Compton in scalar electrodynamics∗

J.G. Abreu, C.B. Compean, M. Cordero, Y. de Boer, A. De Santis, F.V. Flores, L.A. Muñoz, F. Rodrigues, K. Vervink Student Discussion Group

Abstract The goal of this project is to compute the Compton scattering cross-section for scalar electrodynamics. A comparison with the experimental result is also made.

1 The project In our group we were concerned about the possibility of predicting the spin nature of by just measuring scattered against free electrons. So, we decided to compute the total cross-section for Compton scattering with scalar electrodynamics and compare it with experimental proof. The starting point is the gauge invariant Lagrangian which describes the interaction of a scalar particle with a massless vector boson.

1 2 µ µ = (∂ A ∂ A ) + [(∂ + ieA )ϕ]† [(∂ + ieA )ϕ] , L −4 µ ν − ν µ µ µ µ µ 2 µ = + ieA (ϕ∗∂ ϕ ϕ∂ ϕ∗) + e A A ϕ∗ϕ . (1) Lfree µ − µ We can extract the vertices using the rules given by Ref. [1]:

– For each interaction in the Lagrangian, we introduce a conservation delta, (2π)4δ( p). – Introduce a factor coming from the degeneracy of identical particles in the interaction and the P couplings coming from i . Lint – For each field derivative ∂ φ, a ip factor is associated with the momentum of the incoming µ − µ particle.

Then, for the -scalar interaction, we have

4 4 ie(p + p0 )(2π) δ (p p0 k) , (2) − µ µ − − and for the 2-photon-scalar interaction

2 4 4 2ie g (2π) δ (p p0 k k0) . (3) µν − − − The next step consists of applying the recipe.

– Draw all topologically distinct diagrams. – For each internal scalar field of momentum k, we attach the (k) = i/(k 2 m2 + iε). D − – For each external photon line, we attach a polarization vector µ.

From Fig. 1, we obtain the following amplitude

*Work performed as a student project under the supervision of C. Rojas.

293 J.G.ABREU,C.B.COMPEAN,M.CORDERO, Y. DE BOER,A.DE SANTIS, F.V. FLORES,...

2 µ ν i = e 0 p0 + k0 + p0 (p + k + p)  (p + k) A − µ ν D 2 µ ν e µ0 p k0 + p p0 k + p0 ν (p k0) − µν− − D − + 2iµ0 g ν ,   2 µν ie 0  , (4) ≡ − µT ν where

(2p + k )µ(2p + k)ν (2p k )µ(2p k)ν µν = 0 0 + − 0 0 − 2gµν , T (p + k)2 m2 (p k )2 m2 − − − 0 − (2p + k )µ(2p + k)ν (2p k )µ(2p k)ν = 0 0 + − 0 0 − 2gµν 2p k 2p k − · · 0 Aµν + Bµν 2gµν . (5) ≡ −

Fig. 1: The Feynman diagrams for the interacion

In order to compute the scattering cross-section, we take the absolute square value of the ampli- tude. Since we measure only the number of photons, the final photon polarizations are summed and the initial ones are averaged.

4 1 2 e µν ρσ = 0  0  ∗ , 2 |A| 2 µT ν ρT σ pol pol X X     e4 e4 = µν ρσg g = µν , 4 T T µρ νσ 4 T Tµν = (Aµν + Bµν 2gµν ) (A + B 2g ) − µν µν − µν = Aµν A + BµνB + 2Aµν B 4Aµ 4Bµ 8 . (6) µν µν µν − µ − µ − Here we used the orthogonality relation

∗ (k) (k) = g . (7) µ ν − µν Xpol We end with

1 1 1 2 1 1 2 = 2e4 m4 2m2 + 2 . (8) 2 |A| " p k − p k0 − p k − p k0 # Xpol  · ·   · ·  For computing the cross-section, we need the phase space factors in a certain reference frame. We choose the laboratory frame, where the incoming photon hits an at rest.

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p = m(1, 0, 0, 0) , k = ω(1, 0, 0, 1) ,

k0 = ω0(1, sin θ, 0, cos θ) , (9)

p = (m + ω ω0, ω0 sin θ, 0, ω ω0 cos θ) , − − − then, p k = mω and p k = mω . So, we have the well-known relation · · 0 0 1 1 1 = (1 cos θ) , (10) ω − ω0 m − ending with the expression

1 2 = 2e4(1 + cos2 θ) . (11) 2 |A| Xpol So, the differential cross-section is

3 3 1 2 m d k0 md p0 4 dσ = 3 3 (2π) δ(p0 + k0 p k) 2 |A| 4(p k) 2ω0(2π) 2p0(2π) − − Xpol · e4 ω 2 = 0 (1 + cos2 θ)dΩ 2(4π)2m2 ω   α2 (1 + cos2 θ) = dΩ . (12) 2m2 (1 + ω/m(1 cos θ))2 − Integrating out, we obtain the total cross-section (which can also be found in Ref. [2])

πα2 2 ω + 4 ω2 + 2 ω3 ln (1 + 2 ω) 3 ω ln (1 + 2 ω) 2 ω2 ln (1 + 2 ω) σ = 2 − − − . (13) T m2 (1 + 2 ω) ω3   At the end, in Fig. 2, we compare the scalar cross-section with the ‘experimental’ one (see, for example, Ref. [1]).

Fig. 2: The percentage ratio between the scalar cross-section and the real one as a function of the photon in units of the electron mass

We can conclude that for low there is no evidence of spin.

3 295 J.G.ABREU,C.B.COMPEAN,M.CORDERO, Y. DE BOER,A.DE SANTIS, F.V. FLORES,...

References [1] Claude Itzykson and Jean Claude Zuber, , 2nd ed. (McGraw-Hill, New York, 1984). [2] Paolo Franzini, Elementary Particle Physics, Lecture Notes, Spring 2005 (University of Rome, La Sapienza), available at the URL http://www.lnf.infn.it/~paolo/.

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