We investigate the possibility of mixing occurring between the families and the conjugate families of SO(n) grand unified theories. Such a mixing alters the V-A structure of the usual charged weak currents. By comparing with the data on muon and pion decays, we set limits on the corresponding mixing angles. We consider the separate cases corresponding to the conjugate neutrinos being either light or heavy

Nandi, S.

Stern, A.

Sudarshan, E. C. G.

English

Publications of the IAS Fellows

* Fermi National Accelerator Laboratory 
FERMILAB-Pub-81/53-THY 
July 1981 
PHENOMENOLOGICAL IMPLICATIONS OF THE EXISTENCE 
OF CONJUGATE FAMILIES ON THE V-A STRUCTURE OF WEAK INTERACTIONS 
S. Nandi+ 
Fermi National Accelerator Laboratory, Batavia, Illinois 60510 
and 
Center for Particle Theory 
University of Texas, Austin, Texas 78712 
and 
A. Stern and E. C. G. Sudarshan 
Center for Particle Theory 
University of Texas, Austin, Texas 78712 
*Summer visitor at Fermilab 
Operated by Unlverrities Research Association Inc. under contract with the United States Department of Energy 
FEKMILAB-Pub-81/53-THY 
PHENOMENOLOCICAL IMPLICATIONS OF THE EXISTENCE 
OF CONJUGATE FAMILIES ON THE V-A STRUCTURE OF WEAK INTERACTIONS 
S. Nandi 
Fermi National Accelerator Laboratory, Batavia, Illinois 60510 
and 
Center for Particle Theory 
University of Texas, Austin, Texas 78712 
and 
A. Stern and E. C. G. Sudarshan 
Center for Particle Theory 
University of Texas, Austin, Texas 78712 
ABSTRACT 
We investigate the possibility of mixing occurring between 
the families and the conjugate families of SO(n) Grand Unified 
Theories. Such a mixing alters the V-A structure of the usual 
charged weak currents. By comparing with the data on muon and 
pion decays, we set limits on the corresponding mixing angles. 
We consider cases where the conjugate neutrinos are either light 
or heavy. 
2 FERMILAB-Pub-81/53-THY 
There have been many proposals recently112 for SO(n), n > 10 
models which can incorporate many families into a single irreduc- 
ible spinor representation of the group. For such models, it is 
known that for every family contained in the representation, 
there exists a corresponding conjugate family. Conjugate fami- 
lies are identical to ordinary families except that they have V+A 
weak interactions.lm3 For example, in the SO(l1) or SO(12) 
models, there exists one family and one conjugate family. So in 
addition to the leptons e and ve, we have E and N, (having 
identical electric charges as e and ve, respectively) which are 
members of the conjugate family. The corresponding charged weak 
current has the form 
J 
a 
= coy (1 - Y )vO 
a 5 e 
+ E"va(l + Y 
5 
)Ni, (1) 
where the superscript "0" denotes "current eigenstates" in terms 
of which the currents assume the simplest form. Upon enlarging 
the group to SO(13) or SO(14), one obtains two families, which we 
can identify as the electron and muon families, and two conjugate 
families. In this case, analogous muon currents must be added to 
(1). 
The charged fermions of the conjugate families are assumed 
to be heavy to be in agreement with observation. However, the 
neutral leptons (e.g., II,) can be light or heavy. In this 
Letter, we study the effects on the low energy weak interaction 
phenomena due to the presence of conjugate families (more pre- 
cisely, due to the mixing of conjugate families with ordinary 
families). We shall consider the separate cases in which the 
3 FERMILAB-Pub-81/53-TRY 
neutral leptons (in the weak doublet) belonging to the conjugate 
families are either light or heavy. 
After spontaneous symmetry breaking, the fermions in these 
theories acquire masses via Yukawa interactions, and, in general, 
the "mass eigenstates" need not be identical to the "current 
eigenstates." Consequently, a mixing can occur amongst particles 
of the same charge.4 Assuming, for simplicity, a real mass 
matrix, we have for the case of the SO(l1) or SO(12) models 
e" = e cos $ + E sin Be, E" = -e sin 0 e + E CDS Be, 
(2) 
YO = " e e cos @e f Ne sin +e, N," = --v e sin be + Ne cos + e' 
where no superscript "0" denotes "mass eigenstates." Substitut- 
ing (2) into (1) we find 
J a = Ja(eve) + Ja(eNe) + Ja(Eue) + Ja(ENe), (3) 
where, for example, 
Ja(eVe) = cos ( Be - 0,) GY,Ve - cos 
Ja(eNe) = -sin(ae - 0,) SyuNe - sin 
( ‘e + 0,) er Y " I a 5e 
(4) 
tee + @,) ~Y~Y~N~. 
Analogous expressions for the muon family and its conjugate 
family must be included in (3) upon considering groups such as 
SO(13) or SO(14). For the purposes of this Letter, we shall 
ignore the Cabibbo-type mixing among the families and only con- 
sider the mixing between a family and its corresponding conjugate 
family. 
Upon setting all the angles in (4) equal to zero, we recover 
the left chiral V-A charged weak current.5J6 In general, 
4 FERMILAB-Pub-81/53-THY 
however, the angles need not all vanish. We obtain below con- 
straints on the angles 8,, $e, and a,,, #II (the corresponding muon 
and muon neutrino mixing angles) by examining the data on (A) 
Muon Decay and (B) Pion Decay. 
A. Muon Decay. The differential decay distribution for 
muon decay is given by7 
dr = 
musine de p,E,dE, 
48x3 
3(W - Ee) + 2p (; Ee - W - ; $ ) 
+ 3 ? n (W - Ee) - 5 cos$W-E,) + 2,($ Ee- W - ;e$$5) 
where 0 is the angle between the electron momentum p, and the 
muon spin direction, E, is the electron energy, K is defined in 
terms of the Fermi coupling constant, and W = (tni + mz)/2mV. In 
the derivation of (5) it has been assumed that the neutrino mass 
is negligible. The parameters p, 5, and 6 will be given below in 
terms of the mixing angles Be, a,,, Q,, and $u. For our purposes, 
II = 0. 
B. Pion Decay. The ratio of the widths T(r + e)/r(r + M) 
is given by 
r(n + e)/r(n + u) = (me/mu)2[(n~ - m,2)/(m; - 10;)]2 (1 + r)tc, (6) 
where the radiative correction r to lowest order is apg 
r = (3alx) ~nbe/mu), and K will be given below in terms of the 
mixing angles Be, a,,, $e, and $J~. 
5 FERMILAB-Pub-81/53-THY 
In computing the parameters in A and B we shall consider the 
following three possibilities for .the neutrino masses: (i) mN , 
e 
"N u 
<< me, (ii) mN 
e 
<< me, mN > mU, and (iii) mN , mN 
u e u 
> mn. 
(i) mNeB mNu << me. Here the effective four-fermi inter- 
action Hamiltonian for muon decay is 
2 [J”( eve) + Ja(eNe)l[J~(uu,) + J~(uN,,)l (7) 
where the currents involving the electron are given in (4). From 
(7) we find 
P = (3/8)[1 + cos 28 e ~0.3 2eul, 5 = 2 cos 2ee - cos 2eu, 
(8) 
6 = (3/B) (COS 28 e + ~0s 2eu)/(2 cos 2ee - cos 2eu). 
The interaction Hamiltonian relevant for pion decay is 
3 Jh 
J2 a C 
Ja'(eve) + JaC(eNe) + Jac(~vn) + Ja+(nNU-j, (9) 
where Jh, is the hadronic weak interaction current. From (9) we 
find K = 1. 
The prediction for the chiral V-A theory is recovered when 
ee = eu = 0. We note that the neutrino mixing angles 6, and +,, 
do not appear in (8). Therefore they may be arbitrarily large 
and still consistent with the data on muon and pion decay. Note 
also that 0 < P C 314. 
Good agreement of the data [cf. Table III0 with the values 
given by V-A theory restrict the angles Be and a,, to be small. 
6 FERMILAB-Pub-81/53-THY 
Upon expanding (8) around a, = av = 0 up to second order, we 
find, p = (3/4)[1 - (ei + et)], E, = 1 - 48: + 2eE, and 6 =(3/4)[1 
+ 3(e; - e:)]. Allowing up to two standard deviations in the 
data for p, 5, and 6, we find 1 Be] < 0.07, and ] au] < 0.06. 
(ii) "N e 
<< me, mN 
v 
> m,,. Here the current Ju(pN,,) no 
longer plays an important role in the muon and pion decay 
processes. Upon removing this current from the expressions (7) 
and (9), we now obtain 
P = (3/8)[1 + 2oucos 2eel, 5 = 2[cos 2ee - uv] 
6 = (3/16)[(cos 2Be + 20~)/(cos 2ee - au)], 
I: = 2/[c0s2( 8 P - 4,) + c0s2(eu + o,)l. 
(10) 
where oi= [cos(ei+$) ~~s(e~-~~)l/[cos~(~~+ $i) + cos2(ai- $,)I, 
i = e,u. 
The prediction for the V-A theory is recovered when Be = a* 
= $lv= 0. Since +e does not appear in the expression (10) it may 
be arbitrarily large and still consistent with the data on muon 
and pion decay. Here -314 c p c 918 and K ) 1. 
Upon expanding (10) around Be = au = $u = 0 up to second 
order, P = (3/4)(1 - ez), 5 = 1 - 48:’ 6 = (3/4)(1 + 3ez), and 
K = 1 + e’, + 9;. Allowing up to two standard deviations in the 
data for p, 5, 6, and K we find 1 BeI < 0.07, and 1 a,,/, 1 $u] 
< 0.27. 
(iii) m Ne' mNll ' mv' Now JG(uNu) and Ja(eNe) no longer 
play an important role in the decay processes and must be removed 
from (7) and (9). This yields 
7 FERMILAB-Pub-81/53-THY 
P = (3/8)[1 + 4ueeJ, 5 = 2[2ue - 0~1, 
6 = (3/B)t(oe + o,,)/(20e - a,,)]. (11) 
I( = [c0s2(ee-+e) + c0s2(ee+~,)]/[c0s2(e~-~~) + cOs*(ep++u)]. 
From (ll), the prediction for the V-A theory is recovered 
when 0 = u = e P l/2. This relation along with K = 1 leads to the 
following possibilities: (a) ep = 8, = 0, I +,I = I +pI = =, 03) 
$e= $u= 0, I eel = I e,l = a, CC) eu = oe = 0, I +,I = I eel = aa 
and (d) +,, = Be = 0, I au] = I$,1 = a, where Q is an arbitrary 
(positive) angle. As in case (i) we again have 0 c p < 314. 
Good agreement of the data [cf. Table I] with the values 
given by V-A theory restrict at least two of the angles (a,,,e,), 
(O,,$e,), (e,,o,), or ($,.e,) to be small. Upon expanding (11) 
around (a), i.e, ee = eu = 0 with I+eI = I g,,l = G held fixed, up 
to second order we find, p = (3/4) [l - (e,' + et) tan2a], 5 
= 1 - 2(282 - aE)tan2., 6 = (3/4)[1 + 3(8: - e:)tan2,], and K = 1 
-(e; - e$(l - tan*a), (I f *n/2. Allowing up to two standard 
deviations in the data for p, 5, 6, and K, we find the following 
inequalities for Be, au, and Q: I eel ltan aI < 0.07, Ie,l ltan al 
C 0.06, and - 0.02 c .3* - a* G 0.07. !J e The analogous expressions 
corresponding to expanding around (b), (c), and (d) are obtained 
by interchanging a,,, $,, a,, be appropriately in the above 
inequalities. 
In conclusion, we have pointed out that mixing between par- 
ticles of a family with those of a conjugate family induces a 
V + A admixture into low energy weak interactions. In applying 
8 FERMILAB-Pub-81/53-THY 
this to pion and muon decays, we find that although the data is 
in very good agreement with the predictions of the chiral V-A 
theory, a certain amount of mixing can be tolerated [cf. Table 
II]. In fact some of the angles may be arbitrarily large. We 
have investigated three separate cases arising from whether the 
conjugate neutrinos are either light or heavy. Accurate measure- 
ments of the parameters P and k may help to distinguish the three 
cases. Finally, in some specific models" mixing angles are 
given by (mi/Mi)1'2, where mi corresponds to the mass of a 
particle in an ordinary family and Mi corresponds to the mass of 
the particle in the conjugate family with which it mixes. The 
bounds in Table II can then be used to give lower limits on the 
masses of some of the leptons in the conjugate families. For 
example, using I a,, < 0.06 [cf. Table II (case i)] and mu = 0.105 
GeV, we obtain 30 GeV as a lower bound on the mass of the 
conjugate muon. Consequently, searching for heavy leptons with 
V + A weak interaction may prove worthwhile. 
We wish to thank D. A. Dicus and A. Buras for helpful dis- 
cussions. S. Nandi wishes to thank the members of Theoretical 
Physics Department of Fermilab for the warm hospitality extended 
to him. This research was supported by the U. S. Department of 
Energy, Grant DE-AS05-76ER03992. 
NOTE ADDED: While this work was being completed we received a 
preprint by K. Enqvist, K. Mursula, J. Maalampi, and M. Roos, 
University of Helsinki report, where a similiar analysis is 
9 FERMILAB-Pub-81/53-THY 
carried out. Unlike us, they restrict their discussions to the 
case where both of the conjugate neutrinos are heavy [case (iii) 
in our paper]. 
10 FERMILAB-Pub-81/53-THY 
REFERENCES 
1. M. Gell-Mann, P. Ramond, and R. Slansky, Rev. Mod. Phys. 50, 
721 (1978). 
2. F. Wilczek and A. Zee, Princeton preprint (1979). 
3. R. N. Mohapatra and B. Sakita, Phys. Rev. D21, 1062 (1980). 
4. Within the context of the SO(12) model, a mixing between the 
family and the conjugate family is obtained by giving non- 
zero vacuum expectation values to the neutral, colorless 
components of a 495-plet (and a singlet) representation of 
Higgs particles. Examples of such components are 
+11> 12>9, 1~0’ $11, 12, 73 8’ +9, 10, 7, 8 
and 4i,j, 1,~ + 4i,j,3,4 
+ $i,j,5,6 where (i,j) = (11,12), (s,io), (7,8) and we are 
using the conventions of S. Nandi, A. Stern, and 
E. C. G. Sudarshan, Structure of Proton Decay in SO(n) Family 
Unification, University of Texas preprint (1981). 
5. E. C. G. Sudarshan and R. E. Marshak, "Proc. Padua-Venice 
Conf. on Mesons and Recently Discovered Particles" (1957) 9 
and Phys. Rev. 109, 1860 (1958); reprinted in P. K. Kabir, 
"Development of Weak Interaction Theory," pp. 118-28, Gordon 
& Breach (1963). 
6. R. P. Feynman and M. Gell-Mann, Phys. Rev. 109, 193 (1958). 
7. See for example, R. E. Marshak, Riazuddin and C. P. Ryan 
"Theory of Weak Interactions," Wiley-Interscience (1969). 
8. W. J. Marciano and A. Sirlin, Phys. Rev. Lett. 36, 1425 
(1976). 
9. T. Goldman and W. J. Wilson, Phys. Rev. D15, 709 (1977). 
10. Review of Particle Properties, Rev. Mod. Phys. 52 (1980) No. 
2, Part II. 
11 FERMILAB-Pub-81/53-THY 
11. In the ~context of Cabibbo-mixing see e.g., H. Fritzch and 
P. Minkowski, "Flavordynamics of Quarks and Leptons" Max- 
Plank Institut preprint (1980). 
12 FERMILAB-Pub-81/53-THY 
Table I. Experimental data for the muon decay parameters. 
P 0.7517 ?: 0.0026 
5 0.972 + 0.013 
6 0.7551 -+ 0.0085 
K 1.03 k 0.02 
13 FERMILAB-Pub-81/53-THY 
Table II. Estimated values for the upper bounds on the mix- 
ing angles ee, 8,,, $,, and +,, for case (i) and (ii). No simple 
bounds on the mixing angles emerge in case (iii). 
case (i) case (ii) 
I eel < 0.07 < 0.07 
I e,l < 0.06 < 0.27 
I +el undetermined undetermined 
I +uI undetermined < 0.27 


Phenomenological implications of the existence of conjugate families on the V-A structure of weak interactions

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Phenomenological implications of the existence of conjugate families on the V-A structure of weak interactions

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