Spin correlations and new physics in τ -lepton decays at the LHC
aa r X i v : . [ h e p - ph ] J u l Spin correlations and new physics in τ -lepton decays at theLHC Alper Hayreter ∗ and German Valencia † Department of Natural and Mathematical Sciences,Ozyegin University, 34794 Istanbul Turkey. and Department of Physics, Iowa State University, Ames, IA 50011. (Dated: July 17, 2018)
Abstract
We use spin correlations to constrain anomalous τ -lepton couplings at the LHC includingits anomalous magnetic moment, electric dipole moment and weak dipole moments. Singlespin correlations are ideal to probe interference terms between the SM and new dipole-type couplings as they are not suppressed by the τ -lepton mass. Double spin asymmetriesgive rise to T -odd correlations useful to probe CP violation purely within the new physicsamplitudes, as their appearance from interference with the SM is suppressed by m τ . Wecompare our constraints to those obtained earlier on the basis of deviations from theDrell-Yan cross-section. ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION There exists a very active τ -lepton program at the ATLAS and CMS collab-orations. They have already studied the reconstruction of the Z -resonance in thedi-tau mode [1, 2] and placed limits on physics beyond the standard model (BSM).Searches for heavy resonances such as Z ′ bosons have excluded such particles in thedi-tau channel up to masses around 1 TeV [3–5]. There is also an active programto observe the Higgs boson in this channel [6–9], as well as searches for additionalneutral scalars that decay to τ + τ − [10]. In this paper we propose the use of spincorrelations to constrain τ anomalous couplings.An effective Lagrangian for BSM physics with a complete catalog of operatorsup to dimension six exists in the literature [11, 12] for the case in which the observed126 GeV state is the Higgs boson in the SM. In a recent paper [13] we studied thesubset of dimension six operators that describe the τ -lepton dipole-type couplings,as well as two dimension eight operators which couple tau leptons directly to gluonsand are thus enhanced by the gluon luminosities [14]. In particular we discussedthe constraints that can be imposed on these operators by studying deviations fromthe Drell-Yan cross section at the LHC as well as by bounding the cross sectionfor production of τ -leptons in association with a Higgs boson. We now extend thatstudy to include spin correlations measured in the angular distributions of muonsor electrons in leptonic decay modes.The couplings involved in the study are the τ -lepton anomalous magnetic mo-ment and electric dipole moment given by a γτ and d γτ respectively, L = e τ σ µν ( a γτ + iγ d γτ ) τ F µν (1)and its corresponding weak dipole moments a Zτ and d Zτ , L = g θ W ¯ τ σ µν (cid:0) a Zτ + iγ d Zτ (cid:1) τ Z µν (2)The τ -lepton dipole moments have been studied many times before in the literature[15–26].These anomalous couplings, Eq. 1 and Eq. 2, originate in the gauge invariantdimension six operators, in the notation of [11], L = g d ℓW Λ ¯ ℓσ µν τ i e φW iµν + g ′ d ℓB Λ ¯ ℓσ µν e φB µν + h . c . (3)The correspondence between these gauge invariant operators and the anomalousmagnetic moment, electric dipole moment (EDM) and weak dipole moment (ZEDM)2f the leptons is given by a γℓ = √ v Λ Re ( d ℓB − d ℓW ) , a Zℓ = − √ v Λ Re (cid:0) d ℓW + sin θ W ( d ℓB − d ℓW ) (cid:1) d γℓ = √ v Λ Im ( d ℓB − d ℓW ) , d Zℓ = − √ v Λ Im (cid:0) d ℓW + sin θ W ( d ℓB − d ℓW ) (cid:1) (4)where v ∼
246 GeV is the Higgs vacuum expectation value, θ W is the usual weakmixing angle and Λ is the scale of new physics, which we take as 1 TeV for ournumerical study.As argued in [14] the usual power counting for new physics operators is alteredfor dimension eight operators that couple a lepton pair directly to gluons due tothe larger parton luminosities. This motivates our inclusion of the “lepton-gluonic”couplings for the τ in this study L = g s Λ (cid:16) d τG G Aµν G Aµν ¯ ℓ L ℓ R φ + d τ ˜ G G Aµν ˜ G Aµν ¯ ℓ L ℓ R φ (cid:17) + h . c . (5)Here G Aµν is the gluon field strength tensor and ˜ G Aµν = (1 / ǫ µναβ G Aαβ its dual.If we allow for CP violating phases in the coefficients, d τG and d τ ˜ G , the resultinggluon-lepton couplings take the form L = v √ g s Λ (cid:16) Re( d τG ) G Aµν G Aµν + Re( d τ ˜ G ) G Aµν ˜ G Aµν (cid:17) ¯ ℓℓ + i v √ g s Λ (cid:16) Im( d τG ) G Aµν G Aµν + Im( d τ ˜ G ) G Aµν ˜ G Aµν (cid:17) ¯ ℓγ ℓ (6)In Table I we summarize the 1 σ constraints that we obtained on the τ -leptonanomalous magnetic moment, electric dipole moment and weak dipole momentsassuming a 14% measurement of the Drell-Yan cross-section at LHC14 in Ref. [13].We compare them to the best existing constraints from Delphi [27], Belle [28] andAleph [29]. The results can be interpreted as a sensitivity to a NP scale Λ ∼ . ∼ m τ a Vτ LHC-14 m τ a Vτ existing m τ d Vτ LHC-14 m τ d Vτ existing V = γ (-0.0054,0.0060) (-0.026,0.007) Delphi (-0.0057,0.0057) (-0.002,0.0041) Belle V = Z (-0.0018,0.0020) (-0.0016,0.0016) Aleph (-0.0017,0.0017) (-0.00067,0.00067) AlephTABLE I: Summary of constraints for 1 σ bounds that can be placed on the τ -leptonanomalous magnetic moment, electric dipole moment and weak dipole moments at LHC14from Ref. [13] compared to existing bounds. There is a typo in Eq. 4 of Ref. [13] that propagates to the conversion of our bounds from thegauge invariant basis to the tau anomalous couplings corrected in an errata. We also use here adifferent, more convenient, normalization for d ℓW .
3o obtain the constraints in Table I, are approximately quadratic in the anomalouscouplings indicating that the interference with the SM is very small. This is, ofcourse, due to the fact that the interference between the SM and the dipole-typecouplings is suppressed by the τ -mass.In this paper we extend our previous study considering constraints that arisefrom spin correlations. These spin correlations evade the helicity suppression ofthe interference terms in the cross-section and produce observables linear in the newphysics couplings. In this way it is possible to improve the constraints on the electricdipole moments and to study their CP violating nature through T -odd asymmetries. II. SPIN CORRELATIONS
Spin correlations in τ -pair production including anomalous dipole type couplingshave been studied in Ref. [21]. In that paper, the spin density matrix for productionof τ -pairs in e + e − colliders in the CM frame was constructed and combined withthe decay matrix for polarized τ in its rest frame. That formalism exhibits the spincorrelations explicitly but is not suited for our calculation. We want to construct(Lorentz scalar) correlations in terms of observable momenta at the LHC, namely,the muon (or electron) momenta and the beam momentum. Furthermore, we wantto measure the correlations with event simulations using MadGraph5 [30]. The mainadvantage of this approach is the ease in introducing different types of new physicswith the aid of
FeynRules [31]. In this paper we limit ourselves to dilepton decaysof the τ pairs, but in a future publication we will address the hadronic decay modes. A. CP violating couplings The imaginary part of the effective couplings gives rise to electric and weakdipole moments of the τ -lepton. These dipole moments are known to produce adouble spin correlation linear in the anomalous coupling, of the form O s ∼ m τ d Z,γτ ǫ µ,ν,α,β p µτ + p ντ − s ατ + s βτ − (7)This correlation originates in the interference between the CP violating edm ampli-tude and the CP conserving SM amplitude. In this case, however, the interferencerequires a fermion helicity flip and is therefore proportional to the τ -lepton mass,resulting in a large suppression at the LHC. On the other hand, contributions thatare quadratic in the new physics couplings do not suffer from this suppression, andEq. 7 is useful to probe terms of the form ∼ Re( d τV ) Im( d τV ), that is, O s ∼ d Z,γτ a Z,γτ ǫ µ,ν,α,β p µτ + p ντ − s ατ + s βτ − (8)and similar terms proportional to both the real and imaginary parts of the couplings d τG and d τ ˜ G . 4ith the muons (or electrons) in leptonic tau decay acting as spin analyzersthis is measurable as O ss = ǫ µ,ν,α,β p µτ + p ντ − p αµ + p βµ − (9)which requires at least partial reconstruction of one τ -momentum direction and maybe better suited for hadronic decay channels.To probe the anomalous couplings that violate CP with terms in the differentialcross-section that are linear in Im( d τV ), but not proportional to the τ mass, we resortto single spin correlations. For example, for the parton level process q ¯ q → τ + τ − onefinds that the Z exchange diagram leads to the CP -odd correlation O s ∼ d Zτ g A (ˆ t − ˆ u ) ǫ µ,ν,α,β ( p − p ) µ p ντ + p ατ − ( s τ − − s τ + ) β (10)where ˆ t , ˆ u and ˆ s are the parton level Mandelstam variables, g A is the axial vectorcoupling of the Z to the charged leptons and we have neglected the smaller vectorcoupling, g V . Note that the parton momenta p , appear in a symmetric combination(from the two antisymmetric factors, (ˆ t − ˆ u ), and the explicit ( p − p )) and thereforethis correlation does not vanish after the symmetrization of p , that follows fromthe convolution with the parton distribution functions for the LHC pp initial state.In order to write T -odd correlations that are sensitive to Eq. 10 and are expressedonly in terms of observable momenta we note that: • In leptonic τ decay, the spin is analyzed by the muon (or the electron) mo-mentum. The simplest way to compute this is using the method of Ref. [32],which shows that for leptonic τ decay, Eq. 10 becomes O ℓℓ s ∼ (ˆ t − ˆ u ) (cid:16) p τ − · p µ − ǫ µ,ν,α,β p µ p ν p ατ − p βµ + + p τ + · p µ + ǫ µ,ν,α,β p µ p ν p αµ − p βτ + (cid:17) (11) • In the lab frame at the LHC the τ -leptons are highly boosted so their three-momenta are very close to that of the muons. Further, in leptonic τ decay it isnot possible to reconstruct the τ momentum completely. We then replace the τ momenta with the corresponding muon momenta in the lab frame obtaining O ℓℓ s lab −→∝ (ˆ t − ˆ u ) ǫ µ,ν,α,β p µ p ν p αµ − p βµ + (12) • In the lab frame, the sum and difference of the proton momenta are just thecenter of mass energy and the beam direction, and the two parton momentaappearing in Eq. 12 have to be expressed in terms of these: P = √ S , , , , P = √ S , , , − P ≡ P + P = √ S (1 , , , , q beam ≡ P − P = √ S (0 , , ,
1) (13) • This leaves us with two possibilities: O = [ ~q beam · ( ~p µ + − ~p µ − ) ~q beam · ( ~p µ + × ~p µ − )] lab O = [ ~q beam · ( ~p µ + + ~p µ − ) ~q beam · ( ~p µ + × ~p µ − )] lab (14)5he CP properties of these two forms merit discussion. Since these correlationsinvolve the beam momentum, a property of the initial state, they do not have definitetransformation properties under CP . In fact, CP transforms LHC correlationsinto anti-LHC (¯ p ¯ p collider) ones. If we consider the same correlations for a p ¯ p collider instead, we see that in this case the first one is CP -odd and can only beproduced by the electric dipole moments. The second one, however, is CP -evenand cannot be produced by electric dipole moments at a p ¯ p collider. This is a novelfeature for colliders, that does not occur at the parton level where the pre-factor q · ( p τ + + p τ − ) = ( p − p ) · ( p + p ) vanishes. We illustrate this with an example inTable VIII.Noting again that the symmetry of the initial pp state at the LHC forbids termslinear in q beam , we use the correlation O test = [ ~q beam · ( p µ + × p µ − )] lab (15)to gauge the statistical significance of our asymmetries. Interestingly, as seen inTable VIII, this asymmetry does not vanish in p ¯ p colliders and would in fact be themost sensitive one to use in that case. B. CP conserving couplings The CP conserving dipole couplings interfere with the SM but this contributionto the cross-section is suppressed by the τ lepton mass as well. It is also possibleto find terms that are linear in the CP conserving anomalous couplings and thatare not helicity suppressed by looking at single spin correlations. One such term isgiven by O spin ∼ a Zτ g A (cid:0) ˆ s (ˆ t − ˆ u ) ( p − p ) · ( s τ − + s τ + ) + (ˆ t − ˆ u ) ( p τ + s τ − − p τ − s τ + ) (cid:1) (16)To study Eq. 16 using only the beam and muon momenta we found the followingtwo observables: the first one is the muon charge asymmetry [33] defined by O C = ∆ | y | ≡ | y µ + | − | y µ − | (17)The charge asymmetry is C -odd and therefore changes sign at ¯ p ¯ p collider and van-ishes at a p ¯ p collider as seen in the example in Table VIII. The second possibility issimply O p T = ~q beam · ( ~p µ + − ~p µ − ) ~q beam · ( ~p µ + + ~p µ − ) (18)which can also be written as the difference in transverse momentum of the twomuons.To measure any of the correlations discussed above we use the fully integratedcounting asymmetries normalized to the standard model cross-section, A i = (cid:18) N + − N − N + + N − (cid:19) (cid:18) σσ SM (cid:19) (19)6here N + = σ ( O i > N − = σ ( O i < III. NUMERICAL STUDY
For our numerical study we generate multiple event samples for the process pp → τ + τ − → ℓ + ℓ − ν τ ¯ ν τ ν µ ¯ ν µ (where ℓ = µ, e but will be a muon for most of ourstudy) at 14 TeV center of mass energy that we summarize in the Appendix. Theanomalous couplings are implemented in MadGraph5 [30] with the aid of
FeynRules [31] . We use the resulting UFO model files to generate events for several values of d τW , d τB , d τG and d τ ˜ G in a range motivated by our previous results from Ref. [13].The events preserve all spin correlations between production and decay of the τ -leptons as they are generated for the complete process. In each case we generateevent samples with one million dimuon or dilepton events after cuts, implying a 1 σ statistical sensitivity to all asymmetries at the ( σ/σ SM × . A. High energy dilepton pairs
The cuts used in our event generation are: • m ττ >
120 GeV implemented in the cuts.f file. The purpose of this cut is toexclude the Z resonance region from consideration, as this will be discussedseparately. We use this idealized cut for simplicity although it may not be pos-sible to implement experimentally for leptonic tau decays. In a more realisticsimulation removal of the Z region can be effectively accomplished with analternative cut on m µµ /E T . A few tests suggested the cuts give similar resultsbut the more realistic one requires much longer event generation time. • p T ℓ >
15 GeV for both muons and electrons. This is a standard acceptancecut in the LHC experiments. The asymmetries due to new physics increasewith an increasing p T ℓ cut at the cost of statistical sensitivity. The numberwe use is a good compromise for million event samples. • | η ℓ | < . The code is available from the authors upon request.
7n order to preserve the spin correlations it is important to calculate matrixelements for the full process. The numerical implementation of this calculation issignificantly complicated by the very narrow τ -lepton width. It is however possibleto do it with the current version of Madgraph, which has the necessary numericalprecision, at the cost of long event generation times. A simple trick to alleviate thisproblem is to use a fictitious (and much larger) τ -lepton width during the eventgeneration, and to then rescale the resulting cross-sections by the narrow-widthapproximation factor Γ τ − fict / Γ τ for each τ propagator , explicitly σ = σ (Γ τ − fict ) × (cid:18) Γ τ − fict Γ τ − exp (cid:19) × (cid:18) Γ τ − SM Γ τ ( d τW ) (cid:19) (20)The first factor is the cross-section calculated by MadGraph5 using as an input Γ τ − fict ,typically 2 . × − GeV. The second factor corrects this by 10 by rescaling tothe experimental width. Finally, the last factor takes into account the dependenceof the τ -width on d τW . To check that this trick does not distort the kinematicdistributions of final state leptons to the extent of affecting our asymmetries, werepeated a few calculations using different fictitious values of the τ width spanningseveral orders of magnitude. We show the results of this exercise in Table X.With the tables presented in the Appendix, we obtain the following approximatenumerical fits for the most relevant observables: σσ SM = 1 + 0 . | d τW | + 0 . | d τB | + 0 . d τW ) A = − .
014 Im( d τW ) − . d τB ) A = 0 .
010 Im( d τW ) + 0 . d τB ) A ss = 0 . d τW ) Im( d τW ) + 0 .
031 Re( d τG ) Im( d τG ) + 0 .
031 Re( d τ ˜ G ) Im( d τ ˜ G ) A C = − . − . × − Re( d τW ) A p T = − . − . × − Re( d τW ) (21)The salient features of these fits are summarized below. • As discussed in Ref. [13], terms in the cross-section linear in the real part ofthe anomalous couplings are suppressed by the τ mass at LHC energies andthis is confirmed both by our fit and by the symmetry of Figure 1. Our fits arenot quite the same as the ones we presented in Ref. [13] due to the different p T ℓ cut used there and the different normalization for d τW . With the cuts usedhere, the bounds placed on the anomalous couplings by measurements of thecross-section (using the same procedure as in Ref. [13]) are shown in Figure 1.Taking only one parameter to be non-zero at a time we find, | Im( d τW ) | < ∼ . , | Im( d τB ) | < ∼ . − . < ∼ Re( d τW ) < ∼ . , | Re( d τB ) | < ∼ . We thank Olivier Mattelaer for this suggestion. m( d τV )Re( d τV ) m τ d Vτ m τ a Vτ γ FIG. 1: Regions of d τV (left) and the corresponding d γ,Zτ , a γ,Zτ (right) allowed by a maxi-mum 14% deviation from the SM cross-section with the cuts described in the text. or equivalently, | m τ d Zτ | < ∼ . , | m τ d γτ | < ∼ . − . < ∼ m τ a Zτ < ∼ . , − . < ∼ m τ a γτ < ∼ . . • A glance at the Feynman diagrams for q ¯ q → τ + τ − → µ + µ − ν τ ¯ ν τ ν µ ¯ ν µ revealsthat the cross-section should be quadratic in d τB and a polynomial of order6 in d τW because the latter also appears in the τ decay vertex as implied bythe gauge invariant form of the operators, Eq. 3. Our numerical calculationindicates that the cross-section has a sensitivity to d τW at most quadratic,in other words the precision of our simulations makes it difficult to allowfor the higher order terms. This is because our procedure is a form of thenarrow width approximation (but keeping spin correlations): the dependenceof σ ( pp → τ + τ − ) on d τW is quadratic, and the τ -lepton branching ratiosremain approximately independent of d τW .Interestingly, the τ -width itself depends on d τW and we could use that to findan additional constraint. In the approximation in which we treat the hadronic τ -decay as decay into free quarks, we findΓ τ ( d τW ) ≈ Γ τ − SM (1 + 0 . d τW ) + · · · ) (24)where · · · stands for much smaller quadratic corrections, and this is the precisefactor we use in Eq. 20. Of course this approximation does not calculate the This precision corresponds to the largest systematic error in the CMS analysis of high invariantmass τ -pairs [3] for the size of thecorrections introduced by d τW . Taken literally, and using from the particledata book that the τ mean life is (290 . ± . × − s [34], it implies | Re( d τW ) | < ∼ . • The T -odd correlations A , exhibit a linear dependence on the imaginary partof the anomalous couplings that is not suppressed by the τ mass, as is expectedfor single spin correlations. We find that this process is about six times moresensitive to Im( d τW ) than to Im( d τB ). • The T -odd and CP -odd asymmetry A ss receives contributions quadratic inthe anomalous couplings that are not suppressed by the τ mass. As discussedabove, they arise from double spin asymmetries produced in the interferenceof the new physics amplitudes with themselves. • The independence on the τ -mass for two asymmetries originating in the inter-ference between new physics Im( d τW ) = 10 and the SM (since Re( d τW ) = 0)is shown in Table VII. • The T and CP -even asymmetries A C,p T exhibit the linear dependence on thereal part of the anomalous couplings implied by the single spin correlation. • At the level of our study, the final dimuon channel can be replaced withthe dilepton channel (including muons and electrons) and this increases thestatistics by a factor of four without affecting the asymmetries. We explicitlycompute the asymmetries for the dilepton channel for one value of d τW inTable IX obtaining the same asymmetries as in the dimuon channel given inthe other Tables. The asymmetries in the dilepton channel are generalizedfrom the dimuon case based on the lepton charge (so we also include the µ + e − and µ − e + final states).The statistical sensitivity to any of the asymmetries in the dilepton channel with100 fb − is 0.005. This translates into the following future constraints | Im( d τW ) | < ∼ . , | Im( d τB ) | < ∼ . | m τ d Zτ | < ∼ × − , | m τ d γτ | < ∼ × − | Re( d τW ) | < ∼ , | m τ a Zτ | < ∼ . , | m τ a γτ | < ∼ . | Re( d τW )Im( d τW ) | < ∼ , | Re( d τG, ˜ G )Im( d τG, ˜ G ) | < ∼ .
16 (26)It is worth noting here that whether we include Eq.24 or not in Eq.20 only affectsthe constraint we find for Re( d τW ), increasing it to 7 from 5. Recall that this approximation used for the SM results in a τ -width that is only about 10%smaller than the experimental width. . Dilepton pairs in the Z -resonance region The Z -resonance region is selected with the cut 60 < m ττ <
120 GeV, withthe same caveats as before. Keeping the remaining cuts unchanged and generatingadditional samples we obtain the following approximate fits. σ Z σ SM = 1 + 0 . d τW ) + 0 . d τW ) + 0 . | d τB | + 0 . d τW ) A = − .
013 Im( d τW ) − . d τB ) A = 0 . d τW ) + 0 . d τB ) A ss = 0 . d τW ) Im( d τW ) + 0 .
001 Re( d τG ) Im( d τG ) + 0 .
001 Re( d τ ˜ G ) Im( d τ ˜ G ) A C = − . − . × − Re( d τW ) A p T = − . − . × − Re( d τW ) (27)We have not included Re( d τB ) for the asymmetries due to the smaller sensitivityalready observed in the previous case. Our main observations in this case are: • Constraints arising from the cross-section are shown in Figure 2. In this casewe have assumed the cross-section can be measured to 7% accuracy, the currentsystematic uncertainty, following Ref. [35]. Taking only one parameter to be
Im( d τV )Re( d τV ) m τ d Vτ m τ a Vτ γ FIG. 2: Regions of d τV (left) and the corresponding d γ,Zτ , a γ,Zτ (right) allowed by a maxi-mum 7% deviation from the SM cross-section with the cuts described in the text. non-zero at a time we find, | Im( d τW ) | < ∼ . , | Im( d τB ) | < ∼ . − . < ∼ Re( d τW ) < ∼ . , | Re( d τB ) | < ∼ . | m τ d Zτ | < ∼ . , | m τ d γτ | < ∼ . − . < ∼ m τ a Zτ < ∼ . , − . < ∼ m τ a γτ < ∼ . Assuming again 100 fb − implies a much better statistical sensitivity of 0.0009when using both electron and muon channels. This better sensitivity is ofcourse due to the much larger cross-section and results in the potential con-straints | Im( d τW ) | < ∼ . , | Im( d τB ) | < ∼ . | m τ d Zτ | < ∼ . × − , | m τ d γτ | < ∼ . × − | Re( d τW ) | < ∼ . , | m τ a Zτ | < ∼ . , | m τ a γτ | < ∼ . | Re( d τW )Im( d τW ) | < ∼ . , | Re( d τG, ˜ G )Im( d τG, ˜ G ) | < ∼ . C. Background
We end this section with a brief discussion of background and how it wouldaffect the constraints estimated so far. For the dilepton channel in τ -pair productionboth τ -leptons in the pair undergo leptonic decay into muons or electrons: pp → τ + τ − → ℓ + ℓ − /E T , ℓ = µ, e where the missing transverse energy, /E T is due to invisibleneutrinos. If one lepton is a muon and the other one an electron the dominantbackground arises from t ¯ t , W + W − or ZZ production. If the two leptons have thesame flavor there is an additional direct Drell-Yan production of ℓ + ℓ − . In addition,as discussed for example in Ref. [4], contributions from processes where a jet or aphoton is misidentified as a lepton are very small.The different handles to control this background have been identified by theexperimental collaborations. Requiring a minimum missing E T can effectively re-move the direct Dell-Yan background, we will use /E T >
20 GeV. To suppress t ¯ t background experiments require at most one jet and no b tagged jets. These re-quirements are hard to implement at the level of our analysis but they should bekept in mind. The requirement that the two leptons be back to back in the trans-verse plane provides additional suppression against top-pairs and W and Z pairs.This is implemented as [3] cos ∆ φ ( ℓ − , ℓ + ) < − .
95 (31)where ∆ φ ( ℓ − , ℓ + ) is the difference in azimuthal angle between lepton pairs. And tofurther suppress the contamination from W products, events are selected with anadditional requirement that the signature being consistent with that of a particledecaying into two τ leptons. With the following projection variables [3] p visξ = ~p T ℓ + · ˆ ξ + ~p T ℓ − · ˆ ξ,p ξ = p visξ + −−−→ E miss T · ˆ ξ (32)we require p ξ − (1 . × p visξ ) > −
10, where ˆ ξ is a unit vector along the bisector ofthe momenta of the two leptons. 12enerating MonteCarlo samples for each of the background processes with Mad-graph and applying all the preceding cuts to background and signal samples resultsin cross sections: σ ( pp → τ + τ − → µ + µ − /E T ) = 20 .
28 fb σ ( pp → t ¯ t → b ¯ bµ + µ − /E T ) = 120 . σ ( pp → W + W − → µ + µ − /E T ) = 18 .
16 fb σ ( pp → ZZ → µ + µ − /E T ) = 1 .
75 fb (33)These numbers allow us to quantify the effect of background as follows. First, thecuts needed to isolate the signal reduce its cross-section by a factor of 4.7 whichresults in a factor of 2.2 loss in statistical sensitivity. In addition the final eventsample will contain background events that, under our assumptions, are not affectedby the new physics. If we use the CMS [36] b-tagging efficiency between 70-85% weexpect less than 11 fb of σ ( pp → t ¯ t ) background to remain. In this case the T-oddasymmetries are reduced by about 2.5.Bounds estimated from the cross-sections do not depend so much on the back-ground cross-section as on its uncertainty and this has already been taken intoaccount when we use the experimental estimates for the precision they can achievein their cross-section measurements. The effect of background on the T-even asym-metries is much harder to estimate, but these do not improve the bounds on Re( d τV )significantly over bounds obtained from cross-sections in any case. IV. SUMMARY
We have examined the possible limits that can be placed on certain anomalouscouplings of τ -leptons at the LHC14 with 100 fb − . We have considered the fourdipole-type couplings that appear at dimension six in the effective Lagrangian aswell as the two τ -gluon couplings that appear at dimension eight. We have found thestatistical sensitivity of single and double spin asymmetries in the dilepton channelto these couplings and compared them to the statistical sensitivity from measur-ing deviations from the SM cross-section at the 14% level. We find that T oddasymmetries can improve the bounds on the CP violating couplings but that single-spin asymmetries do not seem to improve the bounds on the anomalous magneticmoments. Acknowledgments
This work was supported in part by the DOE under contract numberde-sc0009974. We thank David Atwood for useful discussions. G.V. thanks thetheory group at CERN for their hospitality and partial support while this work wascompleted. 13 ppendix A: Tables
All the tables are produced from million event samples for the process pp → τ + τ − → µ + µ − ν τ ¯ ν τ ν µ ¯ ν µ at 14 TeV obtained with the following cuts: p T µ ≥
15 GeV, | η | µ ≤ . m ττ >
120 GeV. With the exceptions noted explicitly below, the τ widthwas set to 2 . × − GeV and the resulting cross-sections were then scaled asdescribed in the main text.
Re( d τW ) Im( d τW ) σ (fb) A A A test A C A p T T -odd correlations A , and T -even correlations A C,p T for severalvalues of Re( d τW ) and Im( d τW ). A test should vanish in all cases and gives us an estimateof the statistical error. In Table II we compute the single-spin asymmetries chosen above for a seriesof values of Re( d τW ) and Im( d τW ) along with A test which should be zero up tostatistical error. In Figure 3 we plot the T -odd asymmetries which exhibit theexpected behavior linear in Im( d τW ). The figure also suggests that they have verysmall dependence on Re( d τW ). In Figure 4 we plot the T -even asymmetries, thesituation is less clear in this case and we need to study other tables to reach anyconclusions.In Table III we set Re( d τW ) = 0 and fit the T -odd correlations to a linearequation. This is shown in Figure 5 and the fits are consistent with the previousones from Table II. We tabulate A test again to assess the size of statistical fluctuationsusing an asymmetry that should be zero. Finally we also tabulate results for the14 tMC Im( d τW ) A Im( d τW ) A FIG. 3: A linear fit to Im( d τW ) for A (left) and A (right) in Table II is supported by thedata. The separation between the three points corresponding to three values of Re( d τW )suggests a very small contribution of the form Re( d τW )Im( d τW ). MC Re( d τW ) A C Re( d τW ) A p T FIG. 4: Values of A C (left) and A p T (right) in Table II. The separation between thefive points corresponding to the values of Im( d τW ) for each Re( d τW ) suggests sizeablecontributions of the form (Im( d τW )) . T -even asymmetries which should not have a linear dependence on Im( d τW ). Thisis confirmed in Figure 6 within our statistical uncertainty.Next we repeat the previous exercise but for Im( d τB ) instead. The T -odd asym-metries are also linear in this coupling as expected, but smaller than those inducedby Im( d τW ) as can be seen in Figure 7. Figure 8 illustrates that the T -even asym-metries are not affected by this coupling within our numerical sensitivity.We turn our attention to the CP conserving couplings in Tables V and VI. We15 m( d τW ) σ (fb) A A A test A C A p T T -odd correlations A , and T -even correlations A C,p T for severalvalues of Im( d τW ) with Re( d τW ) = 0. A test should vanish in all cases and gives us anestimate of the statistical error. √ ˆ s < √ ˆ s > Im( d τW ) A √ ˆ s < √ ˆ s > Im( d τW ) A FIG. 5: MC simulation compared to a linear fit to Im( d τW ) for A (left) and A (right) forhigh m ττ events from Table III and also for events in the Z -region 60 < m ττ <
120 GeV . also find the expected behavior here: Figure 9 shows that the T -even asymmetries arelinear in Re( d τW ), and Table V shows that the T -odd asymmetries are all consistentwith zero. The dependence of the T -even asymmetries on Re( d τB ) is not observedwithin our statistical sensitivity as shown in Figure 10.In Table VII we study the dependence of some of the observables on the τ -leptonmass. In order to keep kinematic factors in the τ decay constant, we also set thebottom-quark mass to be always higher than the τ mass and we take the charm,strange, up and down quarks as well as the muon to be massless. We see thatthe Drell-Yan cross section is approximately independent of the τ -lepton mass, asexpected. The width of the τ -lepton exhibits the m τ dependence predicted by theSM when the muon mass is neglected. The fact that A is approximately constant16 p T A C < √ ˆ s < Im( d τW ) A C , p T A p T A C √ ˆ s > Im( d τW ) A C , p T FIG. 6: MC simulation of A C,p T for high m ττ events from Table III and also for events inthe Z -region 60 < m ττ <
120 GeV as a function of Im( d τW ).Im( d τB ) σ (fb) A A A test A C A p T T -odd correlations A , and T -even correlations A C,p T for severalvalues of Im( d τB ) with Re( d τB ) = 0. A test should vanish in all cases and gives us anestimate of the statistical error. in this table supports our interpretation of this result as originating mainly in thesingle spin asymmetry.In Table VIII we illustrate the CP properties of the T -odd asymmetries discussedabove. The asymmetry A is CP -odd for the case of the p ¯ p collider whereas A is CP -even and therefore cannot be induced by the anomalous coupling Im( d τW ). Forthe LHC, a pp collider, both T -odd asymmetries are possible as they transform intoasymmetries in a ¯ p ¯ p collider under a CP transformation. They do so with oppositesigns as can be seen in the Table. The charge asymmetry is C -odd and thereforechanges sign at ¯ p ¯ p collider and vanishes at a p ¯ p collider as seen in the examplein Table VIII. A test on the other hand is not zero for p ¯ p colliders where the beamdirection can be defined unambiguously.17 ˆ s < √ ˆ s > Im( d τB ) A √ ˆ s < √ ˆ s > Im( d τB ) A FIG. 7: MC simulation compared to a linear fit to Im( d τB ) for A (left) and A (right) forhigh m ττ events from Table IV and also for events in the Z -region 60 < m ττ <
120 GeV . A p T A C < √ ˆ s < Im( d τB ) A C , p T A p T A C √ ˆ s > Im( d τB ) A C , p T FIG. 8: MC simulation of A C,p T for high m ττ events from Table IV and also for events inthe Z -region 60 < m ττ <
120 GeV as a function of Im( d τB ). Table IX illustrates that replacing the dimuon channel with the dilepton channelsimply increases the statistics by a factor of four and does not change the fourasymmetries we have been discussing.In Table X we demonstrate that the trick of using a fictitious τ -lepton width inthe simulations does not affect the cross-section or the asymmetries A and A C (italso does not affect the other asymmetries).In Table XI, we set Re( d τW ) = Im( d τW ) to look for the double spin asymmetrythrough A ss . The result of the fit is as shown in Figure 11. In Table XII, we setRe( d τG ) = Im( d τG ) to look for the double spin asymmetry through A ss . The resultof the fit is as shown in Figure 11.In Table XIII, we set Re( d τ ˜ G ) = Im( d τ ˜ G ) to look for the double spin asymmetrythrough A ss . The result of the fit is as shown in Figure 11.18 e( d τW ) σ (fb) A A A test A C A p T -20 806.2 -0.0059 -0.0033 -0.0208 -0.1022 -0.0899-16 548.8 -0.0045 0.0007 0.0034 -0.0998 -0.0876-12 349.3 0.0000 -0.0017 -0.0034 -0.1008 -0.0807-8 206.9 0.0015 0.0013 0.0004 -0.1078 -0.0915-4 122.3 0.0017 -0.0001 0.0002 -0.1105 -0.09350 95.45 0.0003 -0.0000 0.0019 -0.1125 -0.09554 126.4 -0.0026 0.0011 -0.0007 -0.1155 -0.09998 215.1 -0.0020 -0.0005 0.0001 -0.1235 -0.106012 362.1 0.0045 0.0000 0.0014 -0.1236 -0.102316 566.9 0.0039 0.0036 0.0017 -0.1273 -0.110520 830.3 0.0145 0.0079 0.0019 -0.1221 -0.1127TABLE V: Single spin T -odd correlations A , and T -even correlations A C,p T for severalvalues of Re( d τW ) with Im( d τW ) = 0. A test should vanish in all cases and gives us anestimate of the statistical error. √ ˆ s < √ ˆ s > Re( d τW ) A C √ ˆ s < √ ˆ s > Re( d τW ) A p T FIG. 9: MC simulation compared to a linear fit to Re( d τW ) for A C (left) and A p T (right)for high m ττ events from Table V and also for events in the Z -region 60 < m ττ <
120 GeV. e( d τB ) σ (fb) A A A test A C A p T -20 165.9 -0.0002 -0.0007 0.0016 -0.1155 -0.0950-16 140.7 -0.0007 -0.0028 -0.0006 -0.1139 -0.0981-12 121.1 0.0001 0.0003 -0.0017 -0.1142 -0.0962-8 107.1 0.0008 -0.0010 0.0005 -0.1112 -0.0950-4 98.54 0.0005 0.0000 0.0005 -0.1128 -0.09560 95.45 0.0003 -0.0000 0.0019 -0.1125 -0.09554 97.95 -0.0000 -0.0016 -0.0008 -0.1126 -0.09588 105.9 -0.0009 -0.0000 0.0014 -0.1135 -0.096212 119.5 0.0003 0.0002 -0.0002 -0.1137 -0.098316 138.6 0.0017 -0.0004 0.0002 -0.1147 -0.098120 163.1 -0.0035 -0.0026 -0.0026 -0.1176 -0.1002TABLE VI: Single spin T -odd correlations A , and T -even correlations A C,p T for severalvalues of Re( d τB ) with Im( d τB ) = 0. A test should vanish in all cases and gives us anestimate of the statistical error. MC Re( d τB ) A C Re( d τB ) A p T FIG. 10: MC simulation compared to a quadratic fit to Re( d τB ) for A C (left) and A p T (right) in Table VI. τ (GeV) Γ τ ( m τ )Γ τ (2 . · (cid:18) . m τ (cid:19) Br ( τ + → µ + ν µ ¯ ν τ ) σ ( τ + τ − ) σ ( µ + µ − ν ′ s ) A A C τ mass of cross-section and two asymmetries A and A C for Im( d τW )=10. To remove any kinematic dependence from the decay vertices we use m u = m d = m c = m s = 0, m b = 10 in all cases.Collider σ (fb) A A A test A C pp p ¯ p p ¯ p T -odd and T -even asymmetries with Re( d τW )=0,Im( d τW )=10 for different colliders to exhibit their transformation properties under CP .Im( d τW ) σ (fb) A A A C A p T
10 1111.0 -0.1363 0.0974 -0.1100 -0.0956TABLE IX: Selected asymmetries in the dilepton channel ( pp → τ + τ − → ℓ + ℓ − ν ′ s) withRe( d τW )=0. In this case σ SM = 385 . τ (GeV) Im( d τW ) σ (fb) A A C . × −
10 276.0 0.1019 -0.11312 . × −
10 276.0 0.0952 -0.11562 . × −
10 275.8 0.0960 -0.11792 . × −
10 275.9 0.0953 -0.11692 . × −
10 276.0 0.0975 -0.1155TABLE X: Effect of changing the τ -lepton width in the MC simulation. After the rescalingdescribed in the text the cross-section as well as the asymmetries are seen to be indepen-dent of Γ τ within our numerical accuracy. In all cases we took Re( d τW )=0. e( d τW ) Im( d τW ) σ (fb) A ss A ss induced by interference between Re( d τW ) andIm( d τW ). Re( d τG ) Im( d τG ) σ (fb) A ss A C A p T A ss induced by interference between Re( d τG ) andIm( d τG ). No discernible effect from these couplings is found in other asymmetries.Re( d τ ˜ G ) Im( d τ ˜ G ) σ (fb) A ss A C A p T A ss induced by interference between Re( d τ ˜ G ) andIm( d τ ˜ G ). No discernible effect from these couplings is found in other asymmetries. ˆ s < √ ˆ s > Re( d τW )=Im( d τW ) A ss √ ˆ s < √ ˆ s > Re( d τG )=Im( d τG ) A ss √ ˆ s < √ ˆ s > Re( d τ ˜ G )=Im( d τ ˜ G ) A ss FIG. 11: Comparison of MC and a fit linear in Re( d τW )Im( d τW ) (up), Re( d τG )Im( d τG ),(left) and Re( d τ ˜ G )Im( d τ ˜ G ) (right), for A ss shown separately for events with high m ττ andfor events in the Z -region 60 < m ττ <
120 GeV .Re( d τG ) Im( d τ ˜ G ) σ (fb) A ss A C A p T d τ ˜ G ) Im( d τG ) σ (fb) A ss A C A p T d τG and d τ ˜ G do notshow interference effects.
1] S. Chatrchyan et al. [CMS Collaboration], JHEP , 117 (2011) [arXiv:1104.1617[hep-ex]].[2] G. Aad et al. [ATLAS Collaboration], Phys. Rev. D , 112006 (2011)[arXiv:1108.2016 [hep-ex]].[3] S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B , 82 (2012)[arXiv:1206.1725 [hep-ex]].[4] G. Aad et al. [ATLAS Collaboration], Phys. Lett. B , 242 (2013) [arXiv:1210.6604[hep-ex]].[5] G. Aad et al. [ATLAS Collaboration], arXiv:1502.07177 [hep-ex].[6] G. Aad et al. [ATLAS Collaboration], JHEP , 070 (2012) [arXiv:1206.5971 [hep-ex]].[7] S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B , 68 (2012)[arXiv:1202.4083 [hep-ex]].[8] S. Chatrchyan et al. [CMS Collaboration], JHEP , 104 (2014) [arXiv:1401.5041[hep-ex]].[9] G. Aad et al. [ATLAS Collaboration], arXiv:1501.04943 [hep-ex].[10] V. Khachatryan et al. [CMS Collaboration], JHEP , 160 (2014) [arXiv:1408.3316[hep-ex]].[11] W. Buchmuller and D. Wyler, Nucl. Phys. B , 621 (1986).[12] B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, JHEP , 085 (2010)[arXiv:1008.4884 [hep-ph]].[13] A. Hayreter and G. Valencia, Phys. Rev. D , no. 1, 013015 (2013) [arXiv:1305.6833[hep-ph]].[14] H. Potter and G. Valencia, Phys. Lett. B , 95 (2012) [arXiv:1202.1780 [hep-ph]].[15] J. F. Donoghue, Phys. Rev. D , 1632 (1978).[16] D. J. Silverman and G. L. Shaw, Phys. Rev. D , 1196 (1983).[17] S. M. Barr and W. J. Marciano, Adv. Ser. Direct. High Energy Phys. , 455 (1989).[18] F. del Aguila and M. Sher, Phys. Lett. B , 116 (1990).[19] S. Goozovat and C. A. Nelson, Phys. Lett. B , 128 (1991) [Erratum-ibid. B ,468 (1991)].[20] F. del Aguila, F. Cornet and J. I. Illana, Phys. Lett. B , 256 (1991).[21] W. Bernreuther, O. Nachtmann and P. Overmann, Phys. Rev. D , 78 (1993).[22] F. Cornet and J. I. Illana, Phys. Rev. D , 1181 (1996) [hep-ph/9503466].[23] J. Vidal, J. Bernabeu and G. Gonzalez-Sprinberg, Nucl. Phys. Proc. Suppl. , 221(1999) [hep-ph/9812373].[24] G. A. Gonzalez-Sprinberg, A. Santamaria and J. Vidal, Nucl. Phys. B , 3 (2000)[hep-ph/0002203].[25] J. Bernabeu, G. A. Gonzalez-Sprinberg and J. Vidal, Nucl. Phys. B , 87 (2004)[hep-ph/0404185].[26] J. Bernabeu, G. A. Gonzalez-Sprinberg and J. Vidal, JHEP , 062 (2009)[arXiv:0807.2366 [hep-ph]].[27] J. Abdallah et al. [DELPHI Collaboration], Eur. Phys. J. C , 159 (2004)[hep-ex/0406010].
28] K. Inami et al. [Belle Collaboration], Phys. Lett. B , 16 (2003) [hep-ex/0210066].[29] A. Heister et al. [ALEPH Collaboration], Eur. Phys. J. C , 291 (2003)[hep-ex/0209066].[30] T. Stelzer and W. F. Long, Comput. Phys. Commun. , 357 (1994)[arXiv:hep-ph/9401258]; J. Alwall et al. , JHEP , 028 (2007) [arXiv:0706.2334[hep-ph]]; J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer, T. Stelzer, JHEP ,128 (2011). [arXiv:1106.0522 [hep-ph]].[31] N. D. Christensen and C. Duhr, Comput. Phys. Commun. , 1614 (2009)[arXiv:0806.4194 [hep-ph]].[32] O. Antipin and G. Valencia, Phys. Rev. D , 013013 (2009) [arXiv:0807.1295 [hep-ph]].[33] S. K. Gupta and G. Valencia, Phys. Rev. D , 036009 (2011) [Erratum-ibid. D ,119901 (2012)] [arXiv:1102.0741 [hep-ph]].[34] K. A. Olive et al. [Particle Data Group Collaboration], Chin. Phys. C , 090001(2014).[35] G. Aad et al. [ATLAS Collaboration], Phys. Rev. D , no. 5, 052005 (2015)[arXiv:1407.0573 [hep-ex]].[36] S. Chatrchyan et al. [CMS Collaboration], JINST , P04013 (2013) [arXiv:1211.4462[hep-ex]]., P04013 (2013) [arXiv:1211.4462[hep-ex]].