Hard Matching for Boosted Tops at Two Loops

Cross sections for top quarks provide very interesting physics opportunities, being both sensitive to new physics and also perturbatively tractable due to the large top quark mass. Rigorous factorization theorems for top cross sections can be derived in several kinematic scenarios, including the boosted regime in the peak region that we consider here. In the context of the corresponding factorization theorem for $e^+e^-$ collisions we extract the last missing ingredient that is needed to evaluate the cross section differential in the jet-mass at two-loop order, namely the matching coefficient at the scale $\mu\simeq m_t$. Our extraction also yields the final ingredients needed to carry out logarithmic resummation at next-to-next-to-leading logarithmic order (or N$^3$LL if we ignore the missing 4-loop cusp anomalous dimension). This coefficient exhibits an amplitude level rapidity logarithm starting at $\mathcal{O}(\alpha_s^2)$ due to virtual top quark loops, which we treat using rapidity renormalization group (RG) evolution. Interestingly, this rapidity RG evolution appears in the matching coefficient between two effective theories around the heavy quark mass scale $\mu\simeq m_t$.


Introduction
The top quark mass is one of the most important parameters in the Standard Model. As the heaviest observed fermion, the top quark provides an important probe for the Higgs sector, and gives dominant contributions to many electroweak observables, thus providing strong benchmark constraints for extensions of the Standard Model. Furthermore, the mass of the top quark and the Higgs boson represent crucial parameters in studies of the stability of the Standard Model vacuum [1][2][3][4][5][6]. Precision measurements of the top quark mass are a difficult task due to challenges from both experimental and theoretical sides, mainly related to the fact that the top quark is a colored particle. The current value of the top quark mass from a combined analysis of Tevatron and LHC data is m t = 173.34 ± 0.76 GeV [7], see also [8,9]. The precision obtained in this result relies on Monte Carlo (MC) based template and matrix element methods, which aim to account for essentially all of the kinematic final state information in the top quark events. However, this approach does not account for the relation of the extracted MC top quark parameter to an unambiguous field theoretic QCD top mass definition [10][11][12]. At the time of writing, no procedure to systematically quantify and improve this relation exists. While it seems unlikely that the template and matrix element analyses can be based on first principle QCD calculations which can be systematically improved to specify the JHEP12(2015)059 top mass scheme unambiguously, it is quite plausible that other highly sensitive top mass observables can be devised which can clarify the issue by making high precision theoretical calculations feasible.
One method to determine m t in a well-defined mass scheme from a kinematic spectrum with small uncertainties has been discussed in refs. [10,13,14]. Here the hemisphere dijet invariant mass distribution in the peak region for the production of boosted tops in electronpositron annihilation was suggested as an observable and it was shown that hadron level predictions of the double differential distribution can be carried out in a stable manner within a constrained set of top quark mass schemes. It was in particular demonstrated that the location of the peak of the distribution is highly sensitive to the top quark mass, and that only specific low-scale short-distance mass definitions are suitable for high-precision extractions. Although the effective theory setup developed therein was devised for the context of a future e + e − collider, the approach can be extended to the environment at hadron colliders taking into account the complications related to initial state radiation, underlying event, parton distribution functions and dependence on jet algorithms and jet radius [15]. In refs. [13,14] the calculation for e + e − annihilation was carried out at Nextto-Leading Logarithmic (NLL) accuracy with the perturbative ingredients at O(α s ). In this paper we provide a result for the O(α 2 s ) matching correction at the scale µ m t for the e + e − -collider setup. Taken together with the known O(α 2 s ) results for the jet function in the heavy-quark limit from ref. [16], for the massless soft function from refs. [17][18][19], and input from previous form factor calculations for massless quark production [20,21], our result provides the last missing ingredient needed to extend the e + e − boosted top jet analysis to O(α 2 s ). In turn, with known results, these fixed order contributions can be accompanied with resummation of logarithms up to next-to-next-to-leading logarithmic order (NNLL). Up to the missing four loop cusp anomalous dimension, which is known to give a very small correction (see e.g. [22,23]), all ingredients are also available for N 3 LL.
Boosted top quark production with subsequent decays in the peak region of the invariant mass distributions involves physical effects in a range of widely separated energy scales. The hierarchy between the production energy Q, the top mass m t , the decay width Γ t and the hadronization scale Λ QCD is given by Q m t Γ t > Λ QCD . Given this hierarchy of scales, the cross section contains large logarithms of ratios of these scales which spoil the perturbative expansion in α s . This necessitates to replace fixed order computations by resummed calculations. The Effective Field Theory (EFT) setup devised in ref. [13,14] disentangles the fluctuations at the different scales and allows us to resum the logarithms through renormalization group evolution (RGE). 1 We are interested in the peak region where each of the jet invariant masses, for the top s t and antitop st, is close to the top quark mass, i.e.,

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QCD SCET (6) bHQET (5) H For this kinematic region both of the hierarchiesŝ t,t ∼ Γ t andŝ t,t Γ t are allowed. The sequence of the EFTs and the corresponding modes relevant for this problem are displayed in figure 1. First, hard modes with fluctuations with virtualities of order ∼ Q are integrated out in QCD. The corresponding low-energy theory containing collinear and soft modes is Soft Collinear Effective Theory (SCET) [26][27][28][29], which allows to resum large logarithms between Q and m t . In a second step all fluctuations with virtualities of order ∼ m t are integrated out, and SCET is thus matched onto boosted Heavy Quark Effective Theory (bHQET), an EFT with ultracollinear and ultrasoft modes at a lower invariant mass scale, which allows to resum logarithms between m t andŝ t,t . The factorization theorem for the double differential cross section in e + e − collisions reads Here σ 0 denotes the tree level cross section for e + e − → qq. The terms H Q and H m are hard functions related to the matching from QCD to SCET at the scale µ ∼ Q and from SCET to bHQET at the scale µ ∼ m, respectively. The terms J B and S denote the jet and soft functions, respectively, which are nonlocal matrix elements in bHQET. Note that we use JHEP12(2015)059 J B for the heavy-quark jet function, rather than the symbol B employed in refs. [13,14,16].
Here J B describes the dynamics of the ultracollinear radiation inside the t ort jet at the scale µ ∼ŝ t . The function S incorporates the ultrasoft cross talk between the two jets at the scale µ ∼ mŝ t /Q, which is O(Λ QCD ) in the peak region, and perturbative in the tail above the peak. In eq. (1.2) the RGE between the characteristic scale of each function and the common renormalization scale µ are implicit. We stress that in SCET the top quark is considered as dynamical and hence the RGE takes place with six active flavors, while for the ingredients that arise in bHQET there are only five dynamical flavors in the evolution. Note that it is possible that the O(m t α s /Q) power corrections indicated in eq. (1.2) are absent, but we are not aware of a rigorous proof at this time.
It is through the residual mass term δm appearing in the bHQET jet functions J B that the top quark mass scheme is specified unambiguously beyond tree-level. For order-by-order stable perturbative behavior, the top quark mass scheme employed should be free of the O(Λ QCD ) renormalon ambiguity, thus excluding the pole mass (specified by δm = 0) as a choice. Furthermore, the parametric scaling of higher order corrections defining the mass scheme must be set by scales associated to the measurement, namelyŝ t,t , Γ t m t , in order not to violate the power counting required for the factorization. This excludes employing the MS mass where these corrections scale as δm ∼ α s m t . Valid options include the jet mass scheme [13,14,16] or the MSR mass scheme [10,16] which matches continuously onto MS. These two mass schemes have an adjustable cutoff parameter R which controls the scaling of higher order corrections.
The exact algorithm to determine the two jet regions and the precise form of the observable is irrelevant for the structure of eq. (1.2) as long as parametrically s t ∼ st, but does matter for the explicit perturbative expressions of its ingredients. The restriction s t ∼ st avoids large logarithms of the form ln(s t /st), and is satisfied by variables designed to study the peak region of both jets, such as thrust. In the analysis of ref. [14] all particles were assigned to either of the two top jets depending on which hemisphere with respect to the thrust axis they enter. Thus the observable considered was physically close to eventshape distributions. The analysis of ref. [14] for this inclusive jet observable was carried out at NLL , i.e. including perturbative ingredients at O(α s ) and NLL resummation. At the time of writing the hard function H Q , the bHQET jet function and the soft function are already known up to O(α 2 s ) [16,17,20] or beyond, while resummation can be carried out to N 3 LL. 2 Thus, the only relevant correction missing to perform a N 3 LL analysis for the double hemisphere invariant mass distribution and similar observables in the peak region is the hard function H m at O(α 2 s ). This correction will affect the normalization of the differential cross section, while the shape of the cross section is determined mainly by the jet and soft functions. Here NNLL refers to NNLL resummation with O(α 2 s ) fixed-order matching and matrix element corrections.
In this paper we carry out the computation of the O(α 2 s ) correction to H m . In section 2 we outline two methods to perform the computation. Instead of directly calculating the JHEP12(2015)059 current matching factor between bHQET and SCET, we can also exploit the knowledge of the QCD heavy quark form factor calculated in refs. [30,31] and various properties of the EFT to extract the hard function. In section 3 we carry out the computation at O(α 2 s ) using this method and show how to handle issues associated with the number of active quark flavors. This yields the result given in eq. (3.8) in terms of the pole mass. In the two loop expression for H m we find terms of the form The large logarithm ln(Q 2 /m 2 ) is induced by the separation in rapidity of soft mass-shell fluctuations with the scaling (p + , p − , p ⊥ ) ∼ (m, m, m) from collinear mass-shell fluctuations with (p + , p − , p ⊥ ) ∼ (m 2 /Q, Q, m). It can not be eliminated by a choice of µ or summed by the RGE in µ. This effect is directly related to virtual top quark loops which first appear at O(α 2 s ), and has been discussed in detail in refs. [32,33] together with other subtleties concerning the incorporation of a massive quark in primary massless jet production in SCET. In section 4 we will explicitly carry out the matching calculation for the O(α 2 s C F T F ) correction with primary massive top quarks, and demonstrate how the amplitudes factorize into collinear and soft components which each involve a single rapidity scale. We show that this factorization is the same as that for massless external quarks, computed in ref. [33], up to a different constant term that appears in the collinear corrections. The direct computation of the SCET soft and collinear diagrams at O(α 2 s C F T F ) can be performed elegantly by first computing the virtual correction for the radiation of a "massive gluon" at one-loop and performing in a second step a dispersion integral. In section 5 we show how to resum the type of rapidity logarithm in eq. (1.3) using the framework of the rapidity renormalization group established in refs. [34,35]. We also demonstrate that the residual scale dependence of H m on µ significantly decreases when employing the complete two-loop correction, and assess the impact of the rapidity logarithm. We conclude in section 6.

Setup and notation
As described in refs. [13,14] for the description of the peak region we first match QCD onto SCET, and then SCET onto bHQET. The relevant current operators needed to define the hard functions in eq. (1.2) are where Γ µ v = γ µ and Γ µ a = γ µ γ 5 . The jet fields χ n = W † n ξ n and χn = W † n ξn describe the collinear radiation in SCET, and contain the massive collinear quarks ξ n and ξn [36,37] and Wilson lines W n,n where in position space W † n (x) = P exp ig ∞ 0 dsn·A n (ns+x) . The ultracollinear radiation in bHQET is described by the heavy quark fields h v +,− and by W n,n . The wide-angle radiation in SCET is described by soft Wilson lines S n,n , where in position JHEP12(2015)059 space S † n (x) = P exp ig ∞ 0 ds n·A s (ns+x) , and ultrasoft Wilson lines Y n,n are the analogs with ultrasoft gluon fields in bHQET. The difference between the SCET fields and bHQET fields is that SCET still contains soft and collinear fluctuations at the top mass scale, i.e. the SCET fields contain mass mode fluctuations which scale as (p + , p − , p ⊥ ) ∼ (m, m, m) and (Q, m 2 /Q, m) or (m 2 /Q, Q, m) which are absent in bHQET. This makes our EFT above the top mass scale an SCET II type theory. There are six flavors in the MS running coupling in QCD and SCET, and five flavors in bHQET.
The notation above differs from ref. [14] which used a hybrid of SCET I and SCET II , where the current operator was written as Here the Wilson lines S n,n describe exclusively soft mass mode fluctuations and have ultrasoft zero-bin subtractions. In eq. (2.1) the SCET operator only describes soft fluctuations above and of order of the mass scale m, and not far below m. This simplifies the setup for the matching coefficient calculation, which in particular can be viewed as going from a six flavor theory to a five flavor theory. The matching coefficients between these effective theories are defined by Here both the currents and Wilson coefficients refer to the renormalized quantities. When we refer to the bare objects we will indicate this explicitly as e.g. in J (bare,n l +1) SCET . For all quantities we consider we use the renormalized coupling constant. When we want to separate the color structures of the matching coefficients we will do so in the following way: .

(2.5)
In all the objects above the coupling is renormalized in the MS scheme with the number of dynamical flavors, n f , being either n l or (n l + 1) as indicated by the superscript. Here n l is the number of light quarks, and the additional flavor indicates the heavy quark (here the top quark). The choice for the number of flavors in each of the expressions above is motivated by the scales at which these objects live compared to the top mass. Note that we have kept the number of flavors appearing in C m unspecified, as it can be expressed in either the n l -or the (n l + 1)-flavor scheme. We will be explicit about which scheme we are using in the equations below. The hard functions in eq. (1.2) are related to the Wilson coefficients via Here the dependence on Q in the hard function H m appears due to the boost factor Q/m.

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In eq. (1.2) all the functions live at their respective scales and are evolved to a common scale µ final through renormalization group running. While the jet and the soft functions have convolution running [14], the large logarithms of the hard matching coefficients are summed by multiplicative evolution factors, for µ Q Q, µ m m and µ final < µ m . On the l.h.s. the dependence on µ Q and µ m only comes from higher order corrections when the objects in eq. (2.7) are truncated at a given order in resummed perturbation theory. The same is true for the rapidity scales ν Q and ν m , which are induced by the rapidity RGE that will be discussed further below and in section 5.1. We will frequently drop these arguments that appear after the semicolon. The evolution factors here obey the RG equations is the anomalous dimension for the squared current in bHQET. Eqs. (2.3) and (2.4) suggest two different methods that one can use to calculate the O(α 2 s ) piece of C m or equivalently H m .
1) Indirect calculation using the known result for C Q and the matrix elements for the QCD and bHQET current operators in pure dimensional regularization. Using eq. (2.3) and (2.4), and taking matrix elements of the operators with onshell top-quark states as in [13], we have bHQET . (2.9) Using the relation between bare and renormalized bHQET currents we get Note that the terms on the r.h.s. involve objects with different flavor number schemes for the strong coupling, which must all be converted to n l -flavors to get C (n l ) m . Here we work in dimensional regularization for both UV and IR divergences and renormalize the quantities in the MS scheme. With this regulator we can use the known two loop result for the heavy form factor J QCD given in refs. [30,31]. The result for C Q is also JHEP12(2015)059 known [20,21] in MS, and the result for Z (n l ) bHQET can be determined by RG consistency as discussed below. Loop graphs in bHQET factorize into ultrasoft and ultra-collinear contributions, and in general each involve at most a single dimensionful scale. The use of dimensional regularization for both the UV and IR, and employing onshell external quarks, imply that these loop corrections in bHQET are scaleless and vanish, such that J (bare,n l ) bHQET = 1. In general, the IR divergences in the QCD and bHQET matrix elements will match up, and the UV divergences in J bHQET . Thus we can use the simpler relation 2) Direct calculation by matching the SCET and bHQET current operators. Using eq. (2.4) we can also just directly compute the Wilson coefficient from a matching calculation, computing partonic matrix elements using the same IR regulator in SCET and bHQET, These matrix elements are form factors in the respective theories which we denote by F . We will use the same notation for the color structures in the perturbative expansion of F SCET and F bHQET as in eq. (2.5). We define the relation between bare and renormalized SCET currents by (2.14) As usual the bare currents are µ-independent, so from eqs. (2.10), (2.13) and (2.14) the µ-RG equation for C (n l ) m can be written as where the current anomalous dimensions are computed order-by-order from the counterterms in the standard fashion The anomalous dimension for the SCET current is known to 3-loop order [38]. Up to two loops the result reads The bHQET anomalous dimension can be derived using one of the consistency relations [14] for the factorization theorem in eq. (1.2): where γ S indicates the soft function anomalous dimension for one hemisphere. Using the results for γ J B given in eq. (41) of ref. [16] and for γ S given in eq. (19) of ref. [39] (which can be derived via consistency from the two-loop jet function anomalous dimension [40]) we find Expanding the recently calculated anomalous dimension in HQET at O(α 3 s ) [41,42] we extract in appendix B also the three-loop coefficient, which has -to our knowledge -not yet been displayed in literature.
As mentioned above, the two-loop expression of C m contains large logarithms of the form which cannot be resummed using the RGE in µ. They are rapidity logarithms and originate from a separation of the soft and collinear mass modes which have the same invariant mass but different rapidity. These rapidity logarithms only appear inside H m , and not for the other soft, jet, and hard functions in eq. (1.2). Our focus here will be on the leading rapidity logarithms, which start contributing with the O(α 2 s C F T F ) piece. The latter comes from virtual top quark loops, and hence we only need to compute the correction F (C F T F , n l +1) SCET , while the bHQET graphs give no contribution for this color structure, i.e. F (C F T F ,n l ) bHQET = 0. To set up the stage for rapidity resummation we can factorize the current operators and its matrix elements into products of soft and collinear diagrams, where the {n, s,n} labels in bHQET indicate n-ucollinear, ultrasoft, andn-ucollinear contributions respectively. Note that in order to split up these corrections we must choose an IR regulator which preserves the SCET II nature of the theory. We will regulate the IR divergences using a gluon mass Λ, which thus differs from the use of pure dimensional regularization discussed above for method 1. In SCET II the individual soft and collinear diagrams have rapidity divergences, and using the regulator of refs. [34,35] the coefficients will depend on a rapidity renormalization scale ν. Thus eq. (2.13) can be decomposed into individual contributions involving n-collinear,n-collinear, and soft amplitudes,

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This leads to where we included the dependence on scales and renormalization parameters. Thus we see that the logarithmic dependence on the Q/m boost variable is factorized by the rapidity regularization parameter ν into collinear factors that depend on Q and a soft factor which does not. To sum the rapidity logarithms we can follow the standard approach of matching and running.
We define hard functions H The individual Wilson coefficient and hard functions obey related RG equations, The ν-anomalous dimensions appearing here can be computed directly from the SCET and bHQET counterterms and depend only on m and µ. Taking eqs. (2.10) and (2.14) and introducing individual counterterm factors for each of the collinear and soft component amplitudes, noting that the bare coefficients are ν-independent, and using eq. (2.21) we get As we will see in detail below, individual contributions on the right hand side of eq. (2.24) contain IR divergences, but they will always cancel to leave an IR finite result for the γ Cm ν, i , when we fully expand in either the n l -flavor or (n l + 1)-flavor scheme for the strong coupling.

Two loop determination of H m from QCD heavy form factor
In this section we use the first method outlined in section 2 to determine the bHQET matching coefficient, C m at two loops. From eq. (2.12) the ingredients we need are the UV renormalized QCD two-loop heavy quark form factor, J (n l +1) QCD , in dimensional regularization and the SCET matching coefficient, C (n l +1) Q . In the following we abbreviate the appearing logarithms as From refs. [30,31] we extract the renormalized two loop QCD heavy quark form factor result in the high energy limit, Q 2 m 2 , evaluated at an arbitrary scale µ m, Note that we keep the O( ) part of the one loop piece in F (1,n l +1) QCD since it yields a contribution when considering the cross terms in the expansion of the ratio in eq. (2.12). (One can avoid considering these cross terms and obtain the same answer by taking the logarithm of eq. (2.12).) We remark that in these expressions the pole mass scheme has been used for the top quark mass m.
The other ingredient we need is the well known two-loop expression for C Q , widely used in the SCET literature, and obtained with the aid of the massless form factor calculation JHEP12(2015)059 of refs. [20,21], The remaining quantities in eq. (2.12) are the coefficient C (n l ) m we wish to determine, and the counterterm Z (n l ) bHQET . The contributions to these two quantities can be easily distinguished since Z (n l ) bHQET only has terms with powers of 1/ , whereas C (n l ) m is given by the finite O( 0 ) contribution. Therefore, it is straightforward to distinguish these two quantities unambiguously. Since we wish to determine these with n l active flavors, we must convert the strong coupling in J (n l +1) QCD and C (n l +1) Q to the n l -flavor scheme using the decoupling relation where the one-loop vacuum polarization at zero momentum transfer for a massive quark pair is given by We need to keep terms up to O( ) in eq. (3.4) since they contribute in the dimensional regularization scheme we are using when multiplying O(α s / ) IR divergent terms in eq. (2.12). Using these results in eq. (2.12) we find the following expression for Z This result can also be extracted from earlier literature using the consistency relation for RG running between H m , and the soft and the jet functions in eq. (1.2). In particular, the 1/ 2 terms in eq. (3.6) are given by a term involving the lowest order β-function, and the square of the one-loop result (due to non-abelian exponentiation), while the 1/ terms are directly related to the two-loop anomalous dimension given in eq. (2.19). This provides a key cross-check for Z Finally we arrive at the main result of this section -the result for H m = |C m | 2 in the n l -flavor scheme with the top-mass in the pole scheme (α (3.8)

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As anticipated, all of the logarithms in this expression are minimized for µ m, except for the contributions in the last line that involve the rapidity logarithm α 2 s C F T F ln(Q 2 /m 2 ). To understand the origin of this type of logarithm in the context of the renormalization group requires a further factorization of H (n l ) m into soft and collinear pieces, as in eq. (2.22). In the next section we will carry out an independent calculation of the O(α 2 m . This sets up the rapidity renormalization group analysis of this term, which can be found in section 5.1. In section 5.2 we present the result for H In this section we perform a direct computation of the α 2 s C F T F piece of the matching coefficient C m (m, Q/m, µ) due to massive quark loops using the second method from section 2. We carry out the calculation in analogy to refs. [32,33], where the corresponding contribution to the matching coefficient at the mass scale for massless external quarks (in the following called "primary") was computed. In this section we extend the calculation to the case of primary massive quarks.
Starting from eq. (2.13) we note that for the α 2 s C F T F massive quark term, the bHQET graphs expressed in the usual n l -flavor scheme do not give any contribution. The SCET graphs do contribute, and should be expressed in the same scheme for the strong coupling. Using the decoupling relation in eq. (3.4) we obtain in the notation of eq. (2.5) The second term on the right hand side accounts for the coupling conversion of the SCET form factor from (n l + 1) to n l flavors. 4 As discussed in detail below, we will use a massive gluon as an IR regulator Λ, such that O( ) terms in the coupling conversion in eq. (3.4) can be dropped. For the remainder of this section we will drop the superscript (n l + 1) on the SCET form factors.
We adopt the calculational method of refs. [32,33], where the two loop graphs containing a "secondary" massive quark bubble are calculated by starting with one-loop graphs describing the radiation of a massive gluon with mass M and applying in a second step dispersion relations to account for the gluon splitting into a pair of secondary massive 4 Note that the subscript "α

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quarks with masses m. The corresponding relation can be written as Here Π(m 2 , p 2 ) is the gluonic vacuum polarization due to the massive quark-antiquark bubble, with the imaginary part in d = 4 − 2 dimensions given by

(4.4)
We note that the same method can be applied to account for any kind of secondary particles by a corresponding choice of the polarization function Π. Eq. (4.2) allows us to express the contribution to the SCET form factor due to the massive quark loops as where the "on-shell" form factor is  In eq. (4.5) Λ denotes the gluon mass acting as our IR regulator, which we distinguish from the gluon mass M used in the dispersion integration. Since total bare quantities are µ-independent, we do not add µ as an argument to the components of bare quantities at a specific order. In F (OS,bare) SCET the massive quark contributions to the coupling are renormalized with the onshell subtraction, i.e. F (OS,bare) SCET is given in the scheme with n l dynamic flavors. In eq. (4.5) the second term accounts for the change to n l + 1 dynamic flavors. The form factor itself is still unrenormalized, as indicated by the (bare) superscript. We perform the MS renormalization for the SCET current using eq. (2.14). Incorporating eqs. (4.5) and (2.14) into eq. (4.1) the result for C (C F T F , n l ) m can be written as SCET (Q, m, Λ, µ) . Here the 1-loop form factor F (1,bare) SCET is a UV and IR divergent amplitude, and Z

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is the SCET current counterterm in the (n l + 1)-flavor scheme. Using the explicit form of Π(m 2 , 0) in eq. (3.5) one can rewrite eq. (4.7) as where we see explicitly that the dependence on the IR regulator is canceled. Note that we could have also carried out the computation employing the (n l + 1)-flavor scheme to determine C , which involves converting the bHQET form factor from the n l to (n l + 1)-flavor scheme. In this case the cancellation of IR divergences occurs in a different manner, and involves the O(α s ) bHQET form factor. This approach is discussed in appendix A.
Note that nothing in eq. (4.8) depends on the low energy bHQET theory. Therefore the result applies equally well to the case where one integrates out the heavy quark loop without approaching the jet invariant mass threshold s t → m 2 and matches onto a n l -flavor SCET theory instead of bHQET. In this case the matching coefficient only contains the contribution from the massive quark loop and receives corrections starting at O(α 2 s C F T F ), so switching between the n l and (n l +1)-flavor schemes only affects the corrections at O(α 3 s ) and beyond. This is in close analogy to the case of primary massless quarks discussed in detail in refs. [32,33].

One-loop computation for secondary massive gluons
Having laid out the basic framework in the previous section we now start with calculating the one loop SCET heavy quark form factors for a top-quark of mass m with a massive gluon of mass M to be used in the dispersion relation. The complete unrenormalized SCET result for the current form factor at O(α s ) can be written as
Due to the eikonal structure the result for the soft diagram, F (1,bare) s , is same as that for primary massless quarks [hereμ 2 = µ 2 e γ E /(4π)], For the n-collinear diagram we get We can decompose this contribution into a correction corresponding to the diagram with primary massless quarks, and a UV and IR-finite difference of terms which can be computed in 4 dimensions, . (4.14) After performing a contour integration in k + , carrying out the k ⊥ -integration and rescaling the label momentum as k − ≡ zQ, the finite correction due to the mass of the primary quark yields

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In SCET loop graphs include soft 0-bin subtractions [46] which ensure that there is no double counting of infrared regions. For the soft 0-bin subtraction of F (1,bare) n the dependence on the primary quark mass drops out, and we obtain the same result as for primary massless quarks, which is therefore fully contained in F (1,bare) n,m=0 . Note that the result in eq. (4.15) does not contain any rapidity divergences, so that rapidity logarithms arise only in the computation of F (1) n,m=0 . This can be understood from the fact that the corrections due to soft modes are the same for massless and massive primary quarks, so that the rapidity divergences in the soft sector and, by consistency, also in the collinear sectors have to agree in both cases.
Then-collinear diagram corresponds to switching k − and k + in eq. (4.13). We perform a decomposition analogous to eq. (4.14), . (4.17) The difference correction due to the primary quark mass is again UV and IR-finite and does not contain any rapidity divergences. Thus it yields for any choice of regulator the same result as the n-collinear correction, i.e. Finally, we also have to consider the wave function corrections. In analogy to the computation in ref. [14] we have Using p 2 = m 2 + ∆ 2 and decomposing the integrals into elementary one-and two-point functions we obtain which uses the loop integrals The wave function renormalization constant Z (1) ξ is defined by taking the on-shell limit ∆ → 0 ξ can be written in terms of the wavefunction correction for primary massless quarks and a UV and IR finite remainder, (4.23) The remainder contribution in d = 4 dimensions reads where a was given above in eq. (4.16). The complete finite correction at one-loop, which accounts for the mass of the primary quark is given by the sum of the terms from eqs. (4.15) and (4.24), This result will be used for our two-loop computation in the next section.

Two-loop computation for secondary massive quarks
In this section we use the one-loop results from section 4.2 to calculate the two-loop graph with the massive quark loop, and to determine the C F T F contribution to C m . First we compute F (OS,C F T F ,bare) SCET via eq. (4.6) using the one-loop result in eq. (4.9). Again we can decompose the two loop SCET form factor into a primary massless component and a correction for primary massive top quarks: The calculation for primary massless quarks has already been performed in ref. [33]. We display the result here for convenience: The contribution from the two-loop MS counterterm is known from the massless quark case and reads in the last term of eq. (4.5). The δF term can be computed by inserting the one-loop massive gluon correction term of eq. (4.25) into the dispersive integral (4.6) which can be performed in four dimensions. The result reads Thus the only modification in the massive quark loop contributions to the form factor for primary massive quarks with respect to primary massless quarks is a simple constant term. In particular no additional rapidity logarithm ∼ ln(Q 2 /m 2 ) appears, which can be again traced back to the universality of the soft corrections for massless and massive primary quarks.
Assembling all the pieces above in eq. (4.8) we get the following result for C : which matches exactly with the C F T F result we obtained above in eq. (3.7). In the next section we decompose the SCET form factor result into soft and collinear pieces in order to find the terms needed for the rapidity RGE analysis.

Two loop ingredients for the rapidity renormalization group
In order to determine the ingredients needed for the rapidity renormalization group analysis, we now calculate the O(α 2 s C F T F ) SCET form factor contributions for the individual collinear and soft sectors using dispersion relations. We will employ the symmetric η-regulator [34,35] to regulate the rapidity divergences in the individual sectors. This corresponds to modifying the Wilson lines in the respective sectors according to and similarly for Wn and Sn. Here P µ denotes the label momentum operator and w(ν) is a dimensionless book keeping coupling parameter satisfying The one-loop form factor corrections for the radiation of a massive gluon have been already calculated in ref. [35] for massless quarks. Including the modification due to the quark mass = α In the collinear results we have included the wave function contributions Z ξn /2 and Z ξn /2. The soft-bin subtractions in the collinear diagrams vanish for the η-regulator.
In direct analogy to eq. (4.5) the corresponding two-loop expressions for the individual soft and collinear sectors read for i = n,n, s. Note that for this relation to make sense also the one-loop form factor corrections with a massless gluon have to be decomposed according to eq. (2.21). To achieve this goal we use a gluon mass Λ m as an infrared regulator which allows us to use the results in eq. (4.33). As discussed in section 2, we absorb all divergences of the form 1/η, η 0 / n in the form factors into separate counterterms Z (C F T F ) SCET, i for each sector, so that The explicit results for the counterterms at one-loop are given by 5 SCET,n (Q, m, Λ, µ, ν) , Although the full -dependence in the expression proportional to 1/η should be in principle kept unexpanded, this is only relevant to ensure that the coefficient of the 1/η pole is explicitly µ-independent, which is also true order by order in its expansion. Therefore we show here only the terms up to O( 0 ) which contain the information we need later for the anomalous dimensions.

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while at two-loop they read (4.37) Note that the sum Z SCET, s reproduces the result for the SCET current counterterm Z (C F T F ) SCET in eq. (4.28). These results for the individual collinear and soft counterterms provide the necessary ingredients for determining the rapidity RGE for the collinear and soft sectors below in section 5.1.

Rapidity renormalization group evolution
In our result for the matching coefficient between bHQET and SCET at O(α 2 s ), given above in eq. (3.8), we encountered a large logarithm α 2 s C F T F ln(m 2 /Q 2 ). We discussed the setup for the resummation of such logarithms above in section 2. As shown in section 4 these rapidity logarithms are only related to contributions of the virtual massive quarks that appear in the gluon vacuum polarization, and hence are the same as in the threshold corrections for massless primary quarks in ref. [33]. There it was anticipated that they can be resummed by exponentiation, as is common for these kinds of logarithms. For example, for the radiation of a massive gauge boson the rapidity renormalization group implies that this exponentiation occurs to all orders in perturbation theory [32,35,43,44]. The difference in our case is that the rapidity logarithms start at two-loops, and hence involve the additional issue of one-loop induced corrections due to the scheme change in the coupling constant.
Here we will show explicitly how to treat the rapidity logarithms at O(α 2 s C F T F ) in a rapidity renormalization group framework, and subsequently demonstrate that they indeed exponentiate. We start from eq. (2.24). Up to O(α 2 s ) we only have a contribution from the C F T F dependent terms,

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where the second term accounts for coupling conversion from the (n l + 1)-flavor to n l -flavor scheme. As before, in the n l -flavor scheme the bHQET graphs give no contribution. The results from section 4.4 can now be used to compute this ν-anomalous dimension. Using eq. (4.36) we can calculate the one-loop correction, which exhibits dependence on the infrared gluon-mass regulator Λ. The two-loop term above can be calculated using eq. (4.37) which gives where L m is defined in eq. (3.1). Together these results determine the ν-anomalous dimensions: Note that the IR regulator has canceled out, and that here the coupling [α s (µ)] 2 can be taken in either the n l or (n l + 1)-flavor scheme since the anomalous dimension starts at O(α 2 s ) and the difference is higher order. This result suffices for solving the ν-RGE equations at NNLL order. Using eq.
and contain no large logarithms for µ m, and for ν Q Q and ν m m, respectively. The evolution factor V RRG sums the rapidity logs between ν m and ν Q , and is defined as follows The general result for V RRG , and the result at NNLL, will be given below.
Similarly to the ν-anomalous dimensions, we can also determine individual µ-anomalous dimensions for the collinear and soft sectors, i = n, s,n, Repeating the steps below eq. (5.1) we find with L m and L Q defined in eq. (3.1). Eqs. (2.7) and (5.6) together include the evolution connected to H m in the 2-dimensional µ-ν plane, including that from invariant mass scales µ m to µ Q , that from invariant mass scales µ m to µ final , and that from rapidity scales ν Q to ν m . As demonstrated in ref. [35] the combined µ-ν evolution can be performed along any path and the path independence implies the consistency equation: However, similar to the example of the massive Sudakov form factor considered in ref. [35] we can see from eq. (5.4) that γ Cm ν, s contains potentially large logarithms ln(µ/m) for an arbitrary path in µ-ν-space. This is resolved by a prior resummation exploiting the fact that the derivatives in eq. (5.13) are proportional to the cusp anomalous dimension. Since C m is a matching coefficient between a (n l + 1)-flavor and n l -flavor theory, we can express eq. (5.13) in terms of the difference between the cusp anomalous dimensions Γ cusp [α s ] in the (n l + 1) and n l -flavor schemes. So for γ Cm ν,s we obtain which can be checked using the explicit perturbative expression of Γ cusp [α s ] up to two loops, Integrating eq. (5.14) in µ we obtain the resummed result for γ Cm ν, s ,  For our numerical analysis of H evol we employ scale choices that are appropriate to the peak region of the differential cross section within bHQET. We fix Q = µ Q = 1 TeV, which is a possible c.m. energy for a future linear collider, and µ final = 5 GeV corresponding to the scale of the soft radiation. We do not vary these two scales here since their impact and associated uncertainties have been analyzed elsewhere [14]. They matter only for the overall normalization and thus cancel in the normalized spectrum. In addition we use the MS massm t (m t ) = 163 GeV or pole mass m t = 171.8 GeV using the two-loop conversion, and α (5) s (m Z ) = 0.114 [23,49] and using two-loop conversion at µ =m t to obtain α (6) s (µ). For results with RG evolution that sums large logarithms we use the so called primed counting, i.e. our results at NLL and NNLL include NLL and NNLL evolution kernels together with the hard function boundary conditions at O(α s ) and O(α 2 s ), respectively. 7 For the rapidity evolution we use the expression in eq. (5.17), and the default rapidity scales ν Q = Q and ν m = m t , where m t is either the MS massm t (m t ) or the pole mass.
To determine the impact on the normalization we first note that the two-loop fixed order corrections to H

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In the top-left panel of figure 3 we display the evolved hard function H evol at the first three orders in resummed perturbation theory for values of µ m in the rangem t /2 < µ m < 2m t . We use the MS mass scheme and the expressions for H (n l +1) m,n , H (n l +1) m,n and H (n l +1) m,s from eqs. (5.21) and (5.22). As already observed in ref. [14], there is a significant correction when going from LL to NLL order which more than doubles H evol . From NLL to NNLL we observe that the correction is notably smaller, indicating that the series has stabilized. Although the magnitude of these corrections is not captured by the µ m variation, it is of the size expected from studying the uncertainty associated to the µ final variation. The complete study of the µ final variation requires including the jet and soft functions, which cancel the µ final dependence of H evol to the order one is working. We leave this for future work rather than taking it up here. We observe that the µ m dependence significantly decreases as we go to higher order. This behavior is shown best in the top-right panel of figure 3, where the same curves are plotted, but now normalized to H evol (µ m =m t ) at the respective order. The two-loop result for the hard function H (n l +1) m plays a key role in this reduction of the scale dependence at NNLL . Note that the size of the µ m variation of the blue dashed curve at 2% correlates well with the size of the NNLO fixed order correction in eq. (5.24), which gives a +2% correction. Therefore it is reasonable to take the µ m variation of the solid red curve in this figure as an estimate of the O(α 3 s ) correction in eq. (5.24), which we take to be ±0.2%.
In the lower-left panel of figure 3 we compare the dependence on µ m at NNLL for the MS mass with the corresponding result for the pole mass. In the pole mass case we employ eqs. m,s . We see that the pole mass exhibits a larger sensitivity to the renormalization scale µ m implying a slightly slower convergence of the perturbative series, potentially related to IR renormalon effects.
Finally, we can analyze the impact of the terms related to rapidity logarithms. For µ m =m t (m t ), these terms yield a numerical contribution of −0.0014 in the fixed-order full hard function H (n l +1) m (m t , Q/m t , µ m =m t ) in eq. (5.24). Due to a relatively small coefficient, they do not give a significant correction in comparison with the remaining twoloop contributions which give a numerical correction of 0.0166. Therefore, we anticipate the dependence on the rapidity scales ν Q and ν m to be rather mild. In the lower-right panel of figure 3 we plot H evol at NNLL for the MS mass as a function of µ m , but now with three choices for ν Q /ν m . To obtain these results we varied ν Q up and down by a factor of two, but we note that equivalent results are obtained by instead varying ν m by a factor of two. We see that varying ν Q /ν m by a factor of 2 gives a negligible effect compared to the residual µ m dependence at this order. Therefore, we conclude that including an uncertainty from ν-variation is not necessary to obtain an estimate of the overall perturbative uncertainty of the cross section.

Conclusions
In the context of EFT factorization for boosted top quark production, we have extracted the hard function H m = |C m | 2 describing virtual fluctuations at the top-mass scale, completely at two-loop order using earlier results from refs. [30,31]. This result provides the last JHEP12(2015)059 missing ingredient needed to make N 3 LL resummed predictions (up to the 4-loop cusp anomalous dimension) for the invariant mass distribution of top-jets in the peak region using the factorization theorem of refs. [13,14] given in eq. (1.2). Particular focus was given to the contributions to H m from heavy quark loops, which induce terms with a large logarithm α 2 s C F T F ln(Q 2 /m 2 ) that can not be treated with standard RG evolution in µ. These terms were computed once more directly using collinear and soft matrix elements in SCET, and we have shown how they can be factorized using a rapidity cutoff ν, and RG evolved using rapidity renormalization group equations. Interestingly, this factorization and RG evolution occurs within the Wilson coefficient C m and hence at the amplitude level. Using our result for H m we have assessed the remaining perturbative uncertainty associated to the top-mass scale, µ m m, and estimate it to be very small, ±0.2%, predicting that the two-loop result for H m provides a very accurate result for this function. The total normalization uncertainty in the differential cross section is expected to now be dominated by that from O(α 3 s ) perturbative corrections to the low-scale soft and jet functions, which could be estimated by a dedicated study of the residual µ final dependence at NNLL order.

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and hence is fully consistent with determining C (C F T F ,n l +1) m from eq. (4.8) and then simply applying the coupling conversion in eq. (A.1) in the result. Note that in this (n l + 1)-flavor scheme approach the bHQET one-loop amplitude contributes and plays an important role in obtaining the scheme conversion term involving C To extent the resummation of large logarithms in the factorization theorem in eq. (1.2) from NNLL to N 3 LL the only missing ingredient -besides the cusp anomalous dimension at four-loops -is the O(α 3 s ) noncusp anomalous dimension of the bHQET jet function or equivalently of the bHQET current (which are related to each other via eq. (2.18) with the known three loop result for γ S ). The latter has not been so far given in the literature, but can be extracted from a recent result for the three-loop anomalous dimension of a cusped Wilson loop [41,42], which is equivalent to the full anomalous dimension in HQET. Expanding their result in the lightlike limit x ∼ m/Q → 0, we obtain with the help of the Mathematica package HPL [50] cusp , while the non-logarithmic ingredient of eq. (B.1) represents the noncusp part. Together with the corresponding anomalous dimension of the SCET current this enables one to predict the logarithmic structure of H m at three loops by solving eq. (2.15). Furthermore it allows one to extract the last missing ingredient to predict the full IR-divergent structure of the three-loop full QCD form factor for massive quarks for m Q, which is for example in ref. [31] the coefficient K (3) in eq. (63).
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