General helicity formalism for Single and Double Spin Asymmetries in p p --> pion + X

0.12 MB·English

by  M. Anselmino

GENERAL HELICITY FORMALISM FOR SINGLE AND DOUBLE SPIN
ASYMMETRIES IN pp π +X∗
→
M. Anselmino1, M. Boglione1, U. D’Alesio2, E. Leader3, S. Melis2 and F. Murgia2
(1) Dipartimento di Fisica Teorica, Universita` di Torino and INFN, Sezione di Torino,
Via P. Giuria 1, I-10125 Torino, Italy
(2) Dipartimento di Fisica, Universita` di Cagliari and INFN, Sezione di Cagliari,
C.P. 170, I-09042 Monserrato (CA), Italy
6 (3) Imperial College London, Prince Consort Road, London SW7 2BW, U.K.
0
0
2
Abstract
n
a
We consider within a generalized QCD factorization approach, the high energy
J
inclusive polarized process pp π + X, including all intrinsic partonic motions.
5 →
2 Several new spin and k⊥-dependent soft functions appear and contribute to cross
sections and spin asymmetries. We present here formal expressions for transverse
1
single spin asymmetries and double longitudinal ones. The transverse single spin
v
5 asymmetry, A , is considered in detail, and all contributions are evaluated numer-
N
0 ically. It is shown that the azimuthal phase integrations strongly suppress most
2
contributions, leaving at work mainly the Sivers effect.
1
0
6
0
/
h 1 Introduction and formalism
p
-
p
In the last years we have developed an approach to study inclusive pion production in
e
h (un)polarized hadroniccollisions, at highenergies andmoderately largep [1,2,3]. Inour
T
v: approachwetakeintoaccountthepartonintrinsicmotion,k , inthepartonicdistribution
⊥
i
X functions (PDF’s), in the fragmentation sector (FF) and in the elementary scattering
r processes allowing for a full non collinear kinematics. In writing cross sections we adopt
a
a generalization of the QCD factorization scheme. However, at present, a factorized
formula for inclusive hadron production processes, with the inclusion of k -effects, has to
⊥
beconsideredasareasonablephenomenologicalmodel,sinceacorrespondingfactorization
theorem has not been proved yet.
Let us consider the process (A,S )+(B,S ) C +X, where A and B are two spin
A B
→
one-half hadrons in a pure polarization state, denoted respectively by S and S . The
A B
cross section for this process can be given at leading order (LO) as a convolution of all
possible elementary hard scattering processes, ab cd, with soft, leading twist, spin and
→
k -dependent PDF’s and FF’s, namely:
⊥
dσ(A,SA)+(B,SB)→C+X = ρa/A,SAfˆ (x ,k ) ρb/B,SB fˆ (x ,k )
λa,λ′a a/A,SA a ⊥a ⊗ λb,λ′b b/B,SB b ⊥b
a,b,Xc,d,{λ}
Mˆ Mˆ∗ DˆC (z,k ). (1)
⊗ λc,λd;λa,λb λ′c,λd;λ′a,λ′b ⊗ λc,λ′c ⊥C
∗
Talk deliveredby S. Melis atthe “XI Workshopon High EnergySpin Physics”,SPIN 05,September
27 - October 1, 2005, Dubna, Russia.
1
The notation λ implies a sum over all helicity indices, while a,b,c,d stand for a
{ }
sum over all partonic contributions. ρa/A,SA and ρb/B,SB are the helicity density matrices
λa,λ′a λb,λ′b
for parton a and b respectively, describing the polarization state of partons inside parent
hadrons. We denote by fˆ (x ,k ) the PDF of parton a inside hadron A, similarly
a/A,SA a ⊥a
for parton b. The elementary scattering process is described by means of the helicity
amplitudes Mˆ to becomputed inthehadronc.m. frame. Dueto thenoncollinear
λc,λd;λa,λb
partonic configuration, their expressions do not coincide anymore with the well known
helicity amplitudes calculated in the usual canonical partonic c.m frame, denoted by
Mˆ0 . In fact, contrary to the collinear configuration case, the hadronic production
λc,λd;λa,λb
planedoesnotcoincidewiththepartonicscatteringplane. TheamplitudesMˆ can
λc,λd;λa,λb
becomputed fromtheMˆ0 onesbymeans oftheLorentztransformationconnecting
λc,λd;λa,λb
the partonic c.m frame to the hadronic one; this involves complicated azimuthal phases,
denoted by ϕ , which depend in a highly non trivial way on all kinematical variables
i
of the convolution integral in Eq. (1) [2]. For massless partons there are at most three
independent helicity amplitudes for each partonic subprocess:
Mˆ = Mˆ0 eiϕ1 Mˆ0eiϕ1 Mˆ = Mˆ0 eiϕ2 Mˆ0eiϕ2
++;++ ++;++ ≡ 1 −+;−+ −+;−+ ≡ 2
Mˆ = Mˆ0 eiϕ3 Mˆ0eiϕ3, (2)
−+;+− −+;+− ≡ 3
wherethephases ϕ (i = 1,2,3)alsodependonthehelicity indices. Finally, DˆC (z,k )
i λ ,λ′ ⊥C
c c
is the product of the soft helicity fragmentation amplitudes for the process c C +X.
→
It can be shown that Dˆ (z,k ) DC (z,k ) = DC (z,k ) corresponds to the
C/c ⊥C ≡ ++ ⊥C −− ⊥C
usual unpolarized k -dependent FF, while ∆NDˆ (z,k ) 2ImDC (z,k ) is related
⊥ C/c↑ ⊥C ≡ +− ⊥C
to the Collins function. On-shell gluons cannot be transversely spin polarized; therefore
there is not a gluon Collins function. However one can define a “Collins-like” function for
gluons: ∆NDˆC/T1g(z,k⊥C) ≡ 2ReD+C−(z,k⊥C), related to the fragmentation of a linearly
polarized gluon into an unpolarized hadron.
Eq. (1) contains all possible combinations of spin and k -dependent PDF’s/FF’s. In
⊥
order to give a partonic interpretation of the spin and k -dependent PDF’s, we consider
⊥
in detail the explicit expressions for the quark and gluon helicity density matrices.
k
2 Spin and -dependent PDF’s
⊥
Let us first consider a quark parton a. Its helicity density matrix can be written in terms
of its polarization vector components, Pa, as given in its helicity frame. We can write:
i
1 1+Pa Pa iPa
ρa/A,SA fˆ (x ,k ) = z x − y fˆ (x ,k ). (3)
λa,λ′a a/A,SA a ⊥a 2 Pxa +iPya 1−Pza ! a/A,SA a ⊥a
For a given polarization state of hadron A we can define the following functions:
Pafˆ (x ,k ) = ∆fˆa = fˆa fˆa ∆fˆa (x ,k ) (4)
i a/A,SZ a ⊥a si/SZ si/+ − −si/+ ≡ si/+ a ⊥a
Pafˆ (x ,k ) = ∆fˆa = fˆa fˆa ∆fˆa (x ,k ) (5)
i a/A,SY a ⊥a si/SY si/↑ − −si/↑ ≡ si/↑ a ⊥a
1
fˆ (x ,k ) = fˆ (x ,k )+ ∆fˆ (x ,k ), (6)
a/A,SY a ⊥a a/A a ⊥a 2 a/SY a ⊥a
2
where S and S denote respectively the direction of hadron A polarization in its helic-
Z Y
ity frame. At LO, ∆fˆa (x ,k ) and ∆fˆa (x ,k ) can be easily interpreted as the
si/+ a ⊥a si/↑ a ⊥a
number density of partons, polarized along the i-direction (in the parton helicity frame),
inside respectively a longitudinally or transversely polarized hadron. fˆ (x ,k ) is
a/A,SY a ⊥a
related to unpolarized partons and can be split into two PDF’s: the usual k -dependent
⊥
unpolarized PDF and the Sivers function. The latter can be interpreted as the probability
to find an unpolarized parton inside a transversely polarized hadron. Moreover also the
function ∆fˆa (x ,k ) can be split into two terms:
sy/SY a ⊥a
∆fˆa (x ,k ) = ∆fˆa (x ,k )+∆−fˆa (x ,k ) (7)
sy/SY a ⊥a sy/A a ⊥a sy/SY a ⊥a
1
∆−fˆa (x ,k ) ∆fˆa (x ,k ) ∆fˆa (x ,k ) .
sy/SY a ⊥a ≡ 2 sy/↑ a ⊥a − sy/↓ a ⊥a
h i
∆fˆa (x ,k ) is the Boer-Mulders function which gives the probability to find a trans-
sy/A a ⊥a
verselypolarizedquarkinsideanunpolarizedhadron,whereas∆−fˆa (x ,k )isrelated
sy/SY a ⊥a
to the transversity function (see Appendix B of Ref. [3])
We can also write the analogue of Eq. (3) for gluon partons:
1 1+Pg g i g
ρa/A,SA fˆ (x ,k ) = z T1 − T2 fˆ (x ,k ), (8)
λa,λ′a a/A,SA a ⊥a 2 T1g +iT2g 1−Pzg ! a/A,SA a ⊥a
where g and g are related to the Stokes parameters for linearly polarized gluons.
T1 T2
Similarly to the quark case we can define the functions:
gfˆ (x ,k ) ∆fˆg (x ,k ) gfˆ (x ,k ) ∆fˆg (x ,k ) (9)
T1 g/A,SY a ⊥a ≡ T1/↑ a ⊥a T2 g/A,SY a ⊥a ≡ T2/↑ a ⊥a
gfˆ (x ,k ) ∆fˆg (x ,k ) gfˆ (x ,k ) ∆fˆg (x ,k ) (10)
T1 g/A,SZ a ⊥a ≡ T1/+ a ⊥a T2 g/A,SZ a ⊥a ≡ T2/+ a ⊥a
Pgfˆ (x ,k ) ∆fˆg (x ,k ) Pgfˆ (x ,k ) ∆fˆg (x ,k ).(11)
z g/A,SY a ⊥a ≡ sz/↑ a ⊥a z g/A,SZ a ⊥a ≡ sz/+ a ⊥a
Eqs. (9) and (10) are related respectively to the probability to find linearly polarized
gluons inside a transversely or longitudinally polarized hadron A. Eq. (6) holds also for
unpolarized gluons. Furthermore, in analogy to Eq. (7) we can write:
∆fˆg (x ,k ) = ∆fˆg (x ,k )+∆−fˆg (x ,k ) (12)
T1/↑ a ⊥a T1/A a ⊥a T1/↑ a ⊥a
1
∆−fˆg (x ,k ) = ∆fˆg (x ,k ) ∆fˆg (x ,k ) ,
T1/↑ a ⊥a 2 T1/↑ a ⊥a − T1/↓ a ⊥a
h i
where ∆fˆg (x ,k ) is the analogue of the Boer-Mulders function and is related to
T1/A a ⊥a
linearly polarized gluons inside an unpolarized hadron. ∆−fˆg is the analogue of the
T1/↑
transversity and is related to linearly polarized gluons inside a transversely polarized
hadron.
3 Spin Asymmetries
From Eq. (1) and using the PDF’s defined in Sec. 2, we can calculate unpolarized cross
sections, single(SSA)anddouble(DSA)spinasymmetries. Asanapplication, weconsider
the numerator of the transverse SSA for the process A↑B C +X, limiting ourselves to
→
3
show the results only for two partonic subprocesses: q q q q and g g g g . For
a b c d a b c d
→ →
the q q q q contribution we have:
a b c d
→
[dσ(A↑B C +X) dσ(A↓B C +X)]qaqb→qcqd
→ − → ∝
1
∆fˆ (x ,k )fˆ (x ,k ) Mˆ0 2 + Mˆ0 2 + Mˆ0 2 Dˆ (z,k )
2 a/A↑ a ⊥a b/B b ⊥b | 1| | 2| | 3| C/c ⊥C
h i
+ 2 ∆−fˆa (x ,k ) cos(ϕ ϕ ) ∆fˆa (x ,k ) sin(ϕ ϕ )
sy/↑ a ⊥a 3 − 2 − sx/↑ a ⊥a 3 − 2
h∆fˆb (x ,k )Mˆ0Mˆ0Dˆ (z,k ) i
× sy/B b ⊥b 2 3 C/c ⊥C
+ ∆−fˆa (x ,k ) cos(ϕ ϕ +φH) ∆fˆa (x ,k ) sin(ϕ ϕ +φH)
sy/↑ a ⊥a 1 − 2 C − sx/↑ a ⊥a 1 − 2 C
h fˆ (x ,k )Mˆ0Mˆ0∆NDˆ (z,k ) i (13)
× b/B b ⊥b 1 2 C/c↑ ⊥C
1
+ ∆fˆ (x ,k )∆fˆb (x ,k )Mˆ0Mˆ0∆NDˆ (z,k )cos(ϕ ϕ +φH),
2 a/A↑ a ⊥a sy/B b ⊥b 1 3 C/c↑ ⊥C 1 − 3 C
whereas for the g g g g contribution we have:
a b c d
→
[dσ(A↑B C +X) dσ(A↓B C +X)]gagb→gcgd
→ − → ∝
1
∆fˆ (x ,k )fˆ (x ,k ) Mˆ0 2 + Mˆ0 2 + Mˆ0 2 Dˆ (z,k )
2 g/A↑ a ⊥a g/B b ⊥b | 1| | 2| | 3| C/g ⊥C
h i
+ 2 ∆−fˆg (x ,k ) cos(ϕ ϕ )+∆fˆg (x ,k ) sin(ϕ ϕ )
T1/↑ a ⊥a 3 − 2 T2/↑ a ⊥a 3 − 2
h∆fˆg (x ,k )Mˆ0Mˆ0Dˆ (z,k ) i
× T1/B b ⊥b 2 3 C/g ⊥C
+ ∆−fˆg (x ,k ) cos(ϕ ϕ +2φH)+∆fˆg (x ,k ) sin(ϕ ϕ +2φH)
T1/↑ a ⊥a 1 − 2 C T2/↑ a ⊥a 1 − 2 C
×h fˆg/B(xb,k⊥b)Mˆ10Mˆ20∆NDˆC/T1g(z,k⊥C) i(14)
1
+ 2 ∆fˆg/A↑(xa,k⊥a)∆fˆTg1/B(xb,k⊥b)Mˆ10Mˆ30∆NDˆC/T1g(z,k⊥C)cos(ϕ1 −ϕ3 +2φHC).
Let us briefly comment on Eqs. (13) and (14). There are four terms contributing to
the numerator of the transverse SSA: the Sivers contribution (1st term); the transversity
(“transversity like”in Eq. (14)) Boer-Mulders (“like”) contribution (2nd term); the
⊗
transversity (“like”) Collins(“like”)contribution(3rdterm); theSivers Boer-Mulders
⊗ ⊗
(“like”) Collins (“like”) contribution (4th term). The term “like” is referred to those
⊗
functions in Eq. (14), presented in Sec. 2, which are related to linearly polarized gluons
and are the analogues of those for transversely polarized quarks in Eq. (13).
We have performed a numerical evaluation of the maximal contribution of each (un-
known) term appearing in Eqs. (13) and (14) for the transverse SSA in p↑p π + X.
→
To this purpose, we saturate the positivity bounds for the Sivers and Collins functions
and replace all other spin and k -dependent PDF’s with the corresponding unpolarized
⊥
ones. We sum all possible contributions with the same sign. We assume a Gaussian k
⊥
dependence for PDF’s with k2 1/2 = 0.8 GeV/c while for FF’s k2 (z) 1/2 is taken as
h ⊥i h ⊥C i
in Ref. [1]. For the unpolarized PDF’s and FF we adopt the MRST01 [5] and KKP [6]
sets. For more detail see Refs. [1, 3]. As an example, we consider here the evaluation
of several contributions for the process pp π+ + X in the E704 kinematical regime,
→
√s = 19.4 GeV and p = 1.5 GeV/c, Fig. 1. We can see that the Sivers effect is the dom-
T
inant contribution. The second most relevant contribution is the transversity Collins
⊗
contribution. All other contributions give asymmetries compatible with zero. Such a
result can be understood by looking for instance at Eqs. (13) and (14). All contributions
4
are coupled with complicate non trivial azimuthal phases. Under phase space integration
these azimuthal phases kinematically suppress some contributions. Notice that from our
calculations the Sivers effect alone can explain, in principle, the observed SSA while the
Collins effect cannot. Our approach applies also to DSA. Let us consider as an example
the q q q q contribution to the longitudinal DSA, A :
a b c d LL
→
[dσ(A+B+ C +X) dσ(A+B− C +X)]qaqb→qcqd
→ − → ∝
∆fˆa (x ,k )∆fˆb (x ,k ) Mˆ0 2 Mˆ0 2 Mˆ0 2 Dˆ (z,k )
sz/+ a ⊥a sz/+ b ⊥b | 1| −| 2| −| 3| C/c ⊥C
+ 2∆fˆb (x ,k )Mˆ0Mˆ0Dˆ (z,hk ) i
sx/+ b ⊥b 2 3 C/c ⊥C ×
∆fˆa (x ,k ) cos(ϕ ϕ )+∆fˆa (x ,k ) sin(ϕ ϕ ) (15)
sx/+ a ⊥a 3 − 2 sy/A a ⊥a 3 − 2
fˆ h (x ,k )∆fˆb (x ,k )Mˆ0Mˆ0 sin(ϕ ϕ +φH)∆NDˆ i(z,k ).
− a/A a ⊥a sx/+ b ⊥b 1 3 1 − 3 C C/c↑ ⊥C
The 2nd line of Eq. (15) corresponds to the
usual contribution coming from the helicity PDF’s.
The other terms come from contributions of com-
binations of transversely polarized quarks both in
the PDF/FF sectors. Similarly for gluon channels
with contributions of linearly polarized gluons.
Our formalism can also be applied to the study
of inclusive Λ production in hadronic collisions,
Drell-Yan and SIDIS processes [4]. As a further
application,anumericalevaluationofmaximalcon-
tribution to A is in progress.
LL
References
Figure 1. Different maximized contri-
[1] U. D’Alesio and F. Murgia, Phys. Rev. D70,
butions to A , for the process p↑p
074009 (2004) N →
π+X and E704 kinematics, plotted as
[2] M. Anselmino, M. Boglione, E. Leader, a function of x . The different curves
F
U. D’Alesio and F. Murgia, Phys. Rev. D71, correspondto: solidline=quarkSivers
014002 (2005) mechanism; dashed line = gluon Sivers
mechanism; dotted line = transversity
[3] M. Anselmino, M. Boglione, E. Leader,
Collins. All other contributions are
U. D’Alesio, S. Melis and F. Murgia, Phys. ⊗
much smaller.
Rev. D73, 014020 (2006)
[4] M. Anselmino, M. Boglione, U. D’Alesio
A. Kotzinian, F. Murgia and A. Prokudin,
Phys. Rev. D72,094007 (2005)
[5] A.D. Martin, R.G. Roberts, W.J. Stirling and
R.S. Thorne, Phys Lett. B531, 216 (2002)
[6] B.A. Kniehl, G. Kramer, B. Potter, Nucl.
Phys. B582, 514 (2000)
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