Higgs boson pair production at the Photon Linear Collider in the two Higgs doublet model
Eri Asakawa, Daisuke Harada, Shinya Kanemura, Yasuhiro Okada, Koji Tsumura
aa r X i v : . [ h e p - ph ] A ug Higgs boson pair production at the Photon LinearCollider in the two Higgs doublet model
Eri Asakawa , Daisuke Harada , , Shinya Kanemura , Yasuhiro Okada , and Koji Tsumura
1- Institute of Physics, Meiji Gakuin UniversityYokohama 244-8539, Japan2- KEK Theory Center, Institute of Particle and Nuclear Studies, KEK1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan3- Department of Particle and Nuclear Physics,the Graduate University for Advanced Studies (Sokendai)1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan4- Department of Physics, University of Toyama3190 Gofuku, Toyama 930-8555, Japan5- International Centre for Theoretical PhysicsStrada Costiera 11, 34014 Trieste, ItalyWe calculate the cross section of the lightest Higgs boson pair production at the PhotonLinear Collider in the two Higgs doublet model. We focus on the scenario in which thelightest Higgs boson has the standard model like couplings to gauge bosons. We takeinto account the one-loop correction to the hhh coupling as well as additional one-loopdiagrams due to charged bosons to the γγ → hh helicity amplitudes. We discuss theimpact of these corrections on the hhh coupling measurement at the Photon LinearCollider. The Higgs sector is the last unknown part of the standard model (SM). In the SM, the treelevel Higgs self-coupling λ hhh = 3 m h /v and λ hhhh = 3 m h /v are uniquely determined bythe Higgs boson mass m h , where v is vacuum expectation value (VEV) of the Higgs boson.The effective Higgs potential is written as V = 12 m h h + 13! ˜ λ hhh h + 14! ˜ λ hhhh h + · · · , (1)where the effective Higgs self-couplings ˜ λ hhh and ˜ λ hhhh are given by precision measurementof hhh and hhhh couplings. If the deviation from the SM tree level Higgs self-coupling ( λ hhh and λ hhhh ) is found, it can be regarded as an evidence of new physics beyond the SM. Theorigin of the spontaneous electroweak symmetry breaking (EWSB) would be experimentallytested after the discovery of a new scalar particle by measuring its mass and self-couplings.The Higgs self-coupling measurement is one of main purposes at the International LinearCollider (ILC). The structure of the Higgs potential depends on the scenario of new physicsbeyond the SM, so that precision measurement of the hhh coupling can be a probe of eachnew physics scenario[1, 2]. The 8th general meeting of the ILC physics working group, 1/21, 2009 t is known that the measurement of the triple Higgs boson coupling is rather challengingat the CERN Large Hadron Collider (LHC). At the SLHC with luminosity of 3000 fb − ,the hhh coupling can be determined with an accuracy of 20-30% for 160 GeV ≤ m h ≤
180 GeV[3, 4]. At the ILC, the main processes for the hhh measurement are the doubleHiggs boson production mechanisms via the Higgs-strahlung and the W-boson fusion[5, 6].At the ILC with a center of mass energy of 500 GeV, the double Higgs strahlung process e + e − → Zhh is dominant. On the other hand, W-boson fusion process e + e − → hhν ¯ ν becomes dominant due to its t -channel nature at 1 TeV or higher energies[7]. Sensitivity tothe hhh coupling in these processes becomes rapidly worse for greater Higgs boson masses.In particular, for the intermediate mass range (140 GeV ≤ m h ≤
200 GeV), it has notyet been known how accurately the hhh coupling can be measured by the electron-positroncollision. The Photon Linear Collider (PLC) is an optional experiment of the ILC. Thepossibility of measuring the hhh coupling via the process of γγ → hh has been discussed inRef. [8]. In Ref. [9] the statistical sensitivity to the hhh coupling constant has been studiedespecially for a light Higgs boson mass in relatively low energy collisions.In this paper, we study the double Higgs production process at the PLC. In Sect. 2, wediscuss the statistical sensitivity to the hhh coupling constant via the process of e − e − → γγ → hh at the PLC in the SM. In Sect. 3, we study the new particle effects on the γγ → hh process in the two Higgs doublet model (THDM). hhh coupling constant We study the statistical sensitivity to the hhh coupling constant for wide regions of the Higgsboson masses and the collider energies at the PLC. The γγ → hh process is an one-loopinduced process. The Feynman diagrams for this process in the SM are given in Ref. [8].There are two types of diagrams, which are the pole diagrams and the box diagrams. Theamplitude of the pole diagrams describes as M pole ∝ ˜ λ hhh /s , where √ s is the center ofmass energy of the γγ system. It is suppressed by 1 /s at the high energy region, so thatthe statistical sensitivity to the hhh coupling becomes rapidly worse for this region. On theother hand, the box diagrams do not depend on the hhh coupling.In Fig. 1, we present the statistical sensitivity on the Higgs self-coupling constant atthe PLC. We modify the triple Higgs coupling constant as ˜ λ hhh = λ hhh (1 + δκ ), where δκ represents deviation from the SM prediction. We assume that the efficiency of the particletagging is 100% with an integrated luminosity of 1 / − and E ee is the center of massenergy of the e − e − system. We plot δκ based on statistical error of the event number in the e − e − → γγ → hh process in the SM. Namely, δκ is determined by | N ( δκ ) − N ( δκ = 0) | = p N ( δκ = 0) , (2)for assumed luminosity. Notice that δκ is not symmetric with respect to δκ = 0 becausethere is interference between pole and box diagrams. The cases for δκ > δκ < m h [ E ee ]. It isfound that when the collision energy is limited to be lower than 500-600 GeV the statisticalsensitivity to the hhh coupling can be better for the process in the γγ collision than that inthe electron-positron collision for the Higgs boson with the mass of 160 GeV[10]. The 8th general meeting of the ILC physics working group, 1/21, 2009
00 150 200 250mH (GeV)0102030405060 | δ κ | ( % ) Higgs self coupling sensitivity
Int(L γγ )=1/3ab −1 Efficientcy 100%360GeVRoot(s)=400GeV 500GeV 600GeV 700GeV δκ >0 δκ <0300GeV 300 400 500 600 700 800E ee (GeV)0102030405060 | δ κ | ( % ) Higgs Self Coupling Sensitivity
Int(L γγ )=1/3 ab −1 Efficiency=100%120GeV mH=160GeV 200GeV δκ>0δκ<0
Figure 1: The statistical sensitivity to the hhh coupling constant at the PLC. In the left[right] figure, the statistical sensitivity is shown as a function of m h [ E ee ] for each value of E ee [ m h ]. Solid [Dotted] lines correspond to δκ > δκ <
0] case. γγ → hh process in the THDM We consider the new particle effects on the γγ → hh process in the THDM, in which addi-tional CP-even, CP-odd and charged Higgs boson appear. It is known that non-decouplingloop effect of extra Higgs bosons shift the hhh coupling value from the SM by O (100)%[1].In the γγ → hh helicity amplitudes, there are additional one-loop diagrams by the chargedHiggs boson loop to the ordinary SM diagrams (the W-boson loop and the top quark loop).It is found that both the charged Higgs boson loop contribution to the γγ → hh amplitudesand the non-decoupling effect on the hhh coupling can enhance the cross section from itsSM value significantly[11].In order to study the new physics effect on γγ → hh process, we calculate the helicityamplitudes in the THDM. The THDM Higgs potential is given by V THDM = µ | Φ | + µ | Φ | − ( µ Φ † Φ + h . c . )+ λ | Φ | + λ | Φ | + λ | Φ | | Φ | + λ | Φ † Φ | + λ n (Φ † Φ ) + h . c . o , (3)where Φ and Φ are two Higgs doublets with hypercharge +1 /
2. The Higgs doublets areparametrized as Φ i = (cid:20) ω + i √ ( v i + h i + iz i ) (cid:21) , ( i = 1 , , (4)where VEVs v and v satisfy v + v = v ≃ (246 GeV) . The mass matrices can bediagonalized by introducing the mixing angles α and β , where α diagonalizes the massmatrix of the CP-even neutral bosons, and tan β = v /v . Consequently, we have two CP-even ( h and H ), a CP-odd ( A ) and a pair of charged ( H ± ) bosons. We define α such that h is the SM-like Higgs boson when sin( β − α ) = 1. The 8th general meeting of the ILC physics working group, 1/21, 2009 e concentrate on the case with so called the SM-like limit [sin( β − α ) = 1], where thelightest Higgs boson h has the same tree-level couplings as the SM Higgs boson, and theother bosons do not couple to gauge bosons and behave just as extra scalar bosons. In thislimit, the masses of Higgs bosons are m h = { λ cos β + λ sin β + 2( λ + λ + λ ) cos β sin β } v , (5) m H = M + 18 { λ + λ − λ + λ + λ ) } (1 − cos 4 β ) v , (6) m A = M − λ v , (7) m H ± = M − λ + λ v , (8)where M (= | µ | / √ sin β cos β ) represents the soft breaking scale for the discrete symmetry,and determines the decoupling property of the extra Higgs bosons. When M ∼
0, the extraHiggs bosons H , A and H ± receive their masses from the VEV, so that the masses areproportional to λ i . Large masses cause significant non-decoupling effect in the radiativecorrection to the hhh coupling. On the other hand, when M ≫ v the masses are determinedby M . In this case, the quantum effect decouples for M → ∞ .It is known that in the THDM λ hhh can be changed from the SM prediction by theone-loop contribution of extra Higgs bosons due to the non-decoupling effect (when M ∼ hhh coupling Γ THDM hhh (ˆ s, m h , m h ) is evaluated at the one-loop level as[1]Γ THDM hhh (ˆ s, m h , m h ) ≃ m h v X Φ= H,A,H + ,H − m π v m h (cid:18) − M m (cid:19) − N c m t π v m h . (9)The exact one-loop formula for Γ THDM hhh is given in Ref. [2], which has been used in our actualnumerical analysis.In Fig. 2, we plot the cross sections of γγ → hh for the helicity set (+ , +) as a function ofthe photon-photon collision energy E γγ . The five curves correspond to the following cases,(a) THDM 2-loop: the cross section in the THDM with additional one-loop corrections tothe hhh vertex, Γ THDM hhh .(b) THDM 1-loop: the cross section in the THDM with the tree level hhh coupling con-stant λ hhh .(c) SM 2-loop: the cross section in the SM with additional top loop correction to the hhh coupling Γ SM hhh given in Ref. [2].(d) SM 1-loop: the cross section in the SM with the tree level hhh coupling constant λ SM hhh (= λ hhh for sin( β − α ) = 1).(e) For comparison, we also show the result which corresponds to the SM 1-loop resultwith the effective hhh coupling Γ THDM hhh .In the left figure, there are three peaks in the case (a) (THDM 2-loop). The one at thelowest E γγ is the peak just above the threshold of hh production. There the cross section The 8th general meeting of the ILC physics working group, 1/21, 2009 σ ( γγ → hh ) [f b ] E γγ [GeV]Sub cross section (++) m h = 120 GeV m H ± =m A =m H = 400 GeV sin (α−β) = −1 THDM (2-loop)THDM (1-loop)SM (2-loop)SM (1-loop)SM + Γ hhh THDM σ ( γγ → hh ) [f b ] E γγ [GeV]Sub cross section (++) m h = 160 GeV m H ± =m A =m H = 400 GeV sin (α−β) = −1 THDM (2-loop)THDM (1-loop)SM (2-loop)SM (1-loop)SM + Γ hhh THDM
Figure 2: The cross section ˆ σ (+ , +) for the sub process γγ → hh with the photon helicity set(+ , +) as a function of the collision energy E γγ . In the left [right] figure the parameters aretaken to be m h = 120 [160] GeV for m Φ ( ≡ m H = m A = m H ± ) = 400 GeV, sin( β − α ) = 1,tan β = 1 and M = 0.is by about factor three enhanced as compared to the SM prediction due to the effect of∆Γ THDM hhh / Γ SM hhh ( ∼ γγ → hh . Thesecond peak at around E γγ ∼
400 GeV comes from the top quark loop contribution whichis enhanced by the threshold of top pair production. Around this point, the case (a) can bedescribed by the case (e) (SM+Γ
THDM hhh ). For E γγ ∼ THDM hhh / Γ SM hhh . The third peak at around E γγ ∼
850 GeV is the thresholdenhancement of the charged Higgs boson loop effect, where the real production of chargedHiggs bosons occurs. The contribution from the non-pole one-loop diagrams are dominant.In the right figure, we can see two peaks around E γγ ∼ THDM hhh / Γ SM hhh by several times 100% for E γγ ∼
350 GeV.It also amounts to about 80% for E γγ ∼
400 GeV. For E γγ < H + H − production as in the left figure.In Fig. 3, the full cross section of e − e − → γγ → hh is given from the sub cross sections byconvoluting the photon luminosity spectrum[8]. In our study, we set x = 4 E b ω /m e = 4 . E b is the energy of electron beam, ω is the laser photon energy and m e is theelectron mass. In order to extract the contribution from ˆ σ (+ , +) that is sensitive to the hhh vertex, we take the polarizations of the initial laser beam to be both −
1, and thosefor the initial electrons to be both +0 .
45. The full cross section for m Φ = 400 GeV hassimilar energy dependences to the sub cross section ˆ σ (+ , +) in Fig. 2, where correspondingenergies are rescaled approximately by around √ s ∼ E γγ / . m Φ , the peak around √ s ∼
350 GeV becomes lower because of
The 8th general meeting of the ILC physics working group, 1/21, 2009 σ ( γ γ → hh ) [f b ] √ s [GeV]Full cross section (x=4.8) m h = 120 GeV sin (α−β) = −1 m H + =m A =m H = 400 GeV = 350 GeV= 300 GeV= 250 GeV= 200 GeVSM 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 400 600 800 1000 1200 1400 σ ( γγ → hh ) [f b ] √ s [GeV]Full cross section (x=4.8) m h = 160 GeV sin (α−β) = −1 m H ± =m A =m H = 400 GeV = 350 GeV = 300 GeV= 250 GeV = 200 GeVSM Figure 3: The full cross section of e − e − → γγ → hh as a function of √ s for each valueof m Φ (= m H = m A = m H ± ) with sin( β − α ) = 1, tan β = 1 and M = 0. The case for m h = 120 [160] GeV is shown in the left [right] figure.smaller ∆Γ THDM hhh / Γ SM hhh .In Fig. 4, five curves correspond to the cases (a) to (e) in Fig. 2. In the left figure, onecan see that the cross section is enhanced due to the enlarged Γ THDM hhh for larger values of m Φ which is proportional to m (when M ∼ m Φ (10-20% for m Φ <
300 GeV due to the charged Higgs loop effect)but it becomes rapidly enhanced for greater values of m Φ ( O (100) % for m Φ >
350 GeVdue to the large ∆Γ
THDM hhh ). A similar enhancement for the large m Φ values can be seen inthe right figure. The enhancement in the cross section in the THDM can also be seen for m Φ <
250 GeV, where the threshold effect of the charged Higgs boson loop appears around √ s ∼
600 GeV in addition to that of the top quark loop diagrams. For m Φ = 250-400 GeV,both contributions from the charged Higgs boson loop contribution and the effective hhh coupling are important and enhance the cross section from its SM value by 40-50%. In this paper, we have analysed the new physics loop effects on the cross section of γγ → hh in the THDM with SM-like limit including the next to leading effect due to the extra Higgsboson loop diagram in the hhh vertex. Our analysis shows that the cross section can belargely changed from the SM prediction by the two kinds of contributions; i.e., additonalcontribution by the charged Higgs boson loop effect, and the effective one-loop hhh vertexΓ THDM hhh enhanced by the non-decoupling effect of extra Higgs bosons. The cross sectionstrongly depends on m h and √ s and also on m Φ . The approximation of the full cross The 8th general meeting of the ILC physics working group, 1/21, 2009 σ ( γγ → hh ) [f b ] m H ± =m A =m H [GeV]Full cross section ( √ s = 350 GeV) m h = 120 GeV THDM (2-loop)THDM (1-loop)SM + Γ hhh THDM σ ( γγ → hh ) [f b ] m H ± =m A =m H [GeV]Full cross section ( √ s = 600 GeV) m h = 160 GeVTHDM (2-loop)THDM (1-loop)SM + Γ hhh THDM
SM (2-loop)SM (1-loop)
Figure 4: In the left [right] figure, the full cross section of e − e − → γγ → hh at √ s = 350GeV [600 GeV] for m h = 120 [160] GeV is shown as a function of m Φ (= m H = m A = m H ± )with sin( β − α ) = 1, tan β = 1 and M = 0.section in the case (a) (THDM 2-loop) by using the result in the case (e) (SM+Γ THDM hhh )is a good description for √ s ≪ m Φ / .
8. On the other hand, in a wide region betweenthreshold of top pair production and that of charged Higgs boson pair production, both thecontributions (those from charged Higgs boson loop effect and from Γ
THDM hhh ) are important.In the region below the threshold of the real production of extra Higgs bosons, cross sectionis largely enhanced from the SM value by the effects of the charged Higgs boson loop andthe effective Γ
THDM hhh coupling. These New Physics effects would be detectable at the futurePhoton Linear Collider.
The authors would like to thank all the members of the ILC physics subgroup [12] for usefuldiscussions. This study is supported in part by the Creative Scientific Research Grant No.18GS0202 of the Japan Society for Promotion of Science. The work of S. K. was supportedin part by Grant-in-Aid for Science Research, Japan Society for the Promotion of Science(JSPS), No. 18034004. The work of Y. O. was supported in part by Grant-in-Aid for ScienceResearch, MEXT-Japan, No. 16081211, and JSPS, No. 20244037.
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