' *V E » 'N IC/T6/123

INTERNATIONAL CENTRE FOR

THEORETICAL PHYSICS

IMPROVED PATI-SALAM MODELS

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

Amir Schorr

1976 MIRAWIARE-TRIESTE

* * » • • • — jft*

IC/T6/123

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOE THEORETICAL PHYSICS

IMPROVED PATI-SALAM MODELS *

Amir Schorr

International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

Hew models whicn are closely related to the basic Pati-Salam model

are presented. They contain 36 or ^0 scalar fields compared with the °6

scalars of the basic model and are easier to handle. The vacuum symmetry in

each model is found and predicts at most one GM (Goldstone meson)

compared vlth the eighteen PGM of the basic model. The "colour brightening"

phenomenon hy the leptons,in neutral and charged scattering,ia independent.

The decay mechanisms of the different colour quarts are also independent and

some of the quarks are stable in some of the models (see table).

MIHAMARE - TRIESTE

December 1976

• To be submitted for publication.

I. INTRODUCTION

A few years ago Pati and Salam suggested a unified gauge theory

including strong, e.m. and weak interactions. Their theory is based on the

local group G L = SU(2)l e f t colour

vhere lepton number

is the fourth colour. In order to preserve approximate global SU(3) symmetry

for the vacuum, each scalar field multiplet they choose (A,B,C) containsT P ^

several Identical representations of G . The scalar potential Tr aswell as the Yukawa potential are then taken to be oP = SU(U)r x SU(U)_ x SU{k).

L Tt L

global invariant. In spite of this simplification i t is quite hard to find

the minimum of the scalar potential T in this theory. Since the scalar

mass matrix is complicated, Pati and Salam showed that a "legitimate" extremum

exists for their potential, but they did not prove that their solution actually

corresponds to a minimum point of the Higgs potential. The situation was

changed partially by Ref.2,which predicted the legitimate minimum points which

can be driven from a class of Higgs potentials invariant under the group

This class of potentials,which is denoted

k Higgs scalar representations

of the group G; A ' ^ U 1 I j , i j t l 1 l) (i = l , . . . , k ) . The V°

(i-1) tk-i-1)potential in such a theory is given by:

G = U(N ) x U(HO) x ... x U(IT ).0 x E

by V , is relevant in theories which contain

V ='f1A'

This potential V is the most general renormalizable and invariant potential of

the scalars if k and each N are greater than four. In a case in which the

Higgs potential is given by V , one can predict, using Ref.2, the direction

of the vacuum symmetry. Moreover, by restricting the coupling constants of

V to some domain, one can restrict the vacuum symmetry to be in any

legitimate direction according to the procedure of Ref.2.

In the basic Pati-Salara model the potential Is of the V° type.

Hence, i t was possible to show in Eef-3 that the vacuum symmetry direction

of their model is legitimate* Moreover in Sec.II we prove that the

-2-

potential v t vhich is the general Invariant potential under G (plus one

discrete symmetry A -+ -A ) of the (A } multiplets, can be handled by the

procedure of Ref.2. Hence the same kind of vacuum symmetries arise from V

and ¥ . But it seems to us that the solution of Ref .1 is not a satisfactory

one, mainly because it contains many Higgs scalars (96), of which 18 are pseudo-

Goldstone mesons (PGM). The PGM are the imaginary part of A. , ft.

(i,J » 1,2,3); they are imposed by the enlargement of G to G in r ,

and cannot acquire mass in the tree approximation. These PGM produce AQ f AS

transitions (K •• IT eu, etc.) as well *is other unlikely amplitudes, nance in

order to aTOid those phenomena, some of the PCJM should acquire heavy masses

by radiative corrections. But as the PGM are not natural, hut result from

the invariance of v under G their radiative masses are subject to

renormalization. Hence, one must find all counter-terms in a way that the

minimum point remains almost that of V , and then find the masses of the PGM.

Pati and Salam have argued that on account of the coupling of their Higgs

mesons to W particles (of inass £ 1 0 GeV), with a coupling strength ^ ct,

whatever the complexion of the counter-terms, the finite parts of the masses

of these PGM (arising radiatively) vill be at least of the order of «aMX

53 100 GeV, which is a safe mass for the unwanted processes. However,the•0

radiative corrections might have sin approximate degenerate space in the

region surrounding the minimum point of V • Hence their qualitative argument

would have to be supplemented with a detailed quantitative examination to show

that all the PGM masses can be taken to be as large as the experiments imply.

This is a large undertaking - in view of the large number of terms in the

potential. Other difficulties arise from the PGM, as well as from the presence

of the large number of scalars in the basic F-S model. Among these are the

phenomenologieal analyses of the scalar amplitudes and of the short distance

asymptotic behaviours in the model.' Because of all these difficulties it

seems to ue that, it is worthwhile to find a compact framework for the P-S

unified theory with a minimal number of scalar fields.In this paper we present some models which are closely related to the

fields Lbasic one, but contain fewer scalarsj[3o or kO, and fewer PGM: one or zero. The

local group underlying all these models is G L = SU(2)L * SU(2)R x SU(M C,

the same as in the basic model; but the Higgs scalar field content is different

from this model. (All the models and their characteristic predictions are

shown in the table at the end of this paper.) In Sec.Ill we present model (i)

which contains only three representations of G :

- 3 -

A1 A / ( 2 , 2 , 1 ) , A2 , j ( l , 2 , U ) , A3 « (2 ,1 ,10 . I t i s free of PGM, has a baryon

conservation lav and zero Cabibbo angles . For future purposes we prove t h a t a

general po t en t i a l V gives the sfvme minimum spaces as V .

In Sec.IV,models I I , H and Tr are presented . These models contain

the same mul t ip le t s {A } of model ( I ) , except t ha t the  - i s taken t o be a

complex multipLet ( i t contains two (2 ,2 ,1) representa t ions of G ) . The

sca lar p o t e n t i a l of model I I (V I ) i s Forced to be a GG = U(2), x U(2l x

h i\

invariant. Tlw procedure of Hef.2 is proved to apply for V , and we find

three proper solutions for the vacuvsm symmetry; II , II , I I 'The II

and II solutions .-inch contain a single GM .whereas II ' contains none. In

order to givs these JM3 masses, the N , ff models are presented. These new

models are identical to model3 IIa, IT*", respectively, but their potential V ,

•f contain an additional %eim which breaks G properly. The procedure of

ICflef.2 is relevant ELISO for and we find that the GMs of the IIa, IIC

models acquire masses in the tree approximation level of the N , M models.

In the II and N models the blue quarks are the only ones which can

decay to leptons. This seems to raise difficult questions in the context of

"partial confinement" theory. Hence we present the Tr models,which are

identical to the others, but their potential contains the Re{6 Tr(A A A ))

term. As a result more quarks can decay to leptons in all models (see Table I)

and all of them in model N a • The resulting models have only one local

symmetry and hence contain 16 or 20 scalar particles Of 6 In the P-S model).

The number of PGM is 0 (18 in the P-S model) and the number of relevant

coupling constant (see Sec,IV) is around IS (more in the P-S model). There-

fore a rigorous analysis of the scalar particle spectrum ' and scalar induced

interactions can be made. There are two or three neutral (A ) particles

of which one might be a. light one (few GeV)) and one charged pair {A )

They are the only ones which couple directly to the fermions and their

coupling is proportional to the fermion mass matrix. Hence we expect to find

scalar amplitudes in the (vB) scattering process, mainly in the neutral ones ,

In Sec.V the vector mass matrix is given. The non-diagonRl elements

predicts two interesting phenomena. The first is the "colour brightening" of

gluons in leptonic scattering. It is similar to that in the P-S model in9)neutral amplitudes , but has independent amplitudes which vanish in

some models, in charge processes.The second is the appearance of lepton + quark

•+• 2-lepton transitions and quark decay phenomena. In each model different

-It-

colour q̂ uarks can decay. Unlike the F—S model, these amplitudes are in-

dependent, and some of them vanish in some models. These phenomena, due to

symmetry breaking, should diminish at high energies { > m,r;rO , but may appear

in current energies (5-3QO GeV). The second phenomenon will ~oe reviewed and

related to some of the two- and three-lepton events in (vH) and (e+,e~) inter-8)

actions . At the end of the paper we tabulate all the "improved" models.

II. VACUUM SYMMETRY IN IKE BASIC P-S MODEL

The basic Fati-Salam modelx SU(2)

is invariant under the local group,R •• LJUI-TJJ, . The potential of the Higgs scalars and the

Yukawa interaction are invariant under a larger global symmetry:GG = SU(U)L x su(MR * SU(10c , plus one discrete symmetry; A

3-*- -A3 . Thedistinction between the local ana the global groups arises from the hope toinclude approximate SU(3)C ana SU(3)L,„ symmetries in the theory, but theabsence of eight weak SU(3)T currents and the anomaly problem prevents onefrom enlarging G to contain larger groups. Pati and Salam try to solvethis difficulty by taking the Higgs and Yukawa potentials to be 3 globalinvariant. This basic model contains three complex multiplets of scalarsA , A , A (A,B,C, in the notation of Ref.l), which transform under GG as(lt,U,l), (l,lt ,10, (U,1,M correspondingly. In addition to these scalarsand to the vector bosons there are two fermion multiplets which contain thetvelve coloured quarks of QCD and four, leptons. These multiplets, IJL. a (It ,1 Ji) andty ** (l,U,lt), are given by-(electron and muon might exchange places)

(2.1)

Our interest is in the direction of the symmetry breaking. Hence we shall

concentrate on the full Higgs potential V6 of the "basic" model,which is

sum of the potential v given in Hef.1 ana V̂ ,

-5-

In their original paper, Pati ana Salam motivate by considering the

extremum of their potential that the symmetry breaking of G is fixed by the

YEV of the matrices , which are given by

(2.3)

In Refs.2 and 3 i t was proven that these VEV of the multiplets A can resultP ^

from a p o t e n t i a l which has the form IT . The main goal of t h i s sec t ion is

to prove t h a t the so lu t ion given in (2-3) i s a l eg i t ima te one, i n the frame of V ,

and tha t the IT5 term can change (2.3)only into another l eg i t ima te so lu t ionof V°

In order to show this we must prove the following three points:2)

I ) The s ix matrices D = ( i AA ^ can be

( i 1,2,3) and for each

£ G^, ,*.-, ( f°r a l l

diagonalized simultaneously by an action of the global group G

II) The diagonalized G1, D1

to an"ordered" form; which means D

i either G ^ ^1+1 1+1 (for a ^

J) .

I l l ) All "equivalent elements" of D1 are equal.

The definition of "equivalent elements" is given precisely in Ref.2 and can beexplained roughly in the following way. Writing the potential TT as afunction of a l l the non-vanishing elements in the diagonals of the matrices D ,i t can be separated into^ sum of Bub-functions, where each variable appearsin one sub-function only (from paint I I i t follows that there is one-to-onecorrespondence between D1 and G. elements, hence i t is sufficient to use

the D1 elements only). Two variables which belong to the same diagonal D .J * J

and appear in two separate t>ut identical sub-functions are defined as

"equivalent elements".

-6-

la order to prove tfie3e three points in the frame of (v t V )

ye fol-iov Hat.2, where a l l of then vers prnvcu in the frame of Y . First

Ma transform tlie t ' matrices to diagonal and ordered forms. Then,

entiatlnft the potential 'V with respect to (A ) and aultipiin.^ i t

A1+ , we get,-at the minimum point of VB , the following three equalities:

> -

'the terms C\—— V A > contribute diagonal matrices according toUA ' i

C2.5)

Hence, one can use the procedure of Ref.2 to get the following:

= o (2.6)

a ,J

In t h i s frame we find tha t

The e q u a l i t i e s (2.6) allow the diagonal iza t ion of a l l the matrices G1, D1

s imult aneously.

At t h i s s tage i t I s s t ra ightforward to show tha t the matrices can besimultaneously G % A

transformed / by G e i t h e r in to diagonal form, = 5 a , or_ P R»J K J

ant i -d iagonal form, J

Salam kind of solut ion.)

Hence the minima of the function (IT + IT ) (d ) stay at the same

kind of points as for v and a l l equivalent variables have the same absolute

value at the minimum. Thus we have shown that the three points ( l - I I l ) hold

for the potent ial {T~ + V ) and hence the VEV given in (2.3) Is a legit imate

one in th i s form. I t should be noted that in t h i s case,where 0 » < A? ,/> •s 2 v 2 ^ J)ET '

= < Ag g ' ~ *̂ *_ - , ' t ^he potent ia l V does not give r i se to any newsolutions for the VEV of < A1> and .

The potential IT predicts an approximate global*»ynmsetry SU(3)

which one could r e l a t e to colour and flavour symmetries. But as a consequence

of the i n i t i a l G symmetry of IT there are 9+9 pseudo-Coldstone mesons;

the imaginary part of A. and  , ( i , j = l ,2,3l .which results from the breaking

of eighteen global (non-local) symmetries of v . These PGM Induce1 a l l kinds

of SU(3) . and SU(3)_. scalar interact ions which are not seen incolour flavour

experiment. The most embaraasing ones are the AS ^ AQ semileptonic amplitudes

such as K -* if ejA . I t should be noted that the masses of the A ^ PGM

might r i se by the radiat ive corrections to the order of

" %

a r e t n e vector couplings,R/L

the masses of the right/left

vector masses), according to the specific structure of the higher order terms

in the neighbourhood/ the minimum point of V . But it is quite a difficult

•S-

task to get these masses and keep the minimum point of the whole-n

effective potential close to the minimum point of V . (The problem

of whether such a procedure can be carried out consistently in the P-S unified

theory is beyond the scope of this paper. Some remarks on the difficulties

in such high hierarchy problems can be found in Hef.10.

Taking into account all this and the problems which are presented

by the great number of scalars themselves (96) in the basic model, we vouia

like to find a minimal ana less complicated frame for the P-S theory.

III. KtHIMAL MODEL

In order to avoid the difficulties of the 'basic model, we consider

in the following two different sets of models. In both of them we take the

local group GL to be SU(2)T x SU.(2),j x SU{lt)^ , but in the first model (I)I* . n L

there are only three multiplets A (i = 1,2,3), whereas in the second kind

of models (II,N,Tr) there is an additional A 1 multiplet.

In this section we shall consider model I. The Higgs scalars A

(i = 1,2,3) transform under G as follows:

A1 ~ (1.2..4) Uk 02

(3.1)

1 2A i s a four-dimensional real multiplet and A

A are eight-dimensIonal

are the matrices of

where

complex multiplets. The matrices U., U and U-

SU(2)_ , SU(2)_ and SU(1O_ . The invariant Higgs potential in this modelT 1

is V given by (3.2). We have not chosen to use the hermiticity of A to

simplily the potential. This will enable us to use the result of this

section for the second kind of model and for further analysis of more complicated

models in which A is not hermitian,

-9-

V1

= \T(3.2)

(The potential V is given by (2.1) and has the same functional dependence

on the { A \ multiplets as in the P-S model.) As in the previous section,

our first goal is to prove that the additional terms which appear in the•"•il 0

potential V , other than V , do not change the set of legitimate directions

of the vacuum symmetry which is given in Eef.2. This is done again by

proving the three points (1-3) given in Sec.II which enable us to follow••I

the procedure of Eef.2. The main problem in V is that the aultiplets2 3

A ,A are not represented by square matrices, thus the treatment of the "det"

terms is q.uite difficult.

In order to prove point I, one must show that the terms

VD«T,IT\ ; +

/ • * - •*.

are diagonal , but as the terms det (A2 A2 +) , det ( A3"*"A3 ) and det{A2A3) cannot

be expressed as (det A2 • det A2^) ,4iet A 3 t • det A3) and ( det A 2 .det A3)

correspondingly, i t i s impossible t o prove i t with the help of (2.5).

Therefore we have t o r e ly on the "ordered" form of the diagonal of D ,

which implies that D,3,3

hence the last two columns of A'3t

are Identically zero as shown in the (tollowing:

13-3)

Denoting the square matrix constructed from the f irst two columns of A by

A3 '* , the relevant part of det(A3+A3) 'becomes aet (A3 ' ' ' 'A3 ') = det A3 det A3 ' .

Moreover, denoting the matrix constructed from the f i rs t two columns of A

by A , we can take the term (det A det A3 ) instead of (det A T ) (as the

procedure of Eef.2 is based on taking the terms vith non-vanishing VEV only).

In this way we can use the equality (2.5) for the above two terms, as they

were constructed from square matrices.

For the third problamatic term,det(A A^"),ve make the following2 2

analysis: D is gauged to be diagonal, hence the rowsof are orthogonal.I t i s then straightforward to prove the (3.It) equality for

Therefore it is quite simple to show that these are only two distinct

is*2 "

possibilities. In the first _^det A ^ 4s« zero, hence the order of G

Is fixed by the original potential V ;

.2 '0 and

< >0. The last twoseeond easa < d e t A ^ £ 0 and G. > •= G „

2 "^ 1 1columns of A become irrelevant to our problem and the order of G , G^

0 J-jl •=)0i s f ixed by the o r i g i n a l p o t e n t i a l V . The choice between the two casesi s contingent on the sign of y% and on the r a t i o s of ^ / v y ly •

- 1 1 -

Tlie last point (3) is proved for v in the same way as for V^ .

As all the three points are proved,we could use the results of Ref.2

and list all the legitimate directions of the vacuum symmetry for model (I).

Instead we shall choose those directions which interest us, motivated by some

physical results. The first and most important constraint is as follows.

1) There are no free maseless gluons.

In the light of this condition ve examine the tvo discrete possibilities for

S . First we write the matrices

The second set of results which we should like tc olotain i s :

2) The VSV of AT .If 5i s of the order of 10 -1CT GeV, the

2order of a i s 3-10 GeV and V °2>

are of the order of a few GeV.

These requirements are cons is ten t with, l e f t vector masses of order " kO GeV;gluon masses of a few GeV, and s ix vector bosons X^ with masses of the order

k 5 ; i "10 -10 GeV. (The X^ vector bosons must acquire such a huge mass "because theyintroduce AS = AQ amplitudes ( e . g . \\l~ •* ae ) . )

The s t rongest requirement in t h i s se t of conditions i s tha t b | >> |bg[

which forbids equa l i ty between |'h1l and | b g | . But in model ( I ) A i s

r e a l and i s equal t o , hence there i s no other way to impose

inequa l i ty between |b I and | b . | except by l e t t i n g To,-, be zero. For1 2

the same reasons the equa l i ty between and imposes tha t c 1i s e i t h e r zero or equal t o c g . But i f

is left massless, hence c. should be equal to c

multiplets is given by:

vanishes,a set of four gluons

The VEV of the A1

(3 .8 )

The original symmetry of V is

The overall symmetry of the vacuum is U(l)

x S U ( 2 ) R x S U ( U ) C

u ( i ) RE+Lj

where only one combination of them,U(l) is local and the global

symmetries remain unbroken. Therefore there is no degenerate parameter in

the solution and no degeneracy in the minimum space of v , and hence there

are no pseudo-Goldstone mesonsin this frame.

In order to conclude the discussion on this model,we would like to

present one set of parameters for v which will result in the above situation.

The easiest way to do i t is by fixing the strength of the breaking

= independent of all other conditions. This is

done by taking the couplings U,,a, , which define the strength of the inter-J i»Z

actions Tr(A A )*Tr(A^At'"''), to be much greater than all the other couplings

appearing la r . In this way the vector

is given approximately by

-13-

r = •=/*(3.9)

5 9 2The vector is " ( lO ,10 >10) GeV in this model, and the vector 'bosons will

get their appropriate mass i f the parameters a. u. satisfy (3 .9) . The

other coupling constants fix the appropriate direction of the breaking in

the following way: Ot, should be positive and \^ should be negative, so c«3 3

does not vanish; a should be negative and Y should be positive,2'

so the matrix O

2. v ' 1 '(3.10)

The clearest picture of the scalar field vector is in the U gauge, hence we

remove all the unphysical scalar fields by gauging the A* multiplet to the

form givenin the following:

0

o

(3.11)

where A , B, and C ; C are real fields and the others complex. The mass

matrix in this gauge is given by

(3.12)

where the matrix D is

[D] ̂Lea,

-15-

Choosing the non-diagonal coupling constants appearing in tjie matrix D

to be of order e , and the diagonal ones of order one (the ii parameters

are then chosen according to (3-9)) the three eigenvalues of D are approximatelyP ? 2 1

l6a a ., , Vb (a - «„) , ''c Ca + T-a,) and their corresponding eigen-

vectors are:

B" = 0^

r-1

c° =

There are some unwanted results vhich are unavoidable in the frame of model I.3)

which resultsThe most crucial one is the vanishing of the CaMbbo angle

from the henniticity of A • The second is the conservation of lepton and

This results from the vanishing of b as thebaryon number separately,

transition amplitudes for baryons decaying to leptons is proportional to the

product of b^ and b g (b b ). The conservation of charm and strangeness

in model I makes it unrealistic. But the baryon conservation law misses the10)

spirit of the P-S type of unified theory. It seems that much more natural

models can be represented if the barjran number is constant. (The baryon con-

servation law can be broken explicitly in the potential itself by the term

Re{det(A2A3)}. See remark in Sec.V.B.

IV. IMPROVED MODELS: II ,N AND Tr

A. Model II

Considering the drawbacks of model I we should like to present a new

frame for the Pati-Salam unified theory, that is, enlarging the number of the

scalar particles of model I by adding a new multiplet which transforms like

(2,¥,1)under the local group SU(2)T x SU(2)D x SU(U)_ . The first model ofL n u

this kind is model II. It is invariant under the same local group G and

the discrete symmetry A —••-A as model I. It contains the same A

multiplets of model I, but A is taken to be a complex multiplet. The trans-

formation law under G of the A scalar fields is the same as in model I,

(3.1).

-16-

The invariant potential V of model II is quite eomplicated^ence in

order to make i t possible to find the directions of the vacuum symmetry, we

impose another U(l) global symmetry on V" .

equal on A and on A .) In this frame the potential V±± is given by

(The u(l) charge is

"1"

v11 =

(Hote that by imposing U(l) symmetry, the potential V bscomes invariant

under two other global "accidental" symmetries U(l)1"3 * U(l) ,

where the U(l) charge is one for A and zero for a l l other multiplets.)

As in the previous two sections, we must prove that the predictions ofRef.2 are valid for the new potential v . But since V is contained inV and i t has already been proved in the previous section that these resultsare valid, there i s no need to prove i t again for V , and we can go on tochoose which directions of breaking we want to consider.

Many new legitimate solutions which were forbidden in model I exist

in model I I , since A is a complex multiplet and hence | i - * —»

CU.3)

In addition to C*.3), each one of the models II , I I * , II has another set

of conditions

for model II

' : (U.Ua) for model I I a , (lt.l*b) for model I I a ' C and

0f*° (fc.Ua)

O < D ».Ub)

cy ' (l..l»c)

Conditions C+.3) and and their consequences are the only differences

between the above three models. In order to get the great hierarchy distinction

of the P-S model ( < p 2 > / < p 1 > 2 1Ok , < P 1 > / < P 3 > ^ 10

1*} the. couplings must

have different orders of magnitude- Hence it is far from clear whether these

solutions exist, considering the radiative corrections . One should try

to discover if coupling constants in thia domain, perhaps subject to eigen-

value conditions, can predict asymptotic free theory

Tile vacuum symmetries of models II , II * and IIC differ from eachEM

other. All three are invariant under only one local symmetry, the B(l)

which corresponds to the massless photon- Anyhow,each of the three

spaces is invariant under different sets of global symmetries.

-18-

The potential V • is invariant under three pure global transformations:

U(l) , U(l) , U(l) . (The charge of U(l)1 is the number of the A1 Higgs

particles.) We vould like to find the conserved ones. Observing that at any0 2 "3

minimum space which results from the V type potential, the matrix: ^A ?' -A :

are invariant under the same local group G and the

-19-

v(4.5)

(A

and A

A =

is a h x h matrix constructed with A making up the first two rows,

* the last two. T is the usual Pauli matrix. The aultiplet

A T2 , like A , is a (2,2",l) representation of G .)

The potentials break the three-parameter U(l)1 x U(l) 2 x U(l)3

TT ^ ^ rtglobal symmetry of V to two-parameter symmetries U(l)3 x U(l) and

1 ?+ ^U{l) x U(l) , but they do not contain al l the interactions invariant underthese symmetries. As a result of the over-simple form of V̂ we can

&,c

prove, as in the basic and (I) and (II) models, that the vacuum symmetry's

directions in the H models are only those which can be predicted by the

potential V . The proof is quite similar to that given earlier and hence is

not given here.

We find that the two legitimate solutions of model II which are given

by (4.2) with the conditions (It.3), (k.he.) and (h.ltc) also exist in the newmodels.

In order to obtain the I I C so lu t ion (h.2) v i t h £e = 0 ; a f 0;

f Oj , we use the IT p o t e n t i a l and the same condi t ion of the I I C models

3) and (Itlfc)) but d i s replaced by (X + | | 6 | The t h e r y of t h((4.3) and (It.'fc)), but A^ is replaced by (X + |-|fi| • The theory of the

V potential and the conditions(4.3) and (4.6) will be denoted Tjy NC . The

resulting vacuum symmetry of the Hc model is that of the IIC model,

>o' X

$

model II are broken explicitly in the N , IT models, respectively.

In this way the PGM of model I I C acquires a mass: k\E (|a | + j a j )

in the tree approximation of the Nc version of the (c = 0) solution. The true

GM of the model IIa acquires a mass (-v |C | ( jb *b | + |c e |) i jaj is assumed

to be very small in the tree approximation of the N a version of the (a = 0)

solution. It should be noted that thea/, potentials break explicitlya/ c i

only the symmetries which were already broken spontaneously in the II c

models. Hence the same global symmetries are conserved in the IIa, II a

models as in the Nc , II0 models.

C. The Tr models

8)In the II and H models the Cabibbo angles do not vanish , and thusthe first problem of the minimal model I is solved. The second problem of

model I — conservation of the quark number- is solved only partially in models

II,H as the blue quarks c a n decay to leptons, but the conservation of theR+V

global charge Q . does not allow red and yellow quarks to decay. In a8)

following paper some of the phenomenological consequences followed by

the conservation of the Q global charges will be discussed. In this

section we present new models in which other coloured quarks can decay to

leptons. These new models,which will be referred to as the Tr models, are

similar to the I, II and IT models, but their Higgs potentials include trilinear

interaction terms.

There is only one G invariant trilinear term in our models:

A 3)} . We did not include it in V°, (l. l) as it is quite

complicated to handle and it appeaxs In theories vhere G is a product of

three groups only. In the I, II and N models we eliminated this term by

imposing the discrete symmetry A —• -A

However, such a term changes the vacuum symmetry and we shall add it

to the potentials of these models. Consequently, some of the conserved global

charges can be broken without breaking the one local symmetry which

corresponds to the masBless photon. That is, if the new coupling constant 8

1B bound to a proper domain.

We therefore built four Tr models by adding the interaction term

Ke{0Tr(A1A2A3)i to the potentials of apdels I, R B, II*'C and HC . The

new Tr models vill be denoted by I, & .II ' > N .corresponding to their

respective "original" models listed above. The coupling constant of the Tr

-21-

models are defined to be in the same domains as their corresponding

"original"models.(There is one exception in model ¥ a , where some of the

"original"conditions become Bore restrictive.)

Our aim, in the following, will be to find out the minimum point of

the new Tr potentials. The techniques in all these models are quite similar,

hence we shall concentrate on two models: I and W

The Higgs potential in the "Tr minimal model" is

fU •[ * A *) I

V is given by (3.10). (A1 in model I is four-real-multiplet.)

We simultaneously diagonalized the D , G1 and G matrices by GL

transformation. As A is the (2,2") representation of SU(2)T X SU(2) G1

•> L R

and Er are proportional to the identity matrix. Therefore, we can also diagonal-

ize the D and G matrices, and the matrices can be rotated by G

to the form given by (I4.7). In this, form the rows of the lower half of

are perpendicular

** 0 \ ' 00

>= 0°\

\ =0 . (M)

The coupling constant a is negative, hence b i s zero,and therefore another

SU{U)c transformation can rotate such that c, and e,_ vanish in (U.7).

As the two rows of < A )> are diagonal, thus c i s zero.

6

The domain for 9 in the I model,given by (^ .8) , is chosen in order

to forbid c^ and c to vanish (the new term is linear in c ) .

1 £The VEV of the A , A multiplets has the same structure as in the I model,

where is given by the following:

-22-

«.9)

The non-vanishing of c i s the desired result of the minimal Tr model I . I t

implies the breaking of the "red plus yellow" and leptonic global charges.

Therefore, the red and yellow quarks can decay to leptons. The blue global

charge is conserved.

Proceeding to find the minimum point in the " model we diagonalize

the matrix. As O.1 is negative, is given by (It.10). The

matrix has the same structure as in the I I a model. At the second

stage we rotate by SU(4)c the , matrices to the structure

given by {4.10):

joo \0 O0 b.

f .10)

By using the procedure of Ref.2 i t is straightforward to show that c, = ct- = 0.

In this situation the value of the potential can only diminish if

R+Y+L In the I model the global charges

as a conserved charge.

L, Q, are broken and leave the

The Q is the new "broken charge in the }\ model. The global

blue and leptonic charges £T 8*

are already broken inthei? original model !J shence all q.uarks can decay in the H model to leptons, leaving the fermion

Q H + Y + B + L = qF invariant. In the Tla'c and Hc models the globalnumber charge Q

red charge Q and Q are "broken to Q . The yellow global charge is

conserved and the yellow quarks alone cannot decay to leptons or to other

coloured quarks.

It should be noted that in all the Tr models c can be taken to be

of order 10 GeV. In that way the theory becomes more natural and the

quarks more unstable. In that case 8 should be bound from below (the

Inequality (k.&) becomes "almost" an equality), in order to get 10 GeV2 >,2 2

V. CONSEQUENCES OF THE SYMMETRY BREAKING

A. The vector mass matrices

Using the convention of Hef.l we denote the left/right and colour

vector couplings by g. , fe,, f, respectively. The left/right vectors are0 1)

designated by W , and the SUCO C vectors by V^, X ^ E . The mass term

of these vector basons is given by (5.2). It is expressed by t^e vector'

couplings and the VEV of the scalar fields given in (5.1),

o a

(5.1)

(1-10) 102 GeV 104 GeV.

-25-

*>) ( w*3

i (̂ ^

(5.2)-26-

There are two most interesting phenomena which resul* from the non-diagonal

elements of the mass matrix. The first is the "WX" and "ViC" nixing which

generates the decays of quarks to leptons. The second is the " W " fixing

which generates the production of gluons by leptons (colour "brightening). We

shall consider the second phenomena first.

Beginning with the neutral'sector, we observe that the neutral vector

(W?, W:?, V3 , V S , S°, X°,~X°) mass matrix has in all the original modelsL n

(c, = 0) the same basic features as in the basic P-S model. [The neutral

gluons v «, vp* do n°t m i x with the others in all models.) The five

vectors (W , W , V , V , S ) form in these models five mixture states, amongL D.

them is the one massless photon state.

The VC, W,, are connected to e.11L E all

three gluon vectors, hence the

leptons can scatter to^these gluons. As a result. colour should brighten

in medium energies ((fc ) to (ea ) GeV) in electron-tiositron annihilation1 1

as well as in neutrino elastic scattering. 'The qualitative features of these

phenomena are similar to those predicted by the basic P-S model and are given

in Ref. 9. The fine structure is complicated and depends on the details

of the mass matrix.

The pure X , X vector bosons are eigenstates of the mass matrix in

the original models. This is not the case in the Tr models (c f 0). However,

in the Tr models the diagonal mass matrix elements of these two neutral vectors

are very large compared with the other elements. Hence the eigenstates of-1;

the original mass matrices, are changed only to order ( < p , ^ / ^ P ^ W 10 and the

colour brightening properties of the original and the Tr models are the

same.

In distinction to the situation in the neutral sector, not all the

models predict colour brightening in charged interactions.

In models with cg = 0 (like "5°, Ila'C) there is no "W V~" term and

all colo'irs are conserved in charged interactions. In the other models there

a_re "\f Vp + W Vp" mixing terms, which induced the creation of the Vl and V

gluon in inelastic neutrino scattering. The amplitudes of these interactions

are proportional to the propagator

-27-

,+ V

(5.3)

In this production of the V* gluons, the red and yellow global charges areR+Y >

broken whereas Q is conserved.

The phenomenology of the charged interactions is similar to that

in the P-S basic model and described in Ref .9. It should be noted,bovever,

that all the colour brightening phenomena are characterized by threshold

effects. This threshold is either the gluon mess or the mass of some other

coloured excited state. In addition,most of the charged colour brightening

events are characterized "by charge leptonic events (either (eu?) or (uv ))

produced by the decays of the gluons through \T to non-excited states plus

leptons.

We now proceed to discuss the prediction arising from the "WX" and

"VX" mixing in the various models.

There is one model (l) where none of the quarks can decay to leptons.

This model is the best if there is absolute confinement and quarks cannot

emerge from the colour singlet particles.

In the other three original models,, if, II >C,NC, the blue quarks

are the only ones which can decay to leptons by the W X ~ mixture. (BlueH

lepton transitions are generated by the X'~ vectors, X~ generate yellow lepton

interactions and X generate the red lepton transitions .) The amplitudes

for these decays are proportional to the < W x'"> propagator f

-28-

In the basic P-S model the red and yellow quarks decay to leptons as a

result of the non-zero CabiVbo angles. This is not the case in the three

original models. The red plus yellow global charges are conserved ,

independent of the values of the Cabihbc angles.

The situation in the Tr models is completely different. Red quarks

decay to leptons in all these models, with amplitudes proportional to the

< W L X > propagators and to the propagators

(5.5)

(5.6)

The blue quarks decay to leptons in the Tr models as in their corresponding

original models.

11 is the only model in which a l l the quarks can decay to

leptons. I t is the only one vhere the yellow quarks decay to leptons;

the amplitudes of their decay are proportional to the < W+X"> propagator

and to.the propagator. These propagators are similar to the

< ¥ L X > , propagators, respectively,out c is changed to

C2 •

A s < 0 2 ^ l s m u c h greater than , the most unstable quarks

are the red or the yellow ones. But as the ratio of c ^ c £ , c has not been

fixed by us, the decay rates of the red and yellow quarks were left arbitrary.

Anybtw we prefer to take | c 3 |2 vlO1* GeV2, where I c J 2 + |c , , | 2 should be

of the order of 10 GeV£ . Thus in the subdomain | o | •< p > < I ^ I ' V p Sl c l | 3 3 3 1'

one^eta | — j > 1 , md the red quarks are the most unstable ones, whereas

in the aubdomain f«3l- > IY^I • < P1> , the yellow quarks are the

-29-

most unstable ones. The consequences predicted in these tvo subdomains are

completely different from each other U ) , hence one should define the right

domain according to the experimental results 8 5.

£• Concluding remarks

In order to conclude this paper we wish to consider some points, each

one concerns a different aspect of the "improved" models.

1) In all the "improved" models the number of Higgs fields is reduced

compared with the basic Pati-Salaa, model. As a result we can handle a

potential which predicts the appropriate vector masses and fixed all the

scalar particle masses in the tree approximation {except for models N c, IIC

where a pseudo-Goldstone particle arises). In this approximation the CaMbbo

angles the pattern of coiour brightening through < W V > mixing and the

different rates of quark brightening through and mixing,

are fixed according to the domains of the coupling constant. Moreover, the

small number of scalar particles makes it possible to analyse the scalar

particle spectrum and induced interactions. Taking advantage of all this

it is quite simple to show that in the improved models the scalar-fermion

Yukawa interaction is linearly dependent on the fermion masses 8 ). Hence the

induced scalar interactions, between leptons and hadrons, should increase

wherever heavy charm quarks become relevant (flavour or sea quarks). In a

forthcoming paper we suggest that the (OT) anomaly arises by this

scalar induced phenomenon (see Fig.l).

CAV

N

-30-

2) Another suggest ion which follows n a t u r a l l y in t h i s kind of model i s

t h a t the charged d i l ep ton and th r ee - l ep ton events , seen in lepton-hadron

i n t e r a c t i o n s s are e. consequence of the production and decay of iiquark. s t a t e s -

These diquark a t a t e s s denoted by E1

T a re produced by

Table I

Model

l ) Number of

Higgs fields

and trans,

under G

2) Higgs'

potential

3) Global

symmetry

It) Domain of

coupling

constants

(see Remarks )

1) I

S) T

A1 =• (2,2",1) - It

AS = (1,2,1) - 16

A3 = (2,1,10 - 16

+ dg TrU^A ) + a3 Tr(AJAJ )

+ ay3 Tr(A2+A2A3A3+)

i) ud)2 x ud)3

2) U(l) 2" 3

a2 < 0 a > 3

Y3, >0

1) I l a 2) Ka

3) ? a

A1 •= 2 x (2,7,1) - 8

A 1 As in IA3 J

2) v^ = v l l + Ke{crdet A2"3}

1) U( l ) 1 x U( l ) 2 x U{1)3

2) U( l ) 1 x U(1) S + 3

3) U(l)

* * •

D ^ ) a 3 > °> a2

y °> a i < °

Y2 < 0 ; Y 3 > 0; Y3 < 0

3) As in I I a and

( Y £3 - < P 2 > ) » ( ^ - < P 3 > )

1) I I C 2) NC

3} T

As in I I a

DV11

M TT P T

2) v£ = V11 + Re{4 d e t V }

3) v" = ?N+Ee{fiTr(A1AV)}c c

I ) U ( I ) 1 x ud) 2 x ud) 3

2) ud)2 x ud) 3

3) ud)2"3

As in I I but

* *c^ > 0 , a < 0

plus; (c^oig - (y2)2)> 0

1) I I a > c

2) f l a ' c

As in I I a

Dv11

2) v N

a

but 0- = 01) As In I I a

2) ud)

As in I I a

but

a < 0

eont

Model

5) VEVs

of the

Higgs

fields

j

6) Conser-

ved global

symmetries

7) CM

(Golctstone

mesons)

8) A1

particles

9) Remarks

1)

2)

c + 0 only i

1) QE+Y

2) d ' B

I

I

0

0

c l

0

n (

i

0

0

0.

c l

2)

0

0

0

0

b

0

0

No Cabibbo

A0

angles 8)

Table I

1) I I a

3) I

a i

0

0

0

< A > " (o

c f 0 only ir

3 1 ^ R

(con1

2

6

9_

0

0

0

C2

(3)

0

0

0

H a

0

c

Q

1) 1 coloured t rue GM

2),3) 0

, .0J A j A

1

1) I I C 2) tlc

3) H°

e # 0 on ly

0

0

0

0

in

' - ,

0

0

0

(3>

\

0

0

0

0

0

l ) , 2 ) QR ; Q l Y

3) Q l Y

1) 1 (A1 type) PGM

2 ) , 3 ) 0

2 x A° ; A*

No "colour" brightening in

charged interactions

1) I I a " c

2) T T -

1 _ _ _ - -j

1) As in I l a

butcs = °

2) As in ~Ka

but

c 2 = 0

2! ^ ^

0

2 x A0 ; A*

REMARKS CONCERNING THE TABLE

The number of each remark is the index of the row to which it refers.

In each column there are tvo or three numbered models, their properties are the

same, except if st*ted otherwise by giving them the corresponding number.

1) The local group G is SuC2)T x SUC2),, * SU(1+L . The multiplets1 2 3 L E C

A , A , A correspond to some of the fields appearing in the multiplets A,

B, C of the P-S model, respectively.

2)

i

3) a) The U(l) 1 charge is the number of A1 pa r t i c les .

b) The Yukawa potential breaks the U(l) global symmetry.It) a)

REFERFJICES

1) J.C. Pati and Abdus Salam, Phys. Rev. DIP, 275 (1971*).

2) Amir Schorr, ICTF, Trieste, preprint IC/76/Wt (to appear in J. Math.

Phys. (H.T.)).

3) Amir Schorr, ICTP, Trieste, preprint IC/76/1+5 (to appear in J. Math.

Phjrs. (H.Y. )).

1+) J. Liberman, Phys. Rev. D9_, Ijh9 (197>O .

5) T.P. Cheng, E. Eichter and LrF. Li, Phys. Eev. ££_, 2S59 (.191^)-

6) J. Ellis, M.K. Gaillard and D.V. Nanopoulos, CEEH preprint TH.2O93.

7) ' T.D. Lee,' Phys. Rev. D8., 1226 (1973);

S. Weinberg, Phys. Eev. Letters 21, 657 (1976).

8) Amir Schorr, in preparation.

9) J.C. Pati and Abdus Salam, ICTP, Trieste, preprint IC/76/76 (presented

at the "Neutrino" Conference, Aachen,1976).

10) E. Gildner, Phys. Rev. Dli4_, 1667 (1976).

11) Abdus Salam, ICTP, Trieste, preprint IC/76/21;

See also W.R. Tranklin, Bucl. Phys. Bgl., 160 (1975).

EBRATUM

Page 38: Table 1 5 ) , Model II

(instead of c

II and U sho\ild read 0

in