# Deciphering the CP nature of the 750 GeV resonance111This article is
registered under preprint number: DESY 16-054, arXiv:1604.02029.

Mikael<EMAIL_ADDRESS>Christophe Grojeana,b Marc Riembaua,c
Thibaud Vantalona,c a DESY, Notkestraße 85, D-22607 Hamburg, Germany  b on
leave from ICREA, E-08010 Barcelona, Spain  c IFAE, Barcelona Institute of
Science and Technology (BIST) Campus UAB, E-08193 Bellaterra, Spain

###### Abstract

The recently observed excess in diphoton events at around 750 GeV can be
satisfactorily described in terms of a new spin-0 real singlet with effective
interactions to the gauge bosons. In this letter we first review the current
constraints on this setup. We further explore the production in association
with a gauge boson. We show the potential of this channel to unravel current
flat directions in the allowed parameter space. We then study the potential of
two different asymmetries for disentangling the CP nature of such a singlet in
both gluon fusion and vector-boson fusion. For this matter, we perform an
estimation of the efficiency for selecting signal and background events in
eight different decay modes, namely
$4\ell,2j\,2\ell,2j\ell\,\not{E}_{T},2\gamma\,2j,4\ell\,2j,2\ell\,\gamma\,2j,4j\,2\ell$
and $4j\,\ell\,\not{E}_{T}$. We emphasize that the very different couplings of
this new singlet to the Standard Model particles as well as the larger mass
provide a distinctive phenomenology with respect to Higgs searches. We finally
show that a large region of the parameter space could be tested within the
current LHC run, the dominant channel being $2\gamma\,2j$.

##  1 Introduction

The first bunch of data in proton-proton collisions at $\sqrt{s}=13$ TeV were
successfully delivered by the LHC during last year. Surprisingly, the first
analyses on these data with as few as $\sim 3$ fb-1 have revealed unexpected
results. Indeed, the ATLAS [1] and CMS [2] experiments have pointed out an
excess in diphoton events with an invariant mass of around 750 GeV; the local
significance ranging from 2 to around 3$\sigma$. The reported excess survived
further scrutiny [3] and appears as the best hint in decades for physics
beyond the Standard Model (SM) at colliders. This fact explains the excitement
of the particle physics community, which has translated into a plethora of
papers in a few months [4].

This diphoton excess can be easily interpreted in terms of a spin-0 real
singlet (although explanations in terms of spin-1 and spin-2 particles have
also received some well deserved attention). Both production and decay are
hence to be mediated by heavier states333The model proposed in [5] is an
exception where the diphoton excess originates from a solitary new degree of
freedom without the need for any additional electrically charged particles,
nor new strong dynamics. Alternative non-resonant models with long decay
chains have also been proposed to explain the 750 GeV diphoton excess. In this
letter, we limit ourselves to the simplest interpretation with a single
resonance whose couplings to gluons and photons are mediated by additional
heavier states charged under QCD and QED. whose effects can be encoded in a
small set of effective operators. Throughout this letter we adopt this
approach and we explore the potential of the next run of data to unravel the
CP nature of this candidate, namely whether it is a scalar or a pseudo-scalar
(of course, the 750 GeV singlet could also be an admixture of both scalar and
pseudo-scalar, i.e. it could have some interactions that are not invariant
under CP transformations, but we omit this possibility for now.) We start
considering a generic parameterization in section 2 and discussing the current
constraints. Production via gluon fusion (GF) and vector-boson fusion (VBF)
turn out to be sizable in a large region of the parameter space. However, they
are shown to give flat directions that can be only disentangled if new
production mechanisms are considered. In this respect, we explore the
potential of producing the singlet resonance in association with a Standard
Model (SM) gauge boson in section 3. The rest of the article is structured as
follows. In sections 4 and 5, we introduce two asymmetries in the kinematical
distributions of GF and VBF events. They are intended to differentiate the two
CP hypothesis. The advantage of this approach relies on the fact that most
systematic uncertainties cancel out. Statistical uncertainties are on the
other hand properly taken into account. We perform simulations to estimate the
efficiency for selecting signal and background events in both categories in
eight different decay modes:
$4\ell,2j\,2\ell,2j\ell\,\not{E}_{T},2\gamma\,2j,4\ell\,2j,2\ell\,\gamma\,2j,4j\,2\ell$
and $4j\,\ell\,\not{E}_{T}$. We show that after all cuts, sizable efficiencies
are obtained for most signals while still keeping backgrounds under control.
Despite that we do not attempt to optimize these cuts, all together the eight
channels can probe a wide region of the available parameter space within the
current run of the LHC, as explained in section 6. We conclude in section 7.

##  2 Parameterization and current constraints

Figure 1: Parameter space region in the plane
$c_{\gamma\gamma}\,-\,c_{WW}/c_{BB}$ compatible with the diphoton excess and
current constraints. Regions filled with horizontal lines can not account for
the observed signal. Regions filled with vertical lines are in turn bounded by
direct searches at 8 TeV. The values of $c_{gg}$ are labeled on dashed contour
lines. In the left (right) panel
$\Gamma_{S}=\Gamma_{gg}+\Gamma_{WW}+\Gamma_{BB}$ ($\Gamma_{S}=45$ GeV) is
assumed. The total (additional) width is shown in solid green (brown) lines in
the left (right) panel.

This letter aims mainly to provide a guideline for future efforts on the
analysis of the parity properties of a resonance $S$ with mass $M\sim 750$
GeV. We assume $S$ to be a spin-0 SM gauge singlet. Besides, the production
cross section into diphotons mediated by $S$ is assumed to be 8 fb. The
question of the spin and parity properties of $S$ is made legitimate by the
unexpected character of the excess and thus by the absence of any particular
theoretical prejudice towards one hypothesis. Actually we do not focus on any
particular model nor we attempt to address the effective-field theory of $S$
in full generality. In fact, the relevant Lagrangian for our phenomenological
study can be parameterized as [6]

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{1}{2M}S\bigg{(}g_{3}^{2}c_{gg}G^{2}+g_{2}^{2}c_{WW}W^{2}+g_{1}^{2}c_{BB}B^{2}$
$\displaystyle+g_{3}^{2}\tilde{c}_{gg}G\tilde{G}+g_{2}^{2}\tilde{c}_{WW}W\tilde{W}+g_{1}^{2}\tilde{c}_{BB}B\tilde{B}\bigg{)}.$
(1)

Here, $g_{3},g_{2}$ and $g_{1}$ stand for the SM $SU(3),SU(2)$ and $U(1)$
gauge couplings, respectively. $G$, $W$ and $B$ are the corresponding field-
strength tensors. For a generic $F$, $\tilde{F}$ is defined as
$\tilde{F}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}F^{\alpha\beta}$.
The tilded (non-tilded) coefficients are zero if $S$ is a scalar (pseudo-
scalar). We disregard further couplings to the SM fermions and to the Higgs
doublet since they do not introduce any qualitative change in our analysis.
Actually the latter has anyway to be small to pass the constraints from Higgs
measurements [7] and $ZZ$ resonant searches [8]. The decay width of $S$ into
the different decay modes provided by the interactions above can be easily
computed for $M\gg m_{W,Z}$, with $m_{W(Z)}$ the mass of the $W^{\pm}(Z)$
boson. In this limit, the decay widths to the different pairs of gauge bosons
are given by

$\displaystyle\Gamma_{gg}$ $\displaystyle=$ $\displaystyle
8\pi\alpha_{3}^{2}M\left(c_{gg}^{2}+\tilde{c}_{gg}^{2}\right),\,$
$\displaystyle\Gamma_{\gamma\gamma}$ $\displaystyle=$
$\displaystyle\pi\alpha_{\text{em}}^{2}M\left(c_{\gamma\gamma}^{2}+\tilde{c}_{\gamma\gamma}^{2}\right),$
$\displaystyle\Gamma_{Z\gamma}$ $\displaystyle=$ $\displaystyle
2\pi\alpha_{\text{em}}^{2}M\bigg{[}\left(c_{BB}t_{W}-\frac{c_{WW}}{t_{W}}\right)^{2}$
$\displaystyle+\bigg{(}\tilde{c}_{BB}t_{W}-\frac{\tilde{c}_{WW}}{t_{W}}\bigg{)}^{2}\bigg{]}\,,$
$\displaystyle\Gamma_{ZZ}$ $\displaystyle=$
$\displaystyle\pi\alpha_{\text{em}}^{2}M\bigg{[}\left(c_{BB}t_{W}^{2}+\frac{c_{WW}}{t_{W}^{2}}\right)^{2}$
$\displaystyle+\left(\tilde{c}_{BB}t_{W}^{2}+\frac{\tilde{c}_{WW}}{t_{W}^{2}}\right)^{2}\bigg{]}\,,$
$\displaystyle\Gamma_{WW}$ $\displaystyle=$
$\displaystyle\frac{2\pi\alpha_{\text{em}}^{2}}{s_{W}^{4}}M\left(c_{WW}^{2}+\tilde{c}_{WW}^{2}\right)\,,$
(2)

with $c_{\gamma\gamma}=c_{BB}+c_{WW}$, $t_{W}$ and $s_{W}$ the tangent and
sine of the Weinberg angle, $\alpha_{\text{em}}$ the fine-structure constant
and $\alpha_{3}=g_{3}^{2}/(4\pi)$. The photon field-strength coefficient is
thus given by $4\pi\alpha_{\text{em}}c_{\gamma\gamma}/2M$. The cross section
for the single production of $S$ and the subsequent decay into two photons at
a center of mass energy $\sqrt{s}$ reads

$\sigma^{\gamma\gamma}(s)\,=\,\frac{1}{s}\frac{1}{M\Gamma_{S}}\left(\mathcal{C}_{gg}\Gamma_{gg}+\mathcal{C}_{\gamma\gamma}\Gamma_{\gamma\gamma}\right)\Gamma_{\gamma\gamma},$
(3)

where $\Gamma_{S}$ stands for the total width. $\mathcal{C}_{gg}$ and
$\mathcal{C}_{\gamma\gamma}$ represent instead dimensionless parton
luminosities for gluon and photon fusion, respectively. Their values at 8 (13)
TeV have been found to be approximately 174 (2137) and 11 (54), respectively
[6]. The single production of $S$ via GF at $13$ TeV is thus enhanced with
respect to 8 TeV by a factor of $\sim 5$, which can be in agreement with the
absence of departures from the SM predictions in the first LHC run. This in
fact translates into a bound on $\sigma^{\gamma\gamma}(8\,\text{TeV})\lesssim
2$ fb [9, 10]. This observation is no longer true for single production via
photon fusion. It is only increased by a factor of $\sim 2.9$ and therefore in
tension with current constraints (see for example [11, 12]). The
$c_{\gamma\gamma}$ coupling is bounded from above (below) to avoid too large
(small) a diphoton cross section. In the same vein, experimental searches for
resonant $Z\gamma$ [13], $ZZ$ [14] and $W^{+}W^{-}$ [15] production at 8 TeV
set stringent limits on this setup. This information is summarized in Fig. 1.
The allowed parameter space in the $c_{\gamma\gamma}\,-\,c_{WW}/c_{BB}$ plane
that can explain the excess while evading the current bounds is presented in
this plot. For every point in this plane, $c_{gg}$ has been fixed so that
$\sigma^{\gamma\gamma}(13\,\text{TeV})=8$ fb. The corresponding values are
shown in dashed blue lines. The region filled with vertical lines is excluded
mainly by $W^{+}W^{-}$ and $Z$ searches at 8 TeV. Notice also that the bounds
coming from direct searches would be weaker if
$\sigma^{\gamma\gamma}(13\,\text{TeV})$ was smaller. The solid green (brown)
contour lines stand for the total (additional) width. In the left panel of the
figure we assume that $\Gamma_{S}$ coincides with these contours. In the right
panel we fix it instead to the best-fit value reported by the ATLAS
Collaboration [1], $\Gamma_{S}=45$ GeV, by considering an additional partial
width of $S$ into soft (or partially invisible) particles that escape
detection.

##  3 Associated production

It can be seen from Fig. 1 that resonance searches for massive gauge bosons
are only sensitive to the ratio $c_{WW}/c_{BB}$. It is also apparent that
there are other flat directions, i.e. that the different couplings cannot be
accessed independently from each others. In fact, even if it was possible to
determine $\Gamma_{S}$ experimentally, we would have to measure all $S$ decay
modes to be able to bound each coupling independently. This seems highly
unrealistic, first, because $\Gamma_{S}$ might remain out of the experimental
resolution, and second because it would require to also tag decays into gluons
and (potentially) invisible particles, a notoriously difficult task in the
busy hadronic environment of the LHC. Thus, different strategies should be
considered in this respect. One possibility relies on $S$ production in
association with a gauge boson (a previous study in this direction has been
presented in [16]). The corresponding Feynman diagram is depicted in the left
panel of Fig. 2, while the cross sections are shown in Fig. 3. These have been
computed by using MadGraph v5 [17] (Feynrules v2 [18] has been first used to
implement the interactions of Eq. 2). Other automatic tools have been
developed and used [19] to study models aiming at explaining the diphoton
excess. In the region of parameter space compatible with the reported excess,
the associated production cross sections can be as large as few tens fb. And
even for rare decay modes, (e.g. a branching ratio below 0.001 for
$S\rightarrow ZZ\rightarrow 4\ell$), enough events can still be collected with
large luminosities. Note also that the corresponding backgrounds are almost
negligible (see for example [20] for an experimental study of three photon
final states). Thus, in Fig. 4, we elaborate on the idea of resolving flat
directions using further production modes. To this end, we consider a
hypothetical scenario in which the ratio $c_{WW}/c_{BB}$ has been
experimentally established (this measurement can be performed by just
observing the ratio of $\gamma\gamma$ events over $ZZ$ or $Z\gamma$ events).
Clearly, $c_{\gamma\gamma}$ and $c_{gg}$ cannot just be individually
determined by fitting the diphoton excess. This flat direction in the
$c_{gg}-c_{\gamma\gamma}$ plane is depicted by the orange band in Fig. 4 for
$\Gamma_{S}=45$ GeV and $c_{WW}/c_{BB}=1$. Now in addition if the associated
production $SW^{\pm}\rightarrow 2\gamma\,2j$ is observed to be, for example,
$0.01\pm 0.005$ fb, the degeneracy is broken and $c_{\gamma\gamma}$ can be
constrained independently, as shown by the vertical band in the figure.

Figure 2: Feynman diagrams for $S$ production in association with a gauge
boson $V$. The process on the right panel can only arise for $S$ coupling
directly to the light quarks. It has not been considered in our analysis,
since it can be neglected under simple flavor assumptions (see the text for
details).

The discussion above assumes no substantial direct coupling of $S$ to the SM
fermions. If such couplings exist, another contribution to the associated
production originates when an EW gauge boson is radiated off from one of the
initial quarks (right panel of Fig. 2). However, the cross section is
negligible when linked to light quarks under the assumption that the couplings
obey a minimal flavor violation structure and are therefore naturally expected
to be of the size of the Yukawa couplings. Flavor constraints would be hard to
evade otherwise. If large couplings to the light quarks were nonetheless
present, relying on some cancellation to pass flavor constraints, then a
detailed study of the kinematic would be worth performing in order to
discriminate the various contributions to the associated production. We have
checked that the associated production cross section via an initial $b$ quark
remains smaller than the contributions computed in Fig. 3 in most of the
parameter space. Finally, we have checked that the gluon-fusion associated
production $S+W^{\pm}/Z/\gamma$ together with an extra jet is typically
subdominant too, except in the region of small $c_{\gamma\gamma}$ ($<0.1$)
where it can anyway be reduced by an appropriate cut on the gauge boson
$p_{T}$ and by vetoing the extra jet.

This simple analysis illustrates the importance of considering the associated
production mechanisms. Indeed, the argument does not hold equally well for $S$
production in VBF since it turns out to have a remaining large dependence on
$c_{gg}$. The reason is that, contrary to the Higgs case whose couplings to
the electroweak gauge bosons appear at the tree level, VBF contamination by
gluon initiated processes in the singlet case can be rather large even after
tagging on forward jets [21, 22]. Cuts in this respect are provided in section
5. Nonetheless, it is worth to point out that measurements in VBF together
with the determination of $\Gamma_{S}$ might shed light on possible $S$ hidden
decays [23].

Figure 3: Contour lines of the cross sections (in fb) for the associated
production channel with an electroweak gauge boson in the plane
$c_{\gamma\gamma}-c_{WW}/c_{BB}$. The numbers stand for each group of three
contour lines.

Figure 4: Regions in the $c_{\gamma\gamma}-c_{gg}$ plane constrained by the
diphoton excess for $c_{WW}/c_{BB}=1$ for $\Gamma_{S}=45$ GeV. The vertical
band stands for the value of $c_{\gamma\gamma}$ determined by measurements of
$pp\rightarrow SW^{\pm}\rightarrow 2\gamma\,2j$ (see the text for details).
Dijet constraints from 8 TeV data [28, 29] are also shown.

On top of it, a last comment concerns the spin-1 alternatives for explaining
the diphoton excess. As it has been pointed out in [24], these scenarios rely
on the production of a 750 GeV vector boson that subsequently decays into a
photon and a light scalar. The latter further decays into two collimated
photons that, at the detector level, appear to be a single one. This kind of
setup can not however give rise to sizable amount of three gauge boson events.
Particularly with $W^{\pm}$ in the final state. As a consequence, $S$
production in association with gauge bosons provides a striking signature for
disentangling spin-0 and spin-1 models.

The distinctive kinematics of associated production provides different ways to
inquire the parity of such a scalar. The polar angle of the radiated vector
boson has been highlighted in this respect in the context of Higgs physics
(see for example [25, 26], and [27] for related experimental searches at
Tevatron). The large Higgs coupling to the longitudinal polarization of the
gauge bosons are however instrumental for these studies. The rather small
splitting between the Higgs mass and $m_{Z}$ and $m_{W}$ is also of major
significance for analyses based on the behavior of the cross section near
threshold. Accordingly, this observable is no longer suitable for $S$ physics
(note that $S$ might not even couple to the longitudinal polarization of the
gauge bosons). Related results in this direction have been also pointed out in
[30]. Further observables for Higgs physics have been presented for example in
[31]. Several angles between the Higgs momentum and reconstructed momenta of
the leptons and jets in the decay of the gauge boson produced in association
with the Higgs boson have been identified as discriminating variables to
scrutinize the CP properties of $S$. However, the rather small cross section
in this channel compared to GF and VBF makes the latter much more appropriate
for an early data analysis. We will thus focus on these channels hereafter.

##  4 Gluon fusion

The GF production cross section can be conveniently written as

$\sigma^{\text{GF}}=123\times\left(\frac{c_{gg}}{0.01}\right)^{2}\,\text{fb},$
(4)

as computed at LO using MadGraph. The NN23LO1 [32] parton-distribution
functions (PDFs) have been used. From the computation of the GF production at
higher order in the SM, we expect a large K-factor of order $1.7-2$ at NLO.
This K-factor will anyway drop in the computation of the asymmetry computed
below. We do not include it since a consistent treatment would also require a
NLO estimation of the various backgrounds, which is beyond the scope of our
analysis. Three different decay modes of $S$ are considered in GF, namely
$S\rightarrow ZZ$ in both the fully leptonic ($4\ell$) and the semileptonic
channels ($2j\,2\ell$) as well as $S\rightarrow W^{+}W^{-}$ with semileptonic
decay ($2j\ell\,\not{E}_{T}$). In order to tag these events at the
experimental level, all events are first required to pass the following set of
common cuts. Leptons must have $p_{T}^{\ell}>10$ GeV and $|\eta_{\ell}|<2.5$.
Jets are instead required to have $p_{T}^{j}>20$ GeV and $\eta_{j}<5$. Same-
flavor leptons must be separated by $\Delta R>0.2$ while different-flavor
leptons must fulfill $\Delta R>0.1$. Besides, all leptons must be separated
from other jets by $\Delta R>0.2$, and jets by $\Delta R>0.4$ among
themselves.

Then, exactly two opposite-sign lepton pairs are required in the four-lepton
channel. The two with invariant mass closest to $m_{Z}$ are tagged as coming
from one $Z$, and the other two from the second one. In the semileptonic $ZZ$
decay exactly two opposite-sign leptons and at least two jets must be present.
In the semileptonic $W^{+}W^{-}$ decay exactly one lepton and at least two
jets and $\not{E}_{T}>20$ GeV are instead required. The longitudinal momentum
of the neutrino can be in this case reconstructed by the $W^{\pm}$ on-shell
condition (see for example [34]). Following Ref. [14], we take the smaller in
absolute value among the two possible solutions. There is no further ambiguity
in pairing pair of particles with the corresponding massive gauge bosons in
any of these channels. We therefore require any reconstructed $Z$ ($W^{\pm}$)
mass to be in a window of $\pm 20$ GeV around $m_{Z}$ $(m_{W})$. Besides, each
$Z$ and $W^{\pm}$ is required to have $p_{T}>250$ GeV.

|  $4\ell$ |  $2j\,2\ell$ |  $2j\,\ell\,\not{E}_{T}$
---|---|---|---
$\epsilon$ (%) |  42 |  40 |  30
$\sigma_{b}$ (fb) |  0.04 |  34 |  240
Table 1: Estimated signal efficiencies ($\epsilon$) and background cross
sections ($\sigma_{b}$) for GF events after the cuts described in the text.

On top of this, the invariant mass of the four SM tagged particles is required
to be in the range $[700,800]$ GeV. Finally, the event must not pass the VBF
criteria, to be defined in the next section.

The window around the mass of the gauge bosons in the previous cuts is
required in order to further reduce the background with respect to the signal
(which gets only slightly affected). In particular, pair of jets come mainly
from QCD radiation and hence their invariant masses do not necessarily peak at
the $m_{W}(m_{Z})$ mass. However, if signals with of-shell gauge bosons were
to be studied, the corresponding cut should be relaxed. In this respect, it is
worth mentioning Ref. [33], where the authors claim that the four-lepton
channel can get sizable contributions from processes containing a virtual
photon.

The efficiencies for selecting events in each of these categories are shown in
Table 1. The estimated cross sections for the SM backgrounds after passing all
cuts are also shown. The irreducible backgrounds dominate in all cases.
Therefore, only these have been taking into account. In order to compute all
these quantities we have generated parton-level events with MadGraph v5 which
are subsequently passed through Pythia v6 [35] to account for showering,
hadronization and fragmentation effects. The cuts above are finally
implemented in MadAnalysis v5 [36].

Figure 5: $\theta^{\text{GF}}$ distribution for reconstructed four-lepton
signal events for the scalar (solid blue) and the pseudo-scalar (dashed red)
cases. The background is shown in dotted green. Signal and background
distributions have been independently normalized to unity. Their respective
importance will depend on the parameter space point. Figure 6:
$\mathcal{A}^{\text{GF}}$ distribution for the scalar (solid blue) and the
pseudo-scalar (dashed red) cases with 40 observed events after 10’000 pseudo
experiments. The distance ($d$) between the two central values in terms of the
largest $\sigma$ is also shown.

Having reconstructed the momenta of the four decay products, we can define the
following asymmetry:

$\mathcal{A}^{\text{GF}}=\frac{N(\theta^{\text{GF}}>\pi/4)-N(\theta^{\text{GF}}<\pi/4)}{N(\theta^{\text{GF}}>\pi/4)+N(\theta^{\text{GF}}<\pi/4)},$
(5)

where

$\theta^{\text{GF}}=\bigg{\\{}\begin{array}[]{llllll}\theta&&\text{if}&&&\theta<\pi/2\\\
\pi-\theta&&\text{if}&&&\theta>\pi/2\end{array},$ (6)

and

$\theta=\arccos{\left\\{\frac{(p_{1}\times p_{2})\cdot(p_{3}\times
p_{4})}{|p_{1}\times p_{2}||p_{3}\times p_{4}|}\right\\}},$ (7)

with $p_{1,2}$ and $p_{3,4}$ the three-momenta of the decay products of each
massive gauge boson. This observable has been widely used in Higgs physics
(see for example [37]). However, the small Higgs mass makes some channels
above not suitable for CP studies with this asymmetry, inasmuch as the signal
peaks in the region populated by the SM background. For $S$ decays instead,
the rather large mass allows us to stay in much more suppressed background
regions. Note also that two body $S$ decays could be also considered. As a
matter of fact, photon conversion events have been discussed in the Higgs
literature [38]. The typical opening angle of the lepton products are however
of the order of $m_{e}/E_{\gamma}\sim 10^{-6}$, which is well below any
present or future experimental sensitivity.

Figure 7: One sigma statistical interval for $\mathcal{A}^{\text{GF}}$ as a
function of the total number of observed events for only signal (left panel)
and with as much background as signal (right panel) for the scalar (solid
blue) and pseudo-scalar (dashed red) cases. The distance ($d$) between the two
central values in terms of
$\sigma_{\text{max}}\equiv\text{max}\\{\sigma_{O},\sigma_{E}\\}$ is also
shown.

In four-lepton events, the variable defined in Eq. 6 takes the form shown in
Fig. 5. No significant departures from this shape are found in other channels.
In order to quantify the discrimination power of this asymmetry for a given
number $N_{\text{obs}}$ of observed events, we perform 10’000 pseudo
experiments with $N_{\text{obs}}$ events each. As a matter of example, the
distribution followed by $\mathcal{A}^{\text{GF}}$ for $N_{\text{obs}}=40$ for
signal only is shown in Fig. 6. The one sigma statistical uncertainty is
defined by the symmetric interval around the center of the distribution
containing the 68% of the total area. For the matter of example, this is also
shown in the figure. These 40 events in the $2j\ell\not{E}_{T}$ final state
can be reached, in the minimal width case, for luminosities as low as
$\mathcal{L}\sim 5$ fb-1 for $(c_{\gamma\gamma},c_{WW}/c_{BB})=(2,2)$, while
$(c_{\gamma\gamma},c_{WW}/c_{BB})=(0.1,0.1)$ would require $\mathcal{L}>100$
fb-1. The large background makes the analysis harder and the luminosity needed
to discriminate the two CP hypotheses will be estimated in section 6. Figure 7
gives the asymmetry $\mathcal{A}^{\text{GF}}$ as a function of the total
number of observed events under the assumption of negligeable background (left
panel) and under the assumption of as many background events as signal events
(right panel).

##  5 Vector-boson fusion

The LO cross section for producing $S$ in association with two jets with
$p_{T}$ larger than 10 GeV, separated by at least $\Delta R>0.1$ and with
dijet invariant mass above 400 GeV, can be approximately written as

$\displaystyle\sigma^{\text{VBF}}=\,$
$\displaystyle\bigg{[}45~{}\left(\frac{c_{gg}}{0.01}\right)^{2}+1.2\frac{c_{\gamma\gamma}^{2}}{(1+r)^{2}}$
$\displaystyle+1.7\frac{c_{\gamma\gamma}^{2}r}{(1+r)^{2}}+43\frac{c_{\gamma\gamma}^{2}r^{2}}{(1+r)^{2}}\bigg{]}\,\text{fb},$
(8)

with $r\equiv c_{WW}/c_{BB}$. The coefficients above have been again computed
using MadGraph with the NN23LO1 PDFs. The interference between gluon-initiated
diagrams (proportional to $c_{gg}$) and VBF diagrams is negligible and hence
not shown in this equation. Two example diagrams are depicted in Fig. 8.
Hereafter we denote by $S^{\text{QCD}}$ and $S^{\text{EW}}$ the production
computed using each channel alone. $S^{\text{QCD}}$ in the plane
$c_{\gamma\gamma}-c_{WW}/c_{BB}$ can be easily estimated using this equation
in light of the $c_{gg}$ values provided in Fig. 1. Instead $S^{\text{EW}}$ is
plotted in Fig. 9.

Figure 8: Examples of Feynman diagrams for $S$ production via electroweak
(left) and QCD (right) with two additional jets.

Figure 9: Contour lines in the plane $c_{WW}/c_{BB}-c_{\gamma\gamma}$ of $S$
production cross sections (in fb) with two forward jets initiated by
electroweak gauge bosons and after the parton-level VBF cuts described in the
text.

VBF events can be tagged at the experimental level in five different decay
modes of $S$. These comprise the three possibilities described in the previous
section with two additional forward jets, namely $4\ell\,2j$, $4j\,2\ell$ and
$4j\,\ell\,\not{E}_{T}$, as well as the decay into $\gamma\gamma$ and
$\ell^{+}\ell^{-}\,\gamma$. Events are first selected by imposing the same
common cuts as in GF, while photons should be separated from any other tagged
particle by $\Delta R>0.2$. When more than two jets are present, forward-jet
candidates are selected to be those two jets with invariant mass
$m_{j_{1}j_{2}}$ less similar to $m_{Z}$ (or $m_{W}$) among the four leading
jets. They are subsequently required to fulfill the VBF criteria. This is
defined by $m_{j_{1}j_{2}}>500$ GeV, $\eta_{j_{1}}\eta_{j_{2}}<0$,
$|\Delta\eta_{j_{1}j_{2}}|>3$ and $\Delta R_{j_{1}j_{2}}>0.4$. These cuts are
motivated by previous searches for heavy Higgs bosons [8]. Any reconstructed
$Z$ ($W^{\pm}$) is again required to have a mass within a window of $\pm 20$
GeV around $m_{Z}$ ($m_{W}$). Besides, the $p_{T}$ of the two leading photons
as well as the $p_{T}$ of each reconstructed $Z$ and $W^{\pm}$ must be still
larger than 250 GeV. Finally, we require the invariant mass of the two
reconstructed SM gauge bosons to be in the range [700, 800] GeV. The
efficiencies for selecting events in each of these categories, referred to
events generated using the parton-level cuts in Eq. 5, are shown in Table 2
for $S^{\text{QCD}}$ and $S^{\text{EW}}$. Notice that gluon-initiated VBF can
be dominant due to the rather large coefficient in front of $c_{gg}$ in Eq. 5.
This result contrasts with the Higgs case, the reason being that, unlike the
singlet $S$, the Higgs boson couples to the electroweak gauge bosons at the
tree level.

|  $2\gamma\,2j$ |  $4\ell\,2j$ |  $2\ell\,\gamma\,2j$ |  $4j\,2\ell$ |  $4j\,\ell\,\not{E}_{T}$
---|---|---|---|---|---
$\epsilon_{\text{QCD}}$ |  6 |  12 |  14 |  11 |  9
$\epsilon_{\text{EW}}$ |  18 |  15 |  18 |  15 |  12
$\sigma_{b}\,\,(\text{fb})$ |  0.42 |  0.001 |  0.03 |  1.8 |  15
Table 2: Estimated signal efficiencies ($\epsilon$, in percent) and background
cross sections ($\sigma_{b}$) for VBF events after the cuts described in the
text. Gluon-initiated processes (QCD) can very much contaminate pure
electroweak (EW) VBF.

The estimated background cross sections are also shown in Table 2. The
irreducible backgrounds dominate each category. The channels $4j\,2\ell$ and
$4j\,\ell\,\not{E}_{T}$ are mostly populated by Drell–Yan and $W^{\pm}$
production with radiated jets, rather than by diboson events. These cross
sections are of similar magnitude to those reported in the figures of Ref.
[8], which uses slightly different cuts. We construct the following asymmetry
for VBF events:

$\mathcal{A}^{\text{VBF}}=\frac{N(\theta^{\text{VBF}}>\pi/4)-N(\theta^{\text{VBF}}<\pi/4)}{N(\theta^{\text{VBF}}>\pi/4)+N(\theta^{\text{VBF}}<\pi/4)},$
(9)

where, analogously to the GF case,

$\theta^{\text{VBF}}=\bigg{\\{}\begin{array}[]{llllll}\theta&&\text{if}&&&\theta<\pi/2\\\
\pi-\theta&&\text{if}&&&\theta>\pi/2\end{array},$ (10)

Figure 10: $\theta^{\text{VBF}}$ distribution for reconstructed four-lepton
signal events for the scalar (blue) and the pseudo-scalar (red) cases. Solid
and dashed lines stand for $S^{\text{EW}}$ and $S^{\text{QCD}}$ respectively.
The background is shown in dotted green. Signal and background distributions
have been independently normalized to unity. Their respective importance will
depend on the parameter space point. Figure 11: $\mathcal{A}^{\text{VBF}}$
distribution for the scalar (solid blue) and the pseudo-scalar (dashed red)
cases with 40 observed events and $S^{\text{QCD}}=S^{\text{EW}}$ after 10’000
pseudo experiments. The distance ($d$) between the two central values in terms
of the largest $\sigma$ is shown in the lower panels.

Figure 12: One sigma statistical interval for $\mathcal{A}^{\text{VBF}}$ as a
function of the total number of observed events for only signal (left panel)
and as much background as signal (right panel) for the scalar (solid blue) and
pseudo-scalar (dashed red) cases. $S^{\text{QCD}}=S^{\text{EW}}$ has been
assumed in both panels. The distance ($d$) between the two central values in
terms of $\sigma_{\text{max}}\equiv\text{max}\\{\sigma_{O},\sigma_{E}\\}$ is
shown in the lower panels.

Figure 13: Parameter space regions for which the CP odd and even hypothesis
can be disentangled at the $2\sigma$ level with 300 fb-1. The region where the
CP nature can be determined by the different channels is given by the area
above the corresponding line. The grey striped regions are excluded (see Fig.
1). In the left panel we assume no extra contributions to $\Gamma_{S}$, while
in the right panel we fix $\Gamma_{S}=45$ GeV. The light area enclosed by the
dashed lines stands for the 1.7$\sigma$ region.

with $\theta$ the angle between the $p_{T}$ of the two tagged forward jets.
This observable has been previously considered in the literature in the
context of Higgs studies (see for example [39, 40]). As a matter of example,
we show the distribution for $\theta^{\text{VBF}}$ as reconstructed in four-
lepton signal events in Fig. 10. As in the GF case, the distribution in other
channels does not present significant differences. The discrimination power of
this angle is apparent from the plot. In order to quantify it for a given
number $N_{\text{obs}}$ of observed events we proceed as in the previous
section. Figure 11 shows the distribution followed by
$\mathcal{A}^{\text{VBF}}$ for $N_{\text{obs}}=40$ and
$S^{\text{QCD}}=S^{\text{EW}}$, under the assumption that the background is
negligible. In the $2j2\gamma$ channel, this number of events can be reached
with luminosities of order $\mathcal{L}\sim 60$ fb-1 for
$(c_{\gamma\gamma},c_{WW}/c_{BB})=(2,2)$, while
$(c_{\gamma\gamma},c_{WW}/c_{BB})=(0.1,0.1)$ requiress $\mathcal{L}\sim 200$
fb-1. We plot the one sigma statistical interval as a function of the total
number of observed events in Fig. 12 under the assumption of no background
(left panel) and as much background as signal (right panel). It turns out that
less than 40 (60) events are necessary to start disentangling the CP
properties of $S$ if there is no background (if there is as much background as
signal). Despite this result being apparently much better than the one
obtained in GF (see Fig. 7), in practice VBF is much suppressed (see Eq. 4 and
Fig. 9) and they are hence complementary.

##  6 Results

For each point in the parameter space region and for each of the eight event
categories $i$ defined for GF and VBF, we compute $\mathcal{A}^{\text{GF}}$
and $\mathcal{A}^{\text{VBF}}$ by estimating the number of signal and
background events. For a fixed luminosity, the latter can be derived from
Tables 1 and 2. The number of signal events in each case can be in turn
computed as

$N_{\text{signal}}=\sum_{i}\sigma\times\text{BR}\left(S\rightarrow
i\right)\times\epsilon_{i},$ (11)

where $\epsilon_{i}$ stands for the corresponding experimental efficiency as
provided in Tables 1 and 2, too. We assume these efficiencies to be
independent of the coefficients of the operators in Eq. 2. We have checked
that this is the case in almost the whole parameter space, small variations
arising only in the VBF $2\gamma\,2j$ channel for $c_{WW}\ll
c_{\gamma\gamma}$. At any rate, this region is dominated by $S^{\text{QCD}}$
and therefore not sensitive to these variations. Figure 13 shows the regions
where, with a total luminosity of 300 fb-1, the CP-odd hypothesis can be
excluded at the $2\sigma$ level in favor of the CP-even using the two
asymmetries separately (in the left panel no extra sources for $\Gamma_{S}$
are considered while in the right panel $\Gamma_{S}=45$ GeV instead). These
regions are defined by requiring the mean value of $\mathcal{A}$ in the odd
case to be separated by at least 2$\sigma$ from the mean value of
$\mathcal{A}$ in the even case. For the matter of an example, this separation
($d$) is also shown in Figs. 6 and 11.

The separation between the two hypothesis exceeds 1.5$\sigma$ throughout the
parameter space. It is important to mention that small variations on the
efficiency and cross section of the $2\gamma\,2j$ channel can make the
corresponding region to look notably different for a fixed luminosity. In that
respect, an optimisation of the different cuts, as well as other more
sophisticated analyses like the matrix-element method used [41] in the four-
lepton Higgs decay channel, can help covering larger regions of the parameter
space. At any rate, even with the basic cuts used in our analysis,
luminosities slightly larger than 300 fb-1 will be sufficient to test at the
2$\sigma$ confidence level the whole parameter space compatible with 8 TeV
constraints and 13 TeV data.

The area excluded by searches at 8 TeV (see Fig. 1) has been superimposed.
Note that this area would be smaller if the required diphoton cross section at
13 TeV was smaller than the 8 fb that we are using throughout this letter.
With 30 fb-1 only a small portion of the available parameter space can be
tested.

It can be shown that GF and VBF channels are complementary, the former being
mostly sensitive to the upper region with even small $c_{\gamma\gamma}$. It is
also worth emphasizing the role played by semileptonic $W^{+}W^{-}$ decays.
This is in contrast with Higgs physics, for which considering these final
states is not even possible, inasmuch as the signal peaks in the region
populated by the huge $W^{\pm}+$ jets background. At any rate, the dominant
channel is given by $S\rightarrow\gamma\gamma$ in VBF. Indeed, the fact that
gluon-initiated processes can also contaminate the EW VBF selection makes this
channel sensitive even to regions of small EW couplings, which require large
$c_{gg}$.

##  7 Conclusions

The recently observed diphoton excess around $750$ GeV is triggering a lot of
attention. One of the most widely studied explanations relies on a spin-0 real
singlet with effective interactions to the SM gauge bosons. In this letter we
have thus adopted this setup and discussed the LHC reach for unraveling the CP
properties of such a resonance. First, we reviewed the current constraints and
commented on the possibility of avoiding flat directions (e.g. resolving the
individual couplings to photons and gluons) by considering the associated
production with a gauge boson. We have then studied the LHC potential for
unraveling the CP nature of such a scalar. Two different asymmetries have been
covered in this regard. These are to be constructed out of events produced in
gluon and vector-boson fusion respectively. We have shown that events in these
categories can be efficiently tagged at the experimental level while keeping
backgrounds under control. We have emphasized that as few as $\sim 50$ events
are needed to separate the CP even and odd hypotheses. This number of events
can be reached in different regions of the parameter space during the next LHC
run. In particular, for the full run all the parameter space region is
expected to be probed, relying mainly on the VBF $2\gamma\,2j$ channel.

## Acknowledgements

We are particularly grateful to Pedro Schwaller for enlightening discussions
at the beginning of this project and for useful comments on the manuscript. We
are also thankful to Kai Schmidt-Hoberg, Daniel Stolarski, Alfredo Urbano and
Roberto Vega-Morales for valuable feedback. C.G. is supported by the European
Commission through the Marie Curie Career Integration Grant 631962 and by the
Helmholtz Association. M.R. is supported by la Caixa, Severo Ochoa grant
program. M.R. and T.V. are supported by the Spanish Ministry MEC under grants
FPA2014-55613-P and FPA2011-25948, by the Generalitat de Catalunya grant
2014-SGR-1450, by the Severo Ochoa excellence program of MINECO (grant
SO-2012-0234).

## References

  * [1] The ATLAS collaboration, ATLAS-CONF-2015-081.
  * [2] CMS Collaboration [CMS Collaboration], CMS-PAS-EXO-15-004.
  * [3] The corresponding talks are available in https://indico.in2p3.fr/event/12279/session/12/contribution/163/material/slides/1.pdf and https://indico.in2p3.fr/event/12279/session/12/contribution/218/material/slides/0.pdf.
  * [4] A complete list of references can be found in, http://inspirehep.net/search?p=refersto%3Arecid%3A1410174.
  * [5] Y. Kats and M. Strassler, arXiv:1602.08819 [hep-ph].
  * [6] R. Franceschini et al., arXiv:1512.04933 [hep-ph].
  * [7] The ATLAS and CMS Collaborations, ATLAS-CONF-2015-044.
  * [8] G. Aad et al. [ATLAS Collaboration], Eur. Phys. J. C 76, no. 1, 45 (2016) [arXiv:1507.05930 [hep-ex]].
  * [9] CMS Collaboration [CMS Collaboration], CMS-PAS-HIG-14-006.
  * [10] G. Aad et al. [ATLAS Collaboration], Phys. Rev. D 92, no. 3, 032004 (2015) [arXiv:1504.05511 [hep-ex]].
  * [11] L. A. Harland-Lang, V. A. Khoze and M. G. Ryskin, arXiv:1601.07187 [hep-ph].
  * [12] G. Panico, L. Vecchi and A. Wulzer, arXiv:1603.04248 [hep-ph].
  * [13] G. Aad et al. [ATLAS Collaboration], Phys. Lett. B 738, 428 (2014) [arXiv:1407.8150 [hep-ex]].
  * [14] V. Khachatryan et al. [CMS Collaboration], JHEP 1510, 144 (2015) [arXiv:1504.00936 [hep-ex]].
  * [15] G. Aad et al. [ATLAS Collaboration], JHEP 1601, 032 (2016) [arXiv:1509.00389 [hep-ex]].
  * [16] A. Alves, A. G. Dias and K. Sinha, arXiv:1512.06091 [hep-ph].
  * [17] J. Alwall et al., JHEP 1407, 079 (2014) [arXiv:1405.0301 [hep-ph]].
  * [18] A. Alloul, N. D. Christensen, C. Degrande, C. Duhr and B. Fuks, Comput. Phys. Commun. 185, 2250 (2014) [arXiv:1310.1921 [hep-ph]].
  * [19] F. Staub et al., arXiv:1602.05581 [hep-ph].
  * [20] G. Aad et al. [ATLAS Collaboration], arXiv:1509.05051 [hep-ex].
  * [21] J. D. Bjorken, Phys. Rev. D 47, 101 (1993).
  * [22] V. Del Duca, G. Klamke, D. Zeppenfeld, M. L. Mangano, M. Moretti, F. Piccinini, R. Pittau and A. D. Polosa, JHEP 0610, 016 (2006) [hep-ph/0608158].
  * [23] J. F. Kamenik, B. R. Safdi, Y. Soreq and J. Zupan, arXiv:1603.06566 [hep-ph].
  * [24] M. Chala, M. Duerr, F. Kahlhoefer and K. Schmidt-Hoberg, Phys. Lett. B 755, 145 (2016) [arXiv:1512.06833 [hep-ph]].
  * [25] D. J. Miller, S. Y. Choi, B. Eberle, M. M. Muhlleitner and P. M. Zerwas, Phys. Lett. B 505, 149 (2001) [hep-ph/0102023].
  * [26] S. Y. Choi, D. J. Miller, M. M. Muhlleitner and P. M. Zerwas, Phys. Lett. B 553, 61 (2003) [hep-ph/0210077].
  * [27] T. Aaltonen et al. [CDF and D0 Collaborations], Phys. Rev. Lett. 114, no. 15, 151802 (2015) [arXiv:1502.00967 [hep-ex]].
  * [28] G. Aad et al. [ATLAS Collaboration], Phys. Rev. D 91, no. 5, 052007 (2015) [arXiv:1407.1376 [hep-ex]].
  * [29] CMS Collaboration [CMS Collaboration], CMS-PAS-EXO-14-005.
  * [30] A. Djouadi, J. Ellis, R. Godbole and J. Quevillon, arXiv:1601.03696 [hep-ph].
  * [31] R. Godbole, D. J. Miller, K. Mohan and C. D. White, Phys. Lett. B 730, 275 (2014) [arXiv:1306.2573 [hep-ph]].
  * [32] R. D. Ball et al., Nucl. Phys. B 867, 244 (2013) [arXiv:1207.1303 [hep-ph]].
  * [33] D. Stolarski and R. Vega-Morales, Phys. Rev. D 93, no. 5, 055008 (2016) [arXiv:1601.02004 [hep-ph]].
  * [34] N. F. da Silva Fernandes de Castro, CERN-THESIS-2008-083.
  * [35] T. Sjostrand, S. Mrenna and P. Z. Skands, JHEP 0605, 026 (2006) [hep-ph/0603175].
  * [36] E. Conte, B. Dumont, B. Fuks and C. Wymant, Eur. Phys. J. C 74, no. 10, 3103 (2014) [arXiv:1405.3982 [hep-ph]].
  * [37] The ATLAS collaboration [ATLAS Collaboration], ATLAS-CONF-2015-008.
  * [38] F. Bishara, Y. Grossman, R. Harnik, D. J. Robinson, J. Shu and J. Zupan, JHEP 1404, 084 (2014) [arXiv:1312.2955 [hep-ph]].
  * [39] C. Englert, D. Goncalves-Netto, K. Mawatari and T. Plehn, JHEP 1301, 148 (2013) [arXiv:1212.0843 [hep-ph]].
  * [40] A. Djouadi, R. M. Godbole, B. Mellado and K. Mohan, Phys. Lett. B 723, 307 (2013) [arXiv:1301.4965 [hep-ph]].
  * [41] Y. Chen, E. Di Marco, J. Lykken, M. Spiropulu, R. Vega-Morales and S. Xie, JHEP 1501, 125 (2015) [arXiv:1401.2077 [hep-ex]].


# Visual Guitar Tab Comparison

###### Abstract

We designed a visual interface for comparing different guitar tablature (tab)
versions of the same piece. By automatically aligning the bars of these
versions and visually encoding different metrics, our interface helps
determine similarity, difficulty, and correctness. During our design, we
collected and integrated feedback from musicians and finally conducted a
qualitative evaluation with five guitarists. Results confirm that our
interface effectively supports comparison and helps musicians choose a version
appropriate for their personal skills and tastes. Our source code and online
demo are available at github.com/visvar/visual-guitar-tab-comparison.

##  1 Introduction

Many guitarists learn to play new pieces from tablature files available on the
internet. Such files are abundant on websites such as ultimate-guitar.com,
often with multiple variants or even edit histories. While these different
versions sometimes have user ratings, a musician looking for a tab can not
always assume that the best-rated or most popular one is the most appropriate
for them: Maybe the most correct version exceeds their skills, maybe they
prefer the sound of another one, or maybe the popularity and rating could be
misleading, for example, when the most correct version has been added later
and therefore gotten less attention and fewer ratings. To support guitarists
in this choice, we aim to provide them with a better understanding of the
specific characteristics of each version, as well as explicitly revealing
differences [1] – by visualizing them. Music visualization is a diverse field
[2, 3, 4] and visually augmented sheet music helps analyze rhythm and harmonic
patterns [5] or identifying mistakes in composition [6] and instrument
practice [7]. Our work uses a similarity-based coloring [8] designed to reveal
repeating structures in sheet music. However, to the best of our knowledge,
there is no visualization for comparing alternative versions.

##  2 Design

Tabs usually contain multiple instruments. To make comparison feasible, we
restrict our visualizations to only a single instrument the user chooses for
each version. Following the principle of focus and context, we designed two
main views. Our tab based on alphaTab (alphatab.net) displays all bars of an
instrument in a single row, allowing us to place all versions from top to
bottom, such that the user can scroll through all at the same time. An
additional overview uses the same layout but reduces each bar to a small
rectangle to fit the entire length of all versions onto the screen. Serving as
a minimap, this overview also lets users click on a bar to navigate the
detailed tab view. Since there might be missing or extra bars in some
versions, we automatically align [9] them by inserting empty bars. Based on
feedback from guitarists, we chose different metrics that we visualize using
colors in both views:

Note density. With our density metric, we visualize how many notes each bar
has. Differences between versions can hint at varying difficulty or
correctness, for example, when chords in one version have more notes.

Fret span. Another metric takes the lowest and highest fret played in each bar
to determine how much the guitarist’s fingers have to move while fretting
notes. We added an alternative, where we calculate the distance in millimeters
instead of the number of frets (Figure 1). This value is mapped to a color to
facilitate comparison between but also within versions.

Figure 1: The fret span in mm indicates how much the fretting hand has to move
while playing each bar.

Playing techniques. The annotation of different playing techniques, such as
bends, palm muting, or harmonics, might vary between tab versions. To help
musicians spot versions with techniques they want to learn or avoid, we
designed a visualization with color-coded stacked rectangles that encode the
techniques used in each bar (Figure 2).

Figure 2: Visualization of playing techniques used in each bar of each
version.

Bar similarity. We further wanted to show the similarity between bars, within
and between versions. To indicate this similarity through colors, we apply an
existing similarity-based color mapping [8] to the bars of all versions at
once (Figure 3).

Figure 3: Similar colors indicate similar bars.

Explicit differences. To reveal minor changes, for example, those that occur
when someone uploads a slightly corrected version to a website, we explicitly
encode differences (Figure 4). Such small edits could be barely visible in our
other visualizations.

Figure 4: Explicitly encoded changes (blue) between the top version and
others.

##  3 Qualitative Evaluation

We conducted a pair analytics study [10] with five guitarists (P1–P5) who had
no visualization background, where we spent one hour with each looking at tabs
of their choice together.

Note density. To increase difficulty gradually, P1 would use a version with
fewer notes first and, once learned, one with more notes later. As P5 likes
singing along, she prefers low-density versions where she does not have to
play as much.

Fret span. P4, who plays bass guitar, found our metric for fret span in
millimeters helpful, especially for beginners who cannot yet stretch their
hands enough for the wider gaps between lower frets.

Playing techniques. Visualizing playing techniques in each bar of each version
helped P2 determine which ones were more accurate, as he knew what the song
should sound like. P3 compared the use of techniques between versions to
exclude outliers. According to P5, seeing which techniques a tab contains
could also increase the motivation to play it, as you know which ones you will
practice or avoid.

Bar similarity. Having an overview of the structure could help him learn a new
song faster mentioned P2, as he was able to detect verse, chorus, and bridge
just by looking at the colors. P3 would use similarity to spot and discard
inaccurate versions, as they are more different from the rest.

Explicit differences. P3 suggested using both bar similarity and explicit
differences together, as they complement each other: the former shows overall
structure, and the latter highlights finer details. When making his own
arrangements, P4 sometimes lost track of changes – changes he could easily
spot with our interface.

##  4 Conclusion

Our proposed visual approach effectively supports comparison between versions
of the same piece. As our evaluation indicated, guitarists without prior
visualization experience were able to make helpful findings and even created
their own strategies for estimating quality and difficulty. As a work in
progress, our design also faces limitations: The alignment does not work
precisely for specific cases, for example, when the longest version misses
bars that another one has – such cases require multi-sequence alignment.
Participants missed audio playback to judge how far versions sound different.
Some techniques, such as sweep picking, might not be annotated in tab files
and would require heuristics to detect. Currently, there is no automatic
recommendation or guidance, a feature that could personalized for taste or
skill. In the future, we want to address these limitations and refine our
metrics to better handle chords and different playing techniques. An extension
to large collections of tabs could further support finding suitable pieces or
exercises based on taste and skill.

##  5 Acknowledgments

This work was funded by the Cyber Valley Research Fund and by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID
251654672 – SFB TRR 161, project A08.

## References

  * [1] M. Gleicher, D. Albers, R. Walker, I. Jusufi, C. D. Hansen, and J. C. Roberts, “Visual comparison for information visualization,” _Information Visualization_ , vol. 10, no. 4, pp. 289–309, 2011. [Online]. Available: https://doi.org/10.1177/1473871611416549
  * [2] R. Khulusi, J. Kusnick, C. Meinecke, C. Gillmann, J. Focht, and S. Jänicke, “A survey on visualizations for musical data,” _Computer Graphics Forum (CGF)_ , 2020. [Online]. Available: https://doi.org/10.1111/cgf.13905
  * [3] H. B. Lima, C. G. R. D. Santos, and B. S. Meiguins, “A survey of music visualization techniques,” _ACM Computing Surveys_ , vol. 54, no. 7, 2021\. [Online]. Available: https://doi.org/10.1145/3461835
  * [4] M. Miller, J. Rauscher, D. A. Keim, and M. El-Assady, “CorpusVis: Visual analysis of digital sheet music collections,” _Computer Graphics Forum (CGF)_ , vol. 41, no. 3, pp. 283–294, 2022. [Online]. Available: https://doi.org/10.1111/cgf.14540
  * [5] M. Miller, D. Fürst, H. Hauptmann _et al._ , “Augmenting digital sheet music through visual analytics,” _Computer Graphics Forum (CGF)_ , vol. 41, no. 1, pp. 301–316, 2022. [Online]. Available: https://doi.org/10.1111/cgf.14436
  * [6] R. D. Prisco, D. Malandrino, D. Pirozzi, G. Zaccagnino, and R. Zaccagnino, “Understanding the structure of musical compositions: Is visualization an effective approach?” _Information Visualization_ , vol. 16, no. 2, pp. 139–152, 2017. [Online]. Available: https://doi.org/10.1177/1473871616655468
  * [7] M. Hori, C. M. Wilk, and S. Sagayama, “Piano practice evaluation and visualization by HMM for arbitrary jumps and mistakes,” in _53rd Annual Conference Information Sciences and Systems (CISS)_ , 2019, pp. 1–5. [Online]. Available: https://doi.org/10.1109/CISS.2019.8692813
  * [8] F. Heyen, Q. Q. Ngo, and M. Sedlmair, “Visual overviews for sheet music structure,” in _24th International Society for Music Information Retrieval Conference (ISMIR)_ , 2023. [Online]. Available: https://doi.org/10.48550/arXiv.2308.06140
  * [9] S. B. Needleman and C. D. Wunsch, “A general method applicable to the search for similarities in the amino acid sequence of two proteins,” _Journal of Molecular Biology_ , vol. 48, no. 3, pp. 443–453, 1970. [Online]. Available: https://www.sciencedirect.com/science/article/pii/0022283670900574
  * [10] R. Arias-Hernandez, L. T. Kaastra, T. M. Green _et al._ , “Pair analytics: Capturing reasoning processes in collaborative visual analytics,” in _44th Hawaii International Conference on System Sciences (HICSS)_ , 2011, pp. 1–10. [Online]. Available: https://doi.org/10.1109/HICSS.2011.339


# Theoretical analysis of the leptonic decays
$\bm{B\to\ell\ell\ell\bar{\nu}_{\ell}}$:
Identical leptons in the final state

Mikhail A. Ivanova and Dmitri Melikhova,b,c aJoint Institute for Nuclear
Research, Dubna, 141980, Russia
bD. V. Skobeltsyn Institute of Nuclear Physics, M. V. Lomonosov Moscow State
University, 119991, Moscow, Russia
cFaculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna,
Austria

###### Abstract

We study the effects of the identical leptons in the final state of the
$B^{+}\to\ell^{+}\ell^{-}\ell^{+}\bar{\nu}_{\ell}$ decay. The amplitude of the
process is described by the same form factors as the amplitude of the
${B\to\ell\ell\ell^{\prime}\bar{\nu}^{\prime}_{\ell}}$ decay for non-identical
leptons in the final state. However, the differential distributions are
strongly different, as the
${B^{+}\to\ell^{+}\ell^{-}\ell^{+}\bar{\nu}_{\ell}}$ amplitude contains both
the direct ($M_{a}$) and the exchange ($M_{b}$) diagrams. We calculate a
number of the differential distirbutions. In particular, we propose an
interesting observable that can be readily measured experimentally – the
differential distribution over the invariant mass of the pair of leptons of
the same charge, $\ell^{+}\ell^{+}$. The good news is that the interference
between $M_{a}$ and $M_{b}$, $d{\cal B}_{ab}$, is found to be at the level of
less than 1% in all considered differental distributions and therefore can be
neglected in the full kinematical region of this decay.

##  1 Introduction

This paper extends our recent analysis mi2022  of the $B\to
lll^{\prime}\nu^{\prime}$ decay ($l`\neq l$) to the case of the identical
leptons in the final state ($l^{\prime}=l$). Such reactions are being studied
experimentally exp1 ; exp2 ; exp3 ; exp4 , thus requiring a proper theoretical
understanding. By now, there have been a few theoretical papers sehgal ;
nikitin ; bharucha2021 ; beneke2 ; wang2022 , where $B$-decays into two lepton
pairs have been studied.

The $B\to\gamma^{*}l^{\prime}\nu^{\prime}$ amplitude (see Fig. 1) may be
parametrized via Lorenz-invariant form factors as follows:

$\displaystyle T_{\alpha\nu}(q,q^{\prime}|p)=i\int dx\,e^{iqx}\langle
0|T\\{j^{\rm e.m.}_{\alpha}(x),\bar{u}(0){\cal
O_{\nu}}b(0))\\}|\bar{B}_{u}(p)\rangle=\sum_{i}L^{(i)}_{\alpha\nu}(q,q^{\prime})F_{i}(q^{\prime
2},q^{2})+\ldots,\quad p=q+q^{\prime},$ (1.1)

with $q^{\prime}$ the momentum of the weak $b\to u$ current, and $q$ the
momentum of the electromagnetic current. In Eq. (1.1), ${\cal
O_{\nu}}=\gamma_{\nu},\gamma_{\nu}\gamma_{5}$ and $j_{\alpha}^{\rm e.m.}$ is
the conserved electromagnetic current

$\displaystyle j_{\alpha}^{\rm
e.m.}(0)=eQ_{b}\bar{b}(0)\gamma_{\alpha}b(0)+eQ_{u}\bar{u}(0)\gamma_{\alpha}u(0).$
(1.2)

The quantities $L^{(i)}_{\alpha\nu}(q,q^{\prime})$ represent the transverse
Lorents structures, $q^{\alpha}L^{(i)}_{\alpha\nu}(q,q^{\prime})=0$, and the
dots stand for the longitudinal part which is constrained by the conservation
of the electromagnetic current, $\partial_{\alpha}j_{\alpha}^{\rm e.m.}=0$,
and the equal-time commutation relations.

Fig. 1:  Feynman diagrams describing the amplitude (1.1).

The form factors $F_{i}(q^{\prime 2},q^{2})$ are complicated functions of two
variables, $q^{\prime 2}$ and $q^{2}$; the general properties of these objects
in QCD have been studied recently in ims . Noteworthy, gauge invariance
provides essential constraints on some of the form factors describing the
transition of the $B$-meson into the real photon, i.e., at $q^{2}=0$ bmns2001
; m2002 ; kruger ; kmn2016 .

In the past, theoretical analyses focused on a family of similar reactions,
namely, the $B\to\gamma l^{+}l^{-}$ and $B\to\gamma l\nu$ decays (see, e.g.,
aliev ; korchemsky ; kou ; mn2004 ; beneke1 ; kmn2018 ; beneke2018 ; ivanov ;
zwicky2019 ; bobeth ; zwicky2021 ); these processes are described by the same
form factors as four-lepton $B$-decays, but evaluated at a zero value of one
of the momenta squared. The corresponding form factors depend on one variable
$q^{\prime 2}$, $q^{\prime}$ the momentum of the weak current; for instance,
for radiative leptonic decays $B\to\gamma l^{\prime}\nu^{\prime}$, one needs
the form factors $F_{i}(q^{\prime 2},q^{2}=0)$.

The four-lepton decay of interest, $B\to l^{+}l^{-}l^{\prime}\nu^{\prime}$,
requires the form factors $F_{i}(q^{\prime 2},q^{2})$ for $0<q^{2},q^{\prime
2}<M_{B}^{2}$. The dependence of the form factors on the variable $q^{\prime
2}$ can be predicted reasonably well: there are no hadron resonances in the
full decay region $0<q^{\prime 2}<M_{B}^{2}$, and the $q^{\prime
2}$-dependence of the form factors is determined to a large extent by the
influence of the beauty mesons with the appropriate quantum numbers; all these
mesons are heavier than the $B$-meson and therefore lie beyond the physical
decay region of the variable $q^{\prime 2}$. The calculation of the
$q^{2}$-dependence of the form factors is a much more difficult task: light
vector mesons $V=\rho^{0},\omega,...$ lie in the physical decay region and
should be properly taken into account. At $q^{2}$ in the region of light
vector meson resonances, the form factors cannot be obtained directly in pQCD
ims . Here considerations based on the explicit account of these light vector
resonances—including their finite width effects—are mandatory; the resonance
contributions of interest may be unambiguously expressed via the weak $B\to V$
form factors. Then, at $q^{2}=0$, gauge-invariance constrains the values of
the form factors. These features allow us to calculate the form factors
$F_{i}(q^{\prime 2},q^{2})$ in the region $0<q^{2}\leq 1-2$ GeV2, which
dominates the four-meson decay rates and obtain consistent predictions for the
latter.

A relatively simple case of different lepton flavours $l\neq l^{\prime}$ was
considered in our recent paper mi2022 . In that case one can easily calculate
the differential distribution in $q^{2}$, $q$ the momentum of the $l^{+}l^{-}$
pair, as well as in $q^{\prime 2}$, $q^{\prime}$ the momentum of the
$l^{\prime}\nu^{\prime}$ pair: the angular variables do not enter the form
factors and as the result all angular integrations may be calculated
explicitly yielding the differential distributions in $q^{2}$ and in
$q^{\prime 2}$ in explicit forms.

This paper focuses on the case of the identical leptons in the final state
$l=l^{\prime}$. The amplitude is described by the same form factors as the
case $l\neq l^{\prime}$, so we use the model for these form factors
constructed in mi2022 . However, a specific feature of the case of the
identical leptons is the appearance of the exchange diagrams. For such
diagrams, the variables $q^{2}$ and $q^{\prime 2}$ that determine the form
factors do not coincide with the momenta of the $l^{+}l^{-}$ and $l^{+}\nu$
pairs in the final state and thus the angular variables appear explicitly in
the form factors. As the reuslt, the contribution of the exchange diagrams
cannot be obtained as an explicit analytic expression and a numerical
evaluation of the phase-space integrals is necessary. We provide here all
necessary details for the theoretical description of this reaction and report
the numerical predictions for a number of the differential distributions.

We propose an interesting kinematical variable: the differential distribution
in the momentum of the pair of the same-charge leptons (i.e. the $l^{+}l^{+}$
lepton in the case of the $B^{+}\to l^{+}l^{-}l^{+}\nu$ decay and the
$\mu^{+}e^{+}$ pair in the case of the $B^{+}\to\mu^{+}\mu^{-}e^{+}\nu_{e}$ or
the $B^{+}\to e^{+}e^{-}\mu^{+}\nu_{\mu}$ decay). This distribution can be
measured experimentally in a straightforward way and we obtain predictions for
this differential distribution.

##  2  The $B^{-}\to l^{+}l^{-}l^{\prime-}\bar{\nu}^{\prime}$ form factors

The amplitude of the $B\to lll^{\prime}\nu^{\prime}$ transition for
$l^{\prime}\neq l$ may be parameterized as follows (see also Lattice2021 ) :

$\displaystyle A(B\to
lll^{\prime}\nu^{\prime})=ie^{2}\frac{G_{F}}{\sqrt{2}}\,V_{ub}\cdot\bar{l}\gamma_{\alpha}l\cdot\bar{l}^{\prime}\gamma_{\nu}(1-\gamma_{5})\nu^{\prime}\frac{1}{q^{2}}\bigg{\\{}(g_{\alpha\nu}q^{\prime}q-q^{\prime}_{\alpha}q_{\nu})\frac{F_{1A}}{M_{B}}+q^{\prime}_{\alpha}q_{\nu}\frac{F_{2A}}{M_{B}}+q^{\prime}_{\alpha}q^{\prime}_{\nu}\frac{F^{\prime}_{2A}}{M_{B}}+i\epsilon_{\nu\alpha
q^{\prime}q}\frac{F_{V}}{M_{B}}\bigg{\\}},$

where the form factors satisfy the following constraints

$\displaystyle F_{2A}(q^{\prime 2},q^{2}=0)$ $\displaystyle=$ $\displaystyle
0,$ (2.4) $\displaystyle F^{\prime}_{2A}(q^{\prime 2},q^{2}=0)$
$\displaystyle=$ $\displaystyle\frac{2Q_{B}f_{B}M_{B}}{M_{B}^{2}-q^{\prime
2}}.$ (2.5)

Explicit formulas for the differential distributions in the case
$l^{\prime}\neq l$ have been derived in mi2022 ; we do not repeat these
formulas here but refer to mi2022 .

The same form factors parameterize the amplitude for the case $l^{\prime}\neq
l$; however, one has to take into account the contribution of the lepton
exchange diagrams in which the variables $q^{2}$ and $q^{\prime 2}$ have a
complicated relationship with the momenta of the final lepton pairs. The
details are given in the next Section. We now remind the essential features of
our model of the form factors as developed in mi2022 .

$\bullet$ The contribution of the form factor $F^{\prime}_{2A}(q^{2},q^{\prime
2})$ can be neglected in the case $l=l^{\prime}$ so in what follows we neglect
its contribution.

$\bullet$ For the form factors $F_{1A,2A,V}(q^{\prime 2},q^{2})$ we use
single-subtracted dispersion representations in $q^{2}$. This allows us to
take into account all constraints coming from gauge invariance and from the
known behaviour in the large-energy limit of QCD korchemsky .

$\bullet$ We assume that the spectral densities are saturated by light vector-
meson resonances $\rho^{0}$ and $\omega$ in the $q^{2}$-channel. Since these
resonances emerge in the physical region of the $B$-decay of interest, we take
into account the $q^{2}$-dependent finite widths of these resonances nachtmann
. In the end, we come to the following expressions for the form factors

$\displaystyle F_{1A}(q^{\prime 2},q^{2})$ $\displaystyle=$ $\displaystyle
F_{A}(q^{\prime
2})-\frac{Q_{B}f_{B}M_{B}}{q^{\prime}q}-q^{2}\sum\limits_{V=\rho^{0},\omega}\bigg{(}\frac{1}{M_{V}^{2}}\frac{2M_{B}(M_{B}+M_{V})}{M_{B}^{2}-M_{V}^{2}-q^{\prime
2}}\frac{M_{V}f_{V}}{M_{V}^{2}-q^{2}-i\Gamma_{V}(q^{2})M_{V}}A_{1}^{B\to
V}(q^{\prime 2})\bigg{)},$ $\displaystyle F_{2A}(q^{\prime 2},q^{2})$
$\displaystyle=$
$\displaystyle-q^{2}M_{B}\sum\limits_{V=\rho^{0},\omega}\frac{1}{M_{V}^{2}}\frac{2M_{V}f_{V}}{M_{V}^{2}-q^{2}-i\Gamma_{V}(q^{2})M_{V}}\left[\frac{M_{B}+M_{V}}{M_{B}^{2}-M_{V}^{2}-q^{\prime
2}}A_{1}^{B\to V}(q^{\prime 2})-\frac{A_{2}^{B\to V}(q^{\prime
2})}{(M_{B}+M_{V})}\right]$ (2.7)
$\displaystyle+Q_{B}f_{B}\left(\frac{2M_{B}}{M_{B}^{2}-q^{\prime
2}}-\frac{2M_{B}}{M_{B}^{2}-q^{\prime 2}-q^{2}}\right),$ $\displaystyle
F_{V}(q^{\prime 2},q^{2})$ $\displaystyle=$ $\displaystyle F_{V}(q^{\prime
2})-q^{2}M_{B}\sum\limits_{V=\rho^{0},\omega}\bigg{(}\frac{1}{M_{V}^{2}}\frac{M_{V}f_{V}}{M_{V}^{2}-q^{2}-i\Gamma_{V}(q^{2})M_{V}}\frac{2V^{B\to
V}(q^{\prime 2})}{M_{B}+M_{V}}\bigg{)}.$ (2.8)

$\bullet$ The form factors $F_{A}(q^{\prime 2})$ and $F_{V}(q^{\prime 2})$
describe the $B\to\gamma l^{\prime}\nu^{\prime}$ transition; they emerge as
subtraction terms at $q^{2}=0$ in the $q^{2}$-disperison representations for
the form factors $F_{1A,V}(q^{\prime 2},q^{2})$. The form factors
$F_{A}(q^{\prime 2})$ and $F_{V}(q^{\prime 2})$ are equal to each other at the
leading order of the double $1/E_{\gamma}$
($2M_{B}E_{\gamma}=M_{B}^{2}-q^{\prime 2}$) and $1/M_{B}$ expansions in QCD
korchemsky  but differ in the subleading orders mn2004 ; beneke1 ; beneke2018
:

$\displaystyle F_{A}(q^{\prime 2})$ $\displaystyle=$
$\displaystyle-\frac{Q_{u}f_{B}M_{B}}{2E_{\gamma}\lambda_{B}}+\frac{Q_{b}f_{B}M_{B}}{2E_{\gamma}m_{b}}+O(Q_{u}f_{B}M_{B}/E_{\gamma}^{2}),$
(2.9) $\displaystyle F_{V}(q^{\prime 2})$ $\displaystyle=$
$\displaystyle-\frac{Q_{u}f_{B}M_{B}}{2E_{\gamma}\lambda_{B}}-\frac{Q_{b}f_{B}M_{B}}{2E_{\gamma}m_{b}}+O(Q_{u}f_{B}M_{B}/E_{\gamma}^{2})$
(2.10)

The magnitude of the form factors $F_{A}(q^{\prime 2})$ and $F_{V}(q^{\prime
2})$ is determined to a large extent by the parameter $\lambda_{B}$, the
inverse moment of the $B$-meson light-cone distribution amplitude $\phi_{B}$
korchemsky . Taking into account a large uncertainty in the present knowledge
of the parameter $\lambda_{B}$ beneke1 ; kou ; BraunIvanovKorchemsky2004 ;
kmn2018 ; zwicky2021 ; mn2004 , we use the monopole forms (2.9) and (2.10) in
the full kinematically allowed region of $q^{\prime 2}$ and allow the
variation of $\lambda_{B}$ in the range $\lambda_{B}(1\,{\rm GeV})=(0.5\pm
0.15)$ GeV.

$\bullet$ The contributions of the light vector mesons $V=\rho^{0},\omega$ to
the form factors $F_{1A,2A,V}(q^{\prime 2},q^{2})$ are unambiguous (cf.
beneke2 ) and are expressed via the form factors $A_{1}^{B\to V}(q^{\prime
2})$, $A_{2}^{B\to V}(q^{\prime 2})$, and $V^{B\to V}(q^{\prime 2})$
describing the weak decay $B\to V$. In spite of many efforts to calculate
these form factors in a broad kinematical decay region $0<q^{\prime
2}<M_{B}^{2}$, our knowledge of these quantities is not very accurate, see
e.g. ms2000 ; ballzwicky2005 ; ivanov2 ; gubernari2019 . For our calculations
we use the results from ms2000  and assign to them a 10% uncertainty. The
uncertainties in these form factors, along with the uncertainty in the
parameter $\lambda_{B}$, is the second main source of the uncertainty in the
theoretical predictions for $B\to l^{+}l^{-}l^{\prime}\nu^{\prime}$ decays.

The results presented in the next Section are obtained for our form factor
model described in full detail in Section 5 of mi2022  and for the parameter
$\lambda_{B}=0.65$.

Fig. 2: Two diagrams describing the $B$-decay into
$\mu^{+}\mu^{-}\mu^{+}\bar{\nu}_{\mu}$.

##  3 The decay $B^{+}\to\mu^{+}\mu^{-}\mu^{+}\bar{\nu}_{\mu}$

The case of two identical leptons is technically more involved than the case
of all different leptons, considered in mi2022 . The reason is that an
additional contribution from the interchange of the two $\mu^{+}$ leptons
arises.

The first diagram in Fig. 2, $M_{a}(k_{1},k_{2},k_{3},k_{4})$, is the same as
for the $B$-decay into non-identical leptons (e.g.,
$B^{+}\to\mu^{+}\mu^{-}e^{+}\nu_{e}$). The second diagram is obtained from the
first one by permutation of two final identical leptons:
$M_{b}(k_{1},k_{2},k_{3},k_{4})=M_{a}(k_{3},k_{2},k_{1},k_{4})$. The total
amplitude for the case of two identical leptons in the final state reads

$\displaystyle M_{\rm
tot}=\frac{1}{\sqrt{2}}\,(M_{a}-M_{b})\quad\text{and}\quad|M_{\rm
tot}|^{2}=\frac{1}{2}\left(|M_{a}|^{2}+|M_{b}|^{2}-2\,{\rm
Re}(M_{a}M_{b}^{\ast})\right).$ (3.11)

The factor $1/\sqrt{2}$ in the amplitude corresponds to the factor $1/2$ in
the phase space for the case of two identical leptons. Respectively, we use
the expression for the phase space without the factor $1/2$ correspondng to
the identical particles in the final state.

###  3.1 The differential distribution over the momentum of the
$\mu^{+}\mu^{-}$ pair

In a theoretical consideration, one can calculate the branching fraction and
one-dimensional differential distribution for two kinematical variables
$q_{12}^{2}=(k_{1}+k_{2})^{2}$ (momentum of one of the $\mu^{+}\mu^{-}$ pairs)
and $q_{34}^{2}=(k_{3}+k_{4})^{2}$ (momentum of the $\mu^{+}\nu$ pair). For
diagram $M_{a}$, $q_{12}=q$ and $q_{34}=q^{\prime}$, so that the angular
variables do not enter the form factors; the angular integrals may be taken
analytically. For diagram $M_{b}$, the photon momentum $q$ and the weak-vertex
momentum $q^{\prime}$ do not coincide with $q_{12}$ and $q_{34}$, so that the
angular variables appear in the arguments of the form factors; all angular
integrals should be taken numerically.

|  |
---|---|---
(a) | (b) | (c)
Fig. 3:  The differential distributions in units $10^{-9}$: (a) $d{\cal
B}_{aa}(q_{12}^{2})$ at $0<q_{12}^{2}({\rm GeV}^{2})<2$; (b) $d{\cal
B}_{ab}(q_{12}^{2})$ vs $d{\cal B}_{bb}(q_{12}^{2})$ at $0<q_{12}^{2}({\rm
GeV}^{2})<2$; (c) $d{\cal B}_{bb}(q_{12}^{2})$ in the full range
$4m_{\mu}^{2}<q_{12}^{2}<(M_{B}-m_{\mu})^{2}$.

Obviously, the contribution to branching fraction coming from $|M_{a}|^{2}$
and $|M_{b}|^{2}$ are identically the same due to the symmetry
$k_{1}\leftrightarrow k_{3}$ of the phase-space measure. But verifying this
property is a non-trivial check for numerical evaluation of the five-
dimensional integrals. Fig. 3 shows the $q_{12}^{2}$-differential
distributions. The differential distributions over the variable $q_{23}^{2}$
($q_{23}$ the momentum of another $\mu^{+}\mu^{-}$ pair that may be isolated
in the amplitude) is the same because of the symmetry of the amplitude: the
replacement $k_{1}\to k_{3}$ leads to the replacement $M_{a}\to M_{b}$ and
vice versa.

###  3.2 The differential distribution over the momentum of the
$\mu^{+}\nu_{\mu}$ pair

In a theoretical consideration, we can also calculate the differential
distribution over $q_{34}^{2}=(k_{3}+k_{4})^{2}$ (momentum of the $\mu^{+}\nu$
pair). These distributions are shown in Fig. 4. Obviously, the mixed term may
be neglected in the full range of $q_{34}^{2}$.

|  |
---|---|---
(a) | (b) | (c)
Fig. 4:  The differential distributions in the full range
$m^{2}_{\mu}<q_{34}^{2}<(M_{B}-2m_{\mu})^{2}$ (in units $10^{-9}$): (a)
$d{\cal B}_{aa}(q_{34}^{2})$; (b) $d{\cal B}_{ab}(q_{34}^{2})$; (c) $d{\cal
B}_{bb}(q_{34}^{2})$

###  3.3 The differential distribution over the momentum of the
$\mu^{+}\mu^{+}$ pair

An interesting observable that can be readily measured experimentally is the
differential distribution over the momentum of the $\mu^{+}\mu^{+}$ pair.
Unlike the $\mu^{+}\mu^{-}$ distribitions, one has only one pair of same-
charge leptons in each event. The process is described by the same two
diagrams in Fig. 2 but one has to calculate the distribution over the variable
$q_{13}^{2}=(k_{1}+k_{3})^{2}$. The contributions $d{\cal B}_{aa}(q_{13}^{2})$
and $d{\cal B}_{bb}(q_{13}^{2})$ are equal to each other and coincide with the
distribution $d{\cal B}_{bb}(q_{12}^{2})$ discussed above. Obviously, the
mixed $d{\cal B}_{ab}(q_{13}^{2})$ term can be safely neglected similar to the
case of the distribution in the $l^{+}l^{-}$ momentum $q_{12}^{2}$ considered
above: (i) the integral $\int d{\cal B}_{bb}(q_{13}^{2})dq_{13}^{2}$ comprises
only 1% of $\int d{\cal B}_{aa}(q_{13}^{2})dq_{13}^{2}=\int d{\cal
B}_{bb}(q_{13}^{2})dq_{13}^{2}$. (ii) the distribution $d{\cal
B}_{ab}(q_{13}^{2})$ contains no resonances and is therefore smeared over the
full kinematical $q_{13}^{2}$ range as a small addition to $d{\cal
B}_{aa}(q_{13}^{2})=d{\cal B}_{bb}(q_{13}^{2})$ at the level of 1%. Figure 5
shows our predictions for $d{\cal B}(q_{13}^{2})$.

|
---|---
(a) |  (b)
Fig. 5:  The differential distribution (in units $10^{-9}$) $d{\cal
B}(q_{13}^{2})$ over the momentum of the same-charge lepton pair
$\mu^{+}\mu^{+}$ ($q_{13}=k_{1}+k_{3}$): (a) the full range
$4m_{\mu}^{2}<q_{13}^{2}<(M_{B}-m_{\mu})^{2}$; (b) the range $0<q_{13}^{2}<2$
GeV2.

###  3.4 The branching ratio of the
$B^{+}\to\mu^{+}\mu^{-}\mu^{+}\bar{\nu}_{\mu}$ decay

Table 1 presents our numerical results for the total branching ratio of the
$B^{+}\to\mu^{+}\mu^{-}\mu^{+}\bar{\nu}_{\mu}$ decay and the separate
contributions coming from $|M_{a}|^{2}$, $|M_{b}|^{2}$ and the interference
term $2\times{\rm Re}(M_{a}M_{b}^{\ast})$. We use short-hand notations

$\displaystyle{\cal B}_{aa}$ $\displaystyle=$
$\displaystyle\frac{\tau_{B}}{2M_{B}}\int\\!\\!d\Phi\,|M_{a}|^{2},\quad{\cal
B}_{bb}=\frac{\tau_{B}}{2M_{B}}\int\\!\\!d\Phi\,|M_{b}|^{2},\quad{\cal
B}_{ab}=\frac{\tau_{B}}{2M_{B}}\int\\!\\!d\Phi\,2{\rm Re}(M_{a}M_{b}^{\ast}),$
(3.13) $\displaystyle\hskip 113.81102pt{\cal B}_{\rm
tot}=\frac{1}{2}\Big{(}{\cal B}_{aa}+{\cal B}_{bb}-{\cal B}_{ab}\Big{)}$

where the phase-space measure is given by Eqs. (A.27) and (A.28). One can see
that the contribution of the interference term $2\times{\rm
Re}(M_{a}M_{b}^{\ast})$ is negative and two orders of magnitude less than the
contribution of $|M_{a}|^{2}$ and $|M_{b}|^{2}$. Thus, the interference term
may be neglected. This is a very good news as the calculation of the
interference term is the most time-consuming part of the full calculation.

Mode | $\frac{1}{2}({\cal B}_{aa}+{\cal B}_{bb})$ | $\frac{1}{2}{\cal B}_{ab}$ | ${\cal B}_{\rm tot}$ | ${\cal B}^{\rm exp.cut}_{\rm tot}$
---|---|---|---|---
$\mu^{+}\mu^{-}\mu^{+}\bar{\nu}_{\mu}$ | $2.80\times 10^{-8}$ | $-2.26\times 10^{-10}$ | $2.82\times 10^{-8}$ | $2.73\times 10^{-8}$
Table 1:  Branching ratio of the
$B^{+}\to\mu^{+}\mu^{-}\mu^{+}\bar{\nu}_{\mu}$ decay. Separate contributions
coming from $|M_{a}|^{2}$, $|M_{b}|^{2}$ and the interference term
$2\times{\rm Re}(M_{a}M_{b}^{\ast})$ are also given. The ${\cal B}^{\rm
exp.cut}_{\rm tot}$ is the result obtained by applying the LHCb event
selection criterion (3.14). The results correspond to $\lambda_{B}=0.65$.

We also provide the ${\cal B}^{\rm exp.cut}_{\rm tot}$ which is calculated
making use of the LHCb exp4  event selection criterion: In each event, one can
form two $\mu^{+}\mu^{-}$ pairs; the events are selected on the basis of the
criterion that the lowest of the two $\mu^{+}\mu^{-}$ mass combinations should
be less than 0.98 GeV. In our calculation this corresponds to restricting the
phase-space integration by the condition

$\displaystyle{\rm min}\\{(k_{1}+k_{2})^{2},(k_{3}+k_{2})^{2}\\}\leq
0.96\,{\rm GeV}^{2}.$ (3.14)

##  4 Discussion and Conclusions

Making use of the model for the form factors of mi2022 , we performed a
detailed analysis of the exchange diagrams and the interference effects that
appear in the case of the identical leptons in the final state. We calculate
the differential distributions in various variables: namely, in $q_{12}^{2}$,
the square of the invariant mass of one of the $\mu^{+}\mu^{-}$-pairs (Fig.
3), in $q_{34}^{2}$, the square of the invariant mass of one of the
$\mu^{+}\nu_{\mu}$-pairs (Fig. 5), and in $q_{13}^{2}$, the square of the
invariant mass of one of the $\mu^{+}\mu^{+}$ pair (Fig. 4). The latter
differential distirbution may be readily measured experimentally.

Our findings may be summarized as follows:

$\bullet$ For the differential distribution in $q_{12}^{2}$, $d{\cal B}_{aa}$
has a sharp resonance structure in the region of $\rho$ and $\omega$
resonances. The distribution of $d{\cal B}_{bb}$ spreads over the full range
of $q_{12}^{2}$ and exhibits no resonance structure. Nevertheless, the
integrated differential rates ${\cal B}_{aa}$ and ${\cal B}_{bb}$ are equal to
each other. The interference term $d{\cal B}_{ab}$ contributes at less than 1%
level and may be safely neglected. Noteworthy, the distribution $d{\cal
B}(q_{12}^{2})$ is fully determined by the resonances in all regions of
$q_{12}^{2}$: In the region $0<q_{12}^{2}<1$ GeV2 via $d{\cal B}_{aa}$, and in
the region of $1\,{\rm GeV}^{2}<q_{12}^{2}$ via $d{\cal B}_{bb}$.
Consequently, the perturbative tail of the form factors at $q_{12}^{2}>1$-$2$
GeV2 does not show up in the differential distributions for the identical
leptons in the final state at all. This makes an essential difference with the
case of non-identical leptons, where the region $q^{2}_{12}\gg$ 1 GeV2 is
determined by the pQCD behaviour of the form factors.

$\bullet$ The differential distribution over the momentum of the
$\mu^{+}\nu_{\mu}$ pair, $q_{34}^{2}$, has an interesting shape, different for
$d{\cal B}_{aa}(q_{34}^{2})$ and $d{\cal B}_{bb}(q_{34}^{2})$, and a
numerically negligible interference term $d{\cal B}_{ab}(q_{34}^{2})$. This
differential distrubution is rather interesting theoretically but is unlikely
to be measurable experimentally.

$\bullet$ The differential distribution in $q_{13}^{2}$, the square of the
invariant mass of the $\mu^{+}\mu^{+}$ pair, has a relatively flat non-
resonant structure in the full range of $q_{13}^{2}$. The contribution of the
$M_{a}$ and $M_{b}$ diagrams are equal to each other, $d{\cal
B}_{aa}(q_{13}^{2})=d{\cal B}_{bb}(q_{13}^{2})$. The interference term $d{\cal
B}_{ab}(q_{13}^{2})$ is smeared over the full $q_{13}^{2}$-region as a minor
positive addition at the level of less than 1% and may be safely neglected.

$\bullet$ The good news is that the interference term between the direct
diagram $M_{a}$ and the exchanged diagram $M_{b}$ provides a positive
contribution at the level less than 1% to the differential distribution in all
regions of the kinematical variables and thus can be safely neglected. This
greatly simplifies the calculation procedure as the interference $AB$ term
represents the most time-consuming part of the calculations.

$\bullet$ For ${\cal B}(B\to\mu^{+}\mu^{-}\mu^{+}\nu_{\mu})$, taking into
account all uncertianties, we confirm our result of mi2022 :

$\displaystyle{\rm
Br}(B^{+}\to\mu^{+}\mu^{-}\mu^{+}\bar{\nu}_{\mu})=(3.02\;{{}^{+0.45}_{-0.25}}|_{\lambda_{b}}\pm
0.62|_{\rm\,weak\,ffs})\,10^{-8}.$ (4.15)

Applying the kinematical selection rule for the $\mu^{+}\mu^{-}$ pairs (3.14)
as done by the LHCb collaboration exp4 , leads to a small reduction at the
level of 3% of our theoretical result (4.15).

In summary, we reinforce our previous finding that our theoretical estimate is
only marginally compatible with the upper limits obtained by the LHCb
Collaboration exp4  ${\rm
Br}(B^{+}\to\mu^{+}\mu^{-}\mu^{+}\bar{\nu}_{\mu})\leq 1.6\cdot\,10^{-8}$. see
a small reduction

###### Acknowledgements.

We are grateful to M. Beneke, G. Gagliardi, L. Gladilin, S. Simula, and R.
Zwicky for valuable and interesting discussions.

##  Appendix A Kinematics of the $B$-decay with four leptons in the final
state

We consider the reaction

$\displaystyle
B^{+}(p)\to\ell^{\prime\,+}(k_{3})+\bar{\nu}_{\ell^{\prime}}(k_{4})+\ell^{+}(k_{1})+\ell^{-}(k_{2})\,.$
(A.16)

The two planes of the final particles are shown in Fig. 6.

Fig. 6:  Definition of angles $\theta^{\ast}$, $\theta$ and $\chi$ in the
decay of
$B^{+}\to\ell^{\prime\,+}+\bar{\nu}_{\ell^{\prime}}+\ell^{+}+\ell^{-}$.

The decay amplitude is described by 5 kinematical variables: $\bullet$
$q_{34}^{2}\equiv(k_{3}+k_{4})^{2}$ is the
$\ell^{\prime\,+}\nu_{\ell^{\prime}}$ invariant mass;
$\bullet$ $q_{12}^{2}\equiv(k_{1}+k_{2})^{2}$ is the dilepton invariant mass;
$\bullet$ $\theta^{\ast}$ is the angle of the $\ell^{\prime\,+}$ in the
$\ell^{\prime\,+}\bar{\nu}_{\ell^{\prime}}$ c.m.s w.r.t the
$\ell^{\prime\,+}\bar{\nu}_{\ell^{\prime}}$ flight direction;
$\bullet$ $\theta$ is the angle of the $\ell^{+}$ in the dilepton c.m.s w.r.t
the $\ell^{+}\ell^{-}$ flight direction;
$\bullet$ $\chi$ is the azimuthal angle between
$\ell^{\prime\,+}\nu_{\ell^{\prime}}$ and dilepton planes.

All particles are on their mass shell:

$\displaystyle p^{2}=M_{B}^{2},\quad k_{3}^{2}=m_{3}^{2}\equiv
m_{\ell^{\prime}}^{2},\quad k_{4}^{2}=m_{4}^{2}\equiv
m_{\nu_{\ell^{\prime}}}^{2}=0,\quad k_{1}^{2}=k_{2}^{2}\equiv m_{\ell}^{2}.$
(A.17)

We also introduce the mass notations for the momenta squared:
$m_{12}\equiv\sqrt{q_{12}^{2}}$ and $m_{34}\equiv\sqrt{q_{34}^{2}}$.

The boosted 4-momenta from the $\ell^{+}\ell^{-}$ c.m.s. to the $B$-meson rest
frame are written as

$\displaystyle q_{12}^{\mu}$ $\displaystyle=$ $\displaystyle(E_{12},0,0,|{\bf
k}|),$ (A.18) $\displaystyle k^{\mu}_{1}$ $\displaystyle=$
$\displaystyle\tfrac{1}{2}(E_{12}+v|{\bf
k}|\cos\theta,+vm_{12}\sin\theta\cos\chi,+vm_{12}\sin\theta\sin\chi,|{\bf
k}|+vE_{12}\cos\theta),$ (A.19) $\displaystyle k^{\mu}_{2}$ $\displaystyle=$
$\displaystyle\tfrac{1}{2}(E_{12}-v|{\bf
k}|\cos\theta,-vm_{12}\sin\theta\cos\chi,-vm_{12}\sin\theta\sin\chi,|{\bf
k}|-vE_{12}\cos\theta).$ (A.20)

Here
$v=\lambda(q_{12}^{2},m_{l}^{2},m_{l}^{2})/q_{12}^{4}=\sqrt{1-4m_{l}^{2}/q_{12}^{2}}$
[$\lambda(a,b,c)\equiv(a-b-c)^{2}-4bc$] and

$\displaystyle|{\bf
k}|=\frac{\lambda^{1/2}(M_{B}^{2},q_{34}^{2},q_{12}^{2})}{2M_{B}},\quad
E_{12}=\frac{M_{B}^{2}-q_{23}^{2}+q_{12}^{2}}{2M_{B}},\quad
E_{23}=\frac{M_{B}^{2}+q_{23}^{2}-q_{12}^{2}}{2M_{B}},\quad
E_{12}+E_{23}=M_{B}.$ (A.21)

The boosted momenta from the $\ell^{\prime}\nu_{\ell^{\prime}}$ c.m.s. to the
$B$-meson rest frame read

$\displaystyle q_{34}^{\mu}$ $\displaystyle=$ $\displaystyle(E_{34},0,0,-|{\bf
k}|),$ (A.22) $\displaystyle k^{\mu}_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{E_{34}}(E_{34}E_{3}+|{\bf k}||{\bf
k_{3}}|\cos\theta^{\ast},+E_{34}|{\bf k_{3}}|\sin\theta^{\ast},0,-E_{3}|{\bf
k}|-E_{34}|{\bf k_{3}}|\cos\theta^{\ast}),$ (A.23) $\displaystyle k^{\mu}_{4}$
$\displaystyle=$ $\displaystyle\frac{1}{E_{34}}(E_{34}E_{4}-|{\bf k}||{\bf
k_{3}}|\cos\theta^{\ast},-E_{34}|{\bf k_{3}}|\sin\theta^{\ast},0,-E_{4}|{\bf
k}|+E_{34}|{\bf k_{3}}|\cos\theta^{\ast}\,).$ (A.24)

where

$\displaystyle|{\bf
k_{3}}|=\frac{\lambda^{1/2}(q_{34}^{2},m_{3}^{2},m_{4}^{2})}{2m_{34}},\quad
E_{3}=\frac{q_{34}^{2}+m_{3}^{2}-m_{4}^{2}}{2m_{34}},\quad
E_{4}=\frac{q_{34}^{2}-m_{3}^{2}+m_{4}^{2}}{2m_{34}},\quad
E_{3}+E_{4}=m_{34}.$ (A.25)

The differential decay rate is given by

$\displaystyle
d\Gamma(B\to\ell^{\prime}\nu_{\ell^{\prime}}\ell^{+}\ell^{-})=\frac{1}{2m_{1}}|M(k_{1},\ldots,k_{4})|^{2}\,d\Phi,$
(A.26) $\displaystyle
d\Phi=\frac{1}{(2\pi)^{8}}\delta^{(4)}(p_{1}-k_{3}-k_{4}-k_{1}-k_{2})\,\frac{d^{3}\vec{k}_{3}}{2k^{0}_{3}}\,\frac{d^{3}\vec{k}_{4}}{2k^{0}_{4}}\,\frac{d^{3}\vec{k}_{1}}{2k^{0}_{1}}\,\frac{d^{3}\vec{k}_{2}}{2k^{0}_{2}},\qquad$
(A.27)

where $k^{0}_{i}=\sqrt{m_{\ell}^{2}+\vec{k}_{i}^{\,2}}$ for $(i=1,2)$ and
$k^{0}_{i}=\sqrt{m_{i}^{2}+\vec{k}_{i}^{\,2}}$ for $(i=3,4)$.

The integration over the phase space may be reduced to the integration over
the two kinematical variables $k^{2}$ and $q^{2}$ and three angles
$\theta^{\ast}$, $\theta$ and $\chi$. Then, the differential phase volume in
Eq. (A.27) is given by

$\displaystyle d\Phi=\frac{v}{(4\pi)^{6}}\frac{|{\bf k}|}{M_{B}}\frac{|{\bf
k_{3}}|}{E_{34}}\,dq_{12}2\,dq_{34}^{2}\,d\cos\theta^{\ast}\,d\cos\theta\,d\chi\,\qquad
0\leq\theta^{\ast},\theta\leq\pi,\qquad 0\leq\chi\leq 2\pi.$ (A.28)

The kinematical constrains on the variables $q_{12}^{2}$ and $q_{34}^{2}$ come
from the positivity of the $\lambda$-functions:
$\lambda(q_{34}^{2},m_{\ell^{\prime}}^{2},0)$,
$\lambda(q_{12}^{2},m_{\ell}^{2},m_{\ell}^{2})$ and
$\lambda(M_{B}^{2},q_{12}^{2},q_{34}^{2})$ and read as

$\displaystyle 4m^{2}_{\ell}\leq q_{12}^{2},\quad m^{2}_{\ell^{\prime}}\leq
q_{34}^{2},\quad\sqrt{q_{12}^{2}}+\sqrt{q_{34}^{2}}\leq M_{B}^{2}.$ (A.29)

For calculating single differential distribution in $q_{12}^{2}$ or
$q_{34}^{2}$, we have the following integration limits

$\displaystyle dq_{34}^{2}dq_{12}^{2}:$ $\displaystyle
m_{\ell^{\prime}}^{2}\leq q_{34}^{2}\leq(M_{B}-2m_{\ell})^{2},\quad
4m_{\ell}^{2}\leq q_{12}^{2}\leq(M_{B}-\sqrt{q_{34}^{2}})^{2},$ (A.30)
$\displaystyle dq_{12}^{2}dq_{34}^{2}:$ $\displaystyle 4m_{\ell}^{2}\leq
q_{12}^{2}\leq(M_{B}-m_{\ell^{\prime}})^{2},\quad m_{\ell^{\prime}}^{2}\leq
q_{34}^{2}\leq(M_{B}-\sqrt{q_{12}^{2}})^{2},$ (A.31)

## References

  * (1) M. A. Ivanov and D. Melikhov, Theoretical analysis of the leptonic decays $B\to\ell\ell\ell^{\prime}\bar{\nu}_{\ell^{\prime}}$, Phys. Rev. D 105, 014028 (2022).
  * (2) LHCb Collaboration (R. Aaij et al.), Search for the rare decay $B_{s}^{0}\to\mu^{+}\mu^{-}\mu^{+}\mu^{-}$, Phys. Rev. Lett. 110, 211801 (2013).
  * (3) ATLAS Collaboration (M. Aaboud et al.), Study of the rare decays of $B^{0}_{s}$ and $B_{0}$ into muon pairs from data collected during the LHC Run 1 with the ATLAS detector, Eur. Phys. J. C76, 513 (2016).
  * (4) LHCb Collaboration (R. Aaij et al.), Search for decays of neutral beauty mesons into four muons, JHEP 1703, 001 (2017).
  * (5) LHCb Collaboration (R. Aaij et al.), Search for the rare decay $B^{+}\to\mu^{+}\mu^{-}\mu^{+}\nu_{\mu}$, Eur. Phys. J. C79, 675 (2019).
  * (6) Y. Dincer and L. Sehgal, Electroweak effects in the double Dalitz decay $B(s)\to l^{+}l^{-}l^{\prime+}l^{\prime-}$, Phys. Lett. B556, 169 (2003).
  * (7) A. V. Danilina and N. V. Nikitin, Four-Leptonic Decays of Charged and Neutral $B$ Mesons within the Standard Model, Phys. Atom. Nucl. 81, 347 (2018), Yad. Fiz. 81, 331 (2018); A. Danilina N. Nikitin, and K. Toms, Decays of charged $B$-mesons into three charged leptons and a neutrino, Phys. Rev. D101, 096007 (2020).
  * (8) A. Bharucha, B. Kindra and N. Mahajan, Probing the structure of the $B$ meson with $B\to lll^{\prime}\nu^{\prime}$, ArXiv:2102.03193.
  * (9) M. Beneke, P. Böer, P. Rigatos, and K. K. Vos, QCD factorization of the four-lepton decay $B\to lll\nu$, Eur. Phys. J. C81, 638 (2021).
  * (10) C. Wang, Yu-Ming Wang, Y.-B. Wei, QCD factorization for the four-body leptonic B-meson decays, JHEP 02, 141 (2022).
  * (11) M. A. Ivanov, D. Melikhov, and S. Simula, Form factors for $B\to j_{1}j_{2}$ decays into two currents in QCD, Phys. Rev. D101, 094022 (2020).
  * (12) M. Beyer, D. Melikhov, N. Nikitin, and B. Stech, Weak annihilation in the rare radiative $B\to\rho\gamma$ decay, Phys. Rev. D64, 094006 (2001).
  * (13) D. Melikhov, Dispersion approach to quark binding effects in weak decays of heavy mesons, Eur. Phys. Journal direct 4, 2 (2002) [hep-ph/0110087].
  * (14) F. Kruger and D. Melikhov, Gauge invariance and form-factors for the decay $B\to\gamma l^{+}l^{-}$, Phys. Rev. D67, 034002 (2003).
  * (15) A. Kozachuk, D. Melikhov, and N. Nikitin, Annihilation type rare radiative $B_{(s)}\to V\gamma$ decays, Phys. Rev. D93, 014015 (2016).
  * (16) T. M. Aliev, A. Ozpineci, and M. Savci, $B_{q}\to l^{+}l^{-}\gamma$ decays in light cone QCD, Phys. Rev. D55, 7059 (1997).
  * (17) G. Korchemsky, D. Pirjol, and T.-M. Yan, Radiative leptonic decays of $B$ mesons in QCD, Phys. Rev. D61, 114510 (2000).
  * (18) P. Ball and E. Kou, $B\to\gamma e\nu$ transitions from QCD sum rules on the light cone, JHEP 0304, 029 (2003).
  * (19) D. Melikhov and N. Nikitin, Rare radiative leptonic decays $B_{(d,s)}\to\gamma l^{+}l^{-}$, Phys. Rev. D70, 114028 (2004).
  * (20) M. Beneke and J. Rohrwild, B meson distribution amplitude from $B\to\gamma l\nu$, Eur. Phys. J. C71, 1818 (2011).
  * (21) A. Kozachuk, D. Melikhov, and N. Nikitin, Rare FCNC radiative leptonic $B_{s,d}\to\gamma l^{+}l^{-}$ decays in the Standard Model, Phys. Rev. D97, 053007 (2018).
  * (22) M. Beneke, V. M. Braun, Y. Ji, and Y.-B. Wei, Radiative leptonic decay $B\to\gamma\ell\nu_{\ell}$ with subleading power corrections, JHEP 1807, 154 (2018).
  * (23) S. Dubnicka, A. Z. Dubnickova, M. A. Ivanov, A. Liptaj, P. Santorelli, and C. T. Tran, Study of $B_{s}\to l^{+}l^{-}\gamma$ decays in covariant quark model, Phys. Rev. D99, 014042 (2019).
  * (24) J. Albrecht, E. Stamou, R. Ziegler, and R. Zwicky, Probing flavoured Axions in the Tail of $B_{q}\to\mu^{+}\mu^{-}$, arXiv:1911.05018.
  * (25) M. Beneke, C. Bobeth, and Y.-M. Wang, $B_{d,s}\to\gamma l^{+}l^{-}$ decay with an energetic photon, JHEP 2012, 148 (2020).
  * (26) T. Janowski, B. Pullin, and R. Zwicky, Charged and neutral $\bar{B}_{u,d,s}\to\gamma$ form factors from light cone sum rules at NLO, arXiv:2106.13616.
  * (27) A. Desiderio et al, First lattice calculation of radiative leptonic decay rates of pseudoscalar mesons, Phys. Rev. D103, 014502 (2021).
  * (28) D. Melikhov, O. Nachtmann, V. Nikonov, and T. Paulus, Masses and couplings of vector mesons from the pion electromagnetic, weak, and $\pi\gamma$ transition form-factors, Eur. Phys. J. C34, 345 (2004).
  * (29) V. M. Braun, D. Yu. Ivanov, G. P. Korchemsky, The B meson distribution amplitude in QCD, Phys. Rev. D69, 034014 (2004).
  * (30) D. Melikhov and B. Stech, Weak form-factors for heavy meson decays: an update, Phys. Rev. D62, 014006 (2000).
  * (31) P. Ball and R. Zwicky, $B_{d,s}\to\rho,\omega,K^{*},\phi$ decay form factors from light-cone sum rules reexamined, Phys. Rev. D71, 014029 (2005).
  * (32) M. A. Ivanov, J. G. Körner, S. G. Kovalenko, P. Santorelli, and G. G. Saidullaeva, Form factors for semileptonic, nonleptonic and rare $B(B_{s})$ meson decays, Phys. Rev. D85, 034004 (2012).
  * (33) N. Gubernari, A. Kokulu and D. van Dyk, $B\to P$ and $B\to V$ form factors from $B$-meson light-cone sum rules beyond leading twist, JHEP 1901, 150 (2019).