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Spontaneous CP violation in the minimal supersymmetric standard model at finite temperature
Physics Letters B 306 ( ! 993 ) 67-72 North-Holland
PHYSICS LETTERS B
Spontaneous CP violation in the Minimal Supersymmetric Standard Model at finite temperature Denis Comelli Dipartimento di Fisica Teorica, Universitdt di Trieste, Strada Costiera 1I, Miramare, 1-34014 Trieste, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, 1-34014 Trieste, Italy
and Massimo Pietroni Dipartimento di Fisica, Universit& di Padova, Via Marzolo 8, 1-35100 Padua, Italy and Istituto Nazionale di Fisica Nucleate, Sezione di Padova, 1-35100 Padua, Italy
Received 5 February 1993 Editor: R. Gatto
We show that in the Minimal Supersymmetric Standard Model one-loop effects at finite temperature may lead to a spontaneous breaking of CP invariance in the scalar sector. Requiring that the breaking takes place at the critical temperature for the electroweak phase transition, we find that the parameter space is compatible with a mass of the Higgs pseudoscalar in agreement with the present experimental lower bounds. Possible implications for baryogenesis are discussed.
The possibility o f a spontaneous breakdown o f the C P symmetry in the scalar sector o f the Minimal Supersymmetric Standard Model (MSSM), has been recently investigated [ 1,2 ]. It is well known that, as long as supersymmetry (SUSY) is exact, C P is conserved in the scalar sector o f the MSSM. As a consequence, the only CPviolating effects may arise from the soft SUSY-breaking terms: the scalar masses, the gaugino masses, and the trilinear interactions. If one allows these terms to be complex, then there are two new physical phases in the MSSM which are not present in the Standard Model [ 3 ]. These phases do not appear in the tree-level Higgs potential, but occur in interactions involving the superpartners of the ordinary particles, giving new contributions to the CP-odd observables e, e', and the electric dipole m o m e n t o f the neutron [ 3 ]. If one assumes that all the soft masses and couplings are real, then no new C P violation appears at the tree level. At the one-loop level, the contributions o f graphs with sfermions, charginos, neutralinos and Higgs scalars in the internal lines induce a finite renormalization to the tree-level couplings o f the scalar potential, which may lead to a phase shift between the vacuum expectation values o f the two neutral Higgs fields, i.e. to a spontaneous breaking o f C P in the scalar sector [ 1,2 ]. Unfortunately it comes out that this scenario is not realistic. The point is that spontaneous C P violation can be implemented radiatively only if a pseudoscalar with zero tree-level mass exists, as was shown on general grounds by Georgi and Pais [4]. In the MSSM this implies the existence of a very light Higgs (with a mass o f a few GeV, given by one-loop contributions) [ 2 ], which has been excluded by LEP data [ 5 ]. In the present paper we analyze the possibility o f the spontaneous breakdown of C P at finite temperature in the MSSM. We find that new CP-violating contributions arise from the one-loop corrections at T # 0, so that the effective potential (at T ~ 0) may have a C P non-conserving vacuum, while the pseudoscalar mass, calculated 0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All fights reserved.
67
Volume 306, number 1,2
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27 May 1993
at T = 0, is still compatible with the experimental bounds. As Tgoes to zero the phase of the vacuum expectation values of the Higgs fields vanishes, and the only sources of CP violation remaining are the phase in the CabibboKobayashi-Maskawa matrix and the tTparameter of the QCD vacuum. Nevertheless, this effect may be of great relevance for the electroweak baryogenesis in the MSSM, as it could give rise to a new, time varying, CP-violating phase in the scalar sector, whose size is nearly unbounded by the phenomenology at zero temperature ~. The most general gauge invariant scalar potential for the two-doublet model, along the neutral components, is given by
V=m 2 In112+m 2 I n 2 1 2 - (m2nlH2 +h.c.) +21 In114+22 IH2 ]4"~-23 Inl Iz IH2 [2 dl-24 IHIH21 z + [25(HIH2)2+26 IH~ 12H1//2 +27 In212H1 H2 +h.c. ] ,
(1)
where we assume m32, 25, 26 and 27 to be real, so that CP invariance may be broken only if the vacuum expectation values of the Higgs fields get a non-trivial phase,
64:0, n,
(2)
where we have defined ( H i ) = vl, ( H 2 ) = vzein. Eq. (2) is satisfied if, and only if, 25>0,
(2)
and -- 1
--26V2 --27V2
<
1.
(4)
425vl v2 At the tree level the parameters 2~ (i = 1, ..., 7 ) are fixed by supersymmetry:
2~=22=~(gZ+g~),
23=~(g2Z-g2),
24--
~ 2 ~g2,
25=26=27=0,
(5)
where g2 and gl are the gauge couplings of SU (2)i. and U ( 1 ) r respectively. From eqs. (3) and (5) we immediately read that, at the tree level, CP is not violated in the scalar sector of the MSSM. Since it is the soft breaking of supersymmetry that allows CP violation, the one-loop contributions to the CP-violating couplings 25, 26 and 27 will be proportional to the soft parameters: the gaugino masses M1,2, the sfermion masses r~ 2, the trilinear scalar coupling A, and the bilinear one, B. For this reason, we will include in the one-loop effective potential only those field-dependent mass matrices which contain these parameters. The dominant contributions are given by the stop for the bosonic sector and by chargino and neutralino for the fermionic one. At finite temperature the one-loop contribution to the effective potential can be decomposed into the sum of a T = 0 and a T ~ 0 term: 1
A V r--- o = ~ S t
( ~( 4
l n ~~/2--3)]
AVT,#O
~-AI'rb°s'-k-AI:ferm" t...IVT~ 0 v t..XrT~
(6,7)
0 .
2 2 Defining ab, ff) -=,///b, f f ) / T 2 , where Jlb.~f) is the bosonic (fermionic) mass matrix, the T ¢ 0 contributions may be written as
bos.
4 [ 1 2-
AVT,,o=T Tr'
~ab
1
~
l (a2) 3/2- - - a 4 1 n
64n z
2
( a 2 ~1+21
Ab
_l
a~' --7~3/2 ~ (-1)'~(21+l~)F(l+½) 1=1 (l+1)! \~5n2,/
(ab < 2rt) , at See ref. [6 ] for a discussion on the relationshipbetweenbaryogenesisand explicit CP-violationphenomenologyin the MSSM. 68
(8)
Volume 306, number 1,2
A r2"ferrn.
~-r#o = (af
T"Tr '
PHYSICS LETTERS B
[4~
a~+
1
~
a2
713/2 ~
atln~+-~,_, ~f
t=l
(-1y
1
2-2/-1
(l+ 1 )!
27 May 1993
~(21+ l)r(1+ ½) \ ~ , I
l (9)
,
where Ab= 16Af= 16rt 2 exp ( 3 - 2yE), YE= 0.5772, ( is the Riemann (-function, and Tr' properly counts the degrees of freedom. Eqs. (8) and (9) give an exact representation of the complete one-loop effective potential at finite temperature [ 7 ] for ab < 2Zt and af< 7~, respectively. Taking rh~ = fit 2 the stop mass matrix takes the convenient form
a 2 =a~.l+~t 2,
(10)
where a S =-rh~/T 2, 1 is the identity matrix, and ~2 is the field-dependent part of the mass matrix, i.e. ~t2 ~ 0 as the fields vanish. The only contributions to m z, 25, 26 and 2 7 c o m e from the traces o f ~ 4, a6 and ats; the traces of the higher powers of ~2 contain operators of dimension d > 4, which are suppressed by powers of H2/fft ~ or H2/4~ 2T 2, multiplied by an additional suppressing coefficient coming from the expansion. Inserting eq. ( 10 ) in eq. ( 8 ) and using the binomial expansion for the terms (a 2) z+2 and (a 2 ) 3/2 we can extract the relevant terms from the one-loop effective potential: T4
1 32ha e
1 lnA_~_5 64~5n 2
r4-1 \192na~
~6 [aQ]) Tr' ~6'
+3
--,.~4[a~.]
r'(-\512na~ :!
tr'a 7
(11)
~s [ a ~ ] ) Tr' ~t8 '
(12,13)
where 2
,~2n[aQ]=_n 3/2
~
/=max[l, n--2]
, ,t ( ( 2 / + 1 ) (l+2"~ (a 2e)t+2-", ( - - x ) ~-r~j:~-5-U~F(I+½)\ n ] ~
n=2,3,4,...,
aQ
(14)
In eq. ( 11 ) we have also included the contribution coming from the T = 0 one-loop effective potential, eq. (6). The numerical values of the series ~2, for some values ofaQ are listed in table 1. We now evaluate the traces in eqs. ( I 1 ), (12) and (13) and find the one-loop contribution to m ] and to 25,6,7 coming from the stop:
Am~S)2=3h2AtTau 4Atau
A2tS) = - 12ht
+ ~
2
InA-~-T~ + 3 + 4 ~ 4 [ a ~ ]
I
~ [ a Q ] + 256-1ra
,
(15) (16)
'
Table 1 Values of the series :~2. [a~] for values o f a ~ in the allowed range.
aa
~'4[a~l
,~[a~]
1 2 3 4 5 6
- 4 . 7 4 × 10 -5 - 1.81X 1 0 - 4 --3.8 × 1 0 -4 - 6 . 1 8 X 10 -4 - 8 . 7 6 × 10 -4 - - 1 . 1 4 × 10 -3
-- 1.56× - - 1.42 × - 1.24× --1.05X -8.71 × -7.22×
&Jail 10 - s 10 - s 10 - s 10 - s 10 -6 10 -6
1.24× 10 -7 1.04× 10 -~ 8.01 × 10 -a 5.8 x l 0 -8 4.06× 10-s 2.81 × 1 0 -8
69
Volume 306, number 1,2 A2~*) =
-6h
PHYSICS LETTERS B
2At I au___~_ 2[4~ _ (g
27 May 1993
19fzta~.)+4htau(~[aQ]+l 2 2 2
+g2)(~6[a~ ]
A2~s)=-6hl2A*au[[6h2-](g2+g2)](~6[a~]T
51fna~)]l ,
1"~ 2A 2 __ 1 19fna~)+4ht -~-i(~[a~]+ 512na~)],
(17)
(18)
where we have defined au- tz/T. C h o o s i n g / ? = M 2 = M 2, the squared mass matrices for charginos and neutralinos take a form analogous to that in eq. (10), i.e. they are given by the sum of a multiple of the identity matrix and a field-dependent matrix af
2
= a u2- I
-2
-I-af .
We now insert them in eq. (9) and, following the same strategy we used for the stop, we extract the relevant terms in the one-loop potential, - - ~ + 3 ) + ~ 4 [ a 2] ] T r ' a ~ , T 4 [( 1 6 14 ~2z l n A .~2
T 4 ~ [ a 2 ] T r ' a f 6,
T 4 . ~ t a 2 l T r ' ~ f a,
(19,20,21)
where the values of ~j,[a21-
rt3/2 8
~ I. . . .
[1,n--2l
( - 1 )l 1 - 2 -2t-l (l+2)!
~1[l+2~]
((21+l)F(l+)\ n ]
(a2) l+2-n (n2) '+2
(22)
are listed in table 2. Finally, evaluating the traces in eqs. (19), (20) and (21), we obtain the contributions to the relevant couplings from charginos, Am]C)2 = s i g n ( / l ) g E2T2 /~C)
2[-1(10 au[
.~2
+3)+8~4[a2]] '
(23)
4 4 2 =8gEau:a[au] ,
(24)
A;t~~) = ~ ) =-sign(/l)
4 2u( 3:6 [au] 2 +4au~'s[au] 2 ~2 ) 4g2a
(25)
and from neutralinos, Am~ ")2 = g~ +g2 Am~ c)2, 2g 2
ARtn)
2
-
2
g~ + g l A2(C) "-s2")-2 , ,
i=5,6,7.
(26)
NOW we are ready to discuss the conditions (3) and (4) on the spontaneous CP breaking at finite temperature and their implication on the mass spectrum of the Higgs scalars. The one-loop effective potential at finite temperature has a CP-violating m i n i m u m if (see eqs. (3) and ( 4 ) ) Table 2 Values of the series .~j~[a 2] for values ofa 2 in the allowed range. a.
~4[a2u]
~[a2l
~[a2l
0.5
- 8.29x 10 -5 -3.15×10 -4 -6.55>(10 -4 - 1.06x l0 -a - 1.48x l0 -3 - 1.89x 10-3
- 1.09x 10-4 -9.8 x10 -5 -8.4 ×10 -5 -6.95x l0 -5 - 5.63x l0 -s -4.54x l0 -5
3.84x 10-6 3.21×10 -6 2.45× l0 -6 1.75x 10-6 1.21 x l0 -6 8.18x l0 -7
1
1.5 2 2.5 3 70
Volume 306, number 1,2
PHYSICSLETTERSB
27 May 1993 (27)
A2s>O and 1 --K<
Zi;t6v21(T) + ~0.7vzZ(T)
< 1+ K ,
(28)
where A2s
tanfl(T)
K = 4 A2 6 + A2 7 t a n 2 f l ( T )
Vl,2(T) are the vacuum expectation values at finite temperature, tanfl(T)-v2(T)/v~(T), A2i=A2}s)+ At} c) + A2}") ( i = 5, 6, 7), and r~ 2 is the coefficient of the operator - H I H 2 in the one-loop effective potential at T # 0, that is t ~ ----m~+ Am~c)2 + Am~ ")2 + Arn~')2 .
(29)
The radiatively-corrected mass of the pseudoscalar is given by
m2-
1 +tan2fl/
2
/./2
]./2
t~nfl ~m3-sign(lt) l--~2(3g2+g~)ln~2
_l+tan2fl • tanfl { r~2-slgn(/z)
_3h2AtTau[8_~ae+
2
2 21- 1
(3g~ +gl)T au[l-~n 2
3
2
fft~
16-~2htAtltln-~)
(lna2+~)+4~[a2]]
1 2 f,~,ln ~--~a2o l---~-i~ b +~)+4~4[a~]]},
-~f
(30,
where we have eliminated ~ using eq. (29). From eq. (30) we read that in the limit gb g2, ht~O the pseudoscalar mass vanishes if we require spontaneous CP violation (eq. (28)), so, in agreement with the Georgi-Pais theorem [4], m~ is a one-loop effect. Nevertheless, it does not imply a very light pseudoscalar, as in the case of spontaneous CP violation at T = 0 [ 21. In fact, the contributions at T ¢ 0 may be important, as we can read off from eq. (30): even ifr~l is constrained to be of the same order ofzk~6v~(T) +A~7v,2 (T), as required by spontaneous CP violation, eq. (28), the second and the third terms on the RHS of the last line in eq. (30) may give significant contributions to rn~. The present experimental lower bound on rn~ comes from LEP [ 5 ]; it is about 20 GeV for tan fl= 1 and rapidly saturates the kinematical limit for LEP search, rn~ > 40 GeV, as tan fl becomes greater than about 1.5. In fig. 1 we plot the region in the plane (g, the) in which ma [tan fl/( 1 + tanZfl) ] 1/2 is greater than 14.1 GeV and 17.7 GeV, that is, mA>40 and >50 GeV, respectively, for tanfl=4. We have fixed n~32=~6v~(T)+ Zk~7v~(T), which corresponds to the maximal value for the CP-violating phase (see eq. (4)), cos 8 = 0, and also required that zk~5> 0. As we can see there is a wide region in the parameter space in which CP is broken and ma is compatible with the experimental values, Moreover, we want to stress that our choice/12=M~ =M~ and r~ ~ = r~ ~ has been made only to simplify the derivation in an effective potential approach, but the CP violation is present in a much larger portion of the parameter space. Since we believe that this effect may play a crucial role in the electroweak baryogenesis, we have fixed T = T~.w=O( 150 GeV) [8 ] in fig. 1. As T decreases the values of Vl.2(T), ~ , zMi, and consequently cos 8, change, and at a certain value T = T,~t, the condition (28) is no more fulfilled, i.e. CP is restored. It is worthwhile to mention that in this formalism we are not allowed to take the T--,0 limit in the effective potential, because our formulas (8) and (9) are valid only for T> r~e/2~r,/l/n. Another possibility is that the vacuum expectation values v~.2(T~.w.) in the middle of the electroweak bubbles are too large, so that the condition (28) is not fulfilled (this might follow from the requirement that the anom71
Volume 306, number 1,2
PHYSICS LETTERS B
27 May 1993
GeV fnO 900 800
17.7GeV
/
700 600 500 400 300 200
14.1GeV
100 .'
loo. 15'o. 2o'o. 2go. 3o'0. ago. 4do. 45'o. #
GeV
Fig. 1. The parameter space (above the dashed line) in the plane (#, r~Q) for cos J = 0 , 25>0 (see text). Contours corresponding to mA[ tan/3/( 1+ tan2]3) ] l/2 > 14.1 and 17.7 GeV are plotted. We have fixed At=50 GeV, T= 150 GeV, vl(T)=v2(T)=90 GeV ( v ( T = 0 ) = 174 GeV) and ht= 1.
alous B- and L-violating processes are out of equilibrium [ 9 ] ); in this case CP violation can take place in the bubble walls, where the vacuum expectation values are changing from zero to the value inside the bubble. In order to make more quantitative statements on the role of this effect in the generation of the baryon asymmetry of the Universe, a detailed analysis of the phase transition and of bubble propagation in the MSSM is needed. It will be the subject of a forthcoming publication [ 10 ]. In conclusion, we have shown that the spontaneous CP violation is possible in the Minimal Supersymmetric Standard Model at finite temperature, and still in agreement with the experimental lower bounds on the mass of the Higgs pseudoscalar. The CP breaking may take place soon after the electroweak phase transition, and the CP-violating phase may reach the maximal values c~= + re/2, going to zero as the temperature of the Universe falls down. It is a pleasure to thank A. Brignole, G.R. Dvali, G. Giudice, A. Riotto and G. Senjanovi6 for useful discussions, and A. Masiero and C. Verzegnassi for the continuous encouragement. We are grateful also to M. D'Attanasio, R. De Pietri and G. Marchesini for the kind hospitality at the department of physics of Parma, where part of this work was done.
References [ 1 ] N. Maekawa, Phys. Lett. B 282 (1992) 387. [2 ] A. Pomarol, preprint SCIPP-92-29. [3] J. Ellis, S. Ferrara and D.V. Nanopoulos, Phys. Len. B 114 (1982) 231; W. Buchmiiller and D. Wyler, Phys. Lett. B 121 (1983) 321; M. Dugan, B. Grinstein and L. Hall, Nucl. Phys. B 255 (1985) 413. [4] H. Georgi and A. Pals, Phys. Rev. D 10 (1974) 1246. [ 5 ] M. Davier, preprint LAL-91-48; ALEPH Collah., D. Decamp et al., Phys. Lett. B 265 ( 1991 ) 475. [ 6 ] A.G. Cohen and A.E. Nelson, Phys. Lett. B 297 ( 1992 ) 111. [7] L. Dolan and R. Jackiw, Phys. Rev. D 9 (1974) 3320. [8] G. Giudice, Phys. Rev. D 45 (1992) 3177. [9] A.I. Bocharev, S.V. Kuzmin and M.E. Shaposhnikov, Phys. Lett. B 244 (1990) 275; A.I. Bocharev and M.E. Shaposhnikov, Mod. Phys. Lett. A 2 (1987) 417. [ 10 ] D. Comelli et al., work in progress.
72
PHYSICS LETTERS B
Spontaneous CP violation in the Minimal Supersymmetric Standard Model at finite temperature Denis Comelli Dipartimento di Fisica Teorica, Universitdt di Trieste, Strada Costiera 1I, Miramare, 1-34014 Trieste, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, 1-34014 Trieste, Italy
and Massimo Pietroni Dipartimento di Fisica, Universit& di Padova, Via Marzolo 8, 1-35100 Padua, Italy and Istituto Nazionale di Fisica Nucleate, Sezione di Padova, 1-35100 Padua, Italy
Received 5 February 1993 Editor: R. Gatto
We show that in the Minimal Supersymmetric Standard Model one-loop effects at finite temperature may lead to a spontaneous breaking of CP invariance in the scalar sector. Requiring that the breaking takes place at the critical temperature for the electroweak phase transition, we find that the parameter space is compatible with a mass of the Higgs pseudoscalar in agreement with the present experimental lower bounds. Possible implications for baryogenesis are discussed.
The possibility o f a spontaneous breakdown o f the C P symmetry in the scalar sector o f the Minimal Supersymmetric Standard Model (MSSM), has been recently investigated [ 1,2 ]. It is well known that, as long as supersymmetry (SUSY) is exact, C P is conserved in the scalar sector o f the MSSM. As a consequence, the only CPviolating effects may arise from the soft SUSY-breaking terms: the scalar masses, the gaugino masses, and the trilinear interactions. If one allows these terms to be complex, then there are two new physical phases in the MSSM which are not present in the Standard Model [ 3 ]. These phases do not appear in the tree-level Higgs potential, but occur in interactions involving the superpartners of the ordinary particles, giving new contributions to the CP-odd observables e, e', and the electric dipole m o m e n t o f the neutron [ 3 ]. If one assumes that all the soft masses and couplings are real, then no new C P violation appears at the tree level. At the one-loop level, the contributions o f graphs with sfermions, charginos, neutralinos and Higgs scalars in the internal lines induce a finite renormalization to the tree-level couplings o f the scalar potential, which may lead to a phase shift between the vacuum expectation values o f the two neutral Higgs fields, i.e. to a spontaneous breaking o f C P in the scalar sector [ 1,2 ]. Unfortunately it comes out that this scenario is not realistic. The point is that spontaneous C P violation can be implemented radiatively only if a pseudoscalar with zero tree-level mass exists, as was shown on general grounds by Georgi and Pais [4]. In the MSSM this implies the existence of a very light Higgs (with a mass o f a few GeV, given by one-loop contributions) [ 2 ], which has been excluded by LEP data [ 5 ]. In the present paper we analyze the possibility o f the spontaneous breakdown of C P at finite temperature in the MSSM. We find that new CP-violating contributions arise from the one-loop corrections at T # 0, so that the effective potential (at T ~ 0) may have a C P non-conserving vacuum, while the pseudoscalar mass, calculated 0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All fights reserved.
67
Volume 306, number 1,2
PHYSICSLETTERSB
27 May 1993
at T = 0, is still compatible with the experimental bounds. As Tgoes to zero the phase of the vacuum expectation values of the Higgs fields vanishes, and the only sources of CP violation remaining are the phase in the CabibboKobayashi-Maskawa matrix and the tTparameter of the QCD vacuum. Nevertheless, this effect may be of great relevance for the electroweak baryogenesis in the MSSM, as it could give rise to a new, time varying, CP-violating phase in the scalar sector, whose size is nearly unbounded by the phenomenology at zero temperature ~. The most general gauge invariant scalar potential for the two-doublet model, along the neutral components, is given by
V=m 2 In112+m 2 I n 2 1 2 - (m2nlH2 +h.c.) +21 In114+22 IH2 ]4"~-23 Inl Iz IH2 [2 dl-24 IHIH21 z + [25(HIH2)2+26 IH~ 12H1//2 +27 In212H1 H2 +h.c. ] ,
(1)
where we assume m32, 25, 26 and 27 to be real, so that CP invariance may be broken only if the vacuum expectation values of the Higgs fields get a non-trivial phase,
64:0, n,
(2)
where we have defined ( H i ) = vl, ( H 2 ) = vzein. Eq. (2) is satisfied if, and only if, 25>0,
(2)
and -- 1
--26V2 --27V2
<
1.
(4)
425vl v2 At the tree level the parameters 2~ (i = 1, ..., 7 ) are fixed by supersymmetry:
2~=22=~(gZ+g~),
23=~(g2Z-g2),
24--
~ 2 ~g2,
25=26=27=0,
(5)
where g2 and gl are the gauge couplings of SU (2)i. and U ( 1 ) r respectively. From eqs. (3) and (5) we immediately read that, at the tree level, CP is not violated in the scalar sector of the MSSM. Since it is the soft breaking of supersymmetry that allows CP violation, the one-loop contributions to the CP-violating couplings 25, 26 and 27 will be proportional to the soft parameters: the gaugino masses M1,2, the sfermion masses r~ 2, the trilinear scalar coupling A, and the bilinear one, B. For this reason, we will include in the one-loop effective potential only those field-dependent mass matrices which contain these parameters. The dominant contributions are given by the stop for the bosonic sector and by chargino and neutralino for the fermionic one. At finite temperature the one-loop contribution to the effective potential can be decomposed into the sum of a T = 0 and a T ~ 0 term: 1
A V r--- o = ~ S t
( ~( 4
l n ~~/2--3)]
AVT,#O
~-AI'rb°s'-k-AI:ferm" t...IVT~ 0 v t..XrT~
(6,7)
0 .
2 2 Defining ab, ff) -=,///b, f f ) / T 2 , where Jlb.~f) is the bosonic (fermionic) mass matrix, the T ¢ 0 contributions may be written as
bos.
4 [ 1 2-
AVT,,o=T Tr'
~ab
1
~
l (a2) 3/2- - - a 4 1 n
64n z
2
( a 2 ~1+21
Ab
_l
a~' --7~3/2 ~ (-1)'~(21+l~)F(l+½) 1=1 (l+1)! \~5n2,/
(ab < 2rt) , at See ref. [6 ] for a discussion on the relationshipbetweenbaryogenesisand explicit CP-violationphenomenologyin the MSSM. 68
(8)
Volume 306, number 1,2
A r2"ferrn.
~-r#o = (af
T"Tr '
PHYSICS LETTERS B
[4~
a~+
1
~
a2
713/2 ~
atln~+-~,_, ~f
t=l
(-1y
1
2-2/-1
(l+ 1 )!
27 May 1993
~(21+ l)r(1+ ½) \ ~ , I
l (9)
,
where Ab= 16Af= 16rt 2 exp ( 3 - 2yE), YE= 0.5772, ( is the Riemann (-function, and Tr' properly counts the degrees of freedom. Eqs. (8) and (9) give an exact representation of the complete one-loop effective potential at finite temperature [ 7 ] for ab < 2Zt and af< 7~, respectively. Taking rh~ = fit 2 the stop mass matrix takes the convenient form
a 2 =a~.l+~t 2,
(10)
where a S =-rh~/T 2, 1 is the identity matrix, and ~2 is the field-dependent part of the mass matrix, i.e. ~t2 ~ 0 as the fields vanish. The only contributions to m z, 25, 26 and 2 7 c o m e from the traces o f ~ 4, a6 and ats; the traces of the higher powers of ~2 contain operators of dimension d > 4, which are suppressed by powers of H2/fft ~ or H2/4~ 2T 2, multiplied by an additional suppressing coefficient coming from the expansion. Inserting eq. ( 10 ) in eq. ( 8 ) and using the binomial expansion for the terms (a 2) z+2 and (a 2 ) 3/2 we can extract the relevant terms from the one-loop effective potential: T4
1 32ha e
1 lnA_~_5 64~5n 2
r4-1 \192na~
~6 [aQ]) Tr' ~6'
+3
--,.~4[a~.]
r'(-\512na~ :!
tr'a 7
(11)
~s [ a ~ ] ) Tr' ~t8 '
(12,13)
where 2
,~2n[aQ]=_n 3/2
~
/=max[l, n--2]
, ,t ( ( 2 / + 1 ) (l+2"~ (a 2e)t+2-", ( - - x ) ~-r~j:~-5-U~F(I+½)\ n ] ~
n=2,3,4,...,
aQ
(14)
In eq. ( 11 ) we have also included the contribution coming from the T = 0 one-loop effective potential, eq. (6). The numerical values of the series ~2, for some values ofaQ are listed in table 1. We now evaluate the traces in eqs. ( I 1 ), (12) and (13) and find the one-loop contribution to m ] and to 25,6,7 coming from the stop:
Am~S)2=3h2AtTau 4Atau
A2tS) = - 12ht
+ ~
2
InA-~-T~ + 3 + 4 ~ 4 [ a ~ ]
I
~ [ a Q ] + 256-1ra
,
(15) (16)
'
Table 1 Values of the series :~2. [a~] for values o f a ~ in the allowed range.
aa
~'4[a~l
,~[a~]
1 2 3 4 5 6
- 4 . 7 4 × 10 -5 - 1.81X 1 0 - 4 --3.8 × 1 0 -4 - 6 . 1 8 X 10 -4 - 8 . 7 6 × 10 -4 - - 1 . 1 4 × 10 -3
-- 1.56× - - 1.42 × - 1.24× --1.05X -8.71 × -7.22×
&Jail 10 - s 10 - s 10 - s 10 - s 10 -6 10 -6
1.24× 10 -7 1.04× 10 -~ 8.01 × 10 -a 5.8 x l 0 -8 4.06× 10-s 2.81 × 1 0 -8
69
Volume 306, number 1,2 A2~*) =
-6h
PHYSICS LETTERS B
2At I au___~_ 2[4~ _ (g
27 May 1993
19fzta~.)+4htau(~[aQ]+l 2 2 2
+g2)(~6[a~ ]
A2~s)=-6hl2A*au[[6h2-](g2+g2)](~6[a~]T
51fna~)]l ,
1"~ 2A 2 __ 1 19fna~)+4ht -~-i(~[a~]+ 512na~)],
(17)
(18)
where we have defined au- tz/T. C h o o s i n g / ? = M 2 = M 2, the squared mass matrices for charginos and neutralinos take a form analogous to that in eq. (10), i.e. they are given by the sum of a multiple of the identity matrix and a field-dependent matrix af
2
= a u2- I
-2
-I-af .
We now insert them in eq. (9) and, following the same strategy we used for the stop, we extract the relevant terms in the one-loop potential, - - ~ + 3 ) + ~ 4 [ a 2] ] T r ' a ~ , T 4 [( 1 6 14 ~2z l n A .~2
T 4 ~ [ a 2 ] T r ' a f 6,
T 4 . ~ t a 2 l T r ' ~ f a,
(19,20,21)
where the values of ~j,[a21-
rt3/2 8
~ I. . . .
[1,n--2l
( - 1 )l 1 - 2 -2t-l (l+2)!
~1[l+2~]
((21+l)F(l+)\ n ]
(a2) l+2-n (n2) '+2
(22)
are listed in table 2. Finally, evaluating the traces in eqs. (19), (20) and (21), we obtain the contributions to the relevant couplings from charginos, Am]C)2 = s i g n ( / l ) g E2T2 /~C)
2[-1(10 au[
.~2
+3)+8~4[a2]] '
(23)
4 4 2 =8gEau:a[au] ,
(24)
A;t~~) = ~ ) =-sign(/l)
4 2u( 3:6 [au] 2 +4au~'s[au] 2 ~2 ) 4g2a
(25)
and from neutralinos, Am~ ")2 = g~ +g2 Am~ c)2, 2g 2
ARtn)
2
-
2
g~ + g l A2(C) "-s2")-2 , ,
i=5,6,7.
(26)
NOW we are ready to discuss the conditions (3) and (4) on the spontaneous CP breaking at finite temperature and their implication on the mass spectrum of the Higgs scalars. The one-loop effective potential at finite temperature has a CP-violating m i n i m u m if (see eqs. (3) and ( 4 ) ) Table 2 Values of the series .~j~[a 2] for values ofa 2 in the allowed range. a.
~4[a2u]
~[a2l
~[a2l
0.5
- 8.29x 10 -5 -3.15×10 -4 -6.55>(10 -4 - 1.06x l0 -a - 1.48x l0 -3 - 1.89x 10-3
- 1.09x 10-4 -9.8 x10 -5 -8.4 ×10 -5 -6.95x l0 -5 - 5.63x l0 -s -4.54x l0 -5
3.84x 10-6 3.21×10 -6 2.45× l0 -6 1.75x 10-6 1.21 x l0 -6 8.18x l0 -7
1
1.5 2 2.5 3 70
Volume 306, number 1,2
PHYSICSLETTERSB
27 May 1993 (27)
A2s>O and 1 --K<
Zi;t6v21(T) + ~0.7vzZ(T)
< 1+ K ,
(28)
where A2s
tanfl(T)
K = 4 A2 6 + A2 7 t a n 2 f l ( T )
Vl,2(T) are the vacuum expectation values at finite temperature, tanfl(T)-v2(T)/v~(T), A2i=A2}s)+ At} c) + A2}") ( i = 5, 6, 7), and r~ 2 is the coefficient of the operator - H I H 2 in the one-loop effective potential at T # 0, that is t ~ ----m~+ Am~c)2 + Am~ ")2 + Arn~')2 .
(29)
The radiatively-corrected mass of the pseudoscalar is given by
m2-
1 +tan2fl/
2
/./2
]./2
t~nfl ~m3-sign(lt) l--~2(3g2+g~)ln~2
_l+tan2fl • tanfl { r~2-slgn(/z)
_3h2AtTau[8_~ae+
2
2 21- 1
(3g~ +gl)T au[l-~n 2
3
2
fft~
16-~2htAtltln-~)
(lna2+~)+4~[a2]]
1 2 f,~,ln ~--~a2o l---~-i~ b +~)+4~4[a~]]},
-~f
(30,
where we have eliminated ~ using eq. (29). From eq. (30) we read that in the limit gb g2, ht~O the pseudoscalar mass vanishes if we require spontaneous CP violation (eq. (28)), so, in agreement with the Georgi-Pais theorem [4], m~ is a one-loop effect. Nevertheless, it does not imply a very light pseudoscalar, as in the case of spontaneous CP violation at T = 0 [ 21. In fact, the contributions at T ¢ 0 may be important, as we can read off from eq. (30): even ifr~l is constrained to be of the same order ofzk~6v~(T) +A~7v,2 (T), as required by spontaneous CP violation, eq. (28), the second and the third terms on the RHS of the last line in eq. (30) may give significant contributions to rn~. The present experimental lower bound on rn~ comes from LEP [ 5 ]; it is about 20 GeV for tan fl= 1 and rapidly saturates the kinematical limit for LEP search, rn~ > 40 GeV, as tan fl becomes greater than about 1.5. In fig. 1 we plot the region in the plane (g, the) in which ma [tan fl/( 1 + tanZfl) ] 1/2 is greater than 14.1 GeV and 17.7 GeV, that is, mA>40 and >50 GeV, respectively, for tanfl=4. We have fixed n~32=~6v~(T)+ Zk~7v~(T), which corresponds to the maximal value for the CP-violating phase (see eq. (4)), cos 8 = 0, and also required that zk~5> 0. As we can see there is a wide region in the parameter space in which CP is broken and ma is compatible with the experimental values, Moreover, we want to stress that our choice/12=M~ =M~ and r~ ~ = r~ ~ has been made only to simplify the derivation in an effective potential approach, but the CP violation is present in a much larger portion of the parameter space. Since we believe that this effect may play a crucial role in the electroweak baryogenesis, we have fixed T = T~.w=O( 150 GeV) [8 ] in fig. 1. As T decreases the values of Vl.2(T), ~ , zMi, and consequently cos 8, change, and at a certain value T = T,~t, the condition (28) is no more fulfilled, i.e. CP is restored. It is worthwhile to mention that in this formalism we are not allowed to take the T--,0 limit in the effective potential, because our formulas (8) and (9) are valid only for T> r~e/2~r,/l/n. Another possibility is that the vacuum expectation values v~.2(T~.w.) in the middle of the electroweak bubbles are too large, so that the condition (28) is not fulfilled (this might follow from the requirement that the anom71
Volume 306, number 1,2
PHYSICS LETTERS B
27 May 1993
GeV fnO 900 800
17.7GeV
/
700 600 500 400 300 200
14.1GeV
100 .'
loo. 15'o. 2o'o. 2go. 3o'0. ago. 4do. 45'o. #
GeV
Fig. 1. The parameter space (above the dashed line) in the plane (#, r~Q) for cos J = 0 , 25>0 (see text). Contours corresponding to mA[ tan/3/( 1+ tan2]3) ] l/2 > 14.1 and 17.7 GeV are plotted. We have fixed At=50 GeV, T= 150 GeV, vl(T)=v2(T)=90 GeV ( v ( T = 0 ) = 174 GeV) and ht= 1.
alous B- and L-violating processes are out of equilibrium [ 9 ] ); in this case CP violation can take place in the bubble walls, where the vacuum expectation values are changing from zero to the value inside the bubble. In order to make more quantitative statements on the role of this effect in the generation of the baryon asymmetry of the Universe, a detailed analysis of the phase transition and of bubble propagation in the MSSM is needed. It will be the subject of a forthcoming publication [ 10 ]. In conclusion, we have shown that the spontaneous CP violation is possible in the Minimal Supersymmetric Standard Model at finite temperature, and still in agreement with the experimental lower bounds on the mass of the Higgs pseudoscalar. The CP breaking may take place soon after the electroweak phase transition, and the CP-violating phase may reach the maximal values c~= + re/2, going to zero as the temperature of the Universe falls down. It is a pleasure to thank A. Brignole, G.R. Dvali, G. Giudice, A. Riotto and G. Senjanovi6 for useful discussions, and A. Masiero and C. Verzegnassi for the continuous encouragement. We are grateful also to M. D'Attanasio, R. De Pietri and G. Marchesini for the kind hospitality at the department of physics of Parma, where part of this work was done.
References [ 1 ] N. Maekawa, Phys. Lett. B 282 (1992) 387. [2 ] A. Pomarol, preprint SCIPP-92-29. [3] J. Ellis, S. Ferrara and D.V. Nanopoulos, Phys. Len. B 114 (1982) 231; W. Buchmiiller and D. Wyler, Phys. Lett. B 121 (1983) 321; M. Dugan, B. Grinstein and L. Hall, Nucl. Phys. B 255 (1985) 413. [4] H. Georgi and A. Pals, Phys. Rev. D 10 (1974) 1246. [ 5 ] M. Davier, preprint LAL-91-48; ALEPH Collah., D. Decamp et al., Phys. Lett. B 265 ( 1991 ) 475. [ 6 ] A.G. Cohen and A.E. Nelson, Phys. Lett. B 297 ( 1992 ) 111. [7] L. Dolan and R. Jackiw, Phys. Rev. D 9 (1974) 3320. [8] G. Giudice, Phys. Rev. D 45 (1992) 3177. [9] A.I. Bocharev, S.V. Kuzmin and M.E. Shaposhnikov, Phys. Lett. B 244 (1990) 275; A.I. Bocharev and M.E. Shaposhnikov, Mod. Phys. Lett. A 2 (1987) 417. [ 10 ] D. Comelli et al., work in progress.
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