It is shown how the single spin asymmetry observed in inclusive pion
production is related, in the helicity basis, to the imaginary part of the
product of two different distribution amplitudes, rather than to the usual
quark and gluon distribution functions; there is then no reason why it should
be zero even in massless perturbative QCD, provided the quark intrinsic motion
is taken into account. A simple model is constructed which reproduces the main
features of the data.Comment: (To appear in the proceedings of the XI International Symposium on
  High Energy Spin Physics, September 15-22, Bloomington, Indiana). 5 pages + 1
  figure, LaTeX 2.09, report DFTT47/94, INFNCA-TH-94-21. (figure appended as
  uuencoded, tar-compressed postscript file; uncompressed ps file available
  under request from murgia@vaxca.ca.infn.it

Anselmino, Mauro

Boglione, M. E.

Murgia, F.

English

arXiv

M. Anselmino

M. E. Boglione

F. Murgia

Crossref

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DFTT47/94
INFNCA-TH-94-21
October 1994
Single Spin Asymmetry
In Inclusive Pion Production∗
M. Anselmino†, M.E. Boglione† and F. Murgia‡
†Dipartimento di Fisica Teorica, Università di Torino and
Istituto Nazionale di Fisica Nucleare, Sezione di Torino,
Via P. Giuria 1, I-10125 Torino, Italy
‡Istituto Nazionale di Fisica Nucleare, Sezione di Cagliari,
Via Ada Negri 18, I-09127 Cagliari, Italy
Abstract. It is shown how the single spin asymmetry observed in inclu-
sive pion production is related, in the helicity basis, to the imaginary part
of the product of two different distribution amplitudes, rather than to the
usual quark and gluon distribution functions; there is then no reason why
it should be zero even in massless perturbative QCD, provided the quark
intrinsic motion is taken into account. A simple model is constructed which
reproduces the main features of the data.
Spin physics in large pT inclusive hadronic processes has unique features;
not only it probes the internal structure of hadrons, but, as spin dependent
observables involve delicate interference effects among different amplitudes, it
tests the theory at a much deeper level than unpolarized processes.
We consider here the single spin asymmetry in inclusive pion production
in p− p collisions, p↑+ p → π+X . Let the two protons move along the ẑ-axis
in their c.m. frame and x̂-ẑ be the scattering plane. The proton moving in the
+ẑ direction is polarized transversely to the scattering plane, i.e. along (↑) or
opposite (↓) the ŷ-axis. The single spin asymmetry AN is then defined by:
AN(xF , pT ) =
dσ↑ − dσ↓
dσ↑ + dσ↓
(1)
where dσ is the differential cross section and ↑, ↓ refer to the proton spin
directions; we denote by pL and pT the c.m. longitudinal and transverse pion
∗Talk presented by F. Murgia at the XI International Symposium on High Energy Spin
Physics, September 15-22, Bloomington, Indiana.
1
momentum respectively; xF = 2pL/
√
s is the Feynman variable and
√
s/2 is
the c.m. energy of each incident proton.
Several experimental results are available on AN (for a list of references see
[1]); the E704 Collaboration has produced the most recent, high energy ones
(
√
s/2 ≃ 10 GeV). Two sets of measurements are relevant to our analysis:
i) AN (xF , pT ) for p
↑+p → π±, π0+X , vs. xF in the pT range 0.7 ≤ pT ≤ 2.0
GeV/c [2, 3]; these data show intriguing xF dependence (Fig. 1).
ii) AN(xF , pT ) for p
↑ + p → π0 + X , as a function of pT (up to pT ≃ 4
GeV/c), in the central region (|xF | ≤ 0.1) [4]; in this case no pT dependence
seems to be observed and AN ≃ 0 in the whole pT range (notice that this
updates and corrects some previous results of the same collaboration [5]).
A naive generalization of the QCD-factorization theorem suggests that the
single spin asymmetry can be written qualitatively as:
AN ∼
∑
ab→cd
∆TGa/p ⊗Gb/p ⊗ âN σ̂ab→cd ⊗Dπ/c (2)
where Ga/p is the parton distribution function, that is the number density of
partons a inside the proton, and ∆TGa/p = Ga↑/p↑ − Ga↓/p↑ is the difference
between the number density of partons a with spin ↑ in a proton with spin ↑
and the number density of partons a with spin ↓ in a proton with spin ↑; Dπ/c
is the number density of pions resulting from the fragmentation of parton c;
âN is the single spin asymmetry relative to the a
↑b → cd elementary process
and σ̂ is the cross-section for such process.
The usual argument is then that the asymmetry (2) is bound to be very
small because âN ∼ αsmq/
√
s where mq is the quark mass. This originated
the widespread opinion that single spin asymmetries are essentially zero in
perturbative QCD.
However, it has become increasingly clear in the last years that such conclu-
sion need not be true because subtle spin effects might modify Eq. (2). Such
modifications should take into account the parton transverse motion, higher
twist contributions and possibly non perturbative effects hidden in the spin de-
pendent distribution and fragmentation functions. Several models have been
proposed which differ in practice by which part of AN , Eq. (2), is responsible
for these effects: ∆TGa/p [6, 7, 8], σ̂ [9, 10] or Dπ/c [11, 12].
We briefly discuss here a reformulation of Eq. (2) in the helicity basis, which
is more suitable for applying the factorization theorem and which allows to
formulate a model for the spin dependence of the quark distributions [1]. Our
approach is reminiscent of that of Ref. [6].
In the helicity basis the differential cross-section for the inclusive process
p1(λ1) + p2(λ2) → π + X(λX) can be written in terms of helicity amplitudes
as:
dσ ∼
∑
X,λX
∑
λ1,λ′1,λ2,λ
′
2
MλX ;λ1,λ2 ρλ1,λ2;λ′1,λ′2(p1, p2) M
∗
λX ;λ′1,λ
′
2
(3)
where the sum over X includes also a phase space integral for the undetected
particles and the matrix ρ is the helicity density matrix describing the polar-
2
ization state of the initial protons. In our case p2 is unpolarized, while p1 is
transversely polarized along ±ŷ direction, so that Eq. (1) becomes
AN = 2
∑
X,λX,λ2
Im[MλX ;+,λ2M
∗
λX ;−,λ2
]
∑
X,λX,λ1,λ2
|MλX ;λ1,λ2 |2
· (4)
Eq. (4) shows how a non zero value of AN implies non zero interference effects
between two amplitudes which only differ by one helicity index; its denomi-
nator, instead, proportional to dσ↑ + dσ↓ = 2 dσunp, only depends on moduli
squared of amplitudes and can be written in the parton model as:
dσunp ∼
∑
abcd
∫
dxadxb
1
xc
Ga/p(xa)Gb/p(xb)
dσ̂
dt̂
(ab → cd)Dπ/c(xc) . (5)
In order to express the numerator of Eq. (4) in terms of parton interactions
we have to define Ga/hλXh ,λa;λh(xa,k⊥a) as the helicity distribution amplitude for
the process h(λh) → a(λa) + Xh(λXh), where k⊥a is the transverse momen-
tum of the parton a inside the hadron h; these amplitudes are related to the
unpolarized partonic distribution function by:
Ga/p(xa) =
∑
Xp,λXp
∫
dk⊥a
{
|Ga/pλXp ,+;+(xa,k⊥a)|
2 + |Ga/pλXp ,−;+(xa,k⊥a)|
2
}
. (6)
By applying the same steps which lead to the partonic expression of dσunp
we get for the numerator of AN an expression similar to Eq. (5), with Ga/p(xa)
replaced by
∫
dk⊥a I
a/p
+−(xa,k⊥a), where
I
a/p
+−(xa,k⊥a) ≡
∑
Xp,λXp
Im[Ga/pλXp ,+;+(xa,k⊥a)G
a/p ∗
λXp ,+;−
(xa,k⊥a)] . (7)
Notice that I
a/p
+−(xa,k⊥a) has to vanish for k⊥a = 0, as required by helicity
conservation in the forward direction; moreover, since I
a/p
+−(xa,k⊥a) is an odd
function of k⊥a
1, we must keep into account k⊥a effects also in the partonic
cross sections, otherwise we are left with
∫
dk⊥a I
a/p
+−(xa,k⊥a) = 0. Then
∫
dk⊥aI
a/p
+−(xa,k⊥a)
dσ̃
dt̃
(k⊥a) =
∫
(k⊥a)x>0
dk⊥aI
a/p
+−(xa,k⊥a)
[
dσ̃
dt̃
(+k⊥a)−
dσ̃
dt̃
(−k⊥a)
]
(8)
where dσ̃/dt̃ means that now the partonic cross section includes k⊥a effects.
To give numerical estimates we need a model for the non perturbative
functions I
a/p
+−(x,k⊥); these non diagonal distribution functions play for spin
1This is more easily seen if we observe that our I
a/p
+− (xa,k⊥a) equals ∆
NGa/p↑(xa,k⊥a)=
∑
λa
{
Ga(λa)/p↑(xa,k⊥a) −Ga(λa)/p↑(xa,−k⊥a)
}
defined by Sivers [6].
3
observables the same rôle plaid by the usual diagonal distribution functions
Ga/p in unpolarized cross-sections. We parameterize their x dependence with
simple power behaviours. The dependence on k⊥ is treated, at this stage, in a
simplified way: we replace the integral in Eq. (8) by the value of the integrand
at some average k⊥a = 〈k2⊥a〉1/2. That is we set
∫
dk⊥a I
a/p
+−(xa,k⊥a)
dσ̃
dt̃
(k⊥a) =
k̂⊥
Mh
Nax
αa
a (1− xa)βa
dσ̃
dt̃
(k⊥a) (9)
where Mh is some hadronic mass scale, of the order of 1 GeV, and we assume
k̂⊥ ≃ 0.5 GeV/c. Na can be taken from the usual distribution functions [1].
Our final expression for the single spin asymmetry AN is then
AN =
∑
abcd
∫
dxadxb
1
xc
I
a/p
+−(xa, k⊥a)G
b/p(xb)
[
dσ̃
dt̃
(k⊥a)− dσ̃dt̃ (−k⊥a)
]
Dπ/c(xc)
2
∑
abcd
∫
dxadxb
1
xc
Ga/p(xa)Gb/p(xb)
dσ̂
dt̂
(ab → cd)Dπ/c(xc)
·
(10)
In Eq. (10) we take into account, at lowest perturbative QCD order, all pos-
sible elementary interactions involving quarks and gluons. According to SU(6)
proton wave functions we take I
u/p
+− > 0 for u quarks and I
d/p
+− < 0 for d quarks
(see footnote after Eq. (7)); the sign of I
a/p
+− for the other partonic contributions
is less relevant, and for the moment we assume all these contributions to be
positive. However, a more careful analysis is in progress [1]. The unpolarized
distribution and fragmentation functions are taken from Ref. [13, 14].
In Fig. 1 we compare our results, at pT = 2 GeV/c, with the experimental
data. Most contributions come from qg → qg and gg → gg processes and the
parameters α and β of Eq. (9) yielding these results are αu = αd = αg ≃ −0.6,
βu = βd = 2.5 and βg = 3.5. We also find that, at xF = 0, AN ≃ 0,
independently of pT , in agreement with the most recent data [4].
Our results clearly show how a careful treatment of spin observables and the
inclusion of intrinsic k⊥ effects can yield sizeable values of single spin asymme-
tries in hadronic inclusive pion production, via perturbative QCD dynamics,
contrary to widespread belief.
The off-diagonal distribution functions I
a/p
+− introduced in Eq. (7) contain all
the relevant non perturbative information; similarly to the parton distribution
functions in unpolarized processes, they cannot be computed, but have to be
taken from experiment. This is essentially what we have done here, resulting
in reasonable expressions for I
a/p
+− ; further discussions can be found in Ref.
[1]. Once the non diagonal distribution functions have been obtained from
one set of experiments, they can be used to make genuine perturbative QCD
predictions for other spin observables like single spin asymmetries in π+ p↑ →
π +X and p↑ + p → γ +X . Some experimental results are already available
and more are soon expected.
References
4
[1] Anselmino, M., Boglione, M.E., and Murgia, F., Single spin asymmetry for
p↑ + p → π+X in perturbative QCD, preprint DFTT48/94 and INFNCA-TH-
94-27, October 1994.
[2] Adams, D.L., et al., Z. Phys. C56, 181 (1992).
[3] Adams, D.L., et al., Phys. Lett. B264, 462 (1991).
[4] Nurushev, S.B., Single spin asymmetries and invariant cross-sections of high
pT inclusive π
0 production in p̄− p interactions, these Proceedings.
[5] Adams, D.L., et al., Phys. Lett. B276, 531 (1992).
[6] Sivers, D., Phys. Rev. D41, 83 (1990); Phys. Rev. D43, 261 (1991).
[7] Qiu, J., and Sterman, G., Phys. Rev. Lett. 67, 2264 (1991).
[8] Boros, C., Liang, Z., and Meng, T., Phys. Rev. Lett. 70, 1751 (1993); see also
the talk by T. Meng in these Proceedings.
[9] Szwed, J., Phys. Lett. B105, 403 (1981).
[10] Efremov, A.V. and Teryaev, O.V., Yad. Fiz. 36, 242 (1982) [Sov. J. Nucl. Phys.
36, 140 (1982)].
[11] Collins, J., Nucl. Phys. B396, 161 (1993).
[12] Artru, X., Czyzewski, J., and Yabuki, H., Single spin asymmetry in inclusive
pion production, Collins effect and the string model, preprint LYCEN/9423 and
TPJU 12/94, May 1994.
[13] Brodsky, S.J., Burkardt, M., and Schmidt, E., Perturbative QCD constraints on
the shape of polarized quark and gluon distributions, preprint SLAC-PUB-6087,
January 1994.
[14] Field, R.D., Applications of perturbative QCD, New York: Addison Wesley,
1989, ch. 3.
Figure Caption
Fig. 1 Single spin asymmetry for π+, π0, π−, vs. xF at pT = 2 GeV/c, from
Eq. (10), compared to experimental results [2, 3] (see text for details).
5
This figure "fig1-1.png" is available in "png"
 format from:
http://arxiv.org/ps/hep-ph/9410382v1



Single spin asymmetry in inclusive pion production

Single Spin Asymmetry in Inclusive Pion Production

Single Spin Asymmetry in Inclusive Pion Production

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