Table Of ContentIPM/P-2008/053
A Window on the CP-violating Phases of MSSM from Lepton Flavor Violating
Processes
S. Yaser Ayazi∗ and Yasaman Farzan†
Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran
(Dated: January 18, 2009)
It has recently been shown that by measuring the transverse polarization of the final particles
in the LFV processes µ → eγ, µ → eee and µN → eN, one can derive information on the CP-
9 violating phasesof theunderlyingtheory. Wederiveformulas for thetransversepolarization of the
0 finalparticlesintermsofthecouplingsoftheeffectivepotentialleadingtotheseprocesses. Wethen
0 studythedependenceofthepolarizationsofeandγintheµ→eγandµN →eN ontheparameters
2 oftheMinimalSupersymmetricStandardModel(MSSM).Weshowthatcombiningtheinformation
on various observables in the µ → eγ and µN → eN search experiments with the information on
n
a theelectric dipolemoment oftheelectron can help usto solvethedegeneracies in parameter space
J and to determinethe valuesof certain phases.
8
PACSnumbers: 11.30.Hv,13.35.Bv
1
]
h
p
- I. INTRODUCTION
p
e
h In the framework of the Standard Model (SM), Lepton Flavor Violating (LFV) processes such as µ+ e+γ,
[ µ+ e+e−e+ andµ econversiononnuclei(i.e.,µN eN)areforbidden. WithintheSMaugmentedwithn→eutrino
→ − →
mass and mixing, such processes are in principle allowed but the rates are suppressed by factors of (∆m2/E2 )2 [1]
2 ν W
v and are too small to be probed in the foreseeable future.
3 Variousmodelsbeyondthe SMcangiveriseto LFVraredecaywithbranchingratiosexceedingthepresentbounds
3 [2]:
2
4 Br(µ+ e+γ)<1.2 10−11 Br(µ+ e+e+e−)<1.0 10−12 at 90% C.L.
→ × → ×
.
0 For low scale MSSM (m 100 GeV), these experimentalbounds imply stringent bounds onthe LFV sourcesin
SUSY
1 ∼
the Lagrangian. The MEG experiment at PSI [3], which is expected to release data in summer 2009, will eventually
8
be able to probe Br(µ+ e+γ) down to 10−13. In our opinion, it is likely that the first evidence for physics beyond
0 →
the SM comes from the MEG experiment. If the branching ratio is close to its present bound, the MEG experiment
:
v will detect statistically significant number of such events. As a result, making precision measurement will become a
i
X possibilitywithinafew years. Muonsinthe MEGexperimentareproducedbydecayofthestoppedpions(atrest)so
they are almost 100% polarized. This opens up the possibility of learning about the chiral nature of the underlying
r
a theory by studying the angular distribution of the final particles relative to the spin of the parent particle [4]. In
Ref. [5], it has been shown that by measuring the polarization of the final states in the decay modes µ+ e+γ
and µ+ e+e−e+, one can derive information on the CP-violating sources of the underlying theory. Notic→e that
→
even for the state-of-the-art LHC experiment, it will be quite challenging (if possible at all) to determine the CP-
violating phases in the lepton sector [6]. Suppose the LHC establishes a particular theory beyond the SM such as
supersymmetry. InordertolearnmoreabouttheCP-violatingphases,thewell-acceptedstrategyistobuildyetamore
advancedacceleratorsuchasILC.Consideringthe expensesandchallengesbeforeconstructingsuchanaccelerator,it
is worth to give any alternative method such as the one suggestedin Ref. [5] a thoroughconsideration. In this paper
we elaborate more on this method within the framework of R-parity conserving MSSM.
LFV sourcesin the Lagrangiancanalso give rise to sizeable µ e conversionrate. There are strong bounds on the
−
rates of such processes [4, 7, 8]:
Γ(µTi eTi)
R(µTi eTi) → <6.1 10−13 . (1)
→ ≡ Γ(µTi capture) ×
→
The upper bound on R restricts the LFV sources however, for the time being, the bound from µ eγ is more
→
stringent. The PRISM/PRIMEexperimentis goingto performa new searchforthe µ e conversion[9]. In casethat
−
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
2
thevaluesofLFVparametersareclosetothepresentupperbound,asignificantlylargenumberoftheµ econversion
−
events can be recorded by PRISM/PRIME. Recently it is shown in [10] that if the initial muon is polarized (at least
partially), studying the transverse polarization of the electron yields information on the CP-violating phase. In this
paper, we elaborate more on this possibility taking into account all the relevant effects in the context of R-parity
conserving MSSM.
Intheendofthepaper,westudythepossibilityofeliminatingthedegeneraciesoftheparameterspacebycombining
information from µ eγ and µ e conversionexperiments. We then demonstrate that the forthcoming results from
→ −
d search can help us to eliminate the degeneracies further (cf. Figs. (8-a,8-b)).
e
The paper is organized as follows: In sec. II, using the results of Ref. [5], we calculate the polarization of the final
particles in decay µ eγ in terms of the couplings of the low energy effective Lagrangian (after integrating out the
→
supersymmetricstates). We alsobriefly discussµ eeeandthe challengesofderivingthe CP-violatingphasesby its
→
study. In sec. III, we calculate the transverse polarization of the muon in the µ e conversion experiment in terms
−
of the couplings in the effective Lagrangianwhich give the dominant contributionto µN eN within the MSSM. In
→
sec. IV, we study the overallpatternof the variationof s and P s with phasesand discuss the regionsof the
h T2i h T1 T2i
parameterspacewhere the sensitivity to the phasesaresizeable. In sec. V, wediscuss howby combining information
from the µ eγ and µN eN experiments, we can solve the degeneracies in the parameter space. The conclusions
→ →
are summarized in sec. VI.
II. POLARIZATION OF THE FINAL PARTICLES
The low energy effective Lagrangianthat gives rise to µ eγ can be written as
→
A A A∗ A∗
= Rµ¯ σµνe F + Lµ¯ σµνe F + Re¯ σµνµ F + Le¯ σµνµ F , (2)
R L µν L R µν L R µν R L µν
L m m m m
µ µ µ µ
where σµν = i[γµ,γν] and F is the photon field strength: F = ∂ ε ∂ ε . A and A receive contributions
2 µν µν µ ν − ν µ L R
from the LFV parameters of MSSM at one loop level [4, 11, 12]. In this section we derive the polarizations of the
finalparticlesinthe LFVraredecaysintermsofA andA . Letusdefine the longitudinalandtransversedirections
L R
as follows: ˆl ~p /p~ , Tˆ p~ ~s /p~ ~s and Tˆ Tˆ ˆl. As shown in [5], the partial decay rate of an
e+ e+ 2 e+ µ e+ µ 1 2
≡ | | ≡ × | × | ≡ ×
anti-muon at rest into a positron and a photon with definite spins of~s and~s is
e γ
dΓ[µ+(Pµ+)→e+(Pe+,~se+)γ(Pγ,~sγ)] = mµ α 2 A 2(1+P cosθ)sin2 θs+ (3)
+ L µ
dcosθ 8π (cid:20)| | | | 2
θ
α 2 A 2(1 P cosθ)cos2 s P Re[α α∗A∗A eiφs]sinθsinθ ,
| −| | R| − µ 2 − µ + − L R s(cid:21)
where P is the polarization of the anti-muon, θ is the angle between the directions of the spin of the anti-muon and
µ
the momentum ofthe positron, andθ is the anglebetween the spin ofthe positronand its momentum. In the above
s
formula, φ is the azimuthal angle that the spin of the final positron makes with the plane of spin of the muon and
s
the momentum of the positron. Finally, α and α give the polarization of the final photon:
+ −
α +α α α
~ε Tˆ (Tˆ) ε = + − and ~ε Tˆ (Tˆ) ε = +− −i
1 1 j j 2 2 j j
· ≡ √2 · ≡ √2
X X
j∈{1,2,3} j∈{1,2,3}
where α 2+ α 2 =1. Noticethatforagivenpolarizationofthepositron,thephotonhasadefinitepolarization:
+ −
| | | |
i.e., septting Pµ = 100% and α+ = α−e−iφs(A∗R/A∗L)tanθ/2cotθs/2, we find dΓ/dcosθ = 0. Consider the case that
P =100%andthe positronis emittedinthe directionofthe spinofthe muon; i.e., θ =0. From(3), wefindthatfor
µ
θ =π and α =1, dΓ/dcosθ is maximal. In other words, in this case, the spins of the positron and the photon are
s +
respectivelyalignedinthe directionanti-parallelandparallelto the spinofthe muon. This is expectedbecause when
θ =0 there is a cylindricalsymmetry aroundthe axisparallelto the spinof the muonandtherefore the totalangular
momentum in the direction of the spin does not receive any contribution from the relative angular momentum. This
means the sum of spins in the ˆl direction has to be conserved which in turn implies that the decay rate is maximal
at θ = π and α = 1. Similar consideration also applies to the case that the positron is emitted antiparallel to the
s +
spin of the muon: For θ =π, the emission is maximal at θ =0 and α =1.
s −
3
Summing over the polarization of the final particles in Eq. (3), we obtain
dΓ[µ+(Pµ+)→e+(Pe+,~se+)γ(Pγ,~sγ)] = mµ A 2(1+P cosθ)+ A 2(1 P cosθ) .
L µ R µ
dcosθ 8π | | | | −
~sXγ~se+ (cid:2) (cid:3)
Thus, Γ(µ eγ) is given by (A 2+ A 2). It is convenient to define
L R
→ | | | |
A 2 A 2
L R
R | | −| | . (4)
1 ≡ A 2+ A 2
L R
| | | |
By measuring the total decay rate and the angular distribution of the final particles, one can derive absolute values
A and A . To measure the relative phase of these couplings, the polarization of the final particles also have to be
L R
measured.
Let us define the polarizations of the electron and photon in an arbitrary direction Tˆ respectively as
dΓ µ+ e+(~s = 1Tˆ)γ(~s ) dΓ µ+ e+(~s = 1Tˆ)γ(~s )
s ~sγ h h → e+ 2 γ i− h → e+ −2 γ ii (5)
h Ti≡ P ~sγ~se+ dΓ[µ+ →e+(~se+)γ(~sγ)]
P
and
dΓ µ+ e+(~s )γ(~εqTˆ)
P ~se+ h → e+ i (6)
h Ti≡ P ~sγ~se+ dΓ[µ+ →e+(~se+)γ(~sγ)]
P
where ~ε is the polarization vector of the photon.
From Eq. (3), we find that the polarization of positron (once we average over the polarizations of the photon) is
A 2(1 P cosθ) A 2(1+P cosθ)
R µ L µ
s = s =0 , s = | | − −| | .
h T1i h T2i h li A 2(1 P cosθ)+ A 2(1+P cosθ)
R µ L µ
| | − | |
That is while the linear polarization of the photon (once we sum over the polarization of the positron) is
1
P = P = .
h T1i h T2i 2
Unfortunately, neither the polarization of the positron nor the polarization of the photon carries any information
on the relative phase of A and A . However, the double correlation of the polarization carries such information.
L R
Let us define double correlationas follows
dΓ µ+ e+(~s = 1Tˆ)γ(~εqTˆ′) dΓ µ+ e+(~s = 1Tˆ)γ(~εqTˆ′)
hPT′sTi≡ h → e+ 2 ~sγ~se+ dΓ[µi+−→e+h(~se+→)γ(~sγ)]e+ −2 i (7)
P
where Tˆ and Tˆ′ are arbitrary directions. From Eq. (3), we find
P Re[A∗A ]sinθ
P s = P s = − µ L R (8)
h T1 T1i −h T2 T1i A 2(1 P cosθ)+ A 2(1+P cosθ)
R µ L µ
| | − | |
and
P Im[A∗A ]sinθ
P s = P s = µ L R . (9)
h T1 T2i −h T2 T2i A 2(1 P cosθ)+ A 2(1+P cosθ)
R µ L µ
| | − | |
Thus, as pointed out in [5], to extract the CP-violating phases both polarization and their correlation have to be
measured. Eq. (8) gives the correlation of the polarizations for particles emitted along the direction described by θ.
Averaging over θ, we find
P s = P s = −11PµRe[A∗LAR]sinθdcosθ = −πPµRe[A∗LAR] (10)
h T1 T1i −h T2 T1i −11[|AR|2(1R−Pµcosθ)+|AL|2(1+Pµcosθ)]dcosθ 4(|AL|2+|AR|2)
R
4
and
P s = P s = −11PµIm[A∗LAR]sinθdcosθ = πPµIm[A∗LAR] . (11)
h T1 T2i −h T2 T2i −11[|AR|2(1R−Pµcosθ)+|AL|2(1+Pµcosθ)]dcosθ 4(|AL|2+|AR|2)
R
Notice that to take averageoverangles,one should weighthe polarizationof positronemitted within a giveninterval
(θ,θ+dθ)withthenumberofemissioninthisintervalandthenintegrateoverangles. Thatiswhywehaveintegrated
over dcosθ in both the numerator and denominator of the right-hand side of the ratios in Eqs. (8,9) instead of
calculating P s dcosθ/ dcosθ.
h Ti Tji
From EqsR. (8,9), we find thRat if the polarimeter is located at θ =π/2, the polarization and therefore sensitivity is
maximal. Notice that
4
hPTisTji|θ=π2 = πhPTisTji .
Measurement of P s requires setting polarimeters all around the regionwhere the decay takes place. In sec. IV,
h Ti Tji
we perform an analysis of P s . Up to a factor of 4/π, our results applies to the case that measurement of the
h Ti Tji
polarization is performed only at θ =π/2.
The ratios of the polarizations yield the relative phase of the effective couplings
P s P s P s P s Im[A∗A ]
h T1 T2i = h T1 T2i = h T2 T2i = h T2 T2i = L R .
hPT1sT1i hPT1sT1i hPT2sT1i hPT2sT1i −Re[A∗LAR]
Techniques for the measurement of the transverse polarization of the positron have already been developed and
employedforderivingtheMichelparameters[13]. Measuringthelinearpolarizationofthephotonisgoingtobemore
challenging but is in principle possible [14].
Inthe following,we discuss the LFV processµ+ e+e−e+. The effective LagrangianshowninEq. 2canalsogive
→
risetoLFVraredecayµ+ e+e−e+ throughpenguindiagrams. Moreover,theprocesscanalsoreceivecontributions
→
from the LFV four-fermion terms of the form
C µ¯Γµ(a P +b P )ee¯Γ (c P +d P )e
i i i L i R i,µ i L i R
where a , b , c and d are numbers of order one and Γ = γ or 1. In the framework of R-parity conserving
i i i i i,µ µ
MSSM which is the focus of the present study, the couplings of the four-fermion interaction are suppressed; i.e.,
m2C A . Moreover, the contributions of the A and A terms for the case that the momentum of one of the
µ i ≪ L,R L R
positrons is close to m /2 is dramatically enhanced because in this limit the virtual photon in the corresponding
µ
diagram goes on-shell. In [5], it is shown that by studying the transverse polarization of the positron whose energy
is close to m /2, one can extract information on the phases of the underlying theory. The maximum energy of the
µ
positrons emitted in the decay µ+ e+e−e+ is E m /2 3m2/(2m ). Consider the case that one of the
positrons, e+, has an energy close to→E ; i.e., E max ≃∆Eµ< E− < Ee wµhere ∆E m . Following [5], let us
1 max max− 1 max ≪ µ
define
dΓMax Emax dΓ(µ+ e+e−e+)
= → 1 2 dE dE , (12)
2 1
dcosθdφ Z Z dE dE dcosθdφ
se+2X,se− Emax−∆E 2 1
whereθistheanglebetweenthespinofthemuonandthemomentumofe+(thepositronwhoseenergyisclosetoE )
1 max
and φ is the azimuthal angle that the momentum of e+ makes with the plane made by the momentum of e+ and the
2 1
spinofthemuon. LetussupposethatacutisemployedthatpicksuponlyeventswithE within(E ,E ∆E)
1 max max
−
where 2m <∆E m . Because ofthe enhancement of the amplitude atE E , the number ofevents passing
e µ 1 max
the cut is still sig≪nificant: i.e., ΓMax/Γ (µ+ e+e−e+) = log(m ∆E/4m→2)/ log(m2/4m2) 7/12 > 50%. As
tot → µ e µ e −
shown in [5], (cid:0) (cid:1)
dΓMax αm
= µ A 2 c 2(1+P cosθ)+ A 2 d 2(1 P cosθ) (13)
dcosθdφ 192π3 | L| | e| µ | R| | e| − µ
(cid:2)
m ∆E
+P sinθ(cos(2φ)Re[A A∗d c∗]+sin(2φ)Im[A A∗d c∗])]log µ ,
µ R L e e R L e e 4m2
e
5
where P is the polarization of the initial muon and c and d are the elements of the spinor of e+: v =
µ e e 1 e+
1
√2E (0,d ,c ,0)T where (d 2+ c 2)1/2 =1 and the z-direction is taken to be along the momentum of e+. Using
the a1boveeforemula it is stra|igeh|tforw| aer|dto show that the transverse polarization of e+ is 1
1
P sinθ(cos2φRe[A A∗]+sin2φIm[A A∗])
s = µ R L R L
h T1i A 2(1+P cosθ)+ A 2(1 P cosθ)
L µ R µ
| | | | −
and
P sinθ( cos2φIm[A A∗]+sin2φRe[A A∗])
s = µ − R L R L ,
h T2i A 2(1+P cosθ)+ A 2(1 P cosθ)
L µ R µ
| | | | −
where Tˆ = (~s p~ )/~s p~ and Tˆ =(Tˆ ~p )/Tˆ p~ .
2 − µ× e+1 | µ× e+1| 1 2× e+1 | 2× e+1|
Notice that the averages of s and s over φ vanish, so to extract information on arg[A A∗], one has to
h T1i h T2i R L
measure the azimuthal angle that the momentum of the second positron makes with the plane made by ~s and ~p .
µ e+
1
However, measuring φ will be challenging because when (P P )2 0, the angle between the momenta of the
µ − e+1 →
two emitted positrons converges to π. For general configurationwith (P P )2 m2, the transverse polarization
µ− e+1,2 ∼ µ
of the electron also carries information on the CP-violating phases of the underlying theory. In the framework we
are studying (R-parity conserved MSSM), the rate of µ+ e+e−e+ is small compared to the rate of µ+ e+γ:
Br(µ+ e+e+e−)/Br(µ+ e+γ) α[log(m2/m2) 11/4→] 0.0061.Thus, even if the present bound on µ→ eγ is
→ → ≃ 3π µ e − ≃ →
saturated, the statistics of µ eee will be too low to perform such measurements in the foreseeable future. For this
→
reason, in this paper we will not elaborate on µ eee any further.
→
III. µ−e CONVERSION
In the rangeof parameterspacethat we areinterestedin, the dominantcontributionto the µ e conversioncomes
−
from the γ and Z boson exchange penguin diagrams and the effects of four-Fermi LFV terms can be neglected. The
effective LFV vertex in the penguin diagrams can be parameterized as follows
e
= (H e¯ γµµ +H e¯ γµµ )(Zqq¯ γ q +Zqq¯ γ q )
Leff sinθ cosθ m2 L L L R R R L L µ L R R µ R
W W Z q∈X{u,d}
Q e A∗ A∗
q B∗e¯ γ µ +B∗e¯ γ µ +i Re¯ σµνp µ +i Le¯ σµνp µ (q¯γ q)+H.c. (14)
− p2 (cid:18) L L µ L R R µ R m L ν R m R ν L(cid:19) µ
q∈X{u,d} µ µ
where p = p p is the four-momentum transferred by the photon or Z-boson and Q is the electric charge of the
µ e q
quark. Zq −= T3 Q sin2θ is the coupling of left(right)-handed quark to the Z-boson. H and H are the
L(R) q − q W L R
effective couplings of the Z boson to lepton. A and A are the same couplings that appear in Eq. (2). B (p2) and
L R L
B (p2) vanish for p2 0 so they do not contribute to µ eγ. Let us evaluate and compare the contributions of the
R
→ →
various couplings appearing in Eq. (14). Since A and A flip the chirality, they are suppressed by a factor of m .
L R µ
Ward identity implies that B and B are suppressedby p2 = m2. There is not such a suppressionin H and H ,
R L − µ L R
thus H /m2 B /m2.
L(R) Z ∼ L(R) µ
dΓ(µN eN) 1 P cosθ 1+P cosθ
→ =S − µ aH +b(A∗ +B∗)2+ µ aH +b(A∗ +B∗)2 , (15)
dcosθ (cid:20) 2 | L R L | 2 | R L R | (cid:21)
where S is a numerical factor that includes the nuclear form factor [11] and
e Z(1/2 2sin2θ ) N/2 eZ
W
a= − − and b= (16)
(cid:2) 2m2 sinθ cosθ (cid:3) m2
Z W W µ
in which Z and N are respectively the numbers of protons and neutrons inside the nucleus.
Let us define
K aH +b(A∗ +B∗) (17)
R ≡ L R L
6
and
K aH +b(A∗ +B∗). (18)
L ≡ R L R
From Eq. (15), we observe that the total conversion rate, (dΓ/dcosθ)dcosθ provides us with information on the
sum of KR 2 and KL 2. That is while by studying the anguRlar distribution of the final electron, we can also extract
| | | |
K 2 K 2
R L
R | | −| | . (19)
2 ≡ K 2+ K 2
R L
| | | |
Let us now study what extra information can be extracted by measuring the spin of the final electron.
Similarlytothecaseofµ eγ,letusdefinethedirectionsTˆ andTˆ asfollows: Tˆ =(p~ ~s )/p~ ~s and Tˆ =
1 2 2 e µ e µ 1
→ × | × |
((p~ ~s ) ~p )/(p~ ~s ) p~ . Let us also define
e µ e e µ e
× × | × × |
dΓ µN e(~s = 1Tˆ)N dΓ µN e(~s = 1Tˆ)N
s h → e 2 i i− h → e −2 i i .
h Tii≡ dΓ[µN eN]
~se →
P
It is straightforward to verify that the transverse polarization of the emitted electron in the directions of Tˆ and Tˆ
1 2
are
2Re[K K∗]P sinθ
s = R L µ , (20)
h T1i K 2(1 P cosθ)+ K 2(1+P cosθ)
R µ L µ
| | − | |
2Im[K K∗]P sinθ
s = R L µ . (21)
h T2i K 2(1 P cosθ)+ K 2(1+P cosθ)
R µ L µ
| | − | |
Averaging over the angular distribution, we find
s dΓ dcosθ πRe[K K∗]P
s h T1idcosθ = R L µ , (22)
h T1i≡ R dcdoΓsθdcosθ 2(|KR|2+|KL)|2)
R
and
s dΓ dcosθ πIm[K K∗]P
s h T2idcosθ = R L µ . (23)
h T2i≡ R dcdoΓsθdcosθ 2(|KL|2+|KR|2)
R
The advantage of the study of the µ e conversion over the study of µ eγ is that in the former case there is no
− →
needforperformingthe challengingphotonpolarizationmeasurement. The drawbackis the polarizationofthe initial
muon. While the polarization of muon in the µ eγ experiments is close to 100%, the muons orbiting the nuclei
→
(the muonsin the µ e conversionexperiments)suffer fromlow polarizationof16%orlower[15]. However,there are
−
proposals to “re-”polarize the muon in the muonic atoms by using polarized nuclear targets [16].
In this paper, we take P =20%. For any given value of P , our results can be simply re-scaled.
µ µ
IV. EFFECTS OF THE CP-VIOLATING PHASES OF MSSM
Inthissection,westudythepolarizationsintroducedintheprevioussectionintheframeworkofR-parityconserving
Minimal Supersymmetric Standard Model (MSSM). The part of the superpotential that is relevant to this study can
be written as
W = Y ec L H µ H H (24)
MSSM − i Ri i· d− u· d
c b c c c
where L , H and H are doublets of chiral superfields associated respectively with the left-handed lepton doublets
i u d
andthe twoHiggsdoublets ofthe MSSM. ec is the chiralsuperfieldassociatedwith the right-handedchargedlepton
b c c Ri
field, ec . The index “i” is the flavor index. At the electroweak scale, the soft supersymmetry breaking part of
Ri c
Lagrangianin general has the following form
MSSM = 1/2 M BB+M WW +H.c.
Lsoft − (cid:16) 1 2 (cid:17)
ee ff
0.4 ((mm22RL))Aeeµµµ 0.4 ((mm22RL))Aeeµµµ 7
0.2 0.2
〉PsTT11 0 〉PsTT12 0
〈 〈
-0.2 -0.2
-0.4 -0.4
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
Phases[π] Phases[π]
(a) (b)
1 10
((mm22RL))Aeeµµµ ((mm22RL))Aeeµµµ
8
0.5
-120] 6
1
R1 0 γe)[
>
µBr(-- 4
-0.5
2
-1 0
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
Phases[π] Phases[π]
(c) (d)
FIG. 1: Observable quantities in the µ→ eγ experiment versus the phases of A , (m2) and (m2) . The vertical axes in
µ L eµ R eµ
Figs. (a)-(d)arerespectivelyhPT1sT1i,hPT1sT2i,R1 andBr(µ→eγ). Theinputparameterscorrespond totheP3benchmark
proposed in [20]: |µ|=400 GeV, m0 =1000 GeV, M1/2 =500 GeV and tanβ =10 and we have set |Aµ|=|Ae|=700 GeV. All
theLFVelementsofthesleptonmassmatrixaresettozeroexcept(m2) =2500GeV2 and(m2) =12500GeV2. Wehave
L eµ R eµ
taken P =100%.
µ
((A Y δ +A )ec L H +H.c.) L † (m2) L ec † (m2) ec
− i i ij ij Ri j· d − i L ij j − Ri R ij Rj
m2 H† H m2 H† H ( B H H +H.c.), (25)
− Hu u u− fHfd d d− H fu· d f f g
where the “i” and “j” indices determine the flavor and L˜ consists of (ν˜ e˜ ). Notice that we have divided the
i i Li
trilinearcouplingtoadiagonalflavorpart(A Y δ )andaLFVpart(A withA =0). Termsinvolvingthesquarks
i ei ij ij ii
as well as the gluino mass term have to be added to Eqs. (24,25) but these terms are not relevant to this study.
The Hermiticity of the Lagrangian implies that m2 , m2 , and the diagonal elements of m2 and m2 are all real.
Hu Hd L R
Moreover, without loss of generality, we can rephase the fields to make the parameters M , B as well as Y real.
2 H i
In such a basis, the rest of the above parameters can in general be complex and can be considered as sources of
CP-violation. After electroweak symmetry breaking, A gives rise to LFV masses:
ij
(m2 ) =A H for i=j .
LR ij ijh di 6
Notice that in general A = A and therefore (m2 ) = (m2 ) .
| ij|6 | ji| | LR ij|6 | LR ji|
The CP-violating phases that can in principle show up in the polarizations studied in the previous sections are the
phases of A , the µ-term, M (the Bino mass) and phases of LFV elements of mass matrices in soft supersymmetry
i 1
breaking Lagrangian. The strong bound on the electric dipole moment of the electron implies strong bounds on the
phasesofA ,µandM (see,however[17]). Forthisreason,inthispaper,wesetthephasesoftheseparametersequal
e 1
to zero and focus on the effects of the phases of A and the LFV elements of mass matrices. In the present analysis,
µ
we focus on the effects of the eµ elements. Effects of eτ and µτ elements will be explored elsewhere.
Oncewe turnonthe LFVterms,the phase ofA as wellasthe phasesofthe LFVelements cancontribute to d at
µ e
one loop level [18, 19]. We therefore have to make sure that the bounds on d are satisfied. For the parameters that
e
0.4 ((mm22RL))Aeeµµµ 0.4 ((mm22RL))Aeeµµµ 8
0.2 0.2
〉PsTT11 0 〉PsTT12 0
〈 〈
-0.2 -0.2
-0.4 -0.4
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
Phases[π] Phases[π]
(a) (b)
10
1 ((mm22RL))Aeeµµµ ((mm22RL))Aeeµµµ
8
0.5
-120] 6
1
R1 γe)[
→
0 µ
Br( 4
-0.5 2
-1 0
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
Phases[π] Phases[π]
(c) (d)
FIG. 2: Observable quantities in the µ→ eγ experiment versus the phases of A , (m2) and (m2) . The vertical axes in
µ L eµ R eµ
Figs. (a)-(d)arerespectivelyhPT1sT1i,hPT1sT2i,R1 andBr(µ→eγ). Theinputparameterscorrespond totheP3benchmark
proposed in [20]: |µ|=400 GeV, m0 =1000 GeV, M1/2 =500 GeV and tanβ =10 and we have set |Aµ|=|Ae|=700 GeV. All
theLFV elementsof theslepton mass matrix are set tozero except (m2) =250 GeV2 and (m2) =12500 GeV2. Wehave
L eµ R eµ
taken P =100%.
µ
we have considered in this analysis,the contributions of the phases of eµ elements to d are of order of 10−29 e cm
e
∼
and well below the present bound [2]. The contribution of the phase of A is even lower by one order of magnitude.
µ
In the next section, we shall discuss the role of the forthcoming results of d searches in reducing the degeneracies.
e
As the reference point, we have chosen the mass spectra corresponding to the P3 benchmark which has been
proposed in [20]. We have however let A and A deviate from the corresponding values at the benchmark P3. The
e µ
values ofA andA arechosensuchthatthey satisfy the constraintsfromColorandChargeBreaking(CCB)aswell
i ij
as Unbounded From Below (UFB) considerations [21]. The rest of the bounds and restrictions on the parameters of
supersymmetry are undisturbed by varying A .
i
Figs. (1-3) shows R , Br(µ eγ), P s and P s (see, Eqs. (4,10,11) for definitions) versus the phases of
1 → h T1 T1i h T1 T2i
A andtheLFVelements. Wehaveset A = A howevertheresultsarerobustagainstvaryingthevaluesof A as
µ e µ e
| | | | | |
expected. InFig. (1), wehavetakenA andallthe LFVelements ofthe sleptonmassmatrixotherthan(m2) and
ij L eµ
(m2) equaltozero. Noticethat(m2) and(m2) havebeenchosensuchthatBr(µ eγ)liesclosetoitspresent
expRereimµ ental upper bound. As seen fLromeµFig. (1-Rc),eµfor such choice of (m2) and (m2)→, R is close to zero which
L eµ R eµ 1
means A A . As a result, we expect the transversepolarizationto be sizable. Figs.(1-a,1-b) demonstrate that
L R
| |≈| |
this expectation is fulfilled. From Figs. (1-a,1-b), we also observe that the sensitivity of the transverse polarization
to the phases of (m2) and (m2) is significant so by measuring these polarizations with a moderate accuracy one
L eµ R eµ
can extract information on these phases. However at this benchmark, the sensitivity to the phase of A is quite low.
µ
The input of Fig. (2) is similar to that of Fig. (1) except that a hierarchy is assumed between the left and right
LFV elements: (m2) (m2) . As expected inthis case, R 1 andthe transversepolarizationsare small. To
| L eµ|≪| R eµ| 1 ≈
draw Fig. (3), we have set the LFV elements of m2 and m2 equal to zero and instead we have set A ,A =0. As
L R eµ µe 6
seen in Fig. (3) in this case, the transverse polarizations can be sizeable.
Figs. (4-6) show R , R(µ+Ti e+Ti), s and s (see, Eqs. (19,22,23) for definitions) versus the phases
2 → h T1i h T2i
0.4 ((mm22LLRR))Aeµµµe 0.4 ((mm22LLRR))Aeµµµe 9
0.2 0.2
〉Ts1T1 0 〉Ts1T2 0
P P
〈 〈
-0.2 -0.2
-0.4 -0.4
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
Phases[π] Phases[π]
(a) (b)
1 10
((mm22LLRR))Aeµµµe ((mm22LLRR))Aeµµµe
8
0.5
-120] 6
1
R1 0 γe)[
→
µ
Br( 4
-0.5
2
-1 0
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
Phases[π] Phases[π]
(c) (d)
FIG. 3: Observable quantities in the µ → eγ experiment versus the phases of A , (m2 ) and (m2 ) . The vertical
µ LR eµ LR µe
axes in Figs. (a)-(d) are respectively hPT1sT1i, hPT1sT2i, R1 and Br(µ → eγ). The input parameters correspond to the
P3 benchmark proposed in [20]: |µ| = 400 GeV, m0 = 1000 GeV, M1/2 = 500 GeV and tanβ = 10 and we have set
|A |=|A |=700 GeV. All the LFV elements of the slepton mass matrix are set to zero except (m2 ) (=A hH i)=14 GeV2
µ e LR eµ eµ d
and (m2 ) (=A hH i)=14 GeV2. Wehavetaken P =100%.
LR µe µe d µ
of A and the LFV elements. To draw the figures corresponding to the µ e conversion, we have taken P = 20%.
µ µ
−
If the technical difficulties of polarizing the muon in the µ e conversion experiment is overcome and higher values
of P is achieved, s and s can become larger. Obv−iously, for a given value of P , s and s have to
be rµe-scaled by (Ph/T210i%). AhpaTr2tifrom the polarization, the input parameters in Figs. (µ4,5h,6T)1iare reshpeTc2tiively the
µ
same as the input parameters in Figs. (1,2,3). Notice that in this case, too, the sensitivity to the phase of A is
µ
low. From Fig. (2), we observe that s increases more rapidly with sin[arg[(m2) ]] than with sin[arg[(m2) ]].
|h T2i| L eµ R eµ
For (m2) (m2) cases, at first sight, higher sensitivity to arg[(m2) ] may sound counterintuitive. However,
notic|e tLhaetµ|a≪s |sin(Raregµ[|(m2) ]) increases, R rapidly converges to onLeewµhich means K K and therefore
| R eµ | 2 R ≫ L
s Im[K K∗]/(K 2+ K 2) 0.
h TI2tii∝s remarRkabLle th|atLi|n th|e cRa|se →of Fig. (4) for which (m2) (m2) , R is close to one and the transverse
L eµ ∼ R eµ 2
polarizations is relatively small but in the case of Fig. (5) for which (m2) = 50(m2) , (1 R 1) and the
R eµ L eµ | −| 2|| ∼
transversepolarizationsbecomesizeable. We haveexploredhigherhierarchybetweenthe leftandrightLFVelements
and have found that for (m2) < 500(m2) , s and s diminish. Contrasting Figs. (4,5) with Figs. (1,2),
L eµ ∼ R eµ h T1i h T2i
we find that the polarization studies at the µ eγ and µ e conversion experiments can be complementary. That
→ −
is if 1 R few 0.01, transverse polarization in the µ eγ will become small making the derivation of the
1
−| | ∼ × →
CP-violating phases more challenging. However there is still the hope to derive the phases by polarization studies at
the µ e conversionexperiments. We shall discuss this point in more detail in the description of Fig. 7.
−
Notice that in Figs. (1-6), which all correspond to the benchmark P , sensitivity to the phase of A is low. This
3 µ
is expected because the effect of A is suppressed by tanβ = 10. We have checked for the robustness of this result
µ
and found that for most of the parameter space with large tanβ, sensitivity to the phase of A is low but there are
µ
points at which sensitivity to φ is considerable; e.g., at δ benchmark which has been proposed in [22].
Aµ
0.4 ((mm22RL))Aeeµµµ 0.4 ((mm22RL))Aeeµµµ 10
0.2 0.2
〈〉sT1 0 〈〉sT2 0
-0.2 -0.2
-0.4 -0.4
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
Phases[π] Phases[π]
(a) (b)
60
((mm22RL))Aeeµµµ ((mm22RL))Aeeµµµ
1 50
40
0.5 -140]
1
R2 eTi)[ 30
→
0 Ti
µ
R(
20
-0.5
10
-1 0
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
Phases[π] Phases[π]
(c) (d)
FIG.4: Observablequantitiesintheµ−econversionexperimentversusthephasesofA ,(m2) and(m2) . Theverticalaxes
µ L eµ R eµ
in Figs. (a)-(d) are respectively hsT1i, hsT2i, R2 and R(µTi→ eTi). The input parameters correspond to the P3 benchmark
proposed in [20]: |µ|=400 GeV, m0 =1000 GeV, M1/2 =500 GeV and tanβ =10 and we have set |Aµ|=|Ae|=700 GeV. All
theLFVelementsofthesleptonmassmatrixaresettozeroexcept(m2) =2500GeV2 and(m2) =12500GeV2. Wehave
L eµ R eµ
taken P =20%.
µ
The following remarks are in order:
Inallofthese setsofdiagrams,maximal s correspondsto s =0andvice versa. Thisis expectedfrom
• |h T1i| |h T2i|
Eqs. (22) and (23) because s and s are respectively given by the real and imaginary parts of the same
h T1i h T2i
combinations. For general values of the phases, s 2+ s 2 is solely given by the absolute values of K
|h T1i| |h T2i| L
andK ,andisindependentoftheirrelativephase. Rememberthat K and K canbeextractedbystudying
R R L
| | | |
theangulardistributionofthe electronwithoutmeasuringits spin. Thus,the simultaneousmeasurementofR ,
2
s and s provides a cross-check. A similar consideration holds for R , P s and P s , too.
h T1i h T2i 1 h T1 T1i h T1 T2i
When all the phases are set equal to zero, s and P s vanish but s and P s can be nonzero.
• h T2i h T1 T2i h T1i h T1 T1i
Thus, for the purpose of establishing CP, it will be more convenient to measure s or P s . This is
h T2i h T1 T2i
expected from Eqs. (10,11,22,23).
When(m2 ) =(m2 ) =0,inthe caseofµ eγ,there isasymmetryunder arg[(m2) ] arg[(m2) ]
• LR eµ LR µe → L eµ ↔− R eµ
[seeFigs.(1,2)] butinthe caseoftheµ econversion,thereis notsuchasymmetry [seeFigs.(4,5)]. Moreover,
−
while the dependence of R on the phases is very mild, R can dramatically change with varying some of the
1 2
phases(see,e.g.,Fig.(5-c)). ThiscanbebetterunderstoodinthelimitoftheLFVmassinsertionapproximation.
Remember that observables in the µ eγ decay are given by A and A for (m2 ) = (m2 ) = 0. To
→ L R LR µe LR eµ
leading approximation, A and A are respectively proportional to (m2) and (m2) . As a result, when
L R R eµ L eµ
we vary the phase of (m2) , only the phase of A changes. Similarly varying arg[(m2) ] only changes
R eµ L L eµ
arg[A ]. Since R depends only on the absolute values of A and A , it should not change with varying
R 1 R L
the phases. Remember that P s and P s are given by Re[A A∗] and Im[A A∗] which to leading
h T1 T1i h T1 T2i L R L R
order are proportional to Re[(m2) (m2)∗ ] and Im[(m2) (m2)∗ ]. Thus, there should be a symmetry under
R eµ L eµ R eµ L eµ
