Eur. Phys. J. C (2018) 78:275 Page 3 of 14 275
models regarding s → dν ¯ν transitions, but rather note that
Refs. [27,52] discuss some tree-level effects.
According to the classification in [48], scalar leptoquarks
which are doublets of SU(2)
L
cannot destabilize the pro-
ton via the diquark coupling, since 3B + L = 0, B and L
being baryon and lepton numbers,
4
and therefore can be rel-
atively light. Moreover, it was pointed out in [24,55] that
the weak triplet leptoquark, although having 3B + L = 2,
when embedded in a SU(5) GUT model does not desta-
bilize the proton. The weak doublet leptoquarks may have
weak hypercharges 7/6or1/6. In what follows, differ-
ent leptoquarks are denoted by their transformation under
(SU(3), SU(2)
L
, U (1)
Y
) and we adopt here the notation
introduced in Ref. [48].
2.2.1 R
2
(3, 2, 7/6) scalar LQ
New physics exclusively in semi-leptonic processes natu-
rally evokes LQ frameworks since flavor changing processes
involving four-lepton and four-quark transitions are loop sup-
pressed. Despite that, it might seem sensible to invoke a LQ
model that contributes to b → s process at one-loop level
(with contributions to b → cν at tree-level) and thus mim-
ics the SM amplitudes hierarchy, and may therefore be ade-
quate to address R
K
(∗)
and R
D
(∗)
anomalies simultaneously.
An approach along these lines has been followed in [40]but
large couplings needed to explain R
K
(∗)
at loop-level and
R
D
(∗)
at tree-level are difficult to reconcile with all available
flavor constraints [56]. Here we follow the approach of [41]
where the anomalies in b → s decays are accounted for at
one-loop level.
The Lagrangian describing the interaction of R
2
(3, 2, 7/6)
with quarks and leptons is given by [48,57]:
L
R
2
= (Vg
R
)
ij
¯u
i
P
R
e
j
R
5/3
2
+ (g
R
)
ij
¯
d
i
P
R
e
j
R
2/3
2
+ (g
L
)
ij
¯u
i
P
L
ν
j
R
2/3
2
− (g
L
)
ij
¯u
i
P
L
e
j
R
5/3
2
+ h.c.,
(7)
where P
L(R)
= (1 ∓ γ
5
)/2. The neutrino masses are neg-
ligible in K decays, and then the PMNS matrix reduces to
the identity matrix, 1
3
. In order to explain R
K
(∗)
anomalies it
was suggested in Ref. [41] that the Yukawa matrices g
R
and
g
L
have the following textures
g
R
=
⎛
⎝
00 0
00 0
00(g
R
)
bτ
⎞
⎠
, g
L
=
⎛
⎝
00 0
0 (g
L
)
cμ
(g
L
)
cτ
0 (g
L
)
tμ
(g
L
)
tτ
⎞
⎠
. (8)
Throughout this article, coupling constants are always taken
to be real. Regarding their allowed values, the process
4
For B and L violation in the scalar potential, see [53,54].
τ → μγ receives a LQ contribution proportional to
(g
R
)
bτ
(g
L
)
tμ
V
tb
(m
t
/m
τ
), which is chiraly enhanced, see
[41]. In order to allow for a large (g
L
)
tμ
, (g
R
)
bτ
must be
suppressed and here we set it to zero. Therefore, the only
LQ couplings to fermions are given by g
L
. Moreover, the
couplings g
cτ
L
and g
tτ
L
are strongly constrained by the same
process τ → μγ when g
cμ
L
, g
tμ
L
are both large, of order
O(1).
With the ansatze for g
R
and g
L
in Eq. (8) and the afore-
mentioned constraints, tree-level contributions to b → cν,
and contributions to b → s involving right-handed cur-
rents, are absent. Note, however, that a different coupling
texture was used in Refs. [57,58] to explain the R
D
(∗)
anoma-
lies. See also [59] for a different texture of LQ Yukawa cou-
plings.
In order to further constrain this model, the following
bounds are also important:
(a) in order to explain the R
K
(∗)
anomalies we are compelled
to adjust the NP Wilson coefficient δC
9μ
=−δC
10μ
,
(b) the constraint coming from the difference between the
experimental and theoretical results for the muon anoma-
lous magnetic moment a
exp
μ
− a
SM
μ
= a
μ
= (2.88 ±
0.63±0.49)×10
−9
[60,61] (see, e.g., [62] for the break-
down of uncertainties),
(c) the precise experimental value of Br(Z → μμ), mea-
sured at LEP (cf. Ref. [63]).
These bounds and requirements are summarized in Fig. 1.
Note that the Br(Z → μμ) is sensitive to large values of
g
tμ
L
. In this figure, we limit the couplings to |g
cμ
L
|
2
, |g
tμ
L
|
2
≤
4π in order to stay within the perturbative regime. For
the combined 1σ regions indicated, the predicted anoma-
lous magnetic moment of the muon gets further wors-
ened by 1σ . We have also checked that charmonia
decays [64,65] do not impose important bounds. The expres-
sion for R
(∗)
νν
can be related to loop-induced amplitudes of
b → sμμ transitions, however the resulting constraints are
weak.
Note that LHC constraints of flavored processes at high
energies are becoming sensitive to flavor couplings employed
in low-energy flavor phenomenology, see, e.g., [66]. Their
analysis sets constraints on effective four-fermion opera-
tors contributing to pp → μ
+
μ
−
at the tail of the di-
lepton invariant mass spectrum. Therefore, the results apply
directly for a NP spectrum beyond few TeV, but they may
still remain indicative at the region 1 TeV, in which case a
rough estimate of g
cμ
L
from their study gives |g
cμ
L
| 0.8.
This is not very different from the values used in our anal-
ysis, but for a more precise knowledge of this coupling,
dedicated analyses of LHC data would clarify the situa-
tion.
123