# Relationship between the Electronic Polarization and the Winding Number in
Non-Hermitian Systems

Shohei Masuda and Masaaki Nakamura Department of PhysicsDepartment of Physics
Ehime University Bunkyo-cho 2-5 Ehime University Bunkyo-cho 2-5 Matsuyama
Matsuyama Ehime 790-8577 Ehime 790-8577

###### Abstract

We discuss an extension of the Resta’s electronic polarization to non-
Hermitian systems with periodic boundary conditions. We introduce the
“electronic polarization” as an expectation value of the exponential of the
position operator in terms of the biorthogonal basis. We found that there
appears a finite region where the polarization is zero between two
topologically distinguished regions, and there is one-to-one correspondence
between the polarization and the winding number which takes half-odd integers
as well as integers. We demonstrate this argument in the non-Hermitian Su-
Schrieffer-Heeger model.

So far, non-Hermitian quantum mechanics has been discussed for some specific
situations [1, 2, 3, 4, 5, 6]. In recent years, this field has attracted much
attention in connection with topological phases [7, 8] due to its relevance to
the open quantum systems [9, 10, 11, 12]. The application of topology in
conventional Hermitian quantum systems has relied on the principle of bulk-
edge (-boundary) correspondence [13, 14, 15] where the edge or surface states
in open boundary systems can be characterized by topological invariants
calculated under periodic boundary conditions (PBCs). However, it has been
pointed out that such a bulk-edge correspondence may not be established in
non-Hermitian systems [16, 17, 18]. As a result, a macroscopic number of bulk
states localize at a boundary of the system which is called as the non-
Hermitian skin effect. The skin effect is topologically characterized by the
winding numbers defined by the non-Bloch band theory [19, 20, 21], and that
defined by the Bloch Hamiltonian and the complex spectrum in open systems [22,
23, 24, 25, 26], the biorthogonal polarization [27, 28], and so on.

In this paper, inspired by these studies, we discuss the electronic
polarization in non-Hermitian systems. Throughout of this paper, we consider
only periodic systems, since we need to know fundamental relationship between
topological quantities, before proceeding to deal with open boundary systems.
For conventional Hermitian quantum lattice systems, the electronic
polarization has been introduced by Resta to evaluate the expectation value of
the position operator in periodic systems as $z^{(q)}=\braket{\Psi_{0}}{{\rm
e}^{{\rm i}(2q\pi/L)\hat{X}}}{\Psi_{0}}$ with $q=1$, where
$\hat{X}=\sum_{j=1}^{L}\hat{x}_{j}$ with $\hat{x}_{j}$ being the position
operator at $j$-th site, $L$ is the number of lattices, and $\ket{\Psi_{0}}$
is the ground-state many-body wave function [29, 30]. This quantity can also
be interpreted as the overlap between the ground state and a variational
excited state appearing in the Lieb-Schultz-Mattis (LSM) theorem [31, 32, 33].
$z^{(q)}$ with $q\geq 2$ can be interpreted as extensions of the LSM theorem
for general magnetizations and fillings [34, 35], and by later equivalent
discussions [36]. Resta related $z^{(1)}$ with the electronic polarization as
$\lim_{L\to\infty}(e/2\pi){\rm Im}\;\\!\ln z^{(1)}$. Here and hereafter we
call $z^{(q)}$ itself “polarization”. The polarization $z^{(q)}$ has been
calculated in various systems [37, 36, 38, 39] and is shown to characterize
topological phases and topological transitions in one-dimensional systems.
Recently, an extension of the polarization to higher-dimensional systems has
been proposed based on spiral boundary conditions [40].

As such a fundamental relation, in this paper, we will point out that the
electronic polarization in non-Hermitian systems can be related to a winding
number which may take half integers [17, 41, 42, 43]. We will demonstrate this
argument in the non-Hermitian Su-Schrieffer-Heeger (SSH) model [44, 45, 19].

We introduce the biorthogonal system. The right and left eigenstates for a 1D
quadratic non-Hermitian Hamiltonian in real space satisfy the following
eigenvalue equations:

$\displaystyle\sum_{j^{\prime}}H^{\mathstrut}_{jj^{\prime}}\ket{\psi^{j^{\prime},{\rm
R}}_{k\mu}}=E^{\mathstrut}_{k\mu}\ket{\psi^{j,{\rm R}}_{k\mu}},$ (1a)
$\displaystyle\sum_{j^{\prime}}H^{{\dagger}}_{jj^{\prime}}\ket{\psi^{j^{\prime},{\rm
L}}_{k\mu}}=E^{\ast}_{k\mu}\ket{\psi^{j,{\rm L}}_{k\mu}},$ (1b)

respectively, where
$\ket{\psi^{j,\alpha}_{k\mu}}=\mathcal{U}^{-1}_{jk}\ket{u^{\alpha}_{k\mu}}\,(\alpha={\rm
R},{\rm L})$, $\mu$ is a band index, $\mathcal{U}^{\mathstrut}_{kj}={\rm
e}^{-{\rm i}kx_{j}}/\sqrt{L}$, and $\ket{u^{\alpha}_{k\mu}}$ with the
normalization condition $\braket{u^{\rm L}_{k\mu}}{u^{\rm
R}_{k\nu}}=\delta_{\mu\nu}$ is the eigenstate of the non-Hermitian Bloch
Hamiltonian
$H(k)=\sum_{jj^{\prime}}\mathcal{U}^{\mathstrut}_{kj}H_{jj^{\prime}}\mathcal{U}^{-1}_{j^{\prime}k}$.

Then we define the polarization in non-Hermitian systems as follows;

$z^{(q)}=\braket{\Psi_{\rm L}}{U^{q}}{\Psi_{\rm R}},\quad U=\exp\left[\,{\rm
i}\frac{2\pi}{L}\sum_{j=1}^{L}\hat{x}_{j}\,\right],$ (2)

where $\ket{\Psi_{\rm R(L)}}$ is the Slater determinant constructed by the
occupied right (left) eigenstate, and $\hat{x}_{j}=jn_{j}$, where $n_{j}$ is
the number operator of the electron at the $j$-th unit cell. The definition
(2) for $q=1$ was first introduced by Lee, et al. [46]. They obtained the
right and left many-body ground states by numerical diagonalization in real
space and calculated ${\rm Im}\;\ln z^{(1)}$, which takes $0$ or $\pi$. In
this approach, they could deal with small-size systems, and the behavior of
$z^{(q)}$ itself became unclear.

In order to improve this problem, we extend Resta’s approach for Hermitian
systems [30] to the polarization in non-Hermitian systems, so that we rewrite
Eq. (2) in terms of the right and left one-particle eigenstates:

$\displaystyle z^{(q)}$ $\displaystyle={\rm det}\,S^{(q)},$ (3) $\displaystyle
S^{(q)}_{\mu,\nu}(k_{s},k_{s^{\prime}})$
$\displaystyle=\sum_{j=1}^{L}\braket{\psi^{j,{\rm L}}_{k_{s}\mu}}{{\rm
e}^{{\rm i}\frac{2q\pi}{L}x_{j}}}{\psi^{j,{\rm R}}_{k_{s^{\prime}}\nu}},$ (4)

where $k_{s}=(2\pi/L)(s+\theta/2\pi)$ with $s=0,1,\ldots,L-1$ for the boundary
conditions $\ket{\psi^{j+L,\alpha}_{k\mu}}={\rm e}^{{\rm
i}\theta}\ket{\psi^{j,\alpha}_{k\mu}}$, and the determinant is restricted to
the occupied states. Then the determinant is rewritten as a product of the
biorthogonal overlaps:

$z^{(q)}=(-1)^{qN/L}\prod_{s=0}^{L-1}\,\prod_{\mu\in{\rm
occupied}}\braket{u^{\rm L}_{k_{s+q}\mu}}{u^{\rm R}_{k_{s}\mu}},$ (5)

where $N$ is the number of electrons. The factor $(-1)^{qN/L}$ stems from the
antisymmetry of the determinant [40]. Since Eq. (5) is a simple product of
overlaps between Bloch states with the wave vectors that differ by $k_{q}$,
$z^{(q)}$ can be calculated for large system sizes, and the behavior of
$z^{(q)}$ in the thermodynamic limit will be clarified.

We consider a two-band non-Hermitian system with chiral symmetry
$\tau_{3}H(k)\tau_{3}=-H(k)$. In this case, the Hamiltonian is written as

$H(k)=\begin{pmatrix}&h_{+}(k)\,\\\ \,h_{-}(k)&\end{pmatrix}.$ (6)

Then we define the winding number as follows [13, 14, 15];

$\nu=\frac{{\rm i}}{2\pi}\int^{2\pi}_{0}{\rm
d}k\,r^{-1}(k)\frac{\partial}{\partial k}r(k).$ (7)

where $r(k)=h_{+}(k)/\sqrt{h_{+}(k)h_{-}(k)}$. For Hermitian systems, $r^{-1}$
becomes $r^{\ast}$, and $\nu$ can only take $0$ or $1$. However, for non-
Hermitian systems, the winding number $\nu$ can be half-integers [41, 42, 43].

For the two-band model with the single occupied band, the phase of the
polarization is related to the winding number by Eq. (5) in the $L\to\infty$
limit as follows;

$\ln z^{(1)}={\rm i}\pi+\sum_{s=0}^{L-1}\ln\braket{u^{\rm L}_{k_{s}}}{u^{\rm
R}_{k_{s-1}}}\simeq{\rm i}(1\pm\nu)\pi,$ (8)

which is well-defined unless $\nu$ is half-odd integers, where the signs
“$\pm$” are related to the choice of the eigenstates [15].

Figure 1: (Color online) The non-Hermitian SSH model (9). The dotted boxes
indicate the unit cells.

We concider the non-Hermitian Su-Schrieffer-Heeger (SSH) model with half-
filling [44, 45, 19] as illustrated in Fig. 1. The Hamiltonian in real space
is given by

$\displaystyle\mathcal{H}$
$\displaystyle=t_{1}\sum_{i=1}^{L}c^{\dagger}_{i,{\rm
A}}c^{\mathstrut}_{i,{\rm B}}+t_{2}\sum_{i=1}^{L}c^{\dagger}_{i,{\rm
B}}c^{\mathstrut}_{i+1,{\rm A}}+{\rm H.c.}$ $\displaystyle\
+\gamma\sum_{i=1}^{L}c^{\dagger}_{i,{\rm A}}c^{\mathstrut}_{i,{\rm B}}-{\rm
H.c.},$ (9)

where $c^{\mathstrut}_{i,\alpha}$ $(c^{\dagger}_{i,\alpha})$ denotes the
annihilation (creation) operator of a fermion in sublattice $\alpha$ at the
$i$-th unit cell. When $\gamma=0$, this model is Hermitian, and phase
transitions occur at $t_{1}=\pm t_{2}$. This Hamiltonian can be written as the
quadratic form:

$\mathcal{H}=\sum_{ij,\alpha\beta}c^{\dagger}_{i,\alpha}H_{ij,\alpha\beta}c^{\mathstrut}_{j,\beta},$
(10)

so that it is diagonalized by the Fourier transformation
$c^{\mathstrut}_{k,\alpha}=\sum_{j}\mathcal{U}^{\mathstrut}_{kj}c^{\mathstrut}_{j,\alpha}$
as follows:

$\displaystyle\mathcal{H}$
$\displaystyle=\sum_{k,\alpha\beta}c_{k,\alpha}^{\dagger}H_{\alpha\beta}(k)c_{k,\beta}^{\mathstrut},$
(11) $\displaystyle H(k)$ $\displaystyle=\sum_{\mu=1}^{2}(d_{\mu}+{\rm
i}f_{\mu})\tau_{\mu},$ (12)

where

$\begin{array}[]{ll}d_{1}=t_{1}+t_{2}\cos k,&f_{1}=0,\\\ d_{2}=t_{2}\sin
k,&f_{2}=\gamma,\end{array}$ (13)

and $\tau_{\mu}\ (\mu=1,2,3)$ is Pauli matrices. This model has chiral
symmetry, so that the energy eigenvalues are symmetric about zero energy. The
square of the energy eigenvalue is given by

$E^{2}(k)=t_{1}^{2}+t_{2}^{2}-\gamma^{2}+2t_{1}t_{2}\cos k+{\rm
i}\,2t_{2}\gamma\sin k.$ (14)

The energy gap closes at the exceptional points (EPs)
$(d_{1},d_{2})=(-f_{2},f_{1})$ or $(f_{2},-f_{1})$, which leads to
$t_{1}=-t_{2}\pm\gamma$ for $k=0$ or $t_{1}=t_{2}\pm\gamma$ for $k=\pi$. For
$k=0$ and $\pi$, the energy eigenvalues are
$E=\pm\sqrt{(t_{1}+t_{2})^{2}-\gamma^{2}}$ and
$\pm\sqrt{(t_{1}-t_{2})^{2}-\gamma^{2}}$, respectively, so that they are pure
imaginaries when $t_{1}$ satisfies $-t_{2}-\gamma<t_{1}<-t_{2}+\gamma$ or
$t_{2}-\gamma<t_{1}<t_{2}+\gamma$, respectively [see Fig. 2].

Figure 2: (Color online) (a) The real part of energy eigenvalues, (b) the
polarization and (c) the winding number in the non-Hermitian SSH model (9)
with $t_{2}=1$ and $\gamma=0.5$. The eigenvalues are computed for $L=32$, and
the polarization is calculated for $L=32,256$ and $8192$.

For the non-Hermitian SSH model in real space, we concider the “space
inversion” as $\mathcal{P}c_{j,{\rm A(B)}}\mathcal{P}^{-1}=c_{L+1-j,{\rm
B(A)}}$. Under this transformation, the non-Hermitian Bloch Hamiltonian (12)
satisfies the following relation

$\mathcal{P}H(k)\mathcal{P}^{-1}=H^{\dagger}(k),$ (15)

where $\mathcal{P}=\tau_{1}\mathcal{K}$, and $\mathcal{K}$ is a conjugate
operator. Moreover, this transformation leads to

$z^{(q)}=\braket{\Psi_{\rm R}}{\mathcal{P}U^{q}\mathcal{P}^{-1}}{\Psi_{\rm
L}}={\rm e}^{{\rm i}2q\pi N/L}[z^{(q)}]^{\dagger}.$ (16)

Since $N=L$ in the non-Hermitian SSH model at half-filling, we should choose
$q=1$. Then Eq. (5) is reduced to

$z^{(1)}=(-1)\prod_{s=0}^{L-1}\,\braket{u^{\rm L}_{k_{s+1}-}}{u^{\rm
R}_{k_{s}-}},$ (17)

where $\ket{u^{\rm R(L)}_{k_{s}-}}$ is the right (left) eigenstate of the
occupied band. For $t_{2}+\gamma<t_{1}$, the electrons of the system are
located on unit cells, so that we find that $z^{(1)}=-1$ for even $L$ because
$(2\pi/L)\sum_{x_{j}=1}^{L}x_{j}=\pi(L+1)$. Similarly, for
$-t_{2}+\gamma<t_{1}<t_{2}-\gamma$, the electrons of the system are located on
between a unit cell and the neighboring one, so that we find that $z^{(1)}=1$
for even $L$ because of $(2\pi/L)\sum_{x_{j}=1}^{L}(2x_{j}+1)/2=\pi(L+2)$. On
the other hand, for $-t_{2}-\gamma<t_{1}<-t_{2}+\gamma$ or
$t_{2}-\gamma<t_{1}<t_{2}+\gamma$, the electrons of the system are located
homogeneously, so that $z^{(1)}=0$ for even $L$ because
$(2\pi/L)\sum_{x_{j}=1}^{L}(2x_{j}+1/2)/2=\pi(2L+3)/2$. Here, we have used the
fact that $z^{(1)}$ is always real, since Eq. (16) leads to
$z^{(1)}=[z^{(1)}]^{\dagger}$ in the non-Hermitian SSH model.

The numerical results of the polarization $z^{(1)}$ for different system sizes
are shown in Fig. 2, and $z^{(1)}$ distinguishes three topologically distinct
phases $z^{(1)}=-1,0,+1$, as expected. Here, we have used anti-periodic
boundary conditions $c_{i+L,\alpha}=-c_{i,\alpha}$ to prevent divergences at
EPs, at which phase transitions occur. The results do not depend on the
boundary conditions except for the phase transition points. If the regions
with ${\rm Re}\;E=0$ are interpreted as gapless phases, the results with
$z^{(1)}=0$ are consistent with the LSM theorem. Note that $z^{(1)}$ in a
finite-size system has a finite value, so that $(2\pi)^{-1}{\rm Im}\ln
z^{(1)}$ is not a good quantity to determine topologically different phases,
especially for $z^{(1)}=0$ regions.

Using Eq. (7), we obtain the following winding number of the non-Hermitian SSH
model:

$\nu=\int^{2\pi}_{0}\frac{{\rm d}k}{2\pi}\,\frac{t_{2}(t_{1}\cos k+{\rm
i}\gamma\sin k+t_{2})}{E^{2}},$ (18)

which is well defined except at EPs, where $E^{2}$ is given by Eq. (14). The
winding number is equal to the half of the number of times that, in
$d_{1}$-$d_{2}$ plane, EPs are surrounded by the closed curve when $k$ ranges
from $0$ to $2\pi$ [41].

The numerical results of the winding number $\nu$ are shown in Fig. 2.
Comparing them with the results of the polarization $z^{(1)}$ in the
thermodynamic limit ($L\to\infty$), we find that there are one-to-one
correspondences between $\nu=0,1/2,1$ and $z^{(1)}=-1,0,+1$ for the
topologically different phases. Therefore, Eq.(8) becomes singular at
$z^{(1)}=0$, and two quantities in this case can be well related as
$z^{(1)}=-\cos\nu\pi$.

For comparison to the polarization $z^{(1)}$, we concider the polarization
defined by the occupied right eigenstate. In the non-Hermitian SSH model, this
is given by

$z^{(1)}_{\rm RR}=(-1)\prod_{s=0}^{L-1}\,\braket{u^{\rm R}_{k_{s+1}-}}{u^{\rm
R}_{k_{s}-}},$ (19)

where $\ket{u^{\rm R}_{k\mu}}$ is normalized as $\braket{u^{\rm
R}_{k\mu}}{u^{\rm R}_{k\mu}}=1$, and the numerical results are shown in Fig.
3. Unlike $z^{(1)}$, $z^{(1)}_{\rm RR}$ is generally complex. We find that
${\rm Re}\,z^{(1)}_{\rm RR}\neq 0$ in the region that ${\rm Re}\,E=0$ and
$\nu=1/2$. Therefore, $z^{(1)}_{\rm RR}$ is not a good index to characterize
the different phases comparing with $z^{(1)}$. On the other hand,
$|z^{(1)}_{\rm RR}|$ shows liner behavior in terms of $t_{1}$, and
$|z^{(1)}_{\rm RR}|\neq 1$ or $|z^{(1)}_{\rm RR}|=1$ correspond the phases
with $\nu=1/2$ and $\nu=0,1$, respectively. Thus $|z^{(1)}_{\rm RR}|$ is
better topological index than ${\rm Re}\,z^{(1)}_{\rm RR}$ and ${\rm
Im}\,z^{(1)}_{\rm RR}$, in this case. Comparing the results of the two-types
of polarization $z$ and $z_{\rm RR}$ defined by Eqs. (2) and (19), it is
sufficient to use the former one given by a real number to describe the
topological phases and transitions in the present non-Hermitian system as
discussed in Ref. 6.

Figure 3: (Color online) The real part (orange line), the imaginary part
(orange dashed line) and the absolute value (red line) of the polarization,
which is given by Eq. (19), in the non-Hermitian SSH model (9) with $t_{2}=1$
and $\gamma=0.5$ for $L=8192$.

In wummary, we have studied an extension of the Resta’s electronic
polarization [30] to non-Hermitian systems with chiral symmetry. We have given
the reduction formalism for the polarization (5) which enables us to deals
with large systems close to the thermodynamic limit. Then the relationship
between the polarization and the winding number has been discussed.

We have analyzed the non-Hermitian SSH model and shown numerically that the
polarization defined by the biorthogonal basis (2) takes three different
values $z^{(1)}=-1,0,+1$ for the topologically distinct phases in the
thermodynamic limit. We have also found that there is one-to-one
correspondence with the winding number $\nu=0,1/2,1$ defined in Refs. 17, 41.
Furthermore, this relationship has been shown analytically as Eq. (8). Here we
should be careful to take the thermodynamic limit of $z^{(1)}$: depending on
how to take the thermodynamic limit, the region with $\nu=1/2$ may be lost
[46].

The topological phases for the non-Hermitian skin effect in open systems have
been discussed by a winding number in a different definition [19] from the
present one for periodic systems. This winding number takes only two values,
and the “transition point” where the winding number changes its value appears
to be different. However their difference is only whether the integration is
defined in the generalized Brillouin zone or in the ordinary Brillouin zone
(7). Therefore the present study may contribute to analyze the skin effect by
generalizing the polarization operator. There is another topological number
“biorthogonal polarization” defined in open boundary systems [27, 28]. This
quantity is considered to characterize the number of edge modes [28].
Therefore, despite having a similar appearance, there is no direct
correspondence with the present electronic polarization.

We have also examined the polarization defined only by right eigenstates
$z^{(1)}_{\rm RR}$. This quantity takes a complex value and is not explicitly
related to the winding number. In contrast to this, the polarization defined
by the biorthogonal basis $z^{(1)}$ is a real number [6] and there is a clear
one-to-one correspondence with the winding number, so that $z^{(1)}$ is
considered to be more suitable definition than $z^{(1)}_{\rm RR}$ for the
present closed system.

## Acknowledgment

The authors thank N. Hatano and K. Imura for valuable discussions. M. N.
acknowledges the Visiting Researcher’s Program of the Institute for Solid
State Physics, The University of Tokyo, and the research fellow position of
the Institute of Industrial Science, The University of Tokyo. M. N. is
supported partly by MEXT/JSPS KAKENHI Grant No. JP20K03769.

## References

  * [1] H. Feshbach, Unified theory of nuclear reactions, Ann. Phys. (NY) 5, 357 (1958).
  * [2] N. Hatano and D. R. Nelson, Localization Transitions in Non-Hermitian Quantum Mechanics, Phys. Rev. Lett. 77, 570 (1996).
  * [3] N. Hatano and D. R. Nelson, Vortex pinning and non-Hermitian quantum mechanics, Phys. Rev. B 56, 8651 (1997).
  * [4] N. Hatano and D. R. Nelson, Non-Hermitian delocalization and eigenfunctions, Phys. Rev. B 58, 8384 (1998).
  * [5] C. M. Bender and S. Boettcher, Real Spectra in Non-Hermitian Hamiltonians Having $\mathcal{PT}$ Symmetry, Phys. Rev. Lett. 80, 5243 (1998).
  * [6] C. M. Bender,D. C. Brody, and H. F. Jones, Complex Extension of Quantum Mechanics, Phys. Rev. Lett. 89, 270401 (2002).
  * [7] Y. Ashida, Z. Gong, and M. Ueda, Non-Hermitian physics, Advances in Physics 69, 249 (2020).
  * [8] E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Exceptional topology of non-Hermitian systems, Rev. Mod. Phys. 93, 015005 (2021).
  * [9] T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, Topological photonics, Rev. Mod. Phys. 91, 015006 (2019).
  * [10] L. Lu, J. D. Joannopoulos, and M. Soljacic, Topological photonics, Nat. Photonics 8, 821 (2014).
  * [11] R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Non-Hermitian physics and PT symmetry, Nat. Phys. 14, 11 (2018).
  * [12] S. Longhi, Parity-time symmetry meets photonics: A new twist in non-Hermitian optics, EPL 120, 64001 (2017).
  * [13] F. Wilczek and A. Zee, Appearance of Gauge Structure in Simple Dynamical Systems, Phys. Rev. Lett. 52, 2111 (1984).
  * [14] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Classification of topological insulators and superconductors in three spatial dimensions, Phys. Rev. B 78, 195125 (2008).
  * [15] S. Ryu, A. P. Schnyder, A. Furusaki, and A.W. Ludwig, Topological insulators and superconductors: tenfold way and dimensional hierarchy, New J. Phys. 12, 065010 (2010).
  * [16] K. Esaki, M. Sato, K. Hasebe, and M. Kohmoto, Edge states and topological phases in non-Hermitian systems, Phys. Rev. B 84, 205128 (2011).
  * [17] T. E. Lee, Anomalous Edge State in a Non-Hermitian Lattice, Phys. Rev. Lett. 116, 133903 (2016).
  * [18] H. Shen, B. Zhen, and L. Fu, Topological Band Theory for Non-Hermitian Hamiltonians, Phys. Rev. Lett. 120, 146402 (2018).
  * [19] S. Yao and Z. Wang, Edge States and Topological Invariants of Non-Hermitian Systems, Phys. Rev. Lett. 121, 086803 (2018).
  * [20] K. Yokomizo and S. Murakami, Non-Bloch Band Theory of Non-Hermitian Systems, Phys. Rev. Lett. 123, 066404 (2019).
  * [21] K. Kawabata, N. Okuma, M. Sato, Non-Bloch band theory of non-Hermitian Hamiltonians in the symplectic class, Phys. Rev. B 101, 195147 (2020).
  * [22] Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Topological Phases of Non-Hermitian Systems, Phys. Rev. X 8, 031079 (2018).
  * [23] K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Symmetry and Topology in Non-Hermitian Physics, Phys. Rev. X 9, 041015 (2019).
  * [24] N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Topological Origin of Non-Hermitian Skin Effects, Phys. Rev. Lett. 124, 086801 (2020).
  * [25] P. M. Vecsei, M. M. Denner, T. Neupert, and F. Schindler, Symmetry indicators for inversion-symmetric non-Hermitian topological band structures, Phys. Rev. B 103, L201114, (2021).
  * [26] L. Li, S. Mu, C. H. Lee, and J. Gong, Quantized classical response from spectral winding topology, Nat. Commun. 12, 5294 (2021).
  * [27] F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems, Phys. Rev. Lett. 121, 026808 (2018).
  * [28] E. Edvardsson, F. K. Kunst, T. Yoshida, and E. J. Bergholtz, Phase transitions and generalized biorthogonal polarization in non-Hermitian systems, Phys. Rev. Research 2, 043046 (2020).
  * [29] R. Resta, Macroscopic polarization in crystalline dielectrics: the geometric phase approach, Rev. Mod. Phys. 66, 899 (1994).
  * [30] R. Resta, Quantum-Mechanical Position Operator in Extended Systems, Phys. Rev. Lett. 80, 1800 (1998).
  * [31] E. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain, Ann. Phys. (NY) 16, 407 (1961).
  * [32] I. Affleck and E. H. Lieb, A proof of part of Haldane’s conjecture on spin chains, Lett. Math. Phys. 12, 57 (1986).
  * [33] I. Affleck, Spin gap and symmetry breaking in CuO2 layers and other antiferromagnets, Phys. Rev. B 37, 5186 (1988).
  * [34] M. Oshikawa, M. Yamanaka, and I. Affleck, Magnetization Plateaus in Spin Chains: “Haldane Gap” for Half-Integer Spins, Phys. Rev. Lett. 78, 1984 (1997).
  * [35] M. Yamanaka, M. Oshikawa, and I. Affleck, Nonperturbative Approach to Luttinger’s Theorem in One Dimension, Phys. Rev. Lett. 79, 1110 (1997).
  * [36] A. A. Aligia and G. Ortiz, Quantum Mechanical Position Operator and Localization in Extended Systems, Phys. Rev. Lett. 82, 2560 (1999).
  * [37] R. Resta and S. Sorella, Electron Localization in the Insulating State, Phys. Rev. Lett. 82, 370 (1999).
  * [38] M. Nakamura and J. Voit, Lattice twist operators and vertex operators in sine-Gordon theory in one dimension, Phys. Rev. B 65, 153110 (2002).
  * [39] M. Nakamura and S. Todo, Order Parameter to Characterize Valence-Bond-Solid States in Quantum Spin Chains, Phys. Rev. Lett. 89, 077204 (2002).
  * [40] M. Nakamura, S. Masuda, and S. Nishimoto, Characterization of topological insulators based on the electronic polarization with spiral boundary conditions, Phys. Rev. B 104, L121114, (2021).
  * [41] C. Yin, H. Jiang, L. Li, R. Lü, and S. Chen, Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-Hermitian systems, Phys. Rev. A 97, 052115 (2018).
  * [42] K. Imura and Y. Takane, Generalized bulk-edge correspondence for non-Hermitian topological systems, Phys. Rev. B 100, 165430 (2019).
  * [43] K. Imura and Y. Takane, Generalized Bloch band theory for non-Hermitian bulk-boundary correspondence, Prog. Theor. Exp. Phys. 2020, 12A103.
  * [44] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in Polyacetylene, Phys. Rev. Lett. 42, 1698 (1979).
  * [45] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Soliton excitations in polyacetylene, Phys. Rev. B 22, 2099 (1980).
  * [46] E. Lee, H. Lee, and B.-J. Yang, Many-body approach to non-Hermitian physics in fermionic systems, Phys. Rev. B 101, 121109 (2020).


# Implications of the muon anomalous magnetic moment for the LHC and MUonE

Akanksha Bhardwaj<EMAIL_ADDRESS>School of Physics &
Astronomy, University of Glasgow, Glasgow G12 8QQ, UK
Christoph Englert<EMAIL_ADDRESS>School of Physics &
Astronomy, University of Glasgow, Glasgow G12 8QQ, UK
Panagiotis Stylianou<EMAIL_ADDRESS>School of Physics &
Astronomy, University of Glasgow, Glasgow G12 8QQ, UK

###### Abstract

We consider the anomalous magnetic moment of the muon $a_{\mu}$, which shows a
significant deviation from the Standard Model expectation given the recent
measurements at Fermilab and BNL. We focus on Standard Model Effective Field
Theory (SMEFT) with the aim to identify avenues for the upcoming LHC runs and
future experiments such as MUonE. To this end, we include radiative effects to
$a_{\mu}$ in SMEFT to connect the muon anomaly to potentially interesting
searches at the LHC, specifically Higgs decays into muon pairs and such decays
with resolved photons. Our investigation shows that similar to results for
concrete UV extensions of the Standard Model, the Fermilab/BNL result can
indicate strong coupling within the EFT framework and $a_{\mu}$ is
increasingly sensitive to a single operator direction for high scale UV
completions. In such cases, there is some complementarity between expected
future experimental improvements, yet with considerable statistical challenges
to match the precision provided by the recent $a_{\mu}$ measurement.

###### pacs:

##  I Introduction

The search for new physics beyond the Standard Model (SM), which so far has
been unsuccessful, is one of the highest priorities of the current particle
physics programme. A relevant anomaly in this context might be the recent
measurement of the anomalous muon magnetic moment $a_{\mu}=(g-2)_{\mu}/2$ at
Fermilab Abi et al. (2021), which confirmed the earlier BNL E821 results
Bennett et al. (2004), leading to a $\sim 4\sigma$ tension Aoyama et al.
(2012, 2019); Czarnecki et al. (2003); Gnendiger et al. (2013); Davier et al.
(2017); Keshavarzi et al. (2018); Colangelo et al. (2019); Hoferichter et al.
(2019); Davier et al. (2020); Keshavarzi et al. (2020); Kurz et al. (2014);
Melnikov and Vainshtein (2004); Masjuan and Sanchez-Puertas (2017); Colangelo
et al. (2017); Hoferichter et al. (2018); Gérardin et al. (2019a); Bijnens et
al. (2019); Colangelo et al. (2020); Blum et al. (2020); Colangelo et al.
(2014) (see also Aoyama et al. (2020) for a summary)

$\Delta a_{\mu}=a_{\mu}(\text{exp})-a_{\mu}(\text{\text{SM}})=(25.1\pm
5.9)\times 10^{-10}\,.$ (1)

Anomalous magnetic moments are characterised by dimension-six operators
$\sim\bar{\psi}_{L}\sigma^{\mu\nu}\psi_{R}F_{\mu\nu}+\text{h.c.}$ where $F$
denotes the QED field strength tensor. Therefore, $a_{\mu}$ is directly
sensitive to new interactions in renormalisable field theories and does not
probe any of these theories’ defining parameters. This is part of the reason
why $a_{\mu}$ has received enormous attention from the BSM community, as it
provides a formidably precise tool to constrain the structure of concrete UV
extensions of the SM (see, e.g. Athron et al. (2021) for a recent overview).

In contrast to these model-specific analyses, the null results of a plethora
of new physics analyses at, e.g., the Large Hadron Collider (LHC) have
highlighted model-independent methods based on effective field theory (EFT)
Weinberg (1979) techniques as alternative approaches to search for new physics
effects. EFT is a powerful tool when extra degrees of freedom can be
consistently integrated out Buchalla et al. (2014, 2017); Catà and Jung
(2015); Carmona et al. (2021); Dittmaier et al. (2021) and when measurements
are performed at energy scales that do not violate the scale hierarchies that
are implicitly assumed by the EFT approach Contino et al. (2016). The latter
can be challenging at hadron colliders with their large energy coverage and
significant uncertainties Englert and Spannowsky (2015); Englert et al.
(2020). The extraction of $a_{\mu}$ from data is largely free of such
shortfalls and there has been a range of EFT-based investigations into the
$a_{\mu}$ anomaly Crivellin et al. (2014); Allwicher et al. (2021); Aebischer
et al. (2021); Cirigliano et al. (2021).

Historically, EFT measurements have played a crucial role in shaping the
understanding of physics of the weak scale. A famous example is the muon’s
lifetime providing a measurement of the Fermi constant $G_{F}$, the cut-off of
the low-energy effective theory of the weak interactions. A constraint on the
cut-off gives rise to an upper limit on a more fundamental mass scale. The
latter is an important pointer towards experimental signatures (based on the
assumption of a well behaved perturbative expansion). In the SM

${m\over\Lambda}=\sqrt{G_{F}m_{W}^{2}}\simeq 0.27\ll 1$ (2)

measures the UV completing degrees of freedom of Fermi’s theory in units of
its cut-off.

The Fermi theory shows that new physics could appear at relatively low scales
compared to the cut-off. This is strong motivation to consider $a_{\mu}$ in
EFT to clarify implications for energy scales above the muon mass: In
Allwicher et al. (2021) it was shown that the combination of unitarity
constraints and scale evolution could push the new physics scale to very large
values (an observation that is echoed in concrete UV extensions, see e.g. Baer
et al. (2021); Athron et al. (2021); Frank et al. (2021); Ellis et al.
(2021a); Zhang (2021); Jueid et al. (2021); Altmannshofer et al. (2021);
Chakraborti et al. (2021a, b)). This raises the question whether $a_{\mu}$
merely fixes one parameter of the EFT Lagrangian, perhaps with little
phenomenological implications for UV physics. The comprehensive analysis of
Aebischer et al. (2021) has evaluated the anomalous magnetic moment in
Standard Model Effective Theory (SMEFT) with higher order effects by matching
and evolving $g-2$ in Low Energy Effective Theory (LEFT), including flavour
violating contributions, obtaining results consistent with the previous study
of Ref. Crivellin et al. (2014) which included loop contributions from non-
diagonal contributions of ${\cal{O}}_{le}$ and the dimension-six dipole
operator.

In this work, we employ SMEFT Grzadkowski et al. (2010) to revisit the EFT
context of $a_{\mu}$ with the aim to correlate $a_{\mu}$ with measurements at
the high-luminosity LHC and muon-specific future experiments such as MUonE
Abbiendi et al. (2017) (see also Ref. Coluccio Leskow et al. (2017); Crivellin
et al. (2021); Fajfer et al. (2021); Crivellin and Hoferichter (2021);
Paradisi et al. (2022) for correlations of $a_{\mu}$ with additional modes for
new physics models). To this end, we investigate $a_{\mu}$ at full one-loop
order in SMEFT, building on the results of Dedes et al. (2017, 2020). Our
findings show that as discussed by previous effective interpretations of the
muon $g-2$ Aebischer et al. (2021); Crivellin et al. (2014), the only
operators that could provide explanations for the Fermilab measurement
individually are the dipole operators and ${\cal{O}}_{le}$. To scrutinise
further these operators, we consider a priori sensitive processes, such as the
$h\mu\mu$ signal strength and constraints that can be placed by the future
MUonE experiment. However, none of these avenues can provide comparable
restrictions on the span of the dipole WCs when compared to the Fermilab
measurement.

We organise this work as follows: Section II provides a short overview of
$a_{\mu}$ in SMEFT to make this work self-contained. We explore the impact of
the higher orders at different scales in Section II.1 and additionally discuss
the possibility that $a_{\mu}$ arises as a radiative correction in Section
II.2. Different avenues with the potential to tension the dipole operators are
studied in Section III, focusing on the decay of the $Z$ boson, the
$h\to\mu\mu\gamma$ channel at LHC and also the MUonE experiment. We conclude
in Section IV.

##  II $a_{\mu}$ in SMEFT

Neglecting contributions to the unphysical (longitudinal) anapole moment Czyz
et al. (1988), the vertex function for muon-vector boson interactions can be
expanded as

$\hfill\parbox[h][130.08731pt][c]{108.405pt}{
\includegraphics[scale={0.8}]{./vertex.pdf}
}\vspace{-1.8cm}=-ieQ_{f}\bar{u}(p^{\prime})\bigg{[}\gamma^{\mu}F_{1}(k^{2})+i\frac{\sigma^{\mu\nu}k_{\nu}}{2m_{f}}F_{2}(k^{2})+\frac{\sigma^{\mu\nu}k_{\nu}}{2m_{f}}\gamma_{5}F_{3}(k^{2})\bigg{]}u(p)\;,\hfill$
(3)

where the boson’s momentum is given by $k=p^{\prime}-p$ and
$\sigma_{\mu\nu}=i[\gamma_{\mu},\gamma_{\nu}]/2$ are the usual Lorentz algebra
generators. The fermion legs are on-shell and the electric charge $e$ is
renormalised through the renormalisation condition $F_{1}(0)=1$ at zero
momentum transfer. The anomalous magnetic moment for the muon is then defined
through the form factor as $a_{\mu}=F_{2}(0)$, while $F_{3}$ is related to the
electric dipole moment.

The effective interactions of SMEFT in the Warsaw basis Grzadkowski et al.
(2010) that give rise to these form factors can be modelled with SmeftFR Dedes
et al. (2017, 2020) which employs FeynRules Christensen and Duhr (2009);
Alloul et al. (2014) to obtain the relevant Feynman rules for the SMEFT
Lagrangian truncated at operator-dimension six. Interfacing with FeynArts Hahn
(2001) enables users to enumerate the relevant diagrams, and their respective
amplitudes can be calculated with FormCalc Hahn and Perez-Victoria (1999). The
tensor integrals that appear in these amplitudes are reduced to scalar
Passarino-Veltman functions Passarino and Veltman (1979) (see also Ref. Denner
and Dittmaier (2006)), which can then be evaluated analytically using PackageX
Patel (2015). The form factors can then be extracted from these expressions
with the help of the Gordon identities which re-cast the result into the form
of Eq. (3). For QED, this reduces to the well-known Schwinger result
$a_{\mu}=\alpha/(2\pi)$ Schwinger (1948).

In contrast to the SM, tree-level contributions to the anomalous magnetic
moment are induced in SMEFT by the operators (see Refs. Buchmuller and Wyler
(1986); Alonso et al. (2014))

$\begin{split}{\cal{O}}_{eB}^{f_{1}f_{2}}=&\;(\bar{L}_{f_{1}}\sigma^{\mu\nu}e_{f_{2}})\Phi\,B_{\mu\nu}\;,\\\
{\cal{O}}_{eW}^{f_{1}f_{2}}=&\;(\bar{L}_{f_{1}}\sigma^{\mu\nu}e_{f_{2}})\tau^{I}\Phi\,W^{I}_{\mu\nu}\;,\end{split}$
(4)

where $L$ ($e$) denotes the left-handed (right-handed) lepton, $B_{\mu\nu}$,
$W^{I}_{\mu\nu}$ are the field strength tensors for the $U(1)_{Y}$ and
$SU(2)_{L}$ gauge groups respectively, and $\Phi$ is the Higgs doublet. The
Pauli matrices are denoted by $\tau^{I}$, $I=1,2,3$. The dimension-six
operators induce an anomalous magnetic moment of111We consider only the
diagonal entries of the operators in Eq. (4) and suppress the flavour indices.

$\Delta
a_{\mu}^{\text{tree}}=\frac{\sqrt{2}vm_{\mu}}{e\Lambda^{2}}\Big{[}c_{W}\left(C_{eB}+C_{eB}^{*}\right)-s_{W}\left(C_{eW}+C_{eW}^{*}\right)\Big{]}\,,$
(5)

where $s_{W}$ ($c_{W}$) is the sine (cosine) of the Weinberg angle. Under the
assumption that $C_{eW}$ and $C_{eB}$ are real, any contribution to the
electric dipole moment is removed and the two operators only generate $\Delta
a_{\mu}$ (modulo small SM electroweak radiative effects).

Extending the calculation to one-loop level requires additional
renormalisation constants not present in the SM Alonso et al. (2014); Jenkins
et al. (2014, 2013). While no UV-divergence is induced in the $F_{2}$ and
$F_{3}$ form factors when only SM interactions are present, the additional
SMEFT operators generate divergences that can not be removed with the SM
counterterms, but induce counterterms $\delta C_{eB}$ and $\delta C_{eW}$. By
considering both the $\gamma\mu\mu$ and $Z\mu\mu$ vertices (alongside the
renormalisation of $Z-\gamma$ mixing) we calculate the UV-divergent parts
after dimensional regularisation (in dimensions $d=4-2\epsilon$) and subtract
them in the $\overline{\text{MS}}$ scheme such that both form factors are
renormalised at one-loop order. To this end, we introduce the counterterm
amplitude

$\hfill\parbox[h][130.08731pt][c]{108.405pt}{\includegraphics[scale={0.8}]{./counterterm.pdf}}\vspace{-1.8cm}=-ieQ_{\mu}\bar{u}(p^{\prime})\bigg{[}i\frac{\sigma^{\mu\nu}k_{\nu}}{2m_{f}}\delta
F_{2}+\frac{\sigma^{\mu\nu}k_{\nu}}{2m_{f}}\gamma_{5}(i\delta
F_{3})\bigg{]}u(p)\;,\hfill$ (6)

where $\delta F_{2}$ and $\delta F_{3}$ renormalise the relevant physical
Lorentz structures corresponding to $F_{2}(k^{2})$ and $F_{3}(k^{2})$,
respectively. The factor of $i$ multiplying $\delta F_{3}$ is introduced just
for convenience in expressing the equations later on. We do not consider
divergences relevant to the $F_{1}$ form factor as this is renormalised in the
Thomson limit to correspond to the correct electric charge Denner (1993). We
express $\delta F_{2}$ and $\delta F_{3}$ in terms of the usual SM coupling
and wavefunction renormalisation constants (see Ref. Denner (1993)) and also
introduce $\delta C_{eB}$ and $\delta C_{eW}$ as counterterms for the
${\cal{O}}_{eB}$ and ${\cal{O}}_{eW}$ interactions. This allows us to
calculate

$\begin{split}\delta
F_{2,3}&=\frac{m_{\mu}}{\sqrt{2}e\Lambda^{2}}\bigg{[}C_{eB}K_{1}\pm
C_{eB}^{*}K_{1}^{\prime}+C_{eW}K_{2}\pm
C_{eW}^{*}K_{2}^{\prime}+2c_{W}v(\delta C_{eB}\pm\delta
C_{eB}^{*})-2s_{W}v(\delta C_{eW}\pm\delta C_{eW}^{*})\bigg{]}\;,\\\
K_{1}&=2c_{W}\delta v+c_{W}v(\delta Z_{AA}+\delta Z_{L}^{\mu\mu,*}+\delta
Z_{R}^{\mu\mu})+v(2\delta c_{W}-s_{W}\delta Z_{ZA})\;,\\\ K_{2}&=-2s_{W}\delta
v-s_{W}v(\delta Z_{AA}+\delta Z_{L}^{\mu\mu,*}+\delta
Z_{R}^{\mu\mu})-v(2\delta s_{W}+c_{W}\delta Z_{ZA})\;,\end{split}$ (7)

where the prime indicates $\delta Z_{L}^{\mu\mu,*}\to\delta Z_{L}^{\mu\mu}$
and $\delta Z_{R}^{\mu\mu}\to\delta Z_{R}^{\mu\mu,*}$. In the SM, the Higgs
potential contains

$V(\Phi)\supset v(\mu^{2}+v^{2}\lambda_{H})h=th\;,$ (8)

which is minimised at tree-level via $t=0$. Tadpole diagrams that capture the
shift away from the classical Higgs field value due to higher orders and are
removed by introducing the counterterm (for a detailed discussion see Denner
and Dittmaier (2020))

$\delta t=-m_{h}^{2}\delta v\;,$ (9)

which enters Eq. (7). As there are two independent operators contributing to
$\Delta a_{\mu}$, we repeat the calculation for the $Z\mu\mu$ vertex which
leads to similar counterterm expressions. Subsequently, we determine the
values of $\delta C_{eB}$ and $\delta C_{eW}$ in the $\overline{\text{MS}}$
scheme by simultaneously requiring that the UV-divergences in the $F_{2}$ and
$F_{3}$ form factors for both the $\gamma\mu\mu$ and $Z\mu\mu$ vertices
identically cancel and only finite terms for $d\to 4$ remain.

The one-loop corrections to the magnetic moment in SMEFT also give rise to
soft singularities at finite photon virtuality. These are soft (and universal)
QED corrections to the dimension-six interactions and they cancel against soft
photon emission off the dimension-six vertex of Eq. (3). These soft
singularities vanish in the limit of zero virtuality222We have checked this
explicitly., reflecting the fact that the higher-dimensional operators are a
manifestation of scale-suppressed new physics separated from universal soft
(and collinear) effects in QED (see e.g. Englert et al. (2019) for a general
discussion in the context of QCD). We can therefore omit soft singularities
throughout this calculation.

WC (/$\Lambda^{2}$) | Fermilab/BNL allowed $[1/{\text{TeV}^{2}}]$ | SM allowed $[1/{\text{TeV}^{2}}]$
---|---|---
$C_{eB}$ | $\left[4.73\times 10^{-6},7.63\times 10^{-6}\right]$ | $\left[-1.06\times 10^{-6},1.06\times 10^{-6}\right]$
$C_{eW}$ | $\left[-8.39\times 10^{-6},-1.35\times 10^{-5}\right]$ | $\left[1.88\times 10^{-6},-1.88\times 10^{-6}\right]$
$C_{W}$ | $\left[14.00,22.60\right]$ | $\left[-3.14,3.14\right]$
$C_{\phi B}$ | $\left[0.36,0.58\right]$ | $\left[-0.08,0.08\right]$
$C_{\phi W}$ | $\left[1.06,1.71\right]$ | $\left[-0.24,0.24\right]$
$C_{\phi e}$ | $\left[7.92,12.80\right]$ | $\left[-1.77,1.77\right]$
$C_{\phi l}^{(1)}$ | $\left[-8.27,-13.40\right]$ | $\left[1.85,-1.85\right]$
$C_{\phi l}^{(3)}$ | $\left[-8.27,-13.40\right]$ | $\left[1.85,-1.85\right]$
$C_{le}$ | $\left[-1.69,-2.72\right]$ | $\left[0.38,-0.38\right]$
Table 1: Bounds on WCs in units of $\text{TeV}^{-2}$ for consistency with the
Fermilab measurement and the errors of the SM prediction. We limit ourselves
to WC operator values $\lesssim 4\pi/\text{TeV}^{2}$.

Our aim is to identify operators that give rise to significant corrections to
the SM that push the anomalous magnetic moment to larger values closer to the
Fermilab/BNL result. As such we limit ourselves to operator directions that
are not related to oblique electroweak precision constraints, i.e. we will
neglect the ${\cal{O}}_{\phi WB}$ and ${\cal{O}}_{\phi D}$ operators due to
their relations to the $S$ and $T$ parameters Alonso et al. (2014); Berthier
and Trott (2016); Dedes et al. (2017) (we note that the correlations of
$a_{\mu}$ and electroweak data have been studied elsewhere, e.g. Kanemitsu and
Tobe (2012); Cho et al. (2000)). We have also considered only CP-conserving
operators.333Contributions arise from $C_{\widetilde{W}}$, however these are
of the same order of magnitude as $C_{{W}}$ which can not explain the Fermilab
measurement without approaching the strongly coupled limit. The remaining
Wilson coefficients (WCs) are tabled in Tab. 1 for a renormalisation scale
choice of $\mu=m_{\mu}$ (the scale relevant to the $a_{\mu}$ measurement, we
will discuss the impact of different choices, e.g., $\mu=\Lambda$ further
below) and $\Lambda=1~{}\text{TeV}$.444We have checked our results against the
Renormalisation Group Evolution analysis of Ref. Aebischer et al. (2021) and
find good agreement. As can be seen, only a subset of these operator
constraints is perturbatively meaningful. This reflects the general
observation of the $a_{\mu}$ anomaly in theories with extended particle
spectra Athron et al. (2021); Anisha et al. (2022).

Figure 1: Bounds in the $C_{eW}$-$C_{eB}$ plane with the rest WCs fixed at
zero from the anomalous magnetic moment.

###  II.1 UV-divergent operators: $a_{\mu}$ as input parameter

Figure 2: Individual intervals allowed by the Fermilab measurement for
$C_{eB}$, $C_{eW}$ (left) as a function of the dimensional regularisation
scale $\mu$.  Figure 3: Individual interval allowed by the Fermilab
measurement for $C_{e\gamma}$ including one-loop contributions is shown with
solid lines and filled as a function of $\mu$. When only tree level
contributions are included the corresponding interval is shown with dashed
lines.

From our included operators, the divergent parts of Eq. (3) depend only on
$C_{eB}$, $C_{eW}$, $C_{\phi B}$ and $C_{\phi W}$. The latter two are
constrained mainly from Simplified Template Cross Section (STXS) measurements
and Higgs signal strength measurements at the LHC (see for example Refs. Ellis
et al. (2021b); Anisha et al. (2021)) without much potential to further
clarify the muon anomalous magnetic moment measurement. The tree-level
$C_{eB}$ and $C_{eW}$ insertions on the other hand are hard to constrain due
to chiral suppression. They are a subset of the few possible operators that
can shift significantly the SM expectation when new physics appears at around
the TeV scale. Contours including both operators are shown in Fig. 1.

As any scheme and scale dependent parameter in QFT, the results understood in
terms of Wilson coefficient constraints depend on the (unphysical) dimensional
renormalisation scale $\mu$. From the point of matching calculations a choice
of $\mu=\Lambda$ is intuitive, whilst scale-dependent logarithms are typically
efficiently resummed by choosing an adapted renormalisation scale relative to
the scale of measurement. This is familiar from many calculations in collider
physics (e.g. scale choices of Higgs decays to bottom pairs Dittmaier et al.
(2011)). It is therefore worthwhile to investigate the $\mu$ scale choice as
an indication of the reliability of our calculation with regards to neglected
higher-order effects. We show this in Fig. 2 for $\Lambda=1$ TeV, highlighting
that the effect on the WCs of dipole operators is essentially a shift of the
allowed interval due to the presence of the tree level contribution.
Considering the Fermilab measurement as input, the values of the WCs would
need to increase as $\mu\to\Lambda$. In contrast, the dependence on $C_{\phi
B}$, $C_{\phi W}$ through logarithms $\log(m^{2}/\mu^{2})$ (with $m$
representing a SM mass scale) leads to a different behaviour as a function of
$\mu$ as they contribute only radiatively, which would lead to sign changes
when Eq. (1) is used as input. However as previously stated we are not
considering this due to the strong constraints on these WCs from other
processes. It should be noted that all logarithms are suppressed by the UV
scale $\sim\Lambda^{-2}\log(m^{2}/\mu^{2})$, which as larger scales
$\mu\sim\Lambda$ are considered, decreases slower than the tree level
$\Lambda^{-2}$ contributions.

Figure 4: Impact of $C_{eZ}$ on $C_{e\gamma}$ for $\mu=m_{\mu}$. Relatively
small values of $C_{eZ}$ can cause cancellations rendering the one-loop
contribution of $C_{e\gamma}$ negligible and reproducing the Fermilab $\Delta
a_{\mu}$. For larger sampled values of $C_{eZ}$ the $C_{e\gamma}$ K-factor
increases to keep consistency with the Fermilab measurement.

We evaluate the importance of the $1$-loop contribution to $\Delta a_{\mu}$,
when compared to the tree-level, with the scale choice of the muon mass. To do
so we consider the orthogonal rotation of the dipole operators,

$\begin{split}C_{e\gamma}&=c_{W}C_{eB}-s_{W}C_{eW}\;,\\\
C_{eZ}&=s_{W}C_{eB}+c_{W}C_{eW}\;,\\\ \end{split}$ (10)

such that at tree level only $C_{e\gamma}$ contributes to the muon’s dipole
moments. Taking into account combinations of $C_{e\gamma}$ and an additional
operator, it is possible to evaluate how sizeable the correction factor from
loop corrections (the K-factor) can be when $\Delta a_{\mu}$ lies within the
allowed interval of Fermilab. When no additional operators are present, the
$C_{e\gamma}$ K-factor lies at $\sim 0.52$ for $\mu=m_{\mu}$ and increases
logarithmically as $\mu\to\Lambda=1$ TeV up to $\sim 1.7$. The dependence of
the $C_{e\gamma}$ interval consistent with the Fermilab result is shown in
Fig. 3. This shows that also in EFT (mirroring the findings in concrete UV
models Athron et al. (2021)), the anomalous magnetic moment measurement is
linked to a sizeable coupling. While this is not directly visible from the
tree-level result, the size of the radiative correction (which is much larger
than typical electroweak corrections) indicates a relative strong coupling of
the EFT when the dipole operators alone are considered. The presence of
additional SMEFT operators, however, can reproduce the muon $g-2$ anomaly
without the need of such large radiative corrections. For example, sampling
values of $\Delta a_{\mu}$ and $C_{eZ}$ we can see that even if $C_{eZ}$ is
assumed to of order $10^{-4}$, the anomalous magnetic moment can be reproduced
with small radiative corrections as shown in Fig. 4.

We can repeat this procedure for the rest of the SMEFT operators, calculating
the minimum possible $C_{e\gamma}$ K-factor that is required to obtain
consistency between the sampled $\Delta a_{\mu}$ and the Fermilab measurement,
shown in Fig. 5. Phrased differently, we calculate how large a second SMEFT
contribution needs to be in order to cancel the $1$-loop contribution of
$C_{e\gamma}$ rendering the tree-level as the only relevant contribution. In
general, the radiative correction from the dipole operator remains relevant
for most operators unless they approach the non-perturbative limit. Exceptions
are the $C_{\phi B}$, $C_{\phi W}$ and $C_{le}$ operators which, without
acquiring sizeable values, can introduce cancellations in the NLO part
rendering the loop order negligible.

Figure 5: Plots showing the minimum $C_{e\gamma}$ K-factor possible that can
reproduce the anomalous magnetic moment depending on the range where an
additional operator is sampled.

###  II.2 $a_{\mu}$ as a radiative SMEFT effect

The muon $g-2$ measurement can also be interpreted as an effect of BSM physics
from operators that contribute at one loop level without leading to UV
divergences for the dimension-six truncation. When considering operators
individually, $C_{\phi\square}$, $C_{e\phi}$ and $C_{W}$ cannot produce enough
pull to lift the $g-2$ up to the Fermilab measurement unless their absolute
value divided by $\Lambda^{2}$ exceeds the limit of $4\pi/\text{TeV}^{2}$. For
the electroweak scale of $246$ GeV this would correspond to a WC value of
order one. Electroweak Precision Observables (EWPO) significantly restrict the
allowed values of $C_{\phi l}^{(1)}$, $C_{\phi l}^{(3)}$ and $C_{\phi e}$
Dawson and Giardino (2020). The only remaining currently unconstrained
operator that can provide an explanation of $g-2$ is four-fermion $C_{le}$,
which was also discussed in Refs. Crivellin et al. (2014); Aebischer et al.
(2021). Lepton colliders will be able to place constraints on this operator
Bouzas and Larios (2022); de Blas et al. (2022).

##  III Avenues for tensioning the dipole operators

The dipole operators $C_{eB}$ and $C_{eW}$ are difficult to constrain in
collider environments. In this section, we explore a range of motivated
avenues which could, a priori, provide bounds either with current experimental
uncertainties or in the future. We consider the $Z\to\mu\mu$ decay, expected
to be sensitive to $C_{eZ}$ as well as the muon-Higgs interactions. While the
$\mu\mu\gamma$ and $\mu\mu Z$ vertices are sensitive to the dipole operators
when the Higgs doublet is set to its vacuum expectation value, Higgs physics
such as the 125 GeV Higgs boson’s decay $h\to\mu\mu$ is also sensitive to
interactions that are linked to $a_{\mu}$. Additionally we include resolved
photon emission $h\to\mu\mu\gamma$ which is directly sensitive to the
$a_{\mu}$ operators at the cost of low statistical yield (yet with low
experimental systematics as muons and photons are under very good control at
the LHC). Looking towards the future, we consider the precision environment
provided by the MUonE experiment Abbiendi et al. (2017) proposed at CERN,
aiming to measure the hadronic contributions to the anomalous magnetic moment
in $\mu e$ scattering, whose sensitivity can also be interpreted as bounds on
the SMEFT operators considered in this work.

###  III.1 $Z$ boson decay

The orthogonal rotation of the gauge fields $B_{\mu}$ and $W_{\mu}^{3}$ in the
SM to the physical $Z$ and $\gamma$ also causes the appearance of the $C_{eB}$
and $C_{eW}$ operators in decays of the $Z$ boson to muons, along with
additional BSM interactions. At dimension-six order only the linear
contribution is relevant arising from interference with tree and virtual SM
diagrams. For the discussion of the $Z$ decay we neglect loop contributions
from SMEFT as they are unlikely to significantly impact the (relatively poor)
constraints.

Denoting as ${\cal{M}}_{\text{d6}}$ the tree-level SMEFT amplitude and
${\cal{M}}_{\text{\text{SM}}}^{\text{tree}}$
(${\cal{M}}_{\text{SM}}^{\text{virt}}$) the tree (virtual) SM amplitude, the
departure from the SM decay width is given by

$\delta\Gamma_{Z\to\mu\mu}=\frac{|{\bf{p}}_{1}|}{48\pi
m_{Z}^{2}}\bigg{[}2{\cal{M}}_{\text{d6}}({\cal{M}}_{\text{SM}}^{\text{tree}}+{\cal{M}}_{\text{SM}}^{\text{virt}})\bigg{]}\;,$
(11)

where we average over the initial polarizations of the $Z$ boson and integrate
over the phase space (${\bf{p}}_{1}$ denotes the three-momentum of one of the
final states). The virtual SM amplitude terms are calculated at the
renormalisation scale $\mu=m_{Z}$.

The Particle Data Group (PDG) Zyla et al. (2020) reports an uncertainty of
$0.18$ MeV on the decay width of $Z\to\mu\mu$ which we use to construct a
$\chi^{2}$ and evaluate the allowed bounds at $68\%$ confidence level of each
WC individually for comparability. Results are shown on Tab. 2 where the
dipole operators are essentially unconstrained with bounds exceeding
$4\pi/\text{TeV}^{2}$. The muonic decay of the $Z$ boson can only efficiently
constrain the three operators of the class $\psi^{2}\phi^{2}D$ that contribute
to the anomalous magnetic moment. As with the EWPO that also constrain most of
these operators, we see again that $C_{\phi e}$, $C_{\phi l}^{(1)}$ and
$C_{\phi l}^{(2)}$ contributions can not lift the muon $g-2$ up to the
Fermilab/BNL findings without creating tension with other measurements.

WC ($/\Lambda^{2}$) |  $68\%$ C.L bound
---|---
$C_{\phi e}$ | $\left[-0.019,0.019\right]$
$C_{\phi l}^{(1)}$ | $\left[-0.016,0.016\right]$
$C_{\phi l}^{(3)}$ | $\left[-0.016,0.016\right]$
Table 2: Bounds on WCs at $68$% in units of $\text{TeV}^{-2}$ from the decay
width of the $Z$ boson obtained by constructing a $\chi^{2}$ with the PDG
error (see text). Contributions from dipole operators do not result in
perturbatively meaningful constraints and are not shown.

###  III.2 The $h\to\mu\mu\gamma$ channel

The decay of the Higgs boson to fermions was calculated in Ref. Cullen and
Pecjak (2020) (see also Cullen et al. (2019) for more details on the
calculation) which we utilise in order to identify the sensitivity to the
dipole operators from the signal strength of $h\mu\mu$. The signal strength is
can be expressed numerically as555We obtain the signal strength using the
decay rates given provided in Ref. Cullen and Pecjak (2020).

$\mu=1+\frac{\text{TeV}^{2}}{\Lambda^{2}}(0.67C_{eW}-0.19C_{eB})\;,$ (12)

at a scale $\mu=125$ GeV. Bounds on the WCs are then obtained at $68$% C.L. by
constructing a $\chi^{2}=(\mu-1.19)^{2}/0.35^{2}$ using the PDG expectation
value Zyla et al. (2020) which would correspond to
$C_{eB}\in\left[-2.8,0.8\right]$ and $C_{eW}\in\left[0.24,0.81\right]$ using a
UV scale $\Lambda=1$ TeV. As the sensitivity is still orders of magnitude less
than the anomalous magnetic moment, this motivates looking into alternative
avenues.

Figure 6: Vertex arising from the operators of Eq. (4) that contributes to the
decay of the Higgs.

The dependence of the ${\cal{O}}_{eW}$ and ${\cal{O}}_{eB}$ operators on the
Higgs doublet suggests that $s$-channel processes with propagating Higgs
bosons might provide additional tension on the bounds on $C_{eB}$ and $C_{eW}$
from the anomalous magnetic measurement. The operators generate the diagram
shown in Fig. 6 which provides a final state at the LHC characterised by two
muons and a hard photon, arising from a Higgs resonance (for a detailed
discussion of $\mu\mu\to h\gamma$ process at a future muon collider see Ref.
Paradisi et al. (2022)). As the resonant Higgs is predominantly produced by
gluon fusion at hadron colliders we consider the $p(g)p(g)\to
h\to\mu^{+}\mu^{-}\gamma$ channel. Background contamination arising from the
SM will not be characterised by the same resonance structure which motivates a
bump-hunt analysis around the Higgs peak. While the statistical yield is
small, such data-driven analyses feature largely reduced systematic
uncertainties similar to the Higgs boson’s $h\to\gamma\gamma$ discovery mode.

We use the SMEFTsim Brivio et al. (2017); Brivio (2021) implementation of
SMEFT666We use the effective $ggh$ interaction implemented in SMEFTsim to
model the Higgs production. in the UFO Degrande et al. (2012) format to
generate events with MadEvent Alwall et al. (2011); de Aquino et al. (2012);
Alwall et al. (2014). In the final state we require at least two isolated
leptons identified as muons with transverse momentum $p_{T}(\mu)>15$ GeV in
the central part of the detector (pseudorapidity must satisfy
$|\eta(\mu)|<2.5$). For an isolated muon, the sum of jet transverse momenta in
the region $\Delta R=\sqrt{\Delta\phi^{2}+\Delta\eta^{2}}<0.4$ must be less
than $50\%$ of the lepton’s $p_{T}$,where $\phi$ denotes the azimuthal angle.
Additionally at least one photon must be present with $p_{T}(\gamma)>15$ GeV
and $\eta(\gamma)<2.5$ and along with the leading isolated muons, the Higgs
mass $M_{\text{reco}}$ is reconstructed as the invariant mass of the four
momenta total of the final states. An additional cut $110~{}\text{GeV}\leq
M_{\text{reco}}\leq 135~{}\text{GeV}$ is imposed to select the region close to
the resonant mass of the SM Higgs of $125$ GeV.

Differential cross sections in SMEFT can be expressed as

${\text{d}}\sigma={\text{d}}\sigma_{\text{\text{SM}}}+\frac{C_{i}}{\Lambda^{2}}{\text{d}}\sigma_{i}\;,$
(13)

where the first part arises from pure-SM interactions and the second term
captures the interference of SM and dimension-six operators. We neglect any
term suppressed by $\Lambda^{-4}$. Events are generated independently of the
WC in this linearised setup and an example histogram for the differential
distribution of the reconstructed Higgs mass is shown in Fig. 7. A binned
$\chi^{2}$ can be then constructed as the difference of the events with and
without the EFT interactions squared, weighted by the SM statistical
uncertainty. Performing a fit over the WCs of interest, results in the bounds
shown in Fig. 8 for integrated luminosities of $139$/fb and $3$/ab.

The sensitivity from this channel is once again severely limited compared to
what is indicated from the anomalous magnetic moment.

Figure 7: Example histogram for the reconstructed mass of the Higgs
$M_{\text{reco}}$ showing the shape of the SM contribution and interference of
effective interactions with the SM. Figure 8: Exclusion contours in the
$C_{eW}$-$C_{eB}$ plane from the $h\to\mu\mu\gamma$ channel for integrated
luminosities of $139$/fb and $3$/ab.

###  III.3 MUonE

While we have considered the anomalous magnetic moment of the muon from the
BSM perspective, there is still the possibility that higher order hadronic
contributions from the SM might resolve the $g-2$ anomaly. While these are
calculated with lattice QCD techniques Borsanyi et al. (2018); Blum et al.
(2018); Davies et al. (2020); Gérardin et al. (2019b); Aubin et al. (2020);
Blum et al. (2020); Borsanyi et al. (2021); Lehner and Meyer (2020); Cè et al.
(2022), an alternative approach was proposed through the MUonE experiment
Abbiendi et al. (2017), aiming to determine the hadronic contributions with
high accuracy. Using measurements of the differential cross section
$\text{d}\sigma/\text{d}t$ in elastic $\mu e\to\mu e$ scattering, where $t$ is
the (space-like) squared momentum transfer, the hadronic contributions to the
fine structure constant can be accurately determined. This can then be related
to the respective contributions on the anomalous magnetic moment using a data
driven approach. Scattering off atomic electrons with a muon beam of $150$ GeV
from CERN, the experiment is expected to achieve an integrated luminosity of
${\cal{L}}=1.5\times 10^{7}$/nb for an expected cross section of $\sim
250~{}\mu{\text{b}}$.

While deviations from this cross section could be interpreted as contributions
from QCD, the MUonE experiment would also be able to identify new physics
contributions which could affect the measurement (see for example Ref. Dev et
al. (2020)). The impact of heavy particles can be suppressed by their mass due
to the relatively low centre-of-mass energy, unless they are strongly coupled
to the SM. We consider tree level contributions affecting the MUonE
measurement from SMEFT in order to determine how the experiment can constrain
the allowed ranges of relevant WCs, neglecting flavour-violating contributions
from off-diagonal dimension-six operators.

As before, the $\text{d}\sigma/\text{d}t$ distribution can be written linearly
in terms of higher dimension operators as in Eq. (13), truncated at order
$\Lambda^{-2}$. We calculate the unpolarized $\text{d}\sigma/\text{d}t$
differential distribution for $\mu^{-}e^{-}\to\mu^{-}e^{-}$ for the SM and for
the interference of dimension-six operators with the SM (see Fig. 9). Bounds
on the WCs are obtained with a binned

$\chi^{2}=\sum_{i}{(\delta N_{i})^{2}\over\sigma_{i}^{2}}\;,$ (14)

where $\delta N_{i}$ denotes the deviation in event counts from the SM
$N^{\text{\text{SM}}}_{i}$ and
$\sigma_{i}=\sqrt{\sigma_{\text{stat},i}^{2}+\sigma_{\text{syst},i}^{2}}$ is
the combination of statistical and systematic uncertainties. We use
$\sigma_{\text{stat},i}=\sqrt{N^{\text{\text{SM}}}_{i}}$ and
$\sigma_{\text{syst},i}=10^{-5}N^{\text{\text{SM}}}_{i}$ which is the target
systematic uncertainty by MUonE Abbiendi et al. (2017) (see also Ref. Dev et
al. (2020)).

We show the bounds on the relevant WCs in Tab. 3. The operators $C_{\phi e}$,
$C_{\phi l}^{(1)}$ and $C_{\phi l}^{(3)}$ enter our calculations but their
bounds exceeed $4\pi/\text{TeV}^{2}$; hence we have not included them in Tab.
3. However, aside from the measurement by Fermilab/BNL, the MUonE experiment
can place the most stringent bounds from the approaches we consider in this
paper but as in all the other scenarios these limits are orders of magnitude
less restrictive than the muon $g-2$ result.

Figure 9: Histograms showing the shapes of the $\text{d}\sigma/\text{d}t$ distributions for the contributing WCs and the tree-level SM expectation. For each line the WC is set to unity and the rest to zero. WC (/$\Lambda^{2}$) |  $68\%$ C.L. $[1/\text{TeV}^{2}]$
---|---
$C_{eB}$ | $\left[-0.21,0.21\right]$
$C_{eW}$ | $\left[-0.39,0.39\right]$
Table 3: Bounds on WCs at 68% C.L. obtained with a $\chi^{2}$ analysis on the
$d\sigma/dt$ distribution from MUonE. These correspond to $\lvert
C_{e\gamma}\rvert\leq 0.18$ and $\lvert C_{eZ}\rvert<12.75$.While operators of
the $\psi^{2}\phi^{2}D$ class also contribute to the process, MUonE does not
show significant sensitivity.

##  IV Conclusions

Having calculated the anomalous magnetic moment at one-loop order in SMEFT we
observe that the higher order contributions from the dipole operators $C_{eB}$
and $C_{eW}$ are expected to be sizeable. This supports the general finding
for concrete UV extensions that new states need to be relatively strongly
coupled to the muon when they are heavy (which is an underlying assumption of
the EFT approach). Cancellations can arise from the virtual presence of
additional operators, but they are required to be sizeable in most cases with
the exceptions of $C_{\phi B}$, $C_{\phi W}$ and $C_{le}$. When the operators
are considered on a one-by-one basis, a possible explanation for the measured
value of the muon $g-2$ from Fermilab can only arise from the dipole operators
and the four-lepton interaction quantified by $C_{le}$. This agrees with
previous conclusions of Refs. Crivellin et al. (2014); Aebischer et al. (2021)
(our final $\Delta a_{\mu}$ expression is also in agreement). However, this
leaves little room for a description of $g-2$ arising from new-physics at the
high energy scales without introducing flavour-violating contributions (this
path is explored in Refs. Crivellin et al. (2014); Aebischer et al. (2021)).

Under the conservative assumption that new physics appears specifically in
relation to the muon, we explore the impact of the dipole operators in
different experimental scenarios to evaluate whether it is possible to reach a
similar level of precision as the Fermilab/BNL measurement. While the dipole
operators contribute to the $Z$ decay to muons, the sensitivity of the channel
is poor without any prospect for the dipole operators (though it can provide
constraints on $C_{\phi e}$, $C_{\phi l}^{(1)}$ and $C_{\phi l}^{(3)}$). The
presence of the Higgs doublet in the dipole operators allows us to
additionally assess whether additional tension can be obtained from
interactions of muons with the physical Higgs scalar. The $h\to\mu\mu\gamma$
channel receives SMEFT contributions from $C_{eB}$ and $C_{eW}$ at tree-level
and would therefore be an ideal channel to set bounds directly on
$a_{\mu}$-related interactions. Statistical limitations do not render this
mode competitive at the HL-LHC luminosity of $3$/ab. A precise determination
of the $h\mu\mu$ signal strength can provide improved bounds due to the
appearance of the dipole operators at one-loop level, yet not sensitive enough
to address the Fermilab/BNL tension with the SM.

Our last consideration regarding the dipole operators is the future MUonE
experiment attempting to measure the hadronic contributions in the $g-2$
anomaly arising from the SM. When re-interpreted in terms of SMEFT
interactions, this mode provides a competitive bound on the dipole operator
compared to the other modes considered. However, these are far from the
extreme precision provided by the targeted measurement of the anomalous
magnetic moment at Fermilab/BNL.

Acknowledgements — We thank Anke Biekötter, Christine T.H. Davies and Thomas
Teubner for helpful discussions. A.B. and C.E. are supported by the STFC under
grant ST/T000945/1. C.E. is supported by the Leverhulme Trust under grant
RPG-2021-031 and the IPPP Associateship Scheme. P.S. is funded by an STFC
studentship under grant ST/T506102/1.

## References

  * Abi et al. (2021) B. Abi et al. (Muon g-2), Phys. Rev. Lett. 126, 141801 (2021), eprint 2104.03281.
  * Bennett et al. (2004) G. W. Bennett et al. (Muon g-2), Phys. Rev. Lett. 92, 161802 (2004), eprint hep-ex/0401008.
  * Aoyama et al. (2012) T. Aoyama, M. Hayakawa, T. Kinoshita, and M. Nio, Phys. Rev. Lett. 109, 111808 (2012), eprint 1205.5370.
  * Aoyama et al. (2019) T. Aoyama, T. Kinoshita, and M. Nio, Atoms 7, 28 (2019).
  * Czarnecki et al. (2003) A. Czarnecki, W. J. Marciano, and A. Vainshtein, Phys. Rev. D 67, 073006 (2003), [Erratum: Phys.Rev.D 73, 119901 (2006)], eprint hep-ph/0212229.
  * Gnendiger et al. (2013) C. Gnendiger, D. Stöckinger, and H. Stöckinger-Kim, Phys. Rev. D 88, 053005 (2013), eprint 1306.5546.
  * Davier et al. (2017) M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, Eur. Phys. J. C 77, 827 (2017), eprint 1706.09436.
  * Keshavarzi et al. (2018) A. Keshavarzi, D. Nomura, and T. Teubner, Phys. Rev. D 97, 114025 (2018), eprint 1802.02995.
  * Colangelo et al. (2019) G. Colangelo, M. Hoferichter, and P. Stoffer, JHEP 02, 006 (2019), eprint 1810.00007.
  * Hoferichter et al. (2019) M. Hoferichter, B.-L. Hoid, and B. Kubis, JHEP 08, 137 (2019), eprint 1907.01556.
  * Davier et al. (2020) M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, Eur. Phys. J. C 80, 241 (2020), [Erratum: Eur.Phys.J.C 80, 410 (2020)], eprint 1908.00921.
  * Keshavarzi et al. (2020) A. Keshavarzi, D. Nomura, and T. Teubner, Phys. Rev. D 101, 014029 (2020), eprint 1911.00367.
  * Kurz et al. (2014) A. Kurz, T. Liu, P. Marquard, and M. Steinhauser, Phys. Lett. B 734, 144 (2014), eprint 1403.6400.
  * Melnikov and Vainshtein (2004) K. Melnikov and A. Vainshtein, Phys. Rev. D 70, 113006 (2004), eprint hep-ph/0312226.
  * Masjuan and Sanchez-Puertas (2017) P. Masjuan and P. Sanchez-Puertas, Phys. Rev. D 95, 054026 (2017), eprint 1701.05829.
  * Colangelo et al. (2017) G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer, JHEP 04, 161 (2017), eprint 1702.07347.
  * Hoferichter et al. (2018) M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold, and S. P. Schneider, JHEP 10, 141 (2018), eprint 1808.04823.
  * Gérardin et al. (2019a) A. Gérardin, H. B. Meyer, and A. Nyffeler, Phys. Rev. D 100, 034520 (2019a), eprint 1903.09471.
  * Bijnens et al. (2019) J. Bijnens, N. Hermansson-Truedsson, and A. Rodríguez-Sánchez, Phys. Lett. B 798, 134994 (2019), eprint 1908.03331.
  * Colangelo et al. (2020) G. Colangelo, F. Hagelstein, M. Hoferichter, L. Laub, and P. Stoffer, JHEP 03, 101 (2020), eprint 1910.13432.
  * Blum et al. (2020) T. Blum, N. Christ, M. Hayakawa, T. Izubuchi, L. Jin, C. Jung, and C. Lehner, Phys. Rev. Lett. 124, 132002 (2020), eprint 1911.08123.
  * Colangelo et al. (2014) G. Colangelo, M. Hoferichter, A. Nyffeler, M. Passera, and P. Stoffer, Phys. Lett. B 735, 90 (2014), eprint 1403.7512.
  * Aoyama et al. (2020) T. Aoyama et al., Phys. Rept. 887, 1 (2020), eprint 2006.04822.
  * Athron et al. (2021) P. Athron, C. Balázs, D. H. J. Jacob, W. Kotlarski, D. Stöckinger, and H. Stöckinger-Kim, JHEP 09, 080 (2021), eprint 2104.03691.
  * Weinberg (1979) S. Weinberg, Physica A 96, 327 (1979).
  * Buchalla et al. (2014) G. Buchalla, O. Catà, and C. Krause, Nucl. Phys. B 880, 552 (2014), [Erratum: Nucl.Phys.B 913, 475–478 (2016)], eprint 1307.5017.
  * Buchalla et al. (2017) G. Buchalla, O. Cata, A. Celis, and C. Krause, Nucl. Phys. B 917, 209 (2017), eprint 1608.03564.
  * Catà and Jung (2015) O. Catà and M. Jung, Phys. Rev. D 92, 055018 (2015), eprint 1505.05804.
  * Carmona et al. (2021) A. Carmona, A. Lazopoulos, P. Olgoso, and J. Santiago (2021), eprint 2112.10787.
  * Dittmaier et al. (2021) S. Dittmaier, S. Schuhmacher, and M. Stahlhofen, Eur. Phys. J. C 81, 826 (2021), eprint 2102.12020.
  * Contino et al. (2016) R. Contino, A. Falkowski, F. Goertz, C. Grojean, and F. Riva, JHEP 07, 144 (2016), eprint 1604.06444.
  * Englert and Spannowsky (2015) C. Englert and M. Spannowsky, Phys. Lett. B 740, 8 (2015), eprint 1408.5147.
  * Englert et al. (2020) C. Englert, P. Galler, and C. D. White, Phys. Rev. D 101, 035035 (2020), eprint 1908.05588.
  * Crivellin et al. (2014) A. Crivellin, S. Najjari, and J. Rosiek, JHEP 04, 167 (2014), eprint 1312.0634.
  * Allwicher et al. (2021) L. Allwicher, L. Di Luzio, M. Fedele, F. Mescia, and M. Nardecchia, Phys. Rev. D 104, 055035 (2021), eprint 2105.13981.
  * Aebischer et al. (2021) J. Aebischer, W. Dekens, E. E. Jenkins, A. V. Manohar, D. Sengupta, and P. Stoffer, JHEP 07, 107 (2021), eprint 2102.08954.
  * Cirigliano et al. (2021) V. Cirigliano, W. Dekens, J. de Vries, K. Fuyuto, E. Mereghetti, and R. Ruiz, JHEP 08, 103 (2021), eprint 2105.11462.
  * Baer et al. (2021) H. Baer, V. Barger, and H. Serce, Phys. Lett. B 820, 136480 (2021), eprint 2104.07597.
  * Frank et al. (2021) M. Frank, Y. Hiçyılmaz, S. Mondal, O. Özdal, and C. S. Ün, JHEP 10, 063 (2021), eprint 2107.04116.
  * Ellis et al. (2021a) J. Ellis, J. L. Evans, N. Nagata, D. V. Nanopoulos, and K. A. Olive, Eur. Phys. J. C 81, 1079 (2021a), eprint 2107.03025.
  * Zhang (2021) D. Zhang, JHEP 07, 069 (2021), eprint 2105.08670.
  * Jueid et al. (2021) A. Jueid, J. Kim, S. Lee, and J. Song, Phys. Rev. D 104, 095008 (2021), eprint 2104.10175.
  * Altmannshofer et al. (2021) W. Altmannshofer, S. A. Gadam, S. Gori, and N. Hamer, JHEP 07, 118 (2021), eprint 2104.08293.
  * Chakraborti et al. (2021a) M. Chakraborti, S. Heinemeyer, and I. Saha, in _International Workshop on Future Linear Colliders_ (2021a), eprint 2105.06408.
  * Chakraborti et al. (2021b) M. Chakraborti, S. Heinemeyer, and I. Saha, Eur. Phys. J. C 81, 1114 (2021b), eprint 2104.03287.
  * Grzadkowski et al. (2010) B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek, JHEP 10, 085 (2010), eprint 1008.4884.
  * Abbiendi et al. (2017) G. Abbiendi et al., Eur. Phys. J. C 77, 139 (2017), eprint 1609.08987.
  * Coluccio Leskow et al. (2017) E. Coluccio Leskow, G. D’Ambrosio, A. Crivellin, and D. Müller, Phys. Rev. D 95, 055018 (2017), eprint 1612.06858.
  * Crivellin et al. (2021) A. Crivellin, D. Mueller, and F. Saturnino, Phys. Rev. Lett. 127, 021801 (2021), eprint 2008.02643.
  * Fajfer et al. (2021) S. Fajfer, J. F. Kamenik, and M. Tammaro, JHEP 06, 099 (2021), eprint 2103.10859.
  * Crivellin and Hoferichter (2021) A. Crivellin and M. Hoferichter, JHEP 07, 135 (2021), eprint 2104.03202.
  * Paradisi et al. (2022) P. Paradisi, O. Sumensari, and A. Valenti (2022), eprint 2203.06103.
  * Dedes et al. (2017) A. Dedes, W. Materkowska, M. Paraskevas, J. Rosiek, and K. Suxho, JHEP 06, 143 (2017), eprint 1704.03888.
  * Dedes et al. (2020) A. Dedes, M. Paraskevas, J. Rosiek, K. Suxho, and L. Trifyllis, Comput. Phys. Commun. 247, 106931 (2020), eprint 1904.03204.
  * Czyz et al. (1988) H. Czyz, K. Kolodziej, M. Zralek, and P. Khristova, Can. J. Phys. 66, 132 (1988).
  * Christensen and Duhr (2009) N. D. Christensen and C. Duhr, Comput. Phys. Commun. 180, 1614 (2009), eprint 0806.4194.
  * Alloul et al. (2014) A. Alloul, N. D. Christensen, C. Degrande, C. Duhr, and B. Fuks, Comput. Phys. Commun. 185, 2250 (2014), eprint 1310.1921.
  * Hahn (2001) T. Hahn, Comput. Phys. Commun. 140, 418 (2001), eprint hep-ph/0012260.
  * Hahn and Perez-Victoria (1999) T. Hahn and M. Perez-Victoria, Comput. Phys. Commun. 118, 153 (1999), eprint hep-ph/9807565.
  * Passarino and Veltman (1979) G. Passarino and M. J. G. Veltman, Nucl. Phys. B 160, 151 (1979).
  * Denner and Dittmaier (2006) A. Denner and S. Dittmaier, Nucl. Phys. B Proc. Suppl. 157, 53 (2006), eprint hep-ph/0601085.
  * Patel (2015) H. H. Patel, Comput. Phys. Commun. 197, 276 (2015), eprint 1503.01469.
  * Schwinger (1948) J. S. Schwinger, Phys. Rev. 73, 416 (1948).
  * Buchmuller and Wyler (1986) W. Buchmuller and D. Wyler, Nucl. Phys. B 268, 621 (1986).
  * Alonso et al. (2014) R. Alonso, E. E. Jenkins, A. V. Manohar, and M. Trott, JHEP 04, 159 (2014), eprint 1312.2014.
  * Jenkins et al. (2014) E. E. Jenkins, A. V. Manohar, and M. Trott, JHEP 01, 035 (2014), eprint 1310.4838.
  * Jenkins et al. (2013) E. E. Jenkins, A. V. Manohar, and M. Trott, JHEP 10, 087 (2013), eprint 1308.2627.
  * Denner (1993) A. Denner, Fortsch. Phys. 41, 307 (1993), eprint 0709.1075.
  * Denner and Dittmaier (2020) A. Denner and S. Dittmaier, Phys. Rept. 864, 1 (2020), eprint 1912.06823.
  * Englert et al. (2019) C. Englert, M. Russell, and C. D. White, Phys. Rev. D 99, 035019 (2019), eprint 1809.09744.
  * Berthier and Trott (2016) L. Berthier and M. Trott, JHEP 02, 069 (2016), eprint 1508.05060.
  * Kanemitsu and Tobe (2012) S. Kanemitsu and K. Tobe, Phys. Rev. D 86, 095025 (2012), eprint 1207.1313.
  * Cho et al. (2000) G.-C. Cho, K. Hagiwara, and M. Hayakawa, Phys. Lett. B 478, 231 (2000), eprint hep-ph/0001229.
  * Anisha et al. (2022) Anisha, U. Banerjee, J. Chakrabortty, C. Englert, M. Spannowsky, and P. Stylianou, Phys. Rev. D 105, 016019 (2022), eprint 2108.07683.
  * Ellis et al. (2021b) J. Ellis, M. Madigan, K. Mimasu, V. Sanz, and T. You, JHEP 04, 279 (2021b), eprint 2012.02779.
  * Anisha et al. (2021) Anisha, S. Das Bakshi, S. Banerjee, A. Biekötter, J. Chakrabortty, S. Kumar Patra, and M. Spannowsky (2021), eprint 2111.05876.
  * Dittmaier et al. (2011) S. Dittmaier et al. (LHC Higgs Cross Section Working Group) (2011), eprint 1101.0593.
  * Dawson and Giardino (2020) S. Dawson and P. P. Giardino, Phys. Rev. D 101, 013001 (2020), eprint 1909.02000.
  * Bouzas and Larios (2022) A. O. Bouzas and F. Larios, Adv. High Energy Phys. 2022, 3603613 (2022), eprint 2109.02769.
  * de Blas et al. (2022) J. de Blas, Y. Du, C. Grojean, J. Gu, V. Miralles, M. E. Peskin, J. Tian, M. Vos, and E. Vryonidou, in _2022 Snowmass Summer Study_ (2022), eprint 2206.08326.
  * Zyla et al. (2020) P. A. Zyla et al. (Particle Data Group), PTEP 2020, 083C01 (2020).
  * Cullen and Pecjak (2020) J. M. Cullen and B. D. Pecjak, JHEP 11, 079 (2020), eprint 2007.15238.
  * Cullen et al. (2019) J. M. Cullen, B. D. Pecjak, and D. J. Scott, JHEP 08, 173 (2019), eprint 1904.06358.
  * Brivio et al. (2017) I. Brivio, Y. Jiang, and M. Trott, JHEP 12, 070 (2017), eprint 1709.06492.
  * Brivio (2021) I. Brivio, JHEP 04, 073 (2021), eprint 2012.11343.
  * Degrande et al. (2012) C. Degrande, C. Duhr, B. Fuks, D. Grellscheid, O. Mattelaer, and T. Reiter, Comput. Phys. Commun. 183, 1201 (2012), eprint 1108.2040.
  * Alwall et al. (2011) J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer, and T. Stelzer, JHEP 06, 128 (2011), eprint 1106.0522.
  * de Aquino et al. (2012) P. de Aquino, W. Link, F. Maltoni, O. Mattelaer, and T. Stelzer, Comput. Phys. Commun. 183, 2254 (2012), eprint 1108.2041.
  * Alwall et al. (2014) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli, and M. Zaro, JHEP 07, 079 (2014), eprint 1405.0301.
  * Borsanyi et al. (2018) S. Borsanyi et al. (Budapest-Marseille-Wuppertal), Phys. Rev. Lett. 121, 022002 (2018), eprint 1711.04980.
  * Blum et al. (2018) T. Blum, P. A. Boyle, V. Gülpers, T. Izubuchi, L. Jin, C. Jung, A. Jüttner, C. Lehner, A. Portelli, and J. T. Tsang (RBC, UKQCD), Phys. Rev. Lett. 121, 022003 (2018), eprint 1801.07224.
  * Davies et al. (2020) C. T. H. Davies et al. (Fermilab Lattice, LATTICE-HPQCD, MILC), Phys. Rev. D 101, 034512 (2020), eprint 1902.04223.
  * Gérardin et al. (2019b) A. Gérardin, M. Cè, G. von Hippel, B. Hörz, H. B. Meyer, D. Mohler, K. Ottnad, J. Wilhelm, and H. Wittig, Phys. Rev. D 100, 014510 (2019b), eprint 1904.03120.
  * Aubin et al. (2020) C. Aubin, T. Blum, C. Tu, M. Golterman, C. Jung, and S. Peris, Phys. Rev. D 101, 014503 (2020), eprint 1905.09307.
  * Borsanyi et al. (2021) S. Borsanyi et al., Nature 593, 51 (2021), eprint 2002.12347.
  * Lehner and Meyer (2020) C. Lehner and A. S. Meyer, Phys. Rev. D 101, 074515 (2020), eprint 2003.04177.
  * Cè et al. (2022) M. Cè et al. (2022), eprint 2206.06582.
  * Dev et al. (2020) P. S. B. Dev, W. Rodejohann, X.-J. Xu, and Y. Zhang, JHEP 05, 053 (2020), eprint 2002.04822.


$\mathsf{A}_{123}\,\mathsf{P}_{\\!\\!\scriptscriptstyle\mathsf{A}\mathsf{B}}=\mathsf{P}_{\\!\\!\scriptscriptstyle\mathsf{A}\mathsf{B}}\,\mathsf{B}_{123}$.
The entries in the l.h.s. and r.h.s. are identical so the equation is
satisfied.

For $I=12$, again look at the components of the equation
$\mathsf{A}_{123}\,\mathsf{P}_{\\!\\!\scriptscriptstyle\mathsf{A}\mathsf{B}}=\mathsf{P}_{\\!\\!\scriptscriptstyle\mathsf{A}\mathsf{B}}\,\mathsf{B}_{123}$.
Equating the $(n,k);(k,x)$ component of the l.h.s. and r.h.s. leads to the
difference equation (2.15) with $c_{1}\leftrightarrow c_{3}$ and $N\mapsto
N-k$ (for $0\leq n,x\leq N-k$) hence this equation holds.

Similarly, for $I=23$, rewrite the change of basis equation (C.19) as
$\mathsf{B}_{23}\,\mathsf{P}_{\\!\\!\scriptscriptstyle\mathsf{B}\mathsf{A}}=\mathsf{P}_{\\!\\!\scriptscriptstyle\mathsf{B}\mathsf{A}}\,\mathsf{A}_{23}$.
Equating the $(k,n);(x,k)$ component of the l.h.s. and r.h.s. leads to the
difference equation (2.15) with $N\mapsto N-k$ (for $0\leq n,x\leq N-k$) hence
this equation also holds.

For $I=34$, rewrite the change of basis equation (C.19) as
$\mathsf{A}_{34}\,\mathsf{P}_{\\!\\!\scriptscriptstyle\mathsf{A}\mathsf{B}}=\mathsf{P}_{\\!\\!\scriptscriptstyle\mathsf{A}\mathsf{B}}\,\mathsf{B}_{34}$.
Upon comparing the $(n,k);(k,x)$ components of the l.h.s. and the r.h.s. one
checks that they are indeed equal by virtue of the difference equation (2.15).
Simultaneously, comparing the $(n,k-\varepsilon);(k,x)$ components of the
l.h.s. and the r.h.s. for $\varepsilon=\pm$, one confirms that they are equal
by virtue of the contiguity difference equations (2.22) involving
$\lambda_{\varepsilon}$ respectively. Thus, the equation also holds in this
case.

Finally, an analysis along the same lines as what was done for $I=34$ can be
repeated for $I=234$. Rewriting the change of basis equation (C.19) as
$\mathsf{B}_{234}\,\mathsf{P}_{\\!\\!\scriptscriptstyle\mathsf{B}\mathsf{A}}=\mathsf{P}_{\\!\\!\scriptscriptstyle\mathsf{B}\mathsf{A}}\,\mathsf{A}_{234}$,
one finds that this equation also holds.

With this, we have proven the validity of the change of basis between the
$\mathsf{A}$ and $\mathsf{B}$ bases. The remaining changes of basis follow
immediately by the ${\mathbb{Z}}_{5}$ symmetry. ∎

Again let us remark that these changes of basis are the key ingredient in the
proof of the Biedenharn–Elliot relation for $q$-Racah polynomials (4.6).

## References

  * [1] R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985) 319.
  * [2] E. Bannai and T. Ito, Algebraic combinatorics I, Benjamin-Cummings, Menlo Park, 1984.
  * [3] E. Bannai, H. Kurihara, D. Zhao and Y. Zhu, Multivariate P- and/or Q-polynomial association schemes, arXiv:2305.00707.
  * [4] P. Baseilhac and X. Martin, A bispectral $q$-hypergeometric basis for a class of quantum integrable models, J. Math. Phys. 59 (2018) 011704 and arXiv:1506.06902.
  * [5] G. Bergeron, E. Koelink and L. Vinet, $SU_{q}(3)$ corepresentations and bivariate q-Krawtchouk polynomials, J. Math. Phys. 60 (2019) 051701 and arXiv:1811.02065.
  * [6] P.-A. Bernard, N. Crampe, L. Poulain d’Andecy, L. Vinet and M. Zaimi, Bivariate $P$-polynomial association schemes, Algebraic Combinatorics 7 (2024) 361-382 and arXiv:2212.10824.
  * [7] P.-A. Bernard, N. Crampe, L. Vinet, M. Zaimi and X. Zhang, $m$-distance-regular graphs and their relation to multivariate $P$-polynomial association schemes, arXiv:2309.16016.
  * [8] L.C. Biedenharn, An Identity Satisfied by Racah Coefficients, J. Math. Phys. 31 (1953) 287.
  * [9] H. Chen, On skein algebras of planar surfaces, arXiv:2206.07856.
  * [10] J. Cooke and A. Lacabanne, Higher Rank Askey–Wilson Algebras as Skein Algebras, arXiv:2205.04414.
  * [11] N. Crampe, L. Frappat, J. Gaboriaud, L. Poulain d’Andecy, E. Ragoucy and L. Vinet, The Askey–Wilson algebra and its avatars, J. Phys. A: Math. Theor. 54 (2021) 063001 and arXiv:2009.14815.
  * [12] N. Crampe, L. Frappat, J. Gaboriaud, E. Ragoucy, L. Vinet and M. Zaimi, $\lambda$-Griffiths polynomials: Bispectrality and biorthogonality, arXiv:2311.03256.
  * [13] N. Crampe, L. Frappat, J. Gaboriaud, E. Ragoucy, L. Vinet and M. Zaimi, Griffiths polynomials of Racah type, arXiv:2403.12148.
  * [14] N. Crampe, L. Frappat, L. Poulain d’Andecy and E. Ragoucy, The higher-rank Askey–Wilson algebra and its braid group automorphisms, SIGMA 19 (2023) 077 and arXiv:2303.17677.
  * [15] N. Crampe, L. Frappat and E. Ragoucy, Representations of the rank two Racah algebra and orthogonal multivariate polynomials, Linear Algebra and its Applications 664 (2023) 165–215 and arXiv:2206.01031.
  * [16] N. Crampe, L. Poulain d’Andecy, L. Vinet and M. Zaimi, Askey–Wilson braid algebra and centralizer of $U_{q}(sl_{2})$, Annales Henri Poincaré 24 (2023) 1897–1922 and arXiv:2206.11150.
  * [17] N. Crampe, W. van de Vijver and L. Vinet, Racah problems for the oscillator algebra, the Lie algebra $sl(n)$, and multivariate Krawtchouk polynomials, Annales Henri Poincaré 21 (2020) 3939–3971 and arXiv:1909.12643.
  * [18] N. Crampe, L. Vinet and M. Zaimi, Temperley–Lieb, Birman–Murakami–Wenzl and Askey–Wilson algebras and other centralizers of $U_{q}(\mathfrak{sl}_{2})$, Annales Henri Poincaré 22 (2021) 3499 and arXiv:2008.04905.
  * [19] N. Crampe and M. Zaimi, Factorized $A_{2}$-Leonard pair, arXiv:2312.08312.
  * [20] H. De Bie and H. De Clercq, The $q$-Bannai–Ito algebra and multivariate $(-q)$-Racah and Bannai–Ito polynomials, J. Lond. Math. Soc. 103 (2021) 71–126 and arXiv:1902.07883.
  * [21] H. De Bie, H. De Clercq and W. Van De Vijver, The higher rank $q$-deformed Bannai–Ito and Askey–Wilson algebra, Comm. Math. Phys. 374 (2020) 277–316 and arXiv:1805.06642.
  * [22] H. De Clercq, Higher Rank Relations for the Askey–Wilson and $q$-Bannai–Ito Algebra, SIGMA 15 (2019) 099 and arXiv:1908.11654.
  * [23] P. Diaconis and R. Griffiths, An introduction to multivariate Krawtchouk polynomials and their applications, Journal Statistical Planning and Inference 154 (2014) 39–53 and arXiv:1309.0112.
  * [24] J.J. Duistermaat and F.A. Grünbaum, Differential equations in the spectral parameter, Comm. Math. Phys. 103 (1986) 177–240.
  * [25] C. Dunkl and Y. Xu, Orthogonal polynomials of several variables, Encyclopedia of Math. and its applications, vol. 81, Cambridge Univ. press, 2001.
  * [26] J. P. Elliott, Theoretical studies in nuclear structure V. The matrix elements of non-central forces with an application to the $2p$-shell, Proc. Roy. Soc. A 218 (1953) 345.
  * [27] G. Gasper and M. Rahman, Basic hypergeometric series, Encyclopedia of Math. and its Appl., vol. 35, Cambridge University Press, Cambridge, 1990.
  * [28] G. Gasper and M. Rahman, Some systems of multivariable orthogonal Askey-Wilson polynomials in: Theory and applications of special functions, pp. 209–219, Dev. Math. 13, Springer, New York, 2005.
  * [29] V.X. Genest, P. Iliev and L. Vinet, Coupling coefficients of $su_{q}(1,1)$ and multivariate $q$-Racah polynomials Nuclear Phys. B 927 (2018) 97–123 and arXiv:1702.04626.
  * [30] V.X. Genest, S. Post and L. Vinet, An algebraic interpretation of the multivariate $q$-Krawtchouk polynomials, Ramanujan Journal 43 (2017) 415–445 and arXiv:1508.07770.
  * [31] V.X. Genest, L. Vinet and A. Zhedanov, The multivariate Krawtchouk polynomials as matrix elements of the rotation group representations on oscillator states, J. Phys. A: Math. Theor. 46 (2013) 505203 and arXiv:1306.4256.
  * [32] J. Geronimo and P. Iliev, Bispectrality of multivariable Racah-Wilson polynomials, Constr. Approx. 31 (2010) 417–457 and arXiv:0705.1469.
  * [33] Ya. A. Granovskii and A. S. Zhedanov, Nature of the symmetry group of the $6j$-symbol, Zh. Eksp. Teor. Fiz. 94 (1988) 49–54.
  * [34] Y. I. Granovskii and A. S. Zhedanov, Hidden Symmetry of the Racah and Clebsch-Gordan Problems for the Quantum Algebra $sl_{q}(2)$, Journal of Group Theoretical Methods in Physics 1 (1993) 161–171 and arXiv:hep-th/9304138.
  * [35] R. Griffiths, Orthogonal polynomials on the multinomial distribution, Austral. J. Statist. 13 (1971) 27–35; Corrigenda Austral. J. Statist. 14 (1972) 270.
  * [36] W. Groenevelt, A quantum algebra approach to multivariate Askey–Wilson polynomials, International Mathematics Research Notices 2021 (2021) 3224–3266 and arXiv:1809.04327.
  * [37] W. Groenevelt and C. Wagenaar, An Askey–Wilson algebra of rank 2, SIGMA 19 (2023) 008 and arXiv:2206.03986.
  * [38] M.R. Hoare and M. Rahman, A probabilistic origin for a new class of bivariate polynomials, SIGMA 4 (2008) 89 and arXiv:0812.3879.
  * [39] H.-W. Huang, Finite-dimensional irreducible modules of the universal Askey–Wilson algebra, Comm. Math. Phys. 340 (2015) 959–984 and arXiv:1210.1740.
  * [40] P. Iliev, Bispectral commuting difference operators for multivariable Askey–Wilson polynomials, Trans. Amer. Math. Soc. 363 (2011) 1577–1598 and arXiv:0801.4939.
  * [41] P. Iliev, A Lie theoretic interpretation of multivariate hypergeometric, Compositio Math. 148 (2012) 991–1002 and arXiv:1101.1683.
  * [42] P. Iliev and P. Terwilliger, The Rahman polynomials and the Lie algebras $sl_{3}(\mathbb{C})$, Trans. Amer. Math. Soc. 363 (2012) 4225–4238 and arXiv:1006.5062.
  * [43] E.G. Kalnins, J.M. Kress and W. Miller Jr., Second-order superintegrable systems in con-formally flat spaces. I. Two-dimensional classical structure theory, J. Math. Phys. 46 (2005) 053509.
  * [44] V.K. Khersonskii, A.N. Moskalev and D.A. Varshalovich, Quantum Theory Of Angular Momentum, World Scientific (1988).
  * [45] A.N. Kirillov and N.Yu. Reshetikhin, Representations of the algebra $U_{q}(sl(2))$, $q$-othogonal polynomials and invariants of links, in: Infinite dimensional Lie algebras and groups (World Sci. Publishing, NJ, 1989) 285–339.
  * [46] A.U. Klimyk and N.J. Vilenkin, Representation of Lie groups and special functions, Kluwer Academic, 1991-1993, Dordrecht.
  * [47] R. Koekoek, P.A. Lesky and R.F. Swarttouw, Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer Berlin, Heidelberg (2010).
  * [48] D.A. Leonard, Orthogonal polynomials, duality and association schemes, SIAM J. Math. Anal. 13 (1982) 656–663.
  * [49] A. Messiah, Quantum Mechanics, Volume II, New York: North Holland Publishing (1981).
  * [50] A. Mironov, A. Morozov and A. Sleptsov, On $6j$-symbols for symmetric representations of $U_{q}(su_{N})$, Jetp Lett. 106 (2017) 630–636 and arXiv:1709.02290.
  * [51] H. Mizukawa and H. Tanaka, $(n+1,m+1)$-hypergeometric functions associated to character algebras, Proc. Amer. Math. Soc. 132 (2004) 2613–2618.
  * [52] S. Post, Models of quadratic algebras generated by superintegrable systems in 2D, SIGMA 7 (2011) 036–055 and arXiv:1104.0734.
  * [53] S. Post and A. Walter, A higher rank extension of the Askey–Wilson algebra, arXiv:1705.01860.
  * [54] A.V. Rosenblyum, Spectral analysis of generators of representations of the group $U(3)$, Theoretical and Mathematical Physics 73 (1987) 1352–1356.
  * [55] R. Sasaki, Quantum vs Classical Birth and Death Processes; Exactly Solvable Examples, arXiv:2212.10710.
  * [56] F. Scarabotti, The tree method for multidimensional $q$-Hahn and $q$-Racah polynomials, Ramanujan Journal 25 (2011) 57–91 and arXiv:0912.4357.
  * [57] M.V. Tratnik, Some multivariable orthogonal polynomials of the Askey tableau-discrete families, J. Math. Phys. 32 (1991) 2337–2342.
  * [58] P. Terwilliger, Two linear transformations each tridiagonal with respect to an eigenbasis of the other, Linear Algebra Its Appl. 330 (2001) 149–203 and arXiv:math/0406555.
  * [59] P. Terwilliger, The universal Askey–Wilson algebra, SIGMA 7 (2011) 069 and arXiv:1104.2813.
  * [60] P. Terwilliger and R. Vidunas, Leonard pairs and the Askey–Wilson relations, Journal of Algebra and Its Applications 03 (2004) 411–426 and arXiv:math/0305356.
  * [61] J. Van der Jeugt, Coupling coefficients for Lie algebra representations and addition formulas for special functions, J. Math. Phys. 38 (1997) 2728.
  * [62] J.F. van Diejen and J.V. Stokman, Multivariable q-Racah polynomials, Duke Math. J. 91 (1998) 89–136.
  * [63] W. Wang and S. Yang, On the higher-rank Askey–Wilson algebras, arXiv:2407.10404.
  * [64] J. Wilson, Hypergeometric series recurrence relations and some new orthogonal functions, Ph.D. thesis, University of Wisconsin (1978).
  * [65] A. S. Zhedanov, “Hidden symmetry” of Askey–Wilson polynomials, Theoretical and Mathematical Physics 89 (1992) 1146–1157.
  * [66] A. Zhedanov, $9j$-symbols for the oscillator algebra and the Krawtchouk polynomials in two variables, J. Phys. A: Math. Gen. 30 (1997) 8337.