Lepton Phenomenology of Stueckelberg Portal to Dark Sector

We propose an extension of the Standard Model (SM) with a $U_{A'}(1)$ gauge invariant dark dector connected to the SM via a new portal -- the Stueckelberg portal, arising in the framework of dark photon $A'$ mass generation via Stueckelberg mechanism. This portal opens through the effective dim=5 operators constructed from the covariant term of the auxiliary Stueckelberg scalar field $\sigma$ providing flavor non-diagonal renormalizable couplings of both $\sigma$ and $A'$ to the SM fermions $\psi$. The Stueckelberg scalar plays a role of Goldstone boson in the generation of mass of the Dark Photon. Contrary to the conventional kinetic mixing portal, in our scenario flavor diagonal $A'$-$\psi$ couplings are not proportional to the fermion charges and are, in general, flavor nondiagonal. These features drastically change the phenomenology of dark photon $A'$ relaxing or avoiding some previously established experimental constraints. We focus on the phenomenology of the described scenario of the Stueckelberg portal in the lepton sector and analyze the contribution of the dark sector fields $A'$ to the anomalous magnetic moment of muon $(g-2)_{\mu}$, lepton flavor violating decays $l_{i}\to l_{k}\gamma$ and $\mu-e$ conversion in nuclei. We obtain limits on the model parameters from the existing data on the corresponding observables.

We propose an extension of the Standard Model (SM) with a U A (1) gauge invariant dark dector connected to the SM via a new portal -the Stueckelberg portal, arising in the framework of dark photon A mass generation via Stueckelberg mechanism.This portal opens through the effective dim=5 operators constructed from the covariant term of the auxiliary Stueckelberg scalar field σ providing flavor non-diagonal renormalizable couplings of both σ and A to the SM fermions ψ.The Stueckelberg scalar plays a role of Goldstone boson in the generation of mass of the Dark Photon.Contrary to the conventional kinetic mixing portal, in our scenario flavor diagonal Aψ couplings are not proportional to the fermion charges and are, in general, flavor nondiagonal.These features drastically change the phenomenology of dark photon A relaxing or avoiding some previously established experimental constraints.We focus on the phenomenology of the described scenario of the Stueckelberg portal in the lepton sector and analyze the contribution of the dark sector fields A to the anomalous magnetic moment of muon (g − 2)µ, lepton flavor violating decays li → l k γ and µ − e conversion in nuclei.We obtain limits on the model parameters from the existing data on the corresponding observables.

I. INTRODUCTION
The idea of the dark sector (DS) of the Universe, existing almost independently of the Standard Model (SM) sector, has attracted growing interest in recent years.Originally, DS was thought to be populated by only one dark species, necessary to make up for the lack of matter in the Universe with dark matter (DM).In particular, extensions of DS were motivated by the popular scenario of light sub-GeV DM.It was realized that in this case a dark boson, known as the dark photon, would need to be introduced to prevent the Universe from overclosing.An extended DS can have not only cosmological, but also interesting phenomenological consequences.This DS physics beyond the SM can manifest itself in the phenomena observable experimentally (for a status report see, e.g., Ref. [1]).
An encouraging indication of new physics has recently come from measurements of the anomalous magnetic moment (AMM) of the muon (g −2) µ .The Fermilab Muon g −2 Collaboration published [14] the observation of 4.2 σ deviation of the (g − 2) µ from its SM value and stimulated an explosion of the BSM literature.As is known, measurements of (g − 2) µ are a very sensitive probe of BSM physics.The Fermilab Muon g − 2 result with such unprecedented precision can severely limit or refute many BSM models.
On the other hand, there is no doubt that the SM is an incomplete theory, requiring some physics beyond its scope to explain a number of problems that cannot be addressed in the SM.Among them, the DM problem is one of the most obvious.As we already mentioned, DM hints at the existence of a DS of the Universe, which not only provides DM particle candidates, but is also populated by other particles involved in interactions governed by some dark symmetries.The DS with possible nontrivial physics could have a phenomenological impact on the SM sector through portals such as the well-known kinetic mixing of dark and normal photons.Other hypothetical DS particles can have access to the SM sector through different portals and contribute to various observables, in particular, to (g − 2) µ , allowing one to probe the DS.
Here, we propose an extension of the SM by inclusion of DS with U A (1) symmetry.The corresponding gauge boson A , also known as dark photon, acquires a nonzero gauge invariant mass via the Stueckelberg mechanism [57,58], which implies the existence of a scalar Stueckelberg field σ which is unphysical.
This field opens a new portal from the SM to the dark sector via the effective dimension-5 operator with the covariant derivative of the σ-field.We call it the Stueckelberg portal.In our setup this portal coexists with the conventional kinetic mixing portal and leads to new phenomenological effects in the SM sector, in particular, flavor violation both in lepton and quark sectors.In the present work we focus on the lepton flavor violation (LFV) and the corresponding experimental observables.
We also introduce one dark fermion, χ, charged under U (1) D , which is a viable light DM particle candidate.We postulate that dark scalar boson (DSB) plays important role in this model: (1) generating mass of the dark gauge boson (DGB) or dark photon, via the Stueckelberg mechanism [57,58]; (2) generate a mixing of DS with SM fermion including couplings preserving and violating symmetries of SM (e.g., lepton flavor violation (LFV)).Interaction of DGB and DSB with fermions is based on the idea of a familon (or flavons) [35,45,59].(3) plaing the role auxiliary field for DGB and reduce unphysical degrees of freedom of one.
The paper is organized as follows.In Sec.II we describe our theoretical setup.In Secs.III, IV and V we consider application of the Stueckelberg Portal to phenomenological aspects of the g −2 lepton anomaly, lepton conversion, and rare lepton-flavor-violating decays l i → l k γ which were used to derive limits for couplings occurring in the Stueckelberg portal.In Sec.VI we discuss the boundary to DGB couplings and a possible contribution to g −2 lepton from obtained restrictions for different channels.Section VII is the conclusion.Technical details of our calculations are placed in Appendixes.

II. THEORETICAL SETUP
We consider an extension the SM with a U A (1) gauge invariant dark sector described by the Lagrangian where L SM and L DS are the SM and dark sector Lagrangians.Communication between these two sectors takes place through a portal interactions L port .We suppose that the DS, blind to the SM interactions, is populated with Dirac fermions χ i charged under U A (1).The lightest of which is stable and plays the role of dark matter.By definition, all the SM fields are neutral with respect to this group.The gauge boson, A , of the dark sector U A (1) group is called the dark photon.In the conventional dark photon scenario A acquires its mass m A from spontaneous breaking of the U A (1) group.In contrast, in our approach its mass is a gauge-invariant quantity generated by the Stueckelberg mechanism.This requires the introduction of a scalar Stueckelberg field σ, which plays a role of Goldstone boson in the generation of mass of the dark photon.The U A (1) gauge invariant Stueckelberg Lagrangian L DS with one specie of Dark fermion, χ, reads where, as usual, The Stueckelberg covariant derivative is defined as The U A (1)-symmetry is realized on the dark sector fields according to the transformations where As seen from ( 5), the σ is an axionlike field shift transformed under the U A (1). Quantization of the U (1) A dark sector requires adding to the Lagrangian (2) a gauge-fixing term [32].We choose it in the form where ξ is a gauge parameter.Then the dark sector Lagrangian can be written as In this gauge the Stueckelberg field is decoupled from other fields, making the theory manifestly unitary and renormalizable.Note that the mass of the σ field is proportional to the gauge parameter ξ, signaling that this field is unphysical.
In the gauge (8) the dark photon propagator takes the form Let us turn to the SM-DS portals L port possible in the present model.The well-known example of these is the generic renormalizable portal given by kinetic mixing of the dark and the SM photons, A − A , according to where A is the mixing parameter.It has a rather particular phenomenology, which we comment on latter.In the Stueckelberg framework there exists another specific portal, which relies on the The fields in this expression belong to the following representations of the ; −2; 0).The parameter Λ is the characteristic scale of this effective operator, defining when it opens up in terms of renormalizable interactions of an UV completion.The parameters χ ij and κ ij form 3 ⊗ 3 Hermitian matrices leading to the neutral current flavor violation both in quark and lepton sectors.In the present work we focus only on the lepton sector and make an ad hoc assumption Let us look closely at the effective operator (11).At first glance, it looks a nonrenormalizable operator of dimension 5.However, after the substitution of the expression (3), we find that the gauge-invariant operator (11) generates dimension-4 interactions of dark photon with the SM fermions ψ in the form where vector g V and axial-vector g A dimensionless couplings are defined as As seen, these couplings are linearly scaled with the dark photon mass m A , which is crucial for our analysis of the A contribution to the lepton sector observables and setting limits on the corresponding couplings in function of the intermediate-state mass.The operator (11) also contains the interaction of the unphysical Goldstone-like field with the SM fermions of the form j µ ∂ µ σ.In principle, these interactions should be taken into account in calculations made in the R ξ gauge (8) with the arbitrary parameter ξ.To avoid this complication we select from now on the unitary-type gauge ξ → ∞ in which, as seen from ( 9), the σ becomes infinitely heavy and decouples completely from the observable sector.Therefore, the only physical interactions generated by (11) in this gauge are due to the renormalizable couplings of the dark photon to the SM fermions (12).
These interactions also absorb the kinetic portal (10).In fact, the latter can be removed from the Lagrangian by the conventional field redefinition converting its effect to the flavor diagonal vector interactions of the form (12).As usual, we shift the SM photon field and, then, rescale the dark photon field These redefinitions generate flavor diagonal couplings of the SM fermions ψ to the dark photon: originating from the kinetic mixing term (10).Here T Q is the charge matrix of SM fermions.These interactions feature the characteristic property of the kinetic portal requiring the A couplings to the SM fermions to be proportional to their electric charges.It is clear that terms (16) are completely absorbed by redefinition of the flavor diagonal vector couplings g V ii in (12).Thus, our model contains the following free parameters g ji .Let us highlight two principal differences between the conventional kinetic portal and the Stueckelberg portal scenarios of dark sector.First, in the latter case contrary to the former one the A couplings to the SM fermions are not proportional to the SM fermion electric charges.Second, these couplings are flavor non-diagonal leading to reach LFV phenomenology.Note that the first point can significantly affect the conclusions following from the existing searches of dark photon.In particular, the conventional dark photon from the kinetic portal scenario has been strongly constrained from the data of NA64 experiment at SPS CERN [2,3].For the Stueckelberg dark photon these constraints can be significantly relaxed.
In the subsequent sections we will study contributions of the dark photon A to muon anomalous magnetic moment (g − 2) µ and LFV decays l i → l k γ as well as µ − e conversion in nuclei.

III. ANOMALOUS MAGNETIC MOMENT
In the Stueckelberg portal scenario the SM leptons l receive the dark photon A one-loop contributions to their (g − 2) l shown in Fig. 1.The loop involves A due to its couplings to the l and f SM fermions according to Eqs. (12).We calculate the corresponding the A loop contribution in the unitary gauge, setting ξ → ∞ in (9).In this case the σ-field is decoupled from low-energy theory, as commented in the previous section.
The first calculation of the vector and axial contributions to the lepton anomalous moments in the R ξ gauge (9), taking into account LFV, was made in Ref. [37].As it was noted in Ref. [37], the axial contributions δa A l can obtained from the vector ones δa V l by inverting the sign in front of the mass of the internal lepton m f → −m f .In particular, the δa V l and δa A l contributions due to exchange of dark photon, A , read [37]: where we defined The dimensionless couplings are defined in Eq. (13).For convenience we present details of the calculations of these integrals in Appendix A. The recent experimental measurements of the anomalous magnetic dipole moments of muon and electron a µ,e = (g µ,e −2)/2 demonstrate conspicuous deviation from the predictions of the SM.In particular, defining ∆a = a exp −a SM , one gets ∆a e = (8.7 ± 0.5) × 10 −13 [64] , ∆a e = (4.8± 3.0) × 10 −13 [65] .
In the case of the muon, the value a exp µ was extracted from the combined data of the E821 experiment at BNL [66] and recent Muon (g − 2) measurements at FNAL [14].This experimental result shows 4.2σ deviation from the SM prediction.The value for ∆a e was derived from the recent measurement of the fine-structure constant [65].For completeness we also include in our analysis the same observable for the τ lepton ∆a τ = (2.79)× 10 −4 [67,68]. ( Its precision is significantly worse than for the case of e and µ.This is due to the experimental difficulties in measuring the properties of such a short-lived particle as the τ lepton.For rough estimations we will use the central value of ∆a τ in (21).We compare our theoretical predictions (17) and (18) with the experimental data (19), (20) and (21).First, we extract upper limits on the coupling constants g V,A ij of the dark photon to the SM fermions.Then, in Sec.VI we will discuss the possibility of simultaneous explanation of the muon and the electron (g − 2) in our model, taking into account the limits from LFV processes.FIG.3: Upper bounds on the couplings g V ij of the dark photon A to the SM fermions in the vector+axial-vector channel as a function of its mass m A , derived from the experimental data on (g − 2) l .The shaded area is excluded.We show the plots for different values of the ratio g A ij /g V ij Extracting limits on g V,A ij we apply the conventional simplifying assumption about the presence of only one nonvanishing coupling constant at a time.In Figs. 2 and 3 we show the resulting upper limits for the couplings g V,A le , which are significantly more stringent than for other combinations of flavor indices g V,A lf with f = e due to the factor m l /m f in Eqs.(17) and (18).These latter limits can be approximately obtained from g V,A le with the corresponding rescaling using the mentioned factor.
To estimate the effect of the combined contribution of g V and g A couplings we also studied the upper limits on the coupling g V for different values of the ratio g A /g V .The results are shown in Fig. 3.
Let us to note that the combination of vector and negative axial-vector contributions to (g − 2) µ makes limits less stringent in comparison with the case of the pure vectorial term.This can significantly extend a window in the mass-coupling parameter space for the light dark vector particles.Finally we note that, if the LFV effects in the (g − 2) were not taken into account, then the upper limits for the couplings g V ll would be almost the same as for g A ll .

IV. NUCLEAR LEPTON FLAVOR CONVERSION
We recall again that the Stueckelberg portal model inherently features LFV couplings of the dark sector field A to the SM leptons, described by (12).These LFV couplings can contribute to both flavor-conserving observables (e.g.(g − 2) l studied in the previous section), and to the LFV ones, some of which we consider in what follows.We start with nuclear µ − − e − conversion, which is a LFV process with the participation of a nucleus, It was recently advocated that deep inelastic lepton conversion on nuclei with X denoting all the possible final-state particles, has good prospects for setting limits on the effective couplings of 1 and 2 in the e − µ, τ and µ − τ channels at the fixed-target NA64 experiment [25].However, the process most studied experimentally is coherent µ − − e − conversion in muonic atoms, in which one electron is replaced by a muon.In this case 1 = µ, 2 = e, and X = (A, Z).This process has not yet been discovered experimentally.Presently the best upper limits on its rate R µe have been set by the SINDRUM II experiment on µ − e conversion in 198 Au [69] In the near future the PRISM/PRIME experiment [70] with a titanium 48 Ti target is going to reach the limit In Ref. [71] on the basis of nucleon-meson effective field theory, the above experimental limits were translated into the upper limits on the effective couplings α V,A of the LFV µ − e current to nucleons defined by Lagrangian These limits are In our approach the Lagrangian ( 25) is generated at tree level by the t-channel exchange with A between the µ − e and qq currents, where q = u, d are valence quarks of the nucleon.Note that transitions with q 1 = q 2 do not contribute to coherent µ − e nuclear conversion.Starting from our Lagrangian ( 12) and matching at a certain scale the quark currents with nucleon ones (see for details Ref. [71]) we find the relations where z is a dimensionless constant of order O(1).Thus, from Eqs. ( 26)-( 28) we find the following upper limits: We will use these limits in Sec.VI for our combined analysis of the (g − 2) l and the LFV experimental data.
V. LFV DECAYS li → l k γ The matrix element of this LFV process can be parametrized as Then 4: Feynman diagrams of gauge invariant matrix elements of the interaction lepton with external electromagnetic field accounting LFV effect generated by the A dark photon.
Here we used the Gordon identities listed in Appendix B. Taking into account that m e m µ m τ , we have for the decay width of this process in a very good approximation the expression [74] Γ where F M and F D are the magnetic and dipole form factors, and α = e 2 /(4π) = 1/137.036is the fine-structure constant.The dark sector contributes to l i → l k γ decay only with the dark photon A according to the diagrams in Fig. 4. The corresponding analytical expressions for single LFV coupling read For µ → eγ process with the τ lepton in loop with double LFV coupling we have where Let us note that the diagrams in Figs.4(b) and 4(c) are needed to guarantee gauge invariance of the photon interactions with leptons through loop diagrams induced by the A dark photon.This simultaneously leads to cancellation of a divergence arising from the diagram in Fig. 4(a).
Similarly we can write the A contributions to F M and F D form factors of the for τ → µγ and τ → eγ LFV rare decays for the case when the initial or final leptons are different from the lepton in the loop.We have for τ → µγ decay and for τ → eγ decay

VI. ANALYSIS OF CURRENT LIMITS
In this section we derive experimental bounds on the A couplings g V,A ij for several benchmark scenarios.The current limits for the branchings of the LFV lepton decays l i → l k γ are [68] Br(µ → eγ) < 4.2 × 10 −13 , Br(τ → eγ) < 3.3 × 10 −8 , Br(τ → µγ) < 4.4 × 10 −8 . ( First we analyze the dimensionless LFV couplings g ij by focusing on the scenario of lepton-flavor universality, assuming equal values of their diagonal elements, i.e. g V ii = g V and g A ii = g A for i = e, µ, τ .In this scenario we calculated and estimate coupling from µ → eγ and τ → eγ LFV decays.The results are presented in Fig. 5 for the particular value g ii = 1.Bounds from the lepton (g − 2) l (see Fig. 5) are shown for the case of vanishing flavor-conserving couplings g ii .Note that bounds from τ → µγ are the same as for τ → eγ because in the approximation m e m µ m τ the contributions from the loops are same.Also we omit in our analysis doubly LFV suppressed diagrams with heavy leptons propagating in the loop.
The peaks in Fig. 5 are induced by behavior of the loop integrals h i (x) near the point x = 1 located in the vicinity of the vector boson production threshold.For resolving this problem one needs to include in our analysis finite width Γ A of the decay of dark vector boson to the leptonic pair with Γ A ∼ τ −1 A ∼ g 2 ij in the Breit-Wigner propagator.Limits on the LFV couplings in Fig. 5 include constraints from the lepton (g − 2) and rare LFV l i → l k γ lepton decays.In the case of e−µ LFV transition (see left pictures in Fig. 5) we add bound from e−µ conversion.Suppression for e − µ conversion at heavy masses is induced by heavy bosons exchange in the t channel.The dark A photon keeps a window for possible huge LFV couplings g ij at light masses of leptons.We also would like to note that g ij is different from one will push the limits from LFV processes up.
Using the limit from l i → l k γ decay in the scenario of universal lepton flavor-conserving (LFC) couplings g V ii or g A ii we can deduce the possible lepton contribution to (g − 2) in function of the mass of dark vector photon A .We have estimated the contribution to (g − 2) l of the sum of loops with light leptons e and µ, taking into account the contribution of LFC and LFV couplings, which are constrained by l i → l k γ decay.In this way we write down ∆a li = (∆a li ) LFC + (∆a li ) LFV .Δaμ from bounds μ->eγ/g-2 with g μμ V /g ee V =40 Δaμ from bounds μ->eγ/g-2/e-μ (SINDRUM) with g μμ V /g ee V =40 Δaμ from bounds μ->eγ/g-2/e-μ (SINDRUM) with g μμ V /g ee V =1 10 -6 10 -4 0.01 1 10 -18 For the vector contribution we also included constraints from µ − e conversion.The results are shown in Fig. 6.As can be seen, the dark photon A contribution to (g − 2) e through the vector channel explains the electron anomaly ∆a e for m A < 10 −2 GeV.Attempting to explain both ∆a e and ∆a µ anomalies on account of the A contribution, we find that it is not possible in at least in the lepton universal benchmark scenario where g µµ /g ee = 1 (see the dashed line in the right panel in Fig. 6).Going beyond this simplified scenario we can find the simultaneous solution of ∆a e and ∆a µ for g V µµ > g V ee .A particular solution for g µµ /g ee = 40, properly taking into account the limits from the LFV processes (see Fig. 2), is presented in the right panel of Fig. 6.We note, that this solution is pretty hierarchical requiring separation of two couplings of the similar nature in more than order of magnitude, which looks unnatural.Another possibility avoiding this kind of hierarchy would be to extent the field content of the model amending it with the dark sector (pseudo)scalars providing additional contributions to (g − 2) l .The study of this possibility is beyond the scope of this work.

VII. CONCLUSIONS
We constructed a phenomenological Lagrangian approach which combines SM and DM sectors based on the Stueckelberg mechanism for the generation mass of the dark U D (1) gauge boson (or dark photon).The DM sector contains dark photon, dark scalar, and generic dark fermion fields.Note that the dark scalar generates the mass of the dark photon and plays a role of Goldstone boson in our gauge-invariant formalism.Stueckelberg portal opens new possibilities for study of phenomenology of BSM physics and can be important for running and planning experiments at world-wide facilities (e.g., for the NA64 Experiment at SPS CERN [2,3]).
We derived the limits on the effective couplings of our Lagrangian using data on lepton AMMs, LFV lepton decays l i → γl k , and µ − e conversion.It is known that the latter are very useful because they give more stringent limits on the couplings of effective Lagrangian.We also found that the (g − 2) anomaly cannot be preferably solved within the Stueckelberg portal scenario by the light dark photon in the framework of conservative scenario with taking into account lepton universality.However, the simultaneous explanation of these both anomalies becomes possible once we allow approximately one-order of magnitude hierarchy between the flavor diagonal couplings of A to electron g ee and to muon g µµ , which can be treated as moderately unnatural.We mentioned the possible ways for relaxing this tension with the naturalness.We leave a detailed study of these aspects of our model for the future publications.
In the future we plan to study a possible role of the Stueckelberg portal in different LFV processes including semileptonic decays.We plan include scalar, pseudoscalar dark bosons into the Stueckelberg portal of DM.We also intend to extended our ideas on non-Abelian scenario for the dark sector.
proof of cancellation of the remaining terms is straightforward.Below we list these terms for each diagram using dimensional regularization and for general case of exchange by the boson particle (with spin 0 or 1), i.e. we do not restrict to the exchange of dark photon: Here Γ 1 and Γ 2 are the corresponding Dirac matrices; d Γ1Γ2 (k) = 1 for exchange by scalar/pseudoscalar particles with Γ 1 = I, γ 5 and Γ 2 = I, γ 5 and d µν (k) = −g µν + k µ k ν /m 2 for exchange by vector/axial particles with Γ 1 = γ µ , γ µ γ 5 and Γ 2 = γ ν , γ ν γ 5 .Next, using the Ward identity for inverse fermion propagators q = ( p − m ) − ( p − m ) and free Dirac equations of motion for initial and final leptons we simplify expressions for the individual rest matrix elements as: In case of the diagrams induced by vector particle exchange the logarithmic divergences induced by individual diagrams read (we explicitly show the contribution of transverse and longitudinal part of the vector V or axial A propagator, which are supplied by the subscript T and L, respectively)

FIG. 1 :
FIG.1: Feynman diagrams describing the contributions of the Dark Sector vector A to the anomalous magnetic moments δa l of the leptons, taking into account flavor non-diagonal l − f couplings, where l, f = e, µ, τ .

FIG. 2 :
FIG.2: Upper bounds on the couplings g V of the dark photon A to the SM fermions as a function of its mass m A derived from the data on leptonic (g − 2).The shaded area is excluded.

FIG. 5 :
FIG.5: Limits on vector gij couplings in dependence on masses M A are deduced from an analysis of the following phenomena: g − 2 ratios of leptons, widths of LFV decays µ → eγ and τ → eγ and lepton conversion.The shaded area is excluded by the data.

FIG. 6 :
FIG.6: Estimate of contribution to lepton AMM is made in dependence on masses of dark photon.The limit is established for the benchmark case gee = 10 −5 with taking into account the restriction for LFV couplings

γ µ 2 m 2 m
) Finally, summing Eqs.(C4)-(C6) we get 0, therefore, proving gauge invariance of the set 4(a)-4(c).Now we turn to the discussion of finiteness of the sum of the set of diagrams 4(a)-(c).The logarithmically divergent term in diagram 4(a) is generated by the part of numerator containing two loop momenta: k γ µ k.Applying dimension regularization with D = 4 − 2 it gives the following divergent result in case of exchange by a scalar S or pseudoscalar P particle with Γ 1 = Γ 2 = I or iγ 5 : Figs.4(d) and 4(e) induce the following logarithmic divergencies:M U V ;S/P 4b = i ± 2m j m i − m k .k ± 2m j m k − m i .(C9)respectively.Here and below ± corresponds to exchange of a scalar or pseudoscalar particle.. Summing up divergent contributions of three diagrams we get exact cancellation of the latter: