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2-nd Vienna Central European Seminar, Nov 25-27, 2005
1
Rare Meson Decays in Theories Beyond the
Standard Model
A. Ali (DESY), A. V. Borisov,
M. V. Sidorova (MSU)
2-nd Vienna Central European Seminar, Nov 25-27, 2005
2
MM
2L
SM MM
Beyond the SM: -Massive Majorana neutrinos-R-parity-violating SUSY
2-nd Vienna Central European Seminar, Nov 25-27, 2005
3
Feynman diagrams for the rare meson decay mediated by Majorana neutrinos
MMM M
M M
2q
3q1q
4q
W WN 2q
1q 3q
4q
W
W
Ntree
box
2-nd Vienna Central European Seminar, Nov 25-27, 2005
4
1. Rare Meson Decays mediated by massive Majorana neutrinos
,
)2(22
||)()2()
2
11(
00093
3332)4(4
kkp
kkddpd
m
AApkkp
M
bt
tA - tree diagram - box diagrambA
,)2(
1)(
)(448
ii
i LHqqddA ;,bti
)(iL )(iH - lepton tensor - hadron tensor
The width of the decay:
2-nd Vienna Central European Seminar, Nov 25-27, 2005
5
N
NNNNWiWi
i mUUmp
g
mp
ggL
2222
4
)( 4
,)(2
1
)()()( 5
2222
kmkpmkp
kNiNi
С
LEPTON TENSOR
1Т the relative CP-phases,
,ppt ,ppt ,)(2
1qqpppb .)(
2
1qqpppb
-
ii
iU U - neutrino mixing matrix
2-nd Vienna Central European Seminar, Nov 25-27, 2005
6
HADRON TENSOR
2
1)(
2
1)( 5
435
12)( VqTrVqTrH pp
t
.2
1)(
2
1)( 5
425
13)(
VqVqTrH pp
b
2-nd Vienna Central European Seminar, Nov 25-27, 2005
7
The Bethe-Salpeter Formalism
Mesons as bound states of a quark and an antiquark:
)()()()(0),( 221121 xepMxqxqTN
ixx p
Xipa
a
C
p
X 21 xxx
3,2,1a 3CN
- the center-of-mass - the relative coordinates
- color index
2-nd Vienna Central European Seminar, Nov 25-27, 2005
8
GpMpxiq
p qpxxedq )()1()()( 54
221 /)( MM mmm
Mm
2,1 mm
2)( 21 ppq
pp
21 ppp
- the mass of the meson M
21,qq
- the total 4-momentum of the meson
quark masses-
- the relative 4-momentum
G )()( Cf NSUNSU - -group factor
2-nd Vienna Central European Seminar, Nov 25-27, 2005
9
2
2
2
2122
22
1exp)(1
4)( q
m
qpmq
MMMp
- parameter of the model
)(qp - model dependent scalar function
The definition of the decay constant of the meson:
)}0({)()0()0(0 51
52 xTrNipMqqpif pC
aaM
)()2(
44
4
qqd
Nf pMCM
2-nd Vienna Central European Seminar, Nov 25-27, 2005
10
)(43124
1tMMt LppVVffA
)(2 4213
ppippgppppVVA MMb
)(4
4
4
4
)()()2()2( bpp Lqqqdqd
MM
Amplitude for tree and box diagrams of the meson decay
2-nd Vienna Central European Seminar, Nov 25-27, 2005
11
MN mm Heavy neutrinos
In this case the total decay width is model independent
,||128
212223
74
mKff
mGVMM
MF
N NNNN m
UUm11 - the effective inverse Majorana mass
42134312
1VV
NVVK
CV
2-nd Vienna Central European Seminar, Nov 25-27, 2005
12
The branching ratios ,)(
)(
allM
MMB
.21
m
BС
Using bounds on matrix elements from the others processes one can obtain indirect bounds on the branching ratios.
The bound from of nucleus:
Also there are bounds on other mixing parameters:
decay0 Ge76 181 102.1 GeVmee
32 106.6|| N
eNU32 100.6|| N
NU
2-nd Vienna Central European Seminar, Nov 25-27, 2005
13
From the heavy neutral Majorana lepton detecting experiments:
Assuming one heavy neutrino scenario:
GeVmN 7.90
141 )105.1( GeVm141 )104.1( GeVme
2-nd Vienna Central European Seminar, Nov 25-27, 2005
14
Exp. upper boundson
10104.6 eeK K eK
eeKD KD eKD
c Ind. Boundson
B 2MeV
9100.3 10100.5
10105.8 32109.5 10104.2 24101.1 9100.1
24101.5
4102.1 5103.1 4103.1
9102.2 9100.2 9102.4
31105.1 24109.8 23101.2
Rare decay B
Experimental and indirect bounds on the branching ratios for the rare meson decays mediated by heavy Majorana neutrinos
2-nd Vienna Central European Seminar, Nov 25-27, 2005
15
Light neutrinos mmmN ,
tbbt
N
NNNN mUUm
b-amplitude is model dependent
t-amplitude is model independent
- the effective Majorana mass
2-nd Vienna Central European Seminar, Nov 25-27, 2005
16
tMM
MFt mVVff
mG 22
431222
3
34
16
bMMM
MMFb m
a
aVV
mmG
2
222
421322
8
434
12
tbMMMMM
MMFtb m
a
aVVVVff
mmG
22
421343126
324
14
t
b
tb
- 1-dimensional integral
- 5-dimensional integral
- 6-dimensional integral
The numerical calculations of multidimensional phase integ-rals were carried out using the program VEGAS based on Monte-Carlo mechanism.
2
M
Ma
2-nd Vienna Central European Seminar, Nov 25-27, 2005
17
The bound from :decay0 eVmee 51.0
From cosmological data and neutrino oscillation experiments, assuming the existence of 3 light massive neutrino 321 mmm
eVm 23.03 the constraint
Taking into account i
iU 1|| 2 eVmee 23.0
Using the best fit of mixing matrix from neutrino oscillation experiments
71.059.039.0
71.059.039.0
00.055.084.0
bfU
eVme 15.0
2-nd Vienna Central European Seminar, Nov 25-27, 2005
18
Rare decay
20103.5
tc bc tbc 2MeV 2MeV 2MeV
eeK K eK
eeKD KD eKD
c Ind. Boundson
B 2MeV
20104.1 22102.1 19101.1
22106.3
22102.7
21108.8 20104.4 33103.2 21101.2 20102.1 34102.6 20108.1 20108.8 33100.2
23101.4 23100.4 23102.8
21106.3 21103.3 21103.7
22105.8 22106.7 21107.1
21105.4 21101.4 21101.9
34104.2 34102.2 34100.2
Contribution of tree, box diagrams, their interference and indirect bounds on the branching ratios for the rare meson decays mediated by light Majorana neutrinos
2-nd Vienna Central European Seminar, Nov 25-27, 2005
19
2. Rare Meson Decays in R-parity-violating SUSY Theories
SLBpR 23)1( S, B, L – spin, barion and lepton quantum numbers
The SM fields, including additional Higgs boson fields appearing in the extended gauge models, have Rp=+1 while their superpartners have Rp=-1. This symmetry has been imposed on the minimal supersymmetric SM to ensure B and L number conservation. However, SUSY doesn’t require Rp conservation.
The most general gauge invariant form of the superpotential inminimal SUSY SM is
RPVR WWWp
R-parity is a discrete, multiplicative symmetry defined as
2-nd Vienna Central European Seminar, Nov 25-27, 2005
20
kjiijkkjiijkkjiijkRPV DQUDQLLLLW
- generation indices;kji ,, QL, are lepton and quark doublet superfields
are lepton and up, down quark singlet superfields.DUE ,,
All are the coupling constants. MSSM
proton stability
L B
0,0 0
0,0 0
and
RpAn alternative discrete symmetry with the same physical result, for example:
2,12,1 ,,,,,,,,, HELHELDUQDUQ
0
2-nd Vienna Central European Seminar, Nov 25-27, 2005
21
LLLL gRPV ~
KLagrangian, that yields contributions to the decay
The lepton number violating part of the Lagrangian has the form:
, LLLRPV
.,.)()()~
(~~
2
1 * chjiL jL
ciL
kR
iL
kR
jL
jL
kR
iLijk
..)()~
(~~)()
~(
~~ ** chudduudddddddL jL
ciL
kR
iL
kR
jL
jL
kR
iL
jL
ciL
kR
iL
kR
jL
jL
kR
iLijk
2-nd Vienna Central European Seminar, Nov 25-27, 2005
22
The Lagrangian terms corresponding to gluino and neutralino interactions with fermions and their superpartners are
iiii du ,, iii duq ,
iiii du ~,~
,~~ iii duq~
,~~ ,
.,.)~~~~(2
23
1,,
8
1
)()()(
3~ chqgqqgqgLiba
ibRi
aRi
bLi
aL
abg
..)~)(~)((24
1
3
12 chgL
i
iR
iRR
iL
iLL
Here are 3x3 Gell-Mann matrices.
)(
i - neutralino coupling constants.
,))()((tan)()( 1323 NQTNT WL
,tan)()( 1 NQ WR
Here and are the electric charge and weak isospin of the field .
)(Q )(3 T
,
2-nd Vienna Central European Seminar, Nov 25-27, 2005
23
Feynman diagrams for the rare meson decay
in SUSY model K
K
K
s
du
u
~ ~s
ud
u
K
s
u
u
d
~
~
,0
,0
g~,0
cu~
d~
tree
box
3
2-nd Vienna Central European Seminar, Nov 25-27, 2005
24
4
12~
2~
**
12112222312
223
)2
11(
)()(2(
2)
2
11(
C
LL
K
KK
Nmmmg
ffM
LL
4
12~
2~
*
1211
22
,,,,2~
2~
**1112
)()()
2
11(
mmm
ud
N
g
Nmmm
UU
LRkLLk
kjik
ud
LR
CNkkji CN
jNiNkk
.))(14 2
2~
2~
12112
23
gudC mmmN
g
LR
),1(4)(4)1(3
221 )(
211
2210
20
z
zzz
zzzzzzzzzzdz
,2
2
0
KM
Mz ,2
2
1
KM
mz ,2
2
2
KM
mz ,)1( 2
03 zz
.)(
2
2
KM
ppz
- the 4-momentum of the initial and final mesonspp ,
2-nd Vienna Central European Seminar, Nov 25-27, 2005
25
Using the input parameters for mesons:
Bounds on RPV couplings
and the masses of the various superpartners to be comparable the SUSY mass scale
We found that these R-parity violating contributions could be much larger then those from massive neutrinos.
3''' 10 kjiijk
GeVmSUSY 200
values of neutralino mixing matrix elements (hep-ph/0311123):
MeVf
MeVf
MeVf
D
K
228
,160
,131
.1015.3,1000.9,1043.9,1047.5
,1016.1,1006.6,1002.1,1086.91
422
321
222
12
141
231
121
111
NNNN
NNNN
2-nd Vienna Central European Seminar, Nov 25-27, 2005
26
Rare decay10SUSYmB
eeK K eK
eeKD KD eKD
2112111
7 )(1058.0 2
2122117 )(1020.0
2112211212111
7 )(1019.0 2
11212210 )(1026.0
2212222
10 )(1024.0 2
11222221212210 )(1012.0
B
Bounds on the branching ratios for the rare meson decays in SUSY theories with R-parity non-conservation.
221057.0 221020.0 221073.0 191025.0 191024.0 191047.0