S. I. GodunovaK M. I. Vysotskya'€'d**, E. V. Zhemchugova'd***

" Institute for Theoretical and Experimental Physics 117218, Moscow, Russia

b Novosibirsk State University 630090, Novosibirsk, Russia

cMoscow Institute of Physics and Technology 141700, Dolgoprudny, Moscow Region, Russia

d Moscow Engineering Physics Institute II54O9, Moscow, Russia

Received September 25, 2014

The double Higgs production in the models with isospin-triplet scalars is studied. It is shown that in the see-saw type-II model, the mode with an intermediate heavy scalar, pp H + X 2h + X, may have the cross section that is comparable with that in the Standard Model. In the Georgi-Machacek model, this cross section could be much larger than in the Standard Model because the vacuum expectation value of the triplet can be large.

Cwitribvtiwi for the JETP special issue in honor of V. A. Rubakov's 60th birthday

DOI: 10.7868/S0044451015030040


After the discovery of the Higgs-BE boson at the LHC fl], the next steps to check the Standard Model (SM) are the measurement of the coupling constants of the Higgs boson with other SM particles (tt.WW.ZZ, bb.rf,... ) with better accuracy and the measurement of the Higgs self-coupling that determines the shape of the Higgs potential. In the SM, the triple and quartic Higgs couplings are predicted in terms of the known Higgs rilciSS and vacuum expectation value. Deviations from these predictions would mean the existence of a New Physics in the Higgs potential. The triple Higgs coupling can be measured at the LHC in double Higgs production, in which the gluon fusion dominates: gg —¥ hh. However, the 2h production cross section is very small. According to [2], the cross section at v^ = 14 TeV is aNNLO (gg hh) = 40.2 lb with a 10 15% accuracy. For the final states with rea-

E-mail: sgodunovfflitep.ru

**E-mail: vysotsky'fflitep.ru

E-mail: zhemchugovfflitep.ru

sonable signal/'background ratios (such as hh —¥ blr/j), only at the HL-LHC with the integrated luminosity f Cdt = 3000 fb-1 will the double Higgs production be found and the triple Higgs coupling will be measured [З]1^. We seek the extensions of the SM Higgs sector in which the double Higgs production is enhanced.

One of the well-motivated examples of a nonminimal Higgs sector is provided by the see-saw type-II mechanism of neutrino mass generation [7]. In this mechanism, a scalar isotriplet (Д++, Д+, Д0) with the hypercharge Уд = 2 is added to the SM. The vacuum expectation value (vev) of the neutral component «д generates Majorana masses of the left-handed neutrinos. There are two neutral scalar bosons in the model: the light one, in which the doublet Higgs component dominates and which should be identified with the particle discovered at the LHC (h\ М/, = 125 GeV), and the heavy one, in which the triplet Higgs component dominates (H). The neutrino masses equal ftvд, where /, (i = 1,2,3) originates from Yukawa couplings of the

11 The decays into bbrr and bbW+W final states can be even more promising for measuring the triple Higgs coupling [4, o] (see also [61).

Higgs triplet with the lopton doublets. If neutrinos are light due to a small value of «д while /, are of the order of unity, then H decays into neutrino pairs. Three states, or , and Я, are almost degener-

ate in the model considered in Sec. 2, and the absence of the same-sign dileptons at the LHC from U±=t —¥ decays provides the lower bound m# > 400 GeV [8]. We are interested in the opposite case where г>д reaches the maximum allowed value while neutrinos are light because of small values of /,. In this case, H —¥ /г/г can be the dominant decay mode of a heavy neutral Higgs boson. In this way, we obtain an additional mechanism of the double h production at the LHC.

The bound mH++ > 400 GeV [8] cannot be applied now because

mainly decays into the same-sign diboson [9]. We only need H to be heavy enough for the H —¥ /г/г decay to occur. This case is analyzed in Sec. 2. The invariant mass of the additionally produced /г/г state peaks at (pi + p-iY = ">%■ which is a distinctive feature of the proposed mechanism (see also [10, 111).

H contains a small admixture of the isodoublet state, which makes gluon fusion a dominant mechanism of H production at the LHC. The admixture of the isodoublet component in H equals approximately 2г>д/г\ where v « 250 GeV is the vacuum expectation value of the neutral component of the isodoublet, and in Sec. 2, for y/s = 14 TcV and Мн = 300 GeV, we obtain a (gg —¥ H) « 25 fb. Taking into account that the branching ratio H —¥ /г/г is about 80 %, we obtain a 50 % enhancement of double Higgs production in comparison with the SM.

Since the nonzero value of г>д violates the well-checked equality of the strengths of charged and neutral currents at the tree level,




v>\ should be less than 5 GeV (see Sec. 2). The gg —¥ H cross section was estimated numerically for the maximum allowed value = 5 GeV when the isodoublet admixture is about 5%.

The bound < 5 GeV is removed in the Georgi Machacek (GM) model [12], where, in addition to A, a scalar isotriplet with Y = 0 is introduced. If the vev of the neutral component of this additional field equals i'a, then we have just one in the r.h.s. of Eq. (1): the correction proportional to is canceled. Thus, can be much larger than 5 GeV. The bounds on come from the measurement of the 125 GeV Higgs boson cou-

plings to vector bosons and fermions, which would deviate from their SM values: C; -¥■ C; + «, (i'A/<')2j •

The consideration of an enhancement of 2h production in the GM variant of the see-saw type-II model is presented in Sec. 3. Because the current accuracy of the measurement of C; values in h production and decay is poor, v>\ as large as 50 GeV is allowed and a (.9.9 H) can reach 2 pb, which makes it accessible with the integrated luminosity f Cdt = 300 fb-1 prior to the HL-LHC run. We summarize our results in the Conclusions.


2.1. Scalar sector of the see-saw type-II model

In this subsection, we present the necessary formulas (see [13] for a detailed description). In addition to the SM isodoublet field






(v + ip + ix)


an isotriplet is introduced in the see-saw type-II model: Aa

(Л1 - iA2)

6° = ^(vA+6+iV). (3)

Here, a are the Pauli matrices.

The scalar sector kinetic terms are

(1) ¿kinetic = \D,M2 + Tr (DtlA)\




DtlA = + ge^^A- - Ig'B^A0] -= =

= >i;X'.A>/,//,',A. (6)

The hypercharge = 1 was substituted for the isodoublet and >a = 2 for the isotriplet. The terms quadratic in the vector boson fields are as follows:





The voctor boson masses are

Ml = -j (v2 + 4«;


For the ratio of vector boson masses, neglecting the radiative corrections from the isotriplet (not a bad approximation as far as the heavy triplet decouples), we obtain

Mw Mz


Mz ) sm

1 - ^ V


Comparing the result of the SM fit [14. p. 145] mW1 = 80.381 GeV with the experimental value M^rp = 80.385(15) GeV, we obtain the following upper bound at the 3<r level:

< 5 GeV.


Because the cross sections we are interested in are proportional to we use the upper bound = 5 GeV for numerical estimates in this section.

From the numerical value of the Fermi coupling constant in muon decay, we obtain

v2 + 2i<l = (246 GeV)2


and hence for v>\ < 5 GeV, the value v = 246 GeV can be safely used in deriving (10).

The scalar potential has the form



l"('I>. A) = ~m'i + +

.1/1 Tr [At A] + 4= (■i'/ iViAt-i' + H.c.


which is a truncated version of the most general renor-malizable potential (see, e.g., [15, Eq. (2.6)]). We may simply suppose that the coupling constants that multiply the omitted terms in the potential (A1.A2.A4, and A5) are small2K In the case of the SM, only the first line in (12) remains; the IIlclSS of the Higgs boson equals = 125 GeV while its expectation value is v2 « « m|/A « (246 GeV)2, A « 0.25.

2) We note that a relatively large value of A.-, leads to considerable splitting of the masses of triplet states. If the cascade decay H -»• H+W~ becomes allowed, it greatly diminishes the H —hh branching ratio [16].

At the minimum of (12), the equations 1 2 1,2

->'h\, -Ac - fivA,

1 r2



hold, and hence for vevs of the isodoublet and isotriplet, we obtain

v =

< Ml A- //.2


<-'A =

/im I


2AM2 - 2fi2 2 Ml According to (12), the terms quadratic in tp and 6


V(ip,6) = \m%y2 + \m\62 - (iripô. (16)

Here and below, the terms suppressed by (vA/v)" are omitted.

Denoting the states with definite masses by h and H, we obtain

<p cos a — sin a Y

s sin a cos a H

tan 2a: =



Ml - m% '

M h = 7, ( 'inl+M'l^





Ml =

m%+Ml+J(Mly +4//2<-




Because tan 2a: « 4v&/v -C 1, the mass eigenstate h consists mostly of if, and H consists mostly of 6. We suppose that the particle observed by ATLAS and CMS is /1, so Mf, is about 125 GeV.

The scalar sector of the model, in addition to the massless goldstone bosons that are eaten up by the vector gauge bosons, contains one double-charged field H++, one single-charged field H+, and three real neutral fields ,4, H, and h. H+ is mostly <■>+ with a small admixture, and .4 is mostly with a small \ admixture. All these particles except h are heavy; their masses equal Ma, up to small corrections proportional to v%/MA.

2.2. H decays

The second and fourth terms in potential (12) contribute to H —¥ 2h decays:





^ ($Ti<T2/\t$ + H.c.) -I^S (<p2 - v2) , (21)

where in the second line, \ is dominantly a Goldstone state that forms the longitudinal Z polarization.

With the help of (17), we obtain the expression for the effective Lagrangian that describes the H —¥ 2h decay:

C-Hhh —


= l'A


(MH/Mh) 1

Hli =



Hli2. (22)

In the see-saw type-II model, ne

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