# Spin dependence in the $p$-wave resonance of
$\bm{{{{}^{139}\overrightarrow{\rm{La}}}}+\overrightarrow{n}}$

T. Okudaira  Nagoya University, Furocho, Chikusa, Nagoya 464-8602, Japan
Japan Atomic Energy Agency, 2-4 Shirakata, Tokai, Ibaraki 319-1195, Japan  R.
Nakabe  Nagoya University, Furocho, Chikusa, Nagoya 464-8602, Japan  S. Endo
Nagoya University, Furocho, Chikusa, Nagoya 464-8602, Japan  Japan Atomic
Energy Agency, 2-4 Shirakata, Tokai, Ibaraki 319-1195, Japan  H. Fujioka
Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan  V. Gudkov
University of South Carolina, Columbia, South Carolina 29208, USA  I. Ide
Nagoya University, Furocho, Chikusa, Nagoya 464-8602, Japan  T. Ino  High
Energy Accelerator Research Organization,1-1 Oho, Tsukuba, Ibaraki 305-0801,
Japan  M. Ishikado  Comprehensive Research Organization for Science and
Society, Tokai, Ibaraki 319-1106, Japan  W. Kambara  Japan Atomic Energy
Agency, 2-4 Shirakata, Tokai, Ibaraki 319-1195, Japan  S. Kawamura  Nagoya
University, Furocho, Chikusa, Nagoya 464-8602, Japan  Japan Atomic Energy
Agency, 2-4 Shirakata, Tokai, Ibaraki 319-1195, Japan  R. Kobayashi  Ibaraki
University, 2-1-1 Bunkyo, Mito, Ibaraki 310-8512. Japan  M. Kitaguchi  Nagoya
University, Furocho, Chikusa, Nagoya 464-8602, Japan  T. Okamura  High Energy
Accelerator Research Organization,1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
T. Oku  Japan Atomic Energy Agency, 2-4 Shirakata, Tokai, Ibaraki 319-1195,
Japan  Ibaraki University, 2-1-1 Bunkyo, Mito, Ibaraki 310-8512. Japan  J. G.
Otero Munoz  Indiana University, Bloomington, Indiana 47401, USA  J. D. Parker
Comprehensive Research Organization for Science and Society, Tokai, Ibaraki
319-1106, Japan  K. Sakai  Japan Atomic Energy Agency, 2-4 Shirakata, Tokai,
Ibaraki 319-1195, Japan  T. Shima  Osaka University, Ibaraki, Osaka 567-0047,
Japan  H. M. Shimizu  Nagoya University, Furocho, Chikusa, Nagoya 464-8602,
Japan  T. Shinohara  Japan Atomic Energy Agency, 2-4 Shirakata, Tokai, Ibaraki
319-1195, Japan  W. M. Snow  Indiana University, Bloomington, Indiana 47401,
USA  S. Takada  Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8576
Japan  Japan Atomic Energy Agency, 2-4 Shirakata, Tokai, Ibaraki 319-1195,
Japan  Y. Tsuchikawa  Japan Atomic Energy Agency, 2-4 Shirakata, Tokai,
Ibaraki 319-1195, Japan  R. Takahashi  Japan Atomic Energy Agency, 2-4
Shirakata, Tokai, Ibaraki 319-1195, Japan  S. Takahashi  Ibaraki University,
2-1-1 Bunkyo, Mito, Ibaraki 310-8512. Japan  Japan Atomic Energy Agency, 2-4
Shirakata, Tokai, Ibaraki 319-1195, Japan  H. Yoshikawa  Osaka University,
Ibaraki, Osaka 567-0047, Japan  T. Yoshioka  Kyushu University, 744 Motooka,
Nishi, Fukuoka 819-0395, Japan

###### Abstract

We measured the spin dependence in a neutron-induced $p$-wave resonance by
using a polarized epithermal neutron beam and a polarized nuclear target. Our
study focuses on the 0.75 eV $p$-wave resonance state of 139La+$n$, where
largely enhanced parity violation has been observed. We determined the partial
neutron width of the $p$-wave resonance by measuring the spin dependence of
the neutron absorption cross section between polarized ${}^{139}\rm{La}$ and
polarized neutrons. Our findings serve as a foundation for the quantitative
study of the enhancement effect of the discrete symmetry violations caused by
mixing between partial amplitudes in the compound nuclei.

neutron induced compound nuclei, polarized epithermal neutrons, polarized
nuclear target

##  I Introduction

The spin dependence of the strong interaction between a neutron and a nucleus
can lead to a spin-dependent cross section proportional to
$\bm{\sigma}\cdot\bm{I}$, where $\bm{\sigma}$ and $\bm{I}$ are unit vectors
parallel to the spins of the neutron and the nucleus, respectively. This spin-
dependent cross section can be observed through a spin-dependent transmission
through a polarized nuclear target. At a neutron-nucleus resonance, this
observable can directly determine the spin of compound resonance states.
Consequently, it has been employed in measuring $s$-wave resonances for a
select few nuclides, utilizing both a polarized neutron beam and a polarized
target Keyworth  _et al._ (1973a, b); Alfimenkov  _et al._ (1994). In the case
of $p$-wave resonances, the spin-dependent cross section imparts valuable
information not only regarding the spin of the resonance but also the partial
neutron widths.
These widths enable the exploration of symmetry violation enhancement effects
in the compound nucleus. Enhancements in parity violation, exceeding
magnitudes of nucleon-nucleon interactions by a factor of 106, have been
observed in $p$-wave resonances of nuclei with a mass number of medium-heavy
or heavy nuclei Mitchell  _et al._ (2001). These enhancements can be
understood as a result of parity mixing between $s$\- and $p$\- compound
nuclear resonances. This is referred to as the $s$-$p$ mixing model Mitchell
_et al._ (2001); Sushkov and Flambaum (1982); Bunakov and Gudkov (1983).
Theory suggests that this mechanism can lead to an enhancement of fundamental
time reversal violation, which could be utilized to search for beyond the
Standard Model physics by measuring a T-odd cross section at the $p$-wave
resonance using a polarized target and a polarized neutron beam Gudkov (1991).
We can quantify the enhancement factors associated with both P- and
T-violations by determining the partial neutron width Gudkov (1992); Gudkov
and Shimizu (2018); Okudaira  _et al._ (2018, 2021); Yamamoto  _et al._
(2020); Koga  _et al._ (2022); Endo  _et al._ (2022); Okudaira  _et al._
(2023).
This paper presents the first measurement of the spin-dependent cross section
at the $p$-wave compound resonance, employing a polarized epithermal neutron
beam and a polarized nuclear target. As our target nucleus, we selected 139La,
which displays an exceedingly large enhanced parity violation at the 0.75 eV
$p$-wave resonance Yuan  _et al._ (1991).

##  II Experiment

###  II.1 Experimental setup

The experiment was performed with a pulsed epi-thermal neutron beam at the
RADEN beamline of the Material and Life Science Experimental Facility (MLF) at
the Japan Proton Accelerator Research Complex (J-PARC) Shinohara  _et al._
(2020). The experimental setup is depicted in Fig. 1. The La target is placed
23.0 m from the moderator surface. A 2.0 cm cubic lanthanum metal cooled with
a dilution refrigerator was used as the target. A 6.8 Tesla transverse
magnetic field was applied using a superconducting magnet to polarize the
target nuclei. The neutron beam, collimated to a 3 cm by 3 cm size, was
stripped of thermal neutrons using a cadmium filter upstream of the beamline
to reduce the heat load on the La target induced by the neutron beam. The beam
was polarized with a neutron polarizer using polarized 3He gas (3He spin
filter), located 4.3 m upstream of the polarized target. The 3He spin filter
was polarized using the spin exchange optical pumping method (SEOP) with a 110
W laser system constructed outside of the beamline and then installed on the
beamline with a coil and a double magnetic shield to maintain the 3He
polarization Okudaira  _et al._ (2020). The 3He cell was 45 mm in diameter by
70 mm in length and the pressure was 0.31 MPa. The neutron beam,
longitudinally polarized by the 3He spin filter, was guided using a guide
magnet. The spin direction was adiabatically rotated to the transverse
direction utilizing the stray magnetic field of the superconducting magnet.
The neutron spin was flipped every 30 minutes by flipping the spin of 3He gas
using adiabatic fast passage (AFP) NMR. The loss of the 3He polarization was
$4\times 10^{-5}$ per flip, which was negligibly small. Transmitted neutrons
were recorded in list mode using a 256-pixel lithium glass scintillator
detector located at 24.71 m from the moderator surface Satoh (2015).
Downstream of the La target, another collimator was installed to reduce the
beam divergence. The neutron energy $E_{n}$ was determined using the neutron
time-of-flight (TOF) and the flight path length. The proton beam power was 750
kW during the experiment.

Figure 2 illustrates the configuration around the La target. The La target was
held in place between upper and lower copper holders, fastened using copper
screws. The upper holder was connected to the cold head of the dilution
refrigerator, enabling cooling of the La target through thermal conduction.
The temperature $T$ was monitored with a ruthenium oxide thermometer installed
in the cold head. We performed the experiment in two conditions: (a) low
temperature condition ($T=67~{}\rm{mK}$ ) and (b) high temperature condition
($T=1~{}\rm{K}$). The measurement times were 22 hours and 6 hours,
respectively. In the condition (a), the temperature increase by the beam
irradiation to the La target was approximately 1 mK, indicating that the
temperature difference between the cold head and the La target can be
considered negligible. The temperature fluctuation was also around 1 mK, which
was caused by beam interruptions in the accelerator due to malfunctions.

Figure 1:  Experimental setup.  Figure 2:  Configuration around the La target.
The thermometer was installed in the cold head.

###  II.2 Measurement of the asymmetry

The cross sections for parallel and antiparallel polarized neutron and nucleus
can be written with the spin-independent cross section $\sigma_{0}$ and spin-
dependent cross section $\sigma_{\rm S}$ as

$\displaystyle\sigma_{\pm}=\sigma_{0}\pm\sigma_{\rm{S}},$ (1)

where ${+}$ and ${-}$ denote the parallel and antiparallel spins,
respectively. The asymmetry of neutron counts for parallel and anti-parallel
spins transmitted through the polarized lanthanum target, defined as

$\displaystyle\varepsilon_{\rm{S}}=\frac{N_{-}-N_{+}}{N_{-}+N_{+}},$ (2)

where $N_{-}$ and $N_{+}$ are the neutron counts for parallel and anti-
parallel spins, was measured. The neutron counts $N_{\pm}$ are expressed using
the neutron polarization $P_{n}$ and nuclear vector polarization $P_{I}$ as

$\displaystyle N_{\pm}=\frac{1\pm P_{n}}{2}N\epsilon\exp\left((\sigma_{0}\pm
P_{I}\sigma_{\rm{S}})\rho d\right),$ (3)

where $N$, $\epsilon$, $\rho$, and $d$ are number of incident neutrons,
detection efficiency of a neutron detector, number density of the nuclear
target, and the thickness of the nuclear target, respectively. The spin-
dependent asymmetry $\varepsilon_{\rm S}$ can be described using Eq.3 as

$\displaystyle\varepsilon_{\rm{S}}=P_{n}\tanh\left(P_{I}\sigma_{\rm S}\rho
d\right).$ (4)

In this paper, the measurement and analysis were performed using resonance
parameters of La+n reactions listed in Table 1, which were recently measured
by Endo et al. Endo  _et al._ (2023) using both neutron transmission and ($n$,
$\gamma$) reaction with an intense pulsed neutron beam at J-PARC.

Isotope | $E_{0}\,[{\rm eV}]$ | $J$ | $l$ | $\Gamma^{\gamma}\,[{\rm meV}]$ | $g\Gamma^{n}\,[{\rm meV}]$
---|---|---|---|---|---
$\rm{{}^{139}La}$ | $-38.8\pm 0.4$ | $4$ | $0$ | $60.3\pm 0.5$ | 346$\pm$10
$\rm{{}^{139}La}$ | $0.750\pm 0.001$ | 4 | $1$ | $41.6\pm 0.9$ | $(3.67\pm 0.05)\times 10^{-5}$
$\rm{{}^{138}La}$ | $2.99\pm 0.01$ | 11/2 | $0$ | $95\pm 6$ | $0.65\pm 0.03$
$\rm{{}^{139}La}$ | $72.30\pm 0.01$ | 3 | $0$ | $64.1\pm 3.0$ | $13.1\pm 0.7$
Table 1:  The resonance parameters of La+$n$ for low energy neutrons. The
resonance parameters $E$, $J$, $l$, $\Gamma^{\gamma}$, $g$, and $\Gamma^{n}$
are resonance energy, resonance spin, orbital angular momentum,
$\gamma$-width, g-factor, and neutron width, respectively. The parameters of
138La and 139La are taken from Ref. Mughabghab (2006) and Ref. Endo  _et al._
(2023), respectively.

Figure 3 shows the TOF spectra of the transmitted neutrons and asymmetry
$\varepsilon_{\rm S}$ in conditions (a) and (b). We observed a significant
asymmetry in condition (a), corresponding to a high nuclear polarization,
while the asymmetry disappeared in condition (b) due to the lower nuclear
polarization. The peak and dip structures were observed at the 2.99 eV and
0.75 eV resonances. The global structure observed less than 0.3 ms is
attributed to the spin-dependent cross section of the negative $s$-wave
resonance.

Figure 3:  TOF spectra of the transmitted neutrons (top figure) and the spin-
dependent asymmetries (bottom figure). Black and white points denote the
asymmetries in the conditions (a) and (b), respectively, in the bottom figure.

###  II.3 Neutron polarization

The neutron polarization was obtained using the 3He polarization of the 3He
spin filter. The 3He polarization was determined with the ratio of the
transmitted neutrons for polarized and unpolarized 3He spin filter. The ratio
of the transmitted neutrons is described as

$\displaystyle\frac{N_{\rm{pol}}}{N_{\rm{unpol}}}=\cosh(P_{\rm{He}}(t)\rho_{\rm{He}}d_{\rm{He}}\sigma_{\rm{He}}),$
(5)

where, $P_{\rm{He}}$, $\sigma_{\rm{He}}$, and $\rho_{\rm{He}}d_{\rm{He}}$ are
the 3He polarization, neutron absorption cross section of 3He, and areal
density of 3He gas, respectively. Here, $N_{\rm{pol}}$ is defined as
$N_{+}+N_{-}$ for cancelling the spin-dependent asymmetry derived from the
polarization of the La target. The areal density $\rho_{\rm{He}}d_{\rm{He}}$
was obtained from the measurement of the ratio of transmitted neutrons for
unpolarized 3He spin filter and empty glass cell as 21.4 atm$\cdot$cm. The 3He
polarization was obtained for each flip by fitting the TOF dependence of
$N_{\rm{pol}}/N_{\rm{unpol}}$ using Eq. 5 with a fit parameter of
$P_{\rm{He}}$ as shown in Fig 4. Figure 5 shows the time dependence of the 3He
polarization. The relaxation time of the 3He polarization $\tau$, which was
obtained by fitting with $P_{\rm{He}}(t)=P_{\rm{He}}(0)\exp(-t/\tau)$, was 161
h. The averaged 3He polarization $\bar{P}_{\rm{He}}$ during the measurement
was $(68\pm 1)$%.
The neutron polarization $P_{n}$ transmitted through the 3He spin filter is
determined as

$\displaystyle
P_{n}(t)=-\tanh(P_{\rm{He}}(t)\rho_{\rm{He}}d_{\rm{He}}\sigma_{\rm{He}}).$ (6)

Figure 6 shows an averaged neutron polarization $\bar{P}_{n}$ as a function of
the neutron energy calculated from the averaged 3He polarization. The averaged
neutron polarization at 0.75 eV was $(36.1\pm 0.5)$%.

Figure 4:  Ratio of the counts of transmitted neutrons for the polarized to
the unpolarized 3He spin filter. The curved line shows the best fit.  Figure
5:  3He polarization versus elapsed time from the beginning of the
measurement. The curved line shows the fit result by an exponential function.
The measurement was not conducted from 16 h to 22 h due to a liquid He
transfer for the superconducting magnet.  Figure 6:  Neutron polarization
obtained from the averaged 3He polarization.

###  II.4 Nuclear polarization determined by spin-dependent asymmetry

The 139La nuclear polarization was determined utilizing the spin-dependent
asymmetry at the 2.99 eV $s$-wave resonance of 138La. The spin-dependent
asymmetry at the 2.99 eV resonance, after subtracting of the negative $s$-wave
component, was obtained as

$\displaystyle\varepsilon_{\rm S}=(5.1\pm 0.7)\times 10^{-4}.$ (7)

The spin-dependent cross section of the 2.99 eV resonance $\sigma^{\rm
theo}_{{\rm S},s}$ can be theoretically described using the resonance
parameters listed in Table 1 as

$\displaystyle\sigma^{\rm theo}_{{\rm
S},s}=\frac{5\pi}{11k^{2}}\frac{\Gamma^{n}_{s}\Gamma_{s}}{(E-E_{s})^{2}+(\Gamma_{s}/2)^{2}},$
(8)

where $E_{s}$, $\Gamma^{n}_{s}$, and $\Gamma_{s}$ are the resonance energy,
neutron width, and total width of the 2.99 eV s-wave resonance, respectively.
The nuclear polarization of 138La can be determined using Eqs. 4, 7 and 8,
taking into account its natural abundance and the neutron polarization at 2.99
eV, yielding a value of 4.9$\pm$0.7%. The target temperature $T_{\rm La}$ was
calculated based on a Boltzmann distribution and using the magnetic moment and
nuclear spin listed in Table 2, resulting in
$T_{\rm{La}}=75.7^{+10.2}_{-8.9}~{}\rm{mK}$, which is consistent with the
temperature measured at the cold head of 67 mK. Under the assumption that the
spin temperature of 139La is the same as that of 138La, the corresponding
139La nuclear polarization was determined to be $3.9\pm 0.5\%$.

Isotope | $I^{P}$ | Abundance | $\mu_{0}$
---|---|---|---
$\rm{{}^{139}La}$ | $7/2^{+}$ | $99.91\%$ | $2.78$
$\rm{{}^{138}La}$ | $5^{+}$ | 0.09% | $3.71$
Table 2:  Parameters of lanthanum isotopes. Nuclear spin and parity $I^{P}$,
natural abundance, and nuclear magnetic moment $\mu_{0}$ are listed. The unit
of the nuclear magnetic moment is nuclear magneton.

###  II.5 Spin-dependent cross sections at the resonances

The experimental value of the spin-dependent cross section $\sigma^{\rm
exp}_{\rm S}$ was obtained from the asymmetry $\varepsilon_{\rm S}$ using Eq.
4. The resonance component $\sigma^{\rm{exp}}_{{\rm{S}},r}$ was isolated by
fitting the global structure attributed to the negative $s$-wave component
with a third order polynomial function. The resonance regions listed in the
Table 1 are excluded from the fitting. Figure 7 shows the TOF dependence of
$P_{I}\sigma^{\rm exp}_{\rm S}$ and $P_{I}\sigma^{\rm exp}_{{\rm{S}},r}$. Note
that Fig. 7 was calculated using the areal density of 139La. A $p$-value,
defined as $p=(1-\rm{C.L.})/2$, where C.L. is the confidence level of the non-
zero asymmetry, is also depicted to show the significance of $P_{I}\sigma^{\rm
exp}_{{\rm{S}},r}$ in Fig. 7. The p-value indicates the probability to observe
a non-zero value of $P_{I}\sigma^{\rm exp}_{{\rm{S}},r}$ in the hypothesis of
no asymmetry. A confidence level of over 99.7% corresponds to a p-value less
than $1.35\times 10^{-3}$. The spin-dependence cross section was first
observed at the $p$-wave resonance with over 99.7% C.L. as shown in Fig 7.

The spin-dependent cross section in the $p$-wave resonance region of
$E_{p}-3\Gamma_{p}<E_{n}<E_{p}+3\Gamma_{p}$ after the subtraction of the
negative $s$-wave component, defined as $\sigma^{\rm exp}_{{{\rm{S}},p}}$, is
obtained using the nuclear polarization in Section II.4 as

$\displaystyle\sigma^{\rm exp}_{{\rm{S}},p}=-0.26\pm 0.08~{}\rm{barn},$ (9)

where $E_{p}$ and $\Gamma_{p}$ are the resonance energy and total width of the
$p$-wave resonance, shown in Table 1. Here, the total width is defined as
$\Gamma_{p}=\Gamma_{p}^{\gamma}+\Gamma_{p}^{n}$. The asymmetry of the spin-
dependent cross section relative to the spin-independent cross section of the
$p$-wave component was also obtained as

$\displaystyle A_{\rm S}$ $\displaystyle=$
$\displaystyle\frac{\sigma^{p}_{+}-\sigma^{p}_{-}}{\sigma^{p}_{+}+\sigma^{p}_{-}}=\frac{\sigma^{\rm
exp}_{{\rm S},p}}{\sigma^{\rm theo}_{{\rm 0},p}}$ (10) $\displaystyle=$
$\displaystyle-0.36\pm 0.11.$

The spin-independent cross section $\sigma^{\rm theo}_{{\rm 0},p}$ was
theoretically calculated with a Breit-Wigner formula, defined as,

$\displaystyle\sigma^{\rm{theo}}_{0,p}=\frac{9\pi}{16k^{2}}\frac{\Gamma^{n}_{p}\Gamma_{p}}{(E-E_{p})^{2}+(\Gamma_{p}/2)^{2}}.$
(11)

When using the nuclear polarization calculated from the temperature measured
at the cold head, the differences of $\sigma^{\rm exp}_{{\rm{S}},p}$ and
$A_{\rm S}$ from the values in Eq. 9 and Eq. 10 were $+0.03~{}\rm{barn}$ and
$+0.04$, respectively. These differences were smaller than the statistical
error.

Figure 7:  (a) TOF dependence of $P_{I}\sigma^{\rm exp}_{\rm S}$. The curved
line is the best fit of the global structure derived from the negative
$s$-wave resonance. (b) Resonance component of spin-dependent cross section.
(c) $p$-value for $P_{I}\sigma^{\rm exp}_{{\rm{S}},r}$. The dotted line shows
99.7% confidence level.

##  III Analysis

Under the experimental conditions, the spin-dependent assymetry can be
approximated as

$\displaystyle\varepsilon_{\rm S}\simeq P_{I}P_{n}\rho d\frac{4\pi}{k}{\rm
Im}B^{\prime},$ (12)

as described in the Appendix A, where $B^{\prime}$ is the coefficient in
Eq.(10) in Ref. Gudkov and Shimizu (2020) representing the spin-spin
interaction in the forward angle scattering amplitude. The following
subsections will discuss the implications of the experimental results to the
partial neutron width of the $p$-wave resonance and the spins of the $s$-wave
resonances.

###  III.1 Determination of partial neutron width using spin-dependent cross
section

The partial neutron width can also be extracted from the angular correlations
of $\gamma$-rays emitted from $p$-wave resonances, which arise from
interference between $s$\- and $p$-wave amplitudes Okudaira  _et al._ (2018,
2021); Yamamoto  _et al._ (2020); Koga  _et al._ (2022); Endo  _et al._
(2022). The advantage of using the spin-dependent cross section is that the
neutron partial width can be directly determined without assuming the
interference between partial amplitudes and the final state spin after the
$\gamma$ decay.

The spin-dependent cross section at the $p$-wave resonance can be calculated
using the explicit theoretical expression of $B^{\prime}$ as Gudkov and
Shimizu (2020)

$\displaystyle\sigma^{\rm{theo}}_{{\rm{S}},p}$ $\displaystyle=$
$\displaystyle\frac{4\pi}{k}{\rm{Im}}B^{\prime}=\frac{\pi}{16k^{2}}\frac{\Gamma^{n}_{p}\Gamma_{p}}{(E-E_{p})^{2}+(\Gamma_{p}/2)^{2}}$
(13) $\displaystyle\times\left(-\frac{39}{4}x_{\rm
s}^{2}+\frac{9}{2}\sqrt{\frac{7}{5}}x_{\rm s}y_{\rm s}+\frac{63}{20}y_{\rm
s}^{2}\right),$

where $x_{s}$ and $y_{s}$ are ratios of the neutron partial width of the
channel spin, defined as

$\displaystyle x_{\rm s}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{3}}(-\sqrt{7}x-\sqrt{5}y)$ (14) $\displaystyle
y_{\rm s}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{3}}(\sqrt{5}x-\sqrt{7}y).$ (15)

The neutron partial widths of the neutron total angular momentum $j=1/2$ and
$3/2$ components, denoted as $\Gamma^{n}_{p,j=1/2}$ and
$\Gamma^{n}_{p,j=3/2}$, are expressed by $x$ and $y$ defined as

$\displaystyle x^{2}=\frac{\Gamma^{n}_{p,j=1/2}}{\Gamma^{n}_{p}},\ \
y^{2}=\frac{\Gamma^{n}_{p,j=3/2}}{\Gamma^{n}_{p}},$ (16)

where $x$ and $y$ satisfy $x^{2}+y^{2}=1$. The corresponding mixing angle
$\phi$ can be defined as

$\displaystyle x=\cos\phi,~{}y=\sin\phi,$ (17)

as discussed in Ref Gudkov and Shimizu (2020). The broadening effect by the
pulse shape of the neutron beam at 0.75 eV was negligibly small compared with
the total width of the $p$-wave resonance and the statistical error, and
therefore, the spin-dependent cross section obtained in Eq. 9 can be directly
compared with the theoretical calculation. By calculating the Breit-Wigner
function over the region $E_{p}-3\Gamma_{p}<E_{n}<E_{p}+3\Gamma_{p}$ in Eq.
13, we obtained the following equation.

$\displaystyle-0.26\pm
0.08=0.079\left(-7x^{2}-2\sqrt{35}xy+\frac{2}{5}y^{2}\right)$ (18)

Using Eq. 17 and Eq. 18, we find the solutions for $\phi$ as

$\displaystyle\phi=$ $\displaystyle(74\pm 4)^{\circ},~{}(164\pm 4)^{\circ},$
(19) $\displaystyle~{}(254\pm 4)^{\circ},~{}(344\pm 4)^{\circ}.$

The corresponding $x$ and $y$ values are also obtained as

$\displaystyle(x,y)=$ $\displaystyle(0.28\pm 0.06,~{}0.96\pm 0.02),$ (20)
$\displaystyle(-0.96\pm 0.02,~{}0.28\pm 0.06),$ $\displaystyle(-0.28\pm
0.06,~{}-0.96\pm 0.02),$ $\displaystyle(0.96\pm 0.02,~{}-0.28\pm 0.06).$

The visualization of $\phi$ is shown in Fig. 8. Equation 18 is described as
the curved line in the $xy$ plane. The intersections of the curved lines and
unit circle show the solutions of $\phi$.

Figure 8:  Visualization of the $\phi$ values on the $xy$ plane. The curved
lines, shaded areas, and dashed lines show Eq. 18 and its 1$\sigma$ region of
the statistic error.

The above analysis was also performed using the resonance parameters reported
by other groups in Appendix B. The differences in the analysis results that
arose from differences in the resonance parameters were within the statistical
error. We confirmed that these differences stemming from the resonance
parameters do not affect the conclusions of this paper.

###  III.2 Spin of $s$-wave resonances

For the $s$-wave resonances, the spin $J$ can be directly determined from the
asymmetry. The positive (negative) sign of the asymmetry indicates that
neutrons with parallel (anti-parallel) spin are likely to be absorbed by
nuclei. The sign of the asymmetry in Fig. 7 (a) and (b) implies: $J=4$ for the
negative $s$-wave resonance of 139La, whose spin is $7/2$; $J=11/2$ for the
2.99 eV $s$-wave resonance of 138La, whose spin is 5; and $J=3$ for the 72.3
eV $s$-wave resonance of 139La. The spins of the $s$-wave resonances
determined in this experiment are consistent with the reference values in
Table 1.

##  IV Conclusion

We observed the spin-dependent cross section at the 0.75 eV $p$-wave of
139La+$n$ using a polarized lanthanum target and a polarized pulsed neutron
beam. The partial neutron width of the $p$-wave resonance was determined. In a
separate paper, these results will be compared with other experimental results
of ($n$,$\gamma$) reactions Okudaira  _et al._ (2018, 2021); Yamamoto  _et
al._ (2020); Koga  _et al._ (2022); Endo  _et al._ (2022); Okudaira  _et al._
(2023) in terms of the $s$-$p$ mixing model and will be used for improving a
quantitative understanding the symmetry violation enhancement mechanism.

###### Acknowledgements.

The authors would like to thank the staff of beamline 22 for the maintenance,
the low temperature sample environment team for the operation of the
superconducting magnet and the dilution refrigerator, and MLF and J-PARC for
operating the accelerators and the neutron production target. T. Okudaira
would like to especially thank S. Ohira-Kawamura and M. Matsuura for their
assistance designing the La holder. The neutron scattering experiment was
approved by the Neutron Scattering Program Advisory Committee of IMSS and KEK
(Proposals Nos. 2018S12). The neutron experiment at the Materials and Life
Science Experimental Facility of the J-PARC was performed under a user program
(Proposal No. 2022A0101). This work was supported by JSPS KAKENHI Grant Nos.
20K14495, 23K13122, JST SPRING Grant No. JPMJSP2125, and the US National
Science Foundation PHY-1913789 and PHY-2209481. R. Nakabe acknowledges support
from the Interdisciplinary Frontier Next-Generation Researcher Program of the
Tokai Higher Education and Research System. W. M. Snow acknowledges support
from the Indiana University Center for Spacetime Symmetries. J. G. Otero Munoz
acknowledges support from the National GEM consortium. V. Gudkov acknowledges
support from the U.S. Department of Energy Office of Science, Office of
Nuclear Physics program under Award No. DE-SC0020687.

##  Appendix A Neutron spin behavior in the polarized target and its
approximation

We employ the optical description of the neutron spin behavior in polarized
target material as described in Ref. Kabir (1988); Stodolsky (1986) to
describe the measured asymmetry $\varepsilon_{\rm S}$ as

$\displaystyle\varepsilon_{\rm S}$ $\displaystyle=P_{n}\frac{{\rm
Tr}(\mathfrak{S}^{{\dagger}}\sigma_{x}\mathfrak{S})}{{\rm
Tr}(\mathfrak{S}^{{\dagger}}\mathfrak{S})}=P_{n}\frac{2{\rm Re}A^{\ast}B+{2\rm
Im}C^{\ast}D}{\left|{A}\right|^{2}+\left|{B}\right|^{2}+\left|{C}\right|^{2}+\left|{D}\right|^{2}}.$
(21)

The coefficients $A$, $B$, $C$, and $D$ are related to the forward angle
scattering amplitude given in Ref. Gudkov and Shimizu (2020) as

$\displaystyle A$ $\displaystyle=e^{i\alpha}\cos\beta,\quad
B=ie^{i\alpha}\frac{\sin\beta}{\beta}\beta_{x}$ $\displaystyle C$
$\displaystyle=ie^{i\alpha}\frac{\sin\beta}{\beta}\beta_{z},\quad
D=ie^{i\alpha}\frac{\sin\beta}{\beta}\beta_{y}$ (22)

where

$\displaystyle\frac{\alpha}{Z}$
$\displaystyle=A^{\prime}+P_{1}H^{\prime}(\bm{k}\cdot\bm{I})+P_{2}E^{\prime}((\bm{k}\cdot\bm{I})^{2}-\frac{1}{3})$
$\displaystyle\frac{\beta_{x}}{Z}$
$\displaystyle=P_{1}B^{\prime}+\frac{\mu_{\rm n}m_{\rm
n}}{2\pi\hbar^{2}\rho}B_{\rm
ext}+P_{2}F^{\prime}(\bm{k}\cdot\bm{I})+P_{3}\frac{B_{3}^{\prime}}{3}((\bm{k}\cdot\bm{I})^{2}-1)$
$\displaystyle\frac{\beta_{y}}{Z}$
$\displaystyle=P_{1}D^{\prime}+P_{2}G^{\prime}(\bm{k}\cdot\bm{I})$
$\displaystyle\frac{\beta_{z}}{Z}$
$\displaystyle=C^{\prime}+P_{1}K^{\prime}(\bm{k}\cdot\bm{I})-P_{2}\frac{F^{\prime}}{3}+P_{3}\frac{2B_{3}^{\prime}}{3}(\bm{k}\cdot\bm{I})$
$\displaystyle\beta^{2}$
$\displaystyle=\beta_{x}^{2}+\beta_{y}^{2}+\beta_{z}^{2},\quad
Z=\frac{2\pi\rho d}{k}.$ (23)

Here, $B_{\rm{ext}}$, $\mu_{n}$, and $m_{n}$ denote the external magnetic
field, neutron magnetic moment, and mass, respectively. $\bm{k}$ and $\bm{I}$
are unit vectors parallel to the neutron momentum and the nuclear spin. The
$P_{1}$, $P_{2}$ and $P_{3}$ represent the target nuclear polarization of 1st-
rank (vector), 2nd-rank, and 3rd-rank spherical tensors, respectively, and
amounts $P_{1}=P_{I}=3.9\pm 0.5\%$, $P_{2}=0.10^{+0.03}_{-0.02}\%$,
$P_{3}=(2.1\pm 1.0)\times 10^{-3}\%$ in the present experimental conditions.
Non-zero values of the $(\bm{k}\cdot\bm{I})$ originate from the beam
divergence up to the maximum value of $2\times 10^{-3}$.
The valuables $A^{\prime}$-$G^{\prime}$ are the coefficients of correlation
terms in the forward scattering amplitude for polarized 139La nuclei and
polarized neutrons defined as Gudkov and Shimizu (2020):

$\begin{split}f=&A^{\prime}+P_{1}H^{\prime}({\bm{k}}\cdot{\bm{I}})+P_{2}E^{\prime}\left({(\bm{k}}\cdot{\bm{I}})^{2}-\frac{1}{3}\right)\\\
&+(\bm{\sigma}\cdot{\bm{I}})\left(P_{1}B^{\prime}+P_{2}F^{\prime}({\bm{k}\cdot{\bm{I}}})+P_{3}\frac{B_{3}^{\prime}}{3}\left(({\bm{k}\cdot{\bm{I}}})^{2}-1\right)\right)\\\
&+(\bm{\sigma}\cdot{\bm{k}})\left(C^{\prime}+P_{1}K^{\prime}({\bm{k}}\cdot{\bm{I}})-P_{2}\frac{F^{\prime}}{3}+P_{3}\frac{2B_{3}^{\prime}}{3}({\bm{k}\cdot{\bm{I}}})\right)\\\
&+\bm{\sigma}\cdot({\bm{k}\times{\bm{I}}})\left(P_{1}D^{\prime}+P_{2}G^{\prime}({\bm{k}}\cdot{\bm{I}}).\right)\end{split}$
(24)

The magnitude of coefficients $H^{\prime}$, $E^{\prime}$, $F^{\prime}$,
$B_{3}^{\prime}$, $K^{\prime}$ and $G^{\prime}$ are of the same order or less
than $A^{\prime}$ and $B^{\prime}$ on the basis of the explicit expressions in
Eq. (28)-(37) of Ref. Gudkov and Shimizu (2020). The magnitudes of P-odd,
T-even term $C^{\prime}$ and P-odd, T-odd term $D^{\prime}$ are smaller than
$A^{\prime}$ and $B^{\prime}$ by more than two orders of magnitudes.
Consequently, the value of $\beta$ can be approximated as
$\beta\simeq\beta_{x}$ and we obtain

$\displaystyle\varepsilon_{\rm S}$ $\displaystyle\simeq P_{n}\tanh(2{\rm
Im}\beta_{x}).$ (25)

The numerical value of ${\rm Im}\beta_{x}$ is about $10^{-3}$, which leads to

$\displaystyle\varepsilon_{\rm S}\simeq P_{I}P_{n}\rho d\frac{4\pi}{k}{\rm
Im}B^{\prime}.$ (26)

##  Appendix B Analysis using resonance parameters reported in other
references

For the 0.75 eV $p$-wave resonance, the measurements using neutron
transmission or ($n$, $\gamma$) reaction have been reported by several groups
listed in Table 3 Terlizzi  _et al._ (2007); Alfimenkov  _et al._ (1983); Shwe
_et al._ (1967); J.A.Harvey  _et al._ (1959). The details of each measurement
of the resonance parameters are summarized in Ref. Endo  _et al._ (2023).

| Reference |  $E_{0}$ [eV] |  $\Gamma_{\gamma}$ [meV] |  $g\Gamma_{n}$ [meV]
---|---|---|---|---
| Endo et al. (2023) Endo  _et al._ (2023) | $0.750\pm 0.001$ | $41.6\pm 0.9$ | $(3.67\pm 0.05)\times 10^{-5}$
| Terlizzi et al. (2007) Terlizzi  _et al._ (2007) | $0.758\pm 0.001$ | $40.1\pm 1.9$ | $(5.6\pm 0.5)\times 10^{-5}$
| Alfimenkov et al. (1983) Alfimenkov  _et al._ (1983) | $0.75\pm 0.01$ | $45\pm 5$ | $(3.6\pm 0.3)\times 10^{-5}$
| Shwe et al. (1967) Shwe  _et al._ (1967) | $0.734\pm 0.005$ | $40\pm 5$ | $(3.67\pm 0.22)\times 10^{-5}$
| Harvey et al. (1959) J.A.Harvey  _et al._ (1959) | $0.752\pm 0.011$ | $55\pm 10$ | $(4\pm 1)\times 10^{-5}$
Table 3:  Resonance parameters of 139La $p$-wave resonance reported in each reference. | Reference for
---
resonance parameters
$\sigma^{\rm{exp}}_{{\rm{S}},p}$ [barn] | $A_{\rm S}$ | $\phi~{}[\rm{degree}]$
Endo et al. | $-0.26\pm 0.07$ | $-0.36\pm 0.10$ | $~{}74\pm 4,~{}164\pm 4,~{}254\pm 4,~{}344\pm 4$
Terlizzi et al. | $-0.26\pm 0.07$ | $-0.24\pm 0.06$ | $~{}79\pm 2,~{}159\pm 2,~{}259\pm 2,~{}339\pm 2$
Alfimenkov et al. | $-0.23\pm 0.06$ | $-0.36\pm 0.10$ | $~{}74\pm 4,~{}164\pm 4,~{}254\pm 4,~{}344\pm 4$
Shwe et al. | $-0.26\pm 0.07$ | $-0.35\pm 0.09$ | $~{}75\pm 4,~{}163\pm 4,~{}255\pm 4,~{}343\pm 4$
Harvey et al. | $-0.19\pm 0.06$ | $-0.33\pm 0.10$ | $~{}76\pm 4,~{}163\pm 4,~{}256\pm 4,~{}343\pm 4$
Table 4:  Analysis results using the resonance parameters in each reference.

Tables 4 shows $\sigma^{\rm exp}_{{\rm S},p}$, $A_{\rm S}$, and $\phi$ values
obtained using resonance parameters reported by each group. The central values
for $A_{\rm S}$ show agreement within an accuracy of 10% or less with the
exception of that based on resonance parameters reported by Terilzzi et al..
The central value of $A_{\rm S}$ using this resonance parameters exhibit a
difference of around 30%, which is attributed to Terilzzi et al.’s
$g\Gamma_{n}$ being reported as approximately 30% larger than in other
references. Consequently, the $\phi$ values obtained using resonance
parameters reported by Terilzzi et al. show difference compared to the
analysis using other resonance parameters, as illustrated in Fig. 9. However,
these differences remain consistent within the statistical errors obtained in
the present experiment.

Figure 9:  Visualization of the $\phi$ values obtained using Terilzzi et al.’s
resonance parameters on the $xy$ plane. The curved lines and shaded area shows
Eq. 18 and its 1$\sigma$ region of the statistic error.

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# Impact Parameter Dependence of Inelasticity in $pp$ / $p\overline{p}$
Collisions

P. C. Beggio<EMAIL_ADDRESS>Laboratório de Ciências Matemáticas, Universidade
Estadual do Norte Fluminense Darcy Ribeiro - UENF,
Campos dos Goytacazes - RJ, Brazil.

###### Abstract

We study the impact parameter dependence of inelasticity in the framework of
an updated geometrical model for multiplicity distribution. A formula in which
the inelasticity is related to the eikonal is obtained. This framework permits
a calculation of the multiplicity distributions as well as the inelasticity
once the eikonal function is given. Adopting a QCD inspired parametrization
for the eikonal, in which the gluon-gluon contribution dominates at high
energy and determines the asymptotic behavior of the cross sections, we find
that the inelasticity decreases as collision energy is increased. Our results
predict the KNO scaling violation observed at LHC energies by CMS
Collaboration.

###### keywords:

Eikonal approximation $pp$ / $\bar{p}p$ Inelasticity Multiplicity distribution
13.85.Hd 12.40.Ee 13.85.Ni

##  1 Introduction

Inelasticity is defined as the fraction of the available energy relesead for
multiple particle production in inelastic hadronic interactions. The remaining
part of the incident energy is carried away by the participant’s remnants, so
called leading particles. The energy dependence of inelasticity is a problem
of great interest from both theoretical and experimental standpoints [1].
However, the experimental data are scarce and, on the theoretical side, the
existing models are largely in conflict with each other even in explaining a
simple aspect as the center-of-mass energy dependence of the inelasticity [2],
[3]. For example, the decrease in inelasticity with energy is advocated by
some authors, while others believe that the inelasticity is an increasing
function of energy [1]. Hence the problem remains unsolved. Naturally,
multiplicity distributions are connected to inelasticity ones, so one can
study multiplicity distribution features in order to derive information on the
inelasticity behavior. Following this way, we have updated an existing
phenomenological procedure [4], referred as _Simple One String Model_ , which
allows simultaneous description of several experimental data from elastic and
inelastic channels through the Unitarity Equation [4]. Thus, based on the
_Simple One String Model_ formalism, able to describe the charged multiplicity
distributions from ISR to Collider energies (30.4 - 900 GeV), we have computed
the impact parameter dependence of inelasticity at fixed center-of-mass
energies, $\sqrt{s}$. We have also inferred information on energy dependence
of inelasticity. The plan of the paper is as follows. In the section 2 we
present the basic formalism of the _Simple One String Model_ and the
predictions for $pp/p\overline{p}$ overall multiplicity distributions compared
with the experimental data. In Section 3 we apply the theoretical framework
computing the inelasticity of hadronic reactions. The final remarks are the
content of Section 4.

##  2 _Simple One String Model_ for Multiplicity Distributions

The _Simple One String Model_ has been discussed in references [4], [5] and in
order to define the notation and also update the model, we shall review the
main points here. We work in impact parameter space, b, and to guarantee
unitarity, the inelastic cross sections, $\sigma_{in}$, is calculated via the
relation:

$\sigma_{in}(s)=\int d^{2}b\,Gin(s,b)\,,$ (1)

where

$Gin(s,b)=1-e^{-2\chi_{I}(s,b)}$ (2)

is the Inelastic Overlap Function. In this work we have update the model
adopting the complex eikonal function
$\chi_{pp}^{\overline{p}p}{(s,b)}=\chi_{R}{(s,b)}+i\chi_{I}{(s,b)}$, from Ref.
[6], as will be discussed in subsection 2.1. In Ref. [4] has been adopted the
Henzi Valin parametrization for $Gin(s,b)$. The probabilities of $n$ particle
production, namely multiplicity distribution, $P_{n}$, is the most general
feature of the multiparticle production processes [7] and measurements of
charged particle multiplicity distributions have revealed intrinsic features
in $pp$ / $p\overline{p}$ interactions [8]. The multiplicity distribution is
defined by the formula [9]

$P_{n}(s)={\frac{\sigma_{n}(s)}{\sum_{n=0}^{\infty}\sigma_{n}(s)}}={\frac{\sigma_{n}(s)}{\sigma_{in}(s)}},$
(3)

where $\sigma_{n}$ is the cross section of an $n$-particle process (the so-
called topological cross section). The charged multiplicity distribution, in
the impact parameter formalism, may be constructed by summing contributions
coming from hadron-hadron collisions taking place at fixed impact parameter.
In this way, the idea of a normalized multiplicity distribution at each impact
parameter $b$ is introduced [10]. Thus the multiplicity distribution is
written as

$P_{n}(s)={\frac{\sigma_{n}(s)}{\sigma_{in}(s)}}={\frac{\int
d^{2}b\,G_{in}(s,b)\left[{\frac{\sigma_{n}(s,b)}{\sigma_{in}(s,b)}}\right]}{\int
d^{2}b\,G_{in}(s,b)}}\,,$ (4)

where the topological cross section $\sigma_{n}$ is decomposed into
contributions from each impact parameter $b$ with weight $G_{in}(s,b)$. In the
original formulation [4] the quantity em brackets scales in KNO sense, and the
Eq. (4) can be written as

$P_{n}(s)={\frac{\int
d^{2}b\,{\frac{G_{in}(s,b)}{<n(s,b)>}}\left[<n(s,b)>{\frac{\sigma_{n}(s,b)}{\sigma_{in}(s,b)}}\right]}{\int
d^{2}b\,G_{in}(s,b)}}\,,$ (5)

where $<n(s,b)>$ is the average number of particles produced at $b$ and
$\sqrt{s}$ due to the interactions among hadronic constituents involved in the
collision and, in this model, $<n(s,b)>$ factorizes as [4]

$<n(s,b)>=<N(s)>f(s,b),$ (6)

where $<N(s)>$ is the average multiplicity at $\sqrt{s}$ and $f(s,b)$ is the
so called multiplicity function. Similarly to KNO, is introduced, for each
$b$, the elementary multiplicity distribution

$\psi^{(1)}\left(\frac{n}{<n(s,b)>}\right)=<n(s,b)>{\frac{\sigma_{n}(s,b)}{\sigma_{in}(s,b)}}.$
(7)

Thus, with Eqs. (6) and (7), Eq. (5) becomes

$\Phi(s,z)={\frac{\int
d^{2}b\,{\frac{G_{in}(s,b)}{f(s,b)}}\,\psi^{(1)}\\!\left(\frac{z}{f(s,b)}\right)}{\int
d^{2}b\,G_{in}(s,b)}},$ (8)

where $\Phi(s,z)=<N(s)>P_{n}(s)$ and $z=n/<N(s)>$. Here $z$ represents the
usual KNO scaling variable. Now, to obtain the multiplicity function $f(s,b)$
in terms of the imaginary eikonal $\chi_{I}$, it has been assumed that

  1. 1.

the fractional energy $\sqrt{s^{{}^{\prime}}}$ that is deposited for particle
production in a collision at $b$ is proportional to $\chi_{I}$:

$\sqrt{s^{{}^{\prime}}}=\beta(s)\,\chi_{I}(s,b)\,.$ (9)

The physical motivation of this equation is that eikonal may be interpreted as
an overlap, on the impact parameter plane, of two colliding matter
distributions [10];

  2. 2.

the average number of produced particles depends on the energy
$\sqrt{s^{{}^{\prime}}}$ in the same way as in $e^{-}e^{+}$ annihilations,
which is approximately represented by a power law in $\sqrt{s}$ [4]

$<n(s,b)>=\gamma\,\left(\frac{s^{{}^{\prime}}}{s^{{}^{\prime}}_{0}}\right)^{A},$
(10)

where $s^{{}^{\prime}}_{0}$=1 GeV2. In Ref. [11] a power law energy dependence
of multiplicity in both $pp$ and $Pb+Pb$ collisions has been analyzed based on
the gluon saturation scenario. Now, combining Eqs. (9), (10) and (6), we have

$f(s,b)=\frac{\gamma}{<N(s)>}\left[\frac{\beta(s)}{\sqrt{s^{{}^{\prime}}_{0}}}\right]^{2A}[\chi_{I}(s,b)]^{2A}=\xi(s)[\chi_{I}(s,b)]^{2A},$
(11)

with $\xi(s)$ determined by the usual normalization conditions on $\Phi$ [4]
and that serves to determine $\xi(s)$ as an energy dependent quantity,
explicitly [4]

$\xi(s)=\frac{\int d^{2}b\,G_{in}(s,b)}{\int
d^{2}b\,G_{in}(s,b)[\chi_{I}(s,b)]^{2A}}.\ $ (12)

Thus, adopting an appropriate parametrization for $G_{in}$ and $\psi^{(1)}$,
as well as an adequate value for $A$, we can test the formalism embodied in
Eq. (8) and (12), making direct comparisons with multiplicity distribution
data. In the following, we will discuss the results obtained in the context of
our updated model.

###  2.1 Inputs and Results on Multiplicity Distributions

The _Simple One String Model_ is based on the idea of multiparticle creation
due to the interactions between hadronic constituents in collisions taking
place at $b$. It is assumed that in parton-parton collision there is formation
of a string, in which probably one $q\overline{q}$ has triggered the multitude
of the final particles. In previous analysis the authors [4] considered the
experimental data available on $e^{-}e^{+}$ annihilations as possible source
of information concerning elementary hadronic interactions and, in this work,
we borrow their results. Specifically, assuming a gamma distribution
normalized to 2,

$\psi^{(1)}(z)=2\,\frac{k^{k}}{\Gamma(k)}z^{k-1}e^{-kz},$ (13)

experimental data on $e^{-}e^{+}$ multiplicity distributions were fitted,
obtaining _k_ =10.775 $\pm$ 0.064 $(\chi^{2}/N_{DF}=2.61)$. Also, the average
multiplicity data in $e^{-}e^{+}$ annihilations were fitted by Eq. (10),
giving $A$=0.258 $\pm$ 0.001 and $\gamma$=2.09 $(\chi^{2}/N_{DF}=8.89)$ in the
interval 5.1 GeV $\leq\sqrt{s}\leq$ 183 GeV, and $A$=0.198 $\pm$ 0.004
$(\chi^{2}/N_{DF}=1.7)$ for the set in the interval 10 GeV $<\sqrt{s}\leq$ 183
GeV, respectively [4]. In the analysis done in Ref. [11] the value of $A$=0.11
was obtained within the gluon saturation picture. The difference between the
values of $A$ is, probably, associated with the different sets of experimental
data used in each analysis. We recall that the value of $A$=0.11 was obtained
by using experimental data for average multiplicity of hadrons in the gluon
and quark jets in $e^{-}e^{+}$ annihilation, in the interval of the jet energy
between 0.6 $\sim$ 32 GeV [11]. At the end of the section the _One String
Model_ formalism will be tested using the three $A$ values, above mentioned.
Now is needed a parametrization for the eikonal function, and we have adopted
the QCD-inspired complex eikonal from the work of Block _et al._ [6] in which
the eikonal function is written as a combination of an even and odd eikonal
terms related by crossing symmetry
$\chi_{pp}^{\overline{p}p}{(s,b)}=\chi^{+}{(s,b)}\pm\chi^{-}{(s,b)}$. The even
eikonal is written as the sum of gluon-gluon, quark-gluon and quark-quark
contributions:

$\chi^{+}{(s,b)}=\chi_{qq}{(s,b)}+\chi_{qg}{(s,b)}+\chi_{gg}{(s,b)},$ (14)

while the odd eikonal, that accounts for the difference between $pp$ and
$p\overline{p}$, is parametrized as

$\chi^{-}{(s,b)}=C^{-}\sum\frac{m_{g}}{\sqrt{s}}e^{i\pi/4}W(b;\mu^{-}).$ (15)

The various parameters and functions involved in last two expressions are
discussed in Ref.[6]. By fixing the value of $A$=0.258, $\psi^{(1)}$ given by
Eq. (13) with $k$=10.775, adopting $G_{in}$ from analysis by Block _et al_.
and observing that $\xi(s)$ is obtained by Eq. (12), we have computed the
overall multiplicity distributions arising from $pp/\overline{p}p$ collisions
at energies 52.6, 200, 546 and 900 GeV. The theoretical curves are shown in
Figs. 1, 2, 3 and 4 together with the experimental data. The curves at
$pp$-ISR $52.6$ GeV and CERN $p\overline{p}$ Collider $546$ GeV shows
excellent agreement with the data, Figs. 1 and 3 respectively. At energies
$\sqrt{s}$=200 and 900 GeV, also in CERN $p\overline{p}$ Collider, the
agreement with data seems reasonable since the curves agree with experimental
points for $z^{\prime}\gtrsim 1$, (high multiplicities), Figs. 2 and 4
respectively. In view of the recent results on multiplicity distributions, in
$pp$ collisions at $\sqrt{s}$=0.9, 2.36 and 7 TeV at the Large Hadron Collider
(LHC), reported by CMS Collaboration [15],

Figure 1: Overall scaled multiplicity distribution data for $pp$ at ISR energy
[12], compared to theoretical prediction using the _Simple One String Model_ ,
Eqs. (8) and (12). Figure 2: Overall scaled multiplicity distribution data for
$p\overline{p}$ at Collider energy [13], compared to theoretical prediction
using the _Simple One String Model_ , Eqs. (8) and (12).

we have also computed the overall multiplicity distributions that the _One
String Model_ predicts at LHC energies. The results are shown in Fig. 5 and we
can see violation of KNO scaling, in qualitative agreement with the result
obtained by CMS Collaboration in pseudorapidity interval of $|\eta|<$2.4, as
discussed in Ref. [15]. As mentioned, two fit values of $A$=0.258 and
$A$=0.198 have been obtained in the previous study [4], the first one giving a
better account of lower energy data whereas the second one higher energy data.
As before, we have computed the corresponding hadronic multiplicity
distribution by fixing both the gamma parametrization for $\psi^{(1)}$, Eq.
(13), and the complex eikonal, Eqs. (14) and (15), and considered the two
parametrizations for the average multiplicity, $\sim s^{0.258}$
($\xi(s)$=1.424) and $\sim s^{0.198}$ ($\xi(s)$=1.348). The results at 546 GeV
are shown in Fig. 6 and, for $A$=0.198, we can see the disagreement of the
theoretical curve when compared to the data. As pointed out in [4] the
parametrization $\sim s^{0.198}$ brings information from data at high
energies, while the parametrization $\sim s^{0.258}$ is in agreement with data
at smaller energies. However, the information from the $e^{-}e^{+}$ average
multiplicities at high energies does not reproduce the overall multiplicity
distribution. Hence, by using $A$=0.256 the output seems to be more consistent
with data. In addition, there is no evidence of gluon saturation at CERN
$p\overline{p}$ Collider 546 GeV, however and as a pedagogical exercise, we
have also computed $\Phi$ considering the value of $A$=0.11 ($\xi(s)$=1.211),
Fig. 6. We would like emphasize that when the formalism is applied,
considering the three values of $A$ (0.258, 0.198 and 0.11) at energies 52.6,
200 and 900 GeV, the results are essencially the same obtained in Fig. 6. We
have expressed $\Phi$ in terms of modified the scaling variable
$z^{\prime}=n-N_{o}/<n-N_{o}>$ with $N_{o}$=0.9 representing the average
number of leading particles [4].

##  3 Inelasticity

The concept of inelasticity is essential since it defines the energy available
for particle production in high energy hadronic and nuclear collisions.
However, the impact parameter dependence of the inelasticity is a problem
unsolved. In theoretical works, it is quite natural to assume that the
multiplicity distribution and inelasticity are connected. Indeed, some authors
has defined multiplicity distributions in terms of inelasticity as [1], [16]

$P_{n}(s)=\int_{0}^{1}P(n|K)P[K(s)]dK,$ (16)

where $P[K(s)]$ is the inelasticity distribution and $P(n|K)$ is the
probability of the production of $n$ particles at the given inelasticity $K$.
Thus, based on the connection between multiplicity and inelasticity, we have
explored the parametrization of the _One String Model_ formalism and computed
the impact parameter dependence of inelasticity, as will discuss in next two
subsections.

Figure 3: Overall scaled multiplicity distribution data for $p\overline{p}$ at
Collider energy [14], compared to theoretical prediction using the _Simple One
String Model_ , Eqs. (8) and (12). Figure 4: Overall scaled multiplicity
distribution data for $p\overline{p}$ at Colider energy [13], compared to
theoretical prediction using the S _imple One String Model_ , Eqs. (8) and
(12). Figure 5: Theoretical predictions for overall multiplicity distribution
by using the _Simple One String Model_ , Eqs. (8) and (12), at LHC energies.
Figure 6: Overall scaled multiplicity distribution data for $p\overline{p}$ at
Collider energy [14], compared to theoretical prediction using the _Simple One
String Model_ , Eqs. (8) and (12), considering three different values of $A$,
Eq. (10).

###  3.1 Similarities between $pp/p\overline{p}$ and $e^{+}e^{-}$ Collisions

The idea of a universal hadronization mechanism is not new and similarities
between both processes were indeed observed [16], [17], [18]. For example, the
average multiplicities in $pp/p\overline{p}$ and $e^{+}e^{-}$ collisions
become similar when comparisons are made at the same effective energy for
hadron production. In $pp/p\overline{p}$ collisions the effective energy for
particle production, $E_{eff}$, is the energy left behind by the two leading
protons

$E_{eff}=(\sqrt{s})_{pp}-(E_{leading,1}+E_{leading,2}),$ (17)

or

$E_{eff}=(\sqrt{s})_{pp}-2E_{leading},$ (18)

in the case of symmetric events. (We recall that $(\sqrt{s})_{pp}$ and
$\sqrt{s}$ represents both the center-of-mass energy, however, in this
subsection the notation $(\sqrt{s})_{pp}$ is helpful to differentiate that
from $(\sqrt{s})_{e^{+}e^{-}}$). In $e^{+}e^{-}$ collisions the effective
energy for hadron production coincides with total center-of-mass energy of the
beam

$E_{eff}=(\sqrt{s})_{e^{+}e^{-}}=2E_{beam}.$ (19)

Thus, the same equivalent energy for both $pp/p\overline{p}$ and $e^{+}e^{-}$
collisions can be written as

$(\sqrt{s})_{pp}-2E_{leading}=E_{eff}=(\sqrt{s})_{e^{+}e^{-}}.$ (20)

For the quantitative estimation of the inelasticity, $K$, we can use the
definition [1], [2]

$E_{eff}=K(\sqrt{s})_{pp}\Rightarrow K=\frac{E_{eff}}{(\sqrt{s})_{pp}}.$ (21)

In the following, we will explore the _One String Model_ formalism, able to
describe the multiplicity distributions in wide interval of energy (30.4 - 900
GeV), to obtain information about inelasticity.

###  3.2 Computation of Inelasticity

The Eq. (9) is a key point of the formalism. Physically, it corresponds to the
energy for hadron production deposited at $b$, due to the interactions among
hadronic constituents involved in the collision. Thus, and as discussed in
last section, the fractional energy, $\sqrt{s^{\prime}}$ (Eq. (9)), and the
effective energy for hadron production, $E_{eff}$ (Eq. (21)), represents both
the same physical quantity ($\sqrt{s^{{}^{\prime}}}=E_{eff}$). Now, by using
the Eq. (9), let us write the inelasticity, Eq. (21), as a function of
$\sqrt{s}$ and $b$ as

$2K(s,b)=\frac{\sqrt{s^{{}^{\prime}}}}{(\sqrt{s})_{pp}}=\frac{\beta(s)\chi_{I}(s,b)}{\sqrt{s}}.$
(22)

The factor 2, in the above equation, is due to the fact that the multiplicity
distributions data are normalized to 2. However, we can not calculate the
$K(s,b)$ until the value of $\beta(s)$ is known. To estimate $\beta(s)$ we
note that the parameter $\xi(s)$, which is introduced in Eq. (11), is related
with $\beta(s)$ by

$\xi(s)=\frac{\gamma}{<N(s)>}\left[\frac{\beta(s)}{\sqrt{s^{{}^{\prime}}_{0}}}\right]^{2A}.$
(23)

We recall that $<N(s)>$ is the average multiplicity at $\sqrt{s}$. By using
the values of $A=0.258$, $\gamma$=$2.09$, discussed in subsection 2.1, and the
values of $<N(s)>$ imputed from experiments [12], [13] and [14], and also
observing that $\xi(s)$ is obtained from Eq. (12), we have estimated the
values of $\beta$ at various energies. The results are displayed in Table 1.
We can see clearly that $\beta$ increases as the collision energy also
increases. $\beta$ can be parameterized as $\beta(s)=77.48\sqrt{s}+0.4168$
with $\chi^{2}/N_{DF}$=1.

Table 1:  $\beta(s)$ estimated values at various energies. The values of $<N(s)>$ was imputed from Refs. [12], [13] and [14]. $\sqrt{s}$ GeV | $\xi(s)$ | $<N(s)>$ |  $\beta(s)$ GeV
---|---|---|---
52.6 | $1.612$ | $11.55$ | $69.295$
200 | $1.517$ | $21.4$ | $203.531$
546 | $1.424$ | $27.5$ | $292.729$
900 | $1.377$ | $35.6$ | $452.376$
2360 | $1.286$ | $-$ | $1061.13$
7000 | $1.188$ | $-$ | $2995.08$
14000 | $1.130$ | $-$ | $5912.68$

Now we proceed to compute the impact parameter dependence of inelasticity and
infer some information on its energy dependency.

###  3.3 Results and Discussion

Based on the connection between multiplicity distribution and inelasticity, we
have update and applied the _One String Model_ formalism deriving an
expression, Eq. (22), which allows us to study the impact parameter and energy
dependence of inelasticity. Adopting the Block _et al._ QCD-inspired
parametrization for $\chi_{pp}^{\overline{p}p}{(s,b)}$ [6] and by using the
estimated values of $\beta(s)$, Table 1, we have applied the Eq. (22) by
fixing the collision energy and computed the inelasticity as a function of
$b$. We show, in Fig. 7, the results from our analysis. Naturally, the
inelasticity decreases as a function of impact parameter, $b$. The
inelasticity behavior is essentially the same at energies 546 and 900 GeV. At
energies 52.6 and 200 GeV, we can see appreciable difference just in the
region of $b\sim 0$ (central collisions). It is interesting to note that, from
the range of collision energy 50 $\sim$ 200 GeV to that one 500 $\sim$ 900
GeV, the inelasticity shows a difference about 60 percent in its values for
$b$ $\lesssim 0.5$ _fm_. We can also see that, at fixed $b$, the inelasticity
$K$ decreases as $\sqrt{s}$ increases in the interval 52.6 - 900 GeV. The Eq.
(22) depends on the eikonal and $\beta$ parameter. With the

Figure 7: Impact parameter dependence of the inelasticity by using the formula
obtained in this work, Eq. (22).

eikonal as determined phenomenologically in [6] as input, where high energy
cross sections grow with energy as a consequence of the increasing number of
soft partons populating the colliding particles $(pp/p\overline{p})$, it seems
quite natural to expect that multiparton interactions leads to larger
multiplicities/inelasticities as consequence to the full development of the
gluonic structure. However, looking the same impact parameter dependence of
inelasticity functions at 546 and 900 GeV, Fig. 7, we would be tempted to
conclude that we are observing saturation effects due gluon recombination in
the inelasticity, but there is no evidence of saturation in this range of
energy (52.6 - 900 GeV). We have also applied our approach at energies
$\sqrt{s}$=2.36 and 7 TeV (LHC), as shown in Fig. 7, and the results suggest
that the inelasticity is an increasing function of energy for the interval
2.36 - 7 TeV. As mentioned before, our main purpose is study features of
multiplicity distributions deriving information on inelasticity and our
analysis is based on the model in which is assumed that in parton-parton
collision there is formation of a string. Thus, despite some simplifications
made in the _One String Model_ , the results seems to be consistent with the
multiplicity distributions data in a wide interval of energy (52.6 - 900 GeV).
Hence, the computed inelasticities, in this range of energy, are reliable
results. In counterpart, the results at $\sqrt{s}$=2.36 and 7 TeV are
inconclusive in the context our analysis, because the _One String Model_
probably underestimates the high multiplicities events due to the lack of the
multicomponent structure in its formulation. In fact, recent results reported
by CMS Collaboration pointed out the importance of a multicomponent structure
in hadron-hadron inelastic interactions, in agreement with previous
experimental results (for details see [15]). Inelasticity also has been
studied recently in Ref. [19], where, in the context of both Wdowczyk and
Wolfendale model and UHECR data analysis, it was found that the inelasticity
decreases in very high energy interactions, and, in the same work and by using
the modified Feynman scaling formula, the inelasticity is an increasing
function of the energy. It reflects the subtlety of the theme. We note that at
ISR Energies (30-60 GeV), where the leading particle spectrum could be
measured, the inelasticity is defined to be about 0.5. This value can be
identified with $pp$ collision taking place at $b\sim$ 0.6 _fm_ , Fig. 7. The
_One String Model_ has been used to study the influence on $\Phi$ considering
possible values of $A$ parameter at $\sqrt{s}$=546 GeV, Fig. 6. In addition,
we have also computed $\Phi$ at energies 52.6, 200 and 900 GeV using the
different values of $A$ (0.258, 0.198 and 0.11) and the results, in each
energy, are similar with that obtained in Fig. 6. Finally, we emphasize that
the curves in Figs. 1-4 has not been fit to data, except for the values of $A$
and $k$ (fixed) no experimental information about multiplicity distribution
has gone into the calculation. Hence, the energy evolution of the multiplicity
distributions, from ISR to Collider ($30\sim 900\,$GeV), is correctly
reproduced by changing only the Overlap Function, Eq.(2), without changing the
underlying elementary interaction, in agreement with what could be expected
from QCD.

##  4 Concluding Remarks

Being the impact parameter $b$ an essential variable in a geometrical
description of hadronic collisions, we have investigated the $b$ dependence of
inelasticity and also inferred some information on its energetic behavior. By
using a geometrical model we have derived an expression for $K$ based on the
hypothesis of connection between multiplicity distribution and inelasticity.
We have adopted the Block _et al._ model in our analysis, where the eikonal
functions $\chi_{qq}{(s,b)}$ and $\chi_{qg}{(s,b)}$ are needed to describe the
lower energy forward data, while $\chi_{gg}{(s,b)}$ contribution dominates at
high energy and determines the asymptotic behavior of cross sections. We
believe that the same impact parameter dependence of the inelasticity at
energies 546 and 900 GeV can have important implications for the underlying
gluon-gluon dynamics. In fact, we are testing our formalism by using the QCD
Eikonal Model in which the gluon may develop a dynamical mass [20], [21]. At
energies $\sqrt{s}$=2.36 and 7 TeV the _One String Model_ parametrization can
not be tested, hence the results do not allow for any conclusion. Finally, the
results suggest that there are relationships between the inelasticity and the
eikonal function.

I am grateful to E.G.S. Luna for several instructive discussions.

## References

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# Science based AI model certification for untrained operational environments
with application in traffic state estimation

Daryl Mupupuni1, Anupama Guntu2, Liang Hong3, Kamrul Hasan4 and Leehyun Keel5
Department of Electrical and Computer Engineering, Tennessee State University
Nashville, TN 37209, USA
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The expanding role of Artificial Intelligence (AI) in diverse engineering
domains highlights the challenges associated with deploying AI models in new
operational environments, involving substantial investments in data collection
and model training. Rapid application of AI necessitates evaluating the
feasibility of utilizing pre-trained models in unobserved operational settings
with minimal or no additional data. However, interpreting the opaque nature of
AI’s black-box models remains a persistent challenge. Addressing this issue,
this paper proposes a science-based certification methodology to assess the
viability of employing pre-trained data-driven models in untrained operational
environments. The methodology advocates a profound integration of domain
knowledge, leveraging theoretical and analytical models from physics and
related disciplines, with data-driven AI models. This novel approach
introduces tools to facilitate the development of secure engineering systems,
providing decision-makers with confidence in the trustworthiness and safety of
AI-based models across diverse environments characterized by limited training
data and dynamic, uncertain conditions. The paper demonstrates the efficacy of
this methodology in real-world safety-critical scenarios, particularly in the
context of traffic state estimation. Through simulation results, the study
illustrates how the proposed methodology efficiently quantifies physical
inconsistencies exhibited by pre-trained AI models. By utilizing analytical
models, the methodology offers a means to gauge the applicability of pre-
trained AI models in new operational environments. This research contributes
to advancing the understanding and deployment of AI models, offering a robust
certification framework that enhances confidence in their reliability and
safety across a spectrum of operational conditions.

###### Index Terms:

Artificial Intelligence (AI), Model Certification, Scientific Knowledge,
Physical Inconsistency, Traffic State Estimation (TSE)

##  I Introduction

Artificial intelligence (AI) has experienced tremendous growth and application
across several fields [1, 2, 3]. Application of AI has completely changed the
engineering sector over the last ten years by allowing engineers to design
creative ideas, solve complex problems, and expedite operations in several
engineering disciplines. For instance, various environmental engineering
problems were solved by AI based prediction models such as pollution control
[4], wastewater to improve the performance of several chemical and logical
treatment processes [5]. AI is also useful in autonomous systems and proactive
maintenance in the aerospace and automobile industries [6]. AI is changing
engineering by automating tasks, improving designs, and enabling predictive
analysis. The integration of AI will continue driving innovation in the future
[7].

Although AI models like Artificial Neural Networks, Ensemble Approaches etc.,
continue to perform better than other models in many fields, but their
acceptance in delicate fields like finance and healthcare is questionable
because of the models’ difficulty in being understood and explained [8]. The
lack of interpretability or transparency also known as ”black-box models,” is
a key cause of concern in some AI models and it is challenging to comprehend
how they came to those conclusions [9]. The minimal available of data is
another restriction. For AI models to learn effectively, large, and varied
datasets are necessary to train the models. Owing to the rarity of some events
or the biases present in the data, it can be difficult to obtain balanced
datasets in some areas. As a result, the model’s predictions may be biased or
have skewed representation. This is also reflected in the application of
models developed from these datasets to be employed in untrained operational
situations [10].

Since the AI models have black-box nature, they need to be certified to make
sure it is working efficiently in all environments. This certification can be
done in several ways such as assessment certifying the fulfillment of minimum
ethical requirements and what we describe as nuanced assessment [11]. Some
important applications like robots in work spaces shared with humans [12],
social and environmental [13] need to be certified. This type of certification
study is also important in traffic state estimation (TSE) which is not fully
examined yet. The goal of the certification process is to ascertain whether
the TSE model developed under different traffic state conditions can
accurately anticipate in a new setting while upholding the traffic
conservation rule. By certifying the model, we hope to increase its
interpretability, strengthen its dependability, and demonstrate its
suitability for TSE example that occur in the real world [14]. The findings of
this study have implications for developing the TSE area and advance knowledge
of certifying deep learning models for safety-critical applications.

To fulfill the urgent demand for safety, dependability, and transparency in AI
applications, deep learning models need certification [12]. The contributions
of this paper are the establishment of a thorough science-based AI model
certification methodology that ensures the resilience of deep learning models
in untrained operational situations by thorough examination and the
incorporation of known scientific concepts. This work enables one to ascertain
the extent to which AI models adapt to untrained complex and dynamic
environments, reducing risks and potential accidents by bridging the gap
between conventional data-driven approaches and science-based understanding.
Additionally, the focus on transparency and explainability encourages a better
understanding of model judgments, fostering public confidence and facilitating
regulatory compliance [15],[16]. We seek to increase the precision and
dependability of TSE models, resulting in safer and more effective
transportation systems.

###  I-A Literature Review

Certification of AI models is crucial when implementing them in untrained
operational scenarios, especially where traditional data-driven approaches may
fall short. This is particularly important in applications like autonomous
vehicles, where certification ensures safety and dependability in diverse and
unpredictable conditions such as bad weather and unusual traffic patterns.
Similarly, in healthcare diagnostics, certification is necessary to handle
different patient populations, rare diseases, and unforeseen medical
situations, instilling confidence in the accuracy and clarity of AI-assisted
diagnoses especially in untrained operational environments [17], [18].

In natural disaster prediction and response artificial intelligence models are
used to anticipate natural disasters, such as earthquakes, hurricanes, or
wildfires, need to be certified to function well in disaster scenarios that
haven’t been encountered before [19]. These model certifications guarantees
their ability to adjust to shifting circumstances and deliver accurate early
warnings. In aerospace and aviation AI certification is needed to address the
difficulties of untrained operational contexts, such as air traffic changes,
weather disturbances, and emergency scenarios, certification is required for
autonomous drones and aircraft systems [15].

In the fields of industrial robotics and manufacturing, artificial
intelligence (AI) models must be certified to operate in dynamic production
environments with the possibility of varying raw material supply, equipment
breakdowns, or unforeseen production difficulties.AI models used for
environmental monitoring, such as those that predict weather, ocean currents,
or air quality, must be certified to handle a variety of geographic and
climatic conditions [20].

Certifying AI models is paramount in security and surveillance for operation
in uncontrolled environments, such as public spaces where unforeseen
occurrences and abnormalities may occur [21]. Financial risk assessment also
needs certification owing to financial markets being subject to abrupt
volatility and unforeseen events, AI models for risk assessment and prediction
in finance must be approved for untrained operational settings. For Power Grid
Management, AI models used to manage power grids and energy distribution
systems must be certified to react to unforeseen changes in energy supply,
demand, and potential grid failures [22]. For application in space
exploration, AI models used in space missions need to be certified to
withstand the harsh environments of uncharted space, where data may be
sporadic and unexpected [23].

Certification is necessary in order to check for several challenges that are
encountered when we are trying to employ pre-trained AI models in new
environments. Some of the challenges that may be encountered include models
that may be vulnerable to adversarial assaults, in which malicious inputs are
skillfully created to falsely estimate outcomes [21]. When pre-trained models
are used in untrained environments, privacy concerns may arise because
vulnerable data may unintentionally leak through model outputs. Ethics issues
may also arise if AI models are used in untrained environments without being
properly vetted or knowing their limitations, especially if important choices
are based on their outputs. Pre-trained models can be used in situations where
they are not in compliance with rules or legal requirements, which can result
in legal issues and liability issues [24]. Understanding how pre-trained
models work can be tricky because they are complex sometimes and difficult to
interpret.

Another important application is traffic state estimation where this
certification is mandatory to test the models in different environments. It is
always difficult to acquire huge number of data in all environments since it
involves more cost to collect it. In that scenario, certification of AI model
in other environments is necessary. Thus, it is essential to carry out
thorough testing, validation, and fine-tuning of the pre-trained model in the
particular operating context in order to assure its safety, dependability, and
ethical use in order to reduce these risks. Additionally, keeping an eye on
the model’s performance and user input can aid in spotting and resolving
possible problems [21].

Pre-trained models can be quite effective and helpful in terms of time and
computational power, but they may not always function at their best in novel
and uncharted circumstances thus pausing risks of increased uncertainties.
Pre-trained models that are typically trained on a small number of datasets
may not have a thorough understanding of domains that are unrelated to their
training data. The availability of unbalanced datasets is a frequent problem
in traffic state estimation. A dataset that is uneven in the context of the
traffic state estimation example would be one where some traffic states or
situations are over represented while others are underrepresented or rare
[25]. The imbalance may be caused numerous things, including traffic patterns,
variable levels of congestion, or the scarcity of data from particular places
or times. For instance the traffic dataset that may be readily available is
mainly the NGSIM-CA data from California and the traffic state may not fully
represent the state of traffic in other states say Tennessee. The model may
generate inaccurate or deceptive results in untrained circumstances as a
result of its ignorance of the relevant domain. In some cases the input data
in a context for which the model has not yet been trained differs from that
context, the model may interpret the input incorrectly and produce false
results.

Pre-trained models may inherit biases from their training data, which is known
as bias amplification. These biases may intensify when applied in unknown
environments, producing unfair or discriminating outcomes. It is possible for
pre-trained models to over-fit to specific applications or datasets [26]. They
might not generalize when applied in novel and varied circumstances. Due to
domain variations, the model’s performance may suffer in untrained
circumstances, producing inconsistent results and decreasing user confidence.

It can be very beneficial to use pre-trained AI models in untrained
environments, but doing so requires careful preparation, ongoing monitoring,
and the right tweaks to make sure the model operates properly and safely.
Striking a balance between using the model’s pre-trained skills and tailoring
it to the unique demands of the operational setting is crucial.

###  I-B Methodology

This section of the paper provides background information on traffic flow
physics and its relevance to AI models, particularly in the context of traffic
data certification. It emphasizes the importance of understanding fundamental
traffic concepts, illustrated by the fundamental traffic diagram (Figure 1),
which depicts the nonlinear relationship between traffic speed, flow, and
density [27, 28, 29]. The conservation of vehicles principle, based on mass
conservation, is introduced as a fundamental physics concept governing traffic
flow. Additionally, mathematical models such as the Lighthill-Whitham-Richards
(LWR) model, based on fluid dynamics principles, are discussed for
representing traffic flow and analyzing congestion [30, 31].

Figure 1: Speed against density diagram in traffic flow

The section delves into the relationship between traffic density, flow rate,
and vehicle speed, highlighting the impact of these factors on traffic
conditions [32]. It also introduces equations describing traffic flow,
including the LWR equation. The Greenshields’ fundamental diagram is presented
to summarize the relationship between traffic density, flow rate, and free-
flow speed is mathematically defined by equations 1 and 2 [33, 34].

$q(x,t)=\frac{\partial N(x,t)}{\partial t}$ (1)

$\rho(x,t)=-\frac{\partial N(x,t)}{\partial x}$ (2)

The discussion transitions to the application of Artificial Intelligence (AI)
in traffic state estimation, categorizing it as a data-driven approach. The
potential of AI algorithms, particularly deep learning models, in forecasting
traffic conditions is emphasized, along with the importance of labeled
training data and computational resources [35, 36]. The structure of deep
learning neural networks and the role of the cost function in training are
explained. The paper provides equations for the mean squared error cost
function used in traffic state estimation for the assessment of the
metrics.The cost function, also known as the loss or objective function, is a
crucial mathematical measure in deep learning that quantifies the disparity
between a neural network’s expected output and the actual target values [31].

Accuracy in deep learning is explored, focusing on classification and
regression accuracy calculations and is defined by the equation.

$A_{D_{L}}=\frac{\sqrt{\sum_{x=1}^{X_{m}}\sum_{t=1}^{T_{n}}|\rho(x,t)-\hat{\rho}(x,t)|^{2}}}{\sqrt{\sum_{x=1}^{X_{m}}\sum_{t=1}^{T_{n}}|\rho(x,t)|^{2}}}$
(3)

Equation 3 is used to evaluate the performance of an AI model and the results
obtained from this equation are the ones that plot figure 5.

Regression accuracy is specifically discussed in the context of traffic state
estimation, introducing the Root Mean Squared Error (RMSE) formula.

$C_{D_{L}}=\text{MSE}_{(\rho(x,t),\hat{\rho}(x,t))}=\frac{1}{N}\sum_{i=1}^{N}|\rho(x,t)-\hat{\rho}(x,t)|^{2}$
(4)

Equation 4 is used in the training process. N is the number of estimating
outputs in Mean Squared Error MSE. The estimated vehicle density
$\hat{\rho}(x,t)$ and the actual vehicle density, $\rho(x,t)$ at location x
and time t.

##  II Methodology Discussion: Model Assessment against Science Metrics Laws

To evaluate and certify AI models, the steps shown in figure 2 are followed.

Certifying the suitability of a Traffic State Estimation AI model for
deployment in new environments necessitates the integration of scientific
principles and traffic laws into the model assessment. This involves a
comprehensive examination of the model’s predictions in alignment with
established scientific principles, with a primary focus on the implementation
of the conservation of traffic. This approach serves to foster the development
of dependable models that strictly adhere to physical and mathematical
constraints, thereby establishing a foundational framework for the scientific
validation and acceptance of the AI model. Upon saving the model parameters, a
crucial step ensues, where the models undergo comparison with well-known
physical constraints and physics laws. This comparative analysis aims to
augment the reliability, robustness, and trustworthiness of deep learning
models designed for traffic state estimation.

The pivotal role played by the law of conservation of traffic in this process
is evident as predictions from the model are scrutinized for deviations from
ideality. Ensuring adherence to fundamental principles involves applying
traffic flow conservation rules or laws, validating that the predicted values
for density, velocity, and predicted flow rate align with the actual influx
and outflow of vehicles. The assessment extends to confirming the consistency
of predicted values for flow rate and density with the anticipated connections
suggested by the traffic flow model. An analysis of results is then conducted
to identify anomalies or deviations from the established rules of traffic
flow, scrutinizing expected density values and traffic variables. This
meticulous examination identifies areas, such as instances of congestion or
unexpected correlations between flow rate and density, facilitating a
comprehensive evaluation of the model’s adaptability to new environments.

##  III Experimental Configuration

Assessment of the performance of the AI system’s correctness and dependability
are required for science-based certification of AI models. [14]. This section
introduces a science-based certification framework for AI models, emphasizing
the evaluation of assessment in untrained environments. The certification
process is outlined in Figure 2.

Figure 2: Certification process

In order to certify AI models using physics laws for safety-critical traffic
state estimation The Lax-Hopf method is used to create synthetic traffic
datasets in this paper under no downstream flow and no upstream flow, and it
is specifically tuned to follow the Greenshields’ model and traffic
conservation laws.

Figure 3: Dataset generated with a $v_{f}$ =25

This experiment is done on a road segment defined by 1000 meters (x) observed
over a duration of 50 seconds (t). The traffic density values throughout the
domain were created by the Lax-Hopf method by setting up the following initial
density values $\rho(x,t)$ and initial flow rates.

Initial density: For interval at x in [0, 200] meters is 0.13 vehicle per
meter (veh/m), [200, 500] is 0.06 veh/m and [500, 1000] is 0.03 veh/m at t=0.

Initial Flow rate: For upstream at t in [0, 20] seconds is 0.4 vehicle per
second (veh/s), [20, 40] is 0.01 veh/sec and [40, 50] is 0.2 veh/sec at x=0.
For downstream at t in [0, 30] seconds is 0.3 vehicle per second (veh/s), [30,
35] is 0 veh/sec and [35, 50] is 0.1 veh/sec at x=1000.

We used free flow velocity ($v_{f}$) is 25 meters per second (55 miles per
hour) and jam density ($\rho_{m}$) is 0.15 veh/m. The dataset was created at
the time steps of 0.1 seconds and the distance step is 2 meters. The generated
synthetic dataset is displayed in figure 3.

The deep learning experimental set up uses a 10 hidden layer architecture with
40 neurons each. Two optimization algorithms were used that is the limited
Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) and Adam. The Tensorflow library for
machine learning was implemented in the development of our Traffic State
Estimation.The machine used to run the TSE was a 11th Gen Intel® Core™
i9-11900KF @ 3.50GHz × 16, with a Random Access Memory of 62.5GB.

##  IV Results and Discussions

In conclusion, since the experiment involved first training a machine learning
model to estimate traffic states, speed and flow based on sensor data. Then,
using inputs road geometry and real-time traffic counts, the expected outflows
were calculated by applying the physics-based conservation of traffic formula.
The investigation into the comparison of various $v_{f}$ environments against
the conservation law of traffic flow has yielded valuable insights. By
generating new environments with different $v_{f}$ values while keeping other
parameters constant, we aimed to assess the impact on the conservation law.
While the conservation law provides a fundamental framework for understanding
traffic flow, it is crucial to acknowledge potential deviations in real-world
scenarios, influencing traffic modeling and analysis.

The estimated results from the ML model were then directly compared against
the calculations derived from the established traffic flow physics model. The
differences or errors between the two results were analyzed.

This allowed the paper to systematically validate the ML model’s estimations
and identify any discrepancies by bench marking it against well-established
traffic behavior defined through fundamental physical laws. The conservation
law provided a standardized way to numerically simulate ”ground truth” traffic
states.

Our focus on deviations from the Conservation Law in testing with different
$v_{f}$ environments, all trained on $v_{f}$=25 meter/sec, is depicted in
Figure 4. Notably, the analysis reveals that the lowest deviation occurs when
comparing with a $v_{f}$=25 environment, demonstrating a consistent trend
observed in Figure 5.

Figure 4: DL estimations for $v_{f}$=25.

This section further underscores the significance of certifying deep learning
models with physics laws to enhance predictions in Traffic State Estimation
(TSE). The deep learning model, trained in a specific environment and tested
across diverse settings, particularly aligns with scenarios featuring sensors
strategically positioned within the road network. This approach allows for the
observation and analysis of traffic variables at fixed locations over time,
mirroring real-world setups with fixed sensors.

The monitored traffic conditions and behaviors at predetermined positions
contribute to predicting traffic conditions in new or unobserved areas.
Through this comprehensive exploration, we emphasize the robustness of our
approach, bridging the gap between deep learning models and the underlying
physics governing traffic dynamics for more accurate TSE predictions. The
graphical representation in Figure 4 visually reinforces the trends observed,
validating the effectiveness of our methodology in real-world applications.

The deep learning (DL) model undergoes training using only 6% of the complete
dataset, specifically 15,000 samples, with a fixed $v_{f}$ of 25. The testing
phase employs the remaining dataset, encompassing data from various
environments. During training, we utilized 15,000 iterations with the Adam
optimizer and an additional 50,000 iterations with the L-BFGS optimizer. The
resulting DL error rates for different environments are plotted in figure 5.

As delineated in Figure 5, the lowest DL error occurs at $v_{f}$ = 25 m/sec,
registering at 17.81 % . This alignment with expectations is attributed to the
model’s training and testing being conducted within the $v_{f}$ = 25 m/sec
environment. Notably, this finding underscores the exceptional performance of
the model when trained and tested in the same environment. However, as one
deviates from the conditions of the trained environment, whether in the lower
or higher $v_{f}$ range, the DL performance diminishes, resulting in an
increased DL error. This observation emphasizes the model’s sensitivity to
variations in the training environment, highlighting the importance of
consistent conditions for optimal performance.

$5$$10$$15$$20$$25$$30$$35$$40$$45$$0$$20$$40$$60$$80$$100$VfDL errorDL error
Figure 5: DL error against Vf

##  V Summary and Conclusions

This paper divulged a method of science-based certification of AI models to be
applied in untrained operational environments. We investigated the viability
and efficacy of certifying deep learning models for safety-critical traffic
state estimation using physics rules. Enhancing deep learning models’
dependability and credibility in applications where precise traffic state
estimation is crucial for assuring safety was the goal and data is limited.

The findings of this work pave the door for additional approaches such as
physics-regulated deep learning model and transfer learning to better employ
the models in untrained operating conditions. Furthermore, in this paper we
used only synthetic data, future work should utilize the real world data for
betterment of certification process of the AI models.

## Acknowledgements

This work is supported in part by the National Science Foundation (NSF) under
Grant No. 2130990.

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