#  LAPTH-044/20  The gravimagnetic dipole

Gérard Clément

LAPTh, Université Savoie Mont Blanc, CNRS,
9 chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux cedex, France Email:
<EMAIL_ADDRESS>
(1 February 2021)

We investigate a previously constructed stationary solution of the vacuum
Einstein equations, which represents a system of two non-extreme black holes
with equal masses and opposite NUT charges, connected by a Misner string with
tension. For large separations, the inverse square law force measured by this
tension is attractive or repulsive, according to the relative values of the
masses and NUT charges. For small separations, the force is always repulsive,
so that the system cannot collapse to a single black hole. For given values of
the black hole masses and NUT charges, there is a unique configuration such
that the Misner string is tensionless. This behaves asymptotically as the Kerr
solution, but can be overspinning while remaining free from a ring
singularity, thus evading the usual black hole uniqueness theorems. All double
black hole and string configurations satisfy a generalized first law of black
hole mechanics where the two black holes and the Misner string are treated on
an equal footing.

##  1 Introduction

A recurring question in general relativity is that of the existence of
regular, stationary, asymptotically flat double black hole systems, as
solutions either to the vacuum Einstein equations or to the Einstein-Maxwell
equations. With the exception of linear Majumdar-Papapetrou [1] superpositions
of extreme Reissner-Nordström black holes, these generically can be balanced
only if there is a line of conical singularities (cosmic string) connecting
the two black holes. Recently we have investigated some systems of two extreme
black holes carrying opposite gravimagnetic or NUT charges, and shown that the
parameters can be fine tuned so that the Misner string between the two black
holes is tensionless [2, 3]. Misner strings are usually considered to be
unphysical, but we have shown that they are transparent to geodesic motion,
and argued that they do not lead to observable violations of causality [4].

Non-extreme double black hole solutions of the Einstein-Maxwell equations have
been discussed in [5]. The superposition of two Kerr-NUT solutions was
analyzed in [6], where it was shown that balance could be achieved in the
absence of both cosmic string and Misner string. It was not clear however
whether the two constituent sources were regular black holes, or whether the
solution was free from a ring singularity. A similar problem was later
discussed in [7].

After the double Schwarzschild solution, where balance is clearly impossible,
the simplest asymptotically flat stationary double black hole solution is the
double Schwarzschild-NUT vacuum solution. Such a solution was constructed in
[8] as a non-linear superposition of two sources with equal masses and
opposite NUT charges. The main purpose of the authors of [8] was to show that
their solution reduced to the Kerr black hole when the distance between the
two sources was appropriately chosen. In this paper, we analyze in more detail
this solution, show that it is free from a ring singularity in a large
parameter domain, and determine the two-dimensional surface in the three-
dimensional parameter domain such that the system is weakly balanced, in the
sense that the Misner string connecting the two constituent black holes is
tensionless.

The solution is recalled in the next section. The geometrical structure which
it describes is identified in section 3, where the condition for the absence
of a Kerr-like ring singularity is given, and the horizon characteristics are
determined. The properties of the string connecting the two black holes are
discussed in section 4. In section 5, we evaluate the horizon and string Komar
masses and angular momenta, and show that the dipole system satisfies a
generalized first law of black hole mechanics where the two black holes and
the Misner string are treated on an equal footing. Our conclusions are given
in the final section.

##  2 The solution

The non-linear superposition of two NUT solutions (of the vacuum Einstein
equations)with equal masses $m$ and opposite NUT charges $\nu$, separated by a
distance $2k$ ($k\geq m>0$), leads to the Ernst potential, constructed by
Sibgatullin’s method [9] in [8] :

${\cal E}=\frac{A-B}{A+B},$ (2.1)

with

$\displaystyle A$ $\displaystyle=$
$\displaystyle\left[(m^{2}+\nu^{2})(k^{2}-m^{2})(k^{2}-m^{2}-\nu^{2})-2m^{2}k^{2}\nu^{2}\right](R_{+}-R_{-})(r_{+}-r_{-})$
(2.2)
$\displaystyle-\alpha_{+}\alpha_{-}\left[2(m^{2}+\nu^{2})(k^{2}-m^{2})(R_{+}R_{-}+r_{+}r_{-})\right.$
$\displaystyle\left.+(2m^{4}+(m^{2}+\nu^{2})(k^{2}-m^{2}))(R_{+}+R_{-})(r_{+}+r_{-})\right]$
$\displaystyle-2imk\nu
d\left[(\alpha_{+}-\alpha_{-})(R_{+}r_{+}-R_{-}r_{-})-\right.$
$\displaystyle\left.(\alpha_{+}+\alpha_{-})(R_{+}r_{-}-R_{-}r_{+})\right],$
$\displaystyle B$ $\displaystyle=$ $\displaystyle
4d\left\\{m\alpha_{+}\alpha_{-}\left[(m^{2}-d)(R_{+}+R_{-})-(m^{2}+d)(r_{+}+r_{-})\right]\right.$
(2.3)
$\displaystyle\left.+ik\nu\left[\alpha_{-}(m^{2}-d)(R_{+}-R_{-})-\alpha_{+}(m^{2}+d)(r_{+}-r_{-})\right]\right\\},$

where

$R_{\pm}=\sqrt{\rho^{2}+(z\pm\alpha_{+})^{2}},\quad
r_{\pm}=\sqrt{\rho^{2}+(z\pm\alpha_{-})^{2}},$ (2.4)

$\rho$ and $z$ being the Weyl coordinates, and

$\alpha_{\pm}=\sqrt{m^{2}+k^{2}-\nu^{2}\pm 2d},\quad
d=\sqrt{m^{2}k^{2}+\nu^{2}(k^{2}-m^{2})}.$ (2.5)

The corresponding metric is

$ds^{2}=-f(dt-\omega
d\varphi)^{2}+f^{-1}\left[e^{2\gamma}(d\rho^{2}+dz^{2})+\rho^{2}d\varphi^{2}\right],$
(2.6)

with

$f=\frac{A\bar{A}-B\bar{B}}{(A+B)(\bar{A}+\bar{B})},\quad
e^{2\gamma}=\frac{A\bar{A}-B\bar{B}}{64d^{4}\alpha_{+}^{2}\alpha_{-}^{2}R_{+}R_{-}r_{+}r_{-}},\quad\omega=-\frac{4{\rm
Im}[G(\bar{A}+\bar{B})]}{A\bar{A}-B\bar{B}},$ (2.7) $\displaystyle G$
$\displaystyle=$
$\displaystyle-d[d^{2}+m^{2}(m^{2}+2ik\nu)]\left[(\alpha_{+}-\alpha_{-})(R_{+}r_{+}-R_{-}r_{-})-(\alpha_{+}+\alpha_{-})(R_{+}r_{-}-R_{-}r_{+})\right]$
(2.8)
$\displaystyle+2m^{2}d^{2}\left[(\alpha_{+}+\alpha_{-})(R_{+}r_{+}-R_{-}r_{-})-(\alpha_{+}-\alpha_{-})(R_{+}r_{-}-R_{-}r_{+})\right]$
$\displaystyle-m\alpha_{+}\alpha_{-}(d^{2}+m^{4})(R_{+}+R_{-})(r_{+}+r_{-})+m\left[kd^{2}(k+4i\nu)-(2k^{2}-m^{2})(m^{2}+\nu^{2})^{2}\right.$
$\displaystyle\left.+k^{2}\nu^{4}\right](R_{+}-R_{-})(r_{+}-r_{-})-2m\alpha_{+}\alpha_{-}(m^{2}+\nu^{2})(k^{2}-m^{2})(R_{+}R_{-}+r_{+}r_{-})$
$\displaystyle-2dz\left\\{\alpha_{-}(m^{2}-d)\left[m\alpha_{+}(R_{+}+R_{-})+ik\nu(R_{+}-R_{-})\right]\right.$
$\displaystyle\left.-\alpha_{+}(m^{2}+d)\left[m\alpha_{-}(r_{+}+r_{-})+ik\nu(r_{+}-r_{-})\right]\right\\}$
$\displaystyle+2d\alpha_{+}\alpha_{-}(2m^{2}+ik\nu)\left[m^{2}(R_{+}+R_{-}-r_{+}-r_{-})-d(R_{+}+R_{-}+r_{+}+r_{-})\right]$
$\displaystyle-2md\left[d^{2}-m^{4}-ik\nu(2m^{2}+ik\nu)\right]\left[\alpha_{-}(R_{+}-R_{-})-\alpha_{+}(r_{+}-r_{-})\right]$
$\displaystyle+2md^{2}(m^{2}-k^{2}+\nu^{2}-2ik\nu)\left[\alpha_{-}(R_{+}-R_{-})+\alpha_{+}(r_{+}-r_{-})\right].$

This metric is asymptotically flat, with total mass $M=2m$ and total angular
momentum $J=2k\nu$.

The Ernst potential and the metric, given above in terms of the three
parameters $m$, $k$, and $\nu$, can be reexpressed in terms of the three
parameters $\alpha_{+}$, $\alpha_{-}$, and $m$. The definitions (2.5) can be
inverted to give

$k^{2}-\nu^{2}=\alpha_{\pm}^{2}\mp 2d-m^{2},\quad k^{2}\nu^{2}=(d\pm
m^{2})^{2}-m^{2}\alpha_{\pm}^{2}.$ (2.9)

Other useful identities are

$\displaystyle 4d=\alpha_{+}^{2}-\alpha_{-}^{2}=4m^{2}\sigma\delta,\quad(d\pm
m^{2}\pm m\alpha_{\pm})=m^{2}(\sigma+1)(\delta\pm 1),$
$\displaystyle[(d-m^{2})\alpha_{+}+(d+m^{2})\alpha_{-}]=2m^{3}\delta(\sigma^{2}-1),$
$\displaystyle[(d+m^{2})\alpha_{+}+(d-m^{2})\alpha_{-}+4md]=2m^{3}\delta(\sigma+1)^{2},$
$\displaystyle k^{2}\nu^{2}=m^{4}(\sigma^{2}-1)(\delta^{2}-1),$ (2.10)

where

$\sigma\equiv\frac{\alpha_{+}+\alpha_{-}}{2m},\quad\delta\equiv\frac{\alpha_{+}-\alpha_{-}}{2m}.$
(2.11)

Using these, the three functions $A$, $B$ and $G$ can be reexpressed in terms
of $\alpha_{+}$, $\alpha_{-}$, and $m$ as:

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[(d-m^{2})^{2}\alpha_{+}^{2}+(d+m^{2})^{2}\alpha_{-}^{2}\right](R_{+}-R_{-})(r_{+}-r_{-})$
$\displaystyle-\alpha_{+}\alpha_{-}\left[2(d^{2}-m^{4})(R_{+}R_{-}+r_{+}r_{-})+(d^{2}+m^{4})(R_{+}+R_{-})(r_{+}+r_{-})\right]$
$\displaystyle-2imk\nu
d\left[(\alpha_{+}-\alpha_{-})(R_{+}r_{+}-R_{-}r_{-})-(\alpha_{+}+\alpha_{-})(R_{+}r_{-}-R_{-}r_{+})\right],$
$\displaystyle B$ $\displaystyle=$
$\displaystyle-4d\left\\{(d-m^{2})\alpha_{-}\left[m\alpha_{+}(R_{+}+R_{-})+ik\nu(R_{+}-R_{-})\right]\right.$
$\displaystyle\left.+(d+m^{2})\alpha_{+}\left[m\alpha_{-}(r_{+}+r_{-})+ik\nu(r_{+}-r-)\right]\right\\}$
$\displaystyle G$ $\displaystyle=$ $\displaystyle
d[-(d-m^{2})^{2}\alpha_{+}+(d+m^{2})^{2}\alpha_{-}-2ik\nu
m^{2}(\alpha_{+}-\alpha_{-})](R_{+}r_{+}-R_{-}r_{-})$
$\displaystyle+d[(d-m^{2})^{2}\alpha_{+}+(d+m^{2})^{2}\alpha_{-}+2ik\nu
m^{2}(\alpha_{+}+\alpha_{-})](R_{+}r_{-}-R_{-}r_{+})$
$\displaystyle-m(d^{2}+m^{4})\alpha_{+}\alpha_{-}(R_{+}+R_{-})(r_{+}+r_{-})$
$\displaystyle+\frac{m}{2}\left[(d-m^{2})^{2}\alpha_{+}^{2}+(d+m^{2})^{2}\alpha_{-}^{2}+8ik\nu
d^{2}\right](R_{+}-R_{-})(r_{+}-r_{-})$
$\displaystyle-2m(d^{2}-m^{4})\alpha_{+}\alpha_{-}(R_{+}R_{-}+r_{+}r_{-})$
$\displaystyle-2d(d-m^{2})\alpha_{-}\left[(\alpha_{+}+2m-z)(m\alpha_{+}+ik\nu)R_{+}-(\alpha_{+}-2m+z)(m\alpha_{+}-ik\nu)R_{-}\right]$
$\displaystyle-2d(d+m^{2})\alpha_{+}\left[(\alpha_{-}+2m-z)(m\alpha_{-}+ik\nu)r_{+}-(\alpha_{-}-2m+z)(m\alpha_{-}-ik\nu)r_{-}\right].$

##  3 Structure

For $\nu=0$, $d=km$ and $\alpha_{\pm}=k\pm m$ are real, and the static metric
describes a system of two black holes of mass $m$ connected by a cosmic string
of length $2(k-m)$. The two black holes coalesce when $k=m$ to form a single
Schwarzschild black hole.

The situation is less simple when $\nu\neq 0$, and depends on the domain of
values of $k$. For $k$ large enough, $d$ and $\alpha_{\pm}$ are again real and
the system consists of two black holes connected by a spinning cosmic string
(Misner string). On the $z$ axis, these correspond to three rods, two
symmetrical rods $\alpha_{-}<z<\alpha_{+}$ and $-\alpha_{+}<z<-\alpha_{-}$
corresponding to the two black hole horizons, and the central rod
$-\alpha_{-}<z<\alpha_{-}$ corresponding to the Misner string. We have checked
that on the axis outside the rods ($|z|>\alpha_{+}$), $G$, which is a
polynomial of second order in $z$, vanishes identically, so that the axis
condition $\omega(0,z)=0$ is satisfied. Therefore the solution is
asymptotically flat. The stationary metric now depends on the three parameters
$m$, $k$ and $\nu$, or $m$, $\sigma$ and $\delta$. These are related to the
equal black hole masses $M_{H_{\pm}}$ and opposite black hole angular momenta
$J_{H_{\pm}}$, which shall be evaluated in section 5, and the black hole
separation, related to the string length $2\alpha_{-}=2m(\sigma-\delta)$. The
other string characteristics, mass, angular momentum, and tension, are
dynamically fixed so that the system of two black holes and string is in
stationary equilibrium.

It was shown in [8] that for $k=m$ the solution describes the Kerr black hole
(a single rod) provided $|\nu|<2m$ (and presumably the extreme black hole if
$|\nu|=2m$). In this case, $\alpha_{+}$ is real and $\alpha_{-}$ imaginary.
Clearly, there will be an intermediate domain of values $k>m$ with
$\alpha_{+}$ real and $\alpha_{-}$ imaginary, and it is not clear what will
the solution describe in this case. We have found that $\alpha_{\pm}$ are real
and non-zero provided

$k>k_{+}\;\;{\rm or}\;\;k<k_{-},\quad
k_{\pm}(m,\nu)=\sqrt{m^{2}+2\nu^{2}}\pm|\nu|.$ (3.1)

The reality of the parameters $\alpha_{\pm}$ will be assumed in the following.
We will show in section 4 that the solution for $k=k_{+}(m,\nu)$ is actually
singular for $\nu\neq 0$. It follows that when the separation $2k$ between the
two NUTty black holes with fixed $m$ and $\nu$ is decreased, a singular
configuration with $k=k_{+}(m,\nu)$ is reached before the Kerr configuration
$k=m$.

###  3.1 Absence of ring singularity

The reality condition (3.1) is satisfied in two disjoint parameter domains. We
now prove that the domain $k>k_{+}$ is free from an equatorial ring
singularity, which is always present in the domain $k<k_{-}$ .

The equatorial plane is $z=0$, so that $R_{+}=R_{-}=R$, and $r_{+}=r_{-}=r$,
leading to

$\displaystyle A$ $\displaystyle=$
$\displaystyle-2\alpha_{+}\alpha_{-}[(d^{2}-m^{4})(R+r)^{2}+4m^{4}],$
$\displaystyle B$ $\displaystyle=$
$\displaystyle-8dm\alpha_{+}\alpha_{-}[(d-m^{2})R+(d+m^{2})r].$ (3.2)

For $\rho\to\infty$,

$A+B\simeq-8\alpha_{+}\alpha_{-}d^{2}\rho^{2},$ (3.3)

while, for $\rho=0$,

$A+B=-8m^{6}\alpha_{+}\alpha_{-}\delta^{2}(\sigma+1)^{2}(\sigma^{2}-1)$ (3.4)

so that a necessary condition for the absence of a zero of $A+B$ is
$\sigma>1$, implying $\delta>1$ in view of the last equation (2), and so
$d=m^{2}\sigma\delta>m^{2}$, equivalent to $k>m$. Conversely, it follows from

$d^{2}-m^{4}=(m^{2}+\nu^{2})(k^{2}-m^{2})$ (3.5)

that $k>m$ implies $d>m^{2}$, so that $A<0$ and $B\leq 0$, and therefore $A+B$
cannot vanish, proving that the equatorial ring singularity is absent.

So the necessary and sufficient condition for the absence of an equatorial
ring singularity is $k>m$. This is clearly satisfied in the sector $k>k_{+}$,
but violated in the sector $0<k<k_{-}$, where $k^{2}<m^{2}-2k\nu$. In the
following we will always assume $k>k_{+}$.

###  3.2 Horizons

On the upper horizon ($\rho=0$, $\alpha_{-}<z<\alpha_{+}$),

$R_{\pm}=\alpha_{+}\pm z,\quad r_{\pm}=z\pm\alpha_{-},$ (3.6)

leading to

$\displaystyle A_{H}$ $\displaystyle=$
$\displaystyle-8d\alpha_{-}\left\\{(d+m^{2})[(d-m^{2})\alpha_{+}+(d+m^{2})z]+ik\nu
m(\alpha_{+}^{2}-z^{2})\right\\},$ (3.7) $\displaystyle B_{H}$
$\displaystyle=$
$\displaystyle-8d\alpha_{-}\left\\{m\alpha_{+}[(d-m^{2})\alpha_{+}+(d+m^{2})z]+ik\nu[(d+m^{2})\alpha_{+}+(d-m^{2})z]\right\\}.$

We know that $e^{2\gamma}$ and $\omega$ are constant over each rod (this
should be checked). So their values for a given rod can be obtained by
evaluating the corresponding expressions (2.7) at the axis point $z=0$
(whether or not it belongs to the rod). For the horizon rod, noting
$A_{H}(0,0)=A_{0}$, etc., we obtain:

$\displaystyle A_{0}$ $\displaystyle=$
$\displaystyle-8d\alpha_{+}\alpha_{-}\left[d^{2}-m^{4}+ik\nu
m\alpha_{+}\right],$ $\displaystyle B_{0}$ $\displaystyle=$
$\displaystyle-8d\alpha_{+}\alpha_{-}\left[m\alpha_{+}(d-m^{2})+ik\nu(d+m^{2})\right],$
(3.8) $\displaystyle G_{0}$ $\displaystyle=$
$\displaystyle-4d(d+m^{2})\alpha_{+}\alpha_{-}\left[(\alpha_{+}-2m)(d+m\alpha_{+}+m^{2})+ik\nu(\alpha_{+}+2m)\right],$

leading to the horizon characteristics

$e^{2\gamma_{H}}=-\left(\frac{k\nu
m}{d\alpha_{+}}\right)^{2},\quad\Omega_{H}=\omega_{H}^{-1}=\frac{k\nu
m}{2(d+m^{2})(d+m\alpha_{+}+m^{2})}.$ (3.9)

The surface gravity $\kappa_{H}=\sqrt{|e^{-2\gamma_{H}}|}\,|\Omega_{H}|$ is

$\kappa_{H}=\frac{d\alpha_{+}}{2(d+m^{2})(d+m\alpha_{+}+m^{2})}=\frac{\sigma\delta(\sigma+\delta)}{2m(1+\sigma\delta)(\sigma+1)(\delta+1)},$
(3.10)

from which one can derive the horizon area ${\cal A}_{H}=2\pi
L_{H}/\kappa_{H}$ (where $L_{H}=\alpha_{+}-\alpha_{-}=2m\delta$ is the length
of the horizon rod).

##  4 The string

The two co-rotating black holes, corresponding to the rods ($\rho=0$,
$\alpha_{-}<z<\alpha_{+}$) and ($\rho=0$, $-\alpha_{+}<z<-\alpha_{-}$), are
connected by a finite length spinning cosmic string corresponding to the rod
($\rho=0$, $-\alpha_{-}<z<\alpha_{-}$), On this string,

$R_{\pm}=\alpha_{+}\pm z,\quad r_{\pm}=\alpha_{-}\pm z,$ (4.1)

leading to

$\displaystyle A_{S}$ $\displaystyle=$
$\displaystyle-2\alpha_{+}\alpha_{-}[d^{2}(\alpha_{+}+\alpha_{-})^{2}-m^{4}(\alpha_{+}-\alpha_{-})^{2}],$
$\displaystyle+2[d(\alpha_{+}+\alpha_{-})-m^{2}(\alpha_{+}-\alpha_{-})]^{2}z^{2}-32ik\nu
md^{2}z$ $\displaystyle B_{S}$ $\displaystyle=$
$\displaystyle-8dm\alpha_{+}\alpha_{-}[d(\alpha_{+}+\alpha_{-})-m^{2}(\alpha_{+}-\alpha_{-})]$
(4.2) $\displaystyle-8ik\nu
d[d(\alpha_{+}+\alpha_{-})+m^{2}(\alpha_{+}-\alpha_{-})]z.$

Evaluating these at the string center $z=0$ (where they are real), together
with ${\rm Im}\,G$,

${\rm Im}\,G_{0}=-4k\nu
d\alpha_{+}\alpha_{-}[d(\alpha_{+}+\alpha_{-})-m^{2}(\alpha_{+}-\alpha_{-})],$

we obtain the string characteristics

$\displaystyle e^{2\gamma_{S}}$ $\displaystyle=$
$\displaystyle\left(\frac{[d(\alpha_{+}+\alpha_{-})-m^{2}(\alpha_{+}-\alpha_{-})]^{2}}{4d^{2}\alpha_{+}\alpha_{-}}\right)^{2}\;=\;\left(\frac{(\sigma^{2}-1)^{2}}{\sigma^{2}(\sigma^{2}-\delta^{2})}\right)^{2},$
(4.3) $\displaystyle\omega_{S}$ $\displaystyle=$ $\displaystyle-8k\nu
d\frac{d(\alpha_{+}+\alpha_{-}+4m)+m^{2}(\alpha_{+}-\alpha_{-})}{[d(\alpha_{+}+\alpha_{-})-m^{2}(\alpha_{+}-\alpha_{-})]^{2}}\;=\;-\frac{4k\nu\sigma}{m(\sigma-1)^{2}}.$

From these we can derive the string area and surface gravity,

${\cal A}_{S}=4\pi\alpha_{-}|\omega_{S}|e^{\gamma_{S}}=\frac{16\pi
k\nu(\sigma+1)^{2}}{\sigma(\sigma+\delta)},\quad\kappa_{S}=\frac{4\pi
m(\sigma-\delta)}{{\cal A}_{S}}.$ (4.4)

These characteristics are simply related to the string tension per unit
length, which is $(1-e^{-\gamma_{S}})/4$, and the string spin,
$-\omega_{S}/4$. As discussed in [10], this spin measures the gravimagnetic
flow along the Misner string connecting the two horizons carrying the
effective NUT charges $\mp\omega_{S}/4$ (generically different from the “bare”
NUT charges $\pm\nu$). And the string tension measures the force [11] which is
necessary to balance the gravitational forces between the two sources (black
holes), attractive between the two equal masses and repulsive between the
opposite NUT charges, a negative string tension ($e^{\gamma_{S}}<1$)
corresponding to a net attractive force, and a positive string tension
($e^{\gamma_{S}}>1$) corresponding to a net repulsive force.

For a large separation between the two black holes ($k\gg(m,\,\nu)$,
$\sigma\gg 1$), we find $\sigma\simeq k/m$, $\delta\simeq\mu/m$
($\mu^{2}=m^{2}+\nu^{2}$). The string length $2\alpha_{-}$ is of the order of
$2k$, while the string tension and spin go to

$\frac{1-e^{-\gamma_{S}}}{4}\simeq-\frac{m^{2}-\nu^{2}}{(2k)^{2}},\qquad-\frac{\omega_{S}}{4}\simeq\nu.$
(4.5)

In this quasi-linear newtonian regime, the effective and bare NUT charges
coincide, while the net force is simply the superposition of the inverse
square law gravitational and gravimagnetic forces

On the other hand, when for fixed $m$ and $\nu$ the parameter $k$ is
progressively decreased to its lowest possible limit $k_{+}(m,\nu)$,
corresponding to $\alpha_{-}\to 0$ ($\delta\to\sigma$), $e^{\gamma_{S}}$
diverges while the string length goes to zero, signalling a singularity at
$\rho=z=0$, unless $\sigma$ simultaneously goes to $1$, i.e. $k$ goes to $m$,
which from the condition $k>k_{+}$ is possible only for $\nu=0$. Otherwise,
whatever the relative values of the mass and NUT parameters $m$ and $\nu$, the
regime becomes highly non-linear when the limit $k\to k_{+}$ is approached,
leading to an increasing repulsion which will prevent the two black holes from
coalescing, no matter what external force is exerted.

In the absence of an external force, the stationary double black hole system
is balanced provided the string parameters are constrained by
$e^{\gamma_{S}}=1$, which is achieved by

$\delta^{2}=2-\frac{1}{\sigma^{2}}.$ (4.6)

The regular balanced configurations depend therefore only on two parameters,
which can be chosen for instance as the total mass $M=2m$ and the
dimensionless parameter $\sigma\geq 1$. Recalling that the total angular
momentum is $J\equiv Ma=2k\nu$, for these balanced configurations the ratio

$\frac{|a|}{M}=\frac{k|\nu|}{2m^{2}}=\frac{1}{2}\left(\sigma-\sigma^{-1}\right)$
(4.7)

can take any real non-negative value. Using this relation and the relation

$k^{2}-\nu^{2}=m^{2}(\sigma^{2}+1-\sigma^{-2}),$ (4.8)

one derives the relation between the ratio $\nu/m$ and the parameter $\sigma$,

$\nu^{4}+(\sigma^{2}+1-\sigma^{-2})m^{2}\nu^{2}-(\sigma-\sigma^{-1})^{2}m^{4}=0.$
(4.9)

For $\sigma\geq 1$, $\nu/m$ is bounded above by $1$. More precisely, when
$\sigma$ varies from $1$ to infinity, $\nu/m$ varies from $0$ to $1$.

For fixed mass $M=2m$ and large $\sigma$, one finds

$\nu\simeq m\sigma(1-2\sigma^{-2}),\qquad k\simeq m\sigma(1+\sigma^{-2}).$
(4.10)

The string rod length increases as $L_{S}\simeq M\sigma$, while the horizon
rod lengths go to the finite value $L_{H}\simeq M\sqrt{2}$. The horizon area
and surface gravity go to finite limits, and the horizon angular velocity goes
to zero,

${\cal A}_{H}\simeq
4\pi\,\frac{1+\sqrt{2}}{\sqrt{2}}\,M^{2},\quad\kappa_{H}\simeq\frac{1}{1+\sqrt{2}}\,M^{-1},\quad\Omega_{H}\simeq\frac{1}{\sqrt{2}(1+\sqrt{2})}\,(M\sigma)^{-1},$
(4.11)

in spite of the fact that the total angular momentum diverges as (4.7). Of
course, this large angular momentum is due to the gravimagnetic dipole moment
$L_{S}\simeq 2k\nu$.

In the opposite limit $\sigma\to 1$, one can take $\nu/m\simeq 2a/M$ as
independent dimensionless parameter, in terms of which

$\sigma\simeq 1+\frac{\nu}{2m}+\frac{5\nu^{2}}{8m^{2}},\qquad k\simeq m+\nu.$
(4.12)

The horizon rod lengths are now $L_{H}\simeq M$, while the string rod
contracts to a length $L_{S}\simeq 2a^{2}/M$. The values of the total horizon
area, surface gravity, and horizon angular velocity

$2{\cal A}_{H}\simeq 16\pi
M^{2},\quad\kappa_{H}\simeq\frac{1}{4M},\quad\Omega_{H}\simeq\frac{a}{4M^{2}}$
(4.13)

coincide in lowest order with those of a slowly rotating Kerr black hole with
the same parameters $M$ and $a$.

##  5 Komar observables and first law

Now we compute the horizon and string Komar masses and angular momenta. The
Tomimatsu formulas [12] for these are

$M_{n}=\frac{\omega_{n}}{4}\,[\chi(z_{n+})-\chi(z_{n-})],\quad
J_{n}=\frac{\omega_{n}}{2}\left(M_{n}-\frac{z_{n+}-z_{n-}}{2}\right),$ (5.1)

where $\chi=-{\rm Im}\cal E$111The Ernst potential used in [8] and in the
present paper is the complex conjugate of that used in our previous papers.,
and $z_{n\pm}$ are the upper and lower ends of the rod, $\pm\alpha_{\pm}$ and
$\pm\alpha_{\mp}$ for the upper and lower horizon rods, and $\alpha_{-}$ and
$-\alpha_{-}$ for the string rod. At these ends,

$\displaystyle A(\pm\alpha_{+})$ $\displaystyle=$
$\displaystyle-16d^{2}(d+m^{2})\alpha_{+}\alpha_{-},$ $\displaystyle
B(\pm\alpha_{+})$ $\displaystyle=$
$\displaystyle-16d^{2}\alpha_{+}\alpha_{-}(m\alpha_{+}\pm ik\nu),$
$\displaystyle A(\pm\alpha_{-})$ $\displaystyle=$
$\displaystyle-8d\alpha_{-}\left\\{(d+m^{2})[(d-m^{2})\alpha_{+}+(d+m^{2})\alpha_{-}]\pm
ik\nu 4md\right\\},$ $\displaystyle B(\pm\alpha_{-})$ $\displaystyle=$
$\displaystyle-8d\alpha_{-}\left\\{m\alpha_{+}[(d-m^{2})\alpha_{+}+(d+m^{2})\alpha_{-}]\right.$
(5.2) $\displaystyle\left.\pm
ik\nu[(d+m^{2})\alpha_{+}+(d-m^{2})\alpha_{-}]\right\\}.$

Using these and the identities (2), we obtain,

$\displaystyle\chi(\pm\alpha_{+})$ $\displaystyle=$
$\displaystyle\pm\frac{k\nu}{m^{2}(\sigma+1)(\delta+1)},$
$\displaystyle\chi(\pm\alpha_{-})$ $\displaystyle=$
$\displaystyle\pm\frac{k\nu(\sigma-1)}{m^{2}(\sigma+1)^{2}(\delta+1)},$ (5.3)

Together with the rod angular velocities evaluated in the previous section,
these lead to the horizon and string Komar masses and angular momenta

$\displaystyle M_{H_{\pm}}$ $\displaystyle=$ $\displaystyle
m\,\frac{\sigma\delta+1}{\sigma+1},\quad
J_{H_{\pm}}=-k\nu\,\frac{\sigma\delta+1}{\sigma^{2}-1},$ $\displaystyle M_{S}$
$\displaystyle=$ $\displaystyle-2m\,\frac{\sigma(\delta-1)}{\sigma+1},\quad
J_{S}=2k\nu\,\frac{\sigma(\sigma+\delta)}{\sigma^{2}-1}.$ (5.4)

It is easy to check that these add up to the total mass and angular momentum,

$2M_{H}+M_{S}=2m,\quad 2J_{H}+J_{S}=2k\nu.$ (5.5)

For a large separation between the two black holes ($k\gg(m,\,\nu)$,
$\sigma\gg 1$), the Komar angular momentum is mainly carried by the string,
while the Komar masses of the two horizons and of the string are of the same
order but opposite signs, $M_{H_{\pm}}\simeq M\delta/2$ and
$M_{S}\simeq-M(\delta-1)$ (with $\delta\simeq\sqrt{2}$ for a balanced
configuration). Conversely, when the configuration is balanced and $\nu\to 0$,
the Komar mass is mainly carried by the black holes, the respective horizon
and string Komar angular momenta being of the order of $M^{2}$ but nearly
compensating each other.

By construction, each rod satisfies its own Smarr relation (this is a simple
consequence of the Tomimatsu formulas (5.1)), so that the global Smarr
relation [13]

$M=2\left(2\Omega_{H}J_{H}+\frac{\kappa_{H}}{4\pi}{\cal
A}_{H}\right)+2\Omega_{S}J_{S}+\frac{\kappa_{S}}{4\pi}{\cal A}_{S}$ (5.6)

is trivially satisfied. Note that all the Killing horizons – the two black
hole horizons and the Misner string – contribute equally to this global Smarr
law.

One may surmise that a similar global first law of mechanics

$dM=\sum_{i}\left(\Omega_{H_{i}}\,dJ_{H_{i}}+\frac{\kappa_{H_{i}}}{8\pi}\,d{\cal
A}_{H_{i}}\right)+\Omega_{S}\,dJ_{S}+\frac{\kappa_{S}}{8\pi}\,d{\cal A}_{S}$
(5.7)

($i=1,2$) should hold for all systems of two black holes connected by a Misner
string222This will be proved in [14].. In the present case, careful accounting
shows that the differential first law (5.7) indeed holds for independent
variations of the two dimensionless parameters $\sigma$ and $\delta$ (it is
trivially satisfied for variations of the scale $m$), whether the system is
balanced or unbalanced. It should be stressed that, contrary to the Smarr
relations, the first law is not satisfied independently by the various
interacting components – horizons and string – but only by the composite
system.

The form (5.7) of the first law for two NUTty black holes should be compared
with that recently proposed [15]-[18] for collinear systems of NUTless black
holes connected by cosmic strings, which involves the string tensions. For
instance, the first law for a system of two aligned Kerr black holes reads
[16]

$M=\sum_{i}\left(\Omega_{H_{i}}\,dJ_{H_{i}}+\frac{\kappa_{H_{i}}}{8\pi}\,d{\cal
A}_{H_{i}}\right)+\lambda_{S}\,d\mu_{S},$ (5.8)

with $\mu_{S}$ the tension of the string connecting the two black holes and
$\lambda_{S}$ the conjugate string thermodynamic length [18]. While in the
present case the string tension does not explicitly occur in the first law
(5.7), it is related to the Misner string characteristics by

$\mu_{S}=\frac{1-e^{-\gamma_{S}}}{4},\quad
e^{-\gamma_{S}}=-\frac{\kappa_{S}}{\Omega_{S}}.$ (5.9)

In the NUTless limit $\delta\to 1$ ($\nu\to 0$), the Misner string intensive
quantities $M_{S}$, $J_{S}$ and ${\cal A}_{S}$ all vanish, while the extensive
quantities $\Omega_{S}$ and $\kappa_{S}$ both diverge, their ratio being
related to the string tension by the above relation. We find that in this
limit the string contribution to (5.7) does not vanish, but goes to the limit

$\Omega_{S}\,dJ_{S}+\frac{\kappa_{S}}{8\pi}\,d{\cal
A}_{S}=\frac{m\,d\sigma}{\sigma(\sigma+1)}=\lambda_{S}\,d\mu_{S},$ (5.10)

where the thermodynamic length of the string between the two Schwarzschild
black holes is

$\lambda_{S}=\frac{2m(\sigma+1)(\sigma-1)^{2}}{\sigma^{2}}=2\alpha_{-}\,e^{\gamma_{S}},$
(5.11)

in accordance with the result of [15] (where the distance $R$ between the
centers of the two black-hole rods is related to our parameters by
$R=2m\sigma)$.

##  6 Discussion

We have discussed the properties of an exact asymptotically flat stationary
solution to vacuum gravity, describing a system of two non-extreme black holes
with equal masses and opposite NUT charges, connected by a Misner string with
tension. The solution is free from a ring singularity in a certain parameter
domain. For large separations, the force measured by the string tension is
attractive or repulsive, depending on the relative values of the masses and
NUT charges. For small separations, the force is always repulsive, so that the
system cannot collapse to a single black hole. For given values of the black
hole masses and NUT charges, the distance between the two black holes (the
string length) can always be adjusted so that the string tension vanishes and
the system is balanced. The existence of a Misner string, on which the
(usually enforced) axis condition $\omega(0,z)=0$ is relaxed, is necessary for
this balance to be possible.

This condition being relaxed, the usual black hole uniqueness theorems no
longer apply. The observable charges of the dihole solution (total mass and
angular momentum) take the same values as those of the Kerr solution, while
the overall topology and geometry are different. For small black hole
separations (short dipoles), the angular-momentum-to-mass ratio is small, and
the global horizon characteristics (horizon angular velocity, temperature and
entropy) are roughly the same as those of a slowly rotating Kerr black hole
with the same parameters $M$ and $a=J/M$. On the other hand, for large black
hole separations (long dipoles), the ratio $|a|/M$ becomes arbitrarily large,
corresponding to regular overspinning configurations.

Finally, we have evaluated the horizon and string Komar masses and angular
momenta and other mechanical observables, and showed that the dipole system
satisfies a generalized first law of black hole mechanics (5.7), to which all
the Killing horizons – the two black hole horizons and the Misner string –
contribute equally.

## Acknowledgments

I warmly thank Dmitry Gal’tsov for a critical reading of the manuscript and
useful suggestions.

## References

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# Semi-leptonic form factors of $\Xi_{c}\to\Xi$ in QCD sum rules

Zhen-Xing Zhao1 111Email<EMAIL_ADDRESS>1 School of Physical Science and
Technology,
Inner Mongolia University, Hohhot 010021, China

###### Abstract

The $\Xi_{c}\to\Xi$ form factors are investigated in QCD sum rules in this
work. Contributions from up to dim-5 are considered. For completeness, we also
study the two-point correlation function to obtain the pole residue of $\Xi$,
and higher accuracy is achieved. For the three-point correlation function,
since stable Borel regions can not be found, about $10-20\%$ uncertainties are
introduced for the form factors. The obtained form factors are then used to
predict the decay width of $\Xi_{c}\to\Xi e^{+}\nu$, and our result is larger
than the most recent data of the Belle Collaboration, and still consistent
with the current world average and that of Lattice QCD.

##  I Introduction

Very recently, the Belle Collaboration reported the measurement of the
branching fractions of $\Xi_{c}^{0}\to\Xi^{-}\ell^{+}\nu_{\ell}$ Li:2021uhk :

$\displaystyle{\cal B}(\Xi_{c}^{0}\to\Xi^{-}e^{+}\nu_{e})=(1.72\pm 0.10\pm
0.12\pm 0.50)\%,$ $\displaystyle{\cal
B}(\Xi_{c}^{0}\to\Xi^{-}\mu^{+}\nu_{\mu})=(1.71\pm 0.17\pm 0.13\pm 0.50)\%,$
(1)

which have much smaller uncertainties than the world average $(1.8\pm 1.2)\%$
Zyla:2020zbs . A renaissance is expected in the study of weak decays of singly
charmed baryons.

The study of semi-leptonic decay $\Xi_{c}\to\Xi\ell^{+}\nu$ is of great
phenomenological significance. At the quark level, $\Xi_{c}\to\Xi\ell^{+}\nu$
is induced by the $c\to s$ transition, which provides an ideal place to
constrain the CKM matrix element $V_{cs}$. Furthermore, since the mass of
charm quark is not too large, the decay of charm quark involves significant
non-perturbative effects, which can be used to test some effective theories of
QCD, such as the heavy quark effective theory (HQET).

The $\Xi_{c}\to\Xi$ transition form factors are defined as

$\displaystyle\langle\Xi(p_{2},s_{2})|\bar{s}\gamma_{\mu}(1-\gamma_{5})c|\Xi_{c}(p_{1},s_{1})\rangle$
$\displaystyle=$
$\displaystyle\bar{u}(p_{2},s_{2})\left[\gamma_{\mu}f_{1}(q^{2})+i\sigma_{\mu\nu}\frac{q^{\nu}}{M_{1}}f_{2}(q^{2})+\frac{q_{\mu}}{M_{1}}f_{3}(q^{2})\right]u(p_{1},s_{1})$
$\displaystyle-$
$\displaystyle\bar{u}(p_{2},s_{2})\left[\gamma_{\mu}g_{1}(q^{2})+i\sigma_{\mu\nu}\frac{q^{\nu}}{M_{1}}g_{2}(q^{2})+\frac{q_{\mu}}{M_{1}}g_{3}(q^{2})\right]\gamma_{5}u(p_{1},s_{1}),$
(2)

with $M_{1}=m_{\Xi_{c}}$. In HQET, the form factors $f_{i}$ and $g_{i}$ reduce
to two functions ${\cal F}_{1,2}$ Manohar:2000dt

$\displaystyle f_{1}=g_{1}={\cal F}_{1}+\frac{m_{\Xi}}{m_{\Xi_{c}}}{\cal
F}_{2},$ $\displaystyle f_{2}=f_{3}=g_{2}=g_{3}={\cal F}_{2}.$ (3)

In addition, the results of semi-leptonic form factors are important inputs
for studying the many-body non-leptonic decays using the method of Final-State
Interaction (FSI), as can be seen in Yu:2017zst ; Han:2021azw ; Han:2021gkl .

Some works based on various quark models have been preformed Zhao:2018zcb ;
Faustov:2019ddj ; Geng:2020gjh , while they are highly model dependent. With
SU(3) flavor symmetry, one can also give some good predictions by fitting
experimental data, as can be seen in Geng:2017mxn ; Geng:2018plk ;
Geng:2019bfz . However, SU(3) symmetry breaking is not considered, which
sometimes can bring in uncertainties up to 30%. In Azizi:2011mw , light-con
sum rules (LCSR) are used to study the $\Xi_{c}\to\Xi$ decay, however, the
decay width seems too large compared with other works. Recently, Lattice QCD
calculation of the form factors of $\Xi_{c}\to\Xi$ has been performed
Zhang:2021oja . Compared with the experimental data from Belle, slightly
larger central values of the branching ratios are obtained

$\displaystyle{\cal B}(\Xi_{c}^{0}\to\Xi^{-}e^{+}\nu_{e})=(2.38\pm 0.30\pm
0.33)\%,$ $\displaystyle{\cal
B}(\Xi_{c}^{0}\to\Xi^{-}\mu^{+}\nu_{\mu})=(2.29\pm 0.29\pm 0.31)\%.$ (4)

In this work, we will investigate the form factors of $\Xi_{c}\to\Xi$ in QCD
sum rules (QCDSR). QCDSR is a QCD-based approach to deal with hadronic
parameters. It reveals a direct connection between hadron phenomenology and
QCD vacuum structure via a few universal parameters such as quark condensates
and gluon condensates. The three-point QCDSR has been widely used in studying
the transition form factors. In Shi:2019hbf , we derived the form factors of
doubly heavy baryons to singly heavy baryons using QCDSR for the first time,
and our approach was tested in Zhao:2020mod . In this work, the validity of
the three-point sum rules will be further tested. It is worth noting that HQET
sum rules can be reproduced by taking the heavy quark limit, as can be seen in
Shuryak:1981fza ; Zhao:2020wbw ; MarquesdeCarvalho:1999bqs .

The rest of this paper is arranged as follows. In Sec. II, we will investigate
the two-point correlation function of $\Xi$ to obain its pole residue for
completeness and consistence. 222The pole residue of $\Xi_{c}$ can be found in
our previous work Zhao:2020mod . In Sec. III, we will describe the main idea
of extracting the form factors by the three-point correlation function.
Numerical results of the form factors and their phenomenological applications
will be shown in Sec. IV, and our results will also be compared with other
theoretical predictions and the experimental data. We conclude this paper in
the last section.

##  II Two-point correlation function and pole residue

To access the $\Xi_{c}\to\Xi$ form factors, the pole residues and the
continuum threshold parameters of the initial and final baryons are
indispensable inputs or important references. Therefore, for completeness, we
investigate the two-point correlation functions in this section. Considering
we have performed the analysis of the two-point correlation function of
$\Xi_{c}$ in our previous work Zhao:2020mod , so in this work, we will
concentrate on the two-point correlation function of $\Xi$.

Sum rules start from the definitions of the interpolating currents for
involved hadrons. The following currents are respectively adopted for
$\Xi_{c}$ and $\Xi$ Colangelo:2000dp

$\displaystyle J_{\Xi_{Q}}$
$\displaystyle=\epsilon_{abc}(q_{a}^{T}C\gamma_{5}s_{b})Q_{c},$ $\displaystyle
J_{\Xi}$
$\displaystyle=\epsilon_{abc}(s_{a}^{T}C\gamma^{\mu}s_{b})\gamma_{\mu}\gamma_{5}q_{c},$
(5)

where $q=u/d$, $Q=c$, and $a,b,c$ are the color indices and $C$ is the charge
conjugate matrix. It can be seen from Eq. (5) that: for the antitriplet
$\Xi_{Q}$, the spin and parity for the system of $q$ and $s$ quarks are
$0^{+}$, while those for the whole $\Xi_{c}$ are $1/2^{+}$; for the octet
$\Xi$, the spin and parity for the system of two $s$ quarks are $1^{+}$, while
those for the whole $\Xi$ are $1/2^{+}$.

To arrive at the pole residue, the following two-point correlation function is
defined

$\Pi(p)=i\int d^{4}x\ e^{ip\cdot x}\langle 0|T\\{J(x)\bar{J}(0)\\}|0\rangle.$
(6)

On the hadronic side, after the complete set of hadronic states are inserted,
the correlation function in Eq. (6) is written into

$\Pi^{{\rm
had}}(p)=\lambda_{+}^{2}\frac{\not{p}+M_{+}}{M_{+}^{2}-p^{2}}+\lambda_{-}^{2}\frac{\not{p}-M_{-}}{M_{-}^{2}-p^{2}}+\cdots,$
(7)

where the contribution from the negative parity baryon is also considered, and
$M_{\pm}$ ($\lambda_{\pm}$) stand for the masses (the pole residues) of the
positive- and negative-parity baryons.

On the QCD side, the correlation function in Eq. (6) can be investigated using
OPE technique. In this work, we only consider the contributions from the
dimension-0,3,5 operators and the corresponding diagrams can be found in Fig.
1. These results can be formally written as

$\Pi^{{\rm QCD}}(p)=A(p^{2})\not{p}+B(p^{2}).$ (8)

For practical purpose, the coefficients $A$ and $B$ can be further written
into the following dispersion integrals

$A(p^{2})=\int ds\ \frac{\rho^{A}(s)}{s-p^{2}},\quad B(p^{2})=\int ds\
\frac{\rho^{B}(s)}{s-p^{2}}.$ (9)

Figure 1: The two-point correlation function for $\Xi$, where the numbers
1,2,3 respectively denote one $s$ quark, the other $s$ quark and the $q$
quark. Here we have only shown the perturbative diagram. In this work, we have
also considered the contributions from the quark condensate and quark-gluon
condensate. For the former, there are 3 diagrams: $\langle 1\rangle,\langle
2\rangle,\langle 3\rangle$, and for the latter, 6 diagrams: $\langle
1,2\rangle,\langle 1,3\rangle,\langle 2,1\rangle,\langle 2,3\rangle,\langle
3,1\rangle,\langle 3,2\rangle$. Here $\langle 1\rangle$ stands for the quark
condensate diagram with line 1 disconnected, and $\langle 1,2\rangle$ stands
for the quark-gluon condensate diagram with line 1 disconnected, and line 2
emitting a gluon, and so forth.

Taking advantage of quark-hadron duality and performing the Borel transform,
one can arrive at the sum rule for $1/2^{+}$ baryon

$\displaystyle(M_{+}+M_{-})\lambda_{+}^{2}\exp(-M_{+}^{2}/T_{+}^{2})$
$\displaystyle=\int^{s_{+}}ds\ (M_{-}\rho^{A}+\rho^{B})\exp(-s/T_{+}^{2}),$
(10)

where $T_{+}^{2}$ and $s_{+}$ are the Borel parameter and continuum threshold
parameter, respectively. Differentiating Eq. (10) with respect to
$-1/T_{+}^{2}$, one can obtain the mass for $1/2^{+}$ baryon

$M_{+}^{2}=\frac{\int^{s_{+}}ds\ (M_{-}\rho^{A}+\rho^{B})\ s\
\exp(-s/T_{+}^{2})}{\int^{s_{+}}ds\ (M_{-}\rho^{A}+\rho^{B})\
\exp(-s/T_{+}^{2})}.$ (11)

Eq. (11) can be used to test the sum rule in Eq. (10) given that the mass has
been known in the experiment.

##  III Three-point correlation functions and form factors

In practice, the following simpler parametrization is adopted for
$\Xi_{c}\to\Xi$ transition form factors:

$\displaystyle\langle{\cal
B}_{2}(p_{2},s_{2})|\bar{s}\gamma_{\mu}(1-\gamma_{5})c|{\cal
B}_{1}(p_{1},s_{1})\rangle$ $\displaystyle=$
$\displaystyle\bar{u}(p_{2},s_{2})\left[\frac{p_{1\mu}}{M_{1}}F_{1}(q^{2})+\frac{p_{2\mu}}{M_{2}}F_{2}(q^{2})+\gamma_{\mu}F_{3}(q^{2})\right]u(p_{1},s_{1})$
$\displaystyle-$
$\displaystyle\bar{u}(p_{2},s_{2})\left[\frac{p_{1\mu}}{M_{1}}G_{1}(q^{2})+\frac{p_{2\mu}}{M_{2}}G_{2}(q^{2})+\gamma_{\mu}G_{3}(q^{2})\right]\gamma_{5}u(p_{1},s_{1}),$
(12)

where ${\cal B}_{1,2}$ denote $\Xi{}_{c}$ and $\Xi$, respectively. The form
factors $F_{i}$ and $G_{i}$ are related to $f_{i}$ and $g_{i}$ defined in (2)
through

$\displaystyle F_{1}=f_{2}+f_{3},\quad
F_{2}=\frac{M_{2}}{M_{1}}(f_{2}-f_{3}),\quad
F_{3}=f_{1}-\frac{M_{1}+M_{2}}{M_{1}}f_{2};$ $\displaystyle
G_{1}=g_{2}+g_{3},\quad G_{2}=\frac{M_{2}}{M_{1}}(g_{2}-g_{3}),\quad
G_{3}=g_{1}+\frac{M_{1}-M_{2}}{M_{1}}g_{2}.$ (13)

The three-point correlation functions for $\Xi_{c}\to\Xi$ are defined as

$\Pi_{\mu}^{V,A}(p_{1}^{2},p_{2}^{2},q^{2})=i^{2}\int d^{4}xd^{4}y\
e^{-ip_{1}\cdot x+ip_{2}\cdot y}\langle
0|T\\{J_{\Xi}(y)(V_{\mu},A_{\mu})(0)\bar{J}_{\Xi_{c}}(x)\\}|0\rangle,$ (14)

where $V_{\mu}(A_{\mu})=\bar{s}\gamma_{\mu}(\gamma_{\mu}\gamma_{5})c$ is the
vector (axial-vector) current for $c\to s$ decay. The correlation functions
are then calculated at hadronic level and QCD level. At the hadronic level,
after inserting the complete set of initial and final states and considering
again the contributions from the negative-parity baryons, the vector current
correlation function can be written as

$\displaystyle\Pi_{\mu}^{V,{\rm had}}(p_{1}^{2},p_{2}^{2},q^{2})$
$\displaystyle=$
$\displaystyle\lambda_{f}^{+}\lambda_{i}^{+}\frac{(\not{p}_{2}+M_{2}^{+})(\frac{p_{1\mu}}{M_{1}^{+}}F_{1}^{++}+\frac{p_{2\mu}}{M_{2}^{+}}F_{2}^{++}+\gamma_{\mu}F_{3}^{++})(\not{p}_{1}+M_{1}^{+})}{(p_{2}^{2}-M_{2}^{+2})(p_{1}^{2}-M_{1}^{+2})}$
(15) $\displaystyle+$
$\displaystyle\lambda_{f}^{+}\lambda_{i}^{-}\frac{(\not{p}_{2}+M_{2}^{+})(\frac{p_{1\mu}}{M_{1}^{-}}F_{1}^{+-}+\frac{p_{2\mu}}{M_{2}^{+}}F_{2}^{+-}+\gamma_{\mu}F_{3}^{+-})(\not{p}_{1}-M_{1}^{-})}{(p_{2}^{2}-M_{2}^{+2})(p_{1}^{2}-M_{1}^{-2})}$
$\displaystyle+$
$\displaystyle\lambda_{f}^{-}\lambda_{i}^{+}\frac{(\not{p}_{2}-M_{2}^{-})(\frac{p_{1\mu}}{M_{1}^{+}}F_{1}^{-+}+\frac{p_{2\mu}}{M_{2}^{-}}F_{2}^{-+}+\gamma_{\mu}F_{3}^{-+})(\not{p}_{1}+M_{1}^{+})}{(p_{2}^{2}-M_{2}^{-2})(p_{1}^{2}-M_{1}^{+2})}$
$\displaystyle+$
$\displaystyle\lambda_{f}^{-}\lambda_{i}^{-}\frac{(\not{p}_{2}-M_{2}^{-})(\frac{p_{1\mu}}{M_{1}^{-}}F_{1}^{--}+\frac{p_{2\mu}}{M_{2}^{-}}F_{2}^{--}+\gamma_{\mu}F_{3}^{--})(\not{p}_{1}-M_{1}^{-})}{(p_{2}^{2}-M_{2}^{-2})(p_{1}^{2}-M_{1}^{-2})}$
$\displaystyle+$ $\displaystyle\cdots.$

In Eq. (15), $M_{1(2)}^{+(-)}$ denotes the mass of initial (final) positive
(negative) parity baryon, and $F_{1}^{+-}$ is the form factor $F_{1}$ with the
positive-parity final state and the negative-parity initial state, and so
forth. To arrive at Eq. (15), we have adopted the following definitions of
pole residues for positive- and negative-parity baryons

$\displaystyle\langle 0|J_{+}|{\cal B}_{+}(p,s)\rangle$
$\displaystyle=\lambda_{+}u(p,s),$ $\displaystyle\langle 0|J_{+}|{\cal
B}_{-}(p,s)\rangle$ $\displaystyle=(i\gamma_{5})\lambda_{-}u(p,s),$ (16)

and the following conventions for the 12 form factors $F_{i}^{\pm\pm}$

$\displaystyle\langle{\cal B}_{f}^{+}(p_{2},s_{2})|V_{\mu}|{\cal
B}_{i}^{+}(p_{1},s_{1})\rangle$ $\displaystyle=\bar{u}_{{\cal
B}_{f}^{+}}(p_{2},s_{2})[\frac{p_{1\mu}}{M_{1}^{+}}F_{1}^{++}+\frac{p_{2\mu}}{M_{2}^{+}}F_{2}^{++}+\gamma_{\mu}F_{3}^{++}]u_{{\cal
B}_{i}^{+}}(p_{1},s_{1}),$ $\displaystyle\langle{\cal
B}_{f}^{+}(p_{2},s_{2})|V_{\mu}|{\cal B}_{i}^{-}(p_{1},s_{1})\rangle$
$\displaystyle=\bar{u}_{{\cal
B}_{f}^{+}}(p_{2},s_{2})[\frac{p_{1\mu}}{M_{1}^{-}}F_{1}^{+-}+\frac{p_{2\mu}}{M_{2}^{+}}F_{2}^{+-}+\gamma_{\mu}F_{3}^{+-}](i\gamma_{5})u_{{\cal
B}_{i}^{-}}(p_{1},s_{1}),$ $\displaystyle\langle{\cal
B}_{f}^{-}(p_{2},s_{2})|V_{\mu}|{\cal B}_{i}^{+}(p_{1},s_{1})\rangle$
$\displaystyle=\bar{u}_{{\cal
B}_{f}^{-}}(p_{2},s_{2})(i\gamma_{5})[\frac{p_{1\mu}}{M_{1}^{+}}F_{1}^{-+}+\frac{p_{2\mu}}{M_{2}^{-}}F_{2}^{-+}+\gamma_{\mu}F_{3}^{-+}]u_{{\cal
B}_{i}^{+}}(p_{1},s_{1}),$ $\displaystyle\langle{\cal
B}_{f}^{-}(p_{2},s_{2})|V_{\mu}|{\cal B}_{i}^{-}(p_{1},s_{1})\rangle$
$\displaystyle=\bar{u}_{{\cal
B}_{f}^{-}}(p_{2},s_{2})(i\gamma_{5})[\frac{p_{1\mu}}{M_{1}^{-}}F_{1}^{--}+\frac{p_{2\mu}}{M_{2}^{-}}F_{2}^{--}+\gamma_{\mu}F_{3}^{--}](i\gamma_{5})u_{{\cal
B}_{i}^{-}}(p_{1},s_{1}).$ (17)

At the QCD level, diagrams up to dimension-5 are considered, as can be seen in
Fig. 2. The correlation function is then expressed as a double dispersion
integral

$\Pi_{\mu}^{V,{\rm
QCD}}(p_{1}^{2},p_{2}^{2},q^{2})=\int^{\infty}ds_{1}\int^{\infty}ds_{2}\frac{\rho_{\mu}^{V,{\rm
QCD}}(s_{1},s_{2},q^{2})}{(s_{1}-p_{1}^{2})(s_{2}-p_{2}^{2})},$ (18)

where the spectral function $\rho_{\mu}^{V,{\rm QCD}}(s_{1},s_{2},q^{2})$ can
be obtained by applying Cutkosky cutting rules to the three-point correlation
function. According to the quark-hadron duality, the four pole terms in Eq.
(15) should be equal to

$\int^{s_{1}^{0}}ds_{1}\int^{s_{2}^{0}}ds_{2}\frac{\rho_{\mu}^{V,{\rm
QCD}}(s_{1},s_{2},q^{2})}{(s_{1}-p_{1}^{2})(s_{2}-p_{2}^{2})},$ (19)

where $s_{1(2)}^{0}$ is the continuum threshold parameter of the initial
(final) baryon. Specifically, one can arrive at 12 equations by comparing the
coefficients of different Dirac structures on both sides of the equality.
Solving these equations, and then applying the Borel transform, one can obtain
the following expressions for $F_{i}^{++}$:

$\displaystyle\lambda_{i}^{+}\lambda_{f}^{+}(F_{1}^{++}/M_{1}^{+})\exp\left(-\frac{M_{1}^{+2}}{T_{1}^{2}}-\frac{M_{2}^{+2}}{T_{2}^{2}}\right)$
$\displaystyle=\frac{\\{M_{1}^{-}M_{2}^{-},M_{2}^{-},M_{1}^{-},1\\}.\\{{\cal
B}A_{1},{\cal B}A_{2},{\cal B}A_{3},{\cal
B}A_{4}\\}}{(M_{1}^{+}+M_{1}^{-})(M_{2}^{+}+M_{2}^{-})},$
$\displaystyle\lambda_{i}^{+}\lambda_{f}^{+}(F_{2}^{++}/M_{2}^{+})\exp\left(-\frac{M_{1}^{+2}}{T_{1}^{2}}-\frac{M_{2}^{+2}}{T_{2}^{2}}\right)$
$\displaystyle=\frac{\\{M_{1}^{-}M_{2}^{-},M_{2}^{-},M_{1}^{-},1\\}.\\{{\cal
B}A_{5},{\cal B}A_{6},{\cal B}A_{7},{\cal
B}A_{8}\\}}{(M_{1}^{+}+M_{1}^{-})(M_{2}^{+}+M_{2}^{-})},$
$\displaystyle\lambda_{i}^{+}\lambda_{f}^{+}F_{3}^{++}\exp\left(-\frac{M_{1}^{+2}}{T_{1}^{2}}-\frac{M_{2}^{+2}}{T_{2}^{2}}\right)$
$\displaystyle=\frac{\\{M_{1}^{-}M_{2}^{-},M_{2}^{-},M_{1}^{-},1\\}.\\{{\cal
B}A_{9},{\cal B}A_{10},{\cal B}A_{11},{\cal
B}A_{12}\\}}{(M_{1}^{+}+M_{1}^{-})(M_{2}^{+}+M_{2}^{-})},$ (20)

where $T_{1,2}^{2}$ are the Borel mass parameters, ${\cal B}A_{i}\equiv{\cal
B}_{T_{1}^{2},T_{2}^{2}}A_{i}$ are doubly Borel transformed coefficients, and
$A_{i}$ are coefficients of

$\displaystyle(e_{1,2,3,4})_{\mu}$ $\displaystyle=$
$\displaystyle\\{\not{p}_{2},1\\}\times\\{p_{1\mu}\\}\times\\{\not{p}_{1},1\\},$
$\displaystyle(e_{5,6,7,8})_{\mu}$ $\displaystyle=$
$\displaystyle\\{\not{p}_{2},1\\}\times\\{p_{2\mu}\\}\times\\{\not{p}_{1},1\\},$
$\displaystyle(e_{9,10,11,12})_{\mu}$ $\displaystyle=$
$\displaystyle\\{\not{p}_{2},1\\}\times\\{\gamma_{\mu}\\}\times\\{\not{p}_{1},1\\}$
(21)

in Eq. (19).

Figure 2: Three-point correlation function for $\Xi_{c}\to\Xi$, where the
numbers $1,2,3,1^{\prime}$ respectively denote the $c$ quark, spectator $s$
quark, $q$ quark and final $s$ quark. Here we have only shown the perturbative
diagram. In this work, we have also considered the contributions from the
quark condensate and quark-gluon condensate. For the former, there are 2 non-
zero diagrams: $\langle 2\rangle,\langle 3\rangle$ and for the latter, 6
diagrams: $\langle 2,1\rangle,\langle 2,1^{\prime}\rangle,\langle
2,3\rangle,\langle 3,1\rangle,\langle 3,1^{\prime}\rangle,\langle 3,2\rangle$.
The double line denotes the $c$ quark, and the cross mark is the vertex of
weak interaction.

##  IV Numerical results and phenomenological applications

In this section, two-point and three-point correlation functions are analyzed
respectively. The condensate parameters are taken at $\mu=1$ GeV as
Colangelo:2000dp : $\langle\bar{q}q\rangle=-(0.24\pm 0.01\ {\rm GeV})^{3}$,
$\langle\bar{q}q\rangle=(0.8\pm 0.2)\langle\bar{q}q\rangle$,
$\langle\bar{q}g_{s}\sigma Gq\rangle=m_{0}^{2}\langle\bar{q}q\rangle$,
$\langle\bar{s}g_{s}\sigma Gs\rangle=m_{0}^{2}\langle\bar{s}s\rangle$ and
$m_{0}^{2}=(0.8\pm 0.2)\ {\rm GeV}^{2}$. $\overline{{\rm MS}}$ masses are used
for quarks unless otherwise stated Zyla:2020zbs :

$\displaystyle m_{c}(m_{c})=1.27\pm 0.02\ {\rm GeV},\quad
m_{s}(m_{c})=0.103\pm 0.012\ {\rm GeV},$ $\displaystyle m_{s}(1\ {\rm
GeV})=0.110\pm 0.013\ {\rm GeV}.$ (22)

When arriving at the above masses, the following one-loop evolution formula
has been adopted Peskin:1995ev

$m_{q}(\mu_{0})\left(\frac{\log(\mu_{0}/\Lambda_{{\rm
QCD}}^{(n_{f})})}{\log(\mu/\Lambda_{{\rm
QCD}}^{(n_{f})})}\right)^{4/\beta_{0}},$ (23)

where $\beta_{0}=11-(2/3)n_{f}$ with $n_{f}$ the number of active flavors, and
$\Lambda_{{\rm QCD}}^{(3)}=223\ {\rm MeV}$ Buras:1998raa ; Zhao:2020mod .

The renormalization scales of two-point correlation functions of $\Xi_{c}$ and
$\Xi$ are respectively taken at $\mu=m_{c}$ and $\mu=1\ {\rm GeV}$. For the
three-point correlation function of a charmed baryon decaying into a light
baryon, we take $\mu=m_{c}$.

###  IV.1 The two-point correlation function for $\Xi$

In this work, the leading logarithm (LL) corrections will also be considered.
The Wilson coefficients of local operators derived in Sec. II should be
multiplied by an evolution factor Ioffe:1981kw

$\left(\frac{\log(\mu_{0}/\Lambda_{{\rm
QCD}}^{(n_{f})})}{\log(\mu/\Lambda_{{\rm
QCD}}^{(n_{f})})}\right)^{2\gamma_{J}-\gamma_{O}},$ (24)

where $\gamma_{J}$ and $\gamma_{O}$ are respectively the anomalous dimensions
of the interpolating current $J$ and the local operator $O$. $\mu_{0}\sim 1\
{\rm GeV}$ is the renormalization scale of the low-energy limit, while
$\mu\sim m_{c}$ is the renormalization scale that we choose for the physical
quantity of interest. For the interpolating currents given in Eq. (5), both of
the anomalous dimensions are $\gamma_{J}=2/\beta_{0}$. 333In Ioffe:1981kw , it
seems that there is a sign mistake for the anomalous dimension of
interpolating current
$\epsilon_{abc}(q_{1a}^{T}C\gamma^{\mu}q_{2b})\gamma_{5}\gamma_{\mu}q_{3c}$
when citing the result from Peskin:1979mn . In addition, our anomalous
dimension for $\epsilon_{abc}(q_{1a}^{T}C\gamma_{5}q_{2b})q_{3c}$ is also
different from that derived in Ovchinnikov:1991mu  with
$\gamma_{J}=-1/\beta_{0}$. We intend to discuss this issue in detail in a new
work. The anomalous dimension for $\bar{\psi}\psi$ is given by
$\gamma_{\bar{\psi}\psi}=4/\beta_{0}$. Similar to Ioffe:1981kw , we will only
consider the LL corrections for the perturbative and quark-condensate terms.

Figure 3: The pole residue of $\Xi$ as a function of the Borel parameters $T_{+}^{2}$. The blue and red curves correspond to optimal and suboptimal choices for $s_{+}$. The extreme points of these curves correspond to the optimal choices of $T_{+}^{2}$ on these curves. The explicit values for these $s_{+}$ and $T_{+}^{2}$ can be found in Table 1. Table 1: Our predictions of the pole residue and mass for $\Xi$. For comparison, the experimental value of $\Xi$ mass is also shown. Optimal and suboptimal $s_{+}$ and $T_{+}^{2}$ are given simultaneously. The central values for the pole residue and mass are taken at optimal $s_{+}$ and $T_{+}^{2}$, and the values at suboptimal $s_{+}$ and $T_{+}^{2}$ provide the error estimates. These results are obtained at the renormalization scale $\mu=1$ GeV. $(s_{+}/{\rm GeV}^{2},\ T_{+}^{2}/{\rm GeV}^{2})$ | $\lambda_{+}/{\rm GeV}^{3}$ | $M_{+}/{\rm GeV}$ | $M_{+}^{{\rm exp}}/{\rm GeV}$
---|---|---|---
$(1.70^{2},2.4),\ (1.75^{2},1.8)$ | $0.0364\pm 0.0015$ | $1.317\pm 0.003$ | $1.315$

The pole residue of $\Xi$ is determined using the sum rule in Eq. (10), and
the corresponding results are shown in Fig. 3 and Table 1. Some comments are
in order.

  * •

The experimental data for the masses of $\Xi^{0,-}$ are respectively $1.315$
GeV and $1.322$ GeV. In Table 1, we only list that of $\Xi^{0}$. Considering
our sum rule analysis is blind to the $u$ quark or $d$ quark within the
hadron, our prediction for the mass of $\Xi$ are in very good agreement with
the experimental result.

  * •

For the optimal parameter selection of $s_{+}=(1.70\ {\rm GeV})^{2}$ and
$T_{+}^{2}=2.4\ {\rm GeV}^{2}$, the contribution ratio of dimension-0,3,5 to
$\lambda_{+}^{2}$ is $1:1:0.3$.

###  IV.2 The three-point correlation function

In this subsection, the sum rule in Eq. (20) will be investigated. For the
continuum threshold parameters $s_{1,2}^{0}$, we just borrow them from those
of the two-point correlation functions as possible optimal choices. They are
respectively $(2.85\ {\rm GeV})^{2}$ and $(1.70\ {\rm GeV})^{2}$, which are
respectively taken from Zhao:2020mod  and from Table 1. The next step is to
find out the optimal choices for the Borel parameters $T_{1,2}^{2}$. Tune the
values of $s_{1,2}^{0}$ around the above reference values, and then scan the
$T_{1}^{2}-T_{2}^{2}$ plane. However, we can not find a stable region like the
cases for the two-point correlation functions. Considering the Borel
parameters should be close to the squared masses of the corresponding baryons
Shifman:1978bx ; Ioffe:1981kw , the typical values for $T_{1}^{2}$ and
$T_{2}^{2}$ are then roughly $6.0\ {\rm GeV}^{2}$ and $1.7\ {\rm GeV}^{2}$,
and thereby $T_{1}^{2}/T_{2}^{2}\approx 3.5$. In the following, we are
constrained to consider a line segment $T_{1}^{2}=3.5\ T_{2}^{2}$ with
$T_{2}^{2}\in[1,3]\ {\rm GeV}^{2}$ on the $T_{1}^{2}-T_{2}^{2}$ plane. The
following two criteria will be checked:

  * •

Pole dominance. For example,

$r\equiv\frac{\int^{s_{1}^{0}}ds_{1}\int^{s_{2}^{0}}ds_{2}}{\int^{\infty}ds_{1}\int^{\infty}ds_{2}}\gtrsim
0.25.$ (25)

  * •

OPE convergence. It can be achieved by requiring that dim-5/Total should not
be too large.

These two criteria can be well satisfied on a smaller line segment of
$T_{1}^{2}=3.5\ T_{2}^{2}$ with $T_{2}^{2}\in[1.4,2.2]\ {\rm GeV}^{2}$, as can
be seen in Table 2.

Table 2: The quantitative criteria of the pole dominance and OPE convergence for the line segment of $T_{1}^{2}=3.5\ T_{2}^{2}$ with $T_{2}^{2}\in[1.4,2.2]\ {\rm GeV}^{2}$. Because $F_{2}(0)$ is close to 0, the corresponding criteria are not available. | $F_{1}(0)$ | $F_{2}(0)$ | $F_{3}(0)$ | $G_{1}(0)$ | $G_{2}(0)$ | $G_{3}(0)$
---|---|---|---|---|---|---
$r$ | $>36\%$ | \- - | $>26\%$ | $>29\%$ | $>26\%$ | $>38\%$
dim-5/total | $<27\%$ | \- - | $<23\%$ | $<23\%$ | $<20\%$ | $<26\%$

Note that we also consider the LL corrections for the form factors. Since the
anomalous dimensions for the vector current and axial-vector current vanish,
the Wilson coefficients of local operators derived in Sec. III should be
multiplied by

$\left(\frac{\log(\mu_{0}/\Lambda_{{\rm
QCD}}^{(n_{f})})}{\log(\mu/\Lambda_{{\rm
QCD}}^{(n_{f})})}\right)^{\gamma_{J_{\Xi_{c}}}+\gamma_{J_{\Xi}}-\gamma_{O}}.$
(26)

One more thing should be addressed: the pole residue for $\Xi$ in Table 1,
which is evaluated at $\mu=1\ {\rm GeV}$, should be evolved to $\mu=m_{c}$.

After all the parameters are fixed, central values and uncertainties of the
$\Xi_{c}\to\Xi$ form factors $F_{i}$ and $G_{i}$ at $q^{2}=0$ are then given
in Table 3. In this table, central values of the Borel parameters
$(T_{1}^{2},T_{2}^{2})$ are taken as $(6.3,1.8)\ {\rm GeV}^{2}$. The
uncertainties are respectively from the Borel parameters, the continuum
threshold parameters $s_{1,2}^{0}$ and pole residues $\lambda_{i,f}$. When
determining the uncertainties from $s_{1,2}^{0}$, their suboptimal values are
used.

Table 3: Central values and uncertainties for the form factors $F_{i}$ and $G_{i}$ at $q^{2}=0$. $F$ | Central value | Err from $T_{1,2}^{2}$ | Err from $s_{1}^{0}$ | Err from $s_{2}^{0}$ | Err from $\lambda_{i}$ | Err from $\lambda_{f}$
---|---|---|---|---|---|---
$F_{1}(0)$ | $0.92$ | $0.09$ | $0.01$ | $0.05$ | $0.04$ | $0.04$
$F_{2}(0)$ | $0.00$ | $0.01$ | $0.01$ | $0.01$ | $0.00$ | $0.00$
$F_{3}(0)$ | $-1.43$ | $0.10$ | $0.02$ | $0.08$ | $0.07$ | $0.06$
$G_{1}(0)$ | $1.22$ | $0.08$ | $0.01$ | $0.09$ | $0.06$ | $0.05$
$G_{2}(0)$ | $-0.49$ | $0.01$ | $0.01$ | $0.03$ | $0.02$ | $0.02$
$G_{3}(0)$ | $-0.64$ | $0.05$ | $0.01$ | $0.03$ | $0.03$ | $0.03$

To access the $q^{2}$ dependence of the $\Xi_{c}\to\Xi$ form factors, we
calculate the form factors in a small interval $q^{2}\in[-0.5,0]\ {\rm
GeV}^{2}$, and fit these values to the following simplified $z$-expansion:

$f(q^{2})=\frac{a+b\ z(q^{2})}{1-q^{2}/m_{{\rm pole}}^{2}},$ (27)

where $m_{{\rm pole}}=m_{D_{s}}$,

$z(q^{2})=\frac{\sqrt{t_{+}-q^{2}}-\sqrt{t_{+}-q_{{\rm
max}}^{2}}}{\sqrt{t_{+}-q^{2}}+\sqrt{t_{+}-q_{{\rm max}}^{2}}}$ (28)

with $t_{+}=m_{{\rm pole}}^{2}$ and $q_{{\rm
max}}^{2}=(m_{\Xi_{c}}-m_{\Xi})^{2}$. The fitted results of $(a,b)$ are given
in Table 4. In Table 5, our results are compared with those in other works.
Some comments are in order.

  * •

As can be seen in Table 3 that, about 10-20% uncertainties are introduced for
the form factors of $\Xi_{c}\to\Xi$ at $q^{2}=0$. One can see that our results
are close to those of quark models and far away from those from LCSR.

  * •

For our results, the predictions from HQET in Eq. (3) can be well satisfied
except for $g_{2}$ and $g_{3}$. Presumably $g_{2}$ and $g_{3}$ may receive
sizable contributions from the higher power of $1/m_{c}$.

  * •

LL corrections for dimension-0 and dimension-3 are respectively $6\%$ and
$0\%$. For the latter, $\gamma_{J_{\Xi_{c}}}=\gamma_{J_{\Xi}}=2/9$ and
$\gamma_{\bar{\psi}\psi}=4/9$, so it happens that the evolution factor is 1.

  * •

Compared with the analysis of two-point correlation function, the situation
for the three-point correlation function is far from satisfactory. A similar
situation is encountered in our calculation of the $\Lambda_{b}\to\Lambda_{c}$
form factors in Zhao:2020mod . In this work, it turns out that the
contribution ratio of dimension-0,3,5 to $F_{i}$ and $G_{i}$ is roughly
$1:2:1$. One can see clearly that the non-perturbative effects dominate for
the charm quark decay. Contributions from higher dimension operators should be
considered, especially from those containing products of four quark fields
Ioffe:1981kw . We intend to fix this problem in a new version of this
manuscript.

Table 4: The fitted results of $(a,b)$ for the $\Xi_{c}\to\Xi$ form factors. $F$ | $(a,b)$
---|---
$F_{1}$ | $(1.21,-2.77)$
$F_{2}$ | $(-0.02,0.27)$
$F_{3}$ | $(-1.58,1.46)$
$G_{1}$ | $(1.71,-4.72)$
$G_{2}$ | $(-0.76,2.55)$
$G_{3}$ | $(-0.52,-1.14)$
Table 5: Our predictions of the form factors at $q^{2}=0$ are compared with those from the light-front quark model (LFQM) in Geng:2020gjh , LFQM in Zhao:2018zcb , relativistic quark model (RQM) in Faustov:2019ddj , and light-cone sum rules (LCSR) in Azizi:2011mw . The form factors in other works are multiplied by a minus sign to make all $f_{1}$ and $g_{1}$ have the same sign. $F$ | This work | LFQM Geng:2020gjh  | LFQM Zhao:2018zcb  | RQM Faustov:2019ddj  | LCSR Azizi:2011mw
---|---|---|---|---|---
$f_{1}(0)$ | $-0.71\pm 0.18$ | $-0.77\pm 0.02$ | $-0.567$ | $-0.590$ | $-0.194\pm 0.050$
$f_{2}(0)$ | $0.46\pm 0.06$ | $0.96\pm 0.02$ | $0.305$ | $0.441$ | $0.144\pm 0.037$
$f_{3}(0)$ | $0.46\pm 0.06$ | \- - | \- - | $-0.388$ | $0.187\pm 0.049$
$g_{1}(0)$ | $-0.71\pm 0.08$ | $-0.69\pm 0.01$ | $-0.491$ | $-0.582$ | $-0.311\pm 0.081$
$g_{2}(0)$ | $0.14\pm 0.08$ | $0.01\pm 0.00$ | $0.046$ | $-0.184$ | $0.061\pm 0.015$
$g_{3}(0)$ | $1.07\pm 0.08$ | \- - | \- - | $1.144$ | $0.126\pm 0.033$

###  IV.3 Phenomenological applications

Our form factors are then used to predict the semi-leptonic decay widths. The
formulas of polarized decay widths for ${\cal B}_{1}\to{\cal B}_{2}l\nu$ are
given by

$\displaystyle\frac{d\Gamma_{L}}{dq^{2}}$
$\displaystyle=\frac{G_{F}^{2}|V_{{\rm CKM}}|^{2}q^{2}\ p\
(1-\hat{m}_{l}^{2})^{2}}{384\pi^{3}M_{1}^{2}}\left((2+\hat{m}_{l}^{2})(|H_{-\frac{1}{2},0}|^{2}+|H_{\frac{1}{2},0}|^{2})+3\hat{m}_{l}^{2}(|H_{-\frac{1}{2},t}|^{2}+|H_{\frac{1}{2},t}|^{2})\right),$
(29) $\displaystyle\frac{d\Gamma_{T}}{dq^{2}}$
$\displaystyle=\frac{G_{F}^{2}|V_{{\rm CKM}}|^{2}q^{2}\ p\
(1-\hat{m}_{l}^{2})^{2}(2+\hat{m}_{l}^{2})}{384\pi^{3}M_{1}^{2}}(|H_{\frac{1}{2},1}|^{2}+|H_{-\frac{1}{2},-1}|^{2}),$
(30)

where $\hat{m}_{l}\equiv m_{l}/\sqrt{q^{2}}$, $p=\sqrt{Q_{+}Q_{-}}/(2M_{1})$
with $Q_{\pm}=(M_{1}\pm M_{2})^{2}-q^{2}$ and $M_{1}=m_{\Xi_{c}}$,
$M_{2}=m_{\Xi}$. The helicity amplitudes $H_{\lambda_{2},\lambda_{W}}$, which
are written in terms of the form factors, can be found in Shi:2019hbf .

We arrive at:

$\displaystyle\Gamma$ $\displaystyle=(1.47\pm 0.32)\times 10^{-13}\ {\rm
GeV},$ $\displaystyle{\cal B}$ $\displaystyle=(10.2\pm 2.2)\%,$
$\displaystyle\Gamma_{L}/\Gamma_{T}$ $\displaystyle=1.90\pm 0.38$ (31)

for $\Xi_{c}^{+}\to\Xi^{0}e^{+}\nu_{e}$, and

$\displaystyle\Gamma$ $\displaystyle=(1.45\pm 0.31)\times 10^{-13}\ {\rm
GeV},$ $\displaystyle{\cal B}$ $\displaystyle=(3.4\pm 0.7)\%,$
$\displaystyle\Gamma_{L}/\Gamma_{T}$ $\displaystyle=1.90\pm 0.39$ (32)

for $\Xi_{c}^{0}\to\Xi^{-}e^{+}\nu_{e}$, where $\tau(\Xi_{c}^{+})=(456\pm 5)$
fs and $\tau(\Xi_{c}^{0})=(153\pm 6)$ fs have been used Zyla:2020zbs . The
uncertainties come from those of the form factors. Considering the lifetime of
$\Xi_{c}^{0}$ changing from around 112 fs in PDG2018-2019 Tanabashi:2018oca
to around 153 fs in PDG2020 Zyla:2020zbs , in Table 6, only the decay width is
compared with those from other works. It can be seen that, most theoretical
calculations are larger than the most recent experimental data from Belle and
our result is consistent with the current world average, that of ALICE and
that of Lattice QCD.

Table 6: Our prediction for the $\Xi_{c}\to\Xi e^{+}\nu_{e}$ decay width (in units of $10^{-13}$ GeV) are compared with those from other works. This work | LCSR Azizi:2011mw  | SU(3) Geng:2019bfz  | RQM Faustov:2019ddj  | LFQM Zhao:2018zcb  | LQCD Zhang:2021oja  | PDG2020 Zyla:2020zbs  | ALICE ALICE  | Belle Li:2021uhk
---|---|---|---|---|---|---|---|---
$1.45\pm 0.31$ | $4.26\pm 1.49$ | $1.6\pm 0.1$ | $1.40$ | $0.80$ | $1.02\pm 0.19$ | $0.77\pm 0.52$ | $1.04\pm 0.36$ | $0.740\pm 0.224$

##  V Conclusions

In this work, the form factors for the process $\Xi_{c}\to\Xi$ are
investigated in QCD sum rules. For completeness, we also study the two-point
correlation function to obtain the pole residue of $\Xi$. Contributions from
up to dimension-5 have been considered. We have also considered the leading
logarithm corrections. For the two-point correlation function of $\Xi$, a
stable Borel window for the pole residue can be found, higher accuracy is
achieved. However, for the three-point correlation function, stable Borel
regions can not be found. Somewhat artificial criteria have to be adopted to
select the relatively optimal Borel parameters, and about 10-20% uncertainties
are introduced for the form factors. Then these form factor are used to obtain
the branching ratio of $\Xi_{c}^{0}\to\Xi^{-}e^{+}\nu_{e}$ as

$(3.4\pm 0.7)\%,$ (33)

which is larger than the most recent data from Belle, but still consistent
with the current world average, that of ALICE and that of Lattice QCD.

There are still two defects in this work:

  * •

One can see clearly from the calculation that, for the charm quark decay, the
non-perturbative effects dominate. As stated above, the typical contribution
ratio from dimension-0,3,5 to the form factors is roughly $1:2:1$.
Contributions from higher dimensions should be considered, and the form
factors are expected to change about 20% if these contributions are as large
as that from dimension-5. Therefore, a safer prediction for the branching
ratio of $\Xi_{c}^{0}\to\Xi^{-}e^{+}\nu_{e}$ should be

$(3.4\pm 1.7)\%.$ (34)

We intend to fix this problem in a new version of this paper.

  * •

In Zhao:2018zcb , we performed a comprehensive study for the form factors of
singly heavy baryons to light baryons using light-front quark model. We intend
to give another comprehensive study with QCD sum rules in a new work.

## Acknowledgements

The author is grateful to Yu-Ji Shi, Wei Wang, Yan-Sheng Yang and Fu-Sheng Yu
for valuable discussions. This work is supported in part by National Natural
Science Foundation of China under Grant No. 12065020.

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