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Left-handed neutrino mass scale and spontaneously broken lepton number
Volume 99B, number 5
PItYSICS LETTF.RS
5 March 1981
LEFT-HANDED NEUTRINO MASS SCALE AND SPONTANEOUSLY BROKEN LEPTON NUMBER G.B. GELMINI Sektion Physik, Ludwig-Maximilians-Universitiit, Munich, Fed. Rep. Germany and M. RONCADELLI 1 Max-Planck-Institut ffOrPhysik und Astrophysik, Munich, Fed. Rep. Germany Received 8 December 1980
Neutrino masses are included in the Weinberg- Salam model in the most economical way. They appear as a manifestation of a new low-mass scale at which the global symmetry of lepton number is spontaneously broken. Two new particles, a Goldstone boson - the majoron - and a neutral Higgs with low mass, appear coupled strongly to neutrinos but so weakly to the other fermions that they cannot be detected at present laboratory energies.
The question of neutrino masses has received much attention recently. No doubt, this is largely due to the claims about the experimental discovery of neutrino oscillations [1] and the massiveness of the ve neutrino [2]. Taking the above results at face value, the ve mass turns out to be some eV and another neutrino would possess almost the same mass [3]. In this letter we try to introduce a mass for the neutrinos enlarging the standard minimal SU(2)L × U(1) model of weak and electromagnetic interactions [4] in the most economical way. We do not extend the fermion sector but only the Higgs sector, adding the minimal number of Higgs bosons, i.e. we do not put in any right-handed v R neutrino. Our feeling is that in an ultimate theory the Higgs mechanism will be replaced by a dynamical symmetry breaking and therefore the Higgs bosons should appear as condensates of the fermions already present in the theory [5] ; this supports our option. Whatever will happen to neutrino masses in future experiments, we assume 'here as a prejudice that the ve mass is of the order of eV. We shall also restrict our discussion for simplicity to only one family. 1 Rotary Fellow. On leave of absence from the INFN, Pavia, Italy.
As long as lepton number conservation is required, a Weyl spinor must be massless. This is why in the Weinberg-Salam model the absence of the v R field ensures the masslessness of the neutrino. However, even if a Dirac mass term is forbidden by the absence of the singlet taR, a Weyl spinor can have a lepton number-violating Majorana mass term: £ mass= mv(VL)cv L + b.c. ,
(1)
which can be rewritten as £mass = my
~v,
(2)
with v =- PL + (VL)C = pc. Therefore the neutrino ta is a Majorana particle in our model. The term £mass has to be introduced by the Higgs mechanism through an invariant Yukawa coupling of ~T L ~ (v L, eL) and (~L) c with a new scalar field, 4", a triplet under SU(2)L ,1 There are two choices now, concerning the lepton number. The conventional attitude [61 is to break it explicitly, assigning L = 0 to 4,. In this case the lepton number would be broken at an3, temperature. However, since we are introducing lepton number violation just in trying to give a mass to the neutrinos, we contl Charge conservation rules out the possibility for a singlet.
0 0 3 1 - - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
411
Volume 99B, number 5
PHYSICS LETTERS
sider the possibility that the lepton number is restored in the regime where the neutrino masses are effectively zero more appealing. In this way a new mass scale, (cb}0, appears, which shows up in three different phenomena: the neutrino masses, the scale at which the lepton number symmetry is broken and the violation of the weak A I = 1/2 rule. Clearly this physically appealing scenario cannot happen if the lepton number violation is explicit, therefore we are lead to consider the spontaneous breakdown of lepton number. At first sight this looks physically as an untenable option: no new long range force has been detected so far. The standard way out from this trouble would be to gauge U(1)LEFr, thereby getting rid of the "unwanted" Goldstone bosom In the present situation this strategy does not work: the gauge boson would acquire a mass of Order gL mv and, unless the gauge coupling constant gL is tuned enormously small in an ad hoc fashion, its coupling with leptons would not be negligible and unwanted large cross sections for elastic scattering, for example, would appear. We are, however, not in a cul-de-sac. It has been shown recently within a different model [7], that a Goldstone boson - the majoron +2 _ associated to the global U(1)LEt, T breakdown through a singlet Higgs boson is not ruled out by present experiments. Here we meet with a similar but more complex problem, because our majoron is not a singlet under SU(2)L X U(1) so it couples to all leptons and quarks. Our lagrangian is £ = £F + 2(3 + £F,~ + £V,m + £~,4 ,
(3)
where £F, £G, £F,~ are, respectively, the usual Weinberg-Salam terms of the kinetic energy for the fertalons (with covariant derivatives), the kinetic energy • + for the gauge fields Au, Zu, W~ and the Yukawa coupling of the usual doublet ~T = (~0+, ~o0) with the fermions. The terms £ F , . and £ v , . are defined as (r i are the Pauli matrices): "~F,~P = --OtM(~L)C 7" • (I, ~L + h.c.,
(4)
£so,q, = IDu~12 + IDu~I2 - V(G ~ ) ,
(s)
+2 We wishto make clear that, although it has been invented in Munich, its name does not come from the typical German name Mayer, but rather from the great Italian physicist Majorana, 412
5 March 198l
where the most general SU(2)L X U(1)LEPT invariant potential is: V(G,b) : a]~0]2 + blq)[ 2 +c]~] 4 + d ] ~ ] 4 + e [~121q~12 + f (qst q~t T) (qbTqb) + iefk n h~ot r/~ OP? k (Pn
(6)
The simultaneous conservation of hypercharge y and lepton number L forbids the appearance of trilinear terms in V(~0,~b). The Yukawa coupling £F,,~ written in terms of the charge eigenstates q50 = 2 - 1 / 2 ( ~ 1 + iq32),
q~+ = ,i~3 ,
cb++ = 2-1/2(dPl _ i ~ 2 ) , is: £F,¢' = --O~M{--N/2(VL)C VL+0 + [(eL)ev E + (VL)C eL] 4P+ + X/~(eL)c eL~++}. (7) The requirement of charge conservation fixes uniquely the breaking direction:
~0
2 - l/2(v D + PD + it?D)
~++~
q!) + +
~\
4~+ ~ =
qb+
]
t
q,0 /
(8)
2-1/2(0 T + 0 T + Jr/T
Both vacuum expectation values OD, oT are chosen real by means of two suitable independent SU(2)L and U(1)LEPT global transformations. We see from £F,~ that only the neutrino acquires a mass, while all other fermion masses arise in the usual way from £F,~ through vD. We get: rnu = o~M vT .
(9)
Naturality suggests to take a M of order one, which yields v T ~ m u ~ eV and since vD ~ 250 GeV, then OT/VD ~ 10-11 ,3. However, even choosing c~M
,3 This huge difference between both mass scales reproposes evidently the usual problem of gauge hierarchies but inverted, the new mass scale being lower than the standard one.
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PHYSICS LETTERS
10 6 , 4 , UT would be of order MeV, hence OT/OD 10 - 5 . Therefore we shall work within the approxinration vT "~ v D. Performing in £ ; , . the shift according to eq. (8) we get: £~,~1, = l t g 2 + g ' 2 ) t v 2 + 4v2)Zu Zu
+ l ( g 2 + g ' 2 ) l / 2 ( V 2 + 4v2)l/2Ga. o T M
+ W ~ 3 ' o + ) + ... . (10)
Here, the gauge fields are defined as usual, namely: g ' B , - gW 3 (11) W~+ ~ 2-1/2(Wu1 -T-iW2), Zv - (g2 +g'2)1/2' •
t
a, ..., h, making possible many arbitrary choices for their values. We take the dimensionless constants of order one. The conditions are: a + c v 2 + ½(e - h ) v 2 = 0 ,
(18)
b + d r 2 + ½ ( e - h ) v 2 = O.
(19)
We get the following physical Higgs particles, besides '1~++ and co+:
+ a1 g2(v2 + 2V2) W l W - "
+ ~g(v 2 + 2V2T)1/2(WI31aO
5 March 1981
i
w a t h g , Bu corresponding to U(1)y a n d g , W,, to SU(2)L. Reading the masses off the lagrangian we have: 1 2o + 2V2T)1/2 Mw = ~g(v
(12)
M Z = g l( g 2 +g'2)1/2(V2 + 402) 112
(13)
The violation of the weak A I = 1/2 rule due to ~ is of the order (VT/VD) 2, in any case much smaller than the contribution o f radiative corrections" [8] and far from lhe experimental accuracy 2.6 X 10 - 2 [9]. The fields 0, o + eaten up by Z u0 and W~ respectively, are defined as: 0 = r~D + 2(VT/VD)r?T ,
(14)
O+ = i [tfl+ + (t)T/V D)N/~ (i}+ ] •
( l 5)
The majoron X and the physical singly charged Higgs co+ are, by definition, the fields perpendicular to the previous ones: X = 2(VT/VD)r?D -- 7/T ,
(16)
co+ = i[(VT/VD)N/2~ + - q)+] .
(17)
/?high = PD + [(e - h)/2c](VT/OD)/? T ,
(20)
/?light = [(/7 -- e)/2c](vT/VD) + ,oT .
(21)
The masses are as follows: m2,++ = h v 2 + 2 f v 2 ,
(22)
2 ' - - 2 2V2) moo+ = ~/,/{,vD +
(23)
m oI 2t = 2cv2 + [(e - h)2/2c] v2 ,
(24)
moL2 = [ 2 d - ( e - h ) 2 / 2 c J v
(25)
2.
The only new particles with small masses, as compared to the typical mass scale vD ~ 250 GeV, are the majoron X and the neutral Higgs boson/?L" Clearly, only these can in principle be detected at present energies. Let us therefore investigate their couplings to matter. The majoron X and/?L are coupled to the neutrino through r/T and/?T, and to the other fermions through 7/D and PD, respectively. The various vertices are listed in fig. 1, where f denotes the quark u or d or the electron. In order to estimate the cross section for v - f scattering, we have used the Dirac formalism for the neutrino + s. Forgetting about interference terms, we have: 1 2 2 2 O(p f)x-exchange - 2 rrG V mr m v / s ,
(26)
and, assuming toOL "~ mf, Ev, °(v f) PL-exchange
2 2 _ e - h mfmv -4~c G2
The potential V(~0, ~ ) must now be considered, in order to determine the physical spin zero particles and their masses. Since V(¢, re) d o e s n o t contain trilinear terms, there are only two conditions for the extrema which evidently under-determine the seven paranleters
Observe that both are independent o f a M and (26) is
,4 The option ozM ~ 10-6 would be realized if one were to think that, by some aesthetical reason, the coupling of the two pieces in the Yukawa coupling were to be almost equal.
,s We think this is a correct procedure when the energy in play is E u >>m u, since we also have m u ~ mf and for a massless particle both Dirac and Majorana formalisms coincide•
{ 2 [ m2 \{ (ECM)22 X .1+ ~ m f 1 +-'v +in--I']t . (ECM)2 m OL 2 -v m.L
8]
(27)
413
Volume 99B, number 5
PHYSICS LETTERS
of magnitudes, such as eV ~
9 > ........ X = i ctM~CsVX V
~)------gL V
= O'M~V gL
........ X = i Ef
5 March 1981
(1k4
f'YsfX~
El=
*1 f= u
c)dv, v = (a2/27r) {(4m2r 3
X [3(1~1 • i)(1~ 2 • i ) -
I~1 " 1~2] - e-moLt~r} . (31)
The first term, corresponding to the majoron, is zero for an S-wave state, but the second term remains. This is a Yukawa well which produces a w-bound state a neutrinium - whose radius R and binding energy B are roughly [10] -
Fig. 1. Relevant vertices of the new light particles, the majoron x and the neutral Higgs OL, with matter (f stands for u, d or e).
B ~ 10-2a2moL also independent of any adjustable parameter. We compare these cross sections and the usual one for the electron-v scattering in the approximation E v ~ me. Thus, ln(ECM/mpL)2 >~ 12 and
R = l / % / m v B "~ 10/aM mV'mvmpL .
2 2 2 o(/.,e) x-exchange "" o(v e)o L_exchange -- G F m e m v/Ev ,
B ~ 1 0 - 2 m v ~ 10 - 2 e V ,
(28)
while the contribution due to W and Z exchange is o(ve)w,Z.exchange ~--(G2/37r)meEv(5 sin 2 0 w + 1). (29) Taking the incident neutrino energy E v of order MeV as a typical value in present experiments and m v of some eV we obtain:
o(ve)×.exchange ~-- o(ve)pL.exchange 10-11 X o(ve)w,Z.exchang e ,
(30)
where the combination of constants has been assumed indicatively to be of order 1. Therefore X and PL exchanges in neutrino-electron scattering are n o t detectable at present laboratory energies. However, they could become dominant at threshold. We wish to point out that the relation tT(t.,Ox.exchange ~ o(pf)OL.exchang e still holds true when m o L is changed by various orders 414
(32)
Taking indicatively rnpL ~--my~aM, thus toOL ~ m v for a M = 1 then R~
lO/m v ~ . l O g f m .
(33) As we see, even for a wide range m v <~mOL <~ 106m v ,4, the neutrinium radius turns out to be 10-7 ~
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PHYSICS LETTERS
G2 2 2 mumf ......... X 4 X 1016 ~ 10-18/a 2 ,< 10 - 6 rr~2(mu/1 eV) 2 (34) therefore a M > 10 - 6 , which is consistent with our assumptions. As long as one generation is considered there are no physical consequences of the majoron and the scalar PL exchange at present laboratory energies (even if lhere could be important astrophysical implications). However, when considering various generations our model provides interesting novel features. The neutrino masses have no connection with the masses of the other members of the respective generations. This is a remarkable feature which differentiates this model from most of those which have appeared so far and could provide an appealing description if neutrinos turn out to have almost equal masses, not following the familar hierarchy. If, on the other hand, uu and v r are indeed heavier than ~'e, our model predicts they would decay rapidly through X and possibly PL emission. We plan to come back to these questions as well as on the astrophysical consequences in a future work. We wish to express our deep gratitude to R.D. Peccei and Y. Chikashige for a number of conversations which gave rise to this paper and for many helpfill suggestions and discussions. Also we thank R.N. Mohapatra and A. Masiero for their comments. One of us (M.R.) would like to thank the Maximilianeum Stiftung for the support given during the early stage of this work and finally we would like to express our thanks to the Theory Group of the Max-PlanckInstitute for their hospitality.
5 March 1981
Refere*wes [11 F. Reines, H.W. Sobel and E. Pasierb, Phys. Rev. Lett. 45 (1980) 1307. {2] V.A. Lyubimov, E.G. Novikov, V.Z. Nozik, E.F. Tretyakov and V.S. Kosik, Phys. Lett. 94B (1980) 266. I31 For a review, see: A. De Rfijula and S. Glashow, Nature 286 (1980) 755; an up-to-date discussion is contained in Nature 287 (1980) 481. [4] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, Proc. 8th Nobel Syrup., ed, N. Svartholm (Stockholm, 1968); S. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285. [51 For example, see: S. Weinberg, Phys. Rev. D13 (1976) 974; D19 (1979) 1277; L. Susskind, Phys. Rev. D20 (1979) 2619; E. Eichten and K. Lane, Phys. Lett. 90B (1980) 125 M. Beg, H. Politzer and P. Ramond, Phys. Rev. Lett. 43 (1979) 1701. [61 R,E. Marshak, R.N. Mohapatra and Riazuddin, VPIpreprint (1980); R. Barbieri, D.V. Nanopoulos, G. Morchio and F. Strocchi, Phys. Lett. 90B (1980)91; T.P. Cheng and L.F. Li, COO-3066-152 preprint (June 1980). [71 Y. Chikashige, R.N. Mohapatra and R.D. Peccei, MPIPAE/PTh 36/80 (September 1980); MPI-PAE/PTh40/80 (October 1980). [8] M. Veltman, LAPP-TH-12preprint (January 1980). [91 Particle Data Group, Review of Particle Properties, Rev. Mod. Phys. (April 1980). [10] J.M. Blatt and V. Weisskopf, Theoretical nuclear physics (Wiley, New York, 1963). [11] L. Resnick, M.K. Sundaresan and P.J.S. Watson, Phys. Rev. D8 (1973) 172. [121 G. Feinberg and J. Sucher, Phys. Rev. 20 (1980) 1717.
415
PItYSICS LETTF.RS
5 March 1981
LEFT-HANDED NEUTRINO MASS SCALE AND SPONTANEOUSLY BROKEN LEPTON NUMBER G.B. GELMINI Sektion Physik, Ludwig-Maximilians-Universitiit, Munich, Fed. Rep. Germany and M. RONCADELLI 1 Max-Planck-Institut ffOrPhysik und Astrophysik, Munich, Fed. Rep. Germany Received 8 December 1980
Neutrino masses are included in the Weinberg- Salam model in the most economical way. They appear as a manifestation of a new low-mass scale at which the global symmetry of lepton number is spontaneously broken. Two new particles, a Goldstone boson - the majoron - and a neutral Higgs with low mass, appear coupled strongly to neutrinos but so weakly to the other fermions that they cannot be detected at present laboratory energies.
The question of neutrino masses has received much attention recently. No doubt, this is largely due to the claims about the experimental discovery of neutrino oscillations [1] and the massiveness of the ve neutrino [2]. Taking the above results at face value, the ve mass turns out to be some eV and another neutrino would possess almost the same mass [3]. In this letter we try to introduce a mass for the neutrinos enlarging the standard minimal SU(2)L × U(1) model of weak and electromagnetic interactions [4] in the most economical way. We do not extend the fermion sector but only the Higgs sector, adding the minimal number of Higgs bosons, i.e. we do not put in any right-handed v R neutrino. Our feeling is that in an ultimate theory the Higgs mechanism will be replaced by a dynamical symmetry breaking and therefore the Higgs bosons should appear as condensates of the fermions already present in the theory [5] ; this supports our option. Whatever will happen to neutrino masses in future experiments, we assume 'here as a prejudice that the ve mass is of the order of eV. We shall also restrict our discussion for simplicity to only one family. 1 Rotary Fellow. On leave of absence from the INFN, Pavia, Italy.
As long as lepton number conservation is required, a Weyl spinor must be massless. This is why in the Weinberg-Salam model the absence of the v R field ensures the masslessness of the neutrino. However, even if a Dirac mass term is forbidden by the absence of the singlet taR, a Weyl spinor can have a lepton number-violating Majorana mass term: £ mass= mv(VL)cv L + b.c. ,
(1)
which can be rewritten as £mass = my
~v,
(2)
with v =- PL + (VL)C = pc. Therefore the neutrino ta is a Majorana particle in our model. The term £mass has to be introduced by the Higgs mechanism through an invariant Yukawa coupling of ~T L ~ (v L, eL) and (~L) c with a new scalar field, 4", a triplet under SU(2)L ,1 There are two choices now, concerning the lepton number. The conventional attitude [61 is to break it explicitly, assigning L = 0 to 4,. In this case the lepton number would be broken at an3, temperature. However, since we are introducing lepton number violation just in trying to give a mass to the neutrinos, we contl Charge conservation rules out the possibility for a singlet.
0 0 3 1 - - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
411
Volume 99B, number 5
PHYSICS LETTERS
sider the possibility that the lepton number is restored in the regime where the neutrino masses are effectively zero more appealing. In this way a new mass scale, (cb}0, appears, which shows up in three different phenomena: the neutrino masses, the scale at which the lepton number symmetry is broken and the violation of the weak A I = 1/2 rule. Clearly this physically appealing scenario cannot happen if the lepton number violation is explicit, therefore we are lead to consider the spontaneous breakdown of lepton number. At first sight this looks physically as an untenable option: no new long range force has been detected so far. The standard way out from this trouble would be to gauge U(1)LEFr, thereby getting rid of the "unwanted" Goldstone bosom In the present situation this strategy does not work: the gauge boson would acquire a mass of Order gL mv and, unless the gauge coupling constant gL is tuned enormously small in an ad hoc fashion, its coupling with leptons would not be negligible and unwanted large cross sections for elastic scattering, for example, would appear. We are, however, not in a cul-de-sac. It has been shown recently within a different model [7], that a Goldstone boson - the majoron +2 _ associated to the global U(1)LEt, T breakdown through a singlet Higgs boson is not ruled out by present experiments. Here we meet with a similar but more complex problem, because our majoron is not a singlet under SU(2)L X U(1) so it couples to all leptons and quarks. Our lagrangian is £ = £F + 2(3 + £F,~ + £V,m + £~,4 ,
(3)
where £F, £G, £F,~ are, respectively, the usual Weinberg-Salam terms of the kinetic energy for the fertalons (with covariant derivatives), the kinetic energy • + for the gauge fields Au, Zu, W~ and the Yukawa coupling of the usual doublet ~T = (~0+, ~o0) with the fermions. The terms £ F , . and £ v , . are defined as (r i are the Pauli matrices): "~F,~P = --OtM(~L)C 7" • (I, ~L + h.c.,
(4)
£so,q, = IDu~12 + IDu~I2 - V(G ~ ) ,
(s)
+2 We wishto make clear that, although it has been invented in Munich, its name does not come from the typical German name Mayer, but rather from the great Italian physicist Majorana, 412
5 March 198l
where the most general SU(2)L X U(1)LEPT invariant potential is: V(G,b) : a]~0]2 + blq)[ 2 +c]~] 4 + d ] ~ ] 4 + e [~121q~12 + f (qst q~t T) (qbTqb) + iefk n h~ot r/~ OP? k (Pn
(6)
The simultaneous conservation of hypercharge y and lepton number L forbids the appearance of trilinear terms in V(~0,~b). The Yukawa coupling £F,,~ written in terms of the charge eigenstates q50 = 2 - 1 / 2 ( ~ 1 + iq32),
q~+ = ,i~3 ,
cb++ = 2-1/2(dPl _ i ~ 2 ) , is: £F,¢' = --O~M{--N/2(VL)C VL+0 + [(eL)ev E + (VL)C eL] 4P+ + X/~(eL)c eL~++}. (7) The requirement of charge conservation fixes uniquely the breaking direction:
~0
2 - l/2(v D + PD + it?D)
~++~
q!) + +
~\
4~+ ~ =
qb+
]
t
q,0 /
(8)
2-1/2(0 T + 0 T + Jr/T
Both vacuum expectation values OD, oT are chosen real by means of two suitable independent SU(2)L and U(1)LEPT global transformations. We see from £F,~ that only the neutrino acquires a mass, while all other fermion masses arise in the usual way from £F,~ through vD. We get: rnu = o~M vT .
(9)
Naturality suggests to take a M of order one, which yields v T ~ m u ~ eV and since vD ~ 250 GeV, then OT/VD ~ 10-11 ,3. However, even choosing c~M
,3 This huge difference between both mass scales reproposes evidently the usual problem of gauge hierarchies but inverted, the new mass scale being lower than the standard one.
Volume 99B, number 5
PHYSICS LETTERS
10 6 , 4 , UT would be of order MeV, hence OT/OD 10 - 5 . Therefore we shall work within the approxinration vT "~ v D. Performing in £ ; , . the shift according to eq. (8) we get: £~,~1, = l t g 2 + g ' 2 ) t v 2 + 4v2)Zu Zu
+ l ( g 2 + g ' 2 ) l / 2 ( V 2 + 4v2)l/2Ga. o T M
+ W ~ 3 ' o + ) + ... . (10)
Here, the gauge fields are defined as usual, namely: g ' B , - gW 3 (11) W~+ ~ 2-1/2(Wu1 -T-iW2), Zv - (g2 +g'2)1/2' •
t
a, ..., h, making possible many arbitrary choices for their values. We take the dimensionless constants of order one. The conditions are: a + c v 2 + ½(e - h ) v 2 = 0 ,
(18)
b + d r 2 + ½ ( e - h ) v 2 = O.
(19)
We get the following physical Higgs particles, besides '1~++ and co+:
+ a1 g2(v2 + 2V2) W l W - "
+ ~g(v 2 + 2V2T)1/2(WI31aO
5 March 1981
i
w a t h g , Bu corresponding to U(1)y a n d g , W,, to SU(2)L. Reading the masses off the lagrangian we have: 1 2o + 2V2T)1/2 Mw = ~g(v
(12)
M Z = g l( g 2 +g'2)1/2(V2 + 402) 112
(13)
The violation of the weak A I = 1/2 rule due to ~ is of the order (VT/VD) 2, in any case much smaller than the contribution o f radiative corrections" [8] and far from lhe experimental accuracy 2.6 X 10 - 2 [9]. The fields 0, o + eaten up by Z u0 and W~ respectively, are defined as: 0 = r~D + 2(VT/VD)r?T ,
(14)
O+ = i [tfl+ + (t)T/V D)N/~ (i}+ ] •
( l 5)
The majoron X and the physical singly charged Higgs co+ are, by definition, the fields perpendicular to the previous ones: X = 2(VT/VD)r?D -- 7/T ,
(16)
co+ = i[(VT/VD)N/2~ + - q)+] .
(17)
/?high = PD + [(e - h)/2c](VT/OD)/? T ,
(20)
/?light = [(/7 -- e)/2c](vT/VD) + ,oT .
(21)
The masses are as follows: m2,++ = h v 2 + 2 f v 2 ,
(22)
2 ' - - 2 2V2) moo+ = ~/,/{,vD +
(23)
m oI 2t = 2cv2 + [(e - h)2/2c] v2 ,
(24)
moL2 = [ 2 d - ( e - h ) 2 / 2 c J v
(25)
2.
The only new particles with small masses, as compared to the typical mass scale vD ~ 250 GeV, are the majoron X and the neutral Higgs boson/?L" Clearly, only these can in principle be detected at present energies. Let us therefore investigate their couplings to matter. The majoron X and/?L are coupled to the neutrino through r/T and/?T, and to the other fermions through 7/D and PD, respectively. The various vertices are listed in fig. 1, where f denotes the quark u or d or the electron. In order to estimate the cross section for v - f scattering, we have used the Dirac formalism for the neutrino + s. Forgetting about interference terms, we have: 1 2 2 2 O(p f)x-exchange - 2 rrG V mr m v / s ,
(26)
and, assuming toOL "~ mf, Ev, °(v f) PL-exchange
2 2 _ e - h mfmv -4~c G2
The potential V(~0, ~ ) must now be considered, in order to determine the physical spin zero particles and their masses. Since V(¢, re) d o e s n o t contain trilinear terms, there are only two conditions for the extrema which evidently under-determine the seven paranleters
Observe that both are independent o f a M and (26) is
,4 The option ozM ~ 10-6 would be realized if one were to think that, by some aesthetical reason, the coupling of the two pieces in the Yukawa coupling were to be almost equal.
,s We think this is a correct procedure when the energy in play is E u >>m u, since we also have m u ~ mf and for a massless particle both Dirac and Majorana formalisms coincide•
{ 2 [ m2 \{ (ECM)22 X .1+ ~ m f 1 +-'v +in--I']t . (ECM)2 m OL 2 -v m.L
8]
(27)
413
Volume 99B, number 5
PHYSICS LETTERS
of magnitudes, such as eV ~
9 > ........ X = i ctM~CsVX V
~)------gL V
= O'M~V gL
........ X = i Ef
5 March 1981
(1k4
f'YsfX~
El=
*1 f= u
c)dv, v = (a2/27r) {(4m2r 3
X [3(1~1 • i)(1~ 2 • i ) -
I~1 " 1~2] - e-moLt~r} . (31)
The first term, corresponding to the majoron, is zero for an S-wave state, but the second term remains. This is a Yukawa well which produces a w-bound state a neutrinium - whose radius R and binding energy B are roughly [10] -
Fig. 1. Relevant vertices of the new light particles, the majoron x and the neutral Higgs OL, with matter (f stands for u, d or e).
B ~ 10-2a2moL also independent of any adjustable parameter. We compare these cross sections and the usual one for the electron-v scattering in the approximation E v ~ me. Thus, ln(ECM/mpL)2 >~ 12 and
R = l / % / m v B "~ 10/aM mV'mvmpL .
2 2 2 o(/.,e) x-exchange "" o(v e)o L_exchange -- G F m e m v/Ev ,
B ~ 1 0 - 2 m v ~ 10 - 2 e V ,
(28)
while the contribution due to W and Z exchange is o(ve)w,Z.exchange ~--(G2/37r)meEv(5 sin 2 0 w + 1). (29) Taking the incident neutrino energy E v of order MeV as a typical value in present experiments and m v of some eV we obtain:
o(ve)×.exchange ~-- o(ve)pL.exchange 10-11 X o(ve)w,Z.exchang e ,
(30)
where the combination of constants has been assumed indicatively to be of order 1. Therefore X and PL exchanges in neutrino-electron scattering are n o t detectable at present laboratory energies. However, they could become dominant at threshold. We wish to point out that the relation tT(t.,Ox.exchange ~ o(pf)OL.exchang e still holds true when m o L is changed by various orders 414
(32)
Taking indicatively rnpL ~--my~aM, thus toOL ~ m v for a M = 1 then R~
lO/m v ~ . l O g f m .
(33) As we see, even for a wide range m v <~mOL <~ 106m v ,4, the neutrinium radius turns out to be 10-7 ~
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G2 2 2 mumf ......... X 4 X 1016 ~ 10-18/a 2 ,< 10 - 6 rr~2(mu/1 eV) 2 (34) therefore a M > 10 - 6 , which is consistent with our assumptions. As long as one generation is considered there are no physical consequences of the majoron and the scalar PL exchange at present laboratory energies (even if lhere could be important astrophysical implications). However, when considering various generations our model provides interesting novel features. The neutrino masses have no connection with the masses of the other members of the respective generations. This is a remarkable feature which differentiates this model from most of those which have appeared so far and could provide an appealing description if neutrinos turn out to have almost equal masses, not following the familar hierarchy. If, on the other hand, uu and v r are indeed heavier than ~'e, our model predicts they would decay rapidly through X and possibly PL emission. We plan to come back to these questions as well as on the astrophysical consequences in a future work. We wish to express our deep gratitude to R.D. Peccei and Y. Chikashige for a number of conversations which gave rise to this paper and for many helpfill suggestions and discussions. Also we thank R.N. Mohapatra and A. Masiero for their comments. One of us (M.R.) would like to thank the Maximilianeum Stiftung for the support given during the early stage of this work and finally we would like to express our thanks to the Theory Group of the Max-PlanckInstitute for their hospitality.
5 March 1981
Refere*wes [11 F. Reines, H.W. Sobel and E. Pasierb, Phys. Rev. Lett. 45 (1980) 1307. {2] V.A. Lyubimov, E.G. Novikov, V.Z. Nozik, E.F. Tretyakov and V.S. Kosik, Phys. Lett. 94B (1980) 266. I31 For a review, see: A. De Rfijula and S. Glashow, Nature 286 (1980) 755; an up-to-date discussion is contained in Nature 287 (1980) 481. [4] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, Proc. 8th Nobel Syrup., ed, N. Svartholm (Stockholm, 1968); S. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285. [51 For example, see: S. Weinberg, Phys. Rev. D13 (1976) 974; D19 (1979) 1277; L. Susskind, Phys. Rev. D20 (1979) 2619; E. Eichten and K. Lane, Phys. Lett. 90B (1980) 125 M. Beg, H. Politzer and P. Ramond, Phys. Rev. Lett. 43 (1979) 1701. [61 R,E. Marshak, R.N. Mohapatra and Riazuddin, VPIpreprint (1980); R. Barbieri, D.V. Nanopoulos, G. Morchio and F. Strocchi, Phys. Lett. 90B (1980)91; T.P. Cheng and L.F. Li, COO-3066-152 preprint (June 1980). [71 Y. Chikashige, R.N. Mohapatra and R.D. Peccei, MPIPAE/PTh 36/80 (September 1980); MPI-PAE/PTh40/80 (October 1980). [8] M. Veltman, LAPP-TH-12preprint (January 1980). [91 Particle Data Group, Review of Particle Properties, Rev. Mod. Phys. (April 1980). [10] J.M. Blatt and V. Weisskopf, Theoretical nuclear physics (Wiley, New York, 1963). [11] L. Resnick, M.K. Sundaresan and P.J.S. Watson, Phys. Rev. D8 (1973) 172. [121 G. Feinberg and J. Sucher, Phys. Rev. 20 (1980) 1717.
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