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# Generic diagrams contributing to the self-energy Σ tad .

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The 2-Higgs-Doublet Model (2HDM) belongs to the simplest extensions of the Standard Model (SM) Higgs sector that are in accordance with theoretical and experimental constraints. In order to be able to properly investigate the experimental Higgs data and, in the long term to distinguish between possible models beyond the SM, precise predictions for...

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... application of the renormal- ization scheme worked out in Ref. [29]. In Appendix A we show in detail how this scheme works and in particular we present its extension from the SM case [29] to the 2HDM. The generic diagrams contributing to the self-energies defined in this 'alternative tadpole' scheme, called Σ tad in the following, are shown in Fig. 1. Besides the generic one-particle irreducible (1PI) diagrams depicted by the first two topologies in Fig. 1, they also contain the tadpole diagrams connected to the self-energies through the CP-even Higgs bosons h and H that are represented by the third topology. The application of the tadpole scheme alters the structure of the mass ...
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... this scheme works and in particular we present its extension from the SM case [29] to the 2HDM. The generic diagrams contributing to the self-energies defined in this 'alternative tadpole' scheme, called Σ tad in the following, are shown in Fig. 1. Besides the generic one-particle irreducible (1PI) diagrams depicted by the first two topologies in Fig. 1, they also contain the tadpole diagrams connected to the self-energies through the CP-even Higgs bosons h and H that are represented by the third topology. The application of the tadpole scheme alters the structure of the mass counterterms and of the off-diagonal wave function renormalization constants 4 such that now the ...
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... are neither IR divergences nor real corrections. The generic diagrams for the virtual corrections and the counterterm are depicted in Fig. 9. The 1PI diagrams contributing to the vertex corrections are given by the triangle diagrams with scalars, fermions, massive gauge bosons and ghost particles in the loops, as shown in the first three rows of Fig. 10, and by the diagrams involving four-particle vertices (last four diagrams of Fig. 10). The corrections to the external leg in Fig. 9 (b) vanish due to the OS renormalization of the H. The mixing contributions (c) and (d) vanish because of the Ward identity for the OS Z boson. The counterterm amplitude is given ...
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... corrections and the counterterm are depicted in Fig. 9. The 1PI diagrams contributing to the vertex corrections are given by the triangle diagrams with scalars, fermions, massive gauge bosons and ghost particles in the loops, as shown in the first three rows of Fig. 10, and by the diagrams involving four-particle vertices (last four diagrams of Fig. 10). The corrections to the external leg in Fig. 9 (b) vanish due to the OS renormalization of the H. The mixing contributions (c) and (d) vanish because of the Ward identity for the OS Z boson. The counterterm amplitude is given ...
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... the µ * denote the polarization vectors of the outgoing Z bosons with four-momentum p 3 and p 4 , respectively. If the tadpole scheme is applied, the HZZ vertex is modified by additional Figure 11: Additional vertex diagrams in the tadpole scheme contributing to the decay H → ZZ. ...
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... contributions, which lead to further diagrams, that have to be taken into account in the computation of the decay width. They are shown in Fig. 11. As the formula for the vertex corrections and counterterms in terms of the scalar one-, two-and three-point functions are quite lengthy, we do not display them explicitly ...
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... Figure 12 shows the ξ W dependence of our process, ...
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... Fig. 13 we show the relative NLO corrections for H ± → W ± h, ∆Γ H ± W ± h , as a function of the charged Higgs boson mass for various renormalization schemes. We denote them as proc : process-dependent p ...
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... Fig. 13 (left) we show results for the process-dependent renormalization and for some representatives of the process-independent schemes, the pOS o , the p c and for comparison also the KOSY c scheme. As can be inferred from the left plot, the process-dependent renormalization leads to much larger NLO corrections than the other schemes. The ...
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... can increase the LO width by more than a factor of three. For the process-independent renormalization schemes on the other hand, the NLO corrections are much milder and vary between about −11 to 20% depending on the renormalization scheme and the charged Higgs mass value (and discarding the unphysical KOSY scheme). This can be inferred from Fig. 13 (right) which displays the results for the process-independent schemes, where the β renormalization is performed both through the charged and through the CP-odd sector. 12 Provided that the same choice for the β renormalization is made, the OS tadpole-pinched scheme, pOS, leads to results closer to the KOSY scheme than the p tadpole-pinched ...
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... Fig. 14 we show the relative NLO corrections for H ± → W ± h as a function of the LO width for all generated scenarios compatible with the applied theoretical and experimental constraints. The colours indicate the results for the process-dependent scheme, the p tadpole- pinched schemes, the OS tadpole-pinched schemes and the KOSY c scheme. The ...
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... Fig. 15 we show the relative NLO corrections for the process H ± → W ± H with the parameters given by Scen3, Eq. (5.14). In the plotted m A range the LO decay width, which does not depend on m A , is given by Γ LO = 4.0568 GeV. In the left plot we have included the results for the process-dependent renormalization, for pOS o , p c and KOSY c . ...
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... for the NLO process, Eq. (4.11). The contributions from the angular counterterms δα and δβ come with the factor 1/t β−α , which is numerically very small in the SM-like limit h ≡ H SM . Therefore any difference in the renormalization schemes for the angles will barely manifest itself in the total NLO corrections. The zoomed in region in Fig. 15 (right) again shows that the KOSY scheme is closer to pOS than to the other schemes and that the usage of the OS scale in δβ is less sensitive to a change of the renormalization scheme, while the renormalization of β via the charged sector is less sensitive to a scale change than the one through the CP-odd sector. 14 The small mA mass range is ...
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... Fig. 16 In the left plot the process-dependent renormalization is included. Additionally we show representatives for process-independent schemes, the pOS o , the p c and the KOSY c scheme. Again the counterterm definition via tauonic heavy Higgs decays leads to much larger corrections than the other schemes. In the investigated mass range it ...
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... in the investigated mass range, the theoretical uncertainty due to missing higher order corrections can be estimated to be of less than a percent to around 6% based on a scale change, and it ranges from the permille level to about 4% when estimated from the change of the β renormalization scheme, discarding the numerically unstable process-dependent scheme. Figure 17 shows the relative NLO corrections ∆Γ H→ZZ for H → ZZ as a function of the LO width for all generated scenarios compatible with the applied theoretical and experimental constraints. The colours indicate the results for the various renormalization schemes. ...
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... renormalization conditions for the tadpoles are shown pictorially in Fig. ...
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... applying the renormalization condition depicted diagrammatically in Fig. 18, the shift can be interpreted as a connected tadpole diagram, containing the Higgs tadpole and its propagator at zero momentum ...
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... not belong to the definition of the mass counterterms and wave function renormalization constants. With the redefinition of the 1PI self-energy as iΣ tad These results can be generalized to the gauge boson and fermion sectors. The application of the tadpole scheme hence requires a redefinition of the self-energies as depicted diagrammatically in Fig. 19. In the gauge and fermion sectors this implies that the tadpole diagrams of the scalar Higgs bosons that couple to the gauge boson and fermion, respectively, have to be included in their self-energy. Furthermore, in the scalar sector the tadpole counterterms drop out of the definition of the wave function renormalization constants and ...

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