- Email: [email protected]
Atmospheric neutrino anomaly and supersymmetric inflation
26 November 1998
Physics Letters B 441 Ž1998. 46–51
Atmospheric neutrino anomaly and supersymmetric inflation G. Lazarides a
a,1
, N.D. Vlachos
b,2
Physics DiÕision, School of Technology, Aristotle UniÕersity of Thessaloniki, Thessaloniki GR 540 06, Greece b Department of Physics, Aristotle UniÕersity of Thessaloniki, Thessaloniki GR 540 06, Greece Received 12 July 1998 Editor: R. Gatto
Abstract A detailed investigation of hybrid inflation and the subsequent reheating process is performed within a m-problem solving supersymmetric model based on a left-right symmetric gauge group. The process of baryogenesis via leptogenesis is especially studied. For nm , nt masses from the small angle MSW resolution of the solar neutrino problem and the recent results of the SuperKamiokande experiment, we show that maximal nm-nt mixing can be achieved. The required value of the relevant coupling constant is, however, quite small Ž; 10y6 .. q 1998 Elsevier Science B.V. All rights reserved.
The hybrid inflationary scenario w1x can be easily implemented w2–4x in the context of supersymmetric theories in a ‘natural’ way meaning that a. there is no need for tiny coupling constants, b. the superpotential used is the most general one allowed by gauge and R- symmetries, c. supersymmetry guarantees that radiative corrections do not invalidate inflation, but rather provide a slope along the inflationary trajectory which drives the inflaton towards the supersymmetric vacua, and d. supergravity corrections can be brought under control so as to leave inflation intact. A moderate extension of the minimal supersymmetric standard model ŽMSSM. based on a left-right symmetric gauge group provides w4x a suitable framework for hybrid inflation. The inflaton is associated with the breaking of SUŽ2.R and consists of a
1 2
E-mail: [email protected] E-mail: [email protected]
gauge singlet and a pair of SUŽ2.R doublets. The doublets can decay into right handed neutrinos, after inflation, reheating the universe and providing a mechanism w5x for baryogenesis through a primordial leptogenesis. The gauge singlet, however, has no direct coupling to light matter in the simplest case. Moreover, its coupling to the SUŽ2.R doublets turns out to be unable to ensure its efficient decay. This difficulty can be overcome by introducing w4,6x a direct superpotential coupling of the gauge singlet superfield to the electroweak higgs doublets. This way the gauge singlet scalar can decay into a pair of higgsinos. It has been shown w6x that, in the presence of gravity-mediated supersymmetry breaking, this gauge singlet acquires a vacuum expectation value Žvev. and consequently generates, through its coupling to the ordinary higgs superfields, the m term of MSSM. A coupling of the scalar components of the SUŽ2.R doublets to the electroweak higgses is automatically induced in this scheme, allowing them to decay into a pair of ordinary higgses in addition to
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 1 2 6 - 5
G. Lazarides, N.D. Vlachosr Physics Letters B 441 (1998) 46–51
their useful Žfor baryogenesis. decay to right handed neutrinos. In this paper, we attempt a detailed study of inflation in the above scheme. In particular, we solve the evolution equations of this system and estimate the reheating temperature. The process of baryogenesis via leptogenesis is also considered and its consequences on nm-nt mixing are analyzed. For masses of nm , nt which are consistent with the small angle MSW resolution of the solar neutrino problem and the recent results of the SuperKamiokande experiment w7x, we examine whether maximal nm-nt mixing can be achieved. Let us first describe the main features of the GL R s SUŽ3.c = SUŽ2.R = SUŽ2.L = UŽ1.ByL symmetric model w6x which solves the m problem. The SUŽ2.R = UŽ1.ByL group is broken by a pair of SUŽ2.R doublet chiral superfields l c, l c which acquire a vev M )) m 3r2 ; Ž0.1 y 1. TeV, the gravitino mass. This breaking is achieved by means of a gauge singlet chiral superfield S which plays a crucial three-fold role: 1. it triggers SUŽ2.R breaking; 2. it generates the m term of MSSM after gravity-mediated supersymmetry breaking; and 3. it leads to hybrid inflation w1x. Ignoring the matter fields of the model, the superpotential reads W s S Ž k l c l c q l h2 y k M 2 . ,
constant tree level vacuum energy density Vtree s k 2 M 4 , which is responsible for inflation. Radiative corrections generate a logarithmic slope w3x along the inflationary trajectory that drives the inflaton toward the minimum. The one-loop contribution to this slope comes from the l c, l c and h supermultiplets, which receive at tree level a non-supersymmetric contribution to the masses of their scalar components from the FS term. For < S < F Sc s M, the l c, l c components become tachyonic, compensate the FS term and the system evolves towards the ‘correct’ supersymmetric minimum at h s 0, l c s l c s M. ŽFor k ) l, h would have become tachyonic earlier and the system would have evolved towards the ‘wrong’ minimum at h / 0, l c s l c s 0.. Inflation can continue at least till < S < approaches the instability at < S < s Sc provided that the slow roll conditions w3,8x are violated only ‘infinitesimally’ close to it. This is true for all values of the relevant parameters considered in this work. The cosmic microwave quadrupole anisotropy can be calculated w3x by standard methods and turns out to be 32p 5r2
dT
ž / T
f
Ž2. .
where the chiral superfield h s Ž h ,h belongs to a bidoublet Ž2,2. 0 representation of SUŽ2.L = SUŽ2.R = UŽ1.By L , and h 2 denotes the unique bilinear inŽ2. variant e i j hŽ1. i h j . Note that the parameters k , l and M can be made positive through a suitable redefinition of the superfields. W in Eq. Ž1. has the most general renormalizable form invariant under the gauge group and a continuous UŽ1. R-symmetry under which S carries the same charge as W, while h, l c, l c are neutral. This symmetry is extended w6x to include the matter fields of the model too and implies automatic baryon number conservation. It has been shown w6x that, after supersymmetry breaking, S develops a vev ² S : f ym 3r2rk which generates a m term with m s l² S : f yŽ lrk . m 3r2 . The model has a built-in inflationary trajectory in the field space along which the FS term is constant w3,4x. This trajectory is parametrized by < S <, < S < ) Sc s M for l ) k Žsee below.. All other fields vanish on this trajectory. The FS term provides us with a
3'5
Q
M
3
ž /
ky1 xy1 Q LŽ x Q .
MP
y1
,
Ž 2.
where MP s 1.22 = 10 19 GeV is the Planck scale and
Ž 1. Ž1.
47
l
LŽ x . s
3
l
x 2 y 1 ln 1 y
k
/ ž /ž /ž / ž /ž k
k
l
q
k
x 2 q 1 ln 1 q
k
l
l
xy2
xy2
q Ž x 2 y 1 . ln Ž 1 y xy2 . q Ž x 2 q 1 . ln Ž 1 q xy2 . ,
Ž 3.
with x s < S
3
M
ž / MP
2
ky2
H1x
2 Q
dx 2 x2
LŽ x .
y1
.
Ž 4.
The spectral index of density perturbations turns out to be very close to unity.
G. Lazarides, N.D. Vlachosr Physics Letters B 441 (1998) 46–51
48
After reaching the instability at < S < s Sc , the system undergoes w9x a short complicated evolution during which inflation continues for another e-folding or so. The energy density of the system is reduced by a factor of about 2-3 during this period. The system then rapidly settles in a regular oscillatory phase about the supersymmetric vacuum. Parametric resonance can be ignored in this case w9x. The inflaton Žoscillating system. consists of the two complex scalar fields S and u s Ž df q df .r '2 , where df s f y M, df s f y M, with mass m i n f l s '2 k M. Here f , f are the neutral components of the superfields l c, l c respectively. The scalar fields S and u predominantly decay into ordinary higgsinos and higgses respectively with a common decay width G h s Ž1r16p . l2 m i n f l , as one can easily deduce from the couplings in Eq. Ž1.. Note, however, that u can also decay to right handed neutrinos n c through the non-renormalizable superpotential term Ž Mn c r 2 M 2 . ffn cn c, allowed by the gauge and R- symmetries of the model w4x. Here, Mn c denotes the Majorana mass of the relevant n c. The scalar u decays preferably into the heaviest n c with Mn c F m i n f lr2. The decay rate is given by
Gn c f
1 16p
k 2 min f l a 2 Ž 1 y a 2 .
1r2
,
Ž 5.
where 0 F a s 2 Mn crm i n f l F 1. The subsequent decay of these n c ’s gives rise to a primordial lepton number w5x. The baryon asymmetry of the universe can then be obtained by partial conversion of this lepton asymmetry through sphaleron effects. The energy densities rS , ru , and rr of the oscillating fields S, u , and the ‘new’ radiation produced by their decay to higgsinos, higgses and n c ’s are controlled by the equations:
r˙S s y Ž 3 H q G h . rS , r˙u s y Ž 3 H q G h q Gn c . ru , Ž 6. r˙r s y4H rr q G h rS q Ž G h q Gn c . ru ,
Ž 7.
'8p 1r2 '3 MP Ž rS q ru q rr . ,
ru Ž t . s rS Ž t . eyGn c Ž tyt 0 . ,
p2 rS q ru s rr s
30
g ) Tr4 ,
Ž 10 .
where the effective number of massless degrees of freedom is g ) s 228.75 for MSSM. The lepton number density n L produced by the n c ’s satisfies the evolution equation: n˙ L s y3Hn L q 2 eGn c nu ,
Ž 11 .
where e is the lepton number produced per decaying right handed neutrino and the factor of 2 in the second term of the rhs comes from the fact that we get two n c ’s for each decaying scalar u particle. Eq. Ž11. is easily integrated out to give n LŽ t . ; nu Ž t 0 .
aŽ t .
y3
ž /
2 eGn c
G h q Gn c
aŽ t0 .
, as t ™ ` ,
Ž 12 . where aŽ t . is the scale factor of the universe. The first equation in Eq. Ž6. gives
rS Ž t . s rS Ž t 0 .
aŽ t .
ž / aŽ t0 .
y3
eyG hŽ tyt 0 . .
Ž 13 .
Combining Eqs. Ž12. and Ž13. we get the asymptotic Ž t ™ `. lepton asymmetry
sŽ t .
Ž 8.
Ž 9.
where t 0 is the cosmic time at the onset of the oscillatory phase. The initial values are taken to be rS Ž t 0 . s ru Ž t 0 . f k 2 M 4r6, rr Ž t 0 . s 0 and, for all practical purposes, we put t 0 s 0. The ‘reheat’ temperature Tr is calculated from the equation
n LŽ t .
where Hs
is the Hubble parameter and overdots denote derivatives with respect to cosmic time t. We have assumed that the potential energy density is, to a good approximation, quadratic in the fields S and u and, thus, the oscillating inflaton system resembles the behavior of ‘matter’. Note that the second equation in Eq. Ž6. can be replaced by
;3
15
1r4
ž /
=
8
py1r2 gy1r4 my1 ) in f l
eGn c G h q Gn c
ry3r4 rS e G h t , r
Ž 14 .
G. Lazarides, N.D. Vlachosr Physics Letters B 441 (1998) 46–51
where sŽ t . ;
2p 2 g ) 45
30
3r4
ž /
rr3r4
p 2 g)
Ž 15 .
is the asymptotic entropy density. For MSSM spectrum between 100 GeV and M, the observed baryon asymmetry n Brs is related w10x to n Lrs by n Brs s yŽ28r79.Ž n Lrs .. It is important to ensure that the primordial lepton asymmetry is not erased by lepton number violating 2 ™ 2 scatterings at all temperatures between Tr and 100 GeV. This requirement gives w10x mnt Q 10 eV which is readily satisfied in our case Žsee below.. Assuming hierarchical light neutrino masses, we take mnm f 2.6 = 10y3 eV which is the central value of the m-neutrino mass coming from the small angle MSW resolution of the solar neutrino problem w11x. The t-neutrino mass will be restricted by the atmospheric anomaly w7x in the range 3 = 10y2 eV Q mnt Q 11 = 10y2 eV. Recent analysis w12x of the results of the CHOOZ experiment w13x shows that the oscillations of solar and atmospheric neutrinos decouple. We thus concentrate on the two heaviest families ignoring the first one. Under these circumstances, the lepton number generated per decaying n c is w8,14x
es
1 8p
g
M3
ž / M2
c 2 s 2 sin2 d <² h
Ž1. :< 2
Ž
Ž
2 m 3D 2 y m 2D 2
m 3D 2
.
s
2
q m 2D 2
c2 .
,
Ž 16 . where g Ž r . s r lnŽ1 q ry2 . , <² hŽ1. :< f 174 GeV, c s cos u , s s sin u , and u Ž0 F u F pr2. and d Žypr2F d - pr2. are the rotation angle and phase which diagonalize the Majorana mass matrix of n c ’s, M R , with eigenvalues M2 , M3 ŽG 0. in the basis where the ‘Dirac’ mass matrix of the neutrinos, M D , is diagonal with eigenvalues m 2D , m 3D ŽG 0.. Note that, for the range of parameters considered here, the scalar u decays into the second heaviest right handed neutrino with mass M2 Ž- M3 . and, thus, Mn c in Eq. Ž5. should be identified with M2 . Moreover, M3 turns out to be bigger than m i n f lr2 as it should. We will denote the two positive eigenvalues of the light neutrino mass matrix by m 2 Žs mnm ., m 3 Žs mnt . with m 2 F m 3 . All the quantities here Žmasses, rotation angles and phases. are ‘asymptotic’ Ždefined at the grand unification scale MGU T .. The determinant
49
and the trace invariance of the light neutrino mass matrix imply w14x 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 17 .
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
P
M2 M3
Ž 18 . The m-t mixing angle u 23 Žs umt . lies w14x in the range < w y u D < F u 23 F w q u D , for w q u D F pr2 , < w y u D < F u 23 F p y w y u D , for w q u D G pr2 , Ž 19 . where w Ž0 F w F pr2. is the rotation angle which diagonalizes the light neutrino mass matrix, m s yM˜ D M Ry1 M D , in the basis where the ‘Dirac’ mass matrix is diagonal and u D Ž0 F u D F pr2. is the ‘Dirac’ mixing angle in the 2-3 leptonic sector Ži.e., the ‘unphysical’ leptonic mixing angle in the absence of the Majorana masses of the n c ’s.. The ‘asymptotic’ Dirac masses of nm , nt as well as u D can be related to the quark sector parameters by assuming approximate SUŽ4.c symmetry. We obtain the asymptotic relations: m 2D f m c , m 3D f m t , sin u D f < Vcb < . Renormalization effects must now be taken into account. To this end, we take MSSM spectrum and large tan b f m trm b . The latter follows from the fact that the MSSM higgs doublets form a SUŽ2.R doublet. It turns out w14x that, in this case, renormalization effects can be accounted for by simply substituting in the above formulae the following numerical values: m 2D f 0.23 GeV, m 3D f 116 GeV and sin u D f 0.03. Also, tan2 2 u 23 increases by about 40% from MGU T to MZ . In order to predict the nm-nt mixing, we take a specific MSSM framework w15x where the three
50
G. Lazarides, N.D. Vlachosr Physics Letters B 441 (1998) 46–51
Yukawa couplings of the third generation unify ‘asymptotically’ and, consequently, tan b f m trm b . We choose the universal scalar mass Žgravitino mass. m 3r2 f 290 GeV and the universal gaugino mass M1r2 f 470 GeV. These values correspond w16x to m t Ž m t . f 166 GeV and m A Žthe tree level mass of the CP-odd scalar higgs boson. s MZ . The ‘asymptotic’ higgsino mass m is related w17x to these parameters by < m
Fig. 1. The mass scale M Žsolid line. and the reheat temperature Tr Ždashed line. as functions of k .
Fig. 2. The allowed region Žbounded by the solid lines. in the k-sin2 2 umt plane for mnmf 2.6=10y3 eV and mntf 7=10y2 eV.
the second heaviest n c, into which the scalar u decays partially, is given by M2 s Mn c s a m i n f lr2 and M3 can be found from Eq. Ž17.. We can use the trace condition in Eq. Ž18. to solve for d Ž u . in the interval 0 F u F pr2. The expression for d Ž u . is subsequently substituted in Eq. Ž16. for e . The leptonic asymmetry as a function of the angle u can be found from Eq. Ž14.. To each value of k correspond two values of the angle u satisfying the low deuterium abundance constraint V B h 2 f 0.025. ŽThese values of u turn out to be quite insensitive to the exact value of n Brs.. The corresponding values of the rotation angle w , which diagonalizes the light neutrino mass matrix, are then found and the allowed region of the mixing angle umt in Eq. Ž19. is determined. Taking into account renormalization effects and superimposing all the permitted regions, we obtain the allowed range of sin2 2 umt as a function of k , shown in Fig. 2. We observe that maximal mixing Žsin2 2 umt f 1. is achieved for 1.5 = 10y6 Q k Q 1.8 = 10y6 . Also, sin2 2 umt R 0.8 w7x corresponds to 1.2 = 10y6 Q k Q 3.4 = 10y6 . The analysis above can be repeated for all values of mnt allowed by SuperKamiokande. The allowed regions in the mnt-k plane for maximal nm -nt mixing Žbounded by the solid lines. and sin2 2 umt R 0.8 Žbounded by the dotted lines. are shown in Fig. 3. Notice that, for sin2 2 umt R 0.8, k f Ž0.9 y 7.5. = 10y6 which is rather small. ŽFortunately, supersymmetry protects this coupling from radiative corrections.. The corresponding values of M and Tr can be
G. Lazarides, N.D. Vlachosr Physics Letters B 441 (1998) 46–51
51
The required value of the coupling constant k is, however, quite small Ž; 10y6 .. We would like to thank B. Ananthanarayan and C. Pallis for useful discussions. This work was supported by the research grant PENEDr95 K.A.1795.
References
Fig. 3. The regions on the mnt-k plane corresponding to maximal nm-nt mixing Žbounded by the solid lines. and sin2 2 umt R 0.8 Žbounded by the dotted lines.. Here we consider the range 3= 10y2 eV Q mnt Q11=10y2 eV Ž mnmf 2.6=10y3 eV..
read from Fig. 1. We find 1.3 = 10 15 GeV Q M Q 2.7 = 10 15 GeV and 10 7 GeV Q Tr Q 3.2 = 10 8 GeV. We observe that M turns out to be somewhat smaller than the MSSM unification scale MGU T . ŽIt is anticipated that GL R is embedded in a grand unified theory.. The reheat temperature, however, satisfies the gravitino constraint ŽTr Q 10 9 GeV.. It should be noted that, for the values of the parameters chosen here, the lightest supersymmetric particle ŽLSP. is w16x an almost pure bino with mass m LS P f 0.43 M1r2 f 200 GeV w18x. Its contribution to the mass of the universe turns out w18x to be V LS P h 2 f 1. In summary, we have investigated hybrid inflation and the subsequent reheating process in the framework of a m-problem solving supersymmetric model based on a left-right symmetric gauge group. The process of baryogenesis via leptogenesis is especially considered. For masses of nm , nt consistent with the small angle MSW resolution of the solar neutrino problem and the recent SuperKamiokande data, we showed that maximal nm-nt mixing can be achieved.
w1x A.D. Linde, Phys. Lett. B 259 Ž1991. 38; Phys. Rev. D 49 Ž1994. 748. w2x E.J. Copeland, A.R. Liddle, D.H. Lyth, E.D. Stewart, D. Wands, Phys. Rev. D 49 Ž1994. 6410. w3x G. Dvali, Q. Shafi, R. Schaefer, Phys. Rev. Lett. 73 Ž1994. 1886. w4x G. Lazarides, R. Schaefer, Q. Shafi, Phys. Rev. D 56 Ž1997. 1324. w5x M. Fukugita, T. Yanagida, Phys. Lett. B 174 Ž1986. 45; W. Buchmuller, M. Plumacher, Phys. Lett. B 389 Ž1996. 73; in ¨ ¨ the context of inflation see G. Lazarides, Q. Shafi, Phys. Lett. B 258 Ž1991. 305. w6x G. Dvali, G. Lazarides, Q. Shafi, Phys. Lett. B 424 Ž1998. 259. w7x T. Kajiata, talk given at the XVIIIth International Conference on Neutrino Physics and Astrophysics ŽNeutrino’98., Takayama, Japan, 4–9 June, 1998. w8x G. Lazarides, hep-phr9802415 Žto appear in the proceedings of the 6th BCSPIN Summer School.. ´ w9x J. Garcia-Bellido, A. Linde, Phys. Rev. D 57 Ž1998. 6075. w10x L.E. Ibanez, ´˜ F. Quevedo, Phys. Lett. B 283 Ž1992. 261. w11x A. Smirnov, hep-phr9611465, and references therein. w12x C. Giunti, hep-phr9802201. w13x M. Apollonio et al., Phys. Lett. B 420 Ž1998. 397. w14x G. Lazarides, Q. Shafi, N.D. Vlachos, Phys. Lett. B 427 Ž1998. 53. w15x B. Ananthanarayan, G. Lazarides, Q. Shafi, Phys. Rev. D 44 Ž1991. 1613. w16x B. Ananthanarayan, Q. Shafi, X.M. Wang, Phys. Rev. D 50 Ž1994. 5980. w17x M. Carena, M. Olechowski, S. Pokorski, C. Wagner, Nucl. Phys. B 426 Ž1994. 269. w18x M. Drees, M. Nojiri, Phys. Rev. D 47 Ž1993. 376.
Physics Letters B 441 Ž1998. 46–51
Atmospheric neutrino anomaly and supersymmetric inflation G. Lazarides a
a,1
, N.D. Vlachos
b,2
Physics DiÕision, School of Technology, Aristotle UniÕersity of Thessaloniki, Thessaloniki GR 540 06, Greece b Department of Physics, Aristotle UniÕersity of Thessaloniki, Thessaloniki GR 540 06, Greece Received 12 July 1998 Editor: R. Gatto
Abstract A detailed investigation of hybrid inflation and the subsequent reheating process is performed within a m-problem solving supersymmetric model based on a left-right symmetric gauge group. The process of baryogenesis via leptogenesis is especially studied. For nm , nt masses from the small angle MSW resolution of the solar neutrino problem and the recent results of the SuperKamiokande experiment, we show that maximal nm-nt mixing can be achieved. The required value of the relevant coupling constant is, however, quite small Ž; 10y6 .. q 1998 Elsevier Science B.V. All rights reserved.
The hybrid inflationary scenario w1x can be easily implemented w2–4x in the context of supersymmetric theories in a ‘natural’ way meaning that a. there is no need for tiny coupling constants, b. the superpotential used is the most general one allowed by gauge and R- symmetries, c. supersymmetry guarantees that radiative corrections do not invalidate inflation, but rather provide a slope along the inflationary trajectory which drives the inflaton towards the supersymmetric vacua, and d. supergravity corrections can be brought under control so as to leave inflation intact. A moderate extension of the minimal supersymmetric standard model ŽMSSM. based on a left-right symmetric gauge group provides w4x a suitable framework for hybrid inflation. The inflaton is associated with the breaking of SUŽ2.R and consists of a
1 2
E-mail: [email protected] E-mail: [email protected]
gauge singlet and a pair of SUŽ2.R doublets. The doublets can decay into right handed neutrinos, after inflation, reheating the universe and providing a mechanism w5x for baryogenesis through a primordial leptogenesis. The gauge singlet, however, has no direct coupling to light matter in the simplest case. Moreover, its coupling to the SUŽ2.R doublets turns out to be unable to ensure its efficient decay. This difficulty can be overcome by introducing w4,6x a direct superpotential coupling of the gauge singlet superfield to the electroweak higgs doublets. This way the gauge singlet scalar can decay into a pair of higgsinos. It has been shown w6x that, in the presence of gravity-mediated supersymmetry breaking, this gauge singlet acquires a vacuum expectation value Žvev. and consequently generates, through its coupling to the ordinary higgs superfields, the m term of MSSM. A coupling of the scalar components of the SUŽ2.R doublets to the electroweak higgses is automatically induced in this scheme, allowing them to decay into a pair of ordinary higgses in addition to
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 1 2 6 - 5
G. Lazarides, N.D. Vlachosr Physics Letters B 441 (1998) 46–51
their useful Žfor baryogenesis. decay to right handed neutrinos. In this paper, we attempt a detailed study of inflation in the above scheme. In particular, we solve the evolution equations of this system and estimate the reheating temperature. The process of baryogenesis via leptogenesis is also considered and its consequences on nm-nt mixing are analyzed. For masses of nm , nt which are consistent with the small angle MSW resolution of the solar neutrino problem and the recent results of the SuperKamiokande experiment w7x, we examine whether maximal nm-nt mixing can be achieved. Let us first describe the main features of the GL R s SUŽ3.c = SUŽ2.R = SUŽ2.L = UŽ1.ByL symmetric model w6x which solves the m problem. The SUŽ2.R = UŽ1.ByL group is broken by a pair of SUŽ2.R doublet chiral superfields l c, l c which acquire a vev M )) m 3r2 ; Ž0.1 y 1. TeV, the gravitino mass. This breaking is achieved by means of a gauge singlet chiral superfield S which plays a crucial three-fold role: 1. it triggers SUŽ2.R breaking; 2. it generates the m term of MSSM after gravity-mediated supersymmetry breaking; and 3. it leads to hybrid inflation w1x. Ignoring the matter fields of the model, the superpotential reads W s S Ž k l c l c q l h2 y k M 2 . ,
constant tree level vacuum energy density Vtree s k 2 M 4 , which is responsible for inflation. Radiative corrections generate a logarithmic slope w3x along the inflationary trajectory that drives the inflaton toward the minimum. The one-loop contribution to this slope comes from the l c, l c and h supermultiplets, which receive at tree level a non-supersymmetric contribution to the masses of their scalar components from the FS term. For < S < F Sc s M, the l c, l c components become tachyonic, compensate the FS term and the system evolves towards the ‘correct’ supersymmetric minimum at h s 0, l c s l c s M. ŽFor k ) l, h would have become tachyonic earlier and the system would have evolved towards the ‘wrong’ minimum at h / 0, l c s l c s 0.. Inflation can continue at least till < S < approaches the instability at < S < s Sc provided that the slow roll conditions w3,8x are violated only ‘infinitesimally’ close to it. This is true for all values of the relevant parameters considered in this work. The cosmic microwave quadrupole anisotropy can be calculated w3x by standard methods and turns out to be 32p 5r2
dT
ž / T
f
Ž2. .
where the chiral superfield h s Ž h ,h belongs to a bidoublet Ž2,2. 0 representation of SUŽ2.L = SUŽ2.R = UŽ1.By L , and h 2 denotes the unique bilinear inŽ2. variant e i j hŽ1. i h j . Note that the parameters k , l and M can be made positive through a suitable redefinition of the superfields. W in Eq. Ž1. has the most general renormalizable form invariant under the gauge group and a continuous UŽ1. R-symmetry under which S carries the same charge as W, while h, l c, l c are neutral. This symmetry is extended w6x to include the matter fields of the model too and implies automatic baryon number conservation. It has been shown w6x that, after supersymmetry breaking, S develops a vev ² S : f ym 3r2rk which generates a m term with m s l² S : f yŽ lrk . m 3r2 . The model has a built-in inflationary trajectory in the field space along which the FS term is constant w3,4x. This trajectory is parametrized by < S <, < S < ) Sc s M for l ) k Žsee below.. All other fields vanish on this trajectory. The FS term provides us with a
3'5
Q
M
3
ž /
ky1 xy1 Q LŽ x Q .
MP
y1
,
Ž 2.
where MP s 1.22 = 10 19 GeV is the Planck scale and
Ž 1. Ž1.
47
l
LŽ x . s
3
l
x 2 y 1 ln 1 y
k
/ ž /ž /ž / ž /ž k
k
l
q
k
x 2 q 1 ln 1 q
k
l
l
xy2
xy2
q Ž x 2 y 1 . ln Ž 1 y xy2 . q Ž x 2 q 1 . ln Ž 1 q xy2 . ,
Ž 3.
with x s < S
3
M
ž / MP
2
ky2
H1x
2 Q
dx 2 x2
LŽ x .
y1
.
Ž 4.
The spectral index of density perturbations turns out to be very close to unity.
G. Lazarides, N.D. Vlachosr Physics Letters B 441 (1998) 46–51
48
After reaching the instability at < S < s Sc , the system undergoes w9x a short complicated evolution during which inflation continues for another e-folding or so. The energy density of the system is reduced by a factor of about 2-3 during this period. The system then rapidly settles in a regular oscillatory phase about the supersymmetric vacuum. Parametric resonance can be ignored in this case w9x. The inflaton Žoscillating system. consists of the two complex scalar fields S and u s Ž df q df .r '2 , where df s f y M, df s f y M, with mass m i n f l s '2 k M. Here f , f are the neutral components of the superfields l c, l c respectively. The scalar fields S and u predominantly decay into ordinary higgsinos and higgses respectively with a common decay width G h s Ž1r16p . l2 m i n f l , as one can easily deduce from the couplings in Eq. Ž1.. Note, however, that u can also decay to right handed neutrinos n c through the non-renormalizable superpotential term Ž Mn c r 2 M 2 . ffn cn c, allowed by the gauge and R- symmetries of the model w4x. Here, Mn c denotes the Majorana mass of the relevant n c. The scalar u decays preferably into the heaviest n c with Mn c F m i n f lr2. The decay rate is given by
Gn c f
1 16p
k 2 min f l a 2 Ž 1 y a 2 .
1r2
,
Ž 5.
where 0 F a s 2 Mn crm i n f l F 1. The subsequent decay of these n c ’s gives rise to a primordial lepton number w5x. The baryon asymmetry of the universe can then be obtained by partial conversion of this lepton asymmetry through sphaleron effects. The energy densities rS , ru , and rr of the oscillating fields S, u , and the ‘new’ radiation produced by their decay to higgsinos, higgses and n c ’s are controlled by the equations:
r˙S s y Ž 3 H q G h . rS , r˙u s y Ž 3 H q G h q Gn c . ru , Ž 6. r˙r s y4H rr q G h rS q Ž G h q Gn c . ru ,
Ž 7.
'8p 1r2 '3 MP Ž rS q ru q rr . ,
ru Ž t . s rS Ž t . eyGn c Ž tyt 0 . ,
p2 rS q ru s rr s
30
g ) Tr4 ,
Ž 10 .
where the effective number of massless degrees of freedom is g ) s 228.75 for MSSM. The lepton number density n L produced by the n c ’s satisfies the evolution equation: n˙ L s y3Hn L q 2 eGn c nu ,
Ž 11 .
where e is the lepton number produced per decaying right handed neutrino and the factor of 2 in the second term of the rhs comes from the fact that we get two n c ’s for each decaying scalar u particle. Eq. Ž11. is easily integrated out to give n LŽ t . ; nu Ž t 0 .
aŽ t .
y3
ž /
2 eGn c
G h q Gn c
aŽ t0 .
, as t ™ ` ,
Ž 12 . where aŽ t . is the scale factor of the universe. The first equation in Eq. Ž6. gives
rS Ž t . s rS Ž t 0 .
aŽ t .
ž / aŽ t0 .
y3
eyG hŽ tyt 0 . .
Ž 13 .
Combining Eqs. Ž12. and Ž13. we get the asymptotic Ž t ™ `. lepton asymmetry
sŽ t .
Ž 8.
Ž 9.
where t 0 is the cosmic time at the onset of the oscillatory phase. The initial values are taken to be rS Ž t 0 . s ru Ž t 0 . f k 2 M 4r6, rr Ž t 0 . s 0 and, for all practical purposes, we put t 0 s 0. The ‘reheat’ temperature Tr is calculated from the equation
n LŽ t .
where Hs
is the Hubble parameter and overdots denote derivatives with respect to cosmic time t. We have assumed that the potential energy density is, to a good approximation, quadratic in the fields S and u and, thus, the oscillating inflaton system resembles the behavior of ‘matter’. Note that the second equation in Eq. Ž6. can be replaced by
;3
15
1r4
ž /
=
8
py1r2 gy1r4 my1 ) in f l
eGn c G h q Gn c
ry3r4 rS e G h t , r
Ž 14 .
G. Lazarides, N.D. Vlachosr Physics Letters B 441 (1998) 46–51
where sŽ t . ;
2p 2 g ) 45
30
3r4
ž /
rr3r4
p 2 g)
Ž 15 .
is the asymptotic entropy density. For MSSM spectrum between 100 GeV and M, the observed baryon asymmetry n Brs is related w10x to n Lrs by n Brs s yŽ28r79.Ž n Lrs .. It is important to ensure that the primordial lepton asymmetry is not erased by lepton number violating 2 ™ 2 scatterings at all temperatures between Tr and 100 GeV. This requirement gives w10x mnt Q 10 eV which is readily satisfied in our case Žsee below.. Assuming hierarchical light neutrino masses, we take mnm f 2.6 = 10y3 eV which is the central value of the m-neutrino mass coming from the small angle MSW resolution of the solar neutrino problem w11x. The t-neutrino mass will be restricted by the atmospheric anomaly w7x in the range 3 = 10y2 eV Q mnt Q 11 = 10y2 eV. Recent analysis w12x of the results of the CHOOZ experiment w13x shows that the oscillations of solar and atmospheric neutrinos decouple. We thus concentrate on the two heaviest families ignoring the first one. Under these circumstances, the lepton number generated per decaying n c is w8,14x
es
1 8p
g
M3
ž / M2
c 2 s 2 sin2 d <² h
Ž1. :< 2
Ž
Ž
2 m 3D 2 y m 2D 2
m 3D 2
.
s
2
q m 2D 2
c2 .
,
Ž 16 . where g Ž r . s r lnŽ1 q ry2 . , <² hŽ1. :< f 174 GeV, c s cos u , s s sin u , and u Ž0 F u F pr2. and d Žypr2F d - pr2. are the rotation angle and phase which diagonalize the Majorana mass matrix of n c ’s, M R , with eigenvalues M2 , M3 ŽG 0. in the basis where the ‘Dirac’ mass matrix of the neutrinos, M D , is diagonal with eigenvalues m 2D , m 3D ŽG 0.. Note that, for the range of parameters considered here, the scalar u decays into the second heaviest right handed neutrino with mass M2 Ž- M3 . and, thus, Mn c in Eq. Ž5. should be identified with M2 . Moreover, M3 turns out to be bigger than m i n f lr2 as it should. We will denote the two positive eigenvalues of the light neutrino mass matrix by m 2 Žs mnm ., m 3 Žs mnt . with m 2 F m 3 . All the quantities here Žmasses, rotation angles and phases. are ‘asymptotic’ Ždefined at the grand unification scale MGU T .. The determinant
49
and the trace invariance of the light neutrino mass matrix imply w14x 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 17 .
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
P
M2 M3
Ž 18 . The m-t mixing angle u 23 Žs umt . lies w14x in the range < w y u D < F u 23 F w q u D , for w q u D F pr2 , < w y u D < F u 23 F p y w y u D , for w q u D G pr2 , Ž 19 . where w Ž0 F w F pr2. is the rotation angle which diagonalizes the light neutrino mass matrix, m s yM˜ D M Ry1 M D , in the basis where the ‘Dirac’ mass matrix is diagonal and u D Ž0 F u D F pr2. is the ‘Dirac’ mixing angle in the 2-3 leptonic sector Ži.e., the ‘unphysical’ leptonic mixing angle in the absence of the Majorana masses of the n c ’s.. The ‘asymptotic’ Dirac masses of nm , nt as well as u D can be related to the quark sector parameters by assuming approximate SUŽ4.c symmetry. We obtain the asymptotic relations: m 2D f m c , m 3D f m t , sin u D f < Vcb < . Renormalization effects must now be taken into account. To this end, we take MSSM spectrum and large tan b f m trm b . The latter follows from the fact that the MSSM higgs doublets form a SUŽ2.R doublet. It turns out w14x that, in this case, renormalization effects can be accounted for by simply substituting in the above formulae the following numerical values: m 2D f 0.23 GeV, m 3D f 116 GeV and sin u D f 0.03. Also, tan2 2 u 23 increases by about 40% from MGU T to MZ . In order to predict the nm-nt mixing, we take a specific MSSM framework w15x where the three
50
G. Lazarides, N.D. Vlachosr Physics Letters B 441 (1998) 46–51
Yukawa couplings of the third generation unify ‘asymptotically’ and, consequently, tan b f m trm b . We choose the universal scalar mass Žgravitino mass. m 3r2 f 290 GeV and the universal gaugino mass M1r2 f 470 GeV. These values correspond w16x to m t Ž m t . f 166 GeV and m A Žthe tree level mass of the CP-odd scalar higgs boson. s MZ . The ‘asymptotic’ higgsino mass m is related w17x to these parameters by < m
Fig. 1. The mass scale M Žsolid line. and the reheat temperature Tr Ždashed line. as functions of k .
Fig. 2. The allowed region Žbounded by the solid lines. in the k-sin2 2 umt plane for mnmf 2.6=10y3 eV and mntf 7=10y2 eV.
the second heaviest n c, into which the scalar u decays partially, is given by M2 s Mn c s a m i n f lr2 and M3 can be found from Eq. Ž17.. We can use the trace condition in Eq. Ž18. to solve for d Ž u . in the interval 0 F u F pr2. The expression for d Ž u . is subsequently substituted in Eq. Ž16. for e . The leptonic asymmetry as a function of the angle u can be found from Eq. Ž14.. To each value of k correspond two values of the angle u satisfying the low deuterium abundance constraint V B h 2 f 0.025. ŽThese values of u turn out to be quite insensitive to the exact value of n Brs.. The corresponding values of the rotation angle w , which diagonalizes the light neutrino mass matrix, are then found and the allowed region of the mixing angle umt in Eq. Ž19. is determined. Taking into account renormalization effects and superimposing all the permitted regions, we obtain the allowed range of sin2 2 umt as a function of k , shown in Fig. 2. We observe that maximal mixing Žsin2 2 umt f 1. is achieved for 1.5 = 10y6 Q k Q 1.8 = 10y6 . Also, sin2 2 umt R 0.8 w7x corresponds to 1.2 = 10y6 Q k Q 3.4 = 10y6 . The analysis above can be repeated for all values of mnt allowed by SuperKamiokande. The allowed regions in the mnt-k plane for maximal nm -nt mixing Žbounded by the solid lines. and sin2 2 umt R 0.8 Žbounded by the dotted lines. are shown in Fig. 3. Notice that, for sin2 2 umt R 0.8, k f Ž0.9 y 7.5. = 10y6 which is rather small. ŽFortunately, supersymmetry protects this coupling from radiative corrections.. The corresponding values of M and Tr can be
G. Lazarides, N.D. Vlachosr Physics Letters B 441 (1998) 46–51
51
The required value of the coupling constant k is, however, quite small Ž; 10y6 .. We would like to thank B. Ananthanarayan and C. Pallis for useful discussions. This work was supported by the research grant PENEDr95 K.A.1795.
References
Fig. 3. The regions on the mnt-k plane corresponding to maximal nm-nt mixing Žbounded by the solid lines. and sin2 2 umt R 0.8 Žbounded by the dotted lines.. Here we consider the range 3= 10y2 eV Q mnt Q11=10y2 eV Ž mnmf 2.6=10y3 eV..
read from Fig. 1. We find 1.3 = 10 15 GeV Q M Q 2.7 = 10 15 GeV and 10 7 GeV Q Tr Q 3.2 = 10 8 GeV. We observe that M turns out to be somewhat smaller than the MSSM unification scale MGU T . ŽIt is anticipated that GL R is embedded in a grand unified theory.. The reheat temperature, however, satisfies the gravitino constraint ŽTr Q 10 9 GeV.. It should be noted that, for the values of the parameters chosen here, the lightest supersymmetric particle ŽLSP. is w16x an almost pure bino with mass m LS P f 0.43 M1r2 f 200 GeV w18x. Its contribution to the mass of the universe turns out w18x to be V LS P h 2 f 1. In summary, we have investigated hybrid inflation and the subsequent reheating process in the framework of a m-problem solving supersymmetric model based on a left-right symmetric gauge group. The process of baryogenesis via leptogenesis is especially considered. For masses of nm , nt consistent with the small angle MSW resolution of the solar neutrino problem and the recent SuperKamiokande data, we showed that maximal nm-nt mixing can be achieved.
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