# Falsifiable analog model predictions of $W$ mass in CDF II and ATLAS

R. N. C. Pfeifer  Dunedin, Otago, New Zealand
<EMAIL_ADDRESS>
(Day Month Year; Day Month Year)

###### Abstract

The CDF II measurement of $W$ boson mass is at significant ($6.9\,\sigma$)
tension with the Standard Model, and moderate ($4.0\,\sigma$) tension with
recent results from ATLAS. Any attempt to interpret this as a signature of new
physics requires high-precision, robust, falsifiable predictions. The
Classical Analogue to the Standard Model In pseudo-Riemannian spacetime
(CASMIR) is an analogue model predicting $W$ and $Z$ boson masses of
$80.3587(22)~{}\mathrm{G}\mathrm{eV}/c^{2}$ and
$91.1877(35)~{}\mathrm{G}\mathrm{eV}/c^{2}$ respectively. During baryon
collisions satisfying $\sqrt{s}<3.09~{}\mathrm{T}\mathrm{eV}$, CASMIR predicts
a color-mediated enhancement of $W$ and $Z$ boson masses, becoming
$80.4340(22)~{}\mathrm{G}\mathrm{eV}/c^{2}$ and
$91.1922(35)~{}\mathrm{G}\mathrm{eV}/c^{2}$ respectively. The unenhanced
masses are consistent with ATLAS data collected at
$\sqrt{s}=7~{}\mathrm{T}\mathrm{eV}$, and the enhanced masses are consistent
with CDF II data collected at $\sqrt{s}=1.96~{}\mathrm{T}\mathrm{eV}$.
According to CASMIR, operation of the LHC at centre-of-mass energies small
compared with $3.09~{}\mathrm{T}\mathrm{eV}$ but large enough for $W$ boson
formation should permit ATLAS to replicate the result from CDF II.

###### keywords:

$W$ boson mass, CDF II, Muon $g-2$.

PACS numbers: 14.70.Fm, 14.60.Ef, 12.60.-i

##  1 Introduction

The Standard Model is undeniably the greatest achievement of modern particle
physics, displaying unrivaled congruence between theory and experiment [1, 2,
3, 4, 5, 6]. However, recent results from Fermilab raise the possibility of
detecting physics beyond the Standard Model, with the CDF II measurement of
$W$ boson mass showing a statistically significant discrepancy of
$6.9\,\sigma$ when compared with the Standard Model prediction [4, 5, 6, 7].
Similarly, measurement of muon magnetic anomaly in the Muon $g-2$ experiment
shows a discrepancy of $5.2\,\sigma$ with respect to the Standard Model value
[8, 9, 10].

While tensions between CDF II and ATLAS are smaller than those between CDF II
and the Standard Model ($4.0\,\sigma$ compared with $6.9\,\sigma$) [7, 4, 5,
6, 11], it is highly possible that further data collection at ATLAS might
yield a value distinct from CDF II at the level of $5\,\sigma$ or better,
representing a discovery. Of course, most such discoveries are prosaic, e.g.

  * •

an unrecognized systematic experimental error, or

  * •

an incomplete understanding of background events mimicking the $W$ boson
signal,

and it is far more rare for such a tension to represent an actual novel
interaction or process leading to a true difference in the measured values.
For the latter possibility, the standard of proof must therefore be set
appropriately high before any such species or interaction may seriously be
contemplated.

Indeed, with regards to the muon gyromagnetic anomaly result, although recent
data have increased the precision of the Fermilab measurement of muon $g-2$
such that the tension with the 2020 theoretical prediction now exceeds the
$5.0\,\sigma$ threshold [8], this must be tempered by other recent
developments. In particular, a recent lattice gauge calculation of the Hadron
Vacuum Polarisation (HVP) disagrees with previous data-driven values obtained
from $e^{+}e^{-}$ experiments [12], and a measurement of the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ cross-section at CMD-3 disagrees with
previous $e^{+}e^{-}$ experiments [13]. Attempts to reconcile these findings
with earlier results are ongoing [14]. For the present, the uncertainty in the
theoretical value must be considered larger than previously, such that the
$5.0\,\sigma$ threshold has not yet been decisively crossed.

The present paper approaches the unexpected results from CDF II and Muon $g-2$
through the application of an analogue model, the Classical Analogue to the
Standard Model In pseudo-Riemannian spacetime (CASMIR). This model was
originally developed by the author to assist in exploring certain interactions
involving the Standard Model in a gravitational field. By construction it
displays high congruence with the Standard Model across almost all
experimentally accessible regimes, and key Standard Model parameters not input
into the model are reproduced to within $0.1\,\sigma$ of their observed values
(A). Discrepancies between CASMIR and the Standard Model arise when computing
the masses of the $W$ and $Z$ bosons, with CASMIR predicting twinned mass
eigenvalues for both species.

Remarkably, predictions from CASMIR precisely reproduce the $W$ boson mass
values observed in both ATLAS and CDF II, with precision greater than
experiment and tensions of $0.1\,\sigma$ or better. This occurs through the
presence in CASMIR of eight additional colored $W$ boson mass eigenstates
which are predicted to be excluded from the $W$ boson mass measurement at
ATLAS, but retained at CDF II. In CASMIR the conventional $W$ boson has a
calculated mass of $80.3587(22)~{}\mathrm{G}\mathrm{eV}/c^{2}$, and the
colored boson eigenstates have masses of
$80.4434(22)~{}\mathrm{G}\mathrm{eV}/c^{2}$. The detected value at CDF II is
the r.m.s. mass across the nine $W$-like mass eigenstates, yielding a
predicted value of $80.4340(22)~{}\mathrm{G}\mathrm{eV}/c^{2}$. There are no
fitting parameters in this calculation.

The value of the muon gyromagnetic anomaly also differs between CASMIR and the
Standard Model, with the CASMIR value being $116592071(46)\times 10^{-11}$
compared with $116591810(43)\times 10^{-11}$ in the Standard Model. This
difference arises because under a specific set of conditions the geometry of
CASMIR allows electroweak bosons to also carry a color charge. Again there are
no fitting parameters.

An analogue model is only useful if it forms the basis for robustly (and
preferably rapidly) testable, falsifiable predictions, ideally with existing
equipment and/or datasets. Based on the analysis presented in this paper, the
enhancement of the $W$ boson mass observed in CDF II is energy-dependent, and
may be replicated in ATLAS by reducing the LHC centre-of-mass energy to
$\sqrt{s}\ll 3.09\mathrm{T}\mathrm{eV}$.

The energy scales of these predictions all fall firmly within the projected
domain of validity of the CASMIR model. If verified, these predictions would
provide an intriguing suggestion of novel bosons (or somehow mass-shifted
versions of the existing $W$ boson) at
$80.4434(22)~{}\mathrm{G}\mathrm{eV}/c^{2}$. If refuted this would be less
interesting to the particle physics community at large, but would bound the
domain of correspondence between CASMIR and the Standard Model at $E\ll
m_{W}c^{2}\sim 80~{}\mathrm{G}\mathrm{eV}$ rather than
$3.1~{}\mathrm{T}\mathrm{eV}$ as presently anticipated.

##  2 Overview: Limits of CASMIR

This Section provides an overview of known regimes in which accelerator
physics predictions from CASMIR are anticipated to deviate from the Standard
Model, with brief discussion of the accessibility of each of these regimes. It
provides context for the more in-depth analysis of the accessible
discrepancies and resulting predictions which is provided in Secs. 3–4.

###  2.1 Weak boson masses

A key discrepancy between CASMIR and the Standard Model is the prediction in
CASMIR of paired eigenvalues for the $W$ and $Z$ boson masses, with the
components of each doublet differing by less than
$90~{}\mathrm{M}\mathrm{eV}/c^{2}$. No suggestion of such a doublet has been
seen in any single accelerator dataset, though it is only recently that the
weak boson masses have been measured to sufficient precision to permit
discrimination of such closely-spaced values. It is also noted that in CASMIR
the upper eigenvalues of each doublet are associated with the presence of a
color charge, which carries implications for the environments in which these
eigenvalues may be observed.

In Sec. 3 these properties of CASMIR are assessed for consistency with
existing datasets, in particular LEP, ATLAS, and CDF II. On the basis of the
available information, and with the exception of the $W$ boson mass
measurement in CDF II, existing datasets are not provably able either to
detect or to exclude the upper $W$ and $Z$ boson mass eigenvalues predicted by
CASMIR.

Prospects for additional near-term tests of CASMIR are discussed, including
prospects for reproduction of the CDF II result in ATLAS.

###  2.2 Muon $g-2$

As a consequence of differences in the electroweak sector, when the muon
gyromagnetic anomaly is measured at the muon “magic momentum” [15], CASMIR
predicts a higher value for the muon gyromagnetic anomaly than the Standard
Model. While a full-precision re-evaluation of the muon gyromagnetic anomaly
using CASMIR is beyond the scope of the present paper, the natures of these
changes to the electroweak sector make it straightforward to determine bounds
on the result of such a calculation (see Sec. 4). The resulting value of
$116592071(46)\times 10^{-11}$ shows significantly reduced tension with
experimental results [8] with only a small increase in uncertainty, and this
reduction in tension exceeds that which can be attributed to expansion of the
confidence interval alone. Consequently CASMIR predicts the muon gyromagnetic
anomaly more accurately at the muon magic momentum than the Standard Model,
and hence—despite this discrepancy with the Standard Model—measurements of
muon $g-2$ do not constrain the applicability of CASMIR.

###  2.3 Preferred rest frame and finite cutoff

An unusual property of CASMIR is the existence of a pseudovacuum state with a
preferred rest frame. This pseudovacuum is associated with an energy scale
$\mathcal{E}_{\Omega}\approx 6.18~{}\mathrm{T}\mathrm{eV}$, and is responsible
for the CASMIR analogue of quantum fluctuations in energy and momentum [16,
17]. However, the finite value of $\mathcal{E}_{\Omega}$ implies a similarly
finite UV cutoff of $\frac{1}{2}\mathcal{E}_{\Omega}\approx
3.1~{}\mathrm{T}\mathrm{eV}$ in the isotropy frame of the cosmic microwave
background (CMB).111Energy scales in CASMIR correspond to the inverse of
autocorrelation scales in the pseudovacuum. Since a field excitation of
wavelength $\lambda$ exhibits autocorrelation across regions of length
$\tfrac{1}{2}\lambda$, a CASMIR energy scale $\mathcal{E}$ corresponds to an
excitation energy $E=\tfrac{1}{2}\mathcal{E}$. In this paper, $\mathcal{E}$
denotes a CASMIR energy scale while $E$ denotes the energy of a particle or
field. As a consequence, processes whose dominant terms involve energy or
momentum fluctuations that are large compared with
$\frac{1}{2}\mathcal{E}_{\Omega}$ when evaluated in the rest frame of the CMB
are heavily suppressed in CASMIR. Similarly, the pseudovacuum may cease to
appear homogeneous to particles accelerated to energies of
$\mathrm{O}\left(\frac{1}{2}\mathcal{E}_{\Omega}\right)$ in the rest frame of
the CMB.

I am unaware of any practical proposals to test the ability of quantum
fluctuations to exceed $3.1~{}\mathrm{T}\mathrm{eV}$, or to use the
circulating beamlines of the LHC as a probe for inhomogeneity of the vacuum at
high boost. Thus, although the existence of a preferred rest frame in CASMIR
represents a significant theoretical break from the Standard Model, as yet it
provides no prospects for experimental discrimination between CASMIR and the
Standard Model.

###  2.4 Higher generation weak bosons

In addition to the color-mediated weak boson mass doublets discussed in Sec.
2.1, CASMIR also predicts the existence of higher-generation weak bosons,
starting with a second-generation $W$ boson at
$16.6~{}\mathrm{T}\mathrm{eV}/c^{2}$ (see Table 2.4). However, the energy
scale of even the lightest of these bosons is presently out of reach of the
LHC.

Masses of the first, second, and third generation uncolored weak bosons in
CASMIR.  Mass Boson First Generation ($\mathrm{G}\mathrm{eV}/c^{2}$) Second
Generation ($\mathrm{T}\mathrm{eV}/c^{2}$) Third Generation
($\mathrm{T}\mathrm{eV}/c^{2}$) $W$ 80. 3587(22) 16. 61320(46) 279. 4038(77)
$Z$ 91. 1877(35) 18. 84672(73) 316. 968(12) ${\bm{\mathrm{H}}}$ 125. 2501(49)
25. 8900(10) 435. 422(17)

While these higher-generation bosons may also appear as virtual particles, any
contributions to currently-accessible processes from such particles would be
heavily suppressed due to the high energies of the higher-generation bosons,
and in CASMIR also due to the finite vacuum energy cutoff discussed in Sec.
2.3. These bosons display exotic behaviors in CASMIR, with particle generation
not behaving as a good quantum number at their interaction vertices. However,
no detectable discrepancies are predicted in presently accessible experimental
regimes.

###  2.5 Dark matter

CASMIR also describes three species of dark matter, denoted $N_{\mu}$,
$[N_{2}]_{\mu}$, and $[N_{3}]_{\mu}$ in keeping with the notation of Ref. 17.
These species have inertial masses of
$80.3810(22)~{}\mathrm{G}\mathrm{eV}/c^{2}$,
$16.61320(46)~{}\mathrm{T}\mathrm{eV}/c^{2}$, and
$279.4038(77)~{}\mathrm{T}\mathrm{eV}/c^{2}$ respectively, but break the weak
principle of equivalence, with all having a gravitational mass of only
$1381.486(37)\,m_{e}$ (Ref. 17, Chapter LABEL:ch:gravity). With their only
significant nongravitational interactions being their coupling to the scalar
boson, prospects for direct detection of these species are minimal. Since
negligible attention has been paid to proposals for dark matter which break
the weak principle of equivalence, it is not presently possible to say whether
these species are consistent with cosmological observations. They have no
impact on the accelerator physics considered in the present paper and are
therefore ignored.

###  2.6 Neutrino mass and oscillations

Neutrinos in CASMIR are massless and Weyl, with equilibration between
generations arising not as a consequence of neutrino oscillation, but through
nonconservation of particle generation at $W$ boson vertices (Ref. 17, Sec.
LABEL:sec:neutrinos). Consequently there exists a discrepancy in the area of
neutrino physics between CASMIR and the Standard Model with massive neutrinos
and a seesaw mechanism. Direct detection of neutrino inertial mass would place
limitations on the application of CASMIR, as would a lower bound on the length
scale over which neutrino oscillations take place. Neither of these events
have occurred. Likewise, CASMIR does not support neutrinoless double beta
decay, and neutrinoless double beta decay has also not been observed. As yet,
neutrino physics does not place bounds on the regime of congruency between
CASMIR, experiment, and the Standard Model.

###  2.7 Additional notes on CASMIR and weak sector constraints

The weak sector provides ample opportunities to constrain the validity of
beyond-Standard-Model constructions. As CASMIR is relatively recently
developed, the process of assessment against weak sector constraints is
ongoing. However, the following considerations offer reason to expect a high
degree of congruency.

In Sec. LABEL:sec:weakmix of Ref. 17 it is established that the effective
lepton/$W$ and lepton/$Z$ vector boson interaction terms in the emergent
CASMIR Lagrangian show high congruency with those of the Standard Model.
Further, the preonic substructure of the fermions permits this conclusion to
be generalized to interactions involving quarks, and conserved symmetries (and
construction from preons) likewise permit this conclusion to be generalized to
three-particle and four-particle vector boson vertices. For fermions and
vector bosons, the electroweak sector of the CASMIR model is found to be
wholly consistent with the electroweak sector of the Standard Model below the
$1\,\sigma$ level, including reproduction of decay rates and branching ratios,
up to effects due to colored (chromatic) weak vector bosons discussed in Secs.
3 and 4 of the present paper.

For processes involving Higgs bosons, direct assessment of constraints on
CASMIR have so far largely been qualitative, and so far have also been
consistent with expectations. A few quantitative calculations involving the
Higgs boson have been performed, though these have no counterpart in the
Standard Model. They are:

  * •

Precision calculation of the mass of the Higgs boson (Ref. 17, Sec.
LABEL:sec:scalbosmass).

  * •

Relationships between lepton and vector boson masses, including Higgs boson
vacuum expectation value contributions to the lepton and vector boson masses
and Higgs boson loop corrections to the lepton and vector boson mass
interactions (Ref. 17, Secs. LABEL:sec:VI:bosonmasses and
LABEL:sec:lepmassint).

  * •

Contributions to the calculation of muon $g-2$ (Sec. 4 of the present paper).

The mass calculations yield lepton, vector boson, and Higgs boson masses
consistent with observation (see A), while the CASMIR muon $g-2$ calculation
shows less tension with experiment than the Standard Model (see Sec. 4).

An important test which has not yet been performed is the comparison of
predicted Higgs boson production rates with the Standard Model and with
observation. In CASMIR, Higgs boson production at energy scales small compared
with $3.1~{}\mathrm{T}\mathrm{eV}$ is suppressed by a factor of approximately
$1.7\times 10^{-2}$ per vertex, denoted
$\big{[}k^{(e)}_{1}(\mathcal{E}_{e})N_{0}\big{]}^{-2}$ (Ref. 17, Sec.
LABEL:sec:scalbosint). However, this suppression effect vanishes above the
$3.1~{}\mathrm{T}\mathrm{eV}$ threshold. Its relevance or otherwise is as yet
unconfirmed in scalar-boson-generating collisions at the LHC, where the beam
energy scales are $\sqrt{s}=7~{}\mathrm{T}\mathrm{eV}$ and above, but the
energies of the scalar bosons generated in these collisions are substantially
more modest.

Although further testing of CASMIR is necessary, the model has so far
demonstrated good performance against electroweak constraints both in
congruence with the Standard Model and in precision calculations extending
beyond the Standard Model.

###  2.8 Summary of limits

The discrepancies between CASMIR and the Standard Model which are likely to
impact upon present experimental results are those which arise from the
existence of additional weak sector vector boson mass eigenstates in CASMIR.
These additional eigenstates correspond to colored versions of the $W$ and $Z$
bosons, having masses within a hundred $\mathrm{M}\mathrm{eV}/c^{2}$ of their
colorless counterparts. Incorporating the colored mass eigenstates into the
calculation of $W$ boson mass yields a good match for the result from CDF II,
and existence of these additional mass eigenstates also favorably impacts
calculation of the muon gyromagnetic anomaly.

In light of these discrepancies, Sec. 3 addresses why current evaluation of
the CDF II dataset is predicted by CASMIR to be sensitive to these novel
species while ATLAS is not, and provides a prescription for detection of the
higher-mass eigenstates in ATLAS. Section 4 then adjusts the calculation of
the muon gyromagnetic anomaly to include the CASMIR colored weak boson mass
eigenstates. The CASMIR result is consistent with experiment with tension well
below $1\,\sigma$, demonstrating that results from the Muon $g-2$ experiment
do not constrain the application of CASMIR as an analogue to the Standard
Model in the relevant energy regimes.

##  3 Weak boson masses

###  3.1 Background

At present some of the most stringent tests of the Standard Model come from
large-scale collider experiments such as CMS [18] and ATLAS [19, 20, 21, 11]
at the Large Hadron Collider (LHC), along with ongoing processing of the data
from the discontinued CDF II at Tevatron [7]. Such results may be considered
independently where they possess some unique merit (for example, extraordinary
precision [7] or uniquely high beam energies [20, 21]), or collectively where
construction of a global average may amplify statistical power and reduce
error [6]. Measurement of the mass of the $W$ boson is one of the most
challenging of these tests, as its decay products include an antineutrino (for
$W^{-}$ boson decay) or a neutrino (for $W^{+}$ boson decay) which is
typically lost to detection. The most precise experimental determination of
$W$ boson mass yet performed is that of the CDF collaboration, using 8.8
inverse femtobarns of integrated luminosity collected in proton–antiproton
collisions at $1.96~{}\mathrm{T}\mathrm{eV}$ centre-of-mass energy in the CDF
II collector at the Fermilab Tevatron collider [7]. This data series produced
the surprising result of $m_{W}=80.4335(94)~{}\mathrm{G}\mathrm{eV}/c^{2}$, in
tension with the Standard Model prediction of
$80.356(6)~{}\mathrm{G}\mathrm{eV}/c^{2}$ at $6.9\,\sigma$ [4, 5, 6], the PDG
global average of $80.377(12)~{}\mathrm{G}\mathrm{eV}/c^{2}$ at a level of
$3.7\,\sigma$ [6], and the ATLAS collaboration measurement of
$80.360(16)~{}\mathrm{G}\mathrm{eV}/c^{2}$ at $4.0\,\sigma$ [11]. 222This
recently-announced unpublished result improves over the previously published
measurement of $80.370(19)~{}\mathrm{G}\mathrm{eV}/c^{2}$.[20, 21] No
mechanism has been identified within the Standard Model to explain this
discrepancy, so it is not unreasonable to seek an explanation beyond the
Standard Model.

###  3.2 CASMIR and chromatic bosons—a brief overview

This subsection begins in Secs. 3.2.1–3.2.2 with a brief overview of particle
construction and mass mechanisms in CASMIR. While not intended as a substitute
for the full exposition,[16, 17, 22, 23] it serves to contextualize the
subsequent discussion of $W$ and $Z$ boson masses and detection within the
CASMIR model. This is followed in Sec. 3.2.3 by a discussion of the
environments in which chromatic bosons may be detected, before proceeding to
specific discussion of the chromatic $W$ and $Z$ bosons in Secs. 3.3 and 3.4
respectively.

####  3.2.1 Particles in CASMIR

CASMIR is a heirarchical effective field theory model in which all species
observed in the Standard Model are constructed from nine charged preons, and
these preons themselves emerge as an effective low-energy description of the
gradients of scalar fields on a higher-dimensional manifold. Although the
microscopic origin of the preons is specific to CASMIR, once they are
constructed these fields display the familiar structure of a sequence of
effective descriptions valid over a descending ladder of energy
scales.333There are two principal energy thresholds in CASMIR, namely:
$\mathcal{E}_{\psi}$, below which preons condense into fermions and bosons and
the pseudovacuum admits a (non-homogeneous) continuum description, and
$\mathcal{E}_{0}$, below which the pseudovacuum appears homogeneous and
chromatic bosons and the pseudovacuum structure can no longer be observed.
Both are measured in the isotropy frame of the pseudovacuum.  This sort of
multiscale description is ubiquitous throughout particle physics—for example,
hadrons are (quasi)particles composed from collections of quarks, atomic
nuclei are composed in turn from hadrons, and yet other quasiparticles such as
phonons may emerge from the collective behaviors of atoms [24, 25].

In CASMIR, all species observed in the Standard Model are quasiparticles
constructed from preons carrying charges with respect to a
$\mathrm{GL}(9,\mathbb{R})\otimes\mathrm{SL}(2,\mathbb{C})$ symmetry. The
$\mathrm{GL}(9,\mathbb{R})$ subgroup behaves as an internal local symmetry,
while $\mathrm{SL}(2,\mathbb{C})$ is the double cover of the local space–time
symmetry of ${{\mathbb{R}^{1,3}}}$. There are nine preons, and these are acted
on by the eighty-one bosons of the internal symmetry group
$\mathrm{GL}(9,\mathbb{R})$. However, in the presence of
$\mathrm{SL}(2,\mathbb{C})$, symmetry group $\mathrm{GL}(9,\mathbb{R})$ admits
the decompositions (Ref. 23, Sec. LABEL:sec:symC18)

$\begin{split}\mathrm{GL}(9,\mathbb{R})\cong\,&\mathrm{SU}(9)\oplus\mathrm{GL}(1,\mathbb{R})\\\
\cong\,&[\mathrm{SU}(3)_{A}\oplus\mathrm{GL}(1,\mathbb{R})_{A}]\otimes[\mathrm{SU}(3)_{C}\oplus\mathrm{GL}(1,\mathbb{R})_{C}]\end{split}$
(1)

where the groups $\mathrm{SU}(3)_{B}$ and $\mathrm{GL}(1,\mathbb{R})_{B}$ for
$B\in\\{A,C\\}$ may alternatively be understood as dimension-8 and dimension-1
representations of symmetry $\mathrm{SU}(3)_{B}$ respectively. In the physical
gauge the $\mathrm{SU}(3)_{A}$ symmetry is broken to
$\mathrm{SU}(2)\otimes\mathrm{U}(1)$, with radiative corrections adjusting the
coupling constants to reproduce the electroweak sector of the Standard Model,
up to the inclusion of the additional colored weak vector bosons.[23, 17] The
group $\mathrm{GL}(1,\mathbb{R})_{A}$ thus corresponds to a trivial
representation of the electroweak symmetry
$\mathrm{SU}(2)\otimes\mathrm{U}(1)$, and $\mathrm{GL}(1,\mathbb{R})_{C}$ to a
trivial representation of the color symmetry $\mathrm{SU}(3)_{C}$.

Under this decomposition the nine preons each carry two labels, corresponding
to charge on the $A$ and $C$ sectors, and color-neutral triplets of identical
$A$-charge correspond (up to a normalization factor) to the three families of
leptonic Weyl spinors,444This paper uses Weyl spinor notation for fermions, as
in e.g. Ref. 26.

Spinor triplet |  Lepton family
---|---
$\psi^{1r}\psi^{1g}\psi^{1b}$ |  $\\{\overline{e}_{R},\overline{\mu}_{R},\overline{\tau}_{R}\\}$
$\psi^{2r}\psi^{2g}\psi^{2b}$ |  $\\{e_{L},\mu_{L},\tau_{L}\\}$
$\psi^{3r}\psi^{3g}\psi^{3b}$ |  $\\{\nu_{e},\nu_{\mu},\nu_{\tau}\\}$,

with the relative phases of the three components determining the mass-matrix
eigenvalue and thus particle generation of a given preon triplet (Ref. 17,
Chapter LABEL:ch:fermion). Quarks correspond to triplets with mismatched
$A$-charge, where these are permitted by gauge, and have a residual color due
to variations in the strength of confinement of the differing preons (Ref. 23,
Sec. LABEL:sec:strongint).

Coupling of fermions to the boson field $N_{\mu}$ associated with the trivial
subgroup $\mathrm{GL}(1,\mathbb{R})_{A}\otimes\mathrm{GL}(1,\mathbb{R})_{C}$
is suppressed over length scales large compared with $\mathcal{L}_{0}$ by
gauge choices (LABEL:eq:U1gauge) and (LABEL:eq:GL1RNgauge) of Ref. 23.
However, fermions may continue to couple to bosons carrying trivial charge on
one of the two subgroups, giving rise to the pure $A$-sector and pure
$C$-sector bosons, or carrying trivial charge on neither of the two subgroups,
corresponding to bosons carrying both $A$\- and $C$-sector charges.

With a boson being composed of one preon and one antipreon, it might be
anticipated that the off-diagonal construction
$\frac{1}{\sqrt{3}}\overline{\nu}_{e}^{\dot{c}}\overline{\sigma}_{\mu}\delta_{\dot{c}c}e^{c}_{L}$
intended to function as the $W$ boson would only change the species of one
preon within a fermion triplet. However, taking one fermion as inbound and one
as outbound, a freedom of choice of coordinate frame on the
$\mathrm{GL}(9,\mathbb{R})$ bundle may be fixed by requiring that the
$A$-sector bosons act on the fermions (as well as the preons) as a
representation of $\mathrm{su}(3)_{A}$ (Ref. 23, Sec.
LABEL:sec:EWint_Wintdetail). The resulting frame exhibits a discontinuity
whose precise location admits some flexibility, with any observer being free
to choose its local effects simultaneous with the interaction in their own
reference frame, as a result of which the $A$-charges of the other two members
of the triplet are also transformed in association with the interaction. Thus
the action of the $W$ boson on the lepton sector is through vertices having
the expected form

$W_{\mu}\overline{e}_{L}\overline{\sigma}^{\mu}\nu_{e}\quad\textrm{and h.c.}$
(2)

and similar for higher generations.

If all of the bosons acting purely on one sector are now made explicit,
including those which vanish in the physical gauge, then this yields nine
bosons on each of the two sectors. It then suffices over macroscopic scales
(e.g. approaching the classical limit, or in many-particle scattering
processes such as mass generation) to model the eighty-one bosons of
$\mathrm{gl}(9,\mathbb{R})$ as pairwise products of these sets, e.g.

$\bm{\phi}_{\mu}^{\dot{a}\dot{c}ac}(x)\sim\int\mathrm{d}^{4}y~{}a^{\dot{a}a}_{\nu}(y)\overline{\Psi}(y)\overline{\sigma}^{\nu}\Psi(x)c^{\dot{c}c}_{\mu}(x),$
(3)

where $\Psi(x)$ represents an appropriately-chosen preon triplet. However, in
single-particle processes this representation of the boson sector is
misleading, and to accurately model the transport of charge and momentum
between fields it is necessary to also take into account processes
transferring both $A$\- and $C$-charge on a single particle.

Introduce the rescaled Gell-Mann matrices

$\displaystyle\lambda_{1}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{rrr}0&1&0\\\
1&0&0\\\ 0&0&0\end{array}\right)\quad$
$\displaystyle\lambda_{2}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{rrr}0&-\mathrm{i}&0\\\
\mathrm{i}&0&0\\\ 0&0&0\end{array}\right)\quad$
$\displaystyle\lambda_{3}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{rrr}1&0&0\\\
0&-1&0\\\ 0&0&0\end{array}\right)$ (13)
$\displaystyle\lambda_{4}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{rrr}0&0&1\\\
0&0&0\\\ 1&0&0\end{array}\right)\quad$
$\displaystyle\lambda_{5}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{rrr}0&0&-\mathrm{i}\\\
0&0&0\\\ \mathrm{i}&0&0\end{array}\right)\quad$
$\displaystyle\lambda_{6}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{rrr}0&0&0\\\
0&0&1\\\ 0&1&0\end{array}\right)$ (23)
$\displaystyle\lambda_{7}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{rrr}0&0&0\\\
0&0&-\mathrm{i}\\\ 0&\mathrm{i}&0\end{array}\right)\quad$
$\displaystyle\lambda_{8}=\frac{2}{\sqrt{6}}\left(\begin{array}[]{rrr}\frac{1}{2}&0&0\\\
0&\frac{1}{2}&0\\\ 0&0&-1\end{array}\right)\quad$
$\displaystyle\lambda_{9}=\frac{1}{\sqrt{3}}\left(\begin{array}[]{rrr}1&0&0\\\
0&1&0\\\ 0&0&1\end{array}\right),$ (33)

which satisfy $\mathrm{Tr}\,{({\lambda_{\tilde{c}}}^{2})}=1$ for all
${\tilde{c}}\in\\{1,\ldots,9\\}$, and recognize that the achromatic (i.e.
colorless) $W$ boson is associated with representation $\lambda_{9}$ on the
$C$ sector, denoted $\lambda^{C}_{9}$. This is then necessarily supplemented
by the eight chromatic $W$ bosons,
$W^{{\tilde{c}}}~{}|~{}{{\tilde{c}}\in\\{1,\ldots,8\\}}$. Off-diagonal, it is
often also convenient to work with the representations
$e_{ij}^{C}~{}|~{}i\not=j$, which are associated with the elementary matrices
$e_{ij}$. For example,
$e^{C}_{12}=\frac{1}{\sqrt{2}}(\lambda^{C}_{1}+\mathrm{i}\lambda_{2})$. In
keeping with operator notation, $W^{\dot{c}c}$ is used to denote a negatively-
charged complex chromatic $W$ boson while ${W^{\dot{c}c}}^{\dagger}$ denotes a
positively-charged complex chromatic $W$ boson. It is usually clear from
context whether a notation such as $W$ or $W^{\dot{c}c}$ refers to both the
positively- and negatively-charged species collectively or whether it
specifically represents the negatively-charged species. The explicit charge
notation $W^{-}$ and $W^{+}$ (or $W^{\dot{c}c-}$ and $W^{\dot{c}c+}$) is used
as needed for clarity.

Chromatic $Z$ bosons may likewise be introduced, as may chromatic photons
though the chromatic photons remain massless (Ref. 17, Sec.
LABEL:sec:colouredAmass).

####  3.2.2 Mass mechanisms in CASMIR

To understand the circumstances in which chromatic $W$ bosons are—and are
not—encountered in CASMIR, it is first necessary to understand the mass
mechanism of CASMIR and how it differs from the Standard Model.

In the Standard Model, mass is hypothesized to arise from interaction with a
vacuum state with nonvanishing Higgs expectation value. A suitable scalar
boson has been observed [18, 19], lending support to this construction. In
contrast, mass in CASMIR arises from nonvanishing expectation values of
products of the fermion, photon, and gluon fields of the pseudovacuum as well
as the scalar boson field, with the latter making only a small contribution.
Nevertheless, calculations performed in CASMIR are generally quite capable of
reproducing Standard Model and experimental results, and to this end, CASMIR
may be considered a useful analogue to the Standard Model.

Considering specifically the achromatic and chromatic $W$ bosons, the
achromatic $W$ boson acquires mass from the fermion, photon, and scalar
sectors of the pseudovacuum only, while the chromatic $W$ boson species also
couple to the gluon sector of the pseudovacuum. The chromatic $W$ bosons thus
acquire a slightly higher mass than their achromatic counterparts, though the
contribution which can arise from gluon couplings in CASMIR is known to be
small compared with that from the fermion sector (Ref. 17, Chapter
LABEL:ch:detail).

The energy scale of the pseudovacuum excitations which engage in these mass-
generating couplings is

$\mathcal{E}_{0}=3.5864883(17)~{}\mathrm{G}\mathrm{eV},$ (34)

a parameter which is uniquely determined from the fine structure constant and
the masses of the electron and muon (Ref. 17, Chapter LABEL:ch:detail). The
associated timescale, being the timescale over which the pseudovacuum appears
on-average homogeneous in its isotropy frame, is denoted $t_{0}$:

$\displaystyle\begin{split}t_{0}&=\frac{h}{|e|}\cdot[3.5864883(17)~{}\mathrm{G}\mathrm{eV}]^{-1}\\\
&\approx 1.2\times 10^{-24}~{}\mathrm{s}.\end{split}$ (35)

Excitations in the pseudovacuum overlap,555More precisely, excitations in the
CASMIR pseudovacuum correspond to gradients in the product of a collection of
underlying (“fundamental”) scalar fields. These fields are myriad, but local
behavior is dominated by those excitations with inflection points within a
local region characterized by scale $\mathcal{L}_{0}=ct_{0}$. These regions of
primary influence overlap, with an average of $N_{0}$ such fields exerting
significant influence at any given point in space–time. Evaluation of the mass
interactions of CASMIR yields an optimal value of $N_{0}\approx 192$ for
emulation of the Standard Model.[17] See also A for the equations which fix
the value of $N_{0}$. and the mean time between pseudovacuum scattering events
is denoted $t_{\Omega}$, associated with an energy scale
$\mathcal{E}_{\Omega}$,

$\displaystyle\mathcal{E}_{\Omega}$
$\displaystyle=6.17960(12)~{}\mathrm{T}\mathrm{eV}$ (36)
$\displaystyle\begin{split}t_{\Omega}&=\frac{h}{|e|}\cdot[6.17960(12)~{}\mathrm{T}\mathrm{eV}]^{-1}\\\
&\approx 6.7\times 10^{-28}~{}\mathrm{s}.\end{split}$ (37)

Construction of an effective massive propagator requires sequential couplings
to the pseudovacuum over multiple intervals $t_{\Omega}$ in the pseudovacuum
isotropy frame.

####  3.2.3 Environments supporting chromatic bosons in CASMIR

With the mass of the chromatic $W$ boson anticipated to be only a little
higher than that of the achromatic $W$ boson, access to both chromatic and
achromatic $W$ boson mass eigenstates may be anticipated in high-energy
environments such as the $1.96~{}\mathrm{T}\mathrm{eV}$ collisions of the CDF
II experiment. However, the chromatic $W$ boson is a colored particle whereas
the total color of a $p\overline{p}$ collision vanishes. In an environment
containing few color charges a chromatic $W$ boson is tightly bound to its
color-neutralising counterparts, and gluon exchange distributes energy and
momentum among these colored species over the preon confinement timescale
$t_{\psi}$. These few-particle confinement interactions do not in general
cancel one another out, and if they occur on a similar or shorter timeframe
than pseudovacuum scattering events, they disrupt the construction of massive
propagators and cause confined colored bosons to behave as if massless.

In contrast, in a sufficiently homogeneous and anisotropic color-rich
environment with vanishing net particle color due to a balanced sum over color
charges, confinement is weakened and a colored boson acquires asymptotic
freedom. Provided the extent of the environment is large compared with
$t_{\psi}$, a chromatic $W$ boson may propagate freely through the colored
environment while engaging in multiple sequential mass-generating interactions
with the CASMIR pseudovacuum. Thus the chromatic $W$ bosons acquire mass only
in color-rich environments with lifetimes long compared with $t_{\psi}$.
Regarding $t_{\psi}$ and the associated energy scale $\mathcal{E}_{\psi}$:

  * •

Since the quark color charges are residuals of the preon color charges, it
follows that $\mathcal{E}_{\psi}$ is no smaller than the strong force scale in
CASMIR.

  * •

In CASMIR, quasiparticles arise as perturbations on a collection of scalar
fields whose inflection points have a characteristic space–time separation
$t_{\Omega}$ in the CASMIR isotropy frame. As a consequence the energy scale
of spontaneous particle emissions is bounded from above by
$\mathcal{E}_{\Omega}$ in this frame, and the mean time between spontaneous
emission events is bounded from below by $t_{\Omega}$. It thus follows that
$\mathcal{E}_{\psi}$ in this frame is no larger than $\mathcal{E}_{\Omega}$.

  * •

There exists no mechanism in the Standard Model or CASMIR to fix the strong
scale below $\mathcal{E}_{\Omega}$, nor experimental evidence to suggest this.
To the extent that CASMIR reproduces the behaviors of the Standard Model (or
in case of discrepancy, reproduces experimentally observed behaviors) below
$\mathcal{E}_{\Omega}$, it may be assumed that the preon binding scale in
CASMIR corresponds to the upper energy scale at which spontaneous gluon
exchange may take place, yielding

$\mathcal{E}_{\psi}=\mathcal{E}_{\Omega}\qquad
t_{\psi}=t_{\Omega}=h\cdot\mathcal{E}_{\Omega}^{-1}.$ (38)

Color-rich environments with lifetimes long compared with $t_{\psi}$ are thus
favored by energy scales small compared with $\mathcal{E}_{\Omega}\approx
6.18~{}\mathrm{T}\mathrm{eV}$.

Next recall that all fermion species in CASMIR are composed from three colored
preons, and all bosons are constructed from co-propagating preon/antipreon
pairs.[23] Binding in the fermions is mediated by preon exchange, in
particular the exchange of preon/antipreon pairs with nonvanishing color but
vanishing electroweak charge, which form a representation of
$\mathrm{SU}(3)_{C}$ and act as effective gluons.

Keeping the above in mind, it is now necessary to identify those environments
which may sustain virtual chromatic weak vector bosons, and those which may
sustain real chromatic weak vector bosons. To begin with virtual bosons,
recall that in CASMIR, virtual particles borrow their energy from the
pseudovacuum. The typical energy imparted by a single pseudovacuum interaction
is on order $\mathcal{E}_{0}$, but $\mathrm{O}(N_{0})$ consecutive
interactions within the pseudovacuum autocorrelation time $t_{0}$ may readily
result in net foreground energy fluctuations with magnitudes in the range
$[0,\mathrm{O}(\mathcal{E}_{\Omega})]$ (with rapidly diminishing but
nonvanishing probability of finding fluctuations above this scale). Over
length or time scales small compared with the pseudovacuum autocorrelation
scale $t_{0}$ but large compared with the preon binding scale $t_{\Omega}$,
the environment of these composite particles is naturally color-rich due to
the intrinsic inhomogeneity of the pseudovacuum over these scales, effectively
manifesting as a large number of extant virtual particles arising from
pseudovacuum interactions. However, over timescales larger than $t_{0}$ the
large-scale homogeneity of the pseudovacuum dominates, and the environment
appears net colorless with localized fluctuations in color occuring only over
timescales within the range $[t_{0},t_{\Omega}]$. Thus the pseudovacuum
environment only appears color-rich to species at energy scales $\mathcal{E}$
satisfying

$3.59~{}\mathrm{G}\mathrm{eV}\approx\mathcal{E}_{0}<\mathcal{E}<\mathcal{E}_{\Omega}\approx
6.18~{}\mathrm{T}\mathrm{eV}.$ (39)

It must, however, be noted that $\mathcal{E}$ is not in general the centre-of-
mass energy of the collider. First, recognize that the energy scale of
individual collisions is much smaller than the collider energy scale, with
useful $W$ boson-forming events typically occurring at
sub-$\mathrm{T}\mathrm{eV}$ collision energies in both ATLAS and CDF II [20,
7]. Second, recognize that energy scale $\mathcal{E}_{0}$ corresponds to the
inverse of the threshold scale for autocorrelation of the pseudovacuum,
denoted by $\mathcal{L}_{0}$ in Refs. 16, 17, 22, and 23, and by $t_{0}$ in
Sec. 3.2.2 above, and represents the smallest distance or time which can be
consistently discriminated from zero by measurements performed on the
pseudovacuum state. For a particle energy $E$ in the isotropy frame of the
pseudovacuum, the smallest time over which measurements on the accelerated
particles are similarly consistently correlated (and not anticorrelated) is
given by the half-period

$t_{E}=\frac{h}{2E}.$ (40)

The energy scale $\mathcal{E}$ is therefore best identified with $2E$, and the
energy must satisfy

$3.59~{}\mathrm{G}\mathrm{eV}\approx\mathcal{E}_{0}<2E<\mathcal{E}_{\Omega}\approx
6.18~{}\mathrm{T}\mathrm{eV},$ (41)

i.e.

$1.79~{}\mathrm{G}\mathrm{eV}<E<3.09~{}\mathrm{T}\mathrm{eV}.$ (42)

As a convention, foreground field energy scales are denoted by $E$ to
distinguish them from pseudovacuum energy scales denoted by $\mathcal{E}$.

The above criterion is sufficient for formation of virtual chromatic weak
vector bosons, and is necessary where a color dipole is spontaneously
generated (i.e. as a result of color-inhomogeneous pseudovacuum interactions).
There then exists a second criterion which is necessary for the formation of
real chromatic weak vector bosons: the existence of a real color-rich
environment to which the chromatic boson is exposed for a duration long
compared with the minimum pseudovacuum interaction timescale $t_{\Omega}$. In
practice a duration of $2t_{\Omega}$ suffices, such that for any mass vertex
occurring at a time $t$ within an interval $I=[0,2t_{\Omega})$ there exists a
second correlated vertex, nominally at a separation of $\pm t_{\Omega}$, which
also lies within interval $I$.

The simplest environment providing an appropriate color-rich foreground
environment is the interior of a hadron. With the $W$ bosons in CDF II and
ATLAS being generated close to rest in the centre-of-mass frame, their
durations of exposure to this color-rich environment are determined by the
energies of the protons or antiprotons in this frame,

$E_{p}=\frac{\sqrt{s}}{2}.$ (43)

The requirement that the color-rich environment endures for a duration of
$2t_{\Omega}$ rather than $t_{\Omega}$ tightens the upper bound of Eq. (42) to
yield

${3.59~{}\mathrm{G}\mathrm{eV}\approx\mathcal{E}_{0}<2E_{p}<\frac{\mathcal{E}_{\Omega}}{2}\approx
3.09~{}\mathrm{T}\mathrm{eV}},$ (44)

or equivalently,

${3.59~{}\mathrm{G}\mathrm{eV}\approx\mathcal{E}_{0}<\sqrt{s}<\frac{\mathcal{E}_{\Omega}}{2}\approx
3.09~{}\mathrm{T}\mathrm{eV}}.$ (45)

The protons at Tevatron satisfy Eqs. (44) and (45), having energies
$E_{p}=0.98~{}\mathrm{T}\mathrm{eV}$ and
$\sqrt{s}=1.96~{}\mathrm{T}\mathrm{eV}$. In contrast the $W$ mass dataset at
ATLAS was collected at $\sqrt{s}=7~{}\mathrm{T}\mathrm{eV}$
($E_{p}=3.5~{}\mathrm{T}\mathrm{eV}$) and does not satisfy these constraints.
The chromatic weak vector boson mass eigenstates are therefore accessible to
CDF II but not to ATLAS.

As a further corollary of these considerations, the chromatic weak vector
boson mass eigenstates are also inaccessible to lepton colliders such as LEP
as their collision environments lack a color-rich (hadronic) foreground.

###  3.3 The chromatic $W$ boson

####  3.3.1 In CDF II

Measurement of the mass of $W$ bosons produced in proton–proton or
proton–antiproton collisions relies on the detection of charged leptons
produced during decay processes such as

$W\longrightarrow e_{L}+\overline{\nu}_{e}$ (46)

and higher-generation counterparts. Given a distribution of energies and
momenta on the emitted electron, the rest mass of the (achromatic) $W$ boson
may then be reconstructed. In CASMIR, however, left-helicity electrons may in
theory also be generated by the decay of chromatic $W$ bosons. As noted in
Sec. 3.2.3, the energy scale of these production events in CDF II favors
interpretation of the pseudovacuum as a color-rich environment, and thus when
modeled in CASMIR it is well-suited to the creation of chromatic as well as
achromatic $W$ bosons.

In terms of the parameters of the CASMIR mass relationships (see A or Ref. 17,
Chapter LABEL:ch:detail), the squares of the masses of the achromatic massive
bosons are proportional to factors which may be written in the form

$\left.1+\frac{z}{18\left[k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\right]^{4}}\qquad\right|\qquad
z\in\mathbb{Z}^{+}$ (47)

where $z=19$ for $W$ bosons and $z=55$ for $Z$ bosons. When colored $W$ and
$Z$ bosons couple to the gluon component of the pseudovacuum this increases
the numerator $z$ to $155\frac{11}{16}$ and ${62\frac{11}{64}}$ respectively.
The ratio of masses of the chromatic and achromatic $W$ bosons is consequently

$\left(\frac{m_{W^{{\tilde{c}}}}}{m_{W}}\right)^{2}=\frac{1+\frac{2491}{288\left[k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\right]^{4}}}{1+\frac{19}{18\left[k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\right]^{4}}}\left(1+\mathcal{O}_{b}\right){.}$
(48)

In conjunction with the equations of A, this yields achromatic and chromatic
boson masses

$\displaystyle m_{W}$
$\displaystyle=80.3587(22)~{}\mathrm{G}\mathrm{eV}/c^{2}$ (49) $\displaystyle
m_{W^{{\tilde{c}}}{|_{{\tilde{c}}<9}}}$
$\displaystyle=80.4434(22)~{}\mathrm{G}\mathrm{eV}/c^{2}$ (50)

where $m_{W}$ is the achromatic $W$ boson mass associated with the trivial
$C$-sector representation $\lambda^{C}_{9}$. The other eight chromatic $W$
boson mass eigenvalues $1\leq{\tilde{c}}\leq 8$ corresponding to the
8-dimensional representation of $\mathrm{su}(3)_{C}$ are degenerate, as the
$\mathrm{SU}(3)_{C}$ symmetry of the color sector is unbroken by construction
[23].

In practice the chromatic $W$ boson mass eigenstates $1\leq{\tilde{c}}\leq 8$
are not observed directly. When a $p\overline{p}$ collision generates a
particle carrying $A$-sector representation $e^{A}_{23}$ (corresponding to
some version of the $W$ boson) or its conjugate, this propagates within the
color-rich environment of the collision for a duration $t_{p}$ which is
characterized by the per-hadron energy $\frac{\sqrt{s}}{2}$. In this context a
factor of $\frac{1}{2}$ arises because $W$ boson formation occurs on average
half-way through the time period for which the site of formation lies within
the color-rich environment. The expression for the mean time that the $W$
boson exists within a color-rich environment is therefore given by

$t_{p}=\frac{h}{2}\cdot\frac{2}{\sqrt{s}}=\frac{h}{|e|}\cdot[1.96~{}\mathrm{T}\mathrm{eV}]^{-1}$
(51)

which satisfies

$t_{p}>2t_{\Omega}.$ (52)

During period $t_{p}$ the boson undergoes multiple mass-generating
interactions with the pseudovacuum and mutiple color-exchange interactions
with the foreground environment as illustrated in Fig. 1(i).

Figure 1: (i) Heuristic representation of pseudovacuum (PV) and color-rich
foreground (CRF) interactions of a weak vector boson having $W$-type character
on the $A$ sector in CDF II. (ii) Heuristic representation of PV and CRF
interactions of a weak vector boson having $W$-type character on the $A$
sector in ATLAS. Time $2t_{\Omega}$ (37) is the minimum length scale reliably
supporting a mass-squared interaction, and $t_{p}$ is the timescale associated
with the energy $\frac{1}{2}\sqrt{s}$ of each proton or antiproton beam (51).
In any hadron collision, the confinement scale—and hence the extent of the
CRF—is determined the greater of $t_{\Omega}$ and $t_{p}$ in the pseudovacuum
isotropy frame, which approximates the centre of mass frame. The grey shaded
area represents multiple particle exchanges of unspecified type between the
propagating boson, the pseudovacuum, and (where present) the color-rich
foreground.

While individual mass vertices may be characterized as chromatic or
achromatic, recall that at the preon scale all scattering interactions involve
exchange of particles carrying both $A$-sector and $C$-sector charges, and in
a color-rich foreground environment a boson associated with representation
$\lambda^{C}_{9}$ may undergo $A$-sector-mediated interactions which as a
side-effect change its representation on the $C$ sector. An individual boson
therefore experiences chromatic mass vertex interactions and achromatic mass
vertex interactions $\frac{8}{9}$ and $\frac{1}{9}$ of the time respectively,
being associated with factors of $m_{W^{{\tilde{c}}}}^{2}$ and $m_{W}^{2}$.
For a $W$-type boson in a color-rich environment the mean mass-squared per
vertex over period $t_{p}$ is therefore given by

$\left([m_{W}]_{\mathrm{r.m.s.}}\right)^{2}:=\frac{1}{9}\sum_{{\tilde{c}}\in\\{1,\ldots,9\\}}m_{W^{{\tilde{c}}}}^{2}$
(53)

where $m_{W^{9}}$ corresponds to the achromatic $W$ boson mass $m_{W}$. The
mass $[m_{W}]_{\mathrm{r.m.s.}}$ equates to $m_{W^{\dot{c}c}}$ in Sec.
LABEL:sec:colouredWmass of Ref. 17, and substituting for $m_{W}$ and
$m_{W^{\tilde{c}}}$ yields

$[m_{W}]_{\mathrm{r.m.s.}}=80.4340(22)~{}\mathrm{G}\mathrm{eV}/c^{2}.$ (54)

This expression receives a small energy-dependent correction due to the
difference in mass of the different $W$ species, which is assumed to be
negligible at the energy scales and precision of the current paper.

Since $t_{p}$ in CDF II is short compared with the lifetime of the $W$ boson
[total width $2.085(42)~{}\mathrm{G}\mathrm{eV}$] [6], there is then a
transition at the end of time period $t_{p}$ from a color-rich to a color-poor
environment. Prior to this transition the $W$-type boson undergoes multiple
foreground interactions in which it may gain or lose momentum, and its
preferred mean rest energy is determined by the mass vertices it has
experienced up to this point, corresponding to $[m_{W}]_{\mathrm{r.m.s.}}$.
For the boson to escape confinement, it must be color-neutral on transitioning
from $t<t_{p}$ to $t\gg t_{p}$, a criterion which implies the recapture of
$\frac{8}{9}$ of $W$-type bosons. Although those which escape are now off-
shell, being now subject only to achromatic ($\lambda^{C}_{9}$) mass vertices,
they remain in an excited state because they no longer exist in a context of
multiple foreground scattering interactions, and so are limited in their
ability to shed this $75~{}\mathrm{M}\mathrm{eV}$ of surplus energy. Candidate
processes such as $W^{*}\rightarrow W+\gamma+\gamma$ are characterized by the
timescale of the surplus, and thus are slow when compared with the usual $W$
boson decay modes. The excited $W$-type bosons therefore decay via the usual
channels, including the dilepton channels

$\displaystyle(W^{-})^{*}$ $\displaystyle\longrightarrow
e_{L}+\overline{\nu}_{e}$ (55) $\displaystyle(W^{+})^{*}$
$\displaystyle\longrightarrow\overline{e}_{L}+\nu_{e}.$ (56)

As usual the (anti)neutrino carries away missing transverse momentum, while
the charged lepton energy distribution reflects the off-shell, excited mass of

$[m_{W}]_{\mathrm{CDF\,II}}=[m_{W}]_{\mathrm{r.m.s.}}=80.4340(22)~{}\mathrm{G}\mathrm{eV}/c^{2}.$
(57)

Note that the recapture of $\frac{8}{9}$ of $W$-type bosons cancels the
ninefold increase in number of $W$-boson-generating channels, and hence the
predicted rate of $W$ boson production is not significantly changed from that
of the Standard Model.

####  3.3.2 In ATLAS

Now compare with the measurement of $W$ boson mass in the ATLAS experiment
[20, 21]. Collection of the existing ATLAS dataset took place at a collider
energy of
$E_{\mathrm{ATLAS}}\,{:\\!}={\sqrt{s}=}~{}7~{}\mathrm{T}\mathrm{eV}$, again
with individual $W$ boson formation events being sub-$\mathrm{T}\mathrm{eV}$.
The energy of $W$ boson formation in ATLAS therefore falls within the interval
in which the pseudovacuum appears color-rich,

$3.59~{}\mathrm{G}\mathrm{eV}\approx\mathcal{E}_{0}<2E<\mathcal{E}_{\Omega}\approx
6.18~{}\mathrm{T}\mathrm{eV},{}$ (41)

but the centre-of-mass energy $\sqrt{s}=7~{}\mathrm{T}\mathrm{eV}$ does not
satisfy the color-rich foreground constraint

${3.59~{}\mathrm{G}\mathrm{eV}\approx\mathcal{E}_{0}<\sqrt{s}<\frac{\mathcal{E}_{\Omega}}{2}\approx
3.09~{}\mathrm{T}\mathrm{eV}}.{}$ (45)

As noted in Sec. 3.2.3 confinement in CASMIR operates over a minimum timescale
of $t_{\Omega}$ in the pseudovacuum isotopy frame, so the duration of the
color-rich collision environment in ATLAS is given by

$\frac{h}{|e|}\min{(\sqrt{s},\mathcal{E}_{\Omega})}^{-1}$ (58)

rather than $\frac{h}{|e|}\sqrt{s}$, but even on substituting
$\mathcal{E}_{\Omega}$ for $\sqrt{s}$ in Eq. (45) the lifetime of the color-
rich foreground is seen to be insufficient to support chromatic weak vector
boson mass vertices. The LHC at $7~{}\mathrm{T}\mathrm{eV}$ therefore
generates only achromatic massive weak vector bosons, with CASMIR predicting
an observed $W$ boson mass at ATLAS of

$m_{W}=80.3587(22)~{}\mathrm{G}\mathrm{eV}/c^{2}{}.$ (49)

It is also worth noting a further key difference in the selection of $W$ boson
decay events for the ATLAS calculation. To reduce background contributions at
energy scales of interest, the ATLAS evaluations of $W$ boson mass explicitly
exclude events in which the charged lepton is closely spatially related to a
hadron jet:[11, 20]

  * •

“Additional isolation requirements on the nearby activity in the ID and
calorimeter are applied to improve background rejection. These isolation
requirements are implemented by requiring the scalar sum of the
$p_{\mathrm{T}}$ of tracks in a cone of size $\Delta
R\equiv\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}<0.4$ around the electron,
$p^{e,\mathrm{cone}}_{\mathrm{T}}$, and the transverse energy deposited in the
calorimeter within a cone of size $\Delta R<0.2$ around the electron,
$E^{\mathrm{cone}}_{\mathrm{T}}$, to be small.”[20]

  * •

“Similarly to the electrons, the rejection of multijet background is increased
by applying an isolation requirement : the scalar sum of the $p_{\mathrm{T}}$
of tracks in a cone of size $\Delta R<0.2$ around the muon candidate,
$p^{\mu,\mathrm{cone}}_{\mathrm{T}}$, is required to be less than 10% of the
muon $p_{\mathrm{T}}$.”[20]

If color charges are entrained along the trajectory of the $W$ boson prior to
it crossing the confinement scale, it is possible that this may bias the
direction of jet emission to align with the hypothesized mass-enhanced $W$
bosons. The effect of apparent rest mass enhancement is not anticipated to
depend on the direction of jet emission, and although this cut may result in
exclusion of a larger-than-anticipated number of $W$ boson decay events at
energies $\sqrt{s}<\frac{1}{2}\mathcal{E}_{\Omega}$, it is not anticipated to
have a significant impact on the value measured for the apparent rest mass of
the $W$ boson.

Measurements and predictions for $W$ boson masses are summarized in Table
3.3.2.

Comparison of predictions from CASMIR with $W$ boson mass measurements from
CDF II (Tevatron),[7] ATLAS (LHC),[11] and LEP.[27] The Standard Model (SM)
prediction for the $W$ boson mass is
$80.356(6)~{}\mathrm{G}\mathrm{eV}/c^{2}$,[4, 5, 6] while the CASMIR
predictions for the achromatic $(W)$ and mixed chromatic/achromatic
$(W^{\dot{c}c})$ boson masses are $80.3587(22)~{}\mathrm{G}\mathrm{eV}/c^{2}$
and $80.4340(22)~{}\mathrm{G}\mathrm{eV}/c^{2}$ respectively. As noted in A,
mass uncertainties in CASMIR are likely overestimated. To avoid unfairly
advantaging the CASMIR results, an upper bound on tensions is also obtained by
taking the CASMIR uncertainty to zero; where this upper bound differs to the
calculated tension, it is reported after the tension in brackets. Tensions
smaller than $1\,\sigma$ are displayed in green online.  Experiment Predicted
Mass observed Tension with Tension with Tension with $W$ boson mass
$(\mathrm{G}\mathrm{eV}/c^{2})$ SM $W$ CASMIR $W$ CASMIR $W^{\dot{c}c}$ CDF II
mixed ($W^{\dot{c}c}$) $80.4335(94)$ $6.9\,\sigma$ $7.7\,\sigma~{}(\leq
8.0\,\sigma)$ $0.1\,\sigma$ ATLAS achromatic ($W$) $80.376(33)$ $0.2\,\sigma$
$0.1\,\sigma$ $4.6\,\sigma$ LEP achromatic ($W$) $80.360(16)$ $0.6\,\sigma$
$0.5\,\sigma$ $1.8\,\sigma$

Further comparisons with other measurements of $W$ boson mass are shown in
Fig. 2.

Figure 2: Comparison of $W$ boson mass measurements[7, 11, 28, 29, 27, 30, 31,
32, 33, 34, 35] with predictions of the Standard Model (SM $W$),[4, 5, 6] the
CASMIR achromatic $W$ boson (CASMIR $W$), and the CASMIR mixed
chromatic/achromatic $W$ boson (CASMIR $W^{\dot{c}c}$). “LEP” denotes the
average value from LEP.[27]

If these results are compared with the predictions of CASMIR then with the
exception of L3 at $1.6\,\sigma$, all results have tension $0.7\,\sigma$ or
less with the CASMIR predicted mass for either the achromatic or mixed
chromatic/achromatic $W$ boson species.

####  3.3.3 In LEP

It is next appropriate to ask whether CASMIR might also predict observation of
a mixed chromatic and achromatic $W$ boson signal in the combined pair
production data from LEP (see ALEPH, DELPHI, L3, OPAL, and the combined LEP
value in Fig. 2) [31, 30, 33, 32, 27]. The widths of the LEP data error bars
prohibit clean discrimination between chromatic and achromatic boson signals,
such that even for the combined LEP dataset, which is most consistent with the
achromatic $W$ boson mass $m_{W}$, tension with the mixed chromatic and
achromatic value $m_{W^{\dot{c}c}}$ is just $1.8\,\sigma$, and is dominated by
experimental uncertainty. The LEP datasets therefore make no assessment as to
whether CASMIR provides a qualitatively good effective description of observed
weak boson mass. Recognising that lepton collisions in LEP do not occur within
a spatially extensive color-rich environment as per Sec. 3.2.3, CASMIR
predicts only achromatic massive weak vector boson formation in lepton
colliders such as LEP.

###  3.4 The chromatic $Z$ boson

There is also a further possibility for testing the predictive power of CASMIR
in the weak sector. Similar to the chromatic $W$ bosons described above,
CASMIR also necessarily incorporates the existence of chromatic $Z$ bosons in
color-rich environments at energies below $\mathcal{E}_{\Omega}$. Repeating
the boson mass calculations of Sec. 3.3 for $Z$ bosons while solving the mass
relationships of A yields

$\displaystyle\left(\frac{m_{Z^{{\tilde{c}}}}}{m_{Z}}\right)^{2}$
$\displaystyle=\frac{1+\frac{3979}{1152\left[k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\right]^{4}}}{1+\frac{55}{18\left[k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\right]^{4}}}\left(1+\mathcal{O}_{b}\right)$
(59) $\displaystyle\left([m_{Z}]_{\mathrm{r.m.s.}}\right)^{2}$
$\displaystyle=\frac{1}{9}\sum_{{\tilde{c}}\in\\{1,\ldots,9\\}}m_{Z^{{\tilde{c}}}}^{2}$
(60) $\displaystyle m_{Z^{\tilde{c}}|_{{\tilde{c}}<9}}$
$\displaystyle=91.1927(35)~{}\mathrm{G}\mathrm{eV}/c^{2}$ (61) $\displaystyle
m_{Z}$ $\displaystyle=91.1877(35)~{}\mathrm{G}\mathrm{eV}/c^{2}$ (62)
$\displaystyle[m_{Z}]_{\mathrm{CDF\,II}}$
$\displaystyle=[m_{Z}]_{\mathrm{r.m.s.}}=91.1922(35)~{}\mathrm{G}\mathrm{eV}/c^{2}.$
(63)

As was seen for chromatic $W$ bosons of Sec. 3.3, a chromatic $Z$ boson is not
independently detectable using existing apparatus, but the existence of the
chromatic $Z$ boson mass vertices within a sufficiently long-lived color-rich
collision environment causes $Z$ bosons to be emitted in an excited state. In
contrast with $W$ boson emission, when a $Z$ boson decays via a charged
dilepton channel both decay products may be recovered, permitting higher
precision measurements of boson mass to be achieved. However, the separation
of the normal and excited energy levels is more than an order of magnitude
smaller than for the $W$ boson, at only $4.5~{}\mathrm{M}\mathrm{eV}$.
Comparison of theory and experiment for CDF II and LEP is presented in Table
3.4, but tensions are small and no significant conclusions can be drawn.

Comparison of predictions from CASMIR with $Z$ boson mass measurements from
CDF II (Tevatron)[7]and LEP.[27] CASMIR predictions for the achromatic $(Z)$
and mixed chromatic/achromatic $(Z^{\dot{c}c})$ boson masses are
$91.1877(35)~{}\mathrm{G}\mathrm{eV}/c^{2}$ and
$91.1922(35)~{}\mathrm{G}\mathrm{eV}/c^{2}$ respectively. As noted in A, mass
uncertainties in CASMIR are likely overestimated. To avoid unfairly
advantaging the CASMIR results, an upper bound on tensions is also obtained by
taking the CASMIR uncertainty to zero; where this upper bound differs to the
calculated tension, it is reported after the tension in brackets. Tensions
smaller than $1\,\sigma$ are displayed in green online.  Experiment Predicted
Mass observed Tension with Tension with $Z$ boson mass
$(\mathrm{G}\mathrm{eV}/c^{2})$ CASMIR $Z$ CASMIR $Z^{\dot{c}c}$ CDF II mixed
($Z^{\dot{c}c}$) $91.1920(75)$ $\phantom{<}\;0.5\,\sigma~{}(\leq 0.6\,\sigma)$
$<0.1\,\sigma$ LEP achromatic ($Z$) $91.1876(21)$ $<0.1\,\sigma$
$\phantom{<}\;1.1\,\sigma~{}(\leq 2.2\,\sigma)$

Fascinatingly, in addition to data collected by CDF II there also exists one
other dataset which is already sensitive to signals consistent with chromatic
$Z$ bosons (as well as chromatic $W$ bosons), and which returns a result
consistent with their existence. This is the Muon $g-2$ experiment at Fermilab
[8], which is discussed next.

##  4 Muon $g-2$

###  4.1 Background and Experimental Status

Measurement of the gyromagnetic anomaly of the muon also provides a highly
sensitive test of the Standard Model. Classically, the gyromagnetic ratio of
the muon is expected to take a value of 2. Quantum corrections adjust this
value, with the corrected figure being written

$g=2(1+a_{\mu})\qquad\mathrm{or}\qquad g-2=2a_{\mu}$ (64)

where $a_{\mu}$ is the muon gyromagnetic anomaly. The Standard Model values of
the electron and muon gyromagnetic anomalies have both been calculated to
remarkable precision [36, 37, 9], and as probes of beyond-Standard-Model
physics, the greater mass of the muon provides for greater sensitivity to
corrections arising at the weak boson scale, and the theoretical values of
these corrections in the Standard Model have been calculated to two loops and
estimated to three [38, 39]. Hadron terms are of similar magnitude, and _de
novo_ calculation poses significant computational challenges, so the hadron
contributions to the gyromagnetic anomaly are generally inferred from
measurements on particle decay processes [9]. The outcome of this formidable
calculation is a prediction

$\displaystyle a_{\mu}^{\mathrm{SM}}$ $\displaystyle=116591810(43)\times
10^{-11}$ (65)

to which electroweak terms incorporating massive bosons (i.e. $W$, $Z$, and
Higgs) contribute

$a_{\mu}^{\mathrm{EW}}=153.6(1.0)\times 10^{-11}.$ (66)

Curiously, this result is also in conflict with results from Fermilab. This
time, the conflict is with the Muon $g-2$ experiment at Fermilab [8] which in
combination with the prior dataset from Brookhaven yelds a global average of

$a_{\mu}^{\mathrm{EXP}}=1165920{59(22)}\times 10^{-11}.$ (67)

At $5.2\,\sigma$ tension this conflict surpasses the conventional $5\,\sigma$
tension required to declare a discovery. However, there have been two
important developments since the theoretical value $a^{\mathrm{SM}}_{\mu}$ was
obtained in 2020. These are:

  * •

a lattice calculation of the Hadron Vacuum Polarisation (HVP) which disagrees
with previous data-driven values obtained from $e^{+}e^{-}$ experiments [12],
and

  * •

a measurement of the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ cross-section at
CMD-3 which also disagrees with previous $e^{+}e^{-}$ experiments. [13]

Attempts to reconcile these results are ongoing [14]. Pending this
reconciliation, it is likely that a contemporary re-evaluation of the Standard
Model prediction for $a_{\mu}$ would carry greater uncertainty and thus
smaller tension with experiment. Nevertheless, this result is also considered
a strong candidate to indicate possible beyond-Standard-Model physics.

###  4.2 Relevance to CASMIR and CDF II

It is possible to independently cross-check the predictions of CASMIR using
the results of the Muon $g-2$ experiment, and doing so provides additional
support for the CASMIR model of the CDF II measurement of $W$ boson mass.
Supplementing colorless photons, $W$, and $Z$ bosons with their more numerous
chromatic counterparts yields a correction to the value of
$a_{\mu}^{\mathrm{EW}}$ in CASMIR which is readily calculated at the 1-loop
level and approximated at the 2-loop level. In principle the introduction of
these species also affects the gluon/hadron contributions to $a_{\mu}$.
However, since the values of these contributions are currently derived from
experiment, the measured values are understood to already incorporate any
contributions from all relevant physical processes (see also Sec. 4.3.4).

The resulting value of $a_{\mu}$ is shown in Sec. 4.3 to be

$a_{\mu}^{\mathrm{CASMIR}}=116592071(46)\times 10^{-11}.$ (68)

Tension with experimental results is reduced from ${5.2}\,\sigma$ for the
Standard Model to $0.2\,\sigma$ for CASMIR. This reduction in tension
primarily reflects that the CASMIR central value differs from the experimental
value by only $0.5$ times the experimental uncertainty
$\sigma^{\mathrm{EXP}}$, compared with ${11}\,\sigma^{\mathrm{EXP}}$ for the
Standard Model.

###  4.3 Calculation in CASMIR

####  4.3.1 Overview of massive vector boson exchange in CASMIR

##### Bosons are chromatic:

As noted above, in the full $\mathrm{SU}(9)$ treatment of CASMIR the $W$ and
$Z$ bosons are augmented by eight colored counterparts with masses
$m_{W^{{\tilde{c}}}}=80.4434(22)~{}\mathrm{G}\mathrm{eV}/c^{2}$ and
$m_{Z^{{\tilde{c}}}}=91.{1927}(35)~{}\mathrm{G}\mathrm{eV}/c^{2}$
respectively. These colored species appear in addition to the usual colorless
bosons in color-rich environments (Sec. 3.2.3), and augment the masses of $W$
and $Z$ boson loops (Ref. 17, Sec. LABEL:sec:tW). Similarly, the photon is
supplemented by eight chromatic photons which are massless and have nontrivial
representation on the $C$ sector. Further, all fermion species in CASMIR are
composed from three colored preons which are bound by gluon exchange, and all
vector bosons are constructed from co-propagating preon/antipreon pairs.

In Appendix LABEL:apdx:massloops of Ref. 17 it is argued that loop corrections
to vertex functions in CASMIR are dominated by short-range contributions which
may be identified as those on the order of half the wavelength of the loop
boson(s) or less. The muons in the Fermilab experiment are held in a storage
ring at the “magic momentum” of $3.1~{}\mathrm{G}\mathrm{eV}/c$, corresponding
to an energy (including rest energy) of ${E}=3.2~{}\mathrm{G}\mathrm{eV}$
prior to any fluctuations arising from interaction with the pseudovacuum. As
discussed in Sec. 3.2.3, the timescale for comparison with pseudovacuum
correlations is that associated with half a wavelength,

$\frac{1}{2}h\cdot[3.2~{}\mathrm{G}\mathrm{eV}]^{-1}=h\cdot[6.4~{}\mathrm{G}\mathrm{eV}]^{-1}.$
(69)

This timescale corresponds to an energy scale
$\mathcal{E}=6.4~{}\mathrm{G}\mathrm{eV}$ satisfying
$\mathcal{E}_{0}<\mathcal{E}<\mathcal{E}_{\Omega}$ (39), or equivalently the
storage ring energy of $E=3.2~{}\mathrm{G}\mathrm{eV}$ satisfies the
relationship $1.79~{}\mathrm{G}\mathrm{eV}<E<3.09~{}\mathrm{T}\mathrm{eV}$
(42), permitting the appearance of virtual colored boson species. However, in
contrast to the hadron collisions in CDF II and ATLAS, the circulating muons
do not experience a color-rich foreground environment and therefore there is
no foreground chromatic $W$ or $Z$ boson formation. The role of these species
is therefore restricted to appearances as virtual particles in loop
corrections.

##### Chromatic enhancement of photon emission:

As discussed in Sec. LABEL:sec:colouredAmass of Ref. 17, the emission of
chromatic photons may enhance the electromagnetic field of a charged particle
where appropriate color freedoms exist. The gauge-like constraint that
fermions always exhibit color neutrality (Ref. 17, Sec. LABEL:sec:Csector)
prohibits the direct (tree-level) emission of chromatic photons from a lepton,
but leaves open the prospect of their emission during electroweak loop
corrections to the tree level process.

Consider the photon emission processes shown in Fig. 3.

Figure 3: (i) One-loop $W$ boson correction to the electromagnetic vertex. As
per Sec. LABEL:sec:tW of Ref. 17 the loop boson is denoted ${\tilde{W}}$. (ii)
Color-labeled preon structure of the $W$ boson loop. (iii) Color-labeled preon
structure of the $W$ boson loop with photon emission. If the photon is
permitted to be chromatic, the number of free preon color labels is increased
by one. (iv) One-loop $W$ boson correction to the electromagnetic vertex, with
chromatic photon emission in the $e^{C}_{ij}$ basis, and restoration of lepton
color neutrality through a gluon-like change of coordinate frame. The notation
${\tilde{W}}^{\prime}$ indicates that a color perturbation has acted on the
superposition making up the loop $W$ boson.

In diagram (i) a photon is emitted from a $W$ boson in a 1-loop correction to
the electromagnetic vertex. The preon structure of the $W$ loop is shown in
diagram (ii), with photon emission being added in diagram (iii). Comparing
diagrams (ii) and (iii), if the photon is free to carry color charge then an
additional independent color label is introduced for a multiplicative factor
of three on the contribution of this loop. Separability of $A$ and $C$ sectors
(Ref. 17, Appendix LABEL:apdx:ACseparability) ensures that chromatic photons
contribute to the electromagnetic field in the far-field regime without
needing to explicitly solve for confinement of the color charge (Ref. 17, Sec.
LABEL:sec:colouredAmass). In practice this process must simultaneously render
the photon colorless and restore the color neutrality of the preon triplet
making up the lepton; the latter effect is achieved by an induced change of
coordinate frame as per Sec. LABEL:sec:Csector which only occurs when the $W$
boson is absorbed by the preon triplet. This coordinate change is functionally
equivalent to exchange of a massless gluon with trivial vertices and a delta
function over momentum. By color confinement and $A/C$ separability this may
conveniently be assumed to recombine with the chromatic photon or its products
and restore color neutrality as per Fig. 3(iv), and by identification with a
choice of coordinate frame the vertices and loop coefficient associated with
this boson must evaluate to 1.

Now recognize that the loop boson in diagram (i) is as described in Sec.
LABEL:sec:tW of Ref. 17, having a $\frac{2}{9}$ chance of being of definite
coloration. When coloration is definite at source and sink, the preons at the
$W^{\tilde{c}}W^{{\tilde{c}}\dagger}A$ vertex are constrained to be of
identical color, and no factor of three accrues. For the other $\frac{7}{9}$,
recognize that both preons are in equal superposition of color charges $r$,
$g$, and $b$, and that summing across all elements of the $e^{C}_{ij}$ basis
all color pairs are equally likely, up to those excluded as part of the
$\frac{2}{9}$ for which color is fixed and definite. Further, by symmetry of
muon construction under $\mathrm{SU}(3)_{C}$ the probability of any given
coloration being associated with that $\frac{2}{9}$ fraction is equally
distributed across all possible pairs. Interaction with the chromatic photons
leaves the any $\mathrm{SU}(3)_{C}$-invariant initial state unchanged, and
thus the $\frac{7}{9}$ fraction of loops which are not singly-labeled in the
$e^{C}_{ij}$ basis may emit $\mathrm{SU}(3)_{C}$-invariant superpositions of
chromatic photons while themselves remaining unchanged.

Finally, returning to the $\frac{2}{9}$ chance that the loop $W$ boson is of
definite coloration, recognize that whereas the mass interaction of Sec.
LABEL:sec:tW of Ref. 17 takes place at multiple separated points along a
finite interval which incorporates segments of fermion propagator, the
electromagnetic interaction comprises a pointlike tree-level interaction and
higher-order terms which are written as corrections to this point-like
interaction. Corrections to the EM vertex must therefore explicitly
incorporate factors arising from any fermion crossings which take place. In
the mass interaction, the ability of the colored $W$ boson to be reabsorbed by
either of two color-appropriate preons yielded an absorption probability of
$\frac{2}{3}$ comprising $\frac{1}{3}+\frac{1}{3}$. These two terms differ by
a preon pair exchange. In the mass interaction this exchange, if it occurs, is
absorbed into the definition of the propagator segment. In the electromagnetic
loop correction, on the other hand, the crossing yields a factor of $-1$ which
is not subsumed, and thus

$\frac{1}{3}\times\left(\frac{1}{3}+\frac{1}{3}\right)=\frac{2}{9}\mathrm{~{}~{}~{}becomes~{}~{}~{}}\frac{1}{3}\times\left(\frac{1}{3}-\frac{1}{3}\right)=0.$
(70)

Consequently any diagram emitting a photon from a loop $W$ boson attracts a
net multiplicative factor across all terms of the $W$ boson superposition
given by

$1\times\left[\frac{1}{3}\times\left(\frac{1}{3}-\frac{1}{3}\right)\right]+3\times\left(7\times\frac{1}{3}\times\frac{1}{3}\right)=\frac{7}{3}.$
(71)

Denote this color-related factor by

$\Theta_{c}:=\frac{7}{3}.$ (72)

Note that discrimination between the $\frac{7}{9}$ and $\frac{2}{9}$
components of the $W$ boson superposition (and evaluation of their associated
coefficients) is dependent on the structure of the diagram as a whole, and
arises on a per-loop and not a per-vertex basis.

Next, consider the $Z$ loop shown in Fig. 4(i).

Figure 4:  (i) One-loop $Z$ boson correction to the electromagnetic vertex. As
per (Ref. 17, Sec. LABEL:sec:tW) the loop boson is denoted $\tilde{Z}$. (ii)
One-loop $Z$ boson correction to the electromagnetic vertex, with chromatic
photon emission in the $e^{C}_{ij}$ basis, and restoration of lepton color
neutrality twice through gluon-like changes of coordinate frame. The notation
$\tilde{Z}^{\prime}$ indicates that a color perturbation has acted on the
superposition making up the loop $Z$ boson ($\tilde{Z}$).

As with the loop $W$ boson, the preons within the $Z$ loop exist in a
superposition of color and there is a freedom to act on these preons with
color transformations which transform individual preons provided the
superposition as a whole remains unchanged. If the photon coupling is replaced
with a chromatic photon coupling, the interacting preon changes color. As this
preon belongs to a fermion triplet, and such triplets define color neutrality
as per Sec. LABEL:sec:Csector, neutrality must be immediately restored by a
change of coordinate frame on ${{\mathbb{C}^{\wedge 18}}}$ which may be
represented as a massless boson associated with a loop evaluating to a factor
of 1.666In principle the process of color transfer may be arbitrarily
complicated; however, since this gluon exchange is equivalent to a change in
coordinate frame, there must always exist a choice of parameters in which the
associated loop evaluates to a factor of 1. In the presence of the $Z$ loop,
this coordinate transform may be chosen to transfer the change of color to the
$Z$ boson as shown in Fig. 4(ii). On recombination with the triplet, a further
frame change carries this color transformation out to recombine with the color
of the chromatic photon as discussed for the $W$ boson scenario above.
Although the CASMIR loop $Z$ boson has the same mass as the free $Z$ boson, it
nevertheless retains the same color considerations as the loop $W$ boson and
therefore this color transfer is admissible seven times out of nine with the
other two terms vanishing. Both $W$ and $Z$ loops therefore attract factors of
$\Theta_{c}$.

##### Change in boson masses:

The next factor to be considered is that associated with the $W$ boson mass.
Compared with the Standard Model scenario, $W$ boson masses in loops are
rescaled by a factor

$\Theta_{m}:=\frac{m_{\tilde{W}}^{2}}{m_{W}^{2}}$ (73)

which evaluates in CASMIR as

$\Theta_{m}=\frac{1+\frac{51}{18\left[k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\right]^{4}}}{1+\frac{19}{18\left[k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\right]^{4}}}+\mathrm{O}\left\\{\left[k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\right]^{-8}\right\\}$
(74)

for

$\left[k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\right]^{-4}=2.77906(21)\times
10^{-4}.$ (75)

Since the present paper evaluates $a_{\mu}^{\mathrm{CASMIR}}$ as a correction
to the result of Ref. 39, the value of $\Theta_{m}$ is evaluated in Eq. (73)
using the value of $m_{W}$ adopted in the original calculation, namely
$80.363(13)~{}\mathrm{G}\mathrm{eV}/c^{2}$. The $Z$ and Higgs masses are not
rescaled.

####  4.3.2 1-loop terms

The Standard Model one-loop corrections to the gyromagnetic ratio of the muon
have been well understood for many decades [40, 41, 42, 43, 44], incorporating
contributions from the $W$, $Z$, and Higgs boson. However, for a Higgs boson
of approximately $125~{}\mathrm{G}\mathrm{eV}/c^{2}$ the Standard Model Higgs
contribution is approximately $2.2\times 10^{-14}$ [45], falling below the
current threshold for experimental relevance. In CASMIR all Higgs vertices
below $\frac{1}{2}\mathcal{E}_{\Omega}\approx 3.1~{}\mathrm{T}\mathrm{eV}$ are
suppressed by a further factor of
$\big{[}k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\big{]}^{-2}$ apiece, reducing
this value to $6.1\times 10^{-18}$. The portion of the 1-loop corrections
relevant at current experimental precision therefore takes the form

$a_{\mu}^{\mathrm{EW}(1)}=\frac{G_{F}m_{\mu}^{2}}{\sqrt{2}\cdot
8\pi^{2}}\left[\frac{10}{3}+\frac{1}{3}\left(1-4\sin^{2}\theta_{W}\right)^{2}-\frac{5}{3}\right]$
(76)

where the Fermi constant $G_{F}$ is proportional to $m_{W}^{\;-2}$. Within the
square brackets, $10/3$ arises from the $W$ boson loop correction and the rest
arises from the $Z$ boson loop correction.

As discussed in Sec. 4.3.1, both the $W$ boson and $Z$ boson loop corrections
are augmented by a factor of $\Theta_{c}$ due to additional coupling to
chromatic photons. Further, the value of $G_{F}$ is determined from muon
decays and in CASMIR is therefore associated with the mass of the colorless
$W$ boson. However, in loops the effective $W$ boson mass is given by
$m_{\tilde{W}}^{\;-2}$ and the $W$ boson contributions to
$a_{\mu}^{\mathrm{EW}}$ must therefore be multiplied by a factor of
$\Theta_{m}^{\;-1}$. There is no mass difference in CASMIR between free and
loop $Z$ bosons and hence this factor accrues to the $W$ boson contributions
only.

Finally, expanding

$\sin^{2}\theta_{W}=1-\frac{m_{W}^{2}}{m_{Z}^{2}},$ (77)

recognize that when $m_{W}^{2}$ appears in this expression it does so in the
context of writing the $Z$ boson calculation in terms of the $W$ boson
calculation—which was performed using the Standard Model $W$ boson mass,
equivalent to the CASMIR achromatic $W$ boson mass. Therefore, no correction
to the factors of $m_{W}^{2}$ appearing in $(1-4\sin^{2}\theta_{W})^{2}$ are
required.

The net correction is thus

$\displaystyle a_{\mu}^{\mathrm{EW}(1)}\longrightarrow
a_{\mu}^{\mathrm{EW(1);CASMIR}}:=\,$
$\displaystyle\frac{\Theta_{c}G_{F}m_{\mu}^{2}}{\sqrt{2}\cdot
8\pi^{2}}\left[\frac{10}{3\,\Theta_{m}}+\frac{1}{3}\left(1-4\sin^{2}\theta_{W}\right)^{2}-\frac{5}{3}\right]$
(78) $\displaystyle=\,$
$\displaystyle\Theta_{c}\Theta_{M}^{(1)}\,a_{\mu}^{\mathrm{EW}(1)}$ (79)

where

$\Theta_{M}^{(1)}:=\frac{\frac{10}{3\,\Theta_{m}}+\frac{1}{3}\left(1-4\sin^{2}\theta_{W}\right)^{2}-\frac{5}{3}}{\frac{10}{3}+\frac{1}{3}\left(1-4\sin^{2}\theta_{W}\right)^{2}-\frac{5}{3}}.$
(80)

This increases the value of $a_{\mu}^{\mathrm{EW}(1)}$ from $194.80(1)\times
10^{-11}$ to $454.18(30)\times 10^{-11}$, with the factor $\Theta_{M}^{(1)}$
contributing $-0.35(30)\times 10^{-11}$.

####  4.3.3 2-loop terms and beyond

Whereas it is relatively straightforward to calculate the 1-loop corrections
in CASMIR, full evaluation of the 2-loop corrections is substantially more
involved and is beyond the scope of this paper. However, following Ref. 39 to
write

$\displaystyle\begin{split}a_{\mu}^{\mathrm{EW}}=\;&a_{\mu}^{\mathrm{EW}(1)}+a^{\mathrm{EW}(2)}_{\mu;\mathrm{bos}}+a^{\mathrm{EW}(2)}_{\mu}(e,\mu,u,c,d,s)+a^{\mathrm{EW}(2)}_{\mu}(\tau,t,b)\\\
&+a^{\mathrm{EW}(2)}_{\mu;\textrm{f-rest,H}}+a^{\mathrm{EW}(2)}_{\mu;\textrm{f-rest,no\,H}}+a^{\mathrm{EW}(\geq
3)}_{\mu},\end{split}$ (81)

a reasonable estimate of the correction may be obtained term-by-term at the
cost of only a relatively small increase in uncertainty. In keeping with Ref.
39, uncertainties in these terms are treated as correlated. There are two
relevant sources of corrections to these terms. First, chromatic bosons may
introduce factors of $\Theta_{c}$. Second, terms involving Higgs exchange are
suppressed by factors of
$\big{[}k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\big{]}^{-4}$. There are also
corrections due to replacement of the loop $W$ boson mass
$m_{W}^{\;-2}\rightarrow m_{\tilde{W}}^{\;-2}$, but with this contribution to
the 1-loop correction already right on the edge of significance, it is assumed
that the corresponding contributions to the 2-loop corrections may be safely
neglected.

Begin with the corrections due to colored bosons. In diagrams containing
chromatic $W$ and $Z$ bosons, photon emission may be augmented by the color
sector. However, such diagrams must not contain a color “tadpole”. If color
enters a loop on one particle only, then by conservation of charge on the $C$
sector the color associated with that particle must be trivial. By Refs. 9,
10, 39, and 46 the majority of two-loop diagrams which admit colored bosons
under this criterion appear in term $a^{\mathrm{EW}(2)}_{\mu;\mathrm{bos}}$.
Some diagrams in this term attract a factor of $\Theta_{c}$ and some do not.
Therefore introduce

$\Theta_{A}^{r}:=1+\frac{\Theta_{c}-1}{2}\pm\frac{\Theta_{c}-1}{2}$ (82)

where the superscript r denotes that the values of $\Theta_{A}^{r}$ mark the
extrema of a range from 1 to $\Theta_{c}$. Some diagrams in
$a^{\mathrm{EW}(2)}_{\mu;\mathrm{bos}}$ also involve Higgs exchange, therefore
similarly introduce

$\displaystyle\Theta_{\bm{\mathrm{H}}}^{r}$
$\displaystyle:=f_{\bm{\mathrm{H}}}+\frac{1-f_{\bm{\mathrm{H}}}}{2}\pm\frac{1-f_{\bm{\mathrm{H}}}}{2}$
(83) $\displaystyle f_{\bm{\mathrm{H}}}$
$\displaystyle:=\big{[}k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\big{]}^{-4}.$
(84)

The CASMIR value of $a^{\mathrm{EW}(2)}_{\mu;\mathrm{bos}}$, denoted
$a^{\mathrm{EW}(2);\mathrm{CASMIR}}_{\mu;\mathrm{bos}}$, then falls within the
range bounded by the extrema of
$\Theta_{A}^{r}\Theta_{\bm{\mathrm{H}}}^{r}\,a^{\mathrm{EW}(2)}_{\mu;\mathrm{bos}}$.

It is convenient to replace $\Theta_{A}^{r}$ and
$\Theta_{\bm{\mathrm{H}}}^{r}$ with estimated values $\Theta_{A}$ and
$\Theta_{\bm{\mathrm{H}}}$ having an uncertainty derived from the indicated
ranges. Given full knowledge of the two-loop corrections this value could be
evaluated precisely, excepting only the uncertainties arising from the input
parameters, and (up to these uncertainties arising from the input parameters)
the distributions associated with $\Theta_{A}$ and $\Theta_{\bm{\mathrm{H}}}$
would be the Dirac delta function. On the other hand, assuming no prior
knowledge of the relative frequencies of the boson species in the loop
corrections yields the most pessimistic unimodal distribution, being one which
is flat between the upper and lower bounds and zero outside. For such a
distribution, 68.3% of values lie within 68.3% of the range. Therefore a
maximally conservative approximation is obtained by defining

$\displaystyle\Theta_{A}$ $\displaystyle:=1+\frac{\Theta_{c}-1}{2}$ (85)
$\displaystyle\Delta\Theta_{A}$ $\displaystyle:=\frac{\Theta_{c}-1}{2}\times
0.683$ (86) $\displaystyle\Theta_{\bm{\mathrm{H}}}$
$\displaystyle:=f_{\bm{\mathrm{H}}}+\frac{1-f_{\bm{\mathrm{H}}}}{2}$ (87)
$\displaystyle\Delta\Theta_{\bm{\mathrm{H}}}$
$\displaystyle:=\frac{1-f_{\bm{\mathrm{H}}}}{2}\times 0.683$ (88)

and treating $\Delta\Theta_{A}$ and $\Delta\Theta_{\bm{\mathrm{H}}}$ as the
effective $1\,\sigma$ uncertainties of $\Theta_{A}$ and
$\Theta_{\bm{\mathrm{H}}}$ respectively. The value of
$a^{\mathrm{EW}(2);\mathrm{CASMIR}}_{\mu;\mathrm{bos}}$ is then given by

$\begin{split}a^{\mathrm{EW}(2);\mathrm{CASMIR}}_{\mu;\mathrm{bos}}:=\,&\Theta_{A}\Theta_{\bm{\mathrm{H}}}\,a^{\mathrm{EW}(2)}_{\mu;\mathrm{bos}}\\\
=\,&-17(12)\times 10^{-11}.\end{split}$ (89)

Three other terms differ between CASMIR and the Standard Model. These are
$a^{\mathrm{EW}(2)}_{\mu;\textrm{f-rest,H}}$, which always contains a Higgs
boson and therefore in CASMIR attracts a factor of $f_{\bm{\mathrm{H}}}$,

$\begin{split}a^{\mathrm{EW}(2);\mathrm{CASMIR}}_{\mu;\textrm{f-rest,H}}:=\,&f_{\bm{\mathrm{H}}}\,a^{\mathrm{EW}(2)}_{\mu;\textrm{f-rest,H}}\\\
=\,&-0.0004196(28)\times 10^{-11},\end{split}$ (90)

$a^{\mathrm{EW}(2)}_{\mu;\textrm{f-rest,no\,H}}$, which attracts a factor of
$\Theta_{c}$ on its $ZZ\gamma$ term only, as shown in Fig. 5,

Figure 5: $ZZ\gamma$ term of $a^{\mathrm{EW}(2)}_{\mu;\textrm{f-rest,no\,H}}$
with chromatic $Z$ bosons in the $e^{C}_{ij}$ basis, with the induced
coordinate frame change on ${{\mathbb{C}^{\wedge 18}}}$ represented by the
gluons in grey. When photon emission is chromatic this also is associated with
an induced gluon interaction to restore color neutrality on the muon, and the
chromatic photon likewise returns an induced gluon on reducing to an
achromatic photon over distances large compared with the strong scale. It is
unnecessary to evaluate the contents of the shaded circle as the color
redistribution process as a whole corresponds to nothing more than a change of
coordinate frame on ${{\mathbb{C}^{\wedge 18}}}$, and therefore yields a
numerical factor of 1. Adapted from Fig. 1(e) of Ref. 47.

$\begin{split}a^{\mathrm{EW}(2);\mathrm{CASMIR}}_{\mu;\textrm{f-rest,no\,H}}:=\,&\Theta_{A}\,a^{\mathrm{EW}(2)}_{\mu;\textrm{f-rest,no\,H}}\\\
=\,&-7.7(2.1)\times 10^{-11},\end{split}$ (91)

and $a^{\mathrm{EW}(\geq 3)}_{\mu}$ which is an uncertainty attributed to
three-loop terms. For $a^{\mathrm{EW}(\geq 3);\mathrm{CASMIR}}_{\mu}$ assume
the greatest possible increase in uncertainty,

$\begin{split}a^{\mathrm{EW}(\geq
3);\mathrm{CASMIR}}_{\mu}:=\,&\Theta_{c}\,a^{\mathrm{EW}(\geq 3)}_{\mu}\\\
=\,&0.00(47)\times 10^{-11}.\end{split}$ (92)

All corrections to $a^{\mathrm{EW}(2)}_{\mu}$ and $a^{\mathrm{EW}(\geq
3)}_{\mu}$ are summarized in Table 4.3.3.

Estimated corrections to two-loop and three-loop contributions to the muon
gyromagnetic anomaly in CASMIR. These corrections arise due to emission of
chromatic photons, a correction to the $W$ boson mass in electroweak loops,
and the relative suppression of Higgs vertices in CASMIR. Standard Model
values are as per Ref. 39, with $a^{\mathrm{EW}(2)}_{\mu;\mathrm{bos}}$ and
$a^{\mathrm{EW}(2)}_{\mu;\textrm{f-rest,H}}$ updated as per Ref. 9.  Term
Boson species Relative factor SM value $\times 10^{11}$ CASMIR value $\times
10^{11}$ $a^{\mathrm{EW}(2)}_{\mu;\mathrm{bos}}$ $W/Z,~{}W/Z/A/H$
$\Theta_{A}\Theta_{\bm{\mathrm{H}}}$ $-19.96(1)$ $-17(12)$
$a^{\mathrm{EW}(2)}_{\mu}(e,\ldots)$ $Z,~{}A$ $1$ $-6.91(50)$ $-6.91(50)$
$a^{\mathrm{EW}(2)}_{\mu}(\tau,\ldots)$ $Z,~{}A$ $1$ $-8.21(10)$ $-8.21(10)$
$a^{\mathrm{EW}(2)}_{\mu;\textrm{f-rest,H}}$ $H,~{}Z/A$ $f_{\bm{\mathrm{H}}}$
$-1.51(1)$ $-0.0004196(28)$ $a^{\mathrm{EW}(2)}_{\mu;\textrm{f-rest,no\,H}}$
$W/Z,~{}W/Z/A$ $\Theta_{A}$ $-4.64(10)$ $-7.7(2.1)$ $a^{\mathrm{EW}(\geq
3)}_{\mu}$ Any $\Theta_{c}$ $0.00(20)$ $0.00(47)$

Collecting all terms and assuming fully correlated errors yields

$\begin{split}a^{\mathrm{EW};\mathrm{CASMIR}}_{\mu}=415(16)\times
10^{-11}.\end{split}$ (93)

####  4.3.4 A note on hadronic contributions

In addition to the electromagnetic and electroweak terms, the gyromagnetic
anomaly also receives significant contributions from the hadronic corrections
denoted $a^{\mathrm{HVP}}_{\mu}$ and $a^{\mathrm{HLbL}}_{\mu}$ [10, 9]. In a
first-principles calculation, these terms would also admit corrections due to
the introduction of chromatic $W$ and $Z$ bosons on energy scales between
$\mathcal{E}_{0}$ and $\mathcal{E}_{\Omega}$. However, these first-principles
calculations have not yet been achieved to the level of precision required,
and consequently the values of $a^{\mathrm{HVP}}_{\mu}$ and
$a^{\mathrm{HLbL}}_{\mu}$ are evaluated using data-driven techniques. If
CASMIR provides either a good description of reality (the chromatic $W$ and
$Z$ boson are real) or a good analogue to experiment (other behaviors occur
whose effects on $a_{\mu}$ are approximated by the chromatic bosons), then the
processes which generate input data for these data-driven calculations already
incorporate any necessary effects attributed in CASMIR to the chromatic weak
bosons, assuming that this input data also comes from color-rich environments,
e.g. colliders operating at appropriate energy scales.

Consequently, in the current data-driven approach no corrections are needed to
the values of $a^{\mathrm{HVP}}_{\mu}$ and $a^{\mathrm{HLbL}}_{\mu}$ adopted
in the final calculations of Refs. 9 and 10. Corrections involving chromatic
weak bosons would, however, be anticipated to be required for first-principles
calculations—again see Refs. 9, 10, and references therein.

####  4.3.5 Total muon gyromagnetic anomaly in CASMIR

Incorporating the CASMIR value for $a^{\mathrm{EW}}_{\mu}$ as per Eq. (93)
yields

$a_{\mu}^{\mathrm{CASMIR}}=116592071(46)\times 10^{-11}.{}$ (68)

Despite the approximations employed, this computed result is only fractionally
less precise than the Standard Model figure of
$a_{\mu}^{\mathrm{SM}}=116591810(43)\times 10^{-11},$ and in contrast with the
Standard Model result, is in near-perfect agreement with experiment (tension
$0.2\,\sigma$ with BNL/Fermilab combined value; see Table 4.3.5).

Comparison of muon gyromagnetic anomaly results from combined Fermilab and BNL
data ($a^{\mathrm{EXP}}_{\mu}$) with predictions from the Standard Model
($a^{\mathrm{SM}}_{\mu}$) and CASMIR ($a^{\mathrm{CASMIR}}_{\mu}$).
Differences are expressed as tensions and as multiples of the experimental
uncertainties. Values smaller than $1\,\sigma$ are displayed in green online.
Tension with $\sigma^{\mathrm{EXP}}$ Difference as a multiple of
$\sigma^{\mathrm{EXP}}$ $a^{\mathrm{SM}}_{\mu}$ ${5.2}\,\sigma$
${\phantom{.3}11}\,\sigma$ $a^{\mathrm{CASMIR}}_{\mu}$ $0.2\,\sigma$
${\phantom{1}0.5}\,\sigma$

There is scope for future calculations to further reduce the uncertainty in
the CASMIR value by fully evaluating the two-loop electroweak terms using the
colored $W$ and $Z$ bosons alongside their achromatic counterparts.

##  5 Conclusion

The Classical Analogue to the Standard Model In pseudo-Riemannian spacetime
(CASMIR) effectively reproduces many properties of the Standard Model. In
addition, it predicts the higher $W$ boson mass detected by CDF II with a
tension of $0.1\,\sigma$. [Observed:
$80.4335(94)~{}\mathrm{G}\mathrm{eV}/c^{2}$. Predicted:
$80.4340(22)~{}\mathrm{G}\mathrm{eV}/c^{2}$.] In CASMIR this increase in mass
arises due to the electroweak vector bosons sometimes being able to carry
color charges, with the heavier, colored $W$ boson mass vertices being
excluded from the ATLAS calculation but contributing in CDF II. This mechanism
also alters the value of the muon gyromagnetic anomaly, yielding a value more
consistent with experiment (tension reduced from ${5.2}\,\sigma$ to
$0.2\,\sigma$).

The utility of an analogue model is determined by its ability to make
predictions about experimental results. In addition to the shift in apparent
$W$ boson mass, which occurs only in hadron collisions where $\sqrt{s}$ is
small compared with $3.09~{}\mathrm{T}\mathrm{eV}$, CASMIR also predicts a
shift in apparent $Z$ boson mass under the same conditions, increasing from
$91.1877(35)~{}\mathrm{G}\mathrm{eV}/c^{2}$ to
$91.1922(35)~{}\mathrm{G}\mathrm{eV}/c^{2}$. Existing experimental data lacks
the precision to definitively probe this small
($4.5~{}\mathrm{M}\mathrm{eV}/c^{2}$) correction.

Current and planned future LHC beam energies are large when compared with the
CASMIR chromatic mass vertex suppression threshold of
$\sqrt{s}<3.09~{}\mathrm{T}\mathrm{eV}$, so future measurements at the LHC are
anticipated to yield the lower (achromatic) values for both $W$ boson and $Z$
boson mass. However, it is nevertheless feasible to probe this mass
augmentation effect using existing and near-term forthcoming apparatus. The
optimal configuration would be a high-luminosity hadron collider such as the
LHC or the proposed HL-LHC upgrade [48], operating below the chromatic mass
vertex suppression threshold of $3.09~{}\mathrm{T}\mathrm{eV}$. (The former
Tevatron accelerator satisfied this latter condition, operating at
$\sqrt{s}=1.96~{}\mathrm{T}\mathrm{eV}$.) Appropriate lower-energy runs could
conceivably be performed during the commissioning process for the HL-LHC.

Finally, if the domain of utility for CASMIR is sufficiently broad then it is
possible that observations in astroparticle physics might support or exclude
higher-generation weak bosons at the energies predicted in Table 2.4, or that
some dark matter detection program or study of scalar boson interactions at
the LHC might suggest existence of a species consistent with Sec. 2.5.
However, there is no imminent expectation for existing experiments to test
these latter predictions.

## Acknowledgments

This research was supported in part by the Perimeter Institute for Theoretical
Physics. Research at the Perimeter Institute is supported by the Government of
Canada through Industry Canada and by the Province of Ontario through the
Ministry of Research and Innovation. The author thanks the Ontario Ministry of
Research and Innovation Early Researcher Awards (ER09-06-073) for financial
support. This project was supported in part through the Macquarie University
Research Fellowship scheme. This research was supported in part by the ARC
Centre of Excellence in Engineered Quantum Systems (EQuS), Project No.
CE110001013.

##  Appendix A Mass relations from CASMIR

The geometry of CASMIR fixes a number of relationships between particle masses
in the analogue model. The values of $\alpha$, $m_{e}$, and $m_{\mu}$ may be
taken as input, with the following expressions then yielding derived values
for $m_{W}$, $m_{Z}$, $m_{\bm{\mathrm{H}}}$, $m_{\tau}$, $N_{0}$, and
$m_{{\tilde{W}}}$ (with the latter two being internal parameters of the CASMIR
model):

$\displaystyle\frac{m_{e}^{2}\left[k^{(e)}_{i}(\mathcal{E}_{e_{i}})\right]^{4}\left[1+\Delta_{e}(m_{e_{i}},\mathcal{E}_{e_{i}})\right]}{m_{e_{i}}^{2}\left[k^{(e)}_{1}(\mathcal{E}_{e_{i}})\right]^{4}\left[1+\Delta_{e}(m_{e},\mathcal{E}_{e_{i}})\right]}=1+\mathcal{O}_{e}(m_{e_{i}},\mathcal{E}_{e_{i}})$
(94)
$\displaystyle\begin{split}&\frac{9m_{{\bm{\mathrm{H}}}}^{2}\left[1+\left(64+\frac{3}{2\pi}-f_{Z}\right)\frac{\alpha}{2\pi}\right]}{20m_{\tilde{W}}^{2}\left\\{\left(1-\frac{1}{9N_{0}}\right)^{2}\left[1+\frac{30\alpha}{9\pi}\left(1+\frac{1}{9N_{0}}\right)\right]+\frac{1}{2\pi}\left[1+\frac{30\alpha}{\pi}\left(1-\frac{1}{9N_{0}}\right)\right]\right\\}}\\\
&\qquad\qquad=\frac{1+\frac{39}{18\left[k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\right]^{4}}}{1+\frac{51}{18\left[k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\right]^{4}}}(1+\mathcal{O}_{b})\end{split}$
(95)
$\displaystyle\frac{3m_{Z}^{2}\left[1+\left(64+\frac{3}{2\pi}-f_{Z}\right)\frac{\alpha}{2\pi}\right]\left\\{1+\frac{51}{18\left[k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\right]^{4}}\right\\}}{4m_{\tilde{W}}^{2}\left[1+\left(\frac{401}{12}+\frac{3}{2\pi}\right)\frac{\alpha}{2\pi}\right]\left\\{1+\frac{55}{18\left[k^{(e)}_{1}\,(\mathcal{E}_{e})\,{N_{0}}\right]^{4}}\right\\}}=1+\mathcal{O}_{b}$
(96)
$\displaystyle\frac{18{N_{0}}^{4}m_{e}^{2}\Biggl{[}1+\left(64+\frac{3}{2\pi}-f_{Z}\right)\frac{\alpha}{2\pi}\Biggr{]}\left\\{1+\frac{51}{18\left[k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\right]^{4}}\right\\}}{m_{\tilde{W}}^{2}\left(1+\frac{2}{N_{0}}\right)^{4}\left(1+\frac{1}{N_{0}}\right)^{4}\left[1+\Delta_{e}(m_{e},\mathcal{E}_{e})\right]}=1+\mathcal{O}_{b}+\mathcal{O}_{e}(m_{e},\mathcal{E}_{e})$
(97)
$\displaystyle\frac{m_{W}^{2}\left\\{1+\frac{51}{18\left[k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\right]^{4}}\right\\}}{m_{\tilde{W}}^{2}\left\\{1+\frac{19}{18\left[k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\right]^{4}}\right\\}}=1+\mathcal{O}_{b}$
(98)

with $e_{i}\in\\{\mu,\tau\\}$ corresponding to $i\in\\{2,3\\}$ respectively in
Eq. (94).
Additional symbols appearing in the above expressions are defined as follows:

$\displaystyle\Delta_{e}(m_{e_{i}},\mathcal{E}):=\,$
$\displaystyle\frac{8\alpha}{3N_{0}(3\alpha+2\pi)}\left\\{1+\frac{(10\pi+180\alpha)m_{e_{i}}^{2}}{3\pi\left[m_{c}^{*}(\mathcal{E})\right]^{2}}+\frac{(5-4f_{Z})\alpha
m_{e_{i}}^{2}}{4\pi m_{\tilde{W}}^{2}}\right\\}+\frac{90\alpha
m_{e_{i}}^{2}}{\pi\left[m_{c}^{*}(\mathcal{E})\right]^{2}}$
$\displaystyle+\frac{(5-4f_{Z})\alpha m_{e_{i}}^{2}}{2\pi
m_{\tilde{W}}^{2}}+\frac{5m_{e_{i}}^{2}}{[m_{c}^{*}(\mathcal{E})]^{2}}\left\\{1+\frac{90\alpha
m_{e_{i}}^{2}}{\pi\left[m_{c}^{*}(\mathcal{E})\right]^{2}}+\frac{(25-12f_{Z})\alpha
m_{e_{i}}^{2}}{6\pi m_{\tilde{W}}^{2}}\right\\}$
$\displaystyle+\frac{40m_{e_{i}}^{2}}{3m_{{\bm{\mathrm{H}}}}^{2}\left[k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\right]^{4}}+\mathcal{O}_{e}(m_{e_{i}},\mathcal{E})$
(99)
$\displaystyle\begin{split}\theta_{e}(\mathcal{E}):=-&\frac{3\pi}{4}\bm{\Biggl{(}}1-\frac{4\sqrt{\delta_{e}\\{r[\Delta_{e}(m_{\tau},\mathcal{E})-\Delta_{e}(m_{\mu},\mathcal{E})]\\}}}{3\pi}\bm{\Biggr{)}}\\\
&\times\bm{\left(}1+\frac{4}{3\pi}\sqrt{\delta_{e}\left\\{r\left[\frac{1+\Delta_{e}(m_{e},\mathcal{E})}{1+\Delta_{e}(0,\mathcal{E})}-1\right]\right\\}}\bm{}\end{split}$
(100) $\displaystyle m_{c}^{2}$
$\displaystyle:=m_{\tilde{W}}^{2}\left\\{\frac{1+\frac{131}{18\left[k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\right]^{4}}}{1+\frac{51}{18\left[k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\right]^{4}}}\right\\}\left(1+\mathcal{O}_{b}\right)$
(101) $\displaystyle{\left[m_{c}^{*}(\mathcal{E})\right]^{2}}$
$\displaystyle:=m_{c}^{2}\left(1-\frac{27}{10}\frac{\mathcal{E}^{2}}{m_{c}^{2}c^{4}}\right)$
(102) $\displaystyle f_{Z}$
$\displaystyle:=\frac{4}{3}\left(1-\frac{6m_{\tilde{W}}^{2}}{m_{Z}^{2}}+\frac{4m_{\tilde{W}}^{4}}{m_{Z}^{4}}\right)$
(103) $\displaystyle\mathcal{E}_{\ell}$ $\displaystyle:=m_{\ell}c^{2}$ (104)
$\displaystyle r(n)$ $\displaystyle:=n\cdot\sqrt{1-\frac{n}{9}\;}$ (105)
$\displaystyle k^{(\ell)}_{n}(\mathcal{E})$
$\displaystyle:=1+\sqrt{2}\cos{\left[\theta_{\ell}(\mathcal{E})-\frac{2\pi(n-1)}{3}\right]}$
(106) $\displaystyle\delta_{e}(n)$
$\displaystyle:=\sqrt{1+\frac{\pi^{2}n}{8}\left(1+\frac{\pi^{2}n}{32}\right)+\mathrm{O}(n^{3})}~{}-1$
(107) $\displaystyle\mathcal{O}_{b}$
$\displaystyle:=\mathrm{O}\left\\{\frac{\alpha}{\pi}\left[k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\right]^{-4}\right\\}+\mathrm{O}\left(\frac{\alpha^{2}}{\pi^{2}}\right)$
(108) $\displaystyle\mathcal{O}_{e}(m_{e_{i}},\mathcal{E}):=\,$
$\displaystyle\mathrm{O}\left(\frac{\alpha}{\pi{N_{0}}^{2}}\right)+\mathrm{O}\left(\frac{\alpha^{2}}{\pi^{2}{N_{0}}}\right)+\mathrm{O}\left[\frac{\alpha
m_{e_{i}}^{4}}{\pi
N_{0}[m_{c}^{*}(\mathcal{E})]^{4}}\right]+\mathrm{O}\left\\{\frac{\alpha^{2}m_{e_{i}}^{2}}{\pi^{2}\left[m_{c}^{*}(\mathcal{E})\right]^{2}}\right\\}$
$\displaystyle+\mathrm{O}\left\\{\frac{m_{e_{i}}^{6}}{\left[m_{c}^{*}(\mathcal{E})\right]^{6}}\right\\}+\mathrm{O}\left\\{\frac{\alpha
m_{e_{i}}^{2}}{\pi
m_{\bm{\mathrm{H}}}^{2}\left[k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\right]^{4}}\right\\}$
(109)
$\displaystyle+\mathrm{O}\left\\{\frac{m_{e_{i}}^{2}}{m_{\bm{\mathrm{H}}}^{2}\left[k^{(e)}_{1}(\mathcal{E}_{e})\right]^{4}{N_{0}}^{5}}\right\\}+\mathrm{O}\left\\{\frac{m_{e_{i}}^{4}}{m_{\bm{\mathrm{H}}}^{4}\left[k^{(e)}_{1}(\mathcal{E}_{e})\,{N_{0}}\right]^{4}}\right\\}$

Noting that Eqs. (94)–(98) yield six independent relationships, as Eq. (94)
may be evaluated for both $i=2$ and $i=3$, the derived values for particle
masses in the CASMIR model are given in Table A.

Particle masses and coupling constants in CASMIR: Observed values, values
predicted by CASMIR, and values predicted by the Standard Model (where these
exist). Parameters $N_{0}$ and $m_{\tilde{W}}$ are also fixed by the equations
of A, and evaluate to $N_{0}=191.9470(37)$ and
$m_{\tilde{W}}=80.3785(22)~{}\mathrm{G}\mathrm{eV}/c^{2}$ respectively. Mass
uncertainties in CASMIR are estimates, so tensions in this table are given as
an upper bound obtained by taking the CASMIR uncertainty to zero.  Parameter
Observed value 21-24CASMIR [16, 22, 23, 17] Standard Model Upper bound
Calculated on tension $(\sigma)$ Calculated Tension $(\sigma)$ $m_{\tau}$
$(\mathrm{M}\mathrm{eV}/c^{2})$ 1776.86(12) [6] 1776.867410(43)
$\phantom{<}\,0.1$ — — $m_{W}$ $(\mathrm{G}\mathrm{eV}/c^{2})$ 80.360(16) [11]
80.3587(22) $\phantom{<}\,0.1$ 80.356(6) [6, 4, 5] 0.2 $m_{Z}$
$(\mathrm{G}\mathrm{eV}/c^{2})$ 91.1876(21) [6] 91.1877(35) $<0.1$ — —
$m_{\bm{\mathrm{H}}}$ $(\mathrm{G}\mathrm{eV}/c^{2})$ 125.25(17) [6],∗
125.2501(49) $<0.1$ —† — ∗ The PDG value of
$125.25(17)~{}\mathrm{G}\mathrm{eV}/c^{2}$ is preferred over the more recent
ATLAS-only value of $125.11(11)~{}\mathrm{G}\mathrm{eV}/c^{2}$ pending release
of updated CMS results and incorporation of both into a new global average.
The Higgs mass predicted by CASMIR shows satisfactory agreement with the
updated ATLAS result (tension upper bound $1.3\,\sigma$) [49]. † The mass of
the Higgs in the Standard Model is a measured rather than a derived quantity.
Theoretical considerations placed some broad constraints on the Higgs mass
prior to its detection in 2012, but these are not a calculated value in the
sense considered here.

Regarding confidence intervals for CASMIR in Table A, it would be easy to
assume that unevaulated higher-order terms such as those in $\mathcal{O}_{b}$
(108) and $\mathcal{O}_{e}$ (109) attract coefficients of $\mathrm{O}(1)$ or
less. However, in CASMIR situations frequently arise where $\mathrm{O}(10)$
degenerate channels may reinforce one another (e.g. equivalent loop correction
diagrams due to eight species of colored $W$ bosons and one colorless $W$
boson). The confidence intervals for CASMIR in Table A are therefore evaluated
for coefficients in the range $\pm 10$. While this is a reasonable precaution
in the absence of any exploration of these higher-order terms, it may
potentially overestimate the uncertainty associated with the CASMIR values of
the calculated parameters, artificially reducing tension. As a check on the
values calculated, an upper bound on tension between experiment and the CASMIR
expressions to present order is then obtained by taking the CASMIR uncertainty
to zero in the calculation of tension. This precaution has negligible effect
on the tensions obtained, with all upper bounds on tensions for the CASMIR
model in Table A being $0.1\,\sigma$ or less.

## References

  * [1] S. Glashow, Nucl. Phys. 22, 579 (1961), 10.1016/0029-5582(61)90469-2.
  * [2] A. Salam and J. C. Ward, Phys. Lett. 13, 168 (1964), 10.1016/0031-9163(64)90711-5.
  * [3] S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967), 10.1103/PhysRevLett.19.1264.
  * [4] M. Awramik, M. Czakon, A. Freitas and G. Weiglein, Phys. Rev. D 69, 053006 (2004), 10.1103/PhysRevD.69.053006.
  * [5] J. Erler and M. Schott, Prog. Part. Nucl. Phys. 106, 68 (2019), 10.1016/j.ppnp.2019.02.007.
  * [6] R. L. Workman et al., Prog. Theor. Exp. Phys. 2022, 083C01 (2022) and 2023 update 10.1093/ptep/ptac097.
  * [7] CDF Collaboration (T. Aaltonen et al.), Science 376, 170 (2022), 10.1126/science.abk1781.
  * [8] The Muon $g-2$ Collaboration, D. P. Aguillard et al., Measurement of the positive muon anomalous magnetic moment to 0.20 ppm, arXiv:2308.06230v1 [hep-ex], (2023).
  * [9] T. Aoyama et al., Physics Reports 887, 1 (2020), 10.1016/j.physrep.2020.07.006.
  * [10] A. Keshavarzi, K. S. Khaw and T. Yoshioka, Nuc. Phys. B 975, 115675 (2022), 10.1016/j.nuclphysb.2022.115675.
  * [11] The ATLAS Collaboration, Improved $W$ boson mass measurement using $\sqrt{s}=7~{}\mathrm{TeV}$ proton-proton collisions with the ATLAS detector, Tech. Rep. ATLAS-CONF-2023-004, CERN (2023).
  * [12] S. Borsanyi et al., Nature 593, 51 (2021), 10.1038/s41586-021-03418-1.
  * [13] CMD-3 Collaboration, F. V. Ignatov et al., Measurement of the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ cross-section from threshold to $1.2~{}\mathrm{GeV}$ with the CMD-3 detector, arXiv:2302.08834v2 [hep-ex], (2023).
  * [14] G. Colangelo et al., Prospects for precise predictions of $a_{\mu}$ in the Standard Model, arXiv:2203.15810v1 [hep-ph], (2022).
  * [15] Muon $g-2$ Collaboration (B. Abi et al.), Phys. Rev. Lett. 126, 141801 (2021), 10.1103/PhysRevLett.126.141801.
  * [16] R. N. C. Pfeifer, A classical analogue to the Standard Model, chapter 1: Normalisable quasiparticles on a manifold with anticommuting co-ordinates arXiv:2106.08130 with updates at https://www.academia.edu/65931513, (2022).
  * [17] R. N. C. Pfeifer, A classical analogue to the Standard Model, chapters 4-10: Particle generations and masses, curved spacetimes and gravitation arXiv:2008.05893 with updates at https://www.academia.edu/65931513, (2022).
  * [18] CMS Collaboration (S. Chatrchyan et al.), Phys. Lett. B 716, 30 (2012), 10.1016/j.physletb.2012.08.021.
  * [19] ATLAS Collaboration (G. Aad et al.), Phys. Lett. B 716, 1 (2012), 10.1016/j.physletb.2012.08.020.
  * [20] ATLAS Collaboration (M. Aaboud et al.), Eur. Phys. J. C 78, 110 (2018), 10.1140/epjc/s10052-017-5475-4.
  * [21] ATLAS Collaboration (M. Aaboud et al.), Eur. Phys. J. C 78, 898 (2018), 10.1140/epjc/s10052-018-6354-3.
  * [22] R. N. C. Pfeifer, A classical analogue to the Standard Model, chapter 2: Colour interactions and fermions from scalar fields on $\mathbb{C}^{\wedge 6}$ arXiv:2108.07719 with updates at https://www.academia.edu/65931513, (2022).
  * [23] R. N. C. Pfeifer, A classical analogue to the Standard Model, chapter 3: Standard Model particle spectrum from scalar fields on $\mathbb{C}^{\wedge 18}$ arXiv:0805.3819 with updates at https://www.academia.edu/65931513, (2022).
  * [24] B. D. Serot and J. Dirk, Int. J. Mod. Phys. E 6, 515 (1997), 10.1142/S0218301397000299.
  * [25] G. P. Srivastava, The Physics of Phonons (Taylor & Francis, 1990).
  * [26] J. Wess and J. Bagger, Introduction to gauge theory, in Proceedings, Les Houches Summer School in Theoretical Physics: Gauge Theories in High Energy Physics - Session 37 : Les Houches, France, August 3-September 11, 1981, eds. M. K. Gaillard and R. Stora (North-Holland, Amsterdam, 1983), pp. 1–55.
  * [27] ALEPH and DELPHI and L3 and OPAL and LEP Electroweak Collaborations (S. Schael et al.), Phys. Rep. 532 (2013), 10.1016/j.physrep.2013.07.004.
  * [28] DØ Collaboration (V. M. Abazov et al.), Phys. Rev. Lett. 108, 151804 (2012), 10.1103/PhysRevLett.108.151804.
  * [29] LHCb Collaboration (R. Aaij et al.), J. High Energ. Phys. 2022, 36 (2022), 10.1007/JHEP01(2022)036.
  * [30] The ALEPH Collaboration (S. Schael et al.), Eur. Phys. J. C 47, 309 (2006), 10.1140/epjc/s2006-02576-8.
  * [31] The OPAL Collaboration (G. Abbiendi et al.), Eur. Phys. J. C 45, 307 (2006), 10.1140/epjc/s2005-02440-5.
  * [32] L3 Collaboration (P. Achard et al.), Eur. Phys. J. C 45, 569 (2006), 10.1140/ejpc/s2005-02459-6.
  * [33] The DELPHI Collaboration (J. Abdallah et al.), Eur. Phys. J. C 55, 1 (2008), 10.1140/epjc/s10052-008-0585-7.
  * [34] CDF Collaboration (T. Affolder et al.), Phys. Rev. D 64, 052001 (2001), 10.1103/PhysRevD.64.052001.
  * [35] DØ Collaboration Collaboration (V. M. Abazov et al.), Phys. Rev. D 66, 012001 (2002), 10.1103/PhysRevD.66.012001.
  * [36] T. Aoyama, M. Hayakawa, T. Kinoshita and M. Nio, Phys. Rev. Lett. 109, 111807 (2012), 10.1103/PhysRevLett.109.111807.
  * [37] T. Aoyama, T. Kinoshita and M. Nio, Phys. Rev. D 97, 036001 (2018), 10.1103/PhysRevD.97.036001.
  * [38] A. Czarnecki, W. J. Marciano and A. Vainshtein, Phys. Rev. D 67, 073006 (2003), 10.1103/PhysRevD.67.073006.
  * [39] C. Gnendiger, D. Stöckinger and H. Stöckiner-Kim, Phys. Rev. D 88, 053005 (2013), 10.1103/PhysRevD.88.053005.
  * [40] K. Fujikawa, B. W. Lee and A. I. Sanda, Phys. Rev. D 6, 2923 (1972), 10.1103/PhysRevD.6.2923.
  * [41] R. Jackiw and S. Weinberg, Phys. Rev. D 5, 2396 (1972), 10.1103/PhysRevD.5.2396.
  * [42] G. Altarelli, N. Cabibbo and L. Maiani, Phys. Lett. B 40, 415 (1972), 10.1016/0370-2693(72)90833-7.
  * [43] I. Bars and M. Yoshimura, Phys. Rev. D 6, 374 (1972), 10.1103/PhysRevD.6.374.
  * [44] W. A. Bardeen, R. Gastmans and B. Lautrup, Nucl. Phys. B 46, 319 (1972), 10.1016/0550-3213(72)90218-0.
  * [45] J. P. Miller, E. de Rafael, B. L. Roberts and D. Stöckinger, Annu. Rev. Nucl. Part. Sci. 62, 237 (2012), 10.1146/annurev-nucl-031312-120340.
  * [46] T. Gribouk and A. Czarnecki, Phys. Rev. D 72, 053016 (2005), 10.1103/PhysRevD.72.053016.
  * [47] A. Czarnecki, B. Krause and W. J. Marciano, Phys. Rev. D 52, R2619(R) (1995), 10.1103/PhysRevD.52.R2619.
  * [48] G. Apollinari, I. Béjar Alonso, O. Brüning, M. Lamont and L. Rossi (editors), High Luminosity Large Hadron Collider HL-LHC: Preliminary Design Report, Tech. Rep. CERN-2015-005, CERN (2015).
  * [49] The ATLAS Collaboration, Combined measurement of the Higgs boson mass from the $H\rightarrow\gamma\gamma$ and $H\rightarrow ZZ^{*}\rightarrow 4\ell$ decay channels with the ATLAS detector using $\sqrt{s}=7$, 8, and 13 TeV $pp$ collision data, arXiv:2308.04775v1 [hep-ex], (2023).