Radiative neutrino decay in superstring models

Volume 180, number 4

PHYSICS LETTERS B

20 November 1986

R A D I A T I V E N E U T R I N O DECAY I N S U P E R S T R I N G M O D E L S K. E N Q V I S T l Research Institute for Theoretical Physics, University of HelsinM, SF-O0170 Helsinki 17, Finland

and J. M A A L A M P I 2 CERN, CH-1211 Geneva 23, Switzerland

Received 29 August 1986

The rate for the radiative decay of neutrinos is calculated in E 6 superstring models with light colour triplet isosinglets h and h. In contrast to the standard model, the rate is not suppressed either by the GIM mechanism or by helicity matching. Comparison with astrophysical data allows one to set an upper limit on the Yukawa couplings ~QLfi and 2d%Ch. Taking Mh.~= O (100 GeV) it is found that 7~.~,)~e.,< 1 0 - 3 10-4, compatible with mvo,my, in the range 10-100 eV. From the cosmological requirement of not photo-ionizing light elements after nucleosynthesis, it follows that the v. must lie in the same mass range and consequently 2,, ~T
The heterotic superstring theory [ 1,2 ] yields, after (2,2) compactification o f the six surplus d i m e n s i o n s to a C a l a b i - Y a u m a n i f o l d with S U ( 3 ) holonomy, an E 6 gauge theory coupled to N = 1 supergravity [ 3,4]. Breaking o f E6 by the Wilson-loop m e c h a n i s m [3,5 ] necessarily implies a low-energy gauge s y m m e t r y larger than the standard SU (3)c X SU (2) L X U (1) r- F o r example, in the no-scale m o d e l [ 6 ], which arises as the effective d = 4 supergravity theory o f superstrings after simple truncation o f the ten-dimensional theory [7 ], the low-energy s y m m e t r y is G=SU(3)c X SU(2)L XU(1 ) XU(I )'.

(1)

In this model [ 6 ] there are no i n t e r m e d i a t e mass scales between the Planck scale Mp-~ 1019 G e V a n d the d y n a m ically-induced s c a l e / 1~- 102-103 GeV, where first the extra U(1 )' invariance a n d then the s t a n d a r d electroweak s y m m e t r y S U ( 2 ) L X U ( I ) r are broken. This structure o f the s y m m e t r y breaking implies that the whole m a t t e r multiplets, having q u a n t u m n u m b e r s o f a f u n d a m e n t a l 27 representation o f E6, survive with all their i m p o r t a n t phenomenological consequences to low energies. One phenomenological aspect o f the superstring theories, quite essential for m o d e l building, is the question o f neutrino mass. In E 6 superstring models fermions can acquire mass, at the tree level, only from the v a c u u m expectation values o f 27 multiplets. As a consequence, the s t a n d a r d see-saw m e c h a n i s m [8 ] where the righth a n d e d neutrino obtains a very large M a j o r a n a mass is not possible in these models. Several scenarios have recently been suggested to resolve this problem. They include, e.g., models where a small mass is i n d u c e d to the observed neutrinos by an i n t e r m e d i a t e mass scale through higher-order terms in the lagrangian [ 9 ], or through a conspiring mixing pattern among neutral fermions [ 10] (there are altogether five neutral states in a 27; see the classification below), or where the j o b is done by E6-singlet zero mass fields [ 11 ] which a p p e a r in (2,0) Address after 1 September: Physics Department, University of Wisconsin, Madison WI 53706, USA. 2 Address after 1 September: Research Insitute for Theoretical Physics, University of Helsinki, SF-00170 Helsinki 17, Finland. 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )

347

Volume 180, number 4

PHYSICS LETTERS B

20 November 1986

(N) i

,,

iI

i

VL

Fig. 1. Characteristic diagram generating a Dirac mass for the neutrino in the E6 superstring model. The h and h are colour triplet isosinglet scalars with their mass in the range 10L10 3 GeV. The supersymm_etriccounterpart would be the diagram where a c squarks and h, h quarks propagate in the ioop.

"~X

d~

i cl~ X ,

k (H)

vR

compactifications [ 12 ]. These scenarios, however, do not seem phenomenologically viable [ 13 ]. It is also possible that some mirror particles, i.e., the particles living in 27", remain massless down to the scale/z; in that case, one can try [ 14] to solve the neutrino mass problem due to mixing of neutrinos and mirror neutrinos with no need of an intermediate scale. However, there is the difficulty that in superstring models there are no renormalizable couplings between 27 and 27*. In the no-scale superstring models [ 6 ], non o f the above mechanisms are conceivable. These models do not develop intermediate scales [ 15 ], and absence of lepton-number-violating processes seems to require massless neutrinos at the tree level. This situation can follow in superstring theories as a result of certain discrete symmetries originating, e.g., in the topological properties o f the internal manifold. Although the well-known nucleosynthesis limit on light neutrinos [ 16 ] may need some revision [ 17 ], there remains the uncomfortable possibility of having too many light neutrinos. If, however, the colour triplet isosinglet fields, leptoquarks, belonging to the representation 10 in the SO(10) decomposition 27 = 16 + 10 + 1 remain light, as they will in the no-scale models, a Dirac mass which is tiny but still lying in an interesting range can be naturally generated through loop corrections [18 ], as shown in fig. 1. In the present paper we will study in detail a question closely related to the neutrino mass problem, namely radiative stability of neutrinos in the E 6 superstring model. We will consider loop diagrams of a similar type to those generating neutrino masses which will allow, as is pointed out also in ref. [ 19 ], for a radiative decay v ~ v ' y of a neutrino many orders of magnitude more rapid than that in the standard model (see fig. 2). The reason for the rapidity of this decay in the superstring model is the fact that diagrams involving leptoquarks, in contrast to the ones mediated by the standard electroweak gauge interactions, are not helicity-suppressed, which yields an enhancement of mJmv in the amplitude. We will confront the ensuing radiative lifetime z ( v ~ v ' y ) with the existing astrophysical and cosmological contraints to derive bounds for the leptoquark Yukawa couplings. Let us start by exhibiting the most general cubic 273 superpotential which describes the interactions among the various components of the supermultiplet 27: P = L H e c + QIZldc + Q H u c + LHvC + HIZlN + h f i N + Q Q h + u ~ d ¢ f i + Q L f i +dCvCh+uCeCh.

/z

\

/

vj kj

%~

d~ ~? (a)

348

(2)

iI

vi

vj "~j

\

d~ (b)

~.i Vi

Fig. 2. Diagrams for radiative neutrino decay vj~vjy. Comparing decay rate to astrophysical data allows us to derive upper limits for the Yukawa couplings 2 and ~. The indices i, j, r, s indicate flavour.

Volume 180,number 4

PHYSICSLETTERSB

20 November 1986

The fields appearing here have their transformation properties under the group G = SU (3) × SU (2) × U (1) r ×U(1 )' as follows:

Q=

du )

L= (v)

=(3,2)1/6.1/3,

= (1,2)-

uC=(3*, 1) 2/31/3 . , d c = ( 3*, 1 ) 1/3,- 1/6,

1/2.- 1/6 ,

e~=(1,1)1,1/3,

vC(1,1)o.5/6,

h = (3,1 )_ 1/3.2/3 , fi= (3",1)1/3._ 1/6, H = ( H°_ ) = (1,2)- 1/2,1/6,

17"I=

("+) ISiO

= ( 1 , 2 ) 1/2,-2/3 ,

N=

(3)

(1,1)o,5/6 •

All the couplings given in (2) cannot obviously be nonvanishing; a simultaneous appearance of the last five terms would contradict the observed stability of the proton since the colour triplet fields h and h are assumed to be light. It is a specific virtue of the superstring models (not shared by the conventional GUTs) that one may assume the diquark couplings of h and h, QQh + uCdCh, to vanish while allowing their leptoquark couplings QLh + d~vCh+ u~eCh and the rest of the superpotential (3) to survive [ 15]. The part of the superpotential (2) relevant for the neutrino mass and radiative decay is, in a more complete notation, given by -k Ot f l - k 2o~aaQ i L~h

k c c k +2ijdiv)h +h.c.,

(4)

where i,j, k= 1..... n~ are flavour indices and or, fl= 1,2 are SU(2) indices. The Yukawa couplings X and 2 are determined by the properties of the internal manifold and, in the absence of detailed knowledge of that manifold, must be taken as free parameters. There are, of course, many phenomenological constraints on these couplings, in particular on 2s, since the light leptoquark exchanges will contribute to different low-energy processes. These have been studied in refs. [ 18-20]. Assuming Mh.~ --~100 GeV, one obtains generically 2 < 10-2-10-3. In general, the diagonal couplings are less severely constrained than the nondiagonal ones. For the decay v ~ v ' T , nonvanishing nondiagonal Yukawa couplings are not necessary, as long as there is flavour mixing in the down quark and leptoquark h, fi systems. In the following, we will for simplicity assume that the couplings really are diagonal and denote ¢~)~ = ¢~)i. Let us call U, Vand Vthe matrices which give the down quarks, leptoquarks h and leptoquarks fi, respectively, in terms of the corresponding mass eigenstates. We can then derive the following expressions for the amplitudes of the two diagrams given in figs. 2a and 2b: d~(vj-,v?/) =-i(~e)E~,~

~j~iVjrVirUsjU~ts

r,s

xJ(--~)4 --

aj(p-q)(l-~,5) t-O-m----~

16-~~-(~e) ~ Ms

(mr) ~

~

(l-~,,)u,(p) ( k _ p ~ _ M ~"

2J2~V'U'jU~'~aJ(P-q)aU~uq"(1-ys)u~(P) ,

(5)

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Volume 180, number 4

PHYSICS LETTERS B

20 November 1986

r,$

~ d4k ( i × j ( 2 ~ ) 4 a j ( p - q ) ( 1 -~'5) p + k - m ,

i

=

m, ( r n , ) _ _

- 16ha ( - le) 2 ~22Fb ~ r,s

s

\

2j2,

i i(2k+q) u ) (1 -ys)U,(P)k2 _ M 2 ( k + q ) 2 _ M 2

VjrVirUsjU~sl~j(p-q)truv~.uqv(1-ys)ui(P)

,

(5 cont'd)

s /

where e and q are the polarization vector and the momentum of the photon and p is the m o m e n t u m of the decaying neutrino vj. The functions Fa and Fb are for small values of the argument ~ = m,/Ms given by Fa(~)=~+~+(l+3~)ln~,

Fb(~)=l+3~+2~ln~.

(6)

A striking fact is that the amplitudes are proportional to the down-quark masses, rather than to the neutrino masses, which would be the case in the standard model [21 ]. This yields a huge enhancement f a c t o r " and the question arises whether the result can still be made compatible, within any sensible range of parameters, with the astrophysical constraints on the radiative neutrino decay. There are many unknown parameters on which the rate F(v:-,v~7 ) depends, i.e., the masses M of the leptoquarks, the leptoquark Yukawa couplings 2, 2 and the flavour-mixing parameters of the leptoquark system. Parameters on which we do have experimental information are the down-quark masses and mixings, the latter being given in the Kobayashi-Maskawa matrix [the matrix U in eq. (5)]. The leptoquark masses originate from soft supersymmetry breaking hfiN couplings [ 15]; they cannot thus exceed the scale M - 102-103 GeV, otherwise the original motivation for supersymmetry would be lost. For the flavour mixing in the h, fi sector, we may, in the absence of any better knowledge, let the phenomenologically observed mixing among the ordinary down quarks guide us, and assume that this is of the same order of magnitude. So that we may obtain an order-of-magnitude estimate, we now consider the decay v,--.v~7 with the simplifying assumptions that only two down quarks, d and s, and two leptoquarks, h~ and h2, are relevant. We further assume that the h~, ha mixing is parametrized by an angle 0n (which we will take equal to the familiar Cabibbo angle 0c). Taking only the leading terms in the ratios rna.~/ML2 into account, the amplitude is given by i (m~ - - m s ) ( M ~ l ~¢a+b(Vo--'VeT) = 12--~-~n2(~e)]~2 e sin 20c sin 20h M~IM~2

av~(P-q)aU~ ~uqv(1 - Y s ) u , , ( p ) •

(7)

Note that the GIM cancellation does not occur with full strength here, since the amplitude is proportional to the masses of the loop particles. The amplitude (7) corresponds to the lifetime z, given by Z-l(V,~VeT) =

9~Z421J sin220c sin220h

(md-ms)2(M~l-M~2)E 3 (MIM2)4

(8)

where 2 - m~e )/2mv, Eo = ( m v,

(9)

is the photon energy. For a numerical estimate we take sin Oh= sin 0c = 0.23, m s - m d ~--0.2 GeV, M~ + M2 = 102 GeV and IMl - M 2 I - 10 GeV, with which the lifetime becomes z -~ = (0.6× 10 ~2 s)-~(2e2~)2(Eo/25 eV) 3 .

(10)

Let us recall that the radiative lifetime in the standard model is z > 1035 s (for Eo=25 eV) [21 ]. There are several observational lower limits for the radiative lifetime of neutrinos from astrophysics [23]. t~ A similar enhancementfactor may arise in the standard model extended to include right-handed interactions; see ref. [22]. 350

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Assuming the neutrino a b u n d a n c e derived from the s t a n d a r d Big Bang cosmology, the most stringent limit for the energy range Eo = 5 - 13 eV (i.e., my, - 10-2 7 eV, mvo = 0), z > 1023 s, is o b t a i n e d from the m e a s u r e m e n t o f the far ultraviolet spectrum o f the b a c k g r o u n d radiation [24]. Above the L y m a n limit E o = 13.6 eV an even more sever limit, r ~> 1024 s, is i m p l i e d by the fact that there exist neutral hydrogen clouds in galactic haloes [25]; for shorter decay times these would have been ionized by the p h o t o n flux ~2. Let us confront these observational limits with our theoretical result (10). Allowing Eo to get values in the range Eo-- 5-50 eV, for example, leads to the following u p p e r limit for the Yukawa couplings: 2e2, < 7)< 1 0 - 6 - 7 ) < 10 -8 ,

(11 )

or generically J,, 2~< 1 0 - 3 - 1 0 -4 .

(12)

It must be stressed that the bounds (11 ) and (12) depend quite considerably on the various assumptions adopted. F o r example, allowing masses an order o f magnitude higher for the leptoquarks ( M ~ + M z - ~ I TeV, IM~ - M21 -~ 0.1 TeV, say) would p e r m i t the Yukawa coupling constants 2, 2 to be o f the size o f a typical gauge coupling constant. On the other hand, assuming Eo > 50 eV would have an effect working into the opposite direction. It is also possible that the level o f mass degeneracy among the leptoquarks is different than assumed here ( IM~ - M 2 1 / ( M ~ + M2) ~- 10%), or that the size o f the flavour mixing in the leptoquark sector is considerably larger or smaller than that among the o r d i n a r y down quarks given by the K M matrix. The above analysis is valid for a general class o f E 6 models with light leptoquarks. In the special case where the loop diagrams depicted in fig. 1 are the only source o f neutrino masses, there will be a d y n a m i c a l relationship between these masses and the radiative decay amplitude, a fact which will affect our conclusions. F r o m fig. 1 we obtain generically [ 18,19 ] mv--~ (22/167~2)mq

(13)

(with 2 ~ 10 - 3 this gives mvo "-~1 eV, rn~, ~ 4 eV a n d mv~ -~ 60 eV) and if we assume that all neutrino masses are o f the same order o f magnitude, we can write F ( v - - , v ' y ) = (am~/M4~) s i n 4 0 × O ( 1 0 - 2 )

,

(14)

where O (10 - 2) takes into account numerical factors a n d sin40 the flavour mixing. [ Incidentally, eqs. (13) and (14) are true also in the case where h-quarks propagate in the loop, because the trilinear scalar coupling ]~CH is suppressed by the d-quark mass. ] The width (14) has the same general form as the corresponding width in the standard model; only the W mass is replaced by Mh, ft. Thus if we assume neutrinos in the 10-100 eV mass range (which means 2 ~ 1 0 - 3 ) , the radiative lifetime will, as we know from the s t a n d a r d case [23], pass the astrophysical limits by several orders o f magnitude. One could still consider the possibility that the tau neutrino v~ m a y become much heavier than the Ve and % due to the fact that the Yukawa couplings 2~, ,~ are not as strongly constrained by the laboratory experiments as the couplings for the other two families. Eq. (13) tells us that masses up to mv~ = 60 M e V are possible if2~, 2~ ~
Volume 180, number 4

PHYSICS LETTERS B

20 November 1986

p r e s e n t case. W e c a n t h u s c o n c l u d e t h a t v~ s h o u l d lie i n t h e e l e c t r o n v o l t r e g i o n a n d t h a t c o n s e q u e n t l y t h e b o u n d 2~, ~ < 1 0 - 3 s h o u l d h o l d . T h e a u t h o r s g r a t e f u l l y a c k n o w l e d g e u s e f u l d i s c u s s i o n s w i t h B. C a m p b e l l , J. Ellis a n d D. N a n o p o u l o s . J . M . was f i n a n c i a l l y s u p p o r t e d b y t h e P a r t i c l e P h y s i c s C o m m i t t e e o f t h e A c a d e m y o f F i n l a n d .

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