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dispersion theoretical analyses have been very successful in the description and understanding of pion nucleon scattering and pion photoproduction already in the 60s . based on unitarity , analyticity , crossing symmetry , gauge invariance and lorentz invariance they provide a powerful tool to investigate the low energy behaviour of the nucleon and the structure of nucleon resonances . during the last few years beams of high current and high duty factor
together with considerably improved particle detection techniques have reduced the statistical errors to the order of a few percent , and promise to keep control of the systematical errors at the same level . to interpret these data with respect to the most interesting features ,
i.e. the threshold behaviour and the electromagnetic excitation of resonances , a partial wave analysis is mandatory . to ensure the consistency and uniqueness of such an analysis
, constraints from unitarity and dispersion relations have to be imposed .
such concepts have proven to be quite successful in pion - nucleon scattering @xcite . in comparison with that field ,
the situation in pion photoproduction is considerably more complex .
the spin and isospin structure leads to twelve independent amplitudes , while in pion - nucleon scattering there are only four such amplitudes . as a consequence
a complete experiment requires the use of many polarization observables .
such a complete experiment has not yet been performed .
however , the new experiments provide an ever increasing amount of precise and new data . at present
, the experimental thrust is mainly on measurements near threshold and around the @xmath7(1232 ) resonance . in the coming years
, a series of experiments at jefferson lab will cover the whole resonance region .
restricting our theoretical investigations to the threshold region and the low - lying resonances , we are lead to choose the method of omns and muskhelishvili to analyse the existing data , because it introduces a natural parametrization and fulfills the constraints of unitarity at the same level .
starting from fixed-@xmath0 dispersion relations for the invariant amplitudes of pion photoproduction , the projection of the multipole amplitudes leads to a well known system of integral equations , @xmath10 where @xmath11 stands for any of the multipoles @xmath12 and @xmath13 for the corresponding ( nucleon ) pole term .
the kernels @xmath14 are known , and the real and imaginary parts of the amplitudes are related by unitarity . in the energy region below two - pion threshold , unitarity is expressed by the final state theorem of watson , @xmath15 where @xmath16 is the corresponding @xmath17 phase shift and @xmath18 an integer .
we have essentially followed the method of schwela et al @xcite to solve eq .
( [ inteq ] ) with the constraint ( [ watson ] ) .
in addition we have taken into account the coupling to some higher states neglected in that earlier reference . at the energies above two - pion threshold up to @xmath19 gev , eq .
( [ watson ] ) has been replaced by an ansatz based on unitarity @xcite . finally , the contribution of the dispersive integrals from @xmath20 gev to infinity has been replaced by @xmath0-channel exchange , parametrized by certain fractions of @xmath21- and @xmath22-exchange .
furthermore , we have to allow for the addition of solutions of the homogeneous equations to the coupled system of eq .
( [ inteq ] ) .
the whole procedure introduces 9 free parameters , which have to be determined by a fit to the data.@xcite    in our data base we have included the recent mami experiments for @xmath2 and @xmath3 production off the proton in the energy range from 160 mev to 420 mev@xcite , both older and more recent data from bonn for @xmath23 production off the proton@xcite , and older frascati@xcite and more recent triumf data@xcite on @xmath4 production off the neutron .
our fit obtained with this data base describes this data very well and in addition it also gives good agreement with data from the world data base ( e.g. said@xcite ) not included in our fit .
in table 1 we give our results for the @xmath24- and @xmath25-wave amplitudes at threshold and compare them to the available information from heavy baryon chiral perturbation theory ( hbchpt ) @xcite and experiment .
the reduced @xmath25-wave amplitudes are defined as usual by @xmath26 , in the limit @xmath27 .
in fact the dependence on the photon energy @xmath28 is not stringent , and in explicit calculations this definition leads to less energy dependence for @xmath29 and @xmath30 , while @xmath31 would vary less without the factor @xmath32 . in general there is a very good agreement between our analysis , hbchpt and experiment . in our analysis the uncertainties are mainly in the neutron channel , especially for @xmath33 , where the lack of precise experimental data , in particular of polarization observables , reflects in the threshold values .
in fact it is interesting to note that the threshold amplitudes given in table 1 are not fitted to the threshold data .
they are only determined from experimental information above @xmath34 mev , therefore , they should be considered as predictions rather than fits . the interplay of the complete knowledge of pion photoproduction at all energies in the framework of dispersion relations can be seen in the individual contributions to the threshold values of the @xmath24-wave amplitudes . for @xmath35
we obtain @xmath36 where the individual contributions are from the pole terms , from @xmath37 and higher multipoles , respectively .
there is special theoretical interest in the @xmath38 amplitude of @xmath39 because it allows for an independent determination of the charge exchange pion - nucleon scattering length via the panofsky ratio , @xmath40 .
this ratio is well determined by experiment , @xmath41 @xcite , and related to the scattering length by time reversal , @xmath42 where @xmath43 and @xmath44 are the @xmath45 momenta of photon and neutral pion at @xmath46 threshold .
using our value of the threshold amplitude , and the measured panofsky ratio , we find @xmath47 .
this has to be compared with the value @xmath48 resulting from a partial wave analysis of pion - nucleon scattering @xcite ( solution kh80 ) .
recently , @xmath49 has also been determined by studying the level spacing of pionic atoms , with a preliminary value of @xmath50 @xcite .    .threshold
amplitudes for pion photoproduction .
the @xmath24-waves @xmath38 are in units of @xmath51 and the reduced @xmath25-wave multipoles are in units of @xmath52 .
our values are compared with results from chiral perturbation theory @xcite and data analysis for charged pion production@xcite and neutral pion production off the proton@xcite . [ cols="<,^,^,^,^,^,^,^,^ " , ]     [ tab : ratios ]    in table 2 we compare our result with two vpi solutions , the solution of the rpi group in a field theoretical lagrangian approach and the experimental analyses of mainz and brookhaven . with the numerical solutions of vpi and rpi we have applied the speed - plot technique in order to separate resonance and background contributions and to determine the pole position and residues . while the `` full '' ratios @xmath53 vary by more than a factor of 2 among these solutions , the ratios @xmath54 are much closer to each others . in particular , the imaginary parts are very stable within only about 30% .    finally , we have determined the photon couplings @xmath55 and @xmath56 of the delta resonance
. from our energy - dependent analysis we get @xmath57 and @xmath58 , both in units of @xmath59 .
however , it should be again noted that both the @xmath53 ratio and the photon couplings , calculated at the k - matrix pole are well - defined quantities but they have no direct connection to quark model calculations of a `` bare '' resonance .
with the new and very precise data obtained at mami in mainz we have obtained a new partial wave analysis for pion photoproduction .
the uncertainties in most multipoles could be considerably improved compared to previous analyses .
very accurate results can be obtained at threshold and in the resonance region . at resonance
we must clearly distinguish between the resonance position @xmath60 mev on the real axis and the pole at @xmath61 mev in the complex plane . at the resonance position , where the phase passes @xmath62
, we obtain an rem ratio of @xmath63 in very good agreement with the experimental analysis@xcite .
this was also recently confirmed in a vpi analysis with a restricted data base@xcite . at the pole in the complex plane
we obtain the ratio of the resonant electric and magnetic multipoles as @xmath64 .
this is a model - independent ratio that can be determined in any analysis or calculation of pion photoproduction . after a long time of confusion about the different ratios that can be defined and constructed out of the measured cross sections or the analysed multipoles @xmath65 and @xmath66
, it now appears that the ratio @xmath54 is the closest one can get to a background subtracted value .
such a ratio must be complex , and it will be a challenge for all microscopic models to determine this ratio .
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a new partial wave analysis of pion photoproduction has been obtained in the framework of fixed-@xmath0 dispersion relations valid form threshold up to 500 @xmath1 . 
 it is based on new mainz data for @xmath2 and @xmath3 production off the proton and both older and more recent data from bonn , frascati and triumf for @xmath3 and @xmath4 . at threshold 
 we obtain a good agreement with the existing data for both charged and neutral pion production . in the resonance region we have precisely determined the electromagnetic properties of the @xmath5 resonance , in particular the e2/m1 ratio @xmath6 . 
 we show that a model independent resonance background subtraction is possible with the speed - plot technique and obtain the @xmath7 pole at @xmath8 mev and the e2/m1 ratio of the residues as @xmath9 .    
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