REVIEW ON SPINOR FIELDS AND APPLICATIONS IN PHYSICS

K. P. S. DE BRITO<EMAIL_ADDRESS>
Department of Physics, Federal University of ABC,
Santo André, SP Brazil

###### Abstract

A review about spinor fields is presented, constructing a outlook through the
last century. Spinor was explored in many contexts more and more in the last
decades. Besides this, more papers about this issue has been produced in the
last decade than in the others before it. As examples, classifications of
spinor fields on the bulk and on compactified 7-manifolds done by the author
are revised and some results about spinor fields on this contexts are
explored, like generating of brane by singular spinors in the bulk, invariants
on five-dimensional black holes and spectral decomposition of a quantum field
on compactified dimensions at low energies.

###### Contents

  1. 1 Introduction
  2. 2 Goal
  3. 3 Bilinear Covariants and Spinor classification on Minkowski space
  4. 4 Bilinear covariants of Spinor Fields on Arbitrary Dimensions and Signatures
  5. 5 Spinor Fields on seven-dimensional Riemmanian spaces
    1. 5.1 Some considerations about Compactifications
  6. 6 Spinor Fields on the Bulk
    1. 6.1 Degeneracy of the flag-dipole Spinor into a dipole spinor generating of brane
    2. 6.2 Invariant on axisymmetric black holes
  7. 7 Conclusion

##  1 Introduction

Many studies around Clifford algebras, and his useful derived objects called
spinor, has been motivated by physical and mathematical reasons. Besides this,
this issue was explored in many contexts. Through the history many progress
took place, since remote times. We know that since the beginning of
civilization, the set of numbers and the set of mathematical objects has been
increased at facing new problems and challenges in human society.

The history maybe starts with the Hamilton’s attempt in construct a geometric
algebra with three dimensions and his discovery about the quaternions as an
extension of the complex number set. After this, other extensions was
constructed. Among them, Grassman built the exterior algebra. Clifford
enumerated his quadratic geometric algebra in a table of classification, which
have periodicity eight, as it was shown by Cartan. Cayley explored a
interacted process to weaken properties in extension of algebras, like real
number are extended to complex number losing the duality property
($\bar{z}=z$) and complex to quaternion losing the commutativity [1].
Geometric constructions inside physical and mathematical scenarios was
facilitated with those new algebraic tools, beside this new intuitions and
insights about such themes arises, because his more geometric nature. Like
example, spin groups was used to represent rotation by Lipschitz, Riesz wrote
the exterior product using the Clifford product.

It is interesting to remember that a Clifford algebra starts from a vector
space and a quadratic form and that the “spinor field” idea, which is used to
describe fermions and started to be explored in the begin of the last century
[2], is constructed from such algebra as a double covering of the special
orthogonal group $SO(n)$ [3, 4]. Besides, they were classified on Minkowski
space by Lounesto [3] and so on many exploration was done at this branch. It
is interesting to salient that one the firsts to mention the term “Lounesto’s
classification” was Romo in a paper of 1993 [5], as it is accused in the
Google Scholar. Thenceforth, many aspects have been explored.

##  2 Goal

Here we are interested in a short review on the last studies about Clifford
algebra and spinor fields, in their classification on some spaces, like the
bulk and compactified spaces, such examples are important to fermions dynamics
in quantum gravity. Besides this, a spinor quantum field is introduced to
compactified spaces at low energies; a type of spinor field on the bulk
degenerate in another that generate the brane; and the classification of
spinor fields on the bulk is corroborated by a verification that the
components of the bilinear form on the bulk fill out all the invariants
construct with the spinor field.

##  3 Bilinear Covariants and Spinor classification on Minkowski space

Bilinear covariants are constructed with spinor fields and can be used to
classify them, through some constraints named Fierz identities and can be
associated to observable of quantum fields. This classification was done in
Minkowski space by Lounesto, where he listed his six classes of spinors, being
three named regular spinors and the others three named singular spinors,
specifically: flagpole, flag-dipole and dipole [3]. A similar work was done in
five-dimensional Lorentzian spaces, in particular in the bulk [6], and in
seven-dimensional Riemannian[7] and Lorentzian [8] spaces, as important
examples can be considered the seven-sphere ($S^{7}$) and the seven-
dimensional Anti-de Sitter space ($AdS_{7}$) respectively. Those
classification, other few results about spinor fields and their applications
on physics are reviewed here.

Given a spinor field $\psi$ on the Minkowski space, bilinear covariants are
written as follows:

$\displaystyle\sigma$ $\displaystyle=$ $\displaystyle\bar{\psi}\psi,$ (1a)
$\displaystyle\mathbf{J}$ $\displaystyle=$ $\displaystyle{{\rm
J}_{\alpha}\theta^{\alpha}=\bar{\psi}\gamma_{\alpha}\psi\,\theta^{\alpha}},$
(1b) $\displaystyle\mathbf{S}$ $\displaystyle=$ $\displaystyle
S_{\alpha\beta}\theta^{\alpha}\wedge\theta^{\beta}=\tfrac{1}{2}i\bar{\psi}\gamma_{\alpha\beta}\psi\,\theta^{\alpha}\wedge\theta^{\beta},$
(1c) $\displaystyle\mathbf{K}$ $\displaystyle=$ $\displaystyle
K_{\alpha}\theta^{\alpha}=i\bar{\psi}\gamma_{5}\gamma_{\alpha}\psi\,\theta^{\alpha},$
(1d) $\displaystyle\omega$ $\displaystyle=$
$\displaystyle-\bar{\psi}\gamma_{5}\psi\,,$ (1e)

where $\gamma_{5}:=i\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}$ is the volume
element in Minkowski spacetime [9]. They can be used to describe the
observables of the electron [10, 11]: the current density corresponds to
$\mathbf{J}$, $\mathbf{S}$ is the spin density, the chiral current is denoted
by $\mathbf{K}$ and $\omega$ gives spinor indications under the CPT symmetry.
Obviously, all them are bilinear on the spinor field $\psi$ and some
constraints result from them, through the Fierz identities, which are written
with such bilinear covariants [3]:

$-(\omega+\sigma\gamma_{5})\mathbf{S}=\mathbf{J}\wedge\mathbf{K},\qquad\mathbf{K}\lrcorner\mathbf{K}+\mathbf{J}\lrcorner\,\mathbf{J}=0=\mathbf{J}\llcorner\mathbf{K},\qquad\mathbf{J}\,\lrcorner\,\mathbf{J}=\omega^{2}+\sigma^{2}\,.$
(2)

These constraint, that are valid only for regular spinors, can be generalised
to encompass the singular spinor case also, transforming the constraints above
in the following expressions with a much better appearance [12]:

$\displaystyle Z\gamma_{\mu}Z=4J_{\mu}Z,\qquad Z^{2}=4\sigma Z,\qquad
iZ\gamma_{\mu\nu}Z=4S_{\mu\nu}Z,$ $\displaystyle-Z\gamma_{5}Z=4\omega Z,\qquad
iZ\gamma_{5}\gamma_{\mu}Z=4K_{\mu}Z.$ (3)

From these constraints, and other not represented here, comes the
classification of the spinors obtained by Lounesto. Many aspects around the
Fierz identities in the relativistic field was explored in some papers
recently: All them are reduced to only one equation through extended Cartan
map [13]. Inversely, Fierz identities are used to reconstruct the spinors
[14]. Completeness and orthogonality relations are obtained through Fierz
identities [15, 16], where it is constructed an equivalence between tensor and
spinor representations. Computations of constraints involving the exotic term
of Dirac equation [19, 17, 18], a general comprehension about the Elko spinor
together its relation with the Dirac spinor [20], a action in supergravity for
each type of spinor [21], some aspects on the Dirac operator [22] and a result
about Hawking radiation obtained from black string tunneling [23, 24].

Summarizing, the Fierz identities provide constraints to the bilinear
covariants, and consequently to the spinor fields $\psi$, splitting all the
set of spinor fields on the Minkowski space into such six classes described
below, besides other three classes with null current density, which can
represent ghost states [25]:

$\displaystyle{\bf regular}\;\left\\{\begin{array}[]{cc}1)\;\omega\neq
0,\;\;\;\sigma\neq 0,\\\ 2)\;\omega=0,\;\;\;\sigma\neq 0,\\\ 3)\;\omega\neq
0,\;\;\;\sigma=0,\end{array}\right\\}$ $\displaystyle-\;\;\boxed{\;\;{\bf
J}\neq 0,\;\;\mathbf{K}\neq 0,\;\;\;\mathbf{S}\neq 0\;\;}$ (7)
$\displaystyle{\bf singular}\;\boxed{\omega=\sigma=0,\;\;{\bf J}\neq 0},$
$\displaystyle\;\left\\{\begin{array}[]{cc}4)\;\;\mathbf{K}\neq
0,\;\;\;\mathbf{S}\neq 0\;\;\text{(flag-dipole spinors)}\\\
\\!\\!\\!\\!\\!\\!5)\;\;\mathbf{K}=0,\;\;\;\mathbf{S}\neq 0\;\;\text{(flagpole
spinors)}\\\ \\!\\!\\!\\!\\!\\!\\!\\!6)\;\;\mathbf{K}\neq
0,\;\;\;\mathbf{S}=0\;\;\text{(dipole spinors)}\end{array}\right.$ (11)
$\displaystyle{\bf``ghosts^{\prime\prime}}\;\boxed{\omega=\sigma=0,\;\;{\bf
J}=0},$ $\displaystyle\left\\{\begin{array}[]{cc}7)\;\;\mathbf{K}\neq
0,\;\;\;\mathbf{S}\neq 0\\\
\\!\\!\\!\\!\\!\\!\;8)\;\;\mathbf{K}=0,\;\;\;\mathbf{S}\neq 0\\\
\\!\\!\\!9)\;\;\mathbf{K}\neq 0,\;\;\;\mathbf{S}=0\end{array}\right.$ (15)

All classes with ${\bf J}\neq 0$ was found to have at least a possible
physical representative. This result was fruit of exploration of those spinor
fields in many contexts by many authors. Flagpoles was considered in the
context of cosmology recently [26] and proposed to be dark matter candidate in
many scenarios [19, 17, 18, 28, 29, 27]. Majorana and Elko [31, 27, 32, 33,
26, 29, 23, 48, 21, 34, 20, 36, 37, 38, 39, 28, 30, 40, 41, 42, 43, 44, 45,
46, 47, 35] spinors fields are examples of flagpoles realisations. In the same
sense, Weyl spinors are examples of dipole spinors and, recently, it was found
a missing example of flag-dipole in ESK models of gravity like solutions of
the Dirac equation in a background with $f(R)$-torsion [49].

##  4 Bilinear covariants of Spinor Fields on Arbitrary Dimensions and
Signatures

The bilinear covariants written with the spinor fields in Minkowski space can
be extended to spaces $M$ of arbitrary dimensions $n$ and signatures $p+q$.
Observables of the spinor $\psi$ are, in the same way that in Minkowski space,
associated to their bilinear covariants, as shown in the previous section. In
such construction, the dual spinor is identified to the conjugation
$\bar{\psi}=\psi^{\dagger}\gamma^{0}$. This is generalised, if it is
considered a hermitian operator $A=A^{\dagger}$, in this way the dual spinor
is written as $\bar{\psi}=\psi^{\dagger}A^{-1}$ and the bilinear covariants
can be expressed by the following expressions [50, 51, 7]:

$\displaystyle B_{0}=\bar{\psi}\psi,$ $\displaystyle\vdots$ $\displaystyle
B_{\alpha_{1}\ldots\alpha_{k}}=\frac{1}{n!}\bar{\psi}\gamma_{\alpha_{1}}\cdots\gamma_{\alpha_{k}}\psi;\;k<n,$
$\displaystyle\vdots$ $\displaystyle B_{1\ldots
n}=\frac{1}{n!}\bar{\psi}\gamma_{1}\ldots\gamma_{n}\psi.$

As it was previously mentioned, the classification of the spinor fields in
arbitrary dimensions and signatures comes from the Fierz identities built with
their respective bilinear covariants. However, besides this, it is necessary
to introduce the homomorphisms $D$ and $J$, which provides structures for the
space of spinors. The homomorphism $D$ is named real structure and implements
the complex conjugation, for other side $J$ is the complex structure that is
equivalent to multiplication by the imaginary unit

$\displaystyle\psi^{*}:=D(\psi),\qquad J^{2}=-{\bf 1}.$ (16)

Depending on the dimension and signature of the space, the structures $D$ and
$J$ can constraint the bilinear covariants [52, 53, 54, 55, 56, 57].

##  5 Spinor Fields on seven-dimensional Riemmanian spaces

It is intended a classification of spinor fields on Riemannian spaces of seven
dimensions, in particular on $S^{7}$, that is an important case in
compactified space of 11-dimensional supergravity ($AdS_{4}\times S^{7}$) [58,
59, 60]. The real and complex structures provide constraints on spinor fields,
that are fixed by the Fierz identities. In such way, Majorana spinor on such
spaces, that are the only class of spinor existent in the literature, are
extended through complexification, resulting in other two new non-trivial
classes of spinor fields [52, 7].

The importance of this classification resides in the physical aspects of the
results that will come from the exploration of spinor fields on seven
dimensional compactified spaces [52, 3, 61, 58, 59, 7], in the same way that
spinor fields classification in Minkowski space in accord to Lounesto has
empowered unexpected discover and a large path of new physical approaches and
new particles [28, 29, 27, 48, 19, 49, 62, 63, 64, 35].

Given a Majorana spinor $\psi$ and a bilinear form $B$. The number of non-null
bilinear covariants can be reduced through the constraint coming from the
property of being Majorana spinor and from the symmetry of $B$. The first
gives that $B(\psi,\gamma^{A}\psi)$ can be different from zero, only if $|A|$
is even, where $A$ is a multi-index and $|A|$ is its degree. The second one
gives $B(\psi,\gamma^{A}\psi)=(-1)^{|A|(|A|+1)/2}B(\psi,\gamma^{A}\psi)$ can
be non-null, only if $|A|=0,3,4,7$. Hence, $B(\psi,\gamma^{A}\psi)$ can be
zero, except if $|A|=0,4.$ Then, it remains only the possible non-vanishing
bilinear covariants $\phi_{0}=\bar{\psi}\psi$ and
$\phi_{4}=\bar{\psi}\gamma_{ijkl}\psi$. If the Fierz identities are imposed,
they are reduced to the following expression:

$\displaystyle(\phi_{4}+1)\circ(\phi_{4}+1)=8(\phi_{4}+1),$ (17)

where $\circ$ is the Clifford product. This fix the inequalities for the
bilinear covariants and it constraints the Majorana spinor to only one class,
as it is knew in the literature [52, 7].

The complex ($J$) structures, which is equal to the volume element
$\gamma_{1}\cdots\gamma_{7}$ in seven dimensions, and the real ($D$) structure
are used to construct other three bilinear forms:

$\displaystyle B(\psi,J\gamma_{A}\psi),\quad B(\psi,D\gamma_{A}\psi),\quad
B(\psi,JD\gamma_{A}\psi).$ (18)

It can be verified that the constraints found previously are identical to the
constraints correspondent to second bilinear form above. It is possible to
infer also that the first and third form are equivalent. Because this, without
redundancies, it remains only the constraints coming from the first form. In
this way, a selection rule and the symmetry of the bilinear form provide that
the bilinear covariant $B(\psi,J\gamma_{A}\psi)$ is different from zero if,
respectively, $|A|$ is odd and
$B(\psi,J\gamma_{A}\psi)=(-1)^{|A|(|A|+1)/2}B(\psi,J\gamma_{A}\psi)$, i.e.,
$|A|=0,3,4,7$. Summarising, $B(\psi,J\gamma_{A}\psi)$ vanishes, except if
$|A|=3,7.$ The case $|A|=3$, that refers to the bilinear form
$(\psi,J\gamma_{ijk}\psi)e^{ijk}$ that has a correspondent in the literature,
because this form can be associated to the three-form torsion gauge, which is
being to much extensively explored in the last decade [58, 65, 66, 67, 68, 60,
59, 61]. However, there is a Hodge duality $(\star\;\cdot\;\;)$ that relates
the two important bilinear forms:
$\frac{1}{|A|!}B(\psi,J\gamma_{A}\psi)e^{A}=\star\frac{1}{|A^{\prime}|!}B(\psi,\gamma_{A}^{\prime}\psi)e^{A^{\prime}}$,
where $|A|+|A^{\prime}|=7.$ Because this fact, the constraints coming from the
bilinear covariant $B(\psi,J\gamma_{A}\psi)$ do not provide any new
information to the classification of the spinor field $\psi$ [52, 7].

Finally, a general classification of the spinor fields on seven-dimensional
Riemannian spaces is obtained, if the bilinear covariant
$B(\psi,\gamma_{A}\psi)$ is complexified into a more general bilinear form, as
follows:

$\displaystyle\beta(\psi,\gamma_{A}\psi)=B(\psi_{R},\gamma_{A}\psi_{R})-B(\psi_{I},\gamma_{A}\psi_{I})+i\left[B(\psi_{R},\gamma_{A}\psi_{I})+B(\psi_{I},\gamma_{A}\psi_{R})\right],$
(19)

where $\psi_{R}$ denotes the real part and $\psi_{I}$ denotes the imaginary
part. In this case, some cancellations enable null and non-null bilinear
covariant values. Indeed, those guarantee the following more general
classification for arbitrary spinor fields on seven-dimensional Riemannian
spaces:

$\displaystyle\upvarphi_{0}=0,\quad\upvarphi_{4}=0,$ (20a)
$\displaystyle\upvarphi_{0}=0,\quad\upvarphi_{4}\neq 0,$ (20b)
$\displaystyle\upvarphi_{0}\neq 0,\quad\upvarphi_{4}=0,$ (20c)
$\displaystyle\upvarphi_{0}\neq 0,\quad\upvarphi_{4}\neq 0\,.$ (20d)

It is noticed two new classes of spinor fields: one regular and other
singular. These new classes can result in new possible candidates to fermion
particles at low energy in spontaneous compactifications of $D=11$ supegravity
[58] and, besides this, it arises interesting possibilities about physical
solutions in the massless sector of gauged $N=8$ supergravity theory [60, 7].
Fot other side, added to the contexts of the Killing spinors and
parallelizable seven-sphere [61, 69], the dynamic of those spinor fields in
each class enumerated above can be explored beginning with the Lagrangian
terms, which in a quaternionic realization assume the form:

$\displaystyle M$ $\displaystyle=$ $\displaystyle tr(\bar{\psi}\psi)$ (21)
$\displaystyle M_{\alpha}$ $\displaystyle=$ $\displaystyle
tr(\bar{\psi}\gamma_{\mu}\psi)$ (22) $\displaystyle K_{\beta\gamma}$
$\displaystyle=$ $\displaystyle
tr(\bar{\psi}\gamma^{\alpha}\gamma_{\beta\gamma}\partial_{\alpha}\psi)$ (23)
$\displaystyle K_{\delta}$ $\displaystyle=$ $\displaystyle
tr(\bar{\psi}\gamma^{\mu}\gamma_{\delta}\partial_{\mu}\psi).$ (24)

Because our classification, we can infer that only the massive term
contributes. Hence, this implies that in last instances in low energies does
not exist dynamics to fermionic fields on the seven-dimensional compactified
spaces [7, 70]. As it was shown by Toppan [70], the terms of the Lagrangian
that need to be considered depend on the realisation taken in consideration.
It is interesting to observe that the quaternionic realisation above is the
most plausible, because the other realisations: real and octonionic, prohibit
at first the existence of kinetic terms. In such way that in such realisations
only appears non-Weyl mass terms of the type
$M_{\perp}=tr(\bar{\psi}_{+}\psi_{-}+\bar{\psi}_{-}\psi_{+})$ [70].

###  5.1 Some considerations about Compactifications

The fermionic fields classification that we proposed play a fundamental role
at compactified models in $D=11$ supergravity. The first compactification
models was proposed in the begin of last century by Klein, who improved a
model of unification in five-dimensional spaces [71, 72, 73], where was
proposed one the first tentatives in unify the gravity force with other force.
The improvement of Klein consisted into decomposing the scalar field in a
Fourier serie through compactifying a dimension into a circle [71]. Similar
procedures can be done in high-dimensional spaces. For example, in a two-
dimensional compactification with sphere form, the scalar field $\varphi$ can
be decomposed in a spherical harmonic serie, and go on for high-dimensional
spheres. This gives the following interpretation: “For $q$ compactified
dimensions, a non-massive theory in $4+q$ dimensions results in a massive
theory in $4$ dimensions” [71], as represented by the equation below.

$\displaystyle\square_{4+q}\varphi=(\square+\nabla_{q})\varphi=(\square+m^{2})\varphi.$
(25)

The next step would be to quantize the field, for this intuit, look that the
fermionic quantum field in Minkowski space takes free value of momentum
(modes) [51]:

$\displaystyle\psi(x)=\int\frac{d^{4}p}{(2\pi)^{4}}\sum_{i}\left(a^{s}_{\bf
p}u^{s}(p)e^{-ip\cdot x}+b^{s\dagger}_{\bf p}v^{s}(p)e^{ip\cdot x}\right).$
(26)

In this way, consider the state $\Psi$ on the compactified space $M^{4}\times
K_{q}$, where $M^{4}$ is the space-time and $K_{q}$ is a q-dimensional
compactified space. It is decomposed like $\Psi(x,y)=\psi(x)\Psi_{q}(y)$.
Hence, observe that in $\Psi_{q}$ there is creation and annihilation operators
associated to each harmonic spherical (mode) in the discrete spectra. The
field factor $\Psi_{q}$ is denoted at low energy on $K_{q}$ by:

$\displaystyle\Psi_{q}(y)=\sum_{I_{n}}\left[Y_{n}^{I_{n}}(y)u_{I_{n}}a_{I_{n}}+\left(Y^{I_{n}}_{n}(y)\right)u^{*}_{I_{n}}b_{I^{*}_{n}}\right],$
(27)

where $I^{*}_{l,m}=I_{l,-m}$ is a multi-index and $a_{I_{n}},b_{I^{*}_{n}}$
are operator valued coefficients.

##  6 Spinor Fields on the Bulk

Given their importance on studies in brane theories, it is intended here to
review the classification of spinor fields on the bulk [74]. A similar
procedure to shown above in four and seven dimensional spaces is done in four
plus one, specifically, in the bulk [6]. Consider a bilinear form $B$ on the
bulk, the deal with spinor fields on the bulk is more elaborated than in seven
dimensions, because it is necessary to introduce quaternionic structures
$J_{i}$, that satisfy the relations [52, 6]

$\displaystyle J_{i}J_{j}=-\delta_{ij}+\epsilon_{ij}^{k}J_{k}.$ (28)

Nevertheless, the proceture to classify spinor fields on the bulk is much
similar to approached in the previous section at seven dimensions. The
constraints come from the form symmetries and selection rule, and classes are
fixed in a classification by the Fierz identities. Then, from the symmetry of
$B$, the bilinear covariants are constrained by

$\phi_{A}:=B(\psi,\gamma_{A}\psi)e^{A}=(-1)^{\frac{|A|(|A|-1)}{2}}B(\psi,\gamma_{A}\psi)e^{A}.$

This implies that $B(\psi,\gamma_{A}\psi)$ is zero, except if $|A|=0,1,4,5$.

The quaternionic structures are used to construct other three bilinear forms
$B(\psi,J_{i}\gamma_{A}\psi)$, from whose symmetries comes other constraints
for the spinor fields on the bulk. Explicitly, the bilinear form is
constrained by

$\hat{\phi}_{A}^{i}:=B(\psi,J_{i}\gamma_{A}\psi)e^{A}=(-1)^{\frac{|A|(|A|-1)}{2}+1}B(\psi,J_{i}\gamma_{A}\psi)e^{A},$

that vanishes, except if $|A|=2,3$. From where arises other possibly non-
vanishing six bilinear covariants. However, the number of bilinear covariants
is cut by half, because the Hodge dualities $\phi_{A}=\star\phi_{A^{\prime}}$
and $\hat{\phi}_{A}^{i}=-\star\hat{\phi}_{A^{\prime}}^{i}$, where
$|A|+|A^{\prime}|=5$. Then, only the following bilinear covariants remain:
$\phi_{0},\;\phi_{1}$ and $\hat{\phi}_{2}^{i}$, where $i=1,2,3$, which are the
only ones possible to be non-null. Besides this, the Fierz identities

$\displaystyle\phi_{1}\circ\phi_{1}-\sum_{i}\hat{\phi}_{2}^{i}\circ\hat{\phi}_{2}^{i}=7(\phi_{0})^{2}+6\phi_{0}\phi_{1},$
(29)
$\displaystyle\phi_{1}\circ\hat{\phi}_{2}^{i}+\hat{\phi}_{2}^{i}\circ\phi_{0}+\sum_{jk}\epsilon_{ijk}\hat{\phi}_{2}^{j}\circ\hat{\phi}_{2}^{k}=6\phi_{0}\hat{\phi}_{2}^{i}$
(30)

fix them to form only one class, as found in the literature [52, 6]. This
class can be extended through a kind of complexification to encompass
arbitrary spinor fields on the bulk

$\displaystyle\beta_{k}(\psi,\gamma_{A}\psi)=B(\psi,\gamma_{A}J_{1}\psi)-iB(\psi,\gamma_{A}J_{3}\psi).$
(31)

Through this process, it is found other six non-trivial class of spinor fields
on five-dimensional Lorentzian spaces like the bulk, where three are regular
spinor fields and the other three are singular [52, 7, 6], as it is shown
below:

$\displaystyle{\bf regular}\;-\;\phi_{0}\neq
0,\left\\{\begin{tabular}[]{cc}$\quad\phi_{1}\neq
0,$&$\left\\{\begin{tabular}[]{c}$\quad\hat{\phi}^{i}_{2}\neq 0,$\\\
$\quad\hat{\phi}^{i}_{2}=0,$\end{tabular}\right.$\\\
$\quad\phi_{1}=0,$&$\left\\{\begin{tabular}[]{c}$\quad\hat{\phi}^{i}_{2}\neq
0,$\\\ $\quad\hat{\phi}^{i}_{2}=0,$\end{tabular}\right.$\end{tabular}\right.$
(32g) $\displaystyle{\bf
singular}\;-\;\phi_{0}=0,\left\\{\begin{tabular}[]{cc}$\quad\phi_{1}\neq
0,$&$\left\\{\begin{tabular}[]{c}$\quad\hat{\phi}^{i}_{2}\neq 0,$\\\
$\quad\hat{\phi}^{i}_{2}=0,$\end{tabular}\right.$\\\
$\quad\phi_{1}=0,$&$\left\\{\begin{tabular}[]{c}$\quad\hat{\phi}^{i}_{2}\neq
0,$\\\ $\quad\hat{\phi}^{i}_{2}=0,$\end{tabular}\right.$\end{tabular}\right.$
(32n) .

It is important to salient that the contributing terms of the Lagrangian for
each class of spinor fields depends on its realization [70, 6], but this is
beyond the scope of this paper.

###  6.1 Degeneracy of the flag-dipole Spinor into a dipole spinor generating
of brane

Spinor constrained to the brane contained in the bulk must obey the
classification for spinor fields in Minkowski space [3, 74, 6]. In this way,
it can be analyzed if a spinor field in a given class generate brane or does
not, calculated the coupling constants and his correspondent energy density,
and how this occur [74, 6]. Given a scalar field $\varphi$ massless on a
Randall-Sundrum background

$\displaystyle ds^{2}=exp(-2A(r))\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dr^{2},$ (33)

the graviton modes can be calculated and their masses [74]. If a fermionic
spinor field represented by $\psi=(a(r),0,b(r),0)^{\intercal}$ is added to
this system, the spinor components can be solved through the Dirac and
Einstein equations

$\displaystyle{R_{a}}^{A}-\frac{1}{2}{e_{a}}^{A}R=\chi{T_{a}}^{A}+{e_{a}}^{A}\Lambda,\qquad[i\not{D}-m+\lambda(\bar{\psi}\psi)]=0,$
(34)

and be expressed in terms of the extra dimension and of the coupling constant.
Additionally, the energy density can be computed, resulting in $T_{0}^{0}=-2$
[74, 6].

If instead to consider the spinor field given above, we do different and we
consider a two-components flag-dipole expressed by $\psi=(a(r),0,0,b(r))$ and
the metric $ds^{2}=\varphi^{2}(r)(\eta_{\mu\nu}dx^{\mu}dx^{\nu})-dr^{2}$, we
surprisingly found a null energy-momentum tensor. Because, such flag-dipole
does not generate brane and does not have existence. However, from the
constraint ${T_{\bar{0}}}^{1}=0=\varphi^{\prime}(r)a(r)b(r)$, we can infer
that the flag-dipole degenerate into a one-component dipole spinor field
$\psi=(a(r),0,0,0)^{\intercal}$, whose component and the correspondent metric
can be computed [65, 74, 6]

$\displaystyle\varphi(r)=\varphi_{0}exp(\pm r\sqrt{\Lambda/6}),\qquad
a(r)=a_{0}exp(\mp r\sqrt{2\Lambda/3}).$ (35)

###  6.2 Invariant on axisymmetric black holes

Consider a five-dimensional axisymmetric and stationary black hole and a
observable Killing vector $\xi^{\mu}$ on it. The results obtained in the begin
of this section can be corroborate here through this example. There is a
current vector $J^{\mu}$ associated and equal to this Killing vector. If a
null Killing vector in the event horizon is translated from the black hole
neighborhood to the spatial infinity it became a time Killing vector, but the
one-form current density is preserved in the fünfbein basis
$e^{A}={e^{A}}_{\mu}dx^{\mu}$ [75, 76, 77, 6]. If it is imposed the contraint
to the spinor field $\psi$: $(\gamma^{0}\pm\gamma^{5})\psi=0$, it is expressed
in components like
$\psi=(\alpha_{1},\alpha_{2},\alpha_{1},\alpha_{2})^{\intercal}$. In a given
representation to the gamma matrices on the bulk with signature four plus one

$\displaystyle\gamma^{0}=i\sigma^{1}\otimes{\bf
1}_{2},\quad\gamma^{k}=-\sigma^{2}\otimes\sigma^{k},\quad\gamma^{4}=\sigma^{3}\otimes{\bf
1}_{2},$ (36)

the Killing vector is represented by:

$\displaystyle\xi^{\mu}=\partial_{t}+\Omega_{1}\partial_{\phi_{1}}+\Omega_{2}\partial_{\phi_{2}},$
(37)

where $\phi_{1},\phi_{2}$ and $\Omega_{1},\Omega_{2}$ denote the related two
azimuth angles and two azimuth angular velocities from the angular coordinates
of the sphere $S^{3}$ correspondent to the five-dimensional black hole.

Remember that there are three bilinear forms on five-dimensional Lorentzian
spaces that can be non-vanishing and can represent observables associated to
the spinor field $\psi$:

$\displaystyle\phi_{0}$ $\displaystyle=$
$\displaystyle\bar{\psi}\psi,\qquad\qquad\qquad\;-\;{\bf normalization}$ (38)
$\displaystyle\phi_{1}$ $\displaystyle=$
$\displaystyle\bar{\psi}\gamma_{\mu}\psi e^{\mu},\qquad\qquad\;-\;{\bf
current\;vector}$ (39) $\displaystyle\hat{\phi}_{2}^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\bar{\psi}J_{i}\gamma_{\mu\nu}\psi\right)e^{\mu\nu}\;\;\,-\;{\bf
dual\;spin}.$ (40)

The constraint imposed on $\psi$ permits to write all the bilinear form in
function of the components of $\psi$. This implies in redundancies on the
bilinear forms components, which the different ones reduces to the following
four terms:

$\displaystyle|\alpha_{1}|^{2}\pm|\alpha_{2}|^{2},\qquad\alpha_{1}^{\dagger}\alpha_{2}\pm\alpha_{2}^{\dagger}\alpha_{1}.$
(41)

It is important to say that these components fill out all the invariants that
can be constructed on the five-dimensional Lorentzian spaces with the spinor
field $\psi$ constrained [6].

##  7 Conclusion

In this work, we reviewed some aspects of the studies about spinor fields on
some spaces like on Minkowski space [3], on seven-dimensional Riemannian
spaces [7], for example $S^{7}$, and on five-dimensional Lorentzian spaces
[6], for example the bulk. Besides a general expression for bilinear
covariants that are listed on spaces of arbitrary dimension and signatures
[7]. Summarizing, the author obtained in previous papers two new non-trivial
classes of spinor fields [7], in particular, on seven-dimensional compactified
spaces, which play a very crucial rule on $D=11$ supergravity, beside six new
non-rivial classes of spinor fields on the bulk [6].

Studies about compactification permit us to write the quantum field to
fermionic particle at low energies on compactified spaces. Besides this, a
view around brane enabled us to verify that the flag-dipole spinor degenerate
into the dipole spinor, which generate brane in the bulk. For other side, an
analysis on the cosmological point view through axisymmetric black hole
permitted us infer, given the constraint for the spinor field:
$(\gamma_{0}\pm\gamma_{5})\psi=0$, that bilinear covariants components fill
out all the invariants, which can be built with a spinor field on five-
dimensional Lorentzian space.

## Acknowledgments

The author thanks to the very prolific discussions with Roldão da Rocha and to
the UFABC, CAPES and CNPq (grant No. 150882/2017-3) grants for support.

## References

  * [1] J. C. Baez, Bull. Am. Math. Soc.  39 (2002) 145.
  * [2] Purkayastha Deb, Anuradha Das, Some quantum theoretical studies in astrophysics and cosmology,(Guwahati, 1900).
  * [3] P. Lounesto, _Clifford Algebras and Spinors_ , (Cambridge Univ. Press, Cambridge, 2002).
  * [4] B. H. Lawson and M.-L. Michelson, Spin Geometry, (Princeton University Press, Princeton, 1990).
  * [5] S. R.-Romo, Found. Phys. 23 (1993) 1535.
  * [6] K. P. S. de Brito and R. da Rocha, J.Phys. A 49 41 (2016) 415403.
  * [7] K. P. S. de Brito, L. Bonora and R. da Rocha, JHEP 02 (2015) 069.
  * [8] L. Bonora and R. da Rocha, JHEP 01 (2016) 133.
  * [9] J. Vaz and R. da Rocha, _An Introduction to Clifford Algebras and Spinors_ , (Oxford Univ. Press, Oxford, 2016).
  * [10] P. A. M. Dirac, Roy. Soc. A 117 (1928) 610.
  * [11] P. A. M. Dirac, Proc. R. Soc. Lond. A 155 (1936) 447.
  * [12] J. P. Crawford, J. Math. Phys. 26 (1985) 1429.
  * [13] F. Reifler, R. Morris, J. Math. Phys. 27 (1986) 2803.
  * [14] Y. Takahashi, Phys. Rev. D 26 (1982) 2169.
  * [15] C. C. Nishi, Am. J. Phys. 73 (2005) 1160.
  * [16] J. F. Nieves, P. B. Pal, Am. J. Phys. 72 (2004) 1100.
  * [17] J. M. Hoff da Silva, C. H. Coronado Villalobos and R. da Rocha, Universe 2 (2016) 8.
  * [18] R. da Rocha and R. T. Cavalcanti, Phys. Atom. Nucl. 80 (2017) 329.
  * [19] R. da Rocha, A. E. Bernardini and J. M. Hoff da Silva, JHEP 04 (2011) 110.
  * [20] R. da Rocha and J. M. Hoff da Silva, J. Math. Phys.  48 (2007) 123517.
  * [21] R. da Rocha and J. M. Hoff da Silva, Int. J. Geom. Meth. Mod. Phys.  6 (2009) 461.
  * [22] D. Vassilevich, J. Math. Phys.  56 (2015) 103511.
  * [23] R. da Rocha and J. M. Hoff da Silva, Europhys. Lett. 107 (2014) 50001.
  * [24] R. T. Cavalcanti and R. da Rocha, Adv. High Energy Phys.  2016 (2016) 4681902.
  * [25] C. H. Coronado Villalobos, J. M. Hoff da Silva and R. da Rocha, Eur. Phys. J. C 75 (2015) 266.
  * [26] A. Basak, J. R. Bhatt, S. Shankaranarayanan and K. V. Prasantha Varma, JCAP 04 (2013) 025.
  * [27] D. V. Ahluwalia, C. -Y. Lee and D. Schritt, Phys. Rev. D 83 (2011) 065017.
  * [28] A. E. Bernardini and R. da Rocha, Phys. Lett. B 717 (2012) 238.
  * [29] D. V. Ahluwalia, C. Y. Lee, D. Schritt and T. F. Watson, Phys. Lett. B 687 (2010) 248.
  * [30] D. V. Ahluwalia, S. P. Horvath, JHEP 11 (2010) 078. [arXiv:1008.0436 [hep-ph]].
  * [31] D. V. Ahluwali, Evidence for Majorana Neutrinos: Dawn of a new era in spacetime structure, in Beyond the Desert, ed. H. V. Klapdor-Kleingrothaus, (Oulu, Finland, 2002) p. 143.
  * [32] I. C. Jardim, G. Alencar, R. R. Landim and R. N. Costa Filho, Phys. Rev. D 91 (2015) 085008. [[
  * [33] Y. X. Liu, X. N. Zhou, K. Yang and F. W. Chen, Phys. Rev. D 86 (2012) 064012.
  * [34] R. da Rocha and J. M. Hoff da Silva, Adv. Appl. Clifford álgebras 20 (2010) 847.
  * [35] M. Dias, F. de Campos and J. M. Hoff da Silva, Phys. Lett. B 706 (2012) 352.
  * [36] R. T. Cavalcanti, J. M. Hoff da Silva and R. da Rocha, Eur. Phys. J. Plus 129 (2014) 246.
  * [37] J. M. Hoff da Silva and R. da Rocha,Int. J. Mod. Phys. A 24 (2009) 3227.
  * [38] L. Fabbri, Phys. Rev. D 85 (2012) 047502.
  * [39] D. V. Ahluwalia and A. C. Nayak,Int. J. Mod. Phys. D 23 (2014) 1430026.
  * [40] K. E. Wunderle and R. Dick, Can. J. Phys. 87 (2009) 909.
  * [41] S. H. Pereira and A. Pinho S. S.,Int. J. Mod. Phys. D 23 (2014) 1444008.
  * [42] A. Basak and S. Shankaranarayanan, JCAP 1505 (2015) 034.[43]
  * [43] A. Alves, M. Dias and F. de Campos,Int. J. Mod. Phys. D 23 (2014) 1444005.[44]
  * [44] L. Fabbri and S. Vignolo,Int. J. Mod. Phys. D 23 (2014) 1444001.
  * [45] A. P. S. S., S. H. Pereira and J. F. Jesus,Eur. Phys. J. C 75 (2015) 36.
  * [46] B. Agarwal, P. Jain, S. Mitra, A. C. Nayak and R. K. Verma, Phys. Rev. D 92 (2015) 075027.
  * [47] S. H. Pereira, A. P. S.S. and J. M. Hoff da Silva, JCAP 1408 (2014) 020.
  * [48] R. da Rocha and W. A. Rodrigues, Jr., Mod. Phys. Lett. A 21 (2006) 65.
  * [49] R. da Rocha, L. Fabbri, J. M. Hoff da Silva, R. T. Cavalcanti and J. A. Silva-Neto, J. Math. Phys. 54 (2013) 102505.
  * [50] I. Benn and R. Tucker, An Introduction to Spinors and Geometry with applications in Physics, (Adam Hilger, Bristol, 1987).
  * [51] M. E. Peskin and D. V. Schoeder, An Introduction to Quantum Field Theory, (Perseus Book, Massachusetts, 1995).
  * [52] C. I. Lazaroiu, E. M. Babalic, I. A. Comam, JHEP 09 (2013) 156.
  * [53] R. A. Mosna and W. A. Rodrigues Jr., J. Math. Phys. 45 (2004) 2945.
  * [54] E. M. Babalic and C. I. Lazaroiu, Rom. Journ. Phys.  58 (2013) 414.
  * [55] L. Bonora, F. F. Ruffino and R. Savelli, Bollettino U. M. I. IV (2012) 9.
  * [56] W. A. Rodrigues and E. Capelas de Oliveira, Lect. Notes Phys.  722 (2007) 1.
  * [57] C. I. Lazaroiu and E. M. Babalic, JHEP 1306 (2013) 054.
  * [58] E. Cremmer, B. Julia and J. Scherk, Phys. Lett. B 76 (1978) 409.
  * [59] F. Englert, M. Rooman, P. Spindel, Phys. Lett. B 127 (1983) 47.
  * [60] B. Biran, F. Englert, B. de Wit and H. Nicolai, Phys. Lett. B 124 (1983) 45.
  * [61] M. Cederwall and C. R. Preitschopf, Comm. Math. Phys.  167 (1995) 373.
  * [62] R. T. Cavalcanti, Int. J. Mod. Phys. D 23 (2014) 1444002.
  * [63] J. M. Hoff da Silva and R. da Rocha, Phys. Lett. B 718 (2013) 1519.
  * [64] R. da Rocha and J. G. Pereira, Int. J. Mod. Phys. D 16 (2007) 1653.
  * [65] A. Gimbrère, Kaluza-Klein spectra from compactified and warped extra dimensions, (IOP, Amsterdam, 2014).
  * [66] E. K. Loginov,Int. J. Geom. Meth. Mod. Phys.  10 (2013) 1320005.
  * [67] M. A. Akivis, Siberian Math. J. 19 (1978) 171.
  * [68] E. K. Loginov, Phys. Rev. D 80 (2009) 124009.
  * [69] R. da Rocha and J. Vaz Jr., J. Algebra 301 (2006) 459.
  * [70] H. L. Carrion, M. Rojas and F. Toppan, JHEP 04 (2003) 040.
  * [71] M. J. Duff, B. E. W. Nilsson and C. N. Pope, Phys. Rept. 130 (1986) 1.
  * [72] H. Nastase, Introduction to the AdS/CFT Correspondence, (Cambridge University Press, Cambridge, 2015).
  * [73] M. J. Duff, [hep-th/9912164]. 86 citations counted in INSPIRE as of 27 Apr 2017
  * [74] V. Dzhunushaliev and V. Folomeev, Gen. Rel. Grav.  43 (2011) 1253.
  * [75] J. Mei, [hep-th/1104.3728].
  * [76] J. Mei, Gen. Rel. Grav.  44, 2191 (2012).
  * [77] H. Lu, J. Mei and C. N. Pope,Nucl. Phys. B 806 (2009) 436.


# Inclusive approach to hunt for the beauty-charmed baryons $\Xi_{bc}$

Qin Qin<EMAIL_ADDRESS>School of Physics, Huazhong University of Science
and Technology, Wuhan 430074, China  Yu-Ji Shi<EMAIL_ADDRESS>Helmholtz-
Institut für Strahlen- und Kernphysik and Bethe Center for Theoretical
Physics, Universität Bonn, 53115 Bonn, Germany  Wei Wang  MOE Key Laboratory
for Particle Physics, Astrophysics and Cosmology, Shanghai Key Laboratory for
Particle Physics and Cosmology, School of Physics and Astronomy, Shanghai Jiao
Tong University, Shanghai, 200240, China  Guo-He Yang  School of Physics,
Huazhong University of Science and Technology, Wuhan 430074, China  Fu-Sheng
Yu  School of Nuclear Science and Technology, Lanzhou University, Lanzhou
730000, China  Center for High Energy Physics, Peking University, Beijing
100871, China  Ruilin Zhu111corresponding authors: Qin Qin, Yu-Ji Shi and
Ruilin Zhu<EMAIL_ADDRESS>Department of Physics and Institute of
Theoretical Physics, Nanjing Normal University, Nanjing, Jiangsu 210023, China

###### Abstract

With a distinctive internal structure from all established hadrons, the
beauty-charmed baryons $\Xi_{bc}$ can provide us with new points of view to
decipher the strong interaction. In this work, we point out that the inclusive
$\Xi_{bc}\to\Xi_{cc}^{++}+X$ decay is a golden channel for the experimental
discovery of $\Xi_{bc}$ at the LHC. A unique feature of this process is that
the $\Xi_{cc}^{++}$ is displaced, which greatly reduces the combinatorial
background. A feasibility analysis is performed on the $\Xi_{bc}^{+}$ search,
which is expected to have a longer lifetime than $\Xi_{bc}^{0}$ and thus a
better displacement resolution. The $\Xi_{bc}^{+}\to\Xi_{cc}^{++}+X$ branching
ratio is calculated within the heavy diquark effective theory. Combining the
$\Xi_{bc}$ production rate and the $\Xi_{cc}^{++}$ detection efficiency, we
anticipate that hundreds of signal events will be collected during LHCb Run 3.

###### pacs:

13.30.-a;12.39.Hg;12.39.St

Introduction. — Doubly heavy baryons, especially the $\Xi_{cc}^{++}$, have
recently received much attention LHCb:2017iph . Different from other baryons
with one or zero heavy quarks, doubly heavy baryons resemble a ‘double-star’
core surrounded by a light ‘planet’. The $\Xi_{cc}^{++}$ discovery also
motivated studies to probe the nature of exotic four quark states or
structures, e.g. cusps or true resonances (see e.g. Karliner:2017qjm ;
Eichten:2017ffp ; Liu:2019zoy ). Very recently, a first doubly heavy
tetraquark candidate $T_{cc}^{+}$ was observed by the LHCb LHCb:2021vvq ;
LHCb:2021auc . However unlike in doubly charmed systems, the ‘double-star’
core in the beauty-charm baryons $\Xi_{bc}$ is imbalanced, resulting in
diverse features. Compared to the charm-charm binary, the beauty-charm core is
expected to have a smaller size, behaving more like a point particle.
Moreover, the beauty-charmed baryons involve more energy scales, the beauty
mass, the charm mass and the nonperturbative QCD scale $\Lambda_{\text{QCD}}$,
so they implicate more affluent dynamics. Thereby, the beauty-charmed baryons
would provide a unique new hadronic platform to decode the strong interaction.

Experimentalists have made abundant efforts to search for the beauty-charmed
baryons $\Xi_{bc}$. However, such searches are much more difficult than those
for $\Xi_{cc}^{++}$. For example, the exclusive channels $\Xi_{bc}^{0}\to
D^{0}pK^{-}$ LHCb:2020iko  and $\Xi_{bc}^{0}\to\Xi_{c}^{+}\pi^{-}$
LHCb:2021xba  were used to search for $\Xi_{bc}$ at the LHCb, but no
definitive evidence was established. With some theoretical and experimental
inputs, the experimental upper limit on the
$\Xi_{bc}^{0}\to\Xi_{c}^{+}\pi^{-}$ branching ratio can be extracted from Ref.
LHCb:2021xba  to be $\mathcal{O}(10^{-4})$. Comparing it to the theoretical
prediction Wang:2017mqp , we find a big gap of about 3 orders of magnitude.
One difficulty in such exclusive searches lies in the limited production rate
for $\Xi_{bc}$ at the LHC, but a bigger challenge is due to the very low
reconstruction efficiency, because a beauty typically decays with fractions of
$\mathcal{O}(10^{-3})$ even to the most abundant exclusive final states
Wang:2017mqp ; Kiselev:2001fw ; Han:2021gkl . To overcome this difficulty, we
propose an approach to search for $\Xi_{bc}$ via an inclusive decay channel
$\Xi_{bc}\to\Xi_{cc}^{++}+X$, where $X$ stands for all possible particles.

Figure 1: Sketch of $\Xi_{bc}$ production and decay at the LHC. The secondary
decay vertex of $\Xi_{bc}\to\Xi_{cc}^{++}+X$ is displaced from the proton-
proton collision vertex, which produces a unique signal: a displaced
$\Xi_{cc}^{++}$.

This inclusive approach to search for $\Xi_{bc}$ has multiple advantages.
Firstly, it has a much larger branching ratio than any exclusive decay
channel. Secondly, the detection efficiency is greatly improved because only
$\Xi_{cc}^{++}$ needs to be reconstructed. Lastly but very importantly,
because the weakly decaying $\Xi_{bc}$ has a relatively long lifetime and can
typically form a sub-millimeter displaced secondary decaying vertex, the
$\Xi_{cc}^{++}$’s generated from $\Xi_{bc}$ do not draw back to the primary
proton-proton collision vertices. This feature characterized by a nonzero
impact parameter (IP) can clearly distinguish the signal events from the main
background, strongly produced $\Xi_{cc}^{++}$’s from the primary vertices. To
clarify this point, a diagrammatic sketch is displayed in Fig. 1. The use of
the IP has been applied at the LHCb for a long time (see e.g. LHCb:2010wqx ),
and it was proposed that displaced $B_{c}$ mesons can be used to search for
$\Xi_{bb}$ in Gershon:2018gda , which greatly inspired the original idea of
this work. According to Gershon:2018gda , the relatively long lifetime of
$\Xi_{bc}^{+}$ Cheng:2019sxr  can ensure it a good IP resolution, while the
situation will be worse for $\Xi_{bc}^{0}$. Apart from the IP, the distance
between the $\Xi_{cc}^{++}$ decay vertex and the primary collision vertex can
also be used in the search to improve the sensitivity.

As the $\Xi_{bc}$ production rates and lifetimes have been evaluated in e.g.
Zhang:2011hi ; Ali:2018xfq  and Cheng:2019sxr  (see also references therein),
respectively, the remaining key issue is the $\Xi_{bc}\to\Xi_{cc}^{++}+X$
branching ratio. Based on the heavy diquark effective theory Brambilla:2005yk
; An:2018cln ; Shi:2020qde , we calculate the branching ratio and find
$\mathcal{B}(\Xi_{bc}^{+}\to\Xi_{cc}^{++}+X)\approx$ 7%. Moreover, because the
signal mainly characterizes in a displaced $\Xi_{cc}^{++}$, the reconstruction
efficiency of the signal events is close to the detection efficiency of
$\Xi_{cc}^{++}$, which can be reliably extracted from previous experiments.
Combining all this information, we find that hundreds of signal events are
expected during LHCb Run 3, with an integrated luminosity of 23 fb-1 by 2024.
Consequently, the proposed inclusive approach for the $\Xi_{bc}$ search is
most feasible and also timely for the LHCb study.

Decay rate. — In the feasibility analysis of this approach, the inclusive
$\Xi_{bc}\to\Xi_{cc}^{++}+X$ decay rate was calculated with the following
several steps. Based on the heavy quark symmetry and the heavy diquark
symmetry, each step of the calculation is trustworthy. Firstly, under the
heavy (di)quark symmetry, it can be demonstrated that the leading contribution
to the inclusive $\Xi_{bc}\to\Xi_{cc}^{++}+X$ decay is from
$\mathcal{X}_{bc}\to\mathcal{X}_{cc}+\bar{f}f^{\prime}$, where
$\mathcal{X}_{QQ^{\prime}}$ stands for a heavy diquark constituted by the
heavy $Q$ and $Q^{\prime}$ quarks and $f^{(\prime)}$ can be any possible
quarks or leptons. Subsequently, the unknown
$\mathcal{X}_{bc}\to\mathcal{X}_{cc}$ diquark transition current was evaluated
by matching from the $b\to c$ transition current. Afterwards, the decay rate
of $\Xi_{bc}\to\Xi_{cc}^{++}+X$ was numerically calculated, with possible
theoretical uncertainties taken into account.

We first validate the treatment of the two heavy quarks $QQ^{\prime}$ as a
point-like object in a doubly heavy baryon.The $QQ^{\prime}$ form a color
anti-triplet and have an attractive potential. As illustrated by
Brodsky:2011zs ; Yan:2018zdt ; Hu:2005gf , the distance between the two heavy
quarks is estimated as $r_{QQ}\sim 1/(m_{Q}v)$ with $v$ being the heavy quark
velocity in the baryon rest frame, while the spatial size of the light quark
in the baryon is $r_{Qq}\sim 1/{\Lambda_{\text{QCD}}}$. Furthermore, it can be
deduced that $v$ is small if $m_{Q}$ is heavy enough Bodwin:1994jh . Numerical
calculations confirm the hierarchy by giving $v_{c}^{2}\sim 0.3$ and
$v_{b}^{2}\sim 0.1$ Bodwin:1994jh ; Quigg:1979vr . It indicates that
$m_{b}v_{b}^{2}\sim m_{c}v_{c}^{2}\sim\Lambda_{\text{QCD}}$, so
$r_{QQ}/r_{Qq}\sim\Lambda_{\text{QCD}}/(m_{Q}v)\ll 1$. In conclusion, the two
heavy quarks can be treated as a point-like diquark compared to the baryon
size. This greatly simplifies the structure of a three-quark system to a bound
state of a heavy diquark and a light quark.

Benefitting from the quark-diquark picture, we can formulate the inclusive
decay of a doubly heavy baryon within the heavy diquark effective theory
Brambilla:2005yk ; An:2018cln ; Shi:2020qde . Performing the operator product
expansion, the inclusive $\Xi_{bc}\to H_{cc}+X$ decay rate can be expanded in
inverse powers of the diquark mass $M_{\mathcal{X}}$, with the leading-power
contribution given by the free diquark decay rate,

$\displaystyle\Gamma(\Xi_{bc}\to
H_{cc}+X)=\sum_{f,f^{\prime}}\Gamma(\mathcal{X}_{bc}\to\mathcal{X}_{cc}\bar{f}f^{\prime})+\mathcal{O}\left({1\over
M_{\mathcal{X}}}\right).$ (1)

The fermion pairs $\bar{f}f^{\prime}$ include $\bar{v}_{\ell}\ell^{-}$
($\ell=e,\mu,\tau$) and $\bar{u}d,\bar{u}s,\bar{c}d,\bar{c}s$. The $H_{cc}$
represents all doubly charmed hadrons, including the ground-state baryons
$\Xi_{cc}^{++(+)}$, $\Omega_{cc}^{+}$ and the tetraquarks $T_{cc}$, and their
excited states as well. As the fragmentation rates to strange baryons and to
tetraquarks are much smaller than those to non-strange baryons (see, e.g.,
LHCb:2014ofc ; LHCb:2021qbv ), the fragmentations to $\Omega_{cc}$ and
$T_{cc}$ and their excited states are neglected in the following discussions.
For the non-strange baryons, the excited states eventually decay strongly (or
electromagnetically) into $\Xi_{cc}$. Therefore, all the $\Xi_{bc}\to
H_{cc}+X$ decay processes produce a displaced $\Xi_{cc}$, half $\Xi_{cc}^{++}$
and half $\Xi_{cc}^{+}$ by the isospin symmetry, i.e., $\Gamma(\Xi_{bc}\to
H_{cc}+X)\approx\Gamma(\Xi_{bc}\to\Xi_{cc}+X)\approx
2\Gamma(\Xi_{bc}\to\Xi_{cc}^{++}+X)$.

In the evaluation of
$\Gamma(\mathcal{X}_{bc}\to\mathcal{X}_{cc}\bar{f}f^{\prime})$ induced by the
weak interaction vertices such as
$\bar{c}\gamma^{\mu}P_{L}b\bar{f}\gamma_{\mu}P_{L}f^{\prime}$ with the left-
handed projector $P_{L}\equiv(1-\gamma_{5})/2$, the $\bar{f}f^{\prime}$ part
can be factorized out at the leading order of the strong coupling constant
$\alpha_{s}$, and the key issue in the calculation is the remaining diquark
current
$\langle\mathcal{X}^{i}_{cc}|\bar{c}\gamma^{\mu}P_{L}b|\mathcal{X}^{l}_{bc}\rangle$,
where $i,l$ are color indices. The S-wave diquark $\mathcal{X}_{bc}$ is either
a scalar or axial-vector, but as implied by studies of the beauty-charmed
baryon spectroscopy Ebert:1996ec ; He:2004px ; Yu:2018com ; Weng:2018mmf ;
Li:2019ekr , an axial-vector $\mathcal{X}_{bc}$ state is dominant in the
$\Xi_{bc}$ baryons. On the other hand, $\mathcal{X}_{cc}$ can only be an
axial-vector due to the flavor and spin symmetries. The calculation of the
diquark current is performed in two different kinematic regions, the large
recoil region and the small recoil region. In the former, perturbative
calculation is applicable because typically a hard gluon exchange between the
spectator quark and the weak interacting quarks is required, as displayed in
FIG. 2. In practice, we adopt the non-relativistic QCD (NRQCD) factorization
for this calculation, which were applied to calculate the
$B_{c}\to\eta_{c},J/\psi$ form factors Qiao:2012vt ; Qiao:2012hp ; Zhu:2017lqu
. In the small recoil region, the so-called soft overlap contribution is
dominant and the perturbative QCD expansion is less trustworthy. However, the
heavy quark symmetry determines the form of the diquark current at the zero
recoil point. For the intermediate region, we use a simplified $z$-series
expansion Bourrely:2008za  to perform the interpolation.

The vector and axial-vector diquark currents can be parametrized as

$\displaystyle\langle\mathcal{X}^{i}_{cc}(v,\epsilon)|\bar{c}\gamma^{\mu}b|\mathcal{X}^{l}_{bc}(v^{\prime},\epsilon^{\prime})\rangle$
(2) $\displaystyle=$
$\displaystyle\delta_{il}\sqrt{2M_{cc}M_{bc}}\Big{[}-a_{0}\epsilon^{*}\cdot\epsilon^{\prime}v^{\prime\mu}-a_{1}\epsilon^{*}\cdot\epsilon^{\prime}v^{\mu}$
$\displaystyle+a_{2}\epsilon^{*}\cdot
v^{\prime}\epsilon^{\prime\mu}+a_{3}v\cdot\epsilon^{\prime}\epsilon^{*\mu}\Big{]}\;,$
$\displaystyle\langle\mathcal{X}^{i}_{cc}(v,\epsilon)|\bar{c}\gamma^{\mu}\gamma_{5}b|\mathcal{X}^{l}_{bc}(v^{\prime},\epsilon^{\prime})\rangle$
$\displaystyle=$
$\displaystyle\delta_{il}\sqrt{2M_{cc}M_{bc}}\Big{[}-ib_{0}\epsilon^{\epsilon^{\prime}\epsilon^{*}v^{\prime}\mu}-ib_{1}\epsilon^{\epsilon^{\prime}\epsilon^{*}v\mu}\Big{]}\;,$

where $v^{(\prime)}$, $\epsilon^{(\prime)}$ and $M_{cc(bc)}$ are the
4-velocity, polarization vector and mass of $\mathcal{X}_{cc(bc)}$. The
functions $a_{i}(q^{2})$’s and $b_{i}(q^{2})$’s of the transfer momentum
squared $q^{2}$ are to be determined. At the zero recoil (maximal
$q^{2}=(M_{bc}-M_{cc})^{2}$) point, they can be obtained by taking the heavy
quark limit, and the results read

$\displaystyle a_{0,1,2,3}(q^{2}_{\rm{max}})=b_{0,1}(q^{2}_{\rm{max}})=1\;.$
(3)

It can be derived in the following way. Due to the heavy quark symmetry, the
ground state $QQ^{\prime}$ diquark is represented by a Lorentz bilinear field
(see e.g. (9) of Shi:2020qde )

$\displaystyle\mathcal{D}_{v}^{QQ^{\prime}}(x)={1+\not{v}\over
2}[\gamma^{\mu}A_{\mu}(x)+i\gamma_{5}S(x)]C\;,$ (4)

where $C\equiv i\gamma^{0}\gamma^{2}$, the axial-vector field $A_{\mu}(x)$
annihilates an axial-vector diquark with a polarization vector
$\epsilon_{\mu}$ and the scalar field $S(x)$ annihilates a scalar diquark. All
of the color indices are hidden for convenience. Its Lorentz transformation
property is
$\mathcal{D}_{v}(x)\to\mathcal{D}^{\prime}_{v^{\prime}}(x^{\prime})=D(\Lambda)\mathcal{D}_{v}(\Lambda^{-1}x)D(\Lambda)^{\text{T}}$,
with $\Lambda$ and $D(\Lambda)$ the Lorentz transformation matrices for
Lorentz vectors and Dirac spinors, respectively. The conjugate field of
$\mathcal{D}_{v}$ can be introduced as
$\bar{\mathcal{D}}_{v}=\gamma^{0}\mathcal{D}_{v}^{\dagger}\gamma^{0}$ which
transforms as
$\bar{\mathcal{D}}_{v}(x)\to\bar{\mathcal{D}}_{v^{\prime}}(x^{\prime})=[D(\Lambda)^{-1}]^{\text{T}}\bar{\mathcal{D}}_{v}(\Lambda^{-1}x)D(\Lambda)^{-1}$.
Then, we can match the quark transition currents to the corresponding diquark
transition currents via

$\displaystyle\bar{c}\Gamma
b=\text{tr}\left[L^{\text{T}}\;\bar{\mathcal{D}}^{cQ}_{v^{\prime}}\;\Gamma\mathcal{D}^{bQ}_{v}\;\right]\;,$
(5)

which is determined by the heavy quark spin symmetry and Lorentz covariance.
The $\Gamma$ matrix represents a general $4\times 4$ matrix, and only
$\gamma^{\mu}$ and $\gamma^{\mu}\gamma_{5}$ are involved in this work. The
Lorentz bispinor $L$ only depends on $v$ and $v^{\prime}$. The general
expression for $L$ with the correct parity and time-reversal properties is
$L=L_{0}+L_{1}\not{v}+L_{2}\not{v}^{\prime}+L_{3}\not{v}\not{v}^{\prime}$,
where the coefficients $L_{i}$ are functions of $w\equiv v\cdot v^{\prime}$.
The property $\not{v}\mathcal{D}_{v}=\mathcal{D}_{v}$ together with
$v\cdot\epsilon=v^{\prime}\cdot\epsilon^{\prime}=0$ simplifies the current (5)
such that $L^{\text{T}}$ can be replaced by a scalar function
$\xi(w)=L_{0}+L_{1}+L_{2}+L_{3}$. Evaluating the trace with
$\Gamma=\gamma^{\mu},\gamma^{\mu}\gamma_{5}$ under (5) with a proper
normalization gives the matrix elements

$\displaystyle{1\over\sqrt{2M_{bc}M_{cc}}}\langle\mathcal{X}_{cc}(v,\epsilon)|\bar{c}\gamma^{\mu}b|\mathcal{X}_{bc}(v^{\prime},\epsilon^{\prime})\rangle$
(6) $\displaystyle=$
$\displaystyle\xi(w)\Big{[}-\epsilon^{*}\cdot\epsilon^{\prime}v^{\prime\mu}-\epsilon^{*}\cdot\epsilon^{\prime}v^{\mu}+\epsilon^{*}\cdot
v^{\prime}\epsilon^{\prime\mu}+v\cdot\epsilon^{\prime}\epsilon^{*\mu}\Big{]}\;,$
$\displaystyle{1\over\sqrt{2M_{bc}M_{cc}}}\langle\mathcal{X}_{cc}(v,\epsilon)|\bar{c}\gamma^{\mu}\gamma_{5}b|\mathcal{X}_{bc}(v^{\prime},\epsilon^{\prime})\rangle$
$\displaystyle=$
$\displaystyle\xi(w)\Big{[}-i\epsilon^{\epsilon^{\prime}\epsilon^{*}v^{\prime}\mu}-i\epsilon^{\epsilon^{\prime}\epsilon^{*}v\mu}\Big{]}\;,$

where the symmetry factor $\sqrt{2}$ is due to the identical $c$ quarks in
$\mathcal{X}_{cc}$, and the masses are introduced to make $\xi(w)$
dimensionless. Replacing
$\mathcal{X}_{cc}(v,\epsilon)\to\mathcal{X}_{bc}(v^{\prime},\epsilon^{\prime})$
and $c\to b$, the above vector-current expression leads to
$\langle\mathcal{X}_{bc}(v^{\prime},\epsilon^{\prime})|\bar{b}\gamma^{\mu}b|\mathcal{X}_{bc}(v^{\prime},\epsilon^{\prime})\rangle/M_{bc}=2\xi(1)\;v^{\prime\mu}=2v^{\prime\mu}$.
It determines that $\xi(w)=1$ at the zero recoil $w=1$, leading to the final
result in (2) and (3).

Figure 2: A sample Feynman diagram of the
$\mathcal{X}_{bc}\to\mathcal{X}_{cc}$ diquark transition induced by the $V-A$
current at large recoil. The double lines denote the heavy quarks. The gluon
line close to the weak vertex denotes a hard gluon. The dashed lines denote
any number of soft gluons which can be absorbed into the initial and final
diquark wave functions. The velocity of $\mathcal{X}_{cc(bc)}$ is
$v^{(\prime)}$. The relative momentum of the two heavy quarks in
$\mathcal{X}_{cc(bc)}$ is $k^{(\prime)}$.

In the large-recoil (small-$q^{2}$) region, the diquark currents are induced
by exchanges of hard gluons. At the leading order with one hard gluon
exchange, a sample Feynman diagram is shown in FIG. 2. The hard gluon leads to
the large recoil, while the soft gluon exchanges can be absorbed into the
initial and final diquark wave functions in the NRQCD framework. The NRQCD
calculation formulates the diquark currents as nonperturbative matrix elements
along with the corresponding Wilson coefficients as

$\displaystyle
a_{i}[{\chi}_{bc}(v^{\prime})\to{\chi}_{cc}(v)]=\sum_{jk}\frac{c^{jk}_{i}(\mu)}{m_{b}^{\frac{d_{j}-4}{2}}m_{c}^{\frac{d_{k}-4}{2}}}$
(7) $\displaystyle\qquad\times\left\langle 0\left|{\cal
K}^{\prime}_{j}(\mu)\right|{\chi}_{bc}(v^{\prime})\right\rangle\left\langle{\chi}_{cc}(v)\left|{\cal
K}_{k}(\mu)\right|0\right\rangle\;,$

where the ${\cal K}^{(\prime)}(\mu)$ are all possible independent bilinear
combinations of two component operators which can be power counted by the
velocity $v^{(\prime)}$. The $c^{jk}_{i}(\mu)$ are the short-distance Wilson
coefficients which can be calculated order by order in series of $\alpha_{s}$.
Following an analogous procedure to obtain (4) in Qiao:2012vt , we calculate
the quark-level hard kernel with one hard gluon exchange, convolute it with
the diquark nonperturbative matrix elements, and obtain the leading-order
result

$\displaystyle a_{2,3}(q^{2})$ $\displaystyle=$ $\displaystyle{\alpha_{s}\over
2(1-w)^{2}\sqrt{w}}{N_{c}+1\over N_{c}}{1\over
m_{c}^{3}}R_{bc}(0)R_{cc}^{*}(0)\;,$ $\displaystyle a_{0}(q^{2})$
$\displaystyle=$ $\displaystyle b_{0}(q^{2})=\bar{\xi}_{2}a_{2,3}(q^{2})\;,$
$\displaystyle a_{1}(q^{2})$ $\displaystyle=$ $\displaystyle
b_{1}(q^{2})=\bar{\xi}_{1}a_{2,3}(q^{2})\;,$ (8)

where $\bar{\xi}_{1}\equiv m_{b}/M_{bc}$, $\bar{\xi}_{2}\equiv m_{c}/M_{cc}$,
and the number of colors $N_{c}=3$. The diquark wave functions at the origin
are defined through the nonperturbative matrix elements

$\displaystyle\varepsilon_{ijk}\frac{\langle
0|\psi_{c,i}^{\text{T}}i\sigma_{2}{\vec{\sigma}}\psi_{b,j}|{\cal
X}_{bc}^{k}(\vec{\epsilon})\rangle}{\sqrt{2M_{bc}}}$ $\displaystyle=$
$\displaystyle N_{c}!\;{R_{bc}(0)\over\sqrt{4\pi}}\vec{\epsilon}\;,$
$\displaystyle\varepsilon_{ijk}\frac{\langle{\cal
X}_{cc}^{k}(\vec{\epsilon})|\psi_{c,i}^{\dagger}i{\vec{\sigma}}\sigma_{2}\psi^{*}_{c,j}|0\rangle}{\sqrt{4M_{cc}}}$
$\displaystyle=$ $\displaystyle
N_{c}!\;{R_{cc}^{*}(0)\over\sqrt{4\pi}}\vec{\epsilon}^{*}\;,$ (9)

where $\psi$’s are two-component spinor fields, $\sigma$’s are Pauli matrices
and $\varepsilon_{ijk}$ is the Levi-Civita symbol in the color space. In
principle, both the next-to-leading order $\alpha_{s}$ corrections and the
subleading power corrections to the diquark transition amplitudes can be
calculated as the calculation for $B_{c}\to J/\psi$ in Qiao:2012hp ;
Zhu:2017lqu . We leave these calculations to future works. To obtain the
numerical result, it requires the input of the diquark wave functions at the
origin $R_{bc,cc}(0)$. They are obtained by solving the nonrelativistic
Schrödinger equations, with the potential $V(r)=-{2\over
3}{\alpha_{s}(\nu_{\text{lat}})\over r}+{c_{2}r+c_{1}\over c_{3}r+1}+\sigma r$
Soto:2020pfa  with $\nu_{\text{lat}}=2.16\ \rm{GeV}\;,\
\sigma=0.21\rm{GeV}^{2},c_{1}=1.948\ \rm{GeV}\;,\ c_{2}=15.782\ \rm{GeV}\;,\
c_{3}=9.580\ \rm{GeV}$, which were fitted from the lattice calculation
Najjar:2009da ; Luscher:2002qv . The quark masses take values of
$m_{c}=1.392(11)$ GeV and $m_{b}=4.749(18)$ GeV. The ground-state solutions to
the Schrödinger equations give ${R}_{cc}(0)=(0.66\pm 0.06)\ \text{GeV}^{3/2}$
and ${R}_{bc}(0)=(0.87\pm 0.09)\ \text{GeV}^{3/2}$. The uncertainties were
estimated from the differences between the results obtained with the above
lattice potentials and the Cornell potentials Bali:2000gf , though the real
uncertainties could be larger.

To interpolate the diquark current in the whole range from the above results
in the small- and large-recoil regions, a simplified $z$-series expansion
Bourrely:2008za  is adopted with the formulation

$\displaystyle
f(q^{2})={f(0)/[1-{q^{2}/m_{B_{c}}^{2}}]}\left[1+b\zeta(q^{2})+c\zeta^{2}(q^{2})\right],$
(10)

for $a_{0}(=b_{0})$, $a_{1}(=b_{1})$ and $a_{2}(=a_{3})$, where
$\zeta(q^{2})=z(q^{2})-z(0)$,
$z(q^{2})=(\sqrt{t_{+}-q^{2}}-\sqrt{t_{+}-t_{0}})/(\sqrt{t_{+}-q^{2}}+\sqrt{t_{+}-t_{0}})$,
$t_{\pm}=(M_{bc}\pm M_{cc})^{2}$, $t_{0}=t_{+}(1-\sqrt{1-t_{-}/t_{+}})$ and
the free parameters $f(0)$, $b$, and $c$ are to be determined. The value of
$t_{0}$ is chosen to minimize $|z|$ to improve the convergence. Unlike Ref.
Bourrely:2008za , we use $\zeta(q^{2})$ instead of $z(q^{2})$ for the
expansion, though these two parameterizations are equivalent. The expansion in
$\zeta(q^{2})$ ensures that $f(0)$ is exactly the value of the form factor at
$q^{2}=0$. Fitting the points of $q^{2}=0,0.1,0.2,0.3,0.4$ GeV2 and $q_{\rm
max}^{2}\approx 11.3$ GeV2, the parameters are extracted as:
$f(0)=0.247,b=-58.6,c=238.2~{}\text{for}\
a_{2}\;;f(0)=0.124,b=-52.9,c=1898.2~{}\text{for}\
a_{0}\;;f(0)=0.191,b=-57.0,c=388.0~{}\text{for}\ a_{1}$. The corresponding
results are plotted in FIG. 3, with the uncertainties transferred from the
diquark wave functions at origin.

Figure 3: The central values (solid curves) and the $\pm 1\sigma$
uncertainties (shadow areas) of the numerical results for $a_{0,1,2}(q^{2})$
appearing in the diquark currents.

Finally, with the numerical results for the diquark currents, the inclusive
$\Xi_{bc}\to H_{cc}+X$ decay rate (1) can be calculated. The leading power
free diquark decay rates were calculated by phase space integration of the
amplitude squares. For example, for the electron channel contribution
$\Gamma(\mathcal{X}_{bc}\to\mathcal{X}_{cc}e^{-}\bar{\nu}_{e})$, the amplitude
is given by the product of
${4G_{F}/\sqrt{2}}V_{cb}\bar{u}(p_{e})\gamma_{\mu}P_{L}v(p_{\nu})$ and the
diquark current
$\langle\mathcal{X}_{cc}|\bar{c}\gamma^{\mu}P_{L}b|\mathcal{X}_{bc}\rangle$,
where $G_{F}$ is the Fermi constant and $V_{qq^{\prime}}$ is the corresponding
Cabbibo-Kobayashi-Maskawa (CKM) matrix element. The calculation is similar for
the other leptonic channels; for the hadronic channels, the replacement
$|V_{cb}|^{2}\to|V_{cb}V^{*}_{UD}|^{2}(3C_{1}^{2}+2C_{1}C_{2}+C_{2}^{2})$
should be performed at the level of amplitude squares with $U=u,c$ and
$D=d,s$, where the Wilson coefficients are defined in Ref. Lu:2000em . Summing
over the contributions from all possible channels with
$\bar{f}f^{\prime}=e\bar{\nu},\mu\bar{\nu},\tau\bar{\nu},\bar{u}d,\bar{u}s,\bar{c}d,\bar{c}s$,
the numerical result for the inclusive decay rate reads

$\displaystyle\Gamma(\Xi_{bc}\to H_{cc}+X)$ (11) $\displaystyle=$
$\displaystyle(1.9\pm 0.1\pm 0.3\pm 0.4)\times 10^{-13}\ \text{GeV}\;.$

Most numerical inputs have been given previously, except that the Wilson
coefficients took values from Ref. Lu:2000em  and the Fermi constant and the
CKM matrix elements took values from Ref. ParticleDataGroup:2020ssz . The
uncertainties in order are from the quark mass variation, the diquark wave
functions at the origin, and the scale dependence, respectively. The former
two were obtained by varying the values of the quark masses and the diquark
wave functions at the origin as listed below (Inclusive approach to hunt for
the beauty-charmed baryons $\Xi_{bc}$). As for the scale, we chose $\mu=m_{b}$
for the central value calculation, and doubled and halved it for the
uncertainty estimation. In addition, one would expect more uncertainties
induced by unknown power corrections. The dominant $v^{2}$ corrections are
expected to potentially modify the result by $\sim 30\%$ Zhu:2017lqu . The
decay rate translates to the branching ratios as

$\displaystyle\mathcal{B}(\Xi^{+(0)}_{bc}\to H_{cc}+X)\approx 14\%\;(3\%)\;,$
(12)

where we have taken $\tau(\Xi_{bc}^{+(0)})\approx 508\;(105)$ fs Cheng:2019sxr
. As analyzed before, the $\Xi^{+,0}_{bc}\to\Xi_{cc}^{++}+X$ branching ratio
is approximately 1/2 of $\mathcal{B}(\Xi^{+,0}_{bc}\to H_{cc}+X)$.

Phenomenology. — Based on the $\Xi_{bc}\to\Xi_{cc}^{++}+X$ branching ratio
calculated above, as well the information on $\Xi_{bc}$ production and the
$\Xi_{cc}^{++}$ detection efficiency, the number of signal events containing a
displaced $\Xi_{cc}^{++}$ can be estimated.

In practice, the inclusive approach to search for $\Xi_{bc}$ depends crucially
on the lifetimes of the $\Xi_{bc}$ baryons. Only if they fly far enough from
the collision vertices before decaying, the displacement of the decay products
$\Xi_{cc}^{++}$’s can be clearly distinguished. According to the study of
$\Xi_{bb}\to B_{c}^{-}+X$ Gershon:2018gda , with the vertex resolution of the
LHCb detector, the $\Xi_{bb}$ particles with lifetimes above about 500 fs can
lead to displaced $B_{c}$’s with significantly higher IP values than those of
the prompt $B_{c}$’s. In contrast, if their lifetimes are much below 500 fs,
the IP values will hardly help separate their decaying $B_{c}$’s from the
prompt ones. Therefore, we will focus on the $\Xi_{bc}^{+}$, which is expected
to have a sufficiently long lifetime Cheng:2019sxr .

The $\Xi_{bc}$ production cross section at the LHC has been theoretically
evaluated in Refs. Zhang:2011hi ; Ali:2018xfq . To reduce systematic
uncertainties, instead of the direct result for the cross section we adopt the
cross section ratio $\sigma(\Xi_{bc})/\sigma(\Xi_{cc})\approx 40\%$
Zhang:2011hi . The signal is determined by a displaced $\Xi_{cc}^{++}$, so its
detection efficiency is expected to be identical to that of a normal
$\Xi_{cc}^{++}$, $\epsilon(\Xi_{cc}^{++})$. With these inputs, the expected
signal yield $N_{s}$ is expressed as

$\displaystyle N_{s}$ $\displaystyle=$ $\displaystyle
N_{p}(\Xi_{bc}^{+})\cdot\mathcal{B}(\Xi_{bc}\to\Xi_{cc}^{++}+X)\cdot\epsilon(\Xi_{cc}^{++})$
(13) $\displaystyle=$ $\displaystyle
N_{d}(\Xi_{cc}^{++})\cdot{\sigma(\Xi_{bc})\over\sigma(\Xi_{cc})}\cdot\mathcal{B}(\Xi_{bc}^{+}\to\Xi_{cc}^{++}+X),$

where $N_{p,d}$ are the number of produced and detected particles.
Quantitatively, it is expected that LHCb Run 3 will collect approximately
$10^{4}$ $\Xi_{cc}^{++}$’s through the $\Lambda_{c}^{+}K^{-}\pi^{+}\pi^{+}$
LHCb:2019epo  and $\Xi_{c}^{+}\pi^{+}$ LHCb:2018pcs  reconstruction. Combining
the inclusive decay branching ratio (12) and the $\Xi_{bc}$ production
information $\sigma(\Xi_{bc})/\sigma(\Xi_{cc})\approx$ 40% Zhang:2011hi , one
finally arrives at the signal yield at the LHCb Run 3, $N_{s}\approx$ 300\. In
a real measurement, some of these events will get swamped by the background of
the primarily produced $\Xi_{cc}^{++}$’s, if, for example, the
$\Xi_{bc}^{+}$’s do not fly far enough before decaying into $\Xi_{cc}^{++}$’s.
Although it will lose some efficiencies, the $\Xi_{bc}$ discovery will still
be hopeful during LHCb Run 3, and will be very promising during LHCb Run 4 and
at the high-luminosity LHC.

As for the background, a displaced $\Xi_{cc}^{++}$ is also possibly produced
from $B_{c}^{+}$ decays, $B_{c}^{+}\to\Xi_{cc}^{++}+X$. However, such
background is negligible because the branching ratio is expected to be tiny
due to the phase-space suppression. The dominant quark-level transition for
such decays is $\bar{b}\to c\bar{c}\bar{s}$, so the least massive final state
is $\Xi_{cc}^{++}\bar{\Xi}_{c}^{-}$, with $\sim$0.18 GeV phase space.
Considering a similar decay channel $B\to\Lambda_{c}+\bar{\Xi}_{c}$ with
$\sim$0.5 GeV phase space having an $\mathcal{O}(10^{-3})$ branching ratio
ParticleDataGroup:2020ssz , the $B_{c}^{+}\to\Xi_{cc}^{++}\bar{\Xi}_{c}^{-}$
branching ratio is expected to be even smaller. It also allows decay processes
with some other final states such as $\Xi_{cc}\bar{\Xi}_{c}\pi$ and
$\Xi_{cc}\bar{\Xi}_{c}^{*}$, but all of them are expected to have similar or
smaller branching ratios due to even smaller phase spaces compared to
$\Xi_{cc}^{++}\bar{\Xi}_{c}^{-}$. As $B_{c}$ and $\Xi_{bc}$ have production
cross sections of the same order at the LHC Ali:2018xfq , the number of the
displaced $\Xi_{cc}^{++}$’s produced via $B_{c}$ decays is smaller than that
of the signal by at least 1 to 2 orders of magnitude. Therefore, this
background source can be safely neglected.

With the above analyses/calculations about aspects of the experimental
$\Xi_{bc}$ search, we can conclude that it will be very hopeful to discover
$\Xi_{bc}$ during LHCb Run 3 via the inclusive approach that we proposed. The
inclusive approach should be more efficient than searches using exclusive
decays. The exclusive channels induced by $b$ quark decaying typically have
branching ratios smaller than $\mathcal{B}(\Xi_{bc}\to\Xi_{cc}^{++}+X)$ by
more than 1 or 2 orders of magnitude Wang:2017mqp . The $bc$ annihilation
channels are power suppressed and are thus even rarer. The $c$ quark decay
channels suffer low reconstruction efficiencies of the $b$-hadrons in their
final states Han:2021gkl .

Conclusion. — We have proposed that the inclusive $\Xi_{bc}$ decay channel
—or, more explicitly, $\Xi_{bc}^{+}\to\Xi_{cc}^{++}+X$—can be used to search
for the $\Xi_{bc}$ baryons, with a very clean and simple signal: a displaced
$\Xi_{cc}^{++}$. By making use of effective theories of QCD, we have
calculated its branching ratio at the leading order and found that it is
approximately 7%, while the radiative and power corrections are left for
future studies. Based on the result for the $\Xi_{bc}^{+}\to\Xi_{cc}^{++}+X$
branching ratio, the $\Xi_{bc}$ production rate, and the $\Xi_{cc}^{++}$
detection efficiency extracted from previous experiments, we estimated that
LHCb Run 3 can accumulate approximately 300 such signal events. The possible
background, the $B_{c}^{+}\to\Xi_{cc}^{++}+X$ decay, has been demonstrated to
be negligible. In conclusion, the inclusive $\Xi_{bc}^{+}\to\Xi_{cc}^{++}+X$
decay is very likely to serve as the discovery channel for the $\Xi_{bc}$
baryons.

Acknowledgement. — The authors are grateful to Ji-Bo He, Xiang-Peng Wang and
Yan-Xi Zhang for useful discussions on the experimental search at the LHC and
the theoretical framework. This work is supported by Natural Science
Foundation of China under Grants No. 12005068, 11735010, U2032102, 11975112,
12075124 and 12125503, and by the DFG and the NSFC through funds provided to
the Sino-German Collaborative Research Center TRR110 “Symmetries and the
Emergence of Structure in QCD” (NSFC Grant No. 12070131001, DFG Project-ID
196253076).

## References

  * (1) R. Aaij et al. [LHCb], Phys. Rev. Lett. 119, no.11, 112001 (2017) [arXiv:1707.01621 [hep-ex]].
  * (2) M. Karliner and J. L. Rosner, Phys. Rev. Lett. 119, no.20, 202001 (2017) [arXiv:1707.07666 [hep-ph]].
  * (3) E. J. Eichten and C. Quigg, Phys. Rev. Lett. 119, no.20, 202002 (2017) [arXiv:1707.09575 [hep-ph]].
  * (4) Y. R. Liu, H. X. Chen, W. Chen, X. Liu and S. L. Zhu, Prog. Part. Nucl. Phys. 107, 237-320 (2019) [arXiv:1903.11976 [hep-ph]].
  * (5) R. Aaij et al. [LHCb], [arXiv:2109.01038 [hep-ex]].
  * (6) R. Aaij et al. [LHCb], [arXiv:2109.01056 [hep-ex]].
  * (7) R. Aaij et al. [LHCb], JHEP 11, 095 (2020) [arXiv:2009.02481 [hep-ex]].
  * (8) R. Aaij et al. [LHCb], Chin. Phys. C 45, no.9, 093002 (2021) [arXiv:2104.04759 [hep-ex]].
  * (9) W. Wang, F. S. Yu and Z. X. Zhao, Eur. Phys. J. C 77, no.11, 781 (2017) [arXiv:1707.02834 [hep-ph]].
  * (10) V. V. Kiselev and A. K. Likhoded, Phys. Usp. 45, 455-506 (2002) [arXiv:hep-ph/0103169 [hep-ph]].
  * (11) J. J. Han, R. X. Zhang, H. Y. Jiang, Z. J. Xiao and F. S. Yu, Eur. Phys. J. C 81, no.6, 539 (2021) [arXiv:2102.00961 [hep-ph]].
  * (12) R. Aaij et al. [LHCb], Phys. Lett. B 694, 209-216 (2010) [arXiv:1009.2731 [hep-ex]].
  * (13) T. Gershon and A. Poluektov, JHEP 01, 019 (2019) [arXiv:1810.06657 [hep-ph]].
  * (14) H. Y. Cheng and F. Xu, Phys. Rev. D 99, no.7, 073006 (2019) [arXiv:1903.08148 [hep-ph]].
  * (15) J. W. Zhang, X. G. Wu, T. Zhong, Y. Yu and Z. Y. Fang, Phys. Rev. D 83, 034026 (2011) [arXiv:1101.1130 [hep-ph]].
  * (16) A. Ali, Q. Qin and W. Wang, Phys. Lett. B 785, 605-609 (2018) [arXiv:1806.09288 [hep-ph]].
  * (17) N. Brambilla, A. Vairo and T. Rosch, Phys. Rev. D 72, 034021 (2005) [arXiv:hep-ph/0506065 [hep-ph]].
  * (18) H. An and M. B. Wise, Phys. Lett. B 788, 131-136 (2019) [arXiv:1809.02139 [hep-ph]].
  * (19) Y. J. Shi, W. Wang, Z. X. Zhao and U. G. Meißner, Eur. Phys. J. C 80, no.5, 398 (2020) [arXiv:2002.02785 [hep-ph]].
  * (20) S. J. Brodsky, F. K. Guo, C. Hanhart and U. G. Meissner, Phys. Lett. B 698, 251-255 (2011) [arXiv:1101.1983 [hep-ph]].
  * (21) M. J. Yan, X. H. Liu, S. Gonzàlez-Solís, F. K. Guo, C. Hanhart, U. G. Meißner and B. S. Zou, Phys. Rev. D 98, no.9, 091502 (2018) [arXiv:1805.10972 [hep-ph]].
  * (22) J. Hu and T. Mehen, Phys. Rev. D 73, 054003 (2006) [arXiv:hep-ph/0511321 [hep-ph]].
  * (23) G. T. Bodwin, E. Braaten and G. P. Lepage, Phys. Rev. D 51, 1125-1171 (1995) [erratum: Phys. Rev. D 55, 5853 (1997)] [arXiv:hep-ph/9407339 [hep-ph]].
  * (24) C. Quigg and J. L. Rosner, Phys. Rept. 56, 167-235 (1979)
  * (25) R. Aaij et al. [LHCb], JHEP 08, 143 (2014) [arXiv:1405.6842 [hep-ex]].
  * (26) R. Aaij et al. [LHCb], Phys. Rev. D 104, no.3, 032005 (2021) [arXiv:2103.06810 [hep-ex]].
  * (27) D. Ebert, R. N. Faustov, V. O. Galkin, A. P. Martynenko and V. A. Saleev, Z. Phys. C 76, 111-115 (1997) [arXiv:hep-ph/9607314 [hep-ph]].
  * (28) D. H. He, K. Qian, Y. B. Ding, X. Q. Li and P. N. Shen, Phys. Rev. D 70, 094004 (2004) [arXiv:hep-ph/0403301 [hep-ph]].
  * (29) Q. X. Yu and X. H. Guo, Nucl. Phys. B 947, 114727 (2019) [arXiv:1810.00437 [hep-ph]].
  * (30) X. Z. Weng, X. L. Chen and W. Z. Deng, Phys. Rev. D 97, no.5, 054008 (2018) [arXiv:1801.08644 [hep-ph]].
  * (31) Q. Li, C. H. Chang, S. X. Qin and G. L. Wang, Chin. Phys. C 44, no.1, 013102 (2020) [arXiv:1903.02282 [hep-ph]].
  * (32) C. F. Qiao and R. L. Zhu, Phys. Rev. D 87, no.1, 014009 (2013) [arXiv:1208.5916 [hep-ph]].
  * (33) C. F. Qiao, P. Sun, D. Yang and R. L. Zhu, Phys. Rev. D 89, no.3, 034008 (2014) [arXiv:1209.5859 [hep-ph]].
  * (34) R. Zhu, Y. Ma, X. L. Han and Z. J. Xiao, Phys. Rev. D 95, no.9, 094012 (2017) [arXiv:1703.03875 [hep-ph]].
  * (35) C. Bourrely, I. Caprini and L. Lellouch, Phys. Rev. D 79, 013008 (2009) [erratum: Phys. Rev. D 82, 099902 (2010)] [arXiv:0807.2722 [hep-ph]].
  * (36) J. Soto and J. Tarrús Castellà, Phys. Rev. D 102, no.1, 014013 (2020) [erratum: Phys. Rev. D 104, no.5, 059901 (2021)] [arXiv:2005.00551 [hep-ph]].
  * (37) J. Najjar and G. Bali, PoS LAT2009, 089 (2009) [arXiv:0910.2824 [hep-lat]].
  * (38) M. Luscher and P. Weisz, JHEP 07, 049 (2002) [arXiv:hep-lat/0207003 [hep-lat]].
  * (39) G. S. Bali, Phys. Rept. 343, 1-136 (2001) [arXiv:hep-ph/0001312 [hep-ph]].
  * (40) C. D. Lu, K. Ukai and M. Z. Yang, Phys. Rev. D 63, 074009 (2001) [arXiv:hep-ph/0004213 [hep-ph]].
  * (41) P. A. Zyla et al. [Particle Data Group], PTEP 2020, no.8, 083C01 (2020)
  * (42) R. Aaij et al. [LHCb], JHEP 02, 049 (2020) [arXiv:1911.08594 [hep-ex]].
  * (43) R. Aaij et al. [LHCb], Phys. Rev. Lett. 121, no.16, 162002 (2018) [arXiv:1807.01919 [hep-ex]].