# Realization of programmable Ising models in a trapped-ion quantum simulator

Yao Lu<EMAIL_ADDRESS>Shenzhen Institute for Quantum Science and
Engineering, Southern University of Science and Technology, Shenzhen 518055,
China  International Quantum Academy, Shenzhen 518048, China  State Key
Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua
University, Beijing 100084, China  Wentao Chen  State Key Laboratory of Low
Dimensional Quantum Physics, Department of Physics, Tsinghua University,
Beijing 100084, China  Shuaining Zhang  State Key Laboratory of Low
Dimensional Quantum Physics, Department of Physics, Tsinghua University,
Beijing 100084, China  Kuan Zhang  State Key Laboratory of Low Dimensional
Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084,
China  Jialiang Zhang  State Key Laboratory of Low Dimensional Quantum
Physics, Department of Physics, Tsinghua University, Beijing 100084, China
Jing-Ning Zhang  Beijing Academy of Quantum Information Sciences, Beijing
100193, China  Kihwan Kim<EMAIL_ADDRESS>State Key Laboratory
of Low Dimensional Quantum Physics, Department of Physics, Tsinghua
University, Beijing 100084, China  Beijing Academy of Quantum Information
Sciences, Beijing 100193, China  Hefei National Laboratory, Hefei 230088, P.
R. China  Frontier Science Center for Quantum Information, Beijing 100084,
People’s Republic of China

###### Abstract

A promising paradigm of quantum computing for achieving practical quantum
advantages is quantum annealing or quantum approximate optimization algorithm,
where the classical problems are encoded in Ising interactions. However, it is
challenging to build a quantum system that can efficiently map any structured
problems. Here, we present a programmable trapped-ion quantum simulator of an
Ising model with all-to-all connectivity with up to four spins. We implement
the spin-spin interactions by using the coupling of trapped ions to multiple
collective motional modes and realize the programmability through phase
modulation of the Raman laser beams that are individually addressed on ions.
As an example, we realize several Ising lattices with different interaction
connectivities, where the interactions can be ferromagnetic or anti-
ferromagnetic. We confirm the programmed interaction geometry by observing the
ground states of the corresponding models through quantum state tomography.
Our experimental demonstrations serve as an important basis for realizing
practical quantum advantages with trapped ions.

##  I Introduction

Significant theoretical and experimental progress has been made toward
realizing practical quantum applications on Noisy Intermediate-Scale Quantum
(NISQ) processors, which could lead to surpassing classical computational
capabilities [1, 2, 3]. Quantum annealing (QA) [4, 5, 6, 7, 8] and quantum
approximate optimization algorithm (QAOA) [9, 10, 11, 12] are representative
algorithms that can bring out quantum advantages in the application of
practical problems such as combinatorial optimization, quantum many-body
physics, and quantum chemistry. In these algorithms, a problem of interest is
mapped to a quantum Hamiltonian, where the solution of the problem can be
found by either adiabatic evolution to the ground state of the Hamiltonian [5]
or time evolution under alternating phase and mixing Hamiltonians with the
assistance of classical parameter search [9]. It has not yet clearly shown the
possibility of surpassing classical computational power with the quantum
processors in the NISQ regime [13, 1, 3]. Currently, many studies are being
conducted heuristically to find quantum speed-ups with existing quantum
processors [14, 15, 16, 17, 18, 19, 20, 21].

However, due to system limitations, many quantum processors are still unable
to fully map classically hard problems. For example, when a combinatorial
optimization problem is encoded to an Ising model, a fully connected network
with programmability is required for general mapping, which has been a major
challenge in experimental development. It has been proposed to realize all-to-
all connectivity from local interactions at the expense of quadratic enlarge
of the qubit numbers [22, 23]. For Rydberg atoms, it was proposed to use
chains of additional atoms, called quantum wires to implement programmable
all-to-all connectivity [24], and an alternative quantum wire scheme, called
the Rydberg quantum wire, with much less demanding experimental requirements,
has been proposed and demonstrated [25]. On the one hand, ion-trap systems
contain naturally all-to-all connectivity [26], which has been shown through
the realization of global entangling gates and parallel gates [27, 28]. It
also has been theoretically proposed to realize the two-dimensional
connectivity of Ising models with a linear chain of ions [29, 30, 31] and a
chiral dynamics with qubits in a magnetic field gradient has been
experimentally demonstrated [32].

In this paper, we implement arbitrarily programmable interactions of Ising
models and prepare the ground states of the corresponding Hamiltonians through
adiabatic evolution using up to four trapped-ion qubits. We realize the
adiabatic evolution digitally [33, 17, 34], where the Hamiltonians for spin-
spin interactions and transverse fields are alternatively applied. The
programmable spin-spin interactions are implemented by coupling trapped ions
to multiple collective motional modes through mainly phase modulation of the
individually addressing Raman laser beams [35, 36, 37, 38, 27, 28, 39]. The
digital simulation suppresses the residual qubit-motion entanglements that
introduce decoherence during the adiabatic evolution [40, 41, 42, 43]. We
confirm the prepared ground states are close to the pure quantum state through
quantum state tomography (QST). Our method would be the basis for an almost
ideal system corresponding to QA or QAOA using trapped ions. This method can
also be applied to 2D ion crystals, which can rapidly scale up the number of
ion qubits beyond the classical computational capabilities [44, 45].

##  II Results

###  II.1 Experimental setup scheme

We employ a trapped-ion-based quantum simulator, which consists of up to four
${}^{171}\mathrm{Yb}^{+}$ ions in a linear-chain configuration, as illustrated
in Fig. 1a. A single effective spin-1/2 is encoded within the ground manifold
of each ion, denoted by $\ket{\downarrow}={{{}^{2}}S_{1/2}}\ket{F=0,m_{F}=0}$
and $\ket{\uparrow}={{}^{2}}S_{1/2}\ket{F=1,m_{F}=0}$, with an energy gap
approximately $\omega_{0}\approx 2\pi\times 12.64~{}\mathrm{GHz}$ [46]. All
spins can be initialized to the $\ket{\downarrow...\downarrow}$ state by
standard optical pumping, and their quantum states are measured via spin
state-dependent resonant fluorescence. By utilizing the multi-channel
photomultiplier tube (PMT) we can perform site-resolved detection on multiple
ions [47]. For more details on our quantum simulator, see the Method section.

Figure 1:  Experimental realization of programmable Ising models. a.
Experimental setup. A chain of ions trapped in the blade trap is coherently
manipulated via the stimulated Raman transition induced by two orthogonal
laser beams. One of the Raman beams is divided into beams array, providing
individual addressing capability on each ion. b. Example of effective spin-
spin interactions. Arbitrarily connected interactions can be achieved by
properly designing the modulation pattern of each individual beam. c. Example
of mapping Ising lattice. Depending on the connectivity graph of the spin-spin
interactions, it can be mapped to certain parts of the Ising lattice.

Coherent manipulations of spin states are achieved through stimulated Raman
transitions. To enable the capability of individual control of each spin, we
split one of Raman beams into a laser array and then let them pass through a
multi-channel acoustic-optical modulator (AOM) [27]. Effective spin-spin
interactions can be engineered by coupling ions to their collective motional
modes, and here we apply bi-chromatic fields with frequencies of
$\omega_{0}\pm\mu$, leading to a spin-dependent force (SDF) on the entire ion
chain [48],

$H_{\mathrm{SDF}}(t)=\sum_{j,m}\dfrac{\eta_{j,m}\Omega_{j}}{2}(a_{m}\mathrm{e}^{-\mathrm{i}(\mu-\nu_{m})t}\mathrm{e}^{\mathrm{i}\phi_{j}}+\mathrm{H.c.})\sigma_{\mathrm{x}}^{(j)}.$
(1)

Here, $\sigma_{\mathrm{x}}^{(j)}$ represents the Pauli matrix of the $j$-th
ion, $\eta_{j,m}$ denotes the scaled Lamb-Dicke parameter associated with
$j$-th ion and $m$-th motional modes, $\nu_{m}$ corresponds to the frequency
of the $m$-th motional modes, and $a_{m}$ is the corresponding annihilation
operator.

The earlier demonstrations of the Ising models within trapped-ion systems
primarily concentrated on adiabatic regimes [49, 50]. However, this approach
exhibits certain limitations, notably the residual spin-motion couplings
inevitably lead to decoherence of the system during long-time evolution. Also,
the interaction graph has limited flexibility in programming. To address the
above issues, we leverage the optimal control method to optimize the control
parameters. This strategy not only allows us to eliminate undesired spin-
motion couplings but also provides the means to program spin-spin
interactions. Specifically, in our work, we primarily modulate the phases
$\phi_{j}$ and amplitudes $\Omega_{j}$ of the SDF over time [27]. Through
carefully designed modulation patterns, we achieve an effective evolution
operator as below,

$U_{\mathrm{Ising}}\left(\left\\{\Theta_{j,j^{\prime}}\right\\}\right)=\mathrm{exp}\left[-\mathrm{i}\sum_{j<j^{\prime}}\Theta_{j,j^{\prime}}\sigma_{\mathrm{x}}^{(j)}\sigma_{\mathrm{x}}^{j^{\prime}}\right].$
(2)

This operator is valid at a pre-determined time $\tau_{g}$. Here, the values
of any $\Theta_{j,j^{\prime}}$ can be arbitrarily tuned through the design of
the modulation pattern. Thus, it offers the possibility of mapping Equation
(2) to an Ising lattice with arbitrary weighted connectivity, as depicted in
Fig. 1b-c.

Figure 2:  Digitized adiabatic evolution. a. Adiabatic evolution. In the
conventional adiabatic evolution, the strengths of the initial (targeted)
Hamiltonians are continuously ramped down (up) (shown in the dashed lines).
For the digitized approach, the continuous ramping is discretized into several
stages (shown in the solid lines). b. Circuits description of the digitized
adiabatic evolution. After initializing the whole ion chain to
$\ket{\downarrow}$ state, $\pi/2$\- rotations are applied to prepare the
ground state of the transverse field. The whole time evolution is decomposed
into a series of single-spin rotations and multi-spin interactions. Note that
the second-order Trotterization is employed in our experiments. c. Results of
the states’ population for each digital step. Here we choose the fully
connected 3-spin ferromagnetic Ising model as an example. The open markers
indicate experimental results, while the dashed lines guide the theoretical
digital evolution. The error bars of one standard deviation are smaller than
the size of the markers. d. Results of the state tomography. We show the real
parts of the density matrices obtained in the experiments. The colored squares
around axes indicate spin states, where the filled blue and empty squares
represent $\ket{\uparrow}$ and $\ket{\downarrow}$ states, respectively. The
left one indicates the initial ground state of the transverse field along the
$\sigma_{\mathrm{y}}$-direction, with an initialization fidelity estimated to
be 0.964(2). The right part corresponds to the final ground state of the
ferromagnetic Ising model. The dashed boxes represent the diagonal components
of the density matrices obtained from the ideal digitized evolution. All the
error bars here and below represent one standard deviation.

To evaluate our approach in simulating arbitrary Ising models, we employ the
adiabatic evolution techniques to prepare their ground states. By first
preparing all spins to the eigenstate of a transverse field, i.e.
$H_{\mathrm{ini}}=-B\sum_{j}\sigma_{\mathrm{y}}^{j}$ ($B>0$), we could finally
reach the ground state after slowly ramping up (down) the Hamiltonian of the
Ising model (transverse field). As the programmed Ising model described in
Equation (2) is only valid at a discrete time of $\tau_{g}$, the digitized
adiabatic evolution is employed here [17], as illustrated in Fig. 2a. Within
this framework, the time evolution is discretized into alternatively applied
spin-spin interactions and transverse field, as shown in Fig. 2b, and the
strengths of the transverse field and the Ising interaction remain constant
within each digitized step. This digitized evolution would approach the
continuous one as the number of the digitized step $S$ is large enough. To
further verify the final state obtained via the digitized adiabatic evolution,
we execute QST circuits after the adiabatic evolution. Subsequently, we
reconstruct the density matrices of the final states based on the results
obtained from measurements in different bases.

###  II.2 Ising models with three sites

We begin our demonstration by focusing on the Ising models consisting of three
spins. In this situation, three ${}^{171}\mathrm{Yb}^{+}$ are confined in the
trap, and the radial motional frequencies along $x$-direction are estimated as
$\\{\nu_{1},\nu_{2},\nu_{3}\\}=2\pi\times\\{2.173,2.118,2.039\\}~{}\mathrm{MHz}$.
To simplify the experimental demonstration, we consider the scenarios where
the interaction strength between any two spins is restricted to values of
$-J,J$ or $0$ ($J>0$), corresponding to ferromagnetic, anti-ferromagnetic or
no interactions, respectively. The three-site models can be organized into a
one-dimensional (1D) line or a two-dimensional (2D) triangular configuration.
Given that 1D cases which require only the nearest neighbor couplings have
been extensively studied and yield ground states similar to GHZ states [49],
we focus only on the exploration of the high-dimensional Ising models.

Figure 3:  Experimental results of the prepared ground states for different
3-spin Ising models. a. Modulation pattern to realize 3-spin Ising model. In
the pulse optimization, we set the frequency of the SDF to be $\mu=2\pi\times
2.078$ MHz, and the operation duration $\tau_{g}$ to be 120 $\mu$s. The values
of $|\Omega_{j}|$ (shown in the black curve) for all ions are set to be the
same in the experimental optimization, while the sign of $\Omega_{j}$ can be
utilized to adjust the sign of the spin-spin interactions. The phase
modulation is performed smoothly and continuously (shown in blue and red
curves), minimizing the noise in the AOM that occurs when the phase is
suddenly changed. b. Characterization of spin-spin interactions. Utilizing the
modulation pattern shown in a, we address each ion pair to prepare the
corresponding maximally entangled state. We then measure the parity
oscillations of these entangled states, with the sign of the oscillations
indicating the sign of the interaction. The legend on the left side denotes
the fitted contrasts. c-e. Real parts of the reconstructed density matrices
for different Ising model ground states. In each subfigure’s inset, a possible
alignment of spins for the ground state is illustrated. Blue (red) lines
connecting spins represent (anti-)ferromagnetic interactions.

In the context of the 2D triangular lattice, when all interactions between
spin pairs are ferromagnetic, the spin pairs between different sites naturally
align in parallel, thereby minimizing the overall energy of the entire system.
Consequently, the ground state of the system adopts the 3-GHZ state, as
depicted in Fig. 2c-d. Here, the digitized evolution is divided into three
steps ($S=3$) to balance the Trotter error and the experimental operational
errors. The modulation patterns for the phases $\phi_{j}$ and absolute
amplitudes $|\Omega_{j}|$ of SDF are outlined in Fig. 3a. Note that the
amplitude modulation primarily serves as pulse shaping to eliminate errors
arising from off-resonant coupling to the carrier transition. By setting the
sign of all $\Omega_{j}$ to be the same, a ferromagnetic Ising model can be
achieved. All the pairs of spin-spin interactions are experimentally
characterized, as summarized in Fig. 3b. In detail, we address targeted spin
pairs, such as the $j$\- and $j^{\prime}$-th ions, to prepare the maximal
entangling states, by utilizing designed modulation patterns with the proper
amplitudes. After applying the analysis rotation,
$\mathrm{exp}[\mathrm{i}\pi(\sigma_{\phi}^{j}+\sigma_{\phi}^{j^{\prime}})/4]$,
to the prepared entangling state, the measured parity oscillation $P(\phi)$
would reveal the sign of $\Theta_{j,j^{\prime}}$, as the relation of
$P(\phi)\propto\mathrm{sign}(\Theta_{j,j^{\prime}})\sin(2\phi)$.

In the digitized evolution, the values assigned to the transverse field $B$
and the spin-spin interaction $J$ are $\pi/2$ and $\pi/3$, respectively. The
QST result yields a fidelity $\mathcal{F}$ of 0.835(4) for the prepared
quantum ground state. Here $\mathcal{F}$ is defined as
$(\mathrm{Tr}(\sqrt{\sqrt{\rho_{\mathrm{th}}}\rho_{\mathrm{exp}}\sqrt{\rho_{\mathrm{th}}}}))^{2}$,
where $\rho_{\mathrm{th}}$ and $\rho_{\mathrm{exp}}$ are the density matrices
obtained from the simulation of ideal digitized evolution and the experimental
QST data, respectively. The observed errors primarily stem from the
depolarization during effective spin-spin interactions. This point can be
verified that if we only consider the diagonal parts of the density matrices
($\rho^{\mathrm{dig}}=\mathrm{Diag}\left[\rho\right]$), the overlap between
the experimental result and the theoretical value, which is denoted as
$\mathcal{F}_{\mathrm{diag}}=(\mathrm{Tr}(\sqrt{\sqrt{\rho^{\mathrm{diag}}_{\mathrm{th}}}\rho^{\mathrm{diag}}_{\mathrm{exp}}\sqrt{\rho^{\mathrm{diag}}_{\mathrm{th}}}}))^{2}$,
increases to 0.944(7). This also suggests that improving the fidelity of spin-
spin interactions can further enhance the fidelity of the prepared ground
state.

Upon exploring the pure ferromagnetic Ising lattice, we proceed to modify
partial interactions to be antiferromagnetic, subsequently preparing the
corresponding ground states. The results are summarized in Fig. 3c-e. A
crucial observation arises when one or all three bonds are changed to anti-
ferromagnetic interactions. In these scenarios, the spin pairs are unable to
hold the lowest energy simultaneously due to the conflict of the interactions,
resulting in geometric frustration [49]. This phenomenon leads to an
augmentation in ground state degeneracy, with six degenerate ground states
present for both the single bond and all bonds antiferromagnetic Ising
lattices, as illustrated in Fig. 3c-d. The fidelities ($\mathcal{F}$) of the
prepared ground states are estimated as 0.858(5) and 0.871(5) for the single
bond and all bonds anti-ferromagnetic Ising lattice, respectively, while the
overlap of the diagonal parts ($\mathcal{F}_{\mathrm{diag}}$) are 0.984(7) and
0.972(7). Except for the aforementioned cases, when two bonds are flipped to
anti-ferromagnetic, the corresponding ground state reverts to a GHZ-like
state. This is because all spin pairs can simultaneously assume their lowest
energy states, as displayed in Fig. 3e. The state fidelity ($\mathcal{F}$) and
diagonal overlap ($\mathcal{F}_{\mathrm{diag}}$) for this scenario measure at
0.844(5) and 0.934(7), respectively.

Note that, the 3-spin Ising model demonstrated above all utilize the same
phase modulation patterns. For the Ising model in Fig. 3d, we reverse the sign
of amplitude $\Omega_{2}$ to make $\Omega_{0}=\Omega_{1}=-\Omega_{2}$,
allowing us to construct the Ising model of
$-\sigma_{\mathrm{x}}^{0}\sigma_{\mathrm{x}}^{1}+\sigma_{\mathrm{x}}^{0}\sigma_{\mathrm{x}}^{2}+\sigma_{\mathrm{x}}^{1}\sigma_{\mathrm{x}}^{2}$.
In the Ising model presented in Fig. 3c, we also set
$\Omega_{0}=\Omega_{1}=-\Omega_{2}$, and in fact the experimentally prepared
state is the highest excited state. Nevertheless, this state is equivalent to
the ground state of the Ising model
$\sigma_{\mathrm{x}}^{0}\sigma_{\mathrm{x}}^{1}-\sigma_{\mathrm{x}}^{0}\sigma_{\mathrm{x}}^{2}-\sigma_{\mathrm{x}}^{1}\sigma_{\mathrm{x}}^{2}$
except for certain phase sign differences, which can be corrected in the
density matrix reconstruction. Similarly, for results showcased in Fig. 3e, we
also prepare the highest excited state of the ferromagnetic Ising model.

###  II.3 Ising models with four sites

Figure 4:  Experimental results of prepared ground states for 4-spin Ising
models. All the spin pairs in each Ising model are addressed to characterize
the sign of the spin-spin interactions, as depicted in a through c. In a, we
can clearly observe a near cancellation of interaction between the spin pair
0-1. While in b, the interactions for both spin pairs 0-3 and 1-2 are
simultaneously canceled, and the interaction between the pair 0-1 is reversed
to be ferromagnetic as well. Then c presents a typical representation of fully
connected anti-ferromagnetic interactions, which can be treated as a three-
dimensional Ising lattice. In d through f, we provide the experimentally
obtained density matrices resulting from digitized adiabatic evolution, for
the interaction configurations depicted in a through c, respectively

We now transition our focus to the Ising models consisting of four spins.
Towards these models, we introduce an additional ion to the trap, increasing
the total number of ions to four. The trap frequencies are measured as
$\\{\nu_{1},\nu_{2},\nu_{3},\nu_{4}\\}=2\pi\times\\{2.179,2.138,2.081,2.005\\}~{}\mathrm{MHz}$.
In order to mitigate Trotterization errors, the digitized step is heightened
to four as well. The inclusion of an extra spin facilitates an extension of 2D
triangular structure shown in Fig. 3 to a larger lattice. To illustrate, the
anti-ferromagnetic triangular lattice showcased in Fig. 3e is extended to a
bi-triangular lattice, depicted in Fig. 4a and Fig. 4d. Evidently, the
contrast in parity oscillation between spins 0 and 1 almost disappears,
signifying the absence of spin-spin interaction in this particular pair. This
newly introduced site reduces the count of degenerate ground states to two,
compelling both triangular sub-structures to maintain a symmetrical alignment
of spins. Experimentally, we achieve the ground state with a fidelity
$\mathcal{F}$ of 0.576(3), accompanied by a diagonal overlap
$\mathcal{F}_{\mathrm{diag}}$ of 0.853(5). In comparison to the results from
three-site situations, the fidelity of the prepared state diminishes due to
the increase of the digital steps and escalated depolarization error during
the spin-spin interactions.

We also attempt to realize a square lattice by simultaneously removing the
interactions between spin pairs of 0-3 and 1-2, as shown in Fig. 4b and Fig.
4e. Furthermore, the interaction between spin pair 0-1 is engineered to be
ferromagnetic, while the remaining interactions retain anti-ferromagnetic.
This configuration also manifests high geometric frustration within its ground
state. The experimental result reveals a fidelity $\mathcal{F}$ of 0.750(4),
while the diagonal overlap $\mathcal{F}_{\mathrm{diag}}$ approximates to be
0.930(4).

An extension into a three-dimensional (3D) Ising lattice with a tetrahedral
structure can be realized through the all-to-all connection, exemplified in
Fig. 4c-f. This structure constitutes a fundamental and pivotal component in
various solid crystals, typified by pyrochlore oxides. The magnetic
characteristics of this configuration have garnered considerable theoretical
and experimental attention over the last few decades. In this scenario, we
prepare the ground state of a solitary tetrahedral lattice featuring
antiferromagnetic interactions, thereby inducing geometric frustration once
more. All the ground states exhibit a consistent feature: the presence of two
spins up and two spins down, mirroring the ice rule (two-in, two-out) in the
spin-ice system. The fidelity of the ground state we prepare measures at
0.719(3), while the diagonal overlap rests at 0.886(4). The modulation
patterns for all the 4-spin Ising models can be found in the Methods section.

All the above results are corrected to remove the detection error (Methods).
Meanwhile, we can briefly estimate error contributions in our demonstration.
Our experiments are conducted using the same setup detailed in a previous
study [27], and as such, the sources of error remain consistent with those
outlined in that work. As demonstrated in that study, the errors associated
with 3-spin and 4-spin operations can be estimated to be approximately 5% and
7%, respectively. Consequently, for the 3-spin Ising models, our three-step
digitized evolution would yield total fidelities at approximately the 85%
level. Similarly, for the 4-spin models, a four-step evolution would yield
total fidelities around the 75% level. Given these error estimates, it becomes
evident that further improvement in multi-spin operations is crucial to
enhance the performance of programmable Ising models.

##  III Conclusion and outlook

In conclusion, our study brings an efficient approach to realize programmable
Ising models featuring arbitrary connectivity, by leveraging the all-to-all
capability among the trapped ions and the individual controllability. We
suggest that our approach offers not only efficient implementation but also
remarkable scalability. Recent research has demonstrated that the optimization
of modulation patterns required for achieving programmable Ising interactions
demands only polynomial computational resources [51], thus affirming the
scalability of our approach. Through the integration of programmable Ising
interactions with digitized adiabatic evolution, we have experimentally
achieved the successful preparation of quantum ground states for various Ising
lattices comprising up to four spin sites. These results mark a significant
step in the realization of practical QA and QAOA algorithms on NISQ devices
based on trapped-ion platforms.

For future research, the studies on more complex spin systems that own i.e. XY
or Heisenberg interactions should included. Generalized spin models with
arbitrary connectivity open up significant opportunities for exploring richer
quantum phenomena. Moreover, the digitized adiabatic evolution employed here
can be accelerated by implementing the techniques of shortcuts to adiabaticity
(STA) [52]. A recent study also suggests a mechanism of self-healing in
digitized adiabatic state preparation [53]. It suggests that the accumulation
of infidelity during digitized adiabatic evolution could scale lower than the
bounds given by general Trotter errors. Both techniques and features hold the
promise of further enhancing the computational power of programmable trapped-
ion quantum simulators, paving the way for novel advancements in near-term
quantum computation and simulation. During the preparation of this paper, we
became aware of related studies about engineering programmable spin models by
utilizing multi-tone drives [54, 55].

## References

  * Preskill [2018] J. Preskill, Quantum Computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  * Moll  _et al._ [2018] N. Moll, P. Barkoutsos, L. S. Bishop, J. M. Chow, A. Cross, D. J. Egger, S. Filipp, A. Fuhrer, J. M. Gambetta, M. Ganzhorn, _et al._ , Quantum optimization using variational algorithms on near-term quantum devices, Quantum Sci. Technol. 3, 030503 (2018).
  * Bharti  _et al._ [2022] K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, _et al._ , Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys. 94, 015004 (2022).
  * Kadowaki and Nishimori [1998] T. Kadowaki and H. Nishimori, Quantum annealing in the transverse ising model, Phys. Rev. E 58, 5355 (1998).
  * Farhi  _et al._ [2001] E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, A quantum adiabatic evolution algorithm applied to random instances of an np-complete problem, Science 292, 472 (2001).
  * Albash and Lidar [2018] T. Albash and D. A. Lidar, Adiabatic quantum computation, Rev. Mod. Phys. 90, 015002 (2018).
  * Hauke  _et al._ [2020] P. Hauke, H. G. Katzgraber, W. Lechner, H. Nishimori, and W. D. Oliver, Perspectives of quantum annealing: Methods and implementations, Rep. Prog. Phys. 83, 054401 (2020).
  * Yarkoni  _et al._ [2022] S. Yarkoni, E. Raponi, T. Bäck, and S. Schmitt, Quantum annealing for industry applications: Introduction and review, Rep. Prog. Phys. (2022).
  * Farhi  _et al._ [2014] E. Farhi, J. Goldstone, and S. Gutmann, A quantum approximate optimization algorithm, arXiv preprint arXiv:1411.4028 (2014).
  * Hadfield  _et al._ [2019] S. Hadfield, Z. Wang, B. O’gorman, E. G. Rieffel, D. Venturelli, and R. Biswas, From the quantum approximate optimization algorithm to a quantum alternating operator ansatz, Algorithms 12, 34 (2019).
  * Zhou  _et al._ [2020] L. Zhou, S.-T. Wang, S. Choi, H. Pichler, and M. D. Lukin, Quantum approximate optimization algorithm: Performance, mechanism, and implementation on near-term devices, Phys. Rev. X 10, 021067 (2020).
  * Sack and Serbyn [2021] S. H. Sack and M. Serbyn, Quantum annealing initialization of the quantum approximate optimization algorithm, Quantum 5, 491 (2021).
  * Rønnow  _et al._ [2014] T. F. Rønnow, Z. Wang, J. Job, S. Boixo, S. V. Isakov, D. Wecker, J. M. Martinis, D. A. Lidar, and M. Troyer, Defining and detecting quantum speedup, Science 345, 420 (2014).
  * Johnson  _et al._ [2011] M. W. Johnson, M. H. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A. J. Berkley, J. Johansson, P. Bunyk, _et al._ , Quantum annealing with manufactured spins, Nature 473, 194 (2011).
  * Boixo  _et al._ [2014] S. Boixo, T. F. Rønnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, Evidence for quantum annealing with more than one hundred qubits, Nat. Phys. 10, 218 (2014).
  * King  _et al._ [2022] A. D. King, S. Suzuki, J. Raymond, A. Zucca, T. Lanting, F. Altomare, A. J. Berkley, S. Ejtemaee, E. Hoskinson, S. Huang, _et al._ , Coherent quantum annealing in a programmable 2,000 qubit ising chain, Nat. Phys. 18, 1324 (2022).
  * Barends  _et al._ [2016] R. Barends, A. Shabani, L. Lamata, J. Kelly, A. Mezzacapo, U. L. Heras, R. Babbush, A. G. Fowler, B. Campbell, Y. Chen, _et al._ , Digitized adiabatic quantum computing with a superconducting circuit, Nature 534, 222 (2016).
  * Pagano  _et al._ [2020] G. Pagano, A. Bapat, P. Becker, K. S. Collins, A. De, P. W. Hess, H. B. Kaplan, A. Kyprianidis, W. L. Tan, C. Baldwin, _et al._ , Quantum approximate optimization of the long-range ising model with a trapped-ion quantum simulator, Proc. Natl. Acad. Sci. U.S.A. 117, 25396 (2020).
  * Mohseni  _et al._ [2022] N. Mohseni, P. L. McMahon, and T. Byrnes, Ising machines as hardware solvers of combinatorial optimization problems, Nat. Rev. Phys. 4, 363 (2022).
  * Ebadi  _et al._ [2022] S. Ebadi, A. Keesling, M. Cain, T. T. Wang, H. Levine, D. Bluvstein, G. Semeghini, A. Omran, J.-G. Liu, R. Samajdar, _et al._ , Quantum optimization of maximum independent set using Rydberg atom arrays, Science 376, 1209 (2022).
  * Zhu  _et al._ [2022] Y. Zhu, Z. Zhang, B. Sundar, A. M. Green, C. H. Alderete, N. H. Nguyen, K. R. Hazzard, and N. M. Linke, Multi-round QAOA and advanced mixers on a trapped-ion quantum computer, Quantum Sci. Technol. 8, 015007 (2022).
  * Choi [2008] V. Choi, Minor-embedding in adiabatic quantum computation: I. the parameter setting problem, Quantum Inf. Process. 7, 193 (2008).
  * Lechner  _et al._ [2015] W. Lechner, P. Hauke, and P. Zoller, A quantum annealing architecture with all-to-all connectivity from local interactions, Sci. Adv. 1, e1500838 (2015).
  * Qiu  _et al._ [2020] X. Qiu, P. Zoller, and X. Li, Programmable quantum annealing architectures with Ising quantum wires, PRX Quantum 1, 020311 (2020).
  * Kim  _et al._ [2022] M. Kim, K. Kim, J. Hwang, E.-G. Moon, and J. Ahn, Rydberg quantum wires for maximum independent set problems, Nat. Phys. 18, 755 (2022).
  * Murali  _et al._ [2019] P. Murali, N. M. Linke, M. Martonosi, A. J. Abhari, N. H. Nguyen, and C. H. Alderete, Full-stack, real-system quantum computer studies: Architectural comparisons and design insights, in  _Proc. Natl. Acad. Sci. U.S.A._ (2019) pp. 527–540.
  * Lu  _et al._ [2019] Y. Lu, S. Zhang, K. Zhang, W. Chen, Y. Shen, J. Zhang, J.-N. Zhang, and K. Kim, Global entangling gates on arbitrary ion qubits, Nature 572, 363 (2019).
  * Figgatt  _et al._ [2019] C. Figgatt, A. Ostrander, N. M. Linke, K. A. Landsman, D. Zhu, D. Maslov, and C. Monroe, Parallel entangling operations on a universal ion-trap quantum computer, Nature 572, 368 (2019).
  * Korenblit  _et al._ [2012] S. Korenblit, D. Kafri, W. C. Campbell, R. Islam, E. E. Edwards, Z.-X. Gong, G.-D. Lin, L.-M. Duan, J. Kim, K. Kim, _et al._ , Quantum simulation of spin models on an arbitrary lattice with trapped ions, New J. Phys. 14, 095024 (2012).
  * Shapira  _et al._ [2020] Y. Shapira, R. Shaniv, T. Manovitz, N. Akerman, L. Peleg, L. Gazit, R. Ozeri, and A. Stern, Theory of robust multiqubit nonadiabatic gates for trapped ions, Phys. Rev. A 101, 032330 (2020).
  * Manovitz  _et al._ [2020] T. Manovitz, Y. Shapira, N. Akerman, A. Stern, and R. Ozeri, Quantum simulations with complex geometries and synthetic gauge fields in a trapped ion chain, PRX Quantum 1, 020303 (2020).
  * Shapira  _et al._ [2023a] Y. Shapira, T. Manovitz, N. Akerman, A. Stern, and R. Ozeri, Quantum simulations of interacting systems with broken time-reversal symmetry, Phys. Rev. X 13, 021021 (2023a).
  * Mezzacapo  _et al._ [2014] A. Mezzacapo, U. Las Heras, J. Pedernales, L. DiCarlo, E. Solano, and L. Lamata, Digital quantum Rabi and Dicke models in superconducting circuits, Sci. Rep. 4, 1 (2014).
  * Parra-Rodriguez  _et al._ [2020] A. Parra-Rodriguez, P. Lougovski, L. Lamata, E. Solano, and M. Sanz, Digital-analog quantum computation, Phys. Rev. A 101, 022305 (2020).
  * Häffner  _et al._ [2008] H. Häffner, C. F. Roos, and R. Blatt, Quantum computing with trapped ions, Phys. Rep. 469, 155 (2008).
  * Blatt and Roos [2012] R. Blatt and C. F. Roos, Quantum simulations with trapped ions, Nat. Phys. 8, 277 (2012).
  * Schneider  _et al._ [2012] C. Schneider, D. Porras, and T. Schaetz, Experimental quantum simulations of many-body physics with trapped ions, Rep. Prog. Phys. 75, 024401 (2012).
  * Monroe  _et al._ [2021] C. Monroe, W. C. Campbell, L.-M. Duan, Z.-X. Gong, A. V. Gorshkov, P. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano, _et al._ , Programmable quantum simulations of spin systems with trapped ions, Rev. Mod. Phys. 93, 025001 (2021).
  * Wang  _et al._ [2022] K. Wang, J.-F. Yu, P. Wang, C. Luan, J.-N. Zhang, and K. Kim, Fast multi-qubit global-entangling gates without individual addressing of trapped ions, Quantum Science and Technology (2022).
  * Wang and Freericks [2012] C.-C. J. Wang and J. Freericks, Intrinsic phonon effects on analog quantum simulators with ultracold trapped ions, Phys. Rev. A 86, 032329 (2012).
  * Wang  _et al._ [2013] C.-C. J. Wang, A. C. Keith, and J. Freericks, Phonon-mediated quantum spin simulator employing a planar ionic crystal in a penning trap, Phys. Rev. A 87, 013422 (2013).
  * Dylewsky  _et al._ [2016] D. Dylewsky, J. K. Freericks, M. L. Wall, A. M. Rey, and M. Foss-Feig, Nonperturbative calculation of phonon effects on spin squeezing, Phys. Rev. A 93, 013415 (2016).
  * Wall  _et al._ [2017] M. L. Wall, A. Safavi-Naini, and A. M. Rey, Boson-mediated quantum spin simulators in transverse fields: XY model and spin-boson entanglement, Phys. Rev. A 95, 013602 (2017).
  * Wang  _et al._ [2020] Y. Wang, M. Qiao, Z. Cai, K. Zhang, N. Jin, P. Wang, W. Chen, C. Luan, B. Du, H. Wang, _et al._ , Coherently Manipulated 2D Ion Crystal in a Monolithic Paul Trap, Adv. Quantum Technol. 3, 2000068 (2020).
  * Qiao  _et al._ [2022] M. Qiao, Z. Cai, Y. Wang, B. Du, N. Jin, W. Chen, P. Wang, C. Luan, E. Gao, X. Sun, _et al._ , Observing frustrated quantum magnetism in two-dimensional ion crystals, arXiv preprint arXiv:2204.07283 (2022).
  * Olmschenk  _et al._ [2007] S. Olmschenk, K. C. Younge, D. L. Moehring, D. N. Matsukevich, P. Maunz, and C. Monroe, Manipulation and detection of a trapped $yb^{+}$ hyperfine qubit, Phys. Rev. A 76, 052314 (2007).
  * Chen  _et al._ [2023] W. Chen, Y. Lu, S. Zhang, K. Zhang, G. Huang, M. Qiao, X. Su, J. Zhang, J.-N. Zhang, L. Banchi, _et al._ , Scalable and programmable phononic network with trapped ions, Nat. Phys. 19, 877 (2023).
  * Haljan  _et al._ [2005] P. C. Haljan, K.-A. Brickman, L. Deslauriers, P. J. Lee, and C. Monroe, Spin-dependent forces on trapped ions for phase-stable quantum gates and entangled states of spin and motion, Phys. Rev. Lett. 94, 153602 (2005).
  * Kim  _et al._ [2010] K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, Quantum simulation of frustrated ising spins with trapped ions, Nature 465, 590 (2010).
  * Islam  _et al._ [2011] R. Islam, E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. Joseph Wang, J. Freericks, _et al._ , Onset of a quantum phase transition with a trapped ion quantum simulator, Nat. Commun. 2, 377 (2011).
  * Grzesiak  _et al._ [2020] N. Grzesiak, R. Blümel, K. Wright, K. M. Beck, N. C. Pisenti, M. Li, V. Chaplin, J. M. Amini, S. Debnath, J.-S. Chen, _et al._ , Efficient arbitrary simultaneously entangling gates on a trapped-ion quantum computer, Nat. Commun. 11, 2963 (2020).
  * Hegade  _et al._ [2021] N. N. Hegade, K. Paul, Y. Ding, M. Sanz, F. Albarrán-Arriagada, E. Solano, and X. Chen, Shortcuts to adiabaticity in digitized adiabatic quantum computing, Phys. Rev. Appl. 15, 024038 (2021).
  * Kovalsky  _et al._ [2023] L. K. Kovalsky, F. A. Calderon-Vargas, M. D. Grace, A. B. Magann, J. B. Larsen, A. D. Baczewski, and M. Sarovar, Self-healing of trotter error in digital adiabatic state preparation, Phys. Rev. Lett. 131, 060602 (2023).
  * Wu  _et al._ [2023] Q. Wu, Y. Shi, and J. Zhang, Qubits on programmable geometries with a trapped-ion quantum processor, arXiv preprint arXiv:2308.10179 (2023).
  * Shapira  _et al._ [2023b] Y. Shapira, J. Markov, N. Akerman, A. Stern, and R. Ozeri, Programmable quantum simulations on a trapped-ions quantum computer with a global drive, arXiv preprint arXiv:2308.16036 (2023b).

##  IV Methods

###  IV.1 Experimental setup and detection error correction

In Extended Data Fig. 1, we provide a detailed view of our experimental setup,
highlighting both the individual laser addressing system for independent ion
control and the imaging system for spin-state readout. Our experimental
apparatus employs a pulsed laser (COHERENT Mira HP, not shown in the figure)
for stimulated Raman transitions with a center wavelength of 377 nm. This
laser is detuned by approximately 10 THz from the transition frequency between
the ${}^{2}S_{1/2}$ and ${}^{2}P_{1/2}$ levels. The pulsed laser has a pulse
width of about 4 ps and operates at a repetition rate of around 76 MHz.

To drive the stimulated Raman transition, the laser beam is divided into two
separate beams. These beams propagate in equal optical paths and overlap at
the ion-chain position with orthogonal wave vectors. Specifically, one of the
laser beams is divided into a beam array using a diffractive optical element
(from Holo.Or). This beam array passes through a multi-channel Acousto-Optic
Modulator (AOM) (Gooch & Housego AOMC 220-5) and is then tightly focused onto
the ion chain. Both ion spacing and beam spacing are adjusted to be
approximately 5 µm. It’s worth noting that our system has the capability to
independently control a maximum of five ions, although, for our experimental
demonstration, we employ up to four ions.

Extended Data Fig. 1:  Experimental setup. The main components for our
programmable quantum simulator are illustrated. The inset figure shows the
energy levels of the ${}^{171}\mathrm{Yb}^{+}$ ion.

During the detection process, fluorescence photons that are emitted by the
whole ion chain are collected by an objective lens with a numerical aperture
(N.A.) of 0.6 (Photon Gear 15470-S) and then counted using a linear array PMT
module (Hamamatsu H11460). To achieve spatial-resolved measurements, the
nearest ions are mapped to the next-nearest channels of the PMT. The total
detection duration lasts approximately 250 µs. However, our detection process
faces certain challenges. The low quantum efficiency of the PMT, approximately
20% at 370 nm, along with significant crosstalk between nearby ions, limits
our detection fidelity for multiple ions. To characterize the detection error
for three-ion and four-ion cases, we prepare serials of input quantum states
and then measure the output state distribution. The experimentally obtained
error matrices are provided in Extended Data Fig. 2.

Extended Data Fig. 2:  Detection error matrices. The average detection
fidelity for the three-ion system is estimated to be 90.4%, while for the
four-ion system, it is estimated to be 86.2%. These value lead to single-ion
dectection fidelities of around $96.7\%$ ($={\sqrt[3]{90.4\%}}$) for the
three-ion system and $96.3\%$ ($={\sqrt[4]{86.2\%}}$) for the four-ion one.

Both error matrices reveal a single-ion detection fidelity of approximately
96.5%. The infidelity includes the errors stemming from state leakage due to
off-resonant coupling during detection (around 2%) and also the crosstalks
between nearby ions (around 1.5%). With these error matrices, we can manually
correct the raw detection results, denoted by
$\mathbf{P}^{\mathrm{meas}}=\\{p^{\mathrm{meas}}_{0...0},...,p^{\mathrm{meas}}_{1...1}\\}$,
to remove these errors. In practice, we can estimate the real state population
using the reverse of the error matrix $\mathbf{M}$, which allows us to obtain
$\mathbf{P}^{\mathrm{real}}=\mathbf{M}^{-1}.\mathbf{P}^{\mathrm{meas}}$, where
$\mathbf{P}^{\mathrm{real}}$ is the estimated real state population. To avoid
non-physical outcomes from direct inversion, we apply the maximum-likelihood
method. This method estimates the real state population by minimizing the
2-norm of
$\|\mathbf{P}^{\mathrm{meas}}-\mathbf{M}.\mathbf{P}^{\mathrm{real}}\|_{2}$.

###  IV.2 Optimization method for pulse modulation

The optimization method employed here to obtain the modulation patterns is
similar to the approach utilized in the prior work[27]. When integrating the
SDF Hamiltonian in Equation (1) over time, we obtain an evolution operator
given by:

$\displaystyle
U(t)=\mathrm{exp}\left[\sum_{j,m}(\alpha_{j,m}(t)a_{m}^{\dagger}-\alpha_{j,m}^{\star}(t)a_{m})\sigma_{\mathrm{x}}^{(j)}\right.$
(3)
$\displaystyle\left.-\mathrm{i}\sum_{j<j^{\prime}}\theta_{j,j^{\prime}}(t)\sigma_{\mathrm{x}}^{j}\sigma_{\mathrm{x}}^{j^{\prime}}\right].$

Here, $\alpha_{j,m}$ represents the trajectories of the $m$-th motional modes
in the phase space contributed by the $j$-th ion, while $\beta_{j,j^{\prime}}$
indicates the effective two-body interaction between the $j$-th and
$j^{\prime}$-th ions. The analytical forms of $\alpha_{j,m}$ and
$\theta_{j,j^{\prime}}$ are given as follows:

$\alpha_{j,m}(\tau)=-\mathrm{i}\eta_{j,m}\int_{0}^{\tau}\dfrac{\Omega_{j}(t)e^{-\mathrm{i}\phi_{j}(t)}}{2}e^{\mathrm{i}(\nu_{m}-\mu)t}dt,$
(4) $\displaystyle\theta_{j,j^{\prime}}(\tau)$ $\displaystyle=$
$\displaystyle-\sum_{m}\int_{0}^{\tau}dt_{2}\int_{0}^{t_{2}}dt_{1}\dfrac{\eta_{j,m}\eta_{j^{\prime},m}\Omega_{j}(t_{2})\Omega_{j^{\prime}}(t_{1})}{2}$
$\displaystyle\sin\left[(\nu_{m}-\mu)(t_{2}-t_{1})-(\phi_{j}(t_{2})-\phi_{j^{\prime}}(t_{1}))\right].$

The target for optimization is minimizing the cost function
$\sum_{j,m}|\alpha_{j,m}(\tau_{g})|$ for a given $\tau_{g}$, while the set of
$\\{\theta_{j,j^{\prime}}(\tau_{g})=\Theta_{j,j^{\prime}}\\}$ yields
constraints for optimization. In practice, we discretize the phase $\phi_{j}$
and amplitude $\Omega_{j}$ into $K$ segments, each with a segment duration of
$\tau_{s}=\tau/K$. In contrast to the modulation patterns shown in the
previous work [27], in this work, we opt for a larger number of segments
exceeding a hundred. For instance, in the case of the three-ion system
presented in Fig. 3a, we choose a value of $K$ equal to 100. We now provide
the modulation patterns for the four-spin Ising models achieved in our
experiments, as shown in Extended Data Fig. 3.

Extended Data Fig. 3:  Modulation patterns for four-spin Ising models. In the
experiments, we set the value of $\mu$ to be $2\pi\times 2.043$ MHz for all
the patterns. Given the symmetry of the Ising models, the phase modulation
patterns are identical for the center two ions and the outer two ions. It’s
worth noting that any seeming discontinuities in the phase modulation pattern
are the results of the modulo $2\pi$ effect. The number of segments in all
four-ion cases is 100.

Data availability: The data that support the findings of this study are
available from the authors upon request.

Acknowledgements: This work was supported by the Innovation Program for
Quantum Science and Technology under Grants No. 2021ZD0301602, and the
National Natural Science Foundation of China under Grants No.92065205,
No.11974200, and and No.62335013. Y. L. also acknowledges support from the
National Science Foundation of China under Grants No. 12004165, the Guangdong
Basic and Applied Basic Research Foundation under Grant No. 2022B1515120021,
and the Shenzhen Science and Technology Program under Grants No.
RCYX20221008092901006.

Author contributions: Y.L., W.C., S.Z. and K.Z. developed the experimental
system conceived the project. Y.L. and J.-N.Z. optimized experimental schemes.
Y.L. took and analysed the experimental data. K.K. supervised the project. All
authors contributed to write the manuscript.

Competing interests: The authors declare that there are no competing
interests.

Author information: Correspondence and requests for materials should be
addressed to Y.L. and K.K.


# Constraining the UV with the electroweak effective action

Iberê Kuntza, Amanda Malagia

aDepartamento de Física, Universidade Federal do Paraná
PO Box 19044, Curitiba – PR, 81531-980, Brazil

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

By considering an arbitrary bare action describing BSM physics, we use the
Barvinsky–Vilkovisky resummation to obtain the most general non-local
electroweak effective action at second order in the field strength. We also
include the contribution of the functional measure to the effective action,
which is found to modify the Higgs potential by shifting its vacuum value. The
resulting effective action provides one-loop corrections to the $W$ and $Z$
boson masses, ultimately leading to the most general expression for the $\rho$
parameter at one-loop. The functional measure plays a pivotal role as it
allows the parameterization of $\rho$ in inverse powers of the scale of new
physics, while containing non-local form factors. The comparison of $\rho$
with the latest data leads to several constraints on the UV particle spectra
of BSM models.

##  1 Introduction

The standard model has been very successful in describing elementary particles
and their interactions [1]. Since its inception about 40 years ago [2, 3, 4],
many experiments have been set to put it through its paces. Along these four
decades, its predictions have all matched the data of accelerators, making the
standard model one of the joys of modern physics.

On the other hand, as a fundamental description of Nature, one would expect
the standard model to account for the dark sector and quantum gravity, but it
fails at that. Although yet quite elusive, both dark matter [5] and quantum
gravity [6] require degrees of freedom not contained in our theory of particle
physics. This suggests that physics beyond the standard model (BSM) must exist
at higher energies. A more fundamental theory of particles, however, remains
unknown.

The search for BSM physics had until very recently clashed with the
experimental success of the standard model. Indeed, there was no significant
evidence to support the existence of some more fundamental theory. In April
2022, however, the Fermilab announced their measurement of the $W$ boson mass,
whose value deviated from the standard model prediction by $\sim 7\sigma$ [7].
The news had been well received by a large portion of the community as a first
evidence for new physics. However, such deviation also implied in a deviation
from the LHC data obtained thus far. Indeed, the latest LHC’s findings from
March 2023 for the $W$ boson mass agreed with its past measurements and hence
with the standard model [8]. At the moment, we thus face an experimental
tension between the measurements performed at the LHC and at the Tevatron [9].

While we await for the future resolution of this issue, minimal extensions of
the standard model can be explored in a model-independent fashion using
effective field theory. This topic has indeed received a lot of attention
lately [10, 11, 12, 13, 14], but a full account of one-loop corrections has so
far been missing. The effective action can indeed reveal non-local and non-
analytical corrections that might be used to account for new physics.
Moreover, the configuration-space geometry also imparts one-loop corrections
via the functional measure, which must be taken into account when one deals
with the Wilsonian effective theory.

The purpose of this paper is to advance in this direction by computing the
most general one-loop correction, at second order in the field strength, to
some electroweak’s oblique observables for a theory with arbitrary fields.
There are mainly two novel aspects in our calculation: i) the contribution of
the most general non-local form factor at second order in the field strength
and ii) the contribution of the functional measure which reflects a non-
trivial configuration space. These contributions, being non-analytic in the
momentum, correspond to the IR portion of some (still unknown) UV completed
theory, truly corresponding to its model-independent part. Our result allows
the parameterization of physics beyond the standard model in terms of a few
free parameters, namely the number of degrees of freedom, the effective
couplings and the scale of new physics. The last one shows up as a direct
consequence of the configuration-space geometry.

This paper is organized as follows. In Sec. 2, we review the Barvinsky-
Vilkovisky formalism for the resummed effective action at one-loop and we use
it to obtain the non-local contributions to the electroweak effective action.
In Sec. 3, we describe the role of the functional measure in the electroweak
sector, which modifies the Higgs potential. We combine the results of Secs. 2
and 3 to study the Higgs mechanism and calculate corrections to the boson
masses and to the $\rho$ parameter in Sec. 4. Here we also discuss the
different constraints imposed by the latest data. Our conclusions are drawn in
Sec. 5.

##  2 The electroweak effective action

The 1PI effective action $\Gamma[\phi]$ is a central object in the functional
formulation of quantum field theory. Albeit containing the same information as
the other generating functionals, such as the partition function $Z[J]$, it
provides a more transparent interpretation in terms of the mean field in place
of the rather abstract source $J$. This is particularly important for the
study of effective equations of motion, where $\Gamma[\phi]$ naturally
generalizes the classical action $S[\phi]$ in Hamilton’s principle.

In Euclidean signature, the effective action is a solution to [15]:

$\exp{-\Gamma[\varphi]}=\int\mathcal{D}\Phi\,\mathcal{M}[\Phi]\exp{-S[\Phi]+(\Phi^{i}-\varphi^{i})\frac{\delta\Gamma[\varphi]}{\delta\varphi^{i}}}\
,$ (2.1)

where $\mathcal{D}\Phi=\prod_{i}\mathrm{d}\Phi^{i}$. Despite its complicated
nonlinear nature, when combined with the background field method Eq. (2.1)
allows for the application of a wide range of approximate techniques, thus
providing a powerful and systhematical formalism to investigate quantum field
theories. Here $\varphi^{i}$ (resp. $\Phi^{i}$) denote arbitrary background
(resp. bare) fields, $S[\Phi]$ is the classical action and $\mathcal{M}[\Phi]$
denotes the functional measure, to be studied in detail in Sec. 3. We also
adopt the DeWitt index notation, where small letters $i=(I,x^{\mu})$ account
for both discrete $I$ and continuum $x^{\mu}$ indices. Repeated DeWitt indices
thus result in summations over $I$ and integrations over spacetime.

At the one-loop level, Eq. (2.1) yields [16, 17, 18]:

$\Gamma[\varphi]=S+\Gamma^{(1)}+\Gamma^{(1)}_{\mathcal{M}}\,,$ (2.2)

where

$\displaystyle\Gamma^{(1)}$
$\displaystyle=\frac{1}{2}\Tr\log\mathcal{H}_{ij}\,,$ (2.3)
$\displaystyle\Gamma^{(1)}_{\mathcal{M}}$
$\displaystyle=-\log\mathcal{M}[\varphi]\,,$ (2.4)

denote the standard correction due to the Hessian

$\mathcal{H}_{ij}=\frac{\delta^{2}S[\varphi]}{\delta\varphi^{i}\delta\varphi^{j}}\,,$
(2.5)

and due to the functional measure, respectively. The trace in Eq. (2.3) is to
be understood in the functional sense:

$\Tr\mathcal{O}_{ij}=\int\mathrm{d}^{4}x\tr\mathcal{O}_{IJ}(x,x)\ ,$ (2.6)

which also includes the finite-dimensional trace $\tr$ over discrete indices.
We note that the functional-measure contribution is one-loop exact.

For canonical Lagrangians, the Hessian can generally be written in the minimal
form:

$\mathcal{H}_{ij}=\left[\delta_{IJ}(\Box-m^{2})+P_{IJ}\right]\delta(x,x^{\prime})\
.$ (2.7)

It depends solely on the underlying gauge connections (via covariant
derivatives) and interactions, thus being fully characterized by potential
terms, hereby denoted $P^{I}_{\ J}$, and the gauge field strength (fiber
bundle curvature):

$[D_{\mu},D_{\nu}]\varphi^{I}=\Omega^{I}_{\ J\mu\nu}\varphi^{J}\ ,$ (2.8)

where $\Omega^{I}_{\ J\mu\nu}=-igF^{a}_{\mu\nu}(T^{a})^{I}_{\ J}$.

The core of the aforementioned approximations relies on the heat kernel
$K(s)=e^{s\mathcal{H}}$, which enters the one-loop effective action through
the Schwinger proper-time method [16, 17, 18]:

$\Gamma^{(1)}=-\frac{1}{2}\int_{1/\Lambda}^{\infty}\frac{\mathrm{d}s}{s}\Tr
K(s),$ (2.9)

where $\Lambda$ is a UV cutoff. Because $K(s)$ solves the heat equation (with
appropriate boundary conditions), the one-loop effective action can be found
from its solutions.

In general, Eq. (2.9) splits up into a divergent and a finite part:

$\Gamma^{(1)}=\Gamma^{div}+\Gamma^{fin}\ .$ (2.10)

Asymptotic expansions on the proper time then allows one to focus on certain
regimes of interest. The standard perturbation theory regime of Feynman
diagrams, for example, is obtained from the Schwinger-DeWitt expansion, which
finds $K(s)$ (hence the effective action) in powers of inverse mass, namely

$\displaystyle\frac{\mathcal{R}}{m^{2}}\ll 1\
,\quad\frac{\nabla^{2}}{m^{2}}\ll 1\ ,$ (2.11)

where $\mathcal{R}=(P^{I}_{\ J},\Omega^{I}_{\ J\mu\nu})$ denotes generalized
curvatures 111Spacetime curvatures also show up, but we omit them since we are
only interested in Minkowski backgrounds., which includes potentials $P^{I}_{\
J}$ and field strengths $\Omega^{I}_{\ J\mu\nu}$. This expansion reveals both
the one-loop divergence structure $\Gamma^{div}$, which is renormalized by the
operators contained in the bare action $S[\varphi]$, and the physical finite
predictions $\Gamma^{fin}$. Because of Eq. (2.11), the Schwinger-DeWitt
expansion can also be seen as a double expansion in field strengths and
derivatives, thus resulting in a local finite part $\Gamma^{fin}$.

More general expansions also exist, which permit the study of quantum field
theory in other regimes. In covariant perturbation theory, for example, one
gives up the second constraint of Eq. (2.11). In this case, the heat kernel
can be expressed as a series in the generalized curvatures with infinitely
many derivatives [19, 20]:

$\displaystyle\Tr K(s)$ $\displaystyle=\frac{1}{(4\pi
s)^{2}}\int\mathrm{d}^{4}x\Big{\\{}\delta^{I}_{\ I}+s\,P^{I}_{\
I}+s^{2}\left[P_{IJ}\,f_{P}(-s\Box)\,P^{IJ}+\Omega_{IJ\mu\nu}\,f_{\Omega}(-s\Box)\,\Omega^{IJ\mu\nu}\right]\Big{\\}}$
(2.12) $\displaystyle+\mathcal{O}(\mathcal{R}^{3})\ ,$

where

$\displaystyle f_{P}(\xi)$ $\displaystyle=\frac{1}{2}f(\xi)\ ,$ (2.13)
$\displaystyle f_{\Omega}(\xi)$
$\displaystyle=-\frac{1}{2}\frac{f(\xi)-1}{\xi}\ ,$ (2.14) $\displaystyle
f(\xi)$ $\displaystyle=\int_{0}^{1}\mathrm{d}\alpha\,e^{-\alpha(1-\alpha)\xi}\
.$ (2.15)

From Eqs. (2.9) and (2.12), one finds 222We have dropped the zeroth and linear
order terms. The former is a constant that does not affect the equations of
motion and the only effect of the latter is to renormalize the global
coefficient in the potential.:

$\Gamma^{fin}=-\frac{1}{2(4\pi)^{2}}\int\mathrm{d}^{4}x\left[P_{IJ}\,\gamma_{P}\left(-\frac{\Box}{m^{2}}\right)\,P^{IJ}+\Omega_{IJ\mu\nu}\,\gamma_{\Omega}\left(-\frac{\Box}{m^{2}}\right)\,\Omega^{IJ\mu\nu}\right]+\mathcal{O}(\mathcal{R}^{3})\
,$ (2.16)

where the form factors are given by:

$\displaystyle\gamma_{P}(u)$
$\displaystyle=-\frac{1}{2}\int_{0}^{1}\mathrm{d}\alpha\log\left[1+u\alpha(1-\alpha)\right]\
,$ (2.17) $\displaystyle\gamma_{\Omega}(u)$
$\displaystyle=\frac{1}{12}-\frac{1}{2}\int_{0}^{1}\mathrm{d}\alpha\left[\frac{1}{u}+u\alpha(1-\alpha)\right]\log\left[1+u\alpha(1-\alpha)\right]\
.$ (2.18)

Eq. (2.16) is seen as a partial resummation of infinitely many terms of the
form $\mathcal{R}\Box^{n}\mathcal{R}$ contained in the finite part of
Schwinger-DeWitt method. It is important to stress that Eq. (2.16) takes this
same form, with the same form factors, for any theory described by the minimal
Hessian of Eq. (2.7). The differences are only reflected on their
coefficients, which result from the field space where the traces of Eq. (2.16)
are taken and thus depend on the number and types of fields being integrated
out in Eq. (2.1). Regardless of the bare action one starts with, Eq. (2.16)
provides corrections to the scalar and gauge sectors. It is this important
result that allows parameterization of new physics in terms of a handful of
free parameters. Bounds on these parameters would thus restrict the number and
types of fields existing in a fundamental theory.

For our purposes, let us assume the unknown bare action contains the
electroweak’s gauge sector:

$S\supset\int\mathrm{d}^{4}x\left[(D^{\mu}\varphi)^{\dagger}(D_{\mu}\varphi)+V(\varphi)+\frac{1}{4}W_{\mu\nu}^{a}W^{a\mu\nu}+\frac{1}{4}B_{\mu\nu}B^{\mu\nu}\right]\
,$ (2.19)

where

$V(\varphi)=\frac{\lambda^{2}}{2}(\varphi^{\dagger}\varphi-\eta^{2})^{2}\,,$
(2.20)

and $\lambda$ and $\eta$ denote the Higgs self-coupling and the vev,
respectively. The $SU(2)\times U(1)$ covariant derivative reads

$D_{\mu}=\partial_{\mu}-igW^{a}_{\mu}T^{a}-ig^{\prime}YB_{\mu}\ ,$ (2.21)

where $g$ and $g^{\prime}$ are, respectively, the corresponding coupling
constants of the gauge fields $W^{a}_{\mu}$ and $B_{\mu}$ to the Higgs sector,
$T^{a}$ is the SU(2) generator and $Y$ is the weak hypercharge. In this case,
the fiber bundle curvature accounts for both the weak and the electromagnetic
field strength:

$(\Omega_{\mu\nu})^{I}_{\ J}=-igW^{a}_{\mu\nu}(T^{a})^{I}_{\
J}-ig^{\prime}YB_{\mu\nu}\delta^{I}_{\ J}\ .$ (2.22)

For each field in the fundamental Lagrangian, there would be a correction of
the type (2.16). All these possible contributions add up to wit 333We note
that cross-terms in Eq. (2.23) vanish identically because they are
proportional to the trace of the $SU(2)$ generators, which is always zero.:

$\displaystyle\mathcal{L}^{(1)}_{EW}=\sum_{i=1}^{N}\Bigg{[}$ $\displaystyle
c_{\varphi}V^{\prime\prime}(\varphi)\gamma_{P}\left(-\frac{\Box}{m_{\varphi_{i}}^{2}}\right)V^{\prime\prime}(\varphi)+c_{W}W^{a}_{\mu\nu}\gamma_{\Omega}\left(-\frac{\Box}{m_{i}^{2}}\right)W^{a\mu\nu}$
(2.23)
$\displaystyle+c_{B_{i}}B_{\mu\nu}\gamma_{\Omega}\left(-\frac{\Box}{m_{i}^{2}}\right)B^{\mu\nu}\Bigg{]}\
,$

where $m_{\varphi_{i}},m_{i}$ are the masses of the particles, present in the
fundamental Lagrangian, running in the loop and

$\displaystyle c_{\varphi}$ $\displaystyle=-\frac{1}{2(4\pi)^{2}}\,,$ (2.24)
$\displaystyle c_{W}$ $\displaystyle=\frac{g^{2}}{4(4\pi)^{2}}\,,$ (2.25)
$\displaystyle c_{B_{i}}$ $\displaystyle=\frac{g^{\prime
2}}{2(4\pi)^{2}}Y_{i}^{2}n_{i}\ .$ (2.26)

Here $N$ denotes the number of particles in the fundamental theory, $Y_{i}$
and $n_{i}$ are the weak hypercharge and the dimension of the representation,
respectively, for each of these particles. The effects of BSM fields and
interactions present in the bare action are fully accounted by
$c_{\varphi},c_{W},c_{B_{i}},m_{\varphi_{i}}$ and $m_{i}$. Moreover, only
$c_{B_{i}}$ carries information on the fundamental particle’s spin,
$c_{\varphi}$ and $c_{W}$ are the same for any spin.

##  3 The functional measure in the electroweak sector

The functional measure $\mathcal{M}[\Phi]$ in Eq. (2.1) plays a fundamental
role in physics. Examples of its uses include the study of anomalies [21, 22,
23], the cancelation of ultralocal divergences [24, 25] and the formal
justification for the Fadeev-Poppov method [26, 27, 28, 29]. Although some
dispute exists regarding its form [30, 31, 32, 33, 34, 35, 36, 37, 38], from a
geometrical perspective $\varphi^{i}$ are mere configuration-space
coordinates, thus a natural definition for the measure would be [39, 15, 31,
40]:

$\mathcal{M}[\varphi]=\sqrt{\text{Det}\,G_{ij}}\ ,$ (3.27)

where $\text{Det}\,G_{ij}$ denotes the functional determinant of the
configuration-space metric $G_{ij}$. This definition makes sense as the
functional generalization of integrations on manifolds. It is also appealing,
for it upholds, together with a connection compatible to $G_{ij}$, invariance
under field redefinitions and gauge transformations [15, 41]. Indeed, Eq.
(3.27) transforms a (functional) scalar density which cancels the Jacobian
from $\mathcal{D}\varphi$. The phenomenological consequences of the functional
measure (3.27) has just started to be investigated [40, 42, 43, 44].

One should note that there is no canonical choice for $G_{ij}$. Different
configuration-space metrics correpond to unequivalent quantization schemes.
Thus to fully specify the quantum theory, one must provide the classical
action furnished with some $G_{ij}$. Fortunately, the same principles
generally used to write down an action can be translated into the
determination of $G_{ij}$. One requires ultralocality, i.e., proportionality
with the Dirac delta (but not its derivatives):

$G_{ij}=G_{IJ}(\varphi)\delta(x,x^{\prime})\,,$ (3.28)

and invariance of $G_{IJ}(\varphi)$ under the same symmetries present in the
Lagrangian. Notice that while $G_{ij}$ is a metric functional in an infinite-
dimensional space, $G_{IJ}(\varphi)$ is a metric function (not functional) of
the fields (but not their derivatives) in a finite-dimensional space. The
Dirac delta in Eq. (3.28) enforces the same $G_{IJ}$ across all spacetime
points and prevent non-local effects.

From Eqs. (2.4), (2.6) and (3.28), one can see that ultralocality is
responsible for extreme divergences $\delta^{(4)}(0)$ in the quantum action.
The effective field theory lore is to declare our ignorance of the UV by
imposing a physical cutoff $\Lambda$, which makes the theory finite and sets
the scale of new physics. We can implement such a cutoff via a Gaussian
regularization:

$\delta^{(4)}(x)=\frac{\Lambda^{4}}{(2\pi)^{2}}e^{\frac{-x^{2}\Lambda^{2}}{2}}\
,$ (3.29)

hence $\delta^{(4)}(0)=\frac{\Lambda^{4}}{(2\pi)^{2}}$. From Eqs. (2.4) and
(3.27)–(3.29), we then find:

$\Gamma^{(1)}_{\mathcal{M}}=-\frac{\Lambda^{4}}{8\pi^{2}}\int\mathrm{d}^{4}x\log\det
G_{IJ}\ ,$ (3.30)

Note that the Dirac delta divergence is polynomial. The correction (3.30) is
thus expected to be UV sensitive.

For the electroweak sector, there is no gauge-invariant combination of the
gauge fields without derivatives, hence $G_{IJ}$ must depend only on the Higgs
field. At second order, the most general metric determinant reads:

$\det
G_{IJ}=A+B\frac{\varphi^{\dagger}\varphi}{\Lambda^{2}}+\mathcal{O}(\varphi^{3})\
,$ (3.31)

where $A$ and $B$ are dimensionless free parameters. Eq. (3.30) then becomes:

$\Gamma^{(1)}_{\mathcal{M}}=-\frac{\Lambda^{4}}{8\pi^{2}}\int\mathrm{d}^{4}x\log(A+B\frac{\varphi^{\dagger}\varphi}{\Lambda^{2}})\
,$ (3.32)

which provides another correction to the Higgs effective potential.

##  4 Higgs mechanism and oblique corrections

The effective action above, including the definition of the functional
measure, were obtained in the Euclidean space. To obtain physical results, we
must return to Minkowski signature. From now on, everything shall be written
in Minkowski space with signature $(+---)$.

The first term in Eq. (2.23) does not contribute to the Higgs potential since
it contain derivatives. The total one-loop Higgs effective potential thus
receives corrections only from the functional measure 444Corrections of the
Coleman-Weinberg type takes place when one gives up the first (rather than the
second) condition in Eq. (2.11), which results in local terms with infinitely
many generalized curvatures. We shall here consider only covariant
perturbation theory where such corrections do not occur.:

$V_{eff}=\frac{\lambda^{2}}{2}(\varphi^{\dagger}\varphi-\eta^{2})^{2}-\frac{\Lambda^{4}}{8\pi^{2}}\log(1+C\frac{\varphi^{\dagger}\varphi}{\Lambda^{2}})\
,$ (4.33)

where we defined the new constant $C=B/A$. This potential determines the
effective Higgs mass:

$M_{H}=\sqrt{3\lambda^{2}\varphi_{0}^{2}-\lambda^{2}\eta^{2}-\frac{\Lambda^{4}C}{8\pi^{2}(1+C\varphi_{0}^{2})}+\frac{\Lambda^{4}C^{2}\varphi_{0}^{2}}{4\pi^{2}(1+C\varphi_{0}^{2})^{2}}}\
.$ (4.34)

The masses of the $W$ and $Z$ bosons, on the other hand, are obtained from the
Higgs mechanism, as usual. The minimum of Eq. (4.33) yields the corrected
vacuum expectation value 555There are two roots when solving for the minimum.
However, only the result in (4.35) returns to the classical value in the limit
$\hbar\to 0$.:

$\varphi_{0}^{2}\equiv\varphi^{\dagger}\varphi=\eta^{2}+\frac{\Lambda^{2}C}{8\pi^{2}\lambda^{2}\left(1+\frac{C\eta^{2}}{\Lambda^{2}}\right)}+\mathcal{O}(\hbar^{2})\
.$ (4.35)

We note that the functional measure shifts the Higgs vacuum expectation value
$\eta\to\varphi_{0}$.

After symmetry breaking, Eqs. (2.19) and (2.23) yield the effective Lagrangian
for the electroweak interaction:

$\displaystyle\mathcal{L}_{1PI}$
$\displaystyle=W^{+}_{\mu}\,\left[\Pi^{\mu\nu}_{W}(\Box)+M_{W,0}^{2}\eta^{\mu\nu}\right]W^{-}_{\nu}+\frac{1}{2}W^{3}_{\mu}\,\left[\Pi^{\mu\nu}_{W}(\Box)+M_{Z,0}^{2}\cos^{2}\theta_{W}\eta^{\mu\nu}\right]W^{3}_{\nu}$
(4.36)
$\displaystyle+\frac{1}{2}B_{\mu}\,\left[\Pi^{\mu\nu}_{B}(\Box)+M_{Z,0}^{2}\sin^{2}\theta_{W}\eta^{\mu\nu}\right]B_{\nu}-M_{Z,0}^{2}\cos\theta_{W}\sin\theta_{W}\,W_{\mu}^{3}B^{\mu}\
,$

where we made the usual definitions:

$\displaystyle\bar{g}=\sqrt{g^{2}+g^{\prime 2}}\ ,$ (4.37)
$\displaystyle\cos\theta_{W}=\frac{g}{\bar{g}}\ ,$ (4.38)
$\displaystyle\sin\theta_{W}=\frac{g^{\prime}}{\bar{g}}\ ,$ (4.39)
$\displaystyle M_{W,0}=\frac{g}{\sqrt{2}}\varphi_{0}\ ,$ (4.40) $\displaystyle
M_{Z,0}=\frac{\bar{g}}{\sqrt{2}}\varphi_{0}\ ,$ (4.41)

and 666Note the different sign inside the form factors in Minkowski space in
contrast to the Euclidean form factors of Sec. 2.

$\Pi^{\mu\nu}_{K}(\Box)=\eta^{\mu\nu}\left[\left(1+4\sum_{i}c_{K_{i}}\gamma_{\Omega}\left(\frac{\Box}{m_{i}^{2}}\right)\right)\Box\right]-\left[1+4\sum_{i}c_{K_{i}}\gamma_{\Omega}\left(\frac{\Box}{m_{i}^{2}}\right)\right]\partial^{\mu}\partial^{\nu}\
,$ (4.42)

for $K=W,B$. The Lagrangian $\mathcal{L}_{1PI}$ corresponds to the gauge
sector of the effective action $\Gamma$, whereas the Higgs sector has been
omitted for simplicity.

To identify the boson masses, one must diagonalize the quadratic terms.
Because of the different coefficients $c_{W}\neq c_{B_{i}}$ of the non-local
terms in Eq. (4.42), the corresponding form factors of each interaction differ
from each other. As a result, a rotation by $\theta_{W}$ can no longer
diagonalize the quadratic terms. Such diagonalization requires a non-local
linear transformation, which at one-loop level reads 777This can be easily
obtained by performing a general linear transformation and setting the
coefficients of the cross-terms so obtained to zero. We also imposed that
$A_{\mu}$ be massless and the $Z$ boson mass be given by its usual expression
(4.41).:

$\displaystyle\begin{pmatrix}W^{3}_{\mu}\\\
B_{\mu}\end{pmatrix}=\begin{pmatrix}\left(1-g(\Box)\sin^{2}\theta_{W}\right)\cos\theta_{W}&\sin\theta_{W}\\\
-\left(1+g(\Box)\cos^{2}\theta_{W}\right)\sin\theta_{W}&\cos\theta_{W}\end{pmatrix}\begin{pmatrix}Z_{\mu}\\\
A_{\mu}\end{pmatrix}\,,$ (4.43)

where

$g(\Box)=4\sum_{i}(c_{W}-c_{B_{i}})\,\gamma_{\Omega}\left(\frac{\Box}{m_{i}^{2}}\right)\
.$ (4.44)

Note that $g(\Box)\to 1$ for $c_{B_{i}}=c_{W}$ or at the classical limit
$\hbar\to 0$ since the form factor is of one-loop order, thus recovering
Weinberg’s rotation by $\theta_{W}$.

Under the transformation (4.43), the Lagrangian transforms to:

$\mathcal{L}_{1PI}=W^{+}_{\mu}\left[\Pi^{\mu\nu}_{W}(\Box)+M_{W,0}^{2}\eta^{\mu\nu}\right]W^{-}_{\nu}+\frac{1}{2}Z_{\mu}\left[\Pi^{\mu\nu}_{Z}(\Box)+M_{Z,0}^{2}\eta^{\mu\nu}\right]Z_{\nu}+\frac{1}{2}A_{\mu}\Pi^{\mu\nu}_{A}(\Box)A_{\nu}\
,$ (4.45)

with

$\displaystyle\Pi^{\mu\nu}_{Z}(\Box)$
$\displaystyle=\cos^{2}\theta_{W}\left(1-2\sin^{2}\theta_{W}g(\Box)\right)\Pi^{\mu\nu}_{W}(\Box)+\sin^{2}\theta_{W}\left(1+2\cos^{2}\theta_{W}g(\Box)\right)\Pi^{\mu\nu}_{B}(\Box)\
,$ (4.46) $\displaystyle\Pi^{\mu\nu}_{A}(\Box)$
$\displaystyle=\sin^{2}\theta_{W}\Pi^{\mu\nu}_{W}(\Box)+\cos^{2}\theta_{W}\Pi^{\mu\nu}_{B}(\Box)\
.$ (4.47)

From Eq. (4.45), we can identify the mass poles in momentum space:

$\displaystyle\left[1+4\,c_{W}\sum_{i}\gamma_{\Omega}\left(\frac{-k^{2}}{m_{i}^{2}}\right)\right]k^{2}-M^{2}_{W,0}=0\
,$ (4.48)
$\displaystyle\left[1+4\sum_{i}\gamma_{\Omega}\left(\frac{-k^{2}}{m_{i}^{2}}\right)\left(c_{W}\cos^{2}\theta_{W}+c_{B_{i}}\sin^{2}\theta_{W}\right)\right]k^{2}-M^{2}_{Z,0}=0\
,$ (4.49)

from which one can read off the physical masses $M_{W}$ and $M_{Z}$, defined
as the poles of the corresponding propagators at $k=M_{W},M_{Z}$:

$\displaystyle M_{W}^{2}$
$\displaystyle=M_{W,0}^{2}-4\,c_{W}M_{W}^{2}\sum_{i}\gamma_{\Omega}\left(\frac{-M_{W}^{2}}{m_{i}^{2}}\right)\,,$
(4.50) $\displaystyle M_{Z}^{2}$
$\displaystyle=M_{Z,0}^{2}-4\,M_{Z}^{2}\sum_{i}\gamma_{\Omega}\left(\frac{-M_{Z}^{2}}{m_{i}^{2}}\right)\left(c_{W}\cos^{2}\theta_{W}+c_{B_{i}}\sin^{2}\theta_{W}\right)\
.$ (4.51)

Since the form factor $\gamma_{\Omega}(u)$ is already of one-loop order, to
this order we can take $M_{i}\approx M_{i,tree}$ on the RHS, hence:

$\displaystyle M_{W}^{2}$
$\displaystyle=M_{W,tree}^{2}\left[1-4\,c_{W}\sum_{i}\gamma_{\Omega}\left(\frac{-M_{W,tree}^{2}}{m_{i}^{2}}\right)+\frac{\Lambda^{2}C}{8\pi^{2}\lambda^{2}\eta^{2}}\right]\
,$ (4.52) $\displaystyle M_{Z}^{2}$
$\displaystyle=M_{Z,tree}^{2}\left[1-4\sum_{i}\gamma_{\Omega}\left(\frac{-M_{Z,tree}^{2}}{m_{i}^{2}}\right)\left(c_{W}\cos^{2}\theta_{W}+c_{B_{i}}\sin^{2}\theta_{W}\right)+\frac{\Lambda^{2}C}{8\pi^{2}\lambda^{2}\eta^{2}}\right]\
,$ (4.53)

where the tree-level masses read:

$\displaystyle M_{W,tree}$ $\displaystyle=\frac{g}{\sqrt{2}}\eta\ ,$ (4.54)
$\displaystyle M_{Z,tree}$ $\displaystyle=\frac{\bar{g}}{\sqrt{2}}\eta\ .$
(4.55)

We note that the usual relation between the boson masses and the Weinberg
angle $\theta_{W}$ no longer holds in general:

$\rho\equiv\frac{M_{W}^{2}}{M_{Z}^{2}\cos^{2}\theta_{W}}\neq 1\ ,$ (4.56)

where the $\rho$-parameter is given by:

$\displaystyle\rho=\frac{1-4\,c_{W}\sum_{i}\gamma_{\Omega}\left(-\frac{M_{W,tree}^{2}}{m_{i}^{2}}\right)+\frac{\Lambda^{2}C}{8\pi^{2}\lambda^{2}\eta^{2}}}{1-4\sum_{i}\gamma_{\Omega}\left(\frac{-M_{Z,tree}^{2}}{m_{i}^{2}}\right)\left(c_{W}\cos^{2}\theta_{W}+c_{B_{i}}\sin^{2}\theta_{W}\right)+\frac{\Lambda^{2}C}{8\pi^{2}\lambda^{2}\eta^{2}}}\
.$ (4.57)

This alleviates the standard model tight constraint and allows for the
accommodation of possible experimental anomalies reflecting BSM physics.

The form factor in Eq. (4.57), given by the integral form in Eq. (2.18),
contains all one-loop effects at second order in the field strength with
infinitely many derivatives. Such generality makes it difficult to obtain
useful bounds from the experimental value for $\rho$, namely [5]:

$\rho=1.00038\ .$ (4.58)

Further assumptions about the fundamental degrees of freedom, particularly
regarding their masses, allow us to massage the form factor into simpler
expressions. We shall now analyse a few different regimes.

###  4.1 Heavy fields

For $m_{i}\gg M_{i,tree}$, the form factor (2.18) can be expanded as [19, 20,
45]:

$\gamma_{\Omega}(u)=-\frac{u}{120}+\frac{u^{2}}{1680}+\mathcal{O}(u^{3})\ .$
(4.59)

This is the usual regime of effective field theory, where heavy fields
decouple, affecting the IR only through local operators [46]. In this case, to
leading order, the masses (4.52)-(4.53) read:

$\displaystyle M_{W}^{2}$
$\displaystyle=M_{W,tree}^{2}\left[1+\frac{c_{W}}{30}\sum_{i}\left(\frac{M_{W,tree}}{m_{i}}\right)^{2}+\frac{\Lambda^{2}C}{8\pi^{2}\lambda^{2}\eta^{2}}\right]\
,$ (4.60) $\displaystyle M_{Z}^{2}$
$\displaystyle=M_{Z,tree}^{2}\left[1+\frac{1}{30}\sum_{i}\left(c_{W}\cos^{2}\theta_{W}+c_{B_{i}}\sin^{2}\theta_{W}\right)\left(\frac{M_{Z,tree}}{m_{i}}\right)^{2}+\frac{\Lambda^{2}C}{8\pi^{2}\lambda^{2}\eta^{2}}\right]\
,$ (4.61)

and Eq. (4.57) becomes:

$\rho=1-\frac{\sin^{2}\theta_{W}M_{Z,tree}^{2}}{30\left(1+\frac{\Lambda^{2}C}{8\pi^{2}\lambda^{2}\eta^{2}}\right)}\sum_{i}\frac{c_{B_{i}}}{m_{i}^{2}}+\mathcal{O}\left(m_{i}^{-4}\right)\
.$ (4.62)

We note that $c_{W}$ drops out, thus the above expression sets a bound on the
combination $c_{B_{i}}/m_{i}$ only. The effects of the functional measure are
contained in the dimensionless parameter $C$. The limit $C\to 0$ corresponds
to a flat configuration space, in which case the functional measure is
trivial. In this case, the only effect is due to the one-loop form factor from
the Barvinsky-Vilkovisky resummation. From Eq. (4.58) and the experimental
values for the other parameters [5]:

$\displaystyle g$ $\displaystyle=0.63,$ (4.63) $\displaystyle\bar{g}$
$\displaystyle=0.72,$ (4.64) $\displaystyle\eta$
$\displaystyle=173.95\,\text{GeV},$ (4.65) $\displaystyle\lambda$
$\displaystyle=0.51,$ (4.66)

we find

$\sum_{i}\frac{c_{B_{i}}}{m_{i}^{2}}=-6.32\times 10^{-6}\,\text{GeV}^{-2}\ .$
(4.67)

On the other hand, because $C$ is dimensionless, one would actually expect
$C\sim 1$, hence the strength of the functional measure contribution is
completely controlled by the physical cutoff $\Lambda$. Assuming $C$ of order
one and $\Lambda\sim 1$ TeV, we rather obtain:

$\frac{1}{C}\sum_{i}\frac{c_{B_{i}}}{m_{i}^{2}}=-1.65\times
10^{-5}\,\text{GeV}^{-2}\ .$ (4.68)

Eqs. (4.67) and (4.68) offer important constraints on UV physics. Should an
independent measurement be performed to determine the coefficients
$c_{B_{i}}$, one could predict the masses of a fundamental model.

###  4.2 Light fields

In the opposite limit, where the fundamental fields are nearly massless
$m_{i}\ll M_{i,tree}$, the form factor becomes [19, 20, 45]:

$\gamma_{\Omega}(u)=\frac{2}{9}-\frac{1}{12}\log(u)+\mathcal{O}(u^{-1})\ .$
(4.69)

In particular, in the strict massless limit, all higher order terms in (4.69)
vanish identically. In this case, and assuming the size of the form factor
becomes larger than the functional measure contribution, Eqs. (4.52), (4.53)
and (4.57) take the form 888We recall that only the real part of the mass is
interpreted as the physical mass, whereas its imaginary part (due to the
negative argument in the logarithmic form factor) measures the particle decay.
We have thus taken the absolute value of the argument of the logarithm.:

$\displaystyle M_{W}^{2}$
$\displaystyle=M_{W,tree}^{2}\left[1-\frac{8}{9}Nc_{W}+\frac{Nc_{W}}{3}\log\left(\frac{M_{W,tree}^{2}}{\mu^{2}}\right)\right]\
,$ (4.70) $\displaystyle M_{Z}^{2}$
$\displaystyle=M_{Z,tree}^{2}\left[1-\left(\frac{8}{9}-\frac{1}{3}\log\left(\frac{M_{Z,tree}^{2}}{\mu^{2}}\right)\right)\sum_{i}(c_{W}\cos^{2}\theta_{W}+c_{B_{i}}\sin^{2}\theta_{W})\right]\,,$
(4.71) $\displaystyle\rho$
$\displaystyle=\frac{1}{9}-\frac{8}{9}Nc_{W}-\frac{1}{3}\left[\sum_{i}c_{B_{i}}\frac{\sin^{4}\theta_{W}}{\cos^{2}\theta_{W}}-Nc_{W}\cos^{2}\theta_{W}\right]\log\left(\frac{M_{W,tree}^{2}}{\mu^{2}}\right)+\mathcal{O}\left((g^{\prime}/g)^{4}\right)\
,$ (4.72)

where in the last equation we used

$\log\left(\frac{M_{Z,tree}^{2}}{\mu^{2}}\right)=\log\left(\frac{M_{W,tree}^{2}}{\mu^{2}}\right)\frac{g^{\prime
2}}{g^{2}}+\mathcal{O}\left((g^{\prime}/g)^{4}\right)\ ,$ (4.73)

which is justifiable since $g^{\prime}/g\sim 10^{-1}$. Here $\mu$ is an
artificial scale used to make the logarithm dimensionless. In the strict
massless limit, the divergence structure correlates with the logarithmic form
factor in such a way that the running of the couplings $c_{W}(\mu)$ and
$c_{B_{i}}(\mu)$ cancels out the $\mu$ dependence in the logarithm. This is
the typical situation of quantum field theory, in which large logarithms are
controlled by the renormalization group. On the other hand, for massive
particles, however light they are, higher orders in Eq. (4.69) also contribute
and no such correlation exists and the situation is more difficult.

In the strict massless limit, the experimental values (4.58) and (4.63)–(4.66)
give:

$\sum_{i}c_{B_{i}}(M_{Z,tree})=1.46\times 10^{2}+9.85\times 10^{-2}N\ ,$
(4.74)

where we evaluated the coefficients at the $Z$ boson mass because that is the
scale at which the experimental values are obtained. By measuring $c_{B_{i}}$,
one is thus able to determine the number of fields $N$ in the fundamental
theory. In particular, assuming the coefficients $c_{B_{i}}$ to be all of the
same order $c_{B_{i}}\sim c_{B}$ results in:

$c_{B}(M_{Z,tree})=\frac{1.46\times 10^{2}}{N}+9.85\times 10^{-2}\ .$ (4.75)

In the large $N$ approximation, this leads to $c_{B}(M_{Z,tree})=9.85\times
10^{-2}$.

###  4.3 Electroweak masses

The borderline between the regimes outlined above lies at $m_{i}\sim
M_{i,tree}$, in which case (see Eq. (2.18)):

$\gamma_{\Omega}(1)=-7.8\times 10^{-3}\,,$ (4.76)

and

$\displaystyle M_{W}^{2}$ $\displaystyle=M_{W,tree}^{2}\left[1+3.12\times
10^{-2}c_{W}N+\frac{\Lambda^{2}C}{8\pi^{2}\lambda^{2}\eta^{2}}\right]\ ,$
(4.77) $\displaystyle M_{Z}^{2}$
$\displaystyle=M_{Z,tree}^{2}\left[1+3.12\times
10^{-2}\left(c_{W}N\cos^{2}\theta_{W}+\sin^{2}\theta_{W}\sum_{i}c_{B_{i}}\right)+\frac{\Lambda^{2}C}{8\pi^{2}\lambda^{2}\eta^{2}}\right]\
.$ (4.78)

In this case, we find:

$\rho=1+3.12\times
10^{-2}\sin^{2}\theta_{W}\left(c_{W}N-\sum_{i}c_{B_{i}}\right)\ ,$ (4.79)

which, under Eq. (4.58), translates into

$\sum_{i}c_{B_{i}}=-0.05+6.28\times 10^{-4}N\ .$ (4.80)

###  4.4 Large $\Lambda$ expansion

The limit $\Lambda\to\infty$ of Eq. (4.57) results in $\rho=1$ as it should.
Indeed, since $\Lambda$ is the scale where new physics kicks in, sending it to
infinity is the same as not having new physics beyond the standard model,
hence one recovers $\rho=1$. To avoid fine-tuning issues with the Higgs mass,
one usually expects $\Lambda\sim 1$ TeV, which would already provide a good
(large) expansion parameter:

$\frac{\Lambda^{2}}{\eta^{2}}\sim 10^{2}\ .$ (4.81)

The confirmation of the $W$ boson mass obtained at the Fermilab would indeed
set the cutoff at the TeV scale. On the other hand, the LHC has not found any
sign of new physics at the TeV region so far, which might suggest that the
cutoff is even higher $\Lambda\gtrsim 1$ TeV and the expansion even better.
Indeed, there is also the possibility that new physics only sets in at the GUT
scale $\Lambda\sim 10^{16}$, where the renormalization group of all
interactions meet.

Expanding Eq. (4.57) around $\Lambda=\infty$ gives:

$\displaystyle\rho$
$\displaystyle=1+\frac{32\pi^{2}\lambda^{2}\eta^{2}}{C\Lambda^{2}}\sum_{i}\left[c_{W}\gamma_{\Omega}\left(\frac{-M_{Z,tree}^{2}}{m_{i}^{2}}\right)-\left(c_{W}\cos^{2}\theta_{W}+c_{B_{i}}\sin^{2}\theta_{W}\right)\gamma_{\Omega}\left(\frac{-M_{W,tree}^{2}}{m_{i}^{2}}\right)\right]$
(4.82) $\displaystyle+\mathcal{O}(\Lambda^{-4})\ .$

The form factor is still quite difficult to deal with in general, but one can
combine Eq. (4.82) with the other aforementioned expansions to obtain more
transparent expressions. In any case, Eq. (4.82) parameterizes new physics as
an inverse power series in $\Lambda$ in terms of calculable form factors. This
should not be confused with the usual procedure for writing down an effective
Lagrangian in terms of local operators. Indeed, the result (4.82) provides a
parameterization of non-local form factors. We also note that $\Lambda$ only
shows up due to the correct treatment of the functional measure, thus such
parameterization is only possible when the functional measure is present.

##  5 Conclusions

Despite the many phenomena that the standard model cannot account for, the LHC
data frustatingly confirms its predictions. The CDF collaboration, on the
other hand, has found evidence for new physics due to the 7$\sigma$ deviation
observed in the $W$ mass, thus contradicting the LHC findings. At this stage,
effective field theory seems to be the best tool for theoretically probing new
physics while we await confirmation from precision tests.

In this paper, we adopted the effective action formalism as a steppingstone to
effective field theory in order to parameterize the UV physics. This is based
on the observation that the Barvinsky–Vilkovisky resummation provides the same
form factors for any canonical Lagrangian whose Hessian involves only minimal
operators. Our ignorance of the UV is then parameterized by the unknown
coefficients of the non-local form factors. These reflect the unknown number
and type of fields in the fundamental bare action, as described by the weak
hypercharge and the dimension of the representation to which the field
belongs.

Covariant perturbation theory does not give any contribution to the Higgs
effective potential, but the functional measure does. From a geometrical
viewpoint, the latter is written in terms of the configuration-space metric,
which must be determined by symmetry arguments. We computed the Higgs
effective potential with the functional measure correction at leading order.
Ultimately, this correction changed the Higgs vev and, after symmetry
breaking, shifted the classical values of the $W$ and $Z$ boson masses. The
joint contributions from the functional measure and the non-local form factor
to the inverse dressed propagator have been obtained, from which the most
general masses, at one-loop, have been read off.

Our result alleviates the tight relation between $M_{Z}$, $M_{W}$ and $\rho$
found at the standard model, thus allowing for the description of the
deviation discovered by Fermilab, should it be confirmed. On the other hand,
the comparison with experimental data has provided important constraints on
the UV spectra. In all cases, there exist constraints relating the number of
fields, the scale of new physics and the details of the fundamental particles
(e.g. masses, hypercharge). In particular, we have found that the number of
fields and their masses are correlated for the heavy mass limit, whereas
relations among the particles details and the number of fields take place at
the other regimes. If after further scrutiny the LHC data turns out to be
correct, our result could still allow for BSM physics, however tightly
constrained.

## Acknowledgments

The authors are grateful to the National Council for Scientific and
Technological Development (CNPq) and the Brazilian Federal Agency for Support
and Evaluation of Graduate Education (CAPES) for financial support. IK is
funded by the grant no. 303283/2022-0, CNPq.

## References

  * [1] S. Eidelman et al. [Particle Data Group], Phys. Lett. B 592 (2004) no.1-4, 1 doi:10.1016/j.physletb.2004.06.001
  * [2] S. Weinberg, Phys. Rev. Lett. 19 (1967), 1264-1266 doi:10.1103/PhysRevLett.19.1264
  * [3] A. Salam, Conf. Proc. C 680519 (1968), 367-377 doi:10.1142/9789812795915_0034
  * [4] S. L. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D 2 (1970), 1285-1292 doi:10.1103/PhysRevD.2.1285
  * [5] R. L. Workman et al. [Particle Data Group], PTEP 2022 (2022), 083C01 doi:10.1093/ptep/ptac097
  * [6] S. Carlip, Rept. Prog. Phys. 64 (2001), 885 doi:10.1088/0034-4885/64/8/301 [arXiv:gr-qc/0108040 [gr-qc]].
  * [7] T. Aaltonen et al. [CDF], Science 376 (2022) no.6589, 170-176 doi:10.1126/science.abk1781
  * [8] G. Aad et al. [ATLAS], [arXiv:2403.15085 [hep-ex]].
  * [9] S. Amoroso, N. Andari, W. Barter, J. Bendavid, M. Boonekamp, S. Farry, M. Gruenewald, C. Hays, R. Hunter and J. Kretzschmar, et al. [arXiv:2308.09417 [hep-ex]].
  * [10] I. Brivio and M. Trott, Phys. Rept. 793 (2019), 1-98 doi:10.1016/j.physrep.2018.11.002 [arXiv:1706.08945 [hep-ph]].
  * [11] G. Isidori, F. Wilsch and D. Wyler, Rev. Mod. Phys. 96 (2024) no.1, 015006 doi:10.1103/RevModPhys.96.015006 [arXiv:2303.16922 [hep-ph]].
  * [12] J. Ellis, doi:10.31526/ACP.BSM-2021.16 [arXiv:2105.14942 [hep-ph]].
  * [13] A. Drozd, J. Ellis, J. Quevillon and T. You, JHEP 03 (2016), 180 doi:10.1007/JHEP03(2016)180 [arXiv:1512.03003 [hep-ph]].
  * [14] J. Chakrabortty, S. U. Rahaman and K. Ramkumar, Nucl. Phys. B 1000 (2024), 116488 doi:10.1016/j.nuclphysb.2024.116488 [arXiv:2308.03849 [hep-ph]].
  * [15] B. S. DeWitt, “ _The global approach to quantum field theory_ ”, Oxford University Press, Oxford, 2003.
  * [16] A. O. Barvinsky and G. A. Vilkovisky, Phys. Rept. 119 (1985), 1-74 doi:10.1016/0370-1573(85)90148-6
  * [17] D. V. Vassilevich, Phys. Rept. 388 (2003), 279-360 doi:10.1016/j.physrep.2003.09.002 [arXiv:hep-th/0306138 [hep-th]].
  * [18] I. G. Avramidi, Lect. Notes Phys. Monogr. 64 (2000), 1-149 Springer, 2000, ISBN 978-3-540-67155-8 doi:10.1007/3-540-46523-5
  * [19] A. O. Barvinsky and G. A. Vilkovisky, Nucl. Phys. B 282 (1987), 163-188 doi:10.1016/0550-3213(87)90681-X
  * [20] A. O. Barvinsky and G. A. Vilkovisky, Nucl. Phys. B 333 (1990), 471-511 doi:10.1016/0550-3213(90)90047-H
  * [21] K. Fujikawa, Phys. Rev. Lett. 42 (1979), 1195-1198 doi:10.1103/PhysRevLett.42.1195
  * [22] K. Fujikawa, Phys. Rev. D 21 (1980), 2848 [erratum: Phys. Rev. D 22 (1980), 1499] doi:10.1103/PhysRevD.21.2848
  * [23] K. Fujikawa, Phys. Rev. Lett. 44 (1980), 1733 doi:10.1103/PhysRevLett.44.1733
  * [24] E. S. Fradkin and G. A. Vilkovisky, Phys. Rev. D 8 (1973) 4241.
  * [25] E. S. Fradkin and G. A. Vilkovisky, Lett. Nuovo Cim. 19 (1977) 47.
  * [26] O. Babelon and C. M. Viallet, Phys. Lett. B 85 (1979), 246-248 doi:10.1016/0370-2693(79)90589-6
  * [27] O. Babelon and C. M. Viallet, Commun. Math. Phys. 81 (1981), 515 doi:10.1007/BF01208272
  * [28] P. Ellicott, G. Kunstatter and D. J. Toms, Annals Phys. 205 (1991), 70-109 doi:10.1016/0003-4916(91)90238-4
  * [29] P. Ellicott, D. J. Toms and G. Kunstatter, Mod. Phys. Lett. A 4 (1989), 2397-2407 doi:10.1142/S0217732389002689
  * [30] R. K. Unz, Nuovo Cim. A 92 (1986) 397.
  * [31] D. J. Toms, Phys. Rev. D 35 (1987) 3796.
  * [32] V. Moretti, Phys. Rev. D 56 (1997) 7797 [arXiv:hep-th/9705060 [hep-th]].
  * [33] M. Hatsuda, P. van Nieuwenhuizen, W. Troost and A. Van Proeyen, Nucl. Phys. B 335 (1990) 166.
  * [34] P. van Nieuwenhuizen, Nucl. Phys. B Proc. Suppl. 16 (1990) 605.
  * [35] C. Armendariz-Picon, J. T. Neelakanta and R. Penco, JCAP 01 (2015) 035 [arXiv:1411.0036 [hep-th]].
  * [36] M. Becker and M. Reuter, Phys. Rev. D 102 (2020) 125001 [arXiv:2008.09430 [gr-qc]].
  * [37] I. L. Buchbinder and S. L. Lyakhovich, Class. Quant. Grav. 4 (1987) 1487.
  * [38] S. Hamamoto and M. Nakamura, Prog. Theor. Phys. 104 (2000) 691 [arXiv:hep-th/0005131 [hep-th]].
  * [39] E. Mottola, J. Math. Phys. 36 (1995) 2470 [arXiv:hep-th/9502109 [hep-th]].
  * [40] R. Casadio, A. Kamenshchik and I. Kuntz, Annals Phys. 449 (2023), 169203 [arXiv:2210.04368 [hep-th]].
  * [41] G. A. Vilkovisky, Nucl. Phys. B 234 (1984) 125.
  * [42] I. Kuntz and R. da Rocha, Nucl. Phys. B 993 (2023), 116258 [arXiv:2211.11913 [hep-th]].
  * [43] J. M. L. de Freitas and I. Kuntz, [arXiv:2307.13803 [hep-th]].
  * [44] R. Casadio, I. Kuntz and R. da Rocha, [arXiv:2403.13099 [gr-qc]].
  * [45] A. Codello and R. K. Jain, Class. Quant. Grav. 33 (2016) no.22, 225006 doi:10.1088/0264-9381/33/22/225006 [arXiv:1507.06308 [gr-qc]].
  * [46] T. Appelquist and J. Carazzone, Phys. Rev. D 11 (1975), 2856 doi:10.1103/PhysRevD.11.2856