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.Comment: 18 pages, 7 figures. v2: one equation added; references received
  after the publication of v1 are adde

Godunov, S. I.

Rozanov, A. N.

Vysotsky, M. I.

Zhemchugov, E. V.

The European Physical Journal C

English

arXiv

18 pages, 7 figures. v2: one equation added; references received after the publication of v1 are addedInternational audienceAn 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

Godunov, S I.

Zhemchugo, E. V.

HAL AMU

Extending the Higgs sector: an extra singlet

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

S I. Godunov

A. N. Rozanov

M. I. Vysotsky

E. V. Zhemchugov

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Eur. Phys. J. C (2016) 76:1DOI 10.1140/epjc/s10052-015-3826-6Regular Article - Theoretical PhysicsExtending the Higgs sector: an extra singletS I. Godunov1,2,a, A. N. Rozanov3, M. I. Vysotsky1,4,5,b, E. V. Zhemchugov1,5,c1 Institute for Theoretical and Experimental Physics, Moscow 117218, Russia2 Novosibirsk State University, Novosibirsk 630090, Russia3 The Center for Particle Physics of Marseilles, 13288 Marseille, France4 Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region 141700, Russia5 Moscow Engineering Physics Institute, Moscow 115409, RussiaReceived: 15 September 2015 / Accepted: 27 November 2015 / Published online: 2 January 2016© The Author(s) 2016. This article is published with open access at Springerlink.comAbstract An extension of the Standard Model with an addi-tional Higgs singlet is analyzed. Bounds on singlet admixturefor the 125 GeV h boson from electroweak radiative correc-tions and data on h production and decays are obtained. Thepossibility of double h production enhancement at 14 TeVLHC due to a heavy Higgs contribution is considered.1 IntroductionAfter the discovery of the Higgs boson [1,2], all fundamentalparticles of the Standard Model (SM) have finally been found,and now even passionate adepts of the SM should look forphysics beyond it. The pattern of particles we have is ratherasymmetric: there are 12 vector bosons, many leptons andquarks with spin 1/2, and only one scalar particle h withmass 125 GeV. Of course, there is only one particle withspin 2 as well, a graviton. However, unlike the spin 2 case,there are no fundamental principles according to which thereshould exist only one fundamental scalar particle. That iswhy it is quite probable that there are other still undiscoveredfundamental scalar particles in Nature. The purpose of thepresent paper is to consider the simplest extension of the SMby adding one real scalar field to it. Such an extension of theSM has attracted considerable attention: relevant referencescan be found in recent papers [3–6]. An extra singlet canprovide the first order electroweak phase transition neededfor electroweak baryogenesis. It can act as a particle whichconnects SM particles to Dark Matter. Not going into these(very interesting) applications (see [7–20]), we will study thedegree of enhancement of double Higgs production at LHCdue to an extra singlet. To do this we should analyze thea e-mail: sgodunov@itep.rub e-mail: vysotsky@itep.ruc e-mail: zhemchugov@itep.rubounds on the mass of the additional scalar particle and itsmixing with the isodoublet state.An enhancement of hh production occurs due to the mix-ing of the SM isodoublet with an additional scalar field whichis proportional to the vacuum expectation value (vev) of thisfield. Thus the isosinglet is singled out: its vev does not vio-late the custodial symmetry and can be large. For higherrepresentations special care is needed; see Ref. [21] wherean introduction of isotriplet(s) in the SM is discussed.The paper is organized as follows: in Sect. 2 we describethe model and find the physical states. In Sect. 3 we getbounds on the model parameters of the scalar sector fromthe experimental data on h production and decays and fromprecision measurements of Z - and W -boson parameters andthe t-quark and h masses. In Sect. 4 we discuss double hproduction at LHC Run 2.2 The modelAdding to the SM a real field X , we take the scalar fieldspotential in the following form:V (Φ, X) = −m2Φ2Φ†Φ + m2X2X2 + λ2(Φ†Φ)2 + μΦ†ΦX,(1)where Φ is an isodoublet.1 Terms proportional to X3, X4,and Φ†ΦX2 are omitted despite that they are allowed by thedemand of renormalizability: we always may assume thatthey are multiplied by small coupling constants.2 Two com-binations of the parameters entering (1) are known exper-1 We are grateful to J. M. Frère who brought to our attention that asimilar model was considered long ago in [22].2 This choice is justified since the sum of a bare coefficient and radiativecorrection to it can be chosen to be arbitrary small. Let us note that one1231 Page 2 of 6 Eur. Phys. J. C (2016) 76 :1Fig. 1 Dependencies of themodel parameters on the mixingangle for mH = 300 GeVμsinαλsinαmΦmXv Xsinαsinαimentally: the mass of one of the two scalar states, h,which equals 125 GeV, and the isodoublet expectation valuevΦ = 246 GeV. The two remaining combinations are deter-mined by the mass of the second scalar, H , and the angle, α,which describes the singlet–doublet admixture:{h = φ cos α + χ sin α,H = −φ sin α + χ cos α,{φ = h cos α − H sin α,χ = h sin α + H cos α.(2)Since we consider the possibility of double h productionenhancement due to the H contribution, H → hh, we takemH > mh . Substituting in (1)Φ =(φ+1√2(vΦ + φ + iη)), X = vX + χ, (3)at the minimum of the potential we get{λv2Φ + 2μvX = m2Φ,2m2XvX + μv2Φ = 0,(4)Footnote 2 continuedalways can choose the signs of the coefficients in front of the omittedterms so that the stability of the potential will not be violated. Thisis possible due to the freedom in the choice of the values of the barecoefficients.so μ is negative. For the mass matrix using (4) we getM =(Vφφ VφχVφχ Vχχ)=(λv2Φ μvΦμvΦ m2X), (5)where Vφχ ≡ ∂2V∂φ∂χ , . . . The eigenvalues of (5) determinethe masses of the scalar particles:m2h,H =12λv2Φ+12m2X∓√(12λv2Φ −12m2X)2+ μ2v2Φ, (6)where “−” corresponds to mh and “+” to mH . The eigen-functions are determined by the mixing angle α:sin 2α = −2μvΦm2H − m2h, tan α = m2h − λv2ΦμvΦ. (7)Equation (7) determines μ and λ for the given mix-ing angle α, while Eq. (6) determines mX for given α aswell. Finally, Eq. (4) determines the values of mΦ and vX .Figure 1 demonstrates the dependencies just described formH = 300 GeV.123Eur. Phys. J. C (2016) 76 :1 Page 3 of 6 13 Bounds from h production at LHC and electroweakprecision observablesATLAS and CMS Collaborations had detected h productionand decays in the reactionspp → h → fi , (8)where fi , i = 1, 2, . . . , 5 designate the so-called “Big five”final state channels: WW ∗, Z Z∗, γ γ , τ τ¯ , bb¯. The cross sec-tions of reactions (8) are equal to the Higgs production crosssection times the branching ratio of the corresponding decaychannel. The quantities μi are introduced according to thefollowing definition:μi ≡ σpp→h · Γh→ fi /Γh(σpp→h · Γh→ fi /Γh)SM. (9)According to the ATLAS and CMS results, all μi are compat-ible with one within experimental and theoretical accuracy.It means that no New Physics are up to now observed in hproduction and decays.In the model with an extra isosinglet, production and decayprobabilities of h equal that in the SM multiplied by a factorcos2 α, which is why we haveμi = cos2 α, (10)and existing bounds on μi are translated into bounds on themixing angle α. Taking into account all measured productionand decay channels, for the average values experimentalistsobtain [23,24]ATLAS: μ = 1.30+0.18−0.17, (11)CMS: μ = 1.00+0.14−0.13= ±0.09(stat.)+0.08−0.07(theor.) ± 0.07(syst.). (12)Let us stress that the theoretical uncertainty in the calculationof pp → h production cross section at LHC does not allowone to reduce substantially the uncertainty in the value of μ.The bounds from electroweak precision observables (EWPO)are not affected by this particular uncertainty.We fit experimental data with the help of LEPTOP pro-gram [25–29] using mh = 125.14 GeV. The result of the SMfit which accounts for the h mass measurement is shown inTable 1. The quality of the fit is characterized by the χ2 valueχ2/nd.o.f. = 19.6/13. (13)Higgs boson contributions to electroweak observables atone loop are described in LEPTOP by the functions Hi (h) =Hi (m2h/m2Z ). In the case of an extra singlet the followingsubstitution should be performed:Hi (h) → cos2 α Hi (h) + sin2 α Hi (H), H = m2H/m2Z .(14)The same substitution should be made for the functionsδ4Vi (t, h), t = m2t /m2Z , which describe two loops radiativecorrections enhanced as m4t .In two loops a quadratic dependence on the Higgs massappears which is described by the functions δ5Vi . The calcu-lation of these corrections in the case of an extra singlet HiggsTable 1 EWPO fit of theStandard Model Observable Experimental value Standard Model PullΓZ (GeV) 2.4952 (23) 2.4966 (14) −0.5895σh (nb) 41.541 (37) 41.475 (14) 1.7746Rl 20.771 (25) 20.744 (18) 1.0831AlFB 0.0171 (10) 0.0165 (2) 0.6572Aτ 0.1439 (43) 0.1484 (7) −1.0452Rb 0.2163 (7) 0.2158 (0) 0.7699Rc 0.1721 (30) 0.1722 (0) −0.0277AbFB 0.0992 (16) 0.1040 (5) −3.0303AcFB 0.0707 (35) 0.0744 (4) −1.0565s2l (QFB) 0.2324 (12) 0.2313 (1) 0.8771ALR 0.1514 (22) 0.1484 (7) 1.3822Ab 0.923 (20) 0.9349 (1) −0.5941Ac 0.670 (27) 0.6685 (3) 0.0567MW (GeV) 80.3846 (146) 80.3725 (67) 0.8322mt (GeV) 173.24 (95) 174.32 (89) −1.13701/α¯ 128.954 (48) 129.023 (37) −1.43781231 Page 4 of 6 Eur. Phys. J. C (2016) 76 :1is not easy. However, even for a 1000 GeV Higgs these cor-rections are small. We checked that corrections to the valuesof sin α due to δ5Vi terms in Fig. 2 are less than 10−3.The bounds from EWPO on the singlet model parame-ters are presented in Fig. 2a. A χ2 minimum is reached atsin α = 0, mH = 150 GeV, which is the minimum valueallowed for mH in the fit. Experimental data are avoidinga heavy Higgs. The value of χ2 at the minimum coincideswith the SM result (13). Lines of constant χ2 correspond toχ2 = 1, 4, 9, . . . The probabilities that (sin α,mH ) valuesare below these lines are 39, 86, 98.9 %, …3Bounds accounting for both EWPO and direct h produc-tion data (11), (12), are shown in Fig. 2b. We see that for aheavy H the bounds from EWPO dominate, while for a lightH the measurement of μ is more important.4 h, H , and hh production at LHCThe main purpose of this section is to find what enhancementof double Higgs production cross section is possible withenlarged Higgs sector. Let us recall that in the SM the doubleh production cross section is very small. According to therecent result [30], at√s = 14 TeV σNNLO(pp → hh) =40 fb with a 10÷15 % accuracy. We will demonstrate that anenlarged Higgs sector allows one to strongly enhance doubleh production.The cross section of H production at LHC equals that forthe SM Higgs production (for (mh)SM = mH ) multiplied bysin2 α. We take the cross section of the SM Higgs productionat NNLO from Table 3 of [31]. In order to obtain the crosssection of resonant hh production in H decays we shouldmultiply the cross section of H production by Br(H → hh).Let us consider H decays. Decays to hh, W+W−, Z Z ,and t t¯ dominate. For the Hhh coupling we obtainLHhh=[32λvΦ cos2 α sin α − μ2cos α(1 − 3 sin2 α)]Hh2= 2m2h + m2H2vΦsin α cos2 α Hh2≡ gHhhHh2, (15)thusΓH→hh = g2Hhh8πmH√1 −(2mhmH)2. (16)3 Let us note that if a subset of the experimental data from Table 1 isfitted, then the allowed domains of the (sin α,mH ) values will be largerthan those presented in Fig. 2a. Here we disagree with the statementmade in [4] that the fit of only one observable (mW ) allows one to setthe strongest constraint on (sin α,mH ).(a)(b)Fig. 2 Bounds on the singlet model parametersDecays to W+W−, Z Z , t t¯ occur through isodoublet admix-ture in H :L = 2m2WvΦsin α HW+W− + m2ZvΦsin α HZ2+mtvΦsin α Htt¯≡ gHWW HW+W− + 12gHZ Z H Z2 + gHtt¯ Ht t¯, (17)thusΓH→W+W− =g2HWWm3H64πm4W[1 − 4m2Wm2H+ 12m4Wm4H]×√1 −(2mWmH)2, (18)123Eur. Phys. J. C (2016) 76 :1 Page 5 of 6 1Fig. 3 Decay widths andbranching ratios of the heavyHiggs boson formH = 300 GeVsinα sinαH → hhH → WWH → ZZ(a) (b)Fig. 4 Contour plot of σ(pp → H → hh) for √s = 14 TeV. In thisfigure we neglect small effects of H → hh∗ΓH→Z Z = g2HZ Zm3H128πm4Z[1 − 4 m2Zm2H+ 12 m4Zm4H](19)×√1 −(2mZmH)2,ΓH→t t¯ =3g2Htt¯mH8π[1 −(2mtmH)2] 32. (20)The dependences of the widths and branching ratios of Hdecays on the mixing angle α for mH = 300 GeV are shownin Fig. 3.For the cross section of the reaction pp → H → hh wehaveσ(pp → H → hh) = σ(pp → h)SM·sin2 α·Br(H → hh),(21)the lines of constant cross section are shown in Fig. 4 (com-pare Fig. 4 in [6]). H → Z Z decay can be used in order tofind H ; its cross section divided by that for the SM Higgsboson with (mh)SM = mH isFig. 5 Contour plot of R ≡ σ(pp→H)Br(H→Z Z)(σ (pp→h)Br(h→Z Z))SM . In the calculationof R we assume mH > 2mhR ≡ σ(pp → H) · Br(H → Z Z)(σ (pp → h) · Br(h → Z Z))SM= sin4 αsin2 α + Γ (H→hh)ΓSM. (22)Contour plot of R is presented in Fig. 5. Let us note that Rdoes not depend on√s. See alsoσ(pp → H)·Br(H → Z Z)for different values of sin α in Fig. 6.5 ConclusionsIn the models with extended Higgs sector strong resonantenhancement of double Higgs production is possible, whichmakes the search of the pp → hh reaction at Run 2 LHCespecially interesting. According to Fig. 4 the cross sectionof the pp → H → hh reaction can be as large as 0.5 pb, tentimes larger than the SM value.The search for a H boson can go in the same way as forthe heavy SM boson h. The probability of H observation is1231 Page 6 of 6 Eur. Phys. J. C (2016) 76 :1Fig. 6 σ(pp → H) · Br(H → Z Z) for different values of the mixingangle for√s = 14 TeVdiminished compared to that of h because of (a) suppressionof H production cross section by the factor sin2 α ≤ 0.2; and(b) suppression of Br(H → Z Z) because of an additionalH → hh decay mode. Taking these two factors into account,we get a suppression of about a factor 10 of the pp → H →Z Z process probability compared to that for the SM Higgsboson (see Fig. 5).Results for the search of the Higgs-like boson in Z Z decaymode by CMS can be found in [32], Figure 18, [33], Figure7, and by ATLAS in [34], Figure 12. Comparing it with ourFig. 5, we observe that the experimental data start to be sen-sitive to the singlet model expectation for maximally allowedvalues of the mixing angle α.Acknowledgments S. G., M. V. and E. Zh. are partially sup-ported under the grants RFBR No. 14-02-00995 and NSh-3830.2014.2.S G. and E. Zh. are also supported by MK-4234.2015.2. In addi-tion, S. G is supported by RFBR under grant 16-32-60115, DynastyFoundation and by the Russian Federation Government under GrantNo. 11.G34.31.0047.OpenAccess This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate creditto the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.Funded by SCOAP3.References1. The ATLAS Collaboration, Phys. Lett. B 716, 1 (2012)2. 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Eur. Phys. J. C (2016) 76:1
DOI 10.1140/epjc/s10052-015-3826-6
Regular Article - Theoretical Physics
Extending the Higgs sector: an extra singlet
S I. Godunov1,2,a, A. N. Rozanov3, M. I. Vysotsky1,4,5,b, E. V. Zhemchugov1,5,c
1 Institute for Theoretical and Experimental Physics, Moscow 117218, Russia
2 Novosibirsk State University, Novosibirsk 630090, Russia
3 The Center for Particle Physics of Marseilles, 13288 Marseille, France
4 Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region 141700, Russia
5 Moscow Engineering Physics Institute, Moscow 115409, Russia
Received: 15 September 2015 / Accepted: 27 November 2015 / Published online: 2 January 2016
© The Author(s) 2016. This article is published with open access at Springerlink.com
Abstract An extension of the Standard Model with an addi-
tional Higgs singlet is analyzed. Bounds on singlet admixture
for the 125 GeV h boson from electroweak radiative correc-
tions and data on h production and decays are obtained. The
possibility of double h production enhancement at 14 TeV
LHC due to a heavy Higgs contribution is considered.
1 Introduction
After the discovery of the Higgs boson [1,2], all fundamental
particles of the Standard Model (SM) have finally been 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 12 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 principles 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 has attracted considerable attention: relevant references
can be found in recent papers [3–6]. An extra singlet can
provide the 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 (see [7–20]), we will study the
degree of enhancement of double Higgs production at LHC
due to an extra singlet. To do this we should analyze the
a e-mail: sgodunov@itep.ru
b e-mail: vysotsky@itep.ru
c e-mail: zhemchugov@itep.ru
bounds on the mass of the additional scalar particle and its
mixing with the isodoublet state.
An enhancement of hh production occurs due to the mix-
ing of the SM isodoublet with an additional scalar field which
is proportional to the vacuum expectation value (vev) of this
field. Thus the isosinglet is singled out: its vev does not vio-
late the custodial symmetry and can be large. For higher
representations special care is needed; see Ref. [21] where
an introduction of isotriplet(s) in the SM is discussed.
The paper is organized as follows: in Sect. 2 we describe
the model and find the physical states. In Sect. 3 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
the t-quark and h masses. In Sect. 4 we discuss double h
production at LHC Run 2.
2 The model
Adding to the SM a real field X , we take the scalar fields
potential in the following form:
V (Φ, X) = −m
2
Φ
2
Φ†Φ + m
2
X
2
X2 + λ
2
(Φ†Φ)2 + μΦ†ΦX,
(1)
where Φ is an isodoublet.1 Terms proportional to X3, X4,
and Φ†ΦX2 are omitted despite that they are allowed by the
demand of renormalizability: we always may assume that
they are multiplied by small coupling constants.2 Two com-
binations of the parameters entering (1) are known exper-
1 We are grateful to J. M. Frère who brought to our attention that a
similar model was considered long ago in [22].
2 This choice is justified since the sum of a bare coefficient and radiative
correction to it can be chosen to be arbitrary small. Let us note that one
123
1 Page 2 of 6 Eur. Phys. J. C (2016) 76 :1
Fig. 1 Dependencies of the
model parameters on the mixing
angle for mH = 300 GeV
μ
sinα
λ
sinα
mΦ
mX
v X
sinαsinα
imentally: the mass of one of the two scalar states, h,
which equals 125 GeV, and the isodoublet expectation value
vΦ = 246 GeV. The two remaining combinations are deter-
mined by the mass of the second scalar, H , and the angle, α,
which describes the singlet–doublet admixture:
{
h = φ cos α + χ sin α,
H = −φ sin α + χ cos α,
{
φ = h cos α − H sin α,
χ = h sin α + H cos α.
(2)
Since we consider the possibility of double h production
enhancement due to the H contribution, H → hh, we take
mH > mh . Substituting in (1)
Φ =
(
φ+
1√
2
(vΦ + φ + iη)
)
, X = vX + χ, (3)
at the minimum of the potential we get
{
λv2Φ + 2μvX = m2Φ,
2m2XvX + μv2Φ = 0,
(4)
Footnote 2 continued
always can choose the signs of the coefficients in front of the omitted
terms so that the stability of the potential will not be violated. This
is possible due to the freedom in the choice of the values of the bare
coefficients.
so μ is negative. For the mass matrix using (4) we get
M =
(
Vφφ Vφχ
Vφχ Vχχ
)
=
(
λv2Φ μvΦ
μvΦ m2X
)
, (5)
where Vφχ ≡ ∂2V∂φ∂χ , . . . The eigenvalues of (5) determine
the masses of the scalar particles:
m2h,H =
1
2
λv2Φ+
1
2
m2X∓
√(
1
2
λv2Φ −
1
2
m2X
)2
+ μ2v2Φ, (6)
where “−” corresponds to mh and “+” to mH . The eigen-
functions are determined by the mixing angle α:
sin 2α = −2μvΦ
m2H − m2h
, tan α = m
2
h − λv2Φ
μvΦ
. (7)
Equation (7) determines μ and λ for the given mix-
ing angle α, while Eq. (6) determines mX for given α as
well. Finally, Eq. (4) determines the values of mΦ and vX .
Figure 1 demonstrates the dependencies just described for
mH = 300 GeV.
123
Eur. Phys. J. C (2016) 76 :1 Page 3 of 6 1
3 Bounds from h production at LHC and electroweak
precision observables
ATLAS and CMS Collaborations had detected h production
and decays in the reactions
pp → h → fi , (8)
where fi , i = 1, 2, . . . , 5 designate the so-called “Big five”
final state channels: WW ∗, Z Z∗, γ γ , τ τ¯ , bb¯. The cross sec-
tions of reactions (8) are equal to the Higgs production cross
section times the branching ratio of the corresponding decay
channel. The quantities μi are introduced according to the
following definition:
μi ≡ σpp→h · Γh→ fi /Γh
(σpp→h · Γh→ fi /Γh)SM
. (9)
According to the ATLAS and CMS results, all μi are compat-
ible 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
cos2 α, which is why we have
μi = cos2 α, (10)
and existing bounds on μi are translated into bounds on the
mixing angle α. Taking into account all measured production
and decay channels, for the average values experimentalists
obtain [23,24]
ATLAS: μ = 1.30+0.18−0.17, (11)
CMS: μ = 1.00+0.14−0.13
= ±0.09(stat.)+0.08−0.07(theor.) ± 0.07(syst.). (12)
Let us stress that the theoretical uncertainty in the calculation
of pp → h production cross section at LHC does not allow
one to reduce substantially the uncertainty in the value of μ.
The bounds from electroweak precision observables (EWPO)
are not affected by this particular uncertainty.
We fit experimental data with the help of LEPTOP pro-
gram [25–29] using mh = 125.14 GeV. The result of the SM
fit which accounts for the h mass measurement is shown in
Table 1. The quality of the fit is characterized by the χ2 value
χ2/nd.o.f. = 19.6/13. (13)
Higgs boson contributions to electroweak observables at
one loop are described in LEPTOP by the functions Hi (h) =
Hi (m2h/m
2
Z ). In the case of an extra singlet the following
substitution should be performed:
Hi (h) → cos2 α Hi (h) + sin2 α Hi (H), H = m2H/m2Z .
(14)
The same substitution should be made for the functions
δ4Vi (t, h), t = m2t /m2Z , which describe two loops radiative
corrections enhanced as m4t .
In two loops a quadratic dependence on the Higgs mass
appears which is described by the functions δ5Vi . The calcu-
lation of these corrections in the case of an extra singlet Higgs
Table 1 EWPO fit of the
Standard Model Observable Experimental value Standard Model Pull
ΓZ (GeV) 2.4952 (23) 2.4966 (14) −0.5895
σh (nb) 41.541 (37) 41.475 (14) 1.7746
Rl 20.771 (25) 20.744 (18) 1.0831
AlFB 0.0171 (10) 0.0165 (2) 0.6572
Aτ 0.1439 (43) 0.1484 (7) −1.0452
Rb 0.2163 (7) 0.2158 (0) 0.7699
Rc 0.1721 (30) 0.1722 (0) −0.0277
AbFB 0.0992 (16) 0.1040 (5) −3.0303
AcFB 0.0707 (35) 0.0744 (4) −1.0565
s2l (QFB) 0.2324 (12) 0.2313 (1) 0.8771
ALR 0.1514 (22) 0.1484 (7) 1.3822
Ab 0.923 (20) 0.9349 (1) −0.5941
Ac 0.670 (27) 0.6685 (3) 0.0567
MW (GeV) 80.3846 (146) 80.3725 (67) 0.8322
mt (GeV) 173.24 (95) 174.32 (89) −1.1370
1/α¯ 128.954 (48) 129.023 (37) −1.4378
123
1 Page 4 of 6 Eur. Phys. J. C (2016) 76 :1
is not easy. However, even for a 1000 GeV Higgs these cor-
rections are small. We checked that corrections to the values
of sin α due to δ5Vi terms in Fig. 2 are less than 10−3.
The bounds from EWPO on the singlet model parame-
ters are presented in Fig. 2a. A χ2 minimum is reached at
sin α = 0, mH = 150 GeV, which is the minimum value
allowed for mH in the fit. Experimental data are avoiding
a heavy Higgs. The value of χ2 at the minimum coincides
with the SM result (13). Lines of constant χ2 correspond to
χ2 = 1, 4, 9, . . . The probabilities that (sin α,mH ) values
are below these lines are 39, 86, 98.9 %, …3
Bounds accounting for both EWPO and direct h produc-
tion data (11), (12), are shown in Fig. 2b. We see that for a
heavy H the bounds from EWPO dominate, while for a light
H the measurement of μ is more important.
4 h, H , and hh production at LHC
The main purpose of this section is to find what enhancement
of double Higgs production cross section is possible with
enlarged Higgs sector. Let us recall that in the SM the double
h production cross section is very small. According to the
recent result [30], at
√
s = 14 TeV σNNLO(pp → hh) =
40 fb with a 10÷15 % accuracy. We will demonstrate that an
enlarged Higgs sector allows one to strongly enhance double
h production.
The cross section of H production at LHC equals that for
the SM Higgs production (for (mh)SM = mH ) multiplied by
sin2 α. We take the cross section of the SM Higgs production
at NNLO from Table 3 of [31]. In order to obtain the cross
section of resonant hh production in H decays we should
multiply the cross section of H production by Br(H → hh).
Let us consider H decays. Decays to hh, W+W−, Z Z ,
and t t¯ dominate. For the Hhh coupling we obtain
LHhh
=
[
3
2
λvΦ cos
2 α sin α − μ
2
cos α(1 − 3 sin2 α)
]
Hh2
= 2m
2
h + m2H
2vΦ
sin α cos2 α Hh2
≡ gHhhHh2, (15)
thus
ΓH→hh = g
2
Hhh
8πmH
√
1 −
(
2mh
mH
)2
. (16)
3 Let us note that if a subset of the experimental data from Table 1 is
fitted, then the allowed domains of the (sin α,mH ) values will be larger
than those presented in Fig. 2a. Here we disagree with the statement
made in [4] that the fit of only one observable (mW ) allows one to set
the strongest constraint on (sin α,mH ).
(a)
(b)
Fig. 2 Bounds on the singlet model parameters
Decays to W+W−, Z Z , t t¯ occur through isodoublet admix-
ture in H :
L = 2m
2
W
vΦ
sin α HW+W− + m
2
Z
vΦ
sin α HZ2
+mt
vΦ
sin α Htt¯
≡ gHWW HW+W− + 1
2
gHZ Z H Z
2 + gHtt¯ Ht t¯, (17)
thus
ΓH→W+W− =
g2HWWm
3
H
64πm4W
[
1 − 4m
2
W
m2H
+ 12m
4
W
m4H
]
×
√
1 −
(
2mW
mH
)2
, (18)
123
Eur. Phys. J. C (2016) 76 :1 Page 5 of 6 1
Fig. 3 Decay widths and
branching ratios of the heavy
Higgs boson for
mH = 300 GeV
sinα sinα
H → hh
H → WW
H → ZZ
(a) (b)
Fig. 4 Contour plot of σ(pp → H → hh) for √s = 14 TeV. In this
figure we neglect small effects of H → hh∗
ΓH→Z Z = g
2
HZ Zm
3
H
128πm4Z
[
1 − 4 m
2
Z
m2H
+ 12 m
4
Z
m4H
]
(19)
×
√
1 −
(
2mZ
mH
)2
,
ΓH→t t¯ =
3g2Htt¯mH
8π
[
1 −
(
2mt
mH
)2] 32
. (20)
The dependences of the widths and branching ratios of H
decays on the mixing angle α for mH = 300 GeV are shown
in Fig. 3.
For the cross section of the reaction pp → H → hh we
have
σ(pp → H → hh) = σ(pp → h)SM·sin2 α·Br(H → hh),
(21)
the lines of constant cross section are shown in Fig. 4 (com-
pare Fig. 4 in [6]). H → Z Z decay can be used in order to
find H ; its cross section divided by that for the SM Higgs
boson with (mh)SM = mH is
Fig. 5 Contour plot of R ≡ σ(pp→H)Br(H→Z Z)
(σ (pp→h)Br(h→Z Z))SM . In the calculation
of R we assume mH > 2mh
R ≡ σ(pp → H) · Br(H → Z Z)
(σ (pp → h) · Br(h → Z Z))SM
= sin
4 α
sin2 α + Γ (H→hh)
ΓSM
. (22)
Contour plot of R is presented in Fig. 5. Let us note that R
does not depend on
√
s. See alsoσ(pp → H)·Br(H → Z Z)
for different values of sin α in Fig. 6.
5 Conclusions
In the models with extended Higgs sector strong resonant
enhancement of double Higgs production is possible, which
makes the search of the pp → hh reaction at Run 2 LHC
especially interesting. According to Fig. 4 the cross section
of the pp → H → hh reaction can be as large as 0.5 pb, ten
times larger than the SM value.
The search for a H boson can go in the same way as for
the heavy SM boson h. The probability of H observation is
123
1 Page 6 of 6 Eur. Phys. J. C (2016) 76 :1
Fig. 6 σ(pp → H) · Br(H → Z Z) for different values of the mixing
angle for
√
s = 14 TeV
diminished compared to that of h because of (a) suppression
of H production cross section by the factor sin2 α ≤ 0.2; and
(b) suppression of Br(H → Z Z) because of an additional
H → hh decay mode. Taking these two factors into account,
we get a suppression of about a factor 10 of the pp → H →
Z Z process probability compared to that for the SM Higgs
boson (see Fig. 5).
Results for the search of the Higgs-like boson in Z Z decay
mode by CMS can be found in [32], Figure 18, [33], Figure
7, and by ATLAS in [34], Figure 12. Comparing it with our
Fig. 5, we observe that the experimental data start to be sen-
sitive to the singlet model expectation for maximally allowed
values of the mixing angle α.
Acknowledgments S. G., M. V. and E. Zh. are partially sup-
ported under the grants RFBR No. 14-02-00995 and NSh-3830.2014.2.
S G. and E. Zh. are also supported by MK-4234.2015.2. In addi-
tion, S. G is supported by RFBR under grant 16-32-60115, Dynasty
Foundation and by the Russian Federation Government under Grant
No. 11.G34.31.0047.
OpenAccess This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
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