Supersymmetric inflation, baryogenesis and νμ−ντ oscillations

14 May 1998

Physics Letters B 427 Ž1998. 53–58

Supersymmetric inflation, baryogenesis and nm y nt oscillations G. Lazarides a , Q. Shafi b, N.D. Vlachos

c

a

Theory DiÕision, CERN, 1211 GeneÕa 23, Switzerland and Physics DiÕision, School of Technology, Aristotle UniÕersity of Thessaloniki, Thessaloniki GR 540 06, Greece b Bartol Research Institute, UniÕersity of Delaware, Newark, DE 19716, USA c Department of Physics, Aristotle UniÕersity of Thessaloniki, Thessaloniki GR 540 06, Greece Received 3 December 1997 Editor: R. Gatto

Abstract We propose a supersymmetric model with a left-right symmetric gauge group where hybrid inflation,baryogenesis and neutrino oscillations are closely linked. This scheme, supplemented by a familiar ansatz for the neutrino Dirac masses and mixing of the two heaviest families and with the MSW resolution of the solar neutrino puzzle, implies that 1 eV Q mnt Q 9 eV. The mixing angle umt is predicted to lie in a relatively narrow range which will be partially tested by the ChorusrNomad experiment. Restrictions on the CP violating phase dmt are also derived. q 1998 Elsevier Science B.V. All rights reserved.

The minimal supersymmetric standard model ŽMSSM. provides a particularly compelling extension of the SUŽ3.c = SUŽ2.L = UŽ1. Y gauge theory. Yet, it seems quite clear that MSSM, in turn, must be part of a larger picture. Let us list some reasons why: i. In MSSM, there is no understanding of how the supersymmetric m term is ; 10 2 –10 3 GeV. In principle, it could be as large as the Planck mass, MP . ii. An important Žand undetermined. new parameter in MSSM is tan b , the ratio of the vacuum expectation values of the two higgs doublets. Among other things, an understanding of this parameter can shed light on the mass of the Weinberg-Salam higgs. iii. It has become increasingly clear that a combination of both ‘cold’ and ‘hot’ dark matter ŽCHDM. provides w1x a good fit to the data on large scale structure formation, especially if the primordial density fluctuations are essentially scale invariant. In the MSSM framework, one can include dimension five terms which lead to non zero neutrino masses of

magnitude MW2 rMP < 1 eV. Consequently, dark matter cannot be accommodated with purely MSSM fields. iv. It has been impossible, so far, to implement inflation within MSSM. v. Last, but not least, it is not easy to generate in MSSM the observed baryon asymmetry through the electroweak sphaleron processes. Remarkably, all these challenges can be overcome in one fell swoop by considering a modest extension of the MSSM gauge symmetry to H ' SUŽ3.c = SUŽ2.L = SUŽ2.R = UŽ1.ByL . Of course, it is anticipated that H is embedded in a grand unified theory such as SO Ž10. or SUŽ3.c = SUŽ3.L = SUŽ3.R . Apart from aesthetics, there are tantalizing hints from the extrapolation of low energy data for the existence of a supersymmetric unification scale ; 10 16 GeV. The details on how the extension of MSSM to an H-based model can resolve the points above will not be discussed here, especially since an inflationary scenario based on H has been considered in some detail

0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 3 0 6 - 2

G. Lazarides et al.r Physics Letters B 427 (1998) 53–58

54

elsewhere w2,3x. This scenario gives rise to an essentially scale invariant spectrum Žspectral index n , 0.98., and contains both ‘cold’ ŽLSP. and ‘hot’ Žmassive neutrinos. dark matter candidates. The observed baryon asymmetry is generated through partial conversion of a primordial lepton asymmetry w4x. Finally, the parameter tan b is close to m trm b , which also explains why the higgs boson of the electroweak theory has not been seen at LEP II. Its tree level mass is MZ which, after radiative corrections, becomes m h8 , 105–120 GeV. The inflationary phase is associated with the gauge symmetry breaking SUŽ2.R = UŽ1.ByL ™ UŽ1. Y . Of course, since H is presumably embedded in some grand unified symmetry, there may well be more than one inflationary epoch. We concentrate on the last and most relevant one. The above breaking is achieved by a pair of SUŽ2.R doublet ‘higgs’ superfields which have the same gauge quantum numbers as the ‘matter’ right handed neutrino superfields. As a consequence, the inflaton decays primarily into ‘matter’ right handed neutrinos via quartic Žor higher order. superpotential couplings. The ‘reheat’ temperature, TR , turns out to be about one order of magnitude smaller than the mass of the heaviest right handed neutrino that the inflaton can decay into. The gravitino constraint on TR ŽQ 10 9 GeV. allows us to restrict the second and third family right handed neutrino masses M2 , M3 in a fairly narrow range. Our approach poses no obvious constraint on the first family right handed neutrino mass M1 , except from M1 F M2 . The constraints on M2 , M3 , however, together with the leptogenesis scenario will enable us to restrict the oscillation parameters of the nm – nt system w5x. We consider the 2 = 2 ‘asymptotic’ mass matrices M L , M D and M R in the weak basis, where the superscripts L,D and R denote the charged lepton, neutral Dirac, and right handed neutrino sectors respectively. M L , M D are diagonalized by the biuniX L X c Lc c tary rotations L s U L , L s U L , n s U nn X , n c s nc cX U n : X

c

M L ™ M L s U˜ L M L U L s X

c

ž

M D ™ M D s U˜ n M D U n s

mm

/

,

m 3D

/

mt

ž

m 2D

Ž 1. ,

Ž 2.

where the diagonal entries are positive. This gives rise to the ‘Dirac’ mixing matrix U n †U L in the leptonic charged currents. Using the remaining freedom to perform arbitrary phase transformations on the components of LX , n X together with X the X compensating ones on the components of Lc , n c so that X X L D M , M remain unaltered, we can bring this matrix to the form U n †U L ™

ž

cos u D ysin u D

sin u D , cos u D

/

Ž 3.

where u D Ž0 F u D F pr2. is the ‘Dirac’ Žnot the physical. mixing angle in the 2–3 leptonic sector. In this basis, depicted with a double prime on the superfields, the Majorana mass matrix can be written as M R s Uy1 M0 U˜y1 ,

Ž 4.

where M0 s diagŽ M2 , M3 ., with M2 , M3 Žboth positive. being the two Majorana masses, and U is a unitary matrix which can be parametrized as Us

ž

sin u eyi d cos u

cos u ysin u e i d

/ž

eia2 eia3

/

,

Ž 5.

with 0 F u F pr2 and 0 F d - p . The light X neutrino mass matrix, to leading order in M R y1 M D , is m s yM˜ D

X

1 M

X

R

MD ,

Ž 6.

X

where M D is defined in Eq. Ž2.. We can express m as ms

ž

eia2 e

ia3

/

C Ž u ,d .

ž

eia2 eia3

/

,

Ž 7.

where C Ž u , d . depends also on M2 , M3 , m 2D , m 3D . We diagonalize m by a unitary rotation n XX s Vn XXX with Vs

ž

eib2 eib3

/ž

cos w

sin w eyi e

ysin w e i e

cos w

/

,

Ž 8.

where 0 F w F pr2 , 0 F e - p . The ‘Dirac’ mixing matrix in Eq. Ž3. is now multiplied by V † on the left and, after suitable phase absorptions, takes the form

ž

cos u 23

sin u 23 eyi d 23

ysin u 23 e i d 23

cos u 23

/

,

Ž 9.

G. Lazarides et al.r Physics Letters B 427 (1998) 53–58

where 0 F u 23 F pr2 , 0 F d 23 - p . Here, u 23 Žor umt . is the physical mixing angle in the 2–3 leptonic sector and its cosine equals the modulus of the complex number cos w cos u D q sin w sin u D e iŽ jy e . ,

Ž 10 .

where yp F j y e s b 2 y b 3 y e F p . Moreover, d 23 Žor dmt . is the associated CP violating phase which is given by d 23 s j q r y l Žmodulo p ., where l Žyp F l F p . and yr Žyp F r F p . are the arguments of the complex numbers in Eq. Ž10. and cos w sin u D y sin w cos u D e iŽ jy e . respectively. Since j remains undetermined Žsee below., the precise values of u 23 and d 23 cannot be found. However,we can determine the range in which u 23 lies: < w y u D < F u 23 F w q u D , for w q u D F pr2 .

Ž 11 . The double valued function d 23 Ž u 23 . is also determined. We will denote the two positive eigenvalues of the light neutrino mass matrix by m 2 Žor mnm ., m 3 Žor mn . with m 2 F m 3 . Recall that all the quantities t here Žmasses, mixings, etc.. are ‘asymptotic’ Ždefined at the GUT scale.. The determinant and the trace invariance of C Ž u , d . †C Ž u , d . provide us with two constraints on the Žasymptotic. parameters which take the form m2 m3 s 2

Ž m 2D m 3D . M2 M3 2

m2 q m3 s

2

,

Ž 12 .

Ž m 2D 2 c 2 q m 3D 2 s 2 . M2 q

2

2

Ž m 3D 2 c 2 q m 2D 2 s 2 . M3

2

2 2

q

2 Ž m 3D 2 y m 2D 2 . c 2 s 2 cos2 d M2 M3

,

Ž 13 . where u , d are defined in Eq. Ž5., c s cos u , s s sin u . We now need information about the quantities m 2D ,m 3D Žthe ‘asymptotic’ Dirac masses of the muon and tau neutrinos. as well as the ‘Dirac’ mixing angle u D. A plausible assumption, inspired by SO Ž10. for instance, is that these quantities are re-

55

lated to the quark sector parameters by the SUŽ4.c symmetry. In other words, we will assume the asymptotic relations: m 2D s m c ,

m 3D s m t ,

sin u D s < Vcb < .

Ž 14 .

Of course, the SUŽ4.c symmetry is not expected to hold in the down sector of the second family. Contact with experiment can be made after renormalization effects have been taken into account. The pair of MSSM higgs doublets is assumed to belong to the Ž2,2. representation of SUŽ2.L = SUŽ2.R , implying that tan b , m trm b both ‘asymptotically’ and at low energies. The light neutrino masses, in this case, can be obtained by dividing the right hand side of Eq. Ž6. by a factor of 2.44 w6x. Eqs. Ž12., Ž13. now hold with m 2 ,m 3 being the low energy neutrino masses and m D ’s replaced by their asymptotic values divided by 1.56. The latter turn out to be m 2D , 0.23 GeV and m 3D , 116 GeV Žwith tan b , m trm b . w7x. Finally, using SUŽ4.c invariance, the asymptotic ‘Dirac’ mixing angle is calculated from sin u D s < Vcb < Žasymptotic. , 0.03. The renormalization of the mixing angle u 23 , with tan b , m trm b , has been considered in Ref. w6x. The net effect is that sin2 2 u 23 increases by about 40% from MGUT to MZ . In view of the lack of a compelling alternative theoretical framework, we will assume the hierarchy m1 < m 2 < m 3 . We will thus restrict m 2 in the range 1.7 = 10y3 eV Q m 2 Q 3.5 = 10y3 eV, as allowed by the small angle MSW solution w8x. We now recall a few salient features of the inflationary scenario associated with the breaking SUŽ2.R = UŽ1.By L ™ UŽ1. Y . As the inflaton Ž SUŽ2.R doublets f , f . oscillates about its minimum, it decays into the appropriate ‘matter’ right handed neutrino Ž n c . via the effective superpotential coupling n cn cff permitted by the gauge symmetry. The ‘reheat’ temperature TR is then related w3x to the mass MH of the heaviest right handed neutrino the inflaton can decay into: TR , MH r9.2. The inflaton mass is given by m infl , 3.4 = 10 13 GeV. If M2 , M3 are smaller than m inflr2, the inflaton decays predominantly into the heaviest of the two. Then, Eq. Ž12. and the cosmological bound m 3 Q 23 eV 1 require the smallest allowed

1

For the consistency of the CHDM model, the Hubble constant is about 50 kmsy1 Mpcy1 .

G. Lazarides et al.r Physics Letters B 427 (1998) 53–58

56

mass of the heaviest right handed neutrino to be , 9.4 = 10 10 GeV giving TR R 10 10 GeV, in clear conflict with the gravitino constraint. Consequently, we find that m inflr2 F M3 Q 2.5 = 10 13 GeV,

Ž 15 .

where the upper bound comes from the requirement that the coupling constant of the non-renormalizable superpotential term n cn cff , which provides mass for the right handed neutrinos, should not exceed unity. Thus, M3 is restricted in a narrow range, and the inflaton decays into the second heaviest right handed neutrino. The lepton asymmetry is generated by the subsequent decay of this neutrino and is given by w4x nL

sy

s

9 TR

X

X

X

X

2

M2 Im Ž UM D M D †U † . 23

8p m infl M3 Õ 2 Ž UM D M D †U † . 22

,

Ž 16 . where Õ is the electroweak VEV at MGU T . Substituting U from Eq. Ž5., we get nL s

s

9 TR

M2 c 2 s 2 sin2 d Ž m 3D 2 y m 2D 2 .

8p m infl M3

Õ 2 Ž m 3D 2 s 2 q m 2D 2 c 2 .

2

.

Ž 17 . Here we can again replace m 2D , m 3D by their ‘asymptotic’ values divided by 1.56 and Õ by 174

GeV. Assuming the MSSM spectrum between 1 TeV and MGU T , the observed baryon asymmetry n Brs is related w9x to n Lrs by n Brs s y28r79 Ž n Lrs .. We take a fixed value of M3 in the allowed range Ž15. and, for any pair of values Ž m 2 , m 3 ., we calculate M2 and TR . The constraint TR F 10 9 GeV yields a lower bound for the product m 2 m 3 excluding the region below a hyperbola on the m 2 ,m 3 plot. Note that the gravitino constraint combined with the MSW restriction on m 2 yields a lower bound for m 3 . Inside the allowed m 2 ,m 3 region, we can use the trace condition Ž13. to solve for d Ž u . in the interval 0 F u F pr2. Given that we need a negative value for n Lrs so that n Brs ) 0, we see that there is at most one useful branch of the function d Ž u . taken to lie in the region ypr2F d Ž u . F 0. The expression for d Ž u . is subsequently substituted in Eq. Ž17. for the leptonic asymmetry, and the range of u satisfying the constraint 0.02 Q V B h 2 Q 0.03 is found. ŽThis constraint is consistent with the low deuterium abundance as well as with structure formation in ‘cold’ plus ‘hot’ dark matter models.. The m 2 ,m 3 pairs, for which this range of u exists, are consistent with the observed baryon asymmetry. In Fig. 1, we depict the areas on the m 2 ,m 3 plane which satisfy both the gravitino and baryogenesis constraints in the two extreme cases of M3 s m inflr2 Žbounded by the thick solid line. and M3 s 2.5 = 10 13 GeV Žbounded

Fig. 1. The allowed regions in the mnm,mnt plane.

G. Lazarides et al.r Physics Letters B 427 (1998) 53–58

Fig. 2. The allowed regions in the nm – nt oscillation plot.

by the thick dashed line.. The lower line, in both cases, corresponds to the gravitino constraint whereas the upper one comes from the baryogenesis constraint. Note that the baryogenesis constraint at m 2 s 1.7 = 10y3 eV provides us with an upper bound for m 3 . In summary, we get an interesting upper as well as a lower bound on m 3 Ži.e. on V H D M !., and, for each m 3 value, we know the allowed range of m 2 . Namely, for M3 s m inflr2, 1.3 eV Q m 3 Q 8.8

57

eV whereas, for M3 s 2.5 = 10 13 GeV, 0.9 eV Q m 3 Q 5.1 eV. Thus, mnt is restricted in the range of 1 to 9 eV. The discussion above can be extended to yield useful information for umt . For each allowed pair m 2 ,m 3 and, for every value of u in the allowed range, we construct w and e in Eq. Ž8.. The phases a 2 and a 3 in Eq. Ž5. remain undetermined by the conditions Ž12., Ž13. and, consequently, b 2 , b 3 in Eq. Ž8. and j in Eq. Ž10. remain also undetermined. This fact does not allow us to predict the value of umt for each value of u , but only its allowed range given in Eq. Ž11.. The union of all these intervals, comprising all allowed values of u and m 2 for a given m 3 , constitutes the allowed range of umt for this m 3 . These ranges, after taking the renormalization of umt into account, are depicted in Fig. 2 for all possible values of m 3 , and constitute the allowed area of the neutrino oscillation parameters. The region bounded by the thick solid line is the allowed area for M3 s m inflr2, whereas the one bounded by the thick dashed line corresponds to M3 s 2.5 = 10 13 GeV. The areas tested Žto be tested. by past Žfuture. experiments are also indicated in Fig. 2. The area excluded by E531 is depicted in Fig. 1 and lies above the thin solid Ždashed. E531 line, for M3 s m inflr2 Ž2.5 = 10 13 GeV.. CDHS does not appear to have any appreciable effect. Furthermore, if CHORUS gives negative

Fig. 3. The function dmt Ž umt . for M3 s m inflr2 and ‘central’ values of m 2 ,m 3 ,n L rs.

58

G. Lazarides et al.r Physics Letters B 427 (1998) 53–58

results, we must further exclude the area above the corresponding thin solid Ždashed. line in Fig. 1, for M3 s m inflr2 Ž2.5 = 10 13 GeV.. A possibly negative CHORUS result implies that the upper bound for mnt drops down to , 3.7 eV. Notice that the upper limit on baryon asymmetry has no effect on Figs. 1 and 2. The CP violating phase dmt as a function of umt , for given values of M3 ,m 2 ,m 3 and V B h 2 , can now be constructed. We choose M3 s m inflr2, m 2 s 2.6 = 10y3 eV, m 3 s 4 eV, V B h 2 s 0.025 and solve for u . We find two solutions and, for each one of them and any j Žyp F j y e F p ., we calculate umt and dmt s j q r y l. Eliminating j , we obtain the function dmt Ž umt . in the region of Eq. Ž11.. This function turns out to be double valued-the sum of the two branches equals 2 e- and is depicted in Fig. 3, for both values of u , after renormalizing umt . The umt ’s excluded by E531, for mnt s 4 eV, are also indicated and lie to the right of the E531 line. In conclusion, we find that a modest extension of MSSM to SUŽ3.c = SUŽ2.L = SUŽ2.R = UŽ1.ByL can yield significant new results by tying together a number of apparently unrelated phenomena. In particular, inflation can be realized, the spectral index n , 0.98, we get both ‘cold’ Žessentially bino. and ‘hot’ Ž nt . dark matter, while the nt mass and nm – nt mixing is within reach of present and planned experiments. In the simplest scheme, the atmospheric neutrino anomaly remains a mystery.

Acknowledgements We thank K.S. Babu for several discussions regarding the renormalization effects, and E. Tsesmelis for information about the experiments. TMR and NATO support under grant numbers ERBFMRXCT960090 and CRG 970149 is gratefully acknowledged.

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