Vector meson production from a polarized nucleon
arXiv:0704.1565v2  [hep-ph]  29 Aug 2007
Preprint typeset in JHEP style - PAPER VERSION DESY-07-049
arXiv:0704.1565 [hep-ph]
Vector meson production from a polarized nucleon
M. Diehl
Deutsches Elektronen-Synchroton DESY, 22603 Hamburg, Germany
Abstract: We provide a framework to analyze the electroproduction process ep → epρ
with a polarized target, writing the angular distribution of the ρ decay products in terms
of spin density matrix elements that parameterize the hadronic subprocess γ∗p → ρp.
Using the helicity basis for both photon and meson, we find a representation in which the
expressions for a polarized and unpolarized target are related by simple substitution rules.
Keywords: Lepton-Nucleon Scattering, Spin and Polarization Effects.
http://arxiv.org/abs/0704.1565v2
Contents
1. Introduction 1
2. Kinematics and target polarization 2
3. Helicity amplitudes and spin density matrix 4
4. The angular distribution 9
5. Natural and unnatural parity 17
6. Positivity constraints 21
7. Mixing between transverse and longitudinal polarization 23
8. A note on non-resonant contributions 27
9. Summary 28
1. Introduction
Exclusive vector meson production has long played an important role in studying the
strong interaction. The seminal work [1, 2] has renewed interest in this process, showing
that in Bjorken kinematics it provides access to generalized parton distributions and thus
to a wealth of information on the structure of the proton. While most theoretical and
experimental studies so far are for an unpolarized proton, the particular interest of target
polarization became clear when it was pointed out that meson production on a transversely
polarized target is sensitive to the nucleon helicity-flip distribution E [3, 4]. This distri-
bution offers unique views on the orbital angular momentum carried by partons in the
proton [5, 6] and on the correlation between polarization and the spatial distribution of
partons [7]. Whereas the corresponding polarization asymmetry in deeply virtual Compton
scattering is under better theoretical control, vector meson production has the advantage
of a greater sensitivity to the distribution of gluons (which in Compton scattering only
enters at next-to-leading order in αs). This holds not only in the high-energy regime but
even in a wide range of fixed-target kinematics [8, 9, 10], where polarization measurements
are feasible at existing or planned experimental facilities.
A different motivation to study polarized exclusive ρ production is that this channel
plays a rather prominent role in semi-inclusive pion production [11, 12, 9], which has become
a privileged tool to study a variety of spin effects, see e.g. [13]. It is important to identify
– 1 –
kinematical regions where the exclusive channel ep → epρ → epπ+π− dominates semi-
inclusive observables, because in these regions great care must be taken when interpreting
the data in terms of semi-inclusive factorization.
Even with an unpolarized target, the spin structure of the process ep → epρ→ epπ+π−
is very rich, because the angular distribution of the final state contains information on the
helicities of the exchanged virtual photon and of the ρ meson, as was worked out in the
classical analysis of Schilling and Wolf [14]. Yet more detailed information is available with
target polarization [15]. Experiments on unpolarized targets have found that s-channel
helicity is approximately conserved in the transition from the γ∗ to the ρ, with helicity
changing amplitudes occurring at most at the 10% level [16, 17, 18, 19, 20]. This greatly
simplifies the spin structure of the process. The aim of the present paper is to provide
an analysis framework for exclusive ρ production on a polarized nucleon target, making as
explicit as possible the relation between the angular dependence of the cross section and
the helicity amplitudes describing the hadronic subprocess γ∗p→ ρp. We will present our
results in a form that emphasizes the close similarity in structure between an unpolarized
and a polarized target. Using the helicity basis for both virtual photon and meson, we also
provide an alternative to the representation of the unpolarized cross section in [14].
The following section gives the definitions of the kinematics and polarization variables
for the reaction under study. In Section 3 we define the helicity amplitudes and the spin
density matrix elements describing the process and discuss some of their general properties.
In Section 4 we express the angular distribution of the polarized cross section in terms of
these spin density matrix elements and point out some salient features of this representa-
tion. The simplifications arising from distinguishing natural and unnatural parity exchange
in the reaction are discussed in Section 5. A number of positivity bounds relating different
spin density matrix elements are given in Section 6. In Section 7 we explain the complica-
tions arising from the distinction between target polarization relative to the momentum of
either the incident lepton or the virtual photon. The role of non-resonant contributions in
π+π− production is briefly discussed in Section 8. Our results are summarized in Section 9.
2. Kinematics and target polarization
Let us consider the electroproduction process
e(l) + p(p) → e(l′) + p(p′) + ρ(q′) (2.1)
followed by the decay
ρ(q′) → π+(k) + π−(k′), (2.2)
where four-momenta are given in parentheses. Throughout this work we use the one-photon
exchange approximation. All or results are equally valid for the production of a φ followed
by the decay φ→ K+K−. They also hold if the scattered proton is replaced by an inclusive
system X with four-momentum p′, as explained at the end of Section 3.
To describe the kinematics we use the conventional variables for deep inelastic pro-
cesses, Q2 = −q2, xB = Q2/(2p · q) and y = (p · q)/(p · l). We neglect the lepton mass
– 2 –
lepton plane
hadron plane
θγ ST
Figure 1: Kinematics of ep → epρ in the target rest frame. ST is the transverse component of
the target spin vector w.r.t. the virtual photon direction.
throughout and denote the longitudinal lepton beam polarization by Pℓ, with Pℓ = +1
corresponding to a purely right-handed and Pℓ = −1 to a purely left-handed beam. Let us
now go to the target rest frame and introduce the right-handed coordinate system (x, y, z)
of Fig. 1 such that q points in the positive z direction and l has a positive x component. In
this system we have l = |l|(sin θγ , 0, cos θγ) and q = |q|(0, 0, 1), where the angle θγ between
l and q is defined to be between 0 and π. In accordance with the Trento convention [21]
we define the angle φ between the lepton and the hadron plane as the azimuthal angle of
q′ in this coordinate system, and φS as the azimuthal angle of the target spin vector S.
Following [22] we write S = (ST cosφS , ST sinφS ,−SL) with 0 ≤ ST ≤ 1 and −1 ≤ SL ≤ 1,
so that ST and SL describe transverse and longitudinal polarization with respect to the
virtual photon momentum, with SL = 1 corresponding to a right-handed proton in the
γ∗p c.m.
*γ c.m.p
boost
π  π+ c.m.−
Figure 2: Kinematics of the hadronic subprocess γ∗p → ρp followed by the decay ρ → π+π−.
The coordinate systems (x, y, z) and (x′, y′, z′) differ from those in Fig. 1.
– 3 –
To describe the target polarization of a given experimental setup, we introduce an-
other right-handed coordinate system (x′, y′, z′) in the target rest frame such that l =
|l|(0, 0, 1) and q = |q|(− sin θγ , 0, cos θγ) as shown in Fig. 1. In this system we write
S = (PT cosψ,PT sinψ,−PL) with 0 ≤ PT ≤ 1 and −1 ≤ PL ≤ 1, following again [22]. PT
and PL describe transverse and longitudinal polarization with respect to the lepton beam
direction, with PL = 1 corresponding to a right-handed proton in the ep c.m. The two sets
of variables describing the target polarization are related by
ST cosφS = cos θγPT cosψ − sin θγPL ,
ST sinφS = PT sinψ ,
SL = sin θγPT cosψ + cos θγPL , (2.3)
which we will use in Sect. 7. In terms of invariants the mixing angle θγ is given by
sin θγ = γ
1− y − 1
1 + γ2
, γ =
2xBMN
, (2.4)
whereMN is the nucleon mass. In Bjorken kinematics γ is small, and so is sin θγ ≈ γ
1− y.
We finally specify the variables describing the vector meson decay (2.2). This is conve-
niently done in the π+π− c.m., which can be obtained from the γ∗p c.m. by a boost in the
direction of the scattered nucleon as shown in Fig. 2. In the π+π− c.m. we introduce the
right-handed coordinate system (x, y, z) shown in Fig. 2, where p′ = |p′|(0, 0,−1) and where
the target momentum p has a positive x component. In this system we define ϑ and ϕ as the
polar and azimuthal angle of the π+ momentum, i.e. k = |k|(sinϑ cosϕ, sinϑ sinϕ, cos ϑ).
The relation between our notation here and the one of Schilling and Wolf is1
φhere = −Φ[14] , ϕhere = φ [14] , ϑhere = θ [14] . (2.5)
3. Helicity amplitudes and spin density matrix
The strong-interaction dynamics of the electroproduction process (2.1) is fully contained
in the helicity amplitudes for the subprocess γ∗p→ ρp. From these we will construct spin
density matrix elements which describe the angular distribution of the overall reaction
ep → ep π+π− and its dependence on the target polarization.
Since we will deal with interference terms we must specify our phase conventions. We
do this in the γ∗p c.m. and use the right-handed coordinate system (x′, y′, z′) shown in
Fig. 2. In this system we have q = |q|(0, 0,−1) and q′ = |q′|(sinΘ, 0,− cos Θ), with the
scattering angle Θ of the vector meson defined to be between 0 and π. Note that the positive
z′ axis points along p rather than q, as is often preferred for theoretical calculations. We
specify polarization states of the target proton by two-component spinors χ+1/2 = (1, 0)
1We remark that the expression for sinΦ given in eq. (13) of [14] is incorrect since it is always positive.
A correct definition is given in [23].
– 4 –
for positive and χ−1/2 = (0, 1) for negative helicity. For the polarization vectors of the
virtual photon we choose
ε+1 = −
0, 1,−i, 0
, ε−1 =
0, 1, i, 0
εα0 = Nε
qα − q
p · q p
, (3.1)
and for the polarization vectors of the ρ
e+1 = −
0, cos Θ,−i, sinΘ
, e−1 =
0, cos Θ, i, sinΘ
eα0 = Ne
q′α − q
p′ · q′ p
, (3.2)
where the subscripts indicate helicities. Nε and Ne are positive constants ensuring the
proper normalization ε20 = 1 and e
0 = −1 of the longitudinal polarization vectors. In the
ρ rest frame and the coordinate system (x, y, z) of Fig. 2, our meson polarization vectors
have the standard form e+1 = −(0, 1, i, 0)/
2, e−1 = (0, 1,−i, 0)/
2 and e0 = (0, 0, 0, 1).
Our phase conventions for the proton and the virtual photon are as in [22].
We now introduce amplitudes T νσµλ for the subprocess γ
∗(µ)+ p(λ) → ρ(ν)+ p(σ) with
definite helicities µ, ν, λ, σ. Since the above phase conventions are defined with reference
only to momentum vectors of this subprocess, the helicity amplitudes only depend on the
photon virtuality, the γ∗p scattering energy and the scattering angle Θ, or equivalently on
Q2, xB and t = (p− p′)2. With our phase conventions they obey the usual parity relations
T−ν−σ−µ−λ = (−1)
ν−µ−σ+λ T νσµλ (3.3)
for equal Q2, xB and t on both sides. With these helicity amplitudes we define
µµ′,λλ′ = (NT + ǫNL)
T νσµλ
. (3.4)
Regarding the upper indices this is the spin density matrix of the vector meson, whereas the
lower indices specify the polarizations in the γ∗p state from which the meson is produced.2
The normalization factors
λ,ν,σ
∣T νσ+λ
, NL =
λ,ν,σ
∣T νσ0λ
(3.5)
are proportional to the differential cross sections dσT /dt and dσL/dt for transverse and
longitudinal photon polarization, respectively, and
1− y − 1
1− y + 1
y2 + 1
(3.6)
2Taking the trace in the meson polarization indices we obtain the relation
ρννµµ′,λλ′ ∝ dσ
µ′µ/dt
between the spin density matrix ρ introduced here and the cross sections and interference terms used in
[22]. Compared with [22] we take the opposite order of indices in ρ, so that ν and ν′ appear in the standard
order for a spin density matrix.
– 5 –
is the usual ratio of longitudinal and transverse photon flux. In addition to Q2, xB and
t, the spin density matrix elements ρνν
µµ′,λλ′ depend on ǫ through the normalization factor
(NT + ǫNL). If one can perform a Rosenbluth separation by measuring at different ǫ but
equal Q2 and xB, it is advantageous to normalize them instead to NT , NL or
NTNL as
was done in [14]. It is straightforward to implement such a change in the formulae we give
in the following.
We find it useful to introduce the combinations
µµ′ =
µµ′,++ + ρ
µµ′,−−
, lνν
µµ′ =
µµ′,++ − ρνν
µµ′,−−
(3.7)
for an unpolarized and a longitudinally polarized target, where for the sake legibility we
have labeled the target polarization by ± instead of ±1
. The combinations
µµ′ =
µµ′,+− + ρ
µµ′,−+
, nνν
µµ′ =
µµ′,+− − ρνν
µµ′,−+
(3.8)
respectively describe transverse target polarization in the hadron plane (“sideways”) and
perpendicular to it (“normal”). One readily finds that the matrices u , l and s are hermi-
tian, whereas n is antihermitian,
µ′µ =
µ′µ =
µ′µ =
µ′µ = −
. (3.9)
The diagonal elements uννµµ, l
µµ and s
µµ are therefore purely real, whereas n
µµ is purely
imaginary. Furthermore, the parity relations (3.3) translate into
u−ν−ν
−µ−µ′
= (−1)ν−µ−ν′+µ′ uνν′µµ′ , l−ν−ν
−µ−µ′
= −(−1)ν−µ−ν′+µ′ lνν′µµ′ ,
n−ν−ν
−µ−µ′ = (−1)
ν−µ−ν′+µ′ nνν
µµ′ , s
−ν−ν′
−µ−µ′ = −(−1)
ν−µ−ν′+µ′ sνν
µµ′ . (3.10)
As a consequence the matrix elements
u−+−+ , u
−+ , u
0 0 , u
−+ (3.11)
are purely real, whereas the corresponding elements of l , s and n are purely imaginary.
Both experiment and theory indicate that s-channel helicity is approximately conserved
in the γ∗ → ρ transition for small invariant momentum transfer t. Correspondingly, one
expects that spin density matrix elements involving the product of two helicity conserving
amplitudes are greater than interference terms between a helicity conserving and a helicity
changing amplitude, and that those are greater than matrix elements involving the product
of two helicity changing amplitudes (where we refer to the helicities of the photon and the
ρ but not of the nucleon). Exceptions to this rule are however possible, since two large
amplitudes can have a small interference term because of their relative phase, and since
there can be cancellation of individually large terms in the linear combinations (3.7) and
(3.8) associated with different target polarizations. With this caveat in mind one can
readily assess the expected size of the spin density matrix elements (3.7) and (3.8) by
comparing the upper with the lower indices.
– 6 –
Let us now investigate the behavior of our matrix elements for Θ → 0, i.e. in the limit
of forward scattering γ∗p→ ρp. To this end we perform a partial wave decomposition
T νσµλ (Θ) =
tνσµλ(J) d
λ−µ,σ−ν(Θ) (3.12)
where we have suppressed the dependence of T and the partial wave amplitudes t(J) on
Q2 and xB . Using the behavior d
m,n(Θ) ∼ Θ|m−n| of the rotation functions for Θ → 0 we
readily find
µµ′ , l
µµ′ ∼ Θp , nνν
µµ′ , s
µµ′ ∼ Θq (3.13)
p ≥ pmin = min
σ,λ=±1/2
∣ν − µ− σ + λ
∣ν ′ − µ′ − σ + λ
q ≥ qmin = min
σ,λ=±1/2
∣ν − µ− σ + λ
∣ν ′ − µ′ − σ − λ
. (3.14)
With Θ ∝ (t0 − t)1/2 for small Θ, we can rewrite (3.13) as
µµ′ , l
µµ′ ∼
(t0 − t)p/2 , nνν
µµ′ , s
µµ′ ∼
(t0 − t)q/2 , (3.15)
where t0 is the value of t for Θ = 0 at given Q
2 and xB. In Tables 1 and 2 we give
the corresponding powers for the linear combinations of spin density matrix elements that
will appear in our results for the cross section in Section 4. We have ordered the entries
according to the hierarchy discussed after (3.11), listing first terms containing the product
of two helicity conserving amplitudes, then terms containing the interference between a
helicity conserving and a helicity changing amplitude, and finally terms which only involve
helicity changing amplitudes (with helicities always referring to the photon and the ρ but
not to the nucleon).
We emphasize that certain partial wave amplitudes tνσµλ(J) in (3.12) may be zero or
negligibly small for dynamical reasons. The actual powers of (t0 − t)1/2 in (3.15) can
thus be larger than the minimum values pmin and qmin required by angular momentum
conservation. If there is for instance no s-channel helicity transferred between the proton-
proton and the photon-meson transitions, then the relevant powers for n and s are given
by q = pmin+1, which is equal to qmin+2 for all but the first four entries in Tables 1 and 2.
A concrete realization of this scenario is the calculation in [24], where the proton-proton
transition is described by the generalized parton distributions H, E and H̃, Ẽ, which do
not allow for helicity transfer to the photon-meson transition.
In the limit of large Q2 at fixed xB and t, the proof of the factorization theorem
in [2] implies that the transition from a longitudinal photon to a longitudinal ρ becomes
dominant, with all other transitions suppressed by powers of 1/Q. In this limit only the
spin density matrix elements u 0 00 0 and n
0 0 survive and can be expressed as convolutions of
hard-scattering kernels with generalized parton distributions and the light-cone distribution
amplitude of the ρ. To leading order in 1/Q one has in particular
Imn 0 00 0
u 0 00 0
t0 − t
1− ξ2 Im
(1− ξ2) |H|2 −
ξ2 + t/(4M2N )
|E|2 − 2ξ2 Re
, (3.16)
– 7 –
matrix elements pmin
u 0 0++ + ǫu
0 0 0
u 0+0+ − u
0+ − l
u++++ + u
++ + 2ǫu
0 0 l
++ + l
u−+−+ l
u 0 00+ l
u 0+++ − u−0++ + 2Re ǫu 0+0 0 l
++ − l−0++ + 2i Im ǫl 0+0 0 1
u 0+−+ l
u 0−0+ − u
0+ − l
u−+++ + ǫu
0 0 l
++ + ǫl
0 0 2
u++−+ l
u++0+ + u
0+ + l
u−+0+ l
l 0 0++ 2
u 0 0−+ l
u+0−+ l
u+−0+ l
u+−−+ l
Table 1: Minimum values of the powers which control the t→ t0 behavior of combinations of spin
density matrix elements u and l as in (3.15). Some of the combinations are purely real or purely
imaginary because of the symmetry relations (3.9) and (3.10), whereas others are complex valued.
where ξ = xB/(2−xB) and the convolution integrals H and E are for instance given in [22].
Experimental results and phenomenological analysis show however that 1/Q2 suppressed
effects can be numerically significant for Q2 of several GeV2, see e.g. [25, 24, 9, 10]. This
concerns both power corrections within u 0 00 0 or n
0 0 and formally power suppressed spin
density matrix elements such as u++++ or u
0+ . The detailed analysis in [2] reveals that
beyond leading-power accuracy in 1/Q, factorization of meson production into a hard-
scattering subprocess and nonperturbative quantities pertaining either to the target or
to the meson may be broken. On the other hand, factorization based approaches which
go beyond leading power in 1/Q and in particular also evaluate transition amplitudes for
transverse polarization of the γ∗ or ρ have been phenomenologically rather successful, see
e.g. [26, 24]
Let us finally generalize our considerations to the process
e(l) + p(p) → e(l′) +X(p′) + ρ(q′) , (3.17)
where the target proton dissociates into a hadronic system X. In analogy to the elastic case
one can introduce helicity amplitudes T
and combine them into spin density matrix
– 8 –
matrix elements qmin
n 0 0++ + ǫn
0 0 1
n 0+0+ − n
0+ − s
n++++ + n
++ + 2ǫn
0 0 s
++ + s
n−+−+ s
n 0 00+ s
n 0+++ − n−0++ + 2i Im ǫn 0+0 0 s
++ − s−0++ + 2i Im ǫs 0+0 0 0
n 0+−+ s
n 0−0+ − n
0+ − s
n−+++ + ǫn
0 0 s
++ + ǫs
0 0 1
n++−+ s
n++0+ + n
0+ + s
n−+0+ s
s 0 0++ 1
n 0 0−+ s
n+0−+ s
n+−0+ s
n+−−+ s
Table 2: As Table 1 but for combinations of spin density matrix elements n and s .
elements
µµ′,λλ′ = (NT + ǫNL)
ν′σ,X
. (3.18)
The normalization factors NT and NL are defined as in (3.5) but with an additional sum
over all hadronic states X of given invariant mass MX , on which ρ
µµ′,λλ′ now depends
in addition to Q2, xB , t and ǫ. The combinations (3.7) and (3.8) for different target
polarization have the same symmetry properties (3.9) and (3.10) as in the elastic case.
Their behavior for t→ t0 can be different, since in (3.14) one must now take the minimum
over all possible helicities σ = ±1
, . . . of the hadronic system X. One finds however
that the powers pmin and qmin for the combinations of spin density matrix elements in
Tables 1 and 2 are the same as in the elastic case. The results in the remainder of this
work only depend on the properties (3.9) and (3.10) and thus immediately generalize to
the case of target dissociation.
4. The angular distribution
The calculation of the cross section for ep → ep π+π− proceeds by using standard methods
and we shall only sketch the essential steps. More details are for instance given in [14, 27,
– 9 –
22]. With our phase conventions the polarization state of the proton target is described by
the spin density matrix
τλλ′ =
1 + SL ST e
−i(φ−φS)
i(φ−φS) 1− SL
, (4.1)
which is to be contracted with the matrix in (3.4). The result is conveniently expressed in
terms of the combinations (3.7) and (3.8) as
τλλ′ ρ
µµ′,λλ′ = u
µµ′ + SL l
µµ′ + ST cos(φ− φS) sνν
µµ′ − ST sin(φ− φS) inνν
µµ′ (4.2)
and describes the subprocess γ∗p → ρp. The decay ρ → π+π− is taken into account by
multiplication with the spherical harmonics,
ρµµ′ =
τλλ′ ρ
µµ′,λλ′ Y1ν(ϕ, ϑ)Y
1ν′(ϕ, ϑ) , (4.3)
where
Y1+1 = −
sinϑ eiϕ , Y10 =
cos ϑ , Y1−1 =
sinϑ e−iϕ . (4.4)
To obtain the cross section for the overall process ep→ epπ+π− one must finally contract
the matrix ρµµ′ in (4.3) with the spin density matrix of the virtual photon.
3 The cross
section can be written as
dψ dφdϕd(cos ϑ) dxB dQ2 dt
(2π)2
dxB dQ2 dt
WUU + PℓWLU + SLWUL + PℓSLWLL + STWUT + PℓSTWLT
(4.5)
dxB dQ2 dt
1− xB
, (4.6)
where dσT /dt and dσL/dt are the usual γ
∗p cross sections for a transverse and longitudinal
photon and an unpolarized proton, with Hand’s convention for virtual photon flux. The
angular distribution is described by the quantities WXY , where X specifies the beam and
Y the target polarization. The normalization of the unpolarized term WUU is
dϕ d(cos ϑ)WUU(φ,ϕ, ϑ) = 1 . (4.7)
To limit the length of subsequent expressions, we further decompose the coefficients ac-
cording to the ρ polarization and write
WXY (φ,ϕ, ϑ)
cos2ϑ WLLXY (φ) +
2 cos ϑ sinϑ WLTXY (φ,ϕ) + sin
2ϑ W TTXY (φ,ϕ)
(4.8)
for X,Y = U,L. The production of a longitudinal ρ is described by WLLXY , the production
3Up to a global factor, the result of this contraction can e.g. be obtained from eq. (3.20) of [27], with
ρµµ′ in the present work corresponding to σ
µ′µ in [27] and φhere = −ϕ[27].
– 10 –
of a transverse ρ (including the interference between positive and negative ρ helicity) by
W TTXY , and the interference between longitudinal and transverse ρ polarization by W
For a transversely polarized target we have in addition a dependence on φS ,
WXT (φS , φ, ϕ, ϑ)
cos2ϑ WLLXT (φS , φ) +
2 cos ϑ sinϑ WLTXT (φS , φ, ϕ) + sin
2ϑ W TTXT (φS , φ, ϕ)
(4.9)
with X = U,L. In addition to the angles, all coefficients WXY depend on Q
2, xB and t,
which we have not displayed for the sake of legibility.
For unpolarized target and beam we have
WLLUU(φ) =
u 0 0++ + ǫu
− 2 cos φ
ǫ(1 + ǫ) Reu 0 00+ − cos(2φ) ǫu 0 0−+ ,
WLTUU (φ,ϕ) = cos(φ+ ϕ)
ǫ(1 + ǫ) Re
u 0+0+ − u
− cosϕ Re
u 0+++ − u−0++ + 2ǫu 0+0 0
+ cos(2φ+ ϕ) ǫRe u 0+−+
− cos(φ− ϕ)
ǫ(1 + ǫ) Re
u 0−0+ − u
+ cos(2φ − ϕ) ǫRe u+0−+ ,
W TTUU (φ,ϕ) =
u++++ + u
++ + 2ǫu
cos(2φ+ 2ϕ) ǫu−+−+
− cosφ
ǫ(1 + ǫ) Re
u++0+ + u
+ cos(φ+ 2ϕ)
ǫ(1 + ǫ) Re u−+0+
− cos(2ϕ) Re
u−+++ + ǫu
− cos(2φ) ǫRe u++−+
+ cos(φ− 2ϕ)
ǫ(1 + ǫ) Reu+−0+ +
cos(2φ− 2ϕ) ǫu+−−+ . (4.10)
Here and in the following we order terms according to the hierarchy discussed after (3.11),
as already done in Table 1. The terms independent of φ and ϕ in WLLUU and W
UU are
related by
u++++ + u
++ + 2ǫu
0 0 = 1−
u 0 0++ + ǫu
, (4.11)
which ensures the normalization condition (4.7). The terms for beam polarization with an
unpolarized target read
WLLLU (φ) = −2 sinφ
ǫ(1− ǫ) Imu 0 00+ ,
WLTLU (φ,ϕ) = sin(φ+ ϕ)
ǫ(1− ǫ) Im
u 0+0+ − u
− sinϕ
1− ǫ2 Im
u 0+++ − u−0++
− sin(φ− ϕ)
ǫ(1− ǫ) Im
u 0−0+ − u
W TTLU (φ,ϕ) = − sinφ
ǫ(1− ǫ) Im
u++0+ + u
+ sin(φ+ 2ϕ)
ǫ(1− ǫ) Imu−+0+
− sin(2ϕ)
1− ǫ2 Imu−+++
+ sin(φ− 2ϕ)
ǫ(1− ǫ) Imu+−0+ . (4.12)
– 11 –
The results for longitudinal target polarization are very similar, with
WLLUL(φ) = −2 sinφ
ǫ(1 + ǫ) Im l 0 00+ − sin(2φ) ǫ Im l 0 0−+ ,
WLTUL (φ,ϕ) = sin(φ+ ϕ)
ǫ(1 + ǫ) Im
l 0+0+ − l
− sinϕ Im
l 0+++ − l−0++ + 2ǫl 0+0 0
+ sin(2φ+ ϕ) ǫ Im l 0+−+
− sin(φ− ϕ)
ǫ(1 + ǫ) Im
l 0−0+ − l
+ sin(2φ− ϕ) ǫ Im l+0−+ ,
W TTUL (φ,ϕ) =
sin(2φ+ 2ϕ) ǫ Im l−+−+
− sinφ
ǫ(1 + ǫ) Im
l++0+ + l
+ sin(φ+ 2ϕ)
ǫ(1 + ǫ) Im l−+0+
− sin(2ϕ) Im
l−+++ + ǫl
− sin(2φ) ǫ Im l++−+
+ sin(φ− 2ϕ)
ǫ(1 + ǫ) Im l+−0+ +
sin(2φ− 2ϕ) ǫ Im l+−−+ (4.13)
for an unpolarized beam, and
WLLLL (φ) = −2 cosφ
ǫ(1− ǫ) Re l 0 00+ +
1− ǫ2 l 0 0++ ,
WLTLL (φ,ϕ) = cos(φ+ ϕ)
ǫ(1− ǫ) Re
l 0+0+ − l
− cosϕ
1− ǫ2 Re
l 0+++ − l−0++
− cos(φ− ϕ)
ǫ(1− ǫ) Re
l 0−0+ − l
W TTLL (φ,ϕ) =
1− ǫ2 1
l++++ + l
− cosφ
ǫ(1− ǫ) Re
l++0+ + l
+ cos(φ+ 2ϕ)
ǫ(1− ǫ) Re l−+0+
− cos(2ϕ)
1− ǫ2 Re l−+++
+ cos(φ− 2ϕ)
ǫ(1− ǫ) Re l+−0+ (4.14)
for beam polarization. In (4.10) to (4.14) we have used the symmetry relations (3.9) and
(3.10) to write our results with a minimal set of matrix elements uνν
µµ′ or l
µµ′ . Although
they are a little lengthy, their structure is quite simple:
1. The combinations u++
+ u−−
, u 0+
− u−0
and u 0−
− u+0
and their analogs for l
always appear together because the corresponding products of spherical harmonics
are identical, Y1+1Y
1+1 = Y1−1Y
1−1 and Y10Y
1+1 = −Y1−1Y ∗10. In some cases the
corresponding sum can be simplified using symmetry relations like u++0 0 + u
0 0 =
2u++0 0 , but in others one remains with a linear combination of matrix elements that
cannot be separated. With the caveats discussed after (3.11) one finds however that
these combinations are dominated by a single term. Exceptions are Re
u 0+++ −u−0++ +
2ǫu 0+0 0
and Im
l 0+++ − l−0++ + 2ǫl 0+0 0
, each of which contains two interference terms
between a helicity conserving and a helicity changing amplitude.
– 12 –
2. An angular dependence through (kφ + mϕ) is associated with the interference be-
tween transverse and longitudinal ρ polarization for |m| = 1, the interference between
positive and negative ρ helicity for |m| = 2, and equal ρ polarization in the amplitude
and its conjugate for m = 0. In the same way |k| = 1, |k| = 2 and k = 0 are related to
the virtual photon polarization. Notice that for m = 0 one can distinguish transverse
and longitudinal ρ production by the ϑ dependence in (4.8), whereas for k = 0 the
separation of terms for transverse and longitudinal photons requires variation of ǫ.
The beam spin asymmetries WLU and WLL contain no terms with |k| = 2, because
there is no term with Pℓ cos 2φ or Pℓ sin 2φ in the spin density matrix of the virtual
photon.
3. The unpolarized or doubly polarized terms WUU and WLL depend on Reu or Re l
and are even under the reflection (φ,ϕ) → (−φ,−ϕ) of the azimuthal angles, whereas
the single spin asymmetriesWLU andWUL depend on Imu or Im l and are odd under
(φ,ϕ) → (−φ,−ϕ). This is a consequence of parity and time reversal invariance.
4. As we have written our results, the angular distribution for longitudinal target po-
larization can be obtained from the one for an unpolarized target by replacing
cos(kφ+mϕ) Re u → sin(kφ+mϕ) Im l ,
sin(kφ+mϕ) Imu → cos(kφ+mϕ) Re l . (4.15)
Terms with k = m = 0 in WUU and WLL are independent of φ and ϕ, and have of
course no counterparts inWUL orWLU . This corresponds to 16 terms with a different
angular dependence in WUU and 14 terms in WUL, and to 10 terms in WLL and 8
terms in WLU .
The symmetry properties (3.9) and (3.10), which we used to obtain (4.10) to (4.14), are
identical for uνν
µµ′ and in
µµ′ , as well as for l
µµ′ and s
µµ′ . According to (4.2) the cross section
for a transversely polarized target can therefore be obtained from the one for longitudinal
and no target polarization by the replacements
Reu → ST sin(φ− φS) Imn , SL Im l → ST cos(φ− φS) Im s ,
Imu → −ST sin(φ− φS) Ren , SLRe l → ST cos(φ− φS) Re s . (4.16)
We thus simply have
WLLUT (φS , φ) = sin(φ− φS)
n 0 0++ + ǫn
− 2 cos φ
ǫ(1 + ǫ) Imn 0 00+ − cos(2φ) ǫ Imn 0 0−+
+ cos(φ− φS)
−2 sinφ
ǫ(1 + ǫ) Im s 0 00+ − sin(2φ) ǫ Im s 0 0−+
WLTUT (φS , φ, ϕ) = sin(φ− φS)
cos(φ+ ϕ)
ǫ(1 + ǫ) Im
n 0+0+ − n
− cosϕ Im
n 0+++ − n−0++ + 2ǫn 0+0 0
+ cos(2φ+ ϕ) ǫ Imn 0+−+
− cos(φ− ϕ)
ǫ(1 + ǫ) Im
n 0−0+ − n
+ cos(2φ− ϕ) ǫ Imn+0−+
– 13 –
+ cos(φ− φS)
sin(φ+ ϕ)
ǫ(1 + ǫ) Im
s 0+0+ − s
− sinϕ Im
s 0+++ − s−0++ + 2ǫs 0+0 0
+ sin(2φ + ϕ) ǫ Im s 0+−+
− sin(φ− ϕ)
ǫ(1 + ǫ) Im
s 0−0+ − s
+ sin(2φ− ϕ) ǫ Im s+0−+
W TTUT (φS , φ, ϕ) = sin(φ− φS)
n++++ + n
++ + 2ǫn
cos(2φ+ 2ϕ) ǫ Imn−+−+
− cosφ
ǫ(1 + ǫ) Im
n++0+ + n
+ cos(φ+ 2ϕ)
ǫ(1 + ǫ) Imn−+0+
− cos(2ϕ) Im
n−+++ + ǫn
− cos(2φ) ǫ Imn++−+
+ cos(φ− 2ϕ)
ǫ(1 + ǫ) Imn+−0+ +
cos(2φ− 2ϕ) ǫ Im n+−−+
+ cos(φ− φS)
sin(2φ+ 2ϕ) ǫ Im s−+−+
− sinφ
ǫ(1 + ǫ) Im
s++0+ + s
+ sin(φ+ 2ϕ)
ǫ(1 + ǫ) Im s−+0+
− sin(2ϕ) Im
s−+++ + ǫs
− sin(2φ) ǫ Im s++−+
+ sin(φ− 2ϕ)
ǫ(1 + ǫ) Im s+−0+ +
sin(2φ− 2ϕ) ǫ Im s+−−+
(4.17)
for an unpolarized beam, and
WLLLT (φS , φ) = sin(φ− φS)
2 sinφ
ǫ(1− ǫ) Ren 0 00+
+ cos(φ− φS)
−2 cos φ
ǫ(1− ǫ) Re s 0 00+ +
1− ǫ2 s 0 0++
WLTLT (φS , φ, ϕ) = sin(φ− φS)
− sin(φ+ ϕ)
ǫ(1− ǫ) Re
n 0+0+ − n
+ sinϕ
1− ǫ2 Re
n 0+++ − n−0++
+ sin(φ− ϕ)
ǫ(1− ǫ) Re
n 0−0+ − n
+ cos(φ− φS)
cos(φ+ ϕ)
ǫ(1− ǫ) Re
s 0+0+ − s
− cosϕ
1− ǫ2 Re
s 0+++ − s−0++
− cos(φ− ϕ)
ǫ(1− ǫ) Re
s 0−0+ − s
W TTLT (φS , φ, ϕ) = sin(φ− φS)
ǫ(1− ǫ) Re
n++0+ + n
− sin(φ+ 2ϕ)
ǫ(1− ǫ) Ren−+0+
+ sin(2ϕ)
1− ǫ2 Ren−+++
− sin(φ− 2ϕ)
ǫ(1− ǫ) Ren+−0+
+ cos(φ− φS)
1− ǫ2 1
s++++ + s
– 14 –
unpolarized beam polarized beam
WUU WUL WUT WLU WLL WLT
Re u Im l Imn Im s Imu Re l Ren Re s
15 14 16 14 8 10 8 10
Table 3: Number of linear combinations of spin density matrix elements describing the angular
distribution of the cross section (4.5). The number of independent combinations for Reu is one
less than for Imn because of the relation (4.11).
− cosφ
ǫ(1− ǫ) Re
s++0+ + s
+ cos(φ+ 2ϕ)
ǫ(1− ǫ) Re s−+0+
− cos(2ϕ)
1− ǫ2 Re s−+++
+ cos(φ− 2ϕ)
ǫ(1− ǫ) Re s+−0+
(4.18)
for beam polarization. With obvious adjustments, the general structure discussed in points
1 to 3 above is found again for a transverse target. Note that the terms u 0 0++ + ǫu
0 0 and
u++++ + u
++ + 2ǫu
0 0 in the unpolarized coefficients W
UU and W
UU add up to 1 according
to (4.11), whereas their counterparts Im
n 0 0++ + ǫn
and Im
n++++ + n
++ + 2ǫn
WLLUT and W
UT are independent quantities. To keep the close similarity between the two
cases we have not used (4.11) to simplify (4.10).
Since there are two independent transverse polarizations relative to the hadron plane
(normal and sideways) we have a rather large number of terms with different angular
dependence in (4.17) and (4.18). The single spin asymmetry WUT contains 16 terms
with Imn and 14 terms with Im s , whereas the double spin asymmetry WLT contains 8
terms with Ren and 10 terms with Re s . Table 3 lists the number of independent linear
combinations of spin density matrix elements describing the angular distribution for the
different combinations of beam and target spin. For reasons discussed in Section 5 it is
useful to consider the spin density matrices n and s separately. It is then natural to work
in the basis of angular functions given by the product of sin(φ − φS) or cos(φ − φS) with
sin(kφ +mϕ) or cos(kφ +mϕ). With the replacement rules (4.15) and (4.16) we obtain
the combinations
sin(φ− φS) cos(kφ+mϕ) Imn + cos(φ− φS) sin(kφ+mϕ) Im s ,
− sin(φ− φS) sin(kφ+mϕ) Ren + cos(φ− φS) cos(kφ+mϕ) Re s (4.19)
in WUT and WLT , respectively.
We conclude this section by giving the relation between our spin density matrix ele-
ments for an unpolarized target and those in the classical work [14] of Schilling and Wolf.
We have
u 0 0++ + ǫu
0 0 = r
u 0+0+ − u
Im r610 − Re r510
u++++ + u
++ + 2ǫu
0 0 = 1− r0400 ,
u−+−+ = r
1−1 − Im r21−1 ,
– 15 –
Re u 0 00+ = −r500/
u 0+++ − u−0++ + 2ǫu 0+0 0
= 2Re r0410 ,
Re u 0+−+ = Re r
10 − Im r210 ,
u 0−0+ − u
Im r610 +Re r
u−+++ + ǫu
= r041−1 ,
Re u++−+ = r
u++0+ + u
2 r511 ,
Re u−+0+ =
Im r61−1 − r51−1
u 0 0−+ = r
Re u+0−+ = Re r
10 + Im r
Re u+−0+ = −
Im r61−1 + r
u+−−+ = r
1−1 + Im r
1−1 (4.20)
u 0+0+ − u
Im r710 +Re r
Imu 0 00+ = r
u 0+++ − u−0++
= −2 Im r310 ,
u 0−0+ − u
Im r710 − Re r810
Imu−+++ = − Im r31−1 ,
u++0+ + u
2 r811 ,
Imu−+0+ =
Im r71−1 + r
Imu+−0+ = −
Im r71−1 − r81−1
2 . (4.21)
The lower indices in the matrix elements of Schilling and Wolf refer to the ρ helicity and
correspond to the upper indices of u in our notation. Their upper indices correspond to a
representation of the virtual photon spin density matrix which refers partly to circular and
partly to linear polarization, whereas we use the helicity basis for the photon throughout.
The consequences of approximate s-channel helicity conservation are more explicit in our
notation: the relation Im r610 ≈ −Re r510 for instance corresponds to
u 0+0+ − u
u 0−0+ − u
∣. Notice also that the simple relation between single-spin asymmetries
and imaginary parts of spin density matrix elements discussed in point 3 above holds in
the helicity basis but not for linear polarization.
We note that our phase convention (3.1) for the helicity states of the virtual photon
differs from the one in [14] by a relative minus sign between transverse and longitudinal
polarization, and that our normalization factors NT and NL in (3.5) differ from those in
[14] by a factor of two. The combinations of helicity amplitudes corresponding to the spin
density matrix elements in (4.20) and (4.21) should be compared according to
NT + ǫNL
T νσµλ
= ηµµ′
NT + ǫNL
Tνσ,µλ T
ν′σ,µ′λ
, (4.22)
– 16 –
where η0± = η±0 = −1 for the interference of transverse and longitudinal photon polar-
ization, and ηµµ′ = +1 in all other cases.
5. Natural and unnatural parity
The exclusive process γ∗p → ρp is described by eighteen independent helicity amplitudes,
and we have already used approximate s-channel helicity conservation to establish a hierar-
chy among these amplitudes and the spin density matrix elements constructed from them.
A further dynamical criterion to order these quantities is given by natural and unnatural
parity exchange, which we shall now discuss.
Following [14] we define amplitudes N for natural and U unnatural parity exchange as
linear combinations
Nνσµλ =
T νσµλ + (−1)ν−µ T−νσ−µλ
T νσµλ + (−1)λ−σ T ν−σµ−λ
Uνσµλ =
T νσµλ − (−1)ν−µ T−νσ−µλ
T νσµλ − (−1)λ−σ T ν−σµ−λ
. (5.1)
With respect to the photon and meson helicity, the amplitudes N have the same symmetry
behavior as the amplitudes for γ∗t→ ρt on a spin-zero target t, whereas the corresponding
relation for the amplitudes U has an additional minus sign,
= (−1)ν−µNνσµλ , U−νσ−µλ = −(−1)
ν−µ Uνσµλ . (5.2)
For the proton helicity we have relations Nν+µ+ = N
µ− and N
µ− = −Nν−µ+ for natural parity
exchange, compared to Uν+µ+ = −Uν−µ− and Uν+µ− = Uν−µ+ for unnatural parity exchange. This
symmetry behavior immediately implies that in a dynamical description using generalized
parton distributions, amplitudes N go with distributions H and E, whereas amplitudes U
go with distributions H̃ and Ẽ. This is explicitly borne out in the calculation of [24]. Since
U 0σ0λ = 0 according to (5.2), unnatural parity exchange amplitudes are power suppressed
at large Q2 and the leading-twist factorization theorem [2] only applies to the natural
parity exchange amplitudes N 0σ0λ . We remark that in the context of low-energy dynamics
t-channel exchange of a pion plays a prominent role for unnatural parity exchange ampli-
tudes, see e.g. [15]. This has a natural counterpart in the framework of generalized parton
distributions, where pion exchange gives an essential contribution to the distribution Ẽ in
the isovector channel [28, 3, 29].
4The correspondence in (4.20) to (4.22) is obtained from comparing our results (4.10) and (4.12) for the
angular distribution with the ones in eqs. (92) and (92a) of [14], together with the relation between spin
density matrix elements and helicity amplitudes specified in eq. (91) and Appendix A of [14]. We have not
found an explicit specification of the phase convention for the virtual photon polarizations used in [14].
– 17 –
For the spin density matrix elements one readily finds
µµ′ = (NT + ǫNL)
Nνσµ+
+ Uνσµ+
µµ′ = (NT + ǫNL)
Nνσµ+
+ Uνσµ+
µµ′ = (NT + ǫNL)
Nνσµ+
+ Uνσµ+
µµ′ = (NT + ǫNL)
Nνσµ+
+ Uνσµ+
. (5.3)
The matrix elements u and n hence involve a product of two natural parity exchange
amplitudes plus a product of two amplitudes for unnatural parity exchange, whereas l
and s involve the interference between natural and unnatural parity exchange [15]. To
the extent that amplitudes U are smaller than their counterparts N , one can thus expect
that matrix elements l and s are small compared with u and n for equal helicity indices.
Exceptions to this guideline are possible since products Nνσµ+
or Nνσµ−
have a small real or imaginary part due to the relative phase between the two amplitudes.
If amplitudes U are smaller than N , one can furthermore neglect the terms
µµ′ = (NT + ǫNL)
Uνσµ+
µµ′ = (NT + ǫNL)
Uνσµ+
(5.4)
involving unnatural parity exchange in the matrix elements u and n . Using the relations
(−1)ν−µ u−νν′
= uνν
µµ′ − 2ũνν
µµ′ (5.5)
following from (5.2) and (5.3), we have in particular
−u 0+−+ = u 0+++ − 2ũ 0+++ , u−+−+ = u++++ − 2ũ++++ ,
−u−+0+ = u
0+ − 2ũ
0+ , u
−+ = u
++ − 2ũ−+++ . (5.6)
This allows us to rewrite
WLTUU = − cosϕ Re
u 0+++ − u−0++ + 2ǫu 0+0 0
− cos(2φ+ ϕ) ǫRe
u 0+++ − 2ũ 0+++
+ . . . cos(φ+ ϕ) + . . . cos(φ− ϕ) + . . . cos(2φ− ϕ) ,
W TTUU =
u++++ + u
++ + 2ǫu
cos(2φ+ 2ϕ) ǫ
u++++ − 2ũ++++
− cosφ
ǫ(1 + ǫ) Re
u++0+ + u
− cos(φ+ 2ϕ)
ǫ(1 + ǫ) Re
u++0+ − 2ũ
− cos(2ϕ) Re
u−+++ + ǫu
− cos(2φ) ǫRe
u−+++ − 2ũ−+++
+ . . . cos(φ− 2ϕ) + . . . cos(2φ− 2ϕ) ,
– 18 –
W TTLU = − sinφ
ǫ(1− ǫ) Im
u++0+ + u
− sin(φ+ 2ϕ)
ǫ(1− ǫ) Im
u++0+ − 2ũ
+ . . . sin(2ϕ) + . . . sin(φ− 2ϕ) , (5.7)
where terms indicated by . . . are the same as in the original expressions (4.10) and (4.12)
and have not been repeated for brevity. We see that the coefficients of adjacent terms in
(5.7) will be approximately equal to the extent that unnatural parity exchange is suppressed
and s-channel helicity approximately conserved. This can be tested experimentally by
measuring the angular distribution of the final-state particles.
The relations (5.6) and their counterparts for other index combinations can also be
used to approximately isolate spin density matrix elements of particular interest. Consider
as an example the leading-twist matrix element u 0 00 0 , which in the angular distribution
appears only in the combination u 0 0++ + ǫu
0 0 , i.e. together with a matrix element that
should be suppressed since it does not conserve s-channel helicity. If unnatural parity
exchange is strongly suppressed, an even better approximation for u 0 00 0 can be obtained
from the linear combination
ǫu 0 00 0 + 2ũ
u 0 0++ + ǫu
+ u 0 0−+ , (5.8)
whose r.h.s. can be extracted from the angular distribution. Similarly, one can approxi-
mately isolate the matrix element Re u 0+0 0 in the combination
ǫRe u 0+0 0 +Re
ũ 0+++ − ũ−0++
u 0+++ − u−0++ + 2ǫu 0+0 0
+Re u 0+−+ +Reu
. (5.9)
Conversely, one can extract from the angular distribution the linear combinations
ũ++++ + ũ
−− + 2ǫũ
0 0 − 2Re ũ
u++++ + u
++ + 2ǫu
u−+−+ − 12 u
u−+++ + ǫu
−Re u++−+ ,
ũ++0+ + ũ
u++0+ + u
u−+0+ +
u+−0+ , (5.10)
which only involve unnatural parity exchange. In a dynamical approach based on gen-
eralized parton distributions, these combinations are interesting because they isolate the
polarized distributions H̃ and Ẽ and in particular involve these distributions for gluons,
which are very hard to access in any other process.5 The price to pay for this is that
the corresponding amplitudes are power suppressed and cannot be calculated with the
theoretical rigor provided by the leading-twist factorization theorem. On the other hand,
phenomenological analysis indicates that a quantitative description of meson production
at Q2 of a few GeV2 requires the inclusion of power-suppressed effects also for the leading
matrix element u 0 00 0 .
The discussion of the matrix elements for transverse target polarization normal to the
hadron plane proceeds in full analogy to the unpolarized case. With
(−1)ν−µ n−νν′
= nνν
µµ′ − 2ñνν
µµ′ (5.11)
5In contrast to their quark counterparts, H̃g and Ẽg do not appear in pseudoscalar meson production
at leading twist and leading order in αs, see e.g. Section 5.1.1 of [30].
– 19 –
we have
−n 0+−+ = n 0+++ − 2ñ 0+++ , n−+−+ = n++++ − 2ñ++++ ,
−n−+0+ = n
0+ − 2ñ
0+ , n
−+ = n
++ − 2ñ−+++ (5.12)
and can write
WLTUT = cos(φ− φS)
. . .
+ sin(φ− φS)
− cosϕ Im
n 0+++ − n−0++ + 2ǫn 0+0 0
− cos(2φ+ ϕ) ǫ Im
n 0+++ − 2ñ 0+++
+ . . . cos(φ+ ϕ) + . . . cos(φ− ϕ) + . . . cos(2φ − ϕ)
W TTUT = cos(φ− φS)
. . .
+ sin(φ− φS)
n++++ + n
++ + 2ǫn
cos(2φ+ 2ϕ) ǫ Im
n++++ − 2ñ++++
− cosφ
ǫ(1 + ǫ) Im
n++0+ + n
− cos(φ+ 2ϕ)
ǫ(1 + ǫ) Im
n++0+ − 2ñ
− cos(2ϕ) Im
n−+++ + ǫn
− cos(2φ) ǫ Im
n−+++ − 2ñ−+++
+ . . . cos(φ− 2ϕ) + . . . cos(2φ − 2ϕ)
W TTLT = cos(φ− φS)
. . .
+ sin(φ− φS)
ǫ(1− ǫ) Re
n++0+ + n
+ sin(φ+ 2ϕ)
ǫ(1− ǫ) Re
n++0+ − 2ñ
+ . . . sin(2ϕ) + . . . sin(φ− 2ϕ)
, (5.13)
where terms denoted by . . . are as in the original expressions (4.17) and (4.18). Again, the
coefficients of adjacent terms should be approximately equal to the extent that unnatural
parity exchange is suppressed and s-channel helicity approximately conserved. The matrix
elements Imn 0 00 0 and Imn
0 0 can be approximately isolated in
ǫ Imn 0 00 0 + 2 Im ñ
++ = Im
n 0 0++ + ǫn
+ Imn 0 0−+ (5.14)
ǫ Imn 0+0 0 + Im
ñ 0+++ − ñ−0++
n 0+++ − n−0++ + 2ǫn 0+0 0
+ Imn 0+−+ + Imn
. (5.15)
In turn, the linear combinations
ñ++++ + ñ
−− + 2ǫñ
0 0 − 2ñ
n++++ + n
++ + 2ǫn
Imn−+−+ − 12 Imn
n−+++ + ǫn
− Imn++−+ ,
ñ++0+ + ñ
n++0+ + n
n−+0+ +
n+−0+ (5.16)
involve only unnatural parity exchange.
– 20 –
6. Positivity constraints
From the definition (3.4) of the spin-density matrix elements one readily finds
ν′µ′λ′
cνµλ ρ
µµ′,λλ′
= (NT + ǫNL)
cνµλ T
≥ 0 (6.1)
for arbitrary complex numbers cνµλ. Hence ρ
µµ′,λλ′ is a positive semidefinite matrix, with
row indices specified by {νµλ} and column indices by {ν ′µ′λ′}. This implies inequalities
among the spin density matrix elements, which extend those given e.g. in [22, 27]. We
do not attempt here to study the bounds following from positivity of the full 18 × 18
matrix ρνν
µµ′,λλ′ , which is quite unwieldy. Instead, we consider the subset of matrix elements
conserving s-channel helicity for the photon-meson transition and derive a number of simple
inequalities, which may be useful in practice. Ordering the row and column indices as
{+++}, {0 0+}, {−−+}, {++−}, {0 0−}, {−−−}, we have a positive semidefinite matrix
C, which can be written in block form as
A+ B+
B− A−
(6.2)
u++++ + η l
u 0+0+ + η l
u−+−+ − η l−+−+
u 0+0+ + η l
0 0 u
0+ − η l
u−+−+ + η l
u 0+0+ − η l
u++++ − η l++++
(6.3)
s++++ + η n
s 0+0+ − η n
)∗ −s−+−+ + η n−+−+
s 0+0+ + η n
0+ η n
0 0 −s 0+0+ + η n
s−+−+ + η n
s 0+0+ + η n
)∗ −s++++ + η n++++
, (6.4)
where η = ±1. Concentrating first on the matrix elements for an unpolarized or longitu-
dinally polarized target, we find that the matrix Aη has eigenvalues whose expressions are
very lengthy and therefore restrict our attention to 2×2 submatrices. The matrix obtained
from the first and third rows and columns of A+ has eigenvalues
u++++ ±
u−+−+
l++++
Im l−+−+
, (6.5)
whose positivity implies a bound
l++++
Im l−+−+
u++++
u−+−+
. (6.6)
Similarly, the matrix obtained from the first and second and the matrix obtained from the
second and third rows and columns of A+ have respective eigenvalues
u++++ + l
++ + u
u++++ + l
++ − u 0 00 0
∣u 0+0+ + l
u++++ − l++++ + u 0 00 0
u++++ − l++++ − u 0 00 0
∣u 0+0+ − l
, (6.7)
– 21 –
whose positivity gives bounds
Reu 0+0+ +Re l
Imu 0+0+ + Im l
)2 ≤ u 0 00 0
u++++ + l
Reu 0+0+ − Re l
Imu 0+0+ − Im l
)2 ≤ u 0 00 0
u++++ − l++++
. (6.8)
A weaker condition is obtained by taking the sum of these two bounds,
Re l 0+0+
Im l 0+0+
)2 ≤ u 0 00 0 u++++ −
Re u 0+0+
Imu 0+0+
. (6.9)
The bounds (6.6) and (6.9) have right-hand sides involving only matrix elements accessi-
ble with an unpolarized target and constrain the matrix elements for longitudinal target
polarization on their left-hand sides.
As a second example let us derive conditions which involve only matrix elements u
and n . To this end we consider the matrix
C′ = 1
C+D†CD
(6.10)
0 0 1 0 0 0
0 1 0 0 0 0
1 0 0 0 0 0
0 0 0 0 0 1
0 0 0 0 1 0
0 0 0 1 0 0
, (6.11)
which is half the sum of the positive semidefinite matrices C and D†CD and hence positive
semidefinite itself. One readily finds that matrix elements l and s drop out in C′, which
reads
u++++
u 0+0+
u−+−+ n
n 0+0+
n−+−+
u 0+0+ u
0 0 u
0 0 n
u−+−+
u 0+0+
u++++ n
n 0+0+
n++++
−n++++
n 0+0+
)∗ −n−+−+ u++++
u 0+0+
u−+−+
−n 0+0+ −n 0 00 0 −n
0 0 u
−n−+−+
n 0+0+
)∗ −n++++ u−+−+
u 0+0+
u++++
. (6.12)
This matrix has three eigenvalues
u++++ − u−+−+ + Imn++++ − Imn−+−+ ,
u++++ + u
−+ + Imn
++ + Imn
−+ + u
0 0 + Imn
u++++ + u
−+ + Imn
++ + Imn
−+ − u 0 00 0 − Imn 0 00 0
∣u 0+0+ − in
(6.13)
and three further eigenvalues obtained by reversing the sign of all matrix elements n . Their
positivity results in the bounds
Imn++++ − Imn−+−+
u++++ − u−+−+
(6.14)
– 22 –
Reu 0+0+ + Imn
Imu 0+0+ − Ren
u 0 00 0 + Imn
u++++ + u
−+ + Imn
++ + Imn
Reu 0+0+ − Imn
Imu 0+0+ +Ren
u 0 00 0 − Imn 0 00 0
u++++ + u
−+ − Imn++++ − Imn−+−+
. (6.15)
Omitting the terms with Imu 0+0+ and Ren
0+ , one obtains bounds involving only matrix
elements that are accessible with an unpolarized lepton beam.
As we have seen in Section 4, s-channel helicity conserving matrix elements can be
extracted from the angular distribution under the approximation that s-channel helicity
changing transitions are suppressed. The bounds derived in this section may be used to
check the consistency of this approximation.
7. Mixing between transverse and longitudinal polarization
So far we have discussed target polarization longitudinal or transverse to the virtual photon
direction in the target rest frame, which is natural from the point of view of the strong-
interaction dynamics. In an experimental setup one has however definite target polarization
with respect to the lepton beam direction. The transformation from one polarization basis
to the other is readily performed using the relations (2.3). For a target having longitudinal
polarization PL with respect to the lepton beam one finds
dφ dϕd(cos ϑ) dxB dQ2 dt
dxB dQ2 dt
WUU + PL
cos θγWUL − sin θγWUT (φS = 0)
+ PℓWLU + PℓPL
cos θγ WLL − sin θγ WLT (φS = 0)
. (7.1)
Note that in this case the azimuthal angle ψ in (4.5) needs to be defined with respect
to some fixed spatial direction in the target rest frame, rather than with respect to the
(vanishing) transverse component of the target polarization relative to the lepton beam.
We have integrated over this angle in (7.1) because the cross section does not depend on it.
For a target having transverse polarization PT with respect to the lepton beam one
dφS dφ dϕd(cos ϑ) dxB dQ2 dt
(2π)2
dxB dQ2 dt
cos θγ
1− sin2θγ sin2φS
WUU + PT
cos θγ WUT + sin θγ cosφS WUL
1− sin2θγ sin2φS
+ PℓWLU + PℓPT
cos θγ WLT + sin θγ cosφSWLL
1− sin2θγ sin2φS
. (7.2)
The factor cos θγ /(1 − sin2θγ sin2φS) comes from the change of variables from dψ to dφS
– 23 –
in the cross section. The relation between these two angles is readily obtained by setting
PL = 0 in (2.3) and given in [22].
It is a straightforward (if somewhat lengthy) exercise to insert our results (4.13), (4.14)
and (4.17), (4.18) into (7.1) and (7.2) and to rewrite the expressions in terms of a suitable
basis of functions depending on the azimuthal angles. Here we only give the combinations
needed in (7.2) for a transversely polarized target and an unpolarized beam,
cos θγ W
UT (φS , φ) + sin θγ cosφS W
UL(φ)
= sin(φ− φS)
cos θγ Im
n 0 0++ + ǫn
− sin θγ
ǫ(1 + ǫ) Im l 0 00+
− cos(2φ)
cos θγ ǫ Imn
−+ − sin θγ
ǫ(1 + ǫ) Im l 0 00+
− 2 cosφ
cos θγ
ǫ(1 + ǫ) Imn 0 00+ +
sin θγ ǫ Im l
+ cos(φ− φS)
− sin(2φ)
cos θγ ǫ Im s
−+ + sin θγ
ǫ(1 + ǫ) Im l 0 00+
− 2 sinφ
cos θγ
ǫ(1 + ǫ) Im s 0 00+ +
sin θγ ǫ Im l
sin θγ sin(φS + 2φ) ǫ Im l
−+ , (7.3)
cos θγW
UT (φS , φ, ϕ) + sin θγ cosφSW
UL (φ,ϕ)
= sin(φ− φS)
cos(φ+ ϕ)
cos θγ
ǫ(1 + ǫ) Im
n 0+0+ − n
sin θγ
l 0+++ − l−0++ + 2ǫl 0+0 0
+ ǫ Im l 0+−+
− cos(φ− ϕ)
cos θγ
ǫ(1 + ǫ) Im
n 0−0+ − n
sin θγ
l 0+++ − l−0++ + 2ǫl 0+0 0
− ǫ Im l+0−+
+cos(2φ+ ϕ)
cos θγ ǫ Imn
−+ − 12 sin θγ
ǫ(1 + ǫ) Im
l 0+0+ − l
+cos(2φ− ϕ)
cos θγ ǫ Imn
sin θγ
ǫ(1 + ǫ) Im
l 0−0+ − l
− cosϕ
cos θγ Im
n 0+++ − n−0++ + 2ǫn 0+0 0
sin θγ
ǫ(1 + ǫ)
l 0+0+ − l
l 0−0+ − l
+ cos(φ− φS)
sin(φ+ ϕ)
cos θγ
ǫ(1 + ǫ) Im
s 0+0+ − s
sin θγ
l 0+++ − l−0++ + 2ǫl 0+0 0
− ǫ Im l 0+−+
– 24 –
− sin(φ− ϕ)
cos θγ
ǫ(1 + ǫ) Im
s 0−0+ − s
sin θγ
l 0+++ − l−0++ + 2ǫl 0+0 0
+ ǫ Im l+0−+
+sin(2φ+ ϕ)
cos θγ ǫ Im s
sin θγ
ǫ(1 + ǫ) Im
l 0+0+ − l
+sin(2φ− ϕ)
cos θγ ǫ Im s
−+ − 12 sin θγ
ǫ(1 + ǫ) Im
l 0−0+ − l
− sinϕ
cos θγ Im
s 0+++ − s−0++ + 2ǫs 0+0 0
sin θγ
ǫ(1 + ǫ)
l 0+0+ − l
l 0−0+ − l
sin θγ
sin(φS + 2φ+ ϕ) ǫ Im l
−+ + sin(φS + 2φ− ϕ) ǫ Im l+0−+
, (7.4)
cos θγW
UT (φS , φ, ϕ) + sin θγ cosφSW
UL (φ,ϕ)
= sin(φ− φS)
cos θγ Im
n++++ + n
++ + 2ǫn
sin θγ
ǫ(1 + ǫ) Im
l++0+ + l
− cos(2φ)
cos θγ ǫ Imn
−+ − 12 sin θγ
ǫ(1 + ǫ) Im
l++0+ + l
− cosφ
cos θγ
ǫ(1 + ǫ) Im
n++0+ + n
sin θγ ǫ Im l
cos(2φ+ 2ϕ)
cos θγ ǫ Imn
−+ − sin θγ
ǫ(1 + ǫ) Im l−+0+
cos(2φ− 2ϕ)
cos θγ ǫ Imn
−+ − sin θγ
ǫ(1 + ǫ) Im l+−0+
− cos(2ϕ)
cos θγ Im
n−+++ + ǫn
sin θγ
ǫ(1 + ǫ)
Im l−+0+ + Im l
+ cos(φ+ 2ϕ)
cos θγ
ǫ(1 + ǫ) Imn−+0+ +
sin θγ
ǫ Im l−+−+ + 2 Im
l−+++ + ǫl
+ cos(φ− 2ϕ)
cos θγ
ǫ(1 + ǫ) Imn+−0+ +
sin θγ
ǫ Im l+−−+ − 2 Im
l−+++ + ǫl
+ cos(φ− φS)
− sin(2φ)
cos θγ ǫ Im s
sin θγ
ǫ(1 + ǫ) Im
l++0+ + l
− sinφ
cos θγ
ǫ(1 + ǫ) Im
s++0+ + s
sin θγ ǫ Im l
sin(2φ+ 2ϕ)
cos θγ ǫ Im s
−+ + sin θγ
ǫ(1 + ǫ) Im l−+0+
sin(2φ− 2ϕ)
cos θγ ǫ Im s
−+ + sin θγ
ǫ(1 + ǫ) Im l+−0+
− sin(2ϕ)
cos θγ Im
s−+++ + ǫs
sin θγ
ǫ(1 + ǫ)
Im l−+0+ − Im l
– 25 –
+ sin(φ+ 2ϕ)
cos θγ
ǫ(1 + ǫ) Im s−+0+ +
sin θγ
ǫ Im l−+−+ − 2 Im
l−+++ + ǫl
+ sin(φ− 2ϕ)
cos θγ
ǫ(1 + ǫ) Im s+−0+ +
sin θγ
ǫ Im l+−−+ + 2 Im
l−+++ + ǫl
sin θγ
sin(φS + 2φ+ 2ϕ) ǫ Im l
−+ + sin(φS + 2φ− 2ϕ) ǫ Im l+−−+
sin θγ sin(φS + 2φ) ǫ Im l
−+ . (7.5)
Compared with (4.17) and (4.18) we have changed the order of terms such that one readily
sees which coefficients cos θγ Imn or cos θγ Im s receive an admixture from the same coef-
ficients sin θγ Im l . The terms in the last lines of (7.3) and (7.4) and in the last two lines
of (7.5) involve only coefficients sin θγ Im l . They come with an angular dependence which
is absent for sin θγ = 0, as is readily seen by rewriting
sin(φS + 2φ+mϕ) = − sin(φ− φS) cos(3φ+mϕ) + cos(φ− φS) sin(3φ +mϕ) . (7.6)
We see in (7.3) to (7.5) that from the angular dependence of the cross section for
transverse target polarization one can extract linear combinations of terms cos θγ Imn and
sin θγ Im l or of cos θγ Im s and sin θγ Im l . To separate these terms requires an additional
measurement with longitudinal target polarization.6 The expressions (7.3) to (7.5) allow
us to see for which terms the admixture of sin θγ Im l terms can be expected to be small, so
that Imn and Im s may be determined with reasonable accuracy without such an additional
measurement. Let us discuss a few examples.
1. The leading-twist matrix element n 0 00 0 appears in the linear combination
c0 = cos θγ Im
n 0 0++ + ǫn
− sin θγ
ǫ(1 + ǫ) Im l 0 00+ (7.7)
in (7.3) and thus has an admixture from l 0 00+ , which involves one s-channel helicity
changing amplitude. According to Section 5 this admixture is additionally suppressed
if unnatural parity exchange is small compared with natural parity exchange. One
may also add to c0 the angular coefficient
c1 = − cos θγ ǫ Imn 0 0−+ + sin θγ
ǫ(1 + ǫ) Im l 0 00+ (7.8)
from (7.3), thus trading the admixture of sin θγ l
0+ for an admixture of cos θγ n
which involves two s-channel helicity changing amplitudes (but lacks the relative
factor tan θγ and is not suppressed by unnatural parity exchange). We remark that
the linear combination of matrix elements in (5.14) corresponds to c0 − c1/ǫ, where
l 0 00+ does not drop out. Whether c0, c0+ c1 or c0− c1/ǫ gives the best approximation
to cos θγ ǫ Imn
0 0 will thus depend on the detailed magnitude of the relevant terms.
In practice one might for instance use the difference between these terms as a measure
for the uncertainty of this approximation.
6A corresponding separation for semi-inclusive pion production ep → eπX has recently been performed
in [31].
– 26 –
2. The s-channel helicity conserving matrix elements n 0+0+ in (7.4) and n
++ , n
−+ in
(7.5) come together with terms involving at least one s-channel helicity changing
amplitude. These admixtures should hence be negligible unless the corresponding
s-channel helicity conserving matrix element is small itself. For Imn 0+0+ this may for
instance happen because of the relative phase between the interfering amplitudes.
3. The matrix element n 0 00+ in (7.3) comes with an admixture from l
−+ , which involves
two s-channel helicity changing amplitudes and should hence again be suppressed. In
addition, one can extract Im l 0 0−+ from the angular dependence itself, given the last
term in (7.3). We remark that the unpolarized analog u 0 00+ of n
0+ has a real part
which is experimentally seen to be nonzero [17, 19], providing evidence that s-channel
helicity is not strictly conserved in electroproduction. (In the notation of Schilling
and Wolf one has r500 = −
2Reu 0 00+ .)
4. The only s-channel helicity conserving matrix elements for sideways transverse target
polarization in (7.3) to (7.5) are s 0+0+ and s
−+ . They come together with terms
involving at least one s-channel helicity changing amplitude, so that the situation
is similar to the one in point 2. Note however that in the present case there is no
additional suppression of the admixture terms due to unnatural parity exchange,
since both s and l contain one unnatural parity exchange amplitude.
In these examples one thus has the favorable situation that the admixture from longitudinal
polarization terms is probably small and in some cases may even be removed or traded for
yet smaller terms. This does not always happen: the matrix elements n 0+−+ and s
−+ in
(7.4) receive for instance an admixture from the s-channel helicity conserving term l 0+0+ ,
which may not be small itself, so that from the coefficients of sin(φ − φS) cos(2φ + ϕ) or
cos(φ−φS) sin(2φ+ϕ) one cannot directly infer on the matrix elements Imn 0+−+ or Im s 0+−+ .
To make a more precise statement about their size one needs independent information on
Im l 0+0+ , for instance from the positivity bound (6.9).
8. A note on non-resonant contributions
So far we have treated the production of two pions in a two-step picture, where a ρ is first
produced in ep → epρ and then decays as ρ→ π+π−. For deriving the angular distribution
and polarization dependence we have used that the pion pair is in the L = 1 partial wave,
as can be seen in (4.3). We did however not use the narrow-width approximation for
the ρ or make any assumption about its line shape. In fact, our results for the angular
distribution can readily be used at any given invariant mass mππ of the pion pair, with the
ep cross sections on the left- and right-hand sides of (4.5) made differential in mππ. The
spin-density matrix ρνν
µµ′,λλ′ and its linear combinations u , l , s , n then depend on mππ and
refer not to γ∗p → ρp but to γ∗p → π+π− p with π+π− in the L = 1 partial wave. No
explicit reference to the ρ resonance needs to be made in this case.
The situation is more complicated if one considers other partial waves of the pion pair,
which can arise from non-resonant production mechanisms. To describe a general π+π−
– 27 –
state, one should replace ρνν
µµ′,λλ′ with the spin-density matrix ρ
νν′,LL′
µµ′,λλ′
for a pion pair with
angular momentum L in the amplitude and L′ in the conjugate amplitude. One then has to
take YLν(ϕ, ϑ)Y
L′ν′(ϕ, ϑ) instead of Y1ν(ϕ, ϑ)Y
1ν′(ϕ, ϑ) in (4.3) and will obviously obtain
a different angular dependence of the ep cross section. The distribution in ϕ and ϑ for a
pion pair with L = 0, 1, 2 has been discussed in [32].
It is quite simple to test for the presence of L = 0 or L = 2 partial waves in data
by using discrete symmetry properties, and for mππ around the ρ mass one can expect
that partial waves with L = 3 or higher are strongly phase space suppressed. Since even
partial waves of the π+π− system have charge conjugation parity C = +1 and odd partial
waves have C = −1, the interference of L = 1 with L = 0 or L = 2 gives rise to terms in
the angular distribution which are odd under interchange of the π+ and π− momenta, i.e.
under the replacement
ϑ→ π − ϑ , ϕ→ ϕ+ π . (8.1)
Simple examples are an angular dependence like cos ϑ or like an odd polynomial in cos ϑ.
Corresponding observables provide a way to study the L = 0 and L = 2 partial waves as a
“signal” interfering with the ρ resonance “background” [33, 34]. This has been used in the
experimental analysis [35], which did see such interference away from the ρ resonance peak,
whereas close to the peak the predominance of the ρ was too strong to observe a significant
contribution from any partial wave with L 6= 1. If on the other hand one is interested in a
precise study of the L = 1 component, one can eliminate its interference with even partial
waves by symmetrizing the angular distribution according to (8.1). One is then left with
contributions from L = 0 and L = 2 in both the amplitude and its conjugate, which should
be very small around the ρ peak.
9. Summary
We have expressed the fully differential cross section for exclusive ρ production on a po-
larized nucleon in terms of spin density matrix element for the subprocess γ∗p → ρp. We
work in the helicity basis for both γ∗ and ρ and obtain very similar forms for the unpolar-
ized and polarized parts of the cross sections, with the substitution rules (4.15) and (4.16).
The terms for transverse target polarization normal to the hadron plane closely resemble
those for an unpolarized target, and in both cases the number of independent spin density
matrix elements is reduced if one neglects unnatural parity exchange compared with nat-
ural parity exchange. The spin density matrix elements for transverse target polarization
in the hadron plane closely resemble those for a longitudinally polarized target, with both
types of matrix elements involving the interference between natural and unnatural parity
exchange. We have given simple positivity bounds which involve only matrix elements for
an unpolarized target and either those for longitudinal target polarization or for transverse
target polarization normal to the hadron plane. Furthermore, we have investigated the
admixture of longitudinal target polarization relative to the virtual photon momentum for
a target polarized transversely to the lepton beam. This admixture should be small for
the spin density matrix elements which conserve s-channel helicity in the transition from
– 28 –
γ∗ to ρ, but it may be important for s-channel helicity changing matrix elements. Finally,
we have briefly discussed how the results obtained in this paper can be used and extended
for analyzing the production of pion pairs not associated with the ρ resonance.
Acknowledgments
It is a pleasure to thank my colleagues from HERMES for their interest in this work and
for many discussions, especially A. Borissov, J. Dreschler, D. Hasch and A. Rostomyan.
I also gratefully acknowledge helpful discussions with P. Kroll and A. Schäfer. This work
is supported by the Helmholtz Association, contract number VH-NG-004.
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