# Improved limits on the tensor-to-scalar ratio using BICEP and Planck

M. Tristram  Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France
A. J. Banday  IRAP, Université de Toulouse, CNRS, CNES, UPS, (Toulouse),
France  K. M. Górski  Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Drive, Pasadena, California, U.S.A.  Warsaw
University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland  R.
Keskitalo  Computational Cosmology Center, Lawrence Berkeley National
Laboratory, Berkeley, California, U.S.A.  Department of Physics and Space
Sciences Laboratory, University of California, Berkeley, California, U.S.A.
C. R. Lawrence  Jet Propulsion Laboratory, California Institute of Technology,
4800 Oak Grove Drive, Pasadena, California, U.S.A.  K. J. Andersen  Institute
of Theoretical Astrophysics, University of Oslo, Blindern, Oslo, Norway  R. B.
Barreiro  Instituto de Física de Cantabria (CSIC-Universidad de Cantabria),
Avda. de los Castros s/n, Santander, Spain  J. Borrill  Computational
Cosmology Center, Lawrence Berkeley National Laboratory, Berkeley, California,
U.S.A.  Space Sciences Laboratory, University of California, Berkeley,
California, U.S.A.  L. P. L. Colombo  Dipartimento di Fisica, Università degli
Studi di Milano, Via Celoria, 16, Milano, Italy  H. K. Eriksen  Institute of
Theoretical Astrophysics, University of Oslo, Blindern, Oslo, Norway  R.
Fernandez-Cobos  Dpto. de Matemàticas, Estadística y Computación, Universidad
de Cantabria, Avda. de los Castros s/n, E-39005 Santander, Spain.  T. S.
Kisner  Computational Cosmology Center, Lawrence Berkeley National Laboratory,
Berkeley, California, U.S.A.  Department of Physics and Space Sciences
Laboratory, University of California, Berkeley, California, U.S.A.  E.
Martínez-González  Instituto de Física de Cantabria (CSIC-Universidad de
Cantabria), Avda. de los Castros s/n, Santander, Spain  B. Partridge
Haverford College Astronomy Department, 370 Lancaster Avenue, Haverford,
Pennsylvania, U.S.A.  D. Scott  Department of Physics & Astronomy, University
of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia,
Canada  T. L. Svalheim  Institute of Theoretical Astrophysics, University of
Oslo, Blindern, Oslo, Norway  I. K. Wehus  Institute of Theoretical
Astrophysics, University of Oslo, Blindern, Oslo, Norway

###### Abstract

We present constraints on the tensor-to-scalar ratio $r$ using a combination
of BICEP/Keck 2018 (BK18) and Planck PR4 data allowing us to fit for $r$
consistently with the six parameters of the $\rm{\Lambda CDM}$ model. We
discuss the sensitivity of constraints on $r$ to uncertainties in the
$\rm{\Lambda CDM}$ parameters as defined by the Planck data. In particular, we
are able to derive a constraint on the reionization optical depth $\tau$ and
thus propagate its uncertainty into the posterior distribution for $r$. While
Planck sensitivity to $r$ is slightly lower than the current ground-based
measurements, the combination of Planck with BK18 and baryon-acoustic-
oscillation data yields results consistent with $r=0$ and tightens the
constraint to $r<0.032$ at 95% confidence.

##  I Introduction

Introduced in order to resolve problems within the Big-Bang cosmological model
(such as the horizon, flatness, and magnetic-monopole problems), inflation
also naturally provides the seeds for generating primordial matter
fluctuations from quantum fluctuations (see for instance Ref. [1] and
references herein).

Measurements of the cosmic microwave background (CMB) allow constraints to be
placed on the amplitude of the tensor perturbations that are predicted to be
generated by primordial gravitational waves during the inflationary epoch,
leaving some imprints on the CMB anisotropies [2, 3, 4, 5]. Over the last
decade, while no primordial signals have been discovered, significant
improvements on the upper limit for the tensor-to-scalar ratio $r$ have
progressively led to the constraint becoming lower than a few percent in
amplitude: $r<0.11$ in 2013 using only temperature data from Planck [6];
$r<0.12$ in 2015 using polarization from BICEP/Keck and Planck [7] to debias
the initially claimed detection from BICEP/Keck in 2014,
$r=0.2^{+0.07}_{-0.05}$ [8]; $r<0.09$ in 2016 using BICEP/Keck and Planck [9];
$r<0.07$ in 2018 using BICEP/Keck 2015 data (BK15, [10]); $r<0.065$ in 2019
using Planck in combination with BK15 [11]; $r<0.044$ in 2021 using Planck in
combination with BK15 [12]; and $r<0.036$ in 2021 using the latest BICEP/Keck
data (BK18, [13]).

Figure 1: History of constraints on the tensor-to-scalar ratio $r$ (Planck PR1
[6], Planck PR1+BK [7], Planck PR2+BK [9], BK15 [10], Planck PR3+BK15 [11],
Planck PR4 [12], Planck PR4+BK15 [12], BK18 [13], Planck PR4+BK18 this work).
Upper limits are given at 95 % CL.

In this paper, we first discuss the impact of uncertainties in $\rm{\Lambda
CDM}$ parameters for constraints on $r$ derived from the latest BICEP/Keck
data (BK18) [13] alone. Then we add in data from the latest Planck release
(PR4) [14] in order to provide the best currently available constraint on the
tensor-to-scalar ratio $r$.

##  II Cosmological model

The cosmological model used in this paper is based on adiabatic, nearly scale-
invariant perturbations. It has been established as the simplest model that is
consistent with the different cosmological probes and in particular with the
CMB [11].

The standard $\rm{\Lambda CDM}$+r model includes 6+1 parameters. Power spectra
for scalar and tensor modes are parameterized by power laws with no running
and so the parameters include the scalar amplitude $A_{\rm s}$ and the
spectral scalar index $n_{\rm s}$, while the spectral index for the tensor
mode $n_{\rm t}$ is set using single-field slow-roll inflation consistency.
The amplitudes and the tensor-to-scalar power ratio, $r\equiv A_{\rm t}/A_{\rm
s}$, are evaluated at a pivot scale of 0.05 Mpc-1. Three other parameters
($\Omega_{\rm b}h^{2}$, $\Omega_{\rm c}h^{2}$, and $\theta_{\ast}$) determine
the linear evolution of perturbations after they re-enter the Hubble scale.
Finally, the reionization is modeled with a widely-used step-like transition
between an essentially vanishing ionized fraction at early times, to a value
of unity at low redshifts. The transition is modeled using a $\tanh$ function
with a non-zero width fixed to $\Delta z=0.5$ [15]. The reionization optical
depth $\tau$ is then directly related to the redshift at which this transition
occurs.

The CMB power spectra are generated using the Boltzmann-solver code camb [16,
17]. We sample the likelihood combinations using the cobaya framework [18]
with fast and efficient Markov chain Monte Carlo sampling methods described in
Refs. [19] and [20]. All the likelihoods that we use are publicly available on
the cobaya web site111cobaya.readthedocs.io and are briefly described in the
next section.

##  III Planck likelihoods

We use the polarized likelihood at large scales, lowlEB, described in Ref.
[12] and available on github.222github.com/planck-npipe Specifically, it is a
Planck low-$\ell$ polarization likelihood based on cross-spectra using the
Hamimeche-Lewis approximation [21, 22]. Using this formalism, the likelihood
function consistently takes into account the two polarization fields $E$ and
$B$ (including $EE$, $BB$, and $EB$ power-spectra), as well as all
correlations between multipoles and modes. It is important to appreciate that
such correlations are relevant at large angular scales where cut-sky effects
and systematic residuals (both from the instrument and from the foregrounds)
are important. The cross-spectra are calculated on component-separated CMB
“detset” maps processed by commander from the Planck PR4 frequency maps, on 50
% of the sky. The sky fraction is optimized in order to obtain maximum
sensitivity (and lowest sample variance), while ensuring low contamination
from residual foregrounds. The covariance matrix is estimated from the PR4
Monte Carlos. The statistical distribution of the recovered $C_{\ell}$s
naturally includes the effect of all components included in the Monte Carlo,
namely the CMB signal, instrumental noise, Planck systematic effects
incorporated in the PR4 simulations (see Ref. [14]), component-separation
uncertainties, and foreground residuals. In the case of Planck, we are not
able to analytically predict the shape of the full covariance matrix for
component-separated maps. However, analytical predictions exist for the
covariance of instrumental noise in low-resolution individual-frequency maps.
Analysis of these matrices highlights non-trivial structures in the harmonic
space noise, whose covariance cannot be approximated as diagonal. Since
component-separated maps are a combination of the input frequency maps, part
of these structures will carry over into the final covariance matrix, on top
of any additional correlations induced by systematic effects, masking, and
foreground residuals that cannot be modeled analytically but only
reconstructed via simulations. Given these considerations, we cannot apply any
type of simplifying “conditioning” (such as setting off-diagonal elements to
zero), as done for some ground-based experiments, nor do we wish to make such
assumptions about the data. In this paper, unlike previous CMB work to our
knowledge, we now marginalize the likelihood over the unknown true covariance
matrix (as proposed in Ref. [23]) in order to propagate the uncertainty in the
estimation of the covariance matrix caused by a limited number of simulations.
This provides us with a likelihood that is unbiased and accurate for the
estimation of the uncertainty. The robustness of the results is discussed
further in the Appendix.

At large angular scales in temperature, we make use of the Planck public
low-$\ell$ temperature-only likelihood, based on the CMB map recovered from
the component-separation procedure (specifically commander) described in
detail in Ref. [24].

At small scales, we use the Planck HiLLiPoP likelihood, which can include the
$TT$, $TE$, and/or $EE$ power spectra computed on the PR4 detset maps at 100,
143, and 217 GHz. The likelihood is a spectrum-based Gaussian approximation,
with semi-analytic estimates of the $C_{\ell}$ covariance matrix based on the
data. The model consists of a linear combination of the CMB power spectrum and
several foreground residuals, including Galactic dust, cosmic infrared
background, thermal and kinetic Sunyaev-Zeldovich (SZ) effects, SZ-CIB
correlations, and unresolved point sources. For details, see Refs. [12] and
[25, 26, 27].

##  IV BICEP/Keck likelihood

We use the publicly available BICEP/Keck likelihood (BK18) corresponding to
the data taken by the BICEP2, Keck Array, and BICEP3 CMB polarization
experiments, up to and including the 2018 observing season [13]. The format of
the likelihood is identical to the one introduced in Refs. [7] and [10]; it is
a Hamimeche-Lewis approximation [21] to the joint likelihood of the ensemble
of $BB$ auto- and cross-spectra taken between the BICEP/Keck (two at 95, one
each at 150 and 220 GHz), WMAP (23 and 33 GHz), and Planck (PR4 at 30, 44,
143, 217, and 353 GHz) frequency maps. The effective coverage is approximately
$400\,{\rm deg}^{2}$ (which corresponds to 1 % of the sky) centered on a
region with low foreground emission. The data model includes Galactic dust and
synchrotron emission, as well as correlations between dust and synchrotron.

In the following, we neglect correlations between the BICEP/Keck and Planck
data sets. This is justified because the BK18 spectra are estimated on 1 % of
the sky, while the Planck analysis is derived from 50 % of the sky.

##  V Impact of $\mathbf{\Lambda}$CDM uncertainty

In the baseline analysis described in Ref. [13], the BICEP/Keck Collaboration
fixed the cosmology to that of best-fitting model from Planck 2018 and quote
an upper limit of $r<0.036$ at 95% CL. Within this baseline, the uncertainty
on $\rm{\Lambda CDM}$ parameters was not propagated, reducing the width of the
posterior for the tensor-to-scalar ratio $r$. We find that when fitting the
BK18 data for $\rm{\Lambda CDM}$ parameters in addition to $r$, the
uncertainty on $r$ slightly increases (as illustrated in Fig. 2) because the
$\rm{\Lambda CDM}$ parameters except for $A_{\rm s}$ are poorly constrained.

Figure 2: Posterior distribution for the tensor-to-scalar ratio $r$, showing
the impact of marginalization over $\rm{\Lambda CDM}$ parameters.

The constraints on $r$ then become

$\displaystyle r=0.014_{-0.011}^{+0.012}$ $\displaystyle\quad\text{(BK18 with
{$\rm{\Lambda CDM}$}\ fixed)},$ (1) $\displaystyle r=0.015_{-0.013}^{+0.015}$
$\displaystyle\quad\text{(BK18 with {$\rm{\Lambda CDM}$}\ free)},$ (2)

all compatible with zero333Uncertainties in Eq. 1 are slightly larger than
those in Ref. [13], despite using the same likelihood. This small difference
could be due to assuming different values for the reference $\rm{\Lambda CDM}$
model parameters (we used Planck2018 TT,TE,EE+lowE+lensing [11]), or might
arise from using different MCMC/Boltzmann solver codes. and resulting in the
following upper-limits at 95 % CL:

$\displaystyle r<0.036$ $\displaystyle\quad\text{(BK18 with {$\rm{\Lambda
CDM}$}\ fixed)};$ (3) $\displaystyle r<0.042$ $\displaystyle\quad\text{(BK18
with {$\rm{\Lambda CDM}$}\ free)}.$ (4)

##  VI Combining Planck and BICEP/Keck

The addition of Planck data allows us to constrain $\rm{\Lambda CDM}$
parameters, thus reducing the uncertainty on $r$. This was mentioned in a
secondary analysis of Ref. [13] (their appendix E.1), yielding an upper limit
on $r$ similar to that of the baseline when using the earlier Planck PR3 data.
In this paper, we update the Planck data to PR4 and add constraints from the
polarized low-$\ell$ likelihood.

With the new BICEP/Keck data set included, the uncertainty on $r$ has
decreased to $\sigma(r)=0.014$. We may compare this to the Planck uncertainty
$\sigma(r)=0.056$ based on polarized low multipoles; this uncertainty reduces
to $\sigma(r)=0.024$ when the $TT+TE+EE$ high multipole data are included as
well. The addition of low-$\ell$ from Planck polarization modes allows the
degeneracy with $\tau$ to be broken and also slightly reduces the width of the
posterior distribution for $r$. This is illustrated in Fig. 3.

Figure 3: Posterior distributions for $\tau$, $A_{\rm s}$, and $r$ using BK18
[13], Planck [12], and the combination of the two.

The resulting constraint on $r$ using a combination of Planck and BK18 data
tightens to

$\displaystyle r=0.014_{-0.009}^{+0.011}$
$\displaystyle\quad\text{(Planck+BK18)},$ (5)

which corresponds to $r<0.034$ at 95 % CL. The reionization optical depth is
found to be

$\displaystyle\tau=0.057\pm 0.007.$ (6)

The combination of the two data sets allows us to cover the full range of
multipoles that are most sensitive to tensor modes. In combination with baryon
acoustic oscillation (BAO [28]) and CMB lensing [29] data, we obtain an
improved upper limit of

$\displaystyle r<0.032\quad\text{(95\% CL).}$ (7)

In the $n_{\rm s}$–$r$ plane (Fig. 4), the constraints now rule out the
expected potentials for single-field inflation (strongly excluding
$V\propto\phi^{2}$, $\phi$, and even $\phi^{2/3}$ at about 5$\,\sigma$).

Figure 4:  Constraints in the tensor-to-scalar ratio $r$ versus scalar
spectral index $n_{\rm s}$ plane for the $\rm{\Lambda CDM}$ model, using CMB
data in combination with baryon acoustic oscillation (BAO) and CMB lensing
data. The CMB data are Planck PR3 (TT,TE,EE+lowE, gray contour), Planck PR4
[12] (TT,TE,EE+lowlEB, green contour), and Planck PR4 joint with BK18 [13]
(blue contour, this paper). These constraints assume the inflationary
consistency relation and negligible running. Dotted lines show the loci of
approximately constant e-folding number $50<N<60$, assuming simple
$V\propto(\phi/m_{\rm Pl})^{p}$ single-field inflation. Solid lines show the
approximate $n_{\rm s}$–$r$ relation for locally power-law potentials, to
first order in slow roll. The solid black line (corresponding to a linear
potential) separates concave and convex potentials. This plot is adapted from
figure 28 in Ref. [11].

##  VII Conclusions

We have derived constraints on the tensor-to-scalar ratio $r$ using the two
most sensitive data sets to date, namely BICEP3 and Planck PR4. The BICEP/Keck
Collaboration recently released a likelihood derived from their data up to the
2018 observing season, demonstrating a sensitivity on $r$ of
$\sigma_{r}=0.013$, covering the multipole range from $\ell=20$ to 300 [13].
Complementary Planck PR4 data released in 2020 [14] provide information on the
large scales, with a polarized likelihood covering the multipole range from
$\ell=2$ to $\ell=150$ [12]. This has poorer sensitivity, with
$\sigma_{r}=0.024$, but offers independent information, with the constraint on
$r$ coming from a combination of $TT$, $TE$, and large-scale $E$ and $B$ data.
It is interesting to note that constraints derived purely from temperature
anisotropies alone are not competitive anymore ($\sigma_{r}=0.1$ [12]), since
those data are dominated by cosmic variance.

The addition of Planck data (including large angular scales in polarization,
as well as small angular scales in $TT$ and $TE$) allows us to increase the
sensitivity on $r$, as well as to break the degeneracy with the usual six
parameters of the $\rm{\Lambda CDM}$ model. We find that other $\rm{\Lambda
CDM}$ parameters are not affected by the addition of BK18 data (Fig. 5).
Combining Planck PR4 and BK18, we find an upper limit of $r<0.034$, which
tightens to $r<0.032$ when adding BAO and CMB lensing data.

Figure 5: Constraint contours (at 68 and 95 % confidence) on parameters of a
$\rm{\Lambda CDM}$+r model using Planck (red) and Planck+BK18 (black).

Ground-based experiments (such as BICEP/Keck, the Simons Observatory [30], and
later CMB-S4 [31]) will observe the sky with ever deeper sensitivity, placing
even stronger constraints on the tensor-to-scalar ratio $r$ (or detecting
primordial $B$ modes of course). However, improved measurements of the
$\rm{\Lambda CDM}$ parameters are essential to achieve strong constraints on
$r$. In particular reionization optical depth require very large scales, which
are extremely difficult to measure from ground. The next generation of
polarized CMB space missions (including LiteBIRD [32]) will be able to deliver
$\tau$ with a precision dominated by cosmic variance.

###### Acknowledgements.

Planck is a project of the European Space Agency (ESA) with instruments
provided by two scientific consortia funded by ESA member states and led by
Principal Investigators from France and Italy, telescope reflectors provided
through a collaboration between ESA and a scientific consortium led and funded
by Denmark, and additional contributions from NASA (USA). We gratefully
acknowledge support from the CNRS/IN2P3 Computing Center for providing
computing and data-processing resources needed for this work. This research
used resources of the National Energy Research Scientific Computing Center,
which is supported by the Office of Science of the U.S. Department of Energy
under Contract No. DE-AC02-05CH11231. Part of the research was carried out at
the Jet Propulsion Laboratory, California Institute of Technology, under a
contract with the National Aeronautics and Space Administration
(80NM0018D0004).

## Appendix: Statistical discussions

The Planck likelihood used in this analysis is described in detail in Ref.
[12]. It is based on the $N=400$ simulations provided with the Planck PR4
data. Those simulations have been shown to be the most realistic description
of the Planck data, including all relevant systematic effects [14]. Using the
Planck data, we expect correlations at very low-$\ell$, related to long-term
systematics, residuals from foregrounds, and cut-sky effects. These should not
be neglected.

The covariance used in the likelihood has been constructed from the
simulations mentioned above, ensuring the propagation of statistical and
systematic uncertainties up to the fitted parameters. Nevertheless, the
limited number of available simulations induces an uncertainty on the
estimated covariance of the order of 5 % ($1/\sqrt{N}$). The robustness of the
covariance matrix has been checked in two different ways.

Firstly, we marginalized over the unknown true covariance matrix, as described
in Ref. [23]. The recovered maximum posterior is unchanged, while the width of
the posterior is slightly enlarged, as expected due to the marginalization
(see left panel of Fig. 6). We also applied the correction on the inverse
covariance estimate, as proposed in Refs. [23] and [33], recovering the same
result.

Secondly, we ran the same chains using covariance estimates based on only 200
simulations (right panel of Fig. 6). The posterior distributions of $r$
reconstructed from the lowlEB likelihood using covariance estimates based on
either the first or the last 200 simulations are compatible, given the
statistical deviations from the covariance matrix estimates.

Figure 6: Posteriors for $r$. Left: after marginalizing over the true
covariance matrix (dashed line) and correcting the inverse covariance matrix
(dotted line), compared to the effective covariance (solid line). Right: using
covariance estimates based on the first or last 200 simulations, compared to
the effective covariance with 400 simulations (solid line).

We built a Monte Carlo toy model in order to check potential biases in the
recovered constraints. We found that the distribution of the maximum a
posteriori probability (MAP) peaks at the input value, ensuring that the
likelihood is not biased. We found that the marginalization over the unknown
covariance matrix ensures that the additional uncertainty coming from the
covariance estimation is properly propagated throughout the parameter
constraints (see Ref. [23]). This is illustrated in Fig. 7, where we show that
the estimated width of the posterior distribution after marginalization is
compatible with the standard deviation of the maximum a posteriori
probability. On the contrary, the Hamimeche & Lewis likelihood without
marginalization significantly underestimates the uncertainty for low $N$.

Figure 7: $1\,\sigma$ uncertainties estimated without (gray solid line) and
with (green solid line) marginalization over the covariance matrix, compared
to the standard deviation of the MAP.

While there may be concern that this could induce a bias in the derived upper
limit, we have verified that the recovered upper limits are underestimated by
more than 10 % compared to upper limits computed with the input covariance in
less than 6 % of the realisations.

We conclude that covariance matrices based on full end-to-end simulations can
be successfully used in likelihoods to infer parameters. The final uncertainty
then depends on the number of simulations used to estimate the covariance, but
this can be properly taken into account after marginalizing over the true
covariance matrix.

## References

  * Kamionkowski and Kovetz [2016] M. Kamionkowski and E. D. Kovetz, Ann. Rev. Astron. Astrophys. 54, 227 (2016), arXiv:1510.06042 [astro-ph.CO] .
  * Seljak [1997] U. Seljak, Astrophys. J. 482, 6 (1997).
  * Kamionkowski  _et al._ [1997] M. Kamionkowski, A. Kosowsky, and A. Stebbins, Phys. Rev. Lett. 78, 2058 (1997).
  * Seljak and Zaldarriaga [1997] U. Seljak and M. Zaldarriaga, Phys. Rev. Lett. 78, 2054 (1997).
  * Hu and White [1997] W. Hu and M. White, New Astron. 2, 323 (1997), arXiv:astro-ph/9706147 [astro-ph] .
  * Planck Collaboration XVI [2014] Planck Collaboration XVI, Astron. astrophys. 571, A16 (2014), arXiv:1303.5076 .
  * BICEP2/Keck Array and Planck Collaborations [2015] BICEP2/Keck Array and Planck Collaborations, Phys. Rev. Lett. 114, 101301 (2015), arXiv:1502.00612 .
  * BICEP2 Collaboration [2014] BICEP2 Collaboration, Phys. Rev. Lett. 112, 241101 (2014), arXiv:1403.3985 .
  * Planck Collaboration XIII [2016] Planck Collaboration XIII, Astron. astrophys. 594, A13 (2016), arXiv:1502.01589 .
  * BICEP2 Collaboration [2018] BICEP2 Collaboration, Phys. Rev. Lett. 121, 221301 (2018), arXiv:1810.05216 .
  * Planck Collaboration VI [2020] Planck Collaboration VI, Astron. astrophys. 641, A6 (2020), arXiv:1807.06209 .
  * Tristram  _et al._ [2021] M. Tristram _et al._ , Astron. astrophys. 647, A128 (2021), arXiv:2010.01139 .
  * Ade  _et al._ [2021] P. A. R. Ade _et al._ (Bicep/Keck Collaboration), Phys. Rev. Lett. 127, 151301 (2021), arXiv:2110.00483 .
  * Planck Collaboration Int. LVII [2020] Planck Collaboration Int. LVII, Astron. astrophys. 643, A42 (2020), arXiv:2007.04997 .
  * Lewis [2008] A. Lewis, Phys. Rev. D 78, 023002 (2008), arXiv:0804.3865 .
  * Lewis  _et al._ [2000] A. Lewis, A. Challinor, and A. Lasenby, Astrophys. J. 538, 473 (2000), arXiv:astro-ph/9911177 .
  * Howlett  _et al._ [2012] C. Howlett, A. Lewis, A. Hall, and A. Challinor, JCAP 1204, 027, arXiv:1201.3654 .
  * Torrado and Lewis [2021] J. Torrado and A. Lewis, JCAP 05, 057, arXiv:2005.05290 .
  * Lewis and Bridle [2002] A. Lewis and S. Bridle, Phys. Rev. D66, 103511 (2002), arXiv:astro-ph/0205436 .
  * Lewis [2013] A. Lewis, Phys. Rev. D87, 103529 (2013), arXiv:1304.4473 .
  * Hamimeche and Lewis [2008] S. Hamimeche and A. Lewis, Phys. Rev. D 77, 103013 (2008), arXiv:0801.0554 .
  * Mangilli  _et al._ [2015] A. Mangilli, S. Plaszczynski, and M. Tristram, Mon. Not. R. Astron. Soc. 453, 3174 (2015), arXiv:1503.01347 .
  * Sellentin and Heavens [2016] E. Sellentin and A. F. Heavens, Mon. Not. R. Astron. Soc. 456, L132 (2016), arXiv:1511.05969 .
  * Planck Collaboration V [2020] Planck Collaboration V, Astron. astrophys. 641, A5 (2020), arXiv:1907.12875 .
  * Planck Collaboration XV [2014] Planck Collaboration XV, Astron. astrophys. 571, A15 (2014), arXiv:1303.5075 .
  * Planck Collaboration XI [2016] Planck Collaboration XI, Astron. astrophys. 594, A11 (2016), arXiv:1507.02704 .
  * Couchot  _et al._ [2017] F. Couchot, S. Henrot-Versillé, O. Perdereau, S. Plaszczynski, B. Rouillé d’Orfeuil, M. Spinelli, and M. Tristram, Astron. astrophys. 602, A41 (2017), arXiv:1609.09730 .
  * Alam  _et al._ [2021] S. Alam _et al._ (eBOSS), Phys. Rev. D 103, 083533 (2021), arXiv:2007.08991 [astro-ph.CO] .
  * Planck Collaboration VIII [2020] Planck Collaboration VIII, Astron. astrophys. 641, A8 (2020), arXiv:1807.06210 .
  * Ade  _et al._ [2019] P. Ade _et al._ (Simons Observatory Collaboration), J. Cosmol. Astropart. Phys. 2019, 056 (2019), arXiv:1808.07445 .
  * Abazajian  _et al._ [2016] K. N. Abazajian _et al._ (CMB-S4 Collaboration), arXiv e-prints , arXiv:1610.02743 (2016), arXiv:1610.02743 .
  * Hazumi  _et al._ [2020] M. Hazumi _et al._ (LiteBIRD), in  _Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series_, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 11443 (2020) p. 114432F, arXiv:2101.12449 [astro-ph.IM] .
  * Hartlap  _et al._ [2007] J. Hartlap, P. Simon, and P. Schneider, Astron. astrophys. 464, 399 (2007), arXiv:astro-ph/0608064 [astro-ph] .


# Transverse asymmetry of individual $\bm{\gamma}$-rays in the
139La($\bm{\vec{n}}$, $\bm{\gamma}$)140La reaction

M. Okuizumi  Nagoya University, Furocho, Chikusa, Nagoya 464-8602, Japan  C.
J. Auton  Indiana University, Bloomington, Indiana 47401, USA  Nagoya
University, Furocho, Chikusa, Nagoya 464-8602, Japan  S. Endo  Nagoya
University, Furocho, Chikusa, Nagoya 464-8602, Japan  Japan Atomic Energy
Agency, 2-1 Shirane, Tokai 319-1195, Japan  H. Fujioka  Tokyo Institute of
Technology, Meguro, Tokyo 152-8551, Japan  K. Hirota  KEK, 1-1 Oho, Tsukuba,
Ibaraki 305-0801, Japan  T. Ino  KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801,
Japan  K. Ishizaki  Nagoya University, Furocho, Chikusa, Nagoya 464-8602,
Japan  A. Kimura  Japan Atomic Energy Agency, 2-1 Shirane, Tokai 319-1195,
Japan  M. Kitaguchi  Nagoya University, Furocho, Chikusa, Nagoya 464-8602,
Japan  J. Koga  Kyushu University, 744 Motooka, Nishi, Fukuoka 819-0395, Japan
S. Makise  Kyushu University, 744 Motooka, Nishi, Fukuoka 819-0395, Japan  Y.
Niinomi  Nagoya University, Furocho, Chikusa, Nagoya 464-8602, Japan  T. Oku
Japan Atomic Energy Agency, 2-1 Shirane, Tokai 319-1195, Japan  Ibaraki
University, 2-1-1 Bunkyo, mito, Ibaraki 310-8512, Japan  T. Okudaira  Nagoya
University, Furocho, Chikusa, Nagoya 464-8602, Japan  Japan Atomic Energy
Agency, 2-1 Shirane, Tokai 319-1195, Japan  K. Sakai  Japan Atomic Energy
Agency, 2-1 Shirane, Tokai 319-1195, Japan  T. Shima  Osaka University, 10-1
Mihogaoka, Ibaraki, Osaka 567-0047, Japan  H. M. Shimizu  Nagoya University,
Furocho, Chikusa, Nagoya 464-8602, Japan  H. Tada  Nagoya University, Furocho,
Chikusa, Nagoya 464-8602, Japan  S. Takada  Tohoku University, 2-1-1 Katahira,
Aoba, Sendai, Miyagi 980-8577, Japan  Japan Atomic Energy Agency, 2-1 Shirane,
Tokai 319-1195, Japan  S. Takahashi  Ibaraki University, 2-1-1 Bunkyo, mito,
Ibaraki 310-8512, Japan  Y. Tani  Tokyo Institute of Technology, Meguro, Tokyo
152-8551, Japan  T. Yamamoto  Nagoya University, Furocho, Chikusa, Nagoya
464-8602, Japan  H. Yoshikawa  Osaka University, 10-1 Mihogaoka, Ibaraki,
Osaka 567-0047, Japan  T. Yoshioka  Kyushu University, 744 Motooka, Nishi,
Fukuoka 819-0395, Japan

###### Abstract

The enhancement of the parity-violating asymmetry in the vicinity of $p$-wave
compound nuclear resonances was observed for a variety of medium-heavy nuclei.
The enhanced parity-violating asymmetry can be understood using the $s$-$p$
mixing model. The $s$-$p$ mixing model predicts several neutron energy-
dependent angular correlations between the neutron momentum $\vec{k}_{n}$,
neutron spin $\vec{\sigma}_{n}$, $\gamma$-ray momentum $\vec{k}_{\gamma}$, and
$\gamma$-ray polarization $\vec{\sigma}_{\gamma}$ in the $(n,\gamma)$
reaction. In this paper, the improved value of the transverse asymmetry of
$\gamma$-ray emissions, corresponding to a correlation term
$\vec{\sigma}_{n}\cdot(\vec{k}_{n}\times\vec{k}_{\gamma})$ in the
${}^{139}\mathrm{La}(\vec{n},\gamma)^{140}\mathrm{La}$ reaction, and the
transverse asymmetries in the transitions to several low excited states of
${}^{140}\mathrm{La}$ are reported.

compound nuclei, partial wave interference, neutron radiative capture reaction

##  I Introduction

A large parity-violating asymmetry of the cross section has been observed in a
$p$-wave resonance of the ${}^{139}\mathrm{La}+n$ compound state Alfimenkov
_et al._ (1983). The magnitude of the parity-violating asymmetry amounts to
$10^{6}$ times larger than that of nucleon-nucleon scattering, which is
dominated by the interference between parity-unfavored partial waves via the
contribution of the weak interaction in the compound nuclear process Potter
_et al._ (1974); Yuan  _et al._ (1986); Adelberger and Haxton (1985); Shimizu
_et al._ (1993). The enhanced parity-violating asymmetry is explained as the
result of the interference between the $p$-wave resonance and neighboring
$s$-wave resonances ($s$-$p$ mixing model) Sushkov and Flambaum (1982). The
$s$-$p$ mixing model predicts angular correlations between neutron momentum,
neutron spin, $\gamma$-ray momentum, and $\gamma$-ray spin depending on
neutron energy in $(n,\gamma)$ reactions Flambaum and Sushkov (1985).
Investigating these spin angular correlations is essential for understanding
the parity violation enhancement mechanism.

Angular correlations in the $(n,\gamma)$ reaction of several nuclei have been
observed for the integral $\gamma$-spectrum Vesna  _et al._ (2015). Recently,
the neutron energy dependence of the angular distribution was measured for
individual $\gamma$-rays emitted from $p$-wave resonances in 139La Okudaira
_et al._ (2018, 2021), 117Sn Koga  _et al._ (2022), and 132Xe Okudaira  _et
al._ (2023a) using an intense pulsed neutron beam at the Material and Life
Science Experimental Facility (MLF) of the Japan Proton Accelerator Research
Complex (J-PARC). The transverse asymmetry was also measured in 139La Yamamoto
_et al._ (2020, 2022) and 117Sn Endo  _et al._ (2022). The transverse
asymmetry corresponds to the correlation term
$a_{2}\vec{\sigma}_{n}\cdot(\vec{k}_{n}\times\vec{k}_{\gamma})$ in Ref.
Flambaum and Sushkov (1985), where $\vec{\sigma}_{n}$, $\vec{k}_{n}$, and
$\vec{k}_{\gamma}$ are unit vectors for the neutron spin, neutron momentum,
and $\gamma$-ray momentum direction, respectively.

We have accumulated more statistics with the setup described in Ref. Yamamoto
_et al._ (2020, 2022). The transverse asymmetry was measured with improved
statistics for $\gamma$-ray transition to the ground state and also low-
excited states.

##  II Experiment and Analysis

###  II.1 Experiment

The measurements were carried out using the Accurate Neutron-Nucleus Reaction
Measurement Instrument (ANNRI) instrument on beamline-04 of MLF at J-PARC
Igashira  _et al._ (2009). The neutron energy was determined using the neutron
time-of-flight (TOF) method. The neutron beam was polarized using a 3He spin-
filter, which makes use of the large spin dependent cross-section of 3He
nuclei.

The 3He spin was flipped approximately every four hours using the adiabatic
fast-passage NMR method Ino  _et al._ (2012). The $\gamma$-ray energy was
measured with germanium detectors. Li-Glass detectors were installed
downstream of the target, which are used for neutron transmission
measurements.

The ANNRI instrument uses two types of Ge detectors: cluster-type and coaxial-
type detectors, each positioned to surround the target in both the vertical
and horizontal directions Takada  _et al._ (2018). For this analysis, only the
up- and down-cluster detectors were utilized. Further details on the
experimental setup and measurements are described in Ref. Yamamoto  _et al._
(2020).

A metal lanthanum plate of $40~{}\mathrm{mm}\times 40~{}\mathrm{mm}\times
3~{}\mathrm{mm}$ with a purity of 99.9% was used as the nuclear target. The
setup around the target is shown in Fig. 1. The total measurement time was 222
hours.

Figure 1:  The sectional view around the target. There are seven Ge detectors
above and below the target.

###  II.2 Analysis

This analysis focuses on the $\gamma$-rays emitted in each transition to final
states with excitation energies of
$E_{\gamma}=0,~{}30~{}\text{or}~{}35,~{}63,~{}272,~{}318,~{}658,~{}745$, and
$771~{}\mathrm{keV}$. The transition scheme from ${}^{139}\mathrm{La}+n$ to
${}^{140}\mathrm{La}$ is depicted in Fig. 2. Figure 3 shows the $\gamma$-ray
spectrum corresponding to the energy range of transitions to low excited
states in 140La. The peaks at 5126 keV and 5131 keV cannot be separated and
were treated as a single peak. Events within the full-width at the quater-
maximum region of the eight photo peaks indicated by the solid lines in Fig. 3
were used for the analysis.

Figure 2:  Transitions from ${}^{139}\mathrm{La}+n$ resonance state to the low
excited states and the ground state of ${}^{140}\mathrm{La}$. The resonance
energy values are taken from Ref. Endo  _et al._ (2023) Figure 3:  The peaks
indicated by solid and dashed lines are full absorption peaks of 139La. We
focused on the eight peaks indicated by solid lines. Peaks indicated by the
dashed line were not analyzed due to an insufficient number of events. Three
peaks around 4600 keV and on the left side of 4390 keV peak are the single
escape peaks from peaks of 5161 keV, 5131 keV, 5126 keV, 5098 keV, and 4889
keV peaks.

The pileup rate of the $\gamma$-ray signal was less than 5%, which was
corrected for. The detection efficiency of each germanium detector was
determined from measurements of the
${}^{14}\mathrm{N}(n,\gamma)^{15}\mathrm{N}$ reaction, which emits $\gamma$
rays isotropically. The Compton scattering background was estimated using a
third-order polynomial fit on both sides of each peak. The $s$-wave resonance
components were subtracted by fitting with a linear function to obtain the
asymmetry of the $p$-wave resonance. Figure 4 shows the TOF spectra after the
corrections with a fit of the $s$-wave component.

Figure 4:  $\gamma$-ray yield in the vicinity of the $p$-wave resonance and
the result of background fitting. The fitted spectra were averaged for both
spin directions. The energy values of photopeaks are shown in the top left of
each histogram.

In a right-hand coordinate system, we define the $z$-axis corresponding to the
neutron beam direction. The neutron spin polarization is along the positive or
negative $x$-axis direction. We label the spin direction along the positive
and negative $x$-axis direction as ”+” and ”-” respectively. We define
$n^{\mathrm{up}}_{\pm}$ and $n^{\mathrm{down}}_{\pm}$ as the corrected
$\gamma$-ray yields obtained with up- and down- cluster detectors. The summed
TOF spectra of the $p$-wave region corresponding to each sign of
$\vec{\sigma}_{n}\cdot(\vec{k}_{n}\times\vec{k}_{\gamma})$ are shown in Fig.
5.

Figure 5:  $\gamma$-ray yield in the vicinity of the $p$-wave resonance for
each spin direction. White points indicate
$n^{\mathrm{up}}_{-}+n^{\mathrm{down}}_{+}$ which correspond to positive sign
of $\vec{\sigma}_{n}\cdot(\vec{k}_{n}\times\vec{k}_{\gamma})$ and black points
indicate $n^{\mathrm{up}}_{+}+n^{\mathrm{down}}_{-}$ which correspond to
negative sign of $\vec{\sigma}_{n}\cdot(\vec{k}_{n}\times\vec{k}_{\gamma})$.

###  II.3 Transverse Asymmetry

The asymmetry is defined as

$\displaystyle\epsilon^{\mathrm{up,down}}_{\gamma}=\frac{N^{\mathrm{up,down}}_{+}-N^{\mathrm{up,down}}_{-}}{N^{\mathrm{up,down}}_{+}+N^{\mathrm{up,down}}_{-}}.$
(1)

where, $N^{\mathrm{up}}_{\pm}$ and $N^{\mathrm{down}}_{\pm}$ are the integral
of the corrected TOF spectra with a range $E_{r}-2\Gamma_{p}\leq E_{n}\leq
E_{r}+2\Gamma_{p}$ obtained from up- and down-cluster detectors respectively,
where $\Gamma_{p}=\Gamma^{\gamma}+\Gamma^{n}$. The resonance parameters of
139La+$n$ are listed in Table 1.

$E_{r}\,[{\rm eV}]$ | $J$ | $l$ | $\Gamma^{\gamma}\,[{\rm meV}]$ | $g\Gamma^{n}\,[{\rm meV}]$
---|---|---|---|---
$0.750\pm 0.001$ | $4$ | $1$ | $41.6\pm 0.9$ | $(3.67\pm 0.5)\times 10^{-5}$
Table 1:  Resonance parameters of the ${}^{139}\mathrm{La}+n$ $p$-wave
resonance state. $E_{r},~{}J,~{}l,~{}\Gamma^{\gamma},~{}g,$ and $\Gamma^{n}$
are the resonance energy, total angular momentum of the resonance state,
orbital angular momentum of the incident neutron, $\gamma$ width, $g$ factor,
and neutron width, respectively. The values are taken from Ref. Endo  _et al._
(2023).

The neutron polarization is determined from the 3He polarization, which was
obtained from measurements of the neutron transmission rate. The corrected
transverse asymmetry with neutron polarization is expressed as:

$\displaystyle{A^{\prime}}^{\mathrm{up,down}}_{\mathrm{LR}}=\frac{2\epsilon^{\mathrm{up,down}}_{\gamma}}{(P^{+}_{n}+P^{-}_{n})-\epsilon^{\mathrm{up,down}}_{\gamma}(P^{+}_{n}-P^{-}_{n})},$
(2)

where $P_{n}^{+}$ and $P_{n}^{-}$ are the average neutron polarization during
measurements for each spin direction.

The effect of scattering in the target was corrected as:

$\displaystyle
A_{\mathrm{LR}}^{\mathrm{up,down}}={A^{\prime}}^{\mathrm{up,down}}_{\mathrm{LR}}\left(1+\frac{N_{s\geqslant
1}}{N_{s=0}}\right),$ (3)

where $N_{s\geq 1}$ is the number of events in which neutrons scattered in the
target and were subsequently absorbed by the nucleus. $N_{s=0}$ is the number
of events that emit $\gamma$ rays without scattering. The ratio of these
quantities was estimated by Monte Carlo simulation in Ref. Yamamoto  _et al._
(2020).

We determined the transverse asymmetry for each up and down-cluster for each
measurement, and their weighted average was used as the final result
$A_{\mathrm{LR}}$. Note that the sign of $A^{\mathrm{up}}_{\mathrm{LR}}$ is
inverted when averaging since $A^{\mathrm{up}}_{\mathrm{LR}}$ corresponds to
$-\vec{\sigma}_{n}\cdot(\vec{k}_{n}\times\vec{k}_{\gamma})$ and
$A^{\mathrm{down}}_{\mathrm{LR}}$ corresponds to
$\vec{\sigma}_{n}\cdot(\vec{k}_{n}\times\vec{k}_{\gamma})$. The results of
$A_{\mathrm{LR}}$ are shown in Table 2. The transverse asymmetry for the
transition to the ground state was obtained with over $5\sigma$ significance.
Non-zero transverse asymmetries were found in the transitions to the excited
states of 63 keV and 772 keV with a confidence level of over 99.7%.

$E_{\gamma}$ [keV] |  $E_{\rm ex}$ [keV] | $F$ | $A_{\mathrm{LR}}$
---|---|---|---
5161 | 0 | 3 | $-0.86\pm 0.10$
5131, 5126 | 30, 35 | 2,5 | $-0.08\pm 0.05$
5098 | 63 | 4 | $1.43\pm 0.39$
4889 | 272 | 4 | $0.21\pm 0.83$
4843 | 318 | 3 | $-0.01\pm 0.37$
4503 | 658 | 3 | $-1.04\pm 0.53$
4416 | 745 | 4 | $-0.17\pm 4.21$
4390 | 771 | 4 or 5 or 6 | $0.90\pm 0.19$
Table 2:  The final state spin $F$ and the result of $A_{\mathrm{LR}}$ for
each photopeak.

##  III Conclusion

We measured the transverse asymmetries for several transitions from the
$p$-wave resonance in ${}^{139}\mathrm{La}+n$ to several low-excited states of
140La. Transverse asymmetries were observed for transitions to three final
states with a confidence level of over 99.7%. There are indications of angular
correlations for other transitions. The consistency of the $a_{1}$ and $a_{2}$
terms Flambaum and Sushkov (1985) was discussed in Ref. Nakabe  _et al._
(2023) using measurement results for 139La Okudaira  _et al._ (2018); Yamamoto
_et al._ (2020, 2022); Okudaira  _et al._ (2023b). The same analysis will be
performed using the improved values in this paper, and the consistency will be
discussed for the transitions to the excited states.

###### Acknowledgements.

The authors would like to thank the staff of ANNRI for the maintenance of the
germanium detectors, and MLF and J-PARC for operating the accelerators and the
neutron-production target. The neutron experiments at the Materials and Life
Science Experimental Facility of J-PARC were performed under the user program
(Proposals No. 2018B0148 and No. 2019A0185). This work was supported by the
Neutron Science Division of KEK as an S-type research project with program
number 2018S12. This work was partially supported by MEXT KAKENHI Grant No.
JP19GS0210 and JSPS KAKENHI Grant Nos. JP17H02889, 20K14495, and 23K13122.

## References

  * Alfimenkov  _et al._ (1983) V. P. Alfimenkov, S. B. Borzakov, V. V. Thuan, Y. D. Mareev, L. B. Pikelner, A. S. Khrykin, and E. I. Sharapov, Nucl. Phys. A 435, 352 (1983).
  * Potter  _et al._ (1974) J. M. Potter, J. D. Bowman, C. F. Hwang, J. L. McKibben, R. E. Mischke, D. E. Nagle, P. G. Debrunner, H. Frauenfelder, and L. B. Sorensen, Phys. Rev. Lett. 33, 1307 (1974).
  * Yuan  _et al._ (1986) V. Yuan, H. Frauenfelder, R. W. Harper, J. D. Bowman, R. Carlini, D. W. MacArthur, R. E. Mischke, D. E. Nagle, R. L. Talaga, and A. B. McDonald, Phys. Rev. Lett. 57, 1680 (1986).
  * Adelberger and Haxton (1985) E. G. Adelberger and W. C. Haxton, Ann. Rev. Nucl. Part. Sci. 35, 501 (1985).
  * Shimizu  _et al._ (1993) H. Shimizu, T. Adachi, S. Ishimoto, A. Masaike, Y. Masuda, and K. Morimoto, Nuclear Physics A 552, 293 (1993).
  * Sushkov and Flambaum (1982) O. P. Sushkov and V. V. Flambaum, Usp. Fiz. Nauk 136, 3 (1982).
  * Flambaum and Sushkov (1985) V. V. Flambaum and O. P. Sushkov, Nucl. Phys. A 435, 352 (1985).
  * Vesna  _et al._ (2015) V. A. Vesna, Y. M. Gledenov, V. V. Nesvizhevsky, P. V. Sedyshev, and E. V. Shul’gina, Phys. Atom. Nucl. 78, 801 (2015).
  * Okudaira  _et al._ (2018) T. Okudaira, S. Takada, K. Hirota, A. Kimura, M. Kitaguchi, J. Koga, K. Nagamoto, T. Nakao, A. Okada, K. Sakai, H. M. Shimizu, T. Yamamoto, and T. Yoshioka, Phys. Rev. C 97, 034622 (2018).
  * Okudaira  _et al._ (2021) T. Okudaira, S. Endo, H. Fujioka, K. Hirota, K. Ishizaki, A. Kimura, M. Kitaguchi, J. Koga, Y. Niinomi, K. Sakai, T. Shima, H. M. Shimizu, S. Takada, Y. Tani, T. Yamamoto, H. Yoshikawa, and T. Yoshioka, Phys. Rev. C 104, 014601 (2021).
  * Koga  _et al._ (2022) J. Koga, S. Takada, S. Endo, H. Fujioka, K. Hirota, K. Ishizaki, A. Kimura, M. Kitaguchi, Y. Niinomi, T. Okudaira, K. Sakai, T. Shima, H. M. Shimizu, Y. Tani, T. Yamamoto, H. Yoshikawa, and T. Yoshioka, Phys. Rev. C 105, 054615 (2022).
  * Okudaira  _et al._ (2023a) T. Okudaira, Y. Tani, S. Endo, J. Doskow, H. Fujioka, K. Hirota, K. Kameda, A. Kimura, M. Kitaguchi, M. Luxnat, K. Sakai, D. C. Schaper, T. Shima, H. M. Shimizu, W. M. Snow, S. Takada, T. Yamamoto, H. Yoshikawa, and T. Yoshioka, Phys. Rev. C 107, 054602 (2023a).
  * Yamamoto  _et al._ (2020) T. Yamamoto, T. Okudaira, S. Endo, H. Fujioka, K. Hirota, T. Ino, K. Ishizaki, A. Kimura, M. Kitaguchi, J. Koga, S. Makise, Y. Niinomi, T. Oku, K. Sakai, H. M. Shimizu, S. Takada, Y. Tani, H. Yoshikawa, and T. Yoshioka, Phys. Rev. C 101, 064624 (2020).
  * Yamamoto  _et al._ (2022) T. Yamamoto, T. Okudaira, S. Endo, H. Fujioka, K. Hirota, T. Ino, K. Ishizaki, A. Kimura, M. Kitaguchi, J. Koga, S. Makise, Y. Niinomi, T. Oku, K. Sakai, T. Shima, H. M. Shimizu, S. Takada, Y. Tani, H. Yoshikawa, and T. Yoshioka, Phys. Rev. C 105, 039901 (2022).
  * Endo  _et al._ (2022) S. Endo, T. Okudaira, R. Abe, H. Fujioka, K. Hirota, A. Kimura, M. Kitaguchi, T. Oku, K. Sakai, T. Shima, H. M. Shimizu, S. Takada, S. Takahashi, T. Yamamoto, H. Yoshikawa, and T. Yoshioka, Phys. Rev. C 106, 064601 (2022).
  * Igashira  _et al._ (2009) M. Igashira, Y. Kiyanagi, and M. Oshima, Nucl. Instrum. Meth. Phys. Res. A600, 332 (2009).
  * Ino  _et al._ (2012) T. Ino, Y. Arimoto, H. M. Shimizu, Y. Sakaguchi, K. Sakai, H. Kira, T. Shinohara, T. Oku, J. ichi Suzuki, K. Kakurai, and L.-J. Chang, Journal of Physics: Conference Series 340, 012006 (2012).
  * Takada  _et al._ (2018) S. Takada, T. Okudaira, F. Goto, K. Hirota, A. Kimura, M. Kitaguchi, J. Koga, T. Nakao, K. Sakai, H. M. Shimizu, T. Yamamoto, and T. Yoshioka, Journal of Instrumentation 13, P02018 (2018).
  * Endo  _et al._ (2023) S. Endo, S. Kawamura, T. Okudaira, H. Yoshikawa, G. Rovira, A. Kimura, S. Nakamura, O. Iwamoto, and N. Iwamoto, Eur. Phys. J. A 59, 288 (2023).
  * Nakabe  _et al._ (2023) R. Nakabe, C. J. Auton, S. Endo, H. Fujioka, V. Gudkov, K. Hirota, I. Ide, T. Ino, M. Ishikado, W. Kambara, S. Kawamura, A. Kimura, M. Kitaguchi, R. Kobayashi, T. Okamura, T. Oku, T. Okudaira, M. Okuizumi, J. G. O. Munoz, J. D. Parker, K. Sakai, T. Shima, H. M. Shimizu, T. Shinohara, W. M. Snow, S. Takada, R. Takahashi, S. Takahashi, Y. Tsuchikawa, and T. Yoshioka, “High sensitivity of a future search for p-odd/t-odd interactions on the 0.75 ev $p$-wave resonance in $\vec{n}+^{139}\vec{\rm la}$ forward transmission determined using pulsed neutron beam,” (2023), arXiv:2312.06115 [nucl-ex] .
  * Okudaira  _et al._ (2023b) T. Okudaira, R. Nakabe, S. Endo, H. Fujioka, V. Gudkov, I. Ide, T. Ino, M. Ishikado, W. Kambara, S. Kawamura, R. Kobayashi, M. Kitaguchi, T. Okamura, T. Oku, J. G. O. Munoz, J. D. Parker, K. Sakai, T. Shima, H. M. Shimizu, T. Shinohara, W. M. Snow, S. Takada, Y. Tsuchikawa, R. Takahashi, S. Takahashi, H. Yoshikawa, and T. Yoshioka, “Spin dependence in the $p$-wave resonance of ${{}^{139}\vec{\rm{La}}+\vec{n}}$,” (2023b), arXiv:2309.08905 [nucl-ex] .