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\begin{document}

\rightline{UH-511-970-00}
\rightline

\vspace*{4cm}
\title{HIGGS TRANSVERSE MOMENTUM AT THE LHC}

\author{ C. BAL\'AZS }

\address{Department of Physics and Astronomy, University of Hawaii,\\
Honolulu, HI 96822, U.S.A.}

\maketitle

\thispagestyle{empty}

\abstracts{
Resummed QCD corrections to the transverse momentum ($Q_T$) distribution 
of the Standard Model Higgs boson, produced at the CERN Large Hadron 
Collider, are presented. The small $Q_T$ factorization formalism is 
reviewed, which is used to extend the standard hadronic factorization 
theorem to the low $Q_T$ region, with emphasis on the matching to the 
standard hadronic factorization. Comparison of the $Q_T$ predictions from 
the extended factorization and the parton shower method is performed.
}


\section{Introduction}

One of the fundamental open problems of the Standard Model (SM) is 
revealing the dynamics of the electroweak symmetry breaking (EWSB). The 
physical remnant of the spontaneous EWSB, the Higgs boson, is the primary 
object of search at present and future colliders.\,\cite{HiggsSearch} 
At the 14 TeV center of mass energy Large Hadron Collider (LHC) at CERN, 
the SM Higgs boson will mainly be produced in proton--proton collisions 
through the partonic subprocess $g g$ (via top quark loop) $\to H 
X$.
%\,\cite{Spira:1997zs} 
To experimentally understand the Higgs signature, 
to enhance the statistical significance of the signal, and to measure its 
basic properties (mass, lifetime, spin, charge, couplings), it is necessary 
to determine the transverse momentum ($Q_T$) of the Higgs bosons. 

To reliably predict the $Q_T$ distribution of Higgs bosons at the LHC, 
especially for the low to medium $Q_T$ region where the bulk of the rate 
is, the effects of the multiple soft--gluon emission have to be included. 
This, and the need for the systematic inclusion of the higher order QCD 
corrections require the extension of the standard hadronic factorization 
theorem to the low $Q_T$ region. With a smooth matching to the usual 
factorization formalism, it is possible to obtain a prediction in the full 
$Q_T$ range. Since Monte Carlo event generators are heavily utilized in the
extraction of the Higgs signal, it is crucial to establish the reliability of 
their predictions. Thus, results of the parton shower formalism are also 
compared to those of the analytic calculation.


\section{Low $Q_T$ Factorization}

% The problem

In this section the low transverse momentum factorization formalism and 
its matching to the usual factorization is described. When calculating 
fixed order QCD corrections to the cross section ${d\sigma}/{dQ dQ_T^2 
dy}$ for the inclusive process $pp \to H X$, the factorization theorem is 
invoked. The relevant variables $Q$, $Q_T$, and $y$, are the invariant 
mass, transverse momentum, and rapidity of the Higgs boson, respectively. 
The standard factorization
\be 
%\frac{d\sigma}{dQ dQ_T^2 dy} = 
d\sigma = 
\sum_{j_1, j_2} \int \frac{d \xi_1}{\xi_1} \frac{d \xi_2}{\xi_2}
f_{j_1/h_1}(x_1,Q)\,
%\frac{d{\hat \sigma}_{j_1j_2}\left(\frac{x_1}{\xi_1},\frac{x_2}{\xi_2}\right)}
%     {dQ dQ_T^2 dy}\,
d{\hat \sigma}_{j_1j_2}\left(\frac{x_1}{\xi_1},\frac{x_2}{\xi_2}\right)\,
f_{j_2/h_2}(x_2,Q) ,
\ee
a convolution in the partonic momentum fractions $x_1$ and $x_2$, fails 
when $Q_T \ll Q$ as a result of multiple soft and soft+collinear 
emission of gluons from the initial state. The ratio of the two very 
different scales in the partonic cross section ${\hat \sigma}_{j_1j_2}$
(identified by the partonic indices $j_i$), 
produces large logarithms of the form $\ln(Q^2/Q_T^2)$ (being singular 
at $Q_T = 0$), which are not absorbed by the parton distribution functions 
$f_{j/h}$, unlike the ones originating from purely collinear parton 
emission. As a result, the Higgs $Q_T$ distribution calculated using the 
conventional hadronic factorization theorem is unphysical in the low $Q_T$ 
region.

% The solution

To resolve the problem, the differential cross section is split into a part 
which contains all the contribution from the logarithmic terms ($W$), and 
into a regular term ($Y$): 
\be 
\frac{d\sigma}{dQ dQ_T^2 dy} = W(Q,Q_T,x_1,x_2) + Y(Q,Q_T,x_1,x_2) ,
\ee
Since $Y$ does not contain potentially large logs, it can be calculated 
using the usual factorization. The $W$ term has to be evaluated 
differently, keeping in mind that failure of the standard factorization 
occurs because it neglects the transverse motion of the incoming partons 
in the hard scattering. As it is 
proven~\cite{CollinsSoperYTerm,CollinsSoperSterman}, small $Q_T$ 
factorization gives the cross section as a convolution of transverse 
momentum distributions 
\be
W(Q,Q_T,x_1,x_2) = 
\sum_{j_1, j_2} \int d^2 \vec k_T \ 
{\tt C}_{j_1/h_1}(Q,\vec k_T,x_1)\,
{\tt H}_{j_1j_2}(Q,Q_T)\,
{\tt C}_{j_2/h_2}(Q,\vec Q_T - \vec k_T,x_2) .
\ee
Here ${\tt H}_{j_1j_2}$ is a hard scattering function, and ${\tt C}_{j/h}$ 
are partonic density distributions depending on both longitudinal ($x$) 
and transverse ($k_T$) momenta, and on the scale of the factorization 
which is set equal to the hard scale $Q$. The convolution simplifies to a 
product in the Fourier conjugate, i.e. transverse position (${\vec b}$) space
\be
\widetilde{W}(Q,b,x_1,x_2) = 
{\cal C}_{j_1/h_1}(Q,b,x_1)\,
{\cal H}_{j_1j_2}(Q,b)\,
{\cal C}_{j_2/h_2}(Q,b,x_2) ,
\ee
where $\widetilde{W}$, ${\cal C}_{j/h}$ and ${\cal H}_{j_1j_2}$ are the 
Fourier transforms of $W$, ${\tt C}_{j/h}$ and ${\tt H}_{j_1j_2}$. The 
generalized parton distributions ${\cal C}_{j/h}$, together with the hard 
scattering function, satisfy an evolution equation somewhat similar to the 
usual DGLAP equations. 
%Since the virtual QCD corrections, connecting ${\cal C}_{j_1/h_1}$, ${\cal 
%C}_{j_2/h_2}$ and ${\cal H}_{j_1j_2}$, can become soft, the evolution equation 
%has to involve both ${\cal C}_{j/h}$ and ${\cal H}_{j_1j_2}$. 
%{\tilde W} is renormalization group invariant, should all order perturbative 
%corrections are included.

% The CSS formula

The evolution equation, for the production of a colorless boson, takes the 
form\,\cite{CollinsSoperSterman}
\be
\frac{\partial}{\partial \ln Q^2}\widetilde{W}(Q,b,x_1,x_2) = 
- \int_{C_1/b^2}^{C_2 Q^2} \frac{d \mu^2}{\mu^2} 
\left[ 
A \left(\alpha_S(\mu),C_1\right) + 
B \left(\alpha_S(Q),C_1,C_2\right)
\right] ,
\ee
where $\alpha_S$ is the strong coupling constant.
It is customary to choose the renormalization scales arising in the 
evolution equation such that $C_1=2e^{- \gamma _E}\equiv C_0$ and $C_2=1$. 
The solution of the above evolution equation leads to the 
expression\,\cite{CollinsSoperSterman}
\be
\widetilde{W}(Q,b,x_1,x_2) = 
{\cal C}_{j_1/h_1}(Q,b,x_1)\,
e^{-{\cal S}(Q,b_*)}\,
{\cal C}_{j_2/h_2}(Q,b,x_2) ,
\ee
with the Sudakov exponent
\begin{equation}
{\cal S}(Q,b_*) = 
\int_{C_0^2/b_*^2}^{Q^2} \frac{d \mu^2}{\mu^2}
\left[
A \left( \alpha_S(\mu) \right) \ln \left( \frac{Q^2}{\mu^2} \right) + 
B \left( \alpha_S(\mu) \right)
\right] ,
\label{Eq:PerturbativeSudakov}
\end{equation}
which resums the large logarithmic terms.\,\footnote{To prevent evaluation 
of the Sudakov exponent in the non--perturbative region, the impact 
parameter $b$ was replaced by $b_* = b/\sqrt{1+(b/b_{\rm max })^2}$.} The 
partonic recoil against soft gluons as well as the intrinsic partonic 
transverse momentum are included in the modified parton distributions
\begin{equation}
{\cal C}_{j/h}(Q,b,x) =
\sum_a \left[ \int_x^1{\frac{d\xi }\xi }
C_{ja}\left( b_*,\frac{x}{\xi},Q \right) f_{a/h}(\xi,Q) \right]
{\cal F}_{a/h}(x,b,Q) .
\label{CalC}
\end{equation}
The $A$ and $B$ functions, and the Wilson coefficients $C_{ja}$ are free 
of logs and safely calculable perturbatively as expansions in the strong
coupling
\begin{equation}
A(\alpha_S) =
\sum_{n=1}^\infty 
\left( \frac{\alpha _S}\pi \right)^n A^{(n)}, ~~~ {\rm etc.}
\end{equation}
The process independent non--perturbative functions ${\cal F}_{a/h}$, 
describing long distance transverse physics, are extracted from low--energy 
experiments.\,\cite{Brock}

% Matching

In the large $Q_T$ region, where $Q_T \sim Q$, the standard factorization 
theorem is applicable. The matching of the small $Q_T$ region to the large 
$Q_T$ result is achieved via the $Y$ piece. To correct the behavior of the 
resummed piece in the intermediate and high $Q_T$ regions, it is defined 
as the difference of the differential cross section calculated from the 
standard factorization formula at a fixed order $n$ of perturbation theory 
and its $Q_T\ll Q$ asymptote:\,\footnote{The expression of the $Y$ term 
for Higgs production can be found elsewhere.\,\cite{Yuan}}
\begin{equation}
Y(Q,Q_T,x_1,x_2)=
      \frac{d\sigma^{(n)}}{dQ^2\,dQ_T^2dy} -
\left.\frac{d\sigma^{(n)}}{dQ^2\,dQ_T^2dy}\right|_{Q_T\ll Q} .  
\label{Y.def}
\end{equation}
Using this definition, the cross section to order $\alpha_S^n$ is written as
\begin{equation}
\frac{d\sigma }{dQ^2\,dQ_T^2dy}=
W(Q,Q_T,x_1,x_2) +
      \frac{d\sigma^{(n)}}{dQ^2\,dQ_T^2dy} -
\left.\frac{d\sigma^{(n)}}{dQ^2\,dQ_T^2dy}\right|_{Q_T\ll Q} .  
\label{Eq:CSSMatched}
\end{equation}
At low $Q_T$, when the logarithms are large, the asymptotic part dominates 
the $Q_T$ distribution, and the last two terms cancel in 
Eq.~(\ref{Eq:CSSMatched}), leaving $W$ well approximating the cross section. 
At $Q_T$ values comparable to $Q$ the logarithms are small, and the expansion 
of the resummed term cancels the logarithmic terms up to higher orders in 
$\alpha_S$.\footnote{The cancellation is higher order than the order at 
which the singular pieces were calculated.} In this situation the first 
and third terms nearly cancel and the cross section reduces to the fixed order 
perturbative result. After matching the resummed and fixed order cross 
sections in such a manner, it is expected that the normalization of the 
cross section (\ref{Eq:CSSMatched}) reproduces the fixed order total rate, 
since when expanded and integrated over $Q_T$ it deviates from the fixed order 
cross section $\sigma^{(n)}$ only in higher order terms. %\,\cite{BalazsYuanWZ}
Further details of the low $Q_T$ factorization formalism and its application to 
Higgs production can be found in the recent 
literature.\,\cite{BalazsYuanWZthesis}

\section{Higgs $Q_T$ at the LHC}

The low $Q_T$ factorization formalism, described in the previous section, 
is utilized to calculate the QCD corrections to the production of Higgs 
bosons at the LHC. %~\cite{BalazsYuanH} 
In the low $Q_T$ region this 
calculation takes into account the effects of the multiple--soft gluon 
emission including the Sudakov exponent ${\cal S}$ and the non--perturbative 
contributions ${\cal F}_{a/h}$. In the Sudakov exponent the 
$A^{(1)}$, $A^{(2)}$, and $B^{(1)}$ coefficients are included. The ${\cal 
O}(\alpha_S^3)$ virtual corrections are also taken into account by 
including the Wilson coefficient ${\cal C}_{gg}^{(1)}$, which ensures the 
${\cal O}(\alpha_S^3)$ total rate. By matching to the ${\cal 
O}(\alpha_S^3)$ fixed order distributions a prediction is obtained for the 
Higgs production cross section in the full $Q_T$ range which is valid up 
to ${\cal O}(\alpha_S^3)$. The details of this calculation are given in an 
earlier work.~\cite{BalazsYuanH} The analytic results are coded in the 
ResBos Monte Carlo event generator.~\cite{BalazsYuanWZthesis}

% Higgs QT
\begin{figure}
\begin{center}
\epsfig{file=fig_pythia.eps,width=8.65cm}
\end{center}
\caption{
Higgs boson transverse momentum distributions calculated by ResBos 
(curves) and PYTHIA (histograms). The default (middle) ResBos curve was 
calculated with the canonical choice of the renormalization constants, and 
the other two with doubled (lower curve) and halved (upper curve) values 
of $C_1$ and $C_2$. For PYTHIA, the original output with default input 
parameters (dashed), the same rescaled by a factor of $K = 2$ (dash-dotted), 
and a curve calculated by the altered input parameter value $Q_{max}^2 = 
s$ (dotted) are shown. The lower portion, with a logarithmic scale, 
also shows the high $Q_T$ region.
\label{Fig:PYTHIA}}
\end{figure}

Fig.~\ref{Fig:PYTHIA} compares the Higgs boson transverse momentum 
distributions calculated by ResBos (curves) and by PYTHIA \cite{PYTHIA} 
(histograms from version 6.122). The middle solid curve is calculated 
using the canonical choice for the renormalization constants in the 
Sudakov exponent: $C_1 = C_0$, and $C_2 = 1$. To estimate the size of the 
uncalculated $B^{(2)}$ term, these renormalization constants are varied. 
The upper solid curve shows the result for $C_1 = C_0/2, C_2 = 1/2$, and 
the lower solid curve for $C_1 = 2C_0, C_2 = 2$. The band between these 
two curves gives the order of the uncertainty originating from the exclusion 
of $B^{(2)}$. The typical size of this uncertainty, e.g. around the peak 
region, is in the order of $\pm 10$ percent. The corresponding uncertainty 
in the total cross section is also in the same order. This uncertainty is 
larger than the uncertainty arising from the non--perturbative sector of 
the formalism (which was estimated to be less than 5 percent in the 
relevant $Q_T$ region).%\,\cite{BalazsYuanH}


\section{Comparison to Parton Showers}

Multiple soft--gluon radiation from the initial state can also be treated 
by the parton shower technique.\,\cite{PYTHIA} This approach is based on 
the usual factorization theorem, giving the probability ${\cal P} = 
e^{-{\cal S}(Q)}$ of the evolution from the scale $Q_0$ to $Q$, with no 
resolvable branchings by the exponent
\bea
{\cal S}(Q) =  
\int_{Q_0^2}^{Q^2} \frac{d \mu^2}{\mu^2} \frac{\alpha_S(\mu)}{2\pi} 
%\int_{\epsilon}^{1-\epsilon} dz \, C(z) \, P_{a\to bc}(z),
\int_{0}^{1} dz \, P_{a\to bc}(z),
\label{eq:ShowerS}
\eea
which is defined in terms of the DGLAP splitting kernels $P_{a\to bc}(z)$.
%\,\footnote{In Eq.(\ref{eq:ShowerS}) C(z) is a color factor.} 
%The construction of the shower involves the 
%iterative solution of the equation $r={\cal S}(Q_0)/{\cal S}(Q)$, where $0 
%\geq r \geq 1$ is a random number. 
The formalism can be extended to soft emissions as well by using angular 
ordering. The distinct difference between the Sudakov exponents of the low 
$Q_T$ factorization and the parton showering approach is apparent. 
A comparison can be made based on the qualitative argument that parton 
showering resums leading logs which depend only on the given initial 
state. These logs correspond to the logs weighted by the $A$ function in 
Eq.(\ref{Eq:PerturbativeSudakov}). 
%Accidentally B(1)...
%... the shape of the $Q_T$ distribution is mostly determined by the Sudakov
%exponent~\cite{BalazsYuanH}

Fig.~\ref{Fig:PYTHIA} shows that the shape of the PYTHIA histogram agrees 
reasonably with the resummed curve in the low and intermediate $Q_T$ 
($\lesssim 125$ GeV) region. For large $Q_T$, the PYTHIA prediction falls 
under the ResBos curve, since ResBos mostly uses the exact fixed order 
${\cal O}(\alpha_S^3)$ matrix elements in that region, while PYTHIA still 
relies on the multi--parton radiation ansatz. PYTHIA can be tuned to agree 
with ResBos in the high $Q_T$ region, by changing the maximal virtuality a 
parton can acquire in the course of the shower (dotted curve). In 
that case, however, the low $Q_T$ region will have disagreement. 

Since showering is attached to a process after the hard scattering takes 
place, and the parton shower occurs with unit probability, it does not 
change the total cross section for Higgs boson production given by the 
hard scattering rate.
%\cite{odorico}. 
Thus, the total rate is given by PYTHIA at ${\cal O}(\alpha_S^2)$. In 
Fig.~\ref{Fig:PYTHIA} the dashed PYTHIA histogram is plotted without 
altering its output. For easier comparison the default PYTHIA histogram is 
also plotted after the rate is multiplied by the factor $K = 2$ (dash-dotted).
A detailed comparison of the results for Higgs boson production from 
ResBos and from event generators based on the parton shower algorithm is 
the subject of a separate work.\,\cite{BalazsHustonPuljak}

\section*{Acknowledgments}

I thank the organizers of the XXXVth Rencontres de Moriond for their 
hospitality and financial support. I am indebted to my collaborators in 
this work: J.C. Collins, J. Huston, I. Puljak, D.E. Soper, and C.--P. Yuan 
for many invaluable discussions. This work was supported in part by the 
DOE under grant DE-FG-03-94ER40833.

\section*{References}
\begin{thebibliography}{99}
\bibitem{HiggsSearch} 
Some talks presented in this topic at the XXXVth Rencontres de Moriond are: \\
E. Ferrer,
http://moriond.in2p3.fr/EW/2000/transparencies/3$\_$Tuesday/am/Ferrer; \\
A. Quadt,
http://moriond.in2p3.fr/EW/2000/transparencies/3$\_$Tuesday/am/Quadt; \\
A. Turcot,
http://moriond.in2p3.fr/QCD00/transparencies/2$\_$monday/pm/turcot; \\
D. Zeppenfeld, .

%\bibitem{Spira:1997zs}
%M.~Spira,
%%``Higgs boson production and decay at future machines,''
%.
%%%CITATION = ;%%

\bibitem{CollinsSoperYTerm}  J.C.~Collins and D.E.~Soper,
%``Back To Back Jets In QCD: Comparison With Experiment,''
%Phys.\ Rev.\ Lett.\ \textbf{48}, 655 (1982);
\Journal{\PRL}{48}{655}{1982};
%%CITATION = PRLTA,48,655;%%
%
%``Back To Back Jets In QCD,''
%Nucl.\ Phys.\ \textbf{B193}, 381 (1981), \textbf{B213}, 545(E)~(1983);
\Journal{\NPB}{193}{381}{1981};
%%CITATION = NUPHA,B193,381;%%
%\bibitem{YTerm}
%J.C. Collins and D.E. Soper,
%``Back To Back Jets: Fourier Transform From B To K-Transverse,''
%Nucl.\ Phys.\ \textbf{B197}, 446 (1982).
\Journal{\NPB}{197}{446}{1982}.
%%CITATION = NUPHA,B197,446;%%

\bibitem{CollinsSoperSterman}  
J.C. Collins, D.E. Soper and G. Sterman, 
\Journal{\NPB}{250}{199}{1985}.
%%CITATION = NUPHA,B250,199;%%

\bibitem{Brock}
F.~Landry, R.~Brock, G.~Ladinsky and C.--P.~Yuan,
%``New fits for the non-perturbative parameters in the CSS resummation 
% formalism,''
.

\bibitem{Yuan}
C.--P.~Yuan, \Journal{\PLB}{283}{395}{1992}.

\bibitem{BalazsYuanWZthesis}
C.~Bal\'azs and C.--P.~Yuan,
%``Soft gluon effects on lepton pairs at hadron colliders,''
\Journal{\PRD}{56}{5558}{1997}; \\
%%CITATION = PHRVA,D56,5558;%%
C.~Bal\'azs, Ph.D. thesis, Michigan State University (1999),
%``Soft gluon effects on electroweak boson production in hadron 
% collisions,''
.

\bibitem{BalazsYuanH}
C.~Bal\'azs and C.--P.~Yuan,
%``Higgs boson production at the LHC with soft gluon effects,''
\Journal{\PLB}{478}{192}{2000}.
%.

\bibitem{PYTHIA}
T.~Sj\"ostrand,
%``High-energy physics event generation with PYTHIA 5.7 and JETSET 7.4,''
Comput.\ Phys.\ Commun.\ {\bf 82}, 74 (1994).
%%CITATION = CPHCB,82,74;%%

\bibitem{BalazsHustonPuljak}
C.~Bal\'azs, J.~Huston and I.~Puljak .

\end{thebibliography}

\end{document}

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Next adjust the
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and remove the declarations of the
lines and any anomalous spacing.

If you prefer to use some other method then it's very important to
leave the correct amount of vertical space in the figure declaration
to accomodate your figure (remove the lines and change the vspace in the
example.) Send the hard copy figures on separate pages with clear
instructions to match them to the correct space in the final hard copy
text. Please ensure the final hard copy figure is correctly scaled to
fit the space available (this ensures the figure is legible.)

The caption heading for a figure should be placed below the figure.

\subsection{Limitations on the Placement of Tables,
Equations and Figures}\label{sec:plac}

Very large figures and tables should be placed on a page by themselves. One
can use the instruction {\em $\backslash$begin\{figure\}$[$p$]$} or
{\em $\backslash$begin\{table\}$[$p$]$}
to position these, and they will appear on a separate page devoted to
figures and tables. We would recommend making any necessary
adjustments to the layout of the figures and tables
only in the final draft. It is also simplest to sort out line and
page breaks in the last stages.

\subsection{Acknowledgments, Appendices, Footnotes and the Bibliography}
If you wish to have
acknowledgments to funding bodies etc., these may be placed in a separate
section at the end of the text, before the Appendices. This should not
be numbered so use {\em $\backslash$section$\ast$\{Acknowledgments\}}.

It's preferable to have no appendices in a brief article, but if more
than one is necessary then simply copy the
{\em $\backslash$section$\ast$\{Appendix\}}
heading and type in Appendix A, Appendix B etc. between the brackets.

Footnotes are denoted by a letter superscript
in the text,\footnote{Just like this one.} and references
are denoted by a number superscript.
We have used {\em $\backslash$bibitem} to produce the bibliography.
Citations in the text use the labels defined in the bibitem declaration,
for example, the first paper by Jarlskog~\cite{ja} is cited using the command
{\em $\backslash$cite\{ja\}}.

If you more commonly use the method of square brackets in the line of text
for citation than the superscript method,
please note that you need  to adjust the punctuation
so that the citation command appears after the punctuation mark.

\subsection{Final Manuscript}

The final hard copy that you send must be absolutely clean and unfolded.
It will be printed directly without any further editing. Use a printer
that has a good resolution (300 dots per inch or higher). There should
not be any corrections made on the printed pages, nor should adhesive
tape cover any lettering. Photocopies are not acceptable.

The manuscript will not be reduced or enlarged when filmed so please ensure
that indices and other small pieces of text are legible.

\section{Sample Text }

The following may be (and has been) described as `dangerously irrelevant'
physics. The Lorentz-invariant phase space integral for
a general n-body decay from a particle with momentum $P$
and mass $M$ is given by:
\begin{equation}
I((P - k_i)^2, m^2_i, M) = \frac{1}{(2 \pi)^5}\!
\int\!\frac{d^3 k_i}{2 \omega_i} \! \delta^4(P - k_i).
\label{eq:murnf}
\end{equation}
The only experiment on $K^{\pm} \ra \pi^{\pm} \pi^0 \gamma$ since 1976
is that of Bolotov {\it et al}.~\cite{bu}
        There are two
necessary conditions required for any acceptable
parametrization of the
quark mixing matrix. The first is that the matrix must be unitary, and the
second is that it should contain a CP violating phase $\delta$.
 In Sec.~\ref{subsec:fig} the connection between invariants (of
form similar to J) and unitarity relations
will be examined further for the more general $ n \times n $ case.
The reason is that such a matrix is not a faithful representation of the group,
i.e. it does not cover all of the parameter space available.
\begin{equation}
\begin{array}{rcl}
\bf{K} & = &  Im[V_{j, \alpha} {V_{j,\alpha + 1}}^*
{V_{j + 1,\alpha }}^* V_{j + 1, \alpha + 1} ] \\
       &   & + Im[V_{k, \alpha + 2} {V_{k,\alpha + 3}}^*
{V_{k + 1,\alpha + 2 }}^* V_{k + 1, \alpha + 3} ]  \\
       &   & + Im[V_{j + 2, \beta} {V_{j + 2,\beta + 1}}^*
{V_{j + 3,\beta }}^* V_{j + 3, \beta + 1} ]  \\
       &   & + Im[V_{k + 2, \beta + 2} {V_{k + 2,\beta + 3}}^*
{V_{k + 3,\beta + 2 }}^* V_{k + 3, \beta + 3}] \\
& & \\
\bf{M} & = &  Im[{V_{j, \alpha}}^* V_{j,\alpha + 1}
V_{j + 1,\alpha } {V_{j + 1, \alpha + 1}}^* ]  \\
       &   & + Im[V_{k, \alpha + 2} {V_{k,\alpha + 3}}^*
{V_{k + 1,\alpha + 2 }}^* V_{k + 1, \alpha + 3} ]  \\
       &   & + Im[{V_{j + 2, \beta}}^* V_{j + 2,\beta + 1}
V_{j + 3,\beta } {V_{j + 3, \beta + 1}}^* ]  \\
       &   & + Im[V_{k + 2, \beta + 2} {V_{k + 2,\beta + 3}}^*
{V_{k + 3,\beta + 2 }}^* V_{k + 3, \beta + 3}],
\\ & &
\end{array}\label{eq:spa}
\end{equation}
where $ k = j$ or $j+1$ and $\beta = \alpha$ or $\alpha+1$, but if
$k = j + 1$, then $\beta \neq \alpha + 1$ and similarly, if
$\beta = \alpha + 1$ then $ k \neq j + 1$.\footnote{An example of a
matrix which has elements
containing the phase variable $e^{i \delta}$ to second order, i.e.
elements with a
phase variable $e^{2i \delta}$ is given at the end of this section.}
   There are only 162 quark mixing matrices using these parameters
which are
to first order in the phase variable $e^{i \delta}$ as is the case for
the Jarlskog parametrizations, and for which J is not identically
zero.
It should be noted that these are physically identical and
form just one true parametrization.
\bea
T & = & Im[V_{11} {V_{12}}^* {V_{21}}^* V_{22}]  \nonumber \\
&  & + Im[V_{12} {V_{13}}^* {V_{22}}^* V_{23}]   \nonumber \\
&  & - Im[V_{33} {V_{31}}^* {V_{13}}^* V_{11}].
\label{eq:sp}
\eea


\begin{figure}
\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
\vskip 2.5cm
\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
%\psfig{figure=filename.ps,height=1.5in}
\caption{Radiative (off-shell, off-page and out-to-lunch) SUSY Higglets.
\label{fig:radish}}
\end{figure}

\section*{Acknowledgments}
This is where one places acknowledgments for funding bodies etc.
Note that there are no section numbers for the Acknowledgments, Appendix
or References.

\section*{Appendix}
 We can insert an appendix here and place equations so that they are
given numbers such as Eq.~\ref{eq:app}.
\be
x = y.
\label{eq:app}
\ee


