# Vacuum Landscaping: Cause of Nonlocal Influences without Signalling

Gerhard Grössing<EMAIL_ADDRESS>http://www.nonlinearstudies.at Siegfried Fussy
Johannes Mesa Pascasio  Herbert Schwabl  Austrian Institute for Nonlinear
Studies, Akademiehof, Friedrichstr. 10, 1010 Vienna, Austria

###### Abstract

In the quest for an understanding of nonlocality with respect to an
appropriate ontology, we propose a “cosmological solution“. We assume that
from the beginning of the universe each point in space has been the location
of a scalar field representing a zero-point vacuum energy that nonlocally
vibrates at a vast range of different frequencies across the whole universe. A
quantum, then, is a nonequilibrium steady state in the form of a “bouncer“
coupled resonantly to one of those (particle type dependent) frequencies, in
remote analogy to the bouncing oil drops on an oscillating oil bath as in
Couder’s experiments. A major difference to the latter analogy is given by the
nonlocal nature of the vacuum oscillations.

We show with the examples of double- and $n$-slit interference that the
assumed nonlocality of the distribution functions alone suffices to derive the
de~Broglie–Bohm guiding equation for $N$ particles with otherwise purely
classical means. In our model, no influences from configuration space are
required, as everything can be described in 3-space. Importantly, the setting
up of an experimental arrangement limits and shapes the forward and osmotic
contributions and is described as vacuum landscaping.

##  1 Introduction: Quantum Mechanics without Wavefunctions

“Emergent Quantum Mechanics” stands for the idea that quantum mechanics is
based on a more encompassing deeper level theory. This counters the
traditional belief, usually expressed in the context of orthodox Copenhagen-
type quantum mechanics, that quantum theory is an “ultimate” theory whose main
features will prevail for all time and will be applicable to all questions of
physics. Note, for example, that even in more recent approaches to spacetime,
the concept of an “emergent spacetime” is introduced as a description even of
space and time emerging from basic quantum mechanical entities. This, of
course, need not be so, considering the fact that there is “plenty of room at
the bottom”, i.e. as Feynman implied, between present-day resolutions and
minimally possible times and distances, which could in principle be way below
resolutions reasonably argued about in present times (i.e. on Planck scales).

One of the main attractive features of the de Broglie–Bohm interpretation of
the quantum mechanical formalism, and of Bohmian mechanics as well, lies in
the possibility to extend its domain into space and/or time resolutions where
modified behaviours different from quantum mechanical ones may be expected. In
other words, there may be new physics involved that would require an
explicitly more encompassing theory than quantum mechanics, i.e. a deeper
level theory. Our group’s approach, which we pursued throughout the last 10
years, is characterized by the search for such a theory under the premise that
even for nonrelativistic quantum mechanics, the Schrödinger equation cannot be
an appropriate starting point, since the wavefunction is still lacking a firm
theoretical basis and its meaning is generally not agreed upon.

For a similar reason, also the de Broglie–Bohm theory cannot be our starting
point, as it is based on the Schrödinger equation and the use of the
wavefunction to begin with. Rather, we aim at an explicit ansatz for a deeper
level theory without wavefunctions, from which the Schrödinger equation, or
the de Broglie–Bohm guiding equation, can be derived. We firmly believe that
we have accomplished this and we can now proceed to study consequences of the
approach beyond orthodox expectations.

Throughout recent years, apart from our own model, several approaches to a
quantum mechanics without wavefunctions have been proposed Deckert.2007quantum
; Poirier.2010bohmian ; Poirier.2012trajectory-based ;
Schiff.2012communication: ; Hall.2014quantum . These refer to “many classical
worlds” which provide Bohm-type trajectories with certain repulsion effects.
From our realistic point of view, the true ontologies of these models,
however, do not become apparent. So let us turn to our model. As every
physical theory is based on metaphysical assumptions, we must make clear what
our assumptions are. Here they are.

We propose a “cosmological solution” in that the Big Bang, or any other model
explaining the apparent expansion of the universe, is essentially related to
the vacuum energy. (The latter may constitute what is called the dark energy,
but we do not need to specify this here.) We assume that from the beginning of
the universe each point in space has been the location of a scalar field
representing a zero-point vacuum energy that vibrates at a vast range of
different frequencies across the whole universe. More specifically, we
consider the universe as an energetically open system where the vacuum energy
not only drives expansion, but also each individual “particle” oscillation
$\omega=E/\hbar$ in the universe. In order to maintain a particular frequency,
any such oscillator must be characterized by a throughput of energy external
to it. In this regard, we have time and again employed the analogy of Couder’s
experiments with bouncing oil drops on a vibrating bath Couder.2005 ;
Couder.2006single-particle ; Couder.2012probabilities ; Bush.2010quantum ;
Bush.2015new ; Bush.2015pilot-wave : The bouncer/particle is always in
resonant interaction with a relevant environment.

Our model, though also largely classical, has a very different ontology from
the “many classical worlds” one. We consider _one_ “superclassical” world
instead: a purely classical world plus “cosmological nonlocality”, i.e. a
nonlocal bath for every oscillator/particle due to the all-pervading vacuum
energy, which – mostly in the context of quantum mechanics – is called the
zero-point energy. So, it is the one classical world together with the
fluctuating environment related to the vacuum energy that enters our
definition of a quantum as an emergent system. The latter consists of a
bouncer and an undulatory/wave-like nonlocal environment defined by proper
boundary conditions.111As an aside we note that this is not related to de
Broglie’s “nonlinear wave mechanics” DeBroglie.1960book , as there the
nonlinear wave, with the particle as soliton-like singularity, is considered
as one ontic entity. In our case, however, we speak of two separate, though
synchronous elements: local oscillators and generally nonlocal oscillating
fields.

In previous work, we have shown how the Schrödinger equation can be derived
from a nonequilibrium sub-quantum dynamics Groessing.2008vacuum ;
Groessing.2010emergence ; Groessing.2011dice ; Groessing.2012doubleslit ,
where in accordance with the model sketched above the particle is considered
as a steady state with a constant throughput of energy. This, then, leads to
the two-momenta approach to emergent quantum mechanics which shall be outlined
in the next section.

##  2 The Two-Momenta Approach to Emergent Quantum Mechanics

We consider the empirical fact that each particle of nature is attributed an
energy $E=\hbar\omega$ as one of the essential features of quantum
systems.222We have also presented a classical explanation for this relation
from our sub-quantum model Groessing.2011explan , but do not need to use the
details for our present purposes. Oscillations, characterized by some typical
angular frequency $\omega$, are described as properties of off-equilibrium
steady-state systems. ”Particles” can then be assumed to be dissipative
systems maintained in a nonequilibrium steady-state by a permanent throughput
of energy, or heat flow, respectively. The heat flow must be described by an
external kinetic energy term. Then the energy of the total system, i.e. of the
particle and it’s thermal context, becomes

$E_{{\rm tot}}=\hbar\omega+\frac{(\delta\mathbf{p})^{2}}{2m}\;,$ (2.1)

where $\delta\mathbf{p}$ is an additional, fluctuating momentum component of
the particle of mass $m$.

We assume that an effect of said thermal context is given by detection
probability distributions which are wave-like in the particle’s surroundings.
Thus, the detection probability density $P(\mathbf{x},t)$ is considered to
coincide with a classical wave’s intensity
$I(\mathbf{x},t)=R^{2}(\mathbf{x},t),$ with $R(\mathbf{x},t)$ being the wave’s
real-valued amplitude

$P(\mathbf{x},t)=R^{2}(\mathbf{x},t)\;,\quad\text{with normalization}\,\int
P\,\mathrm{d}^{n}x=1\;.$ (2.2)

In ref. Groessing.2008vacuum , we combine some results of nonequilibrium
thermodynamics with classical wave mechanics. We propose that the many
microscopic degrees of freedom associated with the hypothesized sub-quantum
medium can be recast into the emergent macroscopic properties of the wave-like
behaviour on the quantum level. Thus, for the relevant description of the
total system one no longer needs the full phase space information of all
microscopic entities, but only the emergent particle coordinates.

For implementation, we model a particle as being surrounded by a heat bath,
i.e. a reservoir that is very large compared to the small dissipative system,
such that that the momentum distribution in this region is given by the usual
Maxwell–Boltzmann distribution. This corresponds to a “thermostatic”
regulation of the reservoir’s temperature, which is equivalent to the
statement that the energy lost to the thermostat can be regarded as heat.
Thus, one can formulate a proposition of emergence Groessing.2008vacuum
providing the equilibrium-type probability (density) ratio

$\frac{P(\mathbf{x},t)}{P(\mathbf{x},0)}={\rm e}^{-\frac{\Delta Q(t)}{kT}}\;,$
(2.3)

with $k$ being Boltzmann’s constant, $T$ the reservoir temperature, and
$\Delta Q(t)$ the heat that is exchanged between the particle and its
environment.

Equations (2.1), (2.2), and (2.3) are the only assumptions necessary to derive
the Schrödinger equation from (modern) classical mechanics. We need to employ
only two additional well-known results. The first is given by Boltzmann’s
formula for the slow transformation of a periodic motion (with period
$\tau=2\pi/\omega$) upon application of a heat transfer $\Delta Q$. This is
needed as we deal with an oscillator of angular frequency $\omega$ in a heat
bath $Q$, and a change in the vacuum surroundings of the oscillator will come
as a heat transfer $\Delta Q$ . The latter is responsible for a change $\delta
S$ of the action function $S$ representing the effect of the vacuum’s “zero-
point” fluctuations. With the action function $S=\int\left(E_{{\rm
kin}}-V\right)\,\mathrm{d}t$, the relation between heat and action was first
given by Boltzmann Boltzmann.1866uber ,

$\Delta Q(t)=2\omega[\delta S(t)-\delta S(0)]\;.$ (2.4)

Finally, the requirement that the average kinetic energy of the thermostat
equals the average kinetic energy of the oscillator is given, for each degree
of freedom, by

$\frac{kT}{2}=\frac{\hbar\omega}{2}\;.$ (2.5)

Combining these two results, Eqs. (2.4) and (2.5), with (2.3) one obtains

$P(\mathbf{x},t)=P(\mathbf{x},0){\rm e}^{-\frac{2}{\hbar}[\delta
S(\mathbf{x},t)-\delta S(\mathbf{x},0)]}\;,$ (2.6)

from which follows the expression for the momentum fluctuation
$\delta\mathbf{p}$ of (2.1) as

$\delta\mathbf{p}(\mathbf{x},t)=\nabla(\delta
S(\mathbf{x},t))=-\frac{\hbar}{2}\frac{\nabla
P(\mathbf{x},t)}{P(\mathbf{x},t)}\;.$ (2.7)

This, then, provides the additional kinetic energy term for one particle as

$\delta E_{{\rm kin}}=\frac{1}{2m}\nabla(\delta S)\cdot\nabla(\delta
S)=\frac{1}{2m}\left(\frac{\hbar}{2}\frac{\nabla P}{P}\right)^{2}\;.$ (2.8)

Thus, writing down a classical action integral for $j=N$ particles in
$m$-dimensional space, including this new term for each of them, yields (with
external potential $V$)

$A=\int L\,\mathrm{d}^{m}x\,\mathrm{d}t=\int P\left[\frac{\partial S}{\partial
t}+\sum_{j=1}^{N}\frac{1}{2m_{j}}\nabla_{j}S\cdot\nabla_{j}S+\sum_{j=1}^{N}\frac{1}{2m_{j}}\left(\frac{\hbar}{2}\frac{\nabla_{j}P}{P}\right)^{2}+V\right]\,\mathrm{d}^{m}x\,\mathrm{d}t\;,$
(2.9)

where the probability density
$P=P(\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{N},t)$.

With the definition of forward and osmotic velocities, respectively,

$\mathbf{v}_{j}:=\frac{\mathbf{p}_{j}}{m_{j}}=\frac{\nabla_{j}S}{m_{j}}\qquad\textrm{
and
}\qquad\mathbf{u}_{j}:=\frac{\delta\mathbf{p}_{j}}{m_{j}}=-\frac{\hbar}{2m_{j}}\frac{\nabla_{j}P}{P},$
(2.10)

one can rewrite (2.9) as

$A=\int L\,\mathrm{d}^{m}x\,\mathrm{d}t=\int P\left[\frac{\partial S}{\partial
t}+V+\sum_{j=1}^{N}\frac{m_{j}}{2}\mathbf{v}_{j}^{2}+\sum_{j=1}^{N}\frac{m_{j}}{2}\mathbf{u}_{j}^{2}\right]\,\mathrm{d}^{m}x\,\mathrm{d}t\;.$
(2.11)

This can be considered as the basis for our approach with two momenta, i.e.
the forward momentum $m\mathbf{v}$ and the osmotic momentum $m\mathbf{u}$,
respectively. At first glance, the Lagrangian in Eq. (2.11) looks completely
classical, with two kinetic energy terms per particle instead of one. However,
due to the particular nature of the osmotic momentum as given in Eq. (2.10),
nonlocal influences are introduced: even at long distances away from the
particle location, where the particle’s contribution to $P$ is practically
negligibly small, the expression of the form $\frac{\nabla_{j}P}{P}$ may be
large and affects immediately the whole fluctuating environment. This is why
the osmotic variant of the kinetic energy makes all the difference to the
usual classical mechanics, or, in other words, is the basis for quantum
mechanics.

Introducing now the Madelung transformation

$\psi=R\;{\rm e}^{\frac{{\rm i}}{\hbar}S}\;,$ (2.12)

where $R=\sqrt{P}$ as in (2.2), one has, with bars denoting averages,

$\overline{\left|\frac{\nabla_{j}\psi}{\psi}\right|^{2}}:=\int\,\mathrm{d}^{m}x\,\mathrm{d}t\left|\frac{\nabla_{j}\psi}{\psi}\right|^{2}=\overline{\left(\frac{1}{2}\frac{\nabla_{j}P}{P}\right)^{2}}+\overline{\left(\frac{\nabla_{j}S}{\hbar}\right)^{2}}\;,$
(2.13)

and one can rewrite (2.9) as

$A=\int
L\,\mathrm{d}^{m}x\,\mathrm{d}t=\int\,\mathrm{d}^{m}x\,\mathrm{d}t\left[|\psi|^{2}\left(\frac{\partial
S}{\partial
t}+V\right)+\sum_{j=1}^{N}\frac{\hbar^{2}}{2m_{j}}|\nabla_{j}\psi|^{2}\right]\;.$
(2.14)

Thus, with the identity $|\psi|^{2}\frac{\partial S}{\partial t}=-\frac{{\rm
i}\hbar}{2}(\psi^{*}\dot{\psi}-\dot{\psi}^{*}\psi)$, one obtains the familiar
Lagrange density

$L=-\frac{{\rm
i}\hbar}{2}(\psi^{*}\dot{\psi}-\dot{\psi}^{*}\psi)+\sum_{j=1}^{N}\frac{\hbar^{2}}{2m_{j}}\nabla_{j}\psi\cdot\nabla_{j}\psi^{*}+V\psi^{*}\psi\;,$
(2.15)

from which by the usual procedures one arrives at the $N$-particle Schrödinger
equation

${\rm i}\hbar\frac{\partial\psi}{\partial
t}=\left(-\sum_{j=1}^{N}\frac{\hbar^{2}}{2m_{j}}\nabla_{j}^{2}+V\right)\psi\;.$
(2.16)

Note also that from (2.9) one obtains upon variation in $P$ the modified
Hamilton–Jacobi equation familiar from the de Broglie–Bohm interpretation,
i.e.

$\frac{\partial S}{\partial
t}+\sum_{j=1}^{N}\frac{(\nabla_{j}S)^{2}}{2m_{j}}+V(\mathbf{x}_{1},\mathbf{x}_{2},\dots,\mathbf{x}_{N},t)+U(\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{N},t)=0\;,$
(2.17)

where $U$ is known as the “quantum potential”

$U(\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{N},t)=\sum_{j=1}^{N}\frac{\hbar^{2}}{4m_{j}}\left[\frac{1}{2}\left(\frac{\nabla_{j}P}{P}\right)^{2}-\frac{\nabla_{j}^{2}P}{P}\right]=-\sum_{j=1}^{N}\frac{\hbar^{2}}{2m_{j}}\frac{\nabla_{j}^{2}R}{R}\;.$
(2.18)

Moreover, with the definitions of $\mathbf{u}_{j}$ in (2.10) one can rewrite
$U$ as

$U=\sum_{j=1}^{N}\left[\frac{m_{j}\mathbf{u}_{j}^{2}}{2}-\frac{\hbar}{2}(\nabla_{j}\cdot\mathbf{u}_{j})\right]\;.$
(2.19)

However, as was already pointed out in ref. Groessing.2008vacuum , with the
aid of (2.4) and (2.6), $\mathbf{u}_{j}$ can also be written as

$\mathbf{u}_{j}=\frac{1}{2\omega_{j}m_{j}}\nabla_{j}Q\;,$ (2.20)

which thus explicitly shows its dependence on the spatial behaviour of the
heat flow $\delta Q$. Insertion of (2.20) into (2.19) then provides the
thermodynamic formulation of the quantum potential as

$U=\sum_{j=1}^{N}\frac{\hbar^{2}}{4m_{j}}\left[\frac{1}{2}\left(\frac{\nabla_{j}Q}{\hbar\omega_{j}}\right)^{2}-\frac{\nabla_{j}^{2}Q}{\hbar\omega_{j}}\right]\;.$
(2.21)

As in our model particles and fields are dynamically interlocked, it would be
highly misleading to picture the quantum potential in a manner similar to the
classical scenario of particle plus field, where the latter can be switched on
and off like an ordinary potential. Contrariwise, in our case the particle
velocities/momenta must be considered as _emergent_. One can illustrate this
with the situation in double-slit interference (Figure 2.1). Considering an
incoming beam of, say, electrons with wave number $\mathbf{k}$ impinging on a
wall with two slits, two beams with wave numbers $\mathbf{k}_{A}$ and
$\mathbf{k}_{B}$, respectively, are created, which one may denote as “pre-
determined” quantities, resulting also in pre-determined velocities
$\mathbf{v}_{\alpha}=\frac{1}{m}\hbar\mathbf{k}_{\alpha}$,
$\alpha=A\>\mathrm{or}\>B$.

However, if one considers that the electrons are not moving in empty space,
but in an undulatory environment created by the ubiquitous zero-point field
“filling” the whole experimental setup. One has to combine all the
velocities/momenta at a given point in space and time in order to compute the
resulting, or emergent, velocity/momentum field
$\mathbf{v}_{i}=\frac{1}{m}\hbar\boldsymbol{\kappa}_{i}$,
$i=1\,\mathrm{or}\,2$ (Figure 2.1), where $i$ is a bookkeeping index not
necessarily related to the particle coming from a particular slit
Fussy.2014multislit . The relevant contributions other than the particle’s
forward momentum $m\mathbf{v}$ originate from the osmotic momentum
$m\mathbf{u}$. The latter is well known from Nelson’s stochastic theory
Nelson.1966derivation , but its identical form has been derived by one of us
from an assumed sub-quantum nonequilibrium thermodynamics Groessing.2008vacuum
; Groessing.2009origin  as it was described above. As shall be shown in the
next section, our model also provides an understanding and deeper-level
explanation of the microphysical, causal processes involved, i.e. of the
guiding law Groessing.2015implications  of the de Broglie–Bohm theory.

Figure 2.1: Scheme of interference at a double-slit. Considering an incoming
beam of electrons with wave number $\mathbf{k}$ impinging on a wall with two
slits, two beams with wave numbers $\mathbf{k}_{A}$ and $\mathbf{k}_{B}$,
respectively, are created, which one may denote as “pre-determined” velocities
$\mathbf{v}_{\alpha}=\frac{1}{m}\hbar\mathbf{k}_{\alpha}\textrm{,\>$\alpha$=$A$\>or\>$B$.}$
Taking into account the influences of the osmotic momentum field
$m\mathbf{u}$, one has to combine all the velocities/momenta at a given point
in space and time in order to compute the resulting, or emergent,
velocity/momentum field
$\mathbf{v}_{i}=\frac{1}{m}\hbar\boldsymbol{\kappa}_{i},\>\emph{i}=1\mathrm{\>or\>}2$.
This, then, provides the correct intensity distributions and average
trajectories (lower plane).

##  3 Derivation of the De Broglie–Bohm Guiding Equation for _$N$_ Particles

Consider at first one particle in an $n$-slit system. In quantum mechanics, as
well as in our emergent quantum mechanics approach, one can write down a
formula for the total intensity distribution $P$ which is very similar to the
classical formula. For the general case of $n$ slits, it holds with phase
differences $\varphi_{ii^{\prime}}=\varphi_{i}-\varphi_{i^{\prime}}$ between
the slits $i$, $i^{\prime}$ that

$P=\sum_{i=1}^{n}\left(P_{i}+\sum_{i^{\prime}=i+1}^{n}2R_{i}R_{i^{\prime}}\cos\varphi_{ii^{\prime}}\right),$
(3.1)

where the phase differences are defined over the whole domain of the
experimental setup. As in our model the “particle” is actually a bouncer in a
fluctuating wave-like environment, i.e. analogously to the bouncers of the
Couder experiments, one does have some (e.g. Gaussian) distribution, with its
centre following the Ehrenfest trajectory in the free case, but one also has a
diffusion to the right and to the left of the mean path which is just due to
that stochastic bouncing. Thus the total velocity field of our bouncer in its
fluctuating environment is given by the sum of the forward velocity
$\mathbf{v}$ and the respective osmotic velocities $\mathbf{u}_{\mathrm{L}}$
and $\mathbf{u}_{\mathrm{R}}$ to the left and the right. As for any direction
$\alpha$ the osmotic velocity
$\mathbf{u}_{\alpha}=\frac{\hbar}{2m}\frac{\nabla P}{P}$ does not necessarily
fall off with the distance, one has long effective tails of the distributions
which contribute to the nonlocal nature of the interference phenomena
Groessing.2013dice . In sum, one has three distinct velocity (or current)
channels per slit in an $n$-slit system.

We have previously shown Fussy.2014multislit ; Groessing.2014relational  how
one can derive the Bohmian guidance formula from our two-momenta approach.
Introducing classical wave amplitudes $R(\mathbf{w}_{i})$ and generalized
velocity field vectors $\mathbf{w}_{i}$, which represent either a forward
velocity $\mathbf{v}$ or an osmotic velocity $\mathbf{u}$ in the direction
transversal to $\mathbf{v}$, we calculate the phase-dependent amplitude
contributions of the total system’s wave field projected on one channel’s
amplitude $R(\mathbf{w}_{i})$ at the point $(\mathbf{x},t)$ in the following
way. We define a _relational intensity_ $P(\mathbf{w}_{i})$ as the local wave
intensity $P(\mathbf{w}_{i})$ in each channel (i.e. $\mathbf{w}_{i}$),
recalling that there are 3 velocity channels per slit:
$\mathbf{u}_{\mathrm{L}}$, $\mathbf{u}_{\mathrm{R}}$, and $\mathbf{v}$. The
sum of all relational intensities, then, is the total intensity, i.e. the
total probability density. In an $n$-slit system, we thus obtain for the
relational intensities and the corresponding currents, respectively, i.e. for
each channel component $\mathit{i}$,

$\displaystyle P(\mathbf{w}_{i})$
$\displaystyle=R(\mathbf{w}_{i})\mathbf{\hat{w}}_{i}\cdot{\displaystyle\sum_{i^{\prime}=1}^{3n}}\mathbf{\hat{w}}_{i^{\prime}}R(\mathbf{w}_{i^{\prime}})$
(3.2) $\displaystyle\mathbf{J}\mathrm{(}\mathbf{w}_{i}\mathrm{)}$
$\displaystyle=\mathbf{w}_{i}P(\mathbf{w}_{i}),\qquad i=1,\ldots,3n$ (3.3)

with unit vectors $\mathbf{\hat{w}}_{i}$ and

$\cos\varphi_{ii^{\prime}}:=\mathbf{\hat{w}}_{i}\cdot\mathbf{\hat{w}}_{i^{\prime}}\,.$
(3.4)

Consequently, the total intensity and current of our field read as

$\displaystyle P_{\mathrm{tot}}=$
$\displaystyle{\displaystyle\sum_{i=1}^{3n}}P(\mathbf{w}_{i})=\left({\displaystyle\sum_{i=1}^{3n}}\mathbf{\hat{w}}_{i}R(\mathbf{w}_{i})\right)^{2}$
(3.5) $\displaystyle\mathbf{J}_{\mathrm{tot}}=$
$\displaystyle\sum_{i=1}^{3n}\mathbf{J}(\mathbf{w}_{i})={\displaystyle\sum_{i=1}^{3n}}\mathbf{w}_{i}P(\mathbf{w}_{i}),$
(3.6)

leading to the emergent total velocity

$\mathbf{v}_{\mathrm{tot}}=\frac{\mathbf{J}_{\mathrm{tot}}}{P_{\mathrm{tot}}}=\frac{{\displaystyle\sum_{i=1}^{3n}}\mathbf{w}_{i}P(\mathbf{w}_{i})}{{\displaystyle\sum_{i=1}^{3n}}P(\mathbf{w}_{i})}\,,$
(3.7)

which represents the probability flux lines.

In Groessing.2012doubleslit ; Fussy.2014multislit  we have shown with the
example of $n=2,$ i.e. a double-slit system, that Eq. (3.7) can equivalently
be written in the form

$\mathbf{v}_{\mathrm{tot}}=\frac{R_{1}^{2}\mathbf{v}_{\mathrm{1}}+R_{2}^{2}\mathbf{v}_{\mathrm{2}}+R_{1}R_{2}\left(\mathbf{v}_{\mathrm{1}}+\mathbf{v}_{2}\right)\cos\varphi+R_{1}R_{2}\left(\mathbf{u}_{1}-\mathbf{u}_{2}\right)\sin\varphi}{R_{1}^{2}+R_{2}^{2}+2R_{1}R_{2}\cos\varphi}\,.$
(3.8)

The trajectories or streamlines, respectively, are obtained according to
$\mathbf{\dot{x}}=\mathbf{v}_{\mathrm{tot}}$ in the usual way by integration.
As we have first shown in Groessing.2012doubleslit , by re-inserting the
expressions for forward and osmotic velocities, respectively, i.e.

$\mathbf{v}_{i}=\frac{\nabla
S_{i}}{m}\,,\qquad\mathbf{u}_{i}=-\frac{\hbar}{m}\frac{\nabla
R_{i}}{R_{i}}\,,$ (3.9)

one immediately identifies Eq. (3.8) with the Bohmian guidance formula.
Naturally, employing the Madelung transformation for each slit $\alpha$
($\alpha=1$ or $2$),

$\psi_{\alpha}=R_{\alpha}{\rm e}^{\mathrm{i}S_{\alpha}/\hbar},$ (3.10)

and thus
$P_{\alpha}=R_{\alpha}^{2}=|\psi_{\alpha}|^{2}=\psi_{\alpha}^{*}\psi_{\alpha}$,
with $\varphi=(S_{1}-S_{2})/\hbar$, and recalling the usual trigonometric
identities such as $\cos\varphi=\frac{1}{2}\left({\rm
e}^{\mathrm{i}\varphi}+{\rm e}^{-\mathrm{i}\varphi}\right)$, one can rewrite
the total average current immediately in the usual quantum mechanical form as

$\begin{array}[]{rl}{\displaystyle\mathbf{J}_{{\rm tot}}}&=P_{{\rm
tot}}\mathbf{v}_{{\rm tot}}\\\
&={\displaystyle(\psi_{1}+\psi_{2})^{*}(\psi_{1}+\psi_{2})\frac{1}{2}\left[\frac{1}{m}\left(-\mathrm{i}\hbar\frac{\nabla(\psi_{1}+\psi_{2})}{(\psi_{1}+\psi_{2})}\right)+\frac{1}{m}\left(\mathrm{i}\hbar\frac{\nabla(\psi_{1}+\psi_{2})^{*}}{(\psi_{1}+\psi_{2})^{*}}\right)\right]}\\\
&={\displaystyle-\frac{\mathrm{i}\hbar}{2m}\left[\Psi^{*}\nabla\Psi-\Psi\nabla\Psi^{*}\right]={\displaystyle\frac{1}{m}{\rm
Re}\left\\{\Psi^{*}(-\mathrm{i}\hbar\nabla)\Psi\right\\},}}\end{array}$ (3.11)

where $P_{{\rm tot}}=|\psi_{1}+\psi_{2}|^{2}=:|\Psi|^{2}$.

Eq. (3.7) has been derived for one particle in an $n$-slit system. However,
for the spinless particles obeying the Schrödinger equation333As we do not yet
have a relativistic model involving spin our results for the many-particle
case cannot account for the difference in particle statistics, i.e. for
fermions or bosons. This will be a task for future work. it is straightforward
to extend this derivation to the many-particle case. Due to the purely
additive terms in the expressions for the total current and total probability
density, respectively, also for _$N$_ particles, Eqs. (3.5) and (3.6) become

$\displaystyle P_{\mathrm{tot,}N}$
$\displaystyle=\sum_{j=1}^{N}\left[{\displaystyle\sum_{i=1}^{3n}}P(\mathbf{w}_{i})\right]_{j}=\sum_{j=1}^{N}\left[\left({\displaystyle\sum_{i=1}^{3n}}\mathbf{\hat{w}}_{i}R(\mathbf{w}_{i})\right)^{2}\right]_{j}\,,$
(3.12) $\displaystyle\mathbf{J}_{\mathrm{tot,}N}$
$\displaystyle=\sum_{j=1}^{N}\left[\sum_{i=1}^{3n}\mathbf{J}(\mathbf{w}_{i})\right]_{j}=\sum_{j=1}^{N}\left[{\displaystyle\sum_{i=1}^{3n}}\mathbf{w}_{i}P(\mathbf{w}_{i})\right]_{j}\,,$
(3.13)

and, analogously, Eq. (3.7),

$\displaystyle\mathbf{v}_{\mathrm{tot,}N}$
$\displaystyle=\frac{\mathbf{J}_{\mathrm{tot}}}{P_{\mathrm{tot}}}=\frac{{\displaystyle\sum_{j=1}^{N}}\left[{\displaystyle\sum_{i=1}^{3n}}\mathbf{w}_{i}P(\mathbf{w}_{i})\right]_{j}}{{\displaystyle\sum_{j=1}^{N}}\left[{\displaystyle\sum_{i=1}^{3n}}P(\mathbf{w}_{i})\right]_{j}}\,,$
(3.14)

where $\mathbf{w}_{i}$ is dependent on the velocities (3.9) with different
$S_{i}$ and $R_{i}$ for every $j$. In quantum mechanical terms the only
difference now is that the currents’ nabla operators have to be applied at all
of the locations of the respective _$N$_ particles, thus providing

${\displaystyle\mathbf{J}_{{\rm
tot}}}\left(N\right)={\displaystyle\sum_{j=1}^{N}}\frac{1}{m_{j}}{\rm
Re}\left\\{\Psi^{*}\left(t\right)(-\mathrm{i}\hbar\nabla_{j})\Psi\left(t\right)\right\\},$
(3.15)

where $\Psi\left(t\right)$ now is the total $N$-particle wave function,
whereas the flux lines are given by

$\mathbf{v}_{j}\left(t\right)=\frac{\hbar}{m_{j}}\mathrm{Im}\frac{\nabla_{j}\Psi\left(t\right)}{\Psi\left(t\right)}\qquad\forall
j=1,...,N.$ (3.16)

In sum, with our introduction of a relational intensity $P(\mathbf{w}_{i})$
for channels $\mathbf{w}_{i}$, which include sub-quantum velocity fields, we
obtain the guidance formula also for $N$-particle systems in real
3-dimensional space._The central ingredient for this to be possible is to
consider the emergence of the velocity field from the interplay of the
totality of all of the system’s velocity channels._

Figure 3.1: Classical computer simulation of the interference pattern:
intensity distribution with increasing intensity from white through yellow and
orange, with trajectories (red) for two Gaussian slits, and with large
dispersion (evolution from bottom to top; $v_{x,1}=v_{x,2}=0$).  Figure 3.2:
Classical computer simulation of the interference pattern: intensity
distribution with increasing intensity from white through yellow and orange,
with trajectories (red) for two Gaussian slits, and with small dispersion
(evolution from bottom to top; $v_{x,1}=-v_{x,2}$).

In Figures 3.1 and 3.2, trajectories (flux lines) for two Gaussian slits are
shown (from ref. Groessing.2012doubleslit ). These trajectories are in full
accordance with those obtained from the Bohmian approach, as can be seen by
comparison with references Holland.1993 , Bohm.1993undivided , and
Sanz.2009context , for example.

##  4 Vacuum Landscaping: Cause of nonlocal influences without signalling

In the foregoing sections, we pointed out how nonlocality appears in our
model. Particularly in discussing Eqs. (2.9)–(2.11), it was shown that the
form of the osmotic momentum

$m\mathbf{u}=-\frac{\hbar}{2}\frac{\nabla P}{P}$ (4.1)

may be responsible for relevant influences. Moreover, if one assumes a
particle at some position $\mathbf{x}$ in space, and with a probability
distribution $P$, the latter is a distribution around $\mathbf{x}$ with long
tails across the whole experimental setup which may be very thin but still
non-zero. Then, even at locations $\mathbf{y}$ very remote from $\mathbf{x}$,
and although the probability distribution $P$ pertaining to the far-away
particle might be minuscule, it still may become effective immediately through
the zero-point field.

The physical reason for bringing in nonlocality is the assumed resonant
coupling of the particle(s) with fluctuations of the zero-point vacuum filling
the whole experimental setup. Take, for example, a typical “Gaussian slit”. We
effectively describe $P$ by a Gaussian with long non-zero tails throughout the
whole apparatus. As we have seen, in order to calculate on-screen
distributions (i.e. total intensities) of particles which went one-at-a-time
through an $n$-slit device, one only needs a two-momentum description and a
calculation which uses the totality of all relational intensities involving
the relative phases determined across the whole apparatus.

In general, we propose a resonant interaction of the bouncing “particle” with
a _relevant environment_.444In a similar vein, Bohm Bohm.1980wholeness  speaks
of a “relatively independent subtotality” of the universe, to account for the
possible neglect of the “rest of the universe” in practical calculations. For
idealized, non-interacting particles, this relevant environment would be the
whole universe, and thus the idealized prototype of the “cosmological
solution” referred to in the introduction.

For any particle in any experimental setup, however, the relevant environment
is defined by the boundary conditions of the apparatus. Whereas the idealized
one-particle scenario would constitute an indefinite order of vibrations
w.r.t. the particle oscillations potentially locking in, the very building up
of an experiment may represent a dynamical transition from this indefinite
order to the establishment of a definite order. The latter is characterized by
the emergence of standing waves between the boundaries of the apparatus (like,
e.g., source and detector), to which the particle oscillations lock in.
Moreover, if an experimenter decides to change the boundary conditions (e.g.,
by altering the probability landscape between source and detector), such a
“switching” would establish yet another definite order. The introduction or
change of boundary conditions, which immediately affects the probability
landscape, and the forward and the osmotic fields, we term “vacuum
landscaping”.

In other words, the change of boundary conditions of an experimental
arrangement constitutes the immediate transition from one cosmological
solution in the relevant environment (i.e. within the old boundary conditions)
to another (i.e. the new ones). The “surfing” bouncer/particle just locally
jumps from the old to the new standing wave solutions, respectively. This is a
process that happens locally for the particle, practically instantaneously
(i.e. within a time span $\propto\nicefrac{{1}}{{\omega}}),$ and nonlocally
for the standing waves, due to the very definition of the cosmological
solutions. The vacuum landscape is thus nonlocally changed without the
propagation of “signals” in a communication theoretical sense.555It is
_exclusively_ the latter that must be prohibited in order to avoid causal
loops leading to paradoxes. See Walleczek and Grössing Walleczek.2014non-
signalling ; Walleczek.2016nonlocal  for an extensive clarification of this
issue.

We have, for example, discussed in some detail what happens in a double-slit
experiment if one starts with one slit only, and when the particle might pass
it, one opens the second slit Groessing.2013dice ; Groessing.2013joaopessoa .
In accordance with Tollaksen et al. Tollaksen.2010quantum  we found that the
opening of the second slit (i.e. a change in boundary conditions) results in
an uncontrollable shift in momentum on the particle passing the first slit.
Due to its uncontrollability (or, the “complete uncertainty” in
Tollaksen.2010quantum ), this momentum shift cannot be used for signalling.
Still, it is necessary to a posteriori understand the final distributions on
the screen which would be incorrect without acknowledging said momentum kick.

Similarly, Aspect-type experiments of two-particle interferometry can be
understood as alterations of vacuum landscapes. Consider, for example, the
case in two-particle interferometry, where Alice and Bob each are equipped
with an interfering device and receive one of the counter-propagating
particles from their common source. If Alice during the time-of-flight of the
particles changes her device by making with suitable mirrors one of the
interferometer arms longer than the other, this constitutes an immediate
switching from one vacuum landscape to another, with the standing waves of the
zero-point field now reflecting the new experimental arrangement. In other
words, the $P$-field has been changed nonlocally throughout the experimental
setup, and therefore also all relational intensities

$P(\mathbf{w}_{i})=R(\mathbf{w}_{i})\mathbf{\hat{w}}_{i}\cdot{\displaystyle\sum_{i^{\prime}}}\mathbf{\hat{w}}_{i^{\prime}}R(\mathbf{w}_{i^{\prime}})$
(4.2)

involved. The latter represent the relative phase shifts
$\delta\varphi_{i,i^{\prime}}=\delta\arccos\mathbf{\hat{w}}_{i}\cdot\mathbf{\hat{w}}_{i^{\prime}}$
occurring due to the switching, and this change is becoming manifest also in
the total probability density

$P_{\mathrm{tot}}={\displaystyle\sum_{i}}P(\mathbf{w}_{i})=\left({\displaystyle\sum_{i}}\mathbf{\hat{w}}_{i}R(\mathbf{w}_{i})\right)^{2},$
(4.3)

with $i$ running through all channels of both Alice and Bob. The quantum
mechanical nonlocal correlations thus appear without any propagation (e.g.,
from Alice to Bob), superluminal or other. As implied by Gisin’s group
Bancal.2012quantum , this violates a “principle of continuity” of propagating
influences from _A_ to _B_ , but its non-signalling character is still in
accordance with relativity and the nonlocal correlations of quantum mechanics.
Practically instantaneous vacuum landscaping by Alice and/or Bob thus ensures
the full agreement with the quantum mechanical predictions without the need to
invoke (superluminal or other) signalling. Our model is, therefore, an example
of nonlocal influencing without signalling, which was recently shown to
provide a viable option for realistic modelling of nonlocal correlations.
Walleczek.2014non-signalling ; Walleczek.2016nonlocal

##  5 Conclusions and outlook

With our two-momentum approach to an emergent quantum mechanics we have shown
that one can in principle base the foundations of quantum mechanics on a
deeper level that does not need wavefunctions. Still, one can derive from this
new starting point, which is largely rooted in classical nonequilibrium
thermodynamics, the usual nonrelativistic quantum mechanical formalism
involving wavefunctions, like the Schrödinger equation or the de Broglie–Bohm
guiding law. With regard to the latter, the big advantage of our approach is
given by the fact that we avoid the troublesome influence from configuration
space on particles in real space, which Bohm himself has called
“indigestible”. Instead, in our model the guiding equation is completely
understandable in real coordinate space, and actually a rather typical
consequence of the fact that the total current is the sum of all particular
currents, and the total intensity, or probability density, respectively, is
the sum of all relational intensities. As we are working with Schrödinger
(i.e. spinless) particles, accounting for differences in particle statistics
is still an open problem.

As shown, we can replicate quantum mechanical features exactly by subjecting
classical particle trajectories to diffusive processes caused by the presence
of the zero point field, with the important property that the probability
densities involved extend, however feebly, over the whole setup of an
experiment. The model employs a two-momenta approach to the particle
propagation, i.e., forward and osmotic momenta. The form of the latter has
been derived without any recurrence to other approaches such as Nelson’s.

The one thing that _is_ to be digested from our model is the fact that the
relational intensities are nonlocally defined, over the whole experimental
arrangement (i.e. the “relevant environment”). This lies at the bottom of our
deeper-level ansatz, and it is the _only_ difference to an otherwise
completely classical approach. We believe that this price is not too high, for
we obtain a logical, realistic picture of quantum processes which is rather
simple to arrive at. Nevertheless, in order to accept it, one needs to
radically reconsider what an “object” is. We believe that it is very much in
the spirit of David Bohm’s thinking to direct one’s attention away from a
particle-centred view and consider an alternative option: that the universe is
to be taken as a totality, which only under very specific and delicate
experimental arrangements can be broken down to a laboratory-sized relevant
environment, even if that laboratory might stretch along interplanetary
distances. In our approach, the setting up of an experimental arrangement
limits and shapes the forward and osmotic contributions and is described as
vacuum landscaping. Accordingly, any change of the boundary conditions can be
the cause of nonlocal influences throughout the whole setup, thus explaining,
e.g., Aspect-type experiments. We argue that these influences can in no way be
used for signalling purposes in the communication theoretic sense, and are
therefore fully compatible with special relativity.

Accepting that the vacuum fluctuations throughout the universe, or at least
within such a laboratory, are a defining part of a quantum, amounts to seeing
any object like an “elementary particle” as nonlocally extended and,
eventually, as exerting nonlocal influences on other particles. For anyone who
can digest this, quantum mechanics is no more mysterious than classical
mechanics or any other branch of physics.

###### Acknowledgements.

We thank Jan Walleczek for many enlightening discussions, and the Fetzer
Franklin Fund of the John E. Fetzer Memorial Trust for partial support of the
current work. Also, we wish to thank Lev Vaidman and several other colleagues
for stimulating discussions. We thank the latter for the exchange of
viewpoints sometimes closely related to our approach, as they also become
apparent in their respective works: Herman Batelaan Batelaan.2016momentum ,
Ana María Cetto and Luis de la Peña Cetto.2014emerging-quantum , Hans-Thomas
Elze Elze.2018configuration , Basil Hiley Hiley.2016structure , Tim Maudlin
Maudlin.2011quantum , Travis Norsen Norsen.2016bohmian , Garnet Ord
Ord.2009quantum , and Louis Vervoort Vervoort.2016no-go .

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[a]Biplab Dey, on behalf of the LHCb collaboration

# Angular analyses of rare decays at the LHC

###### Abstract

Loop-suppressed penguin $b\to s$ transitions are sensitive to heavy New
Physics particles propagating inside the loops. Thanks to the large sample
sizes from the LHC, we are able to perform multidimensional angular analyses
that are sensitive to interferences between the Standard Model and New Physics
terms. This article surveys the latest results, primarily from LHCb, on $b\to
s\mup\mun$ electroweak and $b\to s\gamma$ radiative penguins.

##  1 Introduction and theory

Figure 1: Flavor-changing neutral currents occur only at the loop-level in the
SM but can be enhanced by NP effects, both in loop (penguin) and tree-level
diagrams.

In the Standard Model (SM), the flavor-changing neutral current process
$b\rightarrow s$ is forbidden at the tree-level and proceeds only via loop-
suppressed diagrams as shown in Fig. 1. These provide excellent avenues to
probe New Physics (NP) contributions that can enter either in loop- or tree-
level processes such as via Leptoquarks (LQ) or heavy $Z^{\prime}$ boson, as
shown in Fig. 1. This article focuses on the electroweak (EWP) and radiative
(Rad) penguin diagrams, but gluonic penguins can also be an important NP
source. A convenient theoretical formalism to study such decays is to regard
the SM as a low energy effective field theory containing dimension $d\leq 4$
local operators from renormalizability requirements. Higher dimensional
operators can be added with an appropriate cutoff scale $\mathchar
28931\relax$, as

$\displaystyle\mathcal{L}_{\rm eff}(x)=\displaystyle\mathcal{L}_{\rm
SM}(x)+\sum_{d>4}\frac{C_{i}}{{\mathchar
28931\relax}^{d-4}}\mathcal{O}_{i}^{(d)}(x)$ (1)

whereby the NP amplitudes have $(E/{\mathchar 28931\relax})^{d-4}$ behavior in
the energy, $E$: divergent at high energies, but suppressed at $E\ll{\mathchar
28931\relax}$. Most relevant for rare $b\rightarrow s$ decays are $d=6$
operators that yield $\mathcal{A}_{\rm eff}\sim C^{\rm SM}/m^{2}_{W}+C^{\rm
NP}/{\mathchar 28931\relax}^{2}_{\rm NP}$. The basis comprises 10 operators
[1]: $\mathcal{O}_{1,2}$ (4-quark tree), $\mathcal{O}_{3-6}$ (4-quark
penguins) and $\mathcal{O}_{8}$ (gluonic penguin) that are suppressed for the
EWP/Rad modes. The dominant left-handed contributions are from the
electromagnetic dipole and weak vector (axialvector) operators

$\displaystyle\mathcal{O}_{7\gamma}=\frac{e}{16\pi^{2}}m_{b}(\overline{s}\sigma_{\mu\nu}P_{R}b)F^{\mu\nu},\;\;\;\mathcal{O}_{9V(10A)}=\frac{e^{2}}{16\pi^{2}}(\overline{s}\gamma_{\mu}P_{L}b)(\overline{\ell}\gamma^{\mu}(\gamma_{5})\ell).$
(2)

The corresponding right-handed (quark side) operators are suppressed in the
SM, but can be enhanced in NP scenarios. The dimensionless couplings (Wilson
coefficients) associated with the operators in Eq. 2 encode the short distance
physics. They are calculated at the $m_{W}$ scale by integrating out the heavy
degrees of freedom from the full theory and evolving to the $m_{b}$ scale
using renormalization group equations. The total amplitudes
$\mathcal{A}(i\rightarrow f)=\displaystyle C_{n}(m_{b})\langle
f|\mathcal{O}_{n}(m_{b})|i\rangle_{\rm had}$ also contains the long-distance
physics (QCD/hadronization) which mostly comes from local form-factors (FFs)
that are computed from lattice QCD and other theory tools, but can get
important non-local contributions (rescattering, charm loops) that are hard to
estimate theoretically.

Thanks to the large $b\overline{b}$ samples available at the LHC, a
comprehensive effort on multidimensional angular analyses in $b\rightarrow
s\ell^{+}\ell^{-}$ and $b\rightarrow s\vec{\gamma}$ is ongoing. These offer a
rich set of angular observables sensitive to $\Delta C_{i}\equiv C_{i}^{\rm
SM}-C_{i}^{\rm NP}$. The thrust has been to identify and probe “theoretically
clean observables” with reduced dependence on the QCD contributions that often
form the largest theory uncertainties. The angles for a typical 4-body final
state decay is shown in Fig. 2 for $\vec{{{\mathchar
28931\relax}^{0}_{\mathrm{b}}}}\rightarrow pK^{-}\ell^{+}\ell^{-}$; the two
other kinematic variables being $q^{2}\equiv m^{2}_{\ell^{+}\ell^{-}}$ and
$k^{2}\equiv m^{2}_{pK}$. For an unpolarized parent ${{\mathchar
28931\relax}^{0}_{\mathrm{b}}}$ (or spin-0 $B_{(s)}$ mesons), $\phi_{\ell}$ is
set to 0 and $\chi=\phi_{p}$ is the single azimuthal angle between the
dilepton and dihadron decay planes. Equivalent variables apply for
$B\rightarrow K\pi\ell^{+}\ell^{-}$ and $B_{s}^{0}\rightarrow
K^{+}K^{-}\ell^{+}\ell^{-}$.

Figure 2: The four angular variables for the decay of a polarized
$\vec{{{\mathchar 28931\relax}^{0}_{\mathrm{b}}}}$ decaying to a 4-body final
state, $pK^{-}\ell^{+}\ell^{-}$: $\theta_{l}$ ($\theta_{p}$) is the $\ell^{-}$
(proton) helicity angle. The beam direction, ${\mathchar 28931\relax}^{\ast}$
flight direction and the ${{\mathchar 28931\relax}^{0}_{\mathrm{b}}}$
polarization defines a coordinate system relative to which the dilepton and
dihadron azimuthal angles can be defined as $\phi_{\ell}$ and $\phi_{p}$.

##  2 Electroweak penguins

###  2.1 The golden channel: $B^{0}\rightarrow K^{\ast 0}\ell^{+}\ell^{-}$

Figure 3: $B^{0}\rightarrow K^{\ast 0}\mu^{+}\mu^{-}$: (a) world-average of
the $P^{\prime}_{5}$ measurements compared to theory (filled/hatched boxes).
The situation is representative after publication of the LHCb Run 1 results.
(b) effect on $C^{\rm NP}_{9,10}$ from a recent global fit in Ref. [2].

$B^{0}\rightarrow K^{\ast 0}\ell^{+}\ell^{-}$ has been a gold-plated channel
[1] since the $B$-factory era due to the relatively narrow $K^{\ast 0}(892)$
resonance. Especially in the low $q^{2}$ regime, where the recoiling $K^{\ast
0}$ has a large $\gamma$-factor in the parent $B^{0}$ frame, QCD sum rules on
the light-cone (LCSR) affords control over the FF calculations. However, with
the BaBar/Belle statistics, only 1-dimensional angular analyses in either the
lepton or hadron helicity angles ($\cos\theta_{l}$ or $\cos\theta_{K}$) were
possible [3]. Full 3-dimensional analysis in $d\theta_{l}d\theta_{K}d\chi$ in
$q^{2}$ bins was possible only with the advent of LHCb. For instance, while
the full BaBar dataset had $\mathcal{O}(50)$ $B^{0}\rightarrow K^{\ast
0}(\rightarrow K^{+}\pi^{-})\mu^{+}\mu^{-}$ events, the existing Run1+2 LHCb
dataset already includes $\mathcal{O}(10^{4})$ clean signal events for this
muonic mode. On the other hand, $B$-factories, including Belle II, have
complementary advantages, with better reconstruction for the $\pi^{0}$ isospin
modes as well the dielectron channels.

An important result from the muonic analyses at LHCb is tension with SM
predictions in the angular observable $P^{\prime}_{5}$ [4] (see Fig. 3a).
Similar tensions have also been seen in $B^{+}\rightarrow
K^{\ast+}\mu^{+}\mu^{-}$ [5] and $B_{s}^{0}\rightarrow\phi\mu^{+}\mu^{-}$ [6].
A related tension is observed in the overall branching fractions in several
$b\rightarrow s\mu^{+}\mu^{-}$ modes which tend to consistently lie lower than
the SM predictions [7, 8, 9]. Competitive results have also come from
ATLAS/CMS [10, 11, 12] where the advantage is higher overall luminosity, but
the disadvantage is the limited $B$-physics trigger bandwidth and lack of a
RICH detector for $K^{+}/\pi^{+}/p$ separation. Numerous global fits with
different data subsets, statistical methods and theory priors have been
performed [2] pointing to a preferred negative $C^{\rm NP}_{9}$. The major
point of contention, however, has been the effect of non-factorizable long-
distance contributions due to soft+hard gluons from charm-loops that can mimic
NP effects (see Fig. 3b). To constrain the non-factorizable part in a data-
driven fashion, LHCb has performed an unbinned angular analysis [13, 14] using
the same dataset as in Ref. [5]. The underlying transversity amplitudes are

$\displaystyle\mathcal{A}^{L,R}_{\tiny\lambda=0,\parallel,\perp}=$
$\displaystyle N_{\lambda}\left\\{{\color[rgb]{0,0,0}\left[(C_{9}\pm
C^{\prime}_{9})\mp(C_{10}\pm
C^{\prime}_{10})\right]}\mathcal{F}_{\lambda}+\frac{2m_{b}M_{B}}{q^{2}}\left[(C_{7}\pm
C^{\prime}_{7})\mathcal{F}^{T}_{\lambda}-16\pi^{2}\frac{M_{B}}{m_{b}}{\color[rgb]{0,0,0}\mathcal{H}_{\lambda}}\right]\right\\},$
(3)

where $\mathcal{F}_{\lambda}$ are the usual local FFs (taken from LCSR and
lattice QCD) and $\mathcal{H}_{\lambda}$ are the new non-factorizable part
which are extracted from a $q^{2}$-dependent parameterization. The values of
$C_{i}^{\rm SM}$ are taken from theory, while allowing for $C_{9,10}^{\rm NP}$
to be floated. The results of the fit are shown in Fig. 4. Good consistency is
found in the extracted binned observables compared to Ref. [5]. The overall
tension with the SM is reduced to $\sim 1.8\sigma$ in $C_{9}$, and $\sim
1.4\sigma$ in global fits. The full Run 1+2 analysis and more precise theory
FFs will improve upon these results.

Figure 4: $B^{0}\rightarrow K^{\ast 0}\mu^{+}\mu^{-}$ long-distance
contribution fits [13, 14]: (a) $q^{2}$ projections; (b) effect on $Re(C^{\rm
NP}_{9,10})$.

###  2.2 Access to tensor states in $B\rightarrow X_{s}\mu^{+}\mu^{-}$

Figure 5: Tensor states in (a) $B^{0}\rightarrow K^{+}\pi^{-}\mu^{+}\mu^{-}$,
(b) $B^{0}_{s}\rightarrow K^{+}K^{-}\mu^{+}\mu^{-}$ decays.

While most of the theory and experimental investigations have focused on the
ground state vector states $K^{\ast}(892)$ and $\phi(1020)$ in $B$-meson
decays, LHCb has also probed the excited $K^{\ast}_{0,2}(1430)\rightarrow
K^{+}\pi^{-}$ [15] and $f^{\prime}_{2}(1525)\rightarrow K^{+}K^{-}$ [7],
including an angular moments analysis [16] for the former, to separate the
$S$-, $P$\- and $D$-wave $K\pi$ states. The results are shown in Fig. 5. The
theory interpretation however will require reliable FFs for $B$ decays to
these excited states.

###  2.3 Angular analysis of $B\rightarrow K\mu^{+}\mu^{-}$

For the 3-body final state $B\rightarrow K\mu^{+}\mu^{-}$, only the lepton
helicity angle, $\theta_{l}$, can be defined and the SM predicts an almost
pure $\sin^{2}\theta_{l}$ distribution save for small effects due to the muon
mass. The angular distribution is be sensitive to new scalar and tensor
operators via the new terms $A_{\rm FB}$ and $F_{\rm H}$:

$\displaystyle
d\Gamma/d\cos\theta_{l}\propto\frac{3}{4}(1-{\color[rgb]{0,0,0}F_{\rm
H}})\sin^{2}\theta_{\ell}+\frac{1}{2}{\color[rgb]{0,0,0}F_{\rm
H}}+{\color[rgb]{0,0,0}A_{\rm FB}}\cos\theta_{\ell}.$ (4)

Figure 6 shows the results of the fit to Run 1 data for both $B^{+,0}$ [17],
subject to the constraints $|A_{\rm FB}|\leq F_{\rm H}/2$, $0\leq F_{\rm
H}\leq 3$, such that the rate in Eq. 4 is positive. The extracted $A_{\rm FB}$
and $F_{\rm H}$ in different $q^{2}$ bins are also consistent with SM.

Figure 6: Charged and neutral $B\rightarrow K\mu^{+}\mu^{-}$ angular analysis
[17]: (a) projection in $\cos\theta_{l}$ showing the expected
$\sin^{2}\theta_{l}$ distribution. (b) extracted $A_{\rm FB}$ and $F_{\rm H}$
with uncertainty contours.

###  2.4 ${{\mathchar 28931\relax}^{0}_{\mathrm{b}}}\rightarrow{\mathchar
28931\relax}\mu^{+}\mu^{-}$ moments analysis

LHCb is unique among the $B$-factories to have access to all $b$-hadron
species, including large samples of ${{\mathchar
28931\relax}^{0}_{\mathrm{b}}}$ baryons. Therefore it is possible to probe
$b\rightarrow s$ penguin decays in the baryonic sector as well. For
${{\mathchar 28931\relax}^{0}_{\mathrm{b}}}\rightarrow{\mathchar
28931\relax}^{0}$ transition, the narrow ${\mathchar 28931\relax}^{0}$ state
allows lattice QCD calculations [18] for the FFs. For a given value of
$q^{2}$, the decay rate for polarized $\vec{{{\mathchar
28931\relax}^{0}_{\mathrm{b}}}}$ and $\vec{{\mathchar 28931\relax}^{0}}$
depends on five angles, $\vec{\mathchar
28938\relax}=\\{\theta,\theta_{l},\phi_{\ell},\theta_{h},\phi_{h}\\}$ (see
Fig. 2) and is expanded in an orthonormal angular basis as

$\displaystyle\frac{d^{5}\Gamma(q^{2})}{d\vec{\mathchar
28938\relax}}=\frac{3}{32\pi^{2}}\sum_{i=1}^{34}K_{i}(q^{2})f_{i}(\vec{\mathchar
28938\relax}).$ (5)

The $K_{i}$ moments can be related to more familiar observables such as
forward-backward asymmetries in the angles: $A^{\ell}_{\rm
FB}=[\frac{3}{2}K_{3}](\rightarrow\cos\theta_{l})$, $A^{h}_{\rm
FB}=[K_{4}+\frac{1}{2}K_{5}](\rightarrow\cos\theta_{h})$, $A^{\ell h}_{\rm
FB}=[\frac{3}{4}K_{6}](\rightarrow\cos\theta_{l}\cos\theta_{h})$. If the
${{\mathchar 28931\relax}^{0}_{\mathrm{b}}}$ is unpolarized, only the first 10
moments are non-zero [19]. Due to the long-lived nature of the ${\mathchar
28931\relax}^{0}$, its reconstruction in LHCb is somewhat non-trivial. At low
$q^{2}$, the Run 1 analysis [8] found very few events. Therefore the analysis
using Run 1 + partial Run 2 (collected between 2011-16) [20] focused on the
high-$q^{2}$ region, $q^{2}\in[15,20]$ GeV2. The results are shown in Fig. 7.
The SM predictions are taken from the EOS [19] software package and show a
slight tension with the data in the $K_{6}$ observable.

Figure 7: ${{\mathchar 28931\relax}^{0}_{\mathrm{b}}}\rightarrow{\mathchar
28931\relax}^{0}\mu^{+}\mu^{-}$ angular analysis in $q^{2}\in[15,20]$ GeV2
[20]: (a) first 10 moments from Eq. 5; (b) comparison of the background-
subtracted data and angular moments model.

###  2.5 Differential cross-sections for ${{\mathchar
28931\relax}^{0}_{\mathrm{b}}}\rightarrow\mathchar
28931\relax(1520)\mu^{+}\mu^{-}$

Figure 8: $m(pK^{-})$ spectrum from Run 1 ${{\mathchar
28931\relax}^{0}_{\mathrm{b}}}\rightarrow pK^{-}\mu^{+}\mu^{-}$ LHCb data: (a)
at the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ resonance [21]; (b) in
the non-resonant region [22].

The first observation of ${{\mathchar
28931\relax}^{0}_{\mathrm{b}}}\rightarrow pK^{-}\mu^{+}\mu^{-}$ decay using
LHCb Run 1 data [22] demonstrated a rich $m(pK)$ spectrum. The comparison
between the resonant and non-resonant spectra is shown in Fig. 8. Employing
the full Run 1+2 datasets, LHCb has examined the $m(pK^{-})$ region around the
narrow $\mathchar 28931\relax(1520)$ resonance. Integrating over the angles
and including the resonances ${\mathchar 28931\relax}(1405)(\frac{1}{2}^{-})$,
${\mathchar 28931\relax}(1520)(\frac{3}{2}^{-})$, ${\mathchar
28931\relax}(1600)(\frac{1}{2}^{+})$ and ${\mathchar
28931\relax}(1800)(\frac{1}{2}^{-})$, 1-dimensional fits are performed in
$m(pK^{-})$. Differential cross-sections for the ${{\mathchar
28931\relax}^{0}_{\mathrm{b}}}\rightarrow\mathchar
28931\relax(1520)\mu^{+}\mu^{-}$ decay are also provided [23] that show large
discrepancies with theory calculations in the low $q^{2}$ region (see Fig. 9).

Figure 9: ${{\mathchar 28931\relax}^{0}_{\mathrm{b}}}\rightarrow\mathchar
28931\relax(1520)\mu^{+}\mu^{-}$ [23]: (a) low $q^{2}$, (b) high $q^{2}$, (c)
$d\Gamma/dq^{2}$.

##  3 Radiative penguins

In the SM, due to the left-handed nature of the weak interaction, the photon
from a $b\rightarrow s\gamma$ is almost purely left-handed; the right-handed
component is suppressed ($C^{\prime}_{7}\sim\frac{m_{s}}{m_{b}}C_{7}$) and is
a sensitive NP probe. However, one needs angular analyses to extract the
interferences. LHCb probes this in several ways: very low $q^{2}$ angular
analysis of $B^{0}\rightarrow K^{\ast 0}e^{+}e^{-}$ [24] and
$B^{0}_{s}\rightarrow\phi e^{+}e^{-}$; angular analysis of $B^{+}\rightarrow
K^{+}\pi^{+}\pi^{-}\gamma$ [25],; time-dependent CP violation of
$B^{0}_{s}\rightarrow\phi\gamma$ [26] and $B^{0}\rightarrow
K^{0}_{\scriptscriptstyle S}\pi^{+}\pi^{-}\gamma$; angular analyses of
${{\mathchar 28931\relax}^{0}_{\mathrm{b}}}\rightarrow{\mathchar
28931\relax}^{0(\ast)}\gamma$ [27].

###  3.1 Angular analysis of ${{\mathchar
28931\relax}^{0}_{\mathrm{b}}}\rightarrow{\mathchar 28931\relax}^{0}\gamma$

Figure 10: (a) If the hadronic system has a 3-body decay, the normal to the
plane defines a preferred direction to extract the photon polarization from
the up-down asymmetry in $\cos\theta$ [25]. (b) An exception is the 2-body
decay of $\vec{{\mathchar 28931\relax}}\rightarrow p\pi^{-}$ where the
measurable polarization of the ${\mathchar 28931\relax}^{0}$ provides the
preferred direction.

To measure the photon polarization in $H_{b}\rightarrow H_{s}\vec{\gamma}$,
the hadronic system must undergo a 3-body decay, so that the normal to the
plane defines a preferred direction and the up-down asymmetry is proportional
to the photon polarization, $\lambda_{\gamma}$. For example, this has been
utilized in $B^{+}\rightarrow K^{+}\pi^{+}\pi^{-}\gamma$ [25] as shown in Fig
10a. In such cases, due to poor knowledge of the resonant structures in the
$H_{s}$ system and thereby the hadronic current $\mathcal{J}_{\mu}^{\rm had}$,
the proportionality factor remains unknown and $\lambda_{\gamma}$ can still
not be extracted out. An exception is the 2-body decay of $\vec{{\mathchar
28931\relax}^{0}}\rightarrow p\pi^{-}$, where the self-analyzing nature of the
${\mathchar 28931\relax}^{0}$ polarization provides the preferred direction.
Moreover, the differential rate is
$d\Gamma/d\cos\theta_{p}\propto(1-\alpha_{\mathchar
28931\relax}\lambda_{\gamma}\cos\theta)$ where the ${\mathchar
28931\relax}^{0}$ decay asymmetry parameter $\alpha_{\mathchar 28931\relax}$
is known quite precisely [28] and thereby $\lambda_{\gamma}$ can be extracted
from a fit to the $\cos\theta_{p}$ slope, as shown in Fig 10b. Experimentally,
${{\mathchar 28931\relax}^{0}_{\mathrm{b}}}\rightarrow{\mathchar
28931\relax}^{0}\gamma$ is challenging due to lack of a reconstructible
secondary vertex for the ${{\mathchar 28931\relax}^{0}_{\mathrm{b}}}$ decay.
Employing dedicated reconstruction to reject the high background, the first
observation with Run 2 2016 data was reported in Ref. [29] and the
polarization measurement with full Run 2 was reported in Ref. [27]. The
measured value of
$\alpha_{\gamma}=0.82^{+0.17}_{-0..26}{\rm(stat.)}^{+0.04}_{-0.13}{\rm(syst.)}$
is compatible with the SM expectation of 1 and from global
$C_{7}^{(^{\prime})}$ fits, reducing a 4-fold ambiguity in the $C_{7}^{\rm
NP}$ phase to a 2-fold ambiguity.

###  3.2 Angular analysis of ${{\mathchar
28931\relax}^{0}_{\mathrm{b}}}\rightarrow pK^{-}\gamma$

Figure 11: Fit projections for ${{\mathchar
28931\relax}^{0}_{\mathrm{b}}}\rightarrow pK^{-}\gamma$ in the $m(pK)$
varaiable. The shaded areas depict interferences between ${\mathchar
28931\relax}^{\ast}$ states with the same spin-paiity.

As for the electroweak penguin case in Sec. 2.5, for the radiative
${{\mathchar 28931\relax}^{0}_{\mathrm{b}}}\rightarrow{\mathchar
28931\relax}^{\ast}(pK^{-})\gamma$ decay, the $[pK^{-}]$ system is dominated
by a large number of overlapping resonances. Employing the full Run1+2 data,
LHCb has performed a 2-dimensional angular analysis of this mode in
$\\{m(pK),\cos\theta_{p}\\}$. All well-established ${\mathchar
28931\relax}^{\ast}$ states [28] are included in an isobar model with Breit-
Wigner lineshapes incorporating mass-dependent widths. The only exception is
the ${\mathchar 28931\relax}(1405)$ resonance where a two-pole Flatte-form is
used. To reduce the large set of parameters, the maximum orbital angular
momentum of the ${\mathchar 28931\relax}^{\ast}\rightarrow pK^{-}$ decay is
taken to be $L=3$. The result from such a reduced model is shown in Fig. 11.
The fit fractions and interference fit fractions are reported from the
amplitude analysis.

##  4 Summary and outlook

The large existing Run 1+2 data samples at the LHC has already allowed the
hitherto (pre-LHC) “rare” $b\rightarrow s$ decays to be probed in an
unprecedented fashion. This will continue into the LHCb upgrade era, with Run
3 having already commenced. Multi-dimensional angular analyses in both the
electroweak and radiative penguin sectors have given rise to surprising
tensions. A large and mature angular analysis effort exists at LHCb which will
continue to be evolve. The thrust will be to include time-dependent CP
violation [30] in more rare decay angular analyses, as well as to probe the
further Cabibbo-suppressed $b\rightarrow d$ transitions [31, 32].

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