aainstitutetext: CAS Key Laboratory of Theoretical Physics, Institute of
Theoretical Physics, Chinese Academy of Sciences, Beijing 100190,
P.R.Chinabbinstitutetext: Korea Institute for Advanced Study, Seoul 02455,
Koreaccinstitutetext: Department of Physics and Siyuan Laboratory, Jinan
University, Guangzhou 510632, P.R. China

# Electroweak Phase Transition in 2HDM under Higgs, Z-pole, and W precision
measurements

Huayang Song b , Wei Su c , Mengchao Zhang<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this work we revisit the existence of a strong first order electroweak
phase transition (SFOEWPT) and recent $m_{W}$ precision measurement in the
Type-I and Type-II 2HDMs. The $\mathcal{O}(100)~{}\textrm{GeV}$ new scalars in
2HDMs are favored by SFOEWPT, which is necessary for electroweak baryogenesis,
and observed $m_{W}$ shift as well. We find that under current constraints,
both Type-I and Type-II 2HDM can explain the SFOEWPT, Z-pole, Higgs precision
measurements and $m_{W}$ precision measurement of CDF-II at same time, and all
these precision measurements are sensitive to heavy Higgs mass splitting in
2HDM. The allowed regions are $\Delta
m_{A/C}\in(-400,400)~{}\textrm{GeV},\tan\beta\in(1,50)$, and $\Delta
m_{A/C}\in(-200,300)~{}\textrm{GeV},\tan\beta\in(1,12)$ for Type-I and Type-II
2HDM respectively. Furthermore future letpon collider measurements on Higgs
and $Z$ boson properties can explore this scenario in more detail or even rule
out it.

††preprint:  KIAS–P22020

##  1 Introduction

With the discovery of the Higgs boson at the Large Hadron Collider (LHC) in
2012 Aad:2012tfa ; Chatrchyan:2012xdj , particle physics has been entering a
new era. Due to the lack of direct search result at LHC, precision studies of
particle physics are becoming important. The current measurements of particle
properties seem to be consistent with all other categories of experiments and
can be described by the Standard Model (SM) quite well. Meanwhile there are
compelling arguments, both from theoretical and observational viewpoints, in
favor of new physics beyond the Standard Model (BSM). The CDF collaboration
has recently reported a precise measurement of the $W$ boson mass, which
indicates a significant tension with the previous measurements and the SM
prediction CDF:2022hxs . Although this result needs to be further confirmed by
other experiments, such as D0, ATLAS, and CMS, it is still an exciting
possible signal indicating the existence of new physics at a place not far
above the electroweak (EW) scale 111Recent study on the new CDF result see
Fan:2022dck ; Lu:2022bgw ; Athron:2022qpo ; Yuan:2022cpw ; Strumia:2022qkt ;
Yang:2022gvz ; deBlas:2022hdk ; Zhu:2022tpr ..

Given this possible signal, the following question is which new physics does
this possible signal point to. Among different kinds of BSM new physics,
electroweak baryogenesis (EWBG) Cline:2006ts ; Morrissey:2012db  is likely to
be relevant to the current possible signal. EWBG was proposed to explain the
observed baryon asymmetry of the universe (BAU). Through the baryon number
breaking sphaleron process Manton:1983nd ; Klinkhamer:1984di ; Kuzmin:1985mm
and CP violated scattering with bubble wall, net baryon number can be produced
during the nucleation process of Higgs field. To trigger the nucleation
process and to prevent the generated net baryon being washed out, the
electroweak phase transition (EWPT) needs to be a strong first order phase
transition. However, due to current measured Higgs mass, the SM EWPT is not
even first order Kajantie:1996mn ; Csikor:1998eu .

Therefore new particles are definitely required for a strong first order
electroweak phase transition (SFOEWPT). Furthermore, new particles that help
to trigger SFOEWPT can not be too heavy than EW scale, otherwise they will be
decoupled in the thermal phase transition process and lose effect. Thus, the
new measure W boson mass might be a hint of EWBG.

One method to trigger SFOEWPT is augmenting the SM Higgs sector via additional
scalars, and it has been studied intensively in the literature Carena:2018vpt
; Cline:2012hg ; Cline:2017qpe ; Carena:2018cjh ; Cline:2009sn ; Moore:1998swa
; Coleman:1973jx ; Arnold:1992rz ; Su:2020pjw ; Nielsen:1975fs ;
Kainulainen:2019kyp ; Huang:2015bta ; Huang:2018aja . Extending the Higgs
sector by a singlet scalar seems to be the simplest choice, but it is
difficult for such model to explain the observed $m_{W}$ under current limits
Lopez-Val:2014jva . In this work we choose Two-Higgs-Doublet Models (2HDMs)
Lee:1973iz ; Branco:2011iw , which extend the SM Higgs sector by another
doublet, as the benchmark model to study the relationship between SFOEWPT and
$m_{W}$. After electroweak symmetry breaking, in addition to the SM-like Higgs
boson $h$, there are three non-SM Higgs bosons, $H$/$A$/$H^{\pm}$, which can
have masses below TeV and couple to the SM-like Higgs $h$ to build an energy
barrier between the symmetric and broken phase. Therefore the EW phase
transition can be first order and strong enough for the baryogenesis
Su:2020pjw . Furthermore, these light extra scalar can induce a positive
$m_{W}$ shift Lopez-Val:2012uou  and modify the predictions on the Z-pole
observables like the oblique parameters $S$, $T$ and $U$ via one-loop
contributions to the $W$ and $Z$ self-energies. The mixing between neutral
scalars and extra loop corrections further reduce the Higgs couplings
$\kappa_{i}=g_{hii}^{\rm 2HDM}/g_{hii}^{\rm SM}$ relative to their SM
expectations. Though the LHC measurements still give a large amount of
available phase space of the 2HDM, future Higgs factories, e.g. ILC
Bambade:2019fyw , FCC-ee Abada:2019lih ; Abada:2019zxq  and CEPC
CEPCStudyGroup:2018ghi ; CEPCPhysics-DetectorStudyGroup:2019wir  can measure
them with unprecedented precision to further constrain the model.

In this paper, we study the constraints from precision measurements
(especially the new $W$ boson mass and future Higgs coupling measurements) on
the 2HDM and explore the possible parameter space which could lead to a
SFOEWPT. Our study shows that SFOEWPT is consistent with the new uplifted
$m_{W}$ in a certain parameter space. But due to the close connection between
$m_{W}$ and other precise measurements, the “SFOEWPT + $m_{W}$” scenario is in
slight tension with current limits. Furthermore, the future precision lepton
collider measurements of both Higgs and $Z$ boson properties could fully rule
out the alive parameters to fulfil SFOEWPT and $m_{W}$, provided the measured
central values locate in the SM prediction. Conversely, if the “SFOEWPT +
$m_{W}$” scenario in 2HDM is true, than a clear deviation from SM prediction
will be observed at future lepton colliders.

The rest of the paper is organized as follows. In Section 2 we briefly
introduce 2HDM models and related constraints. Description on EWPT is also
given in Section 2. In Section 3 we perform parameter space scan on a wide
range and present alive points, with future precise measurements included. We
conclude this work in Section 4.

##  2 2HDM

###  2.1 A Review

In this section, we provide a brief review of the aspects of 2HDMs. For
pedagogical introduction, see Ref. Branco:2011iw  and a recent review Ref.
Wang:2022yhm  in light of current experiments. The scalar sector of 2HDMs
consists two ${\rm SU}(2)_{L}$ doublets $\Phi_{i}$, $i=1,2$, which can be
parameterized as below,

$\Phi_{i}=\left(\begin{matrix}\phi_{i}^{+}\\\
(v_{i}+\phi_{i}^{0}+iG_{i})/\sqrt{2}\end{matrix}\right)$ (1)

where $v_{1}$ and $v_{2}$ are the vacuum expectation values (VEVs) of the
neutral components, satisfying the relation
$v\equiv\sqrt{v_{1}^{2}+v_{2}^{2}}=246~{}\textrm{GeV}$. Assuming CP-conserving
and only a soft breaking of a discrete $\mathcal{Z}_{2}$ symmetry allowed, the
most general Higgs potential can be expressed as,

$\displaystyle V^{\text{0}}(\Phi_{1},\Phi_{2})$ $\displaystyle=$
$\displaystyle
m^{2}_{11}\Phi^{\dagger}_{1}\Phi_{1}+m^{2}_{22}\Phi^{\dagger}_{2}\Phi_{2}-m^{2}_{12}\left(\Phi^{\dagger}_{1}\Phi_{2}+h.c.\right)+\frac{\lambda_{1}}{2}\left(\Phi^{\dagger}_{1}\Phi_{1}\right)^{2}+\frac{\lambda_{2}}{2}\left(\Phi^{\dagger}_{2}\Phi_{2}\right)^{2}$
(2)
$\displaystyle+\lambda_{3}\left(\Phi^{\dagger}_{1}\Phi_{1}\right)\left(\Phi^{\dagger}_{2}\Phi_{2}\right)+\lambda_{4}\left(\Phi^{\dagger}_{1}\Phi_{2}\right)\left(\Phi^{\dagger}_{2}\Phi_{1}\right)+\frac{\lambda_{5}}{2}\left[\left(\Phi^{\dagger}_{1}\Phi_{2}\right)^{2}+h.c.\right].$

where there are eight real parameters,
$\\{m_{11}^{2},m_{22}^{2},m_{12}^{2},\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\lambda_{5}\\}$.
After the electroweak symmetry breaking (EWSB), the scalar sector of a 2HDM
consists of five mass eigenstates: a pair of neutral CP-even Higgses, $h$ and
$H$, a CP-odd Higgs, $A$, and a pair of charged Higgses $H^{\pm}$. We can
express these states as,

$\displaystyle\centering h$
$\displaystyle=-s_{\alpha}\,\phi_{1}+c_{\alpha}\,\phi_{2},\quad\quad\quad$
$\displaystyle A=$ $\displaystyle-
s_{\beta}\,\varphi_{1}\,+c_{\beta}\,\varphi_{2},$ (3) $\displaystyle H$
$\displaystyle=\phantom{-}c_{\alpha}\,\phi_{1}+s_{\alpha}\,\phi_{2},\quad\quad\quad$
$\displaystyle H^{\pm}=$ $\displaystyle-
s_{\beta}\,\phi_{1}^{\pm}+c_{\beta}\,\phi_{2}^{\pm}.$

where we will identify $h$ with the discovered SM-like 125 GeV Higgs without
loss of generality.

For convenience, we will parametrize the potential of 2HDMs by the physical
Higgs masses $m_{h}$, $m_{H}$, $m_{A}$ and $m_{H^{\pm}}$, the mixing angle
between the two CP-even Higgses $\alpha$, $\tan\beta\equiv v_{2}/v_{1}$, the
electroweak VEV $v$, and the soft $\mathcal{Z}_{2}$ symmetry breaking
parameter $m_{12}^{2}$. Note that the vacuum expectation value $v$ and the
mass of the SM-like Higgs, $m_{h}$ are fixed to their known values 246 GeV and
125 GeV respectively, leaving the remaining six independent parameters.

Assigning different $\mathcal{Z}_{2}$ parities to the SM fermions, there are
four types of 2HDMs. However, in this study, we focus on the so-called Type-I
and Type-II 2HDMs, where all fermions obtain their masses from a single Higgs
doublet in Type-I model while up- and down-type fermions obtain their masses
from differnt Higgs doublets in Type-II model. In the Type-II model the
couplings between $A/H$ and down-type fermions are enhanced by $\tan\beta$ and
therefore it is usually more constrained by experiments when $\tan\beta$ is
large.

###  2.2 Theoretical Constraints on 2HDMs

The parameter spaces of 2HDMs are already constrained by theoretical
consideration without experimental results.

  * •

Vacuum stability In order to make the vacuum stable, the scalar potential
should be bounded from below Gunion:2002zf :

$\displaystyle\lambda_{1}>0\ ,\ \lambda_{2}>0\ ,\
\lambda_{3}>-\sqrt{\lambda_{1}\lambda_{2}}\ ,\
\lambda_{3}+\lambda_{4}-|\lambda_{5}|>-\sqrt{\lambda_{1}\lambda_{2}}$ (4)

  * •

Perturbativity and unitarity Requiring perturbativity, we must have
$|\lambda_{i}|\leq 4\pi$. And requiring tree-level unitarity of the scattering
in the 2HDM scalar sector imposes the following additional mass constraints
Ginzburg:2005dt :

$\displaystyle\left|3(\lambda_{1}+\lambda_{2})\pm\sqrt{9(\lambda_{1}-\lambda_{2})^{2}+4(2\lambda_{3}+\lambda_{4})^{2}}\right|<16\pi\
,\ $ (5)
$\displaystyle\left|(\lambda_{1}+\lambda_{2})\pm\sqrt{(\lambda_{1}-\lambda_{2})^{2}+4\lambda_{4}^{2}}\right|<16\pi\
,\ $ (6)
$\displaystyle\left|(\lambda_{1}+\lambda_{2})\pm\sqrt{(\lambda_{1}-\lambda_{2})^{2}+4\lambda_{5}^{2}}\right|<16\pi\
,\ $ (7) $\displaystyle\left|\lambda_{3}+2\lambda_{4}\pm
3\lambda_{5}\right|<8\pi\ ,\ \left|\lambda_{3}\pm\lambda_{4}\right|<8\pi\ ,\
\left|\lambda_{3}\pm\lambda_{5}\right|<8\pi$ (8)

To understand the these constraints, it is useful to consider the relations
between the quartic couplings and the physical masses

$\displaystyle v^{2}\lambda_{1}$
$\displaystyle=m_{h}^{2}-\frac{t_{\beta}\left(m_{12}^{2}-m_{H}^{2}s_{\beta}c_{\beta}\right)}{c_{\beta}^{2}}+\left(m_{h}^{2}-m_{H}^{2}\right)\left[c_{\beta-\alpha}^{2}\left(t_{\beta}^{2}-1\right)-2t_{\beta}s_{\beta-\alpha}c_{\beta-\alpha}\right],$
$\displaystyle v^{2}\lambda_{2}$
$\displaystyle=m_{h}^{2}-\frac{m_{12}^{2}-m_{H}^{2}s_{\beta}c_{\beta}}{t_{\beta}s_{\beta}^{2}}+\left(m_{h}^{2}-m_{H}^{2}\right)\left[c_{\beta-\alpha}^{2}\left(t_{\beta}^{-2}-1\right)+2t_{\beta}^{-1}s_{\beta-\alpha}c_{\beta-\alpha}\right],$
$\displaystyle v^{2}\lambda_{3}$
$\displaystyle=m_{h}^{2}+2m_{H^{\pm}}^{2}-2m_{H}^{2}-\frac{m_{12}^{2}-m_{H}^{2}s_{\beta}c_{\beta}}{s_{\beta}c_{\beta}}-\left(m_{h}^{2}-m_{H}^{2}\right)\left[2c_{\beta-\alpha}^{2}+s_{\beta-\alpha}c_{\beta-\alpha}\left(t_{\beta}-t_{\beta}^{-1}\right)\right],$
$\displaystyle v^{2}\lambda_{4}$
$\displaystyle=m_{A}^{2}-2m_{H^{\pm}}^{2}+m_{H}^{2}-\frac{m_{12}^{2}-m_{H}^{2}s_{\beta}c_{\beta}}{s_{\beta}c_{\beta}},$
$\displaystyle v^{2}\lambda_{5}$
$\displaystyle=m_{H}^{2}-m_{A}^{2}-\frac{m_{12}^{2}-m_{H}^{2}s_{\beta}c_{\beta}}{s_{\beta}c_{\beta}}.$
(9)

We can introduce $\lambda v^{2}\equiv
m_{H}^{2}-m^{2}_{12}/(s_{\beta}c_{\beta})$ following Ref. Gu:2017ckc . The
above expression indicates that the unitarity and perturbativity set up upper
bounds on the mass splittings, which can be roughly taken as $\lambda
v^{2}<4\pi v^{2}$, $m_{A}^{2}-m_{H}^{2}\lesssim\mathcal{O}\left(4\pi
v^{2}-\lambda v^{2}\right)$,
$m_{H^{\pm}}^{2}-m_{H}^{2}\lesssim\mathcal{O}\left(4\pi v^{2}-\lambda
v^{2}\right)$ and $\max\\{t_{\beta},\cot\beta\\}\lesssim\sqrt{(8\pi
v^{2})/(3\lambda v^{2})}$. Generally speaking, large mass splitting among non-
SM Higgses are not allowed for large values of $\lambda v^{2}$ and/or non-SM
Higgs masses.

###  2.3 Direct searches at LEP and LHC

The search for pair-produced charged Higgs bosons at the Large Electron-
Positron Collider (LEP) imposes a lower bound of 80 GeV on the mass of the
charged Higgs boson ALEPH:2013htx , and LEP searches for $AH$ production
constrain the sum of the masses $m_{H}+m_{A}>209~{}\textrm{GeV}$ Schael:2006cr
.

LHC are also looking for direct productions of exotic Higgses via including
$A/H\rightarrow\mu\mu$ Sirunyan:2019tkw ; Aaboud:2019sgt , $A/H\rightarrow bb$
Sirunyan:2018taj ; Aad:2019zwb , $A/H\rightarrow\tau\tau$ Sirunyan:2018zut ;
CMS:2019hvr ; Aad:2020zxo , $A/H\rightarrow\gamma\gamma$ Sirunyan:2018aui ;
Sirunyan:2018wnk ; Aad:2014ioa ; Aaboud:2017yyg ; ATLAS:2018xad ,
$A/H\rightarrow tt$ Sirunyan:2019wph , $H\rightarrow ZZ$ Sirunyan:2018qlb ;
Aaboud:2017rel , $H\rightarrow WW$ Sirunyan:2019pqw ; Aaboud:2017gsl ,
$A\rightarrow hZ\rightarrow bb\ell\ell$ Khachatryan:2015lba ; Sirunyan:2019xls
; Aad:2015wra ; Aaboud:2017cxo , $A\rightarrow hZ\rightarrow\tau\tau\ell\ell$
Khachatryan:2015tha ; Sirunyan:2019xjg ; Aad:2015wra , $H\rightarrow hh$
Sirunyan:2017tqo ; Sirunyan:2018two ; Aad:2015xja ; Aad:2019uzh , and
$A/H\\!\rightarrow\\!HZ\\!/\\!AZ$ Aaboud:2018eoy ; Sirunyan:2019wrn . The null
results have already ruled out a significant portion of parameter space of
2HDM. For a typical mass splitting $m_{A}-m_{H}=m_{H^{\pm}}-m_{H}=200$ GeV ,
the exotic decay channel $A\rightarrow HZ$ has already excluded a neutral
Higgs with mass less than $2m_{t}$ for $\tan\beta<5$ in Type-I model and for
$0.5<\tan\beta<15$ in Type-II model. For large $\tan\beta$ region
($\tan\beta>15$) in Type-II model, this channel put the mass of neutral scalar
$H$ to be above 600 GeV. Top quarks search channels, $4t$ and $A/H\rightarrow
tt$, rule out $m_{H}<800~{}\textrm{GeV}$ for $\tan\beta<0.3$ and
$m_{H}<650~{}\textrm{GeV}$ for $\tan\beta<1.1$ in both two types of 2HDMs.
While $A/H\rightarrow\tau\tau,\gamma\gamma$ can exclude the region
$m<350~{}\textrm{GeV},\tan\beta<1$ in Type-I and -II models,
$A/H\rightarrow\tau\tau$ could fully exclude $m_{H}$ larger than 800 GeV when
$\tan\beta>10$ in Type-II 2HDM. For a complete recasting the LHC direct search
results in the 2HDM, we refer the readers to Ref. Kling:2020hmi ; Su:2020pjw ;
Wang:2022yhm .

###  2.4 Z-pole and Higgs precision measurements

Measurements of Z-pole observables at the Large Electron-Positron Collider
(LEP) impose strong constraints on the 2HDM ALEPH:2005ab . Satisfying Z-pole
constraints requires the charged scalar mass to be close to one of the heavy
neutral scalar masses: $m_{H^{\pm}}\simeq m_{H}$ or $m_{H^{\pm}}\simeq m_{A}$.
In our analysis, we simply take the $S,T,U$ data at $95\%$ Confidence Level
(C.L.) in Tab. 1 to capture the dominant contributions from Z-pole
measurements. Note that a global analysis to recast the $S,T,U$ parameters is
needed by including both Z-pole observables and latest $W$ mass measurement
CDF:2022hxs , here for simplicity we just take them as two separate
measurements and are going to discuss more on the $W$ boson mass effects in
Sec. 2.5.

| Current | CEPC | FCC-ee | ILC
---|---|---|---|---
| $\sigma$ | correlation | $\sigma$ | correlation | $\sigma$ | correlation | $\sigma$ | correlation
$S$ | $T$ | $U$ | ($10^{-2}$) | $S$ | $T$ | $U$ | ($10^{-2}$) | $S$ | $T$ | $U$ | ($10^{-2}$) | $S$ | $T$ | $U$
$S$ | $0.04\pm 0.11$ | $1$ | $0.92$ | $-0.68$ | $1.82$ | $1$ | $0.9963$ | $-0.9745$ | $0.370$ | $1$ | $0.9898$ | $-0.8394$ | $2.57$ | $1$ | $0.9947$ | $-0.9431$
$T$ | $0.09\pm 0.14$ | $-$ | $1$ | $-0.87$ | $2.56$ | $-$ | $1$ | $-0.9844$ | $0.514$ | $-$ | $1$ | $-0.8636$ | $3.59$ | $-$ | $1$ | $-0.9569$
$U$ | $-0.02\pm 0.11$ | $-$ | $-$ | $1$ | $1.83$ | $-$ | $-$ | $1$ | $0.416$ | $-$ | $-$ | $1$ | $2.64$ | $-$ | $-$ | $1$

Table 1: Estimated $S$, $T$, and $U$ ranges and correlation matrices
$\rho_{ij}$ from Z-pole precision measurements of the current results
Chen:2018shg .

Figure 1: Current oblique constraints $S,T,U$ in the plane of $\Delta
m_{A}-\Delta m_{C}$ with $m_{H}=700$ GeV, $\tan\beta=3$ in the alignment limit
$\cos(\beta-\alpha)=0$. The colors are for corresponding parameter value.

To reveal the relation between non-SM Higgs spectra and $S,T,U$, we define
following mass splitting parameters:

$\Delta m_{A}=m_{A}-m_{H},\Delta m_{C}=m_{H^{\pm}}-m_{H}$ (10)

In Fig. missing 1 we present the $S,T,U$ deviation from their SM value as
functions of $\Delta m_{A}$ and $\Delta m_{C}$. Current measure uncertainty
for $S,T,U$ are around 10%, so it can be seen that $T$ parameter provides the
most stringent limit on non-SM Higgs mass splitting.

The LHC has also performed high precision tests on the Higgs couplings, which
indicates all the measurable couplings $\kappa_{i}$ are close to their SM
values. Future Higgs factories will further improve the precision of
measurements in the Higgs sector, and we therefore include hypothetical future
lepton collider results in our study. We adopt the Higgs measurements results
presented in Table 3 in Ref. Chen:2019pkq . Note that for future experiments,
we assume there is no deviation from the SM in Higgs measurements.

###  2.5 $m_{W}$ in the 2HDM

As an important observable used in the SM precise test, $m_{W}$ is closely
related to Z boson mass $m_{Z}$, Fermi constant $G_{F}$, and fine structure
constant $\alpha$

$\displaystyle
m^{2}_{W}\left(1-\frac{m^{2}_{W}}{m^{2}_{Z}}\right)=\frac{\pi\alpha}{\sqrt{2}G_{F}}(1+\Delta
r)$ (11)

where $\Delta r$ corresponds to quantum corrections Lopez-Val:2012uou . In the
2HDM, $m_{W}$ correction can be represented by Sirlin:2012mh

$m_{W}^{\rm 2HDM}=m_{W}^{\rm
SM}\left[1+\frac{\alpha{c_{W}}^{2}}{2\left({c_{W}}^{2}-{s_{W}}^{2}\right)}T(1+\delta\rho^{\rm
2HDM})+\frac{{\alpha}}{8{s_{W}}^{2}}U-\frac{{\alpha}}{4\left({c_{W}}^{2}-{s_{W}}^{2}\right)}S\right]$
(12)

to the $\mathcal{O}(\alpha^{2})$. Here $m_{W}^{\rm
SM}=80.357\mathrm{~{}GeV}\pm 4_{\text{inputs }}\pm$ $4_{\text{theory
}}\mathrm{MeV}$ CDF:2022hxs , $S,T,U$ are oblique parameters in the 2HDM
Chen:2018shg , and $\delta\rho^{\rm 2HDM}=\frac{|\lambda_{hhh}^{\rm
2HDM}|^{2}}{16\pi^{2}m_{h}^{2}}$ are higher order 2HDM effects from enhanced
Higgs boson self-interactions. Currently $\kappa_{hhh}=\lambda_{hhh}^{\rm
2HDM}/\lambda_{hhh}^{\rm SM}\in(-1.0,6.6)$ ATLAS:2021tyg  is already strongly
constrained, thus the higher order effect $\delta\rho^{\rm 2HDM}$ up to
$\mathcal{O}(0.01)$ is weak.

Figure 2: (Left): general picture of $\Delta m_{W}^{\rm 2HDM}$ in the plane of
$\Delta T-\Delta S$ with $\Delta U=0$. The colors are values of $\Delta
m_{W}^{\rm 2HDM}$, varying from -50 MeV to 50 MeV. We have 5 black dash-dotted
lines for $\Delta m_{W}^{\rm 2HDM}=-40,-20,0,20,$ and 40 MeV. (Right):
$m_{W}^{\rm 2HDM}$ in the plane of $\Delta m_{A}-\Delta m_{C}$ with same
benchmark spectrum $m_{H}=700$ GeV as Fig. missing 1. The colors are same to
the left panel.

For convenient, we define

$\Delta m_{W}^{\rm 2HDM}=m_{W}^{\rm 2HDM}-m_{W}^{\rm SM}$ (13)

In the left panel of Fig. missing 2 where we take $\Delta U=0$ as a benchmark
case, we show the general picture of $\Delta m_{W}^{\rm 2HDM}$ in the plane of
$\Delta T-\Delta S$ based on the allowed region shown in the Fig. missing 1.
The colors show values of $\Delta m_{W}^{\rm 2HDM}$, varying from -50 MeV to
50 MeV. We have 5 black dash-dotted lines for $\Delta m_{W}^{\rm
2HDM}=-40,-20,0,20,$ and 40 MeV. Generally speaking, $\Delta m_{W}^{\rm 2HDM}$
mainly depends on $\Delta T$, and the larger $\Delta T$ and smaller $\Delta S$
result in large $\Delta m_{W}^{\rm 2HDM}$. This result can be easily
understood with the sizes of the coefficients in front of $S,T,U$ in Eq.
(missing) 12. In the right panel, we take the benchmark spectrum of Fig.
missing 1 with $m_{H}=700$ GeV and $\tan\beta=3$, and show $\Delta m_{W}^{\rm
2HDM}$ in the plane of $\Delta m_{A}-\Delta m_{C}$. We can see that, under
current various constraints, the benchmark spectrum here can supply for the
theoretical correction meeting the new experimental measurement at CDF-II
CDF:2022hxs . Since in the 2HDM, $\Delta m_{W}^{\rm 2HDM}$ is directly
relevant to oblique parameters, and oblique parameters further depend on non-
SM Higgs mass splitting $\Delta m_{A}$ and $\Delta m_{C}$. So it is clear that
$\Delta m_{W}^{\rm 2HDM}$ is sensitive to non-SM Higgs mass splitting.

###  2.6 Flavour constraints

The charged Higgs $H^{\pm}$ boson couples to both up and down type fermions,
which can lead to flavor changing processes strongly constrained by flavor
physics observations. The most stringent of limits comes from the measurements
of B-meson decays (e.g. $b\rightarrow s\gamma$ and $B^{+}\rightarrow\tau\nu$),
which disfavor $m_{H^{\pm}}<580$ GeV and large values of $\tan\beta$
respectively in Type-II 2HDM Misiak:2017bgg ; Arbey:2017gmh , or
$m_{H^{\pm}}<1$ TeV and small values of $\tan\beta$ ($\tan\beta<1$) in Type-I
model Arbey:2017gmh . However, flavor constraints can be alleviated with
contributions from other sectors of the BSM models Han:2013mga . In this
paper, we only focus on the constraints from $B$-physics on the scalar sector.

###  2.7 Phase transition in the 2HDM

To study EWPT, we need to know the thermal effective potential, which is the
free-energy density, as the function of scalar VEVs. The thermal effective
potential $V(\phi_{1},\phi_{2},T)$ can be schematically expressed as:

$\displaystyle
V(\phi_{1},\phi_{2},T)=V^{\text{0}}(\phi_{1},\phi_{2})+V^{\text{CW}}(\phi_{1},\phi_{2})+V^{\text{CT}}(\phi_{1},\phi_{2})+V^{\text{T}}(\phi_{1},\phi_{2},T).$
(14)

Here $\phi_{i}$ are scalar VEVs, $T$ is the temperature of thermal system,
$V^{\text{0}}(\phi_{1},\phi_{2})$ is the tree-level potential,
$V^{\text{CW}}(\phi_{1},\phi_{2})$ and $V^{\text{CT}}(\phi_{1},\phi_{2})$ are
Coleman-Weinberg potential and counter term respectively, and
$V^{\text{T}}(\phi_{1},\phi_{2},T)$ is thermal correction. Detailed formulas
can be found in the literature Coleman:1973jx ; Arnold:1992rz .

In the very early universe, temperature $T$ is much higher than all the
particles’ mass in our model. The large effective thermal mass keep $\phi_{i}$
at zero and thus maintain the EW-symmetry. And when $T$ become much lower than
EW scale, the global minimum position of $V(\phi_{1},\phi_{2},T)$ on
$\phi_{1}-\phi_{2}$ plane must move to a place where
$\phi^{2}_{1}+\phi^{2}_{2}\neq 0$ to break EW-symmetry. To know whether this
phase transition process is first-order, we can track the minimum point with
$T$ decreasing. If the minimum point (which locates in zero point when $T$ is
very large) “jump to” a non-zero point discontinuously at critical temperature
$T_{c}$, then the EWPT should be first-order. This method has been numerically
implemented in public package BSMPT Basler:2018cwe . We will use this package
in this work.

Furthermore, to prevent baryon number being washed out inside Higgs bubble,
the “wash out” parameter Moore:1998swa  $\xi_{c}\equiv{v_{c}}/{T_{c}}$
($v_{c}$ is the Higgs VEV at $T_{c}$) should roughly be larger than 1.
Considering the uncertainty in $\xi_{c}$ calculation Nielsen:1975fs ;
Kainulainen:2019kyp , we use a slightly looser criteria for SFOEWPT:

$\displaystyle\xi_{c}\equiv\frac{v_{c}}{T_{c}}>0.9$ (15)

##  3 Study results

In this section, we try to explore the SFOEWPT under various current and
future constraints. Specially we have a detailed study about the latest
$m_{W}$ result at CDF-II.

We will firstly have a large amount of random scan points, and our study
include the theoretical constraints, $B$-physics, LHC Run-II direct searches,
current precision measurement of Higgs and Z-pole physics. Then the further
study is performed at future Higgs factories, including CEPC, FCC-ee and ILC
as shown in Tab. missing 1, to confront the SFOEWPT and $m_{W}$ anomaly. Our
study of Higgs precision measurements will deep into one-loop level.

###  3.1 Study method

We perform a 6 parameters random scan for both Type-I and Type-II, and the
scan regions are :

$\displaystyle\tan\beta\in(0.2,50),|\cos(\beta-\alpha)|<0.5\,,m_{A/H^{\pm}}\in(10,1500)\text{
GeV}\ ,$ $\displaystyle m^{2}_{12}\in(0,1500^{2})\text{ GeV}^{2},\
m_{H}\in(130,1500)\textrm{GeV}.$ (16)

The number of samples allowed by current various (except for $m_{W}$ from CDF-
II) is more than 1 million ( of 1 billion points in total) . After considering
the SFOEWPT, it is about a few hundreds of thousands points allowed for
Type-I, but much less for Type-II to be shown in Fig. missing 3 as grey dots.

Figure 3: The allowed parameter space in the plane of $m_{H}-\tan\beta$
(left), $\Delta m_{A}-\Delta m_{C}$ (right). The grey points survive all
theoretical constraints, current experimental constraints, and the conditions
of SFOEWPT. The top and bottom panels are for Type-I and Type-II respectively.
The green ones are able to provide a $m_{W}$ by CDF-II, while the red ones are
allowed by future Higgs and Z-pole precision measurements from CEPC. The red
and green points do not cover each other.

To incorporate in the $m_{W}$ at CDF-II, here in the 2HDM based on Eq.
(missing) 12, we take the $m_{W}$ data at $95\%$ Confidence Level (C.L.) with
the $\chi^{2}$ profile-likelihood fit,

$\chi^{2}=\frac{(m_{W}^{\rm{2HDM}}-m_{W}^{\rm{obs}})^{2}}{\sum\sigma_{m_{W}}^{2}}\,,$
(17)

After taking into account of experiment uncertainties, we have

$\Delta m_{W}^{\rm 2HDM}|_{\rm ex}\in(36.3,103.4){\rm~{}MeV},$ (18)

and if considering SM theoretical uncertainties as well, it is $\Delta
m_{W}^{\rm 2HDM}|_{\rm th+ex}\in(31.1,108.9){\rm~{}MeV}$ for the 6 parameter
scan at $95\%$ C.L.. We will only take $\Delta m_{W}^{\rm 2HDM}|_{\rm ex}$ as
condition of $m_{W}$ for CDF-II in the following study222There is no apparent
difference found for $\Delta m_{W}^{\rm 2HDM}|_{\rm ex}$ and $\Delta
m_{W}^{\rm 2HDM}|_{\rm th+ex}$ in our study..

###  3.2 SFOEWPT under LHC measurements

As discussed above, after scanning the entire parameter space of Type-I and
Type-II 2HDM, we obtain the sector which is allowed by current limits and also
satisfy the SFOEWPT requirement.

As shown in Fig. missing 3, the grey points meet all the conditions of current
various measurements(except for $m_{W}$ at CDF-II) and SFOEWPT inboth types.
For Type-I and Type-II, favored mass region are different:

  * •

Type-I: To satisfy SFOEWPT and current limits, $m_{H}$ distributes in region
$(125,1000)$ GeV with $\tan\beta$ varying from 1 to 50. Mass splitting
$m_{H^{\pm}}-m_{H}$ and $m_{A}-m_{H}$ distribute in region $(-400,400)$ GeV.

  * •

Type-II: Compared with Type-I, Type-II is more limited due to the limited
$\tan\beta$ region. $m_{H}$ distributes in region $(125,1000)$ GeV, while
$\tan\beta$ is limited to $(1,12)$. Mass splitting $m_{H^{\pm}}-m_{H}$ and
$m_{A}-m_{H}$ vary in region $(-200,300)$ GeV.

To summarize, SFOEWPT require the mass of non-SM Higgs $H/A/H^{\pm}$ to be
smaller than 1 TeV and a certain mount of splitting between them. Comparing
with the right panel of Fig. missing 2, it is clear that the uplifted $m_{W}$
is consistent with SFOEWPT requirement.

###  3.3 SFOEWPT under Higgs, Z and W precision measurements

As discussed above, SFOEWPT, Higgs precision measurements at one-loop level,
and Z-pole physics (oblique parameter $S,T,U$) are all connected by heavy
Higgs mass splitting. In more detail, Fig. missing 2 and Eq. (missing) 12
tells that non-zero $\Delta S/T$ is needed for uplifting $m_{W}$. But Higgs
and Z-pole physics have strong constraints on the value of $\Delta S/T$. As
presented in Fig. missing 3, the green points meets all these Higgs, Z-pole,
$m_{W}$ and SFOEWPT conditions. Compared to the grey points, we can see the
allowed $\tan\beta,m_{H},\Delta m_{A}$ and $\Delta m_{C}$ region of green
points does not change a lot. Another feature is they mainly locate around the
boundary region, which is is mainly because current electroweak measurements
is not precise enough, so the uplifted $m_{W}$ in need can still be satisfied
within 2HDM framework. As shown in our benchmark case Fig. missing 1, specific
$\Delta T$ is in need. Since oblique parameters is type universal, thus Type-I
and Type-II have similar features for green region.

However, future lepton colliders, such as CEPC, ILC, and FCC-ee, will measure
electroweak parameters to unprecedented precision. As presented in Tab. 1,
uncertainties of oblique measurements in CEPC can be reduced to 1% level,
which one order smaller than current uncertainties. For the Higgs precision
measurements, the works Gu:2017ckc ; Chen:2019pkq ; Chen:2018shg  have
discussed them for the case of 2HDM systematically. Provided that there is no
apparent deviation of Higgs and Z-pole proprieties to the SM predictions
observed, or in other words, the future measurements turn out to be consistent
with SM prediction, we take CEPC precision measurements as an example to study
the impact from future lepton colliders. Finally as shown in Fig. missing 3,
the red points represent spectrum meeting conditions of Higgs, Z-pole
measurements, and SFOEWPT. We can see, the red region is strongly restricted
to $\Delta m_{A}=0$ or $\Delta m_{C}=\Delta m_{A}$ for both types. For Type-I,
it is $\Delta m_{C}=\Delta m_{A},|\Delta m_{A}|\in(150,350)\textrm{GeV}$, or
$\Delta m_{C}=0,|\Delta m_{A}|\in(150,350)\textrm{GeV}$. While for Type-II, it
is $\Delta m_{C}=\Delta m_{A}$ and $\Delta m_{A}\in(150,250)\textrm{GeV}$. In
both types, the red region and green region are separate from each other,
which means Higgs+Z-pole measurements at CEPC can exclude the region for
$m_{W}$ at CDF-II.

On the other hand, if SFOEWPT with uplifted $m_{W}$ in 2HDM is the true BSM
scenario, deviations from SM prediction will be observed at future
measurements with high confidence level.

##  4  Conclusion

In this work, we revisited the existence of a strong first order electroweak
phase transition (SFOEWPT) in the Type-I and Type-II 2HDMs as the grey points
in Fig. missing 3. At the same time, the latest precision measurement of the
$m_{W}$ at CDF-II, indicates possible existence of new particles with mass
around electroweak scale. We studied them all in the framework of 2HDM.

In detail, we carried out a global analysis, including $W$ boson mass $m_{W}$,
SFOEWPT requirements, direct searches of scalar resonances at the LHC, and
current LHC and future Higgs and Z-pole precision measurements at lepton
colliders such as CEPC, ILC, FCC-ee. We found that,

  * 1.

Since in the 2HDM, $\Delta m_{W}^{\rm 2HDM}$ is directly relevant to oblique
parameters as discussed, which is dependent on the heavy Higgs mass splitting
of $\Delta m_{A}=m_{A}-m_{H}$and $\Delta m_{C}=m_{H^{\pm}}-m_{H}$, we can see
$\Delta m_{W}^{\rm 2HDM}$ is sensitive to heavy Higgs mass splitting. As a
result, all these precision measurements and SFOEWPT in 2HDM are sensitive to
non-SM Higgs mass splitting in 2HDM.

  * 2.

Under current constraints, both Type-I and Type-II 2HDM can explain the
SFOEWPT, Z-pole, Higgs precision measurements and $m_{W}$ precision
measurement of CDF-II at same time. In the Fig. missing 3, we have the green
points satisfying all of them, under current various constraints. Generally
the allowed region are

$m_{H}\in(125,950)~{}\textrm{GeV}\ ,\Delta
m_{A/C}\in(-400,400)~{}\textrm{GeV},\tan\beta\in(1,50)$

for Type-I,

$m_{H}\in(125,900)~{}\textrm{GeV}\ ,\Delta
m_{A/C}\in(-200,300)~{}\textrm{GeV},\tan\beta\in(1,12)$

for Type-II.

  * 3.

With future precision measurements at CEPC, ILC, or FCC-ee, if there is no
deviation to SM observed at Higgs or Z-pole physics, SFOEWPT is still allowed,
but $m_{W}$ from CDF-II can not explained anymore by 2HDM. In other words, if
2HDM is the true BSM scenario after lepton colliders run, deviations from SM
prediction will be observed at future measurements with high confidence level.
Such a constrained parameter space points out a clear direction for
experimental studies and also theoretical explorations for explaining other
phenomenology.

## Acknowledgments

We thank Jin Min Yang, Shufang Su, Yang Zhang for useful discussions and
comments. H.S. is supported by the International Postdoctoral Exchange
Fellowship Program. W.S. is supported by KIAS Individual Grant (PG084201) at
Korea Institute for Advanced Study. M.Z. was supported by the National Natural
Science Foundation of China (NNSFC) under Grant No. 12105118 and 11947118.

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# Formalizing Monoidal Categories and Actions for Syntax with Binders

Benedikt Ahrens<EMAIL_ADDRESS>https://orcid.org/0000-0002-6786-4538
Delft University of TechnologyThe Netherlands University of BirminghamUnited
Kingdom ,  Ralph Matthes<EMAIL_ADDRESS>https://orcid.org/0000-0002-7299-2411 IRIT, Université de Toulouse, CNRS,
Toulouse INP, UT3ToulouseFrance and  Kobe Wullaert<EMAIL_ADDRESS>https://orcid.org/0000-0003-4281-2739 Delft University of TechnologyThe
Netherlands

(2018)

###### Abstract.

We discuss some aspects of our work on the mechanization of syntax and
semantics in the UniMath library, based on the proof assistant Coq. We focus
on experiences where Coq (as a type-theoretic proof assistant with decidable
type-checking) made us use more theory or helped us to see theory more
clearly.

Coq, UniMath, initial semantics, monoidal categories, binding signatures,
computer-checked proofs

††copyright: acmcopyright††journalyear: 2018††doi:
XXXXXXX.XXXXXXX††conference: Make sure to enter the correct conference title
from your rights confirmation emai; June 03–05, 2018; Woodstock, NY††price:
15.00††isbn: 978-1-4503-XXXX-X/18/06

## Introduction

We will report on an effort conducted since 2015 to formalize categorical
semantics of syntax with binders in constructive type theory. The
formalization is a contribution to the UniMath library (Voevodsky et al.,
2022), which is a library of Coq vernacular files that uses Coq in a
deliberately restricted way but adds the univalence axiom and deactivates the
checking of universe levels so as to accommodate Voevodsky’s resizing axioms.
Besides the univalence axiom, it is in general informed by univalent
mathematics, in particular as regards the definition of propositions (as
contractible types) and sets (as types where all equality types are
propositions).

This effort entered a new phase in 2021 when the link to more abstract
formulations of (signatures for) syntax in monoidal categories (such as by way
of actions) was explored and formalized and when categorical constructions
were built more modularly by using displayed categorical notions. Furthermore,
bicategories were used for two purposes: firstly, as a means of organizing
families of categories (the usual purpose of bicategories), and, secondly, for
abstracting away from the concrete situation of functors and natural
transformations in functor categories, by referring instead to an abstract
bicategory that can then be instantiated to the bicategory of categories,
functors, and natural transformations.

In 2022, we identified a reformulation of the notion of monoidal categories
that is no longer based on a notion of tensor as a functor from the product of
its underlying category with itself. Instead, it uses a crafted notion of
_whiskered bifunctor_ that allows us to smoothly introduce a displayed notion
of monoidal category.

We will highlight  (1) how the concept of monoidal category is beneficial to
type-check formalization about syntax with binders in intensional type theory,
(2) how the abstraction provided by bicategories simplified our formalization
involving functors and natural transformations,  (3) why it is better to have
a hand-crafted notion of bifunctor playing the role of the tensor product in
monoidal categories, and  (4) what are our preliminary findings about the
benefit of formalizing categorical recursion schemes as regards to having a
minimal set of assumptions.  We briefly discuss these points below.

##  1\. The Need for Monoidal Categories

When we formalize commuting diagrams as equations in Coq, the minimum
requirement is that the source (and the target) objects of the equated
morphisms agree, and this with respect to convertibility, i. e., detectable by
the type-checker. When the morphisms happen to be natural transformations
between functors and functor composition is involved in those source
“objects”, a mathematician will always argue modulo associativity of
composition and implicitly use the identity as its neutral element. However,
UniMath does not provide those latter laws as convertibility. While
$\mathsf{Id}\cdot F\Rightarrow G$ and $F\Rightarrow G$ are convertible types
of natural transformations, the functors $\mathsf{Id}\cdot F$ and $F$ are not
convertible,111Be aware that in UniMath, composition is written in
diagrammatic order instead; details are found in
https://github.com/UniMath/UniMath/blob/7145259b9cf4f4ba6342943f67736ddcfead9a48/UniMath/CategoryTheory/UnitorsAndAssociatorsForEndofunctors.v
and, consequently, neither are $H(\mathsf{Id}\cdot F)$ and $HF$, for $H$ a
functor on the functor category where $F$ lives. This scenario played an
important role in previously published work of the first two authors (Ahrens
and Matthes, 2015).

Our way out is to explicitly add polymorphic functions that are pointwise the
identity morphism and that mediate between those functors. The concept of
monoidal categories gives a rationale for these insertions and abstracts away
from the concrete situation of functors and natural transformations where
those pointwise identities suffice. Thus, monoidal categories provide a clean
(and much more general) mathematical basis for inserting some pointwise
identities to make compositions well-typed.

However, a more interesting use of monoidal categories is that monoids therein
are an abstraction of monads—when the monoidal category is canonically derived
from an endofunctor category, the two concepts coincide. This use of monoidal
categories was advocated already in 1999 (Fiore et al., 1999). A formalization
of such ideas was given in Coq as of 2020 (Lafont, 2022) and further work is
in progress.

##  2\. The Need for Bicategories

Monoidal categories are nothing but one-object bicategories, but using
monoidal categories does not mean using the conceptualization offered by
bicategories. For a fixed category, its endofunctors and their natural
transformations constitute a monoidal category—this is the application
scenario of the previous section. However, in previous work involving the
first and second author (Ahrens et al., 2022) on categorical representation of
multi-sorted syntax with binding, some building blocks of the considered
functors are not endofunctors. Hence a categorical modeling of all categories,
functors and natural transformations seems appropriate, and this led us to
consider the bicategory of categories. For the sake of this abstract, we focus
on an element of feasibility, not on organizing the semantic elements—most
notably actions of monoidal categories on categories—again into bicategories.
Section 4.3 of the cited work argues that proving complicated equations
between chains of composed natural transformations are “incommensurate with
their proofs on paper, the main problem being that there is too much structure
in those concrete functors and natural transformations, so that any attempt at
simplifying the goals at hand makes them unreadably convoluted”. Our way out
there was to abstract the problem and suitably replace functors by one-cells
and natural transformations by two-cells of a(n unspecified) bicategory, even
if we are finally interested only in the bicategory of categories as an
instance.

##  3\. The Need for Hand-crafted Bifunctors

There was an identified yet unresolved gap in the cited previous work (Ahrens
et al., 2022, Section 4.3): parameterized distributivity for actions could not
be obtained from strong monoidal functors into a crafted monoidal target
category; this was due to the lack of a notion of _displayed_ monoidal
categories that would have allowed one to limit the ingredient to sections
instead of those monoidal functors. This displayed notion resisted definition
efforts due to the underlying notion of monoidal category that relies on a
tensor construed as a functor from $\mathcal{C}\times\mathcal{C}$ to
$\mathcal{C}$, with $\mathcal{C}$ the underlying category. For the displayed
notion, it created major difficulties with transport222Transport is used in
univalent foundations to change a composite type of a term by applying an
equality to a part of that type. along components of pairs arising with the
use of this product category $\mathcal{C}\times\mathcal{C}$. To solve this
issue, instead of working with two-place functions (encoded by pairing), we
moved to a curried view that first takes the left argument and then yields a
function that expects the right-hand side argument—which is good for the
object mapping. For the two-place morphism mapping, we instead employ a
symmetric approach, by considering the one-place mappings where the left resp.
right argument is fixed to the identity, which we call the left resp. right
_whiskering_ , respectively. The name “whiskering” comes from the analogous
treatment of horizontal composition in bicategories in UniMath. In the end,
monoidal categories have become the one-object bicategories, but with a
readable syntax. This change opened the path to a usable definition of
displayed monoidal categories, fully solving the aforementioned problem.333To
be found in the UniMath GitHub repository by searching for the section names
containing FromMonoidalSection.

##  4\. Detecting Superfluous Assumptions

Here, we report on our experience with the formalization of (Fiore and
Saville, 2017, Theorem 4.7) by Fiore and Saville that was formulated as an
axiom in the formalization by Lafont of skewed monoidal categories mentioned
in Sect. 1. The theorem is announced as “a conceptual framework for
parametrised iteratively-constructed initial algebras” in the cited paper, and
it is instantiated to provide a proof of Lemma 4.8 there which says that for a
monoidal category $(\mathcal{C},I,\otimes)$ with some extra properties and an
$\omega$-cocontinuous strong bifunctor $F$ (from
$\mathcal{C}\times\mathcal{C}$ to $\mathcal{C}$), initial $F(Z,-)$-algebras
are parametrised initial for every object $Z$. The paper does not mention how
to prove Theorem 4.7, but we found out that it is an instance of the
generalized Mendler iteration scheme whose formalization Anders Mörtberg
contributed to UniMath in 2016. However, the interesting element for our
purposes here is the presence of assumptions that are not needed for proving
Lemma 4.8 and how we discovered that fact just by using Coq for the
formalization: we formulated the relevant part of Theorem 4.7 with our notion
of whiskered bifunctors (Section 3) in a Coq section without proof, but with
all the requirements with Context commands. We then derived from it Lemma 4.8,
which was formulated similarly. However, once the proof of the theorem was
given, its application for Lemma 4.8 did no longer type-check, the simple
reason being that Coq had identified unused assumptions in the proof of the
theorem, which were hence no longer abstracted when closing the Coq section:
they were $\omega$-cocontinuity of parameter $G$ and the condition that
$\mathcal{A}$ has $\omega$-colimits. Also for Lemma 4.8, the formalization
suggested slightly weaker assumptions, but these are the usual benefits of
formalizing results.

###### Acknowledgements.

We gratefully acknowledge the work by the Coq development team in providing
the Coq proof assistant and surrounding infrastructure, as well as their
support in keeping UniMath compatible with Coq.

## References

  * (1)
  * Ahrens and Matthes (2015) Benedikt Ahrens and Ralph Matthes. 2015.  Heterogeneous Substitution Systems Revisited. In _21st International Conference on Types for Proofs and Programs, TYPES 2015, May 18-21, 2015, Tallinn, Estonia_ _(LIPIcs, Vol. 69)_ , Tarmo Uustalu (Ed.). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2:1–2:23.  https://doi.org/10.4230/LIPIcs.TYPES.2015.2
  * Ahrens et al. (2022) Benedikt Ahrens, Ralph Matthes, and Anders Mörtberg. 2022\.  Implementing a category-theoretic framework for typed abstract syntax. In _CPP ’22: 11th ACM SIGPLAN International Conference on Certified Programs and Proofs, Philadelphia, PA, USA, January 17 - 18, 2022_ , Andrei Popescu and Steve Zdancewic (Eds.). ACM, 307–323.  https://doi.org/10.1145/3497775.3503678
  * Borthelle et al. (2020) Peio Borthelle, Tom Hirschowitz, and Ambroise Lafont. 2020\.  A Cellular Howe Theorem. In _LICS ’20: 35th Annual ACM/IEEE Symposium on Logic in Computer Science, Saarbrücken, Germany, July 8-11, 2020_ , Holger Hermanns, Lijun Zhang, Naoki Kobayashi, and Dale Miller (Eds.). ACM, 273–286.  https://doi.org/10.1145/3373718.3394738
  * Fiore and Saville (2017) Marcelo Fiore and Philip Saville. 2017.  List Objects with Algebraic Structure. In _2nd International Conference on Formal Structures for Computation and Deduction, FSCD 2017, September 3-9, 2017, Oxford, UK_ _(LIPIcs, Vol. 84)_ , Dale Miller (Ed.). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 16:1–16:18.  https://doi.org/10.4230/LIPIcs.FSCD.2017.16
  * Fiore et al. (1999) Marcelo P. Fiore, Gordon D. Plotkin, and Daniele Turi. 1999\.  Abstract Syntax and Variable Binding. In _14th Annual IEEE Symposium on Logic in Computer Science, Trento, Italy, July 2-5, 1999_. IEEE Computer Society, 193–202.  https://doi.org/10.1109/LICS.1999.782615
  * Lafont (2022) Ambroise Lafont. 2022\.  Initial Sigma-monoids for skew monoidal categories in UniMath.  https://doi.org/10.5281/zenodo.6622835 This consists of Coq code that was initially a supplement to (Borthelle et al., 2020).
  * Voevodsky et al. (2022) Vladimir Voevodsky, Benedikt Ahrens, Daniel Grayson, et al. 2022\.  UniMath — a computer-checked library of univalent mathematics.  Available at http://unimath.github.io/UniMath/.


# Phase transitions in the Prisoner’s Dilemma game on scale-free networks

Jacek Miȩkisz Institute of Applied Mathematics and Mechanics, University of
Warsaw  Banacha 2, 02-097 Warsaw, Poland<EMAIL_ADDRESS>Javad
Mohamadichamgavi Institute of Applied Mathematics and Mechanics, University of
Warsaw  Banacha 2, 02-097 Warsaw, Poland<EMAIL_ADDRESS>Jakub Ła̧cki
Institute of Applied Mathematics and Mechanics, University of Warsaw  Banacha
2, 02-097 Warsaw, Poland, now at Google Research, New York, USA
<EMAIL_ADDRESS>
([)

###### Abstract

We study stochastic dynamics of the Prisoner’s Dilemma game on random Erdös-
Rényi and Barabási-Albert networks with a cost of maintaining a link between
interacting players. Stochastic simulations show that when the cost increases,
the population of players located on Barabási-Albert network undergoes a sharp
transition from an ordered state, where almost all players cooperate, to a
state in which both cooperators and defectors coexist. At the critical cost,
the population oscillates in time between these two states. Such a situation
is not present in the Erdös-Rényi network. We provide some heuristic
analytical arguments for the phase transition and the value of the critical
cost in the Barabási-Albert network.

###### keywords:

evolutionary game theory, social dilemmas, Prisoner’s Dilemma game, Barabási-
Albert network, cost of links, stochastic imitation dynamics, phase
transitions

††volume: 1

1 2024  1 1 9 9th February 2024]7th April 2023 9th February 2024 ††thanks:
Corresponding author ††Licensed under a Creative Commons Attribution License
(CC-BY)

##  1 Introduction

Cooperation between unrelated individuals in human and animal societies is an
intriguing issue in biology and social sciences [1, 2, 3, 4, 5, 6]. One can
describe it within the framework of evolutionary game theory and especially
the Prisoner’s Dilemma game. In this game, two players simultaneously decide
whether to cooperate or to defect. The mutual cooperation gives both of them
the reward $R$ which is higher than the punishment $P$ resulting from the
mutual defection. However, a cooperating player is tempted to defect to
receive the highest payoff $T$ leaving the other cooperating player with the
lowest payoff $S$. Payoff inequalities $S<P<R<T$ imply that defection gives a
player a higher payoff than cooperation regardless of a strategy adopted by
its opponent. Therefore rational individuals defect in spite of the fact that
they would be better off if they cooperated.

In the framework of evolutionary game theory [7, 8, 9], payoffs are
interpreted as numbers of offspring who inherit strategies of their parents.
The evolution of very large (infinite) populations is usually modeled by
differential or difference replicator equations which describe time changes of
fractions of the population of individuals playing given strategies [10, 11].
In the case of the Prisoner’s Dilemma, the long-run of such dynamics is the
population consisting of just defectors.

In replicator dynamics, players receive average payoffs weighted by
frequencies of strategies in the infinite population. However, real
populations are finite and individuals receive payoffs (not average payoffs)
which result from interactions with random opponents in well-mixed populations
or neighbors in spatially structured populations. In their pioneering paper
[12], Nowak and May located players on regular graphs and allowed them to
interact only with their neighbors. The payoff of any player is then the sum
of payoffs resulting from individual games. In discrete time moments, players
imitate neighbors with the highest payoff obtained in the previous round,
making perhaps mistakes. In stationary states of such stochastic dynamics,
various structures of coexisting cooperators and defectors were observed [13,
14]. Since then various versions of spatial Prisoner’s Dilemma and other games
have been extensively studied, see a review paper [15]. It was shown and
generally understood that cooperation can be maintained in space-structured
populations. Cooperating players tend to form clusters, receive high payoffs
and therefore are immune to invasion by defectors. Recently there appeared
papers indicating that the structure of a network on which players are located
may play a significant role in promoting the cooperation. Various non-regular
and random graphs were investigated. In particular, Santos and Pacheco [16,
17] shown that the scale-free Barabási-Albert network favors cooperation for a
large range of game parameters.

In such a heterogeneous graph, there are vertices with many edges, the so-
called hubs. Players located on hubs interact with many individuals. It was
shown that existence of hubs favors cooperation. However, maintaining social
ties can be costly. It is natural therefore to introduce participation costs
in spatial games. It was shown in [18] that participation costs reduce the
advantage of heterogeneous networks in maintaining a high level of
cooperation.

Here we study the equilibrium behavior of the imitation dynamics of systems of
interacting individuals playing the Prisoner’s Dilemma game on random Erdös-
Rényi [19] and Barabási-Albert networks [20, 21]. The stochastic dynamics in
spatial games are similar to stochastic updating in the Ising and lattice-gas
models in statistical mechanics. However, in spatial games, in general there
does not exist a global order parameter, like the energy or free energy in the
Ising model, which the system wants to optimize. Similarities and differences
between stochastic dynamics in spatial games and in systems of many
interacting particles were discussed in [22, 23, 15].

Critical phenomena in random networks were studied very extensively, for a
review see [24], mean-field approximation in the Ising model on the Barabási-
Albert network was used in [26, 27], phase transitions in voter models were
analysed in [28, 29, 30]. We have performed Monte-Carlo simulations to explore
dependence of the cooperation level in the stationary state of the imitation
dynamics on the participating cost. We report that in the case of the
Barabási-Albert network we observe a critical value of the cost at which a
population changes abruptly from a high to a lower level of cooperation. This
follows up constructions and simulations presented in [31]

##  2 Model

Players are located on vertices of the Erdös-Rényi (ER) [19] and the Barabási-
Albert (BA) networks [20, 21]. We build the ER network by putting with
probability $p$ an edge between every pair of $N=10^{4}$ vertices. It follows
that the average degree of vertices (the average number of neighbors) is equal
to $\alpha=p(N-1).$ The BA network is built by the preferential attachment
procedure. We start with $m_{o}$ fully connected vertices and then we add
$N-m_{o}$ vertices, each time connecting them with $m$ already available
vertices with probabilities proportional to their degrees. If $m_{o}=\alpha+1$
and $m=\alpha/2$, then we get a graph with the average degree equal to
$\alpha$. It is known that such a graph is scale-free with the probability
distribution of degrees given by $p(k)\sim k^{-3}$ [20, 21, 25].

Individuals play with their neighbors the Prisoner’s Dilemma game. For a
general Prisoner’s Dilemma game we introduce a costs $\gamma$ of maintaining a
link payed by both connected players [18] and hence our payoff matrix reads:

$\begin{array}[]{ccc}&C&D\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
C&1-\gamma&S-\gamma\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
D&T-\gamma&P-\gamma\end{array}$

where the entry $ij$ is the payoff of the row player using the $i$-th strategy
while the column player uses the $j$-th one.

At discrete moments of time, all individuals interact with their neighbors and
receive payoffs which are sums with respect to individual games. Then the
imitation process takes place. A randomly chosen player compares its payoff to
payoffs of all its neighbors and with the probability $1-\epsilon$ chooses the
strategy which provided the highest payoff in the previous round and with the
probability $\epsilon$ adopts the other one, we fix $\epsilon=10^{-3}$. We
interpret $\epsilon$ as a measure of irrationality of players or simply the
noise level. This completes one step of the discrete-time dynamics - a Markov
chain with $2^{N}$ states. The presence of noise makes our chain irreducible -
any state can be reached from any other state in a finite number of steps. It
is also aperiodic which follows from the fact that the chain can remain at any
state with a non-zero probability. This means that our Markov Chain is ergodic
- it has a unique stationary probability distribution, the so-called
stationary state, to which any initial probability distribution converges.

To find a cooperation level in the stationary state we perform stochastic
simulations. We start with a completely random initial conditions with the
fraction of cooperators $=1/2$. Then we perform $10^{5}$ Monte-Carlo rounds
followed by $10^{4}$ rounds, in which frequencies of cooperators are computed.
One round consists of $N=10^{4}$ steps, where $N$ is the number of players, so
that in every round, on average each player has the opportunity to update its
strategy. We repeat simulations 50 times and average the results.

##  3 Results

It is usually assumed, see for example [12, 16], that $S=P=0,R=1$ and $T>1$,
such a game is called a weak Prisoner’s Dilemma. We will follow this practice
here. Stationary fractions of cooperators for various average degrees of
vertices $\alpha$ of ER and BA networks as a function of the cost $\gamma$ of
maintaining one link, for the temptation to defect $T=1.5,1.7$, and $1.9$ are
shown in Fig. 1 and as a function of $T$ for $\gamma=0.46$ in Fig. 2. We
observe that the cost $\gamma$ plays the crucial role in the long-run behavior
of our systems. The effect of $\gamma$ is much more pronounced for the BA
network than for the ER one. For small positive values of $\gamma$, the level
of cooperation is much higher for the BA network than for the ER one; for
bigger $\gamma$ the cooperation level is higher for the ER network.

Figure 1: Fraction of cooperators in the stationary state as a function of a
cost of maintaining a link. Figure 2:  Fraction of cooperators in the
stationary state as a function of $T$, $\gamma=0.46$.

Our main result is that in the case of the BA network, when the cost
increases, the population of players undergoes a sharp transition from an
efficient ordered state, where almost all players cooperate, to a disordered
state in which both cooperators and defectors coexist. For $T=1.5$, this
critical value of $\gamma$ is about $0.46$ as is seen in Fig. 1. This is
reminiscent of the first-order discontinuous phase transition present in
statistical mechanics models of interacting particles. In such models, at the
critical point, there coexist two (or more) phases of the system. A typical
example is the presence of two phases, up and down, in the ferromagnetic Ising
model at the zero external magnetic field below the critical Curie
temperature. To see if such a situation may be present here in the model of
interacting players, we looked at the time evolution of the frequency of
cooperation. In Fig. 3 we see that for $\gamma=0.4$ ($T$=1.5 and $\alpha=12$),
that is below a critical value, the population basically stays at an ordered
state where almost all players cooperate, for $\gamma=0.48$, the population
settles at a state in which both cooperators and defectors coexist. For
$\gamma=0.46$, we observe that the system oscillates between these two states.
Again, this is a typical situation in finite systems of interacting particles
with a discontinuous phase transition in the infinite-system limit.

Figure 3:  Fraction of cooperators after each round in a sample simulation for
various values of $\gamma$. Barabási-Albert network, $T=1.5$, $\alpha=12$.

Spatial structure favors cooperation because clusters of cooperators have
relatively high payoffs and therefore are resistant to attacks of defectors
[12]. So it is natural to expect that the cooperation level is a decreasing
function of the linking cost [18]. To understand the existence of an abrupt
decrease of the cooperation level in the Barabási-Albert network, we calculate
the linking cost at which an average payoff of $D-$strategy is bigger than the
average payoff of a $C-$strategy in the neighborhood of a solitary $D-$player.
More precisely, let us assume that we have a single $D-$player in the
population of cooperators. Its average payoff is equal to the expected value
of the vertex degree, $<k>=\alpha$ multiplied by $T-\gamma$. Now choose one of
its $C-$neighbors. It is known that its degree distribution is proportional to
$kp(k)$, where $p(k)$ is the degree distribution of the network [32, 33].
Therefore the expected value of its degree is given by $<k^{2}>/<k>$. We set
the expected value of the $D-$player to be equal to the expected value of its
neighbor - the $C-$ player, that is

(1) $<k>(T-\gamma)=(1-\gamma)(\frac{<k^{2}>}{<k>}-1)-\gamma.$

We solve the above equation for $\gamma$ and get

(2) $\gamma=\frac{\frac{<k^{2}>}{<k>}-<k>T-1}{\frac{<k^{2}>}{<k>}-<k>}.$

For the expected value of the square of vertex degrees we use the following
formula [34]:

(3) $<k^{2}>=<k>^{2}\frac{\ln N}{4}.$

For $<k>=\alpha=8$ and $N=10000$ we obtain critical values of the cost equal
aproximately to $0.519$ for $T=1.5$ and $0.21$ for $T=1.9$ which is in a good
agreement with values obtained in simulations as can be seen in Fig. 1.
Further more rigorous analysis is needed.

Let us now consider a standard Prisoner’s Dilemma game to examine how changing
the payoff for mutual defection, $P$, while holding other payoff values
constant, affects the level of cooperation. We fix payoffs as $T=1.5,R=1$, and
$S=0$, we vary $\gamma$ and observe its effect on cooperation for various
$P^{\prime}s$ in the Barabási-Albert network. The results in Fig. 4
demonstrate that as $P$ increases, the critical $\gamma$ value, where a phase
transition occurs, decreases. This is because, with a larger $P$, defectors
obtain higher payoffs when playing each other, reducing the incentive to
cooperate for any fixed $\gamma$.

Figure 4: Fractions of cooperators depending on the cost of maintaining a link
for different values of $P$. Barabási-Albert network, $T=1.5$, $\alpha=12$.

##  4 Discussion

We investigated how the cost of maintaining links between players affects the
cooperation level in the spatial Prisoner’s Dilemma games. In the case of the
Barabási-Albert network, we observed that when the cost increases, the
population of players undergoes a sharp transition from a high to a lower
level of cooperation. Our numerical simulations of the time evolution of the
frequency of cooperation showed that at the critical cost the population
oscillates between two states. It means that at such a cost there coexist two
population states: an ordered one where almost all players cooperate and one
in which both cooperators and defectors coexist. We provided some heuristic
analytical arguments for the existence of a phase transition and the value of
the citical cost.

Further research is needed to elucidate the nature of this phase transition
and to provide exact or at least an approximate analytical analysis of the
observed behavior.

###### Acknowledgements.

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 955708. J. Miȩkisz would like to thank Bartosz Sułkowski,
constructions and simulations contained in his bachelor thesis written in 2005
have been extended and are presented in this paper. Computer simulations were
made with the support of the Interdisciplinary Center for Mathematical and
Computational Modeling of the University of Warsaw (ICM UW).

References

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