- Email: [email protected]
Electroweak baryogenesis in a left–right supersymmetric model
31 December 1998
Physics Letters B 445 Ž1998. 199–205
Electroweak baryogenesis in a left–right supersymmetric model Tuomas Multamaki ¨ 1, Iiro Vilja
2
Department of Physics, UniÕersity of Turku, FIN-20014 Turku, Finland Received 17 October 1998; revised 10 November 1998 Editor: P.V. Landshoff
Abstract The possibility of electroweak baryogenesis is considered within the framework of a left–right supersymmetric model. It is shown that for a range of parameters the large sneutrino VEV required for parity breaking varies at the electroweak phase transition leading to a production of baryons. The resulting baryon to entropy ratio is approximated to be n Brs ; a 0.7 = 10y8 , where a is the angle that the phase of sneutrino VEV changes at the electroweak phase transition. q 1998 Elsevier Science B.V. All rights reserved.
The Standard Model ŽSM. fulfills all the requirements, including the Sakharov conditions w1x, for electroweak baryogenesis w2x that may be responsible for the observed baryon asymmetry of the universe. However, in the SM the phase transition has been shown to be too weakly first order to preserve the generated asymmetry w3x. Furthermore, the phase of the Kobayashi-Maskawa -matrix leads to an insufficient CP-violation w4x which further indicates that physics beyond the SM is needed to account for the baryon asymmetry. One of the most popular extensions of the SM is the Minimal Supersymmetric Standard Model ŽMSSM. that, with an appropriate choice of parameters, leads to a large enough baryon asymmetry, n B rs ; 10y1 0 w5,6x. Some of the parameter bounds are, however, quite stringent w7x Žalthough Riotto has recently argued w8x that an additional enhancement factor of about 10 2 is present..
1 2
E-mail: [email protected] E-mail: [email protected]
In the MSSM the conservation of R-parity, R s Žy1. 3Ž ByL.q2 S, is assumed and put in by hand. This is quite ad hoc since it is not required for the internal consistency of the model but for the conservation of baryon and lepton numbers. To avoid e.g. proton decay, the R-parity breaking terms, allowed by gauge invariance, are therefore usually omitted in the MSSM or their couplings are assumed to be very small. The supersymmetric left–right model ŽSUSYLR. based on the gauge group SUŽ2.L = SUŽ2.R = UŽ1.By L explicitly conserves R-parity w9x. Considering the MSSM to be a low energy approximation of the SUSYLR it can also account for the baryon asymmetry of the universe. Furthermore, the see-saw mechanism w10x that can be accommodated in the left–right models, can be used to generate light masses for left-handed neutrinos and large masses for right-handed ones. The possibility of a small Žleft-handed. neutrino mass is therefore often seen as another motivation for the SUSYLR model. By viewing the MSSM as a low energy approximation of the SUSYLR it becomes warranted 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 4 4 6 - 4
T. Multamaki, ¨ I. Vilja r Physics Letters B 445 (1998) 199–205
200
question whether some properties of the SUSYLR affect the prospect of electroweak baryogenesis. This question is studied in the context of the present paper and it is argued that electroweak baryogenesis may be significantly larger in the SUSYLR than in the MSSM. The gauge-invariant superpotential for the SUSYLR model with two bidoublets w13x at zero temperature is given by Žomitting all generation and SUŽ2. indices and the Q-terms.: W s hf LTt 2Ft 2 Lc q hx LTt 2 xt 2 Lc q if Ž LTt 2 D L q LcTt 2 D c Lc . q M tr Ž DD q D c D c . q m 1 tr Ž t 2F Tt 2 x . q mX1 tr Ž t 2F Tt 2F . q mXX1 tr Ž t 2 x Tt 2 x . ,
Ž 1.
cŽ
where LŽ2,1,y 1., L 1,2,1. are the matter superfields and DŽ3,1,2., D c Ž1,3,y 2., DŽ3,1,y 2., D cŽ1,3,2., F Ž2,2,0., x Ž2,2,0. are the Higgs superfields with their respective SUŽ2.L = SUŽ2.R = UŽ1.By L quantum numbers. It is worth noting that since one of the bidoublets will be responsible for the electron mass the mechanism considered here requires at least two bidoublet fields to result in a significant production of baryons. The scalar components of the superfields have the following component representations:
dq L˜ s
n˜ , e˜y
ž /
Fs
ž
c
n L˜c s ˜q , e˜
ž /
f 10
fq 2
fy 1
f 20
/ ž ,
xs
Ds
0 q
d
d0
xq 2
xy 1
x 20
/
y
,
'2
Ž 3.
ž /
ADGw s Õw2 s
,
Ž 4.
where g Ž k i . is a numerical coefficient depending on the degrees of freedom, D the effective diffusion rate, Gw s s 6 a w4 T Ž k s 1. the weak sphaleron rate and Õw the speed of the bubble wall. In Ref. w15x it was shown that A ; I1 ' 2 h i j h k j
H0 du sin Ž u y u i
k
. w A i AXk y A k AXi x X
ycos Ž u i y u k . A i A k Ž u i q u k . ey lq u IHe˜ j , Ž 5.
Ž 2.
xq 2 and L is the left-handed chirality y x1 projection operator. Of course similar term proporwhere H s
s
s yg Ž k i .
`
,
and similarly for the rest of the fields. By standard methods w12x it can be shown that the Lagrangian contains the following terms Žomitting generation indices.: hxH˜ cn˜ R) Le,
nB
dqq
'2
x 10
tional to hf is present in the Lagrangian but since hf is proportional to the electron mass it is neglected here w16x. The Yukawa coupling hx , however, is proportional to the neutrino Dirac mass and is hence only bounded by an experimental upper limit w16x. Comparing with Ref. w15x we note that this is exactly of the form that creates a CP-violating source in the diffusion equation at the electroweak ŽEW. phase transition assuming that ² n˜ : does not vanish and is not a constant. It has been shown that in a wide class of SUSYLR models R-parity is necessarily spontaneously broken by a large sneutrino VEV w14x. Thus for electroweak baryogenesis we need to show that ² n˜ : varies at the EW phase transition. Following Ref. w15x the CP-violating source can now be computed by using methods described in Ref. w11x. In electroweak baryogenesis the created baryon to entropy ratio is
(
where lqs Ž Õw q Õw2 q 4 G˜ D .rŽ2 D . and ² n˜ : ' Ae i u . Clearly n B rs vanishes unless A and u change when going over the bubble wall. To show that electroweak baryogenesis is indeed possible in this model we must first consider the associated Higgs potential. The Higgs potential including soft breaking terms Žomitting the Q-terms again. is w13x: V s VF - terms q Vsoft q VD - terms
Ž 6.
T. Multamaki, ¨ I. Vilja r Physics Letters B 445 (1998) 199–205
where VF - terms sN L˜Tt 2 Ž hfF q hx x . t 2 q 2 ifL˜cTt 2 D c N 2 T
q N L˜cTt 2 Ž hfF q hx x . t 2 q 2 ifL˜Tt 2 D N 2 q tr N hf L˜c L˜T q 2 mX1F T q m 1 x T N 2 q tr N hx L˜c L˜T q 2 mXX1 x T q m 1F T N 2
˜ ˜Tt 2 q M D N 2 q tr N ifL˜c L˜cTt 2 q tr N ifLL q M D c N 2 q N M N 2 tr Ž D †D q D c†D c . , Ž 7 . Vsoft s m2l Ž L˜†L˜ q L˜c†L˜c . qŽ
M12 y N M N 2
. tr Ž D D q D
c† c
D .
q M22 Tr Ž D †D q D c†D c . q N h N 2 L˜c†L˜c L˜†L˜ qNfN2
q Ž M X 2 tr Ž DD q D cD c . q h.c. . Mf2 y 4 N mX1 N 2
†
. tr F F q Ž Mx2 y 4 N mXX1 N 2 . tr x †x . 2 q Ž Mfx y 4 N m 1 N 2 . tr F †x q h.c. m22
ž
q
2
mXX2 2
T
tr Ž t 2F t 2 x . q
mX22 2
/
VD - terms s
c
g L2 8
= Ž iÕ D c q iM ) f D c† . L˜c q h.c. .
c†
² L˜c :s
/
Ž 8.
8
ž/
² D c :s 0X
² L˜ :s
l 0
ž/
Ž 11 .
0 0 , ² D :s 0 d
žd /
Ý N L˜c†tm L˜c m
² D c :s
q tr Ž 2 D tm D q 2 D c†tm D c q FtmTF † gX 2
lX , 0
and using SUŽ2.L = SUŽ2.R-invariance these can be chosen real. In Ref. w14x it was shown that for the absolute minimum of the potential, the triplet fields must be of the Q em preserving form
c
qxtmT x † . N 2 q
Ž 10 .
We will now select the VEVs of the slepton fields to be of the form
q tr Ž 2 D †tm D q 2 D †tm D q F †tmF 8
/
ž
m
g R2
/
q L˜Tt 2 Ž iÕ D q iM ) f D † . L˜ q L˜cTt 2
Ý N L˜†tm L˜
qx †tm x . N 2 q
2
.
q M X 2 Tr Ž DD q D cD c q h.c. .
qL˜ t 2 Ž e1F q e 2 x . t 2 L˜ q h.c. , T
L˜
c† c
/
q iÕ L˜Tt 2 D L˜ q L˜cTt 2 D c L˜c
ž ž
†
ž
tr Ž t 2F t 2F .
/
2
2
ž Ž L˜ L˜ . q Ž L˜
q 4 N f N 2 N L˜cTt 2 D c N 2 q N L˜Tt 2 D N 2
T
tr Ž t 2 x Tt 2 x . q h.c.
q
In Ref. w14x it was shown that the spontaneous breakdown of parity cannot occur without spontaneous R-parity breaking in a model with one bidoublet. We shall now examine the Higgs potential using assumptions similar to those in Ref. w14x. We assume that g 2 and g X 2 are much smaller than h 2 and f 2 and to ensure the required hierarchy of parity and R-parity breaking scales, the couplings involving F or x are assumed to be much smaller than those not involving them. With these approximations the potential simplifies to the same form as in Ref. w14x: V s m 2l Ž L˜†L˜ q L˜c†L˜c . q M12 Tr Ž D †D q D c†D c .
†
q Ž M22 y N M N 2 . tr Ž D †D q D c†D c .
qŽ
201
N L˜c†L˜c y L˜†L˜
q 2tr Ž D †D y D c†D c y D †D q D c†D c . N 2 .
Ž 9.
ž
0 0
dX , 0
/
ž
² D :s
0 0
0 , 0
/
d . 0
ž /
Ž 12 .
The doublet and triplet VEVs can now be determined by substituting Ž11. and Ž12. into Ž10. and finding the minimum. The algebra is quite involved and the general solutions are messy so that it is useful to approximate M12 and M X 2 as small pertur-
T. Multamaki, ¨ I. Vilja r Physics Letters B 445 (1998) 199–205
202
bations as was done in Ref. w14x. The potential now takes the form
fields. The F and x fields acquire Q em conserving VEVs,
V s m 2l Ž l 2 q lX 2 . q m22 Ž l 2 q lX 2 .
Fs
2 2
2
4
4
q h l q f Ž l ql .
Ž 13 .
The parity breaking solution is w14x l s d s 0, lX 2 s
Õ
X
d sy
4f 2
X
d sy
,
Ž Õ 2 y 4 f 2 m2l . M22 , 8 f 4 Ž M22 y M 2 .
fMlX 2
,
M22
ž
V s yf 2 1 y
M2 M22
/
Ž 14 . If we assume that only M f 0, we find a parity breaking solution Žgiven that l / 0, lX / 0.:
dsy
dsy
4 f 2 l 12 q M12 l 12 M M22
,
dXsy
,
X
d sy
l 22 M M22
Õ2 2f2
ž
1y
ž
q2 f 2 1y
M12
M2 M22
Õ2
4 f 2 l 22 q M12
,
q
8f4
ž
1y
2 f 2l2
Õ2 2f2
q2 f
2
ž
ž
1y
1y
M12 4 f 2 lX 2
M2
//
l2
M22
q
8f4
q h 2 l q O Ž M14 . s 0.
2
2
2
q m 1 k 2 q < hx lX l q m 1 k 1 < 2 q 4 < mXX1 k 2 < 2 q < f < 2 < l < 4 q < f lX 2 q Md X < 2 q m 2l Ž < l < 2 q < lX < 2 . q M12 < d X < 2 q Ž M2 y < M < 2 . < d X < 2
q
ž
ž
Mx2 y 4 <
m22 2
q
g L2
q
8 g R2
mXX1 < 2
/
2
<
/
2
k 1 k 2 q h.c. q Ž ÕlX 2d X q e 2 l k 2 lX q h.c. .
/
< l < 2 q < k1 < 2 y < k2 < 2
2
< lX < 2 q 2 Ž < d X < 2 y < d X < 2 q < k 1 < 2 y < k 2 < 2 .
2
2
< lX < 2 y < l < 2 q 2 Ž < d X < 2 y < d X < 2 . .
Ž 18 . 8 Furthermore, assuming that f, M,Õ,e 2 , d X , d X are real Žfor simplicity. and setting hf s 0 the terms containing lX are q
/ Õ2
Ž 17 .
V s hx l k 2 q 2 f lXd X q hx lXk 2 q hf llX q 2 mX1 k 1
8 gX 2
q h 2 lX q O Ž M14 . s 0 m2l y
/
where k 12 q k 22 is of the order of ; 100 GeV after the EW phase transition and k iX < k i . k 1 and k 2 can be chosen real using UŽ1.-invariance. To study the behaviour of the sneutrino VEV at the phase transition, we set the fields to their VEVs after the EW breaking Žwe also approximate d s 0, d s 0 and set k 1X s k 2X s 0 for simplicity.. The potential Ž6. simplifies to
q
Ž 15 .
M12
0 , k2
ž
,
/
4 f 2l2
ž
k2 ’ 0
q M X 2 Ž d Xd X q h.c. . q Mf2 y 4 < mX1 < 2 < k 1 < 2
l 22 Õ
and l 2 , lX 2 can be solved from equations m2l y
/
xs
2
lX 4 .
X
l 12 Õ
0 , k 1X
(
q l 2 Ž Õd q fM )d ) . q lX 2 Ž Õd X q fM )d X ) . qc.c .
ž
k1 0
VlX s 4 f 2 < lX < 2d X 2 q 4 f l k 2 d X Re Ž hx lX . q k 2 < hx < 2 < lX < 2
ž
1y
M12 2 f 2 lX 2
//
l
q < hx < 2 < lX < 2 < l < 2 q 2 k 1 l m 1 Re Ž hx lX . q f 2 < lX < 4
X2
Ž 16 .
We find that in this case l generally does not vanish. Numerical examinations, when further requiring that M X / 0, support this result. Note that after parity breaking the acquired VEVs may change as temperature falls and electroweak symmetry is broken by the VEVs of the F and x
q 2 fMd X Re Ž lX 2 . q m 2l < lX < 2 q 2 Õd X Re Ž lX 2 . q 2 e 2 l k 2 Re Ž lX . q
g R2 8 gX 2
< lX < 4 q 2 < lX < 2 Ž 2 d X 2 y 2 d X 2 q k 12 y k 22 .
< lX < 4 q 2 < lX < 2 Ž 2 d X 2 y 2 d X 2 . 8 y2 < lX < 2 < l < 2 . q
Ž 19 .
T. Multamaki, ¨ I. Vilja r Physics Letters B 445 (1998) 199–205
Writing lX s < lX < e i a , hx s he i b and differentiating with respect to the phase of the complex VEV, a , we obtain y
E VlX Ea
s 4 hf l k 2 d X < lX < q 2 h m 1 l < lX < k 1 sin Ž a q b .
q 2 e 2 l k 2 < lX
Ž 20 .
Ž< lX < can be assumed to be approximately constant to ensure parity breaking and the required heavy righthanded particle masses.. We note that for a / 0 we must require that l / 0. If b s 0 we may easily solve VlXra s 0 for a :
ž
a 1 s arccos y
4 Ž f M d X q Õd X . < l X <
/
.
Ž 21 . To examine when this is a minimum we substitute a 1 into Ž19. Žwriting only the a-dependent terms.: VlX Ž a s a 1 . s y
Ž 2 hf l k 2 d X q h m 1 l k 1 q e2 l k 2 . X
4 Ž f M d q Õd
X
At a s 0 Žthe old minimum. Ž19. is Žagain writing only the a dependent part.: X
sin a
lX 2 Ž 1 y cos a . e iarctanŽ cos ay1 . s Ae i u .
(
X
X X2
VlX Ž a s 0 . s 4 f l k 2 d hl q 2 k 1 l m 1 hl q 2 fMd l q 2 Õd X l X 2 q 2 e 2 l k 2 l X .
Ž 23 .
To assure that a 1 is a minimum, we must thus require that VlX Ž a 1 . - VlX Ž 0 .
Ž 24 .
from which it follows that 4 hf l k 2 d X lX q 2 h m 1 l k 1 lX q 2 e 2 l k 2 lX
(
I1 f 48 h 2 lX 2 Ž 1 y cos a . =arccot
ž
`
=
H0 du g
sin a 1 y cos a 2
q
Ž 2 hf l k 2 d q h m 1 l k 1 q e2 l k 2 . 4 Ž f M d X q Õd X .
/
Ž u . g X Ž u . ey lq u ,
Ž 28 .
where g Ž u . s 12 1 y cos Ž uprL . = u Ž u. y u Ž u y L. q u Ž u y L.
Ž 29 .
and h ; h 3 i , i s 1,2,3. For T s 100 GeV, L s 25rT the integral has a value of ; 0.26. To estimate the value of I2 let us assume further that lX ; 1 TeV and h ; 0.1 so that I1 f Ž 1 y cos a . arccot
q 4 f Md X lX 2 q 4Õd X lX 2 X
Ž 27 .
In the EW phase transition, A changes from 0 to lX 2 Ž 1 y cos a . while the change in u is D u s a . arccotŽ 1 ysincos a . We estimate the changes in A and u to be sinusoidal and choose that only one of the sneutrino VEVs is significantly large. In Ref. w15x it was shown that for T f m ; 100 GeV IHe˜ j is approximately y2 regardless of the considered generation so that Ž5. can be written as
2
Ž 22 .
Ž 26 .
from which it follows that
.
y 2 f M d X l X 2 y 2 Õd X l X 2 .
X
electroweak transition the sneutrino VEV, lX , while remaining approximately constant in magnitude rotates in the complex plane by an angle a 1. Expanding around the new minimum, we may write lXEW ' lX e i a s lX q Ae i u ,
q 4 Ž f Md X q Õd X . < lX < 2 sin Ž 2 a .
2 hf l k 2 d X q h m 1 l k 1 q e 2 l k 2
203
ž
sin a 1 y cos a
/
1.2 = 10 5 .
Ž 30 . 2
- 0.
Ž 25 .
If b / 0 i.e. hx is complex, this question cannot be answered analytically and numerical methods must be utilized. Provided that condition Ž25. is satisfied, it is then clear that as F and x acquire their VEVs at the
The effect of adding a mass insertion can easily be calculated. Taking lX to be constant, Ž5. reduces to `
y2 h i j h k j
H0 du cos Ž u y u i
X
k
.
= A i A k Ž u i q u k . ey lq u IHe˜ j .
Ž 31 .
T. Multamaki, ¨ I. Vilja r Physics Letters B 445 (1998) 199–205
204
If only one of the sneutrino fields, A 3 e i u 3 , has a significant VEV ; 1 TeV, Ž31. becomes I2 f 24 h 2 lX 2
`
H0 du u
X y lq u , 3e
Ž 32 .
where lX is the magnitude of the sneutrino VEV and IHe˜ j f y2 as before. The phase of the sneutrino field VEV is again assumed to change sinusoidally during the EW phase transition i.e.
u 3 Ž u. s a g Ž u.
Ž 33 .
so that the integral in Ž32. can be evaluated and is found to be
p2 2 2 2 Ž lq L qp 2 .
Ž 1 q ey lq L . a
Ž 34 .
For L s 25rT, T s 100 GeV and lqf 1.5 this has a value of about 0.83 a . Taking h ; 0.1 and lX ; 1 TeV, we arrive at an estimate for CP-violation during the electroweak phase transition with one mass insertion, I2 f 2 = 10 5 a .
Acknowledgements
Ž 35 . The authors thank R. Mohapatra for discussions.
Clearly, since
Ž 1 y cos a . arccot
left–right model. As in Ref. w15x the crucial property that allows for sufficient baryon creation is the large sneutrino VEV. In SUSYLR this is set quite naturally by the left–right parity breaking scale. As electroweak breaking takes place the sneutrino VEV must necessarily change for baryogenesis to be possible. We have shown that for a zero temperature approximation there exists a part of the parameter space where the sneutrino VEV rotates in the complex plane while conserving its large amplitude. This process creates a CP-violating source that leads to a production of baryons at the electroweak phase transition through a known mechanism w6x. The amount of baryons produced in this manner is obviously dependent on many unknown parameters and may change significantly due to finite temperature corrections but even a change of O Ž10y3 . in the phase of the neutrino VEV may lead to a significant production of baryons.
ž
sin a 1 y cos a
/
Fa
for a g w0,p x, I1 - I2 indicating that further mass insertions may possibly increase the amount of CPviolation even further. In comparison with I2 the estimate for the CPviolation in the MSSM w6,15x is ; 300 Žwithout the suppression factor w7x. that leads to a baryon to entropy ratio of n B rs ; 10y1 1. Because of Ž4. the baryon to entropy ratio is then nB ; a 0.7 = 10y8 Ž 36 . s so that even a change of O Ž10y3 . in the phase of the sneutrino VEV may lead to significant baryon production at the EW phase transition. This result is obtained after quite a number of assumptions and simplifications. However, these are not carefully chosen to obtain a desired result but are used to simplify the calculation process. In this paper we have considered the possibility of electroweak baryogenesis in the supersymmetric
References w1x A.D. Sakharov, JETP Lett. 5 Ž1967. 24. w2x For recent reviews, see A.G. Cohen, D.B. Kaplan, A.E. Nelson, Ann. Rev. Nucl. Part. Phys. 43 Ž1993. 27; M. Quiros, Helv. Phys. Acta 67 Ž1994. 451; M.E. Shaposhnikov, preprint CERN-THr96-13. w3x M. Shaposhnikov, JETP Lett. 44 Ž1986. 465; Nucl. Phys. B 287 Ž1987. 757; B 299 Ž1988. 797. w4x M.B. Gavela, P. Hernandez, ´ J. Orloff, O. Pene, ` C. Quimbay, Mod. Phys. Lett. A 9 Ž94. 795; Nucl. Phys. B 430 Ž94. 382; P. Huet, E. Sather, Phys. Rev. D 51 Ž95. 379. w5x M. Carena, M. Quiros, C.E.M. Wagner, Phys. Lett. B 380 Ž1996. 81. w6x M. Carena, M. Quiros, A. Riotto, I. Vilja, C.E.M. Wagner, Nucl. Phys. B 503 Ž1997. 387. w7x T. Multamaki, ¨ I. Vilja, Phys. Lett. B 411 Ž1997. 301. w8x A. Riotto, hep-phr9803357. w9x R.N. Mohapatra, Phys. Rev. D 34 Ž1986. 3457; A. Font, L.E. Ibanez, F. Quevedo, Phys. Lett. B 228 Ž1989. 79; S.P. Martin, Phys. Rev. D 46 Ž1992. 2769. w10x M. Gell-Mann, P. Ramond, R. Slansky, in: P. van Niewenhuizen, D.Z. Freedman ŽEds.., Supergravity, North Holland, 1979; T. Yanagida, in: O. Sawada, A. Sugamoto ŽEds..,
T. Multamaki, ¨ I. Vilja r Physics Letters B 445 (1998) 199–205 Proceedings of Workshop on Unified Theory and Baryon Number in the Universe, KEK, 1979. w11x A. Riotto, Nucl. Phys. B 518 Ž1998. 339. w12x H. Haber, G. Kane, Phys. Lett. 117 Ž1985. 75. w13x M. Cvetic, J. Pati, Phys. Lett. B 135 Ž57. 1984; R. Mohapatra, Prog. in Part. and Fields 26 Ž1991. 1; R. Francis, M.
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Frank, C. Kalman, Phys. Rev. D 43 Ž1991. 2369; Y. Ahn, Phys. Lett. B 149 Ž1984. 337. w14x R. Kuchimanhci, R. Mohapatra, Phys. Rev. D 48 Ž1993. 4352. w15x T. Multamaki, ¨ I. Vilja, Phys. Lett. B 433 Ž1998. 67. w16x K. Huitu, J. Maalampi, Phys. Lett. B 344 Ž1995. 217.
Physics Letters B 445 Ž1998. 199–205
Electroweak baryogenesis in a left–right supersymmetric model Tuomas Multamaki ¨ 1, Iiro Vilja
2
Department of Physics, UniÕersity of Turku, FIN-20014 Turku, Finland Received 17 October 1998; revised 10 November 1998 Editor: P.V. Landshoff
Abstract The possibility of electroweak baryogenesis is considered within the framework of a left–right supersymmetric model. It is shown that for a range of parameters the large sneutrino VEV required for parity breaking varies at the electroweak phase transition leading to a production of baryons. The resulting baryon to entropy ratio is approximated to be n Brs ; a 0.7 = 10y8 , where a is the angle that the phase of sneutrino VEV changes at the electroweak phase transition. q 1998 Elsevier Science B.V. All rights reserved.
The Standard Model ŽSM. fulfills all the requirements, including the Sakharov conditions w1x, for electroweak baryogenesis w2x that may be responsible for the observed baryon asymmetry of the universe. However, in the SM the phase transition has been shown to be too weakly first order to preserve the generated asymmetry w3x. Furthermore, the phase of the Kobayashi-Maskawa -matrix leads to an insufficient CP-violation w4x which further indicates that physics beyond the SM is needed to account for the baryon asymmetry. One of the most popular extensions of the SM is the Minimal Supersymmetric Standard Model ŽMSSM. that, with an appropriate choice of parameters, leads to a large enough baryon asymmetry, n B rs ; 10y1 0 w5,6x. Some of the parameter bounds are, however, quite stringent w7x Žalthough Riotto has recently argued w8x that an additional enhancement factor of about 10 2 is present..
1 2
E-mail: [email protected] E-mail: [email protected]
In the MSSM the conservation of R-parity, R s Žy1. 3Ž ByL.q2 S, is assumed and put in by hand. This is quite ad hoc since it is not required for the internal consistency of the model but for the conservation of baryon and lepton numbers. To avoid e.g. proton decay, the R-parity breaking terms, allowed by gauge invariance, are therefore usually omitted in the MSSM or their couplings are assumed to be very small. The supersymmetric left–right model ŽSUSYLR. based on the gauge group SUŽ2.L = SUŽ2.R = UŽ1.By L explicitly conserves R-parity w9x. Considering the MSSM to be a low energy approximation of the SUSYLR it can also account for the baryon asymmetry of the universe. Furthermore, the see-saw mechanism w10x that can be accommodated in the left–right models, can be used to generate light masses for left-handed neutrinos and large masses for right-handed ones. The possibility of a small Žleft-handed. neutrino mass is therefore often seen as another motivation for the SUSYLR model. By viewing the MSSM as a low energy approximation of the SUSYLR it becomes warranted 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 4 4 6 - 4
T. Multamaki, ¨ I. Vilja r Physics Letters B 445 (1998) 199–205
200
question whether some properties of the SUSYLR affect the prospect of electroweak baryogenesis. This question is studied in the context of the present paper and it is argued that electroweak baryogenesis may be significantly larger in the SUSYLR than in the MSSM. The gauge-invariant superpotential for the SUSYLR model with two bidoublets w13x at zero temperature is given by Žomitting all generation and SUŽ2. indices and the Q-terms.: W s hf LTt 2Ft 2 Lc q hx LTt 2 xt 2 Lc q if Ž LTt 2 D L q LcTt 2 D c Lc . q M tr Ž DD q D c D c . q m 1 tr Ž t 2F Tt 2 x . q mX1 tr Ž t 2F Tt 2F . q mXX1 tr Ž t 2 x Tt 2 x . ,
Ž 1.
cŽ
where LŽ2,1,y 1., L 1,2,1. are the matter superfields and DŽ3,1,2., D c Ž1,3,y 2., DŽ3,1,y 2., D cŽ1,3,2., F Ž2,2,0., x Ž2,2,0. are the Higgs superfields with their respective SUŽ2.L = SUŽ2.R = UŽ1.By L quantum numbers. It is worth noting that since one of the bidoublets will be responsible for the electron mass the mechanism considered here requires at least two bidoublet fields to result in a significant production of baryons. The scalar components of the superfields have the following component representations:
dq L˜ s
n˜ , e˜y
ž /
Fs
ž
c
n L˜c s ˜q , e˜
ž /
f 10
fq 2
fy 1
f 20
/ ž ,
xs
Ds
0 q
d
d0
xq 2
xy 1
x 20
/
y
,
'2
Ž 3.
ž /
ADGw s Õw2 s
,
Ž 4.
where g Ž k i . is a numerical coefficient depending on the degrees of freedom, D the effective diffusion rate, Gw s s 6 a w4 T Ž k s 1. the weak sphaleron rate and Õw the speed of the bubble wall. In Ref. w15x it was shown that A ; I1 ' 2 h i j h k j
H0 du sin Ž u y u i
k
. w A i AXk y A k AXi x X
ycos Ž u i y u k . A i A k Ž u i q u k . ey lq u IHe˜ j , Ž 5.
Ž 2.
xq 2 and L is the left-handed chirality y x1 projection operator. Of course similar term proporwhere H s
s
s yg Ž k i .
`
,
and similarly for the rest of the fields. By standard methods w12x it can be shown that the Lagrangian contains the following terms Žomitting generation indices.: hxH˜ cn˜ R) Le,
nB
dqq
'2
x 10
tional to hf is present in the Lagrangian but since hf is proportional to the electron mass it is neglected here w16x. The Yukawa coupling hx , however, is proportional to the neutrino Dirac mass and is hence only bounded by an experimental upper limit w16x. Comparing with Ref. w15x we note that this is exactly of the form that creates a CP-violating source in the diffusion equation at the electroweak ŽEW. phase transition assuming that ² n˜ : does not vanish and is not a constant. It has been shown that in a wide class of SUSYLR models R-parity is necessarily spontaneously broken by a large sneutrino VEV w14x. Thus for electroweak baryogenesis we need to show that ² n˜ : varies at the EW phase transition. Following Ref. w15x the CP-violating source can now be computed by using methods described in Ref. w11x. In electroweak baryogenesis the created baryon to entropy ratio is
(
where lqs Ž Õw q Õw2 q 4 G˜ D .rŽ2 D . and ² n˜ : ' Ae i u . Clearly n B rs vanishes unless A and u change when going over the bubble wall. To show that electroweak baryogenesis is indeed possible in this model we must first consider the associated Higgs potential. The Higgs potential including soft breaking terms Žomitting the Q-terms again. is w13x: V s VF - terms q Vsoft q VD - terms
Ž 6.
T. Multamaki, ¨ I. Vilja r Physics Letters B 445 (1998) 199–205
where VF - terms sN L˜Tt 2 Ž hfF q hx x . t 2 q 2 ifL˜cTt 2 D c N 2 T
q N L˜cTt 2 Ž hfF q hx x . t 2 q 2 ifL˜Tt 2 D N 2 q tr N hf L˜c L˜T q 2 mX1F T q m 1 x T N 2 q tr N hx L˜c L˜T q 2 mXX1 x T q m 1F T N 2
˜ ˜Tt 2 q M D N 2 q tr N ifL˜c L˜cTt 2 q tr N ifLL q M D c N 2 q N M N 2 tr Ž D †D q D c†D c . , Ž 7 . Vsoft s m2l Ž L˜†L˜ q L˜c†L˜c . qŽ
M12 y N M N 2
. tr Ž D D q D
c† c
D .
q M22 Tr Ž D †D q D c†D c . q N h N 2 L˜c†L˜c L˜†L˜ qNfN2
q Ž M X 2 tr Ž DD q D cD c . q h.c. . Mf2 y 4 N mX1 N 2
†
. tr F F q Ž Mx2 y 4 N mXX1 N 2 . tr x †x . 2 q Ž Mfx y 4 N m 1 N 2 . tr F †x q h.c. m22
ž
q
2
mXX2 2
T
tr Ž t 2F t 2 x . q
mX22 2
/
VD - terms s
c
g L2 8
= Ž iÕ D c q iM ) f D c† . L˜c q h.c. .
c†
² L˜c :s
/
Ž 8.
8
ž/
² D c :s 0X
² L˜ :s
l 0
ž/
Ž 11 .
0 0 , ² D :s 0 d
žd /
Ý N L˜c†tm L˜c m
² D c :s
q tr Ž 2 D tm D q 2 D c†tm D c q FtmTF † gX 2
lX , 0
and using SUŽ2.L = SUŽ2.R-invariance these can be chosen real. In Ref. w14x it was shown that for the absolute minimum of the potential, the triplet fields must be of the Q em preserving form
c
qxtmT x † . N 2 q
Ž 10 .
We will now select the VEVs of the slepton fields to be of the form
q tr Ž 2 D †tm D q 2 D †tm D q F †tmF 8
/
ž
m
g R2
/
q L˜Tt 2 Ž iÕ D q iM ) f D † . L˜ q L˜cTt 2
Ý N L˜†tm L˜
qx †tm x . N 2 q
2
.
q M X 2 Tr Ž DD q D cD c q h.c. .
qL˜ t 2 Ž e1F q e 2 x . t 2 L˜ q h.c. , T
L˜
c† c
/
q iÕ L˜Tt 2 D L˜ q L˜cTt 2 D c L˜c
ž ž
†
ž
tr Ž t 2F t 2F .
/
2
2
ž Ž L˜ L˜ . q Ž L˜
q 4 N f N 2 N L˜cTt 2 D c N 2 q N L˜Tt 2 D N 2
T
tr Ž t 2 x Tt 2 x . q h.c.
q
In Ref. w14x it was shown that the spontaneous breakdown of parity cannot occur without spontaneous R-parity breaking in a model with one bidoublet. We shall now examine the Higgs potential using assumptions similar to those in Ref. w14x. We assume that g 2 and g X 2 are much smaller than h 2 and f 2 and to ensure the required hierarchy of parity and R-parity breaking scales, the couplings involving F or x are assumed to be much smaller than those not involving them. With these approximations the potential simplifies to the same form as in Ref. w14x: V s m 2l Ž L˜†L˜ q L˜c†L˜c . q M12 Tr Ž D †D q D c†D c .
†
q Ž M22 y N M N 2 . tr Ž D †D q D c†D c .
qŽ
201
N L˜c†L˜c y L˜†L˜
q 2tr Ž D †D y D c†D c y D †D q D c†D c . N 2 .
Ž 9.
ž
0 0
dX , 0
/
ž
² D :s
0 0
0 , 0
/
d . 0
ž /
Ž 12 .
The doublet and triplet VEVs can now be determined by substituting Ž11. and Ž12. into Ž10. and finding the minimum. The algebra is quite involved and the general solutions are messy so that it is useful to approximate M12 and M X 2 as small pertur-
T. Multamaki, ¨ I. Vilja r Physics Letters B 445 (1998) 199–205
202
bations as was done in Ref. w14x. The potential now takes the form
fields. The F and x fields acquire Q em conserving VEVs,
V s m 2l Ž l 2 q lX 2 . q m22 Ž l 2 q lX 2 .
Fs
2 2
2
4
4
q h l q f Ž l ql .
Ž 13 .
The parity breaking solution is w14x l s d s 0, lX 2 s
Õ
X
d sy
4f 2
X
d sy
,
Ž Õ 2 y 4 f 2 m2l . M22 , 8 f 4 Ž M22 y M 2 .
fMlX 2
,
M22
ž
V s yf 2 1 y
M2 M22
/
Ž 14 . If we assume that only M f 0, we find a parity breaking solution Žgiven that l / 0, lX / 0.:
dsy
dsy
4 f 2 l 12 q M12 l 12 M M22
,
dXsy
,
X
d sy
l 22 M M22
Õ2 2f2
ž
1y
ž
q2 f 2 1y
M12
M2 M22
Õ2
4 f 2 l 22 q M12
,
q
8f4
ž
1y
2 f 2l2
Õ2 2f2
q2 f
2
ž
ž
1y
1y
M12 4 f 2 lX 2
M2
//
l2
M22
q
8f4
q h 2 l q O Ž M14 . s 0.
2
2
2
q m 1 k 2 q < hx lX l q m 1 k 1 < 2 q 4 < mXX1 k 2 < 2 q < f < 2 < l < 4 q < f lX 2 q Md X < 2 q m 2l Ž < l < 2 q < lX < 2 . q M12 < d X < 2 q Ž M2 y < M < 2 . < d X < 2
q
ž
ž
Mx2 y 4 <
m22 2
q
g L2
q
8 g R2
mXX1 < 2
/
2
<
/
2
k 1 k 2 q h.c. q Ž ÕlX 2d X q e 2 l k 2 lX q h.c. .
/
< l < 2 q < k1 < 2 y < k2 < 2
2
< lX < 2 q 2 Ž < d X < 2 y < d X < 2 q < k 1 < 2 y < k 2 < 2 .
2
2
< lX < 2 y < l < 2 q 2 Ž < d X < 2 y < d X < 2 . .
Ž 18 . 8 Furthermore, assuming that f, M,Õ,e 2 , d X , d X are real Žfor simplicity. and setting hf s 0 the terms containing lX are q
/ Õ2
Ž 17 .
V s hx l k 2 q 2 f lXd X q hx lXk 2 q hf llX q 2 mX1 k 1
8 gX 2
q h 2 lX q O Ž M14 . s 0 m2l y
/
where k 12 q k 22 is of the order of ; 100 GeV after the EW phase transition and k iX < k i . k 1 and k 2 can be chosen real using UŽ1.-invariance. To study the behaviour of the sneutrino VEV at the phase transition, we set the fields to their VEVs after the EW breaking Žwe also approximate d s 0, d s 0 and set k 1X s k 2X s 0 for simplicity.. The potential Ž6. simplifies to
q
Ž 15 .
M12
0 , k2
ž
,
/
4 f 2l2
ž
k2 ’ 0
q M X 2 Ž d Xd X q h.c. . q Mf2 y 4 < mX1 < 2 < k 1 < 2
l 22 Õ
and l 2 , lX 2 can be solved from equations m2l y
/
xs
2
lX 4 .
X
l 12 Õ
0 , k 1X
(
q l 2 Ž Õd q fM )d ) . q lX 2 Ž Õd X q fM )d X ) . qc.c .
ž
k1 0
VlX s 4 f 2 < lX < 2d X 2 q 4 f l k 2 d X Re Ž hx lX . q k 2 < hx < 2 < lX < 2
ž
1y
M12 2 f 2 lX 2
//
l
q < hx < 2 < lX < 2 < l < 2 q 2 k 1 l m 1 Re Ž hx lX . q f 2 < lX < 4
X2
Ž 16 .
We find that in this case l generally does not vanish. Numerical examinations, when further requiring that M X / 0, support this result. Note that after parity breaking the acquired VEVs may change as temperature falls and electroweak symmetry is broken by the VEVs of the F and x
q 2 fMd X Re Ž lX 2 . q m 2l < lX < 2 q 2 Õd X Re Ž lX 2 . q 2 e 2 l k 2 Re Ž lX . q
g R2 8 gX 2
< lX < 4 q 2 < lX < 2 Ž 2 d X 2 y 2 d X 2 q k 12 y k 22 .
< lX < 4 q 2 < lX < 2 Ž 2 d X 2 y 2 d X 2 . 8 y2 < lX < 2 < l < 2 . q
Ž 19 .
T. Multamaki, ¨ I. Vilja r Physics Letters B 445 (1998) 199–205
Writing lX s < lX < e i a , hx s he i b and differentiating with respect to the phase of the complex VEV, a , we obtain y
E VlX Ea
s 4 hf l k 2 d X < lX < q 2 h m 1 l < lX < k 1 sin Ž a q b .
q 2 e 2 l k 2 < lX
Ž 20 .
Ž< lX < can be assumed to be approximately constant to ensure parity breaking and the required heavy righthanded particle masses.. We note that for a / 0 we must require that l / 0. If b s 0 we may easily solve VlXra s 0 for a :
ž
a 1 s arccos y
4 Ž f M d X q Õd X . < l X <
/
.
Ž 21 . To examine when this is a minimum we substitute a 1 into Ž19. Žwriting only the a-dependent terms.: VlX Ž a s a 1 . s y
Ž 2 hf l k 2 d X q h m 1 l k 1 q e2 l k 2 . X
4 Ž f M d q Õd
X
At a s 0 Žthe old minimum. Ž19. is Žagain writing only the a dependent part.: X
sin a
lX 2 Ž 1 y cos a . e iarctanŽ cos ay1 . s Ae i u .
(
X
X X2
VlX Ž a s 0 . s 4 f l k 2 d hl q 2 k 1 l m 1 hl q 2 fMd l q 2 Õd X l X 2 q 2 e 2 l k 2 l X .
Ž 23 .
To assure that a 1 is a minimum, we must thus require that VlX Ž a 1 . - VlX Ž 0 .
Ž 24 .
from which it follows that 4 hf l k 2 d X lX q 2 h m 1 l k 1 lX q 2 e 2 l k 2 lX
(
I1 f 48 h 2 lX 2 Ž 1 y cos a . =arccot
ž
`
=
H0 du g
sin a 1 y cos a 2
q
Ž 2 hf l k 2 d q h m 1 l k 1 q e2 l k 2 . 4 Ž f M d X q Õd X .
/
Ž u . g X Ž u . ey lq u ,
Ž 28 .
where g Ž u . s 12 1 y cos Ž uprL . = u Ž u. y u Ž u y L. q u Ž u y L.
Ž 29 .
and h ; h 3 i , i s 1,2,3. For T s 100 GeV, L s 25rT the integral has a value of ; 0.26. To estimate the value of I2 let us assume further that lX ; 1 TeV and h ; 0.1 so that I1 f Ž 1 y cos a . arccot
q 4 f Md X lX 2 q 4Õd X lX 2 X
Ž 27 .
In the EW phase transition, A changes from 0 to lX 2 Ž 1 y cos a . while the change in u is D u s a . arccotŽ 1 ysincos a . We estimate the changes in A and u to be sinusoidal and choose that only one of the sneutrino VEVs is significantly large. In Ref. w15x it was shown that for T f m ; 100 GeV IHe˜ j is approximately y2 regardless of the considered generation so that Ž5. can be written as
2
Ž 22 .
Ž 26 .
from which it follows that
.
y 2 f M d X l X 2 y 2 Õd X l X 2 .
X
electroweak transition the sneutrino VEV, lX , while remaining approximately constant in magnitude rotates in the complex plane by an angle a 1. Expanding around the new minimum, we may write lXEW ' lX e i a s lX q Ae i u ,
q 4 Ž f Md X q Õd X . < lX < 2 sin Ž 2 a .
2 hf l k 2 d X q h m 1 l k 1 q e 2 l k 2
203
ž
sin a 1 y cos a
/
1.2 = 10 5 .
Ž 30 . 2
- 0.
Ž 25 .
If b / 0 i.e. hx is complex, this question cannot be answered analytically and numerical methods must be utilized. Provided that condition Ž25. is satisfied, it is then clear that as F and x acquire their VEVs at the
The effect of adding a mass insertion can easily be calculated. Taking lX to be constant, Ž5. reduces to `
y2 h i j h k j
H0 du cos Ž u y u i
X
k
.
= A i A k Ž u i q u k . ey lq u IHe˜ j .
Ž 31 .
T. Multamaki, ¨ I. Vilja r Physics Letters B 445 (1998) 199–205
204
If only one of the sneutrino fields, A 3 e i u 3 , has a significant VEV ; 1 TeV, Ž31. becomes I2 f 24 h 2 lX 2
`
H0 du u
X y lq u , 3e
Ž 32 .
where lX is the magnitude of the sneutrino VEV and IHe˜ j f y2 as before. The phase of the sneutrino field VEV is again assumed to change sinusoidally during the EW phase transition i.e.
u 3 Ž u. s a g Ž u.
Ž 33 .
so that the integral in Ž32. can be evaluated and is found to be
p2 2 2 2 Ž lq L qp 2 .
Ž 1 q ey lq L . a
Ž 34 .
For L s 25rT, T s 100 GeV and lqf 1.5 this has a value of about 0.83 a . Taking h ; 0.1 and lX ; 1 TeV, we arrive at an estimate for CP-violation during the electroweak phase transition with one mass insertion, I2 f 2 = 10 5 a .
Acknowledgements
Ž 35 . The authors thank R. Mohapatra for discussions.
Clearly, since
Ž 1 y cos a . arccot
left–right model. As in Ref. w15x the crucial property that allows for sufficient baryon creation is the large sneutrino VEV. In SUSYLR this is set quite naturally by the left–right parity breaking scale. As electroweak breaking takes place the sneutrino VEV must necessarily change for baryogenesis to be possible. We have shown that for a zero temperature approximation there exists a part of the parameter space where the sneutrino VEV rotates in the complex plane while conserving its large amplitude. This process creates a CP-violating source that leads to a production of baryons at the electroweak phase transition through a known mechanism w6x. The amount of baryons produced in this manner is obviously dependent on many unknown parameters and may change significantly due to finite temperature corrections but even a change of O Ž10y3 . in the phase of the neutrino VEV may lead to a significant production of baryons.
ž
sin a 1 y cos a
/
Fa
for a g w0,p x, I1 - I2 indicating that further mass insertions may possibly increase the amount of CPviolation even further. In comparison with I2 the estimate for the CPviolation in the MSSM w6,15x is ; 300 Žwithout the suppression factor w7x. that leads to a baryon to entropy ratio of n B rs ; 10y1 1. Because of Ž4. the baryon to entropy ratio is then nB ; a 0.7 = 10y8 Ž 36 . s so that even a change of O Ž10y3 . in the phase of the sneutrino VEV may lead to significant baryon production at the EW phase transition. This result is obtained after quite a number of assumptions and simplifications. However, these are not carefully chosen to obtain a desired result but are used to simplify the calculation process. In this paper we have considered the possibility of electroweak baryogenesis in the supersymmetric
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