Supersymmetry and Neutrino Mass

Proc Indian Natn Sci Acad, 70, A No.1, January 2004, pp.239–249 c Printed in India.

SUPERSYMMETRY AND NEUTRINO MASS

BISWARUP MUKHOPADHYAYA Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad - 211 019 (India)

(Received 31 January 2003; Accepted 30 June 2003)

The existence of neutrino mass and mixing is a strong pointer towards physics beyond the Standard Model. An overview of the possibility of having neutrino masses in supersymmetric theories is attempted here. Some of the recent works reviewed suggest Dirac masses, whereas others include Majorana masses as well. Side by side, it is shown how R-parity violating supersymmetry opens new avenues in the neutrino sector. Reference is also made to light sterile neutrinos, nearly degenerate neutrinos and neutrinos acquiring masses from hard supersymmetry breaking terms which are suppressed by the Planck scale. In several of the cases, it is pointed out how the models that give neutrino masses and mixing have independent motivations of their own, and can be tested in accelerator experiments.

Key Words: R-Parity Violation; Light Sterile Neutrinos; Quasidegenerate Neutrinos; Neutrino Masses Sup- pressed by Planck Scale

1 Introduction entails lepton number violation clearly entails new physics, albeit at high scale, the first one can be prima As has been amply established in the other articles facie dismissed as a ‘trivial’ extension in the form of in this volume, there is a strong evidence nowadays a right-handed neutrino component for each family. in favour of neutrino masses. In addition, the solar1 However, the fact that such a right-handed neutrino and atmospheric2 neutrino data have their most obvi- has none of the strong, weak and electromagnetic in- ous explanation in neutrino oscillation, requiring mix- teractions is curious, if not suggestive of some new ing among neutrinos, or, more generally speaking, interaction in which it takes part. The extreme sup- in the leptonic sector, in analogy with quark mix- pression of neutrino Yukawa couplings necessitated ing which is controlled by the Cabibbo-Kobayashi- by sub-eV Dirac masses is also puzzling. Side by Maskawa (CKM) matrix3. However, in contrast to side, if the LSND claim suggesting the disappearance quark mixing, the most favoured explanations of the of νµ ’s is to taken seriously4 , we most likely need a solar and atmospheric neutrino deficits require very fourth light neutrino, sterile in nature. Since the mass large—even close to maximal—mixing between the of a sterile vectorlike neutrino is not protected by any first two families and the last two. Side by side, the symmetry, and since we can hardly think of any new data indicate a hierarchy of mass splitting, the mass- physics scale below a TeV or so, a light sterile neu-

3 2 2 ¡ squared difference being in the range 10 ¡ -10 eV trino, if it is there at all, warrants a drastically novel between the second and the third families, and, most mechanism for its justiﬁcation.

5 4 2 ¡ favourably, 10 ¡ -10 eV between the first and the The new physics scale to which appeal has mostly second. Though such splittings are most often trans- been made to understand neutrino masses is that per- lated into a corresponding hierarchy in the masses taining to Grand Unified Theories (GUT), restricted themselves, the existence of near-degenerate neutri- to be at least5 about 1016 GeV. However, there are nos, too, cannot be ruled out. other motivations for physics beyond the Standard According to many, all this is an indication of Model within the TeV scale itself. One such is the physics beyond the Standard Model. To see why, let so-called naturalness problem which reflects our lack us recall that, thanks to the electrically neutral charac- of understanding why the Higgs mass (and conse- ter of neutrinos, they can have both Dirac and Ma- quently the electroweak scale MEW ) should be stable jorana masses. While the second possibility which against quadratically divergent radiative corrections. 240 BISWARUP MUKHOPADHYAYA

The most popular solution to this problem has been portant goal of accelerator experiments, it should be offered in terms of supersymmetry (SUSY), a sym- really interesting to look for the particular signatures metry between bosons and fermions, which can pro- of such theoretical schemes as are able explain the ob- vide the necessary cancellations to control the large servations in the neutrino sector. In other words, the radiative corrections6 . Most importantly, it is possi- issue of neutrino masses could provide not only useful ble to keep the Higgs mass within acceptable limits guidelines for theorisation, but might also end up pre- even if SUSY is broken in mass, so long as the break- dicting speciﬁc experimental signals in high-energy ing scale (characterising the boson-fermion splitting) colliders. The present article is aimed at discussing is approximately within the TeV scale. Side by side, some of these possibilities. the observation that the threshold effects arising from In very general terms, some of the ways in which Tev scale SUSY breaking ensures better convergence SUSY can be of special signiﬁcance to neutrino of the three coupling constants at the GUT scale pro- masses are as follows:

vides an added impetus to SUSY7. § In the minimal SUSY Standard Model (MSSM)8, The phenomenon of SUSY can provide new the particle spectrum of the Standard Model (SM) gets scales (in addition to that brought by GUT doubled, there being a superpartner for each known in which most SUSY theories are embedded). particle, apart from the necessity of two Higgs dou- These scales open up additional possibilities in blets which lead to three neutral and a pair of mutually the neutrino sector and can be helpful in ex- conjugate singly charged scalars. There is no experi- plaining mass hierarchies. Also, some features mental evidence yet for any of these superparticles; of the SUSY theory might help us in under- collider experiments have set lower bounds of about standing ultra-small Yukawa couplings.

100 GeV upwards on most of them. Further conse- § The extended particle spectrum in SUSY can quences of SUSY also depend on the details of the lead to mechanisms for mass generation, for ex- spectrum which in turn is crucially dependent on the ample, through additional radiative effects. SUSY breaking mechanism. We know that SUSY has to be broken at any rate if it is there, since we do not § The possibility of low-energy lepton number vi- observe degenerate superpartners for the SM particles. olation inbuilt in certain types of SUSY theo- No completely acceptable SUSY breaking scheme has ries might lead to the generation of Majorana been found so far, although most studies depend upon masses.

¢ a scenario based on N 1 supergravity (SUGRA) § SUSY could explain a naturally light sterile where gravitational interactions with a ‘hidden sec- neutrino, in case we need it to explain the ob- tor’ characterised by a high scale (O £¥¤ M M ) lead P EW ¦ served data. to soft SUSY breaking terms in the observable sector. In addition, schemes of SUSY breaking, for example, In section 2 we discuss Dirac masses in presence 10 via gauge interactions of a messenger sector or via of SUSY. Section 3 is devoted to Majorana neutri- 11 anomaly terms have also been investigated. nos in SUSY scenarios, where lepton number viola- The question is: since the search for physics be- tion takes place at high-scale. Section 4 contains a yond the Standard Model has found a strong candidate summary of neutrino mass generation mechanisms in in SUSY, could SUSY also be responsible for neu- R-parity violating SUSY where the low-energy La- trino masses (and mixing), the clue that nature seems grangian has lepton number violation. In section 5 we to dangle so tantalisingly in front of us? If that be discuss respectively the issues of degenerate neutri- so, then the mass patterns answering to the solar and nos in SUSY and neutrino masses from unusual SUSY atmospheric neutrino data should not only depend on breaking terms. We conclude in section 6. certain speciﬁc aspects of the SUSY model, but also impose constraints on it. It may also be more convinc- 2 When Lepton Number is Conserved–Dirac ing if models are built not just to answer questions on Masses in SUSY neutrinos but have independent motivations of their own from the viewpoint of SUSY as well. Side by If one takes the hierarchy in neutrino mass splitting to side, since the the search for SUSY is already an im- be an indication of the hierarchy in the masses them- SUPERSYMMETRY AND NEUTRINO MASS 241

selves, then, assuming that the solar and atmospheric is the gravitino mass), giving mν 10 eV . This

νµ ντ ν νµ ¨ neutrino deﬁcits are due to ¨ and e oscilla- is an unacceptably large value unless one has near-

tions respectively, the two heaviest neutrinos are about degenerate neutrinos. The solution, therefore, lies 10 to 11 orders of magnitude smaller in mass than in having Az Fz, i.e. in the SUSY conserv- the τ and the µ. The simplest extension of the Stan- ing vev being much smaller than the SUSY break- dard Model spectrum that explains the above masses ing one. This can be realised, for example, in an is one right-handed neutrino per generation. However, O’Raifeartaigh-type model, where a hierarchy be- the onus then falls on us to explain the wide disparity tween the scalar and pseudoscalar components can be of Yukawa couplings that is responsible for the huge envisioned upon generating an effective low-energy mass splitting within the same families, as indicated scalar potential for Z through the condensation of above. The question is: can SUSY provide some ex- some chiral superﬁeld in the SUSY breaking sector planation of such disparity? through non-perturbative effects: Normally, with right-handed neutrino superﬁeld

A 16π2kxm (5) ©

z N, one would expect a term in the superpotential of 3 2

the form

with x O 1 . This yields neutrino Dirac masses of ©

W yν N¯LH (1) 2

© N 2 the order of 10 eV—the right order of magnitude! It may be relevant to comment here that Az can di- where H2 is the Higgs doublet giving mass to fermions rectly lead to small neutrino masses even without the with T 1 2. Of course, here one would ﬁnd it 3 © hard to justify the smallness of yν . On the other hand, above mechanism in a gauge mediated SUSY break- one can forbid such a term with the help of some ing (GMSB) scenario, where the gravitino is a much lighter object. In such a case, however, mass split- discrete symmetry Zn, and assume instead a higher- dimensional term12 ting between the electron and muon neutrinos be- comes considerably smaller than what has been re- k

W yν ZN¯LH (2) ported above, and can at most place us in the solu- © N 2 MP tion space corresponding to vacuum oscillation. Since

the current data strongly disfavour such a solution, the where k is a coupling constant O 1 , MP is the Planck mass, and Z is a superﬁeld that is invariant under the GMSB option is perhaps not of much value in this Standard Model gauge group. Then the superpotential context. 13 given in eqn (2) is allowed, as against the one in eqn There is a very similar approach which puts (1), if the various superﬁeld have the following charge the mechanism of neutrino mass generation in a somewhat bigger perspective. It is well-known assignments under Zn: that in MSSM, there is no natural way to keep

Z Z 1; Z N 1;Z f 0 (3) the Higgsino mass parameter , a Supersymmetry-

© © © n n n conserving mass, within the TeV scale. The param- f being any of the chiral superﬁelds in MSSM. Note µ eter occurs in a term H1H2 in the superpotential, and that (2) implies the existence of a non-renormalizable it is not clear why it is not as big as any of the masses term in the superpotential, which, in the SUGRA con- in the SUSY breaking sector. On the other hand, it text, can arise as an effective coupling, duly sup- is highly desirable to have it around the electroweak pressed by MP. scale so that the minimisation condition for the scalar If Az and Fz are respectively the vacuum expecta- potential can be naturally satisﬁed. tion values (vev) of the scalar and auxiliary compo- With the µ-problem in view, one can think of nents of Z (the latter being the SUSY breaking vev), a global symmetry group G protecting the Higgs then the Dirac mass for the neutrino is given by masses, and forbidding the µ-term in the original superpotential. There can again be a gauge singlet su- kAzv2

mν (4) © M perﬁeld X associated with the SUSY breaking sector, P transforming non-trivially under the group G, which If Az is of the same order as the square root of the ﬁnds its way into the superpotential via the term SUSY breaking vev Fz, then, in a SUGRA scenario, 1

11 W XH H (6)

A M m M 10 GeV (where m X

z © 1 2 X 3 2 P 3 2 MP 242 BISWARUP MUKHOPADHYAYA

MEW Remember also that FX , the vev auxiliary component a suppression of m compared to the unsuppressed 2 22 2 H of X, is of the order of MX ( 10 GeV ) as de- Yukawa strengths of the corresponding charged lep- ﬁned above. One can immediately see that this ‘SUSY ton. This suppression can be used to account for the breaking’ vev gives rise to a µ-parameter in the range smallness of the Dirac neutrino masses compared to 2

of M M m . Thus a value of µ in the naturally X P 3 2 those of their charged partners, although a nearly bi- expected range is ensured. maximal texture remains unexplained. Suppose now that the same scenario includes a Before we end this section, some remarks about right-handed neutrino superfield N. If lepton number radiatively generation Dirac masses in SUSY models is conserved, then one can envision a scenario where are in order. Radiative generation is possible through 12 a term in the superpotential of the form LNH2 is disal- diagrams mediated by neutral gauginos . However, lowed by the charge assignments of the corresponding the fact that the right-handed neutrino superfield is a superfields under G. However, the term ENH1 may Standard Model gauge singlet implies that such a di- still be allowed if N, also a gauge singlet, has a dif- agram can contribute only when additional gauginos ferent charge compared to the Standard Model super- are present. An extension of the gauge group is there- fields. In this case, the source of the neutrino Dirac fore a necessity for such a mechanism to be operative. mass is 1

W XLNH (7) N 2 3 ∆L 2 Terms in the Lagrangian: MP Majorana Masses Again, a scalar vev for X on the order of MX leads to inadmissibly large neutrino masses. The interesting If there is lepton number violation at a high scale M,

contribution comes again from the auxiliary compo- it is possible to have ∆L 2 neutrino mass terms via nent which yields a Dirac mass, given by the dimension-5 operator 16 M2 v v2

X 2 2 λ m (8)

D 2 LLHH (9) MP MP 5 M 3 2

which turns out to be around 10 eV. Once more, which gives neutrino masses on the order of v M. one is left with the task of preventing neutrino masses The most obvious model that gives rise to Majorana

from the SUSY-conserving vev of X. This has been masses of this kind has heavy right-handed neutri- done in the literature by introducing additional U 1 nos in the scale M, with both Yukawa couplings with symmetries in the SUSY breaking sector 14, and pre- SU(2) doublets and L-violating mass terms of its own: venting, in a style similar to the one mentioned earlier,

the scalar potential from developing a vev in the low- NN ! y N¯LH (10) N N est order. 2 Of course, the neutrino mixing pattern still needs so that it is possible to generate very small neutrino to be explained, the particular problem being the pos- mass eigenstates without requiring inordinately small sibility of large mixing both between the ﬁrst and sec- Yukawa coupling. This is the essence of the well- ond generations and the second and third. The only known seesaw mechanism16, 17. It is also seen that one reasonable explanation of this can come from a tex- obtains the light neutrino masses in the expected range 16 ture of the XNLH2 coupling. However, there is no when M is in the Grand Uniﬁcation scale of about 10 clear understanding of how a suitable texture can be GeV. Thus it is customary to treat N as a right-handed

naturally ensured. neutrino belonging to the fundamental representation An alternative explanation of small Dirac neutrino of a GUT group such as SO 10 . In addition, there masses has been offered from the assumption that the can be a right-handed neutrino in each generation, so gauge singlet superﬁeld N is prevented by a global that M in general can be a matrix, real and symmetric. from having an NLH2 term, but that the charges are The prediction of two large mixing angles, however, such that a heavy superﬁeld H can replace H2 in the is a dilemma that is yet to be satisfactorily addressed superpotential 15. Now, if there is mixing between in GUT-inspired textures of M.

H2 and H after SUSY breaking, the Yukawa cou- In what way can SUSY contribute to the Majorana pling of N with the resultant physical Higgs can have mass generation mechanism of the above type? Of SUPERSYMMETRY AND NEUTRINO MASS 243 course, there are numerous versions of SUSY GUT’s component of X small. The only thing that needs justi- where various issues related to the requisite texture fication is the Majorana masses for N as well as the µ- have been discussed. The recourse to SUSY GUT’s term only out of the auxiliary component of X. For the has also its motivation in the observation that the con- latter such a condition is essential if one has to have vergence of the three gauge coupling constants at high µ in the electwroweak scale. It has been argued that scale is better achieved in a SUSY scenario where for the former a similar fate is expected since the two new threshold effects become important around the terms are of the same form. A contribution from the TeV scale. Particular SUSY breaking schemes such scalar vev of X would make the Majorana mass much as GMSB have been also invoked to explain the large higher and the lighter eigenvalue much lower than is flavour mixing necessitated by the observed data. An admissible, unless one can again think of a symme- important component, to which most existing studies try to restrict the scalar component to a vev within of the subject owe their richness, is the question of the TeV scale. It is with an argument of this kind compatibility of large mixing with the limits on lep- that the contributions from the scalar component of X µ γ ton flavour violating processes such as "$# e or have sometimes been dropped from the low-energy ef-

τ "%# eγ. In the SUSY context, the mismatch between fective theory, although this may not be totally above the neutrino and sneutrino mass matrices at low-scale criticisms of arbitrariness. is a source of potentially dangerous flavour violation, Once more, the problem of generating two large and thus the parameters of the theory must be subject mixing angles is not solved in a construction of the to strong constraints. A large number of investigations above type. For that, one has to assume specific tex- in this direction can be found in the literature18 . tures in the XNLH couplings, which in turn requires Here we discuss the following question: in ad- appropriate modelling of the SUSY breaking sector. dition to the GUT scale, can the additional scale(s) It has also been shown in several works20 that the made available to us from SUSY breaking be of any above principle can be extended to include a light use in Majorana mass generation? In relation to Dirac sterile neutrino, something that one might require if masses, we have already found an answer in the affir- the claims from LSND are confirmed. An additional mative. Now we include a brief discussion related to gauge singlet superfield S has to be added for this pur- Majorana masses12, 13, 19. pose. It is, however, necessary to suppress the Yukawa One can, for example, extend the picture outlined coupling for S, and allow S to develop a small mass

above by including a ∆L & 2 mass term for the right- via the scalar component of X, devised to be small by handed neutrino N through a term of the form X †NN mechanisms mentioned earlier. This can be ensured in the superpotential. In exact analogy with the situ- through an appropriate assignment of charge for S un- ation where the µ-parameter is generated around the der the group G. electroweak scale, this mass may also be generated Unlike the case of Dirac neutrinos, a Majorana only from the vev of the auxiliary component of the neutrino can have loop-induced masses without any ﬁeld X. The Majorana mass is thus given by extension of the gaugino sector. The second reference in ref.[12] shows the representative diagrams from 2 MX which such contributions can come. The contribution,

m v ( ( ( (11) ' N ' MP for which explicit expressions can be found in the lit- νν v being of the same order as the electroweak scale. erature, depend on the effective ˜ ˜ as well as left-right For the Dirac mass, however, the scalar vev of X may mixing in the sneutrino mass matrix. Such loop con- be used in the term XLNH , yielding tributions, in regions where they are substantial, may 2 be required to explain (a) the mixing pattern, and (b)

mX v2 the mass pattern itself where, for example, the right- ( ( ( mD & (12) MP handed neutrino develops a large Majorana mass from the scalar component of X, making the seesaw mech- so that the seesaw mass for the light neutrino(s) is anism viable.

given by m2 m v2 M , which is in the desired

) ) D N ' P Before we conclude, two comments may be in range. Note that unlike in the case of Dirac neutrinos, order. First, the mechanisms discussed in this sec- here one does not need to invoke a special mechanism

tion and the last one are important, although they + such as a U * 1 symmetry to keep the vev of the scalar 244 BISWARUP MUKHOPADHYAYA might not be uniformly successful in explaining tex- When R-parity is violated, the following addi- tures etc. The reason for this is the fact that in additional terms can be added to the superpotential22 :

tion to the conventional GUT scale, here the scale mX λ c λ c λ c c c

, 7 7 7

W 6 L L E L Q D U D D i 4 is made available to us by the SUSY breaking scheme. R i jk i j k 4 i jk i j k i jk i j k

ε L H (14) 5 5 5 This enables one to explore newer avenues to address 4 i i 2

λ 7 the yet unanswered questions, hopefully by combin- with the 7 -terms causing B-violation, and the re- ing inputs from the SUSY breaking scale with those maining ones, L-violation. The need to avoid proton from the GUT scale. Secondly, the kind of models decay usually prompts one to have only one of the two outlined here favour, among other things, additional types of nonconservation at a time. Since we are con- right-handed sneutrinos in the electroweak scale. In cerned with neutrino masses here, we will consider fact, since this sneutrino mass is not restricted by the only lepton number violating effects. λ λ Z-decay width, it can even become the lightest super- The -and 7 -terms have been widely studied symmetric particle (LSP). This can have considerable in connection with various phenomenological conse- implications in collider phenomenology as well as is- quences, enabling one to impose various kinds of lim- sues related to dark matter13. its on them23. Their contributions to neutrino masses can be only through loops, and their multitude (there ∆ are 36 such couplings altogether) makes the necessary 4 L , 1 Terms in the Lagrangian–R-Parity Violating SUSY adjustments possible for reproducing the requisite values of neutrino masses and mixing angles. We shall Let us next consider the case where neutrinos can ac- come back to these ‘trilinear’ effects later. quire masses through lepton number violating interac- More interesting, however, are the three bilinear 24 ε

tions at low-energy. This is realised in R-parity violat- terms iLiH2. Since there are only three terms of 2 21 3B 2 L 2S ing SUSY , where R-parity is deﬁned as -/.10 . this type, the model looks simpler and more predictive It can be seen from this deﬁnition that all superparti- with them alone as sources of R-parity violation. This

cles have R ,3. 1 whereas R equals 1 for all the Stan- is particularly so because the physical effects of the dard Model particles. It is also clear from above that trilinear terms can be generated from the bilinears by R-parity, a multiplicatively conserved quantity, can be going to the appropriate bases25. In addition, they have violated when B or L is violated. This makes it possi- interesting consequences of their own26, since terms of ε ble for a superparticle to decay into two or more Stan- the type iLiH2 imply mixing between the Higgsinos dard Model particles, thus rendering the LSP unstable. and the charged leptons and neutrinos. In this discus- In SUSY, squarks and sleptons, all spinless ob- sion, we shall assume, without any loss of generality, jects, carry lepton and baryon numbers. It is thus pos- the existence of such terms involving only the second sible to violate one of these numbers by one unit while and third families of leptons. the other is conserved. This is not possible in the Stan- The scalar potential in such a case contains the dard Model due to the gauge structure and particle as- following terms which are bilinear in the scalar ﬁelds: signments. Such a provision in the SUSY scenario 2 2 2 2 2 2 2 2

V , m L˜ m L˜ m H m H

4 4

scal L 3 4 L 2 1 1 2 2 8 8 8 8 8 8 8 makes it free from the danger of destabilising the pro- 3 8 2

BµH H B ε L˜ H B ε L˜ H

4 4 ton. 4 1 2 2 2 2 2 3 3 3 2 µε 8

µε L˜ 9 H L˜ H (15)

5 5 5

4 4;: : : To see how R-parity violation actually takes place 4 3 3 1 2 2 1 in SUSY, let us remember that the MSSM superpoten- where mL denotes the mass of the ith scalar doublet tial is given by i at the electroweak scale, and m1 and m2 are the mass parameters corresponding to the two Higgs doublets. l c d c

W , µH H h L H E h Q H D 4 MSSM 1 2 4 i j i 1 j i j i 1 j B, B2 and B3 are soft SUSY-breaking parameters.

hu Q H U c (13) An immediate consequence of the additional (L- 5 5 5 4 i j i 2 j violating) soft terms in the potential is a set of non- where the last three terms give the Yukawa interac- vanishing vacuum expectation values (vev) for the tions corresponding to the masses of the charged lep- sneutrinos. This gives rise to the mixing of elec- tons and the down-and up-type quarks, and µ is the troweak gauginos with neutrinos (and charged lep- Higgsino mass parameter. tons) through the sneutrino-neutrino-neutralino (and

SUPERSYMMETRY AND NEUTRINO MASS 245 sneutrino-charged lepton-chargino) interaction terms. see-saw type mass at the tree-level. The orthogonal ν The hitherto massless neutrino states enter into the combination 2 still remains massless. Interestingly, neutralino mass matrix through such mixing and ac- now we have a new seesaw mechanism, where the quire see-saw masses, where the high scale is supplied SUSY breaking scale in the observable sector takes by the massive states. massive states. The parame- the place of the GUT scale or the scale mX discussed ters controlling the neutrino sector in particular and in the earlier sections.

ν R-parity violating effects in general are the bilinear The massive state 3 can be naturally used to ε ε coefﬁcients 2 , 3 and the soft parameters B2, B3. account for atmospheric neutrino oscillations, with 2 > 2 For a better understanding, let us perform a ba- ∆m mν J Large angle mixing between the νµ and 3 sis rotation and remove the R-parity violating bilinear the ντ corresponds to the situation where v v . 2 W 3

terms from the superpotential by suitably redefining The tree-level mass here is clearly controlled by V the lepton and Higgs superfields. This, however, does > 2 2 v E v v . This quantity, defined as the ‘effec- 2 R 3 not eliminate the effects of these terms, since they now tive’ sneutrino vev in the basis where the ε’s are ro- take refuge in the scalar potential. The sneutrino vev’s tated away, can be treated as a basis-independent mea- in this rotated basis (which are functions of both and sure of R-parity violation in such theories27. The SK the ε’s and the soft terms in the original basis) trigger

data on atmospheric neutrinos restrict v E to be on the neutrino-neutralino mixing. Consequently, the 6 < 6 order of a few hundred keV’s. However, it should be neutralino mass matrix in this basis has the following E ε ε remembered that v is a function of 2 and 3, both

form: of which can still be as large as on the order of the

?@ FHG

D G

@ gv g v electroweak scale. For example, in SUGRA-based C C µ B

0 B 0 0 G

@ 2 2 G

@ models, it is possible to have a very small value of

D D

gv D g v

C C

@ G µ B

B 0 0 0

@ G

2 2 v E starting from large ε’s, provided that one assumes

@ G

gv gv D gv3 gv2

C C C C

B B B G

@ M 0 >

= 2 2 2 2 the R-conserving and R-violating soft terms (as also

@ G

D D

D g v g v G

@ g v g v 3 2

C C C

C the slepton and Y 1 Higgs mass parameters) to be E B 0 M 2 2 2 2

D the same at the scale of dynamical SUSY breaking at

gv3 g v3

C C

0 0 B 0 0 28 I A 2 2 a high energy .

gv2 g D v2

C C 0 0 B 0 0 2 2 Also, one has to address the question as to whether

J J J ν ν (16) the treatment of 3 and 2 as mass eigenstates is

where the successive rows and columns correspond to proper, from the viewpoint of the charged lepton mass

B ν ν ν ν

(H˜ H˜ B iW˜ iB˜ τ µ ), τ and µ being the neu- matrix being diagonal in the basis used above. In fact,

K K K K 2 K 1 3 ε trino ﬂavour eigenstates in this basis. Also, with the it can be shown that this is strictly possible when 2 is ε sneutrino vev’s denoted by v2 and v3, much smaller than 3, failing which one has to give a further basis rotation to deﬁne the neutrino mass 1 2 2 2 2 >PO m v v eigenstates. However, the observable consequences

Z 2 R 3

ENM M

Q B L L β β v v 2 2 sin cos ¯g 2 S described here are still valid, with the requirement of

near-maximality shifted from the angle θ to the effec-

M M

E L M and M being the SU L 2 and U 1 gaugino mass tive mixing angle. >UT 2 2 parameters respectively, and ¯g g g E .

R Furthermore, a close examination of the scalar po- ν ν One can now deﬁne two states 3 and 2, where tential in such a scenario reveals the possibility of ad-

> ditional mixing among the charged sleptons, whereby

ν cosθ ντ sinθ νµ J J J (17) 3 R ﬂavour-changing neutral currents (FCNC) can be enhanced. It has been concluded after a detailed study and ν is the orthogonal combination, the neutrino 2 that the suppression of FCNC requires one to have mixing angle being given by the ε-parameters to be small compared to the MSSM

> v3 parameter µ (or, in other words, to the electroweak J J J cosθ V (18) 29 ε ε v2 v2 scale) unless there is a hierarchy between 2 and 3. 2 R 3 However, one still needs to ﬁnd a mechanism for ν ν Clearly, the state 3 — which alone develops cross- mass-splitting between the massless state 2 and the terms with the massive gaugino states — develops a electron neutrino, and to explain the solar neutrino 246 BISWARUP MUKHOPADHYAYA puzzle. While there exist studies which attempt to ex- these two-body channels dominate over three-body plain both the puzzles in terms of bilinear terms only, ones over a large region of the parameter space, the

the existence of the various λ and λ X -terms can also effect of which can be observed in colliders such as give rise to loop contributions to the neutrino mass the upgraded Tevatron, the LHC and a proposed high- matrix30. energy electron-positron collider. In addition, super- The generic expression for such loop-induced particles such as the stop can sometimes decay dom- masses is inantly via R-parity violating interactions, thereby

Y altering the observed signals. Different observable

loop 3 d d 1 X

m m m M λ X λ ν Z i j [ 8π2 k p SUSY m2 ikp jpk quantities related to these decays have been studied q˜ in recent times32, 33, 34, 35.

\ 1 l l 1 m m M λ λ ] ] ] (19) Here we would like to stress upon one distinctive 8π2 k p SUSY m2 ikp jpk

l˜ feature of the scenario that purportedly explains the

_ ` ^ SK results with the help of bilinear R-parity violating where md l denote the down-type quark (charged lep- 2 2 terms. It has been found that over almost the entire ton) masses. m˜, mq˜ are the slepton and squark mass l Yba allowed range of the parameter space in this connec- squared. M µ is the effective scale of su- SUSY Z tion, the lightest neutralino is dominated by th Bino. persymmetry breaking. The mass eigenvalues can be A glance at the neutralino mass matrix reveals that de- obtained by including these loop contributions in the cays of the neutralino ( Bino) in such a case should mass matrix. [ be determined by the coupling of different candidate Again, it should be noted that there may be other fermionic fields in the final state with the massive neu- ways of looking at the problem. For example, it has trino field ν which has a cross-term with the Bino. been shown in31 that, if one assumes either purely bi- 3 Large angle neutrino mixing, on the the hand, implies linear or purely trilinear R-violating interactions at a that ν should have comparable strengths of coupling high scale, running of the mass parameters can lead 3 with the muon and the tau. Thus, a necessary con- to significant sneutrino vev’s at low energy, and at the sequence of the above type of explanation of the SK same time generate loop-induced masses. results should be comparable numbers of muons and If we want the mass thus induced for the second tau’s emerging from decays of the lightest neutralino, generation neutrino to be the right one to solve the so- together with a W -boson in each case32, 33. lar neutrino problem, then one obtains some constraint Of course, the event rates in the channel men- on the value of the λ X s as well as λs. In order to gener- tioned above will depend on whether the two-body de- ate a splitting between the two residual massless neu-

δ 2 6 2 cays mentioned above indeed dominate over the three- d trinos, m 5 c 10 eV (which is suggested for an [ body decays. The latter are controlled by the size of

MSW solution ), a SUSY breaking mass of about 500

Y a

4 5 the λ-and λ X -parameters. If these parameters have to

λ X λ d GeV implies 10 d -10 . Z be of the right size to explain the mass-splitting re- An interesting aspect of the scenario described quired by the solar neutrino deﬁcit, then, for large an- above is that it can have distinctive signatures in col- gle MSW case, the decay widths driven by the trilinear lider experiments. The most striking ones among term are smaller than those for the two-body decays them pertain to decays of the lightest neutralino, pro- by at least an order of magnitude. duced either directly or via cascades. In presence of The other important consequence of this picture is only the trilinear R-violating terms in the superpo- a large decay length for the lightest neutralino. We tential, the lightest neutralino can have various three- have already mentioned that the atmospheric neutrino

body decay modes which can be generically described e χ0 e ν ¯ χ0 ¯ results restrict the basis-independent R-violating pa- by f f and l f1 f2, f , f1 and f2 being dif-

rameter v X to the rather small value of a few hundred ferent quark and lepton ﬂavours that are kinematically keV’s. This value affects the mixing angle involved in allowed in the ﬁnal state. calculating the decay width of the neutralino, which Due to the mixing between neutrinos and neutrali- in turn is given by the formula nos as also between charged leptons and charginos,

the bilinears open up additional decay channels for the e

lightest neutralino, namely, χ 0 e lW and χ0 νZ. ¯h p

] ] ]

Y c L f (20) When the neutralino is heavier than at least the W, Γ M χ˜ 0 1 Z SUPERSYMMETRY AND NEUTRINO MASS 247 where Γ is the rest frame decay width of the light- to justify degenerate neutrino scenarios, and we shall est neutralino and p its momentum. The decay length mention only one approach here38. In this work, the decreases for higher neutrino masses, as a result of close degeneracy of the neutrino masses can be a pri- the enhanced flipping probability between the Bino ori postulated to come from the form of the neutrino and a neutrino, when the LSP is dominated by the mass matrix at the Planck scale. Following works, for Bino. Also, a relatively massive neutralino decays example of Georgi and Glashow, the matrix can be faster and hence has a smaller decay length. The in- taken to correspond exactly to bimaximal mixing at teresting fact here is that even for a neutralino as mas- the Planck scale. The evolution of the mass parame- sive as 250 GeV, the decay length is as large as about ters should provide the requisite splittings at low en- 0.1 to 10 millimetres, which should be observable in a ergy. The evolution is crucially controlled by Yukawa detector32 . couplings, and this is where the dependence on tan β, If the lightest neutralino can have two-body the ratio of the vev’s of the two Higgs doublets be- charged current decays, then the Majorana character comes most important. However, it has been shown38 of the latter also leads to the possibility of like-sign that the solution space corresponding to the large mix- dimuons and ditaus from pair-produced neutralinos35 . ing angle (LMA) MSW mechanism yields an inadmis- Modulo the efficiency of simultaneous identification sible mass splitting unless tan β is very small, which of W-pairs, these like-sign dileptons can also be quite is again incompatible with accelerator data. On the useful in verifying the type of theory discussed here. other hand a seesaw approach, with a high-scale Ma- jorana mass in the range of 1010 GeV, leads to accept- 5 Some Other Possibilities able MSW solutions in the LMA regions. This, how- β ever, gives the best fit for tan g 2 which is at the Nearly Degenerate Neutrinos very edge of the phenomenologically viable MSSM parameter space. If the mass ranges to which the neutrino eigenstates belong are represented by mass-squared differ- Neutrino Mass from Unusual SUSY Breaking Terms ences indicated by the solar and atmospheric neutrino deficits, then it is difficult to account for the hot dark We normally agree to have ‘soft’ SUSY breaking matter content of the universe in terms of neutrinos. terms only, the main reason being the need to con- A way to surmount the difficulty is to postulate nearly trol quadratic divergence of scalar masses. However, degenerate neutrinos36 . Degeneracy also helps us un- since the SUSY breaking interaction is usually an ef- derstand large mixing in a somewhat ‘natural’ manner. fective theory, one may expect higher order terms also At the same time, with a sterile neutrino with mass in to creep into the picture. Though such ‘hard’ terms the similar order, it may provide an explanation of the are potential threats to the stability of scalar masses, LSND results if they are substantiated. they are suppressed by some power(s) of the cut-off However, there are problems with degenerate neu- scale for the effective theory, which in this case turns trinos. The limit on the electron neutrino mass out to be the Planck mass mP. Thus the quadratic from tritium beta decay provides the first restriction. corrections effectively shift the scalar masses by very More seriously, if neutrinos are of Majorana charac- small amounts, and the hard terms are usually ignored ter, then degeneracy can come into serious conflicts as phenomenologically insignificant. Such a possi- with constraints imposed from the search for neutrino- bility is conceivable also in the schemes suggested less double-beta decay. There have been efforts to cir- in ref.[38], with an enlarged SUSY breaking sector. cumvent this difficulty by proposing neutrino mixing Also, such terms have sometimes been exploited to matrices which effect a cancellation between different stabilise flat directions of the scalar potential and gen- eigenstates in such decay37. Also, the literature con- erate intermediate scale vev’s. 39 tains proposals of a partial lifting of the degeneracy. It has been suggested that some of these sup- On the whole, these scenarios cannot be completely pressed higher-dimensional terms may be responsible ruled out, though some natural foundation for any of for neutrino masses. This is true in particular if lepton the models is yet to be found. number is violated. Under such circumstances, one In the context of SUSY, too, efforts have been on may, for example, have a gauge invariant term in the 248 BISWARUP MUKHOPADHYAYA

Lagrangian, of the form many interesting approaches left out in this review.

h The point which has been emphasised here is the fact ε 2 l l l

h j L˜ H (21) k hard i i j i 2 j that SUSY notionally brings in additional mass scales ε into low-energy physics, which can have a role to play where i j is the completely antisymmetric rank-2 ten- sor. The dimensionless coupling h in this case de- in the domain of neutrinos. Also, some special status 2 2 n

pends on j m M where n depends on the speciﬁc of the right-handed neutrino superﬁeld with respect to m X P k SUSY breaking mechanism. Note that this term is L- the governing symmetry in the SUSY breaking sec- violating but R-parity conserving. tor might well be responsible for the different nature Such a term generates Majorana neutrino masses of neutrino masses with respect to those of the other at one-loop level, involving virtual sneutrinos and fermions. Such a point of view can be applied to

SU j 2 gauginos. The induced mass has been shown both Dirac and Majorana masses, and also to cases k to be of the form which give rise to light sterile neutrinos. Side by side, the ∆L 1 terms in the superpotential of an R-parity hg2v2 i 2 2 2 l l l

ν j breaking SUSY theory can use the electroweak sym-

m n F M mν (22) m 32π2m 2 ˜ k ν˜ metry breaking scale itself in a spectacular manner to

where M and m are the SU j 2 gaugino and sneutrino explain not only neutrino masses but also their mix- 2 ν˜ k

masses respectively. The function F ranges between ing pattern. Several of the theories discussed above l 0 l 5 and 0 1 for phenomenologically allowed values of have implications in other aspects of electroweak phe- the mass ratio in the argument. nomenology including high-energy collider phenom- Using such an expression, it can be seen that for ena, which, quite desirably, integrates neutrino-related a sneutrino mass in the range of 100 GeV and phe- model-building into a much bigger canvas. Scenarios nomenologically allowed values of the ratio of the with degenerate neutrinos can also be encompassed Higgs vev’s, the induced neutrino mass turns out to by SUSY models. And ﬁnally, there exists the inter- be too small to be consistent with observed results if esting conjecture that the otherwise undesirable hard

n 1, while for n 1 2 it stays a little above the SUSY breaking terms, suppressed by some power(s)

i i acceptable range.a A mechanism of the above kind of the Planck mass, can after all have a role to play in therefore favours SUSY breaking schemes where the neutrino physics. dimension-4 terms shown in eq.4 are suppressed by It should be admitted ﬁnally that ﬂavour mixing, 2 2

some fractional power of the ratio j m M . An addi- especially that of the bimaximal type, still requires m X P k tional problem, of course, is to explain neutrino mix- special model assumptions. A better understanding of ing in this scheme, for which the evolution of the term SUSY breaking schemes is necessary for further in- shown in eq.10 to low energies has to play a role. sight into the matter.

Concluding Remarks Acknowledgements

I have reviewed some of the various ways in which This work was partially supported by the Board of Re- a SUSY scenario can be responsible for the genera- search in Nuclear Sciences, Government of India. tion of neutrino masses. I must admit that there are

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