Patterns of flavour violation in models with vectorlike quarks
Received: October
avour violation in models with vectorlike quarks
Christoph Bobeth 0 1 2 3 4 6
Andrzej J. Buras 0 1 3 4 6
Alejandro Celis 0 1 3 5
Martin Jung 0 1 2 3 6
0 Open Access , c The Authors
1 Boltzmannstr. 2, D85748 Garching , Germany
2 Excellence Cluster Universe, Technische Universitat Munchen
3 Arnold Sommerfeld Center for Theoretical Physics , 80333 Munchen , Germany
4 Physik Department, TU Munchen, JamesFranckStra e , D85748 Garching , Germany
5 LudwigMaximiliansUniversitat Munchen, Fakultat fur Physik
6 TUM Institute for Advanced Study , Lichtenbergstr. 2a, D85748 Garching , Germany
We study the patterns of avour violation in renormalisable extensions of the Standard Model (SM) that contain vectorlike quarks (VLQs) in a single complex representation of either the SM gauge group GSM or G0SM GSM the relative size of Z and Z0induced avourchanging neutral currents, as well as the size of j F j = 2 contributions including the e ects of renormalisation group Yukawa evolution from M to the electroweak scale that turn out to be very important for models with righthanded currents through the generation of leftright operators. In addition to rare ArXiv ePrint: 1609.04783
quarks; Beyond Standard Model; CP violation; Heavy Quark Physics; Kaon Physics

decouple VLQs in the M = (1
10) TeV range and then at the electroweak scale also Z; Z0
gauge bosons and additional scalars to study the phenomenology. The results depend on
decays like P ! ``, P ! P 0``, P ! P 0
patterns and correlations between these observables which taken together should in the
future allow for di erentiating between VLQ models. In particular the patterns in models
with lefthanded and righthanded currents are markedly di erent from each other. Among
the highlights are large Zmediated new physics e ects in Kaon observables in some of the
the data, implying then uniquely the suppression of KL !
) are still possible. We point out that the combination of NP e ects to
. Signi cant enhancements
the masses of VLQs in a given representation independently of the size of VLQ couplings.
1 Introduction 2 The VLQ models 2.2.1
Scalar sectors
Decoupling of VLQs
VLQ representations
Yukawa interactions of VLQs
Yukawa couplings of several representations
Treelevel decoupling and Z and Z0 e ects
GSMmodels
G0SMmodels
Decoupling at oneloop level
Renormalisation group evolution
4 Implications for the downquark sector
4.1 j F j = 2
4.2 j F j = 1: semileptonic dj ! di + (``;
4.3 j F j = 1: hadronic dj ! diqq and "0="
Patterns of avour violation
5.1 j F j = 2
5.2 j F j = 1
GSM models
G0SM( ) model
Determination of M
Kaon and Bmeson systems
Summary and conclusions
A Scalar sectors of G0SMmodels
A.1 G0SM(S) models
A.2 G0SM( ) models
B VLQ decoupling and RG e ects
2'2D operators
2'3 operators
B.3 TopYukawa RG e ects
C Master formulae for K and B decays
C.1 j F j = 2
C.4 dj ! di qq and "0="
D Statistical approach and numerical input
Introduction
Among the simplest renormalisable extensions of the Standard Model (SM) that do not
introduce any additional
ne tunings of parameters are models in which the only new
particles are vectorlike fermions. Such fermions can be much heavier than the SM ones as
they can acquire masses in the absence of electroweak symmetry breaking. If in the process
of this breaking mixing with the SM fermions occurs, the generation of avourchanging
neutral currents (FCNC) mediated by the SM Z boson is a generic implication. If in
addition the gauge group is extended by a second U(1) factor, a new heavy gauge boson Z0
is present and additional heavy scalars are necessary to provide mass for the Z0 and to break
the extended gaugesymmetry group down to the SM gauge group. There is a rich literature
on FCNCs implied by the presence of vectorlike quarks (VLQs), see in particular [1{12].
The goal of the present paper is an extensive study of patterns of avour violation in
models with VLQs that are based on the following gauge groups:
The choice of the particular symmetry group U(1)L
L [13, 14] is phenomenologically
motivated by the fact that it allows in a simple manner to address successfully the LHCb
anomalies [9, 15], while being anomalyfree and containing less parameters than general Z0
models [16].
In our paper we will be guided by the analyses in refs. [3, 11, 17] which identi ed
all renormalisable models with additional fermions residing in a single vectorlike complex
representation of the SM gauge group with a mass M . It turns out that there are 11
models where new fermions have the proper quantum numbers so that they can couple in
a renormalisable manner to the SM Higgs and SM fermions, thereby implying new sources
of avour violation. Our analysis will concentrate on FCNCs in the K, Bd and Bs systems,
therefore only the ve models with couplings to down quarks are relevant for us, as speci ed
in section 2. We call this class of models GSMmodels.
Consequently the models based on the gauge group G0SM are called G0SMmodels. The
VLQs in these models belong to the same representations under GSM as in GSMmodels, but
are additionally charged under U(1)L
L . These models also contain new heavy scalars.
As we will discuss in detail in section 2 and section 5, the patterns of avour violation
in GSMmodels and G0SMmodels di er signi cantly from each other:
In GSMmodels Yukawa interactions of the SM scalar doublet H involving ordinary
quarks and VLQs imply
avourviolating Z couplings to ordinary quarks, which
transitions is much more involved and depends on whether righthanded (RH) or
lefthanded (LH) avourviolating quark couplings to the Z are present. If they
are RH the e ects of renormalisation group (RG) evolution from M (the common
VLQ mass) down to the electroweak scale, EW, generate leftright operators [18] via
topYukawa induced mixing. These operators are strongly enhanced through QCD
RG e ects below the electroweak scale and in the case of the K system through
chirally enhanced hadronic matrix elements. They dominate then new physics (NP)
contributions to "K , but in the Bs;d meson systems for VLQmasses above 5 TeV they
have to compete with contributions from box diagrams with VLQs [11]. If they are
LH the Yukawa enhancement is less important, because leftright operators are not
present and box diagrams play an important role both in the Bs;d and K systems.
In G0SMmodels the pattern of avour violation depends on the scalar sector involved.
We consider only models in which at least one of the additional scalars is charged
under U(1)L
L in such a way that Yukawa couplings between the given VLQ and
ordinary quarks are allowed. If this is the case for a new scalar which is just a singlet
S under the SM group, the latter imply
avourviolating Z0 couplings to ordinary
quarks without any FCNCs mediated by the Z. In the following we refer to these
models as G0SM(S)models. If, on the other hand, such a Yukawa coupling requires the
scalar to be a doublet , both treelevel Z0 and Z contributions to avour observables
will be present. Their relative size depends on the model parameters, speci cally the
Z0 mass. In these cases we introduce again an additional scalar singlet, but without
Yukawa couplings, since otherwise the Z0 mass would have to be of the order of
the electroweak scale, which is phenomenologically very di cult to achieve. In the
following we refer to these models as G0SM( )models.
In this manner we will consider three classes of VLQ models with rather di erent
patterns of avour violation:
G0SM(S) ;
5 TeV by box diagrams with VLQs
and scalar exchanges, while in the G0SM(S) models also treelevel Z0 exchanges can play
an important, sometimes dominant, role. A particular feature of GSM models are the
topare absent in G0SM models.
In [11] an extensive analysis of the GSMmodels has been performed and a subset of
G0SMmodels has been analyzed in [9, 15]. Therefore it is mandatory for us to state what
is new in our article regarding these models:
The authors of [11] concentrated on the derivation of bounds on the Yukawa
couplings as functions of M but did not study the correlations between various avour
observables which is the prime target of our paper. Similar comments apply to [9].
NP contributions to avour observables depend in each model on the products of
complex Yukawa couplings s d
b s for s ! d, b ! d and b ! s
transitions, respectively, as well as the VLQ mass M . This structure allows to set one
of the qphases to zero, such that each model depends on only ve Yukawa
parameters and M , implying a number of correlations between
avour observables. The
strongest correlations are, however, still found between observables corresponding to
the same avourchanging transition, and we concentrate our analysis on them. The
correlations between observables with di erent transitions are weaker, but could turn
out to be useful in the future when the data and theory improve, in particular in the
context of models for Yukawa couplings.
An important novelty of our paper, relative to [9, 11, 15], is the inclusion of the
signi cantly above its SM prediction [19{22]; it is hence of interest to see which of
the models analyzed by us, if any, are capable of addressing this tension and what
the consequences for other observables are.
general is the inclusion of the e ects of RG topYukawa evolution from M to the
electroweak scale that turn out to be very important for models with RH currents through
the generation of leftright operators contributing to these transitions as mentioned
above. This changes markedly the pattern of avour violation in such models relative
to models with LH currents where no leftright operators are generated.
Our paper is organized as follows. In section 2 we present the particle content of the
considered VLQ models, together with the gauge interactions, Yukawa interactions and the
scalar sector. In section 3 we perform the decoupling of the VLQs and construct the e ective
eld theory (G(S0M)EFT) for each model for scales EW <
< M . Section 4 is devoted to the
the scale EW. This results in explicit avourviolating couplings of the Z and Z0 to the SM
quarks. These enter the e ective Lagrangians for the various avourchanging processes,
from which we derive the explicit formulae for the considered observables. In section 5
we describe the patterns of avour violation expected in di erent models, summarizing
them with the help of two DNA tables. In section 6, after formulating our strategy for
the phenomenology, we present numerical results of our study. We conclude in section 7.
Several appendices collect additional information on the models, the decoupling of VLQs,
RG equations in the GSMEFT, the considered decays, some technical details and the input
and statistical procedure used in the numerical analysis.
The VLQ models
Throughout the article we focus on models with vectorlike fermions residing in complex
representations, either of the the SM gauge group GSM or its extension by an additional
L ) symmetry, U(1)L
L . For both models we adapt the usual SM fermion
standard scalar SU(2)L doublet H.
The gauged (L
L ) symmetry is anomalyfree in the SM [13, 14]. The only
non
L ) charges of the SM fermions are introduced as
Q0(L2L) = Q0( R) = Q0`;
Q0(L3L) = Q0( R) =
righthanded singlets. We normalize the (L
L ) charges of the leptons without loss of
gauge boson Z0. However, such couplings are generated in G0SM models through Yukawa
interactions of SM quarks with VLQs that couple directly to Z0.
VLQ representations
As we are mainly interested in the phenomenology of downquark physics, we will restrict
our analysis to SU(3)c triplets and consider the following ve models with SU(2)L singlets,
doublets and triplets:
D(1; 1=3; X);
where the transformation properties are indicated as (SU(2)L; U(1)Y; U(1)L
denotes the charge under U(1)L
L . It is implied that in GSMmodels the U(1)L
charge should be omitted. The representations D, QV , Qd, Td, Tu correspond to the
models V, IX, XI, VII, VIII introduced in ref. [11], where a complete list of renormalisable
models with vectorlike fermions under GSM can be found, see also [3, 17]. Concerning
studied rst in [9].
The kinetic and gauge interactions of the new VLQs are given by
Lkin = D(i D=
MD)D +
Qa(i D=
MQa ) Qa +
Tr T a(i D=
with appropriate covariant derivatives D
and we follow [11] for the triplet representations
as given in (2.13) and (2.14) of that paper. The masses M of the VLQs introduce a new
scale, which we will assume to be signi cantly larger than all other scales. The covariant
derivative is, omitting the SU(3)c part,
= @
with the gauge couplings g2;1 and g0 of SU(2)L, U(1)Y and U(1)L
L , respectively, and
charges Y and Q0 of U(1)Y and U(1)L
L . The Paulimatrices are denoted by
\hat" on Z^0 indicates that we deal here with the gauge eigenstate and not mass eigenstate,
Yukawa interactions of VLQs
The scalar sector consists of the SM scalar doublet H with its usual scalar potential. The
VLQs interact with SM quarks (qL; uR; dR) via Yukawa interactions
LYuk(H) =
iD HyDR + iTd HyT dR + iTu He yT uR q
i 2H . The complexvalued Yukawa couplings iVLQ give rise to mixing with the
SM quarks and avourchanging Zcouplings, which have been worked out in detail [3, 11]
and are discussed in section 3.1.
G0SM(S)
In models with an additional U(1)L
L the scalar sector has to be extended in order
to generate the mass of the corresponding gauge boson Z0. A complex scalar S(1; 0; X)
(SU(3)c singlet) is added in the minimal version. As VLQs are charged under U(1)L
their Yukawa couplings with the SM doublet H are forbidden, but the ones involving S are
allowed for Q0S =
Q0VLQ and given by [9]
LYuk(S) =
iD diR DL + iV QV R qLi S + h.c. :
models as G0SM(S)models. The special feature of these models is that because of the
absence of treelevel Z contributions treelevel Z0 exchanges dominate
F = 1 transitions
and in some part of the parameter space can also compete with contributions from box
diagrams with VLQs and scalars in the case of
F = 2 transitions.
For VLQs with GSM quantum numbers di erent from one of the SM quark elds, the simple
extension by a scalar singlet is not possible. In a nexttominimal version we therefore add
to the scalar sector an additional scalar SU(2)L doublet
(2; +1=2; X), besides the
SMYukawa couplings  see for example [23]  and in consequence there are no LFV Z0
couplings, which are subject to strong constraints at low energies. The vacuum expectation
value (VEV) of
gives an unavoidable contribution to the Z0 mass of the order of the
electroweak scale, contributes to the mass of H and generates potentially large Z
mass mixing e ects. The latter would be strongly constrained by electroweak precision
tests [24], in particular there would be sizeable corrections to the Z couplings to muons. In
order to avoid these di culties,
is accompanied by an additional complex scalar singlet
S(1; 0; Y ), which breaks the U(1)L
L symmetry at the TeV scale. The L
L charge of
scalar sector and to forbid Yukawa couplings of S with SM fermions and VLQs.
The Yukawa interactions of the VLQs with
LYuk( ) =
Td yT dR + iTu eyT uR qLi + iQd eydiR QdL + h.c.;
and we will refer to these models as G0SM( )models. We note that the
structure of couplings equals the one of GSM models given in eq. (2.5) upon H
models FCNCs are mediated by both Z and Z0 but in the case of
F = 2 transitions box
diagrams with VLQs and scalars play the dominant role for su ciently large M .
For ease of notation, we will sometimes refrain below from explicitly labelling the i by
the VLQ representation, as should be done if several of them are considered simultaneously.
Yukawa couplings of several representations
In our numerics we will consider one VLQ representation at a time as this simpli es the
analysis signi cantly. In particular the number of parameters is quite limited. Still it
is useful to make a few comments on the structure of avourviolating interactions and
at various places in our paper to state how our formulae would be modi ed through the
presence of several VLQ representations in a given model.
We plan to return to the
phenomenology of such models in the future.
When admitting several VLQ representations F m and F n simultaneously, potentially
additional locally gaugeinvariant Yukawa couplings
to be included in the case of GSMmodels [3]. They give rise to
avourchanging neutral
Higgs currents at tree level. In the G0SMmodels the U(1)L
L charges of the additional
have been chosen following the criteria explained above, which xes in turn the
emnF L 'mnFRn with 'mn = H have
'mn = S;
G0SM(S) models, only the particular choice of the U(1)L
L charges Q0QV =
ity to replace QV R qi
L ! qLi QV R in eq. (2.6), which maintains gauge invariance since S is a
singlet. On the other hand, in G0SM( ) models such couplings arise for Qd with D and Td.
are not
perL charge. In
Q0D [9] forbids
Another important consequence of the presence of several representations is the
gendiagrams discussed in section 3.2, which is the case when singlets or triplets together with
doublets are present. In the case of a single representation such operators can also be
generated in models with doublets through the topYukawa RG evolution from M to the
electroweak scale, see section 3.3.
Scalar sectors
which provides masses to gauge bosons and standard fermions in the course of spontaneous
symmetry breaking of SU(2)L
U(1)Y ! U(1)em via the VEV v ' 246 GeV, where
hHi = (0; v=p2)T :
MZ20 = g02vS2X2:
Z0gauge boson
In G0SM( )models the doublet 2
h 0ai = pva ;
v =
appendix A.2.
given model.
Further details on the scalar sectors of the G0SM(S) and G0SM( ) models are collected in
appendix A.1 and A.2, respectively. In table 1 we summarize all G0SMmodels and indicate
Decoupling of VLQs
The VLQ models are characterised by the masses M of the VLQs, the various Yukawa
couplings i
see section 2.3. The present lower bound on M from the LHC is in the ballpark of 1 TeV,
while the lower bounds on MZ0 are typically close to 3 TeV if Z0 has a direct coupling
to light quarks. But as emphasized in [9, 15, 25], Z0 of U(1)L
L does not have such
couplings, implying a much weaker lower bound on its mass, which could in fact be as low
1This convention corresponds to that of the Type I 2HDM.
VLQ Representation
Scalar Singlet
Da(3; 1; 1=3; X)
S(1; 1; 0; X)
Scalar Doublets
H(1; 2; 1=2; 0)
Db(3; 1; 1=3; X) S(1; 1; 0; X=2)
1(1; 2; 1=2; X), 2(1; 2; 1=2; 0)
QV (3; 2; +1=6; +X)
S(1; 1; 0; X)
Qd(3; 2; 5=6; X) S(1; 1; 0; X=2)
Td(3; 3; 1=3; X) S(1; 1; 0; X=2)
Tu(3; 3; +2=3; +X) S(1; 1; 0; X=2)
H(1; 2; 1=2; 0)
1(1; 2; 1=2; X), 2(1; 2; 1=2; 0)
1(1; 2; 1=2; X), 2(1; 2; 1=2; 0)
1(1; 2; 1=2; X), 2(1; 2; 1=2; 0)
and j F j = 2 transitions for M
as the electroweak scale and even lower. While it could also be as heavy as the VLQ mass,
we will assume the hierarchy
MZ . MZ0
or equivalently v . vS
in order to simplify the analysis. It is then natural to decouple rst the VLQs and to
consider EFTs for GSM and G0SM valid between the scales
are subsequently matched in one step onto SU(3)c
U(1)eminvariant phenomenological
mb, where mb denotes
the bottom mass. The coe cients determined in the process will indicate which operators
are the most important. In principle one could consider an intermediate EFT which is
constructed by integrating out Z0 and the new scalars before integrating out top quark,
W and Z, but from the point of view of renormalisation group e ects, integrating out all
these heavy elds simultaneously appears to be an adequate approximation.
at the scale
In this section we present the results from the decoupling of the VLQs that are
important for our phenomenological applications within the framework of the G(S0M)EFTs.
EW is given in section 4. The Lagrangian of the G(S0M)EFT consists of the
dimensionfour interactions of the light elds and dimension six interactions generated by
the decoupling of VLQs
which are invariant under either GSM or G0SM, depending on the model. Thus in
GSMmodels Ldim 4 coincides with the SM Lagrangian and the corresponding nonredundant
set of operators of dimension six has been classi ed in ref. [26]. In G0SMmodels operators
that are invariant under G0SM must be added, which involve the Z0boson and the additional
scalar singlets and/or doublets. The Wilson coe cients Ca2 are e ective couplings, which
2The Wilson coe cients of G(S0M)EFTs are denoted with calligraphic Ci, whereas the ones of
phenomenological EFTs with Ci.
when decoupling VLQs. The decoupling proceeds either by explicit matching calculations
starting at treelevel and including subsequently higher orders or by integrating them out
in the path integral method [3]. The treelevel decoupling has been known for a long time
for GSM models [3] and is given for G0SM(S) models in ref. [9].
Within the EFT, RG equations allow to evolve the Wilson coe cients from
M down to
EW. In leading logarithmic approximation and retaining only the rst logarithm (1stLLA)
it has the approximate solution
Ca( EW) =
which holds as long as the second term remains small compared to the rst. The anomalous
dimension matrices (ADM) ab depend in general on couplings of the gauge, Yukawa and
scalar sectors and are known for the GSMEFT [27{29]. Largest contributions might be
expected for the case of ab / YuyYu
t2 mixing due to the topquark Yukawa coupling
generated at 1stLLA order.3 In particular, as we will see below, in the case of models
with righthanded neutral currents leftright operators can be generated in this manner
j F j = 1 observables.
The VLQs have a very limited set of couplings to light elds, which are either via gauge
interactions (2.3) to the gauge bosons or via Yukawa interactions (2.5){(2.7) to light 
At treelevel, this particular structure of interactions can give rise only to avourchanging
Z and Z0 couplings, whereas all other decoupling e ects are loopsuppressed [30].
The decoupling of the VLQs proceeds in the unbroken phase of SU(2)L
U(1)Y, hence
take place within the G(S0M)EFTs and the transformation from
quark elds are avoureigenstates and neutral components of scalar elds are without VEV
at this stage. After the RG evolution from
M to EW, spontaneous symmetry breaking will
avour to masseigenstates
for fermions and gauge bosons can be performed, accounting for the dimension six part in
Treelevel decoupling and Z and Z0 e ects
The couplings of the VLQs permit at tree level only a dimension six contribution from
the generic 4point diagram in gure 1a. Since its dimension ve contribution vanishes [3],
it is equivalent to consider the 5point diagram
gure 1b, where either SU(2)L or U(1)Y
gauge bosons in GSMmodels or in addition a Z^0 in G0SMmodels is radiated o the VLQ [3,
9]. As a consequence, in GSM and G0SMmodels only operators of the type
3Note that the 1stLLA neglects \secondary mixing" e ects that are present in LLA, i.e. summing all
large logarithms, because although operator OA might not have ADM entry with operator OB (no \direct
mixing"), it can still contribute to the Wilson coe cient CB( EW), if it mixes directly with some operator
OC that in turn mixes directly into OB.
= (qL; uR; dR). The gauge boson G
depends on the representation. Treelevel graph (c) requires
two representations Fm;n with a Yukawa coupling via 'c and give rise to
2'3 operators.
(S S)[qLiujRHe ]
(He yiD H)[uiR
(HyH)[qLiujRHe ]
(HyH)[qLidjRH]
We follow the de nitions of [26] for 2'2D operators, except for the signs of gauge
couplings in the covariant derivatives, and ( 2'3 + h.c.) operators in the case of GSM models and
elds denote the generations. These are all operators that could arise from treelevel decoupling of
VLQs, depending on the model.
are projected in part onto
2'3type operators via equation of motions (EOM) [26, 31].
We list the corresponding de nitions of the operators in table 2, following the notation
of [26] in the case of the GSMEFT and extending it to G0SMEFTs.
After spontaneous symmetry breaking the
2'3 operators contribute to the quark
= u; d) at the scale EW via
mij = p
which allows to substitute Yukawa couplings Y in terms of measured m
and new physics
parameters C 2'3 / Y C 2'2D, see appendix B.2. If several representations of VLQs are
present in a given model and two of them Fm;n couple to a scalar 'c4 via Yukawa couplings
emn, a third possibility is allowed at treelevel depicted in
gure 1c, which contributes
2'3 operators and gives rise to avourchanging neutral H i j interactions at
contributions. Their diagonalisation proceeds as usual for the quark elds with the help of
3 unitary rotations in avour space:
R = mdiag;
V = (VLu)yVLd ;
basis for the VLQ Yukawa couplings i
with diagonal up and downquark masses mdiag and the unitary quarkmixing matrix V . In
the limit of vanishing dimensionsix contributions, V will become the
CabibboKobayashiMaskawa (CKM) matrix of the SM. Throughout we will assume for down quarks the weak
xes also the de nition of the Wilson coe cients C 2'2D (for more details see [32]) and the
After spontaneous symmetry breaking the
2'2D operators give rise to
(Z0) = f i h iLj (Z0)
For completeness, we provide the matching conditions for the Wilson coe cients in
appendix B. We note that RG e ects have been neglected in (3.7) and (3.8) since they are
only due to selfmixing of 2'2D operators as listed in appendix B.3.
SM as corrections from NP to them are in GSMmodels oneloop suppressed. This is also
the case of G0SM(S) models where Z does not play any role in FCNCs. In G0SM( ) models
modi cations of the Zf f couplings come from Z
Z0 mixing. These shifts are relevant
for leptons in partial widths of Z ! `` (see appendix A.2) and could be of relevance in
consistency in G0SM( ) models, although they are negligible in comparison to other e ects.
GSMmodels
In the case of GSMmodels, the decoupling of VLQs gives the results for
L;R(Z) couplings
collected for downquarks in table 3, where
Except for the sign in the case of Tu, our results agree with those in [11]. Furthermore, also
nonzero couplings to uptype quarks arise [11] but they will not play any role in our paper.
4As discussed above 'c = H in GSM and G0SMmodels.
ij =2
mn(V y)nj =2
models. Here Vij is the CKM matrix and
u =
G0SMmodels
In the G0SMmodels, the (L
L ) symmetry xes the Z0 coupling to leptons to be
`L`(Z0) =
`R`(Z0) =
` ` (Z0) = g0Q0`;
with Q0` = f0; +1; 1g for ` = fe; ; g
. Here we have neglected Z
Z0 mixing e ects
existing in G0SM( )models. However, for consistency we have to include these e ects in
the couplings of the Z to leptons
`L`(Z) =
`R`(Z) = gZ s2W + g0Q0` ZZ0 ;
to rst order in the small mixing angle ZZ0 (see appendix A.2 for details). On the other
hand, the gauge couplings to quarks are model dependent.
In G0SM(S)models the scalar sector of S and H generates only nonzero quark couplings
to Z0, whereas in G0SM( )models the scalar sector of S, H and
couplings of SM quarks to both Z0 and Z. We de ne
gives rise to nonzero
2 Xigj0 MMZ220 ;
ij de ned in eq. (3.9) and the Z
Z0 mixing angle [see (A.9)]
Kij in eq. (3.12).
cos is a parameter associated with the scalar sector (see (2.10)) of G0SM(
)Z0 mixing, which is phenomenologically
constrained to be small, ZZ0 < 0:1, due to constraints from the Zboson mass, MZ , and
partial widths Z ! `` measured at LEP, as described in more detail in appendix A.2. The
down and upquark couplings to Z0 and Z are collected for these models in table 4. We
con rm previous ndings [9] for the G0SM(S)models.
We note that the Z0 couplings are suppressed/enhanced by the ratio r0 w.r.t. the
Zcouplings. Enhancement takes place for 2 g0X > gZ
0:75, such that for example r0
can be reached with g0X
1:1, still within the perturbative regime. The couplings of Td
and Tu di er just by a sign and factors 1/2. In distinction to Zcontributions in
GSMmodels, both Z and Z0contributions in G0SM( ) models decouple with large tan , see
r0 VimKmn(V y)nj
u r0VimKmn(V y)nj=2
r0Kij=2
[1 r0 ZZ0] Kij=2
[1 r0 ZZ0] VimKmn(V y)nj=2
models. Here Vij is the CKM matrix.
interactions with scalars ' = H; S;
and SM quarks
= (qL; dR; uR). The crossed graph appears
Decoupling at oneloop level
All other decoupling processes proceed via loops. Those that would lead to noncanonical
kinetic terms in the G(S0M)EFTs can be absorbed by a suitable choice of wavefunction
renormalisation constants in the full theory above the scale
M , resulting in nonminimal
renormalisation of interactions and giving rise to
nite threshold e ects of coupling
constants. In G0SMmodels this is the case for kinetic mixing of B and Z^0 , which enters our
analysis only as a higher order e ect.
All other e ects enter as dimension six operators. The ones with four quarks are most
important for quark avour phenomenology. They involve only VLQYukawa interactions,
as depicted in
gure 2a and
gure 2b, and give rise to
4type operators, among which
thereby the intermediate matching to the GSMinvariant form.5 Still, we outline this step
for completeness here. In the VLQ models considered, there are four relevant
4 operators
in G(S0M)EFTs at the VLQ scale
via RG evolution from
M to EW. These are the (LL)(LL) operators
M and a fth operator is generated due to QCD mixing
[Oq(1q)]ijkl = [qLi
[Oq(3q)]ijkl = [qLi
the (LL)(RR) operators
and the (RR)(RR) operator
[Oq(1d)]ijkl = [qLi
[Oq(8d)]ijkl = [qLi
[Odd]ijkl = [diR
electroweak scale EW [32] as
CVijLL =
CLijR;1 =
CVijRR =
A ; CLijR;2 = Nij 1[Cq(8d)
where Nij is given in (C.2). Here we anticipate this matching to the VLQ scale
there are no RG e ects of phenomenological importance for the discussion of Bmeson and
Kaon sectors. For more details see section 3.3, where also QCD mixing is given for these
operators. Since the Wilson coe cients of these operators are generated at
M at oneloop,
their interplay with other sectors in quark avour physics due to RG mixing are considered
higher order and hence beyond the scope of our work.
via box diagrams (see
gures 2a and 2b), which contain two heavy VLQ propagators
(H+; H0)T . These box diagrams yield the general structure of the Wilson coe cients
Caij ( M ) =
at the scale
EFT, see (C.2). The function
f1(Mm; Mn) =
f1(Mm; Mm) =
5Note that the set of 4type operators is the same in all G(S0M) models and a nonredundant set can be
found in ref. [26].
(Fm; Fn)
LR1, 1=4
VLL, 1=8
VRR, 1=4
LR1, 3=8
LR1, 3=8
VLL, 1=8
followed by corresponding mn.
depends on the VLQ masses of representations Fm;n. The couplings imj are
imj = ( im) j
imj = im( jm)
Fm = D; Td; Tu;
Fm = Qd; QV :
The index a of the operator and the numerical factors mn are collected in table 5. Note
crossed, which gives rise to an additional sign w.r.t. the diagram with noncrossed scalar
XI) we nd an additional factor of 2. Concerning QV (model IX) we nd a contribution to
OVRR instead of
OVLL and also opposite sign. For completeness we provide also the
results for Fm 6= Fn.
In G0SM(S) models we consider only VLQs D and QV and their interference
D : CVRR =
CVLL =
CLR1 =
which agrees with [9] except for a minus sign from crossed scalar propagators in the
interference term D
The results for G0SM( ) models can be found straightforwardly from the ones of the
GSM models, bearing in mind that (2.5) and (2.7) are equivalent up to the replacement
The VLQ treelevel exchange in the considered VLQ scenarios generates only
2'2D and
2'3type operators at the scale M with nonvanishing Wilson coe cients (see appendix B)
G0SM(S) :
depending on the VLQ scenario.6 The RG evolution from
M down to
EW can induce
via operator mixing leading logarithmic contributions also to other classes of operators in
G(S0M) EFTs at the scale
and thus imply additional potential constraints.
EW. These operators are possibly related to a variety of processes
The largest enhancements can appear if the ADM
ab in (3.3) is proportional to the
strong coupling 4
1:4 or the topYukawa coupling yt
1. Note that QCD mixing
is avourdiagonal and hence can not give rise to new genuine phenomenological e ects,
i.e. one can not expect qualitative changes. On the other hand, Yukawa couplings are
the main source of avouro diagonal interactions and we will focus on these here. The
SU(2)L gauge interactions induce via ADMs ab / g22 [29] only intragenerational mixing
between u
iL and are parametrically smaller than ytinduced e ects, such that we
do not consider them here. The U(1)Y gauge interactions are only avourdiagonal and
numerically even more suppressed.
Concerning G0SM models, RG e ects due to topYukawa couplings are absent for 2'2D
2'3 operators, because ' = S;
do not have Yukawa couplings to qL; uR; dR, which
are forbidden by their additional U(1)L
L charge. Hence RG e ects as discussed below
are not present in these scenarios.
and we collect the ones involving the Wilson coe cients (3.22) in appendix B.3. The RG
equations of these Wilson coe cients are also coupled with those of SM couplings, such as
the quartic Higgs coupling and quarkYukawa couplings [27], but in 1stLLA they decouple.
The modi cation of SM couplings due to dim6 e ects can be neglected when discussing
the RG evolution of dim6 e ects themselves in rst approximation. Moreover, the quartic
Higgs coupling is irrelevant for the processes discussed here and the quark masses are
determined from lowenergy experiments, i.e. much below
EW. Hence phenomenologically
most interesting are RG e ects of mixing of 2H2D and 2H3 operators into other operator
classes that do not receive treelevel matching contributions at
M . Those classes are
H4D2 (2) ;
where we list in parentheses the number of operators.7
We focus on the
4 operators,
which all turn out to be fourquark operators, because they are most relevant for processes
7Implying footnote 6.
also CHu and CHud must be considered.
of downtype quarks considered here. We comment shortly on the H6 and H4D2 classes
in appendix B.3.
The RG equation (3.3) implies for a speci c a 2
Ca( EW) =
4, see also [18],
novel chiral structure of the
return to this point in section 4.3.
2H2D contributions. Three of the 4 operators (Oq(1q;3) and Oq(1d)) can mediate
downin both SMEFT and phenomenological EFTs, therefore receiving another suppression in
where it competes with the direct oneloop box contribution in VLQ models discussed in
ated directly by
2H2D operators in the next matching step of GSM to phenomenological
EFTs at EW (see section 4 and gure 4), which are therefore enhanced in these processes
compared to the 1stLLA contributions discussed here. Consequently, the 1stLLA is
one4 operators enhances a speci c hadronic observable. We will
Under the transformation from weak to mass eigenstates for uptype quarks (3.5)
the corresponding ADMs of 4 operators in appendix B.3 transform as
2 VLumdUiagV uy =
[YuyYu]ij =
[YuYuy]ij =
with uptype quark mass mk and the de nition of CKMproducts i(jt) given in (4.1). Since
the ADMs are needed here for the evolution of dim6 Wilson coe cients themselves, we
have used treelevel relations derived from the dim4 part of the Lagrangian only, thereby
neglecting dim6 contributions, which would constitute a dim8 corrections in this context.
In the sum over k only the topquark contribution is relevant (mu;c
mt), if one assumes
that the unitary matrix V is equal to the CKM matrix up to dim6 corrections.8
The j F j = 2 mediating
4 operators involve the combination (3.29). We obtain
via (3.26) and explicit matching conditions (B.1)
Caij ( EW) =
EW
are entirely negligible.
and the VLQmodeldependent factor
a = VLL
a = LR; 1
Fm = D; Td; Tu ;
Fm = Qd; QV ;
m =
Fm = (D; Qd; QV ; Td; Tu):
We note the relations
m Mij2 = [CH(1q)
(Fm = D; Td; Tu);
where the relative sign comes from relative signs in (B.23) and (B.24) when inserted
in (3.17) and
m Mij2 = [CHd]ij ;
(Fm = Qd; QV ) :
We point out the di erent avour structure of the 1stLLA contribution (3.30) compared
to the one of the direct boxcontribution (3.18) discussed in the previous section section 3.2:
showing linear versus quadratic dependence on the product of VLQ Yukawa couplings ij .
A detailed comparison of both contributions is given in section 5.
CVLL(VRR)( EW) = 6
CLR;1( EW) = 6 CLR;1( M ) ;
CLR;2( EW) =
The initial conditions of Caij ( M ) from boxdiagrams are collected in (3.18) and (3.21).
s(6)( M )= s(6)( EW).
Implications for the downquark sector
oneloop level at the scale
M has been presented, including the most important e ects
from the RG evolution down to the electroweak scale
EW. In this section we discuss the
decoupling of degrees of freedom of the order of EW by matching onto phenomenological
for avour physics. In fact, as we have shown, large NP e ects in avour observables can
be present for MVLQ = 10 TeV and in the
avourprecision era one is sensitive to even
VLQs in a given representation independently of the size of Yukawa couplings.
Acknowledgments
Genon for providing us an update of a treelevel CKM
t from CKM tter [94]. This
research was done and
nanced in the context of the ERC Advanced Grant project
\FLAVOUR"(267104) and was partially supported by the DFG cluster of excellence
\Origin and Structure of the Universe". The work of A.C. is supported by the Alexander
von Humboldt Foundation. This work is supported in part by the DFG SFB/TR 110
\Symmetries and the Emergence of Structure in QCD".
Scalar sectors of G0SMmodels
G0SM(S) models
The scalar sector in G0SM(S)models with one complex scalar S(1; 0; X) and the SM doublet
H(2; +1=2; 0) is given by
L = jD Hj2 + jD Sj2
with the potential
V = m2HyH +
We parametrise the SM Higgs doublet and the complex scalar as
H =
v + h0 + iG0 = 2
S =
The neutral masseigenstates are given by (h; H)T ' (h0; R0)T with approximate masses
Kinetic mixing of Z and Z0 is caused by VLQexchange and depends on the VLQ
masses M and the U(1)L
L gauge coupling. It will be neglected in the following, see
ref. [9]. Mass mixing does not occur in G0SM(S) models.
G0SM( ) models
doublets 1
(2; +1=2; X) and
H(2; +1=2; 0) is given by
L = jD
with the potential
V = m2a ya a +
We neglect kinetic mixing and parametrise the mass mixing via
@Z^0 A = @
After partial diagonalization of the neutral gauge boson system, the Z and Z0 masses and
their mass mixing are given by [96]
M^ Z20 = (g0X)2 S
2 =
with e = p
= g2s^W = g1c^W = gZ s^W c^W . The Z
Z0 mixing angle
tan 2 ZZ0 =
= c2 4Xg0
is small unless X becomes large. The diagonalisation of the neutral gauge boson mass
matrix gives mass eigenvalues
MZ2;Z0 =
M^ Z20 + M^ Z2
lution for which MZ < MZ0 , i.e. throughout we will implicitly impose that the lighter mass
eigenstate couples predominantly SMlike to quarks and leptons. As a consequence a lower
bound on g0 will be obtained. On the other hand, the decoupling limit g0 ! 0 is not
excluded, but it will lead to MZ0 < MZ , i.e. that the heavier masseigenstate couples
predominantly to SMlike fermions. The tan
dependence of MZ0 becomes irrelevant once vS &
0:5 TeV. The mixing angle ZZ0 can be suppressed with large tan
and MZ0 , since we work
in the part of the parameter space, where the other possibility of g0 ! 0 is not an option.
In G0SM( )models we make use of the fact that photon and W interactions to leptons
are SMlike in order to determine the values of the fundamental gauge couplings g1;2 and
the VEV v from
e(MZ ), GF and the W boson pole mass MW . As the remaining free
parameters we choose tan , g0, X and vS, whereas dependent parameters are MZ;Z0 and
ZZ0 . Note that the latter depend only on the product g0X, such that there are e ectively
only three parameters. We will restrict this parameter space to
2 TeV: (A.11) The lower bound on tan guarantees perturbativity of the topquark Yukawa coupling [64],
whereas vS is bounded from above by the requirements (3.1) and yields MZ0 . 1:5 TeV
within the above limits. Constraints on these parameters arise from the measured value of
MZ , which we impose with an error of
constraining the new physics contributions of the Zlepton couplings (3.11) that depend on
the ZZ0 and g0 due to gauge mixing. We nd a small mixing angle ZZ0 . 0:1 in the above
speci ed parameter space of tan , g0X and vS if we impose the bound on new physics
contributions to the partial widths of Z ! `` from LEP [24], allowing for 5
from the measured central values, together with the bound on MZ . This justi es the
expansion in the small mixing angle as done in table 4.
VLQ decoupling and RG e ects
This appendix contains results of the Wilson coe cients of 2'2D and
2'3 operators in
G(S0M)EFTs after the treelevel decoupling of VLQs at the scale
M . We provide further
the relations to
avourchanging Z and Z0 couplings (3.7) and (3.8) after spontaneous
symmetry breaking at the scale EW (neglecting selfmixing).
2'2D operators
contributions for
The matching in GSM models at the scale M of order of the VLQ mass yields nonvanishing
D : [CH(1q)]ij = [CH(3q)]ij =
Td : [CH(1q)]ij =
3 [CH(3q)]ij =
Tu : [CH(1q)]ij = 3 [CH(3q)]ij =
4 M 2
8 M 2
Qd : [CHd]ij =
QV : [CHd]ij =
2 M 2
; [CHu]ij =
; [CHud]ij =
in agreement with [3], and analogously for G0SM( ) models with H !
G0SM(S) models for VLQs D and QV yields nonvanishing Wilson coe cients
. The matching of
D : [CSd]ij =
2 M 2
QV : [CSq]ij =
2 M 2
The avourchanging Z and Z0 couplings (3.7) and (3.8) after spontaneous symmetry
breaking are given in terms of the Wilson coe cients at the scale
EW. In the case of
GSMmodels, the treelevel calculation of the process fifj Z from GSMEFT (3.2) yields
G0SM(S)models
with the scalar sector of S and H generates only nonzero couplings to Z0. We nd for
with the EFTcoe cients Ci given in (B.1) and FS
m2Z0 =(g0X). The variant of G0SM(
)models with the scalar sector of S, H and
generates nonzero couplings to Z0 and Z.
The results for G0SM( ) models are similar to GSM models, with the di erence that they
Z0 mixings:
where V = Z; Z0 and
2'3 operators
We de ne the SM Yukawa couplings of quarks as in [26]
LYuk = qL Yd H dR + qL Yu He uR + h.c.:
Nonvanishing Wilson coe cients are generated also for
2'3 operators (see table 2 for
de nitions) as a consequence of the application of equations of motion (EOM) in the
treelevel decoupling of VLQs in section 3.1. Due to the application of EOMs, these Wilson
coe cients scale with the corresponding Yukawa coupling as
[CuH ]ij = Yu CHyuD + (CH(1q)D
Note the matrix multiplications w.r.t. the generation indices of Yu;d with the respective
coe cients CH D inside the brackets.
The treelevel matching in GSMmodels gives nonvanishing contributions at
D : [CH(1q)D + CH(3q)D]ij =
Td : [CH(1q)D + CH(3q)D]ij =
Tu : [CH(1q)D + CH(3q)D]ij =
[CHdD]ij =
[CHdD]ij =
2 M 2
4 M 2
2 M 2
CH(3q)D]ij = 0;
CH(3q)D]ij =
CH(3q)D]ij =
2 M 2
4 M 2
[CHuD]ij =
in agreement with [3]. Analogous Wilson coe cients in G0SM( ) are found by H !
In G0SM(S) models analogous relations
[CuS]ij = Yu CSyuD + CSqD Yuiij
hold with nonvanishing
[CSdD]ij =
[CdS]ij = Yd CSydD + CSqD Yd ij
[CSqD]ij =
TopYukawa RG e ects
This appendix collects the ADM entries of the GSMEFT proportional to the uptype quark
Yukawa coupling Yu from [28], i.e. neglecting contributions from Yd;e. We list them only for
operators that receive leading logarithmic contributions at the scale
EW from the initial
Wilson coe cients at the scale
2H3 operators in the 1stLLA via direct
mixing, see footnote 3. For convenience of the reader we keep here also CHu and CHud,
which are absent in the VLQ models D; Tu; Td; Qd, but contribute in QV for
Vu 6= 0.
2 dCH =
12 Tr CuH YuyYuYuy + YuYuyYu CuyH ;
and leads to a shift of the VEV [29].
The H4D2operators OH
= 6 Tr h (1)
C_HD = 24 Tr hCH(1q)YuYuy
Their Wilson coe cients contribute to the Higgsboson mass and the electroweak precision
observable T =
16Note that if the generation indices are not given explicitly on Yukawa couplings and Wilson coe cients
then a matrix multiplication is implied.
CuH =
2 CH(1q)YuYuyYu + 2 YuYuyYuCHu
+ 6 Tr CuH Yuy Yu + 9 Tr YuYuy CuH + 5 CuH YuyYu +
CdH =
12 Tr CH(3q)YuYuy]Yd + 6 CH(3q)YuYuyYd
2 YuYuyYuCHud
2 YuCuyH Yd
CuH YuyYd + 9 Tr YuYuy CdH
CeH =
12 Tr CH(3q)YuYuy]Ye + 6 Tr YuCuH Y
y
2 YuYuyCdH ;
have selfmixing for CuH;dH , and CuH mixes also into CdH;eH . They receive also contributions
from C 2H2D. The C 2H3 enter fermionmass matrices (3.4) and lead also to fermionHiggs
couplings that are in general avouro diagonal.
2H2Doperators (see table 2)
YuYuyCH(3q) + CH(3q)YuYuy
CHud = 6 Tr[YuYuy]CHud + 3YuyYuCHud
C_H(3q) = 6 Tr YuYuy]CH(3q) + YuYuyCH(3q) + CH(3q)YuYuy
YuYuyCH(1q) + CH(1q)YuYuy ;
CHd = 6 Tr YuYuy]CHd ;
CHu =
2YuyCH(1q)Yu + 6 Tr[YuYuy]CHu + 4 YuyYuCHu + CHuYuyYu ;
show a mixing pattern among CHq
(1;3) as well as CH(1q) and CHu. The latter implies that the
LH scenarios D; Tu; Td will generate via mixing also a RH coupling CHu via CHq, which is
however a oneloop e ect compared to the e ects of CHq. Both CHd and CHud have only
In the case of
4operators there are (LL)(LL) operators
[C_q(1q)]ijkl = +
[C_q(3q)]ijkl =
[YuYuy]ij [CH(1q)]kl + [CH(1q)]ij [YuYuy]kl ;
[YuYuy]ij [CH(3q)]kl + [CH(3q)]ij [YuYuy]kl ;
the (LL)(RR) operators
and the (RR)(RR) operators
[C_q(1u)]ijkl = [YuYuy]ij [CHu]kl
[C_q(1d)]ijkl = [YuYuy]ij [CHd]kl;
[C_uu]ijkl =
[C_u(1d)]ijkl =
[YuyYu]ij [CHu]kl
2[YuyYu]ij [CHd]kl;
[CHu]ij [YuyYu]kl;
under the assumption CHu = 0.
F j = 2
Master formulae for K and B decays
The e ective Lagrangian for neutral meson mixing in the downtype quark sector (dj di !
dj di with i 6= j) can be written as [34]
where the normalisation factor and the CKM combinations are
H F =2 = Nij
X Caij Oaij + h.c.;
Nij =
OVijLL = [di
OLijR;1 = [di
OLijR;2 = [diPLdj ][diPRdj ];
OSijLL;1 = [diPLdj ][diPLdj ];
OSijLL;2 =
which are built out of coloursinglet currents [di : : : dj ][di : : : dj ], where ;
denote colour
indices. The chirality ipped sectors VRR and SRR are obtained from interchanging PL $
PR in VLL and SLL. Note that the minus sign in QSLL;2 arises from di erent de nitions of
]=2 in ref. [34] w.r.t.
= i~
used here. The ADM's of the 5 distinct sectors
(VLL, SLL, LR, VRR, SRR) have been calculated in refs. [33, 34] at NLO in QCD, and
numerical solutions are given in ref. [97]. The NLO ADM's are also available for an alternative
In the SM only
CVijLL( EW)jSM = S0(xt);
S0(x) =
11x + x2)
is nonzero at the scale EW, depending on the ratio xt
W boson masses.
mt2=M W2 of the topquark and
are given in table 14.
The j F j = 2 observables of interest
MK; Bd; Bs , K and sin(2 d;s) derive all from
the complexvalued o diagonal elements M1ij2 of the massmixing matrices of the neutral
mesons [99, 100]. For the latter we use the full higherorder SM expressions in combination
with the LO new physics contributions. In particular for M1d2s, we make use of NLO and
in part NNLO QCD corrections cc; tt; ct collected in table 13 and for the hadronic matrix
QCD corrections B to the SM and use for the hadronic matrix elements the latest results
The e ective Lagrangian for dj ! di
(i 6= j) is adopted from ref. [91],
= fe; ; g
In the SM only
It is given by
has nonvanishing contribution at the scale
C depend on the ratio xt
EW, whereas CR = 0. The functions B and
the gaugeindependent linear combination X0(xt)
4B(xt) [101, 102],
OLij;(R) = [di
SM =
X0(x) =
X0 ! XLSM = 1:481
The Br(K+
the local OL
charm quark at c
2), where
e ective Hamiltonian of the SM as [108]
the \top"sector, when decoupling heavy degrees of freedom at EW, which yields directly
mc, which is enhanced due to the strong CKM hierarchy ( (std) /
) receives in the SM the numerically leading contribution from
when including higher order QCD and electroweak corrections [103{106] as extracted in
ref. [107] from original papers.
The theoretical predictions for b ! s
observables de ned in eq. (6.5) are based
on formulae given in ref. [92].
These expressions account for the leptonnonuniversal
contribution of VLQ's w.r.t. the neutrino avour in G0SM models. However, the particular
structure of the gauged U(1)L
L (2.1) leads to a cancellation of the numerically leading
interference contributions of the SM and new physics [9].
He = N
The NP contributions in VLQmodels cannot compete with the SM contribution to the
treelevel processes entering the \charm"sector, since they are suppressed by an additional
Xt = XLSM + XLsd; + XRsd;
+ = rK+
e e) = 0:5173(25)
contains the experimental value Br(K !
e e) and the isospin correction rK+ and has
been evaluated in ref. [110] (table 2) including various corrections. Further
for Emax
20 MeV [110]. If one takes into account the di erent value of s2
EM =
W = 0:231 taken
in ref. [110] compared to our value in table 13, then
+ = 0:5150
10 10 ( =0:225)8.
The sum (C.12) contains the SM contribution and further the interference of SM
and NP NP. Besides Pc at NNLO in the SM contribution, the NLO numerical values
Xce = 10:05
Xc = 6:64
The branching fraction of KL !
is obtained again by averaging over the three
neutrino avours
) =
L =
rKL KL = 2:231(13)
with XLsd;R; given in eq. (4.8), such that the topsector becomes neutrino avour dependent.
The experimental measurement averages over the three neutrino avours,
; (C.12)
with the assumption that (scd)Xc is real. The NNLO QCD results of the functions Xc [108]
together with long distance contributions [109] are combined into
Pc =
3 Xce +
3 Xc
The numerical value is from ref. [110] (table 2) and it decreases to
L = 2:221
Ld!d`` =
O9ij(;`90) = [di
PL (R)dj ][` `];
O1ij0;`(100) = [di
PL (R)dj ][`
whereas scalar OS`;P(S0;P0) and tensorial operators OT`(T5) are not generated in the context
of VLQ models. In the SM the only nonzero Wilson coe cients,
SM =
C1ij0;` SM =
are lepton avour universal and also universal w.r.t. downtype quark transitions, as the
CKM elements have been factored out. All other Wilson coe cients vanish at the scale
EW. The functions B; C; D depend again on the ratio xt
mt2=M W2 of the topquark and
W boson masses and give two gaugeindependent combinations Y0(xt)
and Z0(xt)
Y0(x) =
Z0(x) =
163x3 + 259x2
15x2 + 18x
ln x: (C.23)
In the predictions of Br(Bd;s !
) and the masseigenstate rate asymmetry
) we include for the SM contribution the NNLO QCD [112] and NLO
EW [39] corrections, whereas NP contributions are included at LO. The values of the
decay constants FBd;s are collected in table 13.
The branching fractions Br(B+
are predicted within the framework outlined in refs. [113{115]. We neglect contributions
from QCD penguin operators, which have small Wilson coe cients and the NLO QCD
corrections to matrix elements of the chargedcurrent operators [116, 117], but include the
VubVud(s). The form factors and their uncertainties are adapted from lattice
calculations [118, 119] for B !
and [120] for B ! K with a summary given in [121]. We
add additional relative uncertainties of 15% for missing NLO QCD corrections and 10%
) at high dilepton invariant mass q2
for possible duality violation [114] in quadrature.
The predictions for observables of B ! K
are based on refs. [89] and [122] for
lowand highq2 regions, respectively. The corresponding results for B ! K form factors in
the two regions are from the LCSR calculation [123] and the lattice calculations [124, 125].
The measurement of Br(KL !
) provides important constraints on its
shortdistance (SD) contributions, despite the dominating longdistance (LD) contributions
inducing uncertainties that are not entirely under theoretical control. In particular there
is the issue of the sign of the interference of the SD part
SD of the decay amplitude
with the LD parts. Allowing for both signs implies a conservative bound
3:1 [74]. Relying on predictions of this sign based on the quite general assumptions
stated in [74, 126, 127] one nds
1:7 which we employ in most of this work.
Note, however, that a di erent sign is found17 in [126, 128], implying
In light of this situation, we comment on the impact of the more conservative choice where
appropriate, which includes both sign choices.
17We thank G. D'Ambrosio and JM. Gerard for the discussion on this point.
last column gives Pa for B(1=2) = 0:57 and B(3=2) = 0:76.
6 8
dj ! di qq and "0="
de nition of the operators can be found and here we restrict ourselves to s ! d, i.e.
ij = sd. At the scale EW (Nf = 5) it reads
Ld!dqq =
+ X( va + vaNP)Oa + X va0Oa0
where O1(c;2) denote currentcurrent operators. The sum over a extends over the QCD and
5]. Thereby we assume that VLQ contributions to other operators are strongly
suppressed. The Wilson coe cients are denoted as za, va(NP) and va0, taken at the scale EW.
For the SMpart, CKM unitarity was used,
and we introduced a new physics contribution vaNP as shown above, which is related to the
VLQcontribution (4.13) as
va0 = Cas0d:
The RG evolution at NLO in QCD and QED leads to the e ective Hamiltonian at a
after decoupling of b and cquarks at scales b;c [129], where ya
za and all Wilson
coe cients are at the scale .
z1O1 + z2O2 + X[za + ya + vaNP]Oa + X va0Oa0
+ h.c.; (C.27)
The coe cients are
where the minus sign is due to h(
)I jOajKi =
)I jOa0jKi for the pseudoscalar pions
nal state [130]. For the readers convenience we provide a seminumerical formula
2:58 + 24:01B6(1=2)
12:70B8(3=2)i
Pa = p(a0) + p(a6)B(1=2) + p(a8)B(3=2)
6 8
B(3=2)( ) = 0:76. For this purpose
with p(an) given in table 12, where the last column gives Pa for B(1=2)( ) = 0:57 and
6
EW = MW , b = mb(mb), c = 1:3 GeV and
compared to 1:9
10 4 in [20] due to di erent numerical inputs.
Statistical approach and numerical input
The input quantities included in our analysis are collected in table 13 and table 14. The
CKM parameters have to be determined independently of contributions from the VLQs.
The \treelevel" t carried out by the CKM tter collaboration achieves such a
determination, taking only measurements into account that are una ected in our NP scenarios, i.e.
(semi)leptonic treelevel decays, treelevel determinations of
as a constraint on . The results of this t are again quoted in table 13.
As a statistical procedure, we choose a frequentist approach. The ts include as
parameters of interest the VLQ couplings and in addition nuisance parameters, which constitute
theoretical uncertainties. The nuisance parameters are listed in table 13 and consist of
CKM parameters from a \treelevel" t;18
The 1 and 2dimensional con dence regions (CL) of parameters are obtained by pro
ling over the remaining parameters, i.e. maximisation of the likelihood function over the
subspace of remaining parameters for a
xed value of the (pair of) parameter(s) of
interest. Similarly, correlation plots for pairs of observables are obtained by pro ling over
all parameters and imposing in addition the speci c values for the pair observables. The
2dimensional 68% and 95% con dence regions are determined then for two degrees of
freedom. The SM predictions of observables are found in the same way by setting VLQ
contributions to zero and pro ling only over the CKM and hadronic nuisance parameters.
18We thank Sebastien DescotesGenon for providing us an update of a treelevel CKM t from CKM
t= 0:22544(+3238)
= 0:125(+3108)
mK = 497:614(24) MeV
FK =F = 1:194(5)
BK = 0:750(15)
ct = 0:496(47)
mBd = 5279:61(16) MeV
mBs = 5366:79(23) MeV
FBd = 190:5(42) MeV
FBd (B^Bd )1=2 = 229:4(93) MeV
B = 0:55(1)
A = 0:8207(7)(13)
= 0:382(+2128)
F = 130:41(20) MeV
tt = 0:5765(65)
cc = 1:87(76)
= 1:638(4) ps
Bd = 1:520(4) ps
Bs = 1:505(4) ps
MW = 80:385(15) GeV
GF = 1:16638(1)
(MZ ) = 1=127:9
MZ = 91:1876(21) GeV
sin2 W = 0:23126(13)
s(MZ ) = 0:1185(6)
quark masses
md(2 GeV) = 4:68(16) MeV [131]
ms(2 GeV) = 93:8(24) MeV [131]
mc(mc) = 1:275(25) GeV
mb(mb) = 4:18(3) GeV
mt(mt) = 163(1) GeV
s= s = 0:124(9)
: Calculated by demanding that the uncertainty of the ratio of the decay constants
given above should equal the uncertainty given explicitly for the ratio, also given in ref. [131].
Calculated from information given in ref. [61]. Note that their assumption for the SU(3) breaking
from the charm sea contribution corresponds to the assumption of a 91:8% correlation for this
uncertainty between Bd and Bs.
low [GeV]
entering M1ij2, see [61] and [140] for correlations.
For the Kaon system threshold crossings to
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