Massive Event-Shape Distributions at N 2 LL
PPrepared for submission to JHEP
IFT-UAM/CSIC-20-79, UWThPh 2020-17
Massive Event-Shape Distributions at N LL Alejandro Bris a,b , Vicent Mateu b,c and Moritz Preisser d a Departamento de Física Teórica, Universidad Autónoma de Madrid,Cantoblanco, 28049, Madrid, Spain b Instituto de Física Teórica UAM-CSIC, E-28049 Madrid, Spain c Departamento de Física Fundamental e IUFFyM,Universidad de Salamanca, E-37008 Salamanca, Spain d University of Vienna, Faculty of Physics, Boltzmanngasse 5, A-1090 Wien, Austria
E-mail: [email protected] , [email protected] , [email protected] Abstract:
In a recent paper we have shown how to optimally compute the differen-tial and cumulative cross sections for massive event-shapes at O ( α s ) in full QCD. In thepresent article we complete our study by obtaining resummed expressions for non-recoil-sensitive observables to N LL + O ( α s ) precision. Our results can be used for thrust, heavyjet mass and C-parameter distributions in any massive scheme, and are easily generalizedto angularities and other event shapes. We show that the so-called E- and P-schemescoincide in the collinear limit, and compute the missing pieces to achieve this level ofaccuracy: the P-scheme massive jet function in Soft-Collinear Effective Theory (SCET)and boosted Heavy Quark Effective Theory (bHQET). The resummed expression is subse-quently matched into fixed-order QCD to extend its validity towards the tail and far-tail ofthe distribution. The computation of the jet function cannot be cast as the discontinuityof a forward-scattering matrix element, and involves phase space integrals in d = 4 − ε dimensions. We show how to analytically solve the renormalization group equation for theP-scheme SCET jet function, which is significantly more complicated than its 2-jettinesscounterpart, and derive rapidly-convergent expansions in various kinematic regimes. Fi-nally, we perform a numerical study to pin down when mass effects become more relevant. a r X i v : . [ h e p - ph ] J un ontents s = 0 s = m s = ∞ I P np for s > m – 1 – Introduction
Since the late 70s, a class of observables called event shapes has been used to test anddetermine fundamental properties of QCD (for a review see [1, 2]), most notably to measurethe strong coupling. As the name suggests, these observables contain information about thegeometric momentum distribution of the final-state particle momenta. The (historic) mainfields of application are e + e − collisions and deep inelastic scattering (DIS), but today therealso exist adaptations developed specifically for pp colliders. In high-energy experimentsmost of the time it is sufficient to use the approximation that all particles in the final stateare massless. If one is interested in high-precision calculations or in cases where the quarkmass is a dominant effect, this approximation is no longer valid.While the theoretical computation of event-shape distributions for massless quarksat e + e − colliders has been pushed to unprecedented precision in recent years, includingfixed-order results to O ( α s ) [3–8] and resummation at next-to-next-to-leading-logarithm(N LL) [9–11] and next-to-next-to-next-to-leading-logarithm (N LL) [12–16], computationsfor massive quarks remain at a lower precision. Such high level of resummation for masslessquarks has been achieved in most cases using Soft-Collinear Effective Theory (SCET) [17–21] (or in the equivalent formalism by Collins, Soper and Sterman [22–26]), in which themost singular terms of the distribution are written in a factorized form, and summation oflarge logarithms is carried out with standard renormalization-group evolution. On the otherhand, using the coherent branching formalism [27], the resummation can be automatizedup to N LL with numeric codes [28, 29]. The results have been extended to the case oforiented event shapes in Ref. [30], and can be used to extract the strong coupling with highprecision by comparing the theoretical expressions to LEP data, see e.g. [10, 13, 31–35].A first step towards having a firmer theoretical control over massive event shapes wastaken in Ref. [36]. In that article, the fixed-order differential and cumulative cross sec-tions were computed at O ( α s ) for any event-shape. Since quark masses screen collineardivergences, there are only soft singularities that translate at threshold into two types ofsingular terms: a Dirac delta and a plus function (at higher orders other singular functionsmight appear). In Ref. [36] the coefficients of the delta and plus functions were computedanalytically, providing a closed -dimensional integral form for the former, and showingthat the latter is universal. Furthermore, the delta function coefficient was provided inan analytic form for most event shapes. An algorithm was devised to compute the non-singular terms through a single integral (which can be carried out analytically in a fewcases), a method much faster and accurate than binning the distribution in conjunctionwith a Monte Carlo integrator. In the present paper, we complement those developmentsby adding resummation of large Sudakov logarithms at N LL. This kind of resummationfor boosted quarks has been worked out only for the hemisphere-mass (doubly differential)distribution in Refs. [37, 38], which can be easily marginalized into heavy jet mass and2-jettiness (and with extra little work, also into a variable called C-Jettiness in Ref. [36]).In both cases, and only through recent computations that will be reviewed in the nextparagraph, N LL precision has been achieved. In this article we take the necessary stepsto bring the accuracy to N LL + O ( α s ) for SCET-I type observables in the massive E- and– 2 –-schemes.Although SCET was first developed as an effective theory for massless particles, it wasquickly generalized to include massive quarks in Ref. [39]. Resummation is achieved inSCET by writing the most singular terms of the cross section as the product or convolutionof various pieces: the hard matrix element, which is the square of the matching coefficientbetween SCET and QCD; the jet function, describing radiation collinear to the initiatingpartons; and the soft function, accounting for wide-angle soft radiation. Quark masses areinfrared modes of the effective theory and therefore do not show up in the hard matchingcoefficient, but they do contribute to the jet (starting at one-loop) and soft (starting attwo loops) functions. In the limit in which the difference between the jet and the heavy-quark massess is much smaller than either of those, a new physical scale emerges, requiringadditional resummation of large logarithms. This is achieved by matching SCET intoboosted heavy-quark effective theory (bHQET). For jettiness all the ingredients necessaryto build an N LL-accurate cross section are known: bHQET jet function [40] SCET massivejet function [41], and bHQET matching coefficient [42], as well as the secondary productioncontributions [43, 44] (all up to two-loop level). The computations carried out in thisarticle make possible N LL resummation both in SCET and bHQET for other classes ofevent shapes. Masses can also modify the endpoint of the distribution, which is their onlyeffect at LL and NLL. This threshold modification depends solely on the scheme used todefine the massive event shape.In the approach followed in [45] to build the differential cross section, kinematic powercorrections are taken into account in fixed-order perturbation theory by matching the SCETcross section onto full QCD. This setup was successfully applied to phenomenological anal-yses of massless event shapes, and is crucial to obtain reliable predictions in the tail andfar-tail of the distribution. In the case of massive event shapes the situation is a bit morecomplex for two reasons: a) the partonic threshold is different in full QCD and the EFT,and b) the EFT prediction does not completely reproduce the singular terms at threshold.The reason is simple to understand: since the mass is an infrared mode, it is power-countedtogether with other infrared scales such as the quark virtuality, in such a way that power-suppressed terms include kinematic as well as mass corrections. Since the main focus ofthis article are event shapes with mass-independent thresholds, issue a) is not relevant. Atleading power, SCET and bHQET only predict the leading ˆ m behavior of the singular termsin QCD. It is however possible to modify the hard and jet functions to fully account forthe singular terms in the factorization theorems, such that power corrections are not dis-tributions and behave well close to threshold. We will show how to apply this prescriptionto event shapes in the E- and P-schemes.Non-vanishing quark masses imply that the mass sensitivity can be tuned using differ-ent schemes for the event-shape definition. To the best of our knowledge, this possibilityhas only been studied in the context of fixed-order perturbation theory in Ref. [36], and inthe present work this analysis is continued by considering resummation of large logarithms.Furthermore, we complete the list of ingredients necessary for a full N LL + O ( α s ) compu- Here and in what follows, Q denotes the e + e − center of mass energy and ˆ m = m/Q the reduced mass. – 3 –ation, which are applied to the case study of P-scheme thrust, making our work a usefulreference for upcoming studies. Our result is essential to consistently include bottom-quarkmass effects in analyses that aim to extract α s from fits to LEP data, and to clarify the topquark mass interpretation problem.This article is organized as follows: in Sec. 2 we discuss kinematics in the dijet limit,which is the core for factorizing cross sections; in Sec. 3 we review the concept of massiveschemes, explore how they affect the mass sensitivity of cross sections, and derive the im-plications they have for the SCET power counting; the factorization theorems in SCET andbHQET are presented in Sec. 4, in which also large-logarithm resummation is introduced;we carry out the computation of the SCET massive jet function in Sec. 5, and use this resultin Sec. 6 to write down the fixed-order expression of the cross section in the dijet limit;in Sec. 7 we compute the bHQET jet function, and derive analytic results for the runningof the non-distributional pieces of the SCET jet function in Sec. 8, presenting some usefulexpansions that can be implemented in different regions of the spectrum; kinematic, mas-sive and non-perturbative power corrections are discussed in Sec. 9, while some numericalinvestigations are carried out in Sec. 10; finally, our conclusions are contained in Sec. 11.Some technical aspects of the computations are relegated to Appendices A and B. In this section we study the kinematics of dijet events: two narrow, nearly back-to-back jets,plus additional soft radiation. In this situation the value of the event-shape is not far fromits minimal value, which we call collectively e min , and the SCET power-counting rules apply.Particles in such events can be either soft, n -collinear or ¯ n -collinear, with momenta whoselight-cone coordinates p µ = ( p + , p − , p ⊥ ) scale like p µs ∼ Q ( λ , λ , λ ) , p µn ∼ Q ( λ , , λ ) and p µ ¯ n ∼ Q (1 , λ , λ ) respectively, with λ the SCET power-counting parameter. Since we areinterested in the primary production of heavy quarks and p = m , for consistency one hasthat ˆ m ∼ λ and soft particles are either massless or have ˆ m s (cid:46) λ . The SCET scaling holdsfor momenta defined in the P- and E-schemes as well. In this limit it is possible to write theevent-shape measurement e as a sum of contributions from collinear (in both directions)and soft particles [46, 47]: e = e n + e ¯ n + e s , (2.1)where e denotes the event-shape measurement in the dijet limit at leading power, and weconsider only soft perturbative particles. For SCET-I type observables, the three terms scalelike λ and are equally important. Moreover, e n , e ¯ n and e s can be written as a sum oversingle-particle contributions. If the event shape is already defined by a single sum of final-state particle momenta (as it is the case for thrust or angularities) this statement is trivial.When the event shape correlates momenta of final-state particles [ e.g. the definition involvesa double sum (C-parameter) or there is a single sum squared (jet masses) ] the situation ismore complicated. In the latter case one has to show explicitly that in this the dijet limitthe leading contribution to the event shape can be written in the form of Eq. (2.1), as wasdone e.g. in Ref. [14] for C-parameter. Let us work out this decomposition for hemispheremasses, which are defined as the square of the total four-momentum flowing into one of the– 4 –emispheres, being those delimited by the plane normal to the thrust axis. Taking the z axis in the thrust direction, the mass of the plus hemisphere takes the following form inlight-cone coordinates: Q ρ + = (cid:88) i ∈ + p + i (cid:88) j ∈ + p − j − (cid:18)(cid:88) i ∈ + p ⊥ i (cid:19) = (cid:88) i ∈ + p + i (cid:88) j ∈ + p − j , (2.2)where we have used that the component of the total hemisphere momenta normal to thethrust axis is identically zero. Let us assume the negative direction of the z axis pointingtowards the plus hemisphere such that it does not contain ¯ n -collinear particles (with p z > ).In the dijet limit, particles i and j can be either soft of collinear, but if both are softthe corresponding contribution is ρ s,s ∝ O ( λ ) and therefore power suppressed. Next weconsider that i is soft and j is collinear, such that the leading contribution comes from the p − j term: Q ρ s + = (cid:18)(cid:88) j ∈ n p − j (cid:19)(cid:18) (cid:88) i ∈ s + p + i (cid:19) = 2 (cid:18)(cid:88) j ∈ n E j (cid:19)(cid:18) (cid:88) i ∈ s + p + i (cid:19) = Q (cid:88) i ∈ s + p + i = QP + s + , (2.3)where we have used that up two power corrections in the dijet limit p − i = 2 E i and thetotal available energy Q is carried by collinear particles only and equally divided into eachhemisphere, such that the total minus momentum flowing into the plus hemisphere is Q upto power corrections. The set of soft particles that belong to the plus hemisphere is denotedby s + . With an identical computation we get for the collinear-collinear contribution Qρ n + = (cid:88) i ∈ c + p + i = P + n , (2.4)with c + the set of n-collinear particles. In the dijet limit we have Qρ + = P + s + + P + n , which isequal to the total plus momentum entering the minus hemisphere. An identical reasoningleads to Qρ − = P − s − + P − ¯ n . Since we have not made any assumption on the mass of theparticles, our result is valid for massless and massive quarks. In this section we review the generalization of event-shape measurements for massive parti-cles that go under the name of “massive schemes” (which should not be mistaken with thequantum field theory mass schemes such as pole, PS, 1S, MSR, MS , . . . ). These schemeswere introduced in an article by Salam and Wicke [48] to study the effects of hadron masseson hadronization power corrections, which were further studied in [47]. Both studies con-sider light quarks only, such that massive schemes have no effect on partonic computations,but change the size of non-perturbative power corrections. For massive quarks, however,switching schemes can dramatically change the cross section, in particular its sensitivityto the quark mass, which is obviously of high interest either in cases where very accuratecomputations demand including quark mass corrections, or when they are the leading ef-fect. A propaedeutic study of mass schemes for heavy quarks in event-shape cross sections– 5 –as carried out in Ref. [36], where fixed-order results where computed and subsequentlyanalyzed. A more complete study demands resummation of large logarithms in the peakand tail of the distribution, and we aim to fill in this gap here.Massless particles travel at the speed of light and their four-momentum satisfies p = 0 ,which when translated into energy and momentum in a given frame implies E p = | (cid:126)p | .Therefore on can interchange E p ↔ | (cid:126)p | in the event-shape definition with no visible effect.If particles become massive, p = m and one has E p > | (cid:126)p | , meaning in turn that replacingthe momentum magnitude by the energy (or vice versa) changes the value of the eventshape. In this context, various mass schemes where defined to quantify this possibility,which reproduce the original “massless” definition in the limit m → :1. E-scheme : As indicated by its name, one replaces momenta by energies with thesubstitution ( p i , (cid:126)p i ) → p i (1 , (cid:126)p i / | (cid:126)p i | ) . The scalar product takes the following form: p E · q E = E p E q (cid:18) − (cid:126)p · (cid:126)q | (cid:126)p || (cid:126)q | (cid:19) , (3.1)such that p E = 0 even for massive particles. One advantage of this prescription isthat hadronization corrections become universal. The variables called angularities [49]were originally defined in this scheme.2. P-scheme : Again the name suggests that energy gets replaced by momenta as ( p i , (cid:126)p i ) → | (cid:126)p i | (1 , (cid:126)p i / | (cid:126)p i | ) , and it happens that most of the classical event shapeswere originally defined in this scheme: thrust [50], C-parameter [51, 52] and broad-ening [53]. The scalar product now reads p P · q P = | (cid:126)p || (cid:126)q | − (cid:126)p · (cid:126)q , which again implies p P = 0 for massless and massive particles.3. M-scheme : The name “massive scheme” is used for event shapes that in their originaldefinition where neither in the P- nor in the E-scheme, such as heavy jet mass [54–56]. Their definition contains both energy and momentum and are the most sensitiveto quark masses, in particular because in this scheme the usual relation p = m issatisfied.It is important to realize that four-momenta as defined in the E- and P-schemes are frame-dependent, and that event shapes are usually defined in terms of magnitudes measured inthe center-of-mass frame. The usual light-cone decomposition applies in either scheme Sp S · q S = 12 p + S q − S + 12 q + S p − S − (cid:126)p S, ⊥ · (cid:126)q S, ⊥ , (3.2)with S = E, P . The specific definition of the event shapes just introduced can be founde.g. in the original papers and will not be repeated here unless necessary. They are alsosummarized, including a discussion on massive schemes, in Ref. [36].
When studying the sensitivity of event shapes at parton level the first possible contributionfor e + e − annihilation comes from the production of a heavy quark-antiquark pair without We consider for now partonic final states, assuming stable massive quarks. – 6 – τ a C ρ
M-scheme − β (1 − β ) − a (1 + β ) a
12 ˆ m (1 − ˆ m ) ˆ m P- and E- schemes
Table 1 . Threshold position for various event shapes in the case of primary production of a stablequark-antiquark pair in different massive schemes. We use β = √ − m , the velocity of the quarksat threshold in natural units. additional radiation. In this case, the thrust axis is parallel to the three-momenta of thequarks, which makes trivial to calculate the threshold for two particles in the final statewith equal mass m . Moreover, this simple computation sets the lower threshold even ifadditional gluons and (massless) quarks are radiated. The results in Tab. 1 show that forevents in which a massive stable quark-antiquark pair is produced ( primary production )only the M-scheme is sensitive to the quark mass while P- and E-schemes are not. In mostof the events there will be some extra radiation present which will modify the former dijetinto two fatter jets or an even more isotropic momentum distribution. For the observableswe study, such processes will mainly contribute for event-shape values away from thresholdvalue adding subleading mass sensitivity (i.e. suppressed by a factor of α s ) even in the P-and E-schemes, but will not substantially change the leading sensitivity of the M-schemedefinition since it comes from the tree-level peak position. From this we can concludethat the M-scheme is preferred if the aim is a mass-sensitive observable (e.g. for quarkmass determinations), but in case that one seeks a mass-insensitive observable, the P- andE-schemes are a better choice. Collinear particles in the n direction satisfy E p = ( p + + p − ) / p − / O ( λ ) and also | (cid:126)p | = (cid:113) E p − m = p − / O ( λ ) such that the E-scheme ( Q = (cid:80) i E i ) and P-scheme( Q p ≡ (cid:80) i | (cid:126)p i | ) normalizations are the same at leading power. Let us compute the four-momenta of massive n -collinear particles in the E- and P-schemes. Since we have seen that E p / | (cid:126)p | = 1 + O ( λ ) , the “large” (or label) components p − and p ⊥ are the same in anyscheme. Let us then focus in the small p + momenta, which in the massive scheme takesthe following form (for n -collinear particles the z component of momenta is negative): p + = E p + p z = E p − (cid:112) | (cid:126)p | − | (cid:126)p ⊥ | = E p − (cid:113) E p − E ⊥ (cid:39) E ⊥ E p + O ( λ ) , (3.3)where we have used the so-called perpendicular energy E ⊥ ≡ (cid:112) | (cid:126)p ⊥ | + m . The com-putation of p + in the P-scheme is very similar since one only has to use | (cid:126)p | instead of E p p + P = | (cid:126)p | + p z = | (cid:126)p | − (cid:112) | (cid:126)p | − | (cid:126)p ⊥ | (cid:39) | (cid:126)p ⊥ | E p + O ( λ ) = p + − m p − + O ( λ ) , (3.4) If the massive partons enter the final state via gluon splitting in a massive quark-antiquark pair (that is,through secondary production) the sensitivity to the quark mass will again be subleading (now suppressedby a factor of α s ). – 7 –here we notice that in the second step p + P takes the same analytic form as in Eq. (3.3)with the replacements E p → | (cid:126)p | and E ⊥ → | (cid:126)p ⊥ | , in the third step we use | (cid:126)p | = E p + O ( λ ) ,and in the last step we replace | (cid:126)p ⊥ | = p + p − − m and E p = p − / O ( λ ) . It is importantto notice that in general p + P (cid:54) = p + because m /p + (cid:39) O (1) . Moreover, the mass thatappears in Eq. (3.4) (and in other any event-shape measurement function) comes fromkinematic on-shell considerations and therefore corresponds to the pole scheme. Finally, letus compute the E-scheme p + component, and for that we only need to make the replacement p z → p z E p / | (cid:126)p | in Eq. (3.3): p + E = E p + E p | (cid:126)p | p z = E p | (cid:126)p | p + P = p + P + O ( λ ) , (3.5)where in the last step we again have used E p / | (cid:126)p | = 1 + O ( λ ) . Since for collinear particles(in any direction) p µP = p µE at leading power, we can safely conclude that at this order thecollinear measurements for all event shapes take the same form in the P- or E-schemes, butis in general different from the M-scheme. In what follows we will work out the collinearmeasurement for a few event shapes. Thrust
The original thrust definition is already in the P-scheme and reads τ P = 1 Q P min ˆ t (cid:88) i ( | (cid:126)p i | − | ˆ t · (cid:126)p i | ) , (3.6)with ˆ t the thrust axis. For collinear particles we have that | ˆ t · (cid:126)p i | = − p z and therefore upto power corrections we have Qτ Pc = Qτ Ec = (cid:88) i ∈ + p + P,i = (cid:88) i ∈ + (cid:18) p + i − m i p − i (cid:19) , (3.7)where we already indicate that the collinear measurement is the same in the E-scheme. InRef. [57] an M-scheme generalization of thrust, dubbed 2-jettiness, was introduced τ J = 1 Q min ˆ t (cid:88) i ( E i − | ˆ t · (cid:126)p i | ) , (3.8)such that its collinear limit is the total plus momentum flowing into the plus hemisphere Qτ Jc = (cid:80) i ∈ + p + i . Since the measurement is completely inclusive, the computation of the jetfunction can be carried out as the imaginary part of a forward-scattering matrix element.This is not the case for the E- and P-schemes if quark masses are non-vanishing. Hemisphere Jet Masses
We already worked out the collinear measurement for heavy jet mass in Eq. (2.4), and get-ting the P-scheme measurement is equally simple since Eq. (2.2) still applies with minimalmodifications: Q ρ P + = (cid:18)(cid:88) i ∈ + | (cid:126)p i | (cid:19) − (cid:18)(cid:88) i ∈ + p zi (cid:19) = (cid:20) (cid:88) i ∈ + ( | (cid:126)p i | + p zi ) (cid:21)(cid:20) (cid:88) j ∈ + ( | (cid:126)p j |− p zj ) (cid:21) = (cid:88) i ∈ + p + P,i (cid:88) j ∈ + p − P,j , (3.9)– 8 –here we again use that the total perpendicular momentum vanishes and use the identity a − b = ( a + b )( a − b ) . With this result we can trivially obtain the collinear measurementusing that p − P,j = 2 E j + O ( λ ) : Qρ Pc, + = Qρ Ec, + = (cid:88) i ∈ + (cid:18) p + i − m i p − i (cid:19) , (3.10)that matches the P-scheme thrust result. The total perpendicular momentum does notvanish in the E-scheme, since there is not such thing as E-scheme three-momentum con-servation (in the same way, P-scheme energy is not conserved either). However, in thedijet limit the perpendicular components are already O ( λ ) and therefore p ⊥ E = p ⊥ + O ( λ ) ,making (cid:80) i ∈ + p ⊥ E,i ∝ O ( λ ) , thence power suppressed, such that the result in Eq. (3.10) isalso valid for the E-scheme. C-parameter
In Ref. [14] it was shown how the C-parameter measurement splits into the sum of soft andcollinear contributions in the dijet limit. The proof relied on the particles being massless, soit cannot be taken for granted that it will work when quarks have a non-zero mass. Here wecarry out a similar proof valid for massive particles as well. C-parameter is defined alreadyin the P-scheme as C P = 32 Q P (cid:88) i,j | (cid:126)p i || (cid:126)p j | sin ( θ ij ) = 32 Q P (cid:88) i,j | (cid:126)p i || (cid:126)p j | [1 + cos( θ ij )][1 − cos( θ ij )] (3.11) = 32 Q P (cid:88) i,j ( | (cid:126)p i || (cid:126)p j | + (cid:126)p i · (cid:126)p j )( | (cid:126)p i || (cid:126)p j | − (cid:126)p i · (cid:126)p j ) | (cid:126)p i || (cid:126)p j | , where we simply use sin ( θ ij ) = 1 − cos ( θ ij ) = [1 + cos( θ ij )][1 − cos( θ ij )] and the definitionof the euclidean scalar product to get to the final form. Next one can express the result interms of P-scheme light-cone coordinates using Eq. (3.2) as follows C P = 32 Q P (cid:88) i,j (2 | (cid:126)p i || (cid:126)p j | − p P,i · p P,j ) p P,i · p P,j | (cid:126)p i || (cid:126)p j | (3.12) = 32 Q P (cid:88) i,j ( p + P,i p + P,j + p − P,i p − P,j + 2 (cid:126)p ⊥ ,i · (cid:126)p ⊥ ,j )( p + P,i p − P,j + p − P,i p + P,j − (cid:126)p ⊥ ,i · (cid:126)p ⊥ ,j )( p + P,i + p − P,i )( p + P,j + p − P,j ) , with a similar result in the E-scheme. Arguments analogous to those used in Ref. [14]apply, and we focus on the collinear measurement only. First consider that both i and j are n -collinear such that the SCET scaling implies C Pnn = 3 Q (cid:20)(cid:18)(cid:88) i ∈ n p + P,i (cid:19)(cid:18)(cid:88) j ∈ n p − P,j (cid:19) − (cid:18)(cid:88) i ∈ n (cid:126)p ⊥ ,i (cid:19) · (cid:18)(cid:88) j ∈ n (cid:126)p ⊥ ,j (cid:19)(cid:21) = 3 Q (cid:88) i ∈ n p + P,i + O ( λ ) , (3.13)where once again we use that the total collinear perpendicular momenta flowing into theplus hemisphere is zero up to power corrections. One gets an analogous result for C P ¯ n ¯ n , while– 9 –f i is n -collinear and j is ¯ n -collinear we get (we already include a factor of to account forthe case in which i is ¯ n -collinear and j is n -collinear) C Pn ¯ n = 3 Q (cid:20)(cid:18)(cid:88) i ∈ n p + P,i (cid:19)(cid:18)(cid:88) j ∈ ¯ n p + P,i (cid:19) + (cid:18)(cid:88) i ∈ n p − P,i (cid:19)(cid:18)(cid:88) j ∈ ¯ n p − P,i (cid:19) + 2 (cid:18)(cid:88) i ∈ n (cid:126)p ⊥ ,i (cid:19) · (cid:18)(cid:88) j ∈ ¯ n (cid:126)p ⊥ ,j (cid:19)(cid:21) = 3 Q (cid:18)(cid:88) i ∈ n p + P,i + (cid:88) i ∈ ¯ n p − P,i (cid:19) + O ( λ ) . (3.14)Summing up the contributions in Eqs. (3.13) and (3.14) we obtain the collinear measurementin the P- and E-schemes: QC Pc = QC Ec = 6 (cid:88) i ∈ + p + P,i = 6 (cid:88) i ∈ + (cid:18) p + i − m i p − i (cid:19) , (3.15)identical to that of thrust or the hemisphere masses up to a factor of . The computationfor the E-scheme is identical, relies on arguments already exposed, and therefore will notbe repeated. For the M-scheme C-jettiness variable introduced in Ref. [58] it was shownin Ref. [59] that the collinear measurement takes the simple form QC Jc = 6 (cid:80) i ∈ + p + i .Therefore, for the reduced C-parameter variable (cid:101) C ≡ C/ the collinear measurement forthe three event shapes we consider coincide in every massive scheme. For simplicity we consider the well-known case of thrust (in either massive scheme), whichcan be easily modified to obtain the corresponding factorized results for C-parameter orheavy jet mass. After having shown that in the three schemes considered Eq. (2.1) holds, thederivation of the factorization theorem is obtained following the steps outlined in Ref. [46].The leading hadronization corrections (which are soft) can also be factorized as an extraconvolution with the so-called shape function, and even though they are included in ournumerical analysis, for the sake of conciseness we ignore them in this section. Likewise,we include kinematic and mass power corrections in our final analysis, but postpone theirdiscussion until Sec. 9
The value of the quark mass can have different hierarchies with respect to the (EFT) hard,jet, and soft scales. In Refs. [43, 44] it was extensively discussed how to setup a consistentvariable-flavor number scheme for final-state jets accounting for primarily and secondarilyproduced massive quarks. Four scenarios can be defined for the cases in which m > µ H (scenario I, which is of no interest for primary quarks since there is no energy to producethem), µ H > m > µ J (scenario II, relevant for very boosted heavy quarks, and betterdescribed in bHQET), µ J > m > µ S (scenario III) and m < µ S (scenario IV). Eachof them has a different factorization theorem and renormalization group evolution setup.Even though the heavy quark mass is a fixed parameter, the jet and soft scales dependon the event-shape value and therefore they change along the spectrum, such that several– 10 –cenarios might occur in a given distribution. For simplicity, we assume the quark massis always smaller than the soft scale, such that we stay in scenario IV even in the peak ofthe distribution. In this way, we avoid having to deal with integrating out the heavy quarkmass and the partonic factorization formula reads σ d ˆ σ SCET d τ = Q H ( Q, µ ) (cid:90) Q ( τ − τ min )0 d (cid:96) J τ ( Q τ − Q(cid:96), µ ) S τ ( (cid:96), µ ) , (4.1)with σ the Born or point-like (massless) cross section, H and S τ the hard and soft functions,respectively, and J τ ( s, µ ) ≡ (cid:90) s − s min s min d s (cid:48) J n ( s − s (cid:48) , µ ) J n ( s (cid:48) , µ ) , (4.2)the thrust jet function, which is the convolution of two single-hemisphere jet functions. TheM-scheme hemisphere jet function has support for s > s min = m , what sets the integrationlimits in Eq. (4.2). Accordingly, the thrust jet function has support for s > s min , implyingthat the minimal value for 2-jettiness is τ J min = 2 ˆ m . We shall present the computation ofthe M- and P-scheme SCET jet function in Sec. 5. The definition of the soft function interms of Wilson lines can be found e.g. in Ref. [46] and the corresponding expression forthe jet function will be given in Sec. 5. The factorization formula takes a simpler form inFourier space σ d ˆ σ SCET d τ = Q π H ( Q, µ ) (cid:90) d x e ixp ˜ J τ (cid:18) xQ , µ (cid:19) ˜ S τ ( x, µ ) , (4.3)with p = Q ( τ − τ min ) and ˜ J τ and ˜ S the Fourier transform of the jet and soft functions. Thethrust jet function in position space is the square of its hemisphere counterpart, and canbe computed as follows ˜ J τ ( y, µ ) = (cid:90) ∞ d s e − i ( s − s min ) y J τ ( s + s min , µ ) = ˜ J n ( y, µ ) . (4.4)In Eqs. (4.1) and (4.3) all matrix elements appear evaluated at the same renormalizationscale µ . In order to minimize large logarithms that appear in each of them one should useRGE equations to evaluate them at their respective natural scales, denoted by µ H ∼ Q , µ J ∼ Q √ τ and µ S ∼ Qτ , such that for small τ there is a strict hierarchy among those: µ H > µ J > µ S and the SCET scaling parameter takes the value λ ∼ √ τ . The form andsolution of the renormalization group equations is also simpler in position space. Usingthose and changing variables to y = x/p one arrives at σ d ˆ σ SCET d τ = H ( Q, µ H ) p (cid:18) e γ E µ S p (cid:19) ˜ ω R ( Q, µ i ) (cid:90) d y π e iy ( i y ) ˜ ω ˜ J τ (cid:18) yQp , µ J (cid:19) ˜ S τ (cid:18) yp , µ S (cid:19) , (4.5) Extending our results to take into account different scenarios poses no difficulty. – 11 –here µ i denotes collectively all renormalization scales (including the common µ ) and weuse the following compact notation R ( Q, µ i ) = Q (cid:18) µ H Q (cid:19) − ω H e ˜ k (cid:18) µ J Qµ S (cid:19) ˜ ω J , ˜ k = ˜ k H + ˜ k J + ˜ k S , ˜ ω = ˜ ω J − ω S , ˜ ω S = ˜ ω Γ c ( µ S , µ ) , ˜ ω J = ˜ ω Γ c ( µ J , µ ) , (4.6) ˜ ω H = ˜ ω Γ c ( µ H , µ ) , ˜ k S = ˜ ω γ S ( µ S , µ ) − k Γ c ( µ S , µ ) , ˜ k H = ˜ ω γ H ( µ H , µ ) − k Γ c ( µ H , µ ) , ˜ k J = ˜ ω J ( µ J , µ ) + 4˜ k Γ c ( µ J , µ ) , with ˜ ω and ˜ k the exponential running kernels defined in terms of integrals over the SCETand QCD anomalous dimensions as follows ˜ ω γ ( µ , µ ) = 2 (cid:90) α µ α d α γ ( α ) β QCD ( α ) , (4.7) ˜ k γ ( µ , µ ) = 2 (cid:90) α µ α d α γ ( α ) β QCD ( α ) (cid:90) αα d α (cid:48) β QCD ( α (cid:48) ) . Here γ can refer to cusp or non-cusp anomalous dimensions, and their dependence on α isin the form of perturbative series that define their respective coefficients β QCD ( α ) = − α s (cid:88) n =1 β n − (cid:16) α π (cid:17) n , Γ cusp ( α ) = (cid:88) n =1 Γ n (cid:16) α π (cid:17) n , γ ( α ) = (cid:88) n =1 γ n (cid:16) α π (cid:17) n . (4.8)The integrals in Eq. (4.7) can be solved analytically in terms of the anomalous-dimensioncoefficients if an expansion in α s is carried out. Their explicit form up to N LL can be founde.g. in Ref. [31]. General expressions valid for arbitrarily high order can also be derived andwill be given elsewhere.The jet function of a massive quark contains terms which are distributions, and henceeasy to Fourier transform, plus others which are regular functions, and to the best of ourknowledge it seems impossible to find an analytic expression in position space for them.Up to one loop, the momentum-space hemisphere jet function can be decomposed in thefollowing form: J n ( s + s min , µ ) = δ ( s ) + α s ( µ )4 π C F (cid:104) J dist ( s, µ ) + 1 m J nd (cid:16) sm (cid:17)(cid:105) + O ( α s ) , (4.9) J dist ( s, µ ) = 1 µ J m =0 (cid:16) sµ (cid:17) + 1 m J m (cid:16) sm (cid:17) , where the massive corrections, either with distributions J m or fully non-distributional J nd ,are µ independent dimensionless functions with support for positive values of their (dimen-sionless) arguments. The µ dependence of J dist is entirely determined from the jet andQCD anomalous dimensions, does not depend on the quark mass, and therefore can befully accounted for in the massless jet function of Eq. (4.9). The only piece that needs anexplicit computation in the E- and P-schemes is J nd , since the rest can be obtained usingconsistency conditions and results obtained in Refs. [38] and [36].– 12 –he integral in Eq. (4.5) can be easily solved for all terms involving only distributions,and generic formulas can be found for instance in Ref. [12]. For the non-distributional pieceof the jet function we carry out resummation in momentum space, and at one loop it ismultiplied by the hard and soft functions at tree-level only. Therefore, using Eq. (4.4) in(4.5) and carrying out the y integration, the non-distributional part of the 1-loop partoniccross section reads σ d ˆ σ nd d τ = C F α s ( µ J )2 π R ( Q, µ i )ˆ m Γ( − ˜ ω ) ( Qe γ E µ S ) ˜ ω (cid:90) Q τ s min d s J nd (cid:16) s − s min m (cid:17) ( Q τ − s ) − − ˜ ω = C F α s ( µ J )2 π R ( Q, µ i )ˆ m (cid:20) µ S e γ E Q ( τ − τ min ) (cid:21) ˜ ω I np (cid:16) ˜ ω, τ − τ min ˆ m (cid:17) , (4.10) I np (˜ ω, y ) = y ˜ ω Γ( − ˜ ω ) (cid:90) y d x ( y − x ) − − ˜ ω J nd ( x ) = 1Γ( − ˜ ω ) (cid:90) d z (1 − z ) − − ˜ ω J nd ( zy ) . The lower limit of integration in the first line has been moved to s min since below thatvalue the jet function has no support. In the E- and P-schemes s min = 0 so we have notlost any generality. To get to the second line we have switched variables in the integral to s = xm + 2 s min , and to obtain the second expression for I np (˜ ω, y ) we switch variables to x = zy . For the partonic cumulative distribution one gets instead ˆΣ nd ( τ c ) ≡ σ (cid:90) τ c d τ d σ nd d τ (4.11) = C F α s ( µ J )2 π R ( Q, µ i )ˆ m µ S e γ E Q (cid:20) µ S e γ E Q ( τ − τ m ) (cid:21) ˜ ω − I np (cid:16) ˜ ω − , τ − τ c ˆ m (cid:17) . To make the function I (˜ ω, y ) smooth in the no resummation limit, achieved when ˜ ω → ,one can integrate by parts to obtain I np (˜ ω, y ) = 1Γ(1 − ˜ ω ) (cid:20) y (cid:90) d z (1 − z ) − ˜ ω J (cid:48) np ( zy ) + J np (0) (cid:21) , (4.12)with J (cid:48) np the derivative of the J np function. This form is particularly useful if the integrationhas to be carried out numerically, making it more convergent and defining its analyticcontinuation to values < ˜ ω < . Further integration by parts can be implemented todefine the integral for even larger values of ˜ ω . If a closed analytical form is found, thisprocedure is unnecessary.Although the discussion in this section has been carried out assuming the pole mass forthe heavy quark, it is straightforward to convert the result to a short-distance scheme. In thescenarios in which SCET applies, the MS scheme is perfectly adequate. In scenario II it ismore convenient to employ low-scale short-distance schemes such as the MSR mass [60, 61]. If the heavy quark mass is large enough or if the jet is very narrow one enters scenario II, inwhich the jet and heavy quark masses are close to each other, corresponding to very boostedquarks. In this kinematic situation, a new physical scale emerges µ B ∼ Qτ / ˆ m ∼ µ S / ˆ m ,– 13 –uch that there is a new hierarchy between scales: µ H > m > µ B > µ S . A practical wayto see how this becomes manifest is looking at the structure of the one-loop jet functionin Eq. (4.9). Since the non-distributional terms are power suppressed when s → s min , it isenough to focus on the terms with distributions, which generically read µ J m =0 (cid:16) sµ (cid:17) = A δ ( s ) + Bµ (cid:20) µ s (cid:21) + + Cµ (cid:20) µ log( s/µ ) s (cid:21) + , (4.13) m J m (cid:16) sm (cid:17) = A m δ ( s ) + B m m (cid:20) m s (cid:21) + + C m m (cid:20) m log( s/m ) s (cid:21) + , where the coefficients A, B and C (with or without subindex m ) depend neither in µ nor in m . In scenarios III and IV one has s (cid:46) m and therefore the choice µ ∼ s makes sure thereare no large logarithms in neither term (the massless limit is smooth since J m + J nd → when m → and no new class of large logarithm emerges). On the other hand, if s (cid:28) m the choice µ ∼ s cannot prevent the logarithms in J m from becoming large. The heavy quark carrying momentum p = mv + k with v = 1 gets integrated out asa dynamical degree of freedom giving raise to heavy-quark effective theory [62–65]. Theremaining degrees of freedom are referred to as ultracollinear and carry residual momentum k . In the heavy quark rest frame [ in which v µ = (1 , ,(cid:126) ⊥ ) ] they are soft k µ = ∆(1 , , ,with ∆ (cid:28) m a low-energy scale, being able to interact with each other and with colorsources representing the integrated-out heavy quarks. In the center-of-mass frame thesemomenta get boosted and a hierarchy is generated among their light-cone components v µ + = (cid:18) mQ , Qm ,(cid:126) ⊥ (cid:19) , k µ + ∼ ∆ (cid:18) mQ , Qm , (cid:19) , q µs = m ∆ Q (1 , , , (4.14) v µ − = (cid:18) Qm , mQ ,(cid:126) ⊥ (cid:19) , k µ − ∼ ∆ (cid:18) Qm , mQ , (cid:19) , where we have included momenta q s which is soft in the center-of-mass frame. The twoboosted copies of HQET are matched onto SCET in order to account for global soft radia-tion, such that the heavy quark and ultracollinear particles can interact with soft degreesof freedom. The typical off-shellness of ultracollinear particles is softer than for collineardegrees of freedom which are part of SCET.Using this framework, it is possible to derive a factorization theorem for the partoniccross section [37] which effectively separates physics at the different involved scales. σ d ˆ σ bHQET d τ = Q H ( Q, µ m ) H m (cid:18) m, Qm , µ m , µ (cid:19)(cid:90) d (cid:96) B τ (cid:18) Q ( τ − τ min ) − Q(cid:96)m , µ (cid:19) S τ ( (cid:96), µ ) , (4.15) In practice one can still identify µ B with the jet scale ( µ B = µ J ), and we will do so in what follows. The bHQET limit should not be confused with the threshold limit, yet another interesting physicalsituation in which radiation other than the heavy quark is soft as compared with the quark mass, whilethere is no hierarchy between m and Q . For an unstable top quark, this scale is of the order of its width ∆ ∼ Γ , but for a stable bottom quarkit can be identified with ∆ ∼ Q τ /m ∼ ( Q τ J − m ) /m for thrust and 2-jettiness, respectively. If only two back-to-back quarks are produced, their velocity equals β = √ − m (cid:39) − m , andtherefore the boost factor reads γ = 1 / (2 ˆ m ) . When boosting momenta in light-cone coordinates theplus/minus components get multiplied/divided by γ (1 − β ) (cid:39) / ˆ m . – 14 –here the hard and soft functions are the same as in the SCET factorization theorem,but there is an additional matching coefficient H m between SCET and bHQET. The jetfunction B τ (ˆ s ) is different from J τ in SCET, has support for ˆ s > , its mass dependence isonly through a global /m factor and contains only distributions. It is also the convolutionof two hemisphere bHQET jet functions B n : B τ (ˆ s, µ ) = m (cid:90) ˆ s d ˆ s (cid:48) B n (ˆ s − ˆ s (cid:48) , µ ) B n (ˆ s (cid:48) , µ ) , (4.16)whose operator definition shall be given in section 7. Since H and H m are the same forall event shapes and S τ does not depend on the quark mass (it sees only light degrees offreedom), the anomalous dimension of the B n function is the same in any massive scheme.In turn this implies that all terms in the jet function except for the Dirac delta are fixed byconsistency and hence are the same in the M-, P- and E-schemes. Knowing H , H m and S τ at one loop, the delta function coefficient in B n can be obtained taking the ˆ m → limit ofthe result quoted in Ref. [36] for the full QCD prediction for the threshold delta functioncoefficient. In this sense, our computation in section 7 will be just a sanity check. The H m matching coefficient and bHQET jet function satisfy J τ ( s + s min ) = H m B τ ( s/m )[1 + O ( s/m )] , such that both factorization theorems smoothly join. For the various argumentsexposed in this paragraph, this also implies that the coefficients B m and C m in Eq. (4.13)do not depend on the massive scheme, and they are known from the 2-jettiness computationof Ref. [38]. Therefore, different massive schemes can differ only in A m and the J nd , butknowing the one-loop hard and soft functions, A m can be obtained again from the masslesslimit of the full-QCD threshold result.Carrying out resummation in bHQET is identical to massless SCET. Since we will limitour numerical analysis to situations in which it is sufficient to use SCET with masses, wewill not give further details on how to solve the corresponding RGE equations, which canbe found elsewhere. Moreover, we will not provide a detailed discussion on how to switchto a short-distance mass scheme in this setup. The jet function accounts for the dynamics of collinear particles within the hemisphere.Since the collinear measurement function in the P- and E-schemes is not the total plusmomentum, it cannot be computed as the discontinuity of a forward scattering amplitude,as was done in [37, 38]. Instead, one has to use the definition given in Ref. [46], whichafter a small modification to match the form of our factorization theorem and minimalmanipulations can be cast into the form. J n ( s, µ ) = (cid:90) d (cid:96) + π J n ( s, (cid:96) + ) , (5.1) J n ( s, (cid:96) + ) = 14 N c Tr (cid:90) d d x e i(cid:96)x (cid:104) | /nχ n ( x ) δ ( s − Q ˆ e n ) χ n,Q (0) | (cid:105) , with χ n,Q the jet field with total minus momentum equal to Q , d = 4 − ε the space-time dimension in dimensional regularization, (cid:96) − = Q and (cid:126)(cid:96) ⊥ = (cid:126) due to label momentum– 15 –onservation, and the trace is taken over spin and color indices. The n -collinear event-shape operator ˆ e n acting on some final state | X (cid:105) pulls out the value of the event shape forthe e n ( X ) , the contribution from n -collinear particles to the value e of the event shape: ˆ e n | X (cid:105) = e n ( X ) | X (cid:105) . To simplify the expression of in the second line of Eq. (5.1) we insertthe identity I = (cid:80) X | X (cid:105)(cid:104) X | after the delta function and shift the field χ n,Q ( x ) to x = 0 employing the momentum operator. Using the label operators for the large components ofthe momenta, the sum over X can be carried out and we obtain the following convenientexpression J n ( s, µ ) = (2 π ) d − N C Tr (cid:20) n/ (cid:104) | χ n (0) δ ( s − Q ˆ e n ) δ ( d − ( (cid:126) P ⊥ X ) δ ( ¯ P − Q ) ¯ χ n (0) | (cid:105) (cid:21) . (5.2)For practical computations one inserts a complete set of states after δ ( ¯ P − Q ) (cid:88) X | X (cid:105)(cid:104) X | ≡ (cid:88) n =1 (cid:88) spin (cid:90) n (cid:89) i =1 d d − (cid:126)p i (2 π ) d − (2 E i ) | X n (cid:105)(cid:104) X n | , (5.3)where we exclude the vacuum from the sum because it does not contribute to the jetfunction. Each term in the sum over n can include several contributions, accounting forvarious particle species (heavy/light quarks and gluons), and the sum over polarizationaffects all particles in the final state. The perturbative expansion of the jet function inpowers of α s is obtained by adding more particles to the sum as well as more virtual(loop) contributions to the matrix elements that appear after inserting the identity, whichin compact form can be written as J n ( s, µ ) = (2 π ) d − N C (cid:88) X δ ( d − ( (cid:126)p ⊥ X ) δ ( p − X − Q ) δ [ s − Q e n ( X )]Tr (cid:20) γ n/ |(cid:104) | χ n (0) | X (cid:105)| (cid:21) . (5.4)For the computation of the P-scheme hemisphere jet function one does not need any regu-larization beyond taking the space-time dimension from to d = 4 − ε . In the followingwe carry out the computation of the jet function using Eq. (5.2) for both 2-jettiness andP-scheme thrust. Although the result is already known for the former, it is instructive torepeat its computation to highlight the difference between the two approaches. In a way,the computation that uses Eq. (5.2) can be obtained applying Cutkosky rules to directlycompute the imaginary part of the forward scattering matrix element. + Figure 1 . One-loop diagrams contributing to the wave-function renormalization at O ( α s ) . It is important to remember that χ n in Eq. (5.2) is composed of bare SCET (quark andgluon) fields, and that it is convenient to carry out our computations using perturbationtheory “around” those (that is, we will not use the so-called renormalized perturbation– 16 –heory). For the jettiness computation through the discontinuity of the forward matrixelement, this entails that the wave-function renormalization factor Z ξ = 1 + α s C F π (cid:34) − ε + 6 log (cid:18) mµ (cid:19) − (cid:35) + O ( α s ) , (5.5)computed with the diagrams shown in Fig. 1 (the soft-gluon contribution vanishes), neverappears directly. The mass in Eq. (5.5) should be understood as the pole scheme. Whenusing Eq. (5.2) one needs to account for Z / ξ since this factor is precisely the overlap betweenthe quantum (bare) collinear field ξ n and the physical collinear state | q n (cid:105) : (cid:104) | ξ n | q n ( (cid:126)p, s ) (cid:105) = Z / ξ u s ( (cid:126)p ) , with u a particle spinor in the collinear limit, and s , (cid:126)p the spin and -momentumof the on-shell collinear quark. On the other hand, when using Eq. (5.2) self-energy diagramson external legs are not included, since their effect is already accounted for in the Z / ξ factor, and it is in this way that one has a one-to-one correspondence with the computationthrough the imaginary part of the forward matrix element. p Figure 2 . Lowest order diagram for the jet function.
The computation at leading order is simple enough that can be carried out for the twomassive schemes simultaneously. The corresponding tree-level diagram is shown in Fig. 2,where the double line represents a heavy quark and the dashed line marks which particlesare on-shell. To compute the phase-space integration it is convenient to use the followingparametrization d d − (cid:126)p E p = d p − p − θ ( p − ) d d − (cid:126)p ⊥ , (5.6) (cid:90) d d − (cid:126)p ⊥ = 2 π − ε Γ(1 − ε ) | (cid:126)p ⊥ | − ε (cid:90) d (cid:126)p ⊥ , which implies that p + has to be expressed in terms of the minus and perpendicular com-ponents through the on-shell condition p + = ( m + | (cid:126)p ⊥ | ) /p − , and since the mass appearsthrough on-shell kinematic relations it corresponds always to the pole scheme. In the secondline we have carried out the angular integrals for the perpendicular momentum, assumingthat matrix elements depend only on its magnitude. We then obtain J tree n ( s ) = (cid:90) d p − p − d d − (cid:126)p ⊥ δ ( d − ( (cid:126)p ⊥ ) δ ( p − − Q ) δ [ s − Q e n ( X )] (cid:88) s Tr (cid:20) n/ u s ( p ) u s ( p ) (cid:21) = δ ( s − s min ) , (5.7)where we have used that the trace of the polarization sum equals p − and have integratedall delta functions except the one with the measurement. The color trace cancels the– 17 – /N c prefactor, and the on-shell condition implies p + = m /Q , such that for the -particlecollinear measurement we get e J ( X ) = p + Q = m Q , e τ ( X ) = p + Q − m p − = 0 , (5.8)which correspond to e min . To include the wave-function renormalization at O ( α s ) one onlyneeds to multiply this result by Z ξ . pℓℓ + p + Figure 3 . Virtual diagrams contributing to the jet function at O ( α s ) . The contribution from virtual gluons can be carried out for the two massive schemessimultaneously since the phase-space integration is identical to the tree-level computation.There are two diagrams contributing, as shown in Fig. 3, which yield the same result, sowe will compute only one of them which will be multiplied by a factor of . Pulling out acollinear gluon field from the Wilson line and using the Feynman rules for massive collinearquarks we obtain the following integral for the leftmost diagram: J virt1 = − iC F g s ˜ µ ε (cid:90) d d (cid:96) (2 π ) d ¯ n ( p + (cid:96) )(¯ n(cid:96) ) (cid:96) [( p + (cid:96) ) − m ] ≡ − iC F g s ( I + p − I ) , (5.9)where ˜ µ = µ e γ E / (4 π ) , the factor of comes from the product ¯ nn and the Casimir C F from the color trace with two Gell-Mann matrices. We are left with two master integrals I and I that can be solved using Feynman parameters for the former I = (cid:90) d x (cid:90) d d (cid:96) (2 π ) d ˜ µ ε ( (cid:96) − x m ) = i Γ( ε ) e εγ E π (cid:90) d x (cid:18) xmµ (cid:19) − ε = i (4 π ) Γ( ε ) e εγ E (1 − ε ) (cid:18) mµ (cid:19) − ε . (5.10)and with a combination of Feynman and Georgi parameters for the latter I = 2 (cid:90) ∞ d λ (cid:90) d x (cid:90) d d (cid:96) (2 π ) d (1 − x )˜ µ ε [ (cid:96) − x m − x (1 − x ) λ ] = − i µ ε e γ E (4 π ) Γ(1 + ε ) (cid:90) d x x − − ε (cid:90) ∞ d λ (cid:26) x (1 − x ) p − (cid:20) λ + m x (1 − x ) p − (cid:21)(cid:27) − − ε (5.11) = − i (4 π ) Γ( ε ) e γ E p − (cid:18) mµ (cid:19) − ε (cid:90) d x x − − ε = i (4 π ) Γ( ε )2 εp − (cid:18) mµ (cid:19) − ε . Adding those two results we find a closed expression for J virt1 : J virt1 = α s C F π e εγ E Γ(1 + ε ) ε (1 − ε ) (cid:18) mµ (cid:19) − ε (5.12) = α s C F π (cid:26) ε + 2 ε (cid:20) − log (cid:18) mµ (cid:19)(cid:21) + 4 + π − (cid:18) mµ (cid:19) + 2 log (cid:18) mµ (cid:19)(cid:27) . – 18 –nterestingly, this result is zero in the massless limit (which has to be taken before expandingin ε ). Therefore, using dimensional regularization, only real-radiation diagrams contribute.The m appearing in Eq. (5.12) is, strictly speaking bare, but since we limit our computationto O ( α s ) we can safely take it as the pole mass as the difference between these two is ahigher order correction. Implementing this result to the jet function computation andintegrating the real momentum results in adding a factor of δ ( e n − e min ) . Multiplying by ,expanding in ε and adding the wave-function renormalization, which is obviously a virtualcontribution, we obtain J virt n ( s, µ ) = α s C F π δ ( s − s min ) (cid:20) ε + 1 ε − ε log (cid:16) mµ (cid:17) +4+ π − (cid:16) mµ (cid:17) +4 log (cid:16) mµ (cid:17)(cid:21) , (5.13)which is the final result of this section. Equation (5.13) should be valid also for SCET-IItype observables since the 1-particle phase space is not yet afflicted by rapidity divergences. pq + +( a ) ( b ) ( c ) Figure 4 . Real-radiation diagrams contributing to the jet function at O ( α s ) . Since the phase-space integrals with two particles do not fully collapse with the Diracdelta functions, the real radiation contributions differ depending on the collinear measure-ment. The diagrams that contribute at O ( α s ) are shown in Fig. 4, where we have omittedthe term in which both gluons are radiated from the Wilson line since it vanishes. Diagrams(a) and (b) give identical contributions and therefore we will compute one of them whichwill be multiplied by a factor of . For all the real contributions label-momentum conser-vation implies (cid:126)q ⊥ = − (cid:126)p ⊥ and q − = Q − p − , which together with the Heaviside functionin Eq. (5.6) sets the integration limits for p − between and Q . For the first diagram,after integrating the gluon momenta with the delta functions and carrying out the angularperpendicular integration one gets J real a ( s, µ ) = 4 α s C F Q ˜ µ ε (4 π ) − ε Γ(1 − ε ) (cid:90) Q d p − p − Q − p − | (cid:126)p ⊥ | − ε d | (cid:126)p ⊥ | m ( Q − p − ) + Q | (cid:126)p ⊥ | δ ( s − Q e n ) . (5.14)Since this diagram involves gluons radiated from Wilson lines, one expects /ε poles,implying also harder integrals. On the other hand, since the corresponding Feynman ruleis simpler, the result is also shorter. For the second (symmetric) diagram, which does not– 19 –eed a factor of two, the result reads J real b ( s, µ ) = α s C F Q e εγ E µ ε (2 π )Γ(1 − ε ) (cid:90) Q d p − δ ( s − Q e n ) ( Q − p − )( p − ) | (cid:126)p ⊥ | − ε d | (cid:126)p ⊥ | m ( Q − p − ) + Q | (cid:126)p ⊥ | (5.15) × (cid:20) − ε )( | (cid:126)p ⊥ | + m )( p − ) − − ε ) m Q + 4(2 − ε ) m Qp − (cid:21) . Since Feynman rules are more cumbersome for gluons radiated from a massive quark, theresult is lengthier. On the other hand, since these correspond to boosted QCD processessingle /ε poles are expected to appear, meaning that no special treatment for the integralis necessary. To solve the integrals in the previous section, the thrust measurement must be expressedin terms of p − and | (cid:126)p ⊥ | . Since the gluon is massless and the quark has mass m we have Q τ = Q (cid:18) p + + q + − m p − (cid:19) = Q | (cid:126)p ⊥ | (cid:18) p − + 1 Q − p − (cid:19) = Q | (cid:126)p ⊥ | ( Q − p − ) p − , (5.16)which is mass independent. With this result we can use the measurement delta function tointegrate | (cid:126)p ⊥ | δ ( s − Q τ ) = ( Q − p − ) p − | (cid:126)p ⊥ | Q δ (cid:18) | (cid:126)p ⊥ | − (cid:112) ( Q − p − ) s p − Q (cid:19) . (5.17)Switching variables to p − = Qx we find the following 1-dimensional integral for J a : J real a,P ( s, µ ) = C F α s e εγ E πm Γ(1 − ε ) (cid:18) sµ (cid:19) − ε (cid:90) d x x − ε (1 − x ) − − ε − x (cid:0) − sm (cid:1) . (5.18)The complication arises because the integral diverges as /ε and contains distributions.The divergence comes from the (1 − x ) − − ε factor, but the subtraction around x = 1 behavesas /s , which combined with the s − ε prefactor implies a new divergence and invalidatesthe subtraction. This pathological behavior is usual in two-loop computations involvingdouble integrals, and the standard way to solving it is using sector decomposition [66].To do so, one needs to get rid of distributions by considering the cumulative jet function,which converts Eq. (5.18) into a double integral. In Appendix A we show how to use thisgeneral method to solve the integral, and follow in this section an easier, albeit less general,procedure. Before that, we solve the integral for the massless case, which is not affected bythe problem just described, and is valid for 2-jettiness and thrust: J real a,m =0 ( s, µ ) = − C F α s e γ E (2 π ) µ (cid:18) sµ (cid:19) − − ε Γ(2 − ε ) ε Γ(2 − ε ) (5.19) = C F α s (2 π ) (cid:34)(cid:18) ε + 1 ε + 2 − π (cid:19) δ ( s ) − µ (cid:18) µ s (cid:19) + (cid:18) ε + 1 (cid:19) + 1 µ (cid:18) µ log( s/µ ) s (cid:19) + (cid:35) . – 20 –n the second line we have expanded the result around ε = 0 to obtain distributions usingthe identity x − ε = 1 ε δ ( x ) + (cid:88) n =0 ε n n ! (cid:20) log n ( x ) x (cid:21) + . (5.20)For the m > case we can transform the integral using hypergeometric function iden-tities. Since these special functions will appear also in Sec. (8), we remind here its integraldefinition (the hypergeometric function is symmetric with respect to its first two argu-ments): F ( a, b, c, z ) = Γ( c )Γ( b )Γ( c − b ) (cid:90) d x x b − (1 − x ) c − b − (1 − zx ) − a (5.21) = (1 − z ) c − b − a F ( c − a, c − b, c, z ) , where the second line is the so called Euler transformation. The integral in Eq. (5.18) isalready into this canonical form and therefore we can write J real a,P ( s, µ ) = − Γ(3 − ε ) C F α s (2 π ) m ε Γ(3 − ε ) (cid:18) sµ (cid:19) − ε F (cid:18) , − ε, − ε, − sm (cid:19) (5.23) = − Γ(3 − ε ) C F α s (2 π ) m ε Γ(3 − ε ) (cid:18) sm (cid:19) − − ε (cid:18) µ m (cid:19) ε F (cid:18) − ε, − ε, − ε, − sm (cid:19) = C F α s (2 π ) m Γ(1 − ε ) (cid:18) sm (cid:19) − − ε (cid:18) µ m (cid:19) ε (cid:90) d x (1 − x ) − ε x − − ε (cid:20) − x (cid:18) − sm (cid:19)(cid:21) − ε , where in the second step we have used Euler’s identity and in the third we write thehypergeometric function back as an integral. The integration in the last term can be easilyexpanded in ε using Eq. (5.20), and defining ˜ s ≡ s/m we have I (˜ s ) ≡ (cid:90) d x (1 − x ) − ε x − − ε [1 − x (1 − ˜ s )] − ε = − ε − ˜ s (cid:90) d x [2 − (2 − ˜ s ) x ][1 − (1 − ˜ s ) x ] (5.24) + ε (cid:90) d x − x ) log[1 − x (1 − ˜ s )] + ˜ sx [2 − (2 − ˜ s ) x + 2] log( x ) − (1 − x ) log(1 − x ) x [1 − (1 − ˜ s ) x ] = − ε + ˜ s [1 − ˜ s +(2 − ˜ s ) log(˜ s )](1 − ˜ s ) + εf (˜ s ) ≡ − ε + ˜ s f (˜ s ) + εf (˜ s ) + O ( ε ) , with f ( s ) a function involving a dilogarithm. This result can be reexpanded in ε togetherwith the prefactor ˜ s − − ε , responsible for the appearance of distributions, finding then ˜ s − − ε I (˜ s ) = 2 ε δ (ˆ s ) − ε (cid:18) s (cid:19) + + 2 (cid:18) log(ˆ s )ˆ s (cid:19) + − f (0) δ (ˆ s ) + f (ˆ s ) + O ( ε ) . (5.25) This property can be easily shown as follows: switching variables x → − x and rearranging terms onefinds F ( a, b, c, z ) = (1 − z ) − a F (cid:16) a, c − b, c, zz − (cid:17) = (1 − z ) − b F (cid:16) c − a, b, c, zz − (cid:17) , (5.22)where the second term is obtained using the symmetry a ↔ b on the first equality. Using the first relationfollowed from the second one arrives to the second line of Eq. (5.21). Alternatively one can use the Mathematica package HypExp [67] to expand directly the hypergeometricfunction. – 21 –herefore one only needs f (0) , which takes a simple form f (0) = (cid:90) d x log(1 − x ) x = − π . (5.26)Putting all partial results together and expanding in ε we get the following expression J real a,P ( s, µ ) = C F α s π (cid:26) δ ( s ) (cid:20) ε + 1 ε log (cid:16) mµ (cid:17) + log (cid:16) mµ (cid:17) + π (cid:21) + 2 µ (cid:20) µ log( s/µ ) s (cid:21) + − (cid:20) ε + 2 log (cid:16) mµ (cid:17)(cid:21) µ (cid:18) µ s (cid:19) + − s − m − s − m ( s − m ) log (cid:18) sm (cid:19)(cid:27) . (5.27)The result is divergent for s → m , although at that kinematic point there is no physicalphenomenon that implies a singularity. We therefore expect that the singularity will cancelwhen adding together all real-radiation diagrams.For the cut self-energy diagram in Fig. 4 (c), performing the same change of variableas in Eq. (5.18) we arrive at J real b,P ( s, µ ) = C F α s e εγ E (2 π )Γ(1 − ε ) (cid:18) sµ (cid:19) − ε (5.28) × (cid:90) d x x − ε (1 − x ) − ε [ s (1 − x ) + xm ] { (1 − ε )(1 − x ) xs − m [2(1 − x ) − (1 − ε ) x ] } . To see how the /ε divergence occurs, we compute first the massless limit of J real b , for whichwe get J real b,m =0 ( s, µ ) = C F α s (1 − ε ) e εγ E (2 π ) µ Γ(1 − ε ) (cid:18) sµ (cid:19) − − ε (cid:90) d x x − ε (1 − x ) − ε (5.29) = C F α s e εγ E Γ(2 − ε )(4 π ) µ Γ(2 − ε ) (cid:18) sµ (cid:19) − − ε = C F α s π (cid:20) µ (cid:18) µ s (cid:19) + − (cid:18) ε + 1 (cid:19) δ ( s ) (cid:21) . At the light of this result one can realize that switching variables to x = ys/m exposesthe divergence, factoring it out front the integral: J real b,P ( s, µ ) = C F α s (2 π )Γ(1 − ε ) µ (cid:18) m e γ E µ (cid:19) ε (cid:18) sµ (cid:19) − − ε (cid:90) m s d y y − ε (cid:0) − sm (cid:1) − ε (cid:2) y (cid:0) − sm (cid:1)(cid:3) (5.30) × (cid:20) (1 − ε ) y (1 + y ) s m − (1 − ε ) y s m + 2 sym − (cid:21) = C F α s π (cid:26) δ ( s ) ε Γ(1 − ε ) (cid:18) m e γ E µ (cid:19) ε (cid:90) ∞ d y y − ε (1 + y ) + 1 µ (cid:18) µ s (cid:19) + × (cid:90) m s d y − sm (cid:2) y (cid:0) − sm (cid:1)(cid:3) (cid:20) y (1 + y ) s m − y s m + 2 sym − (cid:21)(cid:27) = C F α s π (cid:26)(cid:20) ε − (cid:16) mµ (cid:17)(cid:21) δ ( s ) − µ (cid:18) µ s (cid:19) + + 12( s − m ) (cid:20) s − m s + 11 m − m ( s − m ) log (cid:16) sm (cid:17)(cid:21)(cid:27) . – 22 –n the one-to-last step we have used Eq. (5.20) to partially expand in ε and in the last stepwe use Eq. (3.26) of Ref. [36]. The Dirac delta function δ ( s ) sets the upper integrationlimit to infinity and s = 0 in the integrand, becoming the integral so simple that no furtherexpansion in ε is necessary for this term. For the contribution proportional to the plusdistribution we can set ε = 0 right away, and solve the integral with standard methods. Inthe last step we have consistently expanded in ε the full result. The expression is againdivergent as s → m , but as anticipated, the full real-radiation contribution is regular inthis limit: J real P ( s, µ ) = C F α s π (cid:26)(cid:20) ε + 1 ε + 2 ε log (cid:16) mµ (cid:17) + π − (cid:16) mµ (cid:17) + 2 log (cid:16) mµ (cid:17)(cid:21) δ ( s ) − (cid:20) ε + 1 + 2 log (cid:16) mµ (cid:17)(cid:21) µ (cid:18) µ s (cid:19) + + 4 µ (cid:20) µ log( s/µ ) s (cid:21) + (5.31) + s − m s − m ) − s (2 s − m )( s − m ) log (cid:16) sm (cid:17)(cid:27) . For completeness, we also provide the real-radiation contribution for the massless case,which coincides with the full jet function. Adding the tree-level result we recover the knownresult J m =0 = δ ( s ) − C F α s e γ E (4 π ) µ (cid:18) sµ (cid:19) − − ε (4 − ε )Γ(2 − ε ) ε Γ(2 − ε ) (5.32) = δ ( s ) + C F α s π (cid:26) δ ( s ) (cid:18) ε + 3 ε + 7 − π (cid:19) − ε µ (cid:18) µ s (cid:19) + + 4 µ (cid:20) µ log( s/µ ) s (cid:21) + (cid:27) . Let us express the jettiness measurement in terms of minus and perpendicular components: Q τ = Q ( p + + q + ) = Q (cid:18) | (cid:126)p ⊥ | + m p − + | (cid:126)p ⊥ | Q − p − (cid:19) = Q | (cid:126)p ⊥ | + m Q ( Q − p − )( Q − p − ) p − , (5.33)which can be used to solve the measurement delta function for the magnitude of the per-pendicular momentum δ ( s − Q τ J ) = ( Q − p − ) p − | (cid:126)p ⊥ | Q δ (cid:18) | (cid:126)p ⊥ | − (cid:112) ( Q − p − )( sp − − Qm ) Q (cid:19) . (5.34)The argument of this delta function can be zero only if sp − > Q m , what sets the lowerlimit of integration. Therefore, changing variables to p − = Q (1 − x ) we obtain for thediagram in which the gluons are radiated from the Wilson line and the quark particle the– 23 –ollowing result J real a,J ( s, µ ) = C F α s (2 π )Γ(1 − ε ) µ ε e εγ E s − m (cid:90) − m s d x (1 − x ) x − − ε [(1 − x ) s − m ] − ε = C F α s (2 π )Γ(1 − ε ) s ε µ ε e εγ E ( s − m ) ε (cid:90) d y (1 − y ) − ε (cid:18) y − − ε − y − ε s − m s (cid:19) = − C F α s e εγ E (2 π ) µ Γ(1 − ε ) ε Γ(2 − ε ) (cid:18) sµ (cid:19) − ε (cid:18) s − m µ (cid:19) − − ε [ s (1 − ε ) − ε m ] (5.35) = C F α s π (cid:26) δ ( s − m ) (cid:20) ε + 1 ε log (cid:16) mµ (cid:17) + log (cid:16) mµ (cid:17) − π (cid:21) − s − log (cid:0) sm (cid:1) ( s − m ) − (cid:20) ε + 2 log (cid:16) mµ (cid:17)(cid:21) µ (cid:18) µ s − m (cid:19) + + 2 µ (cid:20) µ log[( s − m ) /µ ] s − m (cid:21) + (cid:27) , where in the second line we have switched variables to x = y (1 − m /s ) . Performing thesame change of variables in the diagram in which the gluons are radiated from both quarklines one gets J real b,J ( s, µ ) = C F α s (2 π )Γ(1 − ε ) µ ε e εγ E ( s − m ) (cid:90) − m s d x x − ε [(1 − x ) s − m ] − ε (5.36) (cid:8) (1 − ε ) (cid:2) xs + (2 − x ) m (cid:3) − (2 − ε ) m (cid:9) = C F α s π ( s − m ) − − ε s Γ(1 − ε ) (cid:18) s e γ E µ (cid:19) ε (cid:90) d y y − ε (1 − y ) − ε (cid:2) ( s − m ) y (1 − ε ) − m s (cid:3) = C F α s π ( s − m ) − − ε s (cid:18) s e γ E µ (cid:19) ε Γ(1 − ε )Γ(2 − ε ) [(1 − ε )( s − m ) − m s ]= C F α s π (cid:26)(cid:20) ε + 2 +2 log (cid:16) mµ (cid:17)(cid:21) δ ( s − m ) − µ (cid:18) µ s − m (cid:19) + + 5 s − m s (cid:27) . where we have carried out the same manipulations as in Eq. (5.35). Adding the two resultswe obtain the total contribution for 2-jettiness: J real J ( s, µ ) = C F α s π (cid:26)(cid:20) ε + 1 ε + 2 ε log (cid:16) mµ (cid:17) + 2 − π (cid:16) mµ (cid:17) + 2 log (cid:16) mµ (cid:17)(cid:21) δ ( s ) − (cid:20) ε + 1 + 2 log (cid:16) mµ (cid:17)(cid:21) µ (cid:18) µ s (cid:19) + + 4 µ (cid:20) µ log( s/µ ) s (cid:21) + (5.37) + s − m s − s − m log (cid:16) sm (cid:17)(cid:27) . Adding together the contributions from real and virtual corrections one obtains the com-plete jet function. The divergences are now entirely of UV origin and can be renormalizedmultiplicatively (with a convolution). Since in either massive scheme these are the sameas for massless quarks, the renormalization factor is the same, along with the anoma-lous dimension. Therefore we quote directly the result for the renormalized jet functions– 24 – - - - - - -
101 Thrust2 - Jettiness (a) - - - - - -
50 Thrust2 - Jettiness (b)
Figure 5 . Massive corrections to the jet function. Panels (a) and (b) show the differential andcumulative jet functions, respectively. We show with solid lines the non-distributional functions J nd and Σ nd for P- (red) and M- (blue) schemes. The differential J m function is shown multiplied − as a green dashed line in panel (a) (for x > it is common to both schemes), while − Σ m is shownin panel (b) with red and blue dashed lines for P- and M-schemes, respectively. [ α s ≡ α s ( µ ) ]: J Pn ( s, µ ) = δ ( s ) + α s C F π (cid:26)(cid:20) (cid:16) mµ (cid:17) + 8 log (cid:16) mµ (cid:17) + 4 + π (cid:21) δ ( s )+ 8 µ (cid:20) log( s/µ ) s/µ (cid:21) + − µ (cid:20) (cid:16) mµ (cid:17)(cid:21)(cid:18) µ s (cid:19) + + s − m ( s − m ) − s (2 s − m )( s − m ) log (cid:16) sm (cid:17)(cid:27) ,J Jn ( s + m , µ ) = δ ( s ) + α s C F π (cid:26)(cid:20) (cid:16) mµ (cid:17) + 8 log (cid:16) mµ (cid:17) + 8 − π (cid:21) δ ( s )+ 8 µ (cid:20) log( s/µ ) s/µ (cid:21) + − µ (cid:20) (cid:16) mµ (cid:17)(cid:21)(cid:18) µ s (cid:19) + + s ( m + s ) − s log (cid:16) sm (cid:17)(cid:27) . (5.38)From these equations one can easily read out the functional form of J nd ( x ) defined inEq. (4.9). With some manipulations one can also figure out expressions for J m ( x ) definedin the same equation: J m ( x ) = A S δ ( x ) − (cid:18) x (cid:19) + + 4 (cid:20) log( x ) x (cid:21) + ,J J nd ( x ) = x ( x + 1) − x log(1 + x ) , (5.39) J P nd ( x ) = x − x − − x (2 x − x − log( x ) , with A J = 2 π / and A P = 4 π / − . We shall see that J nd ( x → ∞ ) = − J m ( x ) for both schemes, and show this behavior graphically in Fig. 5(a). This is expected sinceit corresponds to taking the massless limit, and therefore mass corrections should vanishsuch that the jet function becomes equal to the (renormalized) massless result of Eq. (5.32).– 25 –ince J nd contains distributions in this limit, it is advantageous to work with the cumulativejet function Σ n ( s c ) ≡ (cid:90) s c d s J n ( s ) , (5.40)which is a regular function. Likewise, one can define the cumulative functions for J nd and J m , which are also shown in Fig. 5(b). The result can be obtained easily and involvespolylogarithms: Σ Jn ( s + m , µ ) = 1 + α s C F π (cid:26) (cid:16) mµ (cid:17) + 8 log (cid:16) mµ (cid:17) + 8 − π (cid:16) sm (cid:17) − ss + m + 4 log (cid:16) sµ (cid:17) − (cid:20) (cid:16) mµ (cid:17)(cid:21) log (cid:16) sµ (cid:17) + 4 Li (cid:16) − sm (cid:17)(cid:27) , (5.41) Σ Pn ( s, µ ) = 1 + α s C F π (cid:26) (cid:16) mµ (cid:17) + 8 log (cid:16) mµ (cid:17) + 4 + π − (cid:20) (cid:16) mµ (cid:17)(cid:21) log (cid:16) sµ (cid:17) + 4 log (cid:16) sµ (cid:17) + 4 Li (cid:16) − sm (cid:17) + 3 ss − m + ( s − m ) s ( s − m ) log (cid:16) sm (cid:17)(cid:27) . As expected, in the m → limit both results reduce to the (same) massless cumulative jetfunction Σ m =0 n ( s, µ ) = 1 + α s ( µ ) C F π (cid:20) − π − (cid:16) sµ (cid:17) + 2 log (cid:16) sµ (cid:17)(cid:21) . (5.42)To take the derivative one needs to recall that the jet function has support only for positive s ,such that it is effectively proportional to an (implicit) Heaviside function θ ( s ) . Using thefollowing relations:d θ ( x ) d x = δ ( x ) , dd x [ θ ( x ) log n ( x )] = n (cid:20) log n − ( x ) x (cid:21) + , (5.43)one readily arrives at Eq. (5.32). For s > one can expand around m = 0 to find thefollowing compact series J Pn ( s > , µ ) = α s C F π s (cid:26) (cid:16) sµ (cid:17) − (cid:88) i =1 (cid:20) − i − (4 + i − i ) log (cid:16) sm (cid:17)(cid:21)(cid:18) m s (cid:19) i (cid:27) ,J Jn ( s > , µ ) = α s C F π s (cid:26) (cid:16) sµ (cid:17) − (cid:88) i =1 ( − i (cid:18) i + 1 + 4 i (cid:19)(cid:18) m s (cid:19) i (cid:27) , (5.44)with similar results for the cumulative jet functions. Since individual pieces of the P-schemethrust jet function have divergences at s = m it is convenient to compute the expansionof J P nd ( x ) around x = 1 , which can be cast as J P nd ( x ) = − (cid:88) i =0 (1 − x ) i i ( i + 1)( i + 2)( i + 3) . (5.45)– 26 – Fixed-order Prediction in SCET
Inserting our result for the jet function into the SCET factorization theorem of Eq. (4.1),setting all renormalization scales equal and using the known results for the hard and softfunction at one loop H ( Q, µ ) = 1 + α s ( µ ) C F π (cid:20) π −
16 + 12 log (cid:18) Qµ (cid:19) − (cid:18) Qµ (cid:19)(cid:21) , (6.1) S ( (cid:96), µ ) = δ ( (cid:96) ) + α s ( µ ) C F π (cid:26) π δ ( (cid:96) ) − µ (cid:20) µ log( (cid:96)/µ ) (cid:96) (cid:21) + (cid:27) . one arrives at the fixed-order prediction for the partonic singular terms of the P-schemethrust differential cross section: σ d ˆ σ SCETFO d τ = δ ( τ ) + α s ( µ ) C F π F SCET1 ( τ, ˆ m ) + O ( α s ) (6.2) F SCET1 ( τ, ˆ m ) = δ ( τ ) (cid:20) π − m ) + 16 log ( ˆ m ) (cid:21) − m )] (cid:18) τ (cid:19) + + 2( τ − m )( τ − ˆ m ) − τ (2 τ − m )( τ − ˆ m ) log (cid:16) τ ˆ m (cid:17) ≡ A SCET ( ˆ m ) δ ( τ ) + B SCETplus ( ˆ m ) (cid:18) τ (cid:19) + + F SCETNS ( τ, ˆ m ) . In the same way, one can get a similar expression for the cumulative distribution Σ SCET P ,which is among other things useful to take the m → limit. The differential cross sectionhas a similar structure in full QCD, although it is different for vector and axial-vectorcurrents as discussed in Ref. [36], and for P-scheme thrust takes the following form σ C dˆ σ C FO d τ = R C ( ˆ m ) δ ( τ ) + C F α s π F QCD C ( τ, ˆ m ) + O ( α s ) , (6.3) F QCD C ( τ, ˆ m ) = A C ( ˆ m ) δ ( τ ) + B C plus ( ˆ m ) (cid:18) τ (cid:19) + + F C NS ( τ, ˆ m ) , where C = V, A labels the type of current and with R C the tree-level massive R-ratio.Analytic results for A C and B plus C can be found in Ref. [36] and we quote here the universalvalue for the latter: B C plus ( ˆ m ) = (cid:18) − β β (cid:19)(cid:20) (1 + β ) log (cid:18) β m (cid:19) − β (cid:21) , (6.4)with β = √ − m , and where the first and second line of the expression in big parenthesescorrespond to the vector and axial-vector currents, respectively. One recovers the SCETresult for small masses, A C ( ˆ m →
0) = A SCET ( ˆ m ) and B C plus ( ˆ m →
0) = B SCETplus ( ˆ m ) , andalso lim τ → F C NS (cid:16) τ, ˆ m = α √ τ (cid:17) → F SCETNS (cid:16) τ, ˆ m = α √ τ (cid:17) , (6.5) The partonic fixed-order bHQET cross section is identical to the SCET one dropping F SCETNS . In the threshold limit one gets the same result as in full QCD dropping F C NS ( τ, ˆ m ) , which is a powercorrection. – 27 – - - - - - - - Figure 6 . Comparison of the O ( α s ) correction to the differential cross section F ( τ, ˆ m ) in QCD(vector current) and SCET. We enforce the SCET power counting by scaling the reduced mass as ˆ m = α √ τ , with α = O (1) . Solid blue lines show SCET analytic results, while red dots correspondto QCD numerical predictions obtained form the computations in Ref. [36]. The numerical valuesof α are shown in the figure. with α ∼ O (1) . Since for thrust F C NS is only known numerically, in Fig. 6 we show acomparison of QCD and SCET results for the NLO corrections scaling the mass as indicatedin Eq. (6.5). Excellent numerical agreement is found as τ → for various values of α between . and . We show only the vector current as for small values of τ it is indistinguishablefrom the axial-vector one. The computation of the bHQET jet function is significantly simpler that for the SCETcounterpart since in this EFT the mass is no longer a dynamical scale and we are left withtadpole-like integrals. As an immediate consequence of that, much as it happened for themassless SCET jet function, all virtual graphs are automatically zero in dimensional regu-larization since they are scaleless (this includes the wave-function renormalization factor).We are then left with the tree-level, which is common for both massive schemes, and real-radiation diagrams. The collinear event-shape measurements are the same in SCET andbHQET, although the contribution of massive particles needs to be power expanded, suchthat using p = mv + k we obtain for thrust and 2-jettiness the following results Q ( τ Jn − ˆ m ) = p + − m Q = mv + − m Q + k + = k + , (7.1) Qτ Pn = p − + m p − = k + − m Q − m Q + k − = k + + ˆ m k − = 0 , where to get to the last expression in the second lines we use the on-shell condition forheavy quarks v · k = 0 , to be discussed later in this section. The field-theoretical definitionof the bHQET jet function can be obtained from the expression given in Eq. (5.2), and– 28 –aking into account that applying the bHQET power counting to the minus component ofmomenta one obtains δ ( ¯ P − Q ) → δ ( k − + (cid:80) q − ) the jet functions can be written as B n (ˆ s ) = (2 π ) d − Q mN C Tr (cid:104) | W † v + (0) h v + (0) δ (cid:20) ˆ s − Q m (ˆ e n − e min ) (cid:21) δ ( d − ( (cid:126) K ⊥ ) δ ( K − )¯ h v + (0) W v + (0) | (cid:105) . (7.2)Here (cid:126) K is an operator that pulls out the residual momenta of the heavy quarks and the(full) momenta of ultra-collinear particles. We have also used (cid:126)v ⊥ = (cid:126) and W v + has the samefunctional form as W n replacing the collinear gluons by ultra-collinear fields: A n → A + : W v + ( x ) ≡ ¯ P exp (cid:20) − ig (cid:90) ∞ d s ¯ n · A + (¯ n s + x ) (cid:21) , (7.3) W † v + ( x ) ≡ P exp (cid:20) ig (cid:90) ∞ d s ¯ n · A + (¯ n s + x ) (cid:21) . The bHQET phase-space integration involving a heavy quark gets also simplified, and usingagain p = mv + k one has that d d p (2 π ) d − δ ( p − m ) θ ( p ) = d p + d p − d d − (cid:126)p ⊥ π ) d − δ (cid:2) p − p + − | (cid:126)p ⊥ | − m (cid:3) θ ( p − + p + ) (7.4) = d k + d k − d d − (cid:126)k ⊥ π ) d − δ (cid:16) Qk + + m Q k − + k (cid:17) θ (cid:16) Q + k − + m Q + k + (cid:17) = d k + d k − d d − (cid:126)k ⊥ π ) d − δ (cid:16) Qk + + m Q k − (cid:17) θ ( Q ) = d k − d d − (cid:126)k ⊥ Q (2 π ) d − , where in the second line we have used Eq. (4.14) and in the third we power-count away the k in the delta function argument along with all terms but Q inside the Heaviside function. Theon-shell condition for heavy quarks written in light-cone coordinates implies v − k + + k − v + =0 , in agreement with the argument of the delta function. When comparing to Eq. (5.6) weobserve that the p − in the denominator got replaced by Q and that the p − integration isnot limited to positive values only. The phase-space integration for ultracollinear particlesstays the same as in SCET.Feynman diagrams look exactly the same in SCET and bHQET, with the replacement p → k for the heavy quark momenta. Let us compute the tree-level contribution as givenin Fig. 2, which is analogous to the corresponding SCET calculation: B tree n ( s ) = (cid:90) d k − m d d − (cid:126)k ⊥ δ ( d − ( (cid:126)k ⊥ ) δ ( k − ) δ (cid:20) ˆ s − Q m (ˆ e n − e min ) (cid:21)(cid:88) s Tr (cid:2) u s ( p ) u s ( p ) (cid:3) = δ (ˆ s ) , (7.5)where we have used that the trace of the polarization sum equals m and have integratedall delta functions except the one with the measurement. The on-shell condition k − = 0 makes both (shifted) measurements coincide at tree-level, see Eq. (7.1). There are somegeneric features to be learned from this diagram: since there is no Dirac structure in thediagram, the trace of the polarization sum will be always m at any loop order, and sincethere is always one heavy quark which brings an inverse power of Q through its phase space– 29 –ne has the following combination: Q m Q Tr (cid:2) u s ( p ) u s ( p ) (cid:3) = 1 , (7.6)which eliminates the spurious dependence on m and Q , ultraviolet scales that should notappear in EFT computations. To make this non-dependence explicit at higher orders onecan rescale the minus component of ultracollinear real particles as q − i = ( Q/m ) (cid:96) i , as weshall do in the rest of the section.We turn our attention now to real-radiation contributions, for which we can simplifythe heavy-quark propagator using v · k = 0 . We start with diagram (a) of Fig. 4, that afterapplying the bHQET Feynman rules becomes B real a (ˆ s, µ ) = 2 α s C F ( µ e γ E ) ε π Γ(1 − ε ) (cid:90) d (cid:96) − ( (cid:96) − ) | (cid:126)q ⊥ | − ε d | (cid:126)q ⊥ | v · q θ ( (cid:96) − ) δ (cid:20) ˆ s − Q m (ˆ e n − e min ) (cid:21) . (7.7)The on-shell condition on the ultra-collinear gluon momenta implies in light-cone coordi-nates: v · q = | (cid:126)q ⊥ | Q/ ( mq − ) + mq − /Q = [ | (cid:126)q ⊥ | + ( (cid:96) − ) ] /(cid:96) − . For diagram (b) we getinstead B real b (ˆ s, µ ) = − α s C F ( µ e γ E ) ε π Γ(1 − ε ) (cid:90) d (cid:96) − (cid:96) − | (cid:126)q ⊥ | − ε d | (cid:126)k ⊥ | ( v · q ) θ ( (cid:96) − ) δ (cid:20) ˆ s − Q m (ˆ e n − e min ) (cid:21) . (7.8)Let us work out the measurements for thrust and 2-jettiness Q m ( τ Jn − ˆ m ) = Qm ( q + + k + ) = Qm | (cid:126)q ⊥ | q − + mQ q − = | (cid:126)q ⊥ | (cid:96) − + (cid:96) − , (7.9) Q m τ Pn = Qm q + = Qm | (cid:126)q ⊥ | q − = | (cid:126)q ⊥ | (cid:96) − , where we have used Eq. (7.1), the on-shell condition for heavy quarks and ultra-collinearmassless gluons, and the fact that label momentum conservation implies k − = − q − . Withthis result it is very simple to solve the measurement delta function in terms of the perpen-dicular gluon momenta δ (cid:20) ˆ s − Q m ( τ Jn − ˆ m ) (cid:21) = (cid:96) − | (cid:126)q ⊥ | δ (cid:104) | (cid:126)q ⊥ | − (cid:112) ˆ s (cid:96) − − ( (cid:96) − ) (cid:105) , (7.10) δ (cid:18) ˆ s − Q m τ Pn (cid:19) = (cid:96) − | (cid:126)q ⊥ | δ (cid:16) | (cid:126)q ⊥ | − √ ˆ s (cid:96) − (cid:17) , and we will use these results to compute the jet functions in the next two sub-sections. We start with the diagram in which the gluon is radiated from the Wilson line. Switchingvariables to (cid:96) − = ˆ sx we arrive at B real a,P (ˆ s, µ ) = α s ( µ e γ E ) ε C F π ˆ s − − ε Γ(1 − ε ) (cid:90) ∞ d x x − − ε x = − α s Γ(1 + ε ) C F e εγ E πµ ε (cid:18) ˆ sµ (cid:19) − − ε = α s C F π (cid:20)(cid:18) ε + π (cid:19) δ (ˆ s ) − εµ (cid:18) µ ˆ s (cid:19) + + 4 µ (cid:18) µ log(ˆ s/µ )ˆ s (cid:19) + (cid:21) . (7.11)– 30 –ith an identical change of variables we arrive at the following result for diagram (b): B real b,P (ˆ s, µ ) = − α s ( µ e γ E ) ε C F π ˆ s − − ε Γ(1 − ε ) (cid:90) ∞ d x x − ε (1 + x ) = − α s Γ(1 + ε ) C F e εγ E πµ (cid:18) ˆ sµ (cid:19) − − ε = α s C F π (cid:20) ε δ (ˆ s ) − µ (cid:18) µ ˆ s (cid:19) + (cid:21) . (7.12)Adding both diagrams with the appropriate factors we obtain the final expression for theP-scheme hemisphere jet function: B Pn (ˆ s, µ ) = − α s Γ(2 + ε ) C F e εγ E πµ ε (cid:18) ˆ sµ (cid:19) − − ε (7.13) = α s C F π (cid:20)(cid:18) ε + 2 ε + π (cid:19) δ (ˆ s ) − µ (cid:18) ε + 1 (cid:19)(cid:18) µ ˆ s (cid:19) + + 8 µ (cid:18) µ log(ˆ s/µ )ˆ s (cid:19) + (cid:21) . The Dirac delta function in Eq. (7.10) implies that there is a solution for | (cid:126)q ⊥ | only if (cid:96) − < ˆ s , which can be implemented through a Heaviside function and bounds the upperintegration limit for (cid:96) − . With the change of variables implemented in the previous sectionthe integration limits are mapped to the interval (0 , and we get the following result fordiagram (a): B real a,P (ˆ s, µ ) = α s ( µ e γ E ) ε C F π ˆ s − − ε Γ(1 − ε ) (cid:90) d x x − − ε (1 − x ) − ε = − α s Γ(1 − ε ) C F e εγ E πµ ε Γ(1 − ε ) (cid:18) ˆ sµ (cid:19) − − ε = α s C F π (cid:20)(cid:18) ε − π (cid:19) δ (ˆ s ) − εµ (cid:18) µ ˆ s (cid:19) + + 4 µ (cid:18) µ log(ˆ s/µ )ˆ s (cid:19) + (cid:21) , (7.14)that, as expected, differs from the expression in Eq. (7.11) only in the non-divergent termof the delta-function coefficient. Similarly, we obtain for diagram (b) B real b,P (ˆ s, µ ) = − α s ( µ e γ E ) ε C F π ˆ s − − ε Γ(1 − ε ) (cid:90) d x [ x (1 − x )] − ε = − α s Γ(1 − ε ) C F e εγ E πµ Γ(2 − ε ) (cid:18) ˆ sµ (cid:19) − − ε = α s C F π (cid:20) (cid:18) ε + 2 (cid:19) δ (ˆ s ) − µ (cid:18) µ ˆ s (cid:19) + (cid:21) , (7.15)again almost identical to the corresponding P-scheme computation. Adding twice the firstdiagram plus the second we recover the known result for the 2-jettiness bHQET jet function: B Jn (ˆ s, µ ) = − α s Γ(2 − ε ) C F e εγ E πµ ε Γ(2 − ε ) (cid:18) ˆ sµ (cid:19) − − ε (7.16) = α s C F π (cid:20)(cid:18) ε + 2 ε + 4 − π (cid:19) δ (ˆ s ) − εµ (cid:18) ε + 1 (cid:19)(cid:18) µ ˆ s (cid:19) + + 8 µ (cid:18) µ log(ˆ s/µ )ˆ s (cid:19) + (cid:21) . Both schemes have the same divergent structure and hence their anomalous dimension, asexpected, is the same. Furthermore, the difference between the respective delta coefficientsis the same as the same difference for the SCET jet functions. This result was also expectedsince both theories should smoothly match in the bHQET limit.– 31 –
RG Evolution of the SCET Jet Function
In this section we solve the renormalization group equation for the non-distributional partof the jet function for thrust and 2-jettiness. This amounts to finding an analytic expressionfor the function I np defined in the last line of Eq. (4.10). Even though the result for I J np hasbeen already worked out in Ref. [38], we present here the main steps to find the solution asthey are illustrative. Using the rightmost integral expression of the bottom line in Eq. (4.10)we find I J np (˜ ω, y ) = 1Γ( − ˜ ω ) (cid:90) d z (1 − z ) − − ˜ ω (cid:20) zy (1 + zy ) − zy ) zy (cid:21) . (8.1)While the first term in the last line of Eq. (8.1) is already in a canonical form such thatEq. (5.21) can be directly applied, the second contains a logarithm. Expressing it as anintegral log(1 + zy ) zy = (cid:90) d x
11 + xzy , (8.2)brings the second term also into to a canonical form that we can easily integrate, finding (cid:90) d z (1 − z ) − − ˜ ω log(1 + zy ) zy = (cid:90) d x (cid:90) d z (1 − z ) − − ˜ ω
11 + xzy (8.3) = − ω (cid:90) d x F (1 , , − ˜ ω, − xy ) = − ω F (1 , , , , − ˜ ω, − y ) , where in the last step we have used the integral representation of the F function: F ( a , a , a , b , b , z ) = Γ( b )Γ( a )Γ( b − a ) (cid:90) d t t a − (1 − t ) b − − a F ( a , a , b , tz ) , (8.4)with a = a = a = 1 , b = 1 − ˜ ω and b = 2 . After adding the result for the first term wefind an expression slightly simpler than that quoted in Ref. [38], although fully equivalent: I J np (˜ ω, y ) = 1Γ(2 − ˜ ω ) (cid:2) y F (2 , , − ˜ ω, − y ) − (1 − ˜ ω ) F (1 , , , , − ˜ ω, − y ) (cid:3) , (8.5)which has a smooth ˜ ω → limit. For a numerical implementation, one can use standardroutines to evaluate F hypergeometric functions in programming languages such as Math-ematica, Fortran, Python, or C ++ . For F there are built-in routines in Mathematica andPython, while for other languages one can use a numerical integration over F as shownin Eq. (8.4).For P-scheme thrust we can write the logarithm as a derivative to bring all terms intoa canonical form: I P np (˜ ω, y ) = 1Γ( − ˜ ω ) (cid:90) d z (1 − z ) − − ˜ ω (cid:20) zy − − yz ) + 2 zy (2 zy − − yz ) log( zy ) (cid:21) (8.6) − ˜ ω ) (cid:90) d z (1 − z ) − − ˜ ω (cid:26) zy − − yz ) + 2 yz (2 zy − − yz ) (cid:20) log( y ) + dd ε z ε (cid:21)(cid:27) ε → . For y > each of the terms in the integral diverges when z = 1 /y . We can regularize thedivergence adding a small imaginary part y → y + i(cid:15) . This makes each integral complex,– 32 –lthough the sum is real when (cid:15) → . To express our result in terms of a minimal set ofhypergeometric functions, we use the following identity: ( c − b ) F ( a, b − , c, z ) + ( c − z − F ( a, b, c − , z ) (8.7) + [ z ( a − c + 1) + b − F ( a, b, c, z ) = 0 . Furthermore, one can use an additional identity to make the final result manifestly real alsofor the case y > F ( a, b ; c ; z ) = Γ( c )(1 − z ) c − a − b Γ( a + b − c ) F ( c − a, c − b, c − a − b, − z )Γ( a )Γ( b ) (8.8) + Γ( c )Γ( c − a − b ) F ( a, b, a + b − c + 1 , − z )Γ( c − a )Γ( c − b ) . After solving all integrals, recursively applying Eq. (8.7) and transforming the hypergeo-metric functions using Eq. (8.8), one arrives at the second important result of this article: I P np (˜ ω, y ) = 4 y − ω + 5) y − ˜ ω (3 ˜ ω + 7)(1 − y ) (1 + ˜ ω )Γ(1 − ˜ ω ) (cid:26) ˜ ω dd ε F (1 , ε, ω, − y )+ (cid:18) ˜ ω log( y ) − ˜ ωH − ˜ ω − − ω − ˜ ω (cid:19) F (1 , , ω, − y ) (cid:27) ε → (8.10) − (1 − ˜ ω )˜ ω (3 ˜ ω − y + 7)[ H − ˜ ω − log( y )] − ˜ ω [3 ˜ ω ( y + 1) − y + 14] − y + 7Γ(2 − ˜ ω )(1 − y ) , with H a the harmonic number, which for non-integer values of a can be expressed in termsof the digamma function: H a = ψ (0) (1 + a ) + γ E . Equation (8.10) has been cast in away in which the no-resummation limit ˜ ω → is smooth. The singularities that appearin individual terms of J P nd for x = 1 manifest themselves now as a double pole in I P np at y = 1 , which however is fictitious, as the result is indeed smooth at this value. To solve thisproblem in numerical implementations we provide in Sec. 8.2 an expansion of this resultaround y = 1 at arbitrarily high order. The result in Eq. (8.10) is adequate for a numericalimplementation since the derivative with respect to ε can be taken numerically through finitedifferences. It can be also performed analytically, using Eq. (8.8) in F (1 , ε, ω, − y ) and the following identitydd ε F (1 , ε, − ˜ ω + ε, y ) (cid:12)(cid:12)(cid:12)(cid:12) ε → = − ˜ ω y (1 − y ) − − ˜ ω (1 − ˜ ω ) F (1 − ˜ ω, − ˜ ω, − ˜ ω, − ˜ ω, − ˜ ω, y ) , (8.11) The result as given in this equation is very convenient for a numerical implementation, since one onlyneeds to evaluate two hypergeometric functions (which might be numerically expensive) using the followingapproximations: dd ε F (1 , ε, ω, − y ) (cid:12)(cid:12)(cid:12) ε =0 (cid:39) ε [ F (1 , ε, ω, − y ) − F (1 , − ε, ω, − y )] , (8.9) F (1 , , ω, − y ) (cid:39)
12 [ F (1 , ε, ω, − y ) + F (1 , − ε, ω, − y )] , with a value of ε which can be safely taken as small as − . – 33 –o arrive at the equivalent expression: I P np (˜ ω, y ) = (8.12) ˜ ω y (1 − y ) − − ˜ ω [ 2 y (˜ ω + 5) + ˜ ω (3˜ ω + 7) − y ] F (1 − ˜ ω, − ˜ ω, − ˜ ω, − ˜ ω, − ˜ ω, y )Γ(2 − ˜ ω )(1 − ˜ ω ) − (1 − ˜ ω ) ˜ ω (3 ˜ ω − y + 7)[ H − ˜ ω − log( y ) ] − ˜ ω [ 3 ˜ ω ( y + 1) − y + 14 ] − y + 7Γ(2 − ˜ ω )(1 − y ) + [ 2 y (˜ ω + 5) + ˜ ω (3 ˜ ω + 7) − y ] { (1 − ˜ ω )[ H − ˜ ω − log( y ) ] − } F (1 , , − ˜ ω, y )Γ(2 − ˜ ω )(1 − y ) . The (1 − y ) − − ˜ ω factor and both hypergeometric functions are complex for y > but thecombination is real. To have all terms explicitly real for y > one can use the followingrelationdd ε F (1 , ε, ω, − y ) (cid:12)(cid:12)(cid:12) ε → = 1 − yy (2 + ˜ ω ) F (cid:18) , ω, ω, ω, ω, − y (cid:19) + y ˜ ω (1 − y ) 1 + ˜ ω (2 + ˜ ω ) F (2 + ˜ ω, ω, ω, ω, ω, − y ) , (8.13)which does not rely on numerical derivatives, is manifestly real for all positive values of y but is numerically unstable if y → . This poses no problem in practice, since for y < onecan simply switch to Eq. (8.12). To derive the result in Eq. (8.13) we proceed as follows:dd ε F (1 , ε, ω, − y ) (cid:12)(cid:12)(cid:12) ε → = − y dd b F (cid:18) , b, ω, − y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) b → ω = 1 y dd b (cid:20) F (cid:18) , ω, b + 1 , − y (cid:19) − F (cid:18) , b, b + 1 , − y (cid:19)(cid:21) b → ω (8.14) = 1 y dd b (cid:20) y ω F ( b, ω, b + 1 , − y ) − F (cid:18) , b, b + 1 , − y (cid:19)(cid:21) b → ω where in the first step we use Eq. (5.22), in the second we apply the chain rule on derivatives,and in the third line we use again Eq. (5.22) on the first term. Using the identitydd a F ( a, b, a + 1 , z ) = bz ( a + 1) F ( a + 1 , a + 1 , b + 1 , a + 2 , a + 2 , z ) , (8.15)in the two terms of Eq. (8.14) we arrive at the result quoted in Eq. (8.13). In Appendix Bwe present an alternative (although more complicated) expression for I P np which does notinvolve numerical derivatives and with every term manifestly real for y > . We use thisresult as an additional cross check of our analytic derivations. In any case, we shall see thatfor numerical implementations one never needs to use expressions involving hypergeometricfunctions. s = 0 For numerical implementation purposes, it might be convenient to obtain an analytic ex-pansion of I np (˜ ω, y ) around y = 0 . One can do so by using the known expansions for the– 34 –ypergeometric functions, e.g. F ( a, b, c, z ) = Γ( c )Γ( a )Γ( b ) ∞ (cid:88) i =0 Γ( a + i )Γ( b + i )Γ( c + i )Γ( i + 1) z i , (8.16)but in order to have a relation valid at arbitrarily high orders it is simpler to use theexpansion of J nd around s = 0 J P nd ( x ) = − (cid:88) i =0 [ 6 i + 7 + i (7 + 3 i ) log( x ) ] x i , (8.17) J J nd ( x ) = − (cid:88) i =0 (cid:18) i + 4 i + 1 (cid:19) ( − x ) i , on the leftmost expression of the bottom line in Eq. (4.10) and integrate analytically termby term. It turns out that one can sum up the corresponding series using Eq. (8.16) torecover an expression analytically equivalent to Eq. (8.12). The master integrals that wewill need are y ˜ ω Γ( − ˜ ω ) (cid:90) y d x ( y − x ) − − ˜ ω x i = y i Γ(1 + i )Γ(1 + i − ˜ ω ) , (8.18) y ˜ ω Γ( − ˜ ω ) (cid:90) y d x ( y − x ) − − ˜ ω x i log( x ) = y i Γ(1 + i )Γ(1 + i − ˜ ω ) [ H i − H i − ˜ ω + log( y )] , where the bottom line can be obtained from the top one acting with a derivative withrespect to i . We then arrive at I P np (˜ ω, y ) = − − ˜ ω ) (cid:88) i =0 i !(1 − ˜ ω ) i { (6 i + 7) + i (7 + 3 i )[ H i − H i − ˜ ω + log( y )] } y i , (8.19) I J np (˜ ω, y ) = − − ˜ ω ) (cid:88) i =0 (cid:18) i + 4 i + 1 (cid:19) i !(1 − ˜ ω ) i ( − y ) i . where we have used the Pochhammer symbol ( a ) n = Γ( a + n ) / Γ( a ) since it is convenientfor an optimized numerical implementation. Both series converge well for | y | < , andtherefore apply mainly in the peak of the distribution. For 2-jettiness the series can beeasily summed up using Eq. (8.16) and the series expansion for the F hypergeometricfunction: F ( a, b, c, d, e, z ) = Γ( c )Γ( a )Γ( b ) ∞ (cid:88) i =0 Γ( a + i )Γ( b + i )Γ( c + i )Γ( d + i )Γ( e + i )Γ( i + 1) z i . (8.20)For P-scheme thrust one can convert the term involving harmonic numbers into the deriva-tive of ratios of gamma functions to use Eq. (8.16) and recover the result we already ob-tained with a direct integration. The numerical implementation of Ref. [45] (which dealtwith 2-jettiness) did not use this expansion and the evaluation of the non-distributional jetfunction running was the most severe performance bottleneck for the code.– 35 – .2 Expansion around s = m The results obtained in Eqs. (8.10) and (8.12) are not useful for a numerical implementationin the vicinity of y = 1 . When y is sufficiently close to unity one can switch to a seriesexpansion to arbitrary high power using the change of variables z → − z in the rightmostexpression at the bottom of Eq. (4.10) and the following expansion: J P nd [(1 − z )(1 + y )] = − (cid:20) − z )(3 + 2 z ) z log(1 − z ) + z + 6 z (cid:21) − y (1 − z ) (cid:20) − z − z z × log(1 − z ) + 5 z + 18 z (cid:21) − y (1 − z ) (cid:26) − z )(18 − z − z ) z log(1 − z )+ 36 − z − z z (cid:27) − log(1 − z ) (cid:88) i =3 y i (1 − z ) i z i +3 [3( i + 1)( i + 2) − i + 1) z − z ] (8.21) − (cid:88) i =3 y i (1 − z ) i − z i (cid:8) i + 1)( i + 2) − ( i + 1)(3 i + 10) z − − ( i − i ] z (cid:9) + (cid:88) i =3 y i i − (cid:88) k =0 ( − k + i (1 − z ) k +1 z k +3 ( k + 1)( k + 2)(6 − i + 5 k + 4 z )( i − k − i − k − i − k ) . Terms have been combined such that the coefficient of each power in y has a well-defined z → limit and therefore we can integrate coefficient by coefficient. In practice one canintegrate each piece assuming a non-integer value of i and subsequently convert the gammafunctions that would become divergent if ˜ ω = 0 using the identity Γ( ε − n )Γ(1 + ε ) = ( − n − Γ( − ε )Γ( n + 1 − ε ) . (8.22)As expected, there are large cancellations among the various terms for a given power of y ,but when adding all contributions one gets the following nicely convergent series: I P np (˜ ω, y ) = − ω + 9)˜ ωH − ˜ ω + 7 ˜ ω + 19 ˜ ω + 22 ˜ ω + 18(˜ ω + 1)(˜ ω + 2)(˜ ω + 3)Γ(1 − ˜ ω ) (8.23) − ˜ ω Γ(1 + ˜ ω )Γ(1 − ˜ ω ) (cid:88) i =3 (1 − y ) i i − (cid:88) k =0 ( k + 1)( k + 2)![( k + 2)(5 ˜ ω + 9) − i (˜ ω + 1)]( i − k − i − k − i − k )Γ( k + ˜ ω + 4)+ Γ(1 + ˜ ω )2 Γ(1 − ˜ ω ) (cid:88) i =1 i !(1 − y ) i Γ( i + ˜ ω + 4) (cid:8) ˜ ω ( i + ˜ ω + 1)[2 i + i (˜ ω −
3) + i (˜ ω (5 ˜ ω − − − (˜ ω + 6)(5 ˜ ω + 9)] + [20 i − i ˜ ω ( i (3 ˜ ω + 7) + 7 ˜ ω + 9) + 4(5 ˜ ω + 9)] × [˜ ω ψ (0) ( i + 1) − ˜ ω ψ (0) (1 − ˜ ω ) − (cid:9) , where again special care has been taken to write the expression in a manner in which onecan set ˜ ω = 0 without any worries. The series converges well for | − y | < , and therefore,combined with the expansion worked out in the previous section, for P-scheme thrust onecan use expansions if y < . – 36 – .3 Expansion around s = ∞ Since the numerical evaluation of hypergeometric functions is slow, it is convenient to figureout another series expression (in this case, of asymptotic type) around s = ∞ , which is ofcourse tantamount to m = 0 . This limit is very relevant, since it can be applied in the tailof the distribution and almost everywhere if the heavy quark mass is much smaller than thecenter-of-mass energy, as is the case for bottom quarks at LEP. Such asymptotic expansionwas not known by the time in which Refs. [31, 32, 45] were published, and was significantlyaffecting the performance of the respective numerical codes. Even though one could, inprinciple, use known results for the asymptotic expansions of F and F hypergeometricfunctions, it is in practice simpler and more efficient to compute the series directly on itsintegral form. This is complicated since, as we shall see, the expansions involve powers of log( y ) , and so one cannot simply expand the integrand and integrate term by term, as wedid in Secs. 8.1 and 8.2. We found out that the Mellin-Barnes representation X ) ν = 12 πi (cid:90) c + i ∞ c − i ∞ d t ( X ) − y Γ( t )Γ( ν − t )Γ( ν ) , (8.24)with < c < ν , is optimal to achieve our goal [68]. After applying Eq. (8.24), theasymptotic expansion around X (cid:29) is obtained integrating by residues the poles thatappear on the real axis for t > ν (the poles for t ≤ correspond to the expansion X (cid:28) ). We work out this expansion for thrust and 2-jettiness in the rest of this section, butbefore that we note that the asymptotic expansion is well convergent if /y < , which forP-scheme thrust means that in numerical evaluations one can always use one of the threeexpansions discussed in this section and never needs to evaluate hypergeometric functionswith dedicated routines. For jettiness the same statement is almost true, except in a smallvicinity of y = 1 in which, to the best of our knowledge, no expansion can be used. We start from the integral form given in Eq. (8.1). The only complication in this case isthat we have to deal with a logarithm, which does not have the form in Eq. (8.24). However,it can be brought to the standard form using Eq. (8.2) yz log(1 + zy ) = 12 πi (cid:90) d x (cid:90) d + i ∞ d − i ∞ d t ( xyz ) − t Γ( t )Γ(1 − t ) (8.26) = 12 πi (cid:90) d + i ∞ d − i ∞ d t ( zy ) − t Γ( t )Γ(1 − t )1 − t , This representation can also be used to solve the RG equation exactly. Applying a Mellin transformationto the first line of Eq. (8.6) in the y -variable, solving the z -integral and transforming back one gets a closed(and rather short) analytic expression for I P nd in terms of MeijerG functions, I P nd (˜ ω, y ) = 3 G , , (cid:32) y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , , , , ˜ ω (cid:33) − G , , (cid:32) y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , , , , ˜ ω (cid:33) , (8.25)which are not very convenient for a direct numerical evaluation, but can be related to hypergeometricfunctions. – 37 –ith < d < . Since the denominator of the first term in Eq. (8.1) is squared, whenapplying the Mellin-Barnes representation (8.24) the first pole appears at t = 2 . This isaccompanied by an extra power of y , such that we can nicely map the poles of first terminto those of the second by shifting the integration variable t → t + 1 in the former. Afterintegrating over z we obtain I J np (˜ ω, y ) = 12 πi (cid:90) c + i ∞ c − i ∞ d t y − t Γ(1 − t ) Γ( t )Γ(1 − t − ˜ ω ) t − t + 4 t − , with < c < . The integrand has a triple pole at t = 1 and double poles at natural valuesof t larger than . We compute the triple pole by itself and treat the rest generically using Γ( ε − n ) = 1( n !) (cid:20) ε + 2( H n − γ E ) ε (cid:21) + O ( ε ) . (8.27)With this result we obtain the following asymptotic expansion: I J np (˜ ω, y ) = 1Γ(1 − ˜ ω ) (cid:88) n =1 ( − y ) − n c n [˜ ω, log( y )] , (8.28) c (˜ ω, L ) = − − ω [ H − ˜ ω − L ] − (4 + ˜ ω )[ H − ˜ ω − L ] − [1 + π − ψ (1) (1 − ˜ ω )]˜ ω ,c n> (˜ ω, L ) = (1 + ˜ ω ) n − ( n − ( n − { ( n − n − n + 1) + 6][˜ ω ( H n − − H n +˜ ω − + L ) − cos( π ˜ ω )Γ(1 − ˜ ω )Γ(1 + ˜ ω )] − ( n − n + 1) ˜ ω } , using again the Pochhammer symbol. We have written each coefficient in a form such thatthe ˜ ω → limit, relevant in the far tail of the distribution, is smooth. P-scheme thrust
Applying the Mellin-Barnes representation in Eq. (8.24) to the first line of Eq. (8.6) andintegrating over z we arrive at an expression that involves different powers of y with polesshifted accordingly. Therefore, using the same strategy as in the previous section, we canshift the integration variable by one or two units such that poles and powers of y in eachterm exactly match. This is very important, since the expansion in /y must be carried outconsistently given the large cancellations that take place among the various terms due tothe divergence at x = 1 of individual terms in J P nd (exactly as it happened for the expansionaround s = m ). After some work we arrive at the following expression: I P nd (˜ ω, y ) = 12 πi (cid:90) c + i ∞ c − i ∞ d t y − t cos( πt )Γ(1 − t ) Γ( t )Γ(1 − t − ˜ ω ) (8.29) × { (7 − t ) t [ H − t − H − t − ˜ ω + log( y )] + 6 t − } . We have already implemented a few simplifications because we assume the result is real,and therefore discarded the imaginary parts that would arise from ( − y ) − t . We have checkedthat indeed this is the case as long as one expands strictly in y without mixing any powers.Harmonic numbers are caused by the term in J P nd proportional to log( z ) . The integrandhas now double and triple poles, located at natural values of t , the latter arising precisely– 38 –ecause of the harmonic numbers. There are no poles arising from H − t − ˜ ω because ofthe corresponding gamma function in the denominator which has poles at the same values,making the ratio regular. To compute the residues of the poles we need, on top of Eq. (8.27),the following expansion H ε − n Γ( ε − n ) = 1( n !) (cid:26) − ε + 1 ε [ ψ (0) ( n ) − H n + 3 γ E ] (8.30) + 2 γ E n + 2( nγ E − nψ (0) ( n ) − n ε (cid:27) + O ( ε ) , which can be obtained from the relation between harmonic numbers and the digammafunction and a bit of algebra. Using these results we arrive at the following expression, inwhich again special care has been taken to make the ˜ ω → limit smooth: I P np (˜ ω, y ) = 1Γ(1 − ˜ ω ) (cid:88) n =1 c n [˜ ω, log( y )] y n , (8.31) c n (˜ ω, L ) = (1 + ˜ ω ) n − n − (cid:26) n + ˜ ω [ L (3 n − n ( n + ˜ ω ) − n − n ˜ ω + 7˜ ω ] × [cos( π ˜ ω )Γ(1 − ˜ ω )Γ(1 + ˜ ω ) − ˜ ω ( ψ (0) ( n ) − ψ (0) ( n + ˜ ω + 1))]+ ˜ ωn ( n + ˜ ω ) [2 Ln (3 n ( n + 2˜ ω ) − ω ) − n (˜ ω − n + 7) − ω ] − (7 − n ) n (cid:20) ˜ ω ψ (1) ( n + 1) − ˜ ωψ (1) ( n + ˜ ω ) − ˜ ωL − ˜ ω ( n + ˜ ω ) + [ ψ (0) ( n ) − ψ (0) ( n + ˜ ω + 1)][˜ ω ( ψ (0) (1 + n + ˜ ω ) − ψ (0) ( n ))+ 2 cos( π ˜ ω )Γ(1 − ˜ ω )Γ(˜ ω + 1)] (cid:21)(cid:27) . The resummed SCET cross section can be matched to full QCD such that its validity isextended beyond the peak and tail into the far tail. The usual procedure is to add in fixed-order those terms not included in the factorization theorem. For massless quarks these areusually denoted as non-singular contributions, since singular terms (that is, delta or plusfunctions) are fully accounted for in SCET. For massive quarks, terms not contained in thefactorization theorem can be singular, and therefore these will be referred to as non-SCET.In the far tail one sets all renormalization scales equal due to the large cancellations thattake place between SCET and non-SCET terms around τ ∼ / that would be spoiled byresummation. To ensure this cancellation when including hadronization effects one usuallyconvolves the added terms with the same shape function.As already explained in the introduction and discussed in further detail in Sec. 5.3, thefixed-order QCD prediction contains terms which are singular as τ approaches . The reasonis that when including quark masses there are two kind of power corrections to the leading-order EFT prediction: kinematic and massive. Both are power-counted equally in SCETand therefore, when considering the singular cross section one necessarily neglects higher– 39 –owers of m . Mass power corrections can be singular, while kinematic power correctionsare genuinely non-singular. It is in general desirable to absorb mass power corrections intothe EFT description (although this is not strictly speaking necessary), since resummationturns log n ( τ ) /τ into an integrable singularity. This prescription has been adopted alreadyfor 2-jettiness in Refs. [31, 32, 45], and here we succinctly explain how this is implemented.One can modify the SCET matrix elements to absorb all singular pieces. Since at tree-level all matrix elements are either or Dirac delta functions, one only needs to multiplythe tree-level SCET results by R C to fully account for massive power corrections at thisorder. The O ( α s ) massive corrections can be implemented modifying the hard function,that we write as H C ( Q, ˆ m, µ ) = R C ( ˆ m ) (cid:26) α s ( µ ) C F π (cid:20) π − h Cm ( ˆ m )+12 log (cid:18) Qµ (cid:19) − (cid:18) Qµ (cid:19)(cid:21)(cid:27) , (9.1)such that it includes the tree-level mass modification, and the jet function (at this orderone cannot have mass corrections from soft dynamics). For the latter one only needs tomodify J m , the mass correction with distributions, which we write as J Cm ( x, ˆ m ) = [ j Cm ( ˆ m ) + A P ] δ ( x ) + 4 (cid:20) log( x ) x (cid:21) + − [ 1 − y Cm ( ˆ m ) ] (cid:18) x (cid:19) + . (9.2)Corrections coming from B C plus are easy to implement, since they only come from the jetfunction (the hard function does not contain any distribution). It is convenient to define A C ( ˆ m ) = A SCET ( ˆ m ) + A NS C ( ˆ m ) and B C plus ( ˆ m ) = B SCET ( ˆ m ) + B C NS ( ˆ m ) , with A SCET ( ˆ m ) = 2 π − A P + 4 log( ˆ m ) + 16 log ( ˆ m ) . (9.3)Implementing these modifications into the SCET factorization theorem with fixed scalesone arrives at y Cm ( ˆ m ) = 12 R C ( ˆ m ) { [1 − R C ( ˆ m )] B S ( ˆ m ) + B C NS ( ˆ m ) } ,R C ( ˆ m ) [ h Cm ( ˆ m ) + 2 j Cm ( ˆ m ) ] = 2 { B S ( ˆ m )[1 − R C ( ˆ m )] + B C NS ( ˆ m ) } log( ˆ m ) (9.4) + [1 − R C ( ˆ m )] A S ( ˆ m ) + A C NS ( ˆ m ) ≡ H C corr ( ˆ m ) . Since one cannot compute separately h Cm and j Cm , we make an ansatz and split it accordingto a parameter ξh Cm ( ˆ m ) = 1 − ξR C ( ˆ m ) H C corr ( ˆ m ) , j Cm ( ˆ m ) = ξ R C ( ˆ m ) H C corr ( ˆ m ) . (9.5)This parameter reflects our lack of knowledge on the structure of mass power corrections. Toestimate the associated uncertainty we vary it between and , such that for the extremevalues the correction is fully contained in the hard or jet functions. Once all singular Therefore one has to rescale the non-distributional jet function J nd → J nd /R C ( ˆ m ) as well. – 40 –orrections have been absorbed into the SCET matrix elements we can incorporate thetruly non-singular corrections as an additive term dˆ σ C d τ = dˆ σ C SCET d τ + dˆ σ C NS d τ , (9.6) σ C dˆ σ C NS d τ = α s ( µ ) C F π [ F C NS ( τ, ˆ m ) − F SCETNS ( τ, ˆ m )] , where dˆ σ C SCET / d τ refers to the mass-corrected (resummed) partonic SCET cross section,that is, to Eq. (4.1) with the substitutions H → H C and J m → J Cm and with resummationkernels implemented. A similar strategy can be carried out to add power corrections to thebHQET cross section, but these will be discussed elsewhere.So far we have dealt with partonic cross sections. Although for infrared- and collinear-safe observables partonic predictions are already a good description of the full result, fora precision analysis hadronization cannot be ignored. Here we will be concerned with thedominant effect of hadronization, that comes from soft dynamics and is already containedin the leading-power SCET factorization theorem. It is well known that for Qτ (cid:29) Λ QCD the main effect of soft hadronization is shifting the distribution to the right τ → τ − Ω /Q (due to the operator product expansion or OPE), with Ω a non-perturbative parameterthat can be defined in terms of field-theory matrix elements. In the peak of the distribution,hadronization is more complex and must be taken into account by convolving the partonicresult with a hadronic shape function F ( p ) : d σ C ( τ )d τ = (cid:90) Qτ d p dˆ σ C d τ (cid:18) τ − pQ (cid:19) F ( p ) . (9.7)The shape function has support for p > and is normalized (cid:82) ∞ d pF ( p ) = 1 . It is stronglypeaked at p ∼ Λ QCD and has an exponential tail extending towards infinity that en-sures any of its moments is well defined. This behavior enforces the OPE and one has (cid:82) ∞ d p pF ( p ) = Ω . As discussed in Ref. [69], Ω is afflicted by an u = 1 / renormalon thatcan be removed with appropriate subtractions defined on the partonic soft function [70].Since at the order we are working these effects are not yet relevant they will not be discussedany longer.So far we have presented all our results in the pole scheme for the heavy quark mass m ≡ m p . Expressing our cross section in the MS scheme is trivial at this order, since for P-and E-scheme thrust there is no mass dependence at lowest order except in R C . Thereforeone simply has to substitute m → m ( µ ) everywhere the mass appears: jet function, masspower corrections and fixed-order kinematic power corrections, and add an α s correctionfrom the conversion of R C to the MS scheme. We associate the MS mas renormalizationscale to µ J since the jet function is the main responsible for mass effects at the order weare working.
10 Numerical analysis
We have implemented our result for the cross section as given in Eqs. (9.6) and (9.7) in twoindependent numerical codes, which use Mathematica [71] and Python [72], respectively,– 41 –hat agree with each other within more than significant digits. For the evaluation ofdilogarithms Li , hypergeometric F , F and polygamma functions ψ ( n ) , as well as forinterpolations and numerical integration in Python we use the scipy module [73], thatbuilds on the numpy package [74], which is also used for numerical constants such as π or γ E . In Mathematica we simply use built-in native functions.While for the partonic SCET cross section all ingredients are analytic, the partonicnon-SCET cross section is only known numerically through the results of Ref. [36]. Thealgorithm used in that article allows to determine the fixed-order cross section with high-precision, and in practice numerical errors are negligibly small. Our strategy to parametrizethe fixed-order cross section is based on a combination of fit functions and interpolations.In a first step we make the curve less divergent at threshold by subtracting the knownsingular structures. This leaves integrable log-type singularities that cannot be describedwith an interpolation. To make the curve smoother as m → we also subtract the SCETnon-distributional contribution. We split this subtracted cross section into two regions thatmeet at τ = τ lim = 0 . , and use a fit function below τ lim and an interpolation above,constructed in a way such that the curve is smooth at the junction. The fit function isthe sum of a term linear in log( τ ) multiplied by a degree- polynomial in τ plus a secondpolynomial of the same degree. The logarithm contains the expected behavior of non-singular terms. The coefficients of these two polynomials are functions of the reduced mass,and each one of them is parametrized with a fit function of ˆ m , which again consists on thesum of a th -degree polynomial in ˆ m , and log( ˆ m ) times another polynomial of identicaldegree. For the interpolation we take an evenly spaced grid with . bin-size for τ < . and a finer grid below. While the values of τ in the grid do not depend on the mass, theheight of each node does, and we use fit functions of ˆ m to parametrize this dependence withthe same functional form used for τ < τ min : a logarithm of the mass times a polynomialof degree , plus another polynomial of the same order. To code this parametrization inPython in a way which is flexible and efficient we use object-oriented programming.The convolution of the (now fully analytic) partonic cross sections with the shapefunction is performed numerically. To ensure that resummation is properly implementedin the peak and tail of the distribution, being smoothly switched away in the multi-tail,we employ the profile functions introduced in Ref. [14]. It is reasonable to think that thepresence of a non-zero quark mass should modify the profile functions, but since we considerhere physical situations in which the mass is still small, we stick to mass-independentparametrizations. More sophisticated profile parametrizations depending on the value of m were employed e.g. in Ref. [45]. All plots and analyses carried out in the rest of thissection use the MS scheme for the heavy quark mass. Furthermore, we do not implementgap subtractions since they are not very relevant when matrix elements are used at theone-loop level. Unless stated otherwise, we take n (cid:96) = 4 massless quarks and a massivebottom with m b ( m b ) = 4 . GeV. For the strong coupling we use 4-loop running with theboundary condition α ( n f =5) s ( m Z ) = 0 . .We start our numerical discussion by analyzing the size of each term in Fig. 7, whichshows differential cross sections for Q = 20 GeV and GeV for vector and axial-vector– 42 – .00 0.05 0.10 0.15 0.20 0.25 0.30 - - (a) - - - SCETtotal (b) - (c) - - (d) Figure 7 . Decomposition of the differential cross section at Q = 20 GeV (left panels) and GeV(right panels) in various components for the vector (upper plots) and axial-vector (lower plots)currents. Red and blue correspond to the singular and non-distributional terms, respectively, whiletheir sum defines the SCET cross section, shown in magenta. The massless approximation isshown as a dashed gray line, while massive singular corrections are depicted in cyan. The massivecorrections to the SCET cross section (massive singular plus non-distributional) are shown in pink.Finally, the black solid line is the sum of all contributions. currents. We use only the default parameters for the profiles and set the parameter ξ defined in Eq. (9.5) to its canonical value . . We split the distribution as follows (toalleviate notation, in the remainder of this section we drop the superscript C that indicatesthe current): d σ d τ = d σ SCET d τ + d σ NS d τ ≡ d σ sing d τ + d σ nd d τ + d σ NS d τ (10.1) ≡ d σ sing m =0 d τ + d σ sing m d τ + d σ nd d τ + d σ NS d τ ≡ d σ sing m =0 d τ + d σ SCET m d τ + d σ NS d τ , where each term contains hadronization power corrections computed as a convolution withthe same shape function. In the first equality we split the full cross section (shown as a blacksolid line) in SCET and non-SCET contributions, shown in magenta and green, respectively.The SCET cross section can be further divided into the sum of singular d σ sing m / d τ (shown asa red solid line) and non-distributional d σ nd m / d τ (in solid blue) contributions. In our setup,the singular cross section is defined as the contribution from terms in the SCET factorizationtheorem which are singular at threshold if no resummation is implemented. At N LL, these– 43 – .00 0.05 0.10 0.15 0.20 0.25 0.300.00.10.20.30.40.50.6 (a) (b) (c) (d)
Figure 8 . Uncertainty bands for LL (green), NLL (blue) and N LL (red) for P-scheme thrust crosssections at GeV (two figures on top) and GeV (two figures at the bottom), for vector (twoleft figures) and axial-vector (two right figures) currents. The bands are obtained with profilesgenerated randomly selecting values for the parameters that define them. correspond to the distributions that arise from the hard function, the 1-loop soft function,and the J m =0 , J m pieces of the one-loop jet function, with the modifications discussed inSec. 9 to absorb the relevant mass corrections, integrated against the resummation kernels.The non-distributional terms (shown in blue) are defined as the resummed contributionfrom J nd defined in Eq. (4.10). We observe that while the singular contribution is positive,the non-distributional is negative, and they significantly cancel each other when addedtogether. The singular distribution can be cast as the sum of the massless approximation d σ sing m =0 / d τ (shown as a dashed gray line) and singular massive corrections d σ sing m / d τ (cyansolid line). The massless approximation is quite close to the SCET cross section (speciallyfor the vector current), as expected, since the P-scheme decreases the sensitivity to the quarkmass, and the singular massive corrections are very similar to the non-distributional termup to a global sign. We define the SCET massive corrections d σ SCET m / d τ (pink solid line) asthe sum of the singular massive corrections and the non-distributional terms, which turnsout to be rather small, specially for larger values of the center-of-mass energy. The non-SCET cross section has been defined in Eq. (9.6) and contains non-distributional kinematiccorrections coming from the QCD fixed-order cross section. Interestingly, once we absorb– 44 – .00 0.05 0.10 0.15 0.20 0.25 0.300.00.51.01.5 P - scheme thrust2 - jettiness (a) - - scheme thrust2 - jettiness (b) - - scheme thrust2 - jettiness (c) - scheme thrust2 - jettiness (d) Figure 9 . Differential cross section for massless quarks (green lines), 2-jettiness (blue lines) andP-scheme thrust (red lines) produced through the vector current. Panels (a), (b), (c) and (d)correspond to center-of-mass energies of , , and GeV, respectively. all singular terms into the SCET factorization theorem, the non-SCET corrections areabsolutely negligible everywhere except in the far tail. Within the setup defined in Sec. 9only the non-distributional cross section and the massless approximation is the same forvector and axial-vector currents.We study next the convergence of the (resummed) perturbative series for the differentialcross section. To that end, we generate bands randomly modifying our profile functions viaa flat scan on their parameters, varying them within the ranges specified in Ref. [14], withthe exception of the non-singular scale, for which we use the following continuous variation µ ns = 12 [(2 + ns ) µ H − µ J ] , (10.2)with − ≤ n s ≤ . In our scan we also randomly vary ξ between and . In Fig. 8 we showthe resulting perturbative bands at LL (green), NLL (blue) and N LL + O ( α s ) (red) for thevector and axial-vector currents, at two center-of-mass energies: Q = 40 GeV and GeV.Our curves are not self-normalized, but we nevertheless observe and excellent convergencein all cases (even at low energies) in the tail of the distribution, where higher-order bandsare nicely contained in lower-order ones. In the peak we see a big jump between LL andthe two highest orders, and the convergence is not as good as in the tail, what might– 45 – .05 0.10 0.15 0.20 0.25 0.300.010.1110
Figure 10 . Difference between the vector and axial-vector differential cross sections normalized tothe average of the two currents. We show results for Q = 20 GeV (red), Q = 40 GeV (blue) and Q = 80 GeV (green). indicate that the parameters affecting mainly the peak should be varied in wider ranges.A careful inspection of the error bands reveals that the relative uncertainties for LL andNLL are nearly identical in the whole range, and both monotonically increase as τ grows:at Q = 40 [80] GeV they change from
36 [45]% at τ = 0 . to
84 [80]% at τ = 0 . . On theother hand, at N LL the relative uncertainty is completely flat between . ≤ τ ≤ . ,and smaller than the two lower orders:
36 [30]% for Q = 40 [80] GeV. We observe the samerelative uncertainties for vector and axial-vector currents.In Fig. 9 we compare the 2-jettiness (blue) and thrust (red) cross sections for massivequarks produced through the vector current at various center of mass energies, as indicatedin the caption of the plot. As a reference, we also show in green the massless cross section.We observe that the 2-jettiness cross section has a negative deep which becomes morepronounced at low energies. It is produced by large logarithms which could be summedup by matching SCET to bHQET, as discussed in Sec. 4.2. While the massless crosssection is always quite similar to massive P-scheme thrust, the 2-jettiness distribution getsquite different at low energies, with a higher peak shifted to the right. We will study thisbehavior in further detail later in this section. As energies become larger, the three crosssections become similar to one another, but P-scheme thrust is always closer to the masslessresult. In fig. 10 we plot twice the difference between the vector and axial-vector currents,normalized to the sum of the currents. To make the figure clearer, we use a logarithmicscale on the y axis. We observe, as expected, that for larger energies the difference becomessmaller, since both currents approach the (current-independent) massless result.In our last analysis we study the dependence of the peak position and peak heightwith the heavy quark mass. Since the peak position retains some dependence on Q fromsoft hadronization, we fix the value of the center-of-mass energy to GeV, such that wecan make sure the peak moves only due to changes in the mass. In this case we com-pare the results for thrust and 2-jettiness, since the former is relatively mass insensitivewhile the latter has been designed to measure the top quark mass in future linear collid-ers, see e.g. Ref. [37]. We restrict the values of the bottom quark below m b = 14 GeVto make sure we can still apply SCET and scenario II, which should be described usingbHQET, is unimportant. The results of our study are summarized in Fig. 11, where one– 46 – P - scheme thrust2 - Jettiness (a) P - scheme thrust2 - Jettiness (b)
Figure 11 . Peak position (a) and peak height (b) for 2-jettiness (blue) and P-scheme thrust (red)massive cross section. Results correspond to default profiles, vector current and a center-of-massenergy of GeV, and with m b ≡ m b ( m b ) . We vary the bottom mass between and GeV, suchthat SCET still applies. can clearly observe a flat behavior for P-scheme and an obvious quadratic dependence for2-jettiness. For the latter this is nothing but expected, since the peak position is shifted by τ J min = 1 − √ − m (cid:39) m . In fact, if we perform a fit to the 2-jettiness peak position wefind τ max (cid:39) . .
75 ˆ m , which follows almost exactly the blue line in Fig. 11(a) and isin fair agreement with our expectations [the small disagreement is expected since the peakposition should be computed with m b ( µ J ) and not with m b ( m b ) ]. The dependence of thepeak height on the bottom mass is also much larger in jettiness than P-scheme thrust.
11 Conclusions
When considering heavy quarks in the context of event shapes, depending on the schemeused in their definition the mass sensitivity of the cross section can vary significantly. Thissensitivity manifests itself already at lowest order by setting the threshold to a non-zerovalue, and of course increases as the mass grows. This shifted threshold comes solely fromthe jet function.While in a recent paper we discussed how to compute these distributions in fixed-order at NLO, in this article we have shown how to analytically compute the differentialand cumulative cross sections in the E- and P-schemes at N LL + O ( α s ) accuracy inSCET and bHQET. To achieve this goal, we have calculated the missing pieces, namelythe NLO jet function in those two effective field theories. We have shown that in thecollinear limit the heavy quark momenta expressed in the E- and P-schemes coincide, butare different from the original (massive) definition. This entails that for any event shape,the jet function will be identical in the former two schemes, but in the case of thrust, heavyjet mass and C-parameter the measurement function is no longer completely inclusive,meaning that one needs to compute the jet function with cut diagrams, integrating overphase space rather than loop momenta. We provide an optimized and compact form for thejet function definition in each EFT, written in terms of quantum and kinematic operators,– 47 –hat facilitates their computation. For the bHQET jet function we explain how to rescalethe integrated light-particle momenta such that the heavy quark mass drops out before anyintegration is carried out. In the computation of the P-scheme SCET jet function one needsto use either sector decomposition or hypergeometric function identities to properly expandin ε and extract the distributions that appear in this limit.Shifting its argument to relocate the threshold back to zero, the 1-loop SCET massivejet functions can be written as the sum of the massless jet function plus mass corrections.The latter can be further divided into terms with distributions (or singular) and terms with-out distributions. While carrying out resummation for the former is already well known,the terms with regular functions need to be treated in a case-by-case basis. In this articlewe show how to analytically RG evolve the non-distributional terms for 2-jettiness andP-scheme thrust, and derive rapidly-convergent expansions that can be carried out aroundthe heavy-quark ( p = 0 ), “threshold” ( p = m ) [ only for P- and E-schemes ] and massless( p = ∞ ) limits, which nicely overlap with one another such that in numerical implementa-tions there is no need to explicitly evaluate hypergeometric functions at all (for 2-jettinessthere is a small region around p = m for which one cannot use expansions). Our expan-sions can be carried out up to any order such that the result is also arbitrarily precise. Thisis much faster than a direct evaluation of F and F functions, which were the bottleneckof the analysis carried out in Ref. [45].We show how to absorb into the SCET factorization theorem those mass-suppressedsingular terms that appear in fixed-order corrections by a suitable redefinition of the hardand jet functions. After this procedure is carried out, we complete our resummed expressionwith purely kinematic corrections (which are now entirely non-singular), which becomerelevant in the far tail. Hadronization power corrections can be incorporated in the usualway by convolving with a shape function, and with this complete description we haveperformed some numerical investigations. We have shown that there are strong cancellationstaking place between the two types of massive corrections to the SCET factorization theorem(with or without distributions) everywhere except in the peak, and that the remaining non-singular corrections are immaterial everywhere except in the far tail. The cancellations arestronger at larger energies, where also vector and axial-vector currents yield similar results.We have demonstrated that the P-scheme thrust cross section is much closer to the masslessprediction than for 2-jettiness by comparing cross sections as well as investigating the peakposition and height as a function of the heavy quark mass. We have also observed a niceconvergence of the cross section when adding perturbative orders.These results will be highly important for ongoing and forthcoming research in the fieldof event shapes with massive quarks. They will play a relevant role in the determination of α s with high precision (when the bottom quark mass cannot be neglected any longer) andin the Monte Carlo top quark mass parameter calibration. In addition, the computationswe have carried out will be very valuable for top quark mass measurement carried out atfuture linear colliders. Our computations can be applied to other relevant event-shapes suchas angularities, groomed observables like Soft Drop [75] or even recoil-sensitive observableslike jet-broadening. These will be presented in forthcoming publications. Extending ourcomputations to O ( α s ) is certainly challenging, but at least for the bHQET jet function,– 48 –alculations of similar complexity have been carried out for the soft function e.g. in Refs. [76–79], and even numerical approaches have been devised in Refs. [80, 81]. The massiveSCET jet function at O ( α s ) is definitely much more involved, and so far results only existfor 2-jettiness [41], which is certainly simpler since it can be computed as the imaginarypart of a forward-scattering matrix element, such that the usual machinery for multi-loopcomputations can be applied. Acknowledgments
This work was supported in part by the Spanish MINECO Ramón y Cajal program (RYC-2014-16022), the MECD grant FPA2016-78645-P, the IFT Centro de Excelencia SeveroOchoa Program under Grant SEV-2012-0249, the EU STRONG-2020 project under the pro-gram H2020-INFRAIA-2018-1, grant agreement no. 824093 and the COST Action CA16201PARTICLEFACE. A. B. is supported by an FPI scholarship funded by the Spanish MICINNunder grant no. BES-2017-081399. A. B. thanks the University of Salamanca for hospitalitywhile parts of this work were completed.
A Sector Decomposition
The direct ε expansion of J real a,P becomes much simpler if one does not have to deal withdistributions, therefore we consider the cumulative jet function, and to that end we define Σ a ( s c , µ ) ≡ (cid:90) s c d s J real a,p ( s, µ ) . (A.1)Switching variables to s = ys c in Eq. (A.1) and x → − x in Eq. (5.18) we get Σ a ( s c , µ ) = C F α s π Γ(1 − ε ) (cid:18) s c µ (cid:19) − ε I (cid:18) m s c (cid:19) , (A.2) I ( t ) ≡ (cid:90) d y y − ε (cid:90) d x (1 − x ) − ε x − − ε y (1 − x ) + t x . We apply sector decomposition by splitting the x integration in two segments: (0 , y ) and ( y, . In the former we switch variables to x = zy and in the latter we reverse the order ofintegration, which is followed by the change of variables y = z x , to find I ( t ) = (cid:90) d y y − − ε (cid:90) d z (1 − zy ) − ε z − − ε (1 − zy ) + t z + (cid:90) d x x − − ε (1 − x ) − ε (cid:90) d z z − ε (1 − x ) z + t ≡ I α ( t ) + I β ( t ) . (A.3)Since the original singularities at x = 0 , have been properly separated, mapping the formerat y = 0 and the latter at x = 0 , one can expand in ε before integrating. Let us solve I β first, which has a single pole only, such that we can use Eq. (5.20) on x − − ε to obtain I β ( t ) = − ε (cid:90) d zz + t (cid:2) − ε log( z ) (cid:3) + (cid:90) d x (cid:90) d z t − tx − xz + z ( t + z )( xz − t − z ) (A.4) = − ε log (cid:18) t (cid:19) + 12 Li (cid:18) − t (cid:19) − (1 + t ) log (cid:18) t (cid:19) − Li (cid:18)
11 + t (cid:19) + 1 . – 49 –or I α one must start applying Eq. (5.20) to y − − ε in order to regulate the pole of the z -integral. Taking into account the plus-function prescription and that the upper integrationlimit is we get I α ( t ) = − ε (cid:90) d z z − − ε t z − (cid:90) d y (cid:90) d z tz (2 − yz ) − yz (1 + tz )(1 + tz − yz ) , (A.5)where in the second term we have already set ε = 0 . Using again Eq. (5.20) to expand z − − ε in ε and solving the resulting integrals we arrive at I α ( t ) = 12 ε + 12 ε log(1 + t ) + 12 Li ( − t ) − Li (1 − t ) + Li (cid:16)
11 + t (cid:17) (A.6) − tt − t ) + t log(1 + t ) + log(1 + t ) − . Thus, summing I α and I β we obtain: I ( t ) = 12 ε + 12 ε log( t ) + Li (cid:16) − t (cid:17) + 12 log ( t − −
14 log ( t ) − t − t ) + π . (A.7)To obtain this expression, which facilitates taking the t → ∞ limit (that corresponds to s c → ), we have applied the following identities of dilogarithms:Li ( z ) = − Li (1 − z ) − log(1 − z ) log( z ) + π , (A.8)Li ( z ) = − Li (cid:16) z (cid:17) −
12 log ( − z ) − π , where the second line holds for z / ∈ (0 , only. Now we insert Eq. (A.7) into (A.2) andexpanding again in ε becomes trivial. To compute J real a,P we have to take the derivative of Σ a ( s c ) with respect to s c taking into account that it has support only for s c > : J real a,P ( s, µ ) = dd s (cid:104) θ ( s ) Σ a ( s, µ ) (cid:105) . (A.9)Using the relations in Eq. (5.43) and the identity given in Eq. (3.6) of Ref. [36] one arrivesat the result quoted in Eq. (5.27). B Alternative analytic expression of I P np for s > m In this appendix we present an alternative form of I P np in which all terms are manifestlyreal for y > and where no numerical derivatives are involved. In a first step we express F (1 , ε, ω, − y ) in Eq. (8.10) in terms F (1 , ε, − ˜ ω + ε, y ) through Eq. (8.8), To use these relations the functions multiplying θ ( x ) should be either log n ( x ) or regular at x = 0 .Therefore it is convenient to write log( t − as log( t ) − log(1 − /t ) . – 50 –nd then use that for y > one has F (1 , ε, − ˜ ω + ε, y ) = − ˜ ω π ( y − − − ˜ ω [cot( πε ) + i ] y ˜ ω − ε Γ(1 − ˜ ω + ε )Γ(1 − ˜ ω )Γ( ε + 1) (B.3) + (˜ ω − ε ) yε F (cid:18) , ω − ε, − ε, y (cid:19) . The only involved computation left is finding an analytic expression for the derivative of F (1 , ω − ε, − ε, /y ) with respect to ε in the ε → limit. The complication herearises because there are poles in /ε such that one also needs the second derivative, whichas we shall see implies the appearance of the F function. A practical way of doing thederivative is by Taylor expanding. The first derivative reads dd ε F (1 , ω − ε, − ε, y ) = y ˜ ω (1 − y ) − − ˜ ω (1 − ε ) F (1 − ˜ ω, − ε, − ε, − ε, − ε, y ) . (B.4)For the second one needs the first derivative of the F function: dd ε F (1 − ˜ ω, − ε, − ε, − ε, − ε, y ) = (B.5) − y (1 − ˜ ω )(1 − ε )(2 − ε ) F (2 − ˜ ω, − ε, − ε, − ε, − ε, − ε, − ε, y ) . With these results one can obtain the Taylor expansion of F (1 , ω − ε, − ε, y ) , whichallows to compute the first derivative of F (1 , ε, − ˜ ω + ε, /y )dd ε F (1 , ε, − ˜ ω + ε, y ) (cid:12)(cid:12)(cid:12) ε → (B.6) = ˜ ω ( y − − ˜ ω − y ˜ ω − (cid:26)
12 ˜ ω F (cid:18) , , , − ˜ ω, , , , y (cid:19) − F (cid:18) , , − ˜ ω, , , y (cid:19) y (cid:104) H − ˜ ω − log( y ))(2 iπ − H − ˜ ω + log( y )) − ψ (1) (1 − ˜ ω ) + 5 π (cid:105)(cid:27) . To obtain this relation one simply has to divide the integration path in Eq. (5.21) into the segments (0 , /y ) and (1 /y, . Using [1 − z ( y ± iε )] − a = θ (1 − zy )(1 − yz ) − a + θ ( zy − yz − − a [cos( aπ ) ∓ i sin( aπ )] , (B.1)remapping each segment back to (0 , by a change of variables ( z → z/y in the first segment and z → [1 − (1 − /y ) x ] in the second), and carrying out the integrals one finds the following identity: F ( a, b, c, y ± iε ) = Γ(1 − a ) y − b Γ( c )Γ(1 − a + b )Γ( c − b ) F (cid:18) b, b − c, − a + b, y (cid:19) (B.2) + e ± iπa Γ(1 − a )Γ( c ) y b − c ( y − − a − b + c Γ( b )Γ(1 − a − b + c ) F (cid:18) − b, c − b, − a − b + c, y − y (cid:19) . – 51 –sing this result we arrive at the alternative expression for I P np I P np (˜ ω, y ) = ˜ ω [(2˜ ω + 5) y + ˜ ω (3˜ ω + 7) − y ]Γ(1 − ˜ ω )(1 + ˜ ω )(1 − y ) (cid:20) H − ˜ ω − − ˜ ω − log( y ) (cid:21) F (1 , , ˜ ω + 2 , − y )+ 3˜ ω y + 3˜ ω − ωy + 14˜ ω + y − − ˜ ω )( y − + ( y − − − ˜ ω y ˜ ω − [4 y − ω + 5) y − ˜ ω (3˜ ω + 7)] × (cid:26) ˜ ω F (cid:18) , , , − ˜ ω, , , , y (cid:19) − ˜ ω F (cid:18) , , − ˜ ω, , , y (cid:19) (B.7) × ˜ ω y (cid:20) π − (cid:18) H − ˜ ω − − ˜ ω − log( y ) (cid:19) − ψ (1) (1 − ˜ ω ) (cid:21) + y cos( π ˜ ω )Γ(1 − ˜ ω )Γ(1 + ˜ ω ) F (1 , , ω, − y ) (cid:20) H − ˜ ω − − ˜ ω − log( y ) (cid:21)(cid:27) , References [1] M. Dasgupta and G. P. Salam,
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