Polar Angle Dependence of the Alignment Polarization of Quarks Produced in e^+e^- Annihilation

37 Pages·0.32 MB·English

by  S. Groote

MZ-TH/95-19
FTUV/95-52
IFIC/95-54
January 1996
Polar Angle Dependence of the
Alignment Polarization of Quarks
6
9 Produced in e+e−-Annihilation
9
1

n
a
S. Groote , J.G. K¨orner
J ∗ ∗

4
2
Institut fu¨r Physik, Johannes Gutenberg-Universit¨at

1
v Staudingerweg 7, D-55099 Mainz, Germany.
3
1
3 M.M. Tung
1 ∗∗
0
6
9 Departament de F´ısica Te`orica, Universitat de Val`encia
/
h
and IFIC, Centre Mixte Universitat Val`encia — CSIC,
p
-
p C/ Dr. Moliner, 50, E-46100 Burjassot (Val`encia), Spain.
e
h
:
v
i
X
ABSTRACT
r
a
We calculate one-loop radiative QCD corrections to the three polarized and unpolar-
ized structure functions that determine the beam-quark polar angle dependence of the
alignment (or longitudinal) polarization of light and heavy quarks produced in e+e -
−
annihilations. We present analytical and numerical results for the alignment polarization
and its polar angle dependence. We discuss in some detail the zero-mass limit of our
results and the role of the anomalous spin-flip contributions to the polarization observ-
ables in the zero-mass limit. Our discussion includes transverse and longitudinal beam
polarization effects.
∗Supported in part by the BMFT, FRG, under contract 06MZ566,
and by HUCAM, EU, under contract CHRX-CT94-0579
∗∗Feodor-Lynen Fellow
1 Introduction
This is the fifth and final paper in a series of papers devoted to the O(α ) determination
s
of the polarization of quarks produced in e+e -annihilations. In the first paper of this se-
−
ries [1] we calculated the mean alignment polarization (sometimes also called longitudinal
or helicity polarization) of the quark. i.e. the alignment polarization averaged w.r.t. the
relative beam-event orientation. One of us derived convenient Schwinger-type representa-
tions forthe structure functions appearing inthe polarizationexpressions in [2]. Ina third
paper we determined the longitudinal component of the quark’s alignment polarization,
which vanishes at the Born term level [3]. The fourth paper [4] gave results on the two
transverse components of the quark’s polarization (perpendicular and normal, also some-
times referred to as in-the-plane and out-of-the-plane polarization). In the present final
piece of work we present our results on the full polar angle dependence of the quark’s
alignment polarization w.r.t. the electron beam direction including beam polarization
effects.
The full determination of the alignment polarization of the quark involves the calcu-
lation of three polarized and three unpolarized structure functions which are conveniently
chosen as the helicity structure functions Hℓ and H , respectively. The subscripts
U,L,F U,L,F
U, L and F label the relevant density matrix elements of the exchanged vector boson
(γ ,Z), where the polar angle dependence of the contributions of the three structure
V
functions is given by (1 + cos2θ) for U, sin2θ for L and cosθ for F. We mention that
some of the unpolarized helicity structure functions have been calculated before, i.e. the
vector/vector (VV) contribution to H and H [5] and the vector/axial-vector (VA)
U L †
contribution to H [7]. We have verified the results of [5] and [7] and present new results
F
on the axial-vector/axial-vector (AA) contribution to H and H in this paper, where
U L
the AA-contribution to H = H + H was already written down in [1,2]. As con-
U+L U L
cerns the polarized structure functions our results on Hℓ and Hℓ where given in [1,2,3].
U+L L
This paper includes new results on the polarized structure functions Hℓ and Hℓ that are
U F
necessary to determine the full cosθ-dependence of the quark’s alignment polarization.
†The VV-contribution to HU+L = HU +HL has been given a long time ago in the context of QED
(see e.g. [6])
2
The paper is structured such that we start in Sec. 2 by listing the four independent
tree-graph components of the hadron tensor (VV, AA, VA and AV) including its polar-
ization dependence. The relevant helicity components of the structure functions H
U,L,F
and Hℓ are obtained by covariant projections. We then proceed to integrate the pro-
U,L,F
jected tree-graph contributions over the full three-body phase-space and give analytical
results for the once- and twice-integrated tree-graph cross sections. As is well familiar by
now, the final expressions contain infrared (IR) divergences which we choose to regularize
by introducing a small gluon mass. The tree-graph IR divergencies are cancelled by the
corresponding IR divergencies of the one-loop contributions. Thus by adding in the loop
contributions we finally arrive at finite results relevant for the total inclusive cross section
integrated over the hard and the soft regions of the energy of the gluon as presented
later on. In Sec. 2 we also detail the dependence of the polarization on the electroweak
parameters including a discussion of beam polarization effects.
In Sec. 3 we first focus on the soft-gluon region and give results on the polar an-
gle dependence of the alignment polarization of the quark with their typical logarithmic
dependence on the gluon-energy cut. The hard-gluon contribution is given by the com-
plement of the soft-gluon contribution, i.e. as the difference of the full O(α ) and the
s
soft-gluon contribution. Since we provide numerically stable expressions for the latter
two contributions, the hard-gluon contribution can be evaluated in a numerically stable
way. In Sec. 4 we consider the zero quark mass case and calculate the three relevant
polarized and unpolarized structure functions using helicity methods and dimensional
reduction as regularization method. In this way one can keep out of the way of the no-
torious γ -problem when calculating e.g. the VA-contribution to the F-type unpolarized
5
structure function. We compare the mass-zero results with the mass-zero limit of the
corresponding structure function expressions in Sec. 3 and identify the global anomalous
spin-flip contributions to the QCD(m 0) polarized structure functions. Finally, Sec. 5
→
contains our summary and our conclusions. In Appendix A we catalogue some integrals
that appear in the tree-graph integrations in Sec. 2, in Appendix B we list some standard
O(α ) rate functions needed for the rate expressions in Sec. 3. In Appendix C, finally, we
s
consider the case of polarized and unpolarized quark production from transversely and
longitudinally polarized e+e -beams.
−
3
O α
2 ( ) tree-graph contributions and total rates
s
Forthethreebodyprocess(γ ,Z) q(p )+q(p )+g(p )(seeFig.1)wedefine apolarized
V 1 2 3
→
hadron tensor according to (q = p +p +p )
1 2 3
H (q,p ,p ,s) = qqg j 0 0 j qqg . (1)
µν 1 2 h | µ| ih | ν†| i
q,g spins
X
The hadron tensor depends on the vector (V) and axial-vector (A) composition of the
currents in Eq. (1). One has altogether four independent components Hi (i = 1,2,3,4)
µν
which are defined according to (V: γ , A: γ γ )
µ µ 5
1 1
H1 = (HVV +HAA) H2 = (HVV HAA) (2)
µν 2 µν µν µν 2 µν − µν
i 1
H3 = (HVA HAV) H4 = (HVA +HAV).
µν 2 µν − µν µν 2 µν µν
For the tree-graph contribution one calculates
g2N C
H1 = s C F q2( (2 ξ)(ξ(y+z)2 4yz(1 y z))+4yz(y2+z2))g
µν 2y2z2q2 " − − − − − µν
+16y2zp p +4(ξy2 4yz +2ξyz+2y2z +ξz2 +2yz2)(p p +p p )
1µ 1ν 1µ 2ν 2µ 1ν
−
+16yz2p p +4z( 2y +ξy +2y2 +ξz)(p p +p p )
2µ 2ν 1µ 3ν 3µ 1ν
−
+4y(ξy 2z +ξz +2z2)(p p +p p )
2µ 3ν 3µ 2ν
− #
2img2N C
+ s C F q2(2 ξ)y2ε(µνp s) q2(4yz 2ξyz+2y2z 2yz2)ε(µνp s)
y2z2q4 " − 1 − − − 2
+q2(2y2 ξy2 2yz +2ξyz+ξz2)ε(µνp s)
3
− −
4y2(p +p )ε(νp p s)+4y2(p +p )ε(µp p s)
1µ 3µ 1 2 1ν 3ν 1 2
−
+4y(yp +zp +yp )ε(νp p s) 4y(yp +zp +yp )ε(µp p s) (3)
1µ 2µ 3µ 2 3 1ν 2ν 3ν 2 3
− #
2m2g2N C
H2 = s C F q2(ξ(y +z)2 4yz(1 y z))g +8yzp p
µν y2z2q4 "− − − − µν 3µ 3ν #
2img2N C
+ s C F q2(ξy2 2yz +ξyz +ξz2 +4yz2)ε(µνp s)
y2z2q4 " − 1
q2ξz(y z)ε(µνp s)+q2ξy(y z)ε(µνp s)
2 3
− − −
4yz(p +p )ε(νp p s)+4yz(p +p )ε(µp p s)
1µ 3µ 1 2 1ν 3ν 1 2
−
4yz(p p +p )ε(νp p s)+4yz(p p +p )ε(µp p s) (4)
1µ 2µ 3µ 1 3 1ν 2ν 3ν 1 3
− − − #
4
2img2N C
H3 = s C F 4yz(p s)(p p p p )+4yz(p s+p s)(p p p p )
µν y2z2q4 "− 3 1µ 2ν − 2µ 1ν 2 3 1µ 3ν − 3µ 1ν
q2(ξy2 4yz+2ξyz +2y2z +ξz2 +4yz2)(p s s p )
1µ ν µ 1ν
− − −
2q2yz2(p s s p ) q2(ξy2 2yz +ξyz +2y2z)(p s s p ) (5)
2µ ν µ 2ν 3µ ν µ 3ν
− − − − − #
and
2ig2N C
H4 = s C F q2(ξy2 4yz +2ξyz 2y2z +ξz2 +2yz2)ε(µνp p )
µν y2z2q4 "− − − 1 2
q2(2y2 ξy2 2yz +2ξyz+ξz2)ε(µνp p )+4q2ξy2ε(µνp p )
1 3 2 3
− − −
+8y(yp +zp +yp )ε(νp p p ) 8y(yp +zp +yp )ε(µp p p )
1µ 2µ 3µ 1 2 3 1ν 2ν 3ν 1 2 3
− #
2mg2N C
+ s C F q2((ξy2 4yz +2ξyz+2y2z +ξz2 +2yz2)(p s)
y2z2q4 "− − 2
+( 2y2 +ξy2 2yz+2ξyz +4y2z +ξz2)(p s))g
3 µν
− −
4y2(p s)(p p +p p )+8yz(p s)p p
3 1µ 2ν 2µ 1ν 3 2µ 2ν
−
4y(z(p s)+y(p s))(p p +p p )+2q2y2z(p s +s p )
2 3 2µ 3ν 3µ 2ν 1µ ν µ 1ν
−
+q2(ξy2 4yz+2ξyz +4y2z +ξz2 +2yz2)(p s +s p )
2µ ν µ 2ν
−
+q2( 2yz +ξyz +2y2z +ξz2)(p s +s p ) , (6)
3µ ν µ 3ν
− #
wherewehaveaccountedforthespindependenceofthehadrontensorbyusingthequark’s
spin projector u(p ,s)u¯(p ,s) = (/p +m)1(1+γ /s) when calculating the traceaccording to
1 1 1 2 5
Eq. (1). We have used the energy variables y = 1 2p q/q2 and z = 1 2p q/q2 and the
1 2
− · − ·
abbreviation ξ = 4m2/q2. Note that for the spin dependent contributions, H1,2,3(s) are
µν
antisymmetric in µ and ν, while H4 (s) is symmetric in µ and ν. For the spin independent
µν
pieces, the nonvanishing contributions H1,2 are symmetric, H4 is antisymmetric in µ
µν µν
and ν, and there is no spin independent contribution to H3 .
µν
In this paper we are only concerned with the alignment polarization. The covariant
form of the polarization four-vector associated with the alignment polarization is given by
sℓµ = ξ 1/2( (1 y)2 ξ,0,0,1 y) (sℓsℓµ = 1) which reduces to sℓµ = (0,0,0,1) in
− − − − µ −
q
the rest system of the quark when y = 1 √ξ. We then define unpolarized and polarized
−
5
structure functions Hi and Hiℓ (i = 1,2,3,4) according to
µν µν
Hi = Hi (sℓ)+Hi ( sℓ), (7)
µν µν µν −
Hiℓ = Hi (sℓ) Hi ( sℓ). (8)
µν µν − µν −
In order to determine the polar angle dependence of the polarized and unpolarized cross
section one turns to the helicity structure functions H = H + H , H = H and
U ++ L 00
−−
H = H H whichcanbeobtainedfromthecovariantstructurefunctionsinEqs. (7)
F ++
− −−
and (8) by the appropiate helicity projections of the gauge boson,
H = εµ(λ )H εν (λ ) (9)
λZ,γλZ,γ Z,γ µν ∗ Z,γ
and the same for Hℓ .
µν
A convenient way of obtaining the helicity structure functions H (α = U,L,F) is
α
by covariant projection,
pˆµpˆν pˆµpˆν pˆ q
H = gˆµν 1 1 H H = 1 1H H = iεµναβ 1α β H , (10)
U − − p21z ! µν L p21z µν F p1z√q2 µν
where gˆ = g q q /q2 and pˆ = p (p q)q /q2 are the four-transverse metric
µν µν µ ν 1µ 1µ 1 µ
− − ·
tensor and the four-transverse quark momentum, respectively. Note that the covariant
projectors defined in Eq. (10) are (y,z)-independent and can thus be freely commuted
with the y- and z-integrations to be done later on.
After all these preliminaries let us write down the differential polarized and unpo-
larized three-body e+e -cross sections, differential in cosθ and in the two energy variables
−
y and z. One has
dσℓ 3 dσ4ℓ 3 dσ4ℓ
= (1+cos2θ)g U + sin2θ g L
14 14
dcosθdydz 8 dydz 4 dydz
3 dσ1ℓ dσ2ℓ
+ cosθ g F +g F and (11)
41 42
4 dydz dydz!
dσ 3 dσ1 dσ2
= (1+cos2θ) g U +g U
11 12
dcosθdydz 8 dydz dydz!
3 dσ1 dσ2 3 dσ4
+ sin2θ g L +g L + cosθ g F (12)
11 12 44
4 dydz dydz! 4 dydz
where, in terms of the helicity structure functions defined above, one has
dσi πα2v q2
α = Hi . (13)
dydz 3q4 (16π2v α)
6
(α = U,L,F) and the same for σi σiℓ etc. The g coupling factors appearing in
α → α ij
Eqs. (11) and (12) specify the electroweak structure of the lepton-hadron interaction and
are listed in Appendix C. The y, z and cosθ-dependent polarization Pℓ(cosθ,y,z) is then
given by the ratio of the polarized and unpolarized cross sections in Eqs. (11) and (12).
The generalizationoftheabove crosssection expressions to thecasewhere onestarts
with “longitudinally” polarized beams is straightforward and amounts to the replacement
(see Appendix C)
polarized: g [(1 h h+)g +(h h+)g ] (14)
14 − 14 − 44
→ − −
g [(h h+)g +(1 h h+)g ] (i = 1,2)
4i − 1i − 4i
→ − −
unpolarized: g [(1 h h+)g +(h h+)g ] (i = 1,2) (15)
1i − 1i − 4i
→ − −
g [(h h+)g +(1 h h+)g ]
44 − 14 − 44
→ − −
where h and h+ ( 1 h +1) denote the helicity polarization of the electron and the
− ±
− ≤ ≤
positron beam. Clearly there is no interaction between the beams when h+ = h = 1.
−
±
We have presented the results of calculating the full (U +L) and the longitudinal
piece (L) of the polarized hadron tensor in [1,3]. Here we add the last building block,
namely the polarized hadron tensor projected on its forward/backward component (F),
where the requisite covariant projector has been given in Eq. (10). A straightforward
calculation of the O(α ) tree-graph contributions leads to
s
8πα N C 1
H1ℓ(y,z) = s C F 8 4ξ ξ2 2(2 ξ)(2 3ξ)
F (1 y)2 ξ" − − − − − y
− −
1 1 1
ξ(1 ξ)(2 ξ) + 2ξy 4yz 6ξz 4(2 ξ)(3 2ξ)
− − − y2 z2!− − − − − − z
1 z y
+2(1 ξ)(2 ξ)2 +(4 2ξ ξ2) +(28 14ξ +ξ2)
− − yz − − y − z
y y2 y3 y3 y2
+2ξ(2 ξ) ξ(4 ξ) +4 +2ξ 2(8 3ξ) (16)
− z2 − − z2 z z2 − − z #
8πα N C 1 1 1
H2ℓ(y,z) = s C F ξ 4+ξ 2(2 3ξ) ξ(1 ξ) +
F (1 y)2 ξ " − − y − − y2 z2!
− −
1 1 z
+4z 4(3 2ξ) +2(1 ξ)(2 ξ) +ξ
− − z − − yz y
y y y2 y2
+(12 ξ) +2ξ ξ 4 (17)
− z z2 − z2 − z #
7
16πα N C 1 1 1
H4(y,z) = s C F (4 5ξ) ξ(1 ξ) +
F (1 y)2 ξ"− − y − − y2 z2!
− −
q 1 y z y y2 1
2(4 3ξ) +ξ +2z +2 +6 2 +2(1 ξ)(2 ξ) (18)
− − z z2 y z − z − − yz#
In all three above expressions the denominator vanishes when the quark’s three-
momentum is zero, i.e. when p~ = 0 or, in terms of the y-variable, when y = 1 √ξ. How-
1
−
ever, a careful limiting procedure shows that the three structure functions in Eqs. (16)–
(18) tend to finite limiting values (and not to zero as in the case of the longitudinal
structure functions Hiℓ treated in [3]). In contrast to this, the singularities at y = z = 0
L
constitute true IR-singularities. They can be regularized by introducing a small gluon
mass m = √Λq2 which has the effect to deform the phase-space boundary to
g
y = Λξ +Λ, y = 1 ξ (19)
+
− −
q2y 1 qΛ 1
z (y) = 1 y ξ +Λ+ (y Λ)2 Λξ (1 y)2 ξ . (20)
± 4y +ξ ( − − 2 y ± y − − − − )
q q
Notethattheintroductionof asmallgluonmass isrequired onlytodeformtheintegration
boundary. In the calculation of the matrix elements we need not pay regard to the gluon
mass. With the gluon-mass regulator one can then perform the finite integration over
the two phase-space variables y and z. The integration over z gives the cross section’s
dependence on the quark energy variable y. We obtain
8πα N C (1 ξ)(2 ξ)2
H1ℓ(y) = s C F 2 2 − − 4(3 2ξ)(2 ξ)
F (1 y)2 ξ" ( y − − −
− −
z (y)
+(28 14ξ +ξ2)y 2(8 3ξ)y2+4y3 ln +
− − − ) z (y)!
−
1 8ξ(1 ξ)(2 ξ)y
+ (y Λ)2 Λξ (1 y)2 ξ − −
y − − − − (− ξy2+4Λ(1 y ξ +Λ)
q q − −
(1 ξ)(2 ξ)
8 − − +32(2 ξ) 2(18 5ξ)y+20y2
− y − − −
(4 ξ)(40+3ξ)y (4 ξ)3y
− + − (21)
− 2(4y +ξ) (4y+ξ)2)#
8πα N C (1 ξ)(2 ξ)
H2ℓ(y) = s C F ξ 2 2 − − 4(3 2ξ)
F (1 y)2 ξ " ( y − −
− −
z (y)
+(12 ξ)y 4y2 ln +
− − ) z (y)!
−
1 8ξ(1 ξ)y
+ (y Λ)2 Λξ (1 y)2 ξ −
y − − − − (− ξy2+4Λ(1 y ξ +Λ)
q q − −
8
1 ξ (4 ξ)y
8 − +32 12y 4 − (22)
− y − − 4y +ξ )#
16πα N C (1 ξ)(2 ξ) z (y)
H4(y) = s C F 2 − − (4 3ξ)+3y y2 ln +
F (1 y)2 ξ" ( y − − − ) z (y)!
− − −
q1 4ξ(1 ξ)y
+ (y Λ)2 Λξ (1 y)2 ξ −
y − − − − (− ξy2+4Λ(1 y ξ +Λ)
q q − −
1 ξ 16y (4 ξ)2y
4 − +8 y + − . (23)
− y − − 4y +ξ (4y+ξ)2)#
Let us briefly pause to discuss the y-dependence of the forward/backward (F)-
polarizationcomponentPℓ whichwedefineasthe(2cosθ)-momentofthefullθ-dependent
F
polarization Pℓ(cosθ), i.e.
g H1ℓ +g H2ℓ
2cosθPℓ(cosθ) = 41 F 42 F =: Pℓ. (24)
h i g H1 +g H2 F
11 U+L 12 U+L
The IR-limit y 0 in Eq. (24) is well defined since the potentially singular term propor-
→
tional to 1/y in the numerator is cancelled by the 1/y-terms of the denominator giving a
finite polarization value for y 0. In fact the limiting value of Pℓ can be calculated to
→ F
be
2(2 ξ)g +2ξg
Pℓ(y 0) = − 41 42. (25)
F → (4 ξ)g +3ξg
11 12
−
Turning to the other corner of phase-space y 1 √ξ, the limiting value of the po-
→ −
larization expressions can be obtained by expanding numerator and denominator around
y = 1 √ξ. As mentioned before, the (U +L)- and (L)-components of the polarization
−
vanish in this limit, whereas the (F)-component of the polarization tends to the finite
limiting value
2((2 2√ξ ξ)g +ξg )
Pℓ(y 1 ξ) = − − 41 42 . (26)
F → − −3((4 4√ξ +3ξ)g ξg )
q − 11 − 12
In Fig. 2 we have plotted the y-dependence of Pℓ for the top quark case for the three
F
values √q2 = 380, 500 and 1000 GeV using a top quark mass of m = 180 GeV [8]. The
t ‡
polarization Pℓ can be seen to tend to the two limiting values given in Eqs. (25) and (26)
F
as y 0 and y 1 √ξ, respectively.
→ → −
‡We have chosen the lowest energy q2 = 380 GeV to lie well above the nominal threshold value of
q2 =360GeVforaperturbativecalcuplationtomakesense. Asiswellknown,theproductiondynamics
ipn the threshold region requires the consideration of non-perturbative effects. For a discussion of top
quark polarization effects in the threshold region see [9].
9
Finally, with the integration over y we include the relative three-body/two-body
phase-space factor q2/16π2v and express the result in terms of the rate functions
q2
Hˆi(ℓ)(tree) = dydzHi(ℓ)(y,z) (27)
α 16π2v α
Z
for α = F, which gives
α N C q2
Hˆ1ℓ(tree) = s C F (8 4ξ ξ2)J 2(2 ξ)(2 3ξ)J
F 2πv − − 1 − − − 2
(cid:20)
ξ(1 ξ)(2 ξ)(J +J ) 2ξJ 4J 6ξJ 4(2 ξ)(3 2ξ)J
3 9 4 6 7 8
− − − − − − − − −
+2(1 ξ)(2 ξ)2J +(4 2ξ ξ2)J +(28 14ξ +ξ2)J
10 11 12
− − − − −
+2ξ(2 ξ)J ξ(4 ξ)J +4J +2ξJ 2(8 3ξ)J (28)
13 14 15 16 17
− − − − −
(cid:21)
α N C q2
Hˆ2ℓ(tree) = s C F ξ (4+ξ)J 2(2 3ξ)J ξ(1 ξ)(J +J )
F 2πv 1 − − 2− − 3 9
(cid:20)
+4J 4(3 2ξ)J +2(1 ξ)(2 ξ)J +ξJ
7 8 10 11
− − − −
+(12 ξ)J +2ξJ ξJ 4J (29)
12 13 14 17
− − −
(cid:21)
α N C q2
Hˆ4(tree) = s C F (4 5ξ)S ξ(1 ξ)(S +S ) 2(4 3ξ)S
F πv − − 2− − 3 5 − − 4
(cid:20)
+ξS +2S +2S +6S 2S +2(1 ξ)(2 ξ)S . (30)
6 8 9 10 11 12
− − −
(cid:21)
The integrals S have been computed and listed in [1]. Similar techniques allow one to
i
calculate the integrals J (for details see [1,3]). They are listed in Appendix A. Remember
i
that the IR-singularities in S , S , S , J , J and J are cancelled by the corresponding
3 5 12 3 9 10
IR-singularities of the virtual contributions. This is a consequence of the Lee-Nauenberg
theorem.
3 Soft- and hard-gluon regions
For some applications it is desirable to split the three-body phase-space into a soft and a
hard gluon region. The two regions are defined with respect to a fixed cut-off value of the
gluon energy Ecut-off = λ√q2 such that E /√q2 λ and E /√q2 > λ define the soft and
g g ≤ g
the hard-gluonregions, respectively. Technically theintegration over the soft-gluonregion
is quite simple. The hard-gluon contribution can then be obtained as the complement of
10