- Email: [email protected]
Construction of grand unified theories
ANNALS
OF PHYSICS
205, 309-329 (1991)
Construction
of Grand
Unified
Theories
DARIUSZ K. GRECH University
Insiitute of Theoretical of Wroclaw, PL-52-205
Physics, Wroclaw,
Poland
Received April 14, 1990
Systematic treatment of the theoretical and aesthetic requirements employed in the construction of unifying groups and their representations is given. Formulating the extensive set of conditions and giving reasons for them we derive a much more compact class of unified models than the one already discussed in the literature. It is shown that most models based on nonunitary groups and those with exotic representations are excluded. On the other hand, the permissible class of unitary unifying models is large but well ordered, thus making them useful1 for future applications. (0 1991 Academic Press, Inc.
1. INTRODUCTION Despite many interesting achievements the minimal SU(5)-like grand unified theories (GUTS) [l] still suffer from some troublesome problems. With the exception of multi-generational models incorporating eight families of quarks and leptons they predict values too low for the electroweak mixing angle sin’ 8, and the proton lifetime rP comparing with present experiments [2]. In addition, in its simplest form, the SU(5)-like theories do not accommodate the variety of flavours belonging to different generations. The latter are put into unifying models as a simple repetition of a fermionic multiplet describing one family. The origin and the replication of generations is thus still a mystery. One can hope that both problems might be solved by embedding the standard strong-electroweak symmetry based on the cross product SU(3)C x SU(2)L x U( 1) Y into larger groups than SU(5) with a view to accounting for the variety of families in a rather less adhoc manner. New symmetries and interactions for momenta q, MZ,, < q2< Mf (M, and M, are respectively the electroweak and unifying mass scales) are likely to be produced in this way (like technicolor [3] or horizontal interactions [4,5]) and could be attractive enough to improve the features of SU(5)-like schemes. Therefore it seems to be very reasonable to concentrate on the general construction of symmetry groups and their representations for unified models. The most general class of unified models found in this way can be a subject of more detailed examination in the future with application to any modern unified theory including superstrings. In this article we shall perform such construction. The paper is organised in the following way. In the next section we formulate conditions on unifying groups GGUT and on their admissible representations ~$ou~ 309 0003-4916/91 $7.50 595/205/2-6
Copyright 0 1991 by Academic Press, Inc. All ngbis of reproduction in any form reserved.
310
DARIUS2 K.GRECH
which, if satisfied, determine the whole class of admissible GUTS. Next, in Section 3 the complete set of models is found-consistent with explicitly stated requirements of Section 2. A summary of obtained results is presented in Section 4.
2. CONDITIONS FOR UNIFIED MODEL BUILDING
The criteria used for unified model building are not the subject of universal agreement. Some of them owe their origins to deep theoretical reasons, but others are based on aesthetic preferences which amount to certain assumptions of simplicity. Different authors slightly modify these criteria to their own needs C&9]. We present below the set of conditions already discussed in literature (conditions I-VII), but we add also some new ones yet not considered (conditions VIII-X). In our opinion the set given below is the minimal one satisfying both theoretical and aesthetic requirements. It is the following: I. Unifving group GGUT is simple. This condition demands that we have only one coupling constant in the unified model. One can obtain the same effect also in models based on the direct product of simple groups G, i.e., G, x G, x ... x G,, but in the latter case the incorporation of discrete transformations linking the constituents Gi is necessary to guarantee the uniqueness of the unifying coupling constant. Such a possibility, meant as less aesthetic, is not explored here. Because GoUT is simple we must limit the possible class of unifying groups to [lo]: SU(k + l), k 2 1; SO(2k + l), k 2 2; Sp(2k), k 3 3; SO(2k), k 3 4); and the exceptional groups: Gz, Fct, &, -6, Es. contains SU(3), x W(2), x U( 1) Y as a subgroup. In the light of the 11. GGUT work of Dynkin [ll] one can show that it leads to the groups: SU(k + l), k 2 4; S0(2k+ l), ka4; Sp(2k), k 2 4; S0(2k), k > 5; F4; E6; E,; Es. Thus the list of acceptable groups Gour is here only marginally shortened. III. GGuT has only complex representations describing left-handed particles. This requirement means that in the domain for which GoUr is an exact symmetry group all fermions transforming with respect to such complex representations of the unifying group are massless. A simple explanation of this fact is that we are unable to construct the invariant mass term of the initial lagrangian of the theory using only left-handed complex fields. We impose the above condition, because we believe that the origin of all mass terms should be explained by GUTS and such terms should not be inserted in the theory ad hoc. Therefore the unifying representations di; i.e.,
representation
&uT
must be a sum of irreducible
(1) where 4i are neither orthogonal
nor symplectic [12]
(orthogonal
and symplectic
GRAND
UNIFIED
311
THEORIES
representations are real). It follows then that Gour has to be further restricted to SU(n), n > 5; SO(4h + 2) h > 2 or E6. In addition each irreducible constituent #i of representation (1) must not be equivalent to its complex conjugate $+ Let us also remark that because the mapping from di to 4; is an outer automorphism for GGUT mentioned above, there is no way of distinguishing between models based on dGUT and iGuT. IV. GGUT must be asymptotically free. This is to ensure that the theory is perturbatively safe (i.e., its effective coupling constant cr,,,(q2) < 1 for all momenta q). Thanks to this assumption one can safely use perburbation methods in all field theoretic considerations in GUTS. Therefore the first-order /I function (formulae for /? functions are collected in the Appendix) which determines the rate of growth of mGUT should be positive. From (A.14), (A.7), and (A.6) we thus have
1lc.-4~GUT)>o,
(2)
where Cv z $I( V), V is the adjoint representation of GGUT and no Higgs scalars have been included. The delinition of so-called Z-index and its properties can also be found in the Appendix. Because of (1) and (A.5) we see that it is necessary for asymptotic freedom of 4 GUT that each irreducible constitutent di of it is also asymptotically free. All such irreducible representations for groups satisfying conditions I-III are listed with l-indices in Table I. The notation {Akl”}, (m
k
A A I 83 . . . . . . . El .
.
.
..*...
.
(3)
I.e., it is fully antisymmetric in m indices and fully symmetric in A indices in k rows. The asymptotic freedom condition for GGUT results in further restrictions on orthogonal groups. As shown in Table I we are left only with SO( lo), SO( 14), and SO( 18). V. Theory based on G,,, is anomaly free (renormalisable) [13, 141. Such anomalies are connected with asymmetry between left- and right-handed particles and are generated in the anomalous diagrams shown in Fig. 1. For each representa-
176
132
88
132
SO(18
SO( 14)
SO( 10)
&
(1 : 1)
CA, Kim
[i$
A+
A+
Free
I
27
16 126 144
64
256
1) - 2) 2)(n - 3) 1) 1) - 2) - 2) fn’(n’1) fn(n + l)(n + 2) fn(n - l)(n + 2)
n
fn(n +,n(n - l)(n &n(n - 1 )(n $n(n+ in(n’$n(n’l)(n fn(n + l)(n
SU(n)
Asymptotically dim q5(
Complex,
GOUT
Irreducible,
TABLE
6
4 70 68
16
64
f(n - 2)(3n + 1) fn(n” - 4) +(n + 2)(n + 3) $(n + 2)(3n - 1)
$(n-2)(n2-n-4)
n2--3
n+2
- 3)(n - 4)
1
in GUTS
n-2 i(n-2)(n-3) i(n - 2)(n
Representations
- 6)
0
0 0 0
0
0
fn(n’16) i(n + 3)(n + 6) $(n” + 7n - 2)
n+4 2-9 $(n-4)(n2-n-8) -t(n2-7n-2)
d(n - 3)(n - 4)(n - 8)
1 n-4 $(n - 3)(n
GRAND
UNIFIED
313
THEORIES
FIG. 1. Diagram giving contribution to the anomalies. Wavy lines denote intermediate connected with generators Toxb.’ of a gauge group. Continuous line denotes the fermion odd number of vertices the additional ys matrix is present that causes anomalies.
bosons A;b.c propagator. In
tion 4 one can define the anomaly index A(d) in the following way [13]. First we define the normalisation factor dabcfor the fundamental representation q5f of GGUr with generators T, (qSf), dabc2 W{Ta(4ff),
Then A(4) for an arbitrary
representation
Tb(df)> T,(~Y/)).
(4)
4 is obtained from the formula
44) dabc z Tr({T,(d),
T,(d))
T,(4)).
(5)
Comparing (5) with (4), we see that A(#f) = 1. With the help of the relations (6)
44) = -44) 44,
x 4d = dim h .Nh)
4#+42)=M#r)+Nh)r
+ dim 4~ .4h)
(7) (8)
314
DARIUSZ
K. GRECH
which follow directly from the definition (5), we can find anomaly indices for all irreducible representations bi of Go”=, only using the knowledge that A(bf) = 1. They are listed in Table I. In general, the anomaly-free requirement then is (9)
A(4GUT)=CaiA(4i)=o.
This criterion imposes restrictions on possible values of coefficients ai. VI. Unifying representation dGUT contains complex part after breaking GGUT + SU(3),XSU(2)~X U(l),. This is to have massless fermions of the standard model describing ordinary generations, for they acquire a mass only after breaking to W(3), x U( l)EM. VII. Go,, should have real representation at SU(3), x U( 1 )EM level. This is to ensure masses for all fermions below electroweak symmetry breaking mass scale M,. The criterion is peculiarly restrictive for SU(n) models, because it excludes all the so-called exotic representations (i.e., representations which are not fully antisymmetric in group indices, e.g., {Aklm},). There is only one exception to this rule. It happens in the SU(5) model where we have just one possible exotic representation of the form [9]
i su(5)= Ul,+ or, in dimensional
{2),+
{U}5+ {212},
language, i S(i(-j)= 5 + 15 + a + 45.
VIII. tation &uT.
(10)
(11)
There are at least three generations ( g, = 3) included in unifying represenThey transform with respect to SU(3)= x SU(2), x U(l), according
to the pattern (A.16). IX. SU(3)C contained in GGUT is asymptotically free at least near the electroweak mass M, and below it. Thus the first-order QCD b function must be positive, which means in terms of (A.14), (A.lO) and (A.7), that (12)
where $,s”(3)cxsu(2)Lx u(l’y is the complex part of #oUT under SU(3)=x SU(2), x U(l), (the real part describes superheavy fermions decoupled from the low-energy considerations thanks to the decoupling theorem [15]) and IsLIc3) is the l-index of the SU(3), representations. The asymptotic freedom for QCD is usually required to make this theory perturbatively safe. For momenta q* < A4; it comes as a result of experiments in deep inelastic lepto- and electroproduction [16, 173. To explain the necessity of asymptotic freedom for QCD at momenta q* 2 Mi, let us note the following. For
315
GRAND UNIFIED THEORIES
the minimal fermion content describing g, generations (A. 16), condition (12) gives g, Q 8. If (12) is not satisfied (i.e., g, > 8) we find from the two-loop solution of the renormalisation group equations, i, j= 1, 2, 3,
(13)
for the effective coupling constants ai of the standard model that the QCD coupling constant ~rocb z fx3 app roaches unity before the unification point M, is reached, where c~~(M,)=~~(M.~)=~,(M.~). So the theory is not perturbative for momenta high enough and we are unable to deal with it using a perturbation technique of quantum field theory. Thus (12) must hold. X. The effective coupling constants: a3 of SU(3),, a2 of N(2),, and a1 of converge mutually and satisfy below the umyication point the relation U(l), aj>a2>c11. This condition properly relates the strength of three fundamental
interations. It is important only for GUTS with a partly nonasymptotically free SU(3), x SU(2), x U( 1) Y sector, where higher order effects could be more important than first-order ones. This occurs, for instance, in models with superheavy background fields (e.g., superheavy fermions) decoupled below the mass threshold M,> M,. An example of such unifying models, where the presence of superheavy field sector is due to the horizontal interactions is given in Ref. [S J. For fully asymptotically free theories, this condition is satisfied automatically. The last requirement is thus a way for specific unilied models to evolve from those satisfying criterions I-IX, if the mass spectrum of a theory is already known.
3. CONSTRUCTION
OF UNIFIED
MODELS
As we have pointed out in the previous section, conditions IIIV allow only the following unifying groups: SU(n), n 3 5; E,; SO(10); SO(14); and SO(18) with the unifying representation
4
GUT
=
C
aidi,
(14)
where #i are irreducible representations listed in Table I. The coefficients aj in (14) are positive and integer but it is useful to introduce the notation (15)
Thanks to this, all contributions from conjugate irreducible representations 4 are to be interpreted as terms involving 4 with negative coefficients. Such notation will be utilised everywhere below.
316
DARIUS2
K. GRECH
We will now show how the remaining conditions listed in the previous section specify the unified models. It will be done separately for exceptional, orthogonal, and unitary unifying groups. 3.1. Exceptional
Groups
E, is the only admissible unifying exceptional group. Using information from Table I, we find with the help of (14) the general form of unifying representation as
#E6=a(l
a<22
: 11,
(16)
with the constraint on the coefficient a following from (2). Profiting from the decomposition rules E, + SO( 10) [ 181, (1: 1)=1+16+10
(17)
and, then from the branching rules SO(10) -+ SU(5) [18], (1’3)
1= w5
{ij,+ lo= {q5+ {Q5
16= {O},+
{12},
(19) (20)
with (O}, = 1, { 1 }5 = 5, and { 1 2}s = 10 irreducible obtain the complex part c of (16) under SU(5) as
representations
of SU(5), we
o$61:1)=a({i},+{12}5)=a(S+10).
(21)
The direct comparison of (21) with (A.16) gives a = g,. Because Isvcj,(azG1”)) = 4a (see Table I and (A.5)), the asymptotic freedom of N(3), in Eq. (12) requires a 6 8. It gives, together with the condition VIII, dEg= g,(l
g, = 3, .... 8.
: 11,
(22)
The rest of the conditions are automatically satisfied; in particular, representation is anomaly-free due to A( ( 1 : 1)) = 0. 3.2. Orthogonal
the above
Groups
We have here three candidates: SO(18), SO( 14), and SO( 10). The admissible representations of SO(18) and SO(14) are real if one breaks these symmetries to X’(3), x SU(2), x U(l), and this contradicts condition VI. Indeed, we have for SO( 18) --) SO( lo),
A+SOWX, = 8 (A+sm,
+A-
so(m)) = 8(16
+ 16)
(23)
and for SO( 14) -+ SO( lo), A +so(14)
=
‘3A+socm,+
A-sod
=
2W+%),
(24)
GRAND
UNIFIED
317
THEORIES
The above branching rules prove that in both cases the unifying representation is real before SU(3),x SU(2), x U(l), symmetry is reached. The possible reducible repreesentations of SO(10) satisfying (2) are to be built from A,, Cl’l., and [A, 1] _ representations defined in Table I or from their complex conjugates. There are the following possibilities: 1”
a.A,,
a<22
(25)
2”
Cl’l, +a.A+,
a<4
(26)
3”
[I’],
a<4
(27)
4”
[A, l]-
+a.A+,
a<4
(28)
5”
[A, l]-
+a.A-,
aG4.
(29)
+a.A-,
Let us investigate them respectively looking contained in SO( 10).
at asymptotic
freedom of SU(3),
ad. 1” With the help of (19) and by direct comparison we have a = g,. From (AS) and from Table 1 we then obtain /,,(,,(a. A +) = 4a. Because a. A+ is fully complex under SU( 3), x N(2), x U( 1) ,, thus ( 12) requires a 6 8. Combining this with condition VIII one arrives at 4 SO(IO)= ad. 2”
g, = 3, .... 8.
.A+ >
gN
The branching rule SO(10) + SU(5) for [l’]
+ is [18],
[1~]+=1+J+10+i5++;iS+50 and then for SU(5) --+ SU(3),x
X7(2),
(30)
(31)
x U(l),,
1 = (1, l)(O)
(32)
5=(1,2)(-4)+(3,1)(f)
(33)
10=(1,1)(1)+(3,1)(-5)+(3,2)(~)
(34)
is=(l,3)(-1)+(3,2)(-9+(%,1)(j)
(35)
as= (1,2)($)+
(3,1)(-f)+
(3,3)(-f)+
(3,1)(i)
+ (%2)( - i, + (6, l)( -4, + (8,2)(i) 50 = (1, l)(2) + (3, l)(f) + (3,2)(i) + (6,1)( - $I+ @,2)(
-
(36)
+ (6,3)(f)
41,
(37)
where the last number in the bracket denotes the U( 1) Y normalised charge. Therefore the complex part r~ of [ l’]+ at SU(3), x SU(2)L x U( l)y is 5 5 +(L l)(l)+(L W)+(3, W-5) 40$0, = i Yo:;,, +
(Xl)($)
+
(X1)($
(38)
318
DARIUSZ
K. GRJ3CH
with the exotic part i~:&,=(1,3)(-1)+(6,1,(~)+(6,
l)(-f)+(6,
l)(-4)
+ (6,3)(f) + (3,3)( - $1.
(39)
Using Eqs. (19), (38), and (39) we find the full complex part of (26) as Cl’]+ +aA+ ~SO(IO)
= [Cl’l+
SO(10)
+
(19
l)(2)
+
(5
W)
+ (a+ l)U, 1)(1)+4,
N-$)+a(3,2)(:)
+ (a + 1)(3, l)( - 5) + (a + 1)(3, l)(i).
(40)
Comparing with the one family content (A.16) we see that the content (40) contains a generation (a = g, < 4). The IsU(3,- index for the content (40) is calculated from Table I as I SW31
Cl’]+
(0 SO(10)
+aA+)
=
36 + 4u,
(41)
so that condition (12) is not satisfied and the representation (26) should be rejected. ad. 3” Similar considerations as before but with the branching rule (19) replaced by its complex conjugate lead to the complex part of (27) under SU(3),xSU(2),x U(l), as a~;:;,;““-=(1,1)(2)+(1,3)(-1)+(a,l)(~)+(6,1)(-f)
+ 6 UC-‘$
+ (6,3)(s)+
(3,3)(-f)
+(a-l)U,
l)(-l)+(a-1)(3,1)(f)
+ (3, l)($)
+ (a - 1)(3, 1 )( - $, + a( 1,2)( 9 + u(J, 2)( - d,.
(42)
Comparing (42) with (A.16) one can see that the content (27) contains g, = a - 1 generations and, therefore according to condition VIII, it implies a = 4. In this case we have 1151++~a-) = 48, ~suw(~s0,10,
(43)
so that SU(3)c is not asymptotically free. The conclusion is that the content (27) also should be rejected. ad. 4” Let us decompose [d, l] ~ representation of SO(10) on irreducible representations of the standard group. With the help of branching rules to SU(5)
Cl813 [A, l]-
=3+5+10+15+24+3+45
(44)
GRAND
and then, using decomposition with
UNIFIED
319
THEORIES
laws (32)-(36) to W(3),
x SU(2), x U( l)y together
a=(l,2)(-;)+(3,2)(:)+(3,1)(-$) +(3,3)(-~)+(8,1)(1)+(~,2)(~),
we find the complex part CTof [d, l]-
(45)
under SU(3),x
SU(2), x U(l),,
a~~~j=i~~l’oj+(1,1)(1)+(1,2)(-~) + 3(3,2)(i)
+ m
I)(-
:, + (3,2)@
+ (391 l(f) + (1,2)( - 9 + (3,1 N - $1,
with the exotic part cg;“ii, i&&j
(46)
where = (k3)(1)
+ (%3)( - 3) + (3 3)(f)
+ (6, 1 J(f) + C&1)( - f, + 6 2)(4) +@,W-~)+W)w
(47)
Therefore from (46), (47), and (19) the general form of the complex part of the content (28) at the standard model level is ‘d’lJm+OA+=[$/~j ~.sO(101
+(1,2)(-$)+(3,2)(z)
+(3,1)(-$+(a+
l)U, l)(l)
+ (a + l)(L
2)( - $1
+ (a + m,
l)(f)
+ (a + 3)(3,2)(i)
+(a+2)(3,1)(-3)
(48)
with g, = a + 1 generations described by (A.16). To satisfy condition have a > 2. However, Z,,(,,(a~&~‘-
+“*)
= 56 + 4u,
VIII
we must (49)
which contradicts (12) and we conclude that the representation (28) is not acceptable. ad. 5” As before, using (46) and complex conjugation of (19), one obtains the complex part of (29) at SU(3), x W(2), x U( 1) y level, a&/?oi +““-=[&l~;
+ (1,2)(-3)
+ (3,2)(i)
+(3,1)(-~)+(~-1)(1,1)(-1)
+ (a - 1ML 2)(f) + (a - 1)(3, I)( - 4, + b - 3)(X2)( - ;) + (a - 2)(3, 1 )cf),
(50)
320
DARIUSZ
K. GRECH
which contains only one family for a = 0 or a = 4 (compare with (A.16)). As a result the representation (29) is not permitted, being in contradiction with condition VIII. In addition SU(3)C is not asymptotically free in the theory based on (29) because I SU(3) (CT&&)= 56
(51)
I SU(3) (0$$~1Jj+4Ae)=54 so that condition Finally,
(52)
(12) cannot be satisfied.
the only admissible unifying representation 4 SO(10)
=
gN
A+
for orthogonal
g, = 3, .... 8
5
for the SO(10) unified model. The anomaly-free condition cally because A( A + ) = 0. 3.3. Unitary
groups is
(53) (9) is fulfilled automati-
Groups
Let us first consider the exotic case (10). From the branching rules (33), (35), (36), and (45) one can find that the complex part of (10) under W(3), x SU(2),x U(l),is Qsc/(s)=
(1,3)(l) + (3 3)(f)+ (3 3)(-f) -t(6,1)(-~)+(6,1)(~)+(6,2)(~)+(8,1)(1) + (w(-+)+2(3,2)(i)+ (3,2)(z) +(3,1)(-$)+(W(-3)+(W(-9
(54)
ku,3,(~su,,,) = 52.
(55)
and
Hence the asymptotic freedom for SU(3), cannot be obtained. Furthermore, the content (54) is not adequate to describe even one family (see (A.16)). As a consequence it is not permitted by our requirements VIII and IX. All other unifying representations of W(n) groups are the linear combination of nonexotic irreducible representations { 1 } n, { 1* >n, { 13}n, ( 1” } n and their conjugate partners, i.e., 4 SU(n)zi$l
ai{li>.
(56)
The coefficients ai (i= 1, .... 4) are a subject of constraints following from (2) and (9). Thus we have 4
c aiA({l’})=O i=l
(57)
GRAND
UNIFIED
THEORIES
321
and
j, bil 4{W<
lln.
(58)
It is convenient to present solutions with respect to ai in terms of the number of generations g, included in a given model. This number is after Georgi [6], g,&/=N(J)-N(5)
(59)
or, equivalently, g, = N( 10) - N(iiQ
(60)
where N(e) is the total number of $ multiplets of SU(5) in (56). The equivalence is obvious because N(5)+ N(lO)= N(s) + N(n) to satisfy (9) (see Table I for anomaly indices of 5 and 10). Thanks to decomposition SU(n) + W(5), (1},=5+(n-5)x1 {12}.=10+(n-5)~5+;(n-5)(n-6)~1
{l’},=i@+(n-5)xlO+;(n-5)(n-6) x 5 + ;(n - 5)(n - 6)(n - 7) x 1
(61)
{l”}.=J+(n-5)xm+&(n-5)(n-6)~10 +~(n-5)(n-6)(n-7)~5 +&&~-5)(n-6)(n-7)(n-8)~1,
one obtains, with the help of (60) and (56), g, = u, + u3 (n - 6) + +u4(n - 5)(n - 8).
(62)
After suitable substitution of A-indices from Table I we are able to express the solution of the system (62) and (57) in terms of g,, u3, and a4 coefftcients as parameters: a, = (4 - n) g, + $a3 (n - 5)(n - 6) + 4u4(n - 4)(n - 6)(n - 8)
(63)
az=gN-u3(n-6)-&u4(n-5)(n-8).
(64)
The inequality (58) finally determines the admissible values for g,, u3, and u4. All solutions have been listed in Table II, assuming 3 < g, 6 8 to satisfy conditions VIII and IX.
g,{Ij5+gN{121S
kN{Ij6+gN{12f6
3glv{~~,+gN{12~7 (3g,-1)(J},+(g,-1){12},+{13}, (3gN+l)m7+(gN+ Whf (3g,-2){i),+(g,-2)112},+2{13~, (3g,+2){i),+(g,+2)(12j7+2(i3J, (3gN--3)(i},+(g,-3){12j,+3(13}, (3g,+3)~i~7+(gN+3)112}7+3{i3},
SUI(5)
SW71
g,(l : 1) gNA+
b OUT
=.J(6)
E6 SO( 10)
G OUT
ii%
All Possible
Unifying
II
.. .. 8 .. .. 8 .. .. 8 .. .. 8 .. .. 8 .. .. 8 .__, 7 . ... 8 . ... 5 .. .. 8
5, .... 8
3, . ... 8
3, . ... 8
3, . ... 8 3, . ... 8 3,4, 5
4, 5, 6
3, . ... 7
3, . ..) 7
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3
gh’
Representations
TABLE in GUTS
32 12 48
16
5
{ log,, 96 - 2g,
g, =
6, 7, 8
42, g.N=3 log,, g, = 4, . ... 8 { 72-2g,,g,=3,4,5
10&T, log, tog, + 30
g,>4 { 8g,+20, 84-2g,,g,=4, g,=6 1 8g,+24,
50, g/v=3 1 8g,+l6,g,>3 70-2g,,g,=3,4
8g,+ 8g,+8 Sg, + Sg, + Sg, +
6g, %N 4g.w 6g, fk, f&,+4
1 index
3,4
3,4 3,4 -
3 -
394 3. 4 3.4 3.4 3,4 3,4 3 394 -
(SUS%JTs)
6, 7, 8 5, . .. . 8
3, . .. . 8
3, . ... 7
3, . ... 7
3, . ... 8
3, . ... 8
3. . ..) 7
3,4, 5
3 334 3,4
3, . ..) 7
5, . ... 8 3, . ... 6 3
3, . ... 8
3, . ... 8 3, . ... 8 3,4
g,=3,4
8,>4
g,=3,4
66-3g,, g,=3,4 i 16g, - 24, g, > 4 132-2g, 130 - 2g,
{ 14g,, 16gN
80-2g,,
4% g.w=3 { 14g, - 14, g, > 3 g,=3 { 98, loo-2g,, g,>4
14SN
{ 82, 12g, + 32. g, g!v=3 > 4
{ 12g, + 10, g, a 5 100 12g, + 48 98 - 2g,
80-2g,,
108 - 2g, 12g, + 16 90
1 72 12g,-- 2g,, 12, g,> g, = 3, 5 --., 5
12g,-6 12g, + 48
l-37,
Table continued
3, 4
3, 4
3.4
3, 4
3 -
334 3.4 -
s
12gN{~j,6+gN{121,, (12gN-55)1i),6+(gN--o)112},*+
(n-4)g,{~j,+glv{12).
SU( 16)
Wn)
n&l7
lkiv{~~,5+
ScJ(l5)
g.N{12~,5
logN{~~,4+gN{l*~M
{131,6
3, . ... 8
(9g,-28){i},3+(g,-7)(12j,3+{13},,
SU( 14)
3, .. .. 7
%‘v{~~,3+&%{1*~*3
SU(13)
3, .... 6, n < 35 3, ... . 5, ” 135
3, .. .. 6 5, . ... 8
4, . ... 8
3, ..,, 6
3, . ... 8
3, .. .. 6
3, .... 8
3, .... I
&TN
8g,{~~,~+g,v{l*~,~
$ GUT
II-Continued
SU(12)
G GUT
TABLE
g,=3,4, g,=6,7,
5 8
g,=4 g,>4
150-2g,,
&TN=3 g,r3
1%
(2n - 6) g,
26g, 176 - 2g,
i
24gN
132, 126-2g,,
22g,
2og.4’ 1W g,=3 { 104 - 2gN, g, = 4, ... . I 110, g,=8
84--g,, 18g,-36,
18g,
1 index
3
3
3
-
3
3
3, 4
GRANDLJNIFIEDTHEORIES
325
4. CONCLUSIONS In this article we constructed a systematization of unified models based on some theoretical and aesthetic requirements. The permissible class of GUTS is large, but ordered. The unitary groups are its main part. Only two cases of nonunitary models have been found-based on SO(10) and E, groups, respectively. All models with the proper representation structure are listed in Table II. Although above we considered only nonsupersymmetric models, it is clear that all our criteria apply also to the supersymmetric case. The only difference with sypersymmetric GUTS (SUSY GUTS) is that, due to the presence of additional scalar and fermion fields-superpartners of ordinary particles, the asymptotic freedom of Go,, reads from (A.14)
~(3u7-@,,,))>0, instead of (2). Simultaneously, should have from (65)
to ensure that the theory is perturbatively
(65) safe, one
instead of condition (12). These constraints are stronger than in the nonsupersymmetric case and lead to a much narrower class of admissible models than the one we have just worked out. The last column of Table II gives the choice of accepted values of g, in SUSY GUTS for all models permitted by the nonsupersymmetric unification. Let us notice that for SUSY GUTS the maximal number of generations (ghrhnax = 4. This follows from (66) and (A.16) on the basis of analogical considerations as in nonsupersymmetric case (see the analysis in condition IX). Simultaneously, this is the maximal number of generations satisfying (65). It is worth stressing that our solutions in this paper are based on the assumption that no extra symmetries (global or local) exist simultaneously with SU(3),x W(2), x U(l)* at the momenta q2 w Mi,. Otherwise, the complex part of &uT at M,, which is essential for this analysis because of criterion VI, could be different. The case with such extra symmetries must be considered independently for various specific models. Of particular interest are unified models involving technicolor. They, however, have already been investigated in the literature (see, e.g., Ref. [ 191). The compact set of solutions presented is very promising for any further analysis of nonsupersymmetric or supersymmetric unified theories. It has already been used by us in a recent publication [S], but there are still more applications to come which will be the subject of forthcoming elaborations.
59SJ?OSi2-7
326
DARIUSZ K.GRECH
APPENDIX:
CALCULATION
OF j? FUNCTIONS
For purposes of reference here we state basic formulae for the calculation of firstand second-order renormalisation /I functions in nonsupersymmetric and supersymmetric quantum field theories. Let us first define the I-index for an arbitrary representation 4 of the gauge group G with generators T,(4) as
with the normalisation
condition
for the fundamental representation df of G. The l-index has few important properties. Namely, d,, ia which have a product decomposition
for arbitrary
representations
one obtains
64.4)
where dim 4 stands for the dimension (A.4), one obtains the useful relation f
f ( id
of representation
4i
= >
f ,=
l(di)
4. Comparing
(A.3) with
(A.5)
1
which helps significantly in the calculation of the Z-indices for arbitrary representations if the normalisation condition (A.2) for fundamental representation is assumed. For the general case of a quantum field theory invariant with respect to G if G,xG2x . . . x G, gauge group with the fermion representation I/I, it is convenient to introduce new variables, c(:) g ‘I( v(i))
64.6)
c$’
2 ‘Q$“‘) 2
64.7)
c(F) g C,($Ci’) f ’
(A-8)
2
GRAND
UNIFIED
321
THEORIES
where V”’ and It/F’ are respectively the adjoint and the fundamental representations of Gi (i= 1, .... n), tici) denotes the content II/ after reduction G + Gi, and Cz is the eigenvalue of the second-order Casimir operator C,(4) defined for an arbitrary representation f$ as (A.9)
C,(4) if! c Tm. If Gi = W(n),
it is easy to find that c(i) = n
(A. 10)
c+2-l).
(A.ll)
V
If Gi = U( 1 ), then = 0
(A.12)
C:' = c Q;,
(A.13)
CC’)
”
u where the last sum extends over all charged particles with charge Qr, transforming with respect to the one-dimensional representations of U( 1). In the above notation the first-order (pi) and the second-order (/Iii) renormalisation p functions are deduced from the formulae given in the literature [20] for a pure gauge theory as: p,=f(llcpq’)
(A.14)
pi, = 6, { $p$)’
- iQcyy)>
- 2qcy.
(A.15)
Note that there are contributions to a, from two different groups, Gi and Gj (mixture terms), if i # j. Let us consider the particular case of the standard model based on the cross product SU(3), x SU(2), x U(l),. For G=(G,~U(l),)x(G,rSU(2),)x (G, - SU(3),) with the fermion content transforming under SU(3), x W(2), x u(l),
as ti
~U(3k~~U2)LXU~lh~
-gNC(3,l)(~)t(1,2)(-~)+(5,1)(-3) + (3?2,cd,+
(1, l)U)l,
which describes the g, generations, we obtain the following formulae: (A)
for the nonsupersymmetric
case:
(A.16)
328
DARIUSZ
K. GRECH
(A.17) (A.18) (A.19)
(A.20)
where nH is the number of Higgs doublets. (B) for the supersymmetric case: (s)SU(3)c x SW2k x UC1)Y d’ - pp= -2g,+lH PI (sPW3)cx SW~)LXW~)Y 2 fi’,“’ = 6 _ zg, _ in, 82 83
(A.21) (A.22)
(s)SL’(~~XX’(~~X U~)Y $j 8:“’ = 9 _ 2g,
(A.23)
(A.24)
REFERENCES 1. H. GIDRGI AND S. L. GLASHOW, Phys. Rev. tilt. 32 (1974), 438; Theories,” Benjamin, New York, 1984. 2. D. K. GRECH, Int. J. Mod. Phys. A 4 (1989), 2531. 3. For a review see: E. FARHI AND L. SUSSKIND, Phys. Rep. 74 (1981), Phys. 55 (1983), 449. 4. M. A. B. BY~G AND A. SIRLIN, Phys. Rev. Left. 38 (1977), 1113. 5. D. K. GRECH, Z. Phys. C 42 (1989), 599; 46 (1990). 340. 6. H. GEORGI, Nucl. Phys. B 156 (1979), 126.
G.G.
Ross,
“Grand
279; R. K. KAUL,
Unified
Reo. Mod.
GRAND
UNIFIED
THEORIES
329
7. P. H. FRAMPTON, Phys. Left. B 88 (1979), 299. 8. J. CHAKRABARTI, M. Povovrc, AND R. N. MOHAPATRA, Phys. Rev. D 21 (1980), 32. 9. R. C. KING, Nucl. Phys. B 185 (1981), 133. 10. E. CARTAN, “Sur la structure des groupes de transformation finis et continus,” Navy, Paris, 1894. 11. E. B. DYNKIN, “Amer. Math. Sot. Transl. Ser. 2,” Vol. 6, pp. 11l-244, 245-378, Amer. Math. Sot., Providence, RI, 1957. 12. A. I. MALCEV, “Amer. Math. Sot. Transl. Ser. 1.” Vol. 9, pp. 172-213, Amer. Math. Sot., Providence, RI, 1950. 13. J. BANKS AND H. GEORGI, Phys. Rev. D 14 (1976), 1159. 14. S. OKUBO, Phys. Rev. D 16 (1977), 3528. 15. T. APPELQUISTAND J. CARAZZONE, Phys. Rev. D 11 (1975), 2858. 16. P. V. LANDSHOFF,Quarks in high energy interactions, in “Proceedings, 1978 CERN School of Physics, CERN, Geneva, 1978. 17. G. ALTARELLI, Phys. Rep. 81 (1982), 1. 18. R. SLANSKY,Phys. Rep. 79 (1981), 1. 19. D. K. GRECH, Int. J. Mod. Phys. A 4 (1989), 2173. 20. D. R. T. JONES,Nucl. Phys. B 75 (1974), 531; for a review see, e.g., W. MARCIANO AND H. PAGELS, Phys. Rep. C 36 (1978). 137.
OF PHYSICS
205, 309-329 (1991)
Construction
of Grand
Unified
Theories
DARIUSZ K. GRECH University
Insiitute of Theoretical of Wroclaw, PL-52-205
Physics, Wroclaw,
Poland
Received April 14, 1990
Systematic treatment of the theoretical and aesthetic requirements employed in the construction of unifying groups and their representations is given. Formulating the extensive set of conditions and giving reasons for them we derive a much more compact class of unified models than the one already discussed in the literature. It is shown that most models based on nonunitary groups and those with exotic representations are excluded. On the other hand, the permissible class of unitary unifying models is large but well ordered, thus making them useful1 for future applications. (0 1991 Academic Press, Inc.
1. INTRODUCTION Despite many interesting achievements the minimal SU(5)-like grand unified theories (GUTS) [l] still suffer from some troublesome problems. With the exception of multi-generational models incorporating eight families of quarks and leptons they predict values too low for the electroweak mixing angle sin’ 8, and the proton lifetime rP comparing with present experiments [2]. In addition, in its simplest form, the SU(5)-like theories do not accommodate the variety of flavours belonging to different generations. The latter are put into unifying models as a simple repetition of a fermionic multiplet describing one family. The origin and the replication of generations is thus still a mystery. One can hope that both problems might be solved by embedding the standard strong-electroweak symmetry based on the cross product SU(3)C x SU(2)L x U( 1) Y into larger groups than SU(5) with a view to accounting for the variety of families in a rather less adhoc manner. New symmetries and interactions for momenta q, MZ,, < q2< Mf (M, and M, are respectively the electroweak and unifying mass scales) are likely to be produced in this way (like technicolor [3] or horizontal interactions [4,5]) and could be attractive enough to improve the features of SU(5)-like schemes. Therefore it seems to be very reasonable to concentrate on the general construction of symmetry groups and their representations for unified models. The most general class of unified models found in this way can be a subject of more detailed examination in the future with application to any modern unified theory including superstrings. In this article we shall perform such construction. The paper is organised in the following way. In the next section we formulate conditions on unifying groups GGUT and on their admissible representations ~$ou~ 309 0003-4916/91 $7.50 595/205/2-6
Copyright 0 1991 by Academic Press, Inc. All ngbis of reproduction in any form reserved.
310
DARIUS2 K.GRECH
which, if satisfied, determine the whole class of admissible GUTS. Next, in Section 3 the complete set of models is found-consistent with explicitly stated requirements of Section 2. A summary of obtained results is presented in Section 4.
2. CONDITIONS FOR UNIFIED MODEL BUILDING
The criteria used for unified model building are not the subject of universal agreement. Some of them owe their origins to deep theoretical reasons, but others are based on aesthetic preferences which amount to certain assumptions of simplicity. Different authors slightly modify these criteria to their own needs C&9]. We present below the set of conditions already discussed in literature (conditions I-VII), but we add also some new ones yet not considered (conditions VIII-X). In our opinion the set given below is the minimal one satisfying both theoretical and aesthetic requirements. It is the following: I. Unifving group GGUT is simple. This condition demands that we have only one coupling constant in the unified model. One can obtain the same effect also in models based on the direct product of simple groups G, i.e., G, x G, x ... x G,, but in the latter case the incorporation of discrete transformations linking the constituents Gi is necessary to guarantee the uniqueness of the unifying coupling constant. Such a possibility, meant as less aesthetic, is not explored here. Because GoUT is simple we must limit the possible class of unifying groups to [lo]: SU(k + l), k 2 1; SO(2k + l), k 2 2; Sp(2k), k 3 3; SO(2k), k 3 4); and the exceptional groups: Gz, Fct, &, -6, Es. contains SU(3), x W(2), x U( 1) Y as a subgroup. In the light of the 11. GGUT work of Dynkin [ll] one can show that it leads to the groups: SU(k + l), k 2 4; S0(2k+ l), ka4; Sp(2k), k 2 4; S0(2k), k > 5; F4; E6; E,; Es. Thus the list of acceptable groups Gour is here only marginally shortened. III. GGuT has only complex representations describing left-handed particles. This requirement means that in the domain for which GoUr is an exact symmetry group all fermions transforming with respect to such complex representations of the unifying group are massless. A simple explanation of this fact is that we are unable to construct the invariant mass term of the initial lagrangian of the theory using only left-handed complex fields. We impose the above condition, because we believe that the origin of all mass terms should be explained by GUTS and such terms should not be inserted in the theory ad hoc. Therefore the unifying representations di; i.e.,
representation
&uT
must be a sum of irreducible
(1) where 4i are neither orthogonal
nor symplectic [12]
(orthogonal
and symplectic
GRAND
UNIFIED
311
THEORIES
representations are real). It follows then that Gour has to be further restricted to SU(n), n > 5; SO(4h + 2) h > 2 or E6. In addition each irreducible constituent #i of representation (1) must not be equivalent to its complex conjugate $+ Let us also remark that because the mapping from di to 4; is an outer automorphism for GGUT mentioned above, there is no way of distinguishing between models based on dGUT and iGuT. IV. GGUT must be asymptotically free. This is to ensure that the theory is perturbatively safe (i.e., its effective coupling constant cr,,,(q2) < 1 for all momenta q). Thanks to this assumption one can safely use perburbation methods in all field theoretic considerations in GUTS. Therefore the first-order /I function (formulae for /? functions are collected in the Appendix) which determines the rate of growth of mGUT should be positive. From (A.14), (A.7), and (A.6) we thus have
1lc.-4~GUT)>o,
(2)
where Cv z $I( V), V is the adjoint representation of GGUT and no Higgs scalars have been included. The delinition of so-called Z-index and its properties can also be found in the Appendix. Because of (1) and (A.5) we see that it is necessary for asymptotic freedom of 4 GUT that each irreducible constitutent di of it is also asymptotically free. All such irreducible representations for groups satisfying conditions I-III are listed with l-indices in Table I. The notation {Akl”}, (m
k
A A I 83 . . . . . . . El .
.
.
..*...
.
(3)
I.e., it is fully antisymmetric in m indices and fully symmetric in A indices in k rows. The asymptotic freedom condition for GGUT results in further restrictions on orthogonal groups. As shown in Table I we are left only with SO( lo), SO( 14), and SO( 18). V. Theory based on G,,, is anomaly free (renormalisable) [13, 141. Such anomalies are connected with asymmetry between left- and right-handed particles and are generated in the anomalous diagrams shown in Fig. 1. For each representa-
176
132
88
132
SO(18
SO( 14)
SO( 10)
&
(1 : 1)
CA, Kim
[i$
A+
A+
Free
I
27
16 126 144
64
256
1) - 2) 2)(n - 3) 1) 1) - 2) - 2) fn’(n’1) fn(n + l)(n + 2) fn(n - l)(n + 2)
n
fn(n +,n(n - l)(n &n(n - 1 )(n $n(n+ in(n’$n(n’l)(n fn(n + l)(n
SU(n)
Asymptotically dim q5(
Complex,
GOUT
Irreducible,
TABLE
6
4 70 68
16
64
f(n - 2)(3n + 1) fn(n” - 4) +(n + 2)(n + 3) $(n + 2)(3n - 1)
$(n-2)(n2-n-4)
n2--3
n+2
- 3)(n - 4)
1
in GUTS
n-2 i(n-2)(n-3) i(n - 2)(n
Representations
- 6)
0
0 0 0
0
0
fn(n’16) i(n + 3)(n + 6) $(n” + 7n - 2)
n+4 2-9 $(n-4)(n2-n-8) -t(n2-7n-2)
d(n - 3)(n - 4)(n - 8)
1 n-4 $(n - 3)(n
GRAND
UNIFIED
313
THEORIES
FIG. 1. Diagram giving contribution to the anomalies. Wavy lines denote intermediate connected with generators Toxb.’ of a gauge group. Continuous line denotes the fermion odd number of vertices the additional ys matrix is present that causes anomalies.
bosons A;b.c propagator. In
tion 4 one can define the anomaly index A(d) in the following way [13]. First we define the normalisation factor dabcfor the fundamental representation q5f of GGUr with generators T, (qSf), dabc2 W{Ta(4ff),
Then A(4) for an arbitrary
representation
Tb(df)> T,(~Y/)).
(4)
4 is obtained from the formula
44) dabc z Tr({T,(d),
T,(d))
T,(4)).
(5)
Comparing (5) with (4), we see that A(#f) = 1. With the help of the relations (6)
44) = -44) 44,
x 4d = dim h .Nh)
4#+42)=M#r)+Nh)r
+ dim 4~ .4h)
(7) (8)
314
DARIUSZ
K. GRECH
which follow directly from the definition (5), we can find anomaly indices for all irreducible representations bi of Go”=, only using the knowledge that A(bf) = 1. They are listed in Table I. In general, the anomaly-free requirement then is (9)
A(4GUT)=CaiA(4i)=o.
This criterion imposes restrictions on possible values of coefficients ai. VI. Unifying representation dGUT contains complex part after breaking GGUT + SU(3),XSU(2)~X U(l),. This is to have massless fermions of the standard model describing ordinary generations, for they acquire a mass only after breaking to W(3), x U( l)EM. VII. Go,, should have real representation at SU(3), x U( 1 )EM level. This is to ensure masses for all fermions below electroweak symmetry breaking mass scale M,. The criterion is peculiarly restrictive for SU(n) models, because it excludes all the so-called exotic representations (i.e., representations which are not fully antisymmetric in group indices, e.g., {Aklm},). There is only one exception to this rule. It happens in the SU(5) model where we have just one possible exotic representation of the form [9]
i su(5)= Ul,+ or, in dimensional
{2),+
{U}5+ {212},
language, i S(i(-j)= 5 + 15 + a + 45.
VIII. tation &uT.
(10)
(11)
There are at least three generations ( g, = 3) included in unifying represenThey transform with respect to SU(3)= x SU(2), x U(l), according
to the pattern (A.16). IX. SU(3)C contained in GGUT is asymptotically free at least near the electroweak mass M, and below it. Thus the first-order QCD b function must be positive, which means in terms of (A.14), (A.lO) and (A.7), that (12)
where $,s”(3)cxsu(2)Lx u(l’y is the complex part of #oUT under SU(3)=x SU(2), x U(l), (the real part describes superheavy fermions decoupled from the low-energy considerations thanks to the decoupling theorem [15]) and IsLIc3) is the l-index of the SU(3), representations. The asymptotic freedom for QCD is usually required to make this theory perturbatively safe. For momenta q* < A4; it comes as a result of experiments in deep inelastic lepto- and electroproduction [16, 173. To explain the necessity of asymptotic freedom for QCD at momenta q* 2 Mi, let us note the following. For
315
GRAND UNIFIED THEORIES
the minimal fermion content describing g, generations (A. 16), condition (12) gives g, Q 8. If (12) is not satisfied (i.e., g, > 8) we find from the two-loop solution of the renormalisation group equations, i, j= 1, 2, 3,
(13)
for the effective coupling constants ai of the standard model that the QCD coupling constant ~rocb z fx3 app roaches unity before the unification point M, is reached, where c~~(M,)=~~(M.~)=~,(M.~). So the theory is not perturbative for momenta high enough and we are unable to deal with it using a perturbation technique of quantum field theory. Thus (12) must hold. X. The effective coupling constants: a3 of SU(3),, a2 of N(2),, and a1 of converge mutually and satisfy below the umyication point the relation U(l), aj>a2>c11. This condition properly relates the strength of three fundamental
interations. It is important only for GUTS with a partly nonasymptotically free SU(3), x SU(2), x U( 1) Y sector, where higher order effects could be more important than first-order ones. This occurs, for instance, in models with superheavy background fields (e.g., superheavy fermions) decoupled below the mass threshold M,> M,. An example of such unifying models, where the presence of superheavy field sector is due to the horizontal interactions is given in Ref. [S J. For fully asymptotically free theories, this condition is satisfied automatically. The last requirement is thus a way for specific unilied models to evolve from those satisfying criterions I-IX, if the mass spectrum of a theory is already known.
3. CONSTRUCTION
OF UNIFIED
MODELS
As we have pointed out in the previous section, conditions IIIV allow only the following unifying groups: SU(n), n 3 5; E,; SO(10); SO(14); and SO(18) with the unifying representation
4
GUT
=
C
aidi,
(14)
where #i are irreducible representations listed in Table I. The coefficients aj in (14) are positive and integer but it is useful to introduce the notation (15)
Thanks to this, all contributions from conjugate irreducible representations 4 are to be interpreted as terms involving 4 with negative coefficients. Such notation will be utilised everywhere below.
316
DARIUS2
K. GRECH
We will now show how the remaining conditions listed in the previous section specify the unified models. It will be done separately for exceptional, orthogonal, and unitary unifying groups. 3.1. Exceptional
Groups
E, is the only admissible unifying exceptional group. Using information from Table I, we find with the help of (14) the general form of unifying representation as
#E6=a(l
a<22
: 11,
(16)
with the constraint on the coefficient a following from (2). Profiting from the decomposition rules E, + SO( 10) [ 181, (1: 1)=1+16+10
(17)
and, then from the branching rules SO(10) -+ SU(5) [18], (1’3)
1= w5
{ij,+ lo= {q5+ {Q5
16= {O},+
{12},
(19) (20)
with (O}, = 1, { 1 }5 = 5, and { 1 2}s = 10 irreducible obtain the complex part c of (16) under SU(5) as
representations
of SU(5), we
o$61:1)=a({i},+{12}5)=a(S+10).
(21)
The direct comparison of (21) with (A.16) gives a = g,. Because Isvcj,(azG1”)) = 4a (see Table I and (A.5)), the asymptotic freedom of N(3), in Eq. (12) requires a 6 8. It gives, together with the condition VIII, dEg= g,(l
g, = 3, .... 8.
: 11,
(22)
The rest of the conditions are automatically satisfied; in particular, representation is anomaly-free due to A( ( 1 : 1)) = 0. 3.2. Orthogonal
the above
Groups
We have here three candidates: SO(18), SO( 14), and SO( 10). The admissible representations of SO(18) and SO(14) are real if one breaks these symmetries to X’(3), x SU(2), x U(l), and this contradicts condition VI. Indeed, we have for SO( 18) --) SO( lo),
A+SOWX, = 8 (A+sm,
+A-
so(m)) = 8(16
+ 16)
(23)
and for SO( 14) -+ SO( lo), A +so(14)
=
‘3A+socm,+
A-sod
=
2W+%),
(24)
GRAND
UNIFIED
317
THEORIES
The above branching rules prove that in both cases the unifying representation is real before SU(3),x SU(2), x U(l), symmetry is reached. The possible reducible repreesentations of SO(10) satisfying (2) are to be built from A,, Cl’l., and [A, 1] _ representations defined in Table I or from their complex conjugates. There are the following possibilities: 1”
a.A,,
a<22
(25)
2”
Cl’l, +a.A+,
a<4
(26)
3”
[I’],
a<4
(27)
4”
[A, l]-
+a.A+,
a<4
(28)
5”
[A, l]-
+a.A-,
aG4.
(29)
+a.A-,
Let us investigate them respectively looking contained in SO( 10).
at asymptotic
freedom of SU(3),
ad. 1” With the help of (19) and by direct comparison we have a = g,. From (AS) and from Table 1 we then obtain /,,(,,(a. A +) = 4a. Because a. A+ is fully complex under SU( 3), x N(2), x U( 1) ,, thus ( 12) requires a 6 8. Combining this with condition VIII one arrives at 4 SO(IO)= ad. 2”
g, = 3, .... 8.
.A+ >
gN
The branching rule SO(10) + SU(5) for [l’]
+ is [18],
[1~]+=1+J+10+i5++;iS+50 and then for SU(5) --+ SU(3),x
X7(2),
(30)
(31)
x U(l),,
1 = (1, l)(O)
(32)
5=(1,2)(-4)+(3,1)(f)
(33)
10=(1,1)(1)+(3,1)(-5)+(3,2)(~)
(34)
is=(l,3)(-1)+(3,2)(-9+(%,1)(j)
(35)
as= (1,2)($)+
(3,1)(-f)+
(3,3)(-f)+
(3,1)(i)
+ (%2)( - i, + (6, l)( -4, + (8,2)(i) 50 = (1, l)(2) + (3, l)(f) + (3,2)(i) + (6,1)( - $I+ @,2)(
-
(36)
+ (6,3)(f)
41,
(37)
where the last number in the bracket denotes the U( 1) Y normalised charge. Therefore the complex part r~ of [ l’]+ at SU(3), x SU(2)L x U( l)y is 5 5 +(L l)(l)+(L W)+(3, W-5) 40$0, = i Yo:;,, +
(Xl)($)
+
(X1)($
(38)
318
DARIUSZ
K. GRJ3CH
with the exotic part i~:&,=(1,3)(-1)+(6,1,(~)+(6,
l)(-f)+(6,
l)(-4)
+ (6,3)(f) + (3,3)( - $1.
(39)
Using Eqs. (19), (38), and (39) we find the full complex part of (26) as Cl’]+ +aA+ ~SO(IO)
= [Cl’l+
SO(10)
+
(19
l)(2)
+
(5
W)
+ (a+ l)U, 1)(1)+4,
N-$)+a(3,2)(:)
+ (a + 1)(3, l)( - 5) + (a + 1)(3, l)(i).
(40)
Comparing with the one family content (A.16) we see that the content (40) contains a generation (a = g, < 4). The IsU(3,- index for the content (40) is calculated from Table I as I SW31
Cl’]+
(0 SO(10)
+aA+)
=
36 + 4u,
(41)
so that condition (12) is not satisfied and the representation (26) should be rejected. ad. 3” Similar considerations as before but with the branching rule (19) replaced by its complex conjugate lead to the complex part of (27) under SU(3),xSU(2),x U(l), as a~;:;,;““-=(1,1)(2)+(1,3)(-1)+(a,l)(~)+(6,1)(-f)
+ 6 UC-‘$
+ (6,3)(s)+
(3,3)(-f)
+(a-l)U,
l)(-l)+(a-1)(3,1)(f)
+ (3, l)($)
+ (a - 1)(3, 1 )( - $, + a( 1,2)( 9 + u(J, 2)( - d,.
(42)
Comparing (42) with (A.16) one can see that the content (27) contains g, = a - 1 generations and, therefore according to condition VIII, it implies a = 4. In this case we have 1151++~a-) = 48, ~suw(~s0,10,
(43)
so that SU(3)c is not asymptotically free. The conclusion is that the content (27) also should be rejected. ad. 4” Let us decompose [d, l] ~ representation of SO(10) on irreducible representations of the standard group. With the help of branching rules to SU(5)
Cl813 [A, l]-
=3+5+10+15+24+3+45
(44)
GRAND
and then, using decomposition with
UNIFIED
319
THEORIES
laws (32)-(36) to W(3),
x SU(2), x U( l)y together
a=(l,2)(-;)+(3,2)(:)+(3,1)(-$) +(3,3)(-~)+(8,1)(1)+(~,2)(~),
we find the complex part CTof [d, l]-
(45)
under SU(3),x
SU(2), x U(l),,
a~~~j=i~~l’oj+(1,1)(1)+(1,2)(-~) + 3(3,2)(i)
+ m
I)(-
:, + (3,2)@
+ (391 l(f) + (1,2)( - 9 + (3,1 N - $1,
with the exotic part cg;“ii, i&&j
(46)
where = (k3)(1)
+ (%3)( - 3) + (3 3)(f)
+ (6, 1 J(f) + C&1)( - f, + 6 2)(4) +@,W-~)+W)w
(47)
Therefore from (46), (47), and (19) the general form of the complex part of the content (28) at the standard model level is ‘d’lJm+OA+=[$/~j ~.sO(101
+(1,2)(-$)+(3,2)(z)
+(3,1)(-$+(a+
l)U, l)(l)
+ (a + l)(L
2)( - $1
+ (a + m,
l)(f)
+ (a + 3)(3,2)(i)
+(a+2)(3,1)(-3)
(48)
with g, = a + 1 generations described by (A.16). To satisfy condition have a > 2. However, Z,,(,,(a~&~‘-
+“*)
= 56 + 4u,
VIII
we must (49)
which contradicts (12) and we conclude that the representation (28) is not acceptable. ad. 5” As before, using (46) and complex conjugation of (19), one obtains the complex part of (29) at SU(3), x W(2), x U( 1) y level, a&/?oi +““-=[&l~;
+ (1,2)(-3)
+ (3,2)(i)
+(3,1)(-~)+(~-1)(1,1)(-1)
+ (a - 1ML 2)(f) + (a - 1)(3, I)( - 4, + b - 3)(X2)( - ;) + (a - 2)(3, 1 )cf),
(50)
320
DARIUSZ
K. GRECH
which contains only one family for a = 0 or a = 4 (compare with (A.16)). As a result the representation (29) is not permitted, being in contradiction with condition VIII. In addition SU(3)C is not asymptotically free in the theory based on (29) because I SU(3) (CT&&)= 56
(51)
I SU(3) (0$$~1Jj+4Ae)=54 so that condition Finally,
(52)
(12) cannot be satisfied.
the only admissible unifying representation 4 SO(10)
=
gN
A+
for orthogonal
g, = 3, .... 8
5
for the SO(10) unified model. The anomaly-free condition cally because A( A + ) = 0. 3.3. Unitary
groups is
(53) (9) is fulfilled automati-
Groups
Let us first consider the exotic case (10). From the branching rules (33), (35), (36), and (45) one can find that the complex part of (10) under W(3), x SU(2),x U(l),is Qsc/(s)=
(1,3)(l) + (3 3)(f)+ (3 3)(-f) -t(6,1)(-~)+(6,1)(~)+(6,2)(~)+(8,1)(1) + (w(-+)+2(3,2)(i)+ (3,2)(z) +(3,1)(-$)+(W(-3)+(W(-9
(54)
ku,3,(~su,,,) = 52.
(55)
and
Hence the asymptotic freedom for SU(3), cannot be obtained. Furthermore, the content (54) is not adequate to describe even one family (see (A.16)). As a consequence it is not permitted by our requirements VIII and IX. All other unifying representations of W(n) groups are the linear combination of nonexotic irreducible representations { 1 } n, { 1* >n, { 13}n, ( 1” } n and their conjugate partners, i.e., 4 SU(n)zi$l
ai{li>.
(56)
The coefficients ai (i= 1, .... 4) are a subject of constraints following from (2) and (9). Thus we have 4
c aiA({l’})=O i=l
(57)
GRAND
UNIFIED
THEORIES
321
and
j, bil 4{W<
lln.
(58)
It is convenient to present solutions with respect to ai in terms of the number of generations g, included in a given model. This number is after Georgi [6], g,&/=N(J)-N(5)
(59)
or, equivalently, g, = N( 10) - N(iiQ
(60)
where N(e) is the total number of $ multiplets of SU(5) in (56). The equivalence is obvious because N(5)+ N(lO)= N(s) + N(n) to satisfy (9) (see Table I for anomaly indices of 5 and 10). Thanks to decomposition SU(n) + W(5), (1},=5+(n-5)x1 {12}.=10+(n-5)~5+;(n-5)(n-6)~1
{l’},=i@+(n-5)xlO+;(n-5)(n-6) x 5 + ;(n - 5)(n - 6)(n - 7) x 1
(61)
{l”}.=J+(n-5)xm+&(n-5)(n-6)~10 +~(n-5)(n-6)(n-7)~5 +&&~-5)(n-6)(n-7)(n-8)~1,
one obtains, with the help of (60) and (56), g, = u, + u3 (n - 6) + +u4(n - 5)(n - 8).
(62)
After suitable substitution of A-indices from Table I we are able to express the solution of the system (62) and (57) in terms of g,, u3, and a4 coefftcients as parameters: a, = (4 - n) g, + $a3 (n - 5)(n - 6) + 4u4(n - 4)(n - 6)(n - 8)
(63)
az=gN-u3(n-6)-&u4(n-5)(n-8).
(64)
The inequality (58) finally determines the admissible values for g,, u3, and u4. All solutions have been listed in Table II, assuming 3 < g, 6 8 to satisfy conditions VIII and IX.
g,{Ij5+gN{121S
kN{Ij6+gN{12f6
3glv{~~,+gN{12~7 (3g,-1)(J},+(g,-1){12},+{13}, (3gN+l)m7+(gN+ Whf (3g,-2){i),+(g,-2)112},+2{13~, (3g,+2){i),+(g,+2)(12j7+2(i3J, (3gN--3)(i},+(g,-3){12j,+3(13}, (3g,+3)~i~7+(gN+3)112}7+3{i3},
SUI(5)
SW71
g,(l : 1) gNA+
b OUT
=.J(6)
E6 SO( 10)
G OUT
ii%
All Possible
Unifying
II
.. .. 8 .. .. 8 .. .. 8 .. .. 8 .. .. 8 .. .. 8 .__, 7 . ... 8 . ... 5 .. .. 8
5, .... 8
3, . ... 8
3, . ... 8
3, . ... 8 3, . ... 8 3,4, 5
4, 5, 6
3, . ... 7
3, . ..) 7
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3
gh’
Representations
TABLE in GUTS
32 12 48
16
5
{ log,, 96 - 2g,
g, =
6, 7, 8
42, g.N=3 log,, g, = 4, . ... 8 { 72-2g,,g,=3,4,5
10&T, log, tog, + 30
g,>4 { 8g,+20, 84-2g,,g,=4, g,=6 1 8g,+24,
50, g/v=3 1 8g,+l6,g,>3 70-2g,,g,=3,4
8g,+ 8g,+8 Sg, + Sg, + Sg, +
6g, %N 4g.w 6g, fk, f&,+4
1 index
3,4
3,4 3,4 -
3 -
394 3. 4 3.4 3.4 3,4 3,4 3 394 -
(SUS%JTs)
6, 7, 8 5, . .. . 8
3, . .. . 8
3, . ... 7
3, . ... 7
3, . ... 8
3, . ... 8
3. . ..) 7
3,4, 5
3 334 3,4
3, . ..) 7
5, . ... 8 3, . ... 6 3
3, . ... 8
3, . ... 8 3, . ... 8 3,4
g,=3,4
8,>4
g,=3,4
66-3g,, g,=3,4 i 16g, - 24, g, > 4 132-2g, 130 - 2g,
{ 14g,, 16gN
80-2g,,
4% g.w=3 { 14g, - 14, g, > 3 g,=3 { 98, loo-2g,, g,>4
14SN
{ 82, 12g, + 32. g, g!v=3 > 4
{ 12g, + 10, g, a 5 100 12g, + 48 98 - 2g,
80-2g,,
108 - 2g, 12g, + 16 90
1 72 12g,-- 2g,, 12, g,> g, = 3, 5 --., 5
12g,-6 12g, + 48
l-37,
Table continued
3, 4
3, 4
3.4
3, 4
3 -
334 3.4 -
s
12gN{~j,6+gN{121,, (12gN-55)1i),6+(gN--o)112},*+
(n-4)g,{~j,+glv{12).
SU( 16)
Wn)
n&l7
lkiv{~~,5+
ScJ(l5)
g.N{12~,5
logN{~~,4+gN{l*~M
{131,6
3, . ... 8
(9g,-28){i},3+(g,-7)(12j,3+{13},,
SU( 14)
3, .. .. 7
%‘v{~~,3+&%{1*~*3
SU(13)
3, .... 6, n < 35 3, ... . 5, ” 135
3, .. .. 6 5, . ... 8
4, . ... 8
3, ..,, 6
3, . ... 8
3, .. .. 6
3, .... 8
3, .... I
&TN
8g,{~~,~+g,v{l*~,~
$ GUT
II-Continued
SU(12)
G GUT
TABLE
g,=3,4, g,=6,7,
5 8
g,=4 g,>4
150-2g,,
&TN=3 g,r3
1%
(2n - 6) g,
26g, 176 - 2g,
i
24gN
132, 126-2g,,
22g,
2og.4’ 1W g,=3 { 104 - 2gN, g, = 4, ... . I 110, g,=8
84--g,, 18g,-36,
18g,
1 index
3
3
3
-
3
3
3, 4
GRANDLJNIFIEDTHEORIES
325
4. CONCLUSIONS In this article we constructed a systematization of unified models based on some theoretical and aesthetic requirements. The permissible class of GUTS is large, but ordered. The unitary groups are its main part. Only two cases of nonunitary models have been found-based on SO(10) and E, groups, respectively. All models with the proper representation structure are listed in Table II. Although above we considered only nonsupersymmetric models, it is clear that all our criteria apply also to the supersymmetric case. The only difference with sypersymmetric GUTS (SUSY GUTS) is that, due to the presence of additional scalar and fermion fields-superpartners of ordinary particles, the asymptotic freedom of Go,, reads from (A.14)
~(3u7-@,,,))>0, instead of (2). Simultaneously, should have from (65)
to ensure that the theory is perturbatively
(65) safe, one
instead of condition (12). These constraints are stronger than in the nonsupersymmetric case and lead to a much narrower class of admissible models than the one we have just worked out. The last column of Table II gives the choice of accepted values of g, in SUSY GUTS for all models permitted by the nonsupersymmetric unification. Let us notice that for SUSY GUTS the maximal number of generations (ghrhnax = 4. This follows from (66) and (A.16) on the basis of analogical considerations as in nonsupersymmetric case (see the analysis in condition IX). Simultaneously, this is the maximal number of generations satisfying (65). It is worth stressing that our solutions in this paper are based on the assumption that no extra symmetries (global or local) exist simultaneously with SU(3),x W(2), x U(l)* at the momenta q2 w Mi,. Otherwise, the complex part of &uT at M,, which is essential for this analysis because of criterion VI, could be different. The case with such extra symmetries must be considered independently for various specific models. Of particular interest are unified models involving technicolor. They, however, have already been investigated in the literature (see, e.g., Ref. [ 191). The compact set of solutions presented is very promising for any further analysis of nonsupersymmetric or supersymmetric unified theories. It has already been used by us in a recent publication [S], but there are still more applications to come which will be the subject of forthcoming elaborations.
59SJ?OSi2-7
326
DARIUSZ K.GRECH
APPENDIX:
CALCULATION
OF j? FUNCTIONS
For purposes of reference here we state basic formulae for the calculation of firstand second-order renormalisation /I functions in nonsupersymmetric and supersymmetric quantum field theories. Let us first define the I-index for an arbitrary representation 4 of the gauge group G with generators T,(4) as
with the normalisation
condition
for the fundamental representation df of G. The l-index has few important properties. Namely, d,, ia which have a product decomposition
for arbitrary
representations
one obtains
64.4)
where dim 4 stands for the dimension (A.4), one obtains the useful relation f
f ( id
of representation
4i
= >
f ,=
l(di)
4. Comparing
(A.3) with
(A.5)
1
which helps significantly in the calculation of the Z-indices for arbitrary representations if the normalisation condition (A.2) for fundamental representation is assumed. For the general case of a quantum field theory invariant with respect to G if G,xG2x . . . x G, gauge group with the fermion representation I/I, it is convenient to introduce new variables, c(:) g ‘I( v(i))
64.6)
c$’
2 ‘Q$“‘) 2
64.7)
c(F) g C,($Ci’) f ’
(A-8)
2
GRAND
UNIFIED
321
THEORIES
where V”’ and It/F’ are respectively the adjoint and the fundamental representations of Gi (i= 1, .... n), tici) denotes the content II/ after reduction G + Gi, and Cz is the eigenvalue of the second-order Casimir operator C,(4) defined for an arbitrary representation f$ as (A.9)
C,(4) if! c Tm. If Gi = W(n),
it is easy to find that c(i) = n
(A. 10)
c+2-l).
(A.ll)
V
If Gi = U( 1 ), then = 0
(A.12)
C:' = c Q;,
(A.13)
CC’)
”
u where the last sum extends over all charged particles with charge Qr, transforming with respect to the one-dimensional representations of U( 1). In the above notation the first-order (pi) and the second-order (/Iii) renormalisation p functions are deduced from the formulae given in the literature [20] for a pure gauge theory as: p,=f(llcpq’)
(A.14)
pi, = 6, { $p$)’
- iQcyy)>
- 2qcy.
(A.15)
Note that there are contributions to a, from two different groups, Gi and Gj (mixture terms), if i # j. Let us consider the particular case of the standard model based on the cross product SU(3), x SU(2), x U(l),. For G=(G,~U(l),)x(G,rSU(2),)x (G, - SU(3),) with the fermion content transforming under SU(3), x W(2), x u(l),
as ti
~U(3k~~U2)LXU~lh~
-gNC(3,l)(~)t(1,2)(-~)+(5,1)(-3) + (3?2,cd,+
(1, l)U)l,
which describes the g, generations, we obtain the following formulae: (A)
for the nonsupersymmetric
case:
(A.16)
328
DARIUSZ
K. GRECH
(A.17) (A.18) (A.19)
(A.20)
where nH is the number of Higgs doublets. (B) for the supersymmetric case: (s)SU(3)c x SW2k x UC1)Y d’ - pp= -2g,+lH PI (sPW3)cx SW~)LXW~)Y 2 fi’,“’ = 6 _ zg, _ in, 82 83
(A.21) (A.22)
(s)SL’(~~XX’(~~X U~)Y $j 8:“’ = 9 _ 2g,
(A.23)
(A.24)
REFERENCES 1. H. GIDRGI AND S. L. GLASHOW, Phys. Rev. tilt. 32 (1974), 438; Theories,” Benjamin, New York, 1984. 2. D. K. GRECH, Int. J. Mod. Phys. A 4 (1989), 2531. 3. For a review see: E. FARHI AND L. SUSSKIND, Phys. Rep. 74 (1981), Phys. 55 (1983), 449. 4. M. A. B. BY~G AND A. SIRLIN, Phys. Rev. Left. 38 (1977), 1113. 5. D. K. GRECH, Z. Phys. C 42 (1989), 599; 46 (1990). 340. 6. H. GEORGI, Nucl. Phys. B 156 (1979), 126.
G.G.
Ross,
“Grand
279; R. K. KAUL,
Unified
Reo. Mod.
GRAND
UNIFIED
THEORIES
329
7. P. H. FRAMPTON, Phys. Left. B 88 (1979), 299. 8. J. CHAKRABARTI, M. Povovrc, AND R. N. MOHAPATRA, Phys. Rev. D 21 (1980), 32. 9. R. C. KING, Nucl. Phys. B 185 (1981), 133. 10. E. CARTAN, “Sur la structure des groupes de transformation finis et continus,” Navy, Paris, 1894. 11. E. B. DYNKIN, “Amer. Math. Sot. Transl. Ser. 2,” Vol. 6, pp. 11l-244, 245-378, Amer. Math. Sot., Providence, RI, 1957. 12. A. I. MALCEV, “Amer. Math. Sot. Transl. Ser. 1.” Vol. 9, pp. 172-213, Amer. Math. Sot., Providence, RI, 1950. 13. J. BANKS AND H. GEORGI, Phys. Rev. D 14 (1976), 1159. 14. S. OKUBO, Phys. Rev. D 16 (1977), 3528. 15. T. APPELQUISTAND J. CARAZZONE, Phys. Rev. D 11 (1975), 2858. 16. P. V. LANDSHOFF,Quarks in high energy interactions, in “Proceedings, 1978 CERN School of Physics, CERN, Geneva, 1978. 17. G. ALTARELLI, Phys. Rep. 81 (1982), 1. 18. R. SLANSKY,Phys. Rep. 79 (1981), 1. 19. D. K. GRECH, Int. J. Mod. Phys. A 4 (1989), 2173. 20. D. R. T. JONES,Nucl. Phys. B 75 (1974), 531; for a review see, e.g., W. MARCIANO AND H. PAGELS, Phys. Rep. C 36 (1978). 137.