Scale evolution of gluon TMDPDFs

Scale evolution of gluon TMDPDFs

Miguel G. Echevarria∗1,2,a_{, Tomas Kasemets}1,2,b_{, Piet J. Mulders}1,2,c_{, and Cristian Pisano}1,2,d

1_{Department of Physics and Astronomy, VU University Amsterdam, De Boelelaan 1081, NL-1081 HV Amsterdam, the Netherlands} 2_{Nikhef, Science Park 105, NL-1098 XG Amsterdam, the Netherlands}

Abstract.

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-to-next-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 Higgs-boson qT-spectrum is done by applying the following set of consecutive matchings between effective theories [1]: QCD(nf = 6) → QCD(nf = 5) → SCETqT → SCETΛQCD.

First we integrate out the top quark mass to build the e ffec-tive 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  mHwe can refactorize the gluon TMDPDFs in terms of the collinear gluon/quark collinear PDFs, integrating out the intermedi-ate scale qT.

The effective local interaction that describes the gluon-gluon fusion process is [2–6]

Leff= Ct(m2t, μ) H v αs(μ) 12π F μν,a Fμνa , (1)

where v ≈ 246 GeV is the Higgs vacuum expecta-tion value. The Wilson coefficient Ct is known up to

a_{e-mail: [email protected] (* speaker)} b_{e-mail: [email protected]} c_{e-mail: [email protected]} d_{e-mail: [email protected]} NNNLO [7, 8]. At NNLO it is [9, 10] Ct(m2t, μ) = 1 + αs(μ) 4π (5CA− 3CF) + _α s(μ) 4π 2 27 2 C 2 F+  11lnm 2 t μ2 − 100 3  CFCA −  7lnm 2 t μ2 − 1063 36  C2_A−4 3CFTF− 5 6CATF −  8lnm 2 t μ2 + 5  CFTFnf− 47 9 CATFnf  , (2) which anomalous dimension is given solely by the QCD β-function, γt_(α s(μ))= dlnCt(m2t, μ) dlnμ = α 2 s d dαs β(αs(μ)) αs(μ) . (3)

Using the effective lagrangian just introduced, the differ-ential cross section for Higgs production is factorized as

dσ= 1 2s _α s(μ) 12πv 2 C2_t(m2_t, μ) d 3_q (2π)2_2E q  d4y e−iq·y × X  PSA, ¯PSB FaμνFμν,a(y)|X X| Fαβb Fαβ,b(0) PSA, ¯PSB , (4) where s= (P + ¯P)2_.

In a second step, the effective QCD operator is matched onto the SCET-qTone by

Fμν,aFμνa = −2q2C(−q2, μ) g⊥μνBμ,an⊥  S† nS¯n ab Bν,b ¯n⊥, (5) C

Owned by the authors, published by EDP Sciences, 2015

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.

where q2_{= m}2

HandSn(¯n)represent the soft Wilson lines1. TheB⊥μ_n(¯n)operators, which stand for gauge invariant gluon fields, are given by

Bμn⊥= 1 g[¯n· PWn†iD⊥μn Wn]= i¯nαgμ_⊥βWn†F αβ n Wn = i¯nαgμ_⊥βta(W†n)abF αβ,b n , (6)

and the Wilson lines are

Wn(x)= P exp  ig  0 −∞ds ¯n· A a n(x+ ¯ns)ta  , Sn(x)= P exp  ig  0 −∞ds n· A a s(x+ ns)ta  . (7)

The calligraphic typography means that the Wilson lines are written in the adjoint representation, i.e., the color gen-erators 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) Ct2(m2t, μ)H(m2H, μ) m2 H τs dy d2_q ⊥ (2π)2  d2y_⊥e−iq⊥·y⊥ × 2Jnμν(xA, y⊥; μ) J¯n μν(xB, y⊥; μ) S (y⊥; μ)+ O(qT/mH) , (9) where H(m2 H, μ) = |C(−q2, μ2)|2, xA,B= √ τ e±y_{, τ= (m}2 H+ q2 T)/s and σ0(μ)= m2 Hα2s(μ) 72π(N2 c− 1)sv2 . (10)

The pure collinear and soft matrix elements are defined as

Jμνn (xA, y⊥, μ) = −xAP + 2  dy− 2π e −i1 2xAy−P+ × Xn PSA| Bμ,an⊥(y−, y⊥)|Xn Xn| Bν,an⊥(0)|PSA , S (y_⊥, μ) = 1 N2 c− 1 × Xs 0| S† nS¯n ab (y_⊥)|Xs Xs| S†¯nSn ba (0)|0 . (11) As it is shown in Ref. [1] by performing an explicit NLO perturbative calculation, the collinear and soft ma-trix 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 quan-tities. Extending the work done in Refs. [13, 14] for the

1_{A generic vector v}μ_{is decomposed as v}μ_{= ¯n · v}nμ

2 + n · v ¯nμ 2 + v μ ⊥= (¯n· v, n · v, vμ_⊥)= (v+, v−, vμ_⊥), with n= (1, 0, 0, 1), ¯n = (1, 0, 0, −1), n2= ¯n2_{= 0 and n · ¯n = 2. We also use v}

T= |u⊥|, so that v2⊥= −v2T< 0.

quark case to the gluon case, it can be easily shown that the soft function is split as

˜ S (bT; m2H, μ2)= ˜S−  bT; ζA, μ2;Δ−  ˜ S₊bT; ζB, μ2;Δ+  , ˜ S₋bT; ζA, μ2;Δ−  =  ˜ S  Δ− p+, Δ − ¯p−  , ˜ S₊bT; ζB, μ2;Δ+  =  ˜ S  Δ+ p+, Δ + ¯p−  , (12)

where ζAζB = m4_H. Using this splitting, the gluon TMD-PDFs are then defined as

Gμν_g/A(xA, kn⊥, SA; ζA, μ2;Δ−)=  d2_b ⊥eib⊥·kn⊥ J˜nμν(xA, b⊥, SA; μ2;Δ−) ˜S−(bT; ζA, μ2;Δ−) , Gμν_g/B(xB, k¯n⊥, SB; ζB, μ2;Δ+)=  d2b_⊥eib⊥·k¯n⊥ _J˜μν ¯n (xB, b⊥, SB; μ 2_;Δ+_{) ˜S} +(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 well-defined TMDPDFs: dσ= 2σ0(μ) C2t(m2t, μ)H(m2H, μ) dy d2_q ⊥ (2π)2  d2y_⊥e−iq⊥·y⊥ × ˜Gμν_g/A(xA, y⊥, SA; ζA, μ) ˜Gg/B μν(xB, y⊥, SB; 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), lon-gitudinally 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 Gμν[O]_g/A (x,knT)= gμν_⊥ f₁g(x, k2nT) + ⎛ ⎜⎜⎜⎜⎝kμ n⊥kνn⊥ M2 A + gμν_⊥ knT2 2M2 A ⎞ ⎟⎟⎟⎟⎠h⊥g 1 (x, k 2 nT) , Gμν[L]_g/A (x,knT)= i_⊥μνλ gg_1L(x, k2nT)+ kT{μ ⊥ kν}n⊥ 2M2_A λ h ⊥g 1L(x, k 2 nT) , Gμν[T]_g/A (x,knT)= −gμν_⊥ kTST ⊥ MA f_1T⊥g(x, k2_nT) − i_⊥μνknT· ST MA gg_1T(x, k2_nT) + kT{μ ⊥ kν}n⊥ 2M2 A knT · ST MA h⊥g_1T(x, k2_nT) + kT{μ ⊥ Sν}T + ST{μ ⊥ kν}nT 4MA hg_1T(x, k_nT2 ) . (15)

2 Evolution of gluon TMDPDFs

The evolution of all (un-)polarized gluon TMDPDFs is governed by the same anomalous dimension,

d dlnμln ˜G [pol] g/A (xn, b⊥, SA; ζA, μ) ≡ γG  αs(μ), ln ζA μ2  , (16) which is fixed by the renormalization group (RG) equation applied to the cross-section:

0= 2β (αs(μ)) αs(μ) + 2γt_(α s(μ))+ γH ⎛ ⎜⎜⎜⎜⎝αs(μ), ln m2 H μ2 ⎞ ⎟⎟⎟⎟⎠ + γG  αs(μ), ln ζA μ2  + γG  αs(μ), ln ζB μ2  . (17) Thus we have γG  αs(μ), ln ζA μ2  = −ΓA cusp(αs(μ))ln ζA μ2 − γ nc (αs(μ)) , γG  αs(μ), ln ζB μ2  = −ΓA cusp(αs(μ))ln ζB μ2 − γ nc (αs(μ)) , (18) where the non-cusp piece is

γnc_(α

s(μ))= γg(αs(μ))+ γt(αs(μ))+

β(αs(μ)) αs(μ)

. (19)

The perturbative coefficients of Γcuspand γV are known up

to three loops and can be found in Ref. [1]. On the other hand, we also have

d dlnζA

ln ˜G[pol]_g/A (xn, b⊥, SA; ζA, μ) = −Dg(bT; μ2) , (20) where the RG-equation implies also that

dDg dlnμ = Γ

A

cusp. (21)

Regardless how the non-perturbative contribution of the Dg-term at large bT is parametrized, the evolution of all leading-twist gluon TMDPDFs can be consistently per-formed up to NNLL:

˜

G[pol]_g/A (xn, b⊥, SA; ζA, f, μf)= ˜

G[pol]_g/A (xn, b⊥, SA; ζA,i, μi) ˜Rg 

bT; ζA,i, μi, ζA, f, μf 

, (22) where the evolution kernel ˜Rgis given by

˜ Rg bT; ζA,i, μi, ζA, f, μf= exp  μf μi d ¯μ ¯μ γG  αs( ¯μ), ln ζA, f ¯μ2  _ζ A, f ζA,i _−Dg(bT;μi) . (23) The evolution equation of the Dg-term can be solved

analytically and obtain the resummed Dg (as done in

Ref. [16] in the quark case). Setting a = αs(μi)/(4π) and

0.000 0.05 0.10 0.15 0.20bTGeV1 5 10 15 20 R  NNLL NLL LL Qi 2 GeV Qf 125 GeV

Figure 1. Evolution kernel with Qi= 2 GeV, Qf = 125 GeV and

fixed number of flavours nf = 4. The difference between NLL

and NNLL curves is unappreciable.

X= aβ0L⊥we get DR_g(bT; μi)= − ΓA 0 2β0 ln(1− X) +1 2  _a 1− X ⎡⎢⎢⎢⎢⎣− β1Γ₀A β2 0 (X+ ln(1 − X)) +Γ A 1 β0 X ⎤ ⎥⎥⎥⎥⎦ +1 2  _a 1− X 2⎡ ⎢⎢⎢⎢⎣2d2(0)+ ΓA 2 2β0 (X(2− X)) +β1Γ A 1 2β2 0 (X(X− 2) − 2ln(1 − X)) +β2Γ A 0 2β2 0 X2 +β21Γ0A 2β3₀ (ln 2_{(1− X) − X}2₎ ⎤ ⎥⎥⎥⎥⎦ + ..., (24) The coefficient dg₂(0) is:

dg₂(0)= CACA  404 27 − 14ζ3  −  112 27  CATFnf. (25)

In Fig. 1 we show the evolution kernel for fixed ini-tial and final scales (ζ = μ2 _{= Q}2_{), implementing the}

resummed Dg. For those scales the role of the

non-perturbative contribution to the Dg term at large bT is numerically irrelevant, given the perfect convergence

be-tween different resummation orders. However, this does not mean that for other scale choices that contribution would not be relevant.

3 Gluon TMDPDFs in an unpolarized

hadron

The content of gluons inside an unpolarized hadron is parametrized in terms of two distributions:

Gμν[O]_g/A (x,knT)= −1 2g μν ⊥ f1g(x, k 2 nT)+ 1 2  gμν_⊥ −2k μ n⊥kνn⊥ k2 n⊥  h⊥g₁ (x, k2_nT) . (26)

In impact parameter space we write ˜ Gμν[O]_g/A (x,bT)=  d2kn⊥eikn⊥·b⊥Gμν[O]g/A (x,knT) = −1 2g μν ⊥ f˜1g(x, b 2 T)+ 1 2  gμν_⊥ −2b μ ⊥bν⊥ b2 ⊥  ˜h⊥g₁ (x, b2_T) , (27) and thus the relations between the distributions in momen-tum and IPS are

˜ f₁g(x, b2_T)= 2π  dknTknTJ0(knTbT) f₁g(x, k2nT) , ˜h⊥g₁ (x, b2_T)= −2π  dknTknTJ2(knTbT) h⊥g₁ (x, k2nT) . (28) For bT  Λ−1_QCDwe can refactorize the (renormalized) gluon TMDPDFs of an unpolarized hadron A in terms of (renormalized) collinear quark/gluon distributions:

˜ f₁g/A(x, bT; ζA, μ) =  j=q,¯q,g  1 x d ¯x ¯xC˜ f g← j( ¯x, bT; ζA, μ) fj/A(x/ ¯x; μ)+ O(bTΛQCD) , ˜h⊥g/A₁ (x, bT; ζA, μ) =  j=q,¯q,g  1 x d ¯x ¯xC˜ h g← j( ¯x, bT; ζA, μ) fj/A(x/ ¯x; μ)+ O(bTΛQCD) , (29) where the unpolarized collinear quark and gluon PDFs are defined as fq/A(x; μ)= 1 2  dy− 2π e −i1 2y−xP+ × PSA| ξ¯nWn ! (y−)/¯n 2 W † nξn ! (0)|PSA , fg/A(x; μ)= −xP + 2  dy− 2π e −i1 2y−xP+ × PSA| Bμ,an⊥(y−)Ban⊥μ(0)|PSA . (30) Since the soft function can be written to all orders as ln ˜S = Rs(LT, αs(μ))+ Dg(LT, αs(μ)) lnΔ

+_Δ− m2

Hμ2

, (31)

with some generic functionRs and where the Dgterm is related to the cusp anomalous dimension in the adjoint rep-resentation as in Eq. (21), given Eqs. (13) and (31) one can exponentiate the ζ dependence of the TMDPDFs in the fol-lowing way: ˜ f_1,g/Ag (x, bT; ζA, μ) = ⎛ ⎜⎜⎜⎜⎝ ζAb2T 4e−2γE ⎞ ⎟⎟⎟⎟⎠−Dg(bT;μ) ×  j=q,¯q,g  1 x d ¯x ¯xC˜ Q/ f,g← j( ¯x, bT; μ) fj/A(x/ ¯x; μ) , ˜h⊥g_1,g/A(x, bT; ζA, μ) = ⎛ ⎜⎜⎜⎜⎝ ζAb2_T 4e−2γE ⎞ ⎟⎟⎟⎟⎠−Dg(bT;μ) ×  j=q,¯q,g  1 x d ¯x ¯xC˜ Q/ h,g← j( ¯x, bT; μ) fj/A(x/ ¯x; μ) . (32)

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 Dgterm.

In Ref. [1] we perform an explicit one-loop calculation of the kernel functions ˜CQ_g←i/ (x, bT; μ), which are given by

˜ CQ/_{f,g← j}(x, bT; μ)= δ(1 − x) δgi +αs 2π ⎡ ⎢⎢⎢⎢⎣δ(1 − x)δgi ⎛ ⎜⎜⎜⎜⎝ΓA 0 L2 T 8 + β0 2LT ⎞ ⎟⎟⎟⎟⎠ − Pg← j(x)LT+ T_{g← j}f (x) ⎤ ⎥⎥⎥⎥⎦ , ˜ C_h,gQ/ _{← j}(x, bT; μ)= αs 2πT h g← j(x) , (33)

where LT = ln(μ2b2Te2γE/4), the one-loop DGLAP split-ting kernels are

Pg←g(x)= 2CA  x (1− x)₊+ 1− x x + x(1 − x)  +β0 2δ(1 − x) , Pg←q(x)= CF 1+ (1 − x)2 x , (34)

and the functionsT_{g← j}f (x) andT_{g← j}h (x) are given by Tf g←g(x)= −CA π2 12δ(1 − x) , T f g←q(x)= CFx , Th g←g(x)= −2CA 1− x x , T h g←q(x)= −2CF 1− x x . (35) Under RG-equation one has

d dlnμC˜ Q/ g← j(x, bT; μ)= (ΓcuspA LT− γnc) ˜CQ_{g← j}/ (x, bT; μ) − i  1 x dξ ξC˜ Q/ g←i(ξ, bT; μ)Pi← j(x/ξ) , (36) so that we can partially exponentiate the double logarithms by (see Ref. [17] for the quark case):

˜ C_{g← j}Q/ (x, bT; μ)≡ exp hΓ(bT; μ)− hγ(bT; μ) ! ˜I_{g← j}(x, bT; μ) , (37) where dh_Γ dlnμ = Γ A cuspLT, dhγ dlnμ = γ nc_{. (38)}

Thus, the TMDPDFs can be written as ˜ f_1,g/Ag (x, bT; ζA, μ) = ⎛ ⎜⎜⎜⎜⎝ ζAb2_T 4e−2γE ⎞ ⎟⎟⎟⎟⎠−Dg(bT;μ) ehΓ(bT;μ)−hγ(bT;μ) ×  j=q,¯q,g  1 x d ¯x ¯x ˜If,g← j( ¯x, bT; μ) fj/A(x/ ¯x; μ) , ˜h⊥g_1,g/A(x, bT; ζA, μ) = ⎛ ⎜⎜⎜⎜⎝ ζAb2_T 4e−2γE ⎞ ⎟⎟⎟⎟⎠−Dg(bT;μ) ehΓ(bT;μ)−hγ(bT;μ) ×  j=q,¯q,g  1 x d ¯x ¯x ˜Ih,g← j( ¯x, bT; μ) fj/A(x/ ¯x; μ) . (39) When αsLT is of order 1, the expression above still contains large logarithms LT that need to be resummed.

Solving the evolution equations for h_Γand hγ, in the same

way we have done previously for Dg, we get hR_γ(bT; μ)= DRg(bT; μ) ΓA cusp→γnc (40) and hR_Γ(bT; μ)= ΓA 0(X− (X − 1)ln(1 − X)) 2asβ2₀ +β1Γ A 0  2X+ ln2(1− X) + 2ln(1 − X) 4β3 0 −2β0ΓA1(X+ ln(1 − X)) 4β3 0 + as 4β4 0(1− X)  β2 0Γ A 2X 2_{− β} 0(β1ΓA1(X(X+ 2) + 2ln(1 − X)) +β2Γ0A((X−2)X+2(X−1)ln(1−X)))+β21ΓA0(X+ln(1−X))2  . (41)

4 Phenomenology

So far, the resummation techniques discussed previously only make sense in the perturbative region of small bT. For the large bT region we need to parametrize the non-perturbative contribution of TMDPDFs with a model:

˜ f₁g(x, bT; Q2, Q) = exp  Q Qi d ¯μ ¯μ γG  αs( ¯μ), ln Q2 ¯μ2  ⎛⎜⎜_⎜⎜⎝Q2_b2 T 4e−2γE ⎞ ⎟⎟⎟⎟⎠−DRg(bT;Qi) × ehR Γ(bT;Qi)−hRγ(bT;Qi)  j=q,¯q,g  1 x d ¯x ¯x ˜If,g← j( ¯x, bT; Qi) fj/N(x/ ¯x; Qi) × ˜FNP f (x, bT; Q) , ˜h⊥g₁ (x, bT; Q2, Q) = exp  Q Qi d ¯μ ¯μ γG  αs( ¯μ), ln Q2 ¯μ2  ⎛⎜⎜_⎜⎜⎝Q2_b2 T 4e−2γE ⎞ ⎟⎟⎟⎟⎠−DRg(bT;Qi) × ehR Γ(bT;Qi)−hRγ(bT;Qi)  j=q,¯q,g  1 x d ¯x ¯x ˜Ih,g← j( ¯x, bT; Qi) fj/N(x/ ¯x; Qi) × ˜FNP h (x, bT; Q) , (42)

where FNP_f,h account for the non-perturbative contribution at large bT. For the Higgs distribution we have Q = mH, and we fix the resummation scale Qi= Q0+qT(we choose Q0 = 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 contribu-tion of quark-TMDPDFs. Following the same logic, let us consider scale-independent exponentials for the Higgs boson production: ˜ FNP f (x, bT; Q)= e−λ1bT, ˜ F_hNP(x, bT; Q)= e−λ2bT. (43)

With these models, one can quantify the contribution of linearly polarized gluons to the Higgs boson qT -spectrum [18–20]), taking into account the resummation scheme presented above and analyzing the impact of the non-perturbative parameters λ1,2. We define the ratio

be-tween the contributions of unpolarized and linearly polar-ized gluons to the cross-section by

R(xA, xB, qT; Q)= " d2_{b e}−iqT·bT˜h⊥g 1 (xA, bT; Q 2_{, Q) ˜h}⊥g 1 (xB, bT; Q 2_{, Q)} " d2_{b e}−iqT·bTf˜g 1(xA, bT; Q2, Q) ˜f g 1(xB, bT; Q2, Q) . (44) In Fig. 2 we show the results, where the band comes from varying the resummation scale from 2Qito Qi/2.

Acknowledgements

This research is part of the program of the “Stichting voor Fundamenteel Onderzoek der Materie (FOM)”, which is financially supported by the “Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)”. Part of the re-search is supported by the FP7 EU-programmes Hadron-Physics3 (contract no 283286) and the ERC Advanced Grant program QWORK (contract no 320389).

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0 0.05 0.1 0.15 0.2 0 5 10 15 20 q_T [GeV] R Qf = MH x = MH / 3s 0 0.05 0.1 0.15 0.2 0 5 10 15 20 q_T [GeV] R Qf = MH x = MH / 3s h1 = h2 = 0 0 0.05 0.1 0.15 0.2 0 5 10 15 20 q_T [GeV] R Qf = MH x = MH / 3s 0 0.05 0.1 0.15 0.2 0 5 10 15 20 q_T [GeV] R Qf = MH x = MH / 3s 0 0.05 0.1 0.15 0.2 0 5 10 15 20 q_T [GeV] R Qf = MH x = MH / 3s h1 = 0, h2 = 1 0 0.05 0.1 0.15 0.2 0 5 10 15 20 q_T [GeV] R Qf = MH x = MH / 3s 0 0.05 0.1 0.15 0.2 0 5 10 15 20 q_T [GeV] R Qf = MH x = MH / 3s 0 0.05 0.1 0.15 0.2 0 5 10 15 20 q_T [GeV] R Qf = MH x = MH / 3s h1 = 1, h2 = 0 0 0.05 0.1 0.15 0.2 0 5 10 15 20 qT [GeV] R Qf = MH x = MH / 3s 0 0.05 0.1 0.15 0.2 0 5 10 15 20 qT [GeV] R Qf = MH x = MH / 3s 0 0.05 0.1 0.15 0.2 0 5 10 15 20 qT [GeV] R Qf = MH x = MH / 3s h1 = h2 = 1

Figure 2. Ratio of the contributions of linearly polarized and unpolarized gluons to the Higgs boson qT-spectrum at

√

s= 8 TeV and xA= xBfor different values of the non-perturbative parameters λ1,2.

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