Published for SISSA by

Springer

Received: January 25, Revised: May 2, Accepted: May 7, Published: May 24,

2018 2018 2018 2018

I. Balitskya,b and A. Tarasovc a

Physics Department, Old Dominion University, Norfolk, VA 23529, U.S.A. b Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, U.S.A. c Physics Department, Brookhaven National Laboratory, Upton, NY 11973, U.S.A.

E-mail: [email protected], [email protected] Abstract: A typical factorization formula for production of a particle with a small transverse momentum in hadron-hadron collisions is given by a convolution of two TMD parton densities with cross section of production of the final particle by the two partons. For practical applications at a given transverse momentum, though, one should estimate at what momenta the power corrections to the TMD factorization formula become essential. In this paper we calculate the first power corrections to TMD factorization formula for Z-boson production and Drell-Yan process in high-energy hadron-hadron collisions. At the leading order in Nc power corrections are expressed in terms of leading power TMDs by QCD equations of motion. Keywords: NLO Computations, QCD Phenomenology ArXiv ePrint: 1712.09389

c The Authors. Open Access, Article funded by SCOAP3 .

https://doi.org/10.1007/JHEP05(2018)150

JHEP05(2018)150

Power corrections to TMD factorization for Z-boson production

Contents 1 Introduction

1

2 TMD factorization from functional integral

2 6 6 7 11

2 4 Leading power corrections at s  Q2  q⊥ µ 4.1 Leading contribution and power corrections from JAB (x)JBAµ (0) terms 4.2 Leading power contribution 4.2.1 Parametrization of leading matrix elements µ 4.3 Power corrections from JAB (x)JBAµ (0) terms 4.3.1 Fifth line in eq. (4.19): the leading term in N1c 4.3.2 Parametrization of matrix elements from section 4.3.1 4.3.3 Sixth line in eq. (4.19) 4.3.4 Parametrization of matrix elements from section 4.3.3 4.4 Power corrections from JAµ (x)JBµ (0) terms 4.4.1 Last two lines in eq. (4.56) 4.4.2 Parametrization of TMDs from section 4.4.1

12 13 14 15 16 17 20 21 24 24 25 27

5 Results and estimates

28

6 Power corrections for Drell-Yan process

30

7 Conclusions and outlook

31

A Next-to-leading quark fields

33

B Formulas with Dirac matrices

34

C Subleading power corrections C.1 Second, third, and fourth lines in eq. (4.19) C.2 Second to fifth lines in eq. (4.56) C.3 Gluon power corrections from JAµ (x)JAµ (0) terms C.4 Power corrections from Ψ(1) fields

35 35 37 38 39

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3 Power corrections and solution of classical YM equations 3.1 Power counting for background fields 3.2 Approximate solution of classical equations 3.3 Power expansion of classical quark fields

1

Introduction

A typical analysis of differential cross section of particle production in hadron-hadron collisions at small momentum transfer of the produced particle is performed with the help of TMD factorization [1–10]. However, the question of how small should be the momentum transfer in order for leading power TMD analysis to be successful cannot be resolved at the leading-power level. The sketch of the factorization formula for the differential cross section is [1, 11] Z

d2 b⊥ ei(q,b)⊥ Df /A (xA , b⊥ , η)Df /B (xB , b⊥ , η)σ(f f → H)

f

+ power corrections + Y − terms,

(1.1)

where η is the rapidity, q is the momentum of the produced particle in the hadron frame (see ref. [1]), Df /A (x, z⊥ , η) is the TMD density of a parton f in hadron A, and σ(f f → H) is the cross section of production of particle H in the scattering of two partons. The common 2 of the produced hadron, at first wisdom is that when we increase transverse momentum q⊥ the leading power TMD analysis with (nonperturbative) TMDs applies, then at some point 2 ∼ Q2 , where Q2 = q 2 , they are transformed power corrections kick in, and finally at q⊥ into so-called Y-term making smooth transition to collinear factorization formulas. In this 2 power paper we try to answer the question about the first transition, namely at what q⊥ corrections become significant. q2

In our recent paper [12] we calculated power corrections ∼ Q⊥2 for Higgs boson production by gluon-gluon fusion. The result was a TMD factorization formula with matrix elements of three-gluon operators divided by an extra power of m2H . In this paper we q2

calculate power corrections ∼ Q⊥2 for Z-boson production which are determined by quarkquark-gluon operators. In the leading order Z-boson production was studied in [13–21]. The interesting (and unexpected) result of our paper is that at the leading-Nc level matrix elements of the relevant quark-quark-gluon operators can be expressed in terms of leading power quark-antiquark TMDs by QCD equations of motion (see ref. [22]). The method of calculation is very similar to that of ref. [12] so we will streamline the discussion of the general approach and pay attention to details specific to quark operators. The paper is organized as follows. In section 2 we derive the TMD factorization from the double functional integral for the cross section of particle production. In section 3, which is central to our approach, we explain the method of calculation of power corrections based on a solution of classical Yang-Mills equations. In section 4 we find the leading 2 . In section 5 we power correction to particle production in the region s  Q2  q⊥ perform the order-of-magnitude estimate of power corrections and in section 6 present our result for power corrections to the Drell-Yan cross section. The necessary technical details and discussion of subleading power corrections can be found in appendix.

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X dσ = 2 dηd q⊥

2

TMD factorization from functional integral

We consider Z-boson production in the Drell-Yan reaction illustrated in figure 1: hA (pA ) + hB (pA ) → Z(q) + X → l1 (k1 ) + l2 (k2 ) + X,

(2.1)

where hA,B denote the colliding hadrons, and l1,2 the outgoing lepton pair with total momentum q = k1 + k2 . The relevant term of the Lagrangian for the fermion fields ψi describing coupling between fermions and Z-boson is (sW ≡ sin θW , cW ≡ cos θW ) Z X e LZ = d4 x Jµ Z µ (x), Jµ = − ψ¯i γµ (giV − giA γ5 )ψi , (2.2) 2sW cW i

2 where sum goes over different types of fermions, and coupling constants giV = (tL 3 )i − 2qi sW L and giA = (tL 3 )i are defined by week isospin (t3 )i of the fermion i, see ref. [23]. In this paper we take into account only u, d, s, c quarks and e, µ leptons. We consider all fermions to be massless. The differential cross section of Z-boson production with subsequent decay into e+ e− (or µ+ µ− ) pair is

1 − 4s2W + 8s4W dσ e2 Q2 = [−W (pA , pB , q)], 2 dQ2 dydq⊥ 192ss2W c2W (m2Z − Q2 )2 + Γ2Z m2Z where we defined the “hadronic tensor” W (pA , pB , q) as Z 1 X def W (pA , pB , q) = d4 x e−iqx hpA , pB |Jµ (x)|XihX|J µ (0)|pA , pB i (2π)4 ZX 1 = d4 x e−iqx hpA , pB |Jµ (x)J µ (0)|pA , pB i. (2π)4

–2–

(2.3)

(2.4)

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Figure 1. Z-boson production in hadron-hadron collisions.

P As usual, X denotes the sum over full set of “out” states. It should be mentioned that there is a power correction coming from the leptonic tensor term ∼ q µ q ν . However, if we consider quarks to be massless, the only effect of the q µ q ν term comes from the (square of) axial anomaly which has an extra factor αs2 , and such two-loop factor is beyond our tree approximation. The sum over full set of “out” states in eq. (2.4) can be represented by a double functional integral XZ 4 (2π) W (pA , pB , q) = d4 x e−iqx hpA , pB |J µ (x)|XihX|Jµ (0)|pA , pB i (2.5) ˜ f )=A(tf ) A(t

DA˜µ DAµ

ti →−∞

Z

˜ f )=ψ(tf ) ψ(t

~˜ ), ψ(t ˜¯ ψD ˜ ψDψ ¯ ˜ i )) DψD Ψ∗pA (A(t i

˜ iSQCD (A,ψ) ˜ ˜ ψ) ~˜ ), ψ(t ˜ i ))e−iSQCD (A, ~ i ), ψ(ti ))Ψp (A(t ~ i ), ψ(ti )). ×Ψ∗pB (A(t e Jµ (x)J µ (0)ΨpA (A(t i B

In this double functional integral the amplitude hX|Jµ (0)|pA , pB i is given by the integral over ψ, A fields whereas the complex conjugate amplitude hpA , pB |J µ (x)|Xi is represented ˜ A˜ fields. Also, Ψp (A(t ~ i ), ψ(ti )) denotes the proton wave function at by the integral over ψ, ˜ f ) = ψ(tf ) reflect ˜ f ) = A(tf ) and ψ(t the initial time ti and the boundary conditions A(t the sum over all states X, cf. refs. [24–26]. We use Sudakov variables p = αp1 + βp2 + p⊥ , where p1 and p2 are light-like vectors close to pA and pB , and the notations x• ≡ xµ pµ1 and x∗ ≡ xµ pµ2 for the dimensionless p p light-cone coordinates (x∗ = 2s x+ and x• = 2s x− ). Our metric is g µν = (1, −1, −1, −1) so that p · q = (αp βq + αq βp ) 2s − (p, q)⊥ where (p, q)⊥ ≡ −pi q i . Throughout the paper, the sum over the Latin indices i, j, . . . runs over two transverse components while the sum over Greek indices µ, ν, . . . runs over four components as usual. Following ref. [12] we separate quark and gluon fields in the functional integral (2.5) into three sectors (see figure 2): “projectile” fields Aµ , ψA with |β| < σa , “target” fields Bµ , ψB with |α| < σb and “central rapidity” fields Cµ , ψC with |α| > σb and |β| > σa and get Z Z A(t Z ψ˜A (tf )=ψA (tf ) ˜ f )=A(tf ) 1 4 −iqx ˜ W (pA , pB , q) = d xe DAµ DAµ Dψ¯A DψA (2π)4 ˜ ˜ ~˜ ), ψ˜ (t ))Ψ (A(t ~ ), ψ (t )) × Dψ¯˜ Dψ˜ e−iSQCD (A,ψA ) eiSQCD (A,ψA ) Ψ∗ (A(t A

Z ×

A

˜ f )=B(tf ) B(t

pA

˜µ DBµ DB

Z

i

A

i

pA

i

A

i

ψ˜B (tf )=ψB (tf )

Dψ¯B DψB Dψ˜¯B Dψ˜B

˜ ˜ ~˜ ), ψ˜ (t ))Ψ (B(t ~ i ), ψB (ti )) × e−iSQCD (B,ψB ) eiSQCD (B,ψB ) Ψ∗pB (B(t (2.6) i pB B i Z Z C(t Z Z ψ˜C (tf )=ψC (tf ) ˜ f )=C(tf ) ˜ × DCµ DC˜µ Dψ¯C DψC Dψ˜¯C Dψ˜C J˜µ (x)J µ (0) e−iSC +iSC ,

where SC = SQCD (C + A + B, ψC + ψA + ψB ) − SQCD (A, ψA ) − SQCD (B, ψB ) and similarly for S˜C .1 1

This procedure is obviously gauge-dependent. We have in mind factorization in covariant-type gauge, e.g. Feynman gauge.

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X

Z tf →∞ Z 4 −iqx = lim d xe

Our goal is to integrate over central fields and get the amplitude in the factorized form, i.e. as a product of functional integrals over A fields representing projectile matrix elements (TMDs of the projectile) and functional integrals over B fields representing target matrix elements (TMDs of the target). In the spirit of background-field method, we “freeze” projectile and target fields and get a sum of diagrams in these external fields. Since |β| < σa in the projectile fields and |α| < σb in the target fields, at the tree-level one can set with power accuracy β = 0 for m2
the projectile fields and α = 0 for the target fields — the corrections will be O σaNs and m2
O σbNs , where mN is the hadron’s mass. Beyond the tree level, one should expect that the integration over C fields will produce the logarithms of the cutoffs σa and σb which will cancel with the corresponding logs in gluon TMDs of the projectile and the target. The result of integration over C-fields has the schematic form Z Z C(t Z Z ψ˜C (tf )=ψC (tf ) ˜ f )=C(tf ) ˜ ¯ ˜ DCµ DCµ DψC DψC Dψ˜¯C Dψ˜C J˜µ (x)J µ (0) e−iSC +iSC ˜ ˜ ˜ ˜ ˜ ψ˜A ; B, ψB , B, ˜ ψ˜B ), = eSeff (A,ψA ,A,ψA ;B,ψB ,B,ψB ) O(q, x; A, ψA , A,

(2.7)

˜ ψ˜A ; B, ψB , B, ˜ ψ˜B ) is a sum of diagrams connected to J˜µ (x)J µ (0) where O(q, x; A, ψA , A, and eSeff represents a sum of disconnected diagrams (“vacuum bubbles”) in external fields. As usual, since the rapidities of central C fields and of A, B fields are very different, the result of integration over C fields is expressed in terms of Wilson-line operators made form A and B fields. After integration over C fields the amplitude (2.5) can be rewritten as Z Z A(t Z ψ˜A (tf )=ψA (tf ) ˜ f )=A(tf ) 1 4 −iqx ˜ W (pA , pB , q) = d xe DAµ DAµ Dψ¯A DψA Dψ˜¯A Dψ˜A (2π)4 Z B(t ˜ f )=B(tf ) ˜ ψ˜A ) iSQCD (A,ψA ) ∗ ~ −iSQCD (A, ˜ ˜ ~ ˜µ DBµ × e e ΨpA (A(ti ), ψA (ti ))ΨpA (A(ti ), ψA (ti )) DB Z ×

ψ˜B (tf )=ψB (tf )

˜ ψ˜B ) iSQCD (B,ψB ) ∗ ~˜ ), ψ˜ (t )) ¯B Dψ˜B e−iSQCD (B, Dψ¯B DψB Dψ˜ e ΨpB (B(t i B i

˜ ψ˜A ;B,ψB ,B, ˜ ψ˜B ) ~ i ), ψB (ti ))eSeff (A,ψA ,A, ˜ ψ˜A ; B, ψB , B, ˜ ψ˜B ). × ΨpB (B(t O(q, x; A, ψA , A,

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(2.8)

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Figure 2. Rapidity factorization for particle production.

From integrals over projectile and target fields in the above equation we see that the functional integral over C fields should be done in the background of A and B fields satisfying ˜ f ) = A(tf ), A(t

ψ˜A (tf ) = ψA (tf ) and

˜ f ) = B(tf ), B(t

ψ˜B (tf ) = ψB (tf ).

(2.9)

Combining this with our approximation that at the tree level β = 0 for A, A˜ fields and α = 0 ˜ fields, which corresponds to A = A(x• , x⊥ ), A˜ = A(x ˜ • , x⊥ ) and B = B(x∗ , x⊥ ), for B, B ˜ = B(x ˜ ∗ , x⊥ ), we see that for the purpose of calculation of the functional integral over B central fields (2.7) we can set ψA (x• , x⊥ ) = ψ˜A (x• , x⊥ )

˜ ∗ , x⊥ ), B(x∗ , x⊥ ) = B(x

ψB (x∗ , x⊥ ) = ψ˜B (x∗ , x⊥ ).

and (2.10)

˜ ψ˜ do not depend on x∗ , if they coincide at x∗ = ∞ they In other words, since A, ψ and A, ˜ ψ˜B do not depend on x• , if they should coincide everywhere. Similarly, since B, ψB and B, coincide at x• = ∞ they should be equal. Next, in ref. [12] it was demonstrated that due to eqs. (2.10) the effective action ˜ ψ˜A ; B, ψB , B, ˜ ψ˜B ) vanishes for background fields satisfying conditions (2.9).2 Seff (A, ψA , A, Summarizing, we see that at the tree level in our approximation Z

Z DCµ

˜ f )=C(tf ) C(t

DC˜µ

Z

Dψ¯C DψC

Z

ψ˜C (tf )=ψC (tf )

= O(q, x; A, ψA ; B, ψB ),

˜ Dψ˜¯C Dψ˜C J˜µ (x)J µ (0) e−iSC +iSC

(2.11)

where now SC = SQCD (C + A + B, ψC + ψA + ψB ) − SQCD (A, ψA ) − SQCD (B, ψB ) and S˜C = SQCD (C˜ +A+B, ψ˜C +ψA +ψB )−SQCD (A, ψA )−SQCD (B, ψB ). It is well known that in the tree approximation the double functional integral (2.11) is given by a set of retarded Green functions in the background fields [27–29] (see also appendix A of ref. [12] for the proof). Since the double functional integral (2.11) is given by a set of retarded Green functions (in the background fields A and B), calculation of the tree-level contribution to ¯ µ ψ in the r.h.s. of eq. (2.11), is equivalent to solving YM equation for ψ(x) (and Aµ (x)) ψγ with boundary conditions such that the solution has the same asymptotics at t → −∞ as the superposition of incoming projectile and target background fields. The hadronic tensor (2.8) can now be represented as3 Z 1 ˆ x; A, ˆ ψˆA ; B, ˆ ψˆB )|pA i|pB i, W (pA , pB , q) = d4 x e−iqx hpA |hpB |O(q, (2.12) (2π)4 2

It corresponds to cancellation of so-called “Glauber gluons”, see discussion in ref. [1]. As discussed in ref. [12], there is a subtle point in the promotion of background fields to operators. When we calculate O as the r.h.s. of eq. (2.11) the fields ΦA and ΦB are c-numbers; on the other hand, after functional integration in eq. (2.5) they become operators which must be time-ordered in the right sector and anti-time-ordered in the left sector. Fortunately, as we shall see below, all these operators are separated either by space-like distances or light-cone distances so all of them (anti) commute and thus can be treated as c-numbers. 3

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˜ • , x⊥ ), A(x• , x⊥ ) = A(x

ˆ x; A, ˆ ψˆA ; B, ˆ ψˆB ) should be expanded in a series in A, ˆ ψˆA , B, ˆ ψˆB operators and where O(q, evaluated between the corresponding (projectile or target) states: if XZ ˆ ˆ ˆ ˆ ˆ ˆ A (zm )Φ ˆ B (z 0 ) O(q, x; A, ψA ; B, ψB ) = dzm dzn0 cm,n (q, x)Φ (2.13) n m,n

¯ then (where cm,n are coefficients and Φ can be any of Aµ , ψ or ψ) 1 W = (2π)4

Z

4

d xe

−iqx

XZ

ˆ A (zm )|pA i dzm cm,n (q, x)hpA |Φ

Z

ˆ B (z 0 )|pB i. dzn0 hpB |Φ n

(2.14)

As we will demonstrate below, the relevant operators are quark and gluon fields with Wilson-line type gauge links collinear to either p2 for A fields or p1 for B fields.

3

Power corrections and solution of classical YM equations

3.1

Power counting for background fields

As we discussed in previous section, to get the hadronic tensor in the form (2.12) we need to calculate the functional integral (2.11) in the background of the fields (2.10). Since we integrate over fields (2.10) afterwards, we may assume that they satisfy Yang-Mills equations4 / A ψA = 0, iD

ν a DA Aµν = g

X

/ B ψB = 0, iD

ν a DB Bµν

X

f f ψ¯A γ µ ta ψ A ,

f

=g

f f ψ¯B γ µ ta ψ B ,

(3.1)

f µ where Aµν ≡ ∂µ Aν − ∂ν Aµ − ig[Aµ , Aν ], DA ≡ (∂ µ − ig[Aµ , ) and similarly for B fields. It is convenient to choose a gauge where A∗ = 0 for projectile fields and B• = 0 for target fields. The rotation from a general gauge (Feynman gauge in our case, see footnote 1) to this gauge is performed by the matrix Ω(x∗ , x• , x⊥ ) satisfying boundary conditions

Ω(x∗ , x• , x⊥ )

x∗ →−∞

→

∗ [x• , −∞• ]A x ,

Ω(x∗ , x• , x⊥ )

x• →−∞

→

• [x∗ , −∞∗ ]B x ,

(3.2)

where A∗ (x• , x⊥ ) and B• (x∗ , x⊥ ) are projectile and target fields in an arbitrary gauge and ∗ [x• , y• ]A z denotes a gauge link constructed from A fields ordered along a light-like line: ∗ [x• , y• ]A z = Pe

2ig s

R x• y•

dz• A∗ (z• ,z⊥ )

(3.3)

• and similarly for [x∗ , y∗ ]B z . The existence of matrix Ω(x∗ , x• , x⊥ ) was proved in appendix B of ref. [12] by explicit construction. The relative strength of Lorentz components of projectile and target fields 4

As we mentioned, for the purpose of calculation of integral over C fields the projectile and target fields are “frozen”.

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m,n

in this gauge was found in ref. [12] 5/2

3/2

p /1 ψA (x• , x⊥ ) ∼ m⊥ , √ p /1 ψB (x∗ , x⊥ ) ∼ s m⊥ ,

γi ψA (x• , x⊥ ) ∼ m⊥ , 3/2

γi ψB (x∗ , x⊥ ) ∼ m⊥ ,

A• (x• , x⊥ ) ∼ B∗ (x∗ , x⊥ ) ∼ m2⊥ ,

√ p /2 ψA (x• , x⊥ ) ∼ s m⊥ , 5/2

p /2 ψB (x∗ , x⊥ ) ∼ m⊥ ,

Ai (x• , x⊥ ) ∼ Bi (x∗ , x⊥ ) ∼ m⊥ .

(3.4)

(A)

(B)

where A∗i ≡ F∗i and B•i ≡ F•i are field strengths for A and B fields respectively. Thus, to find TMD factorization formula with power corrections at the tree level we need to calculate the functional integral (2.5) in the background fields of the strength given by eqs. (3.4). 3.2

Approximate solution of classical equations

As we discussed in section 2, the calculation of the functional integral (2.11) over C-fields in the tree approximation reduces to finding fields Cµ and ψC as solutions of Yang-Mills equations for the action SC = SQCD (C +A+B, ψC +ψA +ψB )−SQCD (A, ψA )−SQCD (B, ψB ) f f f / + gB / + g C)(ψ / (i∂/ + g A (3.6) A + ψB + ψC ) = 0, X f f f f f f a Dν Fµν (A + B + C) = g (ψ¯A + ψ¯B + ψ¯C )γµ ta (ψA + ψB + ψC ). f

As we discussed above, the solution of eq. (3.6) which we need corresponds to the sum of set of diagrams in background field A + B with retarded Green functions, see figure 3. The retarded Green functions (in the background-Feynman gauge) are defined as (x|

1 1 1 1 |y) ≡ (x| 2 |y)−g(x| 2 Oµν 2 |y) P 2 g µν +2igF µν +ip0 p +ip0 p +ip0 p +ip0 1 1 1 Oµξ 2 Oξ |y)+. . . , +g 2 (x| 2 p +ip0 p +ip0 ν p2 +ip0

(3.7)

where Pµ ≡ i∂µ + gAµ + gBµ , Oµν

Fµν = ∂µ (A + B)ν − µ ↔ ν − ig[Aµ + Bµ , Aν + Bν ],
≡ {pξ , Aξ + Bξ } + g(A + B)2 gµν + 2iFµν (3.8)

and similarly for quarks.

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Here m⊥ is a scale of order of mN or q⊥ . In general, we consider W (pA , pB , q) in the region 2 , m2 , while the relation between q 2 and m2 and between Q2 and s may where s, Q2  q⊥ N N ⊥ be arbitrary. So, for the purpose of counting of powers of s, we will not distinguish between s and Q2 (although at the final step we will be able to tell the difference since our final expressions for power corrections will have either s or Q2 in denominators). Similarly, for the purpose of power counting we will not distinguish between mN and q⊥ so we introduce m⊥ which may be of order of mN or q⊥ depending on matrix element. Note also that in our gauge Z Z 2 x• 0 2 x∗ 0 0 Ai (x• , x⊥ ) = dx A∗i (x• , x⊥ ), Bi (x∗ , x⊥ ) = dx B•i (x0∗ , x⊥ ), (3.5) s −∞ • s −∞ ∗

Hereafter we use Schwinger’s notations for propagators in external fields normalized according to Z (x|F (p)|y) ≡ d−4 p e−ip(x−y) F (p), (3.9) n

d p where we use space-saving notation d−n p ≡ (2π) n . Moreover, when it will not lead to a confusion, we will use short-hand notation Z 1 0 1 O (x) ≡ d4 z(x| |z)O0 (z). (3.10) O O

The solution of eqs. (3.6) in terms of retarded Green functions gives fields Cµ and ψC that vanish at t → −∞. Thus, we are solving the usual classical YM equations5 X ¯ f ta γ µ Ψ f , / Ψf = 0, Dν Faµν = gΨ P (3.11) f

where f f f Ψ f = ψC + ψA + ψB ,

Aµ = Cµ + Aµ + Bµ , Pµ ≡ i∂µ + gCµ + gAµ + gBµ ,

Fµν = ∂µ Aν − µ ↔ ν − ig[Aµ , Aν ],

(3.12)

with boundary conditions6 Aµ (x) Aµ (x)

x∗ →−∞

=

x• →−∞

=

Aµ (x• , x⊥ ),

Ψ(x)

Bµ (x∗ , x⊥ ),

Ψ(x)

x∗ →−∞

=

x• →−∞

=

ψA (x• , x⊥ ), ψB (x∗ , x⊥ )

(3.13)

t→−∞

following from Cµ , ψC → 0. These boundary conditions reflect the fact that at t → −∞ we have only incoming hadrons with A and B fields. 5

We take into account only u, d, s, c quarks and consider them massless. It is convenient to fix redundant gauge transformations by requirements Ai (−∞• , x⊥ ) = 0 for the projectile and Bi (−∞∗ , x⊥ ) = 0 for the target, see the discussion in ref. [30]. 6

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Figure 3. Typical diagram for the classical field with projectile/target sources. The Green functions of the central fields are given by retarded propagators.

As discussed in ref. [12], for our case of particle production with qQ⊥  1 it is possible to find the approximate solution of (3.11) as a series in this small parameter. We will solve eqs. (3.11) iteratively, order by order in perturbation theory, starting from the zero-order approximation in the form of the sum of projectile and target fields A[0] µ (x) = Aµ (x• , x⊥ ) + Bµ (x∗ , x⊥ ), Ψ[0] (x) = ψA (x• , x⊥ ) + ψB (x∗ , x⊥ )

(3.14)

(1) / [0] = L(0) Lψ ≡ PΨ ψ + Lψ , (0)

(1)

Lψ = gγ i Ai ψB + gγ i Bi ψA ,

Lψ =

2g 2g p / A• ψ B + p / B ∗ ψA , s 2 s 1

(3.15)

Pi = i∂i + gAi + gBi

(3.16)

where P• = i∂• + gA• ,

P∗ = i∂∗ + gB∗ ,

are operators in external zero-order fields (3.14). Here we denote the order of expansion in m2

the parameter s⊥ by (. . .)(n) , and the order of perturbative expansion is labeled by (. . .)[n] as usual. The power-counting estimates for linear term in eq. (3.15) comes from eq. (3.4) in the form 9/2 m (0) 5/2 (1) L ψ ∼ m⊥ , Lψ ∼ ⊥ . (3.17) s The gluon linear term is a ¯ [0] γµ ta Ψ[0] = L(−1)a Laµ ≡ Dξ Fξµ + gΨ + L(0)a + L(1)a µ µ µ , 2g 2g b L(−1)a = p1µ f abc Ab∗j B cj + p2µ f abc B•j Acj ∼ sm2⊥ . µ s s (0)a

(3.18)

(1)a

The explicit form of gluon linear terms Lµ and Lµ is presented in eq. (3.26) from our (−1)a paper [12]. For our purposes we need only the leading term Lµ . With the linear terms (3.15) and (3.18), a couple of first terms in our perturbative series are Z Z 1 1 [1] 4 [2] / [1] (z)Ψ[0] (z) Ψ (x) = − d z(x| |z)Lψ (z), Ψ (x) = −g d4 z(x| |z)A (3.19) / / P P for quark fields and Z

1 d4 z(x| 2 µν |z)ab Lbν (z), P g + 2igF µν " Z 1 0 0 [1]b [2]a 4 Aµ (x) = g d z − i(x| 2 µη P ξ |z)aa f a bc Aξ A[1]cη µη P g + 2igF A[1]a µ (x)

=

  1 aa0 a0 bc [1]bξ [1]cη η [1]c + (x| 2 µη |z) f A D A − D A ξ ξ P g + 2igF µη

–9–

(3.20)

#

JHEP05(2018)150

and improving it by calculation of Feynman diagrams with retarded propagators in the background fields (3.14). The first step is the calculation of the linear term for the trial configuration (3.14). The quark part of the linear term has the form

for gluon fields (in the background-Feynman gauge). Next iterations, like Ψ [3] (x) and [3]a Aµ (x), will give us a set of tree-level Feynman diagrams in the background fields Aµ + Bµ and ψA + ψB . Let us consider the fields in the first order in perturbative expansion:

(3.21)

∂ Here α, β, and p⊥ are understood as differential operators α = i ∂x∂ • , β = i ∂x∂ ∗ and pi = i ∂x i. Now comes the central point of our approach. Let us expand quark and gluon propagators in powers of background fields, then we get a set of diagrams shown in figure 3. The typical bare gluon propagator in figure 3 is

p2

1 1 = . 2 + ip0 αβs − p⊥ + i(α + β)

(3.22)

Since we do not consider loops of C-fields in this paper, the transverse momenta in tree diagrams are determined by further integration over projectile (“A”) and target (“B”) fields in eq. (2.8) which converge on either q⊥ or mN . On the other hand, the integrals over α converge on either αq or α ∼ 1 and similarly the characteristic β’s are either βq or β ∼ 1. 2 , one can expand gluon and quark propagators in powers of Since αq βq s = Q2k  q⊥

p2⊥ αβs

  p2⊥ /s 1 1 = 1+ + ... , (3.23) p2 + ip0 s(α + i)(β + i) (α + i)(β + i)    p p p p p2⊥ /s /1 /2 /⊥ 1 / = + + 1 + + . . . . p2 + ip0 s β + i α + i (α + i)(β + i) (α + i)(β + i) The explicit form of operators

1 1 α+i , β+i ,

s 1 (x| |y) = α + i 2

Z

d−2 p

Z ⊥

and

1 (α+i)(β+i)

is

d−α − −iα(x−y)• −iβ(x−y)∗ +i(p,x−y)⊥ dβ e α + i

s = −i (2π)2 δ (2) (x⊥ − y⊥ )θ(x• − y• )δ(x∗ − y∗ ), Z2 Z 1 s −2 d−β (x| |y) = d p⊥ d−α e−iα(x−y)• −iβ(x−y)∗ +i(p,x−y)⊥ β + i 2 β + i s = −i (2π)2 δ (2) (x⊥ − y⊥ )θ(x∗ − y∗ )δ(x• − y• ), 2 Z Z − 1 s −2 d α d−β (x| |y) = d p⊥ e−iα(x−y)• −iβ(x−y)∗ +i(p,x−y)⊥ (α + i)(β + i) 2 α + i β + i s = − (2π)2 δ (2) (x⊥ − y⊥ )θ(x∗ − y∗ )θ(x• − y• ). (3.24) 2

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1 1 Ψ[1] = − Lψ = − Lψ , 2g / / ⊥ + ip0 P αp /1 + β p /2 + s (B∗ p / 1 + A• p /2 ) + P 1 A[1] Lν µ = 2 µν P g + 2igF µν 1 o i = hn Lν . 2g 2g s 2 µν µν α + s B∗ , β + s A• 2 − P⊥ g + 2igF + ip0

After the expansion (3.23), the dynamics in the transverse space effectively becomes trivial: all background fields stand either at x or at 0. The formula (3.21) turns into expansion       p p p p p /1 p /2 /2 /1 /1 /2 p /2 /1 2g 1 p [1] / / Ψf = − + Lψ + 2 B∗ 2 +A• 2 Lψ + 2 P + P Lψ +. . . , βs αs s α β s β ⊥α α ⊥β 
1 ν 1 1  2 A[1] Lµ + P⊥ −g{α, A• }−g{β, B∗ } gµν −2igFµν L +. . . , (3.25) µ = αβs αβs αβs

p2

the expansion was not justified since actual α’s in the integral are ∼ s⊥ and hence the field was misidentified: we have a propagator of B-field rather than of C-field. Fortunately, at the tree level all propagators are retarded and the pinching of poles never occurs. In the higher orders in perturbation theory Feynman propagators in the loops cannot be replaced 1 by retarded propagators so after the expansion (3.23) we can get (α+iβ)(α+iβ 0 ) . In such case the pinching may occur so one needs to formulate a subtraction program to get rid of pinched poles and avoid double counting of the fields. Note that the background fields are also smaller than typical p2k ∼ s. Indeed, from eq. (3.4) we see that p• = 2s β  A• ∼ m2⊥ ( because α ≥ αq  2  p2 . 8 Also (pi + Ai + Bi )2 ∼ q⊥ k 3.3

m2⊥ s )

and similarly p∗  B∗ .

Power expansion of classical quark fields

Now we expand the classical quark fields in powers of

p2⊥ p2k

∼

m2⊥ s

(the expansion of classical

gluon fields is presented in eqs. (3.35)–(3.38) in ref. [12]). From the previous section it is clear that the leading power correction comes only from the first term displayed in eq. (3.19). Expanding it in powers of p2⊥ /p2k as explained in the previous section, we obtain (0)

(0)

(1)

(1)

Ψ(x) = Ψ[0] (x) + Ψ[1] (x) + Ψ[2] (x) + · · · = ΨA + ΨB + ΨA + ΨB + . . . ,

(3.26)

where (0)

Ξ2A = −

¯ (0) = ψ¯A + Ξ ¯ 2A , Ψ A

¯ 2A Ξ

(0)

ΨB = ψB + Ξ1B ,

Ξ1B

¯ (0) = ψ¯B + Ξ ¯ 1B , Ψ B

¯ 1B Ξ

gp /2

1 ψA , α +i s gp / 1 = − ψ¯A γ i Bi 2 , α − i s gp /1 i 1 =− γ Ai ψB , s β + i   gp / 1 ¯ = − ψB γ i Ai 1 . β − i s

ΨA = ψA + Ξ2A ,

7

γ i Bi

(3.27)

Such cutoffs for integrals over C fields are introduced explicitly in the framework of soft-collinear effective theory (SCET), see review [31]. 8 The only exception is the fields B•i or A∗i which are of order of sm⊥ but we saw in ref. [12] that effectively the expansion in powers of these fields is cut at the second term.

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JHEP05(2018)150

1 1 where α1 and β1 are understood as α+i and β+i respectively. One may question why we do not cut the integrals in eq. (3.24) to |α| > σb and |β| > σa according to the definition of C fields in section 2.7 The reason is that in the diagrams like figure 3 with retarded propagators (3.24) one can shift the contour of integration over α and/or β to the complex plane away to avoid the region of small α or β. It should be mentioned, however, that such shift may not be possible if there is pinching of poles in the 1 integrals over α or β. For example, if after the expansion (3.23) we encounter (α+i)(α−i) ,

In this formula Z x• 1 ψA (x• , x⊥ ) ≡ −i dx0• ψA (x0• , x⊥ ), α + i −∞   Z x• 1 ψ¯A (x• , x⊥ ) ≡ i dx0• ψ¯A (x0• , x⊥ ) (3.28) α − i −∞

1 1 and similarly for β±i . From now on we will denote ψ¯A α1 (x) ≡ ψ¯A α−i (x) and





1 1 1 ψ¯B β1 (x) ≡ ψ¯B β−i (x) while in all other places α1 O ≡ α+i O and β1 O ≡ β+i O . It is easy to see that power counting of these quark fields has the form (0)

3/2

(3.29)

As to quark fields Ψ(1) , we present their explicit form in appendix A and prove in appendix C that their contribution is small in the kinematic region s  Q2 . 2 Leading power corrections at s  Q2  q⊥

4

As we mentioned in the introduction, our method is relevant to calculation of power cor2 , m2 . However, the expressions are greatly simplified in the rections at any s, Q2  q⊥ N 2 which we consider in this paper.9 physically interesting case s  Q2  q⊥ As we noted above, we take into account only u, d, s, c quarks and consider them massless. The hadronic tensor takes the form Z W (pA , pB , q) = d2 x⊥ ei(q,x)⊥ W (αq , βq , x⊥ ), (4.1) Z 1 2 W (αq , βq , x⊥ ) ≡ dx• dx∗ e−iαq x• −iβq x∗ hpA , pB |Jµ (x• , x∗ , x⊥ )J µ (0)|pA , pB i, (2π)4 s where (cW ≡ cos θW , sW ≡ sin θW )10      8 8 e −u ¯γµ 1 − s2W − γ5 u − c¯γµ 1 − s2W − γ5 c Jµ = 4sW cW 3 3      4 2 4 2 ¯ + dγµ 1 − sW − γ5 d + s¯γµ 1 − sW − γ5 s . 3 3

(4.2)

After integration over central fields in the tree approximation we obtain Z 2 W (αq , βq , x⊥ ) ≡ dx• dx∗ e−iαq x• −iβq x∗ hpA |hpB |Jµ (x• , x∗ , x⊥ )J µ (0)|pA i|pB i, (2π)4 s (4.3) where µ µ J µ = JAµ + JBµ + JAB + JBA ,   e ¯ Au γ˘ µ ΨAu − Ψ ¯ Ac γ˘ µ ΨAc + Ψ ¯ Ad γ˘ µ ΨAd + Ψ ¯ As γ˘ µ ΨAs , JAµ = −Ψ 4sW cW   e µ ¯ As γ˘ µ ΨBs , ¯ Au γ˘ µ ΨBu − Ψ ¯ Ac γ˘ µ ΨBc + Ψ ¯ Ad γ˘ µ ΨBd + Ψ JAB = −Ψ 4sW cW 9 10

(4.4)

We also assume that Z-boson is emitted in the central region of rapidity so αq s ∼ βq s  Q2 . We denote the weak coupling constant by e/sW and reserve the notation “g” for QCD coupling constant.

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JHEP05(2018)150

(0)

Ψ A ∼ Ψ B ∼ m⊥ .

µ and similarly for JBµ and JBA . Hereafter we use notation γ˘µ ≡ γµ (a − γ5 ) where a is one 8 2 of au,c = (1 − 3 sW ) or ad,s = (1 − 43 s2W ) depending on quark’s flavor. m2

µ The leading power contribution comes only from product JAB (x)JBAµ (0) (or µ JBA (x)JABµ (0)), while power corrections may come from other terms like JAµ (x)JBµ (0). We will consider all terms in turn. µ Leading contribution and power corrections from JAB (x)JBAµ (0) terms

4.1

µ Power expansion of JAB (x)JBAµ (0) reads (0) (0) ¯ A (x)˘ ¯ B (0)˘ ¯ (0) (x)˘ ¯ (0) (0)˘ Ψ γ µ ΨB (x)Ψ γµ ΨA (0) = Ψ γ µ ΨB (x)Ψ γµ ΨA (0) A B

(4.6)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

¯ (x)˘ ¯ (0)˘ ¯ (x)˘ ¯ (0)˘ +Ψ γ µ ΨB (x)Ψ γµ ΨA (0) + Ψ γ µ ΨB (x)Ψ γµ ΨA (0) A B A B ¯ (x)˘ ¯ (0)˘ ¯ (x)˘ ¯ (0)˘ +Ψ γ µ ΨB (x)Ψ γµ ΨA (0) + Ψ γ µ ΨB (x)Ψ γµ ΨA (0) + . . . A B A B In appendix C.4 we demonstrate that terms ∼ Ψ(1) lead to power corrections ∼ ∼

2 q⊥ βq s

which are much smaller than

2 q⊥ α q βq s

=

2 q⊥ Q2k

∼

2 q⊥ Q2

2 q⊥ αq s

or

if Z-boson is emitted in the central

region of rapidity. Note that since we want to calculate the leading power corrections, 2 this change of variables can hereafter we substitute Q2k with Q2 . In the limit s  Q2  q⊥ only lead to errors of the order of subleading power terms. ¯ (0) (x)γµ Ψ(0) (x)Ψ ¯ (0) (0)γ µ Ψ(0) (0), they can be decomposed using As to terms ∼ Ψ A B B A eq. (3.27) as follows: 



 ¯ 2A (x)˘ ¯ 1B (0)˘ ψ¯A + Ξ γµ ψB + Ξ1B (x) [ ψ¯B + Ξ γ µ ψA + Ξ2A (0) + x ↔ 0   = [ψ¯A (x)˘ γµ ψB (x) ψ¯B (0)˘ γ µ ψA (0) (4.7)     µ µ ¯ ¯ ¯ ¯ + [Ξ2A (x)˘ γµ ψB (x) ψB (0)˘ γ ψA (0) + [ψA (x)˘ γµ Ξ1B (x) ψB (0)˘ γ ψA (0)     µ ¯ 1B (0)˘ + [ψ¯A (x)˘ γµ ψB (x) Ξ γ ψA (0) + [ψ¯A (x)˘ γµ ψB (x) ψ¯B (0)˘ γ µ Ξ2A (0)     ¯ 2A (x)˘ ¯ 1B (0)˘ + [Ξ γµ ψB (x) ψ¯B (0)˘ γ µ Ξ2A (0) + [ψ¯A (x)˘ γµ Ξ1B (x) Ξ γ µ ψA (0)     ¯ 2A (x)˘ ¯ 1B (0)˘ + [Ξ γµ ψB (x) Ξ γ µ ψA (0) + [ψ¯A (x)˘ γµ Ξ1B (x) ψ¯B (0)˘ γ µ Ξ2A (0)     ¯ 1B (0)˘ ¯ 2A (x)˘ + [Ξ γµ Ξ1B (x) ψ¯B (0)˘ γ µ ψA (0) + [ψ¯A (x)˘ γµ ψB (x) Ξ γ µ Ξ2A (0) + x ↔ 0. First, let us consider the leading power term coming from the first term in the r.h.s. of this equation. 11

As we mentioned, we will need only first two terms of the expansion given by eqs. (3.27) and (A.2).

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The quark fields are given by a series in the parameter s⊥ , see eqs. (3.27) and (A.2), where Ψ can be any of u, d, s or c quarks.11 Accordingly, the currents (4.4) can be expressed as a series in this parameter, e.g. h i e (0)µ ¯ (0) γ˘ µ Ψ(0) − Ψ ¯ (0) γ˘ µ Ψ(0) + Ψ ¯ (0) γ˘ µ Ψ(0) + Ψ ¯ (0) γ˘ µ Ψ(0) , JAB = −Ψ As Bs Au Bu Ac Bc Ad Bd 4sW cW h e (1)µ ¯ (1) γ˘ µ Ψ(0) − Ψ ¯ c(0) γ˘ µ Ψc(1) − Ψ ¯ (1) γ˘ µ Ψ(0) − Ψ ¯ (0) γ˘ µ Ψ(1) JAB = −Ψ Au Bu Au Bu Ac Bc Ac Bc 4sW cW i ¯ (1) γ˘ µ Ψd(0) + Ψ ¯ (0) γ˘ µ Ψ(1) + Ψ ¯ (1) γ˘ µ Ψ(0) + Ψ ¯ (0) γ˘ µ Ψ(1) . +Ψ (4.5) As Bs As Bs Ad Bd Ad Bd

4.2

Leading power contribution (0)µ

(0)

As we mentioned, the leading-power term comes from JAB (x)JBAµ (0) and (0)µ

(0)

JBA (x)JABµ (0). Using Fierz transformation (ψ¯A γ µ [a − γ5 ]χA )(χ ¯B γµ [a − γ5 ]ψB ) 2  1+a  ¯ α = (ψA γ ψB )(χ ¯B γα χA ) + (ψ¯A γ α γ5 ψB )(χ ¯ B γα γ5 χA ) 2   − a (ψ¯A γ α ψB )(χ ¯B γα γ5 χA ) + (ψ¯A γ α γ5 ψB )(χ ¯ B γα χA )   + (1 − a2 ) (ψ¯A ψB )(χ ¯B χA ) − (ψ¯A γ5 ψB )(χ ¯ B γ5 χA )

(4.8)

16s2W c2W (0) (0)µ hpA |hpB |JABµ (x)JBA (0) + (x ↔ 0)|pA i|pB i = 2  e
1 + a2u ¯ = hψAu (x)γµ ψAu (0)ihψ¯Bu (0)γ µ ψBu (x)i + γµ ⊗ γ µ ↔ γµ γ5 ⊗ γ µ γ5 2
− au hψ¯Au (x)γµ ψAu (0)ihψ¯Bu (0)γ µ γ5 ψBu (x)i + γµ ⊗ γ µ γ5 ↔ γµ γ5 ⊗ γ µ

Nc


¯ ¯ ¯ ¯ + (1 − hψAu (x)ψAu (0)ihψBu (0)ψBu (x)i − hψAu (x)γ5 ψAu (0)ihψBu (0)γ5 ψBu (x)i n o n o n o   + u↔c + u↔d + u↔s + x↔0 , (4.9) a2u )

where ¯u (x)γµ ψˆu (0)|Ai, hψ¯Au (x)γµ ψAu (0)i ≡ hA|ψˆ

hψ¯Bu (x)γµ ψBu (0)i ≡ hB|ψˆ¯u (x)γµ ψˆu (0)|Bi (4.10)

and similarly for other matrix elements (summation over color and Lorentz indices is implied). As usual, after integration over background fields A and B we promote A, ψA and ˆ A subtle point is that our operators are not under T-product ˆ ψ. B, ψB to operators A, ordering so one should be careful while changing the order of operators in formulas like Fierz transformation. Fortunately, all our operators are separated either by space-like intervals or light-like intervals so they commute with each other. In a general gauge for projectile and target fields these expressions read (see eq. (3.2)) ˆ (x , x )γ [x , −∞ ] [x , 0 ] hA|ψ¯ˆ (x)γ ψˆ (0)|Ai = hA|ψ¯ [−∞ , 0 ] ψˆ (0)|Ai, f

µ f

f

•

⊥

µ

•

• x

⊥

⊥ −∞•

•

• 0 f

¯f (x∗ , x⊥ )γµ [x∗ , −∞∗ ]x [x⊥ , 0⊥ ]−∞ [−∞∗ , 0∗ ]0 ψˆf (0)|Bi (4.11) hB|ψˆ¯f (x)γµ ψˆf (0)|Bi = hB|ψˆ ∗ ¯f (0)γµ ψˆf (x)|Ai and hB|ψˆ¯f (0)γµ ψˆf (x)|Bi. and similarly for hA|ψˆ From parametrization of two-quark operators in section 4.2.1, it is clear that the leading power contribution to W (q) of eq. (4.1) comes from the product of two f10 s in eq. (4.13) and (4.15). It has the form [32] Z   e2 lt W (αq , βq , q⊥ ) = − 2 2 d2 k⊥ (1+a2u ) f1u (αq , k⊥ )f¯1u (βq , q⊥ −k⊥ ) 8sW cW Nc   + f¯1u (αq , k⊥ )f1u (βq , q⊥ −k⊥ ) +{u ↔ c}+{u ↔ d}+{u ↔ s} . (4.12)

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with au,c = (1 − 83 s2W ) and ad,s = (1 − 43 s2W ) one obtains

All other terms in the product of eqs. (4.13) and (4.15) give higher power contributions ∼

2 q⊥ lt s W (q)

(but not ∼

2 q⊥ W lt (q))12 Q2

so they can be neglected at Q2  s. Similarly, the

contribution of two matrix elements in eq. (4.17) is ∼ can be neglected as well. 4.2.1

m2⊥ s

in comparison to W lt (q) so it

Parametrization of leading matrix elements

for quark distributions in the projectile and Z 1 dx• d2 x⊥ eiαx• −i(k,x)⊥ hA|ψˆ¯f (x• , x⊥ )γ µ ψˆf (0)|Ai 16π 3 2m2N ¯f µ ¯f 2 2 2 = −pµ1 f¯1f (α, k⊥ ) − k⊥ f⊥ (α, k⊥ ) − pµ2 f3 (α, k⊥ ), s Z 1 2 dx• d2 x⊥ eiαx• −i(k,x)⊥ hA|ψˆ¯f (x• , x⊥ )ψˆf (0)|Ai = mN e¯f (α, k⊥ ) 16π 3

(4.14)

for the antiquark distributions.13 The corresponding matrix elements for the target are obtained by trivial replacements p1 ↔ p2 , x• ↔ x∗ and α ↔ β: Z 1 dx∗ d2 x⊥ e−iβx∗ +i(k,x)⊥ hB|ψˆ¯f (x∗ , x⊥ )γ µ ψˆf (0)|Bi (4.15) 16π 3 2m2N f µ f 2 2 2 = pµ2 f1f (β, k⊥ ) + k⊥ f⊥ (β, k⊥ ) + pµ1 f3 (β, k⊥ ), s Z 1 2 dx∗ d2 x⊥ e−iβx∗ +i(k,x)⊥ hB|ψˆ¯f (x∗ , x⊥ )ψˆf (0)|Bi = mN ef (β, k⊥ ), 16π 3 and 1 16π 3

Z

dx∗ d2 x⊥ eiβx∗ −i(k,x)⊥ hB|ψˆ¯f (x∗ , x⊥ )γ µ ψˆf (0)|Bi µ ¯f 2 2 = −pµ2 f¯1f (β, k⊥ ) − k⊥ f⊥ (β, k⊥ ) − pµ1

1 16π 3

Z

(4.16)

2m2N ¯f 2 f (β, k⊥ ), s 3

2 dx∗ d2 x⊥ eiβx∗ −i(k,x)⊥ hB|ψˆ¯f (x∗ , x⊥ )ψˆf (0)|Bi = mN e¯f (β, k⊥ ).

12

The trivial but important point is that any f (x, k⊥ ) may have only logarithmic dependence on Bjorken x but not the power dependence ∼ x1 . Indeed, at small x the cutoff of corresponding longitudinal integrals comes from the rapidity cutoff σa , see the discussion in section 2. Thus, at small x one can safely put x = 0 and the corresponding logarithmic contributions would be proportional to powers of αs ln σa (or, in some cases, αs ln2 σa , see e.g. ref. [33]). Also, a more technical version of this argument was presented on page 12. 13 In an arbitrary gauge, there are gauge links to −∞ as displayed in eq. (4.11).

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Let us first consider matrix elements of operators without γ5 . The standard parametrization of quark TMDs reads Z 1 dx• d2 x⊥ e−iαx• +i(k,x)⊥ hA|ψˆ¯f (x• , x⊥ )γ µ ψˆf (0)|Ai (4.13) 16π 3 2m2N f µ f 2 2 2 = pµ1 f1f (α, k⊥ ) + k⊥ f⊥ (α, k⊥ ) + pµ2 f3 (α, k⊥ ), s Z 1 2 dx• d2 x⊥ e−iαx• +i(k,x)⊥ hA|ψˆ¯f (x• , x⊥ )ψˆf (0)|Ai = mN ef (α, k⊥ ) 16π 3

Matrix elements of operators with γ5 are parametrized as follows: Z 1 2 2 dx• d2 x⊥ e−iαx• +i(k,x)⊥ hA|ψˆ¯f (x• , x⊥ )γ µ γ5 ψˆf (0)|Ai = iµνλρ pν1 pλ2 k ρ gf⊥ (α, k⊥ ), 16π 3 s Z 1 2 2 dx• d2 x⊥ eiαx• −i(k,x)⊥ hA|ψˆ¯f (x• , x⊥ )γ µ γ5 ψˆf (0)|Ai = iµνλρ pν1 pλ2 k ρ g¯f⊥ (α, k⊥ ). 3 16π s (4.17)

and similarly for the target with usual replacements p1 ↔ p2 , x• ↔ x∗ and α ↔ β. Note that the coefficients in front of f3 , gf⊥ , h and h⊥ 3 in eqs. (4.13), (4.15), (4.17), and (4.18) contain an extra 1s since pµ2 enters only through the direction of gauge link so the result should not depend on rescaling p2 → λp2 . For this reason, these functions do not contribute to W (q) in our approximation. 4.3

µ Power corrections from JAB (x)JBAµ (0) terms

The terms in eq. (4.7) proportional to Ξ fields are

 ¯ 2A (x) γ˘µ ψB (x) + Ξ1B (x) ψ¯A (x) + Ξ (4.19)
µ
 ¯ 1B (0) γ˘ ψA (x) + Ξ2A (0) + x ↔ 0 × [ ψ¯B (0) + Ξ     tw3 ¯ = [Ξ2A (x)˘ γµ ψB (x) ψ¯B (0)˘ γ µ ψA (0) + [ψ¯A (x)˘ γµ Ξ1B (x) ψ¯B (0)˘ γ µ ψA (0)     ¯ 1B (0)˘ + [ψ¯A (x)˘ γµ ψB (x) Ξ γ µ ψA (0) + [ψ¯A (x)˘ γµ ψB (x) ψ¯B (0)˘ γ µ Ξ2A (0)     ¯ 2A (x)˘ ¯ 1B (0)˘ + [Ξ γµ ψB (x) ψ¯B (0)˘ γ µ Ξ2A (0) + [ψ¯A (x)˘ γµ Ξ1B (x) Ξ γ µ ψA (0)     ¯ 2A (x)˘ ¯ 1B (0)˘ + [Ξ γµ ψB (x) Ξ γ µ ψA (0) + [ψ¯A (x)˘ γµ Ξ1B (x) ψ¯B (0)˘ γ µ Ξ2A (0)     ¯ 2A (x)˘ ¯ 1B (0)˘ + [Ξ γµ Ξ1B (x) ψ¯B (0)˘ γ µ ψA (0) + [ψ¯A (x)˘ γµ ψB (x) Ξ γ µ Ξ2A (0) + x ↔ 0.



First, as we demonstrate in appendix C.1, the terms in the second, third, and fourth lines q2 q2 lead to negligible power corrections ∼ αq⊥s or ∼ βq⊥s , so we are left with contribution of the fifth and sixth lines.

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The corresponding matrix elements for the target are obtained by trivial replacements p1 ↔ p2 , x• ↔ x∗ and α ↔ β similarly to eq. (4.16). Finally, for future use we present the parametrization of time-odd TMDs Z 1 dx• d2 x⊥ e−iαx• +i(k,x)⊥ hA|ψˆ¯f (x• , x⊥ )σ µν ψˆf (0)|Ai 16π 3 1 2mN µ ν µ ν 2 2 = (k⊥ p1 − µ ↔ ν)h⊥ (p1 p2 − µ ↔ ν)hf (α, k⊥ ) 1f (α, k⊥ ) + mN s 2mN µ ν 2 + (k⊥ p2 − µ ↔ ν)h⊥ 3f (α, k⊥ ), s Z 1 dx• d2 x⊥ eiαx• −i(k,x)⊥ hA|ψˆ¯f (x• , x⊥ )σ µν ψˆf (0)|Ai 16π 3 1 µ ν ¯ ⊥ (α, k 2 ) − 2mN (pµ pν − µ ↔ ν)h ¯ f (α, k 2 ) =− (k⊥ p1 − µ ↔ ν)h 1f ⊥ ⊥ 1 2 mN s 2mN µ ν ¯ ⊥ (α, k 2 ) − (k⊥ p2 − µ ↔ ν)h (4.18) 3f ⊥ s

Fifth line in eq. (4.19): the leading term in N1c    Let us start with the term ψ¯A (x)˘ γµ Ξ1B (x) ψ¯B (0)˘ γ µ Ξ2A (0) . Performing Fierz transformation (4.8) we obtain  m  n  µ n ¯ ψ¯A (x)γµ (a − γ5 )Ξm (4.20) 1B (x) ψB (0)γ (a − γ5 )Ξ2A (0) 2  n  1 + a  ¯m α α = ψA (x)γα Ξn2A (0) ψ¯B (0)γ α Ξm 1B (x) + (γα ⊗ γ ↔ γα γ5 ⊗ γ γ5 ) 2  m  n  α α − a ψ¯A (x)γα Ξn2A (0) ψ¯B (0)γ α γ5 Ξm 1B (x) + (γα ⊗ γ γ5 ↔ γα γ5 ⊗ γ )    + (1 − a2 ) ψ¯m (x)Ξn (0) ψ¯n (0)Ξm (x) − (1 ⊗ 1 ↔ γ5 ⊗ γ5 ) 4.3.1

=

2A

B

1B

 1+ m n i i ψ¯A (x)γi Ξn2A (0) ψ¯B (0)γ i Ξm 1B (x) + (γi ⊗ γ ↔ γi γ5 ⊗ γ γ5 ) 2  m  n  + (1 − a2 ) ψ¯A (x)Ξn2A (0) ψ¯B (0)Ξm 1B (x) − (1 ⊗ 1 ↔ γ5 ⊗ γ5 )  8  m  n  m⊥ n i m i i ¯ ¯ − a ψA (x)γi Ξ2A (0) ψB (0)γ γ5 Ξ1B (x) + (γi ⊗ γ γ5 ↔ γi γ5 ⊗ γ ) + O . s 

Next, separating color-singlet contributions m k n l m k n l hA, B|(ψ¯A (Bj )nk ψA )(ψ¯B (Ai )ml ψB )|A, Bi = hA, B|(ψ¯A (Ai )ml ψA )(ψ¯B (Bj )nk ψB )|A, Bi 1 m ml l n nk k = hA|(ψ¯A Ai ψA )|AihB|(ψ¯B Bj ψB )|Bi Nc (4.21)

we get    s2 Nc g −2 ψ¯A (x)˘ γµ Ξ1B (x) ψ¯B (0)˘ γ µ Ξ2A (0) (4.22)     2 1+a j 1 i k1 i i ¯ ¯ = ψA (x)Ak (x)γi p /2 γ ψA (0) ψB (0)Bj (0)γ p /1 γ ψB (x) +(γi ⊗γ ↔ γi γ5 ⊗γ γ5 ) 2 α β     j 1 k1 j k j k ¯ +(1−a2 ) ψ¯A (x)Ak (x)p γ ψ (0) ψ (0)B (0) p γ ψ (x) −(γ ⊗γ ↔ γ γ ⊗γ γ ) /2 /1 A B j B 5 5 α β     j 1 i k1 i i ¯ −a ψ¯A (x)Ak (x)γi γ5 p γ ψ (0) ψ (0)B (0)γ p γ ψ (x) +(γ γ ⊗γ ↔ γ ⊗γ γ ) . /2 /1 A B j B i 5 i 5 α β

Using equations (B.3), (B.4), and (B.8) from appendix B we can rewrite eq. (4.22) as   g −2 Nc ψ¯A (x)γµ (a − γ5 )Ξ1B (x) ψ¯B (0)γ µ (a − γ5 )Ξ2A (0) (4.23)    1 + a2 ¯ 1 1 i i ¯ ˜ ˜ ψA (x)p ψB (0)p = /2 [Ai (x) − iγ5 Ai (x)] ψA (0) /1 [B (0) − iγ5 B (0)] ψB (x) s2 α β   2 1−a ¯ 1 + ψA (x)Ak (x)p /2 γj ψA (0) s2 α   h i1 j k jk i ¯ × ψB (0) B (0)p ψB (x) /1 γ − j ↔ k + g B (0)p / 1 γi β   2a 1 ˜ − 2 ψ¯A (x)p [γ A (x) − i A (x)] ψ (0) /2 5 i i A s α    8 1 i i ¯ ˜ (0)] ψB (x) + O m⊥ . × ψB (0)p /1 [B (0) − iγ5 B β s

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A

a2 

For forward matrix elements we get Z 1ˆ ˆ ˆ ¯ • , x⊥ )p dx• e−iαq x• hA|ψ(x /2 [Aˆi (x• , x⊥ ) − iγ5 A˜i (x• , x⊥ )] ψ(0)|Ai α Z 1 ˆ ˆ ¯ • , x⊥ )p ˆ = dx• e−iαq x• hA|ψ(x /2 [Aˆi (x• , x⊥ ) − iγ5 A˜i (x• , x⊥ )]ψ(0)|Ai, αq Z 1ˆ ˆ ˆ ¯ p dx∗ e−iβq x∗ hB|ψ(0) /1 [Aˆj (0) − iγ5 A˜j (0)] ψ(x ∗ , x⊥ )|Bi β Z 1 ˆ¯ p [Aˆ (0) − iγ Aˆ˜ (0)]ψ(x ˆ ∗ , x⊥ )|Bi =− dx∗ e−iβq x∗ hB|ψ(0) /1 j 5 j βq

(4.24)

2 ˆ¯ p (Aˆi −iγ Aˆ ˆ¯ , x )Aˆj (x , x )p γ ψ(0)|Ai ˆ ∗ , x⊥ )|Bi+ 1−a hA|ψ(x ˆ ˜i )(0)ψ(x ×hB|ψ(0) /1 5 • ⊥ • ⊥ /2 j 2 s   2  m⊥ k ˆ ¯ ˆ ˆ ×hB|ψ(0)A (0)p 1+O . (4.25) /1 γk ψ(x∗ , x⊥ )|Bi s

ˆ¯ ˆ ∗ , x⊥ )|Bi = 0. Also, it Note that for unpolarized hadrons hB|ψ(0)( Aˆj (0)p /1 γ k − j ↔ k)ψ(x is easy to see that the last line of eq. (4.23) 2a ˆ ˆ ˆ¯ p (γ Aˆi (0) − iAˆ˜i (0)]ψ(x)|Bi ¯ p ˆ ˆ hA|ψ(x) ψ(0) (4.26) /2 [Aˆi (x) − iγ5 A˜i (x)]ψ(0)|AihB| /1 5 s2 gives zero contribution. Indeed, let us consider the first term in the r.h.s. of this equation. Since ˆ ˆ ¯ p ˆ hA|ψ(x) ∼ xi , /2 [Aˆi (x) − iγ5 A˜i (x)]ψ(0)|Ai ˆi i ˆ ¯ p ˆ hB|ψ(0) ∼ ij xj , (4.27) /1 (γ5 Aˆ (0) − iA˜ (0)]ψ(x)|Bi −

this term vanishes (and similarly all other terms in the r.h.s. of eq. (4.26) do vanish too). Repeating the same steps for the second term in the fifth line in eq. (4.19) we get   ¯ 2A (x)γµ (a−γ5 )ψB (x) Ξ ¯ 1B (0)γ µ (a−γ5 )ψA (0) Nc g −2 Ξ (4.28)      1 1 1+a2 i i ¯ ¯ ˜ ˜ ψA (x)p ψB (0)p = 2 /2 [Ai (0)+iγ5 Ai (0)]ψA (0) /1 [B (x)+iγ5 B (x)]ψB (x) s α β    1−a2 1 + 2 ψ¯A (x)Ak (0)p γ ψ (0) /2 j A s α    1 j k jk i ¯ × ψB (0)[B (x)p /1 γ −j ↔ k+g B (x)p /1 γi ]ψB (x) β    2a 1 ˜ − 2 ψ¯A (x)p [γ A (0)+i A (0)]ψ (0) /2 5 i i A s α     8 1 i i ¯ ˜ (x)]ψB (x) +O m⊥ . × ψB (0)p /1 [B (x)+iγ5 B β s 14

After specifying the projectile and target matrix elements the “A” and “B” labels of the fields become redundant.

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and similarly for other Lorentz structures in eq. (4.23). The corresponding contribution of the r.h.s. of eq. (4.23) to W (αq , βq , x⊥ ) takes the form14  Z 2 e2 g 2 (2π)−4 −iαq x• −iβq x∗ 1+a ˆ¯ , x )p (Aˆ −iγ Aˆ˜ )(x , x )ψ(0)|Ai ˆ − 2 2 dx dx e hA|ψ(x • ∗ • ⊥ /2 i 5 i • ⊥ s2 8sW cW Nc Q2

and similarly for other Lorentz structures in eq. (4.28). Similarly to eq. (4.25), we get the contribution to W (αq , βq , x⊥ ) in the form Z e2 g 2 − dx• dx∗ e−iαq x• −iβq x∗ (4.30) 8(2π)4 s2W c2W Nc Q2 ( 1 + a2 ˆ ˆ ¯ • , x⊥ )p ˆ × hA|ψ(x /2 [Aˆi (0) + iγ5 A˜i (0)]ψ(0)|Ai 2 s ˆi i ˆ ¯ p ˆ ∗ , x⊥ )|Bi × hB|ψ(0) /1 [Aˆ (x∗ , x⊥ ) + iγ5 A˜ (x∗ , x⊥ )]ψ(x +

1 − a2 j ˆ ˆ ¯ • , x⊥ )Aˆj (0)p hA|ψ(x /2 γ ψ(0)|Ai s2 )

ˆ ¯ Aˆk (x∗ , x⊥ )p ˆ ∗ , x⊥ )|Bi × hB|ψ(0) /1 γk ψ(x



 1+O

m2⊥ s

 .

In section 4.3.2, we demonstrated that the matrix elements of quark-antiquark-gluon operators in eqs. (4.25) and (4.30) reduce to the leading-power TMDs from section 4.2.1. Using parametrizations from section 4.3.2 we obtain the contribution of the 5th line in eq. (4.19) to W (q) in the form: W 5th (αq , βq , q⊥ ) e2 = 2 2 4sW cW Nc Q2

Z

"( (1 + a2u )(k, q − k)⊥ f1u (αq , k⊥ )f¯1u (βq , q⊥ − k⊥ )

d2 k⊥

) 1 2 ¯⊥ + 2 (1 − a2u )k⊥ (q − k)2⊥ h⊥ 1u (αq , k⊥ )h1u (βq , q⊥ − k⊥ ) + (αq ↔ βq ) mN #  2  n o n o n o  m⊥ + u↔c + u↔d + u↔s 1+O , (4.31) s where quark↔antiquark (αq ↔ βq ) term comes from x ↔ 0 contribution in eq. (4.19). As we will demonstrate later, the power corrections which reduce to the leading-power TMDs come with the leading power of N1c in the large-Nc approximation — all other power corrections are ∼ N12 or N13 . c

c

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1 1 Hereafter ψ¯A α1 (x) ≡ ψ¯A α−i (x) and ψ¯B β1 (x) ≡ ψ¯B β−i (x) (see eq. (3.28)) while in



1 1 1 1 all other places α O ≡ α+i O and β O ≡ β+i O . For forward matrix elements this gives   Z 1 ˆ −iαq x• ˆ ¯ dx• e hA| ψ (x• , x⊥ )p /2 [Aˆi (0) + iγ5 A˜i (0)]ψ(0)|Ai α Z 1 ˆ ˆ ¯ • , x⊥ )p = dx• e−iαq x• hA|ψ(x /2 [Aˆi (0) + iγ5 A˜i (0)]ψ(0)|Ai, αq   Z 1 ˆ −iβq x∗ ˆ ¯ ˆ ∗ , x⊥ )|Bi dx∗ e hB| ψ (0)p /1 [Aˆj (x∗ , x⊥ ) + iγ5 A˜j (x∗ , x⊥ )]ψ(x β Z 1 ˆ ˆ ¯ p ˆ ∗ , x⊥ )|Bi, =− dx∗ e−iβq x∗ hB|ψ(0) (4.29) /1 [Aˆj (x∗ , x⊥ ) + iγ5 A˜j (x∗ , x⊥ )]ψ(x βq

4.3.2

Parametrization of matrix elements from section 4.3.1

Using QCD equations of motion (3.1) we can rewrite the r.h.s. of eq. (4.32) as Z   ˆ ˆ¯ , x )p p γ ψ(0)|Ai ¯ • , x⊥ )γj p ˆ ˆ dx• dx⊥ e−iαq x• +i(k,x)⊥ k j hA|ψ(x + αq hA|ψ(x /2 γi ψ(0)|Ai • ⊥ /1 /2 i  Z s −iαq x• +i(k,x)⊥ ˆ¯ , x )p ψ(0)|Ai ˆ¯ , x )γ ψ(0)|Ai ˆ ˆ = dx• dx⊥ e − ki hA|ψ(x + αq hA|ψ(x • ⊥ /2 • ⊥ i 2  2i j j ˆ ˆ ¯ ˆ ¯ ˆ + •∗ij k hA|ψ(x• , x⊥ )p /2 γ5 ψ(0)|Ai − iα•∗ij hA|ψ(x• , x⊥ )γ γ5 ψ(0)|Ai s   2 2 2 = −ki 8π 3 sf1 (αq , k⊥ ) + 8π 3 sαq ki f⊥ (αq , k⊥ ) + g ⊥ (αq , k⊥ ) , (4.33) where we used parametrizations (4.13) and (4.17) for the leading power matrix elements. Now, the second term in eq. (4.33) contains extra αq with respect to the first term, so 2 and we get it should be neglected in our kinematical region s  Q2  q⊥ Z g ˆ f dx• dx⊥ e−iαq x• +i(k,x)⊥ hA|ψˆ¯f (x• , x⊥ )p /2 [Aˆi (x• , x⊥ ) − iγ5 A˜i (x• , x⊥ )]ψˆ (0)|Ai 8π 3 s 2 = −ki f1f (αq , k⊥ ) + O(αq ).

(4.34)

By complex conjugation Z g ˆ dx⊥ dx• e−iαq x• +i(k,x)⊥ hA|ψˆ¯f (x• , x⊥ )p /2 [Aˆi (0) + iγ5 A˜i (0)]ψˆf (0)|Ai 8π 3 s 2 = −ki f1f (αq , k⊥ ).

(4.35)

For the corresponding antiquark distributions we get Z g ˆ dx⊥ dx• e−iαx• +i(k,x)⊥ hA|ψˆ¯f (0)p /2 [Aˆi (x• , x⊥ ) + iγ5 A˜i (x• , x⊥ )]ψˆf (x• , x⊥ )|Ai 8π 3 s Z h 1 j ˆ ˆ¯ = 3 dx• dx⊥ e−iαq x• +i(k,x)⊥ − kj hA|ψ(0)γ /2 γ ψ(x ip • , x⊥ )|Ai 8π s i j ˆ ˆ 2 ˆ ¯ ¯ − ihA|ψ(0)γ (4.36) / γ D ip j ψ(x• , x⊥ )|Ai = −ki f1f (αq , k ) ⊥

2

and g 8π 3 s

Z

ˆ dx⊥ dx• e−iαx• +i(k,x)⊥ hA|ψˆ¯f (0)p /2 [Aˆi (0) − iγ5 A˜i (0)]ψˆf (x• , x⊥ )|Ai 2 = −ki f¯1f (α, k⊥ ).

(4.37)

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In this section we will demonstrate that matrix elements of quark-antiquark-gluon operators from section 4.3.1 can be expressed in terms of leading-power matrix elements from section 4.2.1. Let us start with matrix element (4.24) which can be rewritten as (see ref. [22]) Z ˆ ˆ ¯ • , x⊥ )p ˆ g dx• dx⊥ e−iαq x• +i(k,x)⊥ hA|ψ(x (4.32) /2 [Aˆi (x• , x⊥ ) − iγ5 A˜i (x• , x⊥ )]ψ(0)|Ai Z = dx• dx⊥ e−iαq x• +i(k,x)⊥   ← j j ˆ ˆ ¯ ˆ ¯ ˆ ˆ × k hA|ψ(x• , x⊥ )γj p /2 γi ψ(0)|Ai + ihA|ψ(x• , x⊥ ) D γj p /2 γi ψ(0)|Ai .

The corresponding target matrix elements are obtained by trivial replacements x∗ ↔ x• , αq ↔ βq and p /2 ↔ p /1 . Next, let us consider Z g i ˆ • , x⊥ )p ˆ dx• dx⊥ e−iαq x• +i(k,x)⊥ hA|ψ(x (4.38) /2 γ Aˆi (x• , x⊥ )ψ(0)|Ai 8π 3 s Z 1 = 3 dx• dx⊥ e−iαq x• +i(k,x)⊥ 8π s   ← i ˆ • , x⊥ )γ i p ˆ ˆ ˆ ˆ × ki hA|ψ(x ψ(0)|Ai + ihA| ψ(x , x ) D γ p ψ(0)|Ai . /2 /2 • ⊥ i

Again, only the first term contributes in our kinematical region so we finally get Z 2 k⊥ g −iαq x• +i(k,x)⊥ f i ˆ f 2 ˆ ¯ ˆ dx dx e hA| ψ (x , x ) p γ A (x , x ) ψ (0)|Ai = i h⊥ (αq , k⊥ ). • • ⊥ /2 i • ⊥ ⊥ 8π 3 s mN 1f (4.40) By complex conjugation we obtain Z 2 g −iαq x• +i(k,x)⊥ ˆ¯f (x , x )p γ i Aˆ (0)ψˆf (0)|Ai = i k⊥ h⊥ (α , k 2 ). dx dx e hA| ψ / • • i q ⊥ ⊥ ⊥ 2 8π 3 s mN 1f (4.41) For corresponding antiquark distributions one gets in a similar way Z 2 g −iαq x• +i(k,x)⊥ i ˆ ˆ ¯f (0)p ˆf (x• , x⊥ )|Ai = i k⊥ h ¯ ⊥ (αq , k 2 ), dx dx e hA| ψ γ A (x , x ) ψ / • i • ⊥ ⊥ ⊥ 2 8π 3 s mN 1f Z 2 g −iαq x• +i(k,x)⊥ ˆ¯f (0)p γ i Aˆ (0)ψˆf (x , x )|Ai = i k⊥ h ¯ ⊥ (αq , k 2 ). dx dx e hA| ψ /2 • i • ⊥ ⊥ ⊥ 8π 3 s mN 1f (4.42) The target matrix elements are obtained by usual replacements x∗ ↔ x• , αq ↔ βq and p /2 ↔ p /1 . 4.3.3

Sixth line in eq. (4.19)   ¯ 1B (0)˘ In this section we consider [ψ¯A (x)˘ γµ ψB (x) Ξ γ µ Ξ2A (0) which turns to  n  m m ¯ 1B (0)γ µ (a − γ5 )Ξn2A (0) [ψ¯A (x)γµ (a − γ5 )ψB (x) Ξ (4.43)  n  1 + a2  ¯m m ¯ 1B (0)γ i ψB = ψA (x)γi Ξn2A (0) Ξ (x) + (γi ⊗ γ i ↔ γi γ5 ⊗ γ i γ5 ) 2  m  n  m ¯ 1B (0)ψB + (1 − a2 ) ψ¯A (x)Ξn2A (0) Ξ (x) − (1 ⊗ 1 ↔ γ5 ⊗ γ5 )  8  m  n  n i m i i ¯ ¯ 1B (0)γ γ5 ψB (x) + (γi ⊗ γ γ5 ↔ γi γ5 ⊗ γ ) + O m⊥ − a ψA (x)γi Ξ2A (0) Ξ s

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Using QCD equation of motion and parametrization (4.18), one can rewrite the r.h.s. of this equation as Z h i 1 −iαq x• +i(k,x)⊥ ˆ¯ , x )γ i p ψ(0)|Ai ˆ¯ , x )p p ψ(0)|Ai ˆ ˆ dx dx e k hA| ψ(x + α hA| ψ(x / / / • q • • i ⊥ ⊥ ⊥ 2 1 2 8π 3 s   k2 2 2 2 = i ⊥ h⊥ (4.39) 1 (αq , k⊥ ) + αq mN e(α, k⊥ ) + ih(α, k⊥ ) . mN

after Fierz transformation (cf. eq. (4.8)). After separation of color singlet contributions ln k nk m ¯m ¯l hA, B|(Ψ A (Ai ) ψA )(ΨB (Bj ) ψB )|A, Bi nk k ln m abc c lk a k ¯m ¯l ¯m ¯l b m = hA, B|(Ψ hA, B|(Ψ A (Ai ) ψA )(ΨB (Bj ) ψB )|A, Bi+if A (t ) Ai ψA )(ΨB Bj ψB )|A, Bi

1 ¯ A Ai ψA |AihB|Ψ ¯ B Bj ψB |Bi+2if abc hA|Ψ ¯ A td tc Aai ψA |AihB|Ψ ¯ B td Bjb ψB |Bi hA|Ψ Nc 1 ¯ A Ai ψA |AihB|Ψ ¯ B Bj ψB |Bi =− hA|Ψ Nc (Nc2 −1) =

(4.44)

  ¯ 1B (0)˘ − g −2 Nc (Nc2 − 1)[ψ¯A (x)˘ γµ ψB (x) Ξ γ µ Ξ2A (0) (4.45)     2 1+a 1 1 j i = ψ¯A (x)γi p ψ¯B (0)γ k p /2 γ Ak (0) ψA (0) /1 γ Bj (0)ψB (x) 2 2s α β  i i + (γi ⊗ γ ↔ γi γ5 ⊗ γ γ5 )     a2 − 1 1 1 j k ¯ ¯ + ψA (x)p ψB (0)p /2 γ Ak (0) ψA (0) /1 γ Bj (0)ψB (x) s2 α β  j k j k − (γ ⊗ γ ↔ γ γ5 ⊗ γ γ5 )     a 1 1 k i j ¯ ¯ − 2 ψA (x)γi p ψB (0)γ p /2 γ Ak (0) ψA (0) /1 γ γ5 Bj (0)ψB (x) s α β   8 m⊥ i i + (γi ⊗ γ γ5 ↔ γi γ5 ⊗ γ ) + O , s which can be rewritten as     a2 − 1 1 µ ¯ ¯ ¯ −g − 1)[ψA (x)˘ γµ ψB (x) Ξ1B (0)˘ γ Ξ2A (0) = ψA (x)Ak (0)p /2 γj ψA (0) s2 α     8 m⊥ 1 k jk i ψ¯B (0)(B j (0)p , (4.46) /1 γ − j ↔ k + g B (0)p /1 γi )ψB (x) + O β s −2

Nc (Nc2

where we again used formulas (B.3), (B.5), and (B.8) from appendix B. Next, it is easy to see that α1 and β1 in eq. (4.45) give α1q and − β1q : Z

ˆ ¯ • , x⊥ )ΓAˆi (0) 1 ψ(0)|Ai ˆ (4.47) dx• e−iαq x• hA|ψ(x α   Z Z 0 1 ˆ ¯ • , x⊥ )Γ Aˆi (0)ψ(0) ˆ + 2 Fˆ∗i (0) ˆ 0 , 0⊥ ) |Ai, = dx• e−iαq x• hA|ψ(x dx0• ψ(x • αq s −∞   Z ¯ 1 (0)Aˆi (0)Γψ(x ˆ ∗ , x⊥ )|Bi dx∗ e−iβq x∗ hB| ψˆ β   Z Z 0 1 2ˆ −iβq x∗ 0 ˆ 0 ˆ ¯ ¯ ˆ ∗ , x⊥ )|Bi, ˆ =− dx∗ e hB| ψ(0)Ai (0) + dx∗ ψ(x∗ , 0⊥ ) F•i (0) Γψ(x βq s −∞ where Γ is any of the Dirac matrices in eq. (4.45).

– 22 –

JHEP05(2018)150

we obtain

The corresponding contribution to W (αq , βq , x⊥ ) takes the form e2 g 2 8(2π)4 s2W c2W Nc (Nc2 − 1)Q2

Z

dx• dx∗ e−iαq x• −iβq x∗

(4.48)

  Z 0 a2 − 1 2ˆ j ˆ 0 ˆ 0 ˆ ¯ ˆ × hA|ψ(x• , x⊥ )p dx• ψ(x• , 0⊥ ) |Ai /2 γ Aj (0)ψ(0) + F∗j (0) s2 s −∞    2   Z 0 m⊥ ˆ ˆ¯ 0 , 0 ) 2 Fˆ (0) p γ k ψ(x)|Bi ¯ Aˆk (0) + ˆ × hB| ψ(0) 1 + O , dx0∗ ψ(x /1 •k ∗ ⊥ s s −∞

 hB|

ˆ ¯ Aˆj (0) + ψ(0)

Z

0

dx0∗ −∞

  2ˆ 0 ˆ ¯ ˆ ∗ , x⊥ )|Bi = 0 ψ(x∗ , 0⊥ ) F•j (0) p /1 γk − j ↔ k ψ(x s

(4.49)

for the unpolarized hadron. Similarly,     a2 −1 1 µ ¯ ¯ −g ψB (0)˘ γ ψA (0) = 2 ψA (x)Ak (x)p /2 γj ψA (0) s α   8  m⊥ 1 j k jk i ¯ × ψB (0)(B (x)p , (4.50) /1 γ −j ↔ k+g B (x)p /1 γi ) ψB (x) +O β s −2

¯ 2A (x)˘ Nc (Nc2 −1)[Ξ γµ Ξ1B (x)



so the corresponding contribution to W (αq , βq , x⊥ ) is Z g 2 e2 a2 −1 dx• dx∗ e−iαq x• −iβq x∗ 8(2π)4 s2W c2W Nc (Nc2 −1)Q2 s2   Z x• 2ˆ 0 ˆ 0 j ˆ ˆ ¯ ¯ ˆ ×hA| ψ(x• , x⊥ )Aj (x• , x⊥ )+ dx• ψ(x• , x⊥ ) F∗j (x• , x⊥ ) p /2 γ ψ(0)|Ai s −∞     2  Z x∗ m⊥ k ˆ 0 2 ˆ 0 ˆ ¯ ˆ ˆ ×hB|ψ(0)p dx∗ F•k (x∗ , x⊥ )ψ(x∗ , x⊥ ) |Bi 1+O . /1 γ Ak (x∗ , x⊥ )ψ(x∗ , x⊥ )+ s s −∞ Using parametrizations (4.52) and (4.53) from appendix 4.3.4 we obtain the contribution of the 6th line in eq. (4.19) in the form Z e2 2 W 6th (αq , βq , q⊥ ) = − 2 2 d2 k ⊥ k⊥ (q − k)2⊥ (4.51) 4sW cW Nc (Nc2 − 1)Q2 "( h i 1 2 tw3 ˜¯ tw3 (β , q − k ) ¯ tw3 (βq , q⊥ − k⊥ ) + h ˜ tw3 (αq , k⊥ )h × (a − 1) h (α , k ) h q q ⊥ ⊥ ⊥ u u u u u m2N ) #  2  n o n o n o  m⊥ + (αq ↔ βq ) + u ↔ c + u ↔ d + u ↔ s 1+O , s where quark↔antiquark (αq ↔ βq ) term comes from x ↔ 0 replacement, cf. eq. (4.31).

– 23 –

JHEP05(2018)150

where we have used the fact that

4.3.4

Parametrization of matrix elements from section 4.3.3

and similarly for the target matrix elements. Note that unlike two-quark matrix elements, quark-quark-gluon ones may have imaginary parts which we denote by functions with tildes. By complex conjugation we get ( Z g −iαx• +i(k,x)⊥ dx⊥ dx• e hA| ψˆ¯f (x• , x⊥ )Aˆi (x• , x⊥ ) (4.53) 8π 3 s ) Z x• 2   k⊥ 2 0 ˆ 0 i 2 2 ˜ tw3 + dx• ψ¯f (x• , x⊥ ) Fˆ∗i (x• , x⊥ ) p htw3 /2 γ ψˆf (0)|Ai = i f (α, k⊥ ) − ihf (α, k⊥ ) , s mN −∞ ( Z g dx⊥ dx• e−iαx• +i(k,x)⊥ hA| ψˆ¯f (0)Aˆi (0) 8π 3 s ) Z 0 2  k⊥ 2 i ˜¯ tw3 (α, k 2 ) ¯f (x0 , 0⊥ ) Fˆ∗i (0) p ¯ tw3 (α, k 2 ) − ih + dx0• ψˆ h /2 γ ψˆf (x• , x⊥ )|Ai = i • f ⊥ f ⊥ s mN −∞ and similarly for the target matrix elements. For completeness, let us present the structure of gauge links in an arbitrary gauge, for example: ( ) Z x• 2 0 0 ˆ ˆ hA| ψ¯f (x• , x⊥ )Aˆj (x• , x⊥ ) + dx• ψ¯f (x• , x⊥ ) Fˆ∗j (x• , x⊥ ) p (4.54) /2 γi ψˆf (0)|Ai s −∞ Z n 2 x• 0 ¯f (x• , x⊥ )[x• , x0 ]x Fˆ∗j (x0 , x⊥ )[x0 , −∞]x → dx• hA| ψˆ • • • s −∞ o + ψˆ¯f (x0• , x⊥ )[x0• , x• ]x Fˆ∗j (x• , x⊥ )[x• , −∞]x [x⊥ , 0⊥ ]−∞• [−∞• , 0• ]0⊥ p /2 γi ψˆf (0)|Ai. 4.4

Power corrections from JAµ (x)JBµ (0) terms

Power corrections of the second type come from the terms (0) (0) ¯ A (x)˘ ¯ B (0)˘ ¯ (0) (x)˘ ¯ (0) (0)˘ Ψ γ µ ΨA (x)Ψ γµ ΨB (0) + x ↔ 0 = Ψ γ µ ΨA (x)Ψ γµ ΨB (0) A B

+ +

(0) (0) ¯ (1) (x)˘ ¯ (0) (0)˘ Ψ γ µ ΨA (x)Ψ γµ ΨB (0) A B (0) (0) ¯ (0) (x)˘ ¯ (1) (0)˘ Ψ γ µ ΨA (x)Ψ γµ ΨB (0) A B

+ +

(1) (0) ¯ (0) (x)˘ ¯ (0) (0)˘ Ψ γ µ ΨA (x)Ψ γµ ΨB (0) A B (0) (1) ¯ (0) (x)˘ ¯ (0) (0)˘ Ψ γ µ ΨA (x)Ψ γµ ΨB (0) A B

– 24 –

(4.55)

+ x ↔ 0 + ...

JHEP05(2018)150

In this section we present parametrization of matrix elements from section 4.3.3. Similarly to eqs. (4.40)–(4.42) we define ( Z g i dx⊥ dx• e−iαx• +i(k,x)⊥ hA|ψˆ¯f (x• , x⊥ )p (4.52) /2 γ Aˆi (0)ψˆf (0) 8π 3 s ) Z 0  k2  2ˆ 0 ˆ 0 2 2 ˜ tw3 + F∗i (0) dx• ψf (x• , 0⊥ ) |Ai = i ⊥ htw3 f (α, k⊥ ) + ihf (α, k⊥ ) , s mN −∞ ( Z g i dx⊥ dx• e−iαx• +i(k,x)⊥ hA|ψˆ¯f (0)p /2 γ Aˆi (x• , x⊥ )ψˆf (x• , x⊥ ) 8π 3 s ) Z x•  k 2 ¯ tw3 2ˆ 0 ˆ 0 2 2 ˜¯ tw3 + F∗i (x• , x⊥ ) dx• ψf (x• , x⊥ ) |Ai = i ⊥ h f (α, k⊥ ) + ihf (α, k⊥ ) s mN −∞

As we prove in appendix C.2, the leading power correction comes from last two lines in eq. (4.56). We will consider them in turn. 4.4.1

Last two lines in eq. (4.56)

Using eq. (3.27) and separating color-singlet matrix elements, we rewrite the sixth line in eq. (4.56) as     ¯ 2A (x)˘ ¯ 1B (0)˘ [Ξ γµ ψA (x) ψ¯B (0)˘ γ µ Ξ1B (0) +[ψ¯A (x)˘ γµ Ξ2A (x) Ξ γ µ ψA (0) (4.57)      2 g 1 1 k = ψ¯A (x)γ j p γµp /2 γ˘µ Ak (0)ψA (x) ψ¯B (0)˘ /1 γ Bj (x) ψB (0) (Nc2 −1)s2 α β      1 1 j k ¯ + ψ¯A (x)˘ γµp γ A (0) ψ (x) ψ (0)γ p γ ˘ B (x)ψ (0) +x ↔ 0 /2 /1 µ j k A B B α β      g 2 (a2 −1) 1 i i ¯A 1 (x)p ¯ ˜ ˜ = ψ (A +iγ A )(0)ψ (x) ψ (0) p (B (x)−iγ B )(x) ψ (0) /2 i /1 5 i A B 5 B (Nc2 −1)s2 α β      h i1 1 i ˜ i (x)]ψB (0) +x ↔ 0, (0)p + ψ¯A (x)p /2 Ai (0)−iγ5 A˜i (0) ψA (x) ψ¯B /1 [B (x)+iγ5 B α β

where we used eqs. (B.4) and (B.5). For the forward matrix elements   Z ¯ 1 (x• , x⊥ )p ˆ • , x⊥ )|Ai dx• e−iαq x• hA| ψˆ /2 Aˆi (0)ψ(x α Z Z x• /2 ˆ 1 −iαq x• ˆ¯ 0 , x ) 2p ˆ • , x⊥ )|Ai, =− dx• e dx0• hA|ψ(x F∗i (0)ψ(x • ⊥ αq s −∞ Z 1ˆ ˆ ¯ • , x⊥ )p dx• e−iαq x• hA|ψ(x /2 Aˆi (0) ψ(x • , x⊥ )|Ai α Z Z x• /2 ˆ 1 −iαq x• ˆ¯ , x ) 2p ˆ 0 , x⊥ )|Ai, = dx• e dx0• hA|ψ(x F∗i (0)ψ(x • ⊥ • αq s −∞ Z 1ˆ ˆ ¯ p dx∗ e−iβq x∗ hB|ψ(0) /1 Aˆi (x∗ , x⊥ ) ψ(0)|Bi β Z Z 0 /1 ˆ 1 ˆ¯ 2p ˆ 0 , 0⊥ )|Bi, =− dx∗ e−iβq x∗ dx0∗ hB|ψ(0) F•i (x∗ , x⊥ )ψ(x ∗ βq s −∞   Z ¯ 1 (0)p ˆ dx∗ e−iβq x∗ hB| ψˆ /1 Aˆi (x∗ , x⊥ )ψ(0)|Bi β Z Z 0 /1 ˆ 1 −iβq x∗ ˆ¯ 0 , 0 ) 2p ˆ = dx∗ e dx0∗ hB|ψ(x F•i (x∗ , x⊥ )ψ(0)|Bi. ∗ ⊥ βq s −∞

– 25 –

(4.58)

JHEP05(2018)150

In appendix C.4, we will demonstrate that terms ∼ Ψ(1) are small in our kinematical region 2. s  Q 2  q⊥ Terms ∼ Ψ(0) read 



 ¯ 2A (x)˘ ¯ 1B (0)˘ ψ¯A + Ξ γµ ψA + Ξ2A (x) ψ¯B + Ξ γ µ ψB + Ξ1B (0) + x ↔ 0   = [ψ¯A (x)˘ γµ ψA (x) ψ¯B (0)˘ γ µ ψB (0) (4.56)     ¯ 2A (x)˘ + [Ξ γµ ψA (x) ψ¯B (0)˘ γ µ ψB (0) + [ψ¯A (x)˘ γµ Ξ2A (x) ψ¯B (0)˘ γ µ ψB (0)     ¯ 1B (0)˘ + [ψ¯A (x)˘ γµ ψA (x) Ξ γ µ ψB (0) + [ψ¯A (x)˘ γµ ψA (x) ψ¯B (0)˘ γ µ Ξ1B (0)     ¯ 2A (x)˘ ¯ 1B (0)˘ + [Ξ γµ Ξ2A (x) ψ¯B (0)˘ γ µ ψB (0) + [ψ¯A (x)˘ γµ ψA (x) Ξ γ µ Ξ1B (0)     ¯ 2A (x)˘ ¯ 1B (0)˘ + [Ξ γµ ψA (x) ψ¯B (0)˘ γ µ Ξ1B (0) + [ψ¯A (x)˘ γµ Ξ2A (x) Ξ γ µ ψB (0)     ¯ 2A (x)˘ ¯ 1B (0)˘ + [Ξ γµ ψA (x) Ξ γ µ ψB (0) + [ψ¯A (x)˘ γµ Ξ2A (x) ψ¯B (0)˘ γ µ Ξ1B (0) + x ↔ 0.

The corresponding contribution to W (αq , βq , x⊥ ) takes the form Z g 2 e2 (a2 −1) dx• dx∗ e−iαq x• −iβq x∗ 8(2π)4 s2W c2W (Nc2 −1)Q2 s2 (Z  Z 0 x• 2p   0 0 ˆ ¯ 0 , x⊥ ) /2 Fˆ∗i (0)+iγ5 Fˆ˜∗i (0) ψ(x ˆ • , x⊥ )|Ai × dx• dx∗ hA|ψ(x • s −∞ −∞ ˆ ¯ ×hB|ψ(0)

2p /1  s

ˆ ¯ • , x⊥ ) +hA|ψ(x

(4.59)

 ˆ 0 , 0⊥ )|Bi Fˆ• i (x∗ , x⊥ )−iγ5 Fˆ˜• i (x∗ , x⊥ ) ψ(x ∗ 2p /2 

s )   2  2 p   m⊥ / ˆ i 0 ˆ 1 ˆ i ¯ ˆ ˜ ×hB|ψ(x∗ , 0⊥ ) F• (x∗ , x⊥ )+iγ5 F• (x∗ , x⊥ ) ψ(0)|Bi +x ↔ 0 1+O . s s Similarly, for the seventh line in eq. (4.56) using eqs. (3.27) and (B.6) one obtains     ¯ 2A (x)˘ ¯ 1B (0)˘ [Ξ γµ ψA (x) Ξ γ µ ψB (0) +[ψ¯A (x)˘ γµ Ξ2A (x) ψ¯B (0)˘ γ µ Ξ1B (0) +x ↔ 0 (4.60)        2 g 1 1 µ = ψ¯A (x)γ j p ψ¯B (0)γ k p /2 γ˘µ Ak (0)ψA (x) /1 γ˘ Bj (x)ψB (0) 2 2 (Nc −1)s α β        1 1 j µ k ¯ + ψ¯A (x)˘ γµ p γ A (0) ψ (x) ψ (0)˘ γ p γ B (x) ψ +x ↔ 0 /2 /1 j A B B (0) k α β      h i g2 1 2 ¯ ˜ = (1+a ) ψA (x)p /2 Ai (0)+iγ5 Ai (0) ψA (x) (Nc2 −1)s2 α    1 i i ¯ ˜ × ψB (0)p /1 [B (x)+iγ5 B (x)]ψB (0) β  h i 1   h i 1   i i ¯ ¯ ˜ (x) + ψA (x)p ψA (x) ψB (0)p ψB (0) /2 Ai (0)−iγ5 A˜i (0) /1 B (x)−iγ5 B α β       h i h i 1 1 i i ¯ ¯ ˜ ˜ −2a ψA (x)p ψB (0)p /2 Ai (0)+iγ5 Ai (0) ψA (x) /1 γ5 B (x)+iB (x) ψB (0) α β      h i 1 h i  1    i i ˜ (x) + ψ¯A (x)p ψA (x) ψ¯B (0)p ψB (0) /2 Ai (0)−iγ5 A˜i (0) /1 γ5 B (x)−iB α β +x ↔ 0. Using eq. (4.58) one obtains the contribution to W (αq , βq , x⊥ ) in the form Z g 2 e2 (a2 + 1) dx• dx∗ e−iαq x• −iβq x∗ − 8(2π)4 s2W c2W (Nc2 − 1)Q2 s2 (Z  Z 0 x• 2p   ˆ ¯ 0 , x⊥ ) /2 Fˆ∗i (0) + iγ5 Fˆ˜∗i (0) ψ(x ˆ • , x⊥ )|Ai × dx0• dx0∗ hA|ψ(x • s −∞ −∞ 2p   ˆ ¯ 0 , 0⊥ ) /1 Fˆ i (x∗ , x⊥ ) + iγ5 Fˆ˜ i (x∗ , x⊥ ) ψ(0)|Bi ˆ × hB|ψ(x ∗ • • s  /2  ˆ ˆ , x ) 2p ¯ ˆ 0 , x⊥ )|Ai + hA|ψ(x F∗i (0) − iγ5 Fˆ˜∗i (0) ψ(x • ⊥ • s )  2p  i  / ˆ i 0 ˆ 1 ¯ ˆ , 0⊥ )|Bi + x ↔ 0 . × hB|ψ(0) Fˆ• (x∗ , x⊥ ) − iγ5 F˜• (x∗ , x⊥ ) ψ(x ∗ s

– 26 –

(4.61)

JHEP05(2018)150

 ˆ 0 , x⊥ )|Ai Fˆ∗i (0)−iγ5 F˜ˆ∗i (0) ψ(x •

Here we used the fact that the last term in eq. (4.60) 

ˆ ¯ 0 , x⊥ ) − 2a hA|ψ(x •

2p /2 

∼

2a s2




 ˆ • , x⊥ )|Ai Fˆ∗i (0) + iγ5 Fˆ˜∗i (0) ψ(x

(4.62)

gives no contribution since ˆ ¯ 0 , x⊥ ) hA|ψ(x • ˆ ¯ 0 , 0⊥ ) hB|ψ(x ∗

2p /1  s

2p /2  s

 ˆ • , x⊥ )|Ai ∼ xi , Fˆ∗i (0) ± iγ5 Fˆ˜∗i (0) ψ(x

 ˆ γ5 Fˆ•i (x∗ , x⊥ ) ± iFˆ˜•i (x∗ , x⊥ ) ψ(0)|Bi ∼ ij xj

(4.63)

same as in eq. (4.27). Next, using parametrizations (4.66) from the next section we obtain the contribution of the 6th and 7th lines in eq. (4.56) in the form W

Z hn e2 2 (αq , βq , q⊥ ) = 2 2 d k (k, q−k) 2(1+a2u ) ⊥ ⊥ 8sW cW (Nc2 −1)Q2  tw3  tw3 tw3 tw3 × j1u (αq , k⊥ )j2u (βq , q⊥ −k⊥ )− ˜j1u (αq , k⊥ )˜j2u (βq , q⊥ −k⊥ )  tw3 tw3 tw3 tw3 +(1−a2u ) j1u (αq , k⊥ )j1u (βq , q⊥ −k⊥ )+ ˜j1u (αq , k⊥ )˜j1u (βq , q⊥ −k⊥ ) o  tw3 tw3 tw3 tw3 +j2u (αq , k⊥ )j2u (βq , q⊥ −k⊥ )+ ˜j2u (αq , k⊥ )˜j2u (βq , q⊥ −k⊥ ) +αq ↔ βq  2  n o n o n oi  m⊥ + u↔c + u↔d + u↔s 1+O , (4.64) s

6+7th

where αq ↔ βq contribution comes as usually from the (x ↔ 0) term in eq. (4.59). 4.4.2

Parametrization of TMDs from section 4.4.1

We parametrize TMDs from section 4.4.1 as follows x•  /2  ˆ ˆ¯ 0 , x ) 2p ˆ • , x⊥ )|Ai d x⊥ dx• e dx0• hA|ψ(x F∗i (0) + iγ5 Fˆ˜∗i (0) ψ(x • ⊥ s −∞   2 2 = ki j1tw3 (α, k⊥ ) + i˜j1tw3 (α, k⊥ ) , Z Z x•  /2  ˆ g 2 −iαx• +i(k,x)⊥ ˆ¯ , x ) 2p ˆ 0 , x⊥ )|Ai d x dx e dx0• hA|ψ(x F∗i (0) − iγ5 Fˆ˜∗i (0) ψ(x • • ⊥ ⊥ • 3 8π s s −∞  tw3  2 2 = ki j2 (α, k⊥ ) − i˜j2tw3 (α, k⊥ ) . (4.65)

g 8π 3 s

Z

2

−iαx• +i(k,x)⊥

Z

– 27 –

JHEP05(2018)150

s 2 p   ˆ ¯ 0 , 0⊥ ) /1 γ5 Fˆ i (x∗ , x⊥ ) + iFˆ˜ i (x∗ , x⊥ ) ψ(0)|Bi ˆ × hB|ψ(x ∗ • • s 2p   ˆ ¯ • , x⊥ ) /2 Fˆ∗i (0) − iγ5 Fˆ˜∗i (0) ψ(x ˆ 0 , x⊥ )|Ai + hA|ψ(x • s  2p  /1  ˆ i ˆ i 0 ˆ ¯ ˆ ˜ × hB|ψ(0) γ5 F• (x∗ , x⊥ ) − iF• (x∗ , x⊥ ) ψ(x∗ , 0⊥ )|Bi s

By complex conjugation we get 0 /2 ˆ ¯ˆ 2p ˆ 0 , 0⊥ )|Ai d x⊥ dx• e dx0• hA|ψ(0) [F∗i (x) − iγ5 Fˆ˜∗i (x)]ψ(x • s −∞   2 2 = ki j1tw3 (α, k⊥ ) − i˜j1tw3 (α, k⊥ ) , Z Z 0 2p  /2  ˆ g ¯ˆ 0 2 −iαx• +i(k,x)⊥ ˆ d x dx e dx0• hA|ψ(x F∗i (x) + iγ5 Fˆ˜∗i (x) ψ(0)|Ai • ⊥ • , 0⊥ ) 3 8π s s −∞   2 2 = ki j2tw3 (α, k⊥ ) + i˜j2tw3 (α, k⊥ ) . (4.66)

g 8π 3 s

Z

−iαx• +i(k,x)⊥

2

Z

For completeness let us present the explicit form of the gauge links in an arbitrary gauge: ˆ ¯ 0 , x⊥ )Fˆ∗i (0)ψ(x ˆ • , x⊥ ) → ψ(x • ˆ 0 , x )[x0 , −∞ ] [x , 0 ] ¯ ψ(x •

5

⊥

•

• x

⊥

(4.67) ˆ

ˆ

⊥ −∞• [−∞• , 0]0⊥ F∗i (0)[0, −∞• ]0⊥ [0⊥ , x⊥ ]−∞• [−∞• , x• ]x ψ(x• , x⊥ ).

Results and estimates

Combining eqs. (4.12), (4.31), (4.51), and (4.64) we get the leading term and first power 2 in the form corrections to W (q) in the kinematic region s  Q2  q⊥ e2 W (pA , pB , q) = − 2 2 8sW cW Nc

Z

"( 2

d k⊥

(1+a2u )

×f1u (αz , k⊥ )f¯1u (βz , q⊥ −k⊥ )+2(a2u −1)



(k, q−k)⊥ 1−2 Q2



2 (q−k)2 k⊥ ⊥ ⊥ ¯ ⊥ (βz , q⊥ −k⊥ ) h1u (αz , k⊥ )h 1u m2N Q2

h i 2 (q−k)2 2k⊥ 2 tw3 tw3 tw3 ˜¯ tw3 (β , q −k ) ⊥ ¯ ˜ (a −1) h (α , k ) h (β , q −k )+ h (α , k ) h z ⊥ u z ⊥ z ⊥ u z ⊥ ⊥ ⊥ u u (Nc2 −1)Q2 m2N u  tw3  Nc (k, q−k)⊥  tw3 tw3 tw3 − 2 2(1+a2u ) j1u (αz , k⊥ )j2u (βz , q⊥ −k⊥ )− ˜j1u (αz , k⊥ )˜j2u (βz , q⊥ −k⊥ ) 2 Nc −1 Q  tw3 tw3 tw3 tw3 +(1−a2u ) j1u (αz , k⊥ )j1u (βz , q⊥ −k⊥ )+j2u (αz , k⊥ )j2u (βz , q⊥ −k⊥ )  tw3 tw3 tw3 tw3 + ˜j1u (αz , k⊥ )˜j1u (βz , q⊥ −k⊥ )+ ˜j2u (αz , k⊥ )˜j2u (βz , q⊥ −k⊥ ) ) #  2  n o n o n o  m⊥ +(αz ↔ βz ) + u ↔ c + u ↔ d + u ↔ s 1+O , (5.1) s +

where the momentum of the produced Z-boson is q = αz p1 + βz p2 + q⊥ .

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JHEP05(2018)150

Target matrix elements are obtained by usual substitutions α ↔ β, p /2 ↔ p /1 , x• ↔ x∗ , and ˆ ˆ F∗i ↔ F•i .

k 2 (q − k)2 ¯⊥ × f1u (αz , k⊥ )f¯1u (βz , q⊥ − k⊥ ) + 2(a2u − 1) ⊥ 2 2 ⊥ h⊥ 1u (αz , k⊥ )h1u (βz , q⊥ − k⊥ ) mN Q ) #  2   n o n o n o  m⊥ 1 + (αz ↔ βz ) + u ↔ c + u ↔ d + u ↔ s 1+O +O . s Nc Next, eq. (5.3) is a tree-level formula and for an estimate we should specify the rapidity 2 cutoffs for f1 ’s and h⊥ 1 ’s. As we discussed in section 2, the rapidity cutoff for f1 (αz , k⊥ ) is 2 ) σ , where σ and σ are rapidity bounds for central fields. Since we σa and for f1 (αz , k⊥ a b b calculated only tree diagrams made of C-fields we have σa = βz and σb = αz in eq. (5.1).15 2  m2 where we probe the perturbative Next, power corrections become sizable at q⊥ N 1 1 ⊥ 2  Q2 we can tails of TMD’s f1 ∼ k2 and h1 ∼ k4 [34]. So, as long as m2N  k⊥ ⊥ ⊥ approximate ¯ z) m2N h(αz ) ¯ f (αz ) f¯(αz ) ¯ ⊥ m2N h(α 2 ⊥ 2 f1 (αz , k⊥ )' , h (α , k ) ' , f ' , h1 ' (5.4) z ⊥ 1 1 2 4 2 4 k⊥ k⊥ k⊥ k⊥ (up to logarithmic corrections). Similarly, for the target we can use the estimate ¯ z) m2N h(βz ) ¯ f (βz ) f¯(βz ) ¯ ⊥ m2N h(β 2 ⊥ 2 f1 (βz , k⊥ )' , h (β , k ) ' , f ' , h1 ' z ⊥ 1 1 2 4 2 4 k⊥ k⊥ k⊥ k⊥

(5.5)

2  Q2 . as long as m2N  k⊥ 15

In general, we should integrate over C-fields in the leading log approximation and match the logs to the double-log and/or single-log evolution of TMDs.

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For completeness, let us present our final result in the transverse coordinate space "(  Z 2 e2 d b⊥ i(q,b)⊥ 2 W (pA , pB , q) = − 2 2 e (1+au ) f1u (αz , b⊥ )f¯1u (βz , b⊥ ) 4π 2 8sW cW Nc  2 ⊥ 2(a2 −1) 2 ⊥ ⊥¯ 2 ¯⊥ + 2 ∂i f1u (αz , b⊥ )∂i f1u (βz , b⊥ ) + 2u 2 ∂⊥ h1u (αz , b⊥ )∂⊥ h1u (βz , b⊥ ) Q mN Q h i 2(a2 −1) 2 tw3 2 ¯ tw3 2 ˜ tw3 2˜ ¯ tw3 (βz , b⊥ ) + 2 u 2 2 ∂⊥ hu (αz , b⊥ )∂⊥ hu (βz , b⊥ )+∂⊥ hu (αz , b⊥ )∂⊥ h u (Nc −1)Q mN    Nc tw3 tw3 tw3 tw3 + 2 2(1+a2u ) ∂i⊥ j1u (αz , b⊥ )∂i⊥ j2u (βz , b⊥ )−∂i⊥ ˜j1u (αz , b⊥ )∂i⊥ ˜j2u (βz , b⊥ ) 2 (Nc −1)Q  tw3 tw3 tw3 tw3 +(1−a2u ) ∂i⊥ j1u (αz , b⊥ )∂i⊥ j1u (βz , b⊥ )+∂i⊥ j2u (αz , b⊥ )∂i⊥ j2u (βz , b⊥ )   tw3 tw3 tw3 tw3 +∂i⊥ ˜j1u (αz , b⊥ )∂i⊥ ˜j1u (βz , b⊥ )+∂i⊥ ˜j2u (αz , b⊥ )∂i⊥ ˜j2u (βz , b⊥ ) ) #  2  n o n o n o  m⊥ +(αz ↔ βz ) + u ↔ c + u ↔ d + u ↔ s 1+O , (5.2) s R where f1u (αz , b⊥ ) ≡ d2 k⊥ e−i(k,b)⊥ f1u (αz , k⊥ ) etc. Note that in the leading order in Nc power corrections are expressed in terms of leading power functions f1 and h⊥ 1 . To estimate the order of magnitude of power corrections, one 1 can assume that Nc is a good parameter and leave only first term in the r.h.s. of eq. (5.1): "(   Z e2 (k, q − k)⊥ 2 W (pA , pB , q) = − 2 2 d k⊥ (1 + a2u ) 1 − 2 (5.3) Q2 8sW cW Nc

mN ⊥ 2 2 Here we used the fact that due to the “positivity constraint” h⊥ 1 (x, k⊥ ) ≤ |k⊥ | f1 (x, k⊥ ) [35], we can safely assume that the numbers f (x) and h(x) in eqs. (5.4) and (5.5) are of the same m2

order of magnitude so the last term in the third line in eq. (5.6) ∼ QN2 can be neglected. Thus, the relative weight of the leading term and power correction is determined by the ⊥ factor 1 − 2 (k,q−k) . The integrals over k⊥ are logarithmic and should be cut from below Q2 2 by mN and from above by Q2 so we get an estimate   2 q⊥ πe2 1 1 Q2 W (pA , pB , q) = − 2 2 ln + ln (5.7) 2 2 Q2 q⊥ 4sW cW Nc q⊥ m2N hn o n o n o n oi × (1 + a2u )[fu (αz )f¯u (βz ) + f¯u (αz )fu (βz )] + u ↔ c + u ↔ d + u ↔ s , where we assumed that the first term is determined by the logarithmical region 2  k 2  m2 and the second by Q2  k 2  q 2 . By this estimate, the power corq⊥ N ⊥ ⊥ ⊥ 2 increases, the rection reaches the level of few percent at q⊥ ≥ 20 GeV. Of course, when q⊥ q2

correction becomes bigger, but the validity of the approximation Q⊥2  1 worsens. Moreover, we have ignored all logarithmic (and double-log) evolutions which can significantly change the relative strength of power corrections.

6

Power corrections for Drell-Yan process

In this section we consider γ ∗ contribution to the cross section of the Drell-Yan process which is determined by the hadronic tensor Z Wµν (pA , pB , q) = d2 x⊥ ei(q,x)⊥ Wµν (αq , βq , x⊥ ), (6.1) Z 1 2 Wµν (αq , βq , x⊥ ) ≡ dx• dx∗ e−iαq x• −iβq x∗ hpA , pB |Jµem (x• , x∗ , x⊥ )Jνem (0)|pA , pB i, 16π 4 s where Jµem = eu ψ¯u γµ ψu + ed ψ¯d γµ ψd + es ψ¯s γµ ψs + ec ψ¯c γµ ψc is the electromagnetic current for active flavors in our kinematical region.

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JHEP05(2018)150

Substituting this to eq. (5.1) we get the following estimate of the strength of power corrections for Z-boson production Z e2 1 W (pA , pB , q) = − 2 2 d2 k⊥ 2 (5.6) 8sW cW Nc k⊥ (q − k)2⊥ "(   (k, q − k)⊥ × (1 + a2u ) 1 − 2 [fu (αz )f¯u (βz ) + f¯u (αz )fu (βz )] Q2 ) # n o n o n o 2 m 2 N ¯ u (βz ) + h ¯ u (αz )hu (βz )] + 2(au − 1)[hu (αz )h + u↔c + u↔d + u↔s Q2   Z e2 1 (k, q − k)⊥ 2 '− 2 2 d k⊥ 2 1−2 Q2 8sW cW Nc k⊥ (q − k)2⊥ hn o n o n o n oi × (1 + a2u )[fu (αz )f¯u (βz ) + f¯u (αz )fu (βz )] + u ↔ c + u ↔ d + u ↔ s .

tw3 tw3 tw3 tw3 − j1u (αq , k⊥ )j1u (βq , q⊥ − k⊥ ) − j2u (αq , k⊥ )j2u (βq , q⊥ − k⊥ ) tw3 tw3 − ˜j1u (αq , k⊥ )˜j1u (βq , q⊥

n

o

n

o

−

tw3 tw3 k⊥ ) − ˜j2u (αq , k⊥ )˜j2u (βq , q⊥

n

+ u↔c + u↔d + u↔s

o

i

)

− k⊥ ) + (αq ↔ βq )

# .

(6.2)

Let us present also the large-Nc estimate similar to eq. (5.6) Z 2 1 Wµµ (pA , pB , q) = − d2 k⊥ 2 (6.3) Nc k⊥ (q−k)2⊥ "(   (k, q−k)⊥ 2 × eu 1−2 [fu (αq )f¯u (βq )+ f¯u (αq )fu (βq )] Q2 ) # n o n o n o 2 m ¯ u (βq )+ h ¯ u (αq )hu (βq )] N + u↔c + u↔d + u↔s +2e2u [hu (αq )h Q2   Z 2 1 (k, q−k)⊥ '− d2 k⊥ 2 1−2 Nc Q2 k⊥ (q−k)2⊥  × e2 [fu (αq )f¯u (βq )+ f¯u (αq )fu (βq )]+(u ↔ c)+(u ↔ d)+(u ↔ s) . u

Obviously, the relative strength of leading-twist terms and power corrections is the same as for Z-boson production so from our na¨ıve estimate (5.7) one should expect power corrections of order of few percent starting from q⊥ ∼ 14 Q.

7

Conclusions and outlook

In this paper we have calculated the higher-twist power correction to Z-boson production 2 . Our back-of-the-envelope (and Drell-Yan process) in the kinematical region s  Q2  q⊥ estimation of importance of power corrections tells that they reach a few percent of the leading-twist result at q⊥ ∼ 14 Q which surprisingly agrees with the same estimate made in ref. [21] by comparing leading-order fits to experimental data. 16

The problem of calculating power corrections for Wµν with non-convoluted indices is a separate issue which we hope to address in a different publication.

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JHEP05(2018)150

From the results of the present paper it is easy to extract power corrections to Wµµ .16 We replace constants au in eq. (5.1) by e2f and remove factors “1” from expressions like a2 ± 1. One can formally set au → ∞ in γ˘µ ≡ γµ (au − γ5 ), divide the result (5.1) by a2u , and multiply by e2u . After that, we repeat the procedure for other flavors and get "(   Z 2 (k, q − k)⊥ µ 2 Wµ (pA , pB , q) = − d k⊥ e2u 1 − 2 f1u (αq , k⊥ )f¯1u (βq , q⊥ − k⊥ ) Nc Q2 2 2  2 k⊥ (q − k)⊥ ¯⊥ + 2eu h⊥ 1u (αq , k⊥ )h1u (βq , q⊥ − k⊥ ) m2N Q2 
1 tw3 tw3 tw3 tw3 ˜ ¯ ˜ ¯ + 2 h (αq , k⊥ )hu (βq , q⊥ − k⊥ ) + hu (αq , k⊥ )hu (βq , q⊥ − k⊥ ) Nc − 1 u Nc (k, q − k)⊥ h tw3 tw3 tw3 tw3 − e2u 2 2j1u (αq , k⊥ )j2u (βq , q⊥ − k⊥ ) − 2˜j1u (αq , k⊥ )˜j2u (βq , q⊥ − k⊥ ) Nc − 1 Q2

Of course, we made our estimate without taking into account the TMD evolution, notably the most essential double-log (Sudakov) evolution. One should evolve projectile q2

q2

q2

q2

where the coefficient functions cm,n (q, x) are determined by integrals over C-fields and do not depend on the form of projectile or target. To find these coefficients in the first-loop order, we integrate over C-fields in eq. (2.11) with action SC = SQCD (C + A + B, ψC + ψA + ψB ) − SQCD (A, ψA ) − SQCD (B, ψB ) but without any rapidity restrictions on C-fields, ˆ A (zm )Φ ˆ B (z 0 ) in the background fields A, and subtract matrix elements of the operators Φ n ψA and B, ψB multiplied by tree-level coefficients. Both the integrals over C-fields in ˆ A (zm )Φ ˆ B (zn0 ) will have rapidity divergencies which will eq. (2.11) and matrix elements of Φ be canceled in their sum so what remains are the logarithms (or double logs) of the ratio ˆ A (zm ) of kinematical variables (Q2 in our case) to the rapidity cutoffs σa of the operators Φ 0 ˆ and σb of ΦB (zn ). Using the above logic we hope to avoid the problem of double-counting of fields which arises when integrals over longitudinal momenta of C-fields got pinched at small momenta (see the discussion in the end of secttion 3.2). The work is in progress. It should be mentioned that, as discussed in ref. [12], our rapidity factorization is different from the standard factorization scheme for particle production in hadron-hadron scattering, namely splitting the diagrams in collinear to projectile part, collinear to target part, hard factor, and soft factor [1]. Here we factorize only in rapidity and the Q2 evolution 2 dependence of the rapidity evolution kernels, same as in the BK (and NLO arises from k⊥ BK [36]) equations. Also, since matrix elements of TMD operators with our rapidity cutoffs are UV-finite [37, 38], the only UV divergencies in our approach are usual UV divergencies absorbed in the effective QCD coupling. It is worth noting that recently the treatment of power corrections was performed in the framework of SCET theory (see e.g. refs. [39–41]). However, since our rapidity factorization is different from factorization used by SCET, the detailed comparison of power corrections to Z-boson (or Higgs) production would be possible when SCET result for TMD corrections in the form of m12 times matrix elements of quark-antiquark-gluon Z operators will be available. Let us note that we obtained power corrections for Drell-Yan hadronic tensor convoluted over Lorentz indices. It would be interesting (and we plan) to calculate the highertwist correction to full DY hadronic tensor. Also, it is well known that for semi-inclusive deep inelastic scattering (SIDIS) and for DY process the leading-order TMDs have different directions of Wilson lines: one to +∞ and another to −∞ [42, 43]. We think that the same directions of Wilson lines will stay on in the case of power corrections and we plan to study this question in forthcoming publications.

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TMD from σa = βq to σ ˜a = αq⊥s = βq Q⊥2 , target TMDs from σb = αq to σ ˜b = βq⊥s = αq Q⊥2 , and match to the result of leading-log calculation of integral over central fields in the rapidity interval between σ ˜ a and σ ˜b . To accurately match these evolutions, we hope to use logic borrowed from the operator product expansion. We write down a general formula (2.14) Z Z XZ 1 4 −iqx ˆ ˆ B (zn0 )|pB i, (7.1) W = d xe dzm cm,n (q, x)hpA |ΦA (zm )|pA i dzn0 hpB |Φ (2π)4 m,n

Acknowledgments The authors are grateful to S. Dawson, A. Prokudin, T. Rogers, R. Venugopalan, and A. Vladimirov for valuable discussions. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under contracts DE-AC02-98CH10886 and DE-AC05-06OR23177 and by U.S. DOE grant DE-FG0297ER41028.

Next-to-leading quark fields

In this section we present the explicit expressions for the next-to-leading quark fields Ψ (1) . It is convenient to separate these fields in “left” and “right” components: (1)

(1)

Ψ(1) = Ψ1 + Ψ2 ,

(1)

Ψ1 ≡

p /1 p /2 (1) Ψ , s

(1)

Ψ2 ≡

p /2 p /1 (1) Ψ . s

(A.1)

The next-to-leading term in the expansion of the fields (3.19) has the form: gγ i 1 1 j 2g 1 [1] (0) p (A.2) /1 p / 2 P i γ Bj ψA − 2 p /1 p /2 (A• ) ψA , 2 sβ s β α s β 2g p /2 p /1 1 gγ i 1 1 j 2g 1 [1] (0) (1) Ψ2A = − B ψ − p / p / P i γ B j ψA − 2 p / p / (A∗ ) ψA ∗ A s2 α s2 2 1 α β s 2 1α 2g 2 γ j gγ i γ j 1 1 1 k − 2 2p / 2 B ∗ B j ψA + 2 p / 2 P i P j γ B k ψA , s α s α β α i g p / gγ 1 1 j 2g 1 [1] (0) (1) Ψ1B = − 2 γ i Ai ψB − 2 p /2 p / 1 P i γ Aj ψ B − 2 p /2 p /1 (A∗ ) ψB , sα s α β s α i 2g p p / / 1 gγ 1 1 2g 1 [1] (0) (1) j 1 2 Ψ2B = − A• ψ B − 2 p /1 p / 2 P i γ Aj ψ B − 2 p /1 p /2 (A• ) ψB 2 s β s β α s β 2 j i j 2g γ gγ γ 1 1 1 k − 2 2p / 1 A• Aj ψ B + 2 p / 1 P i P j γ Ak ψ B , s β s β α β g p / 1 1 gγ i 2g ¯ 1 [1] 1 ¯A γ j Bj p ¯ (1) = −ψ¯A γ i Bi Ψ − ψ p P − 2 ψA (A• )(0) p /2 /1 / p / , i 1A 2 s(β −i) α−i β −i s s β −i 2 1 2g p p 1 1 gγ i 2g ¯ 1 [1] ¯ (1) = −ψ¯A 1 B∗ /1 /2 − ψ¯A γ j Bj p Ψ p P − 2 ψA (A∗ )(0) p / / / p / i 2A 2 2 1 2 α−i s β −i α−i s s α−i 1 2 2g 2 γ j 1 1 1 gγ j γ i k − ψ¯A Bj B∗ p Pj Pi p /2 2 2 + ψ¯A γ Bk /2 2 , s α α−i β −i α−i s i gp /2 1 1 gγ 2g ¯ 1 [1] ¯ (1) = −ψ¯B γ i Ai Ψ − ψ¯B γ j Aj p Pi − 2 ψB (A∗ )(0) p /1 p /2 / p / , 1B 2 s(α−i) β −i α−i s s α−i 1 2 2g p p 1 1 gγ i 2g ¯ 1 [1] ¯ (1) = −ψ¯B 1 A• /2 /1 − ψ¯B γ j Aj p Ψ p P − 2 ψB (A• )(0) p / / / p / i 2B 2 2 2 1 β −i s α−i β −i s s β −i 2 1 2g 2 γ j 1 1 1 gγ j γ i k − ψ¯B Aj A• p , /1 2 2 + ψ¯B γ Ak Pj Pi p / s β β α β 1 s2 (1)

Ψ1A = −

gp /1

γ i B i ψA −

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A

¯ should be read from right where Pi = i∂i + gAi + gBi , see eq. (3.16). The expressions for Ψ to left, e.g. Z 1 1 γi 1 γi 1 j ¯ ψA γ B j p Pi (x) 2 ≡ dz ψ¯A (z)γ j Bj (z)p Pi |x) 2 /2 p /1 /2 p /1 (z| α − i β − i s α − i β − i s (A.3) 1 1 1 (and α1 ≡ α+i β ≡ β+i as usual). It is easy to see that the power counting of quark fields has the form (cf eq. (3.29)):

(1)

(1)

(1)

(0)

m⊥ . s

(A.4)

(0)

The gluon fields A• and A∗ were calculated in ref. [12]: [1]a

g ab jb A B , 2α j g = − Aab B jb 2β j

(0)

[1]

(A• )(0) =

(0)

[1]

(A∗ )(0)

A• = A• + (A• )(0) ,

[1]a

A∗ = B∗ + (A∗ )(0) ,

(A.5)

and their power counting reads gA• ∼ gB∗ ∼ m2⊥ ,

B

gAi ∼ gBi ∼ m⊥ .

(A.6)

Formulas with Dirac matrices

In the gauge A• = 0 the field Ai can be represented as Z 2 x• 0 Ai (x• , x⊥ ) = dx A∗i (x0• , x⊥ ) s −∞ • (see eq. (3.5)). It is convenient to define a “dual” field Z Z x∗ 2 x• 0 ˜ ˜i (x∗ , x⊥ ) = 2 ˜•i (x0∗ , x⊥ ), A˜i (x• , x⊥ ) = dx• A∗i (x0• , x⊥ ), B dx0 B s −∞ s −∞ ∗

(B.1)

(B.2)

where F˜µν = 12 µνλρ F λρ as usual.17 With this definition, we get 2 •∗ij Aj = A˜i , s

2 ˜i ⇒ A˜i ⊗ B ˜ i = −Ai ⊗ B i , A˜i ⊗ B i = Ai ⊗ B ˜i •∗ij B j = −B s

(B.3)

and therefore j i k i ˜ i γ5 ), Ak γ i p / 2 γ ⊗ Bj γ p /1 γ = p /2 (Ai − iA˜i γ5 ) ⊗ p /1 (B − iB k i i ˜ i γ5 ), Ak γ j p / γ i ⊗ Bj γ p / γ =p / (Ai + iA˜i γ5 ) ⊗ p / (B + iB 2

1

2

1

j k i i ˜ i γ5 ), Ak γ i p / 2 γj ⊗ B γ p /1 γ = p /2 (Ai − iA˜i γ5 ) ⊗ p /1 (B + iB j i k i ˜ i γ5 ). Ak γj p / γi ⊗ B γ p / γ =p / (Ai + iA˜i γ5 ) ⊗ p / (B − iB 2

17

i

1

2

1

(B.4)

We use conventions from Bjorken & Drell where 0123 = −1 and γ µ γ ν γ λ = g µν γ λ + g νλ γ µ − g µλ γ ν − γρ γ5 . Also, with this convention σ ˜ µν ≡ 12 µνλρ σ λρ = iσµν γ5 .

µνλρ

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JHEP05(2018)150

7/2

(1)

Ψ1A ∼ Ψ1B ∼ Ψ2A ∼ Ψ2B ∼

We will also need j i k j i k Ak γi p / 2 γ γ 5 ⊗ Bj γ p / 1 γ γ 5 = Ak γ i p / 2 γ ⊗ Bj γ p /1 γ , k i j k i Ak γ j p / 2 γ i γ 5 ⊗ Bj γ p / 1 γ γ 5 = Ak γ p / 2 γ i ⊗ Bj γ p /1 γ , j k i j k i Ak γ i p / 2 γj γ5 ⊗ B γ p /1 γ γ5 = −Ak γi p / 2 γj ⊗ B γ p /1 γ , j i k j i k Ak γ j p / 2 γi γ5 ⊗ B γ p /1 γ γ5 = −Ak γj p / 2 γi ⊗ B γ p /1 γ

(B.5)

and hence

k i Ak γ j p /2 γi (a − γ5 ) ⊗ Bj γ p /1 γ (a − γ5 ) i i ˜ i γ5 ) − 2ap ˜ i ), = (a2 + 1)p /2 (Ai + iA˜i γ5 ) ⊗ p /1 (B + iB /2 (Ai + iA˜i γ5 ) ⊗ p /1 (γ5 B + iB j k i 2 i ˜ i γ5 ), Ak γi p / γj (a − γ5 ) ⊗ B γ p / γ (a − γ5 ) = (a − 1)p / (Ai − iA˜i γ5 ) ⊗ p / (B + iB 2

1

2

1

i ˜ i γ5 ). Ak γ j p /2 γi (a − γ5 ) ⊗ B γ p /1 γ (a − γ5 ) = (a − 1)p /2 (Ai + iA˜i γ5 ) ⊗ p /1 (B − iB j i

k

2

Next, using formula 1 σ ˜µν ⊗ σ ˜αβ = − (gµα gνβ − gνα gµβ )σξη ⊗ σ ξη (B.7) 2 + gµα σβξ ⊗ σνξ − gνα σβξ ⊗ σµξ − gµβ σαξ ⊗ σνξ + gνβ σαξ ⊗ σµξ − σαβ ⊗ σµν we get j k j Ak p / 2 γj ⊗ B p / 1 γk − A p / 2 γj γ5 ⊗ B p / 1 γk γ5

(B.8) s s j k j j ξη k j i = Ak p / 2 γj ⊗ B p / 1 γk + A p / 2 γk ⊗ B p /1 γj − A σξη ⊗ Bj σ − A σki ⊗ B σj 4 2 s i 2 k k jk k j − Aj p / 1 γ k ⊗ Bj p /2 γ + A σjk ⊗ Bi σ − A σ∗• ⊗ Bk σ∗• − 2A p / 2 γ j ⊗ Bk p /1 γ . 2 s

For appendix C we also need 2 ˜i γ5 ) − γ i γ5 ⊗ pˆ1 γ5 (Bi + iB ˜i γ5 ), (ˆ p2 γ i pˆ1 ⊗ pˆ1 Bi + pˆ2 γ i pˆ1 γ5 ⊗ pˆ1 γ5 Bi ) = −γ i ⊗ pˆ1 (Bi + iB s ˜i γ5 )γ i + pˆ2 γ5 ⊗ (Bi + iB ˜i γ5 )γ i γ5 , γk γ i pˆ2 ⊗ Bi γ k + γk γ i pˆ2 γ5 ⊗ Bi γ k γ5 = pˆ2 ⊗ (Bi + iB i j i j i i ˜ i γ5 )p ˜ i γ5 )p p / γ γ ⊗p / Bj + p / γ5 γ γ ⊗ p / γ 5 Bj = p / ⊗ (B + iB / + γ5 p / ⊗ γ5 (B + iB / , 2

1

2

1

2

1

2

1

i ˜ i γ5 ) + p ˜ i γ5 ). p /2 γ γ ⊗ p / 1 Bj + p / 2 γ5 γ γ ⊗ p / 1 γ 5 Bj = p /2 ⊗ p /1 (B − iB / 2 γ5 ⊗ p /1 γ5 (B − iB (B.9) i j

C C.1

i j

i

Subleading power corrections Second, third, and fourth lines in eq. (4.19)

In this appendix we show that second, third, and fourth lines in eq. (4.19) yield subleading power corrections and can be neglected in our approximation.

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JHEP05(2018)150

j i k Ak γ i p (B.6) /2 γ (a − γ5 ) ⊗ Bj γ p /1 γ (a − γ5 ) i i ˜ i γ5 ) − 2ap ˜ i ), = (a2 + 1)p /2 (Ai − iA˜i γ5 ) ⊗ p /1 (B − iB /2 (Ai − iA˜i γ5 ) ⊗ p /1 (γ5 B − iB

Let us consider for example the last term in the third line of eq. (4.19). The Fierz transformation (4.8) yields   [ψ¯A (x)˘ γµ ψB (x) ψ¯B (0)˘ γ µ Ξ2A (0)  n  1 + a2  ¯m m = ψA (x)γα Ξn2A (0) ψ¯B (0)γ α ψB (x) + (γα ⊗ γ α ↔ γα γ5 ⊗ γ α γ5 ) 2  m  n  m − a ψ¯A (x)γα Ξn2A (0) ψ¯B (0)γ α γ5 ψB (x) + (γα ⊗ γ α γ5 ↔ γα γ5 ⊗ γ α )    + (1 − a2 ) ψ¯m (x)Ξn (0) ψ¯n (0)ψ m (x) − (1 ⊗ 1 ↔ γ5 ⊗ γ5 ) A

B

B

 n  1+ m m ψ¯A (x)γi Ξn2A (0) ψ¯B (0)γ i ψB (x) + (γi ⊗ γ i ↔ γi γ5 ⊗ γ i γ5 ) 2  m  n  m − a ψ¯A (x)γi Ξn2A (0) ψ¯B (0)γ i γ5 ψB (x) + (γi ⊗ γ i γ5 ↔ γi γ5 ⊗ γ i )  8  m  n  m⊥ 2 n m ¯ ¯ + (1 − a ) ψA (x)Ξ2A (0) ψB (0)ψB (x) − (1 ⊗ 1 ↔ γ5 ⊗ γ5 ) + O . s

(C.1)

Sorting out the color-singlet contributions18 we get m k n m m k n m hA, B|(ψ¯A (Bj )nk ψA )(ψ¯B ψB )|A, Bi = hA, B|(ψ¯A ψA )(ψ¯B (Bj )nk ψB )|A, Bi 1 l l n nk k = hA|(ψ¯A ψA )|AihB|(ψ¯B Bj ψB )|Bi Nc

(C.2)

and therefore    − Nc ψ¯A (x)˘ γµ ψB (x) ψ¯B (0)˘ γ µ Ξ2A (0) (C.3)    2   1+a j1 i i i ¯ ¯ = ψA (x)γi p /2 γ ψA (0) ψB (0)gBj (0)γ ψB (x) + (γi ⊗ γ ↔ γi γ5 ⊗ γ γ5 ) 2s α    2   1−a j1 ¯ ¯ + ψA (x)p /2 γ ψA (0) ψB (0)gBj (0)ψB (x) − (1 ⊗ 1 ↔ γ5 ⊗ γ5 ) s α      a j1 i i i ¯ ¯ − ψA (x)γi γ5 p /2 γ ψA (0) ψB (0)gBj (0)γ ψB (x) + (γi γ5 ⊗ γ ↔ γi ⊗ γ γ5 ) , s α 1 where α1 ≡ α+i , see eq. (3.28). For the forward matrix elements we get Z 1 ˆ¯ , x )p Γ 1 ψ(0)|Ai ˆ dx⊥ dx• e−iαq x• +i(k,x)⊥ hA|ψ(x • ⊥ /2 s α Z 1 1 2 ˆ¯ , x )p Γψ(0)|Ai ˆ = dx⊥ dx• e−iαq x• +i(k,x)⊥ hA|ψ(x ∼ fΓ (αq , k⊥ ), • ⊥ /2 αq s αq

(C.4)

where Γ is any of γ-matrices with transverse indices. Next, consider Z ˆ ¯ ˆ ∗ , x⊥ )|Bi ˆi (0)ψ(x dx⊥ dx∗ e−iβq x∗ +i(k,x)⊥ hB|ψ(0)g B (C.5) Z Z 0 2 −iβq x∗ +i(k,x)⊥ ˆ¯ ˆ ∗ , x⊥ )|Bi ∼ ki f tw3 (βq , k 2 ), = dx⊥ dx∗ e dx0∗ hB|ψ(0)g Fˆ•i (x0∗ , 0⊥ )ψ(x ⊥ s m −∞ 18

Recall that after the promotion of background fields to operators we can still move those operators freely since all of them commute, see footnotes 2, 10 and 11.

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=

2A

a2 

2 ) is some function of order one (by power counting (3.4) this matrix where f tw3 (βq , k⊥ element (C.5) is ∼ 1). Also, this function may have only logarithmical singularities in βq as βq → 0 but not the power behavior β1q .19 The corresponding contribution to W (q) of eq. (4.1) is proportional to Z q2 q2 1 2 d−2 k⊥ fΓi (αq , k⊥ )ki f tw3 (βq , k⊥ ) ∼ ⊥ W lt  ⊥2 W lt (C.6) αq s αq s Q 2 q⊥ W lt (recall that we assume Q2 2 2 2 q⊥ q⊥ q⊥ √ αq s ' Q s  Q2 ). In a similar

so it can be neglected in comparison to the contributions ∼

2A

2A

B

B

¯ 2A in eq. (3.27) we see that the last term in the r.h.s. From the explicit form of Ξ2A and Ξ vanishes while the first two are small. Indeed,  n  2 ¯m n m hA, B| Ξ (C.8) /1 Ξ2A (0) ψ¯B (0)p /2 ψB (x) |A, Bi 2A (x)p s    p  n  / 2 1 l k 1 m ˆ km (x)B ˆ nl (0)p = hA, B| ψ¯A (x)γ i 2 γ j ψA (0) ψ¯B (0)g 2 B /2 ψB (x) |A, Bi i j s α s α    8 p m⊥ 2 1 i /2 j 1 ˆ 2 ˆ ˆ ˆ ¯ ¯ ˆ = hA| ψ (x)γ γ ψ(0)|AihB|ψ(0)g Aj (0)Aˆi (x)p , /2 ψ(x)|Bi ∼ O sNc α s α s so the contribution to W is of order of C.2

m4⊥ W lt . s2

Second to fifth lines in eq. (4.56)

Here we show that second to fifth lines in eq. (4.56) either vanish or can be neglected. Obviously, matrix element of the operator in the second line vanishes. Formally, Z ˆ ˆ¯ γ ψ(0)|Ai, ¯ • , x⊥ )˘ ˆ • , x⊥ )|Ai = δ(αq )hA|ψ(0)˘ ˆ dx• e−iαq x• hA|ψ(x γµ ψ(x µ Z ˆ¯ γ ψ(0)|Bi ˆ¯ γ ψ(0)|Bi ˆ ˆ dx∗ e−iβq x∗ hB|ψ(0)˘ = δ(βq )hB|ψ(0)˘ (C.9) µ µ 19

Large x∗ correspond to low-x domain where matrix elements can be calculated in a shock-wave background of the target particle. The typical propagator in the shock-wave external field has a factor p2 ⊥

e−i αs (x−z)∗ where z∗ is a position of a shock wave and p⊥ is of order of characteristic transverse momen
p2 ⊥ −1 tum [44, 45]. The integration over large x∗ gives then βq + αs and since the integration over α is restricted from above by σa , such terms cannot give β1q (cf. refs [37, 38]).

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that Z-boson is produced in a central range of rapidity so way one can show that the remaining three terms in the second and third lines of eq. (4.19) give small contributions to W (q). Next, it is easy to see that the matrix element of the fourth line of eq. (4.19) vanishes. Indeed, let us consider the first term in the fourth line and perform Fierz transformation (4.8):   ¯ 2A (x)˘ [Ξ γµ ψB (x) ψ¯B (0)˘ γ µ Ξ2A (0)  n  1 + a2  ¯ m m = Ξ2A (x)γα Ξn2A (0) ψ¯B (0)γ α ψB (x) + (γα ⊗ γ α ↔ γα γ5 ⊗ γ α γ5 ) 2  m   ¯ (x)γα Ξn (0) ψ¯n (0)γ α γ5 ψ m (x) + (γα ⊗ γ α γ5 ↔ γα γ5 ⊗ γ α ) −a Ξ 2A 2A B B  m   ¯ (x)Ξn (0) ψ¯n (0)ψ m (x) − (1 ⊗ 1 ↔ γ5 ⊗ γ5 ) . + (1 − a2 ) Ξ (C.7)

From the power counting (3.4) we see that this term is ∼ contribution of the last two lines in eq. (4.56). C.3

m8⊥ s ,

so we are left with the

Gluon power corrections from JAµ (x)JAµ (0) terms

There is one more contribution which should be discussed and neglected: e2 16s2W c2W 



 ¯ 2A (x) γ˘µ ψA (x) + Ξ2A (x) ¯ 2A (0) γ˘ µ ψA (0) + Ξ2A (0) × ψ¯A (x) + Ξ ψ¯A (0) + Ξ       e2 ¯ 2A (x)˘ ¯ 2A (0)˘ Ξ γµ ψA (x) ψ¯A (0)˘ γ µ Ξ2A (0) + ψ¯A (x)˘ γµ Ξ2A (x) Ξ γ µ ψA (0) = 2 2 16sW cW       ¯ 2A (x)˘ ¯ 2A (0)˘ + Ξ γµ ψA (x) Ξ γ µ ψA (0) + ψ¯A (x)˘ γµ Ξ2A (x) ψ¯A (0)˘ γ µ Ξ2A (0) , (C.11)

JAµ (x)JAµ (0) =

where we neglected terms which cannot contribute to W due to the reason discussed after eq. (C.9), i.e. that one hadron (“A” or “B”) cannot produce Z-boson on its own. Let us consider the first term in the r.h.s. of this equation    ¯ 2A (x)˘ Ξ γµ ψA (x) ψ¯A (0)˘ γ µ Ξ2A (0) (C.12)   2 p /2 ˆ g f.int. ˆ 1 (x)γ p ¯ˆ γ µ /2 γj 1 ψ(0)|AihB| ˆ → − hA| ψ¯ γ˘µ ψ(x)ψ(0)˘ Aˆai (x)Aˆaj (0)|Bi i 2 2Nc (Nc − 1) α s s α   p /2 ˆ /2 1 ˆ g2 1 ˆ ˆ¯ γ k p ¯ =− hA| ψ (x)γ γ ˘ ψ(x) ψ(0)˘ γj ψ(0)|Ai i k 2Nc (Nc2 − 1) α s s α  8 m⊥ ai aj × hB|Aˆ (x)Aˆ (0)|Bi + O , s f.int.

where → denotes functional integration over A and B fields in eq. (2.8). The matrix element hB|Aˆai (x)Aˆaj (0)|Bi for unpolarized hadrons can be proportional either to 2xi xj + x2⊥ g ij or to g ij . Since the former structure does not contribute due to k (2xi xj + x2⊥ g ij )γi p / 2 γ ⊗ γk p / 2 γj = 0

– 38 –

(C.13)

JHEP05(2018)150

and, non-formally, one hadron cannot produce Z-boson on his own. For a similar reason, matrix elements of the operators in the third and fourth lines in eq. (4.56) vanish — either projectile or target matrix element will be of eq. (C.9) type. In addition, from the explicit form Ξ’s in eq. (3.27) it is easy to see that the fifth line in eq. (4.56) can be rewritten as follows:     ¯ 2A (x)˘ ¯ 1B (0)˘ [Ξ γµ Ξ2A (x) ψ¯B (0)˘ γ µ ψB (0) +[ψ¯A (x)˘ γµ ψA (x) Ξ γ µ Ξ1B (0) +x ↔ 0 (C.10)    p p   / / 1 1 = ψ¯A (x)γ i gBi (x) 2 γ˘µ 2 γ k gBk (x) ψA (x) ψ¯B (0)˘ γ µ ψB (0) α s s α    p  /1 µ p /1 k 1 1 i ¯ ¯ +[ψA (x)˘ γµ ψA (x) ψB (0)γ gAi (0) γ˘ γ gAk (0) ψB (0) +x ↔ 0 β s s β    p   / 1 1 = 2 ψ¯A (x)γ i gBi (x) 22 (a−γ5 )γ k gBk (x) ψA (x) ψ¯B (0)p /2 (a−γ5 )ψB (0) α s α    p  /1 1 1 i k ¯ ¯ +2[ψA (x)p ψB (0)γ gAi (0) 2 (a−γ5 )γ gAk (0) ψB (0) +x ↔ 0. /1 (a−γ5 )ψA (x) β s β

we get    ¯ 2A (x)˘ hA, B| Ξ γµ ψA (x) ψ¯A (0)˘ γ µ Ξ2A (0) |A, Bi =−

(C.14)

g2 1 ˆ¯ p (a − γ ) 1 ψ(0)|Ai ˆ ψ(0) ˆ hA|ψˆ¯ (x)p /2 (a − γ5 )ψ(x) /2 5 2Nc (Nc2 − 1)s2 α α  8 m⊥ a ai × hB|Aˆi (x)Aˆ (0)|Bi + O . s

where we used parametrization (4.6) from ref. [12]. Since the gluon TMD Dg (xB , x⊥ ) behaves only logarithmically as xB → 0 [38], the contribution of eq. (C.14) to W (q) is of m2

m2

order of βq⊥s  Q⊥2 (note that the projectile TMD in the r.h.s. of eq. (C.12) does not have 1 αq terms for the same reason as in eq. (C.22)). Similarly, all other terms in eq. (C.11) are either

m2⊥ βq s

C.4

Power corrections from Ψ(1) fields

or

m2⊥ αq s

so they can be neglected.20

First, let us notice that terms like (1) ψ¯A (x)˘ γµ ψA (x)ψ¯A (0)˘ γ µ ΨA (0),

(1) ψ¯A (x)˘ γµ ψA (x)ψ¯B (0)˘ γ µ ΨB (0)

(C.16)

give zero contribution since ψ¯A (x)˘ γµ ψA (x) does not depends on x∗ so R −iβ x q ∗ dx∗ e = 2πδ(βq ). µ Let us consider now the last two lines in the power expansion (4.6) of JAB (x)JBAµ (0): (0) (0) (1) (0) ¯ (1) (x)˘ ¯ (0) (0)˘ ¯ (0) (x)˘ ¯ (0) (0)˘ Ψ γ µ ΨB (x)Ψ γµ ΨA (0) + Ψ γ µ ΨB (x)Ψ γµ ΨA (0) A B A B

(C.17)

(0) (0) (0) (1) ¯ (0) (x)˘ ¯ (1) (0)˘ ¯ (0) (x)˘ ¯ (0) (0)˘ +Ψ γ µ ΨB (x)Ψ γµ ΨA (0) + Ψ γ µ ΨB (x)Ψ γµ ΨA (0) + . . . A B A B

After Fierz transformation (4.8) the first term in the above equation turns to (0) (1) ¯ (0) (x)˘ ¯ (0) (0)˘ Ψ γ µ ΨB (x)Ψ γµ ΨA (0) (C.18) A B 2  n(0)  1 + a  ¯ m(0) n(1) α m(0) ¯ ΨA (x)γα ΨA (0) Ψ (x) + (γα ⊗ γ α ↔ γα γ5 ⊗ γ α γ5 ) = B (0)γ ΨB 2  m(0)  n(0)  n(1) m(0) α ¯ ¯ −a Ψ (x)γα ΨA (0) Ψ (x) + (γα ⊗ γ α γ5 ↔ γα γ5 ⊗ γ α ) A B (0)γ γ5 ΨB  m(0)  n(0)  n(1) m(0) ¯ ¯ + (1 − a2 ) Ψ (x)ΨA (0) Ψ (x) − (1 ⊗ 1 ↔ γ5 ⊗ γ5 ) A B (0)ΨB    n  1 + a2 2  ¯m n(1) m = ψ (x)p /2 Ψ1A (0) ψ¯B (0)p /1 ψB (x) + (p /2 ⊗ p /1 ↔ p / 2 γ5 ⊗ p / 1 γ5 ) 2 s A   8   n  m⊥ 2  ¯m n(1) m ¯ ψA (x)p . −a /2 Ψ1A (0) ψB (0)p /1 γ5 ψB (x) + (p /2 ⊗ p / 1 γ5 ↔ p / 2 γ5 ⊗ p /1 ) + O s s 20

It is worth mentioning that if Z-boson is produced in the region of rapidity close to the projectile, the contribution (C.15) may be the most important since gluon parton densities at small xB are larger than quark ones.

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For the forward target matrix element one obtains Z dx∗ e−iβq x∗ hB|Aˆai (x)Aˆai (0)|Bi (C.15) Z Z x∗ Z 0 4 a = 2 dx∗ e−iβq x∗ dx0∗ dx00∗ hB|Fˆ•i (x0∗ , x⊥ )Fˆ•ai (x00∗ , 0⊥ )|Bi s −∞ −∞ Z 4 1 a = 2 2 dx∗ e−iβq x∗ hB|Fˆ•i (x∗ , x⊥ )Fˆ•ai (0)|Bi = − 8π 2 αs Dg (βq , x⊥ ), βq s βq

Since (1)

p /2 Ψ1A = −

gp /2 p /1 sβ

γ i Bi Ψ A +

gγ i 1 1 j 2g 1 [1] (0) p / 2 Pi γ Bj Ψ A − p / (A• ) ΨA s β α s 2β

(C.19)

we get

B

2a + 2g s

("



m i ψ¯A (x) p /2 p /1 γ



1 Bi β

1 B



2

1

2

1

1 1 [1] (0) i j1 +p / 2 γ γ Pi Bj + 2 p /2 (A• ) β α β

nk

# k ψA (0)

) n m × ψ¯B (0)p /1 γ5 ψB (x) + (p /2 ⊗ p / 1 γ5 ↔ p / 2 γ5 ⊗ p /1 ) .





(C.20)

Let us start with the first term in parentheses in the second line of eq. (C.20). Using eq. (B.9) the corresponding matrix element can be rewritten as   1 + a2 1 ˆ ˆ iˆ ˆ ˆ ¯ ¯ ˆ ˜ − ghA|ψ(x)γ ψ(0)|AihB|ψ(0)p (Ai + iγ5 Ai ) (0)ψ(x)|Bi /1 2sNc β   1 + a2 1 ˆ˜ ) (0)ψ(x)|Bi. i ˆ ˆ¯ p ¯ ˆ ˆ ˆ − ghA|ψ(x)γ γ5 ψ(0)|AihB| ψ(0) (γ A + i A (C.21) /1 5 i i 2sNc β Let us consider   Z 1 ˆ ˆ −iβq x∗ ˆ ¯ ˆ ∗ , x⊥ )|Bi ˜ dx∗ e hB|ψ(0)p (Ai + iγ5 Ai ) (0)ψ(x (C.22) /1 β Z Z 0 2i ˆ¯ p Fˆ (x0 , 0 ) + iγ Fˆ˜ (x0 , 0 )ψ(x ˆ ∗ , x⊥ )|Bi, = dx∗ e−iβq x∗ dx0∗ x0∗ hB|ψ(0) /1 •i ∗ ⊥ 5 •i ∗ ⊥ s −∞ where we used 1 ˆ Ak (z∗ , z⊥ ) = −i β + i

Z

z∗ −∞

dz∗0

2i Aˆk (z∗0 , z⊥ ) = − s

Z

z∗ −∞

dz∗0 (z − z 0 )∗ Fˆ•k (z∗0 , z⊥ ).

Let us compare this matrix element to that of eq. (4.46):   Z 1 −iβq x∗ ˆ ¯ ˆ ∗ , x⊥ )|Bi dx∗ e hB| ψ (0)Aˆi (0)Γψ(x (C.23) β − i   Z Z 0 1 ˆ ˆ¯ 0 , 0 ) 2 Fˆ (0) Γψ(x ¯ Aˆi (0) + ˆ ∗ , x⊥ )|Bi. =− dx∗ e−iβq x∗ hB| ψ(0) dx0∗ ψ(x •i ∗ ⊥ βq s −∞ We see that β1z in eq. (C.23) is traded for an extra x0∗ ∼ 1 in eq. (C.22) (recall that x0∗ in the target matrix elements is inversely proportional to characteristic β’s in the target which are of order 1). Consequently, power correction due to matrix element (C.22) can be neglected

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(0) (1) ¯ (0) (x)˘ ¯ (0) (0)˘ Ψ γ µ ΨB (x)Ψ γµ ΨA (0) A B (" #    nk 1 + a2 1 1 1 1 [1] m i j (0) k i =− 2 g ψ¯A (x) p Bi + p ψA (0) /2 p /1 γ / 2 γ γ Pi Bj + 2 p /2 (A• ) s β β α β )  n  m × ψ¯ (0)p / ψ (x) + (p / ⊗p / ↔p / γ5 ⊗ p / γ5 )

m2

m2

Using eq. (B.9) we get " #  nk   1+a2 1 1 i j m − 2 g ψ¯am (x)p Pi Bj ψak (0) ψ¯bn (0)p /2 γ γ /1 ψb (x) +(p /2 ⊗ p /1 ↔ p / 2 γ5 ⊗ p / 1 γ5 ) s β α   2 1 1 ˆi f.int. 1+a ˆ i ˆ ˆ ¯ ˆ ¯ ˆ ˜ → − ghA|ψ(x)p (A −iA γ5 )(0) ψ(x)|Bi /2 i∂i ψ(0)|AihB|ψ(0)p /1 Nc s 2 α β   1+a2 1ˆ 1 ˆi ˆ 2 i ˆ ˆ ¯ ¯ ˆ ˆ ˜ + g hA|ψ(x)p (A −iA γ5 )(0) ψ(x)|Bi /2 Ai ψ(0)|AihB|ψ(0)p /1 Nc (Nc2 −1)s2 α β   1+a2 1 1 i i ˆ ˆ ¯ ¯ ˆ ˆ ˆ ˆ − ghA|ψ(x)p (i∂i A +g Ai A )(0) ψ(x)|Bi /2 ψA (0)|AihB|ψ(0)p /1 Nc s 2 α β   2(1+a2 ) 1 1 ˆ ˆ ˆ ¯ p ¯ p ˆ − ghA|ψ(x) F˜∗• (0) ψ(x)|Bi /2 ψA (0)|AihB|ψ(0) / 1 γ5 Nc s 3 α β +(p /2 ⊗ p /1 ↔ p / 2 γ5 ⊗ p /1 γ5 ).

(C.25)

It is easy to see that projectile matrix elements lead to terms ∼ α1q after integration over x• , for example Z Z Z x• 2 1ˆ −iαq x• iαq x• ˆ ¯ ˆ dx• e hA|ψ(x)p dx• e dx0• (C.26) /2 Ai (0) ψ(0)|Ai = α αq s −∞ ˆ ¯ p ˆ 0 , −x⊥ ) + Fˆ∗i (x0 , −x⊥ )ψ(x ˆ • , −x⊥ )|Ai. × hA|ψ(0) / [Fˆ∗i (x• , −x⊥ )ψ(x 2

•

– 41 –

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in our kinematic region since the matrix element (C.21) is ∼ s⊥ rather than ∼ αq β⊥q s . Similarly, the second term in eq. (C.21) does not contribute in our kinematical region. This is the same reason why we neglected power corrections (C.6). In general, as we discussed in ref. [12], the way to figure out integrations that give β1q is very simple: R x∗ take βq → 0 and check if there is an infinite integration of the type −∞ dx0∗ without any integrand. Similarly, the factor α1q can be figured out from (possible) unrestricted integrals over x0• after one sets αq = 0 in the relevant matrix element. Note that to get the terms ∼ Q12 = αq1βq s we need to find contributions which satisfy both of the above conditions. Next, consider the second term in parentheses in the second line of eq. (C.20). Using eq. (4.44) the corresponding matrix element can be rewritten as " #  nk  n  1+a2 1 1 m i j k m (0) ψ¯B (0)p − 2 g ψ¯A (x)p Pi Bj ψA /2 γ γ /1 ψB (x) s β α     nk nk 1+a2 ¯m 1 1 1 1 k i j k m i j ¯ = − 2 g ψA (x)p Bj i∂i ψA (0)+g ψA (x)p ψ (0) /2 γ γ /2 γ γ Ai Bj s β α β α A 
nk 1 k  n  m i j1 m + ψ¯A (x)p γ γ i∂ B +gB B ψ (0) ψ¯B (0)p /2 /1 ψB (x) i j i j A β α   2 1 ˆ f.int. 1+a i j1 ˆ ˆ ¯ ˆ ¯ ˆ → − ghA|ψ(x)p Aj (0)ψ(x)|Bi /2 γ γ i∂i ψ(0)|AihB|ψ(0)p /1 Nc s 2 α β   1+a2 1 ˆ 2 i j ˆ 1 ˆ ˆ ˆ ¯ ¯ ˆ + g hA|ψ(x)p Aj (0) ψ(x)|Bi /2 γ γ Ai ψ(0)|AihB|ψ(0)p /1 Nc (Nc2 −1)s2 α β   1+a2 1 i j1 ˆ ˆ ¯ ¯ ˆ ˆ ˆ ˆ − ghA|ψ(x)p (i∂i Aj +g Ai Aj )(0) ψ(x)|Bi. (C.24) /2 γ γ ψA (0)|AihB|ψ(0)p /1 Nc s 2 α β

On the other hand, the target matrix elements cannot give a β1q factor. For the first two lines in the r.h.s. of eq. (C.25) we proved this in eq. (C.22) above. As to the last lines in eq. (C.25), the target matrix element can be rewritten as Z dx∗ e

−iβq x∗

ˆ¯ p ×hB|ψ(0) /1

 Z Z 0 1 2i i i −iβq x∗ ˆ ˆ ˆ ˆ (i∂i A +g Ai A )(0) ψ(x∗ , x⊥ )|Bi = − dx∗ e dx0∗ β s −∞ #! Z x00∗ 000 ˆ ∗ , x⊥ )|Bi ˆ i Fˆ•i (x00 , 0⊥ )+ 2g Fˆ i (x00 , 0⊥ ) ˆ iD dx000 ψ(x ∗ ∗ F•i (x∗ , 0⊥ ) s • ∗ −∞

ˆ ¯ p hB|ψ(0) /1 " Z 0 x∗

−∞

dx00∗



Z

  1 ˆ˜ ˆ ¯ ˆ ∗ , x⊥ )|Bi dx∗ e hB|ψ(0)p F∗• (0) ψ(x / 1 γ5 β Z Z 0 ˆ¯ p γ Fˆ˜ (x0 , 0 )ψ(x = −i dx∗ e−iβq x∗ dx0∗ hB|ψ(0) /1 5 ∗• ∗ ⊥ ˆ ∗ , x⊥ )|Bi. −iβq x∗

(C.27)

−∞

We see that at βq = 0 there are no unrestricted integration over any longitudinal variable so the r.h.s. of these equations cannot give β1q factor and therefore the contribution to W (q) is ∼

m2⊥ αq s



m2⊥ . Q2

Finally, we should consider the third term in eq. (C.20)    n  1 + a2 1 [1]nl (0) l m m ¯ − 2 2 g ψA (x)p /2 (A• ) ψA (0) ψ¯B (0)p /1 ψB (x) s β #" # "  nk  kl 2 1 + a 2 ¯m 1 1 l n j m = − 2 g ψA (x)p Aj (0) ψA (0) ψ¯B (0)p B (0) ψB (x) /2 /1 s α β " #" #  kl  nk 1 + a2 2 ¯m 1 1 l n j m + g ψA (x)p Aj (0) ψA (0) ψ¯B (0)p B (0) ψB (x) , /2 /1 s2 α β

(C.28)

[1]

where we used eq. (A.5) for (A• )(0) . Taking projectile and target matrix elements and separating color-singlet contributions using eq. (4.44), we obtain (1 + a2 )Nc 2 ˆ ¯ p − 2 2 g hA|ψ(x) /2 s (Nc − 1)



   1 ˆ 1 ˆj ˆ ˆ ¯ ˆ Aj (0) ψ(0)|AihB|ψ(0)p A (0) ψ(x)|Bi. /1 α β

(C.29)

It has been demonstrated in eq. (C.22) that such matrix elements cannot give α1q and 1 βq so their contribution to W (q) is small in our kinematical region. Moreover, since the above arguments do not depend on presence (or absence) of γ5 , we proved that all terms in eq. (C.20) give small contributions at αq , βq  1. In a similar way, one can demonstrate that the other three terms in eq. (C.17) can be neglected.

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and

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References [1] J. Collins, Foundations of perturbative QCD, Cambridge University Press (2013). [2] J.C. Collins and D.E. Soper, Parton Distribution and Decay Functions, Nucl. Phys. B 194 (1982) 445 [INSPIRE].

[4] X.-d. Ji, J.-p. Ma and F. Yuan, QCD factorization for semi-inclusive deep-inelastic scattering at low transverse momentum, Phys. Rev. D 71 (2005) 034005 [hep-ph/0404183] [INSPIRE]. [5] M.G. Echevarria, A. Idilbi and I. Scimemi, Factorization Theorem For Drell-Yan At Low qT And Transverse Momentum Distributions On-The-Light-Cone, JHEP 07 (2012) 002 [arXiv:1111.4996] [INSPIRE]. [6] C.W. Bauer, S. Fleming, D. Pirjol and I.W. Stewart, An Effective field theory for collinear and soft gluons: Heavy to light decays, Phys. Rev. D 63 (2001) 114020 [hep-ph/0011336] [INSPIRE]. [7] C.W. Bauer, D. Pirjol and I.W. Stewart, Soft collinear factorization in effective field theory, Phys. Rev. D 65 (2002) 054022 [hep-ph/0109045] [INSPIRE]. [8] C.W. Bauer, S. Fleming, D. Pirjol, I.Z. Rothstein and I.W. Stewart, Hard scattering factorization from effective field theory, Phys. Rev. D 66 (2002) 014017 [hep-ph/0202088] [INSPIRE]. [9] T. Becher and M. Neubert, Drell-Yan Production at Small qT , Transverse Parton Distributions and the Collinear Anomaly, Eur. Phys. J. C 71 (2011) 1665 [arXiv:1007.4005] [INSPIRE]. [10] J.-Y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, A Formalism for the Systematic Treatment of Rapidity Logarithms in Quantum Field Theory, JHEP 05 (2012) 084 [arXiv:1202.0814] [INSPIRE]. [11] J. Collins and T. Rogers, Understanding the large-distance behavior of transverse-momentum-dependent parton densities and the Collins-Soper evolution kernel, Phys. Rev. D 91 (2015) 074020 [arXiv:1412.3820] [INSPIRE]. [12] I. Balitsky and A. Tarasov, Higher-twist corrections to gluon TMD factorization, JHEP 07 (2017) 095 [arXiv:1706.01415] [INSPIRE]. [13] F. Landry, R. Brock, G. Ladinsky and C.P. Yuan, New fits for the nonperturbative parameters in the CSS resummation formalism, Phys. Rev. D 63 (2001) 013004 [hep-ph/9905391] [INSPIRE]. [14] J.-w. Qiu and X.-f. Zhang, Role of the nonperturbative input in QCD resummed Drell-Yan QT distributions, Phys. Rev. D 63 (2001) 114011 [hep-ph/0012348] [INSPIRE]. [15] F. Landry, R. Brock, P.M. Nadolsky and C.P. Yuan, Tevatron Run-1 Z boson data and Collins-Soper-Sterman resummation formalism, Phys. Rev. D 67 (2003) 073016 [hep-ph/0212159] [INSPIRE].

– 43 –

JHEP05(2018)150

[3] J.C. Collins, D.E. Soper and G.F. Sterman, Transverse Momentum Distribution in Drell-Yan Pair and W and Z Boson Production, Nucl. Phys. B 250 (1985) 199 [INSPIRE].

[16] S. Mantry and F. Petriello, Transverse Momentum Distributions in the Non-Perturbative Region, Phys. Rev. D 84 (2011) 014030 [arXiv:1011.0757] [INSPIRE]. [17] P. Sun, J. Isaacson, C.P. Yuan and F. Yuan, Universal Non-perturbative Functions for SIDIS and Drell-Yan Processes, Int. J. Mod. Phys. A 33 (2018) 1841006 [arXiv:1406.3073] [INSPIRE]. [18] U. D’Alesio, M.G. Echevarria, S. Melis and I. Scimemi, Non-perturbative QCD effects in qT spectra of Drell-Yan and Z-boson production, JHEP 11 (2014) 098 [arXiv:1407.3311] [INSPIRE].

[20] A. Bacchetta, F. Delcarro, C. Pisano, M. Radici and A. Signori, Extraction of partonic transverse momentum distributions from semi-inclusive deep-inelastic scattering, Drell-Yan and Z-boson production, JHEP 06 (2017) 081 [arXiv:1703.10157] [INSPIRE]. [21] I. Scimemi and A. Vladimirov, Analysis of vector boson production within TMD factorization, Eur. Phys. J. C 78 (2018) 89 [arXiv:1706.01473] [INSPIRE]. [22] P.J. Mulders and R.D. Tangerman, The Complete tree level result up to order 1/Q for polarized deep inelastic leptoproduction, Nucl. Phys. B 461 (1996) 197 [Erratum ibid. B 484 (1997) 538] [hep-ph/9510301] [INSPIRE]. [23] Particle Data Group collaboration, C. Patrignani et al., Review of Particle Physics, Chin. Phys. C 40 (2016) 100001 [INSPIRE]. [24] I.I. Balitsky and V.M. Braun, Nonlocal Operator Expansion for Structure Functions of e+ e− Annihilation, Phys. Lett. B 222 (1989) 123 [INSPIRE]. [25] I.I. Balitsky and V.M. Braun, The Nonlocal operator expansion for inclusive particle production in e+ e− annihilation, Nucl. Phys. B 361 (1991) 93 [INSPIRE]. [26] I.I. Balitsky and V.M. Braun, Valleys in Minkowski space and instanton induced cross-sections, Nucl. Phys. B 380 (1992) 51 [INSPIRE]. [27] F. Gelis and R. Venugopalan, Large mass q q¯ production from the color glass condensate, Phys. Rev. D 69 (2004) 014019 [hep-ph/0310090] [INSPIRE]. [28] F. Gelis and R. Venugopalan, Particle production in field theories coupled to strong external sources, Nucl. Phys. A 776 (2006) 135 [hep-ph/0601209] [INSPIRE]. [29] F. Gelis, T. Lappi and R. Venugopalan, High energy scattering in Quantum Chromodynamics, Int. J. Mod. Phys. E 16 (2007) 2595 [arXiv:0708.0047] [INSPIRE]. [30] A.V. Belitsky, X. Ji and F. Yuan, Final state interactions and gauge invariant parton distributions, Nucl. Phys. B 656 (2003) 165 [hep-ph/0208038] [INSPIRE]. [31] I.Z. Rothstein and I.W. Stewart, An Effective Field Theory for Forward Scattering and Factorization Violation, JHEP 08 (2016) 025 [arXiv:1601.04695] [INSPIRE]. [32] R.D. Tangerman and P.J. Mulders, Intrinsic transverse momentum and the polarized Drell-Yan process, Phys. Rev. D 51 (1995) 3357 [hep-ph/9403227] [INSPIRE]. [33] Y.V. Kovchegov and M.D. Sievert, Calculating TMDs of a Large Nucleus: Quasi-Classical Approximation and Quantum Evolution, Nucl. Phys. B 903 (2016) 164 [arXiv:1505.01176] [INSPIRE].

– 44 –

JHEP05(2018)150

[19] S. Catani, D. de Florian, G. Ferrera and M. Grazzini, Vector boson production at hadron colliders: transverse-momentum resummation and leptonic decay, JHEP 12 (2015) 047 [arXiv:1507.06937] [INSPIRE].

[34] J. Zhou, F. Yuan and Z.-T. Liang, Hyperon Polarization in Unpolarized Scattering Processes, Phys. Rev. D 78 (2008) 114008 [arXiv:0808.3629] [INSPIRE]. [35] A. Bacchetta, M. Boglione, A. Henneman and P.J. Mulders, Bounds on transverse momentum dependent distribution and fragmentation functions, Phys. Rev. Lett. 85 (2000) 712 [hep-ph/9912490] [INSPIRE]. [36] I. Balitsky and G.A. Chirilli, Next-to-leading order evolution of color dipoles, Phys. Rev. D 77 (2008) 014019 [arXiv:0710.4330] [INSPIRE]. [37] I. Balitsky and A. Tarasov, Rapidity evolution of gluon TMD from low to moderate x, JHEP 10 (2015) 017 [arXiv:1505.02151] [INSPIRE].

[39] I. Moult, I.W. Stewart and G. Vita, A subleading operator basis and matching for gg → H, JHEP 07 (2017) 067 [arXiv:1703.03408] [INSPIRE]. [40] R. Goerke and M. Luke, Power Counting and Modes in SCET, JHEP 02 (2018) 147 [arXiv:1711.09136] [INSPIRE]. [41] M. Beneke, M. Garny, R. Szafron and J. Wang, Anomalous dimension of subleading-power N-jet operators, JHEP 03 (2018) 001 [arXiv:1712.04416] [INSPIRE]. [42] S.J. Brodsky, D.S. Hwang and I. Schmidt, Final state interactions and single spin asymmetries in semiinclusive deep inelastic scattering, Phys. Lett. B 530 (2002) 99 [hep-ph/0201296] [INSPIRE]. [43] J.C. Collins, Leading twist single transverse-spin asymmetries: Drell-Yan and deep inelastic scattering, Phys. Lett. B 536 (2002) 43 [hep-ph/0204004] [INSPIRE]. [44] I. Balitsky, Operator expansion for high-energy scattering, Nucl. Phys. B 463 (1996) 99 [hep-ph/9509348] [INSPIRE]. [45] I. Balitsky, High-energy QCD and Wilson lines, hep-ph/0101042 [INSPIRE].

– 45 –

JHEP05(2018)150

[38] I. Balitsky and A. Tarasov, Gluon TMD in particle production from low to moderate x, JHEP 06 (2016) 164 [arXiv:1603.06548] [INSPIRE].