%Paper: 
%From: Gabriel Lopez Castro <Gabriel.Lopez@fis.cinvestav.mx>
%Date: Thu, 20 Apr 1995 15:04:04 -0600 (CST)


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\title{{\bf Determination of the Higgs Boson Mass by the Cancellation of
Ultraviolet
Divergences in the $SU(2)_L \otimes U(1)$ Theory}}

\author{Gabriel L\'opez Castro \\
Departamento de F\'\i sica,
CINVESTAV del IPN \\
Apartado Postal 14-740, M\'exico  07000, D.F., M\'exico\and
Jean Pestieau \\
Institut de Physique Th\'eorique, Universit\'e catholique de Louvain \\
Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium}



\date{}

\begin{document}


\thispagestyle{empty}
\maketitle



\begin{center}
{\large In memory of Roger Decker}
\end{center}

\vspace*{5mm}

\begin{abstract}



We assume the vanishing of the quadratic divergences  in the
$SU(2)_L \otimes U(1)$ electroweak theory. Using $\,$ the $\,$ top $\,$
 mass $\,$ value reported recently
by the CDF Collaboration $m_t = 176 \pm 8 \pm 10 GeV$, we
predict the  mass of the Higgs boson to be
$m_H = 321 \pm 29 GeV$.
If we assume the vanishing of both quadratic and loga-\-rithmic
divergencies of the top self-mass,
we predict $m_t = 170.5 \pm 0.3 GeV $ and $m_H = 308.6 \pm 0.7 GeV.$
\end{abstract}


\newpage
\par\noindent
{\bf 1.} In the $SU(2)_L \otimes U(1)$ theory, quadratic divergences, at
the one loop level, are
universal (i.e. they are the same for all physical quantities). We assume
they vanish \cite{D}-\cite{O}, namely

\vspace*{2mm}
$$
m^2_e + m^2_\mu + m^2_\tau + 3(m^2_u + m^2_d + m^2_c + m^2_s + m^2_t + m^2_b) =
\frac{3}{2} m^2_W + \frac{3}{4} m^2_Z + \frac{3}{4} m^2_H
\eqno{(1)}
$$

\vspace*{5mm}
\par\noindent
In the following we shall neglect the fermion masses other than $m_t$ and
$m_b$.\\
{}From the recent discovery \cite{A} of the top quark, with a mass
\ba
\setcounter{equation}{2}
m_t &=& 176 \pm 8 \pm 10   GeV \nonumber\\
&=& 176 \pm 13 GeV
%\eqno{(2)}
\ea
obtained by the CDF collaboration at FNAL, we predict from Eq (1), the
Higgs mass
$$m_H = 321 \pm 29 GeV
\eqno{(3)}
$$where we have used \cite{C}
\ba \setcounter{equation}{4}
m_Z &=& 91.1887 \pm 0.0044 GeV \\
m_W &=& 80.23 \pm 0.18 GeV
\ea
\vspace*{3mm}
\par\noindent
{\bf 2.} We assume $m_t$ and $m_H$ to be determined
[1-4,7-9] by requiring the ultraviolet divergencies to vanish for the
(on-mass shell) self-energy of the quark top, namely
$$
\Sigma^{\mbox{div}}_t = - m_t \frac{\alpha}{4 \pi} \; \frac{m^2_Z}{m^2_Z -
m^2_W} F_t (\Lambda) = 0.
\eqno{(6)}
$$
\vspace*{3mm}
\par\noindent
for any $\Lambda \cdot F_t (\Lambda)$, the function containing the cutoff
$\Lambda$ has the general form \cite{H}
\vspace*{3mm}
\ba \setcounter{equation}{7}
&&F_t(\Lambda) \equiv\frac{3}{4} \; \frac{1}{m^2_W m^2_H}\left[ \Lambda^2
\left\{ m^2_H + m^2_Z + 2 m^2_W - 4 m^2_t \right.\right.
\nonumber\\
&&\left.-4 m^2_b \right\}
+ \ln \Lambda^2 \left\{ - \frac{1}{2} m^4_H - m^4_Z - 2 m^4_W
\right.\nonumber\\
&&+ 4 m^4_t + 4 m^4_b + \frac{1}{2} m^2_H (m^2_t-m^2_b) \nonumber\\
&&\left.\left.- \frac{4}{9} m^2_H (m^2_Z - m^2_W)\right\} \right]
\ea

\par\noindent
Eqs(6) and (7) imply the two following relations:
\ba
&&m^2_H + m^2_Z + 2 m^2_W - 4 m^2_t - 4 m^2_b = 0 \\
& & \nonumber \\
&& \frac{1}{2} m^4_H + m^4_Z + 2 m^4_W - 4 m^4_t - 4 m^4_b - \frac{1}{2}
m^2_H (m^2_t - m^2_b)
\nonumber\\
&&+ \frac{4}{9} m^2_H (m^2_Z - m^2_W) = 0
\ea
Of course, Eqs(1) and (8) are identical due to the universality of
quadratic divergences
\cite{D}-\cite{P}. By using Eqs(4), (5), (8) and (9), we predict:
\ba
m_t &=& 170.5 \pm 0.3  GeV \\
m_H &=& 308.6 \pm 0.7 GeV
\ea
The predicted mass for the top quark is in agreement with the central value
reported by the
CDF Collaboration \cite{A}, Eq.(2). We have used $m_b = 5 GeV$.  With $m_b
= 0$, our predictions are
$m_t = 171.1 \pm 0.3 GeV$ and $m_H = 309.7 \pm 0.7 GeV$.\\
Eqs(10) and (11) can be compared with the fitted value of $m_t$  derived
from the
LEP data \cite{C}, $m_t=173 \pm 13 GeV$ if $m_H = 300 GeV$.
Finally, let us mention that the others solutions to Eqs. (8) and (9) given
above namely,  $m_t = 78 GeV $ and $m_H = 58 GeV$,  are excluded by the
present data.
It is interesting to observe that $(2 m_t - m_H) \approx 3(m_Z - m_W)$ and
$m_t \approx m_Z + m_W$.


\vspace*{25mm}


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M. Veltman Acta Phys. Pol. {\bf B12} (1981) 437.
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See also S. Abachi {\em et al}, D$\oslash$ Collaboration, {\it Observation
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FERMILAB-PUB 95/028-E (1995), where $m_t$ = 199 $\pm$ 30 GeV is given.
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\end{document}









