A supersymmetric Weinberg-Salam model

Volume 84B, number 1

PHYSICS LETTERS

4 June 1979

A SUPERSYMMETRIC WEINBERG-SALAM MODEL P.H. DOND! and P.D. JARVIS

Department of Physics, The University, Southampton S09 5NH, UK Received 27 February 1979

We formulate a unified Weinberg-Salam model in terms of the Yang-MiUs theory of SU(2/1) over a Minkowski superspace. The gauge potentials comprise vector bosons and Higgsscalars. Fermions transform in a spinor representation of the space-time supersymmetry.

The spin-statistics connection has hitherto obviated the construction of internal supersymmetries. Ne'eman [1] has recently attempted to circumvent this, in the context of an SU(2/1) gauge theory, by interpreting Higgs bosons as Lorentz scalar forms, rather than conventional fields. Alternatives have been considered [2, 3] in which the Higgs bosons are associated with the scalar components of the gauge multiplets of a sixdimensional Yang-Mills theory. In the present note, we formulate the WeinbergSalam model as a Yang-Mills theory of the graded Lie group SU(2/1) over a graded space-time manifold (xU, ~a), with fermionic directions parametrized by two anticommuting (but Lorentz scalar) parameters ~a, in addition to the normal Minkowski space. The spinstatistics problem is solved, in the sense that the physical vector, scalar and spinor fields of the model are all conventional. The supersymmetry is carried in addition on fields with unconventional statistics, which are interpreted as unphysical. The group SU(2/1)has an eight-dimensional graded Lie algebra with generators satisfying

[Rm, Rn ] = iemnlRl ,

[Rm, Qa] = -½ (Om)a b Qb,

[Rm, ~a] = +½0_b(Om)ba, {Qa, Q_.b}=-(Rmom)a b,

(1)

(Qa, Q b ) = O = {Q.a,Q_.b},

where m,n=O, 1,2,3;

R m a m = Roo 0 - R • a , emn l = +1 ( - 1 )

if (mnl) an even (odd) permutation

of (1 2 3), =0

otherwise.

The fundamental three-dimensional representation is

R 0 - -~

Ri=~

0(:) 0

°1:(o o

1

(2)

The space-time supersymmetry group which we use has generators Juv, Pu' .Ea~, II~ and IIua, of which Juv and Pu generate the Poincare group, the -a# are generators of an Sp(2) group acting on the odd coordinates, and the Ha, l-lua are Lorentz scalar and Lorentz vector supertranslation generators. The commutation relations are

[Juv, Jpol = i01#oJvo + ~lvpJ#o - r~.pJvo - rlvoJ#p) , a,b=l,2;

Om=(l,~ )

and 75

Volume 84B, number 1

PHYSICS LETTERS

[su~, Po ] : i ( ~ , . P u - ~ u o e , ) ,

4 June 1979

transforming in the same manner as X. The operation of exterior differentation is defined

[]uv' Hoo~] = i(vvo liue - ~uoIIve),

OX = (dxU0 u + d~=0e)X.

(8)

[z~t3, II~] = e&rIl e + %~IIt3, The differentials dxU and d~e are one-forms for which a graded wedge product is defined [4] such that

[Nan , IIp~,] = e3~Hpe + ee,yHp~ , {Hue' Ha} = -iee/~Pu '

[Pu'

n~el = -irluvli e

{Hue , nv~ } : ~quvXc~3 - i % [ J . v ,

,

dxUAdx v=-dx vAdxu, (3)

dxU A d~ e = - d ~ e A dxU,

and all others zero. Here ~uv is the usual Minkowski metric with signature (+, - , - , - ) , and ee3 is the twodimensional antisymmetric metric io 2. This algebra has a differential representation on superfields ~(x, ~) given by

Thus the graded two-form

p . = -iO u ,

corresponds to the usual Yang-Mills field strength, and has an expansion

J.~ = - i ( x u a ~ - x ~ a u ) ,

Hue = +i(~e0 u +xuO~) ,

lie= -i0,~,

_-"~ = ~eO~ + ,~O,~,

(4)

which is unitary provided ~2 is related to ~1 by complex conjugation, viz. (~1, ~2) = (~, ~). Gauge theories are most elegantly formulated in terms of the structure equations of differential geometry [4]. In the present case, the connection becomes an algebra valued graded one-form, with an expansion • (x, ~) = dxU Vu(x , ~) + d~aq~e(x , ~),

(5)

where each of the superfields Vu, Oe has components in the basis of the algebra:

vu(x, o : V: m +

+

¢ e ( X ' ~) = 4> --m u R m + ¢a~Qa + ¢o~aO_,a .

d~e A d~3 = +d~t3 A d~ a .

~r: dcb - igcPcI)

(9)

(10)

9r = a dxudxVFuv + dxUd~eFue + ld~ed~t3Fet3 , Fur

O;,Vv - OvVu - ig[ Vu, Vv] ,

:

Fo~ = 0,.% + O~e - ig{¢,~, % } .

(] l)

The commutation and anticommutation properties of the component fields (6) are precisely those which respect the internal graded algebra, and guarantee that ~r is a graded-algebra valued quantity. Finally, defining dual forms by *dxU = - ( 1 / 3 !) e uuo° d x u d x o d x a d2~,

,

*d~ e

=

- ( 1 / 1 !) e e3 d~t3 d 4 x ,

(12)

(6)

The connection transforms under finite group elements according to the law

the Yang-Mills action is

alp' : U - 1 4 p U _ (i/g) d U - 1 U .

SyM = f _ ,

In order that
The connection with the Weinberg-Salam model is established by taking for the gauge potentials solutions of the following form (up to a gauge transformation):

cOX = d X -

76

igcbX ,

(7)

~r A ~r.

V.(x, 0 = V.(x),

(13)

Ca(x, ~) = Ca(x),

a =q~ =0,

v . = w . . R - 3-1/2 u . R 0

Volume 84B, number 1

PHYSICS LETTERS

491 = 2-1/249aQa

4 June 1979

We take the successive lepton generations to be triplets under SU(2/1):

('/

= 2-1/2 4,1

i f = ( f f i ) = fL ' 492

with the conventionally-assigned charge and hypercharge: Q=R3+R

= 2-1/2 l - -

~2. .

(14)

The action (13), for Vu(x ) and Ca(x) alone, reduces to

- f{

_

1

(15)

where D u 49 = 0 u 49 - i g [ Vu , 491 , and the SU(2) × U(1) invariant trace, rather than the SU(2/1) invariant supertrace, has been taken. The next stage is the introduction of matter, as fermions, into the theory. There are two distinct exterior operators for spinor-valued forms if over an orthogonal group: the ordinary derivative d, and the multiplication of forms by elements P of the Clifford algebra: r i f = dxUr~if.

(16)

Now the Heisenberg algebra [%, a#] = ea#,

(17)

of a set of boson creation and annihilation operators plays the same role for symplectic groups as does the Clifford algebra in the orthogonal case. In fact Pu = 7**X 1, A a = T5 X a~ generate an auxiliary orthosymplectic algebra which contains the usual 0(3, 2) algebra. We therefore extend (16) to 1 ~ = (dxUTu + d~aAa) i f ,

(1 8)

while the action is:

=f~(r~

0 ,

Y=2R 0 .

(21)

There is now the same freedom to effect a separation into commuting arid anticommuting felds as in the gauge sector. Here this is achieved on the two-component chiral projections of the spinor superfields:

1 Ui.t v . U ~ v

+ iDu4912 _ 5g 1 2 [49~12}d 4 x ,

SF = f * p i f

(20)

\ fR/

(°2 = 2-1/2~aQa

Sy

f =e'/J'r'''''

fL= fR

'

fR '

where the if(x, ~) are chosen to be anticommuting, and the t~(x, ~) commuting, corresponding to physical and unphysical components, respectively. From (2), it is clear that this choice is respected by the gauge transformations. Thus far, the if(x, ~) are simultaneously Dirac spinors, and spinors in the Fock space of the a a. In order to decouple the latter, we take a reduced superfield of the form qffx, ~) = if(x, ~) X(~),

(23)

with ~k a Dirac spinor, and X a Fock spinor. This decomposition is in fact consistent with both the internal and space-time supersymmetries. The connection with the Weinberg-Salam model is established by taking for if solutions of the following form (up to a gauge transformation): i(x,

=

= o,

XX= 1 = constant,

i = 1,2, 3,

2 - 1 / 2 ~ a a X = h a = constant, (24)

in addition to (14). The action (19), for t~i(x), becomes (after absorbing the phases o f h 1 = h, h 2 = h into fR, fR):

=f {

^ iq)if

) ) d4x.

- gh( 49¢, +

(25)

The neutral coupling (from (14)) is

• iD u - A a • iDa)if d4x d2~,

9 • 3-1/2PLZ0VL _ _ 9 f A 0 f - ~9 • 3-1/2f'Z075f,

where D u i f = (0 u - i g V u ) i f ,

VL= UR '

(26)

where Daif = (a a - ig~a)if.

(19)

Z 0 = ½x/~W 3 + ½0,

1

A0 = -~W 3 + -~x/3-U.

(27) 77

Volume 84B, number 1

PHYSICS LETTERS

The lagrangian (15) plus (25) is therefore of conventional Weinberg-Salam type [5], with a unique gauge coupling g (since SU(2/1) is a simple group) and Weinberg angle 0w = 30 °. Before further symmetry breaking (which is expected to occur dynamically) all particles are massless. The Higgs self-coupling is determined also to be ~_g2 (since this occurs in the F 2 term). In the present formulation, however, it appears that the Fock spinor × confers an arbitrariness on the Higgs-fermion coupling strength hg, and there seems no reason that h should be constrained to unity [ 1 ] or zero [3 ]. Finally, it should be emphasized that the invariance of the ancestral action SyM + S~ under reflections of Vu, ~ and ffi, ensures the existence of solutions of the required form, (14) and (24). We hope to report elsewhere in more detail on other aspects of the theory. Hadronic matter may be introduced via successive generations of quarks, which are accommodated in four-dimensional multiplets of SU(2/1), each containing an SU(2) doublet and two singlets:

78

4 June 1979

/q,Lk

XIIq

--?j (q)(:)<) qR

'

--

' ( S .....

(28)

~,rR/ An extension of the model to include the strong interactions is in progress. It is a pleasure to thank Professor K.J. Barnes for discussions and constructive criticism of this work.

References [1] [2] [3] [4]

Y. Ne'eman, Phys. Lett. 81B (1979) 190. D.B. Faidie, J. Phys. G5 (1979) L55. E.J. Squires, Phys. Lett. 82B (1979) 395. J. Wess, Supersymmetry-supergmvity, Lectures given at the VIIIth GIFT Intern. Seminar on Theoretical physics (Faculty of Sciences, Salamanca, 1977). [5] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, Proc. 8th Nobel Symp., ed. N. Svartholm (Almquist and Wiksells, 1968).