# Dilution induced magnetic localization in Rb(Co1-xNix)2Se2 single crystals

Hui Liu  Mengwu Huo  Chaoxin Huang  Xing Huang  Hualei Sun  Lan Chen  Center
for Neutron Science and Technology, Guangdong Provincial Key Laboratory of
Magnetoelectric Physics and Devices, School of Physics, Sun Yat-Sen
University, Guangzhou, 510275, China  Juping Xu  Institute of High Energy
Physics, Chinese Academy of Sciences, Beijing 100049, China  Spallation
Neutron Source Science Center, Dongguan 523803, China  Wen Yin  Institute of
High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
Spallation Neutron Source Science Center, Dongguan 523803, China  Runxia Li
<EMAIL_ADDRESS>Materials Science and Engineering, Dongguan University of
Technology, Dongguan 523808, China  Meng Wang<EMAIL_ADDRESS>Center for Neutron Science and Technology, Guangdong Provincial Key Laboratory
of Magnetoelectric Physics and Devices, School of Physics, Sun Yat-Sen
University, Guangzhou, 510275, China

###### Abstract

We report experimental studies on a series of Rb(Co1-xNix)2Se2 (0.02 $\leq
x\leq$ 0.9) powder and single crystal samples using x-ray diffraction, neutron
diffraction, magnetic susceptibility, and electronic transport measurements.
All compositions are metallic and adopt the body-centered tetragonal structure
with $I4/mmm$ space group. Anisotropic magnetic susceptibilities measured on
single crystal samples suggest that Rb(Co1-xNix)2Se2 undergo an evolution from
ferromagnetism to antiferromagnetism, and finally to paramagnetism with
increasing Ni concentration. Neutron diffraction measurements on the samples
with $x$ = 0.1, 0.4, and 0.6 reveal an $A$-type antiferromagnetic order with
moments lying in the $ab$ plane. The moment size changes from 0.69 ($x=0.1$)
to 2.80$\mu_{B}$ ($x=0.6$) per Co ions. Our results demonstrate that dilution
of the magnetic Co ions by substitution of nonmagnetic Ni ions induces
magnetic localization and evolution from itinerant to localized magnetism in
Rb(Co1-xNix)2Se2.

##  I INTRODUCTION

Layered transition metal chalcogenides with the ThCr2Si2-type structure show
versatile characteristics, such as superconductivity, heavy fermion behavior,
and various magnetic orders due to the sensitivity of the magnetic and
electronic correlations to the atomic distances and compositions.Dai (2015);
Gegenwart et al. (2008); Berger et al. (2002); Huan and Greenblatt (1989);
Ronneteg et al. (2010); Narasimhan et al. (1975); Ying et al. (2012); Yang et
al. (2013) The iron chalcogenide compounds $A_{x}$Fe2Se2 ($A=$ K, Rb, Cs, and
Tl) exhibit superconductivity with a transition temperature at $\sim$30 K.Guo
et al. (2010); Fang et al. (2010) While by inducing iron vacancies, for
compositions with $\delta=0.4$ and 0.5, $A_{x}$Fe2-δSe2 become insulating and
show iron vacancy and antiferromagnetic (AFM) orders.Bao et al. (2011); Zhao
et al. (2012); Wang et al. (2016) In the insulating states, the Fe2+ with
fully localized electrons are in a high spin state. The Co-doping in
$A_{x}$Fe2Se2 suppresses the superconductivity rapidly and induces metallic
magnetic ground states.Huang et al. (2021); Lizárraga et al. (2004); Liu et
al. (2018); Yang et al. (2013); Ryu et al. (2015a)

By replacing Fe for Ni, the $A$Ni2Se2 compounds show Pauli paramagnetic (PM),
multi-band characteristics, and superconductivity with superconducting
transition temperatures at $0.8\sim 3.7$ K.Lei et al. (2014); Chen et al.
(2016); Wang et al. (2013); Ryu et al. (2015b); Liu et al. (2022) While the
isostructural compounds KCo2Se2 and RbCo2Se2 exhibit ferromagnetic (FM) order
with Curie temperatures of $T_{\text{C}}=74$ and 83 K, respectively.Huang et
al. (2021); Yang et al. (2013) CsCo2Se2 is an $A$-type AFM metal with moments
lying in the $ab$ plane.Von Rohr et al. (2016); Liu et al. (2018) The size of
the in-plane ordered moments of Co ions in RbCo2Se2 is $\sim$0.60$\mu_{B}$
determined from neutron diffraction measurement, indicating an itinerant
origin.Huang et al. (2021) As a simple expectation, Ni substitution on the Co
sites would progressively tune the ground states of Rb(Co1-xNix)2Se2 from FM
order to PM state. However, in previous studies of Tl(Co1-xNix)2Se2 and
Tl(Co1-xNix)2S2, the AFM order was found to exist in a large region of the Ni
doping level. Although the magnetic order of TlCo2Se2 is AFM and that of
TlCo2S2 is FM.Newmark et al. (1989); Huan et al. (1989) These results indicate
that the effect of Ni substitution on the magnetism in
$A$(Co1-xNix)${}_{2}X_{2}$ ($X=$ Se and S) has a unique mechanism, motivating
us to explore the magnetic and electronic properties of Rb(Co1-xNix)2Se2
single crystals.

In this work, we report detailed studies on the crystallographic, magnetic
structure, magnetic susceptibility, and electronic transport properties of
Rb(Co1-xNix)2Se2 single crystals by utilizing powder x-ray diffraction (XRD),
neutron powder diffraction (NPD), magnetic susceptibility, and resistance
measurements. We find by increasing the Ni concentration, (1) the FM order
transforms to AFM order at around $x=0.06$, (2) the AFM order extends to the
compositions with $x=0.8$, (3) the anisotropic in-plane and out-of-plane
magnetic susceptibilities reverse magnitudes at $x\approx 0.3$, and (4) the
$x=0.9$ compound is paramagnetic. Neutron diffraction measurements reveal that
the AFM phase corresponds to a universal $A$-type magnetic structure with
spins aligned in the $ab$ plane. While the moment size is increased from
0.69(2) for $x=0.1$ to 2.80(6)$\mu_{B}$ for $x=0.6$. The residual resistivity
ratio ($RRR$) shows an intimate relationship with magnetism. The results
suggest that the dilution of magnetic Co by nonmagnetic Ni has increased the
localization of Co 3$d$ electrons. The magnetism evolves from itinerant to
localized in the AFM region of the Rb(Co1-xNix)2Se2 phase diagram. Spin glass
state is also identified in the compounds we explored.

Figure 1: (a) XRD patterns of Rb(Co1-xNix)2Se2 powder samples with $0.02\leq
x\leq 0.9$. (b) EDS determined concentrations of Ni and Co as a function of
the nominal concentration of Ni, $x$. Error bars are the uncertainty of the
EDS results.

##  II EXPERIMENTAL DETAILS

Single crystals of Rb(Co1-xNix)2Se2 with compositions of $x$ = 0.02, 0.04,
0.06, 0.08, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9 were synthesized
by the Bridgman method with two-step reacting. Co1-xNixSe precursors were
prepared by using Ni, Co powders, and Se shots to react at $500^{\circ}$C in
an evacuated quartz tube. The process was repeated two times for homogeneity.
Then, stoichiometric Rb ingot and Co1-xNixSe powders were placed in an
aluminum crucible and sealed in an evacuated quartz tube. The mixtures were
heated to $1070^{\circ}$C and kept for 10 h. After the samples were slowly
cooled down to $700^{\circ}$C at a rate of $3.7^{\circ}$C/$h$, the furnace was
shut down.Liu et al. (2022) We obtained plate-like single crystals with a
typical size of $5\times 5\times 1$ mm3. As the Ni content increases, the
samples have a metallic luster with colors from golden to pink. All samples
are sensitive to oxygen and moisture. They were saved and handled in an argon-
filled glove box.

Powder XRD measurements were conducted on an x-ray diffractometer (Empyrean)
at 300 K. NPD experiment was carried out on the multi-physics instrument (MPI)
installed at the China Spallation Neutron Source (CSNS). Powder samples ground
from single crystals of Rb(Co1-xNix)2Se2 with $x$ = 0.1, 0.4, and 0.6 were
sealed in cylindrical vanadium cans with helium atmosphere and measured at 5,
50, and 200 K. NPD data were refined by employing the Rietveld method using
the FULLPROF Suite software.Rietveld (1969); Rodríguez-Carvajal (1993) The
composition analysis was conducted on an energy dispersive x-ray spectroscopy
(EDS) (EVO, Zeiss). Resistance, dc, and ac susceptibility were measured on a
commercial physical property measurement system (PPMS, Quantum Design). The
in-plane resistivity measurements were performed using the standard four-probe
method.

Figure 2: Temperature dependence of the in-plane resistivity $\rho_{ab}(T)$
for Rb(Co1-xNix)2Se2 with (a) 0.02 $\leq x\leq$ 0.2 and (b) 0.3 $\leq x\leq$
0.9. The red-solid lines are linear fits to the resistivity above the magnetic
transition temperature.

##  III RESULTS AND DISCUSSION

Figure 3: Temperature dependence of the magnetic susceptibilities $\chi(T)$
for Rb(Co1-xNix)2Se2 ($x$ = 0.02, 0.04, and 0.06), measured in ZFC and FC
conditions with H = 0.1 T. The magnetic fields are parallel to the $ab$ plane
in (a), (c), and (e), and parallel to the $c$ axis in (b), (d), and (f).

Figure 1(a) shows powder XRD patterns at room temperature for the compositions
of Rb(Co1-xNix)2Se2 (0.02 $\leq x\leq$ 0.9). The XRD patterns can be fitted
well by the tetragonal ThCr2Si2-type structure with the space group $I4/mmm$.
The fitted lattice constants for different compositions do not show a regular
change, consistent with the expectation for the approximate ionic radii of Co
and Ni. The average lattice parameters $a$ and $c$ from XRD measurements are
3.902(3) and 14.12(2) Å, respectively. The actual compositions determined by
the EDS for single crystals are plotted in Fig. 1(b). The content of Se was
normalized to be 2 in the analysis. The linear relationships between the
nominal and actual contents of Ni and Co suggest that the substitution is
successful for all compositions.

The resistivity [$\rho_{ab}$($T$)] measured in the $ab$-plane on single
crystals of Rb(Co1-xNix)2Se2 in the temperature range from 2 to 300 K is shown
in Fig. 2. All of the samples display metallic behavior. The resistivity
decreases linearly as decreasing temperature before a drop appears at 86 and
72 K for the compounds with $x=0.02$ and 0.04, respectively [Fig. 2(a)]. The
drop in resistivity could be attributed to a FM transition, as observed in FM
RbCo2Se2 and Tl1-xKxCo2Se2, which can be understood by suppression of the
magnetic scattering on conduction electrons in the FM state.Huan and
Greenblatt (1989); Huang et al. (2021) The anomalies can also be observed in
the resistivity of Rb(Co1-xNix)2Se2 with $x=$ 0.06, 0.08, 0.1, 0.2, and 0.3 at
higher temperatures. However, the resistivity shows a hump below the magnetic
transition reminiscent of the behavior in CsCo2Se2, which undergoes an AFM
transition at low temperatures.Yang et al. (2019) Thus, the Ni substitution on
the Co site may induce a FM to AFM ground state transition. The Ni ions
exhibit a paramagnetic characteristic in RbNi2Se2.Liu et al. (2022) Further
increasing the Ni content, the magnetic scattering would be reduced due to the
decrease of the magnetic Co ions. The anomalies in resistivity for the
compounds with $0.4\leq x\leq 0.8$ indeed become weaker, as shown in Fig.
2(b). For the composition with $x$ = 0.9, the $\rho_{ab}$($T$) does not show
any anomaly in the measured temperature range, similar to RbNi2Se2.

Figure 4: Temperature dependence of $\chi(T)$ measured in ZFC and FC
conditions for H $\|$ ab and H $\|$ c with H = 0.1 T for Rb(Co1-xNix)2Se2 with
(a) $x=0.08$, (b) $x=0.1$, (c) $x=0.2$, (d) $x=0.3$, (e) $x=0.4$, (f) $x=0.5$,
(g) $x=0.7$, (h) $x=0.8$, and (i) $x=0.9$. The Inset in (c) is the temperature
dependence of the $\chi^{\prime}(T)$ measured at various $ac$ magnetic field
frequencies between 300 and 9885 Hz for the compound with $x$ = 0.20. The $ac$
magnetic field is parallel to the $ab$ plane. The arrow indicates a shift in
the freezing temperatures.

To investigate the magnetic properties, we measured the magnetic
susceptibilities [$\chi$($T$)] for all the compounds, as shown in Figs. 3 and
4 . Figure 3 shows the $\chi$($T$) under zero-field-cooled (ZFC) and field-
cooled (FC) conditions for the Rb(Co1-xNix)2Se2 crystals with $x$ = 0.02,
0.04, and 0.06. The susceptibility data were collected with the magnetic field
applied along the $ab$ plane [$\chi_{ab}$] and the $c$ axis [$\chi_{c}$],
respectively. The abrupt enhancement of $\chi_{ab}$ below 100 K reveals the
development of a long-range FM order, as shown in Fig. 3(a), (c), and (e) for
the compounds $x$ = 0.02, 0.04, and 0.06. The enhancement could also be
observed on the $\chi_{c}$. However, the strength of the magnetic response
within the layer [$\chi_{ab}$] shows an order of magnitude larger than that of
the out-of-layer [$\chi_{c}$], yielding that the ferromagnetically ordering
moments are aligned in the $ab$ plane. For the compound with $x$ = 0.06, the
out-of-plane susceptibility displays a cusp-like peak around 96 K [Fig. 3(f)],
in contrast to the in-plane FM behavior, implying that an AFM order may occur
along the $c$ axis.

The magnetic susceptibilities for $x$ = 0.08, 0.1, 0.2, 0.3, 0.4, 0.5, 0.7,
0.8, and 0.9 are shown in Figs. 4(a)-4(i) with $H\|ab$ and $H\|c$,
respectively. The cusp-like anomalies are more obvious in the susceptibilities
of Rb(Co1-xNix)2Se2 with 0.08 $\leq x\leq$ 0.8. The magnitudes of $\chi_{ab}$
and $\chi_{c}$ are comparable and consistent with an AFM order. Interestingly,
there is an obvious magnitude change between the $\chi_{ab}$ and $\chi_{c}$ in
the composition range of $x=0.08$ to $x=0.8$, where the magnitude reverses at
$x=0.3$. It is difficult to determine the origin of the changes in magnetic
susceptibility because of the unknown AFM order and the substitution of the
magnetic Co ions.

Comparing the ZFC to FC magnetic susceptibilities, a pronounced bifurcation
can be observed below a transition temperature for the compositions with 0.06
$\leq x\leq$ 0.9. The irreversible behavior implies a spin-glass (SG) state,
which may be related to the doping of the nonmagnetic Ni ions. The frequency-
dependent $ac$ susceptibility measurements were adopted to verify the SG state
for the compound with $x$ = 0.2. The magnetic field was applied parallel to
the $ab$ plane. As shown in the inset of Fig. 4(c), the real part of the $ac$
susceptibilities $\chi^{\prime}$($T$) exhibit a clear frequency dependence.
With increasing frequency, the peak position shifts to higher temperatures
while the peak amplitude decreases. The Mydosh parameter ($\delta T_{f}$)
represents the relative shift in freezing temperature per decade of frequency,
which can be calculated by $\delta T_{f}$ = $\Delta T_{f}$ / ($T_{f}$
$\Delta$log$f$).Ryu et al. (2015b) The spin freezing temperature $T_{f}$ is
defined by the peak position from the ZFC curve. The obtained $\delta T_{f}$
value is 0.031, consistent with the values of 0.0045 $\leq$ $\delta T_{f}$
$\leq$ 0.08 for a typical SG state. For the compound with $x$ = 0.9 in Fig.
4(i), the magnetic transition disappears in the temperature dependence of
susceptibility. The $\chi(T)$ curves exhibit the Pauli PM behavior without
long-range magnetic order, similar to the characteristic of RbNi2Se2.Liu et
al. (2022)

Figure 5: NPD patterns for Rb(Co1-xNix)2Se2 at 5 K with (a) $x$ = 0.1, (b) $x$
= 0.4, and (c) $x$ = 0.6. The insets are temperature dependence of the
magnetic diffraction peak corresponding to the (001) reflection at 5, 50, and
200 K. The errors in the insets indicate one standard deviation.

To obtain accurate magnetic structures, NPD data were collected for samples
with $x$ = 0.1, 0.4, and 0.6. The reflection patterns and the Rietveld method
refinements are shown in Fig. 5. By comparing the data at 5, 50, and 200 K, a
magnetic peak at $d=14.2$ Å can be identified. The magnetic peak can be
indexed with the propagation vector k = $[1,0,0]$ in the notation of the
$I4/mmm$ space group with $A$-type AFM order, as shown in Fig. 6(c). The
magnetic diffraction peak observed at $d$ = 14.2 Å corresponds to the magnetic
Bragg reflection $(H,K,L)=(001)$. Here, we assume the magnetism arising from
the Co ions. The ordered moments are refined to be 0.69(2), 1.98(4), and
2.80(6) $\mu_{\text{{B}}}$ per Co at 5 K for the compounds with $x$ = 0.1,
0.4, and 0.6, respectively. The increased ordered moment for the $x=0.6$
compound is close to the fully localized high spin state of Co2+ with $S=3/2$.

Knowing the $A$-type magnetic order and moment sizes for the compounds with
0.08 $\leq x\leq$ 0.8, the evolution of the magnetic susceptibilities can be
understood. By increasing the content of Ni, the magnetism evolves from
itinerant to localized, accompanied by the enhancement of the ordered moment
size. The AFM intralayer coupling between the FM layers will be increased
accordingly. For the compounds with $x$ = 0.08, 0.1, and 0.2, the $\chi_{ab}$
sharply drops at the magnetic transition temperature $T_{\text{N}}$. The
$\chi_{c}$ has a weak temperature dependence below $T_{\text{N}}$, typically
conforming the $A$-type AFM structure with the magnetic moments lying along
the $ab$-plane. The value of in-plane susceptibility $\chi_{ab}$ is enhanced
by the $ab$-plane collinear ferromagnetic layers, larger than the out-of-plane
susceptibility $\chi_{c}$. A similar magnetic structure has been reported in
CsCo2Se2, which exhibits the approximate magnetic behavior.Yang et al. (2013,
2019) When $x$ = 0.3 and 0.4, the susceptibilities exhibit a crossover between
the $\chi_{ab}$ and $\chi_{c}$. Such a crossover could be ascribed to the
dilution of the magnetic Co ions in the $ab$ plane and the enhancement of the
AFM intralayer coupling.

Figure 6: (a) A phase diagram of Rb(Co1-xNix)2Se2 with the FM transition
temperature [$T_{\text{C}}$], AFM transition temperature [$T_{\text{N}}$,
$T_{\text{N}}^{\ast}$, $T_{\text{N}}^{\ast\ast}$], spin glass transition
temperature [$T_{\text{SG}}$], and the residual resistivity ratio [$RRR$].
$T_{\text{N}}^{\ast}$ represents the AFM transition temperature for the
critical concentration $x$ = 0.6 with FM behavior on $\chi_{ab}$ and AFM
behavior on $\chi_{c}$. $T_{\text{N}}^{\ast\ast}$ is the AFM transition
temperature for $x$ = 0.3 and 0.4, where the anisotropic susceptibility curve
has a crossover below $T_{\text{N}}$. Schematic diagrams of (b) the FM
structure, (c) the AFM structure, and (d) the PM structure in the phase
diagram.

By combining the experimental results on Rb(Co1-xNix)2Se2, we construct a
comprehensive phase diagram in Fig. 6(a), summarizing the evolutions of the
magnetic order, spin glass state, and $RRR$ as a function of the Ni content
$x$. For $x$ = 0.02 and 0.04, the anisotropic magnetic susceptibilities reveal
a FM order with moments aligned in the $ab$ plane, as shown in Fig. 6(b). Both
the FM and AFM transitions could be identified in the compound with $x$ =
0.06, suggesting that the FM to AFM transformation is a first-order transition
and $x$ = 0.06 is close to the critical composition. The AFM structure is
shown in Fig. 6(c). For the compounds with AFM order, the magnetic
susceptibility changes from $\chi_{ab}>\chi_{c}$ to $\chi_{ab}<\chi_{c}$ at
the compositions with $x=$ 0.3 and 0.4, resulting from the dilution of the
magnetic Co ions and localization of the magnetic moments. The AFM transition
vanishes from the susceptibility measurements for the compound with $x$ = 0.9
[Fig. 6(d)]. The SG transition temperature [$T_{\text{SG}}$] is defined as the
bifurcation point between the ZFC and FC susceptibilities. The $T_{\text{SG}}$
and $RRR$ for each composition are plotted in Fig. 6(a). The evolution of the
electronic behavior ($RRR$) is strongly correlated with the magnetic order.
The abrupt increase of $RRR$ in the AFM region could be attributed to the
enhancement of the localization of Co ions. To further understand the nature
of magnetic and transport properties, angle-resolved photoemission
spectroscopy studies on the electronic band structures are required.

##  IV SUMMARY

In conclusion, we have grown the single crystals of Rb(Co1-xNix)2Se2 with 0.02
$\leq x\leq$ 0.9 and comprehensively investigated the physical properties via
the XRD, EDS, NPD, magnetic susceptibility, and electronic transport methods.
By combining these results, we present a phase diagram of Rb(Co1-xNix)2Se2.
All compositions adopt the space group $I4/mmm$ and show a slight difference
in lattice parameters. The magnetism evolves from FM to AFM, then to Pauli
paramagnetism with increasing the concentration of Ni. The electronic
transport behaviors are closely correlated with the magnetic order. The
magnetic structure in the AFM region is confirmed to be the $A$-type AFM order
by NPD. The AFM moment size is increased significantly as increasing the Ni
concentration, demonstrating that the magnetic moments of Co ions become more
localized as the dilution effect by the Ni substitution.

##  V ACKNOWLEDGMENTS

Work was supported by the Guangdong Basic and Applied Basic Research
Foundation (Grants No. 2021B1515120015, 2020B151520065), the National Natural
Science Foundation of China (Grant No. 12174454), the Guangzhou Basic and
Applied Basic Research Foundation (Grant No. 202201011123), Guangdong
Provincial Key Laboratory of Magnetoelectric Physics and Devices (Grant No.
2022B1212010008), and National Key Research and Development Program of China
(Grant No. 2019YFA0705702).

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# Exploring kaon induced reactions for unraveling the nature of the scalar
meson $a_{0}(1817)$

Xiao-Yun Wang<EMAIL_ADDRESS>Department of physics, Lanzhou University of
Technology, Lanzhou 730050, China  Lanzhou Center for Theoretical Physics, Key
Laboratory of Theoretical Physics of Gansu Province, Lanzhou University,
Lanzhou, Gansu 730000, China  Hui-Fang Zhou  Department of physics, Lanzhou
University of Technology, Lanzhou 730050, China  Xiang Liu
<EMAIL_ADDRESS>School of Physical Science and Technology, Lanzhou
University, Lanzhou 730000, China  Lanzhou Center for Theoretical Physics, Key
Laboratory of Theoretical Physics of Gansu Province, Lanzhou University,
Lanzhou, Gansu 730000, China  Key Laboratory of Quantum Theory and
Applications of MoE, Lanzhou University, Lanzhou 730000, China  Frontiers
Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, China
Research Center for Hadron and CSR Physics, Lanzhou University and Institute
of Modern Physics of CAS, Lanzhou 730000, China

###### Abstract

In this study, we comprehensively investigate the production of isovector
scalar meson $a_{0}(1817)$ using the effective Lagrangian approach.
Specifically, we employ the Reggeized $t$ channel Born term to calculate the
total and differential cross sections for the reaction $K^{-}p\rightarrow
a_{0}(1817)\Lambda$. Our analysis reveals that the optimal energy range for
detecting the $a_{0}(1817)$ meson lies between $W=3.4$ GeV and $W=3.6$ GeV,
where the predicted total cross section reaches a minimum value of 112 nb.
Notably, the $t$ channel, as predicted by the Regge model, significantly
enhances the differential cross sections, particularly at extreme forward
angles. Furthermore, we investigate the Dalitz processes of $2\rightarrow 3$
and discuss the feasibility of detecting the $a_{0}(1817)$ meson in
experiments such as J-PARC.

##  I Introduction

The $a_{0}(1817)$ meson has attracted significant attention in the field of
light hadron physics, as its study provides valuable insights into the
intricacies of constructing the light flavor hadron spectroscopy. Recent
experimental findings have added to the intrigue surrounding this meson. The
BaBar Collaboration, through the $\eta_{c}\rightarrow\eta\pi^{+}\pi^{-}$
reaction, discovered a new state named $a_{0}(1700)$, which has a measured
mass of $1704\pm 5(\text{stat.})\pm 2(\text{syst.})$ MeV and a width of
$\Gamma=110\pm 15(\text{stat.})\pm 11(\text{syst.})$ MeV Babar2021 .
Additionally, the BESIII Collaboration observed a state denoted as the
$a_{0}(1710)^{0}$ in the $D_{S}^{+}\rightarrow K_{S}^{0}K_{S}^{0}\pi^{+}$
reaction. However, in this detection process, it was not possible to
differentiate between the $a_{0}(1710)^{0}$ and $f_{0}(1710)$, leading to a
generalization of both states as $S_{0}(1710)$ BESIII:2021anf . It was later
resolved in a subsequent article using the isospin theorem, which
distinguished the isospin $I=1$ state $a_{0}(1710)$ from the isospin $I=0$
state $f_{0}(1710)$ Dai:2021owu . Subsequently, the BESIII Collaboration
conducted another experiment to study the $a_{0}(1710)^{+}$ state with the
quantum numbers $I(J^{P})=1^{+}(0^{+})$. The observation of
$a_{0}(1710)^{+}\rightarrow K^{0}_{S}K^{+}$ was made through an investigation
of the $D^{+}_{S}\rightarrow K^{0}_{S}K^{+}\pi^{0}$ decay BESIII:2022npc .
This experiment reported the mass and decay width of the newly discovered
meson as $M=1.817\pm 0.008(\text{stat.})\pm 0.020(\text{syst.})$ GeV and
$\Gamma=0.097\pm 0.022(\text{stat.})\pm 0.015(\text{syst.})$ GeV,
respectively. In accordance with the designation proposed by the Lanzhou group
et al. Guo:2022xqu , we adopt the name $a_{0}(1817)$ for this newly discovered
isovector scalar meson in our work.

However, there exist discrepancies in the measured mass and decay width of the
$a_{0}(1817)$ meson as observed by the BaBar experiment Babar2021  and the
BESIII experiment Dai:2021owu . Moreover, due to the limited number of
relevant experiments and available experimental data, further observations of
the $a_{0}(1817)$ meson in alternative experiments are necessary. These
observations would facilitate the measurement of pertinent resonance
parameters and provide a more comprehensive understanding of the properties
associated with the $a_{0}(1817)$ meson.

Recent research by the Lanzhou group Guo:2022xqu  has established the
significance of the $a_{0}(1817)$ meson as a reference point in the
construction of scalar meson families. Its primary decay channels include
$\pi\eta(1295)$, $\pi\eta^{\prime}$, $\pi\eta$, $\pi\eta(1475)$, $\pi
b_{1}(1235)$, $K\bar{K}$, and others, with specific details provided in Table
1. Theoretical conjectures propose that the $a_{0}(1450)$ and $a_{0}(1817)$
represent the first and second radial excitations, respectively, of the
$a_{0}(980)$ meson Wang:2022pin . Additionally, it is predicted that the
$a_{0}(2115)$ serves as the third radial excitation, contributing
significantly to the expanding landscape of the light hadron spectrum
Guo:2022xqu . Understanding the composition and meson structure of the
$a_{0}(1817)$ is crucial, as it sheds light on the structural characteristics
of scalar mesons in the realm of light quarks, while also addressing other
pertinent issues currently under debate in the field of hadron physics
Oset:2023hyt ; Zhu:2022wzk ; Zhu:2022guw . Previous studies have considered
the possibility of the $a_{0}(980)$ as a tetraquark candidate Humanic:2022hpq
, and the $a_{0}(1450)$ as a hybrid state comprising a combination of double
and quadruple quarks Fariborz:2022nws . The $f_{0}(1710)$ meson, serving as
the isovector partner of the $a_{0}(1817)$, does not rule out the possibility
of being a scalar glueball. However, given the limited information available
on the structure of the $a_{0}(1817)$, further resonance measurements are
imperative. Consequently, the pressing task at hand involves detecting the
$a_{0}(1817)$ in other experimental settings.

Upon consulting the Particle Data Group PDG , we find that the $K^{-}p$
scattering experiment is particularly noteworthy. Since the discovery of
$K$-mesons Rochester:1947mi , kaon beams have naturally emerged as a powerful
tool for exploring strange hadrons and hypernuclei Wang:2019uwk . Experimental
facilities such as J-PARC Jparc  and OKA@U-70 Obraztsov:2016lhp  offer
excellent opportunities for such investigations. Several literature sources
have presented the production of the $a_{0}(980)$ in the reaction
$K^{-}p\rightarrow\Lambda\eta\pi^{+}\pi^{-}$ Ammar:1968zur ; Wells:1975tf ;
Amsterdam-CERN-Nijmegen-Oxford:1978mws ; Flatte:1976xu . The discovery of the
$\phi(1020)$ meson has been achieved in the reaction $K^{-}p\rightarrow KKn$
Borenstein:1972sb . Moreover, the $a_{1}(1260)$ and $D(1285)$ mesons have been
observed in the reactions $K^{-}p\rightarrow\Sigma^{-}\pi^{+}\pi^{+}\pi^{-}$
Amsterdam-CERN-Nijmegen-Oxford:1977eoy  and
$K^{-}p\rightarrow\Lambda\eta\pi^{+}\pi^{-}$ Amsterdam-CERN-Nijmegen-
Oxford:1978mws , respectively. These examples provide further support for the
possibility of observing $a_{0}(1817)$ in $K^{-}p$ scattering experiments. In
our previous work, we successfully calculated the production of the
$\phi(2170)$ meson via the reaction $K^{-}p\rightarrow\phi(2170)\Lambda$
Wang:2019uwk , the $X_{0}(2900)$ state in
$K^{+}p\rightarrow\Sigma_{c}^{++}X_{0}(2900)$ Lin:2022eau , and the
$\eta_{1}(1855)$ meson through $K^{-}p\rightarrow\eta_{1}(1855)\Lambda$
Wang:2022sib  using efficient Lagrangian methods and the Regge trajectory
model. The numerical results obtained from these calculations provide valuable
insights for future experimental endeavors.

In this study, we explore the production mechanism of the scalar meson
$a_{0}(1817)$ in $K^{-}p$ scattering utilizing an efficient Lagrangian
approach, focusing on meson-induced reactions with $K$-meson exchange solely
in the $t$-channel. Detailed information regarding our methodology will be
presented in the subsequent section. The calculation of both the total cross
section and differential cross section for the $K^{-}p\rightarrow
a_{0}(1817)\Lambda$ reaction holds significant relevance for future high-
precision experimental investigations in this field.

This paper is structured as follows: In Section II, we present the efficient
Lagrangian method and the Regge trajectory model employed for the analysis of
the $a_{0}(1817)$. The numerical results for the total and differential cross
sections are presented in Section III. Finally, we summarize our findings and
draw conclusions in Section IV.

##  II Formalism

The production of the scalar meson $a_{0}(1817)$ through kaon-induced
reactions on a proton target, with $t$ channel $K^{+}$ meson exchange, is
illustrated in Fig. 1. In this study, we neglect the contribution from the $s$
channel with nucleon pole, as it is known to be negligibly small. Typically,
the contribution of the $u$ channel with nucleon exchange is also minimal and
can be neglected at low energies. Moreover, at high energies, the Reggeized
treatment of the $u$ channel renders its contribution to the total cross
section small and negligible. Therefore, we do not include the contributions
from nucleon resonances in the $u$-channel in the current calculation.

Figure 1: Feynman diagram for the $K^{-}p\rightarrow\ a_{0}(1817)\Lambda$ reaction. Table 1:  The partial decay widths of the $a_{0}(1817)$ predicted in Ref. Guo:2022xqu .  channel | $\pi\eta$ | $\pi\eta^{\prime}$ | $\pi\eta(1295)$ | $\pi\eta(1475)$
---|---|---|---|---
$\Gamma$ (MeV) |  22.4$\sim$24.9 |  27.8$\sim$36.2 |  18.0$\sim$47.5 |  6.2$\sim$34.2
channel | $\pi b_{1}(1235)$ | $K\bar{K}$ | $\pi f_{1}(1285)$ | $\rho\omega$
$\Gamma$ (MeV) |  10.8$\sim$19.2 |  7.5$\sim$12.9 |  0.2$\sim$9.5 |  0.1$\sim$3.3

Studying the strong interaction at the quark-gluon level in the energy region
where the resonance can be detected poses significant challenges. Therefore,
in this study, we employ the effective Lagrangian method to perform the
necessary calculations. In the case of kaon-induced production of the
$a_{0}(1817)$, the relevant Lagrangians for the $t$ channel are given by
Wang:2015zcp ; Ryu:2012tw ; Liu:2008qx ; Xiang:2020phx

$\displaystyle\mathcal{L}_{a_{0}KK}$ $\displaystyle=$
$\displaystyle\frac{f_{a_{0}KK}}{2m_{K}}a_{0}\partial_{\mu}\overrightarrow{K}\cdot\partial^{\mu}\overrightarrow{K},$
(1) $\displaystyle\mathcal{L}_{KN\Lambda}$ $\displaystyle=$ $\displaystyle
ig_{KN\Lambda}\bar{N}\gamma_{5}\Lambda K{\ +}\text{ H.c.},$ (2)

where the $a_{0}$, ${K}$, $N$ and $\Lambda$ stand for the $a_{0}(1817)$, $K$,
nucleon and $\Lambda$ fields, respectively.

The coupling constant $g_{KN\Lambda}=-13.24$, which can be determined
Wang:2015zcp  based on the SU(3) flavor symmetry relation Oh:2006hm ;
Oh:2006in , plays a crucial role in our analysis. Additionally, the coupling
constant $f_{a_{0}KK}$ can be determined from the decay width
$\Gamma_{a_{0}\rightarrow\bar{K}K}$, as indicated by the calculation results
using the Nijmegen potential Stoks:1999bz

$\displaystyle\Gamma_{a_{0}\rightarrow K^{+}K^{-}}$ $\displaystyle=$
$\displaystyle\frac{2}{3}\Gamma_{a_{0}\rightarrow\bar{K}K}$ $\displaystyle=$
$\displaystyle\left(\frac{f_{a_{0}KK}}{2m_{K}}\right)^{2}\frac{(M_{a_{0}}^{2}-2m_{K}^{2})^{2}}{32\pi
M_{a_{0}}^{2}}|\vec{p}_{K}^{~{}\mathrm{c.m.}}|$

with

$|\vec{p}_{K}^{~{}\mathrm{c.m.}}|=\frac{\lambda^{1/2}(M_{a_{0}}^{2},m_{K}^{2},m_{K}^{2})}{2M_{a_{0}}}.$
(3)

Here, $\lambda$ represents the Källen function, defined as
$\lambda(x,y,z)=\sqrt{(x-y-z)^{2}-4yz}$. $M_{a_{0}}$ and $m_{K}$ denote the
masses of $a_{0}(1817)$ and the kaon meson, respectively. By considering the
decay width $\Gamma_{a_{0}\rightarrow K^{+}K^{-}}$ to be 5 MeV, we find that
the corresponding coupling constant $f_{a_{0}KK}$ is determined to be 0.52.

Based on the aforementioned Lagrangians, the amplitude for the production of
the $a_{0}(1817)$ through $t$ channel $K^{+}$ exchange in $K^{-}p$ scattering
can be expressed as follows:

$\displaystyle\mathcal{M}_{K}$ $\displaystyle=$ $\displaystyle
i\frac{f_{a_{0}KK}}{2m_{K}}g_{KN\Lambda}F(q^{2})\bar{u}_{\Lambda}(p_{2})$ (4)
$\displaystyle\times\gamma_{5}\frac{1}{t-m_{K}^{2}}(q_{\mu}\cdot
k_{1}^{\mu})u_{N}(p_{1}).$

In the above expression, $\bar{u}_{\Lambda}$ and $u_{N}$ represent the Dirac
spinors of the $\Lambda$ hyperon and nucleon, respectively. For the $t$
channel meson exchange Liu:2008qx , a form factor
$F(q^{2})=(\Lambda_{t}^{2}-m^{2})/(\Lambda_{t}^{2}-q^{2})$ is utilized. Here,
$t=q^{2}=(k_{1}-k_{2})^{2}$ represents the Mandelstam variable. The parameter
$\Lambda_{t}$, the only free parameter in the form factor, will be discussed
in detail in Section III.

The Regge trajectory model has proven to be successful in analyzing hadron
production at high energies Wang:2015xwa ; Haberzettl:2015exa ; Ozaki:2009wp ;
Wang:2017plf . It provides a framework to study the spectral behavior of
traditional light mesons Anisovich:2000kxa . In this model, the Reggeization
procedure is performed by replacing the $t$ channel propagator in the Feynman
amplitudes (Eq. (4)) with the Regge propagator, which can be expressed as
follows:

$\frac{1}{t-m_{K}^{2}}\rightarrow\left(\frac{s}{s_{scale}}\right)^{\alpha_{K}(t)}\frac{\pi\alpha_{K}^{\prime}}{\Gamma[1+\alpha_{K}(t)]\sin[\pi\alpha_{K}(t)]}.$
(5)

Here, the factor $s_{scale}$ is equal to 1 GeV. In addition, the Regge
trajectory $\alpha_{K}(t)$ read as Wang:2015zcp ,

$\alpha_{K}(t)=0.70(t-m_{K}^{2}).\quad\ \ $ (6)

It is note that no additional parameter is introduced after the Reggeized
treatment applying.

##  III Numerical results

###  III.1 Cross section

In the following calculations, we can determine the cross section of the
$K^{-}p\rightarrow a_{0}(1817)\Lambda$ reaction. The differential cross
section in the center of mass (c.m.) frame is given by:

$\frac{d\sigma}{d\cos\theta}=\frac{1}{32\pi
s}\frac{\left|\vec{k}_{2}^{{~{}\mathrm{c.m.}}}\right|}{\left|\vec{k}_{1}^{{~{}\mathrm{c.m.}}}\right|}\left(\frac{1}{2}\sum\limits_{\lambda}\left|\mathcal{M}\right|^{2}\right),$
(7)

where the variable $s=(k_{1}+p_{1})^{2}$ represents the squared center of mass
energy, and $\theta$ represents the angle between the outgoing $a_{0}(1817)$
meson and the direction of the kaon beam in the center of mass frame.
$\vec{k}_{1}^{\mathrm{c.m.}}$ and $\vec{k}_{2}^{\mathrm{c.m.}}$ represent the
three-momenta of the initial kaon beam and the final $a_{0}(1817)$ meson,
respectively.

Since there is no available experimental data for the $K^{-}p\rightarrow
a_{0}(1817)\Lambda$ reaction, we provide predictions for the cross section
based on our calculations, as shown in Fig. 2. The cutoff parameter in the
form factor serves as the only free parameter in these calculations. In
previous studies, different values of the cutoff parameter have been used in
related processes. For instance, in the $\pi^{-}p\rightarrow
K^{\ast}\Sigma^{\ast}$ scattering process based on Reggeized $t$ channel
$K^{(\ast)}$ exchange Xiang:2020phx , a value of $\Lambda_{t}=1.67\pm 0.04$
GeV was employed. In another study Ozaki:2009wp , to achieve better agreement
with experimental data, a value of $\Lambda_{t}=1.55$ GeV was chosen for the
Reggeized $t$ channel with $K$ and $K$ exchange. Additionally, in the kaon-
induced reaction $K^{-}p\rightarrow\eta_{1}(1855)\Lambda$, a cutoff value of
$1.6\pm 0.3$ GeV for the $t$-channel $K^{+}$ exchange was considered
Wang:2022sib . In this work, we adopt $\Lambda_{t}=1.6\pm 0.3$ GeV to ensure a
more reliable and feasible conclusion.

Figure 2: The energy dependence of the total cross section for production of
the $a_{0}(1817)$ and through $t$ channel with cutoff $\Lambda_{t}=1.6\pm 0.3$
GeV. The Full (red) line is for the $K^{-}p\rightarrow a_{0}(1817)\Lambda$
reaction. The bands stand for the error bar of cutoff $\Lambda_{t}.$ Figure 3:
The differential cross section $d\sigma/d\cos\theta$ of the $a_{0}(1817)$ and
$a_{0}(1817)$ production at different center-of-mass (c.m.) energies
$W=3.5,3.6,4.0,6.0$ GeV.

In Fig. 2, the total cross section exhibits a clear variation trend within the
energy range of $W=2$ to $10$ GeV. Notably, there is a prominent peak between
$W=3.4$ and $3.6$ GeV, indicating the potential for observing the
$a_{0}(1817)$ resonance through $K^{-}p$ interactions within this energy
range. The increase in the total cross section is steep leading up to the
peak, with a value of $113$ nb at a center-of-mass energy of $3.5$ GeV.
Following the peak, the downward trend becomes less pronounced. Taking into
account the range of $\Lambda_{t}$ as $1.6\pm 0.3$ GeV and considering the
error band, the total cross section varies by 67 nb from the value at $W=3.5$
GeV.

In Fig. 3, the predicted differential cross section of the $K^{-}p\rightarrow
a_{0}(1817)\Lambda$ reaction is presented based on the Regge trajectory model,
using a cutoff value of $\Lambda_{t}=1.6\pm 0.3$ GeV. It is evident from this
work that the differential cross section is highly dependent on the scattering
angle $\theta$. As the energy increases, the reaction exhibits a strong
forward scattering and gradually strengthens. Therefore, the reinforcement
treatment can be effectively validated through forward angle measurements.

Figure 4 shows the $t$-distribution for the $K^{-}p\rightarrow
a_{0}(1817)\Lambda$ reaction. It can be observed that the differential cross
sections gradually decrease with increasing momentum transfer $t$. However, as
$t$ becomes smaller, the differential cross section values continue to
increase, and this phenomenon requires further experimental verification.

Figure 4: The $t$-distribution for the $K^{-}p\rightarrow a_{0}(1817)\Lambda$
and $K^{-}p\rightarrow a_{0}(1817)\Lambda$ reactions at different c.m.
energies $W=3.5$, $3.6$, $4.0$, $6.0$ GeV. Here, the notations are as that in
Fig. 2.

###  III.2 Dalitz process

According to Table 1, it is evident that the $a_{0}(1817)$ meson frequently
appears as an intermediate state in various decay processes. The minimum decay
width for $a_{0}\rightarrow K\bar{K}$ is $\Gamma_{a_{0}\rightarrow
K\bar{K}}=7.5$ MeV, while the minimum decay width for
$a_{0}\rightarrow\pi\eta$ is $\Gamma_{a_{0}\rightarrow\pi\eta}=22.4$ MeV. In
this study, we aim to calculate the Dalitz process for $K^{-}p\rightarrow
a_{0}(1817)\Lambda\rightarrow K^{+}K^{-}\Lambda$ and $K^{-}p\rightarrow
a_{0}(1817)\Lambda\rightarrow\pi\eta\Lambda$, respectively. The Dalitz process
is of great importance and can provide valuable insights for future
experimental investigations. Generally, the invariant mass distribution for
the Dalitz process can be defined based on the two-body process Kim:2017nxg

$\frac{d\sigma_{K^{-}p\rightarrow a_{0}\Lambda\rightarrow
K^{+}K^{-}\Lambda}}{dM_{K^{+}K^{-}}}\approx\frac{2M_{a_{0}}M_{K^{+}K^{-}}}{\pi}\frac{\sigma_{K^{-}p\rightarrow
a_{0}\Lambda}\Gamma_{a_{0}\rightarrow
K^{+}K^{-}}}{(M_{K^{+}K^{-}}^{2}-M_{a_{0}}^{2})^{2}+M_{a_{0}}^{2}\Gamma_{a_{0}}^{2}},$
(8) $\frac{d\sigma_{K^{-}p\rightarrow
a_{0}\Lambda\rightarrow\pi\eta\Lambda}}{dM_{\pi\eta}}\approx\frac{2M_{a_{0}}M_{\pi\eta}}{\pi}\frac{\sigma_{K^{-}p\rightarrow
a_{0}\Lambda}\Gamma_{a_{0}\rightarrow\pi\eta}}{(M_{\pi\eta}^{2}-M_{a_{0}}^{2})^{2}+M_{a_{0}}^{2}\Gamma_{a_{0}}^{2}}.$
(9)

Here, the total width of $a_{0}$ meson, denoted as $\Gamma_{a_{0}}$, is 97
MeV. For the partial width $\Gamma_{a_{0}\rightarrow K^{+}K^{-}}$, we consider
a value of 5 MeV, and for $\Gamma_{a_{0}\rightarrow\pi\eta}$, we use the value
of 22.4 MeV. Based on these parameters, we calculate the invariant-mass
distributions $d\sigma_{K^{-}p\rightarrow a_{0}\Lambda\rightarrow
K^{+}K^{-}\Lambda}/dM_{K^{+}K^{-}}$ and $d\sigma_{K^{-}p\rightarrow
a_{0}\Lambda\rightarrow\pi\eta\Lambda}/dM_{\pi\eta}$ for center-of-mass
energies ranging from $W=3.5$ GeV to $W=6$ GeV. The results are shown in Fig.
5 and Fig. 6. It can be observed from these figures that there is a peak near
the center-of-mass energy of approximately 1.82 GeV, which has direct
implications for the experimental detection of the $a_{0}(1817)$.

To further assess the feasibility of detecting $a_{0}(1817)$ in $K^{-}p$
interactions, we calculate the ratio $\sigma(K^{-}p\rightarrow
a_{0}(1817)\Lambda\rightarrow K^{+}K^{-}\Lambda)/\sigma(K^{-}p\rightarrow
K^{+}K^{-}\Lambda)$. In Fig. 2, the cross section for $a_{0}(1817)$ production
in $K^{-}p$ scattering is estimated to be approximately 110 nb at $W=3.37$
GeV. Assuming a branching ratio of $BR(a_{0}(1817)\rightarrow
K^{+}K^{-})\approx 5.2\%$, we obtain a total cross section of
$\sigma_{K^{-}p\rightarrow a_{0}(1817)\Lambda\rightarrow
K^{+}K^{-}\Lambda}\approx 5.72$ nb at $W=3.37$ GeV.

In Ref. Mott:1969hwi , a total cross section of 35 $\mu$b is reported for
$W=3.37$ GeV. Based on this value, the ratio at $W=3.37$ GeV can be calculated
as follows:

$\frac{\sigma(K^{-}p\rightarrow a_{0}(1817)\Lambda\rightarrow
K^{+}K^{-}\Lambda)}{\sigma(K^{-}p\rightarrow K^{+}K^{-}\Lambda)}\approx
0.016\%.$ (10)

Considering the current experimental landscape, we are optimistic about the
potential of the J-PARC experiment in detecting the $a_{0}(1817)$ in $K^{-}p$
scattering Nagae:2008zz ; Jparc . The experimental conditions at J-PARC are
well-suited for this purpose. Based on the specifications of the J-PARC
experiment, it is estimated that approximately 42,000 events of
$K^{+}K^{-}\Lambda$ can be generated in 100 days, among which about several
events are expected to involve the $a_{0}(1817)$. By performing calculations,
we find that at $W=3.37$ GeV, the cross section for $K^{-}p\rightarrow
a_{0}(1817)\Lambda\rightarrow\pi\eta\Lambda$ is approximately 25.40 nb,
considering a branching ratio of $BR(a_{0}(1817)\rightarrow\pi\eta)\approx
23.1\%$. These events can be reliably detected at J-PARC every 100 days, with
dozens of events specifically related to the $a_{0}(1817)$. Consequently, the
$a_{0}(1817)$ can be confidently observed from the $K^{-}p\rightarrow
a_{0}(1817)\Lambda\rightarrow\pi\eta\Lambda$ reaction under the current
experimental conditions. Therefore, with the future upgrade of J-PARC, there
is a promising opportunity to discover and study the $a_{0}(1817)$ in greater
detail.

Figure 5: The invariant-mass distribution $d\sigma_{K^{-}p\rightarrow
a_{0}\Lambda\rightarrow K^{+}K^{-}\Lambda}/dM_{K^{+}K^{-}}$ reactions at
different c.m. energies $W=3.5,3.6,4.0,6.0$ GeV. Figure 6: The invariant-mass
distribution $d\sigma_{K^{-}p\rightarrow
a_{0}(1817)\Lambda\rightarrow\pi\eta\Lambda}/dM_{\pi\eta}$ reactions at
different c.m. energies $W=3.5,3.6,4.0,6.0$ GeV.

##  IV SUMMARY

In the past two years, significant progress has been made in the study of the
isovector scalar meson $a_{0}(1817)$ by the BaBar and BESIII Collaborations.
However, the measured resonance parameters of the $a_{0}(1817)$ differ between
these experiments, and there is still a lack of sufficient data to fully
understand its structure PDG . In order to further investigate the intrinsic
properties of the $a_{0}(1817)$, we propose to explore its characteristics
through $K^{-}p$ scattering.

There are several reasons for choosing the $K^{-}p$ interaction to search for
the $a_{0}(1817)$. Firstly, the ground state particle of the $a_{0}(1817)$,
$a_{0}(980)$, has already been observed in the $K^{-}p$ scattering process.
Secondly, the decay channel $a_{0}(1817)\rightarrow K^{+}K^{-}$ plays a
significant role in the overall decay of the $a_{0}(1817)$. By employing the
effective Lagrangian method and the Regge trajectory model in quantum field
theory, we calculate the total and differential cross sections of
$K^{-}p\rightarrow a_{0}(1817)\Lambda$.

Our results indicate that the total cross section exhibits a peak at a center-
of-mass energy of $W=3.4\sim 3.6$ GeV, suggesting that this energy range is
ideal for detecting the $a_{0}(1817)$ through the $K^{-}p\rightarrow
a_{0}(1817)\Lambda$ reaction. Moreover, the differential cross section is
highly sensitive to the scattering angle $\theta$ and the minimum momentum
transfer $t$. Therefore, high-precision data from experimental facilities such
as J-PARC, OKA@U-70, and SPS@CERN, which provide suitable kaon beams, are
eagerly awaited.

Based on the current experimental conditions, the detection of the
$a_{0}(1817)$ from $K^{-}p\rightarrow\pi\eta\Lambda$ is considered more
feasible compared to $K^{-}p\rightarrow K^{+}K^{-}\Lambda$. The theoretical
insights obtained in this study will provide valuable information for future
experiments aimed at identifying and characterizing the $a_{0}(1817)$ state.

## ACKNOWLEDGMENTS

This work is supported by the National Natural Science Foundation of China
under Grants No. 12065014, No. 12047501 and No. 12247101, and by the Natural
Science Foundation of Gansu province under Grant No. 22JR5RA266. We
acknowledge the West Light Foundation of The Chinese Academy of Sciences,
Grant No. 21JR7RA201. X. L. is also supported by the China National Funds for
Distinguished Young Scientists under Grant No. 11825503, National Key Research
and Development Program of China under Contract No. 2020YFA0406400, the 111
Project under Grant No. B20063, the Fundamental Research Funds for the Central
Universities, and the project for top-notch innovative talents of Gansu
province.

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]Minimum quantum degrees with Maya diagrams

Ryan M. Shifler

Department of Mathematical Sciences,
Henson Science Hall,
Salisbury University,
Salisbury, MD 21801

[2010]Primary 14N35; Secondary 14N15, 14M15

We use Maya diagrams to refine the formula by Fulton and Woodward for the smallest powers of the quantum parameter $q$ that occur in a product of Schubert classes in the (small) quantum cohomology of partial flags. Our approach using Maya diagrams yields a combinatorial proof that the minimal quantum degrees are unique for partial flags.

§ INTRODUCTION

Let $\Fl$ be a partial flag. Let $\QH^*(\Fl)$ be the small quantum cohomology with Schubert classes $\sigma_w$, $w \in W^{P_I}$. The set $W^{P_I}$ is defined in Section <ref>.  The small quantum cohomology ring $\QH^*(\Fl)$ is a graded $\mathbb{Z}[q]$-module. The multiplication is given by

\[ \sigma^v \star \sigma_w= \sum_{u,d\geq 0} c_{v^\vee,w}^{u,d}q^d\sigma_u\]
where $c_{v^\vee,w}^{u,d}$ is the Gromov-Witten invariant that enumerates the degree $d$ rational curves that intersect general translates of $\sigma^v$, $\sigma_w$, and $\sigma_u$ where $\sigma^v$ is the Poincare dual of $\sigma_v$. See <cit.> for details. Given any element $\tau \in \QH^*(\Fl)$, we say that $q^d$ occurs in $\tau$ if the coefficient of $q^d\sigma_w$ is not zero for some $w$.

The purpose of the article is to use Maya diagrams to refine a formula by Fulton and Woodward in <cit.> for the smallest power of the quantum parameter $q$ that occur in a product of Schubert classes in the (small) quantum cohomology of partial flags. The results in this article are combinatorial in the sense that we use Maya diagrams to describe the chains that Fulton and Woodward defined in <cit.> (see Proposition <ref>). Minimum quantum degrees are also studied in studied in postnikov:qbruhat, postnikov:affine,Buch,yong, Belkale,Bar,BCLM:EulerChar,ShiflerWithrow.

Postnikov proved that the minimum quantum degree is unique for $G/B$ in <cit.>. This result was later extended to general homogeneous space $G/P$ in <cit.> using geometric techniques. Our approach using Maya diagrams yields a combinatorial proof that the minimal quantum degrees are unique for partial flags. We will briefly discuss the utility of Maya diagrams next which are defined in Definition <ref>.

Maya diagrams for partial flags give a manifest characterization of the Bruhat order by slightly modifying a theorem by Proctor in <cit.> and stated herein as Proposition <ref>. Furthermore, Maya diagrams provide a good picture to generalize the notion of a rim hook for Grassmannians to partial flags as stated in Definition <ref>. With these two key notions, there is a canonical lower bound for the minimal quantum degrees as stated in Lemma <ref>. We then show that this lower bound is achieved in Theorem <ref>. Next we begin with preliminaries to state and prove our main results.

Acknowledgements. I would like to thank Hiroshi Naruse for very useful correspondences.

§ PRELIMINARIES

Let $I=\{i_0:=0 < i_1<i_2< \cdots<i_k < i_{k+1}:=n\}$. Let $\Fl:=\Fl(I;n)$ denote the partial flag given by
\[ \Fl(I;n):=\{0 \subset V_1 \subset V_2 \subset \cdots \subset V_k \subset \mathbb{C}^n: \dim V_j=i_j \}.\]

Consider the root system of type $A_{n-1}$ with positive roots $R^+=\{e_a-e_b: 1 \leq a < b \leq n \}$ and the subset of simple roots $\Delta=\{\alpha_i:=e_a-e_{a-1}:1 \leq a \leq n-1 \}$. The associated Weyl group $W$ is $S_n.$ For $1 \leq i \leq n-1$ denote by $s_i$ the simple reflection corresponding to the root $e_i-e_{i+1}$. Each $I=\{i_0:=0 < i_1<i_2< \cdots<i_k <i_{k+1}:=n\}$ determines a parabolic subgroup $P_I$ with Weyl group $W_P=\left<s_i: i \neq i_j \right>$ generated by reflections with indices not in $I$. Let $\Delta_P:=\{ \alpha_{i_s}: i_s \notin \{i_1, \cdots, i_r \} \}$ and $R^+_P:=\mbox{Span} \Delta_P \cap R^+$; these are the positive roots of $P_I$. Let $\ell(W) \rightarrow \mathbb{N}$ be the length function and denote by $W^{P_I}$ the set of minimal length representatives of the cosets of $W/W_{P_I}$. The length function descends to $W/W_{P_I}$ by $\ell(uW_{P_I})=\ell(u')$ where $u' \in w^P$ is the minimal length representative for the coset $uW_{P_I}$. We have a natural ordering $1<2< \cdots <n$. Then the minimal length representatives $W^{P_I}$ have the form

\[(w(1)<\cdots<w(i_1)| w(i_1+1)<\cdots<w(i_2)| \cdots|w(i_{k-1}+1)<\cdots<w(i_k)|w(i_k+1)<\cdots<w(n) ).\]
Since $w(i_k+1)<\cdots<w(n)$ are determined then we will identify the elements of $W^{P_I}$ with
\[\left(w(1)<\cdots<w(i_1)| w(i_1+1)<\cdots<w(i_2)| \cdots|w(i_{k-1}+1)<\cdots<w(i_k)\right).\]

§.§ Chains

Here will follow the exposition of <cit.> and specialize to the case of partial flags. We will say that two unequal elements $v$ and $w$ in $W^{P_I}$ are adjacent if there is a reflection $s_{e_{{a}}-e_{b}} \in W$ such that $w=vs_{e_a-e_b}$. If $i_{q-1}+1 \leq a\leq i_q$ and $i_{t-1}+1 \leq b\leq i_{t}$ then define

\[d(v,w)=e_a-e_b+\Delta_P=(e_{i_q}-e_{i_q+1})+\cdots+(e_{i_{t-1}}-e_{i_{t-1}+1})+\Delta_P=0^{q-1}1^{t-q}0^{k+1-t}.\]

Define a chain $\mathcal{C}$ from $w$ to $v$ in $W^{P_I}$ to be a sequence $u_0,u_1, \cdots, u_r$ in $W^{P_I}$ such that $u_{i-1}$ and $u_i$ are adjacent for $1 \leq i \leq r$, and, in addition $u_0 \leq v$ and $w \leq u_r$. For any chain $u_0,u_1, \cdots, u_r$ we define the degree of the chain $\mathcal{C}$, denoted $\deg_{\mathcal{C}}(v,w)$, to be the component-wise sum of the degrees $d(u_{i-1},u_i)$ for $1 \leq i \leq r$. Note that there is a chain of degree 0 between $v$ and $w$ exactly when $w \leq v$.

§.§ Quantum Cohomology

Let $\QH^*(\Fl)$ denote the quantum cohomology ring of $\Fl$. The Schubert classes $\sigma_w$, $w \in W^{P_I}$, form a basis. Let $\sigma^w:=\sigma_w^\vee$ be the Poincare dual of $\sigma_w$ for any $w\in W^{P_I}$.Take a variable $q_j$ for each $i_j \in I$ with $1 \leq j \leq k$, and let $\mathbb{Z}[q]$ be the polynomial ring with these $q_j$ as indeterminants where $\deg q_j=i_{j+1}-i_{j-1}$. For a degree $d=\sum_{j=1}^{k}d_j \sigma_{s_{i_j}} \in H_2(\Fl)$, we write $q^d=\Pi_{j=1}^{k}q_j^{d_j}$. The small quantum cohomology ring $\QH^*(\Fl)$ is a graded $\mathbb{Z}[q]$-module. The multiplication is given by

\[ \sigma^v \star \sigma_w= \sum_{u,d\geq} c_{v^\vee,w}^{u,d}q^d\sigma_u\]
where $c_{v^\vee,w}^{u,d}$ is the Gromov-Witten invariant that enumerates the degree $d$ rational curves that intersect general translates of $\sigma^v$, $\sigma_w$, and $\sigma_u$. See <cit.> for details.

§.§ Fulton and Woodward's formula for minimal quantum degrees
Given any element $\tau \in \QH^*(\Fl)$, we say that $q^d$ occurs in $\tau$ if the coefficient of $q^d\sigma_w$ is not zero for some $w$. The following is from <cit.>

Let $v,w \in W^{P_I}$, and let $d$ be a degree. The following are equivalent:

* There is a degree $c \leq d$ such that $q^c$ occurs in $\sigma^v \star \sigma_w$.

* There is a chain of degree $c \leq d$ between $v$ and $w$.

§.§ Hecke Product
The Weyl group $\Wlarge$ admits a partial ordering $\leq$ given by the Bruhat order. Its covering relations are given by $w < ws_\alpha$ where $\alpha \in R^+$ is a root and $\ell(w) < \ell(w s_\alpha)$. We will use the Hecke product on the Weyl group $\Wlarge$. For a simple reflection $s_i$ the product is defined by

\begin{align*}
w \cdot s_i =   \begin{cases}
w s_i & \text{ if $\ell(ws_i)>\ell(w)$;} \\
w & \text{otherwise.}
\end{cases}
\end{align*}

The Hecke product gives $\Wlarge$ a structure of an associative monoid; see e.g. <cit.>*3 for more details.
For any parabolic group $P$, the Hecke product determines a left action $W \times W/\Wlarge_P \to W/\Wlarge_{P}$ defined by

\[
u \cdot (w \Wlarge_{P}) = (u \cdot w) \Wlarge_P.
\]

We will need the technical lemma from <cit.>.

Let $u,v \in W$. Then $(u \cdot v)v^{-1} \leq u$.

§ MAYA DIAGRAMS

We begin this section by defining Maya diagrams in Definition <ref>. Then give the Bruhat order in terms of Maya diagrams in Subsection <ref>.

Let $w \in W^{P_I}$. The Maya diagram $M^w$ corresponding to $w$ is an $(k+1)\times n$ grid with an $x$ in the $(j,w(i))$ positions for $1 \leq i \leq i_j$ and $1 \leq j \leq k+1$. We choose the southwest corner to be $(1,1)$ box. We color the bottom $x$ of each column black with all other $x$'s blue. We denote the row indexed by $j$ as $m^w_j$.

We will need the following technical lemma.

Let $w \in W^{P_I}$ and let $1 \leq j \leq k$. Let $\pi_j:W^{P_I} \rightarrow W^{P_{i_j}}$ denote the natural projection. Then $M^{\pi_j(w)}$ is Maya diagram with two rows and $n$ columns with the top row having an $x$ in each position and the second row is $m^w_j$.

The result is clear.

§.§ Bruhat order with Maya diagrams

We begin the Subsection with technical definitions.

Let $M^w$ be the Maya diagram that corresponds to $w \in W^{P_I}$. Define

\[ f(M^w,a,b)=\begin{cases}
x & a=j, b=w(i), \mbox{ for some } i,j \mbox{ with } 1 \leq i \leq i_j \mbox{ and }1\leq j \leq k+1; \\
0 & \mbox{otherwise}
\end{cases}
\]

Define $S_j(M^w,b)=\#\{q: f(M^w,j,q)=x \mbox{ for }1 \leq q \leq b \}$.

Let $w,v \in W^{P_I}$. We say that $M^w \leq M^v$ if and only if

\[S_j(M^w,b) \geq S_j(M^v,b) \mbox{ for all }j \mbox{ and }b \mbox{ such that }1 \leq b \leq n \mbox{ and } 1 \leq j \leq k+1. \]

The next proposition is a modification of <cit.>.

Let $w,v \in W^{P_I}$. Then $w \leq v$ if and only if $M^w \leq M^v$.

The minimal length representatives of $w=(1|5<9|10<11|4<6|2<7)$ and $v=(2|7<11|10<12|8<9|1<5)$ corresponds to the tableaux

\[ M^w=\ytableausetup{centertableaux}
\begin{ytableau}
\bx   & \bx & x  & \bx& \bx   &\bx& \bx & x & \bx & \bx &\bx & x \\
\bx   & x &   & \bx& \bx   &\bx&x &  & \bx & \bx &\bx &  \\
\bx   &   &   & x  & \bx   & x &  &  & \bx &\bx  &\bx &\\
\bx   &   &   &    & \bx   &   &  &  & \bx &x    &x   &\\
\bx   &   &   &    & x     &   &  &  & x   &     &    &\\
x   &   &   &    &       &   &  &  &     &     &    &
\end{ytableau}\ytableausetup{centertableaux}  \mbox{ and }
\begin{ytableau}
\bx  & \bx & x  & x &\bx  & x  & \bx &\bx  &\bx  &\bx & \bx   &\bx \\
x  & \bx &   &  &x  &   & \bx &\bx  &\bx  &\bx & \bx   &\bx \\
& \bx  &   &  &   &   & \bx &x    &x    &\bx & \bx   &\bx \\
& \bx  &   &  &   &   & \bx &     &     &x   & \bx   &x   \\
& \bx  &   &  &   &   & x   &     &     &    & x     &    \\
& x    &   &  &   &   &     &     &     &    &       &
\end{ytableau}\ytableausetup{centertableaux}\]

Here $M^w \leq M^v$.

§ MAYA DIAGRAMS COMBINATORICS

In Subsection <ref> we give the definition and two examples of the generalized rim hook rule. In Subsection <ref> we use Maya diagrams apply Proposition <ref>.

§.§ Generalized rim hook rule

Maya diagrams give a way to see a generalized rim hook rule manifestly. We begin with two definitions to define this generalized rim hook rule.

Let $v \in W^{P_I}$ and let $M^v$ be the corresponding Maya diagram. Let $1 \leq q \leq t \leq k$. For $1 \leq r \leq k$ define
\[ \phi(M^v,r)=\min \{b:f(M^v,r,b)=x \mbox{ and } f(M^v,r-1,b)=0 \}\]

\[\psi(M^v,r)=\max \{b:f(M^v,r+1,b)=x \mbox{ and } f(M^v,r,b)=0 \}.\]

The generalized $qt$-rim hook rule is given as follows. Let $M^v$ be a Maya diagram corresponding to $v \in w^{P_I}$.

* Let $M^v_{\uparrow q}:=M^v$.

* For $q \leq j \leq t-1$, define $M^v_{\uparrow j+1}$ from $M^v_{\uparrow j}$ by removing the $x$ in position $(j,\phi(M^v_{\uparrow j},j)).$

* Let $M^v_{\downarrow t}:=M^v_{\uparrow t}$.

* For $t-1 \geq j \geq q$, define $M^v_{\downarrow j}$ from $M^v_{\downarrow j+1}$ by adding an $x$ to position $(j,\psi(M^v_{\downarrow j},j))$.

* The completed generalized $qt$-rim hook is given by $M^v_{\downarrow q}$.

Definition <ref> corresponds to the rim hook for Grassmannians. This example is in $\Fl(8;12)$ with $v=(1<2<3<5<8<9<11<12)$. The symbols $\uparrow$ and $\downarrow$ describe the movement of the $x$'s.

\begin{eqnarray*}
\begin{ytableau}
\bx&\bx&\bx & x   & \bx     &  x & x &\bx &\bx  &\bx & \bx&\bx  \\
x&x&x &    & x     &   &  &x & x & & x&x
\end{ytableau}\ytableausetup{centertableaux} &\overset{\strut}{\longrightarrow}&
\ytableausetup{centertableaux}
\begin{ytableau}
x&\bx& \bx       & x   & \bx & x  & x &\bx &\bx  &\downarrow & \bx         &\bx  \\
\uparrow &x& x       &    &  &   &  &x & x  & x & x         &x
\end{ytableau}\ytableausetup{centertableaux}
\end{eqnarray*}

\begin{eqnarray*}
\begin{ytableau}
x&\bx& x     & x  & \bx         & x  & x &\bx &\bx  &\bx & \bx &\bx \\
&\bx& x      &    & \bx         &   &  &\bx &\bx  &\bx & \bx &\bx
\end{ytableau}\ytableausetup{centertableaux}
\end{eqnarray*}

The following is an example of a $26$-generalized rim hook in $\Fl(1,3,5,7,9;12)$. Here $v=(2|3<8|10<12|9<11|1<5)$.

\begin{eqnarray*}
\begin{ytableau}
\bx&\bx&\bx & x   & \bx     &  x & x &\bx &\bx  &\bx & \bx&\bx  \\
x&\bx&\bx &    & x     &   &  &\bx &\bx  &\bx & \bx&\bx  \\
&\bx&\bx&    &       &   &  &\bx & x   & \bx& x  &\bx \\
&\bx&\bx&    &       &   &  &\bx&     & x  &    &x\\
&\bx& x &    &       &   &  &x  &     &    &    & \\
&x  &    &    &       &   &  &    &     &    &    &
\end{ytableau}\ytableausetup{centertableaux} &\overset{\strut}{\longrightarrow}&
\ytableausetup{centertableaux}
\begin{ytableau}
x&\bx& \bx       & x   & \bx & x  & \downarrow &\bx &\bx  &\bx & \bx         &\bx  \\
\uparrow&\bx& x       &    & x &   & \downarrow &\bx &\bx  &\bx & \bx         &\bx  \\
&\bx&\uparrow &    &            &   & x &\bx & x   & \bx& \downarrow &\bx \\
&\bx&\uparrow &    &             &   &  &\bx&     & x  & x           & \downarrow \\
&\bx&\uparrow&    &             &   &  &x  &     &    &             &x \\
&x  &         &    &             &   &  &    &     &    &             &
\end{ytableau}\ytableausetup{centertableaux}
\end{eqnarray*}

\begin{eqnarray*}
\begin{ytableau}
x&\bx& \bx       & x   & \bx & x  & \bx &\bx &\bx  &\bx & \bx         &\bx  \\
&\bx& x       &    & x &   & \bx &\bx &\bx  &\bx & \bx         &\bx  \\
&\bx& &    &            &   & x &\bx & x   & \bx& \bx &\bx \\
&\bx& &    &             &   &  &\bx&     & x  & x           & \bx \\
&\bx& &    &             &   &  &x  &     &    &             &x \\
&x  &         &    &             &   &  &    &     &    &             &
\end{ytableau}\ytableausetup{centertableaux}
\end{eqnarray*}

§.§ Chains in terms of Maya diagrams

The first lemma connects the generalized rim hook to the Hecke product and gives the existence of an appropriate chain.

Let $M^v$ be a Maya diagram that corresponds to $v \in W^{P_I}$. Apply the $qt$-rim hook rule to $M^v$ and call the resulting Maya diagram $M^w$ where $w \in W^{P_I}$. Then we have the follow:

* The following Hecke product holds:

\[ w=v \cdot s_{i_{q-1}+1}s_{i_{q-1}+2} \cdots s_{i_t-2}s_{i_t-1}s_{i_t-2} \cdots s_{i_{q-1}+2}s_{i_{q-1}+1}.\]

* We also have that
\[ s_{e_{i_{q-1}+1}-e_{i_t}}=s_{i_{q-1}+1}s_{i_{q-1}+2} \cdots s_{i_t-2}s_{i_t-1}s_{i_t-2} \cdots s_{i_{q-1}+2}s_{i_{q-1}+1}.\]

* There is a chain $\mathcal{C}$ from $v$ to $w$ where

\[\deg_{\mathcal{C}}(v,w) \leq 0^{q-1}1^{t-q}0^{k+1-t}.\]

We will first prove part (1)Let $M^v$ be a Maya diagram that corresponds to $v \in W^{P_I}$. Then $M^a=M_{\uparrow j}^v$ and $M^b=M_{\uparrow j+1}^v$ for some $a,b \in W^{P_I}$. Then
\[ b=a \cdot s_{i_{j-1}+1}s_{i_{j-1}+2} \cdots s_j.\]

Similarly, $M^a=M_{\downarrow j}^v$ and $M^b=M_{\downarrow j+1}^v$ for some $a,b \in W^{P_I}$. Then
\[ b= a \cdot s_{i_{j+1}-1}s_{i_{j+1}-1} \cdots s_j.\]
Part (1) follows.

Next we prove Part (2).[This is inspired by the proof of <cit.>.] Let $\mathcal{C}$ be a chain from $v$ to $w$ with degree $\deg_{\mathcal{C}}(v,w)$. Let $\alpha=e_{i_{q-1}+1}-e_{i_t}$. Set $u=(v \cdot s_\alpha)s_\alpha$. By Lemma <ref> we have that $u \leq v$. Also, $u$ and $v \cdot s_\alpha$ are adjacent and
\[d(u,v \cdot s_{\alpha})=0^{q-1}1^{t-q}0^{k+1-t}.\]
The result follows.

The next two definitions will give conditions on with generalized rim hooks to apply when we produce a chain of minimum degree.

Let $v,w \in W^{P_I}$ with corresponding Maya diagrams $M^v$ and $M^w$. We say that position $(a,b)$ in $M^v$ is Bruhat order incompatible with $M^w$ if $S_a(M^v,b) > S_a(M^w,b)$. Otherwise, we say position $(a,b)$ in $M^v$ is Bruhat order compatible with $M^w$. We say that row $m_j^v$ is Bruhat order incompatible with $M^w$ if at least one position in $M_j^v$ is Bruhat order incompatible with $M^w$.

Let $M^v$ and $M^w$ be Maya diagrams with $v,w \in W^{P_I}$. Abusing notation let
\[ \displaystyle  R_{(v,w)}:=0^{q-1}1^{t-q}0^{k+1-t}\]
denote the generalized $qt$-rim hook where

* we have that rows
\[m_q^v,m_{q+1}^v, \cdots, m_{t-1}^v\]
are Bruhat order incompatible with $M^w$;

* and $t-q$ is maximum.

The next lemma gives an upper bound of a particular chain produced using the two previous definitions.

Let $M^v$ and $M^w$ be a Maya diagram corresponding to $v,w \in W^{P_I}.$ Let $v_0:=v$. Then there is a chain in $W^{P_I}$ in terms of Maya diagrams given by

\begin{eqnarray*}
\mathcal{C}:M^{v_0} \overset{R_{(v_0,w)}}{\longrightarrow} M^{v_1} \overset{R_{(v_1,w)}}{\longrightarrow} M^{v_2} \overset{R_{(v_{2},w)}}{\longrightarrow}\cdots \overset{R_{(v_{r-1},w)}}{\longrightarrow} M^{v_r}
\end{eqnarray*}

where $M^{v_j} \ngeq M^w$ for $1 \leq j \leq r-1$ and $M^{v_r} \geq M^w$.
\[\displaystyle \deg_{\mathcal{C}}(v,w) \leq \sum_{j=0}^{r-1} R_{(v_j,w)}.\]

This follows from $r$ applications of Lemma <ref>.

The next definition is necessary to state a Lemma <ref> with is a lower bound for the minimal degrees of appropriate chains.

Let $M^v$ and $M^w$ be a Maya diagram corresponding to $v,w \in W^{P_I}.$ Let $1 \leq j \leq k$ and let $\pi_j:W^{P_I} \rightarrow W^{P_{i_j}}$ be the natural projection. Let $v_0=v$. Let $\mathcal{C}$ be a chain in terms Maya diagrams given by

\begin{eqnarray*}
\mathcal{C}:M^{\pi_j(v_0)} \overset{R_{(\pi_j(v_0),\pi_j(w))}}{\longrightarrow} M^{\pi_j(v_1)} \overset{R_{(\pi_j(v_1),\pi_j(w))}}{\longrightarrow} M^{\pi_j(v_2)} \overset{R_{(\pi_j(v_2),\pi_j(w))}}{\longrightarrow}\cdots \overset{R_{(\pi_j(v_{r-1}),\pi_j(w))}}{\longrightarrow} M^{\pi_j(v_r)}
\end{eqnarray*}

where $M^{\pi_j(v_j)} \ngeq M^{\pi_j(w)}$ for $1 \leq j \leq r-1$ and $M^{\pi_j(v_r)} \geq M^{\pi_j(w)}$ and $r$ is minimum. Then define $\deg_j(v,w):=r.$

Let $M^w$ and $M^v$ be a Maya diagram corresponding to $w,v \in W^{P_I}.$ Let $\mathcal{C}$ be any chain from $M^v$ to $M^w$. Then

\[ (\deg_1(v,w), \cdots, \deg_k(v,w)) \leq \deg_{\mathcal{C}}(v,w). \]

This is clear by Definition <ref>.

Lemmas <ref>, <ref> , and <ref> address the technicality that generalized rim hooks do not necessarily remove the first $x$ in a row and place an $x$ in the last open position in a particular row.

Let $M^v$ and $M^w$ be a Maya diagram corresponding to $v,w \in W^{P_I}.$ Let $v_0=v$. Consider

\[M^{v_0} \overset{R_{(v_0,w)}}{\longrightarrow} M^{v_1} \]

\[ \displaystyle R_{(v,w)}=0^{q-1}1^{t-q}0^{k+1-t}.\]

Consider the $j$th rows where $q \leq j \leq t-1$. Suppose that the new $x$ in $m_j^{v_1}$ that is not in $m_j^{v_0}$ is in position $(j,b_0)$. It follows that $S_{j}(M^{v_1},b) \leq S_{j}(M^{w},b)$ for $b \geq b_0$.

For a contradiction assume that $S_j(M^{v_1},b)>S_j(M^{w},b)$ for $b \geq b_0$. Then by the $qt$-rim hook rule we must have that
\[S_{j+1}(M^{v_1},b)=S_j(M^{v_1},b)+(i_{j+1}-i_j).\]
\[ S_j(M^{w},b)+(i_{j+1}-i_j) \geq S_{j+1}(M^{w},b).\]
It follows that $S_{j+1}(M^{v_1},b)>S_{j+1}(M^{w},b).$ Therefore $S_{t}(M^{v_1},b)>S_{t}(M^{w},b)$ be repeating the previous argument. This is a contradiction since $m_t^{v_1} \geq m_t^{w}$. It follows that $S_{j}(M^{v_1},b) \leq S_{j}(M^{w},b)$ for $b \geq b_0$.

Let $M^v$ and $M^w$ be a Maya diagram corresponding to $v,w \in W^{P_I}.$ Let $v_0:=v$. Consider

\[M^{v_0} \overset{R_{(v_0,w)}}{\longrightarrow} M^{v_1} \]

\[ \displaystyle R_{(v,w)}=0^{q-1}1^{t-q}0^{k+1-t}.\]

Consider the $j$th rows where $q \leq j \leq t-1$. Suppose that the $x$ in $m_j^{v_0}$ that is not in $m_j^{v_1}$ is in position $(j,b_0)$. If $f(M^{v_0},j,b)=x$ for some $b<b_0$ then $S_j(M^{v_1},b)\leq S_j(M^{w},b)$.

Let  $f(M^{v_0},j,b)=x$ for some $b<b_0$. It follows that $q>1$. Then we have the following that follows from $t-q$ being maximum:

* We have that $f(M^{v_0},q-1,b)=x$;

* We have that $S_{j}(M^{v_1},b)=S_{q-1}(M^{v_1},b)$;

* We have that $S_{q-1}(M^{v_1},b)\leq S_{q-1}(M^{w},b)$.

The result follows since $S_{q-1}(M^{w},b) \leq S_j(M^{w},b)$.

We use Lemmas <ref>, <ref> to produce the following result.

Let $M^v$ and $M^w$ be a Maya diagram corresponding to $v,w \in W^{P_I}.$ Let $v_0:=0$. If $\mathcal{C}$ is a chain in $W^{P_I}$ in terms of Maya diagrams given by

\begin{eqnarray*}
\mathcal{C}:M^{v_0} \overset{R_{(v_0,w)}}{\longrightarrow} M^{v_1} \overset{R_{(v_1,w)}}{\longrightarrow} M^{v_2} \overset{R_{(v_{2},w)}}{\longrightarrow}\cdots \overset{R_{(v_{r-1},w)}}{\longrightarrow} M^{v_r}
\end{eqnarray*}

where $M^{v_j} \ngeq M^w$ for $1 \leq j \leq r-1$ and $M^{v_r} \geq M^w$. It follows that
\[ SR_j =\deg_j(v,w)\]
where $\displaystyle SR_j$ is the  $j$th component of $\displaystyle \sum_{j=0}^{r-1} R_{(v_j,w)}$.

Let $\mathcal{C}$ be a chain in $W^{P_I}$ in terms of Maya diagrams given by

\begin{eqnarray*}
\mathcal{C}:M^{v_0} \overset{R_{(v_0,w)}}{\longrightarrow} M^{v_1} \overset{R_{(v_1,w)}}{\longrightarrow} M^{v_2} \overset{R_{(v_{2},w)}}{\longrightarrow}\cdots \overset{R_{(v_{p-1},w)}}{\longrightarrow} M^{v_p}.
\end{eqnarray*}

where $SR_j=\deg_j(v,w)$. It follows from Lemmas <ref> and <ref> that $m_j^w \leq m_j^{v_r}$. The result follows.

§ MAIN RESULT

Let $M^v$ and $M^w$ be a Maya diagram corresponding to $v,w \in W^{P_I}.$ If $\mathcal{C}$ is a chain in $W^{P_I}$ in terms of Maya diagrams given by

\begin{eqnarray*}
\mathcal{C}:M^{v_0} \overset{R_{(v_0,w)}}{\longrightarrow} M^{v_1} \overset{R_{(v_1,w)}}{\longrightarrow} M^{v_2} \overset{R_{(v_{2},w)}}{\longrightarrow}\cdots \overset{R_{(v_{r-1},w)}}{\longrightarrow} M^{v_r}
\end{eqnarray*}

where $M^{v_j} \ngeq M^w$ for $1 \leq j \leq r-1$ and $M^{v_r} \geq M^w$ then

\[ (\deg_1(v,w), \cdots, \deg_k(v,w)) = \deg_{\mathcal{C}}(v,w)=\sum_{j=0}^{r-1} R_{(v_j,w)}. \]

Let $\displaystyle SR_j$ denote the the $j$th component of $\displaystyle \sum_{j=0}^{r-1} R_{(v_j,w)}$ and $d_j\mathcal{C}$ denote the $j$ component of $\deg_{\mathcal{C}}(v,w)$. By Lemma <ref>,  $ \displaystyle SR_j = \deg_j(v,w)$. By Lemma <ref> we have that $ \displaystyle \deg_{\mathcal{C}}(v,w) \leq SR_j$.  Lemma <ref> states that $\deg_j(v,w) \leq d_j\mathcal{C}$. Then
\[ \displaystyle SR_j =\deg_j(v,w) \leq d_j\mathcal{C} \leq SR_j. \]

The result follows.

Here we give an example of Theorem <ref>. Consider the following two Maya diagrams.

\[ M^w=\ytableausetup{centertableaux}
\begin{ytableau}
\bx &  \bx &\bx  & x   &  \bx  &\bx&\bx & x &\bx  &\bx  &\bx &x& x \\
\bx &  x &x  &    &  \bx  &\bx&\bx &  &\bx  &\bx  &\bx &&  \\
\bx &    &   &    & \bx   &x  &x   &  &\bx  &\bx  &\bx &&\\
\bx &    &   &    & x   &   &    &  &\bx  &x    &x   &&\\
\bx &    &   &    &      &   &    &  &x    &x     &    &&\\
x   &    &   &    &       &   &    &  &     &     &    &&
\end{ytableau}\ytableausetup{centertableaux}  \mbox{\&}
\begin{ytableau}
\bx&\bx&\bx&x    & \bx     & x  & x &\bx &\bx  &\bx & \bx&x&\bx  \\
x&\bx&\bx&    & x     &   &  &\bx &\bx  &\bx & \bx&&\bx  \\
&\bx&\bx&    &       &   &  &\bx & x   & \bx& x  &&\bx \\
&\bx&\bx&    &       &   &  &\bx &     & x  &    &&x\\
&\bx&x  &    &       &   &  &x   &     &    &    && \\
&x  &   &    &       &   &  &  &     &    &    &&
\end{ytableau}\ytableausetup{centertableaux}.\]

Then we have the following where $!$ indicate the Bruhat order incompatible positions and $\uparrow$ and $\downarrow$ describe the movement of the $x$'s.

\begin{eqnarray*}
\begin{ytableau}
\bx&\bx&\bx & x   & \bx     & x  & x &\bx &\bx  &\bx & \bx&x&\bx  \\
x&\bx&\bx &    & x     &   &  &\bx &\bx  &\bx & \bx&&\bx  \\
&\bx&\bx!&    &       &   &  &\bx & x   & \bx& x  &&\bx \\
&\bx&\bx!&    &       &   &  &\bx!&     & x  &    &&x\\
&\bx& x! &    &       &   &  &x!  &     &    &    && \\
&x  &    &    &       &   &  &    &     &    &    &&
\end{ytableau}\ytableausetup{centertableaux} &\overset{0^{1}1^{3}0^{1}}{\longrightarrow}&
\ytableausetup{centertableaux}
\begin{ytableau}
\bx&\bx& \bx       & x   & \bx & x  & x &\bx &\bx  &\bx & \bx         &x&\bx  \\
x&\bx& x       &    & \downarrow &   &  &\bx &\bx  &\bx & \bx         &&\bx  \\
&\bx&\uparrow &    & x           &   &  &\bx & x   & \bx& \downarrow &&\bx \\
&\bx&\uparrow &    &             &   &  &\bx&     & x  & x           && \downarrow \\
&\bx&\uparrow&    &             &   &  &x  &     &    &             &&x \\
&x  &         &    &             &   &  &    &     &    &             &&
\end{ytableau}\ytableausetup{centertableaux}\\
\end{eqnarray*}

\begin{eqnarray*}
\hspace{20pt}=\ytableausetup{centertableaux}
\begin{ytableau}
\bx&\bx& \bx       & x   & \bx         & x  & x &\bx &\bx  &\bx & \bx        &x&\bx \\
x&\bx& x       &    & \bx         &   &  &\bx &\bx  &\bx & \bx        &&\bx \\
&\bx&         &    & x           &   &  &\bx & x   & \bx& \bx       & &\bx \\
&\bx&         &    &             &   &  &\bx&     & x  & x          & &\bx \\
&\bx&         &    &             &   &  &x!  &     &    &           &  &x   \\
&x  &         &    &             &   &  &   &     &    &            & &
\end{ytableau}\ytableausetup{centertableaux}
\hspace{6pt}\overset{0^{1}1^{1}0^{3}}{\longrightarrow}\hspace{6pt}\ytableausetup{centertableaux}
\begin{ytableau}
\bx&\bx& \bx       & x   & \bx         & x  & x &\bx &\bx  &\bx & \bx        &x&\bx \\
x&\bx& x       &    & \bx         &   &  &\bx &\bx  &\bx & \bx        &&\bx \\
&\bx&         &    & x           &   &  &\bx & x   & \bx& \bx        &&\bx \\
&\bx&         &    &             &   &  &x&     & x  & \downarrow          & &\bx \\
&\bx&         &    &             &   &  &\uparrow  &     &   &  x          & &x   \\
&x  &         &    &             &   &  &   &     &    &             &&
\end{ytableau}\ytableausetup{centertableaux}\\
\end{eqnarray*}

\begin{eqnarray*}
\begin{ytableau}
\bx&\bx& \bx       & x   & \bx         & x  & x &\bx &\bx  &\bx & \bx        &x&\bx \\
x&\bx& x       &    & \bx         &   &  &\bx &\bx  &\bx & \bx        &&\bx \\
&\bx&         &    & x           &   &  &\bx & x   & \bx& \bx       & &\bx \\
&\bx&         &    &             &   &  &x&     & x  & \bx          & &\bx \\
&\bx&         &    &             &   &  &  &     &   &  x           &&x   \\
&x  &         &    &             &   &  &   &     &    &            & &
\end{ytableau}\ytableausetup{centertableaux}.
\end{eqnarray*}

Thus the minimum quantum degree of $\sigma^v \star \sigma_w$ is $0^11^30^1+0^11^10^3=0^12^11^20^1$.

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