# Inclusive weak-annihilation decays and lifetimes of beauty-charmed baryons

Guo-He<EMAIL_ADDRESS>En-Pei<EMAIL_ADDRESS>Corresponding author, Qin Qin1333<EMAIL_ADDRESS>Corresponding author ,
Kang-Kang<EMAIL_ADDRESS>1School of Physics, Huazhong University
of Science and Technology, Wuhan 430074, China
2School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000,
China

###### Abstract

Imbalanced beauty-charmed baryons $\Xi_{bc}^{+,0}$ are of great significance
to the development of heavy flavor physics. In this work, we study the
inclusive weak-annihilation decays of $\Xi_{bc}^{+,0}$ and their contributions
to the $\Xi_{bc}^{+,0}$ lifetimes. For the calculation of the inclusive
$\Xi_{bc}^{+,0}\to X_{cs}$ decay width where $X_{cs}$ stands for the sum of
the final states with charm number +1 and strange number -1, we work in the
heavy diquark effective theory which provides us with a convenient technical
tool to construct the operator product expansion. The $\Xi_{bc}^{+,0}$ is
considered to be a superposition of two states with one containing a spin-0
$bc$ diquark and the other one containing a spin-1 $bc$ diquark. It is found
that both the $\Xi_{bc}^{+,0}$ lifetimes and the $\Xi_{bc}^{+,0}\to X_{cs}$
branching ratios are very sensitive to the $bc$ spin in $\Xi_{bc}^{+,0}$. The
$\Xi_{bc}^{+,0}\to X_{cd}$ results are also presented. As $\Xi_{bc}^{+}$ has a
longer lifetime than $\Xi_{bc}^{0}$ and bigger branching ratios of similar
decay channels, the exclusive decays $\Xi_{bc}^{+}\to D^{(*)+}\Lambda$,
$\Xi_{bc}^{+}\to\Lambda_{c}^{+}\bar{K}^{(*)0}$, $\Xi_{bc}^{+}\to
D^{(*)+}K^{-}p$, and $\Xi_{bc}^{+}\to D^{0}\bar{K}^{(*)0}p$ are more promising
for experimental searches of $\Xi_{bc}^{+,0}$ at the LHC comparing with
exclusive decay $\Xi_{bc}^{0}\to D^{0}pK^{-}$.

###### pacs:

13.30.-a;14.20.Mr;12.39.Hg;12.39.St

##  I Introduction

Doubly heavy hadrons provide a new platform to decipher the strong
interaction. The discovery of the first doubly charmed baryon $\Xi_{cc}^{++}$
LHCb:2017iph  has motivated many further studies. The LHCb collaboration has
precisely measured its lifetime LHCb:2018zpl , mass LHCb:2019epo  and
production LHCb:2019qed . Besides the discovery channel proposed by Yu:2017zst
, $\Xi_{cc}^{++}$ has also been searched via other channels LHCb:2018pcs ;
LHCb:2019ybf ; LHCb:2022rpd . Theoretically, the $\Xi_{cc}^{++}$ decay
properties have been studied extensively by e.g. Wang:2017mqp ; Hu:2017dzi ;
Zhao:2018mrg ; Xing:2018lre ; Shi:2019hbf ; Shi:2019fph ; Xing:2021enr .
Experimental efforts have also been put into searching for the flavor SU(3)
partners of $\Xi_{cc}^{++}$ LHCb:2021rkb ; LHCb:2021eaf , but none of them has
been discovered yet. On the other hand, the experimental discovery potential
of doubly charmed tetraquarks has been theoretically studied in Qin:2020zlg ,
and thereafter the first of them, $T_{cc}^{+}$, was discovered at the LHCb
LHCb:2021vvq ; LHCb:2021auc .

Unlike doubly charmed hadrons, the beauty-charmed baryons $\Xi_{bc}^{+,0}$
contains an imbalanced heavy quark pair, resulting in diverse features and
deserving special attention. Experimental searches for the beauty-charmed
baryons have been performed via the exclusive channels $\Xi_{bc}^{0}\to
D^{0}pK^{-}$ LHCb:2020iko  and $\Xi_{bc}^{0}\to\Xi_{c}^{+}\pi^{-}$
LHCb:2021xba  however no significant signal was found. Very recently, the LHCb
observed two peaking structures using $\Xi_{bc}^{+}\to J/\psi\Xi_{c}^{+}$
LHCb:2022fbu  with a local (global) significance of 4.3 (2.8) and 4.1 (2.4)
standard deviations at masses of 6571 MeV and 6694 MeV, respectively, which
might be very close to the discovery. Besides, it was proposed by Qin:2021wyh
that the inclusive decay $\Xi_{bc}\to\Xi_{cc}^{++}+X$ may serve as the
discovery channel of $\Xi_{bc}^{+,0}$ by making use of the displacement
information of $\Xi_{cc}^{++}$, since plenty of the $\Xi_{bc}$ baryons will be
produced at the LHC Ali:2018ifm ; Ali:2018xfq .

We study in this work the inclusive weak-annihilation decays of
$\Xi_{bc}^{+,0}$, especially $\Xi_{bc}^{+,0}\to X_{cs}$, where $X_{cs}$ stands
for the sum of the final states with charm number +1 and strange number -1.
One example exclusive channel contributing to this inclusive decay is
$\Xi_{bc}^{0}\to D^{0}pK^{-}$, which has been used for $\Xi_{bc}^{0}$ search
LHCb:2020iko . The inclusive decay rate can help evaluate whether the
corresponding exclusive channels such as $\Xi_{bc}^{0}\to D^{0}pK^{-}$ and
$\Xi_{bc}^{+}\to D^{(*)+}\Lambda$,
$\Xi_{bc}^{+}\to\Lambda_{c}^{+}\bar{K}^{(*)0}$, $\Xi_{bc}^{+}\to
D^{(*)+}K^{-}p$, and $\Xi_{bc}^{+}\to D^{0}\bar{K}^{(*)0}p$ are potential
discovery channels of the beauty-charmed baryons. From the theoretical point
of view, this process has a simple structure that the heavy diquark part can
be factorized out at the lowest order of the strong coupling $\alpha_{s}$.
Therefore, the calculation with lower model dependencies can make the
inclusive decay $\Xi_{bc}^{+,0}\to X_{cs}$ as a test of the heavy diquark
effective theory (HDET) Shi:2020qde ; Qin:2021wyh .

Previously, the inclusive $\Xi_{bc}^{+,0}\to X_{cs}$ decay has been studied in
Kiselev:1999kh ; Kiselev:2001fw  where the lifetimes of $\Xi_{bc}^{+,0}$ were
investigated under the assumption that the beauty and charm quarks form a
scalar $bc$ diquark inside $\Xi_{bc}^{+,0}$. On the other hand, the $bc$
diquark was treated as an axial-vector state in Cheng:2019sxr  and the results
for both the $\Xi_{bc}^{+,0}\to X_{cs}$ decay widths and the $\Xi_{bc}^{+,0}$
lifetimes are very different. We take into account the possibility that the
$bc$ diquark in $\Xi_{bc}^{+,0}$ is a superposition of the scalar and axial-
vector states, $\mathcal{S}_{bc}$ and $\mathcal{X}_{bc}$ Roberts:2008wq . It
confirms that the scalar and axial-vector contributions are extremely
different, so the $\Xi_{bc}^{+,0}$ lifetimes and the $\Xi_{bc}^{+,0}\to
X_{cs}$ decay rates can be used to determine the diquark constituent. This
study also improves the calculation in the following aspects: (i) We revisit
the inclusive channel more systematically in the heavy diquark effective
theory Shi:2020qde ; Qin:2021wyh , including power corrections of
$1/m_{bc}^{n}$, where $m_{bc}$ is the $bc$ diquark mass. (ii) We consider the
contribution from the QCD penguin operators. The inclusive $\Xi_{bc}^{+,0}\to
X_{cd}$ results are obtained by the corresponding Cabbibo-Kobayashi-Maskawa
(CKM) matrix element replacement.

The rest of the paper is organized as follows. In section 2, we introduce the
HDET and perform the matching for involved operators. In section 3, the
inclusive decay rate is formulated in the framework of operator product
expansion, and it turns out that the result is restricted by the
reparametrization invariance. The numerical results and relevant
phenomenological discussions are given in section 4. We conclude with section
5.

##  II Heavy Diquark Effective Theory

The $\Xi_{bc}$, as a doubly heavy baryon, has a heavy diquark - light quark
structure, like a “double-star” core surrounded by a light “planet”. The
effective distance between the two heavy quarks is much smaller than that
between any one of them and the light quark, so the two heavy quarks can be
regarded as a point-like heavy diquark if the physics above the QCD scale
$\Lambda_{\rm{QCD}}$ is integrated out. (See Yin:2019bxe ; Yin:2021uom  for a
different description.) In the heavy diquark limit as its mass
$m_{QQ^{\prime}}\to\infty$, the heavy diquark system can be treated as a
static $\bar{3}$ color source, playing the same role as a heavy anti-quark in
a heavy meson Georgi:1990ak .

The spin of the heavy diquark $J_{bc}$ is a good quantum number in the heavy
diquark limit. Therefore, we can use $J_{bc}$ to label different diquark
states, spin-0 scalar $\mathcal{S}_{bc}$ and spin-1 axial-vector
$\mathcal{X}_{bc}$. The ground state $\Xi_{bc}$ is spin-1/2, so it can be
composed of either $\mathcal{S}_{bc}$ or $\mathcal{X}_{bc}$ with light degrees
of freedom $q$ with $J_{q}=1/2$. In principle, $\Xi_{bc}$ is a superposition
of the two states $\Xi_{\mathcal{S}}(\mathcal{S}_{bc}q)$ and
$\Xi_{\mathcal{X}}(\mathcal{X}_{bc}q)$, formulated as

$\displaystyle\left|\Xi_{bc}^{+,0}\right\rangle=\cos\theta\left|\Xi_{\mathcal{X}}\right\rangle+\sin\theta
e^{i\phi}\left|\Xi_{\mathcal{S}}\right\rangle,$ (1)

where $\theta$ is the mixing angle and $\phi$ is the relative phase.

Both the scalar $\mathcal{S}_{bc}$ and axial-vector $\mathcal{X}_{bc}$ can be
described systematically in the HDET, with the Lagrangian given by Shi:2020qde

$\displaystyle\mathcal{L}_{\mathcal{S}}$ $\displaystyle=$ $\displaystyle
i\mathcal{S}_{v}^{\dagger}v\cdot
D\mathcal{S}_{v}-\frac{1}{2m_{\mathcal{S}}}\mathcal{S}_{v}^{\dagger}D^{2}\mathcal{S}_{v}+\mathcal{O}(\frac{1}{m_{\mathcal{S}}^{2}})\;,$
$\displaystyle\mathcal{L}_{\mathcal{X}}$ $\displaystyle=$
$\displaystyle-i\mathcal{X}_{v\mu}^{\dagger}v\cdot
D\mathcal{X}_{v}^{\mu}+\frac{1}{2m_{\mathcal{X}}}\mathcal{X}_{v\mu}^{\dagger}D^{2}\mathcal{X}_{v}^{\mu}$
(2)
$\displaystyle+\frac{ig}{2m_{\mathcal{X}}}\mathcal{X}_{v\mu}^{\dagger}\bar{G}^{\mu\nu}\mathcal{X}_{v\nu}+\mathcal{O}(\frac{1}{m_{\mathcal{X}}^{2}})\;,$

where $D_{\mu}=\partial_{\mu}-igA_{\mu}^{a}\overline{t^{a}}$, $g$ is the
effective coupling constant between the diquark and the gluon,
$\overline{t^{a}}$ is the $\bar{3}$ generator of the color $SU(3)$ group, and
the gluon tensor is defined by
$\bar{G}_{\mu\nu}=\frac{-i}{g}[D_{\mu},D_{\nu}]=G_{\mu\nu}^{a}\overline{t^{a}}$.
The $v$-subscripted scalar and axial-vector fields are related to the original
fields by the definition

$\displaystyle\mathcal{S}(x)$ $\displaystyle=$
$\displaystyle\exp[-im_{\mathcal{S}}v\cdot
x]\mathcal{S}_{v}(x)/\sqrt{2m_{\mathcal{S}}}\;,$
$\displaystyle\mathcal{X}^{\mu}(x)$ $\displaystyle=$
$\displaystyle\exp[-im_{\mathcal{X}}v\cdot
x](\mathcal{X}_{v}^{\mu}(x)+Y_{v}^{\mu}(x))/\sqrt{2m_{\mathcal{X}}}\;,\ \ \ $
(3)

with the large momentum $mv$ subtracted, where the reference velocity $v$ is
often chosen to be the baryon velocity. The static part
$\mathcal{X}_{v}^{\mu}$ satisfies $v\cdot\mathcal{X}_{v}=0$ and the residual
part $Y_{v}^{\mu}$ is suppressed by powers of
$\Lambda_{\rm{QCD}}/m_{\mathcal{X}}$. The leading terms in (II) have the heavy
diquark spin-flavor symmetry. Compared to Shi:2020qde , the fields are
different by a factor of mass square root to make the heavy diquark flavor
symmetry manifest.

The effective Hamiltonian involved in the $bc\to cs$ processes is
Buchalla:1995vs

$\displaystyle\mathcal{H}_{\text{eff}}=\frac{4G_{F}}{\sqrt{2}}V_{cs}^{*}V_{cb}\sum_{i=1}^{6}C_{i}O_{i}$
(4) $\displaystyle
O_{1}=\bar{c}_{\lambda}\gamma^{\mu}P_{L}b_{\rho}\bar{s}_{\rho}\gamma_{\mu}P_{L}c_{\lambda}\;,\
O_{2}=\bar{c}\gamma^{\mu}P_{L}b\;\bar{s}\gamma_{\mu}P_{L}c\;,$ $\displaystyle
O_{3}=\bar{s}\gamma^{\mu}P_{L}b\;\bar{c}\gamma_{\mu}P_{L}c\;,\
O_{4}=\bar{s}_{\lambda}\gamma^{\mu}P_{L}b_{\rho}\bar{c}_{\rho}\gamma_{\mu}P_{L}c_{\lambda}\;,$
$\displaystyle O_{5}=\bar{s}\gamma^{\mu}P_{L}b\;\bar{c}\gamma_{\mu}P_{R}c\;,\
O_{6}=\bar{s}_{\lambda}\gamma^{\mu}P_{L}b_{\rho}\bar{c}_{\rho}\gamma_{\mu}P_{R}c_{\lambda}\;,$

where $P_{L,R}$ are the left- and right-handed projectors, respectively. To
match these operators to the diquark-quark operators, we write the scalar and
axial-vector diquark states as Shi:2020qde

$\displaystyle\left|\mathcal{S}_{bc}^{i}(v)\right\rangle=$
$\displaystyle\frac{\sqrt{E_{\mathcal{S}}}}{4\sqrt{2m_{b}m_{c}}}\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\phi^{*}(\mathbf{k})\epsilon_{ijk}\left[C\gamma_{5}(1+\not{v})\right]_{\beta\gamma}^{\dagger}$
(5) $\displaystyle
Q_{j\beta}^{\dagger}(v,\mathbf{k})Q_{k\gamma}^{\prime\dagger}(v,-\mathbf{k})|0\rangle\;,$
$\displaystyle\left|\mathcal{X}_{bc}^{i}(v,\epsilon)\right\rangle=$
$\displaystyle\frac{-\sqrt{E_{\mathcal{X}}}}{4\sqrt{2m_{b}m_{c}}}\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\phi^{*}(\mathbf{k})\epsilon_{ijk}[C\not{\epsilon}(1+\not{v})]_{\beta\gamma}^{\dagger}$
(6) $\displaystyle
Q_{j\beta}^{\dagger}(v,\mathbf{k})Q_{k\gamma}^{\prime\dagger}(v,-\mathbf{k})|0\rangle\;,$

where the heavy quark operators are defined by

$\displaystyle Q(v,\mathbf{k})=\sum_{s}u^{s}(p)a_{\mathbf{p}}^{s},\text{ with
}\mathbf{p}\equiv m_{Q}\mathbf{v}+\mathbf{k}\;.$ (7)

The function $\phi(\mathbf{k})$ is the distribution of the relative momentum
$\mathbf{k}$ of the quarks in the heavy diquark bound states, and $\mathbf{k}$
is typically small compared to the heavy quark masses. In the heavy quark
limit, the distribution $\phi(\mathbf{k})$ is concentrated at $\mathbf{k}=0$.
The wave functions in the spinor space
$\left[C\gamma_{5}(1+\not{v})\right]_{\beta\gamma}$ and
$\left[C\not{\epsilon}(1+\not{v})\right]_{\beta\gamma}$ are anti-symmetric and
symmetric, respectively, with $\epsilon$ the polarization vector of axial-
vector diquark field and $C\equiv i\gamma^{0}\gamma^{2}$. The antisymmetric
tensor $\epsilon_{ijk}$ is the wave function in the color spaces with $i,j,k$
color indices. The projector $(1+\not{v})$ approximately selects the particle
component and eliminates the antiparticle component. The expressions are not
well defined from the first principle of QCD, but they can give the correct
matching results in the leading power of inverse heavy quark mass.

After performing the match by calculating the $\mathcal{S}_{bc}\to cs$ and
$\mathcal{X}_{bc}\to cs$ transition amplitudes at the lowest order of
$\alpha_{s}$ and $\mathbf{k}$, we obtain the scalar diquark-quark interaction
terms in the Hamiltonian.

$\displaystyle\mathcal{H}_{\mathcal{S}}$ $\displaystyle\ni$ $\displaystyle
A_{12}\epsilon_{ijk}\;\mathcal{S}^{i}\;\bar{c}_{j}P_{R}C\bar{s}^{T}_{k}+A_{34}\epsilon_{ijk}\;\mathcal{S}^{i}\;\bar{c}_{j}P_{R}C\bar{s}^{T}_{k}$
(8)
$\displaystyle+A_{56}/m_{bc}\;\epsilon_{ijk}\;(iD_{\mu}\mathcal{S}^{i})\;\bar{c}_{j}\gamma^{\mu}P_{R}C\bar{s}^{T}_{k}\;,$
$\displaystyle A_{12}$ $\displaystyle=$ $\displaystyle
4\sqrt{m_{bc}}G_{F}V_{cs}^{*}V_{cb}(C_{2}-C_{1})\psi_{bc}(0)\;,$
$\displaystyle A_{34}$ $\displaystyle=$ $\displaystyle
4\sqrt{m_{bc}}G_{F}V_{cs}^{*}V_{cb}(C_{4}-C_{3})\psi_{bc}(0)\;,$
$\displaystyle A_{56}$ $\displaystyle=$ $\displaystyle
2\sqrt{m_{bc}}G_{F}V_{cs}^{*}V_{cb}(C_{5}-C_{6})\psi_{bc}(0)\;,$

and the axial-vector diquark-quark interaction term

$\displaystyle\mathcal{H}_{\mathcal{X}}$ $\displaystyle\ni$ $\displaystyle
B_{5,6}\epsilon_{ijk}\bar{c}_{j}\gamma^{\mu}P_{R}C\bar{s}^{T}_{k}\mathcal{X}_{v,\mu}^{i}\;,$
(9) $\displaystyle B_{56}$ $\displaystyle=$ $\displaystyle-A_{56}\;.$

The $\psi_{bc}(0)$ is the Fourier transformed of $\phi(\mathbf{k})$ at the
zero point.

##  III Operator product expansion

The decay rate of the inclusive $\Xi_{bc}^{+,0}\to X_{cs}$ channel can be
expressed as

$\displaystyle\Gamma(\Xi_{bc}^{+,0}\to X_{cs})$ $\displaystyle=$
$\displaystyle\sum_{X_{cs}}(2\pi)^{4}\delta^{(4)}(p_{\Xi}-p_{X}){|\langle
X_{cs}|\mathcal{H}_{I}|\Xi_{bc}^{+,0}\rangle|^{2}\over 2m_{\Xi}}$ (10)
$\displaystyle=$
$\displaystyle{\text{Im}\langle\Xi_{bc}^{+,0}|\mathcal{T}|\Xi_{bc}^{+,0}\rangle\over
m_{\Xi}}\;,$

where the operator product $\mathcal{T}$ is given by

$\displaystyle\mathcal{T}=i\int
d^{4}x\;T\\{\mathcal{H}_{I}^{\dagger}(x)\mathcal{H}_{I}(0)\\}\;,\mathcal{H}_{I}=\mathcal{H}_{\mathcal{S}}+\mathcal{H}_{\mathcal{X}}\;,$
(11)

by making use of the optical theorem. The nonlocal operator product
$\mathcal{T}$ is further expanded as

$\displaystyle\mathcal{T}=\sum_{i}F_{i}Q^{(n)}_{i}\;,$ (12)

with the technique called local operator product expansion (OPE), where the
$Q^{(n)}_{i}$ are local operators with dimension $n$ and $F_{i}$ are the
corresponding coefficients. Contributions from higher dimensional operators
are suppressed, in this case, by powers of $\Lambda_{\text{QCD}}/m_{bc}$.

Figure 1: The leading order diagram for the diquark-to-diquark amplitudes.

The OPE is performed by requiring that the quark-diquark matrix elements of
the left-handed side and the right-handed side of (12) match each other.
Starting with the scalar diquark, the matching is performed by calculating the
$\mathcal{S}_{bc}\to cs\to\mathcal{S}_{bc}$ amplitude at the least order of
$\alpha_{s}$, as displayed in FIG. 1. The imaginary part of the amplitude is
calculated as

$\displaystyle{\rm{Im}}\;\mathcal{M}(S_{bc}\to
S_{bc})=A\frac{(p^{2}-m_{c}^{2})^{2}}{8\pi p^{2}}\;,$ (13) $\displaystyle
A=|A_{12}+A_{34}+{m_{c}^{2}\over m_{bc}^{2}}A_{56}|^{2}\;,$

where the diquark has momentum $p=m_{bc}v+k$, with the residual momentum
$k\sim\Lambda_{\text{QCD}}$, and the expansion is performed by powers of $k$.
The $k^{0}$ term in the expansion of (13) is reproduced by the dimension-3
operator with the coefficient

$\displaystyle
Q^{s(3)}_{1}=S^{\dagger}_{v}S_{v}\;,\;F^{s}_{1}=\frac{A(m_{bc}^{2}-m_{c}^{2})^{2}}{16\pi
m_{bc}^{3}}\;.$ (14)

The $k^{1}$ term and the $k^{2}$ term correspond to the dimension-4 operator
$S^{\dagger}_{v}(iv\cdot D)S_{v}$ and the dimension-5 operator
$S_{v}^{\dagger}(iD)^{2}S_{v}$, respectively. These two operators are related
by the equation of motion $[iv\cdot D+{(iD)^{2}}/(2m_{bc})]S_{v}=0$. Taking
their coefficients ${A(m_{bc}^{4}-m_{c}^{4})}/({8\pi m_{bc}^{4}})$ and
${A(m_{bc}^{4}-m_{c}^{4})}/({16\pi m_{bc}^{5}})$, it turns out that their
contribution exactly cancel each other. This is actually protected by the
reparametrization invariance proved by Mannel:2018mqv  555The decomposition of
$p$ into $v$ and $k$ is not unique. A small change in the four-velocity of the
order of $\Lambda_{\mathrm{QCD}}/m_{bc}$ can be compensated by a change in the
residual momentum: $v\rightarrow v+\delta v,k\rightarrow k-m\delta v$..
Therefore, up to $\mathcal{O}(\Lambda_{\text{QCD}}^{2}/m_{bc}^{2})$, only the
matrix element of $Q^{(3)}_{1}$ needs to be evaluated. (The chromomagnetic
operator does not exist in the scalar case.) It can be parameterized as
Mannel:2018mqv

$\displaystyle\left\langle\Xi_{S}|S^{\dagger}_{v}S_{v}|\Xi_{S}\right\rangle\equiv
2m_{\Xi}\mu_{3}=2m_{\Xi}(1-{\mu_{\pi}^{2}\over 2m_{bc}^{2}})\;,$ (15)
$\displaystyle\text{with}\
\left\langle\Xi_{S}|S^{\dagger}_{v}(iD)^{2}S_{v}|\Xi_{S}\right\rangle=-2m_{\Xi}\mu_{\pi}^{2}\;.$
(16)

The nonperturbative parameter $\mu_{\pi}^{2}$ takes the value $0.43\pm 0.24$
Bernlochner:2022ucr  in the numerical analysis, which is extracted from the
inclusive $B$ meson decay. We expect that the $\mu_{\pi}^{2}$ values for
$\Xi_{bc}^{+,0}$ and $B$ mesons are the same owing to the heavy quark-diquark
symmetry.

The imaginary part of the $\mathcal{X}_{bc}\to cs\to\mathcal{X}_{bc}$
amplitude corresponding to FIG. 1 is

$\displaystyle{\rm
Im}\;\mathcal{M}_{5,6}(\mathcal{X}_{bc}\to\mathcal{X}_{bc})=B\frac{(p^{2}-m_{c}^{2})^{2}}{8\pi
p^{2}}\;,$ (17) $\displaystyle B=|B_{56}|^{2}(2+{m_{c}^{2}\over
m_{bc}^{2}})\;.$

Because the whole amplitude is suppressed by the penguin Wilson coefficients
$(C_{5}-C_{6})^{2}$, we only perform the OPE to the leading power,

$\displaystyle\mathcal{T}\ni F_{1}^{a}Q_{1}^{a(3)}\;,$ (18) $\displaystyle
Q_{1}^{a(3)}=\mathcal{X}_{v}^{\mu\dagger}\mathcal{X}_{v\mu}\;,\;F^{a}_{1}=\frac{B(m_{bc}^{2}-m_{c}^{2})^{2}}{16\pi
m_{bc}^{3}}\;.$

Analogous to the scalar case, the hadronic matrix element of $Q_{1}^{a(3)}$ is

$\displaystyle\left\langle\Xi_{\mathcal{X}}|\mathcal{X}_{v}^{\mu\dagger}\mathcal{X}_{v\mu}|\Xi_{\mathcal{X}}\right\rangle\equiv
2m_{\Xi}\mu_{3}\;.$ (19)

The $S_{bc}\to cs\to X_{bc}$ amplitude at the leading order vanishes because
of angular momentum conservation, so there would be no interference between
the scalar and axial-vector constituents in the $\Xi_{bc}^{+,0}$ baryons.

##  IV Numerical analysis

If the $bc$ diquark in $\Xi_{bc}$ is purely scalar, the leading-order
inclusive decay width is given by

$\displaystyle\Gamma(\Xi_{S}\to X_{cs})$ (20) $\displaystyle=$
$\displaystyle\frac{2G_{F}^{2}}{\pi
m_{bc}^{2}}[C_{2}-C_{1}+C_{4}-C_{3}+\frac{m_{c}}{2m_{bc}}(C_{5}-C_{6})]^{2}$
$\displaystyle|V_{cs}^{*}V_{cb}|^{2}|\psi_{bc}(0)|^{2}(m^{2}_{bc}-m^{2}_{c})^{2}(1-{\mu_{\pi}^{2}\over
2m_{bc}^{2}})$ $\displaystyle\simeq$ $\displaystyle(5.7\pm 2.1)\times
10^{-13}\;\text{GeV}\;.$

where the Fermi constant, quark masses, and the CKM matrix elements take
values from ParticleDataGroup:2020ssz  and the Wilson coefficients take values
from Lu:2000em . The uncertainty arise from the variation of the quark masses,
the Wilson coefficients, the CKM matrix elements, the diquark wave function at
the origin $\psi_{bc}(0)=\frac{0.87\pm 0.09}{\sqrt{4\pi}}\mathrm{GeV}^{3/2}$
Qin:2021wyh , the nonperturbative parameter $\mu_{\pi}^{2}=0.43\pm 0.24$
Bernlochner:2022ucr  and we have include additional $\sim 30\%$ uncertainty
from possible power corrections Zhu:2017lqu . It can be checked that the
leading power result recovers the free diquark decay rate. The $\Xi_{S}\to
X_{cd}$ result can be obtained if we replace $V_{cs}$ with $V_{cd}$, and it
reads $\Gamma(\Xi_{S}\to X_{cd})=(2.7\pm 1.0)\times 10^{-14}$ GeV. This
procedure also applies to the rest results.

Figure 2: The next-to-leading order Feynman diagrams contributing to
$\Xi_{\mathcal{X}}\to X_{cs}$ with a real gluon emitted from the diquark.

If the diquark in $\Xi_{bc}$ is purely axial-vector, the leading order
inclusive decay width is given by

$\displaystyle\Gamma_{1}(\Xi_{\mathcal{X}}\to X_{cs})$ (21) $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}}{6\pi
m^{2}_{bc}}|V_{cs}^{*}V_{cb}|^{2}(C_{5}-C_{6})^{2}|\psi_{bc}(0)|^{2}$
$\displaystyle(m^{2}_{bc}-m^{2}_{c})^{2}(2+\frac{m_{c}^{2}}{m_{bc}^{2}})(1-{\mu_{\pi}^{2}\over
2m_{bc}^{2}})$ $\displaystyle\simeq$ $\displaystyle(9.0\pm 3.4)\times
10^{-17}\;\text{GeV}\;,$

where the uncertainties are from the same sources as (20). The leading power
result again recovers the free diquark decay rate. Practically, the leading-
order result for the $\Xi_{\mathcal{X}}$ decay rate is highly suppressed by
the Wilson coefficients, such that the next-to-leading-order (NLO)
contributions might be larger. We consider the NLO correction with a real
gluon emitted from the diquark, as shown in FIG.2. Taking into account the
tree operators in (4), we obtain their NLO contribution

$\displaystyle\Gamma_{2}(\Xi_{\mathcal{X}}\to X_{cs})$ (22) $\displaystyle=$
$\displaystyle\frac{\alpha_{s}}{36\pi^{2}m_{bc}^{4}}[(C_{1}^{2}+C_{2}^{2})(x^{2}-xy+y^{2})+C_{1}C_{2}(x^{2}-$
$\displaystyle
4xy+y^{2})]G_{F}^{2}\left|V_{cs}^{*}V_{cb}\right|^{2}|\psi_{bc}(0)|^{2}$
$\displaystyle\int^{m_{b}^{2}}_{0}ds_{12}\bigg{[}\frac{1}{2}\left(-m_{c}^{2}-s_{12}+m_{bc}^{2}\right)$
$\displaystyle\sqrt{-2m_{c}^{2}\left(s_{12}+m_{bc}^{2}\right)+\left(m_{bc}^{2}-s_{12}\right)^{2}+m_{c}^{4}}\bigg{]}$
$\displaystyle\simeq$ $\displaystyle(4.7\pm 1.8)\times
10^{-16}\;\text{GeV}\;,$

where $x=m_{bc}/m_{b},y=m_{bc}/m_{c}$. The strong interaction constant
$\alpha_{s}$ takes values from ParticleDataGroup:2020ssz . The uncertainties
are from the same sources as (20). Indeed, it is larger than the leading-order
contribution (21) owing to enhanced Wilson coefficients. The virtual
correction requires systematical renormalization of the HDET, which is beyond
the scope of this study and is left for future works. To estimate the size of
the virtual correction, we refer to Cox:1985dj , in which the $B\to J/\psi+X$
decay is studied with the same weak interaction vertex as our process. Their
results show that the vertex correction is several times smaller than the hard
bremsstrahlung correction, and approximately 10% of the leading-order
contribution. Therefore, we expect that the virtual correction might cause
another uncertainty $\sim 10\%$. As for multi-particle contributions such as a
quark-anti-quark pair emission $\mathcal{X}_{bc}\to csq\bar{q}$, they are
highly suppressed by $\alpha_{s}^{2}$ and the phase space Huber:2018gii  and
are thus neglected.

Combining these two contributions, the inclusive $\Xi_{\mathcal{X}}$ decay
rate is calculated to be

$\displaystyle\Gamma(\Xi_{\mathcal{X}}\to X_{cs})\simeq(5.6\pm 2.2)\times
10^{-16}\;\text{GeV}\;.$ (23)

The corresponding $X_{cd}$ result is $\Gamma(\Xi_{\mathcal{X}}\to
X_{cd})=(2.7\pm 1.0)\times 10^{-17}$ GeV.

In a general case, $\Xi_{bc}$ might be a superposition of $\Xi_{\mathcal{S}}$
and $\Xi_{\mathcal{X}}$ Roberts:2008wq ; Weng:2018mmf ; Roberts:2007ni . Then,
the inclusive $\Xi_{bc}^{+,0}\to X_{cs}$ decay width is expressed as

$\displaystyle\Gamma(\Xi_{bc}^{+,0}\to X_{cs})$ (24) $\displaystyle=$
$\displaystyle\sin^{2}\theta\;\Gamma(\Xi_{S}\to
X_{cs})+\cos^{2}\theta\;\Gamma(\Xi_{\mathcal{X}}\to X_{cs})\;.$

Note that the interference between $\Xi_{\mathcal{S}}$ and $\Xi_{\mathcal{X}}$
vanishes because of angular momentum conservation.

Besides the $bc$ annihilation, the $bq$ and $cq$ annihilation contributions to
the $\Xi_{bc}^{+,0}$ lifetimes also depend on the $bc$ diquark spin
Cheng:2019sxr ; Kiselev:1999kh ; Kiselev:2001fw . The corresponding analytical
results are given in Appendix. Accepting the hadronic inputs in Cheng:2019sxr
, we numerically calculate these contributions with $\Xi_{bc}^{+,0}$ either
being $\Xi_{\mathcal{X}}$ or $\Xi_{\mathcal{S}}$, as listed in TABLE 1, where
the other contributions independent on the $bc$ spin are also listed. Then,
the mixing angle-dependent results for the total decay widths of
$\Xi_{bc}^{+,0}$ are given by

$\displaystyle\Gamma(\Xi_{bc}^{+,0})=$
$\displaystyle\sin^{2}\theta\;\Gamma_{\mathcal{S}}^{\text{an}}+\cos^{2}\theta\;\Gamma_{\mathcal{X}}^{\text{an}}$
(25)
$\displaystyle+2\sin\theta\cos\theta\cos\phi\Gamma_{\mathcal{SX}}^{\text{an}}+\Gamma^{\text{other}}\;,$

which can be translated to lifetimes shown in FIG. 3. With the lifetimes, the
branching ratios of the inclusive $\Xi_{bc}^{+,0}\to X_{cs}$ decays are
obtained, as shown in FIG. 4.

| $\Gamma_{\mathcal{S}}^{\text{an}}$ | $\quad\Gamma_{\mathcal{X}}^{\text{an}}$ | $\Gamma_{\mathcal{SX}}^{\text{an}}$ | $\Gamma^{\text{other}}$
---|---|---|---|---
$\Xi_{bc}^{0}$ | $1.87\pm 0.12$ | $3.81\pm 0.30$ | $2.20\pm 0.17$ | $1.50\pm 0.31$
$\Xi_{bc}^{+}$ | $0.61\pm 0.22$ | $0.035\pm 0.008$ | $0.020\pm 0.005$ | $2.94\pm 0.73$
Table 1: Various contributions to the total decay widths of $\Xi_{bc}^{+,0}$
in units of $10^{-12}\mathrm{GeV}$, including the {$bc$,$bu$} and {$bc$,$cd$}
weak-annihilation contributions to $\Xi_{bc}^{+}$ and $\Xi_{bc}^{0}$,
respectively. The subscript $\mathcal{S}$ and $\mathcal{X}$ represent the
cases with the $bc$ diquark being scalar or axial-vector. The other
contributions $\Gamma^{\text{other}}$ contain the spectator decay contribution
$\Gamma^{\text{dec}}$ and the Pauli interference $\Gamma^{\text{int}}$
Cheng:2019sxr , which are independent on the diquark spin.

The mixing angle has been studied by many previous studies, e.g.
Roberts:2008wq ; Weng:2018mmf ; Roberts:2007ni . As the predictions for the
mixing angle $\theta$ are very diverse, we vary $\theta$ from 0 to $\pi/2$ and
display the corresponding results in FIG. 3. The lifetime of $\Xi_{bc}^{0})$
is heavily influenced by the phase angle, and the lifetime of $\Xi_{bc}^{+})$
has almost no effect. We also take $\cos\phi=0$, $\sin\theta\simeq 0.39$ and
$\cos\theta\simeq 0.92$ Roberts:2008wq  as a reference, and obtain the
corresponding $\Xi_{bc}^{+,0}\to X_{cs}$ branching ratios,

$\displaystyle\mathcal{B}(\Xi_{bc}^{0}\to
X_{cs})\simeq(1.4^{+0.9}_{-0.6})\%\;,$ (26)
$\displaystyle\mathcal{B}(\Xi_{bc}^{+}\to
X_{cs})\simeq(5.4^{+3.8}_{-2.6})\%\;.$ (27)

The branching ratio of $\Xi_{bc}^{+,0}\to X_{cd}$ are
$(0.26^{+0.18}_{-0.10})\%$ and $(0.07^{+0.04}_{-0.02})\%$, respectively.

It is observed that the lifetimes and the inclusive branching ratios are very
sensitive to the mixing angle, so their measurements in future experiments can
be used to determine whether the diquark in the ground state $\Xi_{bc}^{+,0}$
is scalar or axial-vector, or how they mix each other.

Figure 3: The lifetimes of $\Xi_{bc}^{+,0}$ as functions of the mixing angle
$\theta$. The solid, dotted and dashed curves correspond to the choices of the
phase angle $\cos\phi=0$, $\cos\phi=1$, and $\cos\phi=-1$, respectively. The
vertical line corresponds to the reference value $\sin\theta$ = 0.39
Roberts:2008wq . Figure 4: The branching ratios $\mathcal{B}(\Xi_{bc}^{+,0}\to
X_{cs})$ as functions of the mixing angle $\theta$ with $\cos\phi=0$, where
the vertical line corresponds to the reference value $\sin\theta$ = 0.39
Roberts:2008wq .

As for the $\Xi_{bc}^{+,0}$ search, it was estimated in Kiselev:2001fw  that
the $\mathcal{O}(10\%)$ inclusive $\Xi_{bc}^{+,0}\to X_{cs}$ decay branching
ratio would indicate the typical exclusive channels $\Xi_{bc}\to
D^{(*)}K^{(*)}p$ have branching ratios of several permil. With some estimated
theoretical and experimental inputs, the result of LHCb:2020iko  can be
converted to the upper limit $\mathcal{B}(\Xi^{0}_{bc}\to D^{0}K^{-}p)\lesssim
0.3\%$ assuming that the $\Xi^{0}_{bc}$ lifetime is 100 fs. The experimental
upper limit is very close to the theoretical estimation, so it is quite
hopeful to discover $\Xi_{bc}$ by this channel with more data collected. In
fact, considering the lifetime hierarchy between $\Xi_{bc}^{0}$ and
$\Xi_{bc}^{+}$, searches via an analogous $\Xi_{bc}^{+}$ decay channel would
be more promising. Taking the reference with $\sin\theta\simeq 0.39$ as an
example, the $\Xi_{bc}^{+}$ branching ratio would be larger by a factor of
approximately 4 than $\Xi_{bc}^{0}$, and the detection efficiency at the LHCb
would also be improved by a similar factor owing to more displaced decay
vertices LHCb:2020iko , so it is expected that approximately 16 times events
can be collected by using $\Xi_{bc}^{+}$ decays than using a similar
$\Xi_{bc}^{0}$ decay. Therefore, we propose that $\Xi_{bc}^{+}\to
D^{(*)+}\Lambda$, $\Xi_{bc}^{+}\to\Lambda_{c}^{+}\bar{K}^{(*)0}$,
$\Xi_{bc}^{+}\to D^{(*)+}K^{-}p$, and $\Xi_{bc}^{+}\to D^{0}\bar{K}^{(*)0}p$
should be searched experimentally with priority. They can be searched in the
$p2\pi^{+}2\pi^{-}$, $p2\pi^{+}\pi^{-}K^{-}$ and $p2\pi^{+}2K^{-}$ final
states.

##  V Conclusion

We have calculated the inclusive $\Xi_{bc}^{+,0}\to X_{cs}$ decay widths
making use of the local OPE technique in the framework of HDET. The $\Xi_{bc}$
is treated as the superposition of $\Xi_{\mathcal{S}}$ containing a scalar
$bc$ diquark and $\Xi_{\mathcal{X}}$ containing an axial-vector $bc$ diquark.
The mixing angle $\theta$ dependent results for both the $\Xi_{bc}^{+,0}$
lifetimes and the inclusive $\Xi_{bc}^{+,0}\to X_{cs}$ branching ratios have
been obtained. The corresponding $\Xi_{bc}^{+,0}\to X_{cd}$ results are also
presented. It turns out that these observables are very sensitive to $\theta$,
and thus their measurements can help determine the $bc$ diquark spin property
in $\Xi_{bc}^{+,0}$. With the reference mixing angle $\sin\theta$ = 0.39
Roberts:2008wq , the $\Xi_{bc}^{+}$ lifetime is approximately four times
longer than that of $\Xi_{bc}^{0}$, and the branching ratio
$\mathcal{B}(\Xi_{bc}^{+}\to X_{cs})$ is approximately four times bigger than
$\mathcal{B}(\Xi_{bc}^{0}\to X_{cs})$. We propose that the decay channels
$\Xi_{bc}^{+}\to D^{(*)+}\Lambda$,
$\Xi_{bc}^{+}\to\Lambda_{c}^{+}\bar{K}^{(*)0}$, $\Xi_{bc}^{+}\to
D^{(*)+}K^{-}p$, and $\Xi_{bc}^{+}\to D^{0}\bar{K}^{(*)0}p$ should be used in
$\Xi_{bc}$ searches.

Acknowledgement. — This work is supported by the Natural Science Foundation of
China under grant No. 12005068.

##  Appendix A Appendix: Weak-annihilation contributions to $\Xi_{bc}$ total
widths

The transition operators in the $bq$ and $cq$ weak-annihilation contributions
to the $\Xi_{bc}$ decay widths are

$\displaystyle\mathcal{T}^{an}_{bq}=\frac{G_{F}^{2}\left|V_{cb}\right|^{2}}{2\pi
m_{bu}^{2}}\left(m_{bu}^{2}-m_{c}^{2}\right)^{2}\left(C_{1}-C_{2}\right)^{2}$
$\displaystyle\left(\bar{b}\gamma_{\mu}(1-\gamma_{5})b\right)\left(\bar{q}\gamma^{\mu}(1-\gamma_{5})q\right),$
(28)
$\displaystyle\mathcal{T}^{an}_{cq}=\frac{G_{F}^{2}\left|V_{cs}\right|^{2}}{2\pi
m_{cd}^{2}}\left(m_{cd}^{2}-m_{s}^{2}\right)^{2}\left(C_{1}-C_{2}\right)^{2}$
$\displaystyle\left(\bar{c}\gamma_{\mu}(1-\gamma_{5})c\right)\left(\bar{q}\gamma^{\mu}(1-\gamma_{5})q\right),$
(29)

where the diquark masses $m_{bu}$ and $m_{cd}$ are approximately the heavy
quark mass $m_{b}$ and $m_{c}$ respectively. The hadronic matrix elements of
these operators are estimated in the non-relativistic potential model
Cheng:2019sxr  as,

$\displaystyle\left\langle\Xi_{\mathcal{S}}^{+}\left|\left(\bar{b}\gamma_{\mu}(1-\gamma_{5})b\right)\left(\bar{q}\gamma^{\mu}(1-\gamma_{5})q\right)\right|\Xi_{\mathcal{S}}^{+}\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle\Xi_{\mathcal{S}}^{0}\left|\left(\bar{c}\gamma_{\mu}(1-\gamma_{5})c\right)\left(\bar{q}\gamma^{\mu}(1-\gamma_{5})q\right)\right|\Xi_{\mathcal{S}}^{0}\right\rangle$
$\displaystyle=$ $\displaystyle 2m_{bc}\left|\psi_{q,bc}(0)\right|^{2},$
$\displaystyle\left\langle\Xi_{\mathcal{X}}^{+}\left|\left(\bar{b}\gamma_{\mu}(1-\gamma_{5})b\right)\left(\bar{q}\gamma^{\mu}(1-\gamma_{5})q\right)\right|\Xi_{\mathcal{X}}^{+}\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle\Xi_{\mathcal{X}}^{0}\left|\left(\bar{c}\gamma_{\mu}(1-\gamma_{5})c\right)\left(\bar{q}\gamma^{\mu}(1-\gamma_{5})q\right)\right|\Xi_{\mathcal{X}}^{0}\right\rangle$
$\displaystyle=$ $\displaystyle 6m_{bc}\left|\psi_{q,bc}(0)\right|^{2},$
$\displaystyle\left\langle\Xi_{\mathcal{S}}^{+}\left|\left(\bar{b}\gamma_{\mu}(1-\gamma_{5})b\right)\left(\bar{q}\gamma^{\mu}(1-\gamma_{5})q\right)\right|\Xi_{\mathcal{X}}^{+}\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle\Xi_{\mathcal{S}}^{0}\left|\left(\bar{c}\gamma_{\mu}(1-\gamma_{5})c\right)\left(\bar{q}\gamma^{\mu}(1-\gamma_{5})q\right)\right|\Xi_{\mathcal{X}}^{0}\right\rangle$
$\displaystyle=$ $\displaystyle
2\sqrt{3}m_{bc}\left|\psi_{q,bc}(0)\right|^{2}.$ (32)

Then, the leading-order weak-annihilation inclusive decay widths are given by

$\displaystyle\Gamma_{\mathcal{S}}^{bq}=\frac{G_{F}^{2}\left|V_{cb}\right|^{2}}{2\pi
m_{bu}^{2}}|\psi_{q,bc}(0)|^{2}\left(m_{bu}^{2}-m_{c}^{2}\right)^{2}(C_{1}-C_{2})^{2},$
$\displaystyle\Gamma_{\mathcal{S}}^{cq}=\frac{G_{F}^{2}\left|V_{cs}\right|^{2}}{2\pi
m_{cd}^{2}}|\psi_{q,bc}(0)|^{2}\left(m_{cd}^{2}-m_{s}^{2}\right)^{2}(C_{1}-C_{2})^{2},$
$\displaystyle\Gamma_{\mathcal{X}}^{bq}=3\Gamma_{\mathcal{S}}^{bq},\Gamma_{\mathcal{SX}}^{bq}=\sqrt{3}\Gamma_{\mathcal{S}}^{bq}$
$\displaystyle\Gamma_{\mathcal{X}}^{cq}=3\Gamma_{\mathcal{S}}^{cq},\Gamma_{\mathcal{SX}}^{cq}=\sqrt{3}\Gamma_{\mathcal{S}}^{cq}.$
(33)

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# Bright squeezed vacuum for two-photon spectroscopy: simultaneously high
resolution in time and frequency, space and wavevector.

Paula Cutipa  Max-Planck Institute for the Science of Light, Staudtstraße 2,
Erlangen D-91058, Germany  University of Erlangen-Nuremberg, Staudtstraße
7/B2, 91058 Erlangen, Germany  Corresponding author<EMAIL_ADDRESS>Maria V. Chekhova  Max-Planck Institute for the Science of Light, Staudtstraße
2, Erlangen D-91058, Germany  University of Erlangen-Nuremberg, Staudtstraße
7/B2, 91058 Erlangen, Germany

Entangled photons offer two advantages for two-photon absorption spectroscopy.
One of them, the linear scaling of two-photon absorption rate with the input
photon flux, is only valid at very low photon fluxes and is therefore
impractical. The other is the overcoming of the classical constraints for
simultaneous resolution in time-frequency and in space-wavevector. Here we
consider bright squeezed vacuum (BSV) as an alternative to entangled photons.
The efficiency increase it offers in comparison with coherent light is modest,
but it does not depend on the photon flux. Moreover, and this is what we show
in this work, BSV also provides simultaneously high resolution in time and
frequency, and in space and wavevector. In our experiment, we measure the
widths of the second-order correlation functions in space, time, frequency,
and angle, and demonstrate the violation of the constraint given by the
Fourier transformation, also known as the Mancini criterion of entanglement.

Entangled photons play an important role in different applications such as
quantum imaging, microscopy, spectroscopy, and metrology [1, 2]. In
particular, energy-time and position-momentum entanglement permits the
enhancement of applications based on two-photon absorption (TPA), such as TPA
spectroscopy for complex sample analysis or cellular imaging [3, 4, 5, 6, 7,
8]. One of the advantages of using entangled photons is that it improves the
TPA rate, which then scales with the photon flux linearly, and not
quadratically, as in the case of coherent light [9, 10]. This happens because
the TPA rate is proportional to the second-order correlation function (CF) of
the incident light [11] at zero delay and displacement, $g^{(2)}(0)$, which
for a weak flux of photon pairs scales inversely with the mean photon number
[9]. This enables the analysis of biological samples at low light intensities,
avoiding damage to the sample [3, 10]. However, this linear dependence is only
valid at very low photon fluxes, where it can be hardly used in practice [6,
12].

Another advantage of using photon pairs for TPA spectroscopy is the
simultaneously high and independent resolution in time ($\delta\tau$) and
frequency ($\delta\omega$) they provide [4]. Indeed, photons entangled in time
and frequency violate the classical relation $\delta\omega\delta\tau\geq
2\pi$, also known as the Mancini separability bound [13]. The same situation
is valid for the transverse wavevector ($\delta k$) and space ($\delta\xi$)
[14]. This enables simultaneously high resolution in time, space, frequency,
and wavevector.

Entangled photons are usually generated by spontaneous parametric down-
conversion (PDC). When strongly pumped, PDC produces a brighter state of
light, bright squeezed vacuum (BSV) [15, 16, 17, 18]. For this state,
$g^{(2)}(0)=3$, which provides only threefold enhancement of two-photon
absorption compared to the case of coherent light excitation [19], but this
enhancement is independent of the photon flux. Meanwhile, the mean number of
photons per one mode of BSV, $N=\sinh^{2}(G)$, where $G$ is called the
parametric gain, can be as high as for laser sources. This makes BSV a
prospective source for practical use in multiphoton absorption. Below we show
that the second advantage of entangled photons, the simultaneously high
resolution in time, space, frequency, and wavevector, is also valid for BSV.

The time, frequency, space, and wavevector resolution for two-photon
transitions is determined by the second-order CF. The CF, in the general case,
is a nonfactorable function of four variables: two time moments and two space
positions (in one dimension, for simplicity),
$g^{(2)}(t,t^{\prime};x,x^{\prime})$ [17, 20, 21]. Alternatively, the CF can
be considered as a function of frequencies and transverse wavevectors,
$g^{(2)}(\omega,\omega^{\prime};k,k^{\prime})$ [22].

At low parametric gain, $G\ll 1$, $g^{(2)}(0)\approx 1/G^{2}$ and is very
high. The resolution provided by entangled photons in two-photon spectroscopy
is given by its widths with respect to the corresponding variables: for
instance, $\delta\tau$ is the width of $g^{(2)}(t,t^{\prime};x,x)$ at fixed
$t^{\prime},x$, and similarly with the other variables.

At high gain, $G\gtrsim 1$, photon pairs overlap, and their accidental
coincidences, as well as thermal correlations of photons within one mode,
start to play a role. Then, the value of $g^{(2)}$ reduces [9, 18], but its
widths in time $\delta\tau$, space $\delta\xi$, frequency $\delta\omega$, and
transverse wavevector $\delta k$, further called correlation widths, do not
change much.

In this work we measure all these four widths. To measure $\delta\tau$ and
$\delta\xi$, we build an ultra-fast autocorrelator based on sum-frequency
generation (SFG) [23, 17] in a non-phase-matched crystal. By using type-II SFG
we get rid of the first-order interference fringes [24]. The values of
$\delta\omega$ and $\delta k$ we obtain from the measurement of the covariance
of BSV intensity [16, 25].

Figure 1: Experimental setup for measuring the second-order CF. We produced
BSV in the BBO crystal whose output face was imaged on beam splitter $BS_{2}$
by lens $L_{1}$ and then on the quartz crystal by lens $L_{2}$. In the
autocorrelator, retroreflector $R_{1}$ scanned the time delay and beamsplitter
$BS_{2}$ scanned the spatial displacement. In one arm, a HWP changed the
polarisation, so that type-II SFG was generated in quartz at the output and
registered, after polarizer P, by a spectrometer. The insets show the
wavelength and angular spectra of BSV emitted via collinear degenerate PDC (a,
b), the dependence of the SFG signal on the input polarization with one arm
blocked (c) and the SFG spectrum (d).

The experimental setup is displayed in Fig. 1. We generated BSV with an
amplified Ti-sapphire laser (wavelength $800$ nm, pulse duration $1.6$ ps,
repetition rate $5$ kHz, mean power $1W$, waist $0.8$ mm) in a $10$ mm
$\beta$-barium-borate (BBO) crystal cut for type-I phase matching. The
experiment was run at gain $G=10$, both under collinear (the pump wavevector
at $19.87^{o}$ to the optic axis) and non-collinear (the pump wavevector at
$19.93^{o}$ to the optic axis) phase matching. In both cases, the angular-
frequency spectrum of BSV is bounded, however, it is factorable only in the
collinear configuration [26]. The optic axis of the BBO crystal was in the y-z
plane (where walk-off occured), the pump was polarized along the y-direction,
and the BSV along the x-direction. The pump radiation was rejected with
dichroic mirrors DM and the BSV was collected with lens $L_{1}$ (focal length
$15$ cm), its Fourier plane shown by a dotted line (F). At this position we
placed a spectrometer to measure the wavelength spectrum of the BSV. It was
centered at 1600 nm, and under collinear phase matching it had a bandwidth of
$\Delta\lambda=360$ nm (Fig. 1 (a)). The angular spectrum, measured with a
camera at the same position, is shown in Fig. 1 (b). In all further
measurements, to get rid of the walk-off effect we reduced the angular
spectrum in the y-z plane to $10$ mrad by placing a 1.6 mm slit in plane F
along the x-direction.

The BSV beam was then sent into the autocorrelator through a 50:50 beam
splitter and the time delay was introduced in one arm by retroreflector
$R_{1}$. In the other arm, a broadband half-wave plate HWP changed the
polarisation from ‘x’ to ‘y’. This modification eliminated the first-order
interference at the output of the second beamsplitter $BS_{2}$, where the two
beams were combined. The spatial displacement was generated by moving $BS_{2}$
transversely to the beam propagation direction. Noteworthy, lens $L_{1}$
imaged the output face of the BBO crystal on $BS_{2}$ with an $M=8$
magnification, which we took into account while measuring the width of the
spatial correlation.

Further, lens $L_{2}$ imaged $BS_{2}$ on the SFG crystal, which was a $500$
$\mu$m thick z-cut $\alpha$-quartz. The SFG was non-phasematched [27], which
made it broadband and let all frequency components of BSV contribute [17]. The
dependence of the SFG signal on the input polarization (Fig. 1 (c)) is typical
for type-II SFG, and it originates from the structure of the nonlinear tensor
of quartz. As a result, there was no SFG from either arm of the
autocorrelator, which had ‘x’ ($0^{o}$) or ‘y’ ($90^{o}$) polarization, but
only from both. The SFG radiation was then collected and sent, after polarizer
P, to a spectrometer. The SFG spectrum obtained for a perfectly balanced
autocorrelator is shown in Fig 1 (d). The peak, known as the coherent
contribution, is formed by merging photons from the same pair, and its area
(red) corresponds to the real coincidences in a Hanbury Brown – Twiss (HBT)
measurement. The background is caused by photons from different pairs and its
area (blue) corresponds to the ‘thermal’ and ‘accidental’ coincidences in a
HBT setup [28, 29].

## Frequency and wavevector correlations

Figure 2: Intensity covariance measured versus wavelengths (a) and angles (d)
and its cross-correlation part calculated for the same cases (b,e). Panels (c)
and (f) show, respectively, cross-sections of panels (a,d) (points) and panels
(b,e) (lines) at $\lambda=1.6\>\mu$m and at $\theta=0^{o}$ (white dashed
lines), normalized to the maximum values

.

To find the frequency correlation width $\delta\omega$ of BSV, we obtained its
spectra by placing a multimode fiber tip, connected to a spectrometer, in the
Fourier plane (F in Fig 1). Instead of the CF
$g^{(2)}(\omega,\omega^{\prime})$ mentioned above, we measured the covariance
[16, 30] of the intensity $I(\lambda)$ at different wavelengths
$\lambda,\lambda^{\prime}$:

$\hbox{Cov}(\lambda,\lambda^{\prime})=\langle
I(\lambda)I(\lambda^{\prime})\rangle-\langle I(\lambda)\rangle\langle
I(\lambda^{\prime})\rangle.$ (1)

This value is related to the CF $g^{(2)}(\lambda,\lambda^{\prime})$ as
$\hbox{Cov}(\lambda,\lambda^{\prime})=\langle I(\lambda)\rangle\langle
I(\lambda^{\prime})\rangle\big{(}g^{(2)}(\lambda,\lambda^{\prime})-1\big{)}$.
The frequency correlation width at wavelength $\lambda$ is then
$\delta\omega=2\pi c\delta\lambda/\lambda^{2}$, where $c$ is the speed of
light and $\delta\lambda$ is the FWHM of the covariance distribution. Figure
2(a) shows the covariance calculated according to Eq. (1) from 4000 spectra
obtained in the collinear configuration, each one acquired over 5 pulses. The
shape seen along the main diagonal ($\lambda=\lambda^{\prime}$) is caused by
the auto-correlation of each frequency mode with itself. The shape along the
anti-diagonal demonstrates the cross-correlations between signal and idler
photons. The maxima follow the energy conservation condition,
$\omega_{p}=\omega+\omega^{\prime}$ ($\omega_{p}$ is the pump frequency),
i.e., $\lambda^{-1}+\lambda^{\prime-1}=\hbox{const}$. The normalization is
chosen such that for both cross-correlation and anti-correlation part, the
maximal value is unity. The FWHM of the covariance cross-section (panel (c),
red points) yields the range of correlations $\delta\lambda=12$ nm,
corresponding to $\delta\omega=8.8\cdot 10^{12}$ Hz.

To calculate $\delta\omega$, we use the standard expression for the two-photon
probability amplitude $F(\omega,\omega^{\prime},k,k^{\prime})$, where, to take
into account the high-gain regime, the ‘sinc’ function is replaced by its
hyperbolic analog [31, 32]:

$F(\omega,\omega^{\prime},k,k^{\prime})=F_{p}(\omega+\omega^{\prime}-\omega_{p})F_{p}(k+k^{\prime})\frac{G}{\tilde{G}}\sinh\tilde{G}.$
(2)

Here, $F_{p}(\omega)$ and $F_{p}(k)$ are the pump Gaussian spectral and
transverse wavevector amplitudes, respectively [33],
$\tilde{G}\equiv\sqrt{G^{2}-(\Delta(\omega,\omega^{\prime},k,k^{\prime}))^{2}L^{2}/4}$,
$\Delta(\omega,\omega^{\prime},k,k^{\prime})$ is the wavevector mismatch and
$L$ is the crystal length. In the high-gain regime of PDC, the pump pulse
duration is effectively reduced (by a factor of 3.2 for $G=10$) [25] because
BSV is mainly generated around the pulse peak, and we take it into account by
assuming that the pump spectrum broadens 3.2 times.

The cross-correlation part of the covariance was calculated as
$|F(\omega,\omega^{\prime},0,0)|^{2}$ and plotted in Fig. 2(b). Its cross-
section (panel (c), blue line) gives the value of $\delta\lambda=10$ nm. The
somewhat larger value obtained in experiment is due to the finite resolution
of the spectrometer ($7$ nm).

A similar procedure was performed to find the angular width of correlations.
We replaced the spectrometer with a camera preceded by a $12$ nm bandpass
filter centered at $1600$ nm. From 9000 2D frames, each one acquired over 5
pulses, we selected the central rows to pass to 1D angular spectra and
calculated the intensity covariance. The result is shown in Fig. 2 (d). As in
the case of wavelengths, the angular distribution of the covariance shows the
auto-correlation part (diagonal) and the cross-correlation part (anti-
diagonal). Panel (e) of the same figure shows the calculated cross-
correlation. Similarly to the case of wavelengths, the covariance was
calculated as $|F(\omega_{p}/2,\omega_{p}/2,k,k^{\prime})|^{2}$, with the
effective narrowing of the pump beam waist by a factor of $3.2$ taken into
account. Panel (f) shows the cross-sections of the experimental (red points)
and the calculation (blue line) distributions. The measured angular width of
correlations is $\delta\theta=4.1$ mrad, somewhat larger than the calculated
value $3.4$ mrad. The resulting width in the transverse wavevector is $\delta
k=0.016\>\mu$m-1.

## Spatiotemporal correlations

We used the autocorrelator to measure the dependence of
$g^{(2)}(t,t^{\prime},x,x^{\prime})$ on $\tau\equiv t-t^{\prime}$ and
$\xi\equiv x-x^{\prime}$. For a fixed position of $BS_{2}$ we scanned the time
delay in one arm with $R_{1}$, similar to Ref. [34]. For each time delay
$\tau$ between the two arms, we obtained a spectrum as shown in Fig. 1 (d).
The height of the peak strongly depends on $\tau$ and becomes zero at large
delays, indicating that only photons that belong to different pairs merge into
a sum-frequency photon. From the spectra, we calculated the value of the CF,
according to Ref. [35], as $g^{(2)}(\tau)=2(B(\tau)/B_{max}+R(\tau)/B(\tau))$,
where $R(\tau)$ is the integral spectrum of the peak (red area in Fig. 1 (d)),
$B(\tau)$ the integral spectrum of the background (blue area in Fig. 1 (d)),
and $B_{max}$ its maximal value, which is achieved at $\tau=0$.

Figure 3: Second-order CF $g^{(2)}(\tau,\xi)$ measured (b,c) and calculated
(e,f) for the collinear (b,e) and non-collinear (c,f) PDC. Panels (a) and (b)
show the profiles of panels (b) (points) and (e) (line) at $\xi=0$ and
$\tau=0$, respectively.

This is how we obtained the 1-dimensional plot shown in Fig.3 (a) for $\xi=0$.
The correlation time $\delta\tau$ is equal to its full width at half maximum
(FWHM). We repeated this measurement for each spatial displacement $\xi$ made
with $BS_{2}$. Importantly, each spatial displacement generated an extra time
delay, which we compensated by retroreflector $R_{1}$. By stacking together
$g^{(2)}(\tau)$ distributions at different $\xi$, we obtained the two-
dimensional distribution $g^{(2)}(\tau,\xi)$ (Fig. 3) for both collinear (b)
and non-collinear (c) PDC. Panels (e) and (f) show the corresponding
theoretical distributions, calculated as squared two-dimensional Fourier
transforms of Eq.(2). The distributions for the non-collinear configuration
are narrower due to a broader wavelength-angular spectrum. In agreement with
the theoretical calculation, the correlation time and distance for the
collinear case are, respectively, $18\pm 2$ fs and $45\pm 6$ $\mu$m, and for
the non-collinear case, $12\pm 2$ fs and $33\pm 6$ $\mu$m. The peak value
$g^{(2)}(0,0)$ obtained is $2.2$ and not $3$, as expected, due to the group-
velocity dispersion in the optical elements. Panel (d) shows the space profile
of the CF at $\tau=0$ for collinear PDC: the points and the line are the
cross-sections of panels (b) and (e), respectively.

Figure 4: The CF $g^{(2)}(\tau)$ (a) and $g^{(2)}(\xi)$ (b) measured over
larger time and space scales. The pedestals follow Gaussian distributions
(blue lines) given by the BSV pulse duration (a) and beam waist (b).

Figure 3 shows the CF $g^{(2)}(\tau,\xi)$ distributions only in the vicinity
of its maximum at $\tau=\xi=0$. We also measured its behavior at a larger
scale, with the results shown in Fig. 4 (a,b). The background follows a
Gaussian distribution (blue line) whose widths in time and space are
determined, respectively, by the BSV pulse duration and the beam waist. In
agreement with the above-mentioned effect of the high gain, both widths are
indeed $3.2$ times smaller than those for the pump [36, 18, 25].

Finally, we calculate the time-frequency and space-wavevector products. For
collinear degenerate PDC, we get $\delta\tau\delta\omega=0.16$ and
$\delta\xi\delta k=0.72$. Both values are far below the Mancini (or Fourier)
bound $2\pi$. We note that the second condition, $\delta\xi\delta k\ll 2\pi$,
was also demonstrated recently for BSV obtained through four-wave mixing,
using noise reduction measurement [37].

In conclusion, we measured the the second-order CF for BSV as a function of
space and time. We obtained the temporal and spatial correlation widths of
$\delta\tau=18$ fs and $\delta\xi=45\,\mu$m for the collinear degenerate
configuration and $\delta\tau=12$ fs and $\delta\xi=33\,\mu$m for the non-
collinear non-degenerate configuration, respectively. These values define the
resolution for two-photon spectroscopy with BSV. The high temporal and spatial
resolution is useful for analyzing complex samples and extracting information
about their energy levels. In addition, we found the correlation widths of BSV
in frequency and transverse wavevector, $\delta\omega=8.8\cdot 10^{12}$ Hz,
$\delta k=0.016\>\mu$m-1, respectively. As a result, we have demonstrated that
BSV enables simultaneously high resolution in time and frequency, and space
and wavevector.

We thank Tomás Santiago-Cruz for the help in experiment and Denis Kopylov for
helpful discussions.

## Funding.

P. Cutipa acknowledges the funding provided by the National Agency for
Research and Development (ANID) / DOCTORADO BECAS CHILE/2017 - 72180453.

## Disclosures.

The authors declare no conflicts of interest.

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