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\textbf{Multiparticle phase shift analysis in peripheral production 
experiments: \newline
Lessons from the Past}
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\begin{center}
Geoffrey Fox
\end{center}

\begin{center}
\textit{Indiana University}
\end{center}

\begin{center}
\textit{Computer Science, Informatics and Physics}
\end{center}

\begin{center}
\textit{Community Grid Computing Laboratory, }
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\begin{center}
\textit{501 N Morton Suite 224, Bloomington IN 47404}
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\begin{center}
\underline {gcf@indiana.edu}
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\textbf{Abstract}

We recall some of our understanding of meson resonance production in high 
energy collisions from 20 years ago and its implications for a new 
generation of high statistics experiments. We discuss S-matrix theory, 
duality and Regge exchanges. We briefly mention some computing issues.





\textbf{Introduction}

Some 20 years ago, I switched from particle physics to largely working in 
computer science. At that time, I was part of teams completing analysis of 
three experiments at Fermilab; one [1] E260 saw hadron jets at ``high'' 
transverse momentum (perhaps 5 GeV/c); another [2] E350 studied triple-Regge 
theory and measured the $\rho $ and A$_{2}$ trajectories out to --t = 2 
(GeV/c)$^{2}$; a third [3] E110 studied meson resonance production at 50 to 
200 GeV/c with $\pi ^{ - }$ and K$^{ - }$ beams. Progress has of course 
been dramatic in the area of the first experiment with results in general 
agreement with QCD now seen up to some 50 GeV/c transverse jet momentum. In 
the final two experiments, we were studying the soft limit of strong 
interactions. Here there has been some progress in understanding resonance 
structure -- both from high statistics experiments and more importantly 
major advances in understanding quark model spectroscopy. However progress 
in understanding peripheral production has been limited except for 
diffractive reactions where the Pomeron phenomenology [4] has advanced based 
on deep inelastic diffractive data. In reading this article, please note I 
am recalling understandings that I had some 20 years ago. I apologize for 
any mistakes.





\textbf{Some Partial Wave Analysis Issues}

In examining resonance M* production in reactions like $\gamma $ p $ \to $ 
M* p, one can study both the resonance's parameters (including its 
existence) and its production characteristics. Currently we have no profound 
theoretical understanding of production dynamics but do have a deeper 
understanding and interest in meson spectroscopy and so the latter is our 
primary goal. Nevertheless good phenomenology of the production dynamics 
appears essential, as it is critical in the formulation of the partial wave 
analysis (PWA). These intrinsically parameterize amplitudes and the spin, 
phase and baryon vertex structure of the amplitudes is strongly affected by 
the production dynamics. This implies that PWA is technically hard and it is 
very difficult to produce results with minimum and perhaps more importantly 
understandable bias.

Study of resonances in electron-positron collisions e$^{ + }$ e$^{ - } \quad  
\to $ M* is clean compared to the hadron case but suffers from both limited 
statistics and selection rules that imply that many important resonances 
cannot be produced in this fashion. Study of M* in e$^{ + }$ e$^{ - } \quad  \to 
$ M* X does not have selection rule restrictions but has many of the 
obfuscatory effects of the hadron case compounded by modest statistics. Thus 
for much spectroscopy, study of it via production in Baryon-Baryon and 
Photon-Baryon collisions is attractive.





We can divide such spectroscopy in three broad classes of increasing 
analysis complexity:

\textbullet \textbf{2 ? 2 reactions:} These are clearly very clean but 
limited to a few well known reactions such as $\pi $ N ? $\pi $ N , $\gamma 
$ N ? $\pi $ N and $\pi $ N ? $\eta $ N. Correspondingly this approach can 
only tackle baryon resonances. Analyses of 2 $ \to $ 2 reactions with good 
enough statistics are largely uncontroversial at the partial wave analysis 
level. However the interpretation in terms of resonances may be impossible 
if they have small couplings to the two-body channels.

\textbullet \textbf{2 ? 3 or more final particle reactions where we are 
interested in resonances decaying to full final state: }Examples are $\pi $ 
N ? $\pi  \quad \pi $ N, $\overline p p $annihilation etc. (in this case, one is 
looking at M*X but with well defined s-channel quantum numbers) where the 
full final state is partial wave analyzed. Here we have several new 
technical complications such as a large increase in the number of degrees of 
freedom. In particular realistic ways approaches to 3 particle final states 
usually require resonance assumptions which have intrinsic uncertainties. 
There are some dynamical effects (such as $\pi $ exchange) which can not be 
studied in $\pi $ N ? $\pi $ N and so their effect on the PWA is unclear. 
They could be serious as $\pi $ exchange populates higher partial waves than 
the exchanges seen in $\pi $ N ? $\pi $ N.

\textbullet \textbf{2 ?(state to be analyzed) plus one or more other 
particles (peripherally produced) -- }this in particular includes case of 
peripheral production of meson states by $\pi $ and $\gamma $ beams.\textbf{ 
}Here there are a further set of problems due to confusion of 
``other processes apart from resonance of interest'' forming full final 
state. Further it is not very easy to unambiguously define amplitudes to be 
analyzed. The latter remark assumes the very large number of degrees of 
freedom precludes solving all these problems by a general parameterization 
and ``just fitting''.





The latter difficult case is the one we are interested in! We comment on 
some of the issues in the following.

\begin{figure}[htbp]
\centerline{\includegraphics{pwa1.eps}}
\label{fig1}
\end{figure}




\textbf{S-Matrix Unitarity and Phenomenology}

The S-Matrix approach was very popular 30 years ago and established a set of 
``true principles'' which appeared to be very strong constraints on the 
scattering amplitude. The dream that these constraints could be strong 
enough to determine the amplitude proved impossible to turn into reality. 
Further no way was discovered to guarantee that known ``truths'' were 
satisfied. For example we understood as in figure 1, that a single analytic 
function described the three reactions of the s t and u channels. The 
resonances in the s channel were ``created'' by the forces in the t and u 
channel whose resonances become Regge Poles and not traditional (fixed) 
poles when viewed as exchanged particles or forces. The Veneziano model gave 
an elegant sample amplitude (see paper by Donnachie in these proceedings) 
illustrating how the Regge Poles (resonances) in one channel are not added 
to the exchange poles but rather they are different ways at looking at the 
same amplitude. This ``duality'' is clearest for mesons like the A$_{1}$ 
produced in reactions like $\pi ^{ - }$ p ? $\pi ^{ - } \quad \pi ^{ - 
}\pi ^{ + }$ p. As shown in figure 2, the meson production (figure 2(a)) 
and the $\pi $ exchange (figure 2(b)) descriptions are not necessarily 
alternatives; they could be different (dual) ways of looking at the same 
physics with the $\pi $ 

\begin{figure}[htbp]
\centerline{\includegraphics{pwa2.eps}}
\label{fig2}
\end{figure}


\noindent
exchange force producing the A$_{1 }$in elastic $\pi  \quad \rho $ scattering. 
Formulating duality as a precise rule is not easy. One good but incomplete 
way is to impose finite energy sum rules. Another less quantitative 
condition comes from quantum number connections between dual trajectories. 
The Pomeron and classic quark model exchanges are dual to different families 
of resonances (background for the Pomeron). Further for leading and daughter 
trajectories there are clear expectations of spin and other quantum numbers 
for Regge poles related by duality.

\begin{figure}[htbp]
\centerline{\includegraphics{pwa3.eps}}
\label{fig3}
\end{figure}




The constraint of unitarity is well understood mathematically but there are 
no straightforward ways of parameterizing amplitudes consistent with both 
unitarity and analyticity. In the two-body case (figure 3(a)), many 
formalisms such as the K matrix can guarantee unitarity but not analyticity 
and particular exchange particles. For the 3-body case, figure 3(b), Fadeev 
equations can guarantee unitarity in the sub channels but it is not clear 
this is ``a good idea'' as perhaps it is more important to respect 
analyticity as opposed to unitarity. Such ``conflicts'' and formulations 
that express ``part of the truth'' are typical of S Matrix theory. There are 
similar ``difficulties'' in the spin formalism -- namely in choosing the 
frames to be used for quantizing particle helicities. There are t-channel 
(reasonable for analyticity and respects exchanges), s-channel (best for 
analyticity) and transversity (best for selection rules but mixes amplitudes 
with different kinematic and hence analyticity factors.

\begin{figure}[htbp]
\centerline{\includegraphics{pwa4.eps}}
\label{fig4}
\end{figure}
\begin{figure}[htbp]
\centerline{\includegraphics{pwa5.eps}}
\label{fig5}
\end{figure}
\begin{figure}[htbp]
\centerline{\includegraphics{pwa6.eps}}
\label{fig6}
\end{figure}




We are looking at mesons M* produced by a mechanism similar to that of 
figure 4 with typically a multiparticle decay (here 3 particles). The 
A$_{1}$ production of Figure 2(a) is a special case of this. One will do 
better if this particular diagram is ``big'' as it would be if production 
exchange is one of the well understood trajectories with either a large 
coupling ($\pi $ exchange) or an intercept $\alpha $(0) that is higher than 
background exchanges. Pomeron ($\alpha $(0) = 1) and $\rho $, $\omega $, 
A$_{2}$ ($\alpha $(0) = 0.5) fall into this second class. Even though this 
diagram is hard enough to analyze, there are many backgrounds such as that 
corresponding to diagrams like those of figure 5. Here one of the supposed 
decay products of the top vertex really comes from the bottom baryon vertex. 
Actually these are not necessarily distinct for multiperipheral diagrams 
like that of fig. 6 link the ``bottom'' and ``top'' vertex. This is labeled 
for A$_{1}$ production diagram but the role of multiperipheral diagrams is 
general. The intercept of the exchanged Reggeons in figure 6 is important; 
if as drawn the poles are the Pomeron and the $\pi $, then the ``middle 
produced particle labeled 2'' will be pushed to the top vertex. In contrast, 
if both exchanged links had the same intercept ($f$ exchange for ``production 
exchange'' and A$_{2}$ for ``another exchange'' for example) then the middle 
produced particle 2 is truly equally associated with both vertices. 

\begin{figure}[htbp]
\centerline{\includegraphics{pwa7.eps}}
\label{fig7}
\end{figure}




There are many other difficulties concerning unitarity. As mentioned above, 
we have ways of enforcing this but they do not respect crossing (duality) 
and analyticity very precisely in complicated cases. Unitarity suggests one 
looks at the rescattering diagrams like figure 7 where the rescatter can 
either be in ``production process'' or within final state decay or with some 
mixture. It is not well understood if the Reggeon expansion should be 
applied before or after ``unitarization''. If one unitarizes a Regge 
amplitude, then one generates cuts in the angular momentum plane. However 
Reggeons are properties of the full amplitude and are not necessarily ``Born 
terms'' in some potential theory. Thus maybe the cuts generated by box 
diagrams such as those in figure 7 are deceptive. There is well known 
evidence suggesting cuts; examples include measured polarization in $\pi 
^{ - }$ p ? $\pi ^{0}$ n and less convincingly phenomenology of $\pi $ 
exchange. For partial wave analysis, diagrams such as those of figure 7 are 
extremely difficult to include in an amplitude parameterization. 

\begin{figure}[htbp]
\centerline{\includegraphics{pwa8.eps}}
\label{fig8}
\end{figure}


The $\pi ^{ - }$ p ? $\rho ^{0}$ n reaction is dominated by $\pi $ 
exchange but measurements of the $\rho $ density matrix elements violate the 
predictions of a na\"{\i}ve factorizable $\pi $ exchange. One can trace 
difficulties to vanishing of helicity-flip $\pi \overline p n $at t=0. The 
related reaction $\pi ^{ + }$ p ? $\rho ^{0} \quad \Delta ^{ + + }$ has a 
very clean $\pi $ exchange description compatible with simple Regge pole 
predictions. The features of the first reaction can be gotten by 
``absorption'' (Regge cut) calculations but it is not easy to explain the 
normalization of the observed corrections. Instead a simple analyticity 
argument explains the data with proper normalization. This model (due to 
Williams) formally implies a companion pole (the conspirator) of opposite 
naturality in the angular momentum plane. It is not clear if the conspirator 
is truly a pole or rather as is perhaps more likely a phenomenological 
approximation to a different structure such as angular momentum plane cuts. 
The situation is sort of clear but frustrating. There are a set of 
``truths'' which we do not understand how to make seamlessly compatible. In 
different reactions and different kinematic regions, one must agilely use 
the appropriate truth.





In summary, we have a set of basic truths and a phenomenology of how to 
assemble them to address any one reaction. Some ``truths'' are easier to 
apply and are more reliable than others. The quark model and QCD can give 
useful predictions as to both particle spectra and branching ratios. 
Selection rules are always good including those reflecting conservation of 
quark content. Regge theory has a few spectacular successes in describing 
high energy scattering. For example the linear $\rho $ and A$_{2}$ exchange 
degenerate trajectories with $\alpha $=0 dip for $\rho $ and spin flip 
nucleon coupling typically behave largely as simple Regge poles should. 
Recent results on the Pomeron and indeed a deeper discussion of nearly all 
the issues discussed here can be found in [4]. Some areas like $\pi $ 
exchange, polarization (sensitive to phases of the amplitude) and absorption 
(cuts) are more phenomelogical in their description. Some exchanges are 
known to be big or small and this phenomenology is uniformly successful. 
Finite energy sum rules can be used to express truths such as small high 
energy amplitudes for reactions like backward $\pi ^{ + } \quad \pi ^{ - }$ 
scattering) must translate to (amplitude) cancellation at low $\pi  \quad \pi $ 
energies ($\rho $ A$_{0}$ interference in the s channel)





In parameterizing amplitudes and fitting them to data (whether for PWA or 
otherwise), one must remember that it will be difficult to believe results 
sensitive to uncertainties in these ``partial'' theories.





\textbf{Partial Wave Analysis}

\begin{figure}[htbp]
\centerline{\includegraphics{pwa9.eps}}
\label{fig9}
\end{figure}




Let us see in more detail how the above ``wisdom'' can be applied to PWA. 
One can of course analyze production either in terms of a density matrix or 
an amplitude. The density matrix makes no assumptions but it is not possible 
to use it for detailed analysis to disentangle produced resonances. Thus 
most high statistics experiments expect to use an amplitude expansion. 
Amplitudes require assumptions but are the only easy way to express 
``truths'' like analyticity; they also enforce rank and positivity 
conditions for the density matrix elements. The down side is that any 
realistic amplitude expansion makes assumptions and these could lead to 
biased results. The nature of our ``truths'' is that they do not allow 
amplitudes to be formulated in a way that allows increasing sophistication 
to be ``switched on'' by increasing the richness of parameterization. As 
already discussed we don't know how to simultaneously satisfy constraints. 
Rather one must make reasonable assumptions and check a posteriori how other 
``truths'' are satisfied. Hardest of all one must estimate the impact of 
violations on ones conclusions. This requires multiple fits, good 
visualizations to examine validity of a given parameterization and new ways 
to synthesize them together. Just increasing number of partial waves is not 
in general a satisfactory approach as it doesn't directly translate into 
``satisfying truths''.





It is useful to consider the amplitude factorized so what you see 
corresponds to some reaction like Beam + Reggeon ? 1 + 2 + 3 as shown in 
figure 9. Then the parameterized amplitudes depend on t (mass$^{2}$ of 
Reggeon) and helicities of beam, 1 2 and 3. This picture has obvious 
complications from multiple exchanges and cuts which would change expected 
form of final density matrix. One can expect this picture and indeed 
amplitude-based production PWA analysis to be most reliable when the 
underlying Reggeon 

\begin{figure}[htbp]
\centerline{\includegraphics{pwa10.eps}}
\label{fig10}
\end{figure}


\noindent
phenomenology is convincing. This is the usual culprits: $\rho  \quad \omega $ 
and exchange degenerate A$_{2}$ f$_{2}$ families as well as the diffractive 
Pomeron are probably OK. The $\pi $ exchange with adjustment for its 
conspirator should also be reliable. In contrast B$_{1}$ exchange ($\pi $ 
exchange degenerate partner) is not expected to be very clean. Note the 
factorization picture says it doesn't matter what happens at the bottom 
vertex. Indeed ref. [2] showed that the same Regge $\rho $ exchange picture 
describes $\pi ^{ - }$ p ? $\pi ^{0}$ n, $\pi ^{ - }$ p ? $\pi ^{0}$ 
inclusive, and $\pi ^{ - }$ p ? $\pi ^{0}$ plus any neutral. It is 
perhaps useful to increase statistics by loosening vertex related trigger.





In designing the PWA analysis process, we should have a framework that 
allows all ``truths'' to be examined and selected ``truths'' to be included 
explicitly. This requires attention to spin formalism, unitarity 
(rescattering), Regge exchange phenomenology and the PWA expansion and 
parameterization. In the spin formalism, one must consider:

\textbullet \textit{Amplitude Parameterization} -- is this practical if (potential) rank high as with photon 
beams and/or non-factorizing exchanges?

\textbullet \textit{Density Matrix Formalism} -- how does this cope with explicit contributions, analyticity 
etc.

\textbullet \textit{Transversity versus helicity and s versus t} spin formalism needs to be considered -- there is a trade-off 
of analyticity versus selection rules

For unitarity, one can use current approaches building in two-particle 
unitarity (final state interactions) but their integration with other 
``truths'' is not easy. One should use both explicitly unitary and 
explicitly analytic (etc.) approaches and compare results. For a non-unitary 
parameterization, one can estimate the violation and examine the impact of 
this on the fit. When considering exchange contributions, one can identify 
all allowed (by normal Regge phenomenology) exchanges and catalog where they 
are expected to be large due to coupling constant values and/or values of 
$\alpha $(t,u). One can use the usual duality type arguments to identify 
related s t and u channel exchanges i.e. find where you might expect the 
direct and crossed descriptions to be related. One should develop models for 
exchange contributions using simple phenomenological Regge theory but I 
would have little hope this is very reliable except for Pomeron, $\rho $ and 
A$_{2}$ etc. exchanges but even these won't be so good as to be used 
quantitatively at low direct channel energies. One should identify all $\pi 
$ exchange contributions and expect these to be reliable (with 
``conspirator'' added) near t=0 but unreliable away from there. The status 
of $\pi $ as a Regge pole is problematic. Coming to the PWA analysis itself, 
one can use Cutkosky style acceleration techniques to maximize convergence. 
This is certainly sound as it exploits fully analytic structure but possibly 
it is easier to explicitly include high partial waves rather than choose an 
expansion that maximizes convergence. This is particularly useful if we can 
estimate magnitude from known couplings at exchanged particle poles. 
Considering the Beam-Reggeon reaction of interest, this will often include 
internal exchanges such as the $\pi $ exchange for which we have little 
experience from traditional $\pi $N direct channel analyses. Considering 
dispersion Relations and other analyticity constraints, we should certainly 
enforce ``helicity dependent'' kinematic zeroes. Further we should use 
visual inspection and finite energy sum rules to look at duality 
predictions. We could present data and fits in a way to highlight effects 
(e.g. we could look at fixed u cross sections for reactions like $\pi ^{ + 
} \quad \pi ^{ - }$elastic scattering with no u channel exchanges).





\textbf{Physics Summary}





Unfortunately many confusing effects exist and there is no fundamental 
(correct) way to remove most of them. Theory has failed to provide 
convincing parameterizable amplitudes that one can use to fit/explain data. 
Rather the theory provides some quantitative constraints ($\pi $ pole, 
unitarity, kinematics \ldots ), and many qualitative truths which overlap 
and whose effect can be estimated with errors from 10 to 100{\%}. So one 
should try to minimize effect of the hard (insoluble) problems such as 
``particles from wrong vertex'', ``unestimatable exchange effects'', 
sensitivity to slope of unclear Regge trajectories, absorption etc. One can 
note many of the effects (exchanges) are intrinsically more important in 
multiparticle case than in the well studied $\pi $ N ? $\pi $ N reaction. We 
must try to estimate impact of uncertainties from each effect on results. 
This needs systematic very high statistic studies of relatively clean cases 
such as $\pi $ N ? $\pi  \quad \pi $ N where spectroscopy may be less topical 
but where one can examine uncertainties. We need to first clarify impact of 
difficult issues and then apply this wisdom to new reactions and new meson 
resonances. One can be optimistic as we have orders of magnitude more data 
than when these issues were last looked at carefully.





\textbf{Computational Issues in Analyzing PWA Data}

We finish with a few remarks on computing issues associated with PWA 
analysis. The Grid techniques (\underline {http://www.grid2002.org}) being 
developed for LHC physics will be useful but these largely address 
management of the data reduction problem. This is important but as not as 
critical as the difficult physics analysis discussed above. In considering 
PWA computing, we can perhaps assume unlimited computer time, disk space and 
network bandwidth or perhaps more precisely that there are no significant 
constraints from these issues.





One are of possible importance is the optimization (fitting) technique to 
match parameterized amplitudes and experiment. Actually physics has been 
ahead historically of the computer science field in area of multi-parameter 
fitting. It is possible that ``unlimited computer time'' and better 
management of results could lead to new approaches such as ``ensemble 
fitting techniques'' (e.g. genetic algorithms) which explore parameter space 
more completely than traditional (Taylor expansion) least squares methods. 
We need to develop techniques that are not specially aimed at finding 
``new/best solutions'' but rather on better depicting error band of a given 
``solution''. For example we could have ``histograms'' which automatically 
display ensemble and not a single ``best fit'' -- we have plenty of 
memory/disk space to store the data to be displayed. More generally, we can 
think of our problem as data mining where we need to develop a suite of 
analysis programs that look for and catalog anomalies in experiment, theory 
or the discrepancy between them. We need to link this to the data management 
system so as to more systematically categorize different fits. Perhaps we 
could develop new multi-dimensional visualization (human data-mining) 
methods that improve dramatically on traditional histograms. Perhaps we 
could develop some better ``standard'' displays supported in a modern 
computer approach as a Web Services with portlet interfaces (this suggestion 
betrays my current expertise). Here the printed version of Phys. Rev. 
Letters still has something like a histogram playing role of an ``icon'', 
which is active in on-line version and can access the ``new nifty display''.





In summary we can exploit modern computing technology for more reliable 
optimization methods, the management of data before (as in GriPhyn and PPDG 
[5]) and especially in our case after analysis. We should explore powerful 
multi-dimensional visualization techniques and data mining to discover 
anomalies in data and/or fits and/or discrepancies between fits and data. We 
need to design and build such an environment and as part of the physics 
imperative apply to relatively well understood reactions to clarify the 
difficult conflicting physics issues.





\textbf{References}

1. Experimental groups at Caltech, UCLA, Chicago Circle, Fermilab and 
Indiana, ``Jet Production in High-Energy Hadron-Proton Collisions,'' 
\textit{Nuclear Physics}, \textbf{B171}, 38 (1980). 

2. Fox, G. C., Kennett R. G., Stone A. E. A., Barnes A. V., Walker, R. L., 
Dahl, O. I., Kenney, R. W., Ogawa, A, Pripstein, M, ``Inclusive $\pi ^{0}$ 
and $\eta $ Production from Kaon, Proton and Antiproton Beams in the Triple 
Regge Region,'' \textit{Nuclear Physics}, \textbf{B177}, 1 (1981). 

3. Experimental groups at Caltech, Fermilab, Chicago Circle and Indiana, ``A 
Study of the Reaction $\pi ^{ - }$p$ \to  \quad \pi ^{ - } \quad \pi ^{ + }$n 
at l00 and 175~GeV/c'' \textit{Nuclear Physics}, \textbf{B232}, 189 (1984). 

4. Sandy Donnachie, G\"{u}nter Dosch, Peter Landshoff, Otto Nachtmann, 
,,Pomeron Physics and QCD``, Cambridge University Press, November 2002.

5. Physics Grid at \underline {http://www.griphyn.org/} and Particle Physics 
Data Grid \underline {http://www.ppdg.net/ }





\end{document}
