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) ${\displaystyle {\bar{m}}_Z=\frac{A_0}{\sin \bar{\theta }\cos \bar{\theta }}}$

(2) ${\displaystyle {\bar{m}}_W=\frac{A_0}{\sin \bar{\theta }}}$

with ⁽²⁾

(3) ${\displaystyle A_0={\left(\frac{\pi \alpha }{\sqrt{2}{G}_F}\right)}^{1/2}=37.28057(8)\hspace{0.5em}\text{GeV}}$

(4) ${\displaystyle \sin \bar{\theta }=\sqrt{\frac{3}{14}}.}$

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

Then

(5) ${\displaystyle {\bar{m}}_Z=90.85560(19)\hspace{0.5em}\text{GeV}}$

(6) ${\displaystyle {\bar{m}}_W=80.53524(17)\hspace{0.5em}\text{GeV}}$

to be compared with their experimental values ⁽²⁾

(7) ${\displaystyle m_Z=91.1876(21)\hspace{0.5em}\text{GeV}}$

(8) ${\displaystyle m_W=80.398(25)\hspace{0.5em}\text{GeV}}$

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

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

A.

(9) ${\displaystyle m_Z={\bar{m}}_Z(1+\frac{\alpha }{2})=91.18750(19)\hspace{0.5em}\text{GeV}}$

(10) ${\displaystyle m_W={\bar{m}}_W(1+\frac{\alpha }{2}{)}^{-1/2}=80.38871(17)\hspace{0.5em}\text{GeV}}$

B.

(11) ${\displaystyle m_Z={\bar{m}}_Z{\left(\frac{\cos \bar{\theta }}{\cos {\theta }_W}\right)}^{2/3}=91.18757(19)\hspace{0.5em}\text{GeV}}$

(12) ${\displaystyle m_W={\bar{m}}_W{\left(\frac{\cos {\theta }_W}{\cos \bar{\theta }}\right)}^{1/3}=80.38868(17)\hspace{0.5em}\text{GeV}}$

with ⁽³⁾

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

(14) ${\displaystyle \cos {\theta }_W\equiv \frac{m_W}{m_Z}.}$

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

(15) ${\displaystyle 1-{\text{tan}}^2(\frac{\pi }{4}-{\theta }_W)=3e.}$

Note that

(16) ${\displaystyle 1-{\text{tan}}^2(\frac{\pi }{4}-{\theta }_W)=\frac{4\sin {\theta }_W\cos {\theta }_W}{(\sin {\theta }_W+\cos {\theta }_W{)}^2}.}$

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

(17) ${\displaystyle m_Z=\frac{A_0}{\sin {\theta }_W\cos {\theta }_W}\hspace{0.5em}\frac{1}{(1-\Delta r{)}^{1/2}}}$

we write Eq. (11) as

(18) ${\displaystyle m_Z=\frac{A_0}{\sin {\theta }_W\cos {\theta }_W}\left(\frac{\sin {\theta }_W}{\sin \bar{\theta }}\right){\left(\frac{\cos {\theta }_W}{\cos \bar{\theta }}\right)}^{1/3}.}$

Then

(19) ${\displaystyle \frac{1}{(1-\Delta r{)}^{1/2}}=\left(\frac{\sin {\theta }_W}{\sin \bar{\theta }}\right){\left(\frac{\cos {\theta }_W}{\cos \bar{\theta }}\right)}^{1/3}}$

in the current context.

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

${\displaystyle \begin{array}{cccc}\multicolumn{1}{c}{m_Z}& =\hfill & \frac{1}{\sin {\theta }_W+\cos {\theta }_W}{\left(\frac{\cos \bar{\theta }}{\cos {\theta }_W}\right)}^{23/48}\frac{v_F}{2}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(20)\\ \multicolumn{1}{c}{}& =\hfill & \frac{A_0}{\sin {\theta }_W\cos {\theta }_W}\frac{3}{4}(\sin {\theta }_W+\cos {\theta }_W){\left(\frac{\cos \bar{\theta }}{\cos {\theta }_W}\right)}^{23/48}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(21)\\ \multicolumn{1}{c}{}& \hfill & \hfill \\ \multicolumn{1}{c}{}& =\hfill & 91.18756(19)\hspace{0.5em}\text{GeV}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(22)\end{array}}$

where we have used

(23) ${\displaystyle A_0=\frac{e\hspace{0.5em}{v}_F}{2}}$

and Eqs (15-16).

V. About e and α

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

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

when $x=0.430\pm 0.365$.

For example,

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{\text{if}\hspace{0.5em}\hspace{0.5em}x}& =\hfill & 0.75\hspace{0.5em},\hspace{0.5em}\text{then}\hspace{0.5em}{\alpha }^{-1}=137.035999039\hfill \\ \multicolumn{1}{c}{\text{if}\hspace{0.5em}\hspace{0.5em}x}& =\hfill & 0.50\hspace{0.5em},\hspace{0.5em}\text{then}\hspace{0.5em}{\alpha }^{-1}=137.035999074\hfill \\ \multicolumn{1}{c}{\text{if}\hspace{0.5em}\hspace{0.5em}x}& =\hfill & 0.25\hspace{0.5em},\hspace{0.5em}\text{then}\hspace{0.5em}{\alpha }^{-1}=137.035999109\hfill \\ \multicolumn{1}{c}{}\end{array}}$

Comparing Eqs (15) and (24), we get

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

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

It is worthwhile to note that

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

is satisfied when

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

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.

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Footnote

${}^1$    jean.pestieau[*—at—*]uclouvain.be