NNLO QCD corrections to the polarized top quark decay $t(\uparrow) \to X_b+W^+$

We compute the next-to-next-to-leading order (NNLO) QCD corrections to the decay $t(\uparrow) \to X_b +W^+$ of a polarized top quark. The spin-momentum correlation in this quasi two-body decay is described by the polar angle distribution $\mathrm{d}\Gamma/\mathrm{d}\cos\theta_P=\frac{\Gamma}{2}(1+P_t\, \alpha_P\, \cos\theta_P)$ where $P_t$ is the polarization of the top quark and $\alpha_P$ denotes the asymmetry parameter of the decay. For the latter we find $\alpha^{\mathrm{NNLO}}_P=0.3792\pm 0.0037$.


Introduction
The number of single top quark events reported by the LHC collaborations ATLAS and CMS in Run 1 and 2 is ever increasing. More and more single top quark events have been and are being seen at the LHC [1,2,3,4]. The present situation concerning both ATLAS and CMS results on single top production is nicely summarized in a review article by N. Faltermann [5]. After Run 3 the LHC will operate in the High Luminosity Mode with a projected total luminosity of 3 ab −1 which corresponds to approximately 10 9 single top quark events. In the dominating t-channel process, which is a weak production process, single top quarks are produced with a large longitudinal polarization P t ≃ 0.9 in the direction of the spectator jet in the top quark rest frame, and a slightly smaller polarization of P t ≃ 0.8 for antitop quarks [6,7,8,9]. 1 Since the top quark decays so rapidly, it retains its polarization from birth when it decays. The dominant decay mode is the quasi-twobody mode t(↑) → X b + W + mediated by the quark level transition t → b proportional to the CKM matrix element V tb ≈ 1.
In this paper we study top quark polarization effects in the quasi two-body decay t(↑) → X b + W + at next-to-next-to-leading order (NNLO) in QCD. The NNLO results are obtained in the form of a power series expansion in terms of the ratio x = m W /m t , where m W and m t are the masses of the W boson and the top quark, and we include terms up to x 10 . This analysis can be considered to be complementary to the decay part of the recent numerical NNLO evaluation of polarized top production and decay [11,12].
Since the decay is weak, the top quark is self-analyzing. The angular decay distribution reads 1 Γ dΓ d cos θ P = 1 2 (1 + P t α P cos θ P ), where θ P is the angle between the polarization direction of the top quark and the momentum direction of the W + (see Fig. 1). The analyzing power for the polarization of the decay is given by the asymmetry parameter α P where, at leading order (LO), one has α LO P = (1 − 2x 2 )/(1 + 2x 2 ) = 0.398. Here and throughout this paper we set the bottom quark mass to zero.
The measurements suggested here require the reconstruction of the momentum direction of the W boson which is not simple experimentally. However, the experimentalists 1 Close to maximal values of the polarization of top quarks can be achieved with moderate tuning of the longitudinal beam polarization at the ILC (see e.g. Ref. [10]). Figure 1: Definition of the polar angle θ P in the decay t(↑) → X b + W + .
have devised sophisticated tools to reconstruct the W -boson momentum direction for their analysis of the helicity fractions in unpolarized top quark decays which can also be used in this analysis.
This paper is organized as follows. In the next Section we outline the calculational methods used to obtain our result. In Sec. 3 we provide a numerical analysis of the decay rate and the asymmetry parameter. A summary and outlook are given in Sec. 4. Analytical results for the decay rate can be found in the Appendix.

Calculation
Our calculation follows the approach used in Refs. [13,14,15] for the calculation of the total unpolarized decay rate and in Ref. [16] for the so-called helicity fractions of the W boson.
Using the optical theorem, we compute the top-quark decay width from the imaginary part of self-energy diagrams, where Σ t is computed from one-particle irreducible self-energy diagrams of the top quark.
We sum over the spin degrees of freedom of the W boson, i.e. we do not specify its helicity components as has been done in Ref. [16]. Thus, we use the unitary gauge form for the spin sum: It is clear that the polar angle distribution is sensitive to the longitudinal polarization vector of the top quark, s ℓ,µ t . We have where iΣ is the sum of the top-quark self-energy diagrams. In the rest frame of the top quark the polarization vector reads s ℓ,µ t = (0; 0, 0, 1), i.e. the three-dimensional polarization vector points into the direction of the momentum of the W boson (z direction in Fig. 1).
For our calculation we require a covariant representation of the longitudinal polarization four-vector s ℓ t which is given by where | q | = q 2 0 − q 2 . The polarization four-vector s ℓ,µ t can be seen to satisfy p t · s ℓ t = 0 and s ℓ t · s ℓ t = −1 where we use the fact that p t · q = m t q 0 in the rest system of the top quark. Just as in the case of the helicity fractions, we find that due to the polarization vector we have to deal with the modulus of the W momentum three-vector in the denominator of the expressions for the self-energy diagrams.
There are altogether 38 three-loop diagrams. Since we use the unitary gauge for the W boson there is no need to include Goldstone bosons in the Feynman diagrams. For the gluons we use the covariant R ξ gauge with the spin sum IP µν (R ξ ) = −g µν +ξ k µ k ν /k 2 , where ξ is an arbitrary gauge parameter. We have checked that the gauge-parameter dependence cancels in the final result. Since we only require traces involving an even number of γ 5 matrices, we can work with a naively anticommuting γ 5 [17,18].
After setting the bottom-quark mass to zero, the Feynman integrals corresponding to the top-quark self-energy diagrams depend on two scales: the hard scale m t and the soft scale m W . We then employ the method of regions (see e.g. Ref. [19]) to construct an expansion around the limit where the ratio x = m W /m t of the two scales tends to zero.
Here, we have to consider two regions for each loop momentum (the loop momenta are chosen to be the momenta of the gluons and the W boson). In the so-called hard region, all components of a loop momentum k scale like the hard scale k µ ∼ m t for µ ∈ {0, 1, 2, 3} and in the so-called soft region all components scale like the soft scale k µ ∼ m W . In each region we then expand the integrand according to the scaling of all loop momenta. If the momentum of a gluon is soft, the corresponding loop integral becomes scaleless and is set to zero in dimensional regularization. We are therefore left with only two contributions for each integral: one where all loop momenta are hard and one where the gluon momenta are hard, but the momentum of the W boson is soft.
This expansion makes it also easier to deal with the unwieldy normalization factor 1/| q | appearing in the covariant representation (5). In the hard region, we can express it in terms of a power series in 1/N 2 where N = (p t + q) 2 − m 2 t = 2p t q + q 2 is the denominator of a top-quark propagator with momentum p t + q. Using again p t · q = m t q 0 we find which then leads to the expansion [16] In our calculation of the Feynman diagrams, we are only interested in the imaginary part due to a cut through the W -boson line. Thus, we can replace q 2 by m 2 W in Eq. (7). The series is then truncated at the desired order in x.
In the soft region, it is not possible to construct an expansion of | q |, since | q | 2 = q 2 0 −m 2 W and q 0 ∼ m W in the soft region. However, in this region the loop containing the W boson factorizes from the remaining diagram due to the expansion. Therefore, the only integrals that have to be modified are one-loop massive tadpole integrals, which are relatively simple.
After the expansion, all remaining integrals depend only on a single scale and are thus easier to compute. However, the denominators of the expanded propagators are now raised to higher powers. We use the program rows [20], which implements the so-called Laporta algorithm [21,22], to reduce all of these integrals to a small set of so-called master integrals.
Compared to the calculation of the unpolarized decay rate and the helicity fractions, we do not encounter any new master integrals.

Numerical results
Our analytical results can be found in the Appendix to this paper. For the numerical evaluation of the analytical expression we use the values m t = 173.1 ± 0.6 GeV, m W = 80.385 ± 0.015 GeV and α (5) s (m Z ) = 0.1182 ± 0.0012 [23]. The strong coupling constant is then evolved to the required scale using five-loop running. Note that our result is expressed in terms of the strong coupling constant with six active flavors, whereas the initial value α (5) s (m Z ) is defined with only five. Thus, we also have to use the (four-loop) decoupling relation to translate the latter into the former. All of this is achieved with the help of version 3 of the program RunDec [24,25]. Our central value is α (6) s (m t ) = 0.1078. We present our results in terms of the reduced helicity ratesΓ α defined by The total unpolarized and polarized rates are denoted by α = U + L and α = (U + L) P , where L refers to the longitudinal and U to the unpolarized-transverse polarization of the W boson (the latter is the sum of the two transverse polarizations). We then expand the reduced rates up to the second order in the strong coupling constant α s aŝ where α s ≡ α (6) s (m t ) is defined with six active flavors and evaluated at the renormalization scale µ = m t . Furthermore, we define the coefficients in the x = m W /m t expansion bŷ ForΓ (2) U +L we use the result of Ref. [14]. Note that the coefficients ofΓ U +L contain logarithms of x. In principle, the sums run up to infinity, but in practice we only calculated the terms up to O(x 10 ). This is sufficient to provide a reliable approximation of the full result. Up to the order O(x n ) we then calculate the NLO and NNLO values of the asymmetry parameter according to the ratio whereΓ NLO α is defined as in Eq. (9), but withΓ (2) α set to zero. In Tab. 1 we give numerical results for the coefficients of the reduced rates and the asymmetry parameter. Analytical results are given in Appendix A. For the reduced rates, we find that the absolute values of the coefficients in the power series in x decrease when the power of x increases. The convergence of the x expansion is also illustrated in Figs. 2 and 3. Fig. 2 shows the O(α s ) and O(α 2 s ) contributions toΓ NNLO (U +L) P as functions of x. We observe in both cases that adding terms beyond x 6 leads only to small changes at the physical value of x. Furthermore, the results truncated after x 8 and x 10 are visually Table 1: Numerical values for coefficients in the x expansion of the unpolarized and polarized reduced ratesΓ NNLO U +L andΓ NNLO (U +L) P (cf. Eq. (10)). The results for the rates are given in the last line. In the third and fourth column we list the values of the asymmetry parameter at NLO and NNLO at a given order n in the x expansion (cf. Eq. (11)).  In order to determine the precision of our final result for the asymmetry parameter, we consider the following sources of uncertainties: • The uncertainty in the mass of the top quark. This is the largest source of uncertainty in our result. We note that in our calculation we have employed the pole mass definition for the top quark, whereas the numerical value corresponds to the so- P,i are defined analogously to the ones in Eq. 10, but forΓ (k) (U +L) P instead ofΓ NNLO (U +L) P . Our central value for α (6) s (m t ) is used throughout. The vertical line indicates the physical value of x. Note that the lines for n = 8 lie below the ones for n = 10.
to our result, which is, however, currently not precisely known and not included in our analysis. (Recent efforts to determine this difference can be found in Refs. [26,27].) • Higher orders in QCD. We estimate the size of unknown higher order corrections by taking half the difference between α NNLO P (10) and α NLO P (10). 2 • The strong coupling constant. In addition to the uncertainty in the value of α (5) s (m Z ), we also vary the decoupling scale at which the five-flavor value is translated to the six-flavor one. However, the effect of the latter is completely negligible.
• The uncertainty in the mass of the W boson.
• The truncation of the series in x. We estimate this effect by taking the difference between α NNLO P (10) and α NNLO P (8). As can be seen from Tab. 1, this uncertainty is very small.
• Non-zero bottom-quark mass. We estimate the error due to setting m b to zero by taking the difference to α NNLO P (10) computed as before, but with m b = 5 GeV in the Born-level contributionsΓ In the last line, we have added the different uncertainties in quadrature. It is important to note that the above result includes only QCD corrections. However, at this level of precision, electroweak corrections can also play a role. Since the electroweak NLO corrections toΓ (U +L) P are currently unknown, we make an estimate of their size by looking at the known corrections to the helicity fractions, where they increase the Born-level results by roughly 2% [28]. The total decay rate is shifted by a similar amount [29,30]. Taking both of these corrections into account changes our result for α NNLO P only at the permille level, which is well within our uncertainty estimate.

Summary and outlook
We It is interesting to observe that the power series expansion of the parity-odd polarized rateΓ (i) P contains both even and odd powers while the parity-even unpolarized rate contains only even powers of x. This follows the pattern observed in the NNLO calculation of the helicity fractions [16]. We regret to say that we are lacking a deep understanding of this pattern. We mention that the electroweak NLO corrections to the structure functions do not follow this pattern.
In this paper we have summed over the three helicities of the W boson. It would be interesting to repeat the calculation for the three helicity components of the W boson separately. The corresponding decay distribution is given by It should be clear that all three asymmetries parameters α P j =Γ P j /Γ j (j = +, −, L) must satisfy the positivity condition |α P j | ≤ 1. The LO Born term values for the unpolarized and polarized structure functions are given by [31,32]Γ The LO asymmetry parameter α P j is undetermined for the transverse-plus rate and maximal for the transverse-minus and the longitudinal rate. This has to be compared to the total LO asymmetry parameter α LO P = 0.398 which is far from being maximal. Including the NLO corrections one obtains |α P,NLO j | < 1 for all three asymmetry parameters [32]. This is very gratifying from the point of view that the O(α s ) asymmetry parameters satisfy the necessary positivity condition |α P j | ≤ 1. We expect that the inclusion of NNLO results in the calculation of the asymmetry parameter will retain this feature.

A Analytical results
In this Appendix we provide the analytical results for the reduced rates defined in Eqs. (8,9).
NNLO α 2 s -corrections We present our results in terms of the color-flavor decomposition where C A = N c = 3, T F = 1/2, N L = 5 and N H = 1. The coefficients ofΓ (2) U +L were calculated in Ref. [14] and are presented here for completeness. We havê Γ x 6 ζ(4),