Infrared Properties of Hadronic Structure of Nucleonin Neutron Beta Decays to Order O(\alpha/\pi) in Standard V-A Effective Theorywith QED and Linear Sigma Model of Strong Low–Energy Interactions

# Infrared Properties of Hadronic Structure of Nucleon in Neutron Beta Decays to Order O(α/π) in Standard V−A Effective Theory with QED and Linear Sigma Model of Strong Low–Energy Interactions

A. N. Ivanov Atominstitut, Technische Universität Wien, Stadionallee 2, A-1020 Wien, Austria    R. Höllwieser Atominstitut, Technische Universität Wien, Stadionallee 2, A-1020 Wien, Austria Department of Physics, Bergische Universität Wuppertal, Gaussstr. 20, D-42119 Wuppertal, Germany    N. I. Troitskaya Atominstitut, Technische Universität Wien, Stadionallee 2, A-1020 Wien, Austria    M. Wellenzohn Atominstitut, Technische Universität Wien, Stadionallee 2, A-1020 Wien, Austria FH Campus Wien, University of Applied Sciences, Favoritenstraße 226, 1100 Wien, Austria    Ya. A. Berdnikov Peter the Great St. Petersburg Polytechnic University, Polytechnicheskaya 29, 195251, Russian Federation
July 13, 2019
###### Abstract

Within the standard theory of weak interactions, Quantum Electrodynamics (QED) and the linear –model (LM) of strong low–energy hadronic interactions we analyse infrared properties of hadronic structure of the neutron and proton in the neutron –decays to leading order in the large nucleon mass expansion. We confirm validity and high confidence level of contributions of hadronic structure of the nucleon to the radiative corrections, calculated by Sirlin (Phys. Rev. 164, 1767 (1967)) to leading order in the large nucleon mass expansion. At the level of order relative to Sirlin’s infrared divergent contribution to the neutron radiative –decay (inner bremsstrahlung) we find an infrared divergent contribution, induced by hadronic structure of the nucleon through the one–pion–pole exchange, to the rate of the neutron lifetime from the neutron radiative –decay, which should be cancelled by contributions of virtual photon exchanges to the neutron –decay. Following Ivanov et al. 1805.09702 [hep-ph] we argue that a consistent analysis of such a cancellation may be carried out well in the combined quantum field theory including the Standard Electroweak Model (SEM) and the LM of strong low–energy interactions, where the effective hadron–lepton current–current vertex is caused by the –electroweak–boson exchange.

###### pacs:
11.10.Ef, 11.10.Gh, 12.15.-y, 12.39.Fe

It is well–known that the neutron radiative –decay (inner bremsstrahlung) plays an important for cancellation of infrared divergences, caused by virtual photon exchanges, to the neutron –decay to order Berman1958 ()-Ivanov2017a (), where is the fine–structure constant PDG2018 (). As a physical process the neutron radiative –decay has been investigated theoretically to order in Gaponov1996 ()Ivanov2017 () (see also Ivanov2013 ()) and to order in Gardner2012 (); Gardner2013 (); Ivanov2017b (), respectively, and experimentally in Nico2006 ()Bales2016 (). Recently Ivanov2018b () we have analysed gauge properties of hadronic structure of the nucleon in the neutron –decays within the combined quantum field theory including the standard effective theory of weak interactions Feynman1958 ()Marshak1969 (), Quantum Electrodynamics (QED) and the linear –model (the LM) of strong low–energy interactions GellMann1960 ()DeAlfaro1973 (), which is renormalizable Bernstein1960 (); Lee1969 (); Gervais1969 (); Mignaco1971 (); Strubbe1972 () and in the infinite limit of the scalar –meson mass reproduces the results of the current algebra Weinberg1967 (); Gasiorowicz1969 (). We have shown that in the limit , to leading order in the large nucleon mass expansion and after renormalization such a combined quantum field theory defines the standard Lorentz structure of the matrix element of the hadronic transition of the neutron –decay including vector, axial–vector and pseudoscalar terms Leitner2006 (); Ivanov2018 (), where the contributions of strong low–energy interactions are defined by the axial coupling constant and the one–pion–pole exchange, and the gauge invariant amplitude of the neutron radiative –decay, where the contributions of strong low–energy interactions are presented in terms of the axial coupling constant and one–pion– and two–pion–pole exchanges. The Feynman diagrams of the amplitude of the neutron –decay, calculated to leading order in the large nucleon mass expansion, are shown in Fig. 1.

The amplitude of the neutron –decay is defined by Ivanov2018b ()

 (1)

where is the vector weak coupling constant, and and are the Fermi weak coupling constant and the matrix element of the Cabibbo–Kobayashi–Maskawa (CKM) mixing matrix PDG2018 (), respectively, and is the matrix element of the leptonic current. The matrix element of the hadronic transition calculated in the limit , to leading order in the large nucleon mass expansion and after renormalization, takes the form Ivanov2018b ()

 (2)

where the contributions of strong low–energy interactions are presented by the axial coupling constant and the one–pion–pole exchange. The contribution of the one–pion–pole exchange is necessary for local conservation of the axial hadronic current in the limit Feynman1958 (); Nambu1960 ()

 limmπ→0qμ⟨p(→kp,σp)|J+μ(0)|n(→kn,σn)⟩=0. (3)

The one–pion–pole exchange contribution appears also in the current algebra approach Marshak1969 () (see also Adler1968 ()). The term with the Lorentz structure , where is the nucleon mass, describes the contribution of the weak magnetism Bilenky1959 (); Wilkinson1982 () with the isovector anomalous magnetic moment of the nucleon .

The Feynman diagrams of the amplitude of the neutron radiative –decay are shown in Fig. 2.

The analytical expression of the amplitude of the neutron radiative –decay, defined by the Feynman diagrams in Fig. 2 is given by Ivanov2018b ()

 M(n→pe−¯νeγ)λ=eGV ×{[¯up(→kp,σp)γμ(1−gAγ5)un(→kn,σn)][¯ue(→ke,σe)12ke⋅kQe,λγμ(1−γ5)vν(→kν,+12)] −[¯up(→kp,σp)Qp,λ12kp⋅kγμ(1−gAγ5)un(→kn,σn)][¯ue(→ke,σe)γμ(1−γ5)vν(→kν,+12)]
 −2gAmNqμm2π−q2−i0[¯up(→kp,σp)Qp,λ12kp⋅kγ5un(→kn,σn)][¯ue(→ke,σe)γμ(1−γ5)vν(→kν,+12)] (4)

where and are given by Ivanov2013 (); Ivanov2017 (); Ivanov2017b ()

 Qe,λ=2ε∗λ(k)⋅ke+^ε∗λ(k)^k,Qp,λ=2ε∗λ(k)⋅kp+^ε∗λ(k)^k. (5)

Here is the polarization vector of the photon with the 4–momentum and in two polarization states , obeying the constraint . For the derivation of Eq.(1) we have used the Dirac equations for the free proton and electron. The first two terms in Eq.(2) are given by the Feynman diagrams in Fig. 1a and Fig. 1b, whereas the last four terms correspond to the contributions of the Feynman diagrams in Fig. 2c - Fig. 2f, respectively. As has been shown in Ivanov2018b () the contributions of the sum of the Feynman diagrams in Fig. 1a and Fig. 2b and the sum of Fig. 2c - Fig. 2f are invariant independently under a gauge transformation of the photon wave function , where is an arbitrary constant. The contributions of strong low–energy interactions are presented by the axial coupling constant and the one–pion– and two–pion–pole exchanges in the Feynman diagrams Fig. 2c - Fig. 2f. In the amplitude of the neutron radiative –decay Eq.(Infrared Properties of Hadronic Structure of Nucleon in Neutron Beta Decays to Order in Standard Effective Theory with QED and Linear Sigma Model of Strong Low–Energy Interactions) we have omitted the contribution of the weak magnetism. The contribution of this term together with the proton recoil has been consistently taken into account in the rate of the neutron radiative -decay in Ivanov2017 ().

The contribution of the first two terms in Eq.(Infrared Properties of Hadronic Structure of Nucleon in Neutron Beta Decays to Order in Standard Effective Theory with QED and Linear Sigma Model of Strong Low–Energy Interactions) to the rate of the neutron radiative –decay, taken to leading order in the large nucleon mass expansion with photon from the energy region , has been calculated in Gaponov1996 (); Bernard2004 (); Ivanov2013 (); Ivanov2017 (). It takes the form Ivanov2013 (); Ivanov2017 ()

 λβγ(ωmax,ωmin) = απ(1+3g2A)|GV|2π3∫ωmaxωmindωω∫E0−ωmedEe√E2e−m2eEeF(Ee,Z=1)(E0−Ee−ω)2 (6) ×{(1+ωEe+12ω2E2e)[1βℓn(1+β1−β)−2]+ω2E2e},

where is the well–known relativistic Fermi function of the Coulomb proton–electron final–state interaction, and are the energy and velocity of the decay electron, is the end–point energy of the electron–energy spectrum Ivanov2013 (). In the experimental energy region Bales2016 () the rate Eq.(6) defines the branching ratio , calculated for the neutron lifetime calculated in Ivanov2013 () at Abele2008 () (see also Mund2013 ()). Such a branching ratio does not contradict the experimental value Bales2016 () within two standard deviations. The values of the neutron lifetime and axial coupling constant agree well with recent values of the neutron lifetime and axial coupling constant , which were recommended by Czarnecki et al. Sirlin2018 () as favoured.

The contribution of these two terms, taken to leading order in the large nucleon mass expansion, to the radiative corrections to the rate of the neutron –decay has been calculated in Berman1958 ()Abers1968 (); Gudkov2006 () (see also and Ivanov2013 ()) within finite–photon mass regularization and takes the form Ivanov2013 ()

 λβγ(E0,μ) = απ(1+3g2A)|GV|2π3∫E0medEe√E2e−m2eEeF(Ee,Z=1)(E0−Ee)2 (7) ×{[2ℓn(2(E0−Ee)μ)−3+23E0−EeEe(1+18E0−EeEe)][12βℓn(1+β1−β)−1]+1 +112(E0−Ee)2E2e+12βℓn(1+β1−β)−14βℓn2(1+β1−β)−1βLi2(2β1+β)},

where is the Polylogarithmic function and is an infinitesimal photon mass, which should be taken in the limit Berman1958 ()Abers1968 (); Gudkov2006 () (see also Ivanov2013 ()). The rate of the neutron –decay , taking into account radiative corrections, caused by virtual photon exchanges, calculated within the finite–photon mass regularization, is given by (see Appendix D of Ivanov2013 ())

 λβ(E0,μ) =
 ×[12βℓn(1+β1−β)−1]+32ℓn(mpme)−118−1βLi2(2β1+β)−14βℓn2(1+β1−β)+β2ℓn(1+β1−β)}}. (8)

Summing up the contributions of the rates and and taking the limit we arrive at the rate of the neutron decay

 λn(E0)=(1+3g2A)|GV|2π3∫E0medEe√E2e−m2eEeF(Ee,Z=1)(E0−Ee)2(1+απ¯gn(Ee)), (9)

where the function , defining the radiative corrections to the neutron lifetime, is equal to Sirlin1967 () (see also Appendix D of Ivanov2013 ())

 ¯gn(Ee) = 32ℓn(mpme)−38+2[12βℓn(1+β1−β)−1][ℓn(2(E0−Ee)me)−32+13E0−EeEe] (10) − 2βLi2(2β1+β)+12βℓn(1+β1−β)[(1+β2)+112(E0−Ee)2E2e−ℓn(1+β1−β)].

The contribution of the electroweak–boson exchanges together with QCD corrections has been calculated by Czarnecki et al. Sirlin2004 () (see also Sirlin1986 ()). This defines the the radiative corrections to the neutron lifetime, which are described by the function , where (see Appendix D of Ivanov2013 ()) is caused by electroweak–boson exchanges and QCD corrections Sirlin2004 ().

The infrared divergent contribution of hadronic structure of the nucleon to the rate of the neutron radiative –decay, induced by the term in the rate of the neutron radiative –decay Ivanov2013 (); Ivanov2017 (); Ivanov2017b (), is equal to

 λ(h.s.)βγ(E0,μ)=απ(1+3g2A)|GV|2π3∫E0medEe√E2e−m2eEeF(Ee,Z=1)(E0−Ee)2Δ¯g(h.s.)n(Ee,μ). (11)

Thus, the contribution of hadronic structure of the nucleon to the radiative corrections of order to the neutron lifetime is equal to

 Δ¯g(h.s.)n(Ee) = −2g2A1+3g2Am2em2π{2ℓn(2(E0−Ee)μ)[12βℓn(1+β1−β)−1]+1+12βℓn(1+β1−β)−14βℓn2(1+β1−β) (12) − 1βLi2(2β1+β)}.

Relative to Sirlin’s infrared divergent contribution Eq.(7) the order of this correction is of about . This confirms validity and high confidence level of the contribution of hadronic structure of the nucleon to the radiative corrections of the neutron lifetime, calculated by Sirlin Sirlin1967 () (see Eq.(10)), who proved a factorization of strong low–energy and electromagnetic interactions to leading order in the large nucleon mass expansion and, practically, dealt with structureless neutron and proton.

The main problem of the infrared divergent correction Eq.(12) can be related only to a cancellation of such a divergence by the contribution of virtual photon exchanges, that should be similar to cancellation of the infrared divergence in Eq.(7) by the infrared divergence in Eq.(8). An impossibility to cancel such an infrared divergent correction Eq.(12), calculated to leading order in the large nucleon mass expansion, by virtual photon exchanges in the neutron –decay should testify a certain inconsistency of the approach. Following Ivanov2018b () we may argue that a problem of possible non–cancellation of the infrared divergence Eq.(12) by virtual photon exchanges in the neutron –decay can be related to the use of the standard effective theory of weak interactions. The point is that the effective vertex of nucleon–lepton current–current interaction, accounting for both baryonic and mesonic currents coupled to leptonic current, is not the vertex of the combined quantum field theory including QED and LM or any other theory of strong low–energy interactions. For correct account for contributions of hadronic structure of the nucleon to radiative corrections to the neutron –decays one has to use a combined quantum field theory including the Standard Electroweak Model (SEM) and a theory of strong low–energy interactions (e.g. the LM). However, first of all in our forthcoming publication we are planning to perform an analysis of cancellation of the divergent correction Eq.(12) by virtual photon exchanges in the neutron –decay within the combined quantum field theory described in Ivanov2018b () and discussed in this paper.

We thank Hartmut Abele for discussions stimulating the work under this paper as a step towards the analysis of the SM corrections of order Ivanov2017a (); Ivanov2017b (); Ivanov2018a (). The work of A. N. Ivanov was supported by the Austrian “Fonds zur Förderung der Wissenschaftlichen Forschung” (FWF) under contracts P26781-N20 and P26636-N20 and “Deutsche Förderungsgemeinschaft” (DFG) AB 128/5-2. The work of R. Höllwieser was supported by the Deutsche Forschungsgemeinschaft in the SFB/TR 55. The work of M. Wellenzohn was supported by the MA 23 (FH-Call 16) under the project “Photonik - Stiftungsprofessur für Lehre”.

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