QCD Corrections to Dilepton Production near Partonic Threshold in proton-antiproton Scattering

0.15 MB·English

by  Hirotaka Shimizu

BNL-NT-06/7
HUPD-0604
RBRC-585
YITP-SB-06-03
QCD Corrections to Dilepton Production near Partonic Threshold in p¯p
6
Scattering∗
0
0
2 H. Shimizua, G. Stermanb, W. Vogelsangc and H. Yokoyad

n a Department of Physics, Hiroshima University, Higashi-Hiroshima 739-8526,Japan
a
J b C.N. Yang Institute for Theoretical Physics, Stony Brook University,

1
Stony Brook, New York 11794 – 3840, U.S.A.
3
c Physics Department and RIKEN-BNL Research Center, BrookhavenNational Laboratory,
1
Upton, New York 11973, U.S.A.
v
1
d Department of Physics, Niigata University, Niigata 950-2181,Japan
6
2
1 WepresentarecentstudyoftheQCDcorrectionstodileptonproductionnearpartonicthresholdintransversely
0 polarized p¯p scattering. We analyze the role of the higher-order perturbative QCD corrections in terms of the
6 available fixed-order contributions as well as of all-order soft-gluon resummations for the kinematical regime of
0 proposed experiments at GSI-FAIR. We find that perturbative corrections are large for both unpolarized and
/ polarized cross sections, but that the spin asymmetries are stable. The role of the far infrared region of the
h
p momentum integral in theresummed exponent and theeffect of the NNLLresummation are briefly discussed.
-
p
e
h 1. INTRODUCTION measurementsofATT inpolarizedppcollisionsat
: theBNL-RHICcollider[5]. However,sincetheδf
v
i A polarized antiproton beam of energy Ep¯ = forseaquarksareexpectedtobesmall,theasym-
X 15−22 GeV may be available in future experi- metryisestimatedtobeatmostafewpercent[6].
r ments at the GSI-FAIR project. Measurements In contrast, since for the Drell-Yan process in p¯p
a
of dilepton production in transversely polarized collisionsthescatteringoftwovalencequarkden-
p¯p collisionsare the main motivationfor the pro- sities contributes, and since in addition the kine-
posed GSI-PAX [1] and GSI-ASSIA [2] experi- matical regime of the planned GSI experiments
ments. The measurements would be carried out is such that rather large parton momentum frac-
usingatransverselypolarizedfixedprotontarget, tions x ∼ 0.5 are relevant, a very large A of
TT
or a proton beam of moderate energy Ep = 3.5 order 40%or more is expected [7,8,9]. Therefore,
GeV. unique informationontransversityinthe valence
Measurements of the transverse double-spin region may be obtained from the GSI measure-
asymmetry ments, and information from RHIC and the GSI
would be complementary.
δσ σ↑↑−σ↑↓
A ≡ = , (1) Here we give a brief report on a recent study
TT σ σ↑↑+σ↑↓
ofperturbative-QCDcorrectionstothe crosssec-
defined as the ratio of transverselypolarized and tions and to ATT for Drell-Yan dilepton produc-
unpolarized cross sections, may provide informa- tion at GSI-FAIR [8]. We discuss the available
tionofthetransverselypolarizedpartondistribu- fixed order corrections as well as all-order soft-
tion functions of the proton, dubbed “transver- gluon “threshold” resummation.
sity” δf [3,4]. Transversity will be probed by
∗Presented by H. Yokoya at the “7th International Sym-
posiumon Radiative Corrections (RADCOR 2005)”, Oc-
tober2-7,2005,ShonanVillage,Japan
1
2
2. DRELL-YAN CROSS SECTIONS allowed in this case. The large corrections ex-
ponentiate when Mellin moments of the partonic
By virtue of the factorization theorem, the
cross section, defined as
cross section for the Drell-Yan process at large
lepton pair invariant mass M can be written 1
(δ)ω(k),N(r)= dzzN−1(δ)ω(k)(z,r), (5)
in terms of a convolution of parton distribution qq¯ qq¯
Z0
functions and partonic scattering cross sections:
are taken. To next-to-leading logarithmic (NLL)
d(δ)σ 1 accuracy one then has for the resummed cross
= dx (δ)f (x ,µ2) (2)
dM2dφ a a a section [12,13]:
a,b Zτ
X
1 d(δ)σˆ λ p (δ)ωres,N(r,α (µ))=exp[C (r,α (µ))] (6)
× dx (δ)f (x ,µ2) ab +O , qq¯ s q s
Zτ/xa b b b dM2dφ (cid:18)M(cid:19) 1 zN−1−1
×exp 2 dz
1−z
where τ = M2/S with S the hadronic c.m. en- (cid:26) Z0
ergy, and where φ is the azimuthal angle of one (1−z)2M2 dk2
× TA (α (k )) ,
of the leptons. µ is the factorization scale. As Zµ2 kT2 q s T )
indicated in Eq. (2), there are corrections sup-
pressed with some power p and some hadronic where
scaleλ. Thesecorrectionswillbecome important
α α 2
for small M and in particular for lower-energy Aq(αs)= πsA(q1)+ πs A(q2)+... , (7)
collisions. (cid:16) (cid:17)
with A(1) =C and [14]:
q F
2.1. Fixed-order perturbative calculation
ThepartoniccrosssectioniscalculatedinQCD C 67 π2 5
A(2) = F C − − N , (8)
perturbation theory as a series in αs; q 2 A 18 6 9 f
(cid:20) (cid:18) (cid:19) (cid:21)
d(δ)σˆab =(δ)σˆ(0) ω(0)(z)+ αs(δ)ω(1)(z,r) where Nf is the number of flavors and CA = 3.
dM2dφ ab ab π ab The coefficient C (r,α (µ)) collects mostly hard
q s
h
α 2 virtual corrections. It is given as
+ s (δ)ω(2)(z,r)+... , (3)
π ab
(cid:16) (cid:17) (cid:21) C (r,α )= αs −4+2π2+3lnr +O(α2). (9)
wherez =M2/sˆ,sˆ=x x S andr =M2/µ2. For q s π 3 2 s
a b (cid:18) (cid:19)
the unpolarized cross section the calculation has
We note that it was shown in [15] that these co-
been performed up to O(α2) [10], for the trans-
s efficient functions also exponentiate.
versely polarized case to O(α ) [11]. The lowest
s Eq. (6) is ill-defined because of the divergence
order gives
in the perturbative running coupling α (k ) at
s T
2α2e2 α2e2 kT = ΛQCD. The perturbative expansion of the
σˆq(0q¯) = 9M2sqˆ, δσˆq(0q¯) = 9M2qsˆcos2φ (4) expression shows factorial divergence, which in
QCDcorrespondstoapower-likeambiguityofthe
with ω(0) =δ(1−z). The higher-order functions series. It turns out, however, that the factorial
qq¯
may be found in the literature [10,11]. divergence appears only at nonleading powers of
momentumtransfer. Thelargelogarithmsweare
2.2. Threshold resummation resummingariseintheregion[13]z ≤1−1/N¯ in
Threshold resummation addresses large loga- the integrand in Eq. (6). Therefore to NLL they
rithmic perturbative corrections to the partonic are contained in the simpler expression
cross section that arise when the initial partons
have just enough energy to produce the lepton M2 dk2 N¯k
2 TA (α (k ))ln T (10)
pair. Only emission of relatively soft gluons is ZM2/N¯2 kT2 q s T M
3
for the second exponent in (6). Here we have 103
btchheeolosrween,suµims=musMeedd.efTxophrois“nmfeonirntm.im,taol”wehxicphanwseiownisll[r1e6t]uronf σσ / LO102 NNf123468usnrtttLNhhhldtdl O oLrreOdseurm emxpeadn sion σσ / LO101 NNf123468usnrtttLNhhhldtdl O oLrreOdseurm emxpeadn sion
101
Forthe NLL expansionofthe resummedexpo-
nent one finds from Eqs. (6),(10) [16]: 1002.5 3 3.5 4 4.5 5 1002 4 6 8 10 12
M (GeV) M (GeV)
lnδωres,N(r,α (µ))=C (r,α (µ)) (11) Figure 1. K-factors as defined in Eqs. (14), (15)
qq¯ s q s
for the Drell-Yan cross section as a function of
+2lnN¯ h(1)(λ)+2h(2)(λ,r),
leptoninvariantmassM,inp¯pcollisionwithS =
where 30 GeV2 (left), and S = 210 GeV2 (right).
λ=b α (µ)lnN¯ . (12) up order by order in perturbation theory. We
0 s
expand the resummedformula to next-to-leading
Theexplicitexpressionsforthefunctionsh(1)and order (NLO) and beyond and define the “soft-
h(2) can be found e.g. in Refs. [16,8]. gluon K-factors”
The hadronic cross section is obtained by per-
dσ(res)/dMdφ
forming an inverse Mellin transformation of the Kn ≡ O(αns) , (15)
resummed partonic cross section, multiplied by dσ(LO)/dM(cid:12)dφ
(cid:12)
theappropriatemomentsoftwopartondensities:
which for n = 1,2,... give the effects due to the
d(δ)σres dN O(αn) terms in the resummed formula. The re-
= τ−N s
dM2dφ 2πi sults for K1−8 are also shown in Fig. 1. One can
ZC
d(δ)σˆres,N see that there are very large contributions even
× (δ)fN(δ)fN ab . (13) beyond NNLO, in particular at the higher M.
a b dM2dφ
ab Clearly, the full resummation given by the solid
X
line receives contributions from high orders. We
In order to perform the inverse Mellin integral,
stressthattheO(α )andO(α2)expansionsofthe
weneedto specify aprescriptionfor dealingwith s s
resummed result are in excellent agreement with
the singularity in the perturbative strong cou-
the full NLO and NNLO ones, respectively (cir-
plingconstantinEq.(6). Wewillusetheminimal
cle and square symbols in Figure 1). This shows
prescription developed in Ref. [16], which relies
that the higher-order corrections are really dom-
on use of the NLL expanded form involving the
inated by the threshold logarithms and that the
hi(λ),andonchoosingacontourtotheleftofthe
resummation is accurately collecting the latter.
Landau singularity at λ=1/2 in the complex-N
plane. 2.3. Far infrared cut-off
Figure 1 shows the effects of the higher or-
There is good reason to believe that the
ders generated by resummation for S =30 GeV2
largeenhancementfromsoft-gluonradiationseen
and S = 210 GeV2. We define a resummed “K-
above is only partly physical. The large correc-
factor”astheratiooftheresummedcrosssection
tions arise from a region where the integral in
to the leading order (LO) cross section,
the exponent becomes sensitive to the behavior
of the integrand at small values of k . As long
dσ(res)/dMdφ T
K(res) = , (14) as Λ ≪M/N¯ ≪M, the use of perturbation
dσ(LO)/dMdφ QCD
theory may be justified, but when |N| becomes
which is shown by the solid line in Fig. 1. As very large, k will reach down to nonperturba-
T
can be seen, K(res) is very large, meaning that tive scales. We seek a modificationof the pertur-
resummation results in a dramatic enhancement bative expressionin Eq. (6) that excludes the re-
overLO,sometimes byovertwoordersofmagni- gioninwhichtheabsolutevalueofk islessthan
T
tude for the collisions at lower energy. It is then some nonperturbative scale µ . To implement
0
interesting to see how this enhancement builds this idea, we will adopt a modified resummed
4
hard scattering, which reproduces NLL logarith- presented before. The ratios of the infrared-
mic behavior in the moment variable N so long regulated resummed cross sections to LO show a
as M/N¯ >µ , but “freezes” once M/N¯ <µ . If smoother increase than the “purely minimally”
0 0
nothing else, this will test the importance of the resummed ones. The difference is particularly
region k ≤ Λ for the resummed cross sec- marked at the lower center of mass energy in
T QCD
tion. IfN wererealandpositive,wecouldsimply Fig. 2 (left), with only a modest enhancement
replace the resummed exponent in (10) by overNNLOremaining. Weinterprettheseresults
toindicateastrongsensitivitytononperturbative
M dk N¯k
4 TA (α (k ))ln T , (16) dynamics at the lower energies, and much less at
q s T
Zρ(M/N¯,µ0) kT M the higher.
where ρ(a,b) = max(a,b), and where µ then
0 3. SPIN ASYMMETRY A
serves to cut off the lower logarithmic behav- TT
ior. To provide an expression that can be an- Toperformnumericalstudiesoftheasymmetry
alytically continued to complex N, we choose A weneedtomakeamodelforthetransversity
TT
ρ(a,b) = (ap +bp)1/p, with integer p. This sim- densities in the valence region. Here, guidance is
pleformisconsistentwiththeminimalexpansion provided by the Soffer inequality [17]
given above, and it also allows for a straightfor-
ward analysis of the ensuing branch cuts in the 2 δq(x,Q2) ≤q(x,Q2)+∆q(x,Q2), (18)
complex-N plane. For definiteness, we choose
w(cid:12)hich gives(cid:12)an upper bound for each δq. Fol-
p = 2. We will continue to use the expansions (cid:12) (cid:12)
lowing [6] we utilize this inequality by saturating
in Eq. (11), but redefine λ in Eq. (12) by
the bound at some low input scale Q ≃0.6GeV
0
1 N¯2µ2 using the NLO GRV [18] and GRSV (“standard
λ=b0αs(µ)lnN¯−2b0αs(µ)ln 1+ M20 . (17) scenario”) [19] densities q(x,Q20) and ∆q(x,Q20),
(cid:18) (cid:19) respectively. For Q > Q the transversity densi-
0
Of course, different choices of µ0 give different ties δq(x,Q2) are then obtained using the NLO
results, but we should think of µ as a kind of
0 evolution equations [11].
factorization scale, separating perturbative con-
Figure 3 shows that A is very robust un-
TT
tributions from nonperturbative. Thus changes
der the QCD corrections,including resummation
inµ wouldbecompensatedbychangesinanon-
0 with and without a cutoff. This is expected to
perturbative function. Our interest here, how-
some extent because the emission of soft-gluons
ever, is simply to illustrate the modification of
does not change the spin of the parent parton.
the perturbative sector,whichwe do by choosing
µ =0.3 GeV and µ=0.4 GeV.
0 0.4 0.4
Results for the “K-factor”with these values of
µ are shown in Fig. 2, compared to the same 0.35
0 0.3
NLO, NNLO and resummed cross sections as ATT0.3 ATT
LO 0.2 LO
0.25 NLO NLO
pert. resummed pert. resummed
103 µ0=0.3 GeV µ0=0.3GeV
σσ / LO102 rrrNNeeeLNsss...O L(((µµµO000===000)..34 GGeeVV)) σσ / LO101 rrrNNeeeLNsss...O L(((µµµO000===000)..34 GGeeVV)) NFiL0g.2O2u.5rean33d. fSo3.5Mrp (GitenVh)4easNy4.5LmLm-5reetsru0y.12mAmTe4Td(φca6M=s e(GeV0a)8)t Sat1=0LO301,2
101 GeV2 (left) and S =210 GeV2 (right).
1002.5 3 3.5 4 4.5 5 100 4 6 8 10 12
M (GeV) M (GeV)
Figure 2. K-factors as in Fig. 1, at S =30 GeV2 4. NNLL RESUMMATION
(left) and S = 210 GeV2 (right). The dashed
Thanks to the recent calculation of the three-
(dot-dashed)linesshowtheeffectsofalowercut-
loopsplittingfunctionsbyMoch,Vermaserenand
off µ =300 MeV (400 MeV) for the k integral
0 T
in the exponent.
5
104 105 Acknowledgments
103 104
32σM d/dM (pb GeV)111111000000---202311 LNNNNOLNLNOLLLOL 32σM d/dM (pb GeV)1111100000-02311 LNNNNOLNLNOLLLOL bPWyHT.VYht.-he0ies0gw9rN8oa5rat2ket7ifou,onlPfatHolGYRS.-IS0cK.i3eE5nw4Nc7ae,7sB6F,NsoauLunpndapdnoPadrHtttieoYhdne-,0Ui3n.5gS4r.pa8Dna2ert2ts-.
10-42.5 3 3.5 4 4.5 5 10-22 4 6 8 10 12
M (GeV) M (GeV) partment of Energy(contractnumber DE-AC02-
Figure4. UnpolarizedcrosssectionM3dσ/dM at
98CH10886) for providing the facilities essential
S =30 GeV2 (left) and S =210 GeV2 (right) at
for the completionofhis work. The workofH.Y.
LO, NLO, NNLO, and NLL-, NNLL-resummed,
wassupportedinpartbyaResearchFellowshipof
as function of lepton pair invariant mass M.
the Japan Society for the Promotion of Science.
Vogt [20], we can now perform the threshold re-
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