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COMPOSITE MODELS OF THE WEAK INTERACTIONS
E. Farhi
To cite this version:
E. Farhi. COMPOSITE MODELS OF THE WEAK INTERACTIONS. Journal de Physique Collo-
ques, 1982, 43 (C3), pp.C3-289-C3-292. �10.1051/jphyscol:1982357�. �jpa-00221912�
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CoZZoque C3, suppZ6ment au n o 12, Tome 43, de'cembre 1982 page C3-289
COMPOSITE MODELS OF T H E WEAK I N T E R A C T I O N S
Center for TheoreticaZ Physics, Laborat03 for NueZear Science, Department of Physics, Massachusetts I n s t i t u t e of TechnoZogy, Cambridge, MA 02139, U.S.A.
I have been asked t o t a l k about composite models where t h e s c a l e of t h e binding f o r c e i s l e s s t h a n 1 TeV. These models have i n t e r a c t i o n s whose s c a l e i s l e s s t h a n 1 TeV which i s very n e a r t h e weak i n t e r a c t i o n s c a l e of
G,"~ =Z 300 GeV. T h e r e f o r e I w i l l assume t h a t t h e weak i n t e r a c t i o n s a r e i n f a c t a r e s i d u e of t h e underlying s t r o n g i n t e r a c t i o n binding f o r c e .
Models of t h i s t y p e have t h r e e g e n e r a l f e a t u r e s :
5.) Quarks and l e p t o n s a r e composite on t h e s c a l e Gz1/2 = 300 GeV.
+ -
ri i ) The weak i n t e r a c t i o n bosons W
,
W,
and Z a r e a l s o composite on t h i s s c a l e .i i i ) The observed weak i n t e r a c t i o n s a r e a remnant of t h e c o n f i n i n g i n t e r a c t i o n .
These i d e a s have appeared i n t h e l i t e r a t u r e i n v a r i o u s schemes1. However most of t h e models i n t r o d u c e d a r e n o t based on s e n s i b l e dynamical reasoning.
For a model t o have a glimmer of hope of being c o r r e c t , i t must c o n f r o n t t h r e e i s s u e s .
i ) Spectrum
-
The quantum numbers of t h e bound s t a t e s a r e determined by t h e quantum numbers of t h e c o n s t i t u e n t s . The bound s t a t e s must have t h e observed quantum numbers of t h e quarks and l e p t o n s . I f bound s t a t e s w i t h t h e wrong quantum numbers can be formed, t h e i r absence from t h e spectrum must be explained.i i ) L i g h t n e s s of fermions
-
The observed fermions a r e much l i g h t e r t h a n t h e binding s c a l e . This i s only p o s s i b l e i f t h e bound s t a t e fermions s a t i s f y t h e ' t Hoof t anomaly c o n d i t i o n s .i i i ) C o r r e c t weak i n t e r a c t i o n phemomenology
-
The r e s i d u a l i n t e r a c t i o n s induced by t h e c o n f i n i n g f o r c e must look l i k e t h e known f o u r f e r m i i n t e r a c t i o n a t ~ 2 <% G-1. p r o p e r t i e s of t h e i n t e r a c t i o n i n c l u d e V-A charged c u r r e n t s , u n i v e r s a l i t y and t h e famous n e u t r a l c u r r e n t s . FOne model which seems t o work, a l t h o u g h i t does have some problems w i t h p o i n t ( i i i ) , i s t h e one i n t r o d u c e d by L. F. Abbott and m y s e l f 2 . The key f e a t u r e of t h i s model i s that t h e dynamics a r e determined by t h e same Lagrangian a s t h e s t a n d a r d Glashow-Weinberg-Salam t h e o r y . However t h e parameters a r e a d j u s t e d s o t h a t t h e r e i s no spontaneous symmetry breaking and t h e SU(2)L gauge i n t e r a c t i o n becomes s t r o n g a t 2 300 GeV. The t h e o r y looks v e r y much l i k e QCD and our a n a l y s i s u s e s many of t h e l e s s o n s l e a r n e d from hadronic p h y s i c s .
The i n g r e d i e n t s i n t h e Lagrangian a r e t h e SU(2)L doublet f i e l d s
xC1
(a = l,N), 2N r i g h t handed s U ( 2 ) ~ s i n g l e t s and a fundamental s c a l a r doublet @. The U(l) quantum numbers a r e t h e u s u a l ones w i t h @ a s s i g n e d-
112 e t c . A t t h e c o n f i n i n g s c a l e , gauge s i n g l e t bound s t a t e s form. For each d o u b l e t fennionxC1
we can form two fermionic bound s t a t e s w i t h@.
These a r e
+*XU
andEX^
and correspond t o t h e two ways t o make a s i n g l e t o u t of two SU(2) d o u b l e t s . These bound s t a t e s have U(1) quantum numbers which equal t h e e l e c t r i c charges of t h e observed quarks and l e p t o n s . Theses t a t e s , which a r e l e f t handed, a l s o s a t i s f y t h e I t Hooft anomaly c o n d i t i o n s and we expect them t o be l i g h t on t h e binding s c a l e . The r i g h t handed p a r t n e r s of t h e s e bound s t a t e s a r e t h e r i g h t handed f i e l d s i n t r o d u c e d i n t o
t h e Lagrangian. These r i g h t handed f i e l d s a r e s i n g l e t s under t h e c o n f i n i n g
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982357
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group and have t h e c o r r e c t c h a r g e assignments a s long a s we u s e t h e v a l u e s g i v e n by t h e Weinberg-Salam t h e o r y .
Thus, t h e spectrum and l i g h t n e s s of t h e f e r m i o n i c bound s t a t e s works o u t p e r f e c t l y in t h i s model. I w i l l now d i s c u s s t h e form of t h e e f f e c t i v e low energy i n t e r a c t i o n induced by t h e s t r o n g i n t e r a c t i o n s U ( 2 ) ~ . A t o r n e a r t h e s c a l e G i l l 2 300 GeV t h e r e e x i s t s a v a r i e t y of bound s t a t e s h e l d t o g e t h e r by t h e s t r o n g f o r c e . Three of t h e s e bound s t a t e s have t h e quantum numbers of t h e w + , W- and Z. They a r e formed by t a k i n g s p i n one, gauge s i n g l e t , combinations of $*$*,$ $, and $*$. The exchange of t h e s e p a r t i c l e s and a l l t h e o t h e r p o s s i b l e bound s t a t e s w i l l g i v e r i s e t o an e f f e c t i v e i n t e r a c t i o n between t h e l e f t handed bound s t a t e fermions. The r i g h t handed fermions a r e p o i n t l i k e and do n o t p a r t i c i p a t e i n t h e s e i n t e r a c t i o n s .
A t Q~ <<
Gpl
t h e dominant o p e r a t o r i n t h i s e f f e c t i v e i n t e r a c t i o n i s a four-fermi i n t e r a c t i o n between t h e l e f t handed f i e l d s . The form of t h i s o p e r a t o r can b e g r e a t l y r e s t r i c t e d by i n s i s t i n g t h a t i t r e s p e c t s t h e symmetries of t h e u n d e r l y i n g Lagrangian. I f we n e g l e c t U(1) and c o l o r a t t h e s c a l eGp1I2
t h e n t h e Lagrangian has a g l o b a l SU(N)XSU(2) symmetry.The SU(N) comes from t h e N l e f t handed d o u b l e t s (For t h r e e g e n e r a t i o n s N=12).
The SU(2) i s an e x t r a symmetry which a c t s only on t h e s c a l a r f i e l d $. Now t h e r e a r e only two p o s s i b l e four-fermi i n t e r a c t i o n s between t h e l e f t handed bound s t a t e s which r e s p e c t t h e s e symmetries:
where
Here Ya i s a d o u b l e t under t h e g l o b a l SU(2). I t s two components a r e t h e two d i f f e r e n t gauge s i n g l e t s made from 41 and Xa.
The f i r s t i n t e r a c t i o n i s most of t h e observed four-fermi i n t e r a c t i o n : (4~,/fi) {J* J'-*
-
s i n 2 e J ~ ~ ),
~ )iiL. L 'd
except f o r t h e p a r t involving s i n 2 8 . We can a l s o g e t t h i s i f we assume v e c t o r meson dominance. This i d e a was f i r s t discussed by Hung and S a k u r a i and by Bjorken3. Imagine t h a t t h e t h r e e bound s t a t e s @, W- and Z0
p r e v i o u s l y mentioned a r e p r i m a r i l y r e s p o n s i b l e f o r t h e i n t e r a c t i o n . Also imagine t h a t t h e r e i s a mixing between t h e photon and t h e X much l i k e t h e mixing between t h e photon and t h e pO. This mixing w i l l g i v e r i s e t o t h e e x t r a n e u t r a l c u r r e n t p a r t s of t h e Lagrangian. The observed s i n 2 % i s t h e n p r o p o r t i o n a l t o t h e amount of t h i s mixing. The observed low energy f o u r - f e r m i i n t e r a c t i o n produced by t h e s t a n d a r d weak i n t e r a c t i o n theory i s t h e n reproduced by t h i s c o n f i n i n g model of t h e weak i n t e r a c t i o n s .
Are t h e r e o t h e r i n t e r a c t i o n s , i n a d d i t i o n t o t h e ones j u s t d i s c u s s e d , induced by t h e c o n f i n i n g f o r c e ? I n t h e l i m i t where we n e g l e c t c o l o r and e l e c t r i c charge t h e s e a d d i t i o n a l i n t e r a c t i o n s can only have t h e form
JO .JP0
This term i s n o t observehlLatLthe 5% l e v e l . Now t h e W+ W- and Z form a t r i p l e t under t h e g l o b a l SU(2) group and a r e p r i m a r i l y r e s p o n s i b l e f o r t h e d e s i r e d t r i p l e t exchange term. The unwanted term could a r i s e from t h e exchange of t h e SU(2) s i n g l e t p a r t n e r of t h e t r i p l e t . However t h i s s p i n one s t a t e can n o t be made o u t of two s c a l a r s , a s t h e W+ W- and Z a r e c o n s t r u c t e d . S i n c e t h e s i n g l e t exchange term i s n e c e s s a r i l y d i f f e r e n t i n o r i g i n perhaps i t s suppression i s n o t s u r p r i s i n g .
Another s o u r c e of p o s s i b l e d e v i a t i o n s from t h e d e s i r e d four-fermi i n t e r a c t i o n s a r e o r d e r a s t r o n g ( G F ~ / ~ ) / ~ T c o r r e c t i o n s t o u n i v e r s a l i t y which a r e induced when we no l o n g e r n e g l e c t c o l o r . This has been emphasized by Susskind and Veltman. I n f a c t , S i r l i n has c a l c u l a t e d t h e s e c o r r e c t i o n s i n t h e s t a n d a r d model and t h e y a g r e e w i t h experiment t o a f r a c t i o n of a p e r c e n t . We need a b e t t e r understanding of t h e s e c o r r e c t i o n s i n c o n f i n i n g models.
The g r e a t e s t d i f f i c u l t y f a c e d by a l l c o n f i n i n models of t h e weak i n t e r a c t i o n i s t h e l a r g e v a l u e of t h e observed s i n g e
-
.22. I f t h i s i s a measure of t h e photon-
X mixing then i t i s d i f f i c u l t t o understand whyi t i s n o t more t y p i c a l l y e l e c t r o m a g n e t i c i n magnitude. F o r example, t h e r h o meson photon c o u p l i n g , measured i n p + e+e-, would produce a "s?'n28" o f .02.
Some enhancement mechanism must b e d i s c o v e r e d .
S t i l l , I b e l i e v e t h a t t h e weak i n t e r a c t i o n s may be due t o a s t r o n g i n t e r a c t i o n . Can t h i s be e x p e r i m e n t a l l y t e s t e d ? F i r s t a t low Q~ t h e r e c o u l d s t i l l b e a v e r y s m a l l SU(2) s i n g l e t exchange term w a i t i n g t o b e d i s c o v e r e d . Also i f t h e r e a r e e x c i t e d Z ' s c o n t r i b u t i n g t o t h e n e u t r a l c u r r e n t , t h e s e c a n add an a d d i t i o n a l Jem JVem
I-r
t e r m t o t h e n e u t r a l c u r r e n t a t low Q ~ .
More s p e c t a c u l a r l y , t h e W+, W- and X produced a t h i g h e n e r g y machines, would have p r o p e r t i e s v e r y d i f f e r e n t from t h o s e p r e d i c t e d by s p o n t a n e o u s l y broken gauge t h e o r i e s . The W and Z would b e h e a v i e r and w i d e r w i t h masses between 1 0 0 and 1 7 0 GeV. The r e l a t i o n s h i p
would n o t h o l d . I n f a c t t h e W t o Z mass r a t i o would b e c l o s e r t o u n i t y . The w i d t h s would f o l l o w a s i m p l e s c a l i n g law:
however t h e b r a n c h i n g r a t i o ' s would be v e r y c l o s e t o t h o s e i n t h e s t a n d a r d model.
Another e x c i t i n g e f f e c t 4 would b e t h e p r o d u c t i o n of a h o s t of u n u s u a l r e s o n a n c e s . These would come i n t h r e e v a r i e t i e s :
i ) E x c i t e d W's and Z's.
i i ) E x c i t e d bound s t a t e f e r m i o n s
i i i ) E x o t i c s - t h o s e bound s t a t e s which d o n o t have t h e quantum numbers of (i) and ( i i ) . For example, a c o l o r e d f e r m i o n i c preon and a l e p t o n i c f e r m i o n i c p r e o n c o u l d b i n d t o form a s p i n one, c h a r g e -213, c o l o r
3,
r e s o n a n c e . T h i s s t a t e would decay i n t o a l e p t o n and a n t i q u a r k and c o u l d most e a s i l y be made i n an ep machine. The o b s e r v a t i o n of t h e s e r e s o n a n c e s would h e l p d e t e r m i n e t h e e x a c t form of t h e u n d e r l y i n g s t r o n g i n t e r a c t i o n .A s a f i n a l remark, I would l i k e t o s a y t h a t a l t h o u g h t h e s e composite models of t h e weak i n t e r a c t i o n s a r e a t t r a c t i v e and amusing, t h e y a r e v e r y d i f f i c u l t t o u n d e r s t a n d i n t h e c o n t e x t of s t a n d a r d grand u n i f i e d models.
The minimal GUT p r e d i c t s s i n 2 8 e x a c t l y and we would h a v e t o i n t e r p r e t t h i s a s s e r e n d i p i t y . Our whole u n d e r s t a n d i n g of GUT'S i s b a s e d on
s p o n t a n e o u s symmetry b r e a k i n g and i f t h e weak i n t e r a c t i o n s have a d i f f e r e n t o r i g i n , t h e n t h e l a r g e r p i c t u r e i s thrown i n t o d o u b t . The d i s c o v e r y of a S having t h e p r o p e r t i e s I o u t l i n e d above would r a d i c a l l y change t h e wonder- f u l u n i f i e d i d e a s developed i n t h e l a s t f i f t e e n y e a r s .
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113; R. B a r b i e r i , R. Mohapatra and A . Masiero, Phys. L e t t . E B (1981) 369; L. Abbott, E. F a r h i and A. Schwimmer, Nucl. Phys. B E (1982) 493;F. B o r d i , R. C a s a l b u o n i , D. Dominici and R. G a t t o , Univ. of Geneva p r e p r i n t (1982); Y.-P. Kuang and S.-H. H . Tye, Phys. Rev. D ( t o b e p u b l i s h e d ) ; B. Schrempp and F. Schrempp, Hamburg p r e p r i n t (1982).
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3) J. D. Bjorken, Phys. Rev. D 9 , 335 (1979); P. Q. Hung and J. J. S a k u r a i , Nucl. Phys. B x (1978) 81.
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