# Proton Compton Scattering from Linearly Polarized Gamma Rays

X. Li<EMAIL_ADDRESS>Department of Physics, Duke University, Durham, North
Carolina 27708-0308, USA  Triangle Universities Nuclear Laboratory, Durham,
North Carolina 27708-0308, USA  M.W. Ahmed  Triangle Universities Nuclear
Laboratory, Durham, North Carolina 27708-0308, USA  Department of Mathematics
and Physics, North Carolina Central University, Durham, North Carolina, 27707,
USA  A. Banu  Department of Physics and Astronomy, James Madison University,
Harrisonburg, Virginia 22807, USA  C. Bartram  Triangle Universities Nuclear
Laboratory, Durham, North Carolina 27708-0308, USA  University of North
Carolina at Chapel Hill, Chapel Hill, North Carolina 27516, USA  B. Crowe
Triangle Universities Nuclear Laboratory, Durham, North Carolina 27708-0308,
USA  Department of Mathematics and Physics, North Carolina Central University,
Durham, North Carolina, 27707, USA  E.J. Downie  Institute for Nuclear
Studies, Department of Physics, The George Washington University, Washington
DC 20052, USA  M. Emamian  Triangle Universities Nuclear Laboratory, Durham,
North Carolina 27708-0308, USA  G. Feldman  Institute for Nuclear Studies,
Department of Physics, The George Washington University, Washington DC 20052,
USA  H. Gao  Department of Physics, Duke University, Durham, North Carolina
27708-0308, USA  Triangle Universities Nuclear Laboratory, Durham, North
Carolina 27708-0308, USA  D. Godagama  Department of Physics and Astronomy,
University of Kentucky, Lexington, Kentucky 40506, USA  H.W. Grießhammer
Institute for Nuclear Studies, Department of Physics, The George Washington
University, Washington DC 20052, USA  Department of Physics, Duke University,
Durham, North Carolina 27708-0308, USA  C.R. Howell  Department of Physics,
Duke University, Durham, North Carolina 27708-0308, USA  Triangle Universities
Nuclear Laboratory, Durham, North Carolina 27708-0308, USA  H.J. Karwowski
Triangle Universities Nuclear Laboratory, Durham, North Carolina 27708-0308,
USA  University of North Carolina at Chapel Hill, Chapel Hill, North Carolina
27516, USA  D.P. Kendellen  Department of Physics, Duke University, Durham,
North Carolina 27708-0308, USA  Triangle Universities Nuclear Laboratory,
Durham, North Carolina 27708-0308, USA  M.A. Kovash  Department of Physics and
Astronomy, University of Kentucky, Lexington, Kentucky 40506, USA  K.K.H.
Leung  Department of Physics, Duke University, Durham, North Carolina
27708-0308, USA  Triangle Universities Nuclear Laboratory, Durham, North
Carolina 27708-0308, USA  Department of Physics and Astronomy, Montclair State
University, Montclair, New Jersey 07043, USA  D.M. Markoff  Triangle
Universities Nuclear Laboratory, Durham, North Carolina 27708-0308, USA
Department of Mathematics and Physics, North Carolina Central University,
Durham, North Carolina, 27707, USA  J.A. McGovern  Theoretical Physics Group,
School of Physics and Astronomy, University of Manchester, Manchester M13 9PL,
UK  S. Mikhailov  Triangle Universities Nuclear Laboratory, Durham, North
Carolina 27708-0308, USA  R.E. Pywell  Department of Physics and Engineering
Physics, University of Saskatchewan, Saskatoon, Saskatchewan, S7N 5E2, Canada
M.H. Sikora  Institute for Nuclear Studies, Department of Physics, The George
Washington University, Washington DC 20052, USA  Triangle Universities Nuclear
Laboratory, Durham, North Carolina 27708-0308, USA  J.A. Silano  Triangle
Universities Nuclear Laboratory, Durham, North Carolina 27708-0308, USA
University of North Carolina at Chapel Hill, Chapel Hill, North Carolina
27516, USA  R.S. Sosa  Department of Mathematics and Physics, North Carolina
Central University, Durham, North Carolina, 27707, USA  M.C. Spraker
Department of Physics and Astronomy, University of North Georgia, Dahlonega,
Georgia 30597, USA  G. Swift  Triangle Universities Nuclear Laboratory,
Durham, North Carolina 27708-0308, USA  P. Wallace  Triangle Universities
Nuclear Laboratory, Durham, North Carolina 27708-0308, USA  H.R. Weller
Department of Physics, Duke University, Durham, North Carolina 27708-0308, USA
Triangle Universities Nuclear Laboratory, Durham, North Carolina 27708-0308,
USA  C.S. Whisnant  Department of Physics and Astronomy, James Madison
University, Harrisonburg, Virginia 22807, USA  Y.K. Wu  Department of Physics,
Duke University, Durham, North Carolina 27708-0308, USA  Triangle Universities
Nuclear Laboratory, Durham, North Carolina 27708-0308, USA  Z.W. Zhao
Department of Physics, Duke University, Durham, North Carolina 27708-0308, USA
Triangle Universities Nuclear Laboratory, Durham, North Carolina 27708-0308,
USA

###### Abstract

Differential cross sections for Compton scattering from the proton have been
measured at scattering angles of $55\degree$, $90\degree$, and $125\degree$ in
the laboratory frame using quasimonoenergetic linearly (circularly) polarized
photon beams with a weighted mean energy value of 83.4 MeV (81.3 MeV). These
measurements were performed at the High Intensity Gamma-Ray Source facility at
the Triangle Universities Nuclear Laboratory. The results are compared to
previous measurements and are interpreted in the chiral effective field theory
framework to extract the electromagnetic dipole polarizabilities of the
proton, which gives $\alpha_{E1}^{p}=13.8\pm 1.2_{\rm stat}\pm 0.1_{\rm
BSR}\pm 0.3_{\rm theo},\beta_{M1}^{p}=0.2\mp 1.2_{\rm stat}\pm 0.1_{\rm
BSR}\mp 0.3_{\rm theo}$ in units of 10-4 fm3.

The static electric and magnetic dipole polarizabilities of the proton
$\alpha_{E1}^{p}$ and $\beta_{M1}^{p}$, respectively, reveal the internal
dynamics of the proton. They parametrize the response of the proton’s internal
degrees of freedom to an external electromagnetic field. Considerable efforts
have been taken to study the proton polarizabilities both experimentally and
theoretically Schumacher:2005an ; Griesshammer:2012we ; Hagelstein:2015egb .
In addition, $\beta_{M1}^{p}$ has been shown to be a crucial input in the
determination of the two-photon-exchange contribution to the Lamb shift in
muonic hydrogen Birse:2012eb . For the nucleon in general, the isovector
difference $\beta_{p}-\beta_{n}$ has been connected to the nucleon
electromagnetic mass difference, most recently in Refs. Gasser:2015dwa ;
Walker-Loud:2019qhh . The calculation of nucleon polarizabilities is also an
aim of lattice QCD, and several groups now have published results, albeit
almost all at large pion masses; see Ref. Detmold:2019ghl  and references
therein.

Compton scattering has proven to be a powerful tool to probe the proton
polarizabilities. At incident photon energies far below the pion-production
threshold, Compton scattering from the proton can be described as the elastic
scattering of the photon from a pointlike charged particle with an anomalous
magnetic moment. As the photon energy increases, the effect of the
electromagnetic polarizabilities as the leading-order structure-dependent
contribution to the Compton scattering cross section becomes more significant.
Compton scattering experiments below the pion-production threshold using
liquid hydrogen targets and tagged or untagged bremsstrahlung photon beams
have been performed at several gamma-ray source facilities in the last three
decades. The Compton scattering cross sections on the proton have been
extracted with significantly improved precision. To determine
$\alpha_{E1}^{p}$ and $\beta_{M1}^{p}$ from Compton scattering, rigorous
theoretical calculations are needed to fully describe the process. Chiral
effective field theory ($\chi$EFT) has proven to be successful in interpreting
Compton scattering data in terms of low-energy degrees of freedom
Griesshammer:2012we . Currently the best determination of the Baldin sum rule
(BSR) provides the constraint Gryniuk:2015eza

$\alpha_{E1}^{p}+\beta_{M1}^{p}=14.0\pm 0.2,$ (1)

where the polarizabilities are given here and throughout this Letter in units
of 10-4 fm3. Applying this sum-rule constraint, the latest $\chi$EFT fit to
the global database of proton Compton scattering below 200 MeV gives
McGovern:2012ew ; Griesshammer:2015ahu

$\begin{split}\alpha_{E1}^{p}&=10.75\pm 0.35_{\rm stat}\pm 0.1_{\rm BSR}\pm
0.3_{\rm theo},\\\ \beta_{M1}^{p}&=3.25\mp 0.35_{\rm stat}\pm 0.1_{\rm BSR}\mp
0.3_{\rm theo}.\end{split}$ (2)

(This is an update of the published result which used a slightly different BSR
value deLeon:2001dnx .)

The advent of the free-electron-laser (FEL)-based High Intensity Gamma-ray
Source (HIGS) opens up new opportunities for nuclear Compton scattering
Griesshammer:2012we ; Hagelstein:2015egb ; Weller:2009zza ; Myers:2012xw ;
Myers:2014qaa ; Sikora:2017rfk ; Li:2019irp . This Letter reports the first
proton Compton scattering experiment at HIGS using polarized photons, and the
first analysis of absolute differential cross sections of proton Compton
scattering using linearly polarized gamma-ray beams below the pion-production
threshold. A $\chi$EFT fit has been performed on the Compton scattering cross
sections on the proton obtained in the present experiment to extract the
proton polarizabilities. The results are compared to the previously extracted
values. The influence of this new extraction of the proton polarizabilities
and the potential of further high-precision measurements are also discussed.

The experiment was performed at the HIGS facility at the Triangle Universities
Nuclear Laboratory Weller:2009zza . The HIGS facility utilizes a FEL to
produce intense, quasimonoenergetic, and nearly 100% polarized gamma-ray beams
via Compton backscattering Yan:2019bru ; Wu:2015hta ; Wu:2006zzc . The gamma-
ray beam pulses are about 10 ns wide separated by 179 ns, allowing for clear
identification of beam-related events and rejection of backgrounds in the
detection system. The gamma-ray beam was collimated by a 25.4-mm-diameter lead
collimator. The collimated beam flux was measured and continuously monitored
using a collection of plastic scintillators with an internal radiator Pywell09
by detecting the charged particles produced in the radiator. In this
experiment, the gamma-ray beams were linearly (circularly) polarized at an
intensity-weighted average energy of 83.4 (81.3) MeV with an energy spread of
2.7% (6.5%) FWHM. The on-target intensity of the photon beam was $\approx
10^{7}\,\gamma/\rm{s}$.

The photons were scattered from a cryogenic target of liquid hydrogen
contained in a 20-cm-long conical frustum cell Kendellen16 . The wall and end
caps of the cell were made from 0.125-mm-thick Kapton foil. Scattering data
were taken with the target cell both full of liquid hydrogen with an areal
target density of $(8.40\pm 0.08)\times 10^{23}\,\rm{nuclei}/cm^{2}$ and empty
in order to subtract the empty-target background. The scattered photons were
detected using eight NaI(Tl) detectors (both in plane and out of plane)
located at scattering angles of $\theta=$ 55$\degree$, 90$\degree$, and
125$\degree$ in the laboratory frame. Each detector consisted of a 25.4-cm-
diameter, about 30-cm-long cylindrical core NaI(Tl) crystal surrounded by
7.5-cm-thick NaI(Tl) annular shield segments. The anticoincidence shield was
used to veto cosmic-ray events. The acceptance cone of each detector was
defined by a 15-cm-thick lead collimator installed in the front face of the
NaI(Tl) detector. Five in-plane detectors (one at $\theta=55\degree$, two at
$\theta=90\degree$, the other two at $\theta=125\degree$) were placed on the
tables with their axes and the linear polarization axis of the photon beam
aligned in the same horizontal plane. The other three out-of-plane detectors
(at $\theta=$ 55$\degree$, 90$\degree$, and 125$\degree$, respectively) were
placed beneath this horizontal plane and pointed upward toward the target,
with their axes aligned in a vertical plane perpendicular to the gamma-ray
beam linear polarization direction. The geometry of the experimental apparatus
(see more details in Refs. Li:2019irp ; Li:2020cvm ) was surveyed to a
precision of 0.5 mm and incorporated into a geant4 Agostinelli:2002hh
simulation to determine the effective solid angles of the detectors. The
effective solid angle accounts for the geometric effects due to the extended
target and the finite acceptance of the detectors, as well as the attenuation
of scattered photons in the target cell and the surrounding materials. With
the distance from the front face of each detector collimator aperture to the
target center measuring about 58 cm, the simulated effective solid angles
ranged from 62.7 to 65.6 msr.

Figure 1: Upper: representative energy spectra for full (closed circles) and
empty (open circles) targets at 55$\degree$, 90$\degree$, and 125$\degree$.
Lower: final energy spectra with empty-target events removed. The total fit
(dashed curve) is the sum of the electromagnetic background (dot-dashed curve)
and the geant4 simulated detector response function (solid curve). The
vertical dashed lines indicate the ROI.

The signals from the core and shield NaI detectors were recorded using a
14-bit digitizer with a sampling rate of 500 MHz. A cut was applied on the
energy spectrum from each shield detector to reject cosmic-ray events while
losing a negligible number of Compton events. In addition, the pulsed nature
of the photon beam produced a clear prompt timing peak for beam-related events
atop a flat random background, allowing for a timing cut to veto a large
portion of beam-unrelated backgrounds. The shield-energy cut and the timing
cut on the prompt peak together removed over 99% of the cosmic-ray events
within the region of interest (ROI) in the energy spectrum. After applying
both cuts, the remaining background from random, time-uncorrelated events was
removed by sampling the energy spectrum in the random region and subtracting
it in the prompt region after normalizing to the relative widths of the timing
windows. The above analysis was applied to the full-target and empty-target
data and the representative resulting energy spectra are shown in Fig. 1. For
each detector, the final energy spectrum was obtained by subtracting the
energy spectrum of the empty target from that of the full target after scaling
to the number of incident photons. Typical final energy spectra at the three
scattering angles are shown in Fig. 1.

The aforementioned geant4 simulation of the full experimental apparatus was
used to determine the line shape of the detector. Scattered photons were
propagated from within the target volume to the detectors to simulate for the
detector response functions. The energy of the simulated scattered photon
$E_{\gamma}^{\prime}$ was

$E_{\gamma}^{\prime}=\frac{E_{\gamma}}{1+\frac{E_{\gamma}}{Mc^{2}}(1-\cos\theta)},$
(3)

where $E_{\gamma}$ is the incident photon energy sampled from the beam energy
profile taking into account the energy spread of the photon beam, $\theta$ is
the laboratory scattering angle, $M$ is the proton mass, and $c$ is the speed
of light. The simulated detector response was then convoluted with a Gaussian
smearing function, which accounted for the intrinsic detector resolution, to
obtain the elastic-scattering line shape. As reported in Refs. Myers:2012xw ;
Myers:2014qaa , the in-beam test using the same set of NaI(Tl) detectors has
proven that the measured response function can be accurately reproduced with
such a method. The smeared detector response and the backgrounds were
simultaneously fitted to the final energy spectrum. For the forward-angle
detectors, the background from atomic processes was prominent in the low-
energy region and was fitted with an exponential function. Additional flat
backgrounds from the scattering of the bremsstrahlung photons induced by the
electron beam in the storage ring were fitted for all detectors. The number of
events were summed from the background-subtracted spectrum over the ROI and
then corrected by an efficiency factor to extract the photon yield. The
efficiency factor was defined as the ratio of the integral of the line shape
within the ROI to that of the entire energy range from 0 to 100 MeV. To
determine the differential cross section, the yield was normalized to the
target thickness, the number of incident photons corrected for the average
loss due to target absorption, the bin center correction factors, and the
effective solid angles.

The systematic uncertainties were grouped into two categories, point to point
and overall normalization. The point-to-point uncertainties resulted from
placing cuts in the timing and shield-energy spectra and locating boundaries
of the ROI and the fitting window, which varied between individual detectors.
Additional point-to-point systematic uncertainties were assigned to the two
out-of-plane detectors at the scattering angles of $55^{\circ}$ and
$125^{\circ}$ due to the insufficient measurement of the distances between
them and the target. All contributions were summed in quadrature to obtain the
total point-to-point systematic uncertainty for each data point ranging from
4.5% to 13.8%. The normalization uncertainties included the uncertainties from
the number of incident photons (2%) and the target thickness (1%).

The extracted differential cross sections for circularly and linearly
polarized photon beams are listed in Table 1 and plotted in Fig. 2. Note that
the Compton cross section is insensitive to the circular polarization of the
photon beam for an unpolarized target, the results of the present work using
the circularly polarized beam are compared to the cross sections obtained
using unpolarized tagged photon beams from the Saskatchewan Accelerator
Laboratory (SAL) at an incident photon energy of 81.8 MeV MacGibbon:1995in
and the Mainz Microtron (MAMI) facility at 79.2 MeV deLeon:2001dnx  in Fig. 2,
which shows a good agreement. Given that the linear polarization direction of
the incident photon beam was horizontal in the plane of
$\phi=0\degree/180\degree$, the in-plane (out-of-plane) detectors were at
azimuthal angles $\phi=0\degree\text{ or }180\degree$ ($\phi=270\degree$). The
photon beam asymmetry $\Sigma_{3}$ is defined as

$\Sigma_{3}=\frac{\sigma_{\parallel}-\sigma_{\perp}}{\sigma_{\parallel}+\sigma_{\perp}},$
(4)

where $\sigma_{\parallel}$ ($\sigma_{\perp}$) is the in-plane (out-of-plane)
differential cross section. The $\Sigma_{3}$ values at $\theta=55\degree$,
$90\degree$, and $125\degree$ were determined from the measured differential
cross sections listed in Table 1 where $\sigma_{\parallel}$ at
$\theta=90\degree$ and $125\degree$ were assigned the weighted average of
cross sections at $\phi=0\degree$ and $180\degree$. In Fig. 3, the
$\Sigma_{3}$ results from HIGS are compared to those from MAMI
Sokhoyan:2016yrc , which shows a good agreement.

Table 1: Differential cross section results at $81.3$ and $83.4$ MeV together with the statistical, point-to-point systematic, and normalization systematic uncertainties. $\theta_{\text{lab}}$ | $\phi$ | $d\sigma/d\Omega$ | Stat | Point to point | Normalization
---|---|---|---|---|---
|  | (nb/sr) | (nb/sr) | Syst (nb/sr) | Syst (nb/sr)
Circularly polarized gamma-ray beam at $81.3$ MeV
$55^{\circ}$ | $0^{\circ}$ | 11.45 | $\pm 0.90$ | $\pm 0.99$ | $\pm 0.26$
$55^{\circ}$ | $270^{\circ}$ | 11.61 | $\pm 0.86$ | $\pm 1.60$ | $\pm 0.26$
$90^{\circ}$ | $0^{\circ}$ | 8.87 | $\pm 0.73$ | $\pm 0.84$ | $\pm 0.20$
$90^{\circ}$ | $180^{\circ}$ | 10.14 | $\pm 0.76$ | $\pm 1.03$ | $\pm 0.23$
$90^{\circ}$ | $270^{\circ}$ | 9.55 | $\pm 0.63$ | $\pm 0.53$ | $\pm 0.21$
$125^{\circ}$ | $0^{\circ}$ | 11.46 | $\pm 0.68$ | $\pm 0.62$ | $\pm 0.26$
$125^{\circ}$ | $180^{\circ}$ | 12.83 | $\pm 0.73$ | $\pm 1.11$ | $\pm 0.29$
$125^{\circ}$ | $270^{\circ}$ | 15.08 | $\pm 0.69$ | $\pm 1.31$ | $\pm 0.34$
Linearly polarized gamma-ray beam at $83.4$ MeV
$55^{\circ}$ | $0^{\circ}$ | 5.19 | $\pm 1.01$ | $\pm 0.58$ | $\pm 0.12$
$55^{\circ}$ | $270^{\circ}$ | 17.03 | $\pm 1.17$ | $\pm 1.95$ | $\pm 0.38$
$90^{\circ}$ | $0^{\circ}$ | 3.10 | $\pm 0.94$ | $\pm 0.37$ | $\pm 0.07$
$90^{\circ}$ | $180^{\circ}$ | 3.99 | $\pm 0.83$ | $\pm 0.28$ | $\pm 0.09$
$90^{\circ}$ | $270^{\circ}$ | 18.24 | $\pm 0.93$ | $\pm 0.73$ | $\pm 0.41$
$125^{\circ}$ | $0^{\circ}$ | 7.06 | $\pm 0.77$ | $\pm 0.55$ | $\pm 0.16$
$125^{\circ}$ | $180^{\circ}$ | 7.99 | $\pm 0.66$ | $\pm 0.35$ | $\pm 0.18$
$125^{\circ}$ | $270^{\circ}$ | 19.18 | $\pm 0.92$ | $\pm 0.74$ | $\pm 0.43$
Figure 2: Differential cross section results extracted from the present
experiment (closed triangles, crosses, and circles) compared to the results
from SAL (open triangles, note that the data point at
$\theta_{\text{lab}}=90\degree$ is plotted with a slight offset in
$\theta_{\text{lab}}$ for better viewing) MacGibbon:1995in  and MAMI (open
squares) deLeon:2001dnx . The HIGS results in the upper (lower) panel were
obtained using the circularly (linearly) polarized gamma-ray beam at 81.3 MeV
(83.4 MeV). The error bars shown are the statistical and point-to-point
systematic uncertainties added in quadrature. The curves with $1\sigma$ error
bands are the theoretical cross sections implied by our measured
polarizabilities using the $\chi$EFT framework.  Figure 3: $\Sigma_{3}$
obtained in the present experiment (circles) compared to the results from MAMI
(diamonds) Sokhoyan:2016yrc . The error bars include statistical and point-to-
point systematic uncertainties only. The curve with the $1\sigma$ error band
is the theoretical $\Sigma_{3}$ implied by our measured polarizabilities using
the $\chi$EFT framework. (Note that we did not fit to $\Sigma_{3}$ but to the
individual cross sections.)

The cross sections obtained in this work were fitted using heavy-baryon chiral
perturbation theory (HB$\chi$PT) McGovern:2012ew  with and without the BSR
constraint to extract the proton polarizabilities. The beam profiles were used
to generate energy-weighted predictions for the cross sections. For the 16
HIGS data points of Fig. 2, the point-to-point systematic and statistical
uncertainties were added in quadrature, while the normalizations of the
circularly and linearly polarized beams were allowed to float independently.
However, a single-energy fit without floating normalizations gives negligibly
different results.

With the BSR constraint Eq. (1), 16 data points were fitted with three
parameters (one is $\beta$ and the other two are the normalization factors),
which gives

$\begin{split}\alpha_{E1}^{p}&=13.8\pm 1.2_{\rm stat}\pm 0.1_{\rm BSR}\pm
0.3_{\rm theo},\\\ \beta_{M1}^{p}&=0.2\mp 1.2_{\rm stat}\pm 0.1_{\rm BSR}\mp
0.3_{\rm theo},\end{split}$ (5)

with $\chi^{2}=14.7$ for 13 degrees of freedom. The statistical errors here
and in Eq. (6) were obtained from the $\chi_{\text{min}}^{2}$+1 intervals. An
estimate of the theoretical uncertainties follows the method of the
convergence study in Refs. McGovern:2012ew ; Griesshammer:2015ahu , and gives
the same results. A fit without the BSR constraint was also performed on the
16 data points by varying both polarizabilities, which leads to

$\begin{split}\alpha_{E1}^{p}&=15.4\pm 1.8_{\rm stat},\\\
\beta_{M1}^{p}&=2.1\pm 2.0_{\rm stat},\end{split}$ (6)

with $\chi^{2}=13.2$ for 12 degrees of freedom. We do not quote theoretical
uncertainties as this extraction is only given as a consistency check. Figure
4 shows the results of the extraction of $\alpha_{E1}^{p}$ and
$\beta_{M1}^{p}$ with and without the BSR constraint. We see that the one-
parameter fit and its 1$\sigma$ limits are contained within the 1$\sigma$
ellipse of the two-parameter fit, which validates the use of the BSR in the
former.

Figure 4: The 1$\sigma$ ($\chi_{\text{min}}^{2}$+2.3, outer ellipse) and
$\chi_{\text{min}}^{2}$+1 (inner ellipse) contours of the sum-rule-free fit of
$\alpha_{E1}^{p}$ and $\beta_{M1}^{p}$ in this work. The gray band represents
the BSR of Eq. (1). The two dots are the central values with and without the
BSR constraint.

There is some variation between our results and the values of Eq. (2)
extracted from the same theoretical framework based on the world database of
unpolarized data over a much wider energy range. This variation also pertains
to the current PDG values Zyla:2020zbs

$\alpha_{E1}^{p}=11.2\pm 0.4,\quad\beta_{M1}^{p}=2.5\pm 0.4.$ (7)

The pioneering extraction from the $\Sigma_{3}$ data obtained by MAMI
Sokhoyan:2016yrc , while in better agreement with both of these, has a
sufficiently large uncertainty that it is also in agreement with our
extraction. Fitting to cross sections rather than $\Sigma_{3}$ gives our
result a smaller uncertainty, though conducted at a lower average energy.

Recently, there have been other determinations from the full set or a subset
of the same unpolarized data, in different theoretical frameworks; for
examples, see Refs. Lensky:2014efa ; Pasquini:2017ehj ; Krupina:2017pgr . At
lower energies, different theories tend to agree better; therefore, high-
precision data in the energy region of our experiments have an important role
to play in resolving the discrepancy. The present work, as the first nanobarn-
level Compton scattering measurements at HIGS, demonstrated that with the
improved statistics of the HIGS data, one can extract the proton
polarizabilities with better precision.

In summary, new measurements on Compton scattering from the proton were
performed at HIGS below the pion-production threshold. The polarized cross
sections were extracted for the first time, and unpolarized cross sections of
this work are consistent with the global database. The sum-rule-free
extraction of $\alpha_{E1}^{p}$ and $\beta_{M1}^{p}$ using the χEFT framework
is compatible with the BSR. This work provided a novel experimental approach
for Compton scattering from the proton in low energies and strongly motivates
new high-precision measurements at HIGS to improve the accuracy in proton
polarizabilities determinations.

We acknowledge the support of the HIGS accelerator staff for the delivery of
high-quality gamma-ray beams and the help with the experimental setup. We
acknowledge the contributions of Daniel Phillips to this work. This work is
funded in part by the U.S. Department of Energy under Contracts No. DE-
FG02-03ER41231, No. DE-FG02-97ER41033, No. DE-FG02-97ER41041, No. DE-
FG02-97ER41046, No. DE-FG02-97ER41042, No. DE-SC0005367, No. DE-SC0015393, No.
DE-SC0016581, and No. DE-SC0016656, National Science Foundation Grants No.
NSF-PHY-0619183, No. NSF-PHY-1309130, and No. NSF-PHY-1714833, UK Science and
Technology Facilities Council Grants No. ST/L005794/1 and No. ST/P004423/1,
and funds from the Dean of the Columbian College of Arts and Sciences at The
George Washington University, and its vice president for research. We
acknowledge the financial support of the Natural Sciences and Engineering
Research Council of Canada and the support of Eugen-Merzbacher Fellowship.

Note added.–Recently, new results on polarized Compton scattering from the
proton reported by Mornacchi et al. (A2 Collaboration) Mornacchi:2021  have
been posted.

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[a]Lucia Anna Tarasovičová

# Exploring jet fragmentation using two-particle correlations with $\Lambda$
and K${}^{0}_{\rm S}$

###### Abstract

Complementary to jet reconstruction, two-particle correlations in $\Delta\eta$
and $\Delta\varphi$ are used to study jets, in particular their particle
composition. While in Pb–Pb collisions this is done to characterize the
quark–gluon plasma, pp and p–Pb collisions serve as a reference and are of
interest on their own for their input into the understanding of particle
production mechanisms. Recent ALICE results on the production of strange
particles in small systems (pp and p–Pb collisions) reveal the possibility of
having similar strange hadron production mechanisms in all collision systems.
We present here a study of two-particle correlations triggered with strange
hadrons ($\mathrm{K_{S}^{0}}$, $\Lambda$, $\overline{\Lambda}$) in pp
collisions at 13 TeV and 5.02 TeV and in the most central Pb–Pb collisions at
$\sqrt{s_{\mathrm{NN}}}$= 5.02 TeV. The dependence of the per-trigger yields
of primary charged hadrons on the transverse momenta of the trigger and
associated particles, as well as on the event multiplicity, will be presented
for both the near-side and away-side regions. Moreover, the ratios of these
yields to the yields extracted from inclusive hadron-hadron correlations and
the nuclear modification factor $I_{AA}$ will be discussed. The results are
compared among the three hadron species. In addition, a comparison to
different Monte Carlo generators is presented, which allows us to better
understand strangeness production in jets.

##  1 Introduction

Jet properties depend strongly on the original parton. These differences
between quark and gluon jets were observed in $e^{+}+e^{-}$ annihilations at
LEP by the OPAL and DELPHI collaborations [1], [2], [3]. A fragmenting gluon
emits more soft gluons than a fragmenting quark which leads to more particles
and a wider cone in the case of gluon jets. Moreover, the OPAL experiment
measured the relative production of $\mathrm{K_{S}^{0}}$ mesons and $\Lambda$
baryons to the total charged particle production in quark and gluon jets and
observed an enhanced relative production of $\Lambda$ in gluon-driven jets
[4]. Nevertheless, the jet tagging based on the original parton (quark or
gluon) is much more difficult in pp and in Pb–Pb collisions where the
differentiation between them would help to study the interaction between
quarks, gluons and QGP.

In order to study jets, the two-particle correlation method is used here in pp
collisions at 13 TeV and pp and Pb–Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ =
5.02 TeV. The correlations are triggered with strange $\mathrm{V}^{0}$ hadrons
($\mathrm{K_{S}^{0}}$, $\Lambda$, $\overline{\Lambda}$) and charged primary
hadrons (h), dominated by pions. The selection of a (relatively) high
transverse momentum provides a proxy for the jet axis. Thus, the trigger
particles are selected with following requirements: 3 $<$
$p_{\mathrm{T}}^{\mathrm{trigg}}$ $<$ 20 $\mathrm{GeV/}c$ and 8 $<$
$p_{\mathrm{T}}^{\mathrm{trigg}}$ $<$ 16 $\mathrm{GeV/}c$ for the collision
energy 13 TeV and 5.02 TeV, respectively. These are associated with other
charged hadrons with lower transverse momentum (1 $\mathrm{GeV/}c$ $<$
$p_{\mathrm{T}}^{\mathrm{assoc}}$ $<$ $p_{\mathrm{T}}^{\mathrm{trigg}}$). In
order to build a correlation function, angular differences between trigger and
associated particles are calculated in the $\Delta\varphi$, $\Delta\eta$
space. Such two-dimensional correlation function is normalised per trigger
particle., projected on the $\Delta\varphi$ axis and the underlying event is
subtracted. The latter is assumed to be flat in pp collisions and modulated
due to collective flow in Pb–Pb collisions. Two peaks remain, the near- and
away-side, and by integrating them the per-trigger yields can be extracted.
The effect of the QGP on these peaks can be studied by yield ratios in big and
small collisions systems and the different jet fragmentation processes can be
investigated by means of the ratios of yields triggered with different hadron
species.

Because of the neutrality of the $\mathrm{V}^{0}$ hadrons, they need to be
reconstructed from their decays into charged particles:
$\mathrm{K_{S}^{0}}\rightarrow\pi^{+}+\pi^{-}$ (69%), $\Lambda\rightarrow
p+\pi^{-}$ (64%) and $\overline{\Lambda}\rightarrow\overline{p}+\pi^{+}$ (64%)
[5]. The combinatorial background is suppressed by applying selection criteria
based on the decay topology and the daughter tracks are identified through
their specific energy loss in the TPC. The residual background is subtracted
with the help of side-bands of the invariant mass spectrum.

##  2 Results

In Fig. 1, the $\Delta\varphi$ projections of the three types of studied
correlation functions are shown for the pp collisions at 13 TeV [9]. Each of
the projections is compared with three Monte Carlo (MC) models - PYTHIA8
Monash [6], PYTHIA8 with shoving [7] and EPOS LHC [8]. None of the models is
able to describe the magnitude of the correlation function for all three types
of trigger particles in both shown $p_{\mathrm{T}}^{\mathrm{trigg}}$
intervals. EPOS LHC fails in the description in all cases except for the
$\mathrm{K_{S}^{0}}$-h correlation function while PYTHIA8 Monash is closer to
data in the higher $p_{\mathrm{T}}^{\mathrm{trigg}}$ interval. However, there
is almost no difference between the two considered PYTHIA8 models in low
$p_{\mathrm{T}}$, a difference can be observed in the higher
$p_{\mathrm{T}}^{\mathrm{trigg}}$ interval where the shoving model
underestimates the data strongly.

Figure 1: $\Delta\varphi$ projections of h-h (left), $\mathrm{K_{S}^{0}}$-h
(middle) and $(\Lambda+\overline{\Lambda})$-h (right) correlation functions
for two $p_{\mathrm{T}}^{\mathrm{trigg}}$ intervals, 3 $<$
$p_{\mathrm{T}}^{\mathrm{trigg}}$ $<$ 4 $\mathrm{GeV/}c$ (top) and 9 $<$
$p_{\mathrm{T}}^{\mathrm{trigg}}$ $<$ 11 $\mathrm{GeV/}c$ (bottom) compared
with MC models.

The $\Delta\varphi$ projections of the same types of correlation functions for
collision energy 5.02 TeV are plotted in Fig. 2 for both pp and central
(0-10%) Pb–Pb collisions. The modulation of the magnitudes of the peaks can be
seen: the yield of the near-side peaks is enhanced for the correlation
functions measured in Pb–Pb collisions while the yields of away-side peaks are
suppressed due to parton energy loss in the QGP.

Figure 2: $\Delta\varphi$ projections of h-h (left), $\mathrm{K_{S}^{0}}$-h
(middle) and $(\Lambda+\overline{\Lambda})$-h (right) in central (0-10%) Pb–Pb
(top) and pp (bottom) collisions for the centre-of-mass energy 5.02 TeV with
marked underlying event contribution.

###  2.1 Per-trigger yields

Fig. 3 is showing the per-trigger yields for different trigger particles as a
function of $p_{\mathrm{T}}^{\mathrm{trigg}}$ for different multiplicity
classes. An increasing trend with $p_{\mathrm{T}}$ is observed. This is caused
by the more available energy in a jet triggered with hadron with higher
transverse momentum. Quantitatively, the yields are similar for all trigger
particles on both near- and away-side.

Figure 3: Per-trigger yields from h-h (left), $\mathrm{K_{S}^{0}}$-h (middle)
and $(\Lambda+\overline{\Lambda})$-h (right) correlation functions on the
near-side (top) and away-side (bottom) for different multiplicity classes in
pp collisions at 13 TeV.

###  2.2 $I_{AA}$

One of possible ways to study the impact of the QGP to the jet fragmentation
is the calculation of the $I_{AA}$ ratio which is defined as
$\mathrm{Y_{\Delta\varphi}^{Pb-Pb}}/\mathrm{Y_{\Delta\varphi}^{pp}}$. This
ratio is shown in Fig. 4 where a strong enhancement is observed for low
$p_{\mathrm{T}}^{\mathrm{assoc}}$, both on the near- and away-side. For high
$p_{\mathrm{T}}^{\mathrm{assoc}}$, no modification is observed on the near-
side while there is a strong suppression on the away-side. This suppression is
caused by the energy loss in the QGP. [10]. The left panel of Fig. 4 shows
that the near side $I_{AA}$ is slightly depleted for the
$(\Lambda+\overline{\Lambda})$ triggers with respect to the other trigger
particles, but this effect is not significant.

Figure 4: Ratio of the per-trigger near-side (left) and away-side (right)
yields in Pb–Pb collisions to the ones in pp collisions for different trigger
particles.

###  2.3 Comparison of hadron- and strangeness-triggered yields

In order to be able to study the differences between different trigger
particles in more details, the ratios of yields triggered with
$\mathrm{K_{S}^{0}}$ or $(\Lambda+\overline{\Lambda})$ over the ones extracted
from h-h correlation functions are calculated. The results are shown in the
left part part of Fig. 5. Here, the ratios are compared with models which show
better agreement with the data than it was for the correlation functions (Fig.
1). This is caused by similar mismatch between data and models for all trigger
particles. In a detailed observation, one can see a difference between the
ratios triggered with $\mathrm{K_{S}^{0}}$ and $(\Lambda+\overline{\Lambda})$.
The $Y_{\Delta\varphi}^{\mathrm{K_{S}^{0}}-h}/Y_{\Delta\varphi}^{h-h}$ is
rather flat with $p_{\mathrm{T}}^{\mathrm{trigg}}$ and smaller than unity.
This is probably due to the fact that jets triggered with a
$\mathrm{K_{S}^{0}}$ meson fragment in a smaller amount of charged particles
with respect to the jets triggered with a pion and this effect does not depend
on the initial energy of the trigger particle. The distribution does not show
no dependence on the event multiplicity. However, the ratio
$Y_{\Delta\varphi}^{(\Lambda+\overline{\Lambda})-h}/Y_{\Delta\varphi}^{h-h}$
increases with $p_{\mathrm{T}}^{\mathrm{trigg}}$ which means that the jets
triggered with high-$p_{\mathrm{T}}$ $\Lambda$ or $\overline{\Lambda}$ have
more associated particles than jet triggered with a pion with the same
$p_{\mathrm{T}}$. This could point out that triggering with high
$p_{\mathrm{T}}$ $\Lambda$ or $\overline{\Lambda}$ hyperon biases the jet
selection towards gluon jets which have higher particle multiplicity [3],[2].
To prove this assumption, a separate PYTHIA8 study was performed where pure
gluon ($g+g\rightarrow g+g$) or quark ($q+\overline{q}\rightarrow
q+\overline{q}$) hard processes were simulated. The comparison of results from
this study with data in MB is shown in the right part of Fig. 5. There, it is
visible that the ratios from both quark and gluon samples are almost the same
in case of $\mathrm{K_{S}^{0}}$, but the ratio from gluon sample is shifted
towards higher values and increases slightly with
$p_{\mathrm{T}}^{\mathrm{trigg}}$ in the case of
$(\Lambda+\overline{\Lambda})$.

Figure 5: Ratios of integrated per-trigger yield of $\mathrm{K_{S}^{0}}$-h
(left column) or $(\Lambda+\overline{\Lambda})$-h (right column) to h-h as a
function of $p_{\mathrm{T}}^{\mathrm{trigg}}$ for the near-side for different
event multiplicity classes in the left plot and in the right plot MB results
compared with the PYTHIA8 calculation of the quark and gluon jets.

##  3 Summary and conclusions

The correlation functions triggered with charged hadrons, $\mathrm{K_{S}^{0}}$
mesons and $\Lambda$ ($\overline{\Lambda}$) hyperons are measured in pp and
Pb–Pb collisions at 13 TeV and 5.02 TeV, with the ALICE experiment. The per-
trigger yields are extracted and compared among collision systems and
different trigger particles. The $I_{AA}$ ratio shows an enhancement for low
$p_{\mathrm{T}}^{\mathrm{assoc}}$ on both near- and away side while a
suppression on the away-side is visible for high
$p_{\mathrm{T}}^{\mathrm{assoc}}$. Both can be explained by the interaction of
the initial jet parton with the QGP [10]. The ratios between $\mathrm{V}^{0}$
triggered and charged hadron triggered yields show a difference between
$\mathrm{K_{S}^{0}}$ and $\Lambda$ ($\overline{\Lambda}$) being the trigger
particle. This can be understood through a PYTHIA8 study as a bias towards
gluon jets caused by triggering with $\Lambda$ ($\overline{\Lambda}$).

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