# Gauge transformations and Galilean covariance in nonlinear gauge-coupled
quantum fluids

Yvan Buggy  Patrik Öhberg  SUPA, Institute of Photonics and Quantum Sciences,
Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom

###### Abstract

We investigate certain invariance properties of quantum fluids subject to a
nonlinear gauge potential. In particular, we derive the covariant
transformation laws for the nonlinear potentials under a space-time Galilean
boost and consider $U\left(1\right)$ gauge transformations. We find that the
hydrodynamic canonical field equations are form-invariant in the case of
external gauge functions $\chi(\mathbf{r},t)$, but not for nonlinear gauge
functionals $\chi[\rho]$. Hence, nonlinear gauge potentials are non-trivial
potentials which may not be ”gauged-away”. Notably, for a superfluid in
dimension $d=1$, attempting to do so generates the gauge-pressure of the fluid
in the Hamiltonian density. Further, we investigate how the field equations
transform under arbitrary Galilean transformations. We find that the immediate
lack of Galilean covariance is restored under a suitably chosen transformation
rule set for the potentials, which is identical in form to that of a
Schrödinger particle coupled to external scalar and vector potentials.

††preprint: APS/123-QED

##  I Introduction

The pursuit for implementing artificial electromagnetic potentials for charge
neutral systems has received a steady stream of interest over the last two
decades. These potentials are generally engineered through combined-
interactions, such that a system exhibits spatially varying local eigenstates,
which, in turn, impart a geometric phase onto the wavefunction Dalibard  _et
al._ (2011); Goldman  _et al._ (2014); Berry (1984); Peskin and Tonomura
(1989). Although initial implementations involved static synthetic fields,
further proposals have since been put forward for creating ”dynamical” gauge
fields which are described by their own Hamiltonian and not merely imposed on
the system Kapit and Mueller (2011); Wiese (2013); Banerjee  _et al._ (2012);
Tagliacozzo  _et al._ (2013a); Zohar  _et al._ (2013); Tagliacozzo  _et al._
(2013b). One possibility for realising a dynamical gauge potential, albeit not
a field in the strict sense, is to introduce a ”back-action” where the
dynamics of the gauge potential are tied to the motion of the condensate. Such
a prospect has been proposed in Edmonds  _et al._ (2013a, b); Keilmann  _et
al._ (2011); Greschner  _et al._ (2014); Cardarelli  _et al._ (2016);
Greschner and Santos (2015); Sträter  _et al._ (2016), where the effective
field depends on the spatial configuration of the atoms, in particular, the
atomic density. More recently, a density-dependent gauge potential has been
experimentally demonstrated in a two-dimensional optical lattice, by
modulating the interaction strength in synchrony with the lattice shaking
Clark  _et al._ (2018), where the tunnelling rate depends on the occupation
number.

Density-dependent gauge potentials, which we shall refer to as nonlinear gauge
potentials, have been shown to give rise to a number of interesting
properties, such as chiral solitons Edmonds  _et al._ (2013a); Dingwall  _et
al._ (2018); Dingwall and Öhberg (2019) and anyonic structures Aglietti  _et
al._ (1996). The superfluid properties can also be dramatically affected by
the gauge potential, where for instance drag forces will become directionally
dependent Buggy (2019). From a dynamical perspective, perhaps the most
striking feature of a density-dependent gauge potential is the occurrence of
an exotic flow-dependent nonlinear term in the wave equation for the quantum
phase. This is exemplified notably by the stress tensor of the fluid, which
features a ”gauge pressure” term related to the overlap of the gauge potential
and the gauge-covariant current Buggy  _et al._ (2020). One would expect such
a term to carry important implications for the symmetry properties of the
fluid. For instance, a flow-dependent fluid pressure evidently transforms
between Galilean frames. This suggests that additional transformation rules
should be implemented in order to ”restore” the covariance of the dynamical
equations under a Galilean boost.

In this paper, we derive these transformation rules and also investigate the
invariance of the dynamical equations of such systems under both external and
nonlinear gauge transformations. We work within a hydrodynamic canonical
formalism and consider transformations at the level of the field equations,
rather than the wave equations proceeding from these. Furthermore, although
the motivation for our study stems from a well-established microscopic model
proposed in Edmonds  _et al._ (2013a), we follow a general approach adopted
from a previous paper Buggy  _et al._ (2020), which is not tied to any
particular model. Here, we simply assume that a density-dependent gauge
potential $\mathbf{A}\left(\rho\right)$ enters the Hamiltonian functional,
whose components take the form of arbitrary density-dependent functions.
Hence, our concern lies not with the physical origin of the potentials, but
rather, the formal structure of the dynamical equations which result from such
a coupling. We shall refer to this class of fluids as nonlinear gauge-coupled
quantum fluids.

The paper is outlined as follows. In section II, we begin by reviewing the
canonical field formalism and hydrodynamic equations for such fluids, which
will be used subsequently in the text. Further, in section III, we consider
gauge transformations, treating both the case of external gauge functions and
nonlinear gauge functionals. Finally, in section IV, we investigate the
covariance of the canonical field equations under general Galilean
transformations.

##  II Hydrodynamic equations of the fluid

Our study pertains to the class of nonlinear quantum fluids whose effective
meanfield Hamiltonian may be written in the form

$\hat{H}_{MF}=\frac{\left(\hat{\mathbf{p}}-\mathbf{A}\left(\rho\right)\right)^{2}}{2m}+\eta\left(\rho\right),$
(1)

where $\eta$ and $\mathbf{A}$ are nonlinear effective potentials which depend
on the density of the fluid. After performing a Madelung transformation,
$\psi=\sqrt{\rho}e^{i\theta/\hbar}$, on the macroscopic condensate
wavefunction $\psi$, it is possible to show Buggy  _et al._ (2020) that the
dynamics of the field components $\rho$ and $\theta$, may be expressed in the
Hamiltonian form

$\displaystyle\dot{\rho}\left(\mathbf{r}\right)$ $\displaystyle-\frac{\delta
H}{\delta\theta\left(\mathbf{r}\right)}=0,$ (2)
$\displaystyle\dot{\theta}\left(\mathbf{r}\right)$ $\displaystyle+\frac{\delta
H}{\delta\rho\left(\mathbf{r}\right)}=0,$ (3)

where the Hamiltonian functional of the field, is given by

$H\left[\rho,\theta\right]=\int
d^{3}\mathbf{r}\left\\{\rho\left[\frac{\left(\mbox{\boldmath$\nabla$}\theta-\mathbf{A}\left(\rho\right)\right)^{2}}{2m}+\eta\left(\rho\right)\right]+\mathcal{Q}\right\\},$
(4)

with
$\mathcal{Q}=\left(\hbar\mbox{\boldmath$\nabla$}\rho\right)^{2}/\left(8m\rho\right)$
representing a quantum energy density. Note that the derivatives of the
Hamiltonian appearing in Eqs. (2) and (3) are functional, or variational
derivatives. As mentioned previously, a defining feature of a density-
dependent gauge potential is the occurrence of a nonlinear flow term in the
wave equation for $\theta$. Indeed, substituting the above expression for $H$
into the canonical field equations (2) and (3), yields, respectively, the wave
equations

$\displaystyle\partial_{t}\rho$
$\displaystyle+\mbox{\boldmath$\nabla$}\cdot\left(\rho\mathbf{v}\right)=0,$
(5) $\displaystyle\partial_{t}\theta$
$\displaystyle+\frac{1}{2}mv^{2}+\Phi\left(\rho,\mathbf{u}\right)+Q=0,$ (6)

where

$\mathbf{v}=\mathbf{u}-\mathbf{A}/m,$ (7)

is the physical velocity or mechanical flow,
$\mathbf{u}=\mbox{\boldmath$\nabla$}\theta/m$ is the canonical or phase flow
and $Q=-\hbar^{2}\nabla^{2}\sqrt{\rho}/\left(2m\sqrt{\rho}\right)$ is the
quantum potential. Moreover, the effective potentials lead to a non-trivial
nonlinear term

$\Phi=\eta+\rho\frac{\partial\eta}{\partial\rho}-\rho\mathbf{v}\cdot\frac{\partial\mathbf{A}}{\partial\rho},$
(8)

in the wave equation of the phase, which is linear in the canonical flow
$\mathbf{u}$. As a consequence, the fluid pressure also inherits a flow-
dependent term. This may be seen by expressing the hydrodynamic equations (5)
and (6) in the reference frame of the fluid, which yield Buggy  _et al._
(2020), respectively

$\displaystyle\left(\frac{\partial}{\partial
t}+\mathbf{v}\cdot\mbox{\boldmath$\nabla$}\right)\rho+\rho\mbox{\boldmath$\nabla$}\cdot\mathbf{v}=0,$
(9) $\displaystyle m\rho\left(\frac{\partial}{\partial
t}+\mathbf{v}\cdot\mbox{\boldmath$\nabla$}\right)u_{k}=\nabla_{j}\left(-\delta_{jk}P+\sigma_{jk}\right),$
(10)

where

$\sigma_{jk}=-\frac{\hbar^{2}}{4m\rho}\nabla_{j}\rho\nabla_{k}\rho,$ (11)

is the quantum stress tensor, and $P$ is the fluid pressure, given by

$P=-\frac{\hbar^{2}}{4m}\nabla^{2}\rho+\rho\left(\Phi-\eta\right).$ (12)

Note the distinction between $\mathbf{v}$ and $\mathbf{u}$ in equation (10).
As such, this equation describes the transport of canonical momentum rather
than mechanical momentum. Recalling expression (8) for $\Phi$, we see that $P$
depends explicitly on the flow profile of the fluid. In particular, the fluid
pressure may be viewed as a combination of the following three terms:

$P=P_{Q}+P_{\eta}+P_{\mathbf{A}},$ (13)

where, $P_{Q}=-\hbar^{2}\nabla^{2}\rho/(4m)$, is the quantum pressure,
$P_{\eta}=\rho^{2}\partial_{\rho}\eta$, is the pressure contribution from the
nonlinear scalar potential, and,
$P_{\mathbf{A}}=-\rho^{2}\mathbf{v}\cdot\partial_{\rho}\mathbf{A}$, is the
pressure contribution from the nonlinear gauge potential, which we refer to as
the gauge pressure.

##  III Gauge transformations

Let us investigate the implications of a nonlinear gauge potential on the
symmetry properties of the fluid. To begin with, we consider gauge
transformations of the form

$\displaystyle\theta\rightarrow$ $\displaystyle\theta^{\prime}=\theta+\chi,$
(14) $\displaystyle\mathbf{A}\rightarrow$
$\displaystyle\mathbf{A}^{\prime}=\mathbf{A}+\mbox{\boldmath$\nabla$}\chi,$
(15) $\displaystyle\eta\rightarrow$
$\displaystyle\eta^{\prime}=\eta-\partial_{t}\chi.$ (16)

We shall treat both the case in which $\chi$ is an external single-valued
scalar function of space and time, $\chi=\chi\left(\mathbf{r},t\right)$, and a
functional of the density, $\chi=\chi\left[\rho\right]$. Notice that the
latter is required for evaluating whether nonlinear gauge potentials are
physical potentials which may not be trivially gauged away, since any attempt
to do so involves a gauge functional given by the contour integral of some
function of the density. In both cases, the transformed canonical field
equations may be written in the generic form

$\displaystyle\dot{\rho}\left(\mathbf{r}\right)$ $\displaystyle-\frac{\delta
H^{\prime}}{\delta\theta^{\prime}\left(\mathbf{r}\right)}=0,$ (17)
$\displaystyle\dot{\theta}^{\prime}\left(\mathbf{r}\right)$
$\displaystyle+\frac{\delta H^{\prime}}{\delta\rho\left(\mathbf{r}\right)}=0,$
(18)

where the Hamiltonian transforms according to

$H^{\prime}=\int
d^{3}\mathbf{r}\left\\{\rho\left[\frac{\left(\mbox{\boldmath$\nabla$}\theta^{\prime}-\mathbf{A}^{\prime}\right)^{2}}{2m}+\eta^{\prime}\right]+\mathcal{Q}\right\\}.$
(19)

In the Madelung representation of the fluid, it becomes readily apparent that
the combination of transformations (14) and (15) leaves the mechanical flow
$\mathbf{v}$ from Eq. (7) unchanged, i.e. $\mathbf{v}$ is gauge invariant.
Hence, according to Eq. (19), the new and old Hamiltonians are related by

$H^{\prime}\left[\rho,\theta^{\prime}\right]=H\left[\rho,\theta\right]-\int
d^{3}\mathbf{r}\rho\frac{\partial\chi}{\partial t}.$ (20)

###  III.1 External gauge functions $\chi\left(\mathbf{r},t\right)$

Let us examine the situation where $\chi$ is an externally prescribed scalar
function. Clearly, in this case, $\delta\chi=0$ for arbitrary field variations
$\delta\rho$ and $\delta\theta$. Hence, the first variation of Eq. (20),
yields

$\displaystyle\frac{\delta
H^{\prime}}{\delta\theta^{\prime}\left(\mathbf{r}\right)}$
$\displaystyle=\frac{\delta H}{\delta\theta\left(\mathbf{r}\right)},$ (21)
$\displaystyle\frac{\delta H^{\prime}}{\delta\rho\left(\mathbf{r}\right)}$
$\displaystyle=\frac{\delta
H}{\delta\rho\left(\mathbf{r}\right)}-\frac{\partial\chi\left(\mathbf{r},t\right)}{\partial
t},$ (22)

Inserting the first and second of these into, respectively, the transformed
field Eqs. (17) and (18), we immediately recover the original field Eqs. (2)
and (3). Therefore, the canonical equations of the nonlinear fluid are form
invariant under gauge transformations generated by external gauge functions
$\chi\left(\mathbf{r},t\right)$. Inspecting the form of the nonlinear term in
the wave equation for the phase, $\Phi$ from Eq. (8), this is not surprising.
Indeed, the nonlinear flow term results from a particular form of density-
dependence of $\mathbf{A}$, which remains unchanged by an external gauge
function.

###  III.2 Nonlinear gauge functionals $\chi\left[\rho\right]$

Next we examine the case where $\chi$ takes the form of some functional of the
density. To begin with, observe how the canonical structure of the field
equations (2) and (3), implies that the field components $\rho$ and $\theta$
play the role of conjugate variables, where the Poisson bracket of two
dynamical variables $f$ and $g$ on phase space, reads

$\left\\{f\left(\mathbf{x}\right),g\left(\mathbf{y}\right)\right\\}_{\rho,\theta}\\\
=\int d^{3}\mathbf{r}\left(\frac{\delta
f\left(\mathbf{x}\right)}{\delta\rho\left(\mathbf{r}\right)}\frac{\delta
g\left(\mathbf{y}\right)}{\delta\theta\left(\mathbf{r}\right)}-\frac{\delta
f\left(\mathbf{x}\right)}{\delta\theta\left(\mathbf{r}\right)}\frac{\delta
g\left(\mathbf{y}\right)}{\delta\rho\left(\mathbf{r}\right)}\right).$ (23)

Now, for gauge transformations generated by $\chi\left[\rho\right]$, the field
variables transform according to

$\begin{pmatrix}\rho\\\
\theta\end{pmatrix}\rightarrow\begin{pmatrix}\rho^{\prime}=\rho\\\
\theta^{\prime}=\theta+\chi\left[\rho\right]\end{pmatrix}.$ (24)

Let us check whether the new variables form a canonically conjugate pair. To
this end, we evaluate the Poisson bracket of the new variables with respect to
the old variables, which gives

$\left\\{\rho^{\prime}\left(\mathbf{x}\right),\theta^{\prime}\left(\mathbf{y}\right)\right\\}_{\rho,\theta}\\\
=\int
d^{3}\mathbf{r}\left(\frac{\delta\rho^{\prime}\left(\mathbf{x}\right)}{\delta\rho\left(\mathbf{r}\right)}\frac{\delta\theta^{\prime}\left(\mathbf{y}\right)}{\delta\theta\left(\mathbf{r}\right)}-\frac{\delta\rho^{\prime}\left(\mathbf{x}\right)}{\delta\theta\left(\mathbf{r}\right)}\frac{\delta\theta^{\prime}\left(\mathbf{y}\right)}{\delta\rho\left(\mathbf{r}\right)}\right)\\\
=\delta\left(\mathbf{x}-\mathbf{y}\right),$ (25)

on account of $\rho^{\prime}$ being independent of $\theta$. Hence
$\theta^{\prime}$ and $\rho$ may be treated as independent variables. We shall
make use of this property further in section III.3.

Following the same line of reasoning as section III.1, let us evaluate the
first variation of Eq. (20) for the case of a nonlinear gauge functional.
Since $\chi$ is independent of $\theta$, Eq. (21) remains valid and the
transformed canonical equation (17) for $\rho$ again reduces to the original
canonical equation (2). However, the canonical equation for $\theta$ is no
longer form invariant, but transforms according to

$\dot{\theta}^{\prime}+\frac{\delta
H}{\delta\rho}-\frac{\delta}{\delta\rho}\int
d^{3}\mathbf{r}\rho\partial_{t}\chi\left[\rho\right]=0,$ (26)

where an additional term appears due to $\chi\left[\rho\right]$. Although the
form taken by this term will depend on $\chi\left[\rho\right]$, one may see
that current terms will generally be involved, through
$\partial_{t}\chi\left[\rho\right]$. In the following section, we examine one
such gauge functional. We may therefore conclude that density-dependent gauge
potentials are physical potentials which may not be trivially gauged away.

###  III.3 The one-dimensional gauge-coupled superfluid

Typically, one may be interested in gauge functionals which eliminate the
gauge potential from the kinetic energy density. Let us examine one such case.
As our system, we consider the superfluid fraction of particles in an
optically-addressed weakly-interacting dilute Bose gas of two-level atoms,
first proposed in Edmonds  _et al._ (2013a). Confining the dynamics to one of
the condensate components, a density-modulated gauge potential emerges in the
form $\mathbf{A}=\mathbf{a}\rho$, where $\rho$ is the atomic density of the
relevant condensate component, while
$\mathbf{a}=\left(g_{11}-g_{22}\right)\mbox{\boldmath$\nabla$}\phi/\left(8\Omega\right)$
controls the effective strength and orientation of the nonlinear vector
potential. Here, $\Omega$ is the generalised Rabi-frequency of the light-
matter coupling, $\phi$ is the phase of the incident laser field and
$g_{ij}=4\pi\hbar^{2}a_{ij}/m$ denote the coupling constants of the channels
stemming from pairwise interactions between the atoms, where $a_{ij}$ are the
associated s-wave scattering lengths.

The system is described by a wave equation of the form (6), with
$\Phi=-\mathbf{a}\cdot\mathbf{J}+g\rho$ and
$g=\left(g_{11}+g_{22}+2g_{12}\right)/4$. For a superfluid confined to
dimension $d=1$, the wave equation for the phase reads

$\partial_{t}\theta+\frac{1}{2}mv^{2}-aJ+g\rho+Q=0,$ (27)

where, $v=\left(\partial_{x}\theta-a\rho\right)/m$ and $J=\rho v$. We choose
$d=1$, since the last term in Eq. (26) becomes integrable in this case. In
turn, the fluid pressure again takes the form of equation (13), where the
quantum pressure is $P_{Q}=-\hbar^{2}\partial_{x}^{2}\rho/4m$,
$P_{\eta}=g\rho^{2}/2$, and the gauge pressure reads $P_{A}=-JA$.

The gauge functional which eliminates $\mathbf{A}$ from the kinetic energy
density, is

$\chi=-\int_{-\infty}^{x}dya\rho\left(y,t\right).$ (28)

Invoking this relation in Eq. (20), we find that the new and old Hamiltonians
are related by

$H^{\prime}=H-\int dx\rho^{2}av^{\prime},$ (29)

where $v^{\prime}=\partial_{x}\theta^{\prime}/m$. Hence, an additional term
$-\int dxJ^{\prime}A=-\int dxJA$, appears in the Hamiltonian under the
nonlinear gauge transformation, where $J^{\prime}=\rho v^{\prime}$. In other
words, the gauge-pressure, $P_{A}$, is generated in the Hamiltonian density
when one attempts to eliminate the nonlinear gauge potential. Inserting
expression (29) for $H^{\prime}$ into the field Eq. (18), yields the wave
equation

$\partial_{t}\theta^{\prime}+\frac{1}{2}mv^{\prime}v^{\prime}-2aJ^{\prime}+g\rho+Q=0,$
(30)

where we have used $\delta v^{\prime}/\delta\rho=0$, which holds on account of
$\rho$ and $\theta^{\prime}$ being independent (see section III.2). Comparing
Eqs. (27) and (30), we see that the wave equation has transformed under the
nonlinear gauge transformation. In particular, the attempt to absorb the
nonlinear gauge potential into the phase has produced an additional current
term in the wave equation.

##  IV Galilean Transformations

We now turn our attention to a space-time symmetry group associated with
coordinate transformations, namely, Galilean transformations. The coordinate
transformations from some inertial frame of reference $\Sigma$, to another
frame $\Sigma^{\prime}$ moving with a uniform, nonrelativistic relative
velocity $\mathbf{w}$, may be written in the general form Holland (1995);
Arnold  _et al._ (2007); Iro (2015)

$\displaystyle x_{i}$ $\displaystyle\rightarrow
x_{i}^{\prime}=R_{ij}x_{j}-w_{i}t+a_{i},$ $\displaystyle t$
$\displaystyle\rightarrow t^{\prime}=t+t_{0},$ (31)

where $a_{i}$ and $t_{0}$ are constants representing offsets in position and
time respectively, and $R_{ij}$ is a unitary rotational matrix satisfying
$R_{ij}R_{jk}=\delta_{ik}$. Differential operators also transform between
frames under (31), such that

$\displaystyle\nabla_{i}$
$\displaystyle\rightarrow\nabla_{i}^{\prime}=R_{ij}\nabla_{j},$ (32)
$\displaystyle\frac{\partial}{\partial t}$
$\displaystyle\rightarrow\frac{\partial}{\partial
t^{\prime}}=\frac{\partial}{\partial t}+w_{i}R_{ij}\nabla_{j}.$ (33)

In turn, the velocity field, transforms as

$v_{i}\rightarrow v_{i}^{\prime}=R_{ij}v_{j}-w_{i}.$ (34)

Since the energy and momentum of the fluid transforms under a Galilean boost,
the phase also transforms between reference frames. Let us denote this
transformation, by

$\theta\rightarrow\theta^{\prime}=\theta+\chi,$ (35)

where $\chi=\chi\left(\mathbf{r},t\right)$ is a single-valued scalar function
of space and time. Furthermore, we should not exclude the prospect that the
potentials may be subject to transformations in order that the dynamical
equations be form invariant. In fact, this is already the situation for a
nonrelativistic quantum mechanical particle coupled to both external scalar
and vector potentials Takagi (1991); DeWitt (1957); Brown and Holland (1999).
Accordingly, let us denote the transformations of the nonlinear potentials, as

$\displaystyle\eta$ $\displaystyle\rightarrow\eta^{\prime}=\eta+\mu,$ (36)
$\displaystyle A_{i}$ $\displaystyle\rightarrow
A_{i}^{\prime}=R_{ij}A_{j}+G_{i},$ (37)

where $\mu$ and $\mathbf{G}$ are unspecified. However, in view of the
nonlinear character of $\eta$ and $\mathbf{A}$, we allow for the possibility
that these also depend on the density.

In the frame $\Sigma^{\prime}$, the canonical field equations (2) and (3),
read

$\displaystyle\frac{\partial\rho^{\prime}}{\partial t^{\prime}}$
$\displaystyle-\frac{\delta H^{\prime}}{\delta\theta^{\prime}}=0,$ (38)
$\displaystyle\frac{\partial\theta^{\prime}}{\partial t^{\prime}}$
$\displaystyle+\frac{\delta H^{\prime}}{\delta\rho^{\prime}}=0,$ (39)

where the Hamiltonian from Eq. (4), transforms to

$H^{\prime}=\int
d^{3}\mathbf{r}^{\prime}\left[\rho^{\prime}\left(\frac{1}{2}m\mathbf{v}^{\prime}\cdot\mathbf{v}^{\prime}+\eta^{\prime}\right)+\mathcal{Q}^{\prime}\right],$
(40)

with
$\mathbf{v}^{\prime}=\left(\mbox{\boldmath$\nabla$}^{\prime}\theta^{\prime}-\mathbf{A}^{\prime}\right)/m$.
Since $\rho^{\prime}=\rho$ and $\mathbf{v}^{\prime}$ is restricted by Eq.
(34), the Hamiltonians in the two frames are related, by

$H^{\prime}\left[\rho^{\prime},\theta^{\prime}\right]=H\left[\rho,\theta\right]+\int
d^{3}\mathbf{r}\rho\left(\frac{1}{2}mw^{2}-mw_{i}R_{ij}v_{j}+\mu\right).$ (41)

The first variation of the above equation, yields

$\displaystyle\frac{\delta H^{\prime}}{\delta\rho^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\delta H}{\delta\rho}+\frac{1}{2}mw^{2}-mw_{i}R_{ij}v_{j}$
(42)
$\displaystyle+\mu+\rho\left(\frac{\partial\mu}{\partial\rho}+w_{i}R_{ij}\frac{\partial
A_{j}}{\partial\rho}\right),$ $\displaystyle\frac{\delta
H^{\prime}}{\delta\theta^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\delta H}{\delta\theta}+w_{i}R_{ij}\nabla_{j}\rho.$ (43)

Note that in order to obtain Eq. (43), we have substituted Eq. (7) for
$\mathbf{v}$ into Eq. (41) and integrated by parts accordingly. Let us
investigate the implications of these relations on the transformed canonical
equations (38) and (39). Invoking Eqs. (33) and (43) in Eq. (38), we
immediately recover the original field equation for $\rho$. Hence the
canonical equation (2) is form invariant under a Galilean transformation.
Notice here that no demands are being made of $\mu$, $\mathbf{G}$ or $\chi$,
other than the restriction that $\chi$ be independent of $\theta$, since the
violation of the latter introduces additional terms in Eq. (43). Turning our
attention to the field equation for $\theta$, let us substitute Eqs. (33),
(35) and (41) into Eq. (39), which gives

$\dot{\theta}+\frac{\delta H}{\delta\rho}+\xi=0,$ (44)

where we have grouped the additional terms generated by the Galilean
transformation, in the function $\xi=\xi_{\chi}+\xi_{\pi}$, with

$\displaystyle\xi_{\chi}$ $\displaystyle=\left(\frac{\partial}{\partial
t}+w_{i}R_{ij}\nabla_{j}\right)\chi+w_{i}R_{ij}\left(\nabla_{j}\theta-
mv_{j}\right)+\frac{1}{2}mw^{2},$ (45) $\displaystyle\xi_{\pi}$
$\displaystyle=\mu+w_{i}R_{ij}A_{j}+\rho\left(\frac{\partial\mu}{\partial\rho}+w_{i}R_{ij}\frac{\partial
A_{j}}{\partial\rho}\right).$ (46)

In other words, a suitably chosen set of transformations (35), (36) and (37)
which solves $\xi=0$, signals the form invariance of the field equation (3).
Such a prospect is readily achieved by setting both Eqs. (45) and (46) to
zero. The condition, $\xi_{\chi}=0$, is upheld by a suitable phase
transformation, while the imposition, $\xi_{\pi}=0$, leads to a transformation
rule for the potentials. Indeed, notice that the phase factor,
$\chi=mw^{2}t/2-mw_{i}R_{ij}x_{j}$, solves the first of these two conditions,
such that Holland (1995); Brown and Holland (1999)

$\theta\rightarrow\theta+\frac{1}{2}mw^{2}t-mw_{i}R_{ij}x_{j},$ (47)

while the second condition, is ensured for $\mu=-w_{i}R_{ij}A_{j}$. Thus, the
nonlinear potentials transform according to

$\displaystyle\eta$ $\displaystyle\rightarrow\eta-w_{i}R_{ij}A_{j},$ (48)
$\displaystyle A_{i}$ $\displaystyle\rightarrow R_{ij}A_{j}.$ (49)

These transformations are identical to those which appear for a Schrödinger
field coupled to external scalar and vector potentials Brown and Holland
(1999), but represent a nonlinear transformation. Note that the above
transformations also appear for a classical charge, emerging as the magnetic
limit of Galilean electromagnetism Le Bellac and Lévy-Leblond (1973). In turn,
performing transformations (34), (48) and (49) in Eq. (8) for the nonlinear
scalar term in the wave equation for the phase, $\Phi$, yields

$\Phi\rightarrow\Phi-w_{i}R_{ij}A_{j}.$ (50)

Hence $\Phi$ and $\eta$ undergo an identical transformation. Furthermore,
although the fluid pressure $P$ from Eq. (12) also depends explicitly on
$\mathbf{v}$, performing transformations (48) and (50) in Eq. (12) reveals
that the covariant transformations of the potentials between frames ensure
that the pressure is invariant.

##  V Conclusion

The dynamical equations of a nonlinear gauge-coupled quantum fluid are
invariant under local gauge transformations, insofar as the gauge function
$\chi$ takes the form of an external scalar function of $\mathbf{r}$ and $t$.
However, when $\chi$ becomes a functional of the density, the form-invariance
of the dynamical equation for the phase $\theta$ is lost. As a result,
density-dependent gauge potentials are physical, non-trivial potentials which
cannot be absorbed into the phase, since attempting to do so invariably
destroys the form of the dynamical equations. For a gauge-coupled one-
dimensional superfluid, the additional term in the field equation generated by
a nonlinear gauge transformation becomes integrable. In particular, the gauge-
pressure of the nonlinear fluid is generated in the Hamiltonian density for
gauge transformations which eliminate the gauge potential from the kinetic
energy density. Finally, the immediate lack of Galilean covariance of the
nonlinear fluid may be restored by subjecting the potentials to a nonlinear
transformation. As a result, the flow-dependent fluid pressure inherent to
such systems, becomes invariant under a Galilean transformation. These
transformations are identical in form to those of a Schrödinger field subject
to external scalar and vector potentials, but represent nonlinear
transformations.

###### Acknowledgements.

Y.B. acknowledges support from EPSRC CM-CDT Grant No. EP/L015110/1. P.Ö.
acknowledges support from EPSRC EP/M024636/1.

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The Belle II experiment [Belle2TDR2010, ADACHI201846] studies $CP$-violation
in the $B$-meson system and searches for Physics beyond the Standard Model,
including evidence of dark sector particles, in decays of $B$-mesons,
$D$-mesons and tau leptons [Kou2019]. The SuperKEKB collider [Ohnishi2013]
produces particles of interest by colliding electron and positron beams with
asymmetric energies, mainly at the $\Upsilon(4S)$ resonance. SuperKEKB is a
major upgrade of KEKB [nlhep1995, Kurokawa2003, Abe2013] and has been
operational since 2016. The machine has already reached a world-record
luminosity of 4.65e34cm^-2.s^-1 with a vertical betatron function of
$\beta^{*}_{\rm y}=\SI{1.0}{mm}$ at the interaction point (IP), but the goal
is to increase the luminosity by another order of magnitude in the coming
decade, with a current target peak luminosity of 6.3e35cm^-2.s^-1 for
$\beta^{*}_{\rm y}=\SI{0.3}{mm}$. Luminosity increases by increasing beam
currents and reducing the beam size at the IP, utilizing low-emittance
colliding beams and the so-called nano-beam scheme [Baszczyk2013].

Beam particles that deviate from the nominal orbit are eventually lost by
hitting the beam pipe’s inner wall or other machine apparatus. If the loss
position is close to the Belle II, generated shower particles might reach the
detector and increase its dose rate and hit rate. This increase is referred to
as “beam (induced) background” and is one of the most difficult challenges at
SuperKEKB. In the SuperKEKB and Belle II designs, it was estimated that
several Belle II sub-detectors would be subject to close-to-tolerable
backgrounds at the target peak luminosity [Nakayama2015, LEWIS201969]. The
most vulnerable sub-detectors are the Time-of-Propagation (TOP) particle
identification system and the Central Drift Chamber (CDC). In the TOP, higher
hit rates increase the accumulated output charge in the micro-channel-plate
photo-multiplier tubes (MCP-PMTs) used to read out Cherenkov photons
propagated in quartz bars, which can degrade the quantum efficiency of the
PMTs. One key issue in the CDC is that pattern recognition of charged tracks
becomes increasingly difficult as the wire-hit rate increases.

Given the importance of beam background mitigation to the experiment’s
success, we have studied such backgrounds extensively in the early stages of
SuperKEKB running. The Belle II/SuperKEKB project has three major
commissioning phases:

  * •

Phase 1 was carried out in Spring 2016. No beam collisions occurred, as
SuperKEKB was running without the final focusing system. Belle II had not yet
been installed at the IP. Instead, a system of dedicated beam background
detectors, collectively known as BEAST II, was placed around the IP. We found
that the background level around the IP was safe for Belle II to be installed.
Results of the Phase 1 measurements and simulation are reported in Ref.
[LEWIS201969].

  * •

Phase 2 began in March 2018 and concluded in July 2018. The machine group
demonstrated first the $e^{+}e^{-}$ collisions with Belle II (except for the
vertex detector) now installed at the IP. This commissioning phase confirmed
that proceeding and installing the sensitive vertex detector was safe. Details
and results of the Phase 2 beam background study can be found in Ref.
[Liptak2021].

  * •

Phase 3, which started in March 2019, is dedicated to physics data taking with
a fully instrumented Belle II detector and to increasing the instantaneous
luminosity above 1e35cm^-2.s^-1. We aim to accumulate 50ab^-1 of data by the
2030s, anticipating 7–8 months of operation per year, and assuming 70% of that
operation time is spent on physics runs [LumiProjSuperKEKB].

In the rest of Section LABEL:sec:Introduction, we describe the SuperKEKB
collider and the Belle II detector and provide an overview of the main beam-
induced background sources, background countermeasures, and relevant beam
instrumentation. Section LABEL:sec:CurrentBackgroundLevelsAndMargin reports on
the current (early Phase 3) background levels and safety margins of the Belle
II sub-detectors. In Section LABEL:sec:BackgroundSimulation, we describe the
beam background Monte-Carlo (MC) simulation methodology. In Section
LABEL:sec:MeasuredBackgroundComposition, we explain the methodology of
background measurements and modeling. Section
LABEL:sec:SummaryOfCurrentBackgroundLevels reports on the measured background
composition in Belle II. In Section LABEL:sec:SummaryOfCurrentBackgroundLevels
we apply correction factors of each simulated background process in each sub-
detector to enforce a full agreement with measurements. This detailed
understanding is required to reliably extrapolate the current backgrounds to
different beam conditions. Section LABEL:sec:Extrapolations describes an
extrapolation of backgrounds towards higher luminosity and provides expected
detector safety factors111The safety factor is defined as a ratio between the
detector limit and predicted background rate. It shows how much the background
level can increase before reaching the detector limit.. Finally, in Section
LABEL:sec:Conclusion, we summarize and discuss our findings and their
implications.

##  0.1 SuperKEKB and Belle II

Here, we briefly review the collider and detector sub-systems involved in the
beam-induced background analysis. Further details can be found in Refs.
[Belle2TDR2010, ADACHI201846, Ohnishi2013].

SuperKEKB, illustrated in Fig. 1, is an upgrade of the KEKB accelerator. It is
a 3km-circumference asymmetric-energy electron-positron collider with a
center-of-mass (CM) energy of $\sqrt{s}=\SI{10.58}{GeV}$ which corresponds to
the mass of the $\Upsilon(4S)$ resonance. At the IP, 7GeV electrons stored in
the high-energy ring (HER) collide with 4GeV positrons accumulated in the low-
energy ring (LER). To reach collision luminosity of the order of
1e35cm^-2.s^-1, SuperKEKB utilizes the so-called nano-beam scheme, where the
vertical and horizontal beam sizes at the IP are squeezed down to ∼50nm and
∼10m, respectively, with a horizontal crossing angle of 83mrad to avoid the
hour-glass effect. The relatively large crossing angle also allows i) a new
final focusing system with superconducting quadrupole magnets (QCS) to reside
closer to the IP, ii) separate beamlines for the HER and LER, and iii) a
design that avoids combined-functionality IP magnets with large fringe fields.
To eliminate luminosity degradation caused by beam-beam resonances, dedicated
sextupole magnets are used for the Crab-Waist collision scheme implementation
[raimondi2007beambeam], which aligns the vertical waistline of one beam along
the trajectory of the other beam at the IP.

Figure 1: Schematic drawing of the SuperKEKB collider.

The upgrade from KEKB to SuperKEKB included the following major items. We note
that the list is not exhaustive.

  * •

Short LER dipole magnets were replaced with longer ones.

  * •

The interaction region (IR), $\pm\SI{4}{m}$ around the IP, was redesigned.
This region hosts the Belle II detector, the final focusing system and the IR
beam pipe assembly.

  * •

Beam pipes with a titanium nitride (TiN) coating and antechambers were
installed in the LER to reduce the power density of the synchrotron radiation
(SR).

  * •

A damping ring (DR) was constructed to reduce the injected positron beam
emittance.

  * •

The radio-frequency (RF) system was modified to enable higher beam currents.

  * •

The collimation system was upgraded, see Section 0.3.

There are two major upgrades of SuperKEKB planned in the next decade, during
Long Shutdown 1 (LS1), which began in July 2022, and during Long Shutdown 2
(LS2), expected to begin around 2027. Possible future upgrades of the detector
are strongly linked to upgrades of the machine. The most crucial upgrades
under consideration are discussed in Ref. [Natochii2022].

The Belle II detector, shown in Fig. 2, is a general-purpose particle
spectrometer optimized for precise measurements of $B$-meson pairs via their
decay products. The detector must maintain Belle’s level of performance
[Abashian2002, Brodzicka2012], despite a reduced center of mass boost, and
while operating in a much higher-background environment, which tends to reduce
detector performance and longevity. Belle II replaced a number of Belle sub-
systems to satisfy this requirement and to have better vertexing and particle
identification performance than Belle. Belle II consists of several nested
sub-detectors around the 1-cm radius beryllium beam pipe surrounding the IP.
The Belle II sub-detector closest to the IP is the two-layer pixel detector
(PXD). All 16 modules in the first PXD layer (L1), but only 4 out of the 24
modules in the second PXD layer (L2) have been installed to date. During LS1,
we plan to install a new, fully assembled two-layer PXD, which will increase
the detector’s performance and tolerance of hit occupancy due to backgrounds
[Forti2022]. The PXD is surrounded by the four layers (L3-6) of the double-
sided silicon strip vertex detector (SVD). Both PXD and SVD are surrounded by
the CDC, which is filled with a $\rm He(\SI{50}{\%})+C_{2}H_{6}(\SI{50}{\%})$
gas mixture. The CDC consists of 56 layers with 14336 sense wires of either
axial or stereo orientation for precise measurements of charged particle
trajectories. The charged-particle identification system is based on two sub-
detectors: the barrel’s TOP detector and the Aerogel Ring Imaging Cherenkov
counter (ARICH) in the forward endcap region. The TOP is composed of 2-cm-
thick quartz bars viewed by conventional and atomic layer deposition (ALD)
types of MCP-PMTs, which are arranged into 16 readout slots. The ARICH
consists of 4-cm-thick focusing aerogel radiators and 420 Hybrid Avalanche
Photo Detectors (HAPDs), each having 144 readout channels. The HAPDs are
grouped into 18 segments. For precise energy and timing measurements of
particles, an electromagnetic calorimeter (ECL) is installed in the barrel and
both endcaps. It is composed of 8736 CsI(Tl) crystals and is located inside a
superconducting solenoid that provides a 1.5T magnetic field. Outside the
magnet coil, a $K^{\rm 0}_{\rm L}$ and muon detector (KLM) is installed. The
KLM has 12 and 14 scintillator strip layers read out by silicon
photomultipliers in the forward (FWD) and backward (BWD) endcaps,
respectively. The two innermost KLM barrel layers also utilize scintillators,
while the remaining 13 barrel layers consist of glass-electrode resistive
plate chambers (RPCs). A comprehensive overview of Belle II upgrades planned
for LS1 and LS2 can be found in Ref. [Forti2022].

Figure 2: Overview of the Belle II detector.

##  0.2 Background types

Belle II hits generated by background shower particles deteriorate the
detector’s physics performance. The radiation dose and neutron fluence from
the background showers can also damage sensor components in the detector, such
as silicon devices. Below we review the leading background components that are
relevant at SuperKEKB.

### Touschek background

Touschek scattering [Piwinski1999], one of the leading background sources at
SuperKEKB, refers to intrabeam Coulomb scattering between two particles in the
same beam bunch. Such scattering causes the energies of the two scattered
particles to deviate from the nominal beam energy, with one particle gaining,
and the other losing, energy. The Touschek scattering rate is proportional to
the beam current squared and inversely proportional to the number of bunches
in the ring and the beam size. Due to the nano-beam scheme used at SuperKEKB,
the beam size is much smaller than at KEKB, and consequently, the ring-
integrated beam loss rate due to Touschek scattering is expected to be much
higher. However, the Touschek loss rate inside Belle II has been significantly
suppressed by installing horizontal collimators near the IR.

### Beam-gas background

Beam-gas scattering by residual gas atoms in the beam pipe is another major
background source at SuperKEKB. Beam-gas Coulomb scattering changes the
direction of scattered beam particles, while beam-gas bremsstrahlung
scattering reduces their energy. The beam-gas scattering rate is proportional
to the residual gas pressure and to the beam current. The beam-gas Coulomb
loss rate inside Belle II is expected to be quite high due to the small
diameter of the IP beam pipe and the extremely large vertical betatron
function of the QCS. The loss rate in the detector has been greatly reduced by
installing vertical collimators. However, the jaws of those collimators must
be narrowed, which can cause beam instabilities at high beam currents
[Nakayama2012].

### Luminosity background

Luminosity background is caused by beam collisions at the IP. It is
proportional to luminosity and expected to dominate at the target luminosity
of SuperKEKB, which is about 30 times higher than the record of KEKB.

One important luminosity background is from radiative Bhabha scattering
($e^{+}e^{-}\rightarrow e^{+}e^{-}\gamma$), where beam particles lose energy
by emitting photons and therefore deviate from the nominal orbit. At KEKB,
since a shared final focusing magnet scheme was employed, the outgoing beam
orbits were the off-center in the quadrupole magnets. Therefore, off-energy
beam particles were strongly over-bent and easily lost inside the detector.
Unlike KEKB, the final focusing magnets at SuperKEKB are separate for each
ring, which relaxes the loss rate inside the detector. However, a small
fraction of beam particles with high energy losses can still be lost inside
the detector due to i) beam intrinsic angular divergence at the IP, ii)
angular diffusion by the radiative Bhabha process, iii) the kick from the
detector solenoid field, and iv) the leakage field from the other ring’s
quadrupole magnets, especially for electrons as discussed in Ref.
[OHUCHI2022165930]. At high luminosity, radiative Bhabha beam losses inside
the detector dominate over other Belle II backgrounds.

Radiative Bhabha scattering can also give rise to neutron backgrounds incident
upon Belle II from the accelerator tunnel via the following mechanism: photons
emitted in the radiative Bhabha process at the IP propagate along the beam
axis and escape Belle II. Such photons then hit accelerator magnets located
1020m downstream of the IP. Then, neutrons copiously produced via the giant
photo-nuclear resonance [RevModPhys.47.713] scatter back towards the Belle II
detector. This background increases the hit occupancy in the outer layers of
the KLM. A dedicated study of this background component is described in Ref.
[Schueler2021].

In the two-photon process, $e^{+}e^{-}\rightarrow e^{+}e^{-}e^{+}e^{-}$, beam
particles lose energy by emitting low-momentum electron-positron pairs, and
become a source of Belle II background as described for the radiative Bhabha
process. In addition, the emitted electron and positron curl in the Belle II
solenoid field. They can leave multiple hits in the inner Belle II detectors
if they have high enough transverse momentum.

### Synchrotron radiation background

SR emitted from the beams is another source of background in the inner Belle
II detectors. Since the power of SR is proportional to the beam energy squared
and the magnetic field strength squared, the HER electron beam is the main
source of SR background. SR photons leave PXD and SVD hits with energy ranging
from a few keV to several tens of keV. We pay special attention to this
background because the inner layers of the SVD were severely damaged by HER SR
in the early stages of the Belle experiment.

### Injection background

Since the beam lifetime of SuperKEKB is much shorter than an hour, top-up
injections via a betatron injection scheme [Ohnishi2013] are performed during
physics data taking. When the total beam current is below a set threshold
($\sim 99\%$ of the nominal beam current), charge is injected into buckets
with low bunch-current, at a certain repetition rate (125). Newly injected
bunches are perturbed and oscillate in the horizontal plane around the main
stored beam. This causes increased background rates in Belle II for a few
milliseconds (ms) after injection each time when the newly injected bunch
passes the IP. In order to avoid saturation of the readout, special trigger
vetoes are applied, which lead to dead time in the data acquisition and,
consequently, a reduction in recorded luminosity. A comprehensive description
of the Belle II trigger system is given in Ref. [Kou2019].

The amount and time structure of the injection background observed in Belle II
is shared online with the SuperKEKB operators and can be used to optimize the
injection settings to keep backgrounds low. One of the most important and
difficult tasks for SuperKEKB is maintaining stable injection background
conditions for an extended period.

### Large beam loss accidents

The accidental firing of one of the injection kicker magnets may perturb the
stored beam during its 2-$\mathrm{\micro s}$-long waveform towards a
horizontal collimator, causing severe jaw damage.

Furthermore, for unknown reasons, the stored beam sometimes becomes unstable
to the point of causing catastrophic or so-called sudden beam losses. These
losses have already caused several quenches of QCS magnets, damaged sensitive
components of Belle II, and significantly slowed down the planned luminosity
increase. In other cases, the jaws of collimators were severely damaged, and
beam operation was stopped for about a week to replace the jaws. Such events
frequently occur when the beam current increases above 0.5A. A possible cause
of these events is dust trapped in the beam pipe, but this is not yet fully
understood. We are conducting detailed beam abort analysis using the timing
information from the beam loss monitors installed around the ring. Such
analysis may help us to identify the location where the initial beam loss
occurred.

##  0.3 Background mitigation

Here, we briefly review the crucial countermeasures against major, known
background sources.

### Collimators

Movable beam collimators are installed around SuperKEKB rings to stop beam
particles with large transverse deviation from the nominal beam orbit before
they reach the IR and lead to background hits in Belle II, see Fig. 3.
Moreover, the collimators help protect Belle II and the QCS magnets against
large, unexpected beam losses, including those from accidental injection
kicker firing.

Figure 3: Map of the SuperKEKB collimators used in 2021 and 2022. The letters
V and H in the collimator names indicate vertical and horizontal movable jaws,
respectively. There are twelve sections in each ring named D01 through D12.

There are currently 11 collimators in the LER and 20 in the HER, see Fig. 3.
There are two main types of collimators with different geometries: KEKB-type
collimators are asymmetric and have only one jaw, while SuperKEKB-type
collimators are symmetric with jaws on both sides. More details about the
collimators can be found in Refs. [Natochii2021, Ishibashi2020].

Horizontal collimators effectively stop Touschek scattered particles, while
vertical collimators are mainly used to stop beam-gas Coulomb scattered
particles. The vertical collimators must be closed to very small apertures of
the order of 1mm, and therefore require a precise position control system. The
small apertures can induce Transverse Mode Coupling Instabilities (TMCI) of
the stored beam. They contribute to the overall machine impedance, resulting
in an upper limit on the bunch current for stable operation [Chao1999],

$I_{\rm thresh.}=\frac{4\pi f_{\rm s}E/e}{\sum\limits_{j}\beta_{j}k_{j}},$ (1)

where $I_{\rm thresh.}$ is the the bunch current threshold, $f_{\rm s}$ is
equal to 2.13kHz and 2.80kHz for the LER and HER synchrotron frequency,
respectively, $E$ is the beam energy, $e$ is the unit charge, and $\beta_{j}$
and $k_{j}$ are the beta function and kick factor [Natochii2021] of the $j$-th
collimator, respectively. In contrast, wide-open collimators increase beam
losses in the IR, while too-narrow collimators reduce beam lifetime and
injection efficiency. Therefore, each collimator should be set at the aperture
that optimally balances backgrounds, lifetimes, injection performance and
instabilities [Nakayama2012].

In our previous work on beam backgrounds [Natochii2021], the simulation of the
SuperKEKB collimation system was substantially improved, and it is now deemed
reliable.

### Detector shielding

While collimators successfully reduce single-beam losses inside Belle II, some
fraction of stray beam particles still escape the collimators and are lost
inside the detector. To protect the inner detectors from single-beam and
luminosity background showers, tungsten shields are installed just outside the
IP beam pipe and inside the vertex detector. In addition, thick tungsten
shields are also installed around the QCS, where the beam loss rate is
estimated to be the highest due to a large betatron function.

### IP beam pipe

The IP beam pipe of SuperKEKB is carefully designed to reduce the SR
background [Belle2TDR2010, Ohnishi2011]. SR from upstream of the IP is stopped
by a tapered collimation part of the incoming pipe so that SR will not hit the
central beryllium part of the IP beam pipe. Reflected SR will also not reach
the central IP beam pipe, thanks to a ridge structure on the tapered surface
of the incoming beam pipe. In addition, the effect of back-scattered SR is
significantly reduced in SuperKEKB compared to KEKB. Because there is a
separate QCS magnet for each ring, the outgoing beam orbit is almost straight
and does not produce an SR fan.

##  0.4 Beam instrumentation relevant to background measurements

This section lists the essential instrumentation (other than Belle II)
utilized to monitor beam parameters, the vacuum pressure in the beam pipe, and
background levels at SuperKEKB.

### Beam diagnostics

In SuperKEKB, transverse beam sizes are measured by X-ray beam profile
monitors (XRMs) and visible synchrotron radiation monitors (SRMs). For the
analyses reported here, XRM data are used. The X-ray imaging system uses
Cerium-doped yttrium-aluminum-garnet (YAG:Ce) scintillators combined with CMOS
cameras [MULYANI20191]. A coded aperture imaging technique provides turn-by-
turn vertical and horizontal beam size measurements with a spatial resolution
of the order of $\SI{1}{\micro m}$ and $\SI{10}{\micro m}$, respectively
[Mitsuka2022]. The bunch length is measured by a streak camera installed in
both rings, but dedicated machine time is required to perform measurements.
Therefore, bunch length data are measured only once a year to avoid stopping
of Belle II data taking. Instantaneous and integrated luminosity measurements
are provided by the Luminosity On-line Monitor (LOM), which is based on the
rate of Bhabha scattering events measured by the ECL [Kovalenko2020]. At a
counting rate of about 1Hz, the system’s statistic accuracy is 5% at a
luminosity of 1e34cm^-2.s^-1 and the overall systematic uncertainty is
estimated to be at the level of 1.7%.

### Vacuum system

The vacuum system of the collider is designed to effectively mitigate i)
higher order mode (HOM) power losses, ii) heat and gas loads due to the large
SR power and photon density, and iii) the electron cloud and fast ion effects
in the LER and HER, respectively. A distributed pumping system based on
multilayer non-evaporable getter (NEG) strips [SUETSUGU2008153] is used to
keep the vacuum pressure at the level of 100nPa, which is required to achieve
hours-long beam-gas lifetime. To measure the residual gas pressure in the
collider beam pipe, about 300 cold cathode gauges (CCGs) are installed around
each ring in roughly 10m intervals. These CCGs provide ultra-high vacuum
pressure measurements above 10nPa. A dedicated vacuum pressure simulation
shows that in the absence of circulating beams, the so-called base gas
pressure is almost the same at the center of the beam pipe as in the vicinity
of the CCG. However, due to the finite conductance of the vacuum system,
composed of the beam pipe, CCG, and vacuum pump, the dynamic part of the
pressure, which depends on the beam current and gas molecule desorption rate
from the inner beam pipe walls, is approximately three times higher at the
beam axis than at the CCGs. This factor of three is taken into account in the
beam-gas background study discussed later in the text.

SuperKEKB is instrumented with two residual gas analyzers (RGAs) in the D02
and D06 ring sections, see Fig. 3, to measure the molecular composition of
residual gases in the beam pipe. These devices are mass spectrometers
measuring mass-to-charge ratios of gas ion fragments. Due to the small number
of RGAs, reliable information regarding the gas composition distribution
around the collider is currently unavailable. Therefore, in simulation, an
effective atomic number of $Z_{\rm eff.}=7$, mostly due to CO molecules, is
assumed for the residual gas in the beam pipe [Sahu1997].

### Background monitors

Several dedicated detector systems are used to monitor backgrounds in the
SuperKEKB IR and tunnel, and to ensure safe machine and detector operation.

  * •

Diamond sensor-based detectors (Diamonds) [Bacher2021] are used for radiation
dose rate measurements in the IR, as shown in Fig. 4, where rectangles
highlighted in blue and green indicate detectors used for dose rate monitoring
at a 10Hz readout rate and fast beam abort monitoring at 400kHz, respectively.

  * •

The sCintillation Light And Waveform Sensors (CLAWS) detector system
[Gabriel2021] is based on plastic scintillators and silicon photomultipliers.
It monitors beam-induced backgrounds synchronized with the SuperKEKB
injection. There are in total 32 CLAWS modules with 16 on the forward and 16
on the backward side of the IR around the final focusing magnets. The modules
are located in four different longitudinal positions along the beam direction
(approximately 1, 2, 3, and 4m from the IP) and four different azimuth angles
(0∘, 90∘, 180∘ and 270∘) on each of the magnets.

  * •

The BEAST TPC system uses six compact, high-resolution gaseous detectors
[Jaegle:2019jpx] to provide directional and spectral measurements of the fast
neutron flux [Schueler2021]. Currently, the detectors are located in the
accelerator tunnel near Belle II.

  * •

Four ^3He tube detectors [deJong2017], installed around Belle II, count
thermal neutrons with kinetic energy below about 0.025eV through the following
process:
^3_2He + ^1_0n →^3_1H + ^1_1H + 764keV

  * •

PIN photo-diodes [Ikeda2014] installed next to each collimator are used for
fast beam loss monitoring around the movable jaws.

  * •

5-m-long ion chambers [Ikeda2014] are mounted in cable racks on the outer wall
along the accelerator tunnel. These air-filled gaseous detectors are used to
measure beam losses.

  * •

New loss monitors, based on CsI-crystals with photo-multiplier tubes (PMTs)
and electron-multiplier tubes (EMTs), were recently installed near SuperKEKB
collimators. These new systems with good time synchronization capabilities are
now used to pin down the location of sudden beam losses around the rings.

### Beam abort system

A dedicated fast beam abort system is used to dump unstable beams in order to
avoid severe machine or detector damage. During commissioning Phases 1 and 2,
the abort system included Diamonds (green rectangles, see Fig. 4), PIN photo-
diodes and ionization chambers. In Phase 3, the system was augmented by
including the four forward and four backward CLAWS detectors closest to the
IP. These detectors can trigger a beam abort $\sim\SI{10}{\micro s}$ earlier
than Diamonds, on average.

Figure 4: Locations of diamond detectors in the IR. The dashed rectangles show
the distance from the IP along the beam axis for each group of detectors.
Azimuth angles of the detectors are indicated in rectangles. See text for
further details.