aainstitutetext: Institute for Theoretical Physics, KU Leuven, Celestijnenlaan
200D, B-3001 Leuven, Belgium

# C=Anything and the switchback effect in
Schwarzschild-de Sitter space

Sergio E. Aguilar-Gutierrez<EMAIL_ADDRESS>
###### Abstract

We investigate observables within the framework of the codimension-one
C=Anything (CAny) proposal for Schwarzschild-de Sitter (SdS) space under the
influence of shockwave sources. Within the proposal, there is a set of time-
reversal invariant observables that display the same rate of growth at early
and late times for a background with or without shockwave sources. Once we
introduce shockwaves in the weak gravitational coupling regime, there is a
decrease in the late-time complexity growth due to cancellations with early-
time perturbations, known as the switchback effect. The result shows that some
CAny observables in SdS may reproduce the same type of behavior found in anti-
de Sitter black holes. We comment on how our results might guide us to new
explorations in the putative quantum mechanical theory.

##  1 Introduction

Recently, a lot of interest in quantum information theoretic notions has
surfaced in an effort to characterize semiclassical gravitational observables.
Particularly, holographic complexity has been used to study bulk gravitational
dynamics that can probe regions inaccessible to entanglement entropy
Susskind:2014moa ; Susskind:2014rva ; Stanford:2014jda , and whose variation
can also reproduce gravitational equations of motion Czech:2017ryf ;
Caputa:2018kdj ; Susskind:2019ddc ; Pedraza:2021mkh ; Pedraza:2021fgp ;
Pedraza:2022dqi . This notion is also expected to reproduce the properties for
the computational complexity of a quantum circuit model111See Chapman:2021jbh
for a recent review. of the conformal field theory (CFT) dual to
asymptotically Anti-de Sitter (AdS) spacetimes Susskind:2014rva . This
translates to robust features captured by a large family of proposals,
starting with the Complexity=Volume (CV) Stanford:2014jda , Complexity=Action
(CA) Brown:2015bva ; Brown:2015lvg , Complexity=Spacetime Volume (CV2.0)
Couch:2016exn , and a recent generalization known as the Complexity=Anything
(CAny) proposal Belin:2021bga ; Belin:2022xmt . There are two defining
features for the CAny observables in AdS black holes: (i) a late boundary time
linear growth, and (ii) the switchback effect. The latter is a characteristic
decrease in the late-time linear growth once perturbations in the geometry are
introduced due to energy pulses.

Perhaps, one of the most exciting aspects about this family of spacetime
probes is to characterize the properties of cosmological backgrounds, and in
particular for de Sitter (dS) space. The nature of the microscopic degrees of
freedom encoded inside the cosmological horizon remains mysterious (see
Galante:2023uyf  for a recent review).

A recent proposal in dS holography, known as stretched horizon holography
Susskind:2021esx ; Shaghoulian:2022fop ; Susskind:2022dfz ; Lin:2022rbf ;
Lin:2022nss ; Bhattacharjee:2022ave ; Rahman:2022jsf ; Susskind:2023hnj ;
Susskind:2023rxm ; Franken:2023pni ; Lin:2023trc , has sparked many
developments in dS space complexity. The stretched horizon is a region in the
static patch of dS space where the dual theory is conjectured to be located
Susskind:2021esx . One may therefore perform gravitational dressings with
respect to the stretched horizon to probe the dual degrees of freedom.
Although the precise location is not explicit in this approach, it is expected
to be close to the cosmological horizon. This approach has been explored to
study holographic complexity proposals in asymptotically dS space
Jorstad:2022mls ; Auzzi:2023qbm ; Anegawa:2023dad ; Baiguera:2023tpt ;
Anegawa:2023wrk ; Aguilar-Gutierrez:2023zqm ; Aguilar-Gutierrez:2023tic . One
of the striking features originally found in Jorstad:2022mls  for the CV, CA,
and CV2.0 proposals was that the rate of growth of holographic complexity
might diverge a finite times relative to the stretched horizon, denoted by
hyperfast growth. The result was associated with the conjecture where the
double-scaled Sachdev–Ye–Kitaev (SYK) model is identified as the quantum
mechanical dual to dS2 space Susskind:2021esx . One may introduce a cutoff
surface to perform an analytic continuation allowing for late-time evolution
Jorstad:2022mls .

However, it was recently found that hyperfast growth is not a universal
phenomenon in the space of holographic complexity proposals Aguilar-
Gutierrez:2023zqm . Instead, a set of the CAny observables may evolve to
arbitrarily late (or early) static patch times without introducing regulator
surfaces. Notice, however, that it is still unclear what kind of
interpretation the different types of CAny observables might have for dS
space. To properly define holographic complexity proposals one needs a good
understanding of the basic properties that complexity for the dual field
theory side should satisfy. Such understanding of the microscopics associated
with dS space is still lacking. It is important to learn about the different
signals one should look for in a candidate for the dS space holographic model.

Alternatively, one might find clearer interpretations of holographic
complexity when dS space is embedded in a higher dimensional bulk AdS
spacetime. This perspective was recently approached by Aguilar-
Gutierrez:2023tic . They considered a particular type of CAny proposals in a
braneworld model consisting of a higher dimensional AdS bulk geometry capped
off by a pair of dS space end-of-the-world branes. In this setting, the
resulting complexity is associated with the field theory dual living on a
brane near the asymptotic boundary of the AdS space. In this case, the CAny
proposals with the expected late time growth in the AdS bulk are those that
obey the late time growth in the dS braneworld.

On the other hand, one of the defining properties of holographic complexity
proposals in the AdS context, the switchback effect, has been recently
explored in asymptotically dS spacetime Anegawa:2023wrk ; Baiguera:2023tpt
for the CV, CA, CV2.0 proposals. In this case, there are energy pulses
associated with perturbations in the evolution of the putative dual theory
residing in the stretched horizon. This can be an important diagnostic if the
CAny observables can indeed be associated with complexity consistent with
Nielsen’s geometric approach in the dual theory Aguilar-Gutierrez:2023zqm .
However, the switchback effect in the class of CAny observables where the
late-time growth in dS space is allowed has not been studied. This is the main
goal of our work, to learn new lessons for stretched horizon holography.

To study the switchback effect, we describe the shockwave geometry in
asymptotically dS spacetimes. We specialize in Schwazschild-de Sitter (SdS)
space, which describes spherically symmetric vacuum solutions to Einstein
equations with a positive cosmological constant. We work in the perturbative
weak gravity regime to treat the shockwave geometry based on previous findings
in Shenker:2013pqa , and recently discussed in the context of SdS black holes
by Aguilar-Gutierrez:2023ymx . As for the observables, we will work with
codimension-one CAny proposals evaluated in constant mean curvature (CMC)
slices, originally introduced in Belin:2022xmt . The mean curvature is one of
the main factors distinguishing the proposals that display late time growth
from those with hyperfast growth Aguilar-Gutierrez:2023zqm .

Our work is focused on alternating early and late-time perturbations in the
geometry and the resulting growth of the CAny observables. Our findings
indicate that the set of proposals with late-time growth will display the
switchback effect. The rate of growth will be determined by the definition of
the CAny proposal. In general, however, the manifestation of the switchback
effect in the CAny proposals only occurs once we perform a time-reversal
symmetric extension, originally hinted in Aguilar-Gutierrez:2021bns . We
select CMC slices that minimize the CAny proposal and find that the late-time
and early-time contributions to complexity growth will partially cancel out.
Under these conditions, the analysis of the switchback effect shows great
similarity with respect to that of AdS black hole backgrounds.

Interestingly, the late time growth during the switchback phase is unaffected
by the particular location of the stretched horizon.

The structure of the manuscript goes as follows. In Sec. 2 we review
generalities about the shockwave geometries in SdS spacetimes, as well as the
CAny proposals that display the early and late-time growth in SdS spacetime.
We introduce some new results regarding the generalizing the set of proposals
that reproduce the late time growth of complexity in SdS space of arbitrary
mass (below the extremal one). In Sec. 3, we present the results on the
switchback effect in SdS spacetime by performing a series of alternating
shockwave insertions in the background geometry in the weak gravity regime.
Finally, Sec. 4 includes a summary of our findings in this setting and some
interesting directions for future research. For the convenience of the reader,
we provide an App. A containing some of the details about the evaluation of
the late-time growth of the CAny proposals.

##  2 C=Anything in SdS spacetimes

In this section, we briefly review basic notions about SdSd+1 black holes, the
modification in the geometry due to shockwave insertions, and the CAny
proposals that we investigate in this work. New results include extending the
set of CAny proposals studied in Aguilar-Gutierrez:2023zqm  for SdS space, to
more general observables than volumes of CMC slices. This allows for different
types of late-time growth behaviors in $d=2$, $d=3$. We also show that late-
time growth persists for arbitrary mass black holes.

###  2.1 SdS spacetimes

The configuration of interest is SdSd+1 space, described by the line element

$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=-f(r)\mathrm{d}t_{L/R}^{2}+\frac{\mathrm{d}r^{2}}{f(r)}+r^{2}\mathrm{d}\Omega_{d-1}^{2}\,,$
(1) $\displaystyle f(r)$
$\displaystyle=1-\frac{r^{2}}{\ell^{2}}-\frac{2\mu}{r^{d-2}}\leavevmode\nobreak\
,\quad\mu\equiv\frac{16\pi
G_{N}M}{(d-1)\Omega_{d-1}r^{d-2}}\leavevmode\nobreak\ ,$ (2)

with $\ell^{2}=d(d-1)/(2\Lambda)$;222Through the rest of the work, we use
rescaled coordinates where $\ell=1$. $\Lambda>0$ is the cosmological constant;
$M$ parametrizes the mass of the black hole; $G_{N}$ is Newton’s constant;
$\Omega_{d-1}=2\pi^{d/2}/\Gamma(d/2)$ is the volume of a unit ($d-1$)-sphere;
$\mu\in[0,\,\mu_{N}]$. The case where $\mu=\mu_{N}$, with

$\mu_{N}=\tfrac{1}{d}\quantity(\tfrac{d-2}{d})^{\frac{d-2}{2}}\leavevmode\nobreak\
,$ (3)

describes the most massive black hole supported in dS space, which we refer to
as the extremal SdS limit. Meanwhile, $\mu=0$, reproduces dSd+1 space.

We will be interested in describing shockwaves in the geometry. It’s
convenient to use Kruskal coordinates, defined by

$\displaystyle U_{\rm b,\rm c}$
$\displaystyle=\mathrm{e}^{\frac{f^{\prime}(r_{\rm b,\rm
c})}{2}\left(r_{*}(r)-t\right)}\leavevmode\nobreak\ ,$ (4) $\displaystyle
V_{\rm b,\rm c}$ $\displaystyle=-\mathrm{e}^{\frac{f^{\prime}(r_{\rm b,\rm
c})}{2}\left(r_{*}(r)+t\right)}\,.$

where the subindices “b, c” denote coordinates on the black hole and inflating
patches. These patches are centered at the horizons $r_{\rm b,\rm c}$, and
cover the range $0\leq r<r_{\mathcal{O}}$ and $r_{\mathcal{O}}\leq r<\infty$
respectively, with $r_{\mathcal{O}}$ a reference point, which we take as the
location of the static patch observer. In the present work, we will focus on
the inflating region, as we aim to probe it with CAny observables. We replace
$U_{c}$, $V_{c}\rightarrow$ $U$, $V$ in what follows. We can then express (1)
as

$\mathrm{d}s^{2}=-\frac{4f(r)}{f^{\prime}(r_{\rm
c})}\mathrm{e}^{-f^{\prime}(r_{\rm
c})r_{*}(r)}\mathrm{d}U\,\mathrm{d}V+r^{2}\mathrm{d}\Omega_{d-1}^{2}$ (5)

where $r_{\mathcal{O}}\leq r<\infty$, and $r_{*}=\int\frac{\mathrm{d}r}{f(r)}$
is the tortoise coordinate.

Once we add shockwave perturbations, the geometry gets distorted, as well as
the location of the stretched horizon, see Fig. 1.333We represent the Penrose
diagram with the coordinate $r$ being continuous along the shockwave geometry,
while the coordinates $U$ or $V$ can be discontinuous.

Figure 1: Shockwave perturbations (orange wavy line) along $U_{c}=0$ in: SdS3
space (left); and SdSd+1≥4 space (right), illustrated in the extremal SdS
limit. The stretched horizon (in green) is shown at a fixed location $r=r_{\rm
st}$. The cosmological and black hole horizons are shown with the dashed
lines. The Kruskal coordinates $U$, $V$ are displayed with black arrows.

Although stretched horizon holography fixes the location of the dual theory to
be located at a constant $r$ surface in the static patch, it remains unknown
how the theory should behave under shockwave perturbations. We will be
considering the case where the shockwaves are sent through $U=0$ and the
stretched horizon remains fixed at a constant $r=r_{\rm st}$
coordinate444Similar considerations have been carried out in Baiguera:2023tpt
; Anegawa:2023dad , as well as alternative proposals., for which time
evolution along the stretched horizon is continuous.

We are mainly interested in the metric under shockwave perturbations in the
weak gravitational coupling regime to define the notions of energy below. We
consider an SdS black hole with mass $M$ absorbs a shell of matter with mass
$E\ll M$ along the surface

$U=U_{0}=\mathrm{e}^{\frac{f^{\prime}(r_{\rm
c})}{2}\left(r_{*}(r_{\mathcal{O}})-t_{0}\right)}\leavevmode\nobreak\ ,$ (6)

with $t_{0}$ the static time shockwave insertion with respect to
$r_{\mathcal{O}}$.

The SdS black hole after the shockwave has mass $M-E$ in (1) for matter
obeying the null energy condition (NEC) Baiguera:2023tpt ; Aguilar-
Gutierrez:2023ymx . We glue the coordinates along a shell $U,\,V$ to the past
of the shell with those to the future, denoted by $\tilde{U},\,\tilde{V}$. The
resulting cosmological line element for SdS black holes Aguilar-
Gutierrez:2023ymx ; Shenker:2013pqa :

$\mathrm{d}s^{2}=-\frac{4\tilde{f}(r)}{\tilde{f}^{\prime}(r_{\rm
c})}\mathrm{e}^{-\tilde{f}^{\prime}(\tilde{r}_{\rm
c})\tilde{r}_{*}(r)}\mathrm{d}\tilde{U}\mathrm{d}\tilde{V}+r^{2}\mathrm{d}\Omega_{d-1}^{2}$
(7)

where tilded quantities are given by the replacement of $M\rightarrow M-E$ in
the untilded ones. In the inflating patch, the shift along the $V$ coordinate
can be described by a shift in the coordinate

$\tilde{V}=V-\alpha\leavevmode\nobreak\ .$ (8)

The NEC also imposes that $\alpha\geq 0$ Aalsma:2021kle ; while $E\ll M$
guarantees we work in the $\alpha\ll 1$ limit555See Aguilar-Gutierrez:2023ymx
for remarks on the approximation for the extremal SdS limit.. Notice that the
shift is exactly the opposite sign than in a crunching geometry Gao:2000ga .
We will express the shift parameter as

$\displaystyle\alpha=2\mathrm{e}^{-\tfrac{f^{\prime}(r_{c})}{2}(t_{c}^{(*)}\pm
t_{0})}\leavevmode\nobreak\ .$ (9)

Here the $\pm$ sign depends on whether the shockwave is left or right moving,
and the cosmological scrambling time $t_{c}^{(*)}$ will be defined through
this relation.

We will take the shockwave close to the cosmological horizon and set $U_{0}=0$
in the following. (7) transforms into

$\displaystyle\mathrm{d}s^{2}=-2a\quantity(U[V-\alpha\Theta(U)])\mathrm{d}U\mathrm{d}V+b\quantity(U[V-\alpha\Theta(U)])\mathrm{d}\Omega_{d-1}^{2}\leavevmode\nobreak\
,$ (10) $\displaystyle
a\quantity(UV)=-\tfrac{2}{UV}\tfrac{f(r)}{f^{\prime}(r_{c})^{2}}\leavevmode\nobreak\
,\quad b\quantity(UV)=r^{2}\leavevmode\nobreak\ .$ (11)

###  2.2 C=Anything: CMC slices

We are mainly interested in codimension-one observables within the class of
the C=Anything proposal, introduced in Belin:2021bga ; Belin:2022xmt ,

$\mathcal{C}^{\epsilon}\equiv\frac{1}{G_{N}}\int_{\Sigma_{\epsilon}}\mathrm{d}^{d}\sigma\,\sqrt{h}\leavevmode\nobreak\
F[g_{\mu\nu},\,\mathcal{R}_{\mu\nu\rho\sigma},\,\nabla_{\mu}]\leavevmode\nobreak\
,$ (12)

where $F[g_{\mu\nu},\,\mathcal{R}_{\mu\nu\rho\sigma},\,\nabla_{\mu}]$ is an
arbitrary scalar functional of $d+1$-dimensional bulk curvature invariants,
$\Sigma_{\epsilon}$ is a $d$-dimensional spatial slice labeled by
$\epsilon$($=+,\,-$), which is anchored on the stretched horizon, $h$ is the
determinant of the induced metric, $h_{\mu\nu}$, on $\Sigma_{\epsilon}$. We
will let $F[\dots]$ be a general functional throughout the work. The reader is
referred to footnote 9 to verify that these proposals display a switchback
effect in AdS planar black holes in the time-reversal symmetrization in (43).

To define the region of evaluation, we employ a combination of codimension-one
and codimension-zero volumes with different weights, given by

$\displaystyle\mathcal{C}_{\rm CMC}=\tfrac{1}{G_{N}}\biggl{[}$
$\displaystyle\alpha_{+}\int_{\Sigma_{+}}\mathrm{d}^{d}\sigma\,\sqrt{h}+\alpha_{-}\int_{\Sigma_{-}}\mathrm{d}^{d}\sigma\,\sqrt{h}+\alpha_{B}\int_{\mathcal{M}}\mathrm{d}^{d+1}x\sqrt{-g}\biggr{]}$
(13)

where $\mathcal{M}$ is the bulk region; $\alpha_{\pm}$, $\alpha_{B}$ are
constants; and $\Sigma_{+}$, $\Sigma_{-}$ are the future and past boundary
slices in $\partial\mathcal{M}=\Sigma_{+}\cup\Sigma_{-}$, see Fig 2.

Figure 2: Implementation of the codimension-one CAny proposals with CMC slices
as evaluation regions in the unperturbed (S)dS space, following the notation
in Fig. 1.

The extremization of $\mathcal{C}_{\rm CMC}$ reveals that $\Sigma_{\pm}$ are
CMC slices, whose mean curvature is given by:

$K_{\epsilon}\equiv\evaluated{K}_{\Sigma_{\epsilon}}=-\epsilon\frac{\alpha_{B}}{\alpha_{\epsilon}}\leavevmode\nobreak\
,$ (14)

where $K^{\mu\nu}=h^{\mu\alpha}\nabla_{\alpha}n^{\nu}$ is the extrinsic
curvature, and we consider $n^{\mu}$ to be a future pointing normal vector for
both $\Sigma_{\epsilon}$.

To simplify the evaluation of (12), we employ time-symmetric evolution on each
of the static patches, so that we set $t_{L}=t_{R}$ in (1). Moreover, we
introduce Eddington-Finkelstein coordinates in (1),

$\mathrm{d}s^{2}=-f(r)\mathrm{d}v^{2}+2\mathrm{d}v\mathrm{d}r+r^{2}\mathrm{d}\Omega^{2}_{d-1}\leavevmode\nobreak\
,$ (15)

which are related to the Kruskal coordinates (4) by

$U=\mathrm{e}^{-\tfrac{f^{\prime}(r_{c})}{2}u}\leavevmode\nobreak\ ,\quad
V=-\mathrm{e}^{\tfrac{f^{\prime}(r_{c})}{2}v}\leavevmode\nobreak\ .$ (16)

Evaluating (12, 13) with (1), one finds

$\displaystyle\mathcal{C}^{\epsilon}$
$\displaystyle=\tfrac{\Omega_{d-1}}{G_{N}}\int_{\Sigma_{\epsilon}}\mathrm{d}\sigma\,r^{d-1}\sqrt{-f(r)\dot{v}^{2}+2\dot{v}\dot{r}}\,a(r)\,,$
(17) $\displaystyle\mathcal{C}_{\rm CMC}$
$\displaystyle=\tfrac{\Omega_{d-1}}{G_{N}}\sum_{\epsilon}\alpha_{\epsilon}\int_{\Sigma_{\epsilon}}\mathrm{d}\sigma\,\mathcal{L}_{\epsilon}\leavevmode\nobreak\
,$ (18)

where $a(r)$ is a scalar functional corresponding to the evaluation of
$F[g_{\mu\nu},\,\mathcal{R}_{\mu\nu\rho\sigma},\,\nabla_{\mu}]$; $\sigma$ is a
general parametrization of the coordinates $v(\sigma)$, $r(\sigma)$ on the
slice $\Sigma_{\epsilon}$; and

$\mathcal{L}_{\epsilon}\equiv
r^{d-1}\sqrt{-f(r)\dot{v}^{2}+2\dot{v}\dot{r}}-\epsilon\tfrac{K_{\epsilon}}{d}\dot{v}r^{d}\leavevmode\nobreak\
.$ (19)

The details of the evaluation are shown in App. A. The late-time evolution of
complexity results in:

$\lim_{t\rightarrow\infty}\derivative{t}\mathcal{C}^{\epsilon}\simeq\frac{\Omega_{d-1}}{G_{N}}\sqrt{-f(r_{f})r_{f}^{2(d-1)}}\,a(r_{f})\leavevmode\nobreak\
\text{ with }r_{f}\equiv\lim_{t\rightarrow\infty}r_{t}\leavevmode\nobreak\ .$
(20)

Here $r_{f}$ is a local maximum of the effective potential at late times:

$\evaluated{\mathcal{U}}_{r_{f}}=0,\quad\evaluated{\partial_{{r}}\mathcal{U}}_{r_{f}}=0,\quad\evaluated{\partial^{2}_{{r}}\mathcal{U}}_{r_{f}}\leq
0\leavevmode\nobreak\ .$ (21)

These conditions lead to the following relation

$H(r_{f},\,K_{\epsilon})\equiv
4r_{f}f\left(r_{f}\right)\left((d-1)f^{\prime}\left(r_{f}\right)+K_{\epsilon}^{2}r_{f}\right)+4(d-1)^{2}f\left(r_{f}\right){}^{2}+r_{f}^{2}f^{\prime}\left(r_{f}\right){}^{2}=0\leavevmode\nobreak\
.$ (22)

The roots $r_{f}$ of the function $H(r_{f},\,K_{\epsilon})$ can be found
explicitly for pure dS and extremal SdS black hole limits, as originally
derived in Aguilar-Gutierrez:2023zqm ,

$\displaystyle\quantity({r}^{\rm(dS)}_{f})^{2}$
$\displaystyle=\frac{{K_{\epsilon}}^{2}-2d(d-1)\pm|{K_{\epsilon}}|\sqrt{{K_{\epsilon}}^{2}-4(d-1)}}{2({K_{\epsilon}}^{2}-d^{2})}\leavevmode\nobreak\
,\quad\absolutevalue{K_{\epsilon}}\geq 2\sqrt{d-1}\leavevmode\nobreak\ ;$ (23)
$\displaystyle{r}^{\rm(N)}_{f}$
$\displaystyle=\sqrt{\tfrac{d-2}{d}}\leavevmode\nobreak\
,\quad\absolutevalue{K_{\epsilon}}\geq\sqrt{d}\leavevmode\nobreak\ .$ (24)

Let us now show that for a generic SdS black hole spacetimes ($d\geq 3$),
there will always be a real root to (22). One can evaluate
$H(r_{f},\,K_{\epsilon})$ in (22) with the roots in (23, 24) while keeping the
mass of the black hole arbitrary. We will denote
$m\equiv\mu/\mu_{N}\in[0,\,1]$, such that we may express:

$\displaystyle
H\quantity({r}^{\rm(dS)}_{f},\,2\sqrt{d-1})=\frac{4\mu\left((d-2)^{d}d^{2}\mu-4(d-2)^{2}((d-1)d)^{d/2}\right)}{(d-2)^{4-d}(d-1)^{d-2}d^{d}}\leavevmode\nobreak\
,$ (25) $\displaystyle
H\quantity({r}^{\rm(N)}_{f},\,K_{\epsilon})=\tfrac{4(1-m)\quantity(2K_{\epsilon}^{2}(d-2)+d^{2}(1-m))}{d^{2}}\leavevmode\nobreak\
.$ (26)

Notice that (25) is clearly negative for all $d\geq 3$ and $m\in(0,\,1)$,
while (26) is positive. Moreover, as we increase
$\absolutevalue{K_{\epsilon}}>2\sqrt{d-1}$,
$H\quantity({r}^{\rm(dS)}_{f},\,K_{\epsilon})$ becomes more negative in (22).
Then, according to the _intermediate value theorem_ , there will exist at
least a real root ${r}_{f}\in\quantity[{r}^{\rm(dS)}_{f},\,{r}^{\rm(N)}_{f}]$
for general SdSd+1 space.

On the other hand, since we have allowed $a(r)$ to be an arbitrary function in
(20), we see that when $r_{f}\rightarrow\infty$666This condition is satisfied
in (23) when $K_{\epsilon}=d$ in dS2 and (S)dS3 space. there would be
arbitrary types of late-time growth for $C^{\epsilon}$ depending on the
particular choice of $a(r)$. For instance, the case $a(r)=1$ leads to late-
time exponential behavior when $r_{f}\rightarrow\infty$; meanwhile, having a
different degree of divergence in (20) would lead to enhancement or decrease
in the late-time growth. However, for this to be a valid CAny proposal, we
require also a modification, as explained below.

When one evaluates the early time evolution in (20) $t\rightarrow-\infty$,
there is a sign flip in $K_{\epsilon}\rightarrow-K_{\epsilon}$. As a result,
the rate of growth of the CAny observables at early and late times does not
coincide for a given CMC slice. The future or past growth would be given by
(20), while the other generates hyperfast growth.777See Aguilar-
Gutierrez:2023zqm  for comments about possible interpretations in terms of
circuit complexity. Importantly for us, the switchback effect is not respected
in this case, as on requires a cancellation between early and late-time
contributions to the complexity growth. We will make this more explicit below.

##  3 The switchback effect

We will study the set of observables (12, 13) in the shockwave geometry (10).
We begin performing a sequence of an even number of shockwaves, $n$, in the
inflating patch (i.e. $r\in[r_{\mathcal{O}},\,\infty]$). Let us denote
${t}_{1}$, ${t}_{2}$, …, ${t}_{n}$ as the insertion static patch times with
respect to the stretched horizon in alternating insertion order, i.e.
${t}_{2k+1}>{t}_{2k}$, and ${t}_{2k}<{t}_{2k-1}$, restricted to
$\absolutevalue{{t}_{i+1}-{t}_{i}}\gg{t}_{*}$.

Accounting for the sign of shift in the backreacted metric (10), the
functional (12) has an additive property under these insertions in the strong
shockwave limit Belin:2021bga ; Belin:2022xmt ,

$\displaystyle{\mathcal{C}}^{\epsilon}(t_{L},\,t_{R})=$
$\displaystyle\mathcal{C}^{\epsilon}(t_{R},\,V_{1})+\mathcal{C}^{\epsilon}(V_{1}-\alpha_{1},\,V_{2})+\dots$
(27)
$\displaystyle+\mathcal{C}^{\epsilon}(U_{n-1}+\alpha_{n-1},\,V_{n})+\mathcal{C}^{\epsilon}(V_{n}-\alpha_{n},\,t_{L})$

where $\mathcal{C}^{\epsilon}(\cdot,\,\cdot)$ denotes the contributions from
$\Sigma_{\epsilon}$ with two fixed endpoints and all endpoints are located
either on the left/right horizon or on the stretched horizon $r_{\rm st}$.

For the evaluation of $\mathcal{C}^{\epsilon}$, we search for the locations
$u_{R,\,L}$, $v_{R,\,L}$ where $\Sigma_{\epsilon}$ intersect with the
left/right horizon $r_{c}$,

$v_{R}-v_{t}=\int_{r_{c}}^{r_{t}}\mathrm{d}r\tfrac{\dot{v}}{\dot{r}}=\int_{r_{c}}^{r_{t}}\tfrac{\mathrm{d}r}{f(r)}\quantity(1-\frac{P_{v}^{\epsilon}+\frac{\epsilon
K_{\epsilon}}{d}r^{d}}{\sqrt{-\mathcal{U}(P_{v}\epsilon,\,r)}})\leavevmode\nobreak\
,$ (28)

and $v_{t}=v_{R}(r_{t})$. We perform the expansion around the final slice,
such that (21) allows us to approximate

$\lim_{r\rightarrow
r_{f}}\mathcal{U}(P_{v}^{\epsilon},\,r)\simeq\tfrac{1}{2}(r-r_{f})^{2}\mathcal{U}^{\prime\prime}(P_{v}^{\epsilon},\,r)+\mathcal{O}(\absolutevalue{r-r_{f}}^{3})\leavevmode\nobreak\
.$ (29)

We can proceed as outlined in App. A (see (50)) to express (17) as

$\displaystyle{\mathcal{C}^{\epsilon}}=$
$\displaystyle-\tfrac{2\Omega_{d-1}}{G_{N}}a(r_{t})\sqrt{-f(r_{t})r_{t}^{2(d-1)}}\int_{r_{c}}^{r_{t}}\tfrac{\left(P_{v}^{\epsilon}+\frac{\epsilon
K_{\epsilon}}{d}r^{d}\right)\mathrm{d}r}{f(r)\sqrt{-\mathcal{U}(P_{v}^{\epsilon},r)}}$
(30)
$\displaystyle+\tfrac{2\Omega_{d-1}}{G_{N}}\int_{r_{c}}^{r_{t}}\tfrac{a(r)f(r)\,r^{2(d-1)}+a(r_{t})\sqrt{-f(r_{t})r_{t}^{2(d-1)}}\left(P_{v}^{\epsilon}+\frac{\epsilon
K_{\epsilon}}{d}r^{d}\right)}{f(r)\sqrt{-\mathcal{U}(P_{v}^{\epsilon},\,r)}}\leavevmode\nobreak\
.$

Using (29) to approximate the potential near the final turning point $r_{f}$,
we find that

$\displaystyle{\mathcal{C}^{\epsilon}}=\tfrac{\Omega_{d-1}}{G_{N}}P_{\infty}^{\epsilon}v\leavevmode\nobreak\
,\quad P_{\infty}^{\epsilon}=a(r_{f})\sqrt{-f(r_{f})r_{f}^{2(d-1)}}$ (31)

where $r_{f}$ is determined with (22). The results above (31, 20) can be used
to express the contributions in (27) in Kruskal coordinates (16) as:

$\displaystyle{\mathcal{C}^{\epsilon}}(t_{L},\,V_{R})$
$\displaystyle=\frac{\Omega_{d-1}}{G_{N}}P^{\epsilon}_{\infty}\log{\mathrm{e}^{t_{L}}V_{R}}\leavevmode\nobreak\
,$ (32) $\displaystyle{\mathcal{C}^{\epsilon}}(U_{L},\,V_{R})$
$\displaystyle=\frac{\Omega_{d-1}}{G_{N}}P_{\infty}^{\epsilon}\log{U_{L}V_{R}}\leavevmode\nobreak\
,$ (33) $\displaystyle{\mathcal{C}^{\epsilon}}(U_{L},\,t_{R})$
$\displaystyle=\frac{\Omega_{d-1}}{G_{N}}P_{\infty}^{\epsilon}\log{U_{L}\mathrm{e}^{t_{R}}}\leavevmode\nobreak\
.$ (34)

However, there is also an early time contribution in the shockwave
geometry888This was also noticed in Aguilar-Gutierrez:2023zqm ; silke  for the
AdS black hole background., given by the term

${\mathcal{C}^{\epsilon}}(V_{L},\,U_{R})=\frac{\Omega_{d-1}}{G_{N}}P_{-\infty}^{\epsilon}\log{V_{L}U_{R}}\leavevmode\nobreak\
,$ (35)

where

$P_{-\infty}^{\epsilon}=a(r_{I})\sqrt{-f(r_{I})r_{I}^{2(d-1)}}\leavevmode\nobreak\
,\leavevmode\nobreak\ \text{ with }\leavevmode\nobreak\
r_{I}=\lim_{t\rightarrow-\infty}r_{t}\leavevmode\nobreak\ .$ (36)

As mentioned above in Sec. 2.2, for $t\rightarrow-\infty$, there is a sign
flip in $K_{\epsilon}\rightarrow-K_{\epsilon}$. In that case, $r_{I}$ is a
solution to (21, 22) with the appropriate modification of $K_{\epsilon}$.
However, as we also pointed out, the CMC slices that display late time growth
in the far past/future display hyperfast growth in the future/past
respectively. Instead, consider a protocol where we evaluate (17) over
different CMC slices in the past and future, such that there are always
solutions $r_{f}$ and $r_{I}$ with respect to the stretch horizon evolution.
(27) then transforms into

$\displaystyle{\mathcal{C}^{\epsilon}}(t_{L},\,t_{R})\simeq\tfrac{\Omega_{d-1}}{G_{N}}\Bigl{[}$
$\displaystyle
P_{\infty}^{\epsilon}\log(U_{1}\mathrm{e}^{t_{R}})+P_{-\infty}^{\epsilon}\log(U_{1}+\alpha_{1})V_{2}+P_{\infty}^{\epsilon}\log(V_{2}-\alpha_{2})U_{3}+\dots$
(37)
$\displaystyle+P_{\infty}^{\epsilon}\log((V_{n}-\alpha_{n})\mathrm{e}^{t_{L}})\Bigr{]}\leavevmode\nobreak\
.$

We can then extremize (37) with respect to an arbitrary interception point
($V_{i}$, $U_{i}$) in the multiple shockwave geometry,

$\derivative{{\mathcal{C}^{\epsilon}}(t_{L},\,t_{R})}{V_{i}}=0\leavevmode\nobreak\
,\quad\derivative{{\mathcal{C}^{\epsilon}}(t_{L},\,t_{R})}{U_{i}}=0\leavevmode\nobreak\
,$ (38)

which allows us to locate

$\displaystyle
V_{i}^{\epsilon}=\frac{P_{-\infty}^{\epsilon}\alpha_{i}}{P_{\infty}^{\epsilon}+P_{-\infty}^{\epsilon}}\leavevmode\nobreak\
,\quad
U_{i}^{\epsilon}=-\frac{P_{\infty}^{\epsilon}\alpha_{i}}{P_{\infty}^{\epsilon}+P_{-\infty}^{\epsilon}}\leavevmode\nobreak\
.$ (39)

Replacing the interception points into (37) generates:

${\mathcal{C}^{\epsilon}}\simeq\tfrac{\Omega_{d-1}}{G_{N}}\quantity(P_{+\infty}^{\epsilon}(t_{R}+t_{L})+(P_{+\infty}^{\epsilon}+P_{-\infty}^{\epsilon})\quantity(\sum_{k=1}^{n}t_{k}-nt^{(c)}_{*}))\leavevmode\nobreak\
,$ (40)

up to constant terms in terms of $P_{+\infty}^{\epsilon}$,
$P_{-\infty}^{\epsilon}$.

Importantly, it was noticed in Aguilar-Gutierrez:2023zqm  that the CAny
proposals with a generic functional $F[\dots]$ in (18) for an AdS planar black
hole background only satisfy the switchback effect when the rate of growth in
the past and future are the same. This means that for (18) to obey the
definition of holographic complexity in Belin:2021bga ; Belin:2022xmt , we
require999The derivation of this requirement for the AdS planar black hole
case follows the same steps that we have presented for the SdS case, although
some replacements need to be made. This includes inverting the integration
limits in (28, 28); setting $r_{c}\rightarrow r_{b}$;
$K_{\epsilon}\rightarrow-K_{\epsilon}$; and
$\alpha_{i}\rightarrow-\alpha_{i}$.

$P^{\epsilon}_{+\infty}=P^{\epsilon}_{-\infty}\leavevmode\nobreak\ .$ (41)

In that case, the evaluation of (40) reduces to

$\mathcal{C}^{\epsilon}\propto\absolutevalue{t_{R}+t_{1}}+\absolutevalue{t_{2}-t_{1}}+\dots+\absolutevalue{t_{n}-t_{L}}-2nt^{(c)}_{*}\leavevmode\nobreak\
,$ (42)

where the term $-2nt^{(c)}_{*}$ appears due to cancellation in the complexity
growth due to early and late time perturbations.

Notice that a possible way to satisfy (41) in SdS space can be obtained by
setting $K_{-}=-K_{+}$ and selecting a complexity proposal $\mathcal{C}$
as101010However, instead of minimization, one might as well perform a
maximization over the CMC slices, or an averaging, as either of those would
satisfy (41) in AdS planar black holes; although that would reproduce the
hyperfast growth in SdS space.

$\mathcal{C}=\min_{t}\quantity(\mathcal{C}^{+}(t),\,\mathcal{C}^{-}(t))\leavevmode\nobreak\
.$ (43)

See Fig. 3 for an illustration of the evolution of the CMC slices in the
shockwave geometry.

Figure 3: Representative CMC slices evolving in a single shockwave geometry in
SdS space.

We close the section with a few remarks. First, the result (42) reproduces the
same type of behavior as the switchback effect for AdS planar black holes, at
least for the CAny proposals with early and late-time linear growth in SdS
space. Second, as we mentioned in Sec. 2.2 there are fine-tuned situations
where $\mathcal{C}^{\epsilon}$ can have any type of early and late-time growth
behavior for SdS3. It might be interesting to study the modifications in the
switchback in those cases. Lastly, the switchback effect has also been
recovered in a different and more explicit analysis for particular
asymptotically dS backgrounds Anegawa:2023dad ; Baiguera:2023tpt , hinting at
the possibility that this is a rather universal phenomenon in shockwave
geometries.

##  4 Discussion and outlook

In summary, we studied the appearance of the switchback effect in
asymptotically dS spacetimes by studying the late (and early time) evolution
of the codimension-one CAny observables under shockwave insertions. We picked
a set of observables that are evaluated in CMC slices of different curvature
in the past and future boundaries. We proved that under a weakly gravitating
regime, the CAny observables show a reduction in the complexity growth due to
cancellations of the energy perturbations. We also explicitly verified one of
the predictions in Aguilar-Gutierrez:2023zqm , namely that a time-reversal
symmetric protocol would be necessary for the switchback effect to
occur111111On the field theory side, the notion of Nielsen geometric approach
to complexity Nielsen ; nielsen2006quantum ; dowling2006geometry  suggests
that this must be indeed respected.. Moreover, our findings show a great
similarity with respect to the behavior of CAny proposals for AdS black holes
under the switchback effect. However, we reiterate that the CAny observables
in our study do not necessarily represent holographic complexity in
asymptotically dS space. To have a clear notion of holographic complexity, we
might require a quantum circuit interpretation for the observables, which
would also require a reliable quantum circuit model for dS space. Some toy
models allowing much progress in this direction have been studied in
Bao:2017iye ; Bao:2017qmt ; Niermann:2021wco ; Cao:2023gkw .

We comment on some interesting future directions. Our work focused on the
alternating shockwave insertion on the inflating patch of the SdS spacetime.
However, we can also extend the analysis when the SdS spacetime has multiple
patches, to enquire about the black hole interior as well. These types of
geometries have been used as toy models for multiverses in Aguilar-
Gutierrez:2021bns . One of the striking features previously found was that the
information available to the past light cone of an observer in one of the
multiverses would encode the information of the other universes, in the
semiclassical regime. It would be very interesting to see whether the notions
of general codimension-zero CAny observables may also encode such information.
We might be able to learn if some of these observables might have an
interpretation from the point of view of quantum cosmology. It might be also
fruitful to study how the introduction of perturbations in the geometry
affects the coarse-graining of information found in Aguilar-Gutierrez:2021bns
.

An important aspect in the search for the holographic description of dS space
would be obtaining a dual interpretation of the observables that we studied in
this work. It would be interesting to analyze quantum circuit observables that
display similar behaviors to the ones studied within our work to study whether
the effect of perturbations on the stretched horizon might also have the
interpretation of an epidemic type of growth given the insertion of operators.
Moreover, it would be interesting to see what type of signals can be found in
a UV complete description of the stretched horizon, motivated by the DSSYK
model Susskind:2021esx ; Blommaert:2023opb .

Applications to of CAny to dS space braneworld models were recently carried
out in Aguilar-Gutierrez:2023tic  to gain more information about dS
holography. In these models Randall:1999ee ; Randall:1999vf , one includes an
end-of-the-world brane, whose tension determines the cosmological constant in
the effective gravitational theory on the brane. It would be interesting to
incorporate the switchback effect to characterize perturbations in a double
holographic setting, with a clearer field theory dual. This effective theory
might be further modified by adding intrinsic gravity theories on the brane,
leading to a more intricate holographic complexity evolution Hernandez:2020nem
; ACHKKS ; AL . Moreover, the fluctuations associated with the brane location
lead to an effective description as dS JT gravity on one of the branes
Aguilar-Gutierrez:2023tic . It would be interesting to study the switchback
effect of this effective theory.

On the other hand, as we found in Fig. 2, the CAny observables generically
reach a terminal turning $r_{f}$ (as well as a time-reversal version)
determined by the choice of the CMC slices through (22). This implies that the
CAny proposals in the article do not prove the whole cosmological patch of SdS
spacetime. However, we would expect that any notion of static patch holography
should also encode the degrees of freedom of the inflating region, similar to
investigations in asymptotically AdS space Jorstad:2023kmq .121212We thank
Eivind Jorstad for correspondence about this point. Nevertheless, one can
probe more of the geometry outside the cosmological horizon using the
alternating shockwave geometry. It seems that adding perturbations in the
stretched horizon reveals more information, even when its explicit
localization is irrelevant.

Finally, much progress in the dS holography has been made possible through the
study of interpolating geometries in two-dimensional dilaton-gravity theories
Anninos:2017hhn ; Anninos:2018svg ; Anninos:2020cwo ; Anninos:2022hqo ;
Aguilar-Gutierrez:2023odp ; AESV . While the CV proposal has been extensively
studied in Chapman:2021eyy  for certain interpolating geometries; new members
in this set have recently appeared Anninos:2022hqo , and the CAny proposals
have not been treated yet. This might allow for a clearer interpretation of
the properties studied in this work in the context of dS2 space, appearing
from near extremal limits near horizon limits of black hole geometries
Maldacena:2019cbz ; Castro:2022cuo .

## Acknowledgements

It’s my pleasure to thank Stefano Baiguera, Alex Belin, Rotem Berman, Michal
P. Heller, Eivind Jørstad, Edward K. Morvan-Benhaim, Juan F. Pedraza, Andrew
Svesko, Silke Van der Schueren, and Nicolò Zenoni for valuable discussions. I
also thank the University of Amsterdam, and the Delta Institute for
Theoretical Physics for their hospitality and support during various phases of
this project; and to the organizers of the Modave summer school, where part of
the project was completed. The work of SEAG is partially supported by the FWO
Research Project G0H9318N and the inter-university project iBOF/21/084.

##  Appendix A Details on the CAny evaluation

For the choice

$\sqrt{-f(r)\dot{v}^{2}+2\dot{v}\dot{r}}=r^{d-1}\leavevmode\nobreak\ ,$ (44)

the Euler-Lagrange equations corresponding to (18) can be expressed as

$\dot{r}^{2}+\mathcal{U}(P_{v}^{\epsilon},\,r)=0\leavevmode\nobreak\ ,$ (45)

where

$\displaystyle P_{v}^{\epsilon}$
$\displaystyle\equiv\partialderivative{\mathcal{L}_{\epsilon}}{\dot{v}}=\dot{r}-\dot{v}\,f(r)-\epsilon\frac{K_{\epsilon}}{d}\,r^{d}\leavevmode\nobreak\
;$ (46) $\displaystyle\mathcal{U}(P_{v}^{\epsilon},\,r)$
$\displaystyle\equiv-f(r)r^{2(d-1)}-\left(P_{v}^{\epsilon}+\epsilon\frac{K_{\epsilon}}{d}r^{d}\right)^{2}\leavevmode\nobreak\
.$ (47)

Then, we can express (17) as

$\mathcal{C}^{\epsilon}=\tfrac{2\Omega_{d-1}}{G_{N}}\int_{r_{\rm
st}}^{r_{t}}\tfrac{r^{2(d-1)}a(r)}{\sqrt{-\mathcal{U}(P_{v}^{\epsilon},\,r)}}\mathrm{d}r\leavevmode\nobreak\
.$ (48)

In a similar way, the parameter $t$ can be expressed as

$\displaystyle t$
$\displaystyle=\int_{\Sigma_{\epsilon}}\mathrm{d}r\tfrac{\dot{t}}{\dot{r}}=\int_{\Sigma_{\epsilon}}\mathrm{d}r\tfrac{\dot{v}-\dot{r}/{f(r)}}{\sqrt{-\mathcal{U}(P_{v}^{\epsilon},\,r)}}$
(49) $\displaystyle=-2\int_{r_{\rm
st}}^{r_{t}}\tfrac{\mathrm{d}r}{f(r)\,\sqrt{-\mathcal{U}(P_{v}^{\epsilon},r)}}\left(P_{v}^{\epsilon}+\tfrac{\epsilon
K_{\epsilon}}{d}r^{d}\right)\leavevmode\nobreak\ .$

We proceed to evaluate (48) with (49) carefully. Since
$U(P_{v}^{\epsilon},\,r_{t})=0$ by definition, we need to take care of the
denominator in (48, 49) at each of the turning points. We do so by adding a
subtracting a term:

$\displaystyle{\mathcal{C}^{\epsilon}}=$
$\displaystyle-\tfrac{2\Omega_{d-1}}{G_{N}}a(r_{t})\sqrt{-f(r_{t})r_{t}^{2(d-1)}}\int_{r_{\rm
st}}^{r_{t}}\tfrac{\left(P_{v}^{\epsilon}+\frac{\epsilon
K_{\epsilon}}{d}r^{d}\right)\mathrm{d}r}{f(r)\sqrt{-\mathcal{U}(P_{v}^{\epsilon},r)}}$
(50) $\displaystyle+\tfrac{2\Omega_{d-1}}{G_{N}}\int_{r_{\rm
st}}^{r_{t}}\tfrac{a(r)f(r)\,r^{2(d-1)}+a(r_{t})\sqrt{-f(r_{t})r_{t}^{2(d-1)}}\left(P_{v}^{\epsilon}+\frac{\epsilon
K_{\epsilon}}{d}r^{d}\right)}{f(r)\sqrt{-\mathcal{U}(P_{v}^{\epsilon},\,r)}}\leavevmode\nobreak\
.$

Then, we can identify the relationship between time in (49) and complexity in
(50)

$\displaystyle{\mathcal{C}^{\epsilon}}=$
$\displaystyle\tfrac{\Omega_{d-1}}{G_{N}}a(r_{t})\sqrt{-f(r_{t})r_{t}^{2(d-1)}}t$
(51) $\displaystyle+\tfrac{2\Omega_{d-1}}{G_{N}}\int_{r_{\rm
st}}^{r_{t}}\tfrac{a(r)f(r)\,r^{2(d-1)}+a(r_{t})\sqrt{-f(r_{t})r_{t}^{2(d-1)}}\left(P_{v}^{\epsilon}+\frac{\epsilon
K_{\epsilon}}{d}r^{d}\right)}{f(r)\sqrt{-\mathcal{U}(P_{v}^{\epsilon},\,r)}}\leavevmode\nobreak\
.$

We can then straightforwardly take the time derivative,

$\displaystyle{\derivative{\mathcal{C}^{\epsilon}}{t}}=$
$\displaystyle\tfrac{\Omega_{d-1}}{G_{N}}a(r_{t})\sqrt{-f(r_{t})r_{t}^{2(d-1)}}$
(52)
$\displaystyle+\frac{2\Omega_{d-1}}{G_{N}}\derivative{P_{v}^{\epsilon}}{t}\int_{r_{\rm
st}}^{r_{t}}\mathrm{d}r\tfrac{r^{2(d-1)}\left(a(r_{t})\sqrt{-f(r_{t})r_{t}^{2(d-1)}}-a(r)\quantity(P_{v}^{\epsilon}+\tfrac{\epsilon
K_{\epsilon}}{d}r^{d})\right)}{\quantity(-\mathcal{U}(P_{v}^{\epsilon},\,r))^{3/2}}\leavevmode\nobreak\
.$

Thus, in the $t\rightarrow\infty$ regime we then recover (20). Moreover, the
$\derivative{P_{v}^{\epsilon}}{t}$ term vanishes (20) provided that the
effective potential reaches a maximum Jorstad:2023kmq , shown in (21).

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# Intergenerational gauged $B-L$ model and its implication to muon $g-2$
anomaly and thermal dark matter

Nobuchika Okada<EMAIL_ADDRESS>Department of Physics and Astronomy, University
of Alabama, Tuscaloosa, Alabama 35487, USA  Osamu Seto
<EMAIL_ADDRESS>Department of Physics, Hokkaido University,
Sapporo 060-0810, Japan

###### Abstract

We study the flavor dependent $U(1)_{B_{i}-L_{j}}$ models, where an $i$th
generation of quarks and $j(\neq i)$th generation of leptons are charged. By
solving the anomaly free condition for the matter sector of the SM fermions
and three generations of RH neutrinos, we find that the $j$th generation of
right-handed (RH) neutrino is not necessarily charged under the
$U(1)_{B_{i}-L_{j}}$ gauge symmetry with the charge $-1$ and the other
(neither $i$th nor $j$th) generation of RH neutrino can also be. As a general
solution for the anomaly cancellation conditions, the other two RN neutrinos
than the charge $-1$ RH neutrino may have nonvanishing charge and be stable
due to the gauge invariance, and hence it is a candidate for dark matter (DM)
in our Universe. We apply this result to a $B_{3}-L_{2}$ model and consider a
light thermal DM and a solution to the muon $g-2$ anomaly. We identify the
parameter region to have the DM mass range from MeV to sub-GeV and
simultaneously solve the muon $g-2$ anomaly. We also derive the constraints on
the gauge kinetic mixing parameter by using the latest Borexino phase-II data.

††preprint: EPHOU-23-015

##  I Introduction

Introduction of an extra $U(1)$ gauge interaction is one of the promising and
well-defined extensions of the standard model (SM) of particle physics. The
$B-L$ (baryon number minus lepton number) appears to be an accidental global
symmetry in the SM, indicating that this might be a gauge symmetry in a
ultraviolet (UV) completion of the theory Pati:1973uk ; Davidson:1978pm ;
Mohapatra:1980qe ; Mohapatra:1980 . For such the extended gauge group
$G=SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}\times U(1)_{B-L}$, the
cancellation condition for gauge and mixed gauge-gravitational anomaly
requires that the number of right-handed (RH) neutrinos are three as other SM
fermions. We note that the anomaly cancellation of $U(1)_{B-L}$ gauge symmetry
can be realized for each generation of fermions. Thus, even if the gauge
charge are generation (flavor) dependent, theories are anomaly free. A simple
example is the so-called $(B-L)_{3}$ model Babu:2017olk ; Alonso:2017uky ;
Bian:2017rpg ; Cox:2017rgn ; Elahi:2019drj ; Okada:2019sbb  in which only the
third generation are charged, and experimental bounds with the third
generation fermions are relatively weak compared to those with the first and
the second generation fermions.

Anyway, once we abandon the flavor universality of gauge interactions, we can
consider various extra gauged $U(1)_{X}$ models with $X$ being a corresponding
quantum number that fulfills the anomaly free conditions. It is even possible
for the anomaly to be canceled with only leptons. The total anomalies are
canceled between generations in such a leptophilic gauge interaction based on
the $U(1)_{L_{i}-Lj}$ gauge symmetry with $i$ and $j(\neq i)$ are generation
indices, and among them the $U(1)_{L_{\mu}-L_{\tau}}$ gauge symmetry He:1990pn
; Foot:1990mn  has received particular attention because it can reconcile the
discrepancy of the muon anomalous magnetic moment $g_{\mu}$ (muon $g-2$)
between the SM prediction and experimental results Baek:2001kca ; Ma:2001md .
$B-3L_{i}$ Chun:2018ibr ; Greljo:2021xmg ; Barman:2019aku ; Wang:2019byi ;
Barman:2021yaz  corresponds to quark flavor universality, but lepton flavor
dependent charge assignment. Other examples include $B_{3}-L_{1}$ or
$B_{3}+(L_{1}-L_{2}-L_{3})$ Crivellin:2015lwa ; Bonilla:2017lsq ; Ko:2017quv ,
where the total anomaly is also canceled between generations.

In this paper, we study the flavor dependent $U(1)_{B_{i}-L_{j}}$ models where
an $i$th generation of quarks and $j(\neq i)$th generation of leptons are
charged under $U(1)_{B-L}$ Chun:2018ibr . A $B_{3}-L_{2}$ model had been
studied Allanach:2020kss ; Allanach:2022blr ; Ban:2021tos  in the context of
the so-called $\mathcal{R}(K),\mathcal{R}(K^{*})$ anomaly, but the most recent
analysis has shown that the experimental results are consistent with the SM
predictions LHCb:2022qnv . We examine the anomaly free condition for the
matter sector of the SM fermions and three generations of RH neutrinos. We
point out that the $j$th generation of RH neutrino is not necessarily charged
under the $U(1)_{B_{i}-L_{j}}$ gauge symmetry with the charge $-1$ as been
assigned in Refs. Chun:2018ibr ; Allanach:2020kss  and the other (neither
$i$th nor $j$th) generation of RH neutrino can also be. As a general solution
for the anomaly cancellation conditions, the other two RH neutrinos than the
charge $-1$ may have nonvanishing charge. In such cases, the usual neutrino
Dirac mass term between left-handed (LH) and RH neutrinos cannot be formed due
to the gauge invariance, while the two RH neutrinos can form a Dirac fermion.
This Dirac fermion is stable due to the gauge invariance unless another
$U(1)_{B_{i}-L_{j}}$ charged Higgs doublets are introduced, thus it is a
candidate for dark matter (DM) in our Universe. After general discussion, we
focus on $B_{3}-L_{2}$ model because it could solve the the muon $g-2$ problem
Aoyama:2020ynm . We identify the parameter region to solve the muon $g-2$
problem and to realize viable light thermal weakly interacting massive
particle (WIMP) with the mass in the range of MeV to sub-GeV. Since the
mediator $Z^{\prime}$ boson does not couple light quarks and the DM mass is
small, this Dirac DM is free from the constraints of direct DM search
experiments.

This paper is organized as follows. In the next section, we examine the
anomaly free condition without introducing extra fermions except three RH
neutrinos in $U(1)_{B_{i}-L_{j}}$ gauge symmetry. After Sec. III, we focus on
$U(1)_{B_{3}-L_{2}}$ model. We provide the formula for the muon $g-2$ in Sec.
III, and show the favored parameter region of light thermal WIMP and to solve
the muon $g-2$ problem at the same time in Sec. IV. In Sec. V, we derive the
constraints on the gauge kinetic mixing by examining the electron neutrino
scattering experiments. Sec. VI is devoted to summary.

##  II $B_{i}-L_{j}$ Model

As a variant of $U(1)_{(B-L)_{i}}$ gauge symmetry, we consider cases where
different generations of quarks and leptons are charged under the extra
$U(1)_{X}$ gauge symmetry. The total gauge symmetry is based on the gauge
group $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}\times U(1)_{B_{i}-L_{j}}$
Chun:2018ibr . Anomaly free conditions can be fulfilled without introducing
new fermions besides three RH neutrinos. By solving the set of anomaly
cancellation conditions, we find two different solutions with two free real
parameters $x_{H}$ and $x_{N}$. The $U(1)_{B_{3}-L_{2}}$ charge assignment is
shown in Tables 1 and 2. Here, we have two cases: one is that the second
generation of RH neutrino has the charge $-1$ as in Table 1, the other is that
the first of RH neutrino has the charge $-1$ as in Tabble 2. Somewhat
nontrivial fact is the absence of the case that $i$th generation of RH
neutrino $\nu_{R}^{i}$ has the charge $-1$. In both cases, the remaining other
two RH neutrinos may have nonvanishing opposite charge $x_{N}$ and $-x_{N}$
each other111The similar solution had been found in a $B_{3}-3L_{2}$ model
too, however only the $x_{N}=0$ case was investigated Bonilla:2017lsq .. This
opposite charge offers the possibility that this pair may compose a Dirac
fermion, that we consider in the rest of this paper. Another parameter $x_{H}$
denotes the mixing between $U(1)_{B_{i}-L_{j}}$ and $U(1)_{Y}$ as the extra
$U(1)$ gauge symmetry can be a linear combination of those two gauge
symmetries Appelquist:2002mw ; Oda:2015gna ; Das:2016zue . Since our main
interest is $x_{N}$ in this paper, from now on we take $x_{H}=0$, for
simplicity.

Field and representation under |  | Generation |
---|---|---|---
$(SU(3)_{C},SU(2)_{L},U(1)_{Y})$ | 1 | 2 | 3
$Q^{k}{}_{(\mathbf{3},\mathbf{2},1/6)}$ | $0$ | $0$ | $\frac{1}{6}x_{H}+\frac{1}{3}$
$u^{k}_{R}{}_{(\mathbf{3},\mathbf{1},2/3)}$ | $0$ | $0$ | $\frac{2}{3}x_{H}+\frac{1}{3}$
$d^{k}_{R}{}_{(\mathbf{3},\mathbf{1},-1/3)}$ | $0$ | $0$ | $-\frac{1}{3}x_{H}+\frac{1}{3}$
$L^{k}{}_{(\mathbf{1},\mathbf{2},-1/2)}$ | $0$ | $-\frac{1}{2}x_{H}-1$ | $0$
$e^{k}_{R}{}_{(\mathbf{1},\mathbf{1},-1)}$ | $0$ | $-x_{H}-1$ | $0$
$\nu^{k}_{R}{}_{(\mathbf{1},\mathbf{1},0)}$ | $-x_{N}$ | $-1$ | $x_{N}$
Table 1:  An anomaly free $U(1)_{B_{3}-L_{2}}$ charge assignment for SM particles and RH neutrinos.  Field and representation under |  | Generation |
---|---|---|---
$(SU(3)_{C},SU(2)_{L},U(1)_{Y})$ | 1 | 2 | 3
$Q^{k}{}_{(\mathbf{3},\mathbf{2},1/6)}$ | $0$ | $0$ | $\frac{1}{6}x_{H}+\frac{1}{3}$
$u^{k}_{R}{}_{(\mathbf{3},\mathbf{1},2/3)}$ | $0$ | $0$ | $\frac{2}{3}x_{H}+\frac{1}{3}$
$d^{k}_{R}{}_{(\mathbf{3},\mathbf{1},-1/3)}$ | $0$ | $0$ | $-\frac{1}{3}x_{H}+\frac{1}{3}$
$L^{k}{}_{(\mathbf{1},\mathbf{2},-1/2)}$ | $0$ | $-\frac{1}{2}x_{H}-1$ | $0$
$e^{k}_{R}{}_{(\mathbf{1},\mathbf{1},-1)}$ | $0$ | $-x_{H}-1$ | $0$
$\nu^{i}_{R}{}_{(\mathbf{1},\mathbf{1},0)}$ | $-1$ | $-x_{N}$ | $x_{N}$
Table 2:  An anomaly free $U(1)_{B_{3}-L_{2}}$ charge assignment for SM
particles and RH neutrinos.

So far, we have solved the anomaly cancellation conditions of the gauge group
$SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}\times U(1)_{B_{i}-L_{j}}$ with the
particle content of the SM fermions plus three RH neutrinos. On the other
hand, the solution we have obtained is in fact same as a model constructed by
the gauged $B-L$ extension to only one generation as $U(1)_{B_{i}-L_{j}}$,
where one RH neutrino is introduced for the anomaly cancellation while the
other two RH neutrinos are additionally introduced not necessary for the
anomaly cancellation. Here, the $x_{N}=0$ case corresponds to introducing
Majorana fermions, the $x_{N}\neq 0$ case does to introducing one vectorlike
fermion.

###  II.1 Gauge sector

The gauge kinetic terms of our model are

$\displaystyle\mathcal{L}_{\mathrm{gauge}}=$
$\displaystyle-\frac{1}{4}G_{\mu\nu}G^{\mu\nu}-\frac{1}{4}\hat{W}_{\mu\nu}\hat{W}^{\mu\nu}-\frac{1}{4}\hat{B}_{\mu\nu}\hat{B}^{\mu\nu}-\frac{1}{4}\hat{X}_{\mu\nu}\hat{X}^{\mu\nu}+\frac{\sin\epsilon}{2}\hat{B}_{\mu\nu}\hat{X}^{\mu\nu},$
(1)

where fields with the hat stand for those in gauge eigenstate and $\epsilon$
is the gauge kinetic mixing parameter. Here and hereafter, we use the symbol
$X$ as the extra gauge charge and field, for simplicity. We assume that the SM
Higgs field $\Phi$ is singlet under the $U(1)_{X}$ and the mass of $\hat{X}$
gauge boson $M_{\hat{X}}$ is generated by another scalar field. At the
electroweak (EW) breaking vacuum, we have

$\displaystyle\hat{A}_{\mu}$
$\displaystyle=s_{W}\hat{W}^{3}_{\mu}+c_{W}\hat{B}_{\mu},$ (2)
$\displaystyle\hat{Z}_{\mu}$
$\displaystyle=c_{W}\hat{W}^{3}_{\mu}-s_{W}\hat{B}_{\mu},$ (3)

with $s_{W}=\sin\theta_{W}$ and $c_{W}=\cos\theta_{W}$, where $\theta_{W}$ is
the Weinberg angle. The field redefinition by an orthogonal matrix,

$U_{K}=\left(\begin{array}[]{ccc}1&0&t_{\epsilon}c_{W}\\\
0&1&-t_{\epsilon}s_{W}\\\ 0&0&\frac{1}{c_{\epsilon}}\\\ \end{array}\right),$
(4)

resolves the kinetic mixing but induces the mass mixing :

$M^{2}=\left(\begin{array}[]{ccc}0&0&0\\\
0&M_{\hat{Z}}^{2}&-t_{\epsilon}s_{W}M_{\hat{Z}}^{2}\\\
0&-t_{\epsilon}s_{W}M_{\hat{Z}}^{2}&\frac{1}{c_{\epsilon}^{2}}M_{\hat{X}}^{2}+\left(t_{\epsilon}s_{W}\right)^{2}M_{\hat{Z}}^{2}\\\
\end{array}\right),$ (5)

with $M_{\hat{Z}}^{2}=\frac{1}{4}(g_{1}^{2}+g_{2}^{2})v^{2}$,
$c_{\epsilon}=\cos\epsilon$, $t_{\epsilon}=\tan\epsilon$, and
$M_{\hat{Z}}^{2}$ being the mass generated by the vacuum expectation value
(VEV) of the extra $U(1)_{B_{i}-L_{j}}$ breaking scalar field. The additional
field redefinition to the mass eigenstates can be done with a rotation matrix

$U_{M}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&\cos\theta&-\sin\theta\\\
0&\sin\theta&\cos\theta\\\ \end{array}\right),$ (6)

with the angle222The sign of $\theta$ is opposite to that in Ref. Cho:2020mnc
.

$\tan
2\theta=\frac{-2t_{\epsilon}s_{W}M_{\hat{Z}}^{2}}{M_{\hat{Z}}^{2}-\left(t_{\epsilon}s_{W}\right)^{2}M_{\hat{Z}}^{2}-\frac{1}{c_{\epsilon}^{2}}M_{\hat{X}}^{2}}.$
(7)

The mass eigenvalues are given by

$\displaystyle m_{Z}^{2}$ $\displaystyle=M_{\hat{Z}}^{2}\cos{}^{2}\theta-
t_{\epsilon}s_{W}M_{\hat{Z}}^{2}\sin(2\theta)+\left(\frac{1}{c_{\epsilon}^{2}}M_{\hat{X}}^{2}+\left(t_{\epsilon}s_{W}\right)^{2}M_{\hat{Z}}^{2}\right)\sin{}^{2}\theta,$
(8) $\displaystyle m_{Z^{\prime}}^{2}$
$\displaystyle=\left(\frac{1}{c_{\epsilon}^{2}}M_{\hat{X}}^{2}+\left(t_{\epsilon}s_{W}\right)^{2}M_{\hat{Z}}^{2}\right)\cos{}^{2}\theta+M_{\hat{Z}}^{2}\sin{}^{2}\theta+t_{\epsilon}s_{W}M_{\hat{Z}}^{2}\sin(2\theta).$
(9)

Since the hatted field and the unhatted field are related as

$\displaystyle\left(\begin{array}[]{c}\hat{A}_{\mu}\\\ \hat{Z}_{\mu}\\\
\hat{X}_{\mu}\\\
\end{array}\right)=U_{K}U_{M}\left(\begin{array}[]{c}A_{\mu}\\\ Z_{\mu}\\\
Z^{\prime}_{\mu}\\\
\end{array}\right)=\left(\begin{array}[]{ccc}1&c_{W}s_{\theta}t_{\epsilon}&c_{\theta}c_{W}t_{\epsilon}\\\
0&c_{\theta}-s_{\theta}s_{W}t_{\epsilon}&-c_{\theta}s_{W}t_{\epsilon}-s_{\theta}\\\
0&\frac{s_{\theta}}{c_{\epsilon}}&\frac{c_{\theta}}{c_{\epsilon}}\\\
\end{array}\right)\left(\begin{array}[]{c}A_{\mu}\\\ Z_{\mu}\\\
Z^{\prime}_{\mu}\\\ \end{array}\right),$ (22)

by combining with

$\displaystyle\left(\begin{array}[]{c}\hat{W}^{3}_{\mu}\\\ \hat{B}_{\mu}\\\
\hat{X}_{\mu}\\\ \end{array}\right)$
$\displaystyle=\left(\begin{array}[]{ccc}s_{W}&c_{W}&0\\\ c_{W}&-s_{W}&0\\\
0&0&1\\\ \end{array}\right)\left(\begin{array}[]{c}\hat{A}_{\mu}\\\
\hat{Z}_{\mu}\\\ \hat{X}_{\mu}\\\ \end{array}\right),$ (32)

we find

$\displaystyle\left(\begin{array}[]{c}\hat{W}^{3}_{\mu}\\\ \hat{B}_{\mu}\\\
\hat{X}_{\mu}\\\
\end{array}\right)=\left(\begin{array}[]{ccc}s_{W}&c_{W}c_{\theta}&-c_{W}s_{\theta}\\\
c_{W}&-s_{W}c_{\theta}+t_{\epsilon}s_{\theta}&s_{W}s_{\theta}+t_{\epsilon}c_{\theta}\\\
0&\frac{1}{c_{\epsilon}}s_{\theta}&\frac{1}{c_{\epsilon}}c_{\theta}\\\
\end{array}\right)\left(\begin{array}[]{c}A_{\mu}\\\ Z_{\mu}\\\
Z^{\prime}_{\mu}\\\ \end{array}\right).$ (42)

###  II.2 Fermion masses and Higgs sector

In a “flavored” $B-L$ symmetry as in $U(1)_{(B-L)_{3}}$, due to
$U(1)_{B_{i}-L_{j}}$ gauge symmetry, the $U(1)_{B_{i}-L_{j}}$ singlet SM Higgs
field $\Phi$ cannot give the masses of the $U(1)_{B_{i}-L_{j}}$ charged
fermions and thus realistic fermion flavor mixings cannot be reproduced.

####  II.2.1 Quark mass and mixing

To reproduce the realistic quark mass matrices, a few successful UV
completions for $U(1)_{(B-L)_{3}}$ have been proposed: One is an extension of
Higgs sector by Babu et al. in Ref. Babu:2017olk  and another is introduction
of heavy vectorlike fermions with additional scalars by Alonso et al. in Ref.
Alonso:2017uky . The same mechanism work for at least $U(1)_{B_{3}-L_{j}}$.
Since the details of those UV completions are irrelevant for the following
discussion, we will not consider a specific model further.

####  II.2.2 Lepton mass and mixing

The charged lepton masses can be generated by the SM Higgs field, if
$x_{H}=0$. On the other hand, if $x_{H}\neq 0$ , we need to introduce the
second Higgs doublet with $(\mathbf{1},\mathbf{2},-1/2,x_{H}/2)$ to generate
all charged lepton masses. The generation of neutrino mass also depends on the
Higgs sector. Here we consider only the $x_{H}=0$ case for simplicity.

The $x_{N}=0$ case is simple. Since two RH neutrinos are singlet under any
gauge group in this case, the type-I seesaw mechanism Minkowski:1977sc ;
Yanagida:1979as ; GellMann:1980vs ; Mohapatra:1979ia  works through neutrino
Yukawa coupling with the SM Higgs field $\Phi$ and their Majorana mass.

For $x_{N}\neq 0$ , on the other hand, we need to extend the model in order to
generate observed neutrino masses. The simplest extension would be introducing
$SU(2)$ triplet Higgs fields $\Delta$ Schechter:1980gr ; Magg:1980ut ;
Cheng:1980qt , since the Dirac neutrino mass between the extra $U(1)$ charged
RH neutrinos and the extra $U(1)$ uncharged LH neutrinos are not necessary.
The Yukawa couplings are given by

$\displaystyle\mathcal{L}_{\mathrm{Yukawa}}\supset$
$\displaystyle+\sum_{k,l}\left(-\frac{1}{\sqrt{2}}y^{\Delta_{0}}_{kl}\overline{L^{k~{}C}}\cdot\Delta_{0}L^{l}-\frac{1}{\sqrt{2}}y^{\Delta_{1}}_{kj}\overline{L^{k~{}C}}\cdot\Delta_{1}L^{j}\right)$
$\displaystyle-\sum_{k,l}y^{e}_{kl}\overline{L^{k}}\Phi
e_{R}^{l}-y^{D}_{j{j^{\prime}}}\overline{L^{j}}\widetilde{\Phi}\nu_{R}^{j^{\prime}}-\frac{1}{2}y^{\nu_{R}}_{j^{\prime}}\overline{\nu_{R}^{j^{\prime}~{}C}}\phi_{2}\nu_{R}^{j^{\prime}}+\mathrm{H.c.},$
(43)

where the superscript $C$ denotes the charge conjugation, the dot denotes the
antisymmetric product of $SU(2)$, subscripts of $\Delta$ denote these
$U(1)_{X}$ charges, and $k$ and $l$ run from $1$ to $3$ except the $j$th
generation. $\nu_{R}^{j^{\prime}}$ denotes the RH neutrino with the charge
$-1$. For instance, $j^{\prime}=2$ in the model in Table 1 and $j^{\prime}=1$
in the model in Table 2. $y^{\Delta}$ Yukawa matrices have entries as

$\displaystyle
y^{\Delta_{0}}=\left(\begin{array}[]{ccc}y^{\Delta_{0}}_{11}&0&y^{\Delta_{0}}_{13}\\\
0&0&0\\\ y^{\Delta_{0}}_{31}&0&y^{\Delta_{0}}_{33}\\\
\end{array}\right),\qquad
y^{\Delta_{1}}=\left(\begin{array}[]{ccc}0&y^{\Delta_{1}}_{12}&0\\\
0&y^{\Delta_{1}}_{22}&0\\\ 0&y^{\Delta_{1}}_{32}&0\\\ \end{array}\right),$
(50)

for instance for $X=B_{3}-L_{2}$. We note that we may replace $\Delta_{1}$
with $\Delta_{-1}$. Then, the resultant neutrino mass is expressed as

$\displaystyle
m_{\nu}=-\frac{(y^{D}_{j{j^{\prime}}})^{2}v^{2}}{M_{\nu_{R}^{j^{\prime}}}}+y^{\Delta_{0}}v_{\Delta}^{0}+y^{\Delta_{1}}v_{\Delta}^{1}$
(51)

where $M_{\nu_{R}^{j^{\prime}}}=y^{\nu_{R}}_{j^{\prime}}v_{2}/\sqrt{2}$, and
$v_{\Delta}^{0}(v_{\Delta}^{1})$ are the Majorana mass of
$\nu_{R}^{j^{\prime}}$, and the VEV $\Delta_{0}(\Delta_{1})$, respectively.
The first term represents the neutrino mass generated with $j^{\prime}$th RH
neutrino by type-I seesaw mechanism, but only one component is generated.
Thus, the mainly triplet Higgs fields have to generate the neutrino mass as
above. The charge of Higgs fields are summarized in Table 3.

| SU(3)c | SU(2)L | U(1)Y | U(1)${}_{B_{i}-L_{j}}$
---|---|---|---|---
$\Phi$ | 1 | 2 | $1/2$ | $0$
$\Delta_{0}$ | 1 | 3 | $1$ | $0$
$\Delta_{1}$ | 1 | 3 | $1$ | $1$
$\phi_{2}$ | 1 | 1 | $0$ | $+2$
Table 3: The minimal Higgs sector.

In the rest of this paper, we assume that neutrino masses are generated by
mostly the type-II seesaw mechanism as mentioned above. Here we comments on
another possibility with type-I seesaw mechanism with RH neutrinos, instead of
type-II seesaw. To form Dirac neutrino masses and generate Majorana masses, we
need to introduce other Higgs doublets with the charge $({\bf 1},{\bf
2},1/2,\pm x_{N})$ and another $U(1)_{X}$ charged scalar $\phi_{2x_{N}}$ with
the charge $({\bf 1},{\bf 1},0,2x_{N})$. In this case, the scalar spectrum
includes physical Nambu-Goldstone modes, for example, charged massless
scalars, whose existence is excluded by experiments. Hence, we need further
extensions. As we have seen, unless we introduce additional RH neutrinos with
a vanishing $U(1)$ charge, we need complicated extensions of Higgs sector to
avoid unwanted Nambu-Goldstone modes.

##  III Anomalous magnetic moment of muon

###  III.1 $Z^{\prime}$ boson contribution to muon $g-2$

As in the $L_{\mu}-L_{\tau}$ model Baek:2001kca , this $B_{3}-L_{2}$ model can
also reconcile the muon $g-2$ problem through the $Z^{\prime}$ boson loop
contribution, since the muon is charged under the extra $U(1)$. By comparing
the SM prediction with the latest experimental result Muong-2:2021ojo , the
discrepancy is given as

$\Delta a_{\mu}=(25.1\pm 5.9)\times 10^{-10},$ (52)

with $a_{\mu}\equiv(g_{\mu}-2)/2$. The new contribution from $Z^{\prime}$
boson loop corrections is estimated as Baek:2001kca

$\displaystyle
a_{\mu}=\frac{g_{X}^{2}}{4\pi}\int_{0}^{1}dx\frac{2m_{\mu}^{2}x^{2}(1-x)}{x^{2}m_{\mu}^{2}+(1-x)m_{Z^{\prime}}^{2}},$
(53)

with

$\displaystyle m_{Z^{\prime}}^{2}=\frac{1}{2}g_{X}^{2}v_{S}^{2},$ (54)

where we have omitted a negligible kinetic mixing contribution to
$m_{Z^{\prime}}^{2}$.

###  III.2 Constraints from neutrino trident processes

As in the $L_{\mu}-L_{\tau}$ model, the constraint from neutrino trident
processes by the CCFR experiment CCFR:1991lpl  is imposed on this model. The
coupling to the muon in our model is same as that in the usual
$U(1)_{L_{\mu}-L_{\tau}}$ model, we quote the bound from Ref.
Altmannshofer:2019zhy .

##  IV Dirac right-handed neutrino dark matter

A remarkable result of the anomaly cancellation condition we found for the
$U(1)_{B_{i}-L_{j}}$ model is that non-$j$ generations of RH neutrinos can
have nonvanishing opposite charges $x_{N}$ and $-x_{N}$. From now on, for
concreteness, we consider $U(1)_{B_{3}-L_{2}}$ models listed in Tables 1 and
2. Those two RH neutrinos can form the Dirac spinor

$\displaystyle\chi=\left(\begin{array}[]{c}\nu_{R}^{3}\\\
\nu_{R}^{i}{}^{*}\end{array}\right),$ (57)

where $\nu_{R}^{i}(i=1,\,\mathrm{or}\,\,2)$ has the $U(1)_{B_{3}-L_{2}}$
charge $-x_{N}$, and may have the Dirac mass

$\displaystyle\mathcal{L}\supset-m\nu_{R}^{i}{}^{T}\epsilon\nu_{R}^{3}+\mathrm{H.c.}=-\overline{\chi}m\chi.$
(58)

The Lagrangian of the $\chi$ part is read as

$\displaystyle\mathcal{L}=\overline{\chi}\left[i\gamma^{\mu}\left(\partial_{\mu}-ix_{N}g_{X}\left(\frac{s_{\theta}}{c_{\epsilon}}Z_{\mu}+\frac{c_{\theta}}{c_{\epsilon}}Z^{\prime}_{\mu}\right)\right)-m\right]\chi.$
(59)

It is worth noting that $\chi$ has no direct coupling with the SM particles
thanks to its $U(1)_{X}$ charge assignment, and this fact guarantees the
stability of $\chi$. The DM candidate $\chi$ does not couple with scalar
particle either since its mass does not come from the VEV of a scalar field.
These properties are in a remarkable contrast with RH neutrino DM in the
minimal $U(1)_{X}$ model with the standard charge assignment, where the extra
$Z_{2}$ parity has to be introduced by hand to stabilize the DM particle and
the scalar exchange processes are important for DM physics Okada:2010wd ;
Okada:2016gsh ; Okada:2016tci ; Seto:2016pks .333For a review on this class of
models, see e.g., Ref. Okada:2018ktp .

For $U(1)_{X}=U(1)_{B_{3}-L_{2}}$ and $x_{H}=0$, we obtain the decay rate of
the $Z^{\prime}$ gauge boson as

$\displaystyle\Gamma_{Z^{\prime}}=$
$\displaystyle\frac{g_{X}^{2}m_{Z^{\prime}}}{72\pi}\left(2\left(1-\frac{4m_{t}^{2}}{m_{Z^{\prime}}^{2}}\right)^{3/2}+2\left(1-\frac{4m_{b}^{2}}{m_{Z^{\prime}}^{2}}\right)^{3/2}+6\left(1-\frac{4m_{\mu}^{2}}{m_{Z^{\prime}}^{2}}\right)^{3/2}+3\right.$
$\displaystyle\left.+6x_{N}^{2}\left(1-\frac{4m_{\chi}^{2}}{m_{Z^{\prime}}^{2}}\right)^{3/2}+3\left(1-\frac{m_{\nu_{R}}^{2}}{m_{Z^{\prime}}^{2}}\right)\sqrt{1-\frac{4m_{\nu_{R}}^{2}}{m_{Z^{\prime}}^{2}}}\right)+\mathcal{O}(\epsilon),$
(60)

where $m_{t},m_{b},m_{\mu}$ and $m_{\nu_{R}}$ are the mass of top, bottom
quarks, muon, one RH neutrino with the charge $-1$, and $m_{\chi}$ is the
Dirac mass for the other two RH neutrinos. The typical decay branching ratio
is shown in Fig. 1.

Figure 1: The decay branching ratio of the $Z^{\prime}$ boson for $x_{N}=1$,
$m_{\nu_{R}}=0.03$ GeV and $m_{\chi}=10$ MeV.

###  IV.1 Dark matter abundance

Figure 2: An example of the fraction of each annihilation modes for $x_{N}=5$, $m_{\nu_{R}}=0.1$ GeV, and $m_{Z^{\prime}}=3\times 10^{-3}$ GeV.  |
---|---
Figure 3: The contour reproducing DM abundance $\Omega_{\chi}h^{2}\simeq 0.1$,
for $m_{Z^{\prime}}=2$ MeV and $m_{\nu_{R}}=0.1$ GeV. Left: the case with
$x_{N}=5(1)$ for the solid (dashed) curve on the $m_{\chi}-g_{X}$ plane. The
gray shaded region is excluded by the neutrino trident constraint
Altmannshofer:2019zhy . Right: the case with $g_{X}=1.8\times 10^{-4}$ on the
$m_{\chi}-x_{N}$ plane.  Figure 4: The contour reproducing DM abundance
$\Omega_{\chi}h^{2}\simeq 0.1$ with the muon $g-2$ favored region shaded with
orange. The gray shaded region is excluded by the neutrino trident constraint
Altmannshofer:2019zhy . We take $m_{\nu_{R}}=0.1$ GeV. The solid (dashed)
curve corresponds to $m_{\chi}=5(10)$ MeV and $x_{N}=5(10)$.

We estimate the thermal relic abundance of the Dirac DM $\chi$ by solving the
Boltzmann equation,

$\frac{dn}{dt}+3Hn=-\langle\sigma v\rangle(n^{2}-n_{\rm EQ}^{2}),$ (61)

where $n$ is the number density of $\chi$, $n_{\rm EQ}$ is its number density
at thermal equilibrium, $\langle\sigma v\rangle$ is the thermal averaged
products of the annihilation cross section and the relative velocity. The
annihilation channels are $\chi\chi$ to $f\bar{f}$ via $s$-channel
$Z^{\prime}$ exchange and $\chi\chi$ to $Z^{\prime}Z^{\prime}$ via
$t(u)$-channel $\chi$ exchange. The later is dominant if the channel is
kinematically open as shown in Fig. 2. The resultant DM relic abundance is
given by

$\displaystyle\Omega_{\chi}h^{2}=\frac{1.1\times
10^{9}x_{d}\mathrm{GeV}^{-1}}{\sqrt{8\pi g_{*}}M_{P}\langle\sigma v\rangle},$
(62)

where $x_{d}=m_{\chi}/T_{d}$ with the decoupling temperature $T_{d}$
Kolb:1990vq .

The contours reproducing the observed DM abundance $\Omega_{\chi}h^{2}\simeq
0.1$ are shown in Fig. 3. We note that the constraints that DM dominantly
annihilating into muons with the mass of the order of GeV is stringently
constrained from indirect DM searches MAGIC:2016xys ; Hambye:2019tjt . Thus,
for $m_{\chi}\gtrsim 100$ MeV, the $Z^{\prime}$ mass must be smaller than the
twice of muon mass so that the $Z^{\prime}$ boson does not decay dominantly
into muons. The orange strip indicates the parameter region that can solve the
muon $g-2$ problem.

In Fig. 4, we overlay the parameter region to explain the discrepancy of the
muon $g-2$ on the DM abundance contours. This shows that, for example, a set
of $x_{N}=5,m_{\chi}=5$ MeV and $m_{Z^{\prime}}\lesssim$ several MeV or the
vicinity of $Z^{\prime}$ resonance pole ($2m_{\chi}\sim m_{Z^{\prime}}$) is
able to simultaneously explain the DM abundance and the muon $g-2$
discrepancy.

##  V Other constraints

###  V.1 Electron neutrino elastic scattering

The electron neutrino elastic scattering is an effective processes to probe a
new interaction Harnik:2012ni ; Bilmis:2015lja ; Lindner:2018kjo ;
Chakraborty:2021apc ; Asai:2023xxl . In our model, if the gauge kinetic mixing
parameter $\epsilon$ is not vanishing, the electron neutrino scattering is
mediated by not only the SM interaction but also new $U(1)_{B_{i}-L_{j}}$
gauge interaction. As pointed out in Ref. Bilmis:2015lja , we note the
importance of the interference between the SM processes and the $Z^{\prime}$
boson process. The relevant part of Lagrangian is

$\displaystyle\mathcal{L}\supset
e\hat{A}_{\mu}J_{\hat{A}}^{\mu}+g_{2}(W^{+}_{\mu}J_{W^{+}}^{\mu}+W^{-}_{\mu}J_{W^{-}}^{\mu}+\hat{Z}_{\mu}J_{\hat{Z}}^{\mu})+g_{X}\hat{X}_{\mu}J_{\hat{X}}^{\mu},$
(63)

with the currents

$\displaystyle J_{\hat{A}}^{\mu}=$
$\displaystyle\sum_{k}\overline{e^{k}}\gamma^{\mu}(-1)e^{k},$ (64)
$\displaystyle J_{W^{+}}^{\mu}=$
$\displaystyle\sum_{k}\frac{1}{\sqrt{2}}\overline{\nu^{k}}\gamma^{\mu}P_{L}e^{k},$
(65) $\displaystyle J_{W^{-}}^{\mu}=$
$\displaystyle\sum_{k}\frac{1}{\sqrt{2}}\overline{e^{k}}\gamma^{\mu}P_{L}\nu^{k},$
(66) $\displaystyle J_{\hat{Z}}^{\mu}=$
$\displaystyle\sum_{k}\frac{1}{c_{W}}\left[\overline{\nu^{k}}\gamma^{\mu}\left(\frac{1}{2}\right)P_{L}\nu^{k}+\overline{e^{k}}\gamma^{\mu}\left(\left(-\frac{1}{2}+s_{W}^{2}\right)P_{L}+s_{W}^{2}P_{R}\right)e^{k}\right],$
(67) $\displaystyle J_{\hat{X}}^{\mu}=$
$\displaystyle\overline{\nu^{j}}\gamma^{\mu}(-1)P_{L}\nu^{j}+\overline{e^{j}}\gamma^{\mu}(-1)e^{j}.$
(68)

This can be recast for the mass eigenstates of gauge bosons as

$\displaystyle\mathcal{L}_{\mathrm{int}}=$
$\displaystyle\frac{g_{2}}{\sqrt{2}}\sum_{k}\left(W^{+}_{\mu}\overline{\nu^{k}}\gamma^{\mu}P_{L}e^{k}+W^{-}_{\mu}\overline{e^{k}}\gamma^{\mu}P_{L}\nu^{k}\right)$
$\displaystyle+e(A+c_{W}s_{\theta}t_{\epsilon}Z+c_{W}c_{\theta}t_{\epsilon}Z^{\prime})\sum_{k}\overline{e^{k}}\gamma^{\mu}(-1)e^{k}$
$\displaystyle+g_{2}((c_{\theta}-s_{\theta}s_{W}t_{\epsilon})Z+(-c_{\theta}s_{W}t_{\epsilon}-s_{\theta})Z^{\prime})\sum_{k}\frac{1}{c_{W}}\left[\overline{e^{k}}\gamma^{\mu}\left(\left(-\frac{1}{2}+s_{W}^{2}\right)P_{L}+s_{W}^{2}P_{R}\right)e^{k}\right]$
$\displaystyle+g_{X}\left(\frac{s_{\theta}}{c_{\epsilon}}Z+\frac{c_{\theta}}{c_{\epsilon}}Z^{\prime}\right)\left[\overline{e^{j}}\gamma^{\mu}(-1)e^{j}+\overline{\nu^{j}}\gamma^{\mu}(-1)P_{L}\nu^{j}\right]$
$\displaystyle+g_{2}((c_{\theta}-s_{\theta}s_{W}t_{\epsilon})Z+(-c_{\theta}s_{W}t_{\epsilon}-s_{\theta})Z^{\prime})\sum_{k}\frac{1}{c_{W}}\left[\overline{\nu^{k}}\gamma^{\mu}\left(\frac{1}{2}\right)P_{L}\nu^{k}\right].$
(69)

The interaction with $Z^{\prime}$ boson is expressed as

$\displaystyle\mathcal{L}_{Z^{\prime}-\mathrm{int}}=$ $\displaystyle
ec_{W}c_{\theta}t_{\epsilon}Z^{\prime}_{\mu}\sum_{k}\overline{e^{k}}\gamma^{\mu}(-1)e^{k}$
$\displaystyle+\sum_{k}Z^{\prime}_{\mu}\overline{\nu^{k}}\gamma^{\mu}\left[g_{X}\frac{c_{\theta}}{c_{\epsilon}}(-1)\delta_{kj}+(-c_{\theta}s_{W}t_{\epsilon}-s_{\theta})\frac{g_{2}}{c_{W}}\left(\frac{1}{2}\right)\right]P_{L}\nu^{k}$
$\displaystyle=$
$\displaystyle\sum_{k}g_{e^{k}Z^{\prime}}Z^{\prime}_{\mu}\overline{e^{k}}\gamma^{\mu}e^{k}+\sum_{k}g_{\nu_{L}^{k}Z^{\prime}}Z^{\prime}_{\mu}\overline{\nu^{k}}\gamma^{\mu}P_{L}\nu^{k}.$
(70)

Here, we have defined the effective coupling $g_{e^{k}Z^{\prime}}$ and
$g_{\nu_{L}^{k}Z^{\prime}}$ as above. The differential cross section of $k$th
flavor (anti)neutrino scattering with electron is given as

$\displaystyle\frac{d\sigma_{\nu_{k}(\overline{\nu_{k}})}}{dE_{r}}=\frac{1}{64\pi
E_{\nu}^{2}m_{e}}\overline{|\mathcal{M}|^{2}},$ (71)

and the explicit forms are calculated to be

$\displaystyle\frac{d\sigma(e\nu_{e}\rightarrow e\nu_{e})}{dE_{r}}$
$\displaystyle=$ $\displaystyle\frac{m_{e}}{4\pi
E_{\nu}^{2}}\left(2G_{F}^{2}\left(E_{\nu}^{2}\left(2s_{W}^{2}+1\right)^{2}+4s_{W}^{4}(E_{r}-E_{\nu})^{2}-2E_{r}m_{e}\left(2s_{W}^{2}+1\right)s_{W}^{2}\right)\right.$
$\displaystyle\left.+\frac{\sqrt{2}G_{F}g_{\nu_{L}^{k}Z^{\prime}}g_{e^{k}Z^{\prime}}\left(2E_{\nu}^{2}+4s_{W}^{2}\left(2E_{\nu}^{2}+E_{r}^{2}-E_{r}(2E_{\nu}+m_{e})\right)-E_{r}m_{e}\right)}{\left(2E_{r}m_{e}+m_{Z^{\prime}}^{2}\right)}\right.$
$\displaystyle\left.+\frac{g_{\nu_{L}^{k}Z^{\prime}}^{2}g_{e^{k}Z^{\prime}}^{2}\left(2E_{\nu}^{2}+E_{r}^{2}-E_{r}(2E_{\nu}+m_{e})\right)}{\left(2E_{r}m_{e}+m_{Z^{\prime}}^{2}\right)^{2}}\right),$
(72) $\displaystyle\frac{d\sigma(e\nu_{\alpha}\rightarrow
e\nu_{\alpha})}{dE_{r}}$ $\displaystyle=$ $\displaystyle\frac{m_{e}}{4\pi
E_{\nu}^{2}}\left(2G_{F}^{2}\left(E_{\nu}^{2}\left(2s_{W}^{2}-1\right)^{2}+4s_{W}^{4}\left(E_{r}-E_{\nu}\right)^{2}+2E_{r}m_{e}\left(2s_{W}^{2}-1\right)s_{W}^{2}\right)\right.$
$\displaystyle+\frac{\sqrt{2}G_{F}g_{\nu_{L}^{k}Z^{\prime}}g_{e^{k}Z^{\prime}}\left(-2E_{\nu}^{2}+4s_{W}^{2}\left(2E_{\nu}^{2}+E_{r}^{2}-E_{r}(2E_{\nu}+m_{e})\right)+E_{r}m_{e}\right)}{2E_{r}m_{e}+m_{Z^{\prime}}^{2}}$
$\displaystyle\left.+\frac{g_{\nu_{L}^{k}Z^{\prime}}^{2}g_{e^{k}Z^{\prime}}^{2}\left(2E_{\nu}^{2}+E_{r}^{2}-E_{r}(2E_{\nu}+m_{e})\right)}{\left(2E_{r}m_{e}+m_{Z^{\prime}}^{2}\right)^{2}}\right),$
(73) $\displaystyle\frac{d\sigma(e\overline{\nu_{\alpha}}\rightarrow
e\overline{\nu_{\alpha}})}{dE_{r}}$ $\displaystyle=$
$\displaystyle\frac{m_{e}}{4\pi
E_{\nu}^{2}}\left(2G_{F}^{2}\left(4E_{\nu}^{2}s_{W}^{4}+\left(1-2s_{W}^{2}\right)^{2}(E_{r}-E_{\nu})^{2}+2E_{r}m_{e}\left(1-2s_{W}^{2}\right)s_{W}^{2}\right)\right.$
$\displaystyle+\frac{\sqrt{2}G_{F}g_{\nu_{L}^{k}Z^{\prime}}g_{e^{k}Z^{\prime}}\left(-2(E_{\nu}-E_{r})^{2}+4s_{W}^{2}\left(2E_{\nu}^{2}+E_{r}^{2}-E_{r}(2E_{\nu}+m_{e})\right)+E_{r}m_{e}\right)}{2E_{r}m_{e}+m_{Z^{\prime}}^{2}}$
$\displaystyle\left.+\frac{g_{\nu_{L}^{k}Z^{\prime}}^{2}g_{e^{k}Z^{\prime}}^{2}\left(2E_{\nu}^{2}+E_{r}^{2}-E_{r}(2E_{\nu}+m_{e})\right)}{\left(2E_{r}m_{e}+m_{Z^{\prime}}^{2}\right)^{2}}\right),$
(74)

where $\alpha=\mu$ or $\tau$.

####  V.1.1 Borexino constraints

The differential event rate with respect to the recoil energy $E_{r}$ at the
Borexino detector Kumaran:2021lvv  is given by

$\displaystyle\frac{dR}{dE_{r}}=N_{T}\int_{E_{\nu}^{\mathrm{min}}}^{\infty}\frac{d\Phi}{dE_{\nu}}\left(P_{ee}\frac{d\sigma_{\nu_{e}}}{dE_{r}}+P_{e\mu}\frac{d\sigma_{\nu_{\mu}}}{dE_{r}}+P_{e\tau}\frac{d\sigma_{\nu_{\tau}}}{dE_{r}}\right)dE_{\nu},$
(75)

where $\Phi$ is the flux of solar neutrino, $E_{\nu}$ is the incoming neutrino
energy,

$\displaystyle
E_{\nu}^{\mathrm{min}}=\frac{1}{2}\left(E_{r}+\sqrt{E_{r}^{2}+2E_{r}m_{T}}\right),$
(76)

is the minimal neutrino energy to generate the recoil energy $E_{r}$ by
collision with the target with the mass $m_{T}$, $N_{T}$ is the number of
target particles, and neutrino oscillation effects are taken into account by
multiplying the oscillation probability $P_{ij}\equiv
P(\nu_{i}\rightarrow\nu_{j})$ for each flavor Bahcall:2004mz ;
Altmannshofer:2019zhy ; Nunokawa:2006ms . The solar neutrino flux are taken
from Ref. Haxton:2012wfz .

We show the theoretical prediction of models along with the Borexino results
Borexino:2017rsf  (black curve) in Fig. 5. For the benchmark point
$(g_{X}=1.8\times 10^{-4},m_{Z^{\prime}}=1~{}\mathrm{MeV})$, which can explain
the muon $g-2$ anomaly as well as the thermal DM, we calculate the event rate
of $e\nu$ scattering by varying the gauge kinetic mixing parameter $\epsilon$.
As shown by the green line in Fig. 5, we find the Borexino bound on the gauge
mixing parameter to be $|\epsilon|\lesssim\mathcal{O}(10^{-7})$. For the
comparison, the SM prediction is drawn with the blue curve.

Figure 5: The event rate of $e\nu$ scattering as the function of the recoil
energy. The black curve is the latest Borexino bound. The predictions of the
SM and of the $U(1)_{B_{3}-L_{2}}$ with parameters for the muon $g-2$ and DM
explanation are drawn with blue and green curves, respectively.

####  V.1.2 CHARM-II constraint

Since the extra gauge boson of the $U(1)_{B_{3}-L_{2}}$ gauge theory couples
with muon neutrinos than other flavor of neutrinos, experiments on
$e\nu_{\mu}$ or $e\overline{\nu_{\mu}}$ scattering such as CHARM-II CHARM-
II:1993phx ; CHARM-II:1994dzw  would also provide a constraint on our model.
In Fig. 6, we show the $U(1)_{B_{3}-L_{2}}$ model prediction of the
differential cross section for various $\epsilon$, which is compared with the
CHARM-II results CHARM-II:1993phx . We find, for the benchmark point
$(g_{X}=1.8\times 10^{-4},m_{Z^{\prime}}=1\mathrm{MeV})$, the CHARM-II bound
on the gauge kinetic mixing as $-6\times 10^{-5}\lesssim\epsilon\lesssim
2\times 10^{-4}$ which is less stringent than the Borexino bound shown in the
previous subsection.

|
---|---
Figure 6: The CHARM-II measurement of the differential cross section as the
function of $E_{r}/E_{\nu}$ and the prediction for the $U(1)_{B_{3}-L_{2}}$
model with the muon $g-2$ and DM favored points for various $\epsilon$. Left:
$e^{-}\nu_{\mu}$ scattering. Right: $e^{-}\overline{\nu_{\mu}}$ scattering.

##  VI Summary

We have proposed a variation of flavor dependent gauged $U(1)$ extension of
the SM. Motivated by the fact that the gauged $U(1)_{B-L}$ symmetry is anomaly
free for each generation, an $i$th generation of quarks and $j$th generation
of leptons are charged in the $U(1)_{B_{i}-L_{j}}$ model. One generation of RH
neutrinos must be charged under the $U(1)_{B_{i}-L_{j}}$ for the theory to be
the anomaly free. There is another nontrivial aspect of the model that the
other RH neutrinos may also be charged under the symmetry with the
nonvanishing opposite charge $x_{N}$, and hence they form a Dirac neutrino
$\chi$. This Dirac fermion $\chi$ is stable due to the $U(1)$ gauge invariance
and a natural candidate for DM.

Among various possibilities of charge assignments, the $U(1)_{B_{3}-L_{2}}$
model is attractive, as it may explain the discrepancy of the muon $g-2$
between the experimental results and the SM prediction, and, in addition, the
LHC constraints are relatively weak because the $Z^{\prime}$ interacts with
only third generation of quarks. We have shown that, in a certain parameter
region, the muon $g-2$ anomaly and the thermal DM abundance are simultaneously
explained without contradicting other experimental bounds if the gauge kinetic
mixing is small enough.

## Acknowledgments

This work is supported in part by the U.S. DOE Grant No. DE-SC0012447 (N.O.),
the Japan Society for the Promotion of Science (JSPS) KAKENHI Grants No.
19K03860, No. 19K03865, and No. 23K03402 (O.S.).

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