gelmini, nussinov, roncadelli

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Nuclear Physics B209 (1982) 157-173© North-Holland Publishing Company

B O U N D S A N D P R O S P E C T S F O R T H E M A J O R O N M O D E LO F L E F T H A N D E D N E U T R I N O M A S S E S

Graciela B. GELMINI*

Sektion Ph ysik Ludwig-Maximilians-Universitlit Munich F R G

Shmuel NUSSINOV** and Marco RONCADELLI***

Max -Planck-Insti tut f i ir Physik und Astrophysik Munich F R G

Received 16 November 1981

(Revised 16 July 1982)

We analyze further implications of the existence of a massless boson - the Majoron - mainlycoupled to left-handed neutrinos. Finallepton spectrum in pion leptonic decay is affected by Majoronbremsstrahlung, yielding a bound on Maj oron-neutrino coupling from experimental data. Supernovadynamics is another issue discussed here. We address possible effects of the frequentneutrino-neutrino collisions, neutrino interconversion and light boson bremsstrahlung in neutrinoscattering. In particular, we focus on the question of lepton number non-conservation. We find thatdeleptonization of the core and high-entropy collapse can occur, via a subtle interplay of neutrinointerconversion and light boson bremsstrahlung. Then, we propose an unorthodox solution to themissing mass problem in the universe. Finally, we address the problem of galaxy formation, whichappears in a new perspective within our model.

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

A mode l fo r l e f t -handed neu t r ino mass gene r a t ion v ia non-van i sh ing vacuum

expecta t ion value (VEV) of an i sot r ip le t Higgs f ie ld has been recent ly suggested ,

and some of i ts conseque nces were exp lored [1-12] . On e of i ts main fea tures i s

the existe nce of massive charg ed scalar part icles, X ±, X ±±, with masse s of the orde r

of the s t andar d We in be rg -S al am gauge bosons and in the ra t io ~/2, a neutra l l ight

Higgs bos on pt ., and a t ru ly mass less Gol ds t one boso n - the M aj oro n M - associa ted

wi th the spon t aneous b reak i ng o f a g loba l , non -an oma lou s cu r ren t , B - L . Al l these

part icles may man ife st themse lves in Z ° decays, and e÷e - or e e coll isions.

The n eut r in o mass matr ix has the form:

m i i = g i i v (i,/ = e, /~ , ~-), (1)

where v i s the vacuum expecta t ion value of the new Higgs t r ip le t ~ . A poss ib i l i tyis tha t the i n te rgen era t ion al mixings are smal l , in which case eq. (1) takes the

* Present address: ICTP, Trieste, Italy. Address after 1st January 1983: CERN, Geneva, Switzerland.** On leave from Tel-Aviv University, Ramat-Aviv, Tel-Aviv, Israel. Work supported in part by IsraeliAcademy of Science.***Associato INFN, Pavia, Italy. Address after 1st November 1982: Scuola Normale Superiore, Pisa,Italy.

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1 5 8 G B G e l m i n i et a l / B o u n d s a n d p ro s pe c ts f o r M a j o r o n m o d e l

a p p r o x i m a t e f o rm :

ml = g l lv. (2)

A l t e r n a t i v e l y, a l l g i i e l e m e n t s c a n b e c o m p a r a b l e . C o r r e s p o n d i n g l y, t h e n e u t r i n o

m i x i n g a n g l e s a r e l a rg e . T h e a s t r o p h y s i c s o f t h e c o r e s o f r e d g i a n t s t a r s [ 2 ] y i e l d st h e b o u n d *

v < 100 keV . (3 )

D u e t o e q s . ( 1 ) - ( 3 ) , i t f o l lo w s t h a ta l l n e u t r i n o s m u s t b e l ig h t er t h a n a b o u t1 0 0 k e Vi n t h i s m o d e l .

W i t h r e s p e c t t o t h e c o s m o l o g i c a l i m p l i c a t i o n s o f t h e m o d e l , t h e a n n i h i l a t i o ni i

/'~L/-'L ~ M M d e c i m a t e s t h e i t h n e u t r i n o s p e c i es , as s o o n a s t h e r a d i a t io n t e m p e r a t u r eT e q u a l s t h e m a s s o f ui. T h i s p r e v e n t s n e u t r i n o s o f a n y m a s s f r o m p l a y i n g a n y r 61 ei n t h e s o l u t i o n o f t h e m i s s in g m a s s p u z z l e [ 1 5 ] , e v e n i f n e u t r i n o m a s s e s l a rg e r t h a n

3 0 e V a r e c o n f i r m e d v i a e n d o f /3 - d e c a y s p e c t r u m [ 1 6 ] o r d o u b l e [ 3 -d e c a y [1 7 ]m e a s u r e m e n t s . T h e l a s t p r o c e s s y ie l ds th e f u r t h e r b o u n d :

gee ~< 10 -3. (4)

A m u s i n g l y, e q s . ( 2 ) , ( 3) a n d ( 4 ) t o g e t h e r i m p l y a l r e a d y

mvo ~ ge~v< 100 eV , (5 )

a s l o n g a s i n t e rg e n e r a t i o n a l m i x i n g s a r e n e g l e c t e d . N o c o s m o l o g i c a l r e s t r i c t i o n se x i s t o n g ,~ , o r g ,, , si n c e t h e b o u n d [ 1 8 ] Z im v, ~ 5 0 - 1 5 0 e V isn o t val id in th i sm o d e l . F u r t h e r c o n s t r a i n t s o n g~j c o u l d i n p r i n c i p l e b e i n f e r r e d f r o m t h e b o u n d so n r a r e d e c a y s , su c h a s / z ~ e e l , if i t w e r e n o t b e c a u s e o f th e s t r o n g d e p e n d e n c eo f t h e s e p r o c e s s e s o n t h e u n k n o w n i n t e g e n e r a t i o n a l m i x i n g an g l es . I n c i d e n t a l l y, i no r d e r t o s i m p l i f y t h e f o l l o w i n g d i s cu s s io n w e s h a ll o f t e n i g n o r e m i x i n g e f f e c ts .

O u r g o a l s i n t h e p r e s e n t p a p e r a r e t h e f o l l o w i n g :( a ) We p o i n t o u t t h a t d i s t o r t i o n s o f c h a r g e d l e p t o n s p e c t r a d u e t o M a j o r o n

bre m ss t r a h lu ng o f v~ o r v~ a r e ex pe c t e d in p ro ces s e s such a s 7 r ~ txv~ ,, ~ - -~ v~ ,~ e - ,r ~ ~ r u ,, 7 - ~ u , u ~ e , r - ~ u , ~ , t z - .Th i s imp l i e s d i r e c t bou nd s on g~,, ( g , , ~< 10 - z ) andm u c h w e a k e r o n e s o n g , , , b e c a u s e o f l a ck o f d at a . T h e r e f o r e g , , is o n l y re s t r ic t e db y t h e t r i v ia l b o u n d g , , < 1 .

( b ) We d i s c u s s t h e p o s s i b l e i m p a c t o f o u r M a j o r o n m o d e l o n s u p e r n o v a a s t r o -p h y si cs . In p a r t ic u l a r , w e p o i n t o u t t h e r e l e v a n c e o f f r e q u e n t n e u t r i n o - n e u t r i n oc o l l i s i o n s , n e u t r i n o i n t e r c o n v e r s i o n a n d l i g h t b o s o n b r e m s s t r a h l u n g i n n e u t r i n os c a tt e ri n g s. T h e n , w e a d d r e s s th e i m p o r t a n t is su e o f l e p t o n n u m b e r n o n c o n s e r v a -t io n . T h i s h a s b e e n e m p h a s i z e d i n a r e c e n t p a p e r b y K o l be t a l . [ 5] . A l t h o u g h l e p t o nn u m b e r is c o n s e r v e d in s u p e r n o v a c o re sa n d t h e p r o c e s s c o n s i d e r e d b y K o l b , Tu b b sa n d D i c u s v a n i s h e s , w e f i n d t h a t t h e e f f ec t o f c o r e d e l e p t o n i z a t i o n a n d h i g h - e n t r o p yc o l la p s e e n v i s a g e d b y t h e s e a u t h o r sm a y s ti ll h a p p e n T h i s r e q u i r e s a s u b t l e i n t e r p l a y

* V e r y r e c e n t ly, a m o r e c a r e f u l a n a l y s is h a s b e e n p e r f o r m e d b y F u k u g i t aet a l [ 1 3 , 1 4 ] . F r o m t h es u n , t h e y g e t v < 6 0 0 k e V, w h e r e a s f r o m t h e r e d g i a n t s t h e y o b t a i n v < 6 0 k e V. T h e b o u n d e q . ( 3)i s c o n s i s t e n t w i t h t h e s e r e s u l t s .


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o f n e u t r i n o i n t e r c o n v e r s i o n a n d l ig h t b o s o n b r e m s s t r a h l u n g , a n d f u r t h e r m o r e c a no c c u r o n l y if s o m e l o w e r b o u n d s o n t h e Y u k a w a c o u p l in g s in th e n e u t r i n o s e c t o ra r e m e t .

(c) W e a d d a f e w r e m a r k s o n t h e l o n g r a n g e f o r c e s d u e t o M a j o r o n a n d l i g htH i g g s e x c h a n g e s , w h o s e e f f e c t s t u r n o u t h o w e v e r t o b e n e g l ig i b le .

( d) A s m a l l m a s s f o r t h e M a j o r o n m i g h t b e g e n e r a t e d i f t h e g l o b al s y m m e t r yB - L is e x p l i c it l y b r o k e n b y a s m a ll t e r m i n t h e l a g r a n g i a n . W i t h a m a s s o fa p p r o x i m a t e l y 1 0 - 2 0 e V, t h is c o u l d a c c o u n t f o r t h e m i s si n g m a s s o f t h e U n i v e r s e .

( e) W e b r i ef l y c o m m e n t o n t h e p o s s ib l e c o n n e c t i o n b e t w e e n g a l a x y f o r m a t i o na n d t h e d e n s i t y f l u c t u a t i o n s i n d u c e d b y t h e l a t e p h a s e t r a n s i t i o n a t w h i c h l e p t o nn u m b e r is s p o n t a n e o u s l y b r o k e n . T h i s l as t p o i n t w i ll b e o n l y s u p e rf ic i al ly t r e a t e dh e r e .

T h e r e s u l t s o b t a i n e d i n t h e p r e s e n t p a p e r a p p l y a l s o t o t h e v a r i o u s e x t e n s i o n so f o u r m o d e l d u e t o S c h e c h t e r a n d V a l le [ 4] , B a r b i e r i a n d M o h a p a t r a [ 1 9 ] a n dM o h a p a t r a a n d S e n j a n o v i c [2 0 ].

2 . M a j o r o n b r e m s s t r a h l u n g a n d b o u n d s o n t h e c o u p l i n g s

H i g h p r e c i s i o n m e a s u r e m e n t s f o r ~r a n d / z d e c a y s h a v e b e e n s u c c e s sf u ll y c o m -p a r e d w i t h t h e o r e t i c a l c a l c u l at io n s i n c l u d in g r a d i a t i v e c o r r e c t io n s . T h e p s e u d o s c a l a rM a j o r o n c o u p l e s , w i t h s t r e n g t h g . . , t o t h e u . i n t h e f in a l s t a t e s o f t h e s e d e c a y s .T h e p r o d u c e d M a j o r o n s c a n n o t b e d i r e c tl y d e t e c t e d b e c a u s e o f t h e ir e x t r e m e l yw e a k c o u p l in g t o a n y c h a r g e d f e r m i o n F

hFF~--VmF


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a n d t h e t o t a l r a t e

w i t h

G.B. Gelmini et al. / Bounds and prospects or Majoron model

F T r + t z v . M ) = 2 .,",, 2 1'2 3 ~r,1 ~ - ~ 3 g m , t. rF f ~ m ~ 2,COS2 0c (9)

Z 1 , 3 2 ( l - a ) 2~ -~g -I-~ Og - - 7 0 / 3 - t - 0 / 4 + 0 ~ 5 - - 7 0 t 6 + 0 1 2 ( 1 - - 0 t 2 ) 2In z , (10)e

w h e r e a - - m . / m = a n d e - ~ r n * / r n , . m * b e i n g e f f e c t iv e l y t h e b o u n d o n t h e m a s s a sf ix e d b y t h e e x p e r i m e n t a l r e s o l u t i o n (e >~ 0 .1 M e V ) . T h i s r a t e h a s t o b e c o m p a r e dw i t h t h e o r d i n a r y p h o t o n b r e m s s t r a h l u n g , w h o s e b r a n c h i n g r a t i o [ 2 1 ] i s

F r r ~ t x v ~ y )= ( 1 . 2 4 + 0 . 2 5 ) 1 0 - 4 . ( 11 )

F ( ~" - , ~ v ~ )R e q u i r i n g F ( ~ r ~ t z v ~ M ) n o t t o e x c e e d th e u n c e r t a i n t y i n F ( T r -~ / x v ~ y ), F ( T r -*I z v. M ) / F Tr ~ t ~ v. )


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exis tence of low mass sca lars coupled p r imar i ly to neutr inos . In par t icular, an

exper imen ta l a r r an geme nt ve to ing pho tons cou ld be ex t remely use fu l .

3 . N e u t r i n o n e u t r i n o s c a tt e ri n g . W h e r e is i t i m p o r t a n t ?

On e of the character is t ics of our mo del i s the s t r ik ing predic t ion of large neutrino -neutrino cross sections, due to Majo ro n exchange and p roduc t ion* . The in t e rc onver-s ion VeVe VHVrI, for example, has a cross section**:

2 2 2 2geegI-m~g~gHH 10-23 2

tr(v~v~ vi-iVla)~- 24 Ir E~ w2~ • cm , (14)

with t o ~ - E v / 1 Me V (H s tands for heav y , i .e ., /z, r ) . To get an idea about themagn i t ude o f cr(v~v~ ~ VHvH), we can compare it to the typical cross sections of thes tandard weak in terac t ions , for example the cross sec t ion for coherent n e u t r i n o -nucl eon scat ter ing [22]:

c r ( u N c o h ) G s i n E O w E g v At e - 4 5 A 2 2 2~- - ~l v ~ oJvcm . (15)zr

As is clear, ~r(V~V~UHVH) exce eds o'(vN)coh by a fact or of 8 20 gHH, f or oo~ ~ 2 0,2 - 4A ~ 50 and g2 ~ 1 0 - 6 . Wit h the bou nd g,~, ~< 10 and g, , ~ 0(1) me nt io ne d above,

th is fac tor runs o v e r 1 0 4 t o 1 0 8 .This p redom inan ce shou ld be con t ra s t ed wi th the

negl ig ib le ro le of the Ma jo ron exchange in vF***or FF (F ~ v) processes , s te mmin g

from the smal lness of the hFF coupl ings (see eq. (6)).

However, the net effect of a given interaction depends on the number of collisionsper unit length ~ 1/h ~nar, where A is the mean free path (m.f.p.) , n is the numberdensi ty and t r the cross sec t ion considered. U nd er most c i rc umstances the neut r in o

num ber dens ity, n , , i s ve ry smal l i n compar i son wi th the num ber dens i ty of nuc leons

and e lec t rons nN ~ ne =N A (NA being the A voga dro numbe r) . The resul t ing m.f .p .

fo r neu t r in o -n eu t r in o co ll i sions

1A~, ~ - - (16)

n,zr,,~,

* When the temperature T is higher than the non-s tandard VEV., i.e., T ~ v, the mixing between thetriplet and doublet Higgs fields does not exist . Since the global symmetry B-L is restored, noGoldstone boson exists either (see sect. 4). In this case, M and PL are replaced by ~o, the neutralcomponent of the triplet, with light mass ~ T. The contribution of ~o introduces a factor 4 in thecross sections for all new unconventional neutrino-neutrino scatterings.

** The differential cross section does peak for/3 ~ 1 (super-relativistic neutrinos) near 0 = 0 and theintegrated area of the peak behaves as In (1 -/3). Such small angle scatterings do not contribute tothe diffusion and can be ignored. Hence the simple expression given by eq. (14) is applicable.

***For example, the contribution of M and PL exchange to the neutrino-electron scattering cross sectionis about eleven orders of magnitude smaller than the gauge boson exchange contribution forEv = 1 MeV and my = 1 eV [1].


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is t h e n m u c h b i g g e r t h a n t h e c h a r a c t e r i s t ic s i z e o f t h e s y s t e m R , a n d t h e s y s t e ma n d t h e n e u t r i n o s h a r d l y s c a t t e r f r o m e a c h o t h e r b e f o r e e s c a p i n g .

A s a n e x a m p l e , c o n s i d e r t h e s o l a r c o r e f o r w h i c h R ~ - 1 01 ° c m . T h e t o t a l n e u t r i n of lu x @v c a n b e i n f e r r e d f r o m t h e s o l a r l u m i n o s i t y a n d t h e a v e r a g e n e u t r i n o e n e r g ycoy = 1 t o b e qbv ~ 1 0 3 9 u / s e c . T h e n e u t r i n o d e n s i t y i n t h e c o r e i sn~-~@jR2c ~-

10 9 / c m 3. H en ce , ev en fo r g r i n - - 1 we f i nd , f r o m eqs . ( 14 ) a nd (16 ) : A~v >> 1020 c m >>

R . W e c a n l i k e w i se v e r if y t h a t n o s i g ni fi c an t n e u t r i n o - n e u t r i n o s c a t t e ri n g s o c c u rf o r n e u t r i n o s i n r e a c t o r s o r i n a c c e l e r a t o r b e a m s .

W e d o e x p e c t s t r o n g n e w e f f e c t s i n t h o s e s p e c i a l c i r c u m s t a n c e s w h e n A ~ - R system .A n e x a m p l e i s t h e f a t e o f m a s s i v e ( s a ymv ~- 1 0 0 e V ) c o s m o l o g i c a l n e u t r i n o s , a s t h er a d i a ti o n t e m p e r a t u r e d r o p s d o w n t o T~ =m~ [ 2] . N e u t r i n o s i n t e r c o n v e r t a n d a n n i h i -l a t e w i t h c r o s s s e c t i o n s [ 2 ] l i k e t h o s e o f e q . ( 1 4 ): o - = 1 0 - 27 c m 2 f o rgii 10-3( r e q u i r e d i f m , , = 1 0 0 e V ) . T h e n e u t r i n o n u m b e r d e n s i t y s c a le s w i t hT 3 a n d a t1 0 0 e V i s n ~ ~ 3 1 01 9 c m - 3 . T h e c o r r e s p o n d i n g m . f . p , f o r a n n i h i l a t i o n i s Av~ -,M M '~108 c m t o b e c o m p a r e d w i th t h e h o r i z o nc t . = 3 • 10 TM c m , w h e r e t . ~ 1 08 s e c is t h ea g e o f t h e u n i v e r s e w h e n T = m y, ~ 1 00 e V. E q u i v a l e n t l y, o n e c a n c o m p a r e t h ea n n i h i l a ti o n r a t e w i t h 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 = t 1. T h i s is t h e r e a s o nw h y a ll c o s m o l o g i c a l n e u t r i n o s a n n i h i l a t e a t t e m p e r a t u r e s T ~< m ~, i n t h e p r e s e n tm o d e l , e v e n i f g , ~ 1 0 - 6 - 1 0 - 5 .

O u r q u e s t f o r d e n s e n e u t r i n o " t a r g e t s " l e ad s u s t o s u p e r n o v a c o r es . A s w e w i lln e x t s h o w, i t i s h e r e t h a t w e f i n a l ly h a v en~ ~- nN n~ i . e . , n e u t r i n o n u m b e r d e n s i t i e sa p p r o a c h i n g t h o s e o f th e n u c l e o n s ( o r e l e c t r o n s ) in th e m e d i u m .

S o m e a s p e c t s o f s u p e r n o v a e , f o r e x a i n p l e , t h e n u c l e o s y n t h e s i s o f h e a v i e r e l e -m e n t s , a r e q u a l it a t iv e l y w e ll u n d e r s t o o d . T h e r e a r e h o w e v e r s e v e r a l p u z z le s su c ha s t h e m e c h a n i s m o f t h e " b o u n c e " , r e q u i r e d t o e j e c t th e s t a r m a n t l e , l ea v i n g t h ec o r e a s a r e m n a n t n e u t r o n s t a r [ 2 3] . S e v e r a l y e a r s a g o , i t w a s p o i n t e d o u t [ 2 2 ] t h a tt h e n e u t r i n o d i ff ra c ti o n v ia Z ° e x c h a n g e o n n u c le i p r o d u c e s a m o m e n t u m t r a n s f e ro f t h e o r d e r o f t h e e n e r g y o f t h e n e u t r i n o s ,A p ~ E ~ ~ 2 0 M e V . T h e c o h e r e n c e o ft h e s e l o w e n e r g y s c a t te r i n g s i m p l i e s a q u i c k r i se o f th e c r o s s s e c t i o n - g i v e n ju s tb y eq . ( 1 5 ) - w i t h t h e a t o m i c n u m b e r A . I t h a s b e e n s u g g e s t ed [ 2 2 , 2 4 ] t h a t th ec o h e r e n t n e u t r i n o d i f f r a c t i o n o n t h e F e - N i s h e l l s u r r o u n d i n g t h e c o r e a c c o u n t s f o rt h e b o u n c e ( A = 5 6 f o r F e ) . T h i s h y p o t h e s i s h a s b e e n e x t e n s i v e l y te s t e d n u m e r i c a l l y[ 2 5 ] a n d a p p e a r s t o f a ll a b i t s h o r t e r . M o r e r e c e n t l y, a n o t h e r s c e n a r i o f o r s u p e r n o v ad y n a m i c s d u e to B e t h e , B r o w n , A p p l e g a t e a n d L a t t i m e r ( B B A L ) [ 26 ] h a s b e c o m em u c h m o r e f a v o u r e d . I n th i s s c e n a r i o , th e b o u n c e c o m e s f r o m a s ti ff e q u a t i o n o fs t a t e a n d is i n th e f o r m o f a m e c h a n i c a l s h o c k w a v e . P e c u l i a r t o t h is p i c t u r e i s ac o l l a p s e w i t hsmall specif ic entropy c o n t r o l l e d t o a l a r g e e x t e n t b y t h e d e g e n e r a t eF e r m i e n e r g y o f n e u tr i n o s a n d e l e c t ro n s .

T h e b a s i c r e a c t i o n t r i g g e r i n g t h e c o l l a p s e i s e - + p ~ n + u ~. I t c o n v e r t s a l l p r o t o n sw i t h i n a c o r e o f m a s s ~ 1 . 4 M G-~3. 1 0 3 3 g i n t o t h e n e u t r o n s c o n s t i t u t i n g t h e

r e m n a n t n e u t r o n s ta r . T h e t o t a l n u m b e r o f n e u t r i n o s e m i t t e d i n a f a i rl y s h o r t


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G.B. Gelm ini et al . / Bou nds and prospects for Ma]oron m odel

co l l a p se t im e t¢o~ lap~ i s t h e r e f o r e

N ~ = N A " 3 ' 103 3 ~ 105 7 .

163

( 17 )

L e t tr b e t h e a v e r a g e t i m e t h a t a t y p ic a l m e m b e r o f t hi s g ig a n ti c n e u t r in o p o p u l a t i o nr e m a i n s i n s i d e t h e c o r e b e f o r e e s c a p i n g . I f t~ ~ tco ~lap ~, t h e e m i t t e d n e u t r i n o s s t a yi n t h e c o r e t h r o u g h o u t t h e w h o l e c o l l a p s e . S i n c e w e h a v e p r a c t i c a l l y o n e n e u t r i n oe m i t t e d p e r n u c l e o n , t h is w o u l d i m p l y c o m p a r a b l e n u m b e r d e n s i t i e s n ~ = n N. I ng e n e r a l t r < t c o l l a p s e .T h e n e u t r i n o s s t a y in t h e c o r e o n l y a f r a c t i o nf= - tr/tcollapse < 1o f t h e e f f e c t i v e n e u t r i n o - e m i t t i n g p e r i o d , i .e . , t~ ol~ ap~ , a n d t h e r e f o r e t h e i r d e n s i t yis c o r r e s p o n d i n g l y r e d u c e d :

t rn~, ~ fn N ~ - - n N . (18)

tcoUapse

T h e i m p o r t a n t p o i n t i s t h a t i n a ll c o l l a p s e s c e n a r io s , f i s n o t m u c h s m a l l e r t h a n 1 ,f - 1 0 - l - 1 0 - 3 b e i n g a t y p ic a l r a n g e . T h e e f fe c ti v e n e u t r i n o - e m i t t in g p e r i o d -d u r i n g w h ic h s a y 5 0 % o f t h e n e u t r in o s a r e e m i t t e d - c a n b e r e a d o f f f r o m f ig . 6o f re f . [2 4 ], w h i c h r e p r e s e n t s t h e r e s u l ts o b t a i n e d b y Wi l s o n [ 2 5 ] f o r a c o r e d e n s i t y2 . 1 0 a l g / c m 3. T h i s t i m e , t co l la p se - -~ 0 .0 2 -0 .0 4 s ec . F o r a m a s s o f 3 . 1 0 33 g , t h ec o r r e s p o n d i n g s i z e i s R ~ 3 1 /3107 c m--~ 1.2 107 cm.

C l e a r l y, t h e m i n i m a l p e r i o d t h a t a n e u t r i n o s t a y s i n t h e c o r e i sR / c , i .e . ,4 . 1 0 - 4 s e c. T h i s c o r r e s p o n d s t o t h e f a s t e s t ( c o ll is io n l es s ) e s c a p e . T h u st r>~R/c ,

i . e. , t r y>4 10 -4 s ec . and fo r t he ca se a t ha nd f =tJtcoHapse ~ ( 0 . 5 -1 ) 10 -2 .D u e t o s c a t t e r i n g o f t h e o u t g o i n g n e u t r i n o s o n t h e e l e c t r o n s a n d n u c l e o n s i n t h e

c o r e , tr m a y e x c e e dR / c . I f t h e s c a t t e r i n g is c o m p l e t e l y i s o t r o p i c , a n d t h e n e u t r i n os c a tt e rs S t i m e s b e f o r e l e a v in g th e c o r e , t h e n f r o m s t a n d a r d r a n d o m w a l k e s t i m a t e s[27] :

R ~ x /S hv, ( 19a )

S ~ (R /A~ ) 2 , ( 1 9 b )

tr ~ t o t a l v p a t h l e n g t h / cC h C h

F o r t h e a b o v e c o r e p a r a m e t e r s (p = 2 . 1 0 1 1 g c m - 3, T = 3 M e V ) a n d f o r n e u t r i n oe n e r g i e s ~ 1 6 M e V, t h e m . f. p , f o r n e u t r i n o - e l e c t r o n a n d n e u t r i n o - n u c l e o n c o ll is io n sAv e- -~ 8 " 1 0 6 c m , A . N -- ~ 4 " 1 0 5 c ma r e c o m p a r a b l e t o , e v e n s o m e w h a t s m a l l e r ( b ya b o u t a f a c t o r o f 1 0 ) t h a n t h e c o r e r a d i u s . F r o m e q . ( 1 9 c ) w e f in d t h a t tr c o u l de x c e e d i t s m i n i m a l v a l u eR / c b y - 1 0 a n d

tr 10 R

. . . . 0 .1 - 0 .0 5 (2 0 )f -~ /collapse AvtcoU apse


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1 6 4 G B Gelm in i e t a l / B ound s and prospec ts fo r Ma/oro n mode l

A l o w e r b o u n d o n tc on apse is o f f e r e d b y t h e f r e e f a l l t i m e :

tf~ = (2 4 " r rG N p ) - 1 / 2 = 1 0 - 3 s e c . ( 2 1 )

T h e r a d i a l t h e r m a l n e u t r o n v e l o c i t y( v ~ ) l / E = x / - ~ = x / - 1 / 3 0 0 c ~ - O . 0 6 c a l s oy i e l d s a r o u g h e s t i m a t e o f t o o . a p s e= 1 0 - 2 s e c . * v e r y c lo s e to ( 0 . 5 - 1 )1 0 - 2 s e c . t h ev a l u e s u s e d a b o v e . T h u s i n c o n c l u s i o n

n ~ ~ ( 1 0 - 1 - - 1 0 - 3 ) n N ( 2 2 )

a s a c o n s e r v a t i v e e s t i m a t e .

W e n o w t u r n t o th e n o v e l e l e m e n t o f n e u t r i n o - n e u t r i n o s c a tt e ri n g . S i n c e in t h e

s t a n d a r d m o d e l o-w = c r ~ a n d n v < n~ (o r n N ), th e s e a r e s a f e l y n e g l e c t e d i n a n ys c e n a r i o b a s e d o n s t a n d a r d w e a k i n t e ra c t i o n m o d e l s . H o w e v e r , in t h e p r e s e n t m o d e ln e u t r i n o - n e u t r i n o s c a t te r in g s a r epredominan t . S p e c i f i c a l l y, f r o m e q s . ( 1 5 ) ( u s e df o r a c r o s s s e c t i o n p e r n u c l e o n ) a n d ( 1 4 ) w e f i n d t h e r a t i o o f t h e i n t e r c o n v e r s i o nc r o s s s e c t i o n ~ r( v ~ v ~+ v ~ v ~ ) ( u s e d h e r e s i n c e g ~ i s l e a s t r e s t r i c t e d b y p r e v i o u sa n al ys is ) t o t h e s t a n d a r d n e u t r i n o - n u c l e o n c r o ss s e ct io n

o ' ( v ~ v ~ v ~ ' ~ ) 102° 2 2c r (v N ) = - - 7 - g ~ g ~ , ( 23 )tO v

M u l t i p l y i n g b yn~/nN o f e q . ( 2 3 ) w e f i n a l l y a r r i v e a t4 , ' t ~ - - 1 9h . . . . - . . . 6 0 v q t U - - 1 0 - 1 7 )

2 2 (24)

AvN g~¢gT~T h u s , u n l e s s g¢~g~ ~ 1 0 - 7 5 - 1 0 6 5, t h e m . f . p , f o r t h e n e w p r o c e s s i s s m a l l e r t h a nt h e s t a n d a r d o n e f o r E ~ ~ 1 0 M e V , t h e t y p i c a l e n e r g y c o n s i d e r e d . I f g ~ ~ 1 0 - 6 a n dg ~ ~ 1 , t h e f a c t o r is 1 0 _ 9 - 1 0 - 7 , s i n c e , a s w e h a v e s e e n a b o v e , A ~ ~ / ~v N~ O ( R / 1 0 ) ,w e c e r t a i n l y h a v e A ~


9/17

G B G e l m i n i e t a l / B o u n d s a n d p ro s p ec ts f o r M a j o r o n m o d e l 165

l e p t o n n u m b e r v i o l a ti n g p r o c e s s e s c a n c o n v e r t n e u t r i n o s i n to a n t i - n e u t r i n o s ,l e a d i n g e v e n t u a l l y t oalmost complete deleptonization of the core and high entropycollapse.W e w i ll n o w e l a b o r a t e o n t h e s e p o i n t s.

4.1. MAJORONS AND OUTSTREAMING NEUTRINO FLUX

I t i s c l e a r f r o m t h e d i s c u s s io n o f s e c t 3 t h a t n e u t r i n o - n e u t r i n o s c a t t e r in g s a r eh e r e m u c h m o r e f r e q u e n t t h a n i n th e s t a n d a r d w e a k i n t e r a c ti o n m o d e l s.

S i n c e r a d ia l m o m e n t u m is c o n s e r v e d i n t h e s e p r o ce s s e s , th e o u t w a r d n e u t r i n of lo w f r o m t h e c o r e is u n h i n d e r e d e v e n b y t h e m o s t f r e q u e n t n e u t r i n o - n e u t r i n oc o ll is io n s . T h e l a r g e n u m b e r o f n e u t r i n o s e l f - i n te r a c t io n s m a k e s t h e o u t s t r e a m i n g

n e u t r i n o f lu x m u c h m o r ecoherent. S u p p o s e a n o u t g o i n g n e u t r i n o i s t u r n e d b a c kb y a h a r d c o l li s io n w i t h a n u c l e o n . I n t h e u s u a l s c e n a r i o i t w o u l d f o l l o w i ts n e wd i r e c t i o n f o r a l o n g t i m e u n t i l e s c a p i n g o r s u f f e r i n g a s u b s e q u e n t c o l l i s i o n . I n t h ep r e s e n t m o d e l i t w i l l s h a r e h o w e v e r i t s m o m e n t u m w i t h a l l n e i g h b o u r i n g o u t g o i n gn e u t r i n o s . I t tu r n s b a c k a g a i n a n d c o n t i n u e s t o f o l lo w t h e o u t s t r e a m i n g m o t i o n .T h u s , i n s t e a d o f b e h a v i n g l ik e a d i l u t e g a s , t h e n e u t r i n o s f lo w l ik e a r e l a ti v i s ti c fl u id .

4.2. INTERCONVERSION OF NEUTRINO SPECIES

A s e c o n d e f f e c t o f t h e l a rg e n e u t r i n o - n e u t r i n o c r o s s s e c ti o n s is t h a t t h e i n i t ia l lyp r e s e n t v e w i ll b e i m m e d i a t e l y i n t e r c o n v e r t e d i n t o v , , u , a n d l ig h t H i g g s b o s o n s4 °. A s a c o n s e q u e n c e ,the core contains an equ al nu mb er o f al l neutrino speciesand l ight Higgs mesons in thermal equil ibrium.T h e p r e s e n c e o f v, a n d q~0 w i ll b eo f c r u c i a l i m p o r t a n c e f o r t h e p h y s i c s o f s u p e r n o v a e . D e f e r r i n g t h e d i s c u s s i o n o ft h e m a j o r i ss u e to s u b s e c t. 4 . 3 , w e n o t e h e r e t h a t , e v e n i f t h e t o t a l l e p t o n n u m b e ris c o n s e r v e d - w h ic h is i n d e e d t h e c a se ( s ee b e l o w ) - th e n e u t r i n o n u m b e r d e n s it yf o r e a c h s p e c i e s i n t h e c o r e i s r e d u c e d b y a b o u t a f a c t o r o ffour w i t h r e s p e c t t ot h e v e d e n s i t y in t h e u s u a l s c e n a r i o . T h i s f a c t c a n h a v e s e v e r a l i m p l i c a ti o n s .

T h e r a t e o f t h e r e a c t i o n v e + n ~ p + e - ( th e r e v e r s e o f t h e b a s ic n e u t r o n i z a t i o nreac t ion ) i sreduced a n d p o s s ib l e e f f ec t s o f n e u t r i n o d e g e n e r a c y a r esuppressed.

T h e c r o s s s e c ti o n s o f v~, a n d v, o n e l e c t r o n s a r e d i f f e r e n t f r o m t h o s e o f v ~, s i n c et h e c o r r e s p o n d i n g c h a r g e d c u r r e n t a m p l i t u d e s a r e l a c k i n g . A l s o , t h e l i g h t n e u t r a lb o s o n s qb ° h a v e f o u r f o l d e n h a n c e d Z ° c r o ss s e c ti o n s , d u e t o t h e d o u b l e S U ( 2 ) ® U ( 1 )q u a n t u m n u m b e r s ( w it h r e s p e c t to n e u t ri n o s ) .

4.3. LEPTON NUMBER VIOLATION

W e c o m e n o w t o t h e m o s t d e l ic a t e a n d m o s t p o w e r f u l i m p l ic a ti o n o f t h e M a j o r o nt h e o r y in s u p e r n o v a e : t h e e f f e c t o f l e p t o n n u m b e r v i o l a ti o n [ 5] .


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166 G .B. Ge lmin i et al. / Bou nds and prospects for Ma joron mo del

W e f ir st e l u c i d a t e t h e n a t u r e o f l e p t o n n u m b e r v i o l a t i o n in th e m o d e l . T h e b a s i cf e a t u r e i s t h a t t o t a l l e p to n n u m b e r is b r o k e n o n lys p o n t a n e o u s l y b y th e v a c u u me x p e c t a t i o n v a l u e o f a H i g g s t r i p l e t qO c a r r y i n gt w o u n it s o f l e p t o n n u m b e r . T h es e p a r a t e e , / x a n d r l e p t o n n u m b e r s a r e b r o k e n e x p l ic i tl y, s in c e w e i n t ro d u c e o n l yo n e t r i p l e t q b. T h e r e f o r e , p r o c e s s e s l i k eVeV e ~ v~v~a r e a l l o w e d a n d o c c u r w i th c r o s ss e c t io n s l ik e e q . ( 1 4) . H o w e v e r , A L = 4 p r o c e s s e s , l ik ev e v ~ v ~ - v ~ m u s t h a v e a ne x p l i c it f a c t o r o f v 2 i n t h e a m p l i t u d e , s o t h a t t h e i r c r o s s s e c t i o n s g o t o z e r o l i k e v 4

a s v v a n i s h e s . T h i s a s p e c t w a s o v e r l o o k e d in r ef . [5 ] , w h e r e i t w a s a s s u m e d t h a tb ' L / J L ' ~ / , ) ~ / - ~a l s o p r o c e e d s w i t h a c r o s s s e c t i o n l i k e e q . ( 1 4 ) . To g e t t h e r i g h t r e s u l t ,o n e h a s t o c o n s i d e r t h e t w o d i a g r a m s i n v o l v in g t h e e x c h a n g e o f M a n d 0 L, w h i c hc a n c e l e x a c t l y u p t o t h eM - - p L m a s s d i f f e r e n c e . N o w , i n t h e c a s e v = 0 ,rn oL = m Ms in c e PL a n d M a r e r e s p e c t i v e l y t h e r e a l a n d i m a g i n a r y p a r t s o f t h e n e u t r a lc o m p o n e n t qb ° o f th e H i g g s t r ip l e t, s o t h a to r ( / - ' L / . P L ' '~ / . - '~ / J ~ ) =0 , a s e x p e c t e d s i n c et o t a l le p t o n n u m b e r i s n o w c o n s e r v e d . A l t e r n a t i v e l y, i f v # 0, m M = 0 a n d s o w eo b t a in t h e f ol lo w i n g o r d e r - o f - m a g n i t u d e e s t im a t e :

4 4c cO.(FL k, L.... VL VL ) grn.oL8~ 's 3 , (25)

w h e r e g is t h e r e l e v a n t Yu k a w a c o u p l in g a n dm o L = a v [ 1 , 2 ]. H e n c e ,O ( / , 'L / . - ' L - ) / J ~ / / ~ _ )g o e s li k e v 4, a s e x p e c t e d f r o m t h e a b o v e g e n e r a l a r g u m e n t . R e c a l l i n g t h e b o u n dv < 1 0 0 k e V , w e e xp e c tm o L < v s i n c e a i s n a t u r a l l y l e s s t h a n o n e [ 2 ]. Ta k i n gt y p ic a l ly x /s = 1 0 M e V , w e o b t a i n a t th e v e r y l e a s t a n e x t r a s u p p r e s s i o n b y a b o u t1 0 - 8 w i t h r e s p e c t t o t h e e s t i m a t e o f r e f. [ 5 ],

H o w e v e r , e v e n th is is n o t th e e n d o f t h e s to r y T h e c o r e t e m p e r a t u r e ~ 1 - 1 0 M e Vc o n s i d e r a b l y e x c e e d s t h e t r ip l e t V E V v < 1 0 0 k e V. I t is w e l l k n o w n [ 2 8 ] t h a t u n d e rs u ch c i r c u m s t a n c e s a s p o n t a n e o u s l y b r o k e n s y m m e t r y is r e s t o re d . T h e r e f o r e , i ns id et h e c o r e v = 0 a n d n o l e p t o n n u m b e r v i o l a t i o n a t a ll is p r e s e n t . I n p a r t i c u l a r, t h e

c cprocess /JL/- 'L~ VL/JL h a s v a n i s h i n g c r o s s s e c t i o n .T h e b o u n d s o n v w e r e d e r iv e d f r o m c o n s i d e r a ti o n s o f s te l la r o b j e c t s - t h e s u n,

r e d g ia n ts a n d w h it e d w a r f s - w h o s e c o re t e m p e r a t u r e s a r e l o w e r t h a n t he b o u n do b t a i n e d

T h e s u p e r n o v a e a r e t h e f ir st s y s t e m w h e r en e c e s s a r i l y T > vo v e r t h e c o r e r e g i o n .H e n c e , w e h a v e h e r e a ni ce t w o - p h a s e s y s te m : t h e s y m m e t r i c p h a s e i n si de t h ec o r e w h e r e t o t a l le p t o n n u m b e r is c o n s e r v e d , s u r r o u n d e d b y th e c o o l e r m a n t l ew h e r e v # 0 a n d l e p t o n n u m b e r is s p o n t a n e o u s l y b r o k e n . S i n c e t h is is a s e c o n d - o r d e rp h a s e t r a n s i t io n , w e d o n o t n e c e s s a r i l y e x p e c t v e r y d r a m a t i c e f f e c ts a r is i n g f r o ms u p e r h e a t i n g o f t h e c o r e d u r i n g th e c o n t r a c t i o n o f t h e s ta r . We t h i n k h o w e v e r t h a tt h i s d e l i c a t e a n d i n t e r e s t i n g p o i n t s h o u l d d e s e r v e a m o r e t h o r o u g h i n v e s t i g a t i o n .

F i n a l l y, w e c o m e t o t h e m a j o r p o i n t o f t h i s s e c t i o n . N o t w i t h s t a n d i n g a l l t h e

a b o v e , t h e r e s ul ts o b t a i n e d b y K o l be t a l . [ 5 ] m a y s ti ll b e t r u e T h e b a s i c r e a s o nis t h a t a l t h o u g h l e p t o n n u m b e r is c o n s e r v e d , w e m a y h a v e g e n e r a t i o n o f e l e c t ro n -


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G.B. Gelm in i e t a l. / Bou nds an d prospec ts fo r Ma joron m ode l 167

/ \4/ \ \ ,o

/ \

Fig. 1. Al tho ugh lepton num ber is conse rved in a superno va core, a n t i n e u t r i n o scan be produced inv4, ° scatterings via 4 0 bremss trahlu ng, as shown in the above diagram.

a n t i n e u t r i n o s v i a p r o c e s s e s l i k e ( s e e d i a g r a m i n fi g. 1 ):

vT + q b ° ~ v~ + ~ ° + q b ° . (2 6)

W e r e c a l l f r o m s u b s e c t . 4 . 2 t h a t vT a n d @ 0 a r e i n d e e d a v a i l a b l e i n t h e s u p e r n o v ac o r e . W e c o n s i d e r v, b e c a u s e i ts c o u p l i n g g , , t o q t, ° i s l e a s t r e s t r i c t e d b y b o u n d sd e d u c e d f r o m a v a i la b l e d a t a .

O n c e v~ is p r o d u c e d , t h e k e y r e a c t io n :

v - ~ + p ~ n + e + (27 )

c a n o c c u r. T h i s l e a d s t o ala rge n um be r o f f ree neutrons do m inat in g the pressurea n d , m o s t i m p o r t a n t l y, todepletion of electrons via e+e - ann ihilat ion.T h e r e f o r e ,t h e deleptonization of the coree n v i s i o n e d b y K o l b e t a l. [ 5 ] c a n s t il l o c c u r , p r o v i d e dh o w e v e r t h a t t h e n e w r e a c t i o n ( 2 6) h a s a s u ff ic i en t ly la r g e c r o s s s e c t io n . A r o u g he s t i m a t e o f t h a t c r o s s s e c ti o n c a n b e o b t a i n e d b y v i e w i n g su c h a p r o c e s s a s e la s ti cv~qb° s c a t t e r i n g f o l l o w e d b y q b ° b r e m s s t r a h l u n g . W e h a v e t h e n :

2 2, . ~ o c .~o.~o~ ge ~g~o ' t v , ' P v ~ 'v ' v ) ~ . (28 )

S i n c e th e f o l l o w i n g b o u n d h o l d s [ 3 ]:

2 2 2g ~ + g ~ , + g e t < 4 . 5 . 1 0 - 5 (29)

t h e r e s u l t s b y K o l b e t a l . [ 5 ] a b o u t ahigh-entropy col lapsea s o p p o s e d t o t h e B B A Ls c e n a r i o c a n b e t r u e , p r o v i d e d :

g, , ~> 3 10 .3 . (30)

C l e a r l y, t h e i n t e r c o n v e r s i o n o f n e u t r i n o s p e c i e s p l a y s , a c ru c i a l r o l e , s i n c e f o r t h eo r i g in a l l y p r e s e n t v e w e h a v e t h e b o u n d ge~~< 10 - 3 w h i c h i s i n d i s a g r e e m e n t w i t heq . (30) .

To s u m u p , w e h a v e s e e n s e v e r a l n e w a s p e c ts o f s u p e r n o v a p h y s ic s in t h ef r a m e w o r k o f t h e M a j o r o n m o d e l . I t w o u l d r e q u ir e h o w e v e r a m u c h m o r e c o n c e r t e d

a n d c o o r d i n a t e d e f f o r t o f p a r ti c l e a n d a s t r o p h y s i c is t s t o c a r e f u l ly a s s es s th es i g n if i ca n c e o f t h e n e w f e a t u r e s b r o u g h t u p i n o u r d i sc u s s io n .


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168 G.B . G elmini et a l . / B ound s and prospects for Majoron model

5 . L o n g - r a n g e f o r c e s

T h e e x c h a n g e o f th e M a j o r o n a n d t h e l i g ht H i g gs b e t w e e n n o n r e l a ti v is t ic p a rt ic l e sg e n e ra te s t h e s p i n -d e p e n d e n t ( o l . o ' 2 - 3 t r l , f i r 2 .f / r 3 a n d Yu k a w a p o t e n t i a l s

e -mpLr / r. H o w e v e r , d u e t o t h e s m a l ln e s s o f t h e c o u p l i n g s t o c h a r g e d f e r m i o n s ( e q .( 6 ) ) , t h e e f f e c t s o f s u c h p o t e n t i a l s a r e p r a c t i c a l l y u n o b s e r v a b l e .

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 t o n e u t r i n o s , i n p a r t i c u l a r g ~ , c o u l d b e f a i r l yl a rg e ( g ~ ~ 1 is n o t e x c l u d e d ) . N o n e t h e l e s s , i t t u r n s o u t th a t a l so f o r d e n s e n e u t r i n os y s t e m s , p o t e n t i a l s o f t h e a b o v e f o r m p l a y n o r o l e .

S u f f i ci e n tl y l a rg e n e u t r i n o ( a n d u~ i n p a r t ic u l a r ) d e n s i t i e s o c c u r o n l y a t h i g ht e m p e r a t u r e s T ~> M e V > v. A t s u c h h i g h te m p e r a t u r e s t h e s p o n t a n e o u s l y b r o k e ns y m m e t r y i s r e s t o r e d . I t is m e a n i n g l e s s t o t a lk a b o u t t h e m a s s l e s s M o r t h e l i g h tH i g g s P L i n th i s c o n t e x t w h e r e m a s s l e s s n e ss a s a s o u r c e f o r l o n g r a n g e f o r c e s is al l

i m p o r t a n t . I n f a c t , t h e H i g g s b o s o n s d o h a v e u n d e r s u c h c i r c u m s t a n c e s e f f e c t i v em a s s e s o f o r d e r T. T h e n e u t r i n o - n e u t r i n o i n t e ra c t i o n s h a v e t h e r e f o r e a r a n g e o fo r d e r 1 / T a n d o n l y th e n o n c o h e r e n t l a rg e a n g l e n e u t r i n o - n e u t r i n o s c a t t e ri n g s( d is c u s se d i n d e t a i l in s e c t. 3 ) n e e d t o b e c o n s i d e r e d .

6. Expl ic i tB L b r e a k in g m a s s i v e m a j o r o n s a n d th e m i s si n g m a s s p r o b l e m

T h e e s s e n ti a l i d e a o f t h e p r e s e n t m o d e l is th a tB - L i s b r o k e n s p o n t a n e o u s l y. I tw o u l d a p p e a r r a t h e r a r t i f i c i a l t o b r e a kB - L s p o n t a n e o u s l y, a n d i n a d d i t i o n t o a

s m a l l e r e x t e n t a l s o e x p l ic i tl y. H o w e v e r, i n t h e f o l l o w i n g w e w i l l c o n s i d e r e x a c t l yth is case .

T h e c o s m o l o g i c a l i m p l i c a t i o n s o f a t in y m a s s f o r t h e M a j o r o n ( ra m 2raM.


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G.B. Gelm ini et al. / B ounds and prospects for Ma joron model 169

r e s p e c t i v e l y. T h e e q u a l l y l ik e l y o t h e r a l t e r n a t i v e t h a t

m v o > m M - ~1 0 - 2 0 e V ( 3 1b )

i s s ti ll c o n s i s t e n t w i t h t h e d i r e c t e x p e r i m e n t a l u p p e r b o u n d s o n m vo. I n t h i s c a s e

a ll n e u t r i n o s p e c i e s a n d t h e P L w i ll c o n v e r t i n t o M a j o r o n s * .T h e f i r s t c a s e i s a l m o s t i d e n t i c a l t o t h e s t a n d a r d s c e n a r i o i n w h i c h n o M a j o r o n

e f f e c t s e x i s t a t a l l -w i t h , h o w e v e r , o n e r a t h e r i m p o r t a n t d i f f e r e n ce .T h e n e u t r i n o s( a n d PL a n d M ) s ti l l d e c o u p l e f r o m p h o t o n s a n d e l e c t r o n s a t t h e s a m e t e m p e r a t u r eTo a s i n t h e s t a n d a r d m o d e l , To ~ 1 M e V [ 2 3, 2 9 ]. A t t h i s d e c o u p l i n g p o i n t t h et e m p e r a t u r e o f t h e " n e u t r i n o " p l a s m a is, b y d e fi n it io n , e q u a l t o t h a t o f t h e p h o t o n s ,a n d w e h a v e t h e s a m e n u m b e r o f p a r ti c le s p e r d e g r e e o f f r e e d o m in b o t h p l a s m a s .

I n t h e u s u a l s c e n a r i o , t h e p h o t o n s h e a t u p s li g h tl y ( f r o m 1 .9 K t o 2 . 7 K ) a te l e c t r o n - p o s i t r o n r e c o m b i n a t i o n [ 23 , 2 9 ], s o t h a t t h e n u m b e r d e n s i t y ~ 1 0 0 c m -3

o f e a c h n e u t r i n o s p e c ie s a t p r e s e n t is s m a l l e r th a n t h a t o f t h e p h o t o n s ~ 4 0 0cm -3.I n th e p r e s e n t s c e n a r io , h o w e v e r , a si m i la r re h e a t i n g a n d i n c r e a s e o f n u m b e r d e n s it yo c c u r s a l s o f o r t h e v ~, s i n ce a l l n e u t r i n o s p e c i es , t h e M a j o r o n M a n d P L a n n i h i l a t ei n t o r e . T h u s , e v e n i f m ~e is o n l y 1 0 - 2 0 e V , i t c o u l d h a v e i n t h e p r e s e n t s c e n a r i oa n e f f e c t e q u i v a l e n t t o t h a t o f o n e s p e c i e s o f m a s s r o u g h l yf o u r t i m e s h e a v i e r ,i .e . ,4 0 - 8 0 e V. T h i s w o u l d c e r t a i n l y i m p l y a c r u c ia l r o le i n t h e m i s s i n g m a s s p r o b l e m ,a n d d e p e n d i n g o n t h e H u b b l e c o n s t a n t , i t m a y s uf fi ce t o m a k eP ( Ve ) ~ ' P c r i t i c a l ,a l l o w i n g u s to c l o s e t h e u n i v e r s e w i t h n e u t r i n o s .

T h e s e c o n d c a s e is m o r e s u b tl e . A l l d e g r e e s o f f r e e d o m c o n v e r t i n t o M . I f th e

M a j o r o n h a d a l i fe t im e l o n g e r t h a n t h e p r e s e n t a g e o f t h e u n i v e r se

J-M ~>J u ~ 3 1017 s e c . , ( 32 )

t h e y m i g h t s u p p l y t h e s o lu t i on t o th e m i s s in g m a s s p r o b l e m To m a k e s u r e t h a tt h is d o e s i n d e e d h a p p e n , w e h a v e c h e c k e d t h a t t h e a n n i h i l a t io n p r o c e s s M M - -> 3 '3'v i a a t r i a n g l e l o o p o f X ± , X :~± y i e l d s a c r o s s s e c t i o n s m a l l e r t h a n ~ r( ~v ~ M M ) [ 2 ]b y a b o u t 1 02 8 .

T h e o n l y p o s s i b l e d e c a y o f M is M + 3"3". P r e c i s e l y b e c a u s e M i s t h e G o l d s t o n eb o s o n o f t h e n o n - a n o m a l o u s B - L c u r r e n t , t h e t ri a n g u l a r a n o m a l y a n d th e l o w

e n e r g y e x p r e s s i o n f o r th e d e c a y a m p l i t u d e v a n i sh * * . F u r t h e r m o r e , i t t u rn s o u t t h a tt h e c o n t r i b u t i o n t o th is d e c a y a m p l i t u d e f r o m t h e c h a r g e d H i g g s b o s o n l o o p ( s e efig . 2 ) a ls o v a n i s h e s . T h e r e a s o n is t h a t t h e r e l e v a n tM X + X -a n d M X + X - -c o u p l i n g sv i a t h e v a c u u m t a d p o l e a r e a b s e n t i n th e H i g g s p o t en t i al . T h e e x t r e m e s u p p r e s s i o no f M ~ 3"3" is i m p o r t a n t b e c a u s e o f t h e f o l l o w i n g c o n s i d e r a t i o n s .

I f t h e m a s s o f a t yp i ca l g a la x y, M - 1 0 1 X M ® - 1 0 4 4 g , is a c c o u n t ed f o r b y

M a j o r o n s , w i t h m a s s 1 0 e V - - 1 0 -3 2 g , t h e n s u c h a g a l a x y c o n t a i n s- 1 0 7 6 M a j o r o n s .

See previous foo tnote . * Actually, one can explicitly com pute the co ntribution of the low est order graphs with a triangle loop

of charged fermions. It turns out that they are finite and the various contributions cancel exactlyfor each generation of quarks and lepton s. W e thank T. Y anagida and M. Yoshimura for anillum inating discussion on this point.


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170 G B Gelm in i et a l / Bou nds a nd prospec ts fo r Majoro n mode l

I

/.~ X

X - ~ /

--.<Fig. 2 A priori l e a d i n g c o n t r i b u t i o n t o t h e d e c a y M - ~ 7 3', w h i c h w o u l d r u l e o u t t h e p o s s i b il i ty o fa c c o u n t i n g f o r t h e m i s s i n g m a s s b y m a s s i v e M a j o r o n s . I t v a n i s h e s d u e t o t h e s t r u c t u r e o f t h e H i g g s

p o t e n t i a l .

T h e r e f o r e 1 0 7 6 /J 'M M a j o r o n s d e c a y e v e r y s e c o n d . T h e t o ta l r e su l ti n g l u m i n o s i tyi n t h a t e n e r g y r a n g e ( E v 1~ m M = 5 - 1 0 e V ) , p a r t i c u l a r ly f r o m t h e g a l a c t i c h a l o ,w o u l d m u c h e x c e e d o b s e r v e d l i m i t s , u n l e s s t h e s t r i n g e n t b o u n d

J ' M ~ 1 0 1 ° ~ r ' u ~ 1028 sec . (33)

is i m p o s e d . S i m i l a r c o n s t r a i n t s w o u l d r e s u l t i f t h e M a j o r o n s w e r e n o t c l u s t e r e d i nga l ax i e s .

B e f o r e c o n c l u d i n g th is s ec t io n w e w o u l d l i ke t o m a k e o n e m o r e c o m m e n t w h i c his r e l e v a n t if t h e r e i s e i t h e r a n e u t r i n o o r M a j o r o n b a c k g r o u n d . T h e i d e a t h a t as h a r p n e u t r i n o p u l s e e m i t t e d ( p r e s u m a b l y ) in t h e s u p e r n o v a c o l l a p s e w ill b ed e t e c t a b l e o n t h e e a r t h h a s b e e n e n t e r t a i n e d b y m a n y p h y s i c is t s [ 3 0] . In p a r ti c u l a r ,t h e s e p a r a t i o n o f th e ~ 'e , ~'~ w a v e p a c k e t s [ 3 1 ] c o u l d g e n e r a t e t w o p u l s e s , a n d t h et i m e d e l a y b e t w e e n t h e s e w i ll b e a d i re c t m e a s u r e o f A m 2.

H o w e v e r , i n t h e p r e s e n t m o d e l t h e n e u t r i n o s i n t h e p u l s e w il l i n t e r a c t q u i t es t r o n g l y w i th t h e a m b i e n t n e u t r i n o s o r M a j o r o n s i n t h e g a l a x y ( o r i n t h e i n t e r g a l a c ti cs p a c e i f d e t e c t i o n o f a s u p e r n o v a i n a n o t h e r g a l a x y is e n v i s i o n e d . ) T h e r e l e v a n tc .m. energy for the b 'pulsePbackg. . . d ( o r / ~ p u ls e M b a c k g . . . d )co l l i s ion i s to 2M e V 2 ~E~punse" m~o(or E~p . . . m M ) ~ 1 0 - 4 M e V 2.T h e r e f o r e , n e u t r i n o s i n th e p u l se m a y

2 2 10 - -1 9 4 4~- -19s c a t t e r q u i t e s t r o n g l y o ff t h e b a c k g r o u n d : c r - g ~ gee c m 2, or ~r ~ g ~, tu cm 2 ,in th e c a s e o f M b a c k g r o u n d . I f n e u t r i n o s ( o r M a j o r o n s ) a r e c l u s t e r e d i n t h e g a l a x i esa n d a c c o u n t f o r t h e m i s s i n g m a s s , t h e n t h e i r d e n s i t y t h e r e is = 1 0 8 c m - 3 [ 3 2] . T h em . f . p , f o r t h e v p u l s e i n t h e g a l a x y is t h e n1/n~o -~1011 cm/gT~g~e2(o r 1011cm/g~,4i n t h e s e c o n d c a se ) . T h e r a d i u s o f th e g a l a x y is ~ 2 0 k p c ~ 10 23 c m . H e n c e w ee x p e c t complete dispersal of the pulse.T h e u n i q u e c h a n c e o f m e a s u r i n g A m 2 in am a n n e r w h i c h is c o m p l e t e l y i n d e p e n d e n t o f n e u t r i n o m i x i ng m a y t h u s b e c o m p l e t e l y

2 2 4g , ~ ) e x c e e d sp o i l e d o n c eg~g~T ( o r 1 0 - 1 2D e t e c t i n g a n e u t r i n o p u l s e f r o m a s u p e r n o v a w o u l d b e a sc i en t if ic e v e n t o f m a j o r

c o n s e q u e n c e . I t w o u l d r u l e o u t t h e p o s s ib i l it y o f M a j o r o n s w i t h m a s s e s in th e2 2 4c o s m o l o g i c a l l y r e l e v a n t r a n g e f o rg~eg, , o r g , , g r e a t e r t h a n 1 0 -1 2


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G B Gelm in i e t a l / Bou nds a nd prospec ts fo r Ma joron mod e l 171

7 Galaxy formation in the M ajoron m odel

A l o n g s ta n d i n g p r o b l e m in a s t ro p h y s i c s is t h e u n d e r s t a n d i n g o f t h e s o u r c e ,n a t u r e a n d s u b s e q u e n t g r o w t h o f th e f l u c tu a t io n s w h i c h c o n s p i r e d t o tr a n s f o r m a n

i n it ia l ly fa i r ly s m o o t h d e n s i t y i n t o t h e o b s e r v e d s t a r s, g a l a x i e s, c l u s t e rs a n d e v e nl a r g e r s ca le i n h o m o g e n e i t i e s o b s e r v e d r e c e n t l y [ 3 3] .

O n e o r m o r e s t a b le n e u t r i n o s p e c ie s w i t h m a s s e s in t h e 1 0 - 1 0 0 e V r a n g e c o u l da c c o u n t f o r t h e m i ss in g m a s s p u z z l e . H o w e v e r , i n s o fa r a s t h e a b o v e p r o b l e m isc o n c e r n e d , th e " s t r e a m i n g " o u t o f t h e l ig h t n o n i n t e r a c t i n g n e u t r i n o s t e n d s t os m o o t h i n i ti a l f l u c t u a t io n s , m a k i n g c l u s t e r i n g o n a n y s c a le s m a l l e r t h a n M - 1 01 5M ®e x t r e m e l y u n l i k e l y [ 3 4 ]. I n t h i s s e c t i o n w e w i ll b r i e f l y d i sc u s s t h e p o s s i b l e i m p a c to f t h e p r e s e n t m o d e l o n t h i s i ss u e .

(i) V a r i o u s i n g e n i o u s s t a n d a r d w e a k i n t e r a c t io n s c e n a r i o s f o r g a la x y f o r m a t i o n

in a n e u t r i n o d o m i n a t e d u n i v e r s e h a v e b e e n r e c e n t l y s u g g e s t e d [ 35 ] .O ne in t r i gu ing poss ib i l i t y u t i l iz e s a h i e r a r ch i c a l mv~ >>my, >>rn~o= 30 eV mass

p a t t e r n . T h e h e a v i e r s p e c i e s ( s a y v~ ) f o r m s f ir st a p r e - g a l a x y a t r o u g h l y t h e d e s i r e dsca l e , M M j . . . . . T h e l i g h t e r n e u t r i n o s p e c i es a r e t h e n t r a p p e d a l o n g w i t h b a r y o n si n t o t h e p o t e n t i a l w e l l g e n e r a t e d b y t h e p r e g a l a x y. T h e h e a v y s p e c ie s s h o u l dd i s a p p e a r s h o r t l y t h e r e a f t e r , s o a s t o e n s u r e t h a t n e i t h e r ~ ,~, n o r i ts d e c a y p r o d u c t so v e r d o m i n a t e t h e e n e r g y d e n s i t y o f t h e u n i v e r s e . T h i s fi n al ly l e a v e s b e h i n d t h ep r o p e r - s i z e p r e s e n t - d a y g a la x ie s .

E v i d e n t l y s u c h a s c e n a r i o w o u l d b e d r a s t i c a ll y m o d i f i e d , a n d m o s t l i k e l y w i ll

h a v e t o b e a b a n d o n e d , i f t h e p r e s e n t t r i p le t M a j o r o n m o d e l i s c o r r e c t . J u s t r ec a llt h a t o n c e T ~ m y, t h e h e a v y n e u t r i n o s i m m e d i a t e l y i n t e r c o n v e r t i n to t h e l i g h t e rs p e c ie s a n d n o t i m e i s l e f t f o r p r e g a l a x y f o r m a t i o n .

(ii) It h a s b e e n s u g g e s t e d [ 2 9 ] t h a t t h e p h a s e t r a n s i t io n s i n v o l v e d i n th e q u a r kc o n f i n e m e n t i n to h a d r o n s n a t u r a l ly l e a d t o l a rg e f l u c tu a t i o n s w h i c h a r e t h e s o u r c eo f t h e p r e s e n t l y o b s e r v e d l a rg e s c a l e i n h o m o g e n e i t i e s ( g a l a x i e s , e t c . ) .

T h e p h a s e t r a n s it i o n o f t h e p r e s e n t m o d e l i s u n i q u e , b e i n g t h e l a t es t* . I t o c c u r sa t T = T ~ = v ~< 1 0 0 k e V, a t e m p e r a t u r e w h i c h i s 3 - 4 o r d e r s o f m a g n i t u d e l o w e rt h a n T quark ( th a t c o r r e s p o n d i n g t o q u a r k c o n f i n e m e n t ) a n d 6 - 7 o r d e r s l o w e r t h a n

t h e T w s = u = 3 0 0 G e V . I n p a r t ic u l a r , i t h a p p e n s r e l a t iv e l y n e a r t h e r e c o m b i n a t i o nt e m p e r a t u r e T F ~ 1 e V, a t w h i c h g a la x ie s a r e c o n v e n t i o n a l l y b e l i e v e d t o f o r m . I tis t h e r e f o r e c o n c e i v a b l e t h a t in t h e i n te r i m p e r i o d b e t w e e n T = TcM ~ 1 0 - 1 0 0 k e Va n d T F - t h e t e m p e r a t u r e a t w h i c h g a la x ie s a re s u p p o s e d t o f o r m - t h e s e fl u c tu a t i o n sw i ll n o t d i s s ip a t e b u t r a t h e r w i ll c o n t i n u e t o g r o w.

T h i s in p a r t i c u la r w o u l d b e t h e c a s e if o v e r m o s t o f th e i n t e r i m p e r i o d b e t w e e nt M ( d e f i n e d a s t h e t i m e w h e n t h e t e m p e r a t u r e e q u a l s T M ) a n d t F ( w h i c h is th e t i m ew h e n t h e t e m p e r a t u r e e q u a l s T ~) t h e u n i v e r s e i s m a t t e r d o m i n a t e d . F i rs t, t h eh e a v i e s t n e u t r i n o d o m i n a t e s t h e e n e r g y d e n s it y a n d t h e n ( d u e t o i n t e r c o n v e r s i o n

a n d a n n i h i l a t i o n ) t h e l i g h t e r o n e s t a k e o v e r.* On e o f us (S.N.) wo uld like to than k E . W itten for emph asizing this point.


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172 G B Gelm in i e t a l / Bou nds a nd prospec ts fo r Ma joron mo de l

T h e o r i g i n a l l a rg e f l u c t u a t i o n s i n t h e n e u t r a l c o m p o n e n t o f t h e i s o t r i p l e t s c a l a rf i e l d 4 ° c o u l d r e f l e c t e v e n t u a l l y a l s o a s f l u c t u a t i o n s i n n e u t r i n o n u m b e r d e n s i t y,e n e rg y d e n s i t y a n d - v i a g r a v i t a t i o n a l c a p t u r e - a l s o in b a r y o n d e n s i t y.

W e w o u l d l ik e t o t h a n k A . D e R u j u l a , D . Z . F r e e d m a n , S . L . G l a h s o w, M . L i v io ,M . M i l g r o n a n d M . Yo s h i m u r a f o r h e l p f u l d i s cu s s io n s . E s p e c i a l ly, w e w i s h t oe x p r e s s o u r s i n c e re g r a ti t u d e t o E . W. K o l b a n d D . L . Tu b b s f o r t h e i r c o m m e n t s o na n e a r l y v e r s i o n o f t h is p a p e r. F i n a ll y, o n e o f u s ( M . R . ) w o u l d l i k e t o t h a n k D r.K . R i e d l f o r th e i n v i t a ti o n f r o m t h e M a x i m i l i a n e u m S t if tu n g .

e t e r e n c e s

[1] G.B . Gelm ini and M. Ro ncade l l i , Phys. Let t . 99B (1981) 411[2] H. G eorgi , S.L. Glasho w and S. Nussinov, Nucl . Phys. B 193 (1981) 2 97[3] V. Barger, W.Y. Keung and S. Pakvasa, Phys. Rev. D25 (1982) 907[4] J . Schechter and J .W.F. Valle , Phys. Rev. D25 (1982) 774[5] E.W. Kolb, D.L. Tubbs and D.A. Dicus, Astrophys. J . Let t . 225 (1982) L57[6] G.M . Fuller, E.W . Kolb and D.L. Tubbs, (1981), in prepa rat ion[7] A.M . Coo per e t a l ., Ruth er tord Labora tory prepr in t , RL-81 -055 (1981)[8] V. Barger, H . Baer, W.Y. Keu ng and R.J .N . Phill ips , Un iversi ty of Wisconsin preprint ,

M A D /P H /3 7 (1982), Phys. Rev. D , t o appea r[9] T. Goldman, E .W . Kolb and G.L. S tephenson , J r. , Los Alam os Labora tory prepr in t , LA -U R -82 -

1401 (1982), Phys. Rev. D, to appear[10] F. Buccel la , G.B. Gelmini , A. Masiero and M. Roncadel l i , Max-Planck-Inst i tut preprint , MPI-

PAE/PTh 21 /82 (1982 )[11] S .L . Glashow, ta lk g iven a t the Second W orkshop on G rand Uni f ica tion , (Ann Arbo r, M ichigan ,

Apri l 1981); lectures given at the "Cargese Summer Inst i tute" (July 1981)[12] H. Geo rg i , lec tures g iven a t the " Fou r th K yoto Sum mer Sch ool" ( Ju ly 1981)[13] M. Fukugita , S. W atamu ra and M . Yoshimura, Phys. Rev. Let t . 48 (1982) 1522[14] M. Fukugi ta , S . W atamura and M. Yoshimura , KEK prepr in t , KE K-T H-4 1 (1982)[15] D.M . Schram m and G. Steigman, Gen. Rel . Grav. 13 (1981)1 01;

E. W i t ten , t a lk g iven at the F i r st Workshop on G rand Uni f icat ion , (New Hampshi re , A pr i l '1980) ;H. Sa to , Kyoto prepr in t , RIFP -423 (1981);A. Do lgo v and Ya.B. Ze ldovich , Rev . Mo d. Phys. 53 (1981) 1;G. Steigm an, ta lk given at Neutr ino '81 (Maui, Haw aii , July 1981)

[16] V.A . Lyubim ov et al. , Phys. Let t . 94B (1980) 266 ;A. De Rujula, Nucl . Phys. B188 (1981) 414

[17] M. Do i et a l., Osaka preprin t s , OS- GE 80/2 7 (1980) , O S-G E 81/2 8 , 29 (1981) ;W.C . Haxton, G .J . Stephenson, Jr. and D. Strot tema n, Phys. Re v. Let t . 47 (1981) 153; Phys. Re v.D25 (1982) 2360;P. Minkowski , Bern Univers i ty prepr in t, B UT P-1 5/8 1 (1981);M. Doi and T. Kotani , ta lk given at Neutr ino '81 (Maui , Hawaii , July 1981);S.P. Ro sen, ta lk given at Neutr ino '81 (M aui , Hawaii , July 1981)

[18] R. Cow sik and J . MeClel land, Phys. Rev . Let t . 29 (1972) 66 9; Astroph ys. J . 180 (1973) 6;B. Lee and S. W einberg , Phys. R ev. Let t . 39 (1977) 165

[19] R. Barbier i and R.N. Mo hapa tra, Z. Phys. C l l (1981) 175[20] R.N . Moha patra and G. Senjanovic, Phys. Rev . Let t . 49 (1982) 7[21] Re view of part icle prope rt ies , Re v. Mod. Phys. 52 (1980)[22] D.Z . Freedm an, Phys. Rev. D9 (1974) 1 389[23] S. W einberg, G ravitat ion and C osm olog y (J. W iley, New Y ork, 1972)


17/17

G B Gelm ini et a l / Bou nds a nd prospects for Majoron mod el 173

[24] D.Z . Freedm an, D.N . Schram m and D.L. Tubb s, Ann. R ev. Nucl . Sci. 27 (1977) 167[25] J .R. W ilson, (1976), pr ivate com mu nicat ion to the authors of ref . [24][26] H.A . Bethe , G .E. Brown , J. Applega te and J .M. La t t imer, Nucl. Phys. A3 24 (1979) 487[27] W. Fel ler, An Introduct ion to Probabil i ty Theory and i ts Applicat ions, vols . 1 and 2 (J . Wiley,

New York, 1966)[28] S. Weinberg, Phys. Rev. D9 (1974) 3537;

L. Dolan and R. Jackiw, Phys. Rev. D9 (1974) 2904;A.D. Linde, Rep. Progr. Phys. 42 (1979) 389

[29] A. Do lgo v and Ya.B. Zeld ovich , Rev . Mo d. Phys. 53 (1981) 1[30] D. Fargion, Nuov. Cim. Let t . 31 (1981) 499 and references therein.[31] S. Nussinov, Phys. Lett. 63B (1976) 201[32] S. Tremaine and J .E. Gunn, Phys. Rev, Let t . 42 (1979) 407[33] P.J .E. Pe ebles, The Larg e Scale Structure of the Unive rse, (Princeton Un iversi ty Press, Princeton,

N.J., 1980)[34] J .R. B ond, G . Efstathiou and J .Si lk, Phys. R ev. Let t . 45 (1980) 1980[35] M. Davis , M. Lecar, C . Pryor and E . Wi t ten , Smi thsonian Ast rophys ica l Observa tory, Harvard

Co llege Ob serva tory preprint , 1482 (1981)

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