Lepton angular distributions of Drell-Yan process in pQCD and a geometric approach
Wen-Chen Chang Randall Evan McClellan Jen-Chieh Peng Oleg Teryaev
hep-ph nucl-ex nucl-th






§ INTRODUCTION


The Drell-Yan (D-Y) process together with deep inelastic scattering
are the main tools for extracting the parton distributions in
hadrons <cit.>. The polar and azimuthal angular distributions
of leptons produced in unpolarized D-Y process are sensitive to the
underlying reaction mechanisms as well as novel parton
distributions. For example, Boer-Mulders functions <cit.> have
been suggested to account for a violation of the Lam-Tung (L-T)
relation <cit.> observed in the fixed-target experiments with
pion beams, e.g. E615 <cit.>.

In this proceedings, we compare the data of dilepton angular
parameters λ, μ, ν and the L-T violation quantity
1-λ-2ν measured by E615 <cit.> with the fixed-order
pQCD calculations <cit.>. Furthermore we present
the NLO pQCD predictions for the ongoing COMPASS <cit.>
experiments on the dimuon mass Q and Feynman-x (x_F) dependence
of these angular parameters. In addition, we interpret some notable
features of pQCD results using the geometric
model <cit.>. More results and greater details
can be found in Ref. <cit.>.



§ CALCULATIONS OF DILEPTON ANGULAR PARAMETERS IN DYNNLO


We utilize the DYNNLO (version 1.5) package for the calculations. Via
the LHAPDF6 framework, the parton distribution functions (PDFs) used
for the protons and neutrons are “CT14nlo” and “CT14nnlo” in the
NLO and NNLO calculations, respectively, and “GRVPI1” for the pion
PDFs in both NLO and NNLO calculations.

In the rest frame of the virtual photon in the D-Y process, a commonly
used expression for the lepton angular distributions is given
as

    dσ/dΩ∝ 1+ λcos^2θ
    + μsin 2 θcosϕ
    + ν/2sin^2θcos 2 ϕ,

where θ and ϕ refer to the polar and azimuthal angles of
l^- (e^- or μ^-). To obtain the λ, μ, and ν
parameters, we first calculate the A_i parameters in an alternative
expression of the lepton angular distributions of the D-Y process as
follows:

    dσ/dΩ   ∝    (1+cos^2θ)+A_0/2
    (1-3cos^2θ) +A_1 sin 2 θcosϕ + A_2/2sin^2θcos 2 ϕ.

The angular coefficients A_i could be evaluated by the moments of
spherical harmonic polynomial expressed as

    A_0  =  4 - 10 ⟨cos^2θ⟩;   
    A_1  =  5 ⟨sin2θcosϕ⟩;   
    A_2  =  10 ⟨sin^2θcos2ϕ⟩,

where ⟨ f(θ, ϕ) ⟩ denotes the moment of
f(θ, ϕ) , i.e. the weighted average of f(θ, ϕ) by
the cross sections in Eq. (<ref>). It is straightforward to
show that λ, μ, ν in Eq. (<ref>) are related to
A_0, A_1, A_2 via

    λ = 2-3A_0/2+A_0;   μ  =  2A_1/2+A_0;   ν  =  2A_2/2+A_0.

Equation (<ref>) shows that the L-T relation, 1-λ - 2
ν=0, is equivalent to A_0 = A_2.

In Fig. <ref>, we compare the results of λ, μ,
ν, and the L-T violation, 1-λ-2ν, from the fixed-order
pQCD calculations with 252-GeV π^- + W data from E615
experiment <cit.>. The angular parameters are evaluated as a
function of the dimuon's transverse momentum (q_T) in the
Collins-Soper frame. Overall, the calculated λ, μ and
ν exhibit distinct q_T dependencies. At q_T → 0,
λ, μ and ν approach the values predicted by the
collinear parton model: λ = 1 and μ = ν =0. As q_T
increases, Fig. <ref> shows that λ decreases toward
its large-q_T limit of -1/3 while ν increases toward 2/3,
for both qq̅ and qG processes <cit.>. The
q_T dependence of μ is relatively mild compared to λ and
ν. This is understood as a result of some cancellation effect, to
be discussed in Sec. <ref>.

Comparing the results of the NLO with the NNLO calculations, λ (NNLO) is smaller than λ(NLO) while μ and
ν are very similar for NLO and NNLO. The amount of L-T violation,
1-λ-2ν, is zero in the NLO calculation, and nonzero and
positive in the NNLO calculation. As seen in Fig. <ref>, the
pQCD predicts a sizable magnitude for ν, comparable to the
data. Therefore, the pQCD effect should be included in the extraction
of nonperturbative Boer-Mulders effect from the data of ν.



Next we present the results of the angular coefficients λ,
μ and ν as a function of q_T in various bins of Q and
x_F for the ongoing fixed-target COMPASS experiment at
CERN <cit.>. This experiment runs with 190-GeV π^- beam
and transversely-polarized NH_3 target and unpolarized aluminum
(Al) and tungsten (W) nuclear targets. There are three bins for
Q in the range of 4.0–7.0 GeV, as well as three bins for x_F in
the range of 0–0.6. Our results could be convoluted by the COMPASS
spectrometer acceptances later for a direct comparison with
experimental data in the near future. Since there are no significant
difference between the NLO and NNLO results, we present only the
results from the NLO calculation to illustrate the major features.

Figures <ref>(a) and (b) show λ, μ and ν
as a function of q_T for various bins of the dimuon mass, Q, and
Feynman-x, x_F, respectively. The q_T distributions of λ
and ν parameters depend sensitively on Q, but only weakly on
x_F. As for μ, its q_T distribution has strong dependencies on
x_F and on Q. In particular, the magnitude of μ is small when
x_F is close to 0 and its sign could even turn negative at some
q_T region. As x_F increases, the magnitude of μ increases
pronouncedly.





§ GEOMETRIC MODEL



The E615 D-Y data of lepton angular distributions can be reasonably
well described by the NLO and NNLO pQCD calculations. Various salient
features of Q and x_F dependencies are observed in the predicted
results of λ, μ and ν parameters for COMPASS
experiment based on NLO pQCD. It is of interest to check if these
features of pQCD calculations could be understood using the geometric
approach developed in Refs. <cit.>.

In Refs. <cit.>, the hadron plane, the quark plane, and
the lepton plane of collision geometry are defined in the
Collins-Soper γ^* rest frame. A pair of collinear q and q̅ with equal momenta annihilate into a γ^*. The momentum unit
vector of q is defined as ẑ^', and the quark plane is
formed by the ẑ^' and the ẑ axes of Collins-Soper
frame. The angular coefficients A_i in Eq. (<ref>) can be
expressed in term of θ_1 and ϕ_1 as follows:

    A_0 =  ⟨sin^2θ_1⟩,    
    A_1 = 1/2⟨sin 2θ_1cosϕ_1⟩,    
    A_2 =  ⟨sin^2θ_1 cos 2ϕ_1⟩,

where the θ_1 and ϕ_1 are the polar and azimuthal angles
of the natural quark axis ẑ^' of the quark plane in the
Collins-Soper frame. The ⟨···⟩ in
Eq. (<ref>) denotes that the measured values of A_i at a
given kinematic bin are averaged over events having particular values
of θ_1 and ϕ_1.

Up to NLO (𝒪(α_S)) in pQCD, the quark plane coincides
with the hadron plane and ϕ_1=0. Therefore A_0=A_2 or
1-λ-2ν=0, i.e., the L-T relation is satisfied. Higher order
pQCD processes allow the quark plane to deviate from the hadron plane,
i.e., ϕ_1 ≠ 0. This acoplanarity effect leads to the violation
of the L-T relation. For a nonzero ϕ_1, Eq. (<ref>) shows
that A_2 < A_0. Therefore, when the L-T relation is violated, A_0
must be greater than A_2 or, equivalently, 1 - λ - 2ν
>0. This expectation of 1 - λ - 2ν >0 in the geometric
approach agrees with the results of NNLO pQCD calculations shown in
Fig. <ref>. The geometric approach offers a simple
interpretation for this result.

Furthermore the sign of μ could be either positive or negative,
depending on which parton and hadron the gluon is emitted
from <cit.>. Hence, one expects some cancellation
effects for μ among contributions from various processes. Each
process is weighted by the corresponding density distributions for the
interacting partons. At x_F ∼ 0, the momentum fraction carried by
the beam parton (x_B) is comparable to that of the target parton
(x_T). Therefore, the weighting factors for various processes are of
similar magnitude and the cancellation effect could be very
significant, resulting in a small value of μ. As x_F increases
toward 1, x_B becomes much larger than x_T. In this case the
weighting factors are now dominated by fewer processes, resulting in
less cancellation and a larger value of μ. This explains why the
μ parameter exhibits a strong x_F dependence in
Figs. <ref>(b).



§ SUMMARY


We have presented a comparison of the measurements of the angular
parameters λ, μ, ν and 1-λ-2ν of the D-Y
process from the fixed-target E615 experiment with the corresponding
results from the NLO and NNLO pQCD calculations. Qualitatively the
transverse momentum (q_T) dependence of λ, μ and ν
in the data could be described by pQCD. The difference between NLO and
NNLO results becomes visible at large q_T. The L-T violation part
1-λ-2ν remains zero in the NLO pQCD calculation and turns
positive in NNLO pQCD.

The x_F dependence of the angular parameters is well described by
the geometric picture. In particular, the weak rapidity dependencies
of the λ and ν, and the pronounced rapidity dependency for
μ can be explained by the absence or presence of
rapidity-dependent cancellation effects. The occurrence of
acoplanarity between the quark plane and the hadron plane (ϕ_1
≠ 0), for the pQCD processes beyond NLO leads to a violation of
the L-T relation. The predicted positive value of 1-λ-2ν, or
A_0>A_2 when ϕ_1 is nonzero, is consistent with the NNLO pQCD
results.

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