Method of Studying $\mathchar 28931\relax^{0}_{b}$ decays with one missing
particle

Sheldon Stone and Liming Zhang

Physics Department Syracuse University, Syracuse, NY, USA 13244-1130

A new technique is discussed that can be applied to $\mathchar
28931\relax^{0}_{b}$ baryon decays where decays with one missing particle can
be discerned from background and their branching fractions determined, along
with other properties of the decays. Applications include measurements of the
CKM elements $|V_{ub}|$ and $|V_{cb}|$, and detection of any exotic objects
coupling to $b\rightarrow s$ decays, such as the inflaton. Potential use of
$\overline{B}^{0**}\rightarrow\pi^{+}B^{-}$ and
$\overline{B}_{s}^{0**}\rightarrow K^{+}B^{-}$ to investigate $B^{-}$ decays
is also commented upon.

Submitted to Advances in High Energy Physics

##  1 Introduction

Detection of $b$-flavored hadron decays with one missing neutral particle,
such as a neutrino, is important for many measurements and searches. These
include semileptonic decays, such as $\kern
1.79993pt\overline{\kern-1.79993ptB}{}\rightarrow D\mu^{-}\overline{\nu}$,
$\kern
1.79993pt\overline{\kern-1.79993ptB}{}\rightarrow\pi\mu^{-}\overline{\nu}$,
and any exotic long lived particles that could be produced in decays such as
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}\rightarrow X\chi$, where the $X$
is any combination of detected particles and the $\chi$ escapes the
detector.111In this paper mention of a particular decay mode implies the use
of the charge-conjugated mode as well. These measurements are possible at an
$e^{+}e^{-}$ collider operating at the $\Upsilon(4S)$. Since
$\Upsilon(4S)\rightarrow B\overline{B}$, fully reconstructing either the $B$
or the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ determines the negative
of the initial four-momentum of the other. With this information it is
possible to measure final states where one particle is not detected, such as a
neutrino. To implement this procedure, taking the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ to be fully reconstructed, the missing
mass-squared, $m_{x}^{2}$, is calculated including the information on the
initial $B$ four-momentum and measurements of the found $X$ particles as

$m_{x}^{2}=(E_{B}-E_{X})^{2}-(\overrightarrow{p}\\!\\!_{B}-\overrightarrow{p}\\!\\!_{X})^{2},$
(1)

where $E$ and $\overrightarrow{p}$ indicate energy and three-momentum,
respectively. Peaks in $m_{x}^{2}$ would be indicative of single missing
particles in the $B$ decay.

A related example is charm semileptonic decays with a missing neutrino.
Determinations of branching fractions and form-factors have been carried out
in fixed target experiments, exploiting the measured direction of the charmed
hadron and assuming that the missing particle has zero mass, which leads to a
two-fold ambiguity in the neutrino momentum calculation [1]. If the charm
decay particle is a $D^{0}$, extra constraints can be imposed on its decay
requiring it to be produced from a $D^{*+}$ in the decay
$D^{*+}\rightarrow\pi^{+}D^{0}$. This leads to more constraints than unknowns,
and is quite useful for rejecting backgrounds [2, *Agostino:2004na].

Interesting decays of the $\mathchar 28931\relax^{0}_{b}$ baryon also exist,
but investigations are not feasible in the $\Upsilon(4S)$ energy region.
Potential studies include determination of the CKM matrix element $|V_{cb}|$,
possible using $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar
28931\relax_{c}^{+}\ell^{-}\overline{\nu}$ decays and $|V_{ub}|$ using the
$\mathchar 28931\relax^{0}_{b}\rightarrow p\ell^{-}\overline{\nu}$ mode.

Neutral particles that have not yet been seen could be searched for, even if
they are stable or have long enough lifetimes that they would have only a very
small fraction of their decays inside the detection apparatus. One example of
such a possibly long-lived particle is the “inflaton.” This particle couples
to a scalar field and is responsible for cosmological inflaton. Bezrukov and
Gorbunov predicted branching fractions and decay modes of inflatons, $\chi$,
in $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ meson decays [4] using a
specific model, which is a particular version of the simple chaotic inflation
with a quartic potential and having the inflaton field coupled to the SM Higgs
boson via a renormalizable operator. For $\kern
1.79993pt\overline{\kern-1.79993ptB}{}\rightarrow\chi X_{s}$ decays the
branching fraction is

$\displaystyle{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}\rightarrow\chi X_{s})$
$\displaystyle\simeq
0.3\frac{\left|V_{ts}V_{tb}^{*}\right|^{2}}{\left|V_{cb}\right|^{2}}\left(\frac{m_{t}}{M_{W}}\right)^{4}\left(1-\frac{m_{\chi}^{2}}{m_{b}^{2}}\right)^{2}\theta^{2}$
(2) $\displaystyle\simeq
10^{-6}\cdot\left(1-\frac{m_{\chi}^{2}}{m_{b}^{2}}\right)^{2}\left(\frac{\beta}{\beta_{0}}\right)\left(\frac{\rm
300~{}MeV}{m_{\chi}}\right)^{2},$

where $X_{s}$ stands for strange meson channels mostly saturated by a sum of
$K$ and $K^{*}(890)$ mesons, $m_{\chi}$ and $m_{t}$, the inflaton and top
quark masses, respectively. The model parameters are $\theta$, $\beta$ and
$\beta_{0}$, where $\beta/\beta_{0}\approx{\cal{O}}(1).$ Their inflaton
branching fraction predictions are shown in Fig. 1(a). The branching fractions
are quite similar for $\mathchar 28931\relax^{0}_{b}$ decays. The $\mathchar
28931\relax^{0}_{b}\rightarrow pK^{-}\chi$ channel would seem to be the most
favorable, since the $\mathchar 28931\relax^{0}_{b}$ decay point could be
accurately determined from the $pK^{-}$ vertex. The mass dependent inflaton
branching fraction predictions for different decay modes are shown in Fig.
1(a). Collider searches that rely on directly detecting the inflaton decay
products may not be sensitive to lifetimes much above a few times 1 ns [5],
because the particles mostly decay outside of the detector, while searches
that could be done inclusively, e.g., without detecting the inflaton decay
products, would be independent of this restriction.

Figure 1: Predictions from Ref. [4]. (a) Inflaton branching ratios to various
two-body final states as functions of inflaton mass for $m_{\chi}<1.5$ GeV.
Above 2.5 GeV only quark-antiquark and dilepton modes are predicted. In the
intermediate region no reliable prediction is given. (b) Inflaton lifetime
$\tau_{\chi}$ as a function of the inflaton mass $m_{\chi}$. The lifetime can
be up to two times smaller, depending on model-dependent parameters.

Use of $\mathchar 28931\relax^{0}_{b}$ decays in measuring CKM matrix elements
as well as new particle searches has been not as fruitful as in $B$ meson
decays because $e^{+}e^{-}$ machines have access only to the lighter $B$
mesons. In addition, absolute branching fraction determinations have been made
difficult by the relatively large uncertainty on ${\cal{B}}(\mathchar
28931\relax_{c}^{+}\rightarrow pK^{-}\pi^{+})$. Recently, the Belle
collaboration reduced this uncertainty from 25% to about 5%, allowing for
measurements with much better precision [6].

Inclusive decay searches using $\mathchar 28931\relax^{0}_{b}$ baryons can be
made at high energy colliders if it were possible to find a way to estimate
the $\mathchar 28931\relax^{0}_{b}$ momentum. The $\mathchar
28931\relax^{0}_{b}$ direction is measured by using its finite decay distance.
To get an estimate of the $\mathchar 28931\relax^{0}_{b}$ energy we can use
$\mathchar 28931\relax^{0}_{b}$’s that come from $\mathchar
28934\relax_{b}^{\pm}\rightarrow\pi^{\pm}\mathchar 28931\relax^{0}_{b}$ and
$\mathchar 28934\relax_{b}^{*\pm}\rightarrow\pi^{\pm}\mathchar
28931\relax^{0}_{b}$ decays. The $\mathchar 28934\relax_{b}^{(*)\pm}$ states
were found by the CDF collaboration [7]. Their masses and widths are
consistent with theoretical predictions [8].

The $\mathchar 28931\relax^{0}_{b}$ energy is determined from the measurement
of the $\pi^{\pm}$ from the $\mathchar 28934\relax_{b}^{(*)}$ decay along with
the $\mathchar 28931\relax^{0}_{b}$ direction. Let us assume we have a pion
from the $\mathchar 28934\relax_{b}^{(*)}$ decay. Then

$m_{\mathchar 28934\relax_{b}^{(*)}}^{2}=(E_{\pi}+E_{\mathchar
28931\relax^{0}_{b}})^{2}-(\overrightarrow{p}\\!\\!_{\pi}+\overrightarrow{p}\\!\\!_{\mathchar
28931\relax^{0}_{b}})^{2},$ (3)

and after some algebraic manipulations we find

$\displaystyle|p_{\mathchar 28931\relax^{0}_{b}}|$ $\displaystyle=$
$\displaystyle(-b\pm\sqrt{b^{2}-4ac})/(2a)$ (4) $\displaystyle a$
$\displaystyle=$ $\displaystyle 4(E^{2}_{\pi}-p^{2}_{\pi}\cos^{2}\theta)$
$\displaystyle b$ $\displaystyle=$
$\displaystyle-4p_{\pi}\Delta^{2}_{m}\cos\theta$ $\displaystyle c$
$\displaystyle=$ $\displaystyle 4E^{2}_{\pi}m^{2}_{\mathchar
28931\relax^{0}_{b}}-\Delta^{4}_{m}$ $\displaystyle\Delta^{2}_{m}$
$\displaystyle=$ $\displaystyle m^{2}_{\mathchar
28934\relax_{b}^{(*)}}-m^{2}_{\pi}-m^{2}_{\mathchar 28931\relax^{0}_{b}},$

where $\cos\theta$ is the measured angle between the pion and the $\mathchar
28931\relax^{0}_{b}$, and $m_{\mathchar 28934\relax_{b}^{(*)}}$ indicates
either the $\mathchar 28934\relax_{b}$ or $\mathchar 28934\relax_{b}^{*}$
mass. With the measured $\mathchar 28931\relax^{0}_{b}$ direction and
$\mathchar 28931\relax^{0}_{b}$ energy Eq. (1) can now be used to find decays
with any number of detected and one missing particle. Two possible solutions
result because of the $\pm$ sign in the first line of Eq. 4. In similar
studies one solution is often unphysical. This was seen, for example, using
$D^{*+}\rightarrow\pi^{+}D^{0}$, $D^{0}\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$
decays with one missing pion [9]. The resolution in $m_{x}^{2}$ depends on
several quantities including the measurement uncertainties on momentum of the
final state particles and the $\mathchar 28931\relax^{0}_{b}$ direction, so it
may be advantageous for analyses to select long-lived decays at the expense of
statistics. The relatively long $\mathchar 28931\relax^{0}_{b}$ lifetime of
about 1.5 ps is helpful in this respect [10, *Aaij:2014owa].

The $\mathchar 28934\relax_{b}^{(*)\pm}$ states have only been seen by CDF
[7]. Their data are shown in Fig. 2, and listed in Table 1.

Figure 2:  The ${Q=M(\mathchar 28931\relax^{0}_{b}\pi^{\pm})-M(\mathchar 28931\relax^{0}_{b})-m_{\pi}}$ spectrum for candidates with the projection of the corresponding unbinned likelihood fit superimposed, (a) for $\pi^{+}\mathchar 28931\relax^{0}_{b}$ and (b) for $\pi^{-}\mathchar 28931\relax^{0}_{b}$ candidates. (From Ref. [7]). Table 1:  Summary of the results of the fits to the ${Q=M(\mathchar 28931\relax^{0}_{b}\pi^{\pm})-M(\mathchar 28931\relax^{0}_{b})-m_{\pi}}$ spectra from CDF [7].  State |  $Q$ value, | Natural width, | Yield
---|---|---|---
| MeV |  $\Gamma_{0}$, MeV |
${\mathchar 28934\relax_{b}^{-}}$ | ${56.2}_{-0.5}^{+0.6}$ | ${4.9}_{-2.1}^{+3.1}$ | $340_{-70}^{+90}$
${\mathchar 28934\relax_{b}^{*-}}$ | ${75.8}\pm{0.6}$ | ${7.5}_{-1.8}^{+2.2}$ | $540_{-80}^{+90}$
${\mathchar 28934\relax_{b}^{+}}$ | ${52.1}_{-0.8}^{+0.9}$ | ${9.7}_{-2.8}^{+3.8}$ | $470_{-90}^{+110}$
${\mathchar 28934\relax_{b}^{*+}}$ | ${72.8}\pm{0.7}$ | ${11.5}_{-2.2}^{+2.7}$ | $800_{-100}^{+110}$

##  2 Potential measurements

Although there is no measurement of the relative $\mathchar
28934\relax_{b}^{(*)\pm}/\mathchar 28931\relax^{0}_{b}$ production cross-
section, $r_{\mathchar 28934\relax\Lambda}$, one might imagine that the
production ratio would be close to unity. The pions from the $\mathchar
28934\relax_{b}^{(*)\pm}$ decays have relatively low momenta, so their
detection efficiencies could be small. Although CDF does not report a value
for the production ratio, the number of seen signal events gives an observed
value of $r_{\mathchar 28934\relax\Lambda}$ equal to 13%. This is certainly a
useful sample. Backgrounds will be an issue, however, as the CDF data do show
a substantial amount of non-resonant combinations under the signal peaks, but
this will not prevent searches, just limit their sensitivities with a given
data sample.

Measurement of $|V_{cb}|$, determined using $\mathchar
28931\relax^{0}_{b}\rightarrow\mathchar
28931\relax_{c}^{+}\ell^{-}\overline{\nu}$ decays with $\mathchar
28931\relax_{c}^{+}\rightarrow pK^{-}\pi^{+}$ would provide an important
cross-check on this important fundamental parameter, especially when updated
lattice gauge calculations become available [12]. This measurement is not
subject to the uncertainty on ${\cal{B}}(\mathchar
28931\relax_{c}^{+}\rightarrow pK^{-}\pi^{+})$ provided that the total number
of $\mathchar 28931\relax^{0}_{b}$ events in the event sample is determined
using the same branching fraction [13]. The LHCb determination of the ratio of
$\mathchar 28931\relax^{0}_{b}$ to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ production, for example, uses the
$\mathchar 28931\relax_{c}^{+}\rightarrow pK^{-}\pi^{+}$ decay mode [14], and
then the absolute number of $\mathchar 28931\relax^{0}_{b}$ events produced is
found by measuring the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ rate
in a channel with a known branching fraction. The branching ratio for the
channel $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar
28931\relax_{c}^{+}\ell^{-}\overline{\nu}$ can be determined using Eq. (1)
using the measured value for the $\mathchar 28931\relax^{0}_{b}$ energy
determined by using Eq. (4); a signal would appear near $m_{x}^{2}$ equal to
zero. To determine the four-momentum transfer squared from the $\mathchar
28931\relax^{0}_{b}$ to the $\mathchar 28931\relax_{c}^{+}$ a similar
procedure as used in the decay sequence $D^{*+}\rightarrow D^{0}\pi^{+}$,
$D^{0}\rightarrow K^{*-}\ell^{+}\nu$ can be implemented [2, *Agostino:2004na].
In this procedure, the neutrino mass is set to zero,

$(E_{\mathchar 28931\relax^{0}_{b}}-E_{X})^{2}-(\vec{p}_{\mathchar
28931\relax^{0}_{b}}-\vec{p}_{X})^{2}=m_{x}^{2}=0,$ (5)

where $X$ represents the sum of $\mathchar 28931\relax_{c}^{+}$ and $\ell^{-}$
energies and momenta. Eq. (3) and Eq. (5) can be used as two constraint
equations with one unknown variable $|p_{\mathchar 28931\relax^{0}_{b}}|$.

Measurement of $|V_{ub}|$ using $\mathchar 28931\relax^{0}_{b}\rightarrow
p\ell^{-}\overline{\nu}$ decays is subject to the uncertainty on
${\cal{B}}(\mathchar 28931\relax_{c}^{+}\rightarrow pK^{-}\pi^{+})$ , but here
the current precision of 5% on this branching fraction is sufficient.
Theoretical calculations of the decay width from the lattice gauge
calculations done in a limited four-momentum transfer range [15], light cone
sum rules [16, 17, *Azizi:2009wn, *Huang:2004vf], and QCD sum rules [20,
*Huang:1998rq] can be used to extract $|V_{ub}|$. The
$p\ell^{-}\overline{\nu}$ final state is subject to backgrounds from
$N^{*}\ell^{-}\overline{\nu}$, where $N^{*}\rightarrow p\pi^{0}$, that are
difficult to eliminate and thus the use of the $\mathchar
28934\relax_{b}^{(*)\pm}\rightarrow\pi^{\pm}\mathchar 28931\relax^{0}_{b}$
decay sequence may be crucial. The decay sequence constraint can also possibly
help measure the branching fraction for $\mathchar
28931\relax^{0}_{b}\rightarrow\mathchar
28931\relax_{c}^{(*)+}\tau^{-}\overline{\nu}$ decays as measurements in the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ meson system of analogous
decays are somewhat larger than Standard Model predictions [22,
*Adachi:2009qg, *Bozek:2010xy].

Particles characteristic of scalar fields such as inflatons or dilatons can be
searched for in $\mathchar 28931\relax^{0}_{b}$ decays. It is also possible to
search for Majorana neutrinos through a process similar to that used for
searches in $B^{-}\rightarrow\mu^{-}\mu^{-}\pi^{+}$ decays [5, 25,
*Aaij:2011ex], where the Majorana neutrino, $\nu_{M}$, decays into a
$\mu^{-}\pi^{+}$ pair. The initial quark content of the $\mathchar
28931\relax^{0}_{b}$ is $bud$. The $b$-quark can annihilate with a
$\overline{u}$-quark from a $u\overline{u}$ pair arising from the vacuum into
a virtual $W^{-}$ leaving a $uud$ system that can form a $p$. The virtual
$W^{-}$ then can decay into $\mu^{-}$ in association with a Majorana neutrino
that can transform to its own anti-particle and decay into $\mu^{-}$ and a
virtual $W^{+}$. In the analogous case to the
$B^{-}\rightarrow\mu^{-}\mu^{-}\pi^{+}$ decay, the $W^{+}$ would decay into a
$\pi^{+}$, however here we do not have to detect the Majorana decays, so we
can look for the decay $\mathchar 28931\relax^{0}_{b}\rightarrow
p\mu^{-}\nu_{M}$ independently of the $\nu_{M}$ decay mode or lifetime. Other
mechanisms for Majorana neutrino production discussed in Ref. [27] for $B^{-}$
decays when adopted to $\mathchar 28931\relax^{0}_{b}$ decays, would also lead
to the $p\mu^{-}\nu_{M}$ final state.

A more mundane search can be considered for $\mathchar 28931\relax^{0}_{b}$
decays into non-charmed final states containing $\mathchar 28934\relax^{\pm}$
light baryons; these have been proposed for flavor SU(3) tests [28]. Since the
largest decay modes are $\mathchar 28934\relax^{-}\rightarrow n\pi^{-}$, and
$\mathchar 28934\relax^{+}\rightarrow n\pi^{+}$ or $p\pi^{0}$, there is always
a missing neutron in the $\mathchar 28934\relax^{-}$ decay, while the
$\mathchar 28934\relax^{+}$, in principle, can be detected in the $p\pi^{0}$
mode. The method suggested here can be adopted to search for both $\mathchar
28934\relax^{-}$ and $\mathchar 28934\relax^{+}$ baryons in $\mathchar
28931\relax^{0}_{b}$ decays.

Note that similar methods can be applied to $B^{-}$ decays by the use of the
$\overline{B}^{0**}\rightarrow\pi^{+}B^{-}$ decay sequence. The measured
production ratio of
$\left(\overline{B}^{0**}\rightarrow\pi^{+}B^{-}\right)/B^{-}$ is about 15%,
but the $B^{0**}$’s have widths of about 130 MeV which introduces very large
backgrounds that have thus far precluded their use. Another possible source of
tagged $B^{-}$ events is the decays of $\overline{B}_{s}^{0**}$ mesons into a
$K^{+}B^{-}$ that would have the advantage of a charged kaon tag, and have a
much narrower width.

In conclusion, we propose a new method of analyzing $\mathchar
28931\relax^{0}_{b}$ decays into one missing particle, where the $\mathchar
28931\relax^{0}_{b}$ is part of a detected $\mathchar
28934\relax_{b}^{(*)\pm}\rightarrow\pi^{\pm}\mathchar 28931\relax^{0}_{b}$
decay that provides additional kinematic constraints. This method may be
useful for studies of CKM elements and searches for new particles such as
inflatons, dilatons or Majorana neutrinos. Thus, investigations of $\mathchar
28931\relax^{0}_{b}$ decays may present a unique opportunity in the study of
$b$-flavored hadron decays.

## Acknowledgements

We are grateful for the support of the National Science Foundation, and
discussions with Jon Rosner, and many of our LHCb colleagues.

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