Baryogenesis and supersymmetric GUTs

Volume 125B, number 5

PIIYSICS LETTERS

9 June 1983

BARYOGENESIS AND SUPERSYMMETRIC GUTS J.N. FRY and Michael S. Turner Astronomy and Astrophysics Center, The University of Chicago, Chicago, 11. 6063 7, USA Received 21 March 1983

We examine baryogenesis in the context of supersymmetric GUTs. In this regard, there are four important differences between GUTs and supersymmetric GUTs: the latter have double the number of particle species, larger unified coupling constant, possibility of dimension-live operators which violate baryon number and a higher unilication sc',de. The first three of these differences lead to the increased importance of 2 -. 2 B nonconserving scatterings, which, for a given superheavy gauge or ltiggs mass, result in a smaller baryon asymmetry than for a non-supersymmetric GUT. Of these differences, dimension-five operators (if they exist) are the most important, and can lead to a substantially smaller baryon asymmetry than in the non-supersymmetric case.

Introduction. Supersymmetric extensions of Grand Unified Theories have several appealing features which have led to renewed interest in their study [ 1 ]. Besides earlier motivations [2] including the innate appeal of the Fermi/Bose symmetry and the perpetual hope of eventually incorporating gravity into a quantum field theory, supersymmetric theories include fundamental scalars in an essential way, and the no-renorm',dization theorems [3] may allow a resolution o f the "hierarchy problem", that the scale of weak SU(2) X U ( I ) , M w ~ 10 2 GeV, is a factor of 1012 smaller than the unification scale M X ~ 1014 GeV. It is our purpose here to discuss the effects of supersymmetry on the dynamical generation of the cosmological b a r y o n - a n t i b a r y o n asymmetry [4]. The early universe evolution o f n B / s (hi3 = net baryon density and s = specific entropy density in units o f k B ; we work with fi = c = k B = 1, G N = mp-2) has been discussed by many authors [5,6], and it is a triumph o f the standard theory that it is consistent with the observed value nB/s ~ 10 -I1 --10 -10. Because nB/s depends on all of: ~, the gauge (o~ ~ 42g) or Higgs (all ~ 10 -3 - 1 0 6) coupling constant; M, the superheavy gauge (M X ~ 6 X 1014 GeV) or Higgs boson (MII ~ ?) mass; geff, the effective number of degrees of freedom o f all particle species [geff = Y'gB +~Y'gt: ~ 160 in SU(5)] ; and because supersymmetric theories change these pa0 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland

rameters, adding supersyrnmetric partners for each of the original particle species and introducing the possibility of dimension-five (dim-5) baryon number nonconserving operators, such theories yield a priori completely different results. To preview these results, we note that in supersymmetric grand unified models, the gauge coupling a doubles ( !45 ~ . ~ s ) and geff doubles, which, especially with dim-5 operators, greatly increases the importance of 2 ,-, 2 baryon number nonconserving scattering. These scatterings involve 2 incoming and 2 outgoing quarks or leptons (or their supersymmetric partners), are mediated by superheavy bosons for their supersymmetric partners), and tend to erase any b a r y o n antibaryon asymmetry, making it harder to generate an asymmetry. We shall not discuss e, the parameter which describes the C,CP-violation in the decays of the superheavy particles (e -= net baryon number produced by the decay o f a particle--antiparticle pair). At the tree level e = 0, and e 4 : 0 arises due to loop-diagrams, so that e ~< O(~/41r) [7]. In the context of supersymmetric GUTs, e has been discussed in ref. [8] ; their conclusion is that there are no "surprising" cancellations at the one-loop level, and so, as before, e can be as large as O(~47r). We shall focus on just the statistical mechanics aspects of baryogenesis. 379

Volume 125B, number 5

PtlYSICS L H ' T E R S

Review ofbaryogenesis. In this section we abstract the main equations from earlier studies of the Boltzmann equations for baryogenesis [5,6], and we obtain and discuss some asymptotic solutions which we shall use as a framework to discuss the results of numerical integration of the entire network. The interested reader is referred to the earlier papers [5,6] for more detail. The import of the full network of equations (which includes all the quark, lepton, gauge, and Higgs species and their interactions) for the evolution of the baryon asymmetry can be obtained from a simplified set of equations: X '= -7xKz(X

-. X e q ),

B ' = -TB KzB + e T x K z ( X --Xeq ),

(la)

(lb)

where X traces the density of superheavy X, ,X (in this context, either gauge or Higgs) bosons, n x = n 2 ~- XT 3 ~-- Xn 7 ; B traces the asymmetry between baryons and antibaryons, n B - -.rig --= BT 3 ~ Bn 7 ; z = M / T and prime ' denotes d / d z , 7X and "tB are the dinaensionless rates, normalized by the rate czM: Pi = °al4")'i, for X-decay ( P x = coM at rest, c "" 1), and lbr removal of baryon asymmetry by inverse decay (ID) plus "~ ,-- "~ baryon number nonconserving scatterings (S); K is a combination of constants which measures the "'effectiveness" of the X-decay rate versus expansion rate H = R / R = (4zr3geffT4/45mpi)1/2 at T ~ - M , K -~ (Fx/H)IT~_M = ~mpl/2 (4rr3/45geff)l/2M =(3 × 1017 CeV)(c~/M)(160/get.f)ll2 ,

(2)

so that H = c~l/Kz 2 and ( d / d t ) = (oaIl/Kz)(d/dz)', Xeq is the value of X in the thermal equilibrium, and e is a measure o f C , CP-violation: e =- net baryon number produced in the free decay of an X- X pair. We see here the three ingredients required to generate a baryon asymmetry. With no baryon-nu,nber violations, 7 = 0. With no departure from equilibrium, X -= Xeq , while with no C,(~-violation e = 0. In any of these cases there is no source term in eq. ( I ) for B. The functional dependences of the rates, etc., in eq. (1) are

380

9 June 1983

7X = Fx/mM = cKI(z)/K2(z) ~ c ' z -+c,

,

z ~ 1, z>> 1,

(3a)

Tit) = XeqTX = i'lD/O-h4 = -~c 1 z2K 1(z) 1

,

-+ S c z ,

z ~ I ,

-->cOr/8) 112 z 312 e z ,

z >> I,

(3b)

z>~ 1 ,

(3c)

Xeq = 'YlD/YX = ~z2K2(z) -+1,

z'~l

,

- + ( ~ / 8 ) l / 2 z 3/2 e - z ,

7S = FS/O-M ~ °~Az-l( 1 + z2) ' (d ..4) ,

(3d)

where K 1 and K 2 are modified Bessel functions, A measures tile amplitude of B-nonconserving 2 ~ 2 scatterings (mediated by virtual Xs), essentially counting the number of channels (A "" few X 103), and d is the dimensionality of the effective (low energy) baryon number violating operator. At high temperatures (z "~ [) ')'X -~ 71D "" Z -- decays and inverse decays are suppressed by time-dilation effects because the Xs are relativistic. At low temperatures (z >~ 1), IDs are suppressed because typical fermion pairs are not energetic enough to produce an X boson, TiD ~ z3/2e-Z. The rate of 2 "~, 2 scatterings is F S _-2,n(ov) (n ~- T 3 is the number density of a quark or lepton species). At high temperatures (z '~ 1) (or) -~ A a 2 / T 2, and 7S = F s / a M - ~ A a z -1. At low temperatures (z >> 1) (ou) A~2/M 2 (dim-5) or A a 2 T 2 / M 4 (dim-6), and "/s __. i-s/e.M _~ A a z 7 2d. The form of Ts has been interpolated smoothly between the high- and low-temperature forms [6]. This ignores a possible resonance near z = 1, but (see below) ntffs for K >> 1 and K '~ 1 is sensitive only to the large-z dependence of 713. In writing ')'B = ")'[D + ')'S we are in effect expanding m powers of ~, and we find both these terms to be important. We do not, however, need to go to yet higher order, for the following reason: 7S becomes important only because ')'iD falls off exponentially with z, while 78 falls off only as a power. Another term would be suppressed by (a) another power ofol, and (b) higher powers of z - l , Thus, (1) with (3) contains all significant effects. These equations can be solved, e.g. using "integrat-

Volume 125B. number 5

PItYSICS LETTERS

ence, for in the scattering d o m i n a t e d case nB/s falls off e x p o n e n t i a l l y for large K, rather than as a power. Since for IDs freeze out occurs at z 0 ~ 4.2 × (In c K ) 0"6 and for 2 "~"2 scatterings freeze out occurs at z 0 "" (hod() tin , tile transition from " l D - d o m i n a t i o n " to "scattering d o m i n a t i o n " at freeze out should occur for K ~ K , , where

ing factors" [ 9 ] , to give A(z) = X

Xeq = e x p [ - f x ( z ) l

Z

x

9 June 1983

f exp [ f x ( z ' ) ] ( - X ; q ( Z ' ) ) d z ' , 0

4.2(1n c K , ) 0.6 ~-

B(z) = B(0) e x p l - - f B ( Z ) ]

(Aa'K,) l/n

,

(8a)

or Z

+ e exp[-fB(z)]

cK,)-

K,(ln

0

In fig. 1 we show nB/s as a function o f K f o r A a ~ 10 (d = 6 ) , A a ~ - - 10 ( d = 5), and A s = 100 (d = 6). For K ~< 1, B nonconserving processes are never effective, the X's decay out-of-equilibrium and r/B/S ~ e/geff -- the saturation value. F o r K , ~>K ~> 1 ( " I D d o m i n a t i o n " ) , decays and IDs are effective until z = z0, and the value ofnB/s produced falls cc K 1 (In cK)- 0.6. For K >~K,, 2 *, 2 scatterings are i m p o r t a n t (and are d o m i n a n t at freeze out), and nB/s falls e x p o n e n t i a l l y with K. For Ae= lOandd =6,K.~-O(103);increasingAetol00 (e.g., by increasing the n u m b e r o f scattering channels, or by increasing o~ for fixed K ) causes K , to decrease to O(100). Keeping A e = 10, but changing to dim-5 2 ,, 2 scatterings results in K , -~ O(10). The difference

2

fi(z) : f 7i(z)Kz dz .

(4)

0 This allows easy extraction o f limiting behavior. First, f o r K "~ 1, we s e e f B ,~ 1 always. ,5 is most easily obtained from (1) directly. Nothing happens until z >> 1, when X e . -~ 0, so: A ~ X ~-- e x p ( - . ½ 7 x K z 2 ) , B = e l l -- exp(-- ~ 7 x Kz 2)]. Recalling s ~--ge ff T3, we see nB/s = B(o°)/gef f ~ e/gef f. This is the usual " o u t o f equilibrium d e c a y " scenario result. For K >> 1, A ~-- X e q / 7 X K z . The e q u a t i o n for B can then be solved using "steepest d e s c e n t " ,

0.6n ~

4.2n/Aa.

.f exp[fB(Z')lTxKz'A(z ') dz' ,

(8b)

B = e e - f B f e f B X e q dg oo

CI 0 _ j

.

ZO

where "7BKz = I at z = z 0, and a = d(zKTB)/dzlzo. l'hysically, z 0 is the epoch at which B nonconserving processes "freeze o u t " ( F B ~-If). For ")'ID > ' 7 S at z 0 (eA small, "inverse decay d o m i n a t i o n " ) , z 0 ~ 4.2 (In oK) 0.6 , and

(nB/s) ,,- (e/geff)(n/2)e

-~ iO-2

2 ~" 10-4 1 aA:lO chin:6 2 (:IA:iO0 dim:6 [O_E 3 ClA:IO d,m:5

l(zoK )- 1 i

(e/get.f)K

l(ln c K ) -()'6 .

(6)

On the o t h e r hand, for 7S > '711) at z 0 (aA large, "scattering d o m i n a t i o n " ) , z 0 ~ (hc~K) l/n ,attd

(nB/S) ~ (E/geff)(rr/2)pt l/2(o~gZ)2/n × exp{-[n/(n-- I)](c~KA)I/n}, where n = 2(d - 4). This is a very i m p o r t a n t differ-

(7)

I0-3

I

i

10-1

I0

I0 ]

K

Fig. 1. The asymptotic value of B/e ~--(e/geff) -1 (nB/s) for the simplified set of Boltzmann equations [of. eq. (1), (4)]. as a function of K "" 3 X 1017 GeV c~/M. For curve 1, Ae = 10 andd = 6, for curve 2 Aa = 100 and d = 6, and for curve 3 A e = 10 and d = 5. Increasing the importance of 2 -* 2 scatterings decreases the value of K at which the exponential fall-off begins, cf. curves 1 and 2. Changing from dim--6 to dim-5 2 .* 2 scatterings drastically decreases the value of K at which the exponential fall.off begins, cf. curves 1 and 3.

381

V o l u m e 125B, n u m b e r 5

PItYSICS LETTERS

between using 2 ~ 2 scatterings due to dim-5 and dim6 operators is very dramatic, and the regime of powerlaw decrease essentially disappears. This occurs because the low-temperature cross sections, (oo) ~ A a2/ M 2 versus (ou) ~ A o~2 T2/M 4 , behave very differently. The dim-6 cross section is very temperature dependent, and so 2 ~ 2 scattermgs freeze out earlier. Within this framework we will now discuss the numerical results for supersymmetric versus non-supersymmetric GUTs.

Supersymmetric versus non-supersymmetric GUTs. As we have seen in the previous section, the baryon asymmetry produced depends (m (£/geff) times a function of K and Aa. In a non-supersymmetric SU(5) model, with a = 1/45 and M x = b X 1014 GeV, we haveK x = 1 0 , A o ~ 100(A ~ few X 103), a n d K , 100 for gauge bosons, so that 2 "~"2 scatterings are not very important. For Higgs bosons all "" h2[4rt (h = Yukawa coupling to light fermions); if the superheavy t liggs is the color triplet component of 5-dimensional representation in SU(5), then h ~- mf/(x,~Mw)g (mf = fermion mass, M w = W boson mass,g = gauge coupling). Theeffective ell is the generation averaged coupling, which is dominated by the heaviest family. For 3 families and m t ~-Mw/2, o~H ~ agau~e/24 ~- 10-3.With a H ~ 1 0 - 3 , K t l ~< 1 for ~/1f~> 1014 GeV. With A r m I , K ~ 105, so 2 '-. 2 scatterings are unimportant forMit ~ 1010 GeV. There is as yet no uniquely compelling supersymmetric GUT. Nevertheless, there are certain generic features that must be present in all such models. The number of particle species geff doubles, for each fermion/boson there is a supersymmetric boson/ fermion partner. In particular, there must be a scalar partner for each quark and lepton species. This increases the number of decay channels, and more importantly the number of 2 *, 2 scattering channels. In the minimal supersymmetric extension of SU(5) the unification scales increases to few X 1016 GeV, although additional light Higgs nmltiplets can lower it to O(1015 GeV) [10]. The gauge coupling constant at the unification scale roughly doubles, to 2 l/'25. Finally, in the absence of an additional symmetry forbidding it, there exists a dim-5 operator which violates B. conservation. The process involves 2 quarks or leptons and 2 scalar partners of the quarks and leptons, and is mediated by the fermionic partner of the superheavy Higgs color triplet. The existence and importance of 382

9 J u n e 1983

such an operator in the context of proton decay was pointed out by Weinberg [ 11 ] and Sakai and Yanagida [12]. We have taken our code for the non-supersymmetric SU(5) model [6], and doubled the number of degrees of freedom for each particle species (fermions, gauge and Higgs bosons) in order to simulate a supersymmetric SU(5) model. In addition we have considered the possibility of dim-5 2 ~: 2 scatterings (by modifying the low-temperature form of the 2 *- 2 scattering cross section). The results t'rom integrating our full network tbr this "sinmlated'" supersymnretric SU(5) GUT are shown in fig. 2, along with the results for the non-supersymmetric SU(5) model. The effect of using a dim5 operator for the 2 ,--,2 scatlerings is dramatic - as much as 7 orders-of-magnitude. For ~ll --- 1 0 3 the transition to "scattering domination", where the falloff in production is exponential, occurs for K ~- 10 (M H -~2 X 1013 GeV). [Recall that for%l ~ 10 ..3 in the non-supersymmetric GUT K, ~- 105.] The increased number of scattering channels (A increases by a factor O(10)), and most importantly the milder tern~-----

,

1

\. so°:',L t -~ j0-2 ~'

iTa__~x i0-~

b_.

2.o:,o -~

\'

--., 3-cl:4,tlO"3 ~=" 1{34 4-a:,~,lo-* dtm-6 5-a:4xld z i(~6 6-a<_4xlO-4'

if 3

ct .004 M:

\\

\

lO ~ 1 I0 K ~ 2x 10~7OeV{a/M)

lO-2

i

ioleoeV

i

,

a=4~,ld 4

M:

1017OeV

1014C.ev

10:SGeV

IOI]OeV

J

L __~I

iO~6OeV

,

1016OeV

i

M:

I00

,Ol¢OeV

J

J

iO~GeV

Fig. 2. Tile results of o u r full n e t w o r k for nB/S in t h e " s i m u l a t e d " s u p e r s y m m e t r i c SU(5) m o d e l . F o r curves 1- 4, dim-5 2 ~ 2 s c a t t e r i n g s w e r e used; for cur~es 5 , 6 , dim-6 2 ~ 2 scatterings were used. Tile results for n o n - s u p e r s y m m e t r i c SU(5) are also s h o w n , for c~ = 1 / 4 5 (solid line) a n d e < 10 - 3 ( b r o k e n line). The i m p o r t a n c e of dim-5 2 ,-* 2 s c a t t e r i n g s is very apparrent.

PHYSICS LI(T'I'ERS

Volume 125B, number 5

perature dependence of dim-5 scatterings z 3 versus z 5 for dim-6) are responsible for the increased importance of 2 ~ 2 scattering processes. Without dim-5 scattering processes, the change is less pronounced. Due to the increased number of decay channels the transition from out-of-equilibrium decay to "ID domination" occurs for K slightly less than 1. Due to the increased number of scattering channels the transition to "scattering domination" occurs for smaller values of K. Assuming the C, CP-violation arises at one-loop with maximum amplitude (i.e., e -~ o t H / 4 / r ) , we show in fig. 3 the baryon asymmetry nB/s which is produced by a single Higgs species, when the 2 ,-, 2 scatterings are due to a dim-5 operator. The non-supersymmctric SU(5) results are shown for comparison - it is clear that dim-5 scatterings make it much nrore difficult to produce an asymnretry of the desired magnitude (nB/s ~" 10 1 0 - 1 0 - 1 1 )

Discussion and summary. In a non-supersymmetric SU(5) model 2 *, 2 scatterings are unimportant for baryon generation either by superheavy gauge or Higgs bosons. In supersymmetric models 2 ,, 2 scatterings are more important because the gauge coupling is larger, there are more light species which carry B and L number (and hence more scattering channels), and because of the possibility of dim-5 B nonconserving I

--SUSY SU(5) [0-6 . . . .

Su(5)

iO- ' 2

,\

.--J6-~ ~

I

c

tO-6

i __ L._~_..k_

i0 -4

"

10-2

0t H I:ig. 3. The baryon asymmetry nB/S generated by a Higgs in a supersymmetry SU(5) model with dim-5 2 *, 2 seatterings and one-loop, maximal C, C/U-violation (c "" e l l / 4 n ) as a Function o f e H f o r M H = 1012 , 1013 , 1014 , 1015 , and 1016 OeV. For comparison, the non-supersyrnmetric results are also shown (broken curves).

9 June 1983

processes. In the case of baryon generation by a gauge species, the 2 ~, 2 scatterings are still uninrportant for M ~ 1015-.1016 GeV. If there are dim-5 B nonconserving processes, then 2 ~-. 2 scatterings can be a very significant consideration for baryogenesis by a Higgs. For example, if Oql "" 10- 3 then only forMtl ~ 1014 GeV can a baryon asymmetry of the desired magnitude be produced. Although one expects MII to be of the order of the unification mass, there are scenarios where a much lighter tliggs may be required. In the supersymmetric intlation scenario of Albrecht et al. [13] based upon the geometric hierarchy model of Dimopoulos and Raby [ 1], the universe reheats only (if it reheats at all) to a temperature of order 1012 GeV, and it is unlikely that particles significantly more massive than 1012 GeV can be produced in the reheating process ~ 1 We thank S. Dimopoulos for useful discussions. This work was supported in part by the US DOE through contract DE-AC01-80ER10773A003 (at Chicago). 4-! Kolb and Raby have recently considered baryogenesis in the context of supersymmetric GUTs, and arrived at conclusions which are very similar to ours; sec ref. [ 14].

References [1] S. Dimopoulos and S. Raby, Nucl. Phys. B192 (1981) 353; E. Witten, Nucl. Phys. 13188(1981) 513: S. Dimopoulos and H. Georgi, Nucl. Phys. B193 (1981) 150; M. Dine, W. l.ischler and M. Srednicki, Nucl. Phys. B189 (1981) 575. [2] Y.A. Gol'fand and E.P. Likhtman, JETP Lett. 13 (1971) 323; D. Volkov and V.P. Akulov, Phys. Lett. 46B (1973) 109; J. Wess and B. Zumino, Nucl. Phys. B70 (1974) 39. [3] J. Wess and 13. Zumino, Phys. Lett. 49B (1974) 52; J. lliopoulos and B. Zumino, Nucl. Phys. B76 (1974) 310; S. Ferrara, J. lliopoulos and B. Zumino, Nucl. Phys. B77 (1974) 413. [41 A.D. Sakharov, JI'TP Lett. 5 (1967) 24; M. Yoshimura, Phys. Rev. Lett. 41 (1978) 281; A. Ignatiev, N. Krasnikov, V. Kuzmin and A. Tavkhelidze, Phys. Lett. 76B (1978) 486 ; S. Dimopoulos and L. Susskind, Phys. Rev. DI8 (1978) 4500;

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J. Ellis, M.K. Galliard and D.V. Nanopoulos, Phys. Lett. 80B(1978) 360; D. Toussaint, S.B. Treiman, F. Wilczek and A. Zee, Phys. Rev. D19 (1979) 1036; S. Weinberg, Phys. Rev. Left. 42 (1979) 850. [5] J.A. Harvey, E.W. Kolb, D.B. Reiss and S. Wolfram, Phys. Rev. Lett. 47 (1981) 391; E.W. Kolb and S. Wolfram, Phys. Lett. 91B (1979) 217; Nucl. Phys. B172 (1980) 224. [6] J.N. Fry, K.A. Olive and M.S. Turner, Phys. Rev. Lett. 45 (1980) 2074; Phys. Rev. 1)22 (1980) 2953, 2977. [7] S. Barr, G. Segr~ and II.A. Weldon, Phys. Rev. D20 (1979) 2494; A. Yildiz and P. Cox, Phys. Rev. 1)21 (1980) 306;

384

[81 [9] [ 10] [11 ] [ 12] [131

[ 14]

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D.V. Nanopoulos and S. Weinberg, Phys. Rev. D20 (1980) 2484. H.E. Haber, Phys. Roy. D26 (1982) 1317. M.S. Turner and J.N. Fry, Phys. Rev. D24 (1981) 3341. M.B. Einhorn and I).R.T. Jones, Nucl. Phys. B196 (1982) 475. S. Weinberg, Phys. Rev. D26 (1982) 287. N. Sakai and T. Yanagida, Nucl. Phys. B197 (1982) 533. A. Albrecht, W. Fischler, P. Steinhardt, S. l)imopoulos, E. Kolb and S. Raby, University of Pennsylvania preprint (1983). E.W. Kolb and S. Raby, Los Alamos National Laboratory preprint (1983).