Eur. Phys. J. C (2019) 79:159
https://doi.org/10.1140/epjc/s10052-019-6667-x
Regular Article - Theoretical Physics
Emerging M
d
-anomaly from tree-level determinations of |V
cb
|
and the angle γ
Monika Blanke
1,2,a
, Andrzej J. Buras
3,4
1
Institut für Kernphysik, Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany
2
Institut für Theoretische Teilchenphysik, Karlsruhe Institute of Technology, Engesserstraße 7, 76128 Karlsruhe, Germany
3
TUM-IAS, Lichtenbergstr. 2a, 85748 Garching, Germany
4
Physik Department, TUM, 85748 Garching, Germany
Received: 23 December 2018 / Accepted: 9 February 2019 / Published online: 21 February 2019
© The Author(s) 2019
Abstract We point out that the recently increased value
of the angle γ in the unitarity triangle (UT), determined in
tree-level decays to be γ = (74.0
+5.0
−5.8
)
◦
by the LHCb collab-
oration, combined with the most recent value of |V
cb
| implies
an enhancement of M
d
over the data in the ballpark of 30%.
This is larger by roughly a factor of two than the enhance-
ment of M
s
that is independent of γ . This disparity of
enhancements is problematic for models with constrained
minimal flavour violation (CMFV) and also for U (2)
3
mod-
els. In view of the prospects of measuring γ with the pre-
cision of ±1
◦
by Belle II and LHCb in the coming years,
we propose to use the angles γ and β together with |V
cb
|
and |V
us
| as the fundamental parameters of the CKM matrix
until |V
ub
| from tree-level decays will be known precisely.
Displaying M
s,d
as functions of γ clearly demonstrates the
tension between the value of γ from tree-level decays, free
from new physics (NP) contributions, and M
s,d
calculated
in CMFV and U (2)
3
models and thus exhibits the presence
of NP contributions to M
s,d
beyond these frameworks. We
calculate the values of |V
ub
| and |V
td
| as functions of γ and
|V
cb
| and discuss the implications of our results for ε
K
and
rare K and B decays. We also briefly discuss a future strategy
in which β, possibly affected by NP, is replaced by |V
ub
|.
1 Introduction
The F = 2 transitions in the down-quark sector, that is
B
0
s,d
−
¯
B
0
s,d
and K
0
−
¯
K
0
mixings, have been vital in con-
straining the standard model (SM) and in the search for new
physics (NP) for several decades [1,2]. However, theoretical
uncertainties related to the hadronic matrix elements enter-
ing these transitions and their large sensitivity to the CKM
a
e-mail: monika.blanke@kit.edu
parameters made clear cut conclusions about the presence of
NP impossible. As we demonstrate in this paper, this could
change in the near future.
Among the most important flavour observables we have
at our disposal are
M
s
,M
d
, S
ψ K
S
, S
ψφ
,ε
K
(1)
with M
s,d
being the mass differences in B
0
s,d
−
¯
B
0
s,d
mix-
ings and S
ψ K
S
and S
ψφ
the corresponding mixing induced
CP-asymmetries. ε
K
describes the magnitude of indirect CP-
violation in K
0
−
¯
K
0
mixing. M
s,d
and ε
K
are already
known experimentally with impressive precision. The asym-
metries S
ψ K
S
and S
ψφ
are less precisely measured but have
the advantage of being subject to only very small hadronic
uncertainties.
On the other hand the CKM parameters of particular inter-
est are
|V
us
|, |V
cb
|, |V
ub
|,γ,β, (2)
with the first three being the moduli of the most intensively
studied elements of the CKM matrix, and γ and β being two
angles in the unitarity triangle (UT). The angle γ is to an
excellent approximation equal to the sole complex phase in
the standard parametrization of the CKM matrix.
Now, as elaborated in [3], there are many ways to con-
struct the rescaled UT. They all involve only two inputs, but
as quantified in the latter paper, some pairs are particularly
suited for the determination of the apex ( ¯ρ, ¯η) of this trian-
gle, as only moderate precision on them is required to obtain
a satisfactory determination of ¯ρ and ¯η. The clear winners
from this study are the pairs
(β, γ ), (R
b
,γ), (3)
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