# Time-assisted energy reconstruction in a highly-granular hadronic
calorimeter

C. Graf11footnotetext: now at Robert Bosch GmbH, Renningen, Germany. and F.
Simon22footnotetext: Corresponding author.

###### Abstract

The software compensation algorithms developed for the CALICE Analog Hadron
Calorimeter are extended to incorporate time information on the cell level,
and the performance is studied in GEANT4 simulations with a detector model of
a highly-granular SiPM-on-tile calorimeter. The addition of ns-level time
resolution is found to result in significant improvement of the energy
resolution for local software compensation, with further improvement possible
with better resolution. The high correlation of energy density and time
variables show that both provide sensitivity to correlated underlying shower
features, which limits the potential of timing information when used as a
global rather than a local variable.

##  1 Introduction

Highly-granular electromagnetic and hadronic calorimeters are a crucial part
of particle-flow-centred detectors at future $e^{+}e^{-}$ colliders [1, 2, 3].
The Analog Hadron Calorimeter (AHCAL) developed by the CALICE collaboration
represents one concept for a highly-granular hadronic calorimeter [4]. It is a
sampling calorimeter on plastic scintillator basis read out by Silicon
Photomultipliers with cell sizes of
$$3$\times$3\text{\,}\mathrm{c}\mathrm{m}^{2}$$.

While the spatial resolution and the separation of showers inside a particle
jet play a key role in particle flow detectors [1], the energy resolution of
the hadronic calorimeter still adds to the overall detector performance, both
via the improvement of the energy measurement of neutral hadrons, and through
improved cluster-track matching in particle-flow algorithms (PFA) due to more
accurate energy estimates of clusters assigned to charged particles [5]. In
longitudinally and transversely segmented non-compensating hadronic
calorimeters the energy resolution for hadronic showers can be improved by
amplitude or energy-density dependent offline weighting of the measured
signals in individual detector cells [6, 7, 8], exploiting the difference in
response to electromagnetic and purely hadronic components of the showers. The
high granularity of the CALICE calorimeters makes them perfectly suited for
such reconstruction techniques, which have been developed under the term
software compensation [9] for highly-granular calorimeters. For the AHCAL, an
improvement of the hadronic energy resolution of approximately 20% compared to
a reconstruction without such techniques was achieved.

Since the large-scale physics prototypes constructed by CALICE in the
mid-2000s, the technology of highly-granular calorimeters has evolved, both on
the sensor side with improved performance and better device-to-device
uniformity, and in the area of front-end electronics, most notably by
providing cell-by-cell time measurements. This enables the full five-
dimensional reconstruction of particle showers in space, local energy density
or amplitude, and time. The large technological prototype (LTP) of the AHCAL
[10], a 22 000 channel, 0.5 m3 SiPM-on-tile highly-granular hadronic
calorimeter with steel absorbers, which was first operated in beam tests with
hadrons in 2018, has the ability to provide hit-time measurements on the cell
level with nanosecond precision, as demonstrated in preliminary studies [11].
This capability may allow conclusions about the amount of slow neutrons
generated in the shower. Shower energy that is absorbed in the generation of
neutrons during the spallation processes to overcome the nuclear binding
energy is lost for calorimetric purposes. Thus estimating the amount of
neutrons in a given shower may result in an improvement of the energy
measurement. The extension of the software compensation techniques used for
highly-granular calorimeters to include time as an additional dimension is
thus of high interest to explore the potential in this area.

In the following, methods are presented for incorporating the timing
capabilities of a highly-granular hadronic calorimeter into the energy
reconstruction. For this, two different approaches to software compensation, a
global and a local one, are presented, extending the techniques developed in
[9] to include the time domain in addition to purely energy-based observables.
All methods are developed and evaluated on simulated data.

##  2 Detector geometry and simulated data

The detector geometry used for the simulation is a sampling calorimeter with
steel absorber and 60 active layers. The design follows that of the CALICE
AHCAL LTP, but with an increased number of layers to reduce the effect of
longitudinal energy leakage on the energy reconstruction. The active elements
of the calorimeter are
$$3\text{\,}$\times$3\text{\,}\mathrm{c}\mathrm{m}^{2}$$ polystyrene-based
scintillator tiles with the scintillation light read out by SiPMs housed on
electronics boards with integrated front-end ASICs [10]. Each active layer
consists of 576 channels, resulting in a transverse size of
$$72\text{\,}$\times$72\text{\,}\mathrm{c}\mathrm{m}^{2}$$. The total
stainless steel absorber thickness per layer is
$17\text{\,}\mathrm{m}\mathrm{m}$, with $16\text{\,}\mathrm{m}\mathrm{m}$
provided by the mechanical absorber structure itself, and
$1\text{\,}\mathrm{m}\mathrm{m}$ from the cassettes housing the active
elements. The LTP has 38 layers, which is not sufficiently deep to fully
contain higher-energy hadronic showers without an additional tail catcher with
coarser longitudinal sampling, or an electromagnetic calorimeter upstream of
the hadronic calorimeter. To avoid energy-leakage-induced distortions of the
energy resolution studies discussed in the present paper, 60 active layers and
the corresponding absorbers have been simulated. The simulation was performed
using the DD4HEP [12] framework in combination with GEANT4 v10.03 [13]. The
QGSP_BERT_HP physics list was used for all GEANT4 simulation samples discussed
in this study. Scintillator saturation was implemented using a Birks’
parameter of
$k_{B}\,=\,0.07943\,$\text{\,}\mathrm{m}\mathrm{m}\mathrm{M}\mathrm{e}\mathrm{V}^{-1}$$
[14].

The idealized energy depositions generated by GEANT4 are combined per detector
cell, assuming a maximum integration time of $2000\text{\,}\mathrm{ns}$, and
are transformed into hits in a digitization process that mimics the detector
effects. This includes effects of the read-out electronics (i.e., signal
shaping and electronic noise) and of the SiPMs (i.e, statistical smearing and
saturation). The energy depositions in units of $\text{\,}\mathrm{keV}$ are
converted to the MIP scale by a factor of
$477\text{\,}\mathrm{k}\mathrm{eV}\mathrm{/}\text{MIP}$. This factor is
calibrated such that a minimum ionizing particle (MIP) (i.e., a muon in the
$\text{\,}\mathrm{GeV}$ range) has a most probable energy deposition of
$1\text{\,}\text{MIP}$ in a scintillator tile traversed at normal incidence.
Detector cells with an energy deposition below $0.5\text{\,}\text{MIP}$ are
not considered in the analysis, in line with the threshold typically applied
in the analysis of CALICE AHCAL data. For the time measurements a gaussian
smearing with a resolution of $1\text{\,}\mathrm{ns}$ is applied to the
generator-level time stamp of the hit, which is given by the time of the first
energy deposit in the cell.

The CALICE collaboration has observed good agreement between test beam data of
its physics prototype and simulated hadronic showers with respect to the hit
energy distributions and shower shapes [15]. It was shown by the CALICE T3B
Experiment that the time distribution of hadronic showers, as simulated by
GEANT4 with the physics list used in the present study, is in good agreement
with the simulations using a steel absorber [16].

##  3 Correlation of energy and time observables

The overall hit time distribution for $40\text{\,}\mathrm{GeV}$ pions is shown
in 1(a) on two different time scales. Three different regimes can be
identified: The instantaneous part up to about $3\text{\,}\mathrm{ns}$,
governed by the time resolution of the detector, an intermediate,
exponentially falling, part mainly due to hits caused by neutron elastic
scattering, and a late tail dominated by neutron capture events. In the
present study, the fraction of energy deposited later than
$3\text{\,}\mathrm{ns}$ is adopted as the variable to characterize the timing
behaviour of an event. The hits contributing to this fraction are indicated in
red in 1(a). In 1(b) the median value of this fraction is shown over the range
of beam energies considered. The shaded region visualizes the central
$50\text{\,}\mathrm{\char 37\relax}$-interval. The median value is about
$3\text{\,}\mathrm{\char 37\relax}$ for most beam energies and slightly higher
for lower energies. The monotonic decrease of this value is expected, as the
electromagnetic fraction is increasing for higher energies, resulting in a
reduction of the influence of the neutron-dominated later parts of the shower.

(a)

(b)

Figure 1: (a) Hit time distribution for $40\text{\,}\mathrm{GeV}$ pions in the
simulated calorimeter on a larger and smaller time scale. For the smaller time
scale, the hit times above $3\text{\,}\mathrm{ns}$, which are called _late_
throughout this study, are indicated in red. (b) Median of the fraction of
measured late energy deposistions in a shower. The shaded region indicates the
$50\text{\,}\mathrm{\char 37\relax}$ quantile.

2(a) shows the 2D-distribution of the fraction of late energy depositions in
an event and the relative difference of the reconstructed energy to the beam
energy ($\Delta E$) for $40\text{\,}\mathrm{GeV}$ pions. It can be seen that
for low values of the fraction of late energy deposits
($E_{\text{dep}}(T_{\text{hit}}>$3\text{\,}\mathrm{ns}$)$) on average the
reconstructed energy is too high, while for high fractions of late energy
depositions the reconstructed energy tends to be too low. This observation is
in line with the interpretation that a large amount of late energy depositions
indicates a high hadronic fraction and thus a lower calorimeter response. For
comparison in 2(b) $\Delta E$ is shown dependent on $C_{global}$, an
observable that describes the hit energy distribution of an event and which
was developed to correct the reconstructed energy within the AHCAL [9] in the
global software compensation scheme. This variable is defined as the ratio
between the fraction of hits in the calorimeter with energy depositions larger
than $5\text{\,}\text{MIP}$ ($C_{\text{thr}}$) and the fraction of hits with
energy depositions larger than the average hit energy in this shower
($C_{\text{av}}$):

$C_{\text{global}}=\frac{C_{\text{thr}}}{C_{\text{av}}}.$ (3.1)

This variable exhibits a positive correlation with $\Delta E$, which was
exploited in [9] to improve the energy resolution of the calorimeter.

(a)

(b)

Figure 2: Correlation of (a) the fraction of late energy deposits and (b)
$C_{\text{global}}$ with the ratio of the reconstructed to the true beam
energy ($\Delta E$). The amount of entries in the highest density bins holding
$90\text{\,}\mathrm{\char 37\relax}$ of the data are indicated by the color
scale, while the remaining lower density bins are shown as individual data
points (gray). Additionally, contour lines generated by a kernel density
estimation are drawn in black.

In Figure 3 the absolute value of the correlation coefficients between the
fraction of late energy hits and $C_{global}$ with $\Delta E$ are shown over
the full energy range. Significant correlations of both observables with the
reconstructed energy are visible. For $C_{global}$, however, the correlation
is consistently higher than for the time-based observable. For higher energies
the correlation coefficients are slightly higher, because of the lower
statistical uncertainty resulting from the larger number of hits in these
events. The large correlation coefficients observed indicate that both
observables can be used to enhance the energy resolution of the AHCAL.

Figure 3: Pearson correlation coefficients of the reconstructed energy
($E_{\text{std}}$, see section 4) with the fraction of late energy deposits
(green) and $C_{\text{global}}$ (orange).

In order to study the interplay of the correlation of the late energy
depositions and $C_{\text{global}}$ with $\Delta E$, we decompose the
$C_{\text{global}}$ variable and look at the correlations with $\Delta E$
separately. The numerator in the definition of $C_{\text{global}}$ is
$C_{\text{thr}}$ describing the fraction of the hits in a shower that are of
high energy density, thus being sensitive to the electromagnetic part of the
shower. The fraction of late energy deposits is sensitive to the amount of
neutron interaction in the shower. In 4(a) the data of
$30\text{\,}\mathrm{GeV}50\text{\,}\mathrm{GeV}$ pions is binned with respect
to both variables, $C_{\text{thr}}$ and
$E_{\text{dep}}(T_{\text{hit}}>$3\text{\,}\mathrm{ns}$)$. Additionally the
deviation of the reconstructed to the true energy ($\Delta E$) is indicated as
the color scale. Areas of lighter color indicate a too high energy
reconstruction, while areas of darker color indicate a too low reconstructed
energy. The contour lines of the color space are indicated as a yellow dotted
line. It can be seen that the contour lines are not horizontal and thus both
variables are necessary to explain the variance in $\Delta E$. Both
information of the size of the electromagnetic part as well as of the neutron
activity are necessary for an optimal explanation of $\Delta E$.

In 4(b) the data is binned with respect to $C_{\text{global}}$ on the
$y$-axis. In this case, the contour lines are nearly parallel to the $x$-axis
indicating that $C_{\text{global}}$ incorporates information about the
electromagnetic part of the shower as well as the neutron activity and thus
the timing information only adds marginal value for an improved reconstruction
of the energy with respect to the information encoded in $C_{\text{global}}$.

(a)

(b)

Figure 4: Mean relative deviation of the reconstructed and true energy for
$30\text{\,}\mathrm{GeV}50\text{\,}\mathrm{GeV}$ pions in each bin shown by
the color scale. The data is binned with respect to the fraction of late
energy deposits ($E_{\text{dep}}(T_{\text{hit}}>$3\text{\,}\mathrm{ns}$)$) and
(a) $C_{\text{thr}}$ and (b) $C_{\text{global}}$. The yellow dotted line
indicates the smoothed contour lines of the color space.

##  4 Methods

The standard energy reconstruction method of a hadronic shower $i$ is the sum
of all hit energies $e_{j}$:

$E_{\text{std},i}=\sum_{j\,\in\,\text{hits}}e_{j,i}.$ (4.1)

The software compensation methods improve on this simple reconstruction by
learning weights to down- or upweight certain parts of the shower. Two
different methods are discussed for the time assisted energy reconstruction in
the following: A global and a local software compensation technique.

###  4.1 Global software compensation

The global software compensation technique corrects the energy sum of the
standard reconstruction $E_{\text{std}}$ by a global correction factor, which
is a function of an observable $\theta$ of the shower. In previous work the
aforementioned $C_{global}$ variable was used as the observable $\theta$ [9].
The correction factor is a second order polynomial of $\theta$, such that the
reconstructed energy of event $i$ becomes:

$E_{\text{reco},i}^{\text{global}}=E_{\text{std},i}\cdot(a+b\,\theta_{i}+c\,\theta_{i}^{2}).$
(4.2)

The parameters $a$, $b$, and $c$ are determined by an optimization procedure
minimizing the following loss function:

$L=\sum_{i\,\in\,\text{events}}\frac{\left(E_{\text{reco},i}^{\text{global}}-E_{\text{beam},i}\right)^{2}}{E_{\text{beam},i}},$
(4.3)

where the sum runs over all events in the sample and $E_{\text{beam}}$ is the
simulated true beam energy. In order to incorporate the hit time measurements
into this approach, it can be extended by using a second observable $\phi$:

$E_{\text{reco},i}^{\text{global}}=E_{\text{std},i}\cdot(a+b\,\theta_{i}+c\,\phi_{i}+d\,\theta_{i}\phi_{i}),$
(4.4)

where, in this case, $\phi_{i}$ is the fraction of late energy depositions in
a shower. In order to limit the overall number of parameters the quadratic
terms are omitted in this approach, but a correlation term
$\theta_{i}\phi_{i}$ is added.

As the parameters $a$ to $d$ are expected to change with the beam energy they
are determined for every training beam energy separately. The resulting
parameters are then fitted with a quadratic function such that:

$a(E)=a_{0}+a_{1}\,E+a_{2}\,E^{2},$ (4.5)

and for parameters $b$ to $d$ accordingly. While in the training the true beam
energy is used to determine parameters $a_{0}$ to $a_{2}$, for the test data
set the standard energy reconstruction $E_{\text{std},i}$ is used to determine
the parameters $a$ to $d$ for each shower separately. In total the global
software compensation method has nine free parameters in the case of using one
observable (Equation 4.2) and twelve free parameters in the case of using two
observables (Equation 4.4).

###  4.2 Local software compensation

In the local software compensation approach, weights are applied separately to
each hit in the shower following the procedure described in [9]. For each
shower, the hits are binned with respect to their energy $e$ and different
weights $w$ are applied for each bin:

$E_{\text{reco}}^{\text{local}}=\sum_{j\,\in\,\text{hits}}e_{j}\cdot
w(e_{j},E_{\text{std}}).$ (4.6)

The optimal weights depend on the shower energy. Following the approach in
[17], the weights in each bin are parameterized by a second order polynomial
of the energy:

$w(e_{j},E)=e_{j}\cdot(c_{k,0}+c_{k,1}\,E+c_{k,2}\,E^{2}),$ (4.7)

with $c_{k}$ being the set of parameters for the $k$th energy bin of the
corresponding hit energy $e_{j}$. During training the true beam energy
$E_{\text{beam},i}$ is used in the above equation, while for testing the
standard reconstructed energy $E_{\text{std}}$ is used to evaluate the
weights. The set of weights is determined by minimizing the following loss
function:

$L=\sum_{i\,\in\,\text{events}}\frac{\left[\sum_{j\,\in\,\text{hits}}e_{j,i}\cdot
w(e_{j,i},E_{\text{beam},i})-E_{\text{beam},i}\right]^{2}}{E_{\text{beam},i}}.$
(4.8)

In the presented analysis the hit energy bins shown in Table 1) are used.

Table 1: Energy range of bins used in the local software compensation approach. Bin | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---|---
Range [MIP] | $0.51.0$ | $1.01.5$ | $1.52.5$ | $2.55.0$ | $5.015$ | $1550$ | $50100$ | $>100$

In order to incorporate hit time measurements, the total number of bins is
doubled. One set of hit energy bins is used for early hits
($\leq$3\text{\,}\mathrm{ns}$$), and the other one for late hits
($>$3\text{\,}\mathrm{ns}$$).

##  5 Results

The results of using the global and the local software compensation approach
are presented in Figure 5. The figures consist of three panels with the beam
energy on the $x$-axis. The first panel shows the relative resolution measured
by the $\text{RMS}_{90}$, which is defined as the root mean squared of the
smallest $90\text{\,}\mathrm{\char 37\relax}$ interval of the reconstructed
energy distributions333$\text{RMS}_{90}$ is adopted here to provide a robust
measure of the width of the distribution while reducing the sensitivity to
tails. For a perfectly gaussian distribution, $\text{RMS}_{90}$ is 21% smaller
than the $\sigma$ of the distribution.. The second panel shows the improvement
in the $\text{RMS}_{90}$ of the software compensation methods with respect to
the standard reconstruction. The third panel indicates the linearity of the
methods shown by $\Delta E$, which is the relative deviation of the median
reconstructed energy compared to the true beam energy.

In 5(a) it can be seen that correcting the reconstructed energy globally by
means of the fraction of late energy deposits, improvements of up to
$25\text{\,}\mathrm{\char 37\relax}$ over the standard reconstruction are
achieved. This shows that the correlations observed in section 3 can be used
to obtain a significant improvement in energy resolution. However, using the
same approach with the $C_{global}$ observable leads to larger improvements of
up to $35\text{\,}\mathrm{\char 37\relax}$. The combination of both
observables leads only to minor further improvements in the energy range below
$30\text{\,}\mathrm{GeV}$. This is expected because of the observations made
in section 3.

For the local approach to software compensation using only energy bins leads
to improvements of about $35\text{\,}\mathrm{\char 37\relax}$ over the
standard reconstruction. At the highest energy points this is slightly less
than what is achieved for the global approach, however, for energies of 50 GeV
and below the improvements are larger for the local software compensation.
Adding binning with regards to the hit times improves the energy resolution
further by about $3\text{\,}\mathrm{\char 37\relax}4\text{\,}\mathrm{\char
37\relax}$ on average, bringing it to or beyond the level achieved with global
software compensation over the full energy range. This shows that on a local
level the hit time measurements are capable of improving the energy resolution
compared to the standard local software compensation approach used for the
energy reconstruction in the AHCAL.

(a) Global software compensation

(b) Local software compensation

Figure 5: Results of using (a) global and (b) local software compensation
using different observables compared to the standard reconstruction.

In Figure 6 the global and local version of the time assisted software
compensation methods are compared. Besides the highest energy of
$80\text{\,}\mathrm{GeV}$ where both methods show the same performance, the
local approach gives better results with about a $5\text{\,}\mathrm{\char
37\relax}$ higher improvement over the standard reconstruction for most
energies. These results reflect the observation that the local approach to
software compensation is making better use of the additional time information.

Figure 6: Comparison of global and local software compensation using both
energy and time observables.

###  5.1 Weights for Local Software Compensation

Figure 7 shows the weights used for the local software compensation approach
which uses only one set of hit energy bins. The weights of the first four bins
are above one for the whole energy range, meaning hit energies in these bins
get weighted up. The weights for the other bins are consistently below one and
therefore energy depositions in these bins are weighted down.

The weights for the local software compensation approach using two sets of hit
energy bins is shown in Figure 8 for early and late hits. It can be seen that
the low energy hits in the late bins are weighted up much more compared to the
early energy bins. The late energy bins greater than $5\text{\,}\text{MIP}$
include only very few data samples, as high-energetic, late energy depositions
are rare in the detector. It can be concluded that the improvement in the
resolution of the energy reconstruction process for the time assisted local
software compensation is due to the power of distinguishing early and late low
energetic energy depositions and the corresponding increased weighting of the
late energy depositions.

Figure 7: Weights for the local software compensation approach using only one
set of energy bins. The different colors represent the weights for the
different hit energy bins with the lower edge of the corresponding bin
indicated in the legend.

(a) Early hits

(b) Late hits

Figure 8: Weights for the local software compensation approach using energy
bins for (a) early ($\leq$3\text{\,}\mathrm{ns}$$) and (b) late
($>$3\text{\,}\mathrm{ns}$$) hits. The different colors represents the weights
for the different hit energy bins with the start of the corresponding bin
indicated in the legend.

##  6 Impact of a limited time resolution

To explore the impact of the limited time resolution of
$1\text{\,}\mathrm{n}\mathrm{s}$ on the performance of the reconstruction
algorithms discussed here, the reconstruction methods are repeated assuming a
perfect time resolution. In this case, the time of a calorimeter hit is given
by the exact time of the first energy deposition in that particular cell in
the simulations, without additional time smearing in the digitization step.
All other digitization effects are applied. The parameters of the analysis, in
particular the value of $3\text{\,}\mathrm{n}\mathrm{s}$ used to separate
early and late shower components, are the same.

The results are shown in Figure 9. For global software compensation (9(a)) no
improvement of the energy resolution can be observed with an ideal time
resolution compared to a limited time resolution of $1\text{\,}\mathrm{ns}$.
The energy resolution is dominated by the contribution of $C_{\text{global}}$.
The more accurate time information does not add to a better performance. For
local software compensation a clear improvement in energy resolution can be
seen (9(b)). The improvement in energy resolution compared to the standard
reconstruction increases by $2\text{\,}3\text{\,}$ percent points for energies
below $50\text{\,}\mathrm{GeV}$. The better separation between early and late
hits has a direct impact on the performance of the algorithm.

(a) Global software compensation

(b) Local software compensation

Figure 9: Results of (a) global and (b) local software compensation using
different observables, and a currently realistic and a perfect time
resolution, compared to the standard reconstruction.

##  7 Conclusions

Two methods of time assisted energy reconstruction within the CALICE AHCAL
were presented and studied on simulated data. Both methods rely on the time
structure of hadronic showers, where late hits are predominantly caused by
late neutrons. Strong correlations of the fraction of late energy deposits
with the reconstructed energy were observed. In a global software compensation
approach these correlations have been exploited for an improved energy
resolution compared to the standard reconstruction. However, the improvement
is less than what can be achieved using an observable based on the hit energy
distribution. The combination of both time and energy density observables does
not lead to significant improvements over the purely energy-base methods. For
a local software compensation approach the number of bins were doubled and
weights have been derived separately for early and late energy depositions.
With this approach, improvements in the energy resolution compared to the
standard local software compensation approach using only energy density have
been observed. Over most of the energy range, a better energy resolution than
for the global software compensation has been achieved. A perfect time
resolution enables additional improvements compared to a
$1\text{\,}\mathrm{n}\mathrm{s}$ time resolution.

These results show that time information in highly-granular hadronic
calorimeters, while correlated with information provided by energy density, is
beneficial for energy reconstruction, in particular when local information on
the cell level is used. Considering that the local software compensation
technique introduced here uses only two bin in time, it is expected that more
complex algorithms based on advanced machine-learning techniques which can use
the full time information in a five-dimensional reconstruction (space, energy
density, and time) will enable further improvements in energy reconstruction.
Studies such as [18] give first impressions of this potential.

## Acknowledgements

The authors thank the CALICE Collaboration for fruitful discussions and
helpful comments. This work would not have been possible without the use of
the CALICE software framework.

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  * [18] N. Akchurin, C. Cowden, J. Damgov, A. Hussain and S. Kunori, _On the use of neural networks for energy reconstruction in high-granularity calorimeters_ , _JINST_ 16 (2021) P12036, [arXiv:2107.10207].

  *[AHCAL]: Analog Hadron Calorimeter
  *[PFA]: particle-flow algorithms
  *[LTP]: large technological prototype
  *[SiPMs]: Silicon Photomultiplier


# Proton Stability: From the Standard Model to Beyond Grand Unification

Juven Wang<EMAIL_ADDRESS>Center of Mathematical Sciences and
Applications, Harvard University, MA 02138, USA  Zheyan Wan  Yau Mathematical
Sciences Center, Tsinghua University, Beijing 100084, China  Yi-Zhuang You
Department of Physics, University of California, San Diego, CA 92093, USA

###### Abstract

A proton is known for its longevity, but what is its lifetime? While many
Grand Unified Theories predict the proton decay with a finite lifetime, we
show that the Standard Model (SM) and some versions of Ultra Unification
(which replace sterile neutrinos with new exotic gapped/gapless sectors, e.g.,
topological or conformal field theory under global anomaly cancellation
constraints) with a discrete baryon plus lepton symmetry permit a stable
proton. For the 4d SM with Lie group $G_{{\rm SM}_{q}}\equiv\frac{{\rm
SU}(3)\times{\rm SU}(2)\times{\rm U}(1)_{\tilde{Y}}}{\mathbb{Z}_{q}}$ of
$q=1,2,3,6$ and $N_{f}$ families of 15 or 16 Weyl fermions, in addition to the
continuous baryon minus lepton ${\rm U}(1)_{\bf B-L}$ symmetry, there is also
a compatible discrete baryon plus lepton $\mathbb{Z}_{2N_{f},\bf B+L}$
symmetry. The $\mathbb{Z}_{2N_{f},\bf B+L}$ is discrete due to the ABJ anomaly
under the BPST SU(2) instanton. Although both ${\rm U}(1)_{\bf B-L}$ and
$\mathbb{Z}_{2N_{f},\bf B+L}$ symmetries are anomaly-free under the
dynamically gauged $G_{{\rm SM}_{q}}$, it is important to check whether they
have mixed anomalies with the gravitational background field (spacetime
diffeomorphism under Spin group rotation) and higher symmetries (whose charged
objects are Wilson electric or ’t Hooft magnetic line operators) of SM. We can
also replace the ${\rm U}(1)_{\bf B-L}$ with a discrete variant
$\mathbb{Z}_{4,X}$ for $X\equiv 5({\bf B-L})-\frac{2}{3}{\tilde{Y}}$ of
electroweak hypercharge ${\tilde{Y}}$. We explore a systematic classification
of candidate perturbative local and nonperturbative global anomalies of the 4d
SM, including all these gauge and gravitational backgrounds, via a cobordism
theory, which controls the SM’s deformation class. We discuss the proton
stability of the SM and Ultra Unification in the presence of discrete ${\bf
B+L}$ symmetry protection, in particular $({\rm U}(1)_{\bf
B-L}\times\mathbb{Z}_{2N_{f},\bf B+L})/{\mathbb{Z}_{2}^{F}}$ or
$(\mathbb{Z}_{4,X}\times\mathbb{Z}_{2N_{f},\bf B+L})/{\mathbb{Z}_{2}^{F}}$
with the fermion parity $\mathbb{Z}_{2}^{F}$.

##  I Introduction and Summary

Proton is known for its longevity, but what is its lifetime? The observed
universe is about $10^{10}$ years old, while the proton mean lifetime is
experimentally tested to be more than $10^{30}\sim 10^{34}$ years Zyla  _et
al._ (2020); Nath and Fileviez Perez (2007); Senjanovic (2010). To date, all
experiments that attempt to go beyond the Standard Model (SM) Glashow (1961);
Weinberg (1967); Salam (1968) to observe proton decay predicted by Grand
Unified Theories (GUTs) Pati and Salam (1974); Georgi and Glashow (1974);
Georgi  _et al._ (1974) have not yet succeeded. This motivates us to ask the
following questions and seek their resolutions:

1\. Are there alternative routes to test GUT, other than conventionally
seeking GUT as an effective field theory that appeared at a higher energy
unification?

2\. Can the proton be stable with an infinite lifetime? What mechanism
protects the proton from decay?

For the first question, Ref. [Wang and You, 2022, 2021; Wang  _et al._ , 2022;
You and Wang, 2022] recently suggested that instead of only increasing the
energy scale from the SM to higher energy looking for the GUT structure
(imagining tuning the energy scale along a _vertical_ axis in a phase
diagram), we can indeed move from the Standard Model (SM) vacuum to neighbor
GUT vacua via quantum phase transitions Sachdev (2011) (imagining tuning the
vacuum changing parameters along a _horizontal_ axis in a quantum phase
diagram at zero temperature). In Ref. [Wang and You, 2022, 2021; Wang  _et
al._ , 2022; You and Wang, 2022] viewpoint, the 4d SM and other GUTs are in
the same _deformation class_ of quantum field theories Seiberg (2019), labeled
by $(G,{\mathbf{Z}}_{5})$, the symmetry $G$ and its anomaly
${\mathbf{Z}}_{5}$: In the limit when internal symmetry is weakly coupled or
ungauged, the SM and GUTs could be labeled by an enlarged spacetime-internal
symmetry group $G$ and a certain ’t Hooft anomaly ’t Hooft (1980) of the
symmetry $G$. In a modern quantum field theory (QFT) language, ’t Hooft
anomaly of the symmetry $G$ in $d$d spacetime is specified via the anomaly
inflow Callan and Harvey (1985); Witten and Yonekura (2019) by a $(d+1)$d
$G$-symmetric invertible topological quantum field theory (TQFT) denoted as a
cobordism invariant ${\mathbf{Z}}_{d+1}$ Freed and Hopkins (2016). Thus, we
can tune the SM via quantum phase transitions to the neighbor GUT phases that
necessarily allow proton decays. Theoretically, those quantum vacua tuning
parameters can be the sign-flipping of the $r$ coefficient of the GUT-Higgs
potential ${\rm U}(\Phi_{{\rm GUT}})=(r(\Phi_{{\rm
GUT}})^{2}+\lambda(\Phi_{{\rm GUT}})^{4})$ or the sign flipping of the fermion
mass.

For the second question, the proton longevity may be protected by subtle
symmetries of QFT vacuum (low-energy ground states). Those subtle symmetries,
if found, may not be accidental, but be exact — they seem global symmetries at
SM energy scales, but they should be dynamical gauge symmetries Krauss and
Wilczek (1989) when approaching the Planck or quantum gravity scales, due to
“no global symmetries in quantum gravity” reasoning (see a recent overview
Harlow and Ooguri (2021)).

Those subtle symmetries likely are discrete symmetries, subject to the
nontrivial check of their anomaly matching or cancellation that we will
perform. These discrete symmetries and their anomalies were investigated in
the past (e.g., Ibanez and Ross (1991, 1992); Banks and Dine (1992); Csaki and
Murayama (1998)), they are famous for having potential _nonperturbative global
anomalies_ Witten (1985) (classified by finite abelian group $\mathbb{Z}_{\rm
n}$, detectable via large gauge/diffeomorphism transformations that _cannot_
be continuously deformed from the identity), in contrast to the familiar
_perturbative local anomalies_ (classified by integer $\mathbb{Z}$ classes,
detectable via infinitesimal gauge/diffeomorphism transformations continuously
deformable from the identity, captured by Feynman diagram calculations). These
_local_ and _global_ anomalies include _gauge, gravitational_ or _mixed gauge-
gravitational_ anomaly types depending on whether their path integral non-
invariance is due to gauge or gravitational background fields. But only
recently, thanks to the development of cobordism group classification of
anomalies, these global anomalies become systematically computable Freed and
Hopkins (2016); Garcia-Etxebarria and Montero (2019); Hsieh (2018); Wan and
Wang (2019); Davighi  _et al._ (2020); Wan and Wang (2020a).

Discrete symmetries and their global anomalies can drastically challenge the
paradigm that we used to think of QFT vacuum. For example, if the baryon minus
lepton ${\mathbf{B}-\mathbf{L}}$ vector symmetry or more precisely $X\equiv
5({\mathbf{B}-\mathbf{L}})-\frac{2}{3}{\tilde{Y}}$ chiral symmetry Wilczek and
Zee (1979a, b) (with the integer quantized electroweak hypercharge
${\tilde{Y}}$) is a discrete $\mathbb{Z}_{4,X}$ symmetry, although the
$\mathbb{Z}_{4,X}$ has no local anomalies, the $\mathbb{Z}_{4,X}$ imposes
various global anomaly cancellation conditions Garcia-Etxebarria and Montero
(2019); Hsieh (2018); Guo  _et al._ (2020); Wan and Wang (2020a). In
particular, a $\mathbb{Z}_{16}$ class global anomaly of the mixed gauge-
gravity type (variation on the $\mathbb{Z}_{4,X}$ gauge field and the Spin
spacetime diffeomorphism) implies that $15N_{f}$ Weyl fermion SM alone cannot
cancel the $\mathbb{Z}_{16}$ global anomaly — its anomaly cancellation
requires introducing either the 16th Weyl fermion (the right-handed neutrino),
or 4d noninvertible TQFT or interacting conformal field theory (CFT), or 5d
invertible TQFT, or breaking the $\mathbb{Z}_{4,X}$ down to fermion parity
$\mathbb{Z}_{2}^{F}$, etc Wang (2020a, b, 2021). In other words, rephrasing in
terms of quantum phases of QFT language, the QFT vacuum beyond the SM (BSM)
could be more quantumly entangled than the Landau-Ginzburg old paradigm. These
possible exotic BSM phases are analogous to many exotic quantum phases
explored in the contemporary condensed matter community UQM . The SM together
with those exotic BSM phases constrained by nonperturbative global anomaly
cancellation is called Ultra Unification (UU) Wang (2020a, b, 2021).

Another motivation for our present work is expanding the exploration of the
deformation class of SM Wang  _et al._ (2022). In Ref. [Wang  _et al._ ,
2022], we explored the deformation of SM to GUTs. We had included the
spacetime symmetry (Spin group), the internal symmetry (Lie algebra
$su(3)\times su(2)\times u(1)_{\tilde{Y}}$, and four compatible versions Lie
groups $G_{{\rm SM}_{q}}\equiv\frac{{\rm SU}(3)\times{\rm SU}(2)\times{\rm
U}(1)_{\tilde{Y}}}{\mathbb{Z}_{q}}$, with $q=1,2,3,6$ Tong (2017)), and a
continuous or discrete ${\mathbf{B}-\mathbf{L}}$ like symmetry of the SM. In
Ref. [Wang  _et al._ , 2022], we had left the inclusion of the discrete
${\mathbf{B}+\mathbf{L}}$ vector symmetry for future work. (The ${\rm
U}(1)_{\mathbf{B}+\mathbf{L}}$ is explicitly broken down to
$\mathbb{Z}_{2N_{f}\\#,\mathbf{B}+\mathbf{L}}$ due to the SU(2) instanton
Belavin  _et al._ (1975); ’t Hooft (1976a, b) by the Adler-Bell-Jackiw (ABJ)
anomaly Adler (1969); Bell and Jackiw (1969), where $N_{f}$ is the family
number with an extra $q$-dependent factor denoted as $\\#$, see Anber and
Poppitz (2021).) Previously we had excluded the ${\mathbf{B}+\mathbf{L}}$
because it is not a symmetry for many GUTs. Now the discrete
${\mathbf{B}+\mathbf{L}}$, allowed in SM but disobeyed by GUTs, gives us the
exact opportunity to distinguish the SM from other GUTs. Our present work
means to fill this gap left in Wang  _et al._ (2022) to include the discrete
${\mathbf{B}+\mathbf{L}}$ symmetry, examining the ${\mathbf{B}+\mathbf{L}}$
anomaly, and its SM deformation class. We will see that the discrete
${\mathbf{B}+\mathbf{L}}$ is a good symmetry for SM and some versions of UU,
such that it implies the proton stability in those models. Remarkably,
recently Ref. Byakti  _et al._ (2018); Koren (2022) also emphasize that the
discrete ${\mathbf{B}+\mathbf{L}}$ can avoid the proton decay.

##  II Revisit the Standard Model

Now we revisit the Standard Model (SM) and its symmetry, then explicitly
derive the discrete ${\mathbf{B}+\mathbf{L}}$ symmetry. SM is a 4d chiral
gauge theory of local Lie algebra $su(3)\times su(2)\times u(1)_{\tilde{Y}}$
coupling to $N_{f}=3$ families of 15 or 16 Weyl fermions (written as a left-
handed 15 or 16 multiplet $\psi_{L}$) in the following representation

$\displaystyle({\psi_{L}})_{\rm I}=(\bar{d}_{R}\oplus{l}_{L}\oplus
q_{L}\oplus\bar{u}_{R}\oplus\bar{e}_{R})_{\rm I}\oplus n_{\nu_{{\rm
I},R}}{\bar{\nu}_{{\rm I},R}}\sim\big{(}(\overline{\bf 3},{\bf
1})_{2}\oplus({\bf 1},{\bf 2})_{-3}\oplus({\bf 3},{\bf
2})_{1}\oplus(\overline{\bf 3},{\bf 1})_{-4}\oplus({\bf 1},{\bf
1})_{6}\big{)}_{\rm I}\oplus n_{\nu_{{\rm I},R}}{({\bf 1},{\bf
1})_{0}}\quad\quad$ (1)

for each family (family index ${\rm I},{\rm J}=1,2,3$; with ${\psi_{L}}_{1}$
for $u,d,e$ type, ${\psi_{L}}_{2}$ for $c,s,\mu$ type, and ${\psi_{L}}_{3}$
for $t,b,\tau$ type of quarks and leptons) of $su(3)\times su(2)\times
u(1)_{\tilde{Y}}$. We use ${\rm I}=1,2,3$ for
$n_{\nu_{e,R}},n_{\nu_{\mu,R}},n_{\nu_{\tau,R}}\in\\{0,1\\}$ to label either
the absence or presence of electron $e$, muon $\mu$, or tauon $\tau$ types of
sterile neutrinos. Readers should keep in mind that all unitary internal
symmetries that we will discuss below are part of the subgroup of the largest
internal ${\rm U}(15N_{f})$ or ${\rm U}(16N_{f})$ acting on those Weyl
fermions as unitary rotations. The SM lagrangian consists of Yang-Mills terms
(their gauge sector indices ${I}=1,2,3$ for $u(1),su(2),su(3)$), a possible
theta term for $su(3)$, the Weyl fermions coupled to Yang-Mills gauge fields,
Yukawa-Higgs term, and electroweak Higgs kinetic-potential term:

$\displaystyle{\cal L}_{\rm
SM}={\sum_{{I}=1,2,3}}-\frac{1}{4}F_{{I},\mu\nu}^{\mathrm{a}}F_{{I}}^{\mathrm{a}\mu\nu}{-\,\frac{{\theta_{3}}}{{64}\pi^{2}}{g}_{3}^{2}\epsilon^{\mu\nu\mu^{\prime}\nu^{\prime}}F_{3,\mu\nu}^{\mathrm{a}}F_{3,\mu^{\prime}\nu^{\prime}}^{\mathrm{a}}}+{\psi}^{\dagger}_{L}(\hskip
1.0pt\mathrm{i}\hskip
1.0pt\bar{\sigma}^{\mu}{D}_{\mu,A})\psi_{L}-({\psi}^{\dagger}_{L}\phi\psi_{R}+{\rm
h.c.})+|{D}_{\mu,A}\phi|^{2}-{\rm U}(\phi).\quad$ (2)

The ${\cal L}_{\rm YH}={\psi}^{\dagger}_{L}\phi\psi_{R}+{\rm h.c.}$ is a
shorthand of ${\cal L}_{\rm YH}^{d}+{\cal L}_{\rm YH}^{u}+{\cal L}_{\rm
YH}^{e}=\lambda^{d}_{\rm IJ}{{q}^{\rm I\dagger}_{L}}\phi d_{R}^{\rm
J}+\lambda^{u}_{\rm IJ}\epsilon^{ab}{{q}^{\rm
I\dagger}_{La}}\phi_{b}^{*}u_{R}^{\rm J}+\lambda^{e}_{\rm IJ}{{l}^{\rm
I\dagger}_{L}}\phi e_{R}^{\rm J}+{\rm h.c.}$ with $a,b$ the $su(2)$
fundamental’s index, and the h.c. as hermitian conjugate. Diagonalization of
Yukawa-Higgs term of quark sector implies that the $W^{\pm}$ boson induces a
flavor-changing current mixing between different families, thus we only have a
${\rm U}(1)_{\bf B}$ symmetry for all quarks (instead of an individual ${\rm
U}(1)$ for each quark family), at least _classically_. The diagonalization of
Yukawa-Higgs term of lepton sector without neutrino mass term ${\cal L}_{\rm
YH}^{\nu}=\lambda^{\nu}_{\rm IJ}\epsilon^{ab}{{l}^{\rm
I\dagger}_{La}}\phi_{b}^{*}\nu_{R}^{\rm J}+{\rm h.c.}$ implies that ${\cal
L}_{\rm SM}$ has individual ${\rm U}(1)_{e}$, ${\rm U}(1)_{\mu}$, ${\rm
U}(1)_{\tau}$ for each lepton family. However, established experiments say
each lepton ${\rm U}(1)$ is violated, only the total lepton number ${\rm
U}(1)_{\bf L}$ should be considered, at least _classically_. Thus, we focus on
${\rm U}(1)_{\bf B}$ and ${\rm U}(1)_{\bf L}$ transformations, which send

$\displaystyle{\rm U}(1)_{\bf B}:({\psi_{L}})_{\rm I}$ $\displaystyle\mapsto$
$\displaystyle((\hskip 1.0pt\mathrm{e}^{-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\frac{\alpha_{\bf
B}}{3}}\mathbb{I}_{3}\cdot\bar{d}_{R})\oplus{l}_{L}\oplus(\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{\alpha_{\bf
B}}{3}}\mathbb{I}_{6}\cdot q_{L})\oplus(\hskip 1.0pt\mathrm{e}^{-\hskip
1.0pt\mathrm{i}\hskip 1.0pt\frac{\alpha_{\bf
B}}{3}}\mathbb{I}_{3}\cdot\bar{u}_{R})\oplus\bar{e}_{R})_{\rm I}\oplus
n_{\nu_{{\rm I},R}}{\bar{\nu}_{{\rm I},R}},$ (3) $\displaystyle{\rm U}(1)_{\bf
L}:({\psi_{L}})_{\rm I}$ $\displaystyle\mapsto$
$\displaystyle(\bar{d}_{R}\oplus(\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt{\alpha_{\bf L}}}\mathbb{I}_{2}\cdot{l}_{L})\oplus
q_{L}\oplus\bar{u}_{R}\oplus(\hskip 1.0pt\mathrm{e}^{-\hskip
1.0pt\mathrm{i}\hskip 1.0pt{\alpha_{\bf L}}}\bar{e}_{R}))_{\rm I}\oplus(\hskip
1.0pt\mathrm{e}^{-\hskip 1.0pt\mathrm{i}\hskip 1.0pt{\alpha_{\bf
L}}}n_{\nu_{{\rm I},R}}{\bar{\nu}_{{\rm I},R}}),$ (4)

with ${\alpha_{\bf B}}\in[0,2\pi\cdot 3)$ and ${\alpha_{\bf L}}\in[0,2\pi)$.
The quark’s ${\rm U}(1)_{q}$ is related to baryon’s ${\rm U}(1)_{\bf B}$ via
${\alpha_{q}}={\alpha_{\bf B}}/3\in[0,2\pi)$. Here $\mathbb{I}_{\rm N}$ means
a rank-N diagonal identity matrix that can act on the N-multiplet.

It is well-known that the ${\rm U}(1)_{\bf B}$-${\rm U}(1)_{\tilde{Y}}^{2}$,
${\rm U}(1)_{\bf L}$-${\rm U}(1)_{\tilde{Y}}^{2}$ ${\rm U}(1)_{\bf B}$-${\rm
SU}(2)^{2}$ and ${\rm U}(1)_{\bf L}$-${\rm SU}(2)^{2}$ have local anomalies
captured by triangle Feynman diagrams, with their anomaly coefficients
respectively: $2\cdot 1^{2}-2^{2}-(-4)^{2}=-18$, $2\cdot(-3)^{2}-6^{2}=-18$,
1, and 1. Upon dynamically gauging electroweak $su(2)\times u(1)_{\tilde{Y}}$,
the consequential ABJ anomaly implies that the classical continuous ${\rm
U}(1)_{\bf B}\times{\rm U}(1)_{\bf L}$ symmetry is broken _quantum
mechanically_. Next we check whether any subgroup of the ${\rm U}(1)_{\bf
B}\times{\rm U}(1)_{\bf L}$ still survives under dynamically gauged $G_{{\rm
SM}_{q}}$. The Fujikawa path integral method Fujikawa and Suzuki (2004) shows
that under ${\rm U}(1)_{\bf B}$ and ${\rm U}(1)_{\bf L}$ transformations with
corresponding currents $J_{\bf B}$ and $J_{\bf L}$, the path integral $\bf Z$
changes to

$\displaystyle\int[{\cal D}\psi_{L}][{\cal D}\psi_{L}^{\dagger}]\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\big{(}\int\hskip
1.0pt\mathrm{d}^{4}x\big{(}{\cal L}_{\rm SM}+\alpha_{\bf
B}(\partial_{\mu}J^{\mu}_{\bf B})+\alpha_{\bf L}(\partial_{\mu}J^{\mu}_{\bf
L})\big{)}{{-18(\alpha_{\bf B}+\alpha_{\bf L})N_{f}}n^{(1)}}{-{(\alpha_{\bf
B}+\alpha_{\bf L})N_{f}}n^{(2)}}\big{)}}.$ (5)

Here the instanton numbers $n^{(1)}\equiv\int\hskip
1.0pt\mathrm{d}^{4}x\frac{{g}_{1}^{2}}{{32}\pi^{2}}\epsilon^{\mu\nu\mu^{\prime}\nu^{\prime}}F_{1,\mu\nu}F_{1,\mu^{\prime}\nu^{\prime}}$
and $n^{(2)}\equiv\int\hskip
1.0pt\mathrm{d}^{4}x\frac{{g}_{2}^{2}}{{64}\pi^{2}}\epsilon^{\mu\nu\mu^{\prime}\nu^{\prime}}F_{2,\mu\nu}F_{2,\mu^{\prime}\nu^{\prime}}$
are quantized in integer $\mathbb{Z}$ on spin manifolds.

$\bullet$ ${\rm U}(1)_{{\bf B}-{\bf L}}$ symmetry: When $\alpha_{\bf
B-L}\equiv\alpha_{\bf B}=-\alpha_{\bf L}$, its Ward identity (the derivative
of the partition function ${\bf Z}$ with respect to $\alpha$ variation
$\frac{\delta{\bf Z}}{\delta(\alpha)}\big{|}_{\alpha=0}=0$ vanishes) says that
$\langle\partial_{\mu}(J^{\mu}_{\bf B}-J^{\mu}_{\bf L})\rangle=0$. This shows
${\rm U}(1)_{{\bf B}-{\bf L}}$ is still a symmetry and is anomaly free under
$G_{{\rm SM}_{q}}$.

$\bullet$ $\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ symmetry: The continuous
${\rm U}(1)_{{\bf B}}$, ${\rm U}(1)_{{\bf L}}$ and ${\rm U}(1)_{{\bf B}+{\bf
L}}$ all are broken by ABJ anomaly. But when $\alpha_{\bf
B+L}\equiv\alpha_{\bf B}=\alpha_{\bf L}$, the quantization ${{18(\alpha_{\bf
B}+\alpha_{\bf L})N_{f}}n^{(1)}}={{18\alpha_{\bf B+L}(2N_{f}})n^{(1)}}\in
2\pi\mathbb{Z}$ under U(1) instanton holds when $\alpha_{\bf
B+L}\in\frac{2\pi}{36N_{f}}\mathbb{Z}$; the quantization ${{(\alpha_{\bf
B}+\alpha_{\bf L})N_{f}}n^{(2)}}=\alpha_{\bf B+L}(2N_{f})\in 2\pi\mathbb{Z}$
under SU(2) instanton holds when $\alpha_{\bf
B+L}\in\frac{2\pi}{2N_{f}}\mathbb{Z}$. Overall, this shows
$\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ is still a symmetry and is anomaly
free under $G_{{\rm SM}_{q}}$.

$\bullet$ ${\rm U}(1)_{{\bf B}-{\bf
L}}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$
symmetry: Since ${\rm U}(1)_{{\bf B}-{\bf L}}$ and
$\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ share the fermion parity
$\mathbb{Z}_{2}^{F}$ that acts on fermions $\psi\mapsto-\psi$, the precise
surviving subgroup (which is not broken under ABJ anomaly with dynamically
gauged $G_{{\rm SM}_{q}}$) is ${\rm U}(1)_{{\bf B}-{\bf
L}}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$.
Hereafter we use a standard notation that
${{G_{1}}\times_{{G_{N}}}{{G}_{2}}}\equiv({\frac{{G_{1}}\times{{G}_{2}}}{{G_{N}}}})$
as modding out their common normal subgroup $G_{N}$.

The above result is not affected by whether the $({\psi_{L}})_{\rm I}$
multiplet in (1) includes the 16th Weyl fermion ${\bar{\nu}_{{\rm
I},R}}={({\bf 1},{\bf 1})_{0}}$ for each of the $N_{f}$ families, because
${\bar{\nu}_{{\rm I},R}}$ is sterile to $G_{{\rm SM}_{q}}$ gauge forces.

##  III Proton Stability

We have shown ${\rm U}(1)_{{\bf B}-{\bf L}}$ and
$\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ are good symmetries, anomaly-free
with respect to dynamically gauged $G_{{\rm SM}_{q}}$ for the SM. Precisely
they send the multiplet $({\psi_{L}})_{\rm I}$ to (here ${\alpha_{\bf
B-L}}\in[0,2\pi\cdot 3)$ while $\alpha_{\bf
B+L}\in\frac{2\pi}{2N_{f}}\mathbb{Z}$ is discrete):

$\displaystyle{\rm U}(1)_{\bf B-L}$ $\displaystyle:$ $\displaystyle((\hskip
1.0pt\mathrm{e}^{-\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{\alpha_{\bf
B-L}}{3}}\mathbb{I}_{3}\cdot\bar{d}_{R})\oplus(\hskip 1.0pt\mathrm{e}^{-\hskip
1.0pt\mathrm{i}\hskip 1.0pt{\alpha_{\bf
B-L}}}\mathbb{I}_{2}\cdot{l}_{L})\oplus(\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\frac{\alpha_{\bf B-L}}{3}}\mathbb{I}_{6}\cdot
q_{L})\oplus(\hskip 1.0pt\mathrm{e}^{-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\frac{\alpha_{\bf B-L}}{3}}\mathbb{I}_{3}\cdot\bar{u}_{R})\oplus(\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt{\alpha_{\bf
B-L}}}\bar{e}_{R}))_{\rm I}\oplus(\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt{\alpha_{\bf B-L}}}n_{\nu_{{\rm
I},R}}{\bar{\nu}_{{\rm I},R}}).$ (6)
$\displaystyle\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ $\displaystyle:$
$\displaystyle((\hskip 1.0pt\mathrm{e}^{-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\frac{\alpha_{\bf B+L}}{3}}\mathbb{I}_{3}\cdot\bar{d}_{R})\oplus(\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt{\alpha_{\bf
B+L}}}\mathbb{I}_{2}\cdot{l}_{L})\oplus(\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\frac{\alpha_{\bf B+L}}{3}}\mathbb{I}_{6}\cdot
q_{L})\oplus(\hskip 1.0pt\mathrm{e}^{-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\frac{\alpha_{\bf B+L}}{3}}\mathbb{I}_{3}\cdot\bar{u}_{R})\oplus(\hskip
1.0pt\mathrm{e}^{-\hskip 1.0pt\mathrm{i}\hskip 1.0pt{\alpha_{\bf
B+L}}}\bar{e}_{R}))_{\rm I}\oplus(\hskip 1.0pt\mathrm{e}^{-\hskip
1.0pt\mathrm{i}\hskip 1.0pt{\alpha_{\bf B+L}}}n_{\nu_{{\rm
I},R}}{\bar{\nu}_{{\rm I},R}}).$ (7)

Let us check whether they are symmetries for the familiar GUTs and UU.

  1. 1.

${\rm U}(1)_{{\bf B}-{\bf L}}$ vector or ${\rm U}(1)_{X}$ chiral like
symmetry: ${\rm U}(1)_{{\bf B}-{\bf L}}$ is a factor of the left-right (LR)
$su(3)\times su(2)_{{\rm L}}\times su(2)_{{\rm R}}\times
u(1)_{\frac{\mathbf{B}-\mathbf{L}}{2}}$ model Senjanovic and Mohapatra (1975).
${\rm U}(1)_{{\bf B}-{\bf L}}$ is part of ${\rm SU}(4)$ subgroup of Pati-Salam
(PS) $su(4)\times su(2)_{{\rm L}}\times su(2)_{{\rm R}}$ Pati and Salam
(1974). Thus ${\rm U}(1)_{{\bf B}-{\bf L}}$ is not only an anomaly-free
symmetry but also already dynamically gauged for LR and PS models. Similarly
for the Trinification (Tri) Achiman and Stech (1979); de Rujula  _et al._
(1984); Babu  _et al._ (1986) with the gauged Lie algebra $su(3)\times
su(3)_{{\rm L}}\times su(3)_{{\rm R}}$, the ${\rm U}(1)_{{\bf B}-{\bf L}}$ is
contained in the ${\rm SU}(3)_{{\rm R}}$; thus ${\rm U}(1)_{{\bf B}-{\bf L}}$
is automatically anomaly-free and dynamically gauged in the Trinification.
${\rm U}(1)_{{\bf B}-{\bf L}}$ is not quite correct for Georgi-Glashow $su(5)$
Georgi and Glashow (1974), because ${\rm U}(1)_{{\bf B}-{\bf L}}$ does not act
on the ${\psi_{L}}$ reducibly as $\overline{\bf 5}$, ${\bf 10}$, ${\bf 1}$
multiplets of $su(5)$. But the ${\rm U}(1)_{X}$ with $X\equiv
5({\mathbf{B}-\mathbf{L}})-\frac{2}{3}{\tilde{Y}}$ Wilczek and Zee (1979a, b)
fits the role so that
$(\bar{d}_{R}\oplus{l}_{L})\oplus(q_{L}\oplus\bar{u}_{R}\oplus\bar{e}_{R})\oplus(\bar{\nu}_{R})=\overline{\bf
5}_{-3}\oplus{\bf 10}_{1}\oplus{\bf 1}_{5}$ in $su(5)\times u(1)_{X}$. Neither
${\rm U}(1)_{{\bf B}-{\bf L}}$ nor ${\rm U}(1)_{X}$ is a symmetry for the
flipped $u(5)$ model Barr (1982), but the ${\rm U}(1)_{X_{2}}$ with
$X_{2}\equiv\frac{1}{5}X+\frac{4}{5}\tilde{Y}=({\mathbf{B}-\mathbf{L}})+\frac{2}{3}{\tilde{Y}}$
replaces the role so that
$(\bar{u}_{R}\oplus{l}_{L})\oplus(q_{L}\oplus\bar{d}_{R}\oplus\bar{\nu}_{R})\oplus(\bar{e}_{R})=\overline{\bf
5}_{-3}\oplus{\bf 10}_{1}\oplus{\bf 1}_{5}$ in $su(5)\times u(1)_{X_{2}}$.
Neither ${\rm U}(1)_{{\bf B}-{\bf L}}$, ${\rm U}(1)_{X}$, nor ${\rm
U}(1)_{X_{2}}$ is correct for the $so(10)$ GUT Fritzsch and Minkowski (1975).
The only sensible U(1) factor allowed for the $so(10)$ GUT is an identical
${\rm U}(1)$ phase that acts on the ${\bf 16}$ that together with Spin(10)
forms a $({\rm Spin}(10)\times_{\mathbb{Z}_{4,X}}{\rm U}(1))$ internal
symmetry. But this chiral ${\rm U}(1)$ has a discrete
${\mathbb{Z}_{4,X}}={\mathbb{Z}_{4,X_{2}}}$ subgroup (that replaces the role
of discrete ${{\bf B}-{\bf L}}$) which is a good symmetry (indeed anomaly-free
and gauged) for the $so(10)$ GUT. Finally, UU Wang (2020a, b, 2021) requires a
discrete ${{\bf B}-{\bf L}}$ or $X$ symmetry to enforce a $\mathbb{Z}_{16}$
class global anomaly constraint that can be canceled by replacing the
$\bar{\nu}_{R}$ with TQFT/CFT exotic phases. See (12) for a summary.

  2. 2.

$\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ vector symmetry:
$\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ is a good global symmetry for LR
model, broken down from ${\rm U}(1)_{{\bf B}+{\bf L}}$ by the dynamical ${\rm
SU}(2)_{{\rm L}}$ and ${\rm SU}(2)_{{\rm R}}$ instantons. For other models
such as PS, $su(5)$, flipped $u(5)$ and $so(10)$, the only compatible subgroup
of ${\rm U}(1)_{{\bf B}+{\bf L}}$ is the fermion parity $\mathbb{Z}_{2}^{F}$.
For UU, we have choices to include the $15N_{f}$ Weyl fermion SM only but
without GUT structure plus exotic TQFT/CFT sectors; those UU models can obey
the $\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ symmetry. Other UU which
includes the $su(5)$ GUT allows only the $\mathbb{Z}_{2}^{F}$ subgroup of
$\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$. For Trinification, we can even
choose a generalized unbroken ${\rm U}(1)_{{\bf B}+{\bf L}}$ global symmetry,
which is not only outside the Trinification gauge group but also anomaly-free
under that gauge group with Lie algebra $su(3)\times su(3)_{{\rm L}}\times
su(3)_{{\rm R}}$. See (12) for a summary.

$\displaystyle\begin{tabular}[]{c | c c c c c c c c }\hline\cr&SM&LR \cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{SenjanovicMohapatra1975}{\@@citephrase{(}}{\@@citephrase{)}}}&PS \cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Pati1974yyPatiSalamLeptonNumberastheFourthColor}{\@@citephrase{(}}{\@@citephrase{)}}}&$su(5)$ \cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Georgi1974syUnityofAllElementaryParticleForces}{\@@citephrase{(}}{\@@citephrase{)}}}&flipped $u(5)$ \cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Barr1982flippedSU5}{\@@citephrase{(}}{\@@citephrase{)}}}&$so(10)$ \cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Fritzsch1974nnMinkowskiUnifiedInteractionsofLeptonsandHadrons}{\@@citephrase{(}}{\@@citephrase{)}}}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}Tri} \cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{1979AchimanStechTrinification, 1984deRujulaGeorgiGlashow, BabuHePakvasa1985giTrinification}{\@@citephrase{(}}{\@@citephrase{)}}}&UU \cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{JW2012.15860}{\@@citephrase{(}}{\@@citephrase{)}}}\\\ \hline\cr${{\bf B}-{\bf L}}$ like&${\rm U}(1)_{{\bf B}-{\bf L}}$&${\rm U}(1)_{{\bf B}-{\bf L}}$&${\rm U}(1)_{{\bf B}-{\bf L}}$&${\rm U}(1)_{X}$&${\rm U}(1)_{X_{2}}$&${\mathbb{Z}_{4,X}}$&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}${\rm U}(1)_{{\bf B}-{\bf L}}$}&discrete ${{\bf B}-{\bf L}}$ or $X$\\\ \hline\cr{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}${{\bf B}+{\bf L}}$ like}&$\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$&$\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$&$\mathbb{Z}_{2}^{F}$&$\mathbb{Z}_{2}^{F}$&$\mathbb{Z}_{2}^{F}$&$\mathbb{Z}_{2}^{F}$&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}${\rm U}(1)_{{\bf B}+{\bf L}}$}&$\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ or $\mathbb{Z}_{2}^{F}$\\\ \hline\cr\end{tabular}.\quad\quad$ (12)

We check whether these symmetries avoid the proton $p^{+}$ (or other nucleons
like neutron $n$) decay for some dominant channels. These channels are
constrained to have the lifetime $\tau_{T}$ lower bound around or more than
$10^{33}$ years Zyla  _et al._ (2020); Nath and Fileviez Perez (2007);
Senjanovic (2010). We list down the changes of baryon or lepton number
($\Delta{\bf B}$ or $\Delta{\bf L}$):

$\displaystyle\begin{tabular}[]{c | c | c | c | c | c | c}\hline\cr&$p^{+}\to e^{+}\pi^{0}$&$p^{+}\to\mu^{+}\pi^{0}$&$p^{+}\to\mu^{+}K^{0}$&$p^{+}\to e^{+}K^{0}$&$n\to e^{+}K^{-}$&$n\to e^{-}K^{+}$\\\ \hline\cr$(\Delta{\bf B},\Delta{\bf L})$&$(-1,-1)$&$(-1,-1)$&$(-1,-1)$&$(-1,-1)$&$(-1,-1)$&$(-1,+1)$\\\ \hline\cr\end{tabular}.$ (15)

All these processes have $\Delta({\bf B}-{\bf L})=0$ and $\Delta({\bf B}+{\bf
L})=-2$, except the last one has $\Delta({\bf B}-{\bf L})=-2$ and $\Delta({\bf
B}+{\bf L})=0$ If $\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ is an exact
symmetry of our vacuum, then $\Delta({\bf B}+{\bf L})=0\;{\rm
mod}\;2N_{f}=0\;{\rm mod}\;6$ must hold. Many proton decay channels are thus
forbidden. The last channel $n\to e^{-}K^{+}$ is also unlikely because it
violates even the discrete $\mathbb{Z}_{4,\mathbf{B}-\mathbf{L}}$ which
demands that $\Delta({\bf B}-{\bf L})=0\mod 4$.

Proton decay is forbidden as long as
$\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ is an exact symmetry and
$N_{f}>1$. (Although a single proton decay may be forbidden, the
$\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ symmetry still allows $N_{f}$
protons or baryons together to decay because their $\Delta({\bf B}+{\bf
L})=-2\times N_{f}=-6=0\;{\rm mod}\;6$ is consistent with
$\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ here for $N_{f}=3$.) If all
experiments support that a proton indeed does not decay at all in our vacuum,
then the SM and the UU (that contains only SM without GUT) are viable
candidates deserving further studies, their generalizations with
$\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ symmetry is preferable for future
model buildings. For example, anomaly-free discrete gauge symmetries were
recently discussed in Davighi  _et al._ (2022) (see also references therein),
which serve to exactly stabilize the proton via the all-orders selection rule.

##  IV Cobordism and Higher Anomalies of the Standard Model with
$\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ symmetry

We have checked that ${\rm U}(1)_{{\bf B}-{\bf
L}}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ is
free from perturbative local anomaly under dynamical SM gauge group $G_{{\rm
SM}_{q}}\equiv\frac{{\rm SU}(3)\times{\rm SU}(2)\times{\rm
U}(1)}{\mathbb{Z}_{q}}$. Now we follow the procedure in Wan and Wang (2020a);
Wang  _et al._ (2022) to check whether the full spacetime-internal symmetry
$G$ has any local and global anomalies, including all gauge or gravitational
background fields. The spacetime symmetry is Spin group (the double cover of
the special orthogonal SO group), and the internal symmetry is ${\rm
U}(1)_{\mathbf{B}-\mathbf{L}}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}\times
G_{{\rm SM}_{q}}$, or
$\mathbb{Z}_{4,X}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}\times
G_{{\rm SM}_{q}}$ if the discrete $X$ replacing the continuous
${\mathbf{B}-\mathbf{L}}$ symmetry. We focus on $N_{f}=3$ thus
$\mathbb{Z}_{6,\mathbf{B}+\mathbf{L}}={\mathbb{Z}_{2}^{F}}\times\mathbb{Z}_{3,\mathbf{B}+\mathbf{L}}$.
Given the spacetime-internal $G$, we classify all possible 4d local
($\mathbb{Z}$ class) and global ($\mathbb{Z}_{\rm n}$ class) anomalies via
computing the 5th cobordism group $\mathrm{TP}_{5}(G)$ Freed and Hopkins
(2016):

$\displaystyle\mathrm{TP}_{5}({\rm Spin}\times_{\mathbb{Z}_{2}^{F}}{\rm
U}(1)_{\mathbf{B}-\mathbf{L}}\times\mathbb{Z}_{3,{\mathbf{B}+\mathbf{L}}}\times
G_{{\rm
SM}_{q}})=\big{(}\mathbb{Z}^{11}\big{)}\times\big{(}\mathbb{Z}_{9}\times{\mathbb{Z}_{3}^{7}}\big{)}.$
(16) $\displaystyle\mathrm{TP}_{5}({\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}\times\mathbb{Z}_{3,{\mathbf{B}+\mathbf{L}}}\times
G_{{\rm
SM}_{q}})=\left\\{\begin{array}[]{ll}\big{(}\mathbb{Z}^{5}\times\mathbb{Z}_{2}\times\mathbb{Z}_{4}^{2}\times\mathbb{Z}_{16}\big{)}\times\big{(}\mathbb{Z}_{9}\times\mathbb{Z}_{3}^{4}\big{)},&q=1,3.\\\
\big{(}\mathbb{Z}^{5}\times\mathbb{Z}_{2}^{2}\times\mathbb{Z}_{4}\times\mathbb{Z}_{16}\big{)}\times\big{(}\mathbb{Z}_{9}\times\mathbb{Z}_{3}^{4}\big{)},&q=2,6.\end{array}\right.$
(19)

The anomaly classifications on the right-hand side in the first big bracket
$\big{(}\dots\big{)}$ are obtained in Wan and Wang (2020a); Wang  _et al._
(2022), while those in the second big bracket $\big{(}\dots\big{)}$ are new to
the literature. The $\mathbb{Z}^{11}$ and $\mathbb{Z}^{5}$ classes of local
anomalies are familiar to QFT textbook readers Weinberg (2013). The
$\mathbb{Z}_{2}^{...}\times\mathbb{Z}_{4}^{...}\times\mathbb{Z}_{16}$ classes
of global anomalies were characterized in detail before Wan and Wang (2020a).
These anomalies all canceled, except only $\mathbb{Z}^{2}$ (in
$\mathbb{Z}^{11}$) classes (involving the cubic pure gauge ${\rm
U}(1)_{{\mathbf{B}-\mathbf{L}}}^{3}$ and mixed gauge-gravity ${\rm
U}(1)_{{\mathbf{B}-\mathbf{L}}}$-(gravity)2 anomalies) and the
$\mathbb{Z}_{16}$ class (involving the mixed gauge-gravity global anomaly
between the $\mathbb{Z}_{4,X}$ and the spacetime diffeomorphism), anomalies
can have a nonzero coefficient $-N_{f}+n_{\nu_{R}}\equiv-N_{f}+\sum_{\rm
I}n_{\nu_{{\rm I},R}}$ if the number of family $N_{f}$ is distinct from the
total number of right-handed neutrinos $n_{\nu_{R}}$ Wang  _et al._ (2022).
The nonzero $\mathbb{Z}^{2}$ and $\mathbb{Z}_{16}$ anomalies of
${\mathbf{B}-\mathbf{L}},X$ and gravity background fields as ’t Hooft
anomalies do not imply the sickness of SM as long as
${\mathbf{B}-\mathbf{L}},X$ and gravity are _non-dynamical_. But when
${\mathbf{B}-\mathbf{L}},X$ and gravity are _dynamical_ , to cancel these
anomalies McNamara and Vafa (2019), we can either introduce enough
$n_{\nu_{R}}$, or break ${\mathbf{B}-\mathbf{L}}$, or introduce new exotic
phases from UU Wang (2020a, b, 2021).

The $\mathbb{Z}_{9}\times{\mathbb{Z}_{3}^{7}}$ global anomalies in (16) (which
exactly overlaps the $\mathbb{Z}_{9}\times\mathbb{Z}_{3}^{4}$ in (19) for the
first five generators), involving the discrete $\mathbf{B}-\mathbf{L}$
background field, are new to the literature: they are generated by cobordism
invariants
$\mathfrak{P}_{3}(B^{\prime}_{\mathbb{Z}_{3,\mathbf{B}+\mathbf{L}}})$,
$A^{\prime}_{\mathbb{Z}_{3,\mathbf{B}+\mathbf{L}}}B^{\prime}_{\mathbb{Z}_{3,\mathbf{B}+\mathbf{L}}}c_{1}({\rm
U}(1)),A^{\prime}_{\mathbb{Z}_{3,\mathbf{B}+\mathbf{L}}}c_{1}({\rm
U}(1))^{2}$, $A^{\prime}_{\mathbb{Z}_{3,\mathbf{B}+\mathbf{L}}}c_{2}({\rm
SU}(2))$, $A^{\prime}_{\mathbb{Z}_{3,\mathbf{B}+\mathbf{L}}}c_{2}({\rm
SU}(3))$,
$A^{\prime}_{\mathbb{Z}_{3,\mathbf{B}+\mathbf{L}}}B^{\prime}_{\mathbb{Z}_{3,\mathbf{B}+\mathbf{L}}}c_{1}({\rm
U}(1)_{\mathbf{B}-\mathbf{L}})$,
$A^{\prime}_{\mathbb{Z}_{3,\mathbf{B}+\mathbf{L}}}c_{1}({\rm
U}(1)_{\mathbf{B}-\mathbf{L}})^{2}$, and
$A^{\prime}_{\mathbb{Z}_{3,\mathbf{B}+\mathbf{L}}}c_{1}({\rm
U}(1)_{\mathbf{B}-\mathbf{L}})c_{1}({\rm U}(1))$. Here the generators are
written for the $q=1$ case. Let us explain the notations. Here all cohomology
classes are pulled back to the manifold $M$ along the maps given in the
definition of cobordism groups, e.g. the
$A^{\prime}_{\mathbb{Z}_{3,\mathbf{B}+\mathbf{L}}}\in\operatorname{H}^{1}(\mathrm{B}{\mathbb{Z}_{3,\mathbf{B}+\mathbf{L}}},\mathbb{Z}_{3})$
is pulled back to $\operatorname{H}^{1}(M,\mathbb{Z}_{3})$. The
$B^{\prime}_{\mathbb{Z}_{3,\mathbf{B}+\mathbf{L}}}\equiv(\frac{1}{3}\delta
A^{\prime}_{\mathbb{Z}_{3,\mathbf{B}+\mathbf{L}}}\;{\rm
mod}\;3)\in\operatorname{H}^{2}(\mathrm{B}{\mathbb{Z}_{3,\mathbf{B}+\mathbf{L}}},\mathbb{Z}_{3})$
is pulled back to $\operatorname{H}^{2}(M,\mathbb{Z}_{3})$, defined via the
coboundary operator $\frac{1}{3}\delta$ for cochains (similar to the
differential operator $\hskip 1.0pt\mathrm{d}$ for differential forms) that
maps $\operatorname{H}^{1}$ to $\operatorname{H}^{2}$. The $c_{j}({\rm G})$ is
the $j$th Chern class of the associated vector bundle of the principal
G-bundle. The Postnikov square
$\mathfrak{P}_{3}(B^{\prime}_{\mathbb{Z}_{3,\mathbf{B}+\mathbf{L}}})\equiv\beta_{(9,3)}(B^{\prime}_{\mathbb{Z}_{3,\mathbf{B}+\mathbf{L}}}\cup
B^{\prime}_{\mathbb{Z}_{3,\mathbf{B}+\mathbf{L}}})$ maps
$\operatorname{H}^{2}(...,\mathbb{Z}_{3})\to\operatorname{H}^{5}(...,\mathbb{Z}_{9})$
via the cup product $\cup$ for cochains (similar to the wedge product $\wedge$
for differential forms) and the Bockstein homomorphism $\beta_{(9,3)}$
associated with the induced long exact sequence of cohomology with
coefficients from the short exact sequence
$0\to\mathbb{Z}_{9}\to\mathbb{Z}_{27}\to\mathbb{Z}_{3}\to 0$. Cobordism
invariants for other $q$ are similar Wan and Wang (2020a). For $q=2$, the
${\rm SU}(2)$ and ${\rm U}(1)$ labels change to ${\rm U}(2)$. For $q=3$, the
${\rm SU}(3)$ and ${\rm U}(1)$ labels change to ${\rm U}(3)$. For $q=6$, the
${\rm SU}(2)$ label changes to ${\rm U}(2)$, the ${\rm SU}(3)$ label changes
to ${\rm U}(3)$, and the ${\rm U}(1)$ label changes to either ${\rm U}(2)$ or
${\rm U}(3)$ since $c_{1}({\rm U}(2))=c_{1}({\rm U}(3))$.

Because all these $\mathbb{Z}_{9}\times{\mathbb{Z}_{3}^{7}}$ global anomalies
involve internal symmetries (thus gauge anomalies among
$\mathbf{B}\pm\mathbf{L}$, $X$, and $G_{{\rm SM}_{q}}$) without turning on the
spacetime symmetry background (thus they are not mixed gauge-gravitational or
gravitational anomalies), as long as these internal symmetries are dynamically
gaugeable, we expect all these anomalies are canceled.

Moreover, once $G_{{\rm SM}_{q}}$ is dynamically gauged as in the SM vacuum,
we obtain extra 1-form electric and magnetic symmetries, $G_{[1]}^{e}$ and
$G_{[1]}^{m}$ _kinematically_ as generalized global symmetries Gaiotto  _et
al._ (2015) that act on charged objects (1d electric Wilson lines and 1d
magnetic ’t Hooft lines). By gauging $G_{{\rm SM}_{q}}$, we obtain
$G_{[1]}^{e}\times G_{[1]}^{m}=\mathbb{Z}_{6/q,[1]}^{e}\times{\rm
U}(1)_{[1]}^{m}$ Wan  _et al._ (2020); Anber and Poppitz (2021); Wang and You
(2021) in the dynamical SM gauge theory, for $q=1,2,3,6$ and we define
$q^{\prime}\equiv 6/q\equiv{2^{n_{2}}\cdot 3^{n_{3}}}=6,3,2,1$ as
$(n_{2},n_{3})=(1,1),(0,1),(1,0),(0,0)$ respectively. We compute the potential
’t Hooft anomaly classification of the gauged-$G_{{\rm SM}_{q}}$ SM via
removing $G_{{\rm SM}_{q}}$ and including $G_{[1]}^{e}\times G_{[1]}^{m}$ into
(16) and (19),

$\displaystyle\mathrm{TP}_{5}({\rm Spin}\times_{\mathbb{Z}_{2}^{F}}{\rm
U}(1)_{\mathbf{B}-\mathbf{L}}\times\mathbb{Z}_{3,{\mathbf{B}+\mathbf{L}}}\times\mathbb{Z}_{6/q,[1]}^{e}\times{\rm
U}(1)_{[1]}^{m})=\big{(}\mathbb{Z}^{2}\times\mathbb{Z}_{6/q}\big{)}\times\big{(}\mathbb{Z}_{9}\times(\mathbb{Z}_{3})^{2}\times{(\mathbb{Z}_{3})^{3n_{3}}}\big{)}.$
(20) $\displaystyle\mathrm{TP}_{5}({\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}\times\mathbb{Z}_{3,{\mathbf{B}+\mathbf{L}}}\times\mathbb{Z}_{6/q,[1]}^{e}\times{\rm
U}(1)_{[1]}^{m})=\big{(}\mathbb{Z}_{16}\times(\mathbb{Z}_{4})^{n_{2}}\times\mathbb{Z}_{6/q}\big{)}\times\big{(}\mathbb{Z}_{9}\times(\mathbb{Z}_{3})^{2n_{3}}\big{)}.$
(21)

The anomaly classifications on the right-hand side in the first big bracket
$\big{(}\dots\big{)}$ are obtained recently in Wang  _et al._ (2022), while
those in the second big bracket $\big{(}\dots\big{)}$ are new to the
literature. The $\mathbb{Z}^{2}$ local and $\mathbb{Z}_{16}$ global anomalies
are discussed earlier in (16) and (19), with its anomaly coefficient
proportional to $(-N_{f}+n_{\nu_{R}})$ Wang  _et al._ (2022). The
$\mathbb{Z}_{6/q}$ global anomaly is a mixed anomaly between 1-form symmetries
of $\mathbb{Z}_{6/q,[1]}^{e}$ and ${\rm U}(1)_{[1]}^{m}$, which is identified
to be nonzero in Wang  _et al._ (2022) — which has dynamical constraints on
the SM gauge theories. The $(\mathbb{Z}_{4})^{n_{2}}$ global anomaly is
canceled Wang  _et al._ (2022). The $\mathbb{Z}_{9}$ global anomaly
$\mathfrak{P}_{3}(B^{\prime}_{\mathbb{Z}_{3,\mathbf{B}+\mathbf{L}}})$ and the
$(\mathbb{Z}_{3})^{2}$ global anomalies
$A^{\prime}_{\mathbb{Z}_{3,{\mathbf{B}+\mathbf{L}}}}B^{\prime}_{\mathbb{Z}_{3,{\mathbf{B}+\mathbf{L}}}}c_{1}({\rm
U}(1)_{\mathbf{B}-\mathbf{L}})$ and
$A^{\prime}_{\mathbb{Z}_{3,{\mathbf{B}+\mathbf{L}}}}c_{1}({\rm
U}(1)_{\mathbf{B}-\mathbf{L}})^{2}$ are already canceled to zero in the SM
earlier in (16). The remaining $(\mathbb{Z}_{3})^{3n_{3}}$ in (20) (which
exactly overlaps the $(\mathbb{Z}_{3})^{2n_{3}}$ in (21) for the first two
generators) contains three more extra generators (for $n_{3}=1$):
$A^{\prime}_{\mathbb{Z}_{3,{\mathbf{B}+\mathbf{L}}}}(B^{e}_{\mathbb{Z}_{3,[1]}})^{2}$
,
$A^{\prime}_{\mathbb{Z}_{3,{\mathbf{B}+\mathbf{L}}}}B^{\prime}_{\mathbb{Z}_{3,{\mathbf{B}+\mathbf{L}}}}B^{e}_{\mathbb{Z}_{3,[1]}}$
and $A^{\prime}_{\mathbb{Z}_{3,{\mathbf{B}+\mathbf{L}}}}c_{1}({\rm
U}(1)_{\mathbf{B}-\mathbf{L}})B^{e}_{\mathbb{Z}_{3,[1]}}$. The first two of
these three terms are mixed anomalies between the 0-symmetry
${\mathbf{B}+\mathbf{L}}$ (coupled to 1-form background field
$A^{\prime}_{\mathbb{Z}_{3,{\mathbf{B}+\mathbf{L}}}}$) and the electric
1-symmetry ${\mathbb{Z}_{3,[1]}}$ (coupled to 2-form background field
$B^{e}_{\mathbb{Z}_{3,[1]}}$). It is interesting to check whether these two
mixed $({\mathbf{B}+\mathbf{L}})$-${\mathbb{Z}_{3,[1]}}$ anomalies occur for
SM with $q=1,2$ (thus $n_{3}=1$). We can check this mixed anomaly between
0-symmetry and 1-symmetry via the techniques of the analogous anomaly studied
previously in Gaiotto  _et al._ (2017); Shimizu and Yonekura (2018); Tanizaki
_et al._ (2018); Cordova and Dumitrescu (2018); Wan  _et al._ (2019); Wan and
Wang (2020b). If the $\mathbb{Z}_{3}$ class of
$A^{\prime}_{\mathbb{Z}_{3,{\mathbf{B}+\mathbf{L}}}}(B^{e}_{\mathbb{Z}_{3,[1]}})^{2}$
anomaly is present, then this means that the
$\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}=\mathbb{Z}_{6,\mathbf{B}+\mathbf{L}}$
symmetry should be broken down to a $\mathbb{Z}_{2}^{F}$ subgroup by the ${\rm
PSU}(3)={\rm SU}(3)/\mathbb{Z}_{3}$ fractional instanton. Note that the
instanton number $n^{(N)}\equiv\int\hskip
1.0pt\mathrm{d}^{4}x\frac{{g}_{2}^{2}}{{64}\pi^{2}}\epsilon^{\mu\nu\mu^{\prime}\nu^{\prime}}F_{\mu\nu}F_{\mu^{\prime}\nu^{\prime}}$
for ${\rm SU}(N)$ gauge theory has $n^{(N)}\in\mathbb{Z}$ on spin manifolds,
while turning on 1-form symmetry ${\mathbb{Z}_{N,[1]}}$ background field can
produce ${\rm PSU}(N)$ fractional instantons such that
$n^{(N)}\in\mathbb{Z}/N$ on spin manifolds, and $n^{(N)}\in\mathbb{Z}/N$ (for
odd $N$) or $n^{(N)}\in\mathbb{Z}/2N$ (for even $N$) on non-spin manifolds. So
if ${\rm SU}(3)$ instanton was a source of ${\rm
U}(1)_{\mathbf{B}+\mathbf{L}}\to\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$
symmetry breaking in (5), the ${\rm PSU}(3)$ fractional instanton can further
trigger the breaking
$\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}\to\mathbb{Z}_{2N_{f}/N_{c},\mathbf{B}+\mathbf{L}}=\mathbb{Z}_{2}^{F}$
symmetry, here the family and color numbers match $N_{f}=N_{c}=3$. But it is
the ${\rm SU}(2)$ instanton (not the ${\rm SU}(3)$ instanton) causing the
${\rm
U}(1)_{\mathbf{B}+\mathbf{L}}\to\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$
symmetry breaking. So we conclude that the ${\rm PSU}(3)$ fractional instanton
_cannot_ trigger the further breaking even in the presence of
$\mathbb{Z}_{6/q,[1]}^{e}$-background field (for $q=1,2$). Namely, this
suggests no
$A^{\prime}_{\mathbb{Z}_{3,{\mathbf{B}+\mathbf{L}}}}(B^{e}_{\mathbb{Z}_{3,[1]}})^{2}$
anomaly in the gauged SM.

In comparison, Ref. [Anber and Poppitz, 2021] checked various fractional
instanton contributions also showing a negative result on the mixed
$({\mathbf{B}+\mathbf{L}})$-${\mathbb{Z}_{6/q,[1]}}$ anomalies. Ref. [Anber
and Poppitz, 2021] found that in the presence of various fractional instantons
constructed on a torus with ’t Hooft twisted boundary condition ’t Hooft
(1981), the continuous ${\rm U}(1)_{\mathbf{B}+\mathbf{L}}$ breaks down to
$\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$,
$\mathbb{Z}_{8N_{f},\mathbf{B}+\mathbf{L}}$,
$\mathbb{Z}_{4N_{f},\mathbf{B}+\mathbf{L}}$,
$\mathbb{Z}_{6N_{f},\mathbf{B}+\mathbf{L}}$ for $q=1,2,3,6$. But the familiar
BPST SU(2) instanton Belavin  _et al._ (1975) already breaks ${\rm
U}(1)_{\mathbf{B}+\mathbf{L}}$ to $\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$
shown in (5). Since $\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ is not further
broken down to its subgroup by fractional instantons, Ref. [Anber and Poppitz,
2021] also found no mixed ’t Hooft anomaly between
$\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ and ${\mathbb{Z}_{6/q,[1]}}$
symmetry.

Näively, all these so-called 3-torsion
$\big{(}\mathbb{Z}_{9}\times(\mathbb{Z}_{3})^{2}\times{(\mathbb{Z}_{3})^{3n_{3}}}\big{)}$
and $\big{(}\mathbb{Z}_{9}\times(\mathbb{Z}_{3})^{2n_{3}}\big{)}$ classes
correspond to potential global anomalies in the SM between vector symmetries
(the vector $\mathbf{B}\pm\mathbf{L}$ and the ${\mathbb{Z}^{e}_{3,[1]}}$ from
gauging the vector $su(3)$ color), so there should be no anomalies among
vector symmetries. But the ${\mathbb{Z}^{e}_{3,[1]}}$ also partly descends
from gauging the chiral $u(1)_{\tilde{Y}}$. In particular, the last two terms
in the $(\mathbb{Z}_{3})^{3n_{3}}$, namely
$A^{\prime}_{\mathbb{Z}_{3,{\mathbf{B}+\mathbf{L}}}}B^{\prime}_{\mathbb{Z}_{3,{\mathbf{B}+\mathbf{L}}}}B^{e}_{\mathbb{Z}_{3,[1]}}\equiv
A^{\prime}_{\mathbb{Z}_{3,{\mathbf{B}+\mathbf{L}}}}(\frac{1}{3}\delta
A^{\prime}_{\mathbb{Z}_{3,\mathbf{B}+\mathbf{L}}}\;{\rm
mod}\;3)B^{e}_{\mathbb{Z}_{3,[1]}}$ and
$A^{\prime}_{\mathbb{Z}_{3,{\mathbf{B}+\mathbf{L}}}}c_{1}({\rm
U}(1)_{\mathbf{B}-\mathbf{L}})B^{e}_{\mathbb{Z}_{3,[1]}}$ anomalies, are
subtle anomalies that cannot be checked by the fractional instanton argument
above, thus they deserve further future investigations.

_Conclusion_ — We had identified a continuous ${\mathbf{B}-\mathbf{L}}$ or
discrete $X$, and discrete ${\mathbf{B}+\mathbf{L}}$ symmetry in the SM. For
$N_{f}=3$, we had shown $\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ is free
from anomaly with SM gauged Lie group $G_{{\rm SM}_{q}}$ for all four versions
of $q=1,2,3,6$. We also classify all potential mixed anomalies between
$\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ and gravitational background field
(from Spin diffeomorphism) or higher symmetries
$\mathbb{Z}_{6/q,[1]}^{e}\times{\rm U}(1)_{[1]}^{m}$ of SM in (20) and (21).
We show that $\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ is free from all
anomalies in (16) and (19), and it is free from many potential anomalies in
(20) and (21) involving higher symmetries as well. Although
$\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ is illegal in many GUTs in (12),
$\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ is a perfectly legal symmetry in
the SM and some versions of UU to protect the proton stability. Are there
_other ways_ to protect the proton stability in UU? Notice that UU allows a 4d
TQFT with discrete gauge forces (e.g., two layers of $[\mathbb{Z}_{2}]$ gauge
theories) constructed out of the symmetry extension Wang  _et al._ (2018)
(e.g., the first layer $1\to[\mathbb{Z}_{2}]\to{\rm
Spin}\times{\mathbb{Z}_{4}}\to{\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}{\mathbb{Z}_{4,X}}\to 1$ and the second layer
$1\to[\mathbb{Z}_{2}]\to{\rm Spin}\times{\mathbb{Z}_{8}}\to{\rm
Spin}\times{\mathbb{Z}_{4}}\to 1$) such that the even class of
$\mathbb{Z}_{16}$ global anomaly in the original ${\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}{\mathbb{Z}_{4,X}}$ symmetry is trivialized by
pulled back to the extended ${\rm Spin}\times{\mathbb{Z}_{8}}$ symmetry Hsieh
(2018); Wang (2020a, 2021). Since the quark and baryon all carry odd
${\mathbb{Z}_{4,X}}$ charges, it will be interesting to study the impact of
the _projective symmetry fractionalization_ of ${\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}{\mathbb{Z}_{4,X}}$ on the line or surface
operators of the 4d discrete gauge TQFT on the proton stability.

In the language of _deformation class_ of quantum field theories Seiberg
(2019), our present work together with Ref. [Wang  _et al._ , 2022] suggests
that the 4d SM and GUTs can still be in the larger deformation class of QFT by
including the $\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ symmetry, except
that SM and UU can preserve $\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$, but
many GUTs break $\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ explicitly. It
will be important to understand deeper the implications of the subtle anomaly
cancellation data presented in (16), (19), (20), and (21), so we may use these
results to investigate the SM dynamics better.

_Acknowledgment_ — JW thanks Yuta Hamada, Miguel Montero, and Cumrun Vafa for
encouraging conversations on related topics. During the preparation of this
manuscript, we become aware that Ref. [Koren, 2022] by Seth Koren also
explores the relation between a discrete ${\mathbf{B}+\mathbf{L}}$ symmetry
and the proton stability. JW is supported by Center for Mathematical Sciences
and Applications at Harvard University. ZW is supported by the Shuimu Tsinghua
Scholar Program. YZY is supported by a startup fund at UCSD.

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