# Proposal for PAC 52:
Measurement of $\alpha_{-}$ for $\Lambda\rightarrow p\pi^{-}$

P. Hurck Contact<EMAIL_ADDRESS>School of Physics and Astronomy,
University of Glasgow, United Kingdom  0000-0002-8473-1470 D.I. Glazier
School of Physics and Astronomy, University of Glasgow, United Kingdom  D.G.
Ireland  School of Physics and Astronomy, University of Glasgow, United
Kingdom  K. Livingston  School of Physics and Astronomy, University of
Glasgow, United Kingdom  F. Afzal  HISKP, University of Bonn, Germany  A.
Thiel  HISKP, University of Bonn, Germany  0000-0003-0753-696X Y. Wunderlich
HISKP, University of Bonn, Germany  V. Crede  Florida State University,
Tallahassee, Florida 32306, USA  M. M. Dalton  Jefferson Lab, Newport News,
USA  0000-0001-9204-7559

###### Abstract

We propose to measure the weak decay constant $\alpha_{-}$ for the decay
$\Lambda\rightarrow p\pi^{-}$ using a both circularly and linearly polarized
photon beam with the GlueX spectrometer in Hall D. The measurement will take
advantage of the fact that a measurement with both linear and circular photon
beam polarization results in an over-constrained set of amplitudes which can
be fitted to data and used to extract $\alpha_{-}$ which will be left as a
free parameter in the fit. We expect to determine $\alpha_{-}$ with
statistical uncertainties comparable to existing measurements and independent
systematic uncertainties. This measurement can be performed alongside GlueX-II
running and requires no new hardware or new beam time. The measurement
requires that a sufficient fraction of the electron beam polarization be
longitudinal in the Hall D tagger.

###### Contents

  1. 1 Introduction
  2. 2 Existing measurements
  3. 3 Extraction method
    1. 3.1 Advantage over previous measurement with similar methodology
  4. 4 Photon beam
    1. 4.1 Measurement with elliptically polarized photons
    2. 4.2 Determination of Circular Polarization
      1. 4.2.1 Projection from other measurements
      2. 4.2.2 Physics Reaction in the GlueX Spectrometer
    3. 4.3 Measurement of Absolute Linear Polarization
  5. 5 Proof of concept
    1. 5.1 Existing Data
    2. 5.2 Analysis of $\Lambda$ decay
    3. 5.3 Analysis of $\rho$ decay for Degree of Circular Polarization
  6. 6 Experiment
    1. 6.1 Statistical Uncertainty
    2. 6.2 Systematic Uncertainty
      1. 6.2.1 Polarimetry
      2. 6.2.2 Acceptance
      3. 6.2.3 Background contamination
      4. 6.2.4 Total
  7. 7 Additional physics possible with circularly polarized photons in Hall D
    1. 7.1 $\Sigma^{+}$ Weak Decay Parameter
    2. 7.2 Polarized Spin Density Matrix Elements
    3. 7.3 Amplitude Analysis for Meson Spectroscopy
    4. 7.4 Timelike Compton Scattering
  8. 8 Request
  9. 9 Summary
  10. A Circular polarization degree when using a diamond radiator
  11. B Relationship of S and P waves to the intensity in polarized $\rho$ photoproduction

##  1 Introduction

The decay parameter $\alpha_{-}$ of the parity-violating weak decay
$\Lambda\to p\pi^{-}$ describes the interference between parity-violating $s$
and parity-conserving $p$ waves. Among other things, the parameter
$\alpha_{-}$ of the singly-strange $\Lambda$ hyperon is an important quantity
for the extraction of polarization observables in various experiments. Many
other hyperons exhibit a $\Lambda$ in their decay chain, e.g. in the prominent
decays $\Sigma\to\Lambda\gamma$, $\Xi^{0(-)}\to\Lambda\pi^{0(-)}$, and
$\Omega\to\Lambda K^{-}$, and therefore, the decay parameters of these
hyperons are strongly affected by $\alpha_{-}$. In general, the parameter
$\alpha_{-}$ affects any quantity in which the polarization of the $\Lambda$
is relevant. For this reason, an independent determination of this quantity is
highly desirable given that $\alpha_{-}$ plays an important role in various
fields of physics. For instance, comparing $\alpha_{-}$ with the parameter
$\alpha_{+}$, which originates from the charge-conjugate decay
$\bar{\Lambda}\to\bar{p}\pi^{+}$, provides a test of CP symmetry for strange
baryons and, thus, can potentially shed light on the matter-antimatter
asymmetry in the Universe [1].

Small violations of CP symmetry are predicted by the standard model and are a
well established phenomenon in weak decays of mesons. However, the mechanisms
of the standard model are too specific to yield effects of a size that can
explain the observed matter–antimatter asymmetry of the Universe. Therefore,
CP tests can be considered a promising area to search for physics beyond the
standard model. And so far, no CP-violating effects beyond the standard model
have been observed in the baryon sector. In this respect, a CP violation at
the $3.3\sigma$ level has been found by the LHCb Collaboration in four-body
decays of $\Lambda^{0}_{b}$ and $\bar{\Lambda}^{0}_{b}$ baryons [2]. However,
in the BESIII simultaneous measurement of $\alpha_{-}$ and $\alpha_{+}$ of the
$\Lambda$, no sign of any CP violation was found [3].

Figure 1: Discrepancy between old and new measurements of $\alpha_{-}$. The
refit results based on $\Lambda$ photoproduction at CLAS6 are shown as
posterior densities. Taken from [4] (including caption).

Our goal is to measure the weak decay constant $\alpha_{-}$ for
$\Lambda\rightarrow p\pi^{-}$, using the photoproduction reaction $\gamma p\to
K^{+}\Lambda$. The measurement will be carried out with an elliptically
polarized photon beam with a linear and circular polarization component and
using the GlueX spectrometer in Hall D. Having both linear and circular photon
beam polarization results in an over-constrained set of amplitudes which can
be fitted to the $K^{+}\Lambda$ data and used to extract $\alpha_{-}$, which
will be left as a free parameter in the fit.

We expect that the gathering of this data will have no adverse impacts on the
overall running of GlueX-II and we do not expect to require any additional
dedicated time for systematic studies.

The proposal is structured as follows. In Sec. 2 we survey the landscape of
existing data on $\alpha_{-}$. In Sec. 3 we detail the method for determining
$\alpha_{-}$, which has independent systematic uncertainties from the existing
measurements with highest precision. In Sec. 4 we describe the requirements
for beam circular polarization and polarimetery, which is the only difference
from the approved GlueX-II running. In Sec. 5 we use a small subset of
existing data taken in 2023 to determine a value of $\alpha_{-}$,
demonstrating the feasibility and obtaining a statistical uncertainty to use
in projections. We use the same data set to determine an asymmetry in $\gamma
p\to\rho p$ proportional to the circular polarization. In Sec. 6 we do a
projection of the statistical uncertainty we would obtain from the existing
data and the remainder of the GlueX-II running if we receive an electron beam
polarization of 80%. The projected statistical uncertainty is comparable to
the uncertainties on existing measurements. In Sec. 7 we present some
additional physics opportunities that would become possible with a circularly
polarized photon beam as requested in this proposal.

##  2 Existing measurements

The $\alpha_{-}$ parameter was first measured in the 1960s at Brookhaven in
the reaction $\pi^{-}p\to\Lambda K^{0}$ using a carbon-plate spark chamber.
The value of $\alpha_{-}=0.62\pm 0.07$ was extracted from observing an up-down
asymmetry of the $\Lambda$ decay pions [5], which demonstrated that the
reaction produced polarized $\Lambda$ hyperons. Subsequent experiments in the
late 1960s at the Princeton-Pennsylvania Accelerator [6] using the same
reaction and at the Lawrence Radiation Laboratory [7] using the decay of the
$\Xi$ hyperon into $\Lambda\pi$ provided larger $\Lambda\to p\pi^{-}$ event
samples. The determined values of $0.645\pm 0.017$ and $0.67\pm 0.06$,
respectively, were in good agreement with the earlier measurement. Two
additional measurements in the 1970s at CERN of $0.649\pm 0.023$ [8] and
$0.584\pm 0.046$ [9] provided further results for $\alpha_{-}$, again fairly
consistent with the known values at the time.

More recently, the BESIII Collaboration in 2019 reported a significantly
larger value of $0.750\pm 0.009\pm 0.004$ [3], which was inconsistent with the
Review of Particle Physics (RPP) average value of $0.642\pm 0.013$ quoted
until 2018 [10]. Since these two values had uncertainties at the percent
level, the $5\sigma$ discrepancy rendered these results incompatible. The
about 17 % higher value reported by BESIII in 2019 has triggered a whole new
series of measurements at various laboratories. In the same year, a study by
Ireland et al. [4] based on a sample of photoproduced $K^{+}\Lambda$ events
off the proton collected by the CLAS Collaboration at Jefferson Lab supported
a higher value, but is lower than and in tension with the BESIII value. The
obtained value of $0.721(6)(5)$ was corroborated by multiple statistical tests
as well as a modern phenomenological model, showing that the new value yielded
the best description of the data in question. In 2020, the LHCb Collaboration
presented an analysis of the $\Lambda_{b}^{0}\to J/\psi\Lambda$ angular
distribution and the transverse production polarization of $\Lambda_{b}^{0}$
baryons in proton-proton collisions at centre-of-mass energies of 7, 8 and 13
TeV [11]. The parity-violating asymmetry parameter of the $\Lambda\to
p\pi^{-}$ decay was also determined from the same data and its value of
$0.74^{+0.04}_{-0.03}$ found to be consistent again with the recent 2019
measurement by the BESIII collaboration. Finally, the 2022 report by the
BESIII Collaboration presented the most precise measurements of $\Lambda$
decay parameters and CP asymmetry with a five-dimensional fit to the full
angular distributions of the daughter baryon [12]. The extracted $\alpha_{-}$
value of $0.7519\pm 0.0036\pm 0.0024$ has an unprecedented statistical quality
and the smallest systematic uncertainty reported to date. The decay parameter
$\alpha_{+}$ was also reported. The results are based on a total sample of 10
billion $J/\psi$ events and about 3.2 million quantum-entangled
$\Lambda$-$\bar{\Lambda}$ pairs could be fully reconstructed in the decay
$J/\psi\to\Lambda\bar{\Lambda}$ with $\Lambda(\bar{\Lambda})\to
p\pi^{-}(\bar{p}\pi^{+})$. The most recent analysis by the BESIII provides
further support for the previous BESIII results. A value of $0.757\pm 0.011\pm
0.008$ based on $J/\psi\to\Xi\bar{\Xi}\to\Lambda\bar{\Lambda}\pi\pi$ was
reported in late 2022 [13].

As discussed in Ref. [4], the big discrepancy between the earlier RPP value
based on measurements in the 1960s and 70s, and the more recent results might
be due, for instance, to underestimated systematic effects in the calculation
of correction factors in Ref. [9]. Or in the case of Ref. [8], photographs of
carbon-plate spark chambers were used, and a ten-parameter kinematic fit
applied to each event; several sources of uncertainty were highlighted and
together with the approximate fitting method, there was ample scope for
systematic error. While the previous measurements were all state of the art
when carried out, the RPP 2023 online update lists only some of the more
recent measurements “above the line.”

In a brief summary, several recent measurements of the $\alpha_{-}$ parameter
have addressed the large discrepancy between the old results from the 1960s
and 1970s, and a significantly larger value (by about 15–17 %), first observed
by the BESIII Collaboration in 2019, in support of the larger value. However,
the experimental status of $\alpha_{-}$ is not yet satisfactorily resolved.
The Particle Data Group has not used the 2020 LHCb result [11] for their
average, likely due to the reported fairly large uncertainties. For this
reason, the quoted RPP average is only based on various BESIII measurements
and the result by Ireland et al. based on CLAS data [4]. While the reported
BESIII results are all in good agreement and self-consistent, a smaller, but
significant, discrepancy still persists. Ireland et al. have reported a value
that is about 4 % lower than the averaged BESIII value. To this effect, it is
worth noting that the methodology used by Ireland et al. was different. GlueX
is well positioned to address this remaining discrepancy by providing an
independent cross-check of the methodology discussed in Ref. [4].

##  3 Extraction method

The methodology used for the extraction of $\alpha_{-}$ is based on a
publication by D. Ireland et al [4]. Unfortunately, the literature is plagued
by a variety of different sign conventions for polarization observables for
single-pseudoscalar production. Therefore, we lay out the whole formalism with
all chosen conventions in detail in the following.

The weak decay parameter $\alpha_{-}$ can be extracted by fitting polarization
observables, using an elliptically polarized photon beam. The differential
cross-section for single-pseudoscalar photoproduction using a beam with
circular and linear polarization is given by [14]

$\displaystyle\rho_{f}\frac{\text{d}\sigma}{\text{d}t}=\left.\frac{\text{d}\sigma}{\text{d}t}\right|_{\text{unpolarized}}\\{1$
$\displaystyle+\sigma_{y}P-P^{\gamma}_{L}\cos(2\Phi)(\Sigma+\sigma_{y}T)$
$\displaystyle-P^{\gamma}_{L}\sin(2\Phi)(O_{x}\sigma_{x}+O_{z}\sigma_{z})-P^{\gamma}_{C}(C_{x}\sigma_{x}+C_{z}\sigma_{z})\\}$
(1)

where $P^{\gamma}_{L}$ is the transverse polarization of the beam at an angle
$\Phi$ to the reaction plane and $P^{\gamma}_{C}$ is the degree of right
circular polarization of the beam. The density matrix of the recoiling
particle is denoted as $\rho_{f}=\frac{1}{2}(I+\sigma\cdot P_{f})$ and $P_{f}$
is its polarization.

In order to fit the function using an event-based likelihood, we define the
intensity function

$\displaystyle I(\Phi,\theta_{x^{\prime},y^{\prime},z^{\prime}})$
$\displaystyle=1+{\color[rgb]{0,1,1}\alpha_{-}}\cos\theta_{y^{\prime}}{\color[rgb]{0,0,1}P}-P^{\gamma}_{L}\cos(2\Phi)({\color[rgb]{0,0,1}\Sigma}+{\color[rgb]{0,1,1}\alpha_{-}}\cos\theta_{y^{\prime}}{\color[rgb]{0,0,1}T})$
$\displaystyle\hskip
42.67912pt-P^{\gamma}_{L}\sin(2\Phi)({\color[rgb]{0,1,1}\alpha_{-}}\cos\theta_{x^{\prime}}{\color[rgb]{0,0,1}O_{x^{\prime}}}+{\color[rgb]{0,1,1}\alpha_{-}}\cos\theta_{z^{\prime}}{\color[rgb]{0,0,1}O_{z^{\prime}}})$
$\displaystyle\hskip
42.67912pt-P^{\gamma}_{C}({\color[rgb]{0,1,1}\alpha_{-}}\cos\theta_{x^{\prime}}{\color[rgb]{0,0,1}C_{x^{\prime}}}+{\color[rgb]{0,1,1}\alpha_{-}}\cos\theta_{z^{\prime}}{\color[rgb]{0,0,1}C_{z^{\prime}}})$
(2)

This expression contains seven polarization observables
$O_{j}\in\\{\Sigma,T,P,C_{x^{\prime}},C_{z^{\prime}},O_{x^{\prime}},O_{z^{\prime}}\\}$,
which depend on the angles $\Phi$ and
$\theta_{x^{\prime},y^{\prime},z^{\prime}}$ (c.f. Sec. 5.2), as well as
$P_{L}^{\gamma}$ and $P^{\gamma}_{C}$. The expression also contains the weak
decay parameter $\alpha_{-}$, which we want to determine. Measuring
$\frac{\text{d}\sigma}{\text{d}t}$ allows to extract all $O_{j}$, provided
$\alpha_{-}$ is known. If it is not known, it can be left as a free parameter
in the fit, but this causes Eq. (2) to be under-constrained. To remedy the
situation two so-called Fierz identities which place constraints on the
polarization observables can be exploited [15]:

$\displaystyle\Sigma^{2}-T^{2}+P^{2}+C_{x^{\prime}}^{2}+C_{z^{\prime}}^{2}+O_{x^{\prime}}^{2}+O_{z^{\prime}}^{2}$
$\displaystyle=1$ (3) $\displaystyle\Sigma P-T-
C_{x^{\prime}}O_{z^{\prime}}+C_{z^{\prime}}O_{x^{\prime}}$ $\displaystyle=0$
(4)

Using Eq. (2) and imposing the relations in Eqs. (3) and (4) allows us to
determine $\alpha_{-}$ from a fit to the data.

While writing the cross-section and intensity in terms of polarization
amplitudes nicely illustrates the methodology, it is more convenient to
directly fit the underlying transversity amplitudes. These do automatically
contain the Fierz identities and hence they provide a more straightforward way
to analyze the data without constraining the fits explicitly. They are given
by

$\displaystyle\frac{\text{d}\sigma}{\text{d}t}$
$\displaystyle=|b_{1}|^{2}+|b_{2}|^{2}+|b_{3}|^{2}+|b_{4}|^{2}$ (5)
$\displaystyle\Sigma\frac{\text{d}\sigma}{\text{d}t}$
$\displaystyle=|b_{1}|^{2}+|b_{2}|^{2}-|b_{3}|^{2}-|b_{4}|^{2}$ (6)
$\displaystyle T\frac{\text{d}\sigma}{\text{d}t}$
$\displaystyle=|b_{1}|^{2}-|b_{2}|^{2}-|b_{3}|^{2}+|b_{4}|^{2}$ (7)
$\displaystyle P\frac{\text{d}\sigma}{\text{d}t}$
$\displaystyle=|b_{1}|^{2}-|b_{2}|^{2}+|b_{3}|^{2}-|b_{4}|^{2}$ (8)
$\displaystyle O_{x^{\prime}}\frac{\text{d}\sigma}{\text{d}t}$
$\displaystyle=-2\operatorname{Re}(b_{1}b_{4}^{*}-b_{2}b_{3}^{*})$ (9)
$\displaystyle O_{z^{\prime}}\frac{\text{d}\sigma}{\text{d}t}$
$\displaystyle=-2\operatorname{Im}(b_{1}b_{4}^{*}+b_{2}b_{3}^{*})$ (10)
$\displaystyle C_{x^{\prime}}\frac{\text{d}\sigma}{\text{d}t}$
$\displaystyle=2\operatorname{Im}(b_{1}b_{4}^{*}-b_{2}b_{3}^{*})$ (11)
$\displaystyle C_{z^{\prime}}\frac{\text{d}\sigma}{\text{d}t}$
$\displaystyle=-2\operatorname{Re}(b_{1}b_{4}^{*}+b_{2}b_{3}^{*})$ (12)

In practice, these equations are rearranged for the polarization observable
and inserted into Eq. (2). The resulting intensity can be used for likelihood
fitting. While the amplitudes are under-constrained the resulting polarization
observables are not.

###  3.1 Advantage over previous measurement with similar methodology

As discussed before, our methodology is based on the same idea as used by
Ireland et al. However, they did not have access to actual event level data.
Instead they used published result for the polarization observables $\Sigma$,
$T$, $P$, $O_{x}$, $O_{z}$, $C_{x}$, and $C_{z}$. Of these, only the first
five observables had a common region in $\\{W,\cos\theta\\}$ space, with 314
data points. In order to incorporate $C_{x}$ and $C_{z}$, a Gaussian Process
was used to interpolate the available data. This interpolation made it
possible to estimate $C_{x}$ and $C_{z}$ in the same $\\{W,\cos\theta\\}$
region as the other available data. This data was then used to extract
$\alpha_{-}$ through the constraints provided by the Fierz identities.

Although a large amount of $K^{+}\Lambda$ events went into the measurement of
the data used in their $\alpha_{-}$ estimation, the systematical uncertainties
are potentially large and hard to assess. A reliable uncertainty estimation
requires precise knowledge of all systematic uncertainties present in the
original data, as well as their correlations. Since all used data points were
reported by the CLAS experiment it cannot be ruled out, that they share
dependent systematic errors. It is possible that this could account for the
remaining discrepancy between the Ireland et al and the BESIII result.

Our methodology improves upon this by measuring all polarization observables
in a single measurement. That means that we will be in control of and able to
accurately estimate all systematic uncertainties. This will allow us to
provide a well controlled measurement of $\alpha_{-}$ which is completely
independent of the BESIII methodology.

##  4 Photon beam

This measurement requires the beam photons to have both linear and circular
polarization, here referred to as elliptical polarization. Elliptically
polarized photons can be produced using longitudinally polarized electrons
that are incident on a thin diamond radiator. The degree of linear and
circular polarization of the beam photons will be determined to sufficient
precision using existing hardware as described below.

###  4.1 Measurement with elliptically polarized photons

The amount of circular polarization that a beam photon carries depends on its
energy according to [16]:

$P^{\gamma}_{C}=p_{e}(4x-x^{2})/(4-4x+3x^{2}),$ (14)

where $x=E_{\gamma}/E_{e}$ is the ratio of beam photon $E_{\gamma}$ to
electron beam energy $E_{e}$. Eq. (14) was derived for an amorphous radiator,
when using a diamond radiator there is a small correction downwards [17] (see
Fig. 9 in Appendix A).

The position of the coherent peak will be set for GlueX-II running. We will
obtain a linear polarization of about 38% in the beam region approximately
8.5–9.0 GeV depending on the final electron beam energy. At this energy about
90% of the electron polarization is transferred to the photon.

Recently, the A2 collaboration in Mainz successfully extracted polarization
observables that require linearly or circularly polarized photons from the
same data sample using an elliptically polarized photon beam [17]. We will use
the same approach for the proposed measurement in order to extract all seven
polarization observables given in Eq. (2) from a single data sample.

###  4.2 Determination of Circular Polarization

The degree of circular polarization follows directly from the electron beam
polarization (with a small correction needed for the diamond lattice mostly in
the energy region of the coherent peak, see Appendix A). We do not request any
dedicated beamtime for systematic studies on the effect of the Coherent
Bremsstrahlung on the degree of circular polarization as this will be achieved
using amorphous running for a fraction of the data in line with past
measurements. We will determine the electron beam polarization using
projections from polarization measurements in other halls and using the
reaction $\gamma p\to\rho p$ in the main GlueX spectrometer.

As detailed in Secs. 4.2.2 and 5.3, this data on exclusive $\rho$
photoproduction will be obtained with high statistical precision and will be
available in perpetuity for any period when physics data is available. A full
analysis in terms of partial waves is capable of determining the linear and
circular polarizations independently. A much easier analysis of the helicity
asymmetry is capable of producing a relative polarization, and if it is
calibrated only once using some method, then it can be used for the whole
experiment to provide the absolute polarization over all the running.

We anticipate calibrating this $\rho$ polarimeter by projecting the precession
of the polarization measured in other halls (which is established to 1%
percent accuracy in Halls A and C) through to Hall D. The accuracy with which
this can be done depends on the details of the running conditions. Different
energies in other halls, different linac energies, and particularly different
Wien angles would all contribute to constraining the parameters.

Studies are being done, for the future polarized target program in Hall D,
into a potential future Møller polarimeter in the Hall D tagger—but this would
not be necessary for this proposed measurement to achieve its goals. Such a
device allows an independent check of the other strategies outlined here and
may benefit this program even if it came later.

####  4.2.1 Projection from other measurements

As described in more detail in Sec. 5.1 it is possible to do a combined fit of
polarization data obtained in other halls to fit the beam polarization at the
source, the linac energies, and any Wien angle offset, which allows the
precession to be projected to Hall D. The accuracy that can be achieved using
this approach depends on the details of what data is available for the fit and
the procession angle in Hall D. The more longitudinal the polarization in Hall
D, the more accurate the projection will be. Ideally, if multiple halls
measure the polarization at multiple Wien angles and energies then there are
sufficient independent constraints such that projection can be done with
$<2\%$ uncertainty on the longitudinal polarization.

####  4.2.2 Physics Reaction in the GlueX Spectrometer

The circular polarization of the photons at the target can be measured in the
main GlueX spectrometer using the reaction $\gamma p\to\rho p$. This reaction
has a very high statistics, corresponding to about 10% of the total hadronic
cross section in GlueX. As described in Sec. 5.3, the integrated helicity
asymmetry can be determined with a statistical uncertainty of
$dP^{\gamma}_{C}/P^{\gamma}_{C}=2.7\%$ per 2-hour run at 75% polarization.
This can be a very good relative polarimeter which allows us to monitor the
polarization as a function of time. The full resolution would only be
available later, an online result from $\sim 7\%$ of the data would be
available in real time giving an uncertainty limited to
$dP^{\gamma}_{C}/P^{\gamma}_{C}=3.0\%$ per day for diagnostic purposes.

Sec. 5.3 also describes how a full analysis can be used to determine the beam
polarizations. This requires well calibrated data and a well matched Monte
Carlo simulation and hence will not be available online. Systematic
uncertainties in the degree of circular polarization can be controlled and
minimized by comparing the $\rho$ measurements with polarization values
obtained from other sources. This can be done by doing a combined fit [18] and
leveraging the well understood systematic uncertainties of the Møller
polarimeters in other halls to achieve a polarization systematic uncertainty
in Hall D of $<2\%$. Independently calibrating this “$\rho$ polarimeter” is
essentially equivalent to measuring a new spin-density matrix element (SDME)
for $\gamma p\to\rho p$ reaction, a publishable result.

###  4.3 Measurement of Absolute Linear Polarization

The linear polarization of the beam is measured using the Triplet Polarimeter
(TPol) which uses the process of $e^{+}e^{-}$ pair production on atomic
electrons in a beryllium target foil. On timescales of days this is a
statistics limited measurement. The estimated total systematic uncertainty is
1.5% [19] and over the course of a run a total uncertainty of 2.1% has been
achieved [20].

##  5 Proof of concept

###  5.1 Existing Data

During the running of GlueX-II from January 12 to March 20 2023, the helicity
signal was incorporated into the GlueX DAQ for the first time. This allows
analysis that depends on the helicity of the electron beam to be performed.
During this time, 153 billion events were recorded taking 2,188 TB of space
(runs 120286 to 121207 of the 2023 beamtime in Hall D parlance.)

Table 1: Polarization versus run number for GlueX-II in 2023. The quoted uncertainties are very conservative pending more detailed studies of the precession and the $\rho$ polarimeter. This is sufficient for the “initial subset” of data that is analyzed here. runs | Wien angle | longitudinal polarization
---|---|---
120286 to 120445 | -64.6 | $53.2\pm 4.0\%$
120446 to 121207 | -47.2 | $71.0\pm 4.0\%$

Fortuitously, the longitudinal polarization of the electron beam, when it
arrived in the Hall D tagger, was high, Table 1. The polarization in Hall D
was determined by doing a combined fit of the Møller polarimeter data in Hall
A and Hall B for 2 Wien settings. The fit uses the measured injector energy,
the beam optic elements from a CEBAF Elegant model and takes into account
synchrotron energy loss in the precession. The parameters of the fit are the
absolute beam polarization at the source, the linac energies, and the Wien
angle offset. The result of the fit can be used to be project the beam to Hall
D, see Fig. 2

Figure 2: A fit of the Møller measurements made in early 2023 in Hall A and
Hall B at 2 Wien angles to a global model of the beam energy and polarization
magnitude and launch angle.

An “initial subset” of the GlueX-II 2023 data was chosen to do a preliminary
analysis. This subset is 30 good runs taken over 53 hours between 2023-01-27
to 2023-01-30 with numbers from 120395 to 120438, representing about 5% of the
data during that beam period. This data was given an initial calibration
during 2023 and was fully reconstructed starting in December 2023.

###  5.2 Analysis of $\Lambda$ decay

From the initial subset about 8.6k $\Lambda$ events were extracted. They were
reconstructed in the decay $\Lambda\rightarrow p\pi^{-}$. The resulting
$p\pi^{-}$ invariant mass distribution is shown in Fig. 3.

Figure 3: Invariant mass distribution of $p\pi^{-}$ for $\gamma p\rightarrow
K^{+}p\pi^{-}$ events reconstructed from the “initial subset”, representing
about 5% of the GlueX-II 2023 running period.

The resulting data is analyzed as outlined in Sec. 3.

Figure 4 shows the different reference frames used for this analysis.

Figure 4: Reference frames used in this analysis

The fit variables $\theta_{x^{\prime},y^{\prime},z^{\prime}}$ are defined as
the projections of the proton direction in the $\Lambda$ rest frame, onto the
coordinate system axis. The coordinate system (the so-called primed system) in
the $\Lambda$ rest frame is defined by

$\displaystyle\hat{z}^{\prime}=\frac{\vec{q}}{|\vec{q}|}\hskip
28.45274pt\hat{y}^{\prime}=\frac{\vec{k}\times\vec{q}}{|\vec{k}\times\vec{q}|}\hskip
28.45274pt\hat{x}^{\prime}=\hat{y}^{\prime}\times\hat{z}^{\prime}$ (15)

where $\vec{q}$ denotes the momentum vector of the $K^{+}$ in the centre-of-
mass frame (CMS) and $\vec{k}$ denotes the momentum vector of the beam photon
in the CMS.

The extended maximum likelihood can be written as

$\displaystyle\ln\mathscr{L}=\sum_{i=1}^{N}\ln
I(\Phi,\theta_{x^{\prime},y^{\prime},z^{\prime}})-\int\text{d}\Omega
I(\Phi,\theta_{x^{\prime},y^{\prime},z^{\prime}})\eta(\Phi,\theta_{x^{\prime},y^{\prime},z^{\prime}})$
(16)

where $I(\Phi,\theta_{x^{\prime},y^{\prime},z^{\prime}})$ is the intensity
defined in Eq. (2) and $\eta(\Phi,\theta_{x^{\prime},y^{\prime},z^{\prime}})$
denotes the detector acceptance which has to be taken into account. The sum is
over all $N$ events in the data, while the integral is evaluated by summing
over phase-space MC events which have been passed through a Geant4 based
simulation of the whole GlueX detector setup and then reconstructed and
analyzed like real data. The likelihood function is used in a Markov Chain
Monte Carlo (MCMC) parameter optimization process. Instead of trying to
minimize this multi-dimensional likelihood it is numerically explored.

The result is shown in Fig. 5.

Figure 5: Corner plot visualising the result of the Markov Chain Monte Carlo
parameter estimation to determine $\alpha$ together with the seven
polarization observables $\Sigma$ to $C_{z^{\prime}}$. The overall
normalization is denoted by norm.

The mean and standard deviation extracted from the $\alpha$ posterior
distribution is

$\displaystyle\alpha=0.755\pm 0.115$

which shows very good agreement with the previous data, albeit with large
statistical uncertainty. However, it shows that the proposed method works very
well.

###  5.3 Analysis of $\rho$ decay for Degree of Circular Polarization

This is essentially an extension of measurements of polarized $\rho$ SDMEs
published by GlueX [21] by including circular polarization and constraining
the intensity function with the Partial Wave formalism. Indeed conceptually it
is similar to the proposed measurement of $\alpha$, whereby the circular
polarized intensities are predicted from the linear polarized intensities
providing sensitivity to the degree of polarization from the measured decay
amplitudes.

Following [22] the intensity is written as,

$\displaystyle\mathcal{I}(\Omega,\Phi)=\mathcal{I}^{0}(\Omega)$
$\displaystyle-P^{\gamma}_{L}\mathcal{I}^{1}(\Omega)\cos(2\Phi)-P^{\gamma}_{L}\mathcal{I}^{2}(\Omega)\sin(2\Phi)-P^{\gamma}_{C}\mathcal{I}^{3}(\Omega),$
(17)

The polarized intensity functions can be expanded in moments of Spherical
Harmonics which are then related to the contributing partial waves, which
should be dominated by $P$ waves for $\rho$ decays. This results in an over-
constrained set of relationships allowing us to treat the polarization degrees
as unknown parameters when performing fits of Eq. (17) to the data in terms of
the partial waves. See Appendix B for technical details.

For the same initial subset of GlueX-II runs, we again perform MCMC sampling
of the log likelihood, but now using the intensity of Eq. (17) and with the
polarization degrees left as free parameters. We select the high t-region
($>$$0.5\text{\,}\mathrm{GeV}$), where we expect sensitivity to be largest and
fit 390k random subtracted events. The results of this preliminary analysis
look very promising. The extracted amplitude values agree with expectations
from the $\rho$ SDMEs and are shown in Appendix B, while the resulting
polarizations are indeed consistent with their expected values. The results
are shown in Fig. 6, where cuts have been placed. Using this method
statistical uncertainties on the degrees of polarization will be very small
($<1\%$). Systematic uncertainties will be well understood through measuring
the polarizations in many t and invariant mass bins, as well as using other
channels, such as $\omega$ production.

Figure 6: Degrees of linear and circular polarization extracted in preliminary
fits to 2023 data. The data distributions show MCMC samples. Numerically we
determine $P^{\gamma}_{L}=0.375\pm 0.008$ (stat) and $P^{\gamma}_{C}=0.423\pm
0.027$ (stat), independent of any other polarimetry technique. Systematic
uncertainties must yet be studied.

It would also be possible to use this reaction to give a simple asymmetry with
which to monitor the relative beam polarization on a run-by-run basis, without
needing detailed analysis or acceptance corrections. The helicity dependent
intensity $\mathcal{I}^{3}(\Omega)$ contains spherical harmonic terms which
flip with the beam helicity. In terms of the $\rho$ SDME elements we can
express a 2D asymmetry :

$\displaystyle\mathcal{A}_{3}=$
$\displaystyle\frac{\mathcal{I}(\Omega,h=+1)-\mathcal{I}(\Omega,h=-1)}{\mathcal{I}_{3}(\Omega,h=+1)+\mathcal{I}_{3}(\Omega,h=-1)}=\frac{\mathcal{I}_{3}(\Omega)}{\mathcal{I}_{0}(\Omega)}$
$\displaystyle\approx$
$\displaystyle\frac{3}{4\pi}(\sqrt{2}\operatorname{Im}{\rho^{3}_{10}}\sin{2\theta}\sin{\phi}+\operatorname{Im}{\rho^{3}_{1-1}}\sin^{2}{\theta}\sin{2\phi})/\mathcal{I}_{0}(\Omega)$

We integrate this over $\cos(\theta)$ for +ve and -ve values separately,
projecting onto $\phi$, flipping $\phi$ in the -ve case. The resulting
$\mathcal{A}_{3}(\phi)$, for the initial sample of 30 runs, is shown in Fig.
7(a). It can be approximated by a $\sin(2\phi)$ function due to the dominance
of the $\rho^{3}_{1-1}$ SDME. The asymmetry amplitude is proportional to the
beam polarization. Fitting the integral we get an amplitude of the helicity
asymmetry of $A\approx 6.85\pm 0.05\times 10^{-3}$ for beam polarization of
$53\%$ (Fig. 7(a)). This is a precision on the circular polarization of
$dP^{\gamma}_{C}/P^{\gamma}_{C}=0.73\%$. Such a precision corresponds to
$dP^{\gamma}_{C}/P^{\gamma}_{C}=4.0\%$ per 2-hour run. With a polarization of
$75\%$ the uncertainty would decrease by $0.53/0.75$ to become
$dP^{\gamma}_{C}/P^{\gamma}_{C}=2.7\%$ per run. Integrated over the GlueX-II
running, the statistical uncertainty becomes negligible. Fig. 7(b) shows the
amplitude of the helicity asymmetry plotted for each run in the initial
sample. This shows that we have the statistical power to be sensitive to a
time dependence of the polarization on typical multi-day timescales which has
been previously observed [23].

(a)

(b)

Figure 7: (a) Helicity asymmetry $\mathcal{A}_{3}(\phi)$ from $\rho$
production using the 30 reconstructed runs from 2023. This asymmetry is
proportional to the electron beam polarization. (b) The time dependence of the
amplitude of the helicity asymmetry demonstrating sensitivity to changes in
the polarization. Figure 8: Our proof-of-concept result in comparison to
previous data. Also shown is a projection for the achievable statistical
precision of the proposal as described in Sec. 6.1 (dark green), as well as
our projection for the combined systematic and statistical precision
anticipated for this measurement (c.f. Sec. 6.2.4). The old RPP value and the
LHCb result are not used for calculating the RPP average and fit.

##  6 Experiment

We plan to run alongside the remaining allocated beam time for GlueX-II. We do
not require any changes to the setup of the experiment.

###  6.1 Statistical Uncertainty

Here we use the “initial subset” of data to project a final statistical
uncertainty for the whole of GlueX-II running. Recall that this is 30 of 600
runs from 2023 that had an endpoint circular polarization of $\sim$50% and
gave 8.6k clean $\Lambda\rightarrow p\pi^{-}$ events. The beam time was split
about 100 to 500 into periods of 50% and 70% polarization respectively, so we
project $\sim$28k events with 50% polarization, and $\sim$115k events with 70%
polarization. Assuming that the higher polarization goes directly into our
statistical precision we have already acquired an effective 160k events
(115k*70%/50%) compared to our 30 run subset. GlueX-II still has about 220
days of data taking left, which is about 3.3 times more than what we had in
the 2023 running. Assuming that we get 80% electron beam polarization this is
effectively about 3.8 times more data that we currently have on tape.

In total we project about 28k+160k+700k effective events for GlueX-II running.
Note, these are not total expected events, but scaled up by polarization to
compare to our 30 runs used in the proof-of-principle. That means, in total we
expect more than 100 times more compared to our small 8.6k event subset. This
would result in 10 times smaller statistical uncertainties compared to what we
extracted in the proof-of-principle. This estimate is shown in Fig. 8 as
projection.

The estimate of the statistical accuracy expected for this proposal will be
competitive to the current estimate for $\alpha$ as listed in the RPP. Given
that we are using a completely different method from BESIII and a much
improved methodology over Ireland et al. we will make a crucial contribution
to resolve the remaining tension in the data.

###  6.2 Systematic Uncertainty

####  6.2.1 Polarimetry

As described in Sec. 4.2, a careful analysis of polarization data from other
halls and physics data recorded with the GlueX spectrometer will allow us to
achieve a polarization systematic uncertainty in Hall D of $\sim 2\%$ on the
circular polarization. The linear polarization has already been determined to
$\sim 2\%$ in previous running periods.

To study the effects of these on the extraction of $\alpha_{-}$, MC
simulations were used. The data sample was generated according to the
observables extracted in the proof-of-principle and with a nominal circular
and linear polarization of 0.65 and 0.38, respectively, and then analyzed with
polarizations that were 1$\sigma$ higher. The impact on $\alpha_{-}$ was
determined to be negligible for the circular polarization and adding about
$4\%$ systematic uncertainty in the case of the linear polarization.

As shown in Sec. 5.3, the analysis of specific physics reactions can be used
to determine the polarization. We believe that we can use this to improve the
systematic uncertainty on the linear polarization. Furthermore, the effect of
the linear polarization is only so large because of the values and
correlations for polarization observables $T$ and $P$. In our proof-of-
principle we do not have enough data to bin in momentum transfer $-t$, but it
is expected that the polarization observables will change depending on $-t$.
By binning in $-t$ we would be able to study the extraction with different
sensitivities to the linear polarization allowing us to develop strategies to
minimize this uncertainty. If necessary we could choose a range in $-t$ for
our final analysis that allows us to reduce the systematic uncertainty on the
linear polarization in exchange for some statistical precision.

As such, the numbers quoted above are a conservative upper limit for the
systematic uncertainty of the polarization on our final result and we are
cautiously optimistic that we will be able to halve them in our final result.

####  6.2.2 Acceptance

The measurement of $\alpha_{-}$ relies on an acceptance correction which is
performed by using MC simulations. The GlueX collaboration developed an
excellent understanding of its detector and accurate MC simulations are
achieved through the Geant4 based detector simulation package hdgeant4.
Previous publications indicate that we can assume a systematic effect from the
acceptance correction of less than 2%.

####  6.2.3 Background contamination

One potential large background for $\Lambda$ photoproduction comes from the
$\Sigma^{0}$ photoproduction. The $\Sigma^{0}\rightarrow\Lambda\gamma$ decay
produces soft photons which can be missed due to detector thresholds. Ref.
[24] indicates slightly larger $\Lambda$ than $\Sigma^{0}$ production cross-
sections at $E_{\gamma}=$8\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$. To study
the contamination, simulations for $\gamma p\rightarrow K^{+}\Lambda$ and
$\gamma p\rightarrow K^{+}\Sigma^{0}$ were produced and analyzed. We could
show that strictly requiring no neutral showers in our detector reduces the
$\Sigma^{0}$ background substantially while preserving most of the signal.
Furthermore, one can use the $\phi$-angles of the $K^{+}$ and $\Lambda$ to
improve purity further. A simple requirement that $K^{+}$ and $\Lambda$
$\phi$-angles are back-to-back reduces the $\Sigma^{0}$ contamination further,
in total to less than 2.5%. One could also use them as a discriminatory
variable to perform a background subtraction (e.g. using sWeights) and improve
purity even further. This might actually be the best way to remove the
background contamination.

To study the effect of the background on the extraction of $\alpha_{-}$ we
assume a worst case scenario, where we only use the cut-based approach to
reduce the $\Sigma^{0}$ contamination. Assuming that it is produced exactly
the same way as the signal from $\Lambda$, it would carry some polarization
itself which is passed on to the $\Lambda$ with a factor of $-\frac{1}{3}$
[25], which would then reduce the measured $\Lambda$ polarization and hence
dilute our measurement for $\alpha_{-}$ by about 3%. We expect to correct for
the bias with better than 10% precision so we conservatively assume 0.3%
systematic uncertainty.

In summary, assuming the same amount of $\Lambda/\Sigma^{0}$ being produced,
based on our studies, less than 2.5% contamination from $\Sigma^{0}$
production is expected, which we can correct for. Ultimately, we plan to
subtract the remaining background which will result in a negligible effect
from $\Sigma^{0}$ contamination. In a worst case scenario where we cannot do
the subtraction, it would introduce a systematic bias of about 0.3% on our
extraction of $\alpha_{-}$.

####  6.2.4 Total

A summary of the main systematic uncertainties anticipated for this
measurement is provided in Table 2. For now, we assume that the linear
polarization uncertainty as well as $\Sigma^{0}$ contamination can be taken as
upper limits for how much they will ultimately contribute to the $\alpha_{-}$
measurement. Adding all systematic uncertainties in quadrature, in total we
expect a systematic uncertainty of less than $5.4\%$. With the improvements
planned for the linear polarization measurement, as well as the $\Sigma^{0}$
background subtraction, we are optimistic to reach a total systematic
uncertainty of $2.8\%$. Both numbers are included in Figure 8 to show the
current upper limit and the anticipated total uncertainty expected for our
measurement.

Table 2: Summary of the systematic uncertainties expected in this measurement. Column 2 gives the percentage systematic uncertainty on each quantity and column 3 gives the effect on the final result. Column 4 lists the absolute contribution for our measurement based on a value of $\alpha_{-}=0.75$. The total is calculated for the upper limit as well as our expected systematic uncertainty. | % uncertainty | % contribution | absolute contribution
---|---|---|---
Photon beam circular polarization | 2% |  $<$0.2% |  $<$0.002
Photon beam linear polarization |  $<$2% |  $<$4% |  $<$0.03
Acceptance | 2% | 2% | 0.015
$\Sigma$ contamination |  $<$2.5% |  $<$0.3% |  $<$0.002
Total (current upper limit) |  |  $<$4.5% |  $<$0.034
Total (anticipated) |  | 2.8% | 0.021

##  7 Additional physics possible with circularly polarized photons in Hall D

While the main focus is the weak decay parameter $\alpha_{-}$ for
$\Lambda\rightarrow\pi^{-}p$, the requests described in this proposal, namely
providing circular polarization to Hall D and the mechanisms to measure it,
will allow access to other publishable physics. We outline several of these
below.

###  7.1 $\Sigma^{+}$ Weak Decay Parameter

The same methodology as outlined for the measurement of $\alpha_{-}$ can be
used to analyze the weak decay parameter for $\Sigma^{+}\rightarrow\pi^{0}p$
in the reaction $\gamma p\rightarrow\Sigma^{+}K^{0}_{s}$. The current RPP
value is $-0.982\pm 0.014$ and based on a measurement by BESIII
($J/\psi\rightarrow\Sigma^{+}\bar{\Sigma}^{-}$) and measurements from the
1970s using recoil polarization measurements. Although all previous
measurements agree we would provide a new and methodologically different
measurement.

###  7.2 Polarized Spin Density Matrix Elements

As described in Sec. 4.2, the vector mesons have spin-density matrix elements
(SDMEs) which describe how the photon polarization is transferred to the meson
and give information on the production mechanisms [26]. The presence of a
known amount of circular polarization in the beam will allow the extraction of
additional SDMEs in the photon energy range $6<E_{\gamma}<12$ GeV for $\gamma
p\to\rho p$, $\gamma p\to\omega p$, $\gamma p\to\phi p$ and $\gamma p\to
J/\psi p$. In the latter case these may provide information on the presence of
hadron resonance contributions to the production process.

Similarly, additional SDMEs will be measurable for t-channel baryon production
such as $\gamma p\to\pi\Delta$ or $\gamma p\to K^{+}\Lambda(1520)$.

In this proposal the set of seven polarization observables measured
simultaneously with the $\alpha_{-}$ parameter are essentially an example of
such SDME measurements, though written in a different formalism. They will
provide an additional publication from the same analysis.

These spin observables allow investigation of the mechanisms which contribute
to photoproduction. For example, which particles are exchanged, to what degree
is helicity conserved. This then allows and encourages development of more
sophisticated and better constrained reaction models.

###  7.3 Amplitude Analysis for Meson Spectroscopy

As circular polarization project out additional SDMEs in meson production
reactions, these SDMEs then produce further constraints on contributing
partial waves. These additional constraints can act to reduce possible
ambiguities, in particular removing a complex conjugate ambiguity, and reduce
statistical uncertainties compared to beamtimes without this additional
polarization. This would therefore provide extra assistance for the core GlueX
program.

###  7.4 Timelike Compton Scattering

The helicity asymmetry in Timelike Compton scattering (TCS), measured in the
process $\gamma p\to e^{+}e^{-}p$, is particularly useful as it is an
observable which is zero when there is no TCS contribution. This angular
asymmetry of the decay leptons accesses the real part of the Compton form
factors, important for the Generalized Parton Distribution framework. It has
recently been measured for the first time by the CLAS12 collaboration [27] and
independent measurements at GlueX would provide highly competitive results.

##  8 Request

We request that GlueX be given a share in the beam polarization. In concrete
terms, we request that Hall D be included in the usual calculations to
determine the optimal running conditions that maximize polarization in
multiple halls simultaneously. The optimization is done changing the the linac
energies individually and the Wien angle [28] and is typically able to find a
configuration which provides polarization to all halls.

##  9 Summary

In this document we outline how using the existing GlueX apparatus and
approved beam time we can measure weak decay constant $\alpha_{-}$ for the
decay $\Lambda\rightarrow p\pi^{-}$. This measurement will be able to resolve
a significant discrepancy in the literature. We ask for running in parallel
with GlueX and to be included in the usual calculations to determine the
optimal running conditions that maximize polarization in multiple halls
simultaneously.

## References

  * [1] A.. Sakharov  “Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe”  In _Pisma Zh. Eksp. Teor. Fiz._ 5, 1967, pp. 32–35  DOI: 10.1070/PU1991v034n05ABEH002497
  * [2] R. Aaij  “Measurement of matter-antimatter differences in beauty baryon decays”  In _Nature Phys._ 13, 2017, pp. 391–396  DOI: 10.1038/nphys4021
  * [3] M. Ablikim  “Polarization and Entanglement in Baryon-Antibaryon Pair Production in Electron-Positron Annihilation”  In _Nature Phys._ 15, 2019, pp. 631–634  DOI: 10.1038/s41567-019-0494-8
  * [4] D.. Ireland et al.  “Kaon Photoproduction and the $\Lambda$ Decay Parameter $\alpha_{-}$”  In _Phys. Rev. Lett._ 123.18, 2019, pp. 182301  DOI: 10.1103/PhysRevLett.123.182301
  * [5] J.. Cronin and O.. Overseth  “Measurement of the decay parameters of the Lambda0 particle”  In _Phys. Rev._ 129, 1963, pp. 1795–1807  DOI: 10.1103/PhysRev.129.1795
  * [6] O.. Overseth and R.. Roth  “Time Reversal Invariance in Lambda0 Decay”  In _Phys. Rev. Lett._ 19, 1967, pp. 391–393  DOI: 10.1103/PhysRevLett.19.391
  * [7] P.. Dauber et al.  “Production and decay of cascade hyperons”  In _Phys. Rev._ 179, 1969, pp. 1262–1285  DOI: 10.1103/PhysRev.179.1262
  * [8] W.. Cleland et al.  “A measurement of the beta-parameter in the charged nonleptonic decay of the lambda0 hyperon”  In _Nucl. Phys. B_ 40, 1972, pp. 221–254  DOI: 10.1016/0550-3213(72)90544-5
  * [9] P. Astbury  “Measurement of the Differential Cross-Section and the Spin-Correlation Parameters P,A, and R in the Backward Peak of pi- p –$>$ K0 Lambda at 5-GeV/c”  In _Nucl. Phys. B_ 99, 1975, pp. 30–52  DOI: 10.1016/0550-3213(75)90054-1
  * [10] M. Tanabashi  “Review of Particle Physics”  In _Phys. Rev. D_ 98.3, 2018, pp. 030001  DOI: 10.1103/PhysRevD.98.030001
  * [11] R. Aaij  “Measurement of the $\Lambda^{0}_{b}\rightarrow J/\psi\Lambda$ angular distribution and the $\Lambda^{0}_{b}$ polarisation in $pp$ collisions”  In _JHEP_ 06, 2020, pp. 110  DOI: 10.1007/JHEP06(2020)110
  * [12] M. Ablikim  “Precise Measurements of Decay Parameters and $CP$ Asymmetry with Entangled $\Lambda-\bar{\Lambda}$ Pairs Pairs”  In _Phys. Rev. Lett._ 129.13, 2022, pp. 131801  DOI: 10.1103/PhysRevLett.129.131801
  * [13] M. Ablikim  “Probing CP symmetry and weak phases with entangled double-strange baryons”  In _Nature_ 606.7912, 2022, pp. 64–69  DOI: 10.1038/s41586-022-04624-1
  * [14] I.. Barker, A. Donnachie and J.. Storrow  “Complete Experiments in Pseudoscalar Photoproduction”  In _Nucl. Phys. B_ 95, 1975, pp. 347–356  DOI: 10.1016/0550-3213(75)90049-8
  * [15] A.. Sandorfi, S. Hoblit, H. Kamano and T.-S.. Lee  “Determining pseudoscalar meson photo-production amplitudes from complete experiments”  In _J. Phys. G_ 38, 2011, pp. 053001  DOI: 10.1088/0954-3899/38/5/053001
  * [16] H. Olsen and L.. Maximon  “Photon and Electron Polarization in High-Energy Bremsstrahlung and Pair Production with Screening”  In _Phys. Rev._ 114 American Physical Society, 1959, pp. 887–904  DOI: 10.1103/PhysRev.114.887
  * [17] F. Afzal  “First Measurement Using Elliptically Polarized Photons of the Double-Polarization Observable E for $\gamma$p→p$\pi$0 and $\gamma$p→n$\pi$+”  In _Phys. Rev. Lett._ 132.12, 2024, pp. 121902  DOI: 10.1103/PhysRevLett.132.121902
  * [18] J.. Grames  “Unique electron polarimeter analyzing power comparison and precision spin-based energy measurement” [Erratum: Phys.Rev.ST Accel.Beams 13, 069901 (2010)]  In _Phys. Rev. ST Accel. Beams_ 7, 2004, pp. 042802  DOI: 10.1103/PhysRevSTAB.7.042802
  * [19] M. Dugger  “Design and construction of a high-energy photon polarimeter”  In _Nucl. Instrum. Meth. A_ 867, 2017, pp. 115–127  DOI: 10.1016/j.nima.2017.05.026
  * [20] H. Al Ghoul  “Measurement of the beam asymmetry $\Sigma$ for $\pi^{0}$ and $\eta$ photoproduction on the proton at $E_{\gamma}=9$ GeV”  In _Phys. Rev. C_ 95.4, 2017, pp. 042201  DOI: 10.1103/PhysRevC.95.042201
  * [21] S. Adhikari  “Measurement of spin-density matrix elements in $\rho$(770) production with a linearly polarized photon beam at E$\gamma$=8.2–8.8 GeV”  In _Phys. Rev. C_ 108.5, 2023, pp. 055204  DOI: 10.1103/PhysRevC.108.055204
  * [22] V. Mathieu et al.  “Moments of angular distribution and beam asymmetries in $\eta\pi^{0}$ photoproduction at GlueX”  In _Phys. Rev. D_ 100.5, 2019, pp. 054017  DOI: 10.1103/PhysRevD.100.054017
  * [23] A. Zec  “Ultrahigh-precision Compton polarimetry at 2 GeV”  In _Phys. Rev. C_ 109.2, 2024, pp. 024323  DOI: 10.1103/PhysRevC.109.024323
  * [24] A. Boyarski et al.  “PHOTOPRODUCTION OF K+ LAMBDA AND K+ SIGMA0 FROM HYDROGEN FROM 5-GeV to 16-Gev”  In _Phys. Rev. Lett._ 22, 1969, pp. 1131–1133  DOI: 10.1103/PhysRevLett.22.1131
  * [25] R.. Bradford  “First measurement of beam-recoil observables C(x) and C(z) in hyperon photoproduction”  In _Phys. Rev. C_ 75, 2007, pp. 035205  DOI: 10.1103/PhysRevC.75.035205
  * [26] K. Schilling, P. Seyboth and G.. Wolf  “On the Analysis of Vector Meson Production by Polarized Photons” [Erratum: Nucl.Phys.B 18, 332 (1970)]  In _Nucl. Phys. B_ 15, 1970, pp. 397–412  DOI: 10.1016/0550-3213(70)90070-2
  * [27] P. Chatagnon et al.  “First Measurement of Timelike Compton Scattering”  In _Phys. Rev. Lett._ 127 American Physical Society, 2021, pp. 262501  DOI: 10.1103/PhysRevLett.127.262501
  * [28] D.. Higinbotham  “Electron Spin Precession at CEBAF”  In _AIP Conf. Proc._ 1149.1, 2009, pp. 751–754  DOI: 10.1063/1.3215753

##  Appendix A Circular polarization degree when using a diamond radiator

When using a diamond radiator, the circular polarization degree exhibits a
dependence on the diamond lattice, which leads to a decrease of polarization
at the position of the coherent peaks [17]. The size of the decrease depends
on the beam, diamond and collimator parameters, as well as the chosen coherent
peak position. Fig. 9 shows the calculation for the coherent edge position at
8.6 GeV and an electron polarization degree of $p_{e}=53$% as used in the 2023
running (see Tab. 1). We find that for our case the effects from the diamond
lattice and beam/collimator to have a smaller magnitude than our uncertainty
on the circular polarization with an approximate $1.2$% relative deviation on
average in the range of the coherent edge ($8.0$ GeV$<E_{\gamma}<8.6$ GeV).
Therefore we can reliably use Eq. (14) as a good approximation.

Figure 9: The circular polarization degree as a function of the incoming
photon energy when using a diamond radiator (solid line) with the relevant
beam, collimator and diamond parameters for the GlueX experiment. The relative
deviation between the solid curve and Eq. (14) are shown with the dashed line
at the top.

##  Appendix B Relationship of S and P waves to the intensity in polarized
$\rho$ photoproduction

The intensity for polarized $\rho$ photoproduction can be written as [22],

$\displaystyle\mathcal{I}(\Omega,\Phi)=\mathcal{I}^{0}(\Omega)$
$\displaystyle-P^{\gamma}_{L}\mathcal{I}^{1}(\Omega)\cos(2\Phi)-P^{\gamma}_{L}\mathcal{I}^{2}(\Omega)\sin(2\Phi)-P^{\gamma}_{C}\mathcal{I}^{3}(\Omega),$
(18)

with $\Omega=(\cos(\theta),\phi)$ the decay angles of the $\rho$ to 2 pions,
$\Phi$ is the linear polarization angle with respect to the production plane
of the $\rho$-proton final state, $P^{\gamma}_{L}$ the degree of linear
polarization and $P^{\gamma}_{C}$ the degree of circular polarization. Then,

$\displaystyle\mathcal{I}^{\alpha}=\kappa\sum_{L,M}H^{\alpha}(LM)Y^{m}_{\ell}(\Omega),$
(19)

where $Y^{m}_{\ell}(\Omega)$ are the Spherical Harmonic functions for angular
momentum L and projection M, $H^{\alpha}(LM)$ the moments of the particular
Spherical Harmonic and $\kappa$ a phase space factor.

The moments can be expressed in terms of the spin-density matrix elements
(SDMEs) in the reflectivity basis, labelled by their angular momentum $l$
(equal to the particle spin) and spin projection $m$, and the appropriate
Clebsh-Gordan coefficients $C$,

$\displaystyle
H^{\alpha}(LM)=\sum_{\ell\ell^{\prime},mm^{\prime}}\left(\frac{2\ell^{\prime}+1}{2\ell+1}\right)^{\frac{1}{2}}C^{\ell
0}_{\ell^{\prime}0L0}C^{\ell
m}_{\ell^{\prime}m^{\prime}LM}\rho^{\alpha,\ell\ell^{\prime}}_{mm^{\prime}},$
(20)

where the SDMEs are the sum of the two reflectivity components :

$\displaystyle\rho^{\alpha,\ell\ell^{\prime}}_{mm^{\prime}}=\hskip
5.69046pt^{+}\rho^{\alpha,\ell\ell^{\prime}}_{mm^{\prime}}\hskip
5.69046pt+\hskip 5.69046pt^{-}\rho^{\alpha,\ell\ell^{\prime}}_{mm^{\prime}},$
(21)

which are given by,

$\displaystyle^{\epsilon}\rho^{0,\ell\ell^{\prime}}_{mm^{\prime}}=$
$\displaystyle[\ell]^{(\epsilon)}_{m}[\ell]^{(\epsilon)*}_{m^{\prime}}+(-1)^{m-m^{\prime}}[\ell]^{(\epsilon)}_{-m}[\ell]^{(\epsilon)*}_{-m^{\prime}},$
${}^{\epsilon}\rho^{1,\ell\ell^{\prime}}_{mm^{\prime}}=$
$\displaystyle-\epsilon\left((-1)^{m}[\ell]^{(\epsilon)}_{-m}[\ell]^{(\epsilon)*}_{m^{\prime}}+(-1)^{m^{\prime}}[\ell]^{(\epsilon)}_{m}[\ell]^{(\epsilon)*}_{-m^{\prime}}\right),$
${}^{\epsilon}\rho^{2,\ell\ell^{\prime}}_{mm^{\prime}}=$
$\displaystyle-i\epsilon\left((-1)^{m}[\ell]^{(\epsilon)}_{-m}[\ell]^{(\epsilon)*}_{m^{\prime}}-(-1)^{m^{\prime}}[\ell]^{(\epsilon)}_{m}[\ell]^{(\epsilon)*}_{-m^{\prime}}\right),$
${}^{\epsilon}\rho^{3,\ell\ell^{\prime}}_{mm^{\prime}}=$
$\displaystyle[\ell]^{(\epsilon)}_{m}[\ell]^{(\epsilon)*}_{m^{\prime}}-(-1)^{m-m^{\prime}}[\ell]^{(\epsilon)}_{-m}[\ell]^{(\epsilon)*}_{-m^{\prime}}.$

with $[\ell]$ = S, P , … . For $\rho$ production the P-waves will dominate.

For completeness we give the full expansions of the moments in terms of the S
and P waves (note the Clebsch-Gordan factors are given by numerical
approximations).

$\displaystyle H^{0}(0,0)=$ $\displaystyle
2(^{+}S^{+}S)+2(^{-}S^{-}S)+2(^{+}P_{-1}^{+}P_{-1})+2(^{+}P_{+1}^{+}P_{+1})$
$\displaystyle+2(^{-}P_{-1}^{-}P_{-1})+2(^{-}P_{+1}^{-}P_{+1})+2(^{+}P_{0}^{+}P_{0})+2(^{-}P_{0}^{-}P_{0})$
$\displaystyle H^{1}(0,0)=$ $\displaystyle
2(^{+}S^{+}S)+-2(^{-}S^{-}S)+-4(^{+}P_{+1}^{+}P_{-1})\cos(^{+}\phi_{P+1}-^{+}\phi_{P-1}))$
$\displaystyle+4(^{-}P_{+1}^{-}P_{-1})\cos(^{-}\phi_{P+1}-^{-}\phi_{P-1}))+2(^{+}P_{0}^{+}P_{0})+-2(^{-}P_{0}^{-}P_{0})$
$\displaystyle H^{0}(1,0)=$ $\displaystyle
2.3(^{+}S^{+}P_{0})\cos(^{+}\phi_{S}-^{+}\phi_{P0}))$
$\displaystyle+2.3(^{-}S^{-}P_{0})\cos(^{-}\phi_{S}-^{-}\phi_{P0}))$
$\displaystyle H^{1}(1,0)=$ $\displaystyle
2.3(^{+}S^{+}P_{0})\cos(^{+}\phi_{S}-^{+}\phi_{P0}))$
$\displaystyle-2.3(^{-}S^{-}P_{0})\cos(^{-}\phi_{S}-^{-}\phi_{P0}))$
$\displaystyle H^{0}(1,1)=$
$\displaystyle-1.2(^{+}S^{+}P_{-1})\cos(^{+}\phi_{S}-^{+}\phi_{P-1}))+1.2(^{+}S^{+}P_{+1})\cos(^{+}\phi_{S}-^{+}\phi_{P+1}))$
$\displaystyle-1.2(^{-}S^{-}P_{-1})\cos(^{-}\phi_{S}-^{-}\phi_{P-1}))+1.2(^{-}S^{-}P_{+1})\cos(^{-}\phi_{S}-^{-}\phi_{P+1}))$
$\displaystyle H^{1}(1,1)=$
$\displaystyle-1.2(^{+}S^{+}P_{-1})\cos(^{+}\phi_{S}-^{+}\phi_{P-1}))+1.2(^{+}S^{+}P_{+1})\cos(^{+}\phi_{S}-^{+}\phi_{P+1}))$
$\displaystyle+1.2(^{-}S^{-}P_{-1})\cos(^{-}\phi_{S}-^{-}\phi_{P-1}))+-1.2(^{-}S^{-}P_{+1})\cos(^{-}\phi_{S}-^{-}\phi_{P+1}))$
$\displaystyle H^{2}(1,1)=$
$\displaystyle-1.2(^{+}S^{+}P_{-1})\cos(^{+}\phi_{S}-^{+}\phi_{P-1}))+-1.2(^{+}S^{+}P_{+1})\cos(^{+}\phi_{S}-^{+}\phi_{P+1}))$
$\displaystyle+1.2(^{-}S^{-}P_{-1})\cos(^{-}\phi_{S}-^{-}\phi_{P-1}))+1.2(^{-}S^{-}P_{+1})\cos(^{-}\phi_{S}-^{-}\phi_{P+1}))$
$\displaystyle H^{3}(1,1)=$ $\displaystyle
1.2(^{+}S^{+}P_{-1})\sin(^{+}\phi_{S}-^{+}\phi_{P-1}))+1.2(^{+}S^{+}P_{+1})\sin(^{+}\phi_{S}-^{+}\phi_{P+1}))$
$\displaystyle+1.2(^{-}S^{-}P_{-1})\sin(^{-}\phi_{S}-^{-}\phi_{P-1}))+1.2(^{-}S^{-}P_{+1})\sin(^{-}\phi_{S}-^{-}\phi_{P+1}))$
$\displaystyle H^{0}(2,0)=$
$\displaystyle-0.4(^{+}P_{-1}^{+}P_{-1})+-0.4(^{+}P_{+1}^{+}P_{+1})+-0.4(^{-}P_{-1}^{-}P_{-1})$
$\displaystyle-0.4(^{-}P_{+1}^{-}P_{+1})+0.8(^{+}P_{0}^{+}P_{0})+0.8(^{-}P_{0}^{-}P_{0})$
$\displaystyle H^{1}(2,0)=$ $\displaystyle
0.8(^{+}P_{+1}^{+}P_{-1})\cos(^{+}\phi_{P+1}-^{+}\phi_{P-1}))-0.8(^{-}P_{+1}^{-}P_{-1})\cos(^{-}\phi_{P+1}-^{-}\phi_{P-1}))$
$\displaystyle+0.8(^{+}P_{0}^{+}P_{0})-0.8(^{-}P_{0}^{-}P_{0})$ $\displaystyle
H^{0}(2,1)=$
$\displaystyle-0.7(^{+}P_{0}^{+}P_{-1})\cos(^{+}\phi_{P0}-^{+}\phi_{P-1}))+0.7(^{+}P_{0}^{+}P_{+1})\cos(^{+}\phi_{P0}-^{+}\phi_{P+1}))$
$\displaystyle-0.7(^{-}P_{0}^{-}P_{-1})\cos(^{-}\phi_{P0}-^{-}\phi_{P-1}))+0.7(^{-}P_{0}^{-}P_{+1})\cos(^{-}\phi_{P0}-^{-}\phi_{P+1}))$
$\displaystyle H^{1}(2,1)=$
$\displaystyle-0.7(^{+}P_{0}^{+}P_{-1})\cos(^{+}\phi_{P0}-^{+}\phi_{P-1}))+0.7(^{+}P_{0}^{+}P_{+1})\cos(^{+}\phi_{P0}-^{+}\phi_{P+1}))$
$\displaystyle+0.7(^{-}P_{0}^{-}P_{-1})\cos(^{-}\phi_{P0}-^{-}\phi_{P-1}))-0.7(^{-}P_{0}^{-}P_{+1})\cos(^{-}\phi_{P0}-^{-}\phi_{P+1}))$
$\displaystyle H^{2}(2,1)=$
$\displaystyle-0.7(^{+}P_{0}^{+}P_{-1})\cos(^{+}\phi_{P0}-^{+}\phi_{P-1}))-0.7(^{+}P_{0}^{+}P_{+1})\cos(^{+}\phi_{P0}-^{+}\phi_{P+1}))$
$\displaystyle+0.7(^{-}P_{0}^{-}P_{-1})\cos(^{-}\phi_{P0}-^{-}\phi_{P-1}))+0.7(^{-}P_{0}^{-}P_{+1})\cos(^{-}\phi_{P0}-^{-}\phi_{P+1}))$
$\displaystyle H^{3}(2,1)=$ $\displaystyle
0.7(^{+}P_{0}^{+}P_{-1})\sin(^{+}\phi_{P0}-^{+}\phi_{P-1}))+0.7(^{+}P_{0}^{+}P_{+1})\sin(^{+}\phi_{P0}-^{+}\phi_{P+1}))$
$\displaystyle+0.7(^{-}P_{0}^{-}P_{-1})\sin(^{-}\phi_{P0}-^{-}\phi_{P-1}))+0.7(^{-}P_{0}^{-}P_{+1})\sin(^{-}\phi_{P0}-^{-}\phi_{P+1}))$
$\displaystyle H^{0}(2,2)=$
$\displaystyle-0.98(^{+}P_{+1}^{+}P_{-1})\cos(^{+}\phi_{P+1}-^{+}\phi_{P-1}))$
$\displaystyle-0.98(^{-}P_{+1}^{-}P_{-1})\cos(^{-}\phi_{P+1}-^{-}\phi_{P-1}))$
$\displaystyle H^{1}(2,2)=$ $\displaystyle
0.49(^{+}P_{-1}^{+}P_{-1})+0.49(^{+}P_{+1}^{+}P_{+1})$
$\displaystyle-0.49(^{-}P_{-1}^{-}P_{-1})-0.49(^{-}P_{+1}^{-}P_{+1})$
$\displaystyle H^{2}(2,2)=$ $\displaystyle
0.49(^{+}P_{-1}^{+}P_{-1})-0.49(^{+}P_{+1}^{+}P_{+1})$
$\displaystyle-0.49(^{-}P_{-1}^{-}P_{-1})+0.49(^{-}P_{+1}^{-}P_{+1})$
$\displaystyle H^{3}(2,2)=$ $\displaystyle
0.98(^{+}P_{+1}^{+}P_{-1})\sin(^{+}\phi_{P+1}-^{+}\phi_{P-1}))$
$\displaystyle+0.98(^{-}P_{+1}^{-}P_{-1})\sin(^{-}\phi_{P+1}-^{-}\phi_{P-1}))$

In this case the partial waves are defined by 14 real numbers (2
reflectivities of 1 S wave, 3 P waves ($m=+1,0,-1$), with 2 fixed phases).
While we have 18 moments equations, necessary for being over-constrained.
Hence we may leave $P^{\gamma}_{L}$ and $P^{\gamma}_{C}$ as free parameters in
our fits.

The results of our preliminary analysis are shown in Fig. 6 for the degrees of
polarization. In Figs. 10 and 11 we show the corresponding Partial Wave
Amplitudes extracted in the same fits. These results are well in line with
expectations from the GlueX $\rho$ SDME measurements with a dominant +ve
reflectivity and ${}^{+}P_{+}$ wave.

Figure 10: Magnitudes of partial waves extracted in preliminary fits to 2023
data. The top row shows +ve reflectivty amplitudes and bottom -ve. The data
distributions show MCMC samples.  Figure 11: Phases relative to the $P_{+}$
partial waves extracted in preliminary fits to 2023 data. The top row shows
+ve reflectivty amplitudes and bottom -ve. The data distributions show MCMC
samples. In the case of the -ve reflectivity amplitudes the phases have a
large uncertainty due to their small magnitudes.