Extraction of partonic transverse momentum distributions from semiinclusive deepinelastic scattering, DrellYan and Zboson production
Revised: May
Extraction of partonic transverse momentum distributions from semiinclusive deepinelastic scattering, DrellYan and Zboson production
Alessandro Bacchetta 0 1 2
Filippo Delcarro 0 1 2
Cristian Pisano 0 1 2
Marco Radici 0 1 2
Andrea Signori 0 1 3
0 12000 Jefferson Avenue , Newport News, VA 23606 , U.S.A
1 via Bassi 6, I27100 Pavia , Italy
2 INFN Sezione di Pavia
3 Theory Center, Thomas Jefferson National Accelerator Facility
We present an extraction of unpolarized partonic transverse momentum distributions (TMDs) from a simultaneous fit of available data measured in semiinclusive deepinelastic scattering, DrellYan and Z boson production. To connect data at different scales, we use TMD evolution at nexttoleading logarithmic accuracy. The analysis is restricted to the lowtransversemomentum region, with no matching to fixedorder calculations at high transverse momentum. We introduce specific choices to deal with TMD evolution at low scales, of the order of 1 GeV2. This could be considered as a first attempt at a global fit of TMDs.
aDipartimento di Fisica; Università di Pavia

3
Data analysis
3.1
Semiinclusive DIS data
Semiinclusive DIS
DrellYan and Z production
TMDs and their evolution
3.1.1
3.1.2
Hermes data
Compass data
Lowenergy DrellYan data
Zboson production data
The replica method
2.1
2.2
2.3
3.2
3.3
3.4
4.1
4.2
4.3
4
Results 5
1
Conclusions
Introduction
1 Introduction
2
Formalism
Agreement between data and theory
Transverse momentum dependence at 1 GeV
Stability of our results
Parton distribution functions describe the internal structure of the nucleon in terms of its
elementary constituents (quarks and gluons). They cannot be easily computed from first
principles, because they require the ability to carry out Quantum Chromodynamics (QCD)
calculations in its nonperturbative regime. Many experimental observables in hard
scattering experiments involving hadrons are related to parton distribution functions (PDFs) and
fragmentation functions (FFs), in a way that is specified by factorization theorems (see,
e.g., refs. [
1, 2
]). These theorems also elucidate the universality properties of PDFs and FFs
(i.e., the fact that they are the same in different processes) and their evolution equations
(i.e., how they get modified by the change in the hard scale of the process). Availability of
measurements of different processes in different experiments makes it possible to test
factorization theorems and extract PDFs and FFs through socalled global fits. On the other
side, the knowledge of PDFs and FFs allows us to make predictions for other hard hadronic
processes. These general statements apply equally well to standard collinear PDFs and
FFs and to transversemomentumdependent parton distribution functions (TMD PDFs)
and fragmentation functions (TMD FFs). Collinear PDFs describe the distribution of
partons integrated over all components of partonic momentum except the one collinear to the
parent hadron; hence, collinear PDFs are functions of the parton longitudinal momentum
fraction x. TMD PDFs (or TMDs for short) include also the dependence on the transverse
– 1 –
?
momentum k . They can be interpreted as threedimensional generalizations of collinear
PDFs. Similar arguments apply to collinear FFs and TMD FFs [3].
There are several differences between collinear and TMD distributions. From the formal
point of view, factorization theorems for the two types of functions are different, implying
also different universality properties and evolution equations [4]. From the experimental
point of view, observables related to TMDs require the measurement of some transverse
momentum component much smaller than the hard scale of the process [5, 6]. For
instance, DeepInelastic Scattering (DIS) is characterized by a hard scale represented by the
4momentum squared of the virtual photon ( Q2). In inclusive DIS this is the only scale
If Ph2T
of the process, and only collinear PDFs and FFs can be accessed. In semiinclusive DIS
(SIDIS) also the transverse momentum of the outgoing hadron (PhT ) can be measured [7, 8].
Q2, TMD factorization can be applied and the process is sensitive to TMDs [
2
].
If polarization is taken into account, several TMDs can be introduced [7, 9–12].
At?
tempts to extract some of them have already been presented in the past [13–21]. In this
work, we focus on the simplest ones, i.e., the unpolarized TMD PDF f1q(x; k?2) and the
unpolarized TMD FF D1q!h(z; P 2 ), where z is the fractional energy carried by the detected
hadron h, k? is the transverse momentum of the parton with respect to the parent hadron,
and P
? is the transverse momentum of the produced hadron with respect to the parent
parton. Despite their simplicity, the phenomenology of these unpolarized TMDs presents
several challenges [22]: the choice of a functional form for the nonperturbative
components of TMDs, the inclusion of a possible dependence on partonic flavor [23], the
implementation of TMD evolution [4, 24], the matching to fixedorder calculations in collinear
factorization [25].
We take into consideration three kinds of processes: SIDIS, DrellYan processes (DY)
and the production of Z bosons. To date, they represent all possible processes where
experimental information is available for unpolarized TMD extractions. The only important
process currently missing is electronpositron annihilation, which is particularly important
for the determination of TMD FFs [24]. This work can therefore be considered as the first
attempt at a global fit of TMDs.
The paper is organized as follows. In section 2, the general formalism for TMDs in
SIDIS, DY processes, and Z production is briefly outlined, including a description of the
assumptions and approximations in the phenomenological implementation of TMD evolution
equations. In section 3, the criteria for selecting the data analyzed in the fit are summarized
and commented. In section 4, the results of our global fit are presented and discussed. In
section 5, we summarize the results and present an outlook for future improvements.
2
2.1
Formalism
Semiinclusive DIS
In oneparticle SIDIS, a lepton ` with momentum l scatters off a hadron target N with mass
M and momentum P . In the final state, the scattered lepton momentum l0 is measured
together with one hadron h with mass Mh and momentum Ph. The corresponding reaction
– 2 –
formula is
invariants
where
`(l) + N (P ) ! `(l0) + h(Ph) + X :
The spacelike momentum transfer is q = l
l0, with Q2 =
q2. We introduce the usual
mhN (x; z; jPhT j; Q2) =
d Nh =(dxdzdjPhT jdQ2)
d DIS=(dxdQ2)
;
where d Nh is the differential cross section for the SIDIS process and d DIS is the
corresponding inclusive one, and where PhT is the component of Ph transverse to q (we follow
here the notation suggested in ref. [26]). In the singlephotonexchange approximation, the
multiplicities can be written as ratios of structure functions (see ref. [8] for details):
mhN (x; z; jPhT j; Q2) =
2 jPhT jFUU;T (x; z; Ph2T ; Q2) + 2 "jPhT jFUU;L(x; z; Ph2T ; Q2)
FT (x; Q2) + "FL(x; Q2)
x =
Q2
2 P q
;
y =
P q
P l
;
z =
P Ph
P q
;
=
2M x
Q
:
The available data refer to SIDIS hadron multiplicities, namely to the differential
number of hadrons produced per corresponding inclusive DIS event. In terms of cross sections,
we define the multiplicities as
HJEP06(217)8
(2.1)
(2.2)
(2.3)
;
(2.4)
(2.5)
(2.6)
" =
1
y
4
Q2. In these limits, the structure function
FUU;L of eq. (2.4) can be neglected [27]. The structure function FL in the denominator
contains contributions involving powers of the strong coupling constant
S at an order
that goes beyond the level reached in this analysis; hence, it will be consistently neglected
(for measurements and estimates of the FL structure function see, e.g., refs. [28, 29] and
references therein).
To express the structure functions in terms of TMD PDFs and FFs, we rely on the
factorized formula for SIDIS [
2, 30–37
] (see figure 1 for a graphical representation of the
involved transverse momenta):
FUU;T (x; z; Ph2T ; Q2) =
HUU;T (Q2)
a
X
a
x
Z
?
d2k d2P f
? 1a x; k?2; Q2 D1a~ h z; P?2 ; Q2 (2) zk?
PhT +P
?
+ YUU;T Q2; Ph2T
? (not measured). The struck parton fragments into a hadron,
? (not measured). The total measured
transversemomentum of the final hadron is PhT . When Q2 is very large, the longitudinal components are all
much larger than the transverse components. In this regime, PhT
zk? + P?.
Here, HUU;T is the hard scattering part; f1a(x; k2 ; Q2) is the TMD PDF of unpolarized
?
partons with flavor a in an unpolarized proton, carrying longitudinal momentum fraction
x and transverse momentum k?. The D1a~ h(z; P 2 ; Q2) is the TMD FF describing the
fragmentation of an unpolarized parton with flavor a into an unpolarized hadron h carrying
?
longitudinal momentum fraction z and transverse momentum P
? (see figure 1). TMDs
generally depend on two energy scales [
2
], which enter via the renormalization of ultraviolet
and rapidity divergencies. In this work we choose them to be equal and set them to Q2. The
term YUU;T is introduced to ensure a matching to the perturbative fixedorder calculations
at higher transverse momenta.
In our analysis, we neglect any correction of the order of M 2=Q2 or higher to eq. (2.6).
At large Q2 this is well justified. However, fixedtarget DIS experiments typically collect
a large amount of data at relatively low Q2 values, where these assumptions should be all
tested in future studies. The reliability of the theoretical description of SIDIS at low Q2
has been recently discussed in refs. [39, 40].
Eq. (2.6) can be expanded in powers of S. In the present analysis, we will consider only
the terms at order S0. In this case HUU;T (Q2)
a
corrections include large logarithms L
e2a and YUU;T
log z2Q2=Ph2T , so that SL
0. However, perturbative
1. In the present
analysis, we will take into account all leading and NexttoLeading Logarithms (NLL).1
In these approximations ( S0 and NLL), only the first term in eq. (2.6) is relevant
(often in the literature this has been called W term). We expect this term to provide a
a detailed treatment of the matching to the high Ph2T
good description of the structure function only in the region where Ph2T
Q2 region to future investigations
Q2. We leave
(see, e.g., ref. [25]).
1We remark that formulas at NNLL are available in the literature [41].
– 4 –
transform of the part of eq. (2.6) involving TMDs. The structure function thus reduces to
FUU;T (x; z; Ph2T ; Q2)
2
d T T J0 T jPhT j=z f~1a x; T2 ; Q2 D~ 1a~ h z; T2 ; Q2 ;
where we introduced the Fourier transforms of the TMD PDF and FF according to
f~1a x; T2 ; Q2 =
D~ 1a~ h z; T2 ; Q2 =
Z 1
0
djk?jjk?jJ0 T jk?j f1a x; k?2; Q2 ;
Z 1 djzP2?j jP?jJ0 T jP?j=z D1a~ h z; P?2 ; Q2 :
0
2.2
DrellYan and Z production
In a DrellYan process, two hadrons A and B with momenta PA and PB collide at a
centerofmass energy squared s = (PA + PB)2 and produce a virtual photon or a Z boson plus
hadrons. The boson decays into a leptonantilepton pair. The reaction formula is
A(PA) + B(PB) ! [ =Z + X !]`+(l) + ` (l0) + X:
The invariant mass of the virtual photon is Q2 = q2 with q = l + l0. We introduce the
rapidity of the virtual photon/Z boson
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
where the z direction is defined along the momentum of hadron A (see figure 2).
The cross section can be written in terms of structure functions [42, 43]. For our
purposes, we need the unpolarized cross section integrated over d
and over the azimuthal
angle of the virtual photon,
d
dQ2 dqT2 d
=
0
;Z
FU1U +
21 FU2U :
The elementary cross sections are
0 =
4 2 e2m ;
3Q2s
0Z =
s sin2
2 em
W cos2 W
BR(Z ! `+` ) (Q2
MZ2 );
where
W is Weinberg’s angle, MZ is the mass of the Z boson, and BR(Z ! `+` ) is the
branching ratio for the Z boson decay in two leptons. We adopted the narrowwidth
approximation, i.e., we neglect contributions for Q2 6= MZ2 . We used the values sin2
MZ = 91:18 GeV, and BR(Z ! `+` ) = 3:366 [44]. Similarly to the SIDIS case, in the
W = 0:2313,
kinematic limit qT2
Q2 the structure function FU2U can be neglected (for measurement
and estimates of this structure function see, e.g., ref. [45] and references therein).
The longitudinal momentum fractions of the annihilating quarks can be written in
terms of rapidity in the following way
1
2
=
log
q0 + qz
q0
qz
;
xA = p e ;
Q
s
xB = p e :
Q
s
– 5 –
from two hadrons collide. They have transverse momenta k?A and k?B (not measured). They
produce a virtual photon with (measured) transverse momentum qT = k?A + k?B with respect to
the hadron collision axis.
Some experiments use the variable xF , which is connected to the other variables by the
following relations
p
s xF
Q 2
= sinh 1
;
xA =
r Q2
s
x
2
4
+
F +
xF
2
;
xB = xA
xF :
(2.15)
The structure function FU1U can be written as (see figure 2 for a graphical representation
of the involved transverse momenta)
FU1U (xA; xB; qT2 ; Q2) =
HU1aU (Q2)
X
a
Z
+ YU1U Q2; qT2
+ O M 2=Q2 :
d2k?Ad2k?Bf1a xA; k?2A; Q2 f1a xB; k?2B; Q2 (2) k?A
qT +k?B
PB
q
quark
kA
qT
nucleon
2We remind the reader that the value of weak isospin I3 is equal to +1=2 for u, c, t and 1=2 for d, s, b.
– 6 –
As in the SIDIS case, in our analysis we neglect the YUU term and we consider the hard
coefficients only up to leading order in the couplings, i.e.,
where2
HUU; (Q2)
1a
e
2
a ;
Nc
HUU;Z (Q2)
1a
Va2 + A2a ;
Nc
Va = I3a
2ea sin2
W ;
Aa = I3a :
The structure function can be conveniently expressed as a Fourier transform of the
righthand side of eq. (2.16) as
FU1U (xA; xB; qT2 ; Q2)
2
X
a
HUU
1a Z 1
0
d T T J0 T jqT j f
~1a xA; T2 ; Q2 f~1a xB; T2 ; Q2 :
(2.16)
(2.17)
(2.18)
(2.19)
fe1a(x; T2 ; Q2) =
Ca=i
f1i (x; ; b2
)
De1a!h(z; T2 ; Q2) =
C^a=i
D1i!h (z; ; b2
)
We choose the scale b to be
where E is the Euler constant and
b =
2e E
X
i=q;q;g
X
i=q;q;g
eS( b2;Q2)
eS( b2;Q2)
Q2
2
b
Q2
2
b
( T ; min; max) = max
K( ; b)
K( ; b)
Q2 gK( T )
Q2
0
Q2 gK( T )
Q2
0
fe1NP(x; T2 ) ;
a
De1aN!Ph(z; T2 ) :
;
1
1
e T4 = m4ax !1=4
e T4 = m4in
:
(2.20)
(2.21)
(2.22)
(2.23)
Evolution equations quantitatively describe the connection between different values for the
energy scales. In the following we will set their initial values to
as Q2, so that only Q2 and
b2 need to be specified in a TMD distribution. Following the
formalism of refs. [
2, 34
], the unpolarized TMD distribution and fragmentation functions
in configuration space for a parton with flavor a at a certain scale Q2 can be written as
b2 and their final values
This variable replaces the simple dependence upon T in the perturbative parts of the
TMD definitions of eqs. (2.20), (2.21). In fact, at large T these parts are no longer reliable.
Therefore, the
is chosen to saturate on the maximum value max, as suggested by the CSS
formalism [
2, 34
]. On the other hand, at small T the TMD formalism is not valid and should
be matched to the fixedorder collinear calculations. The way the matching is implemented
is not unique. In any case, the TMD contribution can be arbitrarily modified at small T .
In our approach, we choose to saturate
at the minimum value
min / 1=Q. With the
appropriate choices, for T = 0 the Sudakov exponent vanishes, as it should [46, 47]. Our
choice partially corresponds to modifying the resummed logarithms as in ref. [48] and to
other similar modifications proposed in the literature [25, 49]. One advantage of these kind
of prescriptions is that by integrating over the impact parameter T , the collinear expression
for the cross section in terms of collinear PDFs is recovered, at least at leading order [25].
The choice of the functional form in eq. (2.23) is arbitrary. In the original CSS article [31], a
functional form based on a squareroot saturation was adopted. With our choice,
rapidly
saturates to its limiting values
min and
max, leaving a reasonably wide region where it
is equal to T . This is particularly critical for low values of Q, where
min and
max are
very close. We remind the reader that there are different schemes available to deal with
the high T region, such as the socalled “complex prescription” [50] or an extrapolation
of the perturbative small T calculation to the large T region based on dynamical power
corrections [51].
– 7 –
The values of max and min could be regarded as boundaries between the regions of
“small” and “large” distances. They are not completely arbitrary, but there is also no way
to determine them precisely. We choose to fix them to the values
max = 2e E GeV 1
1:123 GeV 1
;
min = 2e E =Q :
(2.24)
The motivations are the following:
with the above choices, the scale b is constrained between 1 GeV and Q, so that the
collinear PDFs are never computed at a scale lower than 1 GeV and the lower limit
of the integrals contained in the definition of the perturbative Sudakov factor (see
eq. (2.30)) can never become larger than the upper limit;
at Q = Q0 = 1 GeV, max =
min and there are no evolution effects; the TMD is
simply given by the corresponding collinear function multiplied by a nonperturbative
contribution depending on k? (plus possible corrections of order S from the Wilson
coefficients).
At NLL accuracy, for our choice of scales K( ; b) = 0. Similarly, the C and C^ are
perturbatively calculable Wilson coefficients for the TMD distribution and fragmentation
functions, respectively. They are convoluted with the corresponding collinear functions
according to
Ca=i
f1i (x; ; b2) =
C^a=i
D1i!h (z; ; b2) =
Z 1 du
x
u
Ca=i u
x
Z 1 du C^a=i u
z
z u
; ; S
; ; S
2
b
2
b
f1i(u; b2) ;
D1i!h(u; b2) :
In the present analysis, we consider only the leadingorder term in the
S expansion
for C and C^, i.e.,
Ca=i u
x
; ; S
2
b
reduces to
ai (1
x=u);
C^a=i u
z
; ; S
2
b
ai (1
z=u):
(2.27)
As a consequence of the choices we made, the expression for the evolved TMD functions
fe1a(x; T2 ; Q2) = f1a(x; b2) eS( b2;Q2) egK( T ) ln(Q2=Q20) fe1NP(x; T2 ) ;
a
De1a!h(z; T2 ; Q2) = D1a!h(z; b2) eS( b2;Q2) egK( T ) ln(Q2=Q20) De1aN!Ph(z; T2 ) :
The Sudakov exponent S can be written as
S( b2; Q2) =
A
S( 2) ln
+ B
S( 2)
;
Q2
2
where the functions A and B have a perturbative expansions of the form
A
S( 2) =
1
X Ak
k=1
B
S( 2) =
1
X Bk
k=1
S
k
:
Z Q2 d 2
2
b
S
2
k
;
– 8 –
(2.25)
(2.26)
(2.28)
(2.29)
(2.30)
(2.31)
To NLL accuracy, we need the following terms [
31, 52
]
A1 = CF ;
A2 =
CF CA
1
2
67
18
2
6
5
9
Nf ;
3
2
B1 =
CF :
(2.32)
We use the approximate analytic expression for
S at NLO with the
QCD = 340 MeV,
296 MeV, 214 MeV for three, four, five flavors, respectively, corresponding to a value of
S(MZ ) = 0:117. We fix the flavor thresholds at mc = 1:5 GeV and mb = 4:7 GeV. The
integration of the Sudakov exponent in eq. (2.30) can be done analytically (for the complete
expressions see, e.g., refs. [36, 53, 54]).
Following refs. [55–57], for the nonperturbative Sudakov factor we make the traditional
HJEP06(217)8
choice
at low Q2.
with g2 a free parameter. Recently, several alternative forms have been proposed [58, 59].
Also, recent theoretical studies aimed at calculating this term using nonperturbative
methods [60]. All these choices should be tested in future studies. In ref. [61], a good agreement
with data was achieved even without this term, but this is not possible when including data
In this analysis, for the collinear PDFs f1a we adopt the GJR08FFnloE set [62] through
the LHAPDF library [63], and for the collinear fragmentation functions the DSS14 NLO
set for pions [64] and the DSS07 NLO set for kaons [65].3 We will comment on the use of
other PDF sets in section 4.3.
We parametrize the intrinsic nonperturbative parts of the TMDs in the following ways
gK ( T ) =
After performing the antiFourier transform, the f1NP and D1NP in momentum space
correspond to
(2.34)
:
(2.35)
(2.36)
(2.37)
f1NP(x; k?2) =
a
D1aN!Ph(z; P 2 ) =
?
1
1
1 + k
g1a +
g3a!h +
2
?
g1a
1
k
2
?
2 e g1a ;
F =z2 g42a!h
P 2
P 2
e g3a?!h + F z?2 e g4a?!h :
P 2
The TMD PDF at the starting scale is therefore a normalized sum of a Gaussian with
variance g1 and the same Gaussian weighted by a factor k2 . The TMD FF at the starting
scale is a normalized sum of a Gaussian with variance g3 and a second Gaussian with
variance g4 weighted by a factor F P?2 =z2. The choice of this particular functional forms
is motivated by model calculations: the weighted Gaussian in the TMD PDF could arise
?
3After the completion of our analysis, a new set of kaon fragmentation function was presented in ref. [66].
– 9 –
from the presence of components of the quark wave function with angular momentum
L = 1 [67–71]. Similar features occur in models of fragmentation functions [38, 67, 72].
The Gaussian width of the TMD distributions may depend on the parton flavor
a [23, 38, 73]. In the present analysis, however, we assume they are flavor independent.
The justification for this choice is that most of the data we are considering are not
sufficiently sensitive to flavor differences, leading to unclear results. We will devote attention
to this issue in further studies.
Finally, we assume that the Gaussian width of the TMD depends on the fractional
longitudinal momentum x according to
HJEP06(217)8
(2.38)
(2.39)
(2.40)
g1(x) = N1 (1
(1
x) x
x^) x^
;
g3;4(z) = N3;4 (z^ + ) (1
(z + ) (1
z)
z^)
;
where ; ; and N1
functions we have
g1(x^) with x^ = 0:1, are free parameters. Similarly, for fragmentation
where ; ; ; and N3;4
g3;4(z^) with z^ = 0:5 are free parameters.
can be computed analytically:
The average transverse momentum squared for the distributions in eq. (2.36) and (2.37)
k
?2 (x) =
g1(x) + 2 g12(x)
1 + g1(x)
;
P?2 (z) =
g32(z) + 2 F g43(z)
g3(z) + F g42(z)
:
3
Data analysis
The main goals of our work are to extract information about intrinsic transverse momenta,
to study the evolution of TMD parton distributions and fragmentation functions over a large
enough range of energy, and to test their universality among different processes. To achieve
this we included measurements taken from SIDIS, DrellYan and Z boson production from
different experimental collaborations at different energy scales. In this section we describe
the data sets considered for each process and the applied kinematic cuts.
Table 1 refers to the data sets for SIDIS off proton target (Hermes experiment) and
presents their kinematic ranges. The same holds for table 2, table 3, table 4 for SIDIS
off deuteron (Hermes and Compass experiments), DrellYan events at low energy and
Z boson production respectively. If not specified otherwise, the theoretical formulas are
computed at the average values of the kinematic variables in each bin.
3.1
Semiinclusive DIS data
The SIDIS data are taken from Hermes [74] and Compass [75] experiments. Both data
sets have already been analyzed in previous works, e.g., refs. [23, 76], however they have
never been fitted together, including also the contributions deriving from TMD evolution.
The application of the TMD formalism to SIDIS depends on the capability of identifying
the current fragmentation region. This task has been recently discussed in ref. [39], where
the authors point out a possible overlap among different fragmentation regions when the
Cuts
Points
Max. Q2
x range
Hermes Hermes Hermes Hermes
p !
+
p !
Cuts
Cuts
Points
p
s
Q range
Kin. var.
[79]
45
19.4 GeV
49 GeV
=0.40
[79]
45
23.8 GeV
49 GeV
=0.21
qT < 0:2 Q + 0:5 GeV
[79]
78
27.4 GeV
=0.03
E605
[80]
35
38.8 GeV
xF = 0:1
59, 1114 GeV 79, 10.511.5 GeV
with different centerofmass energies.
hard scale Q is sufficiently low. In this paper we do not tackle this problem and we leave
it to future studies. As described in tables 1 and 2, we identify the current fragmentation
region operating a cut on z only, namely 0:2 < z < 0:74.
HJEP06(217)8
Reference
Cuts
Points
p
s
Normalization
[81]
31
Another requirement for the applicability of TMD factorization is the presence of two
separate scales in the process. In SIDIS, those are the Q2 and Ph2T , which should satisfy
the condition Ph2T
Q2, or more precisely Ph2T =z2
Q2. We implement this condition
by imposing PhT < min[0:2 Q; 0:7 Qz] + 0:5 GeV. With this choice, Ph2T is always smaller
than Q2=3, but in a few bins (at low Q2 and z) Ph2T =z2 may become larger than Q2. The
applicability of TMD factorization in this case could be questioned. However, as we will
explain further in section 4.3, we can obtain a fit that can describe a wide region of PhT and
can also perform very well in a restricted region, where TMD factorization certainly holds.
All these choices are summarized in tables 1 and 2.
3.1.1
Hermes data
Hermes hadron multiplicities are measured in a fixed target experiment, colliding a
27:6 GeV lepton beam on a hydrogen (p) or deuterium (D) gas target, for a total of 2688
points. These are grouped in bins of (x; z; Q2; PhT ) with the average values of (x; Q2)
ranging from about (0:04; 1:25 GeV2) to (0:4; 9:2 GeV2). The collinear energy fraction z in
eq. (2.2) ranges in 0:1
z
0:9. The transverse momentum of the detected hadron satisfies
0:1 GeV
jPhT j
1:3 GeV. The peculiarity of Hermes SIDIS experiment lies in the
ability of its detector to distinguish between pions and kaons in the final state, in addition to
determining their momenta and charges. We consider eight different combinations of target
(p; D) and detected charged hadron (
; K ). The Hermes collaboration published two
distinct sets, characterized by the inclusion or subtraction of the vector meson contribution.
In our work we considered only the data set where this contribution has been subtracted.
3.1.2
Compass data
The Compass collaboration extracted multiplicities for chargeseparated but unidentified
hadrons produced in SIDIS off a deuteron (6LiD) target [75]. The number of data points
is an order of magnitude higher compared to the Hermes experiment. The data are
organized in multidimensional bins of (x; z; Q2; PhT ), they cover a range in (x; Q2) from about
(0:005; 1:11 GeV2) to (0:09; 7:57 GeV2) and the interval 0:2
z
0:8. The multiplicities
published by Compass are affected by normalization errors (see the erratum to ref. [75]).
In order to avoid this issue, we divide the data in each bin in (x; z; Q2) by the data point
with the lowest Ph2T in the bin. As a result, we define the normalized multiplicity as
mnorm(x; z; Ph2T ; Q2) =
mhN (x; z; Ph2T ; Q2)
mhN (x; z; min[Ph2T ]; Q2) ;
(3.1)
where the multiplicity mhN is defined in eq. (2.3). When fitting normalized multiplicities,
the first data point of each bin is considered as a fixed constraint and excluded from the
degrees of freedom.
We analyze DrellYan events collected by fixedtarget experiments at lowenergy. These
data sets have been considered also in previous works, e.g., in refs. [56, 57, 77, 78]. We
used data sets from the E288 experiment [79], which measured the invariant dimuon cross
section Ed3 =dq3 for the production of
+
pairs from the collision of a proton beam
with a fixed target, either composed of Cu or Pt. The measurements were performed using
proton incident energies of 200, 300 and 400 GeV, producing three different data sets. Their
respective center of mass energies are p
s = 19:4; 23:8; 27:4 GeV. We also included the set
p
of measurements Ed3 =dq3 from E605 [80], extracted from the collision of a proton beam
with an energy of 800 GeV ( s = 38:8 GeV) on a copper fixed target.
The explored Q values are higher compared to the SIDIS case, as can be seen in table 3.
E288 provides data at fixed rapidity, whereas E605 provides data at fixed xF = 0:1. We
can apply TMD factorization if qT2
Q2, where qT is the transverse momentum of the
intermediate electroweak boson, reconstructed from the kinematics of the final state leptons.
We choose qT < 0:2 Q + 0:5 GeV. As suggested in ref. [79], we consider the target nuclei as
an incoherent ensemble composed 40% by protons and 60% by neutrons.
As we already observed, results from E288 and E605 experiments are reported as Edd33q ;
this variable is related to the differential cross section of eq. (2.12) in the following way:
HJEP06(217)8
Ed3
d3q
=
d
3
d d qT dqT
)
d
2
d d(qT2 ) ;
Ed3
d3q
=
where
is the polar angle of qT and the third term is the average over . Therefore, the
invariant dimuon cross section can be obtained from eq. (2.12) integrating over Q2 and
adding a factor 1= to the result
Numerically we checked that integrating in Q2 only the prefactor q (see eq. (2.13))
introduces only a negligible error in the theoretical estimates. We also assume that em
does not change within the experimental bin. Therefore, for DrellYan we obtain
where Qi;f are the lower and upper values in the experimental bin.
(3.2)
(3.3)
(3.4)
In order to reach higher Q and qT values, we also consider Z boson production in collider
experiments at Tevatron. We analyze data from CDF and D0, collected during Tevatron
collaborations studied the differential cross section for the production of an e+e
pair from
pp collision through an intermediate Z vector boson, namely pp ! Z ! e+e + X.
The invariant mass distribution peaks at the Zpole, Q
MZ , while the transverse
momentum of the exchanged Z ranges in 0 < qT < 20 GeV. We use the same kinematic
cut applied to DrellYan events: qT < 0:2 Q + 0:5 GeV = 18:7 GeV, since Q is fixed to MZ .
The observable measured in CDF and D0 is
apart from the case of D0 Run II, for which the published data refer to 1=
d =dqT . In
order to work with the same observable, we multiply the D0 Run II data by the total cross
section of the process exp = 255:8
16 pb [85]. In this case, we add in quadrature the
uncertainties of the total cross section and of the published data.
We normalize our functional form with the factors listed in table 4. These are the
same normalization factors used in ref. [78], computed by comparing the experimental total
cross section with the theoretical results based on the code of ref. [86]. These factors are
not precisely consistent with our formulas. In fact, as we will discuss in section 4.3 a 5%
increase in these factors would improve the agreement with data, without affecting the
TMD parameters.
3.4
The replica method
Our fit is based on the replica method. In this section we describe it and we give a definition
of the 2 function minimized by the fit procedure. The fit and the error analysis are carried
out using a similar Monte Carlo approach as in refs. [23, 87, 88] and taking inspiration from
the work of the NeuralNetwork PDF (NNPDF) collaboration (see, e.g., refs. [89–91]). The
approach consists in creating M replicas of the data points. In each replica (denoted by
the index r), each data point i is shifted by a Gaussian noise with the same variance as
the measurement. Each replica, therefore, represents a possible outcome of an independent
experimental measurement, which we denote by mhN;r(x; z; Ph2T ; Q2). The number of replicas
is chosen so that the mean and standard deviation of the set of replicas accurately reproduces
the original data points. In this case 200 replicas are sufficient for the purpose. The error for
each replica is taken to be equal to the error on the original data points. This is consistent
with the fact that the variance of the M replicas should reproduce the variance of the
A minimization procedure is applied to each replica separately, by minimizing the
original data points.
following error function:
Er2(fpg) =
X
i
mhN;r(xi; zi; Ph2T i; Qi2)
mhN;theo(xi; zi; Ph2T i; fpg)
2
mhN;2stat +
mhN;2sys (xi; zi; Ph2T i; Qi2)+
mhN;theo(xi; zi; Ph2T i)
2
: (3.6)
The sum runs over the i experimental points, including all species of targets N and
finalstate hadrons h. In each z bin for each replica the values of the collinear fragmentation
functions D1a~ h are independently modified with a Gaussian noise with standard deviation
equal to the theoretical error
D1a~ h. In this work we rely on different parametrizations for
D1a~ h: for pions we use the DSEHS analysis [64] at NLO in
parametrization [65] at LO in
S. The uncertainties
in ref. [92]; they represents the only source of uncertainty in
S; for kaons we use the DSS
D1a~ h are estimated from the plots
mhN;theo. Statistical and
systematic experimental uncertainties
mhN;stat and
mhN;sys are taken from the experimental
collaborations. We do not take into account the covariance among different kinematic bins.
We minimize the error function in eq. (3.6) with Minuit [93]. In each replica we
randomize the starting point of the minimization, to better sample the space of fit parameters.
The final outcome is a set of M different vectors of bestfit parameters, fp0rg; r = 1; : : : M,
with which we can calculate any observable, its mean, and its standard deviation. The
distribution of these values needs not to be necessarily Gaussian. In fact, in this case the 1
values of 2/d.o.f. should be peaked around one.
confidence interval is different from the 68% interval. The latter can simply be computed
for each experimental point by rejecting the largest and the lowest 16% of the M values.
Although the minimization is performed on the function defined in eq. (3.6), the
agreement of the M replicas with the original data is expressed in terms of a 2 function defined
as in eq. (3.6) but with the replacement mhN;r ! mhN , i.e., with respect to the original data
set. If the model is able to give a good description of the data, the distribution of the M
4
Our work aims at simultaneously fitting for the first time data sets related to different
experiments. In the past, only fits related either to SIDIS or hadronic collisions have been
presented. Here we mention a selection of recent existing analyses.
In ref. [23], the authors fitted Hermes multiplicities only (taking into account a total of
1538 points) without taking into account QCD evolution. In that work, a flavor
decomposition in transverse momentum of the unpolarized TMDs and an analysis of the kinematic
dependence of the intrinsic average square transverse momenta were presented. In ref. [76]
the authors fitted Hermes and Compass multiplicities separately (576 and 6284 points
respectively), without TMD evolution and introducing an adhoc normalization for
Compass data. A fit of SIDIS data including TMD evolution was performed on measurements
by the H1 collaboration of the socalled transverse energy flow [55, 94].
Looking at data from hadronic collisions, Konychev and Nadolsky [57] fitted data of
lowenergy DrellYan events and Zboson production at Tevatron, taking into account TMD
evolution at NLL accuracy (this is the most recent of a series of important papers on the
subject [56, 77, 95]). They fitted in total 98 points. Contrary to our approach, Konychev
and Nadolsky studied the quality of the fit as a function of max. They found that the
best value for max is 1:5 GeV 1 (to be compared to our choice
max
1:123 GeV 1, see
section 2.3). Comparisons of bestfit values in the nonperturbative Sudakov form factors are
delicate, since the functional form is different from ours. In 2014 D’Alesio, Echevarria, Melis,
Scimemi performed a fit [78] of DrellYan data and Zboson production data at Tevatron,
Parameters
11
12629
363
1:55
0:05
focusing in particular on the role of the nonperturbative contribution to the kernel of TMD
evolution. This is the fit with the highest accuracy in TMD evolution performed up to
date (NNLL in the Sudakov exponent and O( S) in the Wilson coefficients). In the same
year Echevarria, Idilbi, Kang and Vitev [15] presented a parametrization of the unpolarized
TMD that described qualitatively well some bins of Hermes and Compass data, together
with DrellYan and Zproduction data. A similar result was presented by Sun, Isaacson,
Yuan and Yuan [96].
In the following, we detail the results of a fit to the data sets described in section 3 with
a flavorindependent configuration for the transverse momentum dependence of unpolarized
TMDs. In table 5 we present the total 2. The number of degrees of freedom (d.o.f.) is given
by the number of data points analyzed reduced by the number of free parameters in the
error function. The overall quality of the fit is good, with a global 2/d.o.f. = 1:55
0:05.
Uncertainties are computed as the 68% confidence level (C.L.) from the replica methodology.
4.1
Agreement between data and theory
The partition of the global 2 among SIDIS off a proton, SIDIS off a deuteron, DrellYan
and Z production events is given in table 6, 7, 8, 9 respectively.
Semiinclusive DIS. For SIDIS at Hermes off a proton, most of the contribution to the
2 comes from events with a
+ in the final state. In ref. [23] the high 2 was attributed to
the poor agreement between experiment and theory at the level of the collinear
multiplicities. In this work we use a newer parametrization of the collinear FFs (DSEHS [64]), based
on a fit which includes Hermes collinear pion multiplicities. In spite of this improvement,
the contribution to
2 from Hermes data is higher then in ref. [23], because the present fit
includes data from other experiments (Hermes represents less than 20% of the whole data
set). The bins with the worst agreement are at low Q2. As we will discuss in section 4.3,
we think that the main reason for the large
2 at Hermes is a normalization difference.
This may also be due to the fact that we are computing our theoretical estimates at the
average values of the kinematic variables, instead of integrating the multiplicities in each
bin. Kaon multiplicities have in general a lower 2, due to the bigger statistical errors and
the large uncertainties for the kaon FFs.
For pion production off a deuteron at Hermes the
2 is lower with respect to the
production off a proton, but still compatible within uncertainties. For kaon production off
a deuteron the
2 is higher with respect to the scattering off a proton. The difference is
especially large for K .
h
SIDIS at Compass involves scattering off deuteron only, D ! h , and we identify
. The quality of the agreement between theory and Compass data is better than in
the case of pion production at Hermes. This depends on at least two factors: first, our fit
is essentially driven by the Compass data, which represent about 75% of the whole data
Points
2=points
Hermes
D !
+
190
set; second, the observable that we fit in this case is the normalized multiplicity, defined in
eq. (3.1). This automatically eliminates most of the discrepancy between theory and data
due to normalization.
Figure 3 presents the agreement between the theoretical formula in (2.3) and the
Hermes multiplicities for production of pions off a proton and a deuteron. Different hxi,
hzi and hQ2i bins are displayed as a function of the transverse momentum of the detected
hadron PhT . The grey bands are an envelope of the 200 replica of bestfit curves. For every
point in PhT we apply a 68% C.L. selection criterion. Points marked with different symbols
and colors correspond to different hzi values. There is a strong correlation between hxi and
hQ2i that does not allow us to explore the x and Q2 dependence of the TMDs separately.
Studying the contributions to the 2/points as a function of the kinematics, we notice that
the 2(Q2) tends to improve as we move to higher Q2 values, where the kinematic
approximations of factorization are more reliable. Moreover, usually the
2(z) increases at lower
z values.
Figure 4 has same contents and notation as in figure 3 but for kaons in the final state.
In this case, the trend of the agreement as a function of Q2 is not as clear as for the case
of pions: good agreement is found also at low Q2.
In figure 5 we present Compass normalized multiplicities (see eq. (3.1)) for production
of
off a deuteron for different hxi, hzi, and hQ2i bins as a function of the transverse
momentum of the detected hadron PhT . The open marker around the first PhT point in
each panel indicates that the first value is fixed and not fitted. The correlation between
x and Q2 is less strong than at Hermes and this allows us to study different hxi bins at
fixed hQ2i. For the highest Q2 bins, the agreement is good for all hxi, hzi and Ph2T . In bins
at lower Q2, the descriptions gets worse, especially at low and high z. For fixed hQ2i and
high hzi, a good agreement is recovered moving to higher hxi bins (see, e.g., the third line
from the top in figure 5).
Figure 6 has same content and notation as in figure 5, but for h+
+
. The same
comments on the agreement between theory and the data apply.
〈 〉=
〈 〉=
(
(
(
(
= )
= )
〈 〉=
〈 〉=
= )
= )
〈 〉=
〈 〉=
(
(
= )
= )
hxi, hzi, and hQ2i bins as a function of the transverse momentum of the detected hadron PhT . For
clarity, each hzi bin has been shifted by an offset indicated in the legend.
DrellYan and Z production. The low energy DrellYan data collected by the E288
and E605 experiments at Fermilab have large error bands (see figure 7). This is why the
2 values in table 8 are rather low compared to the other data sets.
The agreement is also good for Z boson production, see table 9. The statistics from
RunII is higher, which generates smaller experimental uncertainties and higher
2,
especially for the CDF experiment.
Figure 7 displays the cross section for DY events differential with respect to the
transverse momentum qT of the virtual photon, its invariant mass Q2 and rapidity y. As for the
〈 〉=
〈 〉=
(
(
(
(
= )
= )
〈 〉=
〈 〉=
= )
= )
〈 〉=
〈 〉=
(
(
= )
= )
hxi, hzi, and hQ2i bins as a function of the transverse momentum of the detected hadron PhT . For
clarity, each hzi bin has been shifted by an offset indicated in the legend.
case of SIDIS, the grey bands are the 68% C.L. envelope of the 200 replicas of the fit
function. The four panels represents different values for the rapidity y or xF (see eq. (2.15)). In
each panel, we have plots for different Q2 values. The lower is Q, the less points in qT we fit
(see also section 3.2). The hard scale lies in the region 4:5 < hQi < 13:5 GeV. This region is
of particular importance, since these “moderate” Q values should be high enough to safely
apply factorization and, at the same time, low enough in order for the nonperturbative
effects to not be shaded by transverse momentum resummation.
〈 〉=
〈 〉=
〈 〉=
〈 〉=
〈 〉=
〈 〉=
〈 〉=
(
(
(
(
(
(
(
= )
= )
= )
= )
= )
= )
= )
〈 〉=〈 〉=
〈 〉=
〈 〉=
different hxi, hzi, and hQ2i bins as a function of the transverse momentum of the detected hadron
PhT . Multiplicities are normalized to the first bin in PhT for each hzi value (see (3.1)). For clarity,
each hzi bin has been shifted by an offset indicated in the legend.
In figure 8 we compare the cross section differential with respect to the transverse
momentum qT of the virtual Z (namely eq. (2.12) integrated over ) with data from CDF
and D0 at Tevatron Run I and II. Due to the higher Q = MZ , the range explored in qT is
much larger compared to all the other observables considered. The tails of the distributions
deviate from a Gaussian behavior, as it is also evident in the bins at higher Q2 in figure 7.
The band from the replica methodology in this case is much narrower, due to the reduced
〈 〉=
〈 〉=
〈 〉=
〈 〉=
〈 〉=
〈 〉=
〈 〉=
(
(
(
(
(
(
(
= )
= )
= )
= )
= )
= )
= )
〈 〉=〈 〉=
〈 〉=
〈 〉=
different hxi, hzi, and hQ2i bins as a function of the transverse momentum of the detected hadron
PhT . Multiplicities are normalized to the first bin in PhT for each hzi value (see (3.1)). For clarity,
each hzi bin has been shifted by an offset indicated in the legend.
sensitivity to the intrinsic transverse momenta at Q = MZ and to the limited range of
bestfit values for the parameter g2, which controls softgluon emission. As an effect of
TMD evolution, the peak shifts from
1 GeV for DrellYan events in figure 7 to
5 GeV in
figure 8. The position of the peak is affected both by the perturbative and the
nonperturbative part of the Sudakov exponent (see section 2.3 and [22]). Most of the contributions
to the 2 comes from normalization effects and not from the shape in qT (see section 4.3).
〈〈〈 〉〉〉===
(
(
(
= )
= )
= )
〈〈〈 〉〉〉===
(
(
(
= )
= )
= )
〈〈〈 〉〉〉===
(
(
(
= )
= )
= )
Points
2=points 0:99
45
E605
35
8
and for different hQi bins. For clarity, each hQi bin has been normalized (the first data point has
been set always equal to 1) and then shifted by an offset indicated in the legend.
E288 [200]
E288 [300]
E288 [400]
energy. The labels in square brackets were introduced in section 3.2.
Transverse momentum dependence at 1 GeV
The variables min and max delimit the range in T where transverse momentum
resummation is computed perturbatively. The g2 parameter enters the nonperturbative Sudakov
exponent and quantifies the amount of transverse momentum due to soft gluon radiation
that is not included in the perturbative part of the Sudakov form factor. As already
explained in section 2.3, in this work we fix the value for min and max in such a way that at
Q = 1 GeV the unpolarized TMDs coincide with their nonperturbative input. We leave g2
Table 10 summarizes the chosen values of min,
max and the bestfit value for g2.
The latter is given as an average with 68% C.L. uncertainty computed over the set of 200
replicas. We also quote the results obtained from replica 105, since its parameters are very
D0) with two different values for the centerofmass energy (ps = 1:8 TeV and p
produced from pp collisions at Tevatron. The four panels refer to different experiments (CDF and
s = 1:96 TeV). In
this case the band is narrow due to the narrow range for the bestfit values of g2.
All replicas
Replica 105
max [GeV 1
]
min [GeV 1
]
g2 [GeV2]
(fixed)
2e E
2e E
(fixed)
2e E =Q
2e E =Q
0:13
0:01
0:128
close to the mean values of all replicas. We obtain a value g2 = 0:13
0:018) obtained in ref. [57], where however no SIDIS data was taken into
consideration, and smaller than the value (g2 = 0:16) chosen in ref. [15]. We stress however
that our prescriptions involving both min and max are different from previous works.
Table 11 collects the bestfit values of parameters in the nonperturbative part of the
TMDs at Q = 1 GeV (see eqs. (2.34) and (2.35)); as for g2, we give the average value over
the full set of replicas and the standard deviation based on a 68% C.L. (see section 3.4),
and we also quote the value of replica 105.
In figure 9 we compare different extractions of partonic transverse momenta. The
hor?
izontal axis shows the value of the average transverse momentum squared for the incoming
parton, k
2 (x = 0:1) (see eq. (2.40)). The vertical axis shows the value of P 2 (z = 0:5),
the average transverse momentum squared acquired during the fragmentation process (see
eq. (2.40)). The white square (label 1) indicates the average values of the two quantities
obtained in the present analysis at Q2 = 1 GeV2. Each black dot around the white square is
an outcome of one replica. The red region around the white square contains the 68% of the
replicas that are closest to the average value. The same applies to the white circle and the
?
TMD PDFs
0:285
N3
[GeV2]
0:212
0:86
0:78
0:39
F
PDFs, hk?2i(x = 0:1), in different phenomenological extractions. (1): average values (white square)
obtained in the present analysis, values obtained from each replica (black dots) and 68% C.L. area
(red); (2) results from ref. [23], (3) results from ref. [97], (4) results from ref. [76] for Hermes data,
(5) results from ref. [76] for Hermes data at high z, (6) results from ref. [76] for normalized Compass
data, (7) results from ref. [76] for normalized Compass data at high z, (8) results from ref. [15].
orange region around it (label 2), related to the flavorindependent version of the analysis
in ref. [23], obtained by fitting only Hermes SIDIS data at an average hQ2i = 2:4 GeV2
and neglecting QCD evolution. A strong anticorrelation between the transverse momenta
is evident in this older analysis. In our new analysis, the inclusion of DrellYan and Z
production data adds physical information about TMD PDFs, free from the influence of TMD
?
FFs. This reduces significantly the correlation between k
?
2 (x = 0:1) and
P 2 (z = 0:5).
The 68% confidence region is smaller than in the older analysis. The average values of
k
2 (x = 0:1) are similar and compatible within error bands. The values of P 2 (z = 0:5)
in the present analysis turn out to be larger than in the older analysis, an effect that is due
mainly to Compass data. It must be kept in mind that the two analyses lead also to
differences in the x and z dependence of the transverse momentum squared. This dependence is
?
?
2.31
0.80
C.L. envelope of the full sets of bestfit curves. The data used in the fit approximately cover the
range 5
10 3 . x . 0:5 and 0:2 . z . 0:7.
HJEP06(217)8
p !
5.18
1.94
p !
D !
2 (x) and figure 10 (b) for P 2 (z). The bands are computed
as the 68% C.L. envelope of the full sets of curves from the 200 replicas. Comparison with
?
other extractions are presented and the legend is detailed in the caption of figure 9.
Stability of our results
In this subsection we discuss the effect of modifying some of the choices we made in our
default fit. Instead of repeating the fitting procedure with different choices, we limit ourselves
to checking how the
2 of a single replica is affected by the modifications.
As starting point we choose replica 105, which, as discussed above, is one of the most
representative among the whole replica set. The global 2=d.o.f. of replica 105 is 1.51. We
keep all parameters fixed, without performing any new minimization, and we compute the
2=d.o.f. after the modifications described in the following.
First of all, we analyze Hermes data with the same strategy as Compass, i.e., we
normalize Hermes data to the value of the first bin in PhT . In this case, the global
2=d.o.f. reduces sharply to 1.27. The partial 2 for the different SIDIS processes measured
at Hermes are shown in table 12. This confirms that normalization effects are the main
contribution to the
2 of SIDIS data and have minor effects on TMDrelated parameters.
In fact, even if we perform a new fit with this modification, the
2 does not improve
significantly and parameters do not change much.
We consider the effect of changing the normalization of the Zboson data: if we increase
the normalization factors quoted in the last row of table 4 by 5%, the 2 quoted in the last
row of table 9 drops to 0.66, 0.52, 0.65, 0.68. This effect is also already visible by eye in
figure 8: the theoretical curves are systematically below the experimental data points, but
the shape is reproduced very well.
We consider the sensitivity of our results to the parameterizations adopted for the
collinear quark PDFs. The
2=d.o.f. varies from its original value 1.51, obtained with the
NLO GJR 2008 parametrization [62], to 1.84 using NLO MSTW 2008 [98], and 1.85 using
NLO CJ12 [99]. In both cases, the agreement with Hermes and Z boson data is not
affected significanlty, the agreement with Compass data becomes slightly worse, and the
agreement with DY data becomes clearly worse.
An extremely important point is the choice of kinematic cuts. Our default choices
are listed in tables 1–4. We consider also more stringent kinematic cuts on SIDIS data:
Q2 > 1:5 GeV2 and 0:25 < z < 0:6 instead of Q2 > 1:4 GeV2 and 0:2 < z < 0:7, leaving
the other ones unchanged. The number of bins with these cuts reduces from 8059 to 5679
and the
2=d.o.f. decreases to the value 1.23. In addition, if we replace the constraint
PhT < Min[0:2 Q; 0:7 Qz] + 0:5 GeV with PhT < Min[0:2 Q; 0:5 Qz] + 0:3 GeV, the number
of bins reduces to 3380 and the 2=d.o.f. decreases further to 0.96. By adopting the even
stricter cut PhT < 0:2 Qz, the number of bins drops to only 477, with a 2/d.o.f. =1.02. We
can conclude that our fit, obtained by fitting data in an extended kinematic region, where
TMD factorization may be questioned, works extremely well also in a narrower region,
where TMD factorization is expected to be under control.
5
In this work we demonstrated for the first time that it is possible to perform a simultaneous
fit of unpolarized TMD PDFs and FFs to data of SIDIS, DrellYan and Z boson production
at small transverse momentum collected by different experiments. This constitutes the
factorization and with the implementation of TMD evolution at NLL accuracy.
first attempt towards a global fit of f1a(x; k?2) and D1a!h(z; P 2 ) in the context of TMD
?
We extracted unpolarized TMDs using 8059 data points with 11 free parameters using
a replica methodology. We selected data with Q2 > 1:4 GeV2 and 0:2 < z < 0:7.
We
restricted our fit to the small transverse momentum region, selecting the maximum value
of transverse momentum on the basis of phenomenological considerations (see section 3).
With these choices, we included regions where TMD factorization could be questioned,
but we checked that our results describe very well the regions where TMD factorization is
supposed to hold. The average
2/d.o.f. is 1:55
0:05 and can be improved up to 1.02
restricting the kinematic cuts, without changing the parameters (see section 4.3). Most of
the discrepancies between experimental data and theory comes from the normalization and
not from the transverse momentum shape.
Our fit is performed assuming that the intrinsic transverse momentum dependence of
TMD PDFs and FFs can be parametrized by a normalized linear combination of a Gaussian
and a weighted Gaussian. We considered that the widths of the Gaussians depend on the
longitudinal momenta. We neglected a possible flavor dependence. For the nonperturbative
component of TMD evolution, we adopted the choice most often used in the literature (see
section 2.3).
We plan to release grids of the parametrizations studied in this work via TMDlib [100]
to facilitate phenomenological studies for present and future experiments.
HJEP06(217)8
In future studies, different functional forms for all the nonperturbative ingredients
should be explored, including also a possible flavor dependence of the intrinsic transverse
momenta. A more precise analysis from the perturbative point of view is also needed,
which should in principle make it possible to relax the tension in the normalization and to
describe data at higher transverse momenta. Moreover, the description at low transverse
momentum should be properly matched to the collinear fixedorder calculations at high
transverse momentum.
Together with an improved theoretical framework, in order to better understand the
formalism more experimental data is needed. It would be particularly useful to extend the
coverage in x, z, rapidity, and Q2. The 12 GeV physics program at Jefferson Lab [101] will
be very important to constrain TMD distributions at large x. Additional data from SIDIS
(at Compass, at a future ElectronIon Collider), DrellYan (at Compass, at Fermilab),
Z=W production (at LHC, RHIC, and at A FixedTarget Experiment at the LHC [
102
])
will be very important. Measurements related to unpolarized TMD FFs at e+e
colliders
(at BelleII, BESIII, at a future International Linear Collider) will be invaluable, since they
are presently missing.
Our work focused on quark TMDs. We remark that at present almost nothing is known
experimentally about gluon TMDs [11, 103], because they typically require higherenergy
scattering processes and they are harder to isolate as compared to quark distributions.
Several promising measurements have been proposed in order to extract both the unpolarized
and linearly polarized gluon TMDs inside an unpolarized proton. The cleanest possibility
would be to look at dijet and heavy quark pair production in electronproton collisions
at a future EIC [104, 105]. Other proposals include isolated photonpair production at
RHIC [106] and quarkonium production at the LHC [107–110].
Testing the formalism of TMD factorization and understanding the structure of
unpolarized TMDs is only the first crucial step in the exploration of the 3D proton structure in
momentum space and this work opens the way to global determinations of TMDs. Building
on this, we can proceed to deepen our understanding of hadron structure via polarized
structure function and asymmetries (see, e.g., refs. [111, 112] and references therein) and,
at the same time, to test the impact of hadron structure in precision measurements at
highenergies, such as at the LHC. A detailed mapping of hadron structure is essential to
interpret data from hadronic collisions, which are among the most powerful tools to look
for footprints of new physics.
Acknowledgments
Discussions with Giuseppe Bozzi and Alexey Vladimirov are gratefully acknowledged. This
work is supported by the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation program (grant agreement No. 647981, 3DSPIN).
AS acknowledges support from U.S. Department of Energy contract DEAC0506OR23177,
under which Jefferson Science Associates, LLC, manages and operates Jefferson Lab. The
work of AS has been funded partly also by the program of the Stichting voor Fundamenteel
Onderzoek der Materie (FOM), which is financially supported by the Nederlandse
Organisatie voor Wetenschappelijk Onderzoek (NWO).
HJEP06(217)8
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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