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IISc-CHEP/10/05 hep-ph/0509070 Signatures of anomalous VVH interactions at a linear collider Sudhansu S. Biswal1,∗, Debajyoti Choudhury2,†, Rohini M. Godbole1,‡, and Ritesh K. Singh1,§ 1Center for High Energy Physics, Indian Institute of Science, Bangalore, 560012, India 2Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India and 7 HarishChandra Research Institute, 0 Chhatnag Road, Jhusi, Allahabad 211019, India 0 2 Weexamine, in a model independentway,thesensitivity of a Linear Collider tothecouplings of alightHiggsbosontogaugebosons. IncludingthepossibilityofCP violation,weconstructseveral n observablesthatprobethedifferentanomalouscouplingspossible. ForanintermediatemassHiggs, a a collider operating at a center of mass energy of 500 GeV and with an integrated luminosity of J 500 fb−1 is shown tobe able to constrain theZZH vertexat thefew percent level, and with even 0 highersensitivityincertaindirections. However,thelackofsufficientnumberofobservablesaswell 3 ascontamination from theZZH vertexlimits theprecision with which theWWH couplingcan be measured. 3 v PACSnumbers: 14.80.Cp,14.70.FM,14.70.Hp 0 7 0 I. INTRODUCTION the minimal SM. For one, the only neutral scalar in the 9 SMis a JCP =0++ state arisingfromaSU(2) doublet 0 L withhypercharge1,whileitsvariousextensionscanhave 5 Although the standard model (SM) has withstood all 0 possible experimental challenges and has been tested several Higgs bosons with different CP properties and / U(1) quantum numbers. The minimal supersymmetric h to an unprecedented degree of accuracy, so far there standardmodel (MSSM), for example, has two CP-even p has been no direct experimental verification of the phe- states and a single CP-odd one [9]. Thus, should a neu- - nomenon of spontaneous symmetry breaking. With the p tral spin-0 state be observed at the LHC, a study of its latterbeing consideredacentralpillarofthis theoryand e CP-propertywouldbe essentialtoestablishitastheSM h itsvariousextensions,thesearchforaHiggsbosonisone Higgs boson [10]. : ofthemainaimsformanycurrentandfuturecolliders[1]. v Since, at an e+e− collider, the dominant production WithintheSM,theonlyfundamentalspin-0objectisthe i X (CP-even) Higgs boson and remains the only particle in modes of a neutral Higgs boson proceed via its coupling with a pair of gauge bosons (VV, V = W,Z), any r theSMspectrumtobefoundyet. Rather,alowerbound a change in the VVH couplings from their SM values can on the mass of the SM Higgs boson, (about 114.5 GeV) be probed via such production processes. Within the isprovidedbythe directsearchesatthe LEPcollider[2]. SM/MSSM, the only (renormalizable) interaction term Electroweakprecisionmeasurements, on the other hand, involving the Higgs boson and a pair of gauge bosons provideanupperboundonitsmassofabout204GeVat is the one arising from the Higgs kinetic term. How- 95%C.L.[3]. Itshouldberealizedthatboththeselimits ever, once we accept the SM to be only an effective low- are model dependent and may be relaxed in extensions energy description, higher-dimensional (and hence non- of the SM. For example, the lower limit can be relaxed renormalizable) terms are allowed. If we only demand in generic 2-Higgs doublet models [4] or in models with Lorentz invariance and gauge invariance, the most gen- CP violation [5]. In the latter case, direct searches at eral coupling structure may be expressed as LEPandelsewherestillallowthelightestHiggsbosonto be as light as 10 GeV [6]. Similarly, the upper bound on b the mass of the (lightest) Higgs in some extensions may Γµν = gV aV gµν + mV2 (k1νk2µ−gµν k1·k2) be substantially higher [7]. The Large Hadron Collider (cid:20) V (LHC) is expected to be capable [8] of searching for the b˜ Higgs boson in the entire mass range allowed. +mV2 ǫµναβ k1αk2β (1) V # It is then quite obvious that just the discovery of the Higgs bosonat the LHC will notbe sufficientto validate where k denote the momenta of the two W’s (Z’s), i gSM = e cotθ M and gSM = 2 e M /sin2θ . In W W Z Z Z W thecontextoftheSM,atthetreelevel,aSM =aSM = 1 W Z while the other couplings vanish identically. At the one- ∗Electronicaddress: [email protected] loop level or in a different theory, effective or otherwise, †Electronicaddress: [email protected] ‡Electronicaddress: [email protected] thesemayassumesignificantlydifferentvalues. Westudy §Electronicaddress: [email protected] thismostgeneralsetofanomalouscouplingsoftheHiggs 2 boson to a pair of Ws and Z at a linear collider (LC) in theprocessese+e− ff¯H,withf beingalightfermion. → e+Te−he varfifo¯uHs,kpinroemceaetdiicnagl dviisatrvibecuttoiornbsofsoorntfhuesiponrocaensds e+ f(cid:22) e+ V f(cid:22) → V Higgsstrahlung, with unpolarized beams has been stud- f V ied in the context of the SM [11]. The effect of beam H polarization has also been investigated for the SM [12]. The anomalous ZZH couplings have been studied in V (cid:0) Refs.[13, 14, 15, 16, 17] for the LC and in Refs.[18, 19] e(cid:0) f e H for the LHC in terms of higher dimensional operators. (a) (b) Ref. [20]investigatesthe possibility to probe the anoma- lousVZH couplings,V =γ,Z,usingtheoptimalobserv- FIG. 1: Feynman diagrams for the process e+e− → ff¯H; able technique [21] for both polarized and un-polarized (a) is t-channel or fusion diagram, while (b) is s-channel or beams. Ref. [22], on the other hand, probes the CP- Bjorken diagram. For f = e,νe both (a) & (b) contribute violating coupling ˜bZ by means of asymmetries in kine- while for theall otherfermions only (b) contributes. maticaldistributionsandbeampolarization. InRef.[23], the VVH vertex is studied in the process of γγ H W+W−/ZZ using angular distributions of th→e dec→ay can arise only at a higher order in a renormalizable products. theory [24]. Furthermore, within models such as the The rest of the paper is organized as follows. In sec- SM/MSSM where the tree-level scalar potential is CP- tion II we discuss the possible sources and symmetries conserving,˜b maybegeneratedonlyatanorderofper- V of the anomalous VVH couplings and the rates of vari- turbation theory higher than that in which the Higgs ousprocessesinvolvingthesecouplings. InsectionIII we sector acquires CP-violating terms. However, in an ef- construct several observables with appropriate CP and fectivetheory,whichsatisfiesSU(2) U(1)symmetry,the T˜ property to probe various ZZH anomalous couplings. couplings b and˜b can arise, at th⊗e lowest order, from V V In section IV we construct similar observables to probe terms suchas F FµνΦ†Φ orF F˜µνΦ†Φ [25] whereΦ µν µν anomalous WWH couplings, which we then use along is the usual Higgs doublet, F is a field strength tensor µν with the ones constructed for the ZZH case. Section V andF˜µν itsdual. Itcanbeeasilyascertainedthattheef- contains a discussion and summary of our findings. fects ofthe higherdimensionalterms inthe trilinearver- tices of interest can be absorbed into b (˜b ) by ascrib- V V ingthemwithnon-trivialmomentum-dependences(form II. THE VVH COUPLINGS factor behavior). Clearly, if the cut-off scale Λ of this theory is much larger than the typicalenergy at which a The anomalous VVH couplings in Eq.(1) can appear scatteringexperimentistobeperformed,thesaiddepen- fromvarious sourcessuch as via higher ordercorrections dence would be weak. In all processes that we shall be to the vertex in a renormalizable theory [24] or from considering, this turns out to be the case. In particular, higher dimensional operators in an effective theory [25]. the Bjorken process (Fig. 1(b)) essentially proceeds at a For example, in the MSSM, the non-zero phases of the fixed center-of-mass energy, hence both b and ˜b are Z Z trilinear SUSY breaking parameter A and the gaug- constant for this process. Even for the other processes ino/higgsino mass parameters can induce CP-violating of interest, namely gauge boson fusion (Fig. 1(a)), the termsinthescalarpotentialatoneloopleveleventhough momentumdependenceofthe form-factorshavearather the tree level potential is CP-conserving. As a conse- minor role to play, especially for Λ>1TeV. This sug- quence, the Higgs-boson mass eigenstates can turn out gests that we can treat a ,b ,˜b a∼s phenomenological to be linear combinations of CP-even and -odd states. V V V and energy-independent parameters. This modifies the effective coupling of the Higgs boson to the known particles from what is predicted in the SM (or even from that within a version of MSSM with no A. Symmetries of anomalous couplings CP-violation accruing from the scalar sector). In a generic multi-doublet model, whether supersym- A consequence of imposing an SU(2) U(1) symme- metric[26]orotherwise[27],the couplingsofthe neutral ⊗ try would be to relate the anomalous couplings, b and Higgs bosons to a pair of gauge bosons satisfy the sum W ˜b , for the WWH vertex with those for the ZZH ver- rule W tex. However, rather than attempting to calculate these a2 =1. couplings within a given model, we shall treat them as VVHi purelyphenomenologicalinputs,whoseeffectonthekine- i X matics of various final states in collider processes can be Thus, while aVVHi for a given Higgs boson can be sig- analyzed. Ingeneral,eachofthesecouplingscanbecom- nificantly smaller than the SM value, any violation of plex, reflecting possible absorptiveparts of the loops, ei- the above sum rule would indicate either the presence ther from the SM or from some new high scale physics of higher SU(2)L multiplets or more complicated sym- beyond the SM. It is easy to see that a non-vanishing metry breaking structures (such as those within higher- value for either (b ) or (˜b ) destroys the hermiticity V V dimensional theories) [27]. The couplings b or ˜b of the effective tℑheory. Suℑch couplings can be envisaged V V 3 the Z. Of these three, the e+e−H channel is consid- TABLE I: Transformation properties of various anomalous erably suppressed (by almost a factor of 10) with re- couplings underdiscrete transformations. spect to the ν ν¯ H channel over a very wide range of e e Trans. aV ℜ(bV) ℑ(bV) ℜ(˜bV) ℑ(˜bV) center-of-mass energies (√s) and Higgs masses. As can CP + + + − − be expected, atlarge√s, the Bjorkenprocesssuffers the T˜ + + − − + usual s-channel suppression and has a smaller cross section compared to that for W/Z-fusion. In fact, even for √s=500GeV and unpolarized beams, the Bjorken process dominates over W-fusion only for relatively large when one goes beyond the Born approximation, whence Higgs masses [29, 30, 31]. theyarisefromfinalstateinteractions,or,inotherwords, In view of this, it might seem useful to concentrate out of the absorptive part(s) of higher order diagrams, first on the dominant channel, viz e+e− νν¯H and presumably mediated by new physics. A fallout of non- thereby constrain b and ˜b . However, i→t is immedi- hermitiantransitionmatricesisnon-zeroexpectationval- W W ately obvious that the Bjorken process too contributes ues of observables which are odd under CPT˜, where T˜ to this final state and hence the couplings ∆a , b and stands for the pseudo-time reversal transformation, one Z Z ˜b have a role to play. Since the total rate is a CP-even which reverses particle momenta and spins but does not Z (aswellasT˜even)observable,itcanreceivecontribution interchange initial and final states. Of course, such non- only from (b ). [Note that a non-zero∆a wouldonly zero expectation values will be indicative of final state ℜ V V rescaletheSMrates.] Theothernon-standardcouplings, interaction only when kinematic cuts are such that the phasespaceintegrationrespectsCPT˜. Notethata too odd under CP and/or T˜, are responsible for various po- V lar and azimuthal asymmetries and contribute nothing can be complex in general and can give an additional T˜-odd contribution. However, for the processes that we to the total rate on integration over a symmetric phase space[35]. This can be understood best by considering willconsider,the phase ofatleastone ofa anda can W Z the square of the invariant amplitude pertaining to on- always be rotated away,and we make this choice for a . Z shell Z-production, namely e+e− ZH : Henceforth,weshallassumethataW andaZ areclose to → theirSMvalue,i.e. ai =1+∆ai,therationalebeingthat ℓ2+r2 E2 p2 cos2θ any departure from aSM and aSM respectively would be 2 a 2 e e 1+ Z − W Z |M| ∝ | Z| 4 m2 the easiest to measure (cid:20) Z (cid:21) (a b∗) ramUentfeorrstumnaatkeilnyg, tahnisasntailllylseisavceusmubsewrsiotmhem.anOynefremeigphat- + ℜ mZ2ZZ (ℓ2e+re2)√sEZ (2) argue that SU(2) U(1) gaugeinvariance wouldpredict (a ˜b∗) ⊗ + ℑ Z Z (ℓ2 r2)√sp cosθ ∆aW = ∆aZ. However, once symmetry breaking effects m2 e− e Z areconsidered,thisdoesnotnecessarilyfollow[24]. Nev- ertheless, we will make this simplifying assumption that with ℓ (r ) denoting the electron’s couplings to the Z, e e ∆aW is real and equal to ∆aZ, i.e. aW = aZ, since the and EZ,p,θ the energy, momentum and scattering an- equality is found to hold true in some specific cases [28] gle of the Z in the c.m.-frame. The proportionality (and would be dictated if SU(2) U(1) were to be an constant includes, alongwith the couplings etc., a factor ⊗ exact symmetry of the effective theory). With this as- s/[(s m2)2+Γ2m2]. Thissuggeststhattheanomalous sumption, we list, in Table I, the CP and T˜ properties contr−ibutZion vanZishZes for large s, inspite of the higher- of such operators. dimensionalnature ofthe coupling. Furthermore,Eq.(2) Finally,keepinginview the higher-dimensionalnature also demonstrates that neither (b ) nor (˜b ) may Z Z ℑ ℜ of all of these couplings, we retain only contributions up maketheirpresencefeltifthepolarizationoftheZ could to the lowest non-trivial order, arising from terms linear simply be summedover. Italsoshowsthatthe contribu- in the additional couplings. tion due to (˜b ) would disappear when integrated over Z ℑ a symmetric part of the phase space as has been men- tionedbefore. Thus,ifwewanttoprobethesecouplings, B. Cross-sections we would need to look at rates integrated only over partial (non-symmetric) phase space. As an example, let us consider the forward-backward asymmetry for the Z- The dominant channels of Higgs production at an boson. As can be seen from Eq.(2), it is proportional to electron-positron colliders are (˜b ) alone. This can be understood by realizing that Z ℑ 1. the 2-body Bjorken process (e+e− ZH); this observable is proportional to the expectation value → of (p~ p~ ) and hence is a CP-odd and T˜-even quantity e Z · 2. in association with a pair of neutrinos (e+e− just as (˜bZ) is. For (bZ) and (˜bZ), which are T˜- ν ν¯ H), i.e. W-fusion; → odd, oneℑhas to look atℑthe azimuthℜal correlationsof the e e final state fermions. Equivalently one can look for par- 3. inassociationwithane+e−pair(e+e− e+e−H), tial cross-section, restricting the azimuthal angles over → i.e. Z-fusion a given range. A discussion of how partial cross-section can be used to probe anomalous couplings is given in Note that the Bjorken-process also contributes to the Appendix A. Further, Eq.(2) also indicates that the another two final states through the subsequent decay of gular distribution of the Higgs (or, equivalently, that of 4 theZ)isdifferentfortheSMpiecethanthatforthepiece namely proportionalto (b );thedifferencegettingaccentuated Z at higher √s. Tℜhis, in principle, could be exploited to Ef,Ef¯≥10GeV, 5◦ ≤θf,θf¯≤175◦. (4) increase the sensitivity to (b ). Z Inthis paperwe restrictℜourselvesto the caseofa first For leptons, i.e. f = ℓ, we require a lepton–lepton sepa- generation linear collider. For such √s, the interference ration: between the W-fusion diagramwith the s-channelone is enhanced for non-zero bZ and ˜bZ. At a first glance, it ∆Rℓ−ℓ+ ≥0.2 (5) may seem that the kinematic difference between the two along with a b jet–lepton isolation: setof diagramscouldbe exploitedandthe twocontribu- − tions separated from each other with some simple cuts. ∆R 0.4 (6) However, in actuality, such a simple approach does not bℓ ≥ suffice to adequately decouple them. It is thus contin- for each of the four jet-lepton pairings. For f = q, i.e. gent upon us to first constrain the non-standard ZZH light quarks, we impose, instead, couplingsfromprocessesthatinvolvejusttheseandonly then to attempt to use WW-fusion process to probe the ∆R 0.7 (7) WWH vertex. q1q2 ≥ for each of the six pairings. On the other hand, if the Z weretodecayintoneutrinos,therequirementsofEqs.(4– III. THE ZZH COUPLINGS 7) are no longer applicable and instead we demand that the events containonly the two b-jets along with a mini- TheanomalousZZH couplingshavebeenstudiedear- mum missing transverse momentum, viz lier in the process e+e− ff¯H in the presence of an → pmiss 15GeV . (8) anomalous γγH coupling [20] making use of optimal ob- T ≥ servables [21]. The CP-violating anomalous ZZH cou- The above set of cuts select the events corresponding to plings alone have also been studied in Ref. [22], which the processofinterest,rejectingmostofthe QED-driven constructs asymmetries for both polarized and unpo- backgrounds. To further distinguish between the role larized beams. We, however, choose to be conserva- of the Bjorken diagram and that due to the ZZ (WW) tive and restrict ourselves to unpolarized beams. And, fusioninthecaseofe+e−H (νν¯H)finalstateweneedto rather than advocating the use of complicated statis- select/de-select the events corresponding to the Z-mass tical methods, we construct various simple observables pole. This is done via an additional cut on the invariant that essentially require only counting experiments. Fur- mass of ff¯, viz. thermore, we include the decay of the Higgs boson, account for b-tagging efficiencies and kinematical cuts to obtain more realistic sensitivity limits. Since we are pri- R1 ≡ mff¯−MZ ≤5ΓZ select Z-pole , (9) m(2amrily inmtereste1d40inGtehVe),inHtermeb¯dbiiastethmeadsosmHinigagnstbdoescoany R2 ≡ (cid:12)(cid:12)mff¯−MZ(cid:12)(cid:12)≥5ΓZ de-select Z-pole. b H ≤ ≤ → (cid:12) (cid:12) mode with a branching fraction > 0.9. Since an e(cid:12)xercise such(cid:12) as the current one would be un- ∼ dertaken only after the Higgs has been discovered and its mass measured to a reasonable accuracy, one may alternatively demand that the energy of the Higgs (re- A. Kinematical Cuts constructed ¯bb pair) is close to (s+m2 m2)/(2√s), H − Z namely For a realistic study of the process e+e− ff¯H(b¯b), → we choose to work with a Higgs boson of mass 120 GeV R1′ E− E E+ and a collider center-of-mass energy of 500 GeV. To en- ≡ H ≤ H ≤ H (10) R2′ E <E− or E >E+ sure detectability of the b-jets, we require, for each, a ≡ H H H H minimumenergyandaminimumangulardeviationfrom the beam pipe. Furthermore,the two jets should be well where EH± = (s+m2H −(mZ ∓5ΓZ)2)/(2√s). This has separated so as to be recognizable as different ones. To the advantage of being applicable to the νν¯H final state be quantitative, we require that as well. The b-jet tagging efficiency is taken to be 0.7. We add the statistical error and a presumed 1% system- Eb,E¯b ≥ 10GeV, ainticquearrdorrat(uarcec.ruiInngofrtohmerlwumoridnso,sitthyemfleuacstuuraetmioenntisnettch.e) 5◦ ≤θb,θ¯b ≤ 175◦ (3) measurement of a cross-sections is assumed to be ∆Rb¯b ≥ 0.7 ∆σ = σ / +ǫ2σ2 , (11) where(∆R)2 (∆φ)2+(∆η)2 with∆φand∆ηdenoting SM L SM ≡ q the separationbetween the two b-jets in azimuthalangle while that for an asymmetry is and rapidity respectively. For events with the Z decaying into a pair of leptons 1 A2 ǫ2 (∆A)2 = − SM + (1 A2 )2. (12) or light quarks, we have similar demands on the latter, σ 2 − SM SM L 5 Here σ is the SM value of cross-section, is the inte- the total rates and thus it is interesting to investigate SM grated luminosity of the e+e− collider andLǫ is the frac- possible correlations between a and (b ). Parame- Z Z ℜ tional systematic error. Since we work in the linear ap- terizing small variations in a as a = (1+∆a ), the Z Z Z proximationfortheanomalouscouplings,anyobservable, cross-sections can be re-expressed as rate or asymmetry, can be written as σ(R1;µ,q) = [20.7 (1+2∆a )+196 (b )] fb(17) Z Z ℜ ( )= O . i i i O {B } B and X Then we define the blind region as the region in the pa- σ(R2;e) = [4.76 (1+2∆a ) 0.147 (b )] fb(18) rameter space for which Z − ℜ Z where σ(R1;µ,q) stands for cross-section with R1 cut ( ) ( 0 ) f δ , (13) |O {Bi} −O { } |≤ O withµandlightquarksq inthefinalstate,i.e.,the combination used to obtain Eq.(16), and, as before, terms wheref isthedegreeofstatisticalsignificance, ( 0 )is O { } quadratic in small parameters have been neglected. Us- theSMvalueof andδ isthestatisticalfluctuationin O O ing these two rates we can obtain simultaneous con- . All the limits and blind regions quoted in this paper O straintsinthe ∆a (b )plane,asexhibitedinFig.2. are obtained using the above relation. Note that in all Z−ℜ Z TheobliquelinesareobtainedusingEq.(17)whereasthe the cases that we will consider, the asymmetries vanish almost vertical lines are obtained using Eq.(18). Thus, identically within the SM. σ(R2;e) alone can constrain a to within a few percent Z of the SM value. It is amusing to consider, at this stage, the possibility that ∆a could have been large, as for B. Cross-sections Z examplehappensintheMSSM.Clearly,theveryformof Eqs.(17 & 18) tells us that this would have amounted to The simplest observable, of course, is the total rate. just a two-fold ambiguity with a second and symmetric Note that (b ), inspite ofbeing T˜-odd, does resultin a non-zero, tℑhouZgh small, contribution to the total cross- allowedregionlyingaroundaZ =−1. Sinceweareinter- ested in the SM like Higgs boson, we constrainourselves section. This isbut aconsequenceofthe absorptivepart to region near a =1 as shown in Fig. 2. in the propagator and would have been identically zero Z in the limit of vanishing widths. Since we retain contributions to the cross-section that are at best linear in 0.015 thecouplings,themajornon-trivialanomalouscontribu- σ(R1; µ, q) tion, on imposition of the R1 cut, emanates from (b ) 0.01 σ(R2; e) Z ℜ with only subsidiary contributions from (b ). For our Z default choice (a 120 GeV Higgs at a macℑhine operating 0.005 at √s = 500GeV), on selecting the Z-pole (R1 cut) the )Z rates, in femtobarns, are e(b 0 R σ(e+e−) = 1.28+12.0 (b )+0.189 (b ) -0.005 Z Z ℜ ℑ σ(µ+µ−) = 1.25+11.9 (b ) Z ℜ (14) -0.01 σ(uu¯/cc¯) = 2 [4.25+40.2 (b )] Z ℜ σ(dd¯/ss¯) = 2 [5.45+51.6 (b )] Z -0.015 ℜ -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 On de-selecting the Z-pole (R2 cut) we obtain, instead ∆a Z σ(e+e−)=[4.76 0.147 (b )] fb. (15) − ℜ Z FIG.2: Theregioninthe∆aZ−ℜ(bZ)planeconsistentwith 3σ variations in the rates of Eqs.(17) and (18) for an inte- Thetotalrates,bythemselves,maybeusedtoputstrin- grated luminosity of 500 fb−1. gent constraints on (b ). For Z decaying into light Z ℜ quarks and µs (with the R1 cut) and for an integrated From Eq.(14) and Eq.(17) it is clear that the total luminosityof500fb−1,thelackofanydeviationfromthe rates,forthe R1cut, dependononlyonecombinationof SM expectations would give a and (b ), i.e. on Z Z ℜ (b ) 0.44 10−2 (16) |ℜ Z |≤ × η1 2 ∆aZ +9.46 (bZ). (19) ≡ ℜ at the 3σ level. We do not use the e+e−H final state Withthiscombinationwecanwriteallthecross-sections in deriving the above constraint as it receives a contri- given in Eqs.(14) and (17) as σ = σSM (1+η1), where bution proportional to (b ) too. This arises from the ℑ Z the limit from Fig. 2 translates to interference of the Bjorken diagram with the ZZ-fusion diagramduetothepresenceoftheabsorptivepartinthe η1 0.042, (20) near-on-shell Z-propagator. | |≤ Thecross-sectionsshowninEqs.(14&15)andthecon- i.e., 4.2% (3σ) variation of the rates. This variation ± straint of Eq.(16) have been derived assuming the SM can also be parameterized with (b ) 0.0044 keep- Z |ℜ | ≤ value for a . Clearly, any variation in a would affect ing ∆a = 0 or with ∆a 0.021 keeping (b ) = 0, Z Z Z Z Z | | ≤ ℜ 6 (i.e., the intercepts of the solid lines in Fig. 2 on the y– and µs. For an integrated luminosity of 500 fb−1, the and x–axes respectively). In other words, the individual corresponding 3σ limit is limit, i.e. the limit obtained keeping only one anomalous coupling non-zero, on ℜ(bZ) is 0.0044 and that on |ℑ(˜bZ)|≤0.038. (23) ∆a is 0.021. On the other hand, if the R2 cut were Z to be operative, we obtain the constraint ∆a 0.034 Z | | ≤ almostindependent of (b ), see Fig. 2. This constraint Z ℜ translatestoa 6.8%(3σ)variationintherateσ(R2;e). 1.2 ± Notethatthecontributionsproportionaltotheabsorp- 1 c H c tivepartoftheZ-propagatorareproportionaltoΓZ away 0.8 cb cH SM fromtheresonance,i.e. fortheR2cut. Henceinthiscase 0.6 b it is a higher order effect and thus ignored in Eqs.(15) 0.4 and(18). Ontheotherhand,neartheZ-resonancethese θ 0.2 s o terms are proportional to 1/Γ and hence of the same c 0 Z d order in the perturbation theory. Thus, in order to be σ/ -0.2 d consistentata givenorderinthe couplingα weretain -0.4 em these contribution only with the R1 cut. -0.6 ∼ -0.8 Im(b ) Z -1 e+e−→ e+e− H (R1 cut) C. Forward-backward asymmetry -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 cosθ The final state constitutes of two pairs of identifiable particles : b¯b coming from the decay of Higgs boson and FIG. 3: Polar angle distribution of the b−quark (thin line) ff¯, where f = b. One can define forward-backward follows that of Higgs boson (thick line) but is a bit smeared 6 asymmetry with respect to all the four fermions. But out. Thisisbecausetheb−quarkissphericallydistributedin we choose, among them, the asymmetries with definite therest frame of Higgs boson. transformation properties under CP and T˜. In the present case we have only one such forward-backward Itisinterestingtospeculateastothesensitivityofthe asymmetry, i.e. the expectation value of (p~e− p~e+) aforward-backwardasymmetryconstructedwithrespect − · (p~f + p~f¯). In other words, it is the forward-backward totheanglesubtendedby,say,theb-jet,ratherthanthat asymmetry with respect to the polar angle of the Higgs for the reconstructed H. Since the b quarks are spheri- boson (up to an overallsign) and given as − callydistributedinthe restframeofHiggs,theirangular distributionstrackthatoftheHiggsboson,modulosome σ(cosθ >0) σ(cosθ <0) A (cosθ )= H − H . (21) smearing [see Fig. 3]. This is as true for the anomalous FB H σ(cosθH >0)+σ(cosθH <0) contribution as for the SM. The smearing only serves to decrease the sensitivity as is evinced by the forward- ThisobservableisCP oddandT˜even,andhenceaprobe backward asymmetries constructed with respect to θ , purely of of (˜bZ) [see Table I]. Note that this asym- the polar angle of the b quark. For events correspondb- metry is propℑortional to (re2 −le2), where re(le) are the ing to the R1 cut, this a−mounts to right-(left-) handed couplings of the electron to the Z- boson. With the R1 cut—see Eq.(9)—operative, a semi- 0.0489 (˜b ) 0.909 (˜b ) analytical expression for this asymmetry, keeping only ℜ Z − ℑ Z (e+e−) 1.28 terms linear in the anomalous couplings, is given by  0.059 ℜ(˜bZ)−1.22 ℑ(˜bZ) (e+e−) AFB(cb)= −103.8.9812.25ℑ(˜b(˜bZ)) (µ+µ−) 1.28 − ℑ Z (qq¯) AFB(cH)= −1.21.2ℑ5(˜bZ) (µ+µ−) Using final states wi1t9h.4µs or light quarks, one may then −18.5 ℑ(˜bZ) (qq¯) use AFB(cb) to probe down to Intheabove,“q”stan1d9s.f4orallfourflavorsoflightqua(2rk2)s |ℑ(˜bZ)|≤0.051 summed over and cH cosθH. Note here, that the con- at3σ level,for anintegratedluminosity of500fb−1,and ≡ tributiontothedenominatorofEq.(21)fromtheanoma- assumingalltheotheranomalouscouplingstobezero. It loustermshavebeendroppedasthe formalismallowsus isnotsurprisingthatthesensitivityislowerascompared to retain terms only upto the first order in these cou- to that of A (c )—see Eq.23—for θ carries only sub- FB H b plings. In any case, their presence would have had only sidiary information leading to a reduction in the size of a miniscule effect on the ensuing bounds. The asymme- theasymmetry. Putdifferently,thedistributionfortheb try corresponding to the R2 cut is very small and is not isidenticaltothatforthe¯b(therebyeliminatingtheneed considered. Omitting the (e+e−) final state on account forchargemeasurement)andeachis drivenprimarilyby of the presence of (˜b ), we use only the light quarks θ . Z H ℜ 7 Note that we have desisted from using the non-zero this in deriving our final limits. Note that we do not ac- forward-backward asymmetry in the polar angle distri- count for any combinatorics in obtaining the above said bution of f(f¯). Such observables do not have the requi- limits. However, we argue that the invariant masses of site CP-propertiesandare,infact, non-zeroevenwithin the b¯b pair coming from Z and H are non-overlapping theSM.Thepresenceofnon-zero (˜b )providesonlyan hence the change in the above limits due to combina- Z ℑ additionalsourceforthe sameandthe limits extractable torics are expected to be small. would be weaker than those we have obtained. Another obvious observable is AUD(φµ−). Using this, an integratedluminosity of 500 fb−1, would lead to a 3σ constraint of D. Up-down asymmetry (˜b ) 0.35. (26) Z ℜ |≤ TheHiggsbeingaspin–0object,itsdecayproductsare The reduced sensitivity of the Z µ+µ− channel as isotropicallydistributedinits restframe. This,however, compared to the Z b¯b channel is→easy to understand. is not true of the Z. Still, CP conservation ensures that AsEq.(B5)demonst→rates,A (φ ) (r2 ℓ2)(r2 ℓ2). the leptons from Z decay are symmetrically distributed UD f ∝ e− e f− f Since r ℓ , this naturally leads to an additional µ µ about the plane of production. Thus, an up-downasym- | | ≈ | | suppression for AUD(φµ−). metry defined as Forthee+e−H case,ontheotherhand,theZZ-fusion diagram leads to a contribution that is proportional to σ(sinφ>0) σ(sinφ<0) AUD(φ)= σ(sinφ>0)−+σ(sinφ<0) (24) (re2+ℓ2e)2 and is, thus, unsuppressed. Accentuating this contribution by employing the R2 cut on m , we have ee can be non-zero for the anomalous couplings. In 5.48 (˜b ) o[(tp~hee−r−wp~eo+r)d×s, p~Ha] · n(p~ofn-−zerp~of¯)eixspeactaCtPionoddval(uaendfoT˜r ARU2D(φe−) = 4.ℜ76 Z (27) odd)observable. Inournotation,itisdrivenbythenon- and this, for an integrated luminosity of 500 fb−1, leads zero real part of ˜bZ, and, for our choice of parameters, to a 3σ constraint of amounts to (˜b ) 0.057. (28) Z 0.354 (˜b ) 0.226 (˜b ) |ℜ | ≤ Z Z AUD(φe−) = − ℜ 1.2−8 ℑ Herewenotethatthe limitonℜ(˜bZ),obtainedusingthe R2cutgivenabove,ismuchbetterthantheoneobtained 0.430 (˜b ) AUD(φµ−) = − 1.2ℜ5 Z (25) 4usbinfignaRl1stcautteianssEuqm.(i2n6g),ao2r0e%vecnhtahrgeeodneetdecetriiovnedeffifrocmientchye. 4.62 (˜b ) Z A (φ ) = − ℜ UD u 4.25 E. Combined polar and azimuthal asymmetries 7.98 (˜b ) Z A (φ ) = − ℜ UD d 5.45 Rather than considering individual asymmetries in- uptolinearorderintheanomalouscouplings. Inobtain- volving the (partially integrated) distributions in either ingEq.25,theR1cuthasbeenimposedontheff¯invari- of the polar or the azimuthal angle, one may attempt to ant mass. Note that, except for the e+e−H final state, combine the information in order to potentially enhance A isaprobepurelyof (˜b ). Thecrosssectionforthe the sensitivity. To this end, we define a momentum cor- UD Z e+e−H final state receiveℜs additional contribution from relation of the form (˜b ) due to the absorptive part of the Z-propagatorin Z ℑthe Bjorken diagram. Although AUD(φu) and AUD(φd) C1 = (p~e− −~pe+)·p~µ− offer much larger sensitivity to ℜ(˜bZ) than do either of (cid:2)[(p~e− −~pe+)×~pH(cid:3)]·(p~µ− −~pµ+) , (29) AUD(φe−) and AUD(φµ−), the former can not be used where the sign(cid:2)of the term inthe firstsquare br(cid:3)acketde- as the measurementofsuchasymmetries requirescharge cides if the µ− is in forward(F)hemisphere with respect determination for light quark jets. However,one can de- to the direction of e− or backward(B). Similarly, the termine the charge of b-quarks [32] and using A (φ ), UD b signofthe term in secondsquarebracketdefines if µ− is fortheb’sresultingfromtheZ decay,wemayobtain,for above(U)orbelow(D)theHiggsproductionplane. Thus an integrated luminosity of 500 fb−1, a 3σ bound of the expectation value of the sign of this correlation is (˜b ) 0.042 same as the combined polar-azimuthalasymmetry given Z |ℜ |≤ by, with 100% charge determination efficiency and only (FU)+(BD) (FD) (BU) A(θ ,φ ) = − − (˜b ) 0.089 µ µ (FU)+(BD)+(FD)+(BU) Z |ℜ |≤ 0.659 (b ) 0.762 (˜b ) if the efficiency were 20%. Note though that the Z b¯b = ℑ Z − ℜ Z (30) → 1.25 final state is beset with additional experimental com- plications (such as final state combinatorics) than the withthe secondequalitybeingapplicablefortheR1cut. semileptonic channels and hence we would not consider Inthe above,(FU)is the partialcross-sectionsfor µ− in 8 the forward-updirection andso on for others. Note that Cd1epiesnT˜d-soodndbboutthdtoheesTñ-ootddhacvoeupalidnegfisnaitseseCePniannEdqh.(e3n0c)e. oTbAsBerLvEabIlIe:sLaitm3iσtsleovnelaantomanailnoutesgZraZteHdlcuomupinlionsgitsyfroofm50v0afrbio−u1s. We do not consider the analogous asymmetry for qq¯H Coupling 3σ Bound Observable used final state as it demands charge determination for light |∆aZ| 0.034 σ with R2cut; f =e− quarks (although Z b¯b may be considered profitably). 0.0044 cSoimntirlaibrulyt,iofnosr ftrhoeme+T→˜e-−evHenficnoaulpsltinagtes,aAs(wθee,llφea)ndrecheeinvcees |ℜ(bZ)| 8>><(0∆.0a1Z2 =0) σ with R1cut; f =µ,q not considered for the analysis. |ℑ(bZ)| >>:0.1(4|∆aZ|=0.034) Acom with R1cut; f =µ−,e− |ℜ(˜bZ)| 0.057 AUD(φe−) with R2 cut |ℑ(˜bZ)| 0.038 AFB(cH) with R1cut; 0.3 f =µ,q 0.2 Thesemi-analyticalexpressionforthisasymmetryforR1 0.1 )Z A(θµ,φµ) cut is given by m(b 0 Acom [e + µ ] 0.766 (b ) I -0.1 ARU2D(φe) Acµom = 1.2ℑ5 Z -0.2 Acom = 0.757ℑ(bZ)−0.048ℜ(bZ) (33) e 1.28 -0.3 14.2 (b ) -0.06 -0.04 -0.02 0 0.02 0.04 0.06 Acom = ℑ Z Re(∼b ) b 5.45 Z FIG. 4: Region in ℜ(˜bZ)−ℑ(bZ) plane corresponding to the Using final states with either electrons or muons, we 3σvariationofasymmetries. SlantlinesareforA(θµ,φµ)and would obtain a 3σ limit of the vertical lines are for AUD with an integrated luminosity of 500 fb−1. The horizontal limit shown are due to Acom for (bZ) 0.14, (34) |ℑ |≤ e− and µ− in the finalstate. for500fb−1 ofintegratedluminosityandmaintainingall the other form-factors to be zero. This limit is better Using this T˜-odd asymmetry along with AUD, (˜bZ) than the one obtained in the preceding subsection. ℜ and (bZ) can be constrained simultaneously. Fig. 4 Once again, inclusion of the Z b¯b channel, i.e. a showℑs the limit on (b ) as a function of (˜b ). measurement of Acom, would impro→ve the situation dra- ℑ Z ℜ Z b matically even for a nominal charge detection efficiency. For example, an efficiency as low as just 20%, is enough to obtain F. Another asymmetry (b ) 0.050. Z |ℑ |≤ We reemphasize though that our final results do not ex- Similartotheprevioussubsection,wecandefineaCP- even and T˜-odd correlation as ploit this possibility. We also note that the sensitivity to the T˜-odd couplings is large for f = q as compare to f = ℓ. As the 2 = [(p~e− p~e+) ~pZ] expressions in the Appendix B demonstrate, AUD, with C − · theR1cutoperational,isproportionalto(l2 r2)(l2 r2) [(p~e− −~pe+)×~pH]·(p~f −p~f¯) (31) and Acom is proportionalto (l2+r2)(l2 re2−).eThufs−, ffor (cid:2) (cid:3) f = ℓ, the asymmetries are preoporetionfa−l tofat least one which is a probe of the CP-even and T˜-odd coupling powerof (l2 r2) where r l and hence are smaller e− e | e|≈|e| (b ). Here, the sign of the term in the first square compared to those for the f =q case. Z ℑ bracket decides whether the Higgs boson is in the forward(F′) or backward(B′) hemisphere, while the sign of the term in the second square bracketindicates if f is G. Summary of Limits on the ZZH couplings above(U) or below(D) the Higgs production plane. The expectationvalue ofthe signof 2 canthus be expressed C In the preceding five subsections we discussed observ- as an asymmetry of the form ableswhichwill be able to probe eachofthe five anoma- lousZZH couplings. Theensuinglimitsaresummarized (F′U)+(B′D) (F′D) (B′U) in Table II. Acom = − − . (32) (F′U)+(B′D)+(F′D)+(B′U) Several points are in order here 9 Recall that, of the five anomalous terms, only two The event selection criteria we use are the same as • viz ∆a and (b ), haveidenticaltransformations in previous section except that the cuts of Eqs. (4 Z Z ℜ underbothCP andT˜. Consequently,thecontribu- & 6) are replaced by that of Eq. (8). Imposing tions proportional to the two are intertwined and ∆aW = ∆aZ ∆a as argued for earlier, the resul- can only be partially separated. In fact, the most tant cross-section≡,for the R1′ cut, can be parameterized generallimits onthesetwoaretobe obtainedfrom as, Fig. 2. σ1 = [7.69 (1+η1) 1.89 (bZ) − ℑ Asfortheotherthreecouplings,wehavebeenable +0.458 (b )+0.786 (b )] fb (35) • ℜ W ℑ W to construct observables that are sensitive to only while the R2′ cut would lead to a single coupling, thereby disengaging each of the corresponding bounds in Table II from contamina- σ2 = [52.1 (1+2 ∆a) 6.99 (bZ) tions from any of the other couplings. − ℜ 0.162 (b ) 19.5 (b )] fb (36) Z W − ℑ − ℜ The polar-azimuthal asymmetry, A(θ,φ), is sensi- • tive to T˜-odd couplings. However, the limits ob- Nfinoetde hinerEeqt.h(1a9t)thapepseaamrseaηb1o=ve2ow∆iang+t9o.4t6heℜa(sbsZu)mapstdioen- tained using µ final state alone are weaker than ∆a = ∆a ∆a. As the contributions proportional W Z the ones obtained by combining ARU2D and Acom to (bV) appea≡r due to interference of the WW fusion (see Fig 4). Inclusion of electrons in the final state diaℑgram with the absorptive part of the Z-pro−pagator will improve the sensitivity of A(θ,φ), but only at in Bjorken diagram, formally, these terms are at one or- thecostofcontaminationbytheT˜-evenZZH cou- der of perturbation higher than the rest. Note that the plings. Thus, in our present analysis, the role of bounds in Table II imply that A(θ,φ) is only a confirmatory one. 6.99 (b )+0.162 (b ) 0.0839 Z Z | ℜ ℑ |≤ Note,however,thatmanyoftheseasymmetriesare • proportional to (l2 r2) = (1 4 sin2θ ). Since and hence the correspondingcontribution to σ2 is at the e − e − W per-mille level. Since we are not sensitive to such small this parameter is known to receive large radiative contributions, we may safely ignore this combination for corrections, the importance of calculating higher- order effects cannot but be under-emphasized. allfurtheranalysis. Lookingatfluctuationsinσ2,the3σ bound would, then, be ObservablesconstrainingT˜-oddcouplingsrequired • charge determination of fermion f in the ff¯H(b¯b) |2 ∆a−(19.5/52.1)ℜ(bW)|≤0.035. (37) final state thereby eliminating (the dominant) f = The limits on (b ) and ∆a are thus strongly corre- W ℜ q final states from the analysis. This explains a latedanddisplayedinFig.5. Notethatacomplementary relatively poor limit on (bZ). For f = ν, the boundon∆ahadalreadybeenobtainedinSectionIIIB. ℑ process involves WWH couplings as well. This is If we assume that, of these two couplings, only one is discussed in the next section. non-zero, the corresponding individual limits would be (b ) 0.097 (if ∆a=0) and, similarly, ∆a 0.017 W |ℜ |≤ | |≤ (if (b ) = 0). Interestingly, the last mentioned bound W IV. THE WWH COUPLINGS is tℜwice as strong as that obtained in Section IIIB. Of course,hadwenotmadetheassumptionof∆a =∆a , W Z As discussed at the beginning of the last section, the ormadeadifferentassumption,theboundsderivedabove contribution from non-standard ZZH couplings to the (and Fig.5) would have looked very different. νν¯H(b¯b) final state is not negligible even if on-shell Z Having constrained (bW), we may now use σ1 to in- ℜ productionisdisallowedbyimposingtheaforementioned vestigate possible bounds on (bW). To this end, it is ℑ R2 cut. With the neutrinos being invisible, we are left useful to define a further subsidiary variable κ1 as withonlytwoobservables: thetotalcross-sectionandthe κ1 7.69 η1 1.89 (bZ), forward-backward asymmetry with respect to the polar ≡ − ℑ (38) angleoftheHiggsboson. ThedeviationfromtheSMex- κ1 0.604 | | ≤ pectations for the cross section depends mainly on ∆a V with the inequality having been derived using Table II. and (b ). Similarly, the forward-backward asymme- V Thecorrespondingconstraintinthe (b )– (b )plane ℜ W W try can be parameterized, in the large, by just ℑ(˜bV). is shownin Fig.6 for variousrepreseℜntativeℑvalues ofκ1. The contribution of the other couplings, viz. (bV) and Clearly, the contamination from the ZZH couplings is ℑ (˜bV), to either of these observables are proportional to very large and inescapable. Any precise measurement of ℜ theabsorptivepartofZ-propagatorandareunderstand- (b ), in the presentcase, requires very accurate deter- W ℑ ably suppressed, especially for the R2 cut. mination of the ZZH vertex. The individual limit on Now, irrespective of the CP properties of the Higgs, (bW), for κ1 = (bW)=0 is given in Table III. ℑ ℜ its decay products are always symmetrically distributed Next, we look at the forward-backward asymmetry in its rest frame. In addition, the momentum of the in- with respect to c which, for our cuts, we find to be H dividual neutrino is not available for the construction of paanlidynigrTe˜sc.−topdrdobaesyomf ℑm(ebtVry).anCdonℜs(ebq˜Vu)e,nit.ley.,twhee Tdõ−nodotdhcaovue- A1FB(cH) = h+−01.2.2904ℜ((˜b˜bZW))−70.1.214ℑ2(˜bZ(˜b)W) /7.69(39) ℜ − ℑ i 10 0.3 0.4 σ(R2; e) 0.2 σ 0.3 R2 0.2 0.1 0.1 )W )W A2FB(cH) Re(b 0 ∼Im(b -0. 10 AFB(cH) [f = µ, q] -0.1 -0.2 -0.2 -0.3 -0.4 -0.3 -0.04 -0.02 0 0.02 0.04 -0.04 -0.02 0 0.02 0.04 ∼ ∆a (=∆aZ=∆aW) Im(bZ) FIG. 5: Region of ∆a−ℜ(bW) plane corresponding to 3σ FIG. 7: Regions of ℑ(˜bZ)−ℑ(˜bW) plane corresponding to variation of σ2 for L=500 fb−1. The vertical line shows the 3σ variation of A2FB(cH). The vertical line denote limits on limit on ∆aZ from Fig. 2. ℑ(˜bZ) from Table II. 1.5 TABLEIII:Individuallimitsonanomalous WWH couplings κ1 = -0.6 fromvariousobservablesat3σlevelatanintegratedluminos- 1 ity of 500 fb−1 0.5 Coupling Limit Observable used b)W 0 |ℜ|(∆bWa|)| ≤≤ 00..001974 σσ22 Im( |ℑ(bW)| ≤ 0.56 σ1 -0.5 κ1 = 0.0 |ℜ(˜bW)| ≤ 1.4 A1FB(cH) |ℑ(˜bW)| ≤ 0.37 A2FB(cH) -1 κ = 0.6 1 -1.5 dence into one variable by defining -0.1 -0.05 0 0.05 0.1 Re(bW) κ3 1.20 (˜bZ)+7.11 (˜bZ) , (41) ≡ ℜ ℑ FIG.6: Region ofℑ(bW)−ℜ(bW) planecorresponding to3σ we have variation of σ1 for κ1=0.0 (big-dashed line), 0.6 (solid line) and −0.6 (small-dashed line). Vertical lines show limit on ℜ(bW) obtained from Fig. 5 for ∆a=0. A1FB(cH) = −κ3+0.294 ℜ(˜bW)−0.242 ℑ(˜bW) /7.69 h i κ3 0.353 . | | ≤ for the R1′ cut, while for the R2′ cut it is The inequality is a consequenceofthe bounds derivedin the previous section (see Table.II). In Fig. 8, we show A2FB(cH) = 3.55 ℑ(˜bZ)+4.00 ℑ(˜bW) /52.1 (40) the limit on (˜bW) as a function of (˜bW) for three rep- ℜ ℑ h i resentative values of κ3, namely κ3 = 0.00, 0.353. The OClnecaerlyagAa2FinB, itshethree oisneatshtartonisgmcoorrereslaentisointivweittohℑ((˜b˜bW)),. tmhoisstfigguenree.ral limit on ℜ(˜bW) can then be g±leaned from Z ℑ whichof coursehas alreadybeen constrained(in Section IIIC) from a consideration of similar forward-backward asymmetries in the b¯bqq¯and b¯bµ−µ+ channels. The re- TABLE IV: Simultaneous limits on anomalous WWH cou- sultantconstraintinthe (˜b )– (˜b )planeisdisplayed plings from various observables at 3σ level at an integrated ℑ W ℑ Z luminosity of 500 fb−1. in Fig. 7. And assuming a vanishing (˜b ), the individ- Z ual limit on (˜b ) is 0.37. ℑ Coupling ∆a=0 ∆a6=0 W ℑ |∆a| ≤ – 0.034 The only coupling that remains to be constrained at this stage is (˜b ). While A1 does depend on this |ℜ(bW)| ≤ 0.097 0.28 parameter, it,ℜunWfortunately, alFsoBdepends on the other |ℑ(bW)| ≤ 1.4 1.4 three CP-odd anomalous coupling as well. However, |ℜ(˜bW)| ≤ 2.8 2.8 since the lackofsufficientkinematic variablespreventus |ℑ(˜bW)| ≤ 0.40 0.40 from constructing another CP-odd observables, we are forced to use A1 alone, despite its low sensitivity to Notethat,unlikeinthecaseoftheZZH couplings,we FB (˜b ). Collecting all the relevant ZZH vertex depen- have largely been unable to construct observables that W ℜ