Resummed QCD Power Corrections to Nuclear Shadowing

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by  Jianwei Qiu

Resummed QCD Power Corrections to Nuclear Shadowing
Jianwei Qiu and Ivan Vitev
Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA
We calculate and resum a perturbative expansion of nuclear enhanced power corrections to the
structure functions measured in deeply inelastic scattering of leptons on a nuclear target. Our
2 A 2
resultsfortheBjorkenx-,Q -andA-dependenceofnuclearshadowinginF2 (x,Q )andthenuclear
A 2
modifications to FL(x,Q ), obtained in terms of the QCD factorization approach, are consistent
with the existing data. We demonstrate that the low-Q2 behavior of these data and the measured
largelongitudinalstructurefunctionpointtoacriticalroleforthepowercorrectionswhencompared
toother theoretical approaches.
5 PACSnumbers: 12.38.Cy;12.39.St;24.85.+p;25.30.-c
0
0
2 Inordertounderstandtheoverwhelmingdatafromthe in [7]. The transverse and longitudinal structure func-
RelativisticHeavyIonCollider(RHIC)andmakepredic- tions arerelatedto the standardDISstructure functions
n
tionsforthefutureElectronIonCollider(EIC)andLarge F , F as follows: F (x,Q2) = F (x,Q2), F (x,Q2) =
a 1 2 T 1 L
J HadronCollider(LHC)intermsofthesuccessfulpertur- F2(x,Q2)/(2x)−F1(x,Q2) if 4x2m2N ≪ Q2 [6]. In DIS
0 bative QCD factorization approach [1] we need precise the exchange photon γ∗ of virtuality Q2 and energy
1 information of the nuclear parton distribution functions ν = Q2/(2xmN) probes an effective volume of trans-
(nPDFs)φA(x,µ2)offlavorf,atomicweightA,momen- verse area 1/Q2 and longitudinal extent ∆z × x /x,
f N N
2 tum fraction x and factorization scale µ. Although the where ∆z is the nucleon size, x = 1/(2r m ) ∼ 0.1
N N 0 N
v µ-dependence of leading twist nPDFs can be calculated and r ∼ 1.2 fm [5]. When Bjorken x ≪ x the lepton-
0 N
4
in terms of pQCD evolution equations, the x- and A- nucleusDIScoversseveralnucleonsinlongitudinaldirec-
9
dependent boundary condition at a scale µ must be tionwhileitislocalizedinthetransverseplane. Although
0 0
9 fixed from existing measurements, mostly of the struc- a hard interaction involving more than one nucleon is
0 turefunctionFA(x,µ2)inlepton-nucleusdeeplyinelastic suppressed by powers of 1/Q2 [5], it is amplified by the
2
3 scattering (DIS) [2]. It was pointed out recently [3] that nuclear size if x ≪ x . In the collinear factorization
N
0 the available fixed-target data contain significant higher approach we evaluate the nuclear enhanced power cor-
/
h twist effects hindering the extraction of the nuclear par- rections to the structure functions to any power in the
p ton distributions. quantity (ξ2/Q2)(A1/3 −1) and keep ξ2 to the leading
- order in α .
p On the other hand, it has been argued that for physi- s
In terms of collinear QCD factorization, the structure
e cal processes where the effective x is very small and the
h typicalmomentumexchangeofthecollisionQ∼µisnot functions at the lowest order in αs are given by [6]
:
v large the number of soft partons in a nucleus may sat-
1
i urate [4]. Qualitatively, the unknown boundary of this F(LT)(x,Q2) = Q2φ (x,Q2)+O(α ), (1)
X T 2 f f s
novel regime in (x,Q) is where the conventional pertur- Xf
r bative QCD factorization approach should fail [5].
a F(LT)(x,Q2) = O(α ), (2)
L s
In this Letter we present a pQCD calculation of the
resummedpowercorrectionstotheDISnuclearstructure where (LT) indicates the leading twist contribution,
f
functionsanddemonstratetheirimportanceinextracting
runs over the (anti)quark flavors, Qf is their fractioPnal
nPDFs. From our results we quantitatively identify the
charge and φ is the leading twist quark distribution:
characteristic scale of these power corrections ξ2 ≪m2 , f
N
fwoirthQn2u≥clemon2NminacslsusmivNe l=ept0o.9n4-nGucelVeu.sWDeIScoinscalubdoveeththaet φf(x,µ2)=Z d2λπ0 eixλ0hp|ψ¯f(0)2γp++ ψf(λ0)|pi (3)
saturationboundaryandcanbetreatedsystematicallyin
the frameworkof the pQCD factorizationapproachwith in the lightcone A+ = nµA = 0 gauge for hadron mo-
µ
resummed high twist contributions. mentum pµ = p+n¯µ, where n¯µ = [1,0,0⊥] and nµ =
Under the approximation of one-photon exchange, [0,1,0⊥] specify the “+” and “−” lightcone directions,
the lepton-hadron DIS cross section dσ /dxdQ2 ∝ respectively. In Eq. (3) the parameter λ =p+y−.
ℓh 0 0
L Wµν(x,Q2), with Bjorken variable x = Q2/(2p·q) Tocomputethenuclearenhancedpowercorrectionswe
µν
and virtual photon’s invariant mass q2 = −Q2. The choose the A+ = 0 gauge and a frame in which the nu-
leptonic tensor L and hadronic tensor Wµν are de- cleusismovinginthe“+”direction,pµ ≡Pµ/A=p+n¯µ,
µν A
fined in [6]. The hadronic tensor can be expressed in and the exchange virtual photon momentum is qµ =
terms of structure functions based on the polarization −xp+n¯µ+Q2/(2xp+)nµ. In this frame the struck quark
states of the exchange virtual photon: Wµν(x,Q2) = propagates along the “−” direction and interacts with
ǫµνF (x,Q2)+ǫµνF (x,Q2), where ǫµν, ǫµν are given the “remnants” of the nucleus. The leading tree level
T T L L L T
2
γν∗ t=oo γµ∗ Since the effective γL∗ coupling is made of short-distance
contact terms of quark propagators [9], the integrals of
λ˜ and λ˜ in Eq. (5) should be localized.
0 n
To derive the gauge invariant and nuclear enhanced
y− y− powercorrectionsatthe treelevel,weaddgluoninterac-
= y0− i (a) j 0− tions to the struck quark and convertthe gluon field op-
γ∗ γ∗ γ∗ eratorsinthehadronicmatrixelementofWµν tothecor-
T x0 , L x0 + L x0 rbeespaonnedvinengfineulmdbsterrenogftihnt[e5r,a7c]t.ioWnsebnoettweetehnatththeerγe∗mcouust-
~xxx0 x~0 pling and the final-state cut because the quark propaga-
torofmomentumx p+q hasonlytwotypesofterms[9],
y0− (b) y0− yy~0− (c) y0− y~0− i(γ+/2p+)/(xi−x±iiǫ)(poleterm−×→)andi(xp+/Q2)γ−
xi−1 ~xi xi = xi−1 xi l(acroinzteadc.t tCeormnse−q|→ue)n,talyn,dththeesgulmuomnastaiorne torfanthsveercsoehlyerpenot-
multiple scattering can be achieved by sequential inser-
tion of a basic unit consisting of a pair of gluon interac-
yy~− y− (d) y− tions,left-sideofFig.1(d),connectedbyaquarkcontact
i i i
term, and a quark pole term to the left (L) or right (R)
if the unit is to the left or right of the final-state cut.
FIG. 1: (a) Tree level multiple final state scattering of the
struck quark in DIS on nuclei; (b) and (c) leading coupling Integrating over the loop momentum fraction x˜i, we can
for transversely and longitudinally polarized photons to the replace this unit by an effective scalar interaction, right-
activepartons,respectively;(d)effectivescalarvertexforthe side of Fig. 1(d), with a rule,
basic two-gluon interactions.
contributionsto Wµν inlepton-nucleus DIS aregivenby (cid:18)x43πQ2α2s(cid:19)Z d2λπi xeii−(xix−ix−i−11−)λiiǫ γγ−+2γγ−+−x−ii−FiˆF1ˆ2−2(λ(xλi+i))iǫ RL .
the Feynman diagrams in Fig. 1(a). The cut-line rep- 2 xi−x−iǫ
(6)
resents the final state quark and the blob connected to 
γ∗ - the lowest order coupling between the photon and InEq.(6)theboost-invariantoperatorFˆ2(λi)isdifferent
the active partons. For transversely polarized photons from Fˆ2 in Eq. (5) and is defined as
λ0
Fig. 1(b) gives the leading twist contribution in Eq. (1).
∗
For longitudinally polarizedγ the leading tree coupling dλ˜ F+α(λ )F +(λ˜)
Fig. 1(c) results in the first power correction to F : Fˆ2(λ )≡ i i α i θ(λ −λ˜). (7)
L i Z 2π (p+)2 i i
1 4π2α
FL(x,Q2) = F(LT)(x,Q2)+ 2 Q2f4(cid:18) 3Q2s(cid:19) Its expectation value can be related to the small-
Xf x limit of the gluon distribution, hp|Fˆ2(λ )|pi ≈
i
× dλ0 eixλ0hp|ψ¯ (0) γ+ ψ (λ )Fˆ2 |pi, (4) limx→0 21xG(x,Q2), and is independent of λi. It is the
Z 2π f 2p+ f 0 λ0 dλiinEq.(6)thatgivestheA1/3-typenuclearenhance-
Rment[5]. Assumingthattheinteractionleavesthenucle-
where the O(α ) F(LT) is given in [8] and the operator ons in a color singlet state, taking all possible final state
s L
cuts [10] and performing the integrals over the incoming
dλ˜ dλ˜ F+α(λ˜ )F +(λ˜ ) partons’momentum fractionswe obtainthe twist 2+2n
Fˆλ20=Z 2nπ 0 (pn+)2α 0 θ(λ˜n)θ(λ˜0−λ0). (5) contribution to the structure functions:
1 4π2α n dλ (iλ )n γ+ n
δF(n) ≈ Q2 x s 0 eixλ0 0 hP |ψ¯ (0) ψ (λ ) dλ θ(λ )Fˆ2(λ ) |P i, (8)
T 2 f(cid:20) 3Q2 (cid:21) Z 2π n! A f 2p+ f 0 (cid:20)Z i i i (cid:21) A
Xf iY=1
1 4 4π2α n dλ (iλ )n−1 γ+ n−1
δF(n) ≈ Q2 x s 0 eixλ0 0 hP |ψ¯ (0) ψ (λ )Fˆ2 dλ θ(λ )Fˆ2(λ ) |P i, (9)
L 2 f x(cid:20) 3Q2 (cid:21) Z 2π (n−1)! A f 2p+ f 0 λ0 (cid:20)Z i i i (cid:21) A
Xf iY=1
with corrections down by powers of the nuclear size. (9) in a model ofa nucleus of constantlabframe density
ρ(r) = 3/(4πr3) and approximate the expectation value
Weevaluatethemulti-fieldmatrixelementsinEqs.(8), 0
3
oftheproductofoperatorstobeaproductofexpectation 1.1
D) CERN-NA37 CERN-NA37
values of the basic operator units in a nucleon state of F(2 1
momentum p=PA/A: A)/
F(2 0.9 QV
n n
hPA|Oˆ0 iY=1Oˆi|PAi=Ahp|Oˆ0|piiY=1hNphp|Oˆi|pii , 0.8 ξ2 = 0.09 - 0.12 Ge4VH2e 6Li
withthenormalizationNp =3/(8πr03mN). Theintegrals ∆D-T 0.01 QV
dλiθ(λi) = (3r0mN/4)(A1/3 −1) are taken such that EKS89
-0.1
tRhe nuclear effect vanishes for A = 1. Resumming the
A1/3-enhanced power corrections Eqs. (8), (9) we find: D) FNAL-E665
F(2 1 CERN-NA37
/
FA(x,Q2) ≈ N A ξ2(A1/3−1) nxn dnFT(LT)(x,Q2) F(A)2 0.9
T n!(cid:20) Q2 (cid:21) dnx 0.8
nX=0 FNAL-E665
xξ2(A1/3−1) 0.7 CERN-NA37 12C 40Ca
≈ AF(LT) x+ ,Q2 , (10)
T (cid:18) Q2 (cid:19) D-T 0.1
∆
N A 4ξ2 0
FA(x,Q2) ≈ AF(LT)(x,Q2)+ -0.1
L L n!(cid:18)Q2 (cid:19)
× (cid:20)ξ2(A1Q/32−1)(cid:21)nnX=x0n dnFT(LdTn)(xx,Q2) F(A)/F(D)22 00..981 FFNNAALL--EE666655 FNAL-E665
≈ AF(LT)(x,Q2)+ 4ξ2 FA(x,Q2), (11) 0.7
L Q2 T 0.6 131Xe 208Pb
where N is the upper limit on the number of quark- 0.2
0.1
nucleon interactions and ξ2 represents the characteristic D-T 0
∆ -0.1
scale of quark-initiated power corrections to the leading
-0.2
order in αs 0.0001 0.001 0.01 0.1 0.001 0.01 0.1
x x
3πα (Q2) B B
ξ2 = s hp|Fˆ2(λ )|pi.
8r02 i FIG. 2: All-twist resummed F2(A)/F2(D) calculation from
Eqs.(10),(11)versusCERN-NA37[14]andFNAL-E665[13]
In deriving Eqs. (10), (11) we have taken hp|Fˆ2 |pi ≈ data on DIS on nuclei. The band corresponds to the choice
(3r m /4)hp|Fˆ2(λ )|pi and N ≈ ∞ because thλe0 effec- ξ2 = 0.09−0.12 GeV2. Data-Theory, where ∆D−T is com-
0 N i putedfor theset presented bycircles, also shows comparison
tive value of ξ2 is relatively small, as shown below.
totheEKS98scale-dependentshadowingparametrization[2].
Eqs. (10), (11) are the main result of this Letter. Im-
portant applications to other QCD processes and ob-
servables that naturally follow from this new approach
overestimatedthe shiftinthe regionxclosetox where
are given in [11]. The overall factor A takes into ac- N
the N ≈ ∞ should fail [11]. In Fig. 2, we impose
count the leading dependence on the atomic weight and
Q2 =m2 for virtualities smaller than the nucleon mass,
the isospin averageover the protons and neutrons in the N
nucleus is implicit. We emphasize the simplicity of the below which high order corrections in αs(Q) need to
be included and the conventional factorization approach
end result, which amounts to a shift of the Bjorken x
by ∆x = xξ2(A1/3 − 1)/Q2 with only one parameter might not be valid. Our result is comparable to the
ξ2 ∝limx→0xG(x,Q2). In the following numerical eval- EKS98 scale-dependent parametrization [2] of existing
uation we use the lowest order CTEQ6 PDFs [12]. dataonthenuclearmodificationtoF2A(x,Q2),asseenin
Fig. 2 shows a point by point in (x,Q2) calculation the∆D−T =Data−TheorypanelsofFig.2. Weempha-
of the process dependent modification to F (A)/F (D) size,however,thatthephysicalinterpretationisdifferent:
2 2
in [2] the effect is attributed to the modification of the
(per nucleon) in the shadowing x<0.1 region compared
to NA37 and E665 data [13, 14]. We find that a value input parton distributions at µ0 =1.5 GeV in a nucleus
of ξ2 = 0.09−0.12 GeV2, which is compatible with the and its subsequent leading twist scale dependence. In
rangefrompreviousanalysis[15] ofDrell-Yantransverse contrast, our resummed QCD power corrections to the
momentum broadening (ξ2 ∼ 0.04 GeV2) and momen- structure functions systematically cover higher twist for
tumimbalanceindijetphotoproduction(ξ2 ∼0.2GeV2), all values of Q≥µ0.
makes our calculations consistent with the both x- and With ξ2 fixed, Fig. 3 shows the predicted Q2 depen-
A-dependence of the data. Our calculations might have dence of F (Sn)/F (C). The Q2 behavior of our result,
2 2
4
C) 1.1 showcomparisontotheEKS98parameterization[2]. The
F(2 Ax=rm0.0es1r2o5 et al. Ax=rm0.0es1t7o5 et al. QCDpowercorrectionsatsmallandmoderateQ2 tothe
n)/ 1 QV leading twist structure functions can also be tested via
S
F(2 theratioofthecrosssectionsforlongitudinallyandtrans-
versely polarized virtual photons
0.9
CERN-NA37 CERN-NA37
σ FA(x,Q2)
D-T 0.1 EKS98 QV ξ2 = 0.09 - 0.12 GeV2 R(x,Q2;A)= σL = FLA(x,Q2) . (12)
∆ T T
0
-0.1
ComparisontoE143datafor12C[18]isgiveninthebot-
C) 3 10 30
F(2 x=0.035 Q2 [GeV2] tomrightpanelofFig.3. Thedashedcurveincludesonly
F(Sn)/2 1 R0. 8= σL /σT SLAC-E143 tahnedbisreimnssesntsriathivluentgo amnoddigfilcuaotnionspsliotftitnhgeFnuL(LcTle)arfrpomart[o8n]
distributions. The gray band represents a calculation of
0.9
0.6 R, FL(LT) R from Eqs. (10), (11).
R, F
CERN-NA37 L Inconclusion,forξ2 =0.09−0.12GeV2 ourcalculated
D-T 0.1 0.4 and resummed power corrections are consistent with the
∆ 0 0.2 x-, Q2- and A-dependence of existing data on the small-
-0.1 12C x nuclear structure functions without any leading twist
1 3 10 30 0 0.04 0.08 0.12 shadowing. Ourresults,therefore,giveanupper limitfor
Q2 [GeV2] xB the characteristic scale ξ2 of these power corrections in
DIS on cold nuclear matter. Furthermore, the enhance-
FIG. 3: CERN-NA37 data [17] on F2(Sn)/F2(C) show evi- ment of the longitudinal structure function FLA(x,Q2)
dence for a power-law in 1/Q2 behavior consistent with the strongly favors a critical role for the higher twist in the
all-twistresummedcalculation. ComparisontoGribov-Regge presentlyaccessible(smallx<0.1,Q2 ∼fewGeV2)kine-
model by Armesto et al. [16] and the EKS98 shadowing matic regime. Less leading twist shadowing and corre-
parametrization [2](in∆D−T)isalsopresented. Thebottom spondinglylessantishadowingthancurrentlyanticipated
right panel illustrates the role of higher twist contributions
will have an important impact on the interpretation of
to FL on the example of R = σL/σT. Data is from SLAC-
the d + A and Au + Au data from RHIC [19]. The
E143 [18].
predictions of this systematic approach to the process-
dependentlowQ2nuclearmodificationinQCDprocesses
will also be soon confronted by copious new data from
distinctly different from a Gribov-Regge model calcu-
lation by Armesto et al. [16], highlights the important RHIC, EIC and the LHC.
role of resummed QCD power corrections, when pre- This work is supported in partby the US Department
sentedversusNA37data[17]. Data-Theorypanelsagain of Energy under Grant No. DE-FG02-87ER40371.
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