Scale evolution of gluon TMDPDFs

By applying the effective field theory machinery we factorize the transverse momentum spectrum of Higgs boson production, where the main hadronic quantities are the gluon transverse momentum dependent parton distribution functions (TMDPDFs). We properly define those quantities, showing explicitly, in the case of an unpolarized hadron, that they are free from rapidity divergences, and extract their evolution properties. It turns out that the evolution for all eight (un-)polarized leading-twist gluon TMDPDFs is driven by the same evolution kernel, for which we derive the necessary ingredients to obtain a resummation of large logarithms at next-tonext-to-leading-logarithmic accuracy. We make predictions for the contribution of linearly polarized gluons to the Higgs boson qT -spectrum. 1 Factorization theorem in terms of well-defined TMDPDFs The derivation of the factorization theorem for the Higgsboson qT -spectrum is done by applying the following set of consecutive matchings between effective theories [1]: QCD(n f = 6) → QCD(n f = 5) → SCETqT → SCETΛQCD . First we integrate out the top quark mass to build the effective coupling ggH. Then we integrate out the mass of the Higgs boson, mH , obtaining a factorization theorem which holds for qT mH . Finally, forΛQCD qT mH we can refactorize the gluon TMDPDFs in terms of the collinear gluon/quark collinear PDFs, integrating out the intermediate scale qT . The effective local interaction that describes the gluongluon fusion process is [2–6] Leff = Ct(mt , μ) H v αs(μ) 12π Fμν,a Fa μν , (1) where v ≈ 246 GeV is the Higgs vacuum expectation value. The Wilson coefficient Ct is known up to ae-mail: m.g.echevarria@nikhef.nl (* speaker) be-mail: kasemets@nikhef.nl ce-mail: mulders@few.vu.nl de-mail: c.pisano@nikhef.nl NNNLO [7, 8]. At NNLO it is [9, 10] Ct(mt , μ) = 1 + αs(μ) 4π (5CA − 3CF) + ( αs(μ) 4π )2 [27 2 C2 F + ( 11ln mt μ2 − 100 3 ) CFCA − ( 7ln mt μ2 − 1063 36 ) C2 A − 4 3 CFTF − 5 6 CATF − ( 8ln mt μ2 + 5 ) CFTFn f − 47 9 CATFn f ] , (2) which anomalous dimension is given solely by the QCD β-function, γ(αs(μ)) = dlnCt(mt , μ) dlnμ = αs d dαs β(αs(μ)) αs(μ) . (3) Using the effective lagrangian just introduced, the differential cross section for Higgs production is factorized as dσ = 1 2s ( αs(μ) 12πv )2 C2 t (m 2 t , μ) d3q (2π)2Eq ∫ d4y e−iq·y × ∑ X 〈 PS A, P̄S B ∣∣∣ Fa μνF(y) |X〉 〈X| Fb αβF(0) ∣∣∣PS A, P̄S B〉 , (4) where s = (P + P̄)2. In a second step, the effective QCD operator is matched onto the SCET-qT one by Fμν,a Fa μν = −2q2C(−q2, μ) gμνB μ,a n⊥ ( SnSn̄ )ab B n̄⊥ , (5) DOI: 10.1051/ C © Owned by the authors, published by EDP Sciences, 2015 / 0200 (2015) 201 epjconf EPJ Web of Conferences 8 8 , 5 50200 5 This is an Open Access article distributed under the terms of the Creative Commons Attribution License 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 5 5 Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20158502005 where q2 = mH and Sn(n̄) represent the soft Wilson lines 1. The B n(n̄) operators, which stand for gauge invariant gluon fields, are given by Bμn⊥ = 1 g [n̄ · PW† n iD ⊥μ n Wn] = in̄αg μ ⊥βW † n F αβ n Wn = in̄αg μ ⊥βt a(W† n)F αβ,b n , (6) and the Wilson lines are Wn(x) = P exp [ ig ∫ 0 −∞ ds n̄ · An(x + n̄s)ta ] , S n(x) = P exp [ ig ∫ 0 −∞ ds n · As(x + ns)ta ] . (7) The calligraphic typography means that the Wilson lines are written in the adjoint representation, i.e., the color generators are given by (ta)bc = −i f abc. Gauge invariance among regular and singular gauges is guaranteed by the inclusion of transverse gauge links (see Refs. [11, 12] for more details). The Wilson matching coefficient C(−q2, μ), which corresponds to the infrared finite part of the gluon form factor, is given at one-loop by C(−q2, μ) = 1 + αsCA 4π [ −ln2 −q2 + i0 μ2 + π2 6 ] . (8) The cross-section at leading order can be then written as dσ = σ0(μ) C2 t (m 2 t , μ)H(m 2 H , μ) mH τs dy dq⊥ (2π)2 ∫ dy⊥ e−iq⊥·y⊥ × 2J n (xA, y⊥; μ) Jn̄ μν(xB, y⊥; μ) S (y⊥; μ) + O(qT /mH) , (9) where H(mH , μ) = |C(−q 2, μ2)|2, xA,B = √ τ e±y, τ = (mH+ qT )/s and σ0(μ) = mH α 2 s(μ) 72π(N2 c − 1)sv2 . (10) The pure collinear and soft matrix elements are defined as J n (xA, y⊥, μ) = − xAP 2 ∫ dy− 2π e−i 1 2 xAy −P+ × ∑ Xn 〈PS A| B n⊥ (y −, y⊥) |Xn〉 〈Xn| B n⊥(0) |PS A〉 , S (y⊥, μ) = 1 N2 c − 1 × ∑ Xs 〈0| ( SnSn̄ )ab(y⊥) |Xs〉 〈Xs| (S†n̄Sn)ba(0) |0〉 . (11) As it is shown in Ref. [1] by performing an explicit NLO perturbative calculation, the collinear and soft matrix elements defined above contain un-cancelled rapidity divergences and thus are ill-defined. Thus, they need to be properly combined to obtain well-defined hadronic quantities. Extending the work done in Refs. [13, 14] for the 1A generic vector vμ is decomposed as vμ = n̄ · v nμ 2 + n · v n̄μ 2 + v μ ⊥ = (n̄ · v, n · v, vμ⊥) = (v, v−, v μ ⊥), with n = (1, 0, 0, 1), n̄ = (1, 0, 0,−1), n2 = n̄2 = 0 and n · n̄ = 2. We also use vT = |u⊥|, so that v⊥ = −vT < 0. quark case to the gluon case, it can be easily shown that the soft function is split as S̃ (bT ; mH , μ 2) = S̃ − ( bT ; ζA, μ2;Δ− ) S̃ + ( bT ; ζB, μ2;Δ+ ) , S̃ − ( bT ; ζA, μ2;Δ− ) = √ S̃ ( Δ− p+ , Δ− p̄− ) , S̃ + ( bT ; ζB, μ2;Δ+ ) = √ S̃ ( Δ+ p+ , Δ+ p̄− ) , (12) where ζAζB = mH . Using this splitting, the gluon TMDPDFs are then defined as G g/A(xA, kn⊥, S A; ζA, μ 2;Δ−) = ∫ db⊥ eib⊥·kn⊥ J̃ μν n (xA, b⊥, S A; μ2;Δ−) S̃ −(bT ; ζA, μ2;Δ−) , G g/B(xB, kn̄⊥, S B; ζB, μ 2;Δ+) = ∫ db⊥ eib⊥·kn̄⊥ J̃ μν n̄ (xB, b⊥, S B; μ 2;Δ+) S̃ +(bT ; ζB, μ2;Δ+) . (13) We emphasize the fact the the hadronic quantities defined above are free from rapidity divegences and thus have well-behaved evolution properties and can be extracted from experiment. Now, we can write the cross-section in terms of welldefined TMDPDFs: dσ = 2σ0(μ) C2 t (m 2 t , μ)H(m 2 H , μ) dy dq⊥ (2π)2 ∫ dy⊥ e−iq⊥·y⊥ × G̃ g/A(xA, y⊥, S A; ζA, μ) G̃g/Bμν(xB, y⊥, S B; ζB, μ) + O(qT /mH) , (14) where the twiddle refers to impact parameter space (IPS). This factorized cross-section is valid for qT mH . The involved gluon TMDPDFs can be separated in terms on the polarization of the hadron, i.e., unpolarized (O), longitudinally polarized (L) and transverse polarized (T). In Ref. [15] the authors obtained the decomposition of collinear correlators at leading-twist. Extending that decomposition to well-defined TMDPDFs, as given in Eq. (13), we have G g/A (x, knT ) = g μν ⊥ f g 1 (x, k 2 nT ) + ⎜⎜⎜⎜⎝kμ n⊥k n⊥ M2 A + g μν ⊥ k2 nT 2M2 A ⎟⎟⎟⎟⎠ h 1 (x, k2 nT ) , G g/A (x, knT ) = i μν ⊥ λ g g 1L(x, k 2 nT ) + kT {μ ⊥ k ν} n⊥ 2M2 A λ h 1L (x, k 2 nT ) , G ] g/A (x, knT ) = −g μν ⊥ kT S T ⊥ MA f 1T (x, k 2 nT ) − i μν ⊥ knT · ST MA g g 1T (x, k 2 nT ) + kT {μ ⊥ k ν} n⊥ 2M2 A knT · ST MA h 1T (x, k 2 nT ) + kT {μ ⊥ S ν} T + S T {μ ⊥ k ν} nT 4MA hg1T (x, k 2 nT ) . (15) EPJ Web of Conferences

First we integrate out the top quark mass to build the effective coupling ggH.Then we integrate out the mass of the Higgs boson, m H , obtaining a factorization theorem which holds for q T m H . Finally, for Λ QCD q T m H we can refactorize the gluon TMDPDFs in terms of the collinear gluon/quark collinear PDFs, integrating out the intermediate scale q T .The effective local interaction that describes the gluongluon fusion process is [2][3][4][5][6] where v ≈ 246 GeV is the Higgs vacuum expectation value.The Wilson coefficient C t is known up to a e-mail: m.g.echevarria@nikhef.nl(* speaker) b e-mail: kasemets@nikhef.nlc e-mail: mulders@few.vu.nl d e-mail: c.pisano@nikhef.nlNNNLO [7,8].At NNLO it is [9,10] which anomalous dimension is given solely by the QCD β-function, Using the effective lagrangian just introduced, the differential cross section for Higgs production is factorized as where s = (P + P) 2 .In a second step, the effective QCD operator is matched onto the SCET-q T one by and the Wilson lines are The calligraphic typography means that the Wilson lines are written in the adjoint representation, i.e., the color generators are given by (t a ) bc = −i f abc .Gauge invariance among regular and singular gauges is guaranteed by the inclusion of transverse gauge links (see Refs. [11,12] for more details).The Wilson matching coefficient C(−q 2 , μ), which corresponds to the infrared finite part of the gluon form factor, is given at one-loop by The cross-section at leading order can be then written as where The pure collinear and soft matrix elements are defined as As it is shown in Ref. [1] by performing an explicit NLO perturbative calculation, the collinear and soft matrix elements defined above contain un-cancelled rapidity divergences and thus are ill-defined.Thus, they need to be properly combined to obtain well-defined hadronic quantities.Extending the work done in Refs.[13,14] for the 1 A generic vector v μ is decomposed as quark case to the gluon case, it can be easily shown that the soft function is split as where Using this splitting, the gluon TMD-PDFs are then defined as We emphasize the fact the the hadronic quantities defined above are free from rapidity divegences and thus have well-behaved evolution properties and can be extracted from experiment.Now, we can write the cross-section in terms of welldefined TMDPDFs: where the twiddle refers to impact parameter space (IPS).This factorized cross-section is valid for q T m H .The involved gluon TMDPDFs can be separated in terms on the polarization of the hadron, i.e., unpolarized (O), longitudinally polarized (L) and transverse polarized (T).In Ref. [15] the authors obtained the decomposition of collinear correlators at leading-twist.Extending that decomposition to well-defined TMDPDFs, as given in Eq. ( 13), we have EPJ Web of Conferences 02005-p.2

Evolution of gluon TMDPDFs
The evolution of all (un-)polarized gluon TMDPDFs is governed by the same anomalous dimension, which is fixed by the renormalization group (RG) equation applied to the cross-section: Thus we have where the non-cusp piece is The perturbative coefficients of Γ cusp and γ V are known up to three loops and can be found in Ref. [1].
On the other hand, we also have where the RG-equation implies also that Regardless how the non-perturbative contribution of the D g -term at large b T is parametrized, the evolution of all leading-twist gluon TMDPDFs can be consistently performed up to NNLL: where the evolution kernel Rg is given by The evolution equation of the D g -term can be solved analytically and obtain the resummed D g (as done in Ref. [16] in the quark case).Setting a = α s (μ i )/(4π) and The coefficient d g 2 (0) is: In Fig. 1 we show the evolution kernel for fixed initial and final scales (ζ = μ 2 = Q 2 ), implementing the resummed D g .For those scales the role of the nonperturbative contribution to the D g term at large b T is numerically irrelevant, given the perfect convergence between different resummation orders.However, this does not mean that for other scale choices that contribution would not be relevant.

Gluon TMDPDFs in an unpolarized hadron
The content of gluons inside an unpolarized hadron is parametrized in terms of two distributions: TRANSVERSITY 2014 In impact parameter space we write and thus the relations between the distributions in momentum and IPS are For b T Λ −1 QCD we can refactorize the (renormalized) gluon TMDPDFs of an unpolarized hadron A in terms of (renormalized) collinear quark/gluon distributions: where the unpolarized collinear quark and gluon PDFs are defined as Since the soft function can be written to all orders as with some generic function R s and where the D g term is related to the cusp anomalous dimension in the adjoint representation as in Eq. ( 21), given Eqs.( 13) and (31) one can exponentiate the ζ dependence of the TMDPDFs in the following way: Notice that the exponentiation of the ζ-dependence above applies in the same way to all (un)-polarized TMDPDFs defined before, through the same factor and D g term.In Ref. [1] we perform an explicit one-loop calculation of the kernel functions CQ / g←i (x, b T ; μ), which are given by where L T = ln(μ 2 b 2 T e 2γ E /4), the one-loop DGLAP splitting kernels are and the functions T f g← j (x) and T h g← j (x) are given by (35) Under RG-equation one has so that we can partially exponentiate the double logarithms by (see Ref. [17] for the quark case): where Thus, the TMDPDFs can be written as When α s L T is of order 1, the expression above still contains large logarithms L T that need to be resummed.

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Solving the evolution equations for h Γ and h γ , in the same way we have done previously for D g , we get

Phenomenology
So far, the resummation techniques discussed previously only make sense in the perturbative region of small b T .
For the large b T region we need to parametrize the nonperturbative contribution of TMDPDFs with a model: where F NP f,h account for the non-perturbative contribution at large b T .For the Higgs distribution we have Q = m H , and we fix the resummation scale Q i = Q 0 +q T (we choose Q 0 = 2 GeV).In Ref. [17] it was shown that for Z-boson production it was enough to take simple scale-independent exponentials to parametrize the non-perturbative contribution of quark-TMDPDFs.Following the same logic, let us consider scale-independent exponentials for the Higgs boson production: With these models, one can quantify the contribution of linearly polarized gluons to the Higgs boson q Tspectrum [18][19][20]), taking into account the resummation scheme presented above and analyzing the impact of the non-perturbative parameters λ 1,2 .We define the ratio between the contributions of unpolarized and linearly polarized gluons to the cross-section by In Fig. 2 we show the results, where the band comes from varying the resummation scale from 2Q i to Q i /2.

Figure 1 .
Figure 1.Evolution kernel with Q i = 2 GeV, Q f = 125 GeV and fixed number of flavours n f = 4.The difference between NLL and NNLL curves is unappreciable.

Figure 2 .
Figure 2. Ratio of the contributions of linearly polarized and unpolarized gluons to the Higgs boson q T -spectrum at √ s = 8 TeV and x A = x B for different values of the non-perturbative parameters λ 1,2 .