# The $e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ cross sections and
the $\Lambda_{c}^{+}$ electromagnetic form factors within the extended vector
meson dominance model

Cheng Chen<EMAIL_ADDRESS>Institute of Modern Physics, Chinese
Academy of Sciences, Lanzhou 730000, China  School of Nuclear Sciences and
Technology, University of Chinese Academy of Sciences, Beijing 101408, China
Bing Yan<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  College of Mathematics and Physics, Chengdu
University of Technology, Chengdu 610059, China  Ju-Jun Xie
<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  School of Nuclear Sciences and Technology,
University of Chinese Academy of Sciences, Beijing 101408, China  Southern
Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics, Chinese
Academy of Sciences, Huizhou 516000, Guangdong Province, China

###### Abstract

Within the extended vector meson dominance model, we investigate the
$e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ reaction and the
electromagnetic form factors of the charmed baryon $\Lambda_{c}^{+}$. The
model parameters are determined by fitting them to the cross sections of the
process $e^{+}e^{-}\rightarrow\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ and the
magnetic form factor $|G_{M}|$ of $\Lambda^{+}_{c}$. By considering four
charmoniumlike states, called $\psi(4500)$, $\psi(4660)$, $\psi(4790)$, and
$\psi(4900)$, we can well describe the current data on the
$e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ reaction from the reaction
threshold up to $4.96\ \mathrm{GeV}$. In addition to the total cross sections
and $|G_{M}|$, the ratio $|G_{E}/G_{M}|$ and the effective form factor
$|G_{\mathrm{eff}}|$ for $\Lambda^{+}_{c}$ are also calculated, and found that
these calculations are consistent with the experimental data. Within the
fitted model parameters, we have also estimated the charge radius of the
charmed $\Lambda_{c}^{+}$ baryon.

##  I Introduction

The information about the electromagnetic structure of hadrons is reflected in
the electromagnetic form factors (EMFFs) Denig and Salme (2013); Pacetti et
al. (2015); Punjabi et al. (2015). In the single photon exchange
approximation, according to the four-momentum squared $q^{2}$ of the photon,
the physical region is categorized into the spacelike region ($q^{2}<0$) and
timelike region ($q^{2}>0$). Experimentally, the spacelike region corresponds
to the scattering channel $e^{-}B\rightarrow e^{-}B$ ($B$ stands for a
baryon), and the timelike region to the annihilation channel
$e^{+}e^{-}\rightarrow B\bar{B}$ ($\bar{B}$ stands for an antibaryon). In the
spacelike region, the EMFFs are associated with the charge and magnetic moment
distribution of the hadrons at small momentum transfers, but they are hardly
measured for the unstable hadrons, such as $\Lambda$ and $\Sigma$ with a
strange quark, $\Xi$ with two strange quarks, and the $\Lambda^{+}_{c}$ with a
charm quark. Therefore, the process $e^{+}e^{-}\rightarrow B\bar{B}$ is the
main channel to measure the EMFFs in the timelike region for the hyperons
Aubert et al. (2007); Ablikim et al. (2019, 2021a); Gong et al. (2023).

For the $e^{+}e^{-}\rightarrow B\bar{B}$ reaction, the effective form factor
$|G_{\mathrm{eff}}|$ can be extracted from the Born cross section, which
represents the dependence of effective coupling strength of the photon-baryon
interaction vertex $\gamma B\bar{B}$, and $|G_{\mathrm{eff}}|$ is a function
of the four-momentum squared $q^{2}$ of the virtual photon. The vector meson
dominance (VMD) model is a very successful tool for studying the nucleon
electromagnetic form factors, in both the spacelike and timelike regions
Iachello et al. (1973); Iachello and Wan (2004); Bijker and Iachello (2004).
In the VMD model, the virtual photon coupling to baryons is through the
intermediate vector mesons, as shown in Fig. 1, where $V$ stands the vector
mesons that have significant couplings to the final states $B\bar{B}$. On the
theoretical side, within the VMD model, the EMFFs of hyperons were
investigated, such as $\Sigma$ Li and Xie (2021); Yan et al. (2023a), $\Xi$
Yan et al. (2023a), and $\Lambda$ Yang et al. (2019); Li et al. (2022).

Figure 1: The Feynman diagram of $e^{+}e^{-}\rightarrow B\bar{B}$ in the VMD
model.

In the charm sector, the $\Lambda_{c}^{+}$ as the lightest single-charm baryon
is interested in both theoretical and experimental studies. The Belle
Collaboration firstly measured total crosss sections of the process of
$e^{+}e^{-}\rightarrow\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$, and a new
charmoniumlike state, denoted as the $Y(4630)$ was discovered Pakhlova et al.
(2008). After that, in 2018, the BESIII Collaboration published the precise
results near the $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ threshold at four
center of mass energies, and the threshold enhancement of the total cross
section was found Ablikim et al. (2018a). These measurements have attracted
some theoretical studies. In Refs. Dai et al. (2017); Cao et al. (2019), the
$\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ final state interactions (FSI) were
considered in the investigation of the
$e^{+}e^{-}\rightarrow\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ reaction. In
addition to the $X(4660)$, the threshold enhancement was well explained by a
virtual pole generated by $\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ attractive
FSI Cao et al. (2019). Moreover, the EMFFs of $\Lambda_{c}^{+}$ were
investigated by virtue of the VMD model in Ref. Wan et al. (2021), where these
charmonium states $\psi(1S)$, $\psi(2S)$, $\psi(3770)$, $\psi(4040)$,
$\psi(4160)$, and $\psi(4415)$ were included in the model. Note that all these
above charmonium states considered in Ref. Wan et al. (2021) are below the
mass threshold of $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ Workman and Others
(2022). Hence, these nonmonotonic structures above the
$\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ threshold cannot be described by the
model Wan et al. (2021).

Very recently, the BESIII Collaboration published the new measurements of the
$e^{+}e^{-}\rightarrow\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ reaction Ablikim
et al. (2023a). It is found that there is no peak around the energy of
$Y(4630)$, but, a long platform behavior from the reaction threshold to $4.66\
\mathrm{GeV}$ was observed, diverging from the Belle measurements around that
energy region. This new experimental result stimulates further research on the
EMFFs of $\Lambda_{c}^{+}$ and the
$e^{+}e^{-}\rightarrow\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ reaction near
threshold Amoroso et al. (2021); Salnikov and Milstein (2023).

In the present work, based on the new measurements by the BESIII Collaboration
on the $e^{+}e^{-}\rightarrow\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ reaction
Ablikim et al. (2023a), we devoted to studying the EMFFs in the timelike
region of $\Lambda_{c}^{+}$ within an extended vector meson dominance model.
In the VMD model, the electromagnetic form factors of the baryons are
attributed to both the couplings of the virtual photon to the vector mesons
and the vector mesons to the baryons. Since the ground states $\omega$ and
$\phi$ are far from the $\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ mass threshold,
we don’t consider these two particles. On the other hand, there are a number
of excited charmoniumlike states that have been experimentally discovered. For
example, the $Y(4660)$ state was observed and confirmed in the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}\psi(2S)$ reaction by the Belle, BABAR,
and BESIII Collaborations Lees et al. (2014); Wang et al. (2007, 2015);
Ablikim et al. (2021b). Moreover, a state with mass $4675.3\pm 29.5\pm 3.5\
\mathrm{MeV}$ and width $218.3\pm 72.9\pm 9.3\ \mathrm{MeV}$ was discovered in
the process of $e^{+}e^{-}\rightarrow D^{*0}D^{*-}\pi^{+}$, which can be
associated to the $Y(4660)$ state Ablikim et al. (2023b).

In 2022, the BESIII Collaboration reported a search on the process
$e^{+}e^{-}\rightarrow K^{+}K^{-}J/\psi$ at the $\rm c.m.$ energies from
$4.127$ to $4.600\ \mathrm{GeV}$, and a bump structure around $\sqrt{s}=4.5\
\mathrm{GeV}$ was observed, which was denoted as the $Y(4500)$ state Ablikim
et al. (2022a). This state (called $\psi(4500)$ in this work) has also been
seen in previous measurements Yuan et al. (2008); Shen et al. (2014); Ablikim
et al. (2018b). Subsequently, BESIII discovered a charmoniumlike state at
$\sqrt{s}=4.7\ \mathrm{GeV}$ with a significance over $5\sigma$ at the
$K^{+}K^{-}J/\psi$ channel, denoted as $Y(4710)$ Ablikim et al. (2023c). The
BESIII experiment has also measured the cross section of
$e^{+}e^{-}\rightarrow K^{0}_{S}K^{0}_{S}J/\psi$, and found a structure
analogous to $Y(4710)$ Ablikim et al. (2023d). In addition, a higher mass
structure around $4.8$ GeV in the process $e^{+}e^{-}\rightarrow
D^{*+}_{s}D^{*-}_{s}$ was found by the BESIII Collaboration Ablikim et al.
(2023e), denoted as $Y(4790)$. Moreover, there is a possible $\psi(4D)$ state
near $4.9\ \mathrm{GeV}$, which was previously studied in Refs. van Beveren
and Rupp (2011, 2009); van Beveren et al. (2009), and also, in the process
$e^{+}e^{-}\rightarrow K^{+}K^{-}J/\psi$ around $5.0$ GeV appear to show a
slightly enhancement Shen et al. (2014), which might implicit a state existing
near the region. Here, we denoted the potential $\psi(4D)$ as $\psi(4900)$ and
$Y(4790)$ as $\psi(4790)$. These charmoniumlike resonances might be related to
the new findings in the $e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$
reaction by the BESIII Collaboration Ablikim et al. (2023a).

Table 1: Masses and widths of the charmoniumlike states considered in this work. State | Mass $M_{R}$ (MeV) | Width $\Gamma_{R}$ (MeV) | Reference
---|---|---|---
$\psi(4500)$ | $4500$ | $125$ | Ablikim et al. (2023c)
$\psi(4660)$ | $4670$ | $115$ | Lees et al. (2014)
$\psi(4790)$ | $4790$ | $100$ | Ablikim et al. (2023e)
$\psi(4900)$ | $4900$ | $100$ | van Beveren and Rupp (2011, 2009); van Beveren et al. (2009)

Based on the above discussions, we take $\psi(4500)$, $\psi(4660)$,
$\psi(4790)$, and $\psi(4900)$ into account in this work. Their masses and
widths are collected in Table 1. 111To minimize the model parameters, we have
fixed their masses and widths. By considering the contributions of these above
excited vector states, we studied the
$e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ reaction within the VMD
model. It is found that the new measurements of the total cross sections
(corresponding to the $\Lambda^{+}_{c}$ effective form factors) from threshold
up $4.95\ \mathrm{GeV}$ and the $\Lambda^{+}_{c}$ electromagnetic form factors
Ablikim et al. (2023a) can be well described.

This article is organized as follows: in the next section, we will show the
theoretical formalism for studying the $\Lambda_{c}^{+}$ electromagnetic form
factors within the VMD model. Numerical results about the total cross sections
of the $e^{+}e^{-}\rightarrow\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ reaction,
the EMFFs of $\Lambda_{c}^{+}$, and the prediction of the $\sin\Delta\Phi$
about the relative phase between electric and magnetic form factors of
$\Lambda^{+}_{c}$ are shown in Sec. III. Finally, a short summary is given in
Sec. IV.

##  II Theoretical formalism

In the reaction $e^{+}e^{-}\rightarrow\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$,
within the one photon exchange approximation and based on electromagnetic
current conservation and Lorentz invariance, the vertex of
$\gamma\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ could be parameterized into two
independent form factors $F_{1}$ and $F_{2}$, which depend on the squared
four-momentum transfer $q^{2}$ of the virtual photon. The form factors $F_{1}$
and $F_{2}$ are so-called the Pauli and Dirac form factors, respectively. Then
the electric $G_{E}$ and magnetic form factor $G_{M}$ can be obtained by
combining the Pauli and Dirac form factors,

$\displaystyle G_{E}(q^{2})$ $\displaystyle=$ $\displaystyle F_{1}(q^{2})+\tau
F_{2}(q^{2}),$ (1) $\displaystyle G_{M}(q^{2})$ $\displaystyle=$
$\displaystyle F_{1}(q^{2})+F_{2}(q^{2}),$ (2)

where $\tau=q^{2}/(4M_{\Lambda^{+}_{c}}^{2})$, with $M_{\Lambda^{+}_{c}}$ the
mass of the charmed baryon $\Lambda^{+}_{c}$. Within the $G_{E}$ and $G_{M}$,
the Born cross section of the annihilation reaction
$e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ is given by

$\displaystyle\sigma=\frac{4\pi\alpha^{2}\beta
C}{3s}\left(|G_{M}(q^{2})|^{2}+\frac{2M^{2}_{\Lambda_{c}^{+}}}{s}|G_{E}(q^{2})|^{2}\right),$
(3)

where $s=q^{2}$ is the invariant mass squared of the $e^{+}e^{-}$ system, and
$\alpha=e^{2}/(4\pi)$ is the electromagnetic fine structure constant, and
$\beta=\sqrt{1-4M^{2}_{\Lambda_{c}^{+}}/s}$ is the velocity of baryon
$\Lambda_{c}^{+}$. The factor $C$ is the $S$-wave Sommerfeld–Gamow factor
corresponding to the final state Coulomb interaction Sakharov (1948), which
is: $C(y)=\frac{y}{1-e^{-y}}$ with
$y=\frac{\alpha\pi}{\beta}\frac{2M_{\Lambda^{+}_{c}}}{\sqrt{s}}$. Therefore,
considering the $C$ factor, it is expected that the cross section of process
$e^{+}e^{-}\rightarrow\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ is nonzero at the
reaction threshold.

According to the total cross section as in Eq. (3), one can also define the
effective form factor $|G_{\mathrm{eff}}|$ as

$\displaystyle|G_{\mathrm{eff}}(q^{2})|=\sqrt{\frac{2\tau|G_{M}(q^{2})|^{2}+|G_{E}(q^{2})|^{2}}{1+2\tau}}.$
(4)

The effective form factor square $|G_{\mathrm{eff}}(q^{2})|^{2}$ is a linear
combination of $|G_{E}(q^{2})|^{2}$ and $|G_{M}(q^{2})|^{2}$, and proportional
to the total cross section of
$e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ reaction. Besides, the
effective form factor $|G_{\mathrm{eff}}(q^{2})|$ also indicates how much the
experimental $e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ cross section
differs from a point-like charmed $\Lambda^{+}_{c}$ baryon.

In the VMD model, the virtual photon couples to
$\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ through charmoniumlike vector mesons,
thus the Dirac and Pauli form factors are parameterized as follows:

$\displaystyle F_{1}$ $\displaystyle=$ $\displaystyle
g(s)\left(f_{1}+\sum_{i=1}^{4}\beta_{i}B_{R_{i}}\right),$ (5) $\displaystyle
F_{2}$ $\displaystyle=$ $\displaystyle
g(s)\left(f_{2}B_{R_{1}}+\sum_{i=2}^{4}\alpha_{i}B_{R_{i}}\right),$ (6)

with

$\displaystyle
B_{R_{i}}=\frac{M_{R_{i}}^{2}}{M_{R_{i}}^{2}-s-iM_{R_{i}}\Gamma_{R_{i}}},$ (7)

where $R_{1}\equiv\psi(4500)$, $R_{2}\equiv\psi(4660)$,
$R_{3}\equiv\psi(4790)$, and $R_{4}\equiv\psi(4900)$. Their masses
($M_{R_{i}}$) and widths ($\Gamma_{R_{i}}$) are shown in Table 1. In addition,
at $s=0$ and setting the widths $\Gamma_{R_{i}}=0$, with the constraints
$G_{E}=1$ and $G_{M}=\mu_{\Lambda^{+}_{c}}$, the coefficients $f_{1}$ and
$f_{2}$ in Eq. (5) and (6) are obtained as

$\displaystyle f_{1}$
$\displaystyle=1-\beta_{1}-\beta_{2}-\beta_{3}-\beta_{4},$ (8) $\displaystyle
f_{2}$
$\displaystyle=\mu_{\Lambda_{c}^{+}}-1-\alpha_{2}-\alpha_{3}-\alpha_{4},$ (9)

where the $\beta_{1}$, $\beta_{2}$, $\beta_{3}$, $\beta_{4}$ and $\alpha_{2}$,
$\alpha_{3}$, $\alpha_{4}$ are model parameters, which are determined from the
fit to the current experimental data on the
$e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ reaction. Besides, we take
the magneton $\mu_{\Lambda_{c}^{+}}=0.24\ \hat{\mu}_{N}$ as the theoretical
prediction in Ref. Shi et al. (2019), which is consistent with the previous
study Wang et al. (2018).

Furthermore, the $g(s)$ is a phenomenological intrinsic form factor that is
usually chosen in the dipole form

$\displaystyle g(s)=\frac{1}{(1-\gamma s)^{2}},$ (10)

with $\gamma$ also a free parameter.

On the other hand, we will use the Flatté type $\rm Flatt~{}\acute{e}$ (1976)
for the $\psi(4500)$ state, since its mass is below but close to the
$\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ threshold, and it may couple strongly
to the $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ channel. To do this, we take
222Only $s$-wave coupling for the $\psi(4500)$ state to the
$\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ channel is taken into account.

$\displaystyle\Gamma_{\psi(4500)}=\Gamma_{0}+g_{\Lambda_{c}}\sqrt{\frac{s}{4}-M_{\Lambda_{c}^{+}}^{2}},$
(11)

with $\Gamma_{0}=125$ MeV and the second part represents the contribution from
the $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ channel. The $g_{\Lambda_{c}}$ is
an unknown $s$-wave coupling constant of the $\psi(4500)$ state to the
$\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ channel. In deed, a
$\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ bound state with quantum numbers
$J^{PC}=1^{--}$ is predicted in Ref. Dong et al. (2021), where the very near
threshold data on the total cross sections of
$e^{+}e^{-}\to\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ reaction can be reasonably
reproduced with it. While in Ref. Song et al. (2022), possible
$\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ molecular states and their productions
in $p\bar{p}$ collision were investigated in a qusipotential Bethe-Salpeter
equation approach.

##  III Numerical results and discussions

Under the above formulations, we perform nine-parameter [$\gamma$,
$g_{\Lambda_{c}}$, $\beta_{i}$ ($i=1,2,3,4$), and $\alpha_{i}$ ($i=2,3,4$)]
$\chi^{2}$ fit to the experimental data on the total cross sections of
$e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ reaction and the magnetic
form factors $|G_{M}|$ of $\Lambda^{+}_{c}$ from the reaction threshold to
$4.96\ \mathrm{GeV}$ measured by the BESIII Collaboration Ablikim et al.
(2018a, 2023a). There are $28$ data points in total. The fitted parameters and
errors are given in Table 2, with a reasonable small
$\chi^{2}/\mathrm{d.o.f}=0.6$. The fitted result for the parameter $\gamma$ is
$0.147\pm 0.017\ \mathrm{GeV}^{-2}$, which is smaller than the typical values
for the light baryons nucleon Dai et al. (2022); Yan et al. (2023b), $\Lambda$
Li et al. (2022); Dai et al. (2022), $\Sigma$ Dai et al. (2022); Yan et al.
(2023a), and $\Xi$ Yan et al. (2023a).

Table 2: Values of the model parameters determined in this work. Parameter | Value | Parameter | Value
---|---|---|---
$g_{\Lambda_{c}}$ | $1.173\pm 0.259$ | $\beta_{4}$ | $-0.141\pm 0.097$
$\beta_{1}$ | $1.883\pm 0.484$ | $\alpha_{2}$ | $1.089\pm 0.297$
$\beta_{2}$ | $-1.101\pm 0.302$ | $\alpha_{3}$ | $0.438\pm 0.192$
$\beta_{3}$ | $-0.439\pm 0.194$ | $\alpha_{4}$ | $0.133\pm 0.096$

In Fig. 2, we show the fitted results for the total cross sections with the
blue solid line compared with the experimental data. 333The data from Belle
Collaboration were not taken into account in our $\chi^{2}$ fit. One can see
that, within the VMD model and by considering the contributions from
$\psi(4500)$, $\psi(4660)$, $\psi(4790)$, and $\psi(4900)$, the total cross
sections of $e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ reaction can
be well described quite in the considered energy regions $4.57\
\mathrm{GeV}<\sqrt{s}<4.96\ \mathrm{GeV}$. From Fig. 2, the cross section for
the $e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ reaction remains
almost constant from the reaction threshold to $4.67\ \mathrm{GeV}$,
exhibiting a long plateau. Here, we propose that the shape is primarily
influenced by the resonances $\psi(4500)$ and $\psi(4660)$ in the VMD model.
Although the structure of $\psi(4660)$ is not significant, the contribution
from it is essential. Similarly, in the higher energy region, the cross
sections from $4.74\ \mathrm{GeV}$ to $4.79\ \mathrm{GeV}$ shows a plateau
with an approximate width of $50\ \mathrm{MeV}$, attributed to the two states
of $\psi(4660)$ and $\psi(4790)$. Moreover, the $\psi(4900)$ state is needed
to explain the small bump structure around $4.92$ GeV.

Figure 2: The obtained total cross sections compared with the experiment data.
The red circle with error bars represents the data published in 2018 by the
BESIII Collaboration Ablikim et al. (2018a), the red rectangles in 2023 by the
BESIII Collaboration Ablikim et al. (2023a), and the green rectangles stand
for the data taken from Belle in 2008 Pakhlova et al. (2008).

Since we can reproduce the total cross sections of
$e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ reaction, it is expected
that the $\Lambda^{+}_{c}$ effective form factor can also be described well,
as shown in Fig. 3 by the red line, compared with the experimental data
Ablikim et al. (2023a). In Fig. 3, we also show the fitted numerical results
for the $\Lambda^{+}_{c}$ magnetic form factor $|G_{M}|$ (the green line)
comparison with the experimental measurements. Again, one can see that the
fitted results are in agreement with the experimental measurements of
$|G_{M}|$ by the BESIII Collaboration Ablikim et al. (2023a). This indicates
that the nonmonotonic line shapes of the
$e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ cross sections and the
$\Lambda^{+}_{c}$ effective form factors can be explained within the VMD
model, where the contributions of the charmoniumlike states are taken into
account.

Figure 3: The obtained $\Lambda^{+}_{c}$ effective form factor $|G_{\rm eff}|$
and the fitted magnetic form factor $|G_{M}|$ compared with experimental data
from BESIII Collaboration Ablikim et al. (2023a).

The effective form factor $|G_{\mathrm{eff}}|$ for $\Lambda^{+}_{c}$, as
stated in Ref. Ablikim et al. (2023a), does not exhibit oscillatory behavior,
in contrast to the findings in the nucleon case Bianconi and Tomasi-Gustafsson
(2015, 2016); Tomasi-Gustafsson et al. (2021); Dai et al. (2022); Yang et al.
(2022); Lin et al. (2022); Qian et al. (2023). This may be attributed to the
limited energy range considered for the
$e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ reaction. The range from
the reaction threshold to 4.95 GeV is only about 350 MeV. More comprehensive
experimental data covering wider energy regions are required to further
confirm the so-called oscillatory behavior of the effective form factor
$|G_{\mathrm{eff}}|$ for $\Lambda^{+}_{c}$. Nonetheless, nonmonotonic
structures in the line shape of $\Lambda^{+}_{c}$ effective form factors are
indeed present, and these can be naturally reproduced within the vector meson
dominance model by considering the contributions from excited vector states.

As discussed above, the Coulomb enhancement factor leads to a nonzero cross
section for the process $e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ at
the reaction threshold $s=4M^{2}_{\Lambda^{+}_{c}}$. Utilizing the fitted
parameters shown in Table 2, the
$e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ cross section at the
threshold can be obtained

$\displaystyle\sigma_{\rm
thre.}=\frac{\pi^{2}\alpha^{3}}{2M_{\Lambda^{+}_{c}}^{2}}|G_{\mathrm{eff}}(4M^{2}_{\Lambda^{+}_{c}})|^{2}=193.1\
\mathrm{pb},$ (12)

and the corresponding $\Lambda^{+}_{c}$ effective form factor is
$|G_{\mathrm{eff}}(4M^{2}_{\Lambda_{c}^{+}})|=1.16$, which is larger than the
value of $|G_{\mathrm{eff}}(4M^{2}_{p})|=1.00\pm 0.05$ for the case of proton
Pacetti et al. (2015). This may indicate that the Coulomb interaction
dominates the $e^{+}e^{-}\to p\bar{p}$ reaction near the threshold Baldini et
al. (2009); Kuraev et al. (2012). However, for the case of the charmed
$\Lambda_{c}^{+}$ baryon, the contributions from the strong interaction is
prominent.

Within the fitted parameters in Table 2, we have calculated the ratio
$|G_{E}/G_{M}|$, and it is shown in Fig. 4. One can clearly see that the
obtained $|G_{E}/G_{M}|$ are in good agreement with experimental data. The
non-monotonic structures (or the so-called oscillating behavior as in Ref.
Ablikim et al. (2023a)) shown in the line shape of the ratio $|G_{E}/G_{M}|$
can be naturally explained by including the contributions from the charmonium
states $\psi(4500)$, $\psi(4660)$, $\psi(4790)$, and $\psi(4900)$. Note that
the $\psi(4900)$ state is crucial to reproduce the experimental measurements
on the $e^{+}e^{-}\rightarrow\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ reaction,
even there are a few data points in the energy regions from $4.85\
\mathrm{GeV}$ to $4.96\ \mathrm{GeV}$. It is expected that more experimental
measurements around $4.9$ GeV can be used to further study the possible
$\psi(4900)$ state Jia et al. (2023).

Figure 4:  The ratio $|G_{E}/G_{M}|$ compared with experimental data from
BESIII Ablikim et al. (2018a, 2023a).

In the timelike region, the electric $G_{E}$ and magnetic form factor $G_{M}$
are complex. The relative phase $\Delta\Phi$ between them is associated with
the spin polarization of the $\Lambda^{+}_{c}$ in the process of
$e^{+}e^{-}\rightarrow\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ Fäldt and Kupsc
(2017); Ablikim et al. (2020, 2022b) . Here, we can write

$\displaystyle G_{E}/G_{M}=e^{i\Delta\Phi}|G_{E}/G_{M}|.$ (13)

Using the parameters in Table 2, we can obtain the functional dependence of
$\sin\Delta\Phi$ on $\sqrt{s}$, which is shown in Fig. 5. At the threshold,
$\sin\Delta\Phi$ is 0 due to the equality of $G_{E}$ and $G_{M}$, subsequently
increasing to higher values. From $4.68$ to $4.78\ \mathrm{GeV}$, the values
of $\sin\Delta\Phi$ are larger than $0.8$. This suggests that the larger
polarizations of $\Lambda^{+}_{c}$ are more likely to be experimentally
observed in this energy region. In fact, given the total cross sections of
$e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ reaction as shown in Fig.
2, we propose that it would be more efficient to conduct polarization
measurements at the center-of-mass energies from $4.68$ to $4.70\
\mathrm{GeV}$.

It is important that the $\sin\Delta\Phi$ reaches 0 again at around
$\sqrt{s}=4.84\ \mathrm{GeV}$ and then decreases steadily to large negative
values, which might also imply a considerable polarization of
$\Lambda^{+}_{c}$. However, it is worthy to mention that only the data in the
range $4.57<\sqrt{s}<4.96\ \mathrm{GeV}$ were included in our fitting, and
thus we do not have the information on the results for the $\sin\Delta\Phi$ at
higher energy region.

Figure 5: The predictions for $\sin\Delta\Phi$ as a function of $\sqrt{s}$.

Next, we turn to the electromagnetic form factors in the spacelike region. The
spacelike electromagnetic form factors can be easily obtained by converting
$q^{2}$ to $-Q^{2}$ and setting the widths $\Gamma_{R_{i}}\ (i=1,2,3,4)$ in
Eqs. (5) and (6) to zero. The obtained $\Lambda^{+}_{c}$ electric form factor
$G_{E}$ in the spacelike region is shown in Fig. 6, where theoretical
calculations in Ref. Kim and Kim (2018) are also shown for comparing. Our
results are somewhat different quantitatively from the calculations in Ref.Kim
and Kim (2018) with the self-consistent SU(3) chiral quark-soliton model. It
is expected that these theoretical calculations can be tested by future
experiments on the EMFFs of the charmed $\Lambda^{+}_{c}$ baryon and thus will
provide new insight into the production of $\Lambda^{+}_{c}$ in the
$e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ reaction.

Then one can also calculate the $\Lambda_{c}^{+}$ mean squared charge radius,
which is given by

$\displaystyle{\left\langle
r_{\mathrm{E}}^{2}\right\rangle}_{\Lambda_{c}^{+}}=\frac{-6}{G_{E}(0)}{\frac{\mathrm{d}G_{E}(Q^{2})}{\mathrm{d}Q^{2}}}\bigg{|}_{Q^{2}=0}.$
(14)

This dictates the charge radius being determined by the slope of $G_{E}$ at
$Q^{2}=0$. $G_{E}(0)=1$ corresponds to the charge value of $\Lambda_{c}^{+}$.
The radius is determined to be ${\left\langle
r_{E}^{2}\right\rangle}_{\Lambda_{c}^{+}}=0.07\ \mathrm{fm}^{2}$ using the
Eqs. (14) and the parameters liseted in Table 2. The value obtained here is
much smaller than the value, ${\left\langle
r_{E}^{2}\right\rangle}_{\Lambda_{c}^{+}}=0.24\ \mathrm{fm}^{2}$, obtained in
Ref. Kim and Kim (2018). This difference can be easily seen from the two
curves shown in Fig. 6.

Figure 6:  The red line is the prediction of electric form factor $G_{E}$ in
the spacelike region. The purple dashed line is the results in Ref. Kim and
Kim (2018).

##  IV Sumarry

In this work, we investigate the electromagnetic form factors of charmed
$\Lambda_{c}^{+}$ baryon and the total cross sections of the
$e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ reaction within the
extended vector meson dominance model. The model parameters are determined by
fitting them to the experimental data on the
$e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ total cross sections and
the $\Lambda^{+}_{c}$ magnetic form factor $|G_{M}|$, as measured by the
BESIII Collaboration Ablikim et al. (2018a, 2023a). By including the
charmoniumlike states $\psi(4500)$, $\psi(4660)$, $\psi(4790)$, and the
possible $\psi(4900)$ state, the current experimental data can be well
reproduced. Notably, the threshold enhancement of the total cross sections of
the $e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ reaction can be
explained by the model, primarily attributed to the $\psi(4500)$ and
$\psi(4660)$ states. Additionally, the possible $\psi(4900)$ state is crucial
to reproduce the small bump structure around the center-of-mass energy
$\sqrt{s}=4.92$ GeV.

Using the fitted model parameters, we calculated the ratio $|G_{E}/G_{M}|$ and
effective form factor $|G_{\mathrm{eff}}|$, and found that these calculations
are consistent with the experimental data. Moreover, the relative phase
$\Delta\Phi$ about $G_{E}$ and $G_{M}$ has been calculated in our work. We
directly provided the numerical results of $\sin\Delta\Phi$, and found that it
is more efficient to measure the spin polarization of $\Lambda^{+}_{c}$ in the
$e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ reaction around the
center-of-mass energy $\sqrt{s}=4.69\ \mathrm{GeV}$.

Finally, we also studied the $\Lambda^{+}_{c}$ electromagnetic form factors in
the spacelike region using the parameters obtained from the fit in the
timelike region. According to the results of $G_{E}$, the mean squared charge
radius is determined to be $0.07\ \mathrm{fm}^{2}$. The theoretical
calculations here can be tested by future experiments and further experimental
measurements will provide new insight into the production of $\Lambda^{+}_{c}$
in the $e^{+}e^{-}\to\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ reaction.

In conjunction with the studies in Ref. Li et al. (2022) for the $\Lambda$, in
Ref. Yan et al. (2023a) for the $\Sigma$ and $\Xi$, and in Ref. Yan et al.
(2023b) for the nucleon, we conclude that the non-monotonic structures
observed in the line shape of the $e^{+}e^{-}\to B\bar{B}$ total cross
sections can be naturally explained within the vector meson dominance model.
Additionally, the $e^{+}e^{-}\to B\bar{B}$ reactions can be used to study the
excited vector states, especially for those exotic states that couple strongly
to the $B\bar{B}$ channels.

## Acknowledgements

This work is partly supported by the National Key R&D Program of China under
Grant No. 2023YFA1606703, and by the National Natural Science Foundation of
China under Grant No. 12075288. It is also supported by the Youth Innovation
Promotion Association CAS.

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# Tight Hilbert Polynomial and F-rational local rings

Saipriya Dubey, Pham Hung Quy and Jugal Verma  Saipriya Dubey
Department of Mathematics, Indian Institute of Technology Bombay,
Mumbai, India, Email-<EMAIL_ADDRESS>
Pham Hung Quy
Department of Mathematics, FPT University,
Hanoi, Vietnam, Email-<EMAIL_ADDRESS>
Jugal Verma
Department of Mathematics, Indian Institute of Technology Bombay,
Mumbai, India, Email-<EMAIL_ADDRESS>
###### Abstract.

Let $(R,\mathfrak{m})$ be a Noetherian local ring of prime characteristic $p$
and $Q$ be an $\mathfrak{m}$-primary parameter ideal. We give criteria for
F-rationality of $R$ using the tight Hilbert function
$H^{*}_{Q}(n)=\ell(R/(Q^{n})^{*})$ and the coefficient $e_{1}^{*}(Q)$ of the
tight Hilbert polynomial
$P^{*}_{Q}(n)=\sum_{i=0}^{d}(-1)^{i}e_{i}^{*}(Q)\binom{n+d-1-i}{d-i}.$ We
obtain a lower bound for the tight Hilbert function of $Q$ for equidimensional
excellent local rings that generalises a result of Goto and Nakamura. We show
that if $\dim R=2$, the Hochster-Huneke graph of $R$ is connected and this
lower bound is achieved then $R$ is F-rational. Craig Huneke asked if the
$F$-rationality of unmixed local rings may be characterized by the vanishing
of $e_{1}^{*}(Q).$ We construct examples to show that without additional
conditions, this is not possible. Let $R$ be an excellent, reduced,
equidimensional Noetherian local ring and $Q$ be generated by parameter test
elements. We find formulas for $e_{1}^{*}(Q),e_{2}^{*}(Q),\ldots,e_{d}^{*}(Q)$
in terms of Hilbert coefficients of $Q$, lengths of local cohomology modules
of $R,$ and the length of the tight closure of the zero submodule of
$H^{d}_{\mathfrak{m}}(R).$ Using these we prove: $R$ is F-rational $\iff
e_{1}^{*}(Q)=e_{1}(Q)\iff 0ptR\geq 2$ and $e_{1}^{*}(Q)=0.$

###### Key words and phrases:

Tight Hilbert polynomial, F-rational rings, parameter test elements,
d-sequences, Local cohomology

The first author is supported by the Senior Research Fellowship of HRDG, CSIR,
Government of India. The second author is partially supported by a fund of
Vietnam National Foundation for Science and Technology Development (NAFOSTED)
under grant number 101.04-2020.10.

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.
The authors have no conflicts of interest to declare. All co-authors have seen
and agree with the contents of the manuscript and there is no financial
interest to report. We certify that the submission is original work and is not
under review at any other publication.

##  1\. Introduction

The theory of tight closure created by Hochster and Huneke in the 1980’s
introduced several types of local rings such as F-regular, weakly F-regular,
F-rational and F-injective local rings, see for example [7],[8],[24], etc. It
is well known that the Hilbert coefficients can be used to characterize
regular, Cohen-Macaulay and Buchsbaum local rings. It is natural to expect
that F-singularities could be characterized using a certain kind of Hilbert
polynomial that involves the tight closure of ideals. The first step in this
direction was taken by Shiro Goto and Y. Nakamura. In response to a conjecture
of K. Watanabe and K. Yoshida [26], Goto and Nakamura [6] proved the following
interesting characterization of F-rational local rings. The length of an
$R$-module $M$ is denoted by $\ell_{R}(M).$ The tight closure of an ideal $I$
is denoted by $I^{*},$ see Section 2 for definitions.

######  Theorem 1.1 (Goto-Nakamura, 2001).

Suppose $R$ has prime characteristic and it is an equidimensional local ring
of dimension $d.$ Suppose that $R$ is a homomorphic image of a Cohen-Macaulay
local ring. Then (1) $e_{0}(Q)\geq\ell_{R}(R/Q^{*})$ for every
$\mathfrak{m}$-primary parameter ideal $Q$ in $R.$
(2) If $\dim R/\mathfrak{p}=d$ for all $\mathfrak{p}\in\operatorname{Ass}(R),$
and $e_{0}(Q)=\ell_{R}(R/Q^{*})$ for some parameter ideal $Q$ in $R,$ then $R$
is a Cohen-Macaulay F-rational local ring.

For a recent treatment of Goto-Nakamura theorem, see [14]. Since $Q^{*}$ is
contained in the integral closure $\overline{Q}$ of $Q,$
$e_{0}(Q)=e_{0}^{*}(Q).$ Therefore the F-rationality of $R$ is a consequence
of the equality $e^{*}_{0}(Q)=\ell(R/Q^{*})$ for rings mentioned in (2) above.
This was an indication that F-singularities could be characterized in terms of
the tight Hilbert function $H^{*}_{Q}(n)=\ell(R/(Q^{n})^{*}).$ Let $I$ be an
$\mathfrak{m}$-primary ideal of $R$ and $R$ be analytically unramified, i.e.
the $\mathfrak{m}$-adic completion $\hat{R}$ is reduced. By a theorem of Rees
[19], $H^{*}_{I}(n)$ is given by a polynomial $P^{*}_{I}(n)$ for large $n.$ We
call it the tight Hilbert polynomial of $I$ and write it as

$P^{*}_{I}(n)=\sum_{i=0}^{d}(-1)^{i}e_{i}^{*}(I)\binom{n+d-1-i}{d-i}.$

The coefficient $e_{0}^{*}(I)$ is the multiplicity $e_{0}(I)$ of $I.$ The
other coefficients $e_{i}^{*}(I)\in\mathbb{Z}$ are called the tight Hilbert
coefficients of $I.$ The tight Hilbert polynomial was introduced in [4] where
it was proved that an analytically unramified Cohen-Macaulay local ring $R$
having prime characteristic is F-rational if and only if $e_{1}^{*}(Q)=0$ for
some ideal $Q$ generated by a system of parameters of $R.$ This paper is
motivated by the following question of Craig Huneke

######  Question 1.2.

Is it true that an unmixed Noetherian local ring $R$ is F-rational if and only
if for some ideal $Q$ of $R$ generated by a system of parameters,
$e_{1}^{*}(Q)=0$?

We provide a negative answer to Question 1.2, see Proposition 5.3. We show
that F-rationality can be characterized by the vanishing of $e_{1}^{*}(Q)$
where $Q$ is an ideal generated by parameter test elements which form a system
of parameters of $R$ where $R$ is reduced, excellent and equidimensional local
Noetherian ring, see Corollary 4.6.

This paper is organized as follows. In Section 2, we review the necessary
background material related to tight closure of ideals, test ideals,
F-rational local rings, excellent rings and the tight closure of the zero
submodule of $H^{d}_{\mathfrak{m}}(R).$ In Section 3, we generalize the result
of Goto-Nakamura [Theorem 1.1 (1)] for equidimensional excellent local rings
by proving a lower bound for the tight Hilbert function.

######  Theorem 1.3.

Let $(R,\mathfrak{m})$ be an equidimensional excellent local ring of prime
characteristic $p$ and $Q$ be an ideal generated by a system of parameters for
$R$. Then for all $n\geq 0,$

$\ell(R/(Q^{n+1})^{*})\geq\ell(R/Q^{*})\binom{n+d}{d}.$

######  Corollary 1.4.

Let $(R,\mathfrak{m})$ be a reduced equidimensional excellent local ring of
prime characteristic $p$ and $Q$ be an ideal generated by a system of
parameters for $R$. Then

$e_{0}(Q)\geq\ell(R/Q^{*}).$

In the next result, we show that if equality holds for some $n$ in Theorem 1.3
then $R$ is F-rational which can be considered as a generalization of Goto-
Nakamura result [Theorem 1.1 (2)] under additional hypothesis.

######  Theorem 1.5.

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d$ and prime
characteristic $p.$ Let $(S,n)$ be a Cohen-Macaulay local ring of dimension
$d$ and $Q(R)$ be the total quotient ring of $R$ such that $R\subseteq
S\subseteq Q(R)$ and $S$ is a finite $R$-module. Let $Q$ be an ideal of $R$
generated by a system of parameters. Suppose that for some fixed $n\geq 0$,

$\ell(R/(Q^{n+1})^{*})=e_{0}(Q)\binom{n+d}{d}.$

Then $R=S.$ In particular $R$ is F-rational.

If $d=2$ and the Hochster-Huneke graph of $R,$ denoted by $\mathcal{G}(R)$, is
connected then we can take $S$ in the above theorem to be the
$S_{2}$-ification of $R$ and obtain the following

######  Corollary 1.6.

Let $(R,\mathfrak{m})$ be a Noetherian local ring with
$\dim(R/\mathfrak{p})=2$ for all $\mathfrak{p}\in\operatorname{Ass}R$ of prime
characteristic $p$ such that $\mathcal{G}(R)$ is connected. If for an ideal
$Q$ generated by a system of parameters for $R$ and for some $n\geq 0,$

$\ell(R/(Q^{n+1})^{*})=e_{0}(Q)\binom{n+2}{2}$

then $R$ is F-rational.

Let $(R,\mathfrak{m})$ be a $d$-dimensional local Noetherian ring and $I$ be
an $\mathfrak{m}$-primary ideal. Then the Hilbert function of $I$ is defined
as $H_{I}(n)=\ell(R/{I^{n}}).$ For large $n,$ it coincides with a polynomial
of degree $d$ called the Hilbert polynomial of $I$ and it is written as

$P_{I}(n)=e_{0}(I)\binom{n+d-1}{d}-e_{1}(I)\binom{n+d-2}{d-1}+\cdots+(-1)^{d}e_{d}(I).$

If $R$ is analytically unramified then by a Theorem of Rees [19], the normal
Hilbert function of an $\mathfrak{m}$-primary ideal $I,$ namely
$\overline{H_{I}}(n)=\ell(R/\overline{I^{n}})$ coincides with a polynomial of
degree $d$ for large $n.$ This polynomial is called the normal Hilbert
polynomial of $I$ and is given by

$\overline{P_{I}}(n)=e_{0}(I)\binom{n+d-1}{d}-\overline{e_{1}}(I)\binom{n+d-2}{d-1}+\cdots+(-1)^{d}\overline{e_{d}}(I).$

In [17] M. Moralés, N. V. Trung and O. Villamayor characterized regular local
rings in terms of the equality $\overline{e_{1}}(Q)=e_{1}(Q)$ for a parameter
ideal $Q$ of an excellent analytically unramified local ring. It is worth
noting that this result was proved in [15] by replacing the excellence
hypothesis of $R$ with its unmixedness. In Section 4 we find an analogous
characterization for F-rational local rings as a consequence of explicit
formulas for the tight Hilbert coefficients in terms of the lengths of local
cohomology modules $H^{j}_{\mathfrak{m}}(R)$ for $0\leq j\leq d-1,$ $e_{i}(Q)$
for $0\leq i\leq d$ and $\ell(0^{*}_{H^{d}_{\mathfrak{m}}(R)}).$

######  Theorem 1.7.

Let $(R,\mathfrak{m})$ be an excellent reduced equidimensional local ring of
prime characteristic $p$ and dimension $d\geq 2.$ Let
$x_{1},x_{2},\ldots,x_{d}$ be parameter test elements and
$Q=(x_{1},x_{2},\ldots,x_{d})$ be $\mathfrak{m}$-primary. Then
(1) $e_{1}^{*}(Q)=e_{0}(Q)-\ell(R/Q^{*})+e_{1}(Q)\text{ and
}e_{j}^{*}(Q)=e_{j}(Q)+e_{j-1}(Q)\text{ for all }2\leq j\leq d,$
(2) $\displaystyle
e_{1}^{*}(Q)=\sum_{i=2}^{d-1}\binom{d-2}{i-2}\ell(H^{i}_{\mathfrak{m}}(R))+\ell(0^{*}_{H^{d}_{\mathfrak{m}}(R)}),$
(3) $\displaystyle
e_{i}^{*}(Q)=(-1)^{i-1}\left[\sum_{j=0}^{d-i}\binom{d-i-1}{j-2}\ell(H^{j}_{\mathfrak{m}}(R))+\ell(H^{d-i+1}_{\mathfrak{m}}(R))\right]\text{
for }i=2,\ldots,d-1$ and
(4) $e_{d}^{*}(Q)=(-1)^{d-1}\ell(H^{1}_{\mathfrak{m}}(R)).$

######  Corollary 1.8.

Let $(R,\mathfrak{m})$ be an excellent reduced equidimensional local ring of
prime characteristic $p$ and dimension $d\geq 2.$ Let
$x_{1},x_{2},\ldots,x_{d}$ be parameter test elements and
$Q=(x_{1},x_{2},\ldots,x_{d})$ be $\mathfrak{m}$-primary. Then the following
are equivalent.

(i) $R$ is F-rational (ii) $e_{1}^{*}(Q)=e_{1}(Q)$ (iii) $e_{1}^{*}(Q)=0$ and
$0ptR\geq 2.$

In Section 5, we construct examples to illustrate some of the above results.

###  1.1. Notation and Conventions

All the rings in this paper are commutative Noetherian rings with
multiplicative identity 1. We use $(R,\mathfrak{m},k)$ to denote local ring
$R$ with unique maximal ideal $\mathfrak{m}$ and the residue field
$k:=R/\mathfrak{m}.$ For basic results on Cohen-Macaulay rings, excellent
rings, tight closure, Hilbert functions and multiplicity we refer the reader
to [3] and [16].

Acknowledgements. J. K. Verma would like to thank Prof. Craig Huneke for
inviting him to University of Virginia in 2019 for discussions and for asking
Question 1.2 which has led to this paper. Thanks are also due to Ian Aberbach
for informing us about his paper [1]. We thank the referees for a careful
reading and several suggestions which improved the paper.

##  2\. Preliminaries

In this section, we set up some notation and recall results needed in later
sections.

###  2.1. Background on tight closure

Let $R$ be a commutative ring and $I$ be an ideal of $R.$ An element $x\in R$
is said to be integral over $I$ if

$x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n}=0$

for some $a_{i}\in I^{i}$ for $1\leq i\leq n.$ The integral closure of $I,$
denoted by $\overline{I}$ is the collection of all elements that are integral
over $I.$
Let $R$ be a Noetherian ring of prime characteristic $p$ and $R^{\circ}$
denote the subset of $R$ consisting of all elements which are not in any
minimal prime ideal of $R.$ For $I=(x_{1},\ldots,x_{n}),$ let
$I^{[p^{e}]}=(x_{1}^{p^{e}},\ldots,x_{n}^{p^{e}}).$ The tight closure of $I,$
denoted by $I^{*},$ is the set of all elements $x$ for which there exists some
$c\in R^{\circ}$ such that $cx^{p^{e}}\in I^{[p^{e}]}$ for all $p^{e}>>0.$ An
ideal $I$ is said to be tightly closed if $I=I^{*}.$ For any ideal $I,$ we
have $I\subseteq I^{*}\subseteq\overline{I}.$

######  Definition 2.1.

The test ideal of $R,$ denoted by $\tau(R)$ is the ideal generated by elements
$c\in R$ which satisfies any of the following equivalent conditions.
(i) $cx^{q}\in I^{[q]}$ for all $q=p^{0},p^{1},p^{2},\ldots,$ whenever $x\in
I^{*}$ for any ideal $I$ of $R.$
(ii) $cx\in I$ whenever $x\in I^{*}$ for any ideal $I$ of $R.$
An element of $\tau(R)\cap R^{\circ}$ is called a test element.

A Noetherian ring $R$ is said to be weakly F-regular if every ideal of $R$ is
tightly closed. Note that the test ideal of $R$ is the unit ideal if and only
if $R$ is weakly F-regular. Recall that a parameter ideal of height $n$ is an
ideal of height $n$ generated by $n$ elements. For excellent local
equidimensional rings, parameter ideals are those generated by a part of a
system of parameters for $R$ [23].

######  Definition 2.2.

The parameter test ideal of $R,$ denoted by $\tau_{par}(R),$ is the ideal
generated by $c\in R$ such that $cI^{*}\subset I$ for all parameter ideals $I$
of $R$ (equivalently, $cx^{q}\in I^{[q]}$ for all $q=p^{e},$
$e=0,1,2,\ldots$). An element of $\tau_{par}(R)\cap R^{\circ}$ is called a
parameter test element.

######  Definition 2.3.

A Noetherian ring $R$ is called F-rational if all parameter ideals are tightly
closed.

Let $(R,\mathfrak{m})$ be a $d$-dimensional local Noetherian ring and
$x_{1},\ldots,x_{d}$ be a system of parameters. Then the local cohomology
module $H^{d}_{\mathfrak{m}}(R)$ can be expressed as the $d^{th}$ cohomology
of the Čech complex with respect to $x:=x_{1},\ldots,x_{d}$ since
$H^{d}_{\mathfrak{m}}(R)\cong H^{d}_{I}(R),$ where $I=(x_{1},\ldots,x_{d}).$
Any element of $H^{d}_{\mathfrak{m}}(R)$ can be represented as
$\eta:=\left[\frac{r}{x_{1}^{i}x_{2}^{i}\cdots x_{d}^{i}}\right].$ Let $R$ be
a ring of characteristic $p>0.$ The Frobenius map $F:R\rightarrow R$ defined
by $F(r)=r^{p}$ naturally induces an action called the Frobenius action on
$H^{d}_{\mathfrak{m}}(R)$ which takes an element
$\eta=\left[\frac{r}{(x_{1}x_{2}\cdots x_{d})^{i}}\right]$ to
$F(\eta)=\left[\frac{r^{p}}{(x_{1}x_{2}\cdots x_{d})^{ip}}\right].$ Similarly,
the eth iteration of the Frobenius map $F^{e}:R\rightarrow R$ defined as
$F^{e}(r)=r^{p^{e}}$ induces a similar action on $H^{d}_{\mathfrak{m}}(R).$

######  Definition 2.4.

Let $(R,\mathfrak{m})$ be a Noetherian local ring of characteristic $p.$ Then

$0^{*}_{H^{d}_{\mathfrak{m}}(R)}=\\{\eta\in
H^{d}_{\mathfrak{m}}(R):\exists\;c\in R^{\circ}\text{ such that
}\;cF^{e}(\eta)=0\text{ for all }e>>0\\}.$

We record a result from [22] which reveals the interplay of tight closure of
the zero submodule of $H^{d}_{\mathfrak{m}}(R)$ with tight closure of ideal
generated by a system of parameters of $R.$

######  Theorem 2.5.

[22, Proposition 3.3(i)] Let $(R,\mathfrak{m})$ be an excellent
equidimensional local ring of dimension $d,$ and let $x_{1},\ldots,x_{d}$ be a
system of parameters. Then any $z\in(x_{1},\ldots,x_{d})^{*}$ uniquely
determines an element $\eta=\left[\frac{z}{x_{1}x_{2}\cdots x_{d}}\right]\in
0^{*}_{H^{d}_{\mathfrak{m}}(R)}.$ Conversely, if
$\eta=\left[\frac{z}{x_{1}x_{2}\cdots x_{d}}\right]\in
0^{*}_{H^{d}_{\mathfrak{m}}(R)},$ then $z\in(x_{1},\ldots,x_{d})^{*}.$

######  Remark 2.6.

Note that if $R$ is Cohen-Macaulay, $\eta=\left[\frac{z}{x_{1}x_{2}\cdots
x_{d}}\right]\in 0^{*}_{H^{d}_{\mathfrak{m}}(R)}$ and $\eta=0$ if and only if
$z\in(x_{1},\ldots,x_{d})$. Therefore Theorem 2.5 implies that an excellent
Cohen-Macaulay local ring $(R,\mathfrak{m})$ of dimension $d$ is F-rational if
and only if $0^{*}_{H^{d}_{\mathfrak{m}}(R)}=0.$

###  2.2. Excellent Rings

Very often, results in this paper and many results for tight closure assume
that the given local ring is excellent. We shall use the following properties
of excellent rings frequently.
(1) Let $(R,\mathfrak{m})$ be an excellent local ring with $\mathfrak{m}$-adic
completion $\hat{R}$ and $I$ be an $\mathfrak{m}$-primary ideal. Then
$I^{*}\hat{R}=(I\hat{R})^{*}$ [3, Proposition 10.3.18].
(2) Any excellent reduced local ring is analytically unramified [16, Theorem
70].
(3) Test elements exist in reduced excellent local rings [8, Theorem 6.1 (a)].
(4) If $R$ is excellent then it is a homomorphic image of Cohen-Macaulay ring
[12, Corollary 1.2].

##  3\. The tight Hilbert function and F-rationality of $R$

In this section, we give a generalization of Goto-Nakamura results [Theorem
1.1] for equidimensional excellent local rings. We provide a lower bound for
tight Hilbert function and show that when the lower bound is achieved then the
ring is F-rational under some additional conditions on $R.$ Let us first prove
a crucial lemma required for this purpose. Lemma 3.1 follows from [9, Theorem
8.20]. However, we are giving a simpler proof of Lemma 3.1(b). We thank the
referee for giving us a clear proof of the next lemma.

######  Lemma 3.1.

Let $(R,\mathfrak{m})$ be an equidimensional excellent local ring of prime
characteristic $p$ and $Q$ be an $\mathfrak{m}$-primary parameter ideal.
(a) Then for all $n\geq 0$ we have $Q^{n}\cap(Q^{n+1})^{*}=Q^{n}Q^{*}$.
(b) $Q^{n}/Q^{n}Q^{*}$ is a free $R/Q^{*}$-module of rank
$\binom{n+d-1}{d-1}$, where $d=\dim R$.

###### Proof.

(b) We note that $Q^{n}$ is a $R$-module generated by monomials of degree $n$
in $x_{1},\ldots,x_{d}$ which form minimal generators of $Q^{n}$ since
$x_{1},\ldots,x_{d}$ are analytically independent [18, Theorem 5]. Let
$A=\mathbb{F}_{p}[x_{1},\ldots,x_{d}]$ be the polynomial subring of $R$
generated by $x_{1},\ldots,x_{d}$. Set $q=(x_{1},\ldots,x_{d})A$. Let
$m_{1},\ldots,m_{t}$ be monomials in the $x_{i}$ of degree $n$ that form a
minimal generating set of the finite $R/Q^{*}$-module $Q^{n}/Q^{n}Q^{*}$
(since any monomial of greater degree will sit in $Q^{n+1}\subseteq
Q^{n}Q^{*}$). Suppose we have $u_{i}\in R$ such that
$z=\sum_{i=1}^{t}u_{i}m_{i}\in Q^{n}Q^{*}$. To show that the $R/Q^{*}$-module
$Q^{n}/Q^{n}Q^{*}$ is free, we must show that each $u_{i}\in Q^{*}$. For each
$1\leq i\leq t$, set $J_{i}:=(m_{1},\ldots,\widehat{m_{i}},\ldots,m_{t})A$.
Then since $Q^{n}Q^{*}\subseteq(Q^{n+1})^{*}$, we have
$u_{i}m_{i}\in(Q^{n+1})^{*}+J_{i}R=(q^{n+1}R)^{*}+J_{i}R\subseteq((q^{n+1}+J_{i})R)^{*}$.
Thus,
$u_{i}\in((q^{n+1}+J_{i})R)^{*}:_{R}m_{i}\subseteq(((q^{n+1}+J_{i}):_{A}m_{i})R)^{*}$
by [2, Theorem 2.3]. But it is easy to see in the polynomial ring $A$ that
$(q^{n+1}+J_{i}):_{A}m_{i}\subseteq q$. Thus, $u_{i}\in(qR)^{*}=Q^{*}$. ∎

######  Theorem 3.2.

Let $(R,\mathfrak{m})$ be an equidimensional excellent local ring of prime
characteristic $p$ and $Q$ be an ideal generated by a system of parameters for
$R$. Then for all $n\geq 0,$

$\ell(R/(Q^{n+1})^{*})\geq\ell(R/Q^{*})\binom{n+d}{d}.$

###### Proof.

We have

$\ell(R/(Q^{n+1})^{*})=\sum_{k=0}^{n}\ell((Q^{k})^{*}/(Q^{k+1})^{*}).$

For each $k$ we have

$\ell\left(\frac{(Q^{k})^{*}}{(Q^{k+1})^{*}}\right)\geq\ell\left(\frac{Q^{k}+(Q^{k+1})^{*}}{(Q^{k+1})^{*}}\right)=\ell\left(\frac{Q^{k}}{Q^{k}\cap(Q^{k+1})^{*}}\right)=\ell\left(\frac{Q^{k}}{Q^{k}Q^{*}}\right).$

Since $Q^{k}$ is minimally generated over $R$ by ${k+d-1}\choose{d-1}$
generators, the base-changed module $Q^{k}/(Q^{k}Q^{*})$ is also generated
over $R/Q^{*}$ by ${k+d-1}\choose{d-1}$ generators. As it must be free on
these generators by Lemma 3.1,

$\ell((Q^{k})^{*}/(Q^{k+1})^{*})\geq\ell(Q^{k}/Q^{k}Q^{*})=\ell(R/Q^{*}){{k+d-1}\choose{d-1}}.$

Therefore

$\ell(R/(Q^{n+1})^{*})\geq\ell(R/Q^{*})\sum_{k=0}^{n}\binom{k+d-1}{d-1}=\ell(R/Q^{*})\binom{n+d}{d}.$

The proof is complete. ∎

######  Corollary 3.3.

Let $(R,\mathfrak{m})$ be a reduced equidimensional excellent local ring of
prime characteristic $p$ and $Q$ be an ideal generated by a system of
parameters for $R$. Then

$e_{0}(Q)\geq\ell(R/Q^{*}).$

###### Proof.

Since $R$ is analytically unramified, by using Theorem 3.2 for $n>>0$ we have,

$\left[e_{0}(Q)-\ell(R/Q^{*})\right]\binom{n+d}{d}-e_{1}^{*}(Q)\binom{n+d-1}{d-1}+\cdots+(-1)^{d}e_{d}^{*}(Q)\geq
0.$

Therefore $e_{0}(Q)\geq\ell(R/Q^{*}).$ ∎

The following lemma provides equivalent conditions for F-rationality of Cohen-
Macaulay rings.

######  Lemma 3.4.

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of prime characteristic
$p.$ Let $Q$ be an ideal of $R$ generated by a system of parameters. Then the
following are equivalent.
(a)  $Q^{*}=Q,$
(b)  $(Q^{n})^{*}=Q^{n}$ for all $n\geq 1.$
(c)  $(Q^{n})^{*}=Q^{n}$ for some $n\geq 1.$

###### Proof.

(a) $\implies$ (b). Observe that, using [4, Proposition 4.2],
$Q^{n}\cap(Q^{n+1})^{*}=Q^{*}Q^{n}$ for all $n\geq 1.$ Let $Q^{*}=Q.$ Apply
induction on $n.$ The $n=1$ case is an assumption. Suppose that
$(Q^{n})^{*}=Q^{n}$ for $n=1,2,\ldots,r.$ As
$(Q^{r+1})^{*}\subset(Q^{r})^{*}=Q^{r},$ we have

$(Q^{r+1})^{*}=(Q^{r+1})^{*}\cap Q^{r}=Q^{*}Q^{r}=Q^{r+1}.$

By induction $(Q^{n})^{*}=Q^{n}$ for all $n\geq 1.$
(b) $\implies$ (c). This is clear.
(c) $\implies$ (a). Let $(Q^{n})^{*}=Q^{n}$ for some $n\geq 1.$ Therefore
$Q^{n}=Q^{n-1}\cap(Q^{n})^{*}=Q^{*}Q^{n-1}.$ Hence $Q^{*}\subseteq
Q^{n}:Q^{n-1}=Q.$ Therefore $Q^{*}=Q.$ ∎

######  Theorem 3.5.

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d$ and prime
characteristic $p.$ Let $(S,n)$ be a Cohen-Macaulay local ring of dimension
$d$ and $Q(R)$ be the total quotient ring of $R$ such that $R\subseteq
S\subseteq Q(R)$ and $S$ is a finite $R$-module. Let $Q$ be an ideal of $R$
generated by a system of parameters. Suppose that for some fixed $n\geq 0$,

$\ell(R/(Q^{n+1})^{*})=e_{0}(Q)\binom{n+d}{d}.$

Then $R=S.$ In particular $R$ is F-rational.

###### Proof.

Using [3, Proposition 10.1.5] we get $(Q^{n}S)^{*}\cap R\subseteq(Q^{n})^{*}.$
Let $f=[S/\mathfrak{n}:R/\mathfrak{m}].$ Then we obtain the following

(1) $\ell_{R}(R/(Q^{n+1})^{*})\leq\ell_{R}(R/(Q^{n}S)^{*}\cap
R)\leq\ell_{R}(S/(Q^{n+1}S)^{*})\leq\ell_{R}(S/Q^{n+1}S),$

(2)
$\ell_{R}(S/Q^{n+1}S)=f\ell_{S}(S/(Q^{n+1}S))=fe_{0}(QS)\binom{n+d}{d}=e_{0}(Q)\binom{n+d}{d}.$

Therefore, if $\ell(R/(Q^{n+1})^{*})=e_{0}(Q)\binom{n+d}{d},$ then
$(Q^{n+1}S)^{*}=(Q^{n+1}S).$ As $S$ is Cohen-Macaulay, using Lemma 3.4 it
follows that $(QS)^{*}=QS$ and therefore $S$ is F-rational. Now consider the
exact sequence of finite $R$-modules

$0\to R\to S\to C\to 0,$

where $C=S/R.$ From (1) and (2), it follows that
$(Q^{n+1})^{*}=(Q^{n+1}S)^{*}\cap R=Q^{n+1}S\cap R.$ Tensor this sequence with
$R/Q^{n+1}$ to get the exact sequence of $R$-modules

$0\to R/(Q^{n+1})^{*}\to S/Q^{n+1}S\to C/Q^{n+1}C\to 0.$

As $\ell(R/(Q^{n+1})^{*})=e_{0}(Q)\binom{n+d}{d},$ using (1) and (2), we get
$\ell_{R}(R/(Q^{n+1})^{*})=\ell_{R}(S/Q^{n+1}S)$ which yields $C=Q^{n+1}C.$ By
Nakayama’s lemma, $C=0.$ This means $R=S.$ In particular $R$ is F-rational. ∎

We discuss a relationship of $e_{1}^{*}(Q)$ with $S_{2}$-ification. Let
$(R,\mathfrak{m},k)$ be a Noetherian local ring of dimension $d.$ We recall a
few facts about $S_{2}$-ification of $R$ from [10].

######  Definitions 3.6.

(1) We say that $R$ is equidimensional if $\dim R/\mathfrak{p}=d$ for all
minimal primes $\mathfrak{p}$ of $R.$ If $R$ is equidimensional and it has no
embedded associated primes, then $R$ is called unmixed.

(2) Let $(R,\mathfrak{m})$ be an equidimensional local ring of dimension $d$.
The Hochster-Huneke graph $\mathcal{G}(R)$ is a graph where the vertices are
the minimal prime ideals of $R$ and the edges are the pairs of prime ideals
$(P_{1},P_{2})$ with $\mathrm{ht}(P_{1}+P_{2})=1$.

(3) Let $(R,\mathfrak{m},k)$ be an equidimensional and unmixed local ring. We
say that a ring $S$ is an $S_{2}$-ification of $R$ if
(i) $S$ lies between $R$ and its total quotient ring,
(ii) $S$ is module-finite over $R$ and is $S_{2}$ as an $R$-module, and
(iii) for every element $s\in S\setminus R,$ the ideal $D(s):=\\{r\in R:rs\in
R\\}$ has height at least two.

If $R$ is $S_{2}$ then $\mathcal{G}(R)$ is connected. Moreover
$\mathcal{G}(R)$ is connected if and only if the $S_{2}$-ification of $R$ is
local [10, Theorem 3.6].

######  Corollary 3.7.

Let $(R,\mathfrak{m})$ be a Noetherian local ring with
$\dim(R/\mathfrak{p})=2$ for all $\mathfrak{p}\in\operatorname{Ass}R$ of prime
characteristic $p$ such that $\mathcal{G}(R)$ is connected. If for an ideal
$Q$ generated by a system of parameters for $R$ and for some $n\geq 0,$

$\ell(R/(Q^{n+1})^{*})=e_{0}(Q)\binom{n+2}{2}$

then $R$ is F-rational.

###### Proof.

By the result above, the $S_{2}$-ification $S$ of $R$ is a Cohen-Macaulay
local ring that is a finite $R$-module. ∎

##  4\. On the equality $e_{1}^{*}(Q)=e_{1}(Q)$ and F-rational local rings

In [17] M. Moralés, N. V. Trung and O. Villamayor proved the following
characterization of regular local rings.

######  Theorem 4.1.

[17, Theorem 1,2] Let $(R,\mathfrak{m})$ be an analytically unramified
excellent local domain and $I$ be an $\mathfrak{m}$-primary parameter ideal.
If $\overline{e}_{1}(I)=e_{1}(I)$ then $R$ is a regular and
$\overline{I^{n}}=I^{n}$ for all $n.$

In this section, we find explicit formulas for the tight Hilbert coefficients
of an ideal $Q$ generated by system of parameters that are parameter test
elements, in terms of the lengths of local cohomology modules
$H^{j}_{\mathfrak{m}}(R)$ for $0\leq j\leq d-1,$ $e_{i}(Q)$ for $0\leq i\leq
d$ and $\ell(0^{*}_{H^{d}_{\mathfrak{m}}(R)}).$ We use these formulas to
characterize F-rationality of the ring in terms of the equality
$e_{1}^{*}(Q)=e_{1}(Q)$ and also in terms of vanishing of $e_{1}^{*}(Q)$ under
the condition that $0ptR\geq 2.$
Let $(R,\mathfrak{m})$ be a local ring of dimension $d$ and $I$ be any
$\mathfrak{m}$-primary parameter ideal of $R.$ It is well known that
$\ell(R/I)\geq e_{0}(I).$ Moreover, $R$ is Cohen-Macaulay if and only if
$\ell(R/I)=e_{0}(I)$ for some (and hence for all) $I.$ Recall that $R$ is
called Buchsbaum if $\ell(R/I)-e_{0}(I)$ is independent of the choice of $I.$

######  Definition 4.2.

Let $(R,\mathfrak{m})$ be a $d$-dimensional Noetherian local ring. An
$\mathfrak{m}$-primary parameter ideal $I$ is said to be standard if

$\ell(R/I)-e_{0}(I)=\sum_{i=0}^{d-1}\binom{d-1}{i}\ell(H^{i}_{\mathfrak{m}}(R)).$

The following result due to Linquan Ma and Pham Hung Quy plays a crucial role
for proving a characterization of F-rationality in terms of vanishing of
$e_{1}^{*}(Q)$ for $\mathfrak{m}$-primary parameter ideals generated by
parameter test elements.

######  Theorem 4.3.

[13, Theorem 4.3] Let $(R,\mathfrak{m})$ be an excellent equidimensional local
ring such that $\tau_{par}(R)$ is $\mathfrak{m}$-primary. Let $Q$ be an ideal
generated by a system of parameters contained in $\tau_{par}(R).$ Then we have

$\ell(Q^{*}/Q)=\sum_{i=0}^{d-1}\binom{d}{i}\ell(H^{i}_{\mathfrak{m}}(R))+\ell(0^{*}_{H^{d}_{\mathfrak{m}}(R)}).$

######  Remark 4.4.

(i) If $Q$ is an ideal generated by a system of parameters of $R$ consisting
of parameter test elements then it is a standard system of parameters of $R$
[11, Remark 5.11] and [21, Proposition 3.8].
(ii) If $Q$ is generated by a standard system of parameters, then the Hilbert
polynomial, infact Hilbert function of $Q$ can be found in [20, Corollary
3.2], [25, Corollary 4.2], [5, Theorem 7], etc. For $n\geq 0,$

$\ell(R/Q^{n})=\sum_{i=0}^{d}(-1)^{i}e_{i}(Q)\binom{n+d-1-i}{d-i},\text{ where
}$
$e_{i}(Q)=(-1)^{i}\sum_{j=0}^{d-i}\binom{d-i-1}{j-1}\ell(H^{j}_{\mathfrak{m}}(R))\mbox{
for all }i=1,2,\ldots,d.$

(iii) If $x_{1},\ldots,x_{d}\in\tau_{par}(R)$ and $Q=(x_{1},\ldots,x_{d})$ is
$\mathfrak{m}$-primary in $(R,\mathfrak{m})$ then $Q\subseteq\tau_{par}(R)$
and taking radicals on both sides, we obtain
$\mathfrak{m}\subseteq\operatorname{rad}(\tau_{par}(R))$ which implies that
$\tau_{par}(R)$ is either $\mathfrak{m}$-primary or $R.$

######  Theorem 4.5.

Let $(R,\mathfrak{m})$ be an excellent reduced equidimensional local ring of
prime characteristic $p$ and dimension $d\geq 2.$ Let
$x_{1},x_{2},\ldots,x_{d}$ be parameter test elements and
$Q=(x_{1},x_{2},\ldots,x_{d})$ be $\mathfrak{m}$-primary. Then
(1) $e_{1}^{*}(Q)=e_{0}(Q)-\ell(R/Q^{*})+e_{1}(Q)\text{ and
}e_{j}^{*}(Q)=e_{j}(Q)+e_{j-1}(Q)\text{ for all }2\leq j\leq d,$
(2)
$e_{1}^{*}(Q)=\sum_{i=2}^{d-1}\binom{d-2}{i-2}\ell(H^{i}_{\mathfrak{m}}(R))+\ell(0^{*}_{H^{d}_{\mathfrak{m}}(R)}),$
(3)
$e_{i}^{*}(Q)=(-1)^{i-1}\left[\sum_{j=0}^{d-i}\binom{d-i-1}{j-2}\ell(H^{j}_{\mathfrak{m}}(R))+\ell(H^{d-i+1}_{\mathfrak{m}}(R))\right]\text{
for }i=2,\ldots,d.$

###### Proof.

(1) By Lemma 3.1, $Q^{n}/Q^{n}Q^{*}$ is a free $R/Q^{*}$-module of rank
$\binom{n+d-1}{d-1}$ for all $n\geq 1$ and by [1, Lemma 3.1],
$(Q^{n+1})^{*}=Q^{n}Q^{*}$ for all $n\geq 1.$ Hence

$\ell(Q^{n}/Q^{n}Q^{*})=\ell(Q^{n}/(Q^{n+1})^{*})=\ell(R/Q^{*})\binom{n+d-1}{d-1}.$

Thus $\ell(R/(Q^{n+1})^{*})=\ell(R/Q^{n})+\ell(R/Q^{*})\binom{n+d-1}{d-1}$ for
all $n\geq 1.$ By Remark 4.4(ii) the tight Hilbert function of $Q$ is given by

$\displaystyle H_{Q}^{*}(n)$
$\displaystyle=e_{0}(Q)\binom{n+d-2}{d}-e_{1}(Q)\binom{n+d-3}{d-1}+\cdots+(-1)^{d}e_{d}(Q)+\ell(R/Q^{*})\binom{n+d-2}{d-1}$
$\displaystyle=\sum_{i=0}^{d}e_{i}(Q)(-1)^{i}\binom{n+d-2-i}{d-i}+\ell(R/Q^{*})\binom{n+d-2}{d-1}$
$\displaystyle=\sum_{i=0}^{d}e_{i}(Q)(-1)^{i}\left[\binom{n+d-1-i}{d-i}-\binom{n+d-2-i}{d-1-i}\right]+\ell(R/Q^{*})\binom{n+d-2}{d-1}$
$\displaystyle=e_{0}(Q)\binom{n+d-1}{d}-[e_{0}(Q)-\ell(R/Q^{*})+e_{1}(Q)]\binom{n+d-2}{d-1}$
$\displaystyle+\sum_{i=2}^{d}(-1)^{i}[e_{i}(Q)+e_{i-1}(Q)]\binom{n+d-i-1}{d-i}.$

Equating like terms on both sides, we obtain the desired formulas.
(2) From (1) we have $e_{1}^{*}(Q)=e_{0}(Q)-\ell(R/Q^{*})+e_{1}(Q).$ On the
other hand, since $Q$ is standard, using Remark 4.4(iii) and Theorem 4.3 we
have

$\displaystyle\ell(R/Q^{*})$ $\displaystyle=$
$\displaystyle\ell(R/Q)-\sum_{i=0}^{d-1}\binom{d}{i}\ell(H^{i}_{\mathfrak{m}}(R))-\ell(0^{*}_{H^{d}_{\mathfrak{m}}(R)})$
$\displaystyle=$ $\displaystyle
e_{0}(Q)+\sum_{i=0}^{d-1}\binom{d-1}{i}\ell(H^{i}_{\mathfrak{m}}(R))-\sum_{i=0}^{d-1}\binom{d}{i}\ell(H^{i}_{\mathfrak{m}}(R))-\ell(0^{*}_{H^{d}_{\mathfrak{m}}(R)})$
$\displaystyle=$ $\displaystyle
e_{0}(Q)-\sum_{i=1}^{d-1}\binom{d-1}{i-1}\ell(H^{i}_{\mathfrak{m}}(R))-\ell(0^{*}_{H^{d}_{\mathfrak{m}}(R)}),$

where the second equality above follows from Remark 4.4(i). Hence

(3)
$e_{1}^{*}(Q)=\sum_{i=1}^{d-1}\binom{d-1}{i-1}\ell(H^{i}_{\mathfrak{m}}(R))+\ell(0^{*}_{H^{d}_{\mathfrak{m}}(R)})+e_{1}(Q).$

Furthermore by Remark 4.4(ii), it follows that

$\displaystyle e_{1}^{*}(Q)$
$\displaystyle=\sum_{i=1}^{d-1}\binom{d-1}{i-1}\ell(H^{i}_{\mathfrak{m}}(R))+\ell(0^{*}_{H^{d}_{\mathfrak{m}}(R)})-\sum_{j=0}^{d-1}\binom{d-2}{j-1}\ell(H^{j}_{\mathfrak{m}}(R))$
$\displaystyle=\sum_{i=1}^{d-1}\left[\binom{d-1}{i-1}-\binom{d-2}{i-1}\right]\ell(H^{i}_{\mathfrak{m}}(R))+\ell(0^{*}_{H^{d}_{\mathfrak{m}}(R)})$
$\displaystyle=\sum_{i=1}^{d-1}\binom{d-2}{i-2}\ell(H^{i}_{\mathfrak{m}}(R))+\ell(0^{*}_{H^{d}_{\mathfrak{m}}(R)})$
$\displaystyle=\sum_{i=2}^{d-1}\binom{d-2}{i-2}\ell(H^{i}_{\mathfrak{m}}(R))+\ell(0^{*}_{H^{d}_{\mathfrak{m}}(R)}).$

(3) Using Remark 4.4(i)-(ii), we obtain

$\displaystyle\ell(R/Q^{n})$
$\displaystyle=\sum_{i=0}^{d}\binom{n+d-1-i}{d-i}(-1)^{i}e_{i}(Q)\mbox{ for
all }n\geq 1,$ $\displaystyle(-1)^{i}e_{i}(Q)$
$\displaystyle=\sum_{j=0}^{d-i}\binom{d-i-1}{j-1}\ell(H^{j}_{\mathfrak{m}}(R))\mbox{
for all }i=1,2,\ldots,d,$ $\displaystyle\ell(R/Q)-e_{0}(Q)$
$\displaystyle=\sum_{j=0}^{d-1}\binom{d-1}{j}\ell(H_{\mathfrak{m}}^{j}(R))$
$\displaystyle e_{d}(Q)$
$\displaystyle=(-1)^{d}\ell(H_{\mathfrak{m}}^{0}(R)).$

In the formulas above, we follow the convention $\binom{n}{-1}=1$ if $n=-1$
and $\binom{n}{-1}=0$ if $n\neq-1.$ By the above formulas and the fact that
$R$ is reduced and equidimensional,

$e_{d}^{*}(Q)=e_{d}(Q)+e_{d-1}(Q)=(-1)^{d-1}\ell(H^{1}_{\mathfrak{m}}(R)).$

Next we find the formulas for $e_{i}^{*}(Q)$ where $i=2,3,\ldots,d$ in terms
of the lengths of the local cohomology modules. Put
$h^{j}=\ell(H_{\mathfrak{m}}^{j}(R)).$

$\displaystyle e_{i}^{*}(Q)$ $\displaystyle=$ $\displaystyle
e_{i}(Q)+e_{i-1}(Q)$ $\displaystyle=$
$\displaystyle(-1)^{i}\sum_{j=0}^{d-i}\binom{d-i-1}{j-1}h^{j}+(-1)^{i-1}\left[\sum_{j=0}^{d-i}\binom{d-i}{j-1}h^{j}+h^{d-i+1}\right]$
$\displaystyle=$
$\displaystyle(-1)^{i-1}\left[\sum_{j=0}^{d-i}\binom{d-i-1}{j-2}h^{j}+h^{d-i+1}\right].$

∎

In the $\dim 1$ case Question 1.2 has an affirmative answer. Let
$(R,\mathfrak{m})$ be a $1$-dimensional analytically unramified local ring and
$I=(a)$ be $\mathfrak{m}$-primary. Since $R$ is reduced and $\dim R=1,$ $R$ is
Cohen-Macaulay. Let

$P_{I}^{*}(n)=e(I)n-e_{1}^{*}(I).$

If $e_{1}^{*}(I)=0$ then $R$ is F-rational. Let $(b)\subseteq\mathfrak{m}$ be
a minimal reduction of $\mathfrak{m}.$ By Briançon-Skoda Theorem,
$\overline{(b)}=(b)^{*}.$ As $R$ is F-rational, $(b)^{*}=(b).$ Thus
$(b)=\overline{(b)}=\mathfrak{m}.$ Hence $R$ is a regular local ring. In the
case $\dim R\geq 2,$ we have answered Huneke’s question with some additional
hypothesis which can be derived as a consequence of Theorem 4.5.

######  Corollary 4.6.

Let $(R,\mathfrak{m})$ be an excellent reduced equidimensional local ring of
prime characteristic $p$ and dimension $d\geq 2.$ Let
$x_{1},x_{2},\ldots,x_{d}$ be parameter test elements and
$Q=(x_{1},x_{2},\ldots,x_{d})$ be $\mathfrak{m}$-primary. Then the following
are equivalent.
(i) $R$ is F-rational.
(ii) $e_{1}^{*}(Q)=e_{1}(Q).$
(iii) $e_{1}^{*}(Q)=0$ and $0ptR\geq 2.$

###### Proof.

(i) $\iff$ (ii): If $R$ is F-rational then $R$ is Cohen-Macaulay. Therefore
$Q^{n}=(Q^{n})^{*}$ for all $n\geq 1$ [4, Corollary 4.3]. Hence
$\ell(R/(Q^{n+1})^{*})=e_{0}(Q)\binom{n+d}{d}$ for all $n\geq 0$ which implies
that $e_{1}^{*}(Q)=e_{1}(Q)=0.$

Conversely, let $e_{1}^{*}(Q)=e_{1}(Q).$ Using Theorem 4.5(1),
$e_{0}(Q)=\ell(R/Q^{*}).$ As $R$ is unmixed, by [6], $R$ is F-rational.
(i) $\iff$ (iii): If $R$ is F-rational then it is Cohen-Macaulay so that (iii)
holds. Conversely, let $e_{1}^{*}(Q)=0$ and $0ptR\geq 2.$ By Theorem 4.5(2),
it follows that $0^{*}_{H^{d}_{\mathfrak{m}}(R)}=0$ and
$H^{i}_{\mathfrak{m}}(R)=0$ for $2\leq i\leq d-1.$ As $0ptR\geq 2,$
$H^{0}_{\mathfrak{m}}(R)=H^{1}_{\mathfrak{m}}(R)=0.$ Hence $R$ is Cohen-
Macaulay ring with $0^{*}_{H^{d}_{\mathfrak{m}}(R)}=0.$ By Remark 2.6, it
follows that $R$ is F-rational. ∎

##  5\. A Counterexample to Huneke’s question

We provide a negative answer to Huneke’s question by constructing examples of
unmixed local rings in which $e_{1}^{*}(Q)=0$ for an ideal $Q$ generated by a
system of parameters but $R$ is not F-rational. The next proposition gives a
class of examples where $0^{*}_{H^{d}_{\mathfrak{m}}(R)}$ vanishes.

######  Proposition 5.1.

Let $(R,\mathfrak{m})$ be an equidimensional reduced local ring of dimension
$d$, and $\mathrm{Ass}R=\\{P_{1},P_{2}\\}$. Suppose $R/P_{1}$ and $R/P_{2}$
are both F-rational and $\dim R/(P_{1}+P_{2})\leq d-2$. Then
$0^{*}_{H^{d}_{\mathfrak{m}}(R)}=0.$

###### Proof.

Consider the long exact sequence of local cohomology arising from the
following short exact sequence.

$0\to R\to R/P_{1}\oplus R/P_{2}\to R/(P_{1}+P_{2})\to 0.$

Since $\dim R/(P_{1}+P_{2})\leq d-2,$ it follows that
$H^{i}_{\mathfrak{m}}(R/(P_{1}+P_{2}))=0$ for $i=d-1,d.$ This implies that
$H_{\mathfrak{m}}^{d}(R)\cong H_{\mathfrak{m}}^{d}(R/P_{1})\oplus
H_{\mathfrak{m}}^{d}(R/P_{2}).$ Clearly, $0^{*}_{H_{\mathfrak{m}}^{d}(R)}\cong
0^{*}_{H_{\mathfrak{m}}^{d}(R/P_{1})}\oplus
0^{*}_{H_{\mathfrak{m}}^{d}(R/P_{2})}.$ Since $R/P_{i}$ is F-rational for
$i=1,2$ we have $0^{*}_{H_{\mathfrak{m}}^{d}(R/P_{i})}=0$ which implies that
$0^{*}_{H_{\mathfrak{m}}^{d}(R)}=0.$ ∎

######  Lemma 5.2.

Let $(R,\mathfrak{m})$ be an equidimensional reduced local ring of dimension
$d$, and $\mathrm{Ass}\;R=\\{P_{1},P_{2}\\}$. Then for any
$\mathfrak{m}$-primary parameter ideal $Q$ in $R,$

$e_{0}(Q)=e_{0}\big{(}(Q+P_{1})/P_{1}\big{)}+e_{0}\big{(}(Q+P_{2})/P_{2}\big{)}.$

###### Proof.

Since $R$ is reduced, $\ell_{R_{P_{i}}}(R_{P_{i}})=1$ for $i=1,2.$ By the
associativity formula for multiplicity we get,

$\displaystyle e_{0}(Q)$
$\displaystyle=e_{0}\big{(}(Q+P_{1})/P_{1}\big{)}\ell(R_{P_{1}})+e_{0}(Q+P_{2}/P_{2})\ell(R_{P_{2}})$
$\displaystyle=e_{0}\big{(}(Q+P_{1})/P_{1}\big{)}+e_{0}\big{(}(Q+P_{2})/P_{2}\big{)}.$

∎

######  Proposition 5.3.

Let $(R,\mathfrak{m})$ be an equidimensional reduced local ring of dimension
$d$ and prime characteristic $p$ with $\mathrm{Ass}\;R=\\{P_{1},P_{2}\\}$.
Suppose $R/P_{1}$ and $R/P_{2}$ are both F-rational and $\dim
R/(P_{1}+P_{2})\leq d-2$. Then $R$ is not Cohen-Macaulay and for any ideal
generated by a system of parameters $Q,$ we have $e_{1}^{*}(Q)=0$.

###### Proof.

Since $R/P_{i}$ is F-rational we have
$(Q^{n+1}R/P_{i})^{*}=(Q^{n+1}+P_{i})/P_{i}$ for $i=1,2.$ Using [7,
Proposition 6.25(a)], we have $(Q^{n+1})^{*}+P_{i}=Q^{n+1}+P_{i}$ for all
$i=1,2$. Thus $(Q^{n+1})^{*}\subseteq(Q^{n+1}+P_{1})\cap(Q^{n+1}+P_{2})$.
Moreover, $x\in(Q^{n+1})^{*}$ if and only if the image of $x$ in $R/P_{i}$ is
contained in $(Q^{n+1}R/P_{i})^{*}=(Q^{n+1}+P_{i})/P_{i}$ for $i=1,2$. Hence
$(Q^{n+1})^{*}=(Q^{n+1}+P_{1})\cap(Q^{n+1}+P_{2})$. Therefore we have the
short exact sequence

$0\to R/(Q^{n+1})^{*}\to R/(Q^{n+1}+P_{1})\oplus R/(Q^{n+1}+P_{2})\to
R/(Q^{n+1}+P_{1}+P_{2})\to 0$

for all $n\geq 0$. Thus we have

$\displaystyle\ell(R/(Q^{n+1})^{*})$
$\displaystyle=\ell\big{(}R/(Q^{n+1}+P_{1})\big{)}+\ell\big{(}R/(Q^{n+1}+P_{2})\big{)}-\ell\big{(}R/(Q^{n+1}+P_{1}+P_{2})\big{)}$
$\displaystyle=\left[e_{0}\big{(}(Q+P_{1})/P_{1}\big{)}+e_{0}\big{(}(Q+P_{2})/P_{2}\big{)}\right]\binom{n+d}{d}-\ell\big{(}R/(Q^{n+1}+P_{1}+P_{2})\big{)}$
$\displaystyle=e_{0}(Q)\binom{n+d}{d}-\ell(R/(Q^{n+1}+P_{1}+P_{2})),$

where the last equality follows from Lemma 5.2.

Since $\ell(R/(Q^{n+1}+P_{1}+P_{2}))$ is a polynomial of degree atmost $d-2,$
$e_{1}^{*}(Q)=0$ for all $Q$. Consider the the short exact sequence of
$R$-modules

$0\to R\to R/P_{1}\oplus R/P_{2}\to R/(P_{1}+P_{2})\to 0.$

Since $0pt(R/P_{1}\oplus R/P_{2})=d>\dim(R/(P_{1}+P_{2})),$ by the depth Lemma
$0ptR\leq d-1.$ Hence $R$ is not Cohen-Macaulay. ∎

We construct an example to show that the condition $0ptR\geq 2$ in Corollary
4.6 is not superfluous for characterization of F-rationality in terms of
vanishing of $e_{1}^{*}(Q).$

######  Example 5.4.

Let $S=\mathbb{F}_{p}[|X,Y,Z,W|]$ and $R=\frac{S}{I\cap J},$ where $I=(X,Y)$
and $J=(Z,W).$ Let the lower case letters denote images of the upper case
letters. Put $\mathfrak{m}=(x,y,z,w).$ Let $a=x+z,\,b=y+w.$ Then $a,b$ is a
system of parameters. Set $Q=(a,b).$ Since $R$ is Buchsbaum

$\ell\left(\frac{R}{Q}\right)-e_{0}(Q)=\sum_{i=0}^{d-1}\binom{d-1}{i}\ell(H_{\mathfrak{m}}^{i}(R))=\ell(H_{\mathfrak{m}}^{1}(R))=1,$

Note that $H_{\mathfrak{m}}^{1}(R)\cong
H_{\mathfrak{m}}^{0}(R/\mathfrak{m})\cong R/\mathfrak{m}.$ Using
$e_{i}(Q)=(-1)^{i}\sum_{j=0}^{d-i}\binom{d-i-1}{j-1}\ell(H_{\mathfrak{m}}^{j}(R)),$
we get $e_{1}(Q)=-\ell(H_{\mathfrak{m}}^{1}(R))=-1,$ $e_{2}(Q)=0.$ Since $R$
is Buchsbaum and $0^{*}_{H^{d}_{\mathfrak{m}}}(R)=0,$ it follows that
$\tau_{par}(R)=\mathfrak{m}.$ Thus by Theorem 4.5(1),
$e_{2}^{*}(Q)=e_{2}(Q)+e_{1}(Q)=-1$ and $e_{1}^{*}(Q)=0$ by Proposition 5.3.
Therefore

$P_{Q}^{*}(n)=2\binom{n+1}{2}-1.$

######  Example 5.5.

We construct a complete local domain of dimension $2$ that is not F-rational
but there exists an ideal $Q$ generated by a system of parameters $Q$ for
which $e_{1}^{*}(Q)=0.$ Let $k$ be a field of prime characteristic $p\geq 3$
and $R=k[[x^{4},x^{3}y,xy^{3},y^{4}]]$. We have the $S_{2}$-ification of $R$
is the local ring $S=k[[x^{4},x^{3}y,x^{2}y^{2},xy^{3},y^{4}]]$. We have
$C:=S/R\cong k$, so that $\ell(C/JC)=1$ for any $\mathfrak{m}$-primary ideal
$J$ of $R.$ Let $Q$ be any $\mathfrak{m}$-primary ideal parameter ideal of
$R$. Consider the short exact sequence,

$0\to R/(Q^{n+1})^{*}\to S/(Q^{n+1}S)^{*}\to C\to 0.$

We have

$\ell(R/(Q^{n+1})^{*})=\ell(S/(Q^{n+1})^{*}S)-1.$

Since $S$ is F-regular,

$\ell(R/(Q^{n+1})^{*})=e_{0}(Q)\binom{n+2}{2}-1$

for all $n\geq 1$. Since $S/\mathfrak{n}\cong R/\mathfrak{m},$
$e_{0}(Q)=e_{0}(QS).$ Hence $e_{1}^{*}(Q)=0$.

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