On Neutrinos and Fermionic Mass Patterns

Paul M. Fishbane
Physics Dept. and Institute for Nuclear and Particle Physics,
Univ. of Virginia, Charlottesville, VA 22903




Peter Kaus
Physics Dept., Univ. of California, Riverside, CA 92521




Abstract


Recent data on neutrino mass differences is consistent with
a hierarchical neutrino mass structure strikingly similar to
what is observed for the other fermionic masses.





1


Recent experiments [1] have provided quantitative results on differences of the
neutrino masses squared,

2 2 2 2
 = m - m and e = m - me . (1a)


These quantities are given numerically by

-3 -6
 = (2  1)  10 eV2 and e = (6  1.5)  10 eV2. (1b)


This data appears to be consistent in the context of three neutrinos with other
experimental information available [2, 3], if not with all of it [4; to accomodate all the
data, one requires additional neutrino(s)see ref. 5].
In thinking about how to use this data to best advantage, we can start with the
conservative assumption that neutrinos come in three families, with no additional species,
sterile or otherwise. Of course the two pieces of data are inadequate to give the three
neutrino masses. We can get further guidance on how to proceed by looking at the other
fermion sectors, both charged leptons and quarks. There is one obvious common feature
of the different sectors, and that is that the masses have a hierarchical structure; that is, m1
<< m2 << m3, where m1 refers respectively to the d-quark, u-quark, and electron in the
down-quark, up-quark, and charged lepton sectors. When the quark masses are run within
the minimal supersymmetric model to unification scales [6] the mass ratios are:


down-quark sector: ms/mb = 0.034, md/ms = 0.049
up-quark sector: mc/mt = 0.0035, mu/mc = 0.0038 (2a)
charged lepton sector: m/m = 0.059, me/m = 0.0048

Not only do the charged fermion sectors have a hierarchical mass structure, but it
is also true that this structure can be expressed as powers of a single parameter , the sine
of the Cabibbo angle, or the Wolfenstein parameter[7], roughly 0.22. Powers of this
quantity appear throughout the CKM matrix with coefficients of order 1, and the same is
true for the mass ratios, which can be written up to coefficients of order 1 as


down-quark sector: mb:ms:md = 1:2:4
up-quark sector: mt:mc:mu = 1:4:8 (2b)
charged lepton sector: m:m:me = 1:2:6

This fact can be quantified in several ways. For example, we can define according to
these ratios and see to what extent the same values of appear. In this way we find





2


           
 
1/ 4 1/ 2 1/ 2
 m
 m
 m

d s d
0 202
. ; 0 184
. ; 0 220
.
db sb ds
m m m
b b s

     
     
1/8 1/ 4 1/ 4
m m m
u c u
= 0 245
. ; = 0 243
. ; = 0 248
. (3)
ut ct uc
m m m
t t c

 



  
      /
   


 
 
1/ 6 1/ 2 1 4
 m m
  m

e e
0 257
. ; 0 244
. ; 0 264
. .
 
e e
m m m
 











The consistent presence of powers of is striking. Even if the particular power of is
different in each case, those powers appear in simply related form. This of course may
only be numerical coincidence, a remark that would be true for the CKM matrix as well. If
it is, we shall find below that the coincidence has been extended to the neutrino sector.
As for absolute values, the charged lepton masses do not appreciably run, so that
we can use the value m = 1.78 GeV. The very stable grand unification relation mb = m
gives us a convenient route to the down quark sector. However, the absolute scale for the
up sector cannot be known reliably, because the top quark mass is at the infrared fixed
point of the theory and is therefore insensitive to an initial value at ultraviolet scales; when
below we have occasion to refer to the top quark mass we use 300 GeV.
We most easily check to what extent the neutrino masses can fit into this picture
by defining the hierarchy parameter R as follows:


me/m = R m/m (4)


Of course, R only parameterizes solutions to Eq. (1a), but it helps to define what we mean
by a "hierarchical solution." A solution to Eqs. (1a) and (4) is hierarchical when R is very
close to 1 or when it is close to a power of a (small) mass ratio, here m /m. We see
from Eq. (2) that the analog to R for both the down and up quarks is very close to unity
(1.43 and 1.09, respectively). The charged leptons are associated with an analog to R of
O(2) = O(m /m).
Is one of the (quadratic) solutions for the square of the masses hierarchical? The
answer is yes. One branch not only gives a small value of m /m but also permits both R
= 1 and R = m /m as well as R = 0 (leading to a massless electron neutrino). Moreover,
for this branch the ratio m /m is nearly independent of R. This is the branch that
interests us. (See Ref. [8] for a the second, nonhierarchical, solution.) The solution on the
branch of interest is





3


1/ 2
- 2
4R2 + + - +
e  e  e

m = (5a)

2 1 - R2


m = + 2
(5b)
 


R2
m 
= (5c)
e + 2
 


with an appropriate continuation for R > 1.






1 +

For the whole range of R that is allowed for this branch 0 < R < e ,
2 e
the ratio m /m is indeed insensitive to R. This is evident in the solution at some special
values. The case R = 1, which the quark sector suggests, gives

0





0





!
' ( ) ;
" # $





0 0

%





# $
e

# $ $





0 0





e 0
!
' ( ) ; (6)
# $ $


"
0 0 e
&





$
e

# $ $





0 0





e e
!



' ( )





" $ $





0 0 0
e
e

# $ $ # $





The hierarchical nature of these forms is clear when we recall that  >> e. We have
indeed used this to get the approximate forms. The other interesting special case is R =
m /m, (i.e., me /m = (m /m)2, as in the charged lepton sector). This case gives the
approximate result


m ()1/2; m (e)1/2; me (e)3/2/. (7)


Here we have used  >> e to approximate an otherwise opaque result. These and 
masses are roughly the same as the R = 1 case, Eq. (6).
Since in this solution the and  masses are given nearly independently of R by
the data of Ref. [1], so is their mass ratio,


m /m (e)1/2/()1/2 = 0.055  0.014. (8)


Here we have taken the numerical values of Eq. (1b) with errors treated by quadrature.





4


This numerical value is very close to the ratio of the two heaviest masses in the
down and charged lepton sectors. If, as those sectors suggest, we define a quantity
through m 2
 /m = , then

= 0.234  0.029. (9)


This value spans the -values defined by the other three sectors, as Fig. 1 illustrates.
We conclude our discussion with some remarks about the masses themselves. We
have stated that the hierarchical solution determines the masses of the - and -neutrinos
themselves; to a good approximation these are given by Eq. (6), and imply the numerical
values

m -2 -3
= (4.5  1.1)  10 eV and m = (2.5  0.3)  10 eV. (10)

-
The m 2
result can be compared to two recent O(10 eV) predictions [9], both of which
extract the scale for the neutrino masses from the unification scale for a supersymmetric
grand unified theory.
The value of me depends of course on R. If the neutrino sector follows the down-
quark pattern, then its mass is 2
times m. If it follows the charged lepton sector then its
mass is 4
times m. We can refer to these possibilities as case a and case b respectively.
In Fig. 2 we plot the logarithms of these masses as a function of family, together with the
same quantity for the other three sectors. This graph reveals just how different the
neutrino sector must be from the other three sectors in absolute value and how similar all
the sectors could be in slopes. The difference may be easier to understand than the
similarity. Even within a purely standard model view of neutrinosthree sets of right-
handed singlets and left-handed doubletsthey differ from the other fermions in that they
have no electrical charge and can get mass either by a Dirac term alone or by Dirac and
Majorana terms that combine through a seesaw mechanism. The appearance of a common
value of in this context appears to us to be remarkable.
We refer the reader elsewhere [8; see also 10] for a discussion of the leptonic
analog to the CKM matrix, as well as of models that could shed light on the results
described here.


Acknowledgments


We thank P. Ramond for many useful conversations, Darrel Smith for much assistance and
the Aspen Center for Physics for its hospitality. PMF is supported in part by the U.S.
Department of Energy under grant number DE-AS05-89ER40518.





5


References



1. Super-Kamiokande collaboration, Y. Fukuda et al.,  and Phys. Lett. B,
to be published; E. Kearns, ; Y. Fukuda et al., v2.


2. Kamiokande collaboration, K. S. Hirata et al., Phys. Lett. B280, 146 (1992); Y. Fukuda
et al., Phys. Lett. B335, 237 (1994); IMB collaboration, R. Becker-Szendy et al., Nucl.
Phys. B (Proc. Suppl.) 38, 331 (1995); Soudan-2 collaboration, W. W. M. Allison et al.,
Phys. Lett. B391, 491 (1997); MACRO collaboration, M. Ambrosio et al., hep-
; CHOOZ collaboration, M. Apollonio et al., Phys. Lett. B420, 397 (1998).


3. J. N. Bahcall and M. H. Pinsonneault, Rev. Mod. Phys. 67, 781 (1995); J. N. Bahcall,
S. Basu, and M. H. Pinsonneault, ; B. T. Cleveland et al., Nucl. Phys. B
(Proc. Suppl.) 38, 47 (1995); Kamiokande collaboration, Y. Fukuda et al., Phys. Rev.
Lett. 77, 1683 (1996); GALLEX collaboration, W. Hampel et al., Phys. Lett. B388, 384
(1996); SAGE collaboration, J. N. Abdurashitov et al., Phys. Rev. Lett. 77, 4708 (1996).


4. Liquid Scintillator Neutrino Detector (LSND) collaboration, C. Athanassopoulos et al.,
Phys. Rev. Lett. 75, 2650 (1996); ibid., 77, 3082 (1996); .


5. V. Barger, S. Pakvasa, T. J. Weiler, and K. Whisnant, .


6. S. Dimopoulos, L. J. Hall, and S. Raby, Phys. Rev. D 45 (1992) 4195; H. Arason, D. J.
Castano, P. Ramond, and E. J. Piard, Phys. Rev. D 47 (1992) 232; P. Ramond, R. G.
Roberts, and G. G. Ross, Proceedings of Orbis Scientiae, Jan 1993, Coral Gables.


7. N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963); L. Wolfenstein, Phys. Rev. Lett. 51, 1945
(1983).


8. P. M. Fishbane and P. Kaus .


9. F. Wilczek, ; G. Altarelli, .


10. J. K. Elwood, N. Irges, and P. Ramond, ; J. K. Elwood, N. Irges, and
P. Ramond, Phys. Lett. B413, 322 (1997); N. Irges, S. Lavignac, and P. Ramond, hep-
 to be published in Phys. Rev. D.





6


0.3

0.25
u lepton

0.2 neutrino

d
0.15


0.1


0.05


0
1
Sector

Figure 1


The values of extracted from mass ratios according to
Eqs. (3) and (9).





7


12
12 8
Log (M)
10 t


10
10
@
7 c b, A
C 
55 8
8
9 s
6 u, d
44 6
6 B e


33 4
4


2
2

F

Generation
22 0
0
E
1 2 3

D


--2
2
D 
--4
4 a
D e
b
--6
6


Figure 2


The logarithms of the fermion masses. Solid lines connect these
masses within a given sector, enabling a visual comparison of the
sect0ors. For the neutrinos, the labels a and b refer to a choice of
R = 1 and R = O(2), according to the text discussion.





8



