Volume 26B, number 7 PHYSICS LETTERS 4March 1968 PARITY DOUBLET CONSPIRACY OF THE PION FROM FINITE ENERGY SUM RULES IN 7r + P H O T O P R O D U C T I O N A. BIETTI Istituto di Fisica dell'Universit& di Rorna lstituto Nazionale di Fisica Nucleate, Sezione di Rorna, Italy P. DI VECCHIA, F. DRAGO Laboratori Nazionali di Fascati del C.N.E.N., Frascati, Italy and M. L. PACIELLO * Istituto di Fisica dell' Universit~ di Rorna, Italy Received 31 January 1968 We test the essential features of the conspiracy of the pion Regge pole with a positive parity pole in 7r + photoproduction, using finite energy sum rules. In particular we find that the pion residue function varies rapidly with the four momentum transfer t vanishing at t = - 0.03 GeV 2 and the conspirator trajectory chooses nonsense at c~ -- 0. One of the most interesting features of the high energy ~+ photoproduction is the peak shown by the differential cross section in the forward direction [1]. Under the assumption of a Regge behaviour for the amplitudes, this fact has been recently interpreted [2-5] in terms of a conspiracy mecha- nism. Amongst others, a very simple model of con- spiracy has been proposed by Ball et al. [5]. They assume that the pion Regge pole conspires with another pole which has the same quantum numbers but possitive parity (parity doublet or class III conspiracy). Such a model was already considered by Phillips [6] and Arbab and Dash [7] to fit the pn charge exchange data. In order to get a good fit for the differential cross-section Ball et al. as- sume that the pion residue function varies rapid- ly with t, whereas they keep the conspirator residue function constant, i.e. equal to the value given by the conspiracy relation at t = 0. Further- more they take (for small values of t) the con- sprator and the pion trajectories degenerate, as- * Sponsored in part by the Air Force Office of Scien- tific Research through the European Office of Aero- space Research, OAR, United States Air Force, un- der contract F61052 67 C 0084. suming therefore that the conspirator trajectory "chooses nonsense" at ~ = 0. In this note we will show how, using finite energy sum rules [8] for ~+ photoproduction, we can confirm essentially all the principal features of the model of Ball et al. We find thus that the pion residue function for small t has the rapidly varying behaviour sug- gested by Ball et al., and in particular that it vanishes in the neighbourhood of t = 0 (more precisely at t = - 0.03 GeV2). This behaviour is expected from the group theoretical approach if the pion belongs to an M = 1 representation of O(4), i.e. conspires in class III [5,9]. Moreover, we show explicitely that the con- spirator trajectory "chooses nonsense" at ~ = 0. Following Ball [5,10], we can write the four conventional [11] amplitudes for pion photopro- duction as tF 1 + 2rnF 3 A 1 - _ t_4rn2 F1 1 I F2 2rnF3 ~ A 2 t - 4rn 2 + )- ~ + t - 4rn 2 J (1) A 3 = _ F 4 2rnF 3 + F 3 A4-- t-4rn 2 ' 457