All-orders behaviour and renormalons in top-mass observables

We study a simplified model of top production and decay, consisting in a virtual vector boson $W^*$ decaying into a massive-massless $t$-$\bar{b}$ quark-antiquark pair. The top has a finite width and further decays into a stable vector boson $W$ and a $b$ quark. We then consider the emission or the virtual exchange of one gluon, with all possible light-quark loop insertions. These are the dominant diagrams in the limit of an infinite number of light flavours. We devise a procedure to compute this process fully, by analytic and numerical methods, and for any infrared-safe final-state observables. We examine the results at arbitrary orders in perturbation theory, and assess the factorial growth associated with renormalons. We look for renormalon effects leading to corrections of order $\Lambda_{\rm QCD}$, that we dub `linear' renormalons, in the inclusive cross section (with and without selection cuts), in the mass of the reconstructed-top system, and in the average energy of the final-state $W$ boson, considering both the pole and the $\overline{\rm MS}$ scheme for the top mass. We find that the total cross section without cuts, if expressed in terms of the $\overline{\rm MS}$ mass, does not exhibit linear renormalons, but, as soon as selection cuts are introduced, jets-related linear renormalons arise in any mass scheme. In addition, we show that the reconstructed mass is affected by linear renormalons in any scheme and that the average energy of the $W$ boson (that we consider as a simplified example of leptonic observable), in any mass scheme, has a renormalon in the narrow-width limit, that is however screened at large orders for finite top widths, provided the top mass is in the $\overline{\rm MS}$ scheme.


Introduction
The top mass is measured quite precisely at the LHC by both the ATLAS [1] and the CMS [2] Collaborations. Up to now, the methods that yield the most accurate results are the so called "direct" methods, that are based upon the reconstruction of the top-decay products. The measurement is performed by fitting kinematic distributions that are closely related to the top mass with those obtained using an event generator, and by extracting the fitted value of the top mass. Current uncertainties are now near 500 MeV [3,4], so that one can worry whether QCD non-perturbative effects may substantially affect the result. In fact, the experimental collaborations estimate these and other effects by varying parameters in the generators, and eventually comparing different generators. This method has been traditionally used in collider physics to estimate theoretical uncertainties due to the modeling of hadronization and underlying events, and also to estimate uncertainties related to higher perturbative orders, as produced by the shower algorithms [5]. As such, it is a valuable method, but it should not be forgotten that it may only provide a lower bound on the associated errors. It is thus important, at the same time, to investigate the associated uncertainties from a purely theoretical point of view. In consideration of our poor knowledge of non-perturbative QCD, these investigations can at most have a qualitative value, but may help us to understand sources of uncertainties that we might have missed. One such work is presented in Ref. [6], where the authors attempt to relate a theoretically well-defined mass parameter with a corresponding shower Monte Carlo one, using as observable the jet mass of a highly boosted top.
In the present work, we consider the interplay of non-perturbative effects with the behaviour of perturbative QCD at large orders in the coupling constant, focusing in particular upon observables that, although quite simple, may be considered of the kind used in "direct measurements".
It is known that in renormalizable field theories, the renormalization group flow of the couplings leads to the so called renormalons, i.e. to the factorial growth of the coefficients of the perturbative expansion as a function of the order [7][8][9][10][11][12][13][14]. Renormalons lead to a divergence of the perturbative expansion, that thus becomes asymptotic. In particular, in the case of infrared renormalons in asymptotically-free field theories, the ambiguity in the summation of the series corresponds to a power suppressed effect.
For top-mass observables, ambiguities of order Λ/m t (where Λ is some hadronic scale and m t is the top mass) are particularly important, since they affect the top-mass measurements by an amount close to the level of the current accuracy.
In the following we will refer to renormalons leading to linear power suppressions as "linear renormalons" (or, unless explicitly specified differently, simply as "renormalons").
The full renormalon structure of QCD is not known. There is however a fully consistent simplified model where higher order corrections are accessible up to all orders in the coupling, namely the large-n f limit of QCD, where the number of flavours n f is taken large and negative (see, for example, Ref. [15]). Very often, estimates of non-perturbative effects are performed starting with the large-n f result, where, at the end of the calculation, one makes the replacement where C A = 3, T R = 1/2 and n l is the number of light flavors. This approach is called "large-b 0 approximation". With such replacement, the β function of the large (negative) n f theory becomes the β function of the full QCD with n l massless flavours.
In the present work we consider a fictitious process W * → tb → W bb, where the W boson has only a vector coupling to quarks, and examine the behaviour of the cross section, of the reconstructed-top mass and of the energy of the W boson, order by order in the strong coupling expansion, taking the large-n f limit. We consider up to one gluon exchange (or emission), and dress this gluon with an arbitrary number of fermion vacuum-polarization insertions. Furthermore, we also consider final states where the gluon has undergone a splitting into a fermion-antifermion pair, corresponding to a cut vacuum polarization diagram. We assume a finite width for the top quark.
We have devised a method that allows us to compute in principle any observable in our process, without further approximations, making use of simple numerical techniques. We can thus compute the perturbative expansion at any finite order and infer its asymptotic nature for any observable, with the only limitation of the numerical precision.
We focus for simplicity upon simple top-mass observables, such as the inclusive cross section with or without cuts, the reconstructed top mass, defined as the mass of a system comprising the W and a b (notb) jet, and, as a simplified example of leptonic observable, the average value of the energy of the final-state W boson. As discussed earlier, we consider our reconstructed top mass as an oversimplified representation of observables of the kind used in the so called "direct" measurements. We also stress that we consider the kinematic region where the top energy is not much larger than its mass, that is the region typically used in direct measurements.

Generalities on renormalons
Infrared renormalons [9,10] provide a connection between the behaviour of the perturbative expansion at large orders in the coupling constant and non-perturbative effects. They arise when the the last loop integration in the (n + 1)-loop order of the perturbative expansion acquires the form (see e.g. [13,14]) where Q is the typical scale involved in the process and b 0 is the first coefficient of the QCD beta function (2. 2) The coefficient b 0 arises because the running coupling is the source of the logarithms in eq. (2.1). A naive justification of the behaviour illustrated in eq. (2.1) can be given by considering the calculation of an arbitrary dimensionless observable, characterized by a scale Q, including the effect of the exchange or emission of a single gluon with momentum l, leading to a correction that, for small l, takes the form where k is an integer greater than zero for the result to be infrared finite. Assuming that higher order corrections will lead to the replacement of α S with the running coupling evaluated at the scale l, given by the geometric expansion substituting eq. (2.4) into eq. (2.3), we obtain the behaviour of eq. (2.1). The coefficients of the perturbative expansion display a factorial growth. The series is not convergent and can at most be interpreted as an asymptotic series. In general, the terms of the series decrease for low values of n, until they reach a minimum, and then they start to increase with the order. The minimum is reached when that corresponds to n ≈ k/(2b 0 α S (Q)), and the size of the minimal term is The value of k depends upon the process under consideration. In this paper, we are interested in linear IR renormalons, corresponding to k = 1, that can lead to ambiguities in the measured mass of the top quark of relative order Λ QCD /m t , i.e. ambiguities of order Λ QCD in the top mass. Larger values of k lead to corrections of relative order Λ k QCD /m k t , that are totally negligible. It is in general not possible to compute the normalization of the tower of factorially growing terms in non-trivial field theories. There is, however, a context where this calculation simplifies to such an extent that it can be carried out exactly. This is the leading number of flavors approximation, in which one considers the corrections given by the exchange of a single gluon, including all possible vacuum-polarization diagrams given by a single fermion loop. Each vacuum polarization diagram yields a factor of α S n f , where we denote by n f a fictitious number of light flavours, so that, in the large n f limit, these contributions are dominant. In order to obtain an estimate of the renormalon effects in the full non-Abelian theory, at the end of the calculation one performs the replacement n f → −11C A /(4T R ) + n l , where n l is the true number of light flavours in the theory. This leads to the correct, non-Abelian running of the coupling constant. This procedure, known as the "large-b 0 approximation", has been used in several contexts [14], and it leads to reasonable results.
In this work, we study renormalon effects on top-mass related observables in the largeb 0 approximation. We know that, in this framework, there are renormalons arising in the computation of the position of the pole in the top propagators, and we also know that there must be renormalons associated to jets requirements. We will also be able to compute the perturbative expansion order by order in perturbation theory, and thus determine explicitly the effects of renormalons in the perturbative expansion.
Our results can be given in terms of the top mass expressed either in the pole or in the MS mass scheme. We know that the expression of the pole mass in terms of the MS mass has a linear renormalon. If the MS mass is considered a fundamental parameter of the theory, this is to be interpreted as an uncertainty of the order of a typical hadronic scale associated to the position of the pole in the top propagator. One may wonder whether the pole mass could instead be used as a fundamental parameter of the theory, which would imply that the MS mass has an uncertainty of the order of a hadronic scale. In fact, it is well known and clear (but nevertheless we wish to stress it again) that this last point of view is incorrect. QCD is characterized by a short distance Lagrangian, and its defining parameters are short distance parameters. Thus, if we compute an observable in terms of the MS mass, and we find that it has no linear renormalons, we can conclude that the observable has no physical linear renormalons, since its perturbative expansion in terms of the parameters of the short distance Lagrangian has no linear renormalons. On the other end, in the opposite case of an observable that has no linear renormalons if expressed in terms of the pole mass, we must conclude that this observable has a physical renormalon, that is precisely the one that is contained in the pole mass. We also stress that it is the MS mass that should enter more naturally in the electroweak fits [16][17][18] and in the calculations relative to the stability of the vacuum [19][20][21][22], although in practice the pole mass if often used also in these contexts.

Description of the calculation
A sample of Feynman diagrams contributing to the process W * → tb → W bb is depicted in Fig. 1. The dashed blob represents the summation of all self-energy insertion in the large-n f limit.
We want to compute a generic observable, function of the final-state kinematics Φ, that we denote with O(Φ). We assume the eventual presence of a set of cuts Figure 1: Feynman diagram for the Born W * → W bb process (a), and samples of Feynman diagrams for the virtual contribution (b), for the real-emission contribution (c) and for W * → W bb qq production (d).
function of the final-state kinematics, and define The average value of O can be written as where the first term represents the Born contribution, the second the virtual one, the third the one due to the emission of a real gluon and the fourth represents the contribution of the real production of n f qq pairs. Equation (3.2) implicitly defines our notation for the different phase space integration volumes. We always imply that the gluon propagator, in the last three contributions, includes the sum of all vacuum-polarization insertions of light-quark loops.
N Θ is a normalization factor, given by We can then rewrite eq. (3.2) as where we have subtracted and added the same quantity to the qq contribution. In the last two lines, O Θ (Φ g * ) is defined in terms of the Φ g * phase space, that is obtained from the Φ qq phase space by clustering the qq pair into a single pseudoparticle, that we denote with g * . We also define

The normalization factor
The factor N Θ appearing in eqs. (3.5)-(3.7) is in fact simply the inverse of the Born cross section, since the quantities it multiplies are already at NLO level. Thus, in these cases, The factor of N Θ in front of the Born term in eq. (3.4), on the other hand, generates extra contributions of the form where This gives rise to a Born term of the form plus an NLO correction equal to (3.12) In summary, eq. (3.4) becomes (3.13)

Final results
In App. B we prove that the full result with the gluon propagator dressed with all fermionic self-energy corrections can be computed in terms of the matrix elements for the process W * → W bb with the real emission or virtual exchange of one massive gluon of mass λ, and the matrix element for the W * → W bbqq tree-level process. The general result for the average value of a generic observable O, in the presence of final-state cuts Θ, obtained by combining the results of Sec. 3.1 and App. A and B, is where and σ (1) g * (λ, Φ g * ) and σ (1) v (λ, Φ b ) are the real/virtual corrections to the process W * → W bb for the emission/exchange of a single gluon with mass λ, and σ (2) qq (Φ qq ) is the tree-level cross section for the process W * → W bbqq. We denote with k the four-momentum of the qq pair. Notice that, in eq. (3.14), α S in T (λ) cancels against the 1/α S in front of the integral, and so that the resummed result for O does not depend upon the value of µ. As discussed in App. B, T (λ) vanishes for large λ, so that the integral in eq. (3.14) is convergent. In order to use the above formulae, we computed analytically the cross sections σ (2) qq (Φ qq ), σ (1) g * (λ, Φ g * ) and σ (1) v (λ, Φ b ). Due to the finite gluon mass, only ultraviolet divergences arise in the intermediate steps of the calculation. These divergences were dealt with in dimensional regularization. After the mass renormalization has been carried out (adopting a complex mass [23,24] in order to account for the finite top width), the UV divergences cancel in the virtual contribution because of the vector nature for the incoming W * current. For reasons that will become clear later, we have also computed the same cross sections for λ = 0. In this case, also soft and collinear divergences are treated in dimensional regularization, and the full result is obtained applying a subtraction method. Notice that, for a finite gluon mass, large logarithms of the mass arise in the real and virtual contributions, that cancel in the sum. In the massless limit, these large cancellations are handled by the subtraction method, and do not affect the accuracy of the result.
We evaluated the scalar integrals using COLLIER [25]. The final numerical implementation has been built using the POWHEG BOX RES framework [26].
We performed the phase-space integral for V , R, ∆ numerically for several values of λ. For small λ both V and R have logs of λ that cancel in the sum, so that one recovers the result corresponding to the NLO corrections to the W * → W bb process involving the exchange or emission of a single massless gluon. The ∆ term is instead finite by itself, and vanishes for small λ. We then combine these results, for each observable, in our function T that we fit as a function of λ. This allows us to compute the coefficients of the perturbative expansion of our observable at any order in perturbation theory, and also to determine its asymptotic behaviour.
We find that, in general, the behaviour of T for small λ is given by a constant plus a linear term in λ. It is this linear term that is associated with linear renormalons. As shown in Sec. 2, these correspond to power suppressed contributions of order Λ p with p = 1, where Λ is a typical hadronic scale. Higher values of p arise from higher powers of λ in the expansion of T . In the present work, we are interested only in p = 1, since, because of the size of the top mass, higher values are suppressed by a further Λ/m t factor.
The inclusive cross section, with or without cuts, is given by formulae similar to the ones from (3.16) to (3.20), setting O = 1 and omitting the normalization factor N (0) Θ . We then write We notice that when computing inclusive quantities or quantities that do not depend upon the jet kinematics, the ∆(λ) and ∆(λ) terms of eqs. (3.20) and (3.27) are zero. In these cases, our results can just be expressed as functions of the NLO differential cross section computed with a non-zero gluon mass. In general, however, the ∆(λ) and ∆(λ) contributions cannot be neglected, since observables built with the full kinematics may differ from those obtained by clustering the qq pair into a massive gluon. This was first discussed in Ref. [27], in the context of e + e − annihilation into jets. 1

Changing the mass scheme
The relation between the pole mass m and the MS mass m in the large-n f limit is discussed in App. C.2. We have where r f and r d are defined in eqs. (C.32) and (C.33) respectively, and r (f) d is the finite part of r d , that does not receive any contribution from linear terms in λ. The r f contribution can be written in the form where (see eq. (C.36)) Note that the µ dependence disappears in the leading term. The O(λ) term in eq. (3.30) is responsible for the presence of a linear renormalon in the relation between the pole mass and the MS one. 2 In the present work we deal with the finite width of the top quark by using the complex mass scheme [23,24]. Thus, in our mass relation, both m and m are complex, and also r f and r d .
Given a result for a quantity O expressed in terms of the pole mass, representing the average value of some kinematic quantity (possibly including cuts and possibly normalized to the total cross section), in order to find its expression in terms of the MS mass we need to Taylor-expand its mass dependence in its leading order expression, and multiply it by the appropriate mass correction. In order to do so, we express O in terms of the pole mass and its complex conjugate, as if they were independent variables (one can think of m appearing in the amplitude, and m * appearing in its complex conjugate). Denoting with O b the LO prediction, we can write where we have neglected α 2 S (α S b 0 ) n terms and we have dropped the µ dependence in m for ease of notation. Notice that, as far as the linear term in λ is concerned, we get the simplified form The relation between the pole and the MS mass in the large-n f limit is well-known (see e.g. Refs. [15,30,31]). Here we have re-derived it so as to put it in a form similar to eqs. (3.14) and (3.22).

Furthermore, we have
Thus, when going from the pole to the MS mass scheme, the definition for T is modified for small λ into One may wonder where the O b subtraction term, that is present by definition in T , is hiding here. In fact, in the case of normalized observables, it should be kept in mind that O b includes a division by the total cross section. When taking the derivative, the denominator is also derived, yielding the O b subtraction term. This procedure is still valid for a generic observable O that does not involve the normalization factor N Θ , like the total cross section, so also in this case we need to replace T with Notice that the same expression holds for T and T . We also stress that eqs. (3.34) and (3.35) also apply to any so called "short distance" mass schemes [32][33][34][35][36][37][38]. These schemes are such that no mass renormalon affects their definition, and of course in order for this to be the case, their small λ behaviour should be the same one of the MS scheme.

Physical objects
The numerical values of the parameters used in our study are given by

Selection cuts
In order to better mimic realistic experimental analyses adopted at hadron colliders, at times we introduce selection cuts for our cross sections, requiring the presence of a b jet and a (separated)b jet, both having energy greater than 30 GeV. Jets are reconstructed using the Fastjet [39] implementation of the anti-k t algorithm [40] for e + e − collisions, for various values of the radius parameter R. W * → tb → W bb, total cross section

Inclusive cross section
The formula for the inclusive cross section is given in eq. (3.22), that will be applied both without and with cuts.

Inclusive cross section without cuts
In the absence of cuts, the expression for T (λ) in eq. (3.24) simplifies, since ∆(λ), given by eq. (3.27), is identically zero. Its small λ behaviour is shown in Fig. 2. From the figure we can see that the error on T (λ) increases for small λ. However, the point at λ = 0 is directly computed with a massless gluon, by dealing with the soft and collinear singularities with the usual dimensional regularization techniques, and has negligible error. As discussed in Sec. 3.3, the same calculation performed in the MS mass scheme would yield, for the total cross section, to the replacement given in eq. (3.35) So, in the same figure, we also plot (in red) the expression Since this coincides with T (λ) for small λ, we infer that the MS result has no linear term in λ, so that no linear renormalons arise for the total cross section in the MS scheme. From  the figure it is also clear that this holds for both λ Γ t and for λ Γ t , where Γ t is the top width. The λ Γ t behaviour is justified by the fact that, because of the finite width, phase-space points where the top is on shell are never reached (see App. D). Thus, no linear renormalon is present unless one uses the pole-mass scheme, that has a linear renormalon in the counterterm.
As far as the λ Γ t limit is concerned, we notice that the λ behaviour should be the same as that of the narrow width approximation (NWA), where the cross section factorizes in terms of the on-shell top-production cross section, and its decay partial width The behaviour of T (λ), computed either exactly or in the NWA, is shown in Fig. 3. The factor σ(W * → tb) is clearly free of linear renormalons, since it is a totally inclusive decay of a colour-neutral system. Although less obvious, this is also the case for the factor Γ(t → W b) (see Refs. [15,41,42]).

Inclusive cross section with cuts
When the selection cuts discussed in Sec. 4.1 are imposed, the cross section depends explicitly upon the jet radius R. We expect that jets requirements will induce the presence of linear renormalons, and thus linear small-λ behaviour of T , with a slope that goes like 1/R for small R [43]. In Fig. 4 we display the small λ behaviour for T (λ) for the inclusive cross section with cuts, for several jet radii. Together with the results of our calculation, we also plot, for each value of R, a polynomial fit to the data. W * → tb → W bb, total cross section with cuts When changing from the pole to the MS-mass scheme, we only expect a mild R dependent correction 3 to the slope of T (λ) at λ = 0, and thus we cannot expect the same benefit that we observed for the cross section without cuts. This is illustrated in Fig. 5 for several jet radii. The 1/R behaviour is clearly visible. In addition, for relatively large-R values, the use of the MS scheme brings about some reduction to the slope of the linear term. This may be due to the fact that the cross section with cuts captures a good part of the cross section without cuts, and thus it partially inherits its benefits when changing scheme. However, it is also clear that linear non-perturbative ambiguities remain important also in the MS scheme when cuts are involved.

Reconstructed-top mass
In this section we consider the average value M , where M is the mass of the system comprising the W boson and the b jet. Such observable is closely related to the top mass, and, on the other hand, is simple enough to be easily computed in our framework. We use the same selection cuts described previously.
We computed M also in the narrow width limit, by simply setting the top width to 10 −3 GeV. In this limit, top production and decay factorize, so that we have an unambiguous assignment of the final state partons to the top decay products. We first compute M in the narrow width limit, using only the top decay products, and without applying any cuts. We then compute it again, still using only the top decay products, but introducing our standard cuts. Finally we compute it again using all decay products and our standard cuts. The results of these calculations are reported in Fig. 6, where the slope at λ = 0 of T for our observable is plotted as a function of the jet radius R. As expected we see the shape proportional to 1/R for small R [43].
In the case of the calculation of M performed using only the top decay products, and without any cuts, we expect that, for large values of R, the average value of M should get closer and closer to the input top pole mass, irrespective of the value of λ. Thus, the slope of T (λ) for λ = 0 should become smaller and smaller. We find in this case that, for the largest value of R we are using (R = 1.5), the slope has a value around 0.09. When cuts are introduced this value becomes even smaller, around 0.04. This curve is fairly close to the one obtained using all final-state particles and including cuts. The large-R value in this case is −0.08.
If we change scheme from the pole mass to the MS one, the corresponding change of T is given by eq. (3.34), and, for the observable at hand, the derivative term is very near 1. The change in slope when going to the MS scheme is roughly −C F /2 ≈ −0.67. Thus, if we insisted in using the MS mass for the present observable, for large jet radii, we would get an ambiguity larger than if we used the pole mass scheme. The same holds even if we employ a finite top width, as shown in Fig. 7, where the R dependence of the T (λ) slope for Γ t = 1.3279 GeV is plotted.
In Fig. 8 we plot the small λ behaviour of T (λ) for the reconstructed-top mass, com- puted with the finite top width, for several values of the jet radius R. It is clear that our observable is strongly affected by the jet renormalon. The same plot for only the three largest values of R is shown in Fig. 9. The figure shows clearly that the slope of T (λ) for small λ computed with Γ t = 1.3279 GeV changes when λ goes below 1 GeV, that is to say, polynomial fit when it goes below the top width. This behaviour is expected, since the top width acts as a cutoff on soft radiation. In the figure we also report the λ behaviour in the narrow-width approximation. It is evident that the slopes computed in this limit are similar to the slopes with Γ t = 1.3279 GeV, for values of λ larger than the top width. It is also clear that the slopes that we find here for the largest R value are considerably smaller than the slope change induced by a change to a short distance mass scheme, that amounts to −0.67. In other words, the pole mass scheme is more appropriate for this observable, irrespective of finite width effects. We notice that, in the present case, for values of R below 1, the MS scheme seems to be better, because of a cancellation of the R dependent renormalon and the mass one. From our study, however, it clearly emerges that such cancellation is accidental, and one should not rely upon it to claim an increase in accuracy.

W boson energy
In this section we study the behaviour of the average value of the W energy, E W . This observable does not depend upon the jet definition, and can thus be considered a representative of pure "leptonic" observables in top-mass measurements. In this study, we do not apply any cut, in order to avoid jet renormalons. Our goal is to see if this observable is free of renormalons in some mass scheme.
In order to change scheme, according to eq. (3.34), we need the derivative of the Born value of the observable with respect to the real part of the top mass. We have computed  Figure 11: Small λ behaviour of T (λ) for E W , for increasingly smaller values of Γ t . The blue and the black solid lines are a parabolic fit of the computed points, the red line is the same one displayed in Fig. 10.
numerically this quantity, and found the value The small-λ dependence of the corresponding T function is shown in Fig. 10. For values of λ much larger than the width, the slope of the curve is roughly 0.45. Thus, under these conditions, a renormalon is clearly present whether we use the pole or the MS scheme, since the correction in slope due to the use of the latter would be −0.098 × C F /2 = −0.065. For λ below the top width we see a reduction in slope, that is too difficult to estimate because of the lack of statistics. In order to check that the change in slope is related to the top finite width, we ran the program with a reduced Γ t , expecting to see a constant slope extending down to smaller values of λ. This is illustrated in Fig. 11. We clearly see that, as Γ t becomes smaller, the slope of the λ dependence remains constant, near the value 0.45 found before, down to smaller values of λ. Since we have that it is clear that, for a vanishing top width, E W has linear renormalons both in the MS and pole mass scheme. We also performed a run with Γ t = 10 GeV and Γ t = 20 GeV, in order to estimate more accurately the value of the slope for λ Γ t . The results are shown in Fig. 12. In Tab. 1 we illustrate the slopes of T (λ) for small λ, obtained from the polynomial interpolation displayed in Fig. 12, and the corresponding value in the MS scheme, obtained by adding ∂ Re(m) to the fitted slope. This shows that the linear sensitivity largely cancels in the MS scheme. One may now wonder if this cancellation is exact, or just accidental. In fact,  we prove in App. D that the cancellation is exact.

All-orders expansion in α S
We consider now the all-orders expansion of various quantities, in order to see how it is affected by the infrared renormalons, both in the pole and in the MS-mass scheme.
One may think that in our framework we could, for example, compute a mass sensitive observable, extract the mass (for a given value of the observable) in different schemes, and finally convert all results to the pole mass scheme, thus assessing the reliability of the methods used to estimate the renormalon ambiguity in the pole mass [35,[44][45][46]. In fact, within the large-n f approximation, if the method adopted to resum the perturbative expansion is linear, as is the case of the Borel transform method, we should find identical results (always in the large-n f sense) in the MS and the pole-mass schemes. In fact, following eq. (3.31), the relation between the pole and MS-mass scheme, for a generic observable, is given by Neglecting subleading terms, this is an identity, since the expansion of O b in the mass difference stops at the first order in the large-n f limit. When performing the calculation in the pole-mass scheme, we need to resum the expansion of O (1) , while if we perform the calculation in the MS scheme, we are resumming the expansion of the sum of terms in the curly bracket. If the resummation method is linear, this last resummation can be performed on the individual terms inside the curly bracket. This is exactly what we would do on the left-hand side if, after the resummation, we wanted to express the same result in the MS scheme. In other words, if one uses the Borel method to perform the resummation, and defines the pole mass to be the sum of the mass relation formula eq. (3.28), all results obtained in the MS scheme would be identical to those obtained in the pole mass scheme up to terms of relative order α S C F , provided the same Borel sum method is used also for the observables. In the following we will estimate the terms of the perturbative expansion by extrapolating our large-n f results to the realistic QCD case, (i.e. in the large-b 0 approximation). In order to do this, we will replace the b 0 of the large-n f theory with the b 0 of QCD, and perform other minor adjustments, detailed later. Needless to say, corrections to the large-b 0 approximation may be non-negligible. We thus expect that, by changing scheme, we will generate potentially important differences. These differences should not therefore be interpreted as due to large ambiguities related to the choice of mass scheme, but rather to the violation of the large-b 0 approximation.
The procedure we adopt in order to compute the terms of the perturbative expansion follows from eq. (3.14). We fit numerically the λ dependence of the appropriate T or T function, and we take the derivative of the fit. The arctangent factor is instead expanded analytically, and the integration is performed numerically for each perturbative order. The details of the procedure that we followed in order to go from the large-n f theory to the large-b 0 approximation are described at the end of App. A, eqs. (A.9) to (A.13).

Mass-conversion formula
The procedure for the calculation of the mass-conversion formula is described in App. C.2.
Here we switch to the realistic b 0 and C values as discussed in the previous section. The and the c i coefficients are tabulated in Tab. 2, with µ 2 = Re(m 2 ) = m 2 0 , where m 0 is given in eq. (4.1). Since we are using the complex-mass scheme, the c i coefficients are complex, with a small imaginary part, and they have a slight dependence upon the ratio Γ t /Re(m). For small Γ t they become independent on m and Γ t , and their imaginary part vanishes.
We have checked that for Γ t = 0 and in the large-n f limit, i.e. setting C A = 0 in our numerical code used to produce the coefficients of Tab. 2, we obtain the same results presented in Ref. [47].

The inclusive cross section
In this section we deal with the perturbative expansion of the inclusive cross section, first without cuts, and then with cuts.
The function T (λ) of eq. (3.24), needed to calculate the integral in eq. (3.22), is obtained as an interpolation of T (λ) computed at several fixed values of λ. We have chosen a polynomial fit, for values of λ less than 5 GeV, of the form and a cubic spline for larger values of λ. The two forms are required to match in value and slope at the joining point. The same approach is adopted to evaluate T (λ) of eq. (3.17), both for the averaged reconstructed-top mass and for the averaged W -boson energy.   The fitting functions that we obtain are seen to represent sufficiently well the numerical results for T , with the only caveat that, for small λ, these have themselves non-negligible errors. These errors strongly affect the coefficient p 1 , and have negligible effects on the other coefficients. In fact, p 0 is obtained directly from the results computed with a massless gluon, and has a totally negligible error. The p 2 and higher-order coefficients are controlled by the larger values of λ, where our computation has a smaller error. We thus propagated the errors of our numerical data to the p 1 coefficient only, and then to the calculation of the coefficients of the perturbative expansion.

Inclusive cross section without cuts
As discussed in Sec. 5.1, T (λ) for the inclusive cross section does not have any term linear in λ, if expressed in terms of the MS mass. It follows that the total cross section computed in the MS scheme should not have any Λ QCD /m renormalon and should display a better behaviour at large orders.
The coefficients c i of the expansion of eq.   are collected in Tab. 3, in the pole (left) and in the MS (right) schemes. At large orders, the MS inclusive cross section receives much smaller contributions than in the pole-mass scheme. On the other hand, in the pole-mass scheme the factorial growth is already visible at the N 3 LO order, and the minimum of the series is reached for i = 8 (that corresponds to an O(α 8 S ) correction), and it is two orders of magnitude larger than the corresponding contribution computed in the MS scheme. We also notice that the MS result has an NLO correction larger than the pole mass result, an NNLO correction that is similar, and smaller N 3 LO and higher order corrections.

Inclusive cross section with cuts
As we have seen in Sec. 5.2, the presence of selection cuts introduces a linear renormalon in the inclusive cross section proportional to 1/R. In Tab. 4 we present the results for the inclusive cross section, in the pole and in the MS-mass scheme, for a small jet radius, R = 0.1, and a more realistic value, R = 0.5. For small radii, the perturbative expansion displays roughly the same bad behaviour, either when we use the pole or the MS-mass scheme. For larger values of R, the size of the coefficients are typically smaller than the corresponding ones with smaller values of R. In particular, if we compare the coefficients for R = 0.1 and R = 0.5, the second ones are one order of magnitude smaller than the first ones. Furthermore, for R = 0.5, the coefficients computed in the MS-mass scheme are roughly half of the ones computed in the pole-mass scheme. This follows from the fact that, for large values of R, the cross section with cuts approaches the total cross section, thus partially inheriting its properties.

Reconstructed-top mass
In this section, we discuss the terms of the perturbative expansion for the average recon- for three values of the R parameter. We apply the cuts of Sec   From the table we can see that, for very small jet radii, the asymptotic character of the perturbative expansion is manifest in both the pole and MS scheme. For the realistic value R = 0.5, the MS scheme seems to behave slightly better. In fact, this is only a consequence of the fact that the jet-renormalon and the mass-renormalon corrections have opposite signs, with the mass correction in the MS scheme largely prevailing at small orders, yielding positive effects.
As the radius becomes very large, the jet renormalon becomes less and less pronounced, in the pole-mass scheme, leading to smaller corrections at all orders. This is consistent with the discussion given in Sec. 6, where we have seen that, for large radii, the reconstructed mass becomes strongly related to the top pole mass, since it approaches what one would reconstruct from the "true" top decay products.

W boson energy
The coefficients of the perturbative expansion of the average energy of the W boson (8.6) in the pole and MS-mass schemes, are displayed in Tab. 6. We notice that the perturbative expansions are similarly behaved in both schemes up to i ≈ 6, while, for higher orders, the MS scheme result displays a better convergence. This supports the observation, discussed in Sec. 7, that the top width screens the renormalon effects if the MS mass is used. In fact, the 6 th order renormalon contribution is dominated by scales of order m t e −5 ≈ 1.16, as illustrated in Sec. 2, very near the top width.

Conclusions
In this work we have examined non-perturbative corrections related to infrared renormalons relevant to typical top-quark mass measurements, in the simplified context of a W * → tb → W bb process, with an on-shell final-state W boson and massless b quarks. As a further simplification, we have considered only vector-current couplings. We have however fully taken into account top finite-width effects.
We have investigated non-perturbative corrections that arise from the resummation of light-quark loop insertions in the gluon propagator, corresponding to the so called large-n f limit of QCD. The large-n f limit result can be turned into the so called large-b 0 approximation, by replacing the large-n f beta function coefficient with the true QCD one. This approximation has been adopted in several contexts for the study of non-perturbative effects (see e.g. Refs. [15,31,41,42,47,48]).
In this paper we have developed a method to compute the large-n f results exactly, using a combination of analytic and numerical methods. The latter is in essence the combination of four parton level generators, that allowed us to compute kinematic observables of arbitrary complexity. We stress that, besides being able to study the effect of the leading renormalons, we can also compute numerically the coefficients of the perturbative expansion up and beyond the order at which it starts to diverge.
Although our findings have all been obtained in the simplified context just described, we can safely say that all effects that we have found are likely to be present in the full theory, although we are not in a position to exclude the presence of other effects related to the non-Abelian nature of QCD, or to non-perturbative effects not related to renormalons.
Our findings can be summarized as follows: • The total cross section for the process at hand is free of physical linear renormalons, i.e. its perturbative expansion in terms of a short distance mass is free of linear renormalons. This result holds both for finite top width and in the narrow-width limit. In the former case, the absence of a linear renormalon is due to the screening effect of the top finite width, while, in the latter case, it is a straightforward consequence of the fact that both the top production cross section and the decay partial width are free of physical linear renormalons.
By examining the perturbative expansion order by order, we find that, already at the NNLO level, the MS scheme result for the cross section is much more accurate than the pole-mass-scheme one.
We stress that our choice of 300 GeV for the incoming energy corresponds to a momentum of 100 GeV for the top quark, that in turn roughly corresponds to the peak value of the transverse momentum of the top quarks produced at the LHC. Thus, the available phase space for soft radiation at the LHC is similar to the case of the process considered here, so that it is reasonable to assume that our result gives an indication in favour of using the MS scheme for the total cross section without cuts at the LHC.
• As soon as jet requirements are imposed on the final state, corrections of order Λ QCD arise. They have a leading behaviour proportional to 1/R, where R is the jet radius, for small R [43]. These corrections are present irrespective of the top-mass scheme being used. They are however reduced if the efficiency of the cuts is increased, for example by increasing the jet radius, giving an indication in favour of the use of the MS scheme for the total cross section calculation also in the presence of cuts. It should be stressed, however, that with a typical jet radius of 0.5 the behaviour of the perturbative expansion in the MS and pole-mass schemes are very similar, with a rather small advantage of the first one over the latter.
• The reconstructed-top mass, defined as the mass of the system comprising the W and the b jet, has the characteristic power correction due to jets, with the typical 1/R dependence. No benefit, i.e. reduction of the power corrections, seems to be associated with the use of a short-distance mass. In particular, at large jet radii, when the jet renormalon becomes particularly small, in the pole-mass scheme the linear renormalon coefficient is smaller. This observation is justified if one considers that, in the narrow-width limit, the production and decay processes factorize to all orders in the perturbative expansion, yielding a clean separation of radiation in production and decay. In this limit, the system of the top-decay products is well defined, and its mass is exactly equal to the pole mass. Consistently with this observation, we have shown that, for very large jet radii, the linear renormalon coefficient for the reconstructed top mass is quite small (if the observables is expressed in terms of the pole mass). One may then worry that, when reconstructing the top mass from the full final state, renormalons associated with soft emissions in production from the top and from theb quark may affect the reconstructed mass, since these soft emissions may enter the b-jet cone. By comparing the reconstructed mass obtained using only the top-decay products to the one obtain using all final-state particles, we have shown that these effects are in fact small.
We should also add, however, that the benefit of using very large jet radii cannot be exploited at hadron colliders, since we expect other renormalon effects, due to softgluon radiation in production entering the jet cone. This problem can in principle be investigated with our approach, by applying it to the process of tt production in hadronic collisions.
• We have considered, as a prototype for a leptonic observable relevant for top mass measurement, the average energy of the W boson. We have found two interesting results: -In the narrow-width limit, this observable has a linear renormalon, irrespective of the mass scheme being used for the top. This finding does not support the frequent claim that leptonic observables should be better behaved as far as nonperturbative QCD corrections are concerned. It also reminds us that, even if we wanted to measure the top-production cross section by triggering exclusively upon leptons, we may induce linear power corrections in the result that cannot be eliminated by going to the MS scheme.
The presence of renormalons in leptonic observables seems to be in contrast with what is found in inclusive semileptonic decays of heavy flavours [41,42].
We have however verified that there is no contradiction with this case. If the average value of the W energy is computed in the top rest frame (which makes it fully analogous to a leptonic observable in B decay) then no renormalon is present if the result is expressed in terms of the MS mass.
-For finite widths, if a short-distance mass is used, there is no linear renormalon. We verified this numerically, and furthermore we were also able to give a formal proof of this finding. What this means in practice is that the perturbative expansion for this quantity will have factorial growth up to an order n ≈ 1 + log(m/Γ t ), that will stop for higher orders. In practice, for realistic values of the width, this turns out to be a relatively large order. Thus, although in principle we cannot exclude a useful direct determination of the top short-distance mass from leptonic observables, it seems clear that finite-order calculations should be carried out at relatively high orders (up to the fourth or fifth order) in order to exploit it. Although it seems unlikely that results at these high orders may become available in the foreseeable future, perhaps it is not impossible to devise methods to estimate their leading renormalon contributions, still allowing a viable mass measurement.
In this work we have made several simplifying assumptions. These assumptions were motivated by the fact that the calculational technique is new, and we wanted to make it as simple as possible. Some of these restrictions may be removed in future works. For example, we could consider hadronic collisions, the full left-handed coupling for the W , the W finite width and the effects of a finite b mass. Although removing these limitations can lead to interesting results, we should not forget that our calculation does not exhaust all sources of non-perturbative effects that can affect the mass measurement. As an obvious example, we should consider that confinement effects are not present in our large-b 0 approximation, while, on the other hand, it is not difficult to show that they may give rise to linear power corrections. It is clear that theoretical problems of this sort should be investigated by different means.
where we have dropped all the longitudinal terms. In the limit of large number of flavours, i.e. considering only light-quark loops, the exact d-dimensional expression of Π(k 2 , µ 2 ) is given by where T F = n l T R , T R = 1/2, iη is a small imaginary part attached to k 2 in order to perform the analytic continuation, γ E is the Euler-Mascheroni constant, and where we have made the replacement µ 2 → µ 2 e γ E /(4π), according to the MS scheme. In the following, we also need an expansion in of eq. (A.3) from which we can read its counterterm in the MS scheme The renormalized gluon propagator dressed with the sum of all quark-loop insertions is then given by The above expressions are exact in the large-n f limit. For our phenomenological estimate of the contribution to the vacuum polarization coming from the insertion of gluon loops, we naively assume that this contribution can be written as and we add it to eq. (A.3) to get whose expansion in is given by is the first coefficient of the QCD β function. The counterterm of eq. (A.5) is now replaced by By setting C g to the value our formula becomes appropriate to describe a QCD effective coupling, as given in Ref. [49].

B Calculation of the large-n f all-order corrections to an infrared-safe observable
In this section we describe the calculation in the large-n f all-order corrections that has led to the results for a generic infrared-safe observable O illustrated in Sec. 3. We separate the calculation into different contributions O qq receives contributions only from the real graphs with a final state W bbqq, where qq is a pair of light quarks as depicted in Fig. 1 (d). We denote with k 2 the invariant mass of the qq pair. Starting from its O(α 2 S ) tree-level cross section, that we indicate with σ (2) qq (Φ qq ), with no vacuum polarization insertions in the gluon propagator, we obtain the differential cross section σ qq (Φ qq ) with the insertion of all the light-quark bubbles by simply replacing the bare gluon propagators with the dressed one of eq. (A.6) (B.6) From eq. (B.5), we get (B.7) We define (B.9) For inclusive observables and for observables that only depend upon leptonic variables, ∆(λ) is obviously identically zero. For generic observables involving jets, we have found that ∆(λ) ∝ λ for small λ. The following considerations hold for infrared-safe observables that satisfy this property. We now make use of the following replacement This works correctly, since the imaginary part of 1/(λ 2 + iη) in the square bracket leads, for small λ, to a contribution in eq. (B.9) of the form that, under the assumption that ∆(λ) vanishes as λ for small λ, is zero. Equation (B.9) becomes and integrating by parts The boundary terms are absent, because ∆(λ) vanishes for small (by assumption) and large (for kinematic reasons) values of λ.

B.2 The O g contribution
The O g term of eq. (B.4), receives contributions from final states with both a single real gluon or a qq pair. Both these contributions have collinear divergences related to the qq splitting, that cancel in the sum.

B.2.1 The gluon contribution
The first contribution of eq. (B.14) can be computed starting from σ (1) g , the tree-level cross section for the emission of a single (massless) gluon. In general, this contribution will produce soft and collinear singularities, the latter due to the fact that we consider massless b quarks. We must assume that we are using dimensional regularization for this contribution. In order to make the discussion more transparent, it is convenient to introduce as regulator a small mass m q for the quarks in the self-energy corrections, that we denote with Π(λ 2 , m 2 q , µ 2 ). We then have Notice that from the integration of the cross section we may get terms of order 1/ 2 in d = 4 − 2 dimensions. Thus, in the denominator of the last factor in eq. (B.15), although the 1/ pole of the UV divergence in Π cancels against the one in Π ct we should imagine to keep also terms up to order 2 at this stage, since they may yield finite contributions when combined with the double pole of the integration. We will see that, at the end, when combining all contributions, these terms actually cancel.

B.2.2 The qq contribution
We begin by splitting the real phase space dΦ qq into the product of the phase space for the production of a gluon with virtuality λ, that we call dΦ g * , and its decay into a qq pair, that we call dΦ dec dΦ qq = dλ 2 2π dΦ dec dΦ g * . (B.16) Using the optical theorem, we easily obtain the relation where σ (2) qq (Φ qq ) is the tree-level cross section for W * → W bbqq, and σ (1) g * (λ, Φ g * ) is the tree-level cross section for the process W * → W bbg * , where g * is a gluon with mass λ. We have again given a small mass m q to the light quarks, in order to match what we did in App. B.2.1. Thus the second term on the right-hand side of eq. (B.14) can be written as where we have inserted the dressed gluon propagators. As long as m q > 0, no divergences arise from this contribution, since the imaginary part of Π vanishes for λ < 2m q .

B.3 Combination of the gluon and qq contributions
Defining we can combine eq. (B.15) and (B.18) and get . (B.20) With the notation R ( ) (0) we remind that for λ = 0 there are infrared divergences in R that are regulated in dimensional regularization.

B.4 The O v contribution
The virtual contribution with all polarization insertions can be obtained by performing the replacement 1 in the computation of the NLO virtual corrections for W * → W bb, where k is the momentum flowing in the virtual gluon propagator. Ultraviolet divergences arise in individual diagrams, and soft and collinear singularities also arise, so that the calculation must be performed in d = 4 − 2 dimensions. For a generic complex k 2 , using the residue theorem, we can write 5 where Γ is the contour depicted in Fig. 13. Notice that when we write Π(λ 2 , m 2 q , µ 2 ) for real λ 2 we imply, consistently with eq. (A.3), that a positive tiny imaginary part should be added to λ 2 .
From eq. (B.23), we see that we get two contributions to the virtual corrections, corresponding to its two terms where we have set the lower integration limit to 0, since the imaginary part is zero for λ 2 < 4m 2 q , and where V (λ) stands for the virtual contribution to our observable computed with the substitution 1 in all the NLO virtual diagrams. For λ > 0, this corresponds to replace the massless gluon propagator with the propagator of a gluon with mass λ, while for λ = 0 nothing is changed. . The cut of Π(λ 2 , m 2 q , µ 2 ), starting at 4m 2 q , is also shown.
As before we have added the superscript ( ) to V (0), to remind us that this quantity contains poles in due to collinear and soft singularities. As before, in the denominator of the factor multiplying V ( ) (0) we should keep terms up to order 2 . On the other hand, for λ > 0, V (λ) is finite if a mass counterterm has been included in the calculation, and the appropriate wave-function renormalization of the external legs has been carried out. We also notice that eq. (B.24) is meaningful only if V (λ) vanishes as λ goes to infinity. This turns out to be the case, provided that mass renormalization is carried out in the pole mass scheme.
(B.27) Notice that we have written S(0) = R ( ) (0) + V ( ) (0), since the infrared poles cancel in the sum, provided the observable we are considering is IR safe. Furthermore, for the same reason, S(λ) has a well defined limit for λ → 0, that is equal to S(0). In addition, since S(0) is finite, we can neglect terms of order in the denominator of the factor that multiplies it, so that all dependences cancel in eq. (B.27).
We would like now to take the limit m q → 0 in eq. (B.27). In doing so, we must be careful to handle properly the singularities at λ = 0. We thus split eq. (B.27), by adding and subtracting the same quantity, as follows where the first two terms in eq. (B.28) have been merged in the first term of eq. (B.29), and the last term in eq. (B.28) can be turn into the last term of eq. (B.29) because the imaginary part of 1/(λ 2 + iη) is a δ(λ 2 ) function, that yields a zero when multiplied by the expression in the curly brackets. The notation 0 − for the lower bound of the first integral in eq. (B.29) simply means that the integration range should start slightly below 0, so that the δ(λ 2 ) arising from the imaginary part acts in a well defined way. Notice also that the separation of terms in eq. (B.28) does not spoil the convergence at large λ. We can then take safely the limit m q → 0 in both terms of eq. (B.29). In fact, the first term can be expressed as an integral along the contour Γ of Fig. 14, that in turn can be transformed in the residue at λ 2 = −m 2 , that has a well defined limit for m q → 0, and the second term is not singular in the λ → 0 region. After having taken the limit m q → 0, we make use of the identity (see eqs. (A.4) and (A.5)) 1 λ 2 + iη and rewrite (B.29) as We can now integrate by parts in λ 2 . The boundary term at λ = 0 − in the first integral vanishes, since the imaginary part vanishes for λ < 0, while in the second integral it vanishes because the expression in the curly bracket vanishes. We are left with

B.6 Summary
Using eqs. (A.9) and (A.12), we can write the renormalized polarization contribution Π(λ 2 , µ 2 ), for λ 2 > 0, in the form where we have defined Notice that, in the large-n f limit, C = 5/3 and b 0 = −T F /(3π). We can then write Summarizing our findings, we have where In case one is interested in a normalized observable O , where the normalization factor N Θ is the inverse of the total cross section as given in eq.

C Pole -MS mass conversion with a fully dressed gluon propagator
In order to extract the pole-MS mass relation at all orders in the large-n f limit, we follow a strategy similar to the one described in App. B.4, where the virtual contribution is expressed in terms of the NLO correction computed keeping a finite gluon mass λ. We thus begin by calculating the one-loop top-quark self energy with a massive gluon in App. C.1. This result is then used in App. C.2 to derive the mass conversion formula.

C.1 Pole mass with a massive gluon
The one-loop self energy, depicted in Fig. 15, for a quark with bare mass m b , due to the exchange of a gluon of mass λ, computed in d = 4−2 dimensions, with µ rescaled according to the MS prescription, is where we assume p 2 = m 2 b , and we have defined where x = λ 2 m 2 (C. 13) and (C.14) This leads to Our result is consistent with Ref. [47]. Notice that, for large λ, eq. (C.15) becomes and for small λ We also need the exact d-dimensional expression of r λ (m, µ), for λ = 0 and λ m. For λ = 0 we have For λ m (and µ ≈ λ) we have . (C.23)

C.2 All-orders result
In this section we deal with the computation of the on-shell top self-energy in d = 4 − 2 dimensions, Σ( / p, m b , µ, α S ), with the insertion of an infinite number of light-quark loops in the gluon line. The one-loop self energy with a massless gluon is obtained by setting and we can write where we have performed a Taylor expansion in the second line. By computing the imaginary part of the n-th power of the term in the square brackets, we are lead to evaluate integrals of the form where h is a real number, so that r ∞ d (m, µ, α S ) can be straightforwardly evaluated by computer algebraic means at any fixed order in α S .
We emphasize that r d (m, µ, α S ) has no linear renormalon. Indeed if we perform an expansion and we consider the small λ contribution, by writing dλ 2 = 2λdλ, we notice that the integrand behaves as λ log n (λ). This signals the absence of linear renormalons, that come from terms of the type log n (λ), without any power of λ in front.
From eq. (C.28) we get m(µ) = m 1 − α S r f (m, µ, α S ) + r D Cancellation of the linear sensitivity in the total cross section and in "leptonic" observables In order to discuss the issue of the linear sensitivity cancellation in the total cross section and in E W , it is convenient to use the old-fashioned perturbation theory. One writes the propagators as the sum of an advanced and retarded part while, for unstable particles, we have In this way, each Feynman graph is separated into contributions where the vertexes have all possible time orderings. Each line joining two vertexes has an energy set to its onshell value, with an extra negative sign when considering a retarded propagator. For each time ordering the integration of all the k 0 components yields a product of old-fashioned perturbation theory denominators where E is the total energy and E i is the energy of the state i, given by the sum of the energies flowing in the i th cut of the amplitude, times an overall delta function of energy conservation. In Fig. 16 we show a possible time ordering for one graph contributing to the W * → tb → W bb cross section. The corresponding contribution to the cross section is obtained by setting either one of the 2, 3, 4 intermediate states on the energy shell, and changing the sign of the iη in the denominators to the right of the cut. We then define where E t,i = k 2 t,i + m 2 − imΓ t , (D.11) E l,i = k 2 l,i , for l = b,b, g, (D.12) Notice that the top energy has an imaginary part, so that no iη is needed in the denominators containing it. We never include the corresponding cuts since the top width prevents this particle from being on-shell. Thus, the only intermediate states contributing to cuts will be the ones that do not include the top. Then, in the integrand for the cross section, we have the sum When performing the 3-momentum integral for the loops not including the W line, one can approach the singularity in the denominator. However, if there is a direction in the 9dimensional integration space (corresponding to the three 3-momenta flowing in the loops) such that, integrating along it, it leaves the singularities of D 2 , D 3 and D 4 on the same side of the complex plane, the integration contour can be deformed away from the singularities, so that the denominators cannot contribute to mass singularities. The singularity for small gluon mass m g is thus determined only by the remaining factor d 3 k g k 2 g + m 2 g , (D. 16) that gives a quadratic sensitivity to the gluon mass. The only cases when an appropriate deformation of the contour does not exist correspond to Landau singularities [50]. These are characterised by the presence of configurations with intermediate on-shell particles corresponding to classical propagation over large distances. It can be proven that the Landau singularities arise when several denominators go simultaneously on-shell and the momenta of the particles are such that they meet again after having come apart. If this is not possible, the singularity is an avoidable one. In order to explore the possible Landau configurations, one can start with the graph of Fig. 16 with the top lines shrunk to a point. In fact, the top is always off-shell, and it cannot propagate over large distances. The remaining configuration is shown in Fig. 17. In order for the 2, 3 and 4 intermediate states to be on-shell at the same time, either both the b and theb quarks should be collinear to the gluon, or the gluon should be soft. In the first case, also the b, theb and the gluon are collinear, and are all travelling in the opposite direction with respect to the W . Thus, they cannot meet at the same point on the last vertex to the right. On the other hand, if the gluon is soft, the b, theb and the W produced at the primary vertex have momenta that sum to zero, so, again their velocities will make them diverge. One can try to shrink other propagators to a point. Shrinking one fermion propagator to a point, for example the b one at the intermediate state 2, forces the gluon to be either collinear to theb or soft, and again one would end up with a W , a b and a collinear bg system produced at the vertex on the left and meeting at the vertex on the right, which is impossible. Shrinking the W , the two b or the twob lines to a point shrinks the whole graph to a point, leading to nothing. Shrinking a b and ab line to a point leads again to configuration with two massless system (either b quarks of collinear bg systems) and a W , that again cannot meet at the same point. Thus, no Landau configuration can exist, so one infers that the m g sensitivity of the total cross section is at least quadratic. We can repeat the same reasoning including a factor E W in our Feynman graph. The argument runs as before, and so, even for the average energy of the W boson, one expects that the sensitivity to the gluon mass is at least quadratic. Notice that, in order for this to work, one needs that the E W factor is the same for all cuts, which is in fact the case.
The argument fails if the top width is sent to zero. In fact, even at the Born level (i.e. removing the gluon line) the first and last intermediate states have equal energy, but their iη have opposite signs. Under these condition, the pinch is clearly unavoidable.
As a last point, we recall that the total cross section is free of linear m g sensitivity also in the limit of zero width. This happens because, in the zero-width limit, the cross section factorizes into a production cross section times a decay width, and both of them are free of linear sensitivity to m g if the mass is in a short-distance scheme. The same, however, does not hold for the average E W . In fact, the cancellation of mass singularities in Γ t cannot be proven in the same fashion adopted here, since logarithmic divergences are also present in the wave-function renormalization, and cannot be treated in a straightforward way in the old-fashioned perturbation theory.