# An upper bound and a characterization for Gini’s mean difference based on
correlated random variables

Roberto Vila<EMAIL_ADDRESS>Department of Statistics, University of
Brasília, Brasília, Brazil  Department of Mathematics and Statistics, McMaster
University, Hamilton, Ontario, Canada  Narayanaswamy Balakrishnan
<EMAIL_ADDRESS>Department of Mathematics and Statistics, McMaster
University, Hamilton, Ontario, Canada  Helton Saulo<EMAIL_ADDRESS>Department of Statistics, University of Brasília, Brasília, Brazil

###### Abstract

In this paper, we obtain an upper bound for the Gini mean difference based on
mean, variance and correlation for the case when the variables are correlated.
We also derive some closed-form expressions for the Gini mean difference when
the random variables have an absolutely continuous joint distribution. We then
examine some particular examples based on elliptically contoured
distributions, and specifically multivariate normal and Student-$t$
distributions.

Keywords: Gini’s mean difference $\cdot$ Gini index $\cdot$ Correlation
$\cdot$ Elliptically contoured distributions $\cdot$ Exchangeable variables
$\cdot$ Maximum $\cdot$ Minimum $\cdot$ Skew symmetric distribution.

##  1 Introduction

Based on the random variables $X_{1},\ldots,X_{n}$, the Gini mean difference
(GMD) is defined as

$\displaystyle GMD_{n}=\dfrac{1}{\displaystyle\binom{n}{2}}\sum_{1\leqslant
i<j\leqslant n}{\mathbb{E}(|X_{i}-X_{j}|)},$ (1)

provided the involved expectations exist. When $X_{1},\ldots,X_{n}$ is a
random sample (sequence of independent and identically distributed random
variables), we obtain the classical GMD (see La Haye and Zizler,, 2019;
Schezhtman and Yitzhaki,, 1987), given by

$\displaystyle GMD=\mathbb{E}(|X_{1}-X_{2}|).$ (2)

The GMD is a very useful measure of variability in the presence of non-
normality. In these cases, the GMD is superior to variance, in three different
aspects: stochastic dominance, exchangeability and stratification; see
Yitzhaki, (2003) and Yitzhaki, (2015) for details on other interesting
features of the GMD. This measure has also been reported to be superior than
Pearson and Spearman correlation coefficients for some distributions; see
Schezhtman and Yitzhaki, (1999) and Kattumannil et al., (2022). Based on the
GMD, Schmid and Semeniuk, (2021) introduced tests on positive correlation and
methods for monitoring the correlation structure of a process. Interesting
applications of the GMD can also be found in survival analysis; as discussed
by Bonetti et al., (2009), for example.

This work explores some properties of the GMD. First, in Section 2, we derive
an upper bound for the GMD which holds for all distributions with finite
second moment. This upper bound is a refinement of the one derived by Cerone
and Dragomir, (2005). In addition, the derived upper bound does not require
independent random variables, as required in the works of Cerone and Dragomir,
(2005) and Yin et al., (2022). Next, in Section 3, we provide more informative
results and present in particular methods for the exact calculation of GMD for
jointly distributed absolutely continuous random variables. Finally, in
Section 4, we derive the GMD for the multivariate normal and Student-$t$
distributions to illustrate the results, developed here.

##  2 Upper bound for GMD

The following theorem presents an upper bound for the GMD that holds for all
distributions with finite second moment.

######  Theorem 1.

Let $X_{1},\ldots,X_{n}$ be random variables with finite second moments, and
let $\mu_{i}=\mathbb{E}(X_{i})$, $\sigma_{i}^{2}={\rm Var}(X_{i})$ and
$\rho_{i,j}={\rm Corr}(X_{i},X_{j})$, for $i,j=1,\ldots,n$. Then, we have

$\displaystyle
GMD_{n}\leqslant\dfrac{1}{\displaystyle\binom{n}{2}}\sum_{1\leqslant
i<j\leqslant
n}\left[\sqrt{(\sigma_{i}-\sigma_{j}\rho_{i,j})^{2}+\sigma_{j}^{2}(1-\rho_{i,j}^{2})}+|\mu_{i}-\mu_{j}|\right].$

###### Proof.

Writing $|X_{i}-X_{j}|=|Y_{i}-Y_{j}+\mu_{i}-\mu_{j}|$, where
$Y_{i}=X_{i}-\mu_{i}$ and $Y_{j}=X_{j}-\mu_{j}$, and then applying triangular
inequality, we get $|X_{i}-X_{j}|\leqslant|Y_{i}-Y_{j}|+|\mu_{i}-\mu_{j}|$.
Now, by Jensen’s inequality, we then have

$\displaystyle\mathbb{E}(|X_{i}-X_{j}|)$
$\displaystyle\leqslant\sqrt{\mathbb{E}(|Y_{i}-Y_{j}|^{2})}+|\mu_{i}-\mu_{j}|$
$\displaystyle=\sqrt{{\rm Var}(Y_{i}-Y_{j})}+|\mu_{i}-\mu_{j}|=\sqrt{{\rm
Var}(X_{i}-X_{j})}+|\mu_{i}-\mu_{j}|,$

where we have used the invariance with regard to translations of the variance
in the last equality. Upon using the known identity that ${\rm
Var}(X_{i}-X_{j})=\sigma_{i}^{2}+\sigma_{j}^{2}-2{\rm Cov}(X_{i},X_{j})$, the
expression on the right-hand side can be rewritten as

$\displaystyle=\sqrt{\sigma_{i}^{2}+\sigma_{j}^{2}-2{\rm
Cov}(X_{i},X_{j})}+|\mu_{i}-\mu_{j}|=\sqrt{\sigma_{i}^{2}+\sigma_{j}^{2}-2\sigma_{i}\sigma_{j}\rho_{i,j}}+|\mu_{i}-\mu_{j}|.$

Then, from the definition of the GMD in (1), the proof gets completed. ∎

In Theorem 1, upon taking $\mu_{1}=\mathbb{E}(X_{i})$, $\sigma_{1}^{2}={\rm
Var}(X_{i})$ and $\rho_{i,j}={\rm Corr}(X_{i},X_{j})$, for $i,j=1,\ldots,n$,
we deduce

$\displaystyle
GMD_{n}\leqslant\sqrt{2}\sigma_{1}\,\dfrac{1}{\displaystyle\binom{n}{2}}\sum_{1\leqslant
i<j\leqslant n}\sqrt{1-\rho_{i,j}}.$ (3)

The above inequality has appeared in Schmid and Semeniuk, (2021, Theorem 9.1).
Furthermore, if we take $n=2$ in (3), we obtain
$GMD_{2}={\mathbb{E}(|X_{1}-X_{2}|)}\leqslant\sqrt{2}\sigma_{1}\sqrt{1-\rho_{1,2}}$.
So, for $\rho_{1,2}=0$,

$\displaystyle GMD_{2}\leqslant\sqrt{2}\sigma_{1}.$ (4)

We emphasize that the upper bound in (4) refines the upper bound of
$(4/\sqrt{3})\sigma_{1}$ in Cerone and Dragomir, (2005), but not the one of
$(2/\sqrt{3})\sigma_{1}$ presented in Yin et al., (2022, Corollary 3.1).
Another point worth mentioning is that the bound in (4) is for variables that
are not necessarily independent, as required in the results of Cerone and
Dragomir, (2005) and Yin et al., (2022).

Let us now suppose that $X_{1},\ldots,X_{n},$ are independent and identically
distributed random variables with mean $\mu_{1}$ and variance
$\sigma_{1}^{2}$, and that the variable $Z_{1}$, the standardized version of
$X_{1}$, has a norm in $L^{p}$, denoted by $\lVert
Z\rVert_{p}=[\mathbb{E}(|Z|^{p})]^{1/p}$, $p>1$. Moreover, let us denote
$X_{(1)}=\min\\{X_{1},X_{2}\\}$ and $X_{(2)}=\max\\{X_{1},X_{2}\\}$ for the
smallest and largest of the variables $X_{1}$ and $X_{2}$, respectively. Then,
from Barry, (1985), it follows that
$\mathbb{E}(X_{(2)})\leqslant\mu_{1}+\lVert
Z_{1}\rVert_{p}[({p-1})/({2p-1})]^{(p-1)/p}\sigma_{1},\ p>1,$ and
consequently, the GMD in (2) is such that

$\displaystyle
GMD=\mathbb{E}(X_{(2)}-X_{(1)})=2[\mathbb{E}(X_{(2)})-\mu_{1}]\leqslant
C_{p}\sigma_{1},$ (5)

with $C_{p}=2\lVert Z_{1}\rVert_{p}[(p-1)/(2p-1)]^{(p-1)/p}.$ If we take $p=2$
in (5), the upper bound $C_{2}\sigma_{1}=(2/\sqrt{3})\sigma_{1}$ of Yin et
al., (2022, Corollary 3.1) is obtained. Further, depending on the choice of
distribution of $X_{1}$, it is possible to find values of $1<p<2$ such that
the upper bound $C_{p}\sigma_{1}$ is slightly better than
$(2/\sqrt{3})\sigma_{1}$. For example, it is not difficult to verify that, for
$X_{1}\sim N(0,1)$ and $p=3/2$, we have
$C_{3/2}\sigma_{1}={[\Gamma({1/4})]^{2/3}}\sigma_{1}/({\sqrt{2}\sqrt[3]{\pi}})\approx
1.14\sigma_{1}<C_{2}\sigma_{1}=({2/\sqrt{3}})\sigma_{1}\approx
1.15\sigma_{1}.$

##  3 Closed-form for GMD

In this section, we assume that the random variables $X_{1},\ldots,X_{n}$ are
absolutely continuous and are not necessarily independent. To state the main
result of this section (in Theorem 2), we write

$\displaystyle h_{i,j}(x)=f_{X_{j}}(x)\,{\pi_{i,j}(x)\over R_{i,j}},$ (6)

where $\pi_{i,j}(x)=F_{X_{i}|(X_{j}=x)}(x)$ denotes the conditional cumulative
distribution function (CDF) of $X_{i}|(X_{j}=y)$ evaluated at $y=x$, and
$R_{i,j}=\mathbb{P}(X_{i}\leqslant X_{j})=\mathbb{E}[\pi_{i,j}(X_{j})]$ is the
stress-strength reliability.

######  Proposition 1.

The function $h_{i,j}$ in (6) is a probability density function (PDF).

###### Proof.

A simple algebraic manipulation shows that

$\displaystyle h_{i,j}(x)=f_{X_{i}}(x)\,\dfrac{\pi_{j,i}(x)}{R_{j,i}}$
$\displaystyle=f_{X_{i}}(x)\,\dfrac{\mathbb{P}(X_{i}\geqslant
X_{j}|X_{i}=x)}{R_{j,i}}$ (7)
$\displaystyle=f_{X_{i}}(x)\,\dfrac{\displaystyle\int_{0}^{\infty}f_{X_{i}-X_{j}|(X_{i}=x)}(y){\rm
d}y}{R_{j,i}}$
$\displaystyle=\dfrac{\displaystyle\int_{0}^{\infty}f_{X_{i},X_{i}-X_{j}}(x,y){\rm
d}y}{R_{j,i}}=f_{X_{i}|(X_{i}\geqslant X_{j})}(x).$ (8)

So, from the above identities, it is evident that $h_{i,j}$ is indeed a PDF. ∎

Let $H_{i,j}$ be the CDF corresponding to $h_{i,j}$. From here on, we assume
that all involved expected values exist.

######  Theorem 2.

Let $\bm{X}=(X_{1},\ldots,X_{n})^{\top}$ be an absolutely continuous random
vector. Then the GMD in (1) can be expressed as

$\displaystyle GMD_{n}=\dfrac{1}{\displaystyle\binom{n}{2}}\sum_{1\leqslant
i<j\leqslant n}(2R_{j,i}\mu_{H_{j,i}}+2R_{i,j}\mu_{H_{i,j}}-\mu_{i}-\mu_{j}),$

where $\mu_{i}=\mathbb{E}(X_{i})$ and $\mu_{H_{i,j}}$ denotes the expected
value with respect to the distribution $H_{i,j}$.

###### Proof.

Let $m_{ij}=\min\\{X_{i},X_{j}\\}$ and $M_{ij}=\max\\{X_{i},X_{j}\\}$,
$1\leqslant i<j\leqslant n$. Then the GMD in (1) can be written as

$\displaystyle GMD_{n}=\dfrac{1}{\displaystyle\binom{n}{2}}\sum_{1\leqslant
i<j\leqslant n}\mathbb{E}(M_{ij}-m_{ij}).$ (9)

In order to find a closed-form expression for the GMD, from (9), it is
essential to know the PDFs of $M_{ij}$ and $m_{ij}$. Indeed, the law of total
probability gives

$\displaystyle F_{M_{ij}}(x)$ $\displaystyle=\mathbb{P}(X_{i}\leqslant
x|X_{i}\geqslant X_{j})R_{j,i}+\mathbb{P}(X_{j}\leqslant x|X_{j}\geqslant
X_{i})R_{i,j},$

where $R_{i,j}$ is as given in (6). Then, by using (8), the PDF of $M_{ij}$ is

$\displaystyle f_{M_{ij}}(x)$ $\displaystyle=f_{X_{i}|(X_{i}\geqslant
X_{j})}(x)R_{j,i}+f_{X_{j}|(X_{j}\geqslant X_{i})}(x)R_{i,j}$
$\displaystyle=f_{X_{i}}(x)\pi_{j,i}(x)+f_{X_{j}}(x)\pi_{i,j}(x),$ (10)

where $\pi_{j,i}$ is as defined in (6).

Now, upon using the well-known identities that
$\min\\{x,y\\}=-\max\\{-x,-y\\}$, $f_{-X_{i}}(-x)=f_{X_{i}}(x)$ and
$F_{-X_{j}|(-X_{i}=-x)}(-x)=1-\pi_{j,i}(x)$, we can write the PDF of
$m_{ij}=\min\\{X_{i},X_{j}\\}$ as

$\displaystyle f_{m_{ij}}(x)$ $\displaystyle=f_{\max\\{-X_{i},-X_{j}\\}}(-x)$
$\displaystyle\stackrel{{\scriptstyle\eqref{pdf-2}}}{{=}}f_{-X_{i}}(-x)F_{-X_{j}|(-X_{i}=-x)}(-x)+f_{-X_{j}}(-x)F_{-X_{i}|(-X_{j}=-x)}(-x)$
$\displaystyle=f_{X_{i}}(x)[1-\pi_{j,i}(x)]+f_{X_{j}}(x)[1-\pi_{i,j}(x)]$
$\displaystyle\stackrel{{\scriptstyle\eqref{pdf-2}}}{{=}}f_{X_{i}}(x)+f_{X_{j}}(x)-f_{M_{ij}}(x).$
(11)

Therefore, by (3) and (3), the GMD in (9) can be expressed as

$\displaystyle GMD_{n}$
$\displaystyle=\dfrac{1}{\displaystyle\binom{n}{2}}\sum_{1\leqslant
i<j\leqslant
n}\int_{-\infty}^{\infty}x\,\big{[}f_{M_{ij}}(x)-f_{m_{ij}}(x)\big{]}{\rm d}x$
$\displaystyle=\dfrac{1}{\displaystyle\binom{n}{2}}\sum_{1\leqslant
i<j\leqslant
n}\int_{-\infty}^{\infty}x\,\big{[}2f_{X_{i}}(x)\pi_{j,i}(x)+2f_{X_{j}}(x)\pi_{i,j}(x)-f_{X_{i}}(x)-f_{X_{j}}(x)\big{]}{\rm
d}x$ (12) $\displaystyle=\dfrac{1}{\displaystyle\binom{n}{2}}\sum_{1\leqslant
i<j\leqslant
n}\int_{-\infty}^{\infty}x\,\big{[}2R_{j,i}h_{j,i}(x)+2R_{i,j}h_{i,j}(x)-f_{X_{i}}(x)-f_{X_{j}}(x)\big{]}{\rm
d}x,$

where $R_{i,j}$ and $h_{i,j}$ are as given in (6). This completes the proof of
theorem. ∎

A similar derivation of the PDF of the maximum of $n$ random variables has
also been given by Arellano-Valle and Genton, (2008).

In order to state the next result, we adopt the notation
$g_{i,j}(x)=2f_{X_{j}}(x)\pi_{i,j}(x)$, where
$\pi_{i,j}(x)=F_{X_{i}|(X_{j}=x)}(x)$, $\pi_{i,j}$ is a skewing function,
i.e., it satisfies $0\leqslant\pi_{i,j}(x)\leqslant 1$ and
$\pi_{i,j}(-x)=1-\pi_{i,j}(x)$, and $f_{X_{j}}$ is symmetric around $0$. Under
these conditions, it is clear that $g_{i,j}$ is a skew-symmetric PDF, and let
$G_{i,j}$ be the corresponding CDF.

######  Corollary 1.

Let $\bm{X}=(X_{1},\ldots,X_{n})^{\top}$ be an absolutely continuous random
vector. If $\pi_{i,j}$ is a skewing function and $f_{X_{j}}$ is symmetric
around $0$, then the GMD is given by

$\displaystyle GMD_{n}=\dfrac{1}{\displaystyle\binom{n}{2}}\sum_{1\leqslant
i<j\leqslant n}(\mu_{G_{j,i}}+\mu_{G_{i,j}}-\mu_{i}-\mu_{j}),$

with $\mu_{i}=\mathbb{E}(X_{i})$ and $\mu_{G_{i,j}}$ being the expected value
with respect to the distribution $G_{i,j}$.

###### Proof.

The proof follows directly from the identity in (12). ∎

######  Proposition 2.

Let $\bm{X}=(X_{1},\ldots,X_{n})^{\top}$ be an absolutely continuous
exchangeable random vector. If $\pi_{1,2}$ is a skewing function and
$f_{X_{1}}$ is symmetric around $0$, then the GMD is given by

$\displaystyle GMD_{n}=\dfrac{2}{\displaystyle\binom{n}{2}}\sum_{1\leqslant
i<j\leqslant n}(\mu_{G^{*}_{j,i}}-\mu_{1}),$

where $\mu_{1}=\mathbb{E}(X_{1})$ and $\mu_{G^{*}_{j,i}}$ denotes the expected
value with respect to the skew-symmetric CDF
$G^{*}_{j,i}(x)=\int_{-\infty}^{x}2f_{X_{i}}(t)F_{X_{j}}(t){\rm d}t$.

###### Proof.

Because $\bm{X}$ is exchangeable, we have
$(X_{1},\ldots,X_{n})^{\top}\stackrel{{\scriptstyle\mathscr{D}}}{{=}}(X_{i_{1}},\ldots,X_{i_{n}})^{\top}$,
for each permutation $(i_{1}\ldots,i_{n})$ of $(1,\ldots,n)$. Here,
$\stackrel{{\scriptstyle\mathscr{D}}}{{=}}$ means that the two sides of the
equality have the same distribution. Consequently,
$\mu_{G_{j,i}}=\mu_{G_{i,j}}$ and $\mu_{i}=\mu_{j}=\mu_{1}$, for $1\leqslant
i<j\leqslant n$. Hence, by applying Corollary 2, the required result follows.
∎

As a consequence of Proposition 2, the following result readily follows.

######  Proposition 3.

Let $X_{1},\ldots,X_{n}$ be independent random variables with identical
distributions $F_{X_{1}}$. Then, the classical GMD in (2) is simply

$\displaystyle GMD=\mu_{G_{1}}-\mu_{1}=\int_{0}^{1}(2u-1)F_{X_{1}}^{-1}(u){\rm
d}u,$ (13)

where $\mu_{1}=\mathbb{E}(X_{1})$ and $\mu_{G_{1}}$ is the expected value with
respect to the skew-symmetric CDF
$G_{1}(x)=\int_{-\infty}^{x}2f_{X_{1}}(t)F_{X_{1}}(t){\rm d}t$.

Furthermore, the Gini index, defined by $G=GMD/(2\mu_{1})$, is simply
$G=\left[({\mu_{G_{1}}/\mu_{1}})-1\right]/2$.

Observe that the formula in (13) has appeared recently in the work of Yin et
al., (2022, Proposition 2.1).

##  4 Some examples

In this section, we consider the class of elliptically contoured (EC)
distributions for the $n\times 1$ random vector
$\bm{X}=(X_{1},\ldots,X_{n})^{\top}$. First, we recall that a random vector
$\bm{X}$ has an EC distribution, also called an elliptically symmetric
distribution, if $\bm{X}$ has joint PDF (see Fang et al.,, 1990) as

$\displaystyle
f(\bm{x})=k_{n}|\bm{\Sigma}|^{-1/2}g^{(n)}[(\bm{x}-\bm{\mu})^{\top}\bm{\Sigma}^{-1}(\bm{x}-\bm{\mu})],\quad\bm{x}\in\mathbb{R}^{n}.$

We use the notation $\bm{X}\sim EC_{n}(\bm{\mu},\bm{\Sigma},g^{(n)})$, where
$\bm{\mu}\in\mathbb{R}^{n}$ is a location parameter, $\bm{\Sigma}$ is a scale
matrix (positive-definite real $n\times n$ matrix), $g^{(n)}$ is the PDF
generator and $k_{n}$ is a normalizing constant.

As EC distributions are invariant under marginalization and conditioning (Fang
et al.,, 1990, Theorem 2.16), and our results only require marginal and
conditional laws involving the variables $X_{i}$ and $X_{j}$, for $1\leqslant
i<j\leqslant n$, our analysis is restricted to the marginalization and
conditioning corresponding to the bivariate case. In particular, we now
discuss in detail the multivariate normal and Student-$t$ cases.

###  4.1 Multivariate normal distribution

Let $\bm{X}\sim EC_{n}(\bm{\mu},\bm{\Sigma},g^{(n)})$, where
$g^{(n)}(x)=\exp(-x/2)$ is the PDF generator of the multivariate normal
distribution. It is well-known that the conditional distribution of $X_{i}$,
given $X_{j}=x$, is
$N(\mu_{i}+\rho_{i,j}(x-\mu_{j})\sigma_{i}/\sigma_{j},\sigma_{i}^{2}(1-\rho_{i,j}^{2}))$
and that its unconditional distribution is $X_{j}\sim
N(\mu_{j},\sigma_{j}^{2})$. Let $\phi$ and $\Phi$ be the PDF and CDF of the
standard normal distribution, respectively. By using the standardization of
$X_{i}|(X_{j}=x)$, we have

$\displaystyle\pi_{i,j}(x)=F_{X_{i}|(X_{j}=x)}(x)=\Phi\left({1\over\sqrt{1-\rho_{i,j}^{2}}}\,\left[{{x-\mu_{i}\over\sigma_{i}}-\rho_{i,j}\left({x-\mu_{j}\over\sigma_{j}}\right)}\right]\right).$

Hence, by (6),

$\displaystyle h_{i,j}(x)={1\over
R_{i,j}}\,{1\over\sigma_{j}}\,\phi\left({x-\mu_{j}\over\sigma_{j}}\right)\Phi\left({1\over\sqrt{1-\rho_{i,j}^{2}}}\,\left[{{x-\mu_{i}\over\sigma_{i}}-\rho_{i,j}\left({x-\mu_{j}\over\sigma_{j}}\right)}\right]\right).$
(14)

By using (14) and making the change of variable $z=(x-\mu_{j})/\sigma_{j}$, we
have

$\displaystyle\mu_{H_{i,j}}=\int_{-\infty}^{\infty}x{\rm d}H_{i,j}(x)={1\over
R_{i,j}}\int_{-\infty}^{\infty}(\sigma_{j}z+\mu_{j})\,\phi(z)\Phi\left({1\over\sqrt{1-\rho_{i,j}^{2}}}\,\left[{{\mu_{j}-\mu_{i}\over\sigma_{i}}+\left({\sigma_{j}\over\sigma_{i}}-\rho_{i,j}\right)z}\right]\right){\rm
d}z.$

Using the well-known formulas $\int_{-\infty}^{\infty}x\phi(x)\Phi(a+bx){\rm
d}x=(b/\sqrt{1+b^{2}})\phi(a/\sqrt{1+b^{2}})$ and
$\int_{-\infty}^{\infty}\phi(x)\Phi(a+bx){\rm d}x=\Phi(a/\sqrt{1+b^{2}})$, the
integral on the RHS of the above identity is

$\displaystyle={1\over
R_{i,j}}\left[{\dfrac{\sigma_{j}}{c_{i,j}}\left(\dfrac{\sigma_{j}}{\sigma_{i}}-\rho_{i,j}\right)}\,\phi\left(\dfrac{1}{c_{i,j}}\left(\dfrac{\mu_{j}-\mu_{i}}{\sigma_{i}}\right)\right)+\mu_{j}\Phi\left(\dfrac{1}{c_{i,j}}\left(\dfrac{\mu_{j}-\mu_{i}}{\sigma_{i}}\right)\right)\right],$

where
$c_{i,j}=\sqrt{1-\rho_{i,j}^{2}+[({\sigma_{j}}/{\sigma_{i}})-\rho_{i,j}]^{2}}.$

Thus, by applying Theorem 2, we obtain the following closed-form expression
for the GMD:

$\displaystyle GMD_{n}$
$\displaystyle=\dfrac{2}{\displaystyle\binom{n}{2}}\sum_{1\leqslant
i<j\leqslant
n}\left[{\dfrac{\sigma_{j}}{c_{i,j}}\left(\dfrac{\sigma_{j}}{\sigma_{i}}-\rho_{i,j}\right)}\,\phi\left(\dfrac{1}{c_{i,j}}\left(\dfrac{\mu_{j}-\mu_{i}}{\sigma_{i}}\right)\right)+\mu_{j}\Phi\left(\dfrac{1}{c_{i,j}}\left(\dfrac{\mu_{j}-\mu_{i}}{\sigma_{i}}\right)\right)-{\mu_{j}\over
2}\right]$ $\displaystyle+\dfrac{2}{\displaystyle\binom{n}{2}}\sum_{1\leqslant
i<j\leqslant
n}\left[{\dfrac{\sigma_{i}}{c_{j,i}}\left(\dfrac{\sigma_{i}}{\sigma_{j}}-\rho_{j,i}\right)}\,\phi\left(\dfrac{1}{c_{j,i}}\left(\dfrac{\mu_{i}-\mu_{j}}{\sigma_{j}}\right)\right)+\mu_{i}\Phi\left(\dfrac{1}{c_{j,i}}\left(\dfrac{\mu_{i}-\mu_{j}}{\sigma_{j}}\right)\right)-{\mu_{i}\over
2}\right].$ (15)

######  Proposition 4.

Let $\bm{X}\sim EC_{n}(\bm{\mu},\bm{\Sigma},g^{(n)})$ be an exchangeable
random vector, where $g^{(n)}(x)=\exp(-x/2)$. The GMD is then simply

$\displaystyle
GMD_{n}={2\over\sqrt{\pi}}\,{\sigma_{1}}\,\dfrac{1}{\displaystyle\binom{n}{2}}\sum_{1\leqslant
i<j\leqslant n}{\sqrt{1-\rho_{i,j}}},$

where $\mu_{1}=\mathbb{E}(X_{1})$ and $\sigma_{1}^{2}={{\rm Var}(X_{1})}$.

###### Proof.

As $\bm{X}$ is an exchangeable vector, we have $\mu_{i}=\mu_{j}=\mu_{1}$,
$\sigma_{i}=\sigma_{j}=\sigma_{1}$, $\rho_{i,j}=\rho_{j,i}$, $R_{i,j}=R_{j,i}$
and $c_{i,j}=c_{j,i}=\sqrt{2(1-\rho_{i,j})}$. Then, the required result
follows directly from (4.1). ∎

The expression in Proposition 4 has appeared in the work of Schmid and
Semeniuk, (2021, see below their Theorem 9.1).

###  4.2 Multivariate Student-$t$ distribution

Let $\bm{X}\sim EC_{n}(\bm{\mu},\bm{\Sigma},g^{(n)})$, where
$g^{(n)}(x)=(1+x/\nu)^{-(\nu+n)/2}$ is the PDF generator of the multivariate
Student-$t$ distribution with $\nu$ degrees of freedom. It is well-known that
(see Remark 1 of Saulo et al.,, 2023)

$\displaystyle X_{i}|(X_{j}=x)\sim
t_{\nu+1}\Biggl{(}\mu_{i}+\rho_{i,j}\sigma_{i}\Big{(}{x-\mu_{j}\over\sigma_{j}}\Big{)},{\nu+\left({x-\mu_{j}\over\sigma_{j}}\right)^{2}\over\nu+1}\,\sigma_{i}^{2}(1-\rho_{i,j}^{2})\Biggr{)},$

and that its unconditional distribution is $X_{j}\sim
t_{\nu}(\mu_{j},\sigma_{j}^{2})$. By using the standardization of
$X_{i}|(X_{j}=x)$, we have

$\displaystyle\pi_{i,j}(x)=F_{X_{i}|(X_{j}=x)}(x)$
$\displaystyle=F_{\nu+1}\left(\sqrt{(\nu+1)/(1-\rho_{i,j}^{2})\over\nu+\left({x-\mu_{j}\over\sigma_{j}}\right)^{2}}\,\left[{{x-\mu_{i}\over\sigma_{i}}-\rho_{i,j}\left({x-\mu_{j}\over\sigma_{j}}\right)}\right]\right).$

Hence, by (6), we find

$\displaystyle h_{i,j}(x)={1\over
R_{i,j}}\,{1\over\sigma_{j}}\,f_{\nu}\left({x-\mu_{j}\over\sigma_{j}}\right)F_{\nu+1}\left(\sqrt{(\nu+1)/(1-\rho_{i,j}^{2})\over\nu+\left({x-\mu_{j}\over\sigma_{j}}\right)^{2}}\,\left[{{x-\mu_{i}\over\sigma_{i}}-\rho_{i,j}\left({x-\mu_{j}\over\sigma_{j}}\right)}\right]\right).$
(16)

Here, $f_{\nu}$ and $F_{\nu}$, respectively, denote the PDF and CDF of a
Student-$t$ distribution with $\nu$ degrees of freedom.

So, from the identity in (16), by making the change of variable
$z=(x-\mu_{j})/\sigma_{j}$, it follows that

$\displaystyle\mu_{H_{i,j}}$ $\displaystyle=\int_{-\infty}^{\infty}x{\rm
d}H_{i,j}(x)={1\over
R_{i,j}}\int_{-\infty}^{\infty}(\sigma_{j}z+\mu_{j})f_{\nu}(z)F_{\nu+1}\left(\sqrt{\nu+1\over\nu+z^{2}}\,({\tau_{i,j}+\lambda_{i,j}z})\right){\rm
d}z,$ (17)

where
$\tau_{i,j}=(\mu_{j}-\mu_{i})\big{/}\big{(}\sigma_{i}\sqrt{1-\rho_{i,j}^{2}}\,\big{)}$
and
$\lambda_{i,j}=[(\sigma_{j}/\sigma_{i})-\rho_{i,j}]\big{/}\sqrt{1-\rho_{i,j}^{2}}$.
Multiplying and dividing by
$F_{\nu}\big{(}\tau_{i,j}\big{/}\sqrt{1+\lambda_{i,j}^{2}}\,\big{)}$,
$\mu_{H_{i,j}}$ in (17) can be expressed as

$\displaystyle\mu_{H_{i,j}}={F_{\nu}\left(\tau_{i,j}\Bigl{/}\sqrt{1+\lambda_{i,j}^{2}}\,\right)\over
R_{i,j}}\int_{-\infty}^{\infty}(\sigma_{j}z+\mu_{j})f_{\nu}(z;\lambda_{i,j},\tau_{i,j}){\rm
d}z,$ (18)

with

$\displaystyle
f_{\nu}(z;\lambda_{i,j},\tau_{i,j})=\dfrac{1}{F_{\nu}\left(\tau_{i,j}\Bigl{/}\sqrt{1+\lambda_{i,j}^{2}}\,\right)}\,f_{\nu}(z)F_{\nu+1}\left(\sqrt{\nu+1\over\nu+z^{2}}\,(\tau_{i,j}+\lambda_{i,j}z)\right)$

being the Arellano-Valle and Genton’s extended skewed Student’s
$t$-distribution (see Arellano-Valle and Genton,, 2010). Now, as
$f_{\nu}(\cdot;\lambda_{i,j},\tau_{i,j})$ is a PDF and because (see
Proposition 7 of Arellano-Valle and Genton,, 2010)

$\displaystyle\int_{-\infty}^{\infty}zf_{\nu}(z;\lambda_{i,j},\tau_{i,j}){\rm
d}z={\lambda_{i,j}\over\sqrt{1+\lambda_{i,j}^{2}}}\,{\nu\over\nu-1}\left(1+\dfrac{\tau_{i,j}^{2}/(1+\lambda_{i,j}^{2})}{\nu}\right)\dfrac{f_{\nu}\left(\tau_{i,j}\Bigl{/}\sqrt{1+\lambda_{i,j}^{2}}\,\right)}{F_{\nu}\left(\tau_{i,j}\Bigl{/}\sqrt{1+\lambda_{i,j}^{2}}\,\right)},\quad\nu>1,$

$\mu_{H_{i,j}}$ in (18) can be given as

$\displaystyle\mu_{H_{i,j}}={1\over
R_{i,j}}\left[{\dfrac{\sigma_{j}}{c_{i,j}}\left(\dfrac{\sigma_{j}}{\sigma_{i}}-\rho_{i,j}\right)}\,{\nu\over\nu-1}\left(1+\dfrac{1}{\nu
c_{i,j}^{2}}\left(\dfrac{\mu_{j}-\mu_{i}}{\sigma_{i}}\right)^{2}\right)f_{\nu}\left(\dfrac{1}{c_{i,j}}\left(\dfrac{\mu_{j}-\mu_{i}}{\sigma_{i}}\right)\right)+\mu_{j}F_{\nu}\left(\dfrac{1}{c_{i,j}}\left(\dfrac{\mu_{j}-\mu_{i}}{\sigma_{i}}\right)\right)\right],$

where
$c_{i,j}=\sqrt{1-\rho_{i,j}^{2}+[({\sigma_{j}}/{\sigma_{i}})-\rho_{i,j}]^{2}}.$

Finally, by applying Theorem 2 with $\mu_{H_{i,j}}$ as above, we obtain

(21)

Observe that the GMD corresponding to the multivariate Student-$t$ in (21)
converges to the GMD of multivariate normal law in (4.1) as $\nu\to\infty$,
which is to be expected.

######  Proposition 5.

Let $\bm{X}\sim EC_{n}(\bm{\mu},\bm{\Sigma},g^{(n)})$ be an exchangeable
random vector, where $g^{(n)}(x)=(1+x/\nu)^{-(\nu+n)/2}$ and $\nu>1$. Then,
the GMD is given by

$\displaystyle
GMD_{n}={2\over\sqrt{\pi}}\,\sigma_{1}\,\dfrac{\sqrt{2\nu}\,\Gamma({\nu+1\over
2})}{(\nu-1)\Gamma({\nu\over
2})}\,\dfrac{1}{\displaystyle\binom{n}{2}}\sum_{1\leqslant i<j\leqslant
n}\sqrt{1-\rho_{i,j}},$

where $\mu_{1}=\mathbb{E}(X_{1})$ and $\sigma_{1}^{2}={{\rm Var}(X_{1})}$.

###### Proof.

As $\bm{X}$ is exchangeable, we have $\mu_{i}=\mu_{j}=\mu_{1}$,
$\sigma_{i}=\sigma_{j}=\sigma_{1}$, $\rho_{i,j}=\rho_{j,i}$, $R_{i,j}=R_{j,i}$
and $c_{i,j}=c_{j,i}=\sqrt{2(1-\rho_{i,j})}$. Then, the required result
follows directly from (21). ∎

#### Acknowledgements

Roberto Vila and Helton Saulo gratefully acknowledge financial support from
CNPq, CAPES and FAP-DF, Brazil.

#### Disclosure statement

There are no conflicts of interest to disclose.

## References

  * Arellano-Valle and Genton, (2008) Arellano-Valle, R.B. and Genton, M.G. (2008). On the exact distribution of the maximum of absolutely continuous dependent random variables.  Statistics & Probability Letters, 78:27–35.
  * Arellano-Valle and Genton, (2010) Arellano-Valle, R.B. and Genton, M.G. (2010).  Multivariate extended skew-t distributions and related families.  Metron, 68:201–234.
  * Barry, (1985) Barry, C.A. (1985). p-Norm bounds on the expectation of the maximum of a possibly dependent sample.  Journal of Multivariate Analysis, 17:316–332.
  * Bonetti et al., (2009) Bonetti, M., Gigliarano, C., and Muliere, P. (2009).  The Gini concentration test for survival data.  Lifetime Data Analysis, 15:493-518.
  * Cerone and Dragomir, (2005) Cerone, P. and Dragomir, S.S. (2005). Bounds for the Gini mean difference via the Sonin identity.  Computers and Mathematics with Applications, 50:599–609.
  * Fang et al., (1990) Fang, K.T., Kotz, S. and Ng, K.W. (1990).  Symmetric Multivariate and Related Distributions.  Chapman and Hall, London, UK.
  * Kattumannil et al., (2022) Kattumannil, S.K., Sreelakshmi, N. and Balakrishnan, N. (2022). Non-Parametric Inference for Gini Covariance and its Variants.  Sankhya A, 84:790–807.
  * La Haye and Zizler, (2019) La Haye, R. and Zizler, P. (2019). The Gini mean difference and variance.  Metron, 77:43–52.
  * Saulo et al., (2023) Saulo, H., Vila, R., Cordeiro, S.S. and Leiva, V. (2023).  Bivariate symmetric Heckman models and their characterization.  Journal of Multivariate Analysis, 193:105097.
  * Schezhtman and Yitzhaki, (1987) Schezhtman, E. and Yitzhaki, S. (1987). A measure of association based on Gin’s mean difference.  Communications in Statistics - Theory and Methods, 16:207–231.
  * Schezhtman and Yitzhaki, (1999) Schezhtman, E. and Yitzhaki, S. (1999). On the proper bounds of the Gini correlation.  Economics Letters, 63:133–138.
  * Schmid and Semeniuk, (2021) Schmid, W. and Semeniuk, Y. (2021). A test on correlation based on Gini’s mean difference. In: Mukhopadhyay, N., & Sengupta, P.P. (Eds.).  Gini Inequality Index: Methods and Applications. Chapman and Hall/CRC Press, Boca Raton, Florida.
  * Yin et al., (2022) Yin, X., Balakrishnan, N. and Yin, C. (2022). Bounds for Gini’s mean difference based on first four moments, with some applications.  Statistical Papers (to appear).
https://doi.org/10.1007/s00362-022-01374-0.

  * Yitzhaki, (2003) Yitzhaki, S. (2003).  Gini’s mean difference: a superior measure of variability for non-normal distributions.  Metron, 61:285–316.
  * Yitzhaki, (2015) Yitzhaki, S. (2015).  Gini’s mean difference offers a response to Leamer’s critique.  Metron, 73:31–43.


# Perturbative study of the Gluino-Glue operator in SYM

${}^{,\,a}$, G. Panagopoulosb, H. Panagopoulosa, G. Spanoudes${}^{a,\,c}$, H.
Herodotoua, P. Philippidesa
aDepartment of Physics, University of Cyprus, Nicosia, CY-1678, Cyprus
bDepartment of Physics, Stanford University, CA 94305-2004, USA
cPresent address: Computation-based Science and Technology Research Center,
The Cyprus Institute, 20 Kavafi Str., Nicosia 2121, Cyprus.
E-mail

###### Abstract:

We investigate the renormalization of the Gluino-Glue operator, using both
Lattice Perturbation Theory (LPT) and a Gauge Invariant Renormalization Scheme
(GIRS). The latter scheme involves gauge-invariant Green’s functions of two
operators at different space-time points, which can be also computed via
numerical simulations. There is no need to fix a gauge and the mixing with
gauge noninvariant operators is inconsequential. We calculate perturbatively
the conversion factor relating GIRS with the Modified Minimal Subtraction
scheme. On the other hand, the Gluino-Glue operator being mixes with several
gauge noninvariant operators which have the same quantum numbers. The
determination of the mixing matrix on the lattice demands the calculation of
2-pt and 3-pt Green’s functions with external gluon, gluino and ghost fields
using LPT. We compute at one-loop order the renormalization of the Gluino-Glue
operator and all operator mixing coefficients.

##  1 Introduction

Supersymmetric strongly coupled theories have influenced experimental particle
physics and evidence of SUSY as well as its predictions are still under
investigation. SUSY is expected to emerge at very high energies and it
provides dark matter candidates, arising from the lightest supersymmetric
particles. In addition to that, supersymmetric extensions of the Standard
Model would resolve the hierarchy problem. In this work, we considered the
${\cal N}=1$ supersymmetric Yang-Mills (SYM) theory, which describes the
strong interactions between gluons ($u_{\mu}$) and gluinos ($\lambda$), the
superpartners of the gluons. SYM shares some of the fundamental properties of
supersymmetric theories containing quarks and squarks, while at the same time
it is amenable to high-accuracy non-perturbative investigations; it is thus an
ideal forerunner to the future study of theories containing more superfields.

A fundamental ingredient in these investigations is the “Gluino-Glue”
composite operator, ${\cal O}_{Gg}$. In the present work we study thoroughly
the renormalization and mixing of this operator, to the first perturbative
order. In particular, we present two separate calculations: 1) We compute, in
dimensional regularization, the conversion factors relating the
${\overline{\rm MS}}$ scheme to an intermediate gauge-invariant coordinate-
space scheme. In this second scheme, there is no mixing with gauge variant
operators and no need to fix the gauge. Further, all necessary correlation
functions can be computed nonperturbatively in simulations, again without need
for gauge fixing or ghost fields. However, a downside of this scheme is that
the calculations to order $g^{2n}$ requires evaluation of $(n+1)$-loop Feynman
diagrams. 2) We use lattice perturbation theory and compute, to one loop,
various 2- and 3-point functions of ${\cal O}_{Gg}$. We consider mixing with
all relevant gauge-noninvariant operators, which contain also ghost fields.

For a comprehensive presentation of our results, along with detailed
explanations and a longer list of references, we refer to our publications [1,
2].

##  2 Gluino-Glue Operator

In studying the properties of light bound states, the main observables in SYM
would be particles made up of gluons and gluinos. An important object in such
a study is the Gluino-Glue operator, defined as follows:

${\cal
O}_{Gg}=\sigma_{\mu\nu}\,{\rm{tr}}_{c}(\,u_{\mu\nu}\lambda),\quad\sigma_{\mu\nu}=\frac{1}{2}[\gamma_{\mu},\gamma_{\nu}],\quad
u_{\mu\nu}=\partial_{\mu}u_{\nu}-\partial_{\nu}u_{\mu}+ig[u_{\mu},u_{\nu}]$
(1)

This operator has the lowest possible dimensionality (7/2) compatible with
gauge invariance and being at the same time fermionic. Acting on the vacuum,
${\cal O}_{Gg}$ is expected to excite a light bound state of the theory, which
is a potential supersymmetric partner of the gluinoballs
(${\rm{tr}}_{c}(\bar{\lambda}\lambda)$,
${\rm{tr}}_{c}(\bar{\lambda}\gamma_{5}\lambda)$, $\cdots)$ and the glueballs
(${\rm{tr}}_{c}(u_{\mu\nu}u_{\mu\nu})$,
${\rm{tr}}_{c}(u_{\mu\nu}\tilde{u}_{\mu\nu})$, $\cdots$) [3]. The Gluino-Glue
operator, being composite, could in principle mix with four classes of
operators having the same quantum numbers. The four classes are as follows:
Class G are gauge invariant operators. Class A are operators which are not
gauge invariant but are the BRST variation of some other operators: ${\cal
O}_{A}$. Class B operators vanish by the equations of motion: ${\cal O}_{B}$.
Class C are operators which are not linear combinations of class G, A and B:
${\cal O}_{C}$. Below, we present all candidate operators which can mix with
${\cal O}_{Gg}$ ($\alpha$: gauge fixing parameter, $c$: ghost field):

$\displaystyle{\cal O}_{A1}$ $\displaystyle=$
$\displaystyle\frac{1}{\alpha}{\rm{tr}}_{c}(\lambda\,\partial_{\mu}u_{\mu})-ig\,{\rm{tr}}_{c}(\lambda[c,\bar{c}])$
$\displaystyle{\cal O}_{B1}$ $\displaystyle=$
$\displaystyle{\rm{tr}}_{c}(\not{u}{\not{D}}\lambda)$ $\displaystyle{\cal
O}_{C1}$ $\displaystyle=$
$\displaystyle{\rm{tr}}_{c}(\partial_{\mu}\lambda\,u^{\mu})$
$\displaystyle{\cal O}_{C2}$ $\displaystyle=$
$\displaystyle{\rm{tr}}_{c}(\not{u}\lambda)$ $\displaystyle{\cal O}_{C3}$
$\displaystyle=$ $\displaystyle
ig\,\sigma_{\mu\nu}{\rm{tr}}_{c}(\,\lambda[u_{\mu},u_{\nu}])$
$\displaystyle{\cal O}_{C4}$ $\displaystyle=$ $\displaystyle
ig\,{\rm{tr}}_{c}(\lambda[c,\bar{c}])$

Class C operators cannot contribute in the continuum for the purpose of
${\overline{\rm MS}}$-renormalization. However, they may give finite mixing
coefficients on the lattice. Note also that the operator ${\cal O}_{C2}$ is of
lower dimension and it will not mix with ${\cal O}_{Gg}$ in dimensional
regularization; it may however show up in the lattice formulation. The
presence of symmetries, which are preserved by the SYM action, both in the
continuum and on the lattice, forbids other operators from mixing with the
Gluino-Glue operator. There are no gauge-invariant operators which mix with
${\cal O}_{Gg}$; thus, by applying Gauge Invariant Renormalization Scheme
(GIRS) [4] we only need to calculate one Green’s function (Eq. (2)) to extract
the multiplicative renormalization factor of ${\cal O}_{Gg}$, which
contributes in physical matrix elements.

##  3 Renormalization of Gluino-Glue operator in the GIRS scheme

In the GIRS scheme, renormalization factors are defined via the following
Green’s function, containing a product of Gluino-Glue operators, whose
4-vector positions $x$ and $y$ are distinct:

$G(x-y)\equiv\langle{\cal O}_{Gg}(x){\overline{\cal O}}_{Gg}(y)\rangle.$ (2)

In order to contract gluino fields in Feynman diagrams it is convenient to
choose the charge conjugate operator ${\overline{\cal O}}_{Gg}(y)$, instead of
${\cal O}_{Gg}(y)$, as the second factor in Eq. (2); ${\overline{\cal
O}}_{Gg}(y)$ is defined as (cf. Eq. (1)):

${\overline{\cal
O}}_{Gg}=-\,{\rm{tr}}_{c}(\,\bar{\lambda}u_{\mu\nu})\sigma_{\mu\nu}.$ (3)

Disconnected Feynman diagrams are not present in $G(x-y)$ due to the fact that
${\cal O}_{Gg}$ is not a scalar operator. In the following, we calculate the
Green’s function up to one loop, where we regularize the theory in $d$
dimensions ($d=4-2\,\epsilon$). In the GIRS scheme all non-gauge invariant
operators will not contribute to $G(x-y)$, and one will obtain directly the
multiplicative renormalization of ${\cal O}_{Gg}$, which is the only
renormalization factor which is relevant for physical matrix elements. The
Feynman diagrams for the tree-level and one-loop values of the 2-pt Green’s
function are shown in Fig. 1. By adding tree-level and one-loop contributions,
the bare Green’s function takes the following form:

$\displaystyle G(x-y)^{\rm bare}$
$\displaystyle{=}-2\,\frac{(N_{c}^{2}-1)\>\Gamma(2-\epsilon)^{2}}{\pi^{4-2\epsilon}}(-1+\epsilon)(-3+2\epsilon)\,\not{z}\,(z^{2})^{-4+2\epsilon}\times$
(4) $\displaystyle\Bigg{\\{}1-\frac{g^{2}\
N_{c}}{16\pi^{2}}\left(\bar{\mu}^{2}z^{2}\right)^{\epsilon}\frac{e^{\epsilon\gamma_{E}}\Gamma(-\epsilon)}{4^{\epsilon}\
\epsilon\ (1-\epsilon)^{3}\ (3-2\epsilon)}\times$
$\displaystyle\Bigg{[}(1-\epsilon)(12-48\epsilon+70\epsilon^{2}-39\epsilon^{3}+\epsilon^{4})+\frac{(1-3\epsilon+2\epsilon^{2}+\epsilon^{3})\Gamma(-\epsilon)\Gamma(\epsilon)^{2}\Gamma(4-3\epsilon)}{4(1-2\epsilon)\Gamma(-2\epsilon)^{2}\Gamma(2\epsilon)}\Bigg{]}\Bigg{\\}},$

where $z^{\mu}\equiv y^{\mu}-x^{\mu}$, $\bar{\mu}$ is the ${\overline{\rm
MS}}$ renormalization scale relating the dimensionful coupling constant
$g_{L}$ in the $d$-dimensional Lagrangian to the dimensionless “bare” coupling
constant $g^{B}$: $g_{L}=\mu^{\epsilon}g^{B}$
($\mu=\bar{\mu}\sqrt{e^{\gamma_{E}}/4\pi}$). To this perturbative order, the
distinction between bare and renormalized coupling constants is inessential;
we will thus denote both by $g$.

Figure 1: Tree-level and (eleven) one-loop Feynman diagrams contributing to
the expectation value $\langle{\cal O}_{Gg}(x){\overline{\cal
O}}_{Gg}(y)\rangle$. A wavy (solid) line represents gluons (gluinos). The
dashed line is the ghost field. A cross denotes the insertion of the operator.

A more promising option is to integrate Eq.(2) over three of the four
components of the position vector $(x-y)$, while setting the fourth component
equal to a reference scale $\bar{t}$. Due to the anisotropic lattice employed
in simulations, the temporal direction is a special one. In this sense, a
natural choice for the component $\bar{t}$ is to be temporal; we call this
variant “t-GIRS”. Without loss of generality, we set $x=(x_{1},x_{2},x_{3},0)$
and $y=(0,0,0,\bar{t})$; then the renormalization condition for t-GIRS has the
following form:

${\rm Tr}\Big{[}\Big{(}\int d^{3}\vec{x}\ \langle{\mathcal{O}}^{\rm
t-GIRS}_{Gg}(\vec{x},0){{\overline{\mathcal{O}}}^{\rm
t-GIRS}_{Gg}}(\vec{0},\bar{t})\rangle\Big{)}\gamma_{4}\Big{]}=(Z^{B,{\rm
t-GIRS}}_{Gg})^{2}{\rm Tr}\Big{[}\Big{(}\int d^{3}\vec{x}\
\langle{\mathcal{O}}^{B}_{Gg}(\vec{x},0){{\overline{\mathcal{O}}}^{B}_{Gg}}(\vec{0},\bar{t})\rangle^{\rm
tree}\Big{)}\gamma_{4}\Big{]},$ (5)

where the right-hand side of the above condition is the tree-level Green’s
function with $\epsilon\rightarrow 0$. The conversion factor between
${\overline{\rm MS}}$ and t-GIRS, ${\mathcal{O}}^{{\overline{\rm
MS}}}_{Gg}=C_{Gg}^{{\rm t-GIRS},{\overline{\rm MS}}}{\mathcal{O}}^{\rm
t-GIRS}_{Gg}$, is obtained from the relation:

${\rm Tr}\Big{[}\Big{(}\int d^{3}\vec{x}\ \langle{\mathcal{O}}^{{\overline{\rm
MS}}}_{Gg}(\vec{x},0){{\overline{\mathcal{O}}}^{{\overline{\rm
MS}}}_{Gg}}(\vec{0},\bar{t})\rangle\Big{)}\gamma_{4}\Big{]}=\left(C_{Gg}^{{\overline{\rm
MS}},{\rm t-GIRS}}\right)^{2}{\rm Tr}\Big{[}\Big{(}\int d^{3}\vec{x}\
\langle{\mathcal{O}}^{\rm
t-GIRS}_{Gg}(\vec{x},0){{\overline{\mathcal{O}}}^{\rm
t-GIRS}_{Gg}}(\vec{0},\bar{t})\rangle^{\rm tree}\Big{)}\gamma_{4}\Big{]}.$ (6)

Note that both sides of Eq. (6) are well-defined as $d\to 4$, and thus spatial
integration is performed in $3$ (rather than $d-1$) dimensions. Use of the
integrals:

$\int d^{3}\vec{x}\
\frac{1}{(|\vec{x}|^{2}+t^{2})^{4}}=\frac{\pi^{2}}{8|t|^{5}},\quad\int
d^{3}\vec{x}\
\frac{\ln(|\vec{x}|^{2}+t^{2})}{(|\vec{x}|^{2}+t^{2})^{4}}=\frac{\pi^{2}(-5+12\ln(2)+6\ln(t^{2}))}{48|t|^{5}},$
(7)

in Eq. (6), leads to the following expression for the conversion factor from
t-GIRS to ${\overline{\rm MS}}$:

$C_{Gg}^{{\overline{\rm MS}},{\rm
t-GIRS}}=1+\frac{g^{2}N_{c}}{16\pi^{2}}\Big{(}-\frac{5}{6}+6\gamma_{E}+3\ln(\bar{\mu}^{2}\bar{t}^{2})\Big{)}+\mathcal{O}(g^{4}).$
(8)

Using a standard lattice discretization ${\cal O}_{Gg}^{L}$ of the Gluino-Glue
operator [5], the non-perturbative value of its renormalization factor,
$Z_{Gg}^{L,{\rm t-GIRS}}$, can be found via:

$\displaystyle(Z_{Gg}^{L,{\rm t-GIRS}})^{2}{\rm Tr}\Big{[}\Big{(}\int
d^{3}\vec{x}\
\langle{\mathcal{O}}^{L}_{Gg}(\vec{x},0){{\overline{\mathcal{O}}}^{L}_{Gg}}(\vec{0},\bar{t})\rangle\Big{)}\gamma_{4}\Big{]}$
$\displaystyle=$ $\displaystyle{\rm Tr}\Big{[}\Big{(}\int
d^{3}\vec{x}\,\lim_{\epsilon\to 0}\langle{\cal O}_{Gg}(x){\overline{\cal
O}}_{Gg}(y)\rangle^{{\rm tree}}\Big{)}\gamma_{4}\Big{]}$ (9) $\displaystyle=$
$\displaystyle 3\frac{(N_{c}^{2}-1)|t|}{\pi^{2}t^{5}}.$

##  4 Perturbative study of the Gluino-Glue operator on the lattice

Given that ${\cal O}_{Gg}$ has a relatively high dimensionality, 7/2, there is
a number of other (non-gauge invariant) composite operators with which ${\cal
O}_{Gg}$ can, and will, mix [2]; this becomes apparent when one calculates
Green’s functions with elementary external fields, as is done in a typical
renormalization procedure. A proper treatment of this mixing entails studying
the 2-pt and 3-pt Green’s functions of ${\cal O}_{Gg}$ with external gluino
and gluon fields. The renormalized fields and operators are related to bare
ones through:

$u_{\mu}^{R}=\sqrt{Z_{u}}\,u^{B}_{\mu},\quad\lambda^{R}=\sqrt{Z_{\lambda}}\,\lambda^{B},\quad
c^{R}=\sqrt{Z_{c}}\,c^{B},\quad{\cal O}^{R}_{Gg}=Z_{Gg}{\cal
O}^{B}_{Gg}+z_{A1}{\cal O}^{B}_{A1}+z_{B1}{\cal
O}^{B}_{B1}+\sum_{i=1}^{4}z_{Ci}{\cal O}^{B}_{Ci}$ (10)

We compute the one-loop quantum correction for the relevant Green’s functions
of the Gluino-Glue operator. This allows us to determine renormalization
factors of the operator in the ${\overline{\rm MS}}$ scheme, as well as the
mixing coefficients for the other operators. To this end, our computations are
performed using dimensional and lattice regularizations. We employ a standard
discretization where gluinos are defined on lattice sites and gluons reside on
the links of the lattice; the discretization is based on Wilson’s formulation
of non-supersymmetric gauge theories with clover improvement. Our perturbative
results are analytic expressions depending on the number of colors, $N_{c}$,
the clover coefficient, $c_{\rm SW}$, the Wilson parameter, $r=\pm 1$, the
lattice spacing $a$, and the gauge parameter, $\alpha$.

To obtain $Z_{Gg}$ and all mixing coefficients, $z$, we have calculated, to
one loop and in an arbitrary covariant gauge, the 2-pt (gluino-gluon) and 3-pt
(gluino-gluon-gluon and gluino-ghost-antighost) bare amputated Green’s
functions of ${\cal O}_{Gg}$; these are related to the corresponding
renormalized Green’s functions through:

$\displaystyle\langle u_{\nu}^{R}\,{\cal
O}^{R}_{Gg}\,\bar{\lambda}^{R}\rangle_{amp}$ $\displaystyle=$ $\displaystyle
Z_{\lambda}^{-1/2}\,Z_{u}^{-1/2}Z_{Gg}\langle u_{\nu}^{B}\,{\cal
O}^{B}_{Gg}\,\bar{\lambda}^{B}\rangle_{amp}+z_{A1}\langle u_{\nu}^{B}\,{\cal
O}^{B}_{A1}\,\bar{\lambda}^{B}\rangle_{amp}^{tree}+z_{B1}\langle
u_{\nu}^{B}\,{\cal O}^{B}_{B1}\,\bar{\lambda}^{B}\rangle_{amp}^{tree}$ (11)
$\displaystyle+z_{C1}\langle u_{\nu}^{B}\,{\cal
O}^{B}_{C1}\,\bar{\lambda}^{B}\rangle_{amp}^{tree}+z_{C2}\langle
u_{\nu}^{B}\,{\cal O}^{B}_{C2}\,\bar{\lambda}^{B}\rangle_{amp}^{tree}$
$\displaystyle\langle u_{\nu}^{R}\,u_{\mu}^{R}\,{\cal
O}^{R}_{Gg}\,\bar{\lambda}^{R}\rangle_{amp}$ $\displaystyle=$ $\displaystyle
Z_{\lambda}^{-1/2}\,Z_{u}^{-1}Z_{Gg}\langle u_{\nu}^{B}u_{\mu}^{B}\,{\cal
O}^{B}_{Gg}\,\bar{\lambda}^{B}\rangle_{amp}+z_{B1}\langle
u_{\nu}^{B}u_{\mu}^{B}\,{\cal
O}^{B}_{B1}\,\bar{\lambda}^{B}\rangle_{amp}^{tree}$ (12)
$\displaystyle+z_{C3}\langle u_{\nu}^{B}u_{\mu}^{B}\,{\cal
O}^{B}_{C3}\,\bar{\lambda}^{B}\rangle_{amp}^{tree}$ $\displaystyle\langle
c^{R}\,{\cal O}^{R}_{Gg}\,\bar{c}^{R}\,\bar{\lambda}^{R}\rangle_{amp}$
$\displaystyle=$ $\displaystyle Z_{c}^{-1}\,Z_{\lambda}^{-1/2}Z_{Gg}\langle
c^{B}\,{\cal
O}^{B}_{Gg}\,\bar{c}^{b}\bar{\lambda}^{B}\rangle_{amp}+z_{A1}\langle
c^{B}\,{\cal O}^{B}_{A1}\,\bar{c}^{B}\bar{\lambda}^{B}\rangle_{amp}^{tree}$
(13) $\displaystyle+z_{C4}\langle c^{B}\,{\cal
O}^{B}_{C4}\,\bar{c}^{B}\bar{\lambda}^{B}\rangle_{amp}^{tree}$

The one-loop Feynman diagrams (one-particle irreducible (1PI)) contributing to
these Green’s functions are shown in Figs. 2, 3, 4.

Figure 2: One-loop Feynman diagrams contributing to the two point Green’s
function of the Gluino-Glue operator, $\langle u_{\nu}{\cal
O}_{Gg}\bar{\lambda}\rangle$ . A wavy (dashed) line represents gluons
(gluinos). A cross denotes the insertion of the Gluino-Glue operator. Diagrams
2, 4 do not appear in dimensional regularization; they do however show up in
the lattice formulation.

Figure 3: One-loop Feynman diagrams contributing to the three point Green’s
function of the Gluino-Glue operator, $\langle u_{\nu}u_{\mu}{\cal
O}_{Gg}\bar{\lambda}\rangle$ . A wavy (dashed) line represents gluons
(gluinos). Diagrams 1, 2, 3, 5, 6, 11, and 13 do not appear in dimensional
regularization but they contribute in the lattice regularization. A cross
denotes the insertion of the operator. A mirror version (under exchange of the
two external gluons) of diagrams 3, 4, 5, 6, 8, 10, 14, 15 and 16 must also be
included. Figure 4: One-loop Feynman diagrams contributing to the three point
Green’s function of the Gluino-glue operator, $\langle c\,{\cal
O}_{Gg}\,\bar{c}\,\bar{\lambda}\rangle$ . A wavy (dashed) line represents
gluons (gluinos). A cross denotes the insertion of the operator. The “double
dashed” line is the ghost field. Diagrams 1 and 2 do not appear in dimensional
regularization; they do however show up in the lattice formulation.

We have made a choice of the external momenta for Green’s functions which
allow an unambiguous extraction of all mixing coefficients and renormalization
constants of the operators. We calculate the 2-pt Green’s function $\langle
u_{\nu}^{\alpha_{1}}(-q_{1}){\cal
O}_{Gg}(x)\bar{\lambda}^{\alpha_{2}}(q_{2})\rangle$, for three choices of the
external momenta $q_{1}$ and $q_{2}$: $q_{2}=0$, $q_{1}=0$ and $q_{2}=-q_{1}$.
The 3-pt Green’s functions $\langle c^{\alpha_{3}}(q_{3})\,{\cal
O}_{Gg}\,\bar{c}^{\alpha_{2}}(q_{2})\bar{\lambda}^{\alpha_{1}}(q_{1})\rangle$
and $\langle u_{\nu}^{\alpha_{1}}(-q_{1})u_{\mu}^{\alpha_{2}}(-q_{2})\,{\cal
O}_{Gg}\,\bar{\lambda}^{\alpha_{3}}(q_{3})\rangle$ are calculated at
(${q_{1}=q_{2},\,q_{3}=0}$) and (${q_{2}=0,q_{3}=-q_{1}}$), respectively. We
use the ${\overline{\rm MS}}$ renormalization scheme, in which the left hand
sides of the renormalization conditions in Eqs. (11), (12) and (13) are the
${\overline{\rm MS}}$-renormalized Green’s functions which are calculated by
the elimination of the pole part of the continuum bare Green’s functions. The
right hand sides are the bare latice Green’s functions. The relevant
renormalizations of fields and gauge couplings constant were not all available
for the clover action considered in this work, and had to be calculated as a
prerequisite. Imposing the renormalization conditions we get the results for
the renormalization factor and the mixing coefficients:

$\displaystyle Z_{Gg}^{L,{\overline{\rm MS}}}$ $\displaystyle=$ $\displaystyle
1-\frac{g^{2}N_{c}}{16\pi^{2}}\bigg{(}\frac{9.8696}{N_{c}^{2}}-1.7626-9.9198\,c_{\rm
SW}^{2}+4.9765\,c_{\rm SW}\,r-3\log(a^{2}\,\bar{\mu}^{2})\bigg{)}$ (14)
$\displaystyle z_{B1}^{L,\overline{\textrm{MS}}}$ $\displaystyle=$
$\displaystyle\frac{g^{2}N_{c}}{16\pi^{2}}\bigg{(}0.4241-\frac{3}{2}\log(a^{2}\,\bar{\mu}^{2})\bigg{)}$
(15) $\displaystyle z_{C3}^{L,{\overline{\rm MS}}}$ $\displaystyle=$
$\displaystyle-\frac{g^{2}N_{c}}{16\pi^{2}}0.000114$ (16)

Systematic errors coming from numerical loop integration are much smaller than
the precision presented in the above results. Also, certain mixing
coefficients vanish at one loop:

$z_{A1}^{L,{\overline{\rm MS}}}=z_{C1}^{L,{\overline{\rm
MS}}}=z_{C2}^{L,{\overline{\rm MS}}}=z_{C4}^{L,{\overline{\rm MS}}}=0$ (17)

Further details on this calculation can be found in Ref. [2].

##  5 Summary – Future Plans

In this work we have studied the mixing under renormalization for the Gluino-
Glue operator using lattice perturbation theory. We have calculated the one-
loop renormalization factors and mixing coefficients in the ${\overline{\rm
MS}}$ renormalization scheme. As a prerequisite, we have computed the gluon,
gluino and gauge coupling renormalization. Further, we have evaluated the
conversion factor to ${\overline{\rm MS}}$ for ${\cal O}_{Gg}$ to order
$g^{2}$, using the GIRS renormalization scheme, by calculating the two-loop
diagrams in the correlation function involving a product of two ${\cal
O}_{Gg}$ operators at distinct positions.

Some natural extensions of our work, besides going to higher perturbative
order, are to address SQCD, where the inclusion of quarks and squarks causes
${\cal O}_{Gg}$ to mix with other gauge invariant and noninvariant operators,
even (on the lattice) ones of lower dimensionality. In our ongoing
investigation we plan to address also other improved actions (e.g., overlap
action, Wilson action using stout-smearing in the fermionic action) and other
operators (e.g., Three-gluino operator, Noether Supercurrent operator). In
particular, for detailed information on preliminary perturbative and non-
perturbative results of the Noether supercurrent, see the proceedings by A.
Skouroupathis and I. Soler Calero in this conference [6].

Acknowledgements: M.C. and H.P. acknowledge financial support from the project
“Quantum Fields on the Lattice”, funded by the Cyprus Research and Innovation
Foundation (RIF) under contract number EXCELLENCE/0918/0066.

## References

  * [1] M. Costa, G. Panagopoulos, H. Panagopoulos and G. Spanoudes, “Gauge-invariant Renormalization of the Gluino-Glue operator,” Phys. Lett. B 816 (2021), 136225 doi:10.1016/j.physletb.2021.136225 [arXiv:2102.02036 [hep-lat]].
  * [2] M. Costa, H. Herodotou, P. Philippides and H. Panagopoulos, “Renormalization and mixing of the Gluino-Glue operator on the lattice,” Eur. Phys. J. C 81 (2021) no.5, 401 doi:10.1140/epjc/s10052-021-09173-x [arXiv:2010.02683 [hep-lat]].
  * [3] G. Veneziano and S. Yankielowicz, “An Effective Lagrangian for the Pure N=1 Supersymmetric Yang-Mills Theory,” Phys. Lett. B 113 (1982), 231 doi:10.1016/0370-2693(82)90828-0
  * [4] M. Costa, I. Karpasitis, T. Pafitis, G. Panagopoulos, H. Panagopoulos, A. Skouroupathis and G. Spanoudes, Phys. Rev. D 103 (2021) no.9, 094509 doi:10.1103/PhysRevD.103.094509 [arXiv:2102.00858 [hep-lat]].
  * [5] S. Ali, G. Bergner, H. Gerber, P. Giudice, I. Montvay, G. Münster, S. Piemonte and P. Scior, “The light bound states of $\mathcal{N}=1$ supersymmetric SU(3) Yang-Mills theory on the lattice,” JHEP 03 (2018), 113 doi:10.1007/JHEP03(2018)113 [arXiv:1801.08062 [hep-lat]].
  * [6] G. Bergner, M. Costa, H. Panagopoulos, S. Piemonte, A. Skouroupathis, G. Spanoudes and I. Soler Calero, “Supercurrent renormalization in $\mathcal{N}=1$ supersymmetric Yang-Mills Theory,” PoS LATTICE2021.