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Variations on the Gauge Sector of the Electroweak Model

Jean Pestieau1

Institut de physique théorique,
Université catholique de Louvain,
Belgium

11 August 2009

Abstract

Starting from a 40 year old proposal, new relations between alpha, the fine structure constant, Z and W masses are proposed.

 

I. Forty years ago (1), it has been proposed the following determination of Z and W masses


(1)
m Z = A0 sin θ cos θ


(2)
m W = A0 sin θ

with (2)



(3)
A0 = ( πα 2 GF ) 1/2 = 37.28057 (8) GeV


(4)
sin θ = 314 .

(The weak angle θ in Eqs (1) and (2) is the complementary angle of θ defined in Ref (1): θ + θ = π2 ).

Then

(5) m Z = 90.85560 (19) GeV

(6) m W = 80.53524 (17) GeV

to be compared with their experimental values (2)

(7) mZ = 91.1876 (21) GeV

(8) mW = 80.398 (25) GeV

Let us present variations on Eqs (1) and (2).

II. It is amusing to consider the following simple parametrizations:

A.


(9)
mZ = m Z (1+ α2 )=91.18750 (19) GeV


(10)
m W = m W (1+ α 2 ) -1/2 =80.38871(17) GeV

B.


(11)
mZ = m Z ( cos θ cos θW )2/3 =91.18757(19) GeV


(12)
mW = m W ( cos θW cos θ )1/3 =80.38868(17) GeV

with (3)


(13)
α= e2 4π =[137.035999084(51 )]-1


(14)
cos θW mW mZ .

We used value of cosθW obtained from the empirical relation (4)


(15)
1- tan 2 ( π 4 - θW )=3e.

Note that


(16)
1- tan 2 ( π 4 - θW )= 4sin θW cos θW (sin θW +cos θW )2 .

III. To make contact with a well known parametrization (2)


(17)
mZ = A0 sin θW cos θW 1 (1-Δr )1/2

we write Eq. (11) as


(18)
mZ = A0 sin θW cos θW ( sin θW sin θ ) ( cos θW cos θ )1/3 .

Then


(19)
1 (1-Δr )1/2 =( sin θW sin θ ) ( cos θW cos θ )1/3

in the current context.

IV. It is interesting to note the following empirical formula (4):

mZ = 1 sin θW + cos θW ( cos θ cos θW ) 23/48 vF 2         (20) = A0 sin θW cos θW 34 ( sin θW + cos θW ) ( cos θ cos θW ) 23/48         (21) = 91.18756 (19) GeV         (22)

where we have used


(23)
A0 = e vF 2

and Eqs (15-16).

V. About e and α

With α and e given in Eqs (13), we satisfy the following Equation (4)


(24)
1e -e [1- α4 - ( α4 )2 -x ( α4 )3 ] =3

when x=0.430 ±0.365 .

For example,

ifx = 0.75, then α-1 =137.035999039 ifx = 0.50, then α-1 =137.035999074 ifx = 0.25, then α-1 =137.035999109

Comparing Eqs (15) and (24), we get


(25)
tan2 ( π4 - θW ) = e2 [1- α4 - ( α4 )2 -x ( α4 )3 ]

(In Ref. (4), the following approximation of Eq. (25) is used: tan2 ( π4 - θW ) = e2 ) .

It is worthwhile to note that

1e -e[1- α4 exp ( α4 ) ]=3

is satisfied when

α-1 =137.035999074.

References


    1) J. Pestieau and P. Roy, Phys. Rev. Lett. 23, 349 (1969). See also, H. Terazawa, Phys. Lett. D4, 1579 (1971); J. Pestieau and P. Roy, Lett. Nuovo Cim. 31, 625 (1981); M. Veltman, http://www.lorentz.leidenuniv.nl/history/zeeman/lorentzveltman/Leiden2002lect.pdf (2002).
    2) Review of Particle Physics, C. Amsler, et al., Phys. Lett. B667, 1 (2008).
    3) D. Hanneke, S. Fogwell and G. Gabrielse, Phys. Rev. Lett., 100, 120801 (2008) ; T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio, Phys. Rev. D77, 053012 (2008).
    4) For earlier versions, see G. Lopez Castro and J. Pestieau, Mod. Phys. Lett. A22, 2909 (2007); hep-ph/9804272.

Footnote

1     jean.pestieau[*—at—*]uclouvain.be

 

File translated from the TEX file 40years.tex by TTM, version 3.85 on 15 Aug 2009, 01:23. Further edited with Bluefish 1.0.7.

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