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\begin{document}
\title{Do LEP results suggest that quarks have integer electric charges?}
\author{P.M. Ferreira \\ CFTC, Faculdade de Ci\^encias,\\ 
Universidade de Lisboa, Portugal}
\date{September, 2002} 
\maketitle
\noindent
{\bf Abstract:} We argue that recent results from two-photon processes at LEP
are better explained by quarks possessing integer electric charges.
\vspace{-9cm}
\vspace{10cm}

 
Photon-photon collisions have been very thoroughly studied over the past 
years~\cite{rev}. It has been established~\cite{zer,wit} that the photon may
interact both as a point-like particle, the so-called ``direct" processes 
described by perturbative QED and QCD, and as a composite structure, through
``resolved" processes. In the latter it is believed that one of the photons 
first fluctuates into an hadronic state and those partons interact with the 
second photon. Double resolved processes where both photons behave as having a 
structure are also possible. These processes clearly require non-perturbative 
physics to be described and as such become quite complicated. Two-photon physics
has been successful in reproducing most results from particle accelerators up to
LEP2 energies~\cite{wen} requiring Reggeon and Pomeron 
parametrisations~\cite{pom} to fit, for instance, the total cross section of 
hadron production by two photons~\cite{opal}. However, recent LEP2 results for 
the production of heavy quarks in two-photon collisions~\cite{bb} have raised a 
problem. As can be appreciated in figure~\ref{fig:lep}, we manage to reproduce 
the total cross section for the production of charm pairs at LEP2 by using 
contributions from both direct and resolved processes, at next-to-leading-order 
(NLO) in QCD. In the same plot we see that even with substantial contributions 
from resolved processes the theory is unable to reproduce the data for bottom 
quark production. The Standard Model (SM) prediction is a factor of 3 lower than
the experimental results. 

It is clear from fig.~\ref{fig:lep} that perturbative physics alone is 
insufficient, in the SM, to reproduce the experimental results. This despite the
theoretical predictions having some dependence on the input quark mass and the 
choice of renormalisation scale. A possible explanation for the discrepancy on 
the $b\bar{b}$ cross section was given in ref.~\cite{ber}, requiring a 
supersymmetric model with a light colour octet gluino. In this letter we propose
a different solution: that quarks have integer electric charges and the cross
sections for open charm and bottom production in two-photon processes at LEP2
energies come almost entirely from perturbative processes. Integer charged quark
(ICQ) theories were first proposed by Han and Nambu~\cite{han}. A renormalisable
version was first obtained by Pati and Salam~\cite{pati} and several ICQ 
theories have been obtained via gauge symmetry breaking, differing on the 
initial gauge group, generally larger than the SM's. Witten 
remarked~\cite{wit} that the reaction $e^\pm\,e^\pm \rightarrow e^\pm\, e^\pm\,q
\,\bar{q}$ is the preferential process to establish the character of the quarks'
electric charges.  In fact, as we shall shortly see, the perturbative cross 
sections for this reaction are very different for an ICQ theory or a 
fractionally charged quark (FCQ) one. Detailed measurements of the 
photon structure functions revealed the importance of their non-perturbative 
component.  At low energies or transverse momentum it became impossible to 
disentangle the perturbative contributions from the non-perturbative ones. 
However, it seems a reasonable expectation that at the very high LEP2 energies
the non-perturbative contributions should be small (that is certainly true for 
FCQ theories, see for instance ref.~\cite{dree}). Plus, for charm or bottom 
quark production, the mass of these quarks being so large, we have a ``natural" 
large scale for QCD processes. Once again, it seems reasonable to assume that
perturbative physics should dominate. We will take this as our starting point 
and calculate only perturbative cross sections. 

Single photon reactions give identical results for ICQ or FCQ theories 
(see~\cite{rev}, for instance), the quantity $R = \sigma(e^+ e^- \rightarrow q\,
\bar{q})/\sigma(e^+ e^- \rightarrow \mu^+ \mu^-)$ being the best example: we
do not observe quarks of a given colour individually, rather mesons or baryons 
containing 
them; as such the amplitude of the process $e^+e^- \,\rightarrow\, q\,\bar{q}$ 
is the sum of the amplitudes for each quark's colour,
%%
\begin{equation}
{\cal M}(e^+e^- \,\rightarrow\, q\,\bar{q}) \;=\; \sum_{i=1}^3 \; {\cal M}(e^+
e^- \, \rightarrow\, q^i\,\bar{q}^i) \;\;\; .
\label{eq:amp}
\end{equation}
Now, the probability of ``finding" a quark of a particular colour in the final
state is $1/3$, so the cross section $\sigma(e^+e^- \,\rightarrow\, q\,\bar{q})$
is proportional to the square of the modulus of the amplitude~\eqref{eq:amp} 
multiplied by a factor of $1/3$ and the quantity $R$ is given by
%%
\begin{equation}
R \;=\; \displaystyle{\frac{1}{3}}\,\displaystyle{\sum_q} \,
\left( \displaystyle{\sum_{\mbox{i=colours}}}\, {e_q^i}\right)^2 \;\;\; .
\end{equation} 
It should be noted that in ICQ theories quarks have charges that depend on 
their colour index. An up-type quark, for instance, has charge $(+1, +1, 0)$ for
the quark's colour indices $(1, 2, 3)$. For a down-type quark, the charge is
$(0, 0, -1)$. It is then trivial to see that for both ICQ and FCQ theories
we obtain the same contributions to $R$ ($4/3$ from an up-type quark, $1/3$ from
a down-type one). It can be shown (see for instance ref.~\cite{eu}) that all
processes involving a single photon (or indeed a single $Z^0$) give the same
results in both ICQ and FCQ theories; the basic argument that could also be 
applied to the example we just discussed is that the electromagnetic current $J$
is, in ICQ theories, composed of two pieces, $J\,=\,J_F\,+\,J_8$, where $J_F$ is
the ``usual" current with fractional charges for quarks and $J_8$ the ICQ 
contribution, given by $\lambda_8/\sqrt{3}$. Because the final and initial 
states of any process are colour singlets we must take the trace (in colour 
space) of the amplitude and divide the square by the number of colours. Thus,
for any process involving a single photon (a single current, then), the extra
ICQ contribution is cancelled and the result is the same of an FCQ theory. We 
must therefore have at least two currents $J$ to have different predictions for
ICQ or FCQ theories. An important such case are the decays of $\eta$ mesons into two photons, which have been used in the past as 
evidence against ICQ theories~\cite{chan}. This is based on the fact that a 
simple PCAC analysis of such decays yields for its width the 
expression~\cite{berg}
\begin{equation}
\Gamma_{\gamma\gamma}^X \;=\; \left(\sum_{\mbox{colour}} \,<e_q^2>\right)^2\,
\frac{\alpha^2}{32\pi^3}\,\frac{m_X^3}{f_X^2} \;\;\;\ ,
\end{equation}
where $\alpha$ is the fine structure constant, $m_X$ and $f_X$ are the mass and
decay constant of the meson under consideration and $<e_q^2>$ the mean value of
the squared charges of the quarks in the state given by the meson's 
wavefunction. At this point, in a simple approach, one considers the so-called
``nonet symmetry", that is, the equality of the decay constants for the mesons 
belonging to the same flavour multiplet. The emblematic cases for comparison
between ICQ and FCQ theories are the pion, the $\eta_1$ and the $\eta_8$ states,
so we assume $f_1 = f_8 = f_\pi$. It is simple to show that for the pion and 
$\eta_8$ states the widths are the same for both cases but, for the $\eta_1$, 
the ICQ width is 4 times larger than the FCQ one. The first obvious retort to 
this ``evidence" against ICQ theories is that it relies heavily on untested 
theoretical assumptions, to wit the equality of the decay constants. There is in
fact no {\em a priori} reason why nonet symmetry should hold - in a na\"{\i}ve 
quark model such an assumption might be plausible if the physical mesons $\eta$ 
and $\eta^\prime$ (which result from the mixing of the $\eta_1$ and $\eta_8$ 
states) were ideally mixed but as argued in ref.~\cite{chan} that is hardly the 
case. Further, more elaborate calculations have found serious deviations from 
nonet symmetry - for instance, using a Hidden Local Symmetry model, the authors 
of ref.~\cite{ban} have found $f_1 \,=\,1.4 \, f_8$. Chanowitz~\cite{chan} 
deduced equations for a $\xi$ parameter ($\xi = 1$ for FCQ, $\xi = 2$ for ICQ) 
in terms of experimentally measured quantities which were in principle 
independent of the nonet symmetry hypothesis and seemed to favour FCQ over ICQ. 
The problem there is that from the start those equations were less reliable if 
the theory being tested was ICQ. And it was shown later that even for the 
``normal" FCQ theory Chanowitz's equations could not be applied in a na\"{\i}ve 
manner as they didn't take into account the existence of a more complex mixing 
between $\eta_1$ and $\eta_8$~\cite{ben}. Heavier quarkonia states have similar
problems (for reviews, see for instance~\cite{nov,bes}. For instance, there is
a wealth of data on radiative decays of heavy mesons - these involve dipolar
transitions between different states of the quark bound states, with the
emission of a single photon. And because a single photon is involved both ICQ 
and FCQ theories would have the same predictions for those processes. As for 
two-photon decays, one finds a situation similar to the lighter mesons - there
is a remarkable FCQ prediction~\cite{nov}, 
\begin{equation}
\frac{\Gamma (\eta_c \,\rightarrow\,\gamma \gamma)}{\Gamma (J/\Psi \,\rightarrow
\,e^+\,e^-)} \,=\,\,3\,Q_c^2 \;\frac{|\eta_c(0)|^2}{|J/\Psi(0)|^2}\;\;\; .
\end{equation}
If one {\em assumes} the equality of the meson's wavefunctions at the origin
(the same type of hypothesis that nonet symmetry is based on) the FCQ prediction
is a factor of $4/3$, whereas the ICQ result would be 1. The current 
experimental values favour the FCQ result. However, to obtain this ``prediction"
we needed to make an extra assumption regarding the (unknown) wavefunctions 
of quark bound states. Consider that if the wavefunctions at the origin differ 
by about $15\%$ (and remember that $\eta_c$ and $J/\Psi$ are mesons of different
spin) the experimental results would favour ICQ theories instead. With this in
mind we must ask ourselves, are we testing the character of the quarks's 
electric charges or our models about their bound states? This is but an example 
that shows an unpleasant fact, that to use meson spectroscopy arguments to argue
against ICQ theories one has to make assumptions about the mesons' structure. We
argue, then, that those assumptions at least cast some doubt on the validity of
the final conclusions.  

A further objection is that pseudoscalar meson (or indeed any other quarkonia 
state) decays are an exceedingly difficult field of study, where great doubts 
still persist. For instance, different groups claim evidence for the presence of
a significant gluon component in the $\eta$ and $\eta^\prime$ states~\cite{ball}
or against it~\cite{ben}. The degree of sophistication and complexity involved
in these calculations is immense, and the simple fact is that no work of a 
similar kind has been done for ICQ theories - for one, even the one-loop 
contributions from ICQ theories to processes such as $e^+ e^- \rightarrow e^+ 
e^- q\bar{q}$ (via the two-photon channel) remain to be done, let alone 
non-perturbative contributions. As such it seems impossible at this point to say
whether an ICQ theory, after calculations equally elaborate as those that have 
been performed for the FCQ one, could not do as good a job, if not better, than 
the ``normal" theory in describing meson decays. 

Unlike $R$, the quantity $R_{\gamma\gamma} = \sigma(e^+ e^-\rightarrow e^+ e^- q
\bar{q})/\sigma(e^+ e^- \rightarrow e^+ e^- \mu^+ \mu^-)$ (both processes going 
through the two-photon channel) gives very different results for both types of
theory, namely (at tree-level):
%%
\begin{equation}
\begin{array}{ll} 
R_{\gamma\gamma}^{FCQ} \;= & 3\,\displaystyle{\sum_q}\,e_q^4 \\
R_{\gamma\gamma}^{ICQ}\;= & \displaystyle{\frac{1}{3}}\,\displaystyle{\sum_q} \,
\left( \displaystyle{\sum_{\mbox{i=colours}}}\, {e_q^i}^2\right)^2 \;\;\; .
\end{array}
\label{eq:rgg}
\end{equation}
Notice how the ICQ result reduces to the FCQ one when the quark electric charges
do not depend on the colour. From equation~\eqref{eq:rgg} we deduce immediately 
that for an up-type quark we would have $R_{\gamma\gamma}^{ICQ}/R_{
\gamma\gamma}^{FCQ} \, =\, 9/4$ and for a down-type one, $R_{\gamma\gamma}^{ICQ}
/R_{\gamma\gamma}^{FCQ} \, =\, 9$. Given that the production of muon pairs 
occurs identically in both models the cross section for the production of charm
pairs by photon-photon collisions in an ICQ theory is 9/4 the value of the same 
quantity calculated in an FCQ model. For bottom production, the ICQ cross 
section is 9 times greater than the FCQ one. We follow the procedure of 
ref.~\cite{dree} to calculate 
$\sigma(e^+ e^- \rightarrow e^+ e^- q\bar{q})$ in the Equivalent Photon 
Approximation. We use their suggested spectrum of Weizs\"acker-Williams photons 
and cut-off on the photon's maximum virtuality. We set the free quark production
thresholds at $(3.8\; \mbox{GeV})^2$ (for charm quarks) and $(10.6\; 
\mbox{GeV})^2$ (for bottom quarks). In figure~\ref{fig:icq} we show the 
tree-level cross sections. The dashed lines are the FCQ results, the solid lines
the ICQ ones. The comparison with fig.~\ref{fig:lep} is revealing - the ICQ 
theory not only describes almost as well the production of charm pairs, it also 
succeeds where the SM fails, the production of bottom quarks. 

The QCD corrections to this calculation should improve the agreement, as we 
expect them to increase the tree-level cross section (by no more than 30\% for 
the SM). In ICQ theories gluons generally acquire mass and electric charge; the 
QCD corrections will depend on the particular model we choose. In 
ref.~\cite{god} a model was considered with eight massive gluons of mass $\sim 
0.3$ GeV to explain the $\gamma\gamma$ results from PETRA. More recently massive
gluons of mass about 1 GeV were considered to explain the radiative decays of the $J/\Psi$ and $\Upsilon$ mesons~\cite{fiel}. For such reasonably low gluon 
masses we expect the NLO QCD corrections from the SM to be similar in form to 
those of the ICQ theory. A good estimate of the NLO QCD ICQ cross section is 
therefore to multiply the ``direct" results of fig.~\ref{fig:lep} by factors of 
9/4 and 9 for the charm and bottom quarks respectively. As can be seen from 
table 1 the agreement between data and these estimated ICQ predictions is, 
again, excellent. 
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|c|c|} \hline & & \\
 $\sqrt{s}$ (GeV) & ICQ predictions & Exp. Data  \\ & & \\ \hline & & \\
 91 & 563 - 722 & $460 \pm 92.2 $ \\ & & \\ \hline & & \\
 133 & 680 - 925 & $1360 \pm 300$ \\ & & \\ \hline & & \\
 167 & 770 - 982 & $940 \pm 172.1$ \\ & & \\ \hline & & \\
 183 & 817 - 1125 & $1290 \pm 149.7$ \\ & & \\ \hline & & \\
 194 & 900 - 1181.3 & $1016 \pm 123.7$ \\ & & \\ \hline & & \\
 194 ($b\bar{b}$) & 14 - 16 & $13.1 \pm 3.1$ \\ & & \\ \hline
\end{tabular}
\caption{Comparison of estimated ICQ NLO QCD predictions with experimental
data~\cite{bb}. The ICQ cross sections are obtained from the ``direct" curves of
fig.~\ref{fig:lep}. All cross sections are in picobarn and, except for the last 
line, refer to the reaction $e^+ e^- \rightarrow e^+ e^-\, c\,\bar{c}$.}
\end{center}
\end{table}
Recently~\cite{eu} an ICQ model with three massless gluons and five massive
ones, with masses larger than $\sim 76$ GeV, was proposed. In such a model the
gluon bremsstrahlung corrections will involve only three gluons, not eight. 
Also, given that the gluons' mass is so large, an approximation to the one-loop
cross section would be to consider the contributions of the massless gluons
alone. So, from the results of ref.~\cite{dree}, an estimate of the 
one-loop QCD correction to the FCQ cross section would be
\begin{equation}
\sigma(\gamma\gamma \rightarrow q\,\bar{q})_{1-loop} \; \simeq \;  
\frac{12\pi\alpha^2 e_Q^4}{s_{\gamma\gamma}}\,\beta\,\frac{3\alpha_s}{4\pi}\,
\left[\frac{\pi^2}{2\beta}\,-\, \left(5 - \frac{\pi^2}{4}\right) \,+\, O(\beta) 
\right]\;\;\; ,
\end{equation}
where $s_{\gamma\gamma}$ is the two-photon total energy squared and $\beta = 
(1 - 4\,m_q^2/s_{\gamma\gamma})^{1/2}$, with $m_q$ the quark mass. With three
massless gluons the $SU(2)$ Casimir $3/4$ appears. Repeating the calculations 
that led to fig.~\ref{fig:icq} we see that this estimated correction 
increases the cross sections by no more than $\sim 7\%$. This is probably an 
underestimation of one-loops effects and it improves the agreement with
the experimental results only slightly.

An exact calculation for particular ICQ models, including NLO QCD corrections, 
is clearly necessary at this stage. Such calculation could put bounds on the 
masses of the ICQ gluons (or the colour-breaking vevs of ref.~\cite{eu}). As 
striking as the numerical agreement is the fact that we obtained these results 
from a purely perturbative ICQ theory. More detailed calculations or 
measurements would indicate if there is the need of a (small, surely) 
non-perturbative component to the cross sections at these high energies. A small
resolved component seems necessary to improve the charm cross section, judging 
by our estimated results. 
Let us also emphasise that the analysis that
led to the experimental data presented in refs.~\cite{bb} assumed normal QCD 
backgrounds. With all rigour that analysis would have to be re-made assuming 
background from massive gluons but it seems reasonable to expect the end results
would not be significantly different from those now available. Finally, the 
$b\bar{b}$ cross section is not the only quantity the SM analysis cannot 
reproduce, the high-$P_T$ excess in the cross section for pion production at 
LEP2~\cite{L3}, for the two-photon channel as well, is also unaccounted for. It 
is trivial to see from figure 3 of ref.~\cite{L3} that, even assuming an ICQ
theory, the perturbative results are not enough to explain this discrepancy. But
perhaps the non-perturbative contributions, now taken with integer quark 
charges, will be able to reproduce the data?

In conclusion, we have shown that the experimental results for the total cross 
section for charm and bottom quark production at LEP2, via the two-photon 
channel, are almost perfectly described by a tree-level perturbative 
calculation if one assumes that the quarks have integer charges. We have 
estimated the QCD corrections to these calculations in two different ICQ models 
and found them to be small, as expected, and leading to an improvement in the 
agreement with the data. These results, we believe, must give us pause and 
consider the possibility quarks do, indeed, have integer electric charges. 

\vspace{0.25cm}
{\bf Acknowledgements.} I thank Augusto Barroso and Rui Santos for discussions
and suggestions. This work was supported by a fellowship from Funda\c{c}\~ao 
para a Ci\^encia e Tecnologia, SFRH/BPD/5575/2001. 

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\begin{figure}[htb]
\epsfysize=10cm
\centerline{\epsfbox{sigqq.ps}}
\caption{Cross section for production of $c\bar{c}$ and $b\bar{b}$ pairs at
LEP through the two photon channel from the L3 collaboration~\cite{bb}.}
\label{fig:lep}
\end{figure}
\begin{figure}[htb]
\epsfysize=10cm
\centerline{\epsfbox{sigicq.ps}}
\caption{Tree-level cross sections for production of charm and bottom quarks via
the two-photon channel for FCQ (dotted lines) and ICQ (solid lines) models. The
sets of two lines correspond to quark masses varying in the intervals $1.3 \leq 
m_c \leq 1.7$ GeV, $4.5 \leq m_b \leq 5.0$ GeV.}
\label{fig:icq}
\end{figure}
\end{document} 

