Extending the Higgs sector: an extra singlet

An extension of the Standard Model with an additional Higgs singlet is analyzed. Bounds on singlet admixture in 125 GeV h boson from electroweak radiative corrections and data on h production and decays are obtained. Possibility of double h production enhancement at 14 TeV LHC due to heavy higgs contribution is considered.


I. INTRODUCTION
After the discovery of the Higgs (BEH) boson [1,2], all fundamental particles of the Standard Model (SM) are finally found, and now even passionate adepts of the SM should look for physics beyond it. The pattern of particles we have is rather asymmetric: there are twelve vector bosons, many leptons and quarks with spin 1/2 and only one scalar particle h with mass 125 GeV. Of course, there is only one particle with spin 2 as well, a graviton.
However, unlike the spin 2 case, there are no fundamental principle according to which there should exist only one fundamental scalar particle. That is why it is quite probable that there are other still undiscovered fundamental scalar particles in Nature. The purpose of the present paper is to consider the simplest extension of the SM by adding one real scalar field to it. Such an extension of the SM attracts considerable attention: relevant references can be found in recent papers [3][4][5][6]. Extra singlet can provide first order electroweak phase transition needed for electroweak baryogenesis. It can act as a particle which connects SM particles to Dark Matter. Not going into these (very interesting) applications, we will study the degree of enhancement of double higgs production at LHC due to an extra singlet. To do this we should analyze bounds on the mass of the additional scalar particle and its mixing with isodoublet state.
An enhancement of hh production occurs due to the mixing of the SM isodoublet with additional scalar field which is proportional to the vacuum expectation value (vev) of this field. Thus isosinglet is singled out: its vev does not violate custodial symmetry and can be large. For higher representations special care is needed; see paper [7] where an introduction of isotriplet(s) in the SM is discussed.
The paper is organized as follows: in Section II we describe the model and find the physical states. In Section III we get bounds on the model parameters of the scalar sector from the experimental data on h production and decays and from precision measurements of Z-and W -boson parameters and t-quark and h masses. In Section IV we discuss double h production at LHC Run 2. In Appendix A qualititative description of single and double higgs production at LHC is presented.

II. THE MODEL
Adding to the SM a real field X, we take the scalar fields potential in the following form: where Φ is an isodoublet. 1 Terms proportional to X 3 , X 4 and Φ † ΦX 2 are omitted despite that they are allowed by the demand of renormalizability: we always may assume that they are multiplied by small coupling constants. Two combinations of the parameters entering (1) are known experimentally: it is the mass of one of the two scalar states, h, which equals 125 GeV and the isodoublet expectation value v Φ = 246 GeV. The two remaining combinations are determined by the mass of the second scalar, H (we take m H > m h , though this is not obligatory), and the angle α which describes singlet-doublet admixture: Substituting in (1) at the minimum of the potential we get: so µ is negative. For the mass matrix using (4) we get: where V φχ ≡ ∂ 2 V ∂φ∂χ , . . . Eigenvalues of (5) determine masses of scalar particles: where "−" corresponds to m h and "+"-to m H . Eigenfunctions are determined by the mixing angle α: Equations (7) determine µ and λ for the given mixing angle α, while equations (6) determine m X for given α as well. Finally, equations (4) determine the values of m Φ and v X .
where f i , i = 1, 2, . . . , 5 designate the so-called "Big five" final state channels: W W * , ZZ * , γγ, ττ , bb. Cross sections of reactions (8) are equal to the higgs production cross section times branching ratio of the corresponding decay channel. Quantities µ i are introduced according to the following definition: According to ATLAS and CMS results, all µ i are compatible with one within experimental and theoretical accuracy. It means that no New Physics are up to now observed in h production and decays.
In the model with an extra isosinglet, production and decay probabilities of h equal that in the SM multiplied by a factor cos 2 α, that is why we have: and existing bounds on µ i are translated into bounds on the mixing angle α. Taking into account all measured production and decay channels, for the average values experimentalists obtain [9, 10]: CMS: Let us stress that the theoretical uncertainty in the calculation of pp → h production cross section at LHC does not allow to reduce substantially the uncertainty in the value of µ.
Bounds from electroweak precision observables (EWPO) are not affected by this particular uncertainty.
We fit experimental data with the help of LEPTOP program [11] using m h = 125.14 GeV.
The result of the SM fit which accounts the h mass measurement is shown in Table I. Quality of the fit is characterised by the χ 2 value  Higgs boson contributions to electroweak observables at one loop are described in LEP- In the case of an extra singlet the following substitution should be performed: The same substitution should be made for the functions δ 4 V i (t, h), t = m 2 t /m 2 Z , which describe two loops radiative corrections enhanced as m 4 t . In two loops quadratic dependence on higgs mass appears which is described by functions δ 5 V i . Calculations of these corrections in the case of an extra singlet higgs is not easy. An approximate upper bound has been estimated by assuming that Comparison of two calculations, one with δ 5 V i (h) = cos 2 α δ 5 V i (h), and the other with showed that the correction to the values of sin α in Fig. 2 is less than 10 −3 .
Bounds from EWPO on the singlet model parameters are presented in Fig. 2a. σ NNLO (pp → hh) = 40 fb with a 10 ÷ 15% accuracy. We will demonstrate that enlarged higgs sector allows to strongly enhance double h production.
The cross section of H production at LHC equals that for the SM higgs production (for (m h ) SM = m H ) multiplied by sin 2 α. Cross section of the SM higgs production at NNLO we take from Table 3 of [15]. In order to obtain cross section of resonant hh production in H decays we should multiply cross section of H production by Br(H → hh).
Let us consider H decays. Decays to hh, W + W − , ZZ and tt dominate. For the Hhh coupling we obtain: 2 Let us note that if a subset of experimental data from Table I    thus Decays to W + W − , ZZ, tt occur through isodoublet admixture in H: thus The dependence of the widths and branching ratios of H decays on mixing angle α for m H = 300 GeV are shown in Figure 3.
For the cross section of the reaction pp → H → hh we have:   Fig. 4 (compare to Fig. 4 from [6]). H → ZZ decay can be used in order to find H; its cross section divided by that for the SM higgs Contour plot of R is presented in Fig. 5. Let us note that R does not depend on √ s.

V. CONCLUSIONS
In the models with extended higgs sector strong resonant enhancement of double higgs production is possible which makes the search of pp → hh reaction at Run 2 LHC especially interesting. According to Fig. 4 cross section of pp → H → hh reaction can be as large as 0.5 pb, ten times larger than the SM value.
The search for H boson can also go in the same way as it was for the heavy SM boson Simple analythical formulas which qualititavely describe single and double higgs production in the SM are presented in this section. Let us start with single higgs production in gluon fusion. In the limit m h 2m t , the amplitude of gg → h transition is determined by the top quark contribution into the QCD Gell-Mann-Low function: leading to the well-known result for the production cross section: Here τ =ŝ/s and τ 0 = m 2 h /s; s ≡ (p 1 + p 2 ) 2 is the invariant mass of colliding protons, s = x 1 x 2 s ≡ τ s is the invariant mass of colliding gluons. Integrating over gluons distribution in a proton, we get: Changing the variables from x 1 , x 2 to τ , y according to the following definitions: x 2 = √ τ e −y , and substituting (A2) into (A3), we obtain: where the so-called gluon-gluon luminosity is given by the integral over gluon distributions: where β = 2m t m h

2
, and x = arctan [14] (note that lim mt→∞ F = 1). This adjustment leads to 6% enlargement of σ gg→h Applying all these factors and using PDFs from [12], we obtain numbers presented in  [13], accuracy of the calculated value of σ NNLO pp→h is at the level of ±(10 ÷ 17)% which makes hopes of reducing uncertainty in µ i (and µ) below 10% elusive. In the case of an extra singlet, h and H production cross sections equal the SM one multiplied by cos 2 α and sin 2 α respectively.
Let us turn now to double h production at pp collision in the SM. At the leading order it is described by the two diagrams shown in Fig. 6. According to equations (4) and (11) and Table 1 from [17], the cross section of the double production of the 125 GeV h at 14 TeV LHC in the leading order equals: where the first term in parentheses originates from the square of the triangle diagram, the second-from the square of the box diagram, while the last one is their interference, which diminishes the cross section.
In order to understand result (A7) let us proceed in the following way. In the limit s 4m 2 t the triangle gg → h and box gg → hh amplitudes can be directly extracted from lagrangian (A1), expanding it over h/v Φ : where the first term corresponds to the diagram shown in Fig. 6a, while the second term  Fig. 6b. Triple higgs coupling is given by the following term in the SM lagrangian: which leads to λ hhh = 3m 3 h /v Φ . Hence, for the sum of the triangle and the box diagrams at ŝ which equals zero at threshold whenŝ = (2m h ) 2 [18,19]. For the cross section we get (see Eq. 13 from [18]).
In the high-energy limitŝ 4m 2 t box diagram dominates and the cross section behaves as:σ Normalization constant A is determined by the condition that atŝ = 4m 2 t expressions (A11) and (A12) are equal: Finally, for the cross section of double h production in the SM we obtain the following approximate expression: where Equations (A11)-(A13) should be substituted in (A14) and τ ≡ŝ/s,ŝ being the hh invariant mass. The differential cross section is shown in Fig. 7, while for the total cross section for hh production in the SM we get σ(pp → hh) = 4 fb at √ s = 14 TeV, 3.5 times smaller than the explicit leading order result (A7).