# Strangeness plus-one ($S=+1$) resonance-state $P^{+*}_{0}$ via $K^{+}n\to
K^{*0}p$

Dayoung Lee  0009-0000-4828-3546<EMAIL_ADDRESS>Department of
Physics, Pukyong National University (PKNU), Busan 48513, Korea  Center for
Extreme Nuclear Matters (CENuM), Korea University, Seoul 02841, Korea  Seung-
il Nam∗ 0000-0001-9603-9775<EMAIL_ADDRESS>(corresponding author) Department
of Physics, Pukyong National University (PKNU), Busan 48513, Korea  Center for
Extreme Nuclear Matters (CENuM), Korea University, Seoul 02841, Korea  Asia
Pacific Center for Theoretical Physics (APCTP), Pohang 37673, Korea

###### Abstract

In our current study, we delve into the peak-like structure observed during
the reaction process of $K^{+}n\to K^{0}p$ at approximately $\sqrt{s}\sim 2.5$
GeV. Our focus centers on exploring the potential $S=+1$ resonance
$P^{+*}_{0}\equiv P^{*}_{0}$ as an excited state within the extended vector-
meson and baryon ($VB$) antidecuplet. To achieve this aim, we employ the
effective Lagrangian method in conjunction with the $(u,t)$-channel Regge
approach, operating within the tree-level Born approximation. We thoroughly
examine various spin-parity quantum numbers for the resonance, resulting in a
compelling description of the data, where $M_{P^{*}_{0}}\approx 2.5$ GeV and
$\Gamma_{P^{*}_{0}}\approx 100$ MeV. Furthermore, we propose an experimental
technique to amplify the signal-to-noise ratio ($S/N$) for accurately
measuring the resonance. Notably, our findings reveal that background
interference diminishes significantly within the $K^{*}$ forward-scattering
region in the center-of-mass frame when the $K^{*}$ is perpendicularly
polarized to the reaction plane. Additionally, we explore the recoil-proton
spin asymmetry to definitively determine the spin and parity of the resonance.
This study stands to serve as a valuable reference for designing experimental
setups aimed at investigating and comprehending exotic phenomena in QCD.
Specifically, our insights will inform future J-PARC experiments, particularly
those employing higher kaon beam energies.

Pentaquark, vector-meson and baryon interaction, $S=+1$ production process,
effective Lagrangian method, Regge approach, polarization, resonance, forward-
scattering cross-section.

††preprint: PKNU-NuHaTh-2024

##  I Introduction

Quantum Chromodynamics (QCD) stands as the foundational principle governing
the strong interactions among standard-model particles, portraying various
hadrons through the non-perturbative interactions of quarks and gluons. It
showcases confinement and asymptotic freedom within the framework of non-
Abelian color-SU(3) gauge theory. While the minimal composition for baryons is
represented by $qqq$ and for mesons by $q\bar{q}$, QCD does not forbid more
intricate compositions such as $qqqq\bar{q}$ (pentaquark) and
$q\bar{q}q\bar{q}$ (tetraquark), termed Exotics. Experimental evidence
supporting these exotics has accumulated over decades. The initial observation
of the tetraquark meson $X(3872)$, which defies the simple quark model, dates
back to 2003, reported by the Belle experiment Belle:2003nnu . Recently, its
existence was reaffirmed by the LHCb LHCb:2013kgk  and CMS CMS:2021znk
experiments, specifying $J^{PS}=1^{++}$. Additionally, the Belle experiments
reported evidence for $Y(4660)$ and $Z(4430)$ in 2007 Cotugno:2009ys . Further
tetraquark states have been identified via experiments conducted by LHCb,
Belle, BES III, Fermilab, and others Chen:2022asf . Regarding baryons, LHCb
reported the $P^{+}_{c}(4312,4440,4457)$, observed in decay into $J/\psi$ and
$p$ LHCb:2015yax . Recently, the same facility detected a novel pentaquark
state with the strange-quark $s$ ($udsc\bar{c}$) in the decay of $B^{-}\to
J/\psi\Lambda\bar{p}$ LHCb:2022ogu . While not yet definitively confirmed,
these pentaquarks can potentially be understood as bound states of vector-
meson and baryon ($VB$), denoted as $P^{+}_{c}\sim D^{*}\Sigma_{c}$.

Remarkably, all experimentally observed and confirmed exotics exhibit heavy
(charm) flavor. A theoretical rationale for this peculiarity concerning the
stability of heavy pentaquarks was elucidated through color-spin interactions
among quarks, contrasting with colorless hadronic interactions Park:2018oib .
While the $S=+1$ light-pentaquark $\Theta^{+}\sim uudd\bar{s}$ was reported by
the LEPS collaboration LEPS:2008ghm , supported by the non-topological chiral-
quark soliton model Diakonov:1997mm , its existence remains disputed and
unresolved due to varied outcomes across experimental facilities PDG .
Furthermore, generating the $S=+1$ resonance dynamically proves challenging
within the framework of Weinberg-Tomozawa (WT) type chiral interactions,
primarily due to their repulsive nature. Nevertheless, the suggestion of an
$S=0$ pentaquark-like bound-state $P^{+}_{s}\sim uuds\bar{s}\sim K^{*}\Sigma$,
akin to $P^{+}_{c}$ in heavy flavor, was put forth by one of the present
authors and collaborators Nam:2021ayk ; Khemchandani:2011et , aiming to
elucidate the bump structure observed near $\sqrt{s}\approx 2.1$ GeV in
$\gamma p\to\phi p$ Shim:2024myp . In Ref. Shim:2024myp , the existence of
$P^{+}_{s}$ was found to be crucial for reproducing the nontrivial structure
observed in the spin-density matrix elements (SDMEs), in addition to
successfully describing the angular-dependent cross-sections.

Given the current circumstances outlined above, it becomes of paramount
importance to explore the potential for observing evidence of light-pentaquark
states, either as bound states or resonances within the $VB$ framework, to
further enhance our understanding of QCD regarding exotic particles.
Additionally, even in the $S=+1$ channel, there may exist a resonance or
bound-state pentaquark, despite the lack of support from chiral dynamics KAN .
Hence, in the present study, we undertake a theoretical examination of
relatively aged data pertaining to the $S=+1$ channel reaction process, such
as $K^{+}n\to K^{*0}p$ Bologna-Glasgow-Rome-Trieste:1976hco . Intriguingly,
this reaction process exhibits a peak-like structure around $\sqrt{s}=2.5$
GeV, potentially indicative of an $S=+1$ resonance state, which we
provisionally designate as $P^{+*}_{0}\equiv P^{*}_{0}$. We aim to elucidate
this hypothetical pentaquark resonance $P^{*}_{0}$ in our current
investigation. Drawing upon our previous work Nam:2021ayk , where we assigned
the light-flavor $VB$ pentaquark as $P_{s}\sim qqqs\bar{s}$, assuming it as a
member of the extended $VB$ antidecuplet flavor structure akin to
$\Theta^{+}$, we can extend this assignment to other possible states:
$P_{0}\sim qqqq\bar{s}$, $P_{1}\equiv P_{s}\sim qqqs\bar{s}$, $P_{2}\equiv
P_{ss}\sim qqss\bar{s}$, and $P_{3}\equiv P_{sss}\sim qsss\bar{s}$, where $q$
represents the flavor-SU(2) light quarks ($u$ and $d$), being categorized by
the numbers of the strange quarks they include.

In our previous study Nam:2021ayk , we set the mass of $P_{s}$ to be
approximately $2.071$ GeV, guided by the chiral-unitary approach
Khemchandani:2011et . Assuming $P_{s}(2071)$ as the $S=0$ ground-state
pentaquark of the extended $VB$ antidecuplet and considering the mass
difference between antidecuplets due to the inertia moment of the non-
topological chiral soliton, approximately $\sim 180$ MeV Diakonov:1997mm , the
ground-state $P_{0}$ mass is estimated to be around $1.890$ GeV. Consequently,
we regard the hypothetical resonance in the vicinity of the bump structure at
$2.5$ GeV as a resonance state of $P_{0}$, denoted as $P^{*}_{0}(2500)$.
Furthermore, we highlight several advantageous features of the chosen process:
Firstly, since all hadrons in the final state of $K^{+}n\to
K^{*0}p\to\pi^{\mp}K^{\pm}p$ can be measured experimentally in principle, the
Fermi-motion effects of the deuteron-target experiment can be minimized.
Secondly, in the $s$ channel, it is possible to measure the resonance
simultaneously coupling to $PB$ and $VB$ channels, allowing for the study of
its production mechanism in two distinct dynamics. Lastly, this process
involves the vector meson with polarizations, enabling the selection of
specific quantum numbers for the resonance and control over its production
mechanism.

To achieve this objective, we employ a straightforward tree-level Born
approximation calculation utilizing the effective Lagrangian method. We
account for the Regge trajectories in the $t$ and $u$ channels, incorporating
phenomenological form factors to characterize the spatial extension of the
hadrons. To replicate the observed peak-like structure in the total cross-
section, potentially indicative of the $S=+1$ $VB$ resonance ($\equiv
P^{+*}_{0}$), we introduce a relativistic Breit-Wigner type contribution with
$M_{R}=2.5$ GeV and $\Gamma_{R}=100$ MeV. Our numerical calculations reveal
that, alongside the resonance contribution in the $s$ channel, Reggeized
$\pi$-exchange in the $t$ channel and $\Lambda(1116)$ one in the $u$ channel
dominate the reaction process. We explore various spin-parity quantum numbers
theoretically ($J^{P}=1/2^{\pm},3/2^{\pm}$) since experimental data to
determine them precisely are lacking. Notably, all tested spin-parity quantum
numbers qualitatively reproduce the data well. To mitigate the signal-to-noise
($S/N$) ratio, we propose a specific combination of scattering angle and
polarization ($\theta_{K^{*}}\sim 0$ and $k_{K}\cdot\epsilon_{K^{*}}\sim 0$)
for future experiments, surpassing current programs like the K1.8 beam-line at
J-PARC, which require a kaon beam energy of approximately $3.0$ GeV to access
the $P^{*}_{0}$ region. Additionally, we recommend utilizing the recoil-proton
spin asymmetry to ascertain the spin parity of the resonance.

The current study is structured as follows: Section II presents the
theoretical framework. In Section III, we present the numerical results along
with detailed explanations and discussions. Finally, the concluding section
provides a summary and outlines future directions.

##  II Theoretical Framework

In this section, we will provide a concise overview of the theoretical
framework governing the present reaction process. Relevant Feynman diagrams
for the process are illustrated in Fig. 1, where the four momenta of the
particles involved are also defined. The Yukawa interaction vertices are
described as follows:

$\displaystyle\mathcal{L}_{PBB^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{g_{PBB^{\prime}}}{M_{K}}\bar{B}^{\prime\mu}\Theta_{\mu\nu}(X)(\partial^{\nu}K)\Gamma_{5}B+\rm
h.c.,$ (1) $\displaystyle\mathcal{L}_{VBB^{\prime}}$ $\displaystyle=$
$\displaystyle-\frac{ig_{VBB^{\prime}}}{M_{K^{*}}}\bar{B}^{\prime\mu}\gamma^{\nu}(\partial_{\mu}K^{*}_{\nu}-\partial_{\nu}K^{*}_{\mu})\Gamma_{5}\gamma_{5}B+\rm
h.c.,$ (2) $\displaystyle\mathcal{L}_{PBB}$ $\displaystyle=$ $\displaystyle
ig_{PBB}\bar{B}\Gamma_{5}\gamma_{5}K^{\dagger}B+{\rm h.c.},$ (3)
$\displaystyle\mathcal{L}_{VBB}$ $\displaystyle=$ $\displaystyle
g_{VBB}\bar{B}\gamma^{\mu}\Gamma_{5}V^{\dagger}_{\mu}B+{\rm h.c.},$ (4)
$\displaystyle\mathcal{L}_{PPV}$ $\displaystyle=$ $\displaystyle
ig_{PPV}V^{\mu}[P\partial_{\mu}P^{\dagger}-P^{\dagger}\partial_{\mu}P]+\rm
h.c.,$ (5) $\displaystyle\mathcal{L}_{PBB}$ $\displaystyle=$
$\displaystyle\Big{(}\frac{f_{PBB}}{M_{P}}\Big{)}\bar{B}{\gamma_{5}}(\hbox
to0.0pt{/\hss}{\partial}P)B.$ (6)

In this context, $B^{\prime}$, $B$, $P$, and $V$ denote the fields
corresponding to spin-$3/2$, spin-$1/2$ baryons, pseudoscalar, and vector
mesons, respectively. We utilize the notation $\Gamma_{5}=(I_{4\times
4},\gamma_{5})$ to represent the parity ($+,-$) of $B^{\prime}$. The term
$\Theta_{\mu\nu}(X)$ signifies the off-shell parameter $X$-dependent term
within the Rarita-Schwinger formalism Rarita:1941mf .

$\displaystyle\Theta_{\mu\nu}(X)=g_{\mu\nu}+X\gamma_{\mu}\gamma_{\nu}.$ (7)

As illustrated in Fig. 1, our analysis encompasses the $P^{*}_{0}$ diagram in
the $s$ channel, hyperon ($\Lambda^{(*)}$, $\Sigma^{(*)}$) exchange in the $u$
channel, and $\pi$ exchange in the $t$ channel. Given the absence of
experimentally confirmed $S=+1$ baryon states to date, we exclusively focus on
the hypothetical $VB$ pentaquark $P^{*}_{0}$. Concerning the $t$ channel,
while there could be additional meson exchanges, such as the axial-vector
meson $b_{1}(1235)$, we exclude them due to the scant information available
regarding their decays PDG .

Figure 1: Feynman diagrams pertinent to the process $K^{+}n\to K^{*0}p$ within
the tree-level Born approximation are presented.

Following straightforward calculations, the invariant amplitudes for the
Feynman diagrams depicted in Fig. 1 are presented as follows:

$\displaystyle i\mathcal{M}_{{P^{*}_{0}}(s=1/2)}^{s}$ $\displaystyle=$
$\displaystyle-
g_{K^{*}N{P^{*}_{0}}}g_{KN{P^{*}_{0}}}\bar{u}(p_{2})\Gamma_{5}\hbox
to0.0pt{/\hss}{\epsilon}\frac{\hbox
to0.0pt{/\hss}{q}_{s}+M_{P^{*}_{0}}}{s-M_{P^{*}_{0}}^{2}-iM_{P^{*}_{0}}\Gamma_{P^{*}_{0}}}\gamma_{5}\Gamma_{5}u(p_{1}),$
(8) $\displaystyle i\mathcal{M}_{{P^{*}_{0}}(s=3/2)}^{s}$ $\displaystyle=$
$\displaystyle-\frac{ig_{K^{*}N{P^{*}_{0}}}g_{KN{P^{*}_{0}}}}{M_{K^{*}}M_{K}}\bar{u}(p_{2})\gamma_{5}\Gamma_{5}(k_{2\mu}\epsilon_{\nu}-k_{2\nu}\epsilon_{\mu})\gamma^{\nu}\frac{\left(\hbox
to0.0pt{/\hss}{q}_{s}+M_{P^{*}_{0}}\right)\bar{g}^{\mu\alpha}(q_{s})}{s-M_{P^{*}_{0}}^{2}+iM_{P^{*}_{0}}\Gamma_{P^{*}_{0}}}\big{(}g_{\alpha\beta}+X\gamma_{\alpha}\gamma_{\beta}\big{)}k_{1}^{\beta}\Gamma_{5}u(p_{1}),$
(9) $\displaystyle i\mathcal{M}^{u}_{Y}$ $\displaystyle=$
$\displaystyle-\frac{i}{2}g_{KNY}g_{K^{*}NY}\bar{u}(p_{2})\Gamma_{5}\frac{(\hbox
to0.0pt{/\hss}{q}_{u}+M_{Y})\hbox
to0.0pt{/\hss}{\epsilon}}{u-M_{Y}^{2}-iM_{Y}\Gamma_{Y}}\Gamma_{5}\gamma_{5}u(p_{1}),$
(10) $\displaystyle i\mathcal{M}_{\Sigma^{*}(1385)}^{u}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\frac{g_{K^{*}N\Sigma^{*}}}{M_{K^{*}}}\frac{g_{KN\Sigma^{*}}}{M_{K}}\bar{u}(p_{2})k_{1}^{\nu}\left(g_{\mu\nu}+X\gamma_{\mu}\gamma_{\nu}\right)\frac{\bar{g}^{\mu\alpha}(q_{u})\left(\hbox
to0.0pt{/\hss}{q}_{u}+M_{\Sigma^{*}}\right)}{u-M_{\Sigma^{*}}^{2}+iM_{\Sigma^{*}}\Gamma_{\Sigma^{*}}}\gamma^{\beta}\left(k_{2\alpha}\epsilon_{\beta}-k_{2\beta}\epsilon_{\alpha}\right)\gamma_{5}u(p_{1}),$
(11) $\displaystyle i\mathcal{M}_{\pi}^{t}$ $\displaystyle=$
$\displaystyle-\frac{2if_{\pi NN}\
g_{PPV}}{M_{\pi}}\overline{u}(p_{2}){\gamma_{5}}\frac{\hbox
to0.0pt{/\hss}{q}_{t}(\epsilon\cdot k_{1})}{t-M_{\pi}^{2}}u(p_{1}).$ (12)

Here, the notation $(s,u,t)=(q^{2}_{s},q^{2}_{u},q^{2}_{t})$ denotes the
Mandelstam variables. In our calculations, we utilize the spin-$1/2$ hyperons
$Y=\Lambda(1116,1/2^{+})$, $\Sigma(1192,1/2^{+})$, $\Lambda(1405,1/2^{-})$,
and $\Lambda(1670,1/2^{-})$, along with the hyperon resonance
$\Sigma^{*}(1385,3/2^{+})$. The polarization vector of $K^{*}$ is represented
by $\epsilon_{\mu}$. The spin sum of the Rarita-Schwinger field yields the
following expression:

$\displaystyle\bar{g}^{\mu\nu}(q)=g^{\mu\nu}-\frac{1}{3}\gamma^{\mu}\gamma^{\nu}-\frac{2}{3M^{2}}q^{\mu}q^{\nu}+\frac{q^{\mu}\gamma^{\nu}+q^{\nu}\gamma^{\mu}}{3M}.$
(13)

Here, $M$ and $\Gamma$ represent the mass and full decay width of the particle
corresponding to the field. All relevant couplings are established based on
experimental data and theoretical frameworks PDG ; Khemchandani:2011et ;
Stoks:1994nh , as summarized in Table 1, except for those for $P^{*}_{0}$. To
streamline our analysis, we denote the combined coupling as $g_{P^{*}0}\equiv
g_{K^{*}N{P^{*}_{0}}}g_{KN{P^{*}_{0}}}$ hereafter.

Hyperon $(Y)$ |  $g_{K^{*}NY}$ |  $g_{KNY}$ |  Combined
---|---|---|---
$\Lambda(1116,1/2^{+})$ |  $-4.26$ |  $-13.4$ |  $57.08$
$\Lambda^{*}(1405,1/2^{-})$ |  $0.2-2.7i$ |  $2.5+0.9i$ |  $2.93-6.57i$
$\Lambda^{*}(1670,1/2^{-})$ |  $-0.2+0.8i$ |  $0.2-0.6i$ |  $-0.52-0.04i$
$\Sigma(1192,1/2^{+})$ |  $-2.46$ |  $4.09$ |  $-10.06$
$\Sigma^{*}(1385,3/2^{+})$ |  $-5.48$ |  $-6.94$ |  $38.03$
$\pi$ |  $f_{\pi NN}=0.989$ |  $g_{\pi KK^{*}}=3.76$ |  $3.72$
Table 1: Strong couplings for the hyperon and pion vertices and their
combinations derived from theoretical models and experimental data PDG ;
Khemchandani:2011et ; Stoks:1994nh .

To account for the spatial extension of the hadrons, we incorporate a
phenomenological form factor, defined as follows:

$\displaystyle F(x)=\frac{\Lambda^{4}}{\Lambda^{4}+(x-M^{2}_{x})}.$ (14)

In this expression, $x$ represents the Mandelstam variable, and $M_{x}$
denotes the off-shell mass of the corresponding particle. The cutoff parameter
$\Lambda$ will be adjusted to accurately reproduce the experimental data.

Given that the available data from Ref. Bologna-Glasgow-Rome-Trieste:1976hco
extends up to $\sqrt{s}\sim 5$ GeV, which approaches the asymptotic limit
$s\to\infty$, it becomes imperative to account for the higher-spin states of
the exchange particles. To address this, we incorporate the $t$\- and
$u$-channel Regge trajectories. Following the methodology outlined in Ref.
Kim:2023jij , the $u$-channel amplitudes for the $\Lambda(1116)$,
$\Sigma(1192)$, and $\Sigma^{*}(1385)$ are adjusted using the Regge approach,
wherein each trajectory $\alpha(x)$ is characterized by the intercept
$\alpha^{\prime}$.

$\displaystyle\tilde{\mathcal{M}}_{\Lambda,\Sigma}^{u}(s,u)$ $\displaystyle=$
$\displaystyle C_{\Lambda,\Sigma}(u)\mathcal{M}^{\prime
u}_{\Lambda,\Sigma}\left(\frac{s}{s_{\Lambda,\Sigma}}\right)^{\alpha_{\Lambda,\Sigma}(u)-\frac{1}{2}}\Gamma\left[\frac{1}{2}-\alpha_{\Lambda,\Sigma}(u)\right]\alpha^{\prime}_{\Lambda},$
(15) $\displaystyle\tilde{\mathcal{M}}_{\Sigma^{*}}^{u}(s,u)$ $\displaystyle=$
$\displaystyle C_{\Sigma^{*}}(u)\mathcal{M}^{\prime
u}_{\Sigma^{*}}\left(\frac{s}{s_{\Sigma^{*}}}\right)^{\alpha_{\Sigma^{*}}(u)-\frac{3}{2}}\Gamma\left[\frac{3}{2}-\alpha_{\Sigma^{*}}(u)\right]\alpha^{\prime}_{\Sigma^{*}}.$
(16)

Note that the amplitude $\mathcal{M}^{\prime}$ represents the expression in
Eq. (8) after removing the denominator part of the propagator. The
trajectories are specified as $\alpha_{\Lambda}(u)=-0.65+0.94u$,
$\alpha_{\Sigma}(u)=-0.79+0.87u$, and $\alpha_{\Sigma^{*}}(u)=-0.27+0.9u$.
Additionally, the phenomenological momentum-dependent hadronic coefficient is
defined as follows:

$\displaystyle
C_{Y}(u)=\left[\frac{\epsilon_{Y}\Lambda_{Y}^{2}}{\Lambda_{Y}^{2}-u}\right]^{2}.$
(17)

Finally, the $t$-channel Reggeized amplitude for the $\pi$ reads

$\displaystyle\tilde{\mathcal{M}}_{\pi}^{t}(s,u)$ $\displaystyle=$
$\displaystyle C_{\pi}(u)\mathcal{M}^{\prime
t}_{\pi}\left(\frac{s}{s_{\pi}}\right)^{\alpha_{\pi}(t)}\frac{\pi\alpha^{\prime}_{\pi}}{\Gamma[\alpha_{X}(t)+1]\sin[\pi\alpha_{\pi}(t)]}.$
(18)

Here, $C_{\pi}(t)=a_{\pi}/(1-t/\Lambda_{\pi}^{2})^{2}$. The Regge parameters
and cutoffs for the form factors are provided in Table 2. It is important to
mention that we adopt $X=0$ for simplicity in our analysis.

Hyperon ($Y$) |  $\eta_{Y}$ |  $s_{Y}\,[\rm GeV^{2}]$ |  $\Lambda_{Y}\,[\rm GeV]$
---|---|---|---
$\Lambda(1116,1/2^{+})$ |  $2.60$ |  $1.10$ |  $0.45$
$\Sigma(1192,1/2^{+})$ |  $0.66$ |  $1.10$ |  $0.45$
$\Sigma^{*}(1385,3/2^{+})$ |  $0.66$ |  $1.10$ |  $0.45$
$\pi$ |  $\alpha_{\pi}=0.53$ |  $s_{\pi}=1.00$ |  $\Lambda_{\pi}=1.00$
Table 2: Parameters for the Regge approach and cutoffs in the present work
Kim:2023jij .

##  III Numerical results and Discussions

(a)0.5cm-2cm (b)0.5cm-2cm
---
Figure 2: (a) Total cross-sections for the non-resonant contributions as
functions of $E_{\mathrm{cm}}$, serving as the background (BKG), including the
total (solid), $\pi$ (dotted), $\Lambda(1116)$ (dashed), $\Lambda(1670)$
(long-dashed), $\Lambda(1670)$ (dot-dashed), $\Sigma(1192)$ (long-short-
dashed), and $\Sigma^{*}(1385)$ (long-dot-dashed) components. (b) Total cross-
sections with four different spin-parity quantum numbers of the $P^{*}_{0}$:
$J^{P}=1/2^{+}$ (solid), $J^{P}=1/2^{-}$ (dotted), $J^{P}=3/2^{+}$ (dashed),
and $J^{P}=3/2^{-}$ (long-dashed). Experimental data are obtained from various
sources, including Refs. Bologna-Glasgow-Rome-Trieste:1976hco ; Baker:1975 ;
Goldhaber:1965 ; Buchner:1972 ; Bassompierre:1970 ; Hendrickx:1976 ;
Cords:1971 . Data points that deviate from the background description are
marked with hollow circles.

In this section, we present the numerical results for the reaction process
along with detailed discussions. The cutoffs for the form factors in Eq. (14)
are chosen to be $1.0$ GeV and $0.45$ GeV for the $(s,t)$ and $u$ channels,
respectively, by fitting the data as shown in Table 2. Firstly, in panel (a)
of Fig. 2, we depict the total cross-sections as functions of the center-of-
mass energy (cm)($E_{\mathrm{cm}}$), showcasing the non-resonant $u$\- and
$t$-channel contributions separately. The lines represent contributions from
the total (solid), $\pi$ (dotted), $\Lambda(1116)$ (dashed), $\Lambda(1670)$
(long-dashed), $\Lambda(1670)$ (dot-dashed), $\Sigma(1192)$ (long-short-
dashed), and $\Sigma^{*}(1385)$ (long-dot-dashed). Experimental data are
obtained from both charged (circle) and neutral (square) channels Bologna-
Glasgow-Rome-Trieste:1976hco ; Baker:1975 ; Goldhaber:1965 ; Buchner:1972 ;
Bassompierre:1970 ; Hendrickx:1976 ; Cords:1971 . We observe that the
$\Lambda(1116)$ exchange in the $u$-channel is crucial for reproducing the
strength of the cross-section near the threshold, while the contributions from
other hyperons are almost negligible, as inferred from their coupling
strengths with the Regge coefficients, as depicted in Tables 1 and 2.
Conversely, the $\pi$-exchange contribution in the $t$ channel is significant
for describing the relatively higher-energy region. However, these non-
resonant contributions fail to explain the peak-like structure at
$E_{\mathrm{cm}}\approx 2.5$ GeV. In the figure, data points beyond the
description provided by the background (BKG) are denoted with a hollow circle.
Several explanations can potentially account for the observed peak-like
structure in the cross-sections: 1) Single or combinations of resonances,
which are the focus of our current investigation; 2) Opening of meson-baryon
scattering channels Ahn:2019rdr ; 3) Interferences of scattering amplitudes
LEPS:2016ljn ; and 4) Non-trivial kinematic singularities Szczepaniak:2015eza
.

Considering the scattering of ground-state vector-meson and baryon multiplets
($8,10,\bar{10}$) in the $S=+1$ channel, the possibility of channel opening
appears less likely due to their masses, which are approximately
$M_{\mathrm{max}}=M_{K^{*}}+M_{\Delta}\approx 2.12$ GeV. Although scatterings
involving higher multiplets may contribute, this possibility is beyond the
scope of our current investigation. We also explore the interferences between
the $u$\- and $t$-channel amplitudes with various phase factors, yielding no
significant structures. Another potential source of interference could arise
from crossing the invariant-mass bands on the Dalitz plot for
$K^{+}n\to\pi^{+}K^{-}p$, as a function of $M_{\pi^{+}K^{-}}$ and
$M_{K^{-}p}$, for instance. It’s worth noting that similar considerations
regarding interference possibilities have been explored theoretically and
experimentally for $\gamma p\to\pi^{+}K^{-}p$, with a focus on $\phi$
production LEPS:2016ljn . It was found that interference effects were
negligible. Nontrivial effects, such as the triangle singularity
Szczepaniak:2015eza , could also be introduced. For example, one might
consider a triangle diagram consisting of the $P$-$V$-$B$ internal lines in
the present work, where higher-mass meson-baryon cuts can lead to
singularities. However, achieving a peak at $E_{\mathrm{cm}}\approx 2.5$ GeV
would require very high-mass hadrons. Nonetheless, this remains an intriguing
subject for future exploration, and we leave it for subsequent studies.

Therefore, our focus lies in identifying the presence of a resonance within
the observed structure. It’s important to note that we explore four different
spin-parity scenarios for $P^{*}_{0}$, namely $J^{P}=1/2^{\pm},\,3/2^{\pm}$,
as there is currently no experimental data available for reference. It has
been verified that all these scenarios can qualitatively reproduce the peak-
like structure effectively by adjusting the combined couplings as follows:

$g^{1/2^{+}}_{P^{*}_{0}}=0.65,\,\,\,\,g^{1/2^{-}}_{P^{*}_{0}}=0.15,\,\,\,\,g^{3/2^{+}}_{P^{*}_{0}}=0.70,\,\,\,\,g^{3/2^{-}}_{P^{*}_{0}}=0.23.$
(19)

It’s worth noting that the full decay width of $P^{*}_{0}$ remains consistent
across all spin-parity quantum numbers, set at $\Gamma_{P^{*}_{0}}=100$ MeV.
In panel (b) of Fig. 2, we present the total cross-sections for each case
separately: $J^{P}=1/2^{+}$ (solid line), $J^{P}=1/2^{-}$ (dotted line),
$J^{P}=3/2^{+}$ (dashed line), and $J^{P}=3/2^{-}$ (long-dashed line).

(a)0.5cm-2cm (b)0.5cm-2cm
---

(c)0.5cm-2cm (d)0.5cm-2cm
---

Figure 3: The differential cross-sections for the angular dependence
$d\sigma/d\cos\theta$ [mb] at $E_{\mathrm{cm}}=(2.0,2.5,3.0)$ GeV are depicted
in panels (a), (b), and (c) as functions of $\cos\theta$, illustrating each
contribution. Here, $\theta$ represents the scattering angle of the final
state $K^{*}$ in the cm frame. Panel (d) displays $d\sigma/d\cos\theta$ as a
function of $E_{\mathrm{cm}}$ and $\cos\theta$. In this panel, we exclusively
showcase the results for $P^{*}_{0}(3/2^{-})$, as other quantum-number states
exhibit negligible differences.

In Fig. 3, the numerical results for the differential cross-sections for the
angular dependence $d\sigma/d\cos\theta$ are presented for
$E_{\mathrm{cm}}=(2.0,2.5,3.0)$ GeV in panels (a), (b), and (c), respectively,
illustrating each contribution. Here, $\theta$ denotes the scattering angle of
the final state $K^{*}$ in the cm frame. Notably, strong forward and backward
scatterings are observed from the hyperons in the $u$ channel and from $\pi$
in the $t$ channel, respectively. Conversely, the $s$-channel contribution
yields almost flat curves, as anticipated. Around $E_{\mathrm{cm}}=2.5$ GeV,
corresponding to the resonance region (where we use $3/2^{-}$), a slight yet
discernible angular dependence emerges. As the energy surpasses the resonance
region, the angular dependence markedly separates into forward- and backward-
scattering regions. Panel (d) presents the angular dependence as a function of
$E_{\mathrm{cm}}$ and $\cos\theta$. Notably, the resonance signal becomes more
pronounced around $\cos\theta=0$, where the contributions from the $t$ and $u$
channels are diminished or negligible. These observations suggest a potential
method for experimentally isolating the resonance $P^{*}_{0}$ contribution,
which we will discuss below.

(a)0.5cm-2cm (b)1.5cm3cm
---
Figure 4: (a) Differential cross-sections $d\sigma/d\cos\theta$ for
$\bm{k}_{K}\cdot\bm{\epsilon}_{K^{*}}=0$ for $\theta=0$ (solid), $\pi/4$
(dotted), and $\pi/2$ (dashed) in the cm frame. Here, we only present the
results for the $P^{*}_{0}(3/2^{-})$, as other quantum-number states do not
significantly differ. (b) The same as (a), plotted as a function of
$\cos\theta$ and $E_{\mathrm{cm}}$ for $\bm{k}K\cdot\bm{\epsilon}{K^{*}}=0$ at
$\theta=0$.

Now, we are in a position to discuss how to enhance the $S/N$ for the
$P^{*}_{0}$ resonance. As previously illustrated, the dominant contributions
in the present reaction process originate from the $u$\- and $t$-channel
interactions. Therefore, by focusing solely on the forward-scattering region,
where the hyperon-induced backward scattering is negligible, we can
effectively eliminate this background. The subsequent step involves reducing
the contribution from the $t$-channel $\pi$-exchange. Remarkably, due to its
vector-meson nature, the invariant amplitude for the $\pi$ exchange in Eq. (8)
is directly proportional to $\bm{k}_{K}\cdot\bm{\epsilon}_{K^{*}}$, where
$\bm{k}_{K}$ represents the three-momentum of the initial $K^{+}$ meson.
Consequently, when the polarization of $K^{*}$ is fixed perpendicular to the
reaction plane, where all particle momenta are defined, the $\pi$-exchange
contribution diminishes. In panel (a) of Fig. 4, we display the differential
cross-sections $d\sigma/d\cos\theta$ as a function of $E_{\mathrm{cm}}$ for
$\bm{k}_{K}\cdot\bm{\epsilon}_{K^{*}}=0$ at $\theta=0$ (solid), $\pi/4$
(dotted), and $\pi/2$ (dashed) in the cm frame. Here, we specifically focus on
the result for the $P^{*}_{0}(3/2^{-})$, as other quantum-number states
exhibit minimal variation. At $\theta=0$, where $K^{*}$ is scattered
forwardly, the $P^{*}_{0}$ signal is isolated with minimal background
contamination. As the angle increases, the $S/N$ value worsens, particularly
at $\theta=\pi/4$. Further increasing it to $\theta=\pi/2$ enhances the peak
due to interference between the resonance and the $u$-channel contribution,
although the background contamination also increases. In panel (b) of Fig. 4,
we plot the differential cross-section as a function of $\cos\theta$ and
$E_{\mathrm{cm}}$. Across a wide range of angles ($-0.5\lesssim\cos\theta\leq
1$), clear resonance signals are observed, facilitating their measurement in
experiments. Additionally, we confirm that the unpolarized cross-sections
exhibit minimal dependence on the spin-parity quantum numbers. Therefore, the
discussions provided for Figs. 3 and 4 remain applicable to other quantum-
number cases without loss of generality.

Finally, we would like to propose a physical observable for determining the
spin-parity quantum number of the $P^{*}_{0}$ resonance. The polarizations of
the involved particles, such as the target and recoil nucleons, and $K^{*}$,
are invaluable for this determination. However, achieving a definite neutron
polarization inside the deuteron target poses challenges in experimental
setups. As discussed earlier, to maximize the $S/N$, it’s advantageous to set
the $K^{*}$ polarization as $\bm{k}_{K}\cdot\bm{\epsilon}_{K^{*}}=0$
(perpendicular configuration). Consequently, we rely on controlling the
recoil-proton polarization for quantum-number determination. Hence, we
introduce the recoil-proton spin asymmetry (RSA) defined as follows, with
$\bm{\epsilon}_{K^{*}}$ perpendicular to the reaction plane:

$\displaystyle\Sigma_{\perp}\equiv\frac{d\sigma_{\uparrow}/d\cos\theta-d\sigma_{\downarrow}/d\cos\theta}{d\sigma_{\uparrow}/d\cos\theta+d\sigma_{\downarrow}/d\cos\theta}\Big{|}_{\bm{k}_{K}\cdot\bm{\epsilon}_{K^{*}}=0},$
(20)

where the notation ($\uparrow,\downarrow$) denotes the (up, down)-spin
polarization of the recoil proton, respectively. In panel (a) of Fig. 5, we
illustrate the RSAs as functions of $\cos\theta$ at $E_{\mathrm{cm}}=2.5$ GeV.
Note that the curves mainly comprise contributions from the $s$-channel and
$u$-channel, as the $t$-channel contribution is eliminated by fixing the
polarization as $\bm{k}_{K}\cdot\bm{\epsilon}_{K^{*}}=0$. The spin-$1/2$ cases
exhibit much smaller asymmetry compared to the spin-$3/2$ ones. The
$P^{*}_{0}(1/2^{\pm})$ amplitudes show weak angular dependence and are
relatively small due to their Lorentz structures in the vicinity of
$E_{\mathrm{cm}}=2.5$ GeV as follows:

$\displaystyle
u^{\dagger}(p_{2})\gamma_{0}\gamma_{2}\left(E_{\mathrm{cm}}\gamma_{0}\pm
M_{P^{*}_{0}}\right)\gamma_{5}u(p_{1})\sim A(E_{\mathrm{cm}}\pm
M_{P^{*}_{0}})+B(E_{\mathrm{cm}}\mp
M_{P^{*}_{0}})(\cos\theta,\sin\theta),\,\,\,\,(A,B)\in R.$ (21)

On the contrary, the $P^{*}_{0}(3/2^{\pm})$ amplitudes exhibit highly
nontrivial angular dependencies arising from the spin-sum of the Rarita-
Schwinger field and the kinematic factor $\bm{k}_{K}\cdot\bm{k}_{K^{*}}\propto
t$, as shown in Eq. (8). Consequently, the spin-parity quantum numbers can be
determined from $\Sigma_{\perp}(\cos\theta)$ for the spin-$3/2$ cases, whereas
fixing the parity for the spin-$1/2$ cases proves to be more challenging. In
panel (b) of Fig. 5, the numerical results for the RSA as functions of
$E_{\mathrm{cm}}$ are presented. Each spin-parity case exhibits a distinctive
curve shape above $E_{\mathrm{cm}}\approx 2.2$ GeV. Once more, the spin-$3/2$
cases demonstrate different energy dependence, whereas the spin-$1/2$ ones
remain small and flat due to the same reasons as discussed above.

(a)0.5cm-2cm (b)0.5cm-2cm
---
Figure 5: (a) Recoil-proton spin asymmetry (RSA) in Eq. (20) as functions of
$E_{\mathrm{cm}}$ for different spin-parity quantum numbers of $P^{*}0$, with
$\bm{\epsilon}{K^{*}}$ perpendicular to the reaction plane. (b)
$\Sigma_{\perp}$ as functions of $\cos\theta$ at $E_{\mathrm{cm}}=2.5$ GeV
following the same approach.

##  IV Summary and future perspectives

In our current study, we delve into the peak-like structure observed around
$\sqrt{s}\sim 2.5$ GeV in the reaction $K^{+}n\to K^{*0}p$. We consider the
potential $S=+1$ resonance $P^{+*}_{0}\equiv P^{*}_{0}$ as an excited state
within the extended vector-meson and baryon ($VB$) antidecuplet, particularly
in the $s$ channel. To explore this, we employ the effective Lagrangian method
in conjunction with the Regge approach at the tree-level Born approximation.
We examine various hyperon ($Y$) exchanges in the $u$ channel and pion ($\pi$)
ones in the $t$ channel. Below, we list important findings of the present
studies:

  * •

We find that the currently available experimental data around
$E_{\mathrm{cm}}=2.5$ GeV cannot be adequately explained solely by the tree-
level Born diagrams incorporating Regge contributions as the background (BKG).
However, they are well accounted for by a resonance state characterized by
$M_{P^{*}_{0}}=2.5$ GeV and $\Gamma_{P^{*}_{0}}=100$ MeV, alongside its spin-
parity quantum numbers $J^{P}=1/2^{\pm}$ and $3/2^{\pm}$. To further elucidate
this, we estimate the combined coupling constants for the resonance, denoted
as $g_{P^{*}_{0}}\equiv g_{KN{P^{*}_{0}}}g_{K^{*}N{P^{*}_{0}}}$, by fitting
the data, paving the way for future theoretical studies.

  * •

The angular dependence of the reaction process, represented by
$d\sigma/d\cos\theta$, is primarily influenced by the contributions from the
$t$-channel $\pi$ and $u$-channel $\Lambda(1116)$. These contributions result
in significant enhancements of both forward and backward scattering in the
center-of-mass (cm) frame, which gradually diminish as $E_{\mathrm{cm}}$
increases. However, in contrast, for all spin-parity states of $P^{*}_{0}$, it
becomes evident that the resonance contribution is comparatively smaller than
the background (BKG) and only becomes notable around $E\mathrm{cm}=2.5$ GeV,
particularly in the vicinity of $\cos\theta=0$.

  * •

To accurately study the resonance contribution, we have devised an
experimental approach aimed at enhancing the signal-to-noise ratio ($S/N$) by
taking into account the structure of the predominant background signals
(BKGs). By concentrating on the forward-scattering region and orienting the
polarization of the $K^{*}$ perpendicular to the reaction plane, it becomes
feasible to effectively circumvent the effects originating from the $u$
channel and deactivate those from the $t$ channel, respectively. This strategy
yields a $S/N$ value close to unity. As the scattering angle increases towards
$\cos\theta\approx 0$, the contamination stemming from the $u$-channel becomes
more pronounced, yet the $S/N$ value remains reasonably high.

  * •

Considering the ongoing reaction process, we delve into exploring a polarized
observable known as the recoil-proton spin asymmetry (RSA) with
$\bm{k}_{K}\cdot\bm{\epsilon}_{K^{*}}=0$ to discern the spin-parity quantum
numbers of the $P^{*}_{0}$. We discern distinctive angular dependencies of the
RSAs for the $J^{P}=3/2^{\pm}$ states. However, determining the parity for the
spin-$1/2$ states proves to be challenging due to their weak and minimal
angular dependencies at $E\mathrm{cm}=2.5$ GeV. Furthermore, we analyze the
energy dependencies of the RSA for prospective experiments.

We wish to underscore that the potential $S=+1$ resonance state $P^{*}_{0}$,
proposed to account for the currently available data on $K^{+}n\to K^{*0}p$,
holds promise for illuminating our understanding of exotic phenomena in QCD,
surpassing traditional chiral interaction theories. The conjecture that
$P^{*}_{0}$ pertains to the resonance of the extended $VB$ antidecuplet, akin
to the $PB$ antidecuplet for $\Theta^{+}$, remains inadequately elucidated
within existing theories. Nevertheless, ongoing discourse contemplates
plausible explanations for these antidecuplet resonances NAMYANG .
Furthermore, more realistic theoretical studies will be conducive to future
experimental analyses, i.e., $K^{+}n\to K^{*0}p\to\pi^{0,\mp}K^{0,\pm}p$ for
instance. Such endeavors will yield valuable insights into $P^{*}_{0}$,
particularly in the $K^{0}p$ invariant-mass domain. Relevant investigations
are underway and will be detailed elsewhere.

## Acknowledgment

The authors extend their gratitude for the fruitful discussions with Jung Keun
Ahn (Korea University), Shin Hyung Kim (Kyungpook National University), Ghil-
Seok Yang (Hoseo University), Kanchan Pradeepkumar Khemchandani (Federal
University of São Paulo), and Alberto Martinez Torres (University of São
Paulo). This work received support from the National Research Foundation of
Korea (NRF) grants funded by the Korean government (MSIT) (2018R1A5A1025563,
2022R1A2C1003964, and 2022K2A9A1A06091761).

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aainstitutetext: Vinca Institute of Nuclear Sciences, National Institute of
the Republic of Serbia, University of Belgrade
M. Petrovica Alasa 12-14, Belgrade, Serbiabbinstitutetext: University of
Kragujevac, Faculty of Science, Department of Physics
Radoja Domanovica 12, Kragujevac, Serbia

# Systematic uncertainties in integrated luminosity measurement at CEPC

I. Smiljanic,111Corresponding author. a I. Bozovic Jelisavcic a G. Kacarevic a
N. Vukasinovic a G. Milutinovic Dumbelovic a T. Agatonovic Jovin a I.
Vidakovic a V. Rekovic b J. Stevanovic b and M. Radulovic
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The very forward region of a detector is one of the most challenging regions
to instrument at a future $e^{+}e^{-}$ collider. At CEPC, machine-detector
interface includes, among others, a calorimeter dedicated for precision
measurement of the integrated luminosity at a permille level or better. Here
we review a feasibility of such precision, from the point of view of
systematic effects arising from the luminometer’s mechanical precision and
positioning, beam-related requirements and physics background. A method of the
beam energy spread determination, initially proposed for FCC, is also
discussed for the CEPC beams.

###### Keywords:

$e^{+}$-$e^{-}$ Experiments

##  1 Introduction

The Circular Electron Positron Collider (CEPC) is a large international
scientific facility proposed by the Chinese particle physics community in 2012
to test the validity scale of the Standard Model (SM) through precision
measurements in the Higgs, BSM and EW sectors. These measurements should
provide critical tests of the underlying fundamental physics principles of the
Standard Model and are vital in exploration of New Physics beyond the SM. In
electron-positron collisions, the CEPC is designed to operate at 91.2 GeV as a
$Z^{0}$ factory, at 160 GeV (WW production threshold) and at 240 GeV as a
Higgs factory. The vast amount of bottom quarks, charm quarks and
$\tau$-leptons produced in $Z^{0}$ decays also makes the CEPC an effective
b-factory and $\tau$ and charm factory CEPC_CDR .

In order to achieve precision required for realization of the CEPC physics
program, relative uncertainty of the integrated luminosity measurement should
be of order of $10^{-4}$ at 91.2 GeV and of order of $10^{-3}$ at 240 GeV.
Precision reconstruction of position and energy of electromagnetic showers
generated by Bhabha scattering at a high-energy $e^{+}e^{-}$ collider can be
achieved with finely granulated compact luminometer. The method for integrated
luminosity measurement at CEPC is described in CEPC_CDR . However, the
precision reconstruction of Bhabha scattering doesn’t exhaust the long list of
systematic uncertainties of the integrated luminosity measurement, which
includes detector related uncertainties, beam related uncertainties and
uncertainties originating from physics and machine-related interactions. The
latest, that includes beam-beam interactions and beam-gas scattering, is not
discussed here. In this paper we review the effects of the detector and beam
related uncertainties, namely mechanical uncertainties of the luminometer
positioning and size and uncertainties related to the beam energy, beam
synchronization and interaction point (IP) displacements, as well as the
uncertainty that originates from the miscount of physics background. In
addition, motivated by Janot , we discuss the possibility of CEPC beam energy
spread determination employing the processes in the central tracker.

##  2 Forward region of CEPC

The Machine Detector Interface (MDI) of CEPC CEPC_CDR  is about 6 m far from
the IP. The accelerator components inside the detector without shielding are
within a conical space with an opening angle of arccos 0.993. The two beams
collide at the IP with a crossing angle of 33 mrad in the horizontal plane,
and the final focus length is 2.2 m. Luminometer at CEPC is proposed to cover
the polar angle region between 26 mrad and 105 mrad (with fiducial volume
between 53 mrad and 79 mrad) corresponding to the detector aperture of 28.5 mm
for the inner radius and 100 mm for the outer, at 950 mm distance from the IP.
Luminometer might be supplemented with an additional front layer of tracker in
order to improve $e-\gamma$ separation and calibration of the device. Since
the luminometer will be placed at $z=\pm 950$ mm, shower leakage from the
outer edge of the luminometer, possibly contaminating tracking detectors, have
been studied and proven to be negligible after absorption by a 5 mm iron
filter positioned around the luminometer CEPC_CDR . Layout of the very forward
region at CEPC and the revised beam-pipe design are given in Figure 1 CEPC_CDR
.

Figure 1:  Left: Layout of the very forward region at CEPC; Right: Revised
beam-pipe design.

It is clear that luminometer must be a compact calorimeter providing shower
containment in a longitudinal dimension not larger than 10 cm. Luminometer
technology options are still open. Among the candidates for the luminometer
are CMS-like shashlik type of calorimeter based on Lutetium Yttrium
Orthosilicate ($Lu_{x}Y_{2-x}SiO_{5}:Ce$ \- LYSO) LYSO  and SciFi spaghetti
calorimeter with individually read-out fibers (prototyped for J-PARC KL
experiment JParc ). However, the most compact design currently proposed seems
to be Si-W sandwich type of calorimeter that could provide over 20 $X_{0}$ in
the foreseen longitudinal dimension (Figure 2 lumical ). The prototype of such
a calorimeter has been developed by FCAL Collaboration and has been tested in
several test beam campaigns over the last 10 years Moliere . This
luminometer’s fiducial volume ranges from 53 mrad to 79 mrad if used at CEPC,
with sensors placed in 2 mm air gaps and it should have fine Si-pixel
segmentation (i.e 48/64 azimuthal/radial) and small (effective) Moliere radius
($\sim$2 cm), which would result in excellent resolution in energy and polar
angle of the reconstructed electrons provided by the sampling term
$\alpha$=0.2 and $\sigma_{\theta}\sim 10^{-2}$ mrad lumical .

Figure 2:  Model of Si-W sandwich type of calorimeter developed by FCAL
Collaboration, that might be employed at CEPC.

##  3 Integrated luminosity measurement and systematic uncertainties

Integrated luminosity measurement is a counting experiment based on Bhabha
scattering. It is defined as $L=N_{bh}/\sigma$, where $N_{bh}$ is Bhabha count
in the considered interval of time and the certain phase space within the
detector acceptance (fiducial) region and $\sigma$ is the theoretical cross-
section in the same geometrical and phase space. However, in a real experiment
there are several effects influencing Bhabha count. Here we list some of them
that are addressed at the simulation level, assuming detector geometry as
described in Section 2 and the CEPC beams as in CEPC_CDR : uncertainties from
mechanics (detector manufacturing, positioning and alignment); uncertainties
from the beam properties (center-of-mass energy, beam-energy asymmetry, beam
synchronization, IP displacements) and Bhabha miscounts from two-photon
processes as the main physics background. In order to control integrated
luminosity with the relative precision of $10^{-4}$ ($10^{-3}$), both $N_{bh}$
and $\sigma$ should be known at the same order of magnitude, which means that
all these effects should be controlled with the same precision.

###  3.1 Uncertainties stemming from mechanics and positioning

Systematic uncertainties from detector and machine-detector interface related
effects have been quantified through a simulation study, assuming $10^{7}$
Bhabha scattering events generated using BHLUMI V4.04 Bhabha event generator
BHLUMI , at two center-of-mass energies: 240 GeV and $Z^{0}$ production
threshold. Detector fiducial volume, where the showers are fully contained and
thus the sampling term constant, is assumed as described in Section 2, with
the luminometer placed at ongoing beams. The effective Bhabha cross-section in
this angular range is of order of a few nb. Final state particles are
generated in the polar angle range from 45 mrad to 85 mrad, somewhat broader
than the detector fiducial volume, in order to allow events with non-collinear
Final State Radiation (FSR) to contribute. Close-by particles are summed up to
imitate cluster merging. We assume that the shower leakage from the
luminometer is negligible.

Furthermore, we have applied event selection that is asymmetric in polar angle
acceptance on the left and right arm of the detector, as it has been done at
OPAL OPAL . That is, at one side we consider the full fiducial volume, while
at the other side we shrink the radial acceptance for $\Delta r$. This has
been done subsequently to the left (L) and right (R) side of the luminometer,
on event by event basis. In addition, we require high-energy electrons
carrying above 50% of the available beam energy. Against this type of event
selection for luminosity measurement, we compare the selection based of the
full fiducial volumes on both sides of the detector. An example is given in
Figure 3, illustrating the cancelation of systematics uncertainties caused by
the assumption of L-R symmetry in an event, when asymmetric selection in polar
angle is applied. It is clear that asymmetric selection is advantageous,
requiring a luminometer placed at the outgoing beams.

Figure 3:  Luminosity uncertainty from the longitudinal IP displacements
($\Delta Z_{IP}$) w.r.t. the luminometer, for symmetric (circled) and
asymmetric selection with a radial shrink of the fiducial volume $\Delta r$.

Considered detector-related uncertainties arising from manufacturing,
positioning and alignment are:

$\bullet$

uncertainty of the luminometer inner radius ($\Delta r_{in}$),

$\bullet$

spread of the measured radial shower position w.r.t. the true impact position
on the luminometer front plane ($\sigma_{r}$),

$\bullet$

uncertainty of the longitudinal distance between left and right halves of the
luminometer ($\Delta l$),

$\bullet$

radial and axial ($\sigma_{x_{IP}},\sigma_{z_{IP}}$) mechanical fluctuations
of the luminometer position with respect to the IP, caused by vibrations and
thermal stress,

$\bullet$

twist of the calorimeters corresponding to different rotations of the left and
right detector axis with respect to the outgoing beam ($\Delta\varphi$).

We assumed $10^{-3}$ and $10^{-4}$ contribution to the relative uncertainty of
integrated luminosity from each individual effect, at 240 GeV and $Z^{0}$ pole
respectively. Table 1 gives corresponding requirements of the listed
parameters. It is clear that due to the $\sim 1/\theta^{3}$ dependence of the
Bhabha cross-section from the polar angle, inner aperture of the luminometer
is one of the most demanding mechanical parameters to control. The system for
detector positioning measurement must be such to provide $\sim$80 $\mu$m
uncertainty in distance between L and R halves of luminometer.

Table 1:  Required absolute precision of mechanical parameters contributing to the relative uncertainty of the integrated luminosity of $10^{-3}$ ($10^{-4}$) at 240 GeV center-of-mass energy ($Z^{0}$ pole). parameter | precision @ 240 GeV | precision @ 91 GeV
---|---|---
$\Delta r_{in}$ ($\mu$m) | 10 | 1
$\sigma_{r}$ (mm) | 1.00 | 0.20
$\Delta l$ (mm) | 1.00 | 0.08
$\sigma_{x_{IP}}$ (mm) | 1.0 | 0.5
$\sigma_{z_{IP}}$ (mm) | 10 | 7
$\Delta\varphi$ (mrad) | 6.0 | 0.8

###  3.2 MDI related uncertainties

Several uncertainties are considered that may arise from the beam properties
and its delivery to the interaction point:

$\bullet$

uncertainty of the average net center-of-mass energy ($\Delta E_{CM}$),

$\bullet$

asymmetry in energy of the $e^{+}$ and $e^{-}$ beams $\lvert
E_{e^{+}}-E_{e^{-}}\rvert$ given as the maximal deviation of the beam energy
from its nominal value ($\Delta E$),

$\bullet$

radial ($\Delta x_{IP}$) and axial ($\Delta z_{IP}$) IP position displacements
with respect to the luminometer, caused by the finite transverse beam sizes
and beam synchronization respectively,

$\bullet$

time shift in beam synchronization ($\tau$) leading to IP longitudinal
displacement $\Delta z_{IP}$.

Table 2 gives absolute uncertainties of these parameters contributing to the
relative uncertainty of integrated luminosity in the same way as described for
Table 1.

Table 2:  Required absolute precision of MDI parameters contributing to the relative uncertainty of the integrated luminosity of $10^{-3}$ ($10^{-4}$) at 240 GeV center-of-mass energy ($Z^{0}$ pole). parameter | precision @ 240 GeV | precision @ 91 GeV
---|---|---
$\Delta E_{CM}$ (MeV) | 240 | 9
$\Delta E$ (MeV) | 120 | 5
$\Delta x_{IP}$ (mm) | 1.0 | 0.5
$\Delta z_{IP}$ (mm) | 10.0 | 2.0
$\tau$ (ps) | 15 | 3

Another challenge comes from the uncertainty of energy of individual beams
(and of the effective center-of-mass energy) needed to be controlled at the
level of $\sim 10^{-5}$ with respect to the nominal beam energy at the $Z^{0}$
pole, that is smaller than the foreseen beam energy spread of 0.08% CEPC_CDR .
The current value of the beam energy spread at the $Z^{0}$ pole will
contribute to the relative uncertainty of the Bhabha count maximally as
$1.6\cdot{10^{-3}}$, through the uncertainty of the effective center-of-mass
energy $\Delta E_{CM}$ for the Bhabha cross-section calculation
($\sigma_{Bh}\sim 1/E_{CM}^{2}$), or as the asymmetry in beam energies giving
rise to longitudinal boost of the center-of-mass (collision) system (CM)
w.r.t. the laboratory frame. Figure 4 illustrates counting loss in the
luminometer due to longitudinal boost of the CM frame $\beta_{Z}\sim 2\Delta
E$, at 240 GeV center-of-mass energy.

Figure 4:  Loss of the Bhabha count in the luminometer due to the longitudinal
boost of the CM frame $\beta_{Z}$, where $\beta_{Z}$ is given as $2\Delta
E/E_{CM}$. $\Delta r$ values correspond to different polar angle acceptance of
detector left and right arms. Red dashed line indicates $10^{-3}$ uncertainty
in count required at 240 GeV CEPC.

###  3.3 Two-photon processes as a background

In $e^{+}e^{-}$ collisions there are several 4-fermion proceses
(multiperipheral, annihilation, conversion and bremsstrahlung) representing
possible background for the Bhabha scattering. The multiperipheral (two-
photon) process, given in Figure 5, should be considered due to its large
cross-section ($\sim$nb) and to the fact that spectator electrons are emitted
at very small polar angles. Even though the most of high-energy electron
spectators from these processes go below the luminometer’s angular acceptance
region, some of them can still be misidentified as Bhabha electrons. Here we
quantify their contribution, assuming geometrical parameters as in Section 2.

Figure 5:  Feynman diagram of two-photon process producing $e^{+}e^{-}+2-jet$
final state in electron-positron collisions.

In order to estimate the background to signal ratio at 240 GeV CEPC (where
2-photon processes are more abundant than at the $Z^{0}$ pole, due to $ln(s)$
cross-section dependence), we simulated $10^{5}$ $e^{+}e^{-}\rightarrow
e^{+}e^{-}\mu^{+}\mu^{-}$ events using WHIZARD V2.6 WHIZARD  in the polar
angle range $\lvert\cos\theta\rvert<0.999$. The effective cross-section
$\sigma_{eff}\sim 0.3$ pb is found in the fiducial volume of the luminometer.
$10^{7}$ Bhabha events are simulated using BHLUMI V4.04 in the polar angle
range $\theta>3$ mrad, with $\sigma_{eff}\sim 3.3$ nb in the fiducial volume
of the luminometer. Polar angle distribution for signal and background is
given in Figure 6. Counts are normalized to $5.6$ ab-1, which corresponds to 7
years of data taking at 240 GeV CEPC.

Figure 6:  Normalized polar angle distribution of signal (black line) and
background (red line) events.

From Figure 6 it is clear that the most of spectators go below the
luminometer, while contamination of the signal in the luminometer’s fiducial
volume is of order of $10^{-4}$ even without any event selection. In this
study we’ve looked into spectator electrons in the luminometer fiducial
volume, from the $e^{+}e^{-}\rightarrow e^{+}e^{-}\mu^{+}\mu^{-}$ final state
of 2-photon processes. We found that the total amount of background should be
conservatively scaled by a factor 3 to account for flavor integration. Further
refinements are possible with the coplanarity request between left and right
detector arms ($\lvert\varphi_{e^{+}}-\varphi_{e^{-}}\rvert$), as well as the
cut on relative energy $E_{rel}=\lvert E_{e^{+}}+E_{e^{-}}\rvert/2E_{beam}$
that is also useful to suppress both 2-photon background and off-momentum
particles. Finally, physics background can be always taken as a correction to
the Bhabha count if the uncertainty of its cross-section is available.

##  4 Determination of the beam energy spread

Motivated by the similar work done at FCC-ee Janot , and having in mind that
precision of several EW observables at the $Z^{0}$ pole (such as the $Z^{0}$
mass, width and production cross-section) depends on the precision of beam
energy spread determination, we looked into possibility to measure it at CEPC
using well defined central process like di-muon production
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$. Having in mind the projected
performance of the central tracker to efficiently ($\sim 99\%$) and precisely
(100 $\mu$m position resolution) reconstruct muons CEPC_CDR , process
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$ with a cross-section of 1.5 nb at the
$Z^{0}$ pole seems to be an optimal choice.

We’ll argue that the effective center-of-mass energy ($s^{\prime}$) is
sensitive to variation of the beam energy spread that consequently can be
determined from the population of the peak of the $s^{\prime}$ distribution.
In order to determine $s^{\prime}$ sensitivity to the beam energy spread, we
generated several hundreds thousands $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$
events at 91.2 GeV and 240 GeV center-of-mass energies. Events are generated
using WHIZARD 2.6, in the polar angle range from $8^{o}$ to $172^{o}$, which
is the angular acceptance of the central tracker (TPC) at CEPC CEPC_CDR .
Events are generated simulating individually effects like the Initial State
Radiation (ISR) and detector angular resolution, in order to study their
impact on the $s^{\prime}$ distribution with respect to the concurrent beam
energy spread. Detector energy resolution is simulated by performing Gaussian
smearing of the muons’ polar angles w.r.t. the 0.1 mrad RMS corresponding to
100 $\mu$m position resolution foreseen for TPC at CEPC.

The effective center-of-mass energy, $s^{\prime}$, can be calculated from the
reconstructed muons’ polar angles sprime , as:

$\frac{s^{\prime}}{s}=\frac{\sin\theta^{+}+\sin\theta^{-}-\lvert\sin(\theta^{+}+\theta^{-})\rvert}{\sin\theta^{+}+\sin\theta^{-}+\lvert\sin(\theta^{+}+\theta^{-})\rvert}$
(1)

relying on the excellent TPC spatial resolution. As illustrated in Figure 7
(left), beam energy spread dominates the $s^{\prime}$ shape at energies close
to the nominal center-of-mass energy, while the Figure 7 (right) illustrates
the effect of muon polar angle resolutions of 0.1 mrad and 1 mrad on top of
ISR and the beam energy spread. From Figure 7 (right) is clear that tracker
resolution of 0.1 mrad does not affect the $s^{\prime}$ sensitivity to the
beam energy spread. On the other hand, tracker resolution of 1 mrad
significantly influences the method. We found that central tracker resolution
in polar angle should not be larger than 0.5 mrad, corresponding to the 500
$\mu$m position resolution of TPC. The same stands for the beam energy of 120
GeV, where the nominal beam energy spread of 0.134% is foreseen CEPC_CDR .

Figure 7:  Left: Beam energy spread is the dominant effect to reduce the
number of events at the maximal CM energy; Right: s’ sensitivity to the beam
energy spread and different tracker resolutions.

To exploit $s^{\prime}$ peak sensitivity to the beam energy spread values,
beam energy spread is varied around the nominal value, generating each time
$2.5\cdot 10^{5}$ ($10^{5}$) events per beam energy spread variation at 91.2
GeV (240 GeV). Dependence is illustrated in Figure 8, at the $Z^{0}$ pole and
240 GeV. As expected, larger beam energy spread leads to the corresponding
reduction of the number of di-muon events carrying close-to-maximal available
energy from the collision. Knowing this dependence from simulation enables the
determination of the effective beam energy spread ($\delta^{\prime}$) once the
count of di-muon events is known experimentally.

Figure 8:  Left: Number of di-muon events at the maximal CM energy around
nominal 91.2 GeV w.r.t. the beam energy spread; Right: The same at 240 GeV.

Table 3 shows that with the assumed statistics of collected events at $Z^{0}$
pole, relative variations of the $\delta^{\prime}$ can be measured with 25%
relative systematic uncertainty from the calibration curve and 1.2% relative
statistical uncertainty for only 3 minutes of data taking with $1.02\cdot
10^{36}$ cm-2s-1 instantaneous luminosity. At 240 GeV center-of-mass energy,
$\delta^{\prime}$ can be measured with 15% relative systematic uncertainty
from the calibration curve and 2.3% relative statistical uncertainty but for
approximately 5 days of data taking with instantaneous luminosity of $5.2\cdot
10^{34}$ cm-2s-1.

Table 3: Beam energy spread variations experimentally accessible at CEPC. CEPC |  $\mathcal{L}$ @ IP (cm-2s-1) |  Nominal beam energy spread $\delta$ (%) |  Number of events |  Cross- section $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$ |  Collect. time |  Total relative uncert. of $\delta^{\prime}$ |  Total absolute unc. of $\delta^{\prime}$ (MeV)
---|---|---|---|---|---|---|---
$Z^{0}$ pole | $1.02\cdot 10^{36}$ | 0.080 | $2.5\cdot 10^{5}$ | 1.5 nb | 3 min | 25% | 9
240 GeV | $5.2\cdot 10^{34}$ | 0.134 | $1.0\cdot 10^{5}$ | 4.1 pb | 5 days | 15% | 24

As already mentioned, several precision EW observables at the $Z^{0}$ pole
depend on the beam energy spread uncertainty. Figures 9 and 10 illustrate that
the cross-section for $Z^{0}$ production ($\sigma_{Z}$), $Z^{0}$ total width
($\Gamma_{Z}$) and mass ($m_{Z}$) will receive following contributions from
the foreseen beam energy spread uncertainty: $\delta(\sigma_{Z})\sim 3\cdot
10^{-3}$, $\Delta\Gamma_{Z}\sim$5 MeV and $\Delta m_{Z}\sim$160 keV.

Figure 9:  Impact of the relative precision of the beam energy spread on the
$Z^{0}$ production cross-section.

Figure 10:  Impact of the relative precision of the beam energy spread on the
$Z^{0}$ total width and mass absolute precisions.

Several refinements of this study are possible, yet not expected to
significantly modify the results: effect of ISR (theoretical) uncertainty, FSR
of muons, full detector simulation and impact of similar final state
backgrounds, as well as presence of the beamstrahlung that is however much
less pronounced at circular than at linear machines.

##  5 Conclusions

The study confirmed that the uncertainty of the luminometer inner radius at
the micron level together with the uncertainties of the available center-of-
mass energy and beam energy asymmetry below 10 MeV are posing the most
challenging requirements on measurement of the integrated luminosity with
$10^{-4}$ relative precision at the $Z^{0}$ pole. Permille precision of the
integrated luminosity measurement at 240 GeV CEPC seems to be feasible from
the point of view of the existing technologies.

It is also shown that with the CEPC post CDR design, beam energy spread can be
determined with absolute accuracy corresponding to 9 MeV, in only 3 minutes of
data-taking of $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$ events at the $Z^{0}$
pole, where the statistical uncertainty contributes at the level of several
hundreds of keV. The above translates to relative uncertainty of the $Z^{0}$
production cross-section of $3\cdot 10^{-3}$ and absolute precisions of the
$Z^{0}$ mass of 160 keV and width of 5 MeV respectively.

###### Acknowledgements.

This research is supported by the Ministry of Education, Science and
Technological Development and by the Science Fund of the Republic of Serbia,
through Grant No 7699827, HIGHTONE-P. This paper also contains results
accomplished in in collaboration of the authors with S. Lukic, previously in
the Vinca Institute HEP Group.

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