Three-dimensional imaging of the nucleon and semi-inclusive high-energy reactions

Frontiers of Physics, Dec 2015

We present a short overview of studies of the transverse-momentum-dependent parton distribution functions of the nucleon. The aim of such studies is to provide three-dimensional imaging of the nucleon and a comprehensive description of semi-inclusive high-energy reactions. By summarizing what we have done in constructing the theoretical framework for inclusive deep inelastic lepton–nucleon scattering and one-dimensional imaging of the nucleon, we try to sketch out an outline of what we need to do to construct such a comprehensive theoretical framework for semi-inclusive processes in terms of three-dimensional gauge-invariant parton distributions. Next, we present an overview of what we have already achieved, with an emphasis on the theoretical framework for semi-inclusive reactions in leading-order perturbative quantum chromodynamics but with leading and higher twist contributions. We summarize in particular the results for the differential cross section and azimuthal spin asymmetries in terms of the gauge-invariant transverse-momentum-dependent parton distribution functions. We also briefly summarize the available experimental results on semi-inclusive reactions and the parameterizations of transverse-momentum-dependent parton distributions extracted from them and present an outlook for future studies.

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Three-dimensional imaging of the nucleon and semi-inclusive high-energy reactions

Front. Phys. Three-dimensional imaging of the nucleon and semi-inclusive high-energy reactions Kai-Bao Chen 0 Shu-Yi Wei 0 Zuo-Tang Liang 0 0 School of Physics & Key Laboratory of Particle Physics and Particle Irradiation (Ministry of Education), Shandong University , Jinan 250100 , China We present a short overview of studies of the transverse-momentum-dependent parton distribution functions of the nucleon. The aim of such studies is to provide three-dimensional imaging of the nucleon and a comprehensive description of semi-inclusive high-energy reactions. By summarizing what we have done in constructing the theoretical framework for inclusive deep inelastic lepton-nucleon scattering and one-dimensional imaging of the nucleon, we try to sketch out an outline of what we need to do to construct such a comprehensive theoretical framework for semi-inclusive processes in terms of three-dimensional gauge-invariant parton distributions. Next, we present an overview of what we have already achieved, with an emphasis on the theoretical framework for semi-inclusive reactions in leading-order perturbative quantum chromodynamics but with leading and higher twist contributions. We summarize in particular the results for the differential cross section and azimuthal spin asymmetries in terms of the gauge-invariant transverse-momentum-dependent parton distribution functions. We also briefly summarize the available experimental results on semi-inclusive reactions and the parameterizations of transverse-momentum-dependent parton distributions extracted from them and present an outlook for future studies. transverse momentum dependence; parton distribution function; fragmentation function; collinear expansion; higher twists - tum is also considered, involves not only direct exten- conference [ 3 ]. There are also many other reviews and sions of these distribution functions to include the trans- monographs (e.g., [ 4, 6, 7 ]). The study of higher orders verse momentum dependence, but also many other cor- in pQCD and higher twists seems to be rather difficult, relation functions that describe in particular the correla- and even the factorization properties are unclear [ 5 ]. In tions between the transverse momenta and spins, such as this article, we follow the same line as in the talk [ 2 ] the Sivers function, Boer–Mulders function, and pretze- but briefly summarize progress in the studies on QCD locity. They are generally called transverse-momentum- evolution and refer interested readers to those reviews. dependent (TMD) PDFs. Moreover, higher twist effects also become important and need to be considered consistently. The content of the studies is therefore much more 2 Inclusive DIS & one-dimensional imaging of abundant and more interesting. These TMD PDFs can the nucleon be studied in semi-inclusive reactions and are necessary Our studies on the structure of a fast-moving nucleon forTthheesdtuesdcyriopftitohnreoef-dsiumchenpsriooncaeslsiems.aging of the nucleon sWtaertreedcawll itthhaitn,culunsdivere tDhIeSosnuec-phhaostoen−e+xcNhan→geea−pp+roXx-. is in the rapidly developing stage, and it is not easy to imation, the differential cross section is given by the provide a comprehensive overview of all the different as- Lorentz contraction of the well-known leptonic tensor pects of the studies. Here, we choose to arrange the re- Lμν(l, l , λl) and the hadronic tensor Wμν (q, p, S), i.e., view as follows: First, we will briefly review what we have done in constructing the theoretical framework in one- d3l dimensional case with inclusive deep inelastic lepton– dσ = 2sαQe24m Lμν(l, l , λl)Wμν (q, p, S) 2El . (2.1) nucleon scattering (DIS). In this way, we hope that we can sketch out the main line of what we need to do in the The leptonic tensor is calculable and is given by three-dimensional case. Next, we will try to follow this Lμν(l, l , λl) = 2(lμlν + lνlμ − gμνl · l ) + i2λl μνρσlρqσ. line and summarize the progress already achieved along this direction and what we need to do next. This brief (2.2) review of the one-dimensional case will be presented in Information on the structure of the nucleon is contained Section 2. In Section 3, we will summarize the TMDs in the hadronic tensor, which is defined as defined via the quark–quark correlator. In Section 4, we twioilnl pforresceonntstarubcrtiienfgotvheervtiheeworoeftictahlefraavmaielawbolrekionffosremmai-- Wμν (q, p, S) = 21π X p, S |jμ(0)| X X |jν (0)| p, S inclusive processes. In Section 5, we will summarize the ×(2π)4δ4(p + q − pX ). (2.3) available experimental results and TMD parameterizations extracted from them. Finally, we will summarize Here, l and p denote the four-momenta of the lepton this review in Section 6. and nucleon, respectively, and primes indicate the final This overview article is an extended version of a ple- states; λ stands for the helicity, and S is the polarizanary talk at the 21st International Symposium on Spin tion vector of the nucleon. We use light-cone coordinates Physics (Spin2014) [ 2 ]. It is clear that the simplest and and define the light-cone unit vectors as n¯ = (1, 0, 0⊥), most basic picture is at the leading order (LO) in per- n = (0, 1, 0⊥), and n⊥ = (0, 0, n⊥), so a general fourtthurebreataivree QaClsoD t(wpQoCmDa)joarnddiartectthioenlesaidnintghteworisett.icHaelndcee-, vwehcetroer cAa±n b=e (dAec0o±mpAo3s)e/d√a2s, Aaμnd=AA⊥+ n¯=μ +(0A,0−, nAμ⊥+). AWμ⊥e, velopments toward a comprehensive description of semi- work in the center-of-mass frame of γ∗N and choose the inclusive processes: taking higher-order pQCD into ac- nucleon’s momentum as being in the z direction, so p count and considering higher twist contributions. These and S are decomposed as rcaocnytrbibuuttiaolnsos faorre ciomnpsiosrtteanncty.nTohteomnlayjoforradhvigahnecresatchcaut- pμ = p+ n¯μ + 2Mp+2 nμ, (2.4) ohbfuavttehleebaseedeintnwgmoaadndidreehcinitgiohrneecsr,eonir.ted.ye,reeaiirntshpehrQavaCetDathlosero lifenoaldtlohiwnegeLdtOwoinisnet Sμ = λ pM+ n¯μ + STμ − λ 2Mp+ nμ. (2.5) pQCD but leading and higher twists. The talk [ 2 ] concen- The Bjorken variable is defined as xB = Q2/(2p · q), q = trated mainly on the second direction. For higher-order −xBp+ n¯+ nQ2/(2xBp+); we also define y = (p · q)/(p · l). pQCD contributions involving the evolution of PDFs, an The theoretical framework for inclusive DIS has been overview talk was presented by Daniel Boer at the same constructed in the following steps. First, we studied the qμWμν (q, p, S) = 0, Wμν (q˜, p˜, −S˜) = W μν (q, p, S), Wμ∗ν (q, p, S) = Wνμ(q, p, S), where A˜ denotes the results of A after space reflection, i.e., A˜μ = Aμ. The general form of the hadronic tensor is given by the sum of a symmetric part and an antisymmetric part, Wμν (q, p, S) = Wμ(Sν)(q, p) + iWμ(Aν)(q, p, S), (2.9) where Wμ(Sν)(q, p) and Wμ(Aν)(q, p, S) are given by Wμ(Sν)(q, p) = 2(−gμν + qμq2qν )F1(x, Q2) 1 + xQ2 (qμ + 2xpμ)(qν + 2xpν )F2(x, Q2), Wμ(Aν)(q, p, S) = 2M εμνρσqρ × Sσg1(x, Q2) + (Sσ p · q S · q pσ)g2(x, Q2) , (2.11) − p · q respectively. We found that the hadronic tensor is determined by four independent structure functions, F1, F2, g1, and g2, where the first two describe the unpolarized case, and the latter two are needed for polarized cases. Our knowledge of one-dimensional imaging of the nucleon starts with the “intuitive parton model”, which is very nicely formulated, e.g., in Ref. [ 8 ]. Here, it was argued that, in a fast-moving frame, because of time dilation, quantum fluctuations such as vacuum polarizations can exist for quite a long time. In the infinite momentum frame, such fluctuations exist forever. In this case, a fast-moving nucleon can be viewed as a beam of free “partons”. The probability of the scattering of an electron with a nucleon is taken as the incoherent sum of that of the scattering with each individual parton, more precisely, as a convolution of the number density of partons in the nucleon with the probability of scattering with the parton, i.e., |M(eN → eX )|2 = dxfq(x)|Mˆ (eq → eq)|2, q (2.12) where fq(x) is the number density of partons of flavor q in the nucleon. In this way, we obtained the famous results [ 8 ] kinematics and obtained the general form of the hadronic tensor by applying the basic constraints from the general symmetry requirements such as Lorentz covariance, gauge invariance, parity conservation, and Hermiticity, e.g., (2.8) (2.10) Here, we would like to point out that, with this intuitive parton model, we are doing nothing else but the impulse approximation that we often use in describing a collision process, where we make the following approximations: • during the interaction of the electron with the par ton, interactions between the partons are neglected; • the electron interacts with only one single parton each time; • the scatterings of the electron with different partons are added incoherently. Although the physical picture of the intuitive model is very clear, and the model is elegant and practical, we are not satisfied with the formulation because it is partly qualitative or semiclassical; hence, it is not easy to control the accuracy. A proper formulation should be based on quantum field theory (QFT) and is obtained by starting with the Feynman diagram, as shown in Fig. 1(a). From this diagram, we obtain immediately that Wμ(0ν)(q, p, S) = 1 2π d4k (2π)4 Tr[Hˆ μ(0ν)(k, q)φˆ(0)(k, p, S)], where k is the four-momentum of the parton. Hˆ μ(0ν)(q, k) = γμ(/k + q/)γν(2π)δ+((k + q)2) is a calculable hard part. The matrix element φˆ(0)(k, p, S) = d4zeik·z p, S|ψ¯(0)ψ(z)|p, S is known as the quark–quark correlator and describes the structure of the nucleon. By taking the collinear approximation, i.e., taking k ≈ xp, and neglecting the powersuppressed contributions, i.e., the o(M/Q) terms, we obtain (2.16) (2.17) (2.18) Wμ(0ν)(q, p) ≈ (−gμν + ×fq(x). qμqν q2 ) + This is exactly the same result as that obtained from Eq. (2.12) on the basis of the intuitive parton model. At the same time, we obtain the QFT operator expression of fq(x), defined via the quark–quark correlator given by Eq. (2.18), as fq(x) = dz− eixp+z− p|ψ¯(0) γ+ 2π 2 ψ(z)|p . (2.20) By inserting the expanded expression for the field operator ψ(z) in terms of the plane wave and the creation and/or annihilation operators, we see clearly that fq(x) is indeed the number density of partons in the nucleon. However, from this expression, we also immediately see a severe problem; i.e., this expression is not (local) gauge invariant! We understand that the physical quantity has to be gauge invariant and therefore have to find a solu1 2π 1 2π Wμ(1ν,c)(q, p, S) = (d24πk)14 (d24πk)24 Tr[Hˆ μ(1ν,c)(k1, k2, q)φˆ(ρ1)(k1, k2, p, S)], φˆ(ρ1)(k1, k2, p, S) = d4zd4yeik1z+(k2−k1)y p, S|ψ¯(0)Aρ(y)ψ(z)|p, S , where c in the superscript represents different cuts (left or right) in the diagram. Similarly, corresponding to Fig. 1(c), we have tion for this. The gauge-invariant formulation is obtained by taking (2.19) into account the multiple gluon scattering shown by the diagram series in Figs. 1(a)–(c). This is clear because (local) gauge invariance implies the existence of a gauge interaction that needs to be taken into account. In this way, we obtain where Wμ(jν)(q, p, S) represents the contribution from the diagram with exchange of j gluon(s). They are all expressed as a trace of a calculable hard part and a matrix element depending on the structure of the nucleon. E.g., corresponding to Fig. 1(b), we have j = 1, and Wμ(1ν)(q, p, S) is given by Wμ(1ν)(q, p, S) = Wμ(1ν,c)(q, p, S), c=L,R (2.21) (2.22) (2.23) (2.24) (2.25) (2.26) (2.27) (2.28) (2.29) (2.31) Wμ(2ν)(q, p, S) = Wμ(2ν,c)(q, p, S) = c=L,M,R Wμ(2ν,c)(q, p, S), The matrix element is now a quark–j-gluon(s)–quark correlator. We also immediately see that none of these quark–j-gluon(s)–quark correlators is gauge invariant. To obtain the gauge-invariant form, we need to apply the collinear expansion proposed in [ 9–11 ], which is carried out in the following four steps. (1) Make Taylor expansions of all the hard parts at ki = xip, e.g., Hˆ μ(0ν)(k, q) = Hˆ μ(0ν)(x) + ∂Hˆ μ(0ν)(x) ∂kρ ωρρ kρ + 1 ∂2Hˆ μ(0ν)(x) 2 ∂kρ∂kσ ωρρ kρ ωσσ kσ + · · · , Hˆ μ(1ν,L)ρ(k1, k2, q) = Hˆ μ(1ν,L)ρ(x1, x2) + ∂Hˆ μ(1ν,L)ρ(x1, x2) ωσσ k1σ + ∂Hˆ μ(1ν,L)ρ(x1, x2) ωσσ k2σ + · · · , ∂k1σ ∂k2σ and so on, where ωρρ is a projection operator defined by ωρρ ≡ gρρ − n¯ρnρ . (2) Decompose the gluon field into longitudinal and transverse components, i.e., Aρ(y) = A+(y)n¯ρ + ωρρ Aρ (y). (2.30) (3) Apply the Ward identities, such as ∂Hˆ μ(0ν)(x) ∂kρ ∂Hˆ μ(1ν,L)ρ(x1, x2) ∂k1,σ = −Hˆ μ(1ν)ρ(x, x), = −Hˆ μ(2ν,L)ρσ(x1, x1, x2) −Hˆ μ(2ν,M)ρσ(x1, x1, x2), pρHˆ μ(1ν,L)ρ(x1, x2) = pρHˆ μ(2ν,L)ρσ(x1, x, x2) = , Hμ(1ν,L)σ(x1, x2), (2.32) (2.33) (2.34) pρHˆ μ(2ν,M)ρσ(x1, x, x2) = − x2 − x11 − i Hμ(1ν,L)σ(x1, x2) (4) Sum up all the terms with the same hard part, and we obtain W˜ μ(jν)(q, p, S), d4k (2π)4 Tr Hˆ μ(0ν)(x) Φˆ(0)(k, p, S) , Φˆ(0)(k, p, S) = d4yeiky p, S|ψ¯(0)L(0; y)ψ(y)|p, S , Φˆ(ρ1)(k1, k2, p, S) = Φˆ(ρ2σ)(k1, k2, k, p, S) d4yd4zeik2z+ik1(y−z) p, S|ψ¯(0)L(0; z)Dρ(z)L(z; y)ψ(y)|p, S , D(y) is the covariant derivative and is defined as Dρ(y) = −i∂ρ + gAρ(y). The factor L(0; y) is obtained during the summation of different contributions with the same hard part and is given by W˜ μ(1ν)(q, p, S) = W˜ μ(2ν)(q, p, S) = 1 2π 1 2π d4k1 d4k2 (2π)4 (2π)4 where Φˆ(j)’s are the gauge-invariant unintegrated quark–quark and quark–j-gluon(s)–quark correlators given by where P stands for the path-ordered integral. L(0; y) is nothing else but the well-known gauge link, which makes the quark–quark or quark–j-gluon(s)–quark correlator, and thus the PDFs defined using them, gauge invariant. In this way, we have constructed a theoretical framework for systematically calculating the contributions to the hadronic tensor at the leading order (LO) in pQCD but at leading as well as higher twists. The results are given in terms of the gauge-invariant parton distribution and correlation functions (generally referred to as PDFs). We emphasize the following two further points derived W˜ μ(1ν)(q, p, S) = W˜ μ(2ν)(q, p, S) = 1 2π 1 2π p+dx1p+dx2p+dx Tr[Hˆ μ(2ν,c)ρσ(x1, x2, x)ωρρ ωσσ Φˆ(ρ2σ) (x1, x2, x, p, S)], ∞ y− dξ− ∞ ξ− = 1 − ig dξ−A+(y+, ξ−, y⊥) + (−ig)2 dη−A+(y+, ξ−, y⊥)A+(y+, η−, y⊥) + · · · , (2.44) directly from these expressions. First, we note that after collinear expansion, the hard parts contained in the expressions for W˜ μ(jν)’s, such as those given by Eqs. (2.37)–(2.39), are only functions of the longitudinal component x. They are independent of the other components of the parton momentum k. We can integrate over these components of k and simplify them to where the matrix elements Φˆ ’s are given by (d24πk)14 (d24πk)24 (2dπ4k)4 δ(k1+ − x1p+)δ(k2+ − x2p+)δ(k+ − xp+) Φˆ(2)(k1, k2, k, p, S) dy− dy − dz− eix1p+y−+ixp+(y −−y−)+ix2p+(z−−y −) p, S|ψ¯(0)L(0; y−)Dρ(y−)L(y−; y −) 2π 2π 2π ×Dσ(y −)L(y −; z−)ψ(z−)|p, S . From these expressions, we see explicitly that only xi dependences of the quark–quark and/or quark–j-gluon–quark correlators are involved. This means that only one-dimensional imaging of the nucleon is relevant in inclusive DIS. Second, owing to the existence of the projection operator ωρρ , the hard parts can be further simplified a great deal. They are given by Hˆ μ(0ν)(x) = πhˆ(μ0ν)δ(x − xB ), π Hˆ μ(1ν,L)ρ(x1, x2)ωρρ = 2q · p ˆh(μ1ν)ρωρρ δ(x1 − xB), 2π Hˆ μ(2ν,L)ρσ(x1, x2, x)ωρρ ωσσ = (2q · p)2 n¯ρhˆ(μ1ν)σ + x2 − xB − iε Nˆμ(2ν)ρσ Hˆ μ(2ν,M)ρσ(x1, x2, x)ωρρ ωσσ = 2π hˆ(2)ρσωρρ ωσσ δ(x − xB), (2q · p)2 μν ωρρ ωσσ δ(x1 − xB ), where ˆh(μ0ν) = γμ/nγν /p+, hˆ(μ1ν)ρ = γμ/n¯γρ/nγν , hˆ(μ2ν)ρσ = p+γμ/n¯γρ/nγσ/n¯γν /2, and Nˆμ(2ν)ρσ = q−γμγρ/nγσγν are matrices independent of xi. We insert them into Eqs. (2.45)–(2.47) and obtain the simplified expressions for the hadronic tensor as , Tr hˆ(2)ρσωρρ ωσσ ϕˆρ(2σM)(xB ) , μν where, for explicitness, we omit p, S in the arguments of the correlators. These correlators are defined as p+dy− eixp+y− p, S|ψ¯(0)Dρ(0)L(0; y−)ψ(y−)|p, S , 2π p+dy− eixp+y− p, S|ψ¯(0)Dρ(0)L(0; y−)Dσ(y−)ψ(y−)|p, S , (2.60) 2π p+dy− p+dz− eixp+y−+i(x2−x)p+z− p, S|ψ¯(0)L(0; z−)Dρ(z−)Dσ(z−)L(z−; y−)ψ(y−)|p, S , (2.61) 2π 2π φˆ(2L)(x1) ≡ σ dxdx2 n¯ρ Φˆ(ρ2σ)(x1, x2, x, p, S) = p+dy− eixp+y− p, S|ψ¯(0)D−(0)Dσ(0)L(0; y−)ψ(y−)|p, S . (2.62) 2π Φ(S0)(x) + iγ5Φ(P0S) (x) + γαΦ(α0)(x) +γ5γα Φ˜(α0)(x) + iσαβ γ5Φ(T0α)β (x) . (2.63) The basic Lorentz covariants are constructed from pα, nα, Sα, and εαβρσ. We obtain the following general results: (2.64) (2.65) (2.66) (2.67) (2.68) Φ(S0)(x) = M e(x), Φ(P0S) (x) = λM eL(x), Φ(α0)(x) = p+n¯αf1(x) + M ε⊥αρSTρ fT (x) + Mp+2 nαf3(x), We see explicitly that all the relevant components of independent. This is because we have so far considered the quark–j-gluon–quark correlators depend only on one only the LO pQCD contributions, i.e., the tree diagrams. single parton momentum. This means that only quark–j- To go to higher orders of pQCD, we take the loop digluon–quark correlators that depend on one single parton agrams, gluon radiation, and so on into account. After momentum are relevant in inclusive DIS. proper handling of these contributions, we obtain the fac We emphasize that the results given by Eqs. (2.37)– torized form [ 6 ], in which the PDFs acquire the scale (Q) (2.39) and their simplified forms given by Eqs. (2.55)– dependence governed by the QCD evolution equations. (2.62), including the gauge links, are derived in the In practice, PDFs are parameterized and are given in the collinear expansion. They are just the sum of the con- PDF library (PDFlib). tributions from the diagram series shown in Fig. 1. This In summary, to study one-dimensional imaging of the formalism provides a basic theoretical framework for de- nucleon with inclusive DIS, we take the following steps. scribing inclusive DIS at LO pQCD but at leading and higher twist contributions in terms of gauge-invariant PDFs. The PDFs are defined in terms of QFT operators via these quark–quark correlators by expanding them in terms of γ matrices and basic Lorentz covariants. For example, for Φˆ(0)(x, p, S), we have • General symmetry analysis leads to the general form of the hadronic tensor and/or the cross section in terms of four independent structure functions. • The parton model without QCD interaction leads to LO in pQCD and leading twist results for the structure functions in terms of Q-independent PDFs without (local) gauge invariance. • The parton model with QCD multiple gluon scattering after collinear expansion leads to LO in pQCD and leading and higher twist contributions in terms of Q-independent but gauge-invariant PDFs. • The parton model with QCD multiple gluon scattering and loop diagram contributions after collinear approximation, regularization, and renormalization leads to leading and higher-order pQCD and leading twist contributions in factorized forms in terms of Q-evolved and gauge-invariant PDFs. In the following, we will follow these four steps and summarize what we have achieved in the threedimensional case. As in Ref. [ 2 ], we will focus mainly on the theoretical framework at LO pQCD but consistently take leading and higher twist contributions into account. Before that, we emphasize the following two historical developments that may be helpful in constructing the theoretical framework for the TMD case. First, as mentioned, the study of three-dimensional imaging of the nucleon was triggered by the experimental observation of single-spin left–right asymmetries (SSAs) in the inclusive hadron–hadron collision with a transversely polarized projectile or target. It was known that pQCD leads to negligibly small asymmetry for the hard part [ 12 ], but the observed asymmetry can be as large as 40% [ 13 ]. The hunt for such large asymmetries has lasted for decades, with the following milestones: • In 1991, Sivers introduced [ 14 ] the asymmetric quark distribution in a transversely polarized nucleon, which is now known as the Sivers function. Φ˜(α0)(x) = λp+ n¯αg1L(x) + M ST αgT (x) M 2 +λ p+ nαg3L(x), Φ(T0ρ)α(x) = p+ n¯[ρST α]h1T (x) − M ε⊥ραh(x) +λM n¯[ρnα]hL(x) + M 2 p+ n[ρST α]h3T (x), where ε⊥ρσ ≡ εαβρσn¯αnβ, and the anticommutation symbol A[ρBσ] ≡ AρBσ − AσBρ. The scalar functions f (x), g(x), and h(x) are the corresponding PDFs. There are a total of 12 such functions; 3 of them, i.e., f1(x), g1L(x), and h1T (x), contribute at leading twist and have clear probability interpretations, whereas 6 of them contribute at twist-3, and the other 3 contribute at twist4. We further note that in fact the three time-reversal odd terms eL(x), fT (x), and h(x) vanish in the onedimensional case. We keep them in Eqs. (2.65)–(2.68) for later comparison with fragmentation functions. We also see that the PDFs involved here are all scale • In 1993, Boros, Liang, and Meng proposed [ 15 ] a phenomenological model that provides an intuitive physical picture showing that the asymmetry arises from the orbital angular momenta of quarks and what they called the surface effect caused by the initial- or final-state interactions. • In 1993, Collins published [ 16 ] his proof that the Sivers function has to vanish because of parity and time-reversal invariance. • In 2002, Brodsky, Hwang, and Schmidt calculated [ 17 ] the SSA for semi-inclusive deep inelastic scattering (SIDIS) using an explicit example in which they took the orbital angular momentum of quarks and multiple gluon scattering into account. • In 2002, immediately after [ 17 ], Collins pointed out [ 18 ] that multiple gluon scattering is contained in the gauge link and that the conclusion of his proof in 1993 was incorrect because he did not consider the gauge link. He further showed that by taking the gauge link into account, the same proof leads to the conclusion that the Sivers function for DIS and that for the Drell–Yan process have opposite signs. Belitsky, Ji, and Yuan resolved [ 19, 20 ] the problem of defining the gauge link for a TMD parton density in the light-cone gauge where the gauge potential does not vanish asymptotically. The second historical development concerns the study of azimuthal asymmetry in SIDIS. Georgi and Politzer showed in 1977 [ 21 ] that final-state gluon radiation leads to azimuthal asymmetries and could be used as a “clean test” of pQCD. However, soon after, in 1978, Cahn showed [ 22 ] that similar asymmetries can also be obtained if one includes the intrinsic transverse momenta of partons. The latter (now called the Cahn effect), although power suppressed at higher twist, can be quite significant and cannot be neglected, as the values of the asymmetries themselves are usually not very large. The following two points are particularly valuable lessons that we learned from these historical developments: when studying TMDs, • it is important to take the gauge link into account; • higher twist effects can be important. Both of these points demand that, to describe SIDIS in terms of TMDs, we need the proper QFT formulation rather than the intuitive parton model. 3 TMDs defined via quark–quark correlator The TMD PDFs of quarks are defined via the TMD quark–quark correlator Φ(0)(x, k⊥; p, S) given by Eq. (2.40) (after integration over k−). A systematical study is given in Ref. [ 23 ], and a very comprehensive treatment can also be found in Ref. [ 24 ]. Here, we first expand it in terms of γ matrices and obtain a scalar, a pseudoscalar, a vector, an axial vector, and an antisymmetric and spacereflection odd tensor part, i.e., (3.1) (3.3) (3.4) +γ5γα Φ˜(α0)(x, k⊥; p, S) + iσαβ γ5Φ(T0α)β (x, k⊥; p, S) . The operator expressions of these coefficients are given by the traces of the quark–quark correlator with the corresponding Dirac matrices. For example, for the vector component, we have Φ(α0)(x, k⊥; p, S) = Tr γα Φˆ(0)(x, k⊥; p, S) = dz−d2z⊥ei(xp+z−−k⊥·z⊥) p, S|ψ¯(0)L(0; z) γ2α ψ(z)|p, S . (3.2) We then analyze the Lorentz structure of each part by expressing it in terms of possible “basic Lorentz covariants” and scalar functions. From Φˆ(0)(x, k⊥; p, S), we obtain the results as [ 23 ] Φ(S0)(x, k⊥; p, S) = M e(x, k⊥) − ε⊥ρσk⊥ρSTσ eT⊥(x, k⊥) , M Φ(P0S) (x, k⊥; p, S) = M λeL(x, k⊥) − k⊥M· ST eT (x, k⊥) , Φ(α0)(x, k⊥; p, S) = p+ n¯α f1(x, k⊥) − ε⊥ρσMk⊥ρSTσ f1⊥T (x, k⊥) + k⊥α f ⊥(x, k⊥) − ε⊥ρσMk⊥ρSTσ fT⊥1(x, k⊥) +ε⊥αρk⊥ρ λfL⊥(x, k⊥) − k⊥M· ST fT⊥2(x, k⊥) + Mp+2 nα f3(x, k⊥) − ε⊥ρσMk⊥ρSTσ f3⊥T (x, k⊥) , (3.5) Φ˜(α0)(x, k⊥; p, S) = p+ n¯α λg1L(x, k⊥) − k⊥M· ST g1⊥T (x, k⊥) + M ST αgT (x, k⊥) − ε⊥αβ k⊥βg⊥(x, k⊥) M 2 +k⊥α λgL⊥(x, k⊥) − k⊥M· ST gT⊥(x, k⊥) + p+ nα λg3L(x, k⊥) − k⊥M· ST g3T (x, k⊥) , Φ(T0ρ)α(x, k⊥; p, S) = p+ n¯[ρST α]h1T (x, k⊥)− p+ n¯[ρεM⊥α]βk⊥β h1⊥(x, k⊥)+ p+ n¯M[ρk⊥α] λh1⊥L(x, k⊥)− k⊥M· ST h1⊥T (x, k⊥) +ST [ρk⊥α]hT⊥(x, k⊥) − M ε⊥ραh(x, k⊥) + M n¯[ρnα] λhL(x, k⊥) − k⊥M· ST hT (x, k⊥) + These scalar functions are known as TMD PDFs. There ing basic Lorentz covariant is k⊥ dependent. are a total of 32 such TMD PDFs. Among them, 8 con- Higher twist TMD PDFs are also defined via quark– tribute at leading twist, and they all have clear probabil- j-gluon(s)–quark correlators such as those given by Eqs. ity interpretations such as the number density f1(x, k⊥), (2.59)–(2.62). Many of them, however, are not indepenhelicity distribution g1L(x, k⊥), transversity h1T (x, k⊥), dent, as they are related to those defined via the quark– Sivers function f1⊥T (x, k⊥), and Boer–Mulders function quark correlator through the QCD equation of motion, h1⊥(x, k⊥); 16 contribute at twist-3, and the other 8 con- γ · D(z)ψ(z) = 0. We can obtain relations such as tribute at twist-4. We emphasize that they are all scalar nα functions of x and k⊥; i.e., they depend on x and k2 . xΦ(⊥0ρ)(x, k⊥; p, S) = − p+ Reϕ(α1ρ)(x, k⊥; p, S) ⊥ LoIrfenwtez cinovteagrriaantetsoavreerodd2dki⊥n, kt⊥ervmasniisnh.wEhqisc.h(3t.h3e)–b(3a.s7ic) +ε⊥ρσImϕ˜(α1σ)(x, k⊥; p, S) , (3.8) jtuhset lreeadduincge ttowitsht,e ocnolryre3spoofndthineg8Esqusr.v(i2v.e6:4)t–h(e2.n6u8m).bAert x Φ˜(⊥0ρ)(x, k⊥; p, S) = − pn+α Reϕ˜(α1ρ)(x, k⊥; p, S) density f1(x), helicity distribution g1L(x), and transver- +ε⊥ρσImϕ(α1σ)(x, k⊥; p, S) . (3.9) sity h1T (x). We show the leading twist and twist-3 TMD PDFs in It is interesting to see that [ 35 ], although it is not generTables 1 and 2, respectively. In these tables, we show ally proved, all the twist-3 TMD PDFs that are defined also the results for L = 1, i.e., if we neglect multiple via the quark–gluon–quark correlator ϕ(ρ1) and involved gluon scattering and simply take a nucleon as an ideal in SIDIS are replaced by those defined via the quark– gas system consisting of quarks and antiquarks (see, e.g., quark correlator Φ(0). [ 24 ]). We also note that the conventions used here have We emphasize that fragmentation is just conjugate to the following systematics: f , g, and h represent unpo- parton distribution. A systematic study of the general larized, longitudinally polarized, and transversely polar- structure of the fragmentation function (FF) defined via ized quarks, respectively; the subscript L or T stands for the corresponding quark–quark correlator is presented in longitudinally or transversely polarized nucleons, respec- Ref. [ 26 ]. We should have one-to-one correspondence betively; the subscript 1 indicates leading twist, no number tween TMD PDFs and TMD FFs. E.g., corresponding indicates twist-3, and subscript 3 indicates twist-4; the to the quark–quark correlator Φ(0)(k, p, S) given by Eq. symbol ⊥ in the superscript denotes that the correspond- (2.40) and the expanded form in Eq. (3.1), we have For a spin-1/2 hadron, we have perfect one-to-one correspondence to those given by Eqs. (3.3)–(3.7) for parton distributions in the nucleon, i.e., d4ξe−ikF ξ 0|L†(0, ∞)ψ(0)|hX hX |ψ¯(ξ)L(ξ, ∞)|0 , × × × gT (x) × Comparing them with the results given by Eqs. (3.3)– (3.7), we see clearly the one-to-one correspondence between the FFs and PDFs. As an example, we show the eight leading twist components in Table 3. We do not show the results for L = 1 for FFs. This is because even if we neglect the multiple gluon scattering that leads to the gauge link, final-state interactions can still exist between h and X . In this case, time-reversal invariance does not lead to zero results for the T-odd amplitudes. For spin-1 hadrons, the polarization is described by ρ zΞ(S0)(z, kF ⊥; p, S) = M E(z, kF ⊥) + ε⊥ρσMkF ⊥STσ ET⊥(z, kF ⊥) , zΞ(P0S) (z, kF ⊥; p, S) = M λEL(z, kF ⊥) + kF ⊥M· ST ET (z, kF ⊥) , ρ zΞ(α0)(z, kF ⊥; p, S) = p+n¯α D1(z, kF ⊥) + ε⊥ρσMkF ⊥STσ D1⊥T (z, kF ⊥) + kF ⊥αD⊥(z, kF ⊥) + M ε⊥αρSTρ DT (z, kF ⊥), +ε⊥αρkFρ ⊥ λDL⊥(z, kF ⊥) + kF ⊥M· ST DT⊥(z, kF ⊥) + Mp+2 nα D3(z, kF ⊥) + ε⊥ρσMkFρ ⊥STσ D3⊥T (z, kF ⊥) , (3.14) zΞ˜(α0)(z, kF ⊥; p, S) = p+n¯α λG1L(z, kF ⊥) + kF ⊥M· ST G1⊥T (z, kF ⊥) + M ST αGT (z, kF ⊥) + ε⊥αβkFβ ⊥G⊥(z, kF ⊥) +kF ⊥α λGL⊥(z, kF ⊥) + kF ⊥M· ST GT⊥(z, kF ⊥) + Mp+2 nα λG3L(z, kF ⊥) + kF ⊥M· ST G3T (z, kF ⊥) , β zΞ(T0ρ)α(z, kF ⊥; p, S) = p+ n¯[ρST α]H1T (z, kF ⊥) + p+ n¯[ρεM⊥α]βkF ⊥ H1⊥(z, kF ⊥) + p+n¯[MρkF ⊥α] λH1⊥L(z, kF ⊥) + kF ⊥M· ST H1⊥T (z, kF ⊥) + ST [ρkF ⊥α]HT⊥(z, kF ⊥) + M ε⊥ραH(z, kF ⊥) M 2 +n¯[ρnα] M λHL(z, kF ⊥) + kF ⊥ · ST HT⊥(z, kF ⊥) + p+ + n[ρkF ⊥α] λH3⊥L(z, kF ⊥) + kF ⊥ · ST H3⊥T (z, kF ⊥) . M M β n[ρST α]H3T (z, kF ⊥) + n[ρε⊥Mα]βkF ⊥ H3⊥(z, kF ⊥) (3.10) (3.11) (3.12) (3.13) (3.15) (3.16) the polarization vector S and also the polarization tensor T (see, e.g., [ 25 ] and [ 26 ]). The tensor polarization part has five independent components. They are given by a Lorentz scalar SLL, a Lorentz vector SLμT = (0, SLxT , SLyT , 0), and a Lorentz tensor STμνT , which has two independent nonzero components, STxxT and STxyT , in the rest frame of the hadron. These polarization parameters can be related to the probabilities for particles in different spin states [ 25 ]. In this case, the TMD quark–quark correlator Ξˆ(0)(z, kF ⊥; p, S) is decomposed into a spin-independent part, a vector-polarizationdependent part, and a tensor-polarization-dependent part; i.e., Ξˆ(0)(z, kF ⊥; p, S) = ΞˆU(0)(z, kF ⊥; p, S) + ΞˆV (0)(z, kF ⊥; p, S) + ΞˆT (0)(z, kF ⊥; p, S). The spinindependent and vector-polarization-dependent part ΞˆU+V (0)(z, kF ⊥; p, S) takes exactly the same decomposition as that for the spin-1/2 hadron given by Eqs. (3.12)–(3.16). The tensor-polarization-dependent part is presented in Ref. [ 26 ] and is given by zΞTS (0)(z, kF ⊥; p, S) = M SLLELL(z, kF ⊥) + kF ⊥M· SLT EL⊥T (z, kF ⊥) + kF ⊥ · SMT T2 · kF ⊥ ET⊥T (z, kF ⊥) , zΞTP(S0)(z, kF ⊥; p, S) = M EL⊥T (z, kF ⊥) + kF SLT ⊥ M ⊥kF αkβ STαβT ET⊥T (z, kF ⊥) , M 2 zΞTα(0)(z, kF ⊥; p, S) = p+ n¯α SLLD1LL(z, kF ⊥) + kF ⊥M· SLT D1⊥LT (z, kF ⊥) + kF ⊥ · SMT T2 · kF ⊥ D1⊥T T (z, kF ⊥) +kF ⊥α SLLDLL(z, kF ⊥) + kF ⊥M· SLT DL⊥T (z, kF ⊥) + kF ⊥ · SMT T2 · kF ⊥ DT⊥T (z, kF ⊥) +M SLT αDLT (z, kF ⊥) + kFρ ⊥ST T ραDT⊥T (z, kF ⊥) + Mp+2 nα SLLD3LL(z, kF ⊥) + kF ⊥M· SLT D3⊥LT (z, kF ⊥) + kF ⊥ · SMT T2 · kF ⊥ D3⊥T T (z, kF ⊥) , zΞ˜Tα(0)(z, kF ⊥; p, S) = p+ n¯α εkF⊥SLT ⊥ M G1⊥LT (z, kF ⊥) + ε⊥kF⊥ρMkF2⊥σSTρσT G1⊥T T (z, kF ⊥) +ε⊥ραkFρ ⊥ SLLGL⊥L(z, kF ⊥) + kF ⊥M· SLT GL⊥T (z, kF ⊥) + kF ⊥ · SMT T2 · kF ⊥ GT⊥T (z, kF ⊥) +M ε⊥ραSLρT GLT (z, kF ⊥) + ε⊥αρkF ⊥σSTρσT GT⊥T (z, kF ⊥) M 2 + p+ nα εkF ⊥SLT ⊥ M G3⊥LT (z, kF ⊥) + ε⊥kF⊥ρMkF2⊥σSTρσT G3⊥T T (z, kF ⊥) , zΞTT (ρ0α)(z, kF ⊥; p, S) = p+ n¯[ρεM⊥α]σkFσ ⊥ SLLH1⊥LL(z, kF ⊥) + kF ⊥M· SLT H1⊥LT (z, kF ⊥) (3.17) (3.18) (3.19) (3.20) + kF ⊥ · ST T · kF ⊥ H1⊥T T (z, kF ⊥) + p+n¯[ρε⊥α]σSLσT H1LT (z, kF ⊥) + M 2 p+ n¯[ρε⊥α]σkF ⊥δSTσδT H1⊥T T (z, kF ⊥) M +M ε⊥ρα SLLHLL(z, kF ⊥) + kF ⊥M· SLT HL⊥T (z, kF ⊥) + kF ⊥ · SMT T2 · kF ⊥ HT⊥T (z, kF ⊥) +n¯[ρnα] εk⊥F⊥SLT HL⊥T (z, kF ⊥) + ε⊥kF⊥σMkF ⊥δSTσδT HT⊥T (z, kF ⊥) + For the tensor-polarization-dependent part, we have zΞTS (0)(z; p, S) = M SLLELL(z), zΞTP(S0)(z; p, S) = 0, M 2 zΞTα(0)(z; p, S) = p+n¯αSLLD1LL(z) + M SLT αDLT (z, kF ⊥) + p+ nαSLLD3LL(z), zΞ˜Tα(0)(z; p, S) = M ε⊥ραSLρT GLT (z), zΞTT (ρ0α)(z; p, S) = p+n¯[ρε⊥α]σSLσT H1LT (z) + M ε⊥ραSLLHLL(z) + M 2 p+ n[ρε⊥α]σSLσT H3LT (z). (3.22) (3.23) (3.24) (3.25) (3.26) (3.27) (3.28) (3.29) (3.30) (3.31) We see that, for the spin-independent and vectorpolarization-dependent parts, 12 components survive; 3 of them contribute at twist-2, 6 contribute at twist-3, and the other 3 contribute at twist-4. This is exactly the same as the result for the PDFs for nucleons, and we have exact one-to-one correspondence between the results given by Eqs. (3.22)–(3.26) and those given by Eqs. (2.64)– (2.68). For the tensor-polarization-dependent part, only 8 components survive; 2 of them contribute at twist-2, 4 contribute at twist-3, and the other 2 contribute at twist4. This corresponds to the situation for the PDFs for vector mesons. We should have a one-to-one correspondence between the tensor-polarization-dependent FFs for production of spin-1 hadrons and those PDFs for spin-1 hadrons. We also list the twist-2 components in Table 4. framework. (I) The general forms of hadronic tensors: For all three classes of processes, the general forms of hadronic 4 Accessing the TMDs in high-energy tensors have been studied and obtained. For SIDIS, this reactions has been discussed in Refs. [ 27–30 ], and it has been The TMDs can be studied in semi-inclusive high-energy shown that one needs 18 independent structure funcreactions such as SIDIS (e− + N → e− + h + X ), semi- tions for spinless h. A comprehensive study of Drell–Yan inclusive Drell–Yan processes (h + h → l+ + l− + X ), and processes was made in Ref. [31], and the number of insemi-inclusive hadron production in e+e−-annihilation dependent structure functions is 48 for hadrons with (e+ + e− → h1 + h2 + X ). For SIDIS, we study TMD spin 1/2. A study of e+e− annihilation was presented PDFs and TMD FFs, whereas for Drell–Yan processes in Ref. [ 32 ], and one needs 72 for spin-1/2 h1 and h2. and e+e− annihilation, we study TMD PDFs and TMD The results are systematically presented in those papers, FFs separately. We now follow the same steps as those and we will not repeat them here. However, we present, for inclusive DIS and summarize briefly what we have al- as an example, the general form of the differential cross ready done in constructing the corresponding theoretical section for e−N → e−hX . It is given by dσ dxdydzdψd2ph⊥ y2 FUU = 1 − ε y2 FUL = 1 − ε y2 FLU = 1 − ε y2 FLL = 1 − ε y2 FUT = 1 − ε y2 1 − ε + = αe2m xyQ2 1 + γ2 2x FUU + λlFLU + λFUL + λlλFLL + S⊥FUT + λlS⊥FLT , FUU,T + εFUU,L + 2ε(1 + ε)FUcoUs φh cos φh + εFUcoUs 2φh cos 2φh , 2ε(1 + ε)FUsiLn φh sin φh + εFUsiLn 2φh sin 2φh , 2ε(1 − ε)FLsiUn φh sin φh, 1 − ε2FLL + 2ε(1 − ε)FLcoLs φh cos φh , 2ε(1 + ε)FUsiTn φS sin φS + FUsiTn(,Tφh−φS) + εFUsiTn(,Lφh−φS) sin(φh − φS ) (4.1) (4.2) (4.3) (4.4) (4.5) (4.7) (4.9) 2ε(1 − ε)FLcoTs(2φh−φS) cos(2φh − φS ) , where ε = (1 − y − 41 γ2y2)/(1 − y + 12 y2 + 14 γ2y2), γ = 2M x/Q; the azimuthal angle ψ is that of the outgoing lepton l around the incident lepton beam with respect to an arbitrary fixed direction, which, for a transversely polarized target, is taken as the direction of ST . In the deep inelastic limit, neglecting power-suppressed terms, dψ = dφS . From Eqs. (4.1)–(4.7), we see explicitly that the 18 structure functions F are determined by the different azimuthal asymmetries in different polarization cases. These different azimuthal asymmetries are just defined by the average value of the corresponding trigonometric functions, for example: AsUinT(φh−φS) = sin(φh − φS ) UT = FUsiTn(,Tφh−φS) + εFUsiTn(,Lφh−φS) 2(FUU,T + εFUU,L) , (4.8) AsUinT(φh+φS) = sin(φh + φS ) UT = εFUsiTn(φh+φS) 2(FUU,T + εFUU,L) . We also emphasize that they are the general forms independent of the parton model and are valid at leading and higher twist and also leading and higher order in pQCD. (II) LO in pQCD and leading twist parton model results: These are the simplest parton model results and can be obtained easily. E.g., for SIDIS, dσ(0) dxdydzdφS d2ph⊥ = xαye2Qm2 F U(0U) + λlF L(0U) + λF U(0L) +λlλF L(0L) + S⊥F U(0T) + λlS⊥F L(0T) , (4.10) F U(0U) = A(y)C[f1D1] + 2(1 − y)C[w1h1⊥H1⊥] cos(2φh), +εFUsiTn(φh+φS) sin(φh + φS ) + 2ε(1 + ε)FUsiTn(2φh−φS) sin(2φh − φS ) + εFUsiTn(3φh−φS) sin(3φh − φS ) , (4.6) FLT = 2ε(1 − ε)FLcoTs φS cos φS + 1 − ε2F cos(φh−φS) cos(φh − φS ) LT The weights wi are given by w1(k⊥, kF ⊥) = −2(pˆhT · kF ⊥)(pˆhT · k⊥) + (k⊥ · kF ⊥) , M Mh w2(k⊥, kF ⊥) = − pˆhTM· k⊥ , w3(k⊥, kF ⊥) = − pˆhT · kF ⊥ , Mh w4(k⊥, kF ⊥) = − 2(pˆhT · k⊥)2(pˆhT · kF ⊥) , M 2Mh (pˆhT · k⊥)(k⊥ · kF ⊥) + k⊥2(pˆhT · kF ⊥) M 2Mh where pˆhT = phT /|phT | is the corresponding unit vector. The results can be obtained from those given in, e.g., [ 30 ] by neglecting all the power-suppressed contributions. From Eqs. (4.10)–(4.16), we see in particular that, at leading twist, there exist six nonzero azimuthal asymmetries in different polarization cases, i.e., cos 2φh (U0U) = (1 − y) C[w1h1⊥H1⊥] , A(y) C[f1D1] sin 2φh (U0L) = (1 − y) C[w1h1⊥LH1⊥] , A(y) C[f1D1] sin(φh − φS ) (U0T) = C[w2f1⊥T D1] , 2C[f1D1] sin(φh + φS ) (U0T) = (1 − y) C[w3h1T H1⊥] , A(y) C[f1D1] sin(3φh − φS ) (U0T) = (1 − y) C[w4h1⊥T H1⊥] , A(y) C[f1D1] (4.13) (4.14) (4.15) (4.16) (4.17) (4.27) (4.18) (4.19) (4.20) (4.21) (4.22) (4.23) (4.24) (4.25) (4.26) cos(φh − φS ) (L0T) = C(y) C[−w2g1T D1] . 2A(y) C[f1D1] They are determined by the Boer–Mulders function h⊥ 1 convoluted with the Collins function H1⊥, the wormgear (longi-transversity) h1⊥L convoluted with the Collins function H1⊥, the Sivers function f1⊥T convoluted with D1, the transversity h1T convoluted with the Collins function H1⊥, and the worm-gear (trans-helicity distribution) g1⊥T convoluted with the Collins function H1⊥. The azimuthal asymmetries AsUinT(φh∓φS) are due to the Sivers and Collins effects and are often referred to as Sivers asymmetry and Collins asymmetry, respectively. We emphasize that the results given by Eqs. (4.10)– (4.21) give a complete parton model result at LO in pQCD and leading twist. They can be used to extract the TMDs at this order. Any attempt to go beyond LO in pQCD or to consider higher twists needs to go beyond this expression. (III) LO in pQCD, leading and higher twist results: For semi-inclusive processes in which only one hadron is involved, either in the initial or the final state, it has been shown [ 33–37 ] that collinear expansion can be applied. Such processes include SIDIS [e− +N → e− +q(jet)+X ], and e+e−-annihilation [e+ + e− → h + q¯(jet) + X ]. By applying collinear expansion, we have constructed the theoretical framework for these processes by which leading as well as higher twist contributions can be calculated systematically to LO in pQCD. The complete results up to twist-3 are presented in Refs. [ 35–37 ]. For polarized e− + N → e− + q(jet) + X , the simplified expressions for the hadronic tensor are very similar to those for the inclusive DIS given by Eqs. (2.55)–(2.58), (4.11) (4.12) F L(0U) = 0, F L(0L) = C(y)C[g1LD1], F U(0L) = 2(1 − y)C[w1h1⊥LH1⊥] sin(2φh), F L(0T) = C(y)C[−w2g1T D1] cos(φh − φS ), F U(0T) = A(y)C[w2f1⊥T D1] sin(φh − φS ) + 2(1 − y)C[w3h1T H1⊥] sin(φh + φS ) + 2(1 − y)C[w4h1⊥T H1⊥] sin(3φh − φS ), where A(y) = 1 + (1 − y)2, and C(y) = y(2 − y). C[wif D] denotes the convolution of f and D weighted by wi, i.e., d2k⊥d2kF ⊥δ(2)(k⊥ − kF ⊥ − phT /z)wi(k⊥, kF ⊥, phT )f q(x, k⊥)Dq→hX (z, kF ⊥). W˜ μ(0ν,si)(q, p, S, k⊥) = Tr hˆ(μ0ν) Φˆ(0)(xB , k⊥) , W˜ μ(1ν,L,si)(q, p, S, k⊥) = Tr hˆ(μ1ν)ρωρρ ϕˆ(1,L)(xB , k⊥) , ρ W˜ μ(2ν,L,si)(q, p, S, k⊥) = W˜ μ(2ν,M,si)(q, p, S, k⊥) = , WUU (x, k⊥, φ) = A(y)fq(x, k⊥) − 2xQ|k⊥| B(y)fq⊥(x, k⊥) cos φ, WLU (x, k⊥, φ) = − 2xQ|k⊥| D(y)g⊥(x, k⊥) sin φ, WUT (x, k⊥, φ, φS ) = |k⊥| A(y)f1⊥T (x, k⊥) sin(φ − φS ) M 2xM k2 + B(y) 2M⊥2 fT⊥(x, k⊥) sin(2φ − φS ) − fT (x, k⊥) sin φS , Q WUL(x, k⊥, φ) = − 2xQ|k⊥| B(y)fL⊥(x, k⊥) sin φ, WLL(x, k⊥, φ) = C(y)g1L(x, k⊥) − 2xQ|k⊥| D(y)gL⊥(x, k⊥) cos φ, WLT (x, k⊥, φ, φS ) = |k⊥| C(y)g1⊥T (x, k⊥) cos(φ − φS ) M 2xM − Q k2 D(y) gT (x, k⊥) cos φS − 2M⊥2 gT⊥(x, k⊥) cos(2φ − φS ) , (4.30) (4.31) (4.32) (4.33) (4.34) (4.35) (4.36) (4.37) (4.38) (4.39) (4.40) (4.41) (4.42) (4.43) where B(y) = 2(2 − y)√1 − y, and D(y) = 2y√1 − y. For unpolarized e− + N → e− + q(jet) + X , the results up to twist-4 have also been obtained [ 34 ]: dσUU dxdyd2k⊥ = 2παe2meq 2 Q2y A(y)f1(x, k⊥) − 2B(y) |k⊥| xf ⊥(x, k⊥) cos φ Q 2 −4(1 − y) |kQ⊥2| x[ϕ(31)⊥(x, k⊥) − ϕ˜(31)⊥(x, k⊥)] cos 2φ +8(1 − y) 2x2M 2 Q2 f3(x, k⊥) − 2A(y) |kQ⊥2|2 x[ϕ(32,L)⊥(x, k⊥) − ϕ˜(32,L)⊥(x, k⊥)] . These results are expressed in terms of the gauge-invariant TMD PDFs or FFs and can be used as the basis for measuring these TMDs via the corresponding process at the LO in pQCD. We would like in particular to draw attention to the results for e+ + e− → h + q¯(jet) + X for h with different spins [ 37 ]. Here, for the hadronic tensor, we again obtain very similar formulae for this process; e.g., corresponding to Eqs. (4.28)–(4.30), we have W˜ μ(1ν,)L,si)(q, p, S, k⊥|e+e−) = − 4p1· q Tr hˆ(μ1ν)ρωρρ Ξˆ(ρ1)(zB, k⊥) , W˜ μ(2ν,L,si)(q, p, S, k⊥|e+e−) = W˜ μ(2ν,M,si)(q, p, S, k⊥|e+e−) = 2|pT | SLn(T0)(y, z, pT ) = − 3zM q T0q(y)D1(z, pT ) , q Pq(y)T0q(y)G1⊥LT (z, pT ) Complete twist-3 results for the differential cross sec- the semi-inclusive high-energy reactions mentioned tions, azimuthal asymmetries, and polarizations were ob- above in terms of QCD and the parton model, TMDs tained for hadrons with spin 0, 1/2, and 1 in Ref. [ 37 ]. are needed, and the factorization theorem has to involve For spin-1 hadrons in particular, we see that tensor po- the transverse momentum dependence. A TMD factorlarization is involved, even at the leading twist level; we ization theorem has been established at the leading twist have, for e+e− annihilation at the Z0 pole, for semi-inclusive processes [ 42–49 ]. TMD evolution theory is also developing very rapidly [ 50–63 ]. Boer [ 3 ] gave SL(0L)(y, z, pT ) = q T0q(y)D1LL(z, pT ) , (4.44) an overview at Spin2014, and an annual workshop series 2 q T0q(y)D1(z, pT ) dedicated to this topic was established in 2012. We refer interested readers to these talks and overviews. (4.45) 5 Available data and parameterizations q T0q(y)D1⊥LT (z, pT ) 2|pT | SLt(T0)(y, z, pT ) = − 3zM , (4.46) All three types of semi-inclusive reactions have been inq T0q(y)D1(z, pT ) vestigated experimentally. The results are summarized, STnnT(0)(y, z, pT ) = − 23|pMT 2|2 q qTT0q(0qy()yD)D1⊥T1T(y(,zp, TpT) ) , lea.grs.,kiinanadnRuomsbtoemryoafnp[l6en4,ar6y5]t.aHlkesrea,twSepwini2ll0j1u4stbbyriSetfloysummarize the main data available and then try to sort (4.47) out the available TMD parameterizations. STntT(0)(y, z, pT ) = 2|pT |2 q Pq(y)T0q(y)G1⊥T T (z, pT ) , For SIDIS, measurements have been made by the 3M 2 q T0q(y)D1(y, pT ) HERMES Collaboration [ 66–70 ] at DESY, the COM(4.48) PASS Collaboration [ 71–78 ] at CERN, CLAS [ 79, 80 ], and the Hall A Collaboration [ 81–84 ] at Jefferson Labowhere n and t denote the two transverse directions of the ratory (JLab). We list these SIDIS experiments in Table produced vector meson, normal to and inside the produc- 5 and briefly summarize the results as follows. tion plane, respectively. The coefficient T0q(y) = cq1ce1[(1− At DESY, the single-spin asymmetries for SIDIS were y)2 + y2] − cq3ce3[1 − 2y], where ce1 = (ceV )2 + (ceA)2, and first measured with a longitudinally polarized target by ce3 = 2ceV ceA; y in this reaction is defined as y ≡ l1+/k+. HERMES [66] for production of charged pions and then Pq(y) = T1q(y)/T0q(y) is the polarization of the quark measured for the first time with a transversely polarized produced by Z0 decay, and T1q(y) = −cq3ce1[(1 − y)2 + target in [ 67 ]. They found nonzero Sivers and Collins y2] + cq1ce3[1 − 2y]. This situation has been much less ex- asymmetries sin(φh − φS ) UT and sin(φh + φS ) UT . plored to date and is a worthwhile topic for many further Measurements were then conducted for π0 and kaons [68, studies. 69] and also for azimuthal asymmetries cos φh UU and The three types of semi-inclusive processes mentioned cos(2φh) UU in the unpolarized case [ 70 ]. above always involve two hadrons. How to apply collinear At CERN, COMPASS has measured the Sivers and expansion to such processes has not been demonstrated. Collins asymmetries in reactions with deuteron or proIt is unclear how one can systematically calculate lead- ton targets for production of charged hadrons, pions, and ing and higher twist contributions. Nevertheless, prac- kaons [ 71–78 ], and also cos φh UU and cos(2φh) UU in tical twist-3 calculations have been performed for these the unpolarized case [78]. processes [ 38–41 ] using the following steps: At JLab, CLAS has measured [ 79, 80 ] sin(2φh) UL (i) draw Feynman diagrams with multiple gluon scat- for pions with different charges and sin φh LU for π0. tering to the order of one gluon exchange, The Hall A Collaboration has measured [ 81–84 ] the (ii) insert the gauge link in the correlator wherever Collins and Sivers asymmetries for π± and K±, and needed to make it gauge invariant, and cos(φh − φs) LT for π± and sin(3φh − φs) UT . These (iii) carry out calculations to the order 1/Q. measurements are all summarized in Table 5. Although this method has not been proved, it is inter- In addition to the data for SIDIS, we now have meaesting to see that the results obtained this way reduce surements of the azimuthal asymmetries in e+e− → exactly to those obtained in the corresponding simplified π+π+X made by Belle [ 85–87 ] and the BaBar collaboracases where collinear expansion is applied if we take the tion [ 88 ], and also preliminary results from BES [ 89 ]. For corresponding fragmentation functions as δ functions. Drell–Yan processes, data are available on the azimuthal (IV) TMD factorization and evolution: To describe asymmetries in, e.g., reactions using pion beams [ 90–93 ] 101204-16 and pp or pD collisions [ 94, 95 ]. Table 5 Available measurements on azimuthal asymmetries in SIDIS Collaboration HERMES daAntltthoouggivhetphreedciasteacaornetrsotlilloffatrhferoTmMsDusffiicnievnotlvlyeda,bduinf-- ΔN fq(x, k⊥) = − 2|Mk⊥| f1⊥Tq(x, k⊥). (5.7) ferent sets of TMD parameterizations have already been There already exist different sets such as the Bochum extracted from them. We briefly sort them out as follows. [ 101–103 ], Torino [ 96, 104, 106 ], and Vogelsang–Yuan The first part concerns what people called the first- [ 105 ] fits. One thing seems to be clear: the Sivers funcphase parameterizations, i.e., TMD parameterizations tion is nonzero for protons, and it has different signs for without QCD evolution. Here, the following results in u and d quarks, as shown in Fig. 2. particular are available. We emphasize once more that (3) Transversity and Collins function: A simultaneous all the results, including the figures, are taken from Refs. extraction of the transversity and Collins function from [ 96–112 ]. Interested readers are referred to these refer- SIDIS data on Collins asymmetry obtained by the HERences for more details. MES [ 67–70 ] and COMPASS [ 71–77 ] collaborations and (1) Transverse momentum dependence: This is usu- from e+e− data obtained by Belle [ 85–87 ] has been carally taken as [ 96–100 ] a Gaussian in a factorized form ried out by the Torino group [ 97, 107 ]. A form similar to independent of the longitudinal variable z or x, e.g., that of the Sivers function has been taken, e.g., ons and kaons have been used for parameterization. The Sivers function is usually parameterized [ 96, 101–106 ] in the form of the number density fq(x, k⊥) multiplied by an x-dependent factor Nq(x) and a k⊥-dependent factor h(k⊥), i.e., ΔN fq(x, k⊥) = 2Nq(x)h(k⊥)fq(x, k⊥), (5.3) where Nq(x) is taken as a binomial function of x, Nq(x) = Nqxαq (1 − x)βq (αq + βq)αq+βq /αqαq βqβq , (5.4) and h(k⊥) is taken as a Gaussian, h(k⊥) = √2e(|k⊥|/M1)e−k⊥2/M12 . Here the Sivers function ΔN fq(x, k⊥) is defined via 1 fq/N↑ (x, k⊥) = fq/N (x, k⊥)+ 2 ΔN fq(x, k⊥)S ·(pˆ×kˆ⊥), which is related to the Sivers function f1⊥T (x, k⊥) defined in Eq. (3.5) by (5.5) (5.6) f1(x, k⊥) = f1(x)e−k⊥2/ k⊥2 /π k2 , ⊥ D1(z, kF ⊥) = D1(z)e−kF2 ⊥/ kF2⊥ /π kF2 ⊥ . (5.1) (5.2) The width has been fitted, and the form and flavor dependence and so on have been tested. The typical values okf F2t⊥he =fitt0ed.20wGidetVhs2 a[r9e6,].eR.go.,ugkh⊥2ly s=pea0k. 2in5g,GtehVi2s iasnda quite satisfactory fit. However, it has also been pointed out, e.g., in [ 99 ] for the TMD FF, that the Gaussian form seems to depend on the flavor and even on z, which means that it is only a zeroth-order approximation. (2) Sivers function: All the data available from HERMES [ 67–69 ], COMPASS [ 71–74, 76, 77 ], and JLab Hall A [ 81, 82, 84 ] on the Sivers asymmetries in SIDIS for piFig. 2 Example of the parameterizations of the Sivers functions for u and d flavors at Q2 = 2.4(GeV/c)2 by the Torino group. The figure is taken from [104]. NqT (x) = NqT xα(1 − x)β (α +ααββ)βα+β , NqC (z) = NqC zγ (1 − z)δ (γ +γγδδ)δγ+δ . h(phT ) = √2e |phT | e−p2hT /Mh2 , Mh and it has been determined that the Collins function is nonzero and has different signs, e.g., for u → π+ or d → π+, as shown in Fig. 3. Here, similar to the case for the Sivers function, the Collins function ΔN Dh/q(z, kF ⊥) is defined via Dh/q↑ (z, phT ) = Dq/N (z, phT ) 1 + 2 ΔN Dh/q(z, phT )sq(kˆq × pˆhT ), which is related to the Collins function H1⊥(z, phT ) defined in Eq. (3.16) by ΔN Dh/q(z, phT ) = 2|phT | H1⊥q(z, phT ). zMh (4) Boer–Mulders function: It was pointed out [ 111 ] that the HERMES and COMPASS data on cos 2φ asymmetry [ 70, 78 ] provide the first experimental evidence of the Boer–Mulders effect in SIDIS. Studies in this direction have been made in Refs. [ 110, 111 ] to extract the Boer–Mulders function from the SIDIS data [ 70, 78 ] and in Refs. [ 108, 109, 112 ] to extract it from the Drell–Yan data [ 90–95 ]. A fit to the first moments of the Boer–Mulders functions of the u and d quarks is shown in Fig. 4. The form is again similar to the Sivers function, being the Sivers function just multiplied by a (5.8) (5.9) (5.10) (5.11) (5.12) (5.13) (5.14) Fig. 4 First extractions of the Boer-Mulders function h1⊥u(x) and h1⊥d(x). This figure is taken from Ref. [ 111 ]. constant, e.g., h1⊥q(x, k⊥) = λqf1⊥Tq(x, k⊥). (5.15) However, we note that the cos 2φ asymmetry receives twist-4 contributions because of the Cahn effect [ 22 ]. A proper treatment of these twist-4 effects involves twist-4 TMDs, as shown in Eq. (4.39) and in Ref. [ 34 ]. Because of the multiple gluon scattering shown in Fig. 1, the twist-4 effects could differ greatly from that given in Ref. [ 22 ], the results in which correspond to the case of L = 1. A careful check might change the conclusion obtained in Refs. [ 108–112 ]. Attempts to parameterize other TMDs such as the pretzelocity h1⊥T have also been made [ 113 ]. Although there is not enough data to yield high-accuracy constraints, the obtained qualitative features are also interesting. The second part concerns the QCD evolution of the TMDs. As mentioned earlier, this is a topic that has recently developed very rapidly. Recent dedicated publications include [ 50–63 ]. QCD evolution equations have been constructed, in particular for unpolarized TMD PDFs and also for polarized TMDs such as the Sivers function. The numerical results obtained from the evolution equations show explicitly that QCD evolution is Fig. 3 Example of the Torino parameterizations of the transversity and Collins function. In the left panel, we see the transversities xΔT q(x) = xh1q(x) for q = u, d; in the right panel, we see the first moments of the favored and disfavored Collins functions. The figure is taken from Ref. [107]. Fig. 5 Example showing the TMD evolution of the Gaussian parameterization in the low k⊥-region. The curves show the evolved Bochum Gaussian fits of up quark Sivers function at x = 0.1. This figure is taken from Ref. [ 54 ]. 101204-18 Fig. 6 Example showing the evolved k⊥ dependence in the large k⊥ region. Here we see the up-quark Sivers function at Q = 5 GeV and Q = 91.19 compared with the corresponding Gaussian fits at low-k⊥ region at x = 0.1. This figure is taken from Ref. [ 54 ]. Fig. 7 Example showing the difference between the results of the TMD evolution with a DGLAP evolution for x-dependence only for unpolarized TMD PDF. This figure is taken from Ref. [ 55 ]. very significant for the TMDs. Not only the form of the k⊥ dependence, but also the width of the Gaussian, evolves with Q. More precisely, at small k⊥, Gaussian parameterization can be used, but the width evolves with Q. At larger k⊥, the form of the k⊥ dependence is determined mainly by gluon radiation and deviates greatly from a Gaussian; it also evolve with Q. In Fig. 5, we see an example for the evolution of the Gaussian parameterization at small k⊥; in Fig. 6, we see the evolution of the shape at large k⊥. It is also important to use the comprehensive TMD evolution rather than a separate evolution of the transverse and longitudinal dependences. We show an example in Fig. 7. The last thing we would like to mention regarding TMD parameterizations is the TMD library (TMDlib). We are happy to see that an initial version was created [ 114 ] in 2014 and that it was updated recently. 6 Summary and outlook In summary, by comparing the studies with what we did in constructing the theoretical framework in studying one-dimensional imaging of the nucleon with inclusive DIS, we presented a brief overview of available results on the studies of three-dimensional imaging of the nucleon with SIDIS and other semi-inclusive reactions. We summarized in particular the general form of the TMDs defined via quark–quark correlators for both TMD PDFs and FFs. We emphasized the theoretical framework for semi-inclusive reactions at LO pQCD but with leading and higher twist contributions consistently. This theoretical framework was obtained by applying the collinear expansion technique developed in the 1980s for inclusive DIS to these semi-inclusive processes. We summarized in particular how it applies to all high energy processes involving one hadron. The results obtained in such a framework should be used as starting points for studying TMDs experimentally. Finally, we emphasize that three-dimensional imaging of the nucleon has been a hot and rapidly developing topic in recent years. Many advances have been made, and many questions are open. We see in particular that frameworks at LO pQCD and leading and higher twists for processes involving one hadron can be constructed using collinear expansions. A factorization theorem for leading twist but with LO and higher-order pQCD contributions and QCD evolution equations for unpolarized TMD PDFs and the Sivers functions have also been established. Especially in view of the operational and planned facilities such as electron–ion colliders, we expect even more rapid development in coming years. This overview is far from complete. We apologize for omitting many aspects, such as the generalized parton distributions, the Wigner function, model calculations of TMDs, the nuclear dependences, and the hyperon polarization. Acknowledgements We thank X. N. Wang, Y. K. Song, J. H. Gao, and many other people for collaboration and help in preparing this review. Z. T. Liang thanks also John Collins and Zebo Tang for communications. This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11035003 and 11375104), the Major State Basic Research Development Program in China (Grant No. 2014CB845406) and the CAS Center for Excellence in Particle Physics (CCEPP). Open Access This article is distributed under the terms of the 101204-19 Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. 101204-20 small transverse momentum, J. High Energy Phys. 0702, 093 (2007) 101204-21 101204-22 1. For a recent review of experiments, see e .g., A. D. Krisch , Hard collisions of spinning protons: Past, present and future , Eur. Phys. J. A 31 ( 4 ), 417 ( 2007 ), there are also a number of reviews on this topic, see e .g., Z. T. Liang and C. Boros , Int. J. Mod. Phys. A 15 , 927 ( 2000 ), V. Barone , F. Bradamante , and A. Martin , Transverse-spin and transverse-momentum effects in high-energy processes , Prog. Part. Nucl. Phys . 65 ( 2 ), 267 ( 2010 ) 2. Z. T. Liang , Three dimensional imaging of the nucleon-TMD (theory and phenomenology ), arXiv: 1502.03896 [hep-ph], Plenary talk at the 21st International Symposium on Spin Physics, October 20-24 , 2014 3. D. Boer , Overview of TMD evolution, arXiv: 1502 .00899 [hep-ph], Invited talk at the 21st International Symposium on Spin Physics, October 20-24 , 2014 4. J. Collins, TMD theory, factorization and evolution , Int. J. Mod. Phys. Conf. Ser . 25 , 1460001 ( 2014 ), arXiv: 1307 . 2920 5. There are definitely cases where factorization might break down, see e .g., J. P. Ma , J. X. Wang , and S. Zhao , Breakdown of QCD factorization for P-wave quarkonium production at low transverse momentum , Phys. Lett. B 737 , 103 ( 2014 ), arXiv: 1405 .3373 [ hep-ph], and the references given there 6. J. C. Collins , D. E. Soper , and G. F. Sterman , Factorization of hard processes in QCD, Adv . Ser. Direct. High Energy Phys. 5 , 1 ( 1989 ), arXiv: hep-ph/0409313 7. J. Collins, Foundations of perturbative QCD (Cambridge monographs on particle physics , nuclear physics and cosmology) , Cambridge: Cambridge University Press, 2011 8. R. P. Feynman , Photon Hadron Interactions, W. A. Benjamin , 1972 9. R. K. Ellis , W. Furmanski , and R. Petronzio , Power corrections to the parton model in QCD, Nucl . Phys. B 207 ( 1 ), 1 ( 1982 ) 10. R. K. Ellis , W. Furmanski , and R. Petronzio , Unravelling higher twists, Nucl. Phys. B 212 ( 1 ), 29 ( 1983 ) 11. J. W. Qiu and G. F. Sterman , Power corrections in hadronic scattering (I): Leading 1/Q2 corrections to the Drell-Yan cross section , Nucl. Phys. B 353 ( 1 ), 105 ( 1991 ) ; J. W. Qiu and G. F. Sterman , Power corrections in hadronic scattering (II): Factorization, Nucl . Phys. B 353 ( 1 ), 137 ( 1991 ) 12. G. L. Kane , J. Pumplin , and W. Repko , Transverse quark polarization in large-pT reactions , e+e − jets, and leptoproduction: A test of quantum chromodynamics , Phys. Rev. Lett . 41 ( 25 ), 1689 ( 1978 ) 13. D. L. Adams , et al. (FNAL-E704 Collaboration), Analyzing power in inclusive π+ and π− production at high xF with a 200 GeV polarized proton beam , Phys. Lett. B 264 ( 3-4 ), 462 ( 1991 ) 14. D. W. Sivers, Single-spin production asymmetries from the hard scattering of pointlike constituents , Phys. Rev. D 41 ( 1 ), 83 ( 1990 ); D. W. Sivers, Hard-scattering scaling laws for single-spin production asymmetries , Phys. Rev. D 43 ( 1 ), 261 ( 1991 ) 15. C. Boros , Z. T. Liang , and T. C. Meng , Quark spin distribution and quark-antiquark annihilation in single-spin hadronhadron collisions , Phys. Rev. Lett . 70 ( 12 ), 1751 ( 1993 ) 16. J. C. Collins, Fragmentation of transversely polarized quarks probed in transverse momentum distributions , Nucl. Phys. B 396 ( 1 ), 161 ( 1993 ) 17. S. J. Brodsky , D. S. Hwang , and I. Schmidt , Final-state interactions and single-spin asymmetries in semi-inclusive deep inelastic scattering , Phys. Lett. B 530 ( 1-4 ), 99 ( 2002 ) 18. J. C. Collins, Leading-twist single-transverse-spin asymmetries: Drell-Yan and deep-inelastic scattering , Phys. Lett. B 536 ( 1-2 ), 43 ( 2002 ) 19. X. Ji and F. Yuan , Parton distributions in light-cone gauge: Where are the final-state interactions? Phys . Lett. B 543 ( 1- 2 ), 66 ( 2002 ) 20. A. V. Belitsky , X. Ji , and F. Yuan , Final state interactions and gauge invariant parton distributions , Nucl. Phys. B 656 ( 1-2 ), 165 ( 2003 ) 21. H. Georgi and H. Politzer , Clean tests of quantum chromodynamics in μp scattering , Phys. Rev. Lett . 40 ( 1 ), 3 ( 1978 ) 22. R. N. Cahn , Azimuthal dependence in leptoproduction: A simple parton model calculation , Phys. Lett. B 78 , 269 ( 1978 ) 23. K. Goeke , A. Metz , and M. Schlegel , Parameterization of the quark-quark correlator of a spin-1/2 hadron , Phys. Lett. B 618 ( 1-4 ), 90 ( 2005 ) 24. P. Mulders, invited talk at the 21st International Symposium on Spin Physics, October 20-24 , 2014 , Beijing, China, and lectures in 17th Taiwan nuclear physics summer school , Aug. 25 - 28 , 2014 25. A. Bacchetta and P. J. Mulders , Deep inelastic leptoproduction of spin-one hadrons , Phys. Rev. D 62 ( 11 ), 114004 ( 2000 ), arXiv: hep-ph/0007120 26. K. B. Chen , S. Y. Wei , W. H. Yang , and Z. T. Liang , Three dimensional fragmentation functions from the quark-quark correlator , arXiv: 1505 .02856 [hep-ph] 27. M. Gourdin , Semi-inclusive reactions induced by leptons , Nucl. Phys. B 49 , 501 ( 1972 ) 28. A. Kotzinian , New quark distributions and semi-inclusive electroproduction on polarized nucleons , Nucl. Phys. B 441 ( 1-2 ), 234 ( 1995 ) 29. M. Diehl and S. Sapeta , On the analysis of lepton scattering on longitudinally or transversely polarized protons , Eur. Phys. J. C 41 ( 4 ), 515 ( 2005 ) 30. A. Bacchetta , M. Diehl , K. Goeke , A. Metz , P. J. Mulders and M. Schlegel , Semi-inclusive deep inelastic scattering at 31. S. Arnold , A. Metz , and M. Schlegel , Dilepton production from polarized hadron hadron collisions , Phys. Rev. D 79 ( 3 ), 034005 ( 2009 ) 32. D. Pitonyak , M. Schlegel , and A. Metz , Polarized hadron pair production from electron-positron annihilation , Phys. Rev. D 89 ( 5 ), 054032 ( 2014 ) 33. Z. Liang and X. N. Wang , Azimuthal and single-spin asymmetry in deep-inelastic lepton-nucleon scattering , Phys. Rev. D 75 ( 9 ), 094002 ( 2007 ) 34. Y. Song , J. Gao , Z. T. Liang , and X. N. Wang , Twist-4 contributions to the azimuthal asymmetry in semi-inclusive deeply inelastic scattering , Phys. Rev. D 83 ( 5 ), 054010 ( 2011 ) 35. Y. Song , J. Gao , Z. T. Liang , and X. N. Wang , Azimuthal asymmetries in semi-inclusive deep inelastic scattering with polarized beam and/or target and their nuclear dependences , Phys. Rev. D 89 ( 1 ), 014005 ( 2014 ) 36. S. y. Wei, Y. Song , and Z. Liang , Higher-twist contribution to fragmentation function in inclusive hadron production in e+e− annihilation , Phys. Rev. D 89 ( 1 ), 014024 ( 2014 ) 37. S. Y. Wei , K. Chen , Y. Song , and Z. Liang , Leading and higher twist contributions in semi-inclusive e+e− annihilation at high energies , Phys. Rev. D 91 ( 3 ), 034015 ( 2015 ), arXiv: 1410 .4314 [hep-ph] 38. 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 ( 1-2 ), 197 ( 1996 ) [Erratum, Nucl . Phys. B 484 , 538 ( 1997 )] 39. D. Boer , R. Jakob , and P. J. Mulders , Asymmetries in polarized hadron production in e+e− annihilation up to order 1/Q, Nucl . Phys. B 504 ( 1-2 ), 345 ( 1997 ) 40. Z. Lu and I. Schmidt , Transverse momentum dependent twist-three result for polarized Drell-Yan processes , Phys. Rev. D 84 ( 11 ), 114004 ( 2011 ) 41. A. P. Chen , J. P. Ma , and G. P. Zhang , Twist-3 contributions in semi-inclusive DIS with transversely polarized target , arXiv: 1505 .03217 [hep-ph] 42. J. C. Collins and D. E. Soper , Back-to-back jets in QCD, Nucl . Phys. B 193 ( 2 ), 381 ( 1981 ) [Erratum, Nucl . Phys. B 213 , 545 ( 1983 )] 43. J. C. Collins and D. E. Soper , Parton distribution and decay functions , Nucl. Phys. B 194 ( 3 ), 445 ( 1982 ) 44. 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 ( 1-4 ), 199 ( 1985 ) 45. J. C. Collins , D. E. Soper , and G. F. Sterman , Factorization for short distance hadron-hadron scattering , Nucl. Phys. B 261 , 104 ( 1985 ) 46. X. Ji , J. P. Ma , and F. Yuan , Factorization of large-x quark distributions in a hadron , Phys. Lett. B 610 ( 3-4 ), 247 ( 2005 ) 47. A. Idilbi , X. Ji , J. P. Ma , and F. Yuan , Collins-Soper equation for the energy evolution of transverse-momentum and spin dependent parton distributions , Phys. Rev. D 70 ( 7 ), 074021 ( 2004 ) 48. X. Ji , J. P. Ma , and F. Yuan , QCD factorization for spindependent cross sections in DIS and Drell-Yan processes at low transverse momentum , Phys. Lett. B 597 ( 3-4 ), 299 ( 2004 ) 49. X. Ji , J. Ma, and F. Yuan , QCD factorization for semiinclusive deep-inelastic scattering at low transverse momentum , Phys. Rev. D 71 ( 3 ), 034005 ( 2005 ) 50. A. A. Henneman , D. Boer , and P. J. Mulders , Evolution of transverse momentum dependent distribution and fragmentation functions , Nucl. Phys. B 620 ( 1-2 ), 331 ( 2002 ) 51. J. Zhou , F. Yuan , and Z. T. Liang , QCD evolution of the transverse momentum dependent correlations , Phys. Rev. D 79 ( 11 ), 114022 ( 2009 ) 52. Z. B. Kang , B. W. Xiao , and F. Yuan , QCD resummation for single spin asymmetries , Phys. Rev. Lett . 107 ( 15 ), 152002 ( 2011 ) 53. S. M. Aybat and T. C. Rogers , Transverse momentum dependent parton distribution and fragmentation functions with QCD evolution , Phys. Rev. D 83 ( 11 ), 114042 ( 2011 ) 54. S. M. Aybat , J. C. Collins , J. W. Qiu , and T. C. Rogers, QCD evolution of the Sivers function , Phys. Rev. D 85 ( 3 ), 034043 ( 2012 ) 55. M. Anselmino , M. Boglione , and S. Melis , Strategy towards the extraction of the Sivers function with transverse momentum dependent evolution , Phys. Rev. D 86 ( 1 ), 014028 ( 2012 ) 56. P. Sun and F. Yuan , Transverse momentum dependent evolution: Matching semi-inclusive deep inelastic scattering processes to Drell-Yan and W/Z boson production , Phys. Rev. D 88 ( 11 ), 114012 ( 2013 ) 57. J. P. Ma and G. P. Zhang, QCD corrections of all structure functions in transverse momentum dependent factorization for Drell-Yan processes , J. High Energy Phys . 2014 ( 2 ), 100 ( 2014 ), arXiv: 1308 . 2044 [hep-ph] 58. M. G. Echevarria , A. Idilbi , Z. B. Kang , and I. Vitev , QCD evolution of the Sivers asymmetry , Phys. Rev. D 89 ( 7 ), 074013 ( 2014 ) 59. C. A . Aidala , B. Field , L. P. Gamberg , and T. C. Rogers, Limits on transverse momentum dependent evolution from semi-inclusive deep inelastic scattering at moderate Q, Phys . Rev. D 89 ( 9 ), 094002 ( 2014 ) 60. Z. B. Kang , A. Prokudin , P. Sun , and F. Yuan , Nucleon tensor charge from Collins azimuthal asymmetry measurements , Phys. Rev. D 91 ( 7 ), 071501 ( 2015 ), arXiv: 1410 .4877 [hep-ph] 61. M. G. Echevarria , A. Idilbi , and I. Scimemi , Unified treatment of the QCD evolution of all (un-)polarized transverse momentum dependent functions: Collins function as a study case , Phys. Rev. D 90 ( 1 ), 014003 ( 2014 ) 62. 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 ( 7 ), 074020 ( 2015 ), arXiv: 1412 .3820 [hep-ph] 63. Z. B. Kang , A. Prokudin , P. Sun , and F. Yuan , Extraction of quark transversity distribution and Collins fragmentation functions with QCD evolution , arXiv: 1505 .05589 [hep-ph] 64. A. Rostomyan (for the HERMES collaboration ), “Highlights of HERMES”, Plenary talk at the 21st International Symposium on Spin Physics, October 20-24 , 2014 , Beijing, China 65. M. Stolarski (On behalf of the COMPASS Collaboration), “Latest results from the COMPASS experiment” , Plenary talk at the 21st International Symposium on Spin Physics, October 20-24 , 2014 , Beijing, China 66. A. Airapetian , et al. (HERMES Collaboration), Evidence for a single-spin azimuthal asymmetry in semi-inclusive Pion electroproduction , Phys. Rev. Lett . 84 ( 18 ), 4047 ( 2000 ), arXiv: hep-ex/9910062 67. A. Airapetian , et al. (HERMES Collaboration), Single-spin asymmetries in semi-inclusive deep-inelastic scattering on a transversely polarized hydrogen target , Phys. Rev. Lett . 94 ( 1 ), 012002 ( 2005 ), arXiv: hep-ex/0408013 68. A. Airapetian , et al. (HERMES Collaboration), Observation of the naive-T -odd Sivers effect in deep-inelastic scattering , Phys. Rev. Lett . 103 ( 15 ), 152002 ( 2009 ), arXiv: 0906 .3918 [hep-ex] 69. A. Airapetian , et al. (HERMES Collaboration) , Effects of transversity in deep-inelastic scattering by polarized protons , Phys. Lett. B 693 ( 1 ), 11 ( 2010 ), arXiv: 1006 .4221 [hepex] 70. A. Airapetian , et al. (HERMES Collaboration), Azimuthal distributions of charged hadrons, pions, and kaons produced in deep-inelastic scattering off unpolarized protons and deuterons , Phys. Rev. D 87 ( 1 ), 012010 ( 2013 ), arXiv: 1204 .4161 [hep-ex] 71. V. Y. Alexakhin , et al. (COMPASS Collaboration), First measurement of the transverse spin asymmetries of the deuteron in semi-inclusive deep inelastic scattering , Phys. Rev. Lett . 94 ( 20 ), 202002 ( 2005 ), arXiv: hep-ex/0503002 72. E. S. Ageev , et al. (COMPASS Collaboration), A new measurement of the Collins and Sivers asymmetries on a transversely polarised deuteron target , Nucl. Phys. B 765 ( 1-2 ), 31 ( 2007 ), arXiv: hep-ex/0610068 73. M. Alekseev , et al. (COMPASS Collaboration), Collins and Sivers asymmetries for pions and kaons in muon-deuteron DIS, Phys . Lett. B 673 ( 2 ), 127 ( 2009 ), arXiv: 0802 .2160 [hep-ex] 74. M. G. Alekseev , et al. (COMPASS Collaboration), Measurement of the Collins and Sivers asymmetries on transversely polarised protons , Phys. Lett. B 692 ( 4 ), 240 ( 2010 ), arXiv: 1005 .5609 [hep-ex] 75. C. Adolph , et al. (COMPASS Collaboration), I - Experimental investigation of transverse spin asymmetries in μ-p SIDIS processes: Collins asymmetries , Phys. Lett. B 717 , 376 ( 2012 ), arXiv: 1205 .5121 [hep-ex] 76. C. Adolph , et al. (COMPASS Collaboration), II - Experimental investigation of transverse spin asymmetries in μ-p SIDIS processes: Sivers asymmetries , Phys. Lett. B 717 , 383 ( 2012 ), arXiv: 1205 .5122 [hep-ex] 77. C. Adolph , et al. (COMPASS Collaboration), Collins and Sivers asymmetries in muonproduction of pions and kaons off transversely polarised protons , Phys. Lett. B 744 , 250 ( 2015 ), arXiv: 1408 .4405 [hep-ex] 78. C. Adolph , et al. (COMPASS Collaboration), Measurement of azimuthal hadron asymmetries in semi-inclusive deep inelastic scattering off unpolarised nucleons , Nucl. Phys. B 886 , 1046 ( 2014 ), arXiv: 1401 .6284 [hep-ex] 79. H. Avakian , et al. ( The CLAS Collaboration), Measurement of single- and double-spin asymmetries in deep inelastic pion electroproduction with a longitudinally polarized target , Phys. Rev. Lett . 105 ( 26 ), 262002 ( 2010 ), arXiv: 1003 .4549 [hep-ex] 80. M. Aghasyan , H. Avakian , P. Rossi , E. De Sanctis, D. Hasch , M. Mirazita , D. Adikaram , M. J. Amaryan , et al., Precise measurements of beam spin asymmetries in semi-inclusive π0 production, Phys . Lett. B 704 , 397 ( 2011 ), arXiv: 1106 .2293 [hep-ex] 81. X. Qian , et al. (Jefferson Lab Hall A Collaboration), Single spin asymmetries in charged Pion production from semiinclusive deep inelastic scattering on a transversely polarized He3 target at Q2 = 1.4 - 2.7 GeV2, Phys . Rev. Lett . 107 ( 7 ), 072003 ( 2011 ), arXiv: 1106 .0363 [nucl-ex] 82. J. Huang , et al. (Jefferson Lab Hall A Collaboration), Beamtarget double-spin asymmetry ALT in charged pion production from deep inelastic scattering on a transversely polarized He3 target at 1.4 < Q2 < 2.7 GeV2, Phys . Rev. Lett . 108 ( 5 ), 052001 ( 2012 ), arXiv: 1108 .0489 [nucl-ex] 83. Y. Zhang , et al. (Jefferson Lab Hall A Collaboration), Measurement of “pretzelosity” asymmetry of charged pion production in semi-inclusive deep inelastic scattering on a polarized He3 target , Phys. Rev. C 90 ( 5 ), 055209 ( 2014 ), arXiv: 1312 .3047 [nucl-ex] 84. Y. X. Zhao , et al. (Jefferson Lab Hall A Collaboration), Single spin asymmetries in charged kaon production from semiinclusive deep inelastic scattering on a transversely polarized He3 target , Phys. Rev. C 90 ( 5 ), 055201 ( 2014 ), arXiv: 1404 .7204 [nucl-ex] 85. K. Abe , et al. (Belle Collaboration), Measurement of azimuthal asymmetries in inclusive production of hadron pairs in e+e− annihilation at Belle , Phys. Rev. Lett . 96 , 232002 ( 2006 ), arXiv: hep-ex/0507063 86. R. Seidl , et al. (Belle Collaboration), Measurement of azimuthal asymmetries in inclusive production of hadron pairs in e+e− annihilation at √s = 10.58 GeV, Phys. Rev. D 78 ( 3 ), 032011 ( 2008 ) [ Erratum: Phys. Rev. D 86 , 039905 ( 2012 )], arXiv: 0805 .2975 [hep-ex] 87. A. Vossen , et al. (Belle Collaboration), Observation of transverse polarization asymmetries of charged pion pairs in e+eannihilation near √s= 10.58 GeV, Phys . Rev. Lett . 107 ( 7 ), 072004 ( 2011 ), arXiv: 1104 .2425 [hep-ex] 88. J. P. Lees , et al. (BaBar Collaboration), Measurement of Collins asymmetries in inclusive production of charged pion pairs in e+e− annihilation at BABAR , Phys. Rev. D 90 ( 5 ), 052003 ( 2014 ), arXiv: 1309 .5278 [hep-ex] 89. Y. Guan , I. Garzia, H. Li , X. R. Lyu , and W. Yan , Future results on fragmentation functions from BESIII , EPJ Web Conf. 85 , 02037 ( 2015 ) ; also talk given at the 21st International Symposium on Spin Physics, October 20-24 , 2014 , Beijing, China 90. J. Badier , et al. (NA3 Collaboration), Angular distributions in the dimuon hadronic production at 150 GeV/c, Z. Phys . C 11 , 195 ( 1981 ) 91. S. Falciano , et al. (NA10 Collaboration), Angular distributions of muon pairs produced by 194 GeV/c negative pions , Z. Phys. C 31 , 513 ( 1986 ) 92. M. Guanziroli , et al. (NA10 Collaboration), Angular distributions of muon pairs produced by negative pions on deuterium and tungsten , Z. Phys . C 37 , 545 ( 1988 ) 93. J. S. Conway , C. E. Adolphsen , J. P. Alexander , K. J. Anderson , J. G. Heinrich , J. E. Pilcher , A. Possoz , E. I. Rosenberg , C. Biino , J. F. Greenhalgh , W. C. Louis , K. T. McDonald , S. Palestini , F. C. Shoemaker , and A. J. S. Smith , Experimental study of muon pairs produced by 252-GeV pions on tungsten , Phys. Rev. D 39 ( 1 ), 92 ( 1989 ) 94. L. Y. Zhu , et al. (NuSea Collaboration), Measurement of angular distributions of Drell-Yan dimuons in p+d interactions at 800 GeV/c , Phys. Rev. Lett . 99 ( 8 ), 082301 ( 2007 ), arXiv: hep-ex/0609005 95. L. Y. Zhu , et al. (NuSea Collaboration), Measurement of angular distributions of Drell-Yan dimuons in p+p interactions at 800 GeV/c, Phys. Rev. Lett . 102 ( 18 ), 182001 ( 2009 ), arXiv: 0811 .4589 [nucl-ex] 96. M. Anselmino , M. Boglione , U. D'Alesio , A. Kotzinian , F. Murgia , and A. Prokudin , Role of Cahn and Sivers effects in deep inelastic scattering , Phys. Rev. D 71 ( 7 ), 074006 ( 2005 ) 97. M. Anselmino , M. Boglione , U. D'Alesio , A. Kotzinian , F. Murgia , A. Prokudin , and C. Turk , Transversity and Collins functions from SIDIS and e+e− data , Phys. Rev. D75(5) , 054032 ( 2007 ) 98. P. Schweitzer, T. Teckentrup , and A. Metz , Intrinsic transverse parton momenta in deeply inelastic reactions , Phys. Rev. D 81 ( 9 ), 094019 ( 2010 ) 99. A. Signori , A. Bacchetta , M. Radici , and G. Schnell, Investigations into the flavor dependence of partonic transverse momentum , J. High Energy Phys . 2013 ( 11 ), 194 ( 2013 ) 100. M. Anselmino , M. Boglione , J. O. Gonzalez , S. Melis , and A. Prokudin , Unpolarised transverse momentum dependent distribution and fragmentation functions from SIDIS multiplicities , J. High Energy Phys . 1404 , 005 ( 2014 ), arXiv: 1312 . 6261 101. A. V. Efremov , K. Goeke , S. Menzel , A. Metz , and P. Schweitzer , Sivers effect in semi-inclusive DIS and in the Drell-Yan process , Phys. Lett. B 612 ( 3-4 ), 233 ( 2005 ) 102. J. C. Collins , A. V. Efremov , K. Goeke , S. Menzel , A. Metz , and P. Schweitzer , Sivers effect in semiinclusive deeply inelastic scattering , Phys. Rev. D 73 ( 1 ), 014021 ( 2006 ) 103. S. Arnold , A. V. Efremov , K. Goeke , M. Schlegel , and P. Schweitzer , Sivers effect at Hermes, Compass and Clas12 , arXiv: 0805 .2137 [hep-ph] 104. M. Anselmino , M. Boglione , U. D'Alesio , A. Kotzinian , S. Melis , F. Murgia , A. Prokudin , and C. Turk , Sivers effect for pion and kaon production in semi-inclusive deep inelastic scattering , Eur. Phys. J. A 39 ( 1 ), 89 ( 2009 ) 105. W. Vogelsang and F. Yuan , Single-transverse-spin asymmetries: From deep inelastic scattering to hadronic collisions , Phys. Rev. D 72 ( 5 ), 054028 ( 2005 ) 106. A. Bacchetta and M. Radici , Constraining quark angular momentum through semi-inclusive measurements , Phys. Rev. Lett . 107 ( 21 ), 212001 ( 2011 ) 107. M. Anselmino , M. Boglione , U. D'Alesio , S. Melis , F. Murgia , and A. Prokudin , Simultaneous extraction of transversity and Collins functions from new semi-inclusive deep inelastic scattering and e+e− data , Phys. Rev. D 87 ( 9 ), 094019 ( 2013 ) 108. B. Zhang , Z. Lu , B. Q. Ma , and I. Schmidt , Extracting BoerMulders functions from p + D Drell-Yan processes , Phys. Rev. D 77 ( 5 ), 054011 ( 2008 ) 109. B. Zhang , Z. Lu , B. Q. Ma , and I. Schmidt, cos 2φ asymmetries in unpolarized semi-inclusive DIS , Phys. Rev. D 78 ( 3 ), 034035 ( 2008 ) 110. V. Barone , A. Prokudin , and B. Q. Ma , Systematic phenomenological study of the cos 2φ asymmetry in unpolarized semi-inclusive DIS , Phys. Rev. D 78 ( 4 ), 045022 ( 2008 ) 111. V. Barone , S. Melis , and A. Prokudin , Boer-Mulders effect in unpolarized SIDIS: An analysis of the COMPASS and HERMES data on the cos 2φ asymmetry , Phys. Rev. D 81 ( 11 ), 114026 ( 2010 ) 112. Z. Lu and I. Schmidt , Updating Boer-Mulders functions from unpolarized pd and pp Drell-Yan data , Phys. Rev. D 81 ( 3 ), 034023 ( 2010 ) 113. J. Zhu and B. Q. Ma , Probing the leading-twist transversemomentum-dependent parton distribution function h1⊥T via the polarized proton-antiproton Drell-Yan process , Phys. Rev. D 82 ( 11 ), 114022 ( 2010 ) 114. F. Hautmann , H. Jung , M. Krer , P. J. Mulders , E. R. Nocera , T. C. Rogers, and A. Signori , TMDlib and TMDplotter: Library and plotting tools for transverse-momentumdependent parton distributions , Eur. Phys. J. C 74 ( 12 ), 3220 ( 2014 )


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Kai-Bao Chen, Shu-Yi Wei, Zuo-Tang Liang. Three-dimensional imaging of the nucleon and semi-inclusive high-energy reactions, Frontiers of Physics, 2015, 101204, DOI: 10.1007/s11467-015-0477-x