Revisiting fermion helicity flip in Podolsky's Electromagnetism
Jorge Henrique Sales, Alfredo Takashi Suzuki, Ronaldo Thibes
aa r X i v : . [ h e p - ph ] A ug Revisiting fermion helicity flip in Podolsky’s Electromagnetism
Jorge Henrique Sales , A.T. Suzuki, and Ronaldo Thibes Universidade Estadual de Santa Cruz,Rodovia Jorge Amado km 16CEP 45662-900 - Ilh´eus, BA, Brazil. Instituto de F´ısica Te´orica-UNESP,R. Dr. Bento Teobaldo Ferraz, 271CEP 01140-070 S˜ao Paulo, SP, Brazil and Departamento de Ciˆencias Exatas e Naturais,Universidade Estadual do Sudoeste da Bahia,CEP 45700-000, Itapetinga, BA, Brazil (Dated: September 24, 2018)The spin projection of a massive particle onto its direction of motion is called helicity (or “hand-edness”). It can therefore be positive or negative. When a particle’s helicity changes from positiveto negative (or vice-versa) due to its interaction with other particles or fields, we say there is ahelicity flip. In this work we show that such helicity flip can be seen for an electron of 20
MeV of energy interacting with a charged scalar meson through the exchange of a virtual photon. Thisphoton does not necessarily need to be Podolsky’s proposed photon; in fact, it is independent of it.
PACS numbers:
I. INTRODUCTION
As an introduction, we report to the work of Accioly and Mukai [1] previously published back in 1997. In theirwork they claim that the massive fermions of energy 100
M eV do exhibit helicity flip as the particles interact withPodolsky’s generalized electromagnetic field. Their claim is based on and drawn from the three-dimensional diagramplots shown in figures 7 and 8 in the conclusion section of their article. However, those plots clearly show us that there is no helicity flip since de graph for their P (fermion’s polarization) axis for both diagrams present only positivevalues in the range 0 ≤ P ≤
1. In order to show a helicity filp one has to have in the plot, a graph going from positivevalues to negative values of P .In this work we revisit their calculation for the polarization considering a charged fermion being scattered by acharged scalar pion in the Podolsky’s electrodynamics [2]. To do that, we consider at the tree level the scattering of acharged fermion by a scalar charged pion, initially at rest in the laboratory frame. For a fermion with energy 100 M eV we reproduce their tree-dimensional plot (see Figure 1). This plot does not evidence helicity flip for the fermion.However, as we lower the fermion’s energy to 20
M eV (see Figure 2) we can clearly see the helicity flip for thefermion, from positive values of P going to negative values of P (the bottom right corner of the tree-dimensionalgraph has values around P = − . e − + π + → e − + π + (1)where e − represents the electron traveling with energy E towards a pion π + at rest. After the scattering, thereemerges an electron in a direction that makes an angle θ relative to the original direction. This process reveals achange in the polarization of the electron discussed in [1], using Podolsky’s theory [2–5]. First we obtain the relevantPodolsky’s photon propagator which enter in the calculation of the right-handed and left-handed helicities. We showthat for the electron with 100 M eV , there is no helicity flip; however, for lower energy electron with 20
M eV , helicityflip does really occur.
II. HELICITY FLIP
The covariant Lagrangian density for the theory is given by L = − F µν F µν + a ∂ α F µα ) − λ ( ∂ µ A µ ) + ( ∂ µ φ ) ∗ D µ φ − m π φ ∗ φ + ψ ( i / ∂ − m e ) ψ + eψγ µ A µ ψ ++ ie ( A µ φ∂ µ φ ∗ − A µ φ ∗ ∂ µ φ ) + e A µ φ ∗ A µ φ. (2)In the Lagrangean density above, we have represented the electron mass by m e and the pion mass by m π , electricallycharged with charges − e and + e respectivelly. The relevant Feynman rules that we need are V µA ( p, q ) = ieγ µ is the vertex for the fermions with initial momentum p and final momentum qV µB ( v, w ) = − ie ( v + w ) µ is the vertex for the pions of initial momentum v and final momentum w (3)The propagator for the photon in the Podolsky’s theory can be calculated from the Lagrangean density and is givenby [1] D µν ( k ) = iM k ( k − M + iε ) " g µν − − λ (1 − k M ) k k µ k ν , (4)where k is the momentum of the virtual photon exchanged between the electron and the pion and M . = 1 /a is thecharacteristic Podolsky’s parameter.Without loss of generality, we may consider the effect of negative helicity of an incident electron in a pion at restaccording to the process (1) [1]. The helicity flip for a fermionic particle beam can be calculated by the evaluation ofthe scattered polarization defined by P = 1 − N right N left + N right , (5)where N left is the number of fermions that emerge from the scattering with negative helicity and N right is the numberof scattered fermions with positive helicity.The polarization as defined by Equation (5) ranges between − ≤ P ≤
1. If there is no positive helicity amongthe scattered particles, N right = 0, and consequently P = 1 and all the scattered fermions have the same helicity asthe incident fermions, that is, negative helicity. We then conclude that there is no helicity flip in this case. On theother hand, if N left = 0 (that is, all the scattered fermions have now positive helicity, or opposite helicity as comparedto the incident ones) then P = − P : a change in the sign of the quantity P as defined in Equation(5) signals helicity flip; otherwise no helicity flip.For the process of scattering (1), we have the analytic expression from the Feynman rules, N right = | ¯ u R ( q ) V µA ( p, q ) u L ( p ) D µν ( k ) V νB ( v, w ) | , and N left = | ¯ u L ( q ) V µA ( p, q ) u L ( p ) D µν ( k ) V νB ( v, w ) | , where D µν ( k ) is the photon propagator (4).Spinors u L ( p ) and u R ( p ) satisfy the property (we use Feynman slash notation for contraction with Dirac gammamatrices) u L,R ( p ) = (cid:18) γ / S L,R (cid:19) u L,R ( p ) , (6)and the polarization vectors are given by S µ L,R ( p ) = (cid:18) ∓ | ~p | m e , ∓ p m e ~p | ~p | (cid:19) . (7)We thus have N Right = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ieM k ( k − M ) ¯ u R ( q ) γ µ u L ( p ) ( g µν − − λ (1 − k M ) k k µ k ν ) ( v + w ) ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (8)Using now the following properties for fermions with momentum p and q , we have: u L,R ( p ) | α ¯ u L,R ( p ) | β = (cid:26)(cid:18) / p + m e m e (cid:19) (cid:18) γ / S L,R ( p )2 (cid:19) . (cid:27) αβ The α and β are matrix indices and we may rewrite the expression as N Right = e M ( k ) ( k − M ) T r (cid:26)(cid:18) / p + m e m e (cid:19) (cid:18) γ / S R ( q )2 (cid:19) / Q × (cid:18) / p + m e m e (cid:19) (cid:18) γ / S L ( p )2 (cid:19) / Q (cid:27) , where Q = v + w . Using the trace properties for gamma matrices, we have N Right = P El (cid:26) − m e t lab (cid:20)q E − m e ( E + 2 m π ) − E lab p E − m e cos θ (cid:21) ×× hp E − m e ( − E lab + 2 m π ) + E lab p E − m e cos θ i + (cid:0) m π − t lab (cid:1) m e (cid:18) E q E − m e − E lab p E − m e cos θ (cid:19) (cid:18) E lab p E − m e − E q E − m e cos θ (cid:19) ++ 12 h(cid:0) m π − t lab (cid:1) t lab − ( s lab − u lab ) i p E − m e p E − m e ( EE lab cos θ ) m e ! + (9) − m e ( s lab − u lab ) (cid:20)(cid:18)q E − m e ( E + 2 m π ) − E lab p E − m e cos θ (cid:19) × (cid:18) E lab p E − m e − E q E − m e cos θ (cid:19) ++ (cid:18) E q E − m e − E lab p E − m e cos θ (cid:19) (cid:20)p E − m e ( − E lab + 2 m π ) + E q E − m e cos θ (cid:21)(cid:21)(cid:27) . In Equation (9), for the sake of convenience, we have introduced the shorthand notation for the overall coefficientas P El . = e M t ( t − M ) m e . For N left we have similarly N left = P El (cid:26)(cid:0) m π − t lab (cid:1) t lab − ( s lab − u lab ) + 1 m e t lab (cid:20)q E − m e ( E + 2 m π ) − E lab p E − m e cos θ (cid:21) × hp E − m e ( − E lab + 2 m π ) + E lab p E − m e cos θ i + − (cid:0) m π − t lab (cid:1) m e (cid:18) E q E − m e − E lab p E − m e cos θ (cid:19) × (cid:18) E lab p E − m e − E q E − m e cos θ (cid:19) − h(cid:0) m π − t lab (cid:1) t lab − ( s lab − u lab ) i × p E − m e p E − m e ( EE lab cos θ ) m e ! − m e ( s lab − u lab ) × (10) (cid:20)(cid:18)q E − m e ( E + 2 m π ) − E lab p E − m e cos θ (cid:19) (cid:18) E lab p E − m e − E q E − m e cos θ (cid:19) + − (cid:18) E q E − m e − E lab p E − m e cos θ (cid:19) (cid:20)p E − m e ( − E lab + 2 m π ) + E q E − m e cos θ (cid:21)(cid:21)(cid:27) . In the above expressions, we have used the Mandelstam variables in the laboratory frame s lab = m e + m π + 2 Em π ,t lab = 2 m π ( E lab − E ) ,u lab = 2 (cid:0) m e + m π (cid:1) − s lab − t lab, where E lab = (cid:0) m π E + m e (cid:1) ( E + m π ) + cos θ (cid:0) E − m e (cid:1) q m π − m e sin θ ( E + m π ) + ( E − m e ) cos θ (11)and E is the fermion energy. We note that the scattered fermion polarizarion P as given in Equation (5) does notdepend on the overall coefficient P El that appears in front of the N right and N left . This means that the polarizationdoes not depend on the characteristic Podolsky’s parameter M . = 1 /a . This means our result is independent of thecharacter of the virtual photon propagator, whether it is given by the usual Maxwell’s theory or by the Podolsky’sapproach to electromagnetism.We are working in the reference frame where the pion is at rest, that is, w µ = ( m π , , , E = 100 Mev. In this case, we reproduce exactly the result obtained in reference [1], and our graph inFigure 1 is precisely their Figure 7. Neither their Figure 7 nor Figure 8 of their article show that for fermion mass ofthe order of electron mass, there is a helicity flip as they claimed. Even when the fermion mass gets larger than theelectron mass there is only a change in the direction of the helicity (not exactly paralel to the momentum direction),but still the helicity does not undergo sign change, so there is no flip. FIG. 1: Electron scattering with initial total energy equal to 100 MeV.
In our graph of Figure 2, however, we have a curious effect: If we lower the electron energy from 100 MeV to 20MeV, we observe a range of negative polarization, implying that helicity flip is occuring for increasing mass of thefermion.
III. CONCLUSION
We have shown that the authors of reference [1] have missed their point as far as their claim of helicity flip forelectron interacting with Podolsky’s photon is concerned.Their affirmation of helicity flip for the electron undergoinginteraction with a scalar charged meson with an exchange of Podolsky’s type photon cannot be substantiated. Theirgraph does not corroborate for such an assertion and neither their allegation that this effect is due to the Podolsky’sphoton is warranted.However, we have shown that for a particular lower energy for the electron, i.e., 20 MeV instead of 100 MeVconsidered in reference [1] we clearly see a helicity flip. Morever, this flip has nothing to do with the interactingintermediate photon being a la
Podolsky or not, for the polarization factor does not depend on any Podolsky’s factor a . Acknowledgments:
ATS thanks the hospitality of the Physics Department of NCSU, Raleigh, NC and gratefullyacknowledges a research grant from FAPESP, BPE process 2014/20829-2. J.H. Sales wishes to thank NCSU andIFT for the hospitality while this work was in progress and acknowledges research grants from FAPESB-Propp-
FIG. 2: Electron scattering with initial total energy equal to 20 MeV. [1] A. J. Accioly and H. Mukai, Z. Phys. C75