# Charged Lepton EDM with Extra Yukawa Couplings

Wei-Shu Hou, Girish Kumar and Sven Teunissen  Department of Physics, National
Taiwan University, Taipei 10617, Taiwan

###### Abstract

In a two Higgs doublet model with extra Yukawa couplings, we assess the new
physics contributions to charged lepton electric dipole moment. We focus
especially on muon (and tau) EDM where in the coming decade several
experiments — Muon g-2, J-PARC and PSI (and Belle II) — will push
sensitivities down by several orders of magnitude. With the working assumption
that extra Yukawa couplings are analogous to SM ones in strength and taking
exotic scalar masses in sub-TeV range, we find that $\mu$ and $\tau$ EDM can
be enhanced to values larger than new physics scenarios that scale with lepton
mass. The main effect comes from the flavor-conserving extra top coupling
$\rho_{tt}$ via well-known two-loop diagrams. Deviating from our working
assumption, if the muon $g-2$ anomaly arises from the one-loop diagram, driven
by singly enhanced lepton flavor violating $\rho_{\tau\mu}$ coupling, it can
also induce rather large muon EDM, accessible at upcoming experiments.

##  I Introduction

All charged-parity violation (CPV) observed in the laboratories so far are in
the quark sector. The Cabibbo-Kobayashi-Maskawa (CKM) matrix Cabibbo:1963yz ;
Kobayashi:1973fv  that govern the weak interactions of the Standard Model (SM)
has been impressively successful in accounting for all such effects. But, as
is known, the KM phase does not provide enough CPV to satisfy the Sakharov
condition Sakharov:1967dj  for the disappearance of antimatter in the
Universe. Various new physics (NP) models attempt to make up for this by
supplying new complex phases to hopefully explain the baryon asymmetry of the
Universe (BAU).

The general two Higgs doublet model (g2HDM) Branco:2011iw ; Davidson:2005cw ,
also known as 2HDM Type-III Hou:1991un , is attractive in this context. In
g2HDM, one does not introduce a $Z_{2}$ symmetry to enforce Natural Flavor
Conservation Glashow:1976nt . Thus, the second Higgs doublet brings in extra
Yukawa coupling matrices, which are in general complex. The new phases,
especially those associated with extra top quark couplings, $\rho_{tt}$ and
$\rho_{tc}$, which are likely the stronger in strength, can drive electroweak
baryogengesis (EWBG) and explain BAU Fuyuto:2017ewj ; Fuyuto:2019svr . Complex
extra Yukawa couplings can also contribute to CPV observables accessible in
the laboratory, such as charged lepton electric dipole moment ($\ell$EDM),
which we denote as $d_{\ell}$.

Lepton EDMs are powerful probes of NP CPV. The SM predictions for $\ell$EDM
are extremely tiny (e.g. $|d_{e}|\sim 10^{-38}\;e$ cm Pospelov:2005pr ;
Pospelov:2013sca ; Smith:2017dtz ) because the sole complex phase, the KM
phase, survives only at the four-loop level. With essentially no SM
background, detection of $\ell$EDM will establish the presence of NP of CPV
nature.

Recently, there has been very good progress in the search for $e$EDM. The ACME
experiment exploits the large internal electric field of ThO molecules to
measure $e$EDM. Already in 2013, ACME provided the best upper limit on $e$EDM,
$|d_{e}|<8.7\times 10^{-28}$ $e$ cm ACME:2013pal . Impressively, the bound was
further improved to ACME:2018yjb

$\displaystyle|d_{e}|<1.1\times 10^{-29}\;e\,{\rm cm},~{}~{}\;({\rm
ACME\,2018})$ (1)

which is the best upper limit on any EDM. The bound of Eq. (1) puts stringent
constraint on any BSM model that seek to address the BAU problem.

For g2HDM, consistency with Eq. (1) has helped shed light on the structure of
extra Yukawa couplings. As noted in Ref. Fuyuto:2019svr , one can accommodate
EWBG in g2HDM with flavor conserving coupling $|\rho_{tt}|\gtrsim 0.1$ (see
Ref. Fuyuto:2017ewj  for flavor violating coupling $\rho_{tc}$ which is immune
to $e$EDM constraint), but parameter space with single coupling scenario runs
into trouble with the ACME 2018 bound. This can be avoided by introducing
extra electron Yukawa coupling $\rho_{ee}$, which helps salvage parameter
space favorable for EWBG. Interestingly, the organization of these couplings
follow the pattern Fuyuto:2019svr ,

$\displaystyle|\rho_{ee}/\rho_{tt}|\propto\lambda_{e}/\lambda_{t},$ (2)

suggesting that the extra $\rho$ couplings of g2HDM have similar hierarchical
structure as Yukawa couplings in SM. Eq. (2) provides not only an elegant way
of complying with the ACME 2018 bound, it connects EWBG with tiny CPV effects
in the laboratories, such as $e$EDM.

Seeing that $e$EDM is highly sensitive to extra Yukawa couplings, in this
paper we wish to explore the detailed contributions of extra Yukawa couplings
to $\mu$ and $\tau$ EDM. Such an investigation is well-timed for the muon,
where the Fermilab Muon g-2 experiment Muong-2:2021ojo  has affirmed recently
the significant deviation from SM expectation in a related observable — the
anomalous magnetic moment of the muon. Any NP in muon $g-2$ certainly can have
important implications for $\mu$EDM, as we will show later.

The current bound on $\mu$EDM is by the Muon g-2 experiment at Brookhaven
National Laboratory (BNL) Muong-2:2008ebm ,

$\displaystyle|d_{\mu}|<1.8\times 10^{-19}\;e\,{\rm cm},~{}~{}\;({\rm
BNL\,2008})$ (3)

while an indirect limit, extracted from $e$EDM bound of ACME 2018, is claimed
Ema:2021jds  at $\sim 2\times 10^{-20}\;e\,{\rm cm}$. In the coming years,
several experiments will search for $\mu$EDM and improve Eq. (3). The Fermilab
Muon g-2 Chislett:2016jau  and the J-PARC muon g-2/EDM Abe:2019thb
experiments project sensitivities at $\sim 10^{-21}\;e$ cm. Another experiment
being planned at PSI Adelmann:2021udj  aims at improving by at least another
order of magnitude, eventually reaching a sensitivity of $6\times 10^{-23}\;e$
cm. Thus, prospects for $\mu$EDM in the coming decade appears promising.

lepton ($\ell$) |  Current bound ($e$ cm) |  Future sensitivity ($e$ cm)
---|---|---
electron |  $1.1\times 10^{-29}$ (ACME ACME:2018yjb ) |  -volatile-
muon |  $1.8\times 10^{-19}$ (BNL Muong-2:2008ebm ) |  $\sim 10^{-21}$ (FNAL Chislett:2016jau , J-PARC Abe:2019thb )
|  $\sim 2\times 10^{-20}$ (Pospelov Ema:2021jds ) |  $\sim 6\times 10^{-23}$ – (PSI Adelmann:2021udj )
tau |  $\operatorname{Re}d_{\tau}=(-0.62\pm 0.63)\times 10^{-17}$ (Belle Belle:2021ybo ) |  $\sim 10^{-18}{\rm-}10^{-19}$ (Belle II Belle-II:2018jsg )
|  |  $\sim 0.7\times 10^{-19}$ (Bernreuther Bernreuther:2021elu )
|  $\operatorname{Im}d_{\tau}=(-0.40\pm 0.32)\times 10^{-17}$ (Belle Belle:2021ybo ) |  $\sim 10^{-18}{\rm-}10^{-19}$ (Belle II Belle-II:2018jsg )
|  |  $\sim 0.4\times 10^{-19}$ (Bernreuther Bernreuther:2021elu )
Table 1: Present bounds and future sensitivities for EDM of charged leptons,
$|d_{\ell}|$.

Compared to $e$EDM and $\mu$EDM, $\tau$EDM measurement is hampered by the
short $\tau$ lifetime. The present limits are derived from spin-momentum
correlations of final state decay products in
$e^{+}e^{-}\to\gamma^{\ast}\to\tau^{+}\tau^{-}$, where $\tau$EDM enters the
matrix element of the $\gamma\tau\tau$ vertex. An old limit of the Belle
experiment Belle:2002nla  with $29.5$ fb-1 was very recently updated
Belle:2021ybo  to full dataset of 833 fb-1. These limits on the real and
imaginary parts of $d_{\tau}$, at $\sim 10^{-17}\;e$ cm, can be read off Table
1. An indirect bound extracted from $e$EDM Ema:2021jds  seems much better, but
real and imaginary parts were not separately given. Belle II Belle-II:2018jsg
should improve the bound by more than an order of magnitude. Furthermore, some
improvement is possible by studying “optimal” observables Bernreuther:2021elu
.

If one assumes that the NP contribution scales as $m_{\ell}$, such as models
with DAmbrosio:2002vsn  minimal flavor violation (MFV), one expects
$|d_{\mu}/d_{e}|\simeq m_{\mu}/m_{e}$. Taking the current limit on $e$EDM of
Eq. (1), $d_{\mu}$ is expected at only $\sim 2\times 10^{-27}\;e$ cm. A glance
at Table 1 shows that future sensitivities fall short by more than four orders
of magnitude, and it is even farther away for $d_{\tau}$. But this $m_{\ell}$
scaling may not hold in NP models where the underling flavor dynamics is quite
different Hiller:2010ib ; Crivellin:2018qmi , and the situation need not be as
pessimistic. The g2HDM provides a simple example where extra Yukawa couplings
can give much larger $d_{\mu}$ and $d_{\tau}$. We will show that, in a
scenario favored by the muon $g-2$ anomaly, very large $d_{\mu}$ that can be
probed in the coming years is possible.

This paper is organized as follows. In the next section, we give the extra
Yukawa interactions in g2HDM and our working assumptions for their strength.
We then discuss $\mu$EDM in Sec. III and present results for the one- and two-
loop contributions, while in Sec. IV we analyze $\tau$EDM in a similar
fashion. In Sec. V, we discuss extra Yukawa couplings that violate our working
assumptions, as possibly hinted by muon $g-2$, and discuss implications for
EDM of all three charged leptons. Finally, we offer our conclusion in Sec. VI.

##  II Preliminaries

Before we proceed to the specifics, let us lay out the framework of the model.
The Yukawa interactions in g2HDM are given by Davidson:2005cw ; Hou:2017hiw ,

$\displaystyle\mathcal{L}=-$
$\displaystyle\frac{1}{\sqrt{2}}\sum_{f=u,d,\ell}\bar{f}_{i}\Big{[}\big{(}\lambda^{f}_{i}\delta_{ij}s_{\gamma}+\rho^{f}_{ij}c_{\gamma}\big{)}h$
$\displaystyle+\big{(}\lambda^{f}_{i}\delta_{ij}c_{\gamma}-\rho^{f}_{ij}s_{\gamma}\big{)}H-i\,{\rm
sgn}(Q_{f})\rho^{f}_{ij}A\Big{]}R\,f_{j}$
$\displaystyle-\bar{u}_{i}\left[(V\rho^{d})_{ij}R-(\rho^{u\dagger}V)_{ij}L\right]d_{j}H^{+}$
$\displaystyle-\bar{\nu}_{i}\rho^{\ell}_{ij}R\,\ell_{j}H^{+}+{h.c.},$ (4)

where $L\,(R)=(1\mp\gamma_{5})/2$ are projection operators, $Q_{f}$ is the
charge of fermion $f$ in units of $e$, and $V$ is the CKM matrix. The
shorthand $c_{\gamma}\equiv\cos\gamma$ and $s_{\gamma}\equiv\sin\gamma$
represent mixing between two CP even scalars, $h$ and $H$, which is known to
be very small.

Although $\rho_{ij}$ couplings are generic and do not possess information
about SM Yukawa couplings $\lambda_{i}$, it is intriguing that data suggest
the $\rho$ matrices imitate closely the hierarchical structure of SM Yukawa
matrices (see Eq. (2)), as already mentioned in the Introduction. The $\Delta
F=2$ processes such as $B_{s(d)}-\overline{B}_{s(d)}$ and $K^{0}-\bar{K}^{0}$
mixing constrain off-diagonal entries of down-type $\rho^{d}$ to very small
values Crivellin:2013wna , which indicates that $\rho^{d}$ is nearly diagonal.
On the other hand, the extra top Yukawa coupling $\rho_{tt}$, important for
our study, is allowed to be significant in strength Hou:2020chc . For the
lepton sector, the $\rho_{\ell\ell^{\prime}}$ coupling related to third and
second generation are tightly constrained from LFV decays $h\to\tau\mu$ and
$\tau\to\mu\gamma$, but with $\rho_{tt}\sim 1$, values of
$\rho_{\tau\mu}=\rho_{\mu\tau}\lesssim{\cal O}(\lambda_{\tau})$ are easily
allowed even for $c_{\gamma}\sim 0.1$ Hou:2020tgl .

With these arguments in mind, we assume the following strengths for
$\rho_{ij}$ related to $u$\- and $\ell$-type matrices Hou:2020itz ,

$\displaystyle\rho_{ii}\lesssim{\cal
O}(\lambda_{i});\;\;\rho_{1i}\lesssim{\cal
O}(\lambda_{1});\;\;\rho_{3j}\lesssim{\cal O}(\lambda_{3})\;(j\neq 1),$ (5)

where the first two conditions can also be applied to elements of the
$\rho^{d}$ matrix. Down-type couplings, however, do not play much role in our
discussion, so Eq. (5) will be our working assumption while estimating the
typical size of $\ell$EDM in g2HDM. But we note that deviation from Eq. (5) is
possible. For example, NP in muon $g-2$ may call for large deviation from the
last condition in Eq. (5) for interpretation in 2HDM Hou:2021sfl . We will
discuss in Sec. V the implications of such a scenario for $\ell$EDM and assess
the impact on our results.

It is worth pointing out that with $c_{\gamma}\lesssim 0.1$ our assumptions
$\rho_{\mu\mu}\lesssim{\cal O}(\lambda_{\mu})$ and
$\rho_{\tau\tau}\lesssim{\cal O}(\lambda_{\tau})$ are easily compatible with
the latest data on di-muon and di-tau decays of the $h$ boson, respectively.
These processes provide access to the strength and phase of extra Yukawa
couplings. Both ATLAS ATLAS:2020fzp  and CMS CMS:2020xwi  have reported the
$h\to\mu\mu$ branching fraction relative to SM at $1.2\pm 0.6$ and
$1.19^{+0.40+0.15}_{-0.39-0.14}$, respectively. The $h\mu\mu$ coupling in Eq.
(4) induces the following modification in g2HDM,

$\displaystyle\frac{{\cal B}(h\to\mu\mu)}{{\cal B}(h\to\mu\mu)_{\rm
SM}}=\left|s_{\gamma}+c_{\gamma}\operatorname{Re}\frac{\rho_{\mu\mu}}{\lambda_{\mu}}\right|^{2}+\left|c_{\gamma}\operatorname{Im}\frac{\rho_{\mu\mu}}{\lambda_{\mu}}\right|^{2}.$
(6)

For $c_{\gamma}=0.1$ and
$\operatorname{Re}\rho_{\mu\mu}=\operatorname{Im}\rho_{\mu\mu}=\lambda_{\mu}/\sqrt{2}$,
Eq. (6) gives $\simeq 1.14$, consistent with data.

The $h\to\tau\tau$ process provides an avenue for the study of CP properties
of the $h$ boson at the LHC (see e.g. Refs. DellAquila:1988bko ;
Harnik:2013aja ). Very recently, CMS CMS:2020rpr  has carried out a study of
the CP structure of the $h\tau\tau$ coupling with $137$ fb-1 data, reporting
the phase of the $\tau$ Yukawa coupling, $\phi_{\tau\tau}$, at $4^{\circ}\pm
17^{\circ}(\pm 36^{\circ})$ at the $68\%\,(95\%)$ C.L., with rather large
errors. In the notation of Ref. CMS:2020rpr , the $h\tau\tau$ coupling in Eq.
(4) gives,

$\displaystyle\tan\phi_{\tau\tau}=\frac{c_{\gamma}\operatorname{Im}\rho_{\tau\tau}}{s_{\gamma}\lambda_{\tau}+c_{\gamma}\operatorname{Re}\rho_{\tau\tau}}.$
(7)

For $c_{\gamma}=0.1$ with
$\operatorname{Re}\rho_{\tau\tau}=\operatorname{Im}\rho_{\tau\tau}=\lambda_{\tau}/\sqrt{2}$,
we find $\phi_{\tau\tau}\simeq 3.8^{\circ}$, which is compatible with data. A
more precise measurement of $\phi_{\tau\tau}$ in the future could constrain
this extra $\tau$ Yukawa coupling.

Figure 1: One-loop digram for $\mu$EDM.

##  III $\mu$EDM

The EDM, $d_{\ell}$, of lepton $\ell$ is defined through the effective
Lagrangian,

$\displaystyle\mathcal{L}_{\mathrm{eff}}^{\rm
EDM}=-\frac{i}{2}d_{\ell}\,\bar{\ell}\,\sigma_{\alpha\beta}\gamma_{5}\ell\,F^{\alpha\beta},$
(8)

where $\sigma_{\alpha\beta}=i[\gamma_{\alpha},\gamma_{\beta}]/2$, and
$F^{\alpha\beta}$ is the electromagnetic field strength tensor.

In g2HDM, the first finite contribution to $\ell$EDM appears at one-loop. A
sample one-loop diagram for $\mu$EDM in g2HDM is given in Fig. 1. The dipole
operator in Eq. (8) is chirality violating, so an additional mass insertion is
required on the fermion line to obtain the correct chiral structure. This
means that analogous diagrams with lighter leptons in the loop are chirally
suppressed, and we neglect them. Fig. 1 gives Omura:2015xcg ,

$\displaystyle d_{\mu}|_{\rm 1-loop}\simeq$
$\displaystyle-e\frac{m_{\tau}}{32\pi^{2}}\operatorname{Im}(\rho_{\tau\mu}\rho_{\mu\tau})\left[c_{\gamma}^{2}\,\frac{\log
x_{h\tau}-3/2}{m_{h}^{2}}\right.$
$\displaystyle+\left.s^{2}_{\gamma}\,\frac{\log
x_{H\tau}-3/2}{m_{H}^{2}}-\frac{\log x_{A\tau}-3/2}{m_{A}^{2}}\right],$ (9)

where $x_{ij}=m_{i}^{2}/m_{j}^{2}$.

We note that the contribution of the $h$ boson is negligible in Eq. (9)
because $c_{\gamma}^{2}\ll 1$, while the contributions of CP even and odd
scalars have a relative sign between them, indicating that in the $H$–$A$
degeneracy limit, one-loop contributions will be highly suppressed, which is
indeed the case. For example, taking $c_{\gamma}=0.1$ and $m_{H}=m_{A}=$ 300
GeV, we get $|d_{\mu}|\simeq 4\times
10^{-22}|\operatorname{Im}(\rho_{\tau\mu}\rho_{\mu\tau})|\;e$ cm. With
$|\rho_{\tau\mu}|=|\rho_{\mu\tau}|\lesssim\lambda_{\tau}$, Eq. (5), we obtain
$|d_{\mu}|\lesssim 4\times 10^{-26}\;e$ cm at one-loop, which is three orders
of magnitude below PSI sensitivity. With $m_{A}$ and $m_{H}$ sufficiently
apart, the one-loop contribution increases, but it remains outside of
experimental reach. For example, with $m_{H}=300$ GeV and $m_{A}=500$ GeV and
keeping other values as before, we obtain $|d_{\mu}|\lesssim 7\times
10^{-25}\;e$ cm. Thus, we find that our working assumption of
$\rho_{\ell\ell^{\prime}}$ in Eq. (5) renders the one-loop contribution too
small. Next, we consider two-loop diagrams — also known as Barr-Zee or
Bjorken-Weinberg diagrams Barr:1990vd ; Bjorken:1977vt .

It is well known that, despite having additional loop suppression, certain
two-loop diagrams can dominate over one-loop diagrams. This can be understood
qualitatively by noting that Fig 1 contains three chirality flips while only
one flip in Fig. 2, which can let the latter compensate for the additional
loop factor ($\sim\alpha/\pi$). Barr and Zee Barr:1990vd  calculated neutral
scalar contributions with top quark and gauge bosons in the loop, which was
later extended Leigh:1990kf ; Chang:1990sf ; Kao:1992jv ; Bowser-Chao:1997kjp
; Abe:2013qla  to include other two-loop diagrams.

Figure 2:  Typical two-loop diagram for $\mu$EDM, where $\phi$ ($H^{\pm}$) is
a neutral (charged) scalar, and the gray blob stands for fermion, gauge, or
scalar loop depending on the attached legs.

Following the convention used in Ref. Fuyuto:2019svr , we categorize two-loop
contributions to $\mu$EDM into three parts,

$d_{\mu}|_{\rm 2-loop}=d_{\mu}^{\phi\gamma}+d_{\mu}^{\phi
Z}+d_{\mu}^{H^{+}W^{+}},$ (10)

related to effective $\phi\gamma\gamma$, $\phi\gamma Z$ and $H^{+}\gamma
W^{+}$ vertices, respectively, as in Fig 2.

For $d_{\mu}^{\phi\gamma}$ with fermions in the loop, the dominant
contribution comes from the top quark. For convenience, it is written as the
sum of two terms Fuyuto:2019svr ,

$(d_{\mu}^{\phi\gamma})_{t}=(d_{\mu}^{\phi\gamma})_{t}^{\rm
mix}+(d_{\mu}^{\phi\gamma})_{t}^{\rm extra}$ (11)

where

$\displaystyle\left(d_{\mu}^{\phi\gamma}\right)_{t}^{\rm
mix}=\frac{e\,\alpha\,s_{\gamma}\,c_{\gamma}}{6\sqrt{2}\pi^{3}v}\Bigl{[}\operatorname{Im}\rho_{\mu\mu}\,\Delta
f(x_{th},x_{tH})\Bigr{.}$ $\displaystyle\hskip
64.87224pt+\Bigl{.}\frac{m_{\mu}}{m_{t}}\operatorname{Im}\rho_{tt}\,\Delta
g(x_{th},x_{tH})\Bigr{]},$ (12)
$\displaystyle\left(d_{\mu}^{\phi\gamma}\right)_{t}^{\rm
extra}=\frac{e\,\alpha}{12\pi^{3}m_{t}}$ $\displaystyle\hskip
8.5359pt\times\Bigl{\\{}\operatorname{Im}\rho_{\mu\mu}\operatorname{Re}\rho_{tt}\left[c_{\gamma}^{2}f(x_{th})+s_{\gamma}^{2}f(x_{tH})+g(x_{tA})\right]\Bigr{.}$
$\displaystyle\hskip
14.22636pt+\Bigl{.}\operatorname{Re}\rho_{\mu\mu}\operatorname{Im}\rho_{tt}\left[c_{\gamma}^{2}g(x_{th})+s_{\gamma}^{2}g(x_{tH})+f(x_{tA})\right]\Bigr{\\}}.$
(13)

where $\Delta f(a,b)=f(a)-f(b)$, $\Delta g(a,b)=g(a)-g(b)$, and loop functions
$f(x)$ and $g(x)$ are given in Appendix A. Note that for $x\sim 1$,
$f(x),g(x)\sim x$, and for small x, $f(x),g(x)\sim x(\log x)^{2}/2$, which
further exemplifies why the top quark diagram is dominant.

Eq. (12) is due to mixing of SM Yukawa couplings with extra Yukawa couplings,
and therefore has a linear dependence on $\operatorname{Im}\rho_{ii}$ with no
dependance on $m_{A}$. On the other hand, Eq. (13) contains contributions from
extra Yukawa couplings only. For very small $c_{\gamma}$, the “mix” term
becomes negligible, and the top loop contribution
$(d_{\mu}^{\phi\gamma})_{t}\simeq(d_{\mu}^{\phi\gamma})_{t}^{\rm extra}$ is
governed by Eq. (13). One also notes that, for $\rho_{\mu\mu}$ and $\rho_{tt}$
both purely real or purely imaginary, Eq. (13) does not contribute, which
yields $(d_{\mu}^{\phi\gamma})_{t}\simeq(d_{\mu}^{\phi\gamma})_{t}^{\rm mix}$.

Figure 3:  Different two-loop contributions to $|d_{\mu}|$ as function of
$\rho_{tt}$, with $c_{\gamma}=0.1$, $m_{H}=m_{A}=m_{H^{\pm}}=300$ GeV, and
$\rho_{\mu\mu}=i\lambda_{\mu}$ for illustration. Red (black) curves correspond
to $\phi_{tt}=0\,(\pi/2)$. Solid, dashed, and dot-dashed curves represent
contributions of $|d_{\mu}^{\phi\gamma}|$, $|d_{\mu}^{H^{+}W^{+}}|$, and
$|d_{\mu}^{\phi Z}|$, respectively.

The diagram with $W^{+}$ in the loop induces contribution to
$d_{\mu}^{\phi\gamma}$ for finite $c_{\gamma}$. Defining the $\phi W^{+}W^{-}$
coupling as $i\,gm_{W}C_{\phi WW}$ Gunion:2002zf ; Davidson:2010xv , where
$C_{hWW}=s_{\gamma}$, $C_{HWW}=c_{\gamma}$, and $C_{AWW}=0$ , the diagram
gives,

$\displaystyle\left(d_{\mu}^{\phi\gamma}\right)_{W}=-\frac{e\,\alpha
s_{\gamma}c_{\gamma}}{32\sqrt{2}\pi^{3}v}\operatorname{Im}\rho_{\mu\mu}\left[I_{\gamma}(m_{h})-I_{\gamma}(m_{H})\right],$
(14)

with loop function $I_{\gamma}(a)$ given in Appendix A. This diagram has sign
opposite the top-loop, hence large cancellation can occur in some parameter
space for finite $c_{\gamma}$. The cancellation is, however, sensitive to
complex phases of extra Yukawa couplings.

Adding Eqs. (11) and (14) and taking $c_{\gamma}=0.1$, $m_{H}=m_{A}=300$ GeV,
we illustrate $d_{\mu}^{\phi\gamma}$ as solid lines in Fig. 3 vs $\rho_{tt}$
coupling. We take $|\rho_{\mu\mu}|=\lambda_{\mu}$ in accordance of Eq. (5) and
assume it to be purely imaginary. The red (black) lines correspond to purely
real (imaginary) $\rho_{tt}$. For $\phi_{tt}=0$, as $\rho_{tt}$ increases,
top-loop diagram starts to compete with $W$-loop, resulting in cancellation
between the two, seen as the dip in the plot. For larger values of
$\rho_{tt}$, the top-loop overwhelms the $W$-loop, thereby bringing
$d_{\mu}^{\phi\gamma}$ to $\sim 10^{-24}\;e$ cm. For $\phi_{tt}=\pi/2$,
$(d_{\mu}^{\phi\gamma})_{t}$ is given by Eq. (12) only, where the $\rho_{tt}$
term is suppressed further by $m_{\mu}$, hence $(d_{\mu}^{\phi\gamma})_{t}$ is
feebly sensitive to $\rho_{tt}$ and never overpowers $W$-loop even for large
$|\rho_{tt}|\sim 1$.

For $d_{\mu}^{\phi Z}$, one simply replaces the inner photon propagator (one
connecting loop with muon line) with the $Z$ propagator. By parity, only the
vector part of $Z\mu\mu$ coupling contributes. Its smallness, together with
$1/m_{Z}^{2}$ from the $Z$ propagator, the $d_{\mu}^{\phi Z}$ effect is
typically suppressed with respect to $d_{\mu}^{\phi\gamma}$. The corresponding
formulae are relegated to Appendix A. In Fig. 3, $d_{\mu}^{\phi Z}$ are given
as dot-dashed lines, with color coding as before. For both red and black
lines, $|d^{\phi Z}_{\mu}|$ always lie far lower than
$|d^{\phi\gamma}_{\mu}|$, except near cancellation regions of
$|d^{\phi\gamma}_{\mu}|$.

On the other hand, complex extra Yukawa couplings of charged Higgs boson in
Eq. (4) contribute significantly to $d_{\mu}^{H^{+}W^{+}}$. The main effect
comes from top-bottom in the loop, which gives Bowser-Chao:1997kjp ,

$\displaystyle(d_{\mu}^{H^{+}W^{+}})_{t/b}$
$\displaystyle=-\frac{3e\alpha|V_{tb}|^{2}\operatorname{Im}(\rho_{tt}\rho_{\mu\mu})}{128\pi^{3}m_{t}\sin^{2}\theta_{W}}$
$\displaystyle\hskip-5.69046pt\times\Bigl{[}Q_{t}F_{t}(x_{\phi
t},x_{Wt})+Q_{b}F_{b}(x_{\phi t},x_{Wt})\Bigr{]},$ (15)

where $\phi$ stands for $H^{+}$ only, and functions $F_{q}(a,b)$ are given in
Appendix A.

Diagrams with $W^{+}$ and scalars in the loop were calculated in Ref.
Abe:2013qla . Adapting to our case, we find,

$\displaystyle(d_{\mu}^{H^{+}W^{+}})_{W}$ $\displaystyle=-\,\frac{e\,\alpha
s_{\gamma}c_{\gamma}\operatorname{Im}\rho_{\mu\mu}}{128\sqrt{2}\pi^{3}\,v\sin^{2}\theta_{W}}$
$\displaystyle\hskip-5.69046pt\times\Bigl{[}{\cal
I}_{4}(m_{h},m_{H^{+}})-{\cal I}_{4}(m_{H},m_{H^{+}})\Bigr{]},$ (16)

where ${\cal I}_{4}(a,b)$ is given in Appendix A.

Combining Eqs. (15) and (16), we give $d_{\mu}^{H^{+}W^{+}}$ in Fig. 3 as
dashed lines, taking the same values of scalar masses and $c_{\gamma}$ as
before. For $\phi_{tt}=0$ (red), $d_{\mu}^{H^{+}W^{+}}$ is basically governed
by Eq. (15) and slightly lower than $d_{\mu}^{\phi\gamma}$, but significantly
larger than $d_{\mu}^{\phi W}$. At $\rho_{tt}=0$, the origin,
$d_{\mu}^{H^{+}W^{+}}$ is given by Eq. (16), which lies between
$10^{-27}$–$10^{-26}\;e$ cm. For extremely tiny $\rho_{tt}$, the effects from
Eqs. (15) and (16) tend to cancel. This behavior is similar to the case of
$d_{\mu}^{\phi\gamma}$ discussed previously. For $\phi_{tt}=\pi/2$ (black),
there is no dependence on $\rho_{tt}$ and the total contribution is given by
Eq. (16), which is represented as a horizontal black dashed line.

We see from Fig. 3 that, our working assumption of Eq. (5) suggests $\mu$EDM
in g2HDM can be enhanced to $\sim 10^{-24}\;e$ cm. This, however, lies out of
reach of muon g-2 experiments at FNAL and J-PARC by three orders of magnitude.
Even with improved sensitivity at PSI, shown as cyan band in Fig. 3, it cannot
probe extra Yukawa couplings with strength as given in Eq. (5).

Some comments are in order. There are additional subdominant diagrams related
to charged Higgs loops which depend on trilinear Higgs couplings. The
expressions of these are given in Appendix A. We ignore them since our scope
is to cover effects from Yukawa interactions of Eq. (4). These contribution
can be important in models with CPV scalar potentials, or 2HDM models with
softly broken $Z_{2}$ symmetry, where large contributions such as in Eq. (15)
are negligible (see e.g. Refs. Abe:2013qla ; Inoue:2014nva ; Chun:2019oix ;
Altmannshofer:2020shb ; Egana-Ugrinovic:2018fpy ). For the former case, we
should mention that there are additional so-called “kite diagrams”, calculated
recently in Ref. Altmannshofer:2020shb , which are required to obtain a gauge-
invariant result.

##  IV $\tau$EDM

The diagrams contributing to $\tau$EDM is similar to $\mu$EDM, except a more
significant coupling $\rho_{\tau\tau}\sim\lambda_{\tau}$ is involved, hence
one would expect further enhancement.

One obtains the one-loop effect for $\tau$EDM by replacing $``\mu"\to``\tau"$
in Fig. 1. Diagrams with internal $\mu$ and $e$ are again neglected because of
chiral suppression. Since the flavor conserving $\phi\tau\tau$ vertex is
involved, the SM Yukawa $\lambda_{\tau}$ also contributes via interference
with $\rho_{\tau\tau}$. Using the results of Ref. Crivellin:2013wna , the one-
loop contribution is

$\displaystyle d_{\tau}$ $\displaystyle\simeq-\frac{e\,m_{\tau}}{32\pi^{2}}$
$\displaystyle\,\times\left\\{\left[s_{2\gamma}\lambda_{\tau}\operatorname{Im}\rho_{\tau\tau}+c_{\gamma}^{2}\operatorname{Im}\rho_{\tau\tau}^{2}\right]\frac{\log
x_{h\tau}-3/2}{m_{h}^{2}}\right.$
$\displaystyle\,+\left[-s_{2\gamma}\lambda_{\tau}\operatorname{Im}\rho_{\tau\tau}+s_{\gamma}^{2}\operatorname{Im}\rho_{\tau\tau}^{2}\right]\frac{\log
x_{H\tau}-3/2}{m_{H}^{2}}$ $\displaystyle\hskip
91.6179pt-\operatorname{Im}\rho_{\tau\tau}^{2}\left.\frac{\log
x_{A\tau}-3/2}{m_{A}^{2}}\right\\}.$ (17)

In the small $c_{\gamma}$ limit, $\lambda_{\tau}$ effects decouple. Taking
$c_{\gamma}=0.1$, $m_{H}=m_{A}=300$ GeV, and
$\operatorname{Re}\rho_{\tau\tau}=\operatorname{Im}\rho_{\tau\tau}\leq\lambda_{\tau}/\sqrt{2}$
for $\tau$ coupling following Eq. (5), we find $|d_{\tau}|\lesssim 6\times
10^{-25}\;e$ cm, which is about five orders below Belle II projections. For
$m_{H}=300$ GeV and $m_{A}=500$ GeV, the one-loop contribution increases to
$|d_{\tau}|\lesssim 1.3\times 10^{-24}\;e$ cm, but remains out of Belle II
reach.

The discussion of two-loop contribution to $d_{\tau}$ is analogous to
$d_{\mu}$ in Sec. III. Therefore, similar to Eq. (10), we write for $d_{\tau}$
at two-loop,

$d_{\tau}|_{\rm 2-loop}=d_{\tau}^{\phi\gamma}+d_{\tau}^{\phi
Z}+d_{\tau}^{H^{+}W^{+}},$ (18)

where expressions for each term can be obtained from Eqs. (11) and (14)-(16)
by change of label $``\mu"\to``\tau"$.

Figure 4:  Different two-loop contributions to $|d_{\tau}|$ as function of
$\rho_{tt}$, taking $\rho_{\tau\tau}=i\lambda_{\tau}$. Color coding is the
same as in Fig. 3.

In Fig. 4 we give results for each term in Eq. (18) as functions of
$\rho_{tt}$. The same benchmarks are used as in previous section:
$m_{H}=m_{A}=m_{H^{\pm}}=$ 300 GeV, $c_{\gamma}=0.1$, and we take
$\rho_{\tau\tau}=i\lambda_{\tau}$. Red (black) lines represent
$\phi_{tt}=0\,(\pi/2)$. The characteristics of each type of contribution are
the same as the $\mu$EDM case in Sec. III, except that each contribution is
now larger by $\rho_{\tau\tau}/\rho_{\mu\mu}$. From Fig. 4 and for our
benchmarks, $d_{\tau}$ can reach $\sim 10^{-23}\;e$ cm, but again falls short
of Belle II sensitivity.

Figure 5:  Correlation between muon $g-2$ and $\mu$EDM. Pink (light pink)
region shows parameter space that explain Eq. (19) at $1\sigma\,(2\sigma)$,
while dashed lines are $d_{\mu}$ contours. The left (right) plot is for
$c_{\gamma}=0$, $m_{H}=$ 300 GeV and $m_{A}=340$ (500) GeV. All other
$\rho_{ij}$ are set to zero.

##  V Discussion

With our working assumption for $\rho_{ij}$ given in Eq. (5) and sub-TeV
exotic scalars, we find that $\mu$ and $\tau$ EDM can be enhanced in g2HDM,
but are not large enough to be probed in the coming years. However, recent
progress in muon $g-2$ may suggest good prospects for $\mu$EDM.

The Fermilab Muon g-2 experiment Muong-2:2021ojo  reported recently their
first measurement of the muon anomalous magnetic moment,
$a_{\mu}\equiv(g-2)_{\mu}$, confirming the old BNL result Muong-2:2006rrc .
The combined value of $a_{\mu}^{\rm Exp}=116592061(41)\times 10^{-11}$
disagrees with the SM prediction, $a_{\mu}^{\rm SM}=116591810(43)\times
10^{-11}$ Aoyama:2020ynm , by more than $4\sigma$,

$\displaystyle a_{\mu}^{\rm Exp}-a_{\mu}^{\rm SM}=(251\pm 59)\times 10^{-11}.$
(19)

One of the simplest (and easiest, thanks to the chiral enhancement factor of
$m_{\tau}/m_{\mu}\simeq 17$) NP explanation of Eq. (19) is provided by g2HDM
via the same diagram as in Fig. 1. But it would require significant departure
from Eq. (5) for some couplings. As shown in Ref. Hou:2021sfl , to explain Eq.
(19) at $1\sigma$, one needs $|\rho_{\tau\mu}|=|\rho_{\mu\tau}|\simeq{\cal
O}(20)\lambda_{\tau}$, with $m_{H}$ and $m_{A}$ sub-TeV but nondegenerate.

Several implications, relevant for the present paper, follow. First, one has
to adopt Hou:2021sfl  near-perfect alignment, i.e. $c_{\gamma}\to 0$. Though
odd, it is needed to satisfy the CMS limit of ${\cal B}(h\to\tau\mu)<0.15\%$
CMS:2021rsq , which constrains $|\rho_{\tau\mu}c_{\gamma}|<0.1\lambda_{\tau}$
for $\rho_{\tau\mu}=\rho_{\mu\tau}$. This bound implies $c_{\gamma}<0.005$ for
$1\sigma$ solution of muon $g-2$. Another important implication is a rather
small $\rho_{tt}$. The decay $\tau\to\mu\gamma$ arises from Fig. 2 with
initial $\mu$ replaced by $\tau$, which constrains the product
$\rho_{\tau\mu}\rho_{tt}$. To comply with the Belle bound of ${\cal
B}(\tau\to\mu\gamma)<4.2\times 10^{-8}$ Belle:2021ysv , large $\rho_{\tau\mu}$
means small $\rho_{tt}$. Even more surprising Hou:2021sfl  is that the
collider search for $gg\to H,A\to\tau\mu$ provides a much better constraint.
Combining both flavor and LHC constraints, $\rho_{tt}$ seems limited to small
values $\rho_{tt}\lesssim 0.1$, which is small compared with $\lambda_{t}\cong
1$.

Small $\rho_{tt}$ and $c_{\gamma}$ will suppress the two-loop contributions to
$\ell$EDM: the top-loop diagrams by small $\rho_{tt}$, and $W$-loop diagrams
by $c_{\gamma}\to 0$. But large $\rho_{\tau\mu}(\rho_{\mu\tau})$ values can
give rather large $\mu$EDM via the one-loop diagram.

Defining
$\rho_{\tau\mu}\rho_{\mu\tau}=|\rho_{\tau\mu}\rho_{\mu\tau}|\exp(\phi_{CP})$,
we illustrate in Fig. 5 the range of $d_{\mu}$ accessible in the
$|\rho_{\tau\mu}\rho_{\tau\mu}|$-$\phi_{CP}$ plane while simultaneously
accounting for muon $g-2$, Eq. (19). The left plot is for $m_{H}=$ 300 GeV and
$m_{A}=$ 340 GeV, which are not far from degenerate, while the right plot is
for the same $m_{H}$ but a much heavier $m_{A}=$ 500 GeV. We see from both
plots that, given the complexity of $\rho_{\tau\mu}\rho_{\mu\tau}$, the
solution space for muon $g-2$ projects large $d_{\mu}$ values that are well
within PSI sensitivity! In the more optimistic scenario where $m_{H}$ and
$m_{A}$ are farther apart, the parameter space in the right plot may even
appear to be within reach of Fermilab and J-PARC muon g-2 experiments, namely
$\sim 10^{-21}\;e$ cm. We caution, however, that this would demand very large
$|\rho_{\tau\mu}|\sim|\rho_{\mu\tau}|$ values.

On the other hand, one does not expect such extraordinary enhancements for
$d_{\tau}$. In fact, if muon $g-2$ arises from g2HDM as described, then
$d_{\tau}$ will be much smaller than our estimates in Sec. IV. This is because
the two-loop contribution to $d_{\tau}$ in Eq. (18) becomes suppressed by
small $\rho_{tt}$ and $c_{\gamma}$, as argued previously. Furthermore, one-
loop enhancement similar to $\mu$EDM is not possible because simultaneously
large $\rho_{\tau\tau}$ and $\rho_{\tau\mu}(\rho_{\mu\tau})$ would make
$\tau\to\mu\gamma$ too large at one-loop and violate the Belle bound, as we
showed recently in Ref. Hou:2021qmf .

What about $e$EDM? The leading one-loop results ($\tau$ in the loop) for
$e$EDM are given by Eq. (9), replacing $``\mu"\to``e"$. Therefore, at one
loop, $d_{e}$ is proportional to $\operatorname{Im}(\rho_{\tau
e}\rho_{e\tau})$. The $\mu\to e\gamma$ process constrains $\rho_{\tau
e}\,(\rho_{\tau e})$ severely. For values of $\rho_{\tau\mu}$ consistent with
muon $g-2$ solution, the current MEG MEG:2016leq  bound of ${\cal B}(\mu\to
e\gamma)<4.2\times 10^{-13}$ gives $|\rho_{\tau
e}|=|\rho_{e\tau}|\lesssim{\cal O}(\lambda_{e})$ Hou:2021qmf , which is in
line with our working assumption in Eq. (5). Taking
$|\operatorname{Im}(\rho_{\tau e}\rho_{e\tau})|\simeq\lambda_{e}^{2}$,
$c_{\gamma}=0$, and $m_{H}=$ 300 GeV and $m_{A}=$ 500 GeV, we find
$|d_{e}|\lesssim 5\times 10^{-32}\;e$ cm, which is more than two orders of
magnitude below the current ACME bound, $|d_{e}|<1.1\times 10^{-29}\;e$ cm.
For $m_{H}$ closer to $m_{A}$, $d_{e}$ will be further suppressed. The two-
loop results for $d_{e}$ can be obtained by replacing $``\mu"\to``e"$ in the
corresponding expressions for $\mu$EDM in Sec. III. If one assumes
$\rho_{ee}\sim 0$, then two-loop diagrams for $e$EDM is essentially given by
the second term of Eq. (12) (after obvious change of lepton indices), which
depends on $|s_{2\gamma}\operatorname{Im}\rho_{tt}|$ and $m_{H}$. For $m_{H}=$
300 GeV, we find $|d_{e}|\simeq 2\times
10^{-27}|s_{2\gamma}\operatorname{Im}\rho_{tt}|\;e$ cm. With both parameters
constrained to be small, $c_{\gamma}<0.005$ and $|\rho_{tt}|\lesssim 0.1$, it
is rather easy to satisfy the current ACME bound, without resort to the
cancellation mechanism Fuyuto:2019svr  to evade the experimental bound. It
remains to be seen whether parameter space exists where simultaneously small
$\rho_{tt}$ and $c_{\gamma}$ can still provide solutions to BAU in g2HDM.

##  VI Conclusion

We study one- and two-loop mechanisms for charged lepton EDM in g2HDM, the two
Higgs doublet model that possesses extra Yukawa couplings. We find both
$d_{\mu}$ and $d_{\tau}$ can be enhanced to considerably larger values —
thanks to the extra top Yukawa coupling $\rho_{tt}\sim{\cal O}(1)$.

The question for g2HDM is how it has managed to hide so well from scrutiny, if
the exotics Higgs bosons are sub-TeV in mass. This we have elucidated in a
previous work Hou:2020itz , where we developed Eq. (5) as our working
assumption for the strength of extra $\rho^{f}_{ij}$ couplings, namely
$\rho_{ii}\lesssim{\cal O}(\lambda_{i})$, $\rho_{1i}\lesssim{\cal
O}(\lambda_{1})$, and $\rho_{3j}\lesssim{\cal O}(\lambda_{3})$ for $j\neq 1$.
It reflects the mass-mixing hierarchies observed in the SM sector Hou:1991un ,
and finds further support from the cancellation mechanism Fuyuto:2019svr , or
interplay between $\rho_{ee}$ and $\rho_{tt}$ to satisfy the recent ACME 2018
bound on $d_{e}$, while $\rho_{tt}$ still drives baryogenesis. Flavor changing
couplings of $h(125)$ are further suppressed by the recent emergent phenomenon
of alignment, i.e. small $h$–$H$ mixing.

With our working assumption as above and taking central values, we showed that
$d_{\mu}$ lies in the ballpark range of $10^{-24}$ $e$ cm, while $d_{\tau}$ is
in the $10^{-23}$ $e$ cm range. These numbers could go up by another order of
magnitude as allowed by Eq. (5). But though they are much larger than typical
expectations in MFV-like scenarios, with future sensitivities as stated in
Table 1, it is unlikely that $d_{\mu}$ and $d_{\tau}$ would be observed in the
near future. Thus, Eq. (5) continues to “protect” g2HDM from revealing itself,
and we have advocated elsewhere for LHC direct search Hou:2020chc , as well as
several flavor probes Hou:2020itz .

The recent experimental affirmation of the muon $g-2$ “anomaly”, however,
brings in some hope, but at a cost. In g2HDM, the one-loop mechanism can quite
simply account for the discrepancy, if
$|\rho_{\tau\mu}|\simeq|\rho_{\mu\tau}|={\cal O}(20\lambda_{\tau})\sim 0.2$.
This would grossly violate our working assumption of Eq. (5); but Nature
herself is the judge and the jury, and maybe this is what she has been telling
us through the “anomaly”. The point is, the same one-loop diagram can generate
$d_{\mu}$ through the complexity of $\rho_{\tau\mu}\rho_{\mu\tau}$, which can
be tested with projected experimental sensitivities. But this single-enhanced
coupling scenario would then imply quite small $\rho_{tt}\lesssim 0.1$, while
alignment must be near-perfect, which by themselves are rather peculiar. These
in turn suggest that $d_{\tau}$ will be rather suppressed, and $d_{e}$ can be
much below ACME bound without resort to cancellation mechanism.

Acknowledgments This research is supported by MOST 109-2112-M-002-015-MY3,
110-2639-M-002-002-ASP and 110-2811-M-002-620 of Taiwan, and NTU grants
110L104019 and 110L892101.

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## Appendix A

Below we provide Barr-Zee formulae related to $d_{\mu}^{\phi Z}$ and charged
Higgs loop, together with expressions for all the loop functions used for
calculation of EDMs in this paper.

###  A.1 More Barr-Zee formulae

The Barr-Zee formulae for $d_{\mu}^{\phi Z}$ are as follows. Similar to
$d_{\mu}^{\phi\gamma}$, we decompose the top-loop contributions to
$d_{\mu}^{\phi Z}$ as $(d_{\mu}^{\phi Z})_{t}=(d_{\mu}^{\phi Z})_{t}^{\rm
mix}+(d_{\mu}^{\phi Z})_{t}^{\rm extra}$. The expressions for each term are,

$\displaystyle\left(d_{\mu}^{\phi Z}\right)_{t}^{\rm
mix}=\frac{e\,\alpha\,s_{2\gamma}}{256\pi^{3}m_{t}s_{W}^{2}c_{W}^{2}}(1-4\,s^{2}_{W})(1-\tfrac{8}{3}s^{2}_{W})\left\\{\left[\lambda_{t}\operatorname{Im}\rho_{\mu\mu}\left(\frac{m_{h}^{2}f(x_{th})-m_{Z}^{2}f(x_{tZ})}{m_{h}^{2}-m_{Z}^{2}}-h\to
H\right)\right]\right.$
$\displaystyle+\left.\biggl{[}\lambda_{\mu}\operatorname{Im}\rho_{tt}\,\biggl{(}f\to
g\biggr{)}\biggr{]}\right\\},$ (20) $\displaystyle\left(d_{\mu}^{\phi
Z}\right)_{t}^{\rm
extra}=\frac{e\,\alpha}{128\pi^{3}m_{t}\,s_{W}^{2}c_{W}^{2}}(1-4\,s^{2}_{W})(1-\tfrac{8}{3}s^{2}_{W})\left\\{\left[\operatorname{Im}\rho_{\mu\mu}\operatorname{Re}\rho_{tt}\left(c_{\gamma}^{2}\frac{m_{h}^{2}f(x_{th})-m_{Z}^{2}f(x_{tZ})}{m_{h}^{2}-m_{Z}^{2}}\right.\right.\right.$
$\displaystyle\left.\left.\left.+s_{\gamma}^{2}\frac{m_{H}^{2}f(x_{tH})-m_{Z}^{2}f(x_{tZ})}{m_{H}^{2}-m_{Z}^{2}}+\frac{m_{A}^{2}g(x_{tA})-m_{Z}^{2}g(x_{tZ})}{m_{A}^{2}-m_{Z}^{2}}\right)\right]\right.$
$\displaystyle+\left.\biggl{[}\operatorname{Im}\rho_{tt}\operatorname{Re}\rho_{\mu\mu}\,\biggl{(}f\leftrightarrow
g\biggr{)}\biggr{]}\right\\},$ (21)

The $W$-loop contribution is,

$\displaystyle\left(d_{\mu}^{\phi Z}\right)_{W}=-\frac{e\,\alpha
s_{\gamma}c_{\gamma}(1-4s_{W}^{2})}{128\sqrt{2}\pi^{3}v\,s_{W}^{2}}\operatorname{Im}\rho_{\mu\mu}\Bigl{[}I_{Z}(m_{h})-I_{Z}(m_{H})\Bigr{]},$
(22)

and lastly, the charged Higgs loop diagram gives,

$\displaystyle\left(d_{\mu}^{\phi
Z}\right)_{H^{+}}=-\frac{e\,\alpha\,(c_{W}^{2}-s_{W}^{2})(1-4s_{W}^{2})}{256\sqrt{2}\pi^{3}v\,s_{W}^{2}c_{W}^{2}}\operatorname{Im}\rho_{\mu\mu}\Bigl{[}c_{\gamma}C_{hH^{+}H^{-}}I_{3}^{Z}(m_{H}^{+},m_{h})-s_{\gamma}C_{HH^{+}H^{-}}I_{3}^{Z}(m_{H}^{+},m_{H})\Bigr{]},$
(23)

where the trilinear couplings $\phi H^{+}H^{-}$ are defined as $-ivC_{\phi
H^{+}H^{-}}$ in notation of Refs. Gunion:2002zf ; Davidson:2010xv . Note that
$C_{AH^{+}H^{-}}=0$.

The charged Higgs loop contribution for $d_{\mu}^{\phi\gamma}$ and
$d_{\mu}^{H^{\pm}W^{\pm}}$ are,

$\displaystyle\left(d_{\mu}^{\phi\gamma}\right)_{H^{+}}=-\frac{e\,\alpha}{32\sqrt{2}\pi^{3}v}\operatorname{Im}\rho_{\mu\mu}\Bigl{[}c_{\gamma}C_{hH^{+}H^{-}}I_{3}^{\gamma}(m_{H}^{+},m_{h})-s_{\gamma}C_{HH^{+}H^{-}}I_{3}^{\gamma}(m_{H}^{+},m_{H})\Bigr{]},$
(24)
$\displaystyle\left(d_{\mu}^{H^{+}W^{+}}\right)_{H^{+}}=-\frac{e}{256\sqrt{2}\pi^{4}v}\operatorname{Im}\rho_{\mu\mu}\Bigl{[}c_{\gamma}C_{hH^{+}H^{-}}{\cal
I}_{5}(m_{h},m_{H}^{+},)-s_{\gamma}C_{HH^{+}H^{-}}{\cal
I}_{5}(m_{H},m_{H}^{+},)\Bigr{]}.$ (25)

###  A.2 Loop Functions

The loop functions appearing in various Barr-Zee formulae of lepton EDM are
listed below.

The functions appearing in top quark contribution to $d_{\mu}^{\phi\gamma}$
and $d_{\mu}^{\phi Z}$ are Barr:1990vd ,

$\displaystyle
f(a)=a\int_{0}^{1}dz\frac{1/2-z+z^{2}}{z(1-z)-a}\log\frac{z(1-z)}{a},\quad
g(a)=\frac{a}{2}\int_{0}^{1}dz\frac{1}{z(1-z)-a}\log\frac{z(1-z)}{a}.$ (26)

The functions appearing in $W$-loop contribution to $d_{\mu}^{\phi\gamma}$ and
$d_{\mu}^{\phi Z}$ are Abe:2013qla ,

$\displaystyle
I_{V}(m_{\phi})=\frac{m_{W}^{2}}{m_{\phi}^{2}-m_{V}^{2}}\left\\{\left[6-\frac{m_{V}^{2}}{m_{W}^{2}}+\left(1-\frac{m_{V}^{2}}{2m_{W}^{2}}\right)\frac{m_{\phi}^{2}}{m_{W}^{2}}\right]\right.\left(\frac{m_{\phi}^{2}}{m_{W}^{2}}f(m_{W}^{2}/m_{\phi}^{2})-\frac{m_{V}^{2}}{m_{W}^{2}}f(m_{W}^{2}/m_{V}^{2})\right)$
$\displaystyle-\left[-10+\frac{3m_{V}^{2}}{m_{W}^{2}}+\left(1-\frac{m_{V}^{2}}{2m_{W}^{2}}\right)\frac{m_{\phi}^{2}}{m_{W}^{2}}\right]\left.\left(\frac{m_{\phi}^{2}}{m_{W}^{2}}g(m_{W}^{2}/m_{\phi}^{2})-\frac{m_{V}^{2}}{m_{W}^{2}}g(m_{W}^{2}/m_{V}^{2})\right)\right\\}.$
(27)

The functions appearing in $H^{+}$-loop contribution to $d_{\mu}^{\phi\gamma}$
and $d_{\mu}^{\phi Z}$ are Abe:2013qla ,

$\displaystyle
I_{3}^{V}(m_{H}^{+},m_{\phi})=\frac{v^{2}}{m_{\phi}^{2}-m_{V}^{2}}\left\\{\left[\frac{m_{V}^{2}}{m_{H^{+}}^{2}}\Bigl{(}f(m_{H^{+}}^{2}/m_{V}^{2})-g(m_{H^{+}}^{2}/m_{V}^{2})\Bigr{)}\right]-\biggl{[}V\to\phi\biggr{]}\right\\}.$
(28)

The functions appearing in fermion loop contributions to
$d_{\mu}^{H^{+}W^{+}}$ are Bowser-Chao:1997kjp ; Jung:2013hka ,

$\displaystyle F_{q}(a,b)=\frac{T_{q}(a)-T_{q}(b)}{a-b};~{}~{}q=t,b,$ (29)

where

$\displaystyle T_{b}(a)$
$\displaystyle=\frac{1-3a}{a^{2}}\frac{\pi^{2}}{6}+\left(\frac{1}{a}-\frac{5}{2}\right)\log
a-\frac{1}{a}-\left(2-\frac{1}{a}\right)\left(1-\frac{1}{a}\right){\rm
Li}_{2}(1-a),$ (30) $\displaystyle T_{b}(a)$
$\displaystyle=\frac{2a-1}{a^{2}}\frac{\pi^{2}}{6}+\left(\frac{3}{2}-\frac{1}{a}\right)\log
a+\frac{1}{a}-\frac{1}{a}\left(2-\frac{1}{a}\right){\rm Li}_{2}(1-a).$ (31)

Finally, the functions appearing in diagrams with $W^{+}$ and $H^{+}$ loops
for $d_{\mu}^{H^{+}W^{+}}$ are Abe:2013qla ,

$\displaystyle{\cal
I}_{4(5)}(m_{\phi},m_{H^{+}})=\frac{m_{W}^{2}}{m_{H^{+}}^{2}-m_{W}^{2}}\Bigl{[}I_{4(5)}(m_{W},m_{\phi})-I_{4(5)}(m_{H^{+}},m_{\phi},)\Bigr{]},$
(32)

where,

$\displaystyle I_{4}(m_{i},m_{\phi})$
$\displaystyle=\int_{0}^{1}dz\frac{m_{i}^{2}\left(z(1-z)^{2}-4(1-z)^{2}+\frac{m_{H^{+}}^{2}-m_{\phi}^{2}}{m_{W}^{2}}z(1-z)^{2}\right)}{m_{W}^{2}(1-z)+m_{\phi}^{2}z-m_{i}^{2}z(1-z)}\log\left(\frac{m_{W}^{2}(1-z)+m_{\phi}^{2}z}{m_{i}^{2}z(1-z)}\right),$
(33) $\displaystyle I_{5}(m_{i},m_{\phi})$
$\displaystyle=2\int_{0}^{1}dz\frac{m_{i}^{2}z(1-z)^{2}}{m_{H^{+}}^{2}(1-z)+m_{\phi}^{2}z-m_{i}^{2}z(1-z)}\log\left(\frac{m_{H^{+}}^{2}(1-z)+m_{\phi}^{2}z}{m_{i}^{2}z(1-z)}\right).$
(34)


# On a paucity result in Incidence Geometry ††This work is supported by the
Russian Science Foundation under grant 19–11–00001.

Shkredov I.D

Annotation.

We obtain some asymptotic formulae (with power savings in their error terms)
for the number of quadruples in the Cartesian product of an arbitrary set
$A\subset{\mathbb{R}}$ and for the number of quintuplets in $A\times A$ for
any subset $A$ of the prime field $\mathbb{F}_{p}$. Also, we obtain some
applications of our results to incidence problems in $\mathbb{F}_{p}$.

##  1 Introduction

Incidence Geometry (see, e.g., [13, section 8]) deals with the incidences
among different geometrical objects such as points, lines, curves, surfaces
etc. A typical problem of this area is to estimate the quantity

$\mathcal{I}(\mathcal{P},\mathcal{L}):=|\\{(p,l)\in\mathcal{P}\times\mathcal{L}:\,p\in
l\\}|\,,$ (1)

where the set of points $\mathcal{P}$ and the set of lines $\mathcal{L}$
belong to $\mathbb{F}\times\mathbb{F}$ with $\mathbb{F}$ be a field, say. In
our paper the set of points $\mathcal{P}$ will be the Cartesian product
$A\times B$ for some sets $A,B\subseteq\mathbb{F}$. This particular choice of
${\cal P}$ is very important for the applications see, e.g., [2], [5], [6],
[9], [10], [13] because, basically, Cartesian products are naturally connected
with arithmetic. In this note we study collinear tuples in $A\times B$.
Namely, for any $k\geqslant 3$ we define ${\mathcal{C}}_{k}(A,B)$ to be the
number of collinear $k$–tuples in $A\times B$. Let
${\mathcal{C}}_{k}(A)={\mathcal{C}}_{k}(A,A)$. A consequence of the famous
Szemerédi–Trotter Theorem [12] gives us that for any $A\subset{\mathbb{R}}$
one has

${\mathcal{C}}_{3}(A)\ll|A|^{4}\log|A|$ (2)

(a short proof of this result contains in [10]). Actually, for ${\cal
P}=A\times A$ it is easy to see that bound (2) is equivalent to the
Szemerédi–Trotter Theorem (up to logarithms). In our paper [2] we have
obtained an asymptotic formula for the number of collinear quadruples in the
case of the prime field $\mathbb{F}=\mathbb{F}_{p}$

${\mathcal{C}}_{4}(A)=\frac{|A|^{8}}{p^{2}}+O(|A|^{5}\log|A|)\,.$ (3)

The proof is based on Stevens–de Zeeuw’s Theorem [11]. Both bounds (2), (3)
are known to be tight up to some powers of logarithms as the case
$A=\\{1,\dots,n\\}$ shows, namely,
${\mathcal{C}}_{3}(A)\gg|A|^{4}\log^{c}|A|$, $c>0$ for such $A$ (see
discussion in [2, Pages 603, 633]). The problem of finding sharp asymptotic
formula for the number of collinear triples in $A\times A$,
$A\subseteq\mathbb{F}_{p}$ is a well–known important open question and at the
moment the best result is (see [2], [6])

${\mathcal{C}}_{3}(A)=\frac{|A|^{6}}{p}+O(\min\\{|A|^{9/2},p^{1/2}|A|^{7/2}\\})\,.$
(4)

As for ${\mathcal{C}}_{k}(A)$ with large $k$, then one can easily obtain an
analogues of formulae (2), (3) but, generally speaking, the error term must be
at least $2|A|^{k+1}$ (take horizontal/vertical lines), and hence the
consideration of large $k$ does not give anything new.

In this paper we make a further step and obtain a paucity result (see Theorems
22, 29 below) for higher ${\mathcal{C}}_{k}(A)$. As we said before, in
general, one cannot obtain non–trivial bounds for such quantities and thus we
need some restrictions, which we formulate in terms of energies. For any two
sets $A,B\subseteq\mathbb{F}$ the additive energy of $A$ and $B$ is defined by

$\mathsf{E}^{+}(A,B)=|\\{(a_{1},a_{2},b_{1},b_{2})\in A\times A\times B\times
B~{}:~{}a_{1}-b_{1}=a_{2}-b_{2}\\}|\,.$

If $A=B$, then we simply write $\mathsf{E}^{+}(A)$ for $\mathsf{E}^{+}(A,A)$.
Similarly, one can define the multiplicative energy of $A$ and $B$. Finally,
put
$\overline{\mathsf{E}}^{\times}(A)=\max_{s\in\mathbb{F}}\mathsf{E}^{\times}(A-s)\leqslant|A|^{3}$.

###### Theorem 1

Let $A,B\subset\mathbb{R}$ be sets. Then

${\mathcal{C}}_{4}(A,B)-|B||A|^{4}-|A||B|^{4}\ll|B||A|^{3}+|A||B|^{3}+$
$(\mathsf{E}^{+}(A)\mathsf{E}^{+}(B))^{1/10}{\mathcal{C}}^{2/5}_{3}(A,B)|A|^{8/5}|B|^{6/5}\log^{2/5}|A|+(\overline{\mathsf{E}}^{\times}(A)\overline{\mathsf{E}}^{\times}(B))^{1/4}|A|^{2}|B|^{3/2}\log^{3/2}|A|\,.$
(5)

Further let $p$ be a prime number, $A\subset\mathbb{F}_{p}$. Then

${\mathcal{C}}_{5}(A)-\frac{|A|^{10}}{p^{3}}-2|A|^{6}\lesssim\frac{|A|^{7}}{p}+\frac{|A|^{4}}{p^{2}}{\mathcal{C}}_{3}(A)+(\mathsf{E}^{+}(A))^{1/6}|A|^{11/2}+(\overline{\mathsf{E}}^{\times}(A))^{1/2}|A|^{9/2}\,.$
(6)

###### Corollary 2

Let $A,B\subset{\mathbb{R}}$, $A=B$ and
$\mathsf{E}^{+}(A),\overline{\mathsf{E}}^{\times}(A)\leqslant|A|^{3-c}$, where
$c>0$. Substituting estimate (2) to Theorem 6, we get

${\mathcal{C}}_{4}(A)-2|A|^{5}\ll|A|^{5-c/5}\log^{4/5}|A|\,,$ (7)

and in the case of the prime field and, say, $|A|\leqslant p^{2/3}$, we have

${\mathcal{C}}_{5}(A)-2|A|^{6}\lesssim|A|^{6-c/6}\log^{5/6}|A|\,.$ (8)

Once again the condition $\mathsf{E}^{+}(A)\leqslant|A|^{3-c}$ is required for
(7), (8). One can easily see that a random set $A$ satisfies both assumptions
$\mathsf{E}^{+}(A),\overline{\mathsf{E}}^{\times}(A)\leqslant|A|^{3-c}$, where
$c>0$. Also, there are some concrete constructions of such sets, see section
3, namely, one can take $A=B^{-1}+s$ for any $s\in\mathbb{F}$ and
$B\subseteq\mathbb{F}$ such that $|B+B|\ll|B|$ and then
$\mathsf{E}^{+}(A),\overline{\mathsf{E}}^{\times}(A)\leqslant|A|^{3-c}$. It
seems like that our Theorem 6 is the first paucity result in the area. As for
lower bounds on ${\mathcal{C}}_{k}(A)$ we trivially have $\Omega(|A|^{k})$ in
the error term, counting $k$–tuples, which belong to the diagonal $(x,x)$,
where $x$ runs over $A$.

We obtain some applications of our paucity theorems in section 4. For example,
let us formulate a consequence of our new incidence result in $\mathbb{F}_{p}$
concerning points/lines incidences. The basic regime here is the following:
the sets $A$, $B$ are large and the sets $X,Y$ are small comparable to $p$.

###### Theorem 3

Let $k\geqslant 2$ be an integer, $M\geqslant 1$ be a real number,
$A,B,C,X,Y\subseteq\mathbb{F}_{p}$, $0\notin X$ be sets, $|YC|\leqslant M|Y|$,
and $|C|\gtrsim_{k}M^{2^{k+1}}$, $|C|^{k-1}\geqslant(p/|Y|)^{2}$. Suppose that

$\max\\{\mathsf{E}^{+}(B),\overline{\mathsf{E}}^{\times}(B)\\}\leqslant|B|^{3-\delta}\,,$

where $\delta>0$ is a constant. Then

$|\\{(a,b,x,y)\in A\times B\times X\times
Y~{}:~{}y=bx+a\\}|-\frac{|A||B||X||Y|}{p}\ll\sqrt{|A||B|}|X||Y|\cdot|B|^{-\frac{\delta}{35\cdot
2^{k}}}\,.$ (9)

There are two main advantages of Theorem 9. First of all, it is an asymptotic
formula (and, again, the set $X\times Y$, which corresponds to the lines
$\mathcal{L}$ can be rather small, even of size $p^{\varepsilon}$) but not
just upper bounds for ${\cal I}(\mathcal{P},\mathcal{L})$ as in [11], say.
Some asymptotic formulae for the quantity ${\cal I}(\mathcal{P},\mathcal{L})$
were known before in the specific case of large sets (see [14]) and in the
case of Cartesian products but with large sets of lines, see [6] and [11].
Such asymptotic formulae are important in the problem of estimating
exponential sums (see, e.g., [6]) and in the questions on mixing times of some
Markov chains [9], where one requires to have the set of points $A\times B$ to
be really large. The second advantage of Theorem 9 is that it works even for
$|A|,|B|\gg p$ (for size of the set $B$ see the exact formulation of Theorem
32 below).

##  2 Definitions and preliminaries

By ${\mathbf{G}}$ we denote an abelian group. Sometimes we underline the group
operation writing $+$ or $\times$ in the considered quantities (as the energy,
the representation function and so on, see below). Let $\mathbb{F}$ be the
field ${\mathbb{R}}$ or $\mathbb{F}=\mathbb{F}_{p}=\mathbb{Z}/p\mathbb{Z}$ for
a prime $p$. Let $\mathbb{F}^{*}=\mathbb{F}\setminus\\{0\\}$.

We use the same capital letter to denote set $A\subseteq\mathbb{F}$ and its
characteristic function $A:\mathbb{F}\to\\{0,1\\}$ and in the case of finite
$\mathbb{F}$ we write $f_{A}(x):=A(x)-|A|/|\mathbb{F}|$ for the balanced
function of $A$. Given two sets $A,B\subset{\mathbf{G}}$, define the sumset of
$A$ and $B$ as

$A+B:=\\{a+b~{}:~{}a\in{A},\,b\in{B}\\}\,.$

In a similar way we define the difference sets and higher sumsets, e.g.,
$2A-A$ is $A+A-A$. We write $\dotplus$ for a direct sum, i.e., $|A\dotplus
B|=|A||B|$. For an abelian group ${\mathbf{G}}$ the Plünnecke–Ruzsa inequality
(see, e.g., [13]) holds stating

$|nA-mA|\leqslant\left(\frac{|A+A|}{|A|}\right)^{n+m}\cdot|A|\,,$ (10)

where $n,m$ are any positive integers. We use representation function
notations like $r_{A+B}(x)$ or $r_{A-B}(x)$ and so on, which counts the number
of ways $x\in{\mathbf{G}}$ can be expressed as a sum $a+b$ or $a-b$ with $a\in
A$, $b\in B$, respectively. For example, $|A|=r_{A-A}(0)$.

For any two sets $A,B\subseteq{\mathbf{G}}$ the additive energy of $A$ and $B$
is defined by

$\mathsf{E}(A,B)=\mathsf{E}^{+}(A,B)=|\\{(a_{1},a_{2},b_{1},b_{2})\in A\times
A\times B\times B~{}:~{}a_{1}-b_{1}=a_{2}-b_{2}\\}|\,.$

If $A=B$, then we simply write $\mathsf{E}(A)$ for $\mathsf{E}(A,A)$. More
generally, for sets (real functions) $A_{1},\dots,A_{2k}$
($f_{1},\dots,f_{2k}$) belonging to an arbitrary (noncommutative) group
${\mathbf{G}}$ and $k\geqslant 2$ define the energy
$\mathsf{T}_{k}(A_{1},\dots,A_{2k})$ as

$\mathsf{T}_{k}(A_{1},\dots,A_{2k})=$ $=|\\{(a_{1},\dots,a_{2k})\in
A_{1}\times\dots\times A_{2k}~{}:~{}a_{1}a^{-1}_{2}\dots
a_{k-1}a^{-1}_{k}=a_{k+1}a^{-1}_{k+2}\dots a_{2k-1}a^{-1}_{2k}\\}|\,,$ (11)

and

$\mathsf{T}_{k}(f_{1},\dots,f_{2k})=\sum_{a_{1}a^{-1}_{2}\dots
a_{k-1}a^{-1}_{k}=a_{k+1}a^{-1}_{k+2}\dots
a_{2k-1}a^{-1}_{2k}}f_{1}(a_{1})\dots f_{2k}(a_{2k})\,.$

In the abelian case put for $k\geqslant 2$

$\mathsf{E}^{+}_{k}(A)=\sum_{x}r^{k}_{A-A}(x)=\sum_{\alpha_{1},\dots,\alpha_{k-1}}|A\cap(A+\alpha_{1})\cap\dots\cap(A+\alpha_{k-1})|^{2}\,.$
(12)

Clearly, $|A|^{k}\leqslant\mathsf{E}^{+}_{k}(A)\leqslant|A|^{k+1}$. Having a
function $f:\mathbb{F}\to\mathbb{C}$, we put
$\overline{\mathsf{E}}^{\times}(f)=\max_{s\in\mathbb{F}}\mathsf{E}^{\times}(f-s)$.
In particular, $\overline{\mathsf{E}}^{\times}(A)\leqslant|A|^{3}$ for any
finite set $A$, $A\subseteq\mathbb{F}$.

The signs $\ll$ and $\gg$ are the usual Vinogradov symbols. When the constants
in the signs depend on a parameter $M$, we write $\ll_{M}$ and $\gg_{M}$. All
logarithms are to base $2$. If we have a set $A$, then we will write
$a\lesssim b$ or $b\gtrsim a$ if $a=O(b\cdot\log^{c}|A|)$, $c>0$. Let us
denote by $[n]$ the set $\\{1,2,\dots,n\\}$.

Given an arbitrary line $l$ denote by $i_{A,B}(l)=|l\cap(A\times B)|$ hence in
particular, $\mathcal{I}(A\times B,\mathcal{L})=\sum_{l\in L}i_{A,B}(l)$.
Clearly, for $k\geqslant 3$ one has

${\mathcal{C}}_{k}(A,B)=\sum_{l~{}:~{}i_{A,B}(l)>1}i^{k}_{A,B}(l)+O_{k}(|A|^{2}|B|^{2}\cdot\left(\min\\{|A|,|B|\\}\right)^{k-3}\,.$
(13)

One can think about the first term in (13) as another definition of the
quantity ${\mathcal{C}}_{k}(A,B)$.

We need a result on the energy of an arbitrary set of affine transformations,
see [3, Theorem 2.1]. We consider any set of affine transformations
$\mathcal{L}\subseteq{\rm Aff}(\mathbb{F})$ as a subset of
$\mathbb{F}^{*}\times\mathbb{F}$, i.e., we associate a map $x\to ax+b$ with a
point $(a,b)\in\mathbb{F}^{*}\times\mathbb{F}$.

###### Theorem 4

Let $\mathcal{L}\subseteq{\rm Aff}(\mathbb{F})$ be a set of lines such that no
line in $\mathbb{F}^{*}\times\mathbb{F}$ contains more than $M(\mathcal{L})$
points of $\mathcal{L}$, and no vertical line in
$\mathbb{F}^{*}\times\mathbb{F}$ contains more than $m(\mathcal{L})$ points of
$\mathcal{L}$. Suppose that $m(\mathcal{L})|\mathcal{L}|\leqslant p^{2}$. Then

$\mathsf{E}(\mathcal{L})\ll
m^{1/2}(\mathcal{L})|\mathcal{L}|^{5/2}+M(\mathcal{L})|\mathcal{L}|^{2}\,.$

The following result is contained in [9, Proposition 7]. Here we have
considered the case of the prime field (which is more difficult) and the case
of the real numbers can be treated in analogues way.

###### Proposition 5

Let $A,B\subseteq\mathbb{F}$ be sets and $\mathcal{L}$ be a set of affine
transformations over $\mathbb{F}$. Then for any positive integer $k$ one has

${\cal I}(A\times
B,\mathcal{L})-\frac{|A||B||\mathcal{L}|}{|\mathbb{F}|}\ll\sqrt{|A||B||\mathcal{L}|}\cdot(\mathsf{T}_{2^{k}}(\mathcal{L})|A|\log|A|)^{1/2^{k+2}}\,.$
(14)

More precisely, if $\mathbb{F}={\mathbb{R}}$, then

${\cal I}(A\times
B,\mathcal{L})\ll\sqrt{|A||B|}|\mathcal{L}|^{1/3}\cdot(\mathsf{T}_{2^{k}}(\mathcal{L})|A|\log|A|)^{1/3\cdot
2^{k}}\,.$ (15)

We need the following structural result on the higher energies, see [8,
Theorem 4].

###### Theorem 6

Let ${\mathbf{G}}$ be an abelian group, $A\subseteq{\mathbf{G}}$ be a set,
$\delta,\varepsilon\in(0,1]$ be parameters, $\varepsilon\leqslant\delta$.
$1)~{}$ Then there is
$k=k(\delta,\varepsilon)=\exp(O(\varepsilon^{-1}\log(1/\delta)))$ such that
either $\mathsf{E}_{k}(A)\leqslant|A|^{k+\delta}$ or there is
$H\subseteq{\mathbf{G}}$, $|H|\gtrsim|A|^{\delta(1-\varepsilon)}$,
$|H+H|\ll|A|^{\varepsilon}|H|$ and there exists $Z\subseteq{\mathbf{G}}$,
$|Z||H|\ll|A|^{1+\varepsilon}$ with

$|(H\dotplus Z)\cap A|\gg|A|^{1-\varepsilon}\,.$

$2)~{}$ Similarly, either there is a set $A^{\prime}\subseteq A$,
$|A^{\prime}|\gg|A|^{1-\varepsilon}$ and $P\subseteq{\mathbf{G}}$,
$|P|\gtrsim|A|^{\delta}$ such that for all $x\in A^{\prime}$ one has
$r_{A-P}(x)\gg|P||A|^{-\varepsilon}$ or
$\mathsf{E}_{k}(A)\leqslant|A|^{k+\delta}$ with $k\ll 1/\varepsilon$.

Thus either $\mathsf{E}_{k}(A)\leqslant|A|^{k+\delta}$, or $A$ has some strong
structural properties depending on the parameters $\delta$ and $\varepsilon$.
In the later case we say that $A$ is $\mathsf{E}_{k}$–exceptional with
$\delta,\varepsilon$.

We use a simplified version of [7, Theorem 5], as well as [9, Lemma 9] in our
last section.

###### Theorem 7

Let $k\geqslant 2$ be an integer, $M\geqslant 1$ be a real number,
$A,B\subseteq\mathbb{F}_{p}$ be sets, $|AB|\leqslant M|A|$, and
$|B|\gtrsim_{k}M^{2^{k+1}}$. Then

$\mathsf{T}^{+}_{2^{k}}(A)\lesssim_{k}M^{2^{k+1}}\left(\frac{|A|^{2^{k+1}}}{p}+|A|^{2^{k+1}-1}\cdot|B|^{-\frac{k-1}{2}}\right)\,.$
(16)

###### Lemma 8

Let $A,B\subseteq\mathbb{F}^{*}_{p}$ be sets, and
$\mathcal{L}=\\{(a,b)~{}:~{}a\in A,\,b\in B\\}\subseteq{\rm
Aff}(\mathbb{F}_{p})$. Then for any $k\geqslant 2$ one has

$\mathsf{T}_{k}(f_{\mathcal{L}})\leqslant|A|^{2k-1}\mathsf{T}^{+}_{k}(f_{B})\,.$
(17)

##  3 The proof of the main result

First of all, combining Theorem 4 and Proposition 15, we obtain a result on
”rich” lines (i.e., lines having large intersections with $A\times B$) in
terms of the quantities $m(\mathcal{L})$ and $M(\mathcal{L})$. We follow the
method from [5] and [3]. Given a set $\mathcal{L}$ of affine transformations
over $\mathbb{F}$ and a real number $\tau\geqslant 2$ put

$\mathcal{L}_{\tau}=\mathcal{L}_{\tau}(A):=\\{l\in\mathcal{L}~{}:~{}|l\cap(A\times
B)|\geqslant\tau\\}\,.$

Formulae (18), (19) below can be treated as an alternative (to the
Szemerédi–Trotter type estimates) upper bound for size of
$|\mathcal{L}_{\tau}|$.

###### Corollary 9

Let $A,B\subseteq\mathbb{F}$ be sets and $\mathcal{L}$ be a set of affine
transformations over $\mathbb{F}$. Also, let $\tau\geqslant 1$ be a real
number, $\frac{2|A||B|}{|\mathbb{F}|}\leqslant\tau$ and
$m(\mathcal{L}_{\tau})|\mathcal{L}_{\tau}|\leqslant|\mathbb{F}|^{2}$. Then

$|\mathcal{L}_{\tau}|\ll\tau^{-16/3}m^{1/3}(\mathcal{L}_{\tau})|A|^{10/3}|B|^{8/3}\log^{2/3}|A|+\tau^{-4}M^{1/2}(\mathcal{L}_{\tau})|A|^{5/2}|B|^{2}\log^{1/2}|A|\,.$
(18)

More precisely, if $\mathbb{F}={\mathbb{R}}$, then

$|\mathcal{L}_{\tau}|\ll\tau^{-4}(m(\mathcal{L}_{\tau})|A|^{8}|B|^{6}\log^{2}|A|)^{1/3}+\tau^{-3}M^{1/2}(\mathcal{L}_{\tau})|A|^{2}|B|^{3/2}\log^{1/2}|A|\,.$
(19)

P r o o f. Let $m=m(\mathcal{L}_{\tau})$ and $M=M(\mathcal{L}_{\tau})$. By
Proposition 15 with $k=1$, the definition of the set $\mathcal{L}_{\tau}$ and
our condition $\frac{2|A||B|}{|\mathbb{F}|}\leqslant\tau$, we get

$\tau|\mathcal{L}_{\tau}|\ll\sqrt{|A||B||\mathcal{L}_{\tau}|}\cdot(\mathsf{E}(\mathcal{L}_{\tau})|A|\log|A|)^{1/8}\,.$

We now apply Theorem 4 and obtain

$\tau|\mathcal{L}_{\tau}|\ll\sqrt{|A||B||\mathcal{L}_{\tau}|}(|A|\log|A|)^{1/8}\cdot(m^{1/2}|\mathcal{L}_{\tau}|^{5/2}+M|\mathcal{L}_{\tau}|^{2})^{1/8}\,.$
(20)

The last inequality is equivalent to (18). To obtain estimate (19) we just use
formula (15) instead of (14). This completes the proof. $\hfill\Box$

Now let us give simple upper bounds for the quantities
$m(\mathcal{L}_{\tau})$, $M(\mathcal{L}_{\tau})$.

###### Lemma 10

Let $A,B\subseteq\mathbb{F}$ be sets, $\tau\geqslant 2$ and $\mathcal{L}$ be a
set of affine transformations over $\mathbb{F}$. Then for any integer
$k\geqslant 2$ one has

$m(\mathcal{L}_{\tau})\leqslant\tau^{-k}(\mathsf{E}^{+}_{k}(A)\mathsf{E}^{+}_{k}(B))^{1/2}\quad\quad\mbox{
and }\quad\quad
M(\mathcal{L}_{\tau})\leqslant\tau^{-k}(\overline{\mathsf{E}}^{\times}_{k}(A)\overline{\mathsf{E}}^{\times}_{k}(B))^{1/2}\,.$

P r o o f. Let $m=m(\mathcal{L}_{\tau})$ and $M=M(\mathcal{L}_{\tau})$. By the
definition of the quantity $m$ we can find $m$ lines in $\mathcal{L}_{\tau}$
having the form $\alpha x+\beta$, where $\alpha\neq 0$ is a fixed number and
$\beta$ runs over a set $\mathcal{B}$ of cardinality $m$. Using the definition
of the set $\mathcal{L}_{\tau}$ and the Hölder inequality, we obtain

$\tau m\leqslant\sum_{\beta\in\mathcal{B}}\,\sum_{x\in B}A(\alpha
x+\beta)=\sum_{\beta\in\mathcal{B}}r_{\alpha B-A}(\beta)\leqslant
m^{1-1/k}(\mathsf{E}^{+}_{k}(\alpha B,A))^{1/k}\leqslant
m^{1-1/k}(\mathsf{E}^{+}_{k}(A)\mathsf{E}^{+}_{k}(B))^{1/2k}\,.$

Similarly, by the definition of the quantity $M$ we find $M$ lines concurrent
at a certain point $(x_{0},y_{0})$. We parameterise them by their slopes:
$\beta=\frac{y-y_{0}}{x-x_{0}}$, $\beta\in\mathcal{B}$, $|\mathcal{B}|=M$.
Hence as above

$\tau
M\leqslant\sum_{\beta\in\mathcal{B}}\,r_{(A-y_{0})/(B-x_{0})}(\beta)\leqslant
M^{1-1/k}(\mathsf{E}^{\times}(A-y_{0},B-x_{0}))^{1/2}\leqslant
M^{1-1/k}(\overline{\mathsf{E}}^{\times}_{k}(A)\overline{\mathsf{E}}^{\times}_{k}(B))^{1/2k}\,.$

This completes the proof. $\hfill\Box$

Now we are ready to obtain our main result in the case of the real field and a
result on ${\mathcal{C}}_{5}(A,B)$ for small $A$ and $B$ in the case of the
prime field.

###### Theorem 11

Let $A,B\subset\mathbb{R}$ be sets. Then

${\mathcal{C}}_{4}(A,B)-|B||A|^{4}-|A||B|^{4}\ll|B||A|^{3}+|A||B|^{3}+$
$(\mathsf{E}^{+}(A)\mathsf{E}^{+}(B))^{1/10}{\mathcal{C}}^{2/5}_{3}(A,B)|A|^{8/5}|B|^{6/5}\log^{2/5}|A|+(\overline{\mathsf{E}}^{\times}(A)\overline{\mathsf{E}}^{\times}(B))^{1/4}|A|^{2}|B|^{3/2}\log^{3/2}|A|\,.$
(21)

If $A,B\subseteq\mathbb{F}_{p}$ such that
${\mathcal{C}}^{2}_{4}(A,B)\log|A|\leqslant p|A|^{5}|B|^{4}$ and
$|A|^{5}|B|^{6}\leqslant p^{9}$, then

${\mathcal{C}}_{5}(A,B)-|B||A|^{5}-|A||B|^{5}\ll|B||A|^{4}+|A||B|^{4}+{\mathcal{C}}^{\prime}_{5}(A,B)\,,$

where the error term ${\mathcal{C}}^{\prime}_{5}(A,B)$ is at most

$(\mathsf{E}^{+}(A)\mathsf{E}^{+}(B))^{1/12}{\mathcal{C}}^{1/2}_{4}(A,B)|A|^{5/3}|B|^{4/3}\log^{1/3}|A|+(\overline{\mathsf{E}}^{\times}(A)\overline{\mathsf{E}}^{\times}(B))^{1/4}|A|^{5/2}|B|^{2}\log^{3/2}|A|\,.$
(22)

P r o o f. We start with (21). The term $|B||A|^{4}+|A||B|^{4}$ corresponds to
vertical/horizontal lines plus an error, which is $O(|B||A|^{3}+|A||B|^{3})$.
Let $\mathcal{L}$ be the set of non–vertical and non–horizontal lines, having
at least two points in $A\times B$. Denote by $\mathcal{E}$ the error term in
(22). Let $\Delta>0$ be a parameter, which we will choose later. Having the
definition of the quantity ${\mathcal{C}}_{3}(A,B)$ and using the diadic
Dirichlet principle, we get

$\mathcal{E}\ll\Delta{\mathcal{C}}_{3}(A,B)+\sum_{j~{}:~{}2^{j}\geqslant\Delta}2^{4j}|\mathcal{L}_{2^{j}}|=\Delta{\mathcal{C}}_{3}(A,B)+\mathcal{E}^{\prime}\,.$

Put $m_{*}=(\mathsf{E}^{+}(A)\mathsf{E}^{+}(B))^{1/2}$,
$M_{*}=(\overline{\mathsf{E}}^{\times}(A)\overline{\mathsf{E}}^{\times}(B))^{1/2}$.
Applying bound (19) of Corollary 19, combining with Lemma 10 (with the
parameter $k=2$), we obtain

$\mathcal{E}^{\prime}\ll\sum_{j~{}:~{}2^{j}\geqslant\Delta}2^{4j}\cdot\left(2^{-14j/3}(m_{*}|A|^{8}|B|^{6}\log^{2}|A|)^{1/3}+2^{-4j}M^{1/2}_{*}|A|^{2}|B|^{3/2}\log^{1/2}|A|\right)$
$\ll\Delta^{-2/3}(m_{*}|A|^{8}|B|^{6}\log^{2}|A|)^{1/3}+M^{1/2}_{*}|A|^{2}|B|^{3/2}\log^{3/2}|A|\,.$

Now choosing
$\Delta^{5/3}=(m_{*}|A|^{8}|B|^{6}\log^{2}|A|)^{1/3}{\mathcal{C}}^{-1}_{3}(A,B)$
(one can check that $\Delta\geqslant 2$ thanks to formula (13), say), we get

$\mathcal{E}\ll{\mathcal{C}}^{2/5}_{3}(A,B)m^{1/5}_{*}|A|^{8/5}|B|^{6/5}\log^{2/5}|A|+M^{1/2}_{*}|A|^{2}|B|^{3/2}\log^{3/2}|A|$

as required. To obtain (22) we use the same argument, namely,

$\mathcal{E}\ll\Delta{\mathcal{C}}_{4}(A,B)+\sum_{j~{}:~{}2^{j}\geqslant\Delta}2^{5j}|\mathcal{L}_{2^{j}}|=\Delta{\mathcal{C}}_{4}(A,B)+\mathcal{E}^{\prime}\,,$
(23)

and

$\mathcal{E}^{\prime}\ll\sum_{j~{}:~{}2^{j}\geqslant\Delta}2^{5j}\left(2^{-6j}m^{1/3}_{*}|A|^{10/3}|B|^{8/3}\log^{2/3}|A|+2^{-5j}M^{1/2}_{*}|A|^{5/2}|B|^{2}\log^{1/2}|A|\right)$
(24)
$\ll\Delta^{-1}m^{1/3}_{*}|A|^{10/3}|B|^{8/3}\log^{2/3}|A|+M^{1/2}_{*}|A|^{5/2}|B|^{2}\log^{3/2}|A|\,.$
(25)

Now the optimal choice of $\Delta$ is
$\Delta^{2}=m^{1/3}_{*}|A|^{10/3}|B|^{8/3}\log^{2/3}|A|\cdot{\mathcal{C}}^{-1}_{4}(A,B)$
(again $\Delta\geqslant 2$ thanks to formula (13), say) and hence

$\mathcal{E}\ll{\mathcal{C}}^{1/2}_{4}(A,B)m^{1/6}_{*}|A|^{5/3}|B|^{4/3}\log^{1/3}|A|+M^{1/2}_{*}|A|^{5/2}|B|^{2}\log^{3/2}|A|\,.$

It remains to check that $\Delta\geqslant 2|A||B|/p$ and
$m(\mathcal{L}_{\Delta})|\mathcal{L}_{\Delta}|\leqslant p^{2}$. Since by our
condition
${\mathcal{C}}^{2}_{4}(A,B)\log|A|\leqslant p|A|^{5}|B|^{4}$, it follows that
in view of trivial lower bounds $\mathsf{E}^{+}(A)\geqslant|A|^{2}$,
$\mathsf{E}^{+}(B)\geqslant|B|^{2}$, we get

$\Delta^{2}\gg
m^{1/3}_{*}|A|^{5/6}|B|^{2/3}p^{-1/2}\geqslant|A|^{7/6}|B|p^{-1/2}\,.$

It is easy to see that the last quantity is greater than $(2|A||B|/p)^{2}$
thanks to $|A|^{5}|B|^{6}\leqslant p^{9}$. Finally, using
$|\mathcal{L}_{\Delta}|\leqslant{\mathcal{C}}_{4}(A,B)\Delta^{-4}$, Lemma 10
with $k=2$, the definition of the quantity $\Delta$ and our assumption, we
obtain

$m(\mathcal{L}_{\Delta})|\mathcal{L}_{\Delta}|\leqslant
m_{*}{\mathcal{C}}_{4}(A,B)\Delta^{-6}\leqslant{\mathcal{C}}^{4}_{4}(A,B)|A|^{-10}|B|^{-8}\log^{2}|A|\leqslant
p^{2}\,.$ (26)

This completes the proof. $\hfill\Box$

Of course one can rewrite the condition
${\mathcal{C}}^{2}_{4}(A,B)\log|A|\leqslant p|A|^{5}|B|^{4}$ in terms of
cardinalities of $A$ and $B$, using (3) but we leave this a little bit more
precise condition. Now let us consider a constructive family of sets
satisfying the assumptions of Theorem 22.

Example. Let $A\subseteq\mathbb{F}$, $|A|<\sqrt{|\mathbb{F}|}$, say,
$|A+A|\leqslant K|A|$ and consider $X:=A^{-1}$. Then one can quickly show that
$\mathsf{E}^{+}(X)\ll_{K}|A|^{3-1/4}$ (see [6, Lemma 14]) and
$\mathsf{E}^{\times}(X+s)\ll_{K}|A|^{3-c}$ for a certain $c>0$ and any
$s\in\mathbb{F}$. We give a sketch of the proof of the last estimate. Indeed,
for $s=0$ it immediately follows from Rudnev’s Theorem [4] (with $c=1/2$) and
for $s\neq 0$ (dividing we can suppose that $s=1)$ we see that it is enough to
solve the equation

$\left(\frac{1}{a+b}+1\right)\left(\frac{1}{c+d}+1\right)=\lambda\,,$ (27)

where $\lambda\neq 1$ is a fixed number. If we consider $a,c$ as variables and
$b,d$ as coefficients, then, clearly, (27) determines a family of conics and
the number of the solutions to the equation can be estimated via the main
result of [1], say.

Similarly, we obtain an analogue of estimate (22) without any restrictions on
size of $A$.

###### Theorem 12

Let $A\subseteq\mathbb{F}_{p}$ be a set and $f_{A}(x)=A(x)-|A|/p$. Then

$\sum_{l\in{\rm
Aff}(\mathbb{F}_{p})}\left|\sum_{x}f_{A}(x)f_{A}(lx)\right|^{5}\lesssim(\mathsf{E}^{+}(f_{A}))^{1/6}|A|^{11/2}+(\overline{\mathsf{E}}^{\times}(f_{A}))^{1/2}|A|^{9/2}\,,$
(28)

where the summation over $l$ in the last formula is taken over all affine
transformations having at least two points in $A\times A$. In particular,

${\mathcal{C}}_{5}(A)-\frac{|A|^{10}}{p^{3}}-2|A|^{6}\lesssim\frac{|A|^{7}}{p}+\frac{|A|^{4}}{p^{2}}{\mathcal{C}}_{3}(f_{A})+(\mathsf{E}^{+}(f_{A}))^{1/6}|A|^{11/2}+(\overline{\mathsf{E}}^{\times}(f_{A}))^{1/2}|A|^{9/2}\,.$
(29)

P r o o f. Let $\mathcal{E}$ be the left–hand side of (28). First of all, we
redefine the set $\mathcal{L}_{\tau}$ as

$\mathcal{L}_{\tau}=\mathcal{L}_{\tau}(f_{A}):=\left\\{l\in\mathcal{L}~{}:~{}\left|\sum_{x}f_{A}(x)f_{A}(lx)\right|\geqslant\tau\right\\}\,.$

Then estimate (14) of Proposition 15 with $A=B$ gives us

$\tau|\mathcal{L}_{\tau}|\ll\sqrt{|A||B||\mathcal{L}_{\tau}|}\cdot(\mathsf{E}(\mathcal{L}_{\tau})|A|\log|A|)^{1/8}\,.$
(30)

Indeed,

$\tau|\mathcal{L}_{\tau}|\leqslant\sum_{l\in\mathcal{L}_{\tau}}\left|\sum_{x}f_{A}(x)f_{A}(lx)\right|=\sum_{l\in\mathcal{L}_{\tau}}\varepsilon(l)\sum_{x}f_{A}(x)f_{A}(lx)\,,$

where by $\varepsilon(l)$ we have denoted the sign of
$\left|\sum_{x}f_{A}(x)f_{A}(lx)\right|$. In other words, we consider a new
function
$\mathcal{L}^{\varepsilon}_{\tau}(l):=\mathcal{L}_{\tau}(l)\varepsilon(l)$.
Clearly, $\|\mathcal{L}^{\varepsilon}_{\tau}\|_{1}=|\mathcal{L}_{\tau}|$ and
for any integer $l\geqslant 2$ one has
$\mathsf{T}_{l}(\mathcal{L}^{\varepsilon}_{\tau})\leqslant\mathsf{T}_{l}(\mathcal{L}_{\tau})$.
Thus using the Cauchy–Schwarz inequality as in the proof of [9, Proposition 7]
(alternatively see the proof of Theorem 32 below), we obtain (30) (without any
condition on $\tau$). Thus Proposition 15 (estimate (30)) holds for such
defined $\mathcal{L}_{\tau}$ and hence Corollary 19 takes place as well. As
for an analogue of Lemma 10, we have for $m=m(\mathcal{L}_{\tau})$ and an
arbitrary even $k$ that

$\tau m\leqslant\sum_{\beta\in\mathcal{B}}\,\left|\sum_{x\in B}f_{A}(\alpha
x+\beta)\right|=\sum_{\beta\in\mathcal{B}}|r_{\alpha B-f_{A}}(\beta)|\leqslant
m^{1-1/k}(\mathsf{E}^{+}_{k}(\alpha f_{B},f_{A}))^{1/k}\leqslant$ $\leqslant
m^{1/2}(\mathsf{E}^{+}_{k}(f_{A})\mathsf{E}^{+}_{k}(f_{B}))^{1/4}\,.$

and similarly for $M(\mathcal{L}_{\tau})$. Finally, in [2] it was proved that

$\sigma=\sum_{l\in{\rm
Aff}(\mathbb{F}_{p})}\left|\sum_{x}f_{A}(x)f_{A}(lx)\right|^{4}\ll|A|^{5}\log|A|\,,$
(31)

where again the summation over $l$ in the last formula is taken over all
affine transformations having at least two points in $A\times A$. Actually,
(31) is equivalent to asymptotic formula (3). Thus we can repeat the
calculations in (23)—(25) and obtain

$\mathcal{E}\ll\sigma^{1/2}m^{1/6}_{*}|A|^{3}\log^{1/3}|A|+M^{1/2}_{*}|A|^{9/2}\log^{3/2}|A|\ll$
$\ll m^{1/6}_{*}|A|^{11/2}\log^{5/6}|A|+M^{1/2}_{*}|A|^{9/2}\log^{3/2}|A|\,,$

where $m_{*}=\mathsf{E}^{+}(f_{A})$,
$M_{*}=\overline{\mathsf{E}}^{\times}(f_{A})$. It remains to check the
condition $m({\mathcal{L}}_{\tau})|{\mathcal{L}}_{\tau}|\leqslant p^{2}$ as in
(26). Using these calculations, as well as (31), we see that the condition
$|A|\ll p/\log^{3}|A|$ is enough. Splitting our set $A$ if its needed and
loosing some logarithms, we arrive to (28) for all $A$.

Now let us obtain (29). The term $2|A|^{6}$ corresponds to vertical/horizontal
lines and since
$\mathsf{E}^{+}(f_{A})),\overline{\mathsf{E}}^{\times}(f_{A})\gg|A|^{2}$, it
follows that all appearing terms, which less than $|A|^{35/6}$ are negligible.
Let $i(l)=|l\cap(A\times A)|$ and below we consider just $l$ with $i(l)>1$.
Then $\sum_{l}i^{2}(l)=|A|^{4}-|A|^{2}$ and

$\sum_{l}i^{5}(l)=\sum_{l}i^{2}(l)\left(i(l)-\frac{|A|^{2}}{p}+\frac{|A|^{2}}{p}\right)^{3}=\sum_{l}i^{2}(l)\left(i(l)-\frac{|A|^{2}}{p}\right)^{3}+$
$+\frac{3|A|^{2}}{p}\sum_{l}i^{2}(l)\left(i(l)-\frac{|A|^{2}}{p}\right)^{2}+\frac{3|A|^{4}}{p^{2}}\left(\sum_{l}i^{3}(l)-\frac{|A|^{2}}{p}\sum_{l}i^{2}(l)\right)+\frac{|A|^{10}}{p^{3}}-\frac{|A|^{8}}{p^{3}}\,.$

It is easy to check that

$\sum_{l}i^{3}(l)-\frac{|A|^{2}}{p}\sum_{l}i^{2}(l)\leqslant{\mathcal{C}}_{3}(f_{A})\,.$

The sum $\sum_{l}i^{2}(l)\left(i(l)-\frac{|A|^{2}}{p}\right)^{2}$ was
estimated in [2, page 606], namely, splitting the summation over
$i(l)<2|A|^{2}/p$ and $i(l)\geqslant 2|A|^{2}/p$, as well as using (3), we get

$\sum_{l}i^{2}(l)\left(i(l)-\frac{|A|^{2}}{p}\right)^{2}\ll\left(\frac{|A|^{2}}{p}\right)^{2}p|A|^{2}+|A|^{5}\log|A|\ll|A|^{5}\log|A|$

and hence we have the remaining term $O(p^{-1}|A|^{7}\log|A|)$. Finally,

$\sum_{l}\left(i(l)-\frac{|A|^{2}}{p}+\frac{|A|^{2}}{p}\right)^{2}\left(i(l)-\frac{|A|^{2}}{p}\right)^{3}=\sum_{l}\left(i(l)-\frac{|A|^{2}}{p}\right)^{5}+\frac{2|A|^{2}}{p}\sum_{l}\left(i(l)-\frac{|A|^{2}}{p}\right)^{4}+$
$+\frac{|A|^{4}}{p^{2}}\sum_{l}\left(i(l)-\frac{|A|^{2}}{p}\right)^{3}=\mathcal{E}+O(p^{-1}|A|^{7}\log|A|+\frac{|A|^{4}}{p^{2}}{\mathcal{C}}_{3}(f_{A}))\,.$

Combining all bounds and estimate (28), we obtain (29). This completes the
proof. $\hfill\Box$

##  4 Some applications

The power saving in (28) (for sets with small additive/multiplicative energies
of its shifts) allows us to obtain our new incidence result Theorem 9 from the
introduction. We follow the method of the proof from [9, Proposition 7].

###### Theorem 13

Let $k\geqslant 2$ be an integer, $M\geqslant 1$ be a real number,
$A,B,C,X,Y\subseteq\mathbb{F}_{p}$, $0\notin X$ be sets, $|YC|\leqslant M|Y|$,
and $|C|\gtrsim_{k}M^{2^{k+1}}$, $|C|^{k-1}\geqslant(p/|Y|)^{2}$. Suppose that
$\max\\{\mathsf{E}^{+}(f_{B}),\overline{\mathsf{E}}^{\times}(f_{B})\\}\leqslant|B|^{3-\delta}$,
where $\delta>0$ is a constant. Then

$|\\{(a,b,x,y)\in A\times B\times X\times
Y~{}:~{}y=bx+a\\}|-\frac{|A||B||X||Y|}{p}\ll\sqrt{|A||B|}|X||Y|\cdot|B|^{-\frac{\delta}{35\cdot
2^{k}}}\,.$ (32)

P r o o f. At the beginning we repeat the arguments from the proofs of [9,
Proposition 8] and Proposition 15. Let ${\mathcal{L}}$ be the set of the lines
$\\{l_{x,y}\\}$, $x\in X$, $y\in Y$ defined as $y=bx+a$, so $a,b$ are
variables. Thus the equation from the left–hand side of (32) can be treated as
a question about incidences between points ${\cal P}=A\times B$ and lines
${\mathcal{L}}$. We have

${\cal I}(A\times B,\mathcal{L})=\frac{|A||B||\mathcal{L}|}{p}+\sum_{x\in
A}\sum_{l\in{\mathcal{L}}}f_{B}(lx)=\frac{|A||B||\mathcal{L}|}{p}+\sigma\,.$
(33)

Using the Hölder inequality several times, we get

$\sigma^{2^{k}}\leqslant|A|^{2^{k-1}}|B|^{2^{k-1}-1}\sum_{h}r_{(\mathcal{L}^{-1}\mathcal{L})^{2^{k-1}}}(h)\sum_{x}f_{B}(x)f_{B}(hx)\,.$
(34)

Assume that the summation in the last formula is taken over lines $h$, having
at least two points in $B\times B$, denote the rest as $\sigma_{*}$ and
suppose that $\sigma_{*}\leqslant 2^{-1}\sigma$, say. Then applying the Hölder
inequality one more time, combining with formula (28) of Theorem 29, we obtain

$\sigma^{5\cdot 2^{k}}\leqslant|A|^{5\cdot 2^{k-1}}|B|^{5\cdot
2^{k-1}-5}\mathsf{T}_{2^{k}}({\mathcal{L}})|{\mathcal{L}}|^{3\cdot
2^{k}}\cdot\sum_{h}\left|\sum_{x}f_{B}(x)f_{B}(hx)\right|^{5}\ll$
$\ll|A|^{5\cdot 2^{k-1}}|B|^{5\cdot
2^{k-1}-5}\mathsf{T}_{2^{k}}({\mathcal{L}})|{\mathcal{L}}|^{3\cdot
2^{k}}|B|^{6-\delta/7}\,.$

By Lemma 17, we know that
$\mathsf{T}_{2^{k}}({\mathcal{L}})\leqslant|X|^{2^{k+1}-1}\mathsf{T}^{+}_{2^{k}}(Y)$.
Using our conditions $|C|\gtrsim_{k}M^{2^{k+1}}$,
$|C|^{k-1}\geqslant(p/|Y|)^{2}$, as well as Theorem 16 for $A=Y$ and $B=C$ to
estimate the quantity $\mathsf{T}^{+}_{2^{k}}(Y)$, we obtain

$\sigma\ll\sqrt{|A||B|}|X||Y|(|B|^{1-\delta/7}/(|X|p))^{1/5\cdot
2^{k}}\leqslant\sqrt{|A||B|}|X||Y|\cdot|B|^{-\delta/(35\cdot 2^{k})}\,.$ (35)

It remains to estimate $\sigma_{*}$. Returning to (34), we see that

$\sum_{x}f_{B}(x)f_{B}(hx)=\sum_{x}B(x)B(hx)-\frac{|B|^{2}}{p}$

and hence

$\sigma^{5\cdot 2^{k}}_{*}\ll|A|^{5\cdot 2^{k-1}}|B|^{5\cdot
2^{k-1}-5}|\mathcal{L}|^{5\cdot 2^{k}}\,.$

and this is better than (35). This completes the proof. $\hfill\Box$

We now obtain another application, using our structural Theorem 6 (also, see
the discussion and the definitions after this result). We show that for all
sets having no special form one can obtain a good power saving for
${\mathcal{C}}_{n}(f_{A})$. It allows us to estimate nontrivially (in a rather
strong sense) some mixed energies of shifts of $A$, see Theorem 40 below.

###### Theorem 14

Let $A\subseteq\mathbb{F}$ be a set and $f_{A}(x)=A(x)-|A|/|\mathbb{F}|$.
Suppose that $A$ is not $\mathsf{E}^{+}$–exceptional and for any
$s\in\mathbb{F}$ the set $A-s$ is not $\mathsf{E}^{\times}$–exceptional with
$\delta,\varepsilon$. Then for any number
$n\geqslant\exp(C\varepsilon^{-1}\log(1/\delta))$, where $C>0$ is an absolute
constant one has

$\sum_{l\in{\rm
Aff}(\mathbb{F})}\left|\sum_{x}f_{A}(x)f_{A}(lx)\right|^{n}\ll|A|^{n+2/3+\delta}\,.$
(36)

P r o o f. Suppose that $n\geqslant 6$ is an even number and let
$\mathcal{E}_{n}$ be the left–hand side of (36), i.e. the $n$th moment of the
function $\sum_{x}f_{A}(x)f_{A}(lx)$. We mimic the calculations in (23)—(25)
and obtain

$\mathcal{E}_{n}\ll\sum_{j}2^{nj}|\mathcal{L}_{2^{j}}|\,.$ (37)

Now to estimate $|\mathcal{L}_{2^{j}}|$ we use formula (18) of Corollary 19
and Lemma 10 with $k_{1}=3n-16\geqslant 2$, $k_{2}=2n-8\geqslant 4$ (hence
$k_{1}$, $k_{2}$ are even numbers automatically) to get

$|\mathcal{L}_{\tau}|\ll\tau^{-(k_{1}+16)/3}(\mathsf{E}^{+}_{k_{1}}(f_{A}))^{1/3}|A|^{6}\log^{2/3}|A|+\tau^{-(k_{2}+8)/2}(\overline{\mathsf{E}}^{\times}_{k_{2}}(f_{A}))^{1/2}|A|^{9/2}\log^{1/2}|A|\,.$

Hence summing the last estimate over $j$ and using our basic bound (37), we
get

$\mathcal{E}_{n}\ll\sum_{j}2^{nj-j(k_{1}+16)/3}(\mathsf{E}^{+}_{k_{1}}(f_{A}))^{1/3}|A|^{6}\log^{2/3}|A|+\sum_{j}2^{nj-j(k_{2}+8)/2}(\overline{\mathsf{E}}^{\times}_{k_{2}}(f_{A}))^{1/2}|A|^{9/2}\log^{1/2}|A|$
$\ll(\mathsf{E}^{+}_{3n-16}(f_{A}))^{1/3}|A|^{6}\log^{5/3}|A|+(\overline{\mathsf{E}}^{\times}_{2n-8}(f_{A}))^{1/2}|A|^{9/2}\log^{3/2}|A|\,.$

Now applying Theorem 6 and the condition that $A-s$ is not exceptional for
both energies $\mathsf{E}^{+}$, $\mathsf{E}^{\times}$ with
$\delta,\varepsilon$, we derive

$\mathcal{E}_{n}\ll|A|^{n+2/3+\delta/3}\log^{5/3}|A|\ll|A|^{n+2/3+\delta/10}$

as required. $\hfill\Box$

Of course one can obtain an analogue of asymptotic formula (29) of Theorem 29
for larger $k$ but it requires some calculations and we leave it for the
interested reader.

Having sets $A,B\subset\mathbb{F}$ and an integer $k\geqslant 2$ let
$\tilde{{\mathcal{C}}}_{k}(A,B)$ be
${\mathcal{C}}_{k}(A,B)-|A||B|^{k}-|B||A|^{k}$, that is, we have deleted all
horizontal/vertical lines from our consideration. Also, we put

$\tilde{\mathsf{E}}^{\times}_{k}(A,B)=\left|\left\\{(a_{1},\dots,a_{k},b_{1},\dots,b_{k})\in(A\setminus\\{0\\})^{k}\times(B\setminus\\{0\\})^{k}~{}:~{}\frac{a_{1}}{b_{1}}=\dots=\frac{a_{k}}{b_{k}}\right\\}\right|\,.$

Then from the equation of a line $y=\lambda x+\mu$, intersecting $A\times B$,
we derive

$\tilde{{\mathcal{C}}}_{k}(A,B)+O_{k}(|A|^{2}|B|^{2}\cdot\left(\min\\{|A|,|B|\\}\right)^{k-3})=\sum_{\lambda\neq
0}\mathsf{E}^{+}_{k}(B,\lambda
A)=\sum_{\mu}\tilde{\mathsf{E}}^{\times}_{k}(B-\mu,A)$ (38)

similar to formula (13). As will we see from the proof of Theorem 40 there are
other expressions for ${\mathcal{C}}_{k}(A,B)$ in terms of higher energies
$\mathsf{E}_{k}$. Notice that in the symmetric case $A=B$ we always have
$\tilde{\mathsf{E}}^{\times}_{k}(A,A)\gg|A|^{k}$. In contrast, using our
paucity result, we obtain

###### Theorem 15

Let $A,B\subset\mathbb{F}_{p}$ be sets, $|A|=|B|\leqslant p^{2/3}$, and
$\mathsf{E}^{+}(A)$, $\overline{\mathsf{E}}^{\times}(A)\leqslant|A|^{3-c}$,
where $c\in[0,1)$. Then there are $b_{1},b_{2}\in B$ such that

$\tilde{\mathsf{E}}^{\times}_{4}(A-b_{1},A-b_{2})\ll|A|^{4-\frac{2c}{15}}\log|A|\,.$
(39)

Now suppose that our set $A$ is not $\mathsf{E}^{+}$–exceptional and for any
$s\in\mathbb{F}$ the set $A-s$ is not $\mathsf{E}^{\times}$–exceptional with
$\delta,\varepsilon$. Then for any number
$n\geqslant\exp(C\varepsilon^{-1}\log(1/\delta))$, where $C>0$ is an absolute
constant one can find a pair $b_{1},b_{2}\in B$ with

$\tilde{\mathsf{E}}^{\times}_{n}(f_{A}-b_{1},f_{A}-b_{2})\ll|A|^{n-1/3+\delta}\,.$
(40)

P r o o f. Let $L=\log|A|$. The number of collinear quintuplets
$(b_{1},b_{2}),(a_{1},a^{\prime}_{1}),\dots,(a_{4},a^{\prime}_{4})\in(B\times
B)\times(A\times A)^{4}$ equals the number $\sigma$ of the solutions to the
system

$\frac{a_{1}-b_{1}}{a^{\prime}_{1}-b_{2}}=\dots=\frac{a_{4}-b_{1}}{a^{\prime}_{4}-b_{2}}$

with non–zero numerators and denominators plus quintuplets, which correspond
to vertical/horizontal lines. Thus the sum $\sigma:=\sum_{b_{1},b_{2}\in
B}\tilde{\mathsf{E}}^{\times}_{4}(A-b_{1},A-b_{2})$ can be estimated via
formula (22) of Theorem 22. We cannot write
$\sigma\leqslant\tilde{{\mathcal{C}}}_{5}(A,A)^{4/5}\tilde{{\mathcal{C}}}_{5}(B,B)^{1/5}$
using the Hölder inequality because one can have $i_{A,A}(l)=1$, say.
Nevertheless, the contribution of these terms in view of formula (3) is at
most ${\mathcal{C}}_{4}(A\cup B)\ll|A|^{5}L$ and hence it is negligible. Thus
by estimate (22) and our restrictions $|A|=|B|\leqslant p^{2/3}$, we obtain

$\sum_{b_{1},b_{2}\in
B}\tilde{\mathsf{E}}^{\times}_{4}(A-b_{1},A-b_{2})\ll(|A|^{6-c/6}L^{5/6}+|A|^{5}L)^{4/5}(|A|^{6}L+|A|^{5}L)^{1/5}+|A|^{5}L\ll|A|^{6-2c/15}L$

and we have proved bound (39). To obtain (40) we just apply Theorem 36 instead
of Theorem 22. This completes the proof. $\hfill\Box$

Of course, in a similar way, using formula (38) and our bounds for
$\tilde{{\mathcal{C}}}_{k}(A,B)$, one can derive some upper bounds for
$\mathsf{E}^{+}_{k}(B,\lambda A)$, $\tilde{\mathsf{E}}^{\times}_{k}(B-\mu,A)$
for typical shifts $\lambda,\mu$.

## References

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I.D. Shkredov
Steklov Mathematical Institute,
ul. Gubkina, 8, Moscow, Russia, 119991
<EMAIL_ADDRESS>