JINR-P2-82-900

Anomalous dimensions of quark masses in the three-loop approximation
O.V. Tarasov
Joint Institute for Nuclear Research, Dubna, Russia

###### Abstract

The results of calculation of the three-loop radiative correction to the
renormalization constant of fermion masses for non-abelian gauge theory
interacting with fermions are presented. Dimensional regularization and the
t’Hooft-Veltman minimal subtraction scheme are used. The method of calculation
is described in detail. The renormalization group function $\gamma_{m}$
determining the behavior of the effective mass of fermions is presented. The
anomalous dimensions of fermions for QED and QCD up to three loops are given.
All calculations were performed on a computer with the help of the
SCHOOONSCHIP system for analytical manipulations.

The effects determined by the nonzero quark and lepton masses in quantum
chromodynamics, the standard $SU(2)\times U(1)$ model, usual and
supersymmetric grand unified theories are now attracting now more and more
attention. Considerable efforts were devoted to determining the mass of the
Higgs scalar particle from the LEP experiments. The threshold effects [1], the
light quark masses [2] and the value of the ratio $m_{b}/m_{\tau}$ in the
frame of the grand unified theories [3] are usually investigated by the
renormalization group (RG) method [4]. Application of the RG allows one to sum
the leading momentum-dependent logarithms to all orders in terms of the
limited number of the RG parameters (beta functions and anomalous dimensions)
which are calculated in the framework of perturbation theory.

The most characteristic feature of the mass account is large contributions of
the higher orders in perturbation theory. For example, the two-loop
corrections to the ratio $m_{b}/m_{\tau}$ [3] are 20% from the leading
contribution. In the $SU(5)$ grand unified theory from two-loop analysis it
was concluded [3] that the number of flavors cannot exceed six. The one - loop
approximation gives a less rigid restriction- the number of flavors may be
less than eight. The next approximations - the three loop and higher, can also
change this restriction. Higher corrections to the renormalization group
functions depend more strongly on the number of quarks.

We consider the non-Abelian gauge theory with fermions belonging to the
representation $R$ of the gauge group G:

$L=-\frac{1}{4}G_{\mu\nu}^{a}G_{\mu\nu}^{a}-\frac{1}{2\alpha}\left(\partial_{\mu}A^{a}_{\mu}\right)^{2}-\partial_{\mu}\overline{\eta}^{a}\partial_{\mu}\eta^{a}+\\\
gf^{abc}\overline{\eta}^{a}A^{b}_{\mu}\partial_{\mu}\eta^{c}+i\sum_{l=1}^{f}\overline{\psi}^{l}_{i}\hat{D}\psi^{l}_{i}-\sum_{l=1}^{f}m_{l}\overline{\psi}^{l}_{i}\psi^{l}_{i},$
$\displaystyle
G^{a}_{\mu\nu}=\partial_{\mu}A^{a}_{\nu}-\partial_{\nu}A^{a}_{\mu}+gf^{abc}A_{\mu}^{b}A_{\nu}^{c},$
$\displaystyle
D_{\mu}\psi^{l}_{i}=\partial_{\mu}\psi^{l}_{i}-igR^{a}_{ij}\psi_{j}^{l}A^{a}_{\mu}.$

Here $\eta^{a}$ is the ghost field, $\alpha$ is the gauge parameter and
$f^{abc}$ are totally antisymmetric structure constants of the gauge group
$G$. The indices of the fermion field $\psi_{i}^{m}$ specify color ($i$) and
flavor ($m$), respectively. The matrices $R^{a}$ obey the following relations:

$\left[R^{a},R^{b}\right]_{-}=if^{abc}R^{c},~{}~{}~{}~{}f^{acd}f^{bcd}=C\delta^{ab},$
$R^{a}R^{a}=C_{F}I,~{}~{}~{}~{}~{}tr\left(R^{a}R^{b}\right)=t\delta^{ab}.$

In particular, the values of group invariants $C$, $C_{F}$ and $T$ in the
fundamental (quark) representation of $SU(N)$ are:

$\displaystyle
C=N,~{}~{}~{}~{}~{}~{}C_{F}=\frac{N^{2}-1}{2N},~{}~{}~{}~{}~{}~{}t=\frac{1}{2}.$

In this paper we adopt the renormalization prescription by ’t Hooft [5] the
so-called ”minimal subtraction scheme” (MS), which by definition subtracts
only pole parts in $\varepsilon$ from a given diagram. The renormalization
constant $Z_{\Gamma}$ relating the dimensionally regularized 1PI Green
function with the renormalized one

$\displaystyle\Gamma_{R}\left(\frac{Q^{2}}{\mu^{2}},h,m,\alpha\right)=\lim_{\varepsilon\rightarrow
0}Z_{\Gamma}\left(\frac{1}{\varepsilon},h,\alpha\right)\Gamma\left(Q^{2},h_{B},m_{B},\alpha_{B},\varepsilon\right),$

looks in this scheme like

$\displaystyle
Z_{\Gamma}\left(\frac{1}{\varepsilon},h,\alpha\right)=1+\sum_{\nu=1}^{\infty}c_{\Gamma}^{(\nu)}(h,\alpha)\frac{1}{\varepsilon^{\nu}},$

with $\varepsilon=(4-n)/2$, $n$ being the space-time dimension,
$h=g^{2}/(4\pi)^{2}$ and $\mu$ is the renormalization parameter with the
dimension of mass. The relationship between the bare charge $h_{B}^{2}$ and
the renormalized one includes the product of appropriate $Z$’s. The most
convenient choice of $Z$’s is as follows:

$\displaystyle
h_{B}=(\mu^{2})^{\varepsilon}h\widetilde{Z}_{1}^{2}Z_{3}^{-1}\widetilde{Z}_{3}^{-2}=(\mu^{2})^{\varepsilon}\left[h+\sum_{\nu=1}^{\infty}a^{(\nu)}(h)\frac{1}{\varepsilon^{\nu}}\right].$

Here $\widetilde{Z}_{1}$ is the renormalization constant of the ghost-ghost-
gluon vertex, $Z_{3}$ and $\widetilde{Z}_{3}$ being those of the inverse gluon
and ghost propagators, respectively. Bare parameters $m_{B}$ and $\alpha_{B}$
are connected with the renormalized ones as:

$\displaystyle
m_{B}=Z_{m}m=m\left[1+\sum_{\nu=1}^{\infty}b^{(\nu)}(h)\frac{1}{\varepsilon^{\nu}}\right],$
$\displaystyle\alpha_{B}=Z_{3}\alpha=\alpha\left[1+\sum_{\nu=1}^{\infty}d^{(\nu)}(h,\alpha)\frac{1}{\varepsilon^{\nu}}\right].$
(1)

In Ref.[6] it was shown that $a^{(\nu)}(h)$ and $b^{(\nu)}(h)$ in the
$\overline{MS}$ renormalization scheme are gauge independent. This gives us a
possibility to use the simplest gauge for calculation. We choose to work in
the Feynman gauge $\alpha=1$ throughout this paper.

The Green function
$\Gamma_{R}\left(\frac{Q^{2}}{\mu^{2}},h^{2},m,\alpha\right)$ in the
$\overline{MS}$ scheme satisfies the renormalization group equation [5], [7]:

$\displaystyle\left\\{Q^{2}{\partial\over\partial
Q^{2}}-\beta(h){\partial\over\partial
h}+\left[1-\gamma_{m}(h)\right]m{\partial\over\partial m}\right.$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\delta(h,\alpha)\alpha{\partial\over\partial\alpha}-\gamma_{\Gamma}(h,\alpha)\right\\}\Gamma_{R}\left(\frac{Q^{2}}{\mu^{2}},h,m,\alpha\right)=0.$
(2)

Here

$\displaystyle\beta(h)=\left.\mu^{2}{\partial
h\over\partial\mu^{2}}\right|_{h_{B},m_{B},\alpha_{B}-fixed},$
$\displaystyle\gamma_{m}(h)=\left.\mu^{2}{\partial\ln
m\over\partial\mu^{2}}\right|_{h_{B},m_{B},\alpha_{B}-fixed},$
$\displaystyle\delta(h,\alpha)=\left.\mu^{2}{\partial\ln\alpha\over\partial\mu^{2}}\right|_{h_{B},m_{B},\alpha_{B}-fixed},$
$\displaystyle\gamma_{\Gamma}(h,\alpha)=-\left.\mu^{2}{\partial\ln
Z_{\Gamma}\over\partial\mu^{2}}\right|_{h_{B},m_{B},\alpha_{B}-fixed}.$

The functions $\beta(h),\gamma_{m}(h),\delta(h,\alpha)$ and
$\gamma_{\Gamma}(h,\alpha)$ can be expressed in terms of the coefficients in
front of ${\varepsilon}^{-1}$ in the decompositions of the corresponding
renormalization constants $Z$ [5], [7]:

$\displaystyle\beta(h)=\left(h{\partial\over\partial h}-1\right)a^{(1)}(h)$
$\displaystyle~{}~{}~{}~{}~{}~{}=h[2\widetilde{\gamma}_{1}(h,\alpha)-\gamma_{3}(h,\alpha)-2\widetilde{\gamma}_{3}(h,\alpha)]=-\beta_{1}h^{2}-\beta_{2}h^{3}-\beta_{3}h^{4}-\cdots,$
$\displaystyle\gamma_{m}(h)=h{\partial b^{(1)}(h)\over\partial
h}=-\gamma_{1}h-\gamma_{2}h^{2}-\gamma_{3}h^{3}-\cdots,$
$\displaystyle\delta(h,\alpha)=h{\partial d^{(1)}(h)\over\partial
h}=\gamma_{3}(h,\alpha),$ $\displaystyle\gamma_{\Gamma}(h,\alpha)=h{\partial
c^{(1)}(h)\over\partial h},$ (3)

where $\widetilde{\gamma}_{1}(h)$ is the anomalous dimension of the ghost-
ghost-gluon vertex, $\gamma_{3}$ and $\widetilde{\gamma}_{3}$ are anomalous
dimensions of the gluon and ghost propagators respectively. The coefficients
in front of $1/\varepsilon^{\nu}(\nu\geq 2)$ for the renoramlization constants
$Z_{\Gamma}$ and $Z_{m}$ are related to $c^{(1)}$ and $b^{(1)}$ by [5], [7]:

$\displaystyle\left\\{\beta(h){\partial\over\partial
h}+\gamma_{3}(h,\alpha)\alpha{\partial\over\partial\alpha}+\gamma_{\Gamma}(h,\alpha)\right\\}c^{(\nu)}(h,\alpha)=h{\partial\over\partial
h}c^{(\nu+1)}(h,\alpha),$
$\displaystyle\left\\{\gamma_{m}(h)+\beta(h){\partial\over\partial
h}\right\\}b^{(\nu)}(h)=h{\partial\over\partial h}b^{(\nu+1)}(h).$

These relationships were used to partially verify our calculations.

The goal of this paper is to calculate the three-loop approximation for
$\gamma_{m}(h)$. The two-loop calculations were presented in Refs.[8]. To find
$\gamma_{m}(h)$, it is convenient to use the relation
$\gamma_{m}(h)=-\gamma_{2}(h,\alpha)+\gamma_{\overline{\psi}\psi}(h,\alpha)$
that follows from the equation

$\displaystyle Z_{m}=Z_{\overline{\psi}\psi}Z^{-1}_{2}$

where

$\gamma_{\overline{\psi}\psi}(h,\alpha)=-\left.\mu^{2}{\partial\over\partial\mu^{2}}\ln
Z_{\overline{\psi}\psi}\right|_{h_{B},m_{B},\alpha_{B}-fixed},$

and $Z_{\overline{\psi}\psi}$ is the renormalization constant of the two-point
fermion Green function with the insertion $\overline{\psi}(y)\psi(y)$, i.e

$\langle\overline{\psi}(x)\psi(0)\int d^{n}y\overline{\psi}(y)\psi(y)\rangle.$

According to the method described in [9], for the renormalization constants
$Z_{\overline{\psi}\psi}$ and $Z_{2}$ we shall use the following
representation:

$\displaystyle
Z_{\overline{\psi}\psi}=1-KR^{\prime}\Gamma_{\overline{\psi}\psi},$
$\displaystyle Z^{-1}_{2}=1-KR^{\prime}\Gamma_{2}.$

Here $K$ is an operator that removes regular in $\varepsilon$ terms

$K\sum_{\nu}b_{\nu}\varepsilon^{\nu}=\sum_{\nu<0}b_{\nu}\varepsilon^{\nu},$

and $R^{\prime}$ is the $R$ operation for the minimal subtraction scheme
without the last subtraction applied to the Green function $\Gamma$, i.e

$R=\left(1-K\right)R^{\prime}.$

We apply $K$ and $R^{\prime}$ operations to the diagrams of the following
type:

where $-\\!\\!\bigotimes\\!\\!-$ corresponds to the insertion
$\int\overline{\psi}(y)\psi(y)d^{n}y$.

As an illustrative example we consider calculation of the contribution to
$Z_{\overline{\psi}\psi}$ from one of the three-loop diagrams. For example,

The $R^{\prime}$ for this diagram with the combinatorial factor and
appropriate counterterms can schematically be represented as:

Substituting expressions for the one- and two- loop diagrams and multiplying
them by appropriate $KR^{\prime}$, we get for the diagram, counterterms with
two- and one- loop subdiagrams the following expression:

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+$
$\displaystyle\\!\left[\frac{39}{4\varepsilon^{3}}+\frac{19}{4\varepsilon^{2}}+\frac{231}{16\varepsilon}\right]C_{F}C^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-$
$\displaystyle\\!\left[\frac{39}{4\varepsilon^{3}}+\frac{299}{8\varepsilon^{2}}+\frac{975}{8\varepsilon}-\frac{117}{4\varepsilon}\zeta(3)\right]C_{F}C^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=$
$\displaystyle\\!\left[\frac{13}{4\varepsilon^{3}}-\frac{223}{24\varepsilon^{2}}+\frac{551}{48\varepsilon}+\frac{1}{\varepsilon}\zeta(3)\right]C_{F}C^{2}.$

All calculations were performed on the CDC-6500 computer using the algebraic
manipulation system SCHOONSCHIP [10]. In Appendix, we present contributions to
$Z^{-1}_{2}$ and $Z_{\overline{\psi}\psi}$ from diagrams with one- and two-
loop insertions, factorized diagrams and 1PI diagrams. At the three loop level
we get:

$\displaystyle Z^{-1}_{2}$ $\displaystyle=$ $\displaystyle
1+\frac{1}{\varepsilon}hC_{F}+\left[\frac{1}{\varepsilon^{2}}\left(\frac{1}{2}C_{F}-C\right)+\frac{1}{\varepsilon}\left(\frac{17}{4}C-tf-\frac{3}{4}C_{F}\right)\right]h^{2}C_{F}$
$\displaystyle+\left[\frac{1}{\varepsilon^{3}}\left(\frac{7}{4}C^{2}-\frac{1}{3}Ctf-
CC_{F}+\frac{1}{6}C^{2}_{F}\right)\right.$
$\displaystyle-\frac{1}{\varepsilon^{2}}\left(\frac{104}{9}C^{2}-\frac{56}{9}Ctf+\frac{8}{9}t^{2}f^{2}-\frac{73}{12}CC_{F}+\frac{5}{3}C_{F}tf+\frac{3}{4}C^{2}_{F}\right)$
$\displaystyle-\frac{1}{\varepsilon}\left(\frac{1301}{108}Ctf+\frac{143}{12}CC_{F}-\frac{10559}{432}C^{2}-\frac{20}{27}t^{2}f^{2}\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.\left.-C_{F}tf-\frac{1}{2}C^{2}_{F}-4CC_{F}\zeta(3)+\frac{5}{2}C^{2}\zeta(3)\right)\right]h^{3}C_{F},$

$\displaystyle
Z_{m}=1-3C_{F}h\frac{1}{\varepsilon}+\left[\frac{1}{\varepsilon^{2}}\left(\frac{11}{2}C-2tf+\frac{9}{2}C_{F}\right)+\frac{1}{\varepsilon}\left(-\frac{97}{12}C+\frac{5}{3}tf-\frac{3}{4}C_{F}\right)\right]h^{2}C_{F}$
$\displaystyle+\left[\frac{1}{\varepsilon^{3}}\left(\frac{88}{9}Ctf-\frac{33}{2}CC_{F}-\frac{121}{9}C^{2}-\frac{16}{9}t^{2}f^{2}+6C_{F}tf-\frac{9}{2}C^{2}_{F}\right)\right.$
$\displaystyle+\frac{1}{\varepsilon^{2}}\left(-\frac{484}{27}Ctf+\frac{313}{12}CC_{F}+\frac{1679}{54}C^{2}+\frac{40}{27}t^{2}f^{2}-\frac{29}{3}C_{F}tf+\frac{9}{4}C^{2}_{F}\right)$
$\displaystyle+\frac{1}{\varepsilon}\left(\frac{556}{81}Ctf+\frac{43}{4}CC_{F}-\frac{11413}{324}C^{2}+\frac{140}{81}t^{2}f^{2}+\frac{46}{3}C_{F}tf\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.\left.-\frac{43}{2}C^{2}_{F}+16\zeta(3)tf(C-C_{F})\right)\right]h^{3}C_{F}.$

Using (1) and (3) we get

$\displaystyle\gamma_{m}(h)=-3C_{F}h-\left(\frac{97}{6}C-\frac{10}{3}tf+\frac{3}{2}C_{F}\right)C_{F}h^{2}+\left(\frac{556}{27}Ctf+\frac{129}{4}CC_{F}\right.$
$\displaystyle~{}-\left.\frac{11413}{108}C^{2}+\frac{140}{27}t^{2}f^{2}+46C_{F}tf-\frac{129}{2}C^{2}_{F}+48\zeta(3)tf(C-C_{F})\right)h^{3}C_{F},$
$\displaystyle\gamma_{2}(h,1)=-C_{F}h-\left(\frac{17}{2}C-2tf-\frac{3}{2}C_{F}\right)C_{F}h^{2}-\left(\frac{10559}{144}C^{2}-\frac{1301}{36}Ctf\right.$
$\displaystyle~{}~{}\left.-\frac{143}{4}CC_{F}+\frac{20}{9}t^{2}f^{2}+3C_{F}tf+\frac{3}{2}C^{2}_{F}+12CC_{F}\zeta(3)-\frac{15}{2}C^{2}\zeta(3)\right)h^{3}C_{F}.$
(4)

At the two loop level our results coincide with that presented in [8]. Unlike
the $\beta$\- function, the three-loop contribution to $\gamma_{m}$ depens on
the Riemann function $\zeta(3)=\sum_{l=1}^{\infty}1/l^{3}$. The coefficient in
$\gamma_{m}$ proportional to $C_{F}t^{2}f^{2}$ coincides with that predicted
in [11]. For the quantum chromodynamics ($SU(3)$ gauge group with $C=3$,
$C_{F}=4/3$, $t=1/2$) we have:

$\displaystyle\gamma_{m_{i}}(h)=-4h-\left(\frac{202}{3}-\frac{20}{9}f\right)h^{2}-\left(\frac{3747}{3}-\frac{2216}{27}f-\frac{140}{81}f^{2}\right)h^{3}+\frac{160}{3}\zeta(3)fh^{3},$
$\displaystyle\gamma_{2}(h,1)=-\frac{4}{3}h-\left(\frac{97}{3}+\frac{4}{3}f\right)h^{2}-\left(\frac{24941}{36}-\frac{1253}{18}f+\frac{20}{27}f^{2}-26\zeta(3)\right)h^{3}.$

For completeness we present also expression for the $\beta$\- function [12]:

$\displaystyle\beta(h)=-\left(11-\frac{2}{3}f\right)h^{2}-\left(102-\frac{38}{3}f\right)h^{3}-\left(\frac{2857}{2}-\frac{5033}{18}f+\frac{325}{54}f^{2}\right)h^{4}.$

From (4), one can easily get $\gamma_{2}$ and $\gamma_{m}$ for the quantum
electrodynamics, by setting $C=0$, $C_{F}=1$, $tf=1$:

$\displaystyle\gamma_{m}(h)$ $\displaystyle=$
$\displaystyle-3h+\frac{11}{6}h^{2}-\frac{719}{54}h^{3}-48\zeta(3)h^{3},$
$\displaystyle\gamma_{2}(h)$ $\displaystyle=$
$\displaystyle-h+\frac{7}{2}h^{2}-\frac{121}{18}h^{3}.$

The solutions of the renormalization group equation (2) in the t’Hooft scheme
are expressed in terms of the effective parameters $\overline{h}$,
$\overline{m}$, $\overline{\alpha}$ determined by the system of equations:

$\displaystyle{d\overline{h}\over
dl}=\beta(\overline{h}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}l=\ln\frac{Q^{2}}{\mu^{2}},$
$\displaystyle{d\ln\overline{\alpha}\over
dl}=\delta(\overline{h},\overline{\alpha}),$
$\displaystyle{d\ln\overline{m}_{i}\over dl}=\gamma_{m_{i}}(\overline{h}).$

The expression for $\overline{h}(L)$ up to three loops was given in [12]:

$\displaystyle\overline{h}(L)$ $\displaystyle=$
$\displaystyle\frac{1}{\beta_{1}L}-\frac{\beta_{2}}{\beta_{1}^{3}}\frac{\ln
L}{L^{2}}+{\beta^{2}_{2}(\ln^{2}L-\ln
L)\over\beta_{1}^{5}L^{3}}+{\beta_{3}\beta_{1}-\beta_{2}^{2}\over\beta_{1}^{5}L^{3}}$
$\displaystyle-$
$\displaystyle{\beta_{2}^{3}\ln^{3}L\over\beta_{1}^{7}L^{4}}+{5\beta_{2}^{3}\ln^{2}L\over
2\beta_{1}^{7}L^{4}}+{(-3\beta_{1}\beta_{2}\beta_{3}+2\beta_{2}^{3})\ln
L\over\beta_{1}^{7}L^{4}}+O\left(\frac{1}{L^{4}}\right)$

Here
$L=\ln\frac{Q^{2}}{\Lambda^{2}}\equiv\int\frac{dx}{\beta(x)}+\ln\frac{Q^{2}}{\mu^{2}}$.
An arbitrary integration constant is chosen to cancel the term with $1/L^{2}$
in the decomposition of $\overline{h}(L)$.

The solution for $\overline{m}_{i}$ is:

$\displaystyle\overline{m}_{i}(L)={\hat{m}_{i}(L)}^{-\frac{\gamma_{1}}{\beta_{1}}}\left\\{1+\frac{\left(\gamma_{2}\beta_{1}-\beta_{2}\gamma_{1}\right)}{\beta_{1}^{3}L}-\frac{\beta_{2}\gamma_{1}}{\beta_{1}^{3}}\frac{\ln
L}{L}\right.$
$\displaystyle+\frac{1}{2\beta_{1}^{6}}\Bigl{(}\beta_{1}^{3}\gamma_{3}+\beta_{1}^{2}\gamma_{2}^{2}-\beta_{1}^{2}\beta_{2}\gamma_{2}+\beta_{1}^{2}\beta_{3}\gamma_{1}-\beta_{1}\beta_{2}^{2}\gamma_{1}-2\beta_{1}\beta_{2}\gamma_{1}\gamma_{2}+\beta_{2}^{2}\gamma_{1}^{2}\Bigr{)}\frac{1}{L^{2}}$
$\displaystyle\left.+\frac{1}{\beta_{1}^{6}}\Bigl{(}\beta_{2}^{2}\gamma_{1}^{2}-\beta_{1}^{2}\beta_{2}\gamma_{2}-\beta_{1}\beta_{2}\gamma_{1}\gamma_{2}\Bigr{)}\frac{\ln
L}{L^{2}}+\frac{1}{2\beta_{1}^{6}}\left(\beta_{1}\beta_{2}^{2}\gamma_{1}+\beta_{2}^{2}\gamma_{1}^{2}\right)\frac{\ln^{2}L}{L^{2}}\right\\}.$

At the present time $\delta(h,\alpha)$ in an arbitrary gauge at the three-loop
level is not known. But in the MS scheme physical quantities depend only on
$\overline{h}$, so for them we can use the renormalization group at the three-
loop level.

The most interesting application of our results can be the $m_{b}/m_{\tau}$
ratio in the $SU(5)$ grand unified theory and calculation of $\gamma_{m}(h)$
at three loop level in supersymmetric theories [12].

Acknowledgments

I am very grateful to D.V.Shirkov, A.L. Kataev and D.I.Kazakov for valuable
discussions.

The present text was published in 1982 as a JINR Communication,
JINR-P2-82-900.

##  1 Appendix

The three-loop 1PI diagrams of the fermion propagator.

The contributions to $Z_{2}^{-1}$ from different sets of the diagrams
($KR^{\prime}$):
Fig.1.

$\displaystyle-\frac{C_{F}}{\varepsilon^{3}}\left[-\frac{7}{4}C^{2}-\frac{4}{3}CC_{F}+\frac{1}{3}Ctf\right]$
$\displaystyle-\frac{C_{F}}{\varepsilon^{2}}\left[\frac{219}{24}C^{2}+\frac{35}{6}CC_{F}-\frac{4}{3}C_{F}-\frac{19}{6}Ctf-\frac{4}{3}C_{F}tf\right]$
$\displaystyle-\frac{C_{F}}{\varepsilon}\left[-\frac{233}{12}C^{2}+\frac{17}{2}CC_{F}-\frac{7}{12}C^{2}_{F}+\frac{43}{6}Ctf-\frac{7}{3}C_{F}tf\right]-\frac{C_{F}}{\varepsilon}C\zeta(3)\left(\frac{5}{2}C-2C_{F}\right).$

Fig.2a:

$\frac{C_{F}^{2}}{\varepsilon^{2}}\left(\frac{5}{12}C-\frac{1}{3}tf\right)+\frac{C_{F}^{2}}{\varepsilon}\left(\frac{3}{8}C-\frac{1}{6}tf\right).$

Fig.2b:

$\displaystyle-\frac{C_{F}}{\varepsilon^{2}}\Bigl{[}\frac{175}{72}C^{2}-\frac{55}{18}Ctf+\frac{8}{9}t^{2}f^{2}\Bigr{]}-\frac{C_{F}}{\varepsilon}\left[-\frac{2171}{432}C^{2}+\frac{527}{108}Ctf+2C_{F}tf-\frac{20}{27}t^{2}f^{2}\right].$

Fig.2c:

$\displaystyle-\frac{C_{F}^{2}}{\varepsilon^{3}}\left[\frac{1}{3}C-\frac{1}{6}C_{F}\right]+\frac{C_{F}^{2}}{\varepsilon^{2}}\Bigl{[}3C-\frac{7}{12}C_{F}$
$\displaystyle-$ $\displaystyle\frac{2}{3}tf\Bigr{]}$ $\displaystyle-$
$\displaystyle\frac{C_{F}^{2}}{\varepsilon}\left[\frac{91}{24}C-2\zeta(3)C+\frac{1}{12}C_{F}-\frac{5}{6}tf\right].$

Fig.2d: $~{}~{}~{}~{}~{}~{}~{}~{}-\frac{2}{\varepsilon^{3}}C_{F}^{2}C$
The contributions to $Z_{\overline{\psi}\psi}$ from the diagrams with the
insertion $-\\!\\!\bigotimes\\!\\!-$ were calculated simultaneously with the
diagrams shown in Fig.1 and Fig.2. In fact, for the fermion propagator instead
of $\hat{p}/p^{2}$ we used $(\hat{p}+m)/p^{2}$, and after multiplication of
all propagators we kept only the part without the mass term and the part
linear in $m$. The coefficient in front of $m$ determines the contribution of
the diagrams with the mass insertion. For example, from the diagram

we get the following diagrams with the insertion:

The contributions to $Z_{\overline{\psi}\psi}$ from different sets of diagrams
($KR^{\prime}$)

Fig.1:

$\displaystyle-$
$\displaystyle\frac{C_{F}}{\varepsilon^{3}}\left[\frac{31}{3}C^{2}+\frac{50}{3}CC_{F}+8C_{F}^{2}-4Ctf-\frac{8}{3}C_{F}tf\right]$
$\displaystyle-$
$\displaystyle\frac{C_{F}}{\varepsilon^{2}}\left[-\frac{181}{6}C^{2}-\frac{80}{3}C_{F}C+8C_{F}^{2}+\frac{34}{3}Ctf+\frac{8}{3}C_{F}tf\right]$
$\displaystyle-$
$\displaystyle\frac{C_{F}}{\varepsilon}\left[\frac{181}{4}C^{2}-\frac{145}{3}CC_{F}+17C_{F}^{2}-13Ctf+\frac{20}{3}C_{F}tf\right]$
$\displaystyle-$
$\displaystyle\frac{C_{F}}{\varepsilon}\zeta(3)\left[-\frac{3}{2}C^{2}+20C_{F}C-16C_{F}^{2}-8Ctf\right].$

Fig.2a:

$-\frac{C_{F}^{2}}{\varepsilon^{3}}\left[\frac{10}{3}-\frac{8}{3}tf\right]-\frac{C_{F}^{2}}{\varepsilon^{2}}\left[-\frac{17}{3}C+4tf\right]-\frac{C_{F}^{2}}{\varepsilon}\left[-C+\frac{4}{3}tf\right].$

Fig.2b:

$\displaystyle-$
$\displaystyle\frac{C_{F}}{\varepsilon^{3}}\left[\frac{175}{36}C^{2}-\frac{55}{9}Ctf+\frac{16}{9}t^{2}f^{2}\right]$
$\displaystyle-$
$\displaystyle\frac{C_{F}}{\varepsilon^{2}}\left[-\frac{337}{27}C^{2}+\frac{346}{27}Ctf+4C_{F}tf-\frac{64}{27}t^{2}f^{2}\right]$
$\displaystyle-$
$\displaystyle\frac{C_{F}}{\varepsilon}\left[\frac{18685}{1296}C^{2}-\frac{1915}{324}Ctf-17C_{F}tf-\frac{80}{81}t^{2}f^{2}\right]$
$\displaystyle-$
$\displaystyle\frac{C_{F}}{\varepsilon}\zeta(3)\left[-C^{2}-8Ctf+16C_{F}tf\right]$

Fig.2c:

$\displaystyle-$
$\displaystyle\frac{C_{F}^{2}}{\varepsilon^{3}}\left[6C+\frac{8}{3}C_{F}-\frac{8}{3}tf\right]-\frac{C_{F}^{2}}{\varepsilon^{2}}\left[-17C-10C_{F}+4tf\right]$
$\displaystyle-$
$\displaystyle\frac{C_{F}^{2}}{\varepsilon}\left[\frac{80}{3}C+5C_{F}-\frac{16}{3}tf-16\zeta(3)C+16\zeta(3)C_{F}\right]$

Fig.2d: 0.

## References

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–arXiv:1301.5645 [hep-ph].


# PejorativITy: Disambiguating Pejorative Epithets
to Improve Misogyny Detection in Italian Tweets

###### Abstract

Misogyny is often expressed through figurative language. Some neutral words
can assume a negative connotation when functioning as pejorative epithets.
Disambiguating the meaning of such terms might help the detection of misogyny.
In order to address such task, we present PejorativITy, a novel corpus of
1,200 manually annotated Italian tweets for pejorative language at the word
level and misogyny at the sentence level. We evaluate the impact of injecting
information about disambiguated words into a model targeting misogyny
detection. In particular, we explore two different approaches for injection:
concatenation of pejorative information and substitution of ambiguous words
with univocal terms. Our experimental results, both on our corpus and on two
popular benchmarks on Italian tweets, show that both approaches lead to a
major classification improvement, indicating that word sense disambiguation is
a promising preliminary step for misogyny detection. Furthermore, we
investigate LLMs’ understanding of pejorative epithets by means of contextual
word embeddings analysis and prompting.

Keywords: Word sense disambiguation, Hate speech detection, Pejorative
language

PejorativITy: Disambiguating Pejorative Epithets
to Improve Misogyny Detection in Italian Tweets

Arianna Muti1🖂, Federico Ruggeri2, Cagri Toraman3,
---
Lorenzo Musetti4, Samuel Algherini4, Silvia Ronchi4,
Gianmarco Saretto4, Caterina Zapparoli4, Alberto Barrón-Cedeño1
1DIT, University of Bologna, Forlì, Italy
2DISI, University of Bologna, Bologna, Italy
3CEng, Middle East Technical University, Ankara, Turkey
4Expert AI, Modena, Italy
{arianna.muti2, federico.ruggeri6<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
{lmusetti, sronchi, czapparoli<EMAIL_ADDRESS>

Abstract content

Disclaimer: This paper contains examples of offensive and explicit content.

## 1\. Introduction

Pejorative language refers to a word or phrase that has negative connotations
and is intended to disparage or belittle.111https://www.merriam-
webster.com/dictionary/pejorative An inoffensive word becoming pejorative is a
form of semantic drift known as pejoration; thus, pejorativity is context-
dependent: pejorative words have one primary neutral meaning, and another
negatively connotated meaning. The opposite is known as melioration, which is
when a term begins as pejorative and eventually is adopted in a neutral sense,
like in the case of slur reappropriation Galinsky et al. (2013). Pejorative
words are relevant in misogyny detection since many neutral words are used to
address women in an offensive way, targeting either their physical aspect or
their intelligence. We refer to such terms as pejorative epithets. Some
examples in Italian are balena (whale/fat woman) and gallina (chicken/stupid).
State-of-the-art models struggle to correctly classify misogyny when sentences
contain such terms Fersini et al. (2020). The occurrence of polysemic words
with a pejorative connotation in the training set and a neutral connotation in
the test set results in a great number of false positives Muti and Barrón-
Cedeño (2020). For this reason, we introduce pejorative epithets
disambiguation as a preliminary step to detect misogyny. Our goal is to assess
whether the disambiguation of potentially pejorative epithets improves the
detection of misogynistic language, while reducing the rate of false
positives.

Figure 1: Our pipeline. Step 1: a model identifies the connotation of possibly
pejorative epithets. Step 2: the identified connotation is used to enrich
(CONCAT) and substitute (SUBST) part of the textual input for misogyny
detection.

In this work, we aim to answer three research questions:

RQ1

Which epithets are used in misogynistic language in Italian?

RQ2

Can the disambiguation of such words decrease the error rate in misogyny
detection?

RQ3

Can encoder-based language models and generative LLMs differentiate if a word
in a tweet is pejorative or neutral based on its context?

To address RQ1, we compile a list of pejorative words used online to address
women. We use such words to retrieve new tweets, and build PejorativITy, a
novel corpus of Italian tweets, annotated at the word level for pejorativity,
and at the sentence level for misogyny.

To address RQ2, we fine-tune two BERT-based models: modelpej to identify
whether a word in the context of a tweet is pejorative or neutral, following
Dinu et al. (2021), and modelmis to detect misogyny. We use the output of
modelpej to inform modelmis of whether the target word is pejorative within
that context or not. Figure 1 represents our pipeline.

To address RQ3, we compare the cosine similarity between the contextualised
word embeddings of a BERT-based model and their univocal corresponding words
(anchors) before and after fine-tuning for pejorativity detection.
Additionally, we prompt popular instruction-tuned LLMs to test their ability
to disambiguate potentially pejorative words based on the context.

Our contribution is threefold: (1) we release a corpus manually annotated for
pejorativity at the word level and misogyny at the sentence level; (2) we
develop a transformer-based model for detecting pejorative words, whose
predictions are used to enhance the performance of the model targeting
misogyny detection; and (3) we analyse the performance of SOTA generative
models on pejorative epithets disambiguation.

To the best of our knowledge, this is the first work that proposes word sense
disambiguation to linguistically inform computational models for misogyny
detection.

The PejorativITy dataset and the code for all the experiments are available at
https://github.com/arimuti/PejorativITy.

## 2\. Related Work

Misogyny and sexism detection have been explored in different platforms, such
as Gab and Reddit Kirk et al. (2023); Guest et al. (2021), Twitter Jha and
Mamidi (2017); Anzovino et al. (2018), and blogs Breitfeller et al. (2019) in
English; and in different languages, such as Spanish Anzovino et al. (2018);
Plaza et al. (2023), Arabic Almanea and Poesio (2022), and Turkish Toraman et
al. (2022). In Italian, the reference datasets for the identification of
misogyny are the two compiled in the framework of the two editions of the
Automatic Misogyny Identification shared task (AMI) Fersini et al. (2018,
2020).

Our work takes inspiration from Dinu et al. (2021), who (a) explore pejorative
language on social media for the first time; (b) build a multilingual lexicon
of pejorative terms for English, Spanish, Italian, and Romanian; (c) release a
dataset of tweets annotated for pejorative use; and (d) present an attempt to
automatically disambiguate pejorative words in their dataset. Our contribution
differs since, for the first time, the information about the pejorativity of a
word is leveraged to inform the model for misogyny detection. Moreover, our
pejorative lexicon contains words that are currently used on Twitter to
address women in a misogynistic manner. Whereas Dinu et al.’s lexicon
considers hate speech in general, most gender-based words are outdated or
missing, and it does not focus on the sort of slang typically used online.

Another similar work is Pamungkas et al. (2023), who develop the Swear Words
Abusiveness Dataset (SWAD), where abusive swearing in English tweets is
manually annotated at the word level to address the task of predicting the
abusiveness of a swear word based on its context. While their work focuses on
spotting slurs when used in a neutral way (i.e. meliorations), our aim is to
disambiguate neutral words used in an offensive way (i.e. pejoration).
Moreover, Pamungkas et al. exclude highly ambiguous words when creating their
target word lexicon, whereas we precisely focus on them.

Word | Literal | Pejorative | Neutral anchor | Pejorative anchor
---|---|---|---|---
acida | acid/sour | peevish | aspra | intrattabile, stronza
asina | female donkey | stupid | ciuco | stupida
balena | whale/flash | fat woman | cetaceo, balenare | grassa
bambola | doll | girl (objectifying) | giocattolo | donna attraente
cagna | female dog | bitch | cane femmina, canide | donna di facili costumi, troia
cavalla | female horse | ugly/whore | equino | brutta, alta e grossa
civetta | owl | tease | volatile rapace | donna che cerca attenzioni
cesso | toilet | ugly | water, bagno, toilette | brutta
contadina | farmer | ignorant, illiterate | agricoltore femmina | donna ignorante
cortigiana | court lady | prostitute | dama di corte | prostituta
cozza | mussel | ugly/clingy | mollusco | donna brutta, appiccicosa
femminista | feminist | feminazi | femminista | polemica, fastidiosa
fogna | sewer | skanky | fognatura | schifosa, bocca
gallina | chicken | stupid | pennuto | stupida
grezza | raw | rude woman | non lavorato | rozza
lesbica | lesbian | lesbian (offensive) | donna a cui piacciono le donne | schifosa
lurida | dirty | skanky | sporca | promiscua, troia
maiala | sow | whore | maiale femmina | promiscua, troia
mucca | cow | bitch | bovide | stupida, troia
oca | goose | stupid girl | pennuto | stupida, pettegola
pecora | sheep | doormat | ovino | stupida
strega | witch | hag, unpleasant | maga | crudele
vacca | cow | whore | bovino | donna di facili costumi, troia
zingara | gipsy | shabby | gitana | trasandata

Table 1: Italian pejorative lexicon, their literal and pejorative translations
in English, and their anchors.

## 3\. Corpus Compilation

To provide an overview of which misogynous epithets are commonly used on
Twitter in Italian (RQ1), we compile a novel corpus. The compilation involves
two steps: the creation of a lexicon of polysemic words that can function as
pejorative epithets for women, and the retrieval of tweets containing such
words.

#### Lexicon.

We collect our lexicon by selecting words from three distinct sources. (1) We
ask ten Italian native speakers to provide a list of offensive words used
online to address women. The speakers use social media on a daily basis and
their age ranges between 27 and 39 years. (2) We retrieve the keywords used in
the two Italian corpora for the Automatic Misogyny Identification (AMI) shared
task Fersini et al. (2018, 2020). 3) We consult the ’List of Dirty Naughty
Obscene Bad Words’.222https://github.com/LDNOOBW/List-of-Dirty-Naughty-
Obscene-and-Otherwise-Bad-Words/tree/master, consulted on January 2023. We
only keep polysemic words whose primary meaning is neutral and that are
frequently used on Twitter with both pejorative and neutral connotations. To
ensure the quality of our vocabulary, we qualitatively verify that such words
are used with both connotations by manually searching them on Twitter.333Due
to their exclusive neutral or negative connotation on Twitter, the following
words are discarded: barile, banco, botte, barbona, facile, gatta morta,
passeggiatrice, porca, principessa, privilegiata, psicopatica, scrofa, somara,
travestita.

Table 1 shows our lexicon of 24 words. For each word, we report the English
translation of its literal and pejorative meaning, and their anchors in
Italian. Anchor words refer to the unambiguous words used to define polysemic
words. We call these words anchors because their meaning is univocal and does
not change according to the context. For instance, the word balena (whale) is
used to refer to either a sea mammal or an overweight woman. In contrast, the
anchor words cetaceo (cetacean) and grassa (fat) only refer to the animal in
the first case and to being overweight in the second case, at least as far as
their use in Twitter is concerned.444In this case, the word balena has a third
anchor word, from the verb balenare, which means ’to flash’.

#### Tweets.

We use Twarc555https://twarc-project.readthedocs.io to retrieve tweets from
December 2022 to February 2023 containing words in our lexicon. We select 50
tweets for each word in our lexicon, resulting in 1,200 tweets. We keep a
balance of pejorative and neutral use of lexicon words, although an equal
distribution for each word could not be guaranteed.

ID |  Tweet |  Translation
---|---|---
70019 |  Non voglio una cagna un cane ce l’ho giaaaa |  I don’t want a female dog/bitch, I have a dog already.
30021 |  Wow sei una bambola! |  Wow you’re a beautiful girl/doll!
10010 |  Xchè avrà dato una risposta acida a lui |  Because he/she will have given him a sharp answer
61209 |  Ma come fai a dire che sei una balena sei bellissima |  How can you say you’re a whale/fat, you’re beautiful
Table 2: Examples of tweets with potentially pejorative words (underlined).

## 4\. Data Annotation

We recruit six annotators with a background in linguistics, gender studies,
cognitive sciences, and NLP to label our corpus for pejorative word
disambiguation and misogyny detection.

We first devise a pilot annotation study to explore the complexity of the task
and observe differences in how male and female annotators perceive pejorative
connotations. For this purpose, we follow a descriptive annotation paradigm
Röttger et al. (2022), which encourages annotator subjectivity by not
providing guidelines. We split the annotators into two groups and assign 50
tweets each for labeling. Each group is composed of two women and one man with
ages ranging between 27 and 39 years old.

We use Krippendorff’s alpha Krippendorff (2011) to measure the inter-annotator
agreement (IAA). The IAA of the first group is moderate for both pejorativity
(0.48) and misogyny (0.50), whereas the IAA of the second group is fair for
pejorativity (0.33) and moderate for misogyny (0.50). We observe that, in
terms of gender differences, men tend to consider sexual objectifying
compliments as non-pejorative. Based on annotators’ feedback, we identify five
major areas of disagreement:

#### Lack of context.

Some tweets are very short, lacking enough context to understand the intention
of the author. We decide to label such tweets as neutral. Consider tweet 70019
in Table 2.

Although it is likely that the author uses humour to address a woman as a
cagna (bitch), the context does not allow for a clear interpretation: it is
possible that the author does not want another (female) dog, because he has
already one.

#### Objectifying compliments.

Some tweets are intended to compliment women, by means of objectification.
Thus, we label them as pejorative. In the tweet 30021 in Table 2, the term
bambola is used as a compliment, but it is objectifying and, therefore, should
be considered pejorative.

#### Pejorative epithets towards objects.

Some words are used pejoratively towards inanimate objects, therefore, they
should be labeled as neutral. In the tweet 10010 in Table 2, the term acida
refers to an inanimate thing (an answer), although the term is used
pejoratively.

#### Pejorative epithets towards men.

Words that are used pejoratively against men should be labeled as pejorative,
so that the corpus can be used for the general task of pejorativity detection
regardless of the auxiliary task.

#### Reported Speech.

Some tweets contain pejorative epithets, although the intention is not
harmful, because they are contained in reported speech. We label them as
pejorative, since the annotation refers to the word, not to the whole
sentence. Consider tweet 61209 in Table 2: the word balena is pejorative, but
it is used in a positive way by means of negation.

We devise a second pilot annotation, getting closer to a prescriptive
annotation paradigm Röttger et al. (2022), by providing the above guidelines
to the annotators. We select the top 50 tweets that caused more debate during
the first study annotation phase. The IAA computed on all six annotators is
0.53 (moderate), denoting an improvement over the first pilot study.

#### PejorativITy.

After the pilot studies, we annotate our collected corpus of 1,200 tweets.
Only one person carries out the whole annotation process. We select the
annotator with the most interdisciplinary background, who is an expert in
gender studies, linguistics and NLP, who has been a target of misogyny. This
setting is considered among the best practices for the annotation of phenomena
like pejorative epithets and misogyny Abercrombie et al. (2023).

Table 3 shows the statistics of our corpus. The Pearson correlation between
misogyny and pejorativity labels is 0.70, which is in line with our
expectations. The tweets for which misogyny and pejorativity are not aligned
are mainly reported speech or men-related offensive language. It is worth
noting that some sentences might not be considered misogynous, as they do not
express hate towards women. However, they might be considered sexist. For
instance, the sentence “che bella bambola ciao tesoro”666translation: what a
beautiful doll (girl), hi darling does not express hate but perpetuates the
objectification of women by addressing the target of the tweet as a doll,
falling into the category of benevolent sexism Gothreau et al. (2022).

Class | Training | Test | Total
---|---|---|---
Misogynous | 369 | 28 | 397
Pejorative | 363 | 28 | 391
Not pejorative |  6 | – |  6
Non-misogynous | 735 | 68 | 803
Pejorative | 172 | 18 | 190
Not pejorative | 563 | 50 | 613
Table 3: Statistics of the PejorativITy corpus. The same tweets are annotated
for misogyny and pejorativity, for a total of 1,200 instances. For both the
misogynous and the non-misogynous tweets, we report how many contain a
pejorative word and how many do not.

## 5\. Experiments

To understand the impact of disambiguating pejorative words for misogyny
detection (RQ2), we experiment with AlBERTo Polignano et al. (2019), a popular
BERT-based model trained on $200\,M$ Italian tweets. In particular, we fine-
tune AlBERTo on two downstream tasks: pejorative word disambiguation and
misogyny detection.

For pejorativite word disambiguation, we evaluate AlBERTo only on our corpus.
For misogyny detection, we also consider the two other benchmark datasets for
Italian: AMI-2018 Fersini et al. (2018) and AMI-2020 Fersini et al. (2020). To
the best of our knowledge, these are the only corpora that address misogyny
detection on Italian tweets. Table 4 shows their statistics.

AMI-2018 |  Misogynous |  Not |  Total
---|---|---|---
Train |  1,828 |  2,172 |  4,000
Test |  512 |  488 |  1,000
AMI-2020 |  Misogynous |  Not |  Total
Train |  2,337 |  2,663 |  5,000
Test |  500 |  500 |  1,000
Table 4: Statistics of the AMI 2018 and 2020 corpora Fersini et al. (2018,
2020).

We formulate the disambiguation of pejorative words as a binary classification
task, where a model classifies a word contained in a sentence as pejorative or
neutral. Then, we use the information about the pejorativity of a word to
enrich the input to the model responsible for the detection of misogyny. Since
AMI-2018 and AMI-2020 are not annotated for pejorative word disambiguation, we
use the model fine-tuned on our corpus to determine the connotation of
ambiguous words.

Formally, we devise the following pipeline, where $w\in W$ is a word from our
lexicon $W$ of pejorative words:

  1. (a)

We train modelpej that, given a tweet containing a word $w\in W$, predicts
whether $w$ is being used in a pejorative way.

  2. (b)

We enrich input tweets in all data partitions by injecting knowledge about the
pejorativity of our lexicon words according to modelpej. We try two different
approaches to modify the input data: i) we concatenate the information about
the pejorativity of $w$ at the end of the tweet or ii) we substitute the
ambiguous $w$ with its corresponding anchor word.

  3. (c)

We train modelmis to detect misogyny with the enriched input tweets.

Our pipeline is meant to process any tweet. However, as a first step, we check
whether it contains at least one $w\in W$. In our setup, when testing on the
subset of AMI-2018 and AMI-2020 containing only pejorative words (epithets),
thhat are recognized through string matching after lemmatization.

As hyper-parameters, we use the AdamW optimizer with $\epsilon=1^{-8}$
Loshchilov and Hutter (2017). We fine-tune AlBERTo for 4 epochs with batch
size 16. We report macro and per-class F1-measure as standard metrics for
binary classification tasks, averaged over three individual runs. All the
experiments are run using Google Colab’s GPU.

Approach |  Macro |  Mis. |  Not
---|---|---|---
baseline |  0.68 |  0.56 |  0.79
concatenation |  |  |
w/ gold |  0.83 |  0.78 |  0.88
w/ predictions |  0.75 |  0.68 |  0.82
substitution |  |  |
w/ gold |  0.87 |  0.82 |  0.92
w/ predictions |  0.77 |  0.69 |  0.84
Table 5: Macro and per-class F1-score on PejorativITy concerning misogyny
detection.

## 6\. Results

Regarding pejorative word disambiguation, the fine-tuned AlBERTo model
(modelpej) reaches a macro F1-measure of $0.82\pm 0.03$ on the PejorativITy
test partition.

Table 5 shows the classification performance for misogyny detection on the
PejorativITy test partition. We compare our fine-tuned AlBERTo model
(baseline) against the alternatives that leverage pejorative word
disambiguation. We evaluate the concatenation and substitution approaches
using modelpej (w/ predictions) and annotators’ labels (w/ gold) since our
corpus contains annotations for pejorative word disambiguation. The evaluation
of our proposed approaches with gold labels defines an upper bound to our
pipeline. We observe a notable improvement over the baseline model for
concatenation ($+7$ absolute points) and substitution ($+9$ absolute points)
when using modelpej predictions. The improvement significantly increases when
both approaches consider gold labels, with a maximum gain of $+19$ absolute
points. These results reflect the effectiveness of our approach and
corroborate our initial hypothesis on reducing the false positive rate for
misogyny detection.

Table 6 shows the number of false positives in the three datasets, before and
after the inclusion of pejorative information both by concatenation and
substitution. The decrease of false positives is clear in AMI-2020 and in our
PejorativITy test set. In AMI-2018, no decrease is observed. One of the
reasons for this low impact is that AMI-2018 contains pejorative epithets only
in 34 instances out of 1000 (compared to 192 in AMI-2020), therefore we did
not expect our approach to have a huge impact on that dataset.

Table 7 shows the classification performance for misogyny detection on
AMI-2018 and AMI-2020. To assess the impact of our pipeline on these corpora,
we show the performance of the models both on the test instances that contain
words in our lexicon (epithets) and on the whole corpora. In particular, we
perform fuzzy string matching (Section 3) to filter tweets according to this
criterion, resulting in 389 (355 train, 34 test) tweets for AMI-2018 and 605
(413 train, 192 test) tweets for AMI-2020 in the training and test set
respectively. We observe an F1-measure improvement of $+3$ absolute points in
AMI-2018 and $+4$ absolute points in AMI-2020 with the concatenation approach.
In contrast, the substitution strategy does not lead to any performance gain.
A possible explanation is the quality of substituted anchors. We provide an
example in the next section. Since AMI corpora mainly contain tweets with
explicit misogyny, the limited number of retrieved samples is expected. For
this reason, the observed gain on selected tweets does not impact the overall
performance on the original test partition in both corpora (whole).

Dataset | Baseline | concat. | subst.
---|---|---|---
PejorativiITy |  25 |  16 |  21
AMI 2018 | 107 | 107 | 112
AMI 2020 | 127 | 126 | 121
Table 6: False positive rates comparison. In the PejorativITy the total number
of instance is 96, while in AMI 2018 and 2020 is 1,000.

To sum up, our results suggest that the disambiguation of potentially
pejorative words is helpful in addressing misogyny detection when targeting
ambiguous examples.

### 6.1. Qualitative Error Analysis

We carry out a manual error analysis, by observing misclassified tweets in
AMI-2020 epithets and our corpus for the task of misogyny detection. We
compare misclassified tweets in the three settings: baseline, concatenation,
and substitution.

Regarding the concatenation approach, most of the misclassifications occur
when reported misogyny is concerned. The model struggles to recognise when a
pejorative epithet is used in a reported speech to condemn a misogynistic
attitude and not to address a potential target. It is worth noticing that if a
pejorative connotation is predicted in reported speech, this does not imply
that misogyny is predicted. Consider the following example:

  * Lei è acida perché non ha figli penso che darebbe fastidio a qualsiasi donna. Che schifo.777She’s peevish because she doesn’t have children I think it would bother all women. Disgusting.

AMI-2018 | epithets | whole
---|---|---
Approach | Macro | Mis. | Not | Macro | Mis. | Not
baseline | 0.79 | 0.77 | 0.81 | 0.86 | 0.87 | 0.85
concatenation | 0.82 | 0.81 | 0.83 | 0.86 | 0.88 | 0.85
substitution | 0.79 | 0.79 | 0.80 | 0.86 | 0.87 | 0.84
AMI-2020 | epithets |  whole
Approach | Macro | Mis. | Not | Macro | Mis. | Not
baseline | 0.77 | 0.74 | 0.81 | 0.82 | 0.84 | 0.81
concatenation | 0.81 | 0.77 | 0.84 | 0.83 | 0.84 | 0.82
substitution | 0.77 | 0.73 | 0.81 | 0.82 | 0.84 | 0.81

Table 7: Macro and per-class F1-measure on AMI-2018 and AMI-2020 concerning
misogyny detection. We report metrics for each corpus (whole) and their subset
containing words in our lexicon (epithets).

In this example, the author of the tweet criticises a reported misogynous
sentence. Even if acida is correctly predicted as pejorative, the model still
gets the correct prediction that the sentence is non-misogynous. Another
observed pattern of misclassification is when the target of the pejorative
epithet is a man. In this case, the tweet should not be considered misogynous,
although it contains a pejorative word from our lexicon. This bias is
introduced due to the annotation of pejorative epithets against men as
pejorative. Overall, the overlap between tweets classified as containing
pejorative words and those classified as misogynous is of 26 tweets in the
PejorativITy test set (out of 96), 12 tweets out of 34 in the
AMI2018_epithets, and 67 out of 192 in AMI2020_epithets. We highlight this
aspect to show that model_$mis$ does not necessarily learn to classify
misogyny according to model_$pej$’s outcome.

Regarding the substitution approach, we observe that a wrong pejorative
prediction of lexicon words affects the prediction of misogyny. The following
example:

  * Ma la balena con gli shorts cortissimi invece è vittima del patriarkato e può vestirsi come vuole?888That whale/fat girl with very short pants is a victim of the patriarchy and can dress up as she wants?

is correctly classified by the baseline model. A misclassification of the word
balena,which modelpej predicts as neutral, causes confusion in both enriched
models.

|  | pretrained | Fine-tuned
---|---|---|---
Lexicon | Anchor |  Pejorative | Neutral | Pejorative | Neutral
| aspra | $0.27\pm 0.12$ | $0.27\pm 0.14$ | $0.09\pm 0.12$ | $0.29\pm 0.10$
acida | intrattabile | $0.28\pm 0.12$ | $0.28\pm 0.14$ | $0.28\pm 0.05$ | $0.27\pm 0.07$
| stronza | $0.31\pm 0.14$ | $0.31\pm 0.17$ | $0.53\pm 0.12$ | $0.23\pm 0.15$
balena | balenare | $0.26\pm 0.12$ | $0.30\pm 0.10$ | $0.19\pm 0.10$ | $0.44\pm 0.08$
cetaceo | $0.22\pm 0.12$ | $0.26\pm 0.09$ | $0.04\pm 0.10$ | $0.36\pm 0.10$
grassa | $0.19\pm 0.12$ | $0.22\pm 0.09$ | $0.29\pm 0.09$ | $0.07\pm 0.07$
cagna | canide | $0.43\pm 0.15$ | $0.29\pm 0.15$ | $0.08\pm 0.05$ | $0.25\pm 0.06$
donna di facili costumi | $0.42\pm 0.13$ | $0.27\pm 0.15$ | $0.30\pm 0.04$ | $0.21\pm 0.09$
troia | $0.41\pm 0.16$ | $0.26\pm 0.16$ | $0.57\pm 0.08$ | $0.21\pm 0.10$
cesso | water | $0.37\pm 0.14$ | $0.37\pm 0.13$ | $0.08\pm 0.06$ | $0.26\pm 0.08$
bagno | $0.39\pm 0.14$ | $0.41\pm 0.13$ | $0.07\pm 0.06$ | $0.35\pm 0.10$
toilette | $0.37\pm 0.13$ | $0.39\pm 0.12$ | $0.09\pm 0.05$ | $0.30\pm 0.08$
brutta | $0.39\pm 0.15$ | $0.40\pm 0.13$ | $0.43\pm 0.07$ | $0.16\pm 0.09$
lesbica |  donna a cui piacciono le donne | $0.40\pm 0.13$ | $0.42\pm 0.16$ | $0.28\pm 0.05$ | $0.34\pm 0.09$
schifosa | $0.32\pm 0.15$ | $0.32\pm 0.17$ | $0.30\pm 0.09$ | $0.18\pm 0.06$
vacca | bovino | $0.31\pm 0.14$ | $0.25\pm 0.12$ | $0.10\pm 0.07$ | $0.22\pm 0.07$
donna di facili costumi | $0.35\pm 0.12$ | $0.29\pm 0.12$ | $0.27\pm 0.05$ | $0.20\pm 0.08$
troia | $0.35\pm 0.14$ | $0.29\pm 0.13$ | $0.50\pm 0.09$ | $0.25\pm 0.14$

Table 8: Average cosine similarity between lexicon word embeddings and both
pejorative and  neutral anchor word embeddings in pejorative and neutral
samples. Embeddings extracted from both the pretrained and the fine-tuned
AlBERTo model.

## 7\. Analysis of Contextualised Word Embeddings

To investigate the semantic knowledge of the AlBERTo pretrained language model
Polignano et al. (2019) about the pejorative epithets and to evaluate how
fine-tuning affects its knowledge (RQ3), we extract and analyse the
contextualised word embeddings of our lexicon words.

To extract these embeddings, we perform fuzzy string matching on input tweets
to retrieve the tokenized text span corresponding to lexicon words. We use
fuzzy string matching to address all representations of a lexicon word (e.g.,
balena and balenare). It is worth noticing that the retrieved text span may
contain multiple tokens according to the employed tokenization process. In our
scenario, the AlBERTo model employs the sentencepiece tokenizer Kudo and
Richardson (2018), the common tokenization process for transformer models. For
instance, the lexicon word balena is tokenized to the [balen, ##a] text span.
We then use these text spans to aggregate the corresponding word embeddings.
We define the word embedding of a lexicon word as the average of the AlBERTo
token embeddings in the retrieved text span. Considering balena, we define its
word embedding by extracting the embeddings of balen and ##a and computing
their average.

We compute the average cosine similarity between lexicon words and their
corresponding neutral and pejorative anchors. To carry out our analysis, we
consider lexicon words from PejorativITy with several neutral and offensive
anchors: acida, balena, cagna, cesso, lesbica, and vacca.

Table 8 reports the results on PejorativITy comparing the pretrained AlBERTo
model and its fine-tuned version. The pretrained model does not discriminate
between  neutral and  offensive anchors in pejorative and neutral samples. For
instance, the average cosine similarity between acida and its pejorative
anchor  stronza is 0.31 in both class samples. In contrast, our fine-tuned
AlBERTo model shows relevant discrepancies when considering lexicon word
embeddings in pejorative and neutral samples. For instance, the similarity
between acida and its neutral anchor  aspra is 0.09 in pejorative samples and
0.29 in neutral ones. In contrast, the similarity between acida and its
pejorative anchor  stronza is significantly higher in pejorative samples
(0.53), compared to neutral ones (0.23). This is an indicator that our fine-
tuned model acquired knowledge about acida being a synonym of  stronza when
used in a pejorative way. We observe similar trends in all other selected
lexicon words.

The average similarity with  pejorative anchors in pejorative and neutral
samples using pretrained embeddings is 0.34 and 0.29, respectively. In
contrast, the similarities are 0.39 and 0.20 using fine-tuned embeddings.
These results show  pejorative anchors have higher similarity with lexicon
words in pejorative samples. Likewise,  neutral anchors with lexicon words in
neutral samples: the average similarity in pejorative and neutral samples
using pretrained embedding is 0.33 and 0.34, respectively, while the
similarities are 0.11 and 0.31 using fine-tuned embeddings.

To prevent a biased analysis where our fine-tuned model learns a
representation of anchors similar to that of lexicon words, we compute each
anchor frequency in our corpus. The computed frequency for each anchor is
close to zero, thus excluding potential biases in our analysis.

|  Tweet |  Translation |  Generated Disambiguation
---|---|---|---
1. |  Tra il non normalizzare e il dargli della balena ci sta l’oceano pacifico. |  Between the non-normalisation and calling her whale there is the Pacific Ocean. |  The meaning of the word balena in this sentence is to indicate something large, imposing, and difficult to overcome or face, similar to the Pacific Ocean.
2. |  Non voglio una cagna un cane ce l’ho giaaaa |  I don’t want a female dog/bitch, I have a dog already. |  The meaning of the word cagna in this sentence is “ragazza” (girl). The word cagna is a slang term used to refer to a girl. In this context, the phrase means "I don’t want a girl, I already have a dog."
3. |  Chi sei tu per dare cagna e zoccola alle persone? |  Who are you to address people as bitch and whore? |  In this case, the author of the sentence is calling the other user a bitch probably to express anger or contempt toward her.
4. |  Madonna mia che cozza questa!! |  What an ugly girl! |  The meaning of the word cozza in this phrase is to express surprise or amazement. Specifically, the phrase can be literally translated as "My God, how beautiful this is!".

Table 9: Examples of Mistral pejorative word disambiguation. Lexicon words are
underlined.

## 8\. Analysis of Large Language Models

We prompt open-source generative large language models (LLMs) to investigate
their knowledge of pejorative words. Specifically, we aim to find out whether
they are able to clarify the meaning of our lexicon words (RQ3).

We follow a zero-shot prompting approach without any training data for the
downstream task Liu et al. (2023). We design the following prompt that asks
the meaning of a pejorative _{ word}_ in a given _{ sentence}_. We prompt the
LLMs for each test instance in our corpus. The prompt is given as follows:

  * [INST] Di seguito è riportata un’istruzione che descrive un task. Scrivete una risposta che completi adeguatamente la richiesta.
### Istruzione:
Qual è il significato della parola ’’{word}’’ in questa frase?
’’{sentence}’’[/INST]
### Risposta:

The translation in English would be:

  * [INST] Below there is an instruction describing a task. Write a response that completes the request appropriately.
### Instruction:
What is the meaning of the word "{word}" in this sentence?
’’{sentence}’’[/INST]
### Response:

We use three open-source LLMs for our analysis:

#### LlaMa:

LlaMa is a decoder-based language model pretrained on publicly available data
collections Touvron et al. (2023a). Since LlaMa does not support Italian, we
employ Camoscio999https://github.com/teelinsan/camoscio, an Italian
instruction-tuned LlaMa model.

#### LlaMa2:

LlaMa2101010https://huggingface.co/meta-llama/Llama-2-7b-hf is an optimized
version of LlaMa by increasing context length from 2048 to 4096, and applying
group-query attention Touvron et al. (2023b). The majority of the training
data of LlaMa2 is in English, but it still responds to Italian prompts due to
a small amount of training data in Italian.

#### Mistral:

MistralAI111111https://huggingface.co/mistralai is based on LlaMa2 but
exhibits superior performance due to the employed attention mechanisms such as
group-query attention and sliding-window attention Jiang et al. (2023). The
details of the corpora used in training are not given, yet it responds to
Italian prompts.

For all models, we select the 7b model version with 8-bit weights due to
hardware constraints. We apply Beam Search for text generation with the
following hyperparameters. The temperature is set to 0.2, the number of beams
is set to 4 with a top-p value of 0.75, the output length is set to 300 tokens
with a repetition penalty of 1.8. The analysis is conducted on 4 GPUs (NVIDIA
GeForce RTX 2080 Ti).

### 8.1. Qualitative Analysis

We manually inspect all the responses generated by the three LLMs.

#### Mistral.

We observe that Mistral is the best in disambiguating the connotations of
lexicon words, especially when it comes to neutral senses. For instance, it
correctly disambiguates when balena refers to the animal and when to the verb
balenare.

However, Mistral struggles when the term balena is used pejoratively. Consider
Example 1 in Table 9. Mistral gets the idea that balena is used as a metaphor
for something big, but it does not link its meaning to being overweight.

Mistral is remarkably good at capturing irony as well. Consider Example 2 in
Table 9. While this example caused trouble to human annotators for the lack of
context, Mistral is confident in identifying the pejorative connotation of the
lexicon word cagna.

Although performing very well, Mistral struggles with reported speech, too.
Consider Example 3 in Table 9. While Mistral correctly identifies the
pejorative connotation, it fails to understand that the author of the tweet is
condemning, not enforcing, a misogynistic statement.

Moreover, in some cases, Mistral makes up meanings. For instance, Mistral
defines cavalla (horse / ugly and tall woman) as a “a painful surprise”, while
it defines cozza (mussel / ugly, clingy) as “impatiently waiting”.121212In the
sentence ”Sta cozza non vedeva l’ora”, translated as ”That ugly girl couldn’t
wait” A possible explanation is that Mistral uses the semantics of the whole
sentence to generate a definition of lexicon words. In some other cases,
Mistral generates the opposite meaning. In Example 4 of Table 9, Mistral
defines cozza as “surprisingly beautiful”.

#### Llama and Camoscio.

Neither model shows an adequate performance in disambiguating lexicon words.
In most cases, both models produce the following answer: “the word {word}
means {word}”, which is not useful for disambiguation.

Our analysis suggests that off-the-shelf instruction-tuned LLMs have ample
room for improvement concerning pejorative word disambiguation. A fine-tuning
phase on the task could address the highlighted issues. However, we believe
that a detailed analysis of instruction-tuned LLMs on our proposed pipeline
deserves a separate study. We leave this analysis as future work.

## 9\. Conclusions

We introduce pejorative word disambiguation as a preliminary step for misogyny
detection to reduce the error rate of classification models on polysemic words
that can serve as pejorative epithets. For this purpose, we build a lexicon of
polysemic words with both pejorative and neutral connotations and use it to
compile a novel corpus of 1,200 manually expert-annotated Italian tweets for
pejorative word disambiguation and misogyny detection. We validate our
pipeline by evaluating AlBERTo Polignano et al. (2019) on our corpus and on
two benchmark corpora in Italian: AMI-2018 Fersini et al. (2018) and AMI-2020
Fersini et al. (2020). We explore two approaches to inject pejorativity
information: concatenation and substitution. Our results show that the
disambiguation of potentially pejorative words leads to notable classification
improvements in all testing scenarios. Furthermore, we analyse the word
embedding representation of AlBERTo and show that the encoding of lexicon
words is closer to their ground-truth connotation after fine-tuning. Lastly,
we qualitatively analyse several off-the-shelf instruction-tuned LLMs on
pejorative word disambiguation to evaluate their capabilities, showing that
there is ample room for improvement.

Future research directions include the extension of our pipeline to
automatically extract potentially pejorative words at the span level; the
application of knowledge bases like ConceptNet Speer et al. (2017) for
pejorative word disambiguation; and the implementation of instruction-tuned
LLMs in our pipeline. Moreover, we plan to expand this work towards other
languages, and the cultures behind them. Our aim is to carry out a cross-
cultural analysis on the differences in terms of pejorative terms for misogyny
across cultures with different perspectives towards women rights and feminism.

## 10\. Ethical Considerations

We use publicly available tweets to collect our corpus. All data collection
adheres to Twitter’s terms of service and privacy policies. As this research
involves the analysis of publicly available tweets, we do not seek explicit
consent from individual users. Nevertheless, we make every effort to protect
the anonymity of all individuals mentioned or quoted in this work: Any
reported example is carefully selected to avoid identifying specific users or
victims.

## 11\. Limitations

#### Language.

In our study, we only focus on the Italian language. While this choice does
not limit the applicability of our contributions, we are aware that including
other languages could strengthen the impact of our results. We leave this
extension as future work.

#### Corpus

Although our lexicon covers a wide variety of words that can serve as
pejorative epithets for women, it is not an exhaustive list, as we have
discarded all the terms that are not polysemic and that are used only with one
connotation (either positively or negatively) on Twitter.

Only 100 tweets are annotated by six annotators, while the remaining 1,100 are
labelled by only one annotator. Although we select an expert with an
interdisciplinary background in linguistics, gender studies and NLP to carry
out all the annotations, their personal biases, opinions, or interpretations
can lead to skewed or one-sided data.

#### Approaches.

A limitation of our study concerns the substitution approach. First of all,
some words have more than one neutral anchor words. This is the case of
balena, which has two neutral anchors: balenare (to flash) and cetaceo (sea
mammal). In neutral examples, we substitute balena with both anchors. This
process may alter the semantic meaning of the tweet since only one anchor is
suitable for substitution. Moreover, in some cases, we replace a lexicon word
with anchors that do not have the same meaning. For instance, the neutral
anchor of acida is aspra (sour). However, expressions like sour beer or sour
cream do not have a valid anchor replacement. Therefore, replacing aspra with
acida is not an appropriate substitution.

#### Models.

We only employ AlBERTo to carry out our experiments. However, several other
models might lead to different results. Therefore, our experiments are not
sufficient to generalise the results of our analysis to all encoder-based
models.

#### Prompting.

The most popular generative models —GPT family— have not been included in this
study, although they could have shown promising capacities in disambiguating
the senses of our polysemic words. Nevertheless, we intentionally exclude them
from this study, as we want to focus on open-source models only.

## Acknowledgements

A. Muti’s research is carried out under project “DL4AMI–Deep Learning models
for Automatic Misogyny Identification”, in the framework of Progetti di
formazione per la ricerca: Big Data per una regione europea più ecologica,
‘digitale e resiliente—Alma Mater Studiorum–Universita di ‘Bologna, Ref.
2021-15854. The work of F. Ruggeri is supported by the European Union’s
Horizon Europe research and innovation programme under GA 101070000. We thank
to Umitcan Sahin for his support during corpus compilation.

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