Scales of Mass Generation for Quarks, Leptons and Majorana Neutrinos
\noindent We study \,$2\to n$\, inelastic fermion-(anti)fermion scattering into multiple longitudinal weak gauge bosons and derive universal upper bounds on the scales of fermion mass generation by imposing unitarity of the $S$-matrix. We place new upper limits on the scales of fermion mass generation, independent of the electroweak symmetry breaking scale. We find that the strongest $2\to n$ limits fall in a narrow range, $3-170$\,TeV (with $n=2-24$), depending on the observed fermion masses. %\vspace*{-3mm}
HEPUFFPRL.tex

%%% Version for repost to HEP-PH/0502178
%%% Accepted by Phys.Rev.Lett.   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%  Title:    Scales of Mass Generation for                  %%%
%%%            Quarks, Leptons and Majorana Neutrinos         %%%
%%%  Authors:  Duane A. Dicus and Hong-Jian He                %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%--------------------- RevTex4 Source File ---------------------
%%
%% ****** Start of file template.aps ****** %
%%
%%   This file is part of the APS files in the REVTeX 4 distribution.
%%   Version 4.0 of REVTeX, August 2001
%%
%%
%%   Copyright (c) 2001 The American Physical Society.
%%
%%   See the REVTeX 4 README file for restrictions and more information.
%%
%
% This is a template for producing manuscripts for use with REVTEX 4.0
% Copy this file to another name and then work on that file.
% That way, you always have this original template file to use.
%
% Group addresses by affiliation; use superscriptaddress for long
% author lists, or if there are many overlapping affiliations.
% For Phys. Rev. appearance, change preprint to twocolumn.
% Choose pra, prb, prc, prd, pre, prl, prstab, or rmp for journal
%  Add 'draft' option to mark overfull boxes with black boxes
%  Add 'showpacs' option to make PACS codes appear
%  Add 'showkeys' option to make keywords appear


\documentclass[aps,twocolumn,showpacs,preprintnumbers,amsmath,amssymb]{revtex4}
%\documentclass[aps,prd,twocolumn,groupedaddress,amssymb]{revtex4}
%
%\documentclass[aps,prl,preprint,superscriptaddress]{revtex4}
%\documentclass[aps,prl,twocolumn,groupedaddress]{revtex4}

\usepackage{epsfig}
\usepackage{graphicx}
\usepackage{amssymb} 
\usepackage{here}
\usepackage{eucal}
%
\usepackage{dcolumn}% Align table columns on decimal point
\usepackage{bm}%      bold math



\addtolength{\textheight}{5mm}


\bibliographystyle{unsrt}  % for BibTeX - sorted numerical labels by order of
                           % first citation.


% You should use BibTeX and apsrev.bst for references
% Choosing a journal automatically selects the correct APS
% BibTeX style file (bst file), so only uncomment the line
% below if necessary.
%\bibliographystyle{apsrev}


%------------------- Command Section -------------------------------
\newcommand{\Tr}{\text{\,Tr\,}}
\newcommand{\TeV}{\text{\,TeV}}

\def\cut{\Lambda}
\def\Ucut{\Lambda_{\rm U}^{~}}
\def\O{{\mathcal O}}
\def\C{{\mathcal C}}
\def\N{{\mathcal N}}
\def\T{{\mathcal T}}
\def\hh{\widehat{h}}
%
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\beqa}{\begin{eqnarray}}
\newcommand{\eeqa}{\end{eqnarray}}
\newcommand{\ba}{\begin{array}}
\newcommand{\ea}{\end{array}}
\newcommand{\infinity}{\infty}
%
\newcommand{\mev}{{\rm MeV}}
\newcommand{\gev}{${\rm GeV}$}
\newcommand{\tev}{${\rm TeV}$}
\newcommand{\tr}{{\rm Tr}}
\newcommand{\NAIVE}{na{\"\i}ve~}
\newcommand{\X}{\chi}
\newcommand{\nuL}{\nu^{~}_L}

\def\ep{\epsilon}
\def\vrho{\varrho}
\def\A{{\cal A}}
\def\C{{\cal C}}
%\def\O{{\cal O}}
\def\D{{\cal D}}
\def\R{{\cal R}}
\def\T{{\cal T}}
\def\TT{{\cal T}}
\def\SS{{\cal S}}
\def\V{{\cal V}}
\def\noi{\noindent}
\def\End{\end{document}}
\def\nor{\normalsize}
\def\sq{\sqrt{2}}
\def\del{\partial }
\def\to{\rightarrow}
\def\To{\Rightarrow}
\def\tolr{\leftrightarrow}
\def\tri{\blacktriangleright}
\def\dis{\displaystyle}
\def\f{\frac}
\def\ov{\overline}
\def\ovr{\overrightarrow}
\def\[{\left[}
\def\]{\right]}
\def\({\left(}
\def\){\right)}
\def\br{\blacktriangleright}
\def\ra{\rightarrow}
\def\under{\underline}
\def\G{{\cal G}}
\def\GSM{{\cal G}_{\rm SM}}
\def\Ahh{\widehat{A}}
\def\Ah{\widetilde{A}}
\def\pih{\widetilde{\pi}}
\def\At{\widetilde{A}}
\def\pit{\widetilde{\pi}}
\def\l{{\ell}}
\def\tm{{\widetilde{m}}}
\def\Om{{\Omega}}
\def\W{\mathcal W}
\def\Z{\mathcal Z}
\def\DL{\Delta {\mathcal L}}
\def\a{\alpha}
\def\b{\beta}
\def\GA{\Gamma}
\def\C{\mathcal C}
\def\U1EM{U(1)_{\rm em}}
\def\O{\mathcal O}
\def\R{\mathcal R}
\def\la{\langle}
\def\ra{\rangle}
\def\deg{\circ}
\def\leqq{\leqslant}
\def\geqq{\geqslant}
\def\lan{\langle}
\def\ran{\rangle}

\setlength{\baselineskip}{24pt}

\def\sb{\sin\beta}
\def\cb{\cos\beta}
\def\sbb{\sin^2\!\beta}
\def\cbb{\cos^2\!\beta}
\def\sa{\sin\alpha}
\def\ca{\cos\alpha}
\def\saa{\sin^2\alpha}
\def\caa{\cos^2\alpha}
%
\def\ssb{s_{\beta}^{~}}
\def\ssbb{s_{\beta}^{2}}
\def\ccb{c_{\beta}^{~}}
\def\ccbb{c_{\beta}^{2}}
\def\ssa{s_{\alpha}^{~}}
\def\ssaa{s_{\alpha}^{2}}
\def\cca{c_{\alpha}^{~}}
\def\ccaa{c_{\alpha}^{2}}
%
\def\st{\sin\theta}
\def\ct{\cos\theta}
\def\sp{\sin\phi}
\def\cp{\cos\phi}
\def\cw{c_{\rm w}}
\def\sw{s_{\rm w}}
\def\d{\delta}



\def\gt{\widetilde{g}}
\def\[{\left[}
\def\]{\right]}
\def\dis{\displaystyle}
\def\S{\Sigma}
\def\cut{\Lambda}
\def\cutU{\Lambda_{\rm U}}
\def\Eu{E^{\star}}
\def\LL{\mathcal L}
\def\RE{\Re\mathfrak{e}}
\def\IM{\Im\mathfrak{m}}
\def\MP{M_{\rm Pl}}
\def\MI{\mathfrak{m}_{\rm I}^{~}}
\def\MII{\mathfrak{m}_{\rm I}^{2}}
\def\B{\mathfrak{B}}
\def\nb{\overline{n}}
\def\nL{\nu_L^{~}}
\def\nR{\nu_R^{~}}
\def\nLp{\nu_{L+}^{~}}
\def\nLm{\nu_{L-}^{~}}
\def\nLpm{\nu_{L\pm}^{~}}
\def\fbar{\bar{f}}
%
\def\leqq{\leqslant}
\def\geqq{\geqslant}        
\def\RE{\Re\mathfrak{e}}
\def\IM{\Im\mathfrak{m}}
\def\dif{\partial }
\def\lg{\left\lgroup}
\def\rg{\right\rgroup}
\def\br{\blacktriangleright}
\def\ra{\rightarrow}
\def\under{\underline}
\def\G{{\mathcal G}}
\def\H{{\mathcal H}}
\def\GSM{{\cal G}_{\rm SM}}       
%
\def\thisday{December, 2004}


\begin{document}

%\preprint{\large hep-ph/0502178}
%\preprint{UTHEP-04-16}
\preprint{\large hep-ph/0502178}


% Use the \preprint command to place your local institutional report
% number in the upper righthand corner of the title page in preprint mode.
% Multiple \preprint commands are allowed.
% Use the 'preprintnumbers' class option to override journal defaults
% to display numbers if necessary
%\preprint{}


%Title of paper
\title{Scales of Mass Generation for Quarks, Leptons
       and Majorana Neutrinos
      }



% repeat the \author .. \affiliation  etc. as needed
% \email, \thanks, \homepage, \altaffiliation all apply to the current
% author. Explanatory text should go in the []'s, actual e-mail
% address or url should go in the {}'s for \email and \homepage.
% Please use the appropriate macro foreach each type of information

% \affiliation command applies to all authors since the last
% \affiliation command. The \affiliation command should follow the
% other information
% \affiliation can be followed by \email, \homepage, \thanks as well.
%
\author{{\sc Duane A. Dicus} ~and~ {\sc Hong-Jian He}\,\,}
%\email{dicus@physics.utexas.edu,   hjhe@physics.utexas.edu}
\affiliation{Center for Particle Physics, % and Department of Physics, 
             University of Texas at Austin, TX 78712, USA
}

%\homepage[]{Your web page}
%\thanks{}
%\altaffiliation{}

%Collaboration name if desired (requires use of superscriptaddress
%option in \documentclass). \noaffiliation is required (may also be
%used with the \author command).
%\collaboration can be followed by \email, \homepage, \thanks as well.
%\collaboration{}
%\noaffiliation

%\date{\thisday}


\begin{abstract}
\noindent
We study  \,$2\to n$\, inelastic fermion-(anti)fermion scattering
into multiple longitudinal weak gauge bosons and derive
universal upper bounds on the scales of
fermion mass generation by imposing  unitarity of the $S$-matrix.
We place new upper limits on the scales
of fermion mass generation, independent of the electroweak
symmetry breaking scale.
We find that the strongest $2\to n$ limits fall in a narrow range, 
$3-170$\,TeV (with $n=2-24$),   
depending on the observed fermion masses. 
%\vspace*{-3mm}
\end{abstract}


% insert suggested PACS numbers in braces on next line
\pacs{%\vspace*{-3mm}
      12.15.Ff, 14.60.Pq, 11.80.-m, 12.60.-i \hfill 
      %hep-ph/0502178}  Preprint$^\#$: UTHEP-04-16}
      %December, 2004}
      Phys. Rev. Lett. (2005), in Press.}

%
%ref: 14.80.Cp, 12.60.Cn, 12.60.Fr
% 11.80.Et(partial wave),  12.60.Rc (composite models)

% insert suggested keywords - APS authors don't need to do this
%\keywords{}
%\preprint{UTHEP-04-16}

%\maketitle must follow title, authors, abstract, \pacs, and \keywords
\maketitle


% body of paper here - Use proper section commands
% References should be done using the \cite, \ref, and \label commands
%\section{Introduction}



%\vspace*{5mm}
\noindent  
\under{\it New Puzzle for Scales of Fermion Mass Generation~}
\vspace*{3mm}


Understanding scales of mass generation for 
quarks, leptons and Majorana neutrinos poses a 
great challenge in elementary particle physics.
Mass generation in the standard model (SM)
relies on a single hypothetical Higgs boson.
Although the masses of weak bosons $(W^\pm,Z^0)$ only involve
electroweak gauge couplings $(g,\,g')$ and the vacuum expectation
value $v=(\sqrt{2}G_F)^{-\f{1}{2}}$,  the quark/lepton masses
arise from products of Yukawa couplings $y_f^{~}$ and $v$\,.\, 
Contrary to the gauge interactions, the Yukawa couplings $y_f^{~}$ are  
flavor-dependent and completely arbitrary,
exhibiting a huge empirical hierarchy 
\,$y_e^{~}/y_t^{~}=m_e^{~}/m_t^{~} \simeq 3\times10^{-6}$\,
between the electron and the top quark.
No compelling principle requires the fermion mass generation to
share the same mechanism as the $W/Z$ gauge bosons.
Furthermore, the tiny neutrino masses 
$\,0.05\,{\rm eV}\lesssim m_\nu^{~}\lesssim 1\,$eV\,\cite{superK}
cannot be generated by such a Higgs boson without
losing renormalizability\,\cite{wein5} or extending the SM
particle spectrum\,\cite{nu-seesaw,nu-rad}.
So far, no Higgs boson has been found, nor is any
Yukawa coupling experimentally measured ---  
the origin and scale of
fermion mass generation remain completely unknown. 




What is wrong with just {\it putting all the bare masses into 
the SM Lagrangian by hand?}  
Such bare mass-terms can be made 
gauge-invariant in the nonlinear realization\,\cite{CCWZ},
but are manifestly nonrenormalizable. 
This causes unitarity violation in
high energy scattering at a scale $\Eu$. 
Generically, we define the scale 
$\cut_x$ for generating a mass $m_x^{~}$ 
to be the {\it minimal energy
above which the bare mass term for $m_x^{~}$ has to be replaced 
by a renormalizable interaction} (adding at least one new physical
state to the particle spectrum). Hence, the unitarity violation
scale $\Eu$ puts a {\it universal upper limit on $\cut_x$,}
\,$\cut_x \leqq \Eu$.


A bare mass-term for $(W^\pm ,Z^0)$ will cause 
the high energy \,$2\to 2$\, scattering of 
longitudinal weak bosons
to violate unitarity at a scale\,\cite{DM},
%
%\beq
$\,E^\star_W ~\simeq~ \sqrt{8\pi}v ~\simeq~ 1.2\,{\rm TeV}\,.$\,
%\eeq
%
This puts an upper limit on the scale of 
electroweak symmetry breaking (EWSB) and 
justifies the TeV energy scale for the 
Large Hadron Collider (LHC) at CERN. 
Similarly, by adding bare mass terms for Dirac fermions 
an upper bound on the scale of fermion mass generation 
can be derived from the \,$2\to 2$\, inelastic scattering  
$f\bar{f} \to V_L^{a_1}V_L^{a_2}$ 
($V^a=W^\pm,Z^0$) \cite{AC,MVW},
%
\beq
\label{eq:UBf22}
E_f^\star ~\simeq~ \dis  \f{8\pi v^2}{~\sqrt{N_c}\,m_f^{~}~} \,,
\eeq
%
where $N_c = 3\,(1)$ for quarks (leptons).
Since this limit is proportional to 
$1/m_f^{~}$, it is independent of the  
bound $E^\star_W$ on the EWSB scale.
For all the SM fermions (except the top quark), 
the limit (\ref{eq:UBf22})
is substantially higher than $E^\star_W$.
For the scale of mass generation for Majorana neutrinos, 
Refs.\,\cite{scott-PRL,scott} derived
an analogous $2\to 2$ bound from the scattering
\,$\nL\nL\to V_L^{a_1}V_L^{a_2}$,  
%
\beq
\label{eq:UBnu22}
E_\nu^\star ~\simeq~ \dis  
%\f{~\,4\pi v^2\,}{\,~m_\nu^{~}~} \,,
\,4\pi v^2/m_{\nu}\,,
\eeq
%
which is around \,$10^{16}$\,GeV\, (the GUT scale) for a
typical neutrino mass \,$m_\nu^{~}\sim 0.05$\,eV \cite{superK}.


However, for $2\to n$ inelastic scattering,
$f\bar{f},\,\nL\nL$ $\to nV_L^a$ ($n\geqq 3$), 
Ref.\,\cite{scott} found that the $n$-body phase space
integration contributes a large energy factor, $s^{n-2}$,
to further enhance the cross section 
(in addition to the energy dependence $s^1$ 
from the squared amplitude), where $\sqrt{s}$ is the center of 
mass energy. From the unitarity condition on the total
cross section\,\cite{scott},
%
\beq
\label{eq:UC2n}
\sigma_{\rm inel}[2\to n] ~\leqq~  4\pi/s \,,
\eeq
%
the unitarity limit for 
$f~(\nL)$ would behave as\,\cite{scott}
%
\beq
\label{eq:UBfscot}
E_{f,\nu}^\star \sim \dis 
v\({v}/{\,m_{f,\nu}^{~}}\)^{\f{1}{n-1}}
%v\(v/m_{f(\nu)}\)^{\f{1}{n-1}}
\longrightarrow v\,,~~~({\rm as}\,~n\to{\rm large}),
\eeq
%
which could be pushed arbitrarily close to the weak scale $v$ and
thus become independent of the fermion mass 
$m_{f,\nu}^{~}$ for large enough \,$n$\,.\,
This raises a striking question: 
{\it is there an independent new scale for 
fermion mass generation revealed from the fermion-(anti)fermion
scattering into weak bosons?}  
We find the behavior (\ref{eq:UBfscot}) very counter-intuitive
since the kinematic condition forces
any \,$2\to n$\, unitarity limit $\Eu$
to grow at least linearly with \,$n$\,,
%
\beq
\label{eq:KC}
\dis
\sqrt{s} ~\,>~\, n M_{W(Z)} \,\simeq\, \dis\f{n}{3}v ~,
~~~\Longrightarrow~~~
\f{\,E^\star}{v} ~\,>~\, \f{n}{3} ~,
\eeq
%
which contradicts (\ref{eq:UBfscot}), 
posing a {\it new puzzle.}  

 
%%%%%%%%%%%%%%%%%%%%%%%% Fig.1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure*}
\label{fig:ff-npi}
\begin{center}
\vspace*{-5mm}
\hspace*{-3mm}
\includegraphics[width=18cm,height=6cm]{Figprl1f.eps}
\vspace*{-6mm}
\caption{Typical contributions to
$f\fbar (\nL\nL) \to n\pi^a$
scattering: 
(a) leading ``contact'' diagrams of ${\mathcal O}(E^1)$;
(b) sub-leading ``$t(u)$-channel'' diagrams of
${\mathcal O}(E^0)$  which can be ignored for the unitarity analysis;
(c) ``hybrid''  contact diagrams ($n\geqq 3$) including Goldstone
self-interactions from the EWSB sector;
(d) ``hybrid''  contact diagrams ($n\geqq 3$) including Goldstone
interactions with the EWSB quanta (illustrated here with the SM Higgs
boson $h^0$  as the simplest example of EWSB).
}
\vspace*{-0.5cm}
\end{center}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\vspace*{3mm}
\noindent  
\under{\it Resolution of the New Puzzle~}
\vspace*{2mm}


Consider high energy $2\to n$ inelastic scattering 
$\,f\fbar,$ $\nL\nL\to nV_L^a\,$. 
Its amplitude can be derived from the corresponding scattering 
$\,f\ov{f},\nL\nL\to n\pi^a\,$
with multiple Goldstone bosons ($\pi^a$) via 
the equivalence theorem\,\cite{ET}.
In the nonlinear realization, 
the bare mass-terms for quarks/leptons and Majorana neutrinos
result in contact interactions between these fermions and
Goldstones, of the type \,$f$-$\fbar$-$\pi^n$\, and
\,$\nL$-$\nL$-$\pi^n$\, ($n\geqq 1$) 
whose couplings are proportional 
to the corresponding fermion mass \,$m_{f(\nu)}^{~}$.\,
We classify the relevant contributions in Fig.\,1.
The contact diagram  Fig.\,1(a) gives a universal  
leading amplitude of \,${\mathcal O}(m_{f,\nu}^{~}E/v^n)$\, according to
power counting\,\cite{PC}, while the non-contact diagrams 
in Fig.\,1(b) are all subleading in energy.     
The contributions in Fig.\,1(c,d) involve the Goldstone
self-interactions and the new particle(s) generating the EWSB
(illustrated here by the SM Higgs boson $h^0$) which are 
model-dependent. A quantitative analysis of Fig.\,1(a,d)   
showed\,\cite{DH} that they may enhance the unitarity
bound by a factor of \,$[{\mathcal O}(2-3)]^{\f{1}{n-1}}$\, 
($n\geqq 3$) which is very close to one for large $n$\,.\, 
Thus it is justified to use the universal Fig.\,1(a) as a 
model-independent, conservative estimate for the unitarity bound. 


Since power counting shows the leading amplitude of
$2\to n$ scattering to be
\,${\mathcal O}(1)(m_{f,\nu}^{~})^{2-\d}s^{\d/2}/v^n$\,
for $n\geqq 2$, 
we deduce the generical form of the squared amplitude,
%
\beq
\label{eq:T}
|\T|^2\,=\,
c_0^{~}(\theta_j)(2N_{c})^{2-\d}\,
m_{f,\nu}^{2(2-\d)}s^{\d}/v^{2n}\,,
\eeq
%
where \,$\d = 1\,(2)$\, for 
\,$f\fbar,\,\nL\nL,\,(V_L^{a_1}V_L^{a_2})$ $\to nV_L^a$\,
in the spin/color-singlet channel, and
$c_{0}^{~}(\theta_j)$ is a dimensionless coefficient with possible 
angular dependence. 
Using (\ref{eq:T}) we will compute the corresponding total cross
section $\sigma_{\rm inel}[2\to n]$ 
and determine its unitarity violation
scale from the condition (\ref{eq:UC2n}). 




We observe that the key for resolving the puzzle posed in 
Eqs.\,(\ref{eq:UBfscot}) and (\ref{eq:KC}) is to analyze the  
additional $n$-dependent {\it dimensionless} factors in the 
exact n-body phase space integration; 
these factors will sufficiently
suppress the $E$-power enhancement $s^{n-2}$ in
Eq.\,(\ref{eq:UBfscot}).
Using (\ref{eq:T}), and
including the exact $n$-body phase space for the cross section
computation, we can formally derive the $2\to n$ unitarity limit, 
%
\beqa
\label{eq:UBus}
\dis
E^\star &=& v\[C_0\, 2^{4n-2}\pi^{2(n-1)}
            \varrho\, (n-1)!\,(n-2)!
\(2N_c\)^{\d-2}  
\right. \nonumber
\\
&& \hspace*{7mm} 
\left. \dis \times
\({v}/{\,m_{f,\nu}^{~}\,}\)^{2(2-\d)}\]^{\f{1}{2(n-2+\d)}}
%\(v/m_{f,\nu}\)^{2(2-\delta)}\]^{\f{1}{2(n-2+\delta)}}
%
\eeqa
%
where $\varrho$ is a symmetry factor for 
identical particles in the final state.
The factor $\,C_0$\, originates from the 
coefficient $c_0^{~}(\theta_j)$ whose possible angular
dependence is included in the exact $n$-body phase 
space integration. 
For the leading contact 
contributions in Fig.\,1(a) no such angular dependence exists. 
Since all large $n$ factors are explicitly
counted in (\ref{eq:UBus}),
we expect that, independent of detail, 
$~C_0^{\f{1}{\,2(n-2+\d)\,}} ~\sim~ 1~$ 
becomes increasingly more accurate as $n$ gets larger.
Thus, setting 
$\,\( C_0\,\varrho\)^{1/2(n-2+\d )}\approx 1$\, in (\ref{eq:UBus})
for simplicity, we derive an estimated unitarity limit \,$\Eu$\,
which is shown in Fig.\,2 (where $\,m_{\nu}^{~}=0.05$\,eV\, is chosen).   
Using the Stirling formula 
$\,n!\simeq n^ne^{-n}\sqrt{2\pi n}\,$, we deduce the
correct asymptotic behavior from (\ref{eq:UBus}),
%
\beq
\label{eq:asymp}
\dis
\Eu \,\to\, v\f{4\pi n}{e} ~~>~~ v\f{n}{3} \,,
~~~~~~~~({\rm for}~n\gg 1)\,, 
\eeq
%
which {\it agrees} with our kinematic condition (\ref{eq:KC}).



%%%%%%%%%%%%%%%%%%%%%%%  Fig.2  %%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[H]
\label{fig:Fig2}
\vspace*{-7mm}
\begin{center}
\includegraphics[width=8.5cm,height=6.8cm]{Figprl2f.eps}    
\vspace*{-5mm}
\caption{Realistic estimates of the unitarity limit $E^\star$,
from the scatterings 
\,$V_L^{a_1}V_L^{a_2},\,t\bar{t},\,b\bar{b},\,
   \tau^-\tau^+,\,e^-e^+,\,\nL\nL
   \to nV_L^a$\, (curves from bottom to top),
as a function of integer \,$n\,(\geqq 2)$\,.\,
}
\vspace*{-6mm}
\end{center}
\end{figure}



Fig.\,2 shows that for \,$V_L^{a_1}V_L^{a_2}$\, 
and \,$t\bar{t}$\, initial states
the strongest bound (minimum of the curve) still occurs at $n= n_s=2$;
while for {\it all light fermions including Majorana neutrinos}
the best limit for the scale of mass generation lies at a {\it new minimum}
with  $\,n= n_s > 2 \,$ and $\,\Eu\,$ no higher than about $100\,$TeV.
These limits are substantially tighter than 
the corresponding classic $2\to 2$ bounds.
The more precise calculations below 
will show these estimates are accurate to within a factor of two.
We stress that it is the {\it competition} between the
large asymptotic linear growth (\ref{eq:asymp}) and the strong
power suppression (\ref{eq:UBfscot}) that generates
a genuine {\it new minimum scale $E^\star_{\min}$} 
for all light fermions
at $\,n= n_s>2\,$, {\it independent of the EWSB scale.} 





\vspace*{5mm}
\noindent  
\under{\it Scales of Mass Generation for Quarks and Leptons~}
\vspace*{3mm}


To improve the estimates in Fig.\,2, we will 
precisely compute the dimensionless coefficient $c_0^{~}$
in the leading squared-amplitude (\ref{eq:T}), so we can
determine the constant $C_0$ in Eq.\,(\ref{eq:UBus}) 
for each given \,$f\fbar ,\nL\nL\to nV_L^a$\, process 
(where the possible identical 
particle factor $\varrho$ will be explicitly included).
We first consider a pair of SM Dirac fermions $(f,f')$ which form a 
left-handed $SU(2)_L$ doublet $F_L=(f_L,f'_L)^T$ and two right-handed 
weak singlets $f_R$ and $f_R'$\,.\, We can formulate their bare Dirac 
mass-terms  $\,-m_f^{~}\ov{f}f - m_{f'}^{~}\ov{f'}f'\,$ 
into the gauge-invariant nonlinear form,
%
\beq
\label{eq:f-mass}
{\cal L}_f = \dis
-m_f\ov{F_L}U\(\ba{c} 1 \\ 0 \ea\)f_R
-m_{f'}\ov{F_L}U\(\ba{c} 0 \\ 1 \ea\)f'_R + {\rm H.c.}
\eeq 
%
where \,$U\,=\,\exp[i\pi^{a}\tau^{a}/v]$\,.\,
This gives $\,f$-$\fbar$-$\pi^n$ ($n\geqq 1$) 
contact interactions between fermions and Goldstone bosons.
We compute the leading scattering amplitude for 
$\,|{\rm in}\rangle \to (\pi^+)^k(\pi^-)^{\ell}(\pi^0)^{n-k-\ell}\,$,\,
where the initial state $|{\rm in}\rangle$ consists of two fermions
in the color-singlet channel\,\cite{DH}.\,

%%%%%%%%%%%%%%%%%%%%%%%%   Fig.3  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[H]
\label{fig:Fig3}
\begin{center}
\vspace*{-6mm}
\hspace*{-1.9mm}
\includegraphics[width=8.9cm,height=7.7cm]{Figprl3f.eps} 
\hspace*{5mm}
\vspace*{-9mm}
\caption{Precise unitarity bounds $E^\star_f$ 
from scatterings 
$\,\xi_1^{~}\xi_2^{~}\to n\pi^a~(nV_L^a)$\,
as a function of \,$n$\,,\, where
$\,\xi_1^{~}\xi_2^{~}=t\bar{t},\,b\bar{b},\,c\bar{c},$ 
$ \,\tau^-\tau^+,\,
   s\bar{s},\,\mu^-\mu^+,\,d\bar{d},\,u\bar{u},\,e^-e^+
$\, 
for $n=$~even and
$\,\xi_1^{~}\xi_2^{~}=t_+\bar{b}_+,\,t_-\bar{b}_-,\,c_+\bar{s}_+,\,
   \tau_-^+\nu_-,\, c_-\bar{s}_-,\,\mu_-^+\nu_-,\,u_-\bar{d}_-,\,
    u_+\bar{d}_+,\,e^+_-\nu_- 
$\, for $n=$~odd, (curves from bottom to top).
The subscripts $\pm$ refer to helicity.
%The curves marked by circles represent bounds for leptons and
%those marked by triangles are for quarks; also, for clarity, 
%the up-type, down-type quarks and the leptons are plotted in 
%three different colors.
}
\end{center}
\vspace*{-3mm}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



Imposing unitarity on the $s$-partial wave, we derive 
corresponding unitarity bound for quarks and leptons,
%
\beqa
\label{eq:UBf-final}
E^\star_f ~~&=&~~ v\[\(\f{v}{m_{\widehat{f}}^{~}}\)^2
               \f{4\pi}{\,N_c\,\widehat{\R}_k^{\max}\,}
               \]^{\f{1}{2(n-1)}}  ,
\eeqa
%
where \,$\widehat{f}\in (f,\,f')$,\, and
$\,\widehat{\R}^{\max}_k \in 
         \(2\R_1^{\max},\,\R_2^{\max}\)\,$ 
with
%
\beq
\ba{lcl}
\R_1^{\max} &=&
\dis\f{\(\f{n}{2}!\)^2
}{~2^{3n-4}\pi^{2n-3}\,(n!)^2\,(n-1)!\,(n-2)!~} \,,
\\[5mm]
%\R_2^{\max} &=&
%\dis\f{1}{~2^{4(n-1)}\pi^{2n-3}\,n!\,(n-1)!\,(n-2)!~} \,,
%\\[4mm]
\R_2^{\max} &=&
\dis\f{\(\f{n-1}{2}\)!\(\f{n+1}{2}\)!}
{~2^{3n-4}\pi^{2n-3}\,(n!)^2(n-1)!\,(n-2)!~} \,,
\ea
\eeq
for $\,n=\,$(even,\,odd), respectively. 
We depict the bound $\,E_f^{\star}\,$ in Fig.\,3,
which shows that the best limits for all light fermions
occur at a {\it new minimum} $\,n= n_s>2\,$
with $\,E^\star_b=24.5\,$TeV for $b$ quark and 
     $\,E^\star_e=107\,$TeV for electron. 
These two limits only differ by a factor of \,$\sim\! 4$\,. 




%%%%%%%%%%%%%%%%%%%%%  Fig.4  %%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[H]
\label{fig:Fig4}
\vspace*{-5mm}
\begin{center}
%\hspace*{3mm}
\includegraphics[width=8.5cm,height=7cm]{Figprl4f.eps}
\vspace*{-4.5mm}
\caption{Scales of mass generation for Majorana neutrinos:
the precise unitarity bounds \,$E^\star_\nu$\,
are plotted as a function of \,$n\,(\geqq 2)$\,.\,  
Only integer values of $n$ have a physical meaning.
}
\end{center}
\vspace*{-6mm}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table*}
\vspace*{-4mm}
\label{Tab:Tab1}
\caption{  
Summary of the {\it strongest} unitarity limit 
\,$E^{\star\min}_{2\to n}$\, 
for each scattering $\,\xi^{~}_1\xi^{~}_2\to nV_L^a$\, 
(\,$\xi^{~}_{1,2} = V_L,\,f,\,\fbar,\,\nL$)
and the corresponding number of final state particles 
\,$n=n^{~}_s$\,,\, 
in comparison to the classic $2\to 2$ limit 
\,$E^{\star}_{2\to 2}$\,.
}
\vspace*{-1mm}
\begin{center}
\begin{tabular}{c||c|cccccc|ccc|c}
\hline\hline
&&&&&&&&&&&\\[-2.5mm]
~$\xi_1^{~}\xi_2^{~}$~      
& ~$V_L^{a_1}V_L^{a_2}$~     & ~$t\ov{t}$  &   
$b\ov{b}$       & $c\ov{c}$     &  $s\ov{s}$ &
$d\bar{d}$      & $u\ov{u}$    ~&~ $\tau^-\tau^+$~  & 
~$\mu^-\mu^+$~  & $e^-e^+$     & ~$\nL\nL$~~        
\\ [1.5mm] 
\hline\hline
&&&&&&&&&&&\\[-2.5mm]
~$M_W,m_{f,\nu}^{~}$\,(GeV)~
& 80.4 & 178  & 4.85 & 1.65 & 0.105 & 0.006 & 0.003 
& 1.777 & 0.106 
& $5.11\!\times\! 10^{-4}$ & $5\!\times\! 10^{-11}$
\\[1.5mm]
\hline
&&&&&&&&&&&\\[-2.5mm]
$n_s^{~}$ 
& $2$  & ~$2$ & $4$ & $6$ & $8$ & $10$ & $10$ ~& $6$ & $8$ 
& $12$ & ~$22$~~   
\\[1.2mm]
\hline
&&&&&&&&&&&\\[-2.5mm]
~$E^{\star(\min)}_{2\to n}$\,(TeV)~ 
& $1.2$ & ~\,$3.49$~ & ~$23.4$~ & \,$30.8$\, 
& \,$52.1$\, & \,$77.4$\, & \,$83.6$\,~
& $33.9$ & $56.3$ & $107$ & ~$158$~~   
\\[1.5mm]
\hline\hline
&&&&&&&&&&&\\[-2.5mm]
~$E^{\star}_{2\to 2}$\,(TeV)~ 
& $1.2$ & ~$3.49$ & $128$ & $377$ & $6\!\times\! 10^3$  & $10^5$ 
& $2\!\times\! 10^5$~ & $606$ & $10^4$ & ~$2\!\times\! 10^6$~ 
& ~~$1.1\!\times\! 10^{13}$~~   
\\[1.5mm]
\hline\hline
\end{tabular}
\end{center}
\vspace*{-3mm}
\end{table*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\vspace*{3mm}
\noindent  
\under{\it Scales of Mass Generation for Majorana Neutrinos~}
\vspace*{3mm}



For Majorana neutrinos, we write their
dimension-3 bare mass term as
$\,-\f{1}{2}m_{\nu}^{ij}\nu_{Li}^T\widehat{C}
    \nu_{Lj}^{~}+{\rm H.c.},\,$
which takes the gauge-invariant nonlinear form 
%
\beq
\label{eq:nu-mass-pi}
{\cal L}_\nu ~=~ \dis
-\f{1}{2}m_\nu^{ij}{L^{\alpha}_i}^T \widehat{C}L^{\beta}_j
 \ov{\Phi}^{\alpha'}\ov{\Phi}^{\beta'}
\epsilon^{\alpha\alpha'} \epsilon^{\beta\beta'} + {\rm H.c.}
\,,
\eeq
%
where $\,\widehat{C}=i\gamma^2\gamma^0\,$,\,
$\,\ov{\Phi}=U(0,1)^T$\,,\, 
%$U=\exp[i\pi^a\tau^a/v]$\,
and $\,L_j=(\nu_{Lj}^{~}/\sqrt{2},\,\ell_{Lj})^T\,$.\,
We then quantitatively compute the high energy scattering
\,$\nL\nL \to   
   \(\pi^+\)^{\ell}\(\pi^-\)^{\ell}\(\pi^0\)^{n-2\ell}$\,.\,
In the actual calculation we use the real Majorana 
field $\,\chi =(\nL +\nu_L^c)/\sqrt{2}\,$ 
and compute the amplitude 
%$\,\f{1}{\sqrt{2}}\[
% |\chi_{j+}\ov{\chi}_{j+}\rangle \mp
% |\chi_{j-}\ov{\chi}_{j-}\rangle \] \to
% \(\pi^+\)^{\ell}\(\pi^-\)^{\ell}\(\pi^0\)^{n-2\ell}
%\,$
for three cases: 
(a) $n({\rm even})=2\ell$\,;\,   
(b) $n({\rm even}) > 2\ell$\,;\,
(c) $n({\rm odd})\geqq 3$\,.\,
We derive the unitarity bound,
%
\beqa
\label{eq:UBnu-final}
E^\star_\nu ~~&=&~~ v\[\(\f{v}{m_{\nu j}^{~}}\)^2
                       \f{4\pi}{\,\R_{\nu k}^{\max}\,}
             \,\]^{\f{1}{2(n-1)}}  ,
\eeqa
%
where $\,\R_{\nu k}^{\max}$\, ($k=1,2,3$) is 
a function of $n$ for three cases above\,\cite{DH}.
From Fig.\,4, we find the optimal limits, 
%
\beq
\label{eq:nuB-min}
\hspace*{-3mm}
E^\star_{\min} = 136,158,170\,{\rm TeV}, 
~~\,{\rm at}~\, n=n_s = 20,22,24,
\eeq
%
for typical inputs \,$m_{\nu j}^{~} = 1.0,\, 0.05,\, 0.01$\,eV.\,
This agrees with our estimate in Fig.\,2 to within
a factor of \,$\sim\! 1.7$. 





\vspace*{4mm}
\noindent  
\under{\it Conclusions~}
\vspace*{1mm}


In this work, we have systematically 
analyzed the unitarity limits
from $2\to n$ inelastic scattering; 
this provides a universal upper bound 
on the scales of mass generation for
quarks, leptons and Majorana neutrinos. 
Table\,1 summarizes our results in 
comparison with classic $2\to 2$ limits\,\cite{AC,scott}.
It shows that the scattering
$\,f\bar{f}\to nV_L^a\,$ ($n\geqq 2$) does reveal
a {\it separate scale} for fermion mass generation.
Our new limits from  $2\to n$ scattering with
$\,n>2\,$ establish {\it new scales of mass generation} for all
light fermions including Majorana neutrinos, and 
are substantially stronger than the classic $2\to 2$ limits.
These new $2\to n$ bounds vary within the range of \,$3-160$\,TeV 
when the input fermion masses change from 
$m_t^{~}$ to $m_{\nu}^{~}$, 
as listed in Table\,1 where the masses are pole-masses except
that for $m_{u,d,s}^{~}$ the more precise $\ov{\rm MS}$ values are 
used\,\cite{PDG}. 


In particular, for Majorana neutrinos with typical
masses $\,m_{\nu}^{~}=1.0-0.01\,$eV, 
the best upper limits on the
scale of mass generation fall in the range $136-170$\,TeV
(with $n= 20-24$). This is only a factor \,$\lesssim 7$\,
weaker than the lowest bound for light Dirac fermions 
despite their huge mass hierarchy
$\,m_{\nu}^{~}/m_{f\neq t}^{~} \leqq
m_{\nu}^{~}/m_b^{~}\approx 2\times(10^{-10}-10^{-12})\,$.
Hence, these limits are insensitive to the variation of 
fermion masses. 
Such a strong non-decoupling feature for the scale of new physics 
associated with light fermion mass generation is essentially due to
the {\it chiral structure} of fermion bare mass-terms, i.e., the 
fact that all left-handed SM fermions are weak-doublets but their
right-handed chiral partners are weak singlets (or possibly absent 
for a radiatively generated Majorana neutrino mass).
It is this feature that
makes the coupling strength of fermions to multiple Goldstone 
bosons (or effectively, multiple longitudinal gauge bosons)
{\it proportional to the fermion mass}, 
so the decoupling theorem\,\cite{DT} does not apply. 
For Majorana neutrino masses generated via the usual 
seesaw\,\cite{nu-seesaw} or radiative\,\cite{nu-rad} mechanism, 
this means that our new limits constrain the scale
of the leptonic Higgs Yukawa interaction, which must invoke
extra new fields (such as right-handed neutrinos or Zee-scalars
or triplet Higgs) as needed for generating lepton number violation 
and ensuring renormalizability 
(although our new bounds do not directly constrain
the masses of these new fields themselves 
which obey the decoupling theorem).


Finally, we have further estimated 
the \,$2\to n$\, unitarity limit on 
the EWSB scale via 
$\,V_L^{a_1} V_L^{a_2} \to n V_L^a$ 
($\pi^{a_1} \pi^{a_2} \to n \pi^a$)
with \,$n\geqq 2$,\, and find
that the best limit   
remains  \,$E^{\star}_W \simeq 1.2$\,TeV 
with $\,n=n_s=2\,$.
Also, the current study of \,$f\fbar \to nV_L^a\,(n\pi^a)$\,
scattering assumes $V_L^a$ ($\pi^a$) 
to be local field or remain local up to the limit 
$E^{\star}_{f}$.
If $V_L^a$ ($\pi^a$) becomes composite much below 
$E^{\star}_{f}$, the pure fermion process
$\,f\fbar \to (f\fbar)^n\,$
may be useful for a model-independent analysis.
A systematic expansion of this Letter 
is given in \cite{DH}.



\vspace*{1mm}
%\section*{Acknowledgments}
\noindent
{\bf Acknowledgments}~~
We thank S. Willenbrock for valuable discussions.
This work was supported by U.S. DOE 
under grant DE-FG03-93ER40757.




%
%\vspace*{-3mm}
\begin{thebibliography}{0}
%\vspace*{-3mm}

\bibitem{superK}
%Super-Kamiokande Collaboration, 
%S. Fukuda, {\it et al.}, 
%Phys. Rev. Lett. %{\bf 85}, 3999 (2000);
%{\bf 86}, 5656 (2001);
%{\bf 82}, 1810 (1999);
%{\bf 81}, 1562 (1998);
%{\bf 81}, 1158 (1998); 
%E.g., 
Y. Ashie, {\it et al.}, 
[SuperK-Kamiokande Collaboration],
Phys. Rev. Lett. {\bf 93}, 101801 (2004) 
[{\tt hep-ex/0404034}];
O. Elgaroy and O. Lahav, 
New J.\ Phys.\ {\bf 7}, 61 (2005).
%[{\tt hep-ph/0412075}].


\bibitem{wein5}
S. Weinberg, Phys. Rev. Lett. {\bf 43}, 1566 (1979).  


\bibitem{nu-seesaw}
P. Minkowski, Phys. Lett. B\,{\bf 67}, 421 (1977); 
M. Gell-Mann, P. Ramond, R. Slansky,
in {\it Proceedings of Workshop on Supergravity}, 
eds. F. van Nieuwenhuizen and D. Freedman (Amsterdam, 1979), 
p.315;
T. Yanagida, in {\it Proceedings of Workshop on Unified Theories
and Baryon Number in Universe}, 
eds. O. Sawada and A. Sugamoto (KEK, 1979), 
p.95;
S. L. Glashow, in {\it Quark and Leptons,} 
eds. M. Levy et al.\
(Plenum, NY, 1980);
R. N. Mohapatra and G. Senjanovic, 
Phys. Rev. Lett. {\bf 44}, 912 (1980).


\bibitem{nu-rad}
A. Zee, Phys. Lett. B\,{\bf 93}, 389 (1980);
L. Wolfenstein, Nucl. Phys. B\,{\bf 175}, 92 (1980);
S. T. Petcov, Phys. Lett.   B\,{\bf 115}, 401 (1982).



\bibitem{CCWZ}
C. G. Callen, S. Coleman, J. Wess, and B. Zumino,
Phys. Rev. {\bf 177}, 2247 (1969).
%S. Weinberg, Physica A\,{\bf 96}, 327 (1979).


\bibitem{DM}
D. A. Dicus and V. S. Mathur, Phys. Rev. D\,{\bf 7}, 3111 (1973);
%\bibitem{LQT}
B. W. Lee, C. Quigg and H. B. Thacker, 
Phys. Rev. D\,{\bf 16}, 1519 (1977);
%\bibitem{Vel}
M. Veltman,  Acta. Phys. Polon. B\,{\bf 8}, 475 (1977).


\bibitem{AC}
T. Appelquist and M. S. Chanowitz, 
Phys. Rev. Lett. {\bf 59}, 2405 (1987).


\bibitem{MVW}
W. Marciano, G. Valencia, and S. Willenbrock,
Phys. Rev. D\,{\bf 40}, 1725 (1989).


\bibitem{scott-PRL}
F. Maltoni, J. M. Niczyporuk, and S. Willenbrock,
Phys. Rev. Lett. {\bf 86}, 212 (2001)
[{\tt hep-ph/0006358}].
%%CITATION = HEP-PH 0006358;%%


\bibitem{scott}
F. Maltoni, J. M. Niczyporuk, and S. Willenbrock,
Phys. Rev. D\,{\bf 65}, 033004 (2002)
[{\tt hep-ph/0106281}].
%%CITATION = HEP-PH 0106281;%%


\bibitem{ET}
For a comprehensive review,
H.-J. He, Y.-P. Kuang and C.-P. Yuan,
DESY-97-056 [{\tt hep-ph/9704276}].
%%CITATION = HEP-PH 9704276;%%
%and references therein.
%H.-J. He and W.\,B. Kilgore,
%Phys. Rev. D{\bf 55}, 1515 (1997);  %[{\tt hep-ph/9609326}].
%%CITATION = HEP-PH 9609326;%% 
%H.-J. He, Y.-P. Kuang, C.-P. Yuan,
%Phys.\ Rev.\ D{\bf 51}, 6463 (1995); %[{\tt hep-ph/9410400}],
%%CITATION = HEP-PH 9410400;%%
%H.-J. He, Y.-P. Kuang, X. Li,
%Phys.\ Lett.\ B{\bf 329}, 278 (1994);
%%CITATION = HEP-PH 9403283;%%
%Phys.\ Rev.\  D{\bf 49}, 4842 (1994);
%%CITATION = PHRVA,D49,4842;%%
%Phys.\ Rev.\ Lett.\ {\bf 69}, 2619 (1992);
%%CITATION = PRLTA,69,2619;%%
%for a comprehensive review,  
%H.-J. He, Y.-P. Kuang, C.-P. Yuan, 
%DESY-97-056 [{\tt hep-ph/9704276}],
%and references therein.



\bibitem{PC}
%\vspace*{-0.3mm}
H.-J. He, Y.-P. Kuang and C.-P. Yuan,
Phys. Rev.  D\,{\bf 55}, 3038 (1997); % [{\tt hep-ph/9611316}];\\
%%CITATION = HEP-PH 9611316;%%
Phys. Lett. B\,{\bf 382}, 149 (1996). % [{\tt hep-ph/9604309}].
%%CITATION = HEP-PH 9604309;%%
%and a comprehensive review,
%DESY-97-056 [{\tt hep-ph/9704276}].
%%CITATION = HEP-PH 9704276;%%


\bibitem{DH} 
D.\,A. Dicus and H.-J. He, Phys. Rev. D (2005), in Press,
%PRD(2005),
[{\tt hep-ph/0409131}];
%%CITATION = HEP-PH 0409131;%%
and also, H.-J. He and D. A. Dicus,
presentation at DPF-2004, {\tt hep-ph/0411024}. 
%%CITATION = HEP-PH 0411024;%%

\bibitem{PDG}
Particle Data Group, Phys. Lett. B\,{\bf 592}, 1 (2004)
and {http://pdg.lbl.gov}.


\bibitem{DT}
%\vspace*{-0.3mm}
T. Appelquist and J. Carrazone, 
Phys. Rev. D\,{\bf 11}, 2856 (1975).    




\end{thebibliography}


\end{document}
\end


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
Title:    Scales of Mass Generation for Quarks, Leptons and Majorana Neutrinos
Authors:  Duane A. Dicus,  Hong-Jian He 
Comments: Phys. Rev. Lett.(in press), minor rewording 
\\
We study  2 --> n inelastic fermion-(anti)fermion scattering into multiple 
longitudinal weak gauge bosons and derive universal upper bounds on the 
scales of fermion mass generation by imposing unitarity of the S-matrix.
We place new upper limits on the scales of fermion mass generation, 
independent of the electroweak symmetry breaking scale.
We find that the strongest 2 --> n limits fall in a narrow range, 
3-170 TeV (with n=2-24), depending on the observed fermion masses. 
\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



hep-ph0502178.extracted_bib
\begin{thebibliography}{0}
%\vspace*{-3mm}

\bibitem{superK}
%Super-Kamiokande Collaboration, 
%S. Fukuda, {\it et al.}, 
%Phys. Rev. Lett. %{\bf 85}, 3999 (2000);
%{\bf 86}, 5656 (2001);
%{\bf 82}, 1810 (1999);
%{\bf 81}, 1562 (1998);
%{\bf 81}, 1158 (1998); 
%E.g., 
Y. Ashie, {\it et al.}, 
[SuperK-Kamiokande Collaboration],
Phys. Rev. Lett. {\bf 93}, 101801 (2004) 
[{\tt hep-ex/0404034}];
O. Elgaroy and O. Lahav, 
New J.\ Phys.\ {\bf 7}, 61 (2005).
%[{\tt hep-ph/0412075}].


\bibitem{wein5}
S. Weinberg, Phys. Rev. Lett. {\bf 43}, 1566 (1979).  


\bibitem{nu-seesaw}
P. Minkowski, Phys. Lett. B\,{\bf 67}, 421 (1977); 
M. Gell-Mann, P. Ramond, R. Slansky,
in {\it Proceedings of Workshop on Supergravity}, 
eds. F. van Nieuwenhuizen and D. Freedman (Amsterdam, 1979), 
p.315;
T. Yanagida, in {\it Proceedings of Workshop on Unified Theories
and Baryon Number in Universe}, 
eds. O. Sawada and A. Sugamoto (KEK, 1979), 
p.95;
S. L. Glashow, in {\it Quark and Leptons,} 
eds. M. Levy et al.\
(Plenum, NY, 1980);
R. N. Mohapatra and G. Senjanovic, 
Phys. Rev. Lett. {\bf 44}, 912 (1980).


\bibitem{nu-rad}
A. Zee, Phys. Lett. B\,{\bf 93}, 389 (1980);
L. Wolfenstein, Nucl. Phys. B\,{\bf 175}, 92 (1980);
S. T. Petcov, Phys. Lett.   B\,{\bf 115}, 401 (1982).



\bibitem{CCWZ}
C. G. Callen, S. Coleman, J. Wess, and B. Zumino,
Phys. Rev. {\bf 177}, 2247 (1969).
%S. Weinberg, Physica A\,{\bf 96}, 327 (1979).


\bibitem{DM}
D. A. Dicus and V. S. Mathur, Phys. Rev. D\,{\bf 7}, 3111 (1973);
%\bibitem{LQT}
B. W. Lee, C. Quigg and H. B. Thacker, 
Phys. Rev. D\,{\bf 16}, 1519 (1977);
%\bibitem{Vel}
M. Veltman,  Acta. Phys. Polon. B\,{\bf 8}, 475 (1977).


\bibitem{AC}
T. Appelquist and M. S. Chanowitz, 
Phys. Rev. Lett. {\bf 59}, 2405 (1987).


\bibitem{MVW}
W. Marciano, G. Valencia, and S. Willenbrock,
Phys. Rev. D\,{\bf 40}, 1725 (1989).


\bibitem{scott-PRL}
F. Maltoni, J. M. Niczyporuk, and S. Willenbrock,
Phys. Rev. Lett. {\bf 86}, 212 (2001)
[{\tt hep-ph/0006358}].
%%CITATION = HEP-PH 0006358;%%


\bibitem{scott}
F. Maltoni, J. M. Niczyporuk, and S. Willenbrock,
Phys. Rev. D\,{\bf 65}, 033004 (2002)
[{\tt hep-ph/0106281}].
%%CITATION = HEP-PH 0106281;%%


\bibitem{ET}
For a comprehensive review,
H.-J. He, Y.-P. Kuang and C.-P. Yuan,
DESY-97-056 [{\tt hep-ph/9704276}].
%%CITATION = HEP-PH 9704276;%%
%and references therein.
%H.-J. He and W.\,B. Kilgore,
%Phys. Rev. D{\bf 55}, 1515 (1997);  %[{\tt hep-ph/9609326}].
%%CITATION = HEP-PH 9609326;%% 
%H.-J. He, Y.-P. Kuang, C.-P. Yuan,
%Phys.\ Rev.\ D{\bf 51}, 6463 (1995); %[{\tt hep-ph/9410400}],
%%CITATION = HEP-PH 9410400;%%
%H.-J. He, Y.-P. Kuang, X. Li,
%Phys.\ Lett.\ B{\bf 329}, 278 (1994);
%%CITATION = HEP-PH 9403283;%%
%Phys.\ Rev.\  D{\bf 49}, 4842 (1994);
%%CITATION = PHRVA,D49,4842;%%
%Phys.\ Rev.\ Lett.\ {\bf 69}, 2619 (1992);
%%CITATION = PRLTA,69,2619;%%
%for a comprehensive review,  
%H.-J. He, Y.-P. Kuang, C.-P. Yuan, 
%DESY-97-056 [{\tt hep-ph/9704276}],
%and references therein.



\bibitem{PC}
%\vspace*{-0.3mm}
H.-J. He, Y.-P. Kuang and C.-P. Yuan,
Phys. Rev.  D\,{\bf 55}, 3038 (1997); % [{\tt hep-ph/9611316}];\\
%%CITATION = HEP-PH 9611316;%%
Phys. Lett. B\,{\bf 382}, 149 (1996). % [{\tt hep-ph/9604309}].
%%CITATION = HEP-PH 9604309;%%
%and a comprehensive review,
%DESY-97-056 [{\tt hep-ph/9704276}].
%%CITATION = HEP-PH 9704276;%%


\bibitem{DH} 
D.\,A. Dicus and H.-J. He, Phys. Rev. D (2005), in Press,
%PRD(2005),
[{\tt hep-ph/0409131}];
%%CITATION = HEP-PH 0409131;%%
and also, H.-J. He and D. A. Dicus,
presentation at DPF-2004, {\tt hep-ph/0411024}. 
%%CITATION = HEP-PH 0411024;%%

\bibitem{PDG}
Particle Data Group, Phys. Lett. B\,{\bf 592}, 1 (2004)
and {http://pdg.lbl.gov}.


\bibitem{DT}
%\vspace*{-0.3mm}
T. Appelquist and J. Carrazone, 
Phys. Rev. D\,{\bf 11}, 2856 (1975).    




\end{thebibliography}