Precision tests of the Standard Model with leptonic and semileptonic kaon decays

0.53 MB·English

by  FlaviaNet Working Group

8
0 Precision tests of the Standard Model with leptonic
0
2 and semileptonic kaon decays

n
a
J

1
1

]
h The FlaviaNet Kaon Working Group∗†‡
p
-
p
e
h Abstract: We present a global analysis of leptonic and semileptonic kaon decays data,
[
including all recent results by BNL-E865, KLOE, KTeV, ISTRA+, and NA48. Experi-

1 mental results are critically reviewed and combined, taking into account theoretical (both
v
analytical and numerical) constraints on the semileptonic kaon form factors. This analysis
7
1 leads to a very accurate determination of V and allows us to perform several stringent
us
8
tests of the Standard Model.
1
.
1
0 Keywords: Vus, CKM, Kaon.
8
0
:
v
i
X
r
a
∗WWW access at http://www.lnf.infn.it/wg/vus
†ThemembersoftheFlaviaNetKaonWorkingGroupwhocontributedmoresignificantlytothisnoteare:
M. Antonelli, V.Cirigliano, P. Franzini, S Glazov, R.Hill, G. Isidori, F. Mescia, M. Moulson, M. Palutan,
E. Passemar, M. Piccini, M. Veltri, O. Yushchenko,R.Wanke.
‡The Collaborations each take responsibility for thepreliminary results of theirown experiment.
Contents
1. Introduction 2
2. Theoretical framework 3
2.1 K and K rates within the SM 3
ℓ3 ℓ2
2.2 Parametrization of K form factors 4
ℓ3
2.2.1 Dispersive constraints 5
2.2.2 Analyticity and improved series expansion 7
2.3 K and K decays beyond the SM 8
ℓ3 ℓ2
2.3.1 The s u effective Hamiltonian 8
→
2.3.2 K rates 10
ℓ2
2.3.3 K rates and kinematical distributions 10
ℓ3
3. Data Analysis 12
3.1 K leading branching ratios and τ 12
L L
3.2 K leading branching ratios and τ 14
S S
3.3 K± leading branching ratios and τ± 14
3.4 Measurement of BR(K )/BR(K ) 15
e2 µ2
3.5 Measurements of Kℓ3 slopes 16
3.5.1 Vector form factor slopes from Kℓ3 16
3.5.2 Scalar and Vector form factor slopes from Kℓ3 18
4. Physics Results 20
4.1 Determination of V f (0) and V /V f /f 20
us + us ud K π
| |× | | | |×
4.1.1 Determination of V f (0) 20
us +
| |×
4.1.2 Determination of V /V f /f 21
us ud K π
| | | |×
4.2 The parameters f (0) and f /f 21
+ K π
4.2.1 Theoretical estimates of f (0) 21
+
4.2.2 Theoretical estimates of f /f 24
K π
4.2.3 A test of lattice calculation: the Callan-Treiman relation 24
4.3 Test of Cabibbo Universality or CKM unitarity 27
4.3.1 Bounds on helicity-suppressed amplitudes 28
4.4 Tests of Lepton Flavor Universality 30
4.4.1 Lepton universality in K decays 30
ℓ3
4.4.2 Lepton universality tests in K decays 30
ℓ2
Acknowledgments 31
A. BRS fit procedure 36
1
B. Fit for K BRs and lifetime 37
L
B.1 Results 39
C. Fit for K± BRs and lifetime 41
C.1 Results 43
D. Averages of form-factor slopes 44
D.1 Procedure 44
D.2 Input data 44
D.3 Fit results for Kℓ3 slopes excluding NA48 Kµ3 data 46
E. Error estimates 47
E.1 K decays 47
e3
E.2 K decays 48
µ3
E.3 From the linear to the dispersive parametrization 48
1. Introduction
In the Standard Model, SM, transition rates of semileptonic processes such as di ujℓν,
→
withdi (uj)beingagenericdown(up)quark,canbecomputedwithhighaccuracy interms
of theFermi couplingG and theelements V of theCabibbo-Kobayashi Maskawa (CKM)
F ji
matrix [1]. Measurements of the transition rates provide therefore precise determinations
of the fundamental SM couplings.
A detailed analysis of semileptonic decays offers also the possibility to set stringent
constraints on new physics scenarios. While within the SM all di ujℓν transitions are
→
ruled by the same CKM coupling V (satisfying the unitarity condition V 2 = 1) and
ji k| ik|
G is the same coupling appearing in the muon decay, this is not necessarily true beyond
F P
theSM.SettingboundsontheviolationsofCKMunitarity,violationsofleptonuniversality,
and deviations from the V A structure, allows us to put significant constraints on various
−
new-physics scenarios (or eventually find evidences of new physics).
In the case of leptonic and semileptonic K decays these tests are particularly signifi-
cant given the large amount of data recently collected by several experiments: BNL-E865,
KLOE, KTeV, ISTRA+, and NA48. These data allow to perform very stringent SM tests
which are almost free from hadronic uncertainties (such as the µ/e universality ratio in
K decays). In addition, the high statistical precision and the detailed information on
ℓ2
kinematical distributions have stimulated a substantial progress also on the theory side:
mostof thetheory-dominated errorsassociated tohadronicformfactors have recently been
reduced below the 1% level.
AnillustrationoftheimportanceofsemileptonicK decaysintestingtheSMisprovided
by the unitarity relation
V 2+ V 2+ V 2 = 1+ǫ . (1.1)
ud us ub NP
| | | | | |
2
Here the V are the CKM elements determined from the various di uj processes, having
ji
→
fixed G from the muon life time: G = 1.166371(6) 10−5GeV−2 [2]. ǫ parametrizes
F µ NP
×
possible deviations from the SM induced by dimension-six operators, contributing either
to the muon decay or to the di uj transitions. By dimensional arguments we expect
→
ǫ M2 /Λ2 , where Λ is the effective scale of new physics. The present accuracy on
NP ∼ W NP NP
V , which is the dominant source of error in (1.1), allows to set bounds on ǫ around
us NP
| |
0.1% or equivalently to set bounds on the new physics scale well above 1 TeV.
In this note we report on progress in the verification of the relation (1.1) as well as
on many other tests of the SM which can be performed with leptonic and semileptonic
K decays. The note is organized as follows. The phenomenological framework needed to
describe K and K decays within and beyond the SM is briefly reviewed in Section 2.
ℓ3 µ2
Section3 is dedicated to the combination of the experimental data. The results and the
interpretation are presented in Section 4.
2. Theoretical framework
2.1 K and K rates within the SM
ℓ3 ℓ2
Within the SM the photon-inclusive K and K decay rates are conveniently decomposed
ℓ3 ℓ2
as [3]
G2m5 2
Γ(K ) = F KC S V 2f (0)2Iℓ (λ ) 1+δK +δKℓ , (2.1)
ℓ3(γ) 192π3 K ew| us| + K +,0 SU(2) em
Γ(K± ) V 2 f2m 1 m2/m2 2 (cid:16) (cid:17)
ℓ2(γ) = us K K − ℓ K (1+δ ) , (2.2)
Γ(π± ) V f2m 1 m2/m2 × em
ℓ2(γ) (cid:12) ud(cid:12) π π (cid:18) − ℓ π (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
where C = 1 (1/2)(cid:12)for t(cid:12)he neutral (charged) kaon decays, Iℓ (λ ) is the phase space
K K +,0
integral that depends on the (experimentally accessible) slopes of the form factors (generi-
cally denotedbyλ ), andS =1.0232(3) istheuniversalshort-distanceelectromagnetic
+,0 ew
correction computed in Ref. [4]. Thechannel-dependentlong-distance electromagnetic cor-
rection factors are denoted by δ and δKℓ. In the K case δ = 0.0070(35) [5, 6], while
em em ℓ2 em −
the four δKℓ are given in Table 1, together with the isospin-breaking corrections due to
em
m = m , denoted by δK .
u 6 d SU(2)
The overall normalization of the K rates depends upon f (0), the K π vector
ℓ3 +
→
form factor at zero momentum transfer [t = (p p )2 = 0]. By convention, f (0) is
K π +
−
defined for the K0 π− matrix element, in the limit m = m and α 0 (keeping
u d em
→ →
kaon and pion masses to their physical value). Similarly, f /f is the ratio of the kaon
K π
and pion decay constants defined in the m = m and α 0 limit. The values of these
u d em
→
hadronic parameters, which represent the dominant source of theoretical uncertainty, will
be discussed in Sect. 4.2.
TheerrorsfortheK electromagneticcorrections,giveninTable1,havebeenobtained
ℓ3
within ChPT, estimating higher-order corrections by naive dimensional analysis [7, 8].
Higher-order chiral corrections have a minor impact in the breaking of lepton universality.
3
δK (%) δKℓ(%)
SU(2) em
K0 0 +0.57(15)
e3
K+ 2.36(22) +0.08(15)
e3
K0 0 +0.80(15)
µ3
K+ 2.36(22) +0.05(15)
µ3
Table 1: Summary of the isospin-breaking corrections factors [7, 8]. The electromagnetic correc-
tions factors correspond to the fully-inclusive K rate.
ℓ3(γ)
The errors are correlated as given below:
1.0 0.1 0.8 0.1
−
1.0 0.1 0.8
 −  . (2.3)
1.0 0.1
 
 1.0 
 
 
2.2 Parametrization of K form factors
ℓ3
The hadronic K π matrix element of the vector current is described by two form factors
→
(FFs), f (t) and f (t), defined by
+ 0
π−(k) s¯γµuK0(p) = (p+k)µf (t)+(p k)µf (t)
h | | i + − −
m2 m2 (2.4)
f (t)= K − π f (t) f (t)
− t 0 − +
(cid:0) (cid:1)
where t = (p k)2. By construction, f (0) = f (0).
0 +
−
In order to compute the phase space integrals appearing in Eq. (2.1) we need ex-
perimental or theoretical inputs about the t-dependence of f (t). In principle, Chiral
+,0
Perturbation Theory (ChPT) and Lattice QCD are useful tools to set theoretical con-
straints. However, in practice the t-dependence of the FFs at present is better determined
by measurements and by combining measurements and dispersion relations.
In the physical region, m2 < t < (m m )2 , a very good approximation for the
ℓ K − π
FFs is given by a Taylor expansion up to t2 terms
(cid:0) (cid:1)
2
f (t) t 1 t
f˜ (t) +,0 = 1+λ′ + λ′′ + .... (2.5)
+,0 ≡ f (0) +,0 m2 2 +,0 m2
+ π (cid:18) π(cid:19)
Note that t = (p p )2 = m2 +m2 2m E , therefore the FFs depend only on E .
K − π K π − K π π
The FF parameters can thus be obtained from a fit to the pion spectrum which is of the
form g(E ) f˜(E )2. Unfortunately t is maximum for E = 0, where g(E ) vanishes.
π π π π
×
Still, experimental information about the vector form factor f˜ measured both from
+
K and K data are quite accurate and so far superior to theoretical predictions. A
e3 µ3
pole parametrization, f˜ (t) = M2/(M2 t), with M 892 MeV corresponding to the
+ V V − V ∼
K∗(892)resonanceandwhichpredictsλ′′ = 2(λ′ )2,isingoodagreementwithpresentdata
+ +
(see later). Improvements of this parametrization have been proposed in Refs. [9, 10, 11].
For instance, in Ref. [11], a dispersive parametrization for f˜ , which has good analytical
+
and unitarity properties and a correct threshold behavior, has been built.
4
Thesituation for the scalar form factor f˜(t) is more complex. For kinematical reasons
0
f (t) is only accessible from K data and one has to deal with the correlations between
0 µ3
the two form factors. Moreover, for f (t), the curvature λ′′ cannot be determined from the
0 0
data and different assumptions for the parametrization of f˜ such as linear, quadratic or
0
polar lead to different results for the slope λ′ which cannot bediscriminated from the data
0
alone. Inturn,theseambiguities induceasystematic uncertainty forV , even thoughdata
us
for partial rates by itself are very accurate. For this reason, the parametrization used has
to rely on theoretical arguments being as model-independent as possible and allowing to
measure at least the slope and the curvature of the form factor.
2.2.1 Dispersive constraints
Thevector andscalar formfactors f (t)inEq.(2.4) areanalytic functionsinthecomplex
+,0
t–plane, except for a cut along the positive real axis, starting at the firstphysical threshold
t = (m +m )2, where they develop discontinuities. They are real for t < t .
th K π th
Cauchy’s theorem implies that f (t) can be written as a dispersive integral along the
+,0
physical cut
∞
1 Imf (s′)
f (t) = ds′ +,0 +subtractions, (2.6)
+,0 π (s′ t i0)
Z − −
tth
whereall possibleon-shellintermediate states contribute toits imaginary partImF (s′). A
k
number of subtractions is needed to make the integral convergent. Particularly appealing
is animproved dispersionrelation recently proposedin Ref.[12]wheretwo subtractions are
performed at t = 0 (where by definition, f˜(0) 1) and at the so-called Callan-Treiman
0
≡
point t (m2 m2) leading to
CT ≡ K − π
t
f˜(t) = exp ln f˜(t ) G(t) (2.7)
0 0 CT
t −
(cid:20) CT (cid:21)
t (t (cid:16)t)(cid:16) ∞ ds′ (cid:17) φ(cid:17)(s′)
CT CT
with G(t) = − ,
π s′ (s′ t )(s′ t iǫ)
Ztth − CT − −
assuming that f˜(t) has no zero. Here φ(x), the phase of f˜(t), can be identified in the
0 0
elastic region with the S-wave, I = 1/2 Kπ scattering phase, δ (s), according to Watson
Kπ
theorem.
A subtraction at t has been performed because the Callan-Treiman theorem implies
CT
f 1
f˜(t ) = K +∆ , (2.8)
0 CT CT
f f (0)
π +
where ∆ (m /4πF ) is a small quantity. ChPT estimates at NLO in the isospin
CT u,d π
∼ O
limit [15], obtain
∆ = ( 3.5 8) 10−3 , (2.9)
CT
− ± ×
where the error is a conservative estimate of the high-order corrections to the expansion
in light quark masses [16]. A complete two-loop evaluation of ∆ , consistent with this
CT
estimate, has been recently presented in Ref. [17].
5
Hence, with only one parameter, f˜(t ), one can determine the shape of f˜ by fitting
0 CT 0
the K decay distribution with the dispersive representation of f˜(t), Eq. (2.7). Then, we
µ3 0
can deduce from Eq. (2.7) the three first coefficients of the Taylor expansion, Eq. (2.5), see
Ref. [12]:
m2 m2
λ′ = π ln f˜(t ) G(0) = π ln f˜(t ) 0.0398(40)) , (2.10)
0 ∆ 0 CT − ∆ 0 CT −
Kπ Kπ
h (cid:16) (cid:17) i h (cid:16) (cid:17) i
λ′′ = (λ′)2 2 m4/t G′(0) = (λ′)2+(4.16 0.50) 10−4 , (2.11)
0 0 − π CT 0 ± ×
λ′′′ = (λ′)3 6 m4/t G′(0) λ′ 3m6/t G′′(0)
0 0 − π CT 0− π CT
= (λ′)3+3 (4.16 0.50) 10−4 λ′ +(2.72 0.11) 10−5. (2.12)
0 ± × 0 ± ×
Furthermore, thanks to Eq. (2.8), measuring f˜(t ) provides a significant constraint on
0 CT
f /f /f (0)limitedonlybythesmalltheoreticaluncertaintyon∆ . Aswewilldiscussin
K π + CT
Section 4.2.3, this representsa powerfulconsistency check of presentlattice QCDestimates
of f /f and f (0).
K π +
A similar dispersive parametrization for the vector form factor has been proposed in
Ref. [11] with two subtractions performed at t = 0. This leads to:
t m2t ∞ ds ϕ(s)
f˜ (t)= exp (Λ +H(t)) , where H(t) = π . (2.13)
+ m2 + π s2 (s t iǫ)
h π i ZtKπ − −
In the elastic region, the phase of the vector form factor, ϕ(s), equals the I = 1/2, P-wave
Kπ scattering phase.
Additional tests can be performed using the expression for the scalar form factor f (t)
0
at order p6 in ChPT [18]:
(f /f 1) 8 8
f (t) = f (0)+∆(t)+ K π − t+ (2Cr +Cr )(m2 +m2)t Cr t2, (2.14)
0 + m2 m2 f4 12 34 K π − f4 12
K − π π π
where
8
f (0) = 1+∆(0) (Cr +Cr )(m2 m2)2 (2.15)
+ − f4 12 34 K − π
π
m2 m2 +m2 m2 f 1 1 ∆′(0)
λ′ = 8 π π K (2Cr +Cr )+ π K +m2
0 f4 f (0) 12 34 m2 m2 f f (0) − f (0) π f (0)
(cid:0)π + (cid:1) K − π (cid:18) π + + (cid:19) +
m4 ∆′′(0)
λ′′ = 16 π Cr +m4
0 − f4 f (0) 12 πf (0)
π + +
Here ∆(t) is a function which receives contributions from order p4 and p6, but like ∆(0) it
isindependentoftheCr,andtheorderp4 chiralconstants Lr onlyappearatorderp6. ∆(t)
i i
and ∆(0) have been evaluated in the physical region in Ref. [18] using for the Lr values a
i
fit to experimental data. An analysis has been presented in ref. [19]. However, the fit has
to be reconsidered in light of the new experimental results as for instance considering the
new K analysis from NA48 and the updated value of f /f .
ℓ4 K π
6
2.2.2 Analyticity and improved series expansion
Armed only with the knowledge that the form factor is analytic outside the cut on the real
axis, analyticity provides powerful constraints on the form factor shape without recourse
to model assumptions. In particular, by an appropriate conformal mapping, the series
expansion (2.5) necessarily “resums” into the form
1
f(t)= (a +a z+a z2+...), (2.16)
0 1 2
φ
where φ is an analytic function and
√t t √t t
th th 0
z(t,t )= − − − (2.17)
0
√t t+√t t
th th 0
− −
is the new expansion parameter. In this “z expansion”, the factor z(t,t ) sums an infinite
0
number of terms, transforming the original series, naively an expansion involving t/t .
+
0.3, into a series with a much smaller expansion parameter. For example, the choice
t = t (1 1 (m2 m2)/t ) minimizes the maximum value of z occurring in the
0 th − − K − π th
physical regioqn, and for this choice z(t,t0) . 0.047.
| |
The function φ and the number t may be regarded as defining a “scheme” for the
0
expansion. The expansion parameter z and coefficients a are then “scheme-dependent”
k
quantities, with the scheme dependence dropping out in physical observables such as f(t).
For the vector form factor, a convenient choice for φ is
1 z(t,0) z(t, Q2) 3/2
φ (t,t ,Q2) = −
F+ 0 32π t Q2 t
r − (cid:18) − − (cid:19)
−1/2 −3/4
z(t,t ) z(t,t ) t t
0 − +
− . (2.18)
× t t t t (t t )1/4
(cid:18) 0− (cid:19) (cid:18) −− (cid:19) +− 0
Thischoiceismotivatedbyargumentsofunitarity, wherebythecoefficients canbebounded
by calculating an inclusive production rate in perturbation theory [23]. In fact, a much
more stringent bound is obtained by isolating the exclusive Kπ production rate in the
vector channel from τ decay data [22]. This enforces [20]
∞ a2
k . 170. (2.19)
a2
k=0 0
X
With this choice of φ, andQ2 = 2GeV2, a convenient choice for t is t = 0.39(m m )2.
0 0 K π
−
This choice eliminates correlations in shape parameters a /a and a /a .
1 0 2 0
The bound on the expansion coefficients can be used to bound errors on physical
quantities describing the form factor shape, as discussed below in Sect. 3.5. A similar
expansion can be used for the scalar form factor. Note that error estimates based on (2.19)
are conservative—no single coefficient is likely to saturate the bound. Also, this boundis a
maximum taken over different schemes; more stringent bounds for particular schemes can
be found in [20].
7
In addition to the direct applications in K decays, it is important for other purposes
ℓ3
to constrain the first few coefficients in (2.16), and check whether the series converges as
expected. K decays provide a unique opportunity to do this. For example, the same
ℓ3
parameterization can be used to constrain the form factor shape in lattice calculations
of f(0), with the threshold t adjusted to the appropriate value for the simulated quark
th
masses. Measurements of a in the kaon system can similarly be used to confirm scaling
k
arguments that apply also in the charm and bottom systems [21].
2.3 K and K decays beyond the SM
ℓ3 ℓ2
2.3.1 The s u effective Hamiltonian
→
On general grounds, assuming only Lorentz invariance and neglecting effective operators
of dimension higher than six, ∆S = 1 charged-current transitions are described by 10
independent operators:
G
H∆S=1 = FV cV (s¯γµLu)(ν¯γµLℓ)+cV (s¯γµLu)(ν¯γµRℓ)
su −√2 us LL LR
(cid:2)
+cV (s¯γµRu)(ν¯γµLℓ)+cV (s¯γµRu)(ν¯γµRℓ)
RL RR
+cS (s¯Lu)(ν¯Lℓ)+cS (s¯Lu)(ν¯Rℓ)
LL LR
+cS (s¯Ru)(ν¯Lℓ)+cS (s¯Ru)(ν¯Rℓ)
RL RR
+cT (s¯σµνLu)(ν¯σµνLℓ)+cT (s¯σµνRu)(ν¯σµνRℓ) +h.c. (2.20)
LL RR
(cid:3)
where L = (1 γ ) and R = (1+γ ). Defining this Hamiltonian at the weak scale, the SM
5 5
−
case corresponds to cV (M2 ) = 1 and all the other coefficients set to zero. The universal
LL W
electromagnetic correction factor S appearing in Eq. (2.1) describes the evolution of cV
ew LL
to hadronic scales: cV (M2)/cV (M2 ) = 1+(S 1)/2 S1/2. A similar expression can
LL ρ LL W ew− ≈ ew
also be written for the Hamiltonian regulating u d transitions.
→
In the case of K πℓν decays only six independent combinations of these operators
→
have a non-vanishing tree-level matrix element:
G
(K πℓν)= FV πℓν c (s¯γµu)(ν¯γ ℓ)+c (s¯γµu)(ν¯γ γ ℓ)
us V µ A µ 5
A → √2
(cid:28) (cid:12)
mℓ (cid:12) mℓ
+ c (s¯u)(ν¯(cid:12)ℓ)+i c (s¯u)(ν¯γ ℓ)
M S (cid:12) M P 5
W W
m m m m
+ s ℓcT(s¯σµνu)(ν¯σµνℓ)+ s ℓcT (s¯σµνu)(ν¯σµνγ ℓ)+h.c. K
M2 M2 γ5 5
W W (cid:12) (cid:29)
(cid:12) (2.21)
(cid:12)
where (cid:12)
c = +(cV +cV +cV +cV ) , (2.22)
V LL RL LR RR
c = (cV +cV cV cV ) , (2.23)
A − LL RL− LR− RR
c = +(cS +cS +cS +cS )M /m , (2.24)
S LL RL LR RR W ℓ
8
ic = (cS +cS cS cS )M /m , (2.25)
P − LL RL− LR− RR W ℓ
cT = 2(cT +cT )M2 /(m m ) , cT = 2(cT cT )M2 /(m m ) . (2.26)
LL RR W ℓ s γ5 − LL− RR W ℓ s
Similarly, in the K ℓν case the independent structures are
→
G
(K ℓν)= FV ℓν k (s¯γµγ u)(ν¯γ ℓ)+k (s¯γµγ u)(ν¯γ γ ℓ)
us V 5 µ A 5 µ 5
A → −√2
(cid:28) (cid:12) (2.27)
m (cid:12) m
ℓ (cid:12) ℓ
+ k (s¯γ u)(ν¯ℓ)+ k (s¯γ u)(ν¯γ ℓ)+h.c. K
S 5 (cid:12) P 5 5
M M
W W (cid:12) (cid:29)
(cid:12)
where (cid:12)
(cid:12)
k = (cV cV +cV cV ) , (2.28)
V − LL− RL LR− RR
k = +(cV cV cV +cV ) , (2.29)
A LL− RL − LR RR
k = (cS cS +cS cS )M /m , (2.30)
S − LL− RL LR− RR W ℓ
k = +(cS cS cS +cS )M /m . (2.31)
P LL− RL − LR RR W ℓ
On general grounds, new degrees of freedom weakly coupled at the scale Λ are ex-
NP
pected to generate corrections of (M2 /Λ2 )to the Wilson coefficients of H∆S=1. Focus-
O W NP su
ing on well-motivated new-physics frameworks, the following two scenarios are particularly
interesting:
In two Higgs doublet models of type-II, such as the Higgs sector of the MSSM,
•
sizable contributions are potentially generated by charged-Higgs exchange diagrams
(see e.g. Ref. [24, 25, 26]). These are well described by the following set of initial
conditions for s u transitions,
→
tan2β m m
cV = 1 and cS = ℓ s , (2.32)
LL LR −(1+ǫ tanβ) m2
0 H+
and for u d transitions,
→
tan2β m m
V,ud S,ud ℓ d
c = 1 and c = . (2.33)
LL LR −(1+ǫ tanβ) m2
0 H+
Here tanβ is the ratio of the two Higgs vacuum expectation values and ǫ is a loop
0
function whose detailed expression can be found in Ref. [25]. In presence of sizable
sources of lepton-flavor symmetry breaking, a non-vanishing scalar-current contribu-
tion to the lepton-flavor violating process K eν is also present [26]. The latter
τ
→
can be parametrized by
m m
cS′ = s τ∆31tan2β . (2.34)
LR m2 R
H+
In the Higgs-less model of Ref. [12], non-standard right-handed quark currents could
•
become detectable. These are described by the following set of initial conditions for
both u s and u d transitions
→ →
cV = (1+δ) and cV = ǫ , (2.35)
LL RL s
V,ud V,ud
c = (1+δ) and c = ǫ , (2.36)
LL RL ns
where ε and δ are free parameters of the model. ǫ can reach a few percents if the
x s
hierarchy of the right-handed mixing matrix is inverted.
9