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Construction of grand unified models under maximal subalgebras
ANNALS
OF PHYSICS
358-383 (1984)
155,
Construction of Grand Unified Models under Maximal Subalgebras XENIA Department
of Physics.
DE LA OSSA
C.
University
of Texas,
Austin,
Texas
78712
AND GUY F. DE T~RAMOND Lyman
Laboratory
of Physics,
Received
Harvard
September
University,
6, 1983;
revised
Cambridge,
Massachusetts
November
1, 1983
02138
A construction of grand unified models of the strong, weak and electromagnetic interactions is described based on the transformation properties of the group generators under a maximal subgroup decomposition without recourse to large representation matrices or to the specific algebraic structures of some classical Lie-groups, such as the Clifford algebra associated with the orthogonal groups or the octonionic structure of the exceptional groups. To illustrate the procedure an explicit construction is given of the SU(5) model useful in the discussion of higher rank groups, of SO(10) under the maximal subalgebras SU(2), x SU(2), x SU(4), and SU(5) x U(l), and of the exceptional group E, under SU(3), x SU(3), x SU(3), and SO( 10) x U(l),. The construction procedure can be used as well with any classical Lie-group.
I. INTRODUCTION A deeper understanding of the structure of matter has followed from the description of the fundamental forces in terms of non-Abelian local-symmetry groups [l] and their subsequent unification in larger semi-simple [2] or simple [3-51 Liegroups [6]. The strong, weak and electromagnetic interactions thus appear as different low-energy manifestations of a single underlying force, whose symmetry is manifest at a very large energy scale. The grand unified theories [7,8] lead to such spectacular prediction as proton decay within observable limits, afford a very precise determination of the electroweak angle [9] and provide a framework for the discussion of problems of cosmological importance such as the baryon asymmetry of the universe [lo]. One of the most outstanding aspects of these models is, however, how the different interactions are incorporated into the theory. Furthermore, the quantum number content of a given representation of the unifying gauge group, which is totally arbitrary in the standard model based on a semi-simple group, is determined by the algebraic structure of the 358 0003.4916/84
$7.50
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group generators. Quantum numbers such as isospin I,, hypercharge Y, B - L (baryon minus lepton), electric charge qem, etc., correspond in most of these models to the Lie-generators of the theory, or to linear combinations of them. The particle states are classified according to their transformation properties under a given gauge group, and the quantum number assignment follows from the commutation relations of the fields with the diagonal generators of the theory. In the present paper we give an explicit construction of grand unified models based on the transformation properties of the group generators written as bilinear forms in terms of the representation vectors of the unifying gauge group, in a basis constructed under maximal subalgebras. The interest of the subgroup decomposition arises from the physics itself: there is compelling evidence that the various interactions observed at low energies (below 100 GeV) are gauge interactions described by the semi-simple structure of a direct product of unitary local-groups in the standard model. Plausible group structures at intermediate energies (left-right symmetric theories of electroweak interactions) are also described by a product of unitary groups. A natural basis for the classification of the particle states according to their transformation properties under the interaction subgroups is thus provided. The different interaction subgroups are used in turn to build the unified group by their embedding into a larger structure. It is also of interest that the same construction procedure, which provides a concrete realization for the algebra, can be used with different classical groups without invoking a specific algebraic structure, such as the Clifford algebra for the orthogonal groups or the octonions associated with the exceptional groups. We also avoid the utilization of large representation matrices which, as a matter of fact, do not exist for the exceptional groups based on non-associative algebras. We are not concerned in this paper with the phenomenological issues of these models, the specific mechanisms of spontaneous or dynamical symmetry breaking or more elaborate algebraic constructions. Most of these issues are discussed in the excellent reviews given in Refs. [ 6-8 ]. The standard model of the electromagnetic, weak and strong interactions is based on the local-symmetry group SU(2), x U(l), x SU(3),. The electroweak sector of the theory [ 111 is based on the gauge group SU(2), x U(l), which acts on flavor quantum numbers. The usual assignment of particles into doublets and singlets in the minimal electroweak model according to the particle helicity depends solely on experimental observations. In this model the left-handed fields of the lepton or quark sector are grouped into SU(2), doublets, whereas the right-handed counterparts or, equivalently, the left-handed charge conjugate fields, are singlets under SU(2),. The quantum numbers corresponding to the generator of the U(l), Abelian sector, the hypercharge generator Q,, are assigned in a given representation in order to obtain the electric charge operator as a linear combination of Q,, and the diagonal generator of SU(2),, the weak isospin. The strong interactions are described by the unbroken local-symmetry group SU(3), which acts on color quantum numbers [ 121. Each flavor or type of quark is assigned to the fundamental representation 3 and the antiquarks to the 3. The leptons do not have strong interactions and are thus assigned to singlets under SU(3),. The
360
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TkRAMOND
representation of the gauge group for a given flavor is given in terms of left-handed quark fields qi and the corresponding left-handed charge conjugate fields qci, which transforms as 3 + 3, and the theory is self-conjugate or vectorlike. The first generation of quarks and leptons, ui, di, e- and v, is described by 16 lefthanded fields. The hypercharge of the right-handed neutrino is zero and it decouples from the theory, leaving effectively 15 fields which are classified according to their SU(2), and SU(3), transformation properties. The representation is complex with respect to SU(2), X U(l), x SU(3),. Moreover, this is the minimal set of fields which is free from anomalies [ 131, a condition for renormalizability. An interesting extension of the minimal electroweak model, which is particularly relevant to our discussion of the SO(10) model, is an ambidextrous theory based on SU(2), X SU(2), X U(l),-, [ 141. The right-handed fields, which are singlets under SU(2),, are grouped in this model into SU(2), doublets. The quantum numbers of the new U(1) generator are also assigned arbitrarily in this model so as to build the electric charge operator as a linear combination of the U( 1) generator and the diagonal generators of SU(2), and SU(2),. The quantum numbers of the U(1) generator are found to correspond to B -L (baryon minus lepton). We discuss in Section II the embedding of the standard component model in the simplest grand unified theory, the W(5) model of Georgi and Glashow [3], which is used to decompose other models based on higher rank Lie-groups in the SU(5) basis. The SU(5) generators are built in terms of the fields in the defining representation 5, or 5 [ 151. In Section III we construct the SO(10) model by studying the transformation properties of the spinorial representation 16 and the group generators under its maximal left-right symmetric subalgebra SU(2), x SU(2), x SU(4),, and under SU(5) X U(l), by expressing the generators in the new basis as a linear combination of the generators in the W(2), x W(2), x SU(4), basis. The E, model is constructed in Section IV by studying the transformation properties of the group generators in terms of the fields in the fundamental representation 27 under the left-right symmetric decomposition SU(3), x SU(3), x SU(3), following the discussion of Sikivie and Giirsey [ 161. We also construct E, under the maximal orthogonal subalgebra SO(10) X U(l),, and the new generators are given in terms of the generators in the SU(3), x SU(3), X SU(3), basis. Some concluding remarks are given in Section V. II. THE SU(5) MODEL
The unified theory of the strong, weak and electromagnetic interactions based on the gauge group SU(5) is the lowest rank model (rank four) based on a simple group structure which contains the standard SU(2), x U(l), x SU(3), component model as a subgroup [3]. More elaborate models [4,5] of physical interest discussed in the next sections are of higher rank and contain extra quantum numbers and particle states. The isospin and hypercharge generators in this model are Lie-generators of SU(5), thus the electric charge operator is also a generator of the theory. The hypercharge assignment in a given representation is no longer arbitrary, and the
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361
electric charges of quarks and Ieptons are commensurate. Besides the 12 generators associated with the standard model, there are 12 generators which close the SU(5) algebra. The new generators carry color and flavor quantum numbers and mediate nucleon decay. The corresponding gauge bosons are exceedingly heavy (of the order of lOi GeV) to have an acceptable proton lifetime. The new vector-bosons can acquire such massesby introducing vacuum expectation values which commute with the SU(3) and SU(2) generators, thus leaving the gauge bosons associated with SU(2), x U(l),. X SU(3), massless at the first stage of spontaneous symmetry breaking. Since the unifying group is a simple Lie group, there is a unique gauge couphng and the different couplings observed at low energies are obtained by renormalization group estimates [9] from the unification scale. Having a viable unifying gauge group, we classify the fermions by studying the quantum number content and transformation properties of a given representation under the unitary subgroup decomposition. In SU(5), the known fermions are assignedto the 15dimensional reducible representation 5 + a. The anomalies in the 3 and a are equal and opposite, and thus the model is free from anomalies 13]. The fundamental 5 decomposesunder SU(2), x SU(3), as s = (2, I) + (133X
(2.1)
and the l0, which is the antisymmetric product of 5 x 5, as lo = (193) + (2,3) + (131).
(2.2
The adjoint representation -24 = 5 x 5 - I transforms as 24 = (3, !) + (1,1) + (138) + (2.5, + CL,3).
(2.3
The matter fields in the reducible representation $ + #J are labeled accordingly by {WA1= iv’, W’L {#Dl
=
14’3
#ri,
419
(2.4) (2.5)
where I+/ = v,.,+, and the group generators as
{Q"l = {QF, Q,, QF, Qriy Qril, with indices A, B = 1, 2,..., 5 D, E = 1, 2,...) 10 kp,v=
1, 2,..., 24
i, j, k = 1, 2, 3 a, p, y = 1, 2,..., 8 I r,s=
1,2
a, b, c = 1,2, 3 I
SU(3), indices,
SU(2), indices.
(2.6)
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The group generators are expressed in terms of the space-integral of bilinear forms of the fermion fields in the fundamental representation vA [ 151
Q’ = j d3x w,t(~)(T”),~ v,(x), or, equivalently,
in terms of the fields in the 5 representation Q" = j d3x I/ +(x)(-T"')",
v/"(x),
(2.7)
I# P3)
where the Ta are the representation matrices of the generators in the fermion basis, and (TA)A, z (T ’ )AB. The color generators QF transform as an octet in the adjoint representation of W(3), and the Qf as an SU(2) triplet
Q;=j&xvi~(-~)i Q;=jd3xft
i
(-f)’
$,
(2.9)
I$,
(2.10)
s
where the A” are the eight Gell-Mann matrices of SU(3) and the ra are the Pauli matrices. The generator of the U(l), sector commutes with the SU(2) and SU(3) generators, i.e., it is invariant under SU(2) and SU(3) rotations and hence has the form Q,, = j d3x(a~"$u'
+ /?iyit~').
(2.11)
Since the hypercharge generator is a generator of SU(5), it is traceless and normalized, as any other generator of SU(5), by Tr{ Q”Q,} = fs”, . Therefore,
WQ,) = 0,
WQ,‘) = $,
and we obtain
Qy= ( d3xfi(-
$y't~/' + $y'tt#),
(2.12)
fixing the phases of a and p. The generators Qri and Qri close the algebra associated with the group generators of SU(5). The generators Qri transform as an SU(2) doublet and SU(3) antitriplet, Q,i
= - 1 lb
j d3x
ft$,
(2.13)
CONSTRUCTION
and Qri is the hermitian
OF GRAND
363
MODELS
conjugate of Q,j Qri = (Qri)t
The normalization
UNIFIED
= - +-
1 d3x y/“,“.
(2.14)
condition for these generators has the form (2.15)
Tr { Q,‘Qsj} = fs,.‘~5,‘. It is now straightforward to obtain the Lie commutator
algebra of SU(5) (2.16)
[Q”, Q”] = itAFL,Q”,
where the ta”” are the structure constants corresponding to the vector space of the SU(5) generators in the basis defined by Eq. (2.6). Since the generators that close the SU(5) algebra are not hermitian in this basis, the structure constants are not necessarily real. From the above expressions for the group generators we obtain the following commutation relations:
IQ; yQi I= kbc Qi 9
[Q,‘, Q;l = ip)
[Q:* Q,“l= ifa,,Q,Y,
,Qri,Q+(-l;lii
IQ; 1Q,l = [Q:, Q,l = 0,
[Qri, Q,]
= $4;
LQ;3Q:l = 0,
[Qi, Q,‘l
= 0,
[Q,‘,Qsj]=-dji
’ Qsi5 Qj,
i Q,!,
(2.17)
sQr+a~(~)jiQ:-f6,a,i~~Q,.
(f)
r using the equal-time anticommutation relations for the fermion field operators ( w,~(x), y”(y)} = 6,‘6(x - y). In (2.17) the sObeand&, are the structure constants of SU(2) and SU(3), respectively. The quantum number assignment of the 5 follows from the commutation of the fields W* with the diagonal generators Q3L, Q,, Q3, and Qsr . We list in Table I the quantum number content of the 5 representation and the corresponding fermionic assignment. I,, is the weak isospin, Y, the weak hypercharge normalized by
[w”, Q,l = & Ywv* and s,,,
(2.18)
the electric charge qem= I,, + Y,. Quark and leptons are in the same irreducible representation, and since the trace of the electric charge operator vanishes in any given representation, the electric charge is quantized: the quark and lepton
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TABLE
I
Quantum Number Assignment for the Fermions in the 5 + 10 Representation of SU(5) Particle
4 em
charges are related by a factor of three. The fermion fields in the N from the antisymmetric product of 5 x 5
are obtained
(2.19)
#ri
=
+
4= *
(WrXi
-
(2.20)
WiXrL
(2.21)
&rSWsXrY
where xA transforms as 5 and 8” = -&‘I = 1. The corresponding quantum number assignment follows from the commutation of #D with the diagonal generators of SU(5) written in terms of the 5. The fermionic assignment is given in Table I. The 24 gauge bosons associated with the group generators transform as the adjoint representation of SU(5) {Alp}= and their quantum
numbers
(2.22)
{W”,,B,,A~~,A,i,,A’iu}
are the eigenvalues of (2.17). We list in Table II the TABLE
II
Quantum Number Assignment for the Gauge Bosom Responsible for the Baryon and Lepton Number Violating Interactions in SU(5) A,’
I 31
A,’ A,’
f -4
26 26
9em
Particle
f
X’
f
Y’
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365
quantum number assignment for the gauge bosons. For reasons of brevity, we list only the eigenvalues of the lepto-quark and diquark bosons associated with the generators that close the SU(5) algebra. The interaction Lagrangian is uniquely determined by the requirement of the gauge invariance of the theory and is given in the reducible representation 5 + @ of SU(5 by
il;=g~(WAy'[WA,Q*l+~~,Ul~n~Q"l,A"..
(2.23
The usual expression of the SCr(5) Lagrangian [8] follows directly from the transformation properties of the matter fields IJ/ and #n under the group generators of the theory Q”.
III. THE SO(10)
MODEL
Could SU(5) be embedded in a group of higher symmetry which gives a theory of physical relevance where all the fermions are assigned to an irreducible complex representation free from anomalies? There are four rank-five simple Lie-groups [ 7, 8 ]: SU(6), SO{ 11L Sp( IO) and SO(10). The fermions in SU(6) are assigned to the 27.dimensional reducible representation 0 + t) + 15, which is anomaly free and contains extra fermions in the 15 with unconventional SU(2),, x SU(3), transformation properties. This model provides a suitable basis for the study of E, according to its maximal subgroup decomposition SU(2) X SU(6), and is more appropriately discussed in the context of E,. SO(l1) and @(lo) have only real representations, and we are left with SO(10) as a possible rank-live group for grand unification [4]. The fermions are assignedto the 16-dimensional irreducible spinorial representation which is anomaly free. SO(10) contains SU(5) x U(l), as a maximal subgroup which gives a model similar to W(5), and also SU(2), x SU(2), x SU(4), where the leptons are included as a fourth color [2]. W(4), is broken to SU(3),, which is left unbroken, and to U(l),_, , thus recovering at lower energies the left-right symmetric electroweak model mentioned in the Introduction. A. SO(10) under W(2),
x W(2),
x stun
Under SU(2), x SU(2), x SU(4), the representation of SO( 10) transforms as
16-dimensional irreducible
- 16 = (2, 134)+ (A,2341%
spinorial
(3.1)
and the adjoint representation 45 as
45 = (3, 1, 1) + (A,371) + (1,!,15) + (23276).
(3.2)
366
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TliRAMOND
We label the matter fields vH and the generators Ql as {VHI = iW,,l W%
(3.3) (3.4)
with indices H, I = 1, 2 ,..., 16 t-, q, 8 = 1, 2,...) 45 a,P,r,4~=1,2,3,4 p, u, 7
indices,
SU(4),
= 1, 2 )...) 15 I
a, b, c = 1,2,3
1,2
r,s=
I
SU(2), indices,
d, e,f = 1, 2, 3
SU(2), indices. I The SO(10) group generators are constructed as bilinear forms in terms of the fields in the 16 spinorial representation. The generators Qz and Qi transform as triplets under SU(2), and SU(2), , respectively, 1,2
t,t.l=
and the QF transform as the adjoint representation 15 of SU(4) (3.7)
where the A” are the 15 4 X 4 representation matrices
matrices of SU(4) 1 1
A3= 0 1 -1
’ 0
/I*,-
;
-2
i
0
i 1 1
/1’5 =-
1
h i
’ -3
i
with diagonal
co~srRuc~10N
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367
under the U(l),_, x SU(3), subgroup of SU(4). The 24 generators QFrb,“,which close the SO( 10) algebra, transform as a doublet under SU(2), and SU(2),, and as a 6 under SU(4), which is the antisymmetric part of the tensor product 4 X 4. QFF is given by Qz5 =I d3x @‘G(uiyty+tyr~ + b&ysa,4,&,s&tuWstol,WUD’),
(3.9)
where ea4YS is the totally antisymmetric tensor of SU(4) (c’234 = s1234= l), and a and b are determined from the requirement Q$ = (Q;t)+=
+E’~E’“E,~~~Q;~
(3.10)
and the normalization condition for the 45 generators of SO(lO), Tr{Q”Q,,) We find a* = -2b = -l/d, thus
fixing the phases of a and b. Specifically, the normalization the expression
of the generators Q$ has
Tr{Q$Q”,“s} = BrsstU&a4a’b’&,,4,ys. The Lie commutator
= 26”,.
(3.12)
algebra [ Qs, Q” ] = ics’$ Qe,
where the cs’lg are the structure constants of SOtlO), expressions for the group generators
(3.13) is obtained from the above
(3.14)
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TliRAMOND
We have used the cyclic properties of EaBys to obtain the last commutation relation in Eq. (3.14). The g,,, are the structure constants of SU(4). The flavor quantum numbers follow from the commutation relations with the diagonal generators of SU(2), and SU(2), and the Qb” generator of SU(4) which is normalized by (3.15) with qern= I,, + I,, + 4(B -L). The flavor content of the 16 representation is given in Table III. The leptons are ‘included as a fourth color [2]. The new field which appears in this model corresponds to an SU(5) singlet and has the quantum numbers of a left-handed antineutrino. The 45 gauge bosons of this theory AI transform as the adjoint representation of SO( 10): (3.16) where W” and Vd are the weak bosons associated with an ambidextrous electroweak theory. The 15 gauge fields AP of SU(4), transform under U(l),-, x SU(3), as 15 = 8 + 1 + 3 + 2, and are identified as the color octet of SU(3),, the gauge boson corresponding to U( 1)8 -L and 6 new bosons associated with the generators that close the SU(4) algebra. These lepto-quark bosons are labeled by XSi and XSi. The remaining 24 gauge bosons A$ are associated with the generators which close the SO(10) algebra and are labeled by Xi, Y’, Xii, Y’i and their antiparticles. The corresponding quantum numbers are the eigenvalues of (3.14) and are given in Table IV for the lepto-quark and diquark bosons of the model. B. The SO(10) Model in the SU(5)
x
U(l), Basis
The SO( 10) model under SU(5) X U(l), provides an attractive alternative to the symmetry breaking patterns of SO(10) which reproduces the interesting features of TABLE Flavor
III
Quantum Number Content for the Fermion Representation and Particle Assignment in the SU(2), x SU(2), X SU(4),
I 3L
I 3R
t
0 0
i 1 2
0 I 2
0 -4
12 f
4em I 3
2
-1
6 1
0
-4
0 0 0
B-L
16 of SO(10) Basis
-I -j
-5-1 1 I 3 1
* 1 0 f 1
Particle
CONSTRUCTION
OF GRAND TABLE
UNIFIED
369
MODELS
IV
Quantum Number Assignment for the Lepto-Quark and Diquark Bosons in SO(I0) A’
I 3L
I 3R
B-L
SU(5). In this basis, the fundamental representation 45 decompose as
spinorial
.-
Particle
4 em
representation
16 and the adjoint
s=$+lJ+i,
(3.17) -
4J=B+1+lJ+lJ.
(3.18)
The matter fields and the generators of the theory are thus labeled by (3.19)
Iv*} = w7 $bT WI, @I =
IQ", Q,,,, Q,, QDL
(3.20)
with indices H, I = 1, 2 ,..., 16
<, q, 8 = 1, 2)..., 45 A, B = 1) 2 ,...) 5 D, E = 1, 2,..., 10
SU(5) indices.
ri,p, v = 1, 2 ,..., 24 1 The SU(5) generators Q” in Eq. (3.20) transform under SU(2), x U(l), x SU(3), according to Eq. (2.6), whereas Q,,, is an SU(5) singlet. The generators Q, which transform as a g of SU(5) decompose under W(2), X U(l), X W(3), as
IQ,1 = IQ', Qri, Q,l and QD = (Q,)? It is important to remark that in SO(lO), the embedded in the spinorial representation 16 which the fields in the 16 anticommute with each other, written as bilinear forms in terms of the 16 fields
(3.21)
5 + B representation of SU(5) is is not reducible. Consequently, all and the generators of SO(l0) are vH.
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The generator of the U(l), sector of SU(5), Q,, is invariant SU(3), rotations and has the form
under SU(2),
and
(3.22)
Note that QY has no bilinear term vtyl, since ly is an SU(5) singlet and QY is an SU(5) generator. The generator of the U(l), sector, Q,,.,, is invariant under SU(5) rotations and hence it is given by (3.23)
Q,, and Q,,, are linear combinations of the flavor diagonal generators Qi and QJ’ of the SU(2), x SU(4), sector of SO(10) Y, = q(r)
where q(,.) is normalized
I3R
= 413,
+ f(B
(3.24)
-L),
- 3(B
(3.25)
-L),
by
[v,wQ,r,l=Lqww,. \/40
(3.26)
We have used the SO(10) normalization condition Tr{Q”Q,} = 26”,. Equation (3.24) follows from the definition of the electric charge, and Eq. (3.25) from the invariance properties of Q,,, under SU(5) rotations as expressed in Eq. (3.23). The identification of the fields in the new basis with the corresponding particle states is given in Table V. The SU(5) representations 5, B and 1 are distinguished by their q(,, assignment. TABLE
V
Identification of the Fermion Fields in the SU(5) X U(l), Basis of SO(10) with the Corresponding Particle States in SO(10) under SU(2), X SU(2), X SU(4), SO( 10) 3 W(5) x U(l),
SO( 10) 2 SU(2), x W(2), x W(4),
q(r) 3 3 -1 -1 -1 -5
Particle
CONSTRUCTION
The generators of the SU(5) since the adjoint representation
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sector of SO(lO), QA, are bilinear in I# +@ and did, -24 is found in the tensor products $x5=1+24
and
lJxu=1+24+75.
It is now straightforward to construct and identify the SU(5) using Table V and Eqs. (3.5k(3.7) and (3.11). We obtain
generators,
Eq. (2.6).
(3.27)
(3.28)
Q:=ld’x
[@
(-q);,j+,it
oj+$;i($)i)ri]
(-;:)i
=Q;, I
.i
p= 1, 2,..., 8, Qr&L(
d3x(y/‘+yi
- ~,,i$,t$ - cijk#k+qi5rj) = Qz,
(3.29) (3.30)
6 Qri
=
z
(Qj,’
Q;;
and for the SU(5)
1 = T
(3.3 I )
E’~E~~~Q<;,
singlet
Q,r,= .( d3x&
(3~~ +I//~ - q&h, - St&)
1 = \ i T Q; - 4- f
Q;.‘.
(3.32)
The generators Q,, which close the SO(10) algebra under SU(5) x U(l),, are bilinear in ~‘4~ and 4;~” since the -10 is found in the tensor products 1 x N and Bxj=lJ+@ We find
Qi
=
-
*J
d3X(E,,$JivS + #+w’ + ~~4~) = I,) - 1 fi i 1 4
(Qk’ + iQi*), (Qi’ + iQi”>,
(3.33)
312
DE LA OSSA
AND
DE ThRAMOND
Qi = (Q’)“, Qri=-Lj
(3.34) d3X(E,,Eijk$sfVk + &,,#itWs + Vt#,i)=-+EijkQ::y
(3.35)
\/z Qri
=
= - +
(Q,,)’
Q,=-‘(
G
=
EijkQ;;
+“SQ;;,
d3x(#‘+tyi + t,v+$)= - ’ fi
(3.36)
(Q: - iQ;>,
(3.37) (3.38)
Qm= (Q,)+-
The gauge bosons of SG(10) under W(5) X U(l),. transform according to Eq. (3.20). The new quantum number, q(r), is zero for the bosons of the SU(5) x U(l), sector and 4 (-4) for the bosons transforming as the a (10). The interaction Lagrangian
%=g,oWHV’[vw
QS14
(3.39)
follows from the transformation properties of the fields wH in the spinorial representation 16 under the group generators, Qs, in the SU(2), x SU(2), x SU(4), or SU(5) x U(l),. basis. Although the SO( 10) decomposition in both bases give identical results well above the symmetry breaking scale, they correspond, according to the various symmetry breaking patterns, to different physical theories at low energies [4, 17, 181. For example, the gauge bosons associated to the SU(2), generators could be relatively light in a left-right symmetric model but correspond to superheavy bosons under W(5) x U(l),. However, the neutral boson associated with the U(l), generator, C,, could be as light as 250 to 300 GeV [ 181 leading to an extended electroweak model based on the gauge group SU(2), x U(l), x U(l),.
IV.
THE E, MODEL
In the search for groups of higher symmetry as candidates for unified theories we consider the rank-6 Lie-groups: SU(7), Sp(12), SO( 12), SO(13) and E,. The fermions in SU(7) are placed on a highly reducible representation, and the resulting models have been studied with the purpose of incorporating various families of quarks and leptons [19] which appear in the multiplets of the representation with a host of exotic particles. From the remaining rank-6 Lie-groups, only E, has complex representations and provides an attractive possibility for grand unification [5]. E, contains SO(l0) and SU(5) as subgroups and thus incorporates the salient properties of these models. Since all the exceptional groups G,, F4, E,, E, and E, contain SU(3) as a subgroup, color is naturally embedded in the octonionic structure of the exceptional algebras which are anomaly free. G, contains SU(3), as a maximal
CONSTRUCTION
OF
GRAND
UNIFIED
MODELS
373
subgroup. We need a larger group structure to incorporate a flavor group in order to define the electric charge operator. F, contains an additional SU(3) which could be used as a flavor group, but it is not appropriate to describe the weak interactions of quarks. E, has SU(3), X SU(3), X SU(3), as a maximal subalgebra and we can consider SU(3), X SU(3), as a flavor group. E, is the only exceptional group which admits complex representations and all the fermions are assigned to the lowestdimensional representation, the fundamental 27 [5]. E, also has as maximal subalgebras SU(2) x SU(6) and SO(10) x U(1). A. E, under SU(3), X SU(3), x SU(3), The fundamental representation 21 of E, transforms under SU(3), X SU(3), X SU(3), as - 27 = (593, 1) + (3, 193) + (1, 3331, (4.1) where the leptons are assigned to the (&3, A), which is a color singlet, the quarks - - to the (3, 1,3), a color triplet and SU(3), singlet, and the antiquarks to the (1-9 3-3 3) - *a color antitriplet and SU(3), singlet. The above decomposition is closely related to the octonionic structure of the exceptional observables of Jordan et al. [20] which are exceptional observables in charge space. The matter fields are labeled by IVIF}
=
{W’dr
Vlai,
@‘I*
(4.2)
The adjoint representation 18 transforms as - 1) + (I, 178) + (3? 37 3) + (3,3,3>,
78 = (8, 1, 1) + (I,&
and the corresponding generators QK are identified by their transformation
IQ'1 = {QL”, QR",Q:. QYd,Q:,}-
(4.3) properties (4.4 )
with indices F, G = 1, 2 ,..., 27 K, Tl, 0
=
1,
2,..., 78
a, b, c = 1, 2, 3,
SU(3), indices,
4 e,f = 1, 2,3,
SU(3), indices,
i,j, k = 1, 2,3,
SU(3), indices,
a, p, y = 1, 2 ,..., 8.
The generators Q& Q; and Q: transform as octets under SU(3) (4.5 1
595/155'2
IO
374
DE LA OSSA
AND
DE TiRAMOND
Q;=(d)x[yQ,t(~)d’Woe+ye~+(-~)edWdi], yldi], whereas the QL, transform hence have the form [ 161
as a color antitriplet,
SU(3),
(4.6) (4.7)
and SU(3),
triplets, and
Q6d = 1 d3X(aEWkvdk tvaj + P&,bc vc!i vbd + Y&defl/la>Wei)
(4-g )
and Qs” = (QAd)t. The closure of the Lie algebra of the generators determines the constants a, p and y up to a phase ial* = [PI’= Iy[* = 4, where we have used the E, normalization
condition
Tr{Q”Q=}
= 36”,. We choose
Pll a=-/?=-y=
l/G,
(4.9)
and thus d3X(EUkWdk tWaj - E,bc v!i vbd - EdefWaftWei),
Q;,zi(
with
(4.10)
\/z Tr{Qi,Q;‘}
= 3~5~~8~~~5~‘.
The generators QK satisfy the commutation
(4.11)
relations
[ QK, Qn] = ir”“, Qw,
(4.12)
where Y’(~~ are the E, structure constants. Using the expressions for the E, generators we obtain the commutation relations
[&, Q;l = ($)
bQ6,,
[Qh,, Q;l = ($)
deQ;,, (4.13)
,&,:Q:i
= (-F)i
[t&d, Q’,,] = +
j
&d,
&,bc&de,&iikQ;f,
CONSTRUCTION
The fermion quantum of the matter fields with the electric charge is generators of the SU(3),
375
OF GRAND UNIFIED MODELS
number assignment follows from the commutation relations the diagonal generators of each of the SU(3) subgroups, and expressed as a linear combination of the four diagonal X W(3), flavor group (4.14)
4 em= I,, + I,, t Yt t Y, 1 where Y,,, is normalized
by (4.15)
[wm Q&l= d%,wr
The flavor quantum numbers of the fundamental representation 22 and the corresponding fermionic assignment are given in Table VI. This model contains extra unobserved fermions with unusual transformation properties. The 78 gauge bosons A”, transform as the adjoint representation of E,, and their quantum number assignment follows from the eigenvalues of (4.13). B. E, under SO(10) x U(l), The maximal orthogonal subalgebra decomposition of E, provides a particularly interesting possibility for symmetry breaking along the maximal subgroup chain E, 3 SO(10) x U(l),1
W(5)
(4.16)
x U(l), x U(l),,
TABLE VI Flavor Quantum Number and Fermionic Assignment for the Fundamental Representation 27 of E, under SU(3), x SU(3), x SC43), I 3L
I 3R
-4
i
1 2 1 z
1 2
0 4 1 2 0
f f i
0 0
i -4
0
0
f 1
0
0
0 1 2
0 0 0
0
f 0
y,.
YR
4em
Particle
0
N,,
-1 -1 1 0 0 I 0 0 23 I / I i 3 f f
E,e, El+ -N,, -V, e,’ -v; N”, Uil d,, h,, u;’ d”’I h“’ I
376
DE LA OSSA
AND
DE ThRAMOND
which can be interpreted as an E group series E, + E, -+ E,, with E, isomorphic to SO(I0) and E, isomorphic to SU(5), by removing appropriate circles from the Dynkin diagrams [6, 7, 221. Under SO(lO), the fundamental 21 and the adjoint x representations decompose as 2J=E+i+lo,
(4.17)
B=Q+1+16+16,
(4.18)
and hence the matter fields and the generators are labeled by (4.19)
Iv,} = {u/,?XYY,L {QKl = lQ5, Q,,,, Qm QHL
(4.20)
with indices F, G = 1, 2 ,..., 27
K, z, w = 1, 2 ,..., 78 H, I = 1, 2 ,..., 16
p, 4 = 1, L..., 10
SO( 10) indices.
[, v, e = 1, 2 )...) 45
The known fermions, which are assigned to the spinorial representation 16 of SO( IO), are included in the lowest-dimensional representation of E,, together with an SO(10) singlet and 10 new fermion fields Y,, which transform as the self-conjugate lowest-dimensional representation of SO(lO), lo. Under SU(5), B = 5 + 5 and thus (4.21)
{Y,} = {Y”,QAl.
The transformation properties of the generators of SO(lO), QC, are given by Eq. (3.20). The generator Q,,, transforms as an SO(10) singlet, and the generators QH, which close the E, algebra, transform according to Eq. (3.17),
{QH) = iHA, G,, G,l
(4.22)
{HA} = {H’, Hi},
(4.23)
{GD} = {G’v Gri, G,}*
(4.24)
and QH = (QH)+. Under SU(5)
G, and G, are W(5) and SU(2), x SU(3), singlets, respectively. To identify the fermions, we construct first the generators of the U( l)y
x
U(l),
x
CONSTRUCTION
U(l), by
OF GRAND
UNIFIED
sector. Since QF commutes with the SU(2),
Qy=,f
d3x(ayrty’ +“t-yr+yr
+ byi+@ + + gY’+Y’
MODELS
311
and the color generators, it is given
cgit#i
+f’nfa,
+ d#Ji#,i + e#+# +
(4.25 )
gqq.
This generator has no bilinear terms in ly and x which are SU(5) and SO(10) singlets, respectively. Now, the YA transform as a 51 and a.., as a 5 of SU(5). and therefore a = f = -f ‘, b = g = -g’. The eigenvalues of QY, a, b, c, d and c, are known from their SU(5) assignment. The generator Q,,., commutes with all the SC!/(~) generators and has the form
Q,,,= j
d3x(avA+yl”
+P&,
+ py+y + 6YA+Y4 - &‘$‘,).
(4.26)
We have not included a bilinear term in K since Q,,, is an SO(10) generator. Note that although vA and Y,’ transform both as a 5 of SU(S), they have not necessarily the same eigenvalues for q(,), since Q,,, is not an SU(5) generator. The generator of the U( l), sector of E,, PO,, is invariant under SO(10) rotations, and thus it has the form
Q,,,= I' d3x(c,d Y, + w+x+ c?Y; Y,,, The generators Q,, Q(,.) and Q,,, are linear combinations of the flavor subgroup SU(3), x SU(3), of E, y, =
where q,t, is normalized
I,,
+ Y, + YR 3
(4.27)
of the diagonal generators
(4.28)
9 (r-j = 413, - W’,, + Y,),
(4.29)
q(t) = WY, - YRL
(4.30)
by (4.3 1)
and Tr{Q”Q,} = 3P1,. We give in Table VII the U(l), x r/(l), x U(l), quantum number content of the fermion fields and the identification with the corresponding particle states. The different SO(10) representations are distinguished by their qtr, assignment. In this model the SU(5) singlet w is not necessarily assigned to a righthanded neutrino, but could be paired with the SO(10) singlet x to form a Dirac spinor.
378
DE
LA
OSSA
AND
TABLE
DE
TJiRAMOND
VII
U(l), X U(l), X U(l), Quantum Number Content and Identification of the Fermion Fields in the SO(10) X U(l), Basis of E, with the Corresponding Particle States under SU(3), x SU(3), x SU(3),
EC ~SO(10)
x U(l),
E, = =J(3), x W(3), x W(3),
Y,
v’,
-f
P v”
-;
vri 3 VI 42 3 I3
X Y’ YA
yP %
42
yi
fP
fir
E,,VSI
Oi
V3i
Particle
qw 3
4
3 -1
1f’
-1
1 0
-1 -5
0
0
-f
!3 f
-f
-2
-2
-2
-2 2
-2
2
-2
E-
( -N 1 ,. h” E_”
~
i -NC j L hi,
The generators of the SO( 10) subgroup of E, , Ql, are bilinear in WAvI and Yi Y, since the adjoint representation of SO(lO), 45, is found in the tensor products
and
u xlJ=45,+1+54,, where, under SU(5),
45,=24+1+10,+10,.
-
The generators Qs are thus given by
s
Ys
(4.32)
CONSTRUCTION
OF GRAND
UNIFIED
379
MODELS
(4.33)
(4.34)
d3x(yr+$ - ~,,#,tti - ciik$kt#rj - Yr+Yi + L?,h,) Qri=(Qri)+ =Q;l, (31yA5fA -qg~D-51y+y/-2YA+YA
= Q;, ,
(4.35) (4.36)
+2n;n,> (4.37)
d3x(-&,&yr
QriZ -‘j
d3X(&,,Eijk#Jjvk
+ #+I# + y/+f + t+jkyk+fij)
f E,,#itv’
+ ‘&“#,i + Y’+L’-
=
-Qi3,
(4.38)
Y’+G,)
fi = -E,, Q;‘,
Q,=-Lj d3X($‘+ly’ +ly+$b +E’SY’P2,) =+ (Q:,- iQ;h u/z
(4.39) (4.40)
and QD = (Q,)+. The SO(10) singlet, Q,,,, is written as (4.4 1)
The generators that close the E, algebra, QH, are bilinear in x’y, 16 is found in the tensor products 1 X 16 and
and WA YP since the
380
DE LA OSSA
AND
DE TkRAMOND
We obtain
d3x@,tiini + erS#+SZs+ I/&’
d3x(ciikqbk+i2, + ~r’$, d3x(cijkiyk+.0j d’x(i/+LJ,
(4.42)
- X+W’) =
- ty+Y’ -x+#)
=
- $+Y’ - E,.,&~ Y’ +x+&)
+ y’+.n, - q.,fii+Ys
Qil,
= -Q;,,
t E,.,E~~~#~~Yk -x+&)
Qf3,
= -E,,
(4.45)
I d3x(Pyr+12s G,=-lj
d3x(#+yi !b
(4.44)
+ #i+Yi - ~‘4) = G
- y/‘+y’ + x+y) = - ’ fi
(Qi - iQh (Qi - iQh
(4.46) (4.47)
and Q” = (Q,)+. The 78 gauge bosons of E, transform according to Eq. (4.20) and the quantum number qct, is zero for the bosons associated with the SO(10) X U(l), sector and -3 (3) for the bosons transforming as the 16 (16). The E, Lagrangian follows from
Since the representation of the new fermion fields Y, is real with respect to SU(2), X SU(3),, they are supposed to acquire superheavy masses and decouple from the theory according to Georgi’s survival hypothesis [23]. If the generators associated with the U(l), and U(l), Abelian sectors remain unbroken at the SO(10) and SU(5) symmetry breaking scales, we recover at low energies (if the associated gauge bosons are relatively light) an extended electroweak model based on the gauge group SU(2), x U(l), x U(l), x U(l), [24]. As was mentioned in Section III, the 27-dimensional reducible representation of SU(6) provides an interesting decomposition of the E, model, when it is embedded in the 27 fundamental irreducible representation of E,. Under its maximal subalgebra SU(2) x SU(6), E, has the branching rule - 22 = (236) + (1, Is). We do not give here a detailed discussion along this line, but the model can be built
CONSTRUCTION
OF GRAND UNIFIED MODELS
381
without difficulty with the construction procedure described in this paper. Different symmetry breaking patterns of E, are found in Refs. [ 7 ] and [ 8 1, and references cited therein.
V. CONCLUDING
REMARKS
We have developed in this paper a simple construction procedure of grand unified models of the strong, weak and electromagnetic interactions exploiting the transformation properties of the matter fields and group generators under the different interaction subgroups, using a maximal subalgebra decomposition and the closure of the algebra to obtain a realization of the group generators. We have given an explicit construction for the models based on unitary, orthogonal and exceptional groups of rank 4, 5 and 6, namely, SU(5), SO(l0) and E,, which are of particular physical interest and simplicity, but the method could be used as well with other groups. We have employed the same construction procedure for different Lie groups, avoiding specific algebraic structures such as Clifford algebras, octonions or bases related to the weights of the representations [6, 7, 251, by making an extensive use of the representations of the unitary groups which are more familiar to many physicists. In SU(5), the generators are constructed as bilinear forms of the fermion fields using only the defining representation 3. For the SO(10) model, the generators are built in terms of the irreducible spinorial representation 16, and for E, in terms of the fundamental 2. The algebraic construction for the SO(10) and E, models is simplified considerably by using the left-right symmetric bases SU(2)L x SU(2), x SU(4), and SU(3), X sU(3), X SU(3),, respectively. The generators in the chiral bases SU(5) x U(l), and SO(10) x U(l), are written as linear combinations of the generators in the L-R symmetric basis. The quantum number content of a given representation, the identification of the matter fields with quarks and leptons and the specific form of the interaction Lagrangian for the various models considered follows from the commutation relations of the fields with the group generators. We have considered only groups which are complex with respect to SU(2), x U(l), x SU(3), and real with respect to SU(3), [26]. This restriction poses severe limitations on the incorporation of a family structure. E,, for example, contains three E, families but has only real representations. The spinor representations of the orthogonal series SO{ 10 + 4n), which could otherwise account for the repetitive family structure [27], are real with respect to SO(10). These models introduce mirror V + A families of heavy fermions. Unitary groups of higher rank SU(7), SU(S), SU(9),... have been considered for family unification [8], but the proliferation of fermions with the rank of the group is enormous. An interesting example due to Georgi [23] in which all the irreducible representations appear only once in the fermionic representation is based on the embedding of SU(5) in SU(11). The fermions are assigned to the 561-dimensional reducible representation 11 +5J + 165 + 330. The model contains three 3 + s chiral families which survive at low
382
DE LA OSSA
AND
DE TliRAMOND
energies! This model, however, is not asymptotically free. An understanding of the family structure as well as the fermion masses and mixing angles remains a challenging problem.
ACKNOWLEDGMENTS
The authors thank the Escuela de Fisica at the Universidad de Costa Rica where most of this work was done. We also acknowledge useful conversations with A. Yildiz and P. Frampton. This research is supported in part by the National Science Foundation under Grant PHY-82-15249.
REFERENCES
1. E. S. AYERS AND B. W. LEE, Phys. Rep. 9 (1973), 1; M. A. B. BBC AND A. SIRLIN. Ann. Rev. Nuclear Sci. 24 (1974), 379. 2. J. C. PATI AND A. SALAM, Phys. Rev. D 8 (1973) 1240; IO (1974) 275. 3. H. GEORGI AND S. L. GLASHOW, Phys. Rev. Lett. 32 (1974), 438. 4. H. FRITZSCH AND P. MINKOWSKI, Ann. Phys. (N.Y$ 93 (1975), 193; H. GEORGI, in “Particles and Fields-1974,” Proceedings, 1974 Meeting of APS Division of Particles and Fields (C. Carlson, Ed.), AIP, New York, 1975. 5. F. G~~RSEY,P. RAMOND, AND P. SIKIVIE, Phys. Lett. B 60 (1976), 177; F. G~~RSEYAND P. SIKIVIE, Phys. Rev. Lett. 36 (1976), 775. 6. B. G. WYBURNE, “Classical Groups for Physicists,” Wiley-Interscience, New York, 1974; H. GEORGI, “Lie Algebras in Particle Physics: From Isospin to Unified Theories,” Benjamin, Cummings Publ., 1982. 7. M. GELL-MANN, P. RAMOND, AND R. SLANSKY, Rev. Modern Phys. 50 (1978), 721; R. SLANSKY, Phys. Rep. 79 (1981), 1. 8. P. LANGAKER, Phys. Rep. 72 (1981), 185. 9. H. GEORGI, H. R. QUINN, AND S. WEINBERG, Phys. Rev. Len. 33 (1974), 45 1; A. J. BLJRAS,J. ELLIS. M. K. GAILLARD, AND D. V. NANOPOULOS, Nuclear Phys. B 135 (1978), 66; W. J. MARCIANO, Phys. Rev. D 20 (1979) 274; T. J. GOLDMAN AND D. A. Ross, Nuclear Phys. B 171 (1980), 273; W. J. MARCIANO AND A. SIRLIN, Phys. Rev. Lett. 46 (1981), 163. 10. For a review see, for example, J. ELLIS, M. K. GAILLARD, AND D. V. NANOPOULOS,CERN preprint TH-2858, 1980. 11. S. L. GLASHOW, Nuclear Phys. 22 (1961), 579; S. WEINBERG, Phys. Rev. Lett. 19 (1967), 1264; A. SALAM, in “Elementary Particle Theory: Relativistic Groups and Analyticity” (N. Svartholm, Ed.), p. 367, Nobel Symposium No. 8, Almqvist & Wiksells, Stockholm, 1968. 12. H. FRITZSCH AND M. GELL-MANN, “XVI International Conference on High Energy Physics,” Vol. 2, p. 135, NAL, Batavia, 1972; H. FRITZSCH, M. CELL-MANN, AND H. LEUTWYLER, Phys. Left. B 47 (1973), 365; S. WEINBERG, Phys. Rev. Lett. 31 (1973), 494; Phys. Rev. D 8 (1973), 4482. 13. C. BOUCHIAT, J. ILIOPOULOS, AND PH. MEYER, Phys. Lett. B 38 (1972), 519. 14. S. WEINBERG, Phys. Rev. Lett. 29 (1972), 388; G. SENJANOVII?AND R. N. MOHAPATRA, Phys. Rev. D 12 (1975), 1502; H. FRITZSCH AND P. MINKOWSKI, Nuclear Phys. B 103 (1976), 61; A. DE R~JJULA, H. GEORGI, AND S. L. GLASHOW, Ann. Phys. (N.Y.) 109 (1977), 242; J. C. PATI, S. RAJPOOT, AND A. SALAM, Phys. Rev. D 17 (1978), 131. 15. The algebraic properties and the particle assignment of irreducible representations of higher dimension follow from the transformation properties of the SU(5) generators written in terms of the 5 or 3. For a different construction see C. SAFLIO&LU, Phys. Rev. D 17 (1978), 1598. 16. P. SIKIVIE AND F. GORSEY, Phys. Rev. D 16 (1977), 816.
CONSTRUCTION
OF GRAND
UNIFIED
MODELS
383
17. M. S. CHANOWITZ, J. ELLIS, AND M. K. GAILLARD, Nuclear Phys. B 128 (1977), 506; M. MACHACEK, Nuclear Phys. B 159 (1979), 37; H. GEORGI AND D. V. NANOPOULOS, Phys. Lett. B 82 (1979), 392; Nuclear Phys. B 155 (1979), 52; 159 (1979), 16; R. N. MOHAPATRA AND B. SAKITA. Phys. Rev. D 21 (1980), 1062; S. RAJPOOT, Phys. Rev. D 22 (1980), 2244. 18. A. MASIERO, Phys. Lett. B 93 (1980), 295; N. G. DESHPANDE AND D. ISKANDAR, Phvs. Lett. B 87 (1979), 383: R. W. ROBINE~ AND J. L. ROSNER, Phys. Rev. D 25 (1982), 3036. 19. P. H. FRAMPTON, Phys. Lett. B 88 (1979), 299; M. CLAUDSON, A. YILDIZ, AND P. M. Cox. Phys. Lett. B 97 (1980), 224; S. M. BARR, Phys. Rev. D 21 (1980). 1424; J. E. KIM. Ph.vs. Rev. Left. 45 (1981), 1916; Phys. Rev. D 23 (1981), 2706. 20. P. JORDAN. J. VON NEUMANN, AND E. P. WIGNER, Ann. of Math. 35 (1934), 29. 21. We have followed a different convention from Ref. [ 16 ] for the signs of /, and y in order to recover the results of the SU(5) and SO(10) models. 22. F. G~~RSEY. Yale University report YTP 81-11, 1981. 23. H. GEORGI, Nuclear Phys. B 156 (1979), 126. 24. R. W. ROBINETT AND J. L. ROSNER, Phys. Rev. D 26 (1982), 2396. 25. G. FELDMAN, T. FULTON, AND P. T. MATTHEWS, Johns Hopkins University preprint, JHU-HET 8208 and 8209, 1982 (unpublished). 26. Different embeddings can be found in Ref. [7]. 27. M. GELL-MANN, P. RAMOND, AND R. SLANSKY, in “Supergravity” (P. van Nieuwenhuizen and D. Z. Freedman, Eds.), North-Holland, Amsterdam, 1979; F. WILCZEK. in “Proceedings, 1979 International Symposium on Lepton and Photon Interactions at High Energies, Fermilab” (T. Kirk and H. Abarbanel. Eds.). Fermilab, Batavia, Ill.. 1980.
OF PHYSICS
358-383 (1984)
155,
Construction of Grand Unified Models under Maximal Subalgebras XENIA Department
of Physics.
DE LA OSSA
C.
University
of Texas,
Austin,
Texas
78712
AND GUY F. DE T~RAMOND Lyman
Laboratory
of Physics,
Received
Harvard
September
University,
6, 1983;
revised
Cambridge,
Massachusetts
November
1, 1983
02138
A construction of grand unified models of the strong, weak and electromagnetic interactions is described based on the transformation properties of the group generators under a maximal subgroup decomposition without recourse to large representation matrices or to the specific algebraic structures of some classical Lie-groups, such as the Clifford algebra associated with the orthogonal groups or the octonionic structure of the exceptional groups. To illustrate the procedure an explicit construction is given of the SU(5) model useful in the discussion of higher rank groups, of SO(10) under the maximal subalgebras SU(2), x SU(2), x SU(4), and SU(5) x U(l), and of the exceptional group E, under SU(3), x SU(3), x SU(3), and SO( 10) x U(l),. The construction procedure can be used as well with any classical Lie-group.
I. INTRODUCTION A deeper understanding of the structure of matter has followed from the description of the fundamental forces in terms of non-Abelian local-symmetry groups [l] and their subsequent unification in larger semi-simple [2] or simple [3-51 Liegroups [6]. The strong, weak and electromagnetic interactions thus appear as different low-energy manifestations of a single underlying force, whose symmetry is manifest at a very large energy scale. The grand unified theories [7,8] lead to such spectacular prediction as proton decay within observable limits, afford a very precise determination of the electroweak angle [9] and provide a framework for the discussion of problems of cosmological importance such as the baryon asymmetry of the universe [lo]. One of the most outstanding aspects of these models is, however, how the different interactions are incorporated into the theory. Furthermore, the quantum number content of a given representation of the unifying gauge group, which is totally arbitrary in the standard model based on a semi-simple group, is determined by the algebraic structure of the 358 0003.4916/84
$7.50
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CONSTRUCTION
OF GRAND
UNIFIED
MODELS
359
group generators. Quantum numbers such as isospin I,, hypercharge Y, B - L (baryon minus lepton), electric charge qem, etc., correspond in most of these models to the Lie-generators of the theory, or to linear combinations of them. The particle states are classified according to their transformation properties under a given gauge group, and the quantum number assignment follows from the commutation relations of the fields with the diagonal generators of the theory. In the present paper we give an explicit construction of grand unified models based on the transformation properties of the group generators written as bilinear forms in terms of the representation vectors of the unifying gauge group, in a basis constructed under maximal subalgebras. The interest of the subgroup decomposition arises from the physics itself: there is compelling evidence that the various interactions observed at low energies (below 100 GeV) are gauge interactions described by the semi-simple structure of a direct product of unitary local-groups in the standard model. Plausible group structures at intermediate energies (left-right symmetric theories of electroweak interactions) are also described by a product of unitary groups. A natural basis for the classification of the particle states according to their transformation properties under the interaction subgroups is thus provided. The different interaction subgroups are used in turn to build the unified group by their embedding into a larger structure. It is also of interest that the same construction procedure, which provides a concrete realization for the algebra, can be used with different classical groups without invoking a specific algebraic structure, such as the Clifford algebra for the orthogonal groups or the octonions associated with the exceptional groups. We also avoid the utilization of large representation matrices which, as a matter of fact, do not exist for the exceptional groups based on non-associative algebras. We are not concerned in this paper with the phenomenological issues of these models, the specific mechanisms of spontaneous or dynamical symmetry breaking or more elaborate algebraic constructions. Most of these issues are discussed in the excellent reviews given in Refs. [ 6-8 ]. The standard model of the electromagnetic, weak and strong interactions is based on the local-symmetry group SU(2), x U(l), x SU(3),. The electroweak sector of the theory [ 111 is based on the gauge group SU(2), x U(l), which acts on flavor quantum numbers. The usual assignment of particles into doublets and singlets in the minimal electroweak model according to the particle helicity depends solely on experimental observations. In this model the left-handed fields of the lepton or quark sector are grouped into SU(2), doublets, whereas the right-handed counterparts or, equivalently, the left-handed charge conjugate fields, are singlets under SU(2),. The quantum numbers corresponding to the generator of the U(l), Abelian sector, the hypercharge generator Q,, are assigned in a given representation in order to obtain the electric charge operator as a linear combination of Q,, and the diagonal generator of SU(2),, the weak isospin. The strong interactions are described by the unbroken local-symmetry group SU(3), which acts on color quantum numbers [ 121. Each flavor or type of quark is assigned to the fundamental representation 3 and the antiquarks to the 3. The leptons do not have strong interactions and are thus assigned to singlets under SU(3),. The
360
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TkRAMOND
representation of the gauge group for a given flavor is given in terms of left-handed quark fields qi and the corresponding left-handed charge conjugate fields qci, which transforms as 3 + 3, and the theory is self-conjugate or vectorlike. The first generation of quarks and leptons, ui, di, e- and v, is described by 16 lefthanded fields. The hypercharge of the right-handed neutrino is zero and it decouples from the theory, leaving effectively 15 fields which are classified according to their SU(2), and SU(3), transformation properties. The representation is complex with respect to SU(2), X U(l), x SU(3),. Moreover, this is the minimal set of fields which is free from anomalies [ 131, a condition for renormalizability. An interesting extension of the minimal electroweak model, which is particularly relevant to our discussion of the SO(10) model, is an ambidextrous theory based on SU(2), X SU(2), X U(l),-, [ 141. The right-handed fields, which are singlets under SU(2),, are grouped in this model into SU(2), doublets. The quantum numbers of the new U(1) generator are also assigned arbitrarily in this model so as to build the electric charge operator as a linear combination of the U( 1) generator and the diagonal generators of SU(2), and SU(2),. The quantum numbers of the U(1) generator are found to correspond to B -L (baryon minus lepton). We discuss in Section II the embedding of the standard component model in the simplest grand unified theory, the W(5) model of Georgi and Glashow [3], which is used to decompose other models based on higher rank Lie-groups in the SU(5) basis. The SU(5) generators are built in terms of the fields in the defining representation 5, or 5 [ 151. In Section III we construct the SO(10) model by studying the transformation properties of the spinorial representation 16 and the group generators under its maximal left-right symmetric subalgebra SU(2), x SU(2), x SU(4),, and under SU(5) X U(l), by expressing the generators in the new basis as a linear combination of the generators in the W(2), x W(2), x SU(4), basis. The E, model is constructed in Section IV by studying the transformation properties of the group generators in terms of the fields in the fundamental representation 27 under the left-right symmetric decomposition SU(3), x SU(3), x SU(3), following the discussion of Sikivie and Giirsey [ 161. We also construct E, under the maximal orthogonal subalgebra SO(10) X U(l),, and the new generators are given in terms of the generators in the SU(3), x SU(3), X SU(3), basis. Some concluding remarks are given in Section V. II. THE SU(5) MODEL
The unified theory of the strong, weak and electromagnetic interactions based on the gauge group SU(5) is the lowest rank model (rank four) based on a simple group structure which contains the standard SU(2), x U(l), x SU(3), component model as a subgroup [3]. More elaborate models [4,5] of physical interest discussed in the next sections are of higher rank and contain extra quantum numbers and particle states. The isospin and hypercharge generators in this model are Lie-generators of SU(5), thus the electric charge operator is also a generator of the theory. The hypercharge assignment in a given representation is no longer arbitrary, and the
CONSTRUCTION
OF GRAND
UNIFIED
MODELS
361
electric charges of quarks and Ieptons are commensurate. Besides the 12 generators associated with the standard model, there are 12 generators which close the SU(5) algebra. The new generators carry color and flavor quantum numbers and mediate nucleon decay. The corresponding gauge bosons are exceedingly heavy (of the order of lOi GeV) to have an acceptable proton lifetime. The new vector-bosons can acquire such massesby introducing vacuum expectation values which commute with the SU(3) and SU(2) generators, thus leaving the gauge bosons associated with SU(2), x U(l),. X SU(3), massless at the first stage of spontaneous symmetry breaking. Since the unifying group is a simple Lie group, there is a unique gauge couphng and the different couplings observed at low energies are obtained by renormalization group estimates [9] from the unification scale. Having a viable unifying gauge group, we classify the fermions by studying the quantum number content and transformation properties of a given representation under the unitary subgroup decomposition. In SU(5), the known fermions are assignedto the 15dimensional reducible representation 5 + a. The anomalies in the 3 and a are equal and opposite, and thus the model is free from anomalies 13]. The fundamental 5 decomposesunder SU(2), x SU(3), as s = (2, I) + (133X
(2.1)
and the l0, which is the antisymmetric product of 5 x 5, as lo = (193) + (2,3) + (131).
(2.2
The adjoint representation -24 = 5 x 5 - I transforms as 24 = (3, !) + (1,1) + (138) + (2.5, + CL,3).
(2.3
The matter fields in the reducible representation $ + #J are labeled accordingly by {WA1= iv’, W’L {#Dl
=
14’3
#ri,
419
(2.4) (2.5)
where I+/ = v,.,+, and the group generators as
{Q"l = {QF, Q,, QF, Qriy Qril, with indices A, B = 1, 2,..., 5 D, E = 1, 2,...) 10 kp,v=
1, 2,..., 24
i, j, k = 1, 2, 3 a, p, y = 1, 2,..., 8 I r,s=
1,2
a, b, c = 1,2, 3 I
SU(3), indices,
SU(2), indices.
(2.6)
362
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AND
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The group generators are expressed in terms of the space-integral of bilinear forms of the fermion fields in the fundamental representation vA [ 151
Q’ = j d3x w,t(~)(T”),~ v,(x), or, equivalently,
in terms of the fields in the 5 representation Q" = j d3x I/ +(x)(-T"')",
v/"(x),
(2.7)
I# P3)
where the Ta are the representation matrices of the generators in the fermion basis, and (TA)A, z (T ’ )AB. The color generators QF transform as an octet in the adjoint representation of W(3), and the Qf as an SU(2) triplet
Q;=j&xvi~(-~)i Q;=jd3xft
i
(-f)’
$,
(2.9)
I$,
(2.10)
s
where the A” are the eight Gell-Mann matrices of SU(3) and the ra are the Pauli matrices. The generator of the U(l), sector commutes with the SU(2) and SU(3) generators, i.e., it is invariant under SU(2) and SU(3) rotations and hence has the form Q,, = j d3x(a~"$u'
+ /?iyit~').
(2.11)
Since the hypercharge generator is a generator of SU(5), it is traceless and normalized, as any other generator of SU(5), by Tr{ Q”Q,} = fs”, . Therefore,
WQ,) = 0,
WQ,‘) = $,
and we obtain
Qy= ( d3xfi(-
$y't~/' + $y'tt#),
(2.12)
fixing the phases of a and p. The generators Qri and Qri close the algebra associated with the group generators of SU(5). The generators Qri transform as an SU(2) doublet and SU(3) antitriplet, Q,i
= - 1 lb
j d3x
ft$,
(2.13)
CONSTRUCTION
and Qri is the hermitian
OF GRAND
363
MODELS
conjugate of Q,j Qri = (Qri)t
The normalization
UNIFIED
= - +-
1 d3x y/“,“.
(2.14)
condition for these generators has the form (2.15)
Tr { Q,‘Qsj} = fs,.‘~5,‘. It is now straightforward to obtain the Lie commutator
algebra of SU(5) (2.16)
[Q”, Q”] = itAFL,Q”,
where the ta”” are the structure constants corresponding to the vector space of the SU(5) generators in the basis defined by Eq. (2.6). Since the generators that close the SU(5) algebra are not hermitian in this basis, the structure constants are not necessarily real. From the above expressions for the group generators we obtain the following commutation relations:
IQ; yQi I= kbc Qi 9
[Q,‘, Q;l = ip)
[Q:* Q,“l= ifa,,Q,Y,
,Qri,Q+(-l;lii
IQ; 1Q,l = [Q:, Q,l = 0,
[Qri, Q,]
= $4;
LQ;3Q:l = 0,
[Qi, Q,‘l
= 0,
[Q,‘,Qsj]=-dji
’ Qsi5 Qj,
i Q,!,
(2.17)
sQr+a~(~)jiQ:-f6,a,i~~Q,.
(f)
r using the equal-time anticommutation relations for the fermion field operators ( w,~(x), y”(y)} = 6,‘6(x - y). In (2.17) the sObeand&, are the structure constants of SU(2) and SU(3), respectively. The quantum number assignment of the 5 follows from the commutation of the fields W* with the diagonal generators Q3L, Q,, Q3, and Qsr . We list in Table I the quantum number content of the 5 representation and the corresponding fermionic assignment. I,, is the weak isospin, Y, the weak hypercharge normalized by
[w”, Q,l = & Ywv* and s,,,
(2.18)
the electric charge qem= I,, + Y,. Quark and leptons are in the same irreducible representation, and since the trace of the electric charge operator vanishes in any given representation, the electric charge is quantized: the quark and lepton
364
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TABLE
I
Quantum Number Assignment for the Fermions in the 5 + 10 Representation of SU(5) Particle
4 em
charges are related by a factor of three. The fermion fields in the N from the antisymmetric product of 5 x 5
are obtained
(2.19)
#ri
=
+
4= *
(WrXi
-
(2.20)
WiXrL
(2.21)
&rSWsXrY
where xA transforms as 5 and 8” = -&‘I = 1. The corresponding quantum number assignment follows from the commutation of #D with the diagonal generators of SU(5) written in terms of the 5. The fermionic assignment is given in Table I. The 24 gauge bosons associated with the group generators transform as the adjoint representation of SU(5) {Alp}= and their quantum
numbers
(2.22)
{W”,,B,,A~~,A,i,,A’iu}
are the eigenvalues of (2.17). We list in Table II the TABLE
II
Quantum Number Assignment for the Gauge Bosom Responsible for the Baryon and Lepton Number Violating Interactions in SU(5) A,’
I 31
A,’ A,’
f -4
26 26
9em
Particle
f
X’
f
Y’
CONSTRUCTION
OF
GRAND
UNIFIED
MODELS
365
quantum number assignment for the gauge bosons. For reasons of brevity, we list only the eigenvalues of the lepto-quark and diquark bosons associated with the generators that close the SU(5) algebra. The interaction Lagrangian is uniquely determined by the requirement of the gauge invariance of the theory and is given in the reducible representation 5 + @ of SU(5 by
il;=g~(WAy'[WA,Q*l+~~,Ul~n~Q"l,A"..
(2.23
The usual expression of the SCr(5) Lagrangian [8] follows directly from the transformation properties of the matter fields IJ/ and #n under the group generators of the theory Q”.
III. THE SO(10)
MODEL
Could SU(5) be embedded in a group of higher symmetry which gives a theory of physical relevance where all the fermions are assigned to an irreducible complex representation free from anomalies? There are four rank-five simple Lie-groups [ 7, 8 ]: SU(6), SO{ 11L Sp( IO) and SO(10). The fermions in SU(6) are assigned to the 27.dimensional reducible representation 0 + t) + 15, which is anomaly free and contains extra fermions in the 15 with unconventional SU(2),, x SU(3), transformation properties. This model provides a suitable basis for the study of E, according to its maximal subgroup decomposition SU(2) X SU(6), and is more appropriately discussed in the context of E,. SO(l1) and @(lo) have only real representations, and we are left with SO(10) as a possible rank-live group for grand unification [4]. The fermions are assignedto the 16-dimensional irreducible spinorial representation which is anomaly free. SO(10) contains SU(5) x U(l), as a maximal subgroup which gives a model similar to W(5), and also SU(2), x SU(2), x SU(4), where the leptons are included as a fourth color [2]. W(4), is broken to SU(3),, which is left unbroken, and to U(l),_, , thus recovering at lower energies the left-right symmetric electroweak model mentioned in the Introduction. A. SO(10) under W(2),
x W(2),
x stun
Under SU(2), x SU(2), x SU(4), the representation of SO( 10) transforms as
16-dimensional irreducible
- 16 = (2, 134)+ (A,2341%
spinorial
(3.1)
and the adjoint representation 45 as
45 = (3, 1, 1) + (A,371) + (1,!,15) + (23276).
(3.2)
366
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LA
OSSA
AND
DE
TliRAMOND
We label the matter fields vH and the generators Ql as {VHI = iW,,l W%
(3.3) (3.4)
with indices H, I = 1, 2 ,..., 16 t-, q, 8 = 1, 2,...) 45 a,P,r,4~=1,2,3,4 p, u, 7
indices,
SU(4),
= 1, 2 )...) 15 I
a, b, c = 1,2,3
1,2
r,s=
I
SU(2), indices,
d, e,f = 1, 2, 3
SU(2), indices. I The SO(10) group generators are constructed as bilinear forms in terms of the fields in the 16 spinorial representation. The generators Qz and Qi transform as triplets under SU(2), and SU(2), , respectively, 1,2
t,t.l=
and the QF transform as the adjoint representation 15 of SU(4) (3.7)
where the A” are the 15 4 X 4 representation matrices
matrices of SU(4) 1 1
A3= 0 1 -1
’ 0
/I*,-
;
-2
i
0
i 1 1
/1’5 =-
1
h i
’ -3
i
with diagonal
co~srRuc~10N
OF GRAND
UNIFIED
MODELS
367
under the U(l),_, x SU(3), subgroup of SU(4). The 24 generators QFrb,“,which close the SO( 10) algebra, transform as a doublet under SU(2), and SU(2),, and as a 6 under SU(4), which is the antisymmetric part of the tensor product 4 X 4. QFF is given by Qz5 =I d3x @‘G(uiyty+tyr~ + b&ysa,4,&,s&tuWstol,WUD’),
(3.9)
where ea4YS is the totally antisymmetric tensor of SU(4) (c’234 = s1234= l), and a and b are determined from the requirement Q$ = (Q;t)+=
+E’~E’“E,~~~Q;~
(3.10)
and the normalization condition for the 45 generators of SO(lO), Tr{Q”Q,,) We find a* = -2b = -l/d, thus
fixing the phases of a and b. Specifically, the normalization the expression
of the generators Q$ has
Tr{Q$Q”,“s} = BrsstU&a4a’b’&,,4,ys. The Lie commutator
= 26”,.
(3.12)
algebra [ Qs, Q” ] = ics’$ Qe,
where the cs’lg are the structure constants of SOtlO), expressions for the group generators
(3.13) is obtained from the above
(3.14)
368
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AND
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TliRAMOND
We have used the cyclic properties of EaBys to obtain the last commutation relation in Eq. (3.14). The g,,, are the structure constants of SU(4). The flavor quantum numbers follow from the commutation relations with the diagonal generators of SU(2), and SU(2), and the Qb” generator of SU(4) which is normalized by (3.15) with qern= I,, + I,, + 4(B -L). The flavor content of the 16 representation is given in Table III. The leptons are ‘included as a fourth color [2]. The new field which appears in this model corresponds to an SU(5) singlet and has the quantum numbers of a left-handed antineutrino. The 45 gauge bosons of this theory AI transform as the adjoint representation of SO( 10): (3.16) where W” and Vd are the weak bosons associated with an ambidextrous electroweak theory. The 15 gauge fields AP of SU(4), transform under U(l),-, x SU(3), as 15 = 8 + 1 + 3 + 2, and are identified as the color octet of SU(3),, the gauge boson corresponding to U( 1)8 -L and 6 new bosons associated with the generators that close the SU(4) algebra. These lepto-quark bosons are labeled by XSi and XSi. The remaining 24 gauge bosons A$ are associated with the generators which close the SO(10) algebra and are labeled by Xi, Y’, Xii, Y’i and their antiparticles. The corresponding quantum numbers are the eigenvalues of (3.14) and are given in Table IV for the lepto-quark and diquark bosons of the model. B. The SO(10) Model in the SU(5)
x
U(l), Basis
The SO( 10) model under SU(5) X U(l), provides an attractive alternative to the symmetry breaking patterns of SO(10) which reproduces the interesting features of TABLE Flavor
III
Quantum Number Content for the Fermion Representation and Particle Assignment in the SU(2), x SU(2), X SU(4),
I 3L
I 3R
t
0 0
i 1 2
0 I 2
0 -4
12 f
4em I 3
2
-1
6 1
0
-4
0 0 0
B-L
16 of SO(10) Basis
-I -j
-5-1 1 I 3 1
* 1 0 f 1
Particle
CONSTRUCTION
OF GRAND TABLE
UNIFIED
369
MODELS
IV
Quantum Number Assignment for the Lepto-Quark and Diquark Bosons in SO(I0) A’
I 3L
I 3R
B-L
SU(5). In this basis, the fundamental representation 45 decompose as
spinorial
.-
Particle
4 em
representation
16 and the adjoint
s=$+lJ+i,
(3.17) -
4J=B+1+lJ+lJ.
(3.18)
The matter fields and the generators of the theory are thus labeled by (3.19)
Iv*} = w7 $bT WI, @I =
IQ", Q,,,, Q,, QDL
(3.20)
with indices H, I = 1, 2 ,..., 16
<, q, 8 = 1, 2)..., 45 A, B = 1) 2 ,...) 5 D, E = 1, 2,..., 10
SU(5) indices.
ri,p, v = 1, 2 ,..., 24 1 The SU(5) generators Q” in Eq. (3.20) transform under SU(2), x U(l), x SU(3), according to Eq. (2.6), whereas Q,,, is an SU(5) singlet. The generators Q, which transform as a g of SU(5) decompose under W(2), X U(l), X W(3), as
IQ,1 = IQ', Qri, Q,l and QD = (Q,)? It is important to remark that in SO(lO), the embedded in the spinorial representation 16 which the fields in the 16 anticommute with each other, written as bilinear forms in terms of the 16 fields
(3.21)
5 + B representation of SU(5) is is not reducible. Consequently, all and the generators of SO(l0) are vH.
370
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AND
DE TkRAMOND
The generator of the U(l), sector of SU(5), Q,, is invariant SU(3), rotations and has the form
under SU(2),
and
(3.22)
Note that QY has no bilinear term vtyl, since ly is an SU(5) singlet and QY is an SU(5) generator. The generator of the U(l), sector, Q,,.,, is invariant under SU(5) rotations and hence it is given by (3.23)
Q,, and Q,,, are linear combinations of the flavor diagonal generators Qi and QJ’ of the SU(2), x SU(4), sector of SO(10) Y, = q(r)
where q(,.) is normalized
I3R
= 413,
+ f(B
(3.24)
-L),
- 3(B
(3.25)
-L),
by
[v,wQ,r,l=Lqww,. \/40
(3.26)
We have used the SO(10) normalization condition Tr{Q”Q,} = 26”,. Equation (3.24) follows from the definition of the electric charge, and Eq. (3.25) from the invariance properties of Q,,, under SU(5) rotations as expressed in Eq. (3.23). The identification of the fields in the new basis with the corresponding particle states is given in Table V. The SU(5) representations 5, B and 1 are distinguished by their q(,, assignment. TABLE
V
Identification of the Fermion Fields in the SU(5) X U(l), Basis of SO(10) with the Corresponding Particle States in SO(10) under SU(2), X SU(2), X SU(4), SO( 10) 3 W(5) x U(l),
SO( 10) 2 SU(2), x W(2), x W(4),
q(r) 3 3 -1 -1 -1 -5
Particle
CONSTRUCTION
The generators of the SU(5) since the adjoint representation
OF GRAND
UNIFIED
371
MODELS
sector of SO(lO), QA, are bilinear in I# +@ and did, -24 is found in the tensor products $x5=1+24
and
lJxu=1+24+75.
It is now straightforward to construct and identify the SU(5) using Table V and Eqs. (3.5k(3.7) and (3.11). We obtain
generators,
Eq. (2.6).
(3.27)
(3.28)
Q:=ld’x
[@
(-q);,j+,it
oj+$;i($)i)ri]
(-;:)i
=Q;, I
.i
p= 1, 2,..., 8, Qr&L(
d3x(y/‘+yi
- ~,,i$,t$ - cijk#k+qi5rj) = Qz,
(3.29) (3.30)
6 Qri
=
z
(Qj,’
Q;;
and for the SU(5)
1 = T
(3.3 I )
E’~E~~~Q<;,
singlet
Q,r,= .( d3x&
(3~~ +I//~ - q&h, - St&)
1 = \ i T Q; - 4- f
Q;.‘.
(3.32)
The generators Q,, which close the SO(10) algebra under SU(5) x U(l),, are bilinear in ~‘4~ and 4;~” since the -10 is found in the tensor products 1 x N and Bxj=lJ+@ We find
Qi
=
-
*J
d3X(E,,$JivS + #+w’ + ~~4~) = I,) - 1 fi i 1 4
(Qk’ + iQi*), (Qi’ + iQi”>,
(3.33)
312
DE LA OSSA
AND
DE ThRAMOND
Qi = (Q’)“, Qri=-Lj
(3.34) d3X(E,,Eijk$sfVk + &,,#itWs + Vt#,i)=-+EijkQ::y
(3.35)
\/z Qri
=
= - +
(Q,,)’
Q,=-‘(
G
=
EijkQ;;
+“SQ;;,
d3x(#‘+tyi + t,v+$)= - ’ fi
(3.36)
(Q: - iQ;>,
(3.37) (3.38)
Qm= (Q,)+-
The gauge bosons of SG(10) under W(5) X U(l),. transform according to Eq. (3.20). The new quantum number, q(r), is zero for the bosons of the SU(5) x U(l), sector and 4 (-4) for the bosons transforming as the a (10). The interaction Lagrangian
%=g,oWHV’[vw
QS14
(3.39)
follows from the transformation properties of the fields wH in the spinorial representation 16 under the group generators, Qs, in the SU(2), x SU(2), x SU(4), or SU(5) x U(l),. basis. Although the SO( 10) decomposition in both bases give identical results well above the symmetry breaking scale, they correspond, according to the various symmetry breaking patterns, to different physical theories at low energies [4, 17, 181. For example, the gauge bosons associated to the SU(2), generators could be relatively light in a left-right symmetric model but correspond to superheavy bosons under W(5) x U(l),. However, the neutral boson associated with the U(l), generator, C,, could be as light as 250 to 300 GeV [ 181 leading to an extended electroweak model based on the gauge group SU(2), x U(l), x U(l),.
IV.
THE E, MODEL
In the search for groups of higher symmetry as candidates for unified theories we consider the rank-6 Lie-groups: SU(7), Sp(12), SO( 12), SO(13) and E,. The fermions in SU(7) are placed on a highly reducible representation, and the resulting models have been studied with the purpose of incorporating various families of quarks and leptons [19] which appear in the multiplets of the representation with a host of exotic particles. From the remaining rank-6 Lie-groups, only E, has complex representations and provides an attractive possibility for grand unification [5]. E, contains SO(l0) and SU(5) as subgroups and thus incorporates the salient properties of these models. Since all the exceptional groups G,, F4, E,, E, and E, contain SU(3) as a subgroup, color is naturally embedded in the octonionic structure of the exceptional algebras which are anomaly free. G, contains SU(3), as a maximal
CONSTRUCTION
OF
GRAND
UNIFIED
MODELS
373
subgroup. We need a larger group structure to incorporate a flavor group in order to define the electric charge operator. F, contains an additional SU(3) which could be used as a flavor group, but it is not appropriate to describe the weak interactions of quarks. E, has SU(3), X SU(3), X SU(3), as a maximal subalgebra and we can consider SU(3), X SU(3), as a flavor group. E, is the only exceptional group which admits complex representations and all the fermions are assigned to the lowestdimensional representation, the fundamental 27 [5]. E, also has as maximal subalgebras SU(2) x SU(6) and SO(10) x U(1). A. E, under SU(3), X SU(3), x SU(3), The fundamental representation 21 of E, transforms under SU(3), X SU(3), X SU(3), as - 27 = (593, 1) + (3, 193) + (1, 3331, (4.1) where the leptons are assigned to the (&3, A), which is a color singlet, the quarks - - to the (3, 1,3), a color triplet and SU(3), singlet, and the antiquarks to the (1-9 3-3 3) - *a color antitriplet and SU(3), singlet. The above decomposition is closely related to the octonionic structure of the exceptional observables of Jordan et al. [20] which are exceptional observables in charge space. The matter fields are labeled by IVIF}
=
{W’dr
Vlai,
@‘I*
(4.2)
The adjoint representation 18 transforms as - 1) + (I, 178) + (3? 37 3) + (3,3,3>,
78 = (8, 1, 1) + (I,&
and the corresponding generators QK are identified by their transformation
IQ'1 = {QL”, QR",Q:. QYd,Q:,}-
(4.3) properties (4.4 )
with indices F, G = 1, 2 ,..., 27 K, Tl, 0
=
1,
2,..., 78
a, b, c = 1, 2, 3,
SU(3), indices,
4 e,f = 1, 2,3,
SU(3), indices,
i,j, k = 1, 2,3,
SU(3), indices,
a, p, y = 1, 2 ,..., 8.
The generators Q& Q; and Q: transform as octets under SU(3) (4.5 1
595/155'2
IO
374
DE LA OSSA
AND
DE TiRAMOND
Q;=(d)x[yQ,t(~)d’Woe+ye~+(-~)edWdi], yldi], whereas the QL, transform hence have the form [ 161
as a color antitriplet,
SU(3),
(4.6) (4.7)
and SU(3),
triplets, and
Q6d = 1 d3X(aEWkvdk tvaj + P&,bc vc!i vbd + Y&defl/la>Wei)
(4-g )
and Qs” = (QAd)t. The closure of the Lie algebra of the generators determines the constants a, p and y up to a phase ial* = [PI’= Iy[* = 4, where we have used the E, normalization
condition
Tr{Q”Q=}
= 36”,. We choose
Pll a=-/?=-y=
l/G,
(4.9)
and thus d3X(EUkWdk tWaj - E,bc v!i vbd - EdefWaftWei),
Q;,zi(
with
(4.10)
\/z Tr{Qi,Q;‘}
= 3~5~~8~~~5~‘.
The generators QK satisfy the commutation
(4.11)
relations
[ QK, Qn] = ir”“, Qw,
(4.12)
where Y’(~~ are the E, structure constants. Using the expressions for the E, generators we obtain the commutation relations
[&, Q;l = ($)
bQ6,,
[Qh,, Q;l = ($)
deQ;,, (4.13)
,&,:Q:i
= (-F)i
[t&d, Q’,,] = +
j
&d,
&,bc&de,&iikQ;f,
CONSTRUCTION
The fermion quantum of the matter fields with the electric charge is generators of the SU(3),
375
OF GRAND UNIFIED MODELS
number assignment follows from the commutation relations the diagonal generators of each of the SU(3) subgroups, and expressed as a linear combination of the four diagonal X W(3), flavor group (4.14)
4 em= I,, + I,, t Yt t Y, 1 where Y,,, is normalized
by (4.15)
[wm Q&l= d%,wr
The flavor quantum numbers of the fundamental representation 22 and the corresponding fermionic assignment are given in Table VI. This model contains extra unobserved fermions with unusual transformation properties. The 78 gauge bosons A”, transform as the adjoint representation of E,, and their quantum number assignment follows from the eigenvalues of (4.13). B. E, under SO(10) x U(l), The maximal orthogonal subalgebra decomposition of E, provides a particularly interesting possibility for symmetry breaking along the maximal subgroup chain E, 3 SO(10) x U(l),1
W(5)
(4.16)
x U(l), x U(l),,
TABLE VI Flavor Quantum Number and Fermionic Assignment for the Fundamental Representation 27 of E, under SU(3), x SU(3), x SC43), I 3L
I 3R
-4
i
1 2 1 z
1 2
0 4 1 2 0
f f i
0 0
i -4
0
0
f 1
0
0
0 1 2
0 0 0
0
f 0
y,.
YR
4em
Particle
0
N,,
-1 -1 1 0 0 I 0 0 23 I / I i 3 f f
E,e, El+ -N,, -V, e,’ -v; N”, Uil d,, h,, u;’ d”’I h“’ I
376
DE LA OSSA
AND
DE ThRAMOND
which can be interpreted as an E group series E, + E, -+ E,, with E, isomorphic to SO(I0) and E, isomorphic to SU(5), by removing appropriate circles from the Dynkin diagrams [6, 7, 221. Under SO(lO), the fundamental 21 and the adjoint x representations decompose as 2J=E+i+lo,
(4.17)
B=Q+1+16+16,
(4.18)
and hence the matter fields and the generators are labeled by (4.19)
Iv,} = {u/,?XYY,L {QKl = lQ5, Q,,,, Qm QHL
(4.20)
with indices F, G = 1, 2 ,..., 27
K, z, w = 1, 2 ,..., 78 H, I = 1, 2 ,..., 16
p, 4 = 1, L..., 10
SO( 10) indices.
[, v, e = 1, 2 )...) 45
The known fermions, which are assigned to the spinorial representation 16 of SO( IO), are included in the lowest-dimensional representation of E,, together with an SO(10) singlet and 10 new fermion fields Y,, which transform as the self-conjugate lowest-dimensional representation of SO(lO), lo. Under SU(5), B = 5 + 5 and thus (4.21)
{Y,} = {Y”,QAl.
The transformation properties of the generators of SO(lO), QC, are given by Eq. (3.20). The generator Q,,, transforms as an SO(10) singlet, and the generators QH, which close the E, algebra, transform according to Eq. (3.17),
{QH) = iHA, G,, G,l
(4.22)
{HA} = {H’, Hi},
(4.23)
{GD} = {G’v Gri, G,}*
(4.24)
and QH = (QH)+. Under SU(5)
G, and G, are W(5) and SU(2), x SU(3), singlets, respectively. To identify the fermions, we construct first the generators of the U( l)y
x
U(l),
x
CONSTRUCTION
U(l), by
OF GRAND
UNIFIED
sector. Since QF commutes with the SU(2),
Qy=,f
d3x(ayrty’ +“t-yr+yr
+ byi+@ + + gY’+Y’
MODELS
311
and the color generators, it is given
cgit#i
+f’nfa,
+ d#Ji#,i + e#+# +
(4.25 )
gqq.
This generator has no bilinear terms in ly and x which are SU(5) and SO(10) singlets, respectively. Now, the YA transform as a 51 and a.., as a 5 of SU(5). and therefore a = f = -f ‘, b = g = -g’. The eigenvalues of QY, a, b, c, d and c, are known from their SU(5) assignment. The generator Q,,., commutes with all the SC!/(~) generators and has the form
Q,,,= j
d3x(avA+yl”
+P&,
+ py+y + 6YA+Y4 - &‘$‘,).
(4.26)
We have not included a bilinear term in K since Q,,, is an SO(10) generator. Note that although vA and Y,’ transform both as a 5 of SU(S), they have not necessarily the same eigenvalues for q(,), since Q,,, is not an SU(5) generator. The generator of the U( l), sector of E,, PO,, is invariant under SO(10) rotations, and thus it has the form
Q,,,= I' d3x(c,d Y, + w+x+ c?Y; Y,,, The generators Q,, Q(,.) and Q,,, are linear combinations of the flavor subgroup SU(3), x SU(3), of E, y, =
where q,t, is normalized
I,,
+ Y, + YR 3
(4.27)
of the diagonal generators
(4.28)
9 (r-j = 413, - W’,, + Y,),
(4.29)
q(t) = WY, - YRL
(4.30)
by (4.3 1)
and Tr{Q”Q,} = 3P1,. We give in Table VII the U(l), x r/(l), x U(l), quantum number content of the fermion fields and the identification with the corresponding particle states. The different SO(10) representations are distinguished by their qtr, assignment. In this model the SU(5) singlet w is not necessarily assigned to a righthanded neutrino, but could be paired with the SO(10) singlet x to form a Dirac spinor.
378
DE
LA
OSSA
AND
TABLE
DE
TJiRAMOND
VII
U(l), X U(l), X U(l), Quantum Number Content and Identification of the Fermion Fields in the SO(10) X U(l), Basis of E, with the Corresponding Particle States under SU(3), x SU(3), x SU(3),
EC ~SO(10)
x U(l),
E, = =J(3), x W(3), x W(3),
Y,
v’,
-f
P v”
-;
vri 3 VI 42 3 I3
X Y’ YA
yP %
42
yi
fP
fir
E,,VSI
Oi
V3i
Particle
qw 3
4
3 -1
1f’
-1
1 0
-1 -5
0
0
-f
!3 f
-f
-2
-2
-2
-2 2
-2
2
-2
E-
( -N 1 ,. h” E_”
~
i -NC j L hi,
The generators of the SO( 10) subgroup of E, , Ql, are bilinear in WAvI and Yi Y, since the adjoint representation of SO(lO), 45, is found in the tensor products
and
u xlJ=45,+1+54,, where, under SU(5),
45,=24+1+10,+10,.
-
The generators Qs are thus given by
s
Ys
(4.32)
CONSTRUCTION
OF GRAND
UNIFIED
379
MODELS
(4.33)
(4.34)
d3x(yr+$ - ~,,#,tti - ciik$kt#rj - Yr+Yi + L?,h,) Qri=(Qri)+ =Q;l, (31yA5fA -qg~D-51y+y/-2YA+YA
= Q;, ,
(4.35) (4.36)
+2n;n,> (4.37)
d3x(-&,&yr
QriZ -‘j
d3X(&,,Eijk#Jjvk
+ #+I# + y/+f + t+jkyk+fij)
f E,,#itv’
+ ‘&“#,i + Y’+L’-
=
-Qi3,
(4.38)
Y’+G,)
fi = -E,, Q;‘,
Q,=-Lj d3X($‘+ly’ +ly+$b +E’SY’P2,) =+ (Q:,- iQ;h u/z
(4.39) (4.40)
and QD = (Q,)+. The SO(10) singlet, Q,,,, is written as (4.4 1)
The generators that close the E, algebra, QH, are bilinear in x’y, 16 is found in the tensor products 1 X 16 and
and WA YP since the
380
DE LA OSSA
AND
DE TkRAMOND
We obtain
d3x@,tiini + erS#+SZs+ I/&’
d3x(ciikqbk+i2, + ~r’$, d3x(cijkiyk+.0j d’x(i/+LJ,
(4.42)
- X+W’) =
- ty+Y’ -x+#)
=
- $+Y’ - E,.,&~ Y’ +x+&)
+ y’+.n, - q.,fii+Ys
Qil,
= -Q;,,
t E,.,E~~~#~~Yk -x+&)
Qf3,
= -E,,
(4.45)
I d3x(Pyr+12s G,=-lj
d3x(#+yi !b
(4.44)
+ #i+Yi - ~‘4) = G
- y/‘+y’ + x+y) = - ’ fi
(Qi - iQh (Qi - iQh
(4.46) (4.47)
and Q” = (Q,)+. The 78 gauge bosons of E, transform according to Eq. (4.20) and the quantum number qct, is zero for the bosons associated with the SO(10) X U(l), sector and -3 (3) for the bosons transforming as the 16 (16). The E, Lagrangian follows from
Since the representation of the new fermion fields Y, is real with respect to SU(2), X SU(3),, they are supposed to acquire superheavy masses and decouple from the theory according to Georgi’s survival hypothesis [23]. If the generators associated with the U(l), and U(l), Abelian sectors remain unbroken at the SO(10) and SU(5) symmetry breaking scales, we recover at low energies (if the associated gauge bosons are relatively light) an extended electroweak model based on the gauge group SU(2), x U(l), x U(l), x U(l), [24]. As was mentioned in Section III, the 27-dimensional reducible representation of SU(6) provides an interesting decomposition of the E, model, when it is embedded in the 27 fundamental irreducible representation of E,. Under its maximal subalgebra SU(2) x SU(6), E, has the branching rule - 22 = (236) + (1, Is). We do not give here a detailed discussion along this line, but the model can be built
CONSTRUCTION
OF GRAND UNIFIED MODELS
381
without difficulty with the construction procedure described in this paper. Different symmetry breaking patterns of E, are found in Refs. [ 7 ] and [ 8 1, and references cited therein.
V. CONCLUDING
REMARKS
We have developed in this paper a simple construction procedure of grand unified models of the strong, weak and electromagnetic interactions exploiting the transformation properties of the matter fields and group generators under the different interaction subgroups, using a maximal subalgebra decomposition and the closure of the algebra to obtain a realization of the group generators. We have given an explicit construction for the models based on unitary, orthogonal and exceptional groups of rank 4, 5 and 6, namely, SU(5), SO(l0) and E,, which are of particular physical interest and simplicity, but the method could be used as well with other groups. We have employed the same construction procedure for different Lie groups, avoiding specific algebraic structures such as Clifford algebras, octonions or bases related to the weights of the representations [6, 7, 251, by making an extensive use of the representations of the unitary groups which are more familiar to many physicists. In SU(5), the generators are constructed as bilinear forms of the fermion fields using only the defining representation 3. For the SO(10) model, the generators are built in terms of the irreducible spinorial representation 16, and for E, in terms of the fundamental 2. The algebraic construction for the SO(10) and E, models is simplified considerably by using the left-right symmetric bases SU(2)L x SU(2), x SU(4), and SU(3), X sU(3), X SU(3),, respectively. The generators in the chiral bases SU(5) x U(l), and SO(10) x U(l), are written as linear combinations of the generators in the L-R symmetric basis. The quantum number content of a given representation, the identification of the matter fields with quarks and leptons and the specific form of the interaction Lagrangian for the various models considered follows from the commutation relations of the fields with the group generators. We have considered only groups which are complex with respect to SU(2), x U(l), x SU(3), and real with respect to SU(3), [26]. This restriction poses severe limitations on the incorporation of a family structure. E,, for example, contains three E, families but has only real representations. The spinor representations of the orthogonal series SO{ 10 + 4n), which could otherwise account for the repetitive family structure [27], are real with respect to SO(10). These models introduce mirror V + A families of heavy fermions. Unitary groups of higher rank SU(7), SU(S), SU(9),... have been considered for family unification [8], but the proliferation of fermions with the rank of the group is enormous. An interesting example due to Georgi [23] in which all the irreducible representations appear only once in the fermionic representation is based on the embedding of SU(5) in SU(11). The fermions are assigned to the 561-dimensional reducible representation 11 +5J + 165 + 330. The model contains three 3 + s chiral families which survive at low
382
DE LA OSSA
AND
DE TliRAMOND
energies! This model, however, is not asymptotically free. An understanding of the family structure as well as the fermion masses and mixing angles remains a challenging problem.
ACKNOWLEDGMENTS
The authors thank the Escuela de Fisica at the Universidad de Costa Rica where most of this work was done. We also acknowledge useful conversations with A. Yildiz and P. Frampton. This research is supported in part by the National Science Foundation under Grant PHY-82-15249.
REFERENCES
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CONSTRUCTION
OF GRAND
UNIFIED
MODELS
383
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