Double Transverse-Spin Asymmetries for Small-Q_T Drell-Yan Pair Production in p\bar{p} Collisions

0.07 MB·English

by  Hiroyuki Kawamura

Double Transverse-Spin Asymmetries for
Small-Q Drell-Yan Pair Production
T
in pp¯ Collisions
Hiroyuki Kawamura and Kazuhiro Tanaka†
9 ∗
0
0 DepartmentofMathematicalSciences,UniversityofLiverpool,LiverpoolL693BX,UK
∗
2 †DepartmentofPhysics,JuntendoUniversity,Inba,Chiba270-1695,Japan

n
a
Abstract. We discuss the double-spin asymmetries in transversely polarized Drell-Yan process,
J
calculatingall-ordergluonresummationcorrectionsuptothenext-to-leadinglogarithmicaccuracy.
9
This resummation is relevant when the transverse-momentum Q of the produced lepton pair is
1 T
small,andreproducesthe(fixed-order)next-to-leadingQCDcorrectionsuponintegratingoverQ .
T
] Theresummationcorrectionsin pp¯-collisionbehavedifferentlycomparedwith pp-collisioncases,
h andaresmallatthekinematicsintheproposedGSIexperiments.Thisfactallowsustopredictlarge
p
valueofthedouble-spinasymmetriesatGSI,usingrecentempiricalinformationonthetransversity.
-
p Keywords: Transversity,Drell-Yanprocess,Antiprotons,Softgluonresummation
e
PACS: 12.38.-t,12.38.Cy,13.85.Qk,13.88.+e
h
[

1 The double-spin asymmetry in Drell-Yan (DY) process with transversely-polarized
v protons and antiprotons, p p¯ l+l X, for azimuthal angle f of a lepton measured in
↑ ↑ −
8 therest frameofthedileptonl+→l withinvariantmassQ andrapidityy, is givenby
9 −
7
.2 A = ds ↑↑/dw −ds ↑↓/dw D Tds /dw = cos(2f )(cid:229) qe2qd q(x1,Q2)d q(x2,Q2)+··· ,
1 TT ds /dw +ds /dw ≡ ds /dw 2 (cid:229) e2q(x ,Q2)q(x ,Q2)+
0 ↑↑ ↑↓ q q 1 2 ···
9 (1)
0 where dw dQ2dydf , the summation is over all quark and antiquark flavors with
v: d q(x,Q2)a≡ndq(x,Q2)beingthetransversityandunpolarizedquark-distributionsinside
i a proton, and the ellipses stand for the corrections of next-to-leading order (NLO) and
X
higher in QCD perturbation theory. The scaling variables x represent the momentum
r 1,2
a fractions associated with the partons annihilating via the DY mechanism, such that
Q2 =(x P +x P )2 =x x S and y=(1/2)ln(x /x ), where S =(P +P )2 is the CM
1 1 2 2 1 2 1 2 1 2
energy squared of p p¯ . In the proposed polarization experiments at GSI [1], moderate
↑ ↑
energies, 30 . S . 200 GeV2, allow us to measure (1) for 0.2 . Q/√S . 0.7, and
probetheproductofthetwoquark-transversitiesinthe“valenceregion”.Recently,QCD
correctionsin(1)atGSIkinematicshavebeenstudied:theNLO(O(a ))corrections[2]
s
aswellasthehigherorderonesintheframeworkofthresholdresummation[3]arerather
small,so thattheLOvalueofA ,which turnsoutto belarge, isratherrobust.
TT
When the transverse momentum Q of the final l+l is also observed, we ob-
T −
tain the new double-spin asymmetry as the ratio of the Q -differential cross sections,
T
A (Q ) [D ds /dw dQ ]/[ds /dw dQ ].Thebulkofl+l pairisproducedatsmall
TT T T T T −
Q Q, w≡here the cross sections (D )ds /dw dQ receive large perturbative correc-
T T T
tion≪swithlogarithmsln(Q2/Q2) multiplyinga at each order, by therecoil from gluon
T s
radiations,andthosehavetobetreatedinanall-orderresummationinQCD[4].Thecor-
responding “Q -resummation” has formally some resemblance to the threshold resum-
T
mation[3],butembodiescontributionsfromdifferent“edgeregion”ofphasespace.The
Q -resummation for D ds /dw dQ has been derived recently [5], summing the corre-
T T T
spondinglargelogarithmsuptonext-to-leadinglogarithmic(NLL)accuracy.Combined
withthatfords /dw dQ [4], weget [6](b 2e gE withg theEulerconstant)
T 0 − E
≡
A (Q )= cos(2f )R d2b eib·QTeS(b,Q)(cid:229) qe2qd q(x1,b20/b2)d q(x2,b20/b2)+···, (2)
TT T
2 R d2b eib·QTeS(b,Q)(cid:229) qe2qq(x1,b20/b2)q(x2,b20/b2)+···
where the numerator and denominator are, respectively, reorganized in the impact pa-
rameter b space in terms of the Sudakov factor eS(b,Q) resumming soft and flavor-
conserving collinear radiation, while the ellipses involve the remaining contributions
of the O(a ) collinear radiation, which can be absorbed into the exhibited terms as
s
d q D C d q, q C q+C g using the corresponding coefficient func-
T qq qq qg
tion→s (D )C ;⊗note tha→t there⊗is no “chir⊗al-odd” gluon distribution to participate in the
T ij
numerator of (2) as well as (1). Using universal Sudakov exponent S(b,Q) with the
first nonleading anomalous dimensions in (2), the first three towers of large logarith-
miccontributionstothecrosssections,a nlnm(Q2/Q2)/Q2 (m=2n 1,2n 2,2n 3),
s T T
are resummed to all orders in a , yielding the NLL resummation. I−n addit−ion to t−hese
s
resummed components relevant for small Q , the ellipses in (2) also involve the other
T
termsofthefixed-ordera , which treat theLO processes in thelarge Q region,so that
s T
(2) is the ratio of the NLL+LO polarized and unpolarized cross sections. We include a
Gaussian smearing as usually as S(b,Q) S(b,Q) g b2, corresponding to intrinsic
NP
→ −
transverse momentum of partons inside proton. The integrations of the NLL+LO cross
sectionsD ds /dw dQ andds /dw dQ overQ coincide,respectively,withthe(fixed-
T T T T
order)NLOcross sectionsD ds /dw and ds /dw , associatedwith A of(1)[5, 6].
T TT
The resummation makes 1/b Q the relevant scale in (2), in contrast to Q in
T
(1). Figure 1 shows [6] the nume∼rical evaluation of (2), as well as of D ds /dw dQ
T T
associated with its numerator, at GSI kinematics using the NLO transversities that
saturate the Soffer bound, 2d q(x,m 2) q(x,m 2)+D q(x,m 2), at a low scale m with
D q the helicity distribution. The NLL re≤summed component dominates D ds /dw dQ
T T
in small and moderate Q region, and similarly for ds /dw dQ reflecting universality
T T
of the large Sudakov effects, which leads to almost constant A (Q ), in particular,
TT T
with even flatter behavior than the corresponding asymmetry [5] for the pp-collision
case. Remarkably, A (Q ) at NLL+LO has almost the same value as that at LL;
TT T
this is in contrast to the pp case where the resummation at higher level enhances the
asymmetry [5]. We note that A (Q ) at LL is given by (2) omitting all nonleading
TT T
corrections, i.e., omitting the ellipses, replacing S(b,Q) by that at the LL level, and
replacingthescaleofthepartondistributionsasb2/b2 Q2,sothattheresultcoincides
with A at LO (see (1)). Therefore, at GSI, bot0h A→(Q ) and A are quite stable
TT TT T TT
when including the QCD (resummation and fixed-order) corrections, with A (Q )
TT T
A (0),and ≃
TT
A (Q ) A . (3)
TT T TT
≃
To clarify the relevant mechanism, A (Q ) A (0) allows us to consider the
TT T TT
≃
Q 0 limit of (2): for Q 0, the b integral is controlled by a saddle point b=b ,
T T SP
→ ≈
0.8 40
3V] (a) NNLLLL+LO (b)
dQ [pb/GeT00..46 LLLO mmetry [%]2300
fdyd Asy NLL+LO
2Q NLL
d0.2 10 LL
/ LO
sd
T
D 0 0
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
Q (GeV) Q (GeV)
T T
FIGURE1. (a)D ds /dw dQ (b)A (Q )withGSIkinematics,S=210GeV2,Q=4GeV,y=f =
T T TT T
0,andwithg =0.5GeV2,usingtheNLOtransversitydistributionsthatcorrespondtotheSofferbound.
NP
whichhasthesamevaluebetweenthenumeratoranddenominatorin(2)suchthat[5,6]
A (0) cos(2f )(cid:229) qe2qd q(x1,b20/b2SP)d q(x2,b20/b2SP) , (4)
TT ≃ 2 (cid:229) e2q(x ,b2/b2 )q(x ,b2/b2 )
q q 1 0 SP 2 0 SP
omitting the small corrections from the LO components involved in the ellipses in (2).
This saddle-point evaluation is exact at NLL accuracy; in particular, the O(a ) contri-
s
butions from the coefficients (D )C , e.g., C g associated with gluon distribution,
T ij qg
⊗
completely decouple as Q 0 (see [4, 5]). The simple form of (4) is reminiscent of
T
A of (1) at LO, but is dif→ferent from the latter in the unconventional scale b2/b2 ;
TT 0 SP
the actual position of the saddle point implies b /b 1 GeV, irrespective of the
0 SP
≃
values of Q and g [5, 6]. In the valence region relevant for GSI kinematics, the u-
NP
quarkcontributiondominatesin(4)and(1),sothattheseasymmetriesarecontrolledby
the ratio, d u(x ,m 2)/u(x ,m 2) with m 2 = b2/b2 and Q2, respectively. Actually the
1,2 1,2 0 SP
scaledependenceinthisratioalmostcancelsbetweenthenumeratoranddenominatoras
d u(x,b2/b2 )/u(x,b2/b2 ) d u(x,Q2)/u(x,Q2) (see Fig. 3 in [6]); this implies (3) at
0 SP 0 SP ≃
GSI.Notethatthisisnotthecasefor ppcollisionsbecauseofverydifferentbehaviorof
thesea-quarkcomponentsundertheevolutionbetweentransversityandunpolarizeddis-
tributions[5];indeedA (Q )>A atRHICandJ-PARC[5].Asimilarlogicapplied
TT T TT
to(2)alsoexplainswhyA (Q )at GSI areflatterthan in pp collisions.
TT T
Anotherconsequence of thesimilarlogicis that d u(x ,1 GeV2)/u(x ,1 GeV2) as
1,2 1,2
a function of x directly determines the Q- as well as S-dependence of the value of
1,2
(3) at GSI, through x = (Q/√S)e y. In Fig. 2, using the NLO transversity distribu-
1,2 ±
tions corresponding to the Soffer bound, the symbols “ ” plot A (Q 1 GeV) of
TT T
△ ≃
(2)atNLL+LO[6].Thedashedcurvedrawstheresultusing(4);thissimpleformulain-
deed workswell.Alsoplottedby thetwo-dot-dashedcurveisA of(1)at LO withthe
TT
transversitiescorrespondingtotheSofferboundatLOlevel,todemonstrate(3).TheQ-
andS-dependenceoftheseresultsreflectthattheratiod u(x,1GeV2)/u(x,1GeV2)isan
increasingfunctionofx.TheseresultsusingtheSofferboundshowthe“maximallypos-
sible” asymmetry, i.e., optimisticestimate. A more realistic estimate of (2) (with Q
T
≃
1 GeV) and (4) is shown [6] in Fig. 2 by the symbols “ ” and the dot-dashed curve,
respectively,withtheNLOtransversitydistributionsassu▽mingd q(x,m 2)=D q(x,m 2) at
50 50
%] 40 S=30GeV2 %] 40 S=210GeV2
[ [
y 30 y 30
r r
et et
m m
20 20
m m
y y
As 10 As 10
0 0
2 2.5 3 3.5 4 2 4 6 8 10
Q (GeV) Q (GeV)
FIGURE2. Thedoubletransverse-spinasymmetriesatGSIasfunctionsofQwithy=f =0.
a low scale m , as suggested by nucleon models and favored by the results of empirical
fit for transversity [7]. The new estimate gives smaller asymmetries compared with the
Soffer bound results, but still yields rather large asymmetries [6]. Based on (3), these
resultsinFig. 2may beconsideredas estimateofA of(1).
TT
At present, empirical information of transversity is based on the LO global fit, us-
ing the semi-inclusive DIS data and assuming that the antiquark transversities vanish,
d q¯(x) = 0, so that the corresponding LO parameterization is available only for u and
d quarks [7]. Fortunately, however, the dominance of the u-quark contribution in the
GSI kinematics allows quantitative evaluation of A at LO using only this empirical
TT
information [6]: the upper limit of the error band for the u- and d-quark transversities
obtainedbytheglobalfit[7]yieldsthe“upperbound”ofA shownbythedottedcurve
TT
in Fig. 2; using (3), this result may be considered also as estimate of A (Q ). In the
TT T
small Q region, our full NLL+LO result of A (Q 1 GeV), shown by “ ”, can be
TT T
≃ ▽
consistentwithestimateusingtheempiricalLOtransversity,buttheseresultshaverather
different behavior for increasing Q, because the u-quark transversity for the former lies
slightly outside the error band of the global fit for x&0.3 [6]. Thus, in the large asym-
metries to be observed at GSI, the behavior of A (Q ), A as functions of Q will
TT T TT
allowus todeterminethedetailedshapeoftransversitydistributions.
ACKNOWLEDGMENTS
Thisworkwas supportedbytheGrant-in-AidforScientificResearch No.B-19340063.
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