The Pion Beta Decay Experiment and A Remeasurement of the Panofsky Ratio

6.1.1 Pion Nucleon Scattering

For the derivation of the relation between b₁ and P it is useful to describe SCX and radiative caption (RC) separately and make use of partial wave decomposition.

The pion-nucleon system represents a linear combination of isospin T=3/2 and T=1/2 isospin states, since nucleons form an isospin 1/2 doublet and pions an isospin 1 triplet, which are broken by electric charge. The third isospin component T_z is attributed to the discussed particles as follows:

1

p

0

-
n

-1

In combination they form either one of four T_3/2 or two T_1/2 states, where only the T_z= ±3/2 orientation directly can be attributed to

, respectively. Looking at the two states of interest one finds

and

, where the coefficients are the familiar Clebsch Gordan coefficients following the Condon-Shortley sign convention (see [Mat97] for example).

6.1.1.1 SCX s-wave Scattering

The SCX process can be well defined by elastic scattering at p N- threshold, where a pion plane wave e^ikz is scattered by a solid body with the scattering amplitude f( q ) and transforms into the deflected wave e^ikr/r ,

where q represents the scattering angle. Or in a different form (following [Che57]), when the pion wavefunction is denoted as p (q):

F(q',q) = , where [florin] contains the phase shift d via . Here q and q' represent the pion momenta before and after scattering.

As is well known the differential cross section is given by d s /d W =F(q',q)². It is useful to perform a partial wave decomposition. Then F becomes

, where are the angular momentum J=l±½ projection operators for a given orbital momentum l and P_lcos( q ) Legendre polynoms.

Since the pion is captured at rest we can neglect all orbital momenta other than l=0 (s-waves). So

becomes 1 and 0;

s is the pion spin position. In addition one should take the isospin decomposition into account and make use of the isospin projection operators

, where t represents the nucleon isospin and [tau] the pion isospin.

Thus, projects the total isospin T=1/2 and T=3/2 states out of the p N system. For example:

. The scattering amplitude now simplifies to , since . Finally, one can calculate the cross section s for p N scattering.

In the case of - elastic scattering one obtains

and analogous for SCX ,

having denoted the scattering lengths a₁ for T=1/2 and a₃ for T=3/2. Often is used, as well.

6.1.1.2 Pion Photoproduction

Usually RC is characterized by its time reversed process: photoproduction of pions ( g N -> p N). This process can be described successfully by a partial wave decomposition of the photon wave function. In Dirac notation the matrix element T_fi , which represents the transition operator for the final (pion-nucleon) and the initial (photon-nucleon) state, can be written as , where F consists of a linear combination of the photon multipole amplitudes and Q contains kinematic and spin information. At threshold[27] only

contributes [Che57]. With l representing orbital angular momentum are the magnetic and electric transition amplitudes, respectively. Then the differential cross section becomes

Due to the p³ momentum dependence of the p-wave amplitudes (p⁵ for E₂-, respectively) only the s-wave term is significant in our case, since the is captured at rest. This leads to

Then , the threshold amplitude of pion photoproduction, is defined via

[Kov97].

Hence, the electric dipole amplitudewith l_p=0 and total spin j=1/2 fully describes the pion photoproduction cross section (and therefore the radiative capture, as well). The ratio of the fundamental processes for a stopping in a H-nucleus then consistently is given through s-wave pion scattering at threshold and calculates to

Hence, knowing , the p N scattering length b₁ can be obtained directly by measuring the Panofsky ratio.

Using

and P=1.546 [Spu77], the actual value is ^-0.253/m_p . has been used with the correction for the binding energy of pionic hydrogen B_1s=0.00324 MeV.

[27] Threshold here means vanishing kinetic energy, namely