Hermes Internal Report:
Partial Wave Expansion of Dihadron Fragmentation Functions
and SIDIS Dihadron Cross Section at Next to Leading Twist
S. Gliske
July 8, 2010
Abstract
This internal note documents advances in theoretical issues relevant to the non-collinear (non-Ph⊥ )
SIDIS dihadron analysis. The central aspect of this note is a partial wave expansion of the dihadron
fragmentation functions in the direct sum basis 2 ⊕ 1 ⊕ 0 rather than the (1 ⊕ 0) ⊗ (1 ⊕ 0) basis common in
the literature. Notation for the partial waves of the fragmentation functions in this new basis is defined,
as well as relations with previous notation. In additionally, both the previously computed twist 2 terms,
along with the not previously computed twist 3 terms, of the cross section are given in terms of structure
functions. At present, the cross section and structure functions are limited to the unpolarized twist 2 and
twist 3 terms, and the transverse target twist 2 terms, as these are most relevant for the current Hermes
AU T analysis. The additional terms can be computed based on the procedures of this document, and
may be included in a future internal note. The interpretation of the structure functions with regard to
the partial wave expansion of the fragmentation functions is also given.
Contents
1 Introduction and Motivation
2
2 Cross Section
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Moments in terms of Structure Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3
4
3 Partial Wave Expansion
5
4 Structure Functions
4.1 Unpolarized Beam, Unpolarized Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Unpolarized Beam, Transversely Polarized Target . . . . . . . . . . . . . . . . . . . . . . .
7
8
10
5 Relation with Previous Notation
11
6 Conclusion
12
1
1
Introduction and Motivation
Mesons have two spin 1/2 valence partons: a quark and and an antiquark. Thus the spin states can be
written in the basis of the 12 ⊗ 12 , i.e. states labeled |l1 , m1 i|l2 , m2 i. However, the physical states are
better expressed in the basis of the direct production 1⊕0, i.e. states labeled |l, mi. The change of basis is
given by Clebsch Gordan coefficients. The |0, 0i state is associated with scalar and pseudoscalar mesons,
while the l = 1 states are associated with vector and pseudovector mesons. The |1, 0i state is associated
with longitudinally polarized mesons, while both |1, ±1i states are associated with transversely polarized
mesons.
Note, however, that the above paragraph refers most directly to the scattering amplitude, not the cross
section. The cross section is equal to the amplitude times its complex conjugate. Thus the physically
observable states are in the space of 12 ⊗ 12 ⊗ 12 ⊗ 12 = (1 ⊕ 0) ⊗ (1 ⊕ 0) = 2 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 0 ⊕ 0. While there
exists Racah coefficients and generalized Casimir operators1 to correctly label all these states, there are
not enough physical observables to distinguish between the three l = 1 states nor the two l = 0 states.
Thus, the cross section appears to be in the space of 2 ⊕ 1 ⊕ 0, with the one perceived l = 1 state being
an admixture of the actual three l = 1 states, likewise the one perceived l = 0 is an admixture of the
two true l = 0 states. This detail regarding the l = 0 states has indirectly been noted in the literature
pp
ss
[1], by writing D1,U U = D1,U
U + D1,U U . However, the fact that one of the three true l = 1 state is pure
pp interference is obscured, as all references known to the author label the perceived l = 1 states as sp
interference, when they include both sp and pp interference.2
In general, one is free to choose any basis in the spin space. The common choice in the literature is
(1 ⊕ 0) ⊗ (1 ⊕ 0) [6, 7], where labels such as T T , LL, LT are used to distinguish groupings of states.
This document will list many reasons why the apparent 2 ⊕ 1 ⊕ 0 basis (hereafter denoted the “direct
sum” based) is more desirable for SIDIS dihadron production, as well as introduce notation to describe
the fragmentation functions in this basis.
In order to write the cross section in this direct sum basis, structure functions are introduced, as are
for single hadron production [2]. The expressions for the cross section in terms of structure functions
is valid for both choices of bases: the basis choice only effecting the specific expressions relating the
structure functions to distribution and fragmentation functions. One of the advantages of the direct sum
basis is that it allows one to extrapolate the SIDIS dihadron cross section from the SIDIS single hadron
cross section. Thus, for the first time, the non-collinear (i.e. non-transverse momentum integrated) SIDIS
dihadron cross section is presented at subleading twist. Discussion of the appearance of the 2 ⊕ 1 ⊕ 0
structure in the cross section can also be better discussed after the introduction of these structure
functions.
As additional motivation for the direct sum basis, it should be noted that the Lund or Artru model of
fragmentation makes specific predictions about the sign of the Collins function based on the spin states.
This model assumes that quarks are produced (at the amplitude level) in a 0++ state, i.e. with the same
quantum numbers as the vacuum. This implies that the Collins function for fragmenting into |0, 0i and
|1, 0i states has one sign, while fragmenting into either of the two |1, ±1i states the Collins function has
opposite sign. The present choice of basis in the literature obscures how these specific |l, mi states are
related to the fragmentation functions in the cross section. In order to determine the relation, one must
trace 8 × 8 matrices of fragmentation functions with other matrices. However, in the cross section with
the fragmentation functions expanded in the direct sum basis, the relations between the fragmentation
functions in the cross section and these amplitude level |l, mi states are given by the usual SU (2) selection
rules—relations familiar to all physicists in this subfield.
This document is organized as follows. First the cross section is written in terms of structure functions.
Next, the specific partial wave expansion of the fragmentation functions is introduced and discussed.
Third, the structure functions are written in terms of the new direct sum partial wave expansion, and
finally a listing is given of the relations between fragmentation functions in this notation and in the
previous notation.
1 Although
2 Please
not related to the scope of this document, many details can be found in Ref. [8].
let me know if you are aware if this detail being discussed in a published article.
2
2
2.1
Cross Section
Preliminaries
This section writes the next-to-leading twist cross section for SIDIS dihadron (including vector meson)
production. Structure functions are introduced, as has been done for single hadron production [2].
While the leading twist contribution is purely rewriting previous results [1, 3], the subleading twist terms
have not been previously available. The subleading terms have been computed using the partial wave
expansion described in Section 3. Interpretation of the structure functions in terms of distribution and
fragmentation functions is considered in Section 4.
The cross section is chosen to be differential with respect to x, y, z, Ph⊥ , Mh , cos ϑ, φh , φR , and
φS . The phase space factor is based on the single hadron phase space factor [2]. An extra factor of 4π is
included to account for the additional phase space of cos ϑ and φR . When the cross section is differential
with respect to Mh2 , no additional phase space factor is included [3]. Using Mh instead of Mh2 introduces
a factor of 1/(2Mh ). Also extra factor of Ph⊥ is included to account for using dPh⊥ dφh instead of d2 Ph⊥ .
The final phase space factor is then
„
«
α2 Mh Ph⊥
γ2
(1)
1
+
,
2πxyQ2
2x
with γ = 2xM/Q. Integrating over Ph⊥ reduces the cross section to the so-called collinear case, since
explicit transverse momentum effects are averaged out.
The following functions [2], arising from the quark and virtual photon QED vertex, occur in the cross
section,
A(x, y)
=
≈
B(x, y)
=
≈
C(x, y)
=
≈
V (x, y)
=
≈
W (x, y)
=
≈
1 − y + 12 y 2 + 14 γ 2 y 2
y2
=
2(1 − ²)
1 + γ2
„
«
1
1 − y + y2 ,
2
1 − y − 14 γ 2 y 2
y2
²=
2(1 − ²)
1 + γ2
(1 − y) ,
´
`
y 1 − 21 y
y2 p
2
1−² =
2(1 − ²)
1 + γ2
„
«
1
y 1− y ,
2
r
2(2 − y)
y2 p
1
2²(1 + ²) =
1 − y − γ 2 y2
(1 − ²)
1 + γ2
4
p
2 (2 − y) 1 − y,
r
y2 p
2y
1
2²(1 − ²) =
1 − y − γ 2 y2
(1 − ²)
1 + γ2
4
p
2y 1 − y,
with
²=
1 − y − 14 y 2 γ 2
.
1 − y + 12 y 2 + 41 y 2 γ 2
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
Note, often the approximations, which are only y-dependent, are used, as the difference is 1/Q suppressed.
For this reason, the functions are written as only y dependent, rather than x and y dependent. Note
the Q2 dependence in γ and ² can be rewritten in terms of dependence on x and y. A common rule of
thumb is that for SIDIS production, terms with A(x, y), B(x, y) and C(x, y) are twist 2, while terms
with V (x, y), W (x, y) are twist 3.
For notational convenience, the cross section can be divided into terms (notated d9 σXY ), dependent
on the the beam helicity (X) and target polarization (Y ), with X, Y taking values L, T , or U for
longitudinal, transverse and unpolarized states, respectively.
3
2.2
Moments in terms of Structure Functions
2.2.1
Unpolarized Beam, Unpolarized Target
The unpolarized moments are
«−1
„
2πxyQ2
γ2
1
+
d9 σU U =
α2 Mh Ph⊥
2x
"
P (ϑ)
sin ϑ cos(φh −φR )
ϑ
2
A(x, y) FU U,T + cos ϑFUcos
U,T + P2 (ϑ)FU U,T + sin ϑ cos(φh − φR )FU U,T
#
+ sin 2ϑ cos(φh −
sin 2ϑ cos(φh −φR )
φR )FU U,T
2
+ sin ϑ cos(2φh −
sin2 ϑ cos(2φh −2φR )
2φR )FU U,T
"
P (ϑ) cos 2φh
2φh
ϑ cos 2φh
2
+ B(x, y) cos 2φh FUcos
+ cos ϑ cos 2φh FUcos
+ P2 (ϑ) cos 2φh FU U
U
U
sin ϑ cos(φh +φR )
+ sin ϑ cos(φh + φR )FU U
2
+ sin2 ϑ cos 2φR FUsin
U
ϑ cos 2φR
sin 2ϑ cos(φh +φR )
+ sin 2ϑ cos(φh + φR )FU U
sin ϑ cos(3φh −φR )
+ sin ϑ cos(3φh − φR )FU U
sin 2ϑ cos(3φh −φR )
+ sin 2ϑ cos(3φh − φR )FU U
#
sin2 ϑ cos(4φh −2φR )
+ sin2 ϑ cos(4φh − 2φR )FU U
,
"
cos φh
+ V (x, y) cos φh FU U
cos ϑ cos φh
+ cos ϑ cos φh FU U
sin ϑ cos(2φh −φR )
+ sin ϑ cos(2φh − φR )FU U
sin 2ϑ cos(2φh −φR )
+ sin 2ϑ cos(2φh − φR )FU U
sin2 ϑ cos(3φh −2φR )
+ sin2 ϑ cos(3φh − 2φR )FU U
+
2ϑ cos φR
sin 2ϑ cos φR FUsin
U
P (ϑ) cos φh
2
+ P2 (ϑ) cos φh FU U
ϑ cos φR
+ sin ϑ cos φR FUsin
U
2
+ sin ϑ cos(φh −
sin2 ϑ cos(φh −2φR )
2φR )FU U
#
.
(13)
All indications, both from the single hadron case and the integrated dihadron case, suggest that FU U,L
is zero at twist three for dihadrons. Thus, terms related to FU U,L are not considered.
2.2.2
„
Unpolarized beam, Transversely Polarized target
The unpolarized beam, transverse target moments are
«
„
«−1
1
2πxyQ2
γ2
1
+
d9 σU T =
ST α2 Mh Ph⊥
2x
"
sin(φ −φS )
A(x, y) sin(φh − φS )FU T,T h
sin ϑ sin(2φh −φR −φS )
+ sin ϑ sin(2φh − φR − φS )FU T,T
cos ϑ sin(φh −φS )
+ cos ϑ sin(φh − φS )FU T,T
sin ϑ sin(φR −φS )
+ sin ϑ sin(φR − φS )FU T,T
sin2 ϑ sin(3φh −2φR −φS )
+ sin2 ϑ sin(3φh − 2φR − φS )FU T,T
P
(ϑ) sin(φh −φS )
2,0
+ P2,0 (ϑ) sin(φh − φS )FU T,T
sin 2ϑ sin(2φh −φR −φS )
+ sin 2ϑ sin(2φh − φR − φS )FU T,T
sin 2ϑ sin(φR −φS )
+ sin 2ϑ sin(φR − φS )FU T,T
#
sin2 ϑ sin(φh −2φR +φS )
+ sin2 ϑ sin(φh − 2φR + φS )FU T,T
"
sin(φh +φS )
+ B(x, y) sin(φh + φS )FU T
sin ϑ sin(2φh −φR +φS )
+ sin ϑ sin(2φh − φR + φS )FU T
cos ϑ sin(φh +φS )
+ cos ϑ sin(φh + φS )FU T
sin ϑ sin(φR +φS )
+ sin ϑ sin(φR + φS )FU T
2
sin ϑ sin(3φh −2φR +φS )
+ sin2 ϑ sin(3φh − 2φR + φS )FU T
P
+ P2,0 (ϑ) sin(φh + φS )FU T2,0
(ϑ) sin(φh +φS )
sin 2ϑ sin(φR +φS )
+ sin 2ϑ sin(φR + φS )FU T
2
sin ϑ sin(φh −2φR +φS )
+ sin2 ϑ sin(φh − 2φR + φS )FU T
sin 2ϑ sin(2φh −φR +φS )
+ sin 2ϑ sin(2φh − φR + φS )FU T
4
sin(3φh −φS )
+ sin(3φh − φS )FU T
sin(4φh −φR −φS )
+ sin(3φh − φS )FU T
sin(2φh +φR −φS )
+ sin(5φh − 2φR − φS )FU T
sin(4φh −φR −φS )
+ sin(3φh − φS )FU T
+ sin(4φh − φR − φS )FU T
+ sin(2φh + φR − φS )FU T
+ sin(4φh − φR − φS )FU T
+ sin(2φh + φR −
"
sin(2φ +φ −φ )
φS )FU T h R S
sin(3φh −φS )
sin(5φh −2φR −φS )
sin(3φh −φS )
+ sin(φh − 2φR −
#
sin(φ −2φR −φS )
φS )FU T h
#
+ V (x, y) twist 3 terms .
2.2.3
(14)
Further Comment
Already in the unpolarized moments, the additional spherical harmonic structure is apparent, as commented on in Section 1. Focusing on the terms arising with A(x, y) in Equation 13, one sees the angular
dependence is exactly that of the real spherical harmonics Yl,m , with polar angle ϑ and azimuthal angle ±(φh − φR ). Spherical harmonics for other terms are not as obvious, as the angular dependence
is a product of Yl,m functions and additional trigonometric functions of φh . The entire structure can
be understood, as well as these observations being made perfectly explicit, by relating these structure
functions to the the partial wave expansion of the fragmentation functions in terms of the states of the
two dihadron system.
3
Partial Wave Expansion
Cross sections for single hadron and dihadron production can be factored [2] into leptonic and hadronic
tensors,
dn σ
∝ 2W µν Lµν ,
(15)
dx
where n is 6 (9) in the single hadron (dihadron) case, and x is the n-tuple of independent variables:
The leptonic tensor comes from the basic QED process of the electron emitting a photon. The leptonic
tensor is the same for both single and dihadron final states. The difference in the cross section due to the
different the final states is encoded in the hadronic tensor. At next to leading order, the hadron tensor
can be written as
" 
ff
µν
2M W
= 32zI Tr Φa γ µ ∆a γ ν
n
o
1
e aA α γ µ ∆a + γ µ γ − γ α ∆a γ ν γ 0 Φ
e a† γ 0
√ Tr γ α γ − γ ν Φ
A α
Q 2
#
n
o
1
α + µ ea
ν a
ν + α a µ 0 e a†
0
,
− √ Tr γ γ γ ∆A α γ Φ + γ γ γ Φ γ γ ∆A α γ
Q 2
−
(16)
e A are the distribution correlation functions and ∆, ∆
e A are the fragmentation correlation
where Φ, Φ
functions. This notation follows Ref. [2]. The tilde indicates those which include quark-gluon-quark
correlators, which are zero in the Wandzura-Wilczek approximation. The difference in the cross sections
between SIDIS production of single hadrons, hadron pairs, vector mesons, etc. occur by using different
fragmentation correlation functions. In particular, all additional dihadron fragmentation functions and
cross section terms, not occurring in single hadron production, can be related to additional partial waves
e A.
of these fragmentation correlation functions ∆, ∆
Thus one can identify the dihadron cross section (without any partial wave expansion of the fragmentation functions) with the single hadron cross section, up to the interpretation of the fragmentation
functions, additional variables, and the phase factor. In addition, the cross section for fragmenting into
a dihadron in the pure scalar, |l, mi = |0, 0i partial wave also has identical form to the single hadron
cross section.
Let us follows this identification of the non-expanded dihadron cross section with the single hadron
cross section. Consider the SIDIS dihadron cross section, written identically as the single hadron cross
section, up to the phase factor, and let each fragmentation function depend on additional variables and
be interpreted as a completely non-expanded dihadron fragmentation function. Let each completely nonexpanded fragmentation function be expanded in terms of the partial waves of the two dihadron system
5
in the direct sum basis according to
D
=
∞ X
l
X
Pl,|m| (cos ϑ)eim(φR −φk ) D |l,mi (z, Mhh , |kT |),
(17)
l=1 m=−l
where D is a generic dihadron fragmentation function. In order to avoid any confusion about normalization, the specific polynomials used are
P0,0
=
1,
(18)
P1,0
=
cos ϑ,
(19)
P1,1
=
(20)
P2,0
=
P2,1
=
P2,2
=
sin ϑ,
´
1`
3 cos2 ϑ − 1 ,
2
3
sin 2ϑ,
2
3 sin2 ϑ.
(21)
(22)
(23)
Note, the normalization of these spherical harmonics has been chosen to simplify the expressions of the
structure functions. Note also, the previous notation for dihadron structure functions are initially partially expanded, as below it will be shown that D1 and G⊥
1 are groupings of different partial waves of the
same completely non-expanded fragmentation function. Then each of these initial fragmentation functions are expanded in Legendre polynomials, and a step (which is denoted the partial wave expansion)
relates these expanded fragmentation functions to partial waves in the (1 ⊕ 0) ⊗ (1 ⊕ 0) basis. This document suggests replacing this complicated, multi-step procedure with the one partial wave expansion of
Equation 17. Note, the previous notation never explicitly recognizes the presence of spherical harmonics,
only the presence Legendre polynomials.
The specific results of this direct sum basis expansion are listed in Section 4. However, in the process
|l,mi
of following this expansion, one discovers that the nine D1
functions are linear combinations of the 6
xy
⊥xy
dihadron functions D1,XY and 3 G1,XY (post Legendre-polynomial expansion) common in the literature
[1, 5, 3, 4]. The XY subscript denotes the polarization state of the dihadron and xy denotes whether the
⊥ |l,mi
term arises from ss, sp, or pp contributions. Likewise, the nine H1
functions are linear combinations
sp .
⊥sp
of the standard (post Legendre-polynomial expansion) dihadron functions 6 H1,XY
and 3 H1,XY
As the difference in notion is the difference in the choice of basis, the total number of leading twist
fragmentation functions has not changed—18 in both cases. However, the previous notation forms four
⊥
groups, labeled by D1 , H1⊥ , G⊥
1 , H1 . This suggests that G1 is at least as different from D1 as it is
from H1⊥ —in fact, the ⊥ superscript suggests that G⊥
is
more
like H1⊥ than it is like D1 . However, the
1
xy
direct sum basis approach makes apparent that the 6 D1,XY
’s and 3 G⊥xy
1,XY ’s are just linear combinations
of partial waves of a single completely non-integrated fragmentation function. That they have different
letters is misleading. Likewise, the nine fragmentation functions occurring with symbols H1⊥ and H1 of
the previous notation are linear combinations of the partial waves of the completely non-expanded Collins
function, H1⊥ . Some have suggested the Q2 evolution and other features might be different between H1⊥
and H1 [1], as previously the process contributing to H1 was not well understood. However, this
new expansion shows that both H1⊥ and H1 arise due to the same Collins effect—defining the process
contributing to H1 —and that any different in their evolution can be linked to their |l, mi state.
As both the “completely non-expanded” fragmentation functions and the direct sum basis expansion
are departures from previous notation, a more rigorous definition than simple comparison with the single
hadron cross section is appropriate. Following Ref. [5], rigorous definitions can be given in terms of
traces of the fragmentation correlator functions. Let
˛
Z
˛
z|R|
˛
+
[Γ]
(24)
dk Tr [Γ∆(k, Ph , R)] ˛
∆ (z, Mh , |kT |, cos ϑ, φR − φk ) = 4π
.
˛ − −
16Mh
k
=Ph /z
Define the leading twist distribution functions according to the trace identities
D1
|kT | iφk ⊥
i
e H1
Mh
=
=
∆[γ
−
∆[(σ
(1+iγ 5 )]
1−
−iσ
6
2−
,
(25)
5
)γ ]
= ∆[(γ
2
1
+iγ )γ
− 5
γ ]
.
(26)
The relations for D1 might appear more familiar as
Re (D1 )
Im (D1 )
=
=
−
∆[γ ] ,
∆
[γ − γ 5 ]
(27)
,
Comparing these trace identities with the previous notation yields the relation
˛
»
–
˛
|R||kT |
=
D1 + i
D1 ˛˛
sin ϑ sin(φR − φk ) G⊥
,
1
M1 M2
Gliske
[5]
˛
»
–
˛
|R|
=
H1⊥ +
H1⊥ ˛˛
sin ϑei(φR −φk ) H1
.
|kT |
Gliske
[5]
Other sources [3] replace product of masses M1 M2 in the denominator with Mh , yielding
˛
»
–
˛
|R||kT |
⊥
=
D1 + i
sin
ϑ
sin(φ
−
φ
)
G
,
D1 ˛˛
R
k
1
Mh2
Gliske
[3]
˛
»
–
˛
|R|
=
H1⊥ +
,
H1⊥ ˛˛
sin ϑei(φR −φk ) H̄1
|kT |
Gliske
[3]
˛
|R|2 ˛˛
=
H
,
|kT |2 1 ˛[3]
(28)
(29)
(30)
(31)
(32)
(33)
(34)
A subtlety exists in the definition of H1 . Note that
˛
˛
˛
˛
H1 ˛
= H̄1 ˛ ,
[5]
[3]
»
–
˛
˛
+ |kT | H ⊥
H1,XY
=
H̄1,XY
, XY = {OL, LT, T T }.
˛
1,XY
|R|
[3]
[3]
(35)
(36)
The completely non-expanded fragmentation functions of Equations 25, 26 are be expanded via
|l,mi
⊥ |l,mi
Equation 17 to define the partial wave dihadron fragmentation functions D1
, H1
. Similar rigorous definitions for the subleading twist fragmentation functions can be determined, but such is still in
preparation. Note, such expressions for the non-collinear (non-transverse momentum integrated) are not
currently available even in the standard notation, as non-collinear dihadron fragmentation is beyond the
current scope of the literature.
Either by comparison of the expressions in Equations 31 or by comparison with the structure functions written in both notations, one can write exactly the relations between both notations for all 18
leading twist fragmentation functions. The explicit relations are given in Section 5. As these is the first
presentation of non-collinear twist 3 dihadron fragmentation functions, relations with previous notation
for twist 3 is not applicable.
While it may seem unusual to include a γ 5 in the trace for an unpolarized function, the γ 5 is necessary
to include the imaginary part of D1 . One should remember that D1 is unpolarized with respect to
the nucleon and struck quark polarization, but not unpolarized with respect to the polarization of the
dihadron. The only term unpolarized with respect to the nucleon, struck quark, and dihadron system is
the |0, 0i partial wave which, since it is purely real, receive no contribution from the trace with γ 5 .
This notation does go against the previous general rule of thumb, that functions with ⊥ only occur
in the non-collinear case. However, which partial waves of which fragmentation functions are present the
collinear case depends on the interplay between the distribution functions and fragmentation functions,
and in general, different partial waves will survive for different combinations. The exact determination
is given in Section 4. Because of the polarization of the lepton, target, and final state all are involved,
the simple rule of thumb is no longer appropriate. Likewise, the symbols for the expanded fragmentation
functions are chosen to match the completely non-expanded fragmentation functions, which are chosen
to match the single hadron case.
4
Structure Functions
Often the weight factor occurring in the integral with the distribution and fragmentation functions is
written in terms of dot products of vectors. In the following, these dot products have been rewritten in
7
terms of norms of vectors and a single cosine or sine function. The advantage to this method is that the
prefactor for all |l, m > states arising from the expansion of the same term have the same weight factor,
up to the argument of the cosine or sine function. When writing in terms of dot products of vectors,
this is not the case. In addition, the expressions with dot products of vectors become increasingly more
complicated for higher |l, mi states, which is an unnecessary complication.
For example, the weight for the sin2 θ cos(4φh − 2φR ) term in the unpolarized dihadron cross section
can be written [3]
h
“
”i h
“
”“
”i
|kT |2 − 4 kT · Pbh⊥
(pT · kT ) − 4 pT · Pbh⊥ kT · Pbh⊥
2M Mhh |kT |2
−
“
”“
”3
8 pT · Pbh⊥ kT · Pbh⊥
2M Mhh |kT |2
,
(37)
which can be equivalently written as
|pT ||kT |
cos(4φh − φp − 3φk ).
M Mhh
(38)
Furthermore, writing in terms of dot products also hides similarities between various moments. For
instance, all the leading order, unpolarized terms involving the Boer-Mulder’s function h⊥
1 have a weight
of the form,
|pT ||kT |
cos((2 − m)φh − φp − (1 − m)φk ),
(39)
M Mhh
when written in terms of the cosine of the angles, rather than dot products. Written as dot products, the
weights have very different form, see e.g. Appendix C of Reference [3]. In particular, the m in Equation
39 corresponds to the m in for the |l, mi angular momentum state of the dihadron. Thus writing in
terms of sine or cosine functions not only makes the expressions simpler, this change also highlights
deeper meanings and relationships between the structure functions, i.e. the direct sum basis partial wave
expansion.
4.1
4.1.1
Unpolarized Beam, Unpolarized Target
Leading Twist
The leading order, unpolarized structure functions are
h
i
|0,0i
FU U,T = I f1 D1
,
h
i
|1,0i
ϑ
FUcos
= I f1 D 1
,
U,T
h
“
”i
sin ϑ cos(φh −φR )
|1,1i
FU U,T
= I 4 cos(φh − φk )f1 Re D1
,
h
i
P2,0 (cos ϑ)
|2,0i
FU U,T
= I D1
,
h
“
”i
sin 2ϑ cos(φh −φR )
|2,1i
FU U,T
= I 3 cos(φh − φk )f1 Re D1
,
h
“
”i
2
sin ϑ cos(2φh −2φR )
|2,2i
FU U,T
= I 6 cos(2φh − 2φk )f1 Re D1
,
»
–
|pT ||kT |
⊥ |0,0i
cos 2φ
FU U h = I −
cos(2φh − φp − φk )h⊥
,
1 H1
M Mhh
»
–
|pT ||kT |
sin ϑ cos(3φh −φR )
⊥ |1,−1i
FU U
= I −
cos(3φh − φp − 2φk )h⊥
H
,
1
1
M Mhh
»
–
|pT ||kT |
⊥ |1,0i
cos ϑ cos 2φh
FU U
= I −
cos(2φh − φp − φk )h⊥
,
1 H1
M Mhh
»
–
|pT ||kT |
sin ϑ cos(φh +φR )
⊥ |1,1i
FU U
= I −
cos(φh − φp )h⊥
,
1 H1
M Mhh
»
–
|pT ||kT |
sin2 ϑ cos(4φh −2φR )
⊥ |2,−2i
FU U
= I −3
cos(4φh − φp − 3φk )h⊥
H
,
1
1
M Mhh
»
–
3 |pT ||kT |
sin 2ϑ cos(3φh −φR )
⊥ |2,−1i
FU U
= I −
cos(3φh − φp − 2φk )h⊥
,
1 H1
2 M Mhh
8
(40)
(41)
(42)
(43)
(44)
(45)
(46)
(47)
(48)
(49)
(50)
(51)
P
(ϑ) cos 2φh
=
sin 2ϑ cos(φh +φR )
=
2,0
FU U
FU U
sin2 ϑ cos(2φR )
FU U
»
–
|pT ||kT |
⊥ |2,0i
cos(2φh − φp − φk )h⊥
,
I −
1 H1
M Mhh
–
»
3 |pT ||kT |
⊥ |2,1i
,
I −
cos(φh − φp )h⊥
1 H1
2 M Mhh
»
–
|pT ||kT |
⊥ |2,2i
I −3
cos(φp − φk )h⊥
.
1 H1
M Mhh
=
(52)
(53)
(54)
For all structure functions, for any twist and polarization, the |l, mi state can be determined from
the angular identifier of the structure function, up to the sign of m. The l states are apparent from the
Legendre polynomial of cos ϑ, while the m state (up to the sign) is specified by the coefficient of φR . For
sine functions of the azimuthal angles, previous convention sometimes factors a −1 from the argument of
the sine function, placing the −1 in relation with the structure function, e.g. the |2, 2i partial wave for
the Sivers AU T moment, Equation 73. The cosine is even with respect to its argument, and thus hides
the sign of m.
The coefficient of φh then determines which m states can possibly survive in the collinear (Ph⊥ integrated) case, as the coefficient of φh can be written as ±(m − mcol. ), with mcol. being the coefficient
of φh (up to the sign) for the |0, 0i term, as well as the m state which survives Ph⊥ integration, i.e. the
⊥ |l,mi
m for the collinear case. For example, all h⊥
moments in the unpolarized, leading twist cross
1 H1
section (Equations 46 through 54) have cosine moments of the form cos(mφR − (m − 2)φh ). Thus the
m state surviving Ph⊥ integration is m = 2, and the |0, 0i state has a modulation of cos(2φ). Thus the
structure functions themselves already allow one to readily determine the angular momentum states of
the two dihadron system.
4.1.2
Subleading Twist
The twist 3 unpolarized structure functions are
»
„
«
2M
|kT |
Mhh D̃⊥ |0,0i
⊥ |0,0i
cos φ
FU U h =
I −
cos(φh − φk ) xhH1
f1
+
Q
Mhh
M
z
„
«–
|0,0i
|pT |
H̃
M
|0,0i
−
cos(φh − φp ) xf ⊥ D1
h⊥
,
+
1
M
Mhh
z
»
„
«
|kT |
2M
Mhh D̃⊥ |1,−1i
sin ϑ cos(2φh −φR )
⊥ |1,−1i
FU U
I −
cos(2φh − 2φk ) xhH1
f1
=
+
Q
Mhh
M
z
„
«–
|1,−1i
|pT |
M ⊥ H̃
|1,−1i
−
cos(2φh − φp − φk ) xf ⊥ D1
h1
,
+
M
Mhh
z
»
„
«
|kT |
2M
Mhh D̃⊥ |1,0i
⊥ |1,0i
cos ϑ cos φh
FU U
=
I −
cos(φh − φk ) xhH1
f1
+
Q
Mhh
M
z
„
«–
|1,0i
|pT |
M ⊥ H̃
|1,0i
−
cos(φh − φp ) xf ⊥ D1
h1
,
+
M
Mhh
z
»
„
«
|kT |
2M
Mhh D̃⊥ |1,1i
⊥ |1,1i
ϑ cos φR
FUsin
=
I
−
xhH
f
+
1
1
U
Q
2Mhh
M
z
„
«–
|pT |
M ⊥ H̃ |1,1i
|1,1i
⊥
−
cos(φp − φk ) xf D1
h1
,
+
M
Mhh
z
»
„
«
|kT |
6M
Mhh D̃⊥ |2,−2i
sin2 ϑ cos(3φh −2φR )
⊥ |2,−2i
FU U
=
I −
cos(3φh − 3φk ) xhH1
+
f1
Q
Mhh
M
z
„
«–
|2,−2i
|pT |
M
H̃
|2,−2i
cos(3φh − φp − 2φk ) xf ⊥ D1
+
,
−
h⊥
1
M
Mhh
z
»
„
«
|kT |
3M
Mhh D̃⊥ |2,−1i
sin 2ϑ cos(2φh −φR )
⊥ |2,−1i
FU U
=
I −
cos(2φh − 2φk ) xhH1
+
f1
Q
Mhh
M
z
„
«–
|2,−1i
|pT |
M ⊥ H̃
|2,−1i
−
cos(2φh − φp − φk ) xf ⊥ D1
+
h1
,
M
Mhh
z
»
P (ϑ) cos φh
2
FU U
=
„
⊥ |2,0i
|kT |
2M
Mhh D̃
⊥ |2,0i
I −
f1
cos(φh − φk ) xhH1
+
Q
Mhh
M
z
9
«
(55)
(56)
(57)
(58)
(59)
(60)
(61)
„
«–
|pT |
M ⊥ H̃ |2,0i
|2,0i
cos(φh − φp ) xf ⊥ D1
+
h1
,
M
Mhh
z
»
„
«
|kT |
3M
Mhh D̃⊥ |2,1i
⊥ |2,1i
I −
xhH1
+
f1
Q
Mhh
M
z
„
«–
|pT |
M ⊥ H̃ |2,1i
|2,1i
−
cos(φp − φk ) xf ⊥ D1
+
h1
,
M
Mhh
z
»
„
«
|kT |
6M
Mhh D̃⊥ |2,2i
⊥ |2,2i
I −
cos(φh + φk ) xhH1
+
f1
Q
Mhh
M
z
„
«–
|2,2i
|pT |
M ⊥ H̃
|2,2i
⊥
−
cos(φh − φp + 2φk ) xf D1
+
h1
.
M
Mhh
z
−
2ϑ cos φR
FUsin
U
sin2 ϑ cos(φh −2φR )
FU U
=
=
(62)
(63)
(64)
The form of the cosine modulation for these terms (Equations 55 through 64) is cos((m − 1)φh − mφR ).
Thus among these equations, the two m = 1 states survive Ph⊥ integration.
4.2
4.2.1
Unpolarized Beam, Transversely Polarized Target
Leading Twist
The leading twist unpolarized beam, transversely polarized target structure functions related to the
Sivers distribution function are
»
“
”–
|pT |
|0,0i
sin(φ −φ )
⊥
,
+ ig1T )D1
FU T,T h S
= −I
cos(φh − φp )Re (f1T
(65)
M
»
“
”–
|pT |
sin ϑ sin(2φh −φR −φS )
|1,−1i
⊥
FU T,T
= −I
(66)
cos(2φh − φp − φk )Re (f1T
+ ig1T )D1
,
M
»
–
“
”
|pT |
cos ϑ sin(φh −φS )
|1,0i
⊥
FU T,T
(67)
= −I
cos(φh − φp )Re (f1T
+ ig1T )D1
,
M
»
–
“
”
|pT |
sin ϑ sin(φR −φS )
|1,1i
⊥
FU T,T
= −I
cos(φp − φk )Re (f1T
+ ig1T )D1
(68)
,
M
»
–
“
”
|pT |
sin2 ϑ sin(3φh −2φR −φS )
|2,−2i
⊥
FU T,T
= −I 3
cos(3φh − φp − 2φk )Re (f1T
+ ig1T )D1
(69)
,
M
»
–
“
”
3 |pT |
sin 2ϑ sin(2φh −φR −φS )
|2,−1i
⊥
FU T,T
= −I
cos(2φh − φp − φk )Re (f1T
+ ig1T )D1
(70)
,
2 M
»
“
”–
|pT |
P2,0 (ϑ) sin(φh −φS )
|2,0i
⊥
FU T,T
cos(φh − φp )Re (f1T
+ ig1T )D1
(71)
= −I
,
M
»
–
“
”
3 |pT |
sin 2ϑ sin(φR −φS )
|2,1i
⊥
FU T,T
= −I
cos(φp − φk )Re (f1T
+ ig1T )D1
(72)
,
2 M
»
–
“
”
|pT |
sin2 ϑ sin(φh −2φR +φS )
|2,2i
⊥
FU T,T
= I 3
cos(φh + φp − 2φk )Re (f1T
+ ig1T )D1
.
(73)
M
The form of the sine modulations for these terms (Equations 65 through 73) is sin((1−m)φh −mφR −φS ).
Amount these equations, one would nominally expect the two m = 1 states to survive Ph⊥ integration.
However, Sivers function causes these terms to be zero in the collinear case.
The leading twist unpolarized beam, transversely polarized target structure functions related to
transversity and the Collins function are
»
–
|pT |
sin(φ +φ )
⊥ |0,0i
FU T h S
= −I
cos(φh − φk )h1 H1
,
(74)
M
–
»
|pT |
⊥ |1,−1i
sin ϑ sin(2φh −φR +φS )
cos(2φh − 2φk )h1 H1
,
FU T
= −I
(75)
M
–
»
|pT |
⊥ |1,0i
cos ϑ sin(φh +φS )
cos(φh − φk )h1 H1
,
FU T
= −I
(76)
M
–
»
|pT |
⊥ |1,1i
sin ϑ sin(φR +φS )
h 1 H1
,
FU T
= −I
(77)
2M
–
»
|pT |
⊥ |2,−2i
sin2 ϑ sin(3φh −2φR +φS )
cos(3φh − 3φk )h1 H1
,
FU T
= −I 3
(78)
M
10
»
sin 2ϑ sin(2φh −φR +φS )
FU T
P
=
(ϑ) sin(φh +φS )
=
sin 2ϑ sin(φR +φS )
=
FU T2,0
FU T
sin2 ϑ sin(φh −2φR −φS )
FU T
=
–
3 |pT |
⊥ |2,−1i
,
cos(2φh − 2φk )h1 H1
2 M
–
»
|pT |
⊥ |2,0i
,
−I
cos(φh − φp )h1 H1
M
»
–
3 |pT |
⊥ |2,1i
−I
h 1 H1
,
2 2M
–
»
|pT |
⊥ |2,2i
I 3
cos(φh − φk )h1 H1
,
M
−I
(79)
(80)
(81)
(82)
where the form of the sine modulations for these terms is sin((1 − m)φh − mφR + φS ). Once again, the
two m = 1 states are survive in the collinear case.
The leading twist unpolarized beam, transversely polarized target structure functions related to pretzelocity and the Collins function are
–
»
|pT |
⊥ |0,0i
sin(3φ −φ )
cos(3φh − 2φp − φk )h1 H1
,
FU T h S
= −I
(83)
M
–
»
|pT |
⊥ |1,−1i
sin(4φ −φ −φ )
FU T h R S
= −I
cos(4φh − 2φp − 2φk )h1 H1
,
(84)
M
–
»
|pT |
⊥ |1,0i
sin(3φ −φ )
,
cos(3φh − 2φp − φk )h1 H1
FU T h S
= −I
(85)
M
–
»
|pT |
⊥ |1,1i
sin(2φ +φ −φ )
FU T h R S
= −I
cos(2φh − 2φp )h1 H1
,
(86)
M
»
–
|pT |
sin(5φ −2φR −φS )
⊥ |2,−2i
FU T h
= −I
cos(5φh − 2φp − 3φk )h1 H1
,
(87)
M
»
–
|pT |
sin(4φ −φ −φ )
⊥ |2,−1i
FU T h R S
= −I
cos(4φh − 2φp − 2φk )h1 H1
(88)
,
M
»
–
|pT |
sin(3φ −φ )
⊥ |2,0i
FU T h S
= −I
cos(3φh − 2φp − φk )h1 H1
(89)
,
M
»
–
|pT |
sin(2φ +φ −φ )
⊥ |2,1i
FU T h R S
= −I
cos(2φh − 2φp )h1 H1
(90)
,
M
»
–
|pT |
sin(φ −2φR −φS )
⊥hh(2,2)
FU T h
= −I
cos(φh − 2φp + φk )h1 H1
(91)
.
M
The form of the sine modulations for these terms (Equations 83 through 91) is sin((3−m)φh −mφR −φS ).
Note, one would now expect only the m = −3 states to survive Ph⊥ integration. However, one usually
assumes that no contribution from l > 2 states. In any case, no terms with l ≤ 2 survive Ph⊥ integration.
5
Relation with Previous Notation
Comparing Equation 31 with Equation 57 of Ref. [3] yields the following relations for the unpolarized
fragmentation function. Note that some sources, including Ref. [3], use O for unpolarized instead of U ,
which is used in this document as well as, e.g. [1].
D1
|0,0i
=
D1,OO
|1,0i
D1
=
D1,OL
|1,±1i
D1
=
|2,0i
=
|2,±1i
=
|2,±2i
=
D1
D1
D1
D1,OT
(92)
(93)
|R||kT | ⊥
±
G1,OT
Mh2
1
D1,LL ,
2„
«
|R||kT | ⊥
1
D1,LT ±
G
1,LT
3
Mh2
„
«
1 |R||kT | ⊥
1
D1,T T ±
G
.
1,T
T
3
2 Mh2
(94)
(95)
(96)
(97)
To compute the relations with the Collins fragmentation function, it is easier to compare Equation
31 with Equation 4.57 of Ref. [1]1 . However, the barred notation for H̄1 of Ref. [3] is used, as it the
11
most clear option. Note the subtlety regarding the notation in the literature, discussed in connection
with Equations 35, 36.
⊥ |0,0i
=
⊥
H1,OO
,
⊥ |1,1i
H1
=
⊥
H1,OT
⊥ |1,0i
=
⊥
H1,OL
,
=
⊥
H1,OT
,
=
1
3
H1
H1
⊥ |1,−1i
H1
⊥ |2,2i
H1
⊥ |2,1i
=
⊥ |2,0i
=
⊥ |2,−1i
=
⊥ |2,−2i
=
H1
H1
H1
H1
6
„
⊥
H1,T
T
„
(98)
|R| ⊥
H̄1,OT ,
+
|kT
(99)
(100)
|R| ⊥
+
H̄1,T T
|kT
|R| ⊥
1
⊥
+
H̄1,LT
H1,LT
3
|kT
1 ⊥
H1,LL ,
2
1 ⊥
H1,LT ,
3
1 ⊥
H1,T T .
3
(101)
«
,
(102)
,
(103)
«
(104)
(105)
(106)
Conclusion
This internal note is intended to form the theoretical framework for the ongoing Hermes SIDIS dihadron
analysis, including sufficient detail on the cross section and the relation of its angular moments to
distribution and fragmentation functions. The structure functions and cross section of this document
will also be used to implement dihadron processes in GMC Trans.
This document has highlighted four major advancements: 1) a new partial wave expansion of the
dihadron fragmentation functions, along with applicable notation; 2) writing the previously available
leading twist dihadron SIDIS cross section in terms of structure functions; 3) determine the twist three
cross section in terms of structure functions; 4) interpreting the structure functions for both leading and
subleading twist in terms of distribution functions and fragmentation functions, using the new partial
wave expansion. Relations between the partial waves in the basis of the new expansion and those in the
old basis are given.
The new expansion and notation has many advantages over the old notation and expansion. In addition to the new notation being less bulky, the notation highlights the spherical harmonic structure
present in the previously written cross section, but obscured by the previous notation. The new notation
and expansion are more precise–as the old notation labels all l = 1 states as sp, while they actually
include both sp and pp contributions. The fragmentation functions occurring in the cross section can
be more easily interpreted, and the partial wave expansion is make explicit, instead of having to reference complicated 8 × 8 matrices. This is especially important when looking for the predictions of the
Artru/Lund fragmentation model regarding the sign of the Collin’s function. The new expansion involves fragmentation functions that are either fully expanded or not expanded at all, while the previous
notation includes several partial expansions. Perhaps more important, the new notation highlights that
there are only two non-expanded fragmentation functions at leading twist, rather than four suggested by
the previous notation.
At present, only structure functions and cross section terms for the unpolarized case, at both leading
and subleading twist, and the transverse target, leading twist, are available. The additional terms can
be made available at a later date.
The single most important improvement of this chose of basis is that it highlights the similarities
between the single hadron and dihadron cross sections. The added complexity of the dihadron cross
section is given a single source—the |l, mi spin quantum numbers of the final state. The number of
non-expanded fragmentation functions are identical. This allows the subleading twist dihadron cross
section to be extrapolated from the subleading twist single hadron cross section, in a procedure much
less complicated than direct calculation. The same procedure could also be applied to higher twist
terms, when available. One particular instance where these subleading twist moments are important is
a misprint in the equation 57 of Ref. [3] for H̄1 . Instead of cos(2φk − 2φR ) it should read cos(φk − φR ), to be
consistent with Ref. [1].
1 Note
12
the interplay between acceptance and the subleading twist unpolarized moments, as this can effect the
extracted leading twist transverse target moments.
References
[1] A. Bacchetta. Probing the transverse spin of quarks in deep inelastic scattering. PhD thesis, Vrije
Universiteit, 2002.
[2] Alessandro Bacchetta, Markus Diehl, Klaus Goeke, Andreas Metz, Piet J. Mulders, and Marc
Schlegel. Semi-inclusive deep inelastic scattering at small transverse momentum. JHEP0702, 093,
2007.
[3] Alessandro Bacchetta and Marco Radici. Partial-wave analysis of two-hadron fragmentation functions. Phys. Rev. D, 67(9):094002, May 2003.
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