Lepton Helicity Distributions in Polarized Drell-Yan Process

22 Pages·0.23 MB·English

by  Jiro Kodaira

DESY 03-009
HUPD - 0301
hep-ph/0301228
January 2003
Lepton Helicity Distributions in
Polarized Drell-Yan Process
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2 Jiro KODAIRAa,b and Hiroshi YOKOYAb

n
a
J a Deutsches Elektronen-Synchrotron, DESY

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Platanenallee 6, D 15738 Zeuthen, GERMANY
2

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v b Department of Physics, Hiroshima University
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2 Higashi-Hiroshima 739-8526, JAPAN
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3 Abstract
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/
h TheleptonhelicitydistributionsinthepolarizedDrell-YanprocessatRHIC
p energy are investigated. For the events with relatively low invariant mass
-
p of lepton pair in which the weak interaction is negligible, only the measure-
e ment of lepton helicity can prove the antisymmetric part of the hadronic
h
: tensor. Therefore it might be interesting to consider the helicity distri-
v
butions of leptons to obtain more information on the structure of nucleon
i
X
from the polarized Drell-Yan process. We estimate the QCD corrections
r at (α ) level to the hadronic tensor including both intermediate γ and Z
a s
O
bosons. We present the numerical analyses for different invariant masses
¯
and show that the u(u¯) and d(d) quarks give different and characteristic
contributions to the lepton helicity distributions. We also estimate the
lepton helicity asymmetry for the various proton’s spin configurations.
1 INTRODUCTION
The measurement of the polarized nucleon structure function g (x,Q2) by the Euro-
1
pean Muon Collaboration [1] in the late ’80s has opened the door to the hadron spin
physics. In the last fifteen years, great progress has been made both theoretically and
experimentally that has considerably improved our knowledge of the spin structure of
nucleon. Through these developments, hadron spin physics has grown up as one of
the most active fields attracting considerable attention. Now our interest has spread
out to various processes to explore the spin structure of hadrons. In particular, in
conjunction with new kind of experiments, the RHIC spin project, etc., we are now in
a position to obtain more information on the structure of hadrons and the dynamics
of QCD. The spin dependent quantity is, in general, very sensitive to the structure of
interactions among various particles. Therefore, we will be able to study the detailed
structure of hadrons based on QCD. We also hope that we can find some clue to new
physics beyond the standard model through the new experimental data.
It is now expected that the polarized proton-proton collisions (RHIC-Spin) at BNL
relativistic heavy-ion collider RHIC [2] will provide sufficient experimental data to
unveil the structure of nucleon. Therefore it is important and interesting to investigate
various processes which might be measured in RHIC experiments. One of those will
be the polarized Drell-Yan process [3].
ThepolarizedDrell-Yanprocess hasbeenstudied bymany authorsbothforlongitu-
dinally [4,5,6,7] and transversely [8,9,10,11,12] polarized case. The W,Z productions
from the polarized hadrons are also investigated [13,14,15]. In this article, we discuss
the lepton helicity distributions from the polarized Drell-Yan process at the QCD one-
loop level. The lepton helicity distributions carry more information on the nucleon
structure [16,17,18,19] than the “inclusive” Drell-Yan observable like the invariant
mass distribution of lepton pair. Now let us consider, for simplicity, the virtual γ me-
diated Drell-Yan process. The subprocess cross section dσˆ is written in terms of the
hadronic and leptonic tensors as,
dσˆ WS +WA LSµν +LAµν = WS LSµν +WALAµν .
∝ µν µν µν µν
(cid:16) (cid:17)(cid:16) (cid:17)
The anti-symmetric part of hadronic tensor W contains spin information on the an-
µν
nihilating partons. However, for observables obtained after summing over the helicities
of lepton or integrating out the lepton distributions, this anti-symmetric part drops
out. Furthermore, the chiral structure of QED and QCD interactions tells us that only
1
particular helicity states are selected for the q q¯annihilation. This observation shows
−
that the polarized and unpolarized Drell-Yan processes are governed by essentially the
same dynamics. On the other hand, if we measure the lepton helicity distributions, we
can reveal the whole structure of the hadronic tensor. We will show that the u(u¯) and
¯
d(d) quarks give characteristic contributions to the lepton helicity distributions.
Thearticleisorganizedasfollows. InSec.2, wereproducethetreelevel crosssection
to define our conventions. We present our calculations for the helicity distributions of
lepton at the QCD one-loop level in Sec.3. We adopt the massive gluon scheme to
regularize the infrared and mass singularities avoiding the complexity in the treatment
of γ . The scheme dependence will be discussed in Sec.4 and we will change the scheme
5
to the MS to perform the numerical studies. We give the numerical results in Sec.5.
Finally, Sec.6 contains the conclusions. The explicit forms for the subprocess cross
sections are listed in Appendix A. The invariant mass distribution of lepton pair in
the massive gluon scheme is given in Appendix B which is used to identify the scheme
changing factor.
2 POLARIZED DRELL-YAN AT TREE LEVEL
Inthis section, we reproduce the treelevel result forthe helicity distributions of leptons
in the polarized Drell-Yan process to establish our notation. For the longitudinally
polarized Drell-Yan process,
¯ ′ ′
N (P ,λ )+N (P ,λ ) l(l,λ)+l(l,λ)+X ,
A A A B B B
→
with λ(= ) being the helicity of each particle, we introduce the parton distributions
±
fA(x), ∆fA by,
a a
fA(x) = fA(x,+)+fA(x, ) , ∆fA(x) = fA(x,+) fA(x, ) ,
a a a − a a − a −
where fA(x,+/ ) denotes the distribution of parton type a with positive/negative
a −
helicity in nucleon A with positive helicity. Based on the factorization theorem, the
hadronic cross section for the helicity distribution of lepton is given as the convolution
of the parton distributions with the hard subprocess cross section dσˆab as,
1 1
dσ(λ ,λ ;λ) = dx dx
A B 1 2
a,b Zτ Zτ/x1
X
fA(x )+λ λ ∆fA(x ) fB(x )+λ λ ∆fB(x )
a 1 a A a 1 b 2 b B b 2 dσˆab(λ ,λ ;λ) , (1)
a b
× 2 2
λXa,λb
2
where
Q2 (l+l′)2
τ = .
S ≡ (P +P )2
A B
The helicity dependent cross section dσˆab(λ ,λ ;λ) for the subprocess,
a b
¯ ′ ′
a(p ,λ )+b(p ,λ ) l(l,λ)+l(l ,λ)+X ,
a a b b
→
is a function of the partonic invariant variables,
s = (p +p )2 = (x P +x P )2 = x x S ,
a b 1 A 2 B 1 2
t = (p l)2 , u = (p l)2 ,
a b
− −
and
Q2 = q2 = (l+l′)2 zs .
≡
The tree level polarized cross section for lepton pair production,
¯ ′ ′
q(p ,λ )+q¯(p ,λ ) l(l,λ)+l(l ,λ) ,
q q q¯ q¯
→
is given by,
1 2
dσˆT = MT dΦ ,
2
2N2s
c (cid:12) (cid:12)
(cid:12) (cid:12)
where dΦ is the two particle phase space an(cid:12)d N(cid:12) (= 3) is the color factor. The square
2 c
of the tree amplitude reads,
2
2 4πα 2
MT = δλq,−λq¯δλ,−λ′Nc Q2 ! fλqλ Lµν WµTν , (2)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
where the tree level hadronic and leptonic tensors are defined as,
WT = Tr(ω p γ p γ ) , L = Tr(ω lγ l′γ ) , (3)
µν λq q¯ µ q ν µν λ µ ν
and
Lµν WT = 2[t2 +u2 +λ λ(u2 t2)] .
µν q −
In Eq.(3) and also below, all momentum p under Tr operator are understood to be p
6
and ω (1+λγ )/2. In Eq.(2), α is the QED fine structure constant and the quantity
λ 5
≡
fλqλ depends on the fermion couplings to the photon and Z-boson in the following way,
Q2
fλqλ = e +Qλq Qλ , (4)
− q q l Q2 M2 +iM Γ
− Z Z Z
3
where e is the electric charge of the quark in units of the electron charge e, M is the
q Z
Z mass, Γ is the Z width. The helicity labels λ , λ of quark and lepton take . The
Z q
±
quark and lepton couplings to the Z boson are
1 sinθ
Q− = T3 e sin2θ , Q+ = e W ,
q,l sinθ cosθ q,l − q,l W q,l − q,l cosθ
W W W
(cid:16) (cid:17)
where e = 1, T3 is the third component of the isospin and θ is the Weinberg angle.
l W
−
In the center of mass (CM) frame of annihilating quarks, the cross section becomes,
dσˆdTQ(2λdq,cλoqs¯;θλ) = δλq,−λq¯ 2Nπc Qα2!2 fλqλ 2 1+cos2θ+2λqλcosθ δ(1−z) , (5)
(cid:12) (cid:12) (cid:16) (cid:17)
(cid:12) (cid:12)
(cid:12) (cid:12)
where θ is the scattering angle of produced lepton.
In Eq.(5), the third term which depends on the helicities of quark and lepton comes
fromtheantisymmetricpartofhadronictensorWT . Fortheobservableaftertakingthe
µν
spin sum of lepton, this antisymmetric part appears in the cross section only through
the parity violating Z interaction in Eq.(4). Therefore, for the events with “small”
values of Q2, the information from the antisymmetric part will be completely lost.
Furthermore the chiral structure of Standard Model interaction forces only particular
helicity states to participate in the process as shown in Eq.(5). These observations tell
us that for the spin summed over final states and low Q2 events, the polarized and
unpolarized Drell-Yan processes are governed by the essentially the same dynamics at
least for the qq¯initiated process.
3 QCD ONE-LOOP CALCULATION
The principle result of this section will be the polarized Drell-Yan cross section at
the QCD one-loop level with the helicity of lepton being fixed. At the QCD one-loop
level, infrared and mass singularities appear and we regularize them by giving a non-
zero mass κ to gluon [10,20] to avoid the complexities from the treatment of γ and
5
phase space integrals. To perform the numerical analyses by using the known MS
parameterization for the parton densities, we have to change the scheme. However, it
is well known how to do it [21]. The diagrams to be calculated are given in Fig.1.
The virtual gluon correction (Figs.1a and 1b) to this process yields,
dσˆV dσˆT α 1 Q2 3 Q2 7 π2
= sC ln2 + ln + , (6)
dQ2dcosθ dQ2dcosθ (cid:18) π F(cid:19) "−2 κ2 2 κ2 − 4 6 #
4
where α = g2/4π is the strong coupling constant and C = (N2 1)/2N for SU(N )
s s F c − c c
of color.
The amplitude for the real gluon emission (Fig.1c and 1d),
¯ ′ ′
q(p ,λ )+q¯(p ,λ ) l(l,λ)+l(l ,λ)+g(k) ,
q q q¯ q¯
→
is given by,
e2g
MR = δλq,−λq¯δλ,−λ′ − Q2s! fλqλ u¯λ(l)γµvλ′(l′)
1 1
v¯ (p ) γ γ +γ γ Tau (p )ǫν(k) ,
× λq¯ q¯ " µ p k ν ν p q µ# λq q a
6 q− 6 6 q− 6
where ǫν(k) is the polarization vector of gluon and Ta is the color matrix. We have
a
′
defined q l+l. The expression for the square of the amplitude given below has been
≡
q
γ , Z
q
(a) (b)
(c) (d) (e) (f)
Figure 1: Parton-level subprocesses contributing to the Drell-Yan process at (α ):
s
O
(a,b) virtual correction, (c,d) real gluon emission (e,f) quark-gluon Compton.
summed over the spin and colors of unobserved particles.
2
MR 2 = δλq,−λq¯NcCF 4πQα2gs! fλqλ 2 Lµν WµRν ,
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
where (cid:12) (cid:12) (cid:12) (cid:12)
2
WR = 2(p k)Tr(ω kγ p γ ) κ2Tr(ω p γ p γ )
µν tˆ2 q¯· λq µ q ν − λq q¯ µ q ν
h i
2
+ 2(p k)Tr(ω p γ kγ ) κ2Tr(ω p γ p γ )
uˆ2 q · λq q¯ µ ν − λq q¯ µ q ν
4 h i
+ (2p p 2p k 2p k)WT +(p +p )kαWT
tˆuˆ q · q¯− q · − q¯· µν qµ q¯µ αν
h
+WT kα(p +p ) , (7)
µα qν q¯ν
i
5
The tree level hadronic and leptonic tensors WT , L were defined in the previous
µν µν
section and new invariant variables are introduced,
tˆ= (p q)2 , uˆ = (p q)2 .
q q¯
− −
In Eq.(7), we have dropped terms which do not contribute when κ2 0. It should
→
be noted that the κ2 terms in the first and second lines can not be neglected since the
phase space integral produces 1/κ2 singularity [20].
The cross section is given by,
1 2
dσˆR = MR dΦ , (8)
2N2s 3
c (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where dΦ is the three particle phase space. To integrate out the irrelevant variables
3
in Eq.(8), we take the CM frame of qq¯:
√s √s
p = (1,0,0,1) , p = (1,0,0, 1) .
q q¯
2 2 −
Parametrizing the momenta of lepton l and lepton pair q by,
lµ = l (1, sinθcosϕ, sinθsinϕ, cosθ) ,
| |
qµ = (q0, q sinθˆcosϕˆ, q sinθˆsinϕˆ, q cosθˆ) ,
where
s+Q2 κ2
q0 = − , q 2 = q02 Q2 ,
2√s | | −
Q2
l = ,
| | 2 q0 q (sinθˆsinθcos(ϕˆ ϕ)+cosθˆcosθ)
−| | −
h i
it is easy to write the phase space element dΦ as,
3
q Q2dQ2dcosθˆdϕˆdcosθdϕ
dΦ = | | .
3 2
32(2π)5√s q0 q (sinθˆsinθcos(ϕˆ ϕ)+cosθˆcosθ)
−| | −
h i
After integrating over the angular variables [10] anddropping terms which vanish when
κ2 0, we write the inclusive cross section for the polarized lepton in the following
→
form:
dσˆR(λ ,λ ;λ) dσˆRT(λ ,λ ;λ;z) dσˆRF(λ ,λ ;λ)
q q¯ = q q¯ FR(Q2, z)+ q q¯ , (9)
dQ2dcosθ dQ2dcosθ dQ2dcosθ
6
where the first term contains the infrared and mass singularities as well as terms asso-
ciated with them and,
dσˆRdTQ(λ2qd,cλoq¯s;λθ;z) = δλq,−λq¯ 2Nπc Qα2!2 fλqλ 2 · 21 (10)
(cid:12) (cid:12)
8z2 (1+λ(cid:12)(cid:12) λ)((cid:12)(cid:12)1+cosθ)2 +(1 λ λ)(1 cosθ)2z2
× "(1+cosθ+z(1 cosθ))4 q − q −
− n o
8z2
+ (1+λ λ)(1+cosθ)2z2 +(1 λ λ)(1 cosθ)2 .
(1 cosθ+z(1+cosθ))4 q − q − #
− n o
with
α 1 Q2 π2 1+z2 Q2
FR(Q2, z) = sC ln2 δ(1 z)+ ln
π F " 2 κ2 − 6 ! − (1 z) κ2
− +
ln(1 z) lnz
+2(1+z2) − 2(1+z2) (1 z) .
1 z ! − 1 z − − #
− + −
The second term in Eq.(9) gives the (α ) finite contribution whose explicit form is
s
O
listed in Appendix A. It should be noted that the first (second) term in Eq.(10) is just
the tree level cross section for the qq¯ annihilation with momenta zp and p (p and
q q¯ q
zp ) times z which arises from the difference between the flux normalizations. These
q¯
terms express the processes with collinear gluon emissions. The function FR can be
understood to be a probability to emit a collinear gluon with momentum (1 z)p
q
−
or (1 z)p and does not depend on the helicities of quarks because of the helicity
q¯
−
conservation of QCD interaction.
By adding the tree level Eq.(5) and virtual Eq.(6) contributions to Eq.(9), the
double logarithmic infrared singularities cancel out and the cross section from the qq¯
initial states becomes,
dσˆT+V+R dσˆRT 7 α α Q2
s s
= 1 C δ(1 z)+ P (z)ln
dQ2dcosθ dQ2dcosθ "(cid:18) − 4 π F(cid:19) − π qq κ2
ln(1 z) lnz
+ C 2(1+z2) − 2(1+z2) (1 z)
F
( 1 z ! − 1 z − − )!#
− + −
dσˆRF
+ , (11)
dQ2dcosθ
where
1+z2 3
P (z) = C + δ(1 z) ,
qq F
(1 z)+ 2 − !
−
7
which is the DGLAP one-loop splitting functions P . To obtain above result, we have
qq
used the fact,
dσˆRT(λ ,λ ;λ;z) dσˆT(λ ,λ ;λ)
q q¯ q q¯
δ(1 z) = .
dQ2dcosθ − dQ2dcosθ
Finally, the contribution from the quark-gluon Compton process (Fig.1e and 1f),
¯ ′ ′ ′ ′
q(p ,λ )+g(k,h) l(l,λ)+l(l ,λ)+q(p ,λ ) ,
q q → q q
where h is the helicity of gluon, can be calculated in the same way as before. To avoid
singularities in the physical region, the gluon mass is taken to be κ2 [20] and the spin
−
projection for incoming gluons reads,
1 kγpδ
∗ q
ǫ (k,h)ǫ (k,h) = g +iǫ .
α β 2 "− αβ αβγδ k pq#
·
The amplitude is,
e2g
MC = δλq,λ′q δλ,−λ′ − Q2s! fλqλ u¯λ(l)γµvλ′(l′)
1 1
× u¯λ′q(p′q)"γµp + kγα +γαp qγµ# Tauλq(pq)ǫαa(k,h) ,
6 q 6 6 q− 6
and its square becomes,
2
MC 2 = N C 4παgs fλqλ 2 Lµν WC ,
c F Q2 ! µν
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
where (cid:12) (cid:12) (cid:12) (cid:12)
1 uˆ
WC = (1+λ h) 1+ Tr(ω p′ γ kγ )
µν s q tˆ! λq q µ ν
2uˆ 1 1
′
+ + (1+λ h)+ (1 λ h) Tr(ω p γ p γ )
(stˆ s q tˆ − q ) λq q µ q ν
1 uˆ 4 4
′ ′
(1 λ h) 1+ Tr(ω kγ p γ )+ (1+λ h)p p + (1 λ h)p p
− tˆ − q s! λq µ q ν tˆ q qµ qν s − q qµ qν
κ2 2Q2 s
′
+ 1 λ h − Tr(ω (p k)γ p γ ) ,
tˆ2 − q s ! λq q − µ q ν
with
s = (p +k)2 , tˆ= (p q)2 , uˆ = (k q)2 .
q q
− −
From the cross section formula for this process,
1 2
dσˆC = MC dΦ ,
2N (N2 1)s 3
c c − (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
8
we obtain the polarized lepton distribution and write it in the same form as Eq.(9),
dσˆC(λ ,h;λ) dσˆCT(λ ;λ;z) dσˆCF(λ ,h;λ)
q = q FC(Q2, z, λ , h)+ q . (12)
dQ2dcosθ dQ2dcosθ q dQ2dcosθ
The first term, in this case, is given by,
dσˆCT(λq;λ;z) = π α 2 fλqλ 2 (13)
dQ2dcosθ 2Nc Q2!
(cid:12) (cid:12)
8z2 (cid:12) (cid:12)
(cid:12) (1(cid:12) +λ λ)(1+cosθ)2z2 +(1 λ λ)(1 cosθ)2 ,
× (1 cosθ+z(1+cosθ))4 q − q −
− n o
and
α Q2 1 z 1
FC(Q2, z, λ , h) = s PC(z;λ ,h) ln +ln − (1+λ h(1 2z)) .
q (cid:18)2π(cid:19) " qg q κ2 z2 !− 4 q − #
Thequantity PC isrelatedtotheunpolarizedandpolarizedDGLAPsplittingfunctions
qg
P (z) and ∆P (z) in the following way,
qg qg
1 (1+λ h)(1 z)2 +(1 λ h)z2
PC(z;λ ,h) = (P (z) λ h∆P (z)) = q − − q .
qg q 2 qg − q qg 4
Equation (13) again expresses the tree level cross section for the qq¯annihilation with
momenta p andzk where theq¯is emittedcollinearly fromtheinitialgluon. Incontrast
q
to the qq¯ annihilation case, the function FC does depend on the helicities of initial
quark and gluon since the emitted antiquark should have the helicity λ = λ to
q¯ q
−
annihilate into the electroweak gauge bosons. The remaining finite contribution dσˆCF
can be found in Appendix A.
The cross section for the q¯g subprocess can be obtained by changing (λ ,h) to
q
( λ , h) and µ ν in WC in the above formulas.
− q − ↔ µν
From these results, the unpolarized cross sections are easily obtained. As a check,
we have reproduced the results of Ref. [20] by neglecting the Z boson contribution.
4 FACTORIZATION AND SCHEME CHANGE
Before proceeding to numerical studies, we must factorize the mass singularities into
the parton densities in the MS scheme since the two-loop evolution of them which
should be combined with the one-loop corrections for the hard part, is given in this
scheme.
9