#### Constraints on new physics from \(K \rightarrow \pi \nu {\bar{\nu }}\)

Eur. Phys. J. C
Constraints on new physics from K → π ν ν¯
Xiao-Gang He 0 1 2
German Valencia 3
Keith Wong 3
0 Tsung-Dao Lee Institute, and School of Physics and Astronomy, Shanghai Jiao Tong University , Shanghai 200240 , China
1 Physics Division, National Center for Theoretical Sciences , Hsinchu 30013 , Taiwan
2 Department of Physics, National Taiwan University , Taipei 10617 , Taiwan
3 School of Physics and Astronomy, Monash University , Melbourne, VIC 3800 , Australia
We study generic effects of new physics on the rare decay modes K L → π 0νν¯ and K + → π +νν¯ . We discuss several cases: left-handed neutrino couplings; right handed neutrino couplings; neutrino lepton flavour violating (LFV) interactions; and I = 3/2 interactions. The first of these cases has been studied before as it covers many new physics extensions of the standard model; the second one requires the existence of a new light (sterile) right-handed neutrino and its contribution to both branching ratios is always additive to the SM. The case of neutrino LFV couplings introduces a CP conserving contribution to K L → π 0νν¯ which affects the rates in a similar manner as a right handed neutrino as neither one of these interferes with the standard model amplitudes. Finally, we consider new physics with I = 3/2 interactions to go beyond the Grossman-Nir bound. We find that the rare kaon rates are only sensitive to new physics scales up to a few GeV for this scenario.
1 Introduction
In the standard model (SM), the rare decay modes K →
π νν proceed dominantly via a short distance contribution
from a top-quark intermediate loop. This allows a precise
calculation of the rates in terms of SM parameters [1,2]. The
effective Hamiltonian responsible for these transitions in the
SM is frequently written as
G F 2α
H = √2 π sW2 Vts Vtd X (xt )s¯γμ PL d
ν¯ γ μ PL ν .
(
1
)
It follows that the branching ratios can then be written as (we
use the notation BK + = B K + → π +νν(γ ) and BKL =
B(K L → π 0νν) throughout this paper),
BK + = κ˜+
Im(Vts Vtd Xt ) 2
λ5
BKL = κL
Im(Vts Vtd Xt ) 2
λ5
,
In these equations, the hadronic matrix element of the quark
current is written in terms of the well measured semileptonic
Ke3 rate and is part of the overall constants κ˜+ and κL .
Modern calculations of the parameters in these equations result
in: κ˜+ = 0.517 × 10−10 which includes long distance QED
corrections [3], and κL = 2.23 × 10−10; the Inami-Lim
function for the short distance top-quark contribution [4]
including NLO QCD corrections [5] and the two-loop electroweak
correction [6], result in Xt = 1.48; and all known effects of
the charm-quark contributions [7–10] in Pc = 0.404. Finally,
λ ≈ 0.225 is the usual Wolfenstein parameter.1
Our estimate for these branching ratios within the SM,
using the latest CKMfitter input [11], is
BK + = (8.3 ± 0.4) × 10−11,
BKL = (2.9 ± 0.2) × 10−11.
These numbers are to be compared with the current
experimental results for the charged [12–15] (measured by BNL
787 and BNL 949) and neutral [16] modes (from KEK
E391a),
BK + = (1.73+−11..1055) × 10−10,
BKL ≤ 2.6 × 10−8 at 90% c.l.
An interesting correlation between these two modes was
<
pointed out by Grossman and Nir (GN), namely that BKL ∼
1 Uncertainties for these quantities can be found in the references.
(
2
)
(
3
)
(
4
)
4.4 BK + which is satisfied in a nearly model independent
way [17]. 2
In this paper we revisit these modes in the context of
generic new physics motivated by the new results that are
expected soon for the charged mode from NA62 at CERN
and for the neutral mode from KOTO in Japan. Our paper is
organised in terms of the neutrino interactions as follows: in
Sect. 2 we briefly review extensions of the SM in which the
neutrino interactions are left handed and flavour conserving;
in Sect. 3 we consider extensions of the SM with right-handed
neutrino interactions; in Sect. 4 we discuss the lepton flavour
violating case. In Sect. 5 we study interactions that violate
the GN bound and finally, in Sect. 6, we conclude.
2 New physics with lepton flavour conserving left-handed neutrinos
In this case the effective Hamiltonian describing the effects
of the new physics (NP) takes the form
G F 2α
He f f = √2 π sW2 Vts Vtd X N s¯γμd
ν¯ γ μ PL ν ,
(
5
)
where the parameters encoding the NP are collected in X N
and the overall constants have been chosen for convenience.
Notice that this form is valid for both left-handed and
righthanded quark currents as only the vector current is operative
for the K → π transition. Numerically it is then possible to
obtain the rates from the SM result, Eq. 2, via the substitution
X (xt ) → X (xt ) + X N . This has been done in the literature
for a variety of models [19] so we will not dwell on this case
here. In Fig. 1 we illustrate the results. In general X N ≡
zeiφ and the parameterisation in Eq. 5 implies that φ = 0
corresponds to NP with the same phase as λt = Vts Vtd .
The green curve corresponds to φ = 0 (so called MFV in
[19]) and its two branches correspond to constructive and
destructive interference with the charm-quark contribution
in Eq. 2. The tick marks on the curve mark values of |X N | =
z. If we allow for an arbitrary phase, this type of NP can
populate the entire area below the GN bound, making it nearly
impossible to translate a non-SM measurement into values
of z and φ.
We illustrate two more situations: the blue line shows φ
being minus the phase of λt , which corresponds to CP
conserving NP which does not contribute to the neutral kaon
mode. The red line shows φ being the same as the phase of
λt , which corresponds to NP which doubles the SM phase.
Interestingly this case nearly saturates the GN bound. For
comparison, we show the purple oval representing the 1σ
2 It was recently noted that the GN bound applied to the experimental
result for K + → π +νν needs to treat a possible two body intermediate
state separately [18].
SM allowed region as predicted using the parameters and
uncertainties in CKMfitter [11]. For the NP, however, we
have only included the SM central values in Eq. 5. Allowing
the SM parameters to vary in the rates that include NP, turns
the green line into an arc-shaped region as can be seen in
Ref. [19] for example.
Finally we have included in the plot a vertical red dashed
line which marks a 30% uncertainty from the SM central
value. This number has been chosen as it corresponds to the
statistical uncertainty that can be achieved with 10 events
that agree with the SM, in the ball park of what is expected
from NA62.
3 A light right handed neutrino
In models which contain a light right handed neutrino the
effective Hamiltonian can be written as
G F 2α 1
He f f = √2 π sW2 Vts Vtd 2 s¯γμd
×
Xt
ν¯ γ μ PL ν + X˜ ν¯ R γ μ PR νR ,
(
6
)
where the first term is the SM, the new physics is
parameterised by X˜ and its coupling to quarks can be through either a
left or right handed current. In writing Eq. 6 we have assumed
that there is only one new neutrino and that its mass is
negligible. The rates for the rare kaon decay modes follow
immediately,
κ
˜+
BK + (νR H ) = BK + (S M ) + 3
κL
BKL (νR H ) = BKL (S M ) + 3
λt X˜
λ5
2
Imλt X˜
λ5
2
(
7
)
where the 1/3 accounts for the fact that we have only one
right handed light neutrino (a factor of 3 from summing over
the left-handed neutrinos is hiding in κ˜+ and κL ). In the result,
Eq. 7, we see that this type of NP can only increase the rates,
as it does not interfere with the SM, and this is illustrated in
Fig. 2. As in the previous case, we have chosen a
parameterisation in Eq. 6 in which X˜ ≡ |X˜ |eiφ and φ = 0 corresponds
to the NP having the same phase as λt . The green line in the
figure corresponds to φ = 0 and the tick marks show that
a maximum value of |X˜ | ∼< 5.5 is allowed by the current
BNL 90% c.l. limit on the charged rate, and that this number
can be reduced to |X˜ | ∼< 2 with about ten events. The pink
region covers the parameter space |X˜ | ≤ 5.5 with an arbitrary
phase, and we show two more lines near the boundary of this
region. The red line is obtained for φ +φλt = (π/2 or 3π/2);
(
8
)
whereas the blue line occurs for φ+φλt = (0 or π ), for which
there is no new contribution to the neutral mode.
Within the specific model detailed in the Appendix, the
effect of the additional neutrino contributes both via a flavour
changing tree-level Z exchange and a one-loop Z penguin
and can be written as,
X˜ = −
MM2Z2 cot2 θR
Z
s2
W I (λt , λH ) +
2
π sW4 VRdbs VRdbd .
α Vts Vtd
The overall strength of the Z coupling is parameterised by
cot θR ∼< 20, where the upper limit arises from requiring the
interaction to remain perturbative [20]. This, combined with
the CMS limit on a Z that decays to tau-pairs MZ >∼ 1.7 TeV
[21], implies that the factor in the first bracket of Eq. 8 can
be of order one. The tree-level contribution (second term in
the second bracket) is constrained to be small by Bs -mixing
and Bd -mixing, |VRdbs VRdbd /(Vts Vtd )| <∼ 3 × 10−3 [22]. The
Inami-Lim factor appearing in the Z penguin, I (λt , λH ), is
less constrained and can be of order 10 [24]. All in all, in our
model the magnitude of X˜ can be order one but its phase is
limited by the size of the tree contribution. This provides an
example of NP in which a measurement of the two rates can
be mapped to parameters in the model.
The existence of an additional light neutrino can, in
general, have other observable consequences. As we show in Ref.
[26], the invisible Z width constrains the mixing between the
Z and Z bosons in our model. This mixing, however, does
not alter the leading contributions to X˜ shown in Eq. 8. In
essence the Z width does not constrain this additional light
neutrino because it is sterile as far as the SM interactions
are concerned. A new light right-handed neutrino also
contributes to the effective number of neutrino species Ne f f
which is constrained by cosmological considerations. In Ref.
[27] we show that this constraint can also be evaded if the
new neutrino mixes dominantly with the tau-neutrino and not
with the muon or electron neutrinos.
4 Neutrino lepton flavour violating interactions
Another possibility consists of interactions that violate
lepton flavour conservation in the neutrino sector. These are
particularly interesting because they can yield CP
conserving contributions to the K L → π 0νν¯ decay. In this case it is
convenient to write
G F 2α 1
He f f = √2 π sW2 2
s¯γμd
×
Vts Vtd Xt + λ5W
ν¯ γ μ PL ν
+λ5
i = j
⎞
Wi j ν¯i γ μ PL ν j ⎠
+ h. c.
to normalise the strength of the NP to that of the SM but
without inserting the SM phase into the new couplings. This
then results in
where again a factor of 1/3 compensates for the factor of
3 hiding in κ˜+ and κL . These lepton flavor violating
contributions (proportional to Wi j , i = j ) produce a very similar
pattern of corrections as the case of the right handed
neutrino Eq. 6. This LFV contribution to the neutral mode is
maximised when
Wi j = −W ji ,
and we illustrate this scenario in Fig. 3. The green line
corresponds to the case Weμ = −Wμe and the dots mark values of
|Weμ|. The allowed region when only Weμ,μe is allowed to be
non-zero and satisfying |Weμ,μe| ≤ 6 with arbitrary phases
is shown in pink. The blue line, where the neutral kaon rate
is unaffected, occurs for Wi j = W ji .
Neither the LFV nor the right-handed neutrino
scenarios interferes with the SM amplitude, so they both result in
additive corrections to the rates. We can illustrate the
correspondence between the two cases by considering the red line
of Fig. 2 for which the phase of X˜ plus the phase of λt equals
π/2, 3π/2 and therefore Re(λt X˜ ) = 0. This line matches
the green line of Fig. 3 for Weμ = −Wμe, and |Weμ| ∼ 1 is
equivalent to | X˜ | ∼ 2.3.
Figure 3 indicates that this model can have important
effects for Wi j ∼ O(
1
). In terms of the leptoquark couplings
shown in the Appendix, ci j is of order
G F 2α g2
ci j ∼ √2 π sW2 Vts Vtd Wi j ∼ (83.5 TeV)2
implying that for leptoquark couplings of electroweak
strength, these rare kaon modes are sensitive to leptoquark
masses up to about 80 TeV.
(
9
)
(
10
)
(
11
)
(
12
)
5 Beyond the Grossman-Nir bound
The hadronic transition between a kaon and a pion can be
mediated in general by an operator that changes isospin by
1/2 or by 3/2. The ratio of matrix elements follows from the
Clebsch-Gordan coefficients
and the GN bound follows from the first of these equations,
appropriate for the s¯d isospin structure of dimension six
effective Hamiltonians of the cases discussed so far. Long
distance contributions in the SM can violate this isospin
relation but they are known to be small [36]. Long distance
contributions within the SM can also produce CP conserving
contributions to K L → π 0νν due to different CP properties
of the relevant operators but these effects are also known to
be small [37].
When the K → π transition is mediated by a vector
current, as in the short distance SM of Eq.1, the K L → π 0νν¯
decay is CP violating due to the CP transformation
properties of the current: s¯γμd ←C →P −d¯γ μs. In the same manner
K L → π 0νν¯ is CP conserving when mediated by a scalar
C P
density as s¯d ←→ d¯s [38].3
3 The operator discussed in this reference, s¯R dL ν¯ R νL , can be generated
by leptoquark exchange at tree level in models which also have right
handed neutrinos. Its effects satisfy the GN bound and, as it does not
interfere with the SM, produces changes to the rates similar to the ones
already discussed for LFV interactions.
To construct a S = 1, I = 3/2 operator one needs
at least four quarks, and they have to take a current-current
form which then leads to a CP conserving K L → π 0νν¯ 4.
Operators with these properties can occur beyond the SM
as we parameterise in the appendix, where we show that the
effect on the K → π νν¯ modes can be written as (when added
to the SM)
Im(Vts Vtd Xt )
λ5
,
These relations are illustrated in Fig. 4 where the range
covered by the rates of Eq. 14 is shown in pink along with the
BNL result in green and the GN exclusion in grey. The SM
central values are shown as the large red dot (the one sigma
SM region is small on the scale of this plot) and the dashed
vertical lines correspond to ±3σ from the central SM value
of BK + . The green curve for φκ = 43◦ and the blue curve
for φκ = 38◦ are chosen to illustrate values that can produce
BKL ∼ 10−9 while keeping BK + near its SM value.
When I = 3/2 interactions are present, the GN bound is
no longer valid. In addition, with the pattern of NP appearing
in Eq. 14 and illustrated in Fig. 4, it is possible to keep the
charged rate close to the SM while making the neutral rate
as large as desired. Interestingly, the GN bound is violated
via two different effects: the factor of 2 present in the second
line of Eq. 14 due to the I = 3/2 nature of the operator,
and the fact that the new contribution to the neutral rate is
CP conserving. As such, a I = 1/2 operator of the
currentcurrent form also violates the GN bound as can be checked by
removing the factor of 2 present in the second line of Eq. 14.
In both cases the new operator produces a CP conserving
contribution to the neutral kaon decay and interferes with the
SM. These properties result in a new contribution that can
cancel the SM for the charged mode but not for the neutral
mode.
Considering the dimension eight operator of the appendix,
Eq. C1, the NP coupling reads,
√
κ 23 = gN4P fπ fK m2K r ps
(
15
)
2π sW4 .
αλ5
3
Figure 4 shows that the rare kaon rates are sensitive to κ 2 ∼
1. With gN P ∼ 1 this then implies they are sensitive to a NP
scale of order ∼ 2.3 GeV. Given that this scale is only
4 A four-quark operator of the form current-scalar-density would lead
to a CP violating K L → π 0νν¯, but this has non-vanishing K → π
matrix elements only if it is I = 1/2.
a few times larger than QC D, our result is the same for
the different types of possibilities discussed in the appendix,
and it shows that even though this scenario is possible in
principle, its effects are extremely small. The conclusion is
that a violation of the GN bound is completely implausible
without a new few GeV particle that carries isospin.
6 Conclusions
We have considered how different types of new physics can
affect the rates of the rare kaon decay modes K → π νν¯ . Our
findings can be summarised as follows.
• Lepton flavour conserving left-handed neutrinos. This
case allows interference between the NP and the SM and
can produce values for the rates anywhere below the GN
bound as shown in Fig. 1. Measurement of these rates
can result in clear evidence for NP but an interpretation
of the results in terms of NP parameters will be much
harder.
• A light right-handed neutrino. In this case the NP
cannot interfere with the SM so the resulting rates are always
larger than the SM values. A measurement of the charged
mode by NA62 which agrees with the SM with roughly
ten events would place strong new constraints on the
magnitude of the RH neutrino interactions. It would also result
< 14 × 10−11.
in an upper bound for the neutral rate BKL ∼
• Neutrino lepton flavour violating interactions. These
scenarios are very similar to new right handed neutrinos as
they also do not interfere with the SM. A measurement of
the charged mode by NA62 would thus produce
equivalent constraints as in the case with RH neutrinos. Models
that generate this type of interactions, such as the LQ
discussed in the appendix, are likely to also generate flavour
conserving left-handed neutrino interactions. In that case
there is no clear connection between a measurement and
NP parameters.
• We found two types of interactions that can violate the
GN bound: those with I = 3/2 transitions; or those
with I = 1/2 current-current interactions. The
former modify the isospin relation underpinning the GN
bound and the latter can cancel the SM contribution to
the charged mode while increasing the rate of the neutral
mode through a CP conserving contribution. Both
scenarios would dilute the correlation between charged and
neutral modes requiring a direct measurement of the
neutral mode to constrain it. Our study in terms of effective
operators suggests that only very low values of the new
physics scale, of order a few GeV, would be observable
Acknowledgements This work was supported in part by the
ResearchFirst undergraduate research program at Monash University and by
the Australian Research Council. X.G.H. was supported in part by
the MOST (Grant No. MOST104-2112-M-002-015-MY3 and
1062112-M-002-003-MY3), and in part by Key Laboratory for Particle
Physics, Astrophysics and Cosmology, Ministry of Education, and
Shanghai Key Laboratory for Particle Physics and Cosmology (Grant
No. 15DZ2272100), and in part by the NSFC (Grant Nos. 11575111
and 11735010).
Open Access This article is distributed under the terms of the Creative
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ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: A model with a right handed neutrino
The model has been described in detail elsewhere [20,
23], here we summarise its salient features. The gauge
group is SU (
3
)C × SU (
2
)L × SU (
2
)R × U (
1
)B−L , but
the three generations of fermions are chosen to
transform differently to single out the third generation. In the
weak interaction basis, the first two generations of quarks
Q1L,2, UR1,2, D1,2 transform as (
3, 2, 1
)(1/3), (
3, 1, 1
)(4/3)
R
and (
3, 1, 1
)(−2/3), and the leptons L1L,2, E 1,2
transR
form as (
1, 2, 1
)(−1) and (
1, 1, 1
)(−2). The third
generation, on the other hand, transforms as Q3L (
3, 2, 1
)(1/3),
Q3R (
3, 1, 2
)(1/3), L3L (
1, 2, 1
)(−1) and L3R (
1, 1, 2
)(−1).
In this way SU (
2
)R acts only on the third generation.
To separate the symmetry breaking scales of SU (
2
)L and
SU (
2
)R , there are two Higgs multiplets HL (
1, 2, 1
)(−1)
and HR (
1, 1, 2
)(−1) with respective vevs vL and vR . An
additional bi-doublet φ (
1, 2, 2
)(0) scalar with vevs v1,2 is
needed to provide mass to the fermions. Since both v1 and v2
are required to be non-zero for fermion mass generation, the
WL and WR gauge bosons of S(
2
)L and SU (
2
)R will mix
with each other. In terms of the mass eigenstates W and W ,
the mixing can be parameterized as
WL = cos ξW W − sin ξW W ,
WR = sin ξW W + cos ξW W .
In the mass eigenstate basis the quark-gauge-boson
interactions are given by,
gL U¯ L γ μVK M DL (cos ξW Wμ+ − sin ξW Wμ+)
LW = − √2
gR U¯ R γ μVR DR (sin ξW Wμ+
− √2
+ cos ξW Wμ+) + h. c.,
LZ = g2L tan θW (tan θR + cot θR )(sin ξZ Zμ + cos ξZ Zμ)
×(d¯Ri VRdb∗i VRdbj γ μdR j − u¯ Ri VRut∗i VRut j γ μu R j ),
(A1)
(A2)
(A3)
In our model UL = (ULi j ), URL = (URLi3) and UL R =
(UL R3i ) and UR = (UR33) are 3 × 3, 3 × 1, 1 × 3 and 1 × 1
matrices, respectively.
Rotating the charged leptons from their weak eigenstates
L,R to their mass eigenstates mL,R , with L,R = VL,R mL,R ,
the lepton interaction with W and W becomes
where U = (u, c, t ), D = (d, s, b), VK M is the
Kobayashi-Maskawa mixing matrix and VR ≡ (VRi j ) =
(VRut∗i VRdbj ) with VRui,dj the unitary matrices which rotate the
right handed quarks u Ri and dRi from the weak to the mass
eigenstate basis.
The model has three left-handed neutrinos νLi and one
right-handed neutrino νR3 . Additional scalars L (
1, 3, 1
)(
2
)
and R (
1, 1, 3
)(
2
) with vevs vL,R are needed to
generate neutrino masses. In order for this model to contribute
to the rare kaon decay modes discussed here, we need the
right-handed neutrino to be light and thus requires vL,R to
be small. The mass eigenstates (νLm , (νmR3 )c) are related by a
unitary transformation to the weak eigenstates as
νL
c
νR3
=
UL URL
UL R UR
νLm
(νmR3 )c
.
gL
LW = − √2 (ν¯L γ μU † L + ν¯ cR3γ μUR∗L j3 L j )
×(cos ξW Wμ+ − sin ξW Wμ+)
− √gR2 (ν¯ Lci γ μUL Ri j R j + ν¯ R3γ μUR3 j R j )
× (sin ξW Wμ+ + cos ξW Wμ+) + h. c.,
LZ = g2L tan θW (tan θR + cot θR )(sin ξZ Zμ + cos ξZ Zμ)
×(τ¯Ri VR∗3i VR3 j γ μτR j − ν¯ R3γ μ PR νR3),
where
U † = UL† VL , UR∗L j3 = (UR∗ Li3VLi j ),
UL Ri j = UL R3i VR3 j , UR3 j = UR33VR3 j .
U is approximately the PMNS matrix. From Eqs. A2 and A4
we see that a large gR /gL will enhance the third generation
interactions with W .
In terms of neutrino mass eigenstates,
ν¯ R3γ μνR3 = −(ν¯Lmi UL∗Rki + ν¯ mR3cUR∗33)γ μ
×(UL Rk j νLmj + UR33νmR3c) .
The new operators in this model that contribute to the rare
kaon decay occur at tree level with new FCNC couplings at
one-loop with new Z penguin [24]. They are
(A4)
(A5)
HT = − G√F 2sW2 MM2Z2 cot2 θR VRdbs VRdbd s¯γμ
2 Z
× PR d ν¯ R3γ μ PR νR3
G F2 πα MM2Z2 cot2 θR Vts Vtd I (λt , λH )s¯γμ
HL = − √
Z
× PL d ν¯ R3γ μ PR νR3
Both contributions couple to the right-handed neutrino so
they do not interfere with the SM. In the quark current, only
the vector term contributes to a K → π transition so both
LH and RH contribute in the same manner to Eq. 6.
Appendix B: Models with leptoquarks
The interest of leptoquarks in kaon decays has been recently
revived in connection to the B-anomalies [30,31], here we
will conduct a model independent analysis as in earlier papers
[32,33]. The scalar S and vector V leptoquark couplings to
SM fermions which include a left-handed neutrino νL are,
†
LS = λL S0 q¯ Lc i τ2 L S0
† †
+λL S˜1/2 d¯R L S˜1/2 + λL S1 q¯ Lc i τ2τ · S1 L + h. c. ,
LV = λLV˜1/2 d¯Rc γμ L V˜1†/μ2 + λLV1 q¯L γμτ · V1†μ L + h. c.,
(B1)
(B3)
(B4)
(B5)
where the leptoquark fields and their transformation
properties under the SM group are given by
Of these leptoquarks all but S0 contribute to processes dd¯ →
νν¯ , ud¯ → +ν, uu¯ → + −and dd¯ → + −. This
usually means that their effects in the kaon sector are severely
constrained by K L → μe which places their mass in the
hundreds of TeV for couplings of electroweak strength and
above 1000 TeV for Pati-Salam leptoquarks [34]. On the
other hand S0 does not contribute to dd¯ → + − processes
and its effects in the kaon sector are mostly constrained by
lepton universality in π 2 and K 2 decays, and as we show
here, by K → π νν¯ . Leptoquark models produce both LFC
and LFV interactions in general so their contribution to the
rare kaon rates are generally of the form
BK + =
The Wi j parameters appearing here are versions of the
ci j in Eq. B5 but with a different normalisation, ci j ∼
G F
√2 π2sαW2 Vts Vtd Wi j . In the main text we only consider the
effect of the LFV couplings as the LFC ones fall under the
same type of NP as Eq. 5.
Appendix C:
I = 3/2 transitions
To change the GN relation we construct a I = 3/2
operator to mediate the K → π transition. This requires four
quark fields, and an example of a dimension eight operator
consistent with the symmetries of the SM that accomplishes
this is
LN P =
gN P
4
u¯γν PR s d¯γμ PR u + d¯γμ PR s
× (u¯γν PR u − d¯γν PR d) g Bμν + h. c.
(C1)
in which gN P is complex. On dimensional grounds this low
energy effective operator is dimension eight and was
therefore normalised with 4. In general there are two
possibilities: the operator may arise from a dimension eight
operator describing physics beyond the SM at the electroweak
scale in which case 4 = 4N P ; or it may arise from a
dimension six new physics operator. In the latter case one
of the quark currents may occur from a long distance
photon, for example, and the scale suppression could be smaller,
4 ∼ 2N P 2QC D as in
LN P ∼ i d¯γμ Dν PR s Bμν + h. c.
A possible bosonisation for the four-quark operator in Eq. C1
of the current-current form is, R2μ1 R1ν3 + R23(R1ν1 − R2ν2),
μ
where Rμ = i fπ2U † DμU and U = exp(2i φ/ fπ ) with φ
the pseudoscalar meson octet as usual in chiral perturbation
(C2)
theory [35]. This allows us to write Eq. C1 as, 5
LN P = 2
gN P sW2 fπ fK
4
× cgW Z μν + h. c.
which then leads to matrix elements
M(K + → π +νν¯ )N P
G F 2√2sW2 fπ fK gN P mν2ν 2 pμK
= √2 4
M(K L → π 0νν¯ )N P
G F 4√2sW2 fπ fK Re(gN P ) mν2ν 2 pμK
= √2 4
√
2∂μ K 0∂ν π 0 + ∂μ K +∂ν π −
(C3)
ν¯ γμ PL ν ,
ν¯ γμ PL ν .
(C4)
Notice that this current current operator leads to a CP
conserving contribution to the K L decay. In addition, compared
to the matrix elements of the operators discussed in
previous sections there is an additional mν2ν term in Eq. C4. This
modifies the rates by the factor
r 2ps m4K ≡
d 3 mν2ν pμK ν¯ γμ PL ν
μ 2
d 3 pK ν¯ γμ PL ν
2
≈ (0.171m2K )2
(C5)
so that r ps = 0.171 for K + decay, and r ps = 0.176 for K L
decays.
It is possible to write an analogue of Eq. C1 using
lefthanded quark fields at the expense of higher dimensionality.
An example being,
LN P ∼
gN P
6
q¯2γμτ I PL q1 q¯1γν τ I PL q1
†
−3q¯2γμ PL q1 q¯1γν τ I PL q1
†τ I
g Bμν + h. c.
(C6)
where 1, 2 are generation indices, is the SM scalar doublet,
and additional flavour changing operators involving charm
are also produced.
In principle one could start with an operator at the
electroweak scale with a flavour structure such that it contributes
only to the neutral kaon decay. This would be a mixture of
I = 1/2 and I = 3/2 operators and the two components
would evolve differently under QCD running resulting in a
different flavour structure at the hadronic scale.
5 We are not interested in finding the most general representation of the
operator in chiral perturbation theory, but only in giving an example of
what the matrix element might look like.
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