Neutrino Masses and Sneutrino Mixing in R-Parity Violating Supersymmetry

SLAC-PUB-8173 June 1999

Neutrino masses and sneutrino mixing in R-parity violating supersymmetry

Yuval Grossmana and Howard E.. Haberb

aStanford Linear Accelerator Center, Stanford University, Stanford, CA 94309

bSanta Cruz Institute for Particle Physics, University of California, Santa Cruz, CA 95064

Invited talk presented at American Physical Society (APS) Meeting of the Division of Particles and Fields (DPOF 99), 1/5/99—1/9/99, Los Angeles, CA, USA

Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309 Work supported by Department of Energy contract DEÐAC03Ð76SF00515.

SLAC-PUB-8173

SCIPP-99/24

hep-ph/yymmnnn

June, 1999

Neutrino masses and sneutrino mixing in R-parity violating

sup ersymmetry

a b

Yuval Grossman and Howard E. Hab er

Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309

Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, CA 95064

Abstract

R-parity-violating sup ersymmetry with a conserved baryon number B pro-

vides a framework for particle physics with lepton number L violating in-

teractions. Two imp ortant prob es of the L-violating physics are neutrino

masses and sneutrino-antisneutrino mass-splittings. Weevaluate these quan-

tities in the context of the most general CP-conserving, R-parity-violating

B -conserving extension of the minimal sup ersymmetric standard mo del. In

generic three-generation mo dels, three sneutrino-antisneutrino mass splittings

are generated at tree-level. In contrast, only one neutrino mass is generated

at tree-level; the other two neutrinos acquire masses at one-lo op. In many

mo dels, the dominant contribution to the radiative neutrino masses is induced

by the non-zero sneutrino-antisneutrino mass splitting.

Invited talk presented by Yuval Grossman at the American Physical So ciety APS meeting of

the division of particles and elds DPF99.

YG is supp orted by the U.S. Department of Energy under contract DE-AC03-76SF00515, and

HEH is supp orted in part by the U.S. Department of Energy under contract DE-FG03-92ER40689. 1

I. INTRODUCTION

The solar and atmospheric neutrino anomalies provide strong indications that the neutri-

nos are massive. In particular, the data suggest that there is near-maximal mixing b etween

and but that their masses are hierarchically separated [1]. To accommo date this data,

the Standard Mo del must be extended, either by intro ducing right-handed neutrinos or by

adding Ma jorana neutrino mass terms that violate lepton number by two units.

In low-energy sup ersymmetric extensions of the Standard Mo del, lepton number and

baryon numb er conservation is not automatically resp ected by the most general set of renor-

malizable interactions. However, the constraints on baryon number violation are extremely

severe in order to avoid fast proton decay. If one wants to enforce lepton numb er and baryon

number conservation in the tree-level sup ersymmetric theory, it is sucient to imp ose one

extra discrete symmetry. In the minimal sup ersymmetric standard mo del MSSM, a mul-

tiplicative symmetry called R-parity is intro duced, where the R quantum number of an

[3B L+2S ]

MSSM eld of spin S , baryon number B and lepton number L is given by 1 .

By intro ducing this symmetry, one eliminates all dimension-four lepton numb er and baryon

numb er-violating interactions. Ma jorana neutrino masses can be generated in an R-parity-

conserving extension of the MSSM involving new L = 2 interactions through the sup er-

symmetric see-saw mechanism [2,3].

Such L = 2 interaction have an imp ortant impact on sneutrino phenomena [3{5].

The sneutrino ~ and antisneutrino ~, which are eigenstates of lepton numb er, are no

longer mass eigenstates. The mass eigenstates are therefore sup erp ositions of ~ and ~, and

sneutrino mixing e ects can lead to a phenomenology analogous to that of K { K and B {

B mixing. The mass splitting between the two sneutrino mass eigenstates is related to

the magnitude of lepton number violation, which is typically characterized by the size of

neutrino masses. In some cases the sneutrino mass splitting may b e enhanced by a factor as

large as 10 compared to the corresp onding neutrino mass [3,6]. As a result, the sneutrino

mass splitting is exp ected generally to b e very small. Yet, it can b e detected in many cases,

if one is able to observe the lepton number oscillation [3].

The primary motivation for intro ducing a conserved R-parity ab ove was to imp ose a

conserved baryon number to avoid fast proton decay. However, this can also be achieved

in low-energy sup ersymmetric mo dels where B is conserved but L is violated so that R-

parity is also violated. In this pap er, we fo cus on the B -conserving R-parity-violating

RPV extension of the MSSM. In such a mo del, neutrinos are massive [7{11] and sneutrino{

antisneutrino pairs are no longer mass-degenerate [3{5]. In Section I I, weintro duce the most

general RPV extension of the MSSM with a conserved baryon number and establish our

notation. In Section I I I, we showhow a tree-level mass for one neutrino is generated due to

neutrino{neutralino mixing. In Section IV, we exhibit how tree-level mass splittings for the

three sneutrino{antisneutrino pairs are generated due to sneutrino{Higgs b osons mixing. In

Section V, we calculate the neutrino masses generated at one lo op. In Section VI, we argue

that in many mo dels, the dominant contribution to the one-lo op neutrino masses is induced

by the non-zero sneutrino{antisneutrino mass splittings. A brief summary and conclusions

are given in Section VI I. 2

II. R-PARITY VIOLATION FORMALISM

In RPV low-energy sup ersymmetry, there is no conserved quantum number that distin-

^ ^

guishes the lepton sup ermultiplets L and the down-typ e Higgs sup ermultiplet H . Here,

m D

m is a generation lab el that runs from 1 to 3. Each sup ermultiplet transforms as a Y = 1

weak doublet under the electroweak gauge group. It is therefore convenient to denote the

four sup ermultiplets by one symbol L =0;1;2;3. The Lagrangian of the theory is xed

by the sup erp otential and the soft-sup ersymmetry-breaking terms.

The most general renormalizable sup erp otential resp ecting baryon number is given by:

h i

j j

i i 0 i j i j

^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

L H + W = ; 1 L L E + L Q D h H Q U

ij m m m nm m

nm n U n

^ ^

where H is the up-typ e Higgs sup ermultiplet, the Q are doublet quark sup ermultiplets,

U n

^ ^ ^

U [D ] are singlet up-typ e [down-typ e] quark sup ermultiplets and the E are the sin-

m m m

glet charged lepton sup ermultiplets. Our notational conventions follow those of ref. [4].

Without loss of generality, the co ecients are taken to be antisymmetric under the

interchange of the indices and . Note that the -term of the MSSM is now extended

to an 4-comp onentvector, . Then, the trilinear terms in the sup erp otential prop ortional

to and contain lepton numb er violating generalizations of the down quark and charged

lepton Yukawa matrices.

Next, we consider the most general set of renormalizable soft-sup ersymmetry-breaking

terms. In addition to the usual soft-sup ersymmetry-breaking terms of the R-parity-

conserving MSSM, one must also add new A and B terms corresp onding to the RPV terms

of the sup erp otential. In addition, new RPV scalar squared-mass terms also exist. As

ab ove, we can streamline the notation by extending the de nitions of the co ecients of

the R-parity-conserving soft-sup ersymmetry-breaking terms to allow for an index of typ e

which can run from 0 to 3 while Latin indices m and n run from 1 to 3. Explicitly,

2 i i 2 2

f f e e e e

D D U U +M Q Q +M V =M

mn n mn n mn soft

m m m n

e e e

D U Q

2 2 i i i 2 2

e e e e

E E + m jH j b L H +h:c: L L +M +M

mn n U ij

m U U

e e

E L

j i j i 0 i

e e f e e e e e

U +h:c:] + [ Q D a H Q L L E + a a L

m ij m U nm m m

n U n

i h

a a

f e e f

+ W +M BB +h:c: : 2 M gg +M W

1 3 2

Note that the single B term of the MSSM is extended to a 4-comp onentvector b , and the

squared-mass term for the down-typ e Higgs b oson and the 3 3 lepton scalar squared-mass

matrix are combined intoa44 matrix. In addition, the matrix A-parameters of the MSSM

are extended in the obvious manner [analogous to the Yukawa coupling matrices in eq. 1];

in particular, a is antisymmetric under the interchange of and . It is sometimes

convenient to follow the more conventional notation in the literature and de ne the A and

B parameters as follows:

a A ; a h A ;

m m E m U nm nm U nm

0 0

a A ; b B ; 3

D nm

nm nm

where rep eated indices are not summed over in the ab ove equations. Finally, the Ma jorana

gaugino masses, M , are unchanged from the MSSM.

i 3

The total scalar p otential is given by:

V = V + V + V ; 4

scalar F D soft

where explicit forms for the sup ersymmetric contributions V and V can b e found in ref. [4].

F D

We do not present here the minimization conditions, but only mention that regions of param-

eter space exist where only the neutral color-singlet scalar elds acquire vacuum exp ectation

p p

values: hh i v = 2 and h~ i v = 2.

U u

Up to this p oint, there is no preferred direction in the generalized generation space

spanned by the L . However, It is sometime convenienttocho ose a particular \interaction"

^ ^

basis such that v = 0, in which case v = v . In this basis, we denote L H .

m 0 d 0 D

For simplicity, we shall imp ose CP-invariance on the mo del which implies that all pa-

rameters app earing in eqs. 1 and 2 and the vacuum exp ectation values of the scalar

elds can be taken to be real. The consequences of CP-violating e ects in this mo del will

be considered elsewhere.

III. NEUTRINO MASS AT TREE LEVEL

The neutrino can b ecome massive due to mixing with the neutralinos [7]. This is deter-

e f

mined by the 7 7 mass matrix in a basis spanned by the two neutral gauginos B and W ,

e e

the higgsinos h and h , and 3 generations of neutrinos, . The tree-level fermion

U D 0 m

e f

mass matrix, with rows and columns corresp onding to fB; W ; h ; g is given by [8,9]:

0 1

M 0 m s v =v m s v =v

1 Z W u Z W

B C

0 M m c v =v m c v =v

2 Z W u Z W

B C

M = ; 5

B C

m s v =v m c v =v 0

Z W u Z W u

B C

@ A

m s v =v m c v =v 0

Z W Z W

where c cos , s sin , and 0 is the 4 4 zero matrix. In a basis-indep endent

W W W W

analysis, it is convenient to intro duce:

cos ; 6

where

X X

2 2 2 2 2 2 2 2

; v v ; v = v + v ' 246 GeV : 7

u d

Note that measures the misalignmentof v and .

It is easy to check that M p ossesses 2 zero eigenvalues. We shall identify the cor-

resp onding states with 2 physical neutrinos of the Standard Mo del [8], while one neutrino

acquires mass through mixing. We can evaluate this mass by computing the pro duct of the

n n

[denoted b elowby det M ] ve non-zero eigenvalues of M

n 2 2 2 2

det M = m M cos sin ; 8

Z 4

where M cos M + sin M . We compare this result with the pro duct of the four

~ W 1 W 2

neutralino masses of the R-parity-conserving MSSM obtained by computing the determi-

nant of the upp er 4 4 blo ckofM with , v replaced by , v resp ectively

0 0 d

det M = m M sin 2 M M : 9

~ 1 2

To rst order in the neutrino mass, the neutralino masses are unchanged by the R-parity

violating terms, and we end up with [9]

n 2 2

det M m M cos sin

m = = : 10

m M sin 2 M M

~ 1 2

det M

Thus, m m cos sin , assuming that all the relevant masses are at the electroweak

scale.

A necessary and sucient condition for m 6= 0 at tree-level is sin 6= 0, which implies

that and v are not aligned. This is generic in RPV mo dels. In particular, the alignment

of and v is not renormalization group invariant [9,10]. Thus, exact alignment at the

low-energy scale can only b e implemented by the ne-tuning of the mo del parameters.

IV. SNEUTRINO MASS SPLITTING

In RPV low-energy sup ersymmetry, the sneutrinos mix with the Higgs b osons. Under

the assumption of CP-conservation, we may separately consider the CP-even and CP-o dd

scalar sectors. Consider rst the case of one sneutrino generation. If R-parityis conserved,

0 0

the CP-even scalar sector consists of two Higgs scalars h and H , with m < m and

h H

~ , while the CP-o dd scalar sector consists of the Higgs scalar, A , the Goldstone b oson

which is absorb ed by the Z , and one sneutrino, ~ . Moreover, the ~ are mass degenerate,

2 so that the standard practice is to de ne eigenstates of lepton numb er: ~ ~ + i~ =

and ~ ~ . When R-parity is violated, the sneutrinos in each CP-sector mix with the

corresp onding Higgs scalars, and the mass degeneracy of ~ and ~ is broken. We exp ect

the RPV-interactions to b e small; thus, we can evaluate the resulting sneutrino mass splitting

in p erturbation theory.

The derivation of the CP-even and CP-o dd scalar squared-mass matrices can be found

in ref. [4]. Working in the v = 0 basis and for one generation we nd

1 0

1 1

2 02 2 2 02

g + g v b g + g v v b b cot +

0 u d 1 0

4 4

C B

1 1

2 02 2 02 2

C B

g + g v v b tan + g + g v b tan ; 11 b M =

u d 0 1 0

d C even B

4 4

A @

b b tan m

1 1

~ ~

and

0 1

b cot b b

0 0 1

B C

M = b b tan b tan ; 12

0 0 1

odd

B C

@ A

b b tan m

1 1

~~ 5

where

2 2 2 2 02 2 2

m M + g + g v v : 13

~~ 1 u d

In the R-parity-conserving limit b = = 0, one obtains the usual MSSM tree-level

1 1

masses for the Higgs b osons and the sneutrinos.

In b oth squared-mass matrices [eqs. 11 and 12], b m is a small parameter that

can be treated p erturbatively. We may then compute the sneutrino mass splitting due to

the small mixing with the Higgs b osons. Using second order matrix p erturbation theory to

compute the eigenvalues, we nd:

2 2

b cos sin

2 2

m = m + + ;

~ ~~

+ 2 2 2 2

cos m m m m

0 0

~~ ~ ~

H h

2 2

m = m + : 14

~ ~ ~

2 2

cos m m

Ab ove, we employ the standard notation for the MSSM Higgs sector observables [12]. Note

that at leading order in b , it suces to use the values for the Higgs parameters in the R-

parity-conserving limit. Then at leading order in b for the sneutrino squared-mass splitting,

2 2 2

m m m we nd

~ ~ ~

2 2

2 b m m sin

1 Z

m = : 15

2 2 2 2 2 2

m m m m m m

~ ~ ~~ ~ ~

H A

where m ' 2m m .

~ ~ ~

We now extend the ab ove results to more than one generation of sneutrinos. In a basis

where v = 0, the resulting CP-even and CP-o dd squared mass matrices are obtained from

eqs. 11 and 12 by replacing b with the three-dimensional vector b and m by the

1 m

3 3 matrix

2 02 2 2 2 2

g + g v v : 16 M M +

mn mn mn m n

u d ~ ~

In general, M is not exp ected to be avor diagonal. In this case, the theory would

~ ~

predict sneutrino avor mixing in addition to the sneutrino{antisneutrino mixing exhibited

ab ove. The relative strength of these e ects dep ends on the relative size of the RPV and

avor-violating parameters of the mo del. To analyze the resulting sneutrino sp ectrum, we

2 2

cho ose a basis in which squared-mass matrix M = m is diagonal. In this

mn m mn

~ ~ ~~

basis b is also suitably rede ned. We will continue to use the same symb ols for these

quantities in the new basis. The CP-even and CP-o dd sneutrino mass eigenstates will be

denoted by ~ and ~ resp ectively. It is a simple matter to extend the p erturbative

+ m m

analysis of the scalar squared-mass matrices if the m are non-degenerate. We then

2 2 2

nd that m m m is given by eq. 15, with the replacement of b and

m m m 1

~ ~ ~

2 2

m by b and m , resp ectively:

m m

~~ ~~

2 2

2 b m m sin

~ ~ m

m Z

m = : 17

~ m

2 2 2 2 2 2

m m m m m m

m m m

~~ ~~ ~~

H A

Since eq. 17 has b een derived in the v = 0 basis, it follows that in an arbitrary basis,

all sneutrino{antisneutrino pairs would b e mass-degenerate if b and v were aligned. How-

ever, this alignment is not renormalization-group invariant. Hence we exp ect that all the

sneutrino{antisneutrino pairs are generically split in mass at tree-level. 6

V. ONE-LOOP NEUTRINO MASSES

In contrast to the sneutrino sector, only one neutrino mass is generated at tree-level due

to neutrino mixing with the neutralinos. Masses for the remaining massless neutrinos will b e

generated by one lo op e ects. There are two classes of one lo op diagrams. The rst consists

of fermion{sfermion lo ops and dep ends on the RPV trilinear terms. The second, which in

many cases is the dominant one, consists of sneutrino{neutralino lo ops and dep ends on the

sneutrino{antisneutrino mass splitting. We now discuss b oth of these e ects in turn.

First, consider the fermion{sfermion lo ops. Contributions to the neutrino mass matrix

are generated from diagrams involving a charged lepton-slepton lo op and an analogous down-

typ e quark-squark lo op [7]. In the limit where the fermion masses can be neglected,

2 3

! !

X X

0 0

4 5

m = m sin 2 ln +3 m sin 2 ln ;

qm q`p mp` ` ` d d

qdr mr d

2 2 2

32 M M

p r

`;p 2 d;r 2

where is the mixing angle of the charge slepton down typ e squark squared-mass

` d

matrix,

2A m

` `

: 19 sin 2 =

2 2

2 2 2

L R +4A m

` `

2 2 2 2 2

Here, A A tan , L M +T esin m cos 2 and R M +

` E 0`` 0 `` 3 W ``

e e

L E

e sin m cos 2 , with T = 1=2 and e = 1. For sin 2 , take e = 1=3 and replace

W 3 d

2 2 2 2

M ! M , M ! M and ` ! d in the ab ove formulae.

e e e e

E D L Q

Second, consider the sneutrino induced masses. In general, the existence of a sneutrino{

antisneutrino mass splitting, which is a result of a L =2 interaction, generates a one-lo op

contribution to the neutrino mass. Wehave computed exactly the one-lo op contribution to

the neutrino mass [m ] from neutralino/sneutrino lo ops [3]. In the limit of m ; m m ,

~ ~

the formulae simplify, and we nd in the one generation case

g m

2 1

f y jZ j ; 20 m =

j jZ

2 2

32 cos

2 2 2

y [y 1lny ] =1 y , with y m =m where f y = , and Z Z cos

j j j j j j jZ j2 W

Z sin is the neutralino mixing matrix element that pro jects out the Z eigenstate from

j 1 W

the j th neutralino. One can check that f y < 0:566, and for typical values of y between 0.1

j j

and 10, f y > 0:25. Since Z is a unitary matrix, we exp ect as a rough order of magnitude

estimate

m 10 m : 21

In the three-generation mo del, a similar estimate holds for the lo op contribution to each

neutrino mass. 7

VI. THE NEUTRINO SPECTRUM

The neutrino sp ectrum is determined by the relative size of the di erent RPV couplings

that control the three sources of neutrino masses. In the v = 0 basis these are [for the

m m

tree level mass, eq. 10], b [for the sneutrino induced one lo op masses, eqs. 21 and 17]

and and [for the trilinear RPV induced one lo op masses, eq. 18]. Therefore, in

ij k

order to understand the structure of the neutrino sp ectrum wemust have a framework that

predicts the magnitude of these parameters. Here we give one example: mo dels based on an

ab elian horizontal symmetry [13]. Further details and examples will b e given in ref. [14].

Consider a simple mo del based on a U 1 ab elian horizontal symmetry, which is describ ed

in detail in ref. [13]. For our purp oses, it is sucient to mention that the order of magnitude

of all the mo del parameters are determined by the assigned horizontal charges to the various

elds in the theory. Moreover, many ratios are predicted indep endently of the sp eci c charge

assignment. For the parameters relevant for neutrino masses we nd the following ratios

` 0 d

v m m

b v

i ij k

jk ij k jk

1; ; ; 22

v v v

i i i

` d

where m m is the charged lepton down-typ e quark mass matrix. Inserting the ab ove

jk jk

order of magnitude predictions into eqs. 10, 21, 17 and 18 we nd that the tree level

yields the dominant contribution to the heaviest neutrino mass. The two lighter neutri-

nos get their masses mainly from the sneutrino{neutralino lo ops as a consequence of the

non-zero tree-level sneutrino{antisneutrino mass splitting; these will be referred to as the

sneutrino{induced masses. The trilinear RPV induced masses are suppressed compared to

4 4 6

the sneutrino{induced masses by at least a factor of order 30 m =v 10 and are therefore

completely negligible. Moreover, the following relations hold

m m

1 i

3 2 2

10 sin ; sin : 23

i3 12

m m

3 2

The fact that the sneutrino{induced neutrino mass is the dominant radiative e ect is

not unique to the ab ove example. For example, in mo dels with high energy alignment, we

also nd that these contributions are larger than the one-lo op contribution from fermion{

sfermion lo ops [14].

VI I. SUMMARY AND CONCLUSIONS

Recent exp erimental signals of neutrino masses and mixing may provide the rst glimpse

of the lepton-number violating world. In R-parity violating sup ersymmetric mo dels that

incorp orate lepton number violation, the sneutrino{antisneutrino mass sp ectrum may pro-

vide additional insight to help us unravel the mystery of neutrino masses and mixing. Such

mo dels generally predict a non-trivial neutrino sp ectrum in which there are several sources

for neutrino masses. One neutrino acquires mass at tree level via neutrino{neutralino mix-

ing. The other two neutrinos acquire radiative masses at one lo op. In many mo dels, the

dominant contribution of the radiative neutrino masses is induced by the non-zero sneutrino{

antisneutrino mass splitting. A detailed study of sneutrino phenomenology at future colliders

can complement the present day study of neutrino mass and mixing in order to shed light

on the nature of the underlying lepton-numb er violation. 8

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