underlying symmetry among the quark and lepton mixing angles

8/14/2019 Underlying Symmetry Among the Quark and Lepton Mixing Angles

1/8

JM-PH-2007-727/18/2007

Underlying symmetry among the quarkand lepton mixing angles

J. S. Markovitch

AbstractThe six quark and lepton mixing angles are generated with the aid of three related forms ofsymmetry, and three constants that derive from the fine structure constant.


2/8

I. Introduction

We demonstrate that the six quark and lepton mixing angles can be generated with

the aid of three related forms of symmetry, and three constants that derive from the fmestructure constant.

We begin by defining the leptonic mass mixing matrix conventionally, but without theusual phases [1]

(la)

where sij == sin(tpij) and cij == cos(tpij)' These CPu's represent the leptonic mixing angles.Additionally, we note that

(lb)

We define the quark mass mixing matrix in the same way, but this time we deviate fromconvention both by omitting its phase, and by swapping its c- and t-quarks [2]. Accordingly, welet

2


3/8

"

(2a)

where si j == sin(Oij) and ei j == cos(Oij) ' These By's represent the quark mixing angles.Additionally, we note that

(2b)

which is consistent with the earlier-mentioned swap of the c- and t-quarks,We also define

and

[U;,L= U ~ l

U;l

[ V " ~Q=Vc!

U;2U ~ 2 U;2

r:;

U;3]U ~ 3 U;3

V ~ Vc!

(3a)

(3b)

in order to impose the Principal Symmetry Condition

L LT =N x (Q _QT) (4)

3


4/8

.

on the quark and lepton mixing angles. Note thatN, which represents the number of quarkcolors, equals 3.

In order to fulfill this condition we now define the following parametric equations.

U J2 = ~ M L N Un = ~ M s V12 = ~ M L

These in tum generate the quark and lepton mixing angles as follows

CfJl3 = sin -J (Ul3 x s i n ( e ~ 3 ) )

where

(5a)

F or later use we also define

(5b)

II. Generating the quark and lepton mixing anglesNow, by letting

1M=-L 10 '

4

(6a)


5/8

1 (6b)Ms = 30000 ',and,

(6c)tp23 =4Y ,

and by requiring that the principal symmetry condition be fulfilled, we generate the followinggood approximations of the quark and lepton mixing angles.

tpI2 =33.21 r tpl3 =0.014 tp23 =45023 =92.36T

The reader will note that the high value for ()23 (about 92.3674418) is a result of the earlier swapof the c- and t-quarks, and that subtracting 90 from ()23 produces this angle expressed in theusual way.

III. The primary symmetry's two subordinate symmetriesIt is important to realize that the parametric equations embody two additional

symmetries. To be specific, if

(7a)

then any values chosen for

Ms, N, and B ~ 3

5


6/8

will produce values for the remaining mixing angles tp13 and 013 that are guaranteed to fulfill theprincipal symmetry condition.

Equally, if

then any values chosen for

(7b)

will produce values-in this case for tpl2 and Ol2-that are guaranteed to fulfill the principalsymmetry condition.

These two Subordinate Symmetries explain why the parametric equations take the formthey do: they are designed to guarantee fulfillment of these two additional symmetries. Hencethe form of the parametric equations cannot be regarded as a degree of freedom exploited to fit

the mixing data.

IV. The fine structure constant

It is noteworthy that the values for ML and Msthat fit the mixingdata, also fix U12, U13,V12, and V13 in such a way as to allow them act in concert to produce this close approximation ofthe fine structure constant inverse

(U12-2 -V l 3)3 (-2 -U 1322 = 137.0360000023 . . . . (8) V12

6


7/8

Remarkably, this equation reproduces the fine structure constant inverse to within about twoparts per billion of its corresponding 2006 CODATA value of 137.035999679 (94) [3].

Moreover, if one regards the integers 1 / ML and 1 / Ms as chosen to fit 137.036, then thequark and lepton mixing angles generated earlier may be said to arise from equations exploitingjust one free parameter: tp23 ; which is to say the six quark and lepton mixing angles are fullydetermined by the two fine-structure-related integers (1 / ML and 1 /Ms), the three symmetriesalready explained (the Principal and two Subordinate), and tp23 = 45" .

However, one can carry this one step further. IfEq. (8) is restated purely in terms ofmixing angles

then it is only the equivalence

. ( ,)2 . ( )2SI n tp23 =SIn tp23 (10)

which allows it to be rewritten still more symmetrically as

Accordingly, tp23 =45" is not a truly free parameter, but precisely that value needed to make thismore symmetrical equation possible. It follows that the earlier parametric equations generate the

7


8/8

six quark and lepton mixing angles while exploiting not one free parameter, but essentially nofree parameters.

References

[1] B. Kayser, "Neutrino Mass, Mixing, and Flavor Change," in Yao, W. M., et al. (ParticleData Group), J. Phys. G 33, 1 (2006).

[2] A. Ceccucci, Z. Ligeti, and Y. Sakai, "The CKM Quark-Mixing Matrix," in Yao, W. M., etal. (Particle Data Group), J. Phys. G 33, 1 (2006).

[3] P.J. Mohr, B.N. Taylor, and D.B. Newell (2007), "The 2006 CODATA RecommendedValues of the Fundamental Physical Constants" (Web Version 5.0). This database wasdeveloped by J. Baker, M. Douma, and S. Kotochigova. Available:http://physics.nist.gov/constants [2007, July 12]. National Institute of Standards andTechnology, Gaithersburg, MD 20899.

8
http://physics.nist.gov/constantshttp://physics.nist.gov/constants

underlying symmetry among the quark and lepton mixing angles

Documents