Model-independent measurement of the top quark polarisation

We introduce a new asymmetry in the decay t ->W b ->l nu b, which is shown to be directly proportional to the polarisation of the top quark along a chosen axis, times a sum of W helicity fractions. The latter have already been precisely measured at the Tevatron and the Large Hadron Collider. Therefore, this new asymmetry can be used to obtain a model-independent measurement of the polarisation of top quarks produced in any process at hadron or lepton colliders.


Introduction
Precision measurements of the top quark properties offer an excellent opportunity to explore indirect effects of new physics beyond the Standard Model (SM). Their theoretical interest is motivated by the large top quark mass, which leads to the common belief that this fermion may be quite sensitive to new physics effects. And, on the experimental side, top quark studies are greatly facilitated by the short lifetime of this quark, τ ∼ 4×10 −25 s, which prevents complications from hadronisation effects and allows to study in detail the properties of a "bare" quark. Thus, for example, the W helicity fractions [1] have been precisely measured at the Tevatron and the Large Hadron Collider (LHC) [2][3][4][5], namely the relative fractions of W bosons with helicity ±1, 0 produced in the decay t → W b.
New physics can enter both the production and decay of the top quark. New production mechanisms may be difficult to spot directly, as is the case of wide tt resonances [6] and non-resonant contributions [7], including t-channel flavour-changing processes [8,9]. But the presence of such contributions would generally result in a top polarisation or tt spin correlation different from the SM predictions [10][11][12][13][14][15]. These are not measurable quantities, however, and can only be probed by analysing angular distributions in the top decay t → W b → ℓνb, with ℓ = e, µ. (Hadronic W decays and leptonic decays to taus are sensitive to the top polarisation too, but their experimental measurement is much more difficult.) For example, a well-known method to probe the top polarisation is through the angular distributions of its decay products in the top quark rest frame. These take the being θ X the angle between the momentum p X of the decay product X = ℓ, ν, b, W in the top quark rest frame, and an arbitrary directionẑ chosen to quantise the top spin. In the above equation, P z is the top polarisation along this direction and α X are constants called "spin analysing power" of the particle X, which can be affected by top anomalous couplings [16,17]. Hence, Eq. (1) clearly shows a production-decay interplay in the θ X distributions: the measurable quantities are the products P z α X , which depend on the production (P z ) and decay properties (α X ) of the top quark. This is a general feature: since the top polarisation (as well as the tt spin correlation) can only be measured through top decay distributions, the resulting observables are also sensitive to anomalous contributions to the W tb vertex. 1 A non-trivial but important issue is then to disentangle new physics in production and decay. This, of course, would become crucial in case that a deviation from the SM predictions was found.
Previous literature [18][19][20] has attempted to get rid of the dependence on the top decay vertex by noting that for the charged lepton the spin analysing power α ℓ depends on W tb anomalous couplings only quadratically, so α ℓ should be less sensitive to new physics. This solution is not satisfactory, however, not only because the new physics affecting top production may modify P at quadratic level too, 2 but also because anomalous W tb couplings are not sufficiently constrained from other sources so as to imply that their quadratic contributions are small. In [22] it has been shown that a global fit to several top decay observables (including W helicity fractions, α ℓ and α b ) can be used to extract P z from single top and tt measurements. In this Letter we focus on a more direct measurement of the top polarisation and introduce a "doubly forward-backward" top decay asymmetry where N stands for the number of events; θ is the angle between the W momentum in the top rest frame p W and a chosen top spin quantisation axisẑ; θ * is the angle between the charged lepton momentum in the W rest frame, p * ℓ , and p W . We show that this asymmetry is related to the top polarisation along theẑ direction and the W helicity in full generality. Since the W helicity fractions can be (and have actually been) measured in a model-independent fashion, A tW FB provides a model-independent measurement of the top polarisation along a chosen axis, in any process of top production at hadron or lepton colliders. In addition, we present here an inequality involving α W and W helicity fractions, which can be used to obtain lower bounds on P z from the measurement of the cos θ W distribution.

Top quark decay in the helicity formalism
We use the Jacob-Wick helicity formalism [23] (see also [24]) to describe the decay of the top quark and W boson using general arguments of angular momentum conservation. Let us fix a (x, y, z) coordinate system in the top quark rest frame, with the positive z axis along the direction in which we want to quantise the top spin. The most general spin state of an ensemble of top quarks can be described by a density matrix with P i = 2 S i . We do not specify the orientation of our x and y axes, which is not relevant for our discussion. The amplitudes for the decay t → W b, for a top quark having third spin component M = ±1/2 and the W boson and b quark having helicities λ 1 = ±1, 0, λ 2 = ±1/2, respectively, can be written as being (θ, φ) the polar and azimuthal angles of p W in the (x, y, z) coordinate system, a Wigner function for a rotation R(α, β, γ) parameterised by its Euler angles (explicit expressions for low j can be found in [25]). Hence, we see that all the dependence on M and the direction of p W is encoded in the D because the two remaining helicity combinations imply a total angular momentum ±3/2 of the W b pair along the direction of p W , which is forbidden for a spin-1/2 decaying top quark.
The decay W → ℓν can also be described in a similar fashion, introducing a (x ′ , y ′ , z ′ ) coordinate system in the W boson rest frame, with the z ′ axis in the direction of p W . Then, the full decay amplitude can be written as with λ 3 (λ 4 ) the helicity of the charged lepton (neutrino) and λ = λ 3 − λ 4 ; (θ * , φ * ) are the polar and azimuthal angles of the charged lepton momentum p * ℓ in the W boson rest frame, using the (x ′ , y ′ , z ′ ) coordinate system. (We denote quantities in the W boson rest frame with asterisks, as opposed to quantities in the top quark rest frame.) Notice the coherent sum over W boson helicities λ 1 . In the case of a W + boson decay, the left-handed structure of the vertex implies λ 3 = 1/2 for the positively charged lepton (anti-fermion) and λ 4 = −1/2 for the neutrino, both taken massless.
From Eqs. (4) and (8), the fully differential decay width is with C a constant phase-space factor and Λ ′ = λ ′ 1 − λ 2 . Integrating over the azimuthal angles φ, φ * gives factors 2πδ M M ′ and 2πδ λ 1 λ ′ 1 , respectively, so that the differential width in the two polar angles reads dΓ dcos θ dcos θ * = 4π 2 C|b λ 3 λ 4 | 2 The total width for t → W b is obtained by integration over the remaining angles, We can identify the helicity fractions F ±,0 , as the relative widths for t → W b with λ 1 = ±1, 0, respectively. Denoting for brevity the sum between brackets in Eq. (12) as D, we have Then, integrating Eq. (10) in the four quadrants cos θ ≷ 0, cos θ * ≷ 0 and dividing by Γ we obtain an explicit expression for the asymmetry in Eq. (2), For anti-top decays λ 3 = −1/2 for the negatively charged lepton and λ 4 = 1/2 for the neutrino, so λ = −1 and the resulting asymmetry is with the helicity fractions for anti-top decays (denoted with bars, as the corresponding anti-top decay amplitudes) satisfyingF 0 = F 0 ,F ± = F ∓ [22].
A by-product of our analysis is obtained by integrating Eq. (10) over the full θ * range to obtain the W boson angular distribution, namely Eq. (1) for X = W . The spin analysing power of the W boson is found to be This implies, given Eqs. (13) for the helicity fractions, that This inequality is practically saturated in the SM because amplitudes with λ 2 = 1/2 are suppressed due to the left-handed W t L b L interaction.

Discussion
We have introduced a FB asymmetry using two angles θ, θ * in the top quark and W boson rest frames, respectively, showing that it is related to the W helicity fractions by Eq. (3). Its value in the SM can be computed using previous calculations for the helicity fractions [22]. The most general effective W tb interaction arising from dimension-six operators can be parameterized as [28] being V L = V tb and V R = g L = g R = 0 in the SM. For this general vertex, we obtain with A naive combination of the current helicity fraction measurements [2][3][4][5], not taking into account correlations between different experiments and data sets, gives F + = 0.007 ± 0.027 , in good agreement with the SM tree-level prediction F + ≃ 3 × 10 −4 , F 0 = 0.697, F − = 0.303 for m t = 172.5 GeV. This implies a relatively small asymmetry A tW FB ≃ 0.11P z . At the LHC, single top quarks produced in the t-channel process are highly polarised in the direction of the spectator quark [11], with P z ≃ 0.9 for centre-of-mass energies √ s = 7, 8 TeV [29], so the expected asymmetry A tW FB ≃ 0.1 is measurable. LHC statistics are excellent and this measurement may eventually be dominated by systematics. In that case, for a precise determination of the single top polarisation it may be convenient to measure instead the ratio A tW FB /A FB , being A FB the well-known lepton FB asymmetry in the W rest frame, see Eq. (19). Given the present helicity fraction measurements, which imply F + /F − 0.05 (this ratio is F + /F − ≃ 10 −3 in the SM), one has A tW FB /A FB ≃ −1/2 P z to a good approximation. Alternatively, A tW FB /F − ≃ 3/8 P z can also be measured. The inequality (17) may also be used to obtain relevant bounds on the top polarisation in processes where it is large. Let us consider again single top production at the LHC. The product κ W ≡ α W P z can be determined from the cos θ W distribution, see Eq. (1). Then, if the experimental measurement is consistent with the SM prediction, say κ exp W ≃ 0.36, the inequality (17) implies 3 which is a very stringent bound since P z ≤ 1 by definition. Note, however, that the same result can be achieved with the measurement of κ ℓ ≡ P z α ℓ from the cos θ ℓ distribution, since α ℓ ≤ 1.
Finally, it is worth mentioning that at a future e + e − International Linear Collider top quarks are produced in pairs with a small but non-zero polarisation, P z ≃ 0.14 in the helicity axis for √ s = 500 GeV [30]. The precision expected for asymmetry measurements is excellent [31], and therefore the measurement of this asymmetry may be very useful to complement top-spin-independent observables to probe anomalous contributions to e + e − → tt independently of the decay vertex [32]. In any case, it is clear that the new asymmetry A tW FB introduced here provides a new handle to measure the top polarisation in any process, and test the presence of new physics in the top sector.