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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)
${\stackrel{‾}{m}}_{Z}=\frac{{A}_{0}}{\mathrm{sin}\stackrel{‾}{\theta }\mathrm{cos}\stackrel{‾}{\theta }}$

(2)
${\stackrel{‾}{m}}_{W}=\frac{{A}_{0}}{\mathrm{sin}\stackrel{‾}{\theta }}$

with (2)

(3)
${A}_{0}={\left(\frac{\pi \alpha }{\sqrt{2}{G}_{F}}\right)}^{1/2}=37.28057\left(8\right) \text{GeV}$

(4)
$\mathrm{sin}\stackrel{‾}{\theta }=\sqrt{\frac{3}{14}}.$

(The weak angle $\stackrel{‾}{\theta }$ in Eqs (1) and (2) is the complementary angle of $\theta$ defined in Ref (1): $\theta +\stackrel{‾}{\theta }=\frac{\pi }{2}$).

Then

(5) ${\stackrel{‾}{m}}_{Z}=90.85560\left(19\right) \text{GeV}$

(6) ${\stackrel{‾}{m}}_{W}=80.53524\left(17\right) \text{GeV}$

to be compared with their experimental values (2)

(7) ${m}_{Z}=91.1876\left(21\right) \text{GeV}$

(8) ${m}_{W}=80.398\left(25\right) \text{GeV}$

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

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

A.

(9)
${m}_{Z}={\stackrel{‾}{m}}_{Z}\left(1+\frac{\alpha }{2}\right)=91.18750\left(19\right) \text{GeV}$

(10)
${m}_{W}={\stackrel{‾}{m}}_{W}\left(1+\frac{\alpha }{2}{\right)}^{-1/2}=80.38871\left(17\right) \text{GeV}$

B.

(11)
${m}_{Z}={\stackrel{‾}{m}}_{Z}{\left(\frac{\mathrm{cos}\stackrel{‾}{\theta }}{\mathrm{cos}{\theta }_{W}}\right)}^{2/3}=91.18757\left(19\right) \text{GeV}$

(12)
${m}_{W}={\stackrel{‾}{m}}_{W}{\left(\frac{\mathrm{cos}{\theta }_{W}}{\mathrm{cos}\stackrel{‾}{\theta }}\right)}^{1/3}=80.38868\left(17\right) \text{GeV}$

with (3)

(13)
$\alpha =\frac{{e}^{2}}{4\pi }=\left[137.035999084\left(51{\right)\right]}^{-1}$

(14)
$\mathrm{cos}{\theta }_{W}\equiv \frac{{m}_{W}}{{m}_{Z}}.$

We used value of $\mathrm{cos}{\theta }_{W}$ obtained from the empirical relation (4)

(15)
$1-{\text{tan}}^{2}\left(\frac{\pi }{4}-{\theta }_{W}\right)=3e.$

Note that

(16)
$1-{\text{tan}}^{2}\left(\frac{\pi }{4}-{\theta }_{W}\right)=\frac{4\mathrm{sin}{\theta }_{W}\mathrm{cos}{\theta }_{W}}{\left(\mathrm{sin}{\theta }_{W}+\mathrm{cos}{\theta }_{W}{\right)}^{2}}.$

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

(17)
${m}_{Z}=\frac{{A}_{0}}{\mathrm{sin}{\theta }_{W}\mathrm{cos}{\theta }_{W}} \frac{1}{\left(1-\Delta r{\right)}^{1/2}}$

we write Eq. (11) as

(18)
${m}_{Z}=\frac{{A}_{0}}{\mathrm{sin}{\theta }_{W}\mathrm{cos}{\theta }_{W}}\left(\frac{\mathrm{sin}{\theta }_{W}}{\mathrm{sin}\stackrel{‾}{\theta }}\right){\left(\frac{\mathrm{cos}{\theta }_{W}}{\mathrm{cos}\stackrel{‾}{\theta }}\right)}^{1/3}.$

Then

(19)
$\frac{1}{\left(1-\Delta r{\right)}^{1/2}}=\left(\frac{\mathrm{sin}{\theta }_{W}}{\mathrm{sin}\stackrel{‾}{\theta }}\right){\left(\frac{\mathrm{cos}{\theta }_{W}}{\mathrm{cos}\stackrel{‾}{\theta }}\right)}^{1/3}$

in the current context.

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

$\begin{array}{cccc}\multicolumn{1}{c}{{m}_{Z}}& =\hfill & \frac{1}{\mathrm{sin}{\theta }_{W}+\mathrm{cos}{\theta }_{W}}{\left(\frac{\mathrm{cos}\stackrel{‾}{\theta }}{\mathrm{cos}{\theta }_{W}}\right)}^{23/48}\frac{{v}_{F}}{2}\hfill & \hfill \left(20\right)\\ \multicolumn{1}{c}{}& =\hfill & \frac{{A}_{0}}{\mathrm{sin}{\theta }_{W}\mathrm{cos}{\theta }_{W}}\frac{3}{4}\left(\mathrm{sin}{\theta }_{W}+\mathrm{cos}{\theta }_{W}\right){\left(\frac{\mathrm{cos}\stackrel{‾}{\theta }}{\mathrm{cos}{\theta }_{W}}\right)}^{23/48}\hfill & \hfill \left(21\right)\\ \multicolumn{1}{c}{}& \hfill & \hfill \\ \multicolumn{1}{c}{}& =\hfill & 91.18756\left(19\right) \text{GeV}\hfill & \hfill \left(22\right)\end{array}$

where we have used

(23)
${A}_{0}=\frac{e {v}_{F}}{2}$

and Eqs (15-16).

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

(24)
$\frac{1}{e}-e\left[1-\frac{\alpha }{4}-\left(\frac{\alpha }{4}{\right)}^{2}-x\left(\frac{\alpha }{4}{\right)}^{3}\right]=3$

when $x=0.430±0.365$.

For example,

$\begin{array}{ccc}\multicolumn{1}{c}{\text{if} x}& =\hfill & 0.75 , \text{then} {\alpha }^{-1}=137.035999039\hfill \\ \multicolumn{1}{c}{\text{if} x}& =\hfill & 0.50 , \text{then} {\alpha }^{-1}=137.035999074\hfill \\ \multicolumn{1}{c}{\text{if} x}& =\hfill & 0.25 , \text{then} {\alpha }^{-1}=137.035999109\hfill \\ \multicolumn{1}{c}{}\end{array}$

Comparing Eqs (15) and (24), we get

(25)
${\text{tan}}^{2}\left(\frac{\pi }{4}-{\theta }_{W}\right)={e}^{2}\left[1-\frac{\alpha }{4}-\left(\frac{\alpha }{4}{\right)}^{2}-x\left(\frac{\alpha }{4}{\right)}^{3}\right]$

(In Ref. (4), the following approximation of Eq. (25) is used: ${\mathrm{tan}}^{2}\left(\frac{\pi }{4}-{\theta }_{W}\right)={e}^{2}\right)$.

It is worthwhile to note that

$\frac{1}{e}-e\left[1-\frac{\alpha }{4}\mathrm{exp}\left(\frac{\alpha }{4}\right)\right]=3$

is satisfied when

${\alpha }^{-1}=137.035999074.$

# 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

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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