Multiple production of Higgs particles is essential to study Higgs
self-couplings at future high-energy colliders. In this paper we calculated the
resonance contributions to the production of three lightest neutral
supersymmetric Higgs bosons in gluon fusion at LHC. The cross sections due to
trilinear Higgs couplings is sizeable but the measurement of the quartic
coupling hhhH(h) seems to be impossible.Comment: 8 pages, 3 figure

Cynolter, G.

Lendvai, E.

Pócsik, G.

English

arXiv

Multiple production of Higgs particles is essential to study Higgsself-couplings at future high-energy colliders. In this paper we calculated theresonance contributions to the production of three lightest neutralsupersymmetric Higgs bosons in gluon fusion at LHC. The cross sections due totrilinear Higgs couplings is sizeable but the measurement of the quarticcoupling hhhH(h) seems to be impossible

Cynolter, G

Lendvai, E

Pócsik, G

CERN Document Server

ITP-Budapest 554December 1999Resonance Production of Three Neutral SupersymmetricHiggs Bosons at LHCG. Cynolter, E. Lendvai and G. PocsikInstitute for Theoretical PhysicsEo¨tvo¨s Lorand University, BudapestAbstractMultiple production of Higgs particles is essential to study Higgs self-couplingsat future high{energy colliders. In this paper we calculated the resonance con-tributions to the production of three lightest neutral supersymmetric Higgsbosons in gluon fusion at LHC. The cross sections due to trilinear Higgs cou-plings is sizeable but the measurement of the quartic coupling hhhH(h) seemsto be impossible.1 IntroductionOne of the basic open questions of particle physics is the nature of electroweak symmetrybreaking. Beyond the discovery of the Higgs boson(s), however, reconstructing the Higgspotential will be necessary, and that requires the experimental study of the self{couplingsof the Higgs bosons.Pair production of neutral Higgs particles in gluon fusion sensitive to the trilinearcouplings of Higgs bosons was studied in the SM [1] and the Minimal SupersymmetricStandard Model (MSSM) [2{5]. QCD corrections were included in the limiting case of aheavy top mass [6].In the SM the trilinear and the quartic couplings of the physical Higgs particle arexed by the Higgs mass. There is no resonance in the pair production of SM Higgses,and the cross section is only ’20{50 fb in the intermediate mass range [3]. No resonanceeect is present in the production of three Standard Model Higgs bosons via one Higgsin gluon fusion and it is estimated to be under the discovery limit at LHC.In the MSSM there are ve Higgs bosons (h; H; A; H) and many trilinear and quarticcouplings among them. Two parameters describe the Higgs sector of the MSSM: MA, themass of the CP-odd Higgs boson A, and tan, the ratio of the two vacuum expectationvalues. For a wide range of MA, in pp collision only the cross section of the hh productionis large [2,3]. Possible other processes, such as the gauge boson fusion and Higgs{strahlungo W bosons or heavy quarks (top) provide less event number.In order to provide further tests of trilinear and quartic Higgs self{couplings, in thispaper we calculate the resonance enhanced production of three lightest supersymmetricHiggs bosons (h) in gluon fusionpp! gg ! hhh: (1)Generally four dierent graphs contribute to the production of three Higgses. Gluonscouple to triangle, box or pentagon quark loops emitting 1,2 or 3 Higgs bosons, as is seenin Fig.1. Squark loops are neglected. We have omitted the pentagon graph producing aflat continuum background for the resonance production and we will give the completecross section valid in a wider range in a subsequent publication [7]. We learn that thequartic coupling is not accessible by LHC experiment. The resonance contribution to (1),however, sizeable at LHC For instance at tan = 3 it yields a cross section of about 300fb. This is about 1=10 times the corresponding hh{case [3].The masses, widths and the couplings were calculated using the complete one{loopand the leading two{loop radiative corrections from [8]. The relevant Higgs self{couplingsare the followinghhh = 3 cos(2) sin( + ) +3M2Zcos3 sin Hhh = 2 sin(2) sin( + )− cos(2) cos( + ) + 3M2Zsin  cos2 sin 2HHh = −2 sin(2) cos( + )− cos(2) sin( + ) + 3M2Zsin2  cos sin Hhhh = 3 sin(2) cos(2) +8M2Zcos  sin2 sin2 hhhh = 3 cos2(2) +4M2Zsin4 sin2 (2)The trilinear and quartic couplings are normalized to 3 = [p2GF ]1=2M2Z and 4 =p2GF M2Z , respectively. The couplings depend on  and the mixing angle  of the CP{even Higgs sectortan 2 =M2A + M2ZM2A −M2Z + = cos 2tan 2: (3)The leading m4t one{loop corrections [9] are parametrized by =3GFp22m4tsin2 log"1 +M2Sm2t#: (4)The common squark mass is xed to MS= 1 TeV. The quartic couplings depend on MAand are shown for two values of tan = 3 and 30 in Fig.2.The paper is organized as follows. In the next section we give the cross section of theresonance production of three h’s in gluon fusion and in section 3 we show the numericalresults.2 The cross sectionThree dierent channels contribute to the resonance production of three lightest supersym-metric Higgs particles (h) in gluon fusion. (a) One virtual Higgs boson (h,H) is producedby the heavy quark triangle and it decays into 3 h’s via the quartic coupling Hhhh orhhhh, Fig. 1a. (b) One virtual Higgs boson decays into 3 h’s in two steps testing trilinearcouplings, gg ! H; h ! (H; h)h ! hhh, Fig. 2b. (c) Heavy quark box diagram couplesto two Higgses, one of them decays subsequently into two h’s, gg ! (H; h)h! hhh, Fig.2c. We have omitted the pentagon graphs contributing only to the continuum production.We believe it increases the production at large tan due to the large hbb{coupling, butdoes not change our conclusion about the measureability of the quartic Higgs{couplings.There are two dierent helicity amplitudes contributing to the total cross section: thetotal spin of the two gluons along the collision axis can be Sz = 0 (contribution F) orSz = 2 (contribution G). The triangle graphs give only contributions F3 (Sz = 0) as theyinvolve a single spin{0 Higgs intermediate state. Box graphs give both contributions F4(Sz = 0) and G4 (Sz = 2). F4 and G4 are Lorentz and gauge invariant decompositions ofthe box amplitude. The tensor basis is the followingSz = 0 A = g − p1p2(p1p2)(5)3Sz = 2 B = g +1p2T (p1p2)(p23p1p2 − 2(p2p3)p1p3 − 2(p1p3)p2p3 + 2p3p3(p1p2))where p1; p2 are the momenta of the incoming gluons and p3 is one of the momenta of theoutgoing Higgs bosons (p3; p4; p5). Here p2T = 2(p1p3)(p2p3)(p1p2)−p23 is the transverse momentumof the third particle. Tensors A and B are orthogonal and normalized to 2 [2].The M{matrix of the process isM = (p2GF )3=2ss^4ab ab(FA + GB): (6)The spin and color averaged parton level cross section of the process gg ! hhh isd^(gg ! hhh) =p2G3F2s1024(2)6hjF j2 + jGj2i1=2(s^45; p24; p25)s^45ds^45ds^13dΩCM45 : (7)Here we used the Chew-Low parametrization of the three particle phase space [10], atrivial angle integration was carried out, s^ik = (pi + pk)2, and the usual lambda functionis (x; y; z) =x− (py +pz)2)(x− (py −pz)2. dΩCM45 is the dierential solid anglein the center of mass system of particles 4 and 5.F receives contributions from the triangle and box graphs, while only box diagramsgive contributions Sz = 2 to GF =Xt;b(C3 F3 +Xi=3;4;5C(i)4 F(i)4 )G =Xt;bXi=3;4;5C(i)4 G(i)4 : (8)Here we separated the functions F3; F4; G4 responsible for the heavy quark triangle andboxes, C3; C4 contain the coupling constants and the propagators. The amplitude hasto be summed over all quark flavours. However, the four light quarks can be neglectedhaving very small Higgs couplings. Supersymmetric particles are assumed to be tooheavy to participate in the triangle or box loop. The second sum in the box contributioncorresponds to the outgoing Higgs particle that couples directly to the quark loop.We calculated F3; F4; G4 and found them in agreement with the results of [2], [1]. Forinstance,F3 = 2m2Qs^2 + (4m2Q − s^)C12; s^ = s^12: (9)whereCij =Zd4qi21(q2 −m2Q)h(q + pi)2 −m2Qi h(q + pi + pj)2 −m2Qi : (10)F4; G4 have long analytic expressions which can be found in ref [2]. The box functionsF(i)4 ; G(i)4 in (8) depend on three momenta (p1; p2; pi); i = 3; 4; 5 and F(i)4 = F4(p1; p2; pi)etc.4In case of a large quark mass (m2Q  s^  M2h;H) we get as in [2]F3 =23+Os^=m2Q; F(i)4 = −23Os^=m2Q; G(i)4 = Os^=m2Q(11)In the limit of light quark masses (m2Q  s^  M2h;H) all the form factors vanish toOm2Q=s^.The generalized triangle couplings are the followingC3 = Ch3 + CH3 + Chh3 + ChH3 + CHh3 + CHH3CHa3 = hhhHaM2Zs^−M2Ha + iMHaΓHagHaQ (12)CHaHb = HaHbhM2Zs^−M2Ha + iMHaΓHaHbhhM2Zs^−M2Hb + iMHbΓHbgHaQ ; Ha;b = h=H:The box couplings areC4 = Ch4 + CH4C(h;H)(i)4 = hh(h=H)M2Zs^kl −M2h=H + iMh=HΓh=HghQgh=HQ ; (13)i; k; l are cyclic. gh=HQ denotes the Higgs{quark couplings normalized to the SM Yukawacoupling [p2GF ]1=2mQ,ght = cos = sin; ghb = − sin = cos; gHt = sin = sin ; gHb = cos = cos: (14)3 ResultsNow, we calculate the production cross section of three lightest neutral supersymmetricHiggs bosons pp! hhh + X via gluon fusion at the LHC energy of ps = 14 TeV in theabove picture by folding the parton level fusion cross section with the gluon luminosity.We used the GRV structure functions [11] for the gluon luminosity at Q2 = s^. Thenumerical integration was made by the VEGAS package [12]. The total cross section vs.MA is shown in Fig.3 for two represantative values of tan = 3 and tan = 30. In theplotted range of MA the heavy CP{even Higgs boson (H) is nearly degenerate in masswith A while the mass of the lightest neutral Higgs h is quickly approaches approximately104 GeV from below.For tan = 3 in the range of MA = 200{350 GeV the cross section is enhanced byresonance eect and large. The cross section goes up to nearly 300 fb and slowly decreasesto about 150 fb giving a large number of events at the expected luminosityRLdt ’100fb−1=year. The main contribution comes from the box diagram where the heavyCP{even Higgs boson H couples to the quark loop and decays into two bosons h. Mhslowly increases around 100 GeV when MH ’ MA reaches 200 GeV the H propagator5enters the resonance region. At MH  2mt there is a threshold eect due to the fall{oof the branching ratio BR(H ! hh) and partly because on{shell top quark pairs can beproduced in the quark loop. For large values of MA the cross section reaches a continuumvalue of a few fb. The dip at small MA, similarly to the case of tan = 30, is the resultof the zero in the trilinear couplings (H=h)hh.For tan =30 the main contribution comes from the box diagram where an h couplesto the quark loop, propagates and decays into two h’s via hhh. The Hhh couplingis small compared to the case of tan = 3 while the hhh coupling is larger giving asizeable continuum. There is no observable resonance eect and the cross section isnearly constant, 6 fb, versus MA after the coupling constant changed sign. The K factoris expected to increase the cross section as in other cases [6,3].The contributions of the diagrams containing a heavy quark triangle are also shown inFig.3 at tan=3. The lowest curve corresponds to the diagram with quartic couplings, theone in the middle sums up all the contributions of the graphs involving a quark triangle.There is a clear resonance eect in both curves when MH reaches 3Mh at 310 GeV. Herethe rst propagator of gg ! H ! Hh! hhh becomes resonant too. The rst rise in themiddle curve is the result of the resonance in the second propagator of Fig. 2b, similarlyto the box. The triangle loops yield, however, only a small fraction of the cross sectioneven at their peak at 300-350 GeV.The suppressed contribution of the quartic coupling implies that in pp ! hhh + Xthe measurement of the quartic coupling is not possible. Not only the box contributionis larger by two orders of magnitude but also the other triangle diagrams are 10 timeslarger. For tan = 30 the case is even worse.In conclusion, in this paper we have calculated the resonance contribution to the crosssection of 3h{production at LHC via gluon fusion. The main contribution comes from thetrilinear Higgs coupling while the quartic ones turn out to be negligible. In the resonanceregion at small tan the ratio of 2h and 3h production is about 10.This work was partially supported by Hungarian Science Foundation Grants underContract Nos. OTKA T029803 and F022998.6References[1] E.W.N. Glover and J.J. van der Bij, Nucl. Phys. B309 (1988) 282.[2] T. Plehn, M. Spira and P.M. Zerwas, Nucl. Phys.B479 (1996) 46-64[3] A. Djouadi, W. Kilian, M. Muhlleitner and P. M. Zerwas Eur.Phys.J. C10 (1999)45-49, hep-ph/9904287.[4] A. Belyaev et al. Phys. Rev. D60 (1999) 075008[5] A. Belyaev, M. Drees, J. K. Mizukoshi hep-ph/9909386[6] S. Dawson, S. Dittmaier and M. Spira, Phys. Rev. D 58 (1998) 115012[7] G. Cynolter, E. Lendvai and G. Pocsik, in preparation.[8] M. Carena, J. Espinosa, M. Quiros and C.E.M. Wagner, Phys. Lett. B355 (1995)209; M. Carena, M. Quiros and C.E.M. Wagner, Nucl. Phys. B461 (1996) 461.;H. E. Haber, R. Hempfling and A. H. Hoang, Z. Phys. C75 (1997) 539; A. Djouadi,J. Kalinowski and M. Spira, Comp. Phys. Comm. 108 (1998) 56, S. Heinemeyer, W.Hollik and G. Weiglein, Eur. Phys. J. C9 (1999) 343.[9] H. E. Haber and R. Hempfling, Phys.Rev. Lett. 66 (1991) 1815; Y. Okada M. Ya-maguchi and T, Yamagida, Prog. Theor. Phys. 85 (1991) 1; J. Ellis, G. Ridol andF. Zwirner, Phys. Lett. B257 (1991) 83.[10] e.g. E. Byckling and K Kajantie, Kinematics, Wiley (1973) p.133[11] M. Glu¨ck, E. Reya and A. Vogt, Z. Phys. C53 (1992) 127,Eur.Phys.J. C5 (1998)461-470.[12] G. P. Lepage, J. Comp. Phys. 27 (1978) 192.7FIGURE CAPTIONSFig.1: Generic diagrams contributing to the production of three CP{even MSSM Higgsbosons in gluon{gluon collisions, gg ! hhh: a) triangle with quartic coupling, b)trianglewith trilinear couplings c) box , d) pentagon contributions.Fig.2: Quartic Higgs couplings in the MSSM as functions of the pseudoscalar Higgs massMA for two representative values of tan=3 and 30.Fig.3: Total cross sections for production of three lightest CP{even MSSM Higgs bosonsin gluon{gluon collisions at the LHC for tan = 3 and 30 (upper two curves). The upperaxis presents the scalar Higgs masses Mh for tan = 3 corresponding to the pseudoscalarmasses MA. The lower two dashed curves represent the resonance contributions of thetriangle graphs of Fig. 2a and 2b for tan =3.8Fig. 19-2-1.5-1-0.500.511.5270 100 200 300 400 500 800 MA (GeV) tanβ=30 tanβ=3λHhhhFig. 2a00.511.522.533.544.570 100 200 300 400 500 800 MA (GeV) tanβ=30 tanβ=3 λhhhhFig. 2b101e-050.00010.0010.010.111010010000 100 200 300 400 500 600 700 80082.5 99.8 101.6 102.2 102.4 102.5 102.6σ [fb] MA [GeV] Mh [GeV] (tanβ=3) tanβ=3 tanβ=30 tanβ=3 tanβ=3 pp →  gg →  hhh+ X LHCbox + trianglebox + triangletriangletriangle, only with quartic couplingFig. 311

Resonance Production of Three Neutral Supersymmetric Higgs Bosons at LHC

ELTE Digital Institutional Repository (EDIT)

arXiv:hep-ph/0003008v1  1 Mar 2000ITP-Budapest 554December 1999Resonance Production of Three Neutral SupersymmetricHiggs Bosons at LHCG. Cynolter, E. Lendvai and G. PócsikInstitute for Theoretical PhysicsEötvös Lorand University, BudapestAbstractMultiple production of Higgs particles is essential to study Higgs self-couplingsat future high–energy colliders. In this paper we calculated the resonance con-tributions to the production of three lightest neutral supersymmetric Higgsbosons in gluon fusion at LHC. The cross sections due to trilinear Higgs cou-plings is sizeable but the measurement of the quartic coupling λhhhH(h) seemsto be impossible.1 IntroductionOne of the basic open questions of particle physics is the nature of electroweak symmetrybreaking. Beyond the discovery of the Higgs boson(s), however, reconstructing the Higgspotential will be necessary, and that requires the experimental study of the self–couplingsof the Higgs bosons.Pair production of neutral Higgs particles in gluon fusion sensitive to the trilinearcouplings of Higgs bosons was studied in the SM [1] and the Minimal SupersymmetricStandard Model (MSSM) [2–5]. QCD corrections were included in the limiting case of aheavy top mass [6].In the SM the trilinear and the quartic couplings of the physical Higgs particle arefixed by the Higgs mass. There is no resonance in the pair production of SM Higgses,and the cross section is only ≃20–50 fb in the intermediate mass range [3]. No resonanceeffect is present in the production of three Standard Model Higgs bosons via one Higgsin gluon fusion and it is estimated to be under the discovery limit at LHC.In the MSSM there are five Higgs bosons (h,H,A,H±) and many trilinear and quarticcouplings among them. Two parameters describe the Higgs sector of the MSSM: MA, themass of the CP-odd Higgs boson A, and tanβ, the ratio of the two vacuum expectationvalues. For a wide range of MA, in pp collision only the cross section of the hh productionis large [2,3]. Possible other processes, such as the gauge boson fusion and Higgs–strahlungoff W bosons or heavy quarks (top) provide less event number.In order to provide further tests of trilinear and quartic Higgs self–couplings, in thispaper we calculate the resonance enhanced production of three lightest supersymmetricHiggs bosons (h) in gluon fusionpp → gg → hhh. (1)Generally four different graphs contribute to the production of three Higgses. Gluonscouple to triangle, box or pentagon quark loops emitting 1,2 or 3 Higgs bosons, as is seenin Fig.1. Squark loops are neglected. We have omitted the pentagon graph producing aflat continuum background for the resonance production and we will give the completecross section valid in a wider range in a subsequent publication [7]. We learn that thequartic coupling is not accessible by LHC experiment. The resonance contribution to (1),however, sizeable at LHC For instance at tanβ = 3 it yields a cross section of about 300fb. This is about 1/10 times the corresponding hh–case [3].The masses, widths and the couplings were calculated using the complete one–loopand the leading two–loop radiative corrections from [8]. The relevant Higgs self–couplingsare the followingλhhh = 3 cos(2α) sin(β + α) +3ǫM2Zcos3 αsin βλHhh = 2 sin(2α) sin(β + α)− cos(2α) cos(β + α) +3ǫM2Zsinα cos2 αsin β2λHHh = −2 sin(2α) cos(β + α)− cos(2α) sin(β + α) +3ǫM2Zsin2 α cosαsin βλHhhh = 3 sin(2α) cos(2α) +ǫ8M2Zcosα sin2 αsin2 βλhhhh = 3 cos2(2α) +ǫ4M2Zsin4 αsin2 β(2)The trilinear and quartic couplings are normalized to λ3 = [√2GF ]1/2M2Z and λ4 =√2GFM2Z , respectively. The couplings depend on β and the mixing angle α of the CP–even Higgs sectortan 2α =M2A +M2ZM2A −M2Z + ǫ/ cos 2βtan 2β. (3)The leading m4t one–loop corrections [9] are parametrized byǫ =3GF√2π2m4tsin2 βlog[1 +M2Sm2t]. (4)The common squark mass is fixed to MS= 1 TeV. The quartic couplings depend on MAand are shown for two values of tanβ = 3 and 30 in Fig.2.The paper is organized as follows. In the next section we give the cross section of theresonance production of three h’s in gluon fusion and in section 3 we show the numericalresults.2 The cross sectionThree different channels contribute to the resonance production of three lightest supersym-metric Higgs particles (h) in gluon fusion. (a) One virtual Higgs boson (h,H) is producedby the heavy quark triangle and it decays into 3 h’s via the quartic coupling λHhhh orλhhhh, Fig. 1a. (b) One virtual Higgs boson decays into 3 h’s in two steps testing trilinearcouplings, gg → H, h → (H, h)h → hhh, Fig. 2b. (c) Heavy quark box diagram couplesto two Higgses, one of them decays subsequently into two h’s, gg → (H, h)h → hhh, Fig.2c. We have omitted the pentagon graphs contributing only to the continuum production.We believe it increases the production at large tanβ due to the large hbb–coupling, butdoes not change our conclusion about the measureability of the quartic Higgs–couplings.There are two different helicity amplitudes contributing to the total cross section: thetotal spin of the two gluons along the collision axis can be Sz = 0 (contribution F) orSz = 2 (contribution G). The triangle graphs give only contributions F3 (Sz = 0) as theyinvolve a single spin–0 Higgs intermediate state. Box graphs give both contributions F4(Sz = 0) and G4 (Sz = 2). F4 and G4 are Lorentz and gauge invariant decompositions ofthe box amplitude. The tensor basis is the followingSz = 0 Aµν = gµν − pν1pµ2(p1p2)(5)3Sz = 2 Bµν = gµν +1p2T (p1p2)(p23pν1pµ2 − 2(p2p3)pν1pµ3 − 2(p1p3)pµ2pν3 + 2pµ3pν3(p1p2))where p1, p2 are the momenta of the incoming gluons and p3 is one of the momenta of theoutgoing Higgs bosons (p3, p4, p5). Here p2T = 2(p1p3)(p2p3)(p1p2)−p23 is the transverse momentumof the third particle. Tensors Aµν and Bµν are orthogonal and normalized to 2 [2].The M–matrix of the process isM = (√2GF )3/2αsŝ4πǫµaǫνb δab(FAµν +GBµν). (6)The spin and color averaged parton level cross section of the process gg → hhh isdσ̂(gg → hhh) =√2G3Fα2s1024(2π)6[|F |2 + |G|2]λ1/2(ŝ45, p24, p25)ŝ45dŝ45dŝ13dΩCM45 . (7)Here we used the Chew-Low parametrization of the three particle phase space [10], atrivial angle integration was carried out, ŝik = (pi + pk)2, and the usual lambda functionis λ(x, y, z) =(x− (√y +√z)2)(x− (√y −√z)2). dΩCM45 is the differential solid anglein the center of mass system of particles 4 and 5.F receives contributions from the triangle and box graphs, while only box diagramsgive contributions Sz = 2 to GF =∑t,b(C3 F3 +∑i=3,4,5C(i)4 F(i)4 )G =∑t,b∑i=3,4,5C(i)4 G(i)4 . (8)Here we separated the functions F3, F4, G4 responsible for the heavy quark triangle andboxes, C3, C4 contain the coupling constants and the propagators. The amplitude hasto be summed over all quark flavours. However, the four light quarks can be neglectedhaving very small Higgs couplings. Supersymmetric particles are assumed to be tooheavy to participate in the triangle or box loop. The second sum in the box contributioncorresponds to the outgoing Higgs particle that couples directly to the quark loop.We calculated F3, F4, G4 and found them in agreement with the results of [2], [1]. Forinstance,F3 = 2m2Qŝ(2 + (4m2Q − ŝ)C12), ŝ = ŝ12. (9)whereCij =∫d4qiπ21(q2 −m2Q)[(q + pi)2 −m2Q] [(q + pi + pj)2 −m2Q] . (10)F4, G4 have long analytic expressions which can be found in ref [2]. The box functionsF(i)4 , G(i)4 in (8) depend on three momenta (p1, p2, pi), i = 3, 4, 5 and F(i)4 = F4(p1, p2, pi)etc.4In case of a large quark mass (m2Q ≫ ŝ ∼ M2h,H) we get as in [2]F3 =23+O(ŝ/m2Q), F(i)4 = −23O(ŝ/m2Q), G(i)4 = O(ŝ/m2Q)(11)In the limit of light quark masses (m2Q ≪ ŝ ∼ M2h,H) all the form factors vanish toO(m2Q/ŝ).The generalized triangle couplings are the followingC3 = Ch3 + CH3 + Chh3 + ChH3 + CHh3 + CHH3CHa3 = λhhhHaM2Zŝ−M2Ha + iMHaΓHagHaQ (12)CHaHb = λHaHbhM2Zŝ−M2Ha + iMHaΓHaλHbhhM2Zŝ−M2Hb + iMHbΓHbgHaQ , Ha,b = h/H.The box couplings areC4 = Ch4 + CH4C(h,H)(i)4 = λhh(h/H)M2Zŝkl −M2h/H + iMh/HΓh/HghQgh/HQ , (13)i, k, l are cyclic. gh/HQ denotes the Higgs–quark couplings normalized to the SM Yukawacoupling [√2GF ]1/2mQ,ght = cosα/ sinβ, ghb = − sinα/ cosβ, gHt = sinα/ sin β, gHb = cosα/ cosβ. (14)3 ResultsNow, we calculate the production cross section of three lightest neutral supersymmetricHiggs bosons pp → hhh +X via gluon fusion at the LHC energy of √s = 14 TeV in theabove picture by folding the parton level fusion cross section with the gluon luminosity.We used the GRV structure functions [11] for the gluon luminosity at Q2 = ŝ. Thenumerical integration was made by the VEGAS package [12]. The total cross section vs.MA is shown in Fig.3 for two represantative values of tanβ = 3 and tanβ = 30. In theplotted range of MA the heavy CP–even Higgs boson (H) is nearly degenerate in masswith A while the mass of the lightest neutral Higgs h is quickly approaches approximately104 GeV from below.For tanβ = 3 in the range of MA = 200–350 GeV the cross section is enhanced byresonance effect and large. The cross section goes up to nearly 300 fb and slowly decreasesto about 150 fb giving a large number of events at the expected luminosity∫Ldt ≃100fb−1/year. The main contribution comes from the box diagram where the heavyCP–even Higgs boson H couples to the quark loop and decays into two bosons h. Mhslowly increases around 100 GeV when MH ≃ MA reaches 200 GeV the H propagator5enters the resonance region. At MH ∼ 2mt there is a threshold effect due to the fall–offof the branching ratio BR(H → hh) and partly because on–shell top quark pairs can beproduced in the quark loop. For large values of MA the cross section reaches a continuumvalue of a few fb. The dip at small MA, similarly to the case of tanβ = 30, is the resultof the zero in the trilinear couplings λ(H/h)hh.For tanβ=30 the main contribution comes from the box diagram where an h couplesto the quark loop, propagates and decays into two h’s via λhhh. The Hhh couplingis small compared to the case of tanβ = 3 while the hhh coupling is larger giving asizeable continuum. There is no observable resonance effect and the cross section isnearly constant, 6 fb, versus MA after the coupling constant changed sign. The K factoris expected to increase the cross section as in other cases [6,3].The contributions of the diagrams containing a heavy quark triangle are also shown inFig.3 at tanβ=3. The lowest curve corresponds to the diagram with quartic couplings, theone in the middle sums up all the contributions of the graphs involving a quark triangle.There is a clear resonance effect in both curves when MH reaches 3Mh at ∼310 GeV. Herethe first propagator of gg → H → Hh → hhh becomes resonant too. The first rise in themiddle curve is the result of the resonance in the second propagator of Fig. 2b, similarlyto the box. The triangle loops yield, however, only a small fraction of the cross sectioneven at their peak at 300-350 GeV.The suppressed contribution of the quartic coupling implies that in pp → hhh + Xthe measurement of the quartic coupling is not possible. Not only the box contributionis larger by two orders of magnitude but also the other triangle diagrams are 10 timeslarger. For tanβ = 30 the case is even worse.In conclusion, in this paper we have calculated the resonance contribution to the crosssection of 3h–production at LHC via gluon fusion. The main contribution comes from thetrilinear Higgs coupling while the quartic ones turn out to be negligible. In the resonanceregion at small tanβ the ratio of 2h and 3h production is about 10.This work was partially supported by Hungarian Science Foundation Grants underContract Nos. OTKA T029803 and F022998.6References[1] E.W.N. Glover and J.J. van der Bij, Nucl. Phys. B309 (1988) 282.[2] T. Plehn, M. Spira and P.M. Zerwas, Nucl. Phys.B479 (1996) 46-64[3] A. Djouadi, W. Kilian, M. Muhlleitner and P. M. Zerwas Eur.Phys.J. C10 (1999)45-49, hep-ph/9904287.[4] A. Belyaev et al. Phys. Rev. D60 (1999) 075008[5] A. Belyaev, M. Drees, J. K. Mizukoshi hep-ph/9909386[6] S. Dawson, S. Dittmaier and M. Spira, Phys. Rev. D 58 (1998) 115012[7] G. Cynolter, E. Lendvai and G. Pócsik, in preparation.[8] M. Carena, J. Espinosa, M. Quiros and C.E.M. Wagner, Phys. Lett. B355 (1995)209; M. Carena, M. Quiros and C.E.M. Wagner, Nucl. Phys. B461 (1996) 461.;H. E. Haber, R. Hempfling and A. H. Hoang, Z. Phys. C75 (1997) 539; A. Djouadi,J. Kalinowski and M. Spira, Comp. Phys. Comm. 108 (1998) 56, S. Heinemeyer, W.Hollik and G. Weiglein, Eur. Phys. J. C9 (1999) 343.[9] H. E. Haber and R. Hempfling, Phys.Rev. Lett. 66 (1991) 1815; Y. Okada M. Ya-maguchi and T, Yamagida, Prog. Theor. Phys. 85 (1991) 1; J. Ellis, G. Ridolfi andF. Zwirner, Phys. Lett. B257 (1991) 83.[10] e.g. E. Byckling and K Kajantie, Kinematics, Wiley (1973) p.133[11] M. Glück, E. Reya and A. Vogt, Z. Phys. C53 (1992) 127,Eur.Phys.J. C5 (1998)461-470.[12] G. P. Lepage, J. Comp. Phys. 27 (1978) 192.7FIGURE CAPTIONSFig.1: Generic diagrams contributing to the production of three CP–even MSSM Higgsbosons in gluon–gluon collisions, gg → hhh: a) triangle with quartic coupling, b)trianglewith trilinear couplings c) box , d) pentagon contributions.Fig.2: Quartic Higgs couplings in the MSSM as functions of the pseudoscalar Higgs massMA for two representative values of tanβ=3 and 30.Fig.3: Total cross sections for production of three lightest CP–even MSSM Higgs bosonsin gluon–gluon collisions at the LHC for tanβ = 3 and 30 (upper two curves). The upperaxis presents the scalar Higgs masses Mh for tanβ = 3 corresponding to the pseudoscalarmasses MA. The lower two dashed curves represent the resonance contributions of thetriangle graphs of Fig. 2a and 2b for tanβ=3.8Fig. 19-2-1.5-1-0.500.511.5270 100 200 300 400 500 800 MA (GeV) tanβ=30 tanβ=3λHhhhFig. 2a00.511.522.533.544.570 100 200 300 400 500 800 MA (GeV) tanβ=30 tanβ=3 λhhhhFig. 2b101e-050.00010.0010.010.111010010000 100 200 300 400 500 600 700 80082.5 99.8 101.6 102.2 102.4 102.5 102.6σ [fb] MA [GeV] Mh [GeV] (tanβ=3) tanβ=3 tanβ=30 tanβ=3 tanβ=3 pp →  gg →  hhh+ X LHCbox + trianglebox + triangletriangletriangle, only with quartic couplingFig. 311

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Resonance Production of Three Neutral Supersymmetric Higgs Bosons at LHC

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