Power corrections and renormalons in Transverse Momentum Distributions
PPrepared for submission to JHEP
Power corrections and renormalons in TransverseMomentum Distributions
Ignazio Scimemi a and Alexey Vladimirov b a Departamento de Física Teórica II, Universidad Complutense de Madrid,Ciudad Universitaria, 28040 Madrid, Spain b Institut für Theoretische Physik, Universität Regensburg,D-93040 Regensburg, Germany
E-mail: [email protected] , [email protected] Abstract:
We study the power corrections to Transverse Momentum Distributions (TMDs)by analyzing renormalon divergences of the perturbative series. The renormalon diver-gences arise independently in two constituents of TMDs: the rapidity evolution kerneland the small-b matching coefficient. The renormalon contributions (and consequentlypower corrections and non-perturbative corrections to the related cross sections) have anon-trivial dependence on the Bjorken variable and the transverse distance. We discussthe consistency requirements for power corrections for TMDs and suggest inputs for theTMD phenomenology in accordance with this study. Both unpolarized quark TMD partondistribution function and fragmentation function are considered. a r X i v : . [ h e p - ph ] F e b ontents β approximation and renormalon divergences 6 β approximation 63.2 The TMD in the large- β approximation 73.3 The TMD anomalous dimensions at large- β and renormalon singularities of D β β D The transverse momentum dependent (TMD) distributions are fundamental non-perturbativeobjects that appear in many relevant processes at LHC, EIC, and e + e − colliders, like VectorBoson Production, Higgs production, Semi-Inclusive Deep Inelastic Scattering, e + e − → hadrons. The factorization theorems which establish the definitions of TMD distributions inQCD and/or in effective field theory have been formulated recently in [1–4], using differentregularization schemes.The perturbative properties of unpolarized TMDs, such as evolution and operatorproduct expansion (OPE) in the regime of small transverse momentum separation, havebeen deduced by several groups using different frameworks (see e.g. [1, 2, 4–9]). The explicitdirect calculation of the TMD evolution function D at NNLO has been provided in [10, 11]and recently it was obtained at N LO [12]. Therefore, nowadays the perturbative knowledgeof the unpolarized TMDs parton distribution functions (PDFs) and fragmentation functions(FFs) is comprehensive, thanks to the results obtained by various groups [11, 13–19]On the contrary, the study of the non-perturbative properties of TMDs has been basedmainly on phenomenological arguments which combine the perturbative information onTMDs with their perturbatively incalculable part [4, 8, 20–25]. These works have lead todifferent forms of implementation of TMDs which in general are not easy to compare. Forinstance, on one hand, the well-known phenomenological considerations of Drell-Yan by[26] and [27] (the so-called BLNY model) implement an ansatz within the standard CSS– 1 –pproach with b ∗ -prescription in the impact parameter space (or b -space). They introduce aset of non-perturbative parameters g , , and all these parameters (including the definitionof b ∗ prescription) are fundamental for these fits. The same model is also the core of theRESBOS program package [28] which is widely used in applications. On another hand, theimplementation of TMDPDFs by [29] does not use b ∗ -prescription. They have found thatpart of the non-perturbative corrections (essentially to the TMD evolution kernel) are neg-ligible. They were able to describe the same data with a different shape of non-perturbativeinput parameterized by two parameters λ , . Fits by other groups that limited themselvesto the analysis of Vector Boson Production and Higgs production are less sensitive to thenon-perturbative input (although it is still necessary) [30, 31]. Additional problems arise inthe consideration of TMDFFs which are known to have very different and/or incomparable(in comparison to TMDPDFs) non-perturbative input.This work is devoted to the study of the leading power corrections to TMD distribu-tions. With this aim, we perform an analysis of the leading renormalon structure of TMDdistributions. A renormalon analysis of the perturbative series gives an important checkof theoretical consistency for any phenomenological ansatz, although it cannot give toostringent restrictions on the fitting parameters. The study of renormalon poles allows tounderstand the asymptotic behavior of the perturbative series and to deduce the form ofthe leading non-perturbative corrections [32–35].An explicit analysis of the renormalon structure for TMDs has never been done to ourbest knowledge, although assumptions on its structure were used even before the actualfield-theoretical definition of TMDs. We refer here, for instance, to the seminal work of [33]about the Sudakov factor in differential cross-section which is usually referred to justifya Gaussian behavior for the non-perturbative part of the TMD evolution kernel [25]. Inorder to describe this effect in the modern TMD framework, we recall that the definitionof TMDs requires the combination of the Soft Function matrix element with the transversemomentum dependent collinear function. As we show in this work, the renormalon diver-gences arise in the perturbative consideration of both of these functions. These renormaloncontributions have different physical meaning and should be treated independently. Firstly,the renormalon divergence of the soft factor results to a power correction within the TMDevolution kernel, which are strictly universal for any TMD due to the universality of the softfactor itself. The leading power correction that we derive here is quadratic. The presenceof these corrections has been shown in [36] by the analysis of the corrections to conformalanomaly. Secondly, the renormalon divergences naturally arise within the coefficients of thesmall- b OPE. A study of those contributions gives access to the next twist corrections ofsmall- b matching and specifies the shape and the general scaling of TMD.The paper is built as the following. We provide the necessary concepts and definitions inSec. 2. In Sec. 3 we perform the calculation of various TMD constituents (such as anomalousdimensions and coefficient functions) within the large- β approximation. In the end of thissection we provide a collection of the main lessons, that follows from our results. The impactof the renormalon divergences on the perturbative series and renormalon subtracted seriesare studied in Sec. 4. One of the main outcomes of the study, namely a consistent ansatzfor TMDs is presented in Sec. (4.2). – 2 – Notation and Basic Concepts
Throughout the paper we follow the notation for TMDs and corresponding functions in-troduced in [14]. The quark TMDPDFs and TMDFFs are given by the following matrixelements F q ← N ( x, b ; ζ, µ ) = Z q ( ζ, µ ) R q ( ζ, µ )2 × (cid:88) X (cid:90) dξ − π e − ixp + ξ − (cid:104) N | (cid:26) T (cid:104) ¯ q i ˜ W Tn (cid:105) a (cid:18) ξ (cid:19) | X (cid:105) γ + ij (cid:104) X | ¯ T (cid:104) ˜ W T † n q j (cid:105) a (cid:18) − ξ (cid:19)(cid:27) | N (cid:105) , ∆ q → N ( z, b ) = Z q ( ζ, µ ) R q ( ζ, µ )4 zN c (2.1) × (cid:88) X (cid:90) dξ − π e − ip + ξ − /z (cid:104) | T (cid:104) ˜ W T † n q j (cid:105) a (cid:18) ξ (cid:19) | X, N (cid:105) γ + ij (cid:104) X, N | ¯ T (cid:104) ¯ q i ˜ W Tn (cid:105) a (cid:18) − ξ (cid:19) | (cid:105) , where R q and Z q are rapidity and ultraviolet renormalization constants, q are quark fieldsand W T are Wilson lines, and ξ = { + , ξ − , b } . The TMDs depend on the Bjorken variables( x for TMDPDFs and z for TMDFFs), the impact parameter b and the factorization scales ζ and µ . The considerations of the TMDPDF and TMDFF are similar in many aspects.Therefore, in order to keep the description transparent we mostly concentrate on the caseof the TMDPDFs, while the results for TMDFFs are presented without derivation.The dependence on the factorization scales is given by the evolution equations, whichare the same for TMDPDF and TMDFF, namely dd ln µ F q ← N ( x, b ; ζ, µ ) = γ q ( µ, ζ )2 F q ← N ( x, b ; ζ, µ ) , (2.2) dd ln ζ F q ← N ( x, b ; ζ, µ ) = −D q ( µ, b ) F q ← N ( x, b ; ζ, µ ) . (2.3)Through the article we consider only the quark TMDs, therefore in the following we suppressthe subscript q on the anomalous dimensions. The values for both anomalous dimensionscan be deduced from the renormalization constants [14]. Also γ and D are related to eachother by the cross-derivatives d D ( µ, b T ) d ln µ = − dγ ( µ, ζ ) d ln ζ = Γ cusp , (2.4)where Γ cusp is the honored cusp anomalous dimension.The solution of the evolution equations Eq. (2.2,2.3) is F ( x, b ; ζ f , µ f ) = R ( b ; ζ f , µ f , ζ i , µ i ) F ( x, b ; ζ i , µ i ) , (2.5)where R is the evolution kernel, R ( b ; ζ f , µ f , ζ i , µ i ) = exp (cid:26) (cid:90) µ f µ i dµµ γ (cid:18) α s ( µ ) , ln ζ f µ (cid:19) (cid:27) (cid:18) ζ f ζ i (cid:19) −D ( µ i , b ) . (2.6)The final values of scaling parameters is dictated by the kinematic of the TMD cross-section. The variable ζ f ∼ Q (with Q being a typical hard scale) is the scale of the– 3 –apidity factorization, and the variable µ f is the scale of hard subprocess factorization. Theintriguing point is that the evolution kernel R is not entirely perturbative, but contains anon-perturbative part. An estimate of the non-perturbative contribution to R is necessaryin order to obtain the cross section in the momentum space where it is usually measured.The non-perturbative part of the evolution kernel is encoded in the D -function whichcan be obtained from the rapidity renormalization constant R q . The definition of therapidity renormalization constant differs from scheme to scheme. In this work we use the δ -regularization scheme defined in [10, 14]. In this scheme, the δ -regularization is used toregularize the rapidity divergences, and the dimensional regularization regularizes the restof divergences. Such a configuration appears to be very effective for the TMD calculus.In particularly, the rapidity renormalization factor R q is expressed via the soft factor S as R q = S − / [14]. In the coordinate space the soft factor is given by the following matrixelement ˜ S ( b T ) = Tr c N c (cid:104) | T (cid:104) S T † n ˜ S T ¯ n (cid:105) (0 + , − , b ) ¯ T (cid:104) ˜ S T † ¯ n S Tn (cid:105) (0) | (cid:105) , (2.7)where we explicitly denote the ordering of operators and S T are Wilson lines, as defined in[10]. Considering the relation between renormalization constants one can show [10], that D = 12 d ln ˜ Sd l δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:15) − finite (2.8)where l δ = ln ( µ / | δδδ | ) . Eq. (2.8) can be used as the formal definition of the TMD evolutionfunction D . In this way, a non-perturbative calculation of the SF gives access to thenon-perturbative structure of D . The soft function is perturbatively universal for bothSemi Inclusive Deep Inelastic Scattering and Drell-Yan type processes. Therefore, theperturbative part of the anomalous dimension D is universal for TMDPDF and TMDFF.One can also expect its universality in the non-perturbative regime.The TMDs are entirely non-perturbative functions. They cannot be evaluated in per-turbative QCD, due to the non-perturbative origin of hadron states. The main subject ofthe paper is the dependence of TMDs on the parameter b which is generically unrestrictedsince it is a variable of Fourier transformation. However it is interesting and numericallyimportant to consider the range of small b (here and later b = √ b ). In this range, theTMDs can be matched onto corresponding integrated parton distributions. At the operatorlevel, the small- b matching is given by the leading term of the small- b OPE. The small- b OPE is a formal operator relation, that relates operators with both light-like and space-likefield separation to operators with only light-like field separation. It reads O ( b ) = (cid:88) n C n ( b , µ b ) ⊗ O n ( µ b ) , (2.9)where C n are Wilson coefficient functions, the µ b is the scale of small- b singularities factor-ization or the OPE matching scale (for simplicity we omit in Eq. (2.9) other matching scalesincluded in the definitions of each component of this equation). Generally, the operators O n are all possible operators with proper quantum numbers and can be organized for instance– 4 –ccording to a power expansion, i.e. twists. In this case, the matching coefficients behaveas C n ( b , µ b ) ∼ (cid:18) bB (cid:19) n f ( ln ( b µ b )) , (2.10)where f is some function. The value of the parameter B is unknown, and its origin is entirelynon-perturbative. In other words, the unknown scale B represents some characteristictransverse size of interactions inside a hadron B (cid:39) O (1GeV) . In practice it is reasonableto consider only the leading term ( n = 0 ) of Eq. (2.9) for b (cid:28) B . In this case, f isan integrated parton distribution (or fragmentation function), and coefficient function iscalled the matching coefficient. So far, the power suppressed terms in Eq. (2.9) has beennot considered, to our best knowledge.For completeness, we recall here the renormalization group properties of the TMDWilson coefficients that we use in the following sections. The evolution equations for thematching coefficients (at µ b = µ ) with respect to ζ is dd ln ζ C f ← f (cid:48) ( x, b T ; µ, ζ ) = −D f ( µ, b T ) C f ← f (cid:48) ( x, b T ; µ, ζ ) , (2.11)where f = q, g species, C f ← f (cid:48) are the matching coefficients on PDFs. It is practicallyconvenient to extract the ζ -dependence from the matching coefficient. We introduce thenotation C f ← f (cid:48) ( x, b T ; µ, ζ ) = exp (cid:16) −D f ( µ, b T ) L √ ζ (cid:17) ˆ C f ← f (cid:48) ( x, L µ ) . (2.12)Here and further we use the following notation for logarithms L X = ln (cid:18) X b e − γ E (cid:19) , l X = ln (cid:18) µ X (cid:19) . (2.13)The ζ -free coefficient function ˆ C satisfies the following renormalization group equation µ ddµ ˆ C f ← f (cid:48) ( x, L µ ) = (cid:88) r (cid:90) x dyy ˆ C f ← r (cid:18) xy , L µ (cid:19) K fr ← f (cid:48) ( y, L µ ) , (2.14)where the kernel K is K fr ← f (cid:48) ( x, L µ ) = δ rf (cid:48) δ (1 − x )2 (cid:16) Γ fcusp L µ − γ fV (cid:17) − P r ← f (cid:48) ( x ) , and P ( x ) is the splitting function (DGLAP kernel). The matching coefficient for TMDFF C f → f (cid:48) satisfies the same set of evolution equation with only substitution of PDF splittingfunction P ( x ) by the FF ones, P ( z ) /z [14]. Using these equations one can find the expres-sion for the logarithmic part of the matching coefficients at any given order, in terms of theanomalous dimensions and the finite part of the coefficient functions. The expressions forthe anomalous dimensions, the recursive solution of the RGEs and the explicit expressionsfor the coefficients C and C can be found, e.g. in [14].– 5 – TMD in large- β approximation and renormalon divergences The leading non-perturbative contribution to the perturbative series is commonly associatedwith renormalons. The renormalon contributions were intensively studied for various matrixelements and in different regimes, for review see [37, 38]. A typical signature of renormalonsis the factorial divergence of the perturbative series. These divergences are often discussedin terms of the corresponding singularities in the Borel plane.The best representative and the only stable way to study the renormalon divergencewithin perturbative QCD is the large- β approximation. The large- β expression can beobtained from the large- N f expression through the procedure of "naive Abelianization"[39, 40]. In this section, we present the calculation of large- β correction to TMDs. Sincethe technique of large- N f calculus is well-known, we skip the detailed evaluation (redirectingthe reader to the related literature) and present only intermediate expressions. β approximation The soft function matrix elements is a key structure for the TMD construction and as suchit is a good starting point for the renormalon analysis. The large- β calculation of the softfactor runs in parallel to the calculation of the integrated soft factor for Drell-Yan, whichis presented in [32] (see Sec.5.3). Here we present our results of the evaluation.To begin with, we evaluate the large- N f contribution to the soft factor, which is given bythe "bubble" resummed diagram, shown in Fig.1.A. The expression for the (renormalized)diagram with n -bubble insertion isSF n = − C F β f (cid:32) a s β f − (cid:15) (cid:33) n +1 n (cid:88) k =0 n ! k !( n − k )! (3.1) ( − k n − k + 1 G ( − (cid:15), − ( n + 1 − k ) (cid:15) ) ( L δ − ψ ( − ( n − k + 1) (cid:15) ) − γ E ) , where β f = T r N f , a s = g / (4 π ) , (cid:15) is the parameter of dimension regularization ( d =4 − (cid:15) ), δδδ = | δ + δ − | with δ +( − ) the being parameters of rapidity regularization for Wilsonlines pointing in n ( ¯ n )-direction [14]. The function G is a standard function that appears inthe large- β calculation [32, 37, 39, 40], and is given by the expression G ( (cid:15), s ) = e sγ E BBB − sµ A s/(cid:15) − − (cid:15) Γ(1 + s )Γ(1 − s + (cid:15) ) , (3.2)with A (cid:15) = 6Γ(1 + (cid:15) )Γ (2 − (cid:15) )Γ(4 − (cid:15) ) , BBB µ = b µ e − γ E . Here, the Euler-Mascheroni constant is a result of the MS scheme. For n = 1 , thisexpression agrees with the direct calculation of the soft factor in δ -regularization [10]. Wealso introduce an additional function for the double-pole part ˜ G ( (cid:15), s ) = − G ( (cid:15), s )( ψ ( s ) + γ E ) . (3.3)– 6 –he functions G and ˜ G have the following Taylor series G ( (cid:15), s ) = ∞ (cid:88) j =0 G j ( (cid:15) ) s j = ∞ (cid:88) j =0 s j ∞ (cid:88) k =0 g [ j ] k (cid:15) k , (3.4) ˜ G ( (cid:15), s ) = ∞ (cid:88) j =0 ˜ G j ( (cid:15) ) s j − = ∞ (cid:88) j =0 s j − ∞ (cid:88) k =0 ˜ g [ j ] k (cid:15) k . (3.5)These expressions define the coefficients g [ j ] k and G j . Note, that g [0] k = ˜ g [0] k and g [1] k = ˜ g [1] k .The procedure of "naive Abelianization" consists in the replacement of N f by thecorresponding β expression [39], i.e. β f = 43 T r N f −→ − β = − C A + 43 T r N f . (3.6)In this way, we obtain the large- β expression for the soft factorSF = − ∞ (cid:88) n =0 C F c n +1 s β (cid:34) ( − n n ! (cid:16) L δδδ G n +1 ( − (cid:15) ) + ˜ G n +2 ( − (cid:15) ) (cid:17) (3.7) + ( − n n + 1 (cid:32) − L δδδ G ( − (cid:15) ) (cid:15) n +1 + ˜ G ( − (cid:15) ) (cid:15) n +2 ( ψ ( n + 2) + γ E ) − ˜ G ( − (cid:15) ) (cid:15) n +1 (cid:33) (cid:35) , where we have introduced the large- β coupling constant c s = β a s > . Note, that in Eq. (3.7) the terms suppressed in (cid:15) are dropped.Eq. (3.7) gives access to the anomalous dimension D , which we study in Sec. 3.3, andto the rapidity renormalization factor R q . The factor R q (we recall that it is equal to R q = S − / in the δ -regularization [14]) from the perspective of the large- β approximationhas the same perturbative combinatorics as the one-loop-truncated pertrubation series. Itis given by R q = 1 − SF (3.8)at δ − = ζ/p + and SF given in Eq. (3.7). This expression is used in the next section toextract the large- β expression of the Wilson coefficients of small- b OPE. β approximation To obtain the TMD matching coefficient one should evaluate the diagrams B and C, whichare shown in Fig. 1. The result for the sum of these diagrams and their Hermitian conju-gations is Φ q ← q = 2 C F β f ∞ (cid:88) n =0 ( a s β f ) n +1 ( − (cid:15) ) n +1 n (cid:88) k =0 n ! k !( n − k )! ( − k G ( − (cid:15), − ( n − k + 1) (cid:15) ) n − k + 1 (3.9) (cid:34) ¯ xx ( n − k ) (cid:15) (1 − (cid:15) )(1 + ( n − k ) (cid:15) ) + 2 x n − k ) (cid:15) (1 − x ) + − δ (¯ x ) ln (cid:18) δ + p + (cid:19) (cid:35) , – 7 – igure 1 . Diagrams contributing to the leading order of large- N f limit. The diagram A is thecontribution to the soft factor. Diagrams B and C are contribution to the matching coefficient.The counter term diagrams are not shown. where we have used the same notation as in Eq. (3.1) and ¯ x = 1 − x . The last term insquare brackets represents the rapidity divergence which appears in the diagram C . For n = 0 , this expression reproduces the result of explicit calculation made in [13].Using Eq. (3.8) and Eq. (3.9) we can complete the result for the large- N f expressionof the TMDPDF, R q Φ = Φ q ← q − SF C F β f ∞ (cid:88) n =0 ( a s β f ) n +1 ( − (cid:15) ) n +1 n (cid:88) k =0 n ! k !( n − k )! ( − k G ( − (cid:15), − ( n − k + 1) (cid:15) ) n − k + 1 (3.10) (cid:34) ¯ xx ( n − k ) (cid:15) (1 − (cid:15) )(1 + ( n − k ) (cid:15) ) + 2 x n − k ) (cid:15) (1 − x ) + + δ (¯ x ) ( L µ − l ζ − ψ ( − ( n − k + 1) (cid:15) ) − γ E ) (cid:35) . Here, we observe the cancellation of the rapidity divergences that leaves the residual l ζ dependence.In order to extract the matching coefficient of the TMDPDF onto the PDF one has toproceed to the renormalization of Eq. (3.10). This is greatly simplified in the δ -regularizationscheme, where all virtual graphs and integrated graphs are zero. The only non-zero con-tribution is the UV counterterm which is a pure (cid:15) -singularity. The accounting of this parteliminates terms singular in (cid:15) , leaving the finite part unchanged. The latter provides thecoefficient function. Performing the "naive Abelianization" as in Eq. (3.6) we obtain thelarge- β result C q ← q = 2 C F β ∞ (cid:88) n =0 c n +1 s (cid:40)(cid:34) ¯ x + 2 x (1 − x ) + (cid:35)(cid:34) γ n +1 ( x ) n + 1 + ( − n n ! g [ n +1]0 [ BBB √ xµ ] (cid:35) (3.11) + ¯ xn + 1 (2 γ n ( x ) + γ n − ( x )) − ¯ x ( − n n ! g [ n ]0 [ BBB √ xµ ]+ δ (¯ x ) ( L µ − l ζ ) (cid:34) g [0] n +1 n + 1 + ( − n n ! g [ n +1]0 (cid:35) + δ (¯ x ) (cid:34) ˜ g [0] n +2 ψ ( n + 2) + γ E n + 1 + ˜ g [1] n +1 ( n + 1) + ( − n n !˜ g [ n +2]0 (cid:35)(cid:41) , where BBB √ xµ = xBBB µ , and x (cid:15) G ( (cid:15) ) = ∞ (cid:88) k =0 γ k (cid:15) k . – 8 –he additional variable in the square brackets for the functions g indicates the modifiedvalue of BBB µ to be substituted.The calculation of TMDFFs matching coefficient proceeds in the same way as for TMD-PDFs. The result of the calculation is z C q ← q = 2 C F β ∞ (cid:88) n =0 c n +1 s (cid:40)(cid:34) ¯ z + 2 z (1 − z ) + (cid:35)(cid:34) γ n +1 ( z − ) n + 1 + ( − n n ! g [ n +1]0 [ BBB µ/ √ z ] (cid:35) (3.12) + ¯ zn + 1 (cid:0) γ n ( z − ) + γ n − ( z − ) (cid:1) − ¯ z ( − n n ! g [ n ]0 [ BBB µ/ √ z ]+ δ (¯ z ) ( L µ − l ζ ) (cid:34) g [0] n +1 n + 1 + ( − n n ! g [ n +1]0 (cid:35) + δ (¯ z ) (cid:34) ˜ g [0] n +2 ψ ( n + 2) + γ E n + 1 + ˜ g [1] n +1 ( n + 1) + ( − n n !˜ g [ n +2]0 (cid:35) − n +1 (cid:88) r =1 (cid:18)(cid:18) ¯ z + 2 z − z (cid:19) γ n − r +1 ( z ) + ¯ z (2 γ n − r ( z ) + γ n − r − ( z ) (cid:19) ( − r ln r ( z )( n + 1) r ! (cid:41) . One can see that the expression for TMDPDF Eq. (3.11) is related to the first four linesof the expression for TMDFF Eq. (3.12) by the crossing relation x → z − . The last line ofEq. (3.12) is specific for TMDFF and it is an effect of the expansion of the normalizationfactor z − (cid:15) .One can check that at n = 0 , the expressions (3.11) and (3.12) coincide with the onecalculated in [14]. β and renormalon singularitiesof D In the articles [10, 14] it was shown that in the δ -regularization scheme the anomalousdimension D can be obtained from the rapidity singular part of the soft factor as in (2.8).Considering the Eq. (3.7) we obtain the anomalous dimension D in the large- β approxi-mation D = − C F β ∞ (cid:88) n =0 c n +1 s (cid:32) ( − n n ! g [ n +1]0 + g [0] n +1 n + 1 (cid:33) . (3.13)The first term in the brackets of Eq. (3.13) behaves ∼ n ! at large n , and represents therenormalon singularity.At this point it is convenient to consider the Borel transformation of the result. Wedefine the Borel transformation of a perturbative series in the usual way f ( c s ) = ∞ (cid:88) n =0 f n c n +1 s = ⇒ B [ f ]( u ) = ∞ (cid:88) n =0 f n u n n ! . (3.14)A perturbative series is Borel summable if an integral ˜ f = (cid:90) ∞ due − u/c s B [ f ]( u ) , (3.15)– 9 –xists. Performing the Borel transformation on the D function and applying Eq. (3.15), wefind D = − C F β (cid:18)(cid:90) c s dx G ( x, − x − (cid:90) ∞ du G (0 , − u ) − u e − u/c s (cid:19) . (3.16)The first term is analytical and reproduces the cusp-anomalous dimension at large- β [32] Γ cusp ( c s ) = 4 C F c s β Γ(4 + 2 c s )6Γ (2 + c s )Γ(1 + c s )Γ(1 − c s ) = 4 C F c s β G ( c s , . (3.17)The function which appears in the second term G (0 , − u ) = BBB uµ e ( − γ E ) u Γ(1 − u )Γ(1 + u ) , (3.18)contains a series of poles at u = 1 , , ... which correspond to infrared renormalons. One cancheck explicitly that the relation Eq. (2.4) holds for large- β expression, due to cancellationof the renormalon divergences in the second term of Eq. (3.16) between derivative of couplingconstant (in the Borel exponent) and derivative of BBB µ (in the function G (0 , − u ) ).There are multiple possibilities to define the sum Eq. (3.13), e.g. one can slightlyshift the integration contour for Eq. (3.16) into the complex plane. The difference betweenintegrals passing from the lower and upper sides of poles is called infrared (IR)-ambiguityand is given by a ( − π ) times the residue at the pole. For the anomalous dimension D itreads δ IR {D} = c b Λ , (3.19)where c = πC F β e (cid:39) . . (3.20)The IR-ambiguity represents the typical scale of the error for perturbative series.The same conclusion, namely the presence of a b -correction for D , was made in Ref. [36]using different argumentation. In Ref. [36] the factorized cross-section has been consideredwithin the soft collinear effective field theory (SCET). It has been shown that the powercorrection to the soft factor which arises in the next-to-leading term of large- Q OPE,is proportional to the soft factor matrix element. Exponentiating the power correctionone obtains the same result as presented here. It is an expected agreement because therenormalon calculation is equivalent to the calculation of the correction term of OPE.The anomalous dimension γ V can be extracted from the coefficient function Eq. (3.11).We consider the derivative of coefficient function at l ζ = L µ µ ddµ ˆ C q ← q ( x, L µ ) = (cid:90) x dyy ˆ C q → q (cid:18) xy , L µ (cid:19) (cid:18)
12 (Γ cusp L µ − γ V ) δ (¯ y ) − P q ← q ( y ) (cid:19) , (3.21)where we have dropped the mixing among flavors. The DGLAP kernel at large- β is givenby the expression P q ← q ( x ) = 2 C F β ∞ (cid:88) n =0 c n +1 s (cid:40) x − x γ n ( x ) + ¯ x (2 γ n − ( x ) + γ n − ( x )) (cid:41) + . (3.22)– 10 –onsidering the derivative of Eq. (3.11) and comparing right and left hand sides of Eq. (3.21)we obtain γ V = − C F β (cid:40) c s (cid:18) ψ (1 + c s ) + 2 γ E + 3 − c s (1 + c s )(2 + c s ) (cid:19) G ( c s , (3.23) + ln ( G ( c s , c s )) G ( c s ,
0) + (cid:90) G ( xc s , − G ( c s , − x dx (cid:41) . This expression contains no singularity, and hence it is renormalon-free, as it is usuallyexpected for an ultraviolet anomalous dimension. β Before the evaluation of the sums in Eq. (3.11-3.12) we extract the part related to theanomalous dimension D to obtain the coefficients ˆ C defined in Eq. (2.12). This procedureis important since the function D contains its own renormalon singularities, as describedin Eq. (3.19). The contribution of D is easily recognized in the third lines of (3.11-3.12)(compare with Eq. (3.13)).The result of the Borel transform for the coefficient ˆ C , Eq. (3.11) is ˆ C = 2 C F β (cid:90) ∞ due − u/c s (cid:40) (cid:18) ¯ x + 2 x (1 − x ) + (cid:19) γγγ ( u ) − u + ¯ x (cid:90) dy (2 + u ¯ y ) γγγ ( yu )+ δ (¯ x ) (cid:18) GGG ( u ) − u + (cid:90) dy GGG (cid:48) ( u ) − GGG (cid:48) ( yu ) u (1 − y ) (cid:19) + (cid:18) ¯ x + 2 x (1 − x ) + (cid:19) (cid:18) − G [ BBB √ xµ ](0 , − u ) u (cid:19) − ¯ xG [ BBB √ xµ ](0 , − u )+ δ (¯ x ) (cid:32) G (0 , − u )( ψ ( − u ) + γ E ) u − u − LLL µ + u (cid:33) (cid:41) , (3.24)where by bold font we denote the Borel transformed functions, G i ( u ) = ∞ (cid:88) n =0 g [ i ] n u n n ! , GGG (cid:48) ( x ) = ddxGGG ( x ) , γ ( u ) = ∞ (cid:88) n =0 γ n ( x ) u n n ! . (3.25)The terms in Eq. (3.24) are collected such that every bracket is finite at u → . The expres-sion for TMDFF coefficient function ˆ C can be obtained using the crossing transformation( x → z − ) and the addition of the normalization contribution (the last line in Eq. (3.12)).In the last two lines of Eq. (3.24) we have the infrared renormalon poles in u = 1 , , .. .One can see that the third line contains only first order poles, while the last line containssecond order poles at G (0 , − u ) ψ ( − u ) . Considering the infrared ambiguity at u = 1 weobtain δ IR { ˆ C } = − c ( x b Λ ) (cid:40) x + 2 x (1 − x ) + − δ (¯ x ) (cid:18) L Λ + 23 (cid:19) (cid:41) , (3.26)– 11 –here constant c is given in Eq. (3.20). The x − dependence of this expression exactly reproduces the x − dependence of the leading terms of the next power correction in small- b OPE, see detailed description in [41]. The consideration of ambiguites of higher renormalonpoles gives access to the higher-power corrections. We obtain δ u = nIR { ˆ C } = πC F β (cid:16) − x b Λ e (cid:17) n n ! n ! (cid:18) x (1 − x ) + + ( n + 1)¯ x − δ (¯ x ) (cid:18) L Λ − ψ n +1 − γ E + 53 (cid:19)(cid:19) . (3.27)However, these expressions can be modified by the infrared renormalon contributions of thehigher-twist terms. The most important information of the higher-power corrections is thatthe renormalons scale as x b , but not as b which is a naive assumption. The consequencesof this fact are discussed in the next sessions.The corresponding calculation for TMDFF gives δ IR { z ˆ C } = − c (cid:18) b Λ z (cid:19) (cid:40) z + 2 z (1 − z ) + − δ (¯ z ) (cid:18) L Λ + 23 (cid:19) (cid:41) . (3.28)which is the same as Eq. (3.26) with the crossing change x → /z . One can see that thedifference in normalization which spoils the crossing between TMDPDFs and TMDFFs,disappears in the renormalon contribution. The higher poles ambiguites are provided usingthe crossing relation x → /z in Eq. (3.27). β The Eq. (3.19, 3.26, 3.28) are one of the main results of this work. These expressions rep-resent the leading power correction to the small- b regime, where all perturbative propertiesof TMDs are derived. These expressions give access to a general structure of the next-to-small- b regime. The practical implementation of results Eq. (3.19, 3.26, 3.28) is given inthe next section, while here we collect the most important observation that follows fromthe large- β calculation and which should be taken into account for TMD phenomenology.The first, and the most obvious, observation is that the leading power corrections are ∼ b . It implies that an exponential decay of the TMDs that is sometimes suggested inphenomenological studies (e.g. [42, 43]) can in no way affect the small- b region. Indeed, itwould imply the corrections ∼ √ b to the small- b OPE, that cannot appear without extrascaling parameter. Nonetheless, exponential corrections can occur in the large- b regime,which is inaccessible by perturbative considerations.Second, one can see that the renormalon corrections to TMDPDFs matching coefficientscales like x b , and not as simply b (as it is usually assumed), nor as x b (as suggested byLaguerre polynomial decomposition [9]). Therefore, the contributions of higher-twist termsin small- b OPE for TMDPDF are largely functions of x b . Correspondingly, TMDFFsmatching coefficients are a function of b /z . This is important in respect of the phe-nomenological implementation of the TMDs. For instance, the b ∗ -prescription which isoften adopted does not respect this scaling and so, in this sense, it is not fully consistentwith the estimated higher twist effects. – 12 –hird, the renormalon contributions to the anomalous dimension D and to matchingcoefficients have different physical origins and do not mix with each other. In fact, theanomalous dimension D is an universal object that is the same for all regimes of b andfor TMDs of different quantum numbers [25]. Thus, the renormalon contribution to D represents a generic universal non-perturbative contribution, alike in the case of heavy quarkmasses. On the other hand, the (infrared) renormalon divergences within the matchingcoefficients are to be canceled by the corresponding (ultraviolet) renormalon contributionsof higher twists. Therefore, while Eq. (3.19) represents a size of a universal non-perturbativecontribution, Eq. (3.26, 3.28) give the form of the twist-four contribution to small- b OPE.In other words, Eq. (3.26, 3.28) estimate very accurately the x -behavior of subleadingcorrection to small- b OPE.The consideration of the anomalous dimension D for gluon distributions is identical tothose of quarks (apart of trivial replacing of common the factor C F by C A ). Contrary, thecalculation of the renormalon contribution for gluon and quark-gluon matching coefficientis much more complicated than the one presented here and is beyond the scope of thispaper. In general, we can expect a non-trivial dependence of the renormalon contributionon the Bjorken variables. At present, we cannot find arguments which suggest a locationfor the renormalon poles and an x b scaling different from that of quarks. Our analysis is limited to the quark TMDs only. Nonetheless, we can advance some con-siderations on possible inputs, which are consistent with our findings and evaluate theirimpact on the non-perturbative structure of TMDs. The suggested ansatz for TMDs doesnot pretend to be unique and moreover is inspired by other popular models. We postponeto a future publication a more dedicated study on the subject.We recall here the form of the TMDPDFs which emerges at small- b is F pertq ← N ( x, b ; ζ f , µ f ) = R ( b , ζ f , µ f ; ζ b , µ ) (cid:88) j (cid:90) x dyy ˆ C q ← j (cid:18) xy , b ; µ (cid:19) f j ← N ( y, µ ) , (4.1)where the evolution kernel R is given in Eq. (2.6). The argument ζ b of R is collected fromthe combination of two exponents: the original factor R (2.6) and the exponential prefactorof ˆ C (2.12), and it takes the value ζ b = 4 e − γ E b . The analogue equation for TMDFFs is obtained replacing consistently the PDF f j ← N bythe fragmentation function d j → N and the coefficient function C q ← j by C q → j , while theevolution kernel remains the same. This expression is usually taken as an initial ansatz forTMD phenomenology.As we pointed earlier there are two places where the non-perturbative effects arise. Thefirst one is the evolution kernel D which is a part of the evolution prefactor R , and it iscommon for all TMDs (TMDPDFs and TMDFFs of various polarizations). The second oneis the higher twist corrections to the small- b OPE. These non-perturbative contributions– 13 –re of essentially different origin and should not be mixed. In particular it is important torealize that the non-perturbative contribution of D enters Eq. (4.1) as a prefactor, while thehigher order terms of OPE are added to the convolution integral. Therefore, the structureof non-perturbative corrections to TMD that we keep in mind is the following F q ← N ( x, b ; ζ f , µ f ) = exp (cid:40) (cid:90) µ f µ dµ (cid:48) µ (cid:48) γ (cid:0) µ (cid:48) , ζ f (cid:1) (cid:41) (cid:18) ζ f ζ b (cid:19) −D ( µ, b ) −D NP ( b ) × (4.2) (cid:88) j (cid:90) x dyy ˆ C q ← j (cid:18) xy , b ; µ (cid:19) f j ← N ( y, µ ) + f NPq ← N ( x, b ; µ ) . Here, D NP is the non-perturbative addition to the anomalous dimension D , and f NP is thecumulative effect of the higher twist corrections to the small- b OPE. At small (perturbative) b , the non-perturbative parts should turn to zero, such that Eq. (4.2) reproduces Eq. (4.1).In the following subsections we construct a minimal non-contradicting anzatz for TMDdistributions that respect the study of large- β approximation. D The non-perturbative part of the anomalous dimension D is one of the most studied in theliterature and the one for which a general consensus is achieved. Usually, the anomalousdimension D is assumed to have quadratic behavior in the non-perturbative region. As weshow in Eq. (3.19) the quadratic behavior is also suggested by the large- β approximation.A more subtle issue concerns the amount of non-perturbative correction to D , which can bevery different depending on the implementation of the TMDs. A check of the renormaloncontribution, as provided in this section, gives an estimate of such correction and it is souseful for practical implementations.Let us present the perturbative series for D in the form D ( µ, b ) = C F β ∞ (cid:88) n =1 ( β a s ( µ )) n ( d n ( L µ ) + δ n ( L µ )) , (4.3)where d n ∼ n ! g [ n +1]0 can be obtained from Eq. (3.13) and δ n is the large- β suppressed part.The numerical comparison of the large- β expression Eq. (3.16) and the exact expression for D is given in the Tab. 1. One can see that generally the large- β expression overestimatesthe exact numbers , which is typical for this approximation.In order to study the properties of the large- β series we introduce a function for itspartial sum M N ( µ, b ) = 1 β N (cid:88) n =1 ( β a s ( µ )) n d n ( L µ ) . (4.4)For N → ∞ the sum is divergent, as discussed in Section 3.3. In order to define M ∞ we consider the Borel transform of M N as in Sec. 3.3. To define the Borel integral inEq. (3.16), we shift the integration contour, slightly above the real axis. The real part of– 14 – d n + δ n d n δ n L µ L µ L µ + 2 . L µ − . L µ + 3 . L µ + 3 . − . L µ − . . L µ + 2 . L µ + 0 . L µ − .
41 0 . L µ + 3 . L µ + 5 . L µ +7 . − . L µ − . L µ − . Table 1 . Numerical comparison of the large- β component of the anomalous dimension D to theexact expression. The coefficients d n and δ n are defined in Eq. (4.3). µ =10 GeV b = 0 . b = 1 . b = 3 . M M M M M M M M ∞ ± δM . ± .
001 0 . ± .
072 0 . ± . Table 2 . The values of partial sums M N at several values of b . The estimate converge value M ∞ and its error band δM are obtained as described in the text the integral (i.e. the principal value integral) gives M ∞ , while the imaginary part representsthe errorband for this estimation. The explicit expression for the latter is δM ( µ, b ) = 2 πβ (cid:104) J (cid:16)(cid:112) µ b e − β as ( µ ) (cid:17) − (cid:105) , (4.5)and the leading behavior at small- b for δM is given by the infrared ambiguity Eq. (3.19).We investigate the convergence of the partial sums of M N to its Borel resummed value M ∞ , in order to find the scale at which the non-perturbative corrections associated withrenormalons become important. The numerical values of partial sums at µ = 10 GeV andat several values of b are presented in Tab. 2 . The graphical representation of these valuesis shown in Fig. 2. The convergence of the series is perfect (in the sense that it convergesat M that is far beyond the scope of modern perturbative calculations) for the range of b (cid:46) GeV − , it becomes weaker at b ∼ GeV − , and it is completely lost at b (cid:38) GeV − .These are the characteristic scales for switching the perturbative and non-perturbativeregimes in D . In other words, the perturbative series can be trustful at b (cid:46) GeV − , butcompletely loses its prediction power for b (cid:38) GeV − . The number N at which convergenceis lost depends on the value of µ , however the interval of convergence in b is µ -independent,e.g. at µ = 50 GeV the series converges to M in the region b (cid:46) GeV − , but again losesstability at ∼ GeV − .In order to proceed to an estimate of the non-perturbative part of D we write it in the– 15 – b T (cid:45) M N Figure 2 . The dependence of partial sums M N on b (in GeV − ). The dashed lines represent M N from N = 1 (bottom line) till N = 7 (top line). The bold line is the value of M ∞ . The shaded areais the error band of M ∞ given by δM . form D ( µ, b ) = (cid:90) µµ dµ (cid:48) µ (cid:48) Γ cusp ( µ ) + D P T ( µ , b ) + D NP ( µ , b ) , (4.6)where D P T is given by the perturbative expression at µ scale, D NP encodes the non-perturbative part. The parameter µ depends on b and should be selected such that a s ( µ ) is a reasonably small number. The non-perturbative part D NP is independent on µ (sincethe evolution part of D is renormalon-free) but depends on the choice of µ .In principle, the best value of the parameter µ can be extracted from the large- β calculation. Indeed, the resummation of bubble-diagrams modifies the coupling in theinteraction vertex, such that a loop integral appears to be naturally regularized in theinfrared region. Practically, the effect of such resummation can be presented as a freezingof the coupling constant at large b . Particularly popular is the b ∗ prescription [44] definedas µ = µ b = C b ∗ ( b ) , b ∗ ( b ) = √ b (cid:112) b /b max , C = 2 e − γ . (4.7)At large b the parameter µ approaches C /b max , which should be chosen much less then Λ , i.e. b max (cid:28) C / Λ ∼ GeV − .For large- b (say b (cid:38) GeV) the non-perturbative part of D dominates the perturbativeone. The large- β calculation allows to estimate the leading contribution (from the side ofsmall- b ’s) to D NP from the infrared ambiguity Eq. (3.19), D NP ( b , µ ) = c Λ b g D ( b, µ ) , (4.8)the function g D should be of order of unity at small- b and it depends on the choice ofthe scale µ . Here Λ is the position of Landau pole and it is expected to be of order O (Λ QCD ) ∼ MeV, which implies c Λ = πC F e / β Λ ∼ .
075 GeV . (4.9)– 16 –ince the large- β approximation overestimates the exact values this number can be con-sidered as an upper bound for non-perturbative input.In order to estimate the parameters of the D more accurately, we consider a kind ofrenormalon subtraction scheme for the anomalous dimension D . We construct a renormalonsubtracted expression D ( µ, b ) = D RS ( µ, b ) by explicitly summing the large- β contributionin Eq. (4.3) D RS ( µ, b ) = M ∞ ( µ, b ) + C F β ∞ (cid:88) n =1 ( β a s ) n δ n ( L µ ) . (4.10)The scale µ here should be chosen such that the logarithm L µ is reasonably small, otherwisethe large- β expansion is significantly violated. Using the model Eq. (4.10) we fit theparameters of Eq. (4.6) at µ = 10 GeV in the range b < GeV, with g D = constant ≡ g K ,at all known perturbative orders. It appears that the result is very stable with respect to b max whose best value we find to be b max (cid:39) (1 . ± . GeV − . (4.11)Concerning the non-perturbative part, it appears to be lower then the crude estimationEq. (4.9) and actually consistent with 0, g K (cid:39) (0 . ± . GeV . (4.12)This value is generally smaller then the typical values presented in the literature, e.g. Ref. [8]quotes g K (cid:39) . GeV , Ref. [45] quotes g K (cid:39) . ± . GeV . But Ref. [29] finds g K consistent with 0, which agrees with the present findings. However, one should take intoaccount that contrary to standard fits, the present considerations are purely theoretical.Moreover in fits with experimental data, one should consider the extra non-perturbativepart of the TMD distribution itself (which is discussed in the next section).Finally, we comment on the possibility of a more sophisticated renormalon subtractionscheme as in the MSR scheme of [46]. In this scheme, one provides a subtraction of therenormalon from a perturbative series which depend on an additional scale µ R . The newrenormalon subtraction scale can result into large logarithms which, in turn, should beresummed. Such a consideration can result in more accurate restrictions on parameters. The non-perturbative corrections to the matching coefficients are necessary for all analysiswhich include low energy data. These corrections have not been deeply studied in QCDtheory and up to now, only a phenomenological treatment has been provided. In this section,we present a consistent ansatz that interpolates the perturbative small- b part of a TMDdistribution with an entirely Gaussian exponent at large- b . The presented ansatz takes intoaccount the lessons learned from the study of renormalon singularities and formulated inSec. 3.5.The renormalon contribution accounts the leading power correction (see detailed ex-planation e.g. in [38, 41, 49]). Thus, the small- b expansion of the TMD distribution, that– 17 –ncludes this power correction, has a form ˆ F q ← N ( x, b ; µ ) = (cid:88) j (cid:90) x dyy (cid:16) ˆ C q ← j ( y, b ; µ ) + yg inq b C renq ← j ( y, b ) (cid:17) f j ← N (cid:18) xy , µ (cid:19) + O ( b ) , (4.13) ˆ D q → N ( z, b ; µ ) = (cid:88) j (cid:90) z dyy (cid:32) ˆ C q → j ( y, b ; µ ) + g outq b y C renq → j ( y, b ) (cid:33) d j → N (cid:18) zy , µ (cid:19) + O ( b ) , (4.14)where the LO coefficient function of the renormalon contribution was calculated in Sec. 3.4and reads C renq ← q ( x, b ) = C renq → q ( x, b ) = 2¯ x + 2 x (1 − x ) + − δ (¯ x ) (cid:18) L Λ + 23 (cid:19) . (4.15)The constants g in,outq are of order c Λ within the large- β approximation, however the actualvalue should be estimated from data. The non-perturbative scale Λ is the same as in thecase of the evolution kernel. The contribution presented here is at LO, and as such hasnot µ -dependence. The µ -dependence of higher perturbative orders can in principle becalculated, using the evolution equation for TMD and the related integrated distribution.At larger values of b Eq. (4.14) is corrected by the higher orders of the OPE, and ata particular scale B (which defines the convergence radius of small- b OPE Eq. (2.10)) itis replaced by a single and entirely non-perturbative function. It is commonly assumedthat at large- b the TMD distribution has Gaussian behavior. This is also supported bythe phenomenological studies of low-energy data (see e.g. Ref. [50] for a study dedicatedto this issue). The interpolation of a Gaussian with the small- b matching Eq. (4.13-4.14)should take into account the previously formulated demands on the power corrections. Inparticular, we have the following two guidelines:(i) In order to be consistent with the general structure of OPE, the interpolation shouldbe done under the convolution integral.(ii) According to the structure of renomalon singularities, the powers of b should bealways supplemented by x (for PDF) and z − (for FF).A viable model, which takes into account both these points, can have the form ˆ F q ← N ( x, b ; µ ) = (4.16) (cid:88) j (cid:90) x dyy e − g b y b (cid:18) ˆ C q ← j ( y, b ; µ ) + yg q b (cid:18) C renq ← j ( y, b ) + δ (¯ y ) g b g q (cid:19)(cid:19) f j ← N (cid:18) xy , µ (cid:19) , ˆ D q → N ( z, b ; µ ) = (4.17) (cid:88) j (cid:90) z dyy e − g b b /y (cid:18) ˆ C q → j ( y, b ; µ ) + g q b y (cid:18) C renq → j ( y, b ) + δ (¯ y ) g b g q (cid:19)(cid:19) d j → N (cid:18) zy , µ (cid:19) . – 18 – H GeV - L LONLONNLO F u ¬ N b H GeV - L b max = b max = b max = b max = F u ¬ N Figure 3 . The TMDPDF ˆ F u ← p ( x, L µ ) as in the model of Eq. (4.16) (the up-quark PDF is takenfrom MSTW [47, 48], at x = 0 . , Λ = 0 . GeV, N f = 3 , g b = . GeV − , g q = 0 . GeV − , µ = C /b ∗ ) as a function of the parameter b in GeV − units. On the left panel we show consequentlycurves for LO, NLO and NNLO matching coefficients ( b max = 1 . GeV − is used). On the rightpanel we present NNLO curve at several values of b max in units of GeV − . The inclusion of the perturbative and power corrections modifies the Gaussian shape dif-ferently for PDF and FF kinematics.In the figures 3-5 we illustrate several features of the renormalon consistent ansatz thatwe propose. In all the plots we fix the µ scale at the value µ = µ ∗ = C /b ∗ . In Fig. 3-left weshow that the change of ˆ F with respect to the perturbative order of matching coefficient.On the right hand side of Fig. 3 we show the dependence on the choice of the scale b max ,which we find very mild for GeV − (cid:46) b max (cid:46) GeV − .The shape of the TMDs can strongly depend of the values of the non-perturbativeconstants g b,q for b ≥ GeV − as shown in Fig. 4-5. The values used in plots parametersare inspired by the fit in [29]. However, they can also change in a real fit with the presentmodel. For b ≤ GeV − the non-perturbative model does not really affect the x − behaviorof the TMD. In Fig. 5 we show instead that for instance at b ∼ . GeV − the modelparameter can start to have their impact.The cross-section built from TMDs in the form (4.16-4.17) and the evolution kernel(4.6) is dependent on the parameters g K , g b and g q . While, the parameter g K is stronglyuniversal, the parameters g b and g q are separate for TMDPDFs and TMDFFs, as well as,different for different flavors. Within the cross-section the dependence on these parametersis smoothed to a more-or-less similar shape (especially for parameters g b and g K ). However,the dependence on these parameters is clearly distinguishable at different energies. As anexample, we show the Drell-Yan cross-section in Fig. 6 and the Z-boson cross section inFig. 7 with some typical values of the experimental energies. While the corrections to theZ-boson production are dominated by g K , at low energies all parameters can compete. Inactual experiments the Z-boson production is only minimally affected by non-perturbativeeffects, so in actual fits it may happen that the value of g K is compatible with zero, whilethe other parameters provide the expected minimal correction (this is for instance the caseof the fit in Ref. [29]). This yields that an estimate of the nature of the TMDs non-perturbative part cannot be done just using the Z-boson production, but needs also data– 19 – H GeV - L g q =+ g q = g q =- - F u ¬ N b H GeV - L g b = g b = g b = g b = F u ¬ N Figure 4 . The TMDPDF ˆ F u ← p ( x, L µ ) as in the model of Eq. (4.16) at NNLO (PDF from MSTW[47, 48], and with x = 0 . , Λ = 0 . GeV, N f = 3 , µ = C /b ∗ with b max = 1 . GeV − ) as a functionof the impact parameter b in GeV − units. On the left panel we show several possible choices of g q in GeV at fixed g b = . GeV . On the right panel we show several possible choices of g b in GeV atfixed g q = . GeV . All curves are at NNLO. - - x xF u ¬ N g q =- g q = g q = Figure 5 . The function x ˆ F u ← p ( x, L µ ) as in the model of Eq. (4.16) at NNLO (PDF from MSTW[47, 48], as a function of x . The other inputs are fixed as Λ = 0 . GeV, N f = 3 , µ = C /b ∗ , b = b max = 1 . GeV − and g b = 0 . GeV . We show the curves at different values of g q . from low energy physics. We postpone to a future work a comparison with data of themodel that we have presented here.To conclude this section, we observe that in the literature we have not found anynon-perturbative input for TMDs fully consistent with the demands dictated by the poweranalysis presented here. For instance the b ∗ -prescription which is used in many phenomeno-logical analysis [8, 25, 28] is inconsistent with Eq. (4.13). Within the b ∗ -prescription thehigher-twist corrections are simulated by replacing b → b ∗ , and including an additionalnon-perturbative factor as ˆ F b ∗ -presc. q ← N ( x, b ; µ ) = (cid:88) j e g j/N ( x, b ) (cid:90) x dyy ˆ C q ← j ( y, b ∗ ; µ ) f j ← N (cid:18) xy , µ (cid:19) , (4.18)and similarly for TMDFF. This expression violates both guidelines formulated before Eq. (4.16).– 20 – igure 6 . The plots of Drell-Yan cross-section p + p → γ + X dσ/dQ dydq T at √ s = 100 GeV, Q = 10 GeV and y = 0 , evaluated using the renormalon ansatz. The impact of different parametersis demonstrated. The black line is the reference curve with all parameters set to 0. The other inputsare fixed as Λ = 0 . GeV, N f = 3 , µ = C /b ∗ , b max = 1 . GeV − . All curves are at NNLO. Considering the small- b expansion of ˆ C ( b ∗ ) in Eq. (4.18), ˆ F b ∗ -presc. q ← N ( x, b ; µ ) | small − b (cid:39) (cid:88) j (cid:90) x dyy (cid:104) ˆ C q ← j ( y, b ; µ ) (4.19) + a s ( µ ) C F b b (cid:18) y (1 − y ) + + ¯ y − δ (¯ y ) (cid:18) L µ − (cid:19)(cid:19) + δ (¯ y ) b g (cid:48)(cid:48) j/N ( x, (cid:105) f j ← N (cid:18) xy , µ (cid:19) , one does not reproduce Eq. (4.13). The main difference comes from the general powerscaling, x b vs. b , see point (ii). The point (i) is violated by the non-perturbative exponentthat is generally x -dependent and positioned outside of convolution integral (although, weshould appreciate that in most application it is taken x -independent). In this work, we have studied the non-perturbative properties associated with renormalonsfor the soft function and unintegrated matrix elements. With this aim, we have evaluatedall constituents of TMD distributions (soft factor, matching coefficient and anomalous di-mensions) within the large- β approximation. The (factorial) divergences of the large- β series are associated with the renormalon contribution and allow to estimate the leadingnon-perturbative contributions. We have found two independent renormalon structures inthe perturbative description of TMD: the soft function and small- b matching coefficients.– 21 – igure 7 . The plots of Z-boson production cross-section p + p → γ + X dσ/dQ dydq T at √ s =1 . TeV, Q = M Z = 91 . GeV and y = 0 , evaluated using the renormalon ansatz. The impact ofdifferent parameters is demonstrated. The black line is the reference curve with all parameters setto 0. The other inputs are fixed as Λ = 0 . GeV, N f = 3 , µ = C /b ∗ , b max = 1 . GeV − . Allcurves are at NNLO. The consideration of the soft function allows to fix the power behavior of the evolutionkernel of TMDs. We show the evidence of infrared renormalons at u = 1 , , .. ( u beingthe Borel parameter). Our results agree with the analysis of the power corrections tofactorized cross-section made in [36]. It also supports the popular assumption about aquadratic power correction to the TMD evolution kernel. However, the impact of the non-perturbative corrections is estimated to be not very significative for experiments whereTMDs are evaluated at scales higher than a few GeV.The nature of the renormalon contribution to the evolution kernel is peculiar, in thesense that it is generated by the non-perturbative part of a matrix element. In someaspects, this is very similar to the renormalon contribution to heavy quark masses. Wehave discussed also an ansatz which implements a consistent renormalon subtraction forthe TMD evolution kernel, which can be useful for phenomenology.The most promising conclusion of the paper comes from the analysis of the renormaloncontribution to the small- b expansion of TMDs. The discussion of these results can befound in Sec. 3.5. We demonstrate that the power corrections to small- b behave as afunction of x b for TMDPDFs and as b /z for TMDFFs. This observation should have asignificant impact on the joined TMDPDF – TMDFF phenomenology. Additionally, thelarge- β computation unveils the form of x − dependence for the leading power correctionto the small- b matching. This behavior should be incorporated in realistic and consistent– 22 –odels for TMDs.We have discussed and formulated the demands on a phenomenological ansatz to in-corporate all collected information. We find that typical models for the non-perturbativepart of TMDs, discussed in the literature, are inconsistent with our conclusions, mainly,due to the naive assumption that the combined powers corrections are largely functions of b (contrary to x b ). In eqns.(4.16-4.17) we construct a simple ansatz that interpolatesthe Gaussian low-energy model for TMDs with the perturbative small- b regime accountingformulated demands. We postpone to a future work the fit of available data using thepresented results. Acknowledgements
We thank Vladimir Braun for numerous discussions and useful comments. We thank theErwin Schrödinger International Institute for Mathematics and Physics (ESI, Vienna) forkind hospitality during the summer 2016 and for propitiating nice discussions on this work.I.S. is supported by the Spanish MECD grant FPA2014-53375-C2-2-P and FPA2016-75654-C2-2-P.
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