††thanks: Chin. Phys. C 45, 083104 (2021); DOI:10.1088/1674-1137/ac06ac

# Study of $C$ parity violating and
strangeness changing $J/{\psi}$ ${\to}$ $PP$ weak decays

Yueling Yang  Institute of Particle and Nuclear Physics, Henan Normal
University, Xinxiang 453007, China  Junliang Lu  Institute of Particle and
Nuclear Physics, Henan Normal University, Xinxiang 453007, China  Mingfei Duan
Institute of Particle and Nuclear Physics, Henan Normal University, Xinxiang
453007, China  Jinshu Huang  School of Physics and Electronic Engineering,
Nanyang Normal University, Nanyang 473061, China  Junfeng Sun  Institute of
Particle and Nuclear Physics, Henan Normal University, Xinxiang 453007, China

###### Abstract

Although $J/{\psi}$ weak decays are rare, they are possible within the
standard model of elementary particles. Inspired by the potential prospects of
the future intensity frontier, the $C$ parity violating $J/{\psi}$ ${\to}$
${\pi}{\eta}^{({\prime})}$, ${\eta}{\eta}^{\prime}$ decays and the strangeness
changing $J/{\psi}$ ${\to}$ ${\pi}K$, $K{\eta}^{({\prime})}$ decays are
studied with the perturbative QCD approach. It is found that the $J/{\psi}$
${\to}$ ${\eta}{\eta}^{\prime}$ decays have relatively large branching ratios,
about the order of $10^{-11}$, which might be within the measurement
capability and sensitivity of the future STCF experiment.

##  I Introduction

Today, nearly fifty years after the discovery of the $J/{\psi}$ particle in
1974 prl.33.1404 ; prl.33.1406 , charmonium continues to be an interesting and
exciting subject of research, because they bridge the physics contents between
the perturbative and nonperturbative energy scales and provide a good place to
understand the complex behavior and dynamics of strong interactions. In
addition, the recent observations of exotic resonances beyond our
comprehension, such as XYZs pdg2020 , have caused an upsurge of research on
charmonium-like states and stimulated a lot of experimental and theoretical
activities.

The $J/{\psi}$ particle, a system consisting of the charmed quark and
antiquark pair $c\bar{c}$, is the lowest orthocharmonium state with the well
established quantum number of $J^{PC}$ $=$ $1^{--}$ pdg2020 . With the same
quantum number $J^{PC}$ as the photon, the $J/{\psi}$ particle can be directly
produced by $e^{+}e^{-}$ annihilation. To date, there are more than $10^{10}$
$J/{\psi}$ events available with the BESIII detector dataweb . Considering the
large $J/{\psi}$ production cross section ${\sigma}$ ${\sim}$ $3400$ $nb$
nimpra614.345 , it is expected that more than $10^{13}$ $J/{\psi}$ events will
be accumulated at the planning Super Tau Charm Facility (STCF) with
$3\,ab^{-1}$ on-resonance dataset in the future. The large amount of data
provides a good opportunity for studying the properties of the $J/{\psi}$
particle, understanding the strong interactions and hadronic dynamics,
exploring novel phenomena, and searching for new physics (NP) beyond the
standard model (SM).

The mass of the $J/{\psi}$ particle, $m_{\psi}$ $=$ $3096.9$ MeV pdg2020 , is
below the open charm threshold. The $J/{\psi}$ hadronic decays via the
annihilation of $c\bar{c}$ quark into gluons are of a higher order in the
quark-gluon coupling ${\alpha}_{s}$ and are therefore severely suppressed by
the phenomenological Okubo-Zweig-Iizuka (OZI) rule ozi-o ; ozi-z ; ozi-i . The
OZI suppression results in (1) the electromagnetic decay ratio having the same
order of magnitude as its strong decay ratio, ${\cal
B}r(J/{\psi}{\to}{\gamma}^{\ast}{\to}{\ell}^{+}{\ell}^{-}+\text{hadrons})$
${\approx}$ $25\%$ and ${\cal B}r(J/{\psi}{\to}ggg)$ ${\approx}$ $64\%$
pdg2020 , and (2) a small decay width, ${\Gamma}_{\psi}$ $=$ $92.9{\pm}2.8$
keV pdg2020 . Generally, the more the number of particles in the final states,
the more the effect of compact phase spaces resulting in a relatively less
occurrence probability, and the lower the experimental signal reconstruction
efficiency. The kinematics are simple for the $J/{\psi}$ two-body decays.
Given the conservation of quantum number $J^{P}$ (i.e., the simultaneous
conservation of both angular momentum and $P$ parity) in the strong and
electromagnetic interactions, the $J/{\psi}$ ${\to}$ $PP$, $PV$, $VV$, $SS$,
$SA$, $AA$ decays originate from the $P$-wave contributions corresponding to
the relative orbital angular momentum of the final states ${\ell}$ $=$ $1$,
and additional $F$-wave (${\ell}$ $=$ $3$) contributions to $J/{\psi}$ ${\to}$
$VV$, $AA$ decays, where $P$, $V$, $S$ and $A$ represent the light $SU(3)$
meson nonets, the pseudoscalar meson $P$ with $J^{P}$ $=$ $0^{-}$, vector
meson $V$ with $J^{P}$ $=$ $1^{-}$, scalar meson $S$ with $J^{P}$ $=$ $0^{+}$,
and axial-vector meson $A$ with $J^{P}$ $=$ $1^{+}$. The $J/{\psi}$ ${\to}$
$PA$, $VS$, $VA$ decays emerge from the $S$-wave (${\ell}$ $=$ $0$)
contributions, and additional $D$-wave (${\ell}$ $=$ $2$) contributions to
$J/{\psi}$ ${\to}$ $VS$, $VA$ decays; however, the $J/{\psi}$ ${\to}$ $PS$
decays are forbidden. Usually, the $V$, $S$ and $A$ mesons are unstable and
decay immediately after their productions into many other particles.
Experimentally, the branching ratios of the $J/{\psi}$ ${\to}$ $PV$ decays for
all the possible flavor-conservation combinations of final states, such as
${\pi}{\rho}$, ${\pi}{\omega}$, ${\eta}^{(\prime)}{\rho}$
${\eta}^{(\prime)}{\omega}$, ${\eta}^{(\prime)}{\phi}$ and
$K\overline{K}^{\ast}$, have been well determined, except for the double-OZI
suppression ${\pi}{\phi}$ mode pdg2020 . For the $J/{\psi}$ ${\to}$ $PP$ and
$VV$ decays, when the final states have explicit $C$-parity, the $C$
invariance forbids these processes, and Bose symmetry strictly forbids two
identical particles in the final states. Presently, only five branching ratios
for the $J/{\psi}$ ${\to}$ ${\pi}^{+}{\pi}^{-}$, $K^{+}K^{-}$,
$K_{L}^{0}K_{S}^{0}$, $K^{{\ast}{\pm}}\overline{K}^{{\ast}{\mp}}$ and
$K^{{\ast}0}\overline{K}^{{\ast}0}$ decays have been quantitatively measured
pdg2020 . Theoretically, the $J/{\psi}$ particle is widely regarded as a
$SU(3)$ singlet, and the possible admixture of light quarks is negligible. It
is usually assumed prd14.298 ; prd18.791 ; prd28.2767 ; prd31.1753 ; prd49.275
; prd74.074003 ; plb645.173 ; prd77.014010 ; jpg35.055002 ; cpc34.299 ;
prd85.074015 ; prd91.014010 ; prd14.852 ; prd32.2883 ; prd32.2961 ; zpc32.467
; plb173.97 ; prd38.824 ; prd38.2695 ; pr174.67 ; prd41.1389 ; prd44.175 ;
zpc61.147 ; plb403.339 ; prd55.2840 ; prd57.5717 ; epjc7.271 ; prd60.074029 ;
prd62.074006 ; epjc28.335 ; ijmpa18.3335 ; jhep0710.026 ; jpg36.115006 ;
epjc65.467 ; cpc34.1785 ; cpc37.073103 ; cpc38.063101 ; npb323.75 ; prd42.1577
; prl80.5060 ; npa828.125  that the $J/{\psi}$ decay into two mesons could be
induced by the interferences of (a) $c\bar{c}$ ${\to}$ $ggg$ ${\to}$
$q\bar{q}$, where $q$ denotes light quark. (b) $c\bar{c}$ ${\to}$ ${\gamma}gg$
${\to}$ $q\bar{q}$, (c) $c\bar{c}$ ${\to}$ ${\gamma}^{\ast}$ ${\to}$
$q\bar{q}$, (d) the $c\bar{c}$ ${\leftrightarrow}$ $q\bar{q}$ mixing, and (e)
the virtual process $c\bar{c}$ ${\to}$ $c\bar{q}$ $+$ $\bar{c}q$ ${\to}$
$q\bar{q}$. Based on the quark model or unitary-symmetry schemes, the
$J/{\psi}$ ${\to}$ $PP$, $PV$ decays via the strong and electromagnetic
interactions have been extensively studied using phenomenological models, such
as the vector meson dominance model in Refs. prd14.298 ; prd18.791 ;
prd28.2767 ; prd31.1753 ; prd49.275 ; prd74.074003 ; plb645.173 ; prd77.014010
; jpg35.055002 ; cpc34.299 ; prd85.074015 ; prd91.014010  and various
parametrization of the OZI $J/{\psi}$ process in Refs. prd14.852 ; prd32.2883
; prd32.2961 ; zpc32.467 ; plb173.97 ; prd38.824 ; prd38.2695 ; pr174.67 ;
prd41.1389 ; prd44.175 ; zpc61.147 ; plb403.339 ; prd55.2840 ; prd57.5717 ;
epjc7.271 ; prd60.074029 ; prd62.074006 ; epjc28.335 ; ijmpa18.3335 ;
jhep0710.026 ; jpg36.115006 ; epjc65.467 ; cpc34.1785 ; cpc37.073103 ;
cpc38.063101 .

Currently, the sum of all the measured branching ratio of exclusive $J/{\psi}$
decay modes, which include leptonic, hadronic and radiative decay modes, is
about 66% pdg2020 , therefore there are many other $J/{\psi}$ decay modes
remain to be experimentally determined and studied. Besides the strong and
electromagnetic decays, the $J/{\psi}$ particle can also decay via the weak
interactions within SM, although it is estimated that the branching ratios of
$J/{\psi}$ weak decays could be very small, about
$2/{\tau}_{D}{\Gamma}_{\psi}$ ${\sim}$ ${\cal O}(10^{-8})$, where ${\tau}_{D}$
and ${\Gamma}_{\psi}$ are the lifetime of the charmed $D$ meson and the full
width of the $J/{\psi}$ particle. In principle, the $J/{\psi}$ weak decays are
possible in different ways: (a) the $c\bar{c}$ pair annihilation into a
virtual $Z$ boson cascading into leptons or quarks, which is unsuitable for
measurements owing to the serious pollution of strong and electromagnetic
decays, (b) the $W$ emission, (c) the $W$ exchange, and (d) the flavor-
changing-neutral currents. The characteristic signal of the $W$-emission
$J/{\psi}$ weak decays is a single charmed hadron in the final states. The
semileptonic $J/{\psi}$ ${\to}$ $D_{(s)}^{(\ast)}$ $+$
${\ell}^{+}{\nu}_{\ell}$ decays and hadronic $J/{\psi}$ ${\to}$
$D_{(s)}^{(\ast)}$ $+$ $X$ decays (where $X$ $=$ $P$, $V$) have been
investigated experimentally pdg2020 ; prd90.112014 ; prd96.111101 ; 2104.06628
; prd89.071101  and phenomenologically zpc62.271 ; epjc54.107 ; jpg36.105002 ;
prd92.074030 ; prd78.074012 ; ahep2013.706543 ; jpg44.045004 ; plb252.690 ;
ijmpa14.937 ; epjc55.607 ; ijmpa30.1550094 ; prd94.034029 ; ijmpa31.1650161 ;
ahep2016.5071671 , where several different upper limits on branching ratios at
a 90% confidence level are obtained mainly from BESII and BESIII experiments
prd90.112014 ; prd96.111101 ; 2104.06628 ; prd89.071101 . When the charmed
hadrons are absent from the final states, the $J/{\psi}$ weak decays into
light hadrons could be induced by the $W$ exchange interactions. It is
generally thought that the amplitudes for the $W$-exchange decays are
suppressed, relative to those for the $W$-emission decays. Only a few studies
about the $W$-exchange $J/{\psi}$ weak decays exist prl96.192001 ; cpc33.85 ;
prd96.112001 . The study of the $J/{\psi}$ ${\to}$ $PP$ weak decays is helpful
in testing the non-conservation of the $C$ parity and strangeness quantum
number. Based on the $1.3{\times}10^{9}$ $J/{\psi}$ events collected with the
BESIII detector, the upper limit on branching ratio for the $C$ parity
violating $J/{\psi}$ ${\to}$ $K_{S}^{0}K_{S}^{0}$ decay, $<$
$1.4{\times}10^{-8}$, was recently obtained at a 95% confidence level
prd96.112001 . Inspired by the potentials of BESIII and future STCP
experiments, in this paper, we will study the $J/{\psi}$ decays into two
pseudoscalar mesons via the $W$ exchange interactions. Our study will provide
a ready reference for future experimental investigations to further test SM
and search for NP.

##  II The effective Hamiltonian

The effective Hamiltonian governing the $J/{\psi}$ ${\to}$ $PP$ weak decay is
written as rmp68.1125 ,

${\cal H}_{\rm eff}\ =\
\frac{G_{F}}{\sqrt{2}}\,\sum\limits_{q_{1},q_{2}}\,V_{cq_{1}}\,V_{cq_{2}}^{\ast}\,\big{\\{}C_{1}({\mu})\,O_{1}({\mu})+C_{2}({\mu})\,O_{2}({\mu})\big{\\}}+{\rm
h.c.},$ (1)

where $G_{F}$ ${\simeq}$ $1.166{\times}10^{-5}\,{\rm GeV}^{-2}$ pdg2020  is
the Fermi coupling constant; $V_{cq_{1,2}}$ is the Cabibbo-Kobayashi-Maskawa
(CKM) element, and $q_{1,2}$ ${\in}$ {$d$, $s$}. The latest values of CKM
elements from data are ${|}V_{cd}{|}$ $=$ $0.221{\pm}0.004$ and ${|}V_{cs}{|}$
$=$ $0.987{\pm}0.011$ pdg2020 . The factorization scale ${\mu}$ separates the
physical contributions into short- and long-distance parts. The Wilson
coefficients $C_{1,2}$ summarize the short-distance physical contributions
above the scales of ${\mu}$. They are computable with the perturbative field
theory at the scale of the $W^{\pm}$ boson mass $m_{W}$, and then evolved to a
characteristic scale of ${\mu}$ for the $c$ quark decay based on the
renormalization group equations.

$\vec{C}({\mu})\,=\,U_{4}({\mu},m_{b})\,M(m_{b})\,U_{5}(m_{b},m_{W})\,\vec{C}(m_{W}),$
(2)

where the explicit expression of the evolution matrix
$U_{f}({\mu}_{f},{\mu}_{i})$ and the threshold matching matrix $M(m_{b})$ can
be found in Ref. rmp68.1125 . The operators describing the local interactions
among four quarks are defined as,

$\displaystyle O_{1}$ $\displaystyle=$
$\displaystyle\big{[}\bar{c}_{\alpha}\,{\gamma}_{\mu}\,(1-{\gamma}_{5})\,q_{1,{\alpha}}\big{]}\,\big{[}\bar{q}_{2,{\beta}}\,{\gamma}^{\mu}\,(1-{\gamma}_{5})\,c_{\beta}\big{]},$
(3) $\displaystyle O_{2}$ $\displaystyle=$
$\displaystyle\big{[}\bar{c}_{\alpha}\,{\gamma}_{\mu}\,(1-{\gamma}_{5})\,q_{1,{\beta}}\big{]}\,\big{[}\bar{q}_{2,{\beta}}\,{\gamma}^{\mu}\,(1-{\gamma}_{5})\,c_{\alpha}\big{]},$
(4)

where ${\alpha}$ and ${\beta}$ are color indices.

It should be noted that the contributions of penguin operators being
proportional to the CKM factors $V_{cd}\,V_{cd}^{\ast}$ $+$
$V_{cs}\,V_{cs}^{\ast}$ $=$ $-V_{cb}\,V_{cb}^{\ast}$ ${\sim}$ ${\cal
O}({\lambda}^{4})$, are not considered here, because they are more suppressed
than the tree contributions, where the CKM factors $V_{cs}\,V_{cs}^{\ast}$
${\sim}$ ${\cal O}(1)$, $V_{cs}\,V_{cd}^{\ast}$ ${\sim}$ ${\cal
O}({\lambda})$, $V_{cd}\,V_{cd}^{\ast}$ ${\sim}$ ${\cal O}({\lambda}^{2})$,
and the Wolfenstein parameter ${\lambda}$ ${\approx}$ $0.2$.

The decay amplitudes can be written as,

${\cal
A}(J/{\psi}{\to}PP)\,=\,\frac{G_{F}}{\sqrt{2}}\,\sum\limits_{q_{1},q_{2}}\,V_{cq_{1}}\,V_{cq_{2}}^{\ast}\,\sum\limits_{i=1}^{2}\,C_{i}({\mu})\,{\langle}PP{|}O_{i}({\mu})\,{|}J/{\psi}{\rangle},$
(5)

where the hadron transition matrix elements (HMEs)
${\langle}PP{|}O_{i}({\mu})\,{|}J/{\psi}{\rangle}$ $=$
${\langle}O_{i}{\rangle}$ relate the quark operators with the concerned
hadrons. Owing to the inadequate understanding of the hadronization mechanism,
the remaining and most critical theoretical work is to properly compute HMEs.
In addition, it is not difficult to imagine that the main uncertainties will
come from HMEs containing nonperturbative contributions.

##  III hadron transition matrix elements

In the past few years, several phenomenological models, such as the QCD
factorization (QCDF) prl83.1914 ; npb591.313 ; npb606.245 ; plb488.46 ;
plb509.263 ; prd64.014036  and perturbative QCD (pQCD) approaches prl74.4388 ;
plb348.597 ; prd52.3958 ; prd63.074006 ; prd63.054008 ; prd63.074009 ;
plb555.197 , have been fully developed and widely employed in evaluating HMEs.
According to these phenomenological models, HMEs are usually expressed as the
convolution of scattering amplitudes and the hadronic wave functions (WFs).
The scattering amplitudes and WFs reflect the contributions at the quark and
hadron levels, respectively. The scattering amplitudes describing the
interactions between hard gluons and quarks are perturbatively calculable. WFs
representing the momentum distribution of compositions in hadron are regarded
as process independent and universal, and could be obtained by nonperturbative
methods or from data. A potential disadvantage of the QCDF approach prl83.1914
; npb591.313 ; npb606.245 ; plb488.46 ; plb509.263 ; prd64.014036  in the
practical calculation is that the annihilation contributions cannot be
computed self-consistently, and other phenomenological parameters are
introduced to deal with the soft endpoint divergences using the collinear
approximation. With the pQCD approach prl74.4388 ; plb348.597 ; prd52.3958 ;
prd63.074006 ; prd63.054008 ; prd63.074009 ; plb555.197 , in order to
regularize the endpoint contributions of the QCD radiative corrections to
HMEs, the transverse momentum are suggested to be retained within the
scattering amplitudes on one hand, and on the other hand a Sudakov factor is
introduced expressly for WFs of all involved hadrons. Finally, the pQCD decay
amplitudes are expressed as the convolution integral of three parts : the
ultra-hard contributions embodied by Wilson coefficients $C_{i}$, hard
scattering amplitudes ${\cal H}$ and soft part contained in hadronic WFs
${\Phi}$.

${\cal A}_{i}\,=\,\prod\limits_{j}{\int}dx_{j}\,db_{j}\,C_{i}(t_{i})\,{\cal
H}_{i}(t_{i},x_{j},b_{j})\,{\Phi}_{j}(x_{j},b_{j})\,e^{-S_{j}},$ (6)

where $x_{j}$ is the longitudinal momentum fraction of the valence quark,
$b_{j}$ is the conjugate variable of the transverse momentum, and $e^{-S_{j}}$
is the Sudakov factor. In this paper, we will adopt the pQCD approach to
investigate the $J/{\psi}$ ${\to}$ $PP$ weak decays within SM.

##  IV kinematic variables

For the $J/{\psi}$ ${\to}$ $PP$ weak decays, the valence quarks of the final
states are entirely different from those of the initial state. Only
annihilation configurations exist. Therefore, the $J/{\psi}$ ${\to}$ $PP$ weak
decays provide us with some typical processes to closely scrutinize the pure
annihilation contributions. As an example, the Feynman diagram for the
$J/{\psi}$ ${\to}$ ${\pi}^{-}K^{+}$ decay are shown in Fig. 1.

(a) (b) (c) (d)

Figure 1: Feynman diagrams for the $J/{\psi}$ ${\to}$ ${\pi}^{-}K^{+}$ decay
with the pQCD approach, where (a,b) are factorizable diagrams, and (c,d) are
nonfactorizable diagrams. The dots denote appropriate interactions, and the
dashed circles denote scattering amplitudes.

It is convenient to use the light-cone vectors to define the kinematic
variables. In the rest frame of the $J/{\psi}$ particle, one has

$p_{\psi}\,=\,p_{1}\,=\,\frac{m_{\psi}}{\sqrt{2}}(1,1,0),$ (7)
$p_{K}\,=\,p_{2}\,=\,\frac{m_{\psi}}{\sqrt{2}}(1,0,0),$ (8)
$p_{\pi}\,=\,p_{3}\,=\,\frac{m_{\psi}}{\sqrt{2}}(0,1,0),$ (9)
$k_{1}\,=\,x_{1}\,p_{1}+(0,0,\vec{k}_{1{\perp}}),$ (10)
$k_{2}\,=\,x_{2}\,p_{2}^{+}+(0,0,\vec{k}_{2{\perp}}),$ (11)
$k_{3}\,=\,x_{3}\,p_{3}^{-}+(0,0,\vec{k}_{3{\perp}}),$ (12)
${\epsilon}_{\psi}^{\parallel}\,=\,\frac{1}{\sqrt{2}}(1,-1,0),$ (13)

where $k_{i}$, $x_{i}$ and $\vec{k}_{i{\perp}}$ are respectively the momentum,
longitudinal momentum fraction and transverse momentum; the quark momentum
$k_{i}$ is illustrated in Fig. 1 (a); ${\epsilon}_{\psi}^{\parallel}$ is the
longitudinal polarization vector of the $J/{\psi}$ particle, and satisfies
both the normalization condition
${\epsilon}_{\psi}^{\parallel}{\cdot}{\epsilon}_{\psi}^{\parallel}$ $=$ $-1$
and the orthogonal relation ${\epsilon}_{\psi}^{\parallel}{\cdot}p_{\psi}$ $=$
$0$; all hadrons are on mass shell, i.e., $p_{1}$ $=$ $m_{\psi}^{2}$,
$p_{2}^{2}$ $=$ $0$ and $p_{3}^{2}$ $=$ $0$.

##  V hadronic wave functions

With the convention of Refs. ijmpa31.1650161 ; prd65.014007 ; jhep0605.004 ;
2012.10581 , the WFs and distribution amplitudes (DAs) are defined as follows.

${\langle}\,0\,{|}\,\bar{c}_{\alpha}(0)\,c_{\beta}(z)\,{|}{\psi}(p_{1},{\epsilon}_{\parallel})\,{\rangle}\,=\,\frac{1}{4}\,f_{\psi}\,{\int}dk_{1}\,e^{+i\,k_{1}{\cdot}z}\,\big{\\{}\\!\not{\epsilon}_{\parallel}\big{[}m_{\psi}\,{\phi}_{\psi}^{v}-\\!\not{p}_{1}\,{\phi}_{\psi}^{t}\big{]}\big{\\}}_{{\beta}{\alpha}},$
(14)
$\displaystyle{\langle}\,K(p_{2})\,{|}\,\bar{u}_{\alpha}(0)\,s_{\beta}(z)\,{|}\,0\,{\rangle}$
(15) $\displaystyle=$
$\displaystyle-\frac{i\,f_{K}}{4}\,{\int}dk_{2}\,e^{-i\,k_{2}{\cdot}z}\,\big{\\{}{\gamma}_{5}\,\big{[}\\!\not{p}_{2}\,{\phi}_{K}^{a}+{\mu}_{K}\,{\phi}_{K}^{p}-{\mu}_{K}\,\big{(}\\!\not{n}_{+}\\!\not{n}_{-}-1\big{)}\,{\phi}_{K}^{t}\big{]}\big{\\}}_{{\beta}{\alpha}},$
$\displaystyle{\langle}\,{\pi}(p_{3})\,{|}\,\bar{d}_{\alpha}(0)\,u_{\beta}(z)\,{|}\,0\,{\rangle}$
(16) $\displaystyle=$
$\displaystyle-\frac{i\,f_{\pi}}{4}\,{\int}dk_{3}\,e^{-i\,k_{3}{\cdot}z}\,\big{\\{}{\gamma}_{5}\,\big{[}\\!\not{p}_{3}\,{\phi}_{\pi}^{a}+{\mu}_{\pi}\,{\phi}_{\pi}^{p}-{\mu}_{\pi}\,\big{(}\\!\not{n}_{-}\\!\not{n}_{+}-1\big{)}\,{\phi}_{\pi}^{t}\big{]}\big{\\}}_{{\beta}{\alpha}},$

where $f_{\psi}$, $f_{K}$ and $f_{\pi}$ are decay constants; ${\mu}_{K,{\pi}}$
$=$ $1.6{\pm}0.2$ GeV jhep0605.004  is the chiral mass; $n_{+}$ $=$ $(1,0,0)$
and $n_{-}$ $=$ $(0,1,0)$ are the null vectors; ${\phi}_{P}^{a}$ and
${\phi}_{P}^{p,t}$ are twist-2 and twist-3, respectively. The explicit
expressions of ${\phi}_{\psi}^{v,t}$ and ${\phi}_{P}^{a,p,t}$ can be found in
Ref. ijmpa31.1650161  and Refs. jhep0605.004 ; 2012.10581 , respectively. We
collect these DAs as follows.

${\phi}_{\psi}^{v}(x)=A\,x\,\bar{x}\,{\exp}\big{(}-\frac{m_{c}^{2}}{8\,{\omega}^{2}\,x\,\bar{x}}\big{)},$
(17)
${\phi}_{\psi}^{t}(x)=B\,(\bar{x}-x)^{2}\,{\exp}\big{(}-\frac{m_{c}^{2}}{8\,{\omega}^{2}\,x\,\bar{x}}\big{)},$
(18)
${\phi}_{P}^{a}(x)\,=\,6\,x\,\bar{x}\,\big{\\{}1+a_{1}^{P}\,C_{1}^{3/2}({\xi})+a_{2}^{P}\,C_{2}^{3/2}({\xi})\big{\\}},$
(19) $\displaystyle{\phi}_{P}^{p}(x)$ $\displaystyle=$ $\displaystyle
1+3\,{\rho}_{+}^{P}-9\,{\rho}_{-}^{P}\,a_{1}^{P}+18\,{\rho}_{+}^{P}\,a_{2}^{P}$
(20) $\displaystyle+$
$\displaystyle\frac{3}{2}\,({\rho}_{+}^{P}+{\rho}_{-}^{P})\,(1-3\,a_{1}^{P}+6\,a_{2}^{P})\,{\ln}(x)$
$\displaystyle+$
$\displaystyle\frac{3}{2}\,({\rho}_{+}^{P}-{\rho}_{-}^{P})\,(1+3\,a_{1}^{P}+6\,a_{2}^{P})\,{\ln}(\bar{x})$
$\displaystyle-$
$\displaystyle(\frac{3}{2}\,{\rho}_{-}^{P}-\frac{27}{2}\,{\rho}_{+}^{P}\,a_{1}^{P}+27\,{\rho}_{-}^{P}\,a_{2}^{P})\,C_{1}^{1/2}(\xi)$
$\displaystyle+$
$\displaystyle(30\,{\eta}_{P}-3\,{\rho}_{-}^{P}\,a_{1}^{P}+15\,{\rho}_{+}^{P}\,a_{2}^{P})\,C_{2}^{1/2}(\xi),$
$\displaystyle{\phi}_{P}^{t}(x)$ $\displaystyle=$
$\displaystyle\frac{3}{2}\,({\rho}_{-}^{P}-3\,{\rho}_{+}^{P}\,a_{1}^{P}+6\,{\rho}_{-}^{P}\,a_{2}^{P})$
(21) $\displaystyle-$ $\displaystyle
C_{1}^{1/2}(\xi)\big{\\{}1+3\,{\rho}_{+}^{P}-12\,{\rho}_{-}^{P}\,a_{1}^{P}+24\,{\rho}_{+}^{P}\,a_{2}^{P}$
$\displaystyle\quad+\frac{3}{2}\,({\rho}_{+}^{P}+{\rho}_{-}^{P})\,(1-3\,a_{1}^{P}+6\,a_{2}^{P})\,{\ln}(x)$
$\displaystyle\quad+\frac{3}{2}\,({\rho}_{+}^{P}-{\rho}_{-}^{P})\,(1+3\,a_{1}^{P}+6\,a_{2}^{P})\,{\ln}(\bar{x})\big{\\}}$
$\displaystyle-$ $\displaystyle
3\,(3\,{\rho}_{+}^{P}\,a_{1}^{P}-\frac{15}{2}\,{\rho}_{-}^{P}\,a_{2}^{P})\,C_{2}^{1/2}(\xi),$

where $\bar{x}$ $=$ $1$ $-$ $x$ and ${\xi}$ $=$ $x$ $-$ $\bar{x}$. ${\omega}$
$=$ $m_{c}\,{\alpha}_{s}(m_{c})$ is the shape parameter. The parameters $A$ in
Eq.(17) and $B$ in Eq.(18) are determined by the normalization conditions,

${\int}{\phi}_{\psi}^{v,t}(x)\,dx\,=\,1.$ (22)

For the meaning and definition of other parameters, refer to Refs. 2012.10581
; jhep0605.004 .

##  VI decay amplitudes

When the final states include the isoscalar ${\eta}$ or/and ${\eta}^{\prime}$,
we assume that the components of glueball, charmonium or bottomonium are
negligible. The physical ${\eta}$ and ${\eta}^{\prime}$ states are the
mixtures of the $SU(3)$ octet and singlet states. In our calculation, we will
adopt the quark-flavor basis description proposed in Ref. prd58.114006 , i.e.,

$\left(\begin{array}[]{c}{\eta}\\\
{\eta}^{\prime}\end{array}\right)\,=\,\left(\begin{array}[]{cc}{\cos}{\phi}&-{\sin}{\phi}\\\
{\sin}{\phi}&{\cos}{\phi}\end{array}\right)\,\left(\begin{array}[]{c}{\eta}_{q}\\\
{\eta}_{s}\end{array}\right),$ (23)

where the mixing angle is ${\phi}$ $=$ $(39.3{\pm}1.0)^{\circ}$ prd58.114006 ,
${\eta}_{q}$ $=$ $(u\bar{u}+d\bar{d})/{\sqrt{2}}$ and ${\eta}_{s}$ $=$
$s\bar{s}$. In addition, we assume that DAs of ${\eta}_{q}$ and ${\eta}_{s}$
are the same as those of pion, but with different decay constants and mass
prd58.114006 ; prd76.074018 ; prd89.114019 ,

$f_{q}\,=\,(1.07{\pm}0.02)\,f_{\pi},$ (24)
$f_{s}\,=\,(1.34{\pm}0.06)\,f_{\pi},$ (25)
$m_{{\eta}_{q}}^{2}\,=\,m_{\eta}^{2}\,{\cos}^{2}{\phi}+m_{{\eta}^{\prime}}^{2}\,{\sin}^{2}{\phi}-\frac{\sqrt{2}\,f_{s}}{f_{q}}(m_{{\eta}^{\prime}}^{2}-m_{\eta}^{2})\,{\cos}{\phi}\,{\sin}{\phi},$
(26)
$m_{{\eta}_{s}}^{2}\,=\,m_{\eta}^{2}\,{\sin}^{2}{\phi}+m_{{\eta}^{\prime}}^{2}\,{\cos}^{2}{\phi}-\frac{f_{q}}{\sqrt{2}\,f_{s}}(m_{{\eta}^{\prime}}^{2}-m_{\eta}^{2})\,{\cos}{\phi}\,{\sin}{\phi}.$
(27)

For the $C$ parity violating $J/{\psi}$ decays, the amplitudes are

$\displaystyle{\cal A}(J/{\psi}{\to}{\pi}^{0}{\eta}_{q})$ $\displaystyle=$
$\displaystyle-\frac{G_{F}}{2\,\sqrt{2}}\,V_{cd}\,V_{cd}^{\ast}\,\big{\\{}a_{2}\,\big{[}{\cal
A}_{ab}({\pi},{\eta}_{q})+{\cal A}_{ab}({\eta}_{q},{\pi})\big{]}$ (28)
$\displaystyle\hskip 65.04034pt+C_{1}\,\big{[}{\cal
A}_{cd}({\pi},{\eta}_{q})+{\cal A}_{cd}({\eta}_{q},{\pi})\big{]}\big{\\}},$
${\cal A}(J/{\psi}{\to}{\pi}^{0}{\eta})\,=\,{\cal
A}(J/{\psi}{\to}{\pi}^{0}{\eta}_{q})\,{\cos}{\phi},$ (29) ${\cal
A}(J/{\psi}{\to}{\pi}^{0}{\eta}^{\prime})\,=\,{\cal
A}(J/{\psi}{\to}{\pi}^{0}{\eta}_{q})\,{\sin}{\phi},$ (30) ${\cal
A}(J/{\psi}{\to}{\eta}_{s}{\eta}_{s})\,=\,\sqrt{2}\,G_{F}\,V_{cs}\,V_{cs}^{\ast}\,\big{\\{}a_{2}\,{\cal
A}_{ab}({\eta}_{s},{\eta}_{s})+C_{1}\,{\cal
A}_{cd}({\eta}_{s},{\eta}_{s})\big{\\}},$ (31) ${\cal
A}(J/{\psi}{\to}{\eta}_{q}{\eta}_{q})\,=\,\frac{G_{F}}{\sqrt{2}}\,V_{cd}\,V_{cd}^{\ast}\,\big{\\{}a_{2}\,{\cal
A}_{ab}({\eta}_{q},{\eta}_{q})+C_{1}\,{\cal
A}_{cd}({\eta}_{q},{\eta}_{q})\big{\\}},$ (32) ${\cal
A}(J/{\psi}{\to}{\eta}{\eta}^{\prime})\,=\,\big{\\{}{\cal
A}(J/{\psi}{\to}{\eta}_{q}{\eta}_{q})-{\cal
A}(J/{\psi}{\to}{\eta}_{s}{\eta}_{s})\big{\\}}\,{\sin}{\phi}\,{\cos}{\phi}.$
(33)

For the strangeness changing $J/{\psi}$ decays, the amplitudes are

${\cal
A}(J/{\psi}{\to}{\pi}^{-}K^{+})\,=\,\frac{G_{F}}{\sqrt{2}}\,V_{cs}\,V_{cd}^{\ast}\,\big{\\{}a_{2}\,{\cal
A}_{ab}({\pi},K)+C_{1}\,{\cal A}_{cd}({\pi},K)\big{\\}},$ (34) ${\cal
A}(J/{\psi}{\to}{\pi}^{0}K^{0})\,=\,-\frac{G_{F}}{2}\,V_{cs}\,V_{cd}^{\ast}\,\big{\\{}a_{2}\,{\cal
A}_{ab}({\pi},K)+C_{1}\,{\cal A}_{cd}({\pi},K)\big{\\}},$ (35) ${\cal
A}(J/{\psi}{\to}K^{0}{\eta}_{s})\,=\,\frac{G_{F}}{\sqrt{2}}\,V_{cs}\,V_{cd}^{\ast}\,\big{\\{}a_{2}\,{\cal
A}_{ab}(K,{\eta}_{s})+C_{1}\,{\cal A}_{cd}(K,{\eta}_{s})\big{\\}},$ (36)
${\cal
A}(J/{\psi}{\to}K^{0}{\eta}_{q})\,=\,\frac{G_{F}}{2}\,V_{cs}\,V_{cd}^{\ast}\,\big{\\{}a_{2}\,{\cal
A}_{ab}({\eta}_{q},K)+C_{1}\,{\cal A}_{cd}({\eta}_{q},K)\big{\\}},$ (37)
${\cal A}(J/{\psi}{\to}K^{0}{\eta})\,=\,{\cal
A}(J/{\psi}{\to}K^{0}{\eta}_{q})\,{\cos}{\phi}-{\cal
A}(J/{\psi}{\to}K^{0}{\eta}_{s})\,{\sin}{\phi},$ (38) ${\cal
A}(J/{\psi}{\to}K^{0}{\eta}^{\prime})\,=\,{\cal
A}(J/{\psi}{\to}K^{0}{\eta}_{q})\,{\sin}{\phi}+{\cal
A}(J/{\psi}{\to}K^{0}{\eta}_{s})\,{\cos}{\phi},$ (39)

where coefficient $a_{2}$ $=$ $C_{1}$ $+$ $C_{2}/N_{c}$ and the color number
$N_{c}$ $=$ $3$; The amplitude building blocks ${\cal A}_{ij}$ are listed in
Appendix A. From the above amplitudes, it is foreseeable that if the
$J/{\psi}$ ${\to}$ ${\pi}{\eta}^{({\prime})}$ decays were experimentally
observed, the CKM element $|V_{cd}|$ could be constrained or extracted.

##  VII numerical results and discussion

Table 1: Values of the input parameters, with their central values regarded as
the default inputs unless otherwise specified. mass, width and decay constants
of the particles pdg2020
---
$m_{{\pi}^{0}}$ $=$ $134.98$ MeV, |  $m_{K^{0}}$ $=$ $497.61$ MeV, |  $f_{{\pi}}$ $=$ $130.2{\pm}1.2$ MeV,
$m_{{\pi}^{\pm}}$ $=$ $139.57$ MeV, |  $m_{K^{\pm}}$ $=$ $493.68$ MeV, |  $f_{K}$ $=$ $155.7{\pm}0.3$ MeV,
$m_{\eta}$ $=$ $547.86$ MeV, |  $m_{{\eta}^{\prime}}$ $=$ $957.78$ MeV, |  $f_{\psi}$ $=$ $395.1{\pm}5.0$ MeV ijmpa31.1650161 ,
$m_{c}$ $=$ $1.67{\pm}0.07$ GeV, |  $m_{J/{\psi}}$ $=$ $3096.9$ MeV, |  ${\Gamma}_{\psi}$ $=$ $92.9{\pm}2.8$ keV,
Gegenbauer moments at the scale of ${\mu}$ $=$ 1 GeV jhep0605.004
$a_{1}^{\pi}$ $=$ $0$, $a_{2}^{\pi}$ $=$ $0.25{\pm}0.15$, $a_{1}^{K}$ $=$
$0.06{\pm}0.03$, $a_{2}^{K}$ $=$ $0.25{\pm}0.15$
Table 2: Branching ratios for the $J/{\psi}$ ${\to}$ $PP$ weak decays, where
the uncertainties originate from mesonic DAs, including the parameters of
$m_{c}$, ${\mu}_{P}$ and $a_{2}^{P}$. $C$ parity violating decay modes
---
mode | ${\cal B}r$ | mode | ${\cal B}r$
$J/{\psi}$ ${\to}$ ${\pi}^{0}{\eta}$ | $(2.32^{+0.81}_{-0.24}){\times}10^{-14}$ |  $J/{\psi}$ ${\to}$ ${\eta}{\eta}^{\prime}$ | $(3.01^{+0.78}_{-0.55}){\times}10^{-11}$
$J/{\psi}$ ${\to}$ ${\pi}^{0}{\eta}^{\prime}$ | $(1.45^{+0.50}_{-0.15}){\times}10^{-14}$ |  |
the strangeness changing decay modes
mode | ${\cal B}r$ | mode | ${\cal B}r$
$J/{\psi}$ ${\to}$ ${\pi}^{-}K^{+}$ | $(0.99^{+0.33}_{-0.15}){\times}10^{-12}$ |  $J/{\psi}$ ${\to}$ $K^{0}{\eta}$ | $(0.84^{+0.25}_{-0.22}){\times}10^{-13}$
$J/{\psi}$ ${\to}$ ${\pi}^{0}K^{0}$ | $(0.49^{+0.16}_{-0.08}){\times}10^{-12}$ |  $J/{\psi}$ ${\to}$ $K^{0}{\eta}^{\prime}$ | $(1.94^{+0.30}_{-0.25}){\times}10^{-12}$

The branching ratio is defined as follows.

${\cal B}r\,=\,\frac{p_{\rm
cm}}{24\,{\pi}\,m_{\psi}^{2}\,{\Gamma}_{\psi}}\,{|}{\cal
A}(J/{\psi}{\to}PP){|}^{2},$ (40)

where $p_{\rm cm}$ is the center-of-mass momentum of final states in the rest
frame of the $J/{\psi}$ particle. Using the inputs in Table 1, the numerical
results of branching ratios are obtained and presented in Table 2. Our
comments on the results are presented as follows.

(1) The $J/{\psi}$ ${\to}$ ${\eta}{\eta}^{\prime}$ decays are Cabibbo-favored.
The $J/{\psi}$ ${\to}$ $K{\pi}$ and $K{\eta}^{({\prime})}$ decays are singly
Cabibbo-suppressed. The $J/{\psi}$ ${\to}$ ${\pi}{\eta}^{({\prime})}$ decays
are doubly Cabibbo-suppressed. Therefore, there is a hierarchical structure,
i.e., ${\cal B}r(J/{\psi}{\to}{\eta}{\eta}^{\prime})$ ${\sim}$ ${\cal
O}(10^{-11})$, ${\cal B}r(J/{\psi}{\to}K{\pi},K{\eta}^{({\prime})})$ ${\sim}$
${\cal O}(10^{-12}-10^{-13})$ and ${\cal
B}r(J/{\psi}{\to}{\pi}{\eta}^{({\prime})})$ ${\sim}$ ${\cal O}(10^{-14})$.

(2) Compared with the external $W$-emission induced $J/{\psi}$ ${\to}$
$D_{(s)}M$ decays, the internal $W$-exchange induced $J/{\psi}$ ${\to}$ $PP$
decays are color-suppressed because the two light valence quarks of the
effective operators belong to different final states. In addition, according
to the the power counting rule of the QCDF approach in the heavy quark limit,
the annihilation amplitudes are assumed to be power suppressed, relative to
the emission amplitudes npb591.313 . Therefore, the branching ratios for
$J/{\psi}$ ${\to}$ $PP$ decays are less than those for $J/{\psi}$ ${\to}$
$D_{(s)}M$ decays by one or two orders of magnitude epjc55.607 ;
ijmpa30.1550094 ; prd94.034029 ; ijmpa31.1650161 ; ahep2016.5071671 .

(3) The nonperturbative mesonic DAs are the essential parameters of the
amplitudes with the pQCD approach. One of the main theoretical uncertainties
arising from participating DAs is given in Table 2. In addition, there are
several other influence factors. For example, the decay constant $f_{\psi}$
and width ${\Gamma}_{\psi}$ will bring 2.5% and 3% uncertainties to branching
ratios.

(4) Branching ratios for the $J/{\psi}$ ${\to}$ ${\eta}{\eta}^{\prime}$ decays
can reach up to the order of $10^{-11}$, which are far beyond the measurement
precision and capability of current BESIII experiment; however, they might be
accessible at the future high-luminosity STCF experiment. It will be very
difficult and challenging but interesting to search for the $J/{\psi}$ ${\to}$
$PP$ weak decays experimentally. It could be speculated that branching ratios
for the $J/{\psi}$ ${\to}$ $PP$ weak decays might be enhanced by including
some novel interactions of NP models. For example, it has been shown in Refs.
prd60.014011 ; cpc25.461  that branching ratios for the $W$-emission
$J/{\psi}$ ${\to}$ $D$ $+$ $X$ weak decays could be as large as $10^{-6}$
${\sim}$ $10^{-5}$ with the contributions from NP. An observation of the
phenomenon of an abnormally large occurrence probability would be a hint of
NP.

##  VIII summary

Within the SM, the $C$ parity violating $J/{\psi}$ ${\to}$
${\pi}^{0}{\eta}^{(\prime)}$ and ${\eta}{\eta}^{\prime}$ decays and the
strangeness changing $J/{\psi}$ ${\to}$ $K{\pi}$ and $K{\eta}^{(\prime)}$
decays are solely valid and possible via the weak interactions; however, they
are very rare. In this paper, based on the latest progress and future
prospects of the $J/{\psi}$ physics at high-luminosity collider, we studied
the $J/{\psi}$ ${\to}$ $PP$ weak decays using the pQCD approach for the first
time. It is found that the branching ratios for the $J/{\psi}$ ${\to}$
${\eta}{\eta}^{\prime}$ decays can reach up to the order of $10^{-11}$, which
might be measurable by the future STCF experiment.

## Acknowledgments

The work is supported by the National Natural Science Foundation of China
(Grant Nos. 11705047, 11981240403, U1632109, 11547014, and 11875122) and the
Program for Innovative Research Team in University of Henan Province
(19IRTSTHN018), the Excellent Youth Foundation of Henan Province
(212300410010) and the Chinese Academy of Sciences Large-Scale Scientific
Facility Program (1G2017IHEPKFYJ01).

##  Appendix A Amplitude building blocks

From the definition of Eq.(15) and Eq.(16), it can be clearly observed that
the twist-3 DAs are always accompanied by a chiral mass ${\mu}_{P}$. With the
pQCD approach, a Sudakov factor is introduced for each of the hadronic WFs.

We take the $J/{\psi}$ ${\to}$ $K^{+}{\pi}^{-}$ decay as an example. For
simplicity, we use the following shorthand forms.

${\phi}_{\psi}^{v,t}\,=\,{\phi}_{\psi}^{v,t}(x_{1})\,e^{-S_{\psi}},$ (41)
${\phi}_{K}^{a}\,=\,{\phi}_{K}^{a}(x_{2})\,e^{-S_{K}},$ (42)
${\phi}_{K}^{p,t}\,=\,\frac{{\mu}_{K}}{m_{\psi}}\,{\phi}_{K}^{p,t}(x_{2})\,e^{-S_{K}},$
(43) ${\phi}_{\pi}^{a}\,=\,{\phi}_{\pi}^{a}(x_{3})\,e^{-S_{\pi}},$ (44)
${\phi}_{\pi}^{p,t}\,=\,\frac{{\mu}_{\pi}}{m_{\psi}}\,{\phi}_{\pi}^{p,t}(x_{3})\,e^{-S_{\pi}},$
(45)

where the definitions of Sudakov factors are

$S_{\psi}\,=\,s(x_{1},p_{1}^{+},b_{1})+2\,{\int}_{1/b_{1}}^{t}\frac{d{\mu}}{{\mu}}{\gamma}_{q},$
(46)
$S_{K}\,=\,s(x_{2},p_{2}^{+},b_{2})+s(\bar{x}_{2},p_{2}^{+},1/b_{2})+2\,{\int}_{1/b_{2}}^{t}\frac{d{\mu}}{{\mu}}{\gamma}_{q},$
(47)
$S_{\pi}\,=\,s(x_{3},p_{3}^{-},b_{3})+s(\bar{x}_{3},p_{3}^{-},1/b_{3})+2\,{\int}_{1/b_{3}}^{t}\frac{d{\mu}}{{\mu}}{\gamma}_{q}.$
(48)

The expression of $s(x,Q,b)$ can be found in Ref. prd52.3958 . ${\gamma}_{q}$
$=$ $-{\alpha}_{s}/{\pi}$ is the quark anomalous dimension. In addition, the
decay amplitudes are always the functions of Wilson coefficient $C_{i}$. It
should be understood that the shorthand

$C_{i}\,{\cal
A}_{jk}({\pi},K)\,=\,\frac{{\pi}\,C_{F}}{N_{c}}\,m_{\psi}^{4}\,f_{\psi}\,f_{K}\,f_{\pi}\,\big{\\{}{\cal
A}_{j}(C_{i})+{\cal A}_{k}(C_{i})\big{\\}},$ (49)

where the color factor $C_{F}$ $=$ $4/3$ and the color number $N_{c}$ $=$ $3$.
The subscripts $j$ and $k$ of building block ${\cal A}_{j(k)}$ correspond to
the indices of Fig. 1. The expressions of ${\cal A}_{i}$ are written as
follows.

$\displaystyle{\cal A}_{a}$ $\displaystyle=$
$\displaystyle{\int}_{0}^{1}dx_{2}\,dx_{3}{\int}_{0}^{\infty}db_{2}\,db_{3}\,H_{ab}({\alpha}_{g},{\beta}_{a},b_{2},b_{3})\,{\alpha}_{s}(t_{a})\,C_{i}(t_{a})$
(50) $\displaystyle
S_{t}(\bar{x}_{2})\,\big{\\{}{\phi}_{K}^{a}\,{\phi}_{\pi}^{a}\,\bar{x}_{2}-2\,{\phi}_{\pi}^{p}\,\big{[}{\phi}_{K}^{p}\,x_{2}+{\phi}_{K}^{t}\,(1+\bar{x}_{2})\big{]}\big{\\}},$
$\displaystyle{\cal A}_{b}$ $\displaystyle=$
$\displaystyle{\int}_{0}^{1}dx_{2}\,dx_{3}{\int}_{0}^{\infty}db_{2}\,db_{3}\,H_{ab}({\alpha}_{g},{\beta}_{b},b_{3},b_{2})\,{\alpha}_{s}(t_{b})\,C_{i}(t_{b})$
(51) $\displaystyle
S_{t}(x_{3})\,\big{\\{}{\phi}_{K}^{a}\,{\phi}_{\pi}^{a}\,x_{3}-2\,{\phi}_{K}^{p}\,\big{[}{\phi}_{\pi}^{p}\,\bar{x}_{3}-{\phi}_{\pi}^{t}\,(1+x_{3})\big{]}\big{\\}},$
$\displaystyle{\cal A}_{c}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}\,{\int}_{0}^{1}dx_{1}\,dx_{2}\,dx_{3}{\int}_{0}^{\infty}db_{1}\,db_{2}\,H_{cd}({\alpha}_{g},{\beta}_{c},b_{1},b_{2})\,{\alpha}_{s}(t_{c})\,C_{i}(t_{c})$
(52)
$\displaystyle\big{\\{}{\phi}_{\psi}^{v}\,\big{[}{\phi}_{K}^{a}\,{\phi}_{\pi}^{a}\,(x_{1}-x_{3})+({\phi}_{K}^{p}\,{\phi}_{\pi}^{p}-{\phi}_{K}^{t}\,{\phi}_{\pi}^{t})\,(\bar{x}_{2}-x_{3})$
$\displaystyle\hskip
17.34189pt+({\phi}_{K}^{p}\,{\phi}_{\pi}^{t}-{\phi}_{K}^{t}\,{\phi}_{\pi}^{p})\,(2\,x_{1}-\bar{x}_{2}-x_{3})\big{]}$
$\displaystyle-{\phi}_{\psi}^{t}\big{[}\frac{1}{2}\,{\phi}_{K}^{a}\,{\phi}_{\pi}^{a}+2\,{\phi}_{K}^{p}\,{\phi}_{\pi}^{t}\big{]}\big{\\}}_{b_{2}=b_{3}},$
$\displaystyle{\cal A}_{d}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}\,{\int}_{0}^{1}dx_{1}\,dx_{2}\,dx_{3}{\int}_{0}^{\infty}db_{1}\,db_{2}\,H_{cd}({\alpha}_{g},{\beta}_{d},b_{1},b_{2})\,{\alpha}_{s}(t_{d})\,C_{i}(t_{d})$
(53)
$\displaystyle\big{\\{}{\phi}_{\psi}^{v}\,\big{[}{\phi}_{K}^{a}\,{\phi}_{\pi}^{a}\,(x_{2}-x_{1})+({\phi}_{K}^{p}\,{\phi}_{\pi}^{p}-{\phi}_{K}^{t}\,{\phi}_{\pi}^{t})\,(x_{3}-\bar{x}_{2})$
$\displaystyle\hskip
17.34189pt+({\phi}_{K}^{p}\,{\phi}_{\pi}^{t}-{\phi}_{K}^{t}\,{\phi}_{\pi}^{p})\,(2\,\bar{x}_{1}-\bar{x}_{2}-x_{3})\big{]}$
$\displaystyle-{\phi}_{\psi}^{t}\big{[}\frac{1}{2}\,{\phi}_{K}^{a}\,{\phi}_{\pi}^{a}-2\,{\phi}_{K}^{t}\,{\phi}_{\pi}^{p}\big{]}\big{\\}}_{b_{2}=b_{3}},$
$\displaystyle
H_{ab}({\alpha},{\beta},b_{i},b_{j})\,=\,-\frac{{\pi}^{2}}{4}\,b_{i}\,b_{j}\,\big{\\{}J_{0}(b_{j}\sqrt{{\alpha}})+i\,Y_{0}(b_{j}\sqrt{{\alpha}})\big{\\}}$
$\displaystyle\hskip
17.34189pt\big{\\{}{\theta}(b_{i}-b_{j})\big{[}J_{0}(b_{i}\sqrt{{\beta}})+i\,Y_{0}(b_{i}\sqrt{{\beta}})\big{]}J_{0}(b_{j}\sqrt{{\beta}})+(b_{i}{\leftrightarrow}b_{j})\big{\\}},$
(54) $\displaystyle H_{cd}({\alpha},{\beta},b_{1},b_{2})$ (55)
$\displaystyle=$ $\displaystyle
b_{1}\,b_{2}\,\big{\\{}\frac{i\,{\pi}}{2}\,{\theta}({\beta})\big{[}J_{0}(b_{1}\sqrt{{\beta}})+i\,Y_{0}(b_{1}\sqrt{{\beta}})\big{]}+{\theta}(-{\beta})K_{0}(b_{1}\sqrt{-{\beta}})\big{\\}}$
$\displaystyle\frac{i\,{\pi}}{2}\,\big{\\{}{\theta}(b_{1}-b_{2})\big{[}J_{0}(b_{1}\sqrt{{\alpha}})+i\,Y_{0}(b_{1}\sqrt{{\alpha}})\big{]}J_{0}(b_{2}\sqrt{{\alpha}})+(b_{1}{\leftrightarrow}b_{2})\big{\\}},$

where $I_{0}$, $J_{0}$, $K_{0}$ and $Y_{0}$ are Bessel functions. The
parametrization of the Sudakov factor $S_{t}(x)$ can be found in Ref.
plb555.197 . The virtualities of gluons and quarks are

${\alpha}_{g}\,=\,m_{\psi}^{2}\,\bar{x}_{2}\,x_{3},$ (56)
${\beta}_{a}\,=\,m_{\psi}^{2}\,\bar{x}_{2},$ (57)
${\beta}_{b}\,=\,m_{\psi}^{2}\,x_{3},$ (58)
${\beta}_{c}\,=\,{\alpha}_{g}-m_{\psi}^{2}\,x_{1}\,(\bar{x}_{2}+x_{3}),$ (59)
${\beta}_{d}\,=\,{\alpha}_{g}-m_{\psi}\,\bar{x}_{1}\,(\bar{x}_{2}+x_{3}),$
(60) $t_{a,b}\,=\,{\max}(\sqrt{{\beta}_{a,b}},1/b_{2},1/b_{3}),$ (61)
$t_{c,d}\,=\,{\max}(\sqrt{{\alpha}_{g}},\sqrt{{|}{\beta}_{c,d}{|}},1/b_{1},1/b_{2}).$
(62)

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# Inducing spectral gaps for the cohomological
Laplacians of $\operatorname{SL}_{n}(\mathbb{Z})$ and
$\operatorname{SAut}(F_{n})$

Piotr Mizerka<EMAIL_ADDRESS>Institute of Mathematics of the Polish
Academy of Sciences, Śniadeckich 8, 00-656, Warsaw, Poland

###### Abstract.

The technique of inducing spectral gaps for cohomological Laplacians in degree
zero was used by Kaluba, Kielak and Nowak [KKN21] to prove property (T) for
$\operatorname{Aut}(F_{n})$ and $\operatorname{SL}_{n}(\mathbb{Z})$. In this
paper, we adapt this technique to Laplacians in degree one. This allows to
provide a lower bound for the cohomological Laplacian in degree one for
$\operatorname{SL}_{n}(\mathbb{Z})$ for every unitary representation. In
particular, one gets in that way an alternative proof of property (T) for
$\operatorname{SL}_{n}(\mathbb{Z})$ whenever $n\geq 3$.

##  1\. Introduction

Cohomological Laplacians were introduced by Bader and Nowak [BN20] in the
context of vanishing of group cohomology with unitary coefficients. Such
vanishing of the first cohomology is known to be equivalent to Kazhdan’s
property (T) and an algebraic characterization for it was found by Ozawa
[Oza16] for finitely generated groups. It turns out that property (T) can be
described in terms of a specific degree zero _Laplacian_ $\Delta$: Ozawa
showed that a group $G$ has property (T) if and only if
$\Delta^{2}-\lambda\Delta$ is a sum of squares in the real group ring
$\mathbb{R}G$ for some positive spectral gap $\lambda$. Using this
characterization, Kaluba, Kielak, Nowak, and Ozawa [KNO19, KKN21] were able to
show property (T) for $\operatorname{SL}_{n}(\mathbb{Z})$ and
$\operatorname{Aut}(F_{n})$ for $n\geq 3$ and $n\geq 5$ respectively.
Estimating the spectral gap $\lambda$ from below has another advantage of
prividing a lower bound for $\kappa(G,S)$, the _Kazhdan constant_ associated
to a generating set $S$ of a group $G$. It is known that $G$ has property (T)
if and only if $\kappa(G,S)>0$ and this inequality does not depend on the
generating set. For a symmetric generating set $S=S^{-1}$, the lower bound for
Kazhdan constants is given by the formula (see [BdlHV08, Remark 5.4.7] and
[KN18])

$\sqrt{\frac{2\lambda}{|S|}}\leq\kappa(G,S),$

where $\lambda$ is a positive constant such that $\Delta^{2}-\lambda\Delta$ is
a sum of squares.

We focus on providing lower bounds for the spectral gap of $\Delta_{1}$, the
one-degree Laplacian. The existence of such a positive spectral gap for a
group $G$ is equivalent to vanishing of the first cohomology of $G$ with
unitary coefficients and _reducibility_ of the second (reducibility means that
the images of the differentials in the chain complex of Hilbert spaces
computing the cohomology for a given unitary representation are closed), see
[BN20, The Main Theorem]. Once a positive spectral gap is found for
$\Delta_{1}$, it turns out to be a lower bound for the Ozawa’s spectral gap,
see Remark 2.2. We acknowledge that it has been recently announced by U. Bader
and R. Sauer that the reducibility condition is a consequence of vanishing of
cohomology one degree lower, see [BS23].

Recently, M. Kaluba, P. W. Nowak and the author showed that for $\Delta_{1}$,
the first cohomological Laplacian of $\operatorname{SL}_{3}(\mathbb{Z})$, and
$\lambda=0.32$, the matrix $\Delta_{1}-\lambda I$ is a sum of squares ([KMN24,
Theorem 1.1]). In this paper we show how to use a slightly stronger statement
for $\operatorname{SL}_{3}(\mathbb{Z})$ to induce the spectral gap for
$G_{n}=\operatorname{SL}_{n}(\mathbb{Z})$, whenever $n\geq 3$. Our technique
works as well for the case $G_{n}=\operatorname{SAut}(F_{n})$. We remark
however that in such a case we were unable to obtain such a spectral gap for
any particular $n$. On the other other hand, in his preprint [Nit22] M.
Nitsche shows property (T) for $\operatorname{SAut}(F_{4})$. This, in
combination with the work of Bader and Nowak [BN20] and the announcement of
Bader and Sauer, would yield the desired gap for the Laplacian in degree one
for $\operatorname{SAut}(F_{4})$. Nevertheless, this will still not allow us
to induce the spectral gap since we would need a spectral gap for a specific
summand of the Laplacian of $\operatorname{SAut}(F_{4})$.

Our induction technique is inspired by the idea of decomposing the generating
set of $G_{n}$ into three parts: square, adjacent, and opposite, see [KKN21].
This allows us to decompose $\Delta_{1}$ for any $G_{n}$ into the sum of the
appropriate parts. Denoting the adjacent part by $\operatorname{Adj}_{n}$ for
any $n\geq 3$, and the generating set for $G_{n}$ by $\altmathcal{S}_{n}$, the
main theorem can be stated as follows.

######  Theorem 1.1 (cf. Corollary 5.5).

Suppose $\operatorname{Adj}_{m}-\lambda I_{|\altmathcal{S}_{m}|}$ is a sum of
squares in
$\mathbb{M}_{|\altmathcal{S}_{m}|\times|\altmathcal{S}_{m}|}(\mathbb{R}G_{m})$.
Then

$\displaystyle\operatorname{Adj}_{n}-\frac{n-2}{m-2}\lambda
I_{|\altmathcal{S}_{n}|}$

is a sum of squares in
$\mathbb{M}_{|\altmathcal{S}_{n}|\times|\altmathcal{S}_{n}|}(\mathbb{R}G_{n})$
for any $n\geq m$.

Bounding the spectral gap for the adjacent part $\operatorname{Adj}_{3}$ of
$\operatorname{SL}_{3}(\mathbb{Z})$, we get the following

######  Corollary 1.2 (cf. Corollary 6.2).

Let $\Delta_{1}$ be the cohomological Laplacian in degree one of
$G_{n}=\operatorname{SL}_{n}(\mathbb{Z})$. Then $\Delta_{1}-\lambda
I_{|\altmathcal{S}_{n}|}$ is a sum of squares for $\lambda=0.217(n-2)$.

To our knowledge, the lower bounds for the spectral gaps we obtained are the
first examples of explicit computations for $\Delta_{1}$ for an infinite
family of property (T) groups.

This article is organized as follows. In section 2 we introduce Fox
derivatives which provide a setting to calculate one-degree Laplacians and
define the positivity in the group ring setting in terms of sums of squares.
In the next section we introduce the presentions for $G_{n}$: using the
_elementary matrices_ for $G_{n}=\operatorname{SL}_{n}(\mathbb{Z})$ and
_Nielsen transvections_ for $G_{n}=\operatorname{SAut}(F_{n})$. Next, we
decompose the Laplacian in degree one into the three parts mentioned above. In
section 5 we show Theorem 1.1 using the technique of symmetrization and apply
this result in the section 6 to obtain a lower bound for the spectral gap of
$\Delta_{1}$ for $G_{n}=\operatorname{SL}_{n}(\mathbb{Z})$ whenever $n\geq 3$.

### Acknowledgements

The author would like to express his sincere thanks to Prof. Dawid Kielak for
many valuable suggestions concerning the induction technique. I am also
greatly indebted to Dr Marek Kaluba for his collaboration on the substance of
section 5.2, as well as his careful reading of the whole contents and many
valuable remarks.

This research was supported by National Science Centre, Poland SONATINA 6
grant _Algebraic spectral gaps in group cohomology_ (grant agreement no.
2022/44/C/ST1/00037).

##  2\. Fox derivatives and sums of squares

We describe the model of $\Delta_{1}$ which appears in Theorem 1.1. This can
be achieved by means of _Fox derivatives_. Further details can be found in the
papers of Fox and Lyndon, [Fox53, Fox54, Lyn50, LS77].

Let $G=\langle s_{1},\ldots,s_{n}|r_{1},\ldots,r_{m}\rangle$ be a finitely
presented group. The Fox derivatives $\frac{\partial}{\partial s_{j}}$ are the
endomorphisms of $\mathbb{Z}F_{n}$, the group ring of the free group
$F_{n}=\langle s_{1},\ldots,s_{n}\rangle$, given by the following conditions:

$\displaystyle\frac{\partial 1}{\partial s_{j}}$ $\displaystyle=0,$
$\displaystyle\frac{\partial s_{i}}{\partial s_{j}}$
$\displaystyle=\delta_{ij},$ $\displaystyle\frac{\partial(uv)}{\partial
s_{j}}$ $\displaystyle=\frac{\partial u}{\partial s_{j}}+u\frac{\partial
v}{\partial s_{j}},$

where $\delta_{ij}$ denotes the Kronecker delta. The _Jacobian_ of $G$ is then
the matrix $J=\left[\frac{\partial r_{i}}{\partial
s_{j}}\right]\in\mathbb{M}_{m\times n}(\mathbb{Z}G)$ obtained by quotienting
the Fox derivatives into the group ring $\mathbb{Z}G$. Denoting by $d$ the
vertical vector $\left[1-s_{j}\right]\in\mathbb{M}_{n\times 1}(\mathbb{Z}G)$,
the one-degree Laplacian $\Delta_{1}$ associated to the presentation
$G=\langle s_{1},\ldots,s_{n}|r_{1},\ldots,r_{m}\rangle$ is defined by

$\Delta_{1}=\Delta_{1}^{+}+\Delta_{1}^{-},$

where $\Delta_{1}^{+}=J^{*}J$ and $\Delta_{1}^{-}=dd^{*}$, and, for any matrix
$M$ with entries in $\mathbb{Z}G$, the matrix $M^{*}$ is the composition of
the matrix transposition with the $*$-involution on $\mathbb{Z}G$ given by
$g\mapsto g^{-1}$.

######  Definition 2.1.

We say that a matrix $M\in\mathbb{M}_{k\times k}(\mathbb{R}G)$ is a sum of
squares if there exist matrices $M_{1},\ldots,M_{l}\in\mathbb{M}_{k\times
k}(\mathbb{R}G)$ such that $M=M_{1}^{*}M_{1}+\ldots M_{l}^{*}M_{l}$.

It follows from [BN20, Lemma 14] that $\Delta_{1}^{\pm}$, and therefore
$\Delta_{1}$ as well, are sums of squares.

By saying that a matrix $M\in\mathbb{M}_{k\times k}(\mathbb{R}G)$ has a
positive _spectral gap_ we mean that there exists a positive $\lambda$ such
that $M-\lambda I_{k}$ is a sum of squares.

As noted in [KMN24, Lemma 2.1], it turns out that we have some freedom in
choosing the relations which build up $\Delta_{1}^{+}$. Denote

$J_{i}=\left[\frac{\partial r_{i}}{\partial s_{1}}\cdots\frac{\partial
r_{i}}{\partial s_{n}}\right]$

and put $\Delta_{I}^{+}=\sum_{i\in I}J_{i}^{*}J_{i}$ for any subset of indices
$I\subseteq\\{1,\ldots,m\\}$. Then, once
$\Delta_{I}^{+}+\Delta_{1}^{-}\in\mathbb{M}_{n\times n}(\mathbb{R}G)$ has a
spectral gap then $\Delta_{1}=\Delta_{I}^{+}+\Delta_{\\{1,\ldots,m\\}\setminus
I}^{+}+\Delta_{1}^{-}\in\mathbb{M}_{n\times n}(\mathbb{R}G)$ has at least the
same spectral gap.

Computing lower bounds for sepctral gaps for the first Laplacians provides the
bounds for the Ozawa’s spectral gap:

######  Remark 2.2.

_Suppose the matrix $C\Delta_{I}^{+}+\Delta_{1}^{-}-\lambda I$ is a sum of
squares for some subset of relator indices $I\subseteq\\{1,\ldots,m\\}$ and a
positive constant $C$ and let $\Delta=d^{*}d$. Then
$\Delta^{2}-\lambda\Delta=d^{*}\left(C\Delta_{I}^{+}+\Delta_{1}^{-}-\lambda
I\right)d$ is a sum of squares as well (note that $d^{*}\Delta_{I}^{+}d$ is
always zero). Note that in the case the generating set $\altmathcal{S}$ and
the set of its inverses, $\altmathcal{S}^{-1}$, are disjoint, then $\Delta$ is
the zero degree Laplacian $|\altmathcal{S}|-\sum_{s\in\altmathcal{S}}s$ for
the symmetric generating set $\altmathcal{S}\cup\altmathcal{S}^{-1}$. _

##  3\. Presentations of $\operatorname{SL}_{n}(\mathbb{Z})$ and
$\operatorname{SAut}(F_{n})$

In this section, we introduce the presentations of
$\operatorname{SL}_{n}(\mathbb{Z})$ and $\operatorname{SAut}(F_{n})$ which we
shall use to perform the Fox calculus and prove property (T) for these groups.
The presentations can be found in [CRW92] and [Ger84] for
$\operatorname{SL}_{n}(\mathbb{Z})$ and $\operatorname{SAut}(F_{n})$
respectively. For the latter, we provide an equivalent presentation which is
easy to obtain from [Ger84].

The presentation for $\operatorname{SL}_{n}(\mathbb{Z})$ is given below.

$\displaystyle\operatorname{SL}_{n}(\mathbb{Z})=\big{\langle}E_{ij},1\leq
i\neq j\leq n\big{|}$ $\displaystyle[E_{ij},E_{kl}],i\neq l,j\neq
k,(i,j)\neq(k,l),$ $\displaystyle[E_{ij},E_{jk}]E_{ik}^{-1}\text{ for distinct
}i,j,k,$ $\displaystyle\left(E_{12}E_{21}^{-1}E_{12}\right)^{4}\big{\rangle}.$

The generator $E_{ij}$ represents the elementary matrix with $1$ at the
$(i,j)$-th entry and the diagonal and zeroes elsewhere. Using Remark 2.2, we
shall exclude the relation $\left(E_{12}E_{21}^{-1}E_{12}\right)^{4}$ from
further considerations.

The generators of $\operatorname{SAut}(F_{n})$ are denoted by $\lambda_{ij}$
and $\rho_{ij}$ for $1\leq i\neq j\leq n$ and are called left and right
_Nielsen transvections_ respectively. The relator set contains commutators and
pentagonal relations as well, along with other relations which we shall not
consider, again by Remark 2.2. We only present the commutator and pentagonal
relations therefore. These relations are defined for distinct indices
$i,j,k,l\in\\{1,\ldots,n\\}$ as follows:

$\displaystyle[\lambda_{i,j},\rho_{ij}],\quad[\lambda_{ij},\lambda_{kl}],\quad[\rho_{ij},\rho_{kl}],\quad[\lambda_{ij},\rho_{kl}],$
$\displaystyle\quad[\lambda_{ij},\lambda_{kj}],\quad[\rho_{ij},\rho_{kj}],\quad[\lambda_{ij},\rho_{ik}],\quad[\lambda_{i,j},\rho_{kj}],$
$\displaystyle\quad\lambda_{ik}^{\pm}[\lambda_{jk}^{\pm},\lambda_{ij}^{-1}],\quad\rho_{ik}^{\pm}[\rho_{jk}^{\pm},\rho_{ij}^{-1}],\quad\lambda_{ik}^{\pm}[\rho_{jk}^{\mp},\lambda_{ij}],\quad\rho_{ik}^{\pm}[\lambda_{jk}^{\mp},\rho_{ij}].$

The transvection generators correspond in $\operatorname{SAut}(F_{n})$ to the
following automorphisms of $F_{n}=\langle s_{1},\ldots,s_{n}\rangle$111We take
in $\operatorname{SAut}(F_{n})$ the group operation order given by composition
of automorphisms, i.e. $\alpha*\beta=\alpha\circ\beta$. Note that this is
different than the convention of Gersten [Ger84]. In his paper the convention
is: $\alpha*\beta=\beta\circ\alpha$. We had to reverse the order in pentagonal
relators for that account.:

$\displaystyle\lambda_{ij}(s_{k})=\begin{cases}s_{j}s_{i}&\text{ if $k=i$,}\\\
s_{k}&\text{otherwise,}\end{cases}\quad\rho_{ij}(s_{k})=\begin{cases}s_{i}s_{j}&\text{
if $k=i$,}\\\ s_{k}&\text{otherwise.}\end{cases}$

##  4\. Decomposition into square, adjacent and opposite part

Let $n\geq 3$ be an integer and $G_{n}$ be $\operatorname{SL}_{n}(\mathbb{Z})$
or $\operatorname{SAut}(F_{n})$. We introduce the following decompositions of
the Laplacians $\Delta_{1}^{\pm}$ of $G_{n}$:

$\displaystyle\Delta_{1}^{\pm}=\operatorname{Sq}_{n}^{\pm}+\operatorname{Adj}_{n}^{\pm}+\operatorname{Op}_{n}^{\pm}.$

Our decomposition is inspired by [KKN21, pp. 543 – 545].

The generators of $G_{n}$ take form $\omega_{ij}$ for $1\leq i\neq j\leq n$
and $\omega$ denoting either elementary matrices or Nielsen transvections.
Then, for any $3\leq m\leq n$, there exists a natural embedding
$G_{m}\hookrightarrow G_{n}$ given by $\omega_{ij}\mapsto\omega_{ij}$. From
now on, when talking about the relations of $G_{n}$, we shall restrict
ourselves to commutator and pentagonal relations only.

Let $C_{n}$ be the $(n-1)$-simplex with the vertices $\\{1,\ldots,n\\}$ and
let $F(C_{n})$ denote the set of its faces, that is the subsets of the set
$\\{1,\ldots,n\\}$. Denoting by $\altmathcal{S}_{n}$ the generator set of
$G_{n}$ and by $\operatorname{Free}(\altmathcal{S}_{n})$ the free group on
$\altmathcal{S}_{n}$, one can define the map

$\displaystyle\phi:\operatorname{Free}(\altmathcal{S}_{n})\longrightarrow
F(C_{n})$

by sending an element of $\operatorname{Free}(\altmathcal{S}_{n})$ to the face
on the indices of the generators occurring in that element, i.e.

$\phi(\omega_{i_{1}j_{1}}\ldots\omega_{i_{k}j_{k}})=\\{i_{1},j_{1},\ldots,i_{k},j_{k}\\}.$

In order to define $\operatorname{Sq}_{n}^{\pm}$,
$\operatorname{Adj}_{n}^{\pm}$ and $\operatorname{Op}_{n}^{\pm}$ we
distinguish from $F(C_{n})$ the following three subsets of faces

$\displaystyle\operatorname{Edg}_{n}$ $\displaystyle=\\{f\in
F(C_{n})||f|=2\\},$ $\displaystyle\operatorname{Tri}_{n}$
$\displaystyle=\\{f\in F(C_{n})||f|=3\\},$
$\displaystyle\operatorname{Tet}_{n}$ $\displaystyle=\\{f\in
F(C_{n})||f|=4\\},$

that is, $\operatorname{Edg}_{n}$, $\operatorname{Tri}_{n}$ and
$\operatorname{Tet}_{n}$ constitute respectively the sets of edges, triangles,
and tetrahedra of $C_{n}$.

The decomposition
$\Delta_{1}^{-}=\operatorname{Sq}_{n}^{-}+\operatorname{Adj}_{n}^{-}+\operatorname{Op}_{n}^{-}$
is obtained as follows:

$\displaystyle\left(\operatorname{Sq}^{-}_{n}\right)_{s,t}$
$\displaystyle=\left(\Delta^{-}_{1}\right)_{s,t}\text{ for
}\phi(s)\cup\phi(t)\in\operatorname{Edg}_{n}\text{ and }0\text{ otherwise},$
$\displaystyle\left(\operatorname{Adj}^{-}_{n}\right)_{s,t}$
$\displaystyle=\left(\Delta^{-}_{1}\right)_{s,t}\text{ for
}\phi(s)\cup\phi(t)\in\operatorname{Tri}_{n}\text{ and }0\text{ otherwise},$
$\displaystyle\left(\operatorname{Op}^{-}_{n}\right)_{s,t}$
$\displaystyle=\left(\Delta^{-}_{1}\right)_{s,t}\text{ for
}\phi(s)\cup\phi(t)\in\operatorname{Tet}_{n}\text{ and }0\text{ otherwise}.$

We remark that the group ring elements $d^{*}\operatorname{Sq}^{-}_{n}d$,
$d^{*}\operatorname{Adj}^{-}_{n}d$, and $d^{*}\operatorname{Op}^{-}_{n}d$ are
equal to the elements $\operatorname{Sq}_{n}$, $\operatorname{Adj}_{n}$, and
$\operatorname{Op}_{n}$ as defined in [KKN21].

The decomposition
$\Delta_{1}^{+}=\operatorname{Sq}^{+}_{n}+\operatorname{Adj}^{+}_{n}+\operatorname{Op}^{+}_{n}$
is defined, in turn, by summing over specific relators of $G_{n}$.

$\displaystyle\left(\operatorname{Sq}^{+}_{n}\right)_{s,t}$
$\displaystyle=\sum_{\phi(r)\in\operatorname{Edg}_{n}}\left(\frac{\partial
r}{\partial s}\right)^{*}\frac{\partial r}{\partial t},$
$\displaystyle\left(\operatorname{Adj}^{+}_{n}\right)_{s,t}$
$\displaystyle=\sum_{\phi(r)\in\operatorname{Tri}_{n}}\left(\frac{\partial
r}{\partial s}\right)^{*}\frac{\partial r}{\partial t},$
$\displaystyle\left(\operatorname{Op}^{+}_{n}\right)_{s,t}$
$\displaystyle=\sum_{\phi(r)\in\operatorname{Tet}_{n}}\left(\frac{\partial
r}{\partial s}\right)^{*}\frac{\partial r}{\partial t}.$

Note that $\left(\operatorname{Adj}^{+}_{n}\right)_{s,t}$ and
$\left(\operatorname{Op}^{+}_{n}\right)_{s,t}$ can be non-zero only if $s=t$
or $\phi(s)\cup\phi(t)$ belongs to $\operatorname{Tri}_{n}$ and
$\operatorname{Tet}_{n}$ respectively. We shall call
$\operatorname{Sq}_{n}^{\pm}$, $\operatorname{Adj}_{n}^{\pm}$ and
$\operatorname{Op}_{n}^{\pm}$ the _square_ , _adjacent_ and _opposite_ part
respectively.

######  Example 4.1.

_Let $n=3$ and let us focus on $G_{3}=\operatorname{SL}_{3}(\mathbb{Z})$. We
have six generators of $G_{3}$, namely the elementary matrices $E_{12}$,
$E_{21}$, $E_{13}$, $E_{31}$, $E_{23}$, and $E_{32}$. The relator set $R$ we
take into account contains the following $18$ relators:_

$\displaystyle[E_{12},E_{13}],[E_{13},E_{12}],[E_{12},E_{32}],[E_{32},E_{12}],$
$\displaystyle[E_{13},E_{23}],[E_{23},E_{13}],[E_{21},E_{23}],[E_{23},E_{21}],$
$\displaystyle[E_{21},E_{31}],[E_{31},E_{21}],[E_{31},E_{32}],[E_{32},E_{31}],$
$\displaystyle[E_{12},E_{23}]E_{13}^{-1},[E_{13},E_{32}]E_{12}^{-1},[E_{21},E_{13}]E_{23}^{-1},$
$\displaystyle[E_{23},E_{31}]E_{21}^{-1},[E_{31},E_{12}]E_{32}^{-1},[E_{32},E_{21}]E_{31}^{-1}.$

_The simplex $C_{3}$ is a triangle. Its faces contain three edges – more
precisely, $\operatorname{Edg}_{3}=\\{\\{1,2\\},\\{1,3\\},\\{2,3\\}\\}$, and
only one triangle, i.e. $\operatorname{Tri}_{3}=\\{\\{1,2,3\\}\\}$. There are
no tetrahedra in $C_{3}$._

$1$$2$$3$$\\{E_{13},E_{31}\\}$$\\{E_{23},E_{32}\\}$$\\{E_{12},E_{21}\\}$
Figure 1. The simplex $C_{3}$ with the assignments of the generators of
$\operatorname{SL}_{3}(\mathbb{Z})$ to its edges given by $\phi$.

_The function $\phi$ takes the following values on the generators (the
situation depicted in Figure 1):_

$\displaystyle\phi(E_{12})=\phi(E_{21})=\\{1,2\\},\quad\phi(E_{13})=\phi(E_{31})=\\{1,3\\},\quad\phi(E_{23})=\phi(E_{32})=\\{2,3\\},$

_and is constant on the relators – for every $r\in R$ one has
$\phi(r)=\\{1,2,3\\}$. Thus, denoting by $(i,j)(k,l)$ the element
$(1-E_{ij})(1-E_{kl})^{*}$, we get_

$\displaystyle\operatorname{Sq}_{3}^{-}=\begin{bmatrix}(1,2)(1,2)&0&(1,2)(2,1)&0&0&0\\\
0&(1,3)(1,3)&0&0&(1,3)(3,1)&0\\\ (2,1)(1,2)&0&(2,1)(2,1)&0&0&0\\\
0&0&0&(2,3)(2,3)&0&(2,3)(3,2)\\\ 0&(3,1)(1,3)&0&0&(3,1)(3,1)&0\\\
0&0&0&(3,2)(2,3)&0&(3,2)(3,2)\end{bmatrix}$

_and_

$\displaystyle\operatorname{Adj}_{3}^{-}=\begin{bmatrix}0&(1,2)(1,3)&0&(1,2)(2,3)&(1,2)(3,1)&(1,3)(3,2)\\\
(1,3)(1,2)&0&(1,3)(2,1)&(1,3)(2,3)&0&(1,3)(3,2)\\\
0&(2,1)(1,3)&0&(2,1)(2,3)&(2,1)(3,1)&(2,1)(3,2)\\\
(2,3)(1,2)&(2,3)(1,3)&(2,3)(2,1)&0&(2,3)(3,1)&0\\\
(3,1)(1,2)&0&(3,1)(2,1)&(3,1)(2,3)&0&(3,1)(3,2)\\\
(3,2)(1,2)&(3,2)(1,3)&(3,2)(2,1)&0&(3,2)(3,1)&0\end{bmatrix}.$

_It is apparent that $\operatorname{Op}_{3}^{-}$ is the zero matrix. Since
$\phi$ equals the face $\\{1,2,3\\}$ on all relators, we have_

$\left(\operatorname{Adj}_{3}^{+}\right)_{s,t}=\sum_{r\in
R}\left(\frac{\partial r}{\partial s}\right)^{*}\left(\frac{\partial
r}{\partial t}\right)=\left(\Delta_{1}^{+}\right)_{s,t}$

_for any two generators $s,t$. In particular, $\operatorname{Sq}_{3}^{+}$ and
$\operatorname{Op}_{3}^{+}$ are the zero matrices._

######  Example 4.2.

_This time, let $n=4$ and $G_{4}=\operatorname{SAut}(F_{4})$. The matrices get
significantly bigger for this case and for this account we shall restrict
ourselves to the demonstration of the function $\phi$ only. The following $24$
transvections consitute the generators of $G_{4}$:_

$\displaystyle\lambda_{12},\lambda_{13},\lambda_{14},\lambda_{21},\lambda_{23},\lambda_{24},\lambda_{31},\lambda_{32},\lambda_{34},\lambda_{41},\lambda_{42},\lambda_{43},$
$\displaystyle\rho_{12},\rho_{13},\rho_{14},\rho_{21},\rho_{23},\rho_{24},\rho_{31},\rho_{32},\rho_{34},\rho_{41},\rho_{42},\rho_{43}.$

_Since there are already relatively many relations of $G_{4}$, we will not
list them all. Instead, we comment on the values that the function $\phi$
takes on them. The simplex $C_{4}$ is a tetrahderon. The faces of $C_{4}$
contain six edges, four triangles and one tetrahedron:_

$\displaystyle\operatorname{Edg}_{4}=\\{\\{1,2\\},\\{1,3\\},\\{1,4\\},\\{2,3\\},\\{2,4\\},\\{3,4\\}\\}$
$\displaystyle\operatorname{Tri}_{4}=\\{\\{1,2,3\\},\\{1,2,4\\},\\{1,3,4\\},\\{2,3,4\\}\\}$
$\displaystyle\operatorname{Tet}_{4}=\\{\\{1,2,3,4\\}\\}.$

_The function $\phi$ takes the following values on the generators (see Figure
2):_

$\displaystyle\phi(\lambda_{12})=\phi(\lambda_{21})=\phi(\rho_{12})=\phi(\rho_{21})=\\{1,2\\},$
$\displaystyle\phi(\lambda_{13})=\phi(\lambda_{31})=\phi(\rho_{13})=\phi(\rho_{31})=\\{1,3\\},$
$\displaystyle\phi(\lambda_{14})=\phi(\lambda_{41})=\phi(\rho_{14})=\phi(\rho_{41})=\\{1,4\\},$
$\displaystyle\phi(\lambda_{23})=\phi(\lambda_{32})=\phi(\rho_{23})=\phi(\rho_{32})=\\{2,3\\},$
$\displaystyle\phi(\lambda_{24})=\phi(\lambda_{42})=\phi(\rho_{24})=\phi(\rho_{42})=\\{2,4\\},$
$\displaystyle\phi(\lambda_{34})=\phi(\lambda_{43})=\phi(\rho_{34})=\phi(\rho_{43})=\\{3,4\\}.$

$1$$2$$3$$4$$\\{\lambda_{12},\lambda_{21},\rho_{12},\rho_{21}\\}$$\\{\lambda_{13},\lambda_{31},\rho_{13},\rho_{31}\\}$$\\{\lambda_{14},\lambda_{41},\rho_{14},\rho_{41}\\}$$\\{\lambda_{23},\lambda_{32},\rho_{23},\rho_{32}\\}$$\\{\lambda_{24},\lambda_{42},\rho_{24},\rho_{42}\\}$$\\{\lambda_{34},\lambda_{43},\rho_{34},\rho_{43}\\}$
Figure 2. The simplex $C_{4}$ with the assignments of the generators of
$\operatorname{SAut}(F_{4})$ to its edges given by $\phi$.

_Let $\leq i,j,k,l\leq 4$ be pairwise distinct. The values of $\phi$ on the
relators are the following:_

$\displaystyle\phi([\lambda_{ij},\rho_{ij}])=\\{i,j\\},\quad\phi([\lambda_{ij},\lambda_{kl}])=\phi(\rho_{ij},\rho_{kl})=\phi([\lambda_{ij},\rho_{kl}])=\\{1,2,3,4\\},$
$\displaystyle\phi([\lambda_{ij},\lambda_{kj}])=\phi([\rho_{ij},\rho_{kj}])=\phi([\lambda_{ij},\rho_{ik}])=\phi([\lambda_{i,j},\rho_{kj}])$
$\displaystyle\qquad\qquad\quad\text{
}=\phi(\lambda_{ik}^{\pm}[\lambda_{jk}^{\pm},\lambda_{ij}^{-1}])=\phi(\rho_{ik}^{\pm}[\rho_{jk}^{\pm},\rho_{ij}^{-1}])$
$\displaystyle\qquad\qquad\quad\text{
}=\phi(\lambda_{ik}^{\pm}[\rho_{jk}^{\mp},\lambda_{ij}])=\phi(\rho_{ik}^{\pm}[\lambda_{jk}^{\mp},\rho_{ij}])=\\{i,j,k\\}.$

###  4.1. Sum of squares decomposition for square and opposite part

It turns out that the square and opposite parts behave well with respect to
sum of squares decompositions. The precise statement is given below.

######  Lemma 4.3.

The matrices $\operatorname{Sq}^{-}_{n}$ and
$\operatorname{Op}^{+}_{n}+2\operatorname{Op}^{-}_{n}$ are sums of squares.

###### Proof.

For any $1\leq i\neq j\leq n$, define
$d^{i,j}\in\mathbb{M}_{|\altmathcal{S}_{n}|\times 1}(\mathbb{R}G_{n})$ to be
the column vector indexed by the generators of $G_{n}$ with the only non-zero
entries being $1-s$, indexed by the generators $s$ satisfying
$\phi(s)=\\{i,j\\}$. That $\operatorname{Sq}^{-}_{n}$ is a sums of squares
follows from the decomposition below:

$\operatorname{Sq}^{-}_{n}=\sum_{1\leq i\neq j\leq
n}d^{i,j}\left(d^{i,j}\right)^{*}.$

We have already observed that both $\operatorname{Op}^{+}_{n}$ and
$\operatorname{Op}^{-}_{n}$ (in this case, just by the definition) have
vanishing $(s,t)$-entries whenever
$\phi(s)\cup\phi(t)\in\operatorname{Tri}_{n}$ or the generator $t$ has the
same indices as $s$ but with the reversed order. In addition, the diagonal
vanishes for $\operatorname{Op}^{-}_{n}$ as well. This is not the case for
$\operatorname{Op}^{+}_{n}$. However, since $\operatorname{Op}^{+}_{n}$ is a
sum of squares by its definition, it follows that the same applies to its
diagonal part. On the other hand, the non-diagonal parts of
$\operatorname{Op}^{+}_{n}$ and $2\operatorname{Op}^{-}_{n}$ cancel out – for
distinct $1\leq i,j,k,l\leq n$ we have:

$\displaystyle\left(\operatorname{Op}^{+}_{n}\right)_{\alpha_{ij},\beta_{kl}}$
$\displaystyle=\left(\frac{\partial[\alpha_{ij},\beta_{kl}]}{\partial\alpha_{ij}}\right)^{*}\frac{\partial[\alpha_{ij},\beta_{kl}]}{\partial\beta_{kl}}+\left(\frac{\partial[\beta_{kl},\alpha_{ij}]}{\partial\alpha_{ij}}\right)^{*}\frac{\partial[\beta_{kl},\alpha_{ij}]}{\partial\beta_{kl}}$
$\displaystyle=-2(1-\alpha_{ij})(1-\beta_{kl})^{*}.$

∎

##  5\. Symmetrization of the adjacent part

In this section we show how to get the lower bound for spectral gap of
$\operatorname{Adj}_{n}=\operatorname{Adj}_{n}^{+}+\operatorname{Adj}_{n}^{-}$
for $G_{n}$ once we assume that the lower bound for the corresponding spectral
gap for $G_{m}$ is known for some $n\geq m\geq 3$. Our method is inspired by
the symmetrization method used in [KKN21].

###  5.1. General setting for symmetrization

Note that the symmetric group $\operatorname{Sym}_{k}$ acts by automorphisms
on $G_{k}$ by sending the generator $\omega_{i,j}$ to
$\omega_{\sigma(i),\sigma(j)}$ for any permutation
$\sigma\in\operatorname{Sym}_{k}$. This induces an action of
$\operatorname{Sym}_{k}$ on $\mathbb{R}G_{k}$ by automorphisms which preserves
the $*$-involution. Using this action we can define the action on
$\mathbb{M}_{|\altmathcal{S}_{k}|\times|\altmathcal{S}_{k}|}(\mathbb{R}G_{k})$
by the formula

$\left(\sigma
A\right)_{s,t}=\sigma\left(A_{\sigma^{-1}(s),\sigma^{-1}(t)}\right),$

for any matrix $A$ over $\mathbb{R}G_{k}$ indexed by the generators
$\altmathcal{S}_{k}$. This is also the action by $*$-algebra automorphisms
(the multiplication being the group ring matrix multiplication). Thus, for any
matrix
$A\in\mathbb{M}_{|\altmathcal{S}_{k}|\times|\altmathcal{S}_{k}|}(\mathbb{R}G_{k})$
and $\sigma\in\operatorname{Sym}_{k}$, the matrix $\sigma A$ is a sum of
squares provided $A$ was a sum of squares.

###  5.2. Invariance of Laplacians

Let us show first that the adjacent Laplacians $\operatorname{Adj}_{k}^{+}$
and $\operatorname{Adj}^{-}_{k}$ for $G_{k}$ are
$\operatorname{Sym}_{k}$-invariant. The statement for
$\operatorname{Adj}^{-}_{k}$ is straightforward:

$\displaystyle\left(\sigma\operatorname{Adj}_{k}^{-}\right)_{s,t}$
$\displaystyle=\sigma\left(\left(\operatorname{Adj}^{-}_{k}\right)_{\sigma^{-1}(s),\sigma^{-1}(t)}\right)=\sigma\left(\left(1-\sigma^{-1}(s)\right)\left(1-\sigma^{-1}(t)\right)^{*}\right)$
$\displaystyle=(1-s)(1-t)^{*}=\left(\operatorname{Adj}^{-}_{k}\right)_{s,t}.$

Let us show the invariance of $\operatorname{Adj}_{k}^{+}$. Let
$a=|\altmathcal{S}_{k}|$ be the number of generators of $G_{k}$ and $b$ be the
number of its relations $r$ such that $\phi(r)\in\operatorname{Tri}_{k}$.
Suppose moreover that this set of relations is invariant with respect to the
action of $\operatorname{Sym}_{k}$, that is, for all
$\sigma\in\operatorname{Sym}_{k}$ and $r$ satisfying
$\phi(r)\in\operatorname{Tri}_{k}$, the word $\sigma(r)$ is another relator.
Notice that each permutation $\sigma$ from $\operatorname{Sym}_{k}$ defines
permutations of $\\{1,\ldots,a\\}$ and $\\{1,\ldots,b\\}$ (we fix the order of
generators $\altmathcal{S}_{k}$ and relations $r$ such that
$\phi(r)\in\operatorname{Tri}_{k}$). Having this, we can endow both products
$\left(\mathbb{R}G_{k}\right)^{a}$ and $\left(\mathbb{R}G_{k}\right)^{b}$ with
the action of $\operatorname{Sym}_{k}$ as follows:

$\displaystyle\sigma\left(\xi_{1},\ldots,\xi_{a}\right)$
$\displaystyle=\left(\sigma\xi_{\sigma^{-1}(1)},\ldots,\sigma\xi_{\sigma^{-1}(a)}\right),$
$\displaystyle\sigma\left(\xi_{1},\ldots,\xi_{b}\right)$
$\displaystyle=\left(\sigma\xi_{\sigma^{-1}(1)},\ldots,\sigma\xi_{\sigma^{-1}(b)}\right).$

Finally, denote by $J$ the Jacobian map from
$\left(\mathbb{R}G_{k}\right)^{a}$ to $\left(\mathbb{R}G_{k}\right)^{b}$ given
by the relations $r$ such that $\phi(r)\in\operatorname{Tri}_{k}$, that is
$\operatorname{Adj}_{k}^{+}=J^{*}J$. It turns out that the invariance of
$\operatorname{Adj}_{k}^{+}$ follows from the equivariance of $J$:

######  Lemma 5.1.

For any $\sigma\in\operatorname{Sym}_{k}$ we have
$\sigma\operatorname{Adj}_{k}^{+}=\operatorname{Adj}_{k}^{+}$ provided $J$ is
equivariant.

###### Proof.

Suppose $J$ is equivariant. Let $\sigma\in\operatorname{Sym}_{k}$. Since we
have ordered the generators in $\altmathcal{S}_{k}$, we can identify them with
the corresponding indices from the set $\\{1,\ldots,a\\}$. Thus, we must show
that
$\left(\sigma\operatorname{Adj}_{k}^{+}\right)_{i,j}=\left(\operatorname{Adj}_{k}^{+}\right)_{i,j}$
for any $1\leq i,j\leq a$.

It is easy to check that the equivariance of $J$ is equivalent to the
following condition: $J_{y,\sigma^{-1}(x)}=\sigma^{-1}J_{\sigma(y),x}$ for any
$1\leq x\leq a$ and $1\leq y\leq b$. Using this relationship, we get

$\displaystyle\left(\sigma\operatorname{Adj}_{k}^{+}\right)_{i,j}$
$\displaystyle=\sigma\left(\sum_{l}\left(J_{l,\sigma^{-1}(i)}\right)^{*}J_{l,\sigma^{-1}(j)}\right)=\sigma\left(\sum_{l}\sigma^{-1}\left(\left(J_{\sigma(l),i}\right)^{*}J_{\sigma(l),j}\right)\right)$
$\displaystyle=\sum_{l}\left(J_{\sigma(l),i}\right)^{*}J_{\sigma(l),j}=\left(\operatorname{Adj}_{k}^{+}\right)_{i,j}.$

∎

The following lemma is the key tool to prove the equivariance of $J$.

######  Lemma 5.2.

For any $\sigma\in\operatorname{Sym}_{N}$, $s\in\\{s_{1},\ldots,s_{N}\\}$ and
$r$ a word in the free group $F_{N}$ generated by $\\{s_{1},\ldots,s_{N}\\}$,
the following holds:

$\frac{\partial
r}{\partial(\sigma(s))}=\sigma\left(\frac{\partial\left(\sigma^{-1}(r)\right)}{\partial
s}\right),$

where the action of $\operatorname{Sym}_{N}$ on $F_{N}$ is given by
$\tau(s_{i})=s_{\tau(i)}$ for any $\tau\in\operatorname{Sym}_{N}$.

###### Proof.

We prove the assertion by induction on the word length of the relation $r$.
Note first that both sides of the equation in question vanish for $r=1$.
Suppose $r$ is a single generator $s^{\prime}$ (if $r$ is the inverse of a
generator, the proof is analogous). In the case
$\sigma(s)\notin\\{s^{\prime},(s^{\prime})^{-1}\\}$ both sides of the equation
in question vanish. If $\sigma(s)=s^{\prime}$, then

$\frac{\partial r}{\partial(\sigma(s))}=\frac{\partial s^{\prime}}{\partial
s^{\prime}}=1=\sigma\left(\frac{\partial s}{\partial
s}\right)=\sigma\left(\frac{\partial\left(\sigma^{-1}(r)\right)}{\partial
s}\right),$

while if $\sigma(s)=(s^{\prime})^{-1}$, then

$\frac{\partial r}{\partial(\sigma(s))}=\frac{\partial
s^{\prime}}{\partial(s^{\prime})^{-1}}=-s^{\prime}=\sigma\left(\frac{\partial
s^{-1}}{\partial
s}\right)=\sigma\left(\frac{\partial\left(\sigma^{-1}(r)\right)}{\partial
s}\right).$

Suppose now the assertion follows for all words of length $k\geq 1$. Take a
word $r$ of length $k+1$. Then $r=uv$ for some words $u$ and $v$ of length at
most $k$. Using the induction assumption, we get

$\displaystyle\frac{\partial r}{\partial(\sigma(s))}$
$\displaystyle=\frac{\partial u}{\partial(\sigma(s))}+u\frac{\partial
v}{\partial(\sigma(s))}$
$\displaystyle=\sigma\left(\frac{\partial\left(\sigma^{-1}(u)\right)}{\partial
s}+\sigma^{-1}(u)\frac{\partial\left(\sigma^{-1}(v)\right)}{\partial
s}\right)$
$\displaystyle=\sigma\left(\frac{\partial\left(\sigma^{-1}(uv)\right)}{\partial
s}\right)=\sigma\left(\frac{\partial\left(\sigma^{-1}(r)\right)}{\partial
s}\right).$

∎

######  Corollary 5.3.

The Jacobian homomorphism $J:(\mathbb{R}G)^{a}\rightarrow(\mathbb{R}G)^{b}$ is
an equivariant map.

###### Proof.

By Lemma 5.2, for any $1\leq x\leq a$ and $1\leq y\leq b$, we get

$\displaystyle
J_{y,\sigma^{-1}(x)}=\frac{\partial(r_{y})}{\partial\left(\sigma^{-1}(s_{x})\right)}=\sigma^{-1}\left(\frac{\partial\left(\sigma(r_{y})\right)}{\partial
s_{x}}\right)=\sigma^{-1}J_{\sigma(y),x}.$

∎

###  5.3. Adjacent part symmetrizes well

In this section we show how to express $\operatorname{Adj}_{n}^{\pm}$ in terms
of $\operatorname{Adj}_{m}^{\pm}$ and the action of the symmetric group
$\operatorname{Sym}_{n}$, for any $n\geq m\geq 3$. This allows us to induce
spectral gaps for $\operatorname{Adj}$ parts (cf. Corollary 5.5).

For any $n\geq 3$ and $\theta\in\operatorname{Tri}_{n}$, we denote by
$\operatorname{Adj}^{\pm}_{\theta}\in\mathbb{M}_{|\altmathcal{S}_{n}|\times|\altmathcal{S}_{n}|}(\mathbb{R}G_{n})$
the embedding of the adjacent part given by the traingle $\theta$ of $C_{n}$,
i.e., for any generators $s$ and $t$ of $G_{n}$, we have

$\displaystyle\left(\operatorname{Adj}_{\theta}^{-}\right)_{s,t}=\begin{cases}(1-s)(1-t)^{*}&\text{if
}\phi(s)\cup\phi(t)=\theta\\\ 0&\text{if
}\phi(s)\cup\phi(t)\neq\theta\end{cases},\quad\left(\operatorname{Adj}_{\theta}^{+}\right)_{s,t}=\sum_{\phi(r)=\theta}\left(\frac{\partial
r}{\partial s}\right)^{*}\frac{\partial r}{\partial t}.$

Since it shall be clear from the context, given $m\leq n$, we shall also
denote by $\operatorname{Adj}_{m}^{\pm}$ the canonical embedding of
$\operatorname{Adj}_{m}^{\pm}\in\mathbb{M}_{|\altmathcal{S}_{m}|\times|\altmathcal{S}_{m}|}(\mathbb{R}G_{m})$
into
$\mathbb{M}_{|\altmathcal{S}_{n}|\times|\altmathcal{S}_{n}|}(\mathbb{R}G_{n})$:

$\displaystyle\left(\operatorname{Adj}_{m}^{-}\right)_{s,t}$
$\displaystyle=\begin{cases}(1-s)(1-t)^{*}&\text{if
}\phi(s)\cup\phi(t)\subseteq\\{1,\ldots,m\\}\\\ 0&\text{if
}\phi(s)\cup\phi(t)\nsubseteq\\{1,\ldots,m\\}\end{cases},$
$\displaystyle\left(\operatorname{Adj}_{m}^{+}\right)_{s,t}$
$\displaystyle=\sum_{\phi(r)\subseteq\\{1,\ldots,m\\}}\left(\frac{\partial
r}{\partial s}\right)^{*}\frac{\partial r}{\partial t}.$

Due to the same reason, we decided to remove the grading $n$ from the notation
$\operatorname{Adj}_{\theta}^{\pm}$.

######  Lemma 5.4.

For $n\geq m\geq 3$, one has

$\sum_{\sigma\in\operatorname{Sym}_{n}}\sigma\left(\operatorname{Adj}_{m}^{\pm}\right)=m(m-1)(m-2)(n-3)!\operatorname{Adj}_{n}^{\pm}.$

###### Proof.

The idea of the proof is to transfer the action of the symmetric group
$\operatorname{Sym}_{n}$ from the matrices over group rings to the simplex
$C_{n}$ and apply the invariance of $\operatorname{Adj}_{m}^{\pm}$ under the
action of $\operatorname{Sym}_{m}$. The transfer is possible due to the
following relationships:

(1)
$\sigma\left(\operatorname{Adj}_{\theta}^{\pm}\right)=\operatorname{Adj}_{\sigma(\theta)}^{\pm},$

holding for every permutation $\sigma\in\operatorname{Sym}_{n}$ and a triangle
$\theta\in\operatorname{Tri}_{n}$. Let us show (1) for
$\operatorname{Adj}^{+}$ only since the proof for $\operatorname{Adj}^{-}$ is
a direct application of its definition and the action of
$\operatorname{Sym}_{n}$ on
$\mathbb{M}_{|\altmathcal{S}_{n}|\times|\altmathcal{S}_{n}|}(\mathbb{R}G_{n})$.
Pick two generators $s$ and $t$ of $G_{n}$. Then, it follows by Lemma 5.2 that

$\displaystyle\left(\sigma\left(\operatorname{Adj}_{\theta}^{+}\right)\right)_{s,t}$
$\displaystyle=\sigma\left(\sum_{\phi(r)=\theta}\left(\frac{\partial
r}{\partial\sigma^{-1}(s)}\right)^{*}\frac{\partial
r}{\partial\sigma^{-1}(t)}\right)$
$\displaystyle=\sum_{\phi(r)=\sigma(\theta)}\left(\frac{\partial r}{\partial
s}\right)^{*}\frac{\partial r}{\partial
t}=\operatorname{Adj}_{\sigma(\theta)}^{+}.$

Note that, for any $k\geq 3$, we can express $\operatorname{Adj}_{k}^{\pm}$ by
summing its projections to the triangles of $C_{k}$:

$\displaystyle\operatorname{Adj}_{k}^{\pm}=\sum_{\theta\in\operatorname{Tri}_{k}}\operatorname{Adj}_{\theta}^{\pm}.$

Applying (1) and the orbit-stabilizer theorem applied to the action of
$\operatorname{Sym}_{k}$ on the triangles $\operatorname{Tri}_{k}$, we get

(2)
$\displaystyle\operatorname{Adj}_{k}^{\pm}=\sum_{\sigma\in\operatorname{Sym}_{k}}\frac{1}{|\operatorname{Stab}\\{1,2,3\\}|}\operatorname{Adj}_{\sigma(\\{1,2,3\\})}^{\pm}=\frac{1}{(k-3)!}\sum_{\sigma\in\operatorname{Sym}_{k}}\sigma\left(\operatorname{Adj}_{3}^{\pm}\right).$

Let $\tau_{i}\in\operatorname{Sym}_{n}$, $i=1,\ldots,(n-m)!$ be the
representatives of the cosets $\operatorname{Sym}_{n}/\operatorname{Sym}_{m}$.
Applying (2) first for $k=n$ and then for $k=m$, we get

$\displaystyle(n-3)!\operatorname{Adj}^{\pm}_{n}$
$\displaystyle=\sum_{\sigma\in\operatorname{Sym}_{n}}\sigma\left(\operatorname{Adj}^{\pm}_{3}\right)$
$\displaystyle=\sum_{i}\tau_{i}\sum_{\tau\in\operatorname{Sym}_{m}}\tau\left(\operatorname{Adj}^{\pm}_{3}\right)$
$\displaystyle=(m-3)!\sum_{i}\tau_{i}\operatorname{Adj}^{\pm}_{m}.$

We conclude the proof by applying $\operatorname{Sym}_{m}$-invariance of
$\operatorname{Adj}^{\pm}_{m}$ (cf. Lemma 5.1 and Corollary 5.3):

$\displaystyle(n-3)!\operatorname{Adj}^{\pm}_{n}$
$\displaystyle=(m-3)!\sum_{i}\tau_{i}\operatorname{Adj}^{\pm}_{m}$
$\displaystyle=(m-3)!\sum_{i}\tau_{i}\frac{1}{m!}\sum_{\tau\in\operatorname{Sym}_{m}}\sigma\left(\operatorname{Adj}^{\pm}_{m}\right)$
$\displaystyle=\frac{1}{m(m-1)(m-2)}\sum_{\sigma\in\operatorname{Sym}_{n}}\sigma\left(\operatorname{Adj}^{\pm}_{m}\right).$

∎

For any $k\geq 3$, let
$\operatorname{Adj}_{k}=\operatorname{Adj}_{k}^{+}+\operatorname{Adj}_{k}^{-}$.
We have the following

######  Corollary 5.5 (cf. Theorem 1.1).

The matrix

$\displaystyle\operatorname{Adj}_{n}-\frac{n-2}{m-2}\lambda
I_{|\altmathcal{S}_{n}|}$

is a sum of squares in
$\mathbb{M}_{|\altmathcal{S}_{n}|\times|\altmathcal{S}_{n}|}(\mathbb{R}G_{n})$,
provided $\operatorname{Adj}_{m}-\lambda I_{|\altmathcal{S}_{m}|}$ is a sum of
squares in
$\mathbb{M}_{|\altmathcal{S}_{m}|\times|\altmathcal{S}_{m}|}(\mathbb{R}G_{m})$.

###### Proof.

As in the case of $\operatorname{Adj}^{\pm}_{m}$, let us also use the notation
$I_{|\altmathcal{S}_{m}|}$ for the canonical embedding of
$I_{|\altmathcal{S}_{m}|}\in\mathbb{M}_{|\altmathcal{S}_{m}|\times|\altmathcal{S}_{m}|}(\mathbb{R}G_{m})$
into
$\mathbb{M}_{|\altmathcal{S}_{n}|\times|\altmathcal{S}_{n}|}(\mathbb{R}G_{n})$.
We show first that

(3)
$\displaystyle(n-2)!m(m-1)I_{|\altmathcal{S}_{n}|}=\sum_{\sigma\in\operatorname{Sym}_{n}}\sigma
I_{|\altmathcal{S}_{m}|}.$

Note that the right-hand side of the expression above is a multiplicity of the
identity matrix. Let $\omega_{0}$ be $1$ in the case
$G_{n}=\operatorname{SL}_{n}(\mathbb{Z})$ and $2$ in the case
$G_{n}=\operatorname{SAut}(F_{n})$. Then each summand $\sigma
I_{|\altmathcal{S}_{m}|}$ contributes $|\altmathcal{S}_{m}|=m(m-1)\omega_{0}$
neutral elements to the diagonal which yields $n!m(m-1)\omega_{0}$ neutral
elements altogether. Since the diagonal of $I_{|\altmathcal{S}_{n}|}$ has
$|\altmathcal{S}_{n}|=n(n-1)\omega_{0}$ neutral elements, the multiplication
factor is equal to $(n-2)!m(m-1)$.

Suppose $\operatorname{Adj}_{m}-\lambda I_{|\altmathcal{S}_{m}|}$ is a sum of
squares. Combining Lemma Lemma 5.4 and (3), we conclude that the following
matrix is a sum of squares in
$\mathbb{M}_{|\altmathcal{S}_{n}|\times|\altmathcal{S}_{n}|}(\mathbb{R}G_{n})$:

$m(m-1)(m-2)(n-3)!\operatorname{Adj}_{n}-(n-2)!m(m-1)\lambda
I_{|\altmathcal{S}_{n}|}.$

Dividing the expression above by $(n-3)!m(m-1)$, we get the assertion. ∎

##  6\. Application to $G_{n}=\operatorname{SL}_{n}(\mathbb{Z})$

In this section, let $G_{n}=\operatorname{SL}_{n}(\mathbb{Z})$ for any $n\geq
3$. Using the algorithm for estimating lower bounds for spectral gaps for
matrices over group rings described in [KMN24], we were able to obtain the
following

######  Lemma 6.1.

The matrix $\operatorname{Adj}_{3}-0.217I_{6}$ is a sum of squares in
$\mathbb{M}_{6\times 6}(\mathbb{R}\operatorname{SL}_{3}(\mathbb{Z}))$.

The replication details of the computation justifying the result above have
been desribed in subsection 6.1.

######  Corollary 6.2 (cf. Corollary 1.2).

Let $\Delta_{1}$ be the cohomological Laplacian in degree one of $G_{n}$. Then
$\Delta_{1}-\lambda I_{|\altmathcal{S}_{n}|}$ is a sum of squares in
$\mathbb{M}_{|\altmathcal{S}_{n}|\times|\altmathcal{S}_{n}|}(\mathbb{R}G_{n})$
for $\lambda=0.217(n-2)$.

###### Proof.

Lemma 6.1 together with Corollary 5.5 show that
$\operatorname{Adj}_{n}-0.217(n-2)I_{|\altmathcal{S}_{n}|}$ is a sum of
squares in
$\mathbb{M}_{|\altmathcal{S}_{n}|\times|\altmathcal{S}_{n}|}(\mathbb{R}G_{n})$.
It follows by Lemma 4.3 that the matrices $\operatorname{Sq}_{n}^{-}$ and
$\widetilde{\operatorname{Op}}_{n}=\operatorname{Op}_{n}^{+}+2\operatorname{Op}_{n}^{-}$
are sums of squares. From this we conclude that
$\Delta_{1}-0.217(n-2)I_{|\altmathcal{S}_{n}|}$ is a sum of squares in
$\mathbb{M}_{|\altmathcal{S}_{n}|\times|\altmathcal{S}_{n}|}(\mathbb{R}G_{n})$:

$\Delta_{1}-0.217(n-2)I_{|\altmathcal{S}_{n}|}=\frac{1}{2}\left(2\left(\operatorname{Adj}_{n}-0.217(n-2)I_{|\altmathcal{S}_{n}|}\right)+\widetilde{\operatorname{Op}}_{n}+\operatorname{Op}_{n}^{+}+2\operatorname{Sq}_{n}^{-}\right).$

∎

###  6.1. Replication details for $\operatorname{SL}_{3}(\mathbb{Z})$

The arguments for converting the numerical sum of squares approximation to the
actual one have been described in subsection 3.2 of [KMN24]. Further details
concerning finding the numerical approximation can be also found in [KMN24].

We provide a Julia [BEKS17] code to replicate the result of Lemma 6.1. The
replication has been described at [KM22], in the ”README.md” file, in the
section marked with the arXiv identifier of this preprint. Note that, in order
to access the version of the repository corresponding to this article,
switching to the branch marked with its arXiv identifier is necessary. The
script ”SL_3_Z_adj.jl” contains the replication code for the result of Lemma
6.1. We encourage, however, to use the precomputed solution and run the script
”SL_3_Z_adj_cert.jl” contained in the ”sl3_adj_precom” folder. It should take
approximately $1$ minute on a standard laptop computer to run the
certification script providing the rigorous mathematical proof (the script
containing the whole computation takes in turn about $2$ hours). The
hermitian-squared matrices from the decomposition obtained in Lemma 6.1 have
entries supported on the ball of radius $2$ with respect to the word-length
metric defined by the generating set $\altmathcal{S}_{3}$ consisting of six
elementary matrices $E_{12}$, $E_{13}$, $E_{21}$, $E_{23}$, $E_{31}$, and
$E_{32}$.

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