the beampipe. The monolithic active pixel sensor (MAPS) vertex detector (MVTX)
consists of three layers providing precision position information to identify
the originating vertex of charged-particle tracks. An intermediate silicon
tracker (INTT) consists of four layers which provide two hit measurements. The
INTT has a fast timing resolution and is used to connect the track
trajectories between the MVTX and Time Projection Chamber (TPC). The compact
TPC has a gateless design and provides the large lever arm for the momentum
measurement. The MVTX, INTT, and TPC trackers have full azimuthal acceptance,
and an acceptance in pseudorapidity of $\left|\eta\right|<1.1$ for events with
a collision vertex within $\left|z\right|<10$ cm. Finally, the TPC Outer
Tracker (TPOT) detector consists of eight rectangular panels situated below
the TPC which provide a confirmation hit to aid in the determination of
corrections due to TPC space charge distortions. All the tracking detectors
are read out in a continuous streaming mode. Some aspects of the tracking
reconstruction in sPHENIX are discussed in Ref. [18].

The calorimeter system comprises the electro-magnetic calorimeter (EMCal) and
a hadronic calorimeter split into an Inner (IHCal) and Outer (OHCal) section
on either side of sPHENIX solenoid magnet. The hermetic system covers the full
acceptance in azimuth and $\left|\eta\right|<1.1$. The EMCal is composed of
towers filled with tungsten and scintillating fiber with a 2-D projective
design. The segmentation of the towers is $\Delta\eta\times\Delta\phi\approx
0.025\times 0.025$, and the EMCal provides approximately twenty radiation
lengths for high-resolution measurements of the energy deposited by photons
and electrons. The hadronic calorimeter is composed of aluminum (IHCal) or
steel (OHCal) plates interleaved with scintillating tiles. The IHCal is
particularly important for catching the start of hadronic showers before the
inactive material in the magnet. The IHCal and OHCal together provide
approximately five nuclear interaction lengths, with a segmentation of
$\Delta\eta\times\Delta\phi\approx 0.1\times 0.1$, and good energy resolution
appropriate for calorimeter-based measurements of high-$p_{\mathrm{T}}$
hadrons and jets. The OHCal also doubles as the flux return for the magnet.
Prototypes of the calorimeter system, and their description in simulation,
have been characterized using test beam at the Fermilab Test Beam Facility
[19, 20].

The tracking and calorimeter barrel detectors are complemented with a suite of
forward detectors for minimum-bias triggering and event categorization. These
include the Minimum-Bias Detector (MBD), the sPHENIX Event Plane Detector
(sEPD), and the Zero-Degree Calorimeter (ZDC) which includes a Shower Max
Detector (SMD). The MBD consists of quartz radiator tubes situated on either
side of the detector at $3.51<\left|\eta\right|<4.61$, and will be used for
minimum-bias triggering and as a possible detector for global centrality
determination. The sEPD consists of two wheels of finely-segmented
scintillating plastic tiles situated at $2.0<\left|\eta\right|<4.9$, and is
intended for event plane determination. Two ZDC detectors are situated 18 m
away on either side of the collision point, which measure primarily neutrons
from the fragmenting nuclei and aid in the event selection. Finally, the SMD
consists of layers of scintillating tiles situated behind the first ZDC
module, providing spatial information. It may be used to determine the first-
order event plane ($\Psi_{1}$) in Au+Au data-taking and as a polarimeter in
polarized $p$+$p$ and $p$+Au data-taking.

The sPHENIX scientific collaboration has proposed a three-year data-taking
program which would accomplish the scientific mission of the experiment. That
plan and some examples of the statistical and kinematic reach for specific
physics measurements are detailed in the sPHENIX Beam Use Proposal [21] and
briefly summarized at the start of each section below. The proposed running
scenario begins with Au+Au running in 2023 to commission the detector and
provide first physics data, transversely polatized $p$+$p$ and $p$+Au running
in 2024 to provide the reference data for the Au+Au program and for cold QCD
physics, and high-statistics Au+Au running in 2025 to provide an archival
dataset for QGP physics with unprecedented statistical precision at RHIC.
Furthermore, the sPHENIX collaboration stands ready to capitalize on
additional opportunities for data-taking to deliver physics beyond the core
program, such as the detailed exploration of intermediate collision species at
RHIC [22].

##  3 Jet probes of the QGP

A major scientific motivation for the sPHENIX experiment is its broad program
of reconstructed-jet physics measurements. The large acceptance, hermetic
hadronic calorimetry, high-efficiency tracking, and large data-taking rate of
the detector will result in an enormous data sample of high-$p_{\mathrm{T}}$
reconstructed jets. Measurements of the jet kinematics can be performed using
information from just the calorimeter system, or by further incorporating
information from the trackers. The luminosity during the first three years of
data-taking is projected to provide one million jets above $30$ GeV and fifty
thousand direct photons above $20$ GeV in the most central 0–10% Au+Au events.
These capabilities will enable precision measurements of overall jet
production and intra-event correlations (such as jet $v_{n}$, di-jet
asymmetries, photon-tagged jets), the modification of the internal jet
(sub-)structure, and correlated presence of particles at large angles from the
jet axis. The kinematic coverage will overlap with those of the LHC at the
high-$p_{\mathrm{T}}$ (for example, projections indicate a measurement of the
jet $R_{\mathrm{AA}}$ out to 70 GeV) while at the same, due to the smaller
underlying event, extending into a lower-$p_{\mathrm{T}}$ regime not
previously accessible. Given the large potential of this aspect of the sPHENIX
physics program, a significant focus of the workshop was dedicated towards
predictions for jet probes of the QGP, which are summarized below.

###  3.1 Baseline for jet and hadron nuclear modification factors in sPHENIX

One of the most robust signals of a hot QCD medium created in nuclear
collisions is the suppression of high momentum spectra of single inclusive
hadrons and jets. The commonly used experimental observable for quantifying
these energy loss phenomena is the ratio of hadron $(h)$ or jet $(j)$ spectra
in AA and $pp$ collisions, which is known as the nuclear modification factor
$R^{j,h}_{\mathrm{AA}}$. However even in the absence of a hot QCD medium,
$R^{j,h}_{\mathrm{AA}}$ deviates from unity due to the different partonic
compositions of a proton and a nucleus. These parton distribution functions
(PDFs) are extracted from global fits of perturbative QCD calculations to a
large set of experimental data. With the increasing order of perturbative
calculations and the increasing amount of experimental data, both proton and
nuclear PDFs are constantly being improved. Nevertheless, uncertainties in
nuclear PDFs (nPDFs) remain the dominant uncertainty in the theoretical
calculations of $R^{j,h}_{\mathrm{AA}}$ baseline in the absence of hot QCD
medium [23, 24].

Here the theoretical baseline of jet and hadron $R_{\mathrm{AA}}$ at
$\sqrt{s}=200\,\mathrm{GeV}$ is computed in minimum bias oxygen-oxygen (OO)
and gold-gold (AuAu) collisions.

$\left.R_{\mathrm{AA}}^{h,j}\right|_{\mathrm{min-
bias}}=\frac{1}{A^{2}}\frac{d\sigma^{h,j}_{\mathrm{AA}}/dp_{T}dy}{d\sigma^{h,j}_{pp}/dp_{T}dy}.$
(1)

Three recent extractions of nuclear PDFs are employed: EPPS21 [25], nNNPDF3.0
[26] and TUJU21 [27] with corresponding proton PDFs.

Figure 2: Baseline (no energy loss) jet nuclear modification factor for (left)
oxygen-oxygen and (right) gold-gold collisions. Blue solid band shows the
cancellation of correlated scale uncertainties at leading order. Hatched bands
show 90% confidence intervals of combined nuclear and proton PDFs. Lower
panels show $R^{j}_{\mathrm{AA}}$ normalized to the central EPPS21 line.

Fig. 2 shows the jet nuclear modification factor $R_{\mathrm{AA}}^{j}$, for
which the correlated variation of factorization and renormalization scales
give negligible scale uncertainty band. Although these calculations were done
only at leading order for partonic jets, previous computations at next-to-
leading order have shown that $R^{j}_{\mathrm{AA}}$ is rather insensitive to
the perturbative order and showering effects [23, 24]. For AuAu collisions
Fig. 2 (right) shows that the nPDF uncertainties for nPDF sets are of order of
10% at $p_{T}^{j}=10\,\mathrm{GeV}$ and grow for larger momentum (especially
for nNNPDF3.0). There is a slight tension between EPPS21 and nNNPDF3.0
extractions, while TUJU21 results sit in between. This indicates that even for
large nuclei for which there are collider data available, different nPDF
extractions could result in non-negligible differences. Finally, for
$p_{T}^{j}>25\,\mathrm{GeV}$ there is a downward trend in
$R_{\mathrm{AA}}^{j}$ and therefore any conclusions about the magnitude of jet
quenching in AuAu should take into account this additional suppression. Fig. 2
(left) shows the analogous results for OO collisions. Surprisingly, the
agreement between the central values of different nPDF sets is better for OO
than AuAu. It could be in part due to the general reduction of nuclear
modification for lighter nuclei. However, the nNNPDF3.0 set, which does not
impose particular $A$ dependence in its extraction, results in much wider
uncertainty bands with over 20% above $p_{T}^{j}>20\,\mathrm{GeV}$, which
points to the lack of experimental data for light nuclei in nPDF extractions.
If a (possibly small) jet quenching signal was to be observed in OO
collisions, the uncertainties of nPDF should be reduced.

Figure 3: Baseline (no energy loss) hadron nuclear modification factor for
(left) oxygen-oxygen and (righ) gold-gold collisions. Green and blue solid
bands show the cancellation of correlated scale uncertainties at leading and
next-to-leading order. Hatched bands show 90% confidence intervals of combined
nuclear and proton PDFs. Lower panels show $R^{h}_{\mathrm{AA}}$ normalized to
the central EPPS21 line.

Fig. 3 shows the charged hadron nuclear modification factor
$R_{\mathrm{AA}}^{h}$ computed at next-to-leading order, using the INCNLO code
[28]888http://lapth.cnrs.fr/PHOX_FAMILY/readme_inc.html with LHAPDF grid
support [29]. For the charged hadrons, the sum of BKK pion and kaon
fragmentation functions (FFs) [30] was used. Different choices in the FFs
result in negligible uncertainty compared to the nPDF uncertainty [23, 24].
Similar to the jet $R_{\mathrm{AA}}$, there is a good cancellation of scale
uncertainties both at leading and next-to-leading order. For AuAu again there
is a slight tension between EPPS21 and nNNPDF3.0 results. For AuAu the
uncertainties and consistency between different nPDF sets are better for
hadrons than jets as the 90% confidence intervals are contained in $\pm 25\%$
bands (see lower panel of Fig. 3 (right)) in the studied momentum range. For
OO collisions the uncertainty bands becomes over 20% above
$p_{T}^{h}>15\,\mathrm{GeV}$.

In summary, this contribution presents the baseline calculations of jet and
hadron $R_{\mathrm{AA}}$ in the absence of hot medium effects for OO and AuAu
collisions at $\sqrt{s}=200\,\mathrm{GeV}$. An experimentally measured
deviation from these baselines would be a clear signal of additional physics
to the perturbative vacuum picture of high-energy particle collisions.
Although jet quenching and energy loss phenomena have been observed previously
in AuAu collisions, the quantified theory uncertainties of a perturbative
baseline are important in the precision quantification of hot QCD medium
effects. In the case of OO collisions, a measurement of statistically
significant deviation from the computed bands in Fig. 2 (left) and Fig. 3
(left) would signify the discovery of energy loss phenomena in small collision
systems with an average of $\sim 10$ colliding nucleons.

###  3.2 JetMed predictions for sPHENIX

JetMed is a parton shower in heavy-ion collisions based on the factorization
between vacuum-like emissions (VLEs) and medium-induced radiations as
described in [31, 32]. This picture of jet evolution, which relies on well-
controlled approximations in pQCD, turns out to be in good qualitative
agreement with LHC data on high $p_{t}$ jets [32, 33, 34]. Even though the
underlying approximations are less justified for the lower $p_{t}$ jets
measured at RHIC, it would be interesting to confront this model with future
sPHENIX data. In particular, JetMed includes color coherence effects via the
in-medium phase space for VLEs which is constrained by the coherence angle
$\theta_{c}$. Unraveling the role of color coherence in jet quenching is an
active field of investigation both at RHIC and at the LHC [35, 36, 37, 38].
This $\theta_{c}$ angle scales like $\theta_{c}\sim 2/(\hat{q}L^{3})^{1/2}$
for a dense medium with average quenching parameter $\hat{q}$ and size $L$
[39, 40, 41]. Hence, since $\hat{q}$ at RHIC energies is typically smaller
than $\hat{q}$ at the LHC, $\theta_{c}$ is larger at RHIC than at the LHC (for
a similar colliding system size $L$). In principle, this should facilitate its
measurement.

Figure 4: JetMed predictions for the $R_{\mathrm{AA}}$ for dijet (left) and
$\gamma$-jet (right) processes in sPHENIX.

Jet substructure techniques are ideal tools to probe color (de)coherence and
potentially measure $\theta_{c}$. Among those, the Soft Drop [42] or
Dynamically groomed (DyG) [43] jet radius have shown good sensitivity to
$\theta_{c}$ due to a rather simple physical mechanism [34, 44]. Namely, the
medium acts as a filter which enhances the production of small
$\theta_{g}\leq\theta_{c}$ jets compared to larger $\theta_{g}>\theta_{c}$
jets, as the latter lose more energy than the former due to more resolved
intrajet sources for energy loss. This is shown in the left panel of Fig. 4
which displays the nuclear modification factor for the $\theta_{g}$
distribution obtained from Soft Drop $(\beta=0,z_{\rm cut}=0.2)$ in red and
dynamical grooming with $a=0.7$ in blue (this choice of $a$ maximizes the
sensitivity to $\theta_{c}$ while reducing non-perturbative and background
effects [44]). In this figure, the uncertainty bands correspond to variations
of the unphysical cut-off parameters in the parton shower and variations of
the effective $\hat{q}$ between $0.4$–$1.2$ GeV${}^{2}/$fm. One observes a
clear relative enhancement of small $\theta_{g}$ jets and a suppression of
large $\theta_{g}$ jets with a transition set by the angular scale
$\theta_{c}$ (which lies inside the vertical grey band on the figure). In
comparison, $R_{AA}(\theta_{g})$ in $\gamma$-jet events is shown in the right
panel of Fig. 4 with the event selection performed on the photon $p_{t}$,
following the proposal of [45] to disentangle bias effects from intrinsic
modifications of the shower (besides energy loss). Interestingly,
$R_{AA}(\theta_{g})$ is not equal to one since JetMed also accounts for
relatively hard intrajet medium-induced emissions, whose typical angular scale
is also set by $\theta_{c}$ [40]. Such complementary studies between dijet and
$\gamma$-jet events is a promising opportunity of the sPHENIX detector.

###  3.3 JETSCAPE predictions for sPHENIX

JETSCAPE is a large-scale benchmarking framework for rigorously testing and
validating physics models describing the dynamics of soft and hard sectors in
ultra-relativistic nucleus-nucleus collisions. This section presents JETSCAPE
predictions for upcoming sPHENIX measurements. The soft sector is simulated
using event-by-event TRENTO [46] initial conditions evolved hydrodynamically
with VISHNU (2+1D) code package [47]. The jet evolution is carried out using a
multi-stage approach within the JETSCAPE framework where the high-virtuality
region of the parton shower is modeled using the MATTER [48] event generator
and the low virtuality region is simulated using the LBT [49] event generator.
The details of the full model calculation can be found in Ref. [50].

Figure 5: Nuclear modification factor $R_{\mathrm{AA}}$ for full jets at most
central ($0-10\%$) collisions at $\sqrt{s_{\mathrm{NN}}}=$200 GeV for jet cone
sizes $R=$0.4, 0.6 and 0.8. The calculation is performed using JETSCAPEv3.5
AA22 tune described in Ref. [50]. (a) Anti-kT based inclusive jets (b) Anti-kT
based charged-hadron jets.  Figure 6:  Nuclear modification of jets at most
central ($0-10\%$) collisions at $\sqrt{s_{\mathrm{NN}}}=$200 GeV. The
calculation is performed using JETSCAPEv3.5 AA22 tune described in Ref. [50].
The parameters in the Soft Drop grooming algorithm are $z_{\mathrm{cut}}=0.2$
and $\beta=0$. (a) The distribution of
$z_{\mathrm{g}}=\mathrm{min}(p^{\mathrm{prong}}_{\mathrm{T}1},p^{\mathrm{prong}}_{\mathrm{T}2}))/(p^{\mathrm{prong}}_{\mathrm{T}1}+p^{\mathrm{prong}}_{\mathrm{T}2})$
for charged-jets of cone size $R=$0.2 and 0.4. (b) Groomed jet mass for
charged jets of cone size $R=$0.2 and 0.4. (c) $p_{\mathrm{T}}$ dependence of
jet fragmentation function for inclusive jets of cone size $R=$0.4. (d)
Nuclear modification of photon-triggered inclusive jets of cone size $R=$0.5.

Fig. 5 presents the nuclear modification factor of inclusive jets and charged
jets for jet cone radius values $R$=0.4, 0.6, and 0.8. The predictions show
that the charged jets are more suppressed compared to inclusive jets at high
jet $p_{\mathrm{T}}$ values. Also, in contrast to jets at LHC, RHIC jets do
not show strong jet cone size dependence. Next, predictions are presented for
groomed charged-jet observables computed using the Soft Drop grooming
algorithm with parameters $z_{\mathrm{cut}}=0.2$ and $\beta$=0 in Fig. 6(a)
and (b). The prediction shows that the nuclear modification of
$z_{\mathrm{g}}$ distribution is consistent with unity, indicating that there
is no strong modification in the hardest splitting during in-medium jet
evolution. Moreover, the trend looks similar to ALICE measurements at
$\sqrt{s_{\mathrm{\mathrm{NN}}}}=$5.02 TeV. The predictions for jet mass [Fig.
6(b)] also shows no significant nuclear modification at low groomed jet mass
$m_{g}<4$ GeV.

Fig. 6(c) presents predictions for fragmentation function
[$D(p_{\mathrm{T}})$] as a function of hadron $p_{\mathrm{T}}$ for two
different inclusive $p^{\mathrm{jet}}_{\mathrm{T}}$ ranges. The results show
strong suppression for high $p_{\mathrm{T}}$ jets. In the end, results are
presented for nuclear modification ($I_{\mathrm{AA}}$) of photon-triggered
inclusive jets [Fig. 6(d)]. The results indicate a rise in $I_{\mathrm{AA}}$
below $p^{\mathrm{jet}}_{\mathrm{T}}=12$ GeV, whereas it is strongly
suppressed for $p^{\mathrm{jet}}_{\mathrm{T}}>12$GeV and remains constant for
higher $p^{\mathrm{jet}}_{\mathrm{T}}$.

In addition to the specific predictions above, a large sPHENIX data-set could
provide many additional opportunities to test the jet quenching model and
probe the hardest in-medium splitting, path-length dependence of photon-
triggered jets, and in-medium energy loss of quark and gluon jets.

###  3.4 $R_{\rm AA}$ and $v_{2}$ as jet substructure observables with sPHENIX

Jets are multi-partonic systems that develop before interactions with the QGP
set in and lead to energy loss and to substantial modifications of their
substructure.

Figure 7: Left: Ratio of jet $R_{\rm AA}$ for different $R$ values to the
$R_{\rm AA}$ with $R=0.1$. Right: Difference in $v_{2}$ between different $R$
and $R=0.1$, for a jet $p_{T}\approx 37$ GeV, as a function of centrality.

Jet modification depends on the degree to which the medium can resolve the
internal jet structure that is dictated by the physics of coherence governed
by a critical angle $\theta_{c}$. To leading logarithmic accuracy, parton
splittings that take place within $\theta<\theta_{c}$ will not be resolved by
the medium. Therefore, in order to compute the total energy lost by a jet, one
needs to consider not only where the radiated energy ends up, but also the
resolved phase-space for a given jet $p_{T}$ and cone-size $R$ which would be
affected by quenching effects. Using resummed quenching weights that
incorporate the Improved Opacity Expansion (IOE) framework for medium-induced
radiation [51] and embedding the system into a realistic heavy-ion environment
for AuAu collisions at $\sqrt{s}=200$ GeV, the $R$ dependence of jet
suppression, $R_{\rm AA}(R)$, is computed in the left panel of Fig. 7. The
only two parameters of the model are $g_{\rm med}$, determining the (so far
fixed) coupling between the energetic partons and the constituents of the QGP,
and $R_{\rm rec}$, which estimates the extent to which thermalized energy is
recovered as a function of $R$ within the jet hemisphere. In these preliminary
results, $g_{\rm med}$ is varied within the range $g_{\rm
med}\in\\{2.3,2.4\\}$, and set $R_{\rm rec}=\pi/2$ (corresponding to a flat
redistribution of the energy). The variation of $R_{\rm rec}$ has limited
effect on jets up to $R\sim 0.6$. Results are presented for jet suppression in
terms of a ratio between $R>0.1$ and $R=0.1$. Overall, there is a very mild
$R$-dependence for the range of $R$ studied, similar to what was found at the
LHC [52], with variations up to $\sim 10\%$. At lower jet $p_{T}$, larger $R$
jets tend to be somewhat less suppressed than small-cone jets. This is due to
the fact that the radiated energy is more likely to remain within the jet for
larger cone size jets. However, by increasing jet $p_{T}$, the size of the jet
phase-space increases more for larger $R$, and then the trend is reversed,
with larger $R$ jets more quenched.

Results are also presented for the jet azimuthal anisotropy $v_{2}(R)$ as a
function of the difference between jets with $R>0.1$ and those with $R=0.1$,
as a function of centrality, in the right panel of Fig. 7. Jet $p_{T}$ has
been set to be approximately $p_{T}^{\rm jet}\approx 37$ GeV. The results show
that as centrality is decreased, $v_{2}$ for moderate $R$ jets, such as
$R=0.3$ and $R=0.4$, sequentially collapse towards the result for small
$R=0.1$. The reason of this sequential grouping is the evolution of
$\theta_{c}$ with centrality due to its strong dependence on the in-medium
traversed length, $\theta_{c}\sim 1/\sqrt{\widehat{q}L^{3}}$. For those jets
with $R>\theta_{c}$, traversing shorter lengths within the medium will make a
larger difference than for those jets with $R<\theta_{c}$, since the size of
the resolved phase-space over which quenching weights are resummed will be
reduced. For this reason, $v_{2}(R)$ is quite sensitive to the typical value
of $\theta_{c}$ at a given centrality. Further details on these results will
be presented in a forthcoming publication.

###  3.5 CoLBT-hydro predictions for sPHENIX

The CoLBT-hydro model [53, 54, 55] is developed to simulate jet propagation
and jet-induced medium response in high-energy heavy-ion collisions. It
combines the microscopic linear Boltzmann transport (LBT) model [49] for the
propagation of energetic jets and recoil partons with the event-by-event
(3+1)D CCNU-LBNL viscous (CLVisc) hydrodynamic model [56, 57, 58] for the
evolution of the bulk medium and soft modes of the jet-induced medium
response. LBT and CLVisc are coupled in real time through a source term from
the energy-momentum lost to the medium by jet shower and recoil partons as
well as the particle-holes or “negative partons" from the back-reaction. The
LBT model [49] is based on the Boltzmann equation for both jet shower and
recoil partons with perturbative QCD (pQCD) leading-order elastic scattering
and induced gluon radiation according to the high-twist approach [59, 60, 61,
62]. The CLVisc [57, 56, 58] viscous hydrodynamic model with the default
freeze-out temperature $T_{f}=137$ MeV, specific shear viscosity
$\eta/s=0.15$, the s95p parameterization of the lattice QCD EoS with a rapid
crossover [63] and Trento [46] initial conditions with a longitudinal envelope
at an initial time $\tau_{0}=0.6$ fm/$c$ can reproduce experimental data on
bulk hadron spectra and anisotropic flows at both RHIC and LHC energies. The
Trento model is also used to provide the transverse spatial distribution of
jet production whose initial configurations are generated from PYTHIA8 [64].
Partons from the initial jet showers, as well as MPI, propagate through the
QGP and generate medium response according to the CoLBT-hydro model. The final
hadron spectra include contributions from the hadronization of hard partons
within a parton recombination model [65, 66] and jet-induced hydro response
via Cooper-Frye freeze-out after subtracting the background from the same
hydro event without the $\gamma$-jet. More detailed descriptions of the LBT
and CoLBT-hydro models are given in Refs. [49, 67, 68, 69, 70, 71] and [53,
55, 72].

Figure 8: Nuclear modification factor $I_{AA}$ as a function of $p_{T}^{\rm
jet}$ for $\gamma$-jet with $p_{T}^{\gamma}$ = 15–20 GeV in central 0–10%
Au+Au collisions at $\sqrt{s}=200$ GeV from the CoLBT-hydro (blue) and LBT
models, compared here to preliminary STAR data [73].  Figure 9:
$\gamma$-hadron (blue dashed) and jet-hadron (red solid) correlation in 0–15%
Au+Au collisions at $\sqrt{s}=200$ GeV as a function of the rapidity
difference $\eta_{h}-\eta_{\rm jet}$ in the $\gamma$ azimuthal direction.

Shown in Fig. 8 are the nuclear modification factors of the $\gamma$-jet
process as a function of $p_{T}^{\rm jet}$ in 0–15% central Au+Au collisions
at $\sqrt{s}$ = 200 GeV from CoLBT-hydro (red) and LBT (blue) model
simulations as compared to the STAR preliminary data [73] for two different
jet cone size values of $R=0.5$ and 0.2. Both model results show a weak jet
cone size dependence. Shown in Fig. 9 are $\gamma$-hadron (dashed blue) and
jet-hadron correlations (red solid) for soft charged hadrons ($p_{T}$ = 1–2
GeV/$c$) as a function of the rapidity difference $\eta_{h}-\eta_{\rm jet}$ in
the $\gamma$ azimuthal directions. The dip on top of the MPI Gaussian peak is
caused by jet-induced diffusion wake. The dip becomes deeper for smaller
$\gamma$-jet asymmetry $p_{T}^{\rm jet}/p_{T}^{\gamma}$ or bigger jet energy
loss. These phenomena can be explored with sPHENIX data.

###  3.6 Disentangling jet modification in sPHENIX

The selection of jets in heavy-ion collisions based on their $p_{T}$ after jet
quenching is known to bias towards jets that lost little energy in the quark-
gluon plasma. The work in Ref. [45] studied and quantified the impact of this
selection bias on jet substructure observables so as to isolate effects caused
by the modification of the substructure of jets by quenching. This study was
performed in a simplified Monte Carlo setup, in which it was possible to
identify the same jet before and after quenching. The work then showed
explicitly that jets selected based on their quenched (i.e. observable)
$p_{T}$ have substantially smaller fractional energy loss than those selected
based on the $p_{T}$ that they would have had in the absence of any quenching.
This selection bias has a large impact on jet structure and substructure
observables. As an example, one can consider the angular separation $\Delta R$
of the hardest splitting in each jet, and found that the $\Delta R$
distribution of the (biased) sample of jets selected based upon their quenched
$p_{T}$ is almost unmodified by quenching. In contrast, quenching causes
dramatic modifications to the $\Delta R$ distribution of a sample of jets
selected based upon their unquenched $p_{T}$, with a significant enhancement
at larger $\Delta R$ coming from the soft particles originating from the wake
of the jet in the quark-gluon plasma. The jets which contribute to this
enhancement are those which have lost the most energy and which were,
therefore, left out of the sample selected after quenching. In the second part
of Ref. [45], a more realistic study showed that the same qualitative effects
could all be observed in events where a jet is produced in association with a
$Z$ boson at the LHC. Selecting jets in such events based on either the jet
$p_{T}$ or the $Z$-boson $p_{T}$ provides an experimentally accessible way to
quantify the effects of selection biases in jet observables and separate them
from the modification of jet substructure caused by quenching. In this
section, results are presented from repeating the same study for the sPHENIX
experiment at RHIC using events in which a jet is produced in association with
a direct photon.

Figure 10:  Top: Fractional $p_{T}$ asymmetry between a photon and its
recoiling jet, for jets with jet $p_{T}$ above $50$ GeV (blue) and photon
$p_{T}$ above $50$ GeV (orange) in vacuum (dotted), and in heavy ion
collisions with hadrons originating from medium response artificially excluded
from (dashed) or included in (solid) the reconstructed jets. Bottom left and
right panels show the distribution of $\Delta R$ for the same samples of jets.

The results below are based on simulations of jets produced in association
with a direct photon with pseudorapidity $|\eta^{\gamma}|\leq 0.9$ in Au+Au
collisions at $\sqrt{s}=200$ GeV in the Hybrid Model of jet quenching [74]. It
is experimentally challenging to access direct photons at low
$p_{T}^{\gamma}$, due to backgrounds from photons that are not produced in the
hard process. This study assumes that sPHENIX will be able to isolate a clean
sample of jets produced in association with direct photons down to
$p_{T}^{\gamma}=30$ GeV, and will have sufficient statistics to access hard
processes with $p_{T}\geq 50$ GeV. As in Ref. [45] jets are selected in two
distinct ways, which yield results that are qualitatively different from one
another in interesting ways. Fig. 10 shows the $p_{T}$ asymmetry and $\Delta
R$ distributions of jets in this sample selected in two ways. The blue
histogram shows jets with $p_{T}^{\mathrm{jet}}\geq 50$ GeV recoiling against
a photon satisfying $p_{T}^{\gamma}\geq 30$ GeV, while the orange histogram
shows jets with $p_{T}^{\mathrm{jet}}\geq 30$ GeV recoiling against a photon
with $p_{T}^{\gamma}\geq 50$ GeV. Fig. 10(a) illustrates that jets selected
using the first method lose a much smaller fraction of their energy. This
selection bias is also born out in the substructure of jets selected in each
way. The $\Delta R$ distribution of jets with $p_{T}^{\mathrm{jet}}\geq 50$
GeV (Fig. 10(b)) is much narrower than for those produced in association with
a photon with $p_{T}^{\gamma}\geq 50$ GeV (Fig. 10(c)), due to the selection
bias in the former case against jets that lost more energy (typically, those
with larger $\Delta R$). As in Ref. [45], there is a (small) enhancement at
large $\Delta R$ in Fig. 10 coming from the response of the medium to jets.
However, this effect is more muted than at the LHC, presumably due to the
smaller separation between the minimum $p_{T}$ of direct photons and the
statistical reach of the hard process at RHIC. This effect could be enhanced
in the scenario that sPHENIX is able to access photons of lower
$p_{T}^{\gamma}$ or jet substructure of jets with higher $p_{T}$. This
possibility should be explored more systematically in future work and as the
specific kinematic capabilities in the recorded data become more clear.

###  3.7 Comprehensive SCETG predictions for sPHENIX

Predictions for the sPHENIX program are presented, including the modification
of light and heavy-flavor hadrons and jets [75, 76, 77, 78], photon-tagged
jets and dijets [79, 80], and jet substructure [77, 81].

Figure 11: Left: Nuclear modification of light and heavy flavor without (left
column) and with (right column) QGP effects in 0-10% and 30-50% O+O collisions
at $\sqrt{s_{NN}}=200$ GeV. Right: Transverse energy dependent nuclear
modification factor $R_{AA}^{\rm jet}$ for different cone radii $R$ in 0-10%
Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV. Inserts show ratios of jet cross
sections for different $R$ in nuclear reactions versus $E_{T}$.

The essential prediction for hard probes in heavy ion collisions is the
suppression of the cross sections of light and heavy hadrons and jets. The
suppression of hadrons can be addressed in both the traditional energy loss
approach and using modern QCD evolution techniques, based on the ability to
calculate full in-medium splitting functions in dense nuclear matter [75].
While phenomenologically the results are similar, the latter approach
constitutes a more rigorous and improvable treatment of the problem. The
suppression of light hadrons in central Au+Au collisions is predicted to be
approximately a factor of four to five. At high transverse momenta
$p_{\mathrm{T}}>20$ GeV cold nuclear matter (CNM) effects play an important
role. At small transverse momenta, CNM effects are small and can only be
observed in $p(d)$+Au collisions where they lead to Cronin-like enhancement,
which is more pronounced at backward rapidity. An important question that is
still unresolved is regarding CNM vs QGP effects in small systems. Recent
calculations [76] predict that the difference is most pronounced in small but
symmetric systems, such as O+O in the left of Fig. 11. This can be seen for
both light ($h^{\pm}$) and heavy ($D^{\pm}$) flavor particles. Reconstructed
inclusive jets show similar suppression to light hadrons in the small radius
$R=0.2$ limit, see the right panel of Fig. 11. A clear reduction of
suppression is predicted as $R$ grows. For heavy-flavor-tagged jets, such as
$b$-jets, sPHENIX is expected to have sensitivity to the dead cone effect
[78], and the reduction of the suppression due to the heavy quark mass is
found to play a role for $p_{T}<25$ GeV. Furthermore, while the presence of
gluon to heavy-flavor fragmentation processes was found to play an important
role at the LHC, leading to similar light and heavy-flavor-tagged jet
suppression, the smaller expected gluon contribution at RHIC energies is
instead predicted to result in slightly smaller suppression for $b$-jets than
for inclusive jets.

Figure 12: Left: The isolated photon-tagged jet asymmetry distribution for
different coupling strengths between the jet and the medium for Au+Au
collisions at $\sqrt{s_{NN}}=200$ GeV. Right: Ratios of nuclear modification
factors for $b$-tagged ($R_{AA}^{bb}$) v.s inclusive ($R_{AA}^{jj}$) dijet
production sPHENIX (right) are plotted as a function of dijet invariant mass
$m_{12}$.

Photon-tagged jets and jet correlations provide more differential probes of
in-medium dynamics. The advantage of photon tagging is that it can provide a
reference relative to which the recoil jet energy loss can be studied. One
important prediction is that because of the non-monotonic form of the photon-
jet cross section, the modification factor (sometimes called $I_{AA}$) has a
non-trivial $p_{T}$ dependence where the suppression decreases near the
trigger photon momentum and can even turn into enhancement [79]. In addition,
the calculations predict a significant modification of the photon-tagged jet
imbalance $z_{J\gamma}$ distribution, shown in the left panel of Fig. 12.
Since the reference $p$+$p$ distribution is narrower than at the LHC, the
broadening is more pronounced and the shift in the mean momentum imbalance
$\Delta\langle z_{J\gamma}\rangle=\langle z_{J\gamma}\rangle_{pp}-\langle
z_{J\gamma}\rangle_{AA}$ is very sensitive to the jet-medium coupling. One
important task at RHIC will be to enhance the effects of jet quenching. The
dijet mass (the mass of the dijet system) is a newly proposed observable that
can achieve this goal [80]. It was found that the suppression in the mass
$m_{12}$ distribution of light dijets can differ from unity by an order of
magnitude. At the same time this observable enhances the sensitivity to the
heavy quark mass effect, such that the transverse momentum dependence of
$b$-dijet suppression is very different than the one for light dijets. This
can be seen in the ratio of the nuclear modifications shown in the right panel
of Fig. 12 where the differences due to the $b$-quark mass can be significant
at low $m_{12}$ [80].

Figure 13: Left: The differential jet shape in vacuum $\psi^{\rm vac.}(r,R)$
is contrasted to the medium-induced contribution $\psi^{\rm med.}(r,R)$ by a
$E_{T}=30$ GeV quark in Au+Au and Cu+Cu collisions at $\sqrt{s_{NN}}=200$ GeV.
The insert illustrates a method for studying the characteristics of these
parton showers. Right: The modifications of the splitting functions for heavy-
flavor-tagged jet is shown for $\sqrt{s_{\rm NN}}=200$ GeV Au+Au collisions.
An important feature is the strong quenching effects for prompt $b$-jets
contrasted by the lack of QGP-induced modification for the $g\to Q\bar{Q}$
splitting.

Jet substructure is a direct way to probe the properties of in-medium parton
showers. These studies were pioneered at the LHC and it is expected that
sPHENIX will make valuable contributions to this physics at RHIC. In the left
panel of Fig. 13 the vacuum and medium-induced differential jet shapes for
quark jets are shown [79]. In comparison to the LHC, where gluons dominate,
RHIC will be dominated by quark jets, which are narrower than the gluon jets.
Thus, the enhancement of the jet shape toward the periphery of the jet
$r/R\rightarrow 1$ is expected to be larger than at the LHC even if the narrow
“core” region remains unmodified. Another important substructure observable is
the distribution of the soft dropped momentum-sharing fraction $z_{g}$, and it
is particularly interesting for heavy-flavor jets [81]. For inclusive jets,
the modification is much smaller than at the LHC. Going to lower jet
transverse momenta leads to a unique dependence of the jet momentum sharing
distribution modification in heavy ion collisions - an inversion of the mass
hierarchy of jet quenching effects, which can be tested by sPHENIX. This is
shown in the right panel of Fig. 13 , where ratios of the momentum sharing
distribution for heavy-flavor tagged jets in Au+Au to $p$+$p$ collisions at
$\sqrt{s_{\rm NN}}=200$ GeV are presented for $10<p_{T,j}<30$ GeV jets. These
corroborate the analytic expectations, showing that the magnitude of the
effects is not only large, but particularly so for $b$-quark jets.

###  3.8 Energy correlations in jets with sPHENIX

This calculation is based on a novel approach to jet substructure in heavy-ion
collisions formulated in terms of energy correlation functions [82]. It can be
demonstrated that the scales of the QGP can be isolated as distinct features
in the correlator spectra. For this purpose, the authors analyse the case of
the 2-point correlator
$\langle\mathcal{E}(\vec{n}_{1})\mathcal{E}(\vec{n}_{2})\rangle$, which
introduces a single scale-sensitive angular parameter,
$\cos\theta=\vec{n}_{1}\cdot\vec{n}_{2}$. The $n$-th weighted normalised
2-point correlator can be computed from the inclusive cross-section,
$\sigma_{ij}$, to produce two hadrons ($i,j$) as

$\frac{\mathrm{d}\Sigma^{(n)}}{\mathrm{d}\theta}=\frac{1}{\sigma}\int\mathrm{d}\vec{n}_{1,2}\sum_{ij}\int\frac{\mathrm{d}\sigma_{ij}}{\mathrm{d}\vec{n}_{i}\mathrm{d}\vec{n}_{j}}\frac{E^{n}_{i}E^{n}_{j}}{Q^{2n}}\delta^{(2)}(\vec{n}_{i}-\vec{n}_{1})\delta^{(2)}(\vec{n}_{j}-\vec{n}_{2})\delta(\vec{n}_{2}\cdot\vec{n}_{1}-\cos\theta),$
(2)

where $E_{i}$ is the lab-frame energy of hadron $i$, $\vec{n}_{i}$ is a two-
component vector specifying its direction, and $Q$ is an appropriate hard
scale. The 2-point correlator (EEC) is computed for a quark jet which
propagates through a static medium of finite length $L$ and jet-quenching
parameter $\hat{q}$ within the particular implementation of BDMPS-Z formalism
[83, 84, 85, 86] for semi-hard splittings given in [87, 88]. Vacuum collinear
radiation is resummed using the celestial operator product expansion [89, 90,
89, 91, 92]. The result is that $\mathrm{d}\Sigma^{(n)}$ can be written in the
factorised form [82]:

$\frac{\mathrm{d}\Sigma^{(n)}}{\mathrm{d}\theta}=\frac{1}{\sigma_{qg}}\int\mathrm{d}z\left(g^{(n)}+F_{\rm
med}\right)\frac{\mathrm{d}\sigma^{\rm
vac}_{qg}}{\mathrm{d}\theta\mathrm{d}z}z^{n}(1-z)^{n}\left(1+\mathcal{O}\left(\alpha_{\mathrm{s}}\ln\theta_{\mathrm{L}}^{-1},~{}\frac{\mu_{\rm
s}}{zE}\right)\right)+\mathcal{O}\left(\frac{\mu_{\rm s}}{E}\right),$ (3)

where
$g^{(1)}(\theta,\alpha_{\mathrm{s}})=\theta^{\gamma(3)}+\mathcal{O}(\theta)$
at fixed coupling given $\mathrm{d}\sigma^{\rm vac}_{qg}$ at
$\mathcal{O}(\alpha_{s})$. Here $\gamma(3)$ is the twist-2 spin-3 QCD
anomalous dimension. $F_{\rm med}(z,\theta)$ is the medium-induced
modification given in [88], and $\mathrm{d}\sigma^{\rm vac}_{qg}$ is the
vacuum splitting cross-section. $\mu_{\mathrm{s}}$ is the low scale of
radiation over which $F_{\rm med}$ is inclusive. $Q$ is fixed as $Q=E$, the
initial jet energy, and let $z=E_{g}/E$.

Figure 14: The EEC evaluated using Eq. 3 for the DC (left panel) and PC
regimes (right panel). The bottom panels show the volume normalised medium
contribution to the distribution, defined as
$\mathrm{d}\Sigma^{(n)}_{\mathrm{med}}=(\mathrm{d}\Sigma^{(n)}-\mathrm{d}\Sigma^{(n)}(\hat{q}=0))/\Sigma^{(n)}_{\mathrm{med}}$,
so the shape can be more easily compared. Figure 15: The EEC in JEWEL with
recoils for $\gamma$+jet Au$+$Au events at $\sqrt{s_{\rm NN}}=200$ GeV and
with a $350$ MeV medium temperature. A similar enhancement is found as in Fig.
14, however at larger $\Delta R$. This is consistent with the reduction in the
jet energy of the sample from $100$ GeV in Fig. 14 to $E^{\gamma}_{\rm T}>20$
GeV. While the specific predictions are shown here for $E=100$ GeV, the
qualitative features are also expected for $E>30$ GeV partons at RHIC.

Fig. 14 presents a numerical evaluation of Eq. 3, where the parameters
($E,L,\hat{q}$) have been chosen such that the left panel corresponds to a
limit where one expects the quark-gluon pair to propagate decoherently (DC)
through the medium. At the time of the workshop, the full calculations were
performed only for $E=100$ GeV partons. We have found qualitatively similar
results with jet energies down to $E=30$ GeV which are appropriate for RHIC
kinematics, although these will need a full, quantitative treatment to
understand the specific physics effect.

The right panel corresponds to the regime where one expects the quark-gluon
pair to propagate partially coherently through the medium (PC limit). A
qualitatively different shape in the spectrum is readily observed between the
two limits. Ref. [82] demonstrates with a simple procedure that this
difference can be used to extract the energy scale at which the onset of
coherence occurs, i.e. the “resolution scale” of the QGP. This preliminary
calculation highlights the exceptional potential of correlators to identify
the presence of particular QGP dynamics at a given scale. In complement to
this analysis, the 2-point correlator was also evaluated using the Monte Carlo
parton shower JEWEL with recoils [93, 94, 95, 96] for quark jets recoiling off
photons [97], as shown in Fig. 15. An enhancement at wide angles similar to
that found in the (semi-)analytical analysis is clearly seen. The EEC was also
found to be robust to a $2$ GeV cut on the track $p_{T}$, as that typically
used experimentally to suppress backgrounds. These measurements of the EEC can
be performed using future sPHENIX data.

###  3.9 Bayesian inference with JETSCAPE using sPHENIX data

Bayesian inference provides a mechanism for systematic and agnostic physics
interpretation of the wealth of information contained in jet quenching
measurements at RHIC and the LHC.

The JETSCAPE collaboration previously carried out a proof-of-principle
Bayesian inference analysis using the inclusive charged hadron
$R_{\mathrm{AA}}{}$ at $\sqrt{s_{\mathrm{NN}}}{}=$ 200 GeV at RHIC and
$\sqrt{s_{\mathrm{NN}}}{}=$ 2.76 and 5.02 TeV at the LHC [98]. This analysis
provided new constraints on the dependence of the jet transport coefficient
$\hat{q}{}$ on medium temperature and parton momentum for several physics
models, thereby demonstrating the viability of Bayesian inference to study jet
quenching.

The next step is to expand the analysis to utilize multiple observables as a
multi-messenger description of jet quenching, which encodes the structure and
dynamics of the QGP into modifications of jet observables. However, in this
study only one new observable is added in order to isolate the impact of the
additional information. To this end, a new Bayesian inference was carried out
with inclusive jet and charged hadron $R_{\mathrm{AA}}{}$, including all
available experimental data for fully corrected inclusive jet distributions.
This analysis utilizes a $\hat{q}{}$ parametrization with six parameters, and
reduces the number of interactions at high virtuality due to coherence
effects. Details of the model are reported in Sec. 3.3 above, and are further
described in [50].

Figure 16 shows the preliminary observable posterior distribution for a multi-
stage model using MATTER+LBT. These results only utilize a subset of the
JETSCAPE simulations. The model is able to describe the data fairly well
overall, although there are some regions of tension. Detailed description of
this work-in-progress analysis is available in Ref. [99], with full results
following in a separate publication.

Figure 16: Preliminary posterior distribution of the calibrated model compared
to a selection of inclusive jet and charged hadron $R_{\mathrm{AA}}{}$ data.
The data are in black and the sampled posterior is in blue. Figure 17: A
schematic description of the application of Bayesian sensitivity
quantification to upcoming sPHENIX measurements at RHIC. The combination of an
existing Bayesian analysis with pseudo-data simulations of future data enables
quantification of the impact of proposed measurements.

Bayesian inference can also be used for sensitivity studies, to assess the
impact of future measurements. This approach is referred to as Bayesian
sensitivity quantification, and is a concept that is well established in other
fields, including in neutrino physics [100]. In heavy-ion physics it has been
applied in the soft sector for oxygen-oxygen collisions [101]. These studies
are implemented by calibrating a model using Bayesian inference, generating
pseudo-data by running new simulations according to the extracted parameter
posterior distribution, and then including the pseudo-data in a new Bayesian
inference analysis. By varying the included observables and their precision,
their future impact can be quantified. Figure 17 shows a cartoon of such an
approach. This technique can be applied to jet quenching measurements,
utilizing the full results from the new JETSCAPE Bayesian analysis reported
here to identify which observables will have the largest impact on physics
parameter extraction. This information can help to prioritize measurement
strategies, thereby maximizing the impact of new data from sPHENIX in the
upcoming RHIC runs.

##  4 Heavy flavor quark probes of the QGP

A second major scientific motivation for the sPHENIX experiment is its
comprehensive program of heavy-flavor physics measurements. In the open charm
sector, the precision vertex detectors will provide the capability for
measurements of the production and azimuthal correlation of $D$ meson
resonances (including $v_{1}$), $D^{0}$-tagged jets, and the relative yields
of charmed baryons such as the $\Lambda_{c}$. In the open beauty sector, this
includes measurements of fully reconstructed $B$ hadrons through the
intermediate identification of non-prompt $D$ mesons and the algorithmic
tagging $b$-jets. In the quarkonium sector, sPHENIX will have sufficient
momentum resolution to separate the three Upsilon states, as well as provide a
large-statistics sample of high-$p_{\mathrm{T}}$ $J/\psi$’s for study.
Finally, the streaming readout capability of the trackers will enable high-
statistics measurements of key topics in small systems, such as the presence
or absence of collective motion for heavy flavor in these systems at RHIC
energies. A number of predictions for heavy-flavor hadron and quarkonia probes
of the QGP were presented at the workshop and are summarized below.

###  4.1 DREENA-A predictions for open heavy flavor in sPHENIX

The main idea behind QGP tomography is that when high-$p_{\mathrm{T}}$
particles transverse QGP, they lose energy. This energy loss is sensitive to
QGP properties, and thus comprehensive comparisons between
high-$p_{\mathrm{T}}$ theory and data can be used to infer some bulk QGP
properties. However, to implement this idea, it is crucial to have a reliable
high-$p_{\mathrm{T}}$ parton energy loss model. This contribution describes a
dynamical energy loss formalism [102, 103, 104], developed with the following
unique features: i) The formalism takes into account finite size, finite
temperature QCD medium consisting of dynamical (that is, moving) partons,
contrary to the widely used static scattering approximation and/or medium
models with vacuum-like propagators. ii) The calculations are based on the
finite temperature generalized Hard-Thermal-Loop approach, in which the
infrared divergences are naturally regulated, so there are no artificial
cutoffs. Non-perturbative effects related to screening of the chromo-magnetic
and chromo-electric fields are also included, as well as a running coupling
[105, 106]. iii) Radiative and collisional energy losses are calculated under
the same theoretical framework, applicable to both light and heavy flavor. iv)
Importantly, there are no fitting parameters in the model, and the temperature
($T$) is a natural variable in the model.

The developed framework further needs to include a full, arbitrary, medium
evolution in the dynamical energy loss. All energy loss properties must be
preserved, without additional simplifications in the numerical procedure,
since all of them are necessary to accurately explain the data. The
computational procedure must also be efficient (time-wise) to generate a
comprehensive set of light and heavy flavor suppression predictions through
the same numerical framework and parameter set. Such predictions can then be
compared with the comprehensive set of available experimental data (for
different probes, collision systems, energies, centralities), if needed
iteratively for different combinations of QGP medium parameters, to extract
medium properties consistent with low and high-$p_{\mathrm{T}}$ theory and
data. These goals were achieved by developing a fully optimized DREENA-A
framework [107], where DREENA stands for Dynamical Radiative and Elastic
ENergy loss Approach. “A” stands for Adaptive temperature profile, meaning
that arbitrary medium evolution that can be used as an input.

Figure 18:  DREENA-A $R_{AA}$ (top panels) and $v_{2}$ (bottom panels)
predictions for 200 GeV Au+Au collisions at RHIC are generated for three
different QGP evolution models (indicated in the legend). Predictions are
generated for 20-30% centrality region for charged hadron (left), D meson
(middle) and B meson (right), and compared with the available experimental
data from RHIC. Figure adapted from [107].

The next question is if one can indeed expect different $T$ profiles to lead
to differences in high-$p_{\mathrm{T}}$ observables. To address this question,
representative evolutions were generated (see [107] for more details): i)
Optical Glauber initialization at initial time $\tau_{0}=1.0$ fm, without
initial transverse flow, followed by 3+1D viscous fluid expansion. ii) EKRT
initialisation with $\tau_{0}=0.2$ fm, also followed 3+1D viscous fluid
dynamics. iii) TRENTo initialisation evolved by free streaming until
$\tau_{0}=1.16$ fm, followed by VISH2+1 viscous fluid dynamics. While they all
agree with low-$p_{\mathrm{T}}$, they lead to quite different $T$ profile
evolutions with time, as discussed in [107]. Can high-$p_{\mathrm{T}}$ data
further constrain these evolutions? To address this, in Fig. 18, these
profiles were used as an input to the DREENA-A to generate
high-$p_{\mathrm{T}}$ $R_{AA}$ and $v_{2}$ predictions for charged hadrons, D
and B mesons at RHIC. Both the $R_{AA}$ and the $v_{2}$ calculations show
notable differences for all types of flavor. Consequently, the DREENA-A
framework can clearly differentiate between $T$ profiles by corresponding
differences in high-$p_{\mathrm{T}}$ observables. Furthermore, heavy-flavor
hadrons show higher sensitivity than light flavor, making them even better
suited for exploring the bulk QGP parameters with high-$p_{\mathrm{T}}$ data.
With the expected availability of precision data from the upcoming high-
luminosity experiments at sPHENIX, the DREENA-A framework will provide an
exciting new opportunity [108, 109] for exploring the bulk QGP properties at
heavy ion collisions.

###  4.2 Heavy flavor jets in sPHENIX with LIDO

The LIDO model [110, 111] is a linearized partonic transport model for jet and
heavy-flavor transport in the quark-gluon plasma. The jet parton undergoes
multiple collisions with QGP constituents, treated in a small-angle diffusion
plus large-angle perturbative scattering approach [110]. Medium-induced parton
branchings (including $g\rightarrow g+g$, $q\rightarrow q+g$, $g\rightarrow
q+\bar{q}$, with $q=u,d,s,c,b$) are included both in the incoherent and the
deep LPM limits. Especially, the LPM suppression is implemented dynamically in
the simulation so that the finite-size and expanding medium features of the
radiation pattern are reproduced [111].

To study jets, a matching routine is developed to transit the Pythia vacuum
parton shower generation and the LIDO on-shell parton transport. A simple
medium response to the hard-parton energy deposition is instrumented to
guarantee the energy-momentum conservation in the simulation of the jet event.
The Pythia8+LIDO framework with a 2+1D hydrodynamic simulation of heavy-ion
collisions has been calibrated to the single-inclusive light and charm meson
nuclear modification factor $R_{AA}$ and the $R=0.4$ jet $R_{AA}$ at both RHIC
and LHC energies in central $A$-$A$ collisions [112].

Figure 19: Flavor dependence of full jet suppression. Left: projection in
central Au+Au collisions at $\sqrt{s}$=200 GeV. Right: in central Pb+Pb
collisions at $\sqrt{s}$=5.02 TeV. The flavor hierarchy comes from both the
dead-cone effect and the respective contributions from $Q\rightarrow$ HF jets
(dominated at low $p_{T}^{\rm jet}$) and $g\rightarrow$ HF jets.  Figure 20:
$D$ (left) and $B$ (right) meson fragmentation functions in jets. The Peterson
fragmentation functions are used in both cases for heavy flavor fragmentation
in the vacuum.

The framework is recently applied to study heavy-flavor (HF) jet quenching
[113]: the flavor-dependent jet $R_{AA}$ (Fig. 19) and the heavy-flavor-in-jet
fragmentation function (Fig. 20) at jet $p_{T}$ relevant for the sPHENIX
experiment. Medium modifications of heavy-flavor jets are controlled by not
only the mass dependence of in-medium parton branchings but also the competing
channels of $Q\rightarrow$HF jets and $g\rightarrow$ HF jets. HF jet
production at RHIC energy and low $p_{T}^{\rm jet}$ is dominated by
$Q\rightarrow$ HF jets fragmentation. It is very different from the situation
at the LHC energy, providing an independent constraint to disentangle
quark/gluon contributions and flavor-dependent jet energy loss. Additionally,
HF-in-jet measurements directly allows one to extract both the vacuum and in-
medium fragmentation functions of heavy quarks and provide a more differential
test of the heavy-flavor dynamics in the medium.

###  4.3 KSU and QTraj predictions for Upsilon suppression in sPHENIX

One of the key motivations used for the construction of sPHENIX was to study
the suppression of bottomonium states, e.g., the $\Upsilon(1S)$ and
$\Upsilon(2S)$, in RHIC energy heavy-ion collisions relative to their
production in $pp$ collisions. The strong suppression of such states provides
crucial information about whether or not a quark-gluon plasma has been
created. In addition, such studies can provide constraints on the initial
temperatures generated during the collisions and in-medium bottomonium
transport properties that can be checked against first principles non-
relativistic QCD and lattice calculations. Due to the largeness of the bottom
quark mass, such quarks are created only in the initial hard scatterings and
it is not expected that there will be a significant regeneration/recombination
effect due to the smallness of both open- and closed-bottom production cross-
sections. Additionally, due to their large mass, theoretical calculations
which rely on non-relativistic effective field theories (EFTs) are in much
better control for describing the physics of bottomonium than for charmonium.

Related to this, in the last decade there has been a significant advance in
the understanding of bottomonium dynamics in the QGP through the use of EFTs
and open quantum system methods [114, 115]. Most recently, these methods have
been applied to make phenomenological predictions for bottomonium suppression
and flow at LHC enerigies [116, 117, 118]. In practice, one solves for the
evolution of the heavy-quarkonium reduced density matrix using a quantum
master equation, which at high temperatures can be cast in the form of the
Lindblad equation [119, 120]. This equation can be efficiently solved using a
realistic 3+1D dissipative hydrodynamic background [121] by making use of an
algorithm called the “quantum trajectories algorithm”, which allows for
massive parallel computations. This algorithm has been implemented for heavy-
quarkonium dynamics in a publicly available package called QTraj [122].

Figure 21: Predictions for bottomonium suppression as a function of $N_{\rm
part}$ at RHIC collision energies. The left panel shows the result obtained
using Coulomb-like potentials emerging from an EFT calculation [116, 117, 118]
and the right panel shows the results emerging from the use of a
phenomenological potential which reduces to a Cornell potential in the $T=0$
limit [123, 124]. The bands in both figures are described in the text.

Although the EFT formalism used is most trustworthy at high temperatures, it
can be used to make predictions at the lower temperatures generated at RHIC.
For this purpose, once again one can use 3+1D dissipative hydrodynamics codes
that have been tuned to agree with the experimentally observed identified
soft-hadron production and flow [125]. The left panel of Fig. 21 presents the
predictions of the EFT approach using QTraj at RHIC 200 GeV collision energy.
For comparison, in the right panel of Fig. 21, a different underlying
potential is used that, instead of being Coulomb-like, includes a non-trivial
long range part which allows it to reduce to a Cornell potential in the $T=0$
limit [123, 124]. This second model is labelled as the Kent State University
(KSU) model due to its origins. In the case of the QTraj predictions, the
bands are obtained by varying the heavy-quarkonium transport coefficients
$\kappa$ in the range suggested by lattice calculations and, in the case of
the KSU predictions, the bands are obtained by varying the effective Debye
mass around its leading-order QCD value.

###  4.4 Modeling quarkonium suppression in sPHENIX

This contribution describes a simple framework to include initial shadowing
and medium effects in the computation of quarkonium related observables in
heavy ion collisions [126]. The approach presented here is particularly
suitable when the survival probability of a quarkonium state that traverses
the plasma can be expressed as a simple analytic function of the initial
temperature. In order to compute the nuclear modification factor $R_{AA}$ in
this situation, one needs to answer the following questions: what is the
initial temperature for a given collision type, how does it depend on the
transverse position, and how does the probability to create a quarkonium state
depend on the transverse position.

The following computation is based on the shadowing model discussed in [127].
The Glauber model can be obtained assuming that nucleons interact exchanging
pomerons. This framework can be modified to include shadowing effects by
introducing a triple pomeron vertex. The initial temperature distribution is
computed by assuming that the temperature scales with the energy density to
the power of $1/4^{\mathrm{th}}$ and that the initial energy density scales
with the number of pions. The proportionality constant between the energy
density and the pion production is not known, however, this information is not
needed to compute the ratio of energy densities between two different points
in transverse space and/or impact parameter. It was found that the initial
temperature scales approximately with the density of participants (or wounded
nucleons) at a given point, both for the conditions at RHIC and the LHC. Using
the same shadowing model, the initial distribution of quarkonium is computed,
which scales approximately with the density of binary collisions.

In order to compute the $R_{AA}$, the survival probability obtained in Ref.
[128] is applied. The approach of [128] is valid when thermal effects are a
perturbation, in other words, well below the melting temperature. The model
employs lattice QCD data on the static potential to obtain the binding energy
and the wave function. These two quantities are then used to determine the
decay width applying the Hard Thermal Loop approach. The focus of this model
is on the finite energy gap that exists between singlets and octets. This gap
strongly suppresses the decay width at low temperatures. The results are shown
in Fig. 22 and feature a very mild medium effect on the $R_{AA}$ of
$\Upsilon(1S)$ at sPHENIX. However, one should take into that this computation
refers to the suppression suffered by the direct state and does not take into
account the feed-down effects.

Figure 22: Predictions for the $R_{\mathrm{AA}}$ vs $N_{\mathrm{part}}$ of
$\Upsilon(1S)$ (left) and $\Upsilon(2S)$ (right) in sPHENIX at RHIC, showing
the cold nuclear matter (orange) and thermal (green) contributions to the
total $R_{\mathrm{AA}}$ (blue).

In fact, in order to compare with experimental inclusive data, it is mandatory
to take into account the feed-down contributions. The feed-down fractions for
the $\Upsilon$(1S) can be estimated as: 70% of direct $\Upsilon$(1S), 8% from
$\Upsilon$(2S) decay, 1% from $\Upsilon$(3S), 15% from $\chi_{\rm B1}$, 5%
from $\chi_{\rm B2}$ and 1% from $\chi_{\rm B3}$, while for the $\Upsilon$(2S)
the different contributions would be: 63% direct $\Upsilon$(2S), 4% of
$\Upsilon$(3S), 30% of $\chi_{\rm B2}$ and 3% of $\chi_{\rm B3}$ [129]. Note
also that for the $\Upsilon$(3S), 40% of the contribution will come from
decays of $\chi_{\rm B3}$. Those estimations are based on data measured by the
LHCb Collaboration [130] at low $p_{T}$. The contribution of the excited
states could be even higher according to CDF Collaboration measurements at
$p_{T}>8$ GeV [131].

Thus, in order to make predictions for the upcoming sPHENIX data, the
following approach was developed.

First, consider that within this model $\Upsilon(3S)$ and the excited
$\chi_{\rm B2}$ and $\chi_{\rm B3}$ states are totally dissociated in Au+Au
collision at RHIC energies. In this case, the nuclear modification factor for
inclusive $\Upsilon(2S)$ production will achieve a value around 0.4 for the
most central collisions. Consequently, the estimated value for $\Upsilon(1S)$
will vary between 0.7 (according to LHCb feed-down estimations) and 0.5
(according to CDF feed-down estimations) at high centrality.

This contribution explores the phenomenological consequences of taking into
account the energy gap between singlet and octet states when computing the
decay width of an upsilon bound state in a medium. Moreover, initial shadowing
effects are also included. This energy gap and the shadowing corrections
induce sizeable effects for the conditions in Au+Au collisions at RHIC and
should be taken into account in phenomenological studies.

##  5 Bulk probes of the QGP

Although originally motivated as an experiment for rare probes, the large
tracking acceptance and high rate capabilities of sPHENIX are highly suited to
measurements of soft, collective phenomena such as correlations and
fluctuations of the flowing QGP medium. Additionally, characterizing the QGP
bulk provides important information for dealing with the backgrounds present
in jet and heavy flavor measurements. Below is a summary of one such physics
opportunity presented at the workshop.

###  5.1 Non-Gaussian flow fluctuations at sPHENIX

The sPHENIX program will open a new window onto the initial condition of the
QGP at RHIC via precision measurements of the event-by-event distribution of
the anisotropic flow coefficients, $v_{n}$. In particular, sPHENIX will permit
us to access the non-Gaussianities of these distributions, which are of
fundamental interest as they emerge from the non-Gaussian properties of the
fluctuating energy density field characterizing the QGP initial condition on
an event-by-event basis [132, 133, 134]. While precision measurements of non-
Gaussian flow fluctuations have been achieved in Pb+Pb collisions at the Large
Hadron Collider [135, 136, 137, 138], two key results are currently missing in
the Au+Au data sets at RHIC. The first involves the skewness of the
fluctuations of elliptic flow, $v_{2}$, which emerges in the fine splitting
(of order 1%) between the fourth-order cumulant, $v_{2}\\{4\\}$, and the
sixth-order cumulant, $v_{2}\\{6\\}$, of $v_{2}$ fluctuations. Linear response
to the initial QGP ellipticity, $v_{2}\propto\varepsilon_{2}$ [139, 140],
yields a negatively-skewed distribution of $v_{2}$ in off-central collisions
($>10\%$) due to the bound $\varepsilon_{2}<1$ [141]. Precision measurements
of the ratio $v_{2}\\{6\\}/v_{2}\\{4\\}$ offer, thus, a precision tool to
scrutinize the hydrodynamic response of the QGP.

The second important result that will be accessible to sPHENIX is the kurtosis
of the fluctuations of $v_{3}$, quantified by the four-particle cumulant
$c_{3}\\{4\\}$ [142, 143, 134]. This quantity also emerges from the response
to an initial triangularity, $v_{3}\propto\varepsilon_{3}$ [144, 139, 140],
whose distribution is non-Gaussian (with a negative kurtosis), potentially due
to the positivity constraint on the local energy density field, $e>0$, shaping
event-to-event the initial condition of the QGP [134]. While measurements of
$c_{3}\\{4\\}$ have been reported by the PHENIX collaboration [145], these
have been performed only via correlations of particles present in $1<|\eta|<3$
that show considerable effects of poorly-understood longitudinal fluctuations.
At midrapidity, where the response to the initial geometry is more robustly
understood, the kurtosis obtained in the existing data sets is compatible with
zero [146].

In this section, the linear response, $v_{n}\propto\varepsilon_{n}$, is
exploited to provide solid baselines for non-Gaussian flow fluctuations at
sPHENIX. Ratios of cumulants are constructed that isolate the impact of the
skewness of $v_{2}$ fluctuations and of the kurtosis of $v_{3}$ fluctuations,
namely,

$\frac{v_{2}\\{6\\}}{v_{2}\\{4\\}}\approx\frac{\varepsilon_{2}\\{6\\}}{\varepsilon_{2}\\{4\\}},\hskip
50.0pt\frac{c_{3}\\{4\\}}{c_{3}\\{2\\}^{2}}\approx\frac{\varepsilon_{3}\\{4\\}^{4}}{\varepsilon_{3}\\{2\\}^{4}}.$
(4)

The anisotropies $\varepsilon_{n}$ are obtained with the
T$\mathrel{\raisebox{-2.1pt}{R}}$ENTo model of initial conditions [46], tuned
to reproduce LHC data as in Ref. [147], where the centrality is evaluated from
the entropy of the collisions. No parameters related to the geometry of the
QGP are changed, most notably the size of nucleons, when moving from LHC and
RHIC collisions, to provide a baseline of results that does not contain any
energy-dependent effects. Results are shown up to 40% centrality, a range
where linear hydrodynamic response is an excellent approximation.

Figure 23: Non-Gaussian fluctuations of $v_{2}$ and $v_{3}$ in heavy-ion
collisions. Left: $v_{2}\\{6\\}/v_{2}\\{4\\}$. Right:
$c_{3}\\{4\\}/c_{3}\\{2\\}^{2}$. Experimental data (symbols) are measurements
by the ATLAS Collaboration in 5.02 TeV Pb+Pb collisions [135]. Lines are
estimates with the T$\mathrel{\raisebox{-2.1pt}{R}}$ENTo model of initial
conditions. Solid line: 5.02 TeV Pb+Pb collisions. Dashed line: 200 GeV Pb+Pb
collisions. Dotted line: 200 GeV Au+Au collisions. The red dotted lines
represent predictions for sPHENIX.

The left panel of Fig. 23 shows predictions for the splitting between
$v_{2}\\{4\\}$ and $v_{2}\\{6\\}$. Results are shown for 200 GeV Au+Au
collisions (dotted lines) and 5.02 TeV Pb+Pb collisions (solid lines). For the
latter, the model is observed to be in decent, though not perfect, agreement
with data by the ATLAS collaboration [138]. Of interest for this discussion is
the prediction relative to the difference between LHC and RHIC, which is shown
in the inset panel. Calculations for 200 GeV Pb+Pb collisions (dashed lines)
show that this departure does not come from a change in nucleon-nucleon cross
section in the T$\mathrel{\raisebox{-2.1pt}{R}}$ENTo calculation. Therefore,
in the present setup the splitting between Pb+Pb and Au+Au collisions comes
likely from the fact that 197Au has a larger ground-state quadrupole
deformation than 208Pb, which leads to enhanced non-Gaussian $v_{2}$
fluctuations in central Au+Au collisions [148]. Sizable departures from this
basic prediction in future data would point, then, to effects at high energy,
most probably from the modification of nucleon structure between RHIC and LHC.
Moving on to the fluctuations of $v_{3}$ in the right panel of Fig. 23, the
T$\mathrel{\raisebox{-2.1pt}{R}}$ENTo estimate of the standardized kurtosis of
$v_{3}$ fluctuations turns out to be in excellent agreement with ATLAS data,
strongly supporting the T$\mathrel{\raisebox{-2.1pt}{R}}$ENTo Ansatz for the
initial energy density fluctuations. The prediction for sPHENIX is shown as a
red dotted line. In this case, good agreement is found between the Pb+Pb and
Au+Au results. Significant departures from this prediction will point to
energy-dependent effects, most likely coming from the different structure of
the colliding nucleons. Precision measurements of the kurtosis from sPHENIX
will, hence, pose novel important constraints on the initial condition of the
QGP.

##  6 QCD with polarized protons and cold nuclei

In addition to the physics program aimed at exploring the large region of
high-temperature QGP created in Au+Au collisions, sPHENIX will explore QCD in
two other regimes: the cold but dense regime accessed in $p$+Au collisions,
and the vacuum QCD physics in $p$+$p$ collisions which serves as a baseline
for interpreting measurements in the other systems. These topics furthermore
have a natural connection to physics which will be explored at the future
Electron-Ion Collider [149] sited at BNL. Here, two potential measurements are
highlighted which are aimed at understanding the multi-dimensional parton
distributions in the proton and the parton fragmentation and hadronization
process, both utilizing the particular capabilities of sPHENIX.

###  6.1 Azimuthal angular correlation in dijet events at sPHENIX

Recently, the azimuthal angular correlation of dijets has received much
attention as a way to probe various multi-dimensional parton distributions
inside the nucleon/nucleus. For example, the $\cos 2\phi$ correlation between
the dijet relative momentum $p_{\mathrm{T}}=p_{1\perp}-p_{2\perp}$ and the
dijet total momentum $q_{\perp}=p_{1\perp}+p_{2\perp}$ in the so-called
correlation limit $p_{\mathrm{T}}\gg q_{\perp}$ has been proposed as a signal
of the linearly polarized gluon distribution.

Figure 24: $\cos 2\phi$ angular correlation in dijet production at midrapidity
in UPCs (left, $p_{\mathrm{T}}=10$ GeV) and in inclusive dijet production in
$pp$ (right, $p_{\mathrm{T}}=20$ GeV) at sPHENIX as a function of the dijet
total momentum $q_{\perp}$.

However, the same angular correlation can also be generated by soft gluon
radiations from the incoming and outgoing partons. The resummation of Sudakov
logarithms $\sim\ln p_{\mathrm{T}}/q_{\perp}$ associated with angular-
dependent soft gluon emissions has been systematically studied in [150, 151]
in the TMD framework building on an earlier work [152]. Fig. 24 shows
predictions for the average $\cos 2\phi$
($\phi=\phi_{p_{\mathrm{T}}}-\phi_{q_{\perp}}$) modulation at sPHENIX. The
left plot shows diffractive dijet events in UPCs $\gamma p(A)\to
jjp^{\prime}(A^{\prime})$ and the right plot shows inclusive dijet production
in $pp$ collisions $pp\to jjX$. Since $p_{\mathrm{T}}$ is much smaller than at
the LHC (see a recent measurement by the CMS collaboration [153]), the
asymmetry is more sensitive to the parameter $g_{\Lambda}$ that characterizes
the nonperturbative Sudakov suppression in the final state jets. Therefore,
future measurements at sPHENIX will be useful to constrain this so far poorly
known parameter.

###  6.2 Hadron-in-jet production with sPHENIX

The study of hadron distributions inside jets has received increasing
attention as an effective tool to understand the fragmentation process over
the last few years. They allow us to probe the one-dimensional (i.e.
collinear) and three-dimensional (i.e. transverse momentum dependent)
fragmentation functions [154, 155], and thus provide us with a deep insight
into the elusive mechanism of hadronization. The unpolarized and transversely
polarized proton-proton collisions at the sPHENIX would provide outstanding
opportunities for probing and measuring these functions. Below, predictions
for two such observables are provided. The first case is a study of the
transverse polarization of $\Lambda$ hyperons inside the jets that are
produced in unpolarized $pp$ collisions. The second case is a study of the so-
called Collins azimuthal asymmetry for charged pions inside the jets in
transversely polarized $p^{\uparrow}p$ collisions.

Figure 25: Left: Predictions for the transverse polarization $P_{\Lambda}$ of
$\Lambda$ hyperons inside the jets in unpolarized $pp$ collisions. Right:
Predictions for the Collins asymmetry
$A^{\sin(\phi_{S_{p}}-\hat{\phi}_{h})}_{TU,U}$ for $\pi^{\pm}$ inside the jet
produced in $p^{\uparrow}p$ collisions, with red (blue) curve are for
$\pi^{+}$ ($\pi^{-}$) production.

The left panel of Fig. 25 presents predictions for the $\Lambda$ transverse
polarization, $P_{\Lambda}$, for $\Lambda$ hyperons inside the jets that are
produced in unpolarized $p$+$p$ collisions at the center-of-mass (CM) energy
$\sqrt{s}=200~{}$GeV, $p+p\to(\mathrm{jet}\,\Lambda^{\uparrow})+X$. Jets are
constructed using the anti-$k_{T}$ jet algorithm with the radius $R=0.4$ in
the rapidity region $0<\eta_{J}<1$. In such a collision, the transverse
momentum ${\bf j}_{\perp}$ of the $\Lambda$ with respect to the jet axis and
the transverse spin ${\bf S}_{h\perp}$ of the $\Lambda$ particle correlate
with each other, generating a $\sin(\hat{\phi}_{h}-\hat{\phi}_{S_{h}})$
azimuthal dependence. The asymmetry is about $1-2\%$ and is a very promising
measurement at sPHENIX.

The right panel of Fig. 25 presents predictions of the Collins asymmetry
$A^{\sin(\phi_{S_{p}}-\hat{\phi}_{h})}_{TU,U}$ for unpolarized $\pi^{\pm}$
production inside the jets in transversely polarized $p^{\uparrow}$+$p$
collisions, $p^{\uparrow}+p\to(\mathrm{jet}\,\pi^{\pm})+X$. In this process,
the transverse momentum ${\bf j}_{\perp}$ of $\pi^{\pm}$ with respect to the
jet axis and the transverse spin ${\bf S}_{p}$ of the initial proton correlate
with each other, generating a $\sin(\phi_{S_{p}}-\hat{\phi}_{h})$ azimuthal
dependence [156, 157]. Such an azimuthal asymmetry is referred to as Collins
asymmetry and is sensitive to the quark transversity distribution in the
transversely polarized proton, and the Collins fragmentation function for
$\pi^{\pm}$. The asymmetry is positive for $\pi^{+}$ and negative for
$\pi^{-}$ inside the jet and increases as $z_{h}$ increases. The magnitude of
the asymmetry is about $4\%$, making measurements of this nature a good match
to the expected large-statistics dataset of sPHENIX.

##  7 Conclusion

This manuscript collects physics predictions presented at a RIKEN-BNL Research
Center (RBRC) workshop in July 2022 for the scientific program of sPHENIX, a
next-generation collider detector at the Relativistic Heavy Ion Collider
designed for a broad set of jet and heavy-flavor probes of the Quark-Gluon
Plasma created in heavy ion collisions. sPHENIX is expected to begin
commissioning and first data-taking in 2023, with a proposed running plan to
enable precision measurements of reconstructed jet quenching, heavy flavor and
quarkonia, cold QCD, and bulk physics. The detector features several
experimental capabilities new to heavy-ion program at RHIC, including $b$-jet
tagging, detailed jet sub-structure with very high statistics, and the
potential observation of the $\Upsilon(3S)$ state in Au+Au collisions. The
contributions in this manuscript highlight compelling aspects of the physics
program and, when possible, provide initial predictions in advance of sPHENIX
data-taking to guide the development of the program. Confronting these
predictions with data will be an important step towards elucidating the nature
of the QGP created in heavy-ion collisions and completing the scientific
mission of the RHIC facility.

## Acknowledgements

The authors gratefully acknowledge all workshop attendees for the fruitful
scientific discussion, as well as Brookhaven National Laboratory (BNL) and the
RIKEN-BNL Research Center (RBRC) for financial and logistical support.

Research support for individual authors is further acknowledged below. Funding
from the U.S. Department of Energy (DOE) Office of Science has been provided
under grants DE-SC0012704 (Y.H.), DE-FG02-03ER41244 (D.V.P.), DE-SC0013470
(M.S.), and DE-AC02-05CH11231 (X-N.W.). Funding from the National Science
Foundation (NSF) has been provided under grants 1848162 (M.C.), PHY-1945471
(Z.K. and F.Z.), 2111046 (A.M.S.), and OAC-2004571 (R.E., A.K., and X-N.W.).
The work of M.D. is supported by the European Research Council, grant
ERC-2016-COG: 725741, and by the Ministry of Science and Technological
Development of the Republic of Serbia. The work of M.Á.E is supported by
European Research Council project ERC-2018-ADG-835105 YoctoLHC, by the Maria
de Maetzu excellence program under projects CEX2020-001035-M and
CEX2019-000918-M, the Spanish Research State Agency under projects
PID2020-119632GB-I00 and PID2019-105614GB-C21, the Xunta de Galicia (Centro
singular de investigación de Galicia accreditation 2019-2022; European Union
ERDF). The work of G.G. is supported by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s Excellence Strategy EXC
2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster), SFB 1225
(ISOQUANT) and FL 736/3-1. The work of J.H. is supported in part by the
GLUODYNAMICS project funded by the “P2IO LabEx (ANR-10-LABX-0038)” in the
framework “Investissements d’Avenir” (ANR-11-IDEX-0003-01) managed by the
Agence Nationale de la Recherche (ANR), France. The work of W.K. is supported
by the US Department of Energy through the Office of Nuclear Physics and the
LDRD program at Los Alamos National Laboratory The work of D.P. is supported
by funding from the European Union’s Horizon 2020 research and innovation
program under the Marie Sklodowska-Curie grant agreement No. 754496. The work
of D.V.P. is supported by funding from the Research Corporation for Scientific
Advancement (RCSA). The work of I.V. is supported in part by the LDRD program
at LANL and by the U.S. Department of Energy under Contract No.
89233218CNA000001. The work of Z.Y. is supported by the NSFC under grant Nos.
11935007, 11861131009 and 11890714.

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