# Existence of finite global norm of potential vector field in a Ricci soliton

Absos Ali Shaikh1∗, Chandan Kumar Mondal2 and Prosenjit Mandal3
1,2,3Department of Mathematics,
The University of Burdwan,Golapbag,
Purba Bardhaman-713101,
West Bengal, India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this article, we investigate global norm of potential vector field in Ricci
soliton. In particular, we have deduced certain conditions so that the
potential vector field has finite global norm in expanding Ricci soliton. We
have also showed that if the potential vector field has finite global norm in
complete non-compact Ricci soliton having finite volume, then the scalar
curvature becomes constant.

00footnotetext:  ∗ Corresponding author.
$\mathbf{2010}$ Mathematics Subject Classification: 53C20; 53C21; 53C44.
Key words and phrases: Expanding Ricci soliton, scalar curvature, Global
finite norm, Riemannian manifold.

##  1\. Introduction and preliminaries

A Riemannian manifold $(M,g)$ is called the Ricci soliton [6] if its Ricci
tensor $Ric$ satisfies the following equation:

(1) $2Ric+\pounds_{X}g=2\lambda g,$

where $\pounds_{X}g$ is the lie derivative of the metric tensor $g$ with
respect to the vector field $X$. In tensors of local coordinates system, (1)
can also be expressed in the following form

(2) $2R_{ij}+\nabla_{i}X_{j}+\nabla_{j}X_{i}=2\lambda g_{ij},$

where $R_{ij}$ denotes the components of the Ricci tensor. The Ricci soliton
is said to be expanding (resp., steady and shrinking) if $\lambda<0$ (resp.,
$\lambda=0$ and $\lambda>0$). The vector field $X$ is known as potential
vector field of the Ricci soliton and if $X$ is the gradient of some smooth
function $f\in C^{\infty}(M)$, then $f$ is called the potential function and
in this case the Ricci soliton is called the gradient Ricci soliton. After
normalization, gradient expanding Ricci soliton reduces to

(3) $R_{ij}+\nabla_{i}\nabla_{j}f=-\frac{1}{2}g_{ij}.$

The study of curvature estimation in Ricci soliton is a well known research
area in differential geometry. In 2009, Zhang [13] proved that the scalar
curvature is non-negative in case of steady and shrinking gradient Ricci
soliton and the scalar curvature is bounded below in case of expanding
gradient Ricci soliton. Later, Chow et. al [5] improved this estimation for
shrinking gradient Ricci soliton. Petersen and Wylie [9] proved that a
gradient expanding Ricci soliton with constant scalar curvature $R$ satisfies
$-\frac{n}{2}\leq R\leq 0$ and if $R=-\frac{n}{2}$, the metric becomes
Einstein, and also Pigola-Rimoldi-Setti [10] and Zhang [14] independently
showed that for such a soliton the scalar curvature $R\geq-\frac{n}{2}$.
Again, Mondal and Shaikh [7] studied gradient Ricci soliton having non-
negative Ricci curvature and realizing convex potential which fulfills finite
weighted Dirichlet condition and proved that in such a case scalar curvature
vanishes. Recently, Chen [4] proved that if the Ricci curvature in a gradient
expanding Ricci soliton satisfies

$\lim\limits_{l(x)\rightarrow\infty}l(x)^{2}|Ric|=0,$

where $l(x)$ is the distance of $x$ from a fixed point, then the scalar
curvature is non-negative, and also showed that if the Ricci curvature in a
gradient expanding Ricci soliton is non-positive, then the scalar curvature is
negative. In the present paper, we have deduced certain conditions so that the
potential vector field has finite global norm in expanding Ricci soliton. We
have also showed that if the potential vector field has finite global norm in
complete non-compact Ricci soliton having finite volume, then the scalar
curvature becomes constant. Throughout this article, by the notation $(M,g)$
or $M$ we denote the Riemannian manifold $M$ with the Riemannian metric $g$.

For any $k$, $0\leq k\leq n$, let $\Lambda^{k}(M)$ be the vector space of all
smooth $k$-forms in $M$. For any $\omega,\eta\in\Lambda^{k}(M)$, the local
inner product of $\omega$ and $\eta$ is denoted by $(\omega,\eta)$ and is
defined by, see [8, p. 149],

$(\omega,\eta)=\omega_{i_{1},\cdots,i_{k}}\eta^{i_{1},\cdots,i_{k}},$

where $\omega=\omega_{i_{1},\cdots,i_{k}}dx^{i_{1}}\wedge\cdots\wedge
dx^{i_{k}}$, $\eta=\eta_{i_{1},\cdots,i_{k}}dx^{i_{1}}\wedge\cdots\wedge
dx^{i_{k}}$ and $\eta^{i_{1},\cdots,i_{k}}=g^{i_{1}j_{1}}\cdots
g^{i_{k}j_{k}}\eta_{i_{1},\cdots,i_{k}}$. For a fixed $k\geq 0$, the Hodge
star operator $*:\Lambda^{k}(M)\rightarrow\Lambda^{n-k}(M)$ is defined by

$*\omega=sgn(I,J)\omega_{i_{1},\cdots,i_{k}}dx^{j_{1}}\wedge\cdots\wedge
dx^{j_{n-k}},$

for $\omega=\omega_{i_{1},\cdots,i_{k}}dx^{i_{1}}\wedge\cdots\wedge
dx^{i_{k}}\in\Lambda^{k}(M)$. Here $j_{1}<\cdots<j_{n-k}$ is the rearrangement
of the complements of $i_{1}<\cdots<i_{k}$ in the set $\\{1,...,n\\}$ in
ascending order and $sgn(I,J)$ is the sign of the permutation
$i_{1},...,i_{k},j_{1},...,j_{n-k}$. For an oriented Riemannian manifold $M$,
the global inner product in $\Lambda^{k}(M)$ is defined by

$\langle\omega,\eta\rangle=\int_{M}\omega\wedge*\eta,$

for $\omega,\eta\in\Lambda^{k}(M)$. We define the global norm of
$\omega\in\Lambda^{k}(M)$ by $\|\omega\|^{2}=\langle\omega,\omega\rangle$ and
remark that $\|\omega\|^{2}\leq\infty$. There is a natural adjoint operator of
the exterior derivative $d:\Lambda^{k}(M)\rightarrow\Lambda^{k+1}(M)$ called
the co-differential operator
$\delta:\Lambda^{k}(M)\rightarrow\Lambda^{k-1}(M)$ and is defined by

$\delta=(-1)^{k}*^{-1}d*=(-1)^{n(k+1)+1}*d*,$

so that the following diagram commutes

$\begin{CD}\Lambda^{k}(M)@>{*}>{}>\Lambda^{n-k}(M)\\\
@V{}V{\delta}V@V{}V{d}V\\\
\Lambda^{k-1}(M)@>{(-1)^{k}*}>{}>\Lambda^{n-k+1}(M)\end{CD}$

For a $1$-form $\omega\in\Lambda^{1}(M)$, we have

$\displaystyle(d\omega)_{ij}$ $\displaystyle=$
$\displaystyle\nabla_{i}\omega_{j}-\nabla_{j}\omega_{i}\quad\text{ and }$
$\displaystyle(\delta\omega)$ $\displaystyle=$
$\displaystyle-\nabla^{i}\omega_{i},$

where $\nabla^{i}=g^{ij}\nabla_{j}$. For a detailed discussion on Hodge
operator and co-differential operator see [8]. By the notation
$\Lambda^{k}_{0}(M)$, we denote the subspace of $\Lambda^{k}(M)$ containing
all $k$-forms in $M$ with compact support. Also the completion of
$\Lambda^{k}_{0}(M)$ with respect to the global inner product
$\langle,\rangle$ is denoted by $L^{s}_{2}(M)$.

In this article, a vector field and its dual $1$-form with respect to the
Riemannian metric $g$ will be denoted by the same letter, i.e., for a vector
field $X=X^{i}\partial_{i}$, the dual $1$-form is
$X=X_{i}dx^{i}=g_{ij}X^{j}dx^{i}$.

##  2\. Main results

######  Definition 2.1.

[12] A vector field on a manofold $M$ is said to have finite global norm if
its dual $1$-form with respect to $g$ belongs to
$L^{1}_{2}(M)\cap\Lambda^{1}(M)$.

Let $o\in M$ be a fixed point and $l(x)$ be the distance from $o$ to $x$ for
each $x\in M$. The open ball with center $o$ and radius $r>0$ is denoted by
$B(r)$. Then there exists a Lipschitz continuous function $\omega_{r}$ such
that for some constant $K>0$, (see [11]),

$\displaystyle|d\omega_{r}|\leq\frac{K}{r}\qquad\text{almost everywhere on }M$
$\displaystyle 0\leq\omega_{r}(x)\leq 1\quad\forall x\in M$
$\displaystyle\omega_{r}(x)=1\quad\forall x\in B(r)$ $\displaystyle\text{ supp
}\omega_{r}\subset B(r)$

Then taking limit, we get $\lim\limits_{r\rightarrow\infty}\omega_{r}=1$. Now
we calculate the global norm of $\omega_{r}dX$ and $\omega_{r}\delta X$ when
$R_{ij}(\nabla^{i}X^{j})=-\lambda R$. Then we have

(4) $\displaystyle g(dX,dX)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\\{(\nabla_{i}X_{j}-\nabla_{j}X_{i})(\nabla^{i}X^{j}-\nabla^{j}X^{i})\\}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\\{4(\nabla_{i}X_{j})(\nabla^{i}X^{j})+4R_{ij}(\nabla^{i}X^{j})-4\lambda\nabla^{i}X^{i}\\}$
$\displaystyle=$ $\displaystyle 4g(\nabla X,\nabla
X)+2R_{ij}(\nabla^{i}X^{j})-2\lambda(\lambda n-R)$ $\displaystyle=$
$\displaystyle 4g(\nabla X,\nabla X)-2n\lambda^{2},$

(5) $\displaystyle g(\delta X,\delta X)$ $\displaystyle=$
$\displaystyle(\nabla^{i}X_{i})(\nabla^{j}X_{j})$ $\displaystyle=$
$\displaystyle g(\lambda n-R,\lambda n-R)=|\lambda n-R|^{2}.$

The above two equations (4) and (5) together imply the following lemma:

######  Lemma 2.1.

If $R_{ij}(\nabla^{i}X^{j})=-\lambda R$, then the potential vector field $X$
of the Ricci soliton (2) satisfies the following:

(6) $\|\omega_{r}dX\|^{2}_{B(2r)}=4\|\omega_{r}\nabla
X\|^{2}_{B(2r)}-2n\|\omega_{r}\lambda\|^{2}_{B(2r)},$ (7) $\|\omega_{r}\delta
X\|^{2}_{B(2r)}=\|\omega_{r}(\lambda n-R)\|^{2}_{B(2r)}.$

Combining lemma 2 and lemma 3 of [12], we have

######  Lemma 2.2.

[12] For any $X\in\Lambda^{1}(M)$, we have

(8) $4\langle\omega_{r}d\omega_{r}\otimes X,\nabla
X\rangle_{B(2r)}+\langle\omega_{r}\nabla^{2}X,\omega_{r}X\rangle_{B(2r)}+2\langle\omega_{r}\nabla
X,\omega_{r}\nabla X\rangle_{B(2r)}=0.$ $\displaystyle\langle\omega_{r}\Re
X,\omega_{r}X\rangle_{B(2r)}=\langle\omega_{r}\nabla^{2}X,\omega_{r}X\rangle_{B(2r)}+\langle\omega_{r}dX,\omega_{r}dX\rangle_{B(2r)}$
(9) $\displaystyle+2\langle\omega_{r}dX,d\omega_{r}\wedge
X\rangle_{B(2r)}+\langle\omega_{r}\delta X,\omega_{r}\delta
X\rangle_{B(2r)}-2\langle\omega_{r}\delta
X,*(d\omega_{r}\wedge*X)\rangle_{B(2r)},$

where $(\nabla^{2}X)_{i}=\nabla^{j}\nabla_{j}X_{i}$, $(\nabla
X)_{ij}=\nabla_{i}X_{j}$ and $(\Re X)_{i}=R^{j}_{i}X_{j}$ is the Ricci
transformation on $\Lambda^{1}(M)$.

######  Lemma 2.3.

[1] For any $X\in\Lambda^{k}(M)$, there exists a positive constant $A$
independent of $r$ such that

$\displaystyle\|d\omega_{r}\otimes X\|^{2}_{B(2r)}$ $\displaystyle\leq$
$\displaystyle\frac{A}{r^{2}}\|X\|^{2}_{B(2r)},$
$\displaystyle\|d\omega_{r}\wedge X\|^{2}_{B(2r)}$ $\displaystyle\leq$
$\displaystyle\frac{A}{r^{2}}\|X\|^{2}_{B(2r)}$ $\displaystyle\text{ and
}\|d\omega_{r}\wedge*X\|^{2}_{B(2r)}$ $\displaystyle\leq$
$\displaystyle\frac{A}{r^{2}}\|X\|^{2}_{B(2r)}.$

From Lemma 2.1 and Lemma 2.3, we have the following:

$\displaystyle|2\langle\omega_{r}dX,d\omega_{r}\wedge X\rangle_{B(2r)}|$
$\displaystyle\leq$ $\displaystyle
2\|\omega_{r}dX\|_{B(2r)}\|d\omega_{r}\wedge X\|_{B(2r)}$ $\displaystyle\leq$
$\displaystyle\frac{1}{4}\|\omega_{r}dX\|^{2}_{B(2r)}+4\|d\omega_{r}\wedge
X\|^{2}_{B(2r)}$ $\displaystyle\leq$ $\displaystyle\|\omega_{r}\nabla
X\|^{2}_{B(2r)}-\frac{n}{2}\|\omega_{r}\lambda\|^{2}_{B(2r)}+\frac{4A}{r^{2}}\|X\|_{B(2r)}^{2},$
(10) $\displaystyle|2\langle\omega_{r}\delta
X,*(d\omega_{r}\wedge*X)\rangle_{B(2r)}|$ $\displaystyle\leq$ $\displaystyle
2\|\omega_{r}\delta X\|_{B(2r)}\|d\omega_{r}\wedge*X\|_{B(2r)}$
$\displaystyle\leq$ $\displaystyle\frac{1}{5}\|\omega_{r}\delta
X\|^{2}_{B(2r)}+5\|d\omega_{r}\wedge*X\|^{2}_{B(2r)}$ $\displaystyle\leq$
$\displaystyle\frac{1}{5}\|\omega_{r}(\lambda
n-R)\|^{2}_{B(2r)}+\frac{5A}{r^{2}}\|X\|_{B(2r)}^{2}.$

Thus using Lemma 2.2, we calculate for Ricci soliton (2):

(11) $\displaystyle\langle\omega_{r}\Re X,\omega_{r}X\rangle_{B(2r)}$
$\displaystyle=$ $\displaystyle-4\langle\omega_{r}d\omega_{r}\otimes X,\nabla
X\rangle_{B(2r)}-2\langle\omega_{r}\nabla X,\omega_{r}\nabla X\rangle_{B(2r)}$
$\displaystyle+\langle\omega_{r}dX,\omega_{r}dX\rangle_{B(2r)}+2\langle\omega_{r}dX,d\omega_{r}\wedge
X\rangle_{B(2r)}$ $\displaystyle+\langle\omega_{r}\delta X,\omega_{r}\delta
X\rangle_{B(2r)}-2\langle\omega_{r}\delta
X,*(d\omega_{r}\wedge*X)\rangle_{B(2r)}$ $\displaystyle\geq$
$\displaystyle-\frac{1}{2}\|\omega_{r}\nabla
X\|^{2}_{B(2r)}-\frac{8A}{r^{2}}\|X\|^{2}_{B(2r)}-2\|\omega_{r}\nabla
X\|^{2}_{B(2r)}+4\|\omega_{r}\nabla X\|^{2}_{B(2r)}$
$\displaystyle-2n\|\omega_{r}\lambda\|^{2}_{B(2r)}-\|\omega_{r}\nabla
X\|^{2}_{B(2r)}+\frac{n}{2}\|\omega_{r}\lambda\|^{2}_{B(2r)}-\frac{4A}{r^{2}}\|X\|_{B(2r)}^{2}$
$\displaystyle+\|\omega_{r}(\lambda
n-R)\|^{2}_{B(2r)}-\frac{1}{5}\|\omega_{r}(\lambda
n-R)\|^{2}_{B(2r)}-\frac{5A}{r^{2}}\|X\|^{2}_{B(2r)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\|\omega_{r}\nabla
X\|^{2}_{B(2r)}-\frac{17A}{r^{2}}\|X\|^{2}_{B(2r)}-\frac{3n}{2}\|\omega_{r}\lambda\|^{2}_{B(2r)}+\frac{4}{5}\|\omega_{r}(\lambda
n-R)\|^{2}_{B(2r)}.$

If the Ricci curvature is non-positive then we can write

$\limsup_{r\rightarrow\infty}\langle\omega_{r}\Re
X,\omega_{r}X\rangle_{B(2r)}\leq 0.$

Hence (11) reduces to the following inequality:

$5\|\nabla X\|^{2}-15n\|\lambda\|^{2}+8\|(\lambda n-R)\|^{2}\leq 0,$

which yields

$5\|\nabla X\|^{2}+\int_{M}(-15n\lambda^{2}+8\lambda^{2}n^{2}-16\lambda
nR+8R^{2})dv\leq 0.$

Therefore, for $n\geq 2$, the above inequality gives

(12) $5\|\nabla X\|^{2}+\int_{M}(-16\lambda nR+8R^{2})dv\leq 0.$

Hence we get

(13) $5\|\nabla X\|^{2}\leq 16\lambda n\int_{M}R.$

######  Theorem 2.4.

Let $(M,g,X)$ be an expanding Ricci soliton which is complete and non-compact.
If the Ricci curvature is non-positive and the following conditions hold:
(i) $R_{ij}(\nabla^{i}X^{j})=R/2,$ and
(ii) $\int_{M}R\geq-c$ for some positive constant $c$, then the potential
vector field has finite global norm. Moreover, if the manifold has finite
volume, then

$\int_{M}R^{2}\leq ncVol(M).$

###### Proof.

Since $(M,g,X)$ is a complete non-compact expanding Ricci soliton, we can take
$\lambda=-\frac{1}{2}$. Now putting the value of $\lambda$ in (13) and using
the given conditions, we get our first result. Also, (12) implies that

$\int_{M}R^{2}\leq-n\int_{M}R\leq ncVol(M).$

∎

######  Theorem 2.5.

Let $(M,g,X)$ be a complete non-compact Ricci soliton with finite volume. If
the potential vector field $X$ is of finite global norm, then the scalar
curvature $R$ must be constant.

###### Proof.

For any $r>0$, we have

$\displaystyle\frac{1}{r}\int_{B(2r)}|X|dV$ $\displaystyle\leq$
$\displaystyle\Big{(}\int_{B(2r)}\langle X,X\rangle
dV\Big{)}^{1/2}\Big{(}\int_{B(2r)}\Big{(}\frac{1}{r}\Big{)}^{2}dV\Big{)}^{1/2}$
$\displaystyle\leq$
$\displaystyle\|X\|_{B(2r)}\frac{1}{r}\Big{(}Vol(M)\Big{)}^{1/2},$

where $Vol(M)$ denotes the volume of $M$. Thus we obtain

$\liminf_{\ r\rightarrow\infty}\frac{1}{r}\int_{B(2r)}|X|dV=0.$

Again we have

$\Big{|}\int_{B(2r)}\omega_{r}divXdV\Big{|}\leq\frac{C}{r}\int_{B(2r)}|X|dV,$

for some constant C. In view of $(\ref{y1})$ the last relation yields

$\int_{M}(n\lambda-R)dV=0,$

which gives $R=n\lambda$. This completes the proof. ∎

## Acknowledgment

The third author gratefully acknowledges to the CSIR(File
No.:09/025(0282)/2019-EMR-I), Govt. of India for financial assistance.

## References

  * [1] Andreotti, A., Carleman estimates for the Laplace-Beltrami equation on complex manifold, Inst. Hautes Etudes Sci. Publ. Math., 25 (1965), 313–362.
  * [2] Chen, C. W., On the asymptotic behavior of expanding gradient Ricci solitons, Ann. Global Anal. Geom., 42(2) (2012), 267–277.
  * [3] Chen, C. W. and Deruelle, A., Structure at infinity of expanding gradient Ricci soliton, Asian J. Math., 19(5) (2015), 933–950.
  * [4] Chen, P. Y., Curvature estimates and gap theorems for expanding Ricci solitons, arXiv:2001.11487, (2020).
  * [5] Chow, B., Lu, P. and Yang, B., Lower bounds for the scalar curvatures of noncompact gradient Ricci solitons, C. R. Math. Acad. Sci. Paris, 349(23-24) (2011), 1265-1267.
  * [6] Hamilton, R. S., Three-manifolds with positive Ricci curvature, J. Diff. Geom., 17 (1982), 255–306.
  * [7] Mondal, C. K. and Shaikh, A. A. On Ricci solitons whose potential is convex, Proc. Indian Acad. Sci. (Math. Sci.), 130(55) (2020), 1–7.
  * [8] Morita, S., Geometry of differential forms, Amer. Math. Soc., 2001.
  * [9] Petersen, P. and Wylie, W., Rigidity of gradient Ricci solitons, Pacific J. Math., 241 (2009), 329–345.
  * [10] Pigola, S., Rimoldi, M. and Setti, A., Remarks on noncompact gradient Ricci solitons, Math. Z. 268 (2011), 777–790.
  * [11] Yau, S. T., Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J., 25(7) (1976), 659–670.
  * [12] Yorozu, S., Conformal and Killing vector fields on complete non-compact Riemannian manifolds, Geom. of Geodesics and Related Topics, 3 (1984), 459–472.
  * [13] Zhang, Z. H., On the completeness of gradient Ricci solitons, Proc. Amer. Math. Soc., 137(8) (2009), 2755–2759.
  * [14] Zhang, S. J., On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below, Acta Math. Sinica, English Series, 27(5) (2011), 871–882.


# The cross section of the process $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ in the
vicinity of charmonium $\psi(3770)$ including three-gluon and $D$-meson loop
contributions

Yu.M. Bystritskiy<EMAIL_ADDRESS>Joint Institute for Nuclear Research,
141980 Dubna, Moscow Region, Russia  A. I. Ahmadov<EMAIL_ADDRESS>Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia
Institute of Physics, Azerbaijan National Academy of Science, Baku, Azerbaijan

###### Abstract

The total cross section of the process $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ is
calculated within the energy range close to the mass of $\psi(3770)$
charmonium state. Two different contributions were considered: the $D$-meson
loop and the three gluon charmonium annihilation one. Both of them contribute
noticeably and in sum fairly reproduce the data. Large relative phase for
these contributions are generated with respect to the pure electromagnetic
mechanism. As a by product the fit for the electromagnetic form factor of
$\Lambda$-hyperon is obtained for the large momentum transferred region.

##  I Introduction

The bound state of a pair of charmed quarks are the one of the most clear and
simple system which allows one to study the details of a confinement mechanism
and to refine the conjectures of Quantum Chormodynamics (QCD). During few
recent decades these states are under intensive experimental and theoretical
study [1, 2, 3, 4]. The special interest is focused to the electron–positron
annihilation processes with the production of different mesonic and baryonic
final states thus giving us the clear way to produce and to study the
charmonium in pure $J^{PC}=1^{--}$ state. The binary final states (i.e. with
two particles finally produced) give the possibility to further simplify the
consideration of the processes with charmonium in the intermediate state [5].

Besides it was shown a long time ago that these processes are the excellent
way to measure the electromagnetic form factors of the particles produced [6,
7]. There is enormous set of the measurements of the electromagnetic form
factors of the proton, for example, by the BaBar Collaboration [8, 9] or by
the BES III Collaboration [10]. But the other baryons are also under study:
$\Lambda^{0}$ and $\Sigma^{0}$ baryons [11, 12], $\Sigma^{\pm}$ baryons [13],
$\Xi^{0}$, $\Xi^{-}$ and $\Omega^{-}$ baryons [14]. For a recent review of the
situation on $\Lambda\bar{\Lambda}$ pair production see a review [15]. A lot
of interest paid to the near threshold behaviour of these form factors [16,
17] which demonstrates non-trivial enhancement effect. Many theoretical ideas
to explain this effect were proposed (see, for example, [18, 19]).

In this paper we want to consider the process of electron-positron
annihilation into a pair of $\Lambda$-hyperons which was recently precisely
measured in the vicinity of $\psi(3770)$ charmonium [20]. The charmonium
$\psi(3770)$ is the one of the intriguing states which was studied by many
collaborations (for example, by KEDR-VEPP-4M [21, 22], CLEO [23] and more
recently by BES III [24, 25]). In paper [25] one can find the measurement of
the cross section of the process $e^{+}e^{-}\to p\bar{p}$ with the specific
care to the region near the mass of $\psi(3770)$ which demonstrates the dip
instead of enhancing Breit–Wigner peak. This is the manifestation of the large
relative phase which is generated by the intermediate charmonium state with
respect to the pure electromagnetic background. In papers [26, 27] we showed
that the source of this phase is mostly attributed to the three gluon
mechanism of the charmonium decay. Here we consider the similar process
$e^{+}e^{-}\to\Lambda\bar{\Lambda}$ which also must have large relative phase
coming from the same three gluon mechanism. Since all the calculations was
already done in [27] here we just briefly recall the main formulae and put the
main focus on the details which differ in $\Lambda\bar{\Lambda}$ case.

The paper is organized in the following manner: in Section II the total cross
section of the process $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ in Born
approximation is presented, the electromagnetic form factors of
$\Lambda$-hyperon are discussed; Section III shows how the charmonium in the
intermediate state contributes to the total cross section; two subsequent
Sections IV and V shows the main formulae for the OZI-allowed mechanism with
the $D$-meson loop and for the OZI-violating three gluons mechanism. Section
VI gives some numerical estimations and comparison of our calculation with
experimental data from BES III [20]; Section VII concludes our results and
proposes some possible extension of this work in the future.

##  II Born approximation

We consider the process of electron-positron annihilation into a pair of
$\Lambda$-baryons:

$\displaystyle
e^{+}(q_{+})+e^{-}(q_{-})\to\Lambda(p_{1})+\bar{\Lambda}(p_{2}),$ (1)

where quantities in parenthesis are the 4-momenta of the corresponding
particles. The cross section for the process has the standard way:

$\displaystyle d\sigma=\frac{1}{8s}\sum_{\text{spins}}\left|{\cal
M}\right|^{2}\,d\Phi_{2},$ (2)

where the summation of the amplitude square $\left|{\cal M}\right|^{2}$ runs
over all possible initial and final particles spin states. We systematically
neglect the mass of the electron $m_{e}$ in this paper. The phase volume of
final particles $d\Phi_{2}$ has the form:

$\displaystyle d\Phi_{2}$
$\displaystyle=\frac{1}{\left(2\pi\right)^{2}}\delta\left(q_{+}+q_{-}-p_{1}-p_{2}\right)\frac{d{\bf
p_{1}}}{2E_{1}}\frac{d{\bf p_{2}}}{2E_{2}}=$ $\displaystyle=\frac{\left|{\bf
p}\right|}{2^{4}\pi^{2}\sqrt{s}}\,d\Omega_{\Lambda}=\frac{\beta}{2^{4}\pi}\,d\cos\theta_{\Lambda},$
(3)

where $d\Omega_{\Lambda}=d\phi_{\Lambda}\,d\cos\theta_{\Lambda}$ and
$\phi_{\Lambda}$ and $\theta_{\Lambda}$ are the azimuthal and the polar angles
of the final $\Lambda$-baryon momentum in the center-of-mass reference frame
(center of mass system, c.m.s), i.e. $\theta_{\Lambda}$ is the angle between
3-momenta of the initial electron ${\bf q_{-}}$ and the final $\Lambda$-baryon
${\bf p_{1}}$ (see Fig. 1) and $\left|{\bf p}\right|\equiv\left|{\bf
p_{1}}\right|=\left|{\bf p_{2}}\right|=\sqrt{s}\beta/2$ is the modulus of
3-momenta of the final $\Lambda$ or $\bar{\Lambda}$. Here the quantity
$\beta=\sqrt{1-4M_{\Lambda}^{2}/s}$ is the final $\Lambda$-baryon velocity,
with $M_{\Lambda}$ being the mass of $\Lambda$-baryon.

Figure 1: The definition of the scattering angle $\theta_{\Lambda}$ from (3)
in the center-of-mass reference frame.

Figure 2:  Feynman diagrams of the process $e^{+}e^{-}\to\Lambda\bar{\Lambda}$
in Born approximation (a) and with the charmonium $\psi(3770)$ intermediate
state (b).

In Born approximation (see Fig. 2) the amplitude ${\cal M}={\cal M}_{B}$ in
(2) describes the electron–positron pair annihilation into virtual photon,
which then produces $\Lambda\bar{\Lambda}$ pair. The amplitude ${\cal M}_{B}$
corresponding to this process has the form:

$\displaystyle{\cal
M}_{B}=\frac{1}{s}J^{e\bar{e}\to\gamma}_{\mu}(q)\,J_{\gamma\to\Lambda\bar{\Lambda}}^{\mu}(q),$
(4)

where $s=q^{2}=\left(q_{+}+q_{-}\right)^{2}=\left(p_{1}+p_{2}\right)^{2}$ is
the total invariant mass squared of the lepton pair ($q$ is the momentum of
the intermediate photon). The quantities $J^{e\bar{e}\to\gamma}_{\mu}$ and
$J^{\gamma\to\Lambda\bar{\Lambda}}_{\mu}$ are electromagnetic currents:

$\displaystyle J^{e\bar{e}\to\gamma}_{\mu}(q)$
$\displaystyle=-e\left[\bar{v}(q_{+})\gamma_{\mu}u(q_{-})\right],$ (5)
$\displaystyle J^{\gamma\to\Lambda\bar{\Lambda}}_{\mu}(q)$
$\displaystyle=e\left[\bar{u}(p_{1})\Gamma_{\mu}(q)v(p_{2})\right],$ (6)

where $e$ is the modulus of electron charge $e=\sqrt{4\pi\alpha}$ with
$\alpha$ being the fine structure constant [28]. In general the vertex of the
photon with the $\Lambda$-baryon has the form:

$\displaystyle\Gamma_{\mu}(q)$
$\displaystyle=F_{1}(q^{2})\gamma_{\mu}-\frac{F_{2}(q^{2})}{4M_{\Lambda}}\left(\gamma_{\mu}{\hat{q}}-{\hat{q}}\gamma_{\mu}\right).$
(7)

where we use the notation ${\hat{a}}\equiv a_{\mu}\gamma^{\mu}$. Here the
functions $F_{1,2}(q^{2})$ are the $\Lambda$-baryon electromagnetic form
factors normalized as $F_{1}(0)=0$ and $F_{2}(0)=\mu_{\Lambda}$, where
$\mu_{\Lambda}$ is the $\Lambda$-baryon anomalous magnetic moment.

It was shown in [29] that the non-trivial structure of the baryon starts to
manifest itself even at relatively low $q^{2}$ and thus one must take into
account these effects of the structure. Since experimental data at the moment
do not allow one to extract the electric $G_{E}$ and the magnetic $G_{M}$ form
factors of a baryons separately we utilize the assumption that
$\left|G_{E}\right|=\left|G_{M}\right|$, i.e. $F_{2}(q^{2})=0$. Then the total
cross section in Born approximation has the form:

$\displaystyle\sigma_{B}(s)=\frac{2\pi\alpha^{2}}{3s}\beta\left(3-\beta^{2}\right)\left|F_{1}\left(s\right)\right|^{2}.$
(8)

The form factor $F_{1}$ is chosen to have pQCD inspired form [30, 31], which
takes into account the running of the QCD coupling constant $\alpha_{s}$:

$\displaystyle F_{1}(s)=\frac{C}{s^{2}\log^{2}\left(s/\Lambda_{\mbox{\tiny
QCD}}^{2}\right)},$ (9)

where $\Lambda_{\mbox{\tiny QCD}}$ is the QCD scale and the constant $C$
should be fitted on the experimental data for baryon–antibaryon production in
a wide energy range.

For proton in our energy region this fit was done in [25] giving $C=(62.6\pm
4.1)~{}\mbox{GeV}^{4}$ (using $\Lambda_{\mbox{\tiny QCD}}=300~{}\mbox{MeV}$).
This fit qualitatively agrees with the more recent result of paper [32] where
this constant was fitted to be equal to $C=72~{}\mbox{GeV}^{4}$ with
$\Lambda_{\mbox{\tiny QCD}}=520~{}\mbox{MeV}$.

Figure 3: The total cross section for the process
$e^{+}e^{-}\to\Lambda\bar{\Lambda}$. Black line is the cross section in Born
approximation (8). The curve errors origin from the form factor constant (10)
fitting errors.

In our case of $\Lambda\bar{\Lambda}$ pair production we fix this constant
using the whole range of BES-III measurement [20] presented in Fig. 3. Fitting
the Born cross section from (8) with respect to this data gives us the
following parameter:

$\displaystyle C=\left(43.1\pm 1.4\right)~{}\mbox{GeV}^{4},$ (10)

which we use further for $\Lambda$-baryon electromagnetic form factor (9). We
note that this expression for the form factor $F_{1}$ with the constant $C$
from (10) works for relatively large momentum transfered $q^{2}$. It does not
pretend to work near threshold, since there are many delicate features playing
important role there, such as Coulomb-like enhancement factor [33, 34] or the
manifestation of wavy nature of baryon stabilization after its emerging from
the vacuum [29].

##  III The quarkonium $\psi(3770)$ intermediate state

The main task of our work is to describe the effect of the charmonium
resonance $\psi(3770)$ excitation in the process (1). As one can see in Fig. 3
the experimental point for the cross section at $\sqrt{s}=M_{\psi}$ (where
$M_{\psi}$ is the $\psi(3770)$ mass) flies higher the Born cross section
curve. In [20] there is a fit of this point with the use of some
phenomenologically inspired formula (see eq. (3) in [20]). Further we develop
the model (based on our previous calculations [26, 27]) which reveals the
underlying mechanism for this point to be upstairs.

In the region of charmonium resonance $\psi(3770)$ excitation one should take
into account the additional contribution to the amplitude:

$\displaystyle{\cal M}={\cal M}_{B}+{\cal M}_{\psi},$ (11)

where ${\cal M}_{\psi}$ takes into account the mechanism with charmonium
$\psi(3770)$ in the intermediate state (see Fig. 2) which is enhanced by the
Breit-Wigner factor:

$\displaystyle{\cal
M}_{\psi}=\frac{g^{\mu\nu}-q^{\mu}q^{\nu}/M_{\psi}^{2}}{s-M_{\psi}^{2}+iM_{\psi}\,\Gamma_{\psi}}J^{e\bar{e}\to\psi}_{\mu}(q)J^{\psi\to\Lambda\bar{\Lambda}}_{\nu}(q),$
(12)

where $\Gamma_{\psi}$ is the total decay width of $\psi(3770)$ resonance and
$J^{e\bar{e}\to\psi}_{\mu}$ and $J^{\psi\to\Lambda\bar{\Lambda}}_{\mu}$ are
the currents which describe the transition of lepton pair into $\psi(3770)$
resonance and the transition of the $\psi(3770)$ resonance into
$\Lambda\bar{\Lambda}$ pair correspondingly. Following [26] we assume that
$J^{e\bar{e}\to\psi}_{\mu}$ has the same structure as
$J^{e\bar{e}\to\gamma}_{\mu}$ from (5), i.e.

$\displaystyle
J^{e\bar{e}\to\psi}_{\mu}(q)=g_{e}\,\left[\bar{v}(q_{+})\gamma_{\mu}u(q_{-})\right],$
(13)

where the constant $g_{e}=F_{1}^{\psi\to e\bar{e}}(M_{\psi}^{2})$ is the value
of the form factor of the vertex $\psi\to e\bar{e}$ at the $\psi(3770)$ mass-
shell (here we follow the same approximation as in the Born case and assume
that $F_{2}^{\psi\to e\bar{e}}(M_{\psi}^{2})=0$). We fix this constant $g_{e}$
via total decay width of $\psi\to e^{+}e^{-}$ which is known to be equal to
$\Gamma_{\psi\to e^{+}e^{-}}=261~{}\mbox{eV}$ [28]:

$\displaystyle g_{e}=\sqrt{\frac{12\pi\Gamma_{\psi\to
e^{+}e^{-}}}{M_{\psi}}}=1.6\cdot 10^{-3}.$ (14)

We neglect a possible imaginary part of the vertex $g_{e}$ since it was shown
in [35] that it is small, less then 10 % of the real part. We also note a
mistake in our paper [27] where this constant is claimed to be related with
the $\psi\to p\bar{p}$ decay.

The amplitude ${\cal M}_{\psi}$ from (12) interferes with the Born one ${\cal
M}_{B}$ giving the standard interference contribution to the cross section:

$\displaystyle
d\sigma_{int}=\frac{1}{8s}\sum_{\text{spins}}2\,\mbox{Re}\left[{\cal
M}_{B}^{+}{\cal M}_{\psi}\right]\,d\Phi_{2},$ (15)

which leads to the following form of the interference contribution to the
total cross section:

$\displaystyle\sigma_{int}(s)=\mbox{Re}\left(\frac{S_{i}(s)}{s-M_{\psi}^{2}+iM_{\psi}\,\Gamma_{\psi}}\right),$
(16)

where $S_{i}(s)$ contains all the dynamics of the transformation of charmonium
into $\Lambda\bar{\Lambda}$ pair and has the following explicit form:

$\displaystyle S_{i}(s)=\frac{eg_{e}\beta}{48\pi
s}\int\limits_{-1}^{1}d\cos\theta_{\Lambda}\sum_{s^{\prime}}\left(J_{\gamma\to\Lambda\bar{\Lambda}}^{\alpha}\right)^{*}J^{\psi\to\Lambda\bar{\Lambda}}_{\alpha}.$
(17)

The subscript index $i$ in the expression above denotes the type of mechanism
of this transformation. Since the mass of $\psi(3770)$ is higher than the
threshold of $D$-meson pair production it is natural to expect that the
$D$-meson loop will be the main mechanism in this reaction (see Fig. 4) and we
consider it below in Section IV. However we need also to consider the OZI-
violating three gluon mechanism (see Fig. 5) which we briefly recall in
Section V.

Having the interference contribution (16) with the total relative phase
between Born amplitude ${\cal M}_{B}$ and the charmonium contribution one
${\cal M}_{\psi}$ we can restore the total cross section using the procedure
described in [27] (see eqs. (15) and (16) there).

##  IV D-meson loop mechanism

Figure 4: $D$-meson loop mechanism.

Here we follow exactly to our previous calculations [26, 27] with only one
systematic modification which is needed to describe the $\Lambda\bar{\Lambda}$
pair final state instead of the proton–antiproton one. The $D$-meson loop
mechanism (presented in Fig. 4) contribute to the interference of a charmonium
state with the Born amplitude (see (17)) as:

$\displaystyle S_{D}\left(s\right)$
$\displaystyle=\alpha_{D}\left(s\right)\,Z_{D}\left(s\right),$ (18)

where

$\displaystyle\alpha_{D}\left(s\right)=\frac{\alpha~{}g_{e}}{2^{4}\,3\pi^{2}}\beta\,F_{1}(s),$
$\displaystyle Z_{D}\left(s\right)=\frac{1}{s}\int\frac{dk}{i\pi^{2}}\times$
$\displaystyle\times\frac{SpD(s,k^{2})}{\left(k^{2}-M_{\Xi}^{2}\right)\left((k-p_{1})^{2}-M_{D}^{2}\right)\left((k+p_{2})^{2}-M_{D}^{2}\right)}\times$
$\displaystyle\quad\times G_{\psi
D\bar{D}}(s,(k+p_{2})^{2},(k-p_{1})^{2})\times$ $\displaystyle\quad\times
G_{\Lambda D\Xi}\left(k^{2},(k-p_{1})^{2}\right)G_{\Lambda
D\Xi}\left(k^{2},(k+p_{2})^{2}\right),$ (19)

and $SpD(s,k^{2})$ is the trace of $\gamma$-matrices over the baryon line:

$\displaystyle SpD(s,k^{2})=$
$\displaystyle=\mbox{Sp}\left[({\hat{p}_{1}}+M_{\Lambda})\gamma_{5}({\hat{k}}+M_{\Xi})\gamma_{5}({\hat{p}_{2}}-M_{\Lambda})({\hat{k}}-M_{\Lambda})\right]=$
$\displaystyle=2\left(\left(k^{2}\right)^{2}+k^{2}\left(s-2\left(M_{D}^{2}+M_{\Lambda}M_{\Xi}\right)\right)-\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\quad\left.\frac{}{}-sM_{\Lambda}M_{\Xi}+c_{D}\right),$
(20) $\displaystyle
c_{D}=M_{D}^{4}+2M_{\Lambda}M_{\Xi}M_{D}^{2}+2M_{\Xi}M_{\Lambda}^{3}-M_{\Lambda}^{4}.$
(21)

The quantities $G_{\psi D\bar{D}}$ and $G_{D\Xi\Lambda}$ in (19) are the form
factors for the vertexes $\psi\to D\bar{D}$ and $D\to\Xi\Lambda$.

The details of the calculation of quantity $Z_{D}\left(s\right)$ can be found
in [27], but here we just recall that technically we calculate imaginary part
of this quantity and then restore the real part of it by using the dispersion
relation with one subtraction at $q^{2}=0$. First we need to mention that the
subtraction constant here is also vanish since the $\Lambda$-hyperon (which is
the $uds$ quarks state) do not have open charm and thus the vertex
$\psi\to\Lambda\bar{\Lambda}$ at $q^{2}=0$ is zero.

Next, cutting the diagram by $D$-meson propagators we get the vertex $\psi\to
D\bar{D}$ with the only dependence over charmonium virtuality $q^{2}=s$ since
$D$-meson legs are on-mass-shell:

$\displaystyle G_{\psi D\bar{D}}\left(s,M_{D}^{2},M_{D}^{2}\right)=g_{\psi
D\bar{D}}\,\frac{M_{\psi}^{2}}{s}\,\frac{\log\left(M_{\psi}^{2}/\Lambda_{D}^{2}\right)}{\log\left(s/\Lambda_{D}^{2}\right)},$
(22)

where scale $\Lambda_{D}$ we fix on the characteristic value of the reaction
$\Lambda_{D}=2M_{D}$ and the constant $g_{\psi D\bar{D}}$ is fixed by the
$\psi\to D\bar{D}$ decay width:

$\displaystyle g_{\psi D\bar{D}}$ $\displaystyle\equiv G_{\psi
D\bar{D}}(M_{\psi}^{2},M_{D}^{2},M_{D}^{2})=$
$\displaystyle=4\,\sqrt{\frac{3\pi\,\Gamma_{\psi\to
D\bar{D}}}{M_{\psi}\,\beta_{D}^{3}}}=18.4,$ (23)

where $\beta_{D}=\sqrt{1-4M_{D}^{2}/M_{\psi}^{2}}$ is the $D$-meson velocity
in this decay.

Next we consider function $G_{\Lambda D\Xi}\left(k^{2},p^{2}\right)$ from
(19). Again the only dependence left in the imaginary part of $Z_{D}$ is the
off-mass-shellness of $\Xi$ baryon in $t$-channel since $k^{2}<0$. In [26, 27]
we used the following form of $\Lambda DP$-vertex based on the results of [36,
37]:

$\displaystyle G_{\Lambda
DP}(k^{2},M_{D}^{2})=\frac{f_{D}\,g_{DN\Lambda}}{m_{u}+m_{c}},\qquad k^{2}<0,$
(24)

where $f_{D}\approx 180~{}\mbox{MeV}$ and

$\displaystyle\frac{g_{DN\Lambda}}{\sqrt{4\pi}}=1.9\pm 0.6.$ (25)

For quark masses the following values are used: $m_{u}\approx
280~{}\mbox{MeV}$ and $m_{c}=1.27~{}\mbox{GeV}$ [28]. The $SU(4)$ symmetry
leads us to the same result for $G_{\Lambda D\Xi}$:

$\displaystyle G_{\Lambda D\Xi}(k^{2},M_{D}^{2})=\frac{f_{D}\,g_{\Lambda
D\Xi}}{m_{u}+m_{c}},\qquad k^{2}<0,$ (26)

where

$\displaystyle g_{\Lambda D\Xi}\approx g_{DN\Lambda}=6.7\pm 2.1.$ (27)

##  V Three gluon mechanism

Figure 5: Three gluon mechanism.

The three gluon mechanism first was considered in [26] and in [27] it was
refined and some misprints and minor mistakes were corrected. So here we just
present the final formulae for its contribution to the interference of a
charmonium state with Born amplitude (see (17)):

$\displaystyle
S_{3g}\left(s\right)=\alpha_{3g}\left(s\right)\,Z_{3g}\left(s\right),$ (28)

where

$\displaystyle\alpha_{3g}\left(s\right)$
$\displaystyle=\frac{\alpha\,\alpha_{s}^{3}}{2^{3}\,3}g_{e}\,g_{col}\,\phi\,\beta\,F_{1}\left(s\right)\,G_{\psi}(s),$
(29) $\displaystyle Z_{3g}\left(s\right)$
$\displaystyle=\frac{4}{\pi^{5}s}\int\frac{dk_{1}}{k_{1}^{2}}\frac{dk_{2}}{k_{2}^{2}}\frac{dk_{3}}{k_{3}^{2}}\times$
$\displaystyle\times\frac{Sp3g~{}\delta\left(q-k_{1}-k_{2}-k_{3}\right)}{(\left(p_{1}-k_{1}\right)^{2}-M_{\Lambda}^{2})(\left(p_{2}-k_{3}\right)^{2}-M_{\Lambda}^{2})},$
(30)

where the quantity
$g_{col}=(1/4)\left<\Lambda\right|d^{ijk}~{}T^{i}T^{j}T^{k}\left|\Lambda\right>=15/2$
is the color factor and $Sp3g$ is the product of traces over the
$\Lambda$-hyperon and the $c$-quark lines:

$\displaystyle Sp3g$
$\displaystyle=\mbox{Sp}\left[\hat{Q}_{\alpha\beta\gamma}({\hat{p}_{c}}+m_{c})\gamma^{\mu}({\hat{p}_{\bar{c}}}-m_{c})\right]\times$
$\displaystyle\times\mbox{Sp}\left[({\hat{p}_{1}}+M_{\Lambda})\gamma^{\alpha}({\hat{p}_{1}}-{\hat{k}_{1}}+M_{\Lambda})\gamma^{\beta}\right.\times$
$\displaystyle\qquad\times\left.(-{\hat{p}_{2}}+{\hat{k}_{3}}+M_{\Lambda})\gamma^{\gamma}({\hat{p}_{2}}-M_{\Lambda})\gamma_{\mu}\right],$

with

$\displaystyle\hat{Q}_{\alpha\beta\gamma}$
$\displaystyle=\frac{\gamma_{\gamma}(-{\hat{p}_{\bar{c}}}+{\hat{k}_{3}}+m_{c})\gamma_{\beta}({\hat{p}_{c}}-{\hat{k}_{1}}+m_{c})\gamma_{\alpha}}{((p_{\bar{c}}-k_{3})^{2}-m_{c}^{2})((p_{c}-k_{1})^{2}-m_{c}^{2})}+$
$\displaystyle+\left[\text{gluon permutations}\right],$ (31)

where the permutations over gluon vertices are performed in the gray block in
Fig. 5.

The quantity $\phi$ in (29) is related to the charmonium wave function
$\psi\left({\bf r}\right)$ as:

$\displaystyle\phi=\frac{\left|\psi\left({\bf r}={\bf
0}\right)\right|}{M_{\psi}^{3/2}}=\frac{\alpha_{s}^{3/2}}{3\sqrt{3\pi}},$ (32)

where $\alpha_{s}$ is the QCD coupling constant. We should note that this
three gluon mechanism is very sensitive to this quantity, since it depends on
its value in a rather high degree (see eqs. (29) and (32)). We use the value
$\alpha_{s}(M_{c})=0.28$ which is expected by the QCD evolution of
$\alpha_{s}$ from the $b$-quark scale to the $c$-quark scale, i.e. to $s\sim
M_{c}^{2}$. We note that this value differs from the one for the $J/\psi$
charmonium which tends to feel much smaller value of parameter
$\alpha_{s}(M_{c})=0.19$ [38].

The factor $G_{\psi}(s)$ in (29) is the form factor which describes the
mechanism of transition of three gluons (with total angular momentum equal to
$1$) into final $\Lambda\bar{\Lambda}$ pair. Following to [27] we suggest that
this mechanism has much in common with the proton–antiproton pair production
from the photon:

$\displaystyle\left|G_{\psi}(s)\right|=\frac{C_{\psi}}{s^{2}\log^{2}\left(s/\Lambda^{2}\right)}.$
(33)

For the constant $C_{\psi}$ here we use the same value as it was onbtained in
the case of proton–antiproton production [27] since gluons to not feel the
flavour of the quarks in the final baryons:

$\displaystyle C_{\psi}=\left(45\pm 9\right)~{}\mbox{GeV}^{4}.$ (34)

##  VI Numerical results

Figure 6:  The quantity $Z_{D}\left(s\right)$ from (19) as a function of
$\sqrt{s}$ starting from the threshold $\sqrt{s}=2M_{\Lambda}$. The vertical
dashed line shows the position of $\psi(3770)$.  Figure 7:  The quantity
$Z_{3g}\left(s\right)$ from (30) as a function of $\sqrt{s}$ starting from the
threshold $\sqrt{s}=2M_{\Lambda}$. The vertical dashed line shows the position
of $\psi(3770)$.

The main building blocks for the cross section are the quantities
$Z_{D}\left(s\right)$ from (19) and $Z_{3g}\left(s\right)$ from (30) which
give the corresponding ($D$-meson loop and three gluon) contributions. In Fig.
6 we present the dependence of $Z_{D}\left(s\right)$ as a function of total
energy $\sqrt{s}$ in the range starting from the threshold of the reaction
$\sqrt{s}=2M_{\Lambda}$ up to $4.5~{}\mbox{GeV}$. One can see that the shape
and the numerical values of the real and the imaginary parts of this quantity
remain the same as it were in the case of proton–antiproton final state (see
Fig. 7 (a) in [27]). As for the Fig. 7 presenting real and imaginary parts of
the quantity $Z_{3g}\left(s\right)$ it shows the similar general behavior of
the curves as it were in the case of proton–antiproton final state (see Fig. 7
(b) in [27]) but the numerical difference is much more seeable. Nevertheless
the characteristic large negative values of this quantity still remains thus
giving large relative phase with respect to the Born contribution in the
amplitude.

Figure 8:  The $D$-meson loop contributing to the total cross section with
respect to the BES III data [20].  Figure 9:  The three gluon contributing to
the total cross section with respect to the BES III data [20].  Figure 10:
The total cross section including two mechanisms in comparison with the BES
III data [20].

In Fig. 8 one can see the contributions from pure $D$-meson loop while in Fig.
9 the pure three gluons contribution is present. Both of these contributions
are compared with the data of BES III collaboration [20] in the vicinity of
$\psi(3770)$ resonance. For $D$-meson loop contributions, Fig. 8, the error
bands are provided by the errors of parameter $g_{\Lambda D\Xi}$ from (27).
One can see that the central value of the curve goes lower then the
experimental point while the upper error band touches it. For the three gluons
contribution, Fig. 9, the error bands represent the uncertainty of the
parameter $C_{\psi}$ from (34). We do not include to this error bands the
possible uncertainties due to the strong dependence of this mechanism of
parameter $\alpha_{s}$ (see text after eq. (32)). Here we see a good agreement
of this pure three gluon mechanism with the data point at $\sqrt{s}=M_{\psi}$.

In Fig. 10 we present the total cross section including both of these
mechanisms in comparison with the BES III data [20]. We see rather fair
agreement of our calculation with the data: the point at $\sqrt{s}=M_{\psi}$
is rather close to the curve and the left and right shoulders of the curve
catch the tendency of the data. Here we must remind that we do not do any
extra fit of parameters. All the parameters of our model are fixed by the
calculation for case of the proton–antiproton final state [27].

Figure 11:  The relative phase of total GeV charmonium $\psi(3770)$
contribution as a function of $\sqrt{s}$.

In Fig. 11 one can see the total relative phase $\phi_{\psi}$ of the
charmonium contribution ${\cal M}_{\psi}$ to the amplitude with respect to the
Born contribution ${\cal M}_{B}$ without Breit-Wigner factor, i.e.:

$\displaystyle
S_{D}\left(s\right)+S_{3g}\left(s\right)=\left|S\left(s\right)\right|e^{i\phi_{\psi}},$
(35)

where $S_{D}\left(s\right)$ was defined in (18) and $S_{3g}\left(s\right)$ is
from (28). The error bands on this plot are due to both $g_{\Lambda D\Xi}$ and
$C_{\psi}$ parameters uncertainties. As one can see at the point of
$\psi(3770)$ charmonium the relative phase is rather large:

$\displaystyle\phi_{\psi}\approx 202^{\circ}.$ (36)

It seems that this feature is common for the charmonium decay into two baryons
final state. We showed this in proton–antiproton final state for the
charmonium $\psi(3770)$ in papers [26, 27] and for the charmonium
$\chi_{c2}(1P)(3556)$ in paper [35].

##  VII Conclusion

We considered the process of the electron–positron annihilation into a
$\Lambda\bar{\Lambda}$-pair in the vicinity of charmonium $\psi(3770)$
resonance. Besides the Born mechanism, which is the pure QED, there are two
contributions related with the intermediate charmonium $\psi(3770)$ state. One
of them is the $D$-meson loop and the other is the three gluon mechanism.

It was shown that both mechanisms contribute noticeably and give much of the
final result. The total sum of them gives rather good agreement with the
experimental point at $\sqrt{s}=M_{\psi}$. It is also important that our curve
reproduce the tendency of the experimental points at left and at right
shoulders with respect to the central point. It is worth to notice again that
we do not use any fitting procedure in this calculation. The parameters were
fixed for the proton-antiproton production channel in the paper [27].

It is extremely desirable to make a precise scan over the energy region around
charmonium $\psi(3770)$ resonance with the small steps. This could support the
conclusion that in charmonium decay the phase of $\psi\to p\bar{p}$ and
$\psi\to\Lambda\bar{\Lambda}$ vertexes are large ($\phi_{\psi}\sim
200^{\circ}$) and can be precisely measured in this channels. We showed this
large phase generation in a set of papers [26, 35, 27].

In the future we plan to consider another binary final states production
processes induced by the charmonium annihilation. Polarized effects are also
can be considered since the formalism is ready and the data are present [39].

###### Acknowledgements.

The authors wish to express their gratitude to Dr. V.A. Zykunov and Dr. E.
Tomasi-Gustafsson for intensive and valuable discussions.

## References

  * [1] D.M. Asner et al.,  Int. J. Mod. Phys. A 24, S1 (2009).
  * [2] BESIII, M. Küßner,  J. Phys. Conf. Ser. 1667, 012024 (2020).
  * [3] PANDA, F. Nerling,  PoS CHARM2020, 004 (2021).
  * [4] C.-Z. Yuan,  Natl. Sci. Rev. 8, nwab182 (2021).
  * [5] DM2, D. Pallin et al.,  Nucl. Phys. B 292, 653 (1987).
  * [6] N. Cabibbo and R. Gatto,  Phys. Rev. Lett. 4, 313 (1960).
  * [7] N. Cabibbo and R. Gatto,  Phys. Rev. 124, 1577 (1961).
  * [8] BaBar, J.P. Lees et al.,  Phys. Rev. D 87, 092005 (2013).
  * [9] BaBar, J.P. Lees et al.,  Phys. Rev. D 88, 072009 (2013).
  * [10] BESIII, M. Ablikim et al.,  Phys. Rev. D 91, 112004 (2015).
  * [11] BaBar, B. Aubert et al.,  Phys. Rev. D 76, 092006 (2007).
  * [12] S. Dobbs, K.K. Seth, A. Tomaradze, T. Xiao, and G. Bonvicini,  Phys. Rev. D 96, 092004 (2017).
  * [13] BESIII, M. Ablikim et al.,  Phys. Lett. B 814, 136110 (2021).
  * [14] S. Dobbs, A. Tomaradze, T. Xiao, K.K. Seth, and G. Bonvicini,  Phys. Lett. B 739, 90 (2014).
  * [15] X. Zhou, L. Yan, R.B. Ferroli, and G. Huang,  Symmetry 14, 144 (2022).
  * [16] BESIII, M. Ablikim et al.,  Phys. Rev. D 97, 032013 (2018).
  * [17] BESIII, M. Ablikim et al.,  Phys. Rev. D 103, 012005 (2021).
  * [18] J. Haidenbauer, H.W. Hammer, U.-G. Meissner, and A. Sibirtsev,  Phys. Lett. B 643, 29 (2006).
  * [19] Z.-Y. Li, A.-X. Dai, and J.-J. Xie,  Chin. Phys. Lett. 39, 011201 (2022).
  * [20] BESIII, M. Ablikim et al.,  Phys. Rev. D 104, L091104 (2021).
  * [21] KEDR, V.V. Anashin et al.,  Nucl. Phys. B Proc. Suppl. 181-182, 353 (2008).
  * [22] V.V. Anashin et al.,  Phys. Lett. B 711, 292 (2012).
  * [23] CLEO, J.Y. Ge et al.,  Phys. Rev. D 79, 052010 (2009).
  * [24] BES, M. Ablikim et al.,  Phys. Lett. B 668, 263 (2008).
  * [25] BESIII Collaboration, M. Ablikim et al.,  Phys.Lett. B735, 101 (2014).
  * [26] A.I. Ahmadov, Yu.M. Bystritskiy, E.A. Kuraev, and P. Wang,  Nucl.Phys. B888, 271 (2014).
  * [27] Yu.M. Bystritskiy,  Phys. Rev. D 103, 116029 (2021).
  * [28] Particle Data Group, P. Zyla et al.,  PTEP 2020, 083C01 (2020).
  * [29] E. Tomasi-Gustafsson, A. Bianconi, and S. Pacetti,  Phys. Rev. C 103, 035203 (2021).
  * [30] G. Lepage and S.J. Brodsky,  Phys. Rev. Lett. 43, 545 (1979),  [Erratum: Phys.Rev.Lett. 43, 1625–1626 (1979)].
  * [31] G. Lepage and S.J. Brodsky,  Phys. Rev. D 22, 2157 (1980).
  * [32] A. Bianconi and E. Tomasi-Gustafsson,  Phys. Rev. Lett. 114, 232301 (2015).
  * [33] J. Haidenbauer, X.W. Kang, and U.G. Meißner,  Nucl. Phys. A 929, 102 (2014).
  * [34] A. Amoroso et al.,  Universe 7, 436 (2021).
  * [35] E.A. Kuraev, Yu.M. Bystritskiy, and E. Tomasi-Gustafsson,  Nucl.Phys. A920, 45 (2013).
  * [36] L. Reinders, H. Rubinstein, and S. Yazaki,  Phys.Rept. 127, 1 (1985).
  * [37] F. Navarra and M. Nielsen,  Phys.Lett. B443, 285 (1998).
  * [38] H. Chiang, J. Hufner, and H. Pirner,  Phys. Lett. B 324, 482 (1994).
  * [39] BESIII, M. Ablikim et al.,  Phys. Rev. D 105, L011101 (2022).


# A block theoretic proof of Thompson’s $A\times B$-Lemma

Radha Kessar and Markus Linckelmann

###### Abstract.

We show that Thompson’s $A\times B$-Lemma can be obtained as a consequence of
the Brauer pair version of Brauer’s Third Main Theorem.

Let $k$ be an algebraically closed field of prime characteristic $p$. Brauer’s
Third Main Theorem [2, Theorem 3], rephrased using Brauer pairs (cf. [1,
Theorem 3.13] or [6, Theorem 6.3.14]), states that if $b$ is the principal
block idempotent of a finite group algebra $kG$, then $\mathrm{Br}_{Q}(b)$ is
the principal block idempotent of $kC_{G}(Q)$, for any $p$-subgroup $Q$ of
$G$. Here $\mathrm{Br}_{Q}:(kG)^{Q}\to kC_{G}(Q)$ denotes the Brauer
homomorphism (cf. [3, §1.2] or [5, Theorem 5.4.1]). In particular, if $kG$ has
a unique block, then $kC_{G}(Q)$ has a unique block. We use this to give a
proof of the following result.

######  Theorem 1 (Thompson’s $A\times B$-Lemma; cf. [4, Chapter 5, Theorem
3.4]).

Let $A\times B$ be a subgroup of the automorphism group of a finite $p$-group
$P$, with $A$ a $p^{\prime}$-group and $B$ a $p$-group. If $A$ acts trivially
on $C_{P}(B)$, then $A=1$.

###### Proof.

Consider the group $G=P\rtimes(A\times B)$, where the notation is as in the
statement, and suppose that $A$ acts trivially on $C_{P}(B)$. Note that
$S=P\rtimes B$ is the unique, and hence normal, Sylow $p$-subgroup of $G$, and
that $A$ can be regarded as a $p^{\prime}$-subgroup of the automorphism group
of $S$. Thus $kG$ has a unique block (cf. [6, Corollary 6.2.9]).

By the assumptions, $A$ acts trivially on the group $Q=C_{P}(B)\times B$. That
is, we have $A\leq C_{G}(Q)$, and hence $C_{G}(Q)=C_{S}(Q)\rtimes A$. By the
above, $kC_{G}(Q)$ has a unique block. We show now that $Q$ is self-
centralising in $S$. Let $x\in C_{S}(Q)$. Write $x=yu$ for some $y\in P$ and
$u\in B$. Since $u\in B$, it follows that conjugation by $u$ preserves the
decomposition $Q=C_{P}(B)\times B$. Thus conjugation by $y$ preserves this
decomposition as well. In particular, $y$ normalises $B$. By elementary group
theory, it follows that $y$ centralises $B$. Indeed, if $u\in$ $B$, then
$yuy^{-1}u^{-1}\in P\cap B=1$. This shows that $C_{S}(Q)\leq Q\leq C_{G}(A)$,
so $C_{G}(Q)=C_{S}(Q)\times A$. But then $kC_{G}(Q)$ has as many blocks as $A$
has irreducible characters, and so we conclude that $A=1$. ∎

## References

  * [1] J. L. Alperin, M. Broué, Local Methods in Block Theory, Ann. Math. 110 (1979), 143–157.
  * [2] R. Brauer, Some applications of the theory of blocks of characters of finite groups I, J. Algebra 1 (1964), 152–167.
  * [3] M. Broué, Brauer Coefficients of $p$-Subgroups Associated with a $p$-Block of a Finite Group, J. Algebra 56 (1979), 365–383.
  * [4] D. Gorenstein, Finite Groups, Chelsea Publishing Company, New York (1980).
  * [5] M. Linckelmann, The block theory of finite group algebras I, Cambridge University Press, London Math. Soc. Student Texts 91 (2018).
  * [6] M. Linckelmann, The block theory of finite group algebras II, Cambridge University Press, London Math. Soc. Student Texts 92 (2018).