**The mass ratio between neutrinos and charged leptons (*)**

G. PREPARATA(1_{) (}2

_{) and SHE-SHENG}

_{XUE}

_{(}1

_{) (**)}

(1_{) INFN, Sezione di Milano - Via Celoria 16, Milano, Italy}

(2_{) Dipartimento di Fisica, Università di Milano - Milano, Italy}

(ricevuto il 2 Ottobre 1996; approvato il 23 Dicembre 1996)

**Summary. — In the framework of the recently proposed electroweak theory on a**

Planck lattice, we are able to solve approximately the lattice Dyson equation for the
fermion self-energy functions and show that the large difference of charged lepton
and neutrino masses is caused by their very different gauge couplings. The predicted
mass ratio ( 1025_{–10}26_{) between neutrinos and charged lepton is fully compatible}

with present experiments.

PACS 11.15.Ha – Lattice gauge theory. PACS 11.30.Rd – Chiral symmetries.

PACS 11.30.Qc – Spontaneous and radiative symmetry breaking.

Since their appearance neutrinos have always been extremely peculiar. In the sixty
years of their life, their charge neutrality, their apparent masslessness, their
left-handedness have been at the centre of a conceptual elaboration and an intensive
experimental analysis that have played a major role in donating to mankind the beauty
*of the electroweak theory. V 2A theory and Fermi universality would possibly have*
eluded us for a long time had the eccentric properties of neutrinos, all tied to their
apparent masslessness, not captured the imagination of generations of
experi-mentalists and theorists alike.

However, with the consolidation of the Standard Model (SM) and in particular with the general views on (spontaneous?) mass generation in the SM, the observed (almost) masslessness of the three neutrinos (ne, nm, nt) has recently come to be viewed as a

very problematic and bizarre feature of the mechanism(s) that must be at work to produce the very rich mass spectrum of the fundamental fields of the SM. Indeed, in the (somewhat worrying) proliferation of the Yukawa couplings of fermions to the Higgs fields that characterizes the generally accepted SM, no natural reason can be found why the charge-neutral neutrinos are the fundamental particles of lowest mass;

(*) The authors of this paper have agreed to not receive the proofs for correction. (**) E-mail address: xueHmilano.infn.it

for in the generally accepted minimal Higgs mechanism, the actual values of the fermion masses are in direct relation with the strengths of their couplings to the Higgs doublet, and it appears rather bizarre that nature has chosen to create a very sophisticated mass pattern by the mere fine-tuning of a large number of parameters.

On the other hand we believe that it cannot be a simple dynamical accident that the only charge-neutral fermions that populate the SM are also the lightest. In this letter we wish to show that this most remarkable property of nature can naturally be understood in the framework of the Planck Lattice Standard Model (PLSM) that we have recently introduced [1-3], based on the general hypothesis that, as a result of the violent quantum fluctuations of the metrical (gravitational) field, at the Planck length

*a*pC 10233cm , space-time is no more a 4-dimensional continuum, but can be well

represented by a (random) lattice with (average) lattice constant equal to

*a 4a*p[4]. Here we would only like to stress that in order to avoid a well-known

theorem by Nielsen and Ninomiya [5], which prevents a simple transcription of the usual continuum (SM) Lagrangian to the lattice, we have been forced to add new 4-fermi terms of the Nambu–Jona-Lasinio type (NJL) [6]. Recently, we analyzed the solutions of the Dyson equations involving the NJL interactions only, with the following results [2]:

1) The fundamental chiral symmetry of the full Lagrangian is spontaneously
*broken, and as a consequence only one family in the quark sector, which is identified*
with the top quark and bottom quark doublet, acquires a mass. The lepton sector
remains massless owing to its different colour structure.

2) The consistent solution of the gap equations also produces non-zero
*Wilson-type parameters r*q *and rl* and mass counterterms for each quark and lepton.

The effective action that is left after such massive rearrangement of the vacuum
*is (a 4a*p)
(1) * _{S 4S}*G

*1 S*D

### !

*xF*

*c(x) mFc(x) 2*2 1

*2 a*

*Fxm*

### !

### [

*cF*

_{(x)}_{(}

_{L}F*m*

*(x) 1RmF(x)*

### )

*rFUmF(x) cF(x 1am*) 1h.c.

### ]

+ R ,*where S*G *is the usual Wilson gauge action, S*D *the usual Dirac action, F 4*

### l

( q )*denotes the lepton (quark) sector, mF, rF* are matrices in flavour and weak isospin
space, and the dots denote the NJL interaction and the necessary counterterms.
Gauge links are given by

*LF*
*m(x) 4UmL(x) VY*
*F*
*L*
*m* *(x) ;* *RmF(x) 4*

## u

*VY*

_{R1}F*m*

*(x)*0 0

*VY*

_{R1}F*m*

*(x)*

## v

, (2)*where Um*q*(x) SU*c*( 3 ), UmL(x) SU*L*( 2 ) and Vm(x) UY*( 1 ). Turning to gauge
interactions we approximately analyzed the Dyson equation for the top and bottom
doublet and obtained a mass ratio between the top and bottom quarks [7] in agreement
with recent experimental indications. In this note, we apply an analogous analysis
to the lepton sector. Our aim is, of course, to see whether gauge interactions

Fig. 1. – The diagrammatic form of the Dyson equation for the lepton sector.

are capable of yielding the large observed differences between the masses of the charged leptons and the neutrinos.

When we take into consideration in addition to the NJL interaction the gauge interactions of the action (1), the Dyson equations for the lepton sector have the structure depicted diagrammatically in fig. 1, which can be written as

*SQc( p) 4mQs*

### !

*g*

*C*

*Q*

*g*

*( p) ;*(3)

*s 4*

*2 g*1

*Nc*

*2p*

*p*d4

_{q}*( 2 p)*4 1 sin2

_{q}*m1 [mQa 1r*l

*w(q) ]*2 1 R , (4)

*where SQc( p)*

### (

*ScQ( 0 ) 4mQ*

### )

are the leptons’ self-energy functions; the second term in*the r.h.s. of (3) denotes the contributions of the relevant gauge interactions and s*stands for the contributions (dots for high-order contributions) of the NJL

*interactions, which are supposed to be the same for neutrinos (Q 4 0) and charged*

*leptons (Q 421) because mQa C0, rQ4 rl(Q 40, 21). We remark that the terms that*

*diverge like 1 Oa have all been consistently cancelled by mass counterterms [2].*

*For external momenta pma b 1 , we divide the integration domain over the variable*

*qm* *into two regions: the “continuum” region: 0 GNqmaNGe and the “lattice” region eG*
*NqmaNGp, where NapmN b e b p . With this separation we can write the contribution of*
the gauge interactions as

*CQ*
*g( p) 4C*A*gQ( p , L) 1uQg(e , r*l*) 1dQg(e , r*l) ,
(5)
*C*A*Q*
*g( p , L) 4*
1
*4 p*2

*L*d4

_{q}*lg(q*2 )

*[ ( p 2q)*2

*1 mg*2]

*SQ*

*c(q)*

*[q*2

*1 mQ*2] , (6)

*where the continuum-region integral is up to L 4eLp* *(Lp4 pOa C 10*19GeV ). As for
*the ln e-divergent contribution in the lattice region, one has*

(7) *uQ*
*g(e , L) 4*

*[e , p]*d4

_{l}*16 p*2

*lQ*

*g(l) SQc(l)*sin2

*lm*O2 cos2

*lm*O2 sin2

_{(l) 1}### (

*mQa 1rw(l)*

### )

2 C*C lQg(Lq) mQ*

### g

*c*2

*(rl*) 2 1 2

*ln e*

### h

,*where we take the asymptotic mean-field value SQc*

### (

*NlN F e*

### )

*C mQ*and numerically

*calculate c*2

*(r*l

*), which is plotted in fig. 2. Note that the q*2-dependent renormalized

*Fig. 2. – The function c*2*(r*l*) vs. r*l.

couplings are given by

*lQg(g*2*) 4lQg(mQ*2*) Z*3*(q*2*, mQ*2) , *lQg(mQ*2) 4
3
*p*
*bQ*_{L}*bQ*_{R}*(mQ*2)
*4 p* ,
(8)
*where (bQ*

L , R*)(m) are left- and right-handed gauge couplings to different fermions (Q)*

*on the mass shell m 4mQ, and Z*3*(q*2*, mQ*2) are normal gauge field renormalization
functions in the Abelian and non-Abelian cases

### (

*SU*L( 2 )

### )

. In eq. (6) the regularcontributions from the lattice region are
(9) *dQA , Z(e , r*l) 42

### g

3*p*

*(bQ*

_{L}

_{1 b}Q_{R})2

*(Lp*)

*4 p*

### h

*r*2 l

*[e , p]*d4

*l*

*16 p*2

*SQ*

*c(l)*sin2

*(l) 1*

### (

*mQa 1rw(l)*

### )

2 , (10)*d*( 0 , 21)

*W*6

*(e , r*l)

*j*4 42

### g

3*p*

*(bQ*L

*1 b*

*Q*R) 2

_{(L}*p*)

*4 p*

### h

*r*2 l

*[e , p]*d4

_{l}*16 p*2

### !

*i 41*3

*NV*KM

*ji*N 2

*S*(21, 0)

*c*

*(l)i*sin2

_{(l) 1}### (

*m*

_{(21, 0)}

_{a 1rw(l)}### )

2 ,*where V*KM *is a KM-type matrix for lepton sector. Since (9) and (10) have no ln e*

*divergence, we may safely take the limit e K0.*

*Due to the fact that there are extra terms arising from c*2*(rl*)

### (

eq. (7)### )

*, dgQ(rl*)

### (

eqs. (9), (10)### )

and NJL terms in (3), the gap equation (3) can have consistent massive*solution (mQ*c0 ) even for small gauge couplings [8]. In this preliminary discussion, we take a trivial (unity) KM matrix in (10) and the Landau mean-field approximation

*SQc(q) CmQ* in (6), (7), (9), (10). Thus, for each lepton family, we have from (3)

*m*n*4 m*n*( NJL ) 1m*n*d*n*Z(r*l*) 1mld*n*W(r*l) ,
(11)

*m*l*4 ml( NJL ) 1ml(CA*l*1 CZ*l*) 1ml*

### (

*d*l

*A(rl) 1d*l

*Z(rl*)

### )

*1 m*n

*d*n

*W(rl*) , (12)

*where we can see that the gauge field contributions CgQ( p) in eq. (6) play the essential*
role in distinguishing between neutrinos and charged leptons. We notice that for
*neutrinos C*A*g*( 0 )*4 ug*( 0 )*4 0 (g 4 A*e.m., W6 and Z0*), while for charged leptons C*AW(21)6 4

*u*(21)W6 4 0 . As for the gap equation of charged leptons (12), we have

(13) *CgQ4 C*A*gQ1 uQg*4
1
*4 p*2

*L*p d4

*q*

*lg(q*2

_{)}

*[ ( p 2q)*2

*1 mg*2] 1

*[q*2

*1 mQ*2]

*1 lQg(Lp) c*2

*(rl*) ,

*where the arbitrary e scale disappears, as it should, due to the cancellation of the ln e*
*term between C*A*gQ( p , L)*

### (

eq. (6)### )

*and uQg(e , rl*)

### (

eq. (7)### )

. By using the renormalization*group relations (8) to estimate the values of b*L

*(L*p

*), b*R

*(L*p)

### (

*a(L*p

*) C1.56a(m*e),

sin2*u*w*(L*p) C2.32 sin2*u*w*(m*Z2)

### )

in (7), (9), (10), (13)### (

*a f a(m*e) 41O137, sin2

*u*wf

sin2* _{u}*
w

*(m*Z2) 40.23

### )

*, one gets (C*A

*g*l

*4 Cg*(21)) (14)

### .

### `

### /

### `

### ´

*C*l

*A*C

### y

3 Q 1023### u

ln*L*p

*m*l

### v

2 1 0 .75 ln*L*p

*m*l

*1 1 .5 c*2

*(r*l)

### z

*a ;*

*C*l

*Z*C

### y

3 .6 Q 1023### g

ln*L*p

*m*Z

### h

2 1 0 .37 ln*L*p

*m*Z

*1 0 .18 c*2

*(r*l)

### z

### g

sin2*u*w

*4 p*

### h

2 2### y

1 .2 Q 1023### g

_{ln}

*L*p

*m*Z

### h

2 1 0 .37 ln*L*p

*m*Z

*1 0 .18 c*2

*(r*l)

### z

*a ,*

*where m*Z

*4 91 .2 GeV . It will be seen soon that dQg(rl*),

### (

eqs. (9), (10)### )

, are small numbers (A1025

_{) due to the smallness of gauge couplings and of r}l2*. Neglecting dgQ(r*l)
*in (12) and adopting the four-fermi coupling g*1determined by the top quark mass, one

can show that the gap equation for charged leptons (12) has a consistent solution

*m*l*A O( MeV ) b L*p for small gauge couplings [3]. However, for the time being, we

are not in a position to obtain the correct spectrum of charged leptons [9]. As for the
*gap equation for neutrinos (11), the consistent solution must be m*nb*m*l because
*( 1 2s) AO*

### (

*a ln (L*p

*Oml*)

### )

c*dg*( 0 )

*(r*l). In fact, in eqs. (11) and (12), we have seen the emergence of the hierarchy spectrum of neutrinos and charged leptons due to their very different gauge couplings.

Although we are still not able to calculate the masses of neutrinos and charged leptons based on eqs. (11) and (12), their mass ratio can be estimated. It is easy to see that the solution to (12) is simply

### .

### /

### ´

*d*n

*W(r*l

*) m*l2

*2 2 D*l

*m*l

*m*n

*2 d*l

*W(r*l

*) m*n24 0 ,

*2 D*l

*4 2(CA*l

*1 CZ*l

*) 2d*l

*A(r*l) 1

### (

*d*n

*Z(r*l

*) 2d*l

*Z(r*l)

### )

. (15)*Fig. 3. – The function G(r*l*) vs. r*l.

From the definitions (9), one obtains

*d*n
*Z(r*l*) 2d*l*Z(r*l*) C82.2ar*l2*G(r*l) ;
(16)
*d*l
*A(r*l*) C58.9ar*l2*G(r*l) ,
(17)
*G(r*l4

_{p}*( 2 p)*4 1 sin2

_{(l}*m*) 1

### (

*mQa 1r*l

*w(l)*

### )

2 , (18)*where G(r*l*) as a function of r*l *is plotted in fig. 3, while for dQW(r*l)

### (

eq. (10)### )

, where*b*R*4 0 , b*L*4 g*2Ok*2 and g*2 *is the SU*L( 2 ) gauge coupling, we have

*d*l
*W(rl) 4d*n*W(rl*) 42
*3 p*
2
*a(L*p)
sin2* _{u}*
w

*(L*p)

*r*l2

*G(r*l

*) C213.72ar*l2

*G(r*l) . (19)

Thus, solving (15), we obtain

*m*n4
*D*l1

### o

*(D*l)21

### (

*d*l

*W(r*l)

### )

2*d*l

*W(r*l)

*m*l. (20)

*We see that D*l*E 0 and dW*b*1 are crucial for obtaining m*nb*m*l*. For r*lC 0 .01 –0 .04 (

1

),

(1_{) In [2] we show that the Wilson parameter for quarks (r}

q4 0 .28 –0 .3 ) is determined by

*one gets cA*2*4 0 .65 –0 .62 and G(rl*) 40.62–0.6, which leads approximately to

*m*nlC ( 1 .0 Q 10

25_{–1 .6 Q 10}24_{) m}

l. (21)

*Putting the experimental charged lepton masses (m*e*4 0 .5 MeV , m*m*4 105 MeV , m*t4

1774 MeV ) into (21), we predict the neutrino masses:

### .

### /

### ´

*m*neC ( 5 –128 ) eV ,

*m*nmC ( 1 –26 ) keV ,

*m*ntC ( 17 –284 ) keV , (22)

which are compatible with present experiments, even though our analysis is still at a
rather rudimentary level. Leaving aside the uncertainties of our analysis, which may
well exceed the theoretical uncertainty (2_{) appearing in our predictions, we believe that}

we can say with a certain degree of confidence that:

1) eqs. (22) represent the first (successful?) attempt to derive within a complete closed dynamical scheme the mass ratios between the members of the lepton doublets;

2) these mass ratios are, to the approximation we work in, universal;

3) the large difference in (22) between the masses of the neutral and charged members of the lepton weak doublets finds its natural and convincing origin in the their gauge couplings, and in particular in the neutrino lack of electromagnetic interactions, which most likely is at the root of the mechanism by which charged leptons acquire their masses [9];

*4) the crucial role played by the Wilson parameter r*l in yielding a connection
between the masses of the charged leptons and their neutrinos, through the high
*energy (AL*p) coupling of the right-handed neutrinos to their left-handed

counterparts via the charged leptons and the W6_{-bosons (1), should be viewed as a}

clearly unique dynamical feature of our proposal.

(2* _{) Note that the main uncertainty comes from the uncertainty of the Wilson parameter r}*
l.

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