# Polarized proton structure in the resonance region

A. N. Hiller Blin  Institute for Theoretical Physics, Regensburg University,
93040 Regensburg, Germany  Institute for Theoretical Physics, Tübingen
University, Auf der Morgenstelle 14, 72076 Tübingen, Germany  V. I. Mokeev
Jefferson Lab, Newport News, Virginia 23606, USA

###### Abstract

In view of the precise data available on inclusive polarized electron
scattering off polarized proton targets in the nucleon resonance excitation
region, we compare these results with the coherent sum of resonant
contributions to the polarized structure function $g_{1}$ and virtual photon
asymmetry $A_{1}$. To this goal, we employ the nucleon resonance
electroexcitation amplitudes determined for photon virtualities $Q^{2}$ $<$
5.0 GeV2 from analyses of the CLAS data on exclusive electroproduction off
protons in the resonance region. Most of the well established resonances of
four star PDG status in the mass range up to 1.75 GeV are included. We find
that the resonance-like structures observed in the inclusive $g_{1}$ data are
related to the resonant contributions in the entire range of photon virtuality
$Q^{2}$ where the data on $g_{1}$ are available. In the range of invariant
mass of the final hadron system $W$ $>$ 1.5 GeV, the data on the asymmetry
$A_{1}$ are well reproduced even when accounting for resonant contributions
only, especially for the larger values of $Q^{2}$ and energies analysed. This
observation offers an interesting hint to quark-hadron duality seen in
polarized inclusive electron scattering observables.

††preprint: JLAB-PHY-23-3773

##  I Introduction

Inclusive electron scattering off protons and the exploration of its
polarization observables offer an essential means to obtaining insights about
the ground proton structure Jimenez-Delgado:2013sma ; Gao:2017yyd ;
Ethier:2020way . The extension of these studies to the resonance region will
allow one to understand the proton structure at large values of the fractional
parton momentum $x$ in the resonance region and eventually to shed light on
the strong interaction dynamics which underlies the transition from the
strongly coupled to the perturbative QCD regimes, as well as the associated
characteristics of quark-hadron duality Melnitchouk:2005zr ;
Lagerquist:2022tml ; Bloom:1970xb ; Osipenko:2003bu ; Prok:2014ltt ;
Malace:2009kw ; Christy:2007ve ; Tvaskis:2016uxm ; Liang:2004tj .

There have been impressive advances in measuring inclusive scattering of
polarized electron beams off polarized nucleon targets E143:1998hbs ;
CLAS:2003rjt ; Bosted:2006gp ; CLAS:2006ozz ; RSS:2006tbm ; RSS:2008ceg ;
CLAS:2014qtg ; CLAS:2017qga , which open the path to duality studies in spin-
dependent observables Close:1972ah ; Carlson:1998gf ; Edelmann:1999yp ;
Melnitchouk:2005zr ; Lagerquist:2022tml . In order to improve the theory
approaches describing the connection between resonances and scaling
contributions, considerations need to be made about the role of the
nonresonant background. While such a quantitative description from first
principles is rather challenging, insight may be obtained from
phenomenological analyses.

The experimental program exploring exclusive $\pi^{+}n$, $\pi^{0}p$, $\eta p$,
and $\pi^{+}\pi^{-}p$ electroproduction channels in the resonance region with
the CLAS detector at Jefferson Lab has provided important new information on
the $\gamma^{*}pN^{*}$ electrocouplings of most nucleon resonances in the mass
range $W\leq 1.75$ GeV and for $Q^{2}\leq 5$ GeV2 Aznauryan:2011qj ;
Aznauryan:2009mx ; Mokeev:2012vsa ; Mokeev:2015lda ; Park:2014yea ;
Carman:2020qmb ; Mokeev:2020hhu ; Burkert:2022ioj . These results allow one to
quantitatively evaluate the coherent sum of resonant contributions to
inclusive electron scattering observables, using parameters of the individual
nucleon resonances extracted from data.

In our previous works Blin:2019fre ; Blin:2021twt ; HillerBlin:2022ltm , we
confronted polarized and unpolarized inclusive electron-scattering data with
the computation of resonant contributions in the resonance region. In the
present work, we include updated data on the polarized structure function
$g_{1}$ CLAS:2017qga  and the virtual photon asymmetry $A_{1}$ E143:1998hbs ;
CLAS:2003rjt ; Bosted:2006gp ; CLAS:2006ozz ; RSS:2006tbm ; RSS:2008ceg ;
CLAS:2014qtg ; CLAS:2017qga . In particular, the latter have extended the
coverage in $Q^{2}$ and $W$ in comparison with the data analyzed in our
previous work, therefore permitting more insightful conclusions about the
behavior of this observable in the resonance region.

In Sec. II we give a brief summary of the formalism, referring to our previous
work HillerBlin:2022ltm  for a detailed description. The results of our
computation compared with the available data are discussed in Sec. III. In
Sec. IV we summarize our findings and give an outlook of these studies.

##  II Formalism

The formalism used in the present work follows that thoroughly described in
our previous article HillerBlin:2022ltm .

In terms of cross sections, the virtual photon asymmetries are given by
roberts_1990 ; Dharmawardane:2004yw ; Melnitchouk:2005zr

$\displaystyle A_{1}$
$\displaystyle=\frac{\sigma_{T}^{1/2}-\sigma_{T}^{3/2}}{\sigma_{T}^{1/2}+\sigma_{T}^{3/2}},\qquad
A_{2}=\frac{\sigma_{I}}{\sigma_{T}},$ (1)

where $\sigma_{I}$ is the real part of the interference amplitude for virtual
photons with longitudinal and transverse polarizations. The structure
functions are then related to the virtual photon asymmetries via

$\displaystyle g_{1}$
$\displaystyle=\frac{1}{\rho^{2}}\,F_{1}\Big{(}A_{1}+A_{2}\sqrt{\rho^{2}-1}\Big{)},$
(2a) $\displaystyle g_{2}$
$\displaystyle=\frac{1}{\rho^{2}}\,F_{1}\Big{(}-A_{1}+\frac{A_{2}}{\sqrt{\rho^{2}-1}}\Big{)},$
(2b)

with the kinematic factor $\rho^{2}=1+Q^{2}/\nu^{2}$. Here, $-Q^{2}$ is the
4-momentum transfer squared between the electron and the proton, while $\nu$
is the virtual photon energy in the lab frame. It is related to the invariant
mass $W$ of the virtual photon–target proton system via
$\nu=(W^{2}-M^{2}+Q^{2})/2M$, where $M$ is the nucleon mass.

The coherent sum of contributions from the resonances $R$ to the inclusive
structure functions can be written as Carlson:1998gf ; Melnitchouk:2005zr

$\displaystyle\left(1+\frac{Q^{2}}{\nu^{2}}\right)g_{1}^{\text{res}}=$
$\displaystyle
M^{2}\sum_{IJ\eta}\Bigg{\\{}\bigg{|}\sum_{R^{IJ\eta}}G_{+}^{R^{IJ\eta}}\bigg{|}^{2}-\bigg{|}\sum_{R^{IJ\eta}}G_{-}^{R^{IJ\eta}}\bigg{|}^{2}$
$\displaystyle+\,\frac{\sqrt{2Q^{2}}}{\nu}\,\Re\bigg{[}\bigg{(}\sum_{R^{IJ\eta}}G_{0}^{{R^{IJ\eta}}}\bigg{)}\bigg{(}\sum_{R^{IJ\eta}}(-1)^{J_{R^{IJ\eta}}-\frac{1}{2}}\,\eta_{R^{IJ\eta}}G_{+}^{R^{IJ\eta}}\bigg{)}^{\ast}\bigg{]}\Bigg{\\}},$
(3a) $\displaystyle\left(1+\frac{Q^{2}}{\nu^{2}}\right)g_{2}^{\text{res}}=$
$\displaystyle-M^{2}\sum_{IJ\eta}\Bigg{\\{}\bigg{|}\sum_{R^{IJ\eta}}G_{+}^{R^{IJ\eta}}\bigg{|}^{2}-\bigg{|}\sum_{R^{IJ\eta}}G_{-}^{R^{IJ\eta}}\bigg{|}^{2}$
$\displaystyle-\,\frac{\nu\sqrt{2}}{\sqrt{Q^{2}}}\,\Re\bigg{[}\bigg{(}\sum_{R^{IJ\eta}}G_{0}^{{R^{IJ\eta}}}\bigg{)}\bigg{(}\sum_{R^{IJ\eta}}(-1)^{J_{R^{IJ\eta}}-\frac{1}{2}}\,\eta_{R^{IJ\eta}}G_{+}^{R^{IJ\eta}}\bigg{)}^{\ast}\bigg{]}\Bigg{\\}},$
(3b) for the spin-dependent structure functions, and $\displaystyle
F_{1}^{\text{res}}=$ $\displaystyle
M\sum_{IJ\eta}\Bigg{\\{}\bigg{|}\sum_{R^{IJ\eta}}G_{+}^{R^{IJ\eta}}\bigg{|}^{2}+\bigg{|}\sum_{R^{IJ\eta}}G_{-}^{R^{IJ\eta}}\bigg{|}^{2}\Bigg{\\}},$
(3c) $\displaystyle\left(1+\frac{\nu^{2}}{Q^{2}}\right)F_{2}^{\text{res}}=$
$\displaystyle
M\nu\sum_{IJ\eta}\Bigg{\\{}\bigg{|}\sum_{R^{IJ\eta}}G_{+}^{R^{IJ\eta}}\bigg{|}^{2}+\bigg{|}\sum_{R^{IJ\eta}}G_{-}^{R^{IJ\eta}}\bigg{|}^{2}+2\bigg{|}\sum_{R^{IJ\eta}}G_{0}^{R^{IJ\eta}}\bigg{|}^{2}\Bigg{\\}},$
(3d)

for the spin-averaged structure functions. The outer sums in Eqs. (3) run over
the possible values of the spin $J$, isospin $I$, and intrinsic parity $\eta$,
while the inner sums run over all resonances $R^{IJ\eta}$ that satisfy
$J_{R}=J$, $I_{R}=I$ and $\eta_{R}=\eta$ for the spin, isospin and parity of
the resonance $R$. The amplitudes $G^{R}_{+}$, $G^{R}_{-}$, and $G^{R}_{0}$
describe the contribution from the electroexcitation amplitudes of the
resonance $R$. They are related to the $\gamma^{*}pN^{*}$ electrocouplings
$A_{1/2}$, $A_{3/2}$, and $S_{1/2}$ as detailed in Ref. HillerBlin:2022ltm .
The the $\gamma^{*}pN^{*}$ electrocouplings have become available from the
studies of exclusive meson electroproduction data with the CLAS detector
within the mass range of $W$ $<$ 1.75 GeV and for $Q^{2}$ $<$ 5.0 GeV2
Carman:2020qmb ; Burkert:2022ioj ; Blin:2019fre ; Mokeev:2022xfo .

##  III Results

Figure 1: Proton $g_{1}$ structure function data CLAS:2017qga  (open black
squares): (a) $Q^{2}\approx 1.10$ GeV2, (b) $Q^{2}\approx 2.26$ GeV2, (c)
$Q^{2}\approx 3.18$ GeV2, (d) $Q^{2}\approx 4.51$ GeV2, compared to the
coherent (thick blue curves) and incoherent (thin blue curves) sum of
resonance contributions. The latter are computed at fixed $Q^{2}$
corresponding to the average value of the binned data in each panel. The
contributions from individual $N^{*}$ and $\Delta^{*}$ states are also shown
separately. The uncertainties for the resonant contributions are computed by
propagating the electrocoupling uncertainties via a bootstrap approach
Blin:2019fre .

In Fig. 1, we compare the experimental results on the $g_{1}$ structure
function measured with CLAS with the resonant contributions, computed by
employing resonance electroexcitation amplitudes deduced from exclusive CLAS
electroproduction data Carman:2020qmb ; Burkert:2022ioj ; Blin:2019fre ;
Mokeev:2022xfo . This is outlined in Sec. II CLAS:2017qga . We constrain
ourselves to the range of $W<1.8$ GeV and $Q^{2}<5$ GeV2 where the resonance
electrocouplings are currently available. Both the individual resonance
contributions, as well as the coherent and incoherent sums of resonances are
shown.

One can clearly see that the qualitative dips-and-peaks behavior in the $W$
dependence of the inclusive data is accounted for by the resonant
contributions, in all $Q^{2}$ bins. The dominant contribution in the first
resonance region is that of the $\Delta(1232)\,3/2^{+}$, which in turn is
driven by the $G_{-}$ amplitude (or by the $A_{3/2}$ electrocoupling).
According to Eq. (3), the contribution from the $G_{-}$ amplitude squared
enters $g_{1}$ with a minus sign. This explains the negative values seen both
in the $g_{1}$ data around that peak, as well as in the purely resonant
contributions. For most of the remaining states, it is the $G_{+}$ amplitude,
related to the $A_{1/2}$ resonance electrocouplings, that dominates
Blin:2019fre ; HillerBlin:2022ltm . For this reason, the total resonant
contributions to $g_{1}$ display a sign flip at $W$ values between the first
and second resonance peaks, as is also observed in the $W$-dependence of the
measured $g_{1}$ data CLAS:2017qga , as depicted in Fig. 1. Our analysis
confirms that the resonance contributions are the drivers of this behavior.

Figure 2:  $A_{1}$ asymmetry data from the E143 experiment at SLAC
E143:1998hbs  (green filled diamonds), the RSS collaboration RSS:2006tbm ;
RSS:2008ceg  (red filled triangles), as well as the CLAS experiments EG1a
(green open triangles) CLAS:2003rjt , EG1DVCS (red open triangles)
CLAS:2014qtg , and EG1b (black closed squares) CLAS:2017qga : (a)
$Q^{2}\approx 1.25$ GeV2, (b) $Q^{2}\approx 2.03$ GeV2, (c) $Q^{2}\approx
3.42$ GeV2, (d) $Q^{2}\approx 4.09$ GeV2, compared to the computed purely
resonant contributions (blue curves). The $1\sigma$ uncertainty bands of the
resonant contributions are computed by propagating the electrocoupling
uncertainties via a bootstrap approach Blin:2019fre .

In Fig. 2, we show the computed resonance contributions to the virtual photon
asymmetry $A_{1}$, compared to the data measured both with the large-
acceptance CLAS detector and with other detectors of smaller acceptance in the
resonance region E143:1998hbs ; CLAS:2003rjt ; Bosted:2006gp ; CLAS:2006ozz ;
RSS:2006tbm ; RSS:2008ceg ; CLAS:2014qtg ; CLAS:2017qga . Since the asymmetry
is defined by a cross-section ratio, the resonance structure becomes elusive.

Nevertheless, it is intriguing to find that, for $W$ $>$ 1.5 GeV, the $W$ and
$Q^{2}$ evolution of $A_{1}$ seen in the data is already rather well described
by the inclusion of resonance contributions only. This points to a particular
sensitivity of the $A_{1}$ observable to the resonance contributions at
$W>1.5$ GeV. Such a behavior offers a hint for quark-hadron duality seen in
this inclusive polarized electron scattering observable. This finding
motivates the ongoing and future studies of resonance electrocouplings with
the CLAS12 detector and a possible CEBAF energy increase up to 22 GeV
Achenbach:2023pba , in order to scrutinize whether this behavior holds for
even larger $Q^{2}$ values and for the higher-mass states.

In addition, the studies presented here can and have been extended to $g_{2}$
and $A_{2}$, therefore calling for future high-acceptance measurements of
these observables.

##  IV Summary and outlook

In these proceedings, we present the results on the exploration of the $W$ and
$Q^{2}$ dependence of the coherent and incoherent sums of nucleon resonance
contributions to the spin-dependent $g_{1}$ structure function and the $A_{1}$
virtual-photon asymmetry. These are evaluated from the experimental results on
$\gamma^{*}pN^{*}$ electrocouplings deduced from the analyses of exclusive
meson electroproduction data. As input, we used the electroexcitation
amplitudes extracted from CLAS data in the mass range up to $W=1.75$ GeV
Carman:2020qmb ; Burkert:2022ioj ; Blin:2019fre ; Mokeev:2022xfo .

Our findings provide evidence that the sign-flip behavior in the $g_{1}$ data
is accounted for by the resonance contributions. In addition, the results
point to a particular sensitivity of the $A_{1}$ observable to the resonant
contributions at $W$ $>$ 1.5 GeV. This calls for further measurements at
larger values of $Q^{2}$ and $W$, to investigate up to which QCD scales the
resonant states remain sizeable and relevant.

Further, the need to confirm the findings in this work for $g_{2}$ and $A_{2}$
gives clear motivation for future large-acceptance measurements of these
observables in experiments with polarized electron beams and for both
longitudinal and transverse target polarizations.

###### Acknowledgements.

We thank S. Kuhn, V. Lagerquist, W. Melnitchouk, and P. Pandey for useful
discussions and providing us with the experimental data shown here. This work
was supported by the U.S. Department of Energy contract DE-AC05-06OR23177,
under which Jefferson Science Associates, LLC operates Jefferson Lab, and by
the Deutsche Forschungsgemeinschaft (DFG) through the Research Unit FOR 2926
(project number 40824754).

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Vol.0 (20xx) No.0, 000–000

11institutetext: School of Physics and Astronomy, Yunnan University, Kunming
650500, China<EMAIL_ADDRESS><EMAIL_ADDRESS>
22institutetext: National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100101, China
Received 20XX Month Day; accepted 20XX Month Day

# New continuum observations of the Andromeda galaxy M 31 with FAST

Wenjun Zhang 11 Xiaohui Sun 11 Jie Wang 22

###### Abstract

We present a new total intensity image of M 31 at 1.248 GHz, observed with the
Five-hundred-meter Aperture Spherical radio telescope (FAST) with an angular
resolution of $4\arcmin$ and a sensitivity of about 16 mK. The new FAST image
clearly reveals weak emission outside the ring due to its high sensitivity on
large-scale structures. We derive a scale length of 2.7 kpc for the cosmic ray
electrons and find that the cosmic ray electrons propagate mainly through
diffusion by comparing the scale length at 4.8 GHz. The spectral index of the
total intensity varies along the ring, which can be attributed to the
variation of the spectra of synchrotron emission. This variation is likely
caused by the change of star formation rates along the ring. We find that the
azimuthal profile of the non-thermal emission can be interpreted by an
axisymmetric large-scale magnetic field with varying pitch angle along the
ring, indicating a complicated magnetic field configuration in M 31.

###### keywords:

galaxies: individual: M 31 – galaxies: ISM – radiation mechanisms: non-
thermal, thermal – ISM: cosmic rays – radio continuum: ISM

##  1 Introduction

The Andromeda galaxy (M 31) is a prominent spiral galaxy that is the nearest
to the Milky Way (Beck et al. 1998). Due to its proximity and outstanding
“ring” of star formation, M 31 has been widely studied in various bands, such
as X-ray (Pietsch et al. 2005), UV (Gil de Paz et al. 2007), near-IR (Jarrett
et al. 2003), far-IR (Rice et al. 1988), and optical (Sarkisyan et al. 2020).
These observations have improved our understanding of the formation and
evolution of M 31.

The radio total intensity of M 31 is concentrated in a ring about 7–13 kpc
from the center (Beck et al. 1998). In this ring, there are also
concentrations of cold molecular gas (Nieten et al. 2006), warm neutral gas
(Chemin et al. 2009), warm ionized gas (Devereux et al. 1994), and dust (Smith
et al. 2012). All these indicate that the M 31 ring is an important star-
forming region and contributes most of the brightness of the galaxy.

There have been many studies based on the radio observations of M 31. The
first radio observation of M 31 was conducted with the Cambridge one-mile
radio telescope at 408 and 1407 MHz with resolutions of $80\arcmin\times
120\arcmin$ and $23\arcmin\times 35\arcmin$, respectively (Pooley 1969). With
the continuing improvement of angular resolution and sensitivity, the follow-
up observations of M 31 reveals more detailed emission structures, such as the
observations by the Effelsberg 100-m telescope at 4850 MHz with a resolution
of $2\farcm 6$ (Berkhuijsen et al. 1983). In addition, the combination of
interferometric observations from the Very Large Array and the single-dish
observations from the Effelsberg radio telescope at 1.5 GHz yielded an image
with a high resolution of $45\arcsec$ covering all the spatial scales (Beck et
al. 1998).

The spectrum of M 31 has been studied based on the radio observations. The
radio emission consists of synchrotron emission (non-thermal) and free-free
emission (thermal). At frequencies below 10 GHz, the non-thermal emission is
dominant (Battistelli et al. 2019), and the fraction of thermal emission is
only about 24% at 1 GHz (Berkhuijsen et al. 2003). Based on the observations
at 1.46 GHz and 4.85 GHz, Berkhuijsen et al. (2003) obtained an overall non-
thermal spectral index of $\alpha=-1\pm 0.1$ ($S_{\nu}\propto\nu^{\alpha}$,
with $S_{\nu}$ being the flux density at frequency $\nu$). Beck et al. (2020)
derived a total intensity spectral index of $-0.71\pm 0.02$, and a synchrotron
emission spectral index of $-0.81\pm 0.03$ using the observations at 2.645
GHz, 4.85 GHz, and 8.35 GHz. The spectral index towards the ring increases to
$-$0.5 because the thermal fraction is large and the spectrum thus tends to be
flat (Fatigoni et al. 2021).

The Five-hundred-meter Aperture Spherical radio Telescope (FAST) is well
suited to image extended sources such as M 31 with its high sensitivity (Jiang
et al. 2020). The observations of M 31 Halo is one of the key projects
currently being conducted by FAST with guaranteed time (PI: Jie Wang), which
focuses on HI mapping in the region of about 800 square degree around the M 31
and pulsar searching in the M 31 galaxy. The continuum and polarization data
have also been recorded simultaneously. This allows us to analyse the total
intensity distribution and the spatial variation of spectral index to infer
the properties of M 31.

The paper is organized as the follows: the observations and results are
described in Sect. 2, and the results are discussed in Sect. 3 , and
summarized in Sect. 4.

##  2 Observations and Results

M 31 was observed in drifting-scan mode (Jiang et al. 2020). In this mode, the
19-beam receiver is rotated by $23\fdg 4$ so that a declination strip of about
$22\arcmin$ can be completed in Nyquist sampling for each individual scan. In
total, 11 drifting scans were conducted during October 1 – 18, 2020. In order
to calibrate the system variations, a linearly polarized noise of about 12.5 K
(Jiang et al. 2019) is injected into the system for 2 seconds every 1000
seconds. The frequency range of the receiving system is 1.0 – 1. 5 GHz, which
is divided into 65536 channels. The width of each channel is about 7.63 kHz.
The backend outputs four channels with two for total intensity $I_{1}$ and
$I_{2}$ , and the other two for Stokes $U$ and $V$.

We follow the data processing procedure developed by Sun et al. (2021). To
obtain the images at a single frequency channel, the following steps were
taken: (1) mitigation of radio frequency interference (RFI); (2) correction of
instrumental polarization leakage and angle, and temperature scale with
injected noise; (3) combination of all the 11 scans; (4) conversion to the
brightness temperature scale ($T_{\rm b}$). We repeat this procedure and
obtain the maps of about 29,000 frequency channels. These maps are then
smoothed to a common resolution of $4\arcmin$.

Figure 1: Total intensity ($I$) image of M 31 at 1.248 GHz from FAST. The
resolution is $4\arcmin$ and the rms noise is 16 mK.

###  2.1 Total intensity maps

Figure 2: Total intensity ($I$) image of M 31 from FAST (top panel), from the
combination of VLA and Effelsberg 100 m telescope (middle panel, Beck et al.
(1998)), and from FAST with point sources marked in the upper panel removed
(bottom panel ). The galaxy is rotated by $-53\degr$.

We average all the frequency channel maps by taking the median values and
obtained the total intensity image at 1.248 GHz, shown in Fig. 1. Some
scanning effects indicated by the stripes along declination direction still
remain in the image, which were caused by drifting of the system. It would
require scans in alternative directions to remove these effects, as shown by
Sun et al. (2022). The rms noise is about 16 mK.

A smaller area containing M 31 is cut out from Fig. 1, rotated by $53\degr$
following Beck et al. (1998) and shown in Fig. 2 (top panel). For comparison,
the map by Beck et al. (1998) combining the interferometer and single-dish
observations at 1.46 GHz is also shown in Fig. 2 (middle panel).

As can be seen from Figs. 1 and 2, there is strong radio emission towards the
center and the ring area. The FAST image clearly reveals weak emission
extending outside the ring towards the southwest. In contrast, this extended
emission is almost absent in the image by Beck et al. (1998).

To study the emission from M 31, we remove the bright compact sources marked
in Fig. 2 (top panel) by subtracting Gaussian fittings of these sources. The
resulted image is shown in Fig. 2 (bottom panel), which is used to investigate
the variations of total intensity below. Note that there still remain
background point sources that cannot be resolved with FAST. We retrieve the
flux intensities of these sources from the NVSS survey with a higher
resolution of 45$\arcsec$ (Condon et al. 1998), and find that their
contribution to the intensity of the ring is less than about 5%. Their
influence to the analyses below can thus be ignored.

Figure 3: Intensity $I$ with the angular distance from the galactic center.
The red line indicates a Gaussian fitting with a scale length of 3.7 kpc after
correcting the influence of beam width.

###  2.2 Radial profile of the total intensity

The radial variation of the total intensity can be used to constrain the
propagation of cosmic ray electrons. We derive the average intensity of
$6\arcmin$-width annuli from the center till the radius of about $110\arcmin$.
The intensity versus radius is shown in Fig. 3. The standard deviation from
each annulus is calculated as the error bar.

The sharp peak near radius 0 is contributed by the nucleus of M 31. The broad
peak from the radius of about $27\arcmin$ to the radius of about $57\arcmin$
indicates the ring area with bright emission. With a distance to M 31 of about
780 kpc (Stanek & Garnavich 1998), this angular range corresponds to about 6 –
13 kpc. The total intensity gradually decreases with increasing radius and
flattens out beyond the radius of about $93\arcmin$.

We fit the total intensity radial profile for radii larger than $27\arcmin$ to
a Gaussian, namely $T\propto\exp[-(r-r_{0})^{2}/2L^{2}]$, where $r$ is the
radius, $r_{0}$ is the peak radius, and $L$ is the scale length. After
accounting for the influence of beam width, we obtain the scale length of
about $16\farcm 2\pm 0\farcm 3$ or $3.7\pm 0.1$ kpc.

###  2.3 Azimuthal profile of the total intensity

The variation of total intensity versus direction reflects the structure of
magnetic field. From the Gaussian fitting of the radial profile, we obtain the
peak position of the ring at $r_{0}=40\farcm 9$. We use the annulus of
$r_{0}\pm L$ to derive the azimuthal profile of total intensity in the ring.
The azimuthal angle starts from the major axis towards the north east and
increases eastwards (Fig. 1) or starts from the major axis towards the left
and increases counter-clockwise (Fig. 2). An average intensity is derived for
every $10\degr$ sector, and the profile is shown in Fig. 4.

The emission is the brightest for the azimuthal angle between $30\degr$ and
$90\degr$, and becomes weaker for the azimuthal angle between $150\degr$ and
$270\degr$, as can also be seen from Fig. 2. There is no clear pattern for the
azimuthal profile.

Figure 4: The total intensity in the ring of M 31 versus azimuthal angle.

###  2.4 Spectral index

The spectral index is related with the emission properties, such as non-
thermal and thermal fraction and cosmic ray electrons aging. We do not derive
the spectral index for all the pixels by fitting the brightness temperature
against frequency because of the influence of background emission. Instead, we
divide the whole M 31 area into 8 regions (Fig. 5) and obtain the spectral
index using Temperature-Temperature plot (TT-plot) method (Turtle et al. 1962)
which is immune to the background emission.

We take the median values of the first 10,000 and the last 10,000 channels to
calculate the spectral index with TT-plot. The corresponding frequencies are
1.055 GHz and 1.420 GHz, which has the largest frequency separation and hence
allows for a more accurate determination of spectral index. The TT-plot
results for the 7 regions are shown in Figs. 5 and 6. Note that the brightness
temperature spectral index $\beta$, as we obtain, can be connected to the flux
density spectral index $\alpha$ as $\alpha=\beta+2$.

The spectral index for the total intensity is not uniform for M 31. Towards
the areas marked as 6, 1, 2 in Fig. 5 where the total intensity emission is
weak, the spectra are steep with spectral index $\alpha$ between $-$0.8 and
$-0.7$. In contrast, the areas marked as 3, 4, 5 in Fig. 5 where the emission
is stronger, the spectra are shallower with spectral index $\alpha$ up to
$-0.44\pm 0.09$. There seems a tendency of larger intensity with flatter
spectrum. The nucleus region marked as 7 has the smallest spectral index of
$-0.90\pm 0.03$. The area outside M 31 marked as 8, detected because of the
high sensitivity of FAST, has a spectral index of $\alpha=-0.64\pm 0.08$, and
has not been well studied before.

Figure 5: Area marked for deriving spectral index with TT-plot method and the
resulted flux density spectral indices $\alpha=\beta+2$.

Figure 6: TT-plots for the areas marked in Fig. 5. The resulted brightness
temperature spectral indices $\beta$ are also shown.

##  3 Discussion

###  3.1 Properties of cosmic ray electrons

The total intensity is composed of both thermal emission from free-free
radiation and non-thermal emission from synchrotron radiation. We take the
images of thermal and non-thermal emission derived by Tabatabaei et al. (2013)
and obtain the thermal fraction versus the azimuthal angle for the ring of M
31, shown in Fig. 7. The thermal fraction is up to 20% for the azimuthal angle
between $30\degr$ and $90\degr$, and is about 10% for the rest. The thermal
emission has a flat spectrum with $\alpha_{\rm th}\approx-0.1$ which is much
larger than that of non-thermal emission. Therefore the mixture of thermal and
non-thermal emission will bring the spectral index of the total emission to a
larger value. Typically, for a thermal fraction of 10% and 20%, the spectral
index of total intensity calculated from the frequency pair of 1.055 and 1.420
GHz is increased by 0.1 and 0.2 in comparison with the spectral index of non-
thermal emission $\alpha_{\rm n}$, respectively.

For the emission ring of M 31, the spectra for total intensity towards the
north part in Fig. 1 or the regions 3, 4, 5 on the left part in Fig. 5 are
shallower than those towards the south part. Taking into account the influence
of thermal emission, the spectral index of the non-thermal emission could be
estimated by subtracting 0.1 or 0.2 from the total intensity spectral index
depending on the thermal fraction. For example, the non-thermal spectral index
could reach $\alpha_{\rm n}\approx-0.64$ for region 5 with thermal fraction of
about 20%. This implies that the non-thermal spectral index towards the
northern part is larger than that towards the southern part of M 31.

The asymmetry of the non-thermal spectral index between the northern and
southern part of M 31 was also found by Fatigoni et al. (2021) based on the
observations at frequencies between 1.5 and 6.6 GHz. The star formation rate
is also higher towards the north (Fatigoni et al. 2021). A higher star
formation rate results in a larger number of stars and consequently a larger
number of supernovae and supernova remnants that can produce more cosmic ray
electrons. Therefore the spectrum is flatter towards the north. Moreover, the
presence of higher cosmic ray electron density results in stronger synchrotron
radiation, which explains the correlation between large intensity and large
spectral index.

For area 8 in Fig. 5, it cannot be the extension from the ring. Otherwise its
spectrum should be steeper because of the aging of cosmic ray electrons as
they propagate from the ring. From the Herschel image at 250 $\mu$m (Viaene et
al. 2014), there is strong emission corresponding to this area, which seems to
be part of a large ring structure surrounding M 31. There could be star
forming activities that produce high energy electrons. High resolution
observations are needed to detect radio emission related with the infrared
emission observed by Herschel.

Figure 7: Thermal fraction obtained from Tabatabaei et al. (2013) versus the
azimuthal angle.

The scale length of the total emission is 3.7 kpc. The thermal emission is
highly concentrated in the ring (Tabatabaei et al. 2013), and therefore has
little influence on the scale length of non-thermal emission. In practice, we
derive a scale length of 3.8 kpc for the non-thermal emission after removing
the contribution of thermal emission. This corresponds to an exponential scale
length of $\sqrt{2}\times 3.8\approx 5.4$ kpc. Assuming an energy
equipartition between magnetic field and cosmic rays, the scale length of the
non-thermal emission is equal to that of cosmic ray electrons multiplied by
the factor $(3-\alpha_{\rm n})/2\approx 2$ (Berkhuijsen et al. 2013). The
scale length of cosmic ray electrons is thus about 2.7 kpc.

The diffusion length of cosmic ray electrons varies with frequency as
$\nu^{-\frac{1}{4}}$ (Mulcahy et al. 2016). It is therefore expected that the
scale length of cosmic ray electrons is about 1.9 kpc at 4.8 GHz, which is
consistent with the scale length of about 3.7 kpc for non-thermal emission and
hence the scale length of about 1.9 kpc for cosmic ray electrons derived by
Beck et al. (2020) at 4.8 GHz. This implies that cosmic ray transports
primarily by diffusion and the energy loss is not significant at 4.8 GHz.

###  3.2 The magnetic field in the ring

We obtain the azimuthal profile of the non-thermal emission within the bright
ring after subtracting the thermal emission (Fig. 7) from the total emission
(Fig. 4). The result is shown in Fig. 8. The non-thermal emission is from
synchrotron radiation and its profile thus reflects the magnetic field
configuration in M 31.

Figure 8: The azimuthal profile of the non-thermal emission within the ring.
The lines indicate the profiles expected from an axisymmetric magnetic field
with a pitch angle of $-15\degr$ and $30\degr$, respectively.

The large-scale regular magnetic field in M 31 mainly follows an axisymmetric
pattern (Han et al. 1998; Fletcher et al. 2004). In this configuration, the
magnetic field perpendicular to the line of sight can be derived as,

$B_{\perp}=B\sqrt{1-\cos^{2}(\phi-\xi)\sin^{2}i}$ (1)

where $\phi$ is the azimuth angle, $\xi$ is the pitch angle, and $i$ is the
inclination angle of M 31. The non-thermal intensity ($I$) can be assessed as
(Beck & Krause 2005),

$I\propto B_{\perp}^{3-\alpha_{\rm n}}$ (2)

Here the energy equipartition between magnetic field and cosmic rays is
assumed.

The random magnetic field has recently been classified into ordered field and
isotropic field to interpret the polarization observations of M 31 (Beck et
al. 2020). The ordered field can be generated from shear of differential
rotation or compression of spiral arm shocks (Jaffe 2019). However, there are
no indications of such large-scale structures inside the ring from radio
observations. We therefore only consider the isotropic random magnetic fields.
Since the path length along different line of sight towards the ring is
similar, the synchrotron emission from the random fields is independent on the
azimuthal angle. The azimuthal variation of non-thermal emission is thus
caused by the large-scale regular field.

The intensity of synchrotron radiation can be derived using Eqs 1 and 2. The
inclination angle $i=75\degr$ (Chemin et al. 2009) is used. The pitch angle is
uncertain. Berkhuijsen et al. (2003) obtained a pitch angle of
$\xi=-13\degr\pm 7\degr$ by analyzing polarization observations at 6 cm.
Fletcher et al. (2004) found a radial change of the pitch angle from
$-17\degr$ to $-8\degr$ based on multi-frequency polarization observations.
Whereas, the recent analysis by Beck et al. (2020) suggested a large variation
of the pitch angle versus azimuthal angle for the total magnetic field
including both regular and ordered field.

Assuming a constant pitch angle for M 31, we would expect the separation of
the two intensity peaks to be $180\degr$, which is clearly inconsistent with
the observed profile shown in Fig. 8. This implies variations of the pitch
angle. We thus propose a simple model of the large scale magnetic field with a
pitch angle of $-15\degr$ for the azimuthal angle range of $0\degr$ to
$190\degr$, and a pitch angle of $30\degr$ for the azimuthal angle range of
$190\degr$ to $360\degr$. Additionally, we impose a factor of 0.95 for the
ratio of the two peaks, which we attribute to the variation of cosmic ray
electron density. The resulted profile is presented in Fig. 8, which
reasonably reproduces the observed profile. A more sophisticated model of
magnetic field by requires both total intensity and polarization observations
at multiple frequencies, and is beyond the scope of this paper.

##  4 Summary

We obtain the total intensity images from the FAST legacy survey of M 31,
covering the frequency range of 1.0 - 1.5 GHz. By averaging over all the
frequency channels, we derive the total intensity image at 1.248 GHz, which
reveals weak extended emission outside the ring because of the high
sensitivity. The image allows us to analyze the propagation of cosmic ray
electrons and the magnetic field configuration.

The main conclusions are as follows:

  1. 1.

We derive a scale length of about 2.7 kpc for the cosmic ray electrons by
fitting the radial profile of non-thermal emission. In comparision with the
scale length at higher frequencies, we find that the cosmic ray electrons
propagate mainly through diffusion.

  2. 2.

The spectral index of the total intensity in the ring, calculated from TT-
plots, clearly exhibits an azimuthal variation, which can be attributed to the
spectral variation of synchrotron radiation. The variation is likely caused by
the change of star formation rates along the ring.

  3. 3.

The weak emission detected outside the ring might not be the extended emission
from the ring. Instead, it might be produced by star forming activities.

  4. 4.

The azimuthal profile of non-thermal emission can be reproduced with an
axisymmetric large-scale magnetic field pattern with pitch angle varying with
azimuthal angle, indicating that the magnetic field configuration in the ring
is complicated.

###### Acknowledgements.

W. Z. and X. S. are supported by the National SKA Program of China (Grant No.
2022SKA0120101). We thank the anonymous referee for the comments that help
improve the paper.

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