Lepton flavor-changing processes in R-parity violating MSSM: Z->l_i l_j and γγ->l_i l_j under new bounds from l_i->l_j γ

0.18 MB·English

by  Junjie Cao

arXive:0908.4556
Lepton flavor-changing processes in R-parity violating MSSM:
¯ ¯
Z ℓ ℓ and γγ ℓ ℓ under new bounds from ℓ ℓ γ
i j i j i j
→ → →
Junjie Cao1, Lei Wu1, Jin Min Yang2,3
1 College of Physics and Information Engineering,
Henan Normal University, Xinxiang 453007, China
0 2 Key Laboratory of Frontiers in Theoretical Physics,
1
0
Institute of Theoretical Physics, Academia Sinica, Beijing 100190, China
2

n 3 Kavli Institute for Theoretical Physics China,
a
J
Academia Sinica, Beijing 100190, China

7

]
h
p
-
p
Abstract
e
h
[ We examine the lepton flavor-changing processes in R-parity violating MSSM. First, we update

2
the constraints on the relevant R-violating couplings by using the latest data on the rare decays
v
6
5 ℓi ℓjγ. We find that the updated constraints are much stronger than the old ones from rare
→
5
4 Z-decays at LEP. Then we calculate the processes Z ℓiℓj and γγ ℓiℓ¯j. We find that with the
→ →
.
8
updated constraints the R-violating couplings can still enhance the rates of these processes to the
0
9
0 sensitivity of GigaZ and photon-photon collision options of the ILC.
:
v
i
X PACS numbers: 14.80.Ly,11.30.Fs, 13.66.De
r
a
1
I. INTRODUCTION
The minimal supersymmetric model (MSSM) is a popular extension of the Standard
Model (SM). Inthis model the invariance ofR-parity, defined byR = ( 1)2S+3B+L for a field
−
with spin S, baryon-number B and lepton-number L, is often imposed on the Lagrangian in
ordertomaintaintheseparateconservationofbaryon-number andlepton-number. Although
R-parity plays a crucial role in the phenomenology of the MSSM (e.g., forbid proton decay
and ensure a perfect candidate for cosmic dark matter), it is, however, not dictated by
any fundamental principle such as gauge invariance and there is no compelling theoretical
motivation for it. The most general superpotential of the MSSM consistent with the SM
gaugesymmetry and supersymmetry contains R-violating interactions which aregiven by [1]
1 1
= λ L L Ec +λ L Q Dc + λ ǫabdUc Dc Dc +µ L H , (1)
WR6 2 ijk i j k ′ijk i j k 2 ′i′jk ia jb kd i i 2
where i,j,k are generation indices, c denotes charge conjugation, a, b and d are the color
indices with ǫabd being the total antisymmetric tensor, H is the Higgs-doublet chiral super-
2
field, and L (Q ) and E (U ,D ) are the left-handed lepton (quark) doublet and right-handed
i i i i i
lepton (quark) singlet chiral superfields. The dimensionless coefficients λ (antisymmetric
ijk
in i and j) and λ in the superpotential are L-violating couplings, while λ (antisym-
′ijk ′i′jk
metric in j and k) are B-violating couplings. So far both theorists and experimentalists
have intensively studied the phenomenology of R-parity breaking supersymmetry in various
processes [2, 3] and obtained some bounds [4].
The lepton flavor-changing (LFC) processes, which have been searched in various exper-
iments [5–7], are a sensitive probe for new physics because they are extremely suppressed
in the SM but can be greatly enhanced in new physics models like supersymmetry [8]. In
R-parity breaking supersymmetry, these rare processes may receive exceedingly large en-
hancement since both λ and λ couplings can make contributions. Such enhancement was
′
¯
considered in the decays l l γ [9] and Z ℓ ℓ [10], the µ e conversion in nuclei [11],
i j i j
→ → −
and the di-lepton productions pp¯/pp e µ +X [12] and e+e e µ [13].
± ∓ − ± ∓
→ →
Since the GigaZ and photon-photon collision options of the ILC can precisely measure
¯ ¯
the LFC processes Z ℓ ℓ and γγ ℓ ℓ (i = j and ℓ = e,µ,τ), we in this work study
i j i j i
→ → 6
these processes in R-violating MSSM. Noting that the experimental upper bounds on the
LFC τ-decays became more stringent recently [6], we will first check the constraints on the
relevant R-violating couplings from the latest measurement of ℓ ℓ γ. Then, with the
i j
→
2
¯ ¯
updated bounds on the relevant R-violating couplings, we calculate Z ℓ ℓ and γγ ℓ ℓ
i j i j
→ →
to figure out if they can reach the sensitivity of the GigaZ and photon-photon collision
options of the ILC.
The paper is organized as follows. In Sec. II we describe the calculations for ℓ ℓ γ,
i j
→
¯ ¯
Z ℓ ℓ and γγ ℓ ℓ . In Sec. III we present some numerical results and discussions.
i j i j
→ →
Finally, a conclusion is drawn in Sec. IV.
II. CALCULATIONS
In terms of the four-component Dirac notation, the Lagrangian of the L-violating inter-
action is given by (in our calculations we take the presence of λ as an example)
′ijk
Lλ′ = −λ′ijk ν˜LidkRdjL +d˜jLdkRνLi +(d˜kR)∗(νLi)cdjL
h
˜li dkuj u˜jdkli (d˜k) (li )cuj +h.c. (2)
− L R L − L R L − R ∗ L L
i
The LFC interactions ℓ ℓ¯V (V = γ,Z) are induced at loop level by exchanging a squark u˜j
i j L
or d˜k, which is shown in Fig.1.
R
γ,Z
u˜L γ,Z u˜L u˜L γ,Z
u˜L u˜L j j j
j j
d
k
ℓi dk ℓj ℓi ℓj ℓi ℓi dk ℓj ℓi dk ℓj ℓj
γ,Z
(a) (b) (c) (d)
γ,Z
d˜R γ,Z d˜R d˜R γ,Z
d˜R d˜R k k k
k k
u
j
ℓi uj ℓj ℓi ℓj ℓi ℓi uj ℓj ℓi uj ℓj ℓj
γ,Z
(e) (f) (g) (h)
FIG. 1: Feynman diagrams for ℓ ℓ transition induced by the L-violating couplings at one-loop
i j
−
level.
For the decays ℓ ℓ γ we take µ eγ as an example to show the analytic results. The
i j
→ →
gauge invariant amplitude of µ eγ is given by
→
M(µ eγ) = 2Au¯(p )P (2ǫ p m ǫ/)u(p ), (3)
e R µ µ µ
→ · − ·
3
d (uc)
ℓj ℓi ℓj ℓi ℓj k j ℓi
ℓi ℓj u˜L(d˜R) u˜L(d˜R)
j k j k
γ γ γ γ γ γ
(a) (b) (c)
ℓj ℓi ℓj ℓi ℓj γ
u˜L d u˜L
j k j
d d u˜L u˜L d u˜L
k k j j k j
d u˜L d
k j k
γ γ γ γ γ ℓ
i
(d) (e) (f)
ℓj ℓi ℓj ℓi ℓj γ
d˜R uj d˜R
k k
uj uj d˜R d˜R uj d˜R
k k k
uj d˜R uj
k
γ γ γ γ γ ℓ
i
(g) (h) (i)
FIG. 2: Feynman diagrams for γγ ℓ ℓ¯ induced by the L-violating couplings at one-loop level.
i j
→
The effective γ ℓ ℓ vertex in (a,b) is defined in Fig. 1.
i j
− −
where A is given by (assuming the degeneracy for squark masses)
A = ieλ′1jkλ′2jk mµ f (m2dk)+f (m2uj) (4)
16π2 m2 1 m2 2 m2
q˜ " q˜ q˜ #
with
1 2 6x2lnx 1 6xlnx
f (x) = (2x2 +5x 1 ) (x2 5x 2+ ) , (5)
1 8(x 1)3 3 − − x 1 − 3 − − x 1
− (cid:20) − − (cid:21)
1 1 6x2lnx 2 6xlnx
f (x) = (2x2 +5x 1 ) (x2 5x 2+ ) . (6)
2 8(x 1)3 3 − − x 1 − 3 − − x 1
− (cid:20) − − (cid:21)
The decay branching ratio reads
48π
BR(µ eγ) = A 2. (7)
→ G2m2| |
F µ
¯
For the decays Z ℓ ℓ we calculate the decay rates numerically by using the effective
i j
→
vertex presented in Appendix A. Note that according to the effective vertex method [14],
the external legs of the effective vertex can be on-shell or off-shell and thus the vertex can be
used in any relevant process. The expression in Eqs.(3-6) can be obtained from the effective
vertex in Appendix A by putting both leptons on shell.
¯
For the process γγ ℓ ℓ , besides Fig.2 (a,b) induced by the effective vertex given
i j
→
in Appendix A, more diagrams shown in Fig.2 (c-i) also come into play. The analytic
4
expressions of the amplitudes of these diagrams are given in Appendix B. These amplitudes
contain the Passarino-Veltman one-loop functions, which are calculated by using LoopTools
[26]. We checked that the amplitudes have gauge invariance and the ultraviolet divergence
cancelled.
Sincethephotonbeamsinγγ collisionaregeneratedbythebackwardComptonscattering
of the incident electron- and the laser-beam, the events number is obtained by convoluting
the cross section of γγ collision with the photon beam luminosity distribution:
d
γγ
Nγγ→ℓiℓ¯j = d√sγγd√Lsγγσˆγγ→ℓiℓ¯j(sγγ) ≡ Le+e−σγγ→ℓiℓ¯j(s) (8)
Z
where d /d√s is the photon-beam luminosity distribution and σ (s) ( s is the
Lγγ γγ γγ→ℓiℓ¯j
squared center-of-mass energy of e+e collision) is defined as the effective cross section of
−
¯
γγ ℓ ℓ . In optimum case, it can be written as [15]
i j
→
xmax xmax dx z2
σ (s) = 2zdzσˆ (s = z2s) F (x)F ( ) (9)
γγ→ℓiℓ¯j Z√a γγ→ℓiℓ¯j γγ Zz2/xmax x γ/e γ/e x
where F denotes the energy spectrum of the back-scattered photon for the unpolarized
γ/e
initial electron and laser photon beams given by
1 1 4x 4x2
F (x) = 1 x+ + (10)
γ/e D(ξ) − 1 x − ξ(1 x) ξ2(1 x)2
(cid:20) − − − (cid:21)
with
4 8 1 8 1
D(ξ) = (1 )ln(1+ξ)+ + . (11)
− ξ − ξ2 2 ξ − 2(1+ξ)2
Here ξ = 4E E /m2 (E is the incident electron energy and E is the initial laser photon
e 0 e e 0
energy) and x = E/E with E being the energy of the scattered photon moving along the
0
initial electron direction.
III. NUMERICAL RESULTS AND DISCUSSIONS
In our calculations we take the SM parameters as [16]
m = 0.106 GeV,m = 1.777 GeV,m = 4.2 GeV,α = 1/137,sin2θ = 0.223 (12)
µ τ b W
The top quark mass is taken as the new CDF value m = 172.3 GeV [17]. The relevant
t
SUSY parameters in our calculations are the masses of squarks as well as the R-parity vio-
lating couplings listed in Table I. The strongest bound on squark mass is from the Tevatron
5
experiment. For example, from the search of the inclusive production of squark and gluino
in R-conserving minimal supergravity model with A = 0, µ < 0 and tanβ = 5, the CDF
0
gives abound of 392GeVat the 95% C.L. for degenerate gluinos andsquarks [18]. However,
this bound may be not applicable to the R-violating scenario because the SUSY signal in
case of R-violation is very different from the R-conserving case. The most robust bounds on
sparticle masses come fromthe LEP results, which give a boundof about 100 GeV onsquark
or slepton mass [19]. In our numerical calculations, we assume the presence of the minimal
number of R-violating couplings, i.e., for each process only the two relevant couplings (not
summed over the family indices) are assumed to be present.
For ℓ ℓ γ, the latest experimental data is [7]
i j
→
BR(µ eγ) < 1.2 10 11, (13)
−
→ ×
BR(τ eγ) < 1.1 10 7, (14)
−
→ ×
BR(τ µγ) < 4.5 10 8. (15)
−
→ ×
We use these data to update the bounds on the relevant L-violating couplings. The new
bounds are compared with the old ones in Table I for m = 100 GeV (here we take squark
q˜
mass of 100 GeV for illustration and for heavier squarks the bounds on the L-violating
couplings will become weak, as will be shown later). We can see that the new bounds are
′ ′
much stronger than the old ones. Since the bounds on λ λ (i = j) are weakest, we only
i33 j33 6
′ ′
consider the contribution of λ λ in our following numerical calculations.
i33 j33
Note that the neutrino masses could also constrain the λ couplings, especially λ [20].
′ ′i33
But these constraints depend on more parameters in addition to the squark mass. For
example, the one-loop λ contributions to the neutrino masses are sensitive to the left-right
′
squark mixings and the two-loop contributions further involve the slepton mass. For small
squark mixings with appropriate sign, there may exist a strong cancellation between one-
loop and two-loop effects, and in this case, the constraints from the neutrino masses can be
avoided. Since the aim of our study is the sensitivity of the LFC processes to λ couplings
′
and the λ contributions to these LFC processes are irrelevant to the additional parameters
′
involved in the contributions to the neutrino masses, in our analysis we did not consider
such constraints from the neutrino masses.
6
TABLE I: Our new upper bounds on the L-violating couplings for m = 100 GeV from ℓ ℓ γ
q˜ i j
→
data [7], in comparison with the old ones [4].
couplings New bounds Old bounds [4]
λ λ , λ λ 7.74 10 5 5.7 10 4
′111 ′211 ′112 ′212 × − × −
λ λ 7.85 10 5 5.7 10 4
′113 ′213 × − × −
λ λ , λ λ 7.78 10 5 5.7 10 4
′121 ′221 ′122 ′222 × − × −
λ λ 7.89 10 5 5.7 10 4
′123 ′223 × − × −
λ λ , λ λ 1.27 10 3 7.7 10 3
′131 ′231 ′132 ′232 × − × −
λ λ 1.63 10 3 1.0 10 2
′133 ′233 × − × −
λ λ , λ λ 5.54 10 4 1.2 10 2
′111 ′311 ′112 ′312 × − × −
λ λ 5.56 10 4 1.2 10 2
′113 ′313 × − × −
λ λ , λ λ 5.57 10 4 1.2 10 2
′121 ′321 ′122 ′322 × − × −
λ λ 5.65 10 4 1.2 10 2
′123 ′323 × − × −
λ λ , λ λ 9.06 10 3 1.2 10 2
′131 ′331 ′132 ′332 × − × −
λ λ 1.17 10 2 1.2 10 2
′133 ′333 × − × −
λ λ , λ λ 3.55 10 4
′211 ′311 ′212 ′312 × −
λ λ 3.60 10 4
′213 ′313 × −
λ λ , λ λ 3.56 10 4
′221 ′321 ′222 ′322 × −
λ λ 3.61 10 4
′223 ′323 × −
λ λ , λ λ 5.80 10 3
′231 ′331 ′232 ′332 × −
λ λ 7.48 10 3
′233 ′333 × −
¯
For Z ℓ ℓ , the upper limits from LEP are [21, 22]
i j
→
BR(Z µe) < 1.7 10 6, (16)
−
→ ×
BR(Z τe) < 9.8 10 6, (17)
−
→ ×
BR(Z τµ) < 1.2 10 5. (18)
−
→ ×
The bounds from these LEP data are compared with the bounds from ℓ ℓ γ in Fig.3.
i j
→
One can see that the upper bounds on the couplings from the LEP Z-decay data [21, 22] are
weaker than the ones from ℓ ℓ γ data [7]. Note that the bounds from the LEP Z-decay
i j
→
data were also studied in [10] and our results are consistent with theirs except that in [10]
7
P )
E
→ m t ( L
' ' ll i33 j33 1100 --21 l '1 3 3l '3 33 olr l' 1 '32 3l3 3 l ' 3 '3 3 3 3 3 : ( IZL C→l ) ' 2 e3 lt 3 l' 3 ' 3(l l2 L 3 3 ' 'l E: 3 1 1 l 3 'PZ3 l 3 l3' 3 ) 3 1→ ' ' 33 23l : 3 3m 3 3'Z 3 3t: : 3 3Z Z :l ( t → → G' i1 → 3gel e 3m l t a ' ' eZ 2 g 2 3) 3l(( 3 3 LG:' m 3 Ei 3 g 3:P t →a) Z →)egm g
10 -3 → em ( G i g a Z )
l ' 1 3 3l ' 2 3 3 : Z
200 300 400 500 600 700800900
M~ (GeV)
q
FIG. 3: Various bounds on the L-violating couplings versus the squark mass. The solid, dashed
and dotted curves are the bounds on λ λ , λ λ and λ λ , respectively. Also shown are
′133 ′233 ′133 ′333 ′233 ′333
the 2σ sensitivity from Z-decays at GigaZ and the 3σ sensitivity from γγ e(or µ) τ at the ILC
→
with center-of-mass energy of 500 GeV and a luminosity of 3.45 102fb 1.
−
×
the sum over index k is implied for λ λ with m = 200 GeV.
′i3k ′j3k q˜
The possible sensitivity of GigaZ to the LFC decays of Z-boson could reach [23]
BR(Z µe) 2.0 10 9, (19)
−
→ ∼ ×
BR(Z τe) κ 6.5 10 8, (20)
−
→ ∼ × ×
BR(Z τµ) κ 2.2 10 8 (21)
−
→ ∼ × ×
8
with the factor κ ranging from 0.2 to 1.0. In Fig. 3 we take κ = 1.0 to show the sensitivity.
In contrast to the R-conserving case in which only Z µτ is accessible at the GigaZ [8],
→
the R-violating couplings under the bound from l l γ can still enhance all the channels
i j
→
Z ℓ ℓ to the sensitivity of the GigaZ. This implies that the GigaZ can further strengthen
i j
→
the bounds on λ λ in case of un-observation. These bounds, unlike the constraints from
′i33 ′j33
neutrino masses which involve more parameters, are only dependent on the squark mass.
For the γγ collision results shown in Fig. 3, we fixed the parameters as ξ = 4.8, D(ξ) =
1.83 and x = 0.83 [15]. Since the L-violating couplings relevant to the process γγ eµ¯
max
→
is stringently constrained by µ eγ, we in Fig. 3 only show the results for the channels
→
with a tau lepton in the final states, i.e., γγ eτ¯, µτ¯. The background for γγ eτ¯
→ →
comes from γγ τ+τ τ ν ν¯ e+, γγ W+W τ ν ν¯ e+ and γγ e+e τ+τ ,
− − e τ − − e τ − −
→ → → → →
and we make kinematical cuts [13]: cosθ < 0.9 and pℓ > 20 GeV (ℓ = e,µ), to enhance
| ℓ| T
the ratio of signal to background. With these cuts, the background cross sections from
γγ τ+τ τ ν ν¯ e+, γγ W+W τ ν ν¯ e+ and γγ e+e τ+τ at √s = 500
− − e τ − − e τ − −
→ → → → →
GeV are suppressed respectively to 9.7 10 4 fb, 1.0 10 1 fb and 2.4 10 2 fb (see Table I
− − −
× × ×
of [13]). To get the 3σ observing sensitivity with 3.45 102 fb 1 integrated luminosity [24],
−
×
the production rates of γγ eτ¯,µτ¯ after the cuts must be larger than 2.5 10 2 fb [13].
−
→ ×
We see from Fig. 3 that under the current bounds from l l γ, the L-violating couplings
i j
→
can still be large enough to enhance the productions γγ eτ¯,µτ¯ to the 3σ sensitivity.
→
¯
We also show the cross sections of γγ ℓ ℓ as a function of center-of-mass energy √s
i j
→
of the ILC in Fig.4. We see that with the increasing of the center-of-mass energy, the cross
sections of these processes become smaller. Such a behavior is similar to the results in the
R-conserving MSSM shown in [13].
Finally, we point out that theLFC processes can also put bounds onthe products λ λ
′i31 ′j31
and λ λ , and our numerical results indicate that such bounds are quite similar to those
′i32 ′j32
¯
in Fig.3. We note that these bounds on λ λ and λ λ from Z l l at GigaZ are
′i31 ′j31 ′i32 ′j32 → i j
generally stronger than those from the neutrino masses [20].
IV. CONCLUSION
We evaluated the lepton flavor-changing processes in R-parity violating MSSM. First, we
used the latest data on the rare decays ℓ ℓ γ to update the constraints on the relevant
i j
→
9
-1 M~ = 300GeV
10 q
P > 20GeV
T
(cid:239) cosq(cid:239)< 0.9
gg
-2 → e t
10
gg
) → m t
b
f
(
s
-3
10
gg
→ e m
-4
10
200 400 600 800 1000 1200 1400 1600 1800 2000
Center-of-Mass Energy of ILC (GeV)
FIG. 4: The cross sections of γγ ℓ ℓ¯ as a function of center-of-mass energy √s. The couplings
i j
→
λ λ , λ λ and λ λ are fixed at their upper bounds at M = 300 GeV.
′133 ′233 ′133 ′333 ′233 ′333 q˜
¯ ¯
R-violating couplings. Then we calculated the processes Z ℓ ℓ and γγ ℓ ℓ . We found
i j i j
→ →
that with the updated constraints the R-violating couplings can still enhance the rates of
these processes to the sensitivity of GigaZ and photon-photon collision options of the ILC.
So, the GigaZ and photon-photon collision of the ILC can either observe these λ-induced
′
LFCprocessesorfurtherstrengthentheboundsontheλ couplingsincaseofun-observation.
′
Acknowledgement
This work was supported in part by the National Natural Science Foundation of China
(NNSFC) under grant Nos. 10505007, 10821504, 10725526 and 10635030, and by HASTIT
under grant No. 2009HASTIT004.
10