Challenges and opportunities for heavy scalar searches in the t t ¯ channel at the LHC
FFERMILAB-PUB-16-262-T
Challenges and opportunities for heavy scalar searchesin the t ¯ t channel at the LHC Marcela Carena a,b,c and Zhen Liu a a Theoretical Physics Department, Fermi National Accelerator Laboratory,PO Box 500, Batavia, IL 60510, U.S.A. b Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, U.S.A. c Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, U.S.A.
E-mail: [email protected] , [email protected] Abstract:
Heavy scalar and pseudoscalar resonance searches through the gg → S → t ¯ t process are challenging due to the peculiar behavior of the large interference effects with thestandard model t ¯ t background. Such effects generate non-trivial lineshapes from additionalrelative phases between the signal and background amplitudes. We provide the analyticexpressions for the differential cross sections to understand the interference effects in theheavy scalar signal lineshapes. We extend our study to the case of CP-violation and furtherconsider the effect of bottom quarks in the production and decay processes. We also evaluatethe contributions from additional particles to the gluon fusion production process, such asstops and vector-like quarks, that could lead to significant changes in the behavior of thesignal lineshapes. Taking into account the large interference effects, we perform lineshapesearches at the LHC and discuss the importance of the systematic uncertainties and smearingeffects. We present projected sensitivities for two LHC performance scenarios to probe the gg → S → t ¯ t channel in various models. Keywords:
Higgs, Top, LHC a r X i v : . [ h e p - ph ] D ec ontents The discovery of the Higgs boson is a great triumph of the Standard Model (SM) and hasopened a new era in particle physics. Being the first fundamental scalar particle ever ob-served, the existence of the Higgs boson substantiates the questioning of basic concepts inparticle physics, such as the hierarchy problem, the naturalness problem and the true natureof neutrino masses. It also opens a window for possible connections to dark matter and theorigin of the matter-anti matter asymmetry. Many of these conundrums can be (partially)addressed by some of the best motivated models currently under exploration, such as super-symmetry (SUSY) [1–3], composite Higgs models [4–7], extended gauge symmetries – e.g.grand unification theories [8], and extended Higgs models such as two-Higgs doublet models(2HDM) [9]. Most of these extensions of the SM require additional scalar bosons. This posestwo basic questions: would there be additional scalar bosons at the electroweak scale?; howcan they be sought at the LHC? – 1 –t will be challenging to discover a heavy scalar at the LHC, in particular if its couplingsto electroweak gauge bosons are small compared to its couplings to third generation fermions,as occurs in many extensions of the SM. Hence, we will focus on the decays of a heavy scalarinto the t ¯ t final state. Hierarchical couplings of the heavy scalars to light quarks lead to lowproduction rates through tree-level processes and the t ¯ t final state has large backgrounds fromSM hadronic processes. In addition, as has been noticed in an earlier work [10] and recentlydiscussed in a related context [11–15], the production of a heavy Higgs boson through top-loopinduced gluon-gluon-fusion with its subsequent decay into t ¯ t has a very large interference effectwith the SM background. This large interference effect is further augmented by a non-trivialrelative phase between the signal and the SM background amplitude, leading to a complexstructure of the signal lineshape as a function of the t ¯ t invariant mass. Possible lineshapesvary from a pure bump to bump-dip, dip-bump and pure dip structures depending on thedifferent heavy scalar masses and the possible additional effects of other new particles in theloop. Authors in Ref. [11] studied the gg → S → t ¯ t channel in supersymmetric and LittleHiggs models at the LHC and considered a parton level analysis without taking into accountthe effects of smearing on the reconstructed t ¯ t invariant mass and the systematic uncertainties.More recent works [14, 15] have considered such effects on the signal total rates. In manycases, however, it is necessary to go beyond a parametrization in terms of the total rate sincethis may overlook cancelations between the peak and the dip structures after smearing. Theprevious studies triggered the interest of the community in further investigating the discoverypotential for heavy scalars in t ¯ t final states.In this work we concentrate on the unique features of the interference effects in the gg → S → t ¯ t , to investigate the feasibility of heavy scalar searches at the LHC. In Sec. 2 weprovide a detailed study in the baseline model with only top-quark loops contributing to theproduction vertex. In Sec. 3 we expand our study to consider additional effects in extensionsof the baseline model. In particular, we investigate the effects of two nearly degenerate Higgsbosons, as in 2HDMs, both for CP eigenstates and in the case of CP-violation in the Higgssector. Moreover, in Sec. 3 we also study the effects of additional particles, beyond the topquark, contributing to the production vertex. These include effects from bottom quarks thatbecome relevant in a Type II 2HDMs with sizable ratio of the two Higgs vacuum expectationvalues (tan β ), heavy colored particles such as stops in SUSY models and Vector-Like Quarks(VLQs) that naturally appear in composite Higgs scenarios. Also in Sec. 3, we present astudy to highlight the relevance of interference effects in the t ¯ t final state for a prospective750 GeV scalar that could account for the excess in the di-photon channel observed at theLHC experiments [16, 17]. In Sec. 4 we perform detailed collider analyses to investigate thereach at the 13 TeV LHC in the search for t ¯ t resonances in the presence of large interferenceeffects, emphasizing the importance of smearing effects and systematic uncertainties. Wepropose a a lineshape search at the LHC, taking into account both the excess and deficit aspart of the signal for two LHC performance scenarios. We demonstrate the physics potentialof this new search in examples of the baseline model and a 2HDM, including the possibilityof nearly degenerate bosons with and without CP-violation. We reserve Sec. 5 to summarize,– 2 –nd briefly discuss possible future directions for scalar resonance searches in the t ¯ t final state. The importance of the gg → S → t ¯ t channel well justifies a comprehensive study of all thesubtleties inherent to this signal, in particular the interference effects. In this section, weanalyze the baseline model that only takes into account the top quark contribution to thegluon fusion production process and considers the effects of one additional single heavy scalarat a time. In the following we focus on heavy neutral scalars that are not charged under the standardmodel gauge groups after electroweak symmetry breaking (color and electrically neutral). Inmany beyond the standard model extensions, the additional scalar couplings to fermions arehierarchical, according to the fermion masses. We adopt such simple set-up for the heavyscalar couplings to the SM fermion sector, which renders the production rate from q ¯ q fusionprocess small and, at the same time, makes the gluon fusion process the dominant productionmode.In addition, for example in CP-conserving 2HDMs, one can study the effects of theCP-even or CP-odd heavy Higgs bosons produced via gluon fusion and decaying into toppairs, that destructively interfere with the SM t ¯ t background. The baseline model considersonly top quark contributions to the gluon fusion production process, and this is appropriate,e.g. for a Type II 2HDM at low tan β , but could be otherwise for moderate to large valuestan β , for which the bottom loop becomes relevant. Moreover, generic 2HDMs usually assumeno additional relevant colored particles other than the standard model fermions and gaugebosons.The above consideration motivates us to write down the following interaction terms of ageneral Lagrangian for a heavy scalar after electroweak symmetry breaking: L Yukawa ⊃ y si √ ttS + i ˜ y si √ tγ tS . (2.1)The top-loop in the triangle diagram induces an effective gluon-gluon-scalar vertex. This canalso be expressed by effective interactions, L Yukawa loop − induced ======== ⇒ − g S gg (ˆ s ) G µν G µν S − i g S gg (ˆ s ) ˜ G µν G µν S, (2.2)where ˜ G µν ≡ (cid:15) µναβ G αβ . This expression is given in terms of form factors of the loop-inducedvertices that explicitly depend on ˆ s .We concentrate on the flavor diagonal Yukawa-like couplings between the heavy scalar S and the chiral fermion fields, since only these diagonal terms contribute to the loop-induced– 3 – bsoluteRealImaginary τ I ( τ ) , I ˜ ( τ )( da s hed ) s ( GeV ) for top - loop Figure 1 . Loop functions of the fermion induced gluon-gluon-scalar vertex as a function of theparameter √ τ ≡ √ ˆ s/ (2 m f ), for a CP-even scalar (solid line) and a CP-odd scalar (dashed lines),respectively. The blue, yellow and green lines correspond to the absolute value, real component andimaginary component of the loop functions, respectively. For convenience, we show the correspondingcenter of mass energy √ ˆ s in units of GeV for the case of a top quark loop on the upper edge of thefigure. Sgg couplings. The
Sgg couplings depend on the Yukawa interactions and correspondingfermion masses, g S gg (ˆ s ) = α s √ π y st m t I ( τ t ) , ˜ g S gg (ˆ s ) = α s √ π ˜ y st m t ˜ I ( τ t ) , (2.3)where I ( τ t ) and ˜ I ( τ t ) are the corresponding loop-functions and τ t = ˆ s m t , f ( τ ) = arcsin ( √ τ ) for τ ≤ , − (cid:18) log √ − /τ − √ − /τ − iπ (cid:19) for τ > I / ( τ ) = 1 τ ( τ + ( τ − f ( τ )) , ˜ I / ( τ ) = f ( τ ) τ . (2.4)In the above, y st is the Yukawa coupling of the heavy scalar to the top quark, whose mass isdenoted by m t .In Fig. 1 we show the numerical values of the loop functions. For convenience, we alsolabel the upper edge of the x -axis in the figure with the corresponding center of mass energy √ ˆ s for the case of a top quark loop. Although we are writing these effective form factors Alternatively, these more conventional loop-functions can be written in terms of kinematic variable β asshown and discussed in the Appendix. The kinematic factor β of the final state top quarks is defined as (cid:113) − m t ˆ s . This kinematic factor β is unrelated to tan β ≡ v /v , the ratio of the vacuum expectation valuesof the two Higgs doublets, to be used later on in this paper. – 4 –onsidering only the top quark in the loop, they can be generalized for other fermions byreplacing y st and m t by y sf and m f in Eqs. 2.3 and 2.4. In Fig. 1 one observes a clear jump inthe behavior of the values of the loop functions when √ ˆ s (cid:39) m f , associated with the thresholdeffect from the on-shell top pairs. For the region far below the threshold, τ f ≡ √ ˆ s/ (2 m f ) (cid:28) τ (cid:28) v (cid:39)
246 GeV – the Vacuum Expectation Value (VEV) of the SMHiggs – y f √ m f = v . Considering the case of the SM Higgs, we observe that each generationof heavy chiral fermions, will contribute to the Higgs-gluon coupling, Eq. (2.3), like g h gg (ˆ s ) = α s πv I ( τ f ) ≈ α s πv (1 + 730 τ f + 221 τ f + O ( τ f )) . (2.6)Neglecting corrections of higher order in τ f , each chiral fermion generation contributes thesame amount α s πv to the SM Higgs-gluon coupling.Just after crossing the fermion pair threshold, τ ≥
1, the imaginary part of the loopfunctions (as shown in Fig. 1) rises quickly, and then decreases slowly for increasing values of τ . The real part, instead, decreases monotonically slightly above the fermion pair thresholdand flips its sign for sufficiently large τ . This implies that the phase of the loop functionrapidly grows after crossing the threshold and remains large (of order π/
2) for any value of √ τ > ∼
2. This special behavior drives the unconventional BSM phenomenology discuss in thispaper and we will come back to this in more detail later on.In Fig. 2 we illustrate three components of the lineshapes for the scalar signal, namelythe Breit-Wigner piece (blue, dotted line), the interference piece proportional to the real com-ponent of the scalar propagator (orange, dashed line) and the interference piece proportionalto the imaginary component of the scalar propagator (green, solid line). To understandthe interference effects in a more explicit way, we can parameterize the scalar propagator,conveniently normalized by a factor ˆ s , as:ˆ s (ˆ s − m S ) + i Γ S m S ≈ m S Γ S − i + 1 (2.7)with ∆ ≡ ˆ s − m S m S Γ S ≈ √ ˆ s − m S Γ S for ˆ sm S − (cid:28) . In the above, ∆ basically parameterizes the deviation of the center mass energy √ ˆ s from thescalar mass m S in units of the scalar width Γ S . The denominator of the propagator in theabove equation is positive definite and increases as the deviation | ∆ | increases. This provides For completeness, the expansion for a pseudoscalar at low τ f follows,˜ g a gg (ˆ s ) = α s πv ˜ I ( τ f ) ≈ α s πv (1 + 13 τ f + 845 τ f + O ( τ f )) . (2.5) – 5 – . W .Re. Int.Im. Int.Schematic Lineshape m S s Figure 2 . The schematic lineshapes of the components of the signal, namely, the Breit-Wignerresonance (blue, dotted line), the interference with background proportional to the real componentof the propagator (orange, dashed line), and the interference with background proportional to theimaginary component of the propagator (green, solid line) as a function of the center of mass energy √ ˆ s . an arc-type profile around values of √ ˆ s close to the scalar mass, since the denominator isminimized for ∆ = 0. After squaring and with small modifications from the numerator,this generates the Breit-Wigner lineshape as shown by the blue, dotted line in Fig. 2. Thereal part of the numerator, 2∆, flips its sign when crossing the scalar mass pole, while theimaginary part of the numerator remains negative. Multiplying the numerator by the arc-type profile of the denominator, this leads in Fig. 2 to the lineshapes schematically shown as adip-bump (orange, dashed line) and a dip (green, solid line) for the real and imaginary parts,respectively. The contributions to the signal lineshapes from the real and imaginary parts ofthe propagator can be further modified by the detailed dynamics of the underlying physics. Inparticular if the overall sign is flipped, these lineshapes will change into a bump-dip structureor a pure bump, instead.In standard analyses of tree-level BSM particle resonant production and decays, the BSMamplitudes are real up to an imaginary contribution from the propagator. Given that the SMbackgrounds are real as well, the only part of the propagator that survives is the real one.Moreover, the real part of the propagator is odd around the resonance mass – as illustratedby the orange, dashed line in Fig. 2 – implying that the interference effect does not contributeto the total signal rate. If the BSM amplitude acquires an imaginary piece in addition tothe imaginary part of the propagator, e.g., from loop functions, a new interference piece willemerge. This new interference contribution is even around the resonance mass –as illustratedby the green, solid line in Fig. 2 – and does change the total signal rate. The relevance ofthis interference contribution does not depend on the precise magnitude of the width of theresonance.The signal amplitudes for the specific case of gg → S → t ¯ t , both for a CP-even and– 6 – ( τ ) I ˜ ( τ ) τ A r g ( I ( τ )) / π s ( GeV ) for top - loop Figure 3 . The phase (argument) of the loop functions in units of π as a function of τ for a scalar(red line) and a pseudoscalar (blue, dashed line), respectively. We label the upper edge of the x-axiswith the corresponding center of mass energy √ ˆ s in GeV for the case of a top quark loop. CP-odd heavy scalar S , are proportional to: A even ∝ y t g S gg = y t I ( τ t ) , A odd ∝ ˜ y t ˜ g S gg = ˜ y t ˜ I ( τ t ) , (2.8)where we have omitted the scalar propagator, color factor and strong coupling constantdependence for simplicity. We can then define the phase of the resonant signal amplitude interms of the reduced amplitude ¯ A and the normalized propagator as, A = ˆ s ˆ s − m S + i Γ S m S | ¯ A| e iθ ¯ A , with θ ¯ A ≡ arg( ¯ A ) . (2.9)When θ ¯ A is 0 (or π ), only the real part of the propagator contributes to the interference termyielding a dip-bump (or bump-dip) structure. This is the standard case mostly studied in theliterature, that does not affect the total signal rate. When θ ¯ A is π/ π/ gg → S → t ¯ t in consideration, the loop functions ( I ( τ t ) and ˜ I ( τ t )) are theonly sources of the additional phase θ ¯ A ( θ ¯ A = arg I ( τ ) or θ ¯ A = arg ˜ I ( τ )). We show in Fig. 3the phase of the fermion loop functions both for the scalar (red line) and pseudoscalar (blue,dashed line) cases. These phases follow the numerical values of the loop functions discussed inFig. 1, and they will be useful in analyzing the signal lineshapes later on. Similarly to Fig. 1, For simplicity of notation, from here on we drop the superscript S from the top Yukawa couplings to heavyscalars. The background amplitude is defined to be positive, as one can always rotate the phase of the signal andbackground amplitudes simultaneously without changing the physical results. This uniquely fixes the definitionof the phase θ ¯ A . – 7 –e label the upper edge of the x-axis with the corresponding center of mass energy √ ˆ s in GeVfor the case of a top quark loop. Throughout the whole τ range, the phase for the pseudoscalaris larger than that of the scalar. A phase of π/ π/ After analyzing the generic features of different lineshape contributions in the previous sec-tion, we now concentrate on the baseline model. The background amplitude from QCD t ¯ t production is much larger in magnitude than the baseline signal amplitude. As a result, theinterference terms often are larger in size and more important than the BSM Breit-Wignerterm. Furthermore, as discussed in the previous section, the phase generated by the loopfunction grows rapidly after crossing the threshold. This phase enhances the interferencecontribution proportional to the imaginary part of the scalar propagator, rendering it muchlarger than that proportional to the real part. Although the sign of the interference is notfixed in the general case, the baseline model ensures this interference contribution to bedestructive. Three factors are important here. Firstly, the loop function rapidly becomes(positive) imaginary after crossing the t ¯ t threshold. Secondly, the propagators near the res-onance have a constant (negative) imaginary part. Thirdly, there is an overall minus signfrom the fermion-loop in the signal amplitude relative to the background. These three factorslead to the overall negative sign of the signal amplitude near the resonance relative to thebackground amplitude, generating the destructive interference. This feature makes the searchfor heavy Higgs bosons in this channel rather unconventional and challenging.Specifically, the partonic cross sections for the signals for the CP-even scalars read,ˆ σ evenBSM (ˆ s ; y t )( gg → S → t ¯ t ) = ˆ σ evenB . W . (ˆ s ; y t ) + ˆ σ evenInt . (ˆ s ; y t ) d ˆ σ evenB . W . (ˆ s ; y t ) dz = 3 α s ˆ s π v β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y t I ( τ t )ˆ s − m S + im S Γ S (ˆ s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ˆ σ evenInt . (ˆ s ; y t ) dz = − α s π β − β z Re (cid:34) y t I ( τ t )ˆ s − m S + im S Γ S (ˆ s ) (cid:35) , (2.10)– 8 –hile for the CP-odd scalars are,ˆ σ oddBSM (ˆ s ; ˜ y t )( gg → S → t ¯ t ) = ˆ σ oddB . W . (ˆ s ; ˜ y t ) + ˆ σ oddInt . (ˆ s ; ˜ y t ) d ˆ σ oddB . W . (ˆ s ; ˜ y t ) dz = 3 α s ˆ s π v β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ y t ˜ I ( τ t )ˆ s − m S + im S Γ S (ˆ s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ˆ σ oddInt . (ˆ s ; ˜ y t ) dz = − α s π β − β z Re (cid:34) ˜ y t ˜ I ( τ t )ˆ s − m S + im S Γ S (ˆ s ) (cid:35) , (2.11)where Γ S (ˆ s ) is the energy dependent width for the scalar, detailed in the Appendix in Eq. A.2,and the variable z is the cosine of the scattering angle between an incoming parton and thetop quark. The leading-order expression for the background partonic cross sections from gg → t ¯ t and q ¯ q → t ¯ t are outlined in the Appendix in Eq. A.3. For collider analyses withdetector acceptance, not the full phase space of z can be used equally, we thus provide thedifferential distribution. However, as the top quark is not very boosted and even forward oneswith z = ± z over the range of [ − ,
1] for oursimplified analysis throughout this paper. In all expressions the factors y t I ( τ t ) and ˜ y t ˜ I ( τ t )are basically the dynamical part of the reduced amplitudes A even , odd in Eq. 2.8, written hereexplicitly for direct connection with the phase θ ¯ A from the loop functions. For generalizedcases with additional contributions, the reduced amplitudes are more useful. The superscriptseven and odd refer to the CP properties of the heavy scalar.For a single heavy scalar being non-CP eigenstate, e.g., coupling to top quarks as y t + i ˜ y t ,the resulting parton level cross sections are given by,ˆ σ CP V
BSM (ˆ s ; y t , ˜ y t )( gg → S → t ¯ t ) = ˆ σ CPVB . W . (ˆ s ; y t , ˜ y t ) + ˆ σ CPVInt . (ˆ s ; y t , ˜ y t ) (2.12) d ˆ σ CPVB . W . (ˆ s ; y t , ˜ y t ) dz = 3 α s ˆ s π v β ( y t | I ( τ t ) | + ˜ y t | ˜ I ( τ t ) | )( β y t + ˜ y t ) (cid:12)(cid:12)(cid:12)(cid:12) s − m S + im S Γ S (ˆ s ) (cid:12)(cid:12)(cid:12)(cid:12) ˆ σ CPVInt . (ˆ s ; y t , ˜ y t ) = σ evenInt . (ˆ s ; y t )( gg → S → t ¯ t ) + σ oddInt . (ˆ s ; ˜ y t )( gg → S → t ¯ t ) , where the even and odd interference pieces follow Eqs. 2.10 and 2.11, respectively. TheBreit-Wigner component receives a contribution proportional to y t ˜ y t as a result of CPV.With CP-violation [9, 22, 23] in the heavy scalar-top sector, the coupling between the scalar S and the top quarks can be expressed as, y t + i ˜ y t = | Y t | (cos θ CP + i sin θ CP ) . (2.13)The maximal CP-violation (CPV max ) in this sector is for θ CP = π/ gg → S → t ¯ t as a functionof the t ¯ t invariant mass, m tt = √ ˆ s , for y t = 1. The width of the heavy scalar in this modelvaries from 3 GeV to 48 GeV (12 GeV to 55 GeV) for a 400 GeV and a 1 TeV CP-even (CP-odd) scalar, respectively. Throughout this paper, we use NNPDF3.0LO [24] for the parton– 9 –
00 500 600 700 800 900 1000 - - - - m tt ( Gev ) d σ s i g / d m tt ( pb / G e V )
13 TeV LHC B . W . + InterferenceCP Even × × × × × × ×
400 500 600 700 800 900 1000 - m tt ( Gev ) d σ s i g / d m tt ( pb / G e V )
13 TeV LHC B . W . + InterferenceCP Odd × × × × × × × Figure 4 . Differential cross sections of heavy CP-even (left panel) and CP-odd (right panel) scalarsignals as a function of the t ¯ t invariant mass at the 13 TeV LHC. Each signal is for a specific valueof the scalar mass and includes both the contribution from the Breit-Wigner lineshape as well as thecontributions from the signal-background interference. The vertical grid lines indicate the location ofthe each of the heavy scalar masses, which range from 400 to 1000 GeV in steps of 100 GeV. Thesignal lineshapes are multiplied by factors indicated in the lower part of the figure to render the signallineshapes visible on a same scale. The solid and dashed gray lines represent a systematic uncertaintyof the background at the ±
2% level and a statistical uncertainty evaluated at 300 fb − assuming a10% selection efficiency, respectively. distribution functions and set the factorization scale to be the same as the t ¯ t invariant mass.We show the CP-even and CP-odd scalar lineshapes at LHC 13 TeV in the left panel and rightpanel, respectively. To make the lineshapes for different masses visible, we multiply the signallineshapes by various factors, indicated in the lower part of both panels. We further show thestatistical uncertainty at 300 fb − with 10% selection efficiency and systematic uncertaintiesof ±
2% of the SM background in dashed and solid gray lines, respectively. Both uncertaintiesinclude the QCD background from gg → t ¯ t and q ¯ q → t ¯ t .From Fig. 4 it follows that for the t ¯ t invariant mass above ∼
500 GeV( ∼
400 GeV), theinterference effects are dominant for the CP-even scalar (CP-odd scalar), as indicated by thesize of deviation from the Breit-Wigner lineshape. The loop function behaviors shown in Fig. 1and Fig. 3 determine the lineshape structures. For increasing values of the t ¯ t invariant mass,the imaginary component of the loop functions grows with respect to its real component,inducing a larger phase θ ¯ A . This behavior of the imaginary part explains the increasinglypronounced dip structure in the lineshapes for larger values of the m t ¯ t . Furthermore, the θ ¯ A phase grows faster for the pseudoscalar than the scalar case, yielding the lineshape puredip structure for smaller values of the scalar mass in the former case. Another importantfeature is the off-shell interference effect, and especially for an off-shell heavy scalar at t ¯ t invariant mass around 400 GeV this effect is quite visible. This off-shell interference is moreprominent for the pseudoscalar because of the s -wave nature of the cross section, comparedto the p -wave ( β suppressed) nature of the scalar case, and is further augmented by theslightly larger width of the pseudoscalar. – 10 – P - even θ CP = π / - odd
450 500 550 600 650 - - - m tt ( Gev ) d σ s i g / d m tt ( f b / G e V )
13 TeV LHC B . W . + Interference CP - even θ CP = π / - odd
750 800 850 900 950 - - - m tt ( Gev ) d σ s i g / d m tt ( f b / G e V )
13 TeV LHC B . W . + Interference
Figure 5 . The signal lineshapes as the sum of the Breit-Wigner contribution and the interferencecontributions for the baseline model as a function of the t ¯ t invariant mass at the 13 TeV LHC. The blue,orange dotted and green dashed lines indicate the total BSM lineshapes for CP phases of 0 (CP-even),1/4 π (CPV max ) and 1/2 π (CP-odd), respectively. The gray curves are the Breit-Wigner contributionsto the total lineshapes alone, with the corresponding CP phases. The heavy scalar masses are set at550 GeV and 850 GeV for the left and right panels, respectively. We show in Fig. 5 detailed lineshapes for two representative scalar masses of 550 GeVand 850 GeV. For a 550 GeV CP-even scalar, the phase θ ¯ A is approximately π/ π/
8, as can be read from Fig. 3. For an850 GeV CP-odd scalar, instead, the phase θ ¯ A is approximately π/ π/
5. These two benchmarks highlight the cases ofthe baseline model for which i) the interferences proportional to the real and imaginary partof the propagator are comparable in size (left panel) and ii) the interferences are dominantlyfrom the piece proportional to the imaginary part of the propagator, resulting in a pure dipstructure (right panel).In Fig. 5 the blue, solid lines; green, dashed lines and orange, dotted lines are the totallineshapes for a CP-even scalar; a CP-odd scalar and a scalar in the CPV max ( θ CP = π/
4) case,respectively. These colored lines are the total BSM effects, including both the Breit-Wignercontribution and the interference with the SM background for a scalar-top quark coupling y t = 1. The corresponding the Breit-Wigner contributions alone are shown by the gray lines.For the 550 GeV scalars, the Breit-Wigner contribution is narrower for the CP-even scalarthan for the CP-odd one, due to the β suppression in the former case. For the 850 GeVscalars, the β suppression is negligible, resulting in almost identical widths for the CP-evenand CP-odd scalars. In addition, as shown in Fig. 1, the absolute value of the loop functionfor the CP-even scalar is smaller than the CP-odd one. Consequently, the CP-odd scalarBreit-Wigner lineshapes are higher than the CP-even ones. For both benchmark masses thetotal lineshapes given by the colored curves show a more pronounce dip structure for theCP-odd case than for the CP-even one. The growth and the larger phase θ ¯ A of the CP-oddloop function discussed in the previous section generates this feature. For the CPV case, the– 11 –ineshapes can be viewed as a properly weighted combination of the CP-even and CP-oddlineshapes, following Eq. 2.12. The channel gg → S → t ¯ t at hadron colliders is crucial for heavy Higgs searches, especiallyin the alignment limit [25] (with or without decoupling) favored by current Higgs bosonmeasurements at the LHC. Gluon-gluon-fusion is the dominant production mode of the heavyscalar and t ¯ t is likely to be the dominant decay mode.The baseline model introduced in the previous section helps us to understand the chal-lenges of the gg → S → t ¯ t search. However, general BSM models usually contain moreingredients, adding new features to the baseline case. Firstly, there could be more than oneheavy scalar particle, as in 2HDMs. If their masses are almost degenerate, as for example inthe MSSM, these scalars will provide new contributions to the signal. Secondly, in additionto the top quark, one can consider the effects of other colored fermions or scalars contribut-ing to the gluon-gluon-scalar vertex. This could importantly modify the phase θ ¯ A in severaldifferent ways. Specifically, there could be effects from loops involving bottom quarks and/oradditional BSM colored particles, such as squarks and VLQs. There could also be CPV ef-fects due to the direct couplings between the heavy scalar and SM fermions as well as otherparticles in the loop. These modifications allow for partial cancellations or enhancementsamong the different components of the gluon-gluon-scalar vertex. We shall discuss all thesepossibilities in the following sections. In this section we study the case of two neutral heavy Higgs bosons with similar masses, asituation that occurs in various models. In a 2HDM, large splittings between these scalarbosons are disfavored by low energy measurements such as the oblique parameters [34]. Inthe Minimal-Supersymmetric-Standard-Model (MSSM), in particular, the heavy Higgs bosons(
H, A, H ± ) are nearly degenerate because of the specific supersymmetric structure of thequartic couplings. Even after radiative corrections, the mass difference between the heavyCP-even and CP-odd scalars in the MSSM is at most of a few tens of GeV for heavy scalarmasses in the 500-1000 GeV range.In the CP-conserving case, the CP-even and CP-odd Higgs bosons do not interfere andthe resulting partonic cross section is simply given as the sum of both, σ BSM (ˆ s )( gg → H/A → t ¯ t ) = σ evenBSM (ˆ s )( gg → H → t ¯ t ) + σ oddBSM (ˆ s )( gg → A → t ¯ t ) , (3.1)where the terms in the above expression are given in Eqs. 2.10 and 2.11, with proper replace-ment of the coupling strengths. On the other hand, the results becomes slightly more complex Some alternative channel have been proposed and studied [15, 26–31], for gauge extensions, see e.g. Ref [32,33]. – 12 –nd interesting if the actual scalar mass eigenstates contain an admixture of CP-even and CP-odd components. In terms of the mass eigenstates S and S , the cross section reads, σ CPVBSM (ˆ s )( gg → S , S → t ¯ t ) = σ CPVBSM (ˆ s )( gg → S → t ¯ t ) + σ CPVBSM (ˆ s )( gg → S → t ¯ t )+ σ S − S Int . (ˆ s )( gg → S , S → t ¯ t ) , (3.2)where the cross sections for S and S follow the expressions for CPV scalars given in Eq. 2.12,whereas the additional interference term between the scalars S and S is given by, dσ S − S Int . (ˆ s )( gg → S , S → t ¯ t ) dz = 3 α s ˆ s π v (3.3)Re ( y S t y S t | I ( τ t ) | + ˜ y S t ˜ y S t | ˜ I ( τ t ) | )( β y S t y S t + ˜ y S t ˜ y S t )(ˆ s − m S + im S Γ S (ˆ s ))(ˆ s − m S − im S Γ S (ˆ s )) . The coefficient in the above equation can be further simplified in the alignment limit of aType II 2HDM, ( y S t y S t | I ( τ t ) | + ˜ y S t ˜ y S t | ˜ I ( τ t ) | )( β y S t y S t + ˜ y S t ˜ y S t ) =sin θ CP (cid:18) y SM t tan β (cid:19) ( | ˜ I ( τ t ) | − | I ( τ t ) | )(1 − β ) . (3.4)The corresponding CP-violating couplings in the alignment limit satisfy, y S t + i ˜ y S t = − y SM t tan β (cos θ CP + i sin θ CP ) ,y S t + i ˜ y S t = − y SM t tan β ( − sin θ CP + i cos θ CP ) . (3.5)From Eq. 3.4, it is clear through its dependence on sin θ CP that the interference piece be-tween the two scalars is only relevant in the presence of CPV. Moreover, due to the propagatorsuppression, this contribution is sizable for almost degenerate masses and mostly in the regionbetween the two scalar masses.The t ¯ t signal from the decay of two nearly degenerate scalars allows for a rich phenomenol-ogy. The resulting lineshape now depends on the masses, the separation between the massvalues, the widths, and the CPV phase of the scalars. In Fig. 6, we show the total signallineshapes for the two nearly degenerate scalars, both for the CP-conserving (blue, dashedlines) and the maximally CP-violating (orange, solid lines) cases. We consider scalars massesof 540 GeV and 560 GeV for the left panel, and 840 GeV and 860 GeV for the right panel,where we take the CP-odd scalar A to be 20 GeV heavier than the CP-even scalar H . Thegreen, dotted lines single out the effect of the additional interference term between the scalars.To make this new interference term easily visible in the figure, we multiplied it by a factor often. – 13 – P ConservingCPV θ CP = π / S - S Int. θ CP = π /
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Figure 6 . The signal lineshapes as the sum of the Breit-Wigner contribution and the interferencecontributions for nearly degenerate heavy scalars as a function of the t ¯ t invariant mass at the 13 TeVLHC. The orange, solid and blue, dashed lines correspond to lineshapes of the CP-violation case with θ CP π/ The main features of the two nearly degenerate heavy scalars yielding a t ¯ t signal are: i) thesignals of the two heavy scalars add to each other, almost “doubling” the height of the bumpsand dips; ii) a new contribution from the S and S signal amplitude interference appears inthe CPV case. In the left panel of Fig. 6, the mass separation between the two scalar massesis somewhat larger than their respective widths and a “double dip” structure for the nearlydegenerate scalars at around 550 GeV appears. In the right panel, we consider scalar massesaround 850 GeV and again a mass separation of 20 GeV. In this case the widths of the twoscalars are larger than the mass separation and a single, centrally flat, dip region appears,instead of the previous “double dip”. The CPV lineshapes differ from the CP-conservingones, and in particular they receive the contribution from the new interference term betweenthe two scalars. From Fig. 6 we observe that the new interference term is mainly in the regionbetween the two scalar masses, and this is easily understood due to the kinematic suppressionfrom the two scalar propagators. Moreover, this new interference term is proportional to thereal component of the product of the two scalar propagators, approximately, 1 + 4∆ ∆ ,where ∆ , are the mass differences between the t ¯ t system and the pole masses of each of thetwo scalars, S , , respectively. The product ∆ ∆ is negative whenever √ ˆ s is between the twoscalar masses and positive otherwise. Moreover, when the mass splitting of the two scalarsis smaller than the average of their widths, ∆ ∆ is a small negative quantity, which is notsufficient to flip the sign of the interference term. As a result, the new interference term ispositive for both examples. Furthermore, in the benchmark model shown in the right panel ofthis figure, the CPV case has a deeper overall dip structure, which may open the possibilityof differentiating CPV from CP-conserving scenarios in future high precision measurements. Here ∆ , is defined analogously to ∆ in Eq. 2.7 – 14 – .2 Scenarios with additional contributions to the gluon-fusion process Models with heavy scalar bosons often occur in association with additional colored particlesyielding new contributions to the loop-induced gluon-gluon-scalar vertex. In addition bottomquark effects, not taken into account in the baseline model, may also contribute in specificregions of parameter space.Before proceeding with a detailed discussion of lineshapes, let us comment on some es-sential differences between new particle contributions to the SM Higgs boson gluon fusionproduction with respect to the same production mode for heavy scalars. For the SM Higgsboson, one is entitled to make use of the low energy theorem to include the effects of heavyBSM particle contributions to loop-induced couplings. In such case one can add the newphysics loops directly to the SM top quark loop, since around the SM Higgs boson mass allthese loop-functions are below the thresholds of the heavy particles, and therefore real. Forheavy scalars, instead, the top quark loop-function is no longer real, and the heavy BSMparticle contributions could have various phases depending on the kinematics. Consequently,a relative phase will be generated between the SM fermion contributions and the BSM par-ticle contributions. This effect could lead to drastic changes in the lineshapes for the heavyscalar and demands a careful treatment of the inclusion of BSM effects in the heavy scalarproduction.In the following we discuss several well-motivated scenarios with additional colored par-ticle effects. We focus on heavy scalar lineshapes considering the new contributions fromfermions and scalars that arise in general 2HDMs as well as in models with VLQs or SUSYmodels with squarks.
In the framework of 2HDMs, it is interesting to revisit the relevance of top quark-loops inthe heavy Higgs-gluon fusion production process. The complete 2HDM is only defined afterconsidering the interaction of the Higgs fields to fermions. In a Type I 2HDM, all SM fermionscouple to a single Higgs field and hence the bottom quark-loop scales in the same way as thetop quark-loop. Therefore the dominant contributions will always come from the top-loopand the subsequent t ¯ t decay, regardless of the tan β value. Consequently, the bottom quarkcontribution is merely a small correction to the phase of the gluon-gluon-scalar vertex and willminimally perturbe our previous discussions. In a type II 2HDM, instead, the contributionfrom bottom quark-loops can be sizable for moderate to large values of tan β , and it isalso directly correlated with the additional partial decay width into b ¯ b . More specifically,the heavy Higgs-bottom Yukawa coupling, and hence the bottom quark-loop contribution,scales as tan β , while the top quark one scales as 1 / tan β . The interplay between these twocompeting contributions leads to a rich phenomenology. In fact, in the large tan β regime,where bottom-loop induced gluon-gluon-fusion production and b ¯ b decay are dominant, thesearch strategy changes, and alternative channels such as those with τ + τ − final states becomemore sensitive. Still in the low to intermediate tan β regime it is of interest to explore the– 15 – an β = β = β = β = β =
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Figure 7 . Signal lineshapes as the sum of the Breit-Wigner contribution and the interference con-tributions in Type II 2HDMs as a function of the t ¯ t invariant mass at the 13 TeV LHC, for variousvalues of tan β , and including the bottom quark contributions. The left and right panels correspondto a heavy CP-even scalar with mass 550 GeV and a CP-odd scalar with mass 850 GeV, respectively. gg → S → t ¯ t channel and consider the effects of the bottom quarks. Due to kinematics, thebottom-loop induced ggS coupling will be in the large τ b regime of Eq. 2.4, leading to veryslowly varying loop functions I / ( τ b ) and ˜ I / ( τ b ). The bottom quark- and top quark-loopcontributions could then interfere constructively or destructively, depending on the relativesign between the two corresponding Yukawa couplings to the heavy scalars.In the following, for simplicity, we only consider the CP conserving Type II 2HDM inthe alignment limit. The tan β enhanced bottom quark contribution to the gluon-fusionproduction of the 125 GeV Higgs boson can be tuned away in the alignment or decouplinglimit, therefore avoiding the corresponding precision measurement constraints. The CPVcase can be considered in a similar way as the CPV discussion in Sec. 2.2. Including thecontributions from both top and bottom quarks, the gluon-gluon-scalar interaction for theCP-even Higgs boson from Eq. 2.3 now reads, g Sgg (ˆ s ) = α s πv (cid:18) − β I ( τ t ) + tan βI ( τ b ) (cid:19) , (3.6)and analogously for the CP-odd Higgs.In Fig. 7 we show two benchmark scenarios for a CP-conserving type II 2HDM, one fora CP-even scalar of mass 550 GeV (left panel) and the other for a CP-odd scalar of mass850 GeV (right panel), while considering various values of tan β . From Fig. 5, it follows thatchanging the CP-properties of the scalar for a similar mass window results in similar lineshapesas those shown in each of the corresponding panels of Fig. 7. We choose to vary tan β between0.5 to 7, where 0.5 yields an enhanced scalar top-quark coupling and 7 represents the casewhere the top- and bottom- quark loop induced gluon-gluon-scalar couplings are minimized.Beyond tan β = 7, the t ¯ t decay will be substantially suppressed due to the large couplings ofthe scalar to bottom quarks. The lineshapes in this figure include both the Breit-Wigner andinterference terms for both the bottom- and top-quark contributions to the loop function.– 16 –or the tan β range considered, a lower value of tan β indicates a larger width and a largersignal cross section. From Fig. 7 we observe that the resulting signal phase changes morevisibly with respect to the SM background for a lighter Higgs boson. This can be understoodbecause for heavier scalars the kinematics is such that the phases of the top and bottom-quark contributions are closer to the asymptotic behavior for large values of τ t,b , as shownin Fig. 3. Such feature is unique to light quark contributions to the loop function. Heavyparticles, instead, will only contribute to the real component of the loop-function. Finally,it is also interesting to notice that the height of the peaks does not change much for thetan β regime under consideration. In this regime the height of the peak has two contributingfactors that cancel each other: the on-resonance amplitude is proportional to 1 / Γ from thepropagator and the production rate is proportional to Γ t , which in turn dominates the totalwidth Γ. For higher values of tan β than those considered in this paper, the height will befurther suppressed by the increasing contribution of Γ b to the total width. Vector-like quarks are well motivated in many BSM theories, e. g. composite Higgs mod-els [35–37], flavor models, grand unified theories. The heavy scalar effective couplings togluons can receive sizable contributions from these vector-like quarks, resulting in impor-tant changes to the phenomenology. We shall discuss some of the most relevant featuresin this section by considering the minimal case of one vector-like SU (2) L quark doublet, Q L = ( ψ L N L ) T and Q R = ( ψ R N R ) T , and one vector-like SU (2) L quark singlet, χ R and χ L , respectively. In the context of 2HDMs, the heavy scalar couplings to vector-like quarksare linked to their chiral masses.The vector-like fermion mass matrix, after electroweak symmetry breaking, can be ex-pressed as, (cid:0) ¯ ψ L , ¯ χ L (cid:1) M Ψ (cid:32) ψ R χ R (cid:33) = (cid:0) ¯ ψ L , ¯ χ L (cid:1) (cid:32) M ψ y Ψ v √ y Ψ v √ M χ (cid:33) (cid:32) ψ R χ R (cid:33) , (3.7)where for simplicity we assume the off-diagonal entries to be identical. The subscript L and R always label chirality. The mixing angle, defined for the mass eigenstates of Dirac spinorsΨ and Ψ , follows Ψ = (cid:32) Ψ ,L Ψ ,R (cid:33) = cos θ Ψ (cid:32) ψ L ψ R (cid:33) + sin θ Ψ (cid:32) χ L χ R (cid:33) . (3.8)with Ψ given by the orthogonal combination. Due to the simplified identical chiral massterm, the mixing angles θ Ψ s are identical for the chiral-left and right components, Ψ i,L andΨ i,R , and satisfy: sin 2 θ Ψ = − √ M ψ + M χ ) y Ψ vm − m . (3.9)– 17 – op + VFTop onlyVF only
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750 800 850 900 950 - - - m tt ( Gev ) d σ s i g / d m tt ( f b / G e V ) Figure 8 . Signal lineshapes as the sum of the Breit-Wigner contribution and the interference contri-butions as a function of the t ¯ t invariant mass at the 13 TeV LHC. The blue (solid), orange (dotted)and green (dashed) lines correspond to the sum of top quark and vector-like quark loop contribu-tions, the top quark contribution alone and the vector-like quark contribution alone, respectively. Thevector-like quark contribution is computed for benchmark parameters M ψ =600 GeV, M χ =1200 GeVand Yukawa y Ψ = 2 as defined in the text. The left and right panels correspond to heavy CP-evenscalar masses of 550 GeV and 850 GeV, respectively. In the alignment limit of a type II 2HDM, the heavy scalar coupling to the vector-like quarks g Ψ i can be expressed as: g Ψ i = ∓ β y Ψ √ θ Ψ . (3.10)Consequently, the sum of the vector-like quark contributions to the gluon-gluon-heavy scalarcoupling reads g Ψ ggH = α s π (cid:32) g Ψ m Ψ I ( ˆ s m ) + g Ψ m Ψ I ( ˆ s m ) (cid:33) , (3.11)while the corresponding result for the heavy CP-odd scalar is very similar. In the heavy masslimit of m Ψ , m Ψ (cid:29) m H , the above contribution can be approximated as, g Ψ ggH ≈ α s π tan β ( M Ψ L + M Ψ R ) y vm Ψ m Ψ ( m Ψ + m Ψ ) I ( ˆ s m Ψ m Ψ ) ≈ α s π tan β y vm Ψ m Ψ . (3.12)We can see from Eq. 3.12 that the loop-induced contribution to gluon-gluon-scalar couplingstakes a form very similar to that one obtained from the low energy theorem of the SMHiggs [38]. Although the heavy Higgs doublet does not have a VEV, its couplings to theheavy vector-like fermions are proportional to that of the SM doublet.In Fig. 8, left and right panels, we present the heavy CP-even scalar lineshapes withcontributions from the vector-like fermions for benchmark scalar masses of 550 GeV and 850GeV, respectively. We show the lineshapes from considering only the top quark contribution(orange, dotted lines), only the VLQ contribution (green, dashed lines) and the coherent sumof both contributions (blue, solid lines). The resulting changes to the lineshapes are sizable.– 18 –he vector-like fermions may enhance the production of the heavy scalars with respect to theSM top-quark loop contribution. At the same time, due to the fact that the VLQ inducedloop function is real, there will be no destructive interference with the SM background. Wechoose a benchmark point with mass parameters M ψ and M χ of 600 GeV and 1200 GeV,respectively. The Yukawa coupling is chosen as y Ψ = 2. In such case the masses of theeigenstates are 440 GeV and 1360 GeV, respectively. Consequently, the 850 GeV scalar iscloser to the threshold of the lighter vector-like quark and receives relatively larger correctionsto the lineshapes in comparison to the 550 GeV one. We note that in 2HDMs, the VLQ willalso contribute to the SM Higgs couplings to gluons, and therefore, the current measurementof the SM-like Higgs properties will constrain the size of the allowed contributions from thesenew fermions. However, due to the m h /m Ψ suppression and the current level of accuracy inthe Higgs boson measurements, such constraints do not play a relevant role at present.If the intermediate colored particles are heavy, effective operators will be sufficient todescribe the physics. In such case our loop-induced gluon-gluon-scalar form factor in Eq. 3.12becomes a constant, and can be identified as the Wilson coefficient of the effective field theory(EFT) operators SGG or SG ˜ G . We give an example in Sec. 3.3. The SUSY partners of the SM colored fermions may also contribute to the gluon-gluon-scalareffective coupling. These scalar quarks also modify the predictions for the observed ∼ g ˜ q S gg (ˆ s ) = − α s π (cid:88) q ; i =1 , g ˜ qi vm q i τ ˜ qi (cid:32) − τ ˜ qi f ( τ ˜ qi ) (cid:33) , (3.13)where the subscript i labels the two scalar mass eigenstates with masses m ˜ q i , that are thesuperpartners of the corresponding SM fermion q . Only the diagonal Higgs-squark-squarkcouplings in the mass basis contribute to Eq. 3.13, and thus the Higgs-squark-squark couplings g ˜ qij are labeled g ˜ qi . For the case of τ ˜ qi (cid:28) τ ˜ qi , and the EFT approach is sufficient to describe thephysics results in this channel. However, the scalars we consider are relatively heavy, andcould be close to the squarks threshold. In this case the phenomenology is rich and interestingand we shall keep the full scale dependence to properly account for such possibility.For scalar masses such that 2 m t < m S < m ˜ t , the loop function for gluon-gluon Higgscoupling from top-quark loop is dominantly imaginary, while that from scalar quarks is real.As a result these two contributions do not interfere with each other, in sharp contrast tothe SM Higgs boson case, where m h < m t < m ˜ q . The squark contributions allow for anadditional adjustment of the relative phases between the ggS production vertex and the t ¯ tS decay vertex, enriching the phenomenology.– 19 – bsoluteRealImaginary - τ I ( τ ) , I ˜ ( τ )( da s hed ) , I ( τ )( do tt ed ) I ( τ ) I ˜ ( τ ) I ( τ ) τ R e l a t i v e P ha s e w . r .t B k g ( π ) Figure 9 . Left panel: loop functions of the scalar-gluon pair vertex as a function of √ τ ≡ √ ˆ s/ (2 m ),with m the mass of the new particle in the loop. The orange, green and blue lines correspond tothe real, imaginary and absolute values of these functions. The solid lines represent the values of thesquark loop function, while for the fermion loop contribution the real and imaginary parts are shownin dotted and dashed lines for the scalar and pseudoscalar case, respectively. The squark-loop functionis multiplied by a factor of four to be visible in a common scale with the fermion loop functions. Rightpanel: induced relative phase with respect to the SM background in units of π for the sfermion loop(green line), fermion loop for a scalar (dotted red line) and a pseudoscalar (dashed blue line). In the left panel of Fig. 9 we show in blue, orange and green, solid lines the absolute,real and imaginary values of the corresponding loop-functions for scalar quarks, respectively.Comparing to spin-1/2 loop-functions shown by the dashed and dotted lines for the scalar andpseudoscalar cases, respectively, the squark loop-function rises and falls much more abruptlynear the threshold. Its real component becomes negative right above threshold. We multiplythe squark function by a factor of four to make it more visible. In the right panel of Fig. 9 weshow the phase generated by the different loop functions as a function of the scale parameter √ τ . As discussed in Section 2, the closer the phase is to π/
2, the more important is theinterference proportional to the imaginary part of the propagator with the SM background,rendering the dip structure more prominent. We show the evolution of such phase for thefermion loop for a scalar (dotted red line) and a pseudoscalar (dashed blue line), as well as forthe squark loop (green line). The phase of the squark loop raises much faster comparing tothe fermion-loop cases, and at large √ τ the phase is close to π . The phases from the fermionsapproaches π/ M ˜ q = (cid:32) M Q + m q + D qL m q X q m q X q M q R + m q + D qR (cid:33) , (3.14)– 20 – op + stopstop onlytop only
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Max * Figure 10 . Signal lineshapes as the sum of the Breit-Wigner contribution and the interferencecontributions, including the effects of SUSY stops in the loop for 850 GeV CP-even scalars, as afunction of the t ¯ t invariant mass at the 13 TeV LHC. The left panel corresponds to the SUSY stopscenario with zero L-R mixing, while the right panel corresponds to the SUSY stop scenario mh ∗ max .The green and yellow lines represent the cases with only top quark loops or stop loops, respectively.The blue lines are the total lineshapes including all contributions. In the right panel we show boththe case for a CP-even and a CP-odd scalar for the solid and dashed lines, respectively. and the mixing angles (defined as ˜ q = cos θ ˜ q ˜ q L + sin θ ˜ q ˜ q R ) that satisfy:sin 2 θ ˜ q = 2 m q X q m q − m q , (3.15)with X q , Y q , D qL and D qR for q = u, d defined in Appendix A, Eq. A.4. In the alignment limitand considering only the dominant stop contributions (setting q = t in the above equations), g ˜ ti can be expressed as: g ˜ t , ( S ) v √ m t + cos 2 β ( D tL/R sin θ ˜ t + D tR/L cos θ ˜ t ) ± m t X t sin 2 θ ˜ t , for S = h − m t tan β − sin 2 β ( D tL/R sin θ ˜ t + D tR/L cos θ ˜ t ) ∓ m t Y t sin 2 θ ˜ t , for S = H ∓ m t Y t sin 2 θ ˜ t , for S = A (3.16)In the above expressions the terms proportional to X t and Y t correspond to the off-diagonalcouplings of the light CP-even Higgs and heavy CP-even Higgs to L-R stops, respectively.While the phenomenological studies on the light Higgs boson focus on X t , which is directlyconnected to stop masses and mixing, and correspondingly to the Higgs mass radiative cor-rections, the heavy Higgs boson coupling mainly depends on an orthogonal quantity Y t . Thestop L-R mixing contribution to the heavy Higgs boson coupling to gluons are proportionalto Y t sin 2 θ t , which in turn is proportional to the product of X t Y t , X t Y t = A t tan β − µ tan β − A t µ (1 − β ) . (3.17)In Fig. 10 we show the comparison of the lineshapes for a heavy scalar of mass 850 GeVconsidering stop contributions to the loop function, and for two scenarios for the stop mixing– 21 –arameters X t and Y t . One is the zero L-R mixing case with vanishing X t . The other is avariation of the mh max scenario [39–41] in which we take X t = √ M SUSY ≈ (cid:113) m ˜ Q m ˜ t R and Y t = 2 X t . We named this modified maximal mixing scenario mh ∗ max such that for tan β = 1,it corresponds to A t = 3 µ . The channel gg → H, A → t ¯ t in supersymmetry could be adominant channel in discovering the heavy Higgs bosons in the low tan β regime. Despitethat the observed 125 GeV Higgs mass disfavors the low tan β ( <
3) regime in the MSSM, ex-tensions of the minimal model, such as the next-to-minimal-supersymmetric standard-modelcan work well in this regime. Therefore, for the purpose of demonstrating the t ¯ t channel’sphysics potential and for easier comparison with previous non-SUSY discussions, we choosea benchmark value of tan β = 1 in these figures. The green and orange lines correspond tothe production of heavy scalars with only the SM top quark loop contribution and only theSUSY stop loop contribution, respectively. The blue lines represent the lineshapes with allcontributions taken into account. In both scenarios we choose the lighter stop mass to beclose to half of the the heavy Higgs boson mass and the heavier stop to be around 1 TeV. Thedetailed numerics of our benchmark stop parameters are listed in the Appendix in Eq. A.5.The stops could change the heavy scalar lineshapes in a distinct way depending on the L-R stop mixing. For the case with zero L-R mixing shown in the left panel of Fig. 10, the stopcontribution (orange line) is relatively small compared to the top contribution (green line),due to the smaller value of the squark loop function. In spite of the fact that the stop loopfunction is real and only produces interference through the real part of the propagator, thesmall value of the Breit-Wigner contributions implies that the interference piece is dominant,leading to a bump-dip structure crossing zero at the scalar pole mass. Once both the top andstop loop contributions are summed up the effect of the stop is hardly noticeable. Moreover,in the zero L-R mixing case the CP-odd scalar does not couple to the stops, and hence we donot show those lineshapes for the CP-odd Higgs. For the mh ∗ max scenario shown in the rightpanel of Fig. 10, the stop contribution could be sizable. We show both the lineshapes for theCP-even Higgs boson and the CP-odd Higgs boson in solid and dashed lines, respectively. TheBreit-Wigner contribution from the stop loop shifts the value of t ¯ t invariant mass where thesignal rate is zero slightly above the heavy scalar pole mass, as illustrated by the orange lines.The contribution from the L-R mixing term dominates and changes the pure dip structuresfrom the top only contribution (green lines) into a bump-dip structure (blue lines). Wepurposefully choose the parameters such that the heavy scalar is only slightly below the lightstop pair production threshold, with a light stop mass of about 435 GeV. We observe thatthe stop threshold effect is only minimally visible in the orange and blue lineshapes in bothpanels, through the small discontinuity at a t ¯ t invariant mass of around 870 GeV. The abovediscussion shows that a relatively light stop, depending on the L-R mixing parameters, couldhave a relevant impact on the search strategy and the sensitivity reach of heavy scalars inthe t ¯ t decay channel. – 22 – .3 Special discussion: A (pseudo)scalar from a putative di-photon excess At the end of 2015 both the ATLAS and CMS collaborations reported a diphoton excess atabout 750 GeV that could have been a truly striking signal of new physics beyond the standardmodel [16, 17]. This excess drew significant attention from the theory community. Manytheoretical descriptions to explain a putative diphoton excess also implied the existence ofthe t ¯ t signal [42, 45–56]. Moreover, many of the explanations, involved sizable contributionsfrom heavy particles, vector-like fermions and scalars, in the loop functions for both thegluon-gluon-scalar production vertex and the diphoton-scalar decay vertex. In the following,we focus on some detailed features of the t ¯ t signal lineshapes from a heavy scalar in theframework of an EFT, where heavy particle loop contributions to the gluon-gluon-scalarcoupling compete with the top quark loop one. We further introduce a convenient rescalingfactor to quantify the signal rate after smearing effects to correctly translate current boundson a t ¯ t resonance search, also taking into account the important interference effects. Weconsider as an example a 750 GeV scalar with no special relevance of the precise mass valueas far as it is in the several hundred GeV range.As it is well-known, the t ¯ t -scalar coupling induces at one-loop level the gluon-gluon-scalarand gamma-gamma-scalar effective vertices. If this t ¯ t -scalar coupling is the dominant sourceof the diphoton process, although the production rate will be sizable, the diphoton branchingfraction will be too small to accommodate a sizable diphoton signal at the reach of the LHC.Indeed, the tree-level two-body decay of a several hundred GeV heavy scalar to top quarkpairs is orders of magnitude too large compared to the electromagnetic, loop suppressed scalarto diphoton decay. A possibility is to increase the production rate to compensate such smalldecay branching fraction to diphotons, however, other searches on the hadronic channels willstrongly disfavor such scenario. Instead, an intriguing possibility for a heavy scalar diphotonsignal could be from heavy charged particle dominance in the gluon production as well asin the diphoton decay modes, with suppressed but still very sizable decay to t ¯ t . A verystraightforward example is a neutral heavy scalar that mainly receives its coupling to gluonpairs and photon pairs through multiple heavy top partner loops, while the coupling of thenew heavy scalar to top quarks is controlled by the mixings of the top partners with the topquark.We consider the following minimal interaction Lagrangian for a pseudoscalar S , L int ⊃ Sf ( i ˜ y t ¯ Q L ˜ Ht R + h.c. ) + c G α s πf G SG ˜ G, (3.18)where the coefficient c G captures contributions to the gluon-gluon-scalar coupling by integrat-ing out the heavy colored particles. The total gluon-gluon-fusion rate for the scalar production For a relatively comprehensive study and quasi-review, see e.g., Ref [42–44], and references therein. We changed notation to the standard one for composite scalar models. Analogous treatment holds for thescalar case. – 23 –lso receives contribution, from the top-quark loop and reads σ ( gg → S ) = σ ( gg → H SM750 GeV ) v f (cid:12)(cid:12)(cid:12) ˜ I ( τ t ) + f ˜ y t f G c G (cid:12)(cid:12)(cid:12) | I ( τ t ) | , = 31 (cid:18) TeV f (cid:19) (cid:12)(cid:12)(cid:12) ˜ I ( τ t ) + c g (cid:12)(cid:12)(cid:12) fb , with c g ≡ c G f ˜ y t f G . (3.19)The rate for the SM Higgs is approximately 740 fb at the 13 TeV LHC [57] and the loopfunctions I ( τ t ) and ˜ I ( τ t ) are as defined in Eq. (2.4). However, it is very important toemphasize that using σ ( gg → S ) from Eq. (3.19) multiplied by the Br( S → t ¯ t ) is no longer avalid approach, since the large interference effects should be appropriately taken into account,as discussed in the previous sections.In the lower left panel of Fig. 11 we show how the relative phase θ ¯ A with respect to theSM gg → t ¯ t background varies as a function of c g , as define in Eq. 3.19. The phases for thescalar and pseudoscalar are represented by the red and blue lines, respectively. The solidlines represent the relative phase for positive c g , while the dashed lines represent π minus therelative phase for negative c g . In the case of dominant t ¯ t contribution (low c g ), the relativephase is near π/ π/
5) for pseudoscalar (scalar). For comparable contribution from top-loop and heavy colored particle loop the phase is still as large as π/
4, while when c g is greaterthan 10 the relative phases becomes negligible.In the upper panel of Fig. 11 we show several lineshapes for the differential distributionfor the gg → S → t ¯ t cross section as a function of the t ¯ t invariant mass, for various benchmarkvalues of c g . The example cases of a 750 GeV pseudoscalar and scalar are displayed in theupper left and upper right panels, respectively. For clarity of presentation, we normalize thelineshapes to the Breit-Wigner parton level cross section at the scalar mass pole. We assumethe total width is dominated by the partial decay to t ¯ t , Γ total ≈ Γ t ¯ t . The resulting lineshapebehavior is independent of the precise normalization of the interaction strength v/f , andtherefore we plot the lineshapes in units of the total width Γ ∝ v /f . This can be understoodsince the signal amplitude does not depend on v/f near the scalar mass pole: the numeratorof the signal amplitude scales as scalar-top pair coupling squared, proportional to v /f , dueto the production and decay vertex while the denominator is proportional to the total width,which is also proportional to v /f . Moreover, the overall lineshape is determined by therelative importance between the Breit-Wigner contribution and the interference contribution,which is characterized by the relative strength of the signal amplitude to the backgroundamplitude, independent of v /f .To better understand Fig. 11, let us discuss the different lineshape behaviors for differentvalues of c g . For large values of c g , for which the heavy colored particle loop dominatesin the gluon-gluon-fusion production, the resulting lineshape for the t ¯ t signal is governedby the Breit-Wigner contribution with a smaller contribution from the interference effect note that t ¯ t could still be the dominant decay channel in comparison with the loop-suppressed (e.g., α s / (8 π ) ) decays to gluon pairs. – 24 – + Γ + Γ + Γ - Γ - Γ - Γ - - + Γ + Γ + Γ - Γ - Γ - Γ m tt ( GeV ) ( d σ B W + I n t / d m tt ) / ( d σ B W / d m tt ) c g = c g = c g =- c g = c g =- CP Odd + Γ + Γ + Γ - Γ - Γ - Γ - - + Γ + Γ + Γ - Γ - Γ - Γ m tt ( GeV ) ( d σ B W + I n t / d m tt ) / ( d σ B W / d m tt ) c g = c g = c g =- c g = c g =- CP Even pseudoscalarscalar (cid:200) c g (cid:200) R e l a ti v e P h a s e w . r . t . B kg (cid:72) Π (cid:76) pseudoscalarscalar - - | c g | σ B W + I n t. / ( σ B W × B r ) Figure 11 . Upper panel:
Representative lineshapes of differential distributions at parton level forthe process gg → S → t ¯ t normalized to the value of the Breit-Wigner contribution at the scalar masspole, as a function of the t ¯ t invariant mass, for various values of c g . The corresponding values of c g are labelled on the lines and characterize the relative contribution from heavy colored particles to thegluon-heavy scalar vertex with respect to the top quark one. The heavy particle S is a CP-odd or aCP-even scalar for the left and right panels, respectively. Lower left panel:
Relative phase θ ¯ A inunits of π as a function of c g . The dashed lines represent ( π − relative phase) for negative c g . Lowerright panel: ratio of the total signal rate within a ± S window for a 750 GeV scalar, including bothresonance and interference contributions, over the naive resonance rate for the gg → S → t ¯ t process,see details in the text. For both lower panels, the CP-even and CP-odd cases are represented by thered and blue lines, respectively, while the positive and negative values of c g are represented by thesolid and dashed lines, respectively. For all the figures, the new physics scale f is chosen to be 1 TeV. proportional to the real part of the propagator. This is shown by the red and red, dashedlines for c g = 10 and c g = −
10, respectively. For negligible values of c g , for which thetop-loop dominates the production, the resulting lineshapes for the t ¯ t signal are pure dipsas shown by the black curves for c g = 0. In the limit of large statistics, the bounds frombump search and dip search could be treated more or less equivalently. However, in thesetwo limits, the constraints from the t ¯ t resonance search should be interpreted with caution. We note that the bump search itself is dominated by the systematic uncertainties and thus projections on – 25 – very different behavior occurs when the top-loop and heavy particle-loop contribution arecomparable, resulting in a bump-dip or a dip-bump structure, as shown in the blue lines andblue dashed lines for c g = 1 and c g = −
1, respectively. In such case, the smearing effects fromthe t ¯ t invariant mass reconstruction will flatten the dips and bumps in the lineshapes andrender the experimental search much more challenging, as we shall see in the next section.In the lower right panel of Fig. 11 we plot the ratio of the total gg → S → t ¯ t BSMrate to the naive rate obtained from σ ( gg → S ) × Br( S → t ¯ t ). The total rate includes theinterference effect and is defined by integrating the signal lineshape over the ± S region. Inthis figure we show the ratios for a heavy scalar and a pseudoscalar, with both signs of c g ,with the same line coding as the lower left panel. For low | c g | , all cases are more of a dipstructure and this ratio could be as small as − − .
7) for the pseudoscalar (scalar). Forsufficiently large | c g | ( >
5) , the signal is Breit-Wigner like and the ratio tends to be oneas expected. For c g around unity, large cancellations occur. Furthermore, the sign of c g alsoplays a role in the exact value of c g for which this ratio approaches zero. The negative c g usually requires larger values to be dominant, as the new physics contributions must firstcancel the real component from the top quark-loop. The ratio of the total gg → S → t ¯ t BSM rate to the naive σ ( gg → S ) × Br( S → t ¯ t ) rate provides a crude estimate of the currentcollider constraints for a given 750 GeV scalar model in the t ¯ t channel. One can divide thecurrent constraints on the t ¯ t production rate, which neglect the interference effects, by theabsolute value of this ratio to obtain an estimate of the constraints on the total productionrate.For the process of gg → S → V V and gg → S → aa , where V represents SM electroweakgauge bosons ( γ , W , Z ) and a is the light particle that later fakes the photon, using σ ( gg → S ) × Br( S → V V, aa ) is appropriate for the total BSM effect because of the smallness ofinterfering SM background. Still, the detailed lineshapes could be useful to determine theproperties of the scalar [58–60], although the effect is not very sizable and quite large statisticsis needed.
The search for a new heavy scalar signal in the gg → S → t ¯ t channel at the hadron col-lider is challenging in various ways. The first challenge comes from the non-conventional dip,bump-dip, or dip-bump structures for which the normal bump search is not optimized. Thesecond is related to the top-quark invariant mass reconstruction that smears the signal by alarge amount. The bump and dip become less pronounced due to events in the bump thatwill populate the dip via mis-reconstruction of the invariant mass and vice versa. Indeed,the fact that events in one region are interpreted as events in the other one produces the this channel should be done in a careful way, otherwise, overly aggressive results can be obtained by blindlyassuming statistical uncertainty dominance. A detailed discussion follows in the next section. – 26 – igure 12 . The total and statistical only bin-by-bin relative error as a function of the the t ¯ t invariantmass from the ATLAS 8 TeV analysis [61] shown in gray and blue histograms, respectively. For furtherdetails, see the discussion in the text. smearing that results in a reduced excess or deficit of events and diminishes considerably thesignificance of the lineshape analysis. The third significant challenge is due to the systematicuncertainty associated with the large production cross section of the SM top quark pairs,which is the irreducible background for t ¯ t resonance searches. The background cross sectionstarts to increase quickly once the process is kinematically allowed, reaching its peak at aninvariant mass near 400 GeV at 13 TeV LHC. In Fig. 4, we show that the background statis-tical uncertainty (dashed gray contour) is very small compared to the systematic uncertainty(solid gray contour) that hides the signal lineshapes. Consequently, reducing the systematicaluncertainty is a key task in order to achieve sensitivity in this channel. Due to the difficultiesjust mentioned, the search for a new heavy scalar in the gg → S → t ¯ t channel is basically notconstrained in the entire mass range slightly above the t ¯ t threshold. In the following we shallre-evaluate the above challenges considering various techniques, and discuss their impact onthe LHC reach.The current result for a t ¯ t resonance search performed by ATLAS [61] results in approx-imately 8% (6%) smearing of the reconstructed t ¯ t invariant mass distribution at around 400GeV (1 TeV). For our regions of interest, the signal mainly lies in the resolved-topology se-lection of the ATLAS search, for which the decay products of the hadronically decaying topquark are expected to be reconstructed in three small radius jets, in contrast to the boostedcase. The resolved-topology is of relevance for our study since we focus on the phenomeno-logically interesting region below one TeV. The CMS t ¯ t resonance search at 8 TeV has similarinvariant mass resolution of around 10% [62].In Fig. 12 we show the current total uncertainty (gray band) and statistical uncertainty The numbers are obtained from the auxiliary material of the ATLAS analysis [61] available at http://hepdata.cedar.ac.uk/view/ins1373299 . The current 13 TeV search has already shown better systematiccontrol [63] but the smallness of the systematics prohibits us from extracting the numbers accurately from theplot. CMS 8 TeV analysis has similar but slightly worse systematic uncertainties [62]. – 27 – able 1 . Benchmarks for two LHC performance scenarios for the t ¯ t lineshape search at 13 TeV,motivated by current results from 8 TeV searches and assuming 30 fb − and 3 ab − of data, respec-tively. Scenario A is based on a conservative assumption for the projected t ¯ t invariant mass resolutionand systematic uncertainties, while Scenario B is based on a more aggressive assumption for bothexperimental parameters. ∆ m t ¯ t Efficiency Systematic UncertaintyScenario A 15% 8% 4% at 30 fb − , halved at 3 ab − Scenario B 8% 5% 4% at 30 fb − , scaled with √ L (blue band) achieved by the ATLAS 8 TeV analysis [61]. The systematic uncertainty can becontrolled at the level of about 2% to 4% in the mass range between 240 GeV and 1 TeV.This search exploits the large data sample available from the LHC by marginalizing the nui-sance parameters that characterize the systematic uncertainties. The uncertainties derivedfrom this method use the data more extensively than other more traditional treatments. Thesystematics for a lineshape search that correlates adjacent bins, such as the one we are con-sidering in our study should be comparable or better than that of a single bin. Thereforewe expect that the systematic uncertainty values from the ATLAS study can be applied toour analysis. With higher integrated luminosity, we expect that the systematic uncertaintieswill improve. On one hand, the large amount of t ¯ t events can be used to better understandthe detector performance and reduce the systematic uncertainties. On the other hand, thelarge data set also means that one can afford a lower signal selection efficiency allowing for t ¯ t events with higher quality in terms of invariant mass reconstruction accuracies and system-atic uncertainties. Moreover, alternatively to the Monte-Carlo based method for backgroundmodeling used by the ATLAS study, one could consider the widely used data driven back-ground subtraction method that tends to improve with larger data sets. Many applicationsof this method show great advantage in complex experimental environments. In addition,development in the analysis techniques may help further reduce the systematics [64–66]. Theabove arguments enable us to define scenarios for our study.In Table 1, we consider two scenarios for the t ¯ t lineshape search using the semi-leptonic t ¯ t sample. Scenario A is more conservative, both for the invariant mass resolution and thehigh luminosity projection, while scenario B is more aggressive. Another relevant parameteris the signal selection efficiency. We chose 8% signal selection efficiency (branching fractionincluded) for Scenario A. For scenario B, instead, we consider a lower signal efficiency of 5%,allowing for a possible more strict requirement on data quality to allow for more optimisticassumptions on the smearing effects and the systematic uncertainties. As discussed earlier,the current values of the systematic uncertainties can be as low as 2% with the LHC 8 In scenario B, we take an invariant mass resolution of 8% throughout the mass range, as quoted byATLAS. In scenario A we take a very conservative value of 15%, slightly above the value quoted by CMS. – 28 –
50 500 550 600 6500.00.51.01.5 m ttreco ( GeV ) d σ S i g / d m tt r e c o ( pb / b i n ) Scenario B LHC 13 TeV @ - y t = m S =
550 GeV
700 800 900 1000 - - - m ttreco ( GeV ) d σ S i g / d m tt r e c o ( pb / b i n ) Scenario B LHC 13 TeV @ - y t = m S =
850 GeV
Figure 13 . The binned differential distribution of the signal and background uncertainties in unitsof pb/bin. The red histograms are the binned signal histograms after background subtraction withthe heavy scalar-top quark Yukawa coupling y t = 2. The blue and gray bands are the backgroundstatistical uncertainties and total uncertainties after smearing and binning, respectively, for 3 ab − oftotal integrated luminosity in the semi-leptonic channel for the performance scenario B. The left andright panels show results for a heavy CP-odd scalar mass of 550 GeV and 850 GeV, respectively. TeV data. We assume a flat 4% systematic uncertainty for the whole range 400 − − . In Scenario A we assume the systematics being halved with the full HL-LHCluminosity and in Scenario B we assume the systematics being scaled with the squared root ofthe total integrated luminosity. We also choose a binning size of 5% of the scalar mass in the t ¯ t invariant mass distribution. In most cases the experimental search uses the full informationon each event, hence binning is not necessary. However, in our simplified statistical treatmentbinning is important, and given the size of the smearing effect, we consider a bin size of 5%of the scalar mass appropriate. For illustration purposes we show in Fig. 17 of the Appendixthe signal lineshape before and after smearing and binning, for the case of a pseudoscalar ofmass 550 GeV with a Yukawa coupling y t = 1.As discussed in earlier sections, many models contain a heavy scalar with different featuresand may also include two scalars of similar masses but different CP properties. The resultinglineshapes are very diverse, depending on the relative phase, new contributions to the effectivegluon-gluon-scalar coupling and the precise separation between heavy scalar masses. As afirst step, we propose a search for a single scalar on the lineshape of the t ¯ t system, performinga template fit in the differential distribution of the t ¯ t invariant mass.In Fig. 13 we show, after smearing and binning, the resulting signal lineshapes for a CP-odd scalar with masses of 550 GeV and 850 GeV, in the left and right panels, respectively, forthe baseline model with Yukawa coupling y t = 2. The signal distributions feature, as shownby the red histograms, a bump-dip structure for the 550 GeV case and almost a pure dipstructure for the 850 GeV case. The statistical uncertainty and total uncertainty at 3 ab − are shown in blue and gray histograms for scenario B, respectively. As discussed in earliersections, the systematic uncertainty is the dominant effect and reducing it by upgrading thedetector, using data to calibrate the machine to the best achievable level, and improving the– 29 – ¯ t system mass reconstruction are crucial for further improvements and possible discovery inthis channel.Based on the distributions and uncertainties shown in Table. 1 and Fig. 13 and assuminga null BSM result in the future data, we can project which region of BSM parameter spacecan be probed. We calculate the significance squared of the lineshape in the (1 ± . m S range, that is equivalent to considering a sum over 10 bins with a bin size of 5% of the scalarmass − log( p ) = (cid:88) n n bkg + δ n . (4.1)In the above, n sig is the number of signal events (could be both positive and negative), n bkg is the number of background events and δ sys is the systematic uncertainty. The p-value forthe signal is then the sum of the significance in quadrature of the bins in the mass window of(1 ± . m S . This is the large background limit of the median expected significance for thelikelihood ratio, where we have dropped two small corrections of order | n sig | /n bkg and δ n bkg according to the Asimov approximation [67, 68]. This treatment basically corresponds to atemplate fit in the invariant mass distribution of the t ¯ t system. We then translate this p-value into significance for a given signal model lineshape. We derive the projected limits asa function of the parameter space for specific models by generating a grid of p-values andfinding (multi-dimensional) contours of 2 σ exclusion. Generating a grid of signal lineshapeswith respect to model parameters is necessary for this search, even for the simplest baselinemodel, since the lineshape is a combination of the interference part proportional to y t andthe Breit-Wigner contribution that, when off peak, is proportional to y t .It is worth to highlight that in the region where the SM background shape departs fromsimple polynomials, for example near the SM threshold peak around 400 GeV, additionaluncertainties on the shape will enter. Simulation driven background estimations may becomea better handle and different systematic uncertainties arise. In addition to considering datadriven estimation for the background, high precision SM calculations are evidently of greatimportance. Indeed, in the case of sizable values of the heavy scalar width, there is importantinterference between the signal and background at far off the peak, and this might changethe overall slope of the background estimation using side bands. Such effects could have animpact on the sensitivity derived using the simplified procedure described in this study.To summarize, in this section we propose to perform a lineshape search using the semi-leptonic t ¯ t channel in the resolved sample. We include the two leading effects, namely, smear-ing and the background normalization systematic uncertainties, and adopt an approximatedstatistical treatment given in Eq. (4.1). Further inclusion of the merged channel and otherdecay modes in the t ¯ t searches could improve the sensitivity, whereas the background shapeuncertainties may affect our sensitivity estimation and need to be taken into account in futureanalyses. – 30 –
00 500 600 700 800 900 1000012345 m S ( GeV ) y t S c ena r i o A @ f b - S c ena r i o B @ f b - Scenario A @ - Scenario B @ - CP Even Scalarbaseline modelgg → S → tt
400 500 600 700 800 900 1000012345 m S ( GeV ) y t S c ena r i o A @ f b - S c ena r i o B @ f b - Scenario A @ - Scenario B @ - CP Odd Scalarbaseline modelgg → S → tt Figure 14 . The projected 95% C.L. exclusion limits on the top quark Yukawa coupling of the CP-even(left panel) and CP-odd (right panel) heavy scalar S at the LHC for the baseline model. The red andgray lines correspond to the performance scenarios A and B, tabulated in Table. 1, and the regionsabove the curves are excluded. The solid and dashed lines show results for an integrated luminosityof 3 ab − and 30 fb − , respectively. As an illustration, the shaded bands indicate a variation of 5%in the required significance to derive the limits. In this section, we present the projected sensitivity of the gg → S → t ¯ t lineshape searchin various model configurations, using the benchmark performance scenarios and statisticalmethod depicted in the previous section. We first show the exclusions in the baseline modelfor a heavy CP-even or CP-odd scalar, while later on we discuss the sensitivities in variousscenarios of Type II 2HDMs.In Fig. 14 we show the exclusion limit on the baseline model as a function of the heavyscalar mass and its Yukawa coupling to the top quark. The left panel shows the result for aCP-even scalar while the right panel is for a CP-odd scalar. The red and gray lines correspondto the 2 σ exclusion limit in scenarios A and B as specified in Table. 1, with the dashed andsolid lines corresponding to LHC 13 TeV at 30 fb − and 3 ab − , respectively. The regionsabove the lines are excluded for each specific scenario and integrated luminosities, as labeledin the figure. To illustrate the effects of possible uncertainties in our statistical and binningtreatment, we present, as an example, shaded bands showing a variation of 5% in the requiredsignificance to derive the limits. In both scenarios A and B, the heavy CP-even (CP-odd)scalar in the baseline model can be excluded up to 450 (550) GeV for a Yukawa coupling y t = 3at 30 fb − . For the same value of y t and 3 ab − , in Scenario A the reach increases to 650GeV and beyond 1 TeV for the heavy CP-even and CP-odd scalars, respectively. In Scenario– 31 –
00 500 600 700 800 900 10000.10.51.2.5. m H ( GeV ) t an β S c ena r i o A @ f b - S c ena r i o B @ f b - Scenario B @ - Scenario A @ -
13 TeV LHCType II 2HDMCP - even Scalar
400 500 600 700 800 900 10000.10.51.2.5. m A ( GeV ) t an β S c ena r i o A @ f b - S c ena r i o B @ f b - Scenario B @ - Scenario A @ -
13 TeV LHCType II 2HDMCP - odd Scalar Figure 15 . The 95% C.L. exclusion on the scalar mass–tan β plane for a type II 2HDM, includingthe effects of bottom quarks in the process. The regions below the curves are excluded. The resultfor the CP-even and CP-odd scalars are shown in the left and right panels, respectively. The colorcoding, lines and legends are the same as in Fig. 14. B, the reach increases beyond 1 TeV for both a CP-even and a CP-odd heavy scalar for aheavy scalar-top Yukawa coupling of y t = 1 at 3 ab − . One can also consider the sensitivityfor a fixed scalar mass at different luminosities and compare the exclusion reach in the heavyscalar-top quark Yukawa coupling strength. For example, the limit improves from 4.5 to2.5, and from 4.5 to 0.7, for a CP-even scalar mass of 550 GeV when luminosity increasesin scenario A and B, respectively. Comparing both performance scenarios, we observed thatwith 30 fb − of integrated luminosity, they have comparable reach, because the differencesin signal efficiencies and energy resolutions compensate each other. However, the exclusionlimits in the more aggressive performance scenario B at 3 ab − yields a much better reachthan in the conservative case of scenario A. This demonstrates again the crucial role that thesystematic uncertainty plays in these projections.Beyond the baseline model, we perform numerical studies for the Type II 2HDM includingthe bottom quark corrections in both the production amplitudes and the decay widths. InFig. 15 we show 95% C.L. exclusion contours in the heavy scalar mass–tan β plane. Thelegends are the same as in Fig. 14, but in this case the regions below the curves are excluded.For the CP-even scalar shown in the left panel, the reach in mass is only around 450 GeVfor most scenarios for tan β = 0 .
5. This moderate reach is mainly due to the β suppressionfactor and the smaller value of the loop function. The restricted reach for the CP-even scalarcase is only overcome in the more aggressive scenario B at 3 ab − , probing mass scales up to1 TeV. For the CP-odd scalar shown in the right panel, the exclusion reach is much better– 32 –
00 500 600 700 800 900 10000.10.51.2.5. m A ( GeV ) t an β S c ena r i o A @ f b - S c ena r i o B @ f b - Scenario B @ - Scenario A @ -
13 TeV LHCType II 2HDMNearly Degenerate
400 500 600 700 800 900 10000.10.51.2.5. m S ( GeV ) t an β S c ena r i o A @ f b - S c ena r i o B @ f b - Scenario B @ - Scenario A @ -
13 TeV LHCType II 2HDM Δ m =
20 GeV CPV max
Figure 16 . The 95% C.L. exclusion on the scalar mass–tan β plane for two nearly degenerateheavy neutral scalars in a Type II 2HDM with bottom quarks effects included. The results for theCP-conserving and maximal CP-violating ( θ CP of π/
4) cases are shown in the left and right panels,respectively. The color coding, lines and legends are the same as in Fig. 15. in comparison with the previous case. Masses up to 450 GeV to 600 GeV can be probed fortan β = 0 . − of integrated luminosity. In all cases consideringjust one new heavy scalar at a time the reach is limited to values of tan β <
2, with the smallexception of M A around 400 GeV that can reach up to tan β = 3. For the scalar mass nearthe top-quark pair threshold below 400 GeV, the 2HDM reach for the CP-even scalar is worsethan the baseline model, and this is due to the t ¯ t branching fraction suppression from thescalar decays into b ¯ b .Given that the two heavy scalar bosons often have nearly degenerate masses in many2HDMs, In Fig. 16 we study such case and show both the CP conserving (left panel) andmaximal CPV (right panel) situations for the heavy Higgs sector in Type II 2HDM. For theCP-conserving case, we assume a mass splitting between the two scalars as in the MSSMfollowing , and hence the reach is equivalent to that of a CP-conserving MSSM in the limit ofheavy squarks, in which both scalar signals simply add, as discussed in Sec. 3.1. For the CPVcase, we assume a constant splitting between the two scalars of 20 GeV, and a new interferenceeffect between the two scalars emerges. This effect slightly changes the projected limits. Weshow the exclusion limits in the tan β - m A (- m S ) plane for the CP conserving (violating)case. The labels are the same as in Fig. 15. The reach increases to 480 GeV and 600 GeV fortan β = 0 . − of integrated luminosity, for scenarios A and B, respectively. In theHL-LHC environment masses as high as 1 TeV for tan β = 0 . β = 2 can be probed– 33 –n scenarios A and B, respectively. Values of tan β (cid:39) Heavy scalars are well motivated in many extensions of the standard model. The typicaldominant production and decay mode of a heavy scalar at hadron colliders is via gluon fusionwith the subsequent decay to a top quark pair, gg → S → t ¯ t . In our baseline model for whichthe ggS effective vertex is dominantly mediated by the top-quark triangle diagram, the signalamplitude interferes with the SM background in a complex way. The total signal lineshape ismainly driven by the behavior of the loop-function evaluated at √ ˆ s close to the heavy scalarmass. As a result one can obtain a lineshape that behaves as a pure bump, a bump-dip, ora pure dip structure depending on the value of the scalar masses. In many BSM models,additional corrections come, for example, from non-trivial CP phases associated with theheavy scalar, the existence of nearly degenerate scalars, or additional loop contributions fromstops or vector-like quarks. In this paper we study the relevant features of top pair productionfrom heavy scalars and evaluate the LHC physics potential in various BSM scenarios.We first discuss the behavior of the loop-function and the resulting lineshapes in thebaseline model for a purely CP-even or CP-odd scalar, as well as a scalar that is a mixture ofCP eigenstates. We obtain different behaviors of the lineshapes parametrized by the additionalphases generated by the loop function of the triangle diagram. We consider the case of nearlydegenerate heavy scalars that may exist in a 2HDM, and show that their contributions addto each other in the lineshapes, resulting in an enhancement of the features of the lineshapestructure and providing a good opportunity for detecting the signal. In the case where the twoquasi-degenerate eigenstates are CP admixtures, there is also a small additional interferenceeffect between them that further modifies the lineshape structure.We also study BSM scenarios with additional heavy particles contributing to the gluoninduced loop function, such as scalar-quarks or vector-like quarks. We have analyzed differentillustrative scenarios: one in which the heavy particle contribution dominates over the SM topquark one, and two others in which the new heavy particle effects are comparable or smallerto those of the top quark. In the case that the heavy particle contribution dominates, thelineshape is given by the standard Breit-Wigner resonance bump plus the off peak interferencebump-dip structure, which is proportional to the real part of the propagator. We exemplifythe above behavior for a vector-like quark model with VLQs heavier than half of the heavyresonance mass. Examples of moderate or comparable effects to those induced by the SMtop quark loop are shown in the case of Supersymmetry. When the stops have a negligibleleft-right mixing, their effects are just a small perturbation to the baseline model. In thecase of sizable mixing, instead, the stop loop may yield a visible contribution and change thelineshape significantly.We provide a study for the search of a heavy scalar with additional contributions to theproduction process in the context of EFT. The specific lineshapes could play a crucial role in– 34 –nterpreting the results and projecting the discovery potential in the t ¯ t channel. We find thatif a scalar mass is in the 700 GeV ballpark and the gluon-gluon-fusion process is dominantlyinduced through top-quark loops, the resulting lineshape is a pure dip. If, instead, there arecontributions from additional heavy colored particles comparable to those of the top quark,the resulting lineshape is a bump-dip structure, where large cancellations occur once smearingeffects are taken into account. We define a ratio of the total signal cross section, includinginterference effects, to the naive signal cross section without interference, that serves as apenalty factor in deriving a crude estimate of the collider limits for a heavy scalar particledecaying to top quark pairs.In the final part of this paper we study the LHC sensitivity to the t ¯ t signal from heavyscalars for two plausible LHC performance scenarios. The real challenge resides in the system-atic uncertainties in this channel and one should make use of the large amount of accumulateddata to reduce them through a better detector calibration and advanced analysis techniques.We propose to complement the normal bump search with a lineshape search that makes bet-ter use of the bump-dip structure by counting both the excess and deficit as part of the BSMsignal. We present the results of our proposed lineshape search for various BSM cases. Firstwe consider a heavy scalar in the baseline model and show that a CP-odd scalar with a topYukawa coupling y t = 2 can be excluded at the 95% C.L. up to 500 GeV in both performancescenarios with 30 fb − of data. The reach can be extended all the way up to 1 TeV for botha CP-even and a CP-odd scalar, with a top Yukawa coupling as low as y t = 1, for the mostaggressive performance scenario with 3 ab − . Next we consider 2HDMs for which the bottomquark effects in the loop-induced production mode and the scalar total width become relevantin the intermediate and large tan β regime. We derive the expected 95% C.L. exclusion limitsfor both the CP-even and CP-odd scalars in the tan β -scalar mass plane. Considering oneheavy neutral Higgs boson at a time, values of tan β of order 1 can be probed for the wholemass range up to 1 TeV for the most aggressive performance scenario with 3 ab − of data. Inthe case that the two heavy scalars are nearly degenerate in mass, we consider the combinedsearch of both particles decaying into t ¯ t and show the improved 95% C.L. exclusion limitsboth for the CP-conserving and CPV cases.A few remarks before concluding: Other BSM searches, such as those involving color orweakly interacting scalar octets may also profit from the discussions in this paper. More-over, higher order corrections may affect the large destructive interference effects, due to thepossible reduction of the phase-space overlap between signal and background, as well as thepossible addition of new relative phases. For example, a next-to-leading-order study on the2HDM [69] showed some distortions of the interference effects and a more detailed analysisfocussing on the specific changes due to the higher orders corrections will be of great interest.Finally, there may be other observables for which the interference effects are reduced, pro-viding additional information on the signal. For example, considering angular distributionscould provide additional sensitivity for the gg → S → t ¯ t search. However, our preliminaryinvestigation of these observables shows very limited gain, in agreement with Ref. [15], mainlydue to large systematic uncertainties and smearing effects. Another useful handle could be to– 35 –onsider top quark polarization to reduce the background without significantly affecting thesignal. Provided higher statistics, polarization may also help to identify the CP propertiesof the heavy scalar. We intend to further explore these points in a future study.
Acknowledgments
We thank Y. Bai, N. Craig, S. Dawson, K. Howe, P. Fox, T. Han, R. Harnik, S. Jung, W.-Y.Keung, K.C. Kong, I. Lewis, J. Lykken, S. Martin, S. Su, B. Tweedie, L.T. Wang, C. Williamsand H. Zhang for helpful discussions. Fermilab is operated by Fermi Research Alliance, LLCunder Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy. Z.L. thanksthe Aspen Center for Physics, which is supported by National Science Foundation grantPHY-1066293, for the hospitality during the final stage of this work.
Notes added:
Upon the completion of this work, a study [71] on interference effects partiallyoverlapping with our discussion in Sec. 3.3 and a numerical study [72] focusing on the inter-ference effects in the t ¯ t channel in the framework of a 2HDM appeared. ATLAS has recentlypublished a conference note using the LHC 8 TeV dataset to search for heavy scalars in the t ¯ t channel with the interference effect taken into account [73]. The result shows comparabalelimits to our projections for the baseline model, however assuming a better systematic controland a larger signal mass window. A Additional Formulae and Benchmarks
The loop functions in Eq. 2.3 and Eq. 2.4 I ( τ t ) and ˜ I ( τ t ) can be alternatively written as, τ t = ˆ s m t , β ≡ (cid:114) − m t ˆ s = (cid:112) − /τ t f ( τ ) = arcsin ( √ − β ) , for τ ≤ , − (cid:16) log β − β − iπ (cid:17) , for τ > I / ( τ ) = (cid:112) − β (1 + β f ( τ )) , ˜ I / ( τ ) = (cid:112) − β f ( τ ) . (A.1)Since the imaginary part of the loop functions come only from f ( τ ) when τ >
1, a directcheck using Cutkosky rules indicates that the coefficient in front of f ( τ ) for I ( τ ) should havea factor β suppression with respect to the f ( τ ) coefficient in ˜ I ( τ ). Various closed-form loopfunctions in this paper are cross-checked using the package Program X [74].The energy-dependent heavy scalar partial width is,Γ q (ˆ s )( S → q ¯ q ) = 316 π ( y q β + ˜ y q ) β ˆ sm S , (A.2) For some recent development and an overview of the top quark polarization tagger, see Ref. [70]. – 36 –ith β ≡ (cid:113) − m q ˆ s . The energy dependence of the width has negligible effect for the narrowwidth case but becomes more relevant for the intermediate to large width case.The tree-level expressions for the SM QCD parton level differential cross sections for the t ¯ t background are d ˆ σ ( gg → t ¯ t ) dz = πα s s β (cid:18) ˆ s ˆ u ˆ t − (cid:19) ˆ u + ˆ t ˆ s d ˆ σ ( q ¯ q → t ¯ t ) dz = 2 πα s s β ˆ u + ˆ t ˆ s (A.3)where ˆ s, ˆ t, ˆ u are the Mandelstam variables and z is the cosine of the scattering angle betweenan incoming parton and the top quark. For collider analyses with detector acceptance, theevents from different regions of the phase space cannot be used in equal manner, especiallyfor light jets, we thus provide the differential distributions. However, as the top quark is notvery boosted and even forward events with z = ± z over the full range [ − ,
1] in our simplified analysis.For the scalar quarks the following abbreviations are used in the main text (in particular X u,d and Y u,d are defined in the alignment limit), D uL = 12 m W (1 −
13 tan θ W ) cos 2 βD uR = 23 m W tan θ W cos 2 βD dL = − m W (1 + 13 tan θ W ) cos 2 βD dR = − m W tan θ W cos 2 β (A.4) X u = A u − µ tan βX d = A d − µ tan βY u = A u tan β + µY d = A b tan β + µ, where θ W is the Weinberg angle. The stop parameters used in Fig. 10 are,zero LR mixing : m Q = 900 GeV , m t R = 400 GeV , X t = 0mh ∗ max : m Q = 900 GeV , m t R = 540 GeV , Y t = 2 X t = 3415 GeV (A.5)and the corresponding stop mass eigenstates are,zero LR mixing : m ˜ t = 436 GeV , m ˜ t = 916 GeVmh ∗ max : m ˜ t = 433 GeV , m ˜ t = 987 GeV– 37 –
00 450 500 550 600 650 700 - - - m ttreco ( GeV ) d σ S i g / d m tt r e c o ( pb / b i n ) Scenario B LHC 13 TeV @ - y t = m S =
550 GeV
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