Δr in the Two-Higgs-Doublet Model at full one loop level—and beyond

Abstract

After the recent discovery of a Higgs-like boson particle at the CERN LHC-collider, it becomes more necessary than ever to prepare ourselves for identifying its standard or non-standard nature. The fundamental parameter Δr, relating the values of the electroweak gauge boson masses and the Fermi constant, is the traditional observable encoding high precision information of the quantum effects. In this work we present a complete quantitative study of Δr in the framework of the general Two-Higgs-Doublet Model (2HDM). While the one-loop analysis of Δr in this model was carried out long ago, in the first part of our work we consistently incorporate the higher order effects that have been computed since then for the SM part of Δr. Within the on-shell scheme, we find typical corrections leading to shifts of ∼20–40 MeV on the W mass, resulting in a better agreement with its experimentally measured value and in a degree no less significant than in the MSSM case. In the second part of our study we devise a set of effective couplings that capture the dominant higher order genuine 2HDM quantum effects on the δρ part of Δr in the limit of large Higgs boson self-interactions. This limit constitutes a telltale property of the general 2HDM which is unmatched by e.g. the MSSM.

Appendix

For the sake of completeness, we provide herewith a more detailed analytical account on selected aspects of our calculation. All UV divergences we handle by means of conventional dimensional regularization in the ’t Hooft–Veltman scheme, setting the number of dimensions to d=4−ε. As usual, we introduce an (arbitrary) mass scale μ in front of the loop integrals in order not to alter the dimension of the result in d dimensions with respect to d=4. After renormalization (in the on-shell scheme, in our case) the results for the physical quantities are finite in the limit d→4. Furthermore, in the practical aspect of the calculation all one-loop structures are reduced in terms of standard Passarino–Veltman coefficients in the conventions of Refs. [178, 179].

•One loop functions at zero momentum

the one-loop vacuum integrals that enter the evaluation of the parameter δρ, which is built upon the weak gauge boson self energies at vanishing momenta, cf. Eq. (8), read as follows:

(47)

(48)

(49)

where Δ _ϵ =2/ϵ+1−γ _E +log(4πμ ²) and the function F(x,y) is defined as follows:

$$ F(x,y) = F(y,x) = \begin{cases} \frac{x+y}{2}- \frac{xy}{x-y} \log(\frac{x}{y}) &x \neq y, \\ 0 & x=y. \end{cases} $$

(50)

The tilded notation for the Passarino–Veltman functions, e.g.

(51)

indicates that these integrals are evaluated at zero momentum. The parameters A,B can be identified with the (squared of the) masses of the virtual particles propagating in the loop, $A\equiv m_{1}^{2}$, $B\equiv m_{2}^{2}$.

With these expressions at hand, it is straightforward to write down a compact analytical form for δρ _2HDM at one-loop in the ’t Hooft–Feynman gauge, starting from the definition of Eq. (8):

(52)

From the above equation we can explicitly read off how the size of δρ depends on the mass splitting between the different Higgs bosons, as well as on the strength of the Higgs/gauge boson couplings—which is modulated by tanβ and the mixing angle α. The first two lines of the full expression (52) is the part that we have denoted $\delta\rho_{2\mathrm{HDM}}^{*}$ in Sect. 2.4, see Eq. (15). We remark that for $M_{\mathrm{A}^{0}}\to M_{\mathrm{H}^{\pm}}$ and |β−α|→π/2 (in which the h ⁰ field behaves SM-like) the full δρ _2HDM→0. This is the precise formulation of the decoupling regime for the unconstrained 2HDM.

In the case of the SM the Higgs contribution to the δρ-parameter (8) is not finite if taken in an isolated form. The complete bosonic contribution to Δr is of course finite and gauge invariant, and therefore unambiguous. To define a Higgs part of it is then a bit a matter of convention. What is important is that the complete M _H-dependence is exhibited correctly and coincides in all conventions. After removing the UV-parts which cancel against other bosonic contributions one arrives at

(53)

The explicit dependence on the scale μ is unavoidable in quantities which are not UV-finite by themselves. It is, however, natural to set e.g. the EW scale choice μ=M _W. In the limit $M_{\mathrm{H}}^{2}\gg M_{\text{W}}^{2}$ we can see Veltman’s screening theorem at work in the SM, as there remain no $M_{\mathrm{H}}^{2}$ terms but a logarithmic Higgs mass dependence. Indeed, in that limit the expression (53) reduces to

$$ \delta\rho^{H}_\mathrm{ SM}\simeq- \frac{3\sqrt{2} G_F M_{\text {W}}^2}{16 \pi^2} \frac{s_{\text{W}}^2}{c_{\text{W}}^2} \biggl\{ \ln\frac{M_\mathrm{H}^2}{M_\mathrm{W}^2}-\frac{5}{6} \biggr\} , $$

(54)

which coincides with the result quoted in Eq. (12) of Sect. 2.3.

The SM Higgs boson contribution to δρ can also be retrieved from the 2HDM result (52) by selecting the h⁰ parts of the contributions involving the h⁰ and the gauge bosons, namely in the last line of that equation. By performing the identification H≡h⁰ and removing the trigonometric factors we are led to

(55)

We see that the last expression coincides with Eq. (53) up to finite additive parts, which of course reflects the arbitrariness of the scale setting μ. As we said, this is not important because the full bosonic contribution to Δr is finite and unambiguous. The fact that we can recover the SM result from (52) in such a way suggests that the expression in the first line of (55) should be subtracted from (52) in order to compute the genuine 2HDM effects on δρ, i.e. the Higgs boson quantum effects beyond those associated to the Higgs sector of the SM. This is in fact the practical recipe that we follow in this paper. Finally, let us notice that the $\delta\rho_{2\mathrm{HDM}}^{*}$ part of (52), i.e. the one which is completely unrelated to the SM Higgs contribution, is precisely the part of the full δρ _2HDM that violates the screening theorem in the 2HDM, as is manifest from Eq. (15) of Sect. 2.4.

•2HDM contributions to the gauge boson self-energies

We quote herewith their complete analytical form, in terms of the standard Passarino–Veltman coefficients and following the conventions of Ref. [178, 179]. The self-energies are evaluated for on-shell gauge bosons, e.g. $p^{2} = M^{2}_{V} [V = \mbox{W}^{\pm}, \mathrm {Z}^{0}]$, in the way they enter the calculation of Δr.

• Two Higgs-boson contributions:

  

  (56)
  

  (57)

• Higgs/gauge boson and Higgs/Goldstone boson contributions:

  

  (58)
  

  (59)

Let us notice that, in the last two expressions, we have explicitly removed the overlap with the SM Higgs boson contribution, to wit:

(60)

•Effective Higgs/gauge boson interactions

To better illustrate how we build up in practice the effective Higgs/gauge boson couplings employed in this study, herewith we provide explicit analytical details for the construction of one of such Born-improved interactions. We carry out the calculation with the help of the standard algebraic packages FeynArts and FormCalc [156, 178, 179]. Without loss of generality, let us take the concrete case of the Z boson coupling to the $\mathcal{CP}$-odd and the light $\mathcal{CP}$-even neutral Higgs bosons $[g_{\mathrm{h}^{0}\mathrm{A}^{0}\mathrm{Z}^{0}}]$. A sample of the Feynman diagrams describing the $\mathcal{O}(\lambda^{2}_{5})$ corrections to this coupling is displayed in the upper row of Fig. 8. The general structure of the associated form factor $a_{\mathrm{h}^{0}\mathrm{A}^{0}\mathrm{Z}^{0}}$ may be cast as:

(61)

Notice that we define our form factors to be real, in order to preserve the hermiticity of the Born-improved Lagrangians derived from them. The different building blocks of Eq. (61) correspond to:

• (a) $V_{\mathrm{h}^{0}\mathrm{A}^{0}\mathrm{Z}^{0}}$, the genuine vertex corrections (cf. e.g. the first two diagrams in the upper row of Fig. 8). For illustration purposes, we provide its complete analytical form:

  

  (62)

• (b) The wave-function corrections associated to each of the external Higgs boson legs (including, as we single out in the last term of Eq. (61), the h⁰–H⁰ mixing one-loop diagrams):

  

  (63)
  

  (64)
  

  (65)

In the last equation, the h⁰–H⁰ mixing self-energy $\hat{\varSigma}_{\mathrm{h}^{0}\mathrm{H}^{0}}(q^{2}) \allowbreak = \varSigma _{\mathrm{h}^{0}\mathrm{H}^{0}}(q^{2}) + \delta Z_{\mathrm{h}^{0}\mathrm{H}^{0}}(q^{2}-M^{2}_{\mathrm{h}^{0}})/2 + \delta Z_{\mathrm{h}^{0}\mathrm{H}^{0}}(q^{2}-M^{2}_{\mathrm{H}^{0}})/2 - \delta M^{2}_{\mathrm{h}^{0}\mathrm{H}^{0}}$ involves the renormalization of the mixing angle α, which we anchor via the relation $\operatorname{Re}\hat{\varSigma}_{\mathrm{h}^{0}\mathrm{H}^{0}}(q^{2}) = 0$ according to [113], with the renormalization scale chosen at the average mass $q^{2} \equiv(M^{2}_{\mathrm {h}^{0}}+M^{2}_{\mathrm{H}^{0}})/2$. As mentioned above, the tilded Passarino–Veltman functions are evaluated at vanishing external momentum.

Let us note in passing that, for the case of the g _hVV-type couplings, and due to he fact that just one single scalar leg is present there, only pieces of type (b) shall give rise to $\mathcal{O}(\lambda^{2}_{5})$ contributions. The same holds as well for the Higgs/gauge/Goldstone boson couplings [g _hVG].