arXiv:hep-ph/0410050v2 7 Dec 2004
DESY 04-193
Single-spin asymmetries: the Trento conventions
Alessandro Bacchetta
∗
Institut f¨ur Theoretische Physik, Universit¨at Regensburg, D-93040 Regensburg, Germany
Umberto D’Alesio
†
INFN, Sezione di Cagliari and Dipartimento di Fisica,
Universit`a di Cagliari, I-09042 Monserrato, Italy
Markus Diehl
‡
Deutsches Elektronen-Synchroton DESY, D-22603 Hamburg, Germany
C. Andy Miller
§
TRIUMF, Vancouver, British Columbi a V6T 2A3, Canada
During the workshop “Transversity: New Developments in Nucleon S pin Structure” (ECT
∗
,
Trento, Italy, 14–18 June 2004), a series of recommendations was p ut forward by the participants
concerning definitions and notations for describing effects of intrinsic transverse momentum of par-
tons in semi-inclusive deep inelastic scattering.
I. DEFINITION OF TRANSVERSE-MOMENTUM DEPENDENT FUNCTIONS
A standard set of definitions and notations for transverse-momentum dependent distribution and fragmentation
functions is given in Refs. [1, 2, 3]. We note that the definition of the antisymmetric tensor in those articles and in
the present note is such that
ǫ
0123
= +1. (1)
Transverse-momentum dependent parton distributions of leading twist can be interpreted a s number densities (see
e.g. Refs. [4, 5, 6]). To connect with this interpretation, we take the example of the dis tribution of unpolarized quarks
in a polarized proton, which is given by
1
f
q/p
↑ (x, k
T
) = f
q
1
(x, k
2
T
) − f
⊥q
1T
(x, k
2
T
)
ǫ
µνρσ
P
µ
k
ν
S
ρ
n
σ
M (P · n)
= f
q
1
(x, k
2
T
) − f
⊥q
1T
(x, k
2
T
)
(
ˆ
P × k
T
) · S
M
,
(2)
where f
q
1
is the unpolarized quark density and f
⊥q
1T
describes the Sivers effect [7]. Here P is the momentum of the
proton, S is its covariant spin vector normalized to S
2
= −1, and M is the proton mass. The covariant definition of
parton distributions requires an a ux iliary lightlike vector n, which plays the r ole of a preferre d direction in a given
physical process.
2
Furthermore, k is the momentum of the quark, k
T
its co mponent perpendicular to P and n , and
x = (k ·n)/(P · n) its light-cone momentum fraction. The second expression in (2) holds in any fra me where n and the
direction
ˆ
P of the proton momentum point in opposite directions.
3
Therefore f
⊥q
1T
> 0 corresponds to a preference of
∗
Electronic address: alessandro.bacchetta@physik.uni-r.de
†
Electronic address: umberto.dalesio@ca.infn.it
‡
Electronic address: markus.diehl@desy.de
§
Electronic address: miller@triumf.ca
1
The following expression is obtained from the quark correlation function in Eq. (2) of Ref. [2] by identifying n = n
−
, multiplying with
n//2 and taking the trace.
2
This direction can for instance be taken along the virtual photon momentum in deep inelastic scattering, or along the momentum of the
second incoming hadron in Drell-Yan lepton pair production. Other choices of n are possible, provided that the corresponding changes
in the result are sufficiently suppressed by inverse powers of the large momentum scale.
3
We use the four-vector k
T
and its square as arguments in the distribution functions to emphasize that they are Lorentz i nvariant. One
may instead use k
T
if it is clear from the context to which frame the vectors refer.
2
the quark to move to the left if the proton is moving towards the observer and the proton spin is pointing upwards.
In the convention of Ref. [8] the Sivers effect is described by
f
q/p
↑ (x, k
T
) − f
q/p
↑ (x, −k
T
) = ∆
N
f
q/p
↑ (x, k
2
T
)
(
ˆ
P × k
T
) · S
|k
T
|
(3)
so that
∆
N
f
q/p
↑ (x, k
2
T
) = −
2|k
T
|
M
f
⊥q
1T
(x, k
2
T
). (4)
Either f
⊥q
1T
or ∆
N
f
q/p
↑ may be referred to as the “Sivers function”. It is strongly encouraged tha t authors use one or
the other of these notations, or provide the relation of the functions they might use to the ones disc us sed here.
Let us give the corresponding relation for the Boer-Mulders function, introduced in Ref. [2]. The distribution of
transversely polarized quarks in an unpolarized proton is
4
f
q
↑
/p
(x, k
T
) =
1
2
f
q
1
(x, k
2
T
) − h
⊥q
1
(x, k
2
T
)
ǫ
µνρσ
P
µ
k
ν
S
qρ
n
σ
M (P · n)
=
1
2
f
q
1
(x, k
2
T
) − h
⊥q
1
(x, k
2
T
)
(
ˆ
P × k
T
) · S
q
M
!
,
(5)
where S
q
is the covariant spin vector o f the quark. Introducing
f
q
↑
/p
(x, k
T
) − f
q
↑
/p
(x, −k
T
) = ∆
N
f
q
↑
/p
(x, k
2
T
)
(
ˆ
P × k
T
) · S
q
|k
T
|
(6)
we get the relation
∆
N
f
q
↑
/p
(x, k
2
T
) = −
|k
T
|
M
h
⊥q
1
(x, k
2
T
). (7)
Likewise there are two common notations for the Collins fragmentation function [10]. With the conventions of
Refs. [1, 2, 3] the number density of an unpolarized hadron h in a transversely p olarized quark is
5
D
h/q
↑
(z, P
hT
) = D
q
1
(z, P
2
hT
) − H
⊥q
1
(z, P
2
hT
)
ǫ
µνρσ
P
hµ
k
ν
S
qρ
n
′
σ
M
h
(P
h
· n
′
)
= D
q
1
(z, P
2
hT
) + H
⊥q
1
(z, P
2
hT
)
(
ˆ
k × P
hT
) · S
q
zM
h
,
(8)
where the measure of the density is dz d
2
P
hT
. Here D
q
1
is the unpo larized fragmentation function, P
h
is the hadron
momentum, M
h
its mass, k is the momentum of the quark, S
q
its covariant spin vector, and n
′
an auxiliary lightlike
vector. Furthermore, z = (P
h
· n
′
)/(k · n
′
) is the light-cone momentum fraction of the hadron with respect to the
fragmenting quark, and P
hT
the component of P
h
transverse to k and n
′
. One can trade P
hT
for k
T
= −P
hT
/z, the
component of k transverse to P
h
and n
′
. The second line of (8) ho lds in frames where n
′
and the direction
ˆ
k of the
quark momentum point in opposite directions. Therefore, H
⊥q
1
> 0 corresponds to a preference of the hadron to move
to the left if the quar k is moving away from the observer and the quark spin is pointing upwards. In the notation
of [11] the Collins effect is described by
D
h/q
↑
(z, P
hT
) − D
h/q
↑
(z, −P
hT
) = ∆
N
D
h/q
↑
(z, P
2
hT
)
(
ˆ
k × P
hT
) · S
q
|P
hT
|
(9)
4
The following expression is obtained by identifying n = n
−
, setting S
T
and λ to zero, multiplying Eq. (2) in Ref. [2] with γ
µ
n
µ
/2 +
iσ
µν
γ
5
n
µ
S
ν
q
/2, taking the trace and dividing by 2. See Eq. (11) and (12) of [9] for this connection to the number density interpretation.
5
The following expression is obtained by identifying n
′
= n
+
, setting S
hT
and λ
h
to zero, multiplying Eq. (5) in Ref. [2] with γ
µ
n
′
µ
/2 +
iσ
µν
γ
5
n
′µ
S
ν
q
/2 and taking the trace. See Eqs. (40) and (41) of [9].
3
so that
∆
N
D
h/q
↑ (z, P
2
hT
) =
2|P
hT
|
zM
h
H
⊥q
1
(z, P
2
hT
). (10)
Either H
⊥q
1
or ∆
N
D
h/q
↑ may be referred to as “Collins function”. Our r e lations (4), (7), (10) agree with (4.8.3a),
(4.8.3b), (6.5.11) in Ref. [6 ].
We finally discuss the a nalog of the Sivers function in fragmentation, introduced by Mulders and Tangerman in
Ref. [1 ]. The number density of a polarized spin-half hadron h in an unpolarized quark is
6
D
h
↑
/q
(z, P
hT
) =
1
2
D
q
1
(z, P
2
hT
) − D
⊥q
1T
(z, P
2
hT
)
ǫ
µνρσ
P
hµ
k
ν
S
hρ
n
′
σ
M
h
(P
h
· n
′
)
=
1
2
D
q
1
(z, P
2
hT
) + D
⊥q
1T
(z, P
2
hT
)
(
ˆ
k × P
hT
) · S
h
zM
h
!
,
(11)
where S
h
is the covariant spin vector of the hadron. As indicated in Ref. [12], we can write
D
h
↑
/q
(z, P
hT
) − D
h
↑
/q
(z, −P
hT
) = ∆
N
D
h
↑
/q
(z, P
2
hT
)
(
ˆ
k × P
hT
) · S
h
|P
hT
|
, (12)
which leads to
7
∆
N
D
h
↑
/q
(z, P
2
hT
) =
|P
hT
|
zM
h
D
⊥q
1T
(z, P
2
hT
). (13)
The definition of each parton distribution contains a Wilson line, which describes interac tio ns with the spectator
partons before or after the hard-scattering process. The path of this Wilson line in space-time is selected by the hard
process in which the parton distribution appears. Each such path corresponds to its own set of distribution functions,
which thus give the number of quarks fo und in the presence of the specified spectator interactions. Different paths
can le ad to different distributions, and the path should be specified in the nota tio n when it is not evident from the
context.
8
Using time reversal symmetry one can show [13]
f
DIS
1
(x, k
2
T
) = f
DY
1
(x, k
2
T
), f
⊥DIS
1T
(x, k
2
T
) = −f
⊥DY
1T
(x, k
2
T
), (14)
where the superscripts respectively specify the distributions with Wilson lines appropriate for semi-inclusive dee p
inelastic scattering (SIDIS) and for Drell-Yan lepton pair production.
Wilson lines with a path selected by the proce ss also app e ar in the definition of fragmentation functions. The
relation between the functions relevant for different processes (such as e
+
e
−
annihilation or SIDIS) is currently under
study.
II. AZIMUTHAL ANGLES IN SEMI-INCLUSIVE DEEP INELASTIC SCATTERING
A recommendation is made concerning the azimuthal angles relevant in the semi-inclusive cross section for
ℓ(l) + p(P ) → ℓ(l
′
) + h(P
h
) + X, (15)
where ℓ denotes the beam lepton, p the pr oton target, and h the produced hadron. As usual we define q = l − l
′
and
Q
2
= −q
2
. The azimuthal angle φ
h
between the lepton and the hadron planes should be defined as
cos φ
h
=
(
ˆ
q × l)
|
ˆ
q × l|
·
(
ˆ
q × P
h
)
|
ˆ
q × P
h
|
,
sin φ
h
=
(l × P
h
) ·
ˆ
q
|
ˆ
q × l| |
ˆ
q × P
h
|
,
(16)
6
The following expression is obtained by identifying n
′
= n
+
, multiplying Eq. (5) in Ref. [2] with n/
′
/2, taking the trace and dividing
by 2.
7
Note that there is a factor −2 too much in Eq. (5) of Ref. [12]. This does not affect any results in that work.
8
This has been realized only recently, and the necessary distinction is not made in [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12].
4
with
ˆ
q = q/|q|, where all vectors refer to the tar get rest frame (or to any frame reached from the target rest frame
by a boost along
ˆ
q). Writing the right-hand sides of (16) in a Lorentz invariant form, one has
cos φ
h
= −
g
µν
⊥
l
µ
P
hν
|l
⊥
| |P
h⊥
|
,
sin φ
h
= −
ǫ
µν
⊥
l
µ
P
hν
|l
⊥
| |P
h⊥
|
(17)
with |l
⊥
| =
p
−g
µν
⊥
l
µ
l
ν
and |P
h⊥
| =
p
−g
µν
⊥
P
hµ
P
hν
. Here we introduced per pendicular projection tensors
g
µν
⊥
= g
µν
−
q
µ
P
ν
+ P
µ
q
ν
P · q (1 + γ
2
)
+
γ
2
1 + γ
2
q
µ
q
ν
Q
2
−
P
µ
P
ν
M
2
,
ǫ
ρσ
⊥
= ǫ
µνρσ
P
µ
q
ν
P · q
p
1 + γ
2
(18)
with γ = 2xM/Q, where x is the Bjorken variable and M again the target mass. Evaluating the right-hand sides of (17)
in the target rest frame, one recovers (16). The azimuthal angle φ
S
relevant for specifying the target polarization is
defined in analogy to (16) and (17), with P
h
replaced by the covariant spin vector S of the target. The definitions
of φ
h
and φ
S
are illustrated in Fig. 1. We emphasize that (16), (17), (18) do not depe nd on the choice of coordinate
axes. For definiteness we show in Fig. 1 o ne frequently use d coordinate sys tem. In this system the tensors defined
in Eq. (18) have nonzero components g
11
⊥
= g
22
⊥
= −1 and ǫ
12
⊥
= −ǫ
21
⊥
= −1. Note that two different conventions for
drawing a ngles and interpreting their sign in figures are in general use in the literature:
A. The z axis is spe cified and angles are drawn as arcs with one arrowhead. If an angle is oriented according to
the right-hand rule it is positive, otherwise it is negative. Fig. 1 illustrates the application of this convention.
B. Illustrated angle s are always a ssumed to be positive. Only the location of the arc affects the definition of the
angle. No orientation should be assigned to the arc, and a ny z axis that may be present does not affect the
angle definition.
It is strongly recommended tha t authors avoid placing single arrowheads o n ar c s when using convention B. When
using convention A, an explicit remark in the caption may be useful when the figure illustrates a situation in which
an angle has a negative value.
y
z
x
hadron plane
lepton plane
l
0
l
S
?
P
h
P
h
?
φ
h
φ
S
FIG. 1: Definition of azimuthal angles for the process (15) in the target rest frame. P
h⊥
and S
⊥
are the components of P
h
and
S transverse to the photon momentum.
Theorists often prefer a coordinate system with the same x axis but with y and z axes opposite to those shown
in Fig. 1, so that in the γ
∗
p center of mass the target moves in the positive z directio n (cf. Sect. I). When working
5
in tha t coordinate system in the context of graphical convention A one can conform with the definition of angles
recommended here by using the oppo site orientation for both φ
h
and φ
S
.
We note that the angles φ
h
and φ
S
defined here are opposite to those defined in Refs. [1, 2, 3], which must be taken
into account when using expressio ns for az imuthal asymmetries from these pa pers .
9
III. ASYMMETRIES AND AZIMUTHAL MOMENTS
Longitudinal single-spin asy mmetries in lepton-proton scattering should always be defined so that
A(φ
h
) ≡
dσ
→
(φ
h
) − dσ
←
(φ
h
)
dσ
→
(φ
h
) + dσ
←
(φ
h
)
, (19)
where in the case of a beam spin asymmetry dσ
→
refers to positive helicity of the lepton. In the case of a targ et
spin asymmetry dσ
→
denotes target polarization opposite to the direction either of the lepton beam or of the virtual
photon.
10
Azimuthal moments a ssociated with beam or target spin asymmetries are defined as, e.g.
h sin φ
h
i ≡
R
dφ
h
sin φ
h
[dσ
→
(φ
h
) − dσ
←
(φ
h
)]
R
dφ
h
[dσ
→
(φ
h
) + dσ
←
(φ
h
)]
(20)
and similarly for h sin 2φ
h
i etc. As an alternative notation one may use A
sin φ
h
= 2h sin φ
h
i.
11
If the cross section is
of the form
dσ
→
dφ
h
= a
0
+ a
1
sin φ
h
,
dσ
←
dφ
h
= a
0
− a
1
sin φ
h
,
(21)
then A
sin φ
h
= a
1
/a
0
has values between −1 and +1, as is natural for an asymmetry.
The single spin asymmetry for transverse target polarization can be written as
A(φ
h
, φ
S
) ≡
dσ(φ
h
, φ
S
) − dσ(φ
h
, φ
S
+ π)
dσ(φ
h
, φ
S
) + dσ(φ
h
, φ
S
+ π)
(22)
and as sociated azimuthal moments as, e.g.
h sin(φ
h
+ φ
S
)i ≡
R
dφ
h
dφ
S
sin(φ
h
+ φ
S
) [dσ(φ
h
, φ
S
) − dσ(φ
h
, φ
S
+ π)]
R
dφ
h
dφ
S
[dσ(φ
h
, φ
S
) + dσ(φ
h
, φ
S
+ π)]
(23)
and similarly for h s in(φ
h
−φ
S
)i etc. It should be straightforward to generalize these c onventions to the case of double
spin asymmetries and of |P
h⊥
|-weighted asymmetries [2].
Acknowledgments
We thank the organizers and all the participants of the workshop “Tra nsversity: New Developments in Nucleon
Spin Str ucture ” (ECT
∗
, Trento, Italy, 14–18 June 2004). Special thanks are due to M. Anselmino, D. Boer , A. Metz,
P.J. Mulders, F. Murgia, F. Pijlman, M. Radici, and G. Schnell for valuable input to the manuscript. This work
is part of the EU Integrated Infrastructure Initiative “Study of strongly interacting matter (HadronPhysics)” under
contract numbe r RII3-CT-2004-506078.
[1] P. J. Mulders and R. D. Tangerman, Nucl. Phys. B461, 197 (1996), erratum-ibid. B484, 538 (1996), hep-ph/9510301.
9
There is an inconsistency in Fig. 1 of Ref. [3] and Fig. 1 of Ref . [2]: according to the formulae given in those papers, the azimuthal angle
shown in those figures (which is positive according to graphical convention A) is equal to −φ and not to φ.
10
Note that target polarization opposite to the virtual photon momentum corresponds to positive helicity of the proton in the γ
∗
p center
of mass.
11
In the literature sometimes the factor 2 is not included, a choice that we do not recommend.
