# Contributions to the Muon $g-2$ from a Three-Form Field

Da<EMAIL_ADDRESS>National Astronomical Observatories, Chinese
Academy of Sciences, Beijing, 100012, China  School of Fundamental Physics and
Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou
310024, China  International Centre for Theoretical Physics Asia-Pacific,
Beijing/Hangzhou, China  Yong<EMAIL_ADDRESS>University of Chinese
Academy of Sciences (UCAS), Beijing 100049, China  School of Fundamental
Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study,
UCAS, Hangzhou 310024, China  National Astronomical Observatories, Chinese
Academy of Sciences, Beijing, 100012, China  International Centre for
Theoretical Physics Asia-Pacific, Beijing/Hangzhou, China  Yue-Liang
<EMAIL_ADDRESS>Institute of Theoretical Physics, Chinese Academy of
Sciences, Beijing 100190, China  School of Fundamental Physics and
Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou
310024, China  International Centre for Theoretical Physics Asia-Pacific,
Beijing/Hangzhou, China  University of Chinese Academy of Sciences (UCAS),
Beijing 100049, China

###### Abstract

We examine contributions to the muon dipole moment $g-2$ from a 3-form field
$\Omega$, which naturally arises from many fundamental theories, such as the
string theory and the hyperunified field theory. In particular, by calculating
the one-loop Feynman diagram, we have obtained the leading-order
$\Omega$-induced contribution to the muon $g-2$, which is found to be finite.
Then we investigate the theoretical constraints from perturbativity and
unitarity. Especially, the unitarity bounds are yielded by computing the tree-
level $\mu^{+}\mu^{-}$ scattering amplitudes of various initial and final
helicity configurations. As a result, despite the strong unitarity bounds
imposed on this model of $\Omega$, we have still found a substantial parameter
space which can accommodates the muon $g-2$ data.

##  I Introduction

A three-form field, i.e., an antisymmetric rank-three tensor field, can
generically appear in many fundamental theories trying to unify the General
Relativity and Quantum Field Theory. For example, in the $10$-dimensional
Type-IIA superstring and supergravity theory, a 3-form field occurs as an
important constituent in the Ramond-Ramond sector Green:1987sp ; Green:1987mn
; Polchinski:1998rr . Furthermore, in the recently proposed Hyper-Unified
Field Theory (HUFT) Wu:2015wwa ; Wu:2017rmd ; Wu:2017urh ; Wu:2018xah ;
Wu:2021ign ; Wu:2021ucc ; Wu:2022mzr , such a 3-form field is naturally
identified as a part of the spin gauge field in this 19-dimensional theory.
Unfortunately, the 3-form field in the superstring theory couples to the
ordinary matter with the gravitational strength Green:1987sp ; Green:1987mn ;
Polchinski:1998rr , so it is difficult to be visible at low energies. On the
other hand, in the HUFT, by dynamically breaking the conformal symmetry
Wu:2017urh ; Wu:2021ign ; Wu:2021ucc  and compactifying the theory into our
mundane 4-dimensional spacetime, we expect that a light 3-form field $\Omega$
is left and potentially generates new observable effects Sezgin:1980tp .

In this study, we consider the indirect loop effects induced by a 3-form field
on the muon electromagnetic dipole moment defined by $a_{\mu}=(g-2)_{\mu}/2$,
which might solve the long-standing muon $g-2$ anomaly Workman:2022ynf .
Currently, the comparison between the Standard Model (SM) prediction and the
experimental data shows a remarkable $\sim 4.25\sigma$ discrepancy
Muong-2:2021ojo :

$\displaystyle\Delta a_{\mu}=a_{\mu}^{\rm Exp}-a_{\mu}^{\rm SM}=(251\pm
59)\times 10^{-11},$ (1)

where the experimental value $a_{\mu}^{\rm Exp}=(116\,592\,061\pm 41)\times
10^{-11}$ is obtained by combining the earlier Brookhaven Muong-2:2006rrc  and
the latest Fermilab Muon $g-2$ Muong-2:2021ojo  data, while the SM prediction
is given by $a^{\rm SM}_{\mu}=(116\,591\,810\pm 43)\times 10^{-11}$ based on
the state-of-art evaluations of various contributions Aoyama:2012wk ;
Aoyama:2019ryr ; Czarnecki:2002nt ; Gnendiger:2013pva ; Davier:2017zfy ;
Keshavarzi:2018mgv ; Colangelo:2018mtw ; Hoferichter:2019mqg ; Davier:2019can
; Keshavarzi:2019abf ; Kurz:2014wya ; Melnikov:2003xd ; Masjuan:2017tvw ;
Colangelo:2017fiz ; Hoferichter:2018kwz ; Gerardin:2019vio ; Bijnens:2019ghy ;
Colangelo:2019uex ; Blum:2019ugy ; Colangelo:2014qya  (see e.g. Ref.
Aoyama:2020ynm  for a recent review and references therein). In light of these
precise measurements and calculations, this muon $g-2$ anomaly is usually
regarded as an indication to the new physics beyond the SM. In the literature,
there have already been many new physics attempts to resolve this anomaly (see
e.g., Ref. Athron:2021iuf  for a recent review and references therein). It is
worth to mention that, in the HUFT, the 3-form field $\Omega$, as a part of
the spin gauge field, couples to the SM fermionic fields inevitably. Hence, it
is highly probable to generate an extra contribution to the muon $g-2$ via its
interaction with muons. Here we would like to carefully examine if the one-
loop contributions induced by $\Omega$ could help us to ameliorate or even
eliminate the $(g-2)_{\mu}$ discrepancy. Furthermore, the scenario we are
considering needs a relatively light $\Omega$ and a moderately large coupling
to muons, which might potentially lead to the breakdown of perturbativity and
unitarity. For that reason, we shall also take into account theoretical
constraints from these issues in the framework of effective field theory.

The paper is organized as follows. In Sec. II, we present the effective field
theory framework of the 3-form field $\Omega$. In particular, we show the
Lagrangian and the propagator of $\Omega$, together with a short discussion of
the significance of the $\Omega$ mass term. Then the leading-order
contribution to the muon $g-2$ induced by $\Omega$ is given in Sec. III by
calculating the associated one-loop Feynman diagram. Sec. IV is devoted to
studying the theoretical constraints from perturbativity and unitarity on our
3-form model. The numerical results are shown in Sec. V by carefully examining
the parameter space of interest. Finally, we conclude in Sec. VI, together
with a short discussion of the possible collider signatures of the 3-form
field.

##  II Lagrangian and Conventions

In this section, we set up our notations and conventions for our model of the
3-form field $\Omega=\Omega_{\mu\nu\rho}\,dx^{\mu}\wedge dx^{\nu}\wedge
dx^{\rho}$. In the fundamental theories such as the Type-IIA superstring
theory or the HUFT, $\Omega$, as a gauge field, couples universally to other
fields or particles. However, these fundamental theories are generically
living in the spacetime with dimensions higher than 4. In order to explain the
physics of our ordinary 4-dimensional spacetime world, one usually involves
the compactification technique to wrap the extra dimensions in small compact
spaces Polchinski:1998rr ; Green:1987sp ; Green:1987mn . In this process, due
to the wavefunctions of various fields in the extra dimensional space, the
couplings of $\Omega$ with the SM particles can become non-universal. Here we
concentrate on the following 3-form field coupling to the SM muon leptons
Wu:2017urh ; Wu:2018xah ; Sezgin:1979zf  444Note that, in Eq. (2), the
convention for the $\Omega$ coupling $g_{h}$ to muons differs from that given
in Fig. 4 of Ref. Wu:2018xah  by a factor $1/4$.

$\displaystyle{\cal L}_{\Omega}\supset-\frac{1}{2\cdot
4!}\Omega^{\mu\nu\rho\sigma}\Omega_{\mu\nu\rho\sigma}-\frac{m_{\Omega}^{2}}{2\cdot
3!}\Omega_{\mu\nu\rho}\Omega^{\mu\nu\rho}-ig_{h}\bar{\mu}\gamma^{\mu\nu\rho}\mu\Omega_{\mu\nu\rho}\,,$
(2)

where $\Omega_{\mu\nu\rho\sigma}$ denotes the four-form field strength of
$\Omega_{\mu\nu\rho}$ defined by

$\displaystyle\Omega_{\mu\nu\rho\sigma}\equiv\partial_{\mu}\Omega_{\nu\rho\sigma}-\partial_{\nu}\Omega_{\rho\sigma\mu}+\partial_{\rho}\Omega_{\sigma\mu\nu}-\partial_{\sigma}\Omega_{\mu\nu\rho}\,,$
(3)

while $\gamma^{\mu\nu\rho}$ can be written as the combination of
$\gamma$-matrices as follows

$\displaystyle\gamma^{\mu\nu\rho}$ $\displaystyle\equiv$
$\displaystyle\gamma^{[\mu}\gamma^{\nu}\gamma^{\rho]}=\frac{1}{3!}\left(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}-\gamma^{\nu}\gamma^{\mu}\gamma^{\rho}+\gamma^{\nu}\gamma^{\rho}\gamma^{\mu}-\gamma^{\rho}\gamma^{\nu}\gamma^{\mu}+\gamma^{\rho}\gamma^{\mu}\gamma^{\nu}-\gamma^{\mu}\gamma^{\rho}\gamma^{\nu}\right)$
(4) $\displaystyle=$ $\displaystyle
i\epsilon^{\mu\nu\rho\sigma}\gamma_{\sigma}\gamma^{5}\,,$

where $\gamma^{5}\equiv
i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}=-i\epsilon^{\mu\nu\rho\sigma}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma}/(4!)$.

In Eq. (2), we have included a mass term to the 3-form field $\Omega$, which
explicitly breaks the original gauge symmetry
$\Omega_{\mu\nu\rho}\to\Omega_{\mu\nu\rho}+3\partial_{[\mu}b_{\nu\rho]}$ with
$b$ a two-form gauge parameter field. Here we do not try to specify the
possible origin of this mass term. Rather, we would like to emphasize its
importance to the theory. Firstly, note that, without this mass term, the
gauge symmetry involving $b$ is recovered in this Lagrangian of $\Omega$.
Thus, the lightness of $\Omega$ is technically natural tHooft:1979rat , which
can be understood in connection with this underlying broken gauge symmetry.
Secondly, further significance of this mass term can be seen by analyzing its
particle content in $\Omega$. In the massless case, $\Omega$ does not possess
any physical propagating degree of freedom (dof) in four spacetime dimensions
Sezgin:1980tp . To see this, let us divide the components of $\Omega$ into two
classes, $\Omega_{0ij}$ and $\Omega_{ijk}$ with the Latin indices denoting the
spatial dimensions. On the one hand, due to the antisymmetric nature of the
field strength, the Lagrangian in Eq. (2) cannot give rise to the kinetic term
for $\Omega_{0ij}$ i.e., $\partial_{0}\Omega_{0ij}$ cannot exist. On the other
hand, $\Omega_{ijk}\equiv\epsilon_{ijk}\phi$, actually being a pseudo-scalar
in nature, suffers from the gauge transformation as
$\Omega_{ijk}\to\Omega_{ijk}+3\partial_{[i}b_{jk]}$, so that it is unphysical
either. However, in the model with a massive 3-form field, the gauge
invariance of $\Omega$ is explicitly broken, so that $\Omega_{ijk}$ retains as
true physical propagating dof in the theory. Following the tensor-projection
operator technique developed in Ref. Rivers:1964 ; Sezgin:1979zf ;
Sezgin:1980tp ; VanNieuwenhuizen:1973fi , we can derive the propagator of the
massive 3-form field $\Omega$ as follows

$\displaystyle
i\Delta^{\Omega}_{\mu\nu\lambda,\,\rho\sigma\kappa}(k)=\frac{-i{\cal
P}_{\mu\nu\lambda\,,\rho\sigma\kappa}(k)}{k^{2}-m_{\Omega}^{2}}\,,$ (5)

where

$\displaystyle{\cal
P}_{\mu\nu\lambda,\,\rho\sigma\kappa}(k)\equiv\frac{1}{6}\left[\theta_{\mu\rho}(\theta_{\nu\sigma}\theta_{\lambda\kappa}-\theta_{\nu\kappa}\theta_{\lambda\sigma})-\theta_{\mu\sigma}(\theta_{\nu\rho}\theta_{\lambda\kappa}-\theta_{\nu\kappa}\theta_{\lambda\rho})-\theta_{\mu\kappa}(\theta_{\nu\sigma}\theta_{\lambda\rho}-\theta_{\nu\rho}\theta_{\lambda\sigma})\right]\,,$
(6)

with

$\displaystyle\theta_{\mu\nu}(k)\equiv\eta_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{m_{\Omega}^{2}}\,.$
(7)

It has been well-known that there are many problems in constructing an
interacting theories for higher-spin fields in the flat spacetime (see e.g.,
Ref. Bekaert:2010hw  for a recent review and references therein), which is
hampered by various no-go theorems Weinberg:1964ew ; Coleman:1967ad ;
Haag:1974qh ; Grisaru:1976vm ; Weinberg:1980kq ; Porrati:2008rm ;
Benincasa:2007xk  in the literature. One may evade these no-go theorems by
introducing higher-derivative couplings Metsaev:2005ar ; Boulanger:2006gr ;
Boulanger:2008tg  or placing the theory on the AdS spacetime Fradkin:1986qy ;
Fradkin:1987ks ; Vasiliev:2011knf . Provided that the 3-form field $\Omega$
transforms non-trivially under the Lorentz symmetry group, one may wonder if
our interacting theory might also be constrained by these no-go theorems. Here
we would like to argue that our model of $\Omega$ is not plagued by these
problems. First of all, as shown before, the massive 3-form field actually
contains a single spin-0 pseudoscalar, rather than any higher-spin particles.
Secondly, the previous no-go theorems focused on the massless higher-spin
case, while $\Omega$ is dynamical only when it becomes massive. Thus, $\Omega$
may be decoupled when the energy is lower than its mass, and does not affect
the infrared behavior of the theory. The above two reasons make either the
Weinberg low-energy theorem Weinberg:1964ew  or the Weinberg-Witten no-go
theorem Weinberg:1980kq  not be applicable to the present interacting 3-form
model. It has been shown in the HUFT Wu:2015wwa ; Wu:2017rmd ; Wu:2017urh ;
Wu:2018xah ; Wu:2021ign ; Wu:2021ucc ; Wu:2022mzr  that the no-go theorems can
also be avoided by introducing the concept of biframe spacetime.

##  III The Muon $g-2$ Contribution

Now it is ready to compute the novel contribution to the muon anomalous
magnetic moment induced by $\Omega$. The relevant Feynman diagram is shown in
Fig. 1, where the double-wiggle line represents the massive 3-form field.

Figure 1: One-loop Feynman diagrams that would contribute to the muon $g-2$.

By applying the Feynman rules according to the Lagrangian in Eq. (2), the
corresponding amplitude of Fig. 1 is given by

$\displaystyle i{\cal M}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}l}{(2\pi)^{4}}\left(g_{h}\gamma^{\rho\sigma\kappa}\right)\frac{i}{\not{l}+\not{k}_{2}-m_{\mu}}\left(-ieQ_{\mu}\gamma^{\mu}\right)\frac{i}{\not{l}+\not{k}_{1}-m_{\mu}}\left(g_{\mu}\gamma^{\tau\nu\lambda}\right)\frac{-i{\cal
P}_{\tau\nu\lambda,\,\rho\sigma\kappa}(l)}{l^{2}-m_{\Omega}^{2}}$ (8)
$\displaystyle=$
$\displaystyle{g_{h}^{2}eQ_{\mu}}\int\frac{d^{4}l}{(2\pi)^{4}}\frac{\gamma^{\rho\sigma\kappa}[\not{l}+\not{k}_{2}+m_{\mu}]\gamma^{\mu}[\not{l}+\not{k}_{1}+m_{\mu}]\gamma^{\tau\nu\lambda}{\cal
P}_{\tau\nu\lambda,\,\rho\sigma\kappa}(l)}{[(l+k_{2})^{2}-m_{\mu}^{2}][(l+k_{1})^{2}-m_{\mu}^{2}](l^{2}-m_{\Omega}^{2})}\,,$

$Q_{\mu}=-1$ and $m_{\mu}$ stand for the muon electric charge and mass,
respectively. By completing the square in the denominator with the Feynman
parameters, the denominator can be transformed into

$\displaystyle\frac{1}{[(l+k_{2})^{2}-m_{\mu}^{2}][(l+k_{1})^{2}-m_{\mu}^{2}](l^{2}-m_{\Omega}^{2})}=\int^{1}_{0}dxdy\frac{\Gamma(3)}{[l^{2}+2l\cdot(xk_{1}+yk_{2})-(1-x-y)m_{\Omega}^{2}]^{3}}$
(9) $\displaystyle=$
$\displaystyle\Gamma(3)\int^{1}_{0}dxdy\frac{1}{[(l+xk_{1}+yk_{2})^{2}-\Delta]^{3}}\,,$

where

$\displaystyle\Delta\equiv(1-x-y)m_{\Omega}^{2}+(x+y)^{2}m_{\mu}^{2}-xyq^{2}\,,$
(10)

where $q\equiv k_{2}-k_{1}$ is the external photon momentum flowing into the
vertex. By shifting the loop momentum as $l\to l-xk_{1}-yk_{2}$, we can
transform the above loop integral into the following form

$\displaystyle i{\cal M}$ $\displaystyle=$ $\displaystyle
eQ_{\mu}g_{h}^{2}\Gamma(3)\int dxdy\int\frac{d^{4}l}{(2\pi)^{4}}{\cal
P}_{\tau\nu\lambda,\,\rho\sigma\kappa}(l-xk_{1}-yk_{2})$ (11)
$\displaystyle\frac{\gamma^{\rho\sigma\kappa}[\not{l}-x\not{k}_{1}+(1-y)\not{k}_{2}+m_{\mu}]\gamma^{\mu}[\not{l}+(1-x)\not{k}_{1}-y\not{k}_{2}+m_{\mu}]\gamma^{\tau\nu\lambda}}{[l^{2}-\Delta]^{3}}\,.$

We now contract the Lorentz indices and classify the obtained terms according
to their divergence degrees. As a result, we find that the terms possibly
proportional to the factor $m_{\mu}(-i\sigma^{\mu\nu}q_{\nu})$ are
logarithmically divergent at most, which indicates that those terms would give
rise to the dominant contribution to the muon $(g-2)$. Thus, we firstly focus
on these logarithmically divergent terms, which, after some tedious
computations, gives

$\displaystyle i{\cal M}^{\rm
log}=eQ_{\mu}g_{h}^{2}m_{\mu}(-i\sigma^{\mu\nu}q_{\nu})\int_{0}^{1}dxdy\frac{6\Gamma(3)}{m_{\Omega}^{2}}2\left[2(x+y)^{2}-6(x+y)+3\right]{\cal
I}_{0}(\Delta)\,,$ (12)

where ${\cal I}_{0}(\Delta)$ is a logarithmically divergent integral
regularized with the dimensional regularization tHooft:1972tcz :

$\displaystyle{\cal
I}_{0}(\Delta)=\frac{i}{16\pi^{2}}\left[\frac{2}{\epsilon}-\gamma+\log\frac{\mu^{2}}{\Delta}\right]\,,$
(13)

where $\epsilon\equiv 4-d$, and $\mu^{2}$ is a reference scale. It can be
shown that the Feynman parameter integral over the UV divergent part of this
amplitude vanishes

$\displaystyle\int^{1}_{0}dx\int^{1-x}_{0}dy[2(x+y)^{2}-6(x+y)+3]=0\,.$ (14)

Therefore, the one-loop contribution to the muon $g-2$ induced by $\Omega$ is
finite. By further considering the large mass hierarchy between $\Omega$ and
muons, we expand $\Delta$ in terms of $m_{\mu}^{2}/m_{\Omega}^{2}$ and just
keep the leading-order term. Thus, the remaining part of $i{\cal M}^{\rm log}$
is of ${\cal O}(m_{\mu}q/m_{\Omega}^{2})$, with the result given by

$\displaystyle i{\cal M}^{\rm log}$ $\displaystyle=$ $\displaystyle
eQ_{\mu}g_{h}^{2}m_{\mu}(-i\sigma^{\mu\nu}q_{\nu})\int^{1}_{0}dx\int^{1-x}_{0}dy\frac{12\Gamma(3)}{m_{\Omega}^{2}}[2(x+y)^{2}-6(x+y)+3]\times$
(15) $\displaystyle(-1)\frac{i}{16\pi^{2}}\log(1-x-y)$ $\displaystyle=$
$\displaystyle
i\frac{e}{2m_{\mu}}(i\sigma^{\mu\nu}q_{\nu})\frac{18}{16\pi^{2}}\frac{Q_{\mu}g_{h}^{2}m_{\mu}^{2}}{m_{\Omega}^{2}}\,.$

The above calculation shows that the $\Omega$-induced anomalous contribution
to the muon $g-2$ is dominated by terms of ${\cal
O}(m_{\mu}^{2}/m_{\Omega}^{2})$, so that we need to further consider the
associated terms in the finite integrals. It can be shown by explicit
calculations that these finite terms are given by

$\displaystyle i{\cal M}^{\rm fin}$ $\displaystyle=$ $\displaystyle
eQ_{\mu}g_{h}^{2}m_{\mu}(-i\sigma^{\mu\nu}q_{\nu})\Gamma(3)\int^{1}_{0}dxdy\left(-\frac{i}{16\pi^{2}}\right)\frac{1}{2}(-12)\left[\frac{(x+y)^{2}-5(x+y)+4}{\Delta}\right]$
(16) $\displaystyle\simeq$ $\displaystyle
ieQ_{\mu}g_{h}^{2}m_{\mu}(i\sigma^{\mu\nu}q_{\nu})\frac{\Gamma(3)(-12)}{2(16\pi^{2})m_{\Omega}^{2}}\int^{1}_{0}dx\int^{1-x}_{0}dy(4-x-y)$
$\displaystyle=$ $\displaystyle
i\frac{e}{2m_{\mu}}(i\sigma^{\mu\nu}q_{\nu})\frac{40}{16\pi^{2}}\frac{(-Q_{\mu})g_{h}^{2}m_{\mu}^{2}}{m_{\Omega}^{2}}\,,$

where we also keep terms up to ${\cal O}(qm_{\mu}/m_{\Omega}^{2})$ in the
amplitude.

Note that the amplitude related to the observable muon $g-2$ is usually
parameterized as follows Peskin:1995ev

$\displaystyle i{\cal
M}(\gamma\mu\to\mu)=\bar{u}(k_{2})(-iQ_{\mu})\left[e\gamma_{\mu}F_{1}(q^{2})+\frac{ie\sigma_{\mu\nu}q^{\nu}}{2m_{\mu}}F_{2}(q^{2})\right]u(k_{1})\,,$
(17)

where the measured muon anomalous magnetic moment can be identified as $\Delta
a_{\mu}=F_{2}(0)$. In comparison with this definition, the leading-order
result of the muon $g-2$ generated by $\Omega$ is given by

$\displaystyle\Delta a_{\mu}^{\Omega}=\Delta a^{\rm log}_{\mu}+\Delta a^{\rm
fin}_{\mu}=\frac{11g_{h}^{2}}{8\pi^{2}}\frac{m_{\mu}^{2}}{m_{\Omega}^{2}}\,.$
(18)

##  IV Constraints from Perturbativity and Unitarity

The Lagrangian of the 3-form field $\Omega$ in Eq. (2) suffers from the
theoretical constraints such as perturbativity and unitarity. As a
renormalizable dimension-4 operator, the effective interaction of $\Omega$
with muons should admit the usual perturbativity constraint. Note that the
one-loop correction to this vertex can be estimated as $\delta g_{h}\sim
g_{h}^{3}/(4\pi)^{2}$. The validity of perturbative expansion requires this
one-loop correction be smaller than the tree-level coupling $g_{h}$, which can
be transformed into the upper bound on $g_{h}$ as $|g_{h}|\lesssim 4\pi$
Nebot:2007bc .

Another restriction on the effective action of $\Omega$ is provided by the
perturbative unitarity Gell-Mann:1969cuq ; Weinberg:1971fb ; Lee:1977yc ;
Lee:1977eg  (also see Refs. Glashow:1976nt ; Huffel:1980sk ; Maalampi:1991fb ;
Kanemura:1993hm ; Akeroyd:2000wc ; Das:2015mwa ; Kanemura:2015ska ;
Goodsell:2018tti ; Appelquist:1987cf ; Chaichian:1987zt ; Falkowski:2016glr ;
Banta:2021dek ; SekharChivukula:2019yul ; Chivukula:2020hvi ;
Chivukula:2021xod ; Chivukula:2022tla ; Huang:2022zop  for an incomplete list
for the application of unitarity bounds in Beyond-SM theories). In order to
obtain these unitarity bounds, we shall compute amplitudes of $\mu^{-}\mu^{+}$
elastic 2-to-2 scatterings shown in Fig. 2.

Figure 2: Feynman diagrams which give rise to the scattering process
$\mu^{-}\mu^{+}\to\mu^{-}\mu^{+}$: (a) $s$-channel; (b) $t$-channel.

By explicit calculations of these Feynman diagrams, the amplitudes of $s$\-
and $t$-channels are given by

$\displaystyle i{\cal M}_{s}$ $\displaystyle=$
$\displaystyle\left[\bar{v}(p_{2})(g_{h}\gamma^{\rho\sigma\kappa})u(p_{1})\right]\frac{-i{\cal
P}_{\rho\sigma\kappa,\,\mu\nu\lambda}(Q)}{Q^{2}-m_{\Omega}^{2}}\left[\bar{u}(k_{1})(g_{h}\gamma^{\mu\nu\lambda})v(k_{2})\right]$
$\displaystyle=$
$\displaystyle\frac{i6g_{h}^{2}}{m_{\Omega}^{2}}[\bar{u}(k_{1})\gamma^{\mu}\gamma^{5}v(k_{2})][\bar{v}(p_{2})\gamma_{\mu}\gamma^{5}u(p_{1})]\,,$
$\displaystyle i{\cal M}_{t}$ $\displaystyle=$
$\displaystyle\left[\bar{u}(k_{1})(g_{h}\gamma^{\rho\sigma\kappa})u(p_{1})\right]\frac{-i{\cal
P}_{\rho\sigma\kappa,\,\mu\nu\lambda}(q)}{q^{2}-m_{\Omega}^{2}}\left[\bar{v}(p_{2})(g_{h}\gamma^{\mu\nu\lambda})v(k_{2})\right]$
(19) $\displaystyle=$
$\displaystyle\frac{i6g_{h}^{2}}{m_{\Omega}^{2}}[\bar{u}(k_{1})\gamma^{\mu}\gamma^{5}u(p_{1})][\bar{v}(p_{2})\gamma_{\mu}\gamma^{5}u(k_{2})]\,,$

where we have defined $Q\equiv p_{1}+p_{2}=k_{1}+k_{2}$ and $q\equiv
p_{1}-k_{1}=k_{2}-p_{2}$ and used the on-shell conditions for external muons.
Also, we have worked in the high-energy limit $s\equiv Q^{2}\gg m_{\mu}^{2}$
so that the muon mass can be ignored. In order to obtain the explicit
expressions of amplitudes, we need to specify the helicty states for the four
external muons. Before making explicit calculations, we should mention that
the Lagrangian for $\Omega$ in Eq. (2) is parity-invariant ($P$-invariant), so
that two processes related by the parity transformation should have exactly
the same amplitude. For example, we can have the following identities:

$\displaystyle{\cal M}(\mu^{-}_{R}\mu^{+}_{L}\to\mu^{-}_{R}\mu^{+}_{L})={\cal
M}(\mu^{-}_{L}\mu^{+}_{R}\to\mu^{-}_{L}\mu^{+}_{R})\,,~{}{\cal
M}(\mu^{-}_{R}\mu^{+}_{L}\to\mu^{-}_{L}\mu^{+}_{R})={\cal
M}(\mu^{-}_{L}\mu^{+}_{R}\to\mu^{-}_{R}\mu^{+}_{L})\,,$ (20)

and so on. Therefore, the number of independent non-vanishing amplitudes are
greatly reduced. In the following, we provide the calculation details for
various nonzero independent amplitudes.

###  IV.1 $\mu^{-}_{R}\mu^{+}_{L}\to\mu^{-}_{R}\mu^{+}_{L}$

In the center-of-mass (com) frame, the momenta and spinors of the incoming
particles are given by

$\displaystyle\mu^{-}_{R}:~{}p_{1}=(E,0,0,E)\,,$
$\displaystyle\mu^{+}_{L}:~{}p_{2}=(E,0,0,-E)\,,$ $\displaystyle
u_{R}(p_{1})=\sqrt{2E}\left(\begin{array}[]{c}0\\\ 0\\\ 1\\\
0\end{array}\right)\,,$ $\displaystyle
v_{L}(p_{2})=\sqrt{2E}\left(\begin{array}[]{c}0\\\ 0\\\ 0\\\
-1\end{array}\right)\,,$ (29)

while the outgoing particles are specified by

$\displaystyle\mu_{R}^{-}:~{}k_{1}=(E,E\sin\theta,0,E\cos\theta)\,,$
$\displaystyle\mu_{L}^{+}:~{}k_{2}=(E,-E\sin\theta,0,-E\cos\theta)\,,$
$\displaystyle u_{R}(k_{1})=\sqrt{2E}\left(\begin{array}[]{c}0\\\ 0\\\
\cos\frac{\theta}{2}\\\ \sin\frac{\theta}{2}\end{array}\right)\,,$
$\displaystyle v_{L}(k_{2})=\sqrt{2E}\left(\begin{array}[]{c}0\\\ 0\\\
\sin\frac{\theta}{2}\\\ -\cos\frac{\theta}{2}\end{array}\right)\,,$ (38)

where we have taken the high-energy limit so that external muon masses can be
neglected. By putting these explicit spinors into the general
$\mu^{-}$$\mu^{+}$ elastic scattering amplitudes in Eq. (IV) for the $s$\- and
$t$-channels, we can obtain the total amplitude as follows

$\displaystyle i{\cal
M}(\mu^{-}_{R}\mu^{+}_{L}\to\mu^{-}_{R}\mu^{+}_{L})=i{\cal M}_{s}+i{\cal
M}_{t}=-24ig_{h}^{2}\left(\frac{s+t}{m_{\Omega}^{2}}\right)\,,$ (39)

where we have used the definitions $s=Q^{2}$ and $t=q^{2}$.

###  IV.2 $\mu_{R}^{-}\mu_{L}^{+}\to\mu^{-}_{L}\mu^{+}_{R}$

In this case, the momenta for external muons are still taken as in Eqs. (IV.1)
and (IV.1). Since the incoming $\mu^{-}\mu^{+}$ takes the same helicity
configuration, the corresponding spinors are kept to be the same. However, due
to the helicity flips for the outgoing muon pair, the spinors for these two
particles are modified as follows

$\displaystyle
u_{L}(k_{1})=\sqrt{2E}\left(\begin{array}[]{c}-\sin\frac{\theta}{2}\\\
\cos\frac{\theta}{2}\\\ 0\\\ 0\end{array}\right)\,,$ $\displaystyle
v_{R}(k_{2})=\sqrt{2E}\left(\begin{array}[]{c}\cos\frac{\theta}{2}\\\
\sin\frac{\theta}{2}\\\ 0\\\ 0\end{array}\right)\,.$ (48)

It turns out that the $t$-channel amplitude in Eq. (IV) vanishes which is
caused by the helicity mismatching in the spinor bilinears. On the other hand,
the $s$-channel does contribute to the total amplitude as follows

$\displaystyle i{\cal
M}(\mu^{-}_{R}\mu^{+}_{L}\to\mu^{-}_{L}\mu^{+}_{R})=i{\cal
M}_{s}=12ig_{h}^{2}\left(\frac{t}{m_{\Omega}^{2}}\right)\,.$ (49)

###  IV.3 $\mu^{-}_{R}\mu^{+}_{R}\to\mu^{-}_{R}\mu^{+}_{R}$

In the com frame, the spinors for incoming and outgoing particles can be given
by

$\displaystyle u_{R}(p_{1})=\sqrt{2E}\left(\begin{array}[]{c}0\\\ 0\\\ 1\\\
0\end{array}\right)\,,$ $\displaystyle
v_{R}(p_{2})=\sqrt{2E}\left(\begin{array}[]{c}1\\\ 0\\\ 0\\\
0\end{array}\right)\,,$ (58) $\displaystyle
u_{R}(k_{1})=\sqrt{2E}\left(\begin{array}[]{c}0\\\ 0\\\
\cos\frac{\theta}{2}\\\ \sin\frac{\theta}{2}\end{array}\right)\,,$
$\displaystyle
v_{R}(k_{2})=\sqrt{2E}\left(\begin{array}[]{c}\cos\frac{\theta}{2}\\\
\sin\frac{\theta}{2}\\\ 0\\\ 0\end{array}\right)\,.$ (67)

By exploiting the above explicit expressions of external muon spinors in Eq.
(IV), we can yield the amplitudes for $s$\- and $t$-channels. As a result, it
is found that the $s$-channel ceases to contributing to the amplitude, while
the $t$-channel gives the dominant contribution as follows

$\displaystyle i{\cal
M}(\mu^{-}_{R}\mu^{+}_{R}\to\mu^{-}_{R}\mu^{+}_{R})=i{\cal
M}_{t}=-12ig_{h}^{2}\left(\frac{s}{m_{\Omega}^{2}}\right)\,.$ (68)

###  IV.4 Numerical Analysis of Unitarity Bounds

Due to the parity symmetry of the Lagrangian in Eq. (2), Eqs. (49), (39) and
(68) give all the nonzero independent amplitudes for the process
$\mu^{-}\mu^{+}\to\mu^{-}\mu^{+}$, which can be used to yield the unitarity
bounds on the 3-form field coupling to muons. Note that the unitarity of the
S-matrix gives the following constraint Lee:1977eg ; Goodsell:2018tti ;
Banta:2021dek ; Huang:2022zop  on the $s$-wave projected amplitude

$\displaystyle\mbox{Re}(a_{0})(\sqrt{s})\leqslant\frac{1}{2}\,,$ (69)

where $a_{0}(\sqrt{s})$ is defined as

$\displaystyle a_{0}(\sqrt{s})=\sqrt{\frac{4|{\bf p}_{i}||{\bf
p}_{f}|}{2^{\delta_{i}+\delta_{f}}s}}\frac{1}{32\pi}\int^{1}_{-1}d(\cos\theta){\cal
M}(i\to f)=\sqrt{\frac{4|{\bf p}_{i}||{\bf
p}_{f}|}{2^{\delta_{i}+\delta_{f}}s}}\frac{1}{16\pi}\int^{0}_{-s}\frac{dt}{s}{\cal
M}(i\to f)\,,$ (70)

where the indices $\delta_{i(f)}=1$ if the two particles in the initial
(final) states are identical to one another, otherwise $\delta_{i(f)}=0$.

By making use of the definition of $a_{0}(\sqrt{s})$, we can obtain the
$s$-wave projected amplitudes for different helicity configurations as follows

$\displaystyle|a_{0}(\mu_{R}^{-}\mu_{L}^{+}\to\mu_{R}^{-}\mu_{L}^{+})|$
$\displaystyle=$
$\displaystyle\frac{3g_{h}^{2}}{4\pi}\frac{s}{m_{\Omega}^{2}}\sim\frac{3g_{h}^{2}}{\pi}\frac{\Lambda^{2}}{m_{\Omega}^{2}}\,,$
$\displaystyle|a_{0}(\mu_{R}^{-}\mu_{L}^{+}\to\mu_{L}^{-}\mu_{R}^{+})|$
$\displaystyle=$
$\displaystyle\frac{3g_{h}^{2}}{8\pi}\frac{s}{m_{\Omega}^{2}}\sim\frac{3g_{h}^{2}}{2\pi}\frac{\Lambda^{2}}{m_{\Omega}^{2}}\,,$
$\displaystyle|a_{0}(\mu^{-}_{R}\mu^{+}_{R}\to\mu^{-}_{R}\mu^{+}_{R})|$
$\displaystyle=$
$\displaystyle\frac{3g_{h}^{2}}{4\pi}\frac{s}{m_{\Omega}^{2}}\sim\frac{3g_{h}^{2}}{\pi}\frac{\Lambda^{2}}{m_{\Omega}^{2}}\,,$
(71)

where in the last equality of each equation above we have taken the high-
energy limit $s\sim 4\Lambda^{2}$ with $\Lambda$ as the UV cutoff scale. By
further requiring the upper limit on ${\rm Re}(a_{0})(\sqrt{s})$, we can
obtain the following unitarity bound for each channel

$\displaystyle\mu^{-}_{R}\mu^{+}_{L}\to\mu^{-}_{R}\mu^{+}_{L}/\mu^{-}_{R}\mu^{+}_{R}\to\mu^{-}_{R}\mu^{+}_{R}:$
$\displaystyle|g_{h}|\lesssim\sqrt{\frac{\pi}{6}}\frac{m_{\Omega}}{\Lambda}\,,$
(72) $\displaystyle\mu^{-}_{R}\mu^{+}_{L}\to\mu^{-}_{L}\mu^{+}_{R}:$
$\displaystyle|g_{h}|\lesssim\sqrt{\frac{\pi}{3}}\frac{m_{\Omega}}{\Lambda}\,.$
(73)

Therefore, it is found that the channels
$\mu^{-}_{R}\mu^{+}_{L}\to\mu^{-}_{R}\mu^{+}_{L}$ and
$\mu^{-}_{R}\mu^{+}_{R}\to\mu^{-}_{R}\mu^{+}_{R}$ provide the strongest
constraint with $|g_{h}|\lesssim\sqrt{\pi/6}(m_{\Omega}/\Lambda)$.

##  V Numerical Results

Given the muon $g-2$ expression in Eq. (18) and the strongest unitarity bound
in Eq. (72), we are now ready to explore the parameter space of this model.
The final numerical result is plotted on the $m_{G}$-$|g_{h}|$ plane in Fig.
3, where the cutoff scale entering the unitarity bound is chosen to be
$\Lambda=500$ GeV (left panel) and 1 TeV (right panel). The solid blue band in
each plot corresponds to the parameter space explaining the muon $g-2$ data at
$2\sigma$ CL, while the red shaded region is disfavored by unitarity. Here we
do not show the theoretical limit from perturbativity $|g_{h}|\lesssim 4\pi$,
since it is too weak to be useful in constraining this model. Moreover, in
order to be consistent with the effective field theory treatment, it is
usually required that the mass of $\Omega$ should be much smaller than the
cutoff scale. Thus, we take the upper boundary of the 3-form field $\Omega$
mass to be $m_{\Omega}=300$ GeV (500 GeV) for $\Lambda=500$ GeV (1 TeV) in
Fig. 3.

Figure 3: Parameter spaces on the $m_{\Omega}$-$g_{h}$ plane for the UV cutoff
to be $\Lambda=500$ GeV (left panel) and 1 TeV (right panel), respectively. In
each plot, the blue region represents the parameter space which can explain
the muon $g-2$ at $2\sigma$ CL, while the red region is excluded by the
perturbative unitarity bound. The cyan region on the right panel can account
for at least $50\%$ of the muon $g-2$ anomaly.

In the left panel of Fig. 3 with the cutoff scale $\Lambda=500$ GeV, even
though the unitarity bound begins to restrict the large coupling region, there
is still a considerable amount of parameter spaces to fully explain the muon
$g-2$ anomaly with 150 GeV $\leqslant m_{G}\leqslant 300$ GeV and
$0.1\lesssim|g_{h}|\lesssim 0.5$. On the other hand, when the cutoff scale is
increased to $\Lambda\gtrsim 800$ GeV, it is found that the blue colored
region totally resolving the muon $g-2$ discrepancy is ruled out by unitarity
arguments, which is illustrated in the right panel of Fig. 3 for $\Lambda=1$
TeV. This implies that the current muon $g-2$ data favors a light tensor field
$\Omega$ with $m_{G}\sim{\cal O}$(100 GeV) and a low cutoff scale
$\Lambda\lesssim 800$ GeV. However, if we relax our requirement and only need
our 3-form model to account for at least $50\%$ of the muon $g-2$ anomaly,
more parameter space now opens, which is colored in cyan in the plot for
$\Lambda=1$ TeV. It is seen that some of the cyan region now escapes the
strong unitarity bound, indicating that our model of the 3-form field can
explain at least $50\%$ of the muon $g-2$ discrepancy.

##  VI Conclusions

In the present work, we have considered the possibility to explain the long-
standing muon $g-2$ anomaly by introducing in the SM a 3-form field $\Omega$,
which can naturally appear in many fundamental theories such as the type IIA
string theory and the HUFT. In particular, we have presented the leading-order
analytic expression of the muon $g-2$ by calculating the one-loop Feynman
induced by $\Omega$. We have also taken into account the theoretical
constraints from perturbativity and unitarity. Especially, we have computed
the independent tree-level $\mu^{+}\mu^{-}$ scattering amplitudes of all
initial and final helicty configurations, from which we have seen that the
channels
$\mu^{-}_{R}\mu^{+}_{L}\to\mu^{-}_{R}\mu^{+}_{L}$/$\mu^{-}_{R}\mu^{+}_{R}\to\mu^{-}_{R}\mu^{+}_{R}$
lead to the most stringent bound on this 3-form field model. Then we have
numerically explored the parameter space which is of interest to explain the
muon $g-2$ anomaly. As a result, despite the strong constraint from the
unitarity, we have still found a substantial parameter region which can solve
the muon $g-2$ discrepancy with the benchmark scenario as $m_{\Omega}\sim 250$
GeV, $|g_{h}|\sim 0.3$, and $\Lambda\sim 500$ GeV. However, when the UV cutoff
scale has risen to a larger value $\Lambda\gtrsim 800$ GeV, all parameter
regions fully resolving the $(g-2)_{\mu}$ anomaly have been excluded by
unitarity. Nevertheless, in the case with $\Lambda=1$ TeV, we have also shown
that at least 50$\%$ of the muon $g-2$ discrepancy can be accounted for by our
3-form field theory without conflicting with the strong unitarity bound.

Note that the interpretation of the muon $g-2$ anomaly requires the 3-form
field $\Omega$ to be relatively light and to have a moderately strong coupling
to $\mu$ leptons, so that it should be well tested by the existing collider
experiments. For example, if we extend this model by assuming that $\Omega$
has a universal coupling to all the SM leptons and quarks, the effective
$\Omega$-gluon interaction could be inevitably generated via SM quark loops.
Thus, the 3-form particle $\Omega$ can be produced at hadronic colliders such
as the LHC mainly through the gluon fusion channel. By observing its decay
products such as $t\bar{t}$, dijet, diphoton, diboson, and dileptons, such the
universal coupling model with a light $\Omega$ has already been ruled out by
the current ATLAS ATLAS:2018rvc ; ATLAS:2018qto ; ATLAS:2019nat ;
ATLAS:2019erb ; Wang:2021rvc  and CMS CMS:2018rkg ; CMS:2018mgb ; CMS:2019qem
; CMS:2018ipm ; CMS:2018dqv ; Radburn-Smith:2018wfo  data. On the other hand,
if this 3-form field is leptophilic, i.e., $\Omega$ only couples to the SM
lepton sector strongly, then the LHC constraints cannot be applied here.
Nevertheless, with the assumption that $\Omega$ interacts with $e^{\pm}$ by a
similar coupling to muons, such a model can be falsified in the near-future
$e^{+}e^{-}$ colliders such as CLIC Linssen:2012hp ; Aicheler:2012bya , ILC
Baer:2013cma , CEPC CEPCStudyGroup:2018rmc ; CEPCStudyGroup:2018ghi  and FCC-
ee FCC:2018evy . Note that the cross section of $e^{-}e^{+}\to\mu^{-}\mu^{+}$
mediated by $\Omega$ is given by

$\displaystyle\sigma(e^{-}e^{+}\to\mu^{-}\mu^{+})=\frac{3g_{h}^{4}}{\pi}\frac{s}{m_{\Omega}^{4}}\,,$
(74)

where $s$ is the com energy squared of the $e^{+}e^{-}$ pair. Interestingly,
the cross section in Eq. (74) shows a monotone increase with $s$, and the
absence of any pole structure at $s=m_{\Omega}^{2}$, which is quite unusual in
an $s$-channel process. Such a feature is caused by the fact that
$(s-m_{\Omega}^{2})$ originally appearing in the denominator of the $\Omega$
propagator in Eq. (5) has been canceled exactly by the same factor generated
in the numerator of the $s$-channel amplitude. Furthermore, if we take
benchmark parameters as $m_{\Omega}=250$ GeV, $g_{h}=0.3$, and $\sqrt{s}\sim
300~{}{\rm GeV}$ which is well below the cutoff scale $\Lambda=500$ GeV as
indicated by the left panel of Fig. 3, the cross section for the
$\mu^{-}\mu^{+}$ pair production induced by $\Omega$ can be estimated to be
$\sim 0.7$ ab. Such a large cross section with its unconventional rise with
$s$ indicates that the process $e^{-}e^{+}\to\mu^{-}\mu^{+}$ is a promising
target in searching for the 3-form field $\Omega$.

Finally, we would like to emphasize that our treatment of the 3-form field
$\Omega$ is, at best, effective only at low energies. Indeed, as clearly seen
from Eqs. (39), (49), (68), and (74), the $\mu^{-}\mu^{+}\to\mu^{-}\mu^{+}$
amplitudes of various polarization configurations and the cross section of
$e^{-}e^{+}\to\mu^{-}\mu^{+}$ increases with the com energy squared $s$, and
would break the unitarity at high energies. Such a pathology is why we have
introduced a UV cutoff scale $\Lambda$ in the theory. It is highly possible
that the theory is unitarized at energies around $\Lambda$ by some mechanisms,
such as the Higgs or St$\ddot{\rm u}$ckelberg mechanisms, which might also
give the mass to the 3-form field and make $\Omega$ a propagating degree of
freedom. Or $\Omega$ is just an effective description of the dynamics when $s$
is much lower than $\Lambda$, as the massless counterpart is not dynamical at
all. Another theoretical problem related to $\Omega$ is the renormalizability
of the theory. As shown in Eqs. (5) and (6), the propagator of the 3-form
field $\Omega$ does not obey the conventional power counting rule as the SM
scalar or vector particles, since adding more internal $\Omega$ propagators
would even increase the UV divergences for the loop integrals, rather than
ameliorating the UV behavior. However, in the present work, we concentrate on
explaining the muon $g-2$ in terms of the one-loop effects of the 3-form field
$\Omega$, whose finiteness reveals that its prediction is only determined by
low-energy dynamics, regardless of the high-energy details or the UV
completion of the theory.

## Acknowledgements

DH is supported in part by the National Natural Science Foundation of China
(NSFC) under Grant No. 12005254, the National Key Research and Development
Program of China under Grant No. 2021YFC2203003, and the Key Research Program
of Chinese Academy of Sciences under grant No. XDPB15. YT is supported in part
by NSFC under Grants No. 11851302, the Fundamental Research Funds for the
Central Universities and Key Research Program of the Chinese Academy of
Sciences. YLW is supported in part by the National Key Research and
Development Program of China under Grant No. 2020YFC2201501, and NSFC under
Grants No. 11851303, No. 11690022, No. 11747601, the Strategic Priority
Research Program of the Chinese Academy of Sciences under Grant No.
XDB23030100, and NSFC special fund for theoretical physics under Grant No.
12147103.

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