Phenomenology of QCD threshold resummation for gluino pair production at NNLL
DDESY 13-035 May 2013LPN 13-021SFB/CPP-13-16
Phenomenology of QCD threshold resummationfor gluino pair production at NNLL
Torsten Pfoh
Deutsches Elektronensynchrotron DESYPlatanenallee 6, D–15738 Zeuthen, Germany [email protected]
Abstract
We examine the impact of threshold resummation for the inclusive hadronic production cross sectionof gluino pairs at next-to-next-to-leading-logarithmic accuracy, compared to the exact next-to-leading-order cross section and the next-to-next-to-leading-order approximation. Here, we applyformulas derived recently in the classical Mellin-space formalism. Moreover, we give the analyticinput for the alternative momentum-space formalism and discuss the crucial points of the numericimplementation. We find that soft resummation keeps the hadronic cross section close to the fixednext-to-leading-order result. a r X i v : . [ h e p - ph ] M a y Introduction
Within the search for new physics at the LHC experiments, one hopes to find evidences for heavycolor-charged particles. These appear in many scenarios for physics beyond the Standard Model(SM). A prominent candidate is the gluino which is a Majorana fermion and the superpartner ofthe SM gluon within various realizations of Supersymmetry (SUSY), as one of the most favoritemodels. Unfortunately, the direct search at ATLAS and CMS have only produced exclusion limits so far which, of course, depend on the model under consideration, and the assumptions on the SUSYparticle spectrum.In order to separate possible signals from the SM background, a crucial theoretical input quantityis the inclusive hadronic production cross section σ pp → (cid:101) g (cid:101) gX . As known from standard QuantumChromodynamics (QCD), the latter is given by a sum over the various partonic production channelswhere the partonic cross sections ˆ σ ij → (cid:101) g (cid:101) g are convoluted with their respective parton luminosityfunctions L ij . Explicitly, one has σ pp → (cid:101) g (cid:101) gX ( s, m (cid:101) g , m (cid:101) q , µ f , µ r ) = (cid:88) i,j = q, ¯ q,g (cid:90) m (cid:101) g /s dτ L ij ( τ, µ f ) ˆ σ ij → (cid:101) g (cid:101) g ( τ s, m (cid:101) g , m (cid:101) q , µ f , µ r ) , (1)where the parton luminosities are themselves given by a convolution of the parton distributionfunctions (PDFs) L ij ( τ, µ f ) = (cid:90) dx (cid:90) dx δ ( x x − τ ) f i/p (cid:0) µ f , x (cid:1) f j/p (cid:0) µ f , x (cid:1) , (2)and we have introduced the hadronic center-of-mass (cms) energy s , the partonic cms energy ˆ s = τ s ,the factorization scale µ f , and the renormalization scale µ r . The gluino mass is denoted by m (cid:101) g , andwe assume mass degeneracy among the squarks flavors, therefore using a single scale m (cid:101) q .At leading order (LO) in perturbation theory, gluino pair production is driven by the partonicsub processes of gluon fusion gg → (cid:101) g (cid:101) g and quark-antiquark annihilation q ¯ q → (cid:101) g (cid:101) g, q = d, u, s, c, b .At next-to-leading-order (NLO), also the gq channel opens [3]. Close to the threshold however, itscontribution is suppressed compared to gluon fusion and quark-antiquark annihilation. The fullNLO result has been implemented in the public program Prospino [4].As very well known, the cross section develops powers of large logarithms ln( β ) near the pro-duction threshold, where the velocity β = (cid:113) − m (cid:101) g / ˆ s ≡ (cid:112) − ρ (3)of the produced particle pair goes to zero. The so-called threshold logarithms spoil the validity ofthe perturbative expansion in the strong coupling constant α s . However, they can be resummedsystematically to all orders in perturbation theory, where the quantity α s ln( β ) counts as order one.In this context, threshold logarithms are sometimes called soft logarithms and one talks about softresummation. There are two approaches to soft resummation which we will shortly discuss below.We will refer to them as Mellin-space formalism [5, 6] and momentum-space formalism [7, 8].Another difficulty arises through the exchange of soft gluons between the final state particles.At fixed order in perturbation theory, this gives rise to so-called Coulomb terms proportional topowers of α s /β . Obviously, with decreasing β , the above ratio becomes O (1) and should thereforebe resummed as well. This can be done in the framework of non-relativistic QCD (NRQCD). Ajoined soft and Coulomb resummation has been worked out in the momentum-space formalism [9].In the Mellin-space formalism however, one would need to calculate the Mellin transformations See Refs. [1, 2] for example.
1f the NRQCD expressions analytically (or at least semi-analytically as a function of the Mellinmoments), a problem which has not been solved so far. Therefore, Coulomb terms are included atfixed order only.For any process, soft resummation up to so-called next-to-leading-logarithmic (NLL) accuracyrequires the knowledge of the color-decomposed Born cross section with respect to the SU (3) c -color configuration of the produced (s)particle pair. At next-to-next-to-leading-logarithmic (NNLL)precision, one needs the color decomposition of the NLO cross section near the threshold, wherehigher powers of β are skipped. For gluino pair production, the state of the art in the momentum-space formalism is a combined soft and Coulomb resummation up to NLL accuracy [10]. Withinthe Mellin-space approach, the NLL results of Ref. [11, 12] have been extended to NNLL precisionrecently in Ref. [13]. Bound-state and finite-width effects at fixed order have been discussed inRef. [14, 15]. In the context of NLL resummation, finite-width effects have been discussed recentlyin Ref. [16]. It has been found that for Γ /m (cid:101) g ≈ α s (modulothe factor α s in the Born cross section), one reproduces the threshold logarithms up to next-to-next-to-leading-order (NNLO). One further obtains constant terms which are generically differentat NNLO for the two approaches. However, a proper matching eliminates these constants and oneis left with the threshold approximation (NNLO th ), where, after factorization of the Born crosssection, all NNLO constants are set to zero. At NLO, the constants are kept as these refer to theexact fixed order result.If the gluino is heavy (of the order of 1 TeV), the threshold-enhanced terms give the dominantcontribution at given order in perturbation theory. Combining the full NLO result with the O ( α s )threshold enhanced terms to the NNLO approximation (NNLO approx ), one finds an increase of thecross section of about 20% with respect to the fixed NLO result [13]. The question of interestis how these findings change in the presence of NNLL resummation. Although the resummationformula for soft logarithms has been derived analytical, it has not been implemented yet. In thiscontext, a general question is how to deal with ambiguities related to the choice of scales or thetreatment of the Landau pole in momentum integrals. A different source of uncertainty comes fromthe unknown squark masses. However, the hadronic process is mainly driven by gluon fusion whichhas only a weak dependence on m (cid:101) q . As a consequence, the impact of the squark masses on the crosssection is negligible to good approximation. In the threshold limit, the dependence on m (cid:101) q actuallyvanishes. The impact of higher order contributions is estimated by a variation of the factorizationand renormalization scale. As discussed in Ref. [13], the main source of uncertainty comes from theshape of the gluon PDFs at high momentum fraction x , and the value of α s ( M Z ). The possiblelarge discrepancies are not covered by the individual PDF errors, and give the main uncertaintywhen setting a reliable exclusion limit.In this article, we study the impact of NNLL soft resummation in the Mellin-space approach.In the sections 2 and 3, we briefly review the methods for soft (and Coulomb) resummation in theMellin- and momentum-space formalism. Concerning the latter, we derive the color-decomposedhard function at NLO which is needed for NNLL resummation. We discuss the ingredients of thepotential function which collects all results from NRQCD derived from the NLO non-relativisticpotential. In the sections 4 and 5, we discuss technical aspects of the numerical implementation ofthe resummation formulas in the two approaches, and explicitly apply the Mellin-space formalism.The inclusive hadronic cross section is discussed in section 6 and we give our conclusion in section 7.An explicit formula for the matching of the momentum-space resummation formula onto the NNLOapproximation is provided in the appendix. 2 Threshold resummation in Mellin space
The traditional approach to threshold resummation has been invented in Ref. [5,6], see also Ref. [17–22] for further discussions. Resummation is performed in Mellin space after introducing moments N with respect to the variable ρ = 4 m (cid:101) g / ˆ s of momentum space. Neglecting the dependencies on µ f and µ r , one has ˆ σ Nij ( m (cid:101) g ) = M [ˆ σ ij ( m (cid:101) g )]( N ) = (cid:90) dρ ρ N − ˆ σ ij (ˆ s, m (cid:101) g ) , (4)where the threshold limit β → N → ∞ . The resummation of threshold logarithmsis achieved by the formulaˆ σ res , N ij, I = ˆ σ B, Nij, I g ij, I g Cij, I ( N + 1) exp (cid:104) G ij, I ( N + 1) (cid:105) + O ( N − ln n N ) . (5)All contributions from soft and collinear radiation exponentiate and are collected in the function G ij, I . To NNLL accuracy, it may conveniently be expanded according to G ij, I ( N ) = ln( (cid:101) N ) · g ij ( (cid:101) λ ) + g ij, I ( (cid:101) λ ) + a s g ij, I ( (cid:101) λ ) + . . . , (6)where (cid:101) N ≡ N exp( γ E ) with γ E denoting the Euler-Mascheroni constant, (cid:101) λ ≡ a s β ln (cid:101) N , β asdefined in Ref. [13], and a s ≡ α s / (4 π ). The explicit expressions are calculated by a double integralover a set of anomalous-dimension functions. Some of these depend on the SU (3) c -color configura-tion of the final-state gluino pair which we label by a capital index I . The color-summed partoniccross section for a given channel is simply given byˆ σ ij → (cid:101) g (cid:101) g = (cid:88) I ˆ σ ij, I . (7)The possible final-state color configurations are obtained by a decomposition of the initial colorstates into irreducible representations according to × = s + s + a + + + s , (8) × ¯3 = s + s + a , (9)for gluon fusion and quark-antiquark annihilation, respectively. There are symmetric ( s ) and anti-symmetric ( a ) color states. The corresponding quadratic Casimir operators C I , which show up inthe color-decomposed partonic cross section, take the values C I = { , , , } for I = { , , , } (10)independent from the symmetry properties. The (color-decomposed) Born cross section ˆ σ Bij, I factorsout in Eq. (5) which is also true for the fixed order partonic cross section in the threshold limit ˆ σ th ij, I .The matching constant g ij, I collects all contributions independent of N and has to be determinedorder by order in perturbation theory by expanding the right-hand side of Eq. (5) in α s andmatching onto ˆ σ th ij, I . Coulomb corrections are not resummed by the above ansatz. They are formallytreated as part of the hard scattering function on which the exponent has to be matched. Therefore,one introduces a second matching term g Cij, I [23] as a power series in α s which is actually a functionof N . If Coulomb corrections are not considered, g Cij, I reduces to a factor one. This produces all the threshold logarithms of the form ln k ( N ). S of the produced heavy particle pair[24–28]. These terms are also not treated by the exponent (6) and thus are fully hosted by the hardmatching constant. As they also depend on the Mellin variable N , it is convenient to include theminto g Cij, I ( N ), although they do not arise from Coulomb exchange. Alternatively, one could alsointroduce a third matching constant g NCij, I ( N ). The spin dependence is described by a parameter v spin which is zero for the gluino pair being in a spin singlet (S=0) configuration and − / An alternative approach which features resummation in momentum space has been invented inRef. [7, 8] in the context of deep inelastic scattering. Here, soft and collinear radiation are describedin the framework of soft-collinear effective theory. In Ref. [9], this method has been adopted toheavy (s)particle pair production and extended to the summation of Coulomb corrections. Thestarting point is the observation that near the threshold the partonic production cross sectioncan be factorized into a hard function H , a soft function W which collects soft fluctuations, anda potential function J which sums Coulomb exchange. Below the threshold, one has a discretespectrum of bound-states. If these are taken into account, J should be expressed as a function ofthe energy E = √ ˆ s − m (cid:101) g relative to the threshold [9, 10, 29].At fixed order in α s , the partonic cross section close to the threshold can be written as [9]ˆ σ ij → (cid:101) g (cid:101) g (ˆ s, µ ) = (cid:88) S (cid:88) I H Sij, I ( µ ) (cid:90) ∞ dω J S I ( E − ω W I ( ω, µ ) , (11)where one sums over the spin and color configurations of the final-state particle pair. The spindependence enters the hard and the potential function at NNLO first. In order to account for thedifferent scales of the problem and to achieve resummation of threshold logarithms, one calculatesthe hard function at the hard scale µ h and the soft function at soft scale µ s . Then, one solvesrenormalization group equations for the hard and the soft function which, in the latter case, sumsthe soft logarithms. The solutions are used to evolve H ( µ h ) and W ( µ s ) to the common scale µ f which is used for the convolution of the partonic cross sections with the parton luminosities. Thegeneral all order resummation formula reads [9]ˆ σ res ij → (cid:101) g (cid:101) g (ˆ s, µ f ) = (cid:88) S (cid:88) I H Sij, I ( µ h ) U ij, I ( m (cid:101) g , µ h , µ s , µ f ) (12) × (cid:90) ∞ dω J S I (cid:0) E − ω (cid:1) ω (cid:18) ω m (cid:101) g (cid:19) η ˜ s ij, I (cid:18) (cid:18) ωµ s (cid:19) + ∂ η , µ s (cid:19) e − γ E η Γ(2 η ) , where the summation of Sudakov-double logarithms is included in the evolution function U ij, I whichis given by U ij, I ( m (cid:101) g , µ h , µ s , µ f ) = (cid:32) m (cid:101) g µ h (cid:33) − a Γ ( µ h ,µ s ) (cid:18) µ h µ s (cid:19) η × exp (cid:20) (cid:0) S ( µ h , µ f ) − S ( µ s , µ f ) (cid:1) (13) − a Vij, I ( µ h , µ s ) + 2 a φi ( µ s , µ f ) + 2 a φj ( µ s , µ f ) (cid:35) . s ij, I ( ρ, µ s ) is the Laplace transform of the MS-renormalized soft function W I ( ω, µ )which, to NLO accuracy, is given by˜ s ij, I ( ρ, µ ) = 1 + a s ( µ ) (cid:20) ( C i + C j ) (cid:18) ρ + π (cid:19) − C I ( ρ − (cid:21) + O (cid:0) a s (cid:1) , (14)where we again define a s ≡ α s / (4 π ). The auxiliary variable η in the Eq. (12) is set to η = 2 a Γ ( µ s , µ f )after performing the derivative. It contains single logarithms [10] which can be seen by expanding η = 4 a s ( C i + C j ) ln( µ s /µ f ) + O ( a s ). From the latter expression one also learns that η is negativeand tends towards zero as µ s approaches µ f . This behavior is preserved at higher orders in α s .As a consequence, the integration kernel ( ω/ (2 m (cid:101) g )) η /ω has to be understood in a distributionalsense [8] as will be further discussed in Sec. 5.The quadratic Casimir operators C i and C I depend on the color configuration of the initial andfinal states, respectively. For gluino pair production via gluon fusion we have C i = C j = C A = 3,for quark-antiquark annihilation we have C i = C j = C F = 4 /
3. With µ s ≤ µ f it hence followsthat η ≤
0. The function a Γ ( ν, µ ) and the Sudakov exponent S ( ν, µ ) are given to NNLO precisionin Eq. (86) and Eq. (87) of Ref. [30]. For NNLL resummation, the respective NLO expressionsare sufficient. For i = j , these are functions of expansion coefficients Γ ik of the cusp anomalousdimension Γ i cusp = ∞ (cid:88) k =0 Γ ik a sk +1 ≡ C i γ cusp , (15)and coefficients of the QCD beta-function β k . The latter are listed in Eq. (85) of the above article,whereas the cusp anomalous dimension is given for the quark-antiquark channel in Eq. (81). Forgluon fusion, one just has to multiply these results by C A /C F according to the Casimir scalingindicated in Eq. (15) above. For i (cid:54) = j , one would replaceΓ i cusp → Γ ij cusp = 12 (Γ i cusp + Γ j cusp ) ≡
12 ( C i + C j ) ∞ (cid:88) k =0 γ ( k )cusp a sk +1 . (16)Note also that the coefficients Γ ik coincide with the coefficients A ( l ) i with l = k + 1, used in theMellin-space formalism (see Ref. [13, 31]). The functions a Vij, I and a φi in Eq. (13) are obtained inanalogy to a Γ by a replacement of Γ ij cusp with γ Vij, I = γ i + γ j + γ H, I in the first case, and with( γ φi + γ φj ) / γ i are related to softradiation from the (mass-less) initial-state particles and are collected in appendix A of Ref. [32].The anomalous dimension γ H, I refers to soft radiation connected to a massive final-state particlein the color representation I . NLO results are given in Eqs. (3.31,32) of Ref. [33]. Here, one canalso find expansion coefficients of γ φi which are related to the evolution of the parton distributionfunction. Explicit expressions up to NLO for color-triplets and octets are given in Eqs. (D.17-20).When taking results from the literature, care has to be taken with respect to the meaning of theparameter n f . Typically, n f denotes the number of active flavors which contribute to the evolutionof α s . So it does in the articles listed above where the various coefficients of the QCD beta functionand anomalous dimensions are collected. However, in some papers about gluino pair production atthe threshold n f denotes the total amount of quark (and squark) flavors which, depending on therenormalization scheme, may differ from the number of active flavors.As discussed in Ref. [9], the all-order resummation formula (12) does not depend on the hard andsoft scale µ h and µ s . However, as all ingredients are only known to finite order in α s , a truncation See Ref. [13, 15] for instance.
5f the perturbative expansion induces a residual dependence on these scales. A good perturbativebehavior is obtained by a suitable choice for µ h and µ s . For the case of gluino pair production,these are µ h ≈ k h m (cid:101) g and µ s = k s m (cid:101) g β with k h and k s being numbers of O (1). The choice of arunning soft scale ∼ m (cid:101) g β is also necessary to reproduce the threshold logarithms at NNLO (fixedorder) [29]. In practice, the above choice becomes problematic when β tends zero. Moreover, α s ( µ s )approaches the Landau pole. Therefore, one should introduce a parameter β cut [29]. For β > β cut ,a running soft scale is used such that the perturbative expansion is not spoiled by large logarithms.On the other hand, for β < β cut , the scale is fixed and kept in the perturbative regime. Notethat a special treatment of the Landau pole is also required in the Mellin-space approach whenperforming the Mellin inversion (see Refs. [19, 20] for instance). In the momentum-space approacha proper choice of β cut should keep the ambiguities which arise when matching the resummed to thefixed order cross section as small as possible. For gluino pair production, this has been analyzedin Ref. [10]. Finally, we note that the choice of the soft scale actually determines what is beingresummed. A dedicated discussion can be found in Ref. [34]. Within the Mellin-space formalism onthe other hand, one has the implicit scale choices µ s = m (cid:101) g /N and µ h = µ f , and the two approachesare formally identical up to O (1 /N )-terms (see for instance the discussion in Ref. [30]).Coming back to the momentum-space approach, one requires the potential function J S I ( E ) whichdescribes the exchange of soft gluons between the final-state particles as well as higher order correc-tions coupled to these diagrams. At fixed order in perturbation theory, pure Coulomb corrections(which correspond to ladder diagrams) give rise to terms proportional to ( α s /β ) n . As already men-tioned, they can be treated by methods of NRQCD using the LO non-relativistic potential and havebeen found to be summed by a Sommerfeld factor ∆ C [35]. Starting from NLO, one also has todeal with terms multiplied by α s ( α s /β ) n . Their summation requires the non-relativistic potentialat NLO [24, 26]. The LO and NLO Greens functions of the non-relativistic Schr¨odinger equation, G (0) C, I ( E ) and G (1) C, I ( E ), are given in Ref. [9, 29], and the contributions to the potential function areobtained by taking two times the imaginary parts. For positive values of the energy E relative tothe threshold, one finds the simple leading expression J (0) I ( E ) = m (cid:101) g π (cid:115) Em (cid:101) g ∆ C (cid:18) πα s (cid:114) m (cid:101) g E D I (cid:19) θ ( E ) (17)= m (cid:101) g π (cid:115) Em (cid:101) g (cid:20) − a s π D I (cid:114) m (cid:101) g E + a s π D I m (cid:101) g E + O (cid:0) a s (cid:1)(cid:21) θ ( E ) , where the Sommerfeld factor is given by ∆ C ( x ) = x/ (exp( x ) −
1) , and we have introduced the colorcoefficient D I of the QCD potential D I = 12 C I − C A = {− , − / , , } for I = { , , , } . (18)Note that close to the threshold where E (cid:28) m (cid:101) g , one has (cid:112) m (cid:101) g /E = 1 /β + O ( β ) with β (cid:28) C ( πα s D I /β ) also described in Ref. [11, 13] which makes thesummation of 1 /β -terms evident. For D I < D I > J (1) I ( E ) can not be brought to a simple form. However, it canbe expressed as a linear combination of dilogarithms and nested harmonic sums with a complexargument λ ( E ) = α s ( − D I ) / (2 (cid:112) − E/m (cid:101) g ) [29]. As a first approximation, it may be sufficient toinclude J (1) I at fixed order in α s . It reads J (1) I ( E ) = m (cid:101) g π a s π D I (cid:20) β (cid:18) ln (cid:16) Em (cid:101) g (cid:17) + 2 ln (cid:16) m (cid:101) g µ C (cid:17)(cid:19) − a (cid:21) + O (cid:0) a s (cid:1) (19)6ith a = 31 / C A − / n l . The choice of the scale µ C is discussed in detail in Ref. [29]. Thecomplete fixed order non-relativistic corrections at NNLO are obtained by adding to Eq. (19) theconvolution of the O ( α s )-Sommerfeld term with the NLO threshold approximation (without theCoulomb term), as well as a non-Coulomb contribution discussed below. The full result has beengiven in Ref. [24] first.If the potential is attractive, discrete bound states may develop below the threshold. Ignoringthe gluino widths, these are described by J bound I ( E ) = 2 ∞ (cid:88) n =1 δ ( E − E n ) (cid:18) m (cid:101) g α s ( − D I )2 n (cid:19) (cid:16) α s π δr (cid:17) θ ( − D I ) , ( E < , (20)for S-wave production [9, 29], where the bound-state energies are given by E n = − m (cid:101) g α s D I n (cid:16) α s π e (cid:17) . (21)The correction terms δr and e stem from the NLO Coulomb Greens function [36] and can befound in Ref. [29], Eqs. (B.4) and (B.5). We stress that bound-state production (2 → → S NC , I ( E ) = D I (cid:0) C A − D I (1 + v spin ) (cid:1) θ ( E ) , (22)where the parameter v spin has been defined above. In summary, the potential function required forNNLL resummation is given by J S I ( E ) = J (0) I ( E ) (cid:16) α s ln( β )∆ S NC , I ( E ) (cid:17) + J (1) I ( E ) , (23)where ln( β ) ≈ ln( (cid:112) E/m (cid:101) g ).As last ingredient of the NNLL resummation formula, the hard function H Sij, I ( µ h ) is needed atNLO. At LO, it is related to the partonic Born cross section [9, 10, 29]ˆ σ (0) ij, I ( µ h ) ≈ m (cid:101) g π (cid:115) Em (cid:101) g H (0) ij, I ( µ h ) ≈ m (cid:101) g π β H (0) ij, I ( µ h ) , (24)where the approximations are valid in the limit ˆ s → m (cid:101) g . Near the threshold, the LO cross sectionis factored out from higher order contributions, where, apart from an overall factor β due to theBorn term, all positive powers of β are set to zero. Writing H ij, I ( µ h ) = H (0) ij, I ( µ h ) (cid:104) a s ( µ h ) H (1) ij, I ( µ h ) + a s ( µ h ) H (2) ij, I ( µ h ) + O (cid:0) a s (cid:1)(cid:105) , (25)the required NLO hard functions are obtained by comparing the NLO fixed order threshold resultswith the NLO expansion of Eq. (12). For this purpose, it is sufficient to set J S I = J (0) I , and we7equire the various anomalous dimension functions at LO only. Using γ (0)cusp = 4, γ (0) H, I = − C I , γ φ (0) i = − γ (0) i , and E = m (cid:101) g β , we find in agreement with Eq. (D.3) from Ref. [9]ˆ σ res , exp ij → (cid:101) g (cid:101) g (ˆ s, µ f ) = (cid:88) I m (cid:101) g π β H ij, I ( µ h ) (cid:34) a s (cid:32) − D I π β + 4( C i + C j ) ln (8 β ) (26) − (cid:16) C I + ( C i + C j ) (cid:0) L f ˜ g (cid:1)(cid:17) ln(8 β ) + ( C i + C j ) L f ˜ g +2 (cid:16) C I + ( C i + C j ) (cid:0) − L fh (cid:1)(cid:17) L f ˜ g + 12 C I + ( C i + C j ) (cid:18) − π (cid:19) − (cid:16) C I − C i + C j ) ln(2) − (cid:0) γ (0) i + γ (0) j (cid:1)(cid:17) L fh + ( C i + C j ) L fh (cid:33) + O (cid:0) a s (cid:1)(cid:35) , where we have defined L fh = ln( µ f /µ h ), and L f ˜ g = ln( µ f /m (cid:101) g ). The dependence on the softscale has canceled between the soft function and the expansion of the evolution function. The termsproportional to L fh run the hard function H (1) ij, I ( µ h ) down to the factorization scale µ f , see also thediscussion in Ref. [9].The explicit evaluation of the soft and the potential function at NLO has produced all thresholdenhanced terms proportional to ln k (8 β ) ( k = 1 ,
2) and 1 /β which are easily identified withinthe fixed order cross section in the threshold limit, given in the Refs. [3] and [13]. However, therenormalization scale has been set equal to the factorization scale in the latter articles. Adopting thischoice simplifies the right-hand side of Eq. (26) due to L fh = 0. From now on, we set µ h = µ f = µ , α s ( µ ) ≡ α s , and switch to the notation of Ref. [13], where L f ˜ g ≡ L µ . The NLO hard functions forgluino pair production in the momentum-space formalism then read H (1) gg, I ( µ ) = 4 C gg , I − C I − C A (cid:18) − π (cid:19) (27) − (cid:16) C I − C A ln(2) (cid:17) L µ − C A L µ H (1) q ¯ q, a ( µ ) = 4 C q ¯ q , a − C A − C F (cid:18) − π (cid:19) (28) − (cid:16) C A − C F ln(2) − β + 6 C F (cid:17) L µ − C F L µ in the MS scheme with n l = 5 active light (mass-less) quark flavors. In the gluon fusion channel, onlycolor-symmetric parts contribute near the threshold. Therefore, one has to sum over I = , , .On the other hand, in the quark-antiquark annihilation channel, one only has contributions fromthe anti-symmetric octet to first approximation. The one-loop matching constants C gg , I and C q ¯ q , a are given in Eqs. (39) and (40) of Ref. [13]. They are functions of the ratio r = m (cid:101) q /m (cid:101) g of thesquared squark and gluino masses.If one expands the resummation formula (12) up to O ( α s ), one obtains some constants besidesthe threshold logarithms at NNLO. In order to obtain the full NNLO hard coefficients H (2) ij, I , adedicated two-loop calculation is necessary to determine all constant terms. As these are notavailable at the moment, it is common to keep only threshold enhanced terms. Thus, if one wishesto match the NNLL resummation onto the NNLO approximation, one requires a cancellation of allconstants in the resummation formulas at O ( α s ). Within the Mellin-space approach, this is achievedby a proper choice of g ij, I in Eq. (5), see Ref. [13]. For an analogous treatment in the momentum-space approach, we need to compute the next order in Eq. (26). Keeping only the O ( a s ) contributionof the potential function, m (cid:101) g β/ (2 π ), and the hard function up to O ( a s ), the convolution with the8LO soft function and multiplication with the evolution function reproduces the soft logarithmsln n ( β ) ( n = 1 , ..,
4) and their coefficients at NNLO if, and only if, we choose µ s = k s m (cid:101) g β . Theexplicit fixed-order expressions can be found in Ref. [13, 24]. The NNLO coefficient of the hardfunction has to be chosen such that the constants of the above expansion cancel at O ( α s ). Here,one should keep in mind that also the yet unknown NNLO soft function, if taken into account,produces constant pieces at O ( a s ). In our case, we just arrange the NNLO hard function in sucha way that all NNLO constant pieces cancel. This ensures a proper matching on the approximatedNNLO cross section under usage of the NLO soft function. The explicit result is given in App. A.The relative impact on the total hard function is rather small, typically at the percent level. Finally,we mention that for NNLL+NNLO approx accuracy, one can also set H (2) ij, I = 0, if the matching tothe full fixed order (see discussion in the next section) is performed at the NNLO level. This differsfrom the choice in Eq. (A.1) only in higher order terms beyond NNLL accuracy, and is applied inRef. [29]. For NNLL resummation, all input functions to the resummation formula are needed at O ( α s ). Note that the NLO expression of the Sudakov exponent S ( ν, µ ) requires the coefficients ofthe cusp anomalous dimension and the QCD beta function up to NNLO (see Ref. [30]). All otheranomalous dimension functions are required at NLO only. In this section, we discuss the numerical implementation of the resummation formula (5). Thefollowing procedure is required in general [37]: First, as contributions of higher powers in β becomenumerically important when calculating the inclusive cross section, one has to match the resummedcross section to the full fixed order result. This is achieved by adding the latter, while subtractingall terms up to O ( a s ) (modulo the common prefactor α s ) of the expanded NNLL expression. Inorder to perform the matching on the NNLO approx partonic cross section, one needs to subtractall terms up to O ( a s ) of expanded resummation formula (denoted by NNLL(2)) and add back theapproximated NNLO result [29].If one chooses to resum in Mellin space, an inverse Mellin transformation is required which hasto be done numerically. Due to omitted 1 /N contributions in the resummation formula (5), thematching onto the NNLO approximation will give a slightly different result compared to the NLOmatching . The resummed partonic cross section is given byˆ σ res , NNLO approx ij → (cid:101) g (cid:101) g = M − (cid:104) ˆ σ NNLL , N ij → (cid:101) g (cid:101) g − ˆ σ NNLL(2) , N ij → (cid:101) g (cid:101) g (cid:105) + ˆ σ NNLO approx ij → (cid:101) g (cid:101) g . (29)The numerical Mellin inversion requires an analytic continuation of the resummed cross sectionto the complex plane. In our case, one simply assumes complex arguments of the logarithms. Amore general discussion on that topic is provided in [38, 39], and a good routine is included in theprogram ANCONT [38]. Here, the Mellin inversion is obtained by f ( ρ ) = 1 π (cid:90) ∞ dz Im (cid:2) e i Φ ρ − c ( z ) M [ f ]( c ( z )) (cid:3) , (30)where Mellin- N is identified with c ( z ), and c ( z ) = c + ze i Φ . As the contour integral around thesingularity at N = 0 is symmetric with respect to the x -axis, only the upper half is evaluated andmultiplied by two. The path is chosen to be a truncated line with an angle Φ close to π . Theparameter integral over z is divided logarithmically into 20 pieces, where each segment is performedby the 32-point Gauss formula. We choose the starting point c = 1 . /N -pole, Here, one has to emphasize that the numerically inverted NNLL(2) cross section does not exactly correspond tothreshold terms NNLO th in momentum space, but comes close towards the threshold. = Β= Β= th NNLL H L NNLO approx res,NNLO approx
NNLL s ` (cid:144)H m g Ž L - Σ ` gg ® g ~ g ~ @ pb D Β= Β= Β= nC,th NNLL nC H L NNLO approx res nC ,NNLO approx NNLL nC s ` (cid:144)H m g Ž L - Σ ` gg ® g ~ g ~ @ pb D Figure 1: Partonic cross section for gluon fusion versus the energy above the production threshold normalizedto the gluino pair mass. The renormalization and factorization scales have been set to µ = m (cid:101) g = 800 GeV.We plot the NNLO approximation which is exact up to NLO, and the NNLO threshold limit NNLO th whichcontains only threshold-enhanced contributions and NLO constants. Moreover, we show the NNLL resummedcross section and its expansion in α s up to second order, NNLL(2), as well as the resummed partonic crosssection matched onto the NNLO approximation. First panel: Coulomb corrections are included at fixed orderin the resummation formula. Second panel: Coulomb corrections are neglected during resummation but keptin the NNLO approximation. = Β= approx r es,NNLO approx s ` (cid:144)H m g Ž L - Σ ` gg ® g ~ g ~ @ pb D Figure 2: Comparison of the resummed partonic cross section to the exact NLO result and the NNLOapproximation. The scale is varied within the interval µ ∈ [ m (cid:101) g / , m (cid:101) g ] around µ = m (cid:101) g = 800 GeV. but left to the Landau pole singularity which is excluded from the integration contour according tothe minimal prescription presented in Ref. [19, 20].In Fig. 1, we plot the resummed partonic cross section in the gluon fusion channel against theenergy above the threshold normalized to the gluino pair mass E/ (2 m (cid:101) g ) = √ ˆ s/ (2 m (cid:101) g ) − µ = m (cid:101) g . We also plotthe NNLO approximation (solid blue) and the threshold approximation NNLO th (dotted blue).Concerning the latter, all analytical input is given in Ref. [13]. As a check of the numerical Mellininversion, we further show the inverted expansion of the soft resummation NNLL(2) which comesclose to NNLO th and the NNLO approximation for β < .
6. The inverted soft resummation NNLLmerges with its O ( α s ) expansion away from the threshold. For low velocities ( β (cid:28) . β < . E <
10 GeV for m (cid:101) g ≈ µ ∈ [ m (cid:101) g / , m (cid:101) g ].As evident from the figures, the NNLL soft resummation significantly depletes the partoniccross section compared to the NNLO approximation and brings it down near the level of the NLOcalculation. Close to the threshold, the resummed cross section is enhanced compared to fixed NLObut still well below the NNLO approximation. In the kinematical region β ≈ .
6, the resummedcross section, the NNLO approximation, and the threshold limit NNLO th cross each other closeby. For higher velocities ( β > . m (cid:101) g = 800 GeV and a common squark mass of 640 GeV. Choosing squark massesabove 1 TeV will not change the conclusions drawn above. This is not true for the quark-antiquarkannihilation channel. However, the hadronic production cross section is mainly driven by gluonfusion (about 99%). Thus, we will not have a closer look on q ¯ q -annihilation. Within the momentum-space formalism, no inversion is required and one can remove the operator M − in Eq. (29). However, the ω -integral over the potential function in Eq. (12) requires an analyticcontinuation to negative values of η , where the integrand has to be understood in a distributionalsense such that the integral is convergent. For NNLL resummation of gluino pairs we need tointegrate I ( ω ) = 1 ω (cid:18) ω m (cid:101) g (cid:19) η e − γ E η Γ(2 η ) (cid:16) I (0) ( ω ) + 16 π a s I (0) S ( ω ) + I (1) ( ω ) (cid:17) , (31)where I (0) ( ω ) = J (0) I (cid:0) E − ω/ (cid:1)(cid:32) a s (cid:34) C I (cid:18) ψ (2 η ) + ln (cid:16) µ s m (cid:101) g (cid:17)(cid:19) (32)+ ( C i + C j ) (cid:18) π
24 + ˆ ψ (2 η ) − ψ (cid:48) (2 η ) + 2 ˆ ψ (2 η ) ln (cid:16) µ s m (cid:101) g (cid:17) + ln (cid:16) µ s m (cid:101) g (cid:17)(cid:19) − (cid:26) C I + 2( C i + C j ) (cid:18) ˆ ψ (2 η ) + ln (cid:16) µ s m (cid:101) g (cid:17)(cid:19) (cid:27) ln (cid:18) ω m (cid:101) g (cid:19) + ( C i + C j ) ln (cid:18) ω m (cid:101) g (cid:19) (cid:35)(cid:33) ,I (0) S ( ω ) = J (0) I (cid:0) E − ω/ (cid:1) ln (cid:18) E − ω/ m (cid:101) g (cid:19) , (33) I (1) ( ω ) = J (1) I (cid:0) E − ω/ (cid:1) . (34)Here, we have defined ˆ ψ ( x ) = γ E + ψ ( x ), where ψ denotes the digamma function. Above thethreshold, the integral runs from 0 to 2 E . As outlined in Ref. [8,9] for instance, analytic continuationto negative values of η can be achieved by a replacement of the integration kernel by a so-called stardistribution which contains subtraction terms that render the integral finite for η > −
1. Explicitly,one applies (cid:90) E dω f (cid:0) E − ω (cid:1) (cid:34) ω (cid:18) ω m (cid:101) g (cid:19) η (cid:35) ∗ (35)= (cid:90) E dωω (cid:104) f ( E − ω − f ( E ) + ω f (cid:48) ( E ) (cid:105)(cid:18) ω m (cid:101) g (cid:19) η + (cid:20) f ( E )2 η − f (cid:48) ( E ) E η + 1 (cid:21) (cid:18) Em (cid:101) g (cid:19) η , (cid:90) E dω f (cid:0) E − ω (cid:1) (cid:34) ln (cid:0) ω m (cid:101) g (cid:1) ω (cid:18) ω m (cid:101) g (cid:19) η (cid:35) ∗ (36)= (cid:90) E dωω (cid:104) f ( E − ω − f ( E ) + ω f (cid:48) ( E ) (cid:105) ln (cid:16) ω m (cid:101) g (cid:17)(cid:18) ω m (cid:101) g (cid:19) η + (cid:34) f ( E )2 η (cid:18) L E − η (cid:19) − f (cid:48) ( E ) E η + 1 (cid:18) L E − η + 1 (cid:19)(cid:35)(cid:18) Em (cid:101) g (cid:19) η , E dω f (cid:0) E − ω (cid:1) ln (cid:0) ω m (cid:101) g (cid:1) ω (cid:18) ω m (cid:101) g (cid:19) η ∗ (37)= (cid:90) E dωω (cid:104) f ( E − ω − f ( E ) + ω f (cid:48) ( E ) (cid:105) ln (cid:16) ω m (cid:101) g (cid:17)(cid:18) ω m (cid:101) g (cid:19) η + (cid:34) f ( E )2 η (cid:18) L E − η L E + 12 η (cid:19) − f (cid:48) ( E ) E η + 1 (cid:18) L E − η + 1 L E + 2(2 η + 1) (cid:19)(cid:35)(cid:18) Em (cid:101) g (cid:19) η , with L E ≡ ln( E/m (cid:101) g ) and f ( E − ω/
2) being a smooth test function on the interval ω ∈ [0 , E ]. Forpositive η , one can drop the star-brackets and the above relations become simple identities. Fornegative η , diverging boundary terms at ω = 0 are removed by the star prescription. In derivingthe fixed order expansion (26), we have assumed positive η in order to perform the convolutionover the LO potential function. We want to stress here that there are different methods for theanalytic continuation. An alternative treatment, based on integration by parts, has been proposedin Ref. [29]. However, due to the identity theorem for holomorphic functions, the continuation to η < η .A different ansatz for analytic continuation is required for the non-Coulomb corrections. Here,one may apply (cid:90) E dω J (0) I (cid:0) E − ω (cid:1) ln (cid:18) E − ω/ m (cid:101) g (cid:19) (cid:34) ω (cid:18) ω m (cid:101) g (cid:19) η (cid:35) ∗ (38)= (cid:90) E dωω (cid:104) J (0) I ( E − ω − J (0) I ( E ) + ω J (0) (cid:48) I ( E ) (cid:105)(cid:18) ω m (cid:101) g (cid:19) η ln (cid:18) E − ω/ m (cid:101) g (cid:19) + (cid:34) J (0) I ( E )2 η ( L E − γ E − ψ (1 + 2 η )) − J (0) (cid:48) I ( E ) E η + 1 ( L E − γ E − ψ (2 + 2 η )) (cid:35)(cid:18) Em (cid:101) g (cid:19) η , where the terms in the last two lines have been evaluated for positive η . See also Ref. [29] for aslightly different treatment. The latter relations can also be applied to the logarithm in Eq. (19) ifone includes the NLO corrections from the Coulomb potential at fixed order.The poles at η = − / η = − / Γ(2 η )in formula (31), see also the discussion in Ref. [8, 9]. The pole at η = 0 is only canceled if there isno logarithm ln( ω ) contained in the integral. Otherwise, the prefactor 1 / Γ(2 η ) is not sufficient togive a final result for η → η → µ s → µ f which, using a runningsoft scale as stated above, is achieved for large velocities ( β ≈ . For instance, it is easy to see that some NLOterms in the soft function dominate the LO term, when η approaches zero. Thus, one should notapply resummation for β close to one. This will exclude the problematic case η → f ( E ) L kE E η with k = 0 , , J (0) I ( E ) L E E η in Eq. (38), which in most cases diverge for E →
0. Inour case, only the repulsive potential J (0) and its products with powers of ln( E ) evaluate to zeroat the threshold. For η < − /
2, the convolution with the parton luminosity is ill-defined, see alsothe equivalent discussion about top-quark pair production in Ref. [29]. However, as outlined in thelatter reference, the continuation to negative values of η can be extended to the convolution overthe hadronic cms energy producing a finite hadronic cross section. Here, I want to thank Pietro Falgari for clarifying discussions. The same statement is also true for the corresponding ansatz of the Mellin-space approach in Eq. (5), of course. g ~ =
800 GeVand m g ~ = m g ~ (cid:144) < Μ < m g ~ NNLO approx
NLOLO ABM11_5n_NNLO6000 8000 10 000 12 000 14 00010 - s @ GeV D Σ @ pb D m g ~ =
800 GeVand m g ~ = m g ~ (cid:144) < Μ < m g ~ NNLO approx
NNLL + NNLO approx
ABM11_5n_NNLO6000 8000 10 000 12 000 14 00010 - s @ GeV D Σ @ pb D Figure 3: Inclusive hadronic cross section versus the squared cms energy. The error bars refer only to scalevariation. Left panel: Fixed order for m (cid:101) g = 800 GeV and m (cid:101) g = 1 TeV. Right panel: NNLO approximationand soft NNLL resummation matched onto the latter. At this point, we want to note that something seems to be different compared to the treatmentin Mellin-space. Suppose that we neglect all kind of Coulomb corrections. It follows that the fixedorder partonic cross section has to vanish near the threshold due to the overall factor β in the borncross section. In Mellin space, this behavior is not spoiled by soft resummation. In the momentum-space approach in the absence of Coulomb corrections, the potential function reduces to the zerothorder in Eq.(17) which is proportional to (cid:112) E/m (cid:101) g ≈ β . If this is multiplied by E η , one finds asingular behavior for E → η < − / p runs from zero to Q with Q being the hard scale of the problem.In any case one should state that resummation defines the partonic cross section as a distributiononly irrespective if Coulomb terms are resummed or not . Therefore, there is no meaningful wayto compare with the purely soft resummation in the Mellin-space formalism at the partonic level,where the threshold behavior mimics the fixed order in the absence of Coulomb terms, but adhocassumptions have been made in order to deal with the Landau singularity.We close this discussion by noting that threshold resummation in general suffers from ambiguitiesdealing with the question of how to treat the Landau pole on the one hand, or such related to thechoice of scales and the matching procedure on the other. The explicit implementation of themomentum-space formalism is passed over to the experts on that approach. In this section, we examine the impact of the NNLL resummation onto the inclusive hadronic crosssection for the examples m (cid:101) g = 800 GeV and m (cid:101) g = 1 TeV. The current exclusion bounds suggesta minimum gluino mass of about 1 TeV [1, 2]. However, we want to stress again that these resultsdepend strongly on the chosen PDF set and the value of α s ( M Z ). The libraries CTEQ6.6 [40] andMSTW08 [41] used in Ref. [1] lead to similar results of the hadronic production cross section. Themore recent sets ABM11 [42] for instance differ from the latter at large parton momentum fraction x Here, I want to thank Martin Beneke and Christian Schwinn for clarifying discussions. (cid:101) g [ GeV] √ s [ TeV] σ LO [ pb] σ NLO [ pb] σ NNLO approx [ pb] σ res , NNLO approx [ pb]MSTW 2008 NNLO7 0.0198 (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1)
800 8 0.0480 (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1)
14 0.9184 (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1)
14 0.1837 (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) ABM11 NNLO7 0.0087 (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1)
800 8 0.0224 (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1)
14 0.5439 (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1)
14 0.1006 (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) Table 1: Hadronic cross section at √ s = 7 , , and 14 TeV, evaluated by using MSTW 2008 and ABM11 PDFsets . The ratio m (cid:101) q /m (cid:101) g is kept fixed to the value 4 /
5. The numbers in brackets correspond to the hadroniccross section evaluated at µ = m (cid:101) g / µ = 2 m (cid:101) g . PDF errors are not included. which is of special importance for heavy (s)particle production. As discussed in Ref. [13], the usage ofABM11 with α s ( M Z ) = 0 . ± . α s ( M Z ) = 0 . ± . approx , the actual choice of thePDF set will only play a minor role. In our analysis, we compare the widely used MSTW08 withABM11 PDFs, where we always use NNLO sets also for the LO and NLO cross sections. We furtherassume a fixed ratio m (cid:101) q /m (cid:101) g = 4 / m (cid:101) q = 640 GeV and m (cid:101) q = 800 GeV. Asmentioned above, this choice has also no significant impact on the result. A variation of the squarkmasses would affect the inclusive hadronic cross section at the percent level only.The theoretical error due to higher order corrections is estimated by a variation of the scale µ = µ f = µ h in the range µ ∈ [ m (cid:101) g / , m (cid:101) g ]. Within the momentum-space approach, one furtherhas the possibility for a variation of the soft scale in order to account for the ambiguities withinthe soft resummation, as well as an independent variation of the hard scale. In principle, the latteris also possible in the Mellin-space approach. For illustration of the perturbative accuracy we willnot include the PDF errors. Plots showing these can be found in Ref. [13]. The convolution of the(resummed) partonic cross sections with the parton luminosity functions according to Eq. (1) isperformed by a VEGAS integration routine using LHAPDF grid files [43] for the PDF sets.In the left panel of Fig. 3, we plot the fixed order inclusive cross section as a function of thehadronic cms energy √ s using ABM11 PDF sets. Explicit numbers for √ s = 7 , , and 14 TeVare given in Table 1. For m (cid:101) g = 800 GeV and √ s = 7 TeV, we observe the K -factors K NLO = σ NLO /σ LO = 1 .
55 and K NNLO = σ NNLO approx /σ NLO = 1 .
30, at 14 TeV we have K NLO = 1 .
50 and K NNLO = 1 .
21. For m (cid:101) g = 1 TeV a similar picture emerges with K NLO = 1 .
63 and K NNLO = 1 . √ s = 7 TeV, and K NLO = 1 .
48 and K NNLO = 1 .
23 at 14 TeV.15n the right panel of Fig. 3, we compare the NNLO approximation to the NNLL resummation,matched onto the latter. Explicit numbers are also given in Table 1. In both examples, the K -factors K NNLL = σ res,NNLO approx /σ NLO are slightly above one ( K NNLL ≈ .
05) over the whole rangeof hadronic cms energies. This is a direct consequence of the partonic result shown in Fig. 2, whereafter convolution with the parton luminosities the corrections due to resummation arise mainly fromthe region β ∈ [0 . , . In this article, we have discussed the effect of NNLL soft resummation on the inclusive cross sectionfor gluino pair production at the LHC by the use of the Mellin-space resummation formalism. Here,we have used analytic results of Ref. [13] and performed the Mellin inversion numerically by the useof standard methods. We have found that soft resummation compensates most of the enhancementof the NNLO soft logarithms compared to the fixed NLO result. This in turn suggests that the NLOcomputation already provides a good theoretical input for constructing exclusion limits at the LHCor, hopefully, to construct confidence regions for future evidences. At this point we want to stressagain that the main source of uncertainty is related to the non-perturbative input when differentPDF sets and initial values for α s ( M Z ) are applied. As discussed in Ref. [13], the individual PDFerrors are not sufficient to cover the resulting discrepancies.We further calculated the hard function required for a joined soft and Coulomb resummation inthe momentum-space approach. Here, we recaptured the relevant points concerning the convolutionover the hadronic cms energy which are given in greater detail in the literature. Finally, we statethat threshold resummation always suffers from ambiguities, no matter which approach is applied. Acknowledgments
I would like to thank Ulrich Langenfeld and Sven-Olaf Moch for good advice and also for providingme with useful pieces of code. I further wish to thank Martin Beneke, Pietro Falgari, and Chris-tian Schwinn for very useful discussions. This work has been supported in part by the DeutscheForschungsgemeinschaft in Sonderforschungsbereich/Transregio 9 and by the European Commissionthrough contract PITN-GA-2010-264564 (
LHCPhenoNet ).16
Matching to the NNLO approximation
In this appendix, we give the relevant part of the NNLO hard function which may be used for thematching onto the NNLO approximation. The specific choice given here sets all NNLO constantsto zero within the resummation formula (12). For the generic scale choice µ s = k s m (cid:101) g β , we obtain H (2) matchij, I = − C ij , I (cid:16) C I (cid:0) − ln(2) (cid:1) + 2( C i + C j ) (cid:0) −
96 ln(2) + 72 ln (2) − π (cid:1)(cid:17) (A.1)+144 C I (cid:0) − ln(2) (cid:1) + 2( C i + C j ) C I (cid:0) −
480 ln(2) + 216 ln (2) + 11 π ln(2) − π (cid:1) +2( C i + C j ) (cid:0) −
768 ln(2) + 576 ln (2) + 44 π ln(2) − π − π + 12172 π (cid:1) + L µ (cid:110) − C ij , I (cid:16) C I + 16( C i + C j ) (cid:0) − (cid:1)(cid:17) − C I (cid:0) − n l − κ ij (cid:1)(cid:0) − ln(2) (cid:1) +24 C I − ( C i + C j ) (cid:16) − C I (cid:0) π + 144 ln(2) (cid:0) − (cid:1)(cid:1) + n l (cid:0) −
272 + 408 ln(2) −
216 ln (2) + 11 π (cid:1) + κ ij (cid:0) −
32 + 48 ln(2) −
36 ln (2) + 116 π (cid:1) −
268 ln(2) −
528 ln(2)+396 ln (2) + 12 π ln(2) − π (cid:17) −
23 ( C i + C j ) (cid:0) −
960 ln(2) + 1008 ln (2) −
432 ln(2) + 61 π ln(2) − π − ζ (3) (cid:1) + γ φ (1) i + γ φ (1) j + γ V (1) ij, I (cid:111) + L µ (cid:110) − C ij , I ( C i + C j ) − C I (cid:0) − n l − κ ij (cid:1) + 2 C I + ( C i + C j ) (cid:16) − C I (cid:0) − (cid:1) + 29 n l (cid:0) −
18 ln(2) (cid:1) +4 κ ij (cid:0) − (cid:1) + 66 ln(2) + π (cid:17) + ( C i + C j ) (cid:0) −
32 + 64 ln(2) −
24 ln (2) + 136 π (cid:1)(cid:111) + L µ (cid:110) ( C i + C j )( −
113 + 2 C I + 29 n l + κ ij ) − C i + C j ) ln(2) (cid:111) + 12 ( C i + C j ) L µ , + ln( k s ) (cid:110) − ( γ φ (1) i + γ φ (1) j ) − γ V (1) ij, I + 4 C I (cid:0) − n l (cid:1)(cid:0) − ln(2) (cid:1) + 24 C I (cid:0) − ln(2) (cid:1) + 12 ( C i + C j ) (cid:16) C I (cid:0) − − π (cid:1) − n l (cid:0) −
408 ln(2)+216 ln(2) − π (cid:1) − + 24 π ln(2) − π (cid:17) + 23 ( C i + C j ) (cid:16) − −
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12 ( C i + C j ) (cid:16)
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