Left-right symmetry breaking and fermion masses

L:ETTERE AL NUOVO OI~tENTO VOL. 28, 1~. 18 30 Agost0 1980

Left-Right Symmetry Breaking and Fermlon Masses (*).

J . -M. F I ~ R ~ (**) (***)

:Faeult~ des Sciences, Universit~ ~ibre de Bruxeltes : Bruxelles

(r icevuto il 20 Giugno 1980)

The smallness of the u- and d -quark current masses has often led to the suggest ion t h a t t h e y migh t result ent i re ly f rom rad ia t ive correct ions t h rough Cabibbo mix ing wi th the heav ie r quarks. W e examine this possibi l i ty in the f ramework of models based on dynamica l s y m m e t r y breaking (DSB). We will show t h a t quite str ict l imi ts are imposed on the values ot heav ie r fermion and boson masses, as well as on WL-W n mixing.

Since the neu t ra l weak- in te rac t ion bosons (Z ~ and A) axe insensi t ive to Cabibbo mix ing and the pure le f t -handed W E are unable to t ransfer mass to fermions, we are forced to ex t end the s tandard model of weak in terac t ions at least to SU2L| SU2R| U1. A mix ing of W~. and WR bosons is induced by fermion masses v ia fig. la). The ini t ia l ly massless neu t r inos and u, d quarks then get mass f rom the g raph lb).

I n (~spontaneously~) broken schemes, invo lv ing the scalar fields, t he energy- dependen t fe rmion masses mi(q ~) behave as constants in first order, and the v a c u u m polar izat ion loop (=A(q~)) general ly diverges. Renormal i za t ion requires (infinite) coun te r t e rms f rom the scalar sector of the Lagrangian , and the va lue of A thus becomes a free p a r a m e t e r in the theory. A l though graph lb) is finite, t he presence of tadpoles also p reven t s any de te rmina t ion of t he neut r ino mass (1).

I f we consider instead t h a t fe rmion masses are of dynamica l origin (2) the si tua- t ion becomes ent i re ly different. Since scalars are exc luded f rom the Lagrangian , t he appearance of mass m a y no t p rovoke any new divergences. This imposes in pa r t i cu la r t h a t t he m o m e n t u m - d e p e n d e n t masses mi(q ~) decrease when lq2[--~c~, a common fea ture of all models exhib i t ing DSB. Of course, calculat ions are much more difficult in this f ramework . Fo r instance, the va lue of the mix ing A(0) is now in pr inciple calculable, since the v a c u u m polar izat ion bubble converges, bu t i t involves the values of m~(q 2) for all q2, which are in general no t known explici t ly , and in pract ice all we

(*) This work was supported in part by the Belgium state and the contract ARC 79/83-12. (**) Charg6 de Recherehes du FNRS. (***) Present address: Theory Division, CERN, Geneva. (1) G. C. BRAI~CO and G. SENJANOVIO: Phys. Rev. D, 18, 1621 (1978). (~) F. ENGLERT, J. M. FR~RE and P. NICOLETOPOULOS: Phys. Left. B, 59, 346 (1975); L. SUSS~rIND: SLAC-PUB-2142; S. WEINBERG: Phys. Rev. D, 19, 1227 (1979).

619

6 2 0 J . -U. F I ~ R ~

A

WL W R

.

L ' , , ~ R qL qL m qR qR

J

a) b)

Fig. 1. - a) WL-W R mixing induced by fermion masses; b) fermion mass correction.

4- L ~ R

c a n give is an estimation based on dimensional analysis:

(1) ~gA A(0) _ 4= mJ0)mj(0) .

We will come back to eq. (1) later, but instead of using this crude value, we now turn to experiment and see which constraints actually l imit A(q2).

I t is convenient to re-express the interaction in terms of mass eigenstates of the Yang-Mills bosons. If M L and M R are the masses of the original bosons, the new ones will have masses Mx and Ms, and the diagonalization is caraeterized by a mixing angle ~.

In the limit of weak mixing, we have

A(0) (2) sin 8 _~ - -

Limits of these parameters from scattering and desintegration experiments have been studied by B~G, BUDNu ~OHAPATRA and SIRLIN (3). They obtain

(3) M~IMI > 2.76, - - 0.06< tg ~ < 0.054.

Wi th this estimate in hand we now turn to the evaluation of fermionic mass correc- tions. In the most straightforward S U2L • S U~R • U1 model, we expect to find the two doublets (~e, e)L and (re, e)R.

We thus calculate the neutrino mass implied by this scheme, as shown in fig. lb). Neglecting the q~-dependence of both A(q 2) and mJq2), which is now a good

approximation, since the integral converges quickly, and evaluating the graph for vanishing external momentum, we obtain

(4) g~ 3 sin ~ cos dq~ 2:3(0) = 4= 4-~ (q2_ ~ , ) ( q 2 ~ ) ( q 2 _ ~ ) ,

Where 2:3 refers to the scalar part of the inverse fermionie propagator s-l(p) = p Z I + 2:3; t h e (generalized) Landau gauge (~) was adopted to avoid the appearance of infinite

(*) M, A. BEG, R. BUDNY, R. MOHAPATRA and A. SIRLIN: Phys. Rev. Lett., 38, 1252 (1977). (4) K. JOHNSON, M. BAKER and R. WXLI~Y: Phys. Rev. Sect. B, 136, 1111 (1964).

L ] ~ F T - R I G H T S Y M ~ ] ~ T R Y B R E A K I N G A N D F ] ~ R M I O N M A S S E S 621

renormal iza t ion of 2~ 1 (in fact , Zl(0 ) = 1 in this gauge). I t is qui te in teres t ing to observe t h a t the bounds presented in (3) yield, according to (4) a neutr ino mass j u s t a fac tor of 3 above the exper imenta l l imi t of 60 eV!

W e h a v e indeed

(5)

m 2 I n 'D~b 2

(M~ - - m,2)(m ~ - M~)

3 g~ 2:2(0 ) = - - - - sin ~ cos ~ m ( M i - - M~).

4z~ 4~

~ In M , ~ :~.~ in ~ ~- (m ~ _ ~ ~ ~ + ~ 2 ~ MI)(M I - M~) (M I - M~)(M~ -- m ~)

3 g2 . M~ 105 eV 4--~ ~ sm ~m. In ~ < 2 sin 2 0 w "

Since M ~ / M 1 cannot be lowered, we obta in a s l ight ly more severe l imi ta t ion on }:

(6) sin ~ < ~ [ 3 g2 M~]-I 3.5" 10 -2 2 sin ~ 0 w �9

As far as no mix ing be tween leptons (in fact, no neut r ino bare masses) is considered, i t is readi ly seen t h a t no fur ther const ra in ts arise f rom considering mass correct ions f rom ~, �9 to the i r respect ive neutr inos.

Keep ing (6) in mind, we now t u r n to the eva lua t ion of mass corrections to quarks. Our purpose is to find out whe ther the current mass of t he first doublet of quarks could arise solely form rad ia t ive corrections, by assuming all the quark masses and mixings to be given.

This is possible only for t he l ighter doublet . I f 2 massless doublets were present, t he Cabibbo mix ing of one of t h e m could always be ro t a t ed away.

W e have p lo t t ed in fig. 2 the va lue of the in tegra l appear ing in (4) for different values of m~ and M~, adopt ing the m a x i m u m value of ~ f rom (6).

3/1 is assumed to t ake the c o m m o n l y admi t t ed va lue of 68 GeV. Since the mass of u and d m a y only arise by Cabibbo mixing wi th the o ther doublets, we in t roduce 0~ and 0m, for t he ma t r ix e lements l inking, respect ively, L and R components of the i - t h double t to t h e first one.

A first, qua l i t a t ive observat ion f rom fig. 2 is tha t , as long as the heavy -qua rk masses m~ do no t exceed 21/2, 2:2 grows wi th m~. Since neut ra l currents are not subject to Cabibbo mixing, the only cont r ibut ions to the u -quark mass come f rom the lower members of o ther doublets ; this is consis tent w i th m~/md<. 1 if in heavier doublets, the lowest componen t is always the l ighter (as seems to be t he case for (c, s), and (t, b).)

On the side of orders of magni tude , however , the p ic tu re is less appealing. The induced mass is indeed of order

(7) minting.d= 0i~0Rt~2(0) ,

if we t ake 0~ to be of the same order as the usual Cabibbo mix ing 0Li --~ 0.2. (More f reedom is p e r m i t t e d in th is choice by the possibi l i ty of lep ton mixing, which allows us to re lax the usual restr ict ions (s).)

(6) B. W. LEE and It. E. SHROCK: Phys. Rev. D, 16, 1444 (1977).

622 j.-M. FR~RE

30

rncl. = 7 M eV M 2 = 2500 GeV

20

=1500

rn u = 4MeV

=1000

10

i =150 I I I I I I I

0 500 1000 1500 2000 m i (GeV)

Fig. 2. - ~= = sin ~raax(g'14~)(3/4X~)~(m(Ml - - M~)/((p' 4- m*)(p" 4- MI)(p = + MI))) p ' dp'.

I

2500

To f ind t h e lowest v a l u e of M s c o m p a t i b l e w i t h such a m e c h a n i s m a n d t h e c o m m o n l y a d o p t e d va lues r a n : 4 MeV, m d : 7 MeV we a s s u m e m a x i m u m m i x i n g of t h e r igh t - h a n d lep tons , i . e . O R : 1 a n d o b t a i n

(8a) M~ ~ 2500 G e V .

I n t h i s case, t h e masses of t h e h e a v y doub le t of q u a r k s are found to b e

(8b) role w = 380 G e V , mhlgh= 2500 GeV .

Are the se va lues sufficient to p r o v i d e a d y n a m i c a l or ig in of ~? W e h a v e a s s u m e d t h a t ~ t akes i t s m a x i m u m value , w h i c h in t h i s case is f o u n d f rom eq. (6) to b e 0 . 9 % .

Acco rd ing to (1), (2) a n d t h e va lues (8), however , a n d t a k i n g 3 eolours i n to accoun t , we w o u l d o b t a i n ~ 0 .6%! w h i c h is of t he r i g h t o rde r of m a g n i t u d e .

A l t e r n a t i v e l y , we can look for t h e abso lu te m i n i m u m va lues of mhi~h a n d mlo~; these are o b t a i n e d in t h e l i m i t M s - * oo, a n d are r e a d f r o m fig. 2 to b e

(9) ~ n 200 G e V , rain 350 GeV T t ~ l o w = ~bhig h = .

Thi s c i r c u m s t a n c e suggests a t e s t of t h e a p p l i c a b i l i ~ of t h e p roposed m e c h a n i s m .

LEFT-RIGHT SYMMETRY BR]~AKING AND F~RMI01~ MASSES 6 ~

I t was shown indeed by VV,~TMAN (~) that the occurrence of widely split fermion doublets would modify the relation between m w and m z in a S U~ • U1 model (which we approach effectively, due to the high value of mwa ). If one considers 3 (eoloured) doublets of quarks

Q = m w l m z cos 2 0w (= 1 in and the min imum mass splitting given by eq. (9), the ratio 2 the standard model) would already be affected by a 0.6% correction. This is not too far from the present precision of the measurements (~).

Finally, I would like to stress that at least for small fermion mass the effect studied above only depends on the mass ratio of the left and right-handed bosons, and not on their individual value, as is best seen from eq. (5). For this reason, the presence of right- handed bosons in unified schemes of strong, electromagnetic and weak interactions could have similar consequences, and link together lepton and quark masses, even if the bosons are supcrheavy.

(6) 1~. VELTMAN" Nucl. Phys. B, 123, 89 (1977). (7) I , LIEDE a n d ~vI. R o o s : He l s ink i p r e p r i n t HV-TFT-79/27 (1979).