Threedimensional imaging of the nucleon and semiinclusive highenergy reactions
Front. Phys.
Threedimensional imaging of the nucleon and semiinclusive highenergy reactions
KaiBao Chen 0
ShuYi Wei 0
ZuoTang 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 transversemomentumdependent parton distribution functions of the nucleon. The aim of such studies is to provide threedimensional imaging of the nucleon and a comprehensive description of semiinclusive highenergy reactions. By summarizing what we have done in constructing the theoretical framework for inclusive deep inelastic leptonnucleon scattering and onedimensional 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 semiinclusive processes in terms of threedimensional gaugeinvariant parton distributions. Next, we present an overview of what we have already achieved, with an emphasis on the theoretical framework for semiinclusive reactions in leadingorder 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 gaugeinvariant transversemomentumdependent parton distribution functions. We also briefly summarize the available experimental results on semiinclusive reactions and the parameterizations of transversemomentumdependent 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 transversemomentum 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 & onedimensional imaging of
abundant and more interesting. These TMD PDFs can the nucleon
be studied in semiinclusive reactions and are necessary Our studies on the structure of a fastmoving nucleon
forTthheesdtuesdcyriopftitohnreoefdsiumchenpsriooncaeslsiems.aging of the nucleon sWtaertreedcawll itthhaitn,culunsdivere tDhIeSosnuecphhaostoen−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 wellknown 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
threedimensional 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 onedimensional 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 fourmomenta 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 lightcone coordinates
Physics (Spin2014) [
2
]. It is clear that the simplest and and define the lightcone 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 centerofmass frame of γ∗N and choose the
inclusive processes: taking higherorder 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 higherorder −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 onedimensional 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 fastmoving 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 fastmoving 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 fourmomentum 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, Sψ¯(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− pψ¯(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, Sψ¯(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 gaugeinvariant 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–jgluon(s)–quark
correlator. We also immediately see that none of these
quark–jgluon(s)–quark correlators is gauge invariant.
To obtain the gaugeinvariant 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, Sψ¯(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, Sψ¯(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 gaugeinvariant unintegrated quark–quark and quark–jgluon(s)–quark correlators given by
where P stands for the pathordered integral. L(0; y) is
nothing else but the wellknown gauge link, which makes
the quark–quark or quark–jgluon(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 gaugeinvariant 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, Sψ¯(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–jgluon–quark
correlators are involved. This means that only onedimensional 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, Sψ¯(0)Dρ(0)L(0; y−)ψ(y−)p, S ,
2π
p+dy− eixp+y− p, Sψ¯(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, Sψ¯(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, Sψ¯(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–jgluon–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 onedimensional 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 gaugeinvariant
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 Qindependent
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 Qindependent but gaugeinvariant PDFs.
• The parton model with QCD multiple gluon
scattering and loop diagram contributions after
collinear approximation, regularization, and
renormalization leads to leading and higherorder pQCD
and leading twist contributions in factorized forms
in terms of Qevolved and gaugeinvariant 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 threedimensional
imaging of the nucleon was triggered by the experimental
observation of singlespin 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 twist3, and the other 3 contribute at
twist4. We further note that in fact the three timereversal
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 finalstate interactions.
• In 1993, Collins published [
16
] his proof that the
Sivers function has to vanish because of parity and
timereversal invariance.
• In 2002, Brodsky, Hwang, and Schmidt calculated
[
17
] the SSA for semiinclusive 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 lightcone 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 finalstate 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, Sψ¯(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 jgluon(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 twist3, and the other 8 con γ · D(z)ψ(z) = 0. We can obtain relations such as
tribute at twist4. 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 twist3 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 twist3 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 onetoone correspondence
betively; the subscript 1 indicates leading twist, no number tween TMD PDFs and TMD FFs. E.g., corresponding
indicates twist3, and subscript 3 indicates twist4; 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 spin1/2 hadron, we have perfect onetoone correspondence to those given by Eqs. (3.3)–(3.7) for parton
distributions in the nucleon, i.e.,
d4ξe−ikF ξ 0L†(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 onetoone 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, finalstate interactions can still exist
between h and X . In this case, timereversal invariance does
not lead to zero results for the Todd amplitudes.
For spin1 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 spinindependent part, a
vectorpolarizationdependent part, and a tensorpolarizationdependent
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 vectorpolarizationdependent part
ΞˆU+V (0)(z, kF ⊥; p, S) takes exactly the same
decomposition as that for the spin1/2 hadron given by Eqs.
(3.12)–(3.16). The tensorpolarizationdependent 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 tensorpolarizationdependent 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 spinindependent and
vectorpolarizationdependent parts, 12 components survive; 3
of them contribute at twist2, 6 contribute at twist3, and
the other 3 contribute at twist4. This is exactly the same
as the result for the PDFs for nucleons, and we have
exact onetoone correspondence between the results given
by Eqs. (3.22)–(3.26) and those given by Eqs. (2.64)–
(2.68). For the tensorpolarizationdependent part, only
8 components survive; 2 of them contribute at twist2, 4
contribute at twist3, and the other 2 contribute at
twist4. This corresponds to the situation for the PDFs for
vector mesons. We should have a onetoone
correspondence between the tensorpolarizationdependent FFs for
production of spin1 hadrons and those PDFs for spin1
hadrons. We also list the twist2 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 highenergy 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 semiinclusive highenergy 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
insemiinclusive 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 spin1/2 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 powersuppressed
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 powersuppressed 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 (longitransversity) 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 wormgear (transhelicity
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 semiinclusive 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 twist3 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⊥) − 2xQk⊥ B(y)fq⊥(x, k⊥) cos φ,
WLU (x, k⊥, φ) = − 2xQk⊥ 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) = − 2xQk⊥ B(y)fL⊥(x, k⊥) sin φ,
WLL(x, k⊥, φ) = C(y)g1L(x, k⊥) − 2xQk⊥ 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 twist4 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⊥22 x[ϕ(32,L)⊥(x, k⊥) − ϕ˜(32,L)⊥(x, k⊥)] .
These results are expressed in terms of the gaugeinvariant 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−) =
2pT 
SLn(T0)(y, z, pT ) = − 3zM
q T0q(y)D1(z, pT )
,
q Pq(y)T0q(y)G1⊥LT (z, pT )
Complete twist3 results for the differential cross sec the semiinclusive highenergy 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 spin1 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 semiinclusive 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 )
2pT 
SLt(T0)(y, z, pT ) = − 3zM
, (4.46) All three types of semiinclusive reactions have been
inq T0q(y)D1(z, pT ) vestigated experimentally. The results are summarized,
STnnT(0)(y, z, pT ) = − 23pMT 22 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 ) = 2pT 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 singlespin 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 semiinclusive 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 twist3 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
]
10120416
and pp or pD collisions [
94, 95
].
Table 5 Available measurements on azimuthal asymmetries in
SIDIS
Collaboration
HERMES
daAntltthoouggivhetphreedciasteacaornetrsotlilloffatrhferoTmMsDusffiicnievnotlvlyeda,bduinf ΔN fq(x, k⊥) = − 2Mk⊥ 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 xdependent 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 zerothorder 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 ) = 2phT  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 BoerMulders 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
twist4 contributions because of the Cahn effect [
22
]. A
proper treatment of these twist4 effects involves twist4
TMDs, as shown in Eq. (4.39) and in Ref. [
34
]. Because of
the multiple gluon scattering shown in Fig. 1, the twist4
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 highaccuracy
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
].
10120418
Fig. 6 Example showing the evolved k⊥ dependence in the large
k⊥ region. Here we see the upquark Sivers function at Q = 5 GeV
and Q = 91.19 compared with the corresponding Gaussian fits at
lowk⊥ 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 xdependence 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
onedimensional imaging of the nucleon with inclusive
DIS, we presented a brief overview of available results
on the studies of threedimensional imaging of the
nucleon with SIDIS and other semiinclusive 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
semiinclusive 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 semiinclusive 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 threedimensional 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 higherorder 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
10120419
Creative Commons Attribution License which permits any use,
distribution, and reproduction in any medium, provided the original
author(s) and the source are credited.
10120420
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10120421
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