#### Anomaly-free models for flavour anomalies

Eur. Phys. J. C
Anomaly-free models for flavour anomalies
John Ellis 0 1
Malcolm Fairbairn 1
Patrick Tunney 1
0 Theoretical Physics Department , CERN, 1211 Geneva 23 , Switzerland
1 Theoretical Particle Physics and Cosmology Group, Physics Department, King's College London , London WC2R 2LS , UK
We explore the constraints imposed by the cancellation of triangle anomalies on models in which the flavour anomalies reported by LHCb and other experiments are due to an extra U(1) gauge boson Z . We assume universal and rational U(1) charges for the first two generations of lefthanded quarks and of right-handed up-type quarks but allow different charges for their third-generation counterparts. If the right-handed charges vanish, cancellation of the triangle anomalies requires all the quark U(1) charges to vanish, if there are either no exotic fermions or there is only one Standard Model singlet dark matter (DM) fermion. There are non-trivial anomaly-free models with more than one such 'dark' fermion, or with a single DM fermion if right-handed up-type quarks have non-zero U(1) charges. In some of the latter models the U(1) couplings of the first- and secondgeneration quarks all vanish, weakening the LHC Z constraint, and in some other models the DM particle has purely axial couplings, weakening the direct DM scattering constraint. We also consider models in which anomalies are cancelled via extra vector-like leptons, showing how the prospective LHC Z constraint may be weakened because the Z → μ+μ− branching ratio is suppressed relative to other decay modes.
1 Introduction
The LHCb Collaboration and other experiments have reported
a number of anomalies in semileptonic B decays, including
apparent violations of μ–e universality in B → K (∗) + −
decays [
1–4
], and apparent deviations from the Standard
Model (SM) predictions for the P5 angular variable in
B → K ∗ + − decay [
5–9
] and the mμ+μ− distribution in
Bs → φμ+μ− decay [
10,11
]. These anomalies have reached
a high level of statistical significance. There are ongoing
discussions of the possible systematic effects and the
uncertainties in the SM calculations, so the jury is still out on the
significances of these flavour anomalies [
12
]. In the
meantime, it is interesting to explore possible interpretations and
look for other phenomenological signatures that might
corroborate them [
13–32
].
In the wake of this recent report of μ–e non-universality in
B → K ∗ + − decay [
1,2
], several phenomenological
analyses favour an anomalous non-SM contribution to the
coefficient of the dimension-6 operator O9μ ≡ (s¯γμ PL b)(μ¯ γ μμ),
but do not exclude a smaller non-SM contribution to the
coefficient of O1μ0 ≡ (s¯γμ PL b)(μ¯ γ μγ5μ) or O9μ ≡
(s¯γμ PR b)(μ¯ γ μμ) [
13–19,27
]. A popular interpretation of
this anomaly is that it is due to the exchange of a U(1) gauge
boson Z with non-universal couplings to both quarks and
leptons [
15,22–24,27,28,31,33–52
]. It is possible to take a
purely phenomenological attitude to this possibility, and not
(yet) concern oneself about the theoretical consistency of
such a Z model. However, any gauge theory should be free
of triangle anomalies, which in general need to be cancelled
by fermions with masses comparable to that of the Z . Thus,
not only does this requirement have the potential to
constrain significantly the possible U(1) couplings of both SM
and non-SM particles, but it may also suggest novel
signatures that could confirm or disprove such a Z interpretation
of the LHCb B → K ∗ + − measurements and other flavour
anomalies.
We explored recently the impact of the
anomalycancellation requirement on simplified dark matter (DM)
models, assuming generation-independent U(1) couplings
to quarks and leptons [
53
]. Rather than take a top-down
based on some postulated ultraviolet scenario, we proposed
some minimal benchmark models with desirable
characteristics such as suppressed leptonic couplings (so as to reduce the
impact of unsuccessful LHC searches for massive Z bosons)
or axial coupling to quarks (so as to reduce the impact of
direct searches for DM scattering).
In this paper we follow an analogous strategy for
flavourful Z models with generation-dependent U(1) couplings to
quarks and leptons, treating DM as a possible optional extra.
Motivated by the long-standing discrepancy between
experiment and the SM prediction for the anomalous magnetic
moment of the muon, gμ − 2, we also consider models with
additional leptons L that are vector-like under the SM gauge
group but might have parity-violating U(1) couplings [
54
].
Since the LHCb anomaly could be explained by just a
left-handed flavour-changing quark coupling ∝ s¯γμ PL b,
but not by a coupling ∝ s¯γμ PR b alone, we mainly follow
[
54
] in assuming that the right-handed charge -1/3 quarks
have vanishing U(1) charges.1 However, we also discuss
in the appendix a scenario for anomaly cancellation with
non-vanishing charges to right-handed charge − 1/3 quarks.
In view of the strong upper limits on anomalous
flavourchanging interactions of strange quarks, we also assume [
54
]
that the first two generations of left-handed down-type quarks
dL , sL have identical U(1) charges, whereas the bL charge is
different, opening the way to the suggested flavour-changing
neutral interaction.2 The reported μ–e non-universality and
LEP constraints lead us also to assume that the electron has
vanishing U(1) couplings [
54
], but we allow arbitrary U(1)
charges for the left- and right-handed μ and τ , generalizing
the anomaly-free models discussed previously that assumed
a Lμ − Lτ charge in the lepton sector [
33–36
]. The
semileptonic B decay data suggest that the couplings of the μ to
the Z are predominantly vector-like, corresponding to
dominance by the O9μ operator over O1μ0, but we do not impose
this restriction a priori. We discuss anomaly-free models with
muon couplings that are not completely vector-like, but have
combinations of O9μ and O1μ0 that are nevertheless consistent
with a global fit the flavour anomalies, as seen in Figure 1.
In the absence of U(1) charges for the right-handed charge
2/3 quarks, we show that the anomaly-cancellation
conditions are so restrictive that there are no solutions with
nonvanishing U(1) charges for quarks. This is also the case if
we include a single dark sector particle. However, we do find
acceptable solutions if we allow for a second ‘dark’ fermion,
as illustrated in Table 1.
There are also solutions with a single DM particle if we
allow non-vanishing U(1) charges for the u R , cR (assumed
to be equal) and tR (allowed be different). We have scanned
for all possible triangle anomaly-free models with charges
that can be expressed in the form p/q with p, q ∈ [
− 4, 4
]
when we normalise the left-handed DM charge YχL = 1.
Among these, four have vector-like μ couplings, three have
no couplings to the first two generations of quarks, and three
1 In contrast, the anomaly-free model proposed in [
31
] has a non-zero
U(1) charge for the bR.
2 Models with similar Z couplings to left-handed charge − 1/3 quarks
were considered in [
15,28
], the latter in the context of an anomaly-free
horizontal symmetry motivated by the Pati–Salam [55] model. See also
[
27
], where the possibility of a weakly-coupled light Z boson was also
considered.
have axial couplings for the DM fermion (as required if it is a
Majorana particle, and which would suppress DM scattering
by a relative velocity factor). These models are all distinct,
with the exception of a single model that combines a
vectorlike Z coupling to the muon with an axial coupling to the
DM particle. We display in Table 2 the U(1) charges for this
model, two other models with vector-like muon couplings
and one with vanishing Z couplings for the first two
generations, as benchmarks that illustrate the potential signatures
of anomaly-free models of flavourful Z bosons with DM.
We also explore models in which anomalies are cancelled
by extra vector-like leptons [
54
], exhibiting an example in
which the LHC Z → μ+μ− signal is suppressed because
of a small Z → μ+μ− branching ratio. Such a model may be
able to explain the discrepancy between SM calculations and
the experimental measurement of the anomalous magnetic
moment of the muon, gμ − 2 [
56
].
The construction described in the appendix allows for a
non-vanishing coupling to all types of quarks, including RH
down-type quarks, and solves all the anomaly constraints
without the need for exotic fermions. In this case the
coefficient of O1μ0 vanishes, and there is an admixture of the O9μ
and O9μ operators.
2 Scenarios for anomaly cancellation
The following are the six anomaly-cancellation conditions to
be considered:
(a) [SU(3)C2 ] × [U(1) ], which implies Tr[{T i , T j }Y ] = 0,
(b) [SU(2)2W ] × [U(1) ], which implies Tr[{T i , T j }Y ] = 0,
(c) [U(1)2Y ] × [U(1) ], which implies Tr[Y2Y ] = 0,
(d) [U(1)Y ] × [U(1) 2 ], which implies Tr[Y Y 2] = 0,
(e) [U(1) 3], which implies Tr[Y 3] = 0,
(f) Gauge-gravity, which implies Tr[Y ] = 0.
In general there can be independent U(1) charges for each
of the 3 × 5 = 15 multiplets of the Standard Model,3
which we label qL,i , u R,i , dR,i , lL,i and eR,i where i is a
generation index, as well as charges for any extra particles
beyond the SM. However, as mentioned in the Introduction,
we make simplifying assumptions motivated by
phenomenological considerations.
SM particles only, no Z couplings to electrons or
righthanded quarks
3 Imposing U(1) invariance of Yukawa interactions and allowing for
quark mixing would, in this case, require more than just the single Higgs
multiplet of the SM. Since this does not affect anomaly cancellation,
we do not investigate this issue further here.
Motivated by the indication that the flavour anomalies
originate in the U(1) couplings to left-handed charge − 1/3
quarks, initially we set the charges of all the right-handed
quarks to zero: Yu R,i = 0 = YdR,i , though this is not
mandated by the data. Motivated by the strong upper limits on
non-SM flavour-changing interactions between the first two
generations of charge − 1/3 quarks, we assume that the
lefthanded doublets in the first two generations have identical
charges YqL ,1 = YqL ,2 ≡ Yq [
54
]. These differ from that of
the third left-handed doublet YqL ,3 ≡ Yt , making possible
the desired flavour-changing b¯γμ PL s coupling.
A complete discussion of the implications of constraints
on flavour-changing couplings, e.g., from F = 2 processes,
is beyond the scope of this work, since it would depend on
the structures of the individual matrices that rotate the quark
fields into the mass basis. Experimentally, only the
combination entering into the CKM matrix is known, and the
structures of the individual matrices depend on details of the
Higgs representations and Yukawa coupling matrices that are
independent of the anomaly cancellation conditions that we
consider here.
In order to avoid the experimental constraints from LEP
and other electroweak measurements [
54
], we also assume
that the electron charges vanish: YlL ,1 = 0 = YeR,1.
However, we allow independent left- and right-handed couplings
for the muon and tau. With these assumptions, and in the
absence of any particles beyond the SM, the
anomalycancellation conditions become
2(2Yq + Yt ) = 0
YμL + YτL + 3Yt + 6Yq = 0
2
3
3YμL − 6YμR + 3YτL − 6YτR + Yt + 2Yq
2 −YμL2 + Yμ2R − YτL2 + YτR2 + Yt 2 + 2Yq 2
− Yμ3R − YτR3 + 2 YμL3 + YτL3
+ 6 Y 3
t + 2Yq 3
= 0
= 0,
− YμR − YτR + 2 YμL + YτL + 6 Yt + 2Yq
= 0. (2.6)
Solving the conditions (2.1), (2.2), (2.3) gives the relations
Yt = −2Yq ,
YτL = −YμL ,
YτR = −YμR .
Using these relations to solve the conditions (2.4) then yields
4 −YμL2 + Yμ2R + 3Yq 2
= 0,
(2.1)
(2.2)
(2.3)
(2.4)
= 0,
(2.5)
(2.7)
(2.8)
(2.9)
(2.10)
which has rational solutions for any rational value of Yq , since
any odd number can be written as the difference between two
squares. Equation (2.10) implies that if Yq = 0 the muon
could not have vector-like U(1) couplings as suggested by
the data, but this is a moot point in this scenario, since (2.5)
yields
−36Yq 3 = 0.
(2.11)
Hence Yq = 0, which in turn implies via (2.7) that Yt also
vanishes and the Z decouples from quarks.4 Therefore, we
must relax the assumptions made above if the flavour
anomalies are to be explained by the exchange of a Z boson.
Including one or two ‘dark’ particles
Adding a single DM particle with vanishing SM couplings
does not remedy the situation, as none of the conditions
(2.7), (2.8), (2.9) are affected, and condition (2.6) implies
TrBSM[Y ] = 0. Hence, if there is a single DM particle it
must have vector-like U(1) couplings, and would not change
the fatal condition (2.11).
The next simplest possibility has two SM-singlet ‘dark’
fermions, A and B, in which case we have, in addition to
(2.7, 2.8) and (2.9) the condition that
YAL = YAR − YBL + YBR
(2.12)
and the remaining anomaly conditions to solve are (2.10) and
−36Yq 3 + 3(YBL − YAR )(YBL − YBR )(YAR + YBR ) = 0,
(2.13)
instead of (2.11) As in the single DM particle case, the
condition (2.10) implies that the muon cannot have vector-like
U(1) couplings. Normalizing YAR = 1 and restricting our
attention to anomaly-free models with U(1) charges that can
be expressed as p/q with p, q ∈ [
− 4, 4
], we find several
solutions with YμL /YμR = 2 .5 In these models the ratio of
the vector-like and axial muon couplings YμV /YμA = − 3,
which may be consistent with the ratio of O9μ and O1μ0
coefficients C μ, C1μ0 allowed by the analyses in [
13–19,27
], as
9
indicated in Fig. 1. We see that the green dot-dashed line
corresponding to models with YμV /YμA = − 3 traverses
the region of the (C9μ, C10) plane preferred in the
analy
μ
sis of [
13
] at the 1-σ level. These models have the same
U(1) charges for the SM particles but different values for
YAR,BL,R = 0, ± 1/3, ± 4/3. None of these solutions has a
DM candidate with a purely axial U(1) coupling, though we
4 The possibility that the Z couples only to t quarks and that the LHCb
flavour anomalies are loop-induced was considered in [
22
].
5 We also find models with vanishing YAR and the same ratio
YμL /YμR = 2.
cannot exclude the possibility that the SM-singlet fermions
might mix in such a way that the lighter mass eigenstate does
have an axial coupling. The U(1) charges of one
representative model are shown in Table 1.
The identification of the lighter SM-singlet fermion mass
eigenstate depends on details of the mixing in the dark
sector that we do not discuss here. Various experimental
constraints should be considered for this fermion to be a realistic
DM candidate: the correct thermal relic density should be
obtained, the cross-sections for scattering on nuclei should
be below the sensitivities of current direct detection
experiments, and LHC and indirect detection bounds should be
taken into account where appropriate. Anomaly cancellation
constrains only the Y charges but not the overall magnitude
g of the gauge coupling, which could be fixed by the
requiring the observed abundance of dark matter. When combined
with the Y charges and mixing patterns in specific models,
predictions for the LHC and dark matter experiments could
be made, but such a study lies beyond the scope of this work.
Including couplings for right-handed charge 2/3 quarks
As an alternative way to relax our initial assumptions, we
allow next for non-vanishing U(1) charges for the
righthanded charge 2/3 quarks u R,i . We recall that the flavour
anomalies apparently originate from left-handed b and s
quarks, but there is no reason to forbid U(1) couplings to
right-handed charge 2/3 quarks. In this Section we assume
that the charges of the RH charge − 1/3 quarks vanish, but
we relax this assumption in the appendix. We assume that
the U(1) charges of the first two generations are identical,
i.e., Yu R ,1 = Yu R ,2, again with the motivation of suppressing
flavour-changing neutral interactions.
In the absence of non-SM fermions, one can readily solve
the anomaly conditions (a, b, c) and (f) above, and the
conditions (d) and (e) then take the following forms:
YlL2,2 − Y 2
eR ,2 − 3YqL2,1 + 6Yu R2,1 = 0,
−2YqL3,1 + Yu R3,1 = 0.
These conditions clearly have non-trivial solutions, but (2.15)
does not admit rational values for both YqL ,1 and Yu R ,1. All
the unification scenarios known to us have rational values for
U(1) charges, so these solutions are not acceptable.
Including couplings for right-handed charge 2/3 quarks and
a DM particle
We are therefore led to consider adding a single DM fermion
χ with charges YχL and YχR . Normalizing YχL = 1, the
anomaly conditions (a), (b), (c), (d) and (f) yield the following
expressions for the other charges:
(2.14)
(2.15)
YtR =
Yτ,L =
−2YqL − 3YμL + 8YqR + 4YμR + (Y )q2L + (Y )2μL + 6(Y )q2R − (Y )2μR ,
2YqL + 3YμL − 8YqR − 4YμR
2 YqR 4YμR − 3YμL − 2YqL YqR + 3(Y )q2L − (Y )2μL + 2(Y )q2R + (Y )2μR ,
8YμL YqR − 2YqL YμL − 9 Y
2YqL + 3YμL − 8YqR − 4YμR
2
qL + YμR 4YμL − 3YμR + 18 Y
2YqL + 3YμL − 8YqR − 4YμR
2
qR ,
−2YqL YμR − 12(Y )q2L + YμL 4YμL − 3YμR + 8YqR YμR + 24(Y )q2R ,
2YqL + 3YμL − 8YqR − 4YμR
−6(Y )q2L + 2YqL + 2(Y )2μL + 3YμL + 12(Y )q2R − 8YqR − 2(Y )2μR − 4YμR ,
2YqL + 3YμL − 8YqR − 4YμR
(2.16)
(2.17)
(2.18)
(2.19)
(2.20)
when 2YqL + 3YμL − 8YqR − 4YμR = 0.6 At this stage YqL ,
YqR , YμL and YμR are undetermined, but we have not yet
applied the anomaly condition (e), which yields an additional
constraint that is algebraically complicated and unrevealing.
Scanning over the four undetermined charges, we find a set
of solutions with YqL = YqR = 0 (which would suppress
Z production at the LHC and direct DM scattering) and
either YμL = YμR (as favoured by the data) or YμL = −YμR .
However, these solutions also have vanishing couplings for
the third-generation quarks, i.e., YtL = YtR = 0, so all the
quark charges vanish.
We are therefore forced to make a ‘Solomonic choice’
between models with vector-like couplings to muons, i.e.,
YμL = YμR , and those with vanishing couplings to first- and
second-generation quarks. Scanning over rational values of
U(1) that can be expressed in the form p/q : p, q ∈ [
−4, 4
],
we find 4 models with vector-like muon couplings and 3 that
have vanishing first- and second-generation quark couplings.
One of the models with vector-like muon couplings also has a
DM particle with a purely axial U(1) coupling that could be
a Majorana particle. The U(1) charges of this model (A) are
listed in Table 2, along with the corresponding charges for
some other models that may serve as interesting benchmarks.
The charges in the second and third rows are for models
(B, C) with vector-like muon couplings but non-axial DM
couplings, and the charges in the bottom two rows are for
models (D, E) with vanishing first- and second-generation
quark couplings and a mixture of vector and axial couplings
to the muon.
In models such as (D, E), the Z production mechanisms
via first- and second-generation q¯ q annihilations that are
usually dominant at the LHC are suppressed, and the constraint
on the Z mass coming from production via b¯s +s¯b collisions
6 There are no ‘interesting’ solutions with vector-like muon couplings,
vanishing first- and second-generation couplings or axial DM couplings
when 2YqL + 3YμL − 8YqR − 4YμR = 0.
is much weaker [
24
]. Moreover, the constraint from searches
for direct DM scattering on nuclei is greatly weakened.
Although the Z coupling to muons is not purely vectorial
in model (D), the ratio of the vector and axial muon couplings
is 7 in this model, so the axial coupling might be
acceptably small. As seen in Fig. 1, the data allow a non-vanishing
axial/vector ratio, although they prefer the opposite relative
sign. Models with YμV /YμA = 7 (dashed purple line) are
compatible with the region of the (C9μ, C10) preferred in the
μ
analysis of [
13
] at the 2-σ level.
Model (E) shares the property of having no coupling to the
first two generations of quarks but has a different mixture of
axial/vector coupling, YμV /YμA = − 1, since it has a purely
left-handed muon coupling. This is compatible with the fit
shown in Fig. 1 at the 3 σ level.
We also found a model (not shown) with vanishing
firstand second-generation quark couplings, but with a muon
coupling that is either purely right-handed, which is disfavoured
by the data.7
Including a vector-like lepton
Finally, we consider a scenario proposed in [
54
] in which
the SM particles are not supplemented by DM, but by extra
leptons, a vector-like doublet (ν , ) and a vector-like singlet
μ . We assume that only the left-handed quarks have non-zero
U(1) charges, with those for the first two generations being
the same. We also assume that the Z coupling of the muon is
purely vectorial. The left-handed components of the doublet
and the right-handed component of the singlet are assumed
to have identical values of Y , but the U(1) charges of the
right-handed doublet and left-handed singlet are free a priori.
Thus the free parameters of the model are YqL , YtL , YμL =
YμR , YτL,R , Y L = YμR , Y R and YμL .
7 We also find models with YχL = 0 but with YχR = 0, in which
YμV /YμA = 7, 0 or −1, which appear to be compatible with the data at
the 2-, 0- and 3-σ level, respectively.
Table 2 The U(1) charges in
some benchmark models with
couplings for right-handed
quarks and a single dark matter
particle that have interesting
properties: (A) vector-like μ
coupling and axial DM
coupling, (B, C) vector-like μ
coupling, (D) no first- and
second-generation couplings
and relatively small axial-vector
μ coupling: YμV /YμA = 7, (E)
no first- and second-generation
couplings and YμV /YμA = −1
We have also scanned rational values of these free
parameters that can be expressed in the form p/q : p, q ∈ [
− 4, 4
].
Since one of the objectives of [
54
] is to explain the
discrepancy between SM calculations of gμ −2 and the experimental
measurement [
56
], via a contribution ∝ 1/MZ 2 , it is
desirable to focus on solutions in which the LHC Z → μ+μ−
signal is suppressed. Since the U(1) charges of the first- and
second-generation quarks are non-vanishing, the only way
to suppress the prospective LHC signal is to suppress the
Z → μ+μ− branching ratio. We have found several
models in which the combined branching ratios for other decays
exceed that for Z → μ+μ− by more than an order of
magnitude. Table 3 displays the model in which the branching ratio
for Z → μ+μ− is most suppressed by the U(1) charges
and multiplicities of states, namely to 3/130, assuming that
the masses of the extra leptons can be neglected, as is the
case if all the fermions are much lighter than MZ /2. Since
MZ may be in the TeV range, this is compatible with the
lower limits on the masses of vector-like leptons given by
the Particle Data Group [
57
], which are ∼ 100 GeV, and
with model-dependent recasts of LHC searches [
58
], which
yield limits ∼ a few hundred GeV.
3 Summary and conclusions
We have explored in this paper the constraints on Z
interpretations of the flavour anomalies in B → K (∗) + − decays
imposed by the cancellation of triangle anomalies, namely
the conditions (a) to (f) stated at the beginning of Section 2.
Models with right-handed charge 2/3 quark couplings and one DM fermion
YqL YqR YtL YtR YμL YμR YτL
Vector-like μ coupling and axial DM coupling
(A) 0 1 1 0
Vector-like μ couplings
(B) 1/3 1/3 − 1/3 0
(C) 1/2 0 − 1/2 1
No first- and second-generation couplings
(D) 0 0 1/2 1
(E) 0 0 1/2 1
We find many models that have not been discussed
previously in the literature, and have novel experimental
signatures involving new particles and/or non-trivial combinations
of the operators O9μ and O1μ0 that are consistent with the
reported flavour anomalies.
Motivated by the observed pattern of flavour anomalies,
we considered initially models in which the Z has quark
couplings that are purely left-handed (universal for the first
two generations, non-universal for the third), and it has no
electron coupling. In this case we find no non-trivial solution
of the anomaly-cancellation conditions in the absence of
nonSM particles, and so are led to introduce ‘dark’ fermions
without SM couplings. In the case of a single DM particle,
there is again no non-trivial solution, but we do find solutions
with 2 ‘dark’ fermions. In none of these does the Z have
a purely vector-like muon coupling, but we find a class of
solutions in which YμV /YμA = −3, a ratio that is compatible
with the data at the 1-σ level as seen in Fig. 1. Examples of
these solutions are shown in Table 1.
We then considered models in which the Z couples to
right-handed charge 2/3 quarks, a possibility that is allowed
by the data. In the absence of a dark sector we find no
solution of the anomaly-cancellation conditions with rational
charges, but we do find a number of interesting solutions in
the presence of a DM fermion, and we show some examples
in Table 2. Some of these have vector-like muon couplings—
models (A), (B) and (C)—and in one of these the DM
particle has a purely axial Z coupling—model (A). In models
(D) and (E) there are no Z couplings to first- and
secondgeneration quarks, so production at the LHC is suppressed
and the experimental constraints on the dark mass scale are
correspondingly reduced. Model (D) is one of a class of
models in which YμV /YμA = 7, a ratio that appears compatible
with the data at the 2-σ level, as also seen in Fig. 1. Model
(E) has a purely left-handed muon coupling and so predicts
instead YμV /YμA = −1, which is compatible with the data
at the 3-σ level. We have also considered models in which
the triangle anomalies are cancelled by vector-like leptons,
exhibiting in Table 3 a model with a vector-like Z muon
coupling in which the branching ratio for Z → μ+μ− is
maximally suppressed.
These examples illustrate that anomaly cancellation is
a powerful requirement that could have interesting
phenomenological consequences linking flavour anomalies to
other observables. Anomaly cancellation requires some
extension of the SM spectrum to include, e.g., a dark sector or
a vector-like lepton. Moreover, either the dark sector should
more than just a single DM particle, or some quarks should
have right-handed couplings to the Z boson. Additionally,
we find several classes of models in which YμV /YμA = 0
in a way that is compatible with the present data but could
be explored in the future. Finally, we have shown that it is
possible to cancel the triangle anomalies using vector-like
leptons in such a way as to suppress the LHC Z → μ+μ−
signal, potentially facilitating an explanation of the anomaly
in gμ − 2.
Acknowledgements The work of JE and MF was supported partly by
the STFC Grant ST/L000326/1. MF and PT are funded by the
European Research Council under the European Union’s Horizon 2020
programme (ERC Grant Agreement no. 648680 DARKHORIZONS). We
thank Ben Allanach for discussions.
Open Access This article is distributed under the terms of the Creative
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Funded by SCOAP3.
Note added After the submission of this paper an updated
calculation of the SM prediction for the Bs mass difference
[
59
] appeared, which strongly constrains any new physics
contribution to the b¯s + s¯b coupling. The updated SM
prediction is around 1.8 σ above the experimental measurement,
meaning that new physics scenarios that give a positive
contribution to Ms are strongly constrained at the 2 σ
confidence level. All the models described here, with the exception
of the those in the appendix, give only a positive contribution
to Ms , since they do not have couplings to RH down-type
quarks.
We have examined the impact of these constraints for
the model in Table 1 (denoted “Model 1”) and model D in
Table 2 (denoted “Model 2”). We have found that Model 1
is in significant tension with the new Bs mixing constraint
and LHC dilepton searches for a Z with mass in the TeV
range, whereas Model 2D is viable, by virtue of its vanishing
couplings to first- and second-generation quarks.
More specifically, for Model 1 to fit the combined analysis
of b → sl+l− flavour anomalies shown in Fig. 1 at the 3 σ
level, while also satisfying the new Bs mass mixing bound
at the 3 σ requires MZ < 19.0 TeV for a fixed gauge
coupling g = 6.8 This Z mass is beyond the reach of the latest
ATLAS dilepton resonance search [
60
], but it may well be
inconsistent with the limit in [
60
] on non-resonant effects,
since our couplings are close to the non-perturbative limit,
and the ATLAS limit on an LL contact interaction apparently
excludes MZ < 24 TeV at the 2-σ level.
However, Model 2D can fit the flavour anomalies at the
2 σ level and satisfy simultaneously the updated Bs bound at
the 2 σ level for MZ < 2.8 TeV with a fixed gauge coupling
of 1.5, which is well within the perturbative range. We note,
however, that the required mixing factor in the b¯s coupling is
0.006 for MZ = 2.8 TeV, i.e., smaller than the corresponding
CKM mixing factor. In this case the width of the Z boson
here is 32%, for which the ATLAS analysis [
60
] requires
σ B A < 2.6 × 10−4 fb, where σ is the production cross
section, B is the branching ratio into muons and A is the
detector acceptance.
We leave a more detailed study of this and the other
models to a later study, but note that Z production via b¯s + s¯b
annihilation is suppressed by a small mixing factor ∼ 0.006
in the amplitude, which can be compared with many other
models in the literature where the mixing is ∼ Vts Vtb ∼ 0.04.
Appendix: Non-vanishing couplings to right-handed down-type quarks
For completeness, we investigate in this appendix the effect
of anomaly cancellation on models with couplings to all
quark fields, including charges for the right-handed
downtype quarks. As before, we fix the charges of the first two
quark generations to be equal. However, in order to restrict
the number of unknowns we take a purely vectorial coupling,
μ
Yμ,L = Yμ,R , for which C10 = 0. In this case the anomaly
cancellation conditions read
0 = 2Y3 − Yb − 2Yd + 4Yq − Yt − 2Yu
0 = 3Y3 + 6Yq + Yμ + Yτ,L
2
0 = 3 Y3 − 2Yb − 4Yd
0 = Y3
+2Yq − 8Yt − 16Yu
−3Yμ + 3Yτ,L − 6Yτ,R
2 + Yb 2
+2 Yd
2 + 2 Yq
2 − 2 Yt
2
0 = 6 Y3
−4 Yu
3 − 3 Yb
− Yτ,L
3 − 6 Yd
+ Yτ,R
3
2
2
2
8 This value fixes the muon coupling close to the non-perturbative
limit, which allows for the highest mass Z that can explain the flavour
anomalies while satisfying the new Bs mixing constraint [
59
].
(3.1a)
(3.1b)
(3.1c)
(3.1d)
⎞
+ 8Yd − 4Yq − Yu ⎠
(3.2b)
+ Yq
2 − 2 Yu
2
+ Yq
2
+ 4Yu
Yd
3 − 2 Yq
−
Yd
2
+ Yq
+12 Yq
+2 Yτ,L
3 − 3 Yt
3 − 6 Yu
3
+ Yμ
3
3
−
Yτ,R
3
0 = 6Y3 − 3Yb − 6Yd + 12Yq − 3Yt
−6Yu + Yμ + 2Yτ,L − Yτ,R
with Y3 = Yq,L ,3, Yt = Yu,R,3, Yq = Yq,L ,1/2 and Yu =
Yu,R,1/2, etc. Also, we have Yμ = Yμ,L = Yμ,R .
Tthe anomaly cancellation for all these charges can be
solved completely algebraically. We find the following
constraints for the unconstrained charges Yq , Yu and Yd :
(3.1e)
a right-handed quark coupling is different from that in the
models discussed previously, since it is possible to obtain
a negative contribution to
tion can bring
Ms . Such a negative
contribu
Ms closer to the experimental value than
the current SM prediction, as noted in [
59
], but we leave an
exploration of this issue to a future study.
(3.2a)
(3.2c)
(3.2d)
(3.2e)
(3.2f )
This construction is always possible as long as Y
d
Yq 2 − 2 Yu 2 = 0 and Yd 2 − 2 Yq 2 + Yu 2 = 0.
This class of models features couplings to the first
generation of quarks. However, the Ms bound in models with
+
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