Bamert, Burgess, Mohapatra

8/12/2019 Bamert, Burgess, Mohapatra

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N U C L E A RP H Y S I C S B

EI.SEV1ER Nuclear Physics B 449 (1995) 25-48

M u l t i - m a j o r o n m o d e s f o r n e u t r i n o l e s s d o u b l e - b e t ad e c a y

R B a m e r t a C . E B u r g e s s b R . N . M o h a p a t r a ca lnstitu de Physique, Universitg de Neuchdtel, I Rue A.L. Breguet, CH-2000 Neuchgltel, Switzerlandb Physics D epartment, McG ill Un iversity, 3600 University St., Mo ntreal, Que bec , Canada, H3A 2T8c Department of Physics, University of Maryland, College Park, M D 2074 2, USA

Received 29 December 1994; revised 17 May 1995; accepted 18 May 1995

A b s t r a c tWe construct two new classes of models for double beta decay, each of which leads to an

electron spectrum which differs from the decays which are usually considered. One of the classeshas a spectrum which has not been considered to date, and which is softer than the usualtwo-neutrino decay of the Standard Model We construct illustrative models to show how otherphenomenological bounds can be accommodated. We typically find that, although all other boundscan be satisfied, the predicted double-beta decay rate only in one class of these models is at bestlarge enough to be just detectable in current experiments.

1 I n t r o d u c t i o n

The purpose of this paper is to present two classes of double-beta ( t i f f ) decay, whichhave not be en hitherto considere d by workers in the field, and are potenti ally experi-mentally dis tinguisha ble from the presently known kinds of decays. Besides presentinga general discussion of the phenomenology of t i f f decay in the kinds of theories whichwe consider, we also explore the other kinds of experiments to which these modelsmight be expected to contribute. In so doing we fill in the last missing items in theclassification of t i f f decay that was presen ted i n Ref. [ 1 ].

First, a brief motivation for this calculation.* Research partially supported by the Swiss NationalFoundation,NSERC of Canada, FCAR du Qu6bec, andby National Science Foundationgrant no. PHY-9119745.

0550-3213/95/ 09.50 (~) 1995 Elsevier Science B.V. All rights reservedS S DI 0550-3213 (95)0 0273- 1


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26 P. Bamert et al ./Nuclear Physics B 449 (1995) 25-48T h e p a s t t e n y e a r s h a v e s e e n a g r e a t d e a l o f e f f o r t g o i n t o t h e e x p e r i m e n t a l s e a r c h

f o r / 3/ 3 d e c a y , a n e x t r e m e l y r a r e p r o c e s s i n w h i c h t w o n e u t r o n s s i m u l t a n e o u s l y d e c a yi n t o p r o t o n s w i t h t h e a c c o m p a n i e d e m i s s i o n o f t w o e l ec t ro n s . T h i s p r o c e s s - - w h e n i to c c u r s t o g e t h e r w i t h th e e m i s s i o n o f t w o a n t i n e u tr i n o s ( f l f l 2 ~ ) - - i s p r e d i c t e d t o o c c u r a ts e c o n d o r d e r i n t h e c h a r g e d - c u r r e n t w e a k i n t e r a c t i o n s i n t h e S t a n d a r d M o d e l ( S M ) . T h ee x p e r i m e n t a l e f f o r t h a s b o r n e f r u i t i n r e c e n t y e a r s , w i t h t h e u n a m b i g u o u s o b s e r v a t i o n o ft h is d e c a y i n a n u m b e r o f d i f f e r e n t n u c l e a r i s o t o p e s [ 2 ] .

O f c o u r s e , p a r t o f t h e i n t e r e st i n t h e s e e x p e r i m e n t s w a s o r i g i n a l l y m o t i v a t e d b y t h ep o s s i b i l i t y o f o b s e r v i n g o t h e r d e c a y s , b e s i d e s t h e S M p r o c e s s . T h e r e a r e s e v e r a l o t h e rt y p e s o f d e c a y m o d e s t o w h i c h t h e s e e x p e r i m e n t s a r e s e n s i t i v e , s u c h a s t h e l e p t o n -n u m b e r v i o l a t i n g n e u t r i n o l e s s p r o c e s s ( f l f l 0 ~ ) , o r t h e d e c a y ( f i f i ~ ) i n to two e l ec t ronsp lus a neu t r a l s ca la r , q~. Bo th o f t he se r eac t i ons ha ve been p r ed i c t ed by a num ber o fp l a u s i b l e a l te r n a t iv e s t o t h e S M , s t a r ti n g w i th t h e s i m p l e G e l m i n i - R o n c a d e l l i ( G R )m o d e l o f l e p t o n - n u m b e r b r e ak i n g [ 3 , 4 ] .

T h e e x p e r i m e n t a l q u a n t i t y t h a t i s u s e d t o d i s t i n g u i s h s u c h e x o t i c d e c a y s f r o m t h er u n - o f - t h e - m i l l S M e v e n t s i s t h e s h a p e o f t h e d e c a y r a t e a s a f u n c t i o n o f t h e e n e r g i e s ,e l and / 3 2 of t he two emi t t ed e l ec t rons . Th i s shape i s s imp le s t f o r / 3 /3o~ decay , s i ncein t h i s c a se t he r a t e i s z e ro u n l e s s t he sum o f t he tw o e l ec t ron ene rg i e s , e = e~ + e2 ,t akes on a spec i f i c va lue , Q . Q r ep re sen t s t he t o ta l ene rg y t ha t c an be ca r r i ed o f f by t het w o e l e c t r o n s , a n d i s g i v e n i n t e r m s o f t h e re l e v a n t m a s s e s - - d e n o t e d r e s p e c t i v e l y b y Ma n d M p f o r t h e p a r e n t a n d d a u g h t e r n u c l e i - - b y Q = M - M p. I n a l l o f th e d e c a y s o fi n t e r e s t Q _~ 2 Me V in s i ze . O the r deca ys ( l ike /3 /32~ an d / 3 /3~ ), wh ich i nvo lve o the rr e l a t i v i s t i c decay p roduc t s i n add i t i on t o t he f i na l e l e c t rons , i n s t ead p r ed i c t a con t i nuousd e c a y d i s t r i b u t i o n t h r o u g h o u t t h e e n t i r e i n t e r v a l 2 m e


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P. Bamert et al . /Nuclear Physics B 449 (1995) 25-48 27

Electron Spectrum

zo i :n = 7 n = 5 n = n = l

E =E 1 +E 2

J The table presented here d iffers slightly from the table in Ref. [ I ]. A trivial change is our introduction ofnew categories ID and IE, even though these are in practice indistinguishable from IIC . Also, the index listedhere for category liE reflects the results of the present paper, and d iffers from the preliminary guess for thisvalue which was given in Ref. [ 1 .

e l e m e n t f o r t h e fi n a l n e u t r i n o s i m p l i e s a n i n d e x n ( S M ) = 5 . F o r t h e s c a l a r d e c a y ,f l f l ~ , o f t h e G R m o d e l , o n t h e o t h e r h a n d, t h e p h a s e s p a c e o f t h e l o n e s c al a r i m p l i e sn ( G R ) = 1 . T h e d i f f e r e n c e b e t w e e n t h e s e p r e d i c t i o n s ( c f . F i g . 1 ) f o r m s t h e b a s i s f o rt h e e x p e r i m e n t a l d i s c r i m i n a t i o n o f t h e s e t w o m o d e l s .

T w o d e v e l o p m e n t s h a v e r e c e n t l y p r o v o k e d a t h eo r e ti c a l r e - e v a lu a t i o n [ 5 - 7 ] o f t h ek i n d s o f n e w p h y s i c s t o w h i c h t h e c u r r e n t e x p e r i m e n t s c a n b e s e n s i t i v e . T h e f i r s t o ft h e s e w a s t h e a d v e n t o f p r e c i s io n m e a s u r e m e n t s o f th e p ro p e r t i e s o f t he Z r e s o n a n c ea t L E P [ 8 ] , w h i c h h a s r u l e d o u t t h e o r i g i n a l G R - t y p e m o d e l s f o r n e w p h y s i c s . T h eo t h e r d e v e l o p m e n t h a s c o m e f r o m t h e t i f f e x p e r i m e n t s t h e m s e l v e s , s t a r t i n g w i t h t h ea p p e a r a n c e o f s e v e r a l i n d i c a ti o n s o f a n e x c e s s o f e le c t r o n s j u s t b e l o w t h e e n d p o i n t i ns o m e o f t h e e x p e r i m e n t a l d a t a fo r s e v er a l t y p e s o f d e c a y s [ 9 ] . A l t h o u g h t h e e v id e n c ef o r t h e e x c e s s h a s s i n c e d i m i n i s h e d [ 1 0 ] , t h e c o n c l u s i o n r e a c h e d b y t h e t h e o r e t ic a lr e - e v a l u a t i o n s t il l s t a n d s : v i a b l e t h e o r i e s w h i c h c a n p r o d u c e o b s e r v a b l e t i f f d e c a y a r ep o s s i b l e , a n d c a n b e c o n s i s t e n t w i t h a l l o t h e r , n o n - t i f f , e x p e r i m e n t a l l i m i t s , b u t o n l y i ft h e n e w p h y s i c s h a s r a th e r d i f f e r e n t p r o p e r t i e s th a n h a v e p r e v i o u s l y b e e n a s s u m e d b a s e do n e x p e r i e n c e w i t h G R - t y p e m o d e l s .

O n e o f t h e m a i n s u r p r i s e s w h i c h a r o s e f r o m t h i s th e o r e t i c a l r e - e x a m i n a t i o n h a s b e e nt h e o b s e r v a t i o n t h a t n e w p h y s i c s c a n b e v e r y g e n e r a l l y d i v i d e d i n to a s m a l l n u m b e r o fc l a s s e s - - d e p e n d i n g o n h o w i t a d d r e s s e s a f e w b a s i c i s s u e s - - a n d t h a t t h e e x p e r i m e n t a lf e a t u r e s t h a t a n y m o d e l p r e d i c t s f o r t i f f d e c a y i s a r o b u s t i n d i c a t o r o f t h e c la s s t o w h i c hi t b e l o n g s [ 1 ] . ( T h e s e i s s u e s a n d c a t e g o r i e s a r e s u m m a r i z e d i n T a b l e 1. I )

Fig. 1. The t i f f spectrum as a fun ction of the two electrons' total kinetic energy for various choices of thespectral index n. n = 1 corresponds to the dotted line, n = 3 is the dashed line, n = 5 is the solid lineand n = 7 is the dash-dotted line. All four curves have been arbitrarily assigned the sam e maximal value forpurposes o f com parison.


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28 P . B a m e r t e t a l . / N u c l e a r P h y s i c s B 4 4 9 ( 1 9 9 5 ) 2 5 - 4 8Table 1A list of alternatives for m odelling double beta decay.

L e A new scalar: j S f l o v Dominan t Spec t r a lscalar decay indexIA broken does not exist Yes None N.A.IB broken is not a Goldstone boson Yes f l f l ,o n = 1IC broken is a Goldstone boson Yes f l f l n = 1ID broken is not a Goldstone boson Yes f l f l~ o, n = 3IE broken is a Goldstone boson Yes , S f l~ o n = 3I I A u n b r o k e nIIB unbrokenIIC unbrokenl i D u n b r o k e nl i E u n b r o k e n

does not exist No None N.A.is not a Goldstone boson ( L e = - 2 ) N o flfl~o n = 1is not a Goldstone boson Le = - - 1 No flfl~o~ n = 3is a Goldstone boson ( L e = - 2 ) N o flfl~o n = 3is a Goldstone boson Le = - - 1 No /3/3~o~o n = 7

A s i s a r g u e d i n R e f . [ 1 ] , t h e o b s e r v e d a b s e n c e o f f lf l0 ~ d e c a y s a t t h e e n d p o i n t o f t h ee l e c t ro n s p e c t r u m s t r o n g l y c o n s t ra i n s t h e c la s s e s o f m o d e l s f o r w h i c h l e p t o n n u m b e r i sb r o k e n i . e . c l a s s e s I A - I E ) . I n p a r t i c u l a r, i t i m p l i e s t h a t a l l o f t h e s c a l a r - e m i t t i n g d e c a y si n t h e s e c l a s s e s a r e e s s e n t i a l l y in d i s t i n g u i s h a b l e f r o m t h e i r c o u n t e r p a r t s in t h e l e p t o n -n u m b e r p r e s e r v i n g c l a s s e s . T h a t i s t o s a y, s o l o n g a s fl fl 0v d e c a y s a r e n o t o b s e r v e d ,m o d e l s i n c l a ss I B a n d I C a r e a t p r e s e n t e x p e r i m e n t a l l y i n d i s ti n g u i s h a b l e f r o m t h o s e i nc l a s s II B . S i m i l a r l y , c l a s s e s I D a n d I E c a n n o t b e d i s t i n g u i s h e d f r o m c l a s s I I C .

I n t e r e s t i n g l y , a l l o f t h e m o d e l s w h i c h h a d b e e n c o n s i d e r e d p r e v i o u s l y , a s w e l l a s m o s to f t h e m o r e r e c e n t p r o p o s a l s , f a l l i n t o o n l y a f e w o f t h e s e v e r a l p o s s i b l e c a t e g o r i e s , c a s e sI B a n d I C o f T a b l e 1 . T h e o t h e r, u n o r t h o d o x , c a t e g o r i e s re m a i n e s s e n t i a l l y u n e x p l o r e d .T h e t w o e x c e p t i o n s t o t h i s s t a t e m e n t a r e c a s e I I D , w h i c h w a s d i s c o v e r e d a n d s t u d i e di n R e f . [ 5 ] , a n d c a s e I E , w h i c h c o n t a i n s a s u p e r s y m m e t r i c m o d e l [ 1 1 ] , i n w h i c h t w os c a l a r s a r e e m i t t e d i n t i f f d e c a y : f l f l . U n f o r t u n a t e l y , t h i s la s t m o d e l h a s s i n c e b e e nr u l e d o u t b y t h e m e a s u r e m e n t s a t L E E B o t h o f th e s e n ew c l a ss e s o f m o d e l s p r e d i c t at i t l e s p e c t r u m h a v i n g n = 3 . T h e r e s u l t i n g s h a p e i s i n t e r m e d i a t e b e t w e e n t h e u s u a l S Ma n d G R s p e c t r a , a n d c a n b e d i s t i n g u i s h e d f r o m t h e s e in c u r r e n t e x p e r i m e n t s [ 1 2 , 1 3 ] .T h e p u r p o s e o f t h e p r e s e n t p a p e r i s t o e x p l o r e i n d e t a i l t h o s e c l a s s e s o f t h e o r i e s o fT a b l e 1 w h i c h h a v e n o t b e e n c o n s i d e r e d t o d a t e. K e e p i n g in m i n d t h e i n d i s t i n g u i s h a b i l i t yo f c l a s s e s I B , I C a n d I I B , a s w e l l a s I D , I E a n d I I C , w e s e e t h e r e a r e t w o t y p e s o fm o d e l s w h i c h w e m u s t h e r e c o n s i d e r: t h o s e o f c l a s se s I IC a n d I I E , b o th o f w h i c hi n v o l v e t i f f d e c a y a c c o m p a n i e d b y t h e e m i s s i o n o f t w o , r a t h e r t h a n j u s t o n e , s c a l a rp a r t ic l e s . T w o s c a la r s m u s t b e e m i t te d b e c a u s e o f l e p t o n - n u m b e r c o n s e r v a t io n , a n d t h ec h a r g e a s s i g n m e n t s o f t h e l i g h t p a r t i c l e s . M o d e l s i n t h e s e tw o c a t e g o r i e s d if f e r f r o mo n e a n o t h e r a c c o r d i n g t o w h e t h e r o r n o t t h e l i g h t s c a l a r p a r t i c l e t h a t is e m i t t e d i s aG o l d s t o n e b o s o n .W e f ir s t e x a m i n e m o d e l s o f c a t e g o r y I I C t o s h o w t h a t t h e s e c a n b e a l t e r e d t o e v a d ed e t e c t i o n a t L E P , a n d w e t h e n w o r k t h r o u g h t h e c o m p l e t e l y n e w c a t e g o r y , I I E , w h i c hp r e d i c t s a s p e c t r a l i n d e x , n = 7 , t h a t i s c o m p l e t e l y d i f f e r e n t f r o m a l l p r e v i o u s c a s e s .


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P. Bamert et al./Nuclear Physics B 449 (1995) 25-48 29S ince t h i s i ndex i s l a r g e r t h an t h at f o r t h e S M d e c a y - - n S M ) = 5 - - t h e r e su l ti n gspec t r a l shape i s s o f t e r t h a n t h e p r e s e n t l y o b s e r v e d /3/32~ spec t r a . Th i s k ind o f so f t e rs p e c t r u m i s m u c h m o r e d i f f i c u l t t o d i s t i n g u i s h f r o m t h e e x p e r i m e n t a l b a c k g r o u n d , a n ds o w e e x p e c t t h e e x p e r i m e n t a l l i m i t f o r t h e b r a n c h i n g r a t i o i n t o t h e n e w n = 7 d e c a ym o d e t o b e c o m p a r a t i v e l y w e a k .

On the t heo re t i c a l s i de , we f i nd t ha t t he nuc l ea r f o rm f ac to r s wh ich appea r i n a l l o ft h e n e w / 3 / 3 ~ d e c a y s a r e w e l l u n d e r s t o o d , s i n c e t h e y a r e t h e s a m e a s t h o s e t h a t a p p e a rin t he more we l l -known decay modes , such a s fo r /3 /30~ and /3 /3~o in t he GR mode l . Wef in d t h a t u s i n g t h e s e m a t r i x e l e m e n t s t o e st i m a t e th e p o t e n ti a l si z e o f t h e / 3 / 3 ~ d e c a yra t e t yp i ca l l y g ive s a r e su l t wh ich can be c lo se t o t he p r e sen t expe r imen ta l l im i t f o r t hen = 3 d e c a y s , p r o v i d e d w e r e l a x t h e c o n d i t i o n s t h a t a r e c a s t o n t h e r e l e v a n t m a s s e s b yb ig bang nuc l eosyn thes i s cons t r a in t s . The e s t ima ted r a t e fo r t he n = 7 decays appea r s t obe t oo sma l l t o be de t ec t ab l e .

T h i s p a p e r i s o r g a n i z e d a s f o l lo w s . T h e n e x t se c t io n c o m p u t e s t h e /3 /3 ~ 0 d e c a y r a t e f o rt h e t w o c l a s s e s o f d e c a y s I I C a n d l i E ) . W e d o s o t o v e r i f y t h e ir s p e c t ra l i n d e x - - w h i c hi s t h e i r e x p e r i m e n t a l s i g n a t u r e - - a n d t o d e t e r m i n e h o w t h e r e s u l t i n g r a t e s d e p e n d o n t h ec o u p l i n g s i n u n d e r l y i n g m o d e l s . F o r c la s s I I C h a v i n g n = 3 ) w e c o n s i d e r tw o t y p e s o fd e c a y m e c h a n i s m s : t h o s e w h i c h p r o c e e d d u e t o th e e x c h a n g e o f n ew s t er il e n e u tr i n o s,a n d t h o s e w h i c h p r o c e e d t h r o u g h t h e i n t e r ac t io n s o f n e w s c a l a r p a rt ic l e s. R e p r e s e n t a t i v em ode l s fo r decays i n c l a s s I IC a r e t hen con s t ruc t ed i n Sec ti o l ]s 3 , and t hose fo r c l a s s l iEin S e c t i o n 4 . S e c t i o n 5 th e n s u m m a r i z e s t h e m o s t im p o r t a n t p h e n o m e n o l o g i c a l b o u n d sf o r t h e s e m o d e l s . O u r c o n c l u s i o n s a r e f i n a l l y s u m m a r i z e d i n S e c t i o n 6 .

2. Tw o sca lar / f ie decaysBefo re cons t ruc t i ng mode l s i n de t a i l , we f i r s t pause he re t o r eco rd exp re s s ions fo r

the decay r a t e i n to two e l ec t rons p lu s two sca l a r s , F ( f l f l~ ) . W e d o s o i n o r d e r t od i s p l a y h o w t h e f l f l ~ o r a t e d e p e n d s o n t h e v a r i o u s m a s s e s a n d c o u p l i n g s o f t h e t h e o r y .W e m a y t h e n u s e t h i s d e p e n d e n c e t o s e e w h a t k i n d s o f n e w p a r t i c l e s c a n b e u s e d t og e n e r a t e a n o b s e r v a b l e s i g n a l i n / 3 /3 e x p e r i m e n t s . W e d e f e r t h e d i s c u s si o n o f a ny f u r t h e rp h e n o m e n o l o g y , w h i c h r e q u i r e s t h e c o n s t ru c t i o n o f a n e x p l ic i t m o d e l , t o th e f o l l o w i n gsec t i ons .

W e c o n s i d e r h e r e t h r e e k i n d s o f s c e n a r io s . F i r s t w e c o n s i d e r t w o t y p e s o f m o d e l s f o rw h i c h / 3 / 3 ~ d e c a y i s m e d i a t e d b y th e a d m i x t u r e o f o r d i n a r y n e u t r in o s w i t h n e w s t e r il ef e r m i o n s . T h e t w o k i n d s o f s u c h d e c a y s w h i c h w e c o n s i d e r a re t h o s e f o r w h i c h t h e s c a l a rem is s ion i s e i t he r de r iva t i ve ly supp re s sed i .e . c a se l iE , w i th n = 7 ) , o r no t c a se I ICwi th n = 3 ) . F ina l l y we cons ide r t he a l t e rna t i ve fo r wh ich/3 /3~o~o decay occu r s becau seo f t he i n t e r ac t i ons o f new types o f s ca l a r pa r t i c l e s . Such sca l a r -med ia t ed decays cana l so a r i s e w i th o r w i th ou t de r iva t i ve supp re s s io n fo r t he f ina l s ca l a r em i s s ion c l a s se sI I E o r I I C ) , b u t s i n c e t h e d e c a y r a t e f o r t h e c l a s s I I E m o d e l s a r e g e n e r a l l y v e r y s m a l l ,we do no t pu r sue t hem in de t a i l f o r t he s ca l a r -med ia t ed ca se .


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30 P Bamer t e t a l . /Nuclear Phys ics B 449 (1995) 25-48

11

v

Pe

. . . . . . . . . . . . . t p

. . . . . . . . . . . . . q 0

n

Fig . 2 . The Feyn m an g raph rep re sen t ing the l ead in g o rder con t ribu tion to the two sca la r/3 /3~ decay th roughs te r i l e l ep ton exchange .2 .1 . F e r m i o n - m e d i a t e d n = 3 d e c a y s

C on s ide r f ir s t a theo ry o f neutr inos, /- - i and N~, wh ich respe c t ive ly car ry l ep ton nu m be rLe ( ~ ' i ) = + 1 a n d L e ( N a ) = 0 , and which a re coupled to a sca la r , q~ , which has l ep tonn u m b e r L e ( q S ) = + 1 . T h e m o s t g e n e r a l r e n o r m a l i z a b le a n d L e - c o n s e r v in g Y u k a w acoup l i ngs i nvo l v i ng t he s e f i e l d s i s

/ ] y u k = - -~ i ( Aia~/L + BiaS/R ) Na qb + h . c . , ( 2 )w h e r e A i a a n d B i a r ep r e s en t a r b i t r a r y Yukawa- coup l i ng m a t r i c e s , and Y t and TR d e n o t et he u s ua l p r o j ec t i ons on t o l e f t - and r i gh t - handed s p i no r s . We us e m a j o r ana s p i no r s he r et o r ep r e s en t t he neu t r inos , s o t he con j uga t e wh i ch appea r s i n Eq . ( 2 ) i s F i = v [ C - l ,w h e r e C i s th e c h a r g e - c o n j u g a t i o n m a t r ix . S u p p o s e a l s o t h a t a n u m b e r o f r i g h t -h a n d e dneu t r i nos a r e i nc l uded i n t he t heo r y , i n o r de r t o f o r m gene r i c l ep t on num ber cons e r v i ngm as s e s , m , , , f o r the Le = 1 s ta tes , and tha t the Le = 0 neu t r i nos , N~ , have gene r i cm a j o r ana m as s e s , mN , , .A s l o n g a s a n y o f th e v i ' s pa r t i c i pa t e i n t he cha r ged - cu r r en t weak i n t e r ac t i ons , t hec o u p l i n g s o f E q . ( 2 ) g e n e r i c a ll y p r o d u c e f l fl ~ d e c a y d u e t o t h e F e y n m a n g r a p h o fF i g . 2 . ( Two s ca l a r s m us t be em i t t ed i n t h i s decay due t o cons e r va t i on o f L e . ) E v e nt houg h F i g . 2 a r i s e s a t tr e e l eve l , t he f ou r - m o m e n t um o f t he exchange d neu t r i no i s no td e t e r m i n e d b y e n e r g y a n d m o m e n t u m c o n s e r v a ti o n . T h i s i s b ec a u s e t h e a m p l i tu d e f o rt he decay o f t he t wo i n i t i a l neu t r ons m us t be convo l u t ed wi t h t he wave f unc t i on o f t hei n i t i a l nuc l eus . The i m por t an t m om en t um s ca l e i n t he i n t eg r a t i on ove r t he exchangedneu t r i no i s the r e f o r e s e t by t he s ca l e o f m o m e n t a t o wh i ch t he nuc l eons w i t h i n t he i n i ti a land f ina l nuc le i have access . S ince th i s sca le i s much la rger than the energy, Q, tha t i sava i l ab l e f o r t he f i na l e l ec t r ons and s ca l a r s , i t i s a good app r ox i m a t i on t o neg l ec t t heou t go i ng s ca l a r and e l ec t r on ene r g i e s i n t he decay am pl i t ude . Eva l ua t i ng t he g r aph i nt h is a p p r o x i m a t i o n g i v e s t h e f o l l o w i n g e x p r e s si o n f o r t h e f l f l ~ d e c a y r a te :


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P Bamert et aL/Nuclear Physics B 449 (1995) 25-4 8 31( G F C O S O c ) 4 2( Q - e l - - e 2 ) 3 I - I P k e k F ( e k ) d e k , ( 3 )d r ( f l f l ~ ) - 8( 2~r ) 5 k = l

w h e r e G F i s Fe r m i s cons t an t , and O c i s t he Cabb i bo ang l e . The quan t i t y A ( B B ~ )r ep r e sen t s t he i n t eg r a l

.A(/3/3~o~) = ( 3 ~ ) 1/2f d 4 g V e ~ , i g e v jJ ~ i ja W l a 't z

x Z . . ( 2 r r ) 4 (g 2 + m 2, _ ie) (g2 + mZj _ ie) (g2 + m2a _ ie) ( 4 )lJa

w h er e V e,,, i s t he K oba yash i - M aska w a- t ype m a t r i x f o r t he lep t on i c cha r ged cu r r en t . Thef a c t o r JV~,Tai n the num er a t o r o f Eq . ( 4 ) deno t e s t he f o l l ow i n g exp r e s s i on :

J ~ i . j a = - - - _ g 2 ) [ a i a B . j a m v i t - A j a B i a m v j k B i a B j a m N a ] n - a i a A j a m v l m v j m N . . 5 )Fi na l l y , the t enso r Wu, , i s an i nde pend en t f unc t i on o f t he space - an d t i m e - l i ke pa r ts ,

go and 11[, o f g , , i n t he r e s t f r am e o f the d ecay i ng nu c l eus , w h i ch encod es t he nu c l ea rm a t r i x e l em en t t ha t i s r e l evan t t o t he decay [ 5 ]

Wu,, _= (2~ ) 3 E/ --E- -E7 f [J ~ , ( x ) J , , (o ) ]~ d 4 x ( N I T * IN) e iex. ( 6 )H e r e J u = 2 g y u y L d d e n o t e s t h e a p p r o p r i a t e h a d r o n i c c h a r g e d c u r r e n t . T h e p r e f a c t o ri n v o l v i n g t h e e n e r g y - t o - m a s s r a t i o - - E l M and E ' / M I, f o r t he pa r en t and daugh t e r nu -c l e i r e spec t i ve l y - - i s r equ i r ed i n o r de r t o ensu r e t ha t Wu~ t r ans f o r m s unde r Lor en t zt r ans f o r m a t i ons a s a t enso r . The f ac t o r o f ( 2 ~ ) 3 is pu r e l y conven t i ona l .

The r e a r e s eve r a l f e a t u r e s i n t he se exp r e s s i ons t ha t a r e no t ew or t hy .( i ) Co m par i son o f Eqs . ( 3 ) and ( 1 ) show s t ha t t he spec tr a l i ndex f o r t h is decay i s

n = 3 . T h e a d d i ti o n a l tw o p o w e r s o f ( Q - e ) , i n co m p a r i s o n w i t h t h e G R m o d e l ,a r i s e he r e s i m p l y due t o t he add i t i ona l phase space vo l um e o f t he s econd s ca l a r .

( i i ) Eq . ( 4 ) depe nds on p r ec i s e l y the s am e com b i na t i on o f nuc l ea r f o r m f ac t o rs ,W t Z t * = W F - - W G T , as do f l f l 2 v and f l f lo~ decays , as wel l as flfl~, d e c a y i n G R - t y p em o d e l s [ 5 ] . ( F o r f u t u r e u s e w e e x p r e ss W ~' u i n te r m s o f t h e F e rm i a n d G a m o w -Te l l e r m a t r i x e l em en t s , WF = WOO and w a r = ~ = 1 W i i. Ref . [ 5 ] g i ve s W r and W a ras exp l i c i t exp r e s s i ons i n t e r m s o f t he nuc l ea r m a t r i x e l em en t s o f va r i ous neu t r i nopo t en t i a l s . ) S i nce t he se pa r t i cu l a r com bi na t i ons o f nuc l ea r m a t r i x e l em en t s havebeen w e l l s t ud i ed w i t h i n t he con t ex t o f t he se o t he r p r oces se s [ 14 - 16 ] , t he r e isre la t ive ly l i t t l e uncer ta in ty in the es t imate for the to ta l ra te for th i s type of /3 /3~odecay.

( i i i ) The d i f f e r en t i a l decay d i s t r i bu t i on f o r t h i s decay a s a f unc t i on o f t he open i ngang l e , 0 , be t w een t he t w o e l ec t r ons i s exac t l y t he s am e a s i t i s f o r / ~ / ~ 2 v , /~ / ~ 0 vand /3 /3~ decay s ( o f bo t h t he G R t ype and t he new /3/3~o decays o f t y pe I I D ) . I ti s g i ven exp l i c i t l y ( neg l ec t i ng t he m u t ua l r epu l s i on o f t he ou t go i ng e l ec t r ons ) by


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32 P. Bamert et al./Nuclear Physics B 449 (1995) 25-4 8d F - 1 - - VlV2 c o s O ) , 7 )F d c o s O 2

w h e r e vi = pi /e. i , f o r i = 1 , 2 , i s the s peed o f the co r r e s po nd i ng e l ec t r on in t henuc l ea r r e s t f r am e .

2 .2 . F e r m i o n - m e d i a t e d n = 7 d e c a y sW e nex t cons i de r the ca s e o f dou b l e s ca l a r em i s s i on , bu t i n the s pec i a l c a s e whe r e t he

am pl i t ude f o r em i t t i ng t he s ca l a r s i s s uppr e s s ed by t he s ca l a r ene r gy . Th i s i s t he ca s ew hen t he s ca la r s have pu r e l y de r i va t ive coup l i ngs , s uch a s wh en t he s ca l a r is a Go l ds t o neb o s o n w h i c h c a r r i e s a n u n b r o k e n l e p t o n n u m b e r [ 5 ] . F o r s u c h m o d e l s t h e e x p r e s s i o ng i ven i n Eq . 4 ) f o r t he m a t r i x e l em en t , , ,4 , van i s hes i den t ica l l y , and s o i t i s neces s a r y towor k t o h i ghe r o r de r i n t he ex t e r na l s ca l a r m om en t a i n o r de r t o ob t a i n a non - van i s h i ngr a t e . S i nce i t t u r ns ou t t ha t t he f i r s t non - van i s h i ng con t r i bu t i on t o t he am pl i t ude a r i s e sa t quad r a t i c o r de r i n t he s ca l a r m om en t a , t he ca l cu l a t i on i s qu i t e cum ber s om e whenp e r f o r m e d d i r e c tl y w i t h th e c o u p l i n g s o f E q . 2 ) .

A be t t e r way t o p r oceed i s t o pe r f o r m a chan ge o f va r i ab le s s o t ha t t he de r i va ti vecoup l i ng na t u r e o f t he Go l ds t one bos ons i s m an i f e s t f r om t he ou t s e t . Once t h i s ha s beendone , t he t r i l i nea r coup l i ng t o neu t r i nos o f a Go l ds t one bos on ca r r y i ng Le = 1 , becom es

/~gb = - i Pi~llX X ia')lL 4- Yia~R) Na o31zq~ -} -h.c ., 8 )whe r e t he coe f f i c i en t s Xi,, and Y/a a r e com pu t ab l e [ 5 ] i n te r m s o f Aia and Bia o f E q . 2 ) ,a s we l l a s t he com pon en t s o f the b r o ken - s ym m e t r y gene r a t o r , Q , f o r wh i ch ~b i s theG o l d s t o n e b o s o n .

T h e c o n t r i b u t io n t o/ 3 /3 d e c a y i n t h e se m o d e l s a r is e s f r o m t h e s a m e F e y n m a n g r a p has be f o r e , F i g . 2 , bu t wi t h the i n t e r ac ti on o f Eq . 8 ) a t e ach ve r tex . Eva l ua t i on o ft h i s g r aph g i ve s a f a i r l y com pl i ca t ed exp r e s s i on f o r t he decay r a t e , bu t t h i s exp r e s s i ons imp l i f i es grea t ly in the spec ia l cas e o f pure ly l e f t -h and ed coupl ings , fo r w hich Y/a = 0 .S i nce t h i s c a s e i s s u f f i c ien t t o an a l ys e t he m ode l s we cons i de r i n l a t e r s ec t ions , w e j u s tqu ote h ere th e dec ay ra te in the l im i t Y/a = 0 :

2d F ( f l ~ o ~ ) - (GFcSOC)4g(27r) ~ ( / [ ~ / ~ P ) 2 ( Q - e l - - ~ 2 ) 7 I - ~ P k S k F ( S k ) k = l dSk , 9 )

where the new quant i ty , A /3 /3~o) , r epresent s the in tegra l( 4 ~ 1/2

f d4~ W e~ W evj~ijaWtX'u . ( 1 0 ) 27 r)4 g2 + m~, i e ) ( g 2 + m ~ j - i e ) ( e 2 + m 2 - i e )t J a - - N,,

A p a r t fr o m i ts n o r m a l iz a t io n , E q . 1 0 ) , d i f fe r s f r o m E q . 4 ) , o b t a i n e d p r e v io u s l y , o n l yb y t h e r e p l a c e m e n t .Afiija --4 J~iia, w h e r e


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P. Bamert et al./Nuclear Physics B 449 (1995) 25-48 33J ~ija ~ ( - - g e ) X i a X j a m N ~ . ( l l )

Once aga i n t he r e a r e s eve r a l p r ope r t i e s o f t he above exp r e s s i ons t ha t bea r em phas i s .( i ) Th e s pec t ra l i ndex f o r t h i s decay i s n = 7 . Th e f ou r add i t iona l power s o f ( Q - e ) ,

in com p ar i s on w i t h Eq . ( 3 ) , a r i s e becaus e each s ca la r - em i s s ion ve r t ex o f Eq . ( 8 )i s exp l i c i t l y p r opor t i ona l t o t he s ca l a r f ou r - m om en t um .

( i i ) T h e d e c a y r a t e d e p e n d s o n l y o n t h e fo r m f a c t o r c o m b i n a t i o n W U u = W F -- w 6 r ,j u s t a s f o r t he non- de r i va t i ve l y coup l ed ca l cu la t i on . ( Th i s i s by con t r a s t wi th t hes i t ua t ion f o r / 3 / 3 decay , w he r e t he m a t r i x e l em en t s f o r t he de r i va t i ve ly s uppr e s s eddecay, c lass I ID , d i f fe r f rom thos e for the emiss ion o f a gene r ic sca la r, c l ass I IB. )

( i i i ) The d i f f e r en t i a l decay d i s t r i bu t i on a s a f unc t i on o f t he e l ec t r on open i ng ang l er em a i ns exac t l y t he s am e a s f o r all o f t he o t he r decays , a s in Eq . ( 7 ) .

2 .3 . S c a l a r - m e d i a t e d n = 3 d e c a y sThe p r ev i ous ca l cu l a t i ons p r oduced s i m i l a r exp r e s s i ons f o r t he decay am pl i t udes , c f .

Eqs . ( 4 ) and ( 1 0 ) , l a r ge l y becaus e t he s am e Feynm an g r aph is r e s pons i b l e f o r i t i n bo t hc l a s se s o f m ode l s . I n t h is s ec t i on we exp l o r e an a l t e rna t i ve c l a ss o f m o de l s f o r w h i chf l f l ~ deca y p r oc eeds v i a a d i f f e r en t p r oces s : t he excha nge o f an exo t i c s et o f s ca la r s.

Co ns i de r the r e f o r e s upp l em en t i ng t he SM by a co l l e c t i on o f new s ca la r s , wh i ch w ec l a s s i f y u s i ng t he i r l ep t on and e l ec t r i c - cha r ge quan t um num ber s , Le and Q. Bes i de s t hevery l ight (Q, Le) = (0 , +1 ) par t i c le , ~p , which i s emi t t ed in the decay , we imagine a l soadd i ng t wo o t he r new c l a s s e s o f f i e l d s , X i and A~,, wh ich resp ec t ive ly ha ve the ( Q , L e )a s si g nm e n t s: ( + l , - 1 ) a n d ( - 2 , + 2 ) . T h e s e q u a n tu m n um b e r s p e r m it t h e f o ll o w i n gtypes of L~- invar iant t r i l inear - sca la r , and Yukawa coupl ings :

l [,zi.j t, i = - - ~ , i , ) ( jAa + h.c . , 12)aC.yuk = -- ~ -e( hL yt . + haRyR)e c Aa + h . c . , 1 3 )

i n add i t i on t o t he cha r ged - cu r r en t coup l i ng~ . s c c = - i A i g W t Z ( ) ( i O # ~ o - c g ~ z X i ( p ) + b . c . 1 4 )

H er e e c deno t e s t he D i r ac con j ug a t e o f t he e l ec t r on f ie ld , and ,~i pa r am e t e r i ze s t hes t r eng t h o f t he s ca l a r cha r ged - cu r r en t i n t e r ac t i on r e l a t i ve t o t he u s ua l SUL( 2) gaugecoup l i ng , g . Fo r t he s ake o f keep i ng ou r f i na l exp r e s s i ons s i m p l e , we a s s um e f r om he r eon tha t the index a t akes only on e va lue , so tha t there is on ly one f i eld , A, having theq u a n t u m n u m b e r s ( Q , L e ) = ( - 2 , + 2 ) . W e t h e r e f o r e o m i t th i s i n d e x f r o m t h e c o u p l i n gparameters as these a re def ined above: i . e . ] ~ i j a _ _ _ [ d j ~ j , and h a ,R ~ hL.R. W e a l so deno tet he m as s e i genva l ues f o r X i and A r e s pec t i ve l y by M i and MA.

Wi t h t he s e coup l i ngs and i n t e r ac t i ons f l f l d e c a y is m e d i a t ed b y t h e F e y n m a n g r a p hof F i g . 3 . E va l ua t i on o f t h i s g r aph g i ve s t he f o l l owi n g decay r a t e :


10/24

34 P B a m e r t e t a l . / N u c l e a r P h y s i c s B 4 4 9 ( 1 9 9 5 ) 2 5 - 4 8

wI

_ _

W t ' ' - . q0n

Fig. 3. The Feyn m an graph contributing t o/ 3/ 3~ decay purely due to the self-interactions of the Higgs sector.

( G F C O S O c ) 4 12 2d F s ( f l f l ~ ) - 8(2~r) 5 [ . A s ( f l f l ~ ) ( Q - - e l - - e 2 3 1 - I P k e k F ( e k ) d e k ,k=l

( 1 5 )w h e r e t h e s c a l a r- m e d i a te d d e c a y a m p l it u d e , . A s ( f l f l ~ ) , i s

( 3 2 ~ ' / 2 h R dag AihJ IzO [ WFg~ W |2(27 / ) 4 (eiT-ff{- gT- j-i,) . ( 1 6 )

In th is express ion h = x / I h L I 2 I h R I2 , andsen2 ~ 1/2

R = 1 + (h)2e---~ie2j , ( 1 7 )wi th s = 2Re[hLh ,~] and me deno t ing the e l ec t ron mass .

These exp re s s ions d i f fe r f rom tho se fo r the fe rmion-m ed ia ted decays in a num ber o fin te re s ting w ays .( i ) Th e spec t ra l index for th is decay is de te rm ined by phase space, a nd is n = 3 . This

is because the assu m ed ~o cou pl ing s a re not der iva t ive ly suppressed . A c lass ofs imi la r n = 7 decays can be ob ta ined by impos ing the add i t iona l r equ i remen t o fde r iva tive coup l ing . We do no t cons ide r such decays exp l i c i tly he re.

( i i ) Once mo re the decay ra t e depends o n ly on the fami l i a r fo rm fac to rs, W F a n dw c r , a l though th i s t ime they do no t a r i s e in the pa r t i cu la r l inea r combina t ionW l X t z = W F - - W G T .

( i i i ) I f the A-e lec t ron cou pl ing has a r ight-h and ed com pon ent , hR 4= 0 , then the d i f fe r-en t i a l decay d i s t r ibu t ion a s a func t ion o f the e l ec t ron open ing ang le dif fers f r o mtha t o f a l l o f the o the r decays we have cons ide red . Assuming on ly one spec ie s o fA, i t s expl ic i t express ion is


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P Bamert et al./Nuclear Physics B 449 (1995) 25-48 35l d F l ( l _ V lV2 )F d cos /9 = 2 ~ cos 0 , 18 )

s o t h e d e v i a t i o n o f t h e d e c a y d i s tr i b u ti o n f r o m t h a t o f E q . 7 ) i s c o n t r o l l e d b y t h ed i f f e r e n c e b e t w e e n R a n d u n i t y .

W e n e x t t u r n t o t h e c o n s t r u c t i o n o f s o m e u n d e r l y i n g m o d e l s f o r e a c h o f t h e s e t y p e so f d e c a y s .

3 M o d e l b u i l d i n g : T h e c a s e o f s p e c t r a l i n d e x n = 3

W e f i r s t c o n s i d e r m o d e l s f o r w h i c h t h e d o m i n a n t c o n t r i b u t i o n t o t i f f d e c a y i s t h r o u g ht h e e m i s s i o n o f t w o s c a la r s , b u t f o r w h i c h t h e e m i s s i o n o f t h es e s c a l a r s i s n o t s u p p r e s s e db y p o w e r s o f t h e s c a l a r m o m e n t u m , i . e . m o d e l s o f t y p e I I C . A s w a s m e n t i o n e d e a r l i e r ,and a s i s d i s cus sed i n som e de t a i l i n Re f s . [ 1 , 5 ] , t h i s a l so i n c l u d e s t h o s e m o d e l s f o rw h i c h t h e l i g h t s c al a r i s t h e G o l d s t o n e b o s o n f o r l e p t o n n u m b e r c l a s s I E ) , e v e n t h o u g ht h e c o u p l i n g s o f s u c h a s c a l a r c a n a l w a y s b e p u t i n to th e d e r iv a t i v e f o r m o f E q . 8 ) . 2

I n t h e f o l l o w i n g s e c t i o n s w e c o n s i d e r a f e w r e p r e s e n t a t i v e m o d e l s w h i c h p r o d u c e t h i sk i n d o f d e c a y .

3 . 1. D e c a y s m e d i a t e d b y s t er i le n e u t r in o s

A m e c h a n i s m f o r p r o d u c i n g f l f l~ d e c a y , w i t h o u t r u n n i n g i n t o c o n f l i c t w i t h t h e L E Pbounds , i s t o r equ i r e t he l i gh t s ca l a r t o be an e l ec t roweak s i ng l e t , wh ich coup l e s t o t heu s u a l e l e c t r o w e a k n e u t r i n o s t h r o u g h t h e i r m i x i n g w i t h v a r i o u s s p e c i e s o f s t e r i l e n e u t r i -nos . Th i s r e su l t s i n a va r i a t i on o f t he o ld s i ng l e t -m a jo ron mo de l [ 17 ] . I n t h i s way anu n a c c e p t a b l e c o n t r i b u t io n t o th e i n v i s ib l e Z w i d t h c a n b e a v o i d e d . A s i m p l e r e a li z a t io no f t h i s i d e a r e q u i r e s t h e i n t r o d u c t io n o f t h r e e s p e c i e s o f e l e c t r o w e a k - s i n g l e t, l e f t- h a n d e dneu t r i nos : N+ and No , t oge the r w i th a s i ng l e s ca l a r f i e l d , 4 ) , c a r ry ing l ep ton number+ l . T h e s u b s c r i p t i = 0 , + o n N i i n d i c at e s t h e l e p to n n u m b e r o f t h e c o r r e s p o n d i n gl e f t - handed s t a t e .

I g n o r i n g , f o r s i m p l i c i ty , a ll m i x i n g s w i t h v ~ a n d v r , w e a r e l e d t o th e f o l l o w i n g , m o s tg e n e r a l , l e p t o n - n u m b e r - i n v a r i a n t m a s s a n d Y u k a w a t e r m s i n v o l v i n g t h e n e w p a r t ic l e s:-- m~fim = -M ( N+ TL N - ) - -~ N o T L N 0 ) + h . c . ,

yuk = - A ( N - T L L ) H - g _ ( N _ T L N o ) 4) - g + ( - N + TL N o ) 4)* + h.c . 19 )HHe re L = ~e ) and H= (HO) r e p r e s e n t t h e u s u a l S M l e p t o n a nd H i g g s d o u b l e ts . W h e r e

neces sa ry t he con juga t e H iggs do ub l e t is deno t ed by / -) = io -2H* . ) F ina l l y , r e ca l l tha to u r u s e o f m a j o r a n a s p i n o r s i m p l i e s -Ni = NTi C - 1 .

2 Briefly, with derivativelycoup led variables he dom inant contribution to f i r decay for sm all scalar momen-tum com es from grap hs in which the sca lar is em itted from the electron line, for w hich an infrared singularitycom pensates for the derivativecoupling.


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36 P Bamert et aL /Nu clea r Physics B 449 (1995) 25-48A l t h o u g h , i n g e ne r a l, b o t h H a n d c o u l d a c q u ir e v a c u u m e x p e c t a ti o n v a l u e s ( v e v s ) :

( H ) = v a n d ( ) = u , i n p r a c t i c e i t is a g o o d a p p r o x i m a t i o n t o w o r k i n t h e l i m i t u ~ 0 .T h i s i s b e c a u s e t h e a b s e n c e o f an o b s e r v a b l e f l f l o ~ d e c a y s i g na l i m p l i es t h e c o m b i n a t i o ng i u m u s t b e s m a l l . O n t h e o t h e r h a n d , t h e c o u p l i n g s g i t h e m s e l v e s c a n n o t b e to o s m a l lo r e l s e t h e f l f l ~ d e c a y o f in t e r e s t h e r e w i l l it s e l f b e u n o b s e r v a b l e . A s a r e su l t , t h e v e vu m u s t i t s e lf b e c o n s t r a i n e d t o b e n e g l i g ib l e i n c o m p a r i s o n t o t h e o t h e r m a s s e s o f t h ep r o b l e m . W e t h e r e f o r e n e g l e c t u i n a ll o f w h a t f o l l o w s . 3 W e n e v e r th e l e s s a s s u m e t h a tt h e s c a l a r i s l i g h t e n o u g h t o p e r m i t f l f l ~ d e c a y t o o c c u r . T h i s c a n b e d o n e b y t u n i n gt h e m a s s i n t h e p o t e n t i a l t o b e s m a l l , o r b y p e r m i t t i n g u v ~ 0 b u t w i t h u t u n e d t o b es m a ll e n o u g h t o n o t p r o d u c e t o o m u c h f l f l o ~ d e c a y . E i t h e r c h o i c e i n v o l v e s a c e r t a i na m o u n t o f a r b i t r a r i n e s s .

T h e n e u t r i n o m a s s s p e c t r u m f o r t h i s m o d e l g r e a t l y s i m p l i f i e s i n t h e u = 0 l i m i t .T h e r e i s one Le 0 m a s s e ig e n s t a t e , No , w i th m a s s m ; t h e r e i s a m a s s l e s s s t a t e ,t g t~'~ = ~ 'eCo - N + so ; a n d t h e r e i s a m a s s i v e D i r a c n e u t r i n o , z,s = Y L + + y R N - , w i t hN t + = N + c o + u ~ s o a n d h a v i n g m a s s M s = x /M 2 + a Z v 2 . H e r e co a n d so d e n o t e t h ec o s i n e a n d s i n e o f t h e m i x i n g a n g l e , 0 , w h i c h i s d e f in e d b y t a n 0 = A v / M .

T h e Y u k a w a c o u p l i n g s o f th e s e m a s s e i g e n s ta t e s t a k e th e f o r m o f E q . ( 2 ) , w i t hA~ ;N o = O , AusN o = g - , Bv ,,N o = - -g + s o , Bus No = g + c o . ( 2 0 )

U s i n g t h e s e e x p r e s s i o n s i n t h e g e n e r a l re s u l t, E q . ( 4 ) , t h e n g i v e s

f l , ( f l ~ ) = ~ M ~ s ~f d4l W ~ , ( g 2 m g 2 + 2 g + g _ c o M s g 2 2 2 2- - g + c M s m ) ( 2 1 ) (27.1 .)4 (g2 _ i e ) (g2 + M 2 _ iE )2 (g2 + m 2 _ ie )

T h i s e x p r e s s i o n i m p l i e s t h a t t h e f l f l d e c a y r a t e is m a x i m i z e d f o r la r g e c o u p l i n g sa n d m i x i n g a n g l e s , a n d f o r s te r i l e - n e u t r in o m a s s e s i n t h e v i c i n i ty o f t h e n u c l e a r - p h y s i c ss c a l e , ~ 1 0 - 1 0 0 M e V , w h i c h d e fi n e s th e i m p o r t a n t i n t e g r a ti o n r e g i o n i n E q . ( 2 1 ) . I no r d e r t o s e e h o w b i g t h is r a t e c a n g e t w e t h e r e f o r e h a v e t o s e e h o w c l o s e w e c a ng e t t o th i s o p t i m a l r a n g e u s i n g p h e n o m e n o l o g i c a l l y a c c e p ta b l e v a lu e s f o r th e v a r i o u sp a r a m e t e r s . T h e r e l e v a n t c o n s t r a i n t s a r e s u m m a r i z e d i n S e c t i o n 5 , a n d t h e y s u g g e s tt h a t th e a m p l i t u d e c a n b e m a d e b i g g e s t f o r t h e f o l l o w i n g va l u es o f m a s s e s a n d m i x i n ga n g l e s :

M s ~ 3 5 0 M e V m ~ 1 M e V ; 0 ,-~ 7 . 5 1 0 - 2 ; g + ~ 0 . 2 ( 2 2 )o r, p r o v i d e d w e r e l a x t h e b o u n d o n m c o m i n g f r o m n u c l e o s y n t h e s i s ( a s e x p l a in e d i nS e c t i o n 5 ) ,

M s ~ m , .~ 3 5 0 M e V ; 0 ~ 7 .5 x 1 0 - 2 ; g + ~ 1 . ( 2 3 )3 Notice that, unlike the choice u = 0, a very small but no n-zero value for u is quite difficult to mak e natural,even in the technical sense I5,18].


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P. Bamert et aL /Nu clear Physics B 449 (1995) 25-48 37I n t e r e s ti n g l y , t h e c o u p l i n g c o n s t a n t g _ n e e d n o t b e l a rg e . N o t i c e a l s o t h a t th e s m a l l n e s so f 0 i m p l i e s a v


14/24

3 8 P B a m e r t e t a L / N u c l e a r P h y s i c s B 4 4 9 ( 1 9 9 5 ) 2 5 - 4 8W ith Eq . ( 22 ) (Eq . ( 2 3 ) ) i n m ind , we t ake g+ ,-~ 0 .2 (g + ,-~ 1 ) , so ~ 0 .1 , Q ~ 2 MeV,m ,.o 2 M e V ( m ,.o 3 5 0 M e V ) a n d PF ~ 6 0 M e V i n o b t a i n i n g t h i s n u m b e r . A l t h o u g ha c o u p l i n g o f 1 0 - 8 i s p r o b a b l y h o p e l e s s t o o b s e r v e in p r e s e n t e x p e r i m e n t s ( w h i c h c a nd e t e c t g eff > 1 0 - 4 ) , o n e a t t h e l ev e l o f 1 0 - 6 i s c l o s e e n o u g h t o o b s e r v a b i l i ty t o w a r r a n ta m o r e d e t a i l e d c o m p a r i s o n o f th i s t y p e o f m o d e l w i t h t h e d a ta .3 .2 . De cays m ed ia ted by v i r tua l sca la rs

W e n e x t t u r n t o m o d e l s f o r w h i c h t h e f l f l ~ d e c a y i s m e d i a t e d b y t h e s e l f - c o u p l i n g sin t he s ca l a r s ec to r , w i thou t t he need fo r any ex t r a heavy s t e r i l e l ep tons . We a rgue i nth i s s ec t i on t ha t f l f l~ d e c a y i n th e s e m o d e l s i s a l w a y s c o n s t r a i n e d b y L E P d a t a t o b et o o s m a l l t o b e o b s e r v e d . T o s e e h o w t h i s w o r k s , c o n s i d e r an e x a m p l e o f su c h a m o d e li n w h i c h t h e s t a n d a r d m o d e l i s e x t e n d e d b y t h e i n c l u s i o n o f : ( i ) a s e c o n d S M H i g g sd o u b l e t , X , w h i c h h a s n o n - z e r o l e p t o n n u m b e r L = - 1 ; ( i i ) a s in g l e t H i g g s b o s o n fi el d ,~b , w i th l ep ton number L = - t - 1 , and ( i i i ) a weak i so t r i p l e t s ca l a r f i e ld , z l , w i th L = -2 .T h e s e c h a r g e a s s i g n m e n t s p e r m i t t h e c o u p l i n g s c o n s i d e r e d i n t h e p r e v i o u s s e c t i o n : t h eA Y u k a w a c o u p l i n g s c o n t a i n a t e r m o f t h e f o r m L L A, w h i c h c o u p l e s i t to t h e e l e c tr o n ,a n d t h e H i g g s p o t e n t i a l c o n t a i n s t h e f o l l o w i n g t e r m s w h i c h a r e r e l e v a n t t o flfl~o~o decay :

V tri = IX1Htx d.b + 2 x r A t x + h . c . ( 2 8 )P ro v ide d t h a t t he s i ng l e t f i e ld , ~b, i s su f f i c i en t l y l i gh t, t he d i a g ram o f F ig . 3 t he n

i n du c e s/ 3 1 3~ o d e c a y . I n t h i s m o d e l t h e e x t r a s c a l a r fi e ld s d o n o t h a v e n o n - z e r o v e v s s ot h a t l e p t o n n u m b e r r e m a i n s a g o o d s y m m e t r y a n d t h e u s u a l n e u t r i n o s r e m a i n m a s s l e s s ,a s i n t he SM.

N e g l e c t i n g 2 0 2 W 0FW F i n c o m p a r i s o n t o ~ F o r , w e f in d t h e a p p r o x i m a t e f o r m f o r t h ea m p l i t u d e o f E q . ( 1 6 ) :

1 2 / 32 sZohtz2wOarpFEF. A s ( f l f l ~ ) ~ ( 2 r ) 3 ~ V 3 7 r 2 M 2 ~ ( p y , M x ) , ( 2 9 )w h e r e so oz lXl i s t h e m i x i n g a n g l e w h i c h c o n t r o l s t h e s t r e n g t h o f t h e m i x i n g b e t w e e nthe l ig h t m ass e ig ens ta te ~o and the s in gle t f ie ld ~b. Y r ( pe , M x ) i s t h e f u n c t io n o b t a i n e dby pe r fo r m ing t he g i n t eg ra t i on , w h ich t akes t he va lue s b ,-~ 1 fo r M x < PF a n d

~ ( p F / M x ) 4 f o r M x >> PF. W i t h t h e s e e x p r e s s i o n s w e f i n d0geffs(13,8~ ) ~ ( 0 . 0 2 ) [ w a r l hs 2 ( tz2Q'~o o o \ - - - ~ a ) . Y '( p v, U x ) . ( 3 0 )

As u sua l , t he r a t e i s l a rge s t i f a l l o f t he exo t i c s ca l a r s have coup l ings t ha t a r e a s l a rgea s p o s s i b l e , a n d m a s s e s t h a t a r e i n t h e r a n g e o f 1 0 - 1 0 0 M e V . T h i s r a n g e i s p a r t i c u l a r l yd a n g e r o u s f o r th e p h e n o m e n o l o g y o f t h e p r e s e n t m o d e l , h o w e v e r, s in c e t h e n e w s c a la r sc o u p l e t o th e p h o t o n a n d t o t h e Z . A n y s u c h p a rt i c le s h a v i n g m a s s e s ~< 5 0 G e Vw o u l d c o n t r i b u t e u n a c c e p t a b l y t o e + e - a n n i h il a ti o n , a n d s o a r e r u l e d o u t. T h e e f f e c t iv ec o u p l i n g , g e f f, f o r s c a l a r - m e d i a t e d f l f l~ decay r a t e i s t he r e fo re supp re s sed a t l e a s t bythe f ac to r s ( Q / M , I ) ( p F / M x ) 4 ,-,o 1 0 - 1 6 a n d s o i s m u c h t o o s m a l l t o b e d e t e c t a b l e .


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P B a m e r t e t a l . / N u c l e a r P h y s i c s B 4 4 9 ( 1 9 9 5 ) 2 5 - 4 8 39S i n c e t h e s u p p r e s s i o n f a c t o r d u e t o t h e l a r g e s c a l a r m a s s e s i s s o d e v a s t a t i n g l y s m a l l ,

t h is r u l e s o u t o b s e r v a b l e s c a l a r - m e d i t a t e d f l f l ~ d e c a y s i n a m u c h l a r g e r c la s s o f m o d e l st h a n t h o s e w e h a v e d i r e c t l y c o n s i d e r e d . T h a t i s , o n e m i g h t e n t e r t a i n m o d e l s f o r w h i c hs c a l a r - m e d i a t e d f l f l ~ deca y do es no t i nvo lv e t he W b oson a t a ll . Af t e r a ll , F ig . 3 work se q u a l l y w e l l i f t h e l o n g i t u d i n a l p a r t s o f t h e W l i n es a r e r e p l a c e d b y y e t a n o t h e r e x o t i cs c a l a r w h i c h c o u p l e s d i r e c t l y t o q u a r k s . I n t h i s c a s e t h e n u c l e a r m a t r i x e l e m e n t s c a n b el a r g e r s in c e t h e y c a n i n v o l v e q u a r k o p e r a t o r s o t h e r t h a n t h e e l e c t r o w e a k c h a r g e d c u r re n t .A l s o , t h e Y u k a w a c o u p l i n g o f s u ch a s ca l a r t o q u ar k s c o u ld e a s il y b e o f o rd e r 1 0 - I - - av a l u e w h i c h i s m u c h l a r g e r th a n t h e c o r r e s p o n d i n g v a l u e o f 1 0 - 4 f o r th e S M H i g g s .N e v e r t h e l e s s , n o n e o f t h e s e e n h a n c e m e n t s c a n o v e r c o m e t h e s u p p r e s s i o n d u e t o l a r g es c a l a r m a s s e s .

4 M o d e l b u i l d i n g : T h e c a s e o f s p e c t r a l i n d e x n = 7

W e n o w t u r n t o a m o d e l w h i c h p r o d u c e s t i f f decay wi th spec t ra l index n = 7 , i . e . am o d e l o f t y p e I I E . T o d o s o w e r e q u i r e a m o d e l i n w h i c h l e p t o n n u m b e r i s u n b r o k e n ,a n d w h i c h c o n t a i n s a G o l d s t o n e b o s o n , ~o, t h a t c a r r ie s l e p t o n n u m b e r + 1 . T h i s c a n o n l yh a p p e n i f th e t h e o r y h a s a n o n - a b e l i a n f l a v o u r s y m m e t r y w h i c h a c t s o n l ep t o n s , a n dw h i c h c o n t a i n s o r d i n a r y l e p t o n n u m b e r a s a g e n e r a t o r .

T h e s i m p l e s t c a s e is to t a k e t h e le p t o n - n u m b e r s y m m e t r y g r o u p to b e G = S U ( 2 ) xU L, ( 1 ) , w i t h G b r o k e n d o w n t o t h e U L ( 1 ) o f l e p to n n u m b e r . T h i s c a n b e d o n e b yw o r k i n g w i t h a v a r ia t i o n o f t h e m o d e l o f R e f . [ 5 ] . W e t h e re f o r e a d d t h e f o l lo w i n gN+e l e c t r o w e a k - s i n g l e t , l e f t -h a n d e d f e r m i o n s t o t h e s ta n d a r d m o d e l : N = ( N o ) s o a n d s _ ,w h i c h t r a n s f o r m u n d e r t h e f la v o u r s y m m e t r y , G , as : N , ~ ( 2 , ), so ~ ( 1 , 0 ) a n ds _ ~ ( 1 , - 1 ) . I f th e u n b r o k e n l e p t o n n u m b e r is c h o se n t o b e Le = T3 + U , w h e r eU i s the gen e ra to r o f UL , ( 1 ) , t hen t he subsc r i p t s o f N+ , No , so and s_ g ive t hec o r r e s p o n d i n g p a r t i c l e s l e p t o n c h ar g e . I n o r d e r t o im p l e m e n t t h e s y m m e t r y b r e a k i n gpa t t e rn G - -+ U L( 1 ) , w e a l so i n t rodu ce t he e l ec t row eak - s ing l e t s ca l a r f ie l d , q~ = ( : o )wh ich t r ans fo rm s un de r G a s : q~ ,~ (2 , ).

F o r t h i s m o d e l t h e m o s t g e n e r a l r e n o r m a l i z a b l e G - i n v a r i a n t m a s s a n d Y u k a w a i n t e r -ac t i ons i nvo lv ing the new f i e l d s a r eMm = -- -~ - (S0 ~LS0) + h . c . ,

/~yuk = - A ( ~ - y L L ) H - g - ( - N i Y L S - ) ~ j ~ ij + go (N---/yLs0) c~j e ij + h.c . 31)I f t he s ca l a r f i e ld s a cqu i r e t he v ev s ( / 4 ) = v and (0 ) = u (w h ich we t ake , f o r

s i m p l i c i t y , t o b e r e a l ) , t h e n G b r e a k s t o U L ( 1 ) as r equ i r ed . I f we a l so neg l ec t m ix ing swi th u u an d u~, t he fo l l ow ing neu t r i no spec t ru m i s p roduced . F i r s t, t he r e i s one m ass l e s sLe = +1 s ta te , u~ = UeCo - N+ so, w h e r e t h e m i x i n g a n g l e , O , s a t i s f i e s t a n O = A v / g _ u .Then , t he r e i s a mass ive L~ = + 1 D i r ac s t a t e , Us = yLN ~ ++ yRs - , with N~_ = N + c o + u ~ s o ,a n d w i t h m a s s M ~ ~//~2l)2 q - g 2 _ u 2 . T h e r e a r e a l s o t w o m a j o r a n a Le = 0 s ta tes , P-t-,


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4 0 P. B a m e r t e t a l . / N u c l e a r P h y s i c s B 4 4 9 ( 1 9 9 5 ) 2 5 - 4 8w h i c h r e s p e c t i v e l y h a v e m a s s e s M + = [v /M 2 + 4g~u 2 4- M] . T h e s e m a s s e i g e n s t a t e sa r e r e l a t ed t o No and so by

( v _ ) = ( / c ~ i s ~ ) ( N o ) , ( 3 2 )/ 2 - - S ~o C ~ , , ] S O

w i t h t a n ( 2 ~ ) = 2 g o u / M . T h e f a c t o r s o f i i n t h i s m i x i n g m a t r i x a r e d u e t o t h e c h i r a lr o t a t i o n n e e d e d t o e n s u r e t h a t b o t h m a s s e i g e n v a l u e s , M + , a r e p o s i t i v e .

T h e n e x t s t e p i s t o r o t a t e f i e l d s t o m a k e t h e d e r i v a t i v e c o u p l i n g s o f t h e G o l d s t o n eb o s o n s e x p l i c i t ( t h i s i s d o n e i n d e ta i l f o r a r e l a te d m o d e l i n R e f . [ 5 ] ) . T h e a p p e a r a n c eo f t h e d e r i v a t iv e c o u p l i n g c a n b e s e e n a s f o l lo w s : u s e e x p o n e n t i a l p a r a m e t e r i z a t io no f t he s ca l a r f i e ld s t ha t have non -ze ro vev so t ha t t he Go lds tone f i e ld appea r s i n t heexponen t i a l . Then r ede f ine t he f e rmion f i e ld s so t ha t t he new f e rmion f i e ld i s g ivenb y F ~ - F e i*/v ( i n t h i s c a se F t i s the h eavy neu t r a l l ep to n ) . Af t e r t h i s r ede f in i t i on ,t h e Y u k a w a i n t e ra c t i o n b e c o m e s c o m p l e t e l y i n d e p e n d e n t o f t h e G o l d s t o n e f i el d , w h i c hr e a p p e a r s w i t h a d e r i v a t i v e c o u p l i n g d u e t o t h e f e r m i o n k i n e ti c e n e r g y t e r m . T h e f in a lf o r m o f t h e i n te r a c t i o n g i v e n in E q . ( 8 ) t h e n e m e r g e s o n d i a g o n a l i z a ti o n o f t h e f e r m i o nm a s s t e r m s w i t h c o u p l i n g m a t r i c e s Xia and Yia giv en by Yia = 0, and

o c~ so s~o . o c~ co sX,. , ._ = t - - , X~,,~.+ - , X, ,y_ = - t - - , X. .~+ - (3 3)U U U UW i t h t h e s e e x p r e s s i o n s w e m a y e v a l u a t e t h e / 3 f l ~ m a t r i x e l e m e n t o f E q . ( 1 0 ) I t

b e c o m e s2 2 2 . - 4S o C 0 W I s

A 33 ~) = 4i-6S ~ U2f d4g W lz~ s~M+ c~oM-

x (27.r)4 ((2 _ ie ) (g2 + M2s _ ie )2 (g2 + M2+ _ i e) - (g2 + M2__ _ ie )( 3 4 )

W e q u a n t i f y t h e s i z e o f t h i s r a t e i n a s i m i l a r w a y a s a l r e a d y d o n e f o r t h e m o d e l o fS e c t i o n 3 . 1 , n a m e l y b y m e a n s o f e s t i m a t i n g t h e m a g n i t u d e o f ge ff . W e t h e r e f o r e c h o o s ev a l u e s f o r m a s s e s a n d m i x i n g a n g l e s t h a t a r e c o m p a t i b l e w i t h t h e p h e n o m e n o l o g i c a lb o u n d s o f S e c t i o n 5 , b u t w h i c h a l s o y i e l d a m a x i m a l r a t e . S p e c i f i c a l l y w e t a k e

M s > ~ 3 5 0 M e V ; M + , M _ ~ a f e w M e V ; 0 ~ 7 . 5 1 0-2 ; g _ ,- ~ 0 . 2 . ( 35 )T h e a m p l i t u d e c a n t h e n b e w r i tt e n i n t h e f o l lo w i n g a p p r o x i m a t e f o r m :

2 s 2 c ~ g 2- M +M - ( s ~ M _ 2A ( f l f l~ ) - ~ 7r M2s - c M+)f d4g W/'*~ (3 6)

x (27.r)4 (g2 - - ie ) ( g 2 + M ~ + T i e ) ( e Z + M 2_ - i E )T o s e e t h i s k e e p i n m i n d t h a t i n t h e l i m i t o f s m a l l 0 w e h a v e u ~ M s ~ g - . C o m p a r i n gt h is w i t h t h e a m p l i t u d e o f t h e G R m o d e l g i v e s a n e q u i v a le n t g ef f o f


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P. Bamert e t a l . /Nuc lear Phys ic s B 449 (1995) 25-48 41g e f f 0.0006) $2C2 _2 10 14~ o o g - ~ , (37)

where the values of Eq. (35) have been applied and we take the mixing angle to satisfytan q~ ,-~ 1/V 2 to maximize the rate (i.e. equivalently M+ ~ 2M_ ). Specifically we takeM + ,,~ E F --~ Q ,,~ 2 MeV, g_ -,~ 0.2, so , ,~ 0.1 and M s ,,~ 6 p F ~ 360 MeV.

In the above we have chosen 0.2 as the maximal value for g_, as is naively requiredby kaon decay measurements (see Section 5). It is probable that values as large asg_ _~ 1 would actually prove to be consistent with these measurements in a morecareful treatment, however. This is because the scalar's derivative coupling provides anadditional suppression to its contribution to kaon decays. If so, then the upper bound ongeff can be relaxed to around geff ~ 10 12.

It is clear that this rate is far too small to have any chance of being detected, largelydue to the suppression by powers of Q that originate with the extremely soft electronspectrum. We consider this model therefore only as an example that such an exotic decayspectrum, with spectral index n = 7, can i n p r i n c i p l e exist. Given that this is also thekind of spectrum that is least well constrained experimentally, since cuts usually excludethe lowest-energy electrons which are usually the most contaminated by backgrounds,it would be very interesting to construct a model which predicts a detectable rate. Wehave so far been unable to find such a model.

5 P h e n o m e n o l o g i c a l c o n s t r a in t sThe previous sections have seen us use particular kinds of masses and couplings in or-

der to maximize the/3/3~o decay rate. The present section is meant to just ify the massesand couplings we have chosen. We do so by listing the main phenomenological con-straints which the models we consider must confront. Since the models with the largestf l t 3 ~ signals involve sterile neutrinos coupled to light scalars, we focus principally onthe constraints on these.

5 . 1 . L a b o r a t o r y l i m i t sIn most of the models considered in this paper the SM has been supplemented by a

number of etectroweak-singlet fermion and scalar fields. Of these, it is only the mixingof the sterile neutrinos with (in this case) the electron neutrino, r e , that is boundedexperimentally. Since the sterile neutrinos in the models we consider dominantly decayinto lighter neutrinos and scalars, all of whom are invisible to present day detectors, theonly bounds which need be considered are those which do not assume a decay into avisible mode. The bounds that do apply can be classified into two subclasses accordingto whether they are due to low- or high-energy experiments [20].At low energies we have those experiments which examine the decays of pions [21 ]and kaons [22] at rest, and search for non-standard contributions to the decay rateand the outgoing electron spectrum. These experiments constrain the masses of sterile


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42 P Bamert et al./Nuclear Physics B 449 1995) 25-48neutrinos in a model-independent way from a few MeV almost up to ~ 100 MeV forpion decays, and up to ,~ 350 MeV for kaon decays. Mixing angles, Uei, with theelectron neutrino, ~ e, are st rongly constrained for sterile neutrinos with masses in thisrange. For example, upper bounds on IUeil can be as low as 7 10 -4 for a mass o f 50MeV [21].

These same experiments, as well as measurements of the muon lifetime, also limitthe possibility for the existence of weak decays into very light scalars in addition toneutrinos 4 [23]. Specifically these limits come from precision measurements of theMichel parameter, p, in muon decay, and from using the measured decay spectra of theoutgoing leptons to constrain decays such as K --~ gN~ or 7r --* gNu. The failure toobserve such decays can be expressed as an upper bound on a hypothetical dimensionlessYukawa coupling, ~ eff, of an effective pe-N-cp interaction. In terms of this coupling, thecurrent bounds are geff < 5.7 x 10 -2 (muon decay), < 1.6 x 10 -2 (pion decay) and< 1.3 x 10 -2 (kaon decay) [23] . 5 Of course, these bounds assume that the kinematicspermit the emission of N~p in these decays. For the models of interest here, the boundtherefore applies if the Le = 0 sterile fermions, Na, are sufficiently light compared to the7r, K and /o r At, as is certainly the case for raN, ~ 1 MeV. For mNo in this mass range,the effect ive coupling that is then bounded in this analysis turns out to be ~eff ~ Socog+.As a result, keeping in mind the bound on 0 discussed below, g+ must be smaller than0.18.

It is clear however that provided Na is too heavy to be produced in these kinds ofdecays (i.e. mNo > 350MeV) the bound described above no longer applies, and g+ canbe of order unity. We are led to consider mlv ~ 1MeV because of the specific way wechoose to avoid conflict with nucleosynthesis (see Section 5.2). If another way to evadethis bound should be found, or if we simply accept the comparatively large number ofeffective degrees of freedom contributed at this epoch, then g+ can indeed be as largeas g+ ~ 1.

For masses heavier than 350 MeV the best bounds on sterile neutrinos usually comefrom beam dump experiments. However this type of experiment relies crucially on theassumption that the isosinglet leptons decay via their charged current weak interactions.The same is also true for bounds coming from searches on the Z-pole for exotic decaysof the Z boson, such as Z ~ N~ ~ W*ef, (see e.g. Ref. [25] ). (Here N denotes theisosinglet state, and W* denotes a virtual W boson.) However, since such decays do notdominate in the models we consider--the sterile states instead prefer to decay into lightscalars--these experiments need not be considered further here.

The best limits, for the mass range above ~ 350 MeV, therefore come from thereduction of the effective charged-current coupling of the electron, which is suppressedby the cosine of the mixing angle between pe and the heavy sterile state. This reductionpotentially affects electroweak precision experiments by influencing the measured value

4 For the case of models with flavor violation see e.g. Ref. [24].5 These bounds can be even more stringent if additional model-dependent assumptions for the couplingsinvolved are made.


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P. Bam ert et aL /Nuclear Physics B 449 (1995) 25-48 43o f F e r m i s c o n s ta n t , G F , t h a t i s i n f e r r e d f r o m m u o n d e c ay . I t a ls o s h o w s u p a s a f a i l u r e o fl e p t o n u n i v e r s a l i t y in l o w - e n e r g y w e a k d e c a y s . T h i s le a d s t o t h e b o u n d IUeil < 7 .5 x 10 -2( 2 0 -) f o r m a s s e s a b o v e ~ 3 5 0 M e V [ 2 6 ] . A n o t h e r b o u n d c o m e s f r o m t h e r e d u ct io no f t h e n e u t r a l c u r r e n t c o u p l i n g o f th e e l e ct r o n - n e u tr i n o , l e a d in g t o a d e c r e a s e d n u m b e ro f l i g h t n e u t r in o s p e c i e s a s d e d u c e d f r o m Z s i n v i s i b l e w i d t h . H o w e v e r t h is b o u n d i ss o m e w h a t w e a k e r t h a n t h e a f o r e m e n t i o n e d u n i v e r s a li t y b o u n d , a n d a p p l i e s o n l y t o s t e ri len e u t r i n o s t h a t a r e h e a v i e r t h a n a b o u t 9 0 G e V , h e n c e i t i s o f n o p a r t i c u l a r i m p o r t a n c e i nou r ca se .

T h e s e b o u n d s h a v e i m m e d i a t e i m p l i c a ti o n s f o r t h e m o d e l s c o n s i d e r e d h e re , s u c h a sthose d i s cus sed i n Sec t i ons 3 .1 and 4 . S ince t he se mode l s have on ly one s t e r i l e s t a t e ,t h e h e a v y D i r a c n e u t r i n o ~ s , t h a t m i x e s w i t h ~ e , w e n e e d o n l y c o n s t r a i n t h e m a s s e s a n dm i x i n g s o f t h i s o n e p a r t i c l e . T o m a x i m i z e t h e f l f l ~ r a t e w e p r e f e r a m i x i n g a n g l e , 0 ,t h a t i s a s l ar g e a s p o s s i b l e , a n d s o w e c h o o s e U s t o b e h e a v y e n o u g h t o e v a d e t h e b o u n d sf rom p ion and k aon decay , Ms ~> 350 MeV, and we t ake 0 t o s a tu r a t e t he un ive r sa l i t ybou nd , 0 < 7 .5 x 10 -2 .5 . 2 . N u c l e o s y n t h e s i s

S t a n d a r d b i g - b a n g c o s m o l o g y v e r y s u c c e s s f u l l y e x p l a i n s t h e a b u n d a n c e s o f l i g h t e l -e m e n t s p r o d u c e d d u r i n g t h e n u c l e o s y n t h e s i s e p o c h [ 2 7 ] . I n f a c t , t h i s s u c c e s s l e a v e sl i t t l e r oom fo r t he ex i s t ence o f new spec i e s o f pa r t i c l e s a t t h i s c r i t i c a l t ime . Spec i f i c a l l yi f t he r e w e re ne w pa r t i c l e s p r e sen t a t a t empe ra tu r e o f ~-, 0 . 1 -2 M eV the i r add i t i ona ld e g r e e s o f f r e e d o m w o u l d i n c r e a s e t h e e x p a n s i o n r a t e o f t h e u n i v e r s e l e a d i n g t o a ne a r l ie r f r e e z e - o u t o f n e u t r o n - p r o t o n c o n v e r t in g i n t e ra c t io n s a n d t h u s t o a l a r g e r a m o u n to f n e u t r o n s t h a t e v e n t u a l l y w o u l d b e c o o k e d i n t o 4 H e . T h e b o u n d o n t h e n u m b e r o fa d d i t i o n a l d e g r e e s o f f r e e d o m i s c o n v e n t i o n a l l y s t a t e d i n t e r m s o f t h e m a x i m a l n u m b e ro f a d d i t i o n a l l i g h t n e u t ri n o s p e c i e s a l l o w e d a t n u c l e o s y n th e s i s 8 N ~ [ 2 8 ] :

~N~ ~< 0 .4 . ( 38 )T h e r e a r e t w o s i m p l e w a y s t o s a t i s f y t h i s b o u n d w h e n a d d i t i o n a l p a r t i c l e t y p e s a r e

p re sen t i n a t heo ry . Fo r pa r t i c l e s t ha t a r e heav i e r t han abou t 1 MeV, t he mos t s t r a igh t -f o r w a r d w a y i s t o h a v e t h e m n o t b e p r e s e n t a t a l l d u r i n g n u c l e o s y n t h e s is . T h i s c a n b ea c h i e v e d b y e i t h e r h a v i n g t h e m d e c o u p l e a n d s u b s e q u e n t l y d e c a y s u f f i c i e n t l y e a r l y - -k e e p i n g i n m i n d t h e p o s s i b l e c o n t r i b u t i o n s o f t h e d e c a y p r o d u c t s - - o r b y h a v i n g t h e mr e m a i n i n t h e r m a l e q u i l i b r i u m a s t h e y b e c o m e n o n - r e l a t i v i s t i c , s o t h a t t h e i r a b u n d a n c eb e c o m e s s u p p r e s s e d b y t h e B o l t z m a n n f a c t o r , e - m / T . F o r p a r t i c l e s w h i c h a r e m u c hl igh t e r than 1 MeV , on t he o the r hand , t he con t r i bu t i on t o ~N~ i s s t r ong ly sup p re s sed i ft h e y d e c o u p l e b e f o r e t h e Q C D phase t r ans i t i on a t T ~ 200 MeV. In t h i s c a se t hey be -c o m e s u f f ic i e n tl y d i lu t e d c o m p a r e d t o o r d i n a r y p a r ti c l es a f t e rw a r d s d u e t o t h e r e h e a ti n go f t h e p h o t o n b a t h d u r i n g t h is p h a s e t ra n s i ti o n .U n f o r t u n a t e l y , t h e s e m e t h o d s a l o n e a r e n o t e n o u g h t o a v o i d c o n f l i c t o f t h e m o d e l scons ide red i n t h i s pape r w i th nuc l eosyn thes i s . Th i s i s because t he ve ry l i gh t s ca l a r s , t ha ta r e p r e s e n t i n a l l o f o u r m o d e l s , c o u p l e t h r o u g h d i m e n s i o n l e s s c o u p l i n g s t o o r d i n a r y


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44 P Bamertet al./NuclearPhysicsB 449 (1995)25-48neu t r i nos , su ch a s ~ e . As a r e su lt , t hey r em a in i n equ i l i b r i um wi th t he se neu t r i nos dow nt o t e m p e r a t u r e s t h a t a r e w e l l b e l o w 1 M e V . T h i s m a k e s a c o n f l i c t w i t h n u c l e o s y n t h e s i sg e n e r i c f o r m o d e l s w h i c h c a n p r o d u c e flfl~ and flfl~ decays .

R a t h e r t h a n t h e r e f o r e c o n s i d e r in g t h e s e m o d e l s t o b e i n e v i t a b l y r u l ed o u t , w e d e m o n -s t r a t e h e r e a n e x i s t e n c e p r o o f t h a t t h e n u c l e o s y n t h e s i s b o u n d c a n b e a v o i d e d , g i v e ns u i t a b l e m a s s e s f o r t h e s t e r i l e n e u t r i n o s . W e p r o p o s e t h i s a s a n e x a m p l e o f t h e k i n d so f i n f o r m a t i o n t h a t c o u l d b e i n f e r re d f r o m c o s m o l o g i c a l b o u n d s , s h o u l d o n e o f th e e x -o t i c tiff d e c a y s w e d e s c r i b e e v e r b e d e te c t e d e x p e r i m e n t a l ly . T h e l o o p h o l e w e d e s c r i b ef o l l o w s i t s s im i l a r a p p l i c a t i o n i n R e f . [ 1 8 ] , a n d i s b a s e d o n a n o b s e r v a t i o n o f R e f . [ 2 9 ] .

Sup pose , t hen , tha t a hea vy neu t r i no s t a t e e i t he r deca ys [29 ] o r ann ih i l a t e s [ 18 ] ,i n to pa r t i c l e s t ha t a r e i n equ i l i b r i um wi th o rd ina ry neu t r i nos , r i gh t a f t e r t he se neu t r i nosf r e e z e o u t o f c h e m i c a l e q u i l i b r i u m w i t h th e p h o t o n b a t h ( i. e . T~ ~ 2 . 3 M e V . ) P r o v i d e dt h i s h a p p e n s b e f o r e t h e n e u t r o n - p r o t o n r a t i o f r e e z e s o u t ( a t Tn/p ~ 0 . 7 M e V ) , t h e n t h ea b u n d a n c e o f n e u t r i n o s w i ll b e i n c r e a s e d r e l a ti v e to t h e s t a n d a r d c o s m o l o g i c a l s c e n a r io .T h i s o v e r a b u n d a n c e o f n e u t r i n o s w i l l a c t t o s u p p r e s s t h e n e u t r o n a b u n d a n c e a t f r e e z eo u t , a n d s o d e c r e a s e s t h e p r e d i c t e d p r o d u c t i o n o f 4He. As a r e su l t i t con t r i bu t e s t on u c l e o s y n t h e s i s a s a negative c o n t r i b u t i o n t o 6N~ [ 2 9 ] .

Cons ide r , t he r e fo r e , a s i ng l e ma jo r ana neu t r i no w i th a mass i n t he c r i t i c a l r eg ion ( i . e .1 - 2 M e V ) w h i c h i s s t i l l i n e q u i l i b r i u m w i t h ns spec i e s o f r e a l s ca l a rs i n add i t i on

to t he nF = 3 spec i e s o f o rd in a ry neu t r i nos , w hen i t bec om es non - r e l a t i v i s t i c . Such apa r t i c l e w i l l ann ih i l a t e ou t r i gh t du r ing t he c r i t i c a l epoch , t he r eby hea t i ng t he neu t r i no -s c a l a r b a t h a b o v e t h a t o f t h e p h o t o n b a t h . T h e n t h e m a g n i t u d e o f 6N~ t ha t r e su l t s duet o th e a b o v e m e c h a n i s m t u rn s o u t to b e [ 2 9 ,1 8 ] t~ N~ = - 4 . 6 ~ n , w h e r e

\ 5/3( n ) ( E ) 2 { ( n F + l ) - k- 4 n S )6n = ~ -~o - 1 = ~ n F + 74-ns - - 1 . ( 3 9 )H e r e n/no a n d E/Eo d e n o t e t h e r a t i o s o f t h e e l e c t r o n - n e u t r i n o n u m b e r d e n s i t i e s a n de n e r g i e s a f t e r a n d b e f o r e a n n i h i l a t i o n . T h e s e c o n d e x p r e s s i o n i s o b t a i n e d f r o m t h e f i r s to n e u s i n g e n t r o p y c o n s e r v a t i o n .

W e c a n n o w s e e h o w t h e m o d e l s d e s c ri b e d i n p r e v i o u s se c ti o n s c a n a c c o m m o d a t ethe nuc l eosyn thes i s bound . Take t he mode l o f Sec t i on 3 .1 a s a r ep r e sen t a t i ve ca se . Th i sm o d e l c o n t a i n s a m a s s i v e D i r a c s ta t e, ~ s , w h i c h m i x e s a p p r e c i a b l y w i t h t h e e l e c tr o nn e u t r in o , a n d s o w h i c h m u s t b e h e a v i e r t h a n ,-~ 3 5 0 M e V t o a v o i d t h e l a b o r a t o r y b o u n d sdesc r i bed ea r l ie r . I t a l so con t a in s a mass iv e m a jo r a na neu t r i no , No , a s we l l as a ve ryl i gh t complex s ca l a r .

S i n c e t h e s t e r i l e n e u t r i n o i s k e p t i n t h e r m a l e q u i l i b r i u m w i t h t h e p h o t o n b a t h b yneu t r i no s ca t t e r i ng v i a s ca l a r (~b ) exchange , i t r em a ins i n equ i l i b r i um lo ng a f t e r t het e m p e r a t u r e h a s d r o p p e d b e l o w i t s m a s s . I t s a b u n d a n c e i s t h e r e f o r e s u f f i c i e n t l y s u p -p r e s s e d t o e n s u r e t h a t i t n e v e r p l a y s a r o l e a t n u c l e o s y n t h e s i s . T h e s a m e w o u l d b e t r u ef o r t h e m a j o r a n a n e u t r i n o i f i t w e r e h e a v y , h o w e v e r t h i s w o u l d l e a v e t h e l i g h t s ca l a rs i ne q u i l i b r i u m a t n u c l e o s y n t h e s i s , c o n t r i b u t i n g 6N~ = 8 . The c on t r i bu t i on o f t he se s ca l a r sc a n b e v e r y e f f ic i e n tl y c a n c e ll e d , h o w e v e r , i f N o s h o u l d h a v e a m a s s o f a r o u n d 1 M e V ,


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P. Bamert et aL/Nuclear Physics B 449 1995) 25-48 45s in c e i n t h i s c a s e i t a n n ih i l a t e s r i g h t i n t im e t o r e h e a t t h e n e u t r i n o s e c to r . 6 I n t h i s c a s ew e ca n e m p l o y E q . ( 3 9 ) w i t h nF = 3 a n d ns = 2 , y i e ld in g ~ n = 0 .4 3 a n d s o a t o t a ls h i f t i n t h e e f f e c t i v e n u m b e r o f n e u t r i n o s p e c i e s o f ~ N v ) t o t = 8 _ 4 . 6 ( 0 . 4 3 ) = - 0 . 8 6 .S i n c e ~ g v ) t o t r i s e s t o 8 a s N 0 s m a s s i n c r e a s e s a b o v e T ~ ~ 2 .3 M e V , i t i s p o s s ib l et o fi n d a n e f f e c t iv e n u m b e r o f li g h t n e u tr i n o s w h i c h i s c o m p a t i b l e w i t h o b s e r v a t i o n s .T o s u m m a r i z e t h i s p a r a g r a p h : C o n f l i c t w i t h n u c l e o s y n t h e s i s c a n b e a v o i d e d i f N o h a s am a s s i n t h e r a n g e o f a f ew M e V .

F o l l o w i n g s i m i l a r l i ne s o f r e a s o n i n g o n e c a n a l so m a k e t h e m o d e l o f S e ct i o n 4 s a t is f yt h e n u c l e o s y n t h e s i s b o u n d . I n th i s c a s e t h e a n n i h i l a t io n o f j u s t o n e m a j o r a n a n e u t r i n od u r i n g t h e cr it ic a l e p o c h i s n o t e n o u g h t h o u g h b e c a u s e w e h a v e t w i c e a s m a n y l i g hts c a la r d e g r e e s o f f r e e d o m a s in t h e m o d e l d i s c u s s e d a b o v e . A s s u m i n g h o w e v e r t ha t t h et w o m a j o r a n a m a s s e i g e n s t a te s v + a r e b o t h n o t m u c h h e a v i e r th a n a fe w M e V s o l v est h e p r o b l e m . T o m a k e t h i s p o i n t e x p l ic i t w e j u s t s t a te t h a t i f both u + a n d v _ a n n i h i l a t e do u t r i g h t i n t h e c r i t ic a l e p o c h t h e n t h i s w o u l d r e s u lt i n ~ N v ) t o t = - - 1 .

5.3. Other constraints fr om as trophysics and cosmology

O t h e r a s t r o p h y s i c a l b o u n d s c a n c o n s t r a i n t h e p r o p e r t i e s o f v e r y w e a k l y i n t e r a c t i n gp a r t i c l e s , s u c h a s t h e s t e r i l e n e u t r i n o s a n d s c a l a r s w e a r e e n t e r t a i n i n g . T h e s e t u r n o u t t on o t s i g n i f i c a n t l y c o n s t r a i n t h e m o d e l s w e c o n s i d e r . W e i l l u s t r a t e t h i s p o i n t i n t h i s s e c t i o nb y e x a m i n i n g t h e se b o u n d s f o r t h e m o d e l c o n s i d e r e d i n S e c t io n 3 . 1 .

Stellar evolutionS t e r i le p a r t i c l e s p r o d u c e d i n th e c o r e o f st a rs c a n d i s r u p t o u r u n d e r s t a n d i n g o f s te l la r

e v o l u t i o n s i n c e t h e y c a n p r o v i d e a n e x t r e m e l y e f f i c i e n t a d d i t i o n a l c o o l i n g m e c h a n i s m f o ra s t ar . F o r t h e m o d e l o f S e c t i o n 3 . 1 , t h e s t e ri l e n e u t r i n o s Us a n d N o a r e s i m p l y t o o h e a v yt o b e p r o d u c e d i n s i g n i f i c a n t n u m b e r s i n s i d e t h e c o r e o f a st ar , s in c e t h e i r m a s s e s a r er e s p e c t i v e ly t a k e n t o b e ,-~ 3 5 0 M e V a n d ,-~ 1 M e V . T h e y a r e t h e r e f o r e n o t c o n s t r a in e db y s t e l l a r e v o l u t i o n . T h e s a m e c o n c l u s i o n a l s o h o l d s f o r t h e l ig h t s ca l a rs o f t h i s m o d e l ,a l t h o u g h f o r d i f f e r e n t r e a s o n s . I n t h i s c a s e t h e s c a l a r s a r e l i g h t e n o u g h t o b e p r o d u c e d ,b u t t h e y o n l y c o u p l e a p p r e c i a b l y t o n e u t r i n o s . B u t s i n c e t e m p e r a t u r e s a n d d e n s i t i e si n s i d e a s t a r a r e n o t s u f f i c i e n t l y h i g h t o p r o d u c e a t h e r m a l o r d e g e n e r a t e p o p u l a t i o n o fn e u t r i n o s , s c a l a r s a r e n o t p r o d u c e d i n s i g n i f i c a n t q u a n t i t ie s f r o m t h e i n t e r io r o f t h e s ta r.D i r e c t c o u p l i n g s w i t h o t h e r p a r t i c l e s , s u c h a s e l e c t r o n s , d o a r i s e d u e t o l o o p e f f e c t s , b u tt h e s e a r e t o o s m a l l t o b e s i g n i f i c a n t . 7

SupernovaeP o p u l a t i o n s o f n e u t r in o s are m a i n t a i n e d , h o w e v e r , w i t h i n t h e c o r e s o f su p e r n o v a e , d u e

t o t h e m u c h h i g h e r t e m p e r a t u re s ( ~ 5 0 - 1 0 0 M e V ) a n d d e n s it ie s t ha t a r e f o u n d th e re .6 No rem ains in equilibrium with ue through scalar exchange.7 See, for exam ple, the second pa per of R ef. [ 5], whe re this issue is treated in m ore detail in a similarcontext.


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46 P Bame rt et al./Nuclear Physics B 449 (1995) 25-4 8Both the scalar, ~, and the sterile majorana neutrino, No, are therefore sufficiently lightto be produced in significant numbers. However since both of these particles are also inthermal equilibrium with the ordinary SM neutrinos, they are trapped in the core of thesupernova and so do not lead to premature cooling s.

P r e s e n t - d a y c o s m o l o g yOne last issue is the effect of ~ on structure formation, and on the energy density

of the universe observed today. Since ~o would remain in thermal equilibrium with theelectron neutrinos (through exchange of virtual No) until a temperature in the region ofleV if it was light, it inevitably annihilates out to the neutrino sector once it becomesnon-relativistic. It will therefore never dominate the energy density of the universe andhence also not affect structure formation nor the age of the universe in any significantway. This in turn means that ~ can have any mass below 1 MeV.

6 C o n c l u s i o n s

In this paper we investigate a novel class of neutrinoless double-beta decays, inwhich two (rather than one or no) scalars are emitted in addition to the two observedelectrons. We do so to provide sample models of the two remaining classes of spectrafor t i f f decays, which had not been hitherto explored. We regard it as useful to constructrepresentative models for each class, since such models are required in order to determinethe implications of other experiments, at different energies, for t i f f decay.

We have obtained the following results:(1) We have found that two-scalar decays always lead to a t i f f electron sum-energy

spectrum whose spectral index is either n = 3 or n = 7. The n = 3 spectrum isintermediate between the SM shape (for which n = 5), and the traditional scalar-emitting shape (n = 1), such as is found in the Gelmini-Roncadelli model. The n = 3spectrum is identical to a recently discovered class of models [5] for which only asingle scalar is emitted during the decay.

(2) The n = 7 spectrum is completely new, and is s o f t e r than the observed SMf l f l2~ spectrum. As a result, it is more difficult to experimentally extract, since mostbackgrounds tend to dominantly produce low-energy electrons. It is intriguing that suchan exotic spectrum could in principle be hiding undetected even now in the present f l f l2~data

(3) Viable models can be constructed which predict an n = 3 t i f f decay rate which isreasonably close to the present experimental limit. In the viable models with detectablerates the decay is mediated by the exchange of a number of sterile neutrinos, whosemasses must lie in the 1-100 MeV range. Their couplings must also be as large as ispermitted. The requirement of such large couplings to obtain observable signals suggests

8 Supernovae bounds can arise howeverfor particles that are not in equilibrium with the SM neutrinos [ 30].


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P Bamert et al. /Nuclear Physics B 449 1995) 25-48 47t ha t an obse r ved f l f l~ s ig n a l s h o u l d b e a c c o m p a n i e d b y t h e a p p e a r a n ce o f n e w p h y s i c si n o t he r expe r i m en t s , such a s i n v i o l a t i ons o f l ep t on un i ve r sa l i t y .

A p o t en t i a l p r ob l e m w i t h t h is m ode l i s t ha t it p r ed i c t s an exces s num ber o f deg r ee so f f r e edo m a t nu c l eosyn t hes i s t ha t i s equ i va l en t t o 78- o f a neu t r ino , i n com par i son w i t ht h e p r e s e n t l y q u o t e d b o u n d o f 0 . 4 . A w a y o f e v a d in g t h is n u c l e o s y n t h e s is b o u n d i sde sc r i be d i n Se c t i on 5 , bu t the r equ i r ed coup l i ngs and m asse s f o r t h i s s cena r i o w ou l dp r ed i c t a f l f l~ r a t e w h i ch i s t w o o r de r s o f m agn i t ude sm a l l e r , and so ou t o f r each o fcu r r en t exp e r i m en t s . We d o no t r ega r d t h i s a s su f fi c i en tl y w o r r y i ng t o p r ec l ude s ea r ch i ngf o r t h i s decay i n cu r r en t expe r i m en t s , s ince an expe r i m en t a l s i gnal w ou l d l i ke l y p r om ptm or e i m ag i na t i ve app r oaches t o unde r s t and i ng nuc l eosyn t hes i s i n t he se m ode l s .

4 ) B e c a u s e o f it s s o f t s p e c tr u m , w e w e r e u n a b l e to c o n s t ru c t v ia b l e m o d e l s o fthe n = 7 f l f l~ decay f o r w h i ch t he p r ed i c t ed r a t e i s l a r ge enough t o be de t ec t ab l e .Thus , a l t hough t he ex i s t ence o f an n = 7 decay m ode i s expe r i m en t a l l y t an t a l i z i ng , i t i su n l i k e l y t o b e d i s c o v e r e d i n tiff decay expe r i m en t s f o r t he f o r e seeab l e f u t u r e .

5 ) F o r t h e p u r p o s e s o f c o m p a r i s o n w e a l s o e x a m i n e d a c la s s o f m o d e l s f o r w h i c hf l f l~ a r i s e s due t o t he m u t ua l i n t e r ac t i ons am ongs t new pa r t i c l e s i n t he s ca l a r s ec t o r ,r a t he r t han f r om new s t e r i l e l ep t ons . A l t hough t he decay r a t e t ha t i s p r ed i c t ed f o r t he sem ode l s i s a l so t oo sm a l l t o be s een , t he decays neve r t he l e s s exh i b i t nove l f ea t u r e s .O ne such i s a non - s t anda r d angu l a r d i s t r i bu t i on o f t he t w o e l ec t r ons a s a f unc t i on o ft he open i ng ang l e be t w een t hem , s i m i l a r t o w ha t a r i s e s i n m ode l s w i t h r i gh t - handedcur rent s .

A l l o f t he se r e su lt s s e r ve t o unde r l i ne the t heo r e t ic a l unde r s t and i ng w h i ch ha s em er g edove r t he pa s t f ew yea r s . Sca l a r - em i t t i ng 8 / 3 decays can r ea sonab l y be expec t ed t o beo b s e r v e d , a l t h o u g h t h e p r o p e r t ie s o f t h e p h e n o m e n o l o g i c a l l y v i a b le m o d e l s w h i c h c a ndo so a r e ve r y d i f f e r en t t han w ha t w ou l d be expec t ed ba sed o n i n t u i t ion tha t i s ba sedo n t h e o r i g in a l G R - t y p e m o d e l s w h i c h o r i g in a l l y m o t i v a t ed t h e se e x p e r i m e n t s.

cknowledgementsW e w o u l d l i k e t o t h a n k K a i Z u b e r f o r p r o v o k i n g a r e - e x a m i n a t i o n o f m u l t i - s c a l a r

m o d e s i n d o u b l e b e ta d ec a y . O n e o f u s E B . ) w i

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