Journal of Physics: Conference Series PAPER • OPEN ACCESS Scattering process in the Scalar Duffin-Kemmer- Petiau gauge theory To cite this article: J Beltran et al 2016 J. Phys.: Conf. Ser. 706 052019   View the article online for updates and enhancements. Related content Duffin-Kemmer-Petiau equation under Hartmann ring-shaped potential H. Hassanabadi, M. Kamali, Z. 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Scattering process in the Scalar Duffin-Kemmer-Petiau gauge theory J Beltran1, B M Pimentel1, D E Soto2 1 Instituto de F́ısica Teorica, Universidade Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz 271, S.P, 01140-070, Brazil. 2 Facultad de Ciencias, Universidad Nacional de Ingeniera UNI, Avenida Tupac Amaru S/N apartado 31139 Lima, Per. E-mail: jhosep@ift.unesp.br, pimentel@ift.unesp.br, dsotob@uni.edu.pe Abstract. In this work we calculate the cross section of the scattering process of the Duffin- Kemmer-Petiau theory coupling with the Maxwell’s electromagnetic field. Specifically, we find the propagator of the free theory, the scattering amplitudes and cross sections at Born level for the Moeller and Compton scattering process of this model. For this purpose we use the analytic representation for free propagators and take account the framework of the Causal Perturbation Theory of Epstein and Glaser. 1. Introduction One of the main restriction to construct field equations is the Lorentz covariance and the causality principle. Thus, elementary particles are described by Lorentz-covariant field equations and they do not interact at a distance. Furthermore, in order to analyse the relativistic effects on the physical systems, relativistic wave equations, such the Dirac, Klein-Gordon-Fock (KGF) and Duffin-Kemmer-Petiau (DKP) equations, are constructed. The solutions of the relativistic wave equations with arbitrary spin have a long history. Besides the KGF and Dirac equations, the Dirac-like first order Lorentz invariant Kemmer equation supplemented with the β-matrices algebra, for both spin-0 and spin-1 particles was first derived by Kemmer in 1939 [1] and they are called the DKP1 equations. Exact solutions of the DKP equation in the presence of an external field have been investigated by some authors, such as the quantum oscillator in the framework of the DKP theory [2, 3] during the last few decades. It is known that in the free field case the DKP and KGF theories are equivalent, both in classical and quantum pictures. For instance, it was shown that both theories are equivalent in the classical level for the cases of minimal interaction with electromagnetic [4] and gravitational fields [5]. Strict proofs of equivalence between both theories were also given for cases of interaction of quantized scalar field with classical and quantized electromagnetic, Yang-Mills and external gravitational fields [6]. However, there are still no general equivalent proofs of equivalence between these theories when interactions and decays of unstable particles are taken into account. Perhaps one of the most evident advantages in working with this theory is the fact that derivative couplings do not appear between DKP and the gauge field. This property has been 1 There were two independent previous works given by Duffin and Petiau. XIII International Workshop on Hadron Physics IOP Publishing Journal of Physics: Conference Series 706 (2016) 052019 doi:10.1088/1742-6596/706/5/052019 Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 used by Gribov, who employed the spin-1 DKP theory to study the quark confinement problem [7]. Such property will result in manifestly covariant expressions for the interaction Hamiltonian and the vacuum expectation values of time ordered products of fields. Other advantages are the formal similarity with SQED and QED, what allows an unified treatment of the scalar and vector fields. For spin zero particles it is known as the Duffin-Kemmer-Petiau theory for scalar fields (SDKP). One of the difficulties in working with SQED is the presence of a term of second order in the coupling constant in the interaction Hamiltonian, which causes trouble in proving gauge invariance. In SDKP theory it was achieved, by an effective approach, that this second order term does not contribute to the S-matrix, and thus it can be neglected when we construct the Feynman rules for the theory. The Epstein-Glaser-Scharf causal perturbative method [8, 9] applied to SQED [10, 11] is a finite ultraviolet (UV) theory which differs from the conventional formalism because the perturbative construction starts from the first order interaction, and the quadratic term appears in the process of distribution splitting as a consequence of gauge invariance. Following the causal approach, J.T. Lunardi, B.M. Pimentel and J.S. Valverde [12] studied the SDKP coupled with the electromagnetic field and found the UV free radiative corrections. Moreover, when the causal approach is supplemented with the analytical representation, this framework gives us a powerful method to approach formally Light-front Dynamics. In particular, it has been obtained a Light-Front Quantum Electrodynamics without prescriptions [13]. The same idea was applied for Generalized Quantum Electrodynamics [14], in the usual dynamics, to calculate the Podolsky’s corrections to the Bhabha scattering at the Born level. In this work we will approach the SDKP coupled with the electromagnetic field in the framework of the causal perturbative method supplemented with the analytical representation. Our goals are to calculate the free electromagnetic and scalar propagators and, to calculate the amplitude transitions and the cross section, at Born level, for the Moeller and Compton scattering. 2. The S-Matrix inductive Causal Program The axiomatic construction of perturbative Quantum Field Theory is reviewed from a causal point of view introduced by Epstein and Glaser in 1973 [8]. In this axiomatic approach the scattering operator S can be written in the following formal perturbative series: S rgs “ 1` 8 ÿ n“1 1 n! ż dx1dx2...dxnTn px1, x2, ..., xnq g px1q g px2q ...g pxnq , (1) where we can identify the quantity Tn as an operator-valued distribution and gbn its test function[16]. The operator S can be constructed using Causality and Poincaré invariance as the principal physical properties. From Causality we can prove the following property Tn px1, ..., xm, xm`1, ..., xnq “ Tm px1, ..., xmqTn´m pxm`1, ..., xnq , (2) if x01, ..., x 0 m ( ą x0m`1, ..., x 0 n ( . The first step in the construction of S is to define T1pxq. In the KGF Formalism we define TKG1 pxq ” i : L1o int :, where L1o int is the interacting lagrangian in first order on the coupling constant avoiding the e2 term obtained by minimal coupling prescription for getting gauge invariance[11]. For DKP formalism the e2 term does not exist, and we can follow the definition T1pxq ” i : Lint : as for Quantum Electrodynamics (QED)[9]. The second step is to construct the intermediary distributions A12px1, x2q ” ´T1px1qT1px2q and R12px1, x2q ” ´T1px2qT1px1q with the goal to obtain advanced A2 and retarded R2 XIII International Workshop on Hadron Physics IOP Publishing Journal of Physics: Conference Series 706 (2016) 052019 doi:10.1088/1742-6596/706/5/052019 2 distributions as A2px1, x2q ” ´T1px1qT1px2q ` T2px1, x2q “ A12px1, x2q ` T2px1, x2q, (3) R2px1, x2q ” ´T1px2qT1px1q ` T2px1, x2q “ R12px1, x2q ` T2px1, x2q. (4) We can see from (2) that the support of A2 is the past of light cone centered in x2, and for R2 the support is the future part. Unlike the usual way to get the advanced and retarded part of propagators using the step function, the Epstein-Glasser’s method does not multiply distributions in the same variable because such multiplication is not always well defined. Now, from (3) and (4) we can compute explicitly the causal distribution D2px1, x2q ” R12px1, x2q ´A 1 2px1, x2q “ R2px1, x2q ´A2px1, x2q. In the computation of D2 we can use Wick’s theorem to obtain the following form for the causal distribution D2px1, x2q “ ÿ k : « ź j φ:pxjq ff dk2px1, x2q « ź l φ:pxlq ff :: « ź m Apxmq ff :, (5) where dk2px1, x2q are numerical distributions with the light cone centered in x2 as support. Using the distribution theory, we can compute R2 as a retarded part of D2. This process is called causal splitting and it is not a trivially process. In causal splitting we need to find the order of singularity ω of dk2px1, x2q. In function of ω, we have two solutions for the retarded part r̂2ppq of dk2px1, x2q in momentum space[17] r̂ppq “ i 2π sgnpp0q ż dt d̂ptpq 1´ t` sgnpp0qi0` , when ω ă 0 (6) r̂ppq “ ” i 2π sgnpp0q ż dt d̂ptpq tω`1p1´ t` sgnpp0qi0`q ı ` ω ÿ a“0 Cap a, when ω ě 0 (7) The solution (6) is the trivial splitting using a step function Θpx0q. The second solution (7) is carefully computed with the distribution theory and in that computation all steps are well defined. In the second solution, the constant Ca appears because of the uniqueness of the solution. Those constants will be computed using physical properties as gauge invariance, charge invariance, etc. Finally, we compute the second term T2px1, x2q making the necessary substitutions. With T2px1, x2q we compute all we need from the scattering operator until second order. 3. Free DKP causal propagators The Epstein-Glasser causal approach follows the Heisenberg program[15]. For that reason we work with the causal propagators obtained from the free field theory. In this work we are going to use Wightman’s formalism to compute those propagators[14][17]. Wightman’s formalism tell us that if we know the green function Ĝpkq associated to some free field equation, then the analytical representation of the positive and negative propagators D̂p˘q are A D̂p˘q, ϕ E “ p2πq´2 ¿ c˘ Ĝpkqϕpk0qdk0. (8) where c`p´q is a counterclockwise closed path which contains only the positive (negative) poles of the Green function Ĝpkq, and ϕpk0q are test functions of D̂p˘q. Using (8), we may define the causal propagator: D̂ pkq “ D̂p`q pkq ` D̂p´q pkq , (9) XIII International Workshop on Hadron Physics IOP Publishing Journal of Physics: Conference Series 706 (2016) 052019 doi:10.1088/1742-6596/706/5/052019 3 furthermore, for local field theories, this propagator must have causal support in the configuration space suppD px1, x2q Ď Γ`2 px2q Y Γ´2 px2q . (10) This property must be proven explicitly for each case. Now we are going to apply Wightman’s formalism for DKP’s fields. The free DKP theory is considered as a Lagrangian [4] LDKP “ i 2 ψ̄βµ ÐÑ B µψ ´mψ̄ψ, (11) where ψ is a multi-component wave function, ψ̄ “ ψ:η0, and η0 “ 2 ` β0 ˘2 ´ 1. We have that tβµu are a set of matrices (µ “ 0, 1, 2, 3) satisfying the algebraic relations βµβνβρ ` βρβνβµ “ βµgνρ ` βρgµν . (12) The equations of motion are then D pBqψ “ piβ.B ´mqψ “ 0 , ψ̄ ÐÝD pBq “ ψ̄ ´ iβ. ÐÝ B `m ¯ “ 0. (13) It is known that the algebra of the β matrices has only three irreducible representations, whose degrees are 1, 5 and 10 [1]. The first one is trivial, having no physical content. The second and third ones correspond, respectively, to the scalar and vectorial representations. In this work we shall restrict us to the scalar case. To obtain the causal propagator, we can find that the Green function of the DKP field equation p13q is given by: Ĝ pkq “ 1 m β.p pβ.p`mq p2 ´m2 ´ I m , (14) then, using the analytical representation (8), the positive (PF) and negative (PF) frequency propagators are Ŝp˘q ppq “ 1 m β.p pβ.p`mq D̂p˘qm ppq . (15) where D̂ p˘q m are the PF and NF scalar propagators[17]. These PF and NF propagators are related to the contractions of DKP fields as follows hkkkkkkikkkkkkj ψa pxq ψ̄b pyq ” ” ψp´qa pxq , ψ̄ p`q b pyq ı “ 1 i S p`q ab px´ yq , (16) hkkkkkkikkkkkkj ψ̄a pxqψb pyq ” ” ψ̄p´qa pxq , ψ p`q b pyq ı “ 1 i S p´q ba px´ yq , (17) Moreover, the causal DKP propagator is obtained as Ŝ “ Ŝp`q ` Ŝp´q, with the following result Ŝ ppq “ 1 m β.p pβ.p`mq D̂m ppq , (18) where D̂m is the known scalar massive Pauli-Jordan propagator given by D̂m ppq “ D̂p`qm ` D̂p´qm “ i 2π sgn pp0q δ ` p2 ´m2 ˘ . (19) It is straightforward confirm that Ŝppq has a causal support as required. XIII International Workshop on Hadron Physics IOP Publishing Journal of Physics: Conference Series 706 (2016) 052019 doi:10.1088/1742-6596/706/5/052019 4 For electromagnetic field quantization by Wightman’s formalism we recommend the reference [17]. The most important result for our calculation is the contraction between two electromagnetic fields hkkkkkkikkkkkkj Aµ pxqAν pyq ” ” Ap´qµ pxq , Ap`qν pyq ı “ iDp`qµν px´ yq , (20) where D p`q µν px1 ´ x2q is given by the following expression Dp`qµν pxq “ gµνD p`q 0 pxq , (21) and D0 is given by (19) with m “ 0. 4. Moeller and Compton scattering As we said in section 2. for DKP gauge theory we define T1pxq ” ie : ψ̄pxqβµψpxq : Aµpxq. Then, in the Epstein-Glasser approach the causal distribution D2 has the following structure D2px, yq “ D p1q 2 `D p2q 2 `D p3q 2 `D p4q 2 `D p5q 2 (22) where each term in (22) could be associated with different processes. In this work we are interested in Moeller and Compton scattering Dp1q and Dp2q respectively. Those terms explicitly are D p1q 2 “ ´ie2 : ψpx2qβ µψpx2qψpx1qβ νψpx1q : Dµνpx1 ´ x2q (23) D p2q 2 “ie2 : ψpx1qβ νSpx1 ´ x2qβ µψpx2q :: Aµpx2qAνpx1q : ´ ie2 : ψpx2qβ µSpx2 ´ x1qβ νψpx1q :: Aµpx2qAνpx1q : (24) 4.1. Moeller Scattering The differential cross section for Moeller scattering is obtained from (23). For splitting (23) the computation of the order of singularity gives us ω “ ´2, what means that the retarded part is given by the solution (6). Making the necessary substitutions we get for T p1q 2 T p1q 2 px1, x2q “ ´e 2igµνβ µ abβ ν cd : ψapx2qψbpx2qψcpx1qψdpxq : DF 0 px1 ´ x2q, (25) where DF 0 px1 ´ x2q is the massless Feynman propagator which in the momentum space is DF 0 pkq “ ´p2πq ´2 1 k2 ` i0` (26) Because the distributional nature of S-matrix, we make our computation of cross section smearing S-matrix with the wave packets |ψiy “ ż d3p1d 3q1ψi pp1,q1q a : pp1q a : pq1q |Ωy , (27) |ψf y “ ż d3p2d 3q2ψf pp2,q2q a : pp2q a : pq2q |Ωy , (28) where a:pplq is a creation operator of a DKP particle with 3-momentum pl and if pi and pf are the ingoing and outgoing 3-momentum of the DKP’s particles, then |ψiy and |ψf y are the wave-packet functions sharply peaked in pi and pf respectively [9]. Then, by definition we have for the transition amplitude Pfi XIII International Workshop on Hadron Physics IOP Publishing Journal of Physics: Conference Series 706 (2016) 052019 doi:10.1088/1742-6596/706/5/052019 5 Pfi ” p|ψf y , |ψiyq2 “ |Sfi|2 “ ˇ ˇ ˇ A ψf ˇ ˇ ˇ S p1q 2 ` . . . ˇ ˇ ˇ ψi E ˇ ˇ ˇ 2 . (29) In the laboratory frame, the incoming particles are a beam of particles with a cylinder form of radius R and velocity v, then summing over all final states we obtain [9] ÿ f Pfi pRq “ 1 πR2 „ p2πq2 1 |v| ż d3p2d 3q2 |M ppi, qi, p2, q2q| 2 δ pp2 ` q2 ´ pi ´ qiq  , (30) where M is a distributional quantity related to the S-Matrix as it follows Sfi “ ie2δ ppf ` qf ´ pi ´ qiqM. (31) The scattering cross section in the laboratory frame is then given by σlab ” lim RÑ8 πR2 ÿ f Pif pRq. (32) where ř f Pif pRq is the average of ř f Pif pRq over the cylinder of radius R defined by the incoming particles. Finally, after some no trivial computation, we end with the following differential cross section dσc.m dΩ “ α2 4s ˇ ˇ ˇ ˇ s´ t u ` s´ u t ˇ ˇ ˇ ˇ 2 , (33) where α is the fine-structure constant and ts, t, uu the Mandelstam’s variables. The solution (33) is in accordance with the one obtained using the KGF equation by the causal approach [11] and the one obtained using the usual approach[19][18]. 4.2. Compton Scattering For computing the differential cross section for Compton scattering we use (24). The order of singularity of (24) is ω “ 0. This value tell us that to obtain the retarded part in the causal splitting process we must use the solution (7) where a non fixed constants will appear. Then, making the necessary substitutions we get the 2-points distribution T p2q 2 T p2q 2 “ e2i : ψcpxqβ ν cdp´S F dapx´ yq ` Cdaδpx´ yqqβ µ abψbpyq :: AµpyqAνpxq : ` e2i : ψapyqβ µ abp´S F bcpy ´ xq ´ C 1 bcδpx´ yqβ ν cdψdpxq :: AµpyqAνpxq : (34) where Cda and C 1bc can not be fixed by causality or causal splitting process. To obtain them we need to use other physical properties or symmetries of S-Matrix. Using charge invariance we get C “ C 1 and using Gauge invariance we get C “ I m . We want to emphasize here that if we make the integration in Delta Dirac distribution in the S-matrix we are going to obtain the multiplication of two DKP fields and two electromagnetic fields in the same variable. This result is consistent with the term e2 placed in the usual approach to get gauge invariance by minimal coupling prescription. In order to compute the differential cross section we could follow the same steps as for the Moeller scattering, but here we need to smear the S-Matrix with the following wave packets |ψiy “ ż d3p1d 3k1ψi pp1,q1q a : pp1q εiνb:νpk1q |Ωy , (35) |ψf y “ ż d3p2d 3k2ψf pp2,q2q a : pp2q εfνb:νpk2q |Ωy , (36) XIII International Workshop on Hadron Physics IOP Publishing Journal of Physics: Conference Series 706 (2016) 052019 doi:10.1088/1742-6596/706/5/052019 6 where εiν and εfν are the 4-polarization vectors for the incoming and outgoing photons, and the b:νpklq are the creation operator valued distributions for photons. The other steps in the computation are the same which were used for Moeler scattering. With the help of the polarization conditions εiki “ 0, εfkf “ 0 (37) and using the reference system where piεi “ 0, piεf “ 0 (38) we get dσlab dΩ “ e4 16π2 ω2 f m2ω2 i ” εµf pkf qεiµpkiq ı2 “ α2ω2 f m2ω2 i ” εµf pkf qεiµpkiq ı2 . (39) where α is the fine-structure. The solution (39) is in accordance with the one obtained using the KGF equation by the Causal Approach [11] and the one obtained using the usual approach[19]. 5. Conclusions We could note that the combination of the analytical representation technique with the Epstein- Glasser causal approach gives us a well define theory for the study of the SDKP. We have calculated the DKP free propagator using analytical representation, and the cross section for the Moeller and Compton scattering process for the SDKP. For the Moeller and Compton scattering process we find, at Born level, whose the results of the differential cross sections are identical to that obtained in the usual treatment of Scalar Quantum Electrodynamics. These results provide us an evidence that the SDKP and the KGF theory are equivalent, at Born level, even when they are coupled with the electromagnetic field for these processes. For future works, we will calculate the radiative corrections and determine some important general results such as the renormalizability of the theory. Acknowledgments J Beltrán thanks CAPES for full support. B M Pimentel thanks CAPES and CNPq for partial support. D E Soto thanks CNPq for partial support. References [1] Petiau G 1936 Acad. Roy. de Belg. A. Sci. Mem. Collect. 16 ; Duffin R Y 1938 Phys. Rev. 54, 1114; Kemmer N 1939 Proc. R. Soc. London 137A, 91. [2] Cardoso T R, Castro L B, Castro A S 2010 J. Phys. A: MAth. Theor. 43 055306. [3] Nedjadi Y and Barrett R C 1994 J. Phys. A: Math. Gen. 27 4301. Nedjadi Y and Barrett R C 1998 J. Phys. A: Math. Gen. 31 6717. [4] Lunardi J T, Pimentel B M, Texeira R G and Valverde J S 2000 Phys. Lett. A288, 165; Nowakowski M 1998 Phys. Lett. A244, 329. [5] Lunardi J T, Pimentel B M and Texeira R G 2001 in Procs. 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[13] Bufalo R , Pimentel B M, Soto D E 2014 Annals of Physics 351 10341061; Annals of Physics 351 10621084. [14] Bufalo R, Pimentel B M, Soto D E 2014 The Causal approach for the electron-positron scattering in the Generalized Quantum Electrodynamics, Phys. Rev. D 90, 085012. [15] Heisenberg W 1943 Z. Physik 120, 513. [16] Bogoliubov N N and Shirkov D V 1980 Introduction to the Theory of Quantized Fields (New York: John Wiley & Sons) 3rd ed. [17] Soto D E 2014 Eletrodinâmica Quântica Generalizada a la Teoria de Perturbação Causal PhD[Thesis] São Paulo: Instituto de F́ısica Teorica Universidade Estadual Paulista. [18] Akhiezer A I and Berestetski V B 1965 Quantum Electrodynamics (New York: Inter.) 2nd ed. [19] Itzykson C and Zuber J-C 1980 Quantum Field Theory (McGraw-Hill). XIII International Workshop on Hadron Physics IOP Publishing Journal of Physics: Conference Series 706 (2016) 052019 doi:10.1088/1742-6596/706/5/052019 8