Bilinear R-parity Violation in Rare Meson Decays

0.19 MB·English

by  A. Ali

BILINEAR R-PARITY VIOLATION IN RARE MESON DECAYS
A.Alia
8
Deutsches Elektronen-Synchrotron, DESY, 22607 Hamburg, Germany
0 A. V.Borisovb, M. V. Sidorovac
0
2 Faculty of Physics, Moscow State University, 119991 Moscow, Russia
Abstract. We discuss rare meson decays K+ → π−ℓ+ℓ′+ and D+ → K−ℓ+ℓ′+
n (ℓ,ℓ′ = e,µ) in a supersymmetric extension of the standard model with explicit
a
breakingofR-paritybybilinearYukawacouplingsinthesuperpotential. Estimates
J
ofthebranchingratiosforthesedecaysaregiven. Wealsocompareournumerical
6 resultswithanalogousonespreviouslyobtainedfortwoothermechanismsoflepton
1 numberviolation: exchangebymassiveMajorananeutrinosandtrilinearR-parity
violation.

]
h
In the standard model (SM), the lepton L and baryon B numbers are con-
p
- servedtoallordersofperturbationtheoryduetotheaccidentalU(1)L U(1)B
p symmetry existing at the level of renormalizable operators. But th×e L and
e
B nonconservation is a generic feature of various extensions of the SM [1].
h
[ That is why lepton-number (LN) violating processes have long been recog-
nized as a sensitive tool to put theories beyond the SM to the test. One of
1
the most well known process of such type is neutrinoless double beta decay
v
(A,Z) (A,Z +2)+e− +e− that has been searched for many years (see,
7
→
1 e.g., [2] and references therein).
5 In Refs. [3,4] rare decays of the pseudoscalar mesons K, D, D , and B of
s
2
the type
.
1 M+ M′−ℓ+ℓ′+ (ℓ,ℓ′ =e,µ) (1)
0 →
8
mediated by light (mN mℓ,mℓ′ ) and heavy (mN mM) Majorana neu-
0 ≪ ≫
trinos were investigated. The indirect upper bounds on the branching ratios
:
v for the decays (1) have been derived taking into account the limits on lepton
i
X mixing and neutrino masses obtained from the precision electroweak measure-
ments,neutrinooscillations,cosmologicaldata,andsearchesoftheneutrinoless
r
a double beta decay. These bounds are greatly more stringent than the direct
experimental ones [5].
In Refs. [6,7] we considered another mechanism of the ∆L = 2 decays
(1) based on the minimal supersymmetric extension of the SM with explicit
R-parity violation (RMSSM, for a review, see [8]). R-parity is defined as
6
R = ( 1)3(B−L)+2S, where B, L, and S are the baryon and lepton numbers
−
and the spin, respectively. The SM fields, including additional Higgs boson
fields appearing in the extended models, have R = 1 while R = 1 for their
−
superpartners.
ae-mail: [email protected]
be-mail: [email protected]
ce-mail: [email protected]
The most generalform for the R-parityand lepton number violating part of
the superpotential is given by [8]
1
W =ε λ LαLβE¯ +λ′ LαQβD¯ +ǫ LαHβ . (2)
6R αβ 2 ijk i j k ijk i j k i i u
(cid:18) (cid:19)
Here i,j,k = 1,2,3 are generation indices, L and Q are SU(2) doublets of
left-handed lepton and quark superfields (α,β = 1,2 are isospinor indices),
E¯ and D¯ are singlets of right-handed superfields of leptons and down quarks,
respectively;H isadoubletHiggssuperfield(withhyperchargeY =1);λ (=
u ijk
λ ), λ′ and ǫ are trilinear and bilinear Yukawa couplings, respectively.
− jik ijk i
We assumed in [6,7] that the bilinear couplings are absent at tree level
(ǫ =0 in Eq. (2)). As well known they are generated by quantum corrections
i
[8] but the dominant contribution of the tree-level trilinear couplings to the
phenomenology is expected. In the present report, we investigate the case of
tree-level bilinear couplings: ǫ =0 with λ=0, λ′ =0. For this case, trilinear
i
6
couplings cannot be generated via radiative corrections.
The bilinear terms in the superpotential (2) induce mixing between the SM
leptons and the MSSM charginos and neutralinos χ˜0 in the mass-eigenstate
n
basis and lead to the following ∆L= 1 lepton-quark operators [9,10]:
±
g
= κ W−ℓ¯γµP χ˜0 +√2g βdν¯ P dd˜∗
LLH −√2 n µ L n k k R R
(cid:16)
+βuν¯ P ucu˜ +βℓ ν¯ P ℓcℓ˜ +βcu¯P ℓcd˜ +H.c. (3)
k k R L ki k R Li R L
(cid:17)
Here P =(1 γ5)/2, γ5 =iγ0γ1γ2γ3; the constants κ (n =1,2,3,4), βd,
L,R ∓ n k
βu, βℓ (k,i = 1,2,3) and βc depend on the elements of the mixing matrices
k ki
diagonalizing the neutrino-neutralino and the charged lepton-chargino mass
matrices.
The Lagrangiandescribing the bilinear mechanism of the decays (1) is
= + + + . (4)
LH SM g˜ χ˜
L L L L L
In addition to the LN violating part (3), it includes the SM charged-current
interactions
g
SM = W+µ( ν¯ℓ(x)γµPLℓ(x)+ q¯(x)γµPLVqq′q′(x))+H.c., (5)
L √2
ℓ q,q′
X X
whereℓ=e,µ,τ;q =u,c,t;q′ =d,s,b;Vqq′ istheCKMmatrix,andtheMSSM
gluino-quark-squarkandneutralino-quark(lepton)-squark(slepton)interactions
[11]
(λ )a
= √2g r b(q¯ g˜(r)q˜b q¯ g˜(r)q˜b)+H.c., (6)
Lg˜ − s 2 aL L− aR R
4
=√2g (ǫ (ψ)ψ¯ χ˜0ψ˜ +ǫ (ψ)ψ¯ χ˜0ψ˜ )+H.c. (7)
Lχ˜ Ln L n L Rn R n R
n=1
X
Here (λ )a are the 3 3 Gell-Mann matrices (r =1,...,8) with color indices
r b
×
a,b=1,2,3; the neutralino coupling constants are defined as
ǫ (ψ)= T (ψ)N +tanθ (T (ψ) Q(ψ))N ,ǫ (ψ)=Q(ψ)tanθ N ,
Ln 3 n2 W 3 n1 Rn W n1
− −
where Q(ψ) and T (ψ) are the electric charge and the third component of the
3
weak isospin for the quark (lepton) field ψ, respectively, and N is the 4 4
nm
×
neutralino mixing matrix.
The leading order amplitude of the decay (1) is described by 9 diagrams
shown in Fig. 1. The hadronic parts of the decay amplitude are calculated
Figure1: Feynmandiagramsforthedecay M+→M′−+ℓ++ℓ′+ inthebilinearR6 MSSM.
Bold vertices correspond to Bethe–Salpeter amplitudes for mesons. There are also crossed
diagramswithinterchangedleptonlines.
with the use of a simple model for the Bethe–Salpeter (BS) amplitudes for
mesons as bound states of a quark and an antiquark [12] (see also [3,4]). In
addition, takinginto accountthatthe mesonmass m m ,m ,where
M W SUSY
≪
m is the W bosonmassandm &100GeVis the commonmassscaleof
W SUSY
superpartners, we neglect momentum dependence in the propagatorsof heavy
particles (see Fig. 1) and use the effective low-energy current-current inter-
action. In this approximation the decay amplitude does not depend on the
specific form of the BS amplitude and is expressed through the known decay
constants of the initial and final mesons, f and f′ .
M M
Finally, for the total width of the decay (1) we have obtained
1 g4f2 f2 m7
Γℓℓ′ ≡Γ(M+ →M′−ℓ+ℓ′+)=(1− 2δℓℓ′) M210Mπ3′ MΦbℓℓi′
4 g2 κ2 V V κ βc ǫ∗ (q ) V
n V V + 13 42 + n Ln 2 V + 13
×(cid:12)(cid:12)nX=12mχ˜n "4m4W (cid:18) 12 43 Nc (cid:19) 2m2W m2q˜2L (cid:18) 43 Nc(cid:19)
(cid:12)
(cid:12) ǫ∗ (q ) V βc2ǫ∗ (q )ǫ∗ (q ) 1
(cid:12) + Ln 3 V + 42 Ln 2 Ln 3 1+
m2q˜3L (cid:18) 12 Nc(cid:19)!− m2q˜3Lm2q˜2L (cid:18) Nc(cid:19)#
2
2 g2βc2
s . (8)
−Nc2m2q˜3Lm2q˜2Lmg˜(cid:12)(cid:12)
(cid:12)
(cid:12)
Here (cid:12)
h−
Φbℓℓi′ = dzz2 1−(h++h−)(2z)−1 2 1−(l++l−)(2z)−1
Z
l+ (cid:2) (cid:3) (cid:2) (cid:3)
[(h z)(h z)(l z)(l z)]1/2 (9)
+ − + −
× − − − −
is the reduced phase space integral with h± = (1 mM′/mM)2 and l± =
±
[(mℓ mℓ′)/mM]2; Nc =3 is the number of colors.
±
Forthenumericalestimatesofthebranchingratios(BRs),Bℓℓ′ =Γℓℓ′/Γtotal,
we have used the known values for the SM couplings g and g , meson decay
s
constants, meson and lepton masses [5], and a typical set of supersymmetric
parameters [13]: a) the MSSM parameters: m = 70 GeV, µ = 500 GeV,
0
M = 200 GeV, tanβ = 4; b) the RPV parameters: Λ ( 3 Λ 2)1/2 =
2 | | ≡ i=1| i|
0.1 GeV2 with 10Λ = Λ = Λ , ǫ2 3 ǫ 2 = Λ with ǫ = ǫ = ǫ ;
1 2 3 | | ≡ i=1| i| | | P 1 2 3
M =(g2/g2)M attheelectroweakscale. UsingtheMSSMmassformulas[11]
3 s 2 P
with the gluino mass m =M , the masses of squarksand neutralinos andthe
g˜ 3
elements of the neutralino mixing matrix were calculated numerically for the
abovesetofparameters. TheresultsofthecalculationswiththeuseofEq. (8)
are shown in the fourth column of Table 1. In the second and third columns
of the table, the present direct experimental bounds on the BRs [5] and the
indirect bounds for the Majorana neutrino mechanism of the rare decays [4]
are shown, respectively. We see that the BRs for the bilinear RPV mechanism
aremuchsmallerthantheseupper bounds. Forcomparison,the trilinearRPV
mechanism leads to the upper limits on the BRs of order 10−23 (10−24) for K
(D) rare decays with the use of conservative bounds on the trilinear couplings
λ′ λ′ .10−3[6,7]. Butformorestringentbounds λ′λ′ .5 10−6[8,14],
| ijk i′j′k′| | | ×
Bℓℓ′(triRMSSM).10−28.
6
Table1: ThebranchingratiosBℓℓ′ fortheraremesondecays M+→M′−ℓ+ℓ′+.
Rare decay Exp. upper Ind. bound Bℓℓ′
bound on Bℓℓ′ on Bℓℓ′ (νMSM) (biRMSSM)
6
K+ π−e+e+ 6.4 10−10 5.9 10−32 3.6 10−49
→ × × ×
K+ π−µ+µ+ 3.0 10−9 1.1 10−24 1.0 10−49
→ × × ×
K+ π−e+µ+ 5.0 10−10 5.1 10−24 4.2 10−49
→ × × ×
D+ K−e+e+ 4.5 10−6 1.5 10−31 1.6 10−48
→ × × ×
D+ K−µ+µ+ 1.3 10−5 8.9 10−24 1.5 10−48
→ × × ×
D+ K−e+µ+ 1.3 10−4 2.1 10−23 3.1 10−48
→ × × ×
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