# Exploring the Orbital Alignments of Galactic Close White Dwarf Binaries with
LISA

Naoki Seto  Department of Physics, Kyoto University, Kyoto 606-8502, Japan

###### Abstract

Using the proposed space gravitational wave detector LISA, we will be able to
measure the geometrical configurations of $\sim 10^{4}$ close white dwarf
binaries in our Galaxy. The obtained data will be an entirely new resource to
examine the randomness of their orbital orientations. Partly motivated by a
recent reported on the systematic alignments of bulge planetary nebulae, we
discuss the outlook of the orientational analysis with LISA. We find that a
quadrupole pattern as small as $\sim 0.05$ can be detected for bulge white
dwarf binaries, owing to their large available number. From such a pattern
analysis, we might geometrically explore fossil records in our Galaxy billions
of years ago.

###### pacs:

PACS number(s): 95.55.Ym 98.80.Es,95.85.Sz

##  I introduction

Since 2015, the LIGO-Virgo-Kagra network has detected gravitational wave (GW)
signals from $\sim 100$ merging extra-Galactic binaries in the 10-1000Hz band
LIGOScientific:2016aoc ; LIGOScientific:2017vwq ; KAGRA:2021vkt . Quite
recently, in the nHz band, various theoretical models have been actively
discussed with the advent of new pulsar timing data NANOGrav:2023gor ;
EPTA:2023fyk ; Reardon:2023gzh ; Xu:2023wog . In the 2030s, the Laser
Interferometer Space Antenna (LISA) will be launched and will explore GWs
around 0.1-100mHz LISA:2022yao . LISA has the potential to observe massive
black holes at cosmological distances, although the estimated detection rates
have large uncertainties LISA:2022yao .

More securely, LISA will separately detect $\sim 10^{4}$ close white dwarf
binaries (CWDBs) in our Galaxy, as nearly monochromatic GW sources
LISA:2022yao ; Hils:1990vc ; Nissanke:2012eh ; Seto:2022iuf ; Kupfer:2018jee .
Indeed, for improving the effective sensitivity of LISA, it is essential to
identify these vast number of binaries and subtract their foreground GW
signals Littenberg:2020bxy . A significant fraction ($\sim 30$%) of the
identified CWDBs will be the bulge component, located near the Galactic center
Ruiter:2007xx . The remaining ones will be distributed more broadly around the
Galactic disk. In both cases, the orbital motions of the CWDBs are directly
encoded in the emitted GW signals pw ; Cutler:1997ta ; Takahashi:2002ky .
Using LISA, we can receive the GW signals, decode them and generate a long and
high-quality list for the orbital configurations of the CWDBs, solely based on
the first principles of physics. Note that most of CWDBs are expected to have
negligible eccentricities due to tidal effects (as for the known CWDBs
LISA:2022yao ; Kupfer:2018jee ).

In relation to the orientations (i.e. the directions of angular momentum
vectors) of CWDBs, on the basis of electromagnetic observations, there
recently appeared an interesting report on the bulge planetary nebulae (PNe)
which are (or are inferred to be) specifically associated with short-period
($\lesssim$1 day) binaries (in total of 14 systems) Tan . In contrast to the
whole bulge PN populations, the orientations of the symmetric axes of the
specific PNe show concentrations nearly parallel to the Galactic plane, at
5$\sigma$ significance. PNe are ionized gas ejected at the formation of white
dwarfs kwi  and closely related to the common envelope phases Ivanova:2012vx .
In the report, the orientations of the specific subset are considered to be
parallel to the orbital angular momentum vectors of the associated short-
period binaries (see also hil ). Then, the angular momentum vectors of the
binaries are not randomly oriented but more probable to be nearly parallel to
the Galactic disk (roughly speaking, removing PNe from the triple geometrical
relations between the short-period binaries, PNe and the Galactic plane). In
general, binary formation should involve various physical processes and could
also depend on time and location Duchene:2013cba . The reported alignments
might be embedded already at the formations of the bulge binaries billions of
years ago, e.g. due to ordered strong magnetic fields as argued in the report
Tan .

Regarding the potential alignments of compact binaries in the bulge, we await
further studies, in particular, independent observational analyses (see also
huang  for nearby binaries). Here the long list of CWDBs provided by LISA
could be an invaluable and solid resource. With the large number of the
available sample, we will be able to detect a weak anisotropic pattern
existing in their orientation distribution function.

In this work, we discuss the outlook of such an inquiry with LISA, paying
special attentions to the geometrical aspects of the involved systems. Along
the way, we point out a fourfold degeneracy at determining the polarization
angle $\psi$, which fixes the binary’s orientation around the line-of-sight
direction. This degeneracy is induced by the underlying symmetry of the
measurable gravitational waveform and can be effectively regarded as an
irreversibility in the information transfer (from the encoding to the
decoding), partially hampering our observational analysis. For the data
analysis of actual CWDB sample, we examine a simple dualistic approach to
statistically enhance its anisotropic pattern. The associated detection limit
can be as small as $\sim 0.05$ for a quadupole mode of the spherical harmonic
expansion. Through this pattern analysis, LISA might enable us to
geometrically delve into the ancient history of our Galaxy.

##  II orientations of CWDBs

###  II.1 Axisymmetric Model

As shown in Fig. 1, we first define the relevant unit vectors for describing
the configuration of a circular CWDB. We put its sky direction $\vec{n}$ and
its orientation $\vec{j}$ (parallel to its angular momentum vector). We also
set $\vec{q}$ as the direction of the Galactic rotation axis. The Galactic
plane is normal to $\vec{q}$.

Given the recent report on the bulge PNe Tan , we are primarily interested in
the probability distribution function ${\cal P}(\vec{j})$ for the orientations
$\vec{j}$ of the bulge CWDBs, in particular, its pattern relative to the
rotation axis $\vec{q}$. We thus set $\vec{q}$ as the polar direction for the
spherical harmonic bases $Y_{lm}(\vec{j})$.

We assume that the function ${\cal P}(\vec{j})$ is axisymmetric around the
vector $\vec{q}$. Then the function ${\cal P}(\vec{j})$ depends only on the
polar angle $\theta$ (i.e. only with $m=0$ modes). Given the normalization
condition, the resultant axisymmetric function is expanded as

$\displaystyle{\cal P}_{A}(\cos\theta)=$
$\displaystyle(4\pi)^{-1/2}[Y_{00}(\theta)+a_{10}Y_{10}(\theta)+a_{20}Y_{20}(\theta)$
(1) $\displaystyle+a_{30}Y_{30}(\theta)+a_{40}Y_{40}(\theta)+\cdots]$

defined in the range $0\leq\theta\leq\pi$. We have $Y_{l0}(\theta)\propto
L_{l}(\cos\theta)$ with the Legendre polynomials $L_{l}(x)$ (in an
unconventional notation to prevent confuses with probability distribution
functions),  which satisfy the odd-even identities

$L_{l}(-x)=(-1)^{l}L_{l}(x).$ (2)

We present some of the explicit forms $Y_{00}=(4\pi)^{-1/2}$,
$Y_{10}\propto\cos\theta$, $Y_{20}\propto(3\cos^{2}\theta-1)/2$,
$Y_{30}\propto(5\cos^{3}\theta-3\cos)/2$ and
$Y_{40}\propto(35\cos^{4}\theta-30\cos^{2}\theta+3)/8$. For a even $l$, we
have $Y_{l0}(0)=Y_{l0}(\pi)>Y_{l0}(\pi/2)$, and a negative value $a_{l0}$
resultantly induces a higher concentration to the equatorial directions
$(\theta=\pi/2)$ rather than the polar directions. We have ${\cal
P}(\vec{j})=1/(4\pi)$ for the isotropic (random) orientation distribution with
$a_{10}=a_{20}=\cdots=0$. Later, in Sec. III, we will discuss which parameters
$a_{l0}$ we can determine for the Galactic CWDBs with LISA.

Figure 1: Schematic picture for a binary configuration. All the seven vectors
are unit vectors. The vectors $\vec{n}$ and $\vec{j}$ represent the direction
and orientation of the binary. The Galactic plane is normal to the vector
$\vec{q}$. The angles $I$ and $\theta$ are respectively between
$\vec{j}$-$\vec{n}$ and $\vec{j}$-$\vec{q}$. The gray plane is normal to the
line-of-sight direction $\vec{n}$, and the vectors $\vec{e}_{j}$ and
$\vec{e}_{q}$ are the projections of $\vec{j}$ and $\vec{q}$, with the
polarization angle $\psi$ between them. The two remaining vectors
$\vec{e}_{k}$ and $\vec{e}_{r}$ are respectively perpendicular to
$\vec{e}_{j}$ and $\vec{e}_{q}$.

###  II.2 Coordinate Transformation

For discussing GW observation, it is convenient to introduce the two unit
vectors $({\vec{e}_{j}},{\vec{e}_{k}})$ normal to the binary direction
$\vec{n}$ (see Fig. 1). Here the unit vector $\vec{e}_{j}$ shows the
transverse projection of the orientation vector $\vec{j}$, with the remaining
one $\vec{e}_{k}(={\vec{n}}\times{\vec{e}_{j}})$.

The unit vector $\vec{e}_{q}$ is similarly defined by the transverse
projection of $\vec{q}$ with $\vec{e}_{r}(={\vec{n}}\times{\vec{e}_{q}})$. For
a given binary direction $\vec{n}$, the vectors $\vec{e}_{q}$ and
$\vec{e}_{r}$ are fixed, and we can specify the orientation $\vec{j}$ in terms
of the inclination angle $I$ and the polarization angle $\psi$ (see Fig. 1).
We have the following relations

${\vec{e}}_{k}={\vec{e}}_{r}\cos\psi+{\vec{e}}_{q}\sin\psi,~{}{\vec{e}}_{j}=-{\vec{e}}_{r}\sin\psi+{\vec{e}}_{q}\cos\psi.$
(3)

Our next task is to generate the distribution function $p(I,\psi)$ given in
the observer’s frame, using the aforementioned one ${\cal P}_{A}(\cos\theta)$
given in the Galactic frame. To this end, we first discuss the mismatch
between $\vec{q}$ and $\vec{e}_{q}$. Our solar system is almost on the mid
Galactic plane. The distance to the bulge is $\sim 8.3$kpc and its scale
height is $\sim 0.5$kpc ga . Therefore, the vector $\vec{q}$ is nearly normal
to $\vec{n}$, and we have $\vec{q}\simeq\vec{e}_{q}$ with the typical mismatch
angle (in radians) $\gamma\sim 0.5/8.3\,{\rm}\ll 1$. We then have

$\cos\theta={\vec{j}}\cdot{\vec{q}}\simeq{\vec{j}}\cdot{\vec{e}}_{q}=\sin
I\cos\psi$ (4)

and correspondingly

$p(I,\psi)\simeq{\cal P}_{A}(\sin I\cos\psi).$ (5)

Below, we apply equalities to the relations (4) and (5) (a distant observer
approximation). It is a straightforward but cumbersome task to derive the
function $p(I,\psi)$ without the approximation. Importantly, our approximation
does not introduce artificial anisotropies to $p(I,\psi)$ from the originally
isotropic function ${\cal P}(\vec{j})=1/(4\pi)$. In terms of the harmonic
expansion $p(I,\psi)=\sum_{lm}b_{lm}Y_{lm}(I,\psi)$, the coefficients $b_{lm}$
at $l\lesssim 1/\gamma\sim 15$ will be virtually unaffected by our
approximation (i.e. ignorable at the lower degrees such as $l\lesssim 4$).
Geometrically, Eq. (5) can be regarded as a $90^{\circ}$ rotation of the polar
direction (from $\theta=0$ to $I=0$), thus keeping the degrees $l$ at the
correspondence of their expansion coefficients (see e.g. Sakurai:2011zz ).
Even if we start from an axisymmetric model ${\cal P}_{A}(\cos\theta)$ as in
Eq. (1), the transformed one $p(I,\psi)$ can depend on the polarization
(azimuthal) angle $\psi$.

##  III GW Observation

###  III.1 Waveform Model

We now focus on a nearly monochromatic GW from a circular CWDB with an orbital
frequency $f/2$. We keep the essential aspects for our study, dropping
irrelevant details.

In the lowest quadrupole approximation, the gravitational waveform at a given
position is expressed as

$\displaystyle{\mbox{\boldmath${h}$}}(t,{\vec{n}},I,\psi)$ $\displaystyle=$
$\displaystyle A_{+}(I)\cos(2\pi ft+\alpha){\bf e_{+}}({\vec{n}},\psi)$ (6)
$\displaystyle+A_{\times}(I)\sin(2\pi ft+\alpha){\bf
e_{\times}}({\vec{n}},\psi)$

with the phase constant $\alpha$ and the two amplitudes
$A_{+}(I)\propto(1+\cos^{2}I)$ and $A_{\times}(I)\propto 2\cos I$ (see e.g. pw
). The transverse-traceless tensors
${\mbox{\boldmath${e}$}}_{+,\times}({\vec{n}},\psi)$ are given by

${\mbox{\boldmath${e}$}}_{+}({\vec{n}},\psi)={\vec{e}_{k}}\otimes{\vec{e}_{k}}-{\vec{e}_{j}}\otimes{\vec{e}_{j}},~{}{\mbox{\boldmath${e}$}}_{\times}({\vec{n}},\psi)={\vec{e}_{k}}\otimes{\vec{e}_{j}}+{\vec{e}_{j}}\otimes{\vec{e}_{k}}$
(7)

with the vectors $\vec{e}_{k}$ and $\vec{e}_{j}$ defined in Eq. (3).

From Eqs. (3) and (7), we readily obtain the identities ${\bf
e_{+,\times}}({\vec{n}},\psi+\pi)={\bf e_{+,\times}}({\vec{n}},\psi)$ and
resultantly

${\mbox{\boldmath${h}$}}(t,{\vec{n}},I,\psi+\pi)={\mbox{\boldmath${h}$}}(t,{\vec{n}},I,\psi).$
(8)

This degeneracy between $\psi$ and $\psi+\pi$ is fundamental, originating from
the spin-2 nature of the gravitational radiation.

From Eqs. (3) and (7), we can also confirm the identities ${\bf
e_{+,\times}}({\vec{n}},\psi+\pi/2)=-{\bf e_{+,\times}}({\vec{n}},\psi)$ and
thus

${\mbox{\boldmath${h}$}}(t,{\vec{n}},I,\psi+\pi/2)=-{\mbox{\boldmath${h}$}}(t,{\vec{n}},I,\psi)$
(9)

(see e.g. Cornish:2003vj ; Roulet:2022kot ). Correspondingly, the $\pi/2$\-
rotation of the polarization angle $\psi$ can be effectively absorbed by the
phase shift $\alpha+\pi$ in Eq. (6). Therefore, we observationally have the
degeneracy between $\psi$ and $\psi+\pi/2$ at the lowest Newtonian order. By
observing the higher post-Newtonian waveforms at frequencies $f/2$ and $3f/2$
and measuring their phases relative to that of the Newtonian one (6), we can,
in principle, distinguish the two states at $\psi$ and $\psi+\pi/2$ (see e.g.
Eq. (11.295b) in pw ). Unfortunately, compared with the Newtonian waveform
(6), the higher ones are suppressed by the post-Newtonian parameter
$\beta=O(c^{2}/v^{2})$. For our CWDBs, we have

$\beta=\left(\frac{\pi GM_{t}f}{c^{3}}\right)^{2/3}\sim
10^{-5}\left(\frac{f}{{\rm
5mHz}}\right)^{2/3}\left(\frac{M_{t}}{1M_{\odot}}\right)^{2/3}$ (10)

where $M_{t}$ is the total mass of the binary. Unlike binary black hole
mergers observed by ground-based detectors (see e.g. LIGOScientific:2020stg ),
the small post-Newtonian waveforms of the CWDBs are easily masked by the
measurement noises, and we cannot practically solve the degeneracy between
$\psi$ and $\psi+\pi/2$. From Eqs. (8) and (9), we have a similar degeneracy
between $\psi$ and $\psi+3\pi/2$.

In summary, GW observation is a geometrical measurement and intrinsically has
a good affinity for studying the configurations of the Galactic CWDBs.
However, because of the symmetry of the system, we have the fourfold
degeneracy between the angles $\psi,\psi+\pi/2,\psi+\pi$ and $\psi+3\pi/2$.

###  III.2 Fourfolded Distribution Function

As discussed in the previous subsection, our observable is not the full
distribution function $p(I,\psi)$ but the folded one

${\bar{p}}(I,\psi_{d})\equiv\sum_{k=0}^{3}p(I,\psi_{d}+k\pi/2)$ (11)

defined in the parameter ranges $0\leq I\leq\pi$ and $0\leq\psi_{d}\leq\pi/2$.
Interestingly, unless the order $m$ is a multiple of 4 (e.g. $m=0,\pm 4,\pm
8,\cdots$), the folding operation erases its $\psi_{d}$-dependent pattern

$\sum_{k=0}^{3}Y_{lm}(I,\psi_{d}+k\pi/2)\propto\sum_{k=0}^{3}\exp(imk\pi/2)=0.$
(12)

Correspondingly, the azimuthal patterns of the function $p(I,\psi_{d})$ come
only from $l\geq 4$.

As commented in Sec. II, an $l$-th order pattern $Y_{l0}(\theta)$ in the
original function Eq. (1) generates the term proportional to $L_{l}(\sin
I\cos\psi)$ in the transformed one $p(I,\psi)$ in Eq. (5). From Eq. (2), we
have

$\displaystyle L_{l}[\sin I\cos\psi]$ $\displaystyle+$ $\displaystyle
L_{l}[\sin I\cos(\psi+\pi)]=0$ $\displaystyle L_{l}[\sin I\cos(\psi+\pi/2)]$
$\displaystyle+$ $\displaystyle L_{l}[\sin I\cos(\psi+3\pi/2)]=0$

for odd numbers $l$. Thus, the fourfolding operation (11) cancels out all the
contributions from the odd order patterns in Eq. (1).

Using Eqs. (5) and (11) and keeping the even orders $l\leq 4$, we obtain

$\displaystyle{\bar{p}}(I,\psi_{d})$ $\displaystyle\propto$ $\displaystyle
Y_{00}-\frac{a_{20}}{2}Y_{20}(I)+\frac{3a_{40}}{8}Y_{40}(I)$
$\displaystyle+\frac{a_{40}}{8}\left(\frac{35}{2}\right)^{1/2}\left[Y_{44}(I,\psi_{d})+Y_{4-4}(I,\psi_{d})\right],$

which is symmetric at $I=\pi/2$. We should point out that the full function
$p(I,\psi)$ (generated from Eq. (1) by Eq. (5)) has the components $Y_{2\pm
2}(I,\psi)$, but they disappear at the folding operation as in Eq. (12) for
$m=\pm 2$.

In Fig 2, we plot the folded distribution function (III.2) in the coordinate

$(x,y)=2\sin[I/2](\cos\psi_{d},\sin\psi_{d})$ (14)

identical to the Lambert azimuthal equal-area projection centered on the face-
on direction $I=0$. Given the symmetry at $I=\pi/2$, we present only the range
$0\leq I\leq\pi/2$.

Figure 2:  The fourfolded orientation distributions $\bar{p}(I,\psi_{d})$
(normalized by the factor $\pi^{-1}$). We apply the area preserving projection
$(x,y)=2\sin(I/2)(\cos\psi_{d},\sin\psi_{d})$. The original distribution
functions are axisymmetric model ${\cal P}_{A}(\cos\theta)$ in Eq. (1)
characterized by the two anisotropy parameters $a_{20}$ and $a_{40}$. Panel
(a) for the isotropic model with $(a_{20},a_{40})=(0,0)$, (b) with $(-0.3,0)$,
(c) with $(-0.3,-0.04)$ and (d) with $(-0.3,-0.1)$.

For a hierarchical case $|a_{20}|\gg|a_{40}|$, the azimuthal dependence is
weak, as expected from Eq. (III.2). For $a_{20}<0$ (corresponding to the
equatorial enhancement in Eq. (1)), we have higher probability around the
face-on configuration ($I=0$ and $\pi$).

From the folded distribution function ${\bar{p}}(I,\psi_{d})$, we can easily
evaluate the polarization degree of the associated GW background. The Stokes
parameters $(I_{s},Q_{s},U_{s},V_{s})$ are its conventional measures rp ;
Romano:2016dpx . Using the notation
$\left\\{\cdots\right\\}_{\psi_{d}I}\equiv\int_{0}^{\pi}dI\int_{0}^{\pi/2}d\psi_{d}\sin
I~{}{\bar{p}}(I,\psi_{d})[\cdots]$ for the angular averagings, we obtain the
expressions such as

$\frac{Q_{s}+iU_{s}}{I_{s}}=\frac{\left\\{e^{-4i\psi_{d}}[(1+\cos^{2}I)^{2}-4\cos^{2}I]\right\\}_{\psi_{d}I}}{\left\\{[(1+\cos^{2}I)^{2}+4\cos^{2}I]\right\\}_{\psi_{d}I}}$
(15)

(defined for the axes $\vec{e}_{r}$ and $\vec{e}_{q}$). For the concrete
profile (III.2), we have
$Q_{s}/I_{s}={35a_{40}}/({336-48\sqrt{5}a_{20}+3a_{40}})$ and
$U_{s}/I_{s}=V_{s}/I_{s}=0$.

##  IV Probing anisotropies

We now discuss how to probe the anisotropies of the original function ${\cal
P}(\vec{j})$, by analyzing the observable function ${\bar{p}}(I,\psi_{d})$,
which is sampled by a finite number of CWDBs. For a probability distribution
function defined on a sphere, following e.g. pro , we can deal with the
discrete sampling effects on the spherical harmonic expansion. Here, paying
attention to the roughly concentric profiles in Fig. 2, we rather examine a
simple dualistic approach.

We divide our binary sample (in total $N$) into the following two subsets: (i)
the low-inclination group with $|\cos I|\geq 1/2$ and (ii) the high-
inclination group with $|\cos I|<1/2$. We put their numbers by $N_{L}$ and
$N_{H}$ ($N_{L}+N_{H}=N$) and define the asymmetric ratio by

${\cal A}\equiv\frac{N_{L}-N_{H}}{N}.$ (16)

For the isotropic (random) profile ${\cal P}(\vec{j})=\rm const$, we have the
vanishing mean $\left\langle{\cal A}\right\rangle=0$ and the shot noise
$\Delta{\cal A}=N^{-1/2}$ for the very basic binomial distribution. Below, we
conservatively take the reference number $N\sim 2000$, considering the
expected fraction of bulge CWDBs Ruiter:2007xx .

For an anisotropic profile ${\cal P}(\vec{j})$, we can estimate the mean
fractions such as

$\frac{\left\langle
N_{H}\right\rangle}{N}=\int_{\pi/3}^{2\pi/3}dI\int_{0}^{\pi/2}d\psi_{d}\sin
I~{}{\bar{p}}(I,\psi_{d})$ (17)

and obtain

$\left\langle{\cal A}\right\rangle=-\frac{192\sqrt{5}a_{20}+135a_{40}}{1024}$
(18)

for the function (III.2). Therefore, if we have the condition
$|\left\langle{\cal A}\right\rangle|>\Delta{\cal A}$, we can probe the
intrinsic anisotropy in ${\cal P}(\vec{j})$ (at $1$-$\sigma$ level). Dropping
the term $\propto a_{40}$ in Eq. (18), the inequality can be expressed as

$|a_{20}|>0.053\left(\frac{N}{2000}\right)^{-1/2}.$ (19)

This result would serve as a rough guidance for the detection limit of the
intrinsic alignment of the bulge CWDBs.

We have some comments on the above simple arguments, in view of the selection
effects at actual observational analysis. As shown in Eq. (6), the signal-to-
noise ratio $\rho$ depends on the sky position $\vec{n}$ of the binary and its
orientation angles $(I,\psi)$. For given parameters $({\vec{n}},\psi)$, the
signal-to-noise ratio $\rho$ generally becomes smallest for an edge-on binary
with $\cos I=0$, as indicated by the expressions $A_{+,\times}(I)$. How about
the dependence on the remaining parameters $({\vec{n}},\psi)$ for edge-on
binaries? In fact, due to the annual rotation of the detector plane of LISA,
the dependence is effectively averaged out, and the associated scatter is
largely suppressed. We can quantitatively examine this, by (i) using the
standard framework for nearly monochromatic binaries with LISA-like detectors
Cutler:1997ta  and (ii) randomly sampling the parameters $({\vec{n}},\psi)$.
For an observational period of an integer times 1yr, the minimum signal-to-
noise ratio $\rho$ of the edge-on binaries is only $\sim 10\%$ smaller than
the rms value of the whole edge-on sample (corresponding to the large number
limit of detectors in Yagi:2011wg ). In any case, the selection effect can be
avoided by using CWDBs in the appropriate region in the $(f,{\dot{f}})$-space
so that the Galactic survey is expected to be complete (e.g. $f>3$mHz)
Seto:2022iuf . Alternatively, when preparing the binary sample, we can
introduce dependence on the angular variables (${\vec{n}},I,\psi$) to the
threshold of the signal-to-noise ratios.

Meanwhile, the estimation errors for the parameter $\cos I$ induce
missclassifications of binaries around the boundary $|\cos I|=1/2$, and we
might need numerical studies to examine the potential biases for the
asymmetric ratio $\cal A$. The Fisher matrix analysis roughly gives
$\Delta\cos I\sim 1/\rho$ (except for $\cos I\sim\pm 1$) for a signal-to-noise
ratio $\rho$ (typically at $\sim 10$-$100$ for a Galactic CWDB) Seto:2022iuf .
Therefore, such confusions are relevant for a small fraction ($\sim 1/\rho$)
of the binaries, eventually increasing the shot noise $\Delta{\cal A}$ only
slightly (order of $N^{-1/2}\rho^{-1})$.

##  V Discussion and Summary

So far, we have mainly discussed the orientations of bulge CWDBs. Using LISA,
from the ampluitude and frequency modulations, we can also measure the
directions $\vec{n}$ of the binaries Cutler:1997ta . In addition, we will be
able to estimate the distances to some of the inspiraling CWDBs by analyzing
their orbital decay rates ${\dot{f}}>0$ Takahashi:2002ky . These pieces of
positional information will help us to roughly select the bulge components.
However, alignment studies will be intriguing also for disk components, and we
do not need to stick too much with a separation between the two components. In
any case, if the observed anisotropies ${\bar{p}}(I,\psi_{d})$ are turned out
to be strong, we could additionally explore the spatial correlation of the
orientations, by combining the two- or three-dimensional positional
information.

Let us briefly summarize our study. Compact binaries emit GWs, encoding their
orbital motions. GW observation is intrinsically geometrical and enables us to
measure the configurations of binary sources. Meanwhile, it was recently
reported that, the orientations of the planetary nebulae associated with
short-period bulge binaries are preferentially aligned to the Galactic plane,
possibly reflecting the frozen initial conditions of the binaries Tan . In the
near future, LISA will separately detect $\sim 10^{4}$ Galactic CWDBs and will
become an ideal tool to examine their alignments, based only on the first
principles of physics. In spite of the fourfold degeneracy of the polarization
angles, LISA could enable us to measure the axisymmetric quadrupole pattern
$a_{20}$ as small as $\sim 0.05$ and could serve as an interesting tool for
geometrically delving into the fossil records in our Galaxy.

###### Acknowledgements.

The author would like to thank M. Iye and K. Tomida for valuable discussions.
He also thanks K. Kyutoku and T. Tanaka for useful conversations.

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11institutetext: Instituto de Física Teórica e IPARCOS, Facultad de CC.
Físicas, Plaza de las Ciencias 1, 28040 Madrid, Spain

# Split gluon masses in $SU(N)\times SU(M)$ theories

Julia Gómez Concejo  Felipe J. Llanes-Estrada  Diego María-Almazán  Alexandre
Salas-Bernárdez

###### Abstract

We extend a known mass-gap equation for pure gluodynamics in global colour
models (formulated in equal time quantization in Coulomb gauge) to one in
which gluons split into two sets which may have different masses. If the
theory is $SU(N)\times SU(M)$ with gluons in both groups having identical
couplings (as suggested by Grand Unification arguments at large scales) it is
immediate to see that different masses are generated for each subgroup. This
global symmetry is not broken, but the split masses erase accidental
symmetries that might be present due to the two couplings being the same at
the large scale, such as $SU(N\times M)$ or similar. We also numerically
explore a couple of low-dimensional examples of simple Lie groups, but in
spite of the system having a form that would seem to allow spontaneous
symmetry breaking, it is not triggered for these groups whose algebra has no
ideal, and the dispersion relations for the various gluons converge to the
same form.

###### pacs:

11.15.-qGauge field theories and 11.15.ExSpontaneous breaking of gauge
symmetries

##  1 Introduction

Spontaneous gauge–boson mass generation is at the core of the Standard Model.
Additionally to the Higgs mechanism, the Schwinger mechanism and similar ideas
allow the gauge bosons (henceforth, “gluons”) to acquire a mass without the
assistance of an explicit additional field Cornwall:1981zr ; Cornwall:2015lna
; Horak:2022aqx ; Alkofer:2003jr ; Fischer:2004ym .

Gluon masses are a welcome gauge–fixed feature of Chromodynamics as they raise
glueballs from the low–lying hadron spectrum Buisseret:2009yv ; Vento:2004xx ;
Bugg:2000zy ; Chen:2005mg  where the existing hadrons are well understood.

This is perhaps worth exploring in the context of Grand Unification because
complicated symmetry breaking patterns Lee:2016wiy ; Mohapatra:1979nn  appear
and the scalar Higgs-type mechanisms to break the symmetry can be convoluted
(in fact, in many Grand Unification situations, the Higgs needs to be
described by a composite field from the start Dobson:2022crf ; Dobson:2022ngz
).

We do not have a particular agenda nor unification model in mind, but want to
generically explore a system of coupled gap equations that may allow splitting
the gluon masses into two or more different values. Having this theoretical
mechanism (which we partially achieve as will be explained in detail) would
allow to have additional theoretical tools to explore unification dynamics.
Because of the first theorem of Vafa and Witten Vafa:1983tf , we know that
spontaneous global colour-symmetry breaking is impossible in the quark sector,
so our exploration concentrates on the Yang-Mills sector alone.

Then there is also the question of why the Standard Model is built out of low-
dimensional Lie groups GarciaFernandez:2015jmn ; Llanes-Estrada:2018azk  that
may well have to do with the spontaneous acquisition of large masses
(triggered by very different evolutions of the coupling constants) by
particles charged under the (absent) large dimensional groups, which would
remove such particles from the spectrum.

These gap equations are formulated in Coulomb gauge, but our considerations
should be easy to extend to other gauges such as Landau gauge Cucchieri:2011ig
; Li:2019hyv . Modeling the Coul-omb gauge dynamics with simple global–colour
model does not capture all the interesting phenomena, such as for example the
Gribov divergent gluon mass at low momentum (a very strongly infrared enhanced
propagator) Reinhardt:2018dhg  but they are strong enough to trigger the
generation of gluon masses: and we are not sure that we want to explore
confinement (including Coulomb confinement is a necessary condition to
describe confinement in arbitrary gauges, as the Coulomb potential is an upper
bound for the QCD potential Zwanziger:2002sh ) in this work, that does not
necessarily restrict itself to Quantum Chromodynamics (QCD) with the group
$SU(3)$ but the production of a gap. This modified dispersion relation with a
finite gluon mass is a feature of a more general class of theories.

In this article we review, in section 2, the obtention of the known pure Yang-
Mills gap equation in the North Carolina State Szczepaniak:1995cw  model; we
solve it for various groups, all of which have the same coupling constant at a
low scale in section 3; we then, in section 4, extend the mechanism to allow
for the possibility of different variational wavefunctions for each of the
gauge bosons, which could possibly trigger spontaneous breaking of a global
symmetry. We succeed in doing this for product Lie groups or any other
situation in which the underlying Lie algebra contains an ideal. Afterwards,
we conduct a first numerical exploration for a few simple Lie groups of
low–dimension, reported in section 5 and do not currently find a situation in
which the symmetry breaks. After a brief outlook, we complement the discussion
with an appendix detailing the numerical solution method, the necessary colour
algebra, and an exhaustive list of the structure constant combinations (in the
particular case of $SU(3)$ only).

##  2 Coulomb–gauge gap equation for a singlet condensate

In this section we present the relatively well–known theory of the mass gap
equation leading to a gluon mass in Coulomb gauge with a color–singlet
condensate (note that any gauge boson mass and condensates are necessarily
features of a gauge–fixed picture of the theory) that therefore respects all
global symmetries. We start from a global–color symmetry preserving
Hamiltonian Szczepaniak:1995cw :

$\displaystyle H$ $\displaystyle=\int
d^{3}\mathbf{x}\,(\mathbf{\Pi^{a}}\cdot\mathbf{\Pi^{a}}+\mathbf{B^{a}}\cdot\mathbf{B^{a}})-$
$\displaystyle-\frac{1}{2}\int d^{3}\mathbf{x}\int
d^{3}\mathbf{y}\rho^{a}_{\rm
glue}(\textbf{x})V(|\textbf{x}-\textbf{y}|)\rho^{a}_{\rm glue}(\textbf{y})\ ,$
(1)

where $\mathbf{\Pi^{a}}$ represents the colour electric field,
$\mathbf{B^{a}}$ the chromomagnetic field, and $\rho^{a}_{\rm
glue}=f^{abc}\mathbf{A}^{b}\cdot\mathbf{\Pi}^{c}$ the colour charge density.
The Hamiltonian differs from exact QCD in that the potential
$V(|\textbf{x}-\textbf{y}|)$ is a c-function (like in electrodynamics) given
below in Eq. (16) simplifying the kernel that appears in the full non-Abelian
theory Christ:1980ku ; Schwinger:1962wd .

In the basis of well-defined momentum particles, with creation,
$a^{a\dagger}$, and destruction, $a^{a}$, boson operators the fields take the
form

$\displaystyle A^{a}_{i}(\mathbf{x})$
$\displaystyle=\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\frac{1}{\sqrt{2|\mathbf{k}|}}\left[a^{a}_{i}(\mathbf{k})+a^{a\dagger}_{i}(-\mathbf{k})\right]e^{i\mathbf{k\cdot
x}}$ (2) $\displaystyle\Pi^{a}_{i}(\mathbf{x})$
$\displaystyle=-i\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\sqrt{\frac{|\mathbf{k}|}{2}}\left[a^{a}_{i}(\mathbf{k})-a^{a\dagger}_{i}(-\mathbf{k})\right]e^{i\mathbf{k\cdot
x}}$ (3) $\displaystyle B^{a}_{i}$
$\displaystyle=\epsilon_{ijk}\left(\bigtriangledown_{j}A^{a}_{k}+\frac{g}{2}f^{abc}A^{b}_{j}A^{c}_{k}\right)\;.$
(4)

Because the Coulomb gauge is spatially transverse, adequate commutation rules
that project out the longitudinal gluons are

$[a_{i}^{a}(\mathbf{k}),a_{j}^{b}(\mathbf{q)^{\dagger}}]=(2\pi)^{3}\delta^{ab}\delta^{3}(\mathbf{k}-\mathbf{q})\left(\delta_{ij}-\hat{k}_{i}\hat{k}_{j}\right)\;,$
(5)

where $\hat{\mathbf{k}}\equiv\mathbf{k}/|\mathbf{k}|$.

A gluon condensed vacuum $\ket{\Omega}$ is variationally chosen by minimizing
the expectation value of the Hamiltonian $\langle H\rangle$. The
quasiparticles that will annihilate it will have a dispersion relation $E(k)$
that serves as the actual variational function, controlling the canonical
Bogoliubov rotation Llanes-Estrada:2000cdq

$\displaystyle\alpha^{a}_{i}$
$\displaystyle=\cosh{\theta^{a}_{k}}a^{a}_{i}(\mathbf{k})+\sinh{\theta^{a}_{k}}a^{a\dagger}_{i}(\mathbf{-k})$
(6) $\displaystyle\alpha^{a\dagger}_{i}$
$\displaystyle=\sinh{\theta^{a}_{k}}a^{a}_{i}(\mathbf{k})+\cosh{\theta^{a}_{k}}a^{a\dagger}_{i}(\mathbf{-k})\;,$
(7)

so that the field expansions become

$\displaystyle A^{a}_{i}(\mathbf{x})$
$\displaystyle=\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\frac{1}{\sqrt{2E^{a}_{\mathbf{k}}}}\left(\alpha^{a}_{i}(\mathbf{k})+\alpha^{a\dagger}_{i}(-\mathbf{k})\right)e^{i\mathbf{k\cdot
x}}$ (8) $\displaystyle\Pi^{a}_{i}(\mathbf{x})$
$\displaystyle=-i\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\sqrt{\frac{E^{a}_{\mathbf{k}}}{2}}\left(\alpha^{a}_{i}(\mathbf{k})-\alpha^{a\dagger}_{i}(-\mathbf{k})\right)e^{i\mathbf{k\cdot
x}}\ .$ (9)

The relation between the hyperbolic Bogoliubov angle $\theta^{a}_{k}$ and the
dispersion relation is then

$\tanh{\theta^{a}_{\mathbf{k}}}=\frac{|\mathbf{k}|-E^{a}_{\mathbf{k}}}{|\mathbf{k}|+E^{a}_{\mathbf{k}}}\
.$ (10)

Although it is not directly used in practice, the vacuum state of the
interacting theory satisfying $\alpha_{i}^{a}\ket{\Omega}=0$ is obtained from
the free vacuum via

$\displaystyle\ket{\Omega}=e^{\left({-\int\frac{d^{3}\mathbf{k}}{2(2\pi)^{3}}}\tanh{\theta^{a}_{\mathbf{k}}}(\delta_{ij}-\hat{k}_{i}\hat{k}_{j})a^{a\dagger}_{i}(\mathbf{k})a^{a\dagger}_{j}(-\mathbf{k})\right)}\ket{0}.$

To apply the Rayleigh-Ritz variational principle we require the expectation
value of the Hamiltonian in the family of rotated vacuum states,

$\displaystyle\langle H_{\Pi}\rangle_{\Omega}$
$\displaystyle=(2\pi)^{3}\delta^{3}(0)\sum_{a}\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\frac{E^{a}_{\mathbf{k}}}{2}$
$\displaystyle\langle H_{B}\rangle_{\Omega}$
$\displaystyle=(2\pi)^{3}\delta^{3}(0)\sum_{a}\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\frac{|\mathbf{k}|^{2}}{2E^{a}_{\mathbf{k}}}$
$\displaystyle\langle H_{V}\rangle_{\Omega}$
$\displaystyle=(2\pi)^{3}\delta^{3}(0)\frac{1}{8}\sum_{abc}\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\frac{d^{3}\mathbf{k}^{\prime}}{(2\pi)^{3}}\times$
$\displaystyle\times$
$\displaystyle\left[(1+(\hat{\mathbf{k}}\cdot\hat{\mathbf{k^{\prime}}}))\hat{V}(|\mathbf{k}-\mathbf{k^{\prime}}|)f^{abc}f^{abc}\left(\frac{E^{c}_{\mathbf{k}}}{E^{b}_{\mathbf{k}^{\prime}}}-C_{G}\right)\right]\
.$ (12)

The factor $\delta^{3}(0)$ simply represents the quantization volume: it can
be ignored in minimizing the energy density. The variational principle then
yields

$\frac{\delta\langle H\rangle_{\Omega}}{\delta
E^{d}_{\mathbf{q}}}=\frac{\delta(\langle H_{\Pi}\rangle_{\Omega}+\langle
H_{B}\rangle_{\Omega}+\langle H_{V}\rangle_{\Omega})}{\delta
E^{d}_{\mathbf{q}}}=0$

and this entails the following mass–gap equation for the gauge bosons,

$\displaystyle(E^{d}_{\mathbf{k}})^{2}$
$\displaystyle=|\mathbf{q}|^{2}-\frac{1}{4}\sum_{a,b}f^{abd}f^{abd}\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\times$
$\displaystyle\times\hat{V}(|\mathbf{k}-\mathbf{q}|)(1+(\mathbf{\hat{k}}\cdot\mathbf{\hat{q}})^{2})\left(\frac{(E^{b}_{\mathbf{k}})^{2}-(E^{d}_{\mathbf{q}})^{2}}{E^{b}_{\mathbf{k}}}\right).$
(13)

This is a nonlinear integral equation for $E_{\mathbf{q}}$ that appears also
on the right, so the solution needs to be iterative until a fixed point is
found. We can simplify a bit by noting that $E_{\mathbf{k}}$ does not depend
on the angular variables, so that defining an effective potential that absorbs
the polar integral

$\hat{V}_{\text{eff}}(\mathbf{k},\mathbf{q})=\frac{1}{2\pi}\int
d\Omega\,\hat{V}(|\mathbf{k}-\mathbf{q}|)(1+(\hat{\mathbf{k}}\cdot\hat{\mathbf{q}})^{2}),$
(14)

the radial equation becomes

$\displaystyle(E^{d}_{\mathbf{q}})^{2}$
$\displaystyle=|\mathbf{q}|^{2}-\frac{1}{4}\sum_{a,b}f^{abd}f^{abd}\int_{0}^{\infty}\frac{d|\mathbf{k}|}{(2\pi)^{2}}|\mathbf{k}|^{2}\times$
$\displaystyle\times\hat{V}_{\text{eff}}(\mathbf{k},\mathbf{q})\left(\frac{(E^{b}_{\mathbf{k}})^{2}-(E^{d}_{\mathbf{q}})^{2}}{E^{b}_{\mathbf{k}}}\right)\
.$ (15)

An alternative way to derive this equation is via the functional approach
Watson:2013ghq .

The potential $\hat{V}(|\mathbf{k}-\mathbf{q}|)$ is the Fourier transform of
the potential $V(|\mathbf{x}-\mathbf{y}|)$ in Eq. (1). In a confining theory,
the potential can be approximated by the Cornell linear+Coulomb $1/r$
potential, resulting in

$V(|\mathbf{x}-\mathbf{y}|)=-\frac{\alpha_{s}}{|\mathbf{x}-\mathbf{y}|}+b\,|\mathbf{x}-\mathbf{y}|\,e^{-\Lambda_{\rm
phen}|\mathbf{x}-\mathbf{y}|}\;,$ (16)

where $\alpha_{s}$ is the strong interaction constant and $b$ the string
tension. The term $e^{-\Lambda_{\rm phen}|\mathbf{x}-\mathbf{y}|}$ is a
regulator that tames the strong infrared growth of the linear potential, but
we will not use it and actually employ the computer grid to regulate the
integration.

Since the gluon pairs in the condensate are in a singlet state, all
quasiparticles have the same dispersion relation and no global symmetry is
broken, $E^{d}_{\mathbf{q}}=E_{\mathbf{q}}$ for all $d$. The sum over the
structure constants is the Casimir of the adjoint representation,
$C_{G}=\sum_{a,b}f^{abd}f^{abd}$, that for $SU(N)$ is simply $C_{G}=N$. Eq.
(15) then reduces to

$(E_{\mathbf{q}})^{2}=|\mathbf{q}|^{2}-\frac{C_{G}}{4}\int_{0}^{\infty}\frac{d|\mathbf{k}|}{(2\pi)^{2}}|\mathbf{k}|^{2}\hat{V}_{\text{eff}}(\mathbf{k},\mathbf{q})\left(\frac{E_{\mathbf{k}}^{2}-E_{\mathbf{q}}^{2}}{E_{\mathbf{k}}}\right)\
.$ (17)

##  3 Colour–symmetric gap equation for various groups

Let us now separately study the effect of the terms of the potential in Eq.
(16), starting by the linear potential (first used in a gap equation, to our
knowledge, in Swift:1983fz ; Adler:1984ri ),

$V_{L}(|\mathbf{x}-\mathbf{y}|)=b\,|\mathbf{x}-\mathbf{y}|\;,$

that, after Fourier transform, becomes

$\hat{V}_{L}=\mathcal{F}^{3}(V_{L})=-\frac{8\pi
b}{|\mathbf{k}-\mathbf{q}|^{4}}\;,$ (18)

and handling the angular integrals yields the effective potential for the
radial equation,

$\displaystyle\hat{V}_{\text{eff},\,L}=$
$\displaystyle\int_{-1}^{1}d\theta\frac{-8\pi
b}{(|\mathbf{k}|^{2}+|\mathbf{q}|^{2}-2|\mathbf{k}||\mathbf{q}|\cos{\theta})^{2}}(1+\cos^{2}\theta)=$
$\displaystyle=$ $\displaystyle-8\pi
b\Big{[}\left(\frac{|\mathbf{k}|^{2}+|\mathbf{q}|^{2}}{|\mathbf{k}|^{2}-|\mathbf{q}|^{2}}\right)^{2}\frac{1}{|\mathbf{k}|^{2}|\mathbf{q}|^{2}}+$
$\displaystyle+\frac{|\mathbf{k}|^{2}+|\mathbf{q}|^{2}}{4|\mathbf{k}|^{3}|\mathbf{q}|^{3}}\log\left(\frac{|\mathbf{k}|-|\mathbf{q}|}{|\mathbf{k}|+|\mathbf{q}|}\right)^{2}\Big{]}\;.$
(19)

The $k$ integral in Eq. (15) has a log infrared divergence upon employing the
$1/(k-q)^{4}$. The regulated equation is numerically solved (as detailed in
the appendix) and the solutions are plot in figure 1 for different symmetry
groups. In all cases we see the emergence of a mass, $m=E(0)$, larger with
increasing dimension of the symmetry group due to the $C_{G}$ colour factor in
Eq. (17).

Figure 1: Computed numerical dispersion relations $E(p)$ with the linear
potential from Eq. (19). From bottom to top they correspond to the groups
$SU(3)$ through $SU(10)$, all with the same string tension $b=0.18$ GeV2.

We now turn to the Coulomb potential that is a good description of the actual
potential when interactions are small, that is, at high momentum transfers in
non-Abelian theories. It is

$V_{C}(|\mathbf{x}-\mathbf{y}|)=-\frac{\alpha_{s}}{|\mathbf{x}-\mathbf{y}|}\;,$

with Fourier transform

$\hat{V}_{C}(|\mathbf{k}-\mathbf{q}|)=\mathcal{F}^{3}(V_{C})=-\frac{4\pi\alpha_{s}}{|\mathbf{k}-\mathbf{q}|^{2}}\,,$
(20)

and, because of the absence of a $\phi$–dependence, just as for the linear
potential, both being central, the effective potential for the radial equation
is

$\displaystyle\hat{V}_{\text{eff},\,C}$
$\displaystyle=\int_{-1}^{1}d\theta\frac{-4\pi\alpha_{s}}{|\mathbf{k}|^{2}+|\mathbf{q}|^{2}-2|\mathbf{k}||\mathbf{q}|\cos{\theta}}(1+\cos^{2}\theta)=$
$\displaystyle=4\pi$
$\displaystyle\alpha_{s}\Big{[}\frac{1}{2|\mathbf{q}|^{2}}+\frac{1}{2|\mathbf{k}|^{2}}+$
$\displaystyle+\frac{|\mathbf{k}|^{4}+6|\mathbf{k}|^{2}|\mathbf{q}|^{2}+|\mathbf{q}|^{4}}{8|\mathbf{k}|^{3}|\mathbf{q}|^{3}}\log\left(\frac{|\mathbf{k}|-|\mathbf{q}|}{|\mathbf{k}|+|\mathbf{q}|}\right)^{2}\Big{]}\
.$ (21)

This potential causes no problem in the infrared $k\xrightarrow{}q$ limit, but
the improper integral in the ultraviolet $k\xrightarrow{}\infty$ does not
converge. Since, unlike the linear potential, the Coulombic one is scale–free,
the solutions scale with the regulating cutoff). Since it is not particularly
appealing that the computer grid determines the mass gap (although common
practice in many computer fields), we will eliminate that dependence by a
fixed momentum subtraction (MOM scheme). We therefore detract from Eq. (17)
the same equation but with a fixed value of the momentum scale, $\mu$, that is
now dictating the solution’s mass. The resulting equation,

$\displaystyle(E_{\mathbf{q}})^{2}=(E^{d}_{\mu})^{2}+|\mathbf{q}|^{2}-\mu^{2}-\frac{C_{G}}{4}\int_{0}^{\infty}\frac{d|\mathbf{k}|}{(2\pi)^{2}}\frac{|\mathbf{k}|^{2}}{E_{\mathbf{k}}}\times$
$\displaystyle\times$
$\displaystyle(\hat{V}_{\text{eff}}(\mathbf{k},\mathbf{q})\big{(}(E_{\mathbf{k}})^{2}-(E_{\mathbf{q}})^{2}\big{)}-\hat{V}_{\text{eff}}(\mathbf{k},\mu)\big{(}(E_{\mathbf{k}})^{2}-(E_{\mu})^{2}\big{)}\
,$

has a much better behaviour, and any $k\xrightarrow{}\infty$ integration
divergence is suppressed by the new fixed-point subtraction, with the same
potential but opposite sign. $\mu$ has to be chosen high enough so that the
energy be practically equal to the momentum, that is, $E(\mu)=\mu$, and one is
in a quasifree regime. Beyond that, $\mu$ is arbitrary just as the choice of
cutoff was. Still it allows control of the problem’s scale without regards to
the integration grid.

Upon applying this method to various symmetry groups, the masses generically
diminish, as can be observed in figure 2. The qualitative features of mass
generation are similar to the linear potential, and in both cases the larger
the dimension of the symmetry group, the larger the mass which is generated,
all other things being equal, due to the $C_{G}$ colour factor.

Figure 2: Dispersion relations solving the gauge–boson gap equation (17) for
different $SU(N)$, all with the same strong coupling constant $\alpha_{s}=1$
in Eq. (3), with cutoffs $k_{1}=10$ GeV and $k_{2}=20$ GeV. The top plot
(regularized but unrenormalized equation) shows two bunches of three functions
$E(k)$, with the lower bunch employing the regulator $k_{1}$ and the upper one
using $k_{2}$; in both cases the symmetry groups correspond to $SU(3)$,
$SU(4)$ and $SU(5)$. The bottom plot shows the same solutions but now with a
MOM subtraction and renormalization scale of $\mu=9$ GeV.

Of course, the scales of both plots have been set so that the resulting gluon
masses make sense at the QCD scale with $SU(3)$, yielding glueballs of
reasonable mass Llanes-Estrada:2000ozq , but the reader can easily scale them
as needed to any other energy regime. The resulting dispersion relations are
standard, as would appear in a plasma with a cutoff or upon solving the
Helmholtz equation in a waveguide, and show that a minimum threshold energy is
required to propagate gluons of any momentum.

The gluon masses generated for large–dimensional groups are not exponentially
far from the QCD one, but this is because we start with the same coupling
constant at a low scale. If instead we started with the same coupling constant
at a very large (Grand Unification) scale, the much larger antiscreening of
the Yang-Mills coupling constant for larger groups would yield exponentially
larger masses at a low–scale, effectively removing such theories from the
spectrum, as we have shown elsewhere GarciaFernandez:2015jmn ; Llanes-
Estrada:2018azk .

##  4 Splitting the masses in $SU(N)\\!\\!\times\\!\\!SU(M)$

The mass generation that we have so far fixed forced us to fix the local gauge
(we have adopted the Coulomb one, but similar results have been obtained in
others), but the solutions fully respect the global color symmetry. In this
section we turn to the possibility that the solutions may spontaneously break
some global symmetry and different gluons come with different masses even if
the scale $\mu$ and the coupling $\alpha_{s}$ are the same for all of them.

We will, for the sake of simplicity, explore the partition of the $N^{2}-1$
gluons of $SU(N)$ in two subsets, one with $n$ lighter gluons and another,
containing the rest of them, heavier. We need a bit more notation to
distinguish the sets, and have opted for lowercase letters
$a,b,c,d...=1,...,n$ to denote the colours of the lighter ($L$) gluons, whose
dispersion relation shall be written as $\omega_{\mathbf{q}}$, and uppercase
letters $A,B,C,D...=n,...,N^{2}-1$ for the heavier ($H$) ones, with dispersion
relation naturally chosen as the capital letter $\Omega_{\mathbf{q}}$. To
refer to all the colors simultaneously we adopt the greek indices
$\alpha,\beta,\gamma,\delta...=1,...N^{2}-1$.

The gap equation (15) now formally separates into a $2\times 2$ system for the
two types of gluons,

$\displaystyle(\omega_{\mathbf{q}}^{d})^{2}=|\mathbf{q}|^{2}-\frac{1}{4}\int_{0}^{\infty}\frac{d|\mathbf{k}|}{(2\pi)^{2}}|\mathbf{k}|^{2}\hat{V}_{\text{eff}}(\mathbf{k},\mathbf{q})\sum_{\alpha}$
$\displaystyle\left[\sum_{b}(f^{\alpha
bd})^{2}\frac{(\omega^{b}_{\mathbf{k}})^{2}-(\omega^{d}_{\mathbf{q}})^{2}}{\omega^{b}_{\mathbf{k}}}+\sum_{B}(f^{\alpha
Bd})^{2}\frac{(\Omega^{B}_{\mathbf{k}})^{2}-(\omega^{d}_{\mathbf{q}})^{2}}{\Omega^{B}_{\mathbf{k}}}\right]$
$\displaystyle(\Omega_{\mathbf{q}}^{D})^{2}=|\mathbf{q}|^{2}-\frac{1}{4}\int_{0}^{\infty}\frac{d|\mathbf{k}|}{(2\pi)^{2}}|\mathbf{k}|^{2}\hat{V}_{\text{eff}}(\mathbf{k},\mathbf{q})\sum_{\alpha}$
$\displaystyle\left[\\!\sum_{B}\\!(f^{\alpha
BD})^{2}\\!\frac{(\Omega^{B}_{\mathbf{k}})^{2}\\!-\\!(\Omega^{D}_{\mathbf{q}})^{2}}{\Omega^{B}_{\mathbf{k}}}\\!+\\!\sum_{b}\\!(f^{\alpha
bD})^{2}\\!\frac{(\omega^{b}_{\mathbf{k}})^{2}\\!-\\!(\Omega^{D}_{\mathbf{q}})^{2}}{\omega^{b}_{\mathbf{k}}}\\!\right]\;.$

Each integral contains two terms between brackets. The first depends only on
the dispersion relation being solved for on the left hand side of the equation
(diagonal terms), be it $\omega$ for the light bosons or $\Omega$ for the
heavy ones. The second term depends on both dispersion relations and couples
the equations, pushing the solution towards the symmetric point
$\omega^{a}=\Omega^{A}$ (this can be seen with a little patience from the
combination of signs).

More compactly, and focusing on the colour structure of these equations, this
system reads

$\displaystyle(\omega_{\mathbf{q}}^{d})^{2}$
$\displaystyle=|\mathbf{q}|^{2}-\frac{1}{4}\bigintsss\frac{d|{\mathbf{k}|}}{\left(2\pi\right)^{2}}|\mathbf{k}|^{2}\hat{V}_{\text{eff}}(\mathbf{k},\mathbf{q})\times$
$\displaystyle\times\left(\text{LL}\sum_{\alpha,b}\left(f^{\alpha
bd}\right)^{2}+\text{LH}\sum_{\alpha,B}\left(f^{\alpha
Bd}\right)^{2}\right)\;,$ $\displaystyle(\Omega_{\mathbf{q}}^{D})^{2}$
$\displaystyle=|\mathbf{q}|^{2}-\frac{1}{4}\bigintsss\frac{d|{\mathbf{k}|}}{\left(2\pi\right)^{2}}|\mathbf{k}|^{2}\times$
$\displaystyle\times\left(\text{HL}\sum_{\alpha,b}\left(f^{\alpha
bD}\right)^{2}+\text{HH}\sum_{\alpha,B}\left(f^{\alpha BD}\right)^{2}\right).$

where the various symbols have obvious meaning shortening Eq.
(LABEL:eq:SistemaRoturaForzada).

The observation that gives this paper its title is that, if the combination of
structure constants of these coupling terms would vanish, which can be
achieved by making all constants of the forms $f^{\alpha Bd}$ and $f^{\alpha
bD}$ to vanish, the two equations completely decouple. We then have two copies
of Eq. (15), one for the light dispersion relation $\omega(q)$ and one for the
heavier $\Omega(q)$. The mass in each case is proportional to the size of the
factors $\sum_{\alpha,b}(f^{\alpha bd})^{2}$ or $\sum_{\alpha,B}(f^{\alpha
BD})^{2}$ that appear in the diagonal terms (not all four terms can
simultaneously vanish in a non-Abelian gauge theory, since they are sums of
squares and some structure constants must be nonzero).

The vanishing of the two coupling terms is precisely what happens if the split
of the gauge bosons is done along the lines of two commuting generating
algebras, so that the algebra corresponding to the total group is not simple,
but the direct sum of two ideals, in group–theory parlance
($\mathfrak{su}(N)\oplus\mathfrak{su}(M)\oplus...$).

We have chosen to split the system in two sets, but the reader can easily note
that a larger number of dispersion relations $\omega_{1}$,
$\omega_{2}$,…$\omega_{j}$ is possible, in which case the system of equations
would further split into several. For each ideal in which we can decompose the
algebra, we will obtain one decoupled equation that will provide a different
gauge–boson mass, as long as the dimensions are different, $N\neq M$, which
drive different colour factors.

This splitting happens even in the presence of the same effective potential,
coupling constant and renormalization scale, and is entirely driven by the
colour factors.

In the case of simple Lie algebras such as $\mathfrak{su}(N)$, without proper
ideals, this decoupling cannot take place (because no subset of generators of
the algebra can commute with all of those outside the subset, in which case
some of the mixed–index $f$ structure constants need to be different from
zero), and the various gap equations are necessarily coupled to one another.
To this we turn in the next section.

##  5 Global colour breaking not obvious for a simple Lie algebra

In this section we report an initial exploration of the new system of coupled
equations for a couple of low-dimensional Lie algebras. Because there is no
ideal, at least three of the four terms contribute, independently of how the
partition of the $N^{2}-1$ gluons is taken. Thus, the system remains coupled.

In a first exercise, we attempt to break the symmetry by hand. In a totally
artificial manner, we include a multiplicative factor in the off-diagonal
terms of Eq. (LABEL:eq:SistemaRoturaForzada) that reduces their intensity. The
outcome is exposed in figure 3, both for a strong artificial suppression by
1/10 but also for a modest reduction factor of 8/10. In both cases, the global
colour symmetry is broken and the system converges to two dispersion relations
with different mass. The plots correspond to a global $SU(3)$ colour group in
which an $SU(2)$ subgroup remain light (dispersion relation $\omega$) and the
rest acquire a heavier dispersion relation $\Omega$.

Figure 3: We artificially attenuate the nondiagonal coupling among dispersion
relations in Eq. (LABEL:eq:SistemaRoturaForzada) to show a reaction of the
system breaking global colour. Upper plot: the multiplicative factor is 0.1,
greatly damping the strength of the coupling, and thus showing enhanced
symmetry breaking. Lower plot: the factor is only 0.8, and the explicit
symmetry breaking is still visible. The symmetry breaking pattern is $SU(3)\to
SU(2)$.

However, if we reset the equation to its original form without artificial
factors explicitly breaking the symmetry, it is not so easy to find a solution
with spontaneous breaking of a simple group. We have not yet deployed this
project to a supercomputer; in a tabletop machine we have been able to quickly
examine the following symmetry breaking chains: $SU(2)\to U(1)$, $SU(3)\to
SU(2)$, $SU(4)\to SU(3)$, $SU(4)\to SU(2)$ and $SU(5)\to SU(4)$. For example,
in this last case, of the $5^{2}-1=24$ bosons, $4^{2}-1=15$ were candidates to
remain light and the remaining 9 candidates to become heavy.

As an example analysis, we provide detail for a partition of the 8 gluons of
$SU(3)$ into a group of 3 and a group of 5. Depending on how the first group
is chosen, its three gluons may correspond to a subgroup $SU(2)$.

Table 1 in the appendix lists the sums of squared structure constants of the
nondiagonal, coupling terms, those multiplying $LH$ and $HL$ in Eq. (4) for
possible combinations of three gluons chosen among the eight of $SU(3)$.

As an example, reading the first row of the table, we observe that the
coupling of gluon number 8 is null. This means that, initially, this gluon’s
dispersion relation does not converge towards the others. However, the rest of
the system is coupled to it in a nonvanishing way (because the corresponding
generator $T^{8}$ is not nor does it belong to an ideal of the underlying
algebra), so that the system evolves towards the symmetric solution. This can
be observed in figure 4.

Figure 4: Intermediate steps towards convergence for the dispersion relations
of $SU(3)$ gluons with initially disparate masses. Shown are calculations at
1, at 100 and at 200 iterations. The gluon numbered as 8 is seen to quickly
decouple from the rest even on the upper diagram, the first iteration. With
100, it is clear that the system is already converging towards the symmetric
solution.

To start the iteration we have chosen an initial mass of 42.4 MeV for the
first three gluons and 848 MeV for the rest (the precise choice of these
numbers is immaterial, they here have to do with the units employed in the
program, 424 MeV as the scale of the string tension when the linear potential
is active, not the case in this purely Coulombic computation). After initial
large jumps, that even change the second derivative of the dispersion
relations in intermediate steps, the gluons converge towards the same mass in
the 1 GeV range, in this case.

One could then conjecture that the system of Eq.
(LABEL:eq:SistemaRoturaForzada) has as only fixed point
$\omega^{a}=\Omega^{A}=\omega$ for all gluons, due to some symmetry upon
reorganizing the $f^{abc}$ structure constants among different choices of the
boson partition. Whatever this might be, we have not yet been able to identify
it.

##  6 Outlook

We have worked out the known gap equation for gluodynamics in the North
Carolina State family of global colour models inspired by Coulomb gauge QCD,
then extended it to allow for the possibility of different bosons acquiring
different mass in a spontaneous way. This leads to a coupled system of
equations and we have performed a first investigation of its colour structure.
For simplicity, we have limited ourselves here to split the gluons into two
groups (light and heavy), but we have also made a few exploratory runs in
which each of the gluons might acquire its own different mass. Computations
here are more numerically costly as several independent functions have to be
simultaneously determined (and note that the number of them grows as $N^{2}-1$
with the dimension of the group). We have found nothing different to report so
we omit the discussion for the time being.

What would be extremely interesting is to find an alley for spontaneous
symmetry breaking among simple groups such as $SU(N)\to SU(M)$ but we have not
yet identified an example where the system converges to two sets of gluons
with different mass in this form. If this was possible, one could do away with
complicated Higgs boson representations in Grand Unified Theories that seem
rather ad hoc. Our exploratory study has been limited in scope and we have
only examined a few group breaking patterns.

Because we do not have a clear proof that the system must remain symmetric for
a simple group either, we must limit ourselves to leaving the question of
whether this is possible as open for future investigation. With more computing
power we hope to be able to systematize the choice of the gluons that remain
vs. those that remain heavy, to extend the system beyond binary (with three,
four or more types of dispersion relations with different gluon masses)

The current findings do show how, due to the colour factors alone (with equal
couplings for the two groups), the gauge bosons in an $SU(N)\times SU(M)$
theory with $N\neq M$ acquire different fixed–gauge masses. This entails a
breaking of possible accidental symmetries: for example, fermions in the
fundamental representation carrying indices for both groups $\psi_{nm}$ could
be seen, stretching the index, (such as in 1=red–up, 2=red–down, 3=blue–up
etc.) as belonging to the fundamental representation of $SU(N\times M)$. This
global symmetry is not gauged by definition of the Lagrangian, rather it would
be “accidental”. It ceases making sense when the mass of the gauge bosons of
the two subgroups are different, so it is broken without the resort of a Higgs
boson multiplet. As argued by Dobson and collaborators Dobson:2022crf ;
Dobson:2022ngz , the Higgs in more general theories than the SM should best be
defined in terms of composite, gauge–invariant fields. It is not unconceivable
that in strongly coupled theories, the equivalent field could be made from
modes in the gauge–boson spectrum itself, due to nonlinearities.

## Appendix

### Numerical solution of the gap equation

In this first appendix we comment on the numerical solution of Eq. (15). The
method is easily extended to the system (4).

We proceed by iteration from an initial guess $\tilde{E}^{d}(q)$ for the
dispersion relation that differs from the real function by
$E^{d}(q)=\tilde{E^{d}}(q)+\epsilon^{d}(q)$ (where we define
$q\equiv|\mathbf{q}|$). This we substitute in Eq. (3) to isolate
$\epsilon^{b}(q)$ to linear order in the Taylor expansion (we here omit the
renormalization subtraction for conciseness, but it has been programmed),

$\displaystyle\tilde{E}^{d}(q)^{2}-q^{2}+\frac{1}{4}\sum_{a,b}f^{abd}f^{abd}\int_{0}^{\infty}\frac{dk}{(2\pi)^{2}}k^{2}\hat{V}_{\rm
eff}(k,q)\times$
$\displaystyle\times\left(\frac{\tilde{E}^{b}(k)^{2}-\tilde{E}^{d}(q)^{2}}{\tilde{E}^{b}(k)}\right)\approx$
$\displaystyle-2\tilde{E}^{d}(q)\epsilon^{d}(q)-\frac{1}{4}\sum_{a,b}f^{abd}f^{abd}\int_{0}^{\infty}\frac{dk}{(2\pi)^{2}}k^{2}\hat{V}_{\rm
eff}(k,q)\times$
$\displaystyle\times\left[\left(\frac{\tilde{E}^{b}(k)^{2}+\tilde{E}^{d}(q)^{2}}{\tilde{E}^{b}(k)^{2}}\epsilon^{b}(k)\right)-2\frac{\tilde{E}^{d}(q)\epsilon^{d}(q)}{\tilde{E}^{b}(k)}\right]\
.$

It is convenient to introduce auxiliary functions,

$\displaystyle b^{d}(q)$
$\displaystyle:=\tilde{E}^{d}(q)^{2}-q^{2}+\frac{1}{4}\sum_{a,b}f^{abd}f^{abd}\times$
$\displaystyle\times\int_{0}^{\infty}\frac{dk}{(2\pi)^{2}}k^{2}\hat{V}_{\rm
eff}(k,q)\left(\frac{\tilde{E}^{b}(k)^{2}-\tilde{E}^{d}(q)^{2}}{\tilde{E}^{b}(k)}\right)$
$\displaystyle A^{db}(q,k)$
$\displaystyle:=-2\tilde{E}^{b}(k)\delta^{bd}\delta(k-q)-$
$\displaystyle-\frac{1}{4}\sum_{a}f^{abd}f^{abd}\frac{k^{2}}{(2\pi)^{2}}\hat{V}_{\rm
eff}(k,q)\times$
$\displaystyle\times\left[\left(\frac{\tilde{E}^{b}(k)^{2}+\tilde{E}^{d}(q)^{2}}{\tilde{E}^{b}(k)^{2}}\right)-2\frac{\tilde{E}^{d}(q)\delta^{bd}\delta(k-q)}{\tilde{E}^{b}(k)}\right]\;,$
(27)

to shorten notation, so that Eq. (LABEL:eq:numericGapglue) is recognisable as
a linear system

$b^{d}(q)=\sum_{b}\int_{0}^{\infty}dk\,A^{db}(q,k)\epsilon^{b}(k)\ .$ (28)

We then discretise momenta to make the expression amenable to automation,

$b^{d}_{i}=\sum_{b}\sum_{j}\Delta k_{j}A^{db}_{ij}\epsilon^{b}_{j}\ .$ (29)

As $E(k)$ approaches its linear asymptote for large $k$ and its nontrivial
structure is at low $k$, we skew the discrete grid to have more points towards
low $k$ with the help of a change of variable that introduces a Jacobian.

Here the algorithms for Eq. (15) and (4) diverge, as the global colour
symmetry (all gluons have equal mass) allows the first to take a simpler form.
Because all $\omega^{a}$ are equal, they can be factored out of the colour
factors, so that the colour indices can be summed with the closure relation.
The auxiliary quantities of Eq. (27) then read

$\displaystyle b(q)$
$\displaystyle:=b^{d}(q)=\tilde{E}(q)^{2}-q^{2}+\frac{C_{G}}{4}\int_{0}^{\infty}\frac{dk}{(2\pi)^{2}}k^{2}\times$
$\displaystyle\times\hat{V}_{\rm
eff}(k,q)\left(\frac{\tilde{E}(k)^{2}-\tilde{E}(q)^{2}}{\tilde{E}(k)}\right)$
$\displaystyle A(q,k)$
$\displaystyle:=\sum_{b}A^{db}(q,k)=-2\tilde{E}(k)\delta(k-q)-\frac{C_{G}}{4}\frac{k^{2}}{(2\pi)^{2}}\times$
$\displaystyle\times\hat{V}_{\rm
eff}(k,q)\left[\left(\frac{\tilde{E}(k)^{2}+\tilde{E}(q)^{2}}{\tilde{E}(k)^{2}}\right)-2\frac{\tilde{E}(q)\delta(k-q)}{\tilde{E}(k)}\right]\
.$ (30)

The linear system that allows to extract each Newton step of the algorithm
then takes the form

$b(q)=\int_{0}^{\infty}dk\,A(q,k)\epsilon(k)\;,$ (31)

or discretized,

$b_{i}=\sum_{j}\Delta k_{j}A_{ij}\epsilon_{j}\,,$ (32)

which is solved for $\epsilon$, allowing for the update of $E(k)$. In the case
of nondiagonal colour couplings, Eq. (29) has to be addressed instead.

### Structure constants for the generic Lie algebra $\mathfrak{su}$(N)

The $SU(N)$ has $N^{2}-1$ generators that commute according to the rules of
the $\mathfrak{su}$(N) algebra, $[T^{a},T^{b}]=if^{abc}T^{c}$ with structure
constants $f^{abc}$.

The following formulae give these structure constants in a direct way which
makes them apt for computer programming, as necessary to solve for example Eq.
(13).

The $N^{2}-1$ generators of $SU(N)$ can be split into three subsets. First,
there are the $N-1$ diagonal matrices of the Cartan subalgebra, that commute
and thus provide simultaneous good quantum numbers (such as hypercharge and
the third component of isospin in the case of $SU(3)$). Then, there are
Bertlmann_2008  $N(N-1)/2$ antisymmetric matrices and $N(N-1)/2$ symmetric but
nondiagonal matrices. We then split the color index of the adjoint
representation into three distinct ones for each of this types of matrices:
$D$ for diagonal, $S$ for symmetric and $A$ for antisymmetric. When acting on
the fundamental representation of the fermions they take the explicit form
Bertlmann_2008 :

$\displaystyle T_{S_{nm}}$
$\displaystyle=\frac{1}{2}(\ket{m}\bra{n}+\ket{n}\bra{m})$ (33) $\displaystyle
T_{A_{nm}}$ $\displaystyle=\frac{1}{2}(\ket{m}\bra{n}-\ket{n}\bra{m})$ (34)
$\displaystyle T_{D_{n}}$
$\displaystyle=\frac{1}{\sqrt{2n(n-1)}}\left(\sum_{k=1}^{N-1}\ket{k}\bra{k}+(1-n)\ket{n}\bra{n}\right)\;.$
(35)

The $n$ and $m$ subindices take $N$ different values, the size of the
fundamental representation of $\mathfrak{su}$(N). We can retrieve the values
of the various subindices from the closed formulae

$\displaystyle S_{nm}$ $\displaystyle=n^{2}+2(m-n)-1$ $\displaystyle A_{nm}$
$\displaystyle=n^{2}+2(m-n)$ $\displaystyle D_{n}$ $\displaystyle=n^{2}-1\;,$

that guarantee that no value is repeated and all values from $1$ to $N^{2}-1$
is covered when additionally imposing $1\leq m<n\leq N$. Thus, any
$a\in\\{S_{nm},A_{nm},D_{n}\\}$ and the correspondence between these and the
usual indices is bijective.

For example, in $SU(2)$ one has three generators, and with this convention the
symmetric one is $T_{S_{21}}=T_{1}$ (Pauli’s $\sigma_{x}/2$ matrix,
essentially), the antisymmetric $T_{A_{21}}=T_{2}$ (that is, $\sigma_{y}/2$)
and the diagonal one is $T_{D_{2}}=T_{3}$ (or $\sigma_{z}/2$). If, in turn, we
now apply these indexing rules to $SU(3)$, we reproduce Gell-Mann’s matrices
in the usual order, with diagonal $T_{3}$ and $T_{8}$.

This indexing system and the explicit expression in terms of commutators of
the generators (normalized as $Tr(T^{a}T^{b})=\delta^{ab}/2$)

$f^{abc}=-2i{\rm Tr}\left[[T^{a},T^{b}],T^{c}\right]\;,$ (36)

that makes their total antisymmetry explicit, allows to find directly
programmable expressions reported by Bossion and Huo Bossion:2021zjn ,

$\displaystyle f_{S_{nm}S_{kn}A_{km}}$ $\displaystyle=f_{S_{nm}S_{nk}A_{km}}$
$\displaystyle=f_{S_{nm}S_{km}A_{kn}}$
$\displaystyle=f_{A_{nm}A_{kn}A_{km}}=\frac{1}{2}$ $\displaystyle
f_{S_{nm}A_{nm}D_{m}}$ $\displaystyle=-\sqrt{\frac{m-1}{2m}}$ $\displaystyle
f_{S_{nm}A_{nm}D_{n}}$ $\displaystyle=\sqrt{\frac{n}{2(n-1)}}$ $\displaystyle
f_{S_{nm}A_{nm}D_{k}}$ $\displaystyle=\sqrt{\frac{1}{2k(k-1)}}\hskip
14.22636ptm<k<n$

(other combinations of the different symmetries are null unless obtained by
permutation and antisymmetry, for example if $f_{123}=1/2$ then
$f_{231}=f_{312}=-f_{321}=-f_{132}=-f_{213}=1/2$).

### Sums of squared structure constants for SU(3), partitioning it in 3-gluon
and 5-gluon subsets

Here we list, as an example of the combinations of $\sum f^{2}$ with various
indices that appear in Eq. (4), an exhaustive list of these combinations for
the example in which the eight gluons of $SU(3)$ split into three light ones
(that, in certain cases but not necessarily, can generate an $SU(2)$ subgroup)
and five heavier ones. The number of combinations of the eight gluons taken
three at a time is

$\begin{pmatrix}8\\\ 3\end{pmatrix}=56\;,$

and we list them explicitly in table 1 and following.

Table 1: Sums of squared structure constants necessary for Eq. (4). The rows alternate in shade. The grey shaded ones indicate the $SU(3)$ gluon combination, with the first three gluons corresponding to the light ones with colour index $d$ and the following five ones to the heavy ones with index $D$. The row with white background immediately below lists, in the first three columns, the corresponding $\sum_{\alpha,B}\left(f^{\alpha Bd}\right)^{2}$ to each $d$ entry from the row above. The remaining columns give $\sum_{\alpha,b}\left(f^{\alpha bD}\right)^{2}$, also associated to the index $D$ immediately above each entry. 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
---|---|---|---|---|---|---|---
1.00 | 1.00 | 1.00 | 0.75 | 0.75 | 0.75 | 0.75 | 0.00
1 | 2 | 4 | 3 | 5 | 6 | 7 | 8
1.75 | 1.75 | 2.50 | 2.25 | 1.50 | 0.75 | 0.75 | 0.75
1 | 2 | 5 | 3 | 4 | 6 | 7 | 8
1.75 | 1.75 | 2.50 | 2.25 | 1.50 | 0.75 | 0.75 | 0.75
1 | 2 | 6 | 3 | 4 | 5 | 7 | 8
1.75 | 1.75 | 2.50 | 2.25 | 0.75 | 0.75 | 1.50 | 0.75
1 | 2 | 7 | 3 | 4 | 5 | 6 | 8
1.75 | 1.75 | 2.50 | 2.25 | 0.75 | 0.75 | 1.50 | 0.75
1 | 2 | 8 | 3 | 4 | 5 | 6 | 7
2.00 | 2.00 | 3.00 | 2.00 | 1.25 | 1.25 | 1.25 | 1.25
1 | 3 | 4 | 2 | 5 | 6 | 7 | 8
1.75 | 1.75 | 2.50 | 2.25 | 1.50 | 0.75 | 0.75 | 0.75
1 | 3 | 5 | 2 | 4 | 6 | 7 | 8
1.75 | 1.75 | 2.50 | 2.25 | 1.50 | 0.75 | 0.75 | 0.75
1 | 3 | 6 | 2 | 4 | 5 | 7 | 8
1.75 | 1.75 | 2.50 | 2.25 | 0.75 | 0.75 | 1.50 | 0.75
1 | 3 | 7 | 2 | 4 | 5 | 6 | 8
1.75 | 1.75 | 2.50 | 2.25 | 0.75 | 0.75 | 1.50 | 0.75
1 | 3 | 8 | 2 | 4 | 5 | 6 | 7
2.00 | 2.00 | 3.00 | 2.00 | 1.25 | 1.25 | 1.25 | 1.25
1 | 4 | 5 | 2 | 3 | 6 | 7 | 8
2.50 | 1.75 | 1.75 | 1.50 | 1.50 | 0.75 | 0.75 | 1.50
1 | 4 | 6 | 2 | 3 | 5 | 7 | 8
2.50 | 2.50 | 2.50 | 1.50 | 1.50 | 1.50 | 1.50 | 1.50
1 | 4 | 7 | 2 | 3 | 5 | 6 | 8
2.50 | 2.50 | 2.50 | 1.50 | 1.50 | 1.50 | 1.50 | 1.50
1 | 4 | 8 | 2 | 3 | 5 | 6 | 7
2.75 | 2.00 | 2.25 | 1.25 | 1.25 | 2.00 | 1.25 | 1.25
1 | 5 | 6 | 2 | 3 | 4 | 7 | 8
2.50 | 2.50 | 2.50 | 1.50 | 1.50 | 1.50 | 1.50 | 1.50
1 | 5 | 7 | 2 | 3 | 4 | 6 | 8
2.50 | 2.50 | 2.50 | 1.50 | 1.50 | 1.50 | 1.50 | 1.50
1 | 5 | 8 | 2 | 3 | 4 | 6 | 7
2.75 | 2.00 | 2.25 | 1.25 | 1.25 | 2.00 | 1.25 | 1.25
1 | 6 | 7 | 2 | 3 | 4 | 5 | 8
2.50 | 1.75 | 1.75 | 1.50 | 1.50 | 0.75 | 0.75 | 1.50
Table 2: Continued from table 1 1 | 6 | 8 | 2 | 3 | 4 | 5 | 7
---|---|---|---|---|---|---|---
2.75 | 2.00 | 2.25 | 1.25 | 1.25 | 1.25 | 1.25 | 2.00
1 | 7 | 8 | 2 | 3 | 4 | 5 | 6
2.75 | 2.00 | 2.25 | 1.25 | 1.25 | 1.25 | 1.25 | 2.00
2 | 3 | 4 | 1 | 5 | 6 | 7 | 8
1.75 | 1.75 | 2.50 | 2.25 | 1.50 | 0.75 | 0.75 | 0.75
2 | 3 | 5 | 1 | 4 | 6 | 7 | 8
1.75 | 1.75 | 2.50 | 2.25 | 1.50 | 0.75 | 0.75 | 0.75
2 | 3 | 6 | 1 | 4 | 5 | 7 | 8
1.75 | 1.75 | 2.50 | 2.25 | 0.75 | 0.75 | 1.50 | 0.75
2 | 3 | 7 | 1 | 4 | 5 | 6 | 8
1.75 | 1.75 | 2.50 | 2.25 | 0.75 | 0.75 | 1.50 | 0.75
2 | 3 | 8 | 1 | 4 | 5 | 6 | 7
2.00 | 2.00 | 3.00 | 2.00 | 1.25 | 1.25 | 1.25 | 1.25
2 | 4 | 5 | 1 | 3 | 6 | 7 | 8
2.50 | 1.75 | 1.75 | 1.50 | 1.50 | 0.75 | 0.75 | 1.50
2 | 4 | 6 | 1 | 3 | 5 | 7 | 8
2.50 | 2.50 | 2.50 | 1.50 | 1.50 | 1.50 | 1.50 | 1.50
2 | 4 | 7 | 1 | 3 | 5 | 6 | 8
2.50 | 2.50 | 2.50 | 1.50 | 1.50 | 1.50 | 1.50 | 1.50
2 | 4 | 8 | 1 | 3 | 5 | 6 | 7
2.75 | 2.00 | 2.25 | 1.25 | 1.25 | 2.00 | 1.25 | 1.25
2 | 5 | 6 | 1 | 3 | 4 | 7 | 8
2.50 | 2.50 | 2.50 | 1.50 | 1.50 | 1.50 | 1.50 | 1.50
2 | 5 | 7 | 1 | 3 | 4 | 6 | 8
2.50 | 2.50 | 2.50 | 1.50 | 1.50 | 1.50 | 1.50 | 1.50
2 | 5 | 8 | 1 | 3 | 4 | 6 | 7
2.75 | 2.00 | 2.25 | 1.25 | 1.25 | 2.00 | 1.25 | 1.25
2 | 6 | 7 | 1 | 3 | 4 | 5 | 8
2.50 | 1.75 | 1.75 | 1.50 | 1.50 | 0.75 | 0.75 | 1.50
2 | 6 | 8 | 1 | 3 | 4 | 5 | 7
2.75 | 2.00 | 2.25 | 1.25 | 1.25 | 1.25 | 1.25 | 2.00
2 | 7 | 8 | 1 | 3 | 4 | 5 | 6
2.75 | 2.00 | 2.25 | 1.25 | 1.25 | 1.25 | 1.25 | 2.00
3 | 4 | 5 | 1 | 2 | 6 | 7 | 8
2.50 | 1.75 | 1.75 | 1.50 | 1.50 | 0.75 | 0.75 | 1.50
3 | 4 | 6 | 1 | 2 | 5 | 7 | 8
2.50 | 2.50 | 2.50 | 1.50 | 1.50 | 1.50 | 1.50 | 1.50
3 | 4 | 7 | 1 | 2 | 5 | 6 | 8
2.50 | 2.50 | 2.50 | 1.50 | 1.50 | 1.50 | 1.50 | 1.50
3 | 4 | 8 | 1 | 2 | 5 | 6 | 7
2.75 | 2.00 | 2.25 | 1.25 | 1.25 | 2.00 | 1.25 | 1.25
3 | 5 | 6 | 1 | 2 | 4 | 7 | 8
2.50 | 2.50 | 2.50 | 1.50 | 1.50 | 1.50 | 1.50 | 1.50
Table 3: Continued from tables 1 and 2. 3 | 5 | 7 | 1 | 2 | 4 | 6 | 8
---|---|---|---|---|---|---|---
2.50 | 2.50 | 2.50 | 1.50 | 1.50 | 1.50 | 1.50 | 1.50
3 | 5 | 8 | 1 | 2 | 4 | 6 | 7
2.75 | 2.00 | 2.25 | 1.25 | 1.25 | 2.00 | 1.25 | 1.25
3 | 6 | 7 | 1 | 2 | 4 | 5 | 8
2.50 | 1.75 | 1.75 | 1.50 | 1.50 | 0.75 | 0.75 | 1.50
3 | 6 | 8 | 1 | 2 | 4 | 5 | 7
2.75 | 2.00 | 2.25 | 1.25 | 1.25 | 1.25 | 1.25 | 2.00
3 | 7 | 8 | 1 | 2 | 4 | 5 | 6
2.75 | 2.00 | 2.25 | 1.25 | 1.25 | 1.25 | 1.25 | 2.00
4 | 5 | 6 | 1 | 2 | 3 | 7 | 8
1.75 | 1.75 | 2.50 | 0.75 | 0.75 | 0.75 | 1.50 | 2.25
4 | 5 | 7 | 1 | 2 | 3 | 6 | 8
1.75 | 1.75 | 2.50 | 0.75 | 0.75 | 0.75 | 1.50 | 2.25
4 | 5 | 8 | 1 | 2 | 3 | 6 | 7
1.25 | 1.25 | 1.50 | 0.50 | 0.50 | 0.50 | 1.25 | 1.25
4 | 6 | 7 | 1 | 2 | 3 | 5 | 8
2.50 | 1.75 | 1.75 | 0.75 | 0.75 | 0.75 | 1.50 | 2.25
4 | 6 | 8 | 1 | 2 | 3 | 5 | 7
2.00 | 2.00 | 1.50 | 0.50 | 0.50 | 0.50 | 2.00 | 2.00
4 | 7 | 8 | 1 | 2 | 3 | 5 | 6
2.00 | 2.00 | 1.50 | 0.50 | 0.50 | 0.50 | 2.00 | 2.00
5 | 6 | 7 | 1 | 2 | 3 | 4 | 8
2.50 | 1.75 | 1.75 | 0.75 | 0.75 | 0.75 | 1.50 | 2.25
5 | 6 | 8 | 1 | 2 | 3 | 4 | 7
2.00 | 2.00 | 1.50 | 0.50 | 0.50 | 0.50 | 2.00 | 2.00
5 | 7 | 8 | 1 | 2 | 3 | 4 | 6
2.00 | 2.00 | 1.50 | 0.50 | 0.50 | 0.50 | 2.00 | 2.00
6 | 7 | 8 | 1 | 2 | 3 | 4 | 5
1.25 | 1.25 | 1.50 | 0.50 | 0.50 | 0.50 | 1.25 | 1.25

As can be seen, the off-diagonal combinations do not all vanish
simultaneously, meaning that the Lie algebra has no ideals of either dimension
3 nor 5 (we of course know that the $\mathfrak{su}(3)$ Lie algebra has no
ideal of any dimension, but it is reassuring to see this appear in the
tabulated data. One might entertain the hope that a clever way of splitting
the structure constants could bring about a breaking of the global symmetry
even for a simple Lie algebra, perhaps of large dimension, but we have not
found an example yet, nor do we know of a theorem (such as the no–go theorem
of Vafa and Witten in the fermion sector) that forbids it at this point.

## Acknowledgments

The authors thank early conversations with Lucas Barbero. Work partially
supported by the EU under grant 824093 (STRONG2020); spanish MICINN under
PID2019-108655GB-I00/AEI/10.13039/501100011033,
PID2019-106080GB-C21; Univ. Complutense de Madrid under research group 910309
and the IPARCOS institute.

This preprint has been issued with number IPARCOS-UCM-23-070

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