Grand unified theories (GUTs) seek to unify the three separate gauge groups SU(3), SU(2) and U(I) of the standard model in a simple group G:
G :::) SU(3) x SU(2) x U(1). (4.56) (See [13] for a review.) The GUT hypothesis is that above some high energy (GUT) scale MG,
M G",
>
1015 GeV (4.57) G is an exact symmetry, which is spontaneously broken at the GUT scale to the standard model, which is itself spontaneously broken at the electroweak scale. In this way the (Iow-energy) gauge coupling strengths (aI, a2, a3) of the standard model are all determined from the unknown (high-energy) coupling strength aGof G, by using the renormalization group equations to 'run' between the GUT and the electroweak energy scales. We have discussed this in some detail in [10] but the essential point is that the evolution of the coupling strengths depends upon the matter content of the low-energy theory. Since neither aG nor mG is known a
priori, the GUT hypothesis can only be tested by starting from the values of the coupling strengths measured at the electroweak scale and running to high energies to see whether they converge to a single value (aG).
The known matter content of the standard model consists of three generations
of
QL
= (3.l.~)B-1
-JL=O
- 2
ui
= (3.1. -J)B=-i L=O
df = (3.1,
i) B=-i L=O
(4.58)LL
= (1.2.-!) B=O L=I
ei
= (1.1. I)B=O L
=-1using the notation ("3. "2. Y), where "3 specifies the colour SU(3) representation, "2 the weak S U (2) representation and Y is the weak hypercharge.
Band L are the baryon and lepton numbers (the superfix C indicates the charge conjugate particle). In addition, the electroweak Higgs
hi = (l,l.!) (4.59)
is an essential ingredient of the standard model, whose discovery is currently awaited, hopefully at the LHC. If we assume just this matter content, besides the 12 gauge vector bosons of the three gauge groups, it is found that the coupling strengths converge and reach a point of closest approach but not coincidence, at the energy scale and coupling strength given in (4.41).
It is remarkable that the couplings come as close as they do and this in itself lends general support to the GUT hypothesis. However, the failure to converge precisely to a common value shows that if the GUT hypothesis is correct, then there must be matter additional to that of the standard model.
Remarkably, the supersymmetric standard model, in which all of the matter particles (4.58) have supersymmetric (bosonic) partners (sparticles), all of the gauge bosons have (fermionic) supersymmetric partners (gauginos) and the Higgs doublet h I has a (fermionic) higgsino partner, does produce the convergence sought [14J. The calculated unification scale and coupling constant are given in (4.42). (Supersymmetry requires an additional Higgs doublet
h2
=
(l,l,-!)
(4.60)plus its superpartner.)
This convergence represents the best evidence we have both for the GUT hypothesis and for low-energy supersymmetry and it is, therefore, natural to wonder whether a GUT with this matter content produces baryogenesis at the level needed to produce the observed asymmetry (4.12) or (4.18). In a general GUT, the matter content (4.58) (and Higgs fields) of the standard model. or its supersymmetric extension, constitute partial or complete representations R of the
GUT gauge group G. The coupling to the gauge bosons A~ of the fennionic matter has the standard fonn
C
= L
Ry"'(iiJll - gGA~tA)R (4.61)R
where t A are the matrix representations of G corresponding to the representation R to which the fennions belong. For the SU(5) GUT, each generation of (4.58) belongs to two irreducible representations
5
and 10 of the group, which decompose into representations of SU(3)c x SU(2)L x U(J)y as follows:- - 1 1
5
=
(3, I, j)+
(1,2, -I) = [df, Ld (4.62) 10=
(3, 2, ~)+
(3, 1.-j) +
(I, I, I)=
[QL,uL' ei.l.
(4.63)Evidently the matrix t A couples the gauge boson A~ to fennions in the representations Rand
R,
whereR
contains the complex conjugate representations, to those given in (4.58), i.e.- - I
QL = (3,2, -6) B=-~
L=O
ii£.
= (3,1,j)
B_1 -'JL=O
~ dL = (3,1, -'J) I B-1 -'J
L=O
(4.64)-LL = (1,2, I) 1 B=O L
=-1
et
= (1, 1, -1) B=O L=l.For baryogenesis, we are concerned with those gauge bosons which are coupled to fennions with a net non-zero baryon number. In the case of the SU(5) GUT with the gauge bosons in the adjoint 24 representation, all of the 12 gauge bosons additional to the 12 of the standard model have this property. They transfonn as
5 - 5
(3,2, -6)
+
(3,2, 6) (4.65) representations of SU(3) x SU(2) x U(l). We denote the (colour triplet) SU(2) doublet by (X, Y), and the SU(3) x SU(2) symmetry requires thatmx=my~MG. (4.66) The allowed decay modes
X - dv, ui, dCuC (4.67)
Y _ dl,
uCu
care shown in figure 4.1.
All violate baryon number conservation. In all cases the difference in the baryon number B and the lepton number L of the final state is
B - L =-'J 2 (4.68)
d
v
u
cl
u
d
u
il
Figure 4.1. Baryon-number non-conserving decays of SU(5) GUT gauge bosons.
Baryogenesis in GUTs
5
5
10
10
Figure 4.2. Gauge vector boson vertices in the minimal SU(5) GUT.
so we may consistently assign (X, Y) this value of B - L, which shows that B - L is conserved in all gauge-boson-mediated processes. B, however, is not separately conserved. Indeed couplings (4.67) and their complex conjugates induce the proton decay modes
p -+ 1r°e+, 1l'+v (4.69)
at a rate which, in the non-supersymmetric model, is calculated [15] to be about one hundred times the measured upper bound implied by (4.28). All of these couplings arise from the Feynman diagram vertices shown in figure 4.2.
Higgs bosons are needed in a GUT to generate both the superheavy masses needed for the non-standard model gauge bosons and Higgs particles, as well as electroweak scale masses for the W± and Z gauge bosons. In the minimal non
supersymmetric SUeS) GUT, the electroweak Higgs doublet is accommodated in a 5 representation H, which in addition includes colour triplet scalars H3 transforming as (3, I, -}) of SU(3) x SU(2) x U(l). The Yukawa couplings have the form
I-I IIKLM
ey =
X[IJ)ho1/l H +8 X[IJ)hUX[KL)HM +h.c. (4.70) where1/1
1 , X[II) are the5,
10 representations of SUeS) which include the matter content (4.58) of the standard model. hu, ho are complex matrices (hu,o)jg acting on the (undisplayed) generation-space labels of1/1
1 and X(/ J). Then the colour triplet Higgs particles have decay modes similar to those of the X -bosonH3 -+ uCdc , ul, dv (4.71) all of which violate baryon-numberconservation. These are shown in figure 4.3.
d
Hs- - -
11
u
Hs- - -
l
cl
H
3 - - u
Figure 4.3. Baryon-number non-conserving decays of SU(5) GUT colour triplet Higgs bosons.
As before, in (4.68), the difference in the baryon number B and the lepton number L is the same for all decays:
B-L=-'J 2 (4.72)
so B - L is conserved in all Higgs mediated processes. CP-violation arises from complex phases which cannot be absorbed by field redefinitions but, as we shall see, there is no contribution to t!.B at tree level. These couplings arise from the Feynman vertices shown in figure 4.4.
-
-10 H- ______ hu
10
10 H- _ _ __ hb
5
Figure 4.4. Higgs (5) hoson vertices in the minimal SU(5) GUT. Single lines represent 5 representations. double lines 10 representations.
We have already noted that. in the minimal SU(5) GUT. the requirements (4.40). (4.44) necessary for departure from thennal equilibrium are more likely to be satisfied by the massive. colour-triplet Higgs scalar H3 than by the massive gauge bosons. so X.Y decays will not contribute significantly to the baryon asymmetry of the universe. Nevertheless. for completeness. we consider the contributions to AB defined in (4.49) from both sources. First we note that the tree-level contribution shown in figures 4.1 and 4.2 is zero. This is clear because in the Born approximation the process X -. dv. for example. and
i -. dii
derive from tenns in the Lagrangian which are Hennitian conjugate to each other.
so their amplitudes are complex conjugates. Since the kinematics of the two processes is identical.
r(x -.
dv)IBom= rei -.
dii)IBom (4.73) and the contribution to ll. B given in (4.49) is zero. The same argument applies to the decays of Y and H3.At the next order. we need to include radiative corrections and look for (CP
violating) contributions to ll.B arising from the interference between the Born tenns and these single-loop radiative corrections. For example. consider the radiative correction to the decay
H3 -. uft, (4.74)
shown in figure 4.5 (f. g are generation labels).
u
H3 - - - -
I
Figure 4.5. Radiative correction to H3 -+ uf.
The matrix element has the form
dM ...., (hDhbhu) f /H (4.75)
where I H is the Feynman loop integral involved. Since the mass of the colour
triplet Higgs satisfies
mH3 » mu
+
mt (4.76)IH is complex. To one-loop order, the square of the total matrix element M satisfies
IM 12 - IMol2
~2 Re[dMM~]QC 2 Re[(hDhbhu)
fg(h~)gfIH]
(4.77)(no summation).
Mo
is the amplitude for the Born approximation, shown in figure 4.3(b). For the corresponding antiparticle decay, we just replace all coupling constants by their complex conjugates and the difference between the rates is given byr(H3 -+ Uflg) - r(li3 -+ ufig) QC Im[(hDhbhu) f8(h~)gf] Im(lH). (4.78) Thus, when we sum over the generation labels the contributions cancel, since
tr(hDhbhUh~)
=
real. (4.79)In fact, all such one-loop interference terms are the absorptive parts of one or other of the two-loop diagrams shown in figure 4.6; the absorptive part is obtained by cutting the internal (fermion) loop and placing the cut fermions on the mass shell. The contribution just discussed arises from the absorptive part of the diagram in figure 4.6(g). It is clear that all of the other diagrams also make zero contribution to the baryon asymmetry [16]: the gauge couplings are real and the scalar couplings enter only in the combinations
tr(huh~)
=
realtr(hDhb) = real. (4.80)
Baryogenesis in GUTs 107
(0)
(b)(c) (d)
-
-...-- --
...- -
...-
(e)
(J)
-
- ... -- .... -
(g)
Figure 4.6. Two-loop diagrams giving one-loop radiative corrections to gauge boson and Higgs boson decays.
The first non-zero contributions arise only from the three-loop radiative corrections to H3 decays, and the four-loop corrections to X,Y decays.
For example. there are three-loop radiative corrections to H3 decay w_hose contributions to the difference between the decay widths of H3 and H3 is proportional to ImT. where
- t t t h ht h
T = tr(hDhuhuhuhD DUO) (4.81 )
which, in general, is non-zero [17-19]. The Yukawa couplings hU.D detennine the fennion masses. The fennions in the three families of 10 and
5
representations are unitarily related to the mass eigenstates, the connection being given by~PmuQ
(4.82) hu = .J2mw~RmDS
(4.83) hD = ../2mwwhere P, Q, R, S are unitary 3 x 3 matrices and
mu = diag(m" mc, mu) (4.84) mD = diag(mb, ms, md). (4.85) Then
2Im T
= --s-
gS tr(m~(muAm~At, muBm~Bt]) (4.86) 16mwwhere A and B are the unitary matrices
A
=
ptR B=
QSt . (4.87) It is easy to see that the dominant contribution to the trace is proportional to [20]m:m:mc!(9) sin 8 (4.88)
where /(9) is a real function of the mixing angles characterizing the matrices A, B and 8 is a CP-violating phase. Remembering that the total decay width of the (colour-triplet) Higgs scalar is given by (4.43) with m f
=
m, the heaviestfennion, we conclude that the baryon asymmetry deriving from the minimal SU(5) GUT satisfies
llB '"
< (a
2G)3
m: m,mc ,... 10-15 (4.89)1r m6 •
w
However, using (4.52) and the observational data (4.18), we require that llB ~6N. x 10-10
>
10-7 (4.90)'"
so there is no doubt that this mechanism cannot explain the measured asymmetry.
In any case, we have already noted that this minimal theory gives an unacceptably high proton decay rate.
The foregoing discussion suggests that to increase the predicted value of llB, we need to arrange that the asymmetry can arise via one-loop corrections to the Higgs decays. This entails enlarging the Higgs content. The simplest method is to include a second 5 of Higgs scalars H', with a different mass or lifetime, whose couplings are of the same fonn (4.70) but with the coupling matrices hU.D
Baryogenesis GUTs
q
... H
...
....X,Y
V\I\IV( : :I x~
....
....q .... H
Figure 4.7. CP-violating. baryon-number non-conserving decays of X.Y bosons.
replaced by h
U.
D' Then. besides the radiative correction to H3 decay shown in figure 4.5, there will be a similar diagram involvingH)
exchange. The difference between the decay rates now satisfiesr(H3 -+ Uflg) -
r(l13
-+ "fig} ex Im[h~h1huh~]lm(lH') (4.91) instead of (4.7S). In general. this is non-zero [IS].Unfortunately, such a model is unsatisfactory in other respects. It will have flavour-changing. Higgs-mediated neutral current decays [21] and. in addition.
like the model with a single 5 of Higgs, it continues to possess the strong CP problem. (See section 5.3.1.) We shall see later that the most attactive solution to the latter problem utilizes a Peccei-Quinn U(l) symmetry [22]. This requires additional Higgs doublets. so that one doublet H is coupled only to dR quarks and the other doublet H' only to UR quarks. This is realized in an SU(5) GUT by coupling one Higgs 5 to the matter (5)(10) and the other to the (10)(10) fields [23]. Thus, instead of (4.70). we have
.l.iH-J hi I IJKLM
C Y = X[IJ) h D."
+
X[IJ) UX[KL]HME+
h.c. (4.92)This looks as though the one-loop radiative correction shown in figure 4.5 is now forbidden, since the exchanged Higgs has to have both D-type and U
type couplings. However, it is allowed, since the Higgs mass eigenstates mix H and H'. Nevertheless, there is no CP-violation and, hence. no contibution to the baryon asymmetry, for the reason (4.79) given earlier. Instead, in this model. the (CP-violating) baryon asymmetry arises from the decays of X, Y gauge bosons with a pair of Higgs bosons in the intermediate or final state [24.25]
(see figure 4.7), and a sufficient asymmetry arises for a wide range of parameters [17,26].
Another variant of the SU(5) minimal model is to introduce a Higgs multiplet belonging to a different representation, i.e. not a 5. Since, in SU(5),
"5 x"5 =
10+ 15"5 x
10=
5+4510 x 10
="5 +
45+
50 (4.93)Higgs belonging to 10, 15, 45 or 50 representations can be coupled to the matter fermions. In particular, the introduction of a 45 Higgs representation also allows the generation of sufficient baryon asymmetry for a wide range of parameters [17].