A supergravity model without elementary gauge singlets

Volume 160B, number 1,2,3

PHYSICS LETTERS

3 October 1985

A SUPERGRAVITY MODEL WITHOUT ELEMENTARY GAUGE SINGLETS Matt VISSER

Phystcs Department, Unwersztyof Southern Cahforma, Los Angeles, CA 90089-0484. USA Recexved 22 April 1985 We present some further calculatmns regarding a previously derived model which exh~bxts spontaneous supergravlty

breaking w~thout using elementary gauge smglet scalars The model m xts s~mplest form does not possess a h~dden sector since all partmles have gauge mteractmns The tree level mass matnces are shown to be completely dmgonahsable The resulting tree level spectrum as not phenomenologacally useful Some extensions of the model, which have better behawour, are then d~scussed.

T h e m o d e l w h i c h we shall be interested in was derived in ref. [1]. Let ~' be a c o l l e c u o n of c o m p l e x scalaz fields in N = 1 supergravity. T h e supergravity l a g r a n g i a n we use is that of C r e m m e r et al. [2]. T h e i n d e x u s a g e m o r e closely follows that of Bagger [3], a n d W i t t e n a n d Bagger [4]. F u r t h e r details m a y b e f o u n d m ref. [1]. L e t a gauge group G act on the q~' via a (possibly reducible) set of g e n e r a t o r s t J:. C o n s i d e r the model defined by K=~q~=0'*0',

W=e~'(eoeo)3/S=e~'(eO'eo')3/8, G=

(~0)+31n[(0q~)(~)]

+2~.

It is n o w e a s y to show that the scalar p o t e n t i a l of this supergravity m o d e l is given b y

V=e2%_,o(e~ep)-s/8(ep,)-,/8[(~(~)lle~Oi

3 12+ _34i , l , ~ _

-

~ ¢v~12] + ~(,~t~¢) 1 2.

N o t e t h a t t h e K~ihler g e o m e t r y is flat. g,e. = 0, 0 : . K = 8,e . . T h u s ~, = g,e .q~s * = ~, * sirmlarly ~ , . = q;, ~t,~=¢'.tA,.seOe=q~,tA's~L (t A =hA~t~ where h~ ~ is the n-bein assocmted with the fibre metric f~o = hA,hAa.) W h e n indices are s u p p r e s s e d it is a s s u m e d that all indices are rinsed. It xs f u r t h e r m o r e easy to see t h a t the a b s o l u t e m i n i m u m of V occurs at:

I(q,q,)l = 3

Da = O. V= O, m3/z =

~ = [(~)/(q,q,)]w2q,,

e c/2 = eK/21W I = e ~ ( 3 ) 3 / 8 e 3/8.

W e shall c o m p u t e the mass matrices using results of ref. [1]. In a n y N = 1 s u p e r g r a v i t y m o d e l where (1) the K ~ h l e r m a n i f o l d is R i e m a n n fiat and, (2) D = 0 at the m i n i m u m , we have

(,.0)2=

O,.Os.G]2[

[ O,O:G

0,, D A OjDA +[ O,D~O/0~

("1/2) =

8,3.

0

3'*Ol*Oh*GOkG]}o

-4- OtOjOkG Ok.G

O,.DAO~.DA] O,DAS,.D ~

'

[-,,,~/~(o,O:G + ~o,G O,G) [ i~-0:DA

(m,) 2 = 2(8,DA)(8,.D,) ,

(m3/2) =

e6/2,

if20'DB

]

~m~/2(af)AB " ( m 2 ) = 0.

0 3 7 0 - 2 6 9 3 / 8 5 / $ 03.30 © Elsevier Science P u b h s h e r s B.V. ( N o r t h - H o l l a n d Physics P u b h s h i n g Divasion)

77

Volume 160B, number 1,2,3

PHYSICS LETTERS

3 October 1985

Here

(df )

= ( O,.C)( O,fop)hA°hB

T o begin computations for the case at hand, choose the coordinate system so that at the absolute m i n i m u m ( ~ ) = 3 / 4 = (ff~). Consequently at the absolute minimum we have ~ = ~ (qY = ~'* ----~, = ¢p,. = ~'). Simple computations now show that at the mintmum: 0,G=2~b,,

~,~jG=8,j-2ep,~j/~,

a,~j~kGOk.G=-4~,~jG.

We m a y now diagonalise 0, OjG using orthogonal transformations. 0, OjG = (0#0a'),j. Let # be the number of complex scalar fields = complex dimension of the K~ihler manifold. Then #=-1(91#_

1,

/,2=I#=1~I#_

1.

In this coordinate system ~ ' = 3V/~ 8,o, ~ ~ [0, # - 1]. The only other significant contribution to the masses arises from O,D~, we show that the matrices 0, OjG and O,Do can be simultaneously diagonalized. Consider the identity

6~D,~ - ( 6jr,G )8'D~ + 8,G( 818'0~). This identity, as written with covariant derivatives, is valid for arbitrary K~ihler potential and for all values of the fields. It is a consequence of the simpler identity:

D,~ = (8'D,,)6,G. Particularizing to the case at hand we see

D ~ = q~,t~'j¢j,

¢p,ta', = (O, OjG)t~',qJ k + O,Gt~'j.

O,G = 2~,, consequently at the mmtmum O,D,,=(-O,0,G)(O,.D,~).

At the minimum

Noting that D,~ = 0, .G • O,D,, = 0, we see that a pair of orthogonal matrices as follows: (gD~) =

OaD,~is pure imaginary. Thus ajD~ may be diagonalised by

i(0,p02),~.

Let N be the dimension of the gauge group, then 02 is N × N, 01 is # × # and p is a # × N real positive semidefinlte and pseudodiagonal matrix. Furthermore 01 commutes with # so we can simultaneously satisfy: ¢'= ~-(I.0),

(a,8,G)= - I ( 9 I , _ , =l~,

(O,D,~)= - i ( p ) .

Let r = rank (p). Then r is just the number of massive gauge bosons. Furthermore r ~< N and r ~< # - 1. The mass matrices may now be seen to be

-tx ( m l ) 2--- 2pWp,

I

_pot

(m3/2) = e c/2,

p oT

J

(ml/2) =

~/~-pT

~rn3/2(df)As'

(m2) = 0.

The eigenvalues of the scalar (mass) 2 matrix are easily seen to be

mo=(4m272 + 2ppT) '/a

( # ofthese),

m0=0

( # ofthese).

M a n y of the massless scalars are illusion, they correspond to the would-be Goldstone bosons which are 78

Volume 160B, number 1,2,3

PHYSICS LETTERS

3 October 1985

eaten by the massive gauge bosons. The physical spectrum is, (1) # - r scalars with m 0 = 0, (2) # - r scalars with m 0 = 2m3/2, (3) r scalars with m 0 = [(2m3/2) 2 + (ml)2] 1/2, for a grand total of 2 # - r physical scalars. A similar comment applies to the s p i n - l / 2 mass matrix. The zero eigenmode corresponds to the would-be goldstino, it is eaten by the gravitino. The physical s p i n - l / 2 mass matrix is in fact ( # + N - 1) × ( # + N - 1) dimensional. To complete the diagonalization we will assume that (df),~ B is diagonal, and define ml/2 = (1/2)m3/2df. Let ~ be the nonsingular r × r matrix obtained from p by deleting rows and columns of zeros in the obvious manner. Then the s p i n - l / 2 mass matrix decomposes into a direct sum

(m,/2)1 =

0

(rnl/2)N_r'

v~-~

(m'/:)rJ"

The physical spectrum is now (1) # - r - 1 spinors with ml/2 = ml/2 = ( m l / 2 ) N - r , (3) 2r spinors with masses given by

m3/2,

(2)

N-r

spinors with

ml/2~-l(~-(ml/2-m3/2)4- [(ml/2-~-m3/2)24-am2]1/2} for a grand total of # + N - 1 spinors. Note that for the 2r badly mixed spinors we have bounds: (a) r spinors with m3/2 < ml/2 < m3/24- mr, (b) r spinors with rhl/2 < ml/2 < rhl/2 + m r As a phenomenological particle spectrum the above is clearly inadequate. Experimentally we do not see ( # - r) massless scalars. Neither do we experimentally observe degeneracy among quark, lepton, and gravitino masses. If we wish to retain this model for semi-realistic model building purposes then extensions of the model are called for. A simple extension of the model is to add extra gauge multlplets (~) that do not show up m the superpotential, Le. K ~ K 4- ~ ; W unchanged. The extra fields may easily be seen to satisfy: m 0 = m3/2,

m l / 2 = 0.

This at least gives us some massless fermions to play with. If one wishes to interpret these as the leptons and quarks one could appeal to radiative corrections to generate the observed lepton and quark masses. The use of radiative corrections in a nonrenormalisable theory is of dubious value at best. A more palatable alternative is to include quark and lepton masses at tree level by modifying the superpotentml.

w

Wexp

With such a modification a stationary point occurs at (OqO = 3/4, ~ = O, while

O,G --* O,G + f , ab~"~ b ( =

2~, at extremum),

OaG~ ~a 4- 2fmbdO'~ b (

= 0

at extremum).

It is easy to see that at the stationary point (1) O,OsG is unchanged, (2) OaObG= 2f, abq,' - M a b , (3) O,O~G= 0, (4) O~ObO,G(O, . G ) = 4f, abq¢ = 2M~b. SO the extra fields have mass matrices:

(ml/2)=m3/2M ' (m2)=m2/2[l ÷ MM, 4M

4M I + _MM

l "

D i a g o n a l s i n g these matrices, we see that the extra fields (which we wish to interpret as the leptons, quarks, sleptons and squarks) obey the sum rule

m~= (m2/2 + 4m3/2m,/2 + m2/2)1/2. Observe that keeping m~ positive requires ml/2 < (2 - v~-) m3/2 or real2 > (2 + ~)m3/2. If this condition is violated the stationary point at (q~q~= 3/4, ~ --- O) is actually a saddle point rather than the desired minimum. 79

Volume 160B, number 1,2,3

PHYSICS LETTERS

3 October 1985

This phenomenon, the possibihty of negative mass E among the ~ fields, could also be used to generate a " t w o step" model in the following sense: (1) Use the q~ fields only to simultaneously break supergravity and the grand unified gauge group. Note ^ ^ that the grand unification gauge bosons receive a mass ( m l ) E = g3 F F / 2 v(Ot,tt~eo ). (2) Allow some of the ~ scalars to get a negative (mass) E, by setting appropriate spinor masses m t/2 ~ ((2 - v ~ ) m 3/2, (2 + vr3)rn 3/2)- Interpret these negative (mass)E scalars as the electroweak Higgs particles and use them to drive the spontaneous breakdown of the electroweak interactions. Some comments are in order. (a) The grand unification scale is now naturally of order mp. (b) The electroweak unification scale is naturally of order m3/2, m3/E being arbitrarily tunable. (c) Arranging negative (mass) 2 in the electroweak sector is technically rather difficult. It is essential that we include couplings between the electroweak and grand unification sectors. (d) The sum rule for the electroweak sector gives us a tree level bound, mr~ggs < Vr3m3/2 . (e) We had carefully arranged the tree level cosmological constant to be zero at the grand unification scale. This will no longer be true below the electroweak breaking scale. In conclusion, we have developed what appears to be a workable strategy for generating a physically reasonable mass spectrum. Because we did not use any gauge singlets in the model, supergravity breaking is inextricably bound with grand unification breaking. All the other physics then occurs at masses of order

m3/E. The ratio

(m3/2/mp) 1s unexplained

in this type of model, being put in by hand.

This work was supported by the Division of High Energy Physics of the US Department of Energy under Contract DE-FG03-84ER40168.

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

80

M Vlsser, thesis, LBL-18189 E Crernmer, S Ferrara, L Glrardello and A van Proeyen. Nucl Phys B212 (1983) 413 J Bagger, Nucl Phys B211 (1983) 302 E Wltten and J Bagger, Phys Lett 115B (1982) 202