A pedagogical review on muon $g-2$

Consider a rectangular electric-current coil in an uniform magnetic field, as shown in Fig.1. From Ampere’s law we know the net force on the coil is zero. But the moment of force on the coil is not zero, whose magnitude is given by

Considering the direction, the moment of force takes the form

where $A$ is the coil area, $\vec{e}_{n}$ is the unit vector of the normal direction of the coil, and the magnetic moment $\vec{M}$ is defined as

It is proved in electromagnetism zhao2018ele that the above two formulas apply to planar coils of arbitrary shape.

If a charged lepton like a muon is regarded as a charged rigid body, and assuming that its charge density $\rho_{e}$ is proportional to the mass density $\rho_{m}$, that is $\rho_{e}=\alpha\rho_{m}$, and thus $e=\alpha m$, where $e$ and $m$ are the charge and mass of the lepton, respectively. The spin that the lepton carries corresponds to a rotation in classical mechanics. If this charged rigid body rotates around the $z$-axis with an angular velocity $\omega$, then its magnetic moment can be obtained by an integration:

where $L$ is the rotational angular momentum of this rigid body. Writing the above result in vector form, we obtain

If this magnetic moment is intrinsic, then its magnitude will not change, and its potential energy in the magnetic field is

where $\vec{B}$ is the magnetic induction intensity.

Of course, such a classical way cannot correctly describe the property of a charged lepton and Eq. (5) needs to be re-derived from Dirac equation with electromagnetic interaction or from QED.

In the following we derive such a potential in Eq.(6) from Dirac equation with electromagnetic interaction zeng2007quantum . A charged lepton is described by a Dirac spinor wave function $\psi(x)$ which satisfies Dirac equation. With electromagnetic interaction, the Dirac equation takes the form

where $\phi$ is the electric potential and $c$ is the speed of light. We take $\hbar=1$ but keep $c$ for the convenience to make a non-relativistic approximation in the following derivation. For $c=1$ the above equation can reduce to the form $\left[i\gamma^{\mu}(\partial_{\mu}+ieA_{\mu})-m\right]\psi=0$ in quantum field theory. The relationships between the matrices $\vec{\alpha}$, $\beta$ and the Dirac $\gamma$ are $\beta=(\gamma^{0})^{-1}$ and $\alpha^{i}=\beta\gamma^{i}$. In Pauli-Dirac representation, we have

We express the four-component wave function as two two-component wave functions:

where the static energy of the lepton has been separated. Substitute Eq. (9) into Eq. (7), we obtain

In the non-relativistic limit, the terms that do not contain $c$ in Eq. (11) can be ignored, and thus we have

Substituting this result into Eq. (10) and taking $c=1$, we obtain

Using the relation

we can get

Substituting Eq. (15) into Eq. (13), we have

where $\vec{\sigma}/2$ is the spin $\vec{S}$ of the lepton. Therefore, in non-relativistic approximation,

Now comparing this result with Eq. (6), we obtain

where the factor $g$ is called the Landé $g$ factor.

The potential of a charged lepton with a magnetic moment $\vec{M}$ in magnetic field $\vec{B}$ takes a form in Eq.(6). So the magnetic moment of a charged lepton can be found out from such a potential.

In QED, as depicted in Fig.2, the electromagnetic interaction of a charged lepton takes a form at tree level

where $A_{\mu}(x)$ is the four-vector potential of the electromagnetic field, $e$ is the magnitude of the electric charge of the charged lepton and $\gamma^{\mu}$ is the Dirac algebra representation matrix given by

with $\sigma^{\mu}$ being the Pauli matrices.

Since we only consider the influence of the applied magnetic field, we have $A_{\mu}(x)=\left(0,-\vec{A}(\vec{x})\right)$ and

In Weyl representation we obtain the free plane-wave solution of the Dirac equation of a charged lepton (we omit the plane-wave factor $e^{-ipx}$ for the time being) peskin2018introduction

where $m$ is the mass of the charged lepton, and the normalization constant $1/\sqrt{2E_{p}}$ is to make $u^{\dagger}u=1$. So we have

From the property of the Pauli matrices

we obtain

where $k^{i}=p^{\prime i}-p^{i}$. Herefater we will not consider the first term which does not contain $\sigma$ matrices and thus is irrelevant to spin and magnetic moment. In the second term, $ip^{\prime i}$ and $-ip^{i}$ come from $\partial^{i}\bar{\psi}$ and $\partial^{i}\psi$, respectively. Retaining the plane-wave factor $e^{-ipx}$, we have

where the total derivative term can be neglected. Since the magnectic potential $A^{i}(x)$ and the magnetic field $B^{i}(x)$ are related as

then we have the spin-dependent (SD) interaction

with $\vec{M}$ being the magnetic moment of a charged lepton

and $\vec{S}$ being the spin of a charged lepton (see eq.(3.111) in peskin2018introduction )

In QED beyond tree level, as shown in Fig.3, the interaction vertex of a charged lepton with electromagnetic field takes the effective form

where $\epsilon_{\mu}(k)$ is the polarization vector of the electromagnetic field and $\Gamma^{\mu}$ is an effective form invloving all Lorentz vectors in Fig.3.

The general form of $\Gamma^{\mu}$ includes $\gamma^{\mu}$, $k^{\mu}=p^{\prime\mu}-p^{\mu}$, $P^{\mu}=p^{\prime\mu}+p^{\mu}$, and some contractions of the anti-symmetric tensor $\epsilon^{\mu\nu\alpha\beta}$ with the momentums. Utilizing the relations

we can reform $\Gamma^{\mu}$ not to contain the contrations of $\gamma$ matrics or $\epsilon^{\mu\nu\alpha\beta}$ with the momentums. Finally, $\Gamma^{\mu}$ takes a form

where the factor $1/2m$ ensures the corresponding $A^{\prime}s$ to be dimensionless, and the factor $i$ ensures the corresponding $A^{\prime}s$ to be real so that $\epsilon_{\mu}(k)\bar{u}(p^{\prime})\Gamma^{\mu}u(p)$ is hermitian. As the functions of $k^{2}$, the coefficients $A_{i}$ are Lorentz scalars.

Since $\bar{u}(p^{\prime})\Gamma^{\mu}u(p)$ is a conserved current in QED, we have

which will further restrain the form of $\Gamma^{\mu}$. From the relations

we have

which leads to $A_{3}=0$ and $A_{5}=-A_{4}4m^{2}/k^{2}$. Then $\Gamma^{\mu}$ takes a form

Further, utilizing the Gordon identities

we have

where the renamed $F$ coefficients are called form factors.

Since the parameter $e$ in the Hamiltonian density will be corrected by the $F$ form factors, it is no longer the measured electric charge which needs to be redefined. For this end, we take $A_{\mu}=(\phi(x),\vec{0})$ and then only need to consider $\Gamma^{0}$. Taking the zero-momentum limit and utilizing

and

we obtain

which means we have a term in the Hamiltonian density:

This term is just the electric potential and thus $eF_{E}(0)$ is the physical charge. So we need to redefine the electric charge to be $e$, which means $F_{E}(0)=1$. This is called the renormalization condition of the electric charge. Therefore, the first term of $\Gamma^{\mu}$ is $\gamma^{\mu}+{\cal O}(k^{2})$.

Since we are concerned primarily with the magnetic moment, we focus on the $F_{M}$ term. For simplicity we set $A_{\mu}=\left(0,\vec{A}(\vec{x})\right)$. In the low energy limit $k_{\nu}=(0,-\vec{k})$, for $\sigma^{\mu\nu}$ only the spatial components $\sigma^{ij}$ need to be considered. On the other hand, since the $F_{M}$ term already contains the momentums to the first power, we can use the zero-momentum limit of $u(p)$. Also noticing the relation

we have

Compared with eqs.(28-30), this leads to an additional term in the Hamiltonian

Compared with eq.(31), we have

where

In the calculation of $F_{M}(0)$, we usually need not to derive the whole $\Gamma^{\mu}$. We can use the $\gamma$ algebra to construct a model-independent projection matrix $P_{M}$ and then calculate $tr(\Gamma^{\mu}P_{M})$.

Before ending this section, we digress a little to discuss the form of electric dipole moment of a charged lepton. For this end, consider the $F_{D}(k^{2})$ term in eq.(47) in the low energy limit and set $A_{\mu}=\left(\phi(x),\vec{0}\right)$. Then for $\sigma^{\mu\nu}$ we only need to cosider $\sigma^{0i}$. Noticing the relation

we have

where $E^{i}=\partial^{i}\phi$. Compared with the potential of an electric dipole momemnt in an electric field $V(x)=-\vec{d}\cdot\vec{E}$, we obtain the electric dipole moment of a charged lepton

Due to its magnetic moment $\vec{M}$, an electric-current coil in a magnetic field $\vec{B}$ may have a non-zero moment of force $\vec{L}=\vec{M}\times\vec{B}$ and hence rotate. For a muon, if assumed to be a uniformly charged ball, its spin may lead to a non-zero moment of force which make the spin rotate around the applied magnetic field:

This is called the Larmor precession of a muon, as shown in Fig.4. The angular velocity of the precession is

So we can obtain the value of $g$ from the measurement of the muon spin-change angle in a given period of time. The short lifetime (2.2 $\mu s$) of a muon can be prolonged by its high velocity from special theory of reletivity. On the other hand, technically the moving muons are easier to control than stationary muons. So in experiments the muons are produced in some way and then injected into a storage ring where the spin precessions are measured.

Of course, in experiments the spin changes of the muons are not measured directly. Instead, they are performed via the measurement of its decay products, i.e., the electrons. The technical details are beyond the scope of this review. However, we should note that the $g-2$ measurement mentioned above is discussed assuming the muons to be stationary. In real experiments, the muons are moving in a storage ring. So we need to perform a Lorentz transformation to obtain the formulas in the laboratory frame. The result in the the laboratory frame is given by book:Jegerlehner:2017

where

where $v$ is the velocity of the muon and $\gamma=1/\sqrt{1-v^{2}/c^{2}}$. If the magnetic field is precisely perpendicular to the storage ring and the velocity of the muon satisfies

then $\gamma\simeq 29.3$ is called the magic $\gamma$-factor and $\vec{\omega}_{a}$ can be simplified as

The BNL E821 result is BNL:gmuon

The Fermilab result combined with the BNL result is FNAL:gmuon

The SM predition

consists of the following contributions Aoyama:2012wk ; Aoyama:2019ryr ; Czarnecki:2002nt ; Gnendiger:2013pva ; Davier:2017zfy ; Keshavarzi:2018mgv ; Colangelo:2018mtw ; Hoferichter:2019gzf ; Davier:2019can ; Keshavarzi:2019abf ; Kurz:2014wya ; Melnikov:2003xd ; Masjuan:2017tvw ; Colangelo:2017fiz ; Hoferichter:2018kwz ; Gerardin:2019vio ; Bijnens:2019ghy ; Colangelo:2019uex ; Blum:2019ugy ; Colangelo:2014qya

where the main uncertainties come from the hadronic contributions in HVP and HLBL. The deviation between the experiment and SM is

which is $4.2\sigma$. However, if the lattice simulation result of the hadronic contribution from the BMW group is taken, the deviation between the experiment and SM can be reduced to $1.5\sigma$ gm2-lattice .

The deviation between the experiment and the SM prediction for the muon $g-2$ may be a harpinger of new physics beyond the SM. Among the new physics theories, the low energy supersymmetry (SUSY) is the most popular candidate. In SUSY the smuons or muon sneutrino plus electroweakinos (electroweak gauginos and higgsinos) contribute to muon $g-2$ at one loop level, the same level as the SM. For a common sparticle mass $M_{SUSY}$, the SUSY contribution to muon $g-2$ can be approximated as Moroi:1995yh

where $\tan\beta$ is the ratio of the vacuum expectation values of the two Higgs doublets. Clearly, to generate the required contribution to explain the muon $g-2$ deviation, a low SUSY mass and a large $\tan\beta$ are favored. Combined with other experimental constraints, such as the dark matter detections and the LHC searches of sparticles, the favored parameter space of SUSY can be specified, which can be of some guidance for the future search of spartilces at the HL-LHC.

In the low energy effective minimal SUSY model(MSSM) or called phenomenological MSSM(pMSSM). the masses of bino ($M_{1}$), winos ($M_{2}$), higgsinos ($\mu$) and smuons/sneutrino ($M_{\tilde{\ell}}$) are all independent parameters. As shown in Fig. 5 gm2-gutsusy-01 , considering other constraints including the dark matter relic density, the dark matter direct detection, the vacuum stability and the LHC search for sleptons, the survived MSSM parameter space can allow for the explanation of the muon $g-2$ at $2\sigma$ level gm2-gutsusy-01 ; gm2-mssm-01 ; gm2-mssm-02 ; gm2-mssm-03 ; gm2-mssm-04 ; gm2-mssm-05 ; gm2-mssm-06 ; Iwamoto:2021aaf ; Gu:2021mjd ; Cox:2021gqq .

The SUSY explanation of the muon $g-2$ at $2\sigma$ level requires light electroweakinos and sleptons, which can be most covered at the HL-LHC gm2-mssm-01 ; Aboubrahim:2021ily . For example, in the scenario where the dark matter is bino-like with bino-wino coannihilation to achieve the correct dark matter relic density, the muon $g-2$ at $2\sigma$ level requires light bino and winos, whose pair production at the LHC can be well probed, as shown in Fig.6.

While the muon $g-2$ anomaly can be readily explained in the low energy effective MSSM due to its large number of free soft SUSY-breaking parameters, the explanation is rather challenging in the constrained models with given SUSY breaking mediation mechanisms  gm2-gutsusy-01 ; gm2-gutsusy-02 ; gm2-gutsusy-03 ; Han:2020exx ; Lamborn:2021snt ; Yin:2021mls , such as mSUGRA/CMSSM, GMSB and AMSB. In these fancy models we have boundary conditions for the soft SUSY-breaking terms at some high energy scale, e.g., the GUT scale in mSUGRA/CMSSM. Due to the boundary conditions, the soft masses at the weak scale are correlated. To give a 125 GeV SM-like Higgs boson mass, the stop masses must be above TeV scale and the correlated slepton masses cannot be as light as required by the explanation of the muon $g-2$ anomaly. For the mSUGRA/CMSSM, the tension between the 125 GeV SM-like Higgs boson mass and the muon $g-2$ is shown in Fig.7. In order to accomodate a 125 GeV SM-like Higgs boson mass and the muon $g-2$ at $2\sigma$ level, these models need to improved or extended. For example, one can make colored sparticles much heavier than uncolored sparticles gm2-SUGRA-ext-01 ; gm2-SUGRA-ext-02 ; gm2-SUGRA-ext-03 ; gm2-SUGRA-ext-04 ; gm2-GMSB-AMSB-ext-01 ; gm2-GMSB-AMSB-ext-02 (so stops can be much heavier than sleptons and gluino can be much heavier than electroweakinos at weak scale) or couple the messengers with the Higgs doublets (so the 125 GeV SM-like Higgs boson mass can be achieved without too heavy stops) GMSB-yukawa-higgs .

If we go beyond the minimal framework of SUSY, the accomodation of the 125 GeV SM-like Higgs boson mass and the muon $g-2$ at $2\sigma$ level may becomes easy. For example, in the next-to-minimal SUSY model (NMSSM) a singlet Higgs field is introduced and hence the 125 GeV SM-like Higgs boson mass can be obtained at tree level without the large effects of heavy stops (the relatively light stops make this model more natural than the MSSM nmssm-mssm ). The explanation of the muon $g-2$ can be achieved in the NMSSM gm2-nmssm-01 ; gm2-nmssm-02 ; gm2-nmssm-03 .

Noe that there have been some attempts to explain the muon $g-2$ together with other physical problems (e.g. the proton radius puzzle Zhu:2021vlz ). Especially, if the value of the fine structure constant measured by the Berkeley experiment berkeley is used, the electron $g-2$ predicted by the SM will also deviate from the experimental value Aoyama:2019ryr ; Hanneke:2008tm . A joint explanation of such an electron $g-2$ anomaly and the muon $g-2$ anomaly is shown to be feasible in SUSY joint-01 ; joint-02 ; joint-03 ; joint-04 ; joint-05 ; Yang:2020bmh . If we further consider the correlation between muon $g-2$ and electron EDM, the CP-phases in SUSY can be stringently constrained cp-phase-01 ; cp-phase-02 . In this review we do not cover various non-SUSY explanations of the $g-2$ or EDM in miscellaneous extensions of the SM, say the 2HDM (see, e.g., Keus:2017ioh ; Wang:2016ggf ; Wang:2018hnw ), the 3-3-1 model (see, e.g., Hue:2021zyw ; Li:2021poy ) and flavor models (see, e.g., Calibbi:2020emz ; Han:2020dwo ; Yin:2021yqy ).