Impact of top-Higgs couplings on di-Higgs production at future colliders
aa r X i v : . [ h e p - ph ] D ec Impact of top-Higgs couplings on Di-Higgs production at futurecolliders
Ning Liu , Songlin Hu , Bingfang Yang , and Jinzhong Han College of Physics & Electronic Engineering,Henan Normal University, Xinxiang 453007, China School of Materials Science and Engineering,Henan Polytechnic University, Jiaozuo 454000, China School of Physics and Electromechnical Engineering,Zhoukou Normal University, Zhoukou, 466001, China
Abstract
Measuring the Higgs-self coupling is one of the most crucial goals of the future colliders, suchas the LHC Run-II and the ILC-based photon collider. Since the new physics can affects the di-Higgs production not only from the Higgs self-coupling but also from the top-Higgs coupling, weinvestigate the di-Higgs production in the presence of the non-standard top-Higgs coupling at theLHC and ILC-based photon collider given the recent Higgs data. Due to the changed interferencebehaviors of the top quark loops with itself or W boson loops, we find that the cross section ofdi-Higgs production at the LHC-14 TeV and ILC-500 GeV can be respectively enhanced up tonearly 3 and 2 times the SM predictions within 2 σ Higgs data allowed parameter region.
PACS numbers: . INTRODUCTION In 2012, the ATLAS and CMS collaborations jointly announced that a bosonic resonancewith a mass around 125 GeV was found at the LHC [1, 2]. So far, most measurements ofits properties are compatible with the predictions of the Higgs boson in the Standard Model(SM) [3, 4]. However, due to the current limited statistics, the Higgs couplings with topquarks and with itself are still vacant and remain to be verified at the future colliders.In the SM, the couplings of fermions to the Higgs boson are proportional to their masses.Due to the large mass, top quark has the strongest coupling to the Higgs boson and isspeculated as a sensitive probe to the new flavor dynamics beyond the SM. As a directway to test the top-Higgs coupling, the associated production of the top pair with Higgsboson has been widely studied at the LHC [5–8]. Besides, the search for a single topassociated production with the Higgs boson was proposed to determine the sign of the top-Higgs coupling at the LHC [9–11]. On the other hand, the top-Higgs coupling also plays avital role in other processes involving the Higgs boson through the quantum effects, such asthe di-Higgs production [12]. This makes the top-Higgs coupling inevitably entangled withthe Higgs self-coupling and affects the measurement of the Higgs self-coupling at the LHC.In the renormalizable Lagrangian of the SM, only the quartic Higgs coupling λ ( φ † φ ) allowed by the electroweak gauge symmetry can generate the Higgs self-coupling. Themeasurement of the Higgs self-coupling is essential to reconstruct the Higgs potential andunderstand the electroweak symmetry breaking (EWSB) mechanism. In some extensions ofthe SM, the self-coupling can be significantly distorted by the loop corrections and becomesensitive to the new physics [13–15]. Besides, a large deviation in the self-coupling may be adirect evidence for strong first-order electroweak phase transition in the early universe [16].At the LHC, the di-Higgs production is the only way to measure the Higgs self-coupling andis dominated by the gluon-gluon fusion mechanism, which has been widely studied in recentyears [17–20]. Among various decay channels, although the 4 b final state has the largestfraction, the rare process hh → b ¯ bγγ is expected to have the most promising sensitivity dueto the low QCD backgrounds at the LHC [21].In this work, we will investigate the effect of non-standard top-Higgs coupling in thedi-Higgs production at the LHC and ILC-based photon collider under the current Higgsdata constraints. Whenever examining the Higgs self-coupling at the LHC, one should keep2n mind that, the main process gg → hh can also be triggered by the top-Higgs couplingitself through the box diagrams. These irrelevant processes lead to the strong cancellationwith those involving the self-coupling in the SM, which makes the cross section of di-Higgsproduction nearly 10 times smaller than the single Higgs production at the LHC. So, thetop-Higgs coupling will affect the extraction of the Higgs self-coupling from the measurementof di-Higgs production at the LHC [22].Given the limited precision of the LHC, an e + e − collider is crucial to scrutinize thedetailed properties of the Higgs boson that might uncover the new physics beyond theSM [23]. In addition to the e + e − collisions, high energy photon-photon collisions can beachieved at the ILC by converting the energetic electron beam to a photon beam throughthe backward Compton scattering [24]. Similar to the process gg → hh at the LHC, γγ → hh also occurs at one-loop level. The measurements of the Higgs self-coupling at the photoncollider were discussed in Ref. [25], where the complementarity of the photon and e + e − collider was emphasized. Recently, an extensive study of the feasibility of the di-Higgsproduction with a parameter set of the ILC-based photon collider was reanalysed in Ref.[26], which concluded that that the channel γγ → hh → b ¯ bb ¯ b process can be observed witha statistical significance of about 5 σ for the integrated luminosity corresponding to 5 yearsrunning of the photon collider. Therefore, the photon collider provide an ideal place to studythe new physics effect in the di-Higgs production.The structure of this paper is organized as follows. In Section II, we will briefly introducethe non-standard top-Higgs interaction and set up the calculations. In Section III, wepresent the numerical results and discuss the effects of non-standard top-Higgs coupling inthe di-Higgs production at the LHC and ILC-based photon collider. Finally, we draw ourconclusions in Section IV. II. TOP-HIGGS INTERACTION AND CALCULATIONS
In the SM the top-Higgs interaction can be written as: L SMt ¯ th = − y t SM Q L t R ˜ φ + h . c ., (1)with y t SM = √ m t /v. (2)3here Q L is the third generation SM quark doublet, φ is the Higgs doublet, ˜ φ i = ǫ ij φ j , andHiggs field vacuum expectation value (vev) v ≈
246 GeV. However, in some new physicsmodels, the top-Higgs interaction can be different from the above SM prediction. Thesenew physics effects on t ¯ th coupling can be model-independently parameterized by a gaugeinvariant dimension-six operator [27]. For example, the term L t ¯ th = − C uφ Λ ( φ † φ )( Q L t R ˜ φ ) + h . c .. (3)Here we should note that the Eq.(3) does correct the top quark mass m t by v [Re C uφ +Im C uφ γ ], which has to be reabsorbed into the physical observable m t in Eq.(2). With thisin mind, after the EWSB, we can have a general t ¯ th interaction including the SM top-Higgscouplings and corrections from the dimension-six operator as following, L t ¯ th = − y t √ t (cos θ + i sin θγ ) th, (4)with y t cos θ = y t SM + v Λ Re C uφ , y t sin θ = v Λ Im C uφ . (5)where y t takes the SM value y t SM when Re C uφ = Im C uφ = 0. For convenience, we definetwo reduced couplings: c t = y t cos θ/y t SM and ˜ c t = y t sin θ/y t SM in the following calculations.Although the CP-violating interaction can contribute to the electric dipole moment (EDM),the bounds on the coupling ˜ c t rely on the assumption of Higgs couplings to other lightfermions [28]. Given that these couplings are generally unobservable at the LHC, we do notimpose EDM constraints in this study. For other low-energy physics constraints, such as B s − ¯ B s and B → X s γ , they are still too weak [29]. The most relevant indirect constraintis from the current Higgs data since the non-standard top-Higgs interaction can change theproduction rate of gg → h and decay width of h → γγ through the loop effect. The signalstrengthes µ i can be parameterized through the reduced couplings as following [29], µ hgg ≃ c t + 2 . c t + 0 . c t ( c t − ,µ hγγ ≃ (1 . − . c t ) + (0 . c t ) . (6)We perform the χ fit of anomalous couplings c t and ˜ c t to the Higgs data by using thepackage HiggsSignals-1.2.0 [30]. 4 gg H HHt tt gg H HHt tt gg HHt tt t gg HHt tt t gg HHt tt t gg HHt tt t gg HHt tt t gg HHt tt t FIG. 1: Feynman diagrams of the process gg → hh at the LHC. A. gg → hh In Fig.1, we show the Feynman diagrams of the process gg → hh at the LHC. As abovementioned, the process gg → hh is generated by triangle and box top quark loop diagrams,respectively. By applying the low energy theorem, we can obtain the effective coupling ofany number of neutral scalar Higgs boson to two gluons [31, 32], L hgg = α s π G aµν G aµν log(1 + hv ) = α s π (cid:18) hv − h v + · · · (cid:19) G aµν G aµν . (7)The first two interactions govern the cross section for di-Higgs production via the gluonfusion in the heavy top limit. From Eq.7, we can see that there is a strong cancellationbetween the triangle and box top quark loops diagrams because of the opposite signs of theeffective couplings. So if the new physics can flip the relative sign of them, the cross sectionof process gg → hh may be greatly enhanced. B. γγ → hh In Fig.2, we show the Feynman diagrams of the process γγ → hh , which is governedby W boson and top quark loop diagrams, respectively. At the ILC, the γγ collisions areobtained by the inverse Compton scattering of the incident electron- and the laser-beam,the events number is calculated by convoluting the cross section of γγ collision with the5 γ H H γ γ HHWWW γγ H HHt tt γγ H HHW WW γγ HHt tt t γγ HHW WW W γγ HHt tt t γγ HHW WWW γγ HHt tt t γγ HHW WW W γ γ H HWW γγ H HHWW
FIG. 2: Feynman diagrams of the process γγ → hh at photon collider. photon beam luminosity distribution: N γγ → hh = Z d √ s γγ d L γγ d √ s γγ ˆ σ γγ → hh ( s γγ ) ≡ L e + e − σ γγ → hh ( s ) (8)where d L γγ / d √ s γγ is the photon-beam luminosity distribution and σ γγ → hh ( s ) ( s is thesquared center-of-mass energy of e + e − collision) is defined as the effective cross section of γγ → hh , which can be written as [33] σ γγ → hh ( s ) = Z x max √ a zdz ˆ σ γγ → hh ( s γγ = z s ) Z x max z /xmax dxx F γ/e ( x ) F γ/e ( z x ) (9)where F γ/e denotes the energy spectrum of the back-scattered photon for the unpolarizedinitial electron and laser photon beams given by F γ/e ( x ) = 1 D ( ξ ) (cid:20) − x + 11 − x − xξ (1 − x ) + 4 x ξ (1 − x ) (cid:21) (10)with D ( ξ ) = (1 − ξ − ξ ) ln(1 + ξ ) + 12 + 8 ξ − ξ ) . (11)Here ξ = 4 E e E /m e ( E e is the incident electron energy and E is the initial laser photonenergy) and x = E/E with E being the energy of the scattered photon moving along theinitial electron direction. In the calculations, we fix the parameters as ξ = 4 . D ( ξ ) = 1 . x max = 0 .
83 [33].Similar to the lagrangian L hgg , the effective coupling of any number of neutral scalarHiggs boson to two photons can be given as [32, 34], L hγγ = α em π ( N c Q t −
74 ) F µν F µν log(1 + hv ) = − α em π (cid:18) hv − h v + · · · (cid:19) F µν F µν . (12)6here is also the cancellation between the triangle and box loop diagrams. But differentfrom gg → hh , the contributions to the process γγ → hh are dominated by the W bosonloops. So, the effect of the non-standard top-Higgs coupling on the di-Higgs production atphoton collider may be smaller than that at the LHC.For the loop calculations, we generate and simplify the amplitudes by using the packages FeynArts-3.9 [35] and
FormCalc-8.2 [36]. All the loop functions are numerically calculatedwith the package
LoopTools-2.8 [37].
III. NUMERICAL RESULTS AND DISCUSSIONS
In the numerical calculations, we take the input parameters of the SM as [38] m t = 173 .
07 GeV , m W = 80 . , m Z = 91 .
19 GeV ,m h = 125 . , sin θ W = 0 . , α ( m Z ) − = 127 . . (13)For the strong coupling constant α s ( µ ), we use its 2-loop evolution with QCD parameterΛ n f =5 = 226 MeV and get α s ( m Z ) = 0 . gg → hh [39]. The renormalization scale µ R and factorizationscale µ F are chosen to be µ R = µ F = m h . We numerically checked that all the UV diver-gences in the loop corrections canceled. Since the cross section of di-Higgs production isdetermined by the phase angle θ and the coupling y t , we will firstly assume y t = y t SM andtake θ = 0 , π/ , π for example to illustrate the effects of different phase angles on the di-Higgs production at the LHC and ILC-based photon collider in Fig.3 and Fig.5, respectively.Then, we will vary both of y t and θ and respectively present the ratios of σ gg → hh /σ gg → hhSM and σ γγ → hh /σ γγ → hhSM under the constraint of the Higgs data in Fig.4 and Fig.6. Here it shouldbe noted that when θ = 0, the coupling y t with the SM value can be potentially dangerousbecause such a value may violate perturbative expansion of effective field theory and/orbe inconsistent with the current LHC limit on the scale of new physics. So, in that case, y t < y t SM is usually needed to satisfy the theoretical and experimental bounds, which canbe seen from Fig.4 and Fig.6. For example, when θ = π/ y t /y t SM should be within therange 0 . − .
6. 7 . gg → hh .
10 10010 . . . .
97 37 . .
68 216 . . . . . . . . ( f b ) s (TeV) LHC
FIG. 3: Cross sections of the process gg → hh for non-standard top-Higgs couplings with y t = y t SM and θ = 0 , π/ , π at the LHC with √ s = 8 , , , ,
100 TeV.
In Fig.3, we present the impact of the non-standard top-Higgs couplings with y t = y t SM and θ = 0 , π/ , π on the cross section of the di-Higgs production at the LHC with √ s =8 , , , ,
100 TeV. From Fig.3, we can find that the top-Higgs coupling with an inversesign y t = − y t SM can significantly enhance the di-Higgs cross sections from 24.52 fb to 88.61fb, while the pseudo-scalar top-Higgs coupling can moderately increase the cross section upto 37.58 fb at √ s = 14 TeV. The reason is that these non-standard top-Higgs interactionscan change the interference behavior of the triangle and box top loop diagrams in theprocess gg → hh . To be specific, the amplitudes of these two kinds of Feynman diagramsfor √ s ≫ m t , m h in the SM can be approximately written as, M tbox ∼ y t α s m t v , (14) M ttriangle ∼ − y t λ hhh α s m t v m h ˆ s (cid:20) log (cid:18) m t ˆ s (cid:19) + iπ (cid:21) . (15)where we take the SM Higgs self-coupling λ hhh = 3 m h /v in our study. For our cases, (i)when θ = π , the top-Higgs coupling y t SM becomes − y t SM so that the sign of M triangle issame as M box ; (ii) when θ = π/
2, the SM coupling y t SM changes to iy t SM . So there is nointerference between the M triangle and M box , which will provide a constructive contributionto gg → hh . 8esides, we can see that the cross section of gg → hh becomes larger with the increaseof √ s and can reach about 1 pb in the SM at √ s = 100 TeV, which is about 40 times largerthan the one at √ s = 14 TeV. However, it should be noted that the amplitudes of trianglediagrams is suppressed by center of mass energy ˆ s . The box diagrams will dominate thecontribution to gg → hh at the VLHC. This means that the extraction of the Higgs self-coulping from the measurement of the total cross section of gg → hh will strongly dependon the assumption of the top-Higgs coupling at the VLHC. In this case, a study of thekinematic distributions of the Higgs bosons is needed to identify the sources of new physicsin di-Higgs production. -1.0 -0.5 0.0 0.5 1.0 1.5-1.0-0.50.00.51.0 1.0 3.03.0 gg->hh / gg->hhsm c t c t ~ SM FIG. 4: Ratios of σ gg → hh /σ gg → hhSM at 14 TeV in the plane of ˜ c t − c t , where the dashed contourscorrespond to the 68% C.L. and 95% C.L. limits given by the current Higgs data fitting. In Fig.4, we plot the ratios of σ gg → hh /σ gg → hhSM at 14 TeV in the plane of ˜ c t − c t , where thedashed contours correspond to the 68% C.L. and 95% C.L. limits given by the current Higgsdata fitting. From Fig.4, we can see that the positive reduced scalar couplings c t > . c t > . σ gg → hh /σ gg → hhSM canonly reach about 3 in the 95% C.L. allowed region at 14 TeV LHC. On the other hand, theprecise measurement of gg → hh will further bound these non-standard top-Higgs couplings.9 . ILC-based Photon Collider
300 400 500 600 700 800 900100010 -5 -4 -3 -2 -1 -5 -5 -3 . . . . * - -3 -5 ( f b ) s (GeV) ILC-
FIG. 5: Cross sections of γγ → hh in the presence of the non-standard top-Higgs couplings with y t = y t SM and θ = 0 , π/ , π at the ILC-based photon collider with √ s = 310 , , , , In Fig.5, we show the cross sections of γγ → hh in the presence of the non-standardtop-Higgs couplings with y t = y t SM and θ = 0 , π/ , π at the ILC-based photon collider with √ s = 310 , , , , θ = π/ θ = 0. This is differentfrom the case of gg → hh at the LHC, where only top quark propagates in the loops. Butfor the process γγ → hh , W boson loops will be involved and have an interference with thetop quark loops. To be specific, the amplitudes of the top quark and W boson box diagramswith √ s ≫ m t , m W , m h in the SM can be approximately written as, M tbox ∼ y t Q t α m t v , (16) M Wbox ∼ y W SM Q W α m W v . (17)where Q t,W is the electric charge of top quark and W boson, respectively. y W SM = gM Z / cos θ W denotes the SM Higgs gauge coupling and is fixed in our calculations. For θ = π/
2, the coefficient of the amplitude of the top quark box diagrams will be changedfrom y t SM to − y t SM . So, as comparison with the SM prediction, the relative sign between the W boson box and the top quark box will be inverted, which leads to a cancellation between10hem; For θ = π , the flipped sign of y t SM can increase the cross section of γγ → hh in twosides: one is from the enhancement of those triangle diagrams involving hγγ ; the other oneis from the constructive interference between the top quark triangle and box diagrams. Wealso note that for different non-standard top-Higgs couplings, the cross section of γγ alwaysreach the maximal value at √ s = 500 GeV. This is caused by the threshold effect of the topquark pair in the loop. When √ s becomes larger, the cross section will decrease. -1.0 -0.5 0.0 0.5 1.0 1.5-1.0-0.50.00.51.0 2.02.0 ->hh / ->hh SM c t c t ~ SM FIG. 6: Similar to Fig.4, but ratios of σ γγ → hh /σ γγ → hhSM at ILC-based photon collider with √ s = 500GeV in the plane of ˜ c t − c t . Similar to Fig.4, we plot the ratios of σ γγ → hh /σ γγ → hhSM at ILC-based photon collider with √ s = 500 GeV in the plane of ˜ c t − c t . From Fig.6, we can see that although the crosssection of σ γγ → hh can be about 13 . σ γγ → hh /σ γγ → hhSM can only reach about 2 in the region allowed by the current Higgs dataat 95% C.L.. So, given the latest analysis of the feasibility of γγ → hh → b ¯ bb ¯ b , such anenhancement effect can be observed at the future photon collider. IV. CONCLUSIONS
After LHC Run-I, measurement of Higgs self-coupling is one of the crucial tasks at futurecolliders, such as the LHC Run-II and the ILC-based photon collider. In this paper, giventhe recent Higgs data, we investigate the di-Higgs production in the presence of the non-11tandard top-Higgs coupling at the LHC and ILC-based photon collider. Due to the changedinterference behaviors of the top quark loops with itself or W boson loops, we find that thecross section of di-Higgs production at the LHC-14 TeV and ILC-500 GeV can be respectivelyenhanced up to nearly 3 and 2 times the SM predictions within 2 σ Higgs data allowedparameter region.
Acknowledgement
This work is supported by the National Natural Science Foundation of China (NNSFC)under grants Nos. 11222548, 11275057, 11305049 and 11347140, by Specialised ResearchFund for the Doctoral Program of Higher Education under Grant No. 20134104120002and by the Startup Foundation for Doctors of Henan Normal University under contractNo.11112. [1] G. Aad et al. [ATLAS Collaboration], Phys. Lett. B , 1 (2012), arXiv:1207.7214 [hep-ex].[2] S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B , 30 (2012), arXiv:1207.7235[hep-ex].[3] The ATLAS Collaboration, ATLAS-CONF-2013-012; ATLAS-CONF-2013-034.[4] The CMS Collaboration, CMS-PAS-HIG-13-001; CMS-PAS-HIG-13-005.[5] ATLAS Collaboration [ATLAS Collaboration], ATLAS-CONF-2014-011.[6] CMS Collaboration [CMS Collaboration], CMS-PAS-HIG-14-010.[7] W. J. Marciano and F. E. Paige, Phys. Rev. Lett. , 2433 (1991); J. Dai, J. F. Gunionand R. Vega, Phys. Rev. Lett. , 2699 (1993) [hep-ph/9306271]; J. Goldstein, C. S. Hill,J. Incandela, S. J. Parke, D. 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