The ALP explanation to muon g2 and its test at future Tera- Zand Higgs factories Jia Liu1 2Xiaolin Ma1yLian-Tao Wang3 4zand Xiao-Ping Wang5 6x

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The ALP explanation to muon (g−2)

and its test at future Tera-Zand Higgs factories

Jia Liu,1, 2, ∗Xiaolin Ma,1, †Lian-Tao Wang,3, 4, ‡and Xiao-Ping Wang5, 6, §

1School of Physics and State Key Laboratory of Nuclear Physics and Technology,

Peking University, Beijing 100871, China

2Center for High Energy Physics, Peking University, Beijing 100871, China

3Enrico Fermi Institute and Department of Physics,

The University of Chicago, Chicago, IL 60637, USA

4Kavli Institute for Cosmological Physics,

The University of Chicago, Chicago, IL 60637, USA

5School of Physics, Beihang University, Beijing 100083, China

6Beijing Key Laboratory of Advanced Nuclear Materials and Physics,

Beihang University, Beijing 100191, China

(Dated: May 2, 2023)

Abstract

Models with an Axion Like Particle (ALP) can provide an explanation for the discrepancy

between experimental measurement of the muon anomalous magnetic moment (g−2)µand the

Standard Model prediction. This explanation relies on the couplings of the ALP to the muon and

the photon. We also include more general couplings to the electroweak gauge bosons and incor-

porate them in the calculations up to the 2-loop order. We investigate the existing experimental

constraints and ﬁnd that they do not rule out the ALP model under consideration as a possible

explanation for the (g−2)µanomaly. At the same time, we ﬁnd the future Tera-Zand Higgs fac-

tories, such as the CEPC and FCC-ee, can completely cover the relevant parameter space through

searches with ﬁnal states (γγ)γ, (µ+µ−)γand (µ+µ−)µ+µ−.

∗jialiu@pku.edu.cn

†themapku@stu.pku.edu.cn

‡liantaow@uchicago.edu

§hcwangxiaoping@buaa.edu.cn

arXiv:2210.09335v2 [hep-ph] 1 May 2023

CONTENTS

I. Introduction 2

II. The ALP properties, and contribution to (g−2)µ5

A. The contribution to the (g−2)µ6

III. The constraints from existing searches and projection of future probes 11

A. Constraints from current results of light particle searches 11

1. Constraints from searches for ﬁnal states of a+γ12

2. Constraints from searches for ﬁnal states of a+¯

ff 14

3. Constraints from measurements of Light-by-Light scattering 16

B. The search at future Zand Higgs factories 17

C. Summary of current limits and projected reach, CW W = 0. 18

D. Summary of constraints and projected reaches, CW W 6= 0. 22

IV. Conclusions 25

Acknowledgments 27

References 27

I. INTRODUCTION

The Strong CP problem in the Standard Model (SM) [1–4] can be solved by the Peccei-

Quinn mechanism, leading to the prediction of the existence a pseudo Nambu-Goldstone

boson, the axion [5–9]. The shift symmetry of the axion implies that it only has derivative

couplings except for non-perturbative eﬀects through strong interactions. In addition, other

axion-like particles (ALPs) are ubiquitous in many new physics scenarios. Similar to the

QCD axion, an ALP is a pseudo-Goldstone boson with an approximate shift symmetry.

The corresponding decay constant faand mass macan be free parameters. It shares similar

interactions with the QCD axion at low energy and predicts rich phenomenology to be

explored in various experiments [10–16].

The anomalous magnetic dipole moment of the muon, (g−2)µis a powerful tool to test

the SM and probe the physics beyond the Standard Model [17–21]. The combined results

of the measurements at the Brookhaven National Laboratory [22] and the Fermi National

Accelerator Laboratory [23] suggest a 4.2 σdiscrepancy between SM prediction [24–44] and

experiment measurement, as ∆aµ≡aExp

µ−aSM

µ= (25.1±5.9) ×10−10. It is worth noting

that the status of the theoretical calculation of the SM prediction has not been settled

yet [45, 46]. In this paper, we make the assumption that the apparent discrepancy is due to

the contribution of new physics beyond the Standard Model.

In this work, we focus on a scenario with the contribution of ALP as the explanation

of the apparent deviation in (g−2)µ. Since the ALP is a pseudoscalar, the axion-muon

coupling contributes ∆aµnegatively. Therefore, a model in which this is the dominant

coupling can not explain the (g−2)µanomaly. If the axion-photon coupling have a diﬀerent

sign comparing with the axion-muon coupling, the contribution can be positive [15, 47–55].

Therefore, both of these couplings need to be present at the same time. Since the axion-

photon coupling comes from the interaction with hypercharge gauge ﬁeld and SU(2)Lgauge

ﬁelds, in general, we should include both of them in addition to the muon coupling. Taking

into account these considerations, we have the following eﬀective Lagrangian at weak scale

energy

LD≤5

eﬀ =X

Cff

∂µa

fγµγ5f+g2

16π2CW W

µν ˜

Wµν,i +g02

16π2CBB

Bµν ˜

Bµν ,(1)

where Wand Bare the SU(2)Land U(1)Yﬁeld strength. In this paper, we study the existing

constraints on the parameters in Eq. 1 from collider searches, and propose to exploit searches

with ﬁnal states (γγ)γ, (µ+µ−)γand (µ+µ−)µ+µ−at future electron-positron colliders [56,

57], with runnings in both the Tera-Zand the Higgs factory modes, to search for the ALP.

We found that future Zfactories can cover most of the parameter region of maup to 85

GeV, which can explain the (g−2)µanomaly, while future Higgs factories can extend the

limits to much higher masses.

The (g−2)µand the constraints on the axion couplings have been extensively studied in

Refs. [15, 51, 53–55]. Our work extends these results in the following aspects.

The 2-loop Barr-Zee diagram contributing to the (g−2)µhas a nontrivial counter-term

arising from the axion shift symmetry in the derivative coupling basis, which is clariﬁed

recently in Ref. [54]. Their calculations only consider the photon in the diagram. In our

study, since we are interested in axion mass up to Zmass, the contributions from the W/Z

gauge boson are also included.

Previous studies on the constraints from other experimental searches focus on the eﬀect

of turning on a single coupling. We start with both the fermion coupling Cµµ and the gauge

boson coupling CBB and CW W due to the requirement of (g−2)µ. It opens up some unique

channels, for example Z→(µ+µ−)γ, in the multiple parameters scenario. It diﬀers from a

previous study of Z→(µ+µ−)γin Ref. [51], which only focused on Cµµ and derived CγZ

from the lepton coupling at the 1-loop level. Therefore, we conducted a reanalysis of this

channel in the context of multiple parameters and particularly focused on its implications for

the (g−2)µparameter region at future electron-positron colliders. In addition, we update

the constraints from various existing experiments. For example, the muonic force study on

U(1)Lµ−Lτat CMS [58] can set limits on the axion-muon coupling, but it is missing in the

previous studies. Moreover, with two gauge couplings, one of the couplings Cγγ or CZγ can

be very small, leading to signiﬁcant changes in the axion decay branching ratio and lifetime.

Therefore, some previous studies do not directly apply to this case.

We should also mention the ﬂavor physics measurements can, in principle, oﬀer sensitive

probes to axion couplings. For instance, in a UV completion of the ALP Lagrangian, there

may exist ﬂavor oﬀ-diagonal derivative couplings between the ALP and leptons. Such cou-

plings could trigger charged lepton ﬂavor violation processes such as µ→eγ and π→µe

and are consequently stringently constrained by experimental results [52, 53, 55, 59]. Hence,

we assume that the ALP-lepton coupling is ﬂavor diagonal at low energy scales in the lepton

mass basis. This setting implies that the UV completion of the ALP model must possess a

speciﬁc ﬂavor structure to account for the pronounced suppression of ﬂavor oﬀ-diagonal cou-

plings [54, 55]. Furthermore, the CW W coupling can induce the ﬂavor-violating couplings in

the down-quark sector through the top quark in the loop, which is severely constrained by the

precision meson measurements. The CW W coupling can be constrained by B+→K+a(µµ),

B→K∗a(µµ), B+→π+a(µµ) for ma<5 GeV [55]. The CBB coupling is less constrained

since it comes in at a higher order. For Cµµ coupling, the three exotic Bmeson decay

channels provide similar constraints. There is one more channel Bs→µ+µ−, which can

provide a constraint competitive with the CMS muonic force search [58] even for large ma.

At the same, such constraints depend on the ﬂavor model, which necessarily involves more

parameters. In this paper, we will assume that ﬂavor constraints are not enough to cover

the interesting parameter space.

The paper is organized as follows. In section II, we describe the axion low-energy model

and calculate the (g−2)µat the 2-loop level. In section III, we start without the coupling

between the ALP and the W boson. The signal ﬁnal states are classiﬁed as a+γand a+¯

and the existing constraints from electron-positron colliders like BaBar [60, 61],Belle-II [62]

and LEP [63–66] and the Large Hadron Collider [58, 67, 68] are discussed extensively in

section III A. We then discuss the constraints from exotic ﬁnal states at future Tera-Zand

Higgs factory in section III B and extend the results including the general couplings to W

boson in section III D. Section IV contains our conclusion.

II. THE ALP PROPERTIES, AND CONTRIBUTION TO (g−2)µ

In this section, we present analytical results useful for our numerical study, including the

ALP couplings, relevant decay widths, and their contribution to (g−2)µ. Following the

notation of [54], we begin with the eﬀective Lagrangian on the basis of the fermion masses.

LD≤5

eﬀ =X

Cff

∂µa

fγµγ5f+αCγγ

4π

Fµν ˜

Fµν +αCγZ

2πswcw

Fµν ˜

Zµν +αCZZ

4πs2

wc2

Zµν ˜

Zµν

+αCW W

πs2

µνρσ∂µWν

+∂ρWσ

−+... , (2)

with the coeﬃcients

Cγγ =CW W +CBB, CγZ =c2

wCW W −s2

wCBB, CZZ =c4

wCW W +s4

wCBB,(3)

where sw≡sin θW,cw≡cos θWwith θWbeing the weak mixing angle. This EFT La-

grangian can be valid upto scale Λ = 4πfa[54, 55]. We have omitted the gluon coupling

and interaction vertices containing more than three ﬁelds. In this work, we only consider

f=µfor simplicity.

For illustrative purposes, we start with a simple case CW W = 0 to present our analytic

results and discuss experimental constraints before we present the full numerical results with

CW W 6= 0. This simple choice is also favored by Electroweak Precision Data (EWPD) (see,

e.g., Figure 27 of Ref. [15]). In this case, we have three free parameters

{ma, Cµµ, CBB }.(4)

In this case, the exotic Zdecay Z→γa is induced by CZγ =−s2

wCBB =−s2

wCγγ. The

partial widths of the exotic Zdecay and the two axion decay channels a→µ+µ−and

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TheALPexplanationtomuon(g2)anditstestatfutureTera-ZandHiggsfactoriesJiaLiu,1,2,XiaolinMa,1,yLian-TaoWang,3,4,zandXiao-PingWang5,6,x1SchoolofPhysicsandStateKeyLaboratoryofNuclearPhysicsandTechnology,PekingUniversity,Beijing100871,China2CenterforHighEnergyPhysics,PekingUniversity,Beijing100871,China3...

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The ALP explanation to muon g2 and its test at future Tera- Zand Higgs factories Jia Liu1 2Xiaolin Ma1yLian-Tao Wang3 4zand Xiao-Ping Wang5 6x

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