CP violation and scalar lepton flavor oscillation

David Bowser-Chao and Wai-Yee KeungPhysics Department, University of Illinois at Chicago, Illinois 60607-7059

~Received 3 April 1997!

Lepton flavor violation can be induced in supersymmetry by the mixing of two or more of the leptonic scalar

partners,l → l 8, where l , l 85 e,m, t . Krashnikov and Arkani-Hamedet al. pointed out that this effect maybe observable at the Next Linear Collider, through slepton pair production with subsequent lepton-flavor-violating ~LFV! decays. If the slepton mixing involves all three generations,CP violation from the Cabibbo-

Kobayashi-Maskawa phase could lead to asymmetries between the observed LFV decaysl → l 8 and l → l 8.We lay down the formalism and give simple expressions for theCP-violating asymmetry in the transitionprobabilities, and consider possible signals at future colliders.@S0556-2821~97!02119-X#

PACS number~s!: 11.30.Er, 11.30.Hv, 14.60.Hi, 14.80.Ly

INTRODUCTION

Quark flavor is not conserved in nature, as manifestlydemonstrated by oscillation in theK-K system. There is noevidence to date, however, for the analogous lepton flavorviolation. The branching ratio form→eg, for example, isless than 4.9310211 at the 95% confidence level@1#. This iswell in accord with the standard model, where lepton flavorviolation ~LFV! is expected to be highly suppressed, due tothe smallness ofmn /v, wheremn is the largest neutrino massandv the electroweak breaking scale.

Lepton flavor violation is also suppressed in exactly su-persymmetric extensions of the standard model; supersym-metry, however, is necessarily broken, and soft-symmetry-breaking terms can in general generate significant LFV@2#.Recently, Krashnikov@3# and Arkani-Hamedet al. @4#showed that the resultant mixing between at least two of thecharged slepton generations could produce dramatic lepton-flavor-violating signals at CERN LEP2 or the Next LinearCollider ~NLC!. While detailing the particular case of two-generation mixing, these authors also noted the obvious ex-tension to mixing among all three generations.

Beyond simply complicating the analysis of lepton flavorviolation, moving to three-generation mixing also gives riseto a new source ofCP violation ~CPV!, namely, the singlecomplex phase in the Cabibbo-Kobayashi-Maskawa~CKM!matrix that relates the electroweak flavor basis to the threemass eigenstates. In this paper we examine the consequencesof this CP-violating phase@5#.

The analysis is similar in spirit to CPV from three-family-neutrino oscillation@6#. We shall provide compact formulasfor the CPV transition probability asymmetry

P( l → l 8)2P( l → l 8), where l , l 8 denote the sleptons

e,m, t and l , l 8 are the respective antiparticles. FollowingRef. @6#, we show that the CPV result is identical for theavailable choices (l ,l 8)5(e,m), (m,t), and (t,e) , and thenconsider observation of the asymmetry at future colliders.

FORMALISM

First, we shall assume~as did Refs.@3,4#! ~1! the presenceof LFV soft-symmetry-breaking terms in the electroweak-

scale Lagrangian, without regard for the nature of their ori-gin, and~2! that left-right slepton mixing is small. The sec-ond assumption simplifies analysis of CPV and LPV effects,which then have separate contributions from the left- andright-handed sleptons. This assumption is not strongly modeldependent, since left-right slepton mixing is generally sup-pressed by the ratio of the lepton mass to the left-right slep-ton mass splitting, with the latter set by theD term to be oforder the electroweak scale@7#. Furthermore, we shall seebelow that CPV effects involving left-right mixing are fur-ther suppressed by the ratio of the slepton width to the left-right mass splitting. We shall thus follow Refs.@3,4# andfocus on the right-handed sleptons, while noting that gener-alization of our analysis to the separate left-handed sleptoncontribution to CPV is also possible.

As a result of mixing, the flavor states (e, m, t ) do notevolve in time with a trivial phase. We define the mixingamplitudesUa i , where the flavor state is expressed by su-perposition of mass eigenstatesuSi& of massesmi ,

u l a&5(i

Ua i uSi&, uSi&5(i

Ua i* u l a&. ~1!

The matrix U can be explicitly written using the standardKM parametrization with three mixing anglesu i and onephased,

S e

m

tD 5S c1 s1c3 s1s3

2s1c2 c1c2c31s2s3eid c1c2s32s2c3eid

2s1s2 c1s2c32c2s3eid c1s2s31c2c3eidD3S S1

S2

S3D , ~2!

with si5sinui andci5cosui .If the lightest supersymmetric particle~LSP! is a neu-

tralino x0 ~which we assume to be primarilyB-ino!, the fla-vor eigenstatel can be tagged by the decayl → lx0. The

PHYSICAL REVIEW D 1 OCTOBER 1997VOLUME 56, NUMBER 7

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flavor stateua& (a5e, m, or t) of a slepton produced att50in the rest frame could be resolved asub& at a later timetwith the transition amplitude

A~ a→b ;t !5(i

Ua iUb i* exp~2 imi t212 Gt !. ~3!

For simplicity we assume this to be the only open decaychannel; if not, results below should be multiplied by thedecay branching ratios@4#. We also take all states to have acommon decay widthG and

Dmi j [mi2mj , Dmi j ,G!m[ 13 ~m11m21m3!. ~4!

Further restrictions onDmi j are discussed below.The time-dependent oscillation probability is

p(a→b;t)5uA(a→b ;t)u2; note that we omit the decayproduct neutralino in our notation. The time-averaged prob-ability that is actually measured is given by

P~ a→b!5E0

`

p~ a→b ;t !dtY E0

`

(b

p~ a→b ;t !dt

5Ua iUb i* Ua j* Ub j S G

G1~mi2mj !iD

5P~ b→ a !. ~5!

From the unitarity ofUa i and the second line of the equationabove, it can be seen that lepton flavor violation vanishes@4#asDmi j !G; in the limit of exactly degenerate sleptons, themass basis can be arbitrarily rotated to bringUa i into diag-onal form.

CP violation is reflected in the asymmetry between thetransition probabilities of CP-conjugate channels,

P(a→b)2P( a→ b ). Since P( a→ b )5P(b→a) fromCPT invariance, the difference flips sign under interchangeof a and b and vanishes for the diagonal casea5b. It isconvenient to introduce the antisymmetric symbol«ab suchthat «em5«mt5«te51. We summarize all properties ofasymmetry,

P~ a→b!2P~ a→ b !5«abA,

A524Im~Ue1Um1* Ue2* Um2!Im~f121f231f31!, ~6!

where

f i j 5G

G1~mi2mj !i, Imf i j 5

~mj2mi !G

G21~mi2mj !2 . ~7!

From Eq. ~4!, this asymmetry result could also have beenobtained by using the narrow width approximation. Note thatA vanishes not just forDmi j !G as discussed above, but alsoif Dmi j @G, since in this limit Imf i j vanishes as well. This issimply due to the fact thatCP violation requires interferencebetweenUa i and the final state decay phases in Eq.~5!; inthe limit of very large slepton mass splitting, Eq.~5! be-comes an incoherent sum over the mass eigenstates. ForCPviolation from slepton mixing, we thus requireDmi j ;G. Wecan also see that CPV effects involving left-right mixing areindeed suppressed by the the ratio of the slepton width to theleft-right mass splitting.

The expression forA involves the single universal CPVcombination@8# of mixing amplitudes in the case of threefamilies,

XCP[Im~Ue1Um1* Ue2* Um2!52s12c1s2c2s3c3sind. ~8!

The magnitude of this CPV parameterXCP ranges from zeroto the maximum valueA3/1850.096. Reference@4# shows

FIG. 1. Dependence of the slepton pair production cross section

on the slepton massm, at the Next Lineare1e2 Collider, with

As5500 GeV. The production of the selectron paireR eR ~solidcurves! depends on the neutralino couplings and masses. Here weassume that the lightest neutralino of massMx is purelyB-ino. We

also show s(e2e1→mRmR) ~dashed line!. Note that

s(e2e1→mRmR)5s(e2e1→ t R t R).

FIG. 2. Uncorrelated asymmetry2(Sm2St) versusDm/G forXCP5XCP

max ~solid! or XCPmax50.13XCP

max ~dashed line!, for f 50.75andDm5Dm125Dm23.

56 3925CP VIOLATION AND SCALAR LEPTON FLAVOR OSCILLATION

that the present experimental bound onB(m→eg) puts nosignificant constraint on the mixing angles, with(Dm/m)sin2u&0.01.

One may wonder how the experimental limit@9# on theelectric dipole moment~EDM! of the electron constrains thephase inU. It turns out that at the one-loop level, the EDMrequires both the left- and right-handed sleptons to partici-pate in the loop. Each handedness contributes a Glashow-Iliopoulos-Maiani ~GIM!-like suppression factor,DmL,R /mL,R ~'0.01 for our case!. Together with a factorml /m from the soft-supersymmetry-breakingAl term or thesupersymmetricm term, which mixes theL and R compo-nents, there is sufficient suppression to expect no significantconstraint on the phase~note that the electron EDMdoesconstrain the phase in the left-right slepton mixing@2#, aswell as putting tight limits on a relative phase between thecomplex mass of the gauge fermion and theAl or m term@10#!.

EXPERIMENTS

It is believed that the cleanest environment to search forsleptons is at the proposed Next Lineare1e2 Collider~NLC! at an energy ofAs.500 GeV to 1.5 TeV. The signa-ture for pair production of sleptons is, assuming the neu-tralino decay discussed above, a pair of charged leptons plusmissing energy. In Fig. 1, we illustrate the cross section for

right-handed slepton production. AtAs5500 GeV,s( e e)

is almost as large as 1 pb under the assumption that theneutralino exchanged in thet channel is purelyB-ino withmassMx550 GeV ~with an obvious degradation for an ad-mixture of B-ino!. Reference@4# calculated the backgroundsfrom WW, e6nW7, and (e1e2)W1W2 to total about 12 pb.Through efficient cuts@11#, Ref. @4# estimates a reduction ofthe background to about 5 fb for unpolarized beams, with30% signal acceptance, with further improvement possiblewith right-handed polarized beams. Because the CPV signalsconsidered here are simply asymmetries in the LFV signalsof the sort considered in Ref.@4#, we shall simply rely onthese figures to provide an estimate of the sensitivity toCPviolation from slepton mixing.

We start with a simpleCP-odd observable which ignoresthe full correlation between the leptonl and the antileptonl 8. From the sample of dilepton events that pass the cuts, weconsider the CPV asymmetry between muon and antimuonevents,

Sm5N~m2!2N~m1!

N~all signal!. ~9!

Clearly only unequal-flavor dilepton (m6l 7,lÞm) eventscontribute to this asymmetry.

SinceSm only involves a single-particle count, it can beexpressed directly in terms of the single-particle transitionprobability @Eq. ~5!#,

Sm5s~ e e !@P~ e→m!2P~ e→m !#1s~ t t !@P~ t→m!2P~ t→m !#

s~ e e !1s~mm !1s~ t t !. ~10!

SinceP(m→m)5P( m→m), s(mm) does not contribute tothe above asymmetry. Using Eq.~6!, we obtain

Sm5Af , f [s~ e e !2s~ t t !

s~ e e !1s~mm !1s~ t t !. ~11!

At the tree level,s( t t ) ands(mm) come only from theg-

Z exchange amplitude in thes channel, buts( e e) involvesadditional neutralino exchange amplitude in thet channel.Thus f in Eq. ~11! is generally nonzero. The fractionf canbe close to 1 in the limit that thet-channel diagram domi-nates, which will of course be true given sufficiently highcollider energy. Figure 1 showsf to be 0.75 atAs5500 GeVand m5150 GeV, under the assumption that the exchangegaugino is purelyB-ino with massMx550 GeV. With anintegrated luminosity of 100 fb21, around at least 104 eventsfrom the slepton pair production will survive the dedicated

cuts @4,11# which severely reduce the predictable back-grounds to only about 500m7e6 events and about the samenumber ofm7t6 events. Thus, it is possible to measure theCP asymmetrySm at the level 0.02 with 5s.

The other flavor asymmetries areSe50 @which vanishes

becauses( t t )5s(mm)# andSm52St .In Fig. 2, we show the size ofSm2St versusDm/G for

f 50.75, in the scenario that the mass differences are equal inmagnitude to the width, namely,m12m25m22m35Dm;G.

In the scenario wheres( e e)'s(mm)'s( t t ) ~e.g., ifthe t-channel gaugino mass is relatively high!, we can nolonger use the uncorrelated asymmetryS to probe CPV. Wecan, however, turn to other correlated observables such asSm e , which is defined by replacing the numerator in Eq.~9!by N(m2e1)2N(m1e2).

The overall amplitude for taggingm e at time t and tafter the production comprises two contributions. One is thet-channel gaugino exchange amplitude~for theselectronpro-duction only! which has the time-evolving factor

3926 56DAVID BOWSER-CHAO AND WAI-YEE KEUNG

A~ e→m!A~ e→ e !5(i

UeiUm i* exp~2 imi t212 Gt !

3(j

Ue j* Ue jexp~2 imj t 2 12 G t !.

~12!

The other is the commons-channelg-Z exchange amplitudewhich has a simpler factor due to the unitarity of theUmatrix,

(a

A~ a→m!A~ a→ e !

5(i

Um i* Ueiexp@~2 imi212 G!~ t1 t !#. ~13!

Denoting the asymmetryS0 in the special case that theschannel dominates, we have the simple expression

Sm e0

52 13 34XCPIm~f12

2 1f232 1f32

2 !52Sm t0

5Se t0 .

~14!

The statistics factor13 can be offset by summing all threeflavor asymmetries, as inStot

0 [Sm e0

2Sm t0

1Se t0 .

The asymmetry formula in the general case, where both(s and t) channels are important, is lengthy but quitestraightforward, and which we present next.

The amplitudes fore2e1→ l l consist of a common el-ementM0 from the s-channelg-Z exchange and an addi-tional pieceM 8 for the casel 5e from thet-channel gauginoexchange. Their explicit expressions at the tree level are pro-vided below. From Eqs.~12! and ~13!, we derive

ds~e2e1→ l l →ab1missing neutralinos!

dcosu5KF uM0u2(

i , jUa i* Ub iUa jUb j* f i j

2 1uM 8u2P~ e→a!P~ e→ b !

12ReS M0M 8* (i jk

Ua i* Ub iUek* UakUe jUb j* f ikf i j D G . ~15!

The flux and phase space factorK is given below. The non-zero tree-level amplitudesM0 andM 8 for different channelsare @12#

M0~eL2eR

1→ l R l R!5e2sl1/2sinuS 1

s1

2 12 1xW

~12xW!~s2MZ!D ,

~16!

M0~eR2eL

1→ l R l R!5e2sl1/2sinuS 1

s1

xW

~12xW!~s2MZ! D ,

~17!

M 8~eR2eL

1→ eR eR!5e2sl1/2sinu

~12xW! (i

uVBiu2

t2Mx i

2. ~18!

Here, u is the c.m. polar scattering angle andMx iare the

neutralino masses (i 51 is the lightest!, while uVBiu2 is themixing probability of theB-ino component in thei th neu-

tralino; l1/25A124m2/s is the slepton c.m. velocity,s andt are the usual invariant squares of energy and momentumtransfer, xW is the electroweak parameter, andMW

2 /MZ2512xW . The flux and phase space factor is

K5l1/2/(128ps).The differential rate difference betweenCP-conjugate

channels is

dDRab

dcosu5KS uM0u2Sab

01uM 8u2@P~ e→b!«ea

2P~ e→a!«eb#A24M0M 8

3(i jk

Im~Ua i* Ub iUek* UakUe jUb j* !Im~f ikf i j ! D .

~19!

The asymmetry is

Sab5DRab

s~ e e !1s~mm !1s~ t t !, ~20!

where the summed differential slepton cross section followstrivially from Eq. ~15!,

ds~ e e !1ds~mm !1ds~ t t !

5K~3uM0u21uM 8u212M0M 8!dcosu.

As expected, settinga5m and summingb reproduces theuncorrelated asymmetry in Eq.~11!. Figure 3 shows a com-parison of CP-conjugate eventsm2e1 ~solid line! andm1e2 ~dashed line! due to slepton pair production at theNLC with As5500 GeV.

56 3927CP VIOLATION AND SCALAR LEPTON FLAVOR OSCILLATION

DISCUSSION

Our study can be generalized to other processes. In thefuture Large Hadron Collider~LHC!, gluinos will be copi-ously produced, if the gluino mass is about a few hundredGeV. Through the chain of cascade decays, sleptons mayoccur in the intermediate state and give dilepton events. Withenough CPV slepton flavor oscillation, asymmetries of the

type discussed here might be observable. In the case ofsingle-gluino production, the single-lepton asymmetry in Eq.~6! applies, wherea denotes the first slepton produced in thedecay chain, andb is the second slepton. To avoid washingout of the asymmetry, some reconstruction of the cascadedecay would be necessary to at least statistically identify theprimary and secondary lepton decay products.

We also note that it is possible the NLC could be operatedin a gg mode @13#. In this case, all single-lepton asymme-tries Sl vanish, butSab could be measured, and is given byEq. ~14! or Eq. ~20! with M 8 set to zero. The signal andbackground rates for slepton pair production are similar tothose for intermediate mass charged Higgs production,which are roughly equal after mild cuts@14#. Both thet-channel-dominated limit of thee1e2 collider asymmetrySa and the dilepton asymmetry measured by agg colliderare proportional toXCP , and the respective mass splittingfactors are Im(f121f231f31) and Im(f12

2 1f232 1f32

2 ).Measurement of both asymmetries could help indirectlymeasure the slepton mass-splitting. Finally, we note that ifleft-handed polarization of the electron beam is possible, thee1e2 collider measurement ofSab would yield the sameinformation as that of the gg collider, since

M 8(eL2eR

1→ l R l R) vanishes at the tree level.Note added.While preparing this manuscript, we learned

@15# that the authors of Ref.@4# were extending their work toinclude analysis of lepton flavor violation in the three-generation case, as well as the CPV effects considered here@16#.

ACKNOWLEDGMENTS

This work was supported in part by the U.S. Departmentof Energy under Grant No. DE-FG02-84ER40173.

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FIG. 3. Comparison ofCP-conjugate eventsm2e1 ~solid line!andm1e2 ~dashed line! due to slepton pair production at the NLCwith As5500 GeV. We choose the case of maximal mixingsu15u25u35p/4 andDm125Dm235G. We illustrate the scenarios

of Mx550 or 100 GeV form5150 GeV. Event rates are shownversus the KM phased. For2p,d,0, the event rates can be readby replacingd by udu and reversing the labels betweenm2e1 andm1e2.

3928 56DAVID BOWSER-CHAO AND WAI-YEE KEUNG