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In questo manuale sono illustrate le regole base per la corretta applicazione del marchio Università di Pavia. Il logo, i caratteri tipografici e i colori scelti sono infatti gli elementi che partecipano alla costruzione dell’identità visiva di qualsiasi attore che voglia presentarsi al mercato, sia esso un prodotto mass market, un’Istituzione o un ateneo. Sono il suo volto commerciale ma anche istituzionale, quello che permetterà all’Università di Pavia di essere riconoscibile nel tempo agli occhi del suo pubblico interno, ma anche esterno. Proprio per il ruolo centrale che rivestono, tali elementi devono essere rappresentati e utilizzati
secondo regole precise e inderogabili, al fine di garantire la coerenza e l’efficacia dell’intero sistema di identità visiva. Per questo è importante che il manuale, nella sua forma cartacea o digitale, venga trasmesso a tutti coloro che in futuro si occuperanno di progettare elementi di comunicazione per l’Università di Pavia.
INTRODUZIONE: L’IMMAGINE UNIVERSITÀ DI PAVIA
UNIVERSITÀ DI PAVIA
giuseppe bozzi
in collaboration withAlessandro Bacchetta, Cristian Pisano, Alexei Prokudin, Marco Radici
(Towards a new formalism for the study of) Azimuthal asymmetries in unpolarised SIDIS
GDR Juljan Rushaj
Prog
etta
zion
e Lo
go3D
Spi
n
Colore 01 Colore 03Colore 02
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$W�WKLV�VWDJH�WKLV�OHWWHU�VKRXOG�LQ�QR�ZD\�EH�FRQVLGHUHG�DV�D�FRPPLWPHQW�RI�ILQDQFLDO�VXSSRUW�E\�WKH�(XURSHDQ8QLRQ�
7HO������������������_�)D[������������������_�$JQHV�.8/&6$5#HF�HXURSD�HX�_�KWWS���HUF�HXURSD�HX(5&�([HFXWLYH�$JHQF\��3ODFH�5RJLHU�����&29����������%(������%UXVVHOV��%HOJLXP
Acos 2�hUU =
"F cos 2�hUU
FUU,T + "FUU,L
.Acos�hUU =
p2"(1 + ")F cos�h
UU
FUU,T + "FUU,L
.
Azimuthal Asymmetries
d�
d�hdP 2
h?/ ↵2
Q2
{FUU,T + "FUU,L +
p2 "(1 + ") cos�h F
cos�hUU + " cos(2�h)F
cos 2�hUU }
Differential cross section
y
z
x
hadron plane
lepton plane
ll S
Ph
Phφh
φS
Unpolarised SIDIS
Do they describe the same dynamics or two competing mechanisms
in the intermediate region?
(i.e., interpolation or sum?)
M2 � q2T � Q2
Intermediate
collinear PDF
M2 � q2T
High
TMD
q2T � Q2
Low
q2T
Q2M2
3 physical scales - 2 theoretical tools
Example of a MISMATCH: M2F (qT , Q) ⇠ M2(M2
q4T+
1
Q2) ⇠ l2,4 + l4,2
l2,4 = h4,2 leading for low qT, subleading for high qTl4,2 = h2,4 leading for high qT, subleading for low qT
Example of a MATCH: M2F (qT , Q) ⇠ M2(1
q2T+
1
Q2) ⇠ l2,2 + l4,2
l2,2 = h2,2 leading in both casesl4,2 = h2,4 subleading in both cases
ln,k = hk,nsame twist only if k=n
M⌧qT⌧Q=
1
M2
X
n,k
hn,k (M
qT)n (
qTQ
)k�2F (qT , Q)M⌧qT=
1
M2
X
n
(M
qT)n hn(
qTQ
)High qT
M⌧qT⌧Q=
1
M2
X
n,k
ln,k (qTQ
)n�2 (M
qT)kF (qT , Q)
qT⌧Q=
1
M2
X
n
(qTQ
)n�2 ln(M
qT)Low qT
Matches and mismatches Bacchetta, Boer, Diehl, Mulders JHEP08(2008)
F cos 2�hUU
FUU,T
Mismatch of non-logarithmic terms
No matching in the intermediate region
l2,2
l3,2l2,4
l4,2h2,2
h2,3
h2,4
h2,4
JHEP08(2008)023
low-qT calculation high-qT calculation leading powers
observable twist order power twist order power match
FUU,T 2 αs 1/q2T 2 αs 1/q2
T yes
FUU,L 4 2 αs 1/Q2
F cos φh
UU 3 αs 1/(QqT ) 2 αs 1/(QqT ) yes
F cos 2φh
UU 2 αs 1/q4T 2 αs 1/Q2 no
F sinφh
LU 3 α2s 1/(QqT ) 2 α2
s 1/(QqT ) yes
F sinφh
UL 3 α2s 1/(QqT )
F sin 2φh
UL 2 αs 1/q4T
FLL 2 αs 1/q2T 2 αs 1/q2
T yes
F cos φh
LL 3 αs 1/(QqT ) 2 αs 1/(QqT ) yes
F sin(φh−φS)UT,T 2 αs 1/q3
T 3 αs 1/q3T yes
F sin(φh−φS)UT,L 4 3 αs 1/(Q2 qT )
F sin(φh+φS)UT 2 αs 1/q3
T 3 αs 1/q3T yes
F sin(3φh−φS)UT 2 α2
s 1/q3T 3 αs 1/(Q2 qT ) no
F sinφS
UT 3 αs 1/(Qq2T ) 3 αs 1/(Qq2
T ) yes
F sin(2φh−φS)UT 3 αs 1/(Qq2
T ) 3 αs 1/(Qq2T ) yes
F cos(φh−φS)LT 2 αs 1/q3
T
F cos φS
LT 3 αs 1/(Qq2T )
F cos(2φh−φS)LT 3 αs 1/(Qq2
T )
Table 2: Leading power behavior of SIDIS structure functions in the intermediate region M ≪qT ≪ Q, corresponding to the expansions in (1.2) and (1.4), respectively. Empty fields indicatethat no calculation is available. The specification of twist 4 for FUU,L and F sin(φh−φS)
UT,L reflects thatthese observables are zero when calculated at twist-two and twist-three accuracy.
only conclude that these low-qT results can potentially match with those of a high-qT
calculation at twist-three accuracy.
In table 2 we collect the results for the leading power behavior of all structure functions
we have discussed. We notice that for several observables the twist of the low-qT and the
high-qT calculation is not the same, which is reminiscent of a similar observation we made
for the high-pT behavior of distribution functions in section 5.3.
6.1 Interpolating from low to high qT
Let us now see how one can practically proceed when the leading terms in the low- and high-
qT descriptions of an observable do not match in the intermediate region. As an example
we take the unpolarized structure function F cos 2φh
UU . We denote its low-qT approximation
– 41 –
The situation for unpolarised SIDIS
JHEP08(2008)023
• γ∗g → qq
CUU,T = 2TR!x2 + (1 − x)2
"!z2 + (1 − z)2
" 1 − x
xz2
Q2
q2T
, (4.16)
Ccos φh
UU = −4TR (2x − 1) (2z − 1)1 − x
z
Q
qT, (4.17)
Ccos 2φh
UU = 8TR x (1 − x), (4.18)
CLL = 2TR (2x − 1)!z2 + (1 − z)2
" 1 − x
xz2
Q2
q2T
, (4.19)
Ccos φh
LL = −4TR (2z − 1)1 − x
z
Q
qT(4.20)
with CF = 4/3 and TR = 1/2. The relation CUU,L = 2Ccos 2φh
UU holds for each individual
subprocess. Our results agree with those in [40, 32].
The behavior of the above results in the region q2T ≪ Q2 can be obtained by rewriting
the δ function in eq. (4.4) as [41]
δ
#q2T
Q2−
(1 − x)(1 − z)
xz
$= δ(1 − x) δ(1 − z) ln
Q2
q2T
+x
(1 − x)+δ(1 − z)
+z
(1 − z)+δ(1 − x) + O
#q2T
Q2ln
Q2
q2T
$,
(4.21)
where the plus-distribution is as usual defined by
% 1
zdy
G(y)
(1 − y)+=
% 1
zdy
G(y) − G(1)
1 − y− G(1) ln
1
1 − z. (4.22)
We have written the hard-scattering coefficients in (4.6) to (4.20) in a way that allows for
an easy extraction of the leading power behavior at small qT /Q. The result is
FUU,T =1
q2T
αs
2π2z2
&
a
xe2a
'fa1 (x)Da
1(z)L
#Q2
q2T
$+ fa
1 (x)(Da
1 ⊗ Pqq + Dg1 ⊗ Pgq
)(z)
+(Pqq ⊗ fa
1 + Pqg ⊗ f g1
)(x)Da
1 (z)
*, (4.23)
FUU,L = 2F cos 2φh
UU , (4.24)
F cos φh
UU = −1
QqT
αs
2π2z2
&
a
xe2a
'fa1 (x)Da
1 (z)L
#Q2
q2T
$+ fa
1 (x)(Da
1 ⊗ P ′qq + Dg
1 ⊗ P ′gq
)(z)
+(P ′
qq ⊗ fa1 + P ′
qg ⊗ f g1
)(x)Da
1 (z)
*, (4.25)
F cos 2φh
UU =1
Q2
αs
2π2z2
&
a
xe2a
'fa1 (x)Da
1(z)L
#Q2
q2T
$+ fa
1 (x)(Da
1 ⊗ P ′′qq + Dg
1 ⊗ P ′′gq
)(z)
+(P ′′
qq ⊗ fa1 + P ′′
qg ⊗ f g1
)(x)Da
1 (z)
*, (4.26)
– 17 –
JHEP08(2008)023• γ∗g → qq
CUU,T = 2TR!x2 + (1 − x)2
"!z2 + (1 − z)2
" 1 − x
xz2
Q2
q2T
, (4.16)
Ccos φh
UU = −4TR (2x − 1) (2z − 1)1 − x
z
Q
qT, (4.17)
Ccos 2φh
UU = 8TR x (1 − x), (4.18)
CLL = 2TR (2x − 1)!z2 + (1 − z)2
" 1 − x
xz2
Q2
q2T
, (4.19)
Ccos φh
LL = −4TR (2z − 1)1 − x
z
Q
qT(4.20)
with CF = 4/3 and TR = 1/2. The relation CUU,L = 2Ccos 2φh
UU holds for each individual
subprocess. Our results agree with those in [40, 32].
The behavior of the above results in the region q2T ≪ Q2 can be obtained by rewriting
the δ function in eq. (4.4) as [41]
δ
#q2T
Q2−
(1 − x)(1 − z)
xz
$= δ(1 − x) δ(1 − z) ln
Q2
q2T
+x
(1 − x)+δ(1 − z)
+z
(1 − z)+δ(1 − x) + O
#q2T
Q2ln
Q2
q2T
$,
(4.21)
where the plus-distribution is as usual defined by
% 1
zdy
G(y)
(1 − y)+=
% 1
zdy
G(y) − G(1)
1 − y− G(1) ln
1
1 − z. (4.22)
We have written the hard-scattering coefficients in (4.6) to (4.20) in a way that allows for
an easy extraction of the leading power behavior at small qT /Q. The result is
FUU,T =1
q2T
αs
2π2z2
&
a
xe2a
'fa1 (x)Da
1(z)L
#Q2
q2T
$+ fa
1 (x)(Da
1 ⊗ Pqq + Dg1 ⊗ Pgq
)(z)
+(Pqq ⊗ fa
1 + Pqg ⊗ f g1
)(x)Da
1 (z)
*, (4.23)
FUU,L = 2F cos 2φh
UU , (4.24)
F cos φh
UU = −1
QqT
αs
2π2z2
&
a
xe2a
'fa1 (x)Da
1 (z)L
#Q2
q2T
$+ fa
1 (x)(Da
1 ⊗ P ′qq + Dg
1 ⊗ P ′gq
)(z)
+(P ′
qq ⊗ fa1 + P ′
qg ⊗ f g1
)(x)Da
1 (z)
*, (4.25)
F cos 2φh
UU =1
Q2
αs
2π2z2
&
a
xe2a
'fa1 (x)Da
1(z)L
#Q2
q2T
$+ fa
1 (x)(Da
1 ⊗ P ′′qq + Dg
1 ⊗ P ′′gq
)(z)
+(P ′′
qq ⊗ fa1 + P ′′
qg ⊗ f g1
)(x)Da
1 (z)
*, (4.26)
– 17 –
JHEP08(2008)023
scattering coefficients with collinear parton distribution and fragmentation functions,
FUU,T =1
Q2
αs
(2πz)2
!
a
xe2a
" 1
x
dx
x
" 1
z
dz
zδ
#q2T
Q2−
(1 − x)(1 − z)
x z
$
×%fa1
&x
x
'Da
1
&z
z
'C(γ∗q→qg)
UU,T +fa1
&x
x
'Dg
1
&z
z
'C(γ∗q→gq)
UU,T +f g1
&x
x
'Da
1
&z
z
'C(γ∗g→qq)
UU,T
(,
(4.4)
as we have already seen in section 3.2. We recall that a runs over flavors of quarks and
of antiquarks. Analogous expressions with different kernels C give the structure functions
FUU,L, F cos φh
UU , and F cos 2φh
UU . At order αs (but not at higher order) one finds the relation
FUU,L = 2F cos 2φh
UU . (4.5)
The structure functions FLL and F cos φh
LL for longitudinal target and beam polarization
are also given by expressions analogous to (4.4), with different kernels C and with the
unpolarized parton densities fa1 and f g
1 replaced by their polarized counterparts ga1 and gg
1 .
The hard-scattering coefficients for the partonic processes γ∗q → qg, γ∗q → gq, γ∗g → qq
can be computed from the respective diagrams (a, a′), (b, b′), (c, c′) in figure 2, and those
for γ∗q → qg, γ∗q → gq, γ∗g → qq are identical to their counterparts obtained by charge
conjugation. Process by process we have
• γ∗q → qg
CUU,T = 2CF
#(1 − x)(1 − z) +
1 + x2z2
xz
Q2
q2T
$, (4.6)
Ccos φh
UU = −4CF)xz + (1 − x)(1 − z)
* Q
qT, (4.7)
Ccos 2φh
UU = 4CF xz, (4.8)
CLL = 2CF
#2(x + z) +
x2 + z2
xz
Q2
q2T
$, (4.9)
Ccos φh
LL = −4CF (x + z − 1)Q
qT, (4.10)
• γ∗q → gq
CUU,T = 2CF
#(1 − x) z +
1 + x2(1 − z)2
xz
1 − z
z
Q2
q2T
$, (4.11)
Ccos φh
UU = 4CF)x (1 − z) + (1 − x) z
* 1 − z
z
Q
qT, (4.12)
Ccos 2φh
UU = 4CF x (1 − z), (4.13)
CLL = 2CF
#2x + 2(1 − z) +
x2 + (1 − z)2
xz
1 − z
z
Q2
q2T
$, (4.14)
Ccos φh
LL = 4CF (x − z)1 − z
z
Q
qT, (4.15)
– 16 –
• convolution of PDFs and FFs with hard scattering coefficients
• expansion of delta function for small qT/Q
• extraction of leading behaviour
From high to intermediate qT
JHEP08(2008)023F sin 2φh
UL ∼M2
q4T
αs F!h⊥(1)
1L H⊥(1)1 , . . .
", (5.84)
FLL ∼1
q2T
αs F!g1D1
", (5.85)
F cos φh
LL ∼1
QqTαs F
!g1D1
", (5.86)
F sin(φh−φS)UT,T ∼
M
q3T
αs F!f⊥(1)1T D1, . . .
", (5.87)
F sin(φh+φS)UT ∼
M
q3T
αs F!h1H⊥(1)
1 , . . .", (5.88)
F sin(3φh−φS)UT ∼
M
q3T
α2s F
!h1H⊥(1)
1 , . . .", (5.89)
F sin φS
UT ∼M
Qq2T
αs F!f⊥(1)1T D1, h1H⊥(1)
1 , . . .", (5.90)
F sin(2φh−φS)UT ∼
M
Qq2T
αs F!f⊥(1)1T D1, . . .
", (5.91)
F cos(φh−φS)LT ∼
M
q3T
αs F!g(1)1T D1, . . .
", (5.92)
F cos φS
LT ∼M
Qq2T
αs F#g(1)1T D1, h1
E
z, . . .
$, (5.93)
F cos(2φh−φS)LT ∼
M
Qq2T
αs F!g(1)1T D1, . . .
". (5.94)
Here either the parton distributions or the fragmentation functions are convoluted with
kernels Ki or Li :
F!fD
"=
1
z2
%
a,i
e2a
&'Ki ⊗ f i
((x)Da(z) + fa(x)
'Di ⊗ Li
((z)
), (5.95)
where the sum runs over quarks and antiquarks for a and over quarks, antiquarks and
gluons for i. As we will see in section 8, these kernels contain logarithms of Q/qT . Their
origin is the dependence of f1(x, p2T ) or D1(z, k2
T ) on ζ or ζh, which we tacitly omitted
in (5.75). When resummed to all orders in αs in the way we sketched in section 3, these
logarithms can lead to a substantial modification of the power laws in (5.79) to (5.94).
A numerical study of these effects on azimuthal asymmetries in Drell-Yan production has
been performed in [58].
We note that for the 1/Q suppressed structure functions in (5.79) to (5.94), contribu-
tions from U(l2T ) taken at lT ≈ −qT are power suppressed or have the same power behavior
as contributions where either pT ≈ −qT or kT ≈ qT . For these structure functions, the
power behavior at high qT hence remains the same if we simply ignore the soft factor and
work with the tree-level convolution (5.52) instead of (5.75).
6. Comparing results at intermediate qT
We can now compare the results for the region M ≪ qT ≪ Q obtained in the low-qT
calculation of the previous section with those obtained in the high-qT calculation. As we
– 37 –
JHEP08(2008)023
to our accuracy. Likewise, the integral over lT gives unity up to αs-corrections according
to (5.76). Since we are considering the region where kT and lT are small compared with
qT the integrals over these momenta should be suitably cut off, as is required for (5.21)
and (5.76) to make sense. Repeating these arguments for the cases where kT or lT are
large, we obtain!
d2pT d2kT d2lT δ(2)
"pT − kT + lT + qT
#f(x, p2
T )D(z, k2T )U(l2T )
≈ f(x, q2T )
D(z)
z2+ f(x)D(z, q2
T ) + f(x)D(z)
z2U(q2
T ) .
(5.77)
For nontrivial functions w(pT ,kT ) the calculation is slightly more involved. Instead of
approximating e.g. pT = kT − lT − qT ≈ −qT , we need to Taylor expand the functions of
pT around −qT . We take as an example the convolution C$"
kT ·pT
#h⊥
1 H⊥1
%appearing in
F cos 2φh
UU and consider the region where pT is large. We perform the integral over pT using
the δ function and obtain!
d2kT d2lT H⊥1 (z, k2
T )U(l2T )"k2
T − kT · lT − kT ·qT
#h⊥
1
"x, (kT − lT − qT )2
#
≈!
d2kT d2lT H⊥1 (z, k2
T )U(l2T )
×"k2
T − kT · lT − kT ·qT
# &h⊥
1
"x, q2
T ) − 2"kT ·qT − lT ·qT
# ∂
∂q2T
h⊥1
"x, q2
T )
'+ · · ·
≈!
d2kT H⊥1 (z, k2
T )
&k2
T h⊥1
"x, q2
T ) + 2"kT ·qT
#2 ∂
∂q2T
h⊥1
"x, q2
T )
'+ · · ·
= 2M2h
H⊥(1)1 (z)
z2
&h⊥
1
"x, q2
T ) + q2T
∂
∂q2T
h⊥1
"x, q2
T )
'+ · · · (5.78)
where both terms in square brackets behave as 1/q4T . The . . . represent contributions from
the regions where kT or lT is large, which are of the same order in 1/qT .
As we see in (5.47), (5.48), and (5.50), the leading power behavior of some distribution
or fragmentation functions comes with a factor α2s. At this order, one must also take into
account regions of integration in (5.75) where two out of the three momenta pT , kT , lT are
large, but it turns out that these do not contribute to the α2s terms given in the following.
Using the high-transverse-momentum behavior in (5.44) to (5.48) and (5.50), we obtain
FUU,T ∼1
q2T
αs F$f1D1
%, (5.79)
F cos φh
UU ∼1
QqTαs F
$f1D1
%, (5.80)
F cos 2φh
UU ∼M2
q4T
αs F$h⊥(1)
1 H⊥(1)1 , . . .
%, (5.81)
F sin φh
LU ∼1
QqTα2
s F$f1D1
%, (5.82)
F sin φh
UL ∼1
QqTα2
s F$g1D1
%, (5.83)
– 36 –
JHEP08(2008)023
Compared with their analogs (5.46), the convolutions
F!D
"=
1
z2
#Da ⊗ Kq + Dg ⊗ Kg
$(5.51)
have an additional factor 1/z2, which reflects the corresponding factor in (5.21).
5.4 Results for structure functions
Let us begin this section by recalling the expressions for SIDIS structure functions at low
qT in terms of transverse-momentum-dependent distribution and fragmentation functions.
Extending earlier work in [11, 12], the study [14] has given a complete set of results at
leading and first subleading order in 1/Q, i.e., at twist-two and twist-three accuracy. A
detailed investigation of color gauge invariance and the appropriate choice of gauge links
has been given in [13]. The calculations just quoted take into account tree-level graphs,
where gluons are restricted to be collinear to the target or to the observed hadron h and
only appear when they are attached to the distribution or fragmentation correlators (see
figure 2 in [14]).
For a compact presentation of the results, we introduce the unit vector h = −qT /|qT |and the transverse-momentum convolution
C!wfD
"=
%
a
xe2a
&d2pT d2kT δ
(2)'pT −kT + qT
(w(pT ,kT ) fa(x, p2
T )Da(z, k2T ), (5.52)
where w(pT ,kT ) is an arbitrary function and the sum runs over quarks and antiquarks.
The results for the structure functions appearing in (2.3) then read [14]
FUU,T = C!f1D1
", (5.53)
FUU,L = O)
M2
Q2,q2T
Q2
*, (5.54)
F cos φh
UU =2M
QC+−
h ·kT
Mh
)xhH⊥
1 +Mh
Mf1
D⊥
z
*−
h ·pT
M
)xf⊥D1+
Mh
Mh⊥
1H
z
*,, (5.55)
F cos 2φh
UU = C+−
2'h ·kT
( 'h ·pT
(− kT ·pT
MMhh⊥
1 H⊥1
,, (5.56)
F sin φh
LU =2M
QC+−
h ·kT
Mh
)xeH⊥
1 +Mh
Mf1
G⊥
z
*+
h ·pT
M
)xg⊥D1+
Mh
Mh⊥
1E
z
*,, (5.57)
F sin φh
UL =2M
QC+−
h ·kT
Mh
)xhLH⊥
1 +Mh
Mg1L
G⊥
z
*+
h ·pT
M
)xf⊥
L D1 −Mh
Mh⊥
1LH
z
*,,
(5.58)
F sin 2φh
UL = C+−
2'h ·kT
( 'h ·pT
(− kT ·pT
MMhh⊥
1LH⊥1
,, (5.59)
FLL = C!g1LD1
", (5.60)
F cos φh
LL =2M
QC+h ·kT
Mh
)xeLH⊥
1 −Mh
Mg1L
D⊥
z
*−
h ·pT
M
)xg⊥L D1 +
Mh
Mh⊥
1LE
z
*,,
(5.61)
– 33 –
JHEP08(2008)023
E
z=
E
z−
m
MhD1, (5.73)
H
z=
H
z+
k2T
M2h
H⊥1 . (5.74)
Using (5.50) and neglecting the small contributions proportional to the quark mass m, we
readily see that the behavior for kT ≫ M is the same for corresponding functions with and
without a tilde.
The tree-level calculations in [11, 13, 14] do not take into account soft gluon exchange or
virtual corrections involving hard loops, so that the soft and hard factors we encountered
in (3.3) and (3.22) do not appear in the convolution (5.52). Detailed investigations of
factorization for SIDIS with measured qT have recently been given in [26, 36] and [50, 27],
extending the seminal work of Collins and Soper [24]. The factorization formulae discussed
in these papers have the form (3.22) and are valid at all orders in αs but restricted to the
leading order in 1/Q. Although a number of subtle issues remain to be fully clarified [27], we
will use (3.22) in the following. Since we aim at deriving expressions at lowest nonvanishing
order in αs, we can neglect the hard factor |H|2 = 1 + O(αs). The convolution in (5.52)
should then be extended to
C!wfD
"=
#
a
xe2a
$d2pT d2kT d2lT δ
(2)%pT − kT + lT + qT
&
× w(pT ,kT ) fa(x, p2T )Da(z, k2
T )U(l2T ) .
(5.75)
At high transverse momentum lT ≫ M the soft factor behaves as U(l2T ) ∼ αs/l2T , with a
coefficient we shall give in (8.51) below. Our normalization convention is
$d2lT U(l2T ) = 1 + O(αs) , (5.76)
where it is understood that the integral must be suitably regularized at large lT .
Whether Collins-Soper factorization can be extended to structure functions that are
of order 1/Q is not known. We note that the study of color gauge invariance in [13]
was limited to qT -integrated observables in this case, and that a problem with twist-three
factorization has been found in a spectator model calculation [15]. In the following we
adopt the working hypothesis that the twist-two factorization formula can simply be taken
over at twist-three accuracy, using the convolution (5.75) also for evaluating the high-qT
behavior of the 1/Q suppressed structure functions in (5.53) to (5.70). We will return to
this point at the end of section 8.3.
We now show how to calculate the high-qT behavior of the convolution (5.75). At
order αs, only one of the factors f(x, p2T ), D(z, k2
T ), U(l2T ) can be taken at high transverse
momentum. Let us first consider the simple case where w(pT ,kT ) = 1. In the region where
pT is large, we use the δ function in (5.75) to perform the pT integral and approximate
pT = kT − lT −qT ≈ −qT in f(x, p2T ). The remaining integrals over kT and lT can then be
carried out independently. According to (5.21) and our discussion after (5.16), the integral
over kT gives a collinear fragmentation function, up to αs-corrections that can be neglected
– 35 –
• transverse-momentum convolution of TMD PDFs and FFs with kinematical functions
• Taylor-expand the functions of pT in the integrand and use the delta function to perform the integral
• extraction of leading behaviour
From low to intermediate qT
3
B. Tentative full TMD formula
To take into account pQCD corrections to the tree-level expression, we propose to replace the involved TMDs withthe following expressions AB: the collinear parts of the chiral odd ones are just preliminary guesses. •
bfa
1
(x, ⇠2T
;Q2) =X
i=q,q,g
�
Ca/i
⌦ f i
1
�
(x, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bfa
1NP
(x, ⇠2T
) , (22)
bh?a
1
(x, ⇠2T
;Q2) =X
i=q,q,g
�
�C1a/i
⌦ h?(1)i
1
�
(x, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bh?a
1NP
(x, ⇠2T
) , (23)
x bf?a(x, ⇠2T
;Q2) =1
2
X
i=q,q,g
�
C 0
a/i
⌦ f i
1
�
(x, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bfa?
NP
(x, ⇠2T
) , (24)
xbha(x, ⇠2T
;Q2) =X
i=q,q,g
�
�C2a/i
⌦ h?(1)i
1
�
(x, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bha
NP
(x, ⇠2T
) , (25)
x bf?a
3
(x, ⇠2T
;Q2) =X
i=q,q,g
�
C 00
a/i
⌦ f i
1
�
(x, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bfa?
NP3
(x, ⇠2T
) , (26)
bDa!h
1
(z, ⇠2T
;Q2) =X
i=q,q,g
�
Ca/i
⌦Di!h
1
�
(z, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bDa!h
1NP
(z, ⇠2T
) , (27)
bH?a
1
(z, ⇠2T
;Q2) =X
i=q,q,g
�
�C1a/i
⌦H?(1)i
1
�
(z, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bH?a
1NP
(z, ⇠2T
) , (28)
1
zbD?a!h(z, ⇠2
T
;Q2) =1
2
X
i=q,q,g
�
C 0
a/i
⌦Di!h
1
�
(z, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bD?a!h
NP
(z, ⇠2T
) , (29)
bHa(z, ⇠2T
;Q2) =X
i=q,q,g
�
�C2a/i
⌦H?(1)i
1
�
(z, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bHa
NP
(z, ⇠2T
) , (30)
where the symbol ⌦ denotes the usual convolutions in longitudinal momentum fractions,
�
Ca/i
⌦ f i
�
(x, ⇠T
, µ2
b
) =
Z
1
x
dx
xC
a/i
⇣x
x, ⇠
T
,↵S
�
µ2
b
�
⌘
f i(x;µ2
b
) , (31)
�
Ca/i
⌦Di!h
�
(z, ⇠T
, µ2
b
) =
Z
1
z
dz
zC
a/i
⇣z
z, ⇠
T
,↵S
�
µ2
b
�
⌘
Di!h(z;µ2
b
) . (32)
The coe�cient functions Ca/i
can be expanded as
Ca/i
(x, µb
) = �ai
�(1� x) +1
X
k=1
C(k)
a/i
(x)
✓
↵s
(µb
)
⇡
◆
k
, (33)
and an analogous expression holds for Ca/i
. The Sudakov exponent S reads
S(µ2
b
, Q2) = �1
2
Z
Q
2
µ
2b
dµ2
µ2
A⇣
↵S
(µ2)⌘
ln
✓
Q2
µ2
◆
+B⇣
↵S
(µ2)⌘
�
, (34)
where A and B have a perturbative expansions of the form
A�
↵S
(µ2)�
=1
X
k=1
Ak
✓
↵S
⇡
◆
k
, B�
↵S
(µ2)�
=1
X
k=1
Bk
✓
↵S
⇡
◆
k
, (35)
with, at leading-logarithm (LL) accuracy, A1
= CF
and B1
= �3CF
/2. More explicitly,
S(⇠2T
, Q2) = �↵S
4⇡C
F
✓
ln2Q2⇠2
T
⇠20
� 3 lnQ2⇠2
T
⇠20
◆
+O(↵2
S
) , (36)
where we have substituted µb
= ⇠0
/⇠T
.
3
B. Tentative full TMD formula
To take into account pQCD corrections to the tree-level expression, we propose to replace the involved TMDs withthe following expressions AB: the collinear parts of the chiral odd ones are just preliminary guesses. •
bfa
1
(x, ⇠2T
;Q2) =X
i=q,q,g
�
Ca/i
⌦ f i
1
�
(x, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bfa
1NP
(x, ⇠2T
) , (22)
bh?a
1
(x, ⇠2T
;Q2) =X
i=q,q,g
�
�C1a/i
⌦ h?(1)i
1
�
(x, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bh?a
1NP
(x, ⇠2T
) , (23)
x bf?a(x, ⇠2T
;Q2) =1
2
X
i=q,q,g
�
C 0
a/i
⌦ f i
1
�
(x, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bfa?
NP
(x, ⇠2T
) , (24)
xbha(x, ⇠2T
;Q2) =X
i=q,q,g
�
�C2a/i
⌦ h?(1)i
1
�
(x, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bha
NP
(x, ⇠2T
) , (25)
x bf?a
3
(x, ⇠2T
;Q2) =X
i=q,q,g
�
C 00
a/i
⌦ f i
1
�
(x, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bfa?
NP3
(x, ⇠2T
) , (26)
bDa!h
1
(z, ⇠2T
;Q2) =X
i=q,q,g
�
Ca/i
⌦Di!h
1
�
(z, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bDa!h
1NP
(z, ⇠2T
) , (27)
bH?a
1
(z, ⇠2T
;Q2) =X
i=q,q,g
�
�C1a/i
⌦H?(1)i
1
�
(z, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bH?a
1NP
(z, ⇠2T
) , (28)
1
zbD?a!h(z, ⇠2
T
;Q2) =1
2
X
i=q,q,g
�
C 0
a/i
⌦Di!h
1
�
(z, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bD?a!h
NP
(z, ⇠2T
) , (29)
bHa(z, ⇠2T
;Q2) =X
i=q,q,g
�
�C2a/i
⌦H?(1)i
1
�
(z, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bHa
NP
(z, ⇠2T
) , (30)
where the symbol ⌦ denotes the usual convolutions in longitudinal momentum fractions,
�
Ca/i
⌦ f i
�
(x, ⇠T
, µ2
b
) =
Z
1
x
dx
xC
a/i
⇣x
x, ⇠
T
,↵S
�
µ2
b
�
⌘
f i(x;µ2
b
) , (31)
�
Ca/i
⌦Di!h
�
(z, ⇠T
, µ2
b
) =
Z
1
z
dz
zC
a/i
⇣z
z, ⇠
T
,↵S
�
µ2
b
�
⌘
Di!h(z;µ2
b
) . (32)
The coe�cient functions Ca/i
can be expanded as
Ca/i
(x, µb
) = �ai
�(1� x) +1
X
k=1
C(k)
a/i
(x)
✓
↵s
(µb
)
⇡
◆
k
, (33)
and an analogous expression holds for Ca/i
. The Sudakov exponent S reads
S(µ2
b
, Q2) = �1
2
Z
Q
2
µ
2b
dµ2
µ2
A⇣
↵S
(µ2)⌘
ln
✓
Q2
µ2
◆
+B⇣
↵S
(µ2)⌘
�
, (34)
where A and B have a perturbative expansions of the form
A�
↵S
(µ2)�
=1
X
k=1
Ak
✓
↵S
⇡
◆
k
, B�
↵S
(µ2)�
=1
X
k=1
Bk
✓
↵S
⇡
◆
k
, (35)
with, at leading-logarithm (LL) accuracy, A1
= CF
and B1
= �3CF
/2. More explicitly,
S(⇠2T
, Q2) = �↵S
4⇡C
F
✓
ln2Q2⇠2
T
⇠20
� 3 lnQ2⇠2
T
⇠20
◆
+O(↵2
S
) , (36)
where we have substituted µb
= ⇠0
/⇠T
.
3
B. Tentative full TMD formula
To take into account pQCD corrections to the tree-level expression, we propose to replace the involved TMDs withthe following expressions AB: the collinear parts of the chiral odd ones are just preliminary guesses. •
bfa
1
(x, ⇠2T
;Q2) =X
i=q,q,g
�
Ca/i
⌦ f i
1
�
(x, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bfa
1NP
(x, ⇠2T
) , (22)
bh?a
1
(x, ⇠2T
;Q2) =X
i=q,q,g
�
�C1a/i
⌦ h?(1)i
1
�
(x, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bh?a
1NP
(x, ⇠2T
) , (23)
x bf?a(x, ⇠2T
;Q2) =1
2
X
i=q,q,g
�
C 0
a/i
⌦ f i
1
�
(x, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bfa?
NP
(x, ⇠2T
) , (24)
xbha(x, ⇠2T
;Q2) =X
i=q,q,g
�
�C2a/i
⌦ h?(1)i
1
�
(x, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bha
NP
(x, ⇠2T
) , (25)
x bf?a
3
(x, ⇠2T
;Q2) =X
i=q,q,g
�
C 00
a/i
⌦ f i
1
�
(x, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bfa?
NP3
(x, ⇠2T
) , (26)
bDa!h
1
(z, ⇠2T
;Q2) =X
i=q,q,g
�
Ca/i
⌦Di!h
1
�
(z, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bDa!h
1NP
(z, ⇠2T
) , (27)
bH?a
1
(z, ⇠2T
;Q2) =X
i=q,q,g
�
�C1a/i
⌦H?(1)i
1
�
(z, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bH?a
1NP
(z, ⇠2T
) , (28)
1
zbD?a!h(z, ⇠2
T
;Q2) =1
2
X
i=q,q,g
�
C 0
a/i
⌦Di!h
1
�
(z, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bD?a!h
NP
(z, ⇠2T
) , (29)
bHa(z, ⇠2T
;Q2) =X
i=q,q,g
�
�C2a/i
⌦H?(1)i
1
�
(z, ⇠T
, µ2
b
) eS(µ
2b ,Q
2)
✓
Q2
Q2
0
◆
gK(⇠T )
bHa
NP
(z, ⇠2T
) , (30)
where the symbol ⌦ denotes the usual convolutions in longitudinal momentum fractions,
�
Ca/i
⌦ f i
�
(x, ⇠T
, µ2
b
) =
Z
1
x
dx
xC
a/i
⇣x
x, ⇠
T
,↵S
�
µ2
b
�
⌘
f i(x;µ2
b
) , (31)
�
Ca/i
⌦Di!h
�
(z, ⇠T
, µ2
b
) =
Z
1
z
dz
zC
a/i
⇣z
z, ⇠
T
,↵S
�
µ2
b
�
⌘
Di!h(z;µ2
b
) . (32)
The coe�cient functions Ca/i
can be expanded as
Ca/i
(x, µb
) = �ai
�(1� x) +1
X
k=1
C(k)
a/i
(x)
✓
↵s
(µb
)
⇡
◆
k
, (33)
and an analogous expression holds for Ca/i
. The Sudakov exponent S reads
S(µ2
b
, Q2) = �1
2
Z
Q
2
µ
2b
dµ2
µ2
A⇣
↵S
(µ2)⌘
ln
✓
Q2
µ2
◆
+B⇣
↵S
(µ2)⌘
�
, (34)
where A and B have a perturbative expansions of the form
A�
↵S
(µ2)�
=1
X
k=1
Ak
✓
↵S
⇡
◆
k
, B�
↵S
(µ2)�
=1
X
k=1
Bk
✓
↵S
⇡
◆
k
, (35)
with, at leading-logarithm (LL) accuracy, A1
= CF
and B1
= �3CF
/2. More explicitly,
S(⇠2T
, Q2) = �↵S
4⇡C
F
✓
ln2Q2⇠2
T
⇠20
� 3 lnQ2⇠2
T
⇠20
◆
+O(↵2
S
) , (36)
where we have substituted µb
= ⇠0
/⇠T
.
A new expression for twist-3 TMDs
2
Based on Eq. (2.70) of Ref. [? ], we can also assume the validity of a relation of this kind
xf?
3
= xf?
3
+ f1
(10)
In the so-called Wandzura–Wilczek approximation, the “pure twist-3” functions with a tilde are neglected. In thatcase, the formulae for the structure functions considerably simplify and we obtain
F cos�h
UU
=2M
QC
� h · P?
zMh
k2?
M2
h?
1
H?
1
� h · k?
Mf1
D1
�
, (11)
F cos 2�h
UU
= C
�2�
h · k?
� �
h · P?
�� k?
· P?
zMMh
h?
1
H?
1
+4M2
Q2
2 (h · k?
)2 � k2
?
2M2
f1
D1
�
. (12)
On the other hand, using Eqs. (4) and (32) of [2] and expanding in 1/Q we obtain, up to relative corrections in 1/Q,the following expressions (see also Ref. [3])
F cos�h
UU
= �2M
QC
h · k?
Mf1
D1
�
, (13)
F cos 2�h
UU
=4M2
Q2
C
2 (h · k?
)2 � k2
?
2M2
f1
D1
�
. (14)
Apart from the fact that no chiral-odd (or T-odd) term is present in the previous equations, the rest is the same asEq. (11) and (12).
A. Fourier transform
The structure functions can be written as Fourier transforms in the following way
F cos�h
UU
= �2MMh
QB1
✓
xbh bH?(1)
1
+M
h
Mbf1
bD?(1)
z
◆
+M
Mh
✓
x bf?(1)
bD1
+M
h
Mbh?(1)
1
bH
z
◆�
, (15)
F cos 2�h
UU
= �MMh
B2
bh?(1)
1
bH?(1)
1
�
+M4
Q2
B2
x bf?(2)
3
bD1
�
, (16)
where we introduced the following notation
Bn
⇥
bf bD⇤
= 2⇡X
a
e2a
x
Z
1
0
d⇠T
⇠n+1
T
Jn
�
⇠T
|PhT
|/z� bfa(x, ⇠2T
) bDa(z, ⇠2T
) . (17)
The Fourier transforms of a generic TMD PDF denoted by f and a generic TMD FF denoted by D are defined,respectively, as
bf�
x, ⇠2T
� ⌘ 1
2⇡
Z
d2k?
ei⇠T ·k?f�
x,k2
?
;Q2
�
=
Z
1
0
d|k?
||k?
|J0
�
⇠T
|k?
|�f�x,k2
?
) , (18)
bD�
z, ⇠2T
� ⌘ 1
2⇡
Z
d2P?
z2ei⇠T ·P?/zD
�
z,P 2
?
) =
Z
1
0
d|P?
|z2
|P?
|J0
�
⇠T
|P?
|/z�D�
z,P 2
?
�
. (19)
Furthermore, we have introduced their ⇠2T
-derivatives [4], namely
bf (n)(x, ⇠2T
) = n!
✓
� 2
M2
@
@⇠2T
◆
n
bf(x, ⇠2T
) =n!
M2n
Z
1
0
d|k?
||k?
| |k?
|n|⇠
T
|n Jn
�
⇠T
|k?
|�f�x,k2
?
�
, (20)
bD(n)(z, ⇠2T
) = n!
✓
� 2
M2
h
@
@⇠2T
◆
n
bD(z, ⇠2T
) =n!
M2n
h
Z
1
0
d|P?
|z2
|P?
| |P?
|nzn|⇠
T
|n Jn�
⇠T
|P?
|/z�D�
x,P 2
?
�
. (21)
2
Based on Eq. (2.70) of Ref. [? ], we can also assume the validity of a relation of this kind
xf?
3
= xf?
3
+ f1
(10)
In the so-called Wandzura–Wilczek approximation, the “pure twist-3” functions with a tilde are neglected. In thatcase, the formulae for the structure functions considerably simplify and we obtain
F cos�h
UU
=2M
QC
� h · P?
zMh
k2?
M2
h?
1
H?
1
� h · k?
Mf1
D1
�
, (11)
F cos 2�h
UU
= C
�2�
h · k?
� �
h · P?
�� k?
· P?
zMMh
h?
1
H?
1
+4M2
Q2
2 (h · k?
)2 � k2
?
2M2
f1
D1
�
. (12)
On the other hand, using Eqs. (4) and (32) of [2] and expanding in 1/Q we obtain, up to relative corrections in 1/Q,the following expressions (see also Ref. [3])
F cos�h
UU
= �2M
QC
h · k?
Mf1
D1
�
, (13)
F cos 2�h
UU
=4M2
Q2
C
2 (h · k?
)2 � k2
?
2M2
f1
D1
�
. (14)
Apart from the fact that no chiral-odd (or T-odd) term is present in the previous equations, the rest is the same asEq. (11) and (12).
A. Fourier transform
The structure functions can be written as Fourier transforms in the following way
F cos�h
UU
= �2MMh
QB1
✓
xbh bH?(1)
1
+M
h
Mbf1
bD?(1)
z
◆
+M
Mh
✓
x bf?(1)
bD1
+M
h
Mbh?(1)
1
bH
z
◆�
, (15)
F cos 2�h
UU
= �MMh
B2
bh?(1)
1
bH?(1)
1
�
+M4
Q2
B2
x bf?(2)
3
bD1
�
, (16)
where we introduced the following notation
Bn
⇥
bf bD⇤
= 2⇡X
a
e2a
x
Z
1
0
d⇠T
⇠n+1
T
Jn
�
⇠T
|PhT
|/z� bfa(x, ⇠2T
) bDa(z, ⇠2T
) . (17)
The Fourier transforms of a generic TMD PDF denoted by f and a generic TMD FF denoted by D are defined,respectively, as
bf�
x, ⇠2T
� ⌘ 1
2⇡
Z
d2k?
ei⇠T ·k?f�
x,k2
?
;Q2
�
=
Z
1
0
d|k?
||k?
|J0
�
⇠T
|k?
|�f�x,k2
?
) , (18)
bD�
z, ⇠2T
� ⌘ 1
2⇡
Z
d2P?
z2ei⇠T ·P?/zD
�
z,P 2
?
) =
Z
1
0
d|P?
|z2
|P?
|J0
�
⇠T
|P?
|/z�D�
z,P 2
?
�
. (19)
Furthermore, we have introduced their ⇠2T
-derivatives [4], namely
bf (n)(x, ⇠2T
) = n!
✓
� 2
M2
@
@⇠2T
◆
n
bf(x, ⇠2T
) =n!
M2n
Z
1
0
d|k?
||k?
| |k?
|n|⇠
T
|n Jn
�
⇠T
|k?
|�f�x,k2
?
�
, (20)
bD(n)(z, ⇠2T
) = n!
✓
� 2
M2
h
@
@⇠2T
◆
n
bD(z, ⇠2T
) =n!
M2n
h
Z
1
0
d|P?
|z2
|P?
| |P?
|nzn|⇠
T
|n Jn�
⇠T
|P?
|/z�D�
x,P 2
?
�
. (21)
A new Fourier-transformed expression for the structure functions
Our work
5
The dominant terms in high-transverse momentum region of the structure function F cos�h
UU
in Eq. (15) are given by
F cos�h
UU
= �4⇡M2
Q
X
a
e2a
x
Z
1
0
d⇠T
⇠2T
J1
✓
⇠T
|PhT
|z
◆✓
M2
h
M2
bfa
1
bD?(1)a!h
z+ x bf?(1)a
bDa!h
1
◆
= � 1
QqT
↵S
2⇡2z2
X
a
e2a
x
Z
1
0
d⇠T
J1
✓
⇠T
|PhT
|z
◆✓
2CF
lnQ2⇠2
T
⇠20
� 3CF
◆
fa
1
(x)Da!h
1
(z) , (49)
where, in the last step, we have substituted the leading-order expressions for bfa
1
and bDa!h
1
in Eqs. (43) and (46),
respectively, and the ones for x bf?(1)a and bD?(1)a!h in Eqs. (44) and (47). Moreover, by means of the equalities
Z
1
0
d⇠T
Jn
✓
⇠T
|PhT
|z
◆
=1
qT
, (50)
Z
1
0
d⇠T
J1
✓
⇠T
|PhT
|z
◆
lnQ2⇠2
T
⇠20
=1
qT
lnQ2
q2T
, (51)
with ⇠0
= 2e��E and qT
= |PhT
|/z, we obtain
F cos�h
UU
= � 1
QqT
↵S
2⇡2z2
X
a
e2a
x fa
1
(x)Da!h
1
(z)L
✓
Q2
q2T
◆
, (52)
in agreement with the logarithmic term of Eq. (37).Similarly, from Eq. (16) in the limit |P
hT
|/z � M ,
F cos 2�h
UU
=2⇡M4
Q2
X
a
e2a
x
Z
1
0
d⇠T
⇠3T
J2
✓
⇠T
|PhT
|z
◆
x bf?(2)
bD1
=1
2
1
Q2
↵S
2⇡2z2
X
a
e2a
x
Z
1
0
d⇠T
⇠T
J2
✓
⇠T
|PhT
|z
◆✓
2CF
lnQ2⇠2
T
⇠20
� 5CF
◆
fa
1
(x)Da!h
1
(z)
=1
2
1
Q2
↵S
2⇡2z2
X
a
e2a
xfa
1
(x)Da!h
1
(z)L
✓
Q2
q2T
◆
, (53)
where the final result has been derived by means of the following relations
Z
1
0
d⇠T
⇠T
Jn
✓
⇠T
|PhT
|z
◆
=1
n, (54)
Z
1
0
d⇠T
⇠T
J2
✓
⇠T
|PhT
|z
◆
lnQ2⇠2
T
⇠20
=1
2
✓
lnQ2
q2T
+ 1
◆
. (55)
III. DRELL–YAN
According to [5], the expression for the structure functions assuming TMD factorization is
F cos�
UU
=2
QC(
(h · k1T
)h⇣
(1� c)x1
f?a + c x1
f?a
⌘
f a
1
� M2
M1
h?a
1
⇣
c x2
ha + (1� c)x2
ha
⌘i
(56)
� (h · k2T
)h
fa
1
⇣
c x2
f?a + (1� c)x2
f?a
⌘
� M1
M2
⇣
(1� c)x1
ha + c x1
ha
⌘
ha?
1
i
)
(57)
with c = 0 for the GJ frame and c = 1/2 for the CS frame.The convolution is defined as
C⇥wfaga⇤
=X
a
e2a
Z
d2k1T
d2k2T
�(2) (qT
� k1T
� k2T
)w(k1T
,k2T
)fa(x1
, k21T
)ga(x2
, k22T
) (58)
5
The dominant terms in high-transverse momentum region of the structure function F cos�h
UU
in Eq. (15) are given by
F cos�h
UU
= �4⇡M2
Q
X
a
e2a
x
Z
1
0
d⇠T
⇠2T
J1
✓
⇠T
|PhT
|z
◆✓
M2
h
M2
bfa
1
bD?(1)a!h
z+ x bf?(1)a
bDa!h
1
◆
= � 1
QqT
↵S
2⇡2z2
X
a
e2a
x
Z
1
0
d⇠T
J1
✓
⇠T
|PhT
|z
◆✓
2CF
lnQ2⇠2
T
⇠20
� 3CF
◆
fa
1
(x)Da!h
1
(z) , (49)
where, in the last step, we have substituted the leading-order expressions for bfa
1
and bDa!h
1
in Eqs. (43) and (46),
respectively, and the ones for x bf?(1)a and bD?(1)a!h in Eqs. (44) and (47). Moreover, by means of the equalities
Z
1
0
d⇠T
Jn
✓
⇠T
|PhT
|z
◆
=1
qT
, (50)
Z
1
0
d⇠T
J1
✓
⇠T
|PhT
|z
◆
lnQ2⇠2
T
⇠20
=1
qT
lnQ2
q2T
, (51)
with ⇠0
= 2e��E and qT
= |PhT
|/z, we obtain
F cos�h
UU
= � 1
QqT
↵S
2⇡2z2
X
a
e2a
x fa
1
(x)Da!h
1
(z)L
✓
Q2
q2T
◆
, (52)
in agreement with the logarithmic term of Eq. (37).Similarly, from Eq. (16) in the limit |P
hT
|/z � M ,
F cos 2�h
UU
=2⇡M4
Q2
X
a
e2a
x
Z
1
0
d⇠T
⇠3T
J2
✓
⇠T
|PhT
|z
◆
x bf?(2)
bD1
=1
2
1
Q2
↵S
2⇡2z2
X
a
e2a
x
Z
1
0
d⇠T
⇠T
J2
✓
⇠T
|PhT
|z
◆✓
2CF
lnQ2⇠2
T
⇠20
� 5CF
◆
fa
1
(x)Da!h
1
(z)
=1
2
1
Q2
↵S
2⇡2z2
X
a
e2a
xfa
1
(x)Da!h
1
(z)L
✓
Q2
q2T
◆
, (53)
where the final result has been derived by means of the following relations
Z
1
0
d⇠T
⇠T
Jn
✓
⇠T
|PhT
|z
◆
=1
n, (54)
Z
1
0
d⇠T
⇠T
J2
✓
⇠T
|PhT
|z
◆
lnQ2⇠2
T
⇠20
=1
2
✓
lnQ2
q2T
+ 1
◆
. (55)
III. DRELL–YAN
According to [5], the expression for the structure functions assuming TMD factorization is
F cos�
UU
=2
QC(
(h · k1T
)h⇣
(1� c)x1
f?a + c x1
f?a
⌘
f a
1
� M2
M1
h?a
1
⇣
c x2
ha + (1� c)x2
ha
⌘i
(56)
� (h · k2T
)h
fa
1
⇣
c x2
f?a + (1� c)x2
f?a
⌘
� M1
M2
⇣
(1� c)x1
ha + c x1
ha
⌘
ha?
1
i
)
(57)
with c = 0 for the GJ frame and c = 1/2 for the CS frame.The convolution is defined as
C⇥wfaga⇤
=X
a
e2a
Z
d2k1T
d2k2T
�(2) (qT
� k1T
� k2T
)w(k1T
,k2T
)fa(x1
, k21T
)ga(x2
, k22T
) (58)
we recover the matching of the 1/Q2 at low and high qTF cos 2�hUU
4
C. High transverse momentum
In the high transverse-momentum region (⇤QCD
⌧ qT
) the structure functions can be computed using collinearfactorization and then approximated in the intermediate transverse-momentum region (⇤
QCD
⌧ qT
⌧ Q) as
F cos�h
UU
= � 1
QqT
↵s
2⇡2z2
X
a
xe2a
fa
1
(x)Da
1
(z)L
✓
Q2
q2T
◆
+ fa
1
(x)�
Da
1
⌦ P 0
+Dg
1
⌦ P 0
gq
�
(z)
+�
P 0
⌦ fa
1
+ P 0
qg
⌦ fg
1
�
(x)Da
1
(z)
�
, (37)
F cos 2�h
UU
=1
Q2
↵s
2⇡2z2
X
a
xe2a
fa
1
(x)Da
1
(z)L
✓
Q2
q2T
◆
+ fa
1
(x)�
Da
1
⌦ P 00
+Dg
1
⌦ P 00
gq
�
(z)
+�
P 00
⌦ fa
1
+ P 00
qg
⌦ fg
1
�
(x)Da
1
(z)
�
, (38)
where the factor L is defined as
L
✓
Q2
q2T
◆
= 2CF
lnQ2
q2T
� 3CF
, (39)
and the splitting functions are defined as
P 0
qg
(x) = 2TR
x (2x� 1) ,
P 0
gq
(z) = �2CF
(1� z) , (40)
P 00
(x) = P 0
(x) , P 00
qg
(x) = 4TR
x2 ,
P 00
gq
(z) = 2CF
z. (41)
These results should be compared to the ones obtained from the TMD formula expended at order ↵S
in theintermediate transverse-momentum region (⇤
QCD
⌧ qT
⌧ Q). Specifically, in the small-⇠T
region, ⇠T
⌧ 1/M , usingEq. (36), to the order ↵
S
we obtain
bfa(x, ⇠2T
) =1
2⇡
(
fa(x) +↵S
⇡
X
i
⇣
C(1)
a/i
⌦ f i
⌘
(x, ⇠2T
)
)
eS(⇠
2T ,Q
2) (42)
=1
2⇡
(
fa(x) +↵S
⇡
"
�1
4C
F
✓
ln2Q2⇠2
T
⇠20
� 3 lnQ2⇠2
T
⇠20
◆
fa(x) +X
i
⇣
C(1)
a/i
⌦ f i
⌘
(x, ⇠2T
)
#)
. (43)
Therefore, from Eq. (20) and neglecting the running of ↵S
since it contributes only at the order ↵2
S
, we find
bf (1) a(x, ⇠2T
) ⌘ � 1
M2
1
⇠T
@
@⇠T
bfa(x, ⇠2T
) =1
M2⇠2T
↵S
4⇡2
CF
✓
2 lnQ2⇠2
T
⇠20
� 3
◆
fa(x) , (44)
bf (2) a(x, ⇠2T
) ⌘ 2
M4
✓
1
⇠T
@
@⇠T
◆
2
bfa(x, ⇠2T
) =1
M4⇠4T
↵S
⇡2
CF
✓
2 lnQ2⇠2
T
⇠20
� 5
◆
fa(x) . (45)
In analogy to Eq. (43), for a generic fragmentation function, we write
bDa!h(z, ⇠2T
) =1
2⇡z2
(
Da!h(z) +↵S
⇡
"
�1
4C
F
✓
ln2Q2⇠2
T
⇠20
� 3 lnQ2⇠2
T
⇠20
◆
Da!h(z) +X
i
⇣
C(1)
a/i
⌦Di!h
⌘
(z, ⇠2T
)
#)
,
(46)
while its first two ⇠T
-derivates are given by
bD(1) a!h(z, ⇠2T
) ⌘ � 1
M2
h
1
z2⇠T
@
@⇠T
bDa!h(z, ⇠2T
) =1
z2M2
h
⇠2T
↵S
4⇡2
CF
✓
2 lnQ2⇠2
T
⇠20
� 3
◆
Da!h(z) , (47)
bD(2) a!h(z, ⇠2T
) ⌘ 2
M4
h
✓
1
z2⇠T
@
@⇠T
◆
2
bDa!h(z, ⇠2T
) =1
z2M4
h
⇠4T
↵S
⇡2
CF
✓
2 lnQ2⇠2
T
⇠20
� 5
◆
Da!h(z) . (48)
5
The dominant terms in high-transverse momentum region of the structure function F cos�h
UU
in Eq. (15) are given by
F cos�h
UU
= �4⇡M2
Q
X
a
e2a
x
Z
1
0
d⇠T
⇠2T
J1
✓
⇠T
|PhT
|z
◆✓
M2
h
M2
bfa
1
bD?(1)a!h
z+ x bf?(1)a
bDa!h
1
◆
= � 1
QqT
↵S
2⇡2z2
X
a
e2a
x
Z
1
0
d⇠T
J1
✓
⇠T
|PhT
|z
◆✓
2CF
lnQ2⇠2
T
⇠20
� 3CF
◆
fa
1
(x)Da!h
1
(z) , (49)
where, in the last step, we have substituted the leading-order expressions for bfa
1
and bDa!h
1
in Eqs. (43) and (46),
respectively, and the ones for x bf?(1)a and bD?(1)a!h in Eqs. (44) and (47). Moreover, by means of the equalities
Z
1
0
d⇠T
Jn
✓
⇠T
|PhT
|z
◆
=1
qT
, (50)
Z
1
0
d⇠T
J1
✓
⇠T
|PhT
|z
◆
lnQ2⇠2
T
⇠20
=1
qT
lnQ2
q2T
, (51)
with ⇠0
= 2e��E and qT
= |PhT
|/z, we obtain
F cos�h
UU
= � 1
QqT
↵S
2⇡2z2
X
a
e2a
x fa
1
(x)Da!h
1
(z)L
✓
Q2
q2T
◆
, (52)
in agreement with the logarithmic term of Eq. (37).Similarly, from Eq. (16) in the limit |P
hT
|/z � M ,
F cos 2�h
UU
=2⇡M4
Q2
X
a
e2a
x
Z
1
0
d⇠T
⇠3T
J2
✓
⇠T
|PhT
|z
◆
x bf?(2)
bD1
=1
2
1
Q2
↵S
2⇡2z2
X
a
e2a
x
Z
1
0
d⇠T
⇠T
J2
✓
⇠T
|PhT
|z
◆✓
2CF
lnQ2⇠2
T
⇠20
� 5CF
◆
fa
1
(x)Da!h
1
(z)
=1
2
1
Q2
↵S
2⇡2z2
X
a
e2a
xfa
1
(x)Da!h
1
(z)L
✓
Q2
q2T
◆
, (53)
where the final result has been derived by means of the following relations
Z
1
0
d⇠T
⇠T
Jn
✓
⇠T
|PhT
|z
◆
=1
n, (54)
Z
1
0
d⇠T
⇠T
J2
✓
⇠T
|PhT
|z
◆
lnQ2⇠2
T
⇠20
=1
2
✓
lnQ2
q2T
+ 1
◆
. (55)
III. DRELL–YAN
According to [5], the expression for the structure functions assuming TMD factorization is
F cos�
UU
=2
QC(
(h · k1T
)h⇣
(1� c)x1
f?a + c x1
f?a
⌘
f a
1
� M2
M1
h?a
1
⇣
c x2
ha + (1� c)x2
ha
⌘i
(56)
� (h · k2T
)h
fa
1
⇣
c x2
f?a + (1� c)x2
f?a
⌘
� M1
M2
⇣
(1� c)x1
ha + c x1
ha
⌘
ha?
1
i
)
(57)
with c = 0 for the GJ frame and c = 1/2 for the CS frame.The convolution is defined as
C⇥wfaga⇤
=X
a
e2a
Z
d2k1T
d2k2T
�(2) (qT
� k1T
� k2T
)w(k1T
,k2T
)fa(x1
, k21T
)ga(x2
, k22T
) (58)
we recover the whole leading term of the high qT expansion F cos�hUU
Preliminary results
THANK YOU FOR SURVIVING A TALK WITHOUT PLOTS
• Examine all possible twist-4 contributions
• Recover the whole cos2phi result
• Extend the new formalism to Drell-Yan
• Compare with experiment (azimuthal modulations)
Outlook
