Purely leptonic decays of the ground charged vector mesons
Eur. Phys. J. C
(2021) 81:1110
https://doi.org/10.1140/epjc/s10052-021-09908-w
Regular Article - Theoretical Physics
Purely leptonic decays of the ground charged vector mesons
Yueling Yang1,a , Zhenglin Li1, Kang Li1, Jinshu Huang2 , Junfeng Sun1,b
1 Institute of Particle and Nuclear Physics, Henan Normal University, Xinxiang 453007, China
2 School of Physics and Electronic Engineering, Nanyang Normal University, Nanyang 473061, China
Received: 2 November 2021 / Accepted: 5 December 2021
© The Author(s) 2021
Abstract The study of the purely leptonic decays of the
ground charged vector mesons is very interesting and significant in determining the CKM matrix elements, obtaining
the decay constant of vector mesons, examining the lepton
flavor universality, and searching for new physics beyond
the standard model. These purely leptonic decays of the
ground charged vector mesons are induced by the weak
interactions within the standard model, and usually have
very small branching ratios, B(ρ − →− ν ) ∼ O(10−13 ),
B(K ∗− →− ν ) ∼ O(10−13 ), B(Dd∗− →− ν ) ∼ O(10−10 ),
B(Bu∗− →− ν ) ∼ O(10−10 ), B(Ds∗− →− ν ) ∼ O(10−6 )
and B(Bc∗− →− ν ) ∼ O(10−6 ). Inspired by the potential prospects of LHCb, Belle-II, STCF, CEPC and FCC-ee
experiments, we discussed the probabilities of experimental
investigation on these purely leptonic decays. It is found
that the measurements of these decays might be possible and
feasible with the improvement of data statistics, analytical
technique, and measurement precision in the future. (1) With
the hadron-hadron collisions, the purely leptonic decays of
∗−
∗− mesons might be accessible at LHC
and Bu,c
ρ − , K ∗− , Dd,s
experiments. (2) With the e+ e− collisions, the purely leptonic
∗−
∗− mesons might be measurable with
and Bu,c
decays of Dd,s
12
0
over 10 Z bosons available at CEPC and FCC-ee exper∗−
→ − ν decays could also be
iments. In addition, the Dd,s
studied at Belle-II and SCTF experiments.
1 Introduction
In the quark model [1–3], mesons are generally regarded
as bound states of the valence quark q and antiquark q̄ .
The classifications of mesons are usually based on the spinparity quantum number J P of the q q̄ system. The spin J
of meson is given by the relation |L − S| ≤ J ≤ |L + S|.
The orbital angular momentum and total spin of the q q̄ sysa e-mail: (corresponding author)
b e-mail: (corresponding author)
0123456789().: V,-vol
tem are respectively L and S, where S = 0 for antiparallel
quark spins, and S = 1 for parallel quark spins. By convention, quarks have a positive parity and antiquarks have a
negative parity. Hence, the parity of meson is P = (−1) L+1 .
The L = 0 states are the ground-state pseudoscalars with
J P = 0− and vectors with J P = 1− . Both quarks and leptons are fermions with spin S = 1/2. Mesons are composed
of a pair of fermions – quark and antiquark, therefore, they
could in principle decay into a pair of fermions, for example, lepton and antilepton. The experimental observation of
the two-body purely leptonic decays of mesons could be a
clear and characteristic manifestation of the quark model.
These leptonic decays provide us with valuable opportunities to fully investigate the microstructure and properties of
mesons. The study of two-body purely leptonic decays of
mesons is very interesting and significant.
The valence quarks of the electrically charged mesons
must have different flavors. Within the standard model (SM)
of elementary particles, the purely leptonic decays of the
charged mesons (PLDCM) are typically induced by the treelevel exchange of the gauge bosons W , the quanta of the
weak interaction fields. Up to today, the masses of all the
experimentally observed mesons are much less than those of
W bosons. Consequently, the massive W bosons are virtual
propagators rather than physical particles in the true picture
of PLDCM. Phenomenologically, by integrating out the contributions from heavy dynamical degrees of freedom such
as the W fields, PLDCM can be properly described by the
low-energy effective theory in analogy with the Fermi theory
for β decays. Considering the fact that leptons are free from
the strong interactions, the corresponding effective Hamiltonian [4] for PLDCM could be written as the product of quark
currents and leptonic currents,
GF
Heff = √ Vq1 q2 q̄1 γμ (1 − γ5 )q2 ¯ γ μ (1 − γ5 )ν + h.c.,
2
(1)
where the contributions of the W bosons are embodied in
the Fermi coupling constant G F 1.166×10−5 GeV−2 [1],
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and Vq1 q2 is the Cabibbo–Kobayashi–Maskawa (CKM) [5,6]
matrix element between the quarks in the charged mesons.
The decay amplitudes can be written as,
GF
ν̄ |Heff |M = √ Vq1 q2 ν̄ |¯ γ μ (1 − γ5 )ν |0
2
× 0|q̄1 γμ (1 − γ5 )q2 |M .
(2)
The leptonic part of amplitudes can be calculated reliably with the perturbative theory. The hadronic matrix elements (HMEs) interpolating the diquark currents between
the mesons concerned and the vacuum can be parameterized
by the decay constants.
With the conventions of Refs. [7,8], the HMEs of diquark
currents are defined as,
0|q̄1 (0) γμ q2 (0)|P(k) = 0,
(3)
0|q̄1 (0) γμ γ5 q2 (0)|P(k) = i f P kμ ,
(4)
0|q̄1 (0) γμ q2 (0)|V (k, ) = f V m V μ ,
(5)
0|q̄1 (0) γμ γ5 q2 (0)|V (k, ) = 0,
(6)
where the nonperturbative parameters of f P and f V are the
decay constants of pseudoscalar P and vector V mesons,
respectively; and m V and μ are the mass and polarization
vector, respectively. To the lowest order, the decay widths are
written as,
2
m2
G 2F
(7)
|Vq1 q2 |2 f P2 m P m 2 1 − 2 ,
8π
mP
2
m2
m 2
G2
1+
, (8)
(V → ν̄ ) = F |Vq1 q2 |2 f V2 m 3V 1 − 2
12π
mV
2 m 2V
(P→ ν̄ ) =
where m P and m are the masses of the charged pseudoscalar
meson and lepton, respectively.
It is clearly seen from the above formula that the highly
precise measurements of PLDCM will allow the relatively
accurate determinations of the product of the decay constants and CKM elements, |Vq1 q2 | f P,V . Theoretically, the
decay constants are nonperturbative parameters, and they
are closely related with the q̄1 q2 wave functions at the origin which cannot be computed from first principles. There
still exist some discrepancies among theoretical results of the
decay constants with different methods, such as the potential model, QCD sum rules, lattice QCD, and so on. If the
magnitudes of CKM element |Vq1 q2 | are fixed to the values
of Ref. [1], the decay constants f P,V will be experimentally
measured, and be used to seriously examine the different
calculations on the decay constants with various theoretical models. Likewise, if the decay constants f P,V are well
known to sufficient precision, the magnitudes of the corresponding CKM element will be experimentally determined,
and provide complementary information to those from other
processes. Within SM, the P → ν̄ and V → ν̄ decays are
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(2021) 81:1110
induced by the axial-vector current of Eq. (4) and vector cu (...truncated)
