- Email: [email protected]
Baryogenesis in a supersymmetric model without R-parity
14 May 1998
Physics Letters B 427 Ž1998. 59–64
Baryogenesis in a supersymmetric model without R-parity Rathin Adhikari 1, Utpal Sarkar Theory Group, Physical Research Laboratory, Ahmedabad - 380009, India Received 15 May 1997; revised 2 March 1998 Editor: H. Georgi
Abstract We propose a simple scenario for baryogenesis in supersymmetric models where baryon number is broken alongwith R-parity. The lightest supersymmetric particle Žneutralino. decays to three quarks and CP-violation comes from interference of tree and one loop box diagrams. The bounds on the R-parity breaking couplings from the out-of-equilibrium condition are considerably relaxed in this scenario. q 1998 Published by Elsevier Science B.V. All rights reserved.
The observable universe contains more matter than antimatter although the ratio is very small for baryons. This asymmetry of the universe w1x can be generated at a very high energy, but in most likelihood it will be washed out at a later stage w2x. So a great deal of interest started in scenarios where the baryon asymmetry is generated during the electroweak phase transition w3,4x by making the phase transition to be a weakly first order. However the condition for survival of the generated baryon asymmetry after the phase transition gives a strong bound on the mass of the higgs doublet w5x, which rule out this scenario. This motivates for alternative scenarios for baryogenesis w6-11x. In this article we propose a simple supersymmetric model to generate baryon asymmetry of the universe at low energy. Baryon number violation arises from R-parity breaking. R-parity violating couplings, which also violate lepton number, is assumed to be small Žor even zero if lepton number is conserved.. The lightest supersymmetric particle ŽLSP., which is
one of the neutralinos x 10 , is now unstable since R-parity is broken. There exist new box-type diagrams for the decay of x 10 , which interferes with the tree level diagram to give enough CP-violation, which generates the baryon asymmetry of the universe. The mass of x 10 is much less than the sfermion masses and as a result when x 10 decays the sfermions has already decayed away, so it does not matter if the sfermions decay in equilibrium and have erased the primordial baryon asymmetry. This relaxes the bounds on the R-parity breaking B-violating couplings considerably. In supersymmetric models, R-parity was introduced as a matter of convenience to prevent fast proton decay. It is now realised that the proton lifetime can be made consistent with experiment without invoking R-parity symmetry. If we don’t impose R-parity in the model, then the minimal supersymmetric standard model allows the following B and L violating terms in the superpotenial c
W s l i jk Li L j Ž E k . q lXi jk Li Q j Ž D k . c
1
E-mail address: [email protected]
c
c
q lXXi jk Ž U i . Ž D j . Ž D k . .
0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 3 0 5 - 0
c
Ž 1.
60
R. Adhikari, U. Sarkarr Physics Letters B 427 (1998) 59–64
Here L and Q are the lepton and quark doublet superfields; E c is the lepton singlet superfield and U c and D c are the quark singlet superfields and i, j,k are the generation indices. In the above the first two terms are lepton number violating while the third term violates baryon number. For the stability of the proton, we assume that l i jk and lXi jk are zero Žif some symmetry like lepton number is present.. The couplings lXXi jk can now be considerably large. They are antisymmetric in the last two indices and in general complex. We consider one of the neutralinos Ž x 10 . as the lightest supersymmetric particle having mass of the order of 100–200 GeV and the sfermions have higher mass of the order of 250 GeV to a few TeV. So before the universe cools down to the temperature of the electroweak symmetry breaking scale the sfermions have already decayed away. Since R-parity is broken, some of the sfermions will have baryon number violating decay channels, which may even wash out the primordial baryon asymmetry of the universe. Near the electroweak scale the neutralinos x 10 are the only superparticles left to be decayed. If R-parity is not broken, these particles are stable. However, since R-parity is now broken, the neutralino can also decay to ordinary quarks and leptons. In these decays at least one of the vertex should not conserve R-parity and hence baryon number is broken. In Figs. 1Ža. and 1Žb. the tree level diagrams for the process x 10 ™ u i R d j R d k R are given, while in Figs. 2Ža. and 2Žb. we present the new type of one loop box diagrams. The interference of the tree level and the one loop diagrams give rise to the CP violation. We are not considering the decay modes like x 10 ™ u i L d j R d k R or x 10 ™ u i R d j L d k R as there will be right-squark left-quark neutralino coupling at the tree 2 level which is suppressed in general by mW in comparison to the right-squark right-quark neutralino coupling appearing at the tree level in the figures. The advantage of the box diagrams is that two of the vertices contains couplings of the higgs and hence elements of the CKM matrix contributes to the baryon asymmetry. This gives a large enhancement and hence we can get a large baryon asymmetry even for considerably smaller R-parity breaking couplings. As the B-violating couplings are antisymmetric with the interchange of the d-type quark indices, the
indices j and k are always different for a particular type of decay mode where i, j and k are fixed. Thus changing all the j indices to k and k indices to j there will be an extra diagram at the tree level corresponding to Fig. 1b and two other box diagrams corresponding to Figs. 2a and 2b. For the box diagrams in Fig. 2Ža. and 2Žb., we require u L u˜ R x 10 and d L d˜R x 10 couplings, which comes from the higgsino component of the neutralino and is proportional to the quark mass w13x. We are not considering intergenerational mixing for these couplings as well as for the right-squark rightquark neutralino couplings in Fig. 1Ža. and 1Žb. as that will lead to some suppression due to the presence of CKM type matrix elements. In the box diagrams the mass insertion for the quarks has been considered to change the left -handed quarks to right handed quarks.
Fig. 1. Tree level diagrams for the decay x 10 ™ u i R d j R d k R .
R. Adhikari, U. Sarkarr Physics Letters B 427 (1998) 59–64
61
Fig. 2. Box diagrams for the decay x 10 ™ u i R d j R d k R .
The thermally averaged decay width for such baryon number violating decays are given by ²G Ž
x 10 ™ u i R d j R d k R
.: f
g 2 N lXXi jk N 2 mx5 10 4
10 p
3
Ž
m2q˜ q T 2
.
2
A21 ,
Ž 2. where the factor A1 s N11)
tan uw
comes from the squark-quark-neutralino couplings, Ni j is the 4 = 4 neutralino mixing matrix in the
convention in which all the neutralino masses are positive Žthe notations of Ref. w14x has been followed here. and the temperature T F mx 10 . The decay of the neutralino should now satisfy the out-of-equilibrium condition near T s T0 f mx 10 , which is ² G Ž x 10 ™ u i d j d k . : - 1.7 g ) Ž T 2rMP . .
'
Ž 3.
This can be satisfied with A1 ; 10y1 to 10y2 , which is possible for a wide range of MSSM parameters for neutralino mass ranging from about 100 to 200 GeV
R. Adhikari, U. Sarkarr Physics Letters B 427 (1998) 59–64
62
with N m N about 200 to 1000 GeV and tan b from 2 to 12, and with lXX - 10y2 to 10y3 . In fact A1 can be even lower than that giving the possibility of considering relatively higher value of lXX allowed by the phenomenological constraint w12x. While calculating the baryon asymmetry in the decay of the neutralino x 10 , one has to take into consideration the fact that through the baryon number conserving x 10 q ™ x 10 q scattering some of the neutralinos may thermalize before they decay and will not contribute to the decay asymmetry. This will introduce an additional suppression factor S given by w8x ² G Ž x 10 ™ u i R d j R d k R :
5
N lXXi jk N 2
mx 10
4p g 2 A41 q A42
ž / T0
5
A21
Ž 4.
in the decay asymmetry, where s Ž x 10 q ™ x 10 q . is the cross-section for x 10 scattering, Õ is the relative velocity and n q is the quark thermal number density. Here A 2 corresponds to left-squark left-quark neutralino coupling and is given by w13x A 2 s 16 tan uw N11 y 12 N12 . The amount of decay asymmetry is defined by
Ý e sS
ef
Ý ijk Ž j/k .
m2q˜ m d i tan uw 3 p mW Ž A41 q A42 .
=Im lXXi jk N11) Vi i)
½
G Ž x 10 ™ u i R d j R d k R . y G x 10 ™ u i R d j R d k R
Ž
ijk Ž j/ k .
Ý
G Ž x 10 ™all .
. .
ijk Ž j/ k .
Ž 5. Here G Ž x 10 ™ all . will be dominated by the main decay mode Ž x 10 ™ u i R d j R d k R . as the neutralino x 10 in our case is the lightest supersymmetric particle and we are considering the absence of any lepton number violating interaction. However there will be contribution from the decay mode like Ž x 10 ™ u i L d j R d k R . and Ž x 10 ™ u i R d j L d k R .. But these decay modes will have the contributions suppressed in 2 general by mW relative to the main decay mode. The decay asymmetry e is generated through the interference of the various possible tree level diagrams wFigs. 1Ža. and 1Žb.x with the box diagrams wFigs. 2Ža. and 2Žb.x. There will be other possible diagrams with j and k indices interchanged Žnot
sin b cos 2b
qm u k m d k lXXk i)j Vk k q
/
N14) m u j m d j lXXji)k Vj j
cos b sin2b
ž
N13)
= m u i m u j lXXji)k Vj j q m u i m u k lXXk i)j Vk k
ž
=I Ž mx 10 ,m q˜ ,m q . ,
² s Ž x 10 q ™ x 10 q . Õn q : f
shown in the figure.. Considering all those diagrams and assuming degenerate squark mass it is found
/
5 Ž 6.
where I Ž mx 10 ,m q˜ ,m q . comes from the absorptive part of the loop integral. Vi j is the CKM matrix, which enters from the higgs coupling. To get non-zero e one has to consider i / j and i / k in Eq. Ž6.. It is clear from the expression that the imaginary part of the product of the couplings is invariant under rephasing of all the quark phases. It can be noticed that it is possible to choose a phase convention, so that the l’s are real and hence the CP-violating phase comes entirely from the CKM matrix. In this basis, there will not be any constraint coming from the electron dipole moment of the neutron on the complex part of the couplings of l. This will reduce one more uncertain parameter in the problem. For simplicity in our box diagram calculations we consider the charged higgs mass mfy and the squark mass m q˜ to be of the same order. It is found that for a wide range of neutralino mass from 100 GeV to 400 GeV and the squark mass from 200 GeV to the order of TeV with mx 10 - m q˜ and for various quark masses, I Ž mx 10 ,m q˜ ,m q . f 10y4 . Particularly the box integral contribution increases when the difference of mx 10 and m q˜ is lesser. However it does not differ much from 10y4 . Only the diagonal elements of the CKM matrix appears in the expression of the decay asymmetry giving effectively no suppression. In Eq. Ž6. it is possible to choose the parameters so that the amount of asymmetry generated in this scenario is large enough. For higher values of tan b greater than 1 the first term in the curly bracket in Eq. Ž6. with sin b the factor cos 2 b gives more contribution to the decay asymmetry. As a representative set of values we consider tan b ; 3; mass of the neutralino of the
R. Adhikari, U. Sarkarr Physics Letters B 427 (1998) 59–64
order of 200 GeV and the mass of squarks are in the range of TeV and A1 f A 2 f 10y2 and N11 and N14 of the order of 10y1 . Since top quark in the decay product leads to phase space suppression relative to other quarks we consider charm and strange quarks associated with one charged higgs coupling and i s 1 or 2 in Eq. Ž6.. With these choice of parameters the decay asymmetry e ; lXXi jk lXXji)k . In this case large enhancement comes from the diagonal CKM matrix elements. For several other choices of parameters also we can have a large e in this scenario. To compute bayon asymmetry generated from the decay asymmetry due to CP violation one has to consider the Boltzman equation for the number density of the neutralino and also the Boltzman equation for the evolution of the baryon number density. In general in the equation for the evolution of the baryon number density one has to take care of the inverse decay for neutralino and the B-non conserving scattering process like u i d j ™ d k x 10 which damp the baryon asymmetry. However in our case with suitable choice of lXX it is possible to find decay rate for neutralino which is much less than the expansion rate and the decay processes will occur at far away from equilibrium. Under that situation both the inverse decays and B- non conserving scattering process become impotent and solving those Boltzman equations are straightforward w1x. At the time of the decay when T - mx 10 the neutralino will be overabundant and their number density nx 10 will be of the order of photon number density ng . At this point it is important to note that w16x for nx 10 F ng the contribution to the total radiation density Ž rr . of the universe is not dominated by the neutralinos as the energy density Ž rx . of the neutralino is
rx s mx 10 nx 10 f mx 10 ng f .24 mx 10 T 3 , which is smaller compared to the total energy density
p2
g T 4 f 30T 4 , 30 ) provided that T ) mx 10r150. This is well satisfied in the present scenario. As in each decay of neutralino the baryon number 1 is produced, considering each decay to quark final state as well as the anti-quark final state the baryon number e is produced. So the baryon number density n B ; e nx 10 ; e ng . As the
rr s
63
entropy density is g ) ng the baryon asymmetry B f ge) and considering total degrees of freedom for different particle species g ) of the order of 100 and with the earlier mentioned choice of parameters the baryon asymmetry B is given by B f 10y2lXXi jk lXXji)k .
Ž 7.
We shall now discuss the constraints on the Bviolating R-parity breaking coupling constant lXX . In general one requires that all B-violating decays of the sfermions should be slow enough so as not to erase the baryon asymmetry before the electroweak phase transition w17x. However, in the present model these constraints are no longer valid since we generate baryon asymmetry at the electroweak scale when all the sfermions have already decayed away. The out of equilibrium condition for the interaction u i d j ™ d k x 10 put severe constraint on the values of lXXi jk w17x given by
lXXi jk - 3 = 10y6
ž
m ˜ 1 TeV
1r2
/
'2 g
tan uw 3
y1
N12
.
Ž 8. For the mass of neutralino of the order of 100 to 200 GeV and the squark mass in the TeV range this constraint on lXXi jk is of the order of 10y3 for some region of the parameter space independent of the generation indices. Another source of constraint on lXXi jk comes from the non-observation of nn oscillations. The earlier b o u n d w 1 8 ,1 9 x w a s l XX1 1 k < 2 = 1 0 y 7 w mr100 GeV x 5r2 . However these constraints are ˜ highly model dependent and may be evaded w20,21x and the upper bound may be of the order of 10y3 or higher depending on the choice of SUSY parameters. The coupling lXX113 is almost free of any constraint, when a suppression factor coming from the flavor changing neutral current is included w21x. For lXX112 the upper bound may vary from 10y4 to 10y1 depending on the stop mass from 100 GeV to 500 GeV w21x. In our case the upper bound on these couplings will be still higher. The product of two B violating couplings are constrained from the rare two body non-leptonic decays of B and D mesons w21x. However those constraints are much weaker. For higher squark mass in the TeV range most stringent constraint essentially
64
R. Adhikari, U. Sarkarr Physics Letters B 427 (1998) 59–64
comes from the out of equilibrium conditions w21,22x for most of the B-violating couplings, which is of the order of 10y2 to 10y3 . From Eqs. Ž3. and Ž7. it is seen that even considering lXX ; 10y4 which implies the decay processes are far away from equilibrium it is easily possible to generate the required amount of baryon asymmetry 10y1 0 . As it is seen that apart from the out of equilibrium constraint it is possible to have lXX as high as 10y1 and as satisfying the out of equilibrium condition in Eq. Ž3. one can consider lXX ; 10y3 which generates high baryon asymmetry it is possible to consider the higher values of lXX when the decay processes will be near equilibrium to generate the required amount of baryon asymmetry. In that case one has to take into account the CP violating part of the interaction u i d j ™ d k x 10 and also the inverse decay processes in the generation of baryon asymmetry. Under that situation the cosmological constraint on lXX may be further relaxed. In our scenario the baryon asymmetry will be a Ž B y L. asymmetry, and hence this is not washed out due to sphaleron transition. On the other hand since the baryon asymmetry is not related to the lepton asymmetry of the universe, all constraints on the lepton number violating interactions arising due to the Majorana masses of the neutrinos are no longer valid. To summarize, we presented a simple supersymmetric model of baryogenesis, where baryon number is violated alongwith R-parity. CP-violation comes from an interference of the tree level and the one loop box-type diagrams, such that two of the elements enters from the CKM matrix. The constraints on the B-violating ŽR-parity violating. couplings are considerably relaxed in this scenario.
Acknowledgements We would like to thank R. Rangarajan for helpful discussions and some clarifications.
References w1x E.W. Kolb, M.S. Turner, The Early Universe, Addison-Wesley, Reading, MA, 1989.
w2x G.’t Hooft, Phys. Rev. Lett. 37 Ž1976. 8; V.A. Kuzmin, V.A. Rubakov, M.E. Shaposhnikov, Phys. Lett. B 155 Ž1985. 36. w3x D. Brahm, in: Proc. XXVI Int. Conf. High Energy Physics, Dallas, Texas, August 1992; M.E. Shaposhnikov, Nucl. Phys. B 287 Ž1987. 757; B 299 Ž1988. 797; V.A. Rubakov, M.E. Shaposhnikov, Usp. Fiz. Nauk 166 Ž1996. 493. w4x M.E. Shaposhnikov, Nucl. Phys. B 287 Ž1987. 757; B 299 Ž1988. 797; N. Turok, J. Zadrozny, Phys. Rev. Lett. 65 Ž1990. 2331; A. Kazarian, S. Kuzmin, M.E. Shaposhnikov, Phys. Lett. B 276 Ž1992. 131; L. McLerran, M.E. Shaposhnikov, N. Turok, M. Voloshin, Phys. Lett. B 256 Ž1991. 451. w5x A.I. Bochkarev, S.V. Kuzmin, M.E. Shaposhnikov, Phys. Lett. B 244 Ž1990. 275; M. Dine, P. Huet, R. Singleton Jr., Nucl. Phys. B 375 Ž1992. 625; M.E. Shaposhnikov, Phys. Lett. B 316 Ž1993. 112. w6x M. Fukugita, T. Yanagida, Phys. Lett. B 174 Ž1986. 45. w7x P. Langacker, R.D. Peccei, T. Yanagida, Mod. Phys. Lett. A 1 Ž1986. 541; M.A. Luty, Phys. Rev. D 45 Ž1992. 445; H. Murayama, H. Suzuki, T. Yanagida, J. Yokoyama, Phys. Rev. Lett. 70 Ž1993. 1912; A. Acker, H. Kikuchi, E. Ma, U. Sarkar, Phys. Rev. D 48 Ž1993. 5006; H. Murayama, H. Suzuki, T. Yanagida, J. Yokoyama, Phys. Rev. Lett. 70 Ž1993. 1912; P.J. O’Donnell, U. Sarkar, Phys. Rev. D 49 Ž1994. 2118; M. Flanz, E.A. Paschos, U. Sarkar, Phys. Lett. B 345 Ž1995. 248; M. Plumacher, Z. Phys. C 74 Ž1997. 549; ¨ L. Covi, E. Roulet, F. Vissani, Phys. Lett. B 384 Ž1996. 169; M. Flanz, E.A. Paschos, U. Sarkar, J. Weiss, Phys. Lett. B 389 Ž1996. 693; U. Sarkar, Phys. Lett. B 390 Ž1997. 97. w8x A. Masiero, A. Riotto, Phys. Lett. B 289 Ž1992. 73. w9x B. Campbell, S. Davidson, K.A. Olive, Phys. Lett. B 303 Ž1993. 63; Nucl. Phys. B 399 Ž1993. 111; U. Sarkar, R. Adhikari, Phys. Rev. D 55 Ž1997. 3836. w10x U. Sarkar, report no. hep-phr9609493. w11x J. Cline, S. Raby, Phys. Rev. D 43 Ž1991. 1781; R.J. Scherrer, J. Cline, S. Raby, D. Seckel, Phys. Rev. D 44 Ž1991. 3760; S. Mollerach, E. Roulet, Phys. Lett. B 281 Ž1992. 303; S. Dimopoulos, L. Hall, Phys. Lett. B 196 Ž1987. 135. w12x H. Dreiner, report no. hep-phr9707435, to be published in Perspectives on Supersymmetry G.L. Kane ŽEd.., World Scientific. w13x J.F. Gunion, H.E. Haber, Nucl. Phys. 272 Ž1986. 1. w14x H.E. Haber, G.L. Kane, Phys. Rep. 117 Ž1985. 75. w15x J. Donoghue et al., Phys. Lett. B 128 Ž1983. 55; A. Bouquet et al., Phys. Lett. B 148 Ž1984. 69; M. Duncan, Nucl. Phys. B 221 Ž1993. 285. w16x S. Weinberg, Phys. Rev. Lett. 42 Ž1979. 850. w17x A. Bouquet, P. Salati, Nucl. Phys. B 284 Ž1987. 557; B.A. Campbell, S. Davidson, J. Ellis, K. Olive, Phys. Lett. B 256 Ž1991. 457; W. Fishler, G.F. Giudice, R.G. Leigh, S. Paban, Phys. Lett. B 258 Ž1991. 45. w18x F. Zwirner, Phys. Lett. B 132 Ž1983. 103. w19x R. Barbieri, A. Masiero, Nucl. Phys. B 267 Ž1986. 679. w20x H. Dreiner, G.G. Ross, Nucl. Phys. B 410 Ž1993. 188. w21x J.L. Goity, M. Sher, Phys. Lett. B 346 Ž1995. 69; C.E. Carlson, P. Roy, M. Sher, Phys. Lett. B 357 Ž1995. 99. w22x D. Cheng, W.Y. Keung, Phys. Lett. B 389 Ž1996. 294.
Physics Letters B 427 Ž1998. 59–64
Baryogenesis in a supersymmetric model without R-parity Rathin Adhikari 1, Utpal Sarkar Theory Group, Physical Research Laboratory, Ahmedabad - 380009, India Received 15 May 1997; revised 2 March 1998 Editor: H. Georgi
Abstract We propose a simple scenario for baryogenesis in supersymmetric models where baryon number is broken alongwith R-parity. The lightest supersymmetric particle Žneutralino. decays to three quarks and CP-violation comes from interference of tree and one loop box diagrams. The bounds on the R-parity breaking couplings from the out-of-equilibrium condition are considerably relaxed in this scenario. q 1998 Published by Elsevier Science B.V. All rights reserved.
The observable universe contains more matter than antimatter although the ratio is very small for baryons. This asymmetry of the universe w1x can be generated at a very high energy, but in most likelihood it will be washed out at a later stage w2x. So a great deal of interest started in scenarios where the baryon asymmetry is generated during the electroweak phase transition w3,4x by making the phase transition to be a weakly first order. However the condition for survival of the generated baryon asymmetry after the phase transition gives a strong bound on the mass of the higgs doublet w5x, which rule out this scenario. This motivates for alternative scenarios for baryogenesis w6-11x. In this article we propose a simple supersymmetric model to generate baryon asymmetry of the universe at low energy. Baryon number violation arises from R-parity breaking. R-parity violating couplings, which also violate lepton number, is assumed to be small Žor even zero if lepton number is conserved.. The lightest supersymmetric particle ŽLSP., which is
one of the neutralinos x 10 , is now unstable since R-parity is broken. There exist new box-type diagrams for the decay of x 10 , which interferes with the tree level diagram to give enough CP-violation, which generates the baryon asymmetry of the universe. The mass of x 10 is much less than the sfermion masses and as a result when x 10 decays the sfermions has already decayed away, so it does not matter if the sfermions decay in equilibrium and have erased the primordial baryon asymmetry. This relaxes the bounds on the R-parity breaking B-violating couplings considerably. In supersymmetric models, R-parity was introduced as a matter of convenience to prevent fast proton decay. It is now realised that the proton lifetime can be made consistent with experiment without invoking R-parity symmetry. If we don’t impose R-parity in the model, then the minimal supersymmetric standard model allows the following B and L violating terms in the superpotenial c
W s l i jk Li L j Ž E k . q lXi jk Li Q j Ž D k . c
1
E-mail address: [email protected]
c
c
q lXXi jk Ž U i . Ž D j . Ž D k . .
0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 3 0 5 - 0
c
Ž 1.
60
R. Adhikari, U. Sarkarr Physics Letters B 427 (1998) 59–64
Here L and Q are the lepton and quark doublet superfields; E c is the lepton singlet superfield and U c and D c are the quark singlet superfields and i, j,k are the generation indices. In the above the first two terms are lepton number violating while the third term violates baryon number. For the stability of the proton, we assume that l i jk and lXi jk are zero Žif some symmetry like lepton number is present.. The couplings lXXi jk can now be considerably large. They are antisymmetric in the last two indices and in general complex. We consider one of the neutralinos Ž x 10 . as the lightest supersymmetric particle having mass of the order of 100–200 GeV and the sfermions have higher mass of the order of 250 GeV to a few TeV. So before the universe cools down to the temperature of the electroweak symmetry breaking scale the sfermions have already decayed away. Since R-parity is broken, some of the sfermions will have baryon number violating decay channels, which may even wash out the primordial baryon asymmetry of the universe. Near the electroweak scale the neutralinos x 10 are the only superparticles left to be decayed. If R-parity is not broken, these particles are stable. However, since R-parity is now broken, the neutralino can also decay to ordinary quarks and leptons. In these decays at least one of the vertex should not conserve R-parity and hence baryon number is broken. In Figs. 1Ža. and 1Žb. the tree level diagrams for the process x 10 ™ u i R d j R d k R are given, while in Figs. 2Ža. and 2Žb. we present the new type of one loop box diagrams. The interference of the tree level and the one loop diagrams give rise to the CP violation. We are not considering the decay modes like x 10 ™ u i L d j R d k R or x 10 ™ u i R d j L d k R as there will be right-squark left-quark neutralino coupling at the tree 2 level which is suppressed in general by mW in comparison to the right-squark right-quark neutralino coupling appearing at the tree level in the figures. The advantage of the box diagrams is that two of the vertices contains couplings of the higgs and hence elements of the CKM matrix contributes to the baryon asymmetry. This gives a large enhancement and hence we can get a large baryon asymmetry even for considerably smaller R-parity breaking couplings. As the B-violating couplings are antisymmetric with the interchange of the d-type quark indices, the
indices j and k are always different for a particular type of decay mode where i, j and k are fixed. Thus changing all the j indices to k and k indices to j there will be an extra diagram at the tree level corresponding to Fig. 1b and two other box diagrams corresponding to Figs. 2a and 2b. For the box diagrams in Fig. 2Ža. and 2Žb., we require u L u˜ R x 10 and d L d˜R x 10 couplings, which comes from the higgsino component of the neutralino and is proportional to the quark mass w13x. We are not considering intergenerational mixing for these couplings as well as for the right-squark rightquark neutralino couplings in Fig. 1Ža. and 1Žb. as that will lead to some suppression due to the presence of CKM type matrix elements. In the box diagrams the mass insertion for the quarks has been considered to change the left -handed quarks to right handed quarks.
Fig. 1. Tree level diagrams for the decay x 10 ™ u i R d j R d k R .
R. Adhikari, U. Sarkarr Physics Letters B 427 (1998) 59–64
61
Fig. 2. Box diagrams for the decay x 10 ™ u i R d j R d k R .
The thermally averaged decay width for such baryon number violating decays are given by ²G Ž
x 10 ™ u i R d j R d k R
.: f
g 2 N lXXi jk N 2 mx5 10 4
10 p
3
Ž
m2q˜ q T 2
.
2
A21 ,
Ž 2. where the factor A1 s N11)
tan uw
comes from the squark-quark-neutralino couplings, Ni j is the 4 = 4 neutralino mixing matrix in the
convention in which all the neutralino masses are positive Žthe notations of Ref. w14x has been followed here. and the temperature T F mx 10 . The decay of the neutralino should now satisfy the out-of-equilibrium condition near T s T0 f mx 10 , which is ² G Ž x 10 ™ u i d j d k . : - 1.7 g ) Ž T 2rMP . .
'
Ž 3.
This can be satisfied with A1 ; 10y1 to 10y2 , which is possible for a wide range of MSSM parameters for neutralino mass ranging from about 100 to 200 GeV
R. Adhikari, U. Sarkarr Physics Letters B 427 (1998) 59–64
62
with N m N about 200 to 1000 GeV and tan b from 2 to 12, and with lXX - 10y2 to 10y3 . In fact A1 can be even lower than that giving the possibility of considering relatively higher value of lXX allowed by the phenomenological constraint w12x. While calculating the baryon asymmetry in the decay of the neutralino x 10 , one has to take into consideration the fact that through the baryon number conserving x 10 q ™ x 10 q scattering some of the neutralinos may thermalize before they decay and will not contribute to the decay asymmetry. This will introduce an additional suppression factor S given by w8x ² G Ž x 10 ™ u i R d j R d k R :
5
N lXXi jk N 2
mx 10
4p g 2 A41 q A42
ž / T0
5
A21
Ž 4.
in the decay asymmetry, where s Ž x 10 q ™ x 10 q . is the cross-section for x 10 scattering, Õ is the relative velocity and n q is the quark thermal number density. Here A 2 corresponds to left-squark left-quark neutralino coupling and is given by w13x A 2 s 16 tan uw N11 y 12 N12 . The amount of decay asymmetry is defined by
Ý e sS
ef
Ý ijk Ž j/k .
m2q˜ m d i tan uw 3 p mW Ž A41 q A42 .
=Im lXXi jk N11) Vi i)
½
G Ž x 10 ™ u i R d j R d k R . y G x 10 ™ u i R d j R d k R
Ž
ijk Ž j/ k .
Ý
G Ž x 10 ™all .
. .
ijk Ž j/ k .
Ž 5. Here G Ž x 10 ™ all . will be dominated by the main decay mode Ž x 10 ™ u i R d j R d k R . as the neutralino x 10 in our case is the lightest supersymmetric particle and we are considering the absence of any lepton number violating interaction. However there will be contribution from the decay mode like Ž x 10 ™ u i L d j R d k R . and Ž x 10 ™ u i R d j L d k R .. But these decay modes will have the contributions suppressed in 2 general by mW relative to the main decay mode. The decay asymmetry e is generated through the interference of the various possible tree level diagrams wFigs. 1Ža. and 1Žb.x with the box diagrams wFigs. 2Ža. and 2Žb.x. There will be other possible diagrams with j and k indices interchanged Žnot
sin b cos 2b
qm u k m d k lXXk i)j Vk k q
/
N14) m u j m d j lXXji)k Vj j
cos b sin2b
ž
N13)
= m u i m u j lXXji)k Vj j q m u i m u k lXXk i)j Vk k
ž
=I Ž mx 10 ,m q˜ ,m q . ,
² s Ž x 10 q ™ x 10 q . Õn q : f
shown in the figure.. Considering all those diagrams and assuming degenerate squark mass it is found
/
5 Ž 6.
where I Ž mx 10 ,m q˜ ,m q . comes from the absorptive part of the loop integral. Vi j is the CKM matrix, which enters from the higgs coupling. To get non-zero e one has to consider i / j and i / k in Eq. Ž6.. It is clear from the expression that the imaginary part of the product of the couplings is invariant under rephasing of all the quark phases. It can be noticed that it is possible to choose a phase convention, so that the l’s are real and hence the CP-violating phase comes entirely from the CKM matrix. In this basis, there will not be any constraint coming from the electron dipole moment of the neutron on the complex part of the couplings of l. This will reduce one more uncertain parameter in the problem. For simplicity in our box diagram calculations we consider the charged higgs mass mfy and the squark mass m q˜ to be of the same order. It is found that for a wide range of neutralino mass from 100 GeV to 400 GeV and the squark mass from 200 GeV to the order of TeV with mx 10 - m q˜ and for various quark masses, I Ž mx 10 ,m q˜ ,m q . f 10y4 . Particularly the box integral contribution increases when the difference of mx 10 and m q˜ is lesser. However it does not differ much from 10y4 . Only the diagonal elements of the CKM matrix appears in the expression of the decay asymmetry giving effectively no suppression. In Eq. Ž6. it is possible to choose the parameters so that the amount of asymmetry generated in this scenario is large enough. For higher values of tan b greater than 1 the first term in the curly bracket in Eq. Ž6. with sin b the factor cos 2 b gives more contribution to the decay asymmetry. As a representative set of values we consider tan b ; 3; mass of the neutralino of the
R. Adhikari, U. Sarkarr Physics Letters B 427 (1998) 59–64
order of 200 GeV and the mass of squarks are in the range of TeV and A1 f A 2 f 10y2 and N11 and N14 of the order of 10y1 . Since top quark in the decay product leads to phase space suppression relative to other quarks we consider charm and strange quarks associated with one charged higgs coupling and i s 1 or 2 in Eq. Ž6.. With these choice of parameters the decay asymmetry e ; lXXi jk lXXji)k . In this case large enhancement comes from the diagonal CKM matrix elements. For several other choices of parameters also we can have a large e in this scenario. To compute bayon asymmetry generated from the decay asymmetry due to CP violation one has to consider the Boltzman equation for the number density of the neutralino and also the Boltzman equation for the evolution of the baryon number density. In general in the equation for the evolution of the baryon number density one has to take care of the inverse decay for neutralino and the B-non conserving scattering process like u i d j ™ d k x 10 which damp the baryon asymmetry. However in our case with suitable choice of lXX it is possible to find decay rate for neutralino which is much less than the expansion rate and the decay processes will occur at far away from equilibrium. Under that situation both the inverse decays and B- non conserving scattering process become impotent and solving those Boltzman equations are straightforward w1x. At the time of the decay when T - mx 10 the neutralino will be overabundant and their number density nx 10 will be of the order of photon number density ng . At this point it is important to note that w16x for nx 10 F ng the contribution to the total radiation density Ž rr . of the universe is not dominated by the neutralinos as the energy density Ž rx . of the neutralino is
rx s mx 10 nx 10 f mx 10 ng f .24 mx 10 T 3 , which is smaller compared to the total energy density
p2
g T 4 f 30T 4 , 30 ) provided that T ) mx 10r150. This is well satisfied in the present scenario. As in each decay of neutralino the baryon number 1 is produced, considering each decay to quark final state as well as the anti-quark final state the baryon number e is produced. So the baryon number density n B ; e nx 10 ; e ng . As the
rr s
63
entropy density is g ) ng the baryon asymmetry B f ge) and considering total degrees of freedom for different particle species g ) of the order of 100 and with the earlier mentioned choice of parameters the baryon asymmetry B is given by B f 10y2lXXi jk lXXji)k .
Ž 7.
We shall now discuss the constraints on the Bviolating R-parity breaking coupling constant lXX . In general one requires that all B-violating decays of the sfermions should be slow enough so as not to erase the baryon asymmetry before the electroweak phase transition w17x. However, in the present model these constraints are no longer valid since we generate baryon asymmetry at the electroweak scale when all the sfermions have already decayed away. The out of equilibrium condition for the interaction u i d j ™ d k x 10 put severe constraint on the values of lXXi jk w17x given by
lXXi jk - 3 = 10y6
ž
m ˜ 1 TeV
1r2
/
'2 g
tan uw 3
y1
N12
.
Ž 8. For the mass of neutralino of the order of 100 to 200 GeV and the squark mass in the TeV range this constraint on lXXi jk is of the order of 10y3 for some region of the parameter space independent of the generation indices. Another source of constraint on lXXi jk comes from the non-observation of nn oscillations. The earlier b o u n d w 1 8 ,1 9 x w a s l XX1 1 k < 2 = 1 0 y 7 w mr100 GeV x 5r2 . However these constraints are ˜ highly model dependent and may be evaded w20,21x and the upper bound may be of the order of 10y3 or higher depending on the choice of SUSY parameters. The coupling lXX113 is almost free of any constraint, when a suppression factor coming from the flavor changing neutral current is included w21x. For lXX112 the upper bound may vary from 10y4 to 10y1 depending on the stop mass from 100 GeV to 500 GeV w21x. In our case the upper bound on these couplings will be still higher. The product of two B violating couplings are constrained from the rare two body non-leptonic decays of B and D mesons w21x. However those constraints are much weaker. For higher squark mass in the TeV range most stringent constraint essentially
64
R. Adhikari, U. Sarkarr Physics Letters B 427 (1998) 59–64
comes from the out of equilibrium conditions w21,22x for most of the B-violating couplings, which is of the order of 10y2 to 10y3 . From Eqs. Ž3. and Ž7. it is seen that even considering lXX ; 10y4 which implies the decay processes are far away from equilibrium it is easily possible to generate the required amount of baryon asymmetry 10y1 0 . As it is seen that apart from the out of equilibrium constraint it is possible to have lXX as high as 10y1 and as satisfying the out of equilibrium condition in Eq. Ž3. one can consider lXX ; 10y3 which generates high baryon asymmetry it is possible to consider the higher values of lXX when the decay processes will be near equilibrium to generate the required amount of baryon asymmetry. In that case one has to take into account the CP violating part of the interaction u i d j ™ d k x 10 and also the inverse decay processes in the generation of baryon asymmetry. Under that situation the cosmological constraint on lXX may be further relaxed. In our scenario the baryon asymmetry will be a Ž B y L. asymmetry, and hence this is not washed out due to sphaleron transition. On the other hand since the baryon asymmetry is not related to the lepton asymmetry of the universe, all constraints on the lepton number violating interactions arising due to the Majorana masses of the neutrinos are no longer valid. To summarize, we presented a simple supersymmetric model of baryogenesis, where baryon number is violated alongwith R-parity. CP-violation comes from an interference of the tree level and the one loop box-type diagrams, such that two of the elements enters from the CKM matrix. The constraints on the B-violating ŽR-parity violating. couplings are considerably relaxed in this scenario.
Acknowledgements We would like to thank R. Rangarajan for helpful discussions and some clarifications.
References w1x E.W. Kolb, M.S. Turner, The Early Universe, Addison-Wesley, Reading, MA, 1989.
w2x G.’t Hooft, Phys. Rev. Lett. 37 Ž1976. 8; V.A. Kuzmin, V.A. Rubakov, M.E. Shaposhnikov, Phys. Lett. B 155 Ž1985. 36. w3x D. Brahm, in: Proc. XXVI Int. Conf. High Energy Physics, Dallas, Texas, August 1992; M.E. Shaposhnikov, Nucl. Phys. B 287 Ž1987. 757; B 299 Ž1988. 797; V.A. Rubakov, M.E. Shaposhnikov, Usp. Fiz. Nauk 166 Ž1996. 493. w4x M.E. Shaposhnikov, Nucl. Phys. B 287 Ž1987. 757; B 299 Ž1988. 797; N. Turok, J. Zadrozny, Phys. Rev. Lett. 65 Ž1990. 2331; A. Kazarian, S. Kuzmin, M.E. Shaposhnikov, Phys. Lett. B 276 Ž1992. 131; L. McLerran, M.E. Shaposhnikov, N. Turok, M. Voloshin, Phys. Lett. B 256 Ž1991. 451. w5x A.I. Bochkarev, S.V. Kuzmin, M.E. Shaposhnikov, Phys. Lett. B 244 Ž1990. 275; M. Dine, P. Huet, R. Singleton Jr., Nucl. Phys. B 375 Ž1992. 625; M.E. Shaposhnikov, Phys. Lett. B 316 Ž1993. 112. w6x M. Fukugita, T. Yanagida, Phys. Lett. B 174 Ž1986. 45. w7x P. Langacker, R.D. Peccei, T. Yanagida, Mod. Phys. Lett. A 1 Ž1986. 541; M.A. Luty, Phys. Rev. D 45 Ž1992. 445; H. Murayama, H. Suzuki, T. Yanagida, J. Yokoyama, Phys. Rev. Lett. 70 Ž1993. 1912; A. Acker, H. Kikuchi, E. Ma, U. Sarkar, Phys. Rev. D 48 Ž1993. 5006; H. Murayama, H. Suzuki, T. Yanagida, J. Yokoyama, Phys. Rev. Lett. 70 Ž1993. 1912; P.J. O’Donnell, U. Sarkar, Phys. Rev. D 49 Ž1994. 2118; M. Flanz, E.A. Paschos, U. Sarkar, Phys. Lett. B 345 Ž1995. 248; M. Plumacher, Z. Phys. C 74 Ž1997. 549; ¨ L. Covi, E. Roulet, F. Vissani, Phys. Lett. B 384 Ž1996. 169; M. Flanz, E.A. Paschos, U. Sarkar, J. Weiss, Phys. Lett. B 389 Ž1996. 693; U. Sarkar, Phys. Lett. B 390 Ž1997. 97. w8x A. Masiero, A. Riotto, Phys. Lett. B 289 Ž1992. 73. w9x B. Campbell, S. Davidson, K.A. Olive, Phys. Lett. B 303 Ž1993. 63; Nucl. Phys. B 399 Ž1993. 111; U. Sarkar, R. Adhikari, Phys. Rev. D 55 Ž1997. 3836. w10x U. Sarkar, report no. hep-phr9609493. w11x J. Cline, S. Raby, Phys. Rev. D 43 Ž1991. 1781; R.J. Scherrer, J. Cline, S. Raby, D. Seckel, Phys. Rev. D 44 Ž1991. 3760; S. Mollerach, E. Roulet, Phys. Lett. B 281 Ž1992. 303; S. Dimopoulos, L. Hall, Phys. Lett. B 196 Ž1987. 135. w12x H. Dreiner, report no. hep-phr9707435, to be published in Perspectives on Supersymmetry G.L. Kane ŽEd.., World Scientific. w13x J.F. Gunion, H.E. Haber, Nucl. Phys. 272 Ž1986. 1. w14x H.E. Haber, G.L. Kane, Phys. Rep. 117 Ž1985. 75. w15x J. Donoghue et al., Phys. Lett. B 128 Ž1983. 55; A. Bouquet et al., Phys. Lett. B 148 Ž1984. 69; M. Duncan, Nucl. Phys. B 221 Ž1993. 285. w16x S. Weinberg, Phys. Rev. Lett. 42 Ž1979. 850. w17x A. Bouquet, P. Salati, Nucl. Phys. B 284 Ž1987. 557; B.A. Campbell, S. Davidson, J. Ellis, K. Olive, Phys. Lett. B 256 Ž1991. 457; W. Fishler, G.F. Giudice, R.G. Leigh, S. Paban, Phys. Lett. B 258 Ž1991. 45. w18x F. Zwirner, Phys. Lett. B 132 Ž1983. 103. w19x R. Barbieri, A. Masiero, Nucl. Phys. B 267 Ž1986. 679. w20x H. Dreiner, G.G. Ross, Nucl. Phys. B 410 Ž1993. 188. w21x J.L. Goity, M. Sher, Phys. Lett. B 346 Ž1995. 69; C.E. Carlson, P. Roy, M. Sher, Phys. Lett. B 357 Ž1995. 99. w22x D. Cheng, W.Y. Keung, Phys. Lett. B 389 Ž1996. 294.