Dip or nothingness of a Higgs resonance from the interference with a complex phase
KKIAS-P15020
Dip or nothingness of a Higgs resonancefrom the interference with a complex phase
Sunghoon Jung, ∗ Jeonghyeon Song, † and Yeo Woong Yoon ‡ School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea School of Physics, KonKuk University, Seoul 143-701, Korea
We show that new resonance shapes – a pure dip, nothingness and an enhanced pure peak –can be produced from the interference between resonance and continuum with a relative phase.Production conditions of those new shapes are derived based on a general parameterization of theinterference. The narrow width approximation is modified to work with the non-zero imaginary partof interference, and the correction factor can characterize the resonance shape. We demonstrate thatthe new resonance shapes of heavy Higgs bosons, H and A in the Type II aligned two Higgs doubletmodel, generally show up in gg → H /A → t ¯ t as well as b ¯ b and γγ channels. The pure A resonancedip in the t ¯ t channel is a particularly interesting signal as it can be probed well by the current searchtechniques that do not even take into account interferences; the high-luminosity LHC 14 TeV canperhaps probe a large part of its parameter space. I. INTRODUCTION
Many exciting discoveries in particle physics historycame with excesses above continuum backgrounds. Theexistence of a new resonance such as
J/ψ mesons [1, 2], W bosons [3, 4], Z bosons [5, 6], top quarks [7, 8], and therecently discovered Higgs boson [9, 10], was all confirmedby clear excesses or a resonance peak in the invariant ortransverse mass distribution. Most new physics searchesat the Large Hadron Collider (LHC) are also focused pri-marily on excesses.A pure Breit-Wigner (BW) resonance peak is, however,modified from the interference with a continuum or otherresonance processes. If the interference is purely real(except for the small imaginary part in a BW propaga-tor), a symmetric dip-peak or peak-dip structure appearsand adds to a BW resonance peak. As for the StandardModel (SM) Higgs boson in the γγ channel at the LHC,the real-part interference shifts the location of the reso-nance peak, affecting the pole mass measurements [11].But a resonance peak never disappears from the purelyreal interference.The modification on the pure BW resonance peak canbe much more significant and come with a more varietyof shapes if the interference involves a significant imag-inary part. The dip-like resonance shape induced fromthe imaginary part in elementary particle physics wasfirst theoretically glimpsed for the heavy Higgs boson in gg → H → t ¯ t [12]. The complex phase arises from theloop of top quarks as the Higgs boson is heavier thanthe top pair threshold. More dedicated study in thischannel [13] found that various dip-like resonances canappear for a wide range of the heavy Higgs boson massin the two Higgs doublet model (2HDM) and the minimalsupersymmetric standard model (MSSM). Since then, ∗ [email protected] † [email protected] ‡ [email protected] the resonance-continuum interferences including imagi-nary parts have been calculated in photon collider pro-cesses: γγ → H → W W [14], ZZ [15], t ¯ t [16] and b ¯ b [17] (as well as t ¯ t → ZZ [18]). A variety of resonanceshapes were found to appear, composed of dips and peakswith various depths and heights. On the other hand,hadron collider studies for gg → H → γγ [11, 19, 20], W W [21, 22] and ZZ [22, 23] found only peak-like shapesof the SM-like heavy Higgs boson. The CP violation, an-other source of a relative phase, was also considered inthe resonance-continuum interference [24], but the stud-ies were not focused on the resonance dip. Meanwhile,dip-like structures were observed in hadronic physics (seereferences in Ref.[13]) and were also expected to be pro-duced from time-like form factors in some particle physicsmodels [25].The diversity of resonance shapes due to the imaginarypart of interference complicates new resonance searches.In particular, the usual narrow width approximation(NWA) does not apply with non-zero imaginary parts,further hindering collider studies. Perhaps, even not allpossible resonance shapes have been systematically iden-tified and studied. Although resonance dip structureswere analyzed by J = 0 partial wave amplitudes for cer-tain processes [13, 26], a generalization to other processesis not straightforward without dedicated calculations.In this paper, we first aim to provide a general de-scription of the resonance-continuum interference with arelative phase in Sec. II. Three most striking new reso-nance shapes are identified – pure dips without associ-ated peaks, nothingness and enhanced pure peaks – andtheir mathematical conditions are derived. Based on thedescription, we find it possible to modify the NWA byintroducing a multiplicative correction factor C , whichcan also be used to tell shape characteristics. They willbe defined in Sec. II, and will be used to demonstrateand discuss various new resonance shapes of heavy Higgsbosons, H and A in the CP-conserving Type II 2HDM,in t ¯ t, b ¯ b and γγ channels (Sec. III). In Sec. III C, variousother Higgs processes will also be briefly discussed in re- a r X i v : . [ h e p - ph ] M a y gard of whether pure dips can be produced.The pure resonance dip is a particularly interestingsignal. As the pure dip has the BW shape (with just anegative sign), its signal is well localized and searches areeased; most importantly, it can be probed by the currentsearches that do not even take into account interferenceeffects. In Sec. III A, we will discuss LHC prospects ofpure A resonance dips in t ¯ t channel by using the latestsearch results and the modified NWA. II. GENERAL FORMALISM
We study the interference between the continuum andthe resonance in a 2 → d ˆ σdz = 132 π ˆ s (cid:88) (cid:12)(cid:12)(cid:12)(cid:12) A bg e iφ bg + M ˆ s − M + iM Γ · A res e iφ res (cid:12)(cid:12)(cid:12)(cid:12) , (1)where z = cos θ ∗ , θ ∗ is the scattering angle in the center-of-mass frame, and we properly sum and average overhelicities and colors. The first term is for the continuumbackground and the second is for the resonance with themass M and the width Γ. A i and φ i are the magnitudeand complex phase of each helicity amplitude, respec-tively. We factor out a BW propagator in the resonanceamplitude. The complex phase can be generated by ei-ther loop diagrams or CP-violating interactions.The interference effects are more clearly shown in thefollowing form:ˆ σ = ˆ σ bg + M (ˆ s − M ) + M w (2) × (cid:34) s − M ) M ˆ σ int c φ + ˆ σ res (cid:18) wR s φ (cid:19)(cid:35) , where s φ = sin φ , c φ = cos φ , andˆ σ bg , res = 132 π ˆ s (cid:90) dz (cid:88) A , res , (3)ˆ σ int e iφ = 132 π ˆ s (cid:90) dz (cid:88) A bg A res e i ( φ res − φ bg ) , (4) R = ˆ σ res ˆ σ int , w ≡ Γ M . (5)The w, R and φ are our key parameters. If the interfer-ence is dominated by a certain helicity amplitude and theresonance amplitude does not depend on the scatteringangle z (as is for a scalar resonance), the R and φ can bewell approximated by the dominant helicity amplitude as R (cid:39) A res A bg , φ (cid:39) φ res − φ bg . (6)These imply that R and φ are the relative strength andphase between the resonance and continuum, respec-tively. gg (cid:174)(cid:70)(cid:174) AB, LHC 14TeV c Φ (cid:61) (cid:72) b (cid:76) Φ(cid:61) Π (cid:72) a (cid:76) Φ(cid:61) (cid:45) Π wR (cid:62) (cid:72) c (cid:76) Φ(cid:61) (cid:45) Π wR (cid:61)
360 380 400 420 4400.20.51.02.05.010.0 M AB (cid:64) GeV (cid:68)
FIG. 1. Resonance shapes from the imaginary-part interfer-ences ( c φ = 0) for M = 400 GeV, Γ = 10 GeV and R = 0 . R = 0 .
05 for the case (c)). The vertical axis uses arbitraryunit. Solid lines are the results for (a) a pure dip in Eq. (7),(b) a pure enhanced peak ( φ = π/ φ = − π/ , w/R = 0 . Depending on R , φ and w for a given process, variousresonance shapes can arise. The first term in the secondline of Eq. (2), originated from the real-part interference( c φ ), is odd in ˆ s near M . The second term, the sumof the resonance square and the imaginary-part interfer-ence ( s φ ), is even in √ ˆ s . The relative strength and signbetween the first and second terms will determine theoverall resonance shape, parameterized well by R and φ .The simplest example is the well-known purely real-partinterference with s φ = 0 (no net relative phase); a peak-dip or dip-peak structure arises and shifts the final peakposition depending on the size and sign of ˆ σ int c φ .On the other hand, the non-zero imaginary interfer-ence can produce strikingly different resonance shapes.First of all, the purely imaginary interference can pro-duce a pure resonance dip if the following conditions aresatisfied: Pure dip: φ = − π , wR > . . (7)A pure dip in an example invariant mass distribution isshown in Fig. 1 by the line (a) for gg → Φ → AB at theLHC 14 TeV. The collider signature would be a deficitrather than an excess in the invariant mass distribution.The pure dip condition in Eq. (7) requires the pure neg-ative imaginary interference, a sizable decay width andrelatively small resonance-to-interference strength.We also show other interesting cases of the pure imag-inary interference in Fig. 1. As shown by the line(b), the resonance can be a pure peak if c φ = 0 and(1 + 2 ws φ /R ) > φ = π/
2) or suppressed ( φ = − π/
2) comparedto the result without interference (orange-dashed). Sur-prisingly, the resonance can even disappear if φ = − π/ w/R = 0 . s near the resonance, the imag-inary part survives under the integration of √ ˆ s acrossthe resonance mass, thus contributing to the total cross-section; on the other hand, the real part practically van-ishes under the integration. We, however, note that theimaginary part adds to the ˆ σ res , simply modifying thecoefficient of the BW-square by (1 + 2 wR s φ ) in Eq. (2).Therefore, the factor (1 + 2 wR s φ ) serves as the correctionfactor to the NWA as if the original resonance-square sig-nal were multiplied by the factor. Thus, we suggest the modified NWA as: σ ( ab → Φ → cd ) w / intf = σ ( ab → Φ) · Br(Φ → cd ) · C, (8)where the subscript “ w/ intf ” emphasizes that full inter-ference effects are included, and the correction factor C is defined as C ≡ (cid:18) wR s φ (cid:19) . (9)For narrow resonances, the energy dependences of R, w and φ can be ignored, and we evaluate them at ˆ s = M .The observation of resonance shape can be limited bythe resolution of the invariant mass. If the width is muchsmaller than the experimental resolution, the observablewould just be the resonance signal integrated over theexperimental bin size. In such case, the correction factor C in Eq. (9) can also serve as a useful measure of howthe resonance would be observed: C < , (10) C = 0 : nothingness ,C > . The “nothingness” can be resulted either from the com-plete disappearance of a resonance ( φ = − π/ , C = 0) orfrom remaining symmetric dip-peak or peak-dip struc-tures ( φ (cid:54) = − π/ , C = 0). We will use the sign of the C factor to discuss resonance shapes in the general param-eter space of heavy Higgs bosons in the next section. III. THE ABNORMALITY OF HEAVY HIGGSRESONANCES IN THE ALIGNED 2HDM
The interference effects from a resonance particle be-come significant when the width is not too narrow. Heavyneutral Higgs bosons, which are ubiquitous in many newphysics models with the extended Higgs sector, are goodcandidates for sizeable total width. In addition, theirproduction through the gluon fusion at one-loop level develops a complex phase when the heavy Higgs bo-son masses are above the t ¯ t or b ¯ b thresholds. As oneof the simplest extensions of the SM, we consider theheavy Higgs bosons in the CP-conserving aligned TypeII 2HDM.A 2HDM [27] introduces two complex SU (2) L Higgsdoublet scalar fields, both of which have nonzero vac-uum expectation values v and v . There are five phys-ical Higgs boson degrees of freedom, the light CP-evenscalar h , the heavy CP-even scalar H , the CP-oddpseudoscalar A , and two charged Higgs bosons H ± . Weconsider the resonance shapes of H and A . The SMHiggs field is a mixture of h and H as H SM = sin( β − α ) h + cos( β − α ) H , (11)where α is the mixing angle between h and H andtan β = v /v . Since the current LHC Higgs data prefersa very SM-like Higgs boson [28], we simply assume the ex-act alignment limit, sin( β − α ) = 1. Then we discuss eachphenomenology of H and A in the two-dimensional pa-rameter space of M H/A and tan β . The Yukawa cou-plings normalized by the SM ones becomeˆ y Ht = − y Hb = − β , ˆ y At = 1ˆ y Ab = 1tan β . (12)Note that the H - t -¯ t and H - b -¯ b vertices have the oppositesign while A - t -¯ t and A - b -¯ b vertices have the same sign.Important decay modes of H and A are t ¯ t , b ¯ b , τ + τ − , γγ , γZ , hh and Zh . The ZZ and W W decay modes areforbidden at the tree level in the alignment limit. In thefollowing subsections, we study t ¯ t , b ¯ b and γγ channels tosee whether striking resonance shapes are produced andto classify the parameter space as pure dip, nothingnessand enhanced peak regions. We also briefly discuss otherchannels in the last subsection. A. Dips and nothingness in gg → H/A → t ¯ t . For M H/A ≥ m t , the dip signal can arise in t ¯ t finalstates [12, 13]. We delineate how the dip conditions aresatisfied, and we discuss various other resonance shapesin a general parameter space. In the interference between gg → H/A → t ¯ t and the continuum background gg → t ¯ t ,a sizable complex phase is naturally generated from thetop quark loop. Since the resonance process is one-loopsuppressed compared to the continuum process and theheavy Higgs width is usually O (1) GeV, the R is roughlyof the same order or smaller than w in most of the pa-rameter space. Therefore, the necessary dip condition w/R > . There is another model parameter, the soft Z symmetry break-ing term m . The m affects the Higgs triple couplings, whichdo not play a significant role in the H/A decays into t ¯ t, b ¯ b, γγ . The helicity amplitudes for the one-loop resonance pro-cess gg → H/A → t ¯ t are M H/Aλ = δ ab δ ij (cid:16) α s π (cid:17) ˆ y t m t G F √ s (cid:102) M H/Aλ (13) × ˆ s ˆ s − M + iM Γ (cid:88) q ˆ y q A H/A / ( τ q ) , where λ collectively denotes the helicities of two gluonsand two photons, M = M H/A , τ q = M / m q , and theloop functions A H/A / are given in Ref. [29]. The nor-malised nonzero helicity amplitudes in the chiral repre-sentation [30] are (cid:102) M H ++++ = − (cid:102) M H ++ −− = − β t , (cid:102) M A ++++ = (cid:102) M A ++ −− = − , (cid:102) M H/A −−−− = − (cid:102) M H/A ++++ , (cid:102) M H/A −− ++ = − (cid:102) M H/A ++ −− , (14)where β t = (cid:112) − m t / ˆ s . Obviously, the Higgs contribu-tions have only color-singlet amplitudes.The helicity amplitudes of the tree level continuumprocesses are M bg , λ = δ ab δ ij πα s N c · (cid:102) M bg λ , (15)where we show only color-singlet components, δ ab δ ij .The normalized helicity amplitudes interfering with theHiggs contributions in Eq. (15) are (cid:102) M bg++++ = − (cid:102) M bg −−−− = (1 + β t ) m t ˆ s / ( t − m t ) ( u − m t ) , (16) (cid:102) M bg++ −− = − (cid:102) M bg −− ++ = (1 − β t ) m t ˆ s / ( t − m t ) ( u − m t ) . The dominant interference is through (cid:102) M ±±±± . Pure dips.
We first focus on pure dips and delineatehow and where in parameter space the dip conditionsare satisfied. The main sources of the relative phase are(i) the top-quark loop in the resonance process and (ii)the relative minus sign between the resonance and thecontinuum amplitudes. Interestingly, the dominantly in-terfering helicity amplitudes for continuum, M ++++ and M −−−− , have opposite sign to the resonance amplitudesas shown in Eqs. (14) and (16). We plot the resulting rel-ative phase φ in Fig. 2. The relative phase approachesto − π for M H/A (cid:46) m t with tan β (cid:46) m t and grows while the real term decreases and eventuallyflips its sign. When the real term crosses zero, the con-dition φ = − π/ A and H in Fig. 2: M A = 850and M H = 1170 GeV for tan β = 2. In these solutions,the relative minus sign between the resonance and con-tinuum amplitudes is essential. Obviously, light quark
500 1000 1500 20000 (cid:45) Π (cid:45) Π (cid:45) Π (cid:45)Π
10 20 302 M H (cid:144) A (cid:64) GeV (cid:68) Φ Τ t (cid:61) M H (cid:144) A (cid:144)(cid:72) m t (cid:76) HA tan
Β(cid:61)
2, 5, 10
FIG. 2. The relative phase, φ in Eq. (4), between the contin-uum gg → t ¯ t and the resonance gg → A/H → t ¯ t as a function M H/A for tan β = 2 , , M A = 850 GeV and M H = 1170GeV can achieve φ = − π/ t β = 2. In the upper horizon-tal axis, we also show the corresponding τ t = M H/A / (4 m t )used in A H/A / ( τ t ). contributions are suppressed by small quark masses, andthe tree-level continuum is purely real.For tan β (cid:38)
5, the bottom-quark loop also inducesthe sizable imaginary part. The bottom-loop phase adds(subtracts) to the top-loop phase for A ( H ) becauseˆ y b has the same (opposite)-sign with ˆ y t ; having alreadysign-flipped real part, the bottom-loop cancels (adds to)the top-loop real parts. When the real part of the toploop is canceled by that of the b quark loop, φ = − π/ A ( H ) boson obtains φ = − π/ β (cid:38) − ◦ ≤ φ ≤ − ◦ ,in which region a resonance will almost look like a puredip as will be shown in Fig. 4 left panel. As discussed,any 2 m t ≤ M A (cid:46)
850 GeV can appear as a pure t ¯ t dipwith some larger tan β , whereas only the H heavier thanabout 1170 GeV can appear as a pure dip. The C factoris also shown in Fig. 3, and is, of course, negative inthe pure dip region. We also show the integrated signalrate using the modified NWA as dashed lines. Alongthe φ = − ◦ contour, the pure dip deficit signal ratedecreases with the increasing M A , but the rate increasesagain near M A (cid:38)
800 GeV. The returning increase isbecause tan β becomes very small there; the deficit signalrate is also plotted as a red-solid line in Fig. 5. Usingthe modified NWA rate, we will study pure dip searchprospects in the later part of this subsection. Nothingness and other effects.
Another interesting res-onance shape, nothingness, can also appear in this finalstate. Let us first discuss the A nothingness in the leftpanel of Fig. 3. As M A becomes lighter or tan β becomessmaller from the pure dip region, the phase deviates from φ = − ◦ . As | s φ | becomes small enough so that the (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) P u r e D i p (cid:45)
400 500 600 700 800 900 100051015202530 M A (cid:64) GeV (cid:68) t a n Β C (cid:61) (cid:72) (cid:43) (cid:144) Rs Φ (cid:76) for gg (cid:174) A (cid:174) tt (cid:45) (cid:45) (cid:45)
11 10204060 (cid:45) (cid:45) (cid:45) (cid:45) P u r e D i p (cid:45) (cid:45)
400 600 800 1000 1200 140051015202530 M H (cid:64) GeV (cid:68) t a n Β C (cid:61) (cid:72) (cid:43) (cid:144) Rs Φ (cid:76) for gg (cid:174) H (cid:174) tt FIG. 3. The C factor in Eq. (9) (solid) and the integrated t ¯ t signal rate (dashed) at the 14 TeV LHC obtained by using themodified NWA in Eq. (8). They are shown for A (left) and H (right) Higgs bosons in the aligned Type II 2HDM. Greenregions are for nearly pure dips ( − ◦ ≤ φ ≤ − ◦ ) and red for nearly nothingness ( − . ≤ C ≤ .
546 548 550 552 554900910920930940950960 M t t (cid:64) GeV (cid:68) d Σ (cid:144) d M tt (cid:64) f b (cid:144) G e V (cid:68) M A (cid:61)
550 GeVLHC 14 TeV gg (cid:174) A (cid:174) tt Φ(cid:61)(cid:45) ° Φ(cid:61)(cid:45) ° Φ(cid:61)(cid:45) °
835 840 845 850 855 860 865110115120125130 M t t (cid:64) GeV (cid:68) d Σ (cid:144) d M tt (cid:64) f b (cid:144) G e V (cid:68) M A (cid:61)
850 GeV, (cid:71) A (cid:61) Β(cid:61) (cid:174) A (cid:174) tt Φ(cid:61)(cid:45) ° FIG. 4. Pure dips in the full physical t ¯ t invariant mass distributions from pp → t ¯ t at the 14 TeV LHC; the results include theresonance gg → A → t ¯ t interfering with the continuum gg → t ¯ t as well as the q ¯ q → t ¯ t continuum. We show for M A = 550 GeVwith tan β = 13 . , . , . φ = − ◦ , − ◦ , − ◦ (left) and M A = 850 GeV with tan β = 2 . imaginary-part interference becomes comparable to ˆ σ res but still with relative minus sign in Eq. (9), then the C factor can vanish. The nothingness region spanning − . ≤ C ≤ . φ (cid:54) = − π/ , C = 0), andthe search is likely challenging as the integrated rate issmall unless a careful shape analysis can be done with agood experimental resolution. Likewise, the H boson inthe Fig. 3 right panel can also show up as nothingnessfor M H <
550 GeV and small tan β . This region is alsosomewhat parallel to the pure dip region. For both A and H , in the region between nothingnessand pure dip bands, the C factor is negative. Acrossthe nothingness region, the C factor flips its sign, andexcesses now will be observed in light enough M A/H (cid:46) ,
550 GeV and large enough tan β for H .Although t ¯ t signal estimated from the modified NWAis naively strongest for tan β (cid:46)
5, the imaginary-partinterference introduces potential challenges; complicatedresonance shapes and possibly small integrated rates nearthe nothingness region will need careful shape analysis– the pure dip can be an interesting exception as willbe discussed. The experimental search in these regionsmay be aided by other production and/or decay channelsthat are not much complicated by interference effects; see
400 500 600 700 8000.010.050.100.501.005.0010.00 M A (cid:64) GeV (cid:68) tt d e f i c it r a t e (cid:64) pb (cid:68) L H C T e V , (cid:144) f b L H C T e V , (cid:144) f b L H C T e V , (cid:144) a b FIG. 5. The 95% CL exclusion prospects of the pure t ¯ t resonance dip of A from the latest 8 TeV 20/fb expectedresult [33] and its 14 TeV projections for 300/fb and 3/ab.The modified NWA in Eq. (8) is used to obtain total deficitsignal rates at LO (solid red) and with a constant NLO K =1 . − ◦ ≤ φ ≤ − ◦ as in Fig. 3 and is terminated for tan β ≥ . e.g. Refs. [31, 32] for the study of t ¯ tH/A → t ¯ tt ¯ t and b ¯ bH/A → b ¯ bt ¯ t associated productions. Pure dip demonstration.
We now demonstrate the puredips in the physical M t ¯ t distributions in Fig. 4 for the A boson. Results are for M A = 550 GeV and M A = 850GeV. For M A = 550 GeV shown in the left panel, weshow three results for φ = − ◦ , − ◦ , − ◦ corre-sponding to tan β = 13 . , . , .
6. For M A = 850 GeV,we used tan β = 2 . φ = − ◦ . Indeed, very cleardips are produced. The left panel also demonstrates thatany resonance with − ◦ ≤ φ ≤ − ◦ appear almostlike pure dips. The integrated deficit rates for thosethree 550 GeV shapes are −
27 fb, −
37 fb, −
53 fb, pre-dominantly determined by tan β , not by φ . The smaller φ (such as φ ∼ − ◦ ), the smaller tan β is needed andthe larger signal rates are obtained. The 850 GeV shapehas a similar integrated deficit rate −
37 fb. Although itsshape is broader than the M A = 550 GeV case, its discov-ery prospect is higher due to smaller background as willbe discussed. In the same Fig. 4, we also show the whole pp → t ¯ t continuum result by dashed lines for compari-son. The leading order (LO) results without top decays,kinematic cuts on the top and any Gaussian smearingare used. We have adopted MSTW LO PDF with scales µ R,F = M A . Pure dip LHC prospects.
In general, the collider studyof heavy Higgs t ¯ t resonances is complicated by the com-plex resonance shapes and the large smearing of the toppair invariant mass distribution. The latest experimentalresolution of the invariant mass is about 6–8% [33, 34],which is bigger than any Higgs widths in the parameterspace shown in Fig. 3 ( w (cid:46) not take into accountany interference effects. The current searches model ascalar resonance as a pure BW peak without any inter-ference effects, but they are inherently sensitive to deficitstoo [33, 34]. A pure dip has the pure BW shape with anegative sign as if no interferences existed. Thus, pre-sumably, the current search techniques and results with-out interferences can be applied to pure peaks and dipsequally well.The current searches then take into account all realisticeffects from detector resolution, smearing and selectioncuts, and provide upper limits on the true signal ratesof pure peaks (and dips) before any selection cuts. Wecan use the modified NWA in Eq. (8) to obtain pure dipsignal rates and simply compare them with the reportedexperimental upper bound.The result is shown in Fig. 5. The expected result of thelatest 8 TeV resonance search [33] is shown with its errorbands. The integrated pure dip signal rates are shown asthe thick red line. As discussed, the rate decreases withthe M A , but it rapidly increases back after M A (cid:38) β satisfying the pure dip conditionsbecomes very small. We also show a red band around thepure dip line for − ◦ ≤ φ ≤ − ◦ ; the upper (lower)red band boundary is from φ = − ◦ ( − ◦ ). The redband is terminated to have tan β ≥ .
5. We have used LOresults for signals. But to roughly grasp the potentiallyimportant next-to-leading-order (NLO) effects, we alsoshow the signal rates by multiplying an assumed constantNLO K = 1 . t ¯ t resonance search is characterized most importantly bya single resonance mass scale M so that both signal-to-background ratio and cut efficiencies remain more or lessconstant over a wide range of proton-proton collision en-ergy; see discussions and validations in Refs. [38, 39].Simply speaking, we use the following relation to ob-tain the projected upper bound σ i bound (superscrippedby i ) from the current (superscripped by j ) upper bound σ j bound as σ i bound σ j bound = (cid:115) P igg P jgg (cid:114) L j L i , (17)where the data luminosity L k and the gluon-gluon partonluminosity function ( k = i, j ) P kgg ≡ (cid:90) τ k dx τ k x f g ( x, Q ) f g ( τ k /x, Q ) , (18)for τ k ≡ M H/A /s k . The 14 TeV projections with300 fb − and 3 ab − are shown in Fig. 5. The projections (cid:45) (cid:45) (cid:45) P u r e D i p
200 300 400 500 600 700 80051015202530 M A (cid:64) GeV (cid:68) t a n Β C (cid:61) (cid:72) (cid:43) (cid:144) Rs Φ (cid:76) for gg (cid:174) A (cid:174) bb FIG. 6. The C factor in Eq. (9) (solid) and the integrated b ¯ b signal rate (dashed) at the 14 TeV LHC obtained using themodified NWA in Eq. (8). Green region is for nearly puredips ( − ◦ ≤ φ ≤ − ◦ ) and red for nearly nothingness( − . ≤ C ≤ . imply that ∼ O (100 − M A (cid:38)
600 GeV. So far, we have ignored anykinematic differences between the top quark producedfrom resonance-square and that from the imaginary-partinterference. We will present a detailed study in our fu-ture publication [36].Although the pure dip prospect in Fig. 5 can be af-fected by various others such as systematics and kine-matics, we think it is a good guideline for future searchesand the method used for this study is a remarkable ap-plication of our results.
B. Dip, nothingness, or peak in gg → H/A → b ¯ b, γγ The gg → H/A → b ¯ b process through the triangle dia-gram at one loop level interferes with the gg → b ¯ b back-ground at tree level. The helicity amplitudes of the signaland the background are the same as Eqs. (13) − (16) withthe replacement of m t → m b , ˆ y t → ˆ y b , and β t → β b .In Fig. 6, we show the C factor and the integrated signalrate for A using the modified NWA. In the whole param-eter region of 200 GeV < M A <
800 GeV, the C factor isalways small positive, like 0 . ∼ .
8. We have the smallerpeak than the BW one. Above the t ¯ t threshold, the smallbranching ratio of A → b ¯ b itself suppresses the modifiedsignal rate further. For M A > M A < m t , there is a nearly nothingness regionat tan β (cid:39)
10 having 0 . < C < .
1. In this region φ (cid:39) π/
2, thus almost a complete destruction occurs.This can be understood by the loop function. A A / ( τ t ) isa real positive number of the order of one for M A < m t . A A / ( τ b ) has negative real part and positive imaginarypart, both being of the order of 0 .
01. Since ˆ y t and ˆ y b have the same sign, tan β (cid:39)
10 brings about the cancel-lation of the real-part interference. Then, we have a purenegative imaginary interference ( φ = − π/
2) and suffi-ciently small R value to fulfil the remaining nothingnesscondition ( w/R = 0 . H with opposite sign ˆ y t and ˆ y b , the correction fac-tor C is alway positive in the range of 0 . ∼ .
5. Minorpeaks will appear. The current b ¯ b resonance search for M bb > σ × Br × A about a few hundred fb, where A is the acceptance. Thesmall C factor worsens the poor observability. However,the γγ collider, which is expected to have much highersensitivity, can probe the process γγ → H/A → b ¯ b in-terfering from the continuum γγ → b ¯ b . This has verysimilar features to gg → H/A → b ¯ b except for the colorfactor.Another potentially important decay channel of theheavy H and A is into γγ . The resonance process is atthe two-loop level and the continuum gg → γγ is throughthe box-type diagram at one loop level. Their leadingorder helicity amplitudes are given in Refs. [11, 19]. Wefind that M H/A ±±±± give the dominant interference, whichhave positive sign relative to the background.Figure 7 shows the correction factor C and the mod-ified single rate σ ( pp → H/A → γγ ) by the inter-ference effects at the 14 TeV LHC. Here we consider0 . < tan β < β severely suppressesboth production and decay so that the signal rate be-comes very tiny.Above the t ¯ t threshold, the most parameter space haspositive values of C for A ( H ) if M A <
700 GeV( M H < pp → H/A → γγ withtwo triangle diagrams, there are three kinds of contribu-tions, top-triangle times top-triangle, top-triangle times b -triangle, and b -triangle times b -triangle. When thetop-contribution is dominant, the total complex phase φ is just twice the phase from a single loop function A H/A / . Before (cid:60) e A H/A / ( τ t ) crosses zero, the argument of A H/A / ( τ t ) is in [0 , π/ φ leads to positive C factor.Below the t ¯ t threshold, the C factor for A stays pos-itive. Although the C factor can be as large as ten attan β (cid:39)
7, the signal rate is too small below 0.01 fb, whichis practically impossible to probe. For tan β = 0 . ∼ C factor is almost constant, about one: we do nothave higher peak structures than the BW one. For H , on the contrary, there is a nothingness strip fortan β = 4 ∼ b quark contribution to the loopfunction becomes sizable to cancel the top quark contri-bution. Above the nothingness line, there is a region fordip-like resonances. Very the small modified NWA rate
13 35 57935 79 P u r e P ea k
200 300 400 500 600 700 8001234567 M A (cid:64) GeV (cid:68) t a n Β C (cid:61) (cid:72) (cid:43) (cid:144) Rs Φ (cid:76) for gg (cid:174) A (cid:174)ΓΓ
44 8 (cid:45) (cid:45) P u r e P ea k (cid:45) (cid:45)
200 300 400 500 600 700 8001234567 M H (cid:64) GeV (cid:68) t a n Β C (cid:61) (cid:72) (cid:43) (cid:144) Rs Φ (cid:76) for gg (cid:174) H (cid:174)ΓΓ FIG. 7. The C factor in Eq. (9) (solid) and the integrated γγ signal rate (dashed) obtained by using the modified NWA forthe A (left) and H (right) bosons in the aligned Type II 2HDM at the 14 TeV LHC. Yellow regions are for nearly pure peaks(80 ◦ ≤ φ ≤ ◦ ) and red for nearly nothingness ( − . ≤ C ≤ . makes it almost impossible to probe them. C. Other candidate processes for heavy Higgs dips
We have so far explicitly studied t ¯ t , b ¯ b and γγ channelsin the alignment limit of 2HDM. In this subsection, webriefly discuss what other hadron collider processes andwhat modifications of the aligned 2HDM model can pro-duce Higgs resonance dips. The discussion will also showhow the dip conditions derived from our general formal-ism can be usefully applied to a wide range of processesand models.The gg → H → W W, ZZ with the heavy SM-like Higgs boson were found to produce peak like res-onances [21–23]. The immediate reason is R ∼ w ∼ w/R > . R can besuppressed if the heavy Higgs boson, H in 2HDM, canhave non-zero but suppressed W W, ZZ couplings pro-portional to cos( β − α ) (deviating from the alignmentlimit). The relative phase arises from the quark loops inthe gg → H , the sign of cos( β − α ) and the unknownrelative sign between the resonance and the continuumamplitudes. The loop phase from gg → H can be ± ◦ as discussed. As one can somewhat freely choose the signof cos( β − α ), the unknown overall sign can perhaps bechosen to satisfy the dip condition φ = − ◦ . There-fore, pure ZZ, W W dips may be produced in this kindof models, but a dedicated calculation is needed to checkthis. The gg → H → γγ for the heavy Higgs in the 2HDMwith M H ∼
200 - 300 GeV can also produce pure reso-nance dips if we deviate from the alignment limit (Fig. 7shows that it is not in the alignment limit). Away fromthe alignment limit, the H → γγ decay one-loop diagramcan have both W - and top-loops. The main phase arisesfrom the W -loop, but the real parts can be canceled be-tween the W - and top-loops, depending on cos( β − α ) andtan β . The R is already loop-suppressed as desired; thus,pure dip conditions can be satisfied. We have numeri-cally checked that this is indeed possible. Specifically, M H = 200 GeV, cos( β − α ) (cid:39) − .
15 and tan β (cid:39) . − . (cid:46) cos( β − α ) (cid:46) (cid:46) tan β (cid:46)
10 for M H ∼
200 - 300 GeV. The collidersearch is, however, likely challenging in the near futurebecause the signal rate is small, limited by the Higgs cou-pling constraints on the size of | cos( β − α ) | (cid:46) .
15 [28]and by partial cancelation of real parts.The gg → H → Zγ is similar to the γγ channel. Theresonance is a two-loop process while the continuum is anone-loop process, giving a loop-suppressed R as desired.As soon as the overall sign comes out correctly, Higgsdips can perhaps be produced. But a dedicated study isneeded to check this.Other processes with the light Higgs boson in the fi-nal states such as gg → H → hh and gg → A → Zh may have similar conclusion as the ZZ and W W chan-nels. Both the resonances and continuum are one-loopprocesses, giving R ∼
1; but the cos( β − α ) suppressionmay play a proper role in producing a dip. But the re-constuction of heavy Higgs resonances in these channelsmight be more challenging than in the ZZ channel.Another important Higgs channel is gg → H/A → τ τ . Since the interfering continuum does not exist up tothe one-loop order, the interference is small and R (cid:29) t ¯ tH ,are tree-level processes and may not have large enoughcomplex phases. Without additional CP violations, theimaginary interference effects will be absent. IV. CONCLUSIONS
We have identified new resonance shapes – pure dips,nothingness and enhanced pure peaks – arising fromthe resonance-continuum interference with a relativephase and derived their production conditions. We havedemonstrated that all of these new shapes can generallyshow up from heavy Higgs bosons, A and H , in t ¯ t , b ¯ b and γγ channels. The pure dip conditions are also ap-plied to various other Higgs processes for the discussionof whether Higgs dips can be produced.In addition, we have modified the NWA to work withthe non-zero imaginary part of interference in Eq. (8); itis now possible to easily estimate the total signal rate ofthe resonance including the imaginary-part interference.The multiplicative correction factor C , defined in Eq. (9),could also categorize the resulting collider observables asexcesses, deficits and nothingness.The collider searches of dip-like heavy Higgs resonances are most promising in t ¯ t channel but, in general, needdedicated shape analyses. Remarkably, for pure t ¯ t dips of A bosons, it was useful to apply the modified NWA andthe latest t ¯ t resonance searches (that do not even accountfor any interference effects yet) to obtain LHC constraintsand prospects (Fig. 5). It was possible because a pure diphas the negative BW shape as if no interferences existed.We hope that this pure dip study in t ¯ t can be seriouslyconsidered by LHC experiments and shed more light ongeneral studies of complex t ¯ t Higgs resonances.A resonance dip or nothingness, a special result ofthe imaginary-part interference, is an overlooked phe-nomenon in heavy (Higgs) resonance searches. Theimaginary-part interferences will be more important innew physics models; presumably, heavier resonances fromnew physics beyond the SM are broader and have moresources of imaginary terms from loop corrections andnew CP violations. Combined with other measurements,resonance shapes will add valuable information on newphysics.
ACKNOWLEDGMENTS
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