Power corrections to TMD factorization for Z-boson production

Journal of High Energy Physics, May 2018

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 N c power corrections are expressed in terms of leading power TMDs by QCD equations of motion.

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Power corrections to TMD factorization for Z-boson production

JHE Power corrections to TMD factorization for Z-boson I. Balitsky 0 1 2 4 A. Tarasov 0 1 2 3 0 Upton , NY 11973 , U.S.A 1 Newport News , VA 23606 , U.S.A 2 Norfolk , VA 23529 , U.S.A 3 Physics Department, Brookhaven National Laboratory 4 Physics Department, Old Dominion University 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 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 rst 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. NLO Computations; QCD Phenomenology - HJEP05(218) 1 Introduction 2 3 5 6 7 3.1 3.2 3.3 4.1 4.2 4.3 TMD factorization from functional integral Power corrections and solution of classical YM equations Power counting for background elds Approximate solution of classical equations Power expansion of classical quark elds 4 Leading power corrections at s Q2 q2 ? 4.3.1 4.3.2 4.3.3 4.3.4 4.4.1 4.4.2 Leading contribution and power corrections from JAB(x)JBA (0) terms Leading power contribution 4.2.1 Parametrization of leading matrix elements Power corrections from JAB(x)JBA (0) terms Fifth line in eq. (4.19): the leading term in N1c Parametrization of matrix elements from section 4.3.1 Sixth line in eq. (4.19) Parametrization of matrix elements from section 4.3.3 4.4 Power corrections from JA (x)JB (0) terms Last two lines in eq. (4.56) Parametrization of TMDs from section 4.4.1 Results and estimates Power corrections for Drell-Yan process Conclusions and outlook A Next-to-leading quark elds B Formulas with Dirac matrices C Subleading power corrections C.1 Second, third, and fourth lines in eq. (4.19) C.2 Second to fth lines in eq. (4.56) C.4 Power corrections from (1) elds C.3 Gluon power corrections from JA (x)JA (0) terms { i { Introduction A typical analysis of di erential 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 di erential cross section is [1, 11] d d d2q ? = XZ f d2b?ei(q;b)? Df=A(xA; b?; )Df=B(xB; b?; ) (f f ! H) + power corrections + Y terms; (1.1) where is the rapidity, q is the momentum of the produced particle in the hadron frame (see ref. [1]), Df=A(x; z?; ) is the TMD density of a parton f in hadron A, and (f f ! H) is the cross section of production of particle H in the scattering of two partons. The common wisdom is that when we increase transverse momentum q2 of the produced hadron, at rst the leading power TMD analysis with (nonperturbative) TMDs applies, then at some point ? power corrections kick in, and nally at q2 ? into so-called Y-term making smooth transition to collinear factorization formulas. In this paper we try to answer the question about the rst transition, namely at what q2 power Q2, where Q2 = q2, they are transformed corrections become signi cant. ? In our recent paper [12] we calculated power corrections duction 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 calculate power corrections q 2 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 speci c 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 nd the leading power correction to particle production in the region s perform the order-of-magnitude estimate of power corrections and in section 6 present our TMD factorization from functional integral We consider Z-boson production in the Drell-Yan reaction illustrated in gure 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 elds i describing coupling between fermions and Z-boson is (sW sin W , cW cos W ) LZ = Z d4x J Z (x); J = e X 2sW cW i i (giV giA 5) i; (2.2) where sum goes over di erent types of fermions, and coupling constants giV = (t3L)i 2qis2W and giA = (t3L)i are de ned by week isospin (t3L)i of the fermion i, see ref. [23]. In this paper we take into account only u; d; s; c quarks and e; leptons. We consider all fermions to The di erential cross section of Z-boson production with subsequent decay into e+e d dQ2dydq2 = ? e2Q2 1 192ss2W c2W (m2Z 4s2W + 8s4W Q2)2 + 2Z m2Z where we de ned the \hadronic tensor" W (pA; pB; q) as [ W (pA; pB; q)]; (2.3) W (pA; pB; q) d=ef XZ d4x e iqxhpA; pBjJ (x)jXihXjJ (0)jpA; pBi = 1 Z d4x e iqxhpA; pBjJ (x)J (0)jpA; pBi: (2.4) 1 Z DB~ DB d4x e iqxhpA; pBjJ (x)jXihXjJ (0)jpA; pBi (2.5) HJEP05(218) approximation. functional integral (2 )4W (pA; pB; q) = XZ 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 q . However, if we consider quarks to be massless, the only e ect of the q q term comes from the (square of) axial anomaly which has an extra factor s2, and such two-loop factor is beyond our tree The sum over full set of \out" states in eq. (2.4) can be represented by a double d4x e iqxZ A~(tf )=A(tf ) DA~ DA Z ~(tf )= (tf ) D ~ D ~D D pA (A~~(ti); ~(ti)) pB (A~~(ti); ~(ti))e iSQCD(A~; ~)eiSQCD(A; )J~ (x)J (0) pA (A~(ti); (ti)) pB (A~(ti); (ti)): In this double functional integral the amplitude hXjJ (0)jpA; pBi is given by the integral over ; A elds whereas the complex conjugate amplitude hpA; pBjJ (x)jXi is represented by the integral over ~; A~ elds. Also, p(A~(ti); (ti)) denotes the proton wave function at the initial time ti and the boundary conditions A~(tf ) = A(tf ) and ~(tf ) = (tf ) re ect the sum over all states X, cf. refs. [24{26]. We use Sudakov variables p = p1 + p2 + p , where p1 and p2 are light-like vectors ? close to pA and pB, and the notations x s so that p q = ( p q + q p) 2 (p; q)? where (p; q)? light-cone coordinates (x = p 2s x+ and x = p s x ). Our metric is g 2 = (1; 1; 1; 1) piqi. Throughout the paper, the x p 1 and x x p2 for the dimensionless sum over the Latin indices i; j; : : : runs over two transverse components while the sum over Greek indices ; ; : : : runs over four components as usual. Following ref. [12] we separate quark and gluon elds in the functional integral (2.5) into three sectors (see gure 2): \projectile" elds A ; A with j j < a, \target" elds B ; B with j j < b and \central rapidity" elds C ; C with j j > b and j j > a and get pA (A~~(ti); ~A(ti)) pA (A~(ti); A(ti)) pB (B~~(ti); ~B(ti)) pB (B~ (ti); B(ti)) 1This procedure is obviously gauge-dependent. We have in mind factorization in covariant-type gauge, Our goal is to integrate over central elds and get the amplitude in the factorized form, i.e. as a product of functional integrals over A elds representing projectile matrix elements (TMDs of the projectile) and functional integrals over B elds representing target matrix elements (TMDs of the target). In the spirit of background- eld method, we \freeze" projectile and target elds and j j < get a sum of diagrams in these external elds. Since j j < a in the projectile elds and b in the target elds, at the tree-level one can set with power accuracy the projectile elds and = 0 for the target elds | the corrections will be O and O mb2Ns , where mN is the hadron's mass. Beyond the tree level, one should expect that the integration over C elds will produce the logarithms of the cuto s a and b which will cancel with the corresponding logs in gluon TMDs of the projectile and the target. The m2N as result of integration over C- elds has the schematic form and eSe represents a sum of disconnected diagrams (\vacuum bubbles") in external elds. As usual, since the rapidities of central C elds and of A, B elds are very di erent, the result of integration over C elds is expressed in terms of Wilson-line operators made form A and B elds. Z pB (B~~(ti); ~B(ti)) pB (B~ (ti); B(ti))eSe (A; A;A~; ~A;B; B;B~; ~B)O(q; x; A; A; A~; ~A; B; B; B~; ~B): (2.8) { 4 { From integrals over projectile and target elds in the above equation we see that the functional integral over C elds should be done in the background of A and B elds satisfying A~(tf ) = A(tf ); ~A(tf ) = A(tf ) and B~(tf ) = B(tf ); ~B(tf ) = B(tf ): Combining this with our approximation that at the tree level for B, B~ B~ = B~(x ; x?), we see that for the purpose of calculation of the functional integral over central elds (2.7) we can set and A(x ; x?) = A~(x ; x?); A(x ; x?) = ~A(x ; x?) B(x ; x?) = B~(x ; x?); B(x ; x?) = ~B(x ; x?): (2.10) In other words, since A, and A~, ~ do not depend on x , if they coincide at x = 1 they should coincide everywhere. Similarly, since B, B and B~, ~B do not depend on x , if they coincide at x = 1 they should be equal. Next, in ref. [12] it was demonstrated that due to eqs. (2.10) the e ective action Se (A; A; A~; ~A; B; B; B~; ~B) vanishes for background elds satisfying conditions (2.9).2 Summarizing, we see that at the tree level in our approximation = O(q; x; A; A; B; B); where now SC = SQCD(C + A + B; C + A + B) SQCD(A; A) SQCD(B; B) and SC = SQCD(C~ + A + B; ~C + A + B) SQCD(A; A) SQCD(B; B). 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 elds [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 elds A and B), calculation of the tree-level contribution to in the r.h.s. of eq. (2.11), is equivalent to solving YM equation for (x) (and A (x)) with boundary conditions such that the solution has the same asymptotics at t ! 1 as the superposition of incoming projectile and target background elds. The hadronic tensor (2.8) can now be represented as3 1 Z 2It corresponds to cancellation of so-called \Glauber gluons", see discussion in ref. [1]. 3As discussed in ref. [12], there is a subtle point in the promotion of background elds to operators. When we calculate O as the r.h.s. of eq. (2.11) the elds A and B 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 separated either by space-like distances or light-cone distances so all of them (anti) commute and thus can be treated as c-numbers. { 5 { evaluated between the corresponding (projectile or target) states: if XZ m;n O^(q; x; A^; ^A; B^; ^B) = dzmdzn0cm;n(q; x) ^ A(zm) ^ B(zn0) (2.13) (where cm;n are coe cients and can be any of A , or ) then m;n W = 1 Z As we will demonstrate below, the relevant operators are quark and gluon elds with Wilson-line type gauge links collinear to either p2 for A elds or p1 for B elds. 3 3.1 equations4 Power corrections and solution of classical YM equations Power counting for background elds 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 elds (2.10). Since we integrate over elds (2.10) afterwards, we may assume that they satisfy Yang-Mills iD= A A = 0; iD= B B = 0; DAA a = g X DBBa = g X f A f B f f t t a f a f A B ; ; where A ig[A ; ) and similarly for B elds. It is convenient to choose a gauge where A = 0 for projectile elds and B target elds. The rotation from a general gauge (Feynman gauge in our case, see footnote 1) to this gauge is performed by the matrix (x ; x ; x?) satisfying boundary conditions (x ; x ; x?) x !! 1 [x ; 1 ]xA ; (x ; x ; x?) x !! 1 [x ; 1 ]xB ; where A (x ; x?) and B (x ; x?) are projectile and target elds in an arbitrary gauge and [x ; y ]zA denotes a gauge link constructed from A elds ordered along a light-like line: [x ; y ]zA = P e 2sig Ryx dz A (z ;z?) and similarly for [x ; y ]zB . The existence of matrix are \frozen". (x ; x ; x?) was proved in appendix B of ref. [12] by explicit construction. The relative strength of Lorentz components of projectile and target elds 4As we mentioned, for the purpose of calculation of integral over C elds the projectile and target elds { 6 { (3.1) (3.2) (3.3) 2 Z x s 1 need to calculate the functional integral (2.5) in the background elds 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- elds in the tree approximation reduces to nding elds C and C as solutions of Yang-Mills Bf + f C ) = 0; D F a (A + B + C) = g X( Af + Bf + f C ) ta( Af + Bf + Cf ): in this gauge was found in ref. [12] p = 1 A(x ; x?) p = 1 B(x ; x?) m5=2; ? spm ; ? A (x ; x?) B (x ; x?) ? is a scale of order of mN or q?. In general, we consider W (pA; pB; q) in the region q2 ; m2N , while the relation between q2 and m2N and between Q2 and s may be arbitrary. So, for the purpose of counting of powers of s, we will not distinguish between ? s and Q2 (although at the nal step we will be able to tell the di erence since our nal 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 (3.6) (3.7) (3.8) As we discussed above, the solution of eq. (3.6) which we need corresponds to the sum of set of diagrams in background eld A + B with retarded Green functions, see gure 3. The retarded Green functions (in the background-Feynman gauge) are de ned as 1 +g2(xj p2 +i p0 O 1 +i p0 jy) (xj p2 +i p0 jy) g(xj p2 +i p0 O p2 +i p0 jy) 1 1 p2 +i p0 O p2 +i p0 jy)+: : : ; (xj P2g 1 +2igF where and similarly for quarks. F fp ; A + B g + g(A + B)2 g + 2iF ig[A + B ; A + B ]; f { 7 { 2 Z x s 1 1 1 $ tions of the central elds are given by retarded propagators. Hereafter we use Schwinger's notations for propagators in external elds normalized according to where we use space-saving notation d np confusion, we will use short-hand notation (xjF (p)jy) d 4p e ip(x y)F (p); 1 O O0(x) d4z(xj jz)O0(z): D F a = X g f ta f P= f = 0; Z Z 1 O f ; { 8 { The solution of eqs. (3.6) in terms of retarded Green functions gives elds C and that vanish at t ! 1. Thus, we are solving the usual classical YM equations5 where A P with boundary conditions6 = C + A + B ; f = Cf + Af + f B ; F (3.12) A (x) x != 1 A (x ; x?); A (x) x != 1 B (x ; x?); (x) x != 1 (x) x != 1 A(x ; x?); t! 1 0. These boundary conditions re ect the fact that at t ! ! we have only incoming hadrons with A and B 5We take into account only u; d; s; c quarks and consider them massless. 6It is convenient to x redundant gauge transformations by requirements Ai( 1 ; x?) = 0 for the projectile and Bi( 1 ; x?) = 0 for the target, see the discussion in ref. [30]. (2dn)pn . Moreover, when it will not lead to a (3.9) (3.10) C (3.11) (3.13) L a L ( 1)a = D F 2g a + g [0] t The explicit form of gluon linear terms L(0)a and L(1)a is presented in eq. (3.26) from our paper [12]. For our purposes we need only the leading term L With the linear terms (3.15) and (3.18), a couple of rst terms in our perturbative series are for quark elds and P are operators in external zero-order elds (3.14). Here we denote the order of expansion in the parameter ms2? 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 The gluon linear term is L(0) m5=2; ? As discussed in ref. [12], for our case of particle production with qQ? to nd 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 elds and improving it by calculation of Feynman diagrams with retarded propagators in the background elds (3.14). The rst step is the calculation of the linear term for the trial con guration (3.14). The quark part of the linear term has the form A[1]a(x) = A[2]a(x) = g d4z Z Z d4z(xj P2g " + (xj P2g 1 jz)abLb (z); 1 { 9 { for gluon elds (in the background-Feynman gauge). Next iterations, like A[3]a(x), will give us a set of tree-level Feynman diagrams in the background elds A + B Let us consider the elds in the rst order in perturbative expansion: Here , , and p? are understood as di erential operators Now comes the central point of our approach. Let us expand quark and gluon propagators in powers of background elds, then we get a set of diagrams shown in gure 3. The typical bare gluon propagator in gure 3 is 1 Since we do not consider loops of C- elds in this paper, the transverse momenta in tree diagrams are determined by further integration over projectile (\A") and target (\B") elds in eq. (2.8) which converge on either q ? or mN . On the other hand, the integrals over converge on either q or Since q qs = Q2 1 and similarly the characteristic 's are either q or 2 q2 , one can expand gluon and quark propagators in powers of p? (3.22) s ( + i )( + i ) jy) = s( + i )( + i ) ( + i )( + i ) k 1 s (xj (xj 1 ? 1 p = 1 + i 1 1 + i jy) = + i jy) = + = = = 1 + + d 2p d 2p d 2p ? ? ? Z Z Z p = 2 + i s 2 s 2 s Z 2 s Z 2 s Z 2 s 2 p2?=s p = ? d + i d + i d d + i +1i , +1i , and ( +i )( +i ) is 1 d e i (x y) i (x y) +i(p;x y)? i (2 )2 (2)(x? ? y ) (x y ) (x y ); i (2 )2 (2)(x? (2 )2 (2)(x? ? y ) (x d + i ? y ) (x e i (x y) i (x y) +i(p;x y)? y ) (x y ); e i (x y) i (x y) +i(p;x y)? y ) (x y ): (3.24) After the expansion (3.23), the dynamics in the transverse space e ectively becomes trivial: all background elds stand either at x or at 0. The formula (3.21) turns into expansion elds in section 2.7 The reason is that in the diagrams like gure 3 with retarded propagators (3.24) one can shift the contour of integration over and/or to the complex plane away to avoid the region of small or . It should be mentioned, however, that such shift may not be possible if there is pinching of poles in the integrals over 1 or . For example, if after the expansion (3.23) we encounter ( +i )( i ) , the expansion was not justi ed since actual 's in the integral are was misidenti ed: we have a propagator of B- eld rather than of C- eld. 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 ( +i )( +i 0) . In such case the pinching may occur so one needs to formulate a subtraction program to get rid of p 2 s? and hence the eld pinched poles and avoid double counting of the elds. Note that the background elds are also smaller than typical p2 s. Indeed, from HJEP05(218) Also (pi + Ai + Bi)2 eq. (3.4) we see that p = 2s q 2 ? A p2.8 k ? m2 ( because 3.3 Power expansion of classical quark elds 2 Now we expand the classical quark elds in powers of pp?2 k q ms2? ) and similarly p B . k m2 s? (the expansion of classical gluon elds 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 rst term displayed in eq. (3.19). Expanding it in powers of p2?=p2 as explained in the previous section, we obtain k (x) = [0](x) + [1](x) + [2](x) + = (A0) + (B0) + (A1) + (B1) + : : : ; (3.26) where 7Such cuto s for integrals over C e ective theory (SCET), see review [31]. 8The only exception is the e ectively the expansion in powers of these elds is cut at the second term. elds are introduced explicitly in the framework of soft-collinear elds B i or A i which are of order of sm? but we saw in ref. [12] that A (0) = A (0) = B (0) = B (0) = A + 2A; A + 2A; B + 1B; B + 1B; s s gp=2 iBi gp=1 iAi A B 1 1 i i 1 + i 1 + i A; B; iBi s gp=2 iAi s gp=1 ; : (3.27) 2A = 2A = 1B = 1B = + i 1 i A(x ; x?) (x ; x?) i i Z x Z x 1 dx0 1 dx0 A(x0 ; x?); A(x0 ; x?) It is easy to see that power counting of these quark elds has the form (0) A (0) B m3=2: ? As to quark elds (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. 4 Leading power corrections at s Q2 q2 ? As we mentioned in the introduction, our method is relevant to calculation of power corrections at any s; Q2 physically interesting case s ? Q2 q ?2 which we consider in this paper.9 q2 ; m2N . However, the expressions are greatly simpli ed in the As we noted above, we take into account only u; d; s; c quarks and consider them massless. The hadronic tensor takes the form = JA + JB + JAB + JBA; 9We also assume that Z-boson is emitted in the central region of rapidity so qs qs Q2. 10We denote the weak coupling constant by e=sW and reserve the notation \g" for QCD coupling constant. where (cW cos W , sW sin W )10 J = e where Z 1 d2x? ei(q;x)? W ( q; q; x?); dx dx e i qx i qx hpA; pBjJ (x ; x ; x?)J (0)jpA; pBi; After integration over central elds in the tree approximation we obtain dx dx e i qx i qx hpAjhpBjJ (x ; x ; x?)J (0)jpAijpBi; and similarly for JB and JBA. Hereafter we use notation of au;c = (1 83 s2W ) or ad;s = (1 43 s2W ) depending on quark's avor. where The quark elds are given by a series in the parameter ms2? , see eqs. (3.27) and (A.2), can be any of u; d; s or c quarks.11 Accordingly, the currents (4.4) can be expressed (a 5) where a is one as a series in this parameter, e.g. (0) JAB (1) JAB = = e e The leading power contribution comes only from product JAB(x)JBA (0) (or JBA(x)JAB (0)), while power corrections may come from other terms like JA (x)JB (0). We will consider all terms in turn. Leading contribution and power corrections from JAB(x)JBA (0) terms Power expansion of JAB(x)JBA (0) reads (4.6) q 2 In appendix C.4 we demonstrate that terms q 2 q?s which are much smaller than (1) lead to power corrections q 2 Q?2 if Z-boson is emitted in the central region of rapidity. Note that since we want to calculate the leading power corrections, hereafter we substitute Q2 with Q2. In the limit s Q2 q2 this change of variables can only lead to errors of the order of subleading power terms. ? (A0)(0), they can be decomposed using First, let us consider the leading power term coming from the rst term in the r.h.s. of this equation. 11As we mentioned, we will need only rst two terms of the expansion given by eqs. (3.27) and (A.2). 4.2 JB(0A) (x)JA(0B) (0). Using Fierz transformation As we mentioned, the leading-power term comes from JA(0B) (x)JB(0A) (0) and ( A = [a where is implied). with au;c = (1 83 s2W ) and ad;s = (1 43 s2W ) one obtains 16s2W c2W hpAjhpBjJA(0B) (x)JB(0A) (0) + (x $ 0)jpAijpBi = o n ^u(0)jAi; h Bu(x) Bu(0)i hBj ^u(x) ^u(0)jBi and similarly for other matrix elements (summation over color and Lorentz indices As usual, after integration over background elds A and B we promote A, B, B to operators A^, ^. A subtle point is that our operators are not under T-product 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 elds these expressions read (see eq. (3.2)) hAj ^f (x) hBj ^f (x) ^f (0)jAi = hAj ^f (x ; x?) [x ; ^f (0)jBi = hBj ^f (x ; x?) [x ; 1 ]x[x?; 0?] 1 [ 1 ; 0 ]0 ^f (0)jAi; 1 ]x[x?; 0?] 1 [ 1 ; 0 ]0 ^f (0)jBi (4.11) and similarly for hAj ^f (0) ^f (x)jAi and hBj ^f (0) ^f (x)jBi. 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] e 2 (4.9) (4.10) A and All other terms in the product of eqs. (4.13) and (4.15) give higher power contributions q 2 s? W lt(q) (but not 2 q Q?2 W lt(q))12 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. Parametrization of leading matrix elements m2 s? in comparison to W lt(q) so it Let us rst consider matrix elements of operators without 5. The standard parametrization of quark TMDs reads dx d2x? e i x +i(k;x)? hAj ^f (x ; x?) dx d2x? ei x i(k;x)? hAj ^f (x ; x?) s 2m2N f f ( ; k2 ); 3 ? dx d2x? ei x i(k;x)? hAj ^f (x ; x?) ^f (0)jAi = mN ef ( ; k2 ) ? for quark distributions in the projectile and for the antiquark distributions.13 $ 12The trivial but important point is that any f (x; k?) may have only logarithmic dependence on Bjorken x but not the power dependence 1 . Indeed, at small x the cuto x of corresponding longitudinal integrals comes from the rapidity cuto a, 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 s ln a (or, in some cases, s ln2 a, see e.g. ref. [33]). Also, a more technical version of this argument was presented on page 12. 13In an arbitrary gauge, there are gauge links to 1 as displayed in eq. (4.11). The corresponding matrix elements for the target are obtained by trivial replacements dx d2x? e i x +i(k;x)? hBj ^f (x ; x?) ^f (0)jBi = p2 f1f ( ; k?2) + k f f ( ; k?2) + p1 ? ? s 2m2N f f ( ; k2 ); 3 ? dx d2x? e i x +i(k;x)? hBj ^f (x ; x?) ^f (0)jBi = mN ef ( ; k2 ); ? dx d2x? ei x i(k;x)? hBj ^f (x ; x?) ^f (0)jBi = s 2m2N f f ( ; k2 ); 3 ? dx d2x? ei x i(k;x)? hBj ^f (x ; x?) ^f (0)jBi = mN ef ( ; k2 ): ? (4.14) (4.15) (4.16) Matrix elements of operators with 5 are parametrized as follows: 1 Z 16 3 1 Z 16 3 The corresponding matrix elements for the target are obtained by trivial replacements Finally, for future use we present the parametrization of time-odd TMDs 1 Z 16 3 1 Z 16 3 = = 1 mN (k p ? 1 + 2mN (k p ? 2 1 mN (k p ? 1 2mN (k p ? 2 s s dx d2x? e i x +i(k;x)? hAj ^f (x ; x?) $ )h1?f ( ; k?2) + s 2mN (p1 p2 $ )hf ( ; k?2) dx d2x? ei x i(k;x)? hAj ^f (x ; x?) $ $ $ )h1?f ( ; k?2) )h3?f ( ; k?2) s 2mN (p1 p2 $ )hf ( ; k?2) and similarly for the target with usual replacements p1 $ p2, x $ . Note that the coe cients in front of f3, gf?, h and h3? in eqs. (4.13), (4.15), (4.17), and (4.18) contain an extra 1s since p2 enters only through the direction of gauge link so the result should not depend on rescaling p2 ! not contribute to W (q) in our approximation. p2. For this reason, these functions do 4.3 Power corrections from JAB(x)JBA (0) terms The terms in eq. (4.7) proportional to elds are First, as we demonstrate in appendix C.1, the terms in the second, third, and fourth lines lead to negligible power corrections fth and sixth lines. q?s , so we are left with contribution of the m A (x) (a 1 + a 2 = = + (1 1 + a 2 2 a 2 a 5) 1mB(x) n B(0) (a m A (x) n 2A(0) n B(0) Next, separating color-singlet contributions hA; Bj( A (Bj )nk k m A)( n B(Ai) ml Bl)jA; Bi = hA; Bj( A (Ai) m ml k A)( B(Bj )nk Bl)jA; Bi n = 1 Nc hAj( AmAiml Al)jAihBj( BnBjnk Bk)jBi Fifth line in eq. (4.19): the leading term in Let us start with the term mation (4.8) we obtain 1B(x) 2A(0) . Performing Fierz transfor1+a2 2 +(1 a2) g = 2Nc 1 + a 2 + a 2 s2 s2 1 2a s2 (1 (1 1B(x) 2A(0) B(0) j 1 j 1 2 j 1 A(x)Ak(x) ip=2 A(0) B(0)Bj(0) ip= A(x)Ak(x)p= A(0) B(0)Bj(0)p= a A(x)Ak(x) i 5p=2 A(0) B(0)Bj(0) ip= 1 1 1 k 1 k 1 k 1 B(x) +( i B(x) ( j B(x) +( i 5 i k $ i $ i 5 j 5 $ i Using equations (B.3), (B.4), and (B.8) from appendix B we can rewrite eq. (4.22) as A(x) (a 5) 1B(x) B(0) (a 5) 2A(0) A(x)p= [Ai(x) 2 i 5A~i(x)] 1 A(0) B(0)p= [Bi(0) 1 i 5B~i(0)] 1 B(x) A(x)Ak(x)p=2 j A(0) 1 B(0) hBj (0)p= 1 k 2 A(x)p= [ 5Ai(x) B(0)p= [Bi(0) 1 i 5B~i(0)] B(x) + O j $ k + gjkBi(0)p=1 i i 1 B(x) iA~i(x)] A(0) 1 1 m8 s ? : (4.20) m8 ? s : (4.21) (4.22) i k 5) 5) i 5) : (4.23) For forward matrix elements we get Z Z = = hAj ^(x ; x?)p=2[A^i(x ; x?) i 5A~^i(x ; x?)] 1 ^(0)jAi s k : 1+a2 s2 hAj ^(x ; x?)p=2(A^i i 5A~^i)(x ; x?) ^(0)jAi s2 hAj ^(x ; x?)A^j (x ; x?)p=2 j ^(0)jAi and similarly for other Lorentz structures in eq. (4.23). The corresponding contribution of HJEP05(218) the r.h.s. of eq. (4.23) to W ( q; q; x?) takes the form14 (4.24) (4.25) (4.26) (4.27) (4.28) Note that for unpolarized hadrons hBj ^(0)(A^j (0)p=1 is easy to see that the last line of eq. (4.23) j $ k) ^(x ; x?)jBi = 0. Also, it 2a s2 hAj ^(x)p=2[A^i(x) ^ i 5A~i(x)] ^(0)jAihBj ^(0)p=1( 5A^i(0) iA~^i(0)] ^(x)jBi gives zero contribution. Indeed, let us consider the rst term in the r.h.s. of this equation. hAj ^(x)p=2[A^i(x) hBj ^(0)p=1( 5A^i(0) ^ i 5A~i(x)] ^(0)jAi iA~^i(0)] ^(x)jBi xi; ij xj ; 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 fth line in eq. (4.19) we get 2A(x) (a 5) B(x) 1B(0) (a 5) A(0) Since Ncg 2 = 1+a2 s2 + 1 a 2 s2 2a s2 redundant. A 1 A 1 A 1 B B 1 1 (x)p=2[Ai(0)+i 5A~i(0)] A(0) B 1 (0)p=1[Bi(x)+i 5B~i(x)] B(x) (x)Ak(0)p=2 j A(0) (0)[Bj (x)p=1 k j $ k +gjkBi(x)p=1 i] B(x) (x)p=2[ 5Ai(0)+iA~i(0)] A(0) (0)p=1[Bi(x)+i 5B~i(x)] B(x) +O m8 s ? : 14After specifying the projectile and target matrix elements the \A" and \B" labels of the elds become g Z 8 3s g Z 8 3s + + g Z 8 3s + Z x g Z 8 3s + Z 0 1 1 example: hAj ( ! s 2 Z x 1 s 2 F^ i(0) Z 0 1 s 2 F^ i(x ; x?) Z x 1 ) Z x 1 dx?dx e i x +i(k;x)? hAj ^f (x ; x?)A^i(x ; x?) ) k2 dx0 ^f (x0 ; x?) 2s F^ i(x ; x?) p=2 i ^f (0)jAi = i m?N htfw3( ; k?2) ih~tfw3( ; k?2) ; dx?dx e i x +i(k;x)? hAj ^f (0)A^i(0) k2 dx0 ^f (x0 ; 0?) 2s F^ i(0) p=2 i ^f (x ; x?)jAi = i m?N htfw3( ; k?2) ih~tfw3( ; k?2) and similarly for the target matrix elements. For completeness, let us present the structure of gauge links in an arbitrary gauge, for In this section we present parametrization of matrix elements from section 4.3.3. Similarly to eqs. (4.40){(4.42) we de ne dx0 ^f (x0 ; 0?) jAi = i ? htfw3( ; k?2) + ih~tfw3( ; k?2) ; k 2 mN dx0 ^f (x0 ; x?) jAi = i ? htfw3( ; k?2) + ih~tfw3( ; k?2) 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 ) ( ) ( k 2 mN (A0)(x) ^f (x ; x?)A^j (x ; x?) + dx0 ^f (x0 ; x?) 2s F^ j (x ; x?) p=2 i ^f (0)jAi ) dx0 hAj n ^f (x ; x?)[x ; x0 ]xF^ j (x0 ; x?)[x0 ; + ^f (x0 ; x?)[x0 ; x ]xF^ j (x ; x?)[x ; o 1]x [x?; 0?] 1 [ 1 ; 0 ]0? p=2 i ^f (0)jAi: 4.4 Power corrections from JA (x)JB (0) terms Power corrections of the second type come from the terms A(x) + + (A1)(x) (A0)(x) In appendix C.4, we will demonstrate that terms (1) are small in our kinematical region Q2 Terms 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. Using eq. (3.27) and separating color-singlet matrix elements, we rewrite the sixth line in eq. (4.56) as A(x) B(0) 1B(0) +[ A(x) 2A(x) 1B(0) (4.57) = = + + g 2 where we used eqs. (B.4) and (B.5). For the forward matrix elements dx e i qx hAj ^ 1 (x ; x?)p=2A^i(0) ^(x ; x?)jAi A(0) 1 Z Z Z Z = = = = q 1 1 1 1 dx0 hAj ^(x0 ; x?) s 2p=2 F^ i(0) ^(x ; x?)jAi; dx e i qx hAj ^(x ; x?)p=2A^i(0) 1 ^(x ; x?)jAi dx e i qx hBj ^(0)p=1A^i(x ; x?) 1 ^(0)jBi s 2p=2 F^ i(0) ^(x0 ; x?)jAi; s dx e i qx hBj ^ 1 s 2p=1 F^ i(x ; x?) ^(0)jBi: (4.58) The corresponding contribution to W ( q; q; x?) takes the form 2p=1 F^ i(x ; x?) i 5F~^ i(x ; x?) ^(x0 ; 0?)jBi (Z x 1 dx0 Z 0 1 s +hAj ^(x ; x?) s hBj ^(x0 ; 0?) s 2p=2 F^ i(0)+i 5F~^ i(0) ^(x ; x?)jAi 2p=2 F^ i(0) i 5F~^ i(0) ^(x0 ; x?)jAi 2p=1 F^ i(x ; x?)+i 5F~^ i(x ; x?) ^(0)jBi +x $ 0 Similarly, for the seventh line in eq. (4.56) using eqs. (3.27) and (B.6) one obtains (4.59) ) 1+O : m2 ? s A(x) 1B(0) A 1 A(x)p=2 hAi(0) i 5A~i(0)i 1 B(0)p=1 hBi(x) i 5B~i(x)i 1 2a A 1 (x)p=2 hAi(0)+i 5A~i(0)i A(x) B 1 (0)p=1 h 5Bi(x)+iB~i(x)i B(0) A(x)p=2 hAi(0) i 5A~i(0)i 1 B(0)p=1 h 5Bi(x) iB~i(x)i 1 B (0) B (0) Using eq. (4.58) one obtains the contribution to W ( q; q; x?) in the form g2e2(a2 + 1) 8(2 )4s2W c2W (Nc2 1)Q2s2 Z (Z x 1 dx0 Z 0 1 2p=2 F^ i(0) + i 5F~^ i(0) ^(x ; x?)jAi hBj ^(x0 ; 0?) s + hAj ^(x ; x?) s 2p=1 F^ i(x ; x?) + i 5F~^ i(x ; x?) ^(0)jBi 2p=2 F^ i(0) i 5F~^ i(0) ^(x0 ; x?)jAi s hBj ^(0) 2p=1 F^ i(x ; x?) i 5F~^ i(x ; x?) ^(x0 ; 0?)jBi + x $ 0 : (4.61) ) Here we used the fact that the last term in eq. (4.60) 2a 2a hAj ^(x0 ; x?) 2p=2 F^ i(0) + i 5F^~ i(0) ^(x ; x?)jAi hBj ^(x0 ; 0?) + hAj ^(x ; x?) 2p=1 s 2p=1 s 2p=2 F^ i(0) 5F^ i(x ; x?) + iF~^ i(x ; x?) ^(0)jBi i 5F^~ i(0) ^(x0 ; x?)jAi 5F^ i(x ; x?) iF~^ i(x ; x?) ^(x0 ; 0?)jBi (4.62) (4.63) (4.64) (4.65) gives no contribution since hAj ^(x0 ; x?) hBj ^(x0 ; 0?) 2p=1 s 2p=2 F^ i(0) 5F^ i(x ; x?) i 5F^~ i(0) ^(x ; x?)jAi iF~^ i(x ; x?) ^(0)jBi xi; ij x j same as in eq. (4.27). of the 6th and 7th lines in eq. (4.56) in the form Next, using parametrizations (4.66) from the next section we obtain the contribution W 6+7th( q; q; q?) = e 2 8s2W c2W (Nc2 1)Q2 Z d2k? (k; q k)?hn2(1+a2u) j1twu3( q; k?)j2twu3( q; q? k?) ~j1twu3( q; k?)~j2twu3( q; q? k ) ? +(1 a2u) j1twu3( q; k?)j1twu3( q; q? k?)+~j1twu3( q; k?)~j1twu3( q; q? k ) ? +j2twu3( q; k?)j2twu3( q; q? k?)+~j2twu3( q; k?)~j2twu3( q; q? k?) + q $ q o +nu $ co+nu $ do+nu $ soi 1+O m2 ? s ; where q $ q 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 d2x?dx e i x +i(k;x)? Z x 2p=2 F^ i(0) + i 5F^~ i(0) ^(x ; x?)jAi g Z 8 3s g Z 8 3s = ki j1tw3( ; k?2) + i~j1tw3( ; k?2) ; d2x?dx e i x +i(k;x)? Z x = ki j2tw3( ; k?2) i~j2tw3( ; k?2) : 1 1 s 2p=2 [F^ i(x) i 5F^~ i(x)] ^(x0 ; 0?)jAi dx0 hAj ^(x0 ; 0?) 2p=2 F^ i(x) + i 5F^~ i(x) ^(0)jAi (4.66) HJEP05(218) g Z 8 3s g Z 8 3s F i $ F^ i. ^ gauge: = ki j1tw3( ; k?2) i~j1tw3( ; k?2) ; d2x?dx e i x +i(k;x)? Z 0 Target matrix elements are obtained by usual substitutions , p=2 $ p=1, x $ x , and For completeness let us present the explicit form of the gauge links in an arbitrary ^(x0 ; x?)F^ i(0) ^(x ; x?) ! Results and estimates Combining eqs. (4.12), (4.31), (4.51), and (4.64) we get the leading term and rst power corrections to W (q) in the kinematic region s Q2 ? q2 in the form e 2 d2k ? "( (1+a2u) 1 2 k2 (q k)2 k?)+2(a2u 1) ?m2N Q2 + 2k?2(q k)2 (Nc2 1)Q2m2N (a2u 1) hhtuw3( z; k?)htuw3( z; q? ? Nc2 1 Q2 Nc (k; q k)? 2(1+a2u) j1twu3( z; k?)j2twu3( z; q? ? h1?u( z; k?)h1?u( z; q? k ) ? k?)+h~tuw3( z; k?)h~tuw3( z; q? k ) ? i k?) ~j1twu3( z; k?)~j2twu3( z; q? k ) ? +(1 a2u) j1twu3( z; k?)j1twu3( z; q? k?)+j2twu3( z; k?)j2twu3( z; q? k ) ? +~j1twu3( z; k?)~j1twu3( z; q? k?)+~j2twu3( z; k?)~j2twu3( z; q? ) n 1+O o n o n o # k ) ? m2 ? s ; (5.1) where the momentum of the produced Z-boson is q = zp1 + zp2 + q . ? i (Nc2 1)Q2 2(1+a2u) @i?j1twu3( z; b?)@i?j2twu3( z; b?) @i?~j1twu3( z; b?)@i?~j2twu3( z; b?) +( z $ z) + u $ c + u $ d + u $ s 1+O n n # m2 ? ; (5.2) where f1u( z; b?) R d2k?e i(k;b)? f1u( z; k?) etc. Note that in the leading order in Nc power corrections are expressed in terms of leading power functions f1 and h1?. To estimate the order of magnitude of power corrections, one can assume that N1c is a good parameter and leave only rst term in the r.h.s. of eq. (5.1): 2 8s2W c2W Nc Z "( d2k ? n o n o + n u $ s 1) ? k2 (q o # 2 (k; q k)2 Q2 k)? ? h1?u( z; k?)h1?u( z; q? k ) ? 1 + O + O : m2 ? s 1 Nc For completeness, let us present our nal result in the transverse coordinate space 2 "( (1+a2u) f1u( z; b?)f1u( z; b?) 2m(a2N2u Q12) @?2h1?u( z; b?)@?2h1?u( z; b?) 2 2(a2u 1) (Nc2 1)Q2m2N Nc 2 tails of TMD's f1 approximate Next, eq. (5.3) is a tree-level formula and for an estimate we should specify the rapidity calculated only tree diagrams made of C- elds we have a = z and b = cuto s for f1's and h1?'s. As we discussed in section 2, the rapidity cuto for f1( z; k?2) is a and for f1( z; k?2) b, where a and b are rapidity bounds for central elds. Since we z in eq. (5.1).15 Next, power corrections become sizable at q2 ? m2N where we probe the perturbative ? k12 and h? 1 ? k14 [34]. So, as long as m2N Q2 we can (up to logarithmic corrections). Similarly, for the target we can use the estimate f1( z; k?2) ' f (k2z) ; h1?( z; k?2) ' m2N h( z) ; f1 ' f (k2z) ; h1? ' f1( z; k?2) ' f (k2z) ; h1?( z; k?2) ' m2N h( z) ; f1 ' f (k2z) ; h1? ' k4 ? k4 ? the double-log and/or single-log evolution of TMDs. 15In general, we should integrate over C- elds in the leading log approximation and match the logs to ? ? k 2 ? m2N h( z) k4 ? m2N h( z) k4 ? (5.3) (5.4) (5.5) 2 2 (k; q d k ? k2 (q ? k)2 ? 1)[hu( z)hu( z) + hu( z)hu( z)] mQ2N2 ) o + u $ c + u $ d + u $ s n HJEP05(218) e 2 k)2 ? 1 2 (k; q k)? Q2 hn(1 + a2u)[fu( z)fu( z) + fu( z)fu( z)]o + nu $ co + nu $ do + nu $ soi: Here we used the fact that due to the \positivity constraint" h1?(x; k?2) we can safely assume that the numbers f (x) and h(x) in eqs. (5.4) and (5.5) are of the same jmk?Nj f1?(x; k?2) [ 35 ], order of magnitude so the last term in the third line in eq. (5.6) Thus, the relative weight of the leading term and power correction is determined by the mQ2N2 can be neglected. factor 1 2 (k;q Q2k)? . The integrals over k by m2N and from above by Q2 so we get an estimate ? are logarithmic and should be cut from below Substituting this to eq. (5.1) we get the following estimate of the strength of power corrections for Z-boson production (5.6) o # (5.7) (6.1) hn(1 + a2u)[fu( z)fu( z) + fu( z)fu( z)]o + nu $ co + nu $ do + nu $ soi; where we assumed that the rst term is determined by the logarithmical region q 2 k 2 m2N and the second by Q2 rection reaches the level of few percent at q? 2 correction becomes bigger, but the validity of the approximation Qq2 ? 2 ? q2 . By this estimate, the power cor20 GeV. Of course, when q2 increases, the ? 1 worsens. Moreover, we have ignored all logarithmic (and double-log) evolutions which can signi cantly change the relative strength of power corrections. 6 Power corrections for Drell-Yan process In this section we consider contribution to the cross section of the Drell-Yan process which is determined by the hadronic tensor e 2 4s2W c2W Nc q 1 2 ln q 2 m?2N + 1 Q2 ln q2 Q2 W (pA; pB; q) = where J em = eu u Z 2 1 2 Z 16 4 s for active avors in our kinematical region. d x? ei(q;x)?W ( q; q; x?); dx dx e i qx i qx hpA; pBjJ em(x ; x ; x?)J em(0)jpA; pBi; u + ed d d + es s s + ec c c is the electromagnetic current 2 u Nc2 1 Nc k)2 ? ? ? j1twu3( q; k?)j1twu3( q; q? ~j1twu3( q; k?)~j1twu3( q; q? k ) ? k ) ? j2twu3( q; k?)j2twu3( q; q? k ) ? ~j2twu3( q; k?)~j2twu3( q; q? i k?) + ( q $ o u $ c + u $ d n u $ s # o : Let us present also the large-Nc estimate similar to eq. (5.6) 2u 1 2 ? k2 (q k)2 (k; q k)? [fu( q)fu( q)+fu( q)fu( q)] +2e2u[hu( q)hu( q)+hu( q)hu( q)] ' ? k2 (q k)2 1 2 n o n + u$c + u$d + u$s o n From the results of the present paper it is easy to extract power corrections to W .16 We replace constants au in eq. (5.1) by ef2 and remove factors \1" from expressions like a 2 1. One can formally set au ! 1 in and multiply by e2u. After that, we repeat the procedure for other avors and get (au 5), divide the result (5.1) by a2u, "( 2u 1 2 (k; q Q2 k)? f1u( q; k?)f1u( q; q? k ) ? ? h1?u( q; k?)h1?u( q; q? k ) ? htuw3( q; k?)htuw3( q; q? k?) + h~tuw3( q; k?)h~tuw3( q; q? k ) ? (k; q 1 Q2 k)? h2j1twu3( q; k?)j2twu3( q; q? k ) ? 2~j1twu3( q; k?)~j2twu3( q; q? k ) ? e2u[fu( q)fu( q)+fu( q)fu( q)]+(u $ c)+(u $ d)+(u $ s) : Obviously, the relative strength of leading-twist terms and power corrections is the same as for Z-boson production so from our nave estimate (5.7) one should expect power corrections of order of few percent starting from q 7 Conclusions and outlook 41 Q. In this paper we have calculated the higher-twist power correction to Z-boson production (and Drell-Yan process) in the kinematical region s Q2 q2 . Our back-of-the-envelope estimation of importance of power corrections tells that they reach a few percent of the ? leading-twist result at q? 4 ref. [21] by comparing leading-order ts to experimental data. 1 Q which surprisingly agrees with the same estimate made in 16The problem of calculating power corrections for W with non-convoluted indices is a separate issue which we hope to address in a di erent publication. (6.2) (6.3) o # 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 TMD from a = q to ~a = q 2 q 2 q?s = q Q?2 , target TMDs from b = q to ~b = q 2 q?s = 2 q Qq?2 , and match to the result of leading-log calculation of integral over central elds in the rapidity interval between ~a and ~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) W = 1 (2 )4 Z where the coe cient functions cm;n(q; x) are determined by integrals over C- elds and do not depend on the form of projectile or target. To nd these coe cients in the rst-loop order, we integrate over C- elds in eq. (2.11) with action SC = SQCD(C + A + B; C + A + B) SQCD(B; B) but without any rapidity restrictions on C- elds, and subtract matrix elements of the operators ^ A(zm) ^ B(zn0) in the background elds A, A and B, B multiplied by tree-level coe cients. Both the integrals over C- elds in eq. (2.11) and matrix elements of ^ A(zm) ^ B(zn0) will have rapidity divergencies which will be canceled in their sum so what remains are the logarithms (or double logs) of the ratio of kinematical variables (Q2 in our case) to the rapidity cuto s a of the operators ^ A(zm) and b of ^ B(zn0). Using the above logic we hope to avoid the problem of double-counting of elds which arises when integrals over longitudinal momenta of C- elds 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 ? di erent 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 arises from k 2 dependence of the rapidity evolution kernels, same as in the BK (and NLO BK [36]) equations. Also, since matrix elements of TMD operators with our rapidity cuto s are UV- nite [37, 38], the only UV divergencies in our approach are usual UV divergencies absorbed in the e ective QCD coupling. It is worth noting that recently the treatment of power corrections was performed operators will be available. in the framework of SCET theory (see e.g. refs. [39{41]). However, since our rapidity factorization is di erent 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 m12Z times matrix elements of quark-antiquark-gluon 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 di erent directions of Wilson lines: one to +1 and another to 1 [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. 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, O ce of Science, O ce of Nuclear Physics under contracts DE-AC02-98CH10886 and DE-AC05-06OR23177 and by U.S. DOE grant DE-FG0297ER41028. A Next-to-leading quark elds In this section we present the explicit expressions for the next-to-leading quark elds (1). It is convenient to separate these elds in \left" and \right" components: (1) = (11) + 1 p=1p=2 (1); (1) 2 p=2p=1 (1): The next-to-leading term in the expansion of the elds (3.19) has the form: (11A) = (21A) = (11B) = (21B) = (11A) = (11B) = (21B) = 1 1 1 1 gp=1 iBi A 2gp=2p=1 B gp=2 iAi B 2gp=1p=2 A 1 1 A B s2 2 p=2B Bj A + g i s2 p=1p=2 Pi 1 1 jBj A 1 2g s2 p=1p=2 (A[1])(0) A; g i s2 p=2p=1 Pi 1 jBj A 1 2g s2 p=2p=1 (A[1])(0) A g i j s2 p= 1 g i s2 p=2p=1 Pi 1 1 kBk A; 1 2g s2 p=2p=1 (A[1])(0) B; 1 jAj B 1 2g s2 p=1p=2 (A[1])(0) B 1 kAk B; s2 2 p=1A Aj B + A iBi s( 1 i B A B B iAi s( 1 i A 2gp=1p=2 s2 2g2 j 2gp=2p=1 s2 2g2 j ABjB p=2 s2 2 + A kBk g i s2 p=1p=2 Pi g i j s2 p= A jBjp=2p=1 A jBjp=1p=2 1 1 i Pi 1 1 i Pj 1 B jAjp=1p=2 B jAjp=2p=1 1 1 i Pi 1 i Pi 1 BAjA p=1 s2 2 + B kAk Pj Pi p= g j i 1 s2 ; i Pi 1 i Pi 1 g i 2g i s2 1 g j i i p=2 s2 ; 1 g i 2g i s2 s2 A(A[1])(0) 1 i p=2p=1; s2 A(A[1])(0) 1 i p=1p=2 1 g i 2g i s2 s2 B(A[1])(0) 1 i p=1p=2; 1 g i 2g i s2 s2 B(A[1])(0) 1 i p=2p=1 (A.1) (A.2) where Pi = i@i + gAi + gBi, see eq. (3.16). The expressions for should be read from right to left, e.g. A j Bj p=2p=1 1 i Pi 1 (x) s2 Z dz A(z) j Bj (z)p=2p=1(zj 1 i Pi 1 i jx) s2 (and 1 1 1 +1i as usual). It is easy to see that the power counting of quark elds has the form (cf eq. (3.29)): The gluon elds A(0) and A(0) were calculated in ref. [12]: (1) 1A (1) 1B (1) 2A (1) 2B m7=2 ? : HJEP05(218) and their power counting reads A(0) = A + (A[1])(0); A(0) = B + (A[1])(0); (A[1]a)(0) = (A[1]a)(0) = 2 AjabBjb gA gB m2 ; gAi gBi m : j Ak ip=2 Ak j p=2 i Ak j p=2 i Bj ip= Bj k j k B B 1 k = p=2(Ai p=1 i = p=2(Ai + iA~i 5) p=1 i = p=2(Ai iA~i 5) iA~i 5) iB~i 5); p=1(Bi p=1(Bi + iB~i 5); p=1(Bi + iB~i 5); j ip = 1 k = p=2(Ai + iA~i 5) p=1(Bi iB~i 5): 17We use conventions from Bjorken & Drell where 0123 = 1 and = g + g g 5. Also, with this convention ~ = i 1 2 (A.3) (A.4) (A.5) (A.6) (B.1) (B.2) (B.4) B Formulas with Dirac matrices In the gauge A = 0 the eld Ai can be represented as (see eq. (3.5)). It is convenient to de ne a \dual" eld 2 Z x Ai(x ; x?) = dx0 A i(x0 ; x?) 2 where F~ = 12 and therefore 2 Z x F 1 2 A~i(x ; x?) = dx0 A~ i(x0 ; x?); B~i(x ; x?) = dx0 B~ i(x0 ; x?); as usual.17 With this de nition, we get 2 Z x ij Aj = A~i; ij Bj = Bi ) Ai B~i = Ai i B ; Ai ~ Bi = Ai B (B.3) and hence Ak ip=2 j(a Ak jp=2 i(a Ak ip=2 j(a Ak jp=2 i(a we get 2 s (p^2 ip^1 k ip^2 + g 5) 5) 5) 5) Bj ip=1 k(a Bj kp=1 i(a Bj kp=1 i(a 5) 5) Next, using formula ~ ~ = (g g g g ) Akp=2 j Bjp=1 k Akp=2 j 5 Bjp=1 k + Akp=2 k j A p=1 k Bjp=2 k + 2 s Ai jk Bjp=1 k 5 Bjp=1 j Bi jk For appendix C we also need 5 Bj k p= i 1 ; Bj ip=1 k = (a2 + 1)p=2(Ai iA~i 5) p=1(Bi iB~i 5) 2ap=2(Ai iA~i 5) p=1( 5Bi iB~i); (B.5) (B.6) (B.7) (B.8) s j A 4 2 k A + g Bj Bk 2 s Ak ki 2Akp=2 j B j j i Bkp=1 j: p^1Bi + p^2 ip^1 5 Bi k + k ip^2 5 p=1Bj + p=2 5 p=1Bj + p=2 5 i j i j p^1 5Bi) = p^1(Bi + iB~i 5) i 5 p^1 5(Bi + iB~i 5); Bi k 5 = p^2 (Bi + iB~i 5) i + p^2 5 (Bi + iB~i 5) i 5; p=1 5Bj = p=2 p=1 5Bj = p=2 (Bi + iB~i 5)p=1 + 5p=2 p=1(Bi iB~i 5) + p=2 5 5(Bi + iB~i 5)p=1; p=1 5(Bi iB~i 5): 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. Let us consider for example the last term in the third line of eq. (4.19). The Fierz transformation (4.8) yields 1 + a2 1 + a2 2 2 Am(x) 2nA(0) Bn(0) a2) 2nA(0) Bn(0) Am(x) 2nA(0) Am(x) i 2nA(0) Bn(0) i Bm(x) + ( i Am(x) i 2nA(0) Bn(0) i 5 Bm(x) + ( i a2) Am(x) 2nA(0) Sorting out the color-singlet contributions18 we get (1 (1 $ 5 $ 5 5 5) 1 $ 5 i $ i 5 1 $ 5 i 5) 5) m8 5) + O ? : (C.1) (C.2) (C.3) (C.4) (C.5) hA; Bj( Am(Bj )nk Ak)( Bn Bm)jA; Bi = hA; Bj( Am Ak)( Bn(Bj )nk m B )jA; Bi 1 = Nc hAj( Al Al)jAihBj( BnBjnk Bk)jBi B(x) A(x) ip=2 A(x)p=2 A(x) i 5p=2 B(0) j 1 j 1 j 1 A(0) A(0) A(0) 2A(0) B(0)gBj (0) i B(x) + ( i $ i 5 i 5) B(0)gBj (0) B(x) (1 1 $ 5 5) B(0)gBj (0) i B(x) + ( i 5 $ i i 5) ; and therefore 2s 1 s s 2 Z where 1 +1i , see eq. (3.28). For the forward matrix elements we get dx?dx e i qx +i(k;x)? hAj ^(x ; x?)p=2 dx?dx e i qx +i(k;x)? hAj ^(x ; x?)p=2 1 q f ( q; k?2); where is any of -matrices with transverse indices. Next, consider dx?dx e i qx +i(k;x)? hBj ^(0)gB^i(0) ^(x ; x?)jBi 1 dx?dx e i qx +i(k;x)? Z 0 dx0 hBj ^(0)gF^ i(x0 ; 0?) ^(x ; x?)jBi m ki f tw3( q; k?2); 18Recall that after the promotion of background elds to operators we can still move those operators freely since all of them commute, see footnotes 2, 10 and 11. where f tw3( q; k?2) is some function of order one (by power counting (3.4) this matrix 1). Also, this function may have only logarithmical singularities in q as q ! 0 but not the power behavior 1 .19 The corresponding contribution to W (q) of q eq. (4.1) is proportional to d 2k?f i ( q; k?)kif tw3( q; k?2) ? W lt Q?2 W lt (C.6) so it can be neglected in comparison to the contributions that Z-boson is produced in a central range of rapidity so q 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 rst term in the fourth line and perform Fierz transformation (4.8): 2 q Q?2 W lt (recall that we assume 2 q?s ' Qp ? q 2 2 Qq?2 ). In a similar 2A(0) 1 + a2 2 2mA(x) 2nA(0) Bn(0) (1 1 $ 5 $ 5 $ 5 5 5) : 5) ) From the explicit form of 2A and 2A in eq. (3.27) we see that the last term in the r.h.s. vanishes while the rst two are small. Indeed, 2mA(x)p=1 2nA(0) Bn(0)p=2 Bm(x) jA; Bi k 1 A (x) i p=2 j 1 Al(0) Bn(0)g2B^ikm(x)B^jnl(0)p=2 Bm(x) jA; Bi s p (x) i =2 j 1 ^(0)jAihBj ^(0)g2A^j (0)A^i(x)p=2 ^(x)jBi O ? ; m8 s 2 hA; Bj = hA; Bj 2 sNc hAj 2 (C.8) (C.9) so the contribution to W is of order of ms24? W lt. C.2 Second to fth lines in eq. (4.56) Here we show that second to fth lines in eq. (4.56) either vanish or can be neglected. Obviously, matrix element of the operator in the second line vanishes. Formally, Z hAj ^(x ; x?) ^(x ; x?)jAi = ( q)hAj ^(0) Z ^(0)jBi = ( q)hBj ^(0) 19Large 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 eld has a factor tum [44, 45]. The integration over large x gives then restricted from above by a, such terms cannot give 1q (cf. refs [37, 38]). q + p 2 s 1 and since the integration over is and, non-formally, one hadron cannot produce Z-boson on his own. For a similar reason, projectile or target matrix element will be of eq. (C.9) type. In addition, from the explicit form 's in eq. (3.27) it is easy to see that the fth line in eq. (4.56) can be rewritten as follows: 2A(x) B(0) 1B(0) (C.10) (x) igBi(x) 2 2 kgBk(x) A(x) B(0) B(0) A(x) (0) igAi(0) 1 1 kgAk(0) 1 p = A(x) p = 5) kgBk(x) B 1 (0) igAi(0) s21 (a 1 B(0) +x $ 0 B(0)p=2(a 5) B(0) 5) kgAk(0) 1 B(0) +x $ 0: ms8? , so we are left with the From the power counting (3.4) we see that this term is contribution of the last two lines in eq. (4.56). C.3 Gluon power corrections from JA (x)JA (0) terms There is one more contribution which should be discussed and neglected: 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 rst term in the r.h.s. of this equation A(x) + 2A(x) A(0) + 2A(0) A(0) + 2A(0) 2A(x) A(x) A(x) 2A(0) A(0) 2A(0) + A(x) A(0) + A(x) 2A(x) 2A(x) A(0) 2A(0) A(0) 2A(0) ; (C.11) = 2 A 1 A 1 p = 1 B p = p = (x) igBi(x) s22 (a 5) A(x) JA (x)JA (0) = e 2 16s2W c2W e 2 16s2W c2W 2A(x) 2A(x) f:int: A(x) 2 2Nc(Nc2 2 2Nc(Nc2 1) hAj 1) hAj A(0) 2A(0) (x) i 2 p = ^(x) ^(0) p = s2 j 1 ^(0)jAihBjA^ai(x)A^aj (0)jBi p= p (x) i s2 k ^(x) ^(0) k =s2 j 1 ^(0)jAi hBjA^ai(x)A^aj (0)jBi + O m8 ? ; (C.12) (C.13) ? where f:int: denotes functional integration over A and B elds in eq. (2.8). either to 2xixj + x2 gij or to gij . Since the former structure does not contribute due to The matrix element hBjA^ai(x)A^aj (0)jBi for unpolarized hadrons can be proportional (2xixj + x2 gij ) ip=2 ? k kp=2 j = 0 A(x) 2 hA; Bj 2A(x) A(0) 2A(0) jA; Bi 2Nc(Nc2 1)s2 hAj ^ 1 (x)p=2(a 5) ^(x) ^(0)p=2(a 5) m8 ? : For the forward target matrix element one obtains Z dx e i qx hBjA^ia(x)A^ai(0)jBi (C.14) (C.15) s2 dx e i qx Z x 1 dx00 hBjF^ai(x0 ; x?)F^ai(x00; 0?)jBi dx e i qx hBjF^ai(x ; x?)F^ai(0)jBi = 8 2 sDg( q; x?); 1 (A0)(x) 1 + a2 2 + (1 a a 2 2 quark ones. q?s so they can be neglected.20 either m2 q?s or m2 m2 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 Q?2 (note that the projectile TMD in the r.h.s. of eq. (C.12) does not have 1 terms for the same reason as in eq. (C.22)). Similarly, all other terms in eq. (C.11) are C.4 Power corrections from A(x) A(x) A(0) A(x) elds give zero contribution since A(x) A(x) does not depends on x so R dx e i qx = 2 ( q). Let us consider now the last two lines in the power expansion (4.6) of JAB(x)JBA (0): (A0)(x) (B0)(x) (B0)(0) After Fierz transformation (4.8) the rst term in the above equation turns to (B0)(x) (B0)(0) (A1)(0) A A B m(0)(x) + ( A B 5 $ 5) m(0)(x) nA(1)(0) A Am(x)p=2 1nA(1)(0) n(0)(0) Bm(0)(x) B (1 1 $ 5 5) Am(x)p=2 1nA(1)(0) m8 s ? : 20It 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 gp=2p=1 iBi A + g i 1 2g s p=2 (A[1])(0) 1 + a2 ( " s2 2a ( " s2 Am(x) p=2p=1 i 1 Bi + p= i j 1 2 PiBj 1 1 + 2p=2 (A[1])(0) Am(x) p=2p=1 i 1 Bi + p= i j 1 2 PiBj 1 1 + 2p=2 (A[1])(0) Let us start with the rst term in parentheses in the second line of eq. (C.20). Using eq. (B.9) the corresponding matrix element can be rewritten as 2sNc 2sNc ghAj ^(x) i ^(0)jAihBj ^(0)p=1 ghAj ^(x) i 5 ^(0)jAihBj ^(0)p=1 1 (A^i + i 5A~i) (0) ^(x)jBi ^ 1 ( 5A^i + iA~i) (0) ^(x)jBi: nk p= ) : nk Ak(0) # Ak(0) # (C.19) (C.20) (C.21) (C.22) (C.23) Let us consider where we used 2i Z 1 + i Z Z 1 2i Z z A^k(z ; z?) = dz0 A^k(z0 ; z?) = dz0 (z z0) F^ k(z0 ; z?): Let us compare this matrix element to that of eq. (4.46): dx e i qx (0)A^i(0) ^(x ; x?)jBi dx e i qx dx0 ^(x0 ; 0?) 2s F^ i(0) ^(x ; x?)jBi: We see that 1z 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 's in the target which are of order 1). Consequently, power correction due to matrix element (C.22) can be neglected in our kinematic region since the matrix element (C.21) is ? rather than 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 gure out integrations that give 1 is very simple: take q ! 0 and check if there is an in nite integration of the type R x q 1 over x0 after one sets q = 0 in the relevant matrix element. Note that to get the terms Q12 = q1 qs we need to nd 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 q qs 1+a2 " s2 Am(x)p=2 i j Am(x)p=2 i j 1 PiBj 1 Bj + Am(x)p=2 i j 1 nk 1 Ak(0) nk 1 nk 1 Ak(0) Bn(0)p=1 Bm(x) Bn(0)p=1 Bm(x) Ai Bj nk 1 Ak(0) f:int: 1+a2 Ncs2 ghAj ^(x)p=2 i j 1 i@i ^(0)jAihBj ^(0)p=1 1 ^ Aj (0) ^(x)jBi 1+a2 Nc(Nc2 1)s2 g2hAj ^(x)p=2 i j A^i 1 ^(0)jAihBj ^(0)p=1 1 A^j (0) ^(x)jBi 1+a2 Ncs2 ghAj ^(x)p=2 i j 1 A(0)jAihBj ^(0)p=1 1 (i@iA^j +gA^iA^j )(0) ^(x)jBi: (C.24) Using eq. (B.9) we get 1+a2 " s2 am(x)p=2 i j Ncs2 ghAj ^(x)p=2 1 1 1+a2 Ncs2 ghAj ^(x)p=2 2(1+a2) Ncs3 ghAj ^(x)p=2 1 +(p=2 p=1 $ p=2 5 p=1 5): 1+a2 Nc(Nc2 1)s2 g2hAj ^(x)p=2A^i 1 ^(0)jAihBj ^(0)p=1 1 (A^i iA~^i 5)(0) 1 PiBj nk 1 ak(0) i@i ^(0)jAihBj ^(0)p=1 bn(0)p=1 bm(x) +(p=2 p=1 $ p=2 5 p=1 5) 1 (A^i iA~^i 5)(0) A(0)jAihBj ^(0)p=1 1 (i@iA^i +gA^iA^i)(0) A(0)jAihBj ^(0)p=1 5 1 F^~ (0) It is easy to see that projectile matrix elements lead to terms x , for example Z dx e i qx hAj ^(x)p=2A^i(0) 1 ^(0)jAi = dx ei qx Z x 1 dx0 hAj ^(0)p=2[F^ i(x ; x?) ^(x0 ; x?) + F^ i(x0 ; x?) ^(x ; x?)jAi: 1 after integration over (C.25) (C.26) On the other hand, the target matrix elements cannot give a 1 factor. For the rst 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 (i@iA^i +gA^iA^i)(0) ^(x ; x?)jBi = Z x0 1 2g F^ i(x00; 0?) Z x00 1 1 dx0 dx000F^ i(x000; 0?) ^(x ; x?)jBi 2i Z and Z dx e i qx hBj ^(0)p=1 5 1 F^~ (0) ^(x ; x?)jBi iZ dx e i qx Z 0 1 dx0 hBj ^(0)p=1 5F^~ (x0 ; 0?) ^(x ; x?)jBi: We see that at q = 0 there are no unrestricted integration over any longitudinal variable so the r.h.s. of these equations cannot give 1q factor and therefore the contribution to W (q) is m2 mQ2?2 . Finally, we should consider the third term in eq. (C.20) 1 + a2 s2 s2 g Am(x)p=2 (A[1]nl)(0) Al(0) Bn(0)p=1 Bm(x) Am(x)p=2 Am(x)p=2 1 1 Aj (0) Aj (0) nk kl Al(0) Al(0) # " # " Bn(0)p=1 Bn(0)p=1 1 Bj (0) 1 Bj (0) kl nk Bm(x) Bm(x) ; where we used eq. (A.5) for (A[1])(0). Taking projectile and target matrix elements and separating color-singlet contributions using eq. (4.44), we obtain 1) ^(0)jAihBj ^(0)p=1 1 A^j (0) (C.29) It has been demonstrated in eq. (C.22) that such matrix elements cannot give 1q and 1 so their contribution to W (q) is small in our kinematical region. 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I. Balitsky, A. Tarasov. Power corrections to TMD factorization for Z-boson production, Journal of High Energy Physics, 2018, 150, DOI: 10.1007/JHEP05(2018)150