Deep inelastic scattering as x -> 1 using soft-collinear effective theory

0.39 MB·English

by  Aneesh V. Manohar

Deep inelastic scattering as x 1 using soft-collinear effective theory
→
Aneesh V. Manohar1
1Department of Physics, University of California at San Diego, La Jolla, CA 92093
(Dated: September2003)
Soft-collinear effective theory (SCET) is used to sum Sudakov double-logarithms in the x → 1
endpointregionforthedeepinelasticscatteringstructurefunction. Thecalculationsaredoneinboth
the target rest frame and the Breit frame. The separation of scales in the effective theory implies
that the anomalous dimension of the SCET current is linear in lnµ, and the anomalous dimension
for theNth moment of thestructurefunction is linear in lnN, toall orders in perturbation theory.
TheSCETformulationisshowntobefreeofLandaupolesingularities. Someimportantdifferences
between the deep inelastic structurefunction and the shapefunction in B decay are discussed.
4
I. INTRODUCTION In this paper, Sudakov double logarithms in the end-
0
point region are calculated using soft-collinear effective
0
2 The deep inelastic scattering cross-section is the in- theory (SCET) [3, 4]. SCET allows one to compute the
clusive cross-section for lepton scattering off a hadronic cross-section in the endpoint region in a systematic ex-
n
a target at large momentum transfer. The cross-section is pansion to any desired accuracy. The results are free
J conventionally written in terms of structure functions of of Landau pole singularities. The calculations are de-
6 the momentum transfer Q2 and a dimensionless variable scribed in detail in both the target rest frame and the
0 x 1. Thestructurefunction(s)F(x,Q2)cannotbe Breit frame, and give an instructive example of the use
2 co≤mpu≤tedin perturbationtheory,but itsQ2 dependence of SCET. There are several unusual aspects of SCET
v can. The structure function contains large logarithms which are discussed here; the frame dependence of the
6 ofthe form α lnQ2/Λ2 n,whichcanbe summedby wayinwhichinfrareddivergencescancelbetweenthesoft
7 s QCD and collinear modes, and the relation between structure
1 evolvingF(x,Q2)fromthelargescaleQ2 toalowerscale
(cid:0) (cid:1) functions and local operators which is different for deep
9 µusingtherenormalizationgroupequations. µischosen
inelastic scattering and B decays. Consistency of the ef-
0 to be a few GeV, parametrically of the order of Λ ,
QCD
3 but still large enough that perturbation theory is valid. fective theory implies that the SCET anomalous dimen-
0 sion is linear in lnµ, which leads to the form Eq. (2) of
As x 1, there are additional large logarithms that
/ → theperturbationseries. TheSCETcalculationtotheac-
h need to be summed to get a reliable evaluation of the
curacypresentedheregivesthemomentsofthestructure
p scattering cross-section. The invariant mass of the final
- hadronic state is function including the first two exponentiated series f0,1
p in Eq. (2), as well as all terms of order α which do not
e Q2(1 x) s
h MX2 = x− , (1) vBanishXaγs Nin R→ef∞. [3.].TDheeempeitnheoladstuicsesdcaptaterrailnleglssttrhuacttuforer
: s
v and M2 0 as x 1. The total cross-section is in- fun→ctions in the Breit frame (but not in the endpoint re-
Xi frared fiXni→te, even th→ough the real and virtual emission gion) were considered in Ref. [5]. The SCET formalism
r processes are separately infrared divergent. In the re- used in this paper is described in Refs. [4].
a gion x 1, real gluon emission is suppressed, and the
→
cancellation between real and virtual emission becomes
more delicate, leading to large corrections to the cross-
II. OUTLINE OF CALCULATION
section. The formofthe perturbationseriesis mostcon-
veniently described in moment space, where x 1 cor-
→
responds to large moments, N , with the heuristic The calculation of the deep inelastic scattering cross-
→ ∞
rule 1 x 1/N. As N , the structure func- section will be performed using a sequence of effective
tion m−omen∼ts contain terms→of ∞the form αrlnsN with field theories. The scattering amplitude involves the in-
s
s 2r. These Sudakov double-logarithms are important teraction of a lepton beam with a hadron target via a
≤
in the endpoint region. The summation of these terms virtual photon. The leptonic interactions are calculable
is well-known, and has been discussed extensively in the usingQED,andwillnotbediscussedhere. Thequantity
literature[1]. ThegeneralresultisthattheNth moment of interest is the interaction of the virtual photon with
of the structure function at Q2, FN(Q2) can be written the hadronic target.
as [1, 2] At scales much largerthan Q2, the interaction of pho-
lnF (Q2) = f (α lnN)lnN +f (α lnN) tons with hadrons is described using the electromag-
N 0 s 1 s
netic current of quarks interacting via the full QCD La-
+α f (α lnN)+... . (2)
s 2 s grangian. The hadronic scattering amplitude is the ma-
Theexponentialoff (α lnN)lnN givestheleadingSu- trix element of the electromagnetic current between the
0 s
dakov double-logarithmic series. initial and final hadronic states (see Fig. 1),
2
SCET current is run from µ = Q to this scale using the
k', E' anomalous dimension, which is computed in Sec. V in
both the target and Breit frames.
k, E At the scale Q2(1 x) Q2λ, p2 is treated as large,
− ∼ X
q = k - k' and so the final state can be integrated out. The time-
ordered product of two SCET currents can be replaced
by a bilocal light-cone operator whose targetmatrix ele-
X
(cid:0) mentisthepartondistributionfunction. Thisisdoneby
integrating out the ξ field in SCET; the resulting oper-
(cid:0) n
p atoriswrittenintermsofultrasoftquarkfieldsψu inthe
target frame, and in terms of n¯-collinear quark fields in
the Breit frame. The bilocal operator is closely related
FIG. 1: The deep inelastic scattering process. The incom- to the Collins-Soper operator [6]. The matching at scale
ing lepton with energy E scatters off a hadronic target with Q2λ is discussed in Sec. VI
momentum p to produce thefinal hadronicstate X.
Thelaststep,discussedinSec.VII,istorunthebilocal
operatorsfromQ2(1 x)tosomelowscaleµ,andmatch
−
Aµ = X jµ P . (3) onto local twist-two operators which give the moment
h | | i sum rules for the deep inelastic structure functions.
The cross-section involves the square of the scattering The computations in this paper are given in Feynman
amplitude,andcanbewrittenintermsoftheproductof gauge. The results, however, are gauge invariant, and
two currents, summed over intermediate states, valid in any gauge. In the effective theory, one has sep-
arate gauge invariance for the ultrasoft, n-collinear and
P jµ X X jν P . (4) n¯-collinear gluons. This has been checked by explicit
h | | ih | | i
X computation.
X
The entire analysis is presented for QCD with a sin-
Theinvariantmassofthe finalhadronicstate forgeneric
gle quark flavor of unit charge, to avoid unnecessary in-
values of x is of order Q2. The final hadronic state X
dices. The final answer is given by summing the results
can be integrated out at the scale Q2, and the product
of this paper over all flavorsweighted with the square of
of currents in Eq. (4) is replaced by a sum over local
theirelectromagneticcharges. Inadditiontothecurrents
twist-twooperatorswhenonetakesmomentsofthecross- ξ¯ γµψ andξ¯ γµξ ,one alsohas the hermitianconjugate
section. This is the conventional method of computing cnurrents ψ¯γµnξ an¯nd ξ¯ γµξ , which give the antiquark
the deep inelastic scattering cross-section. n n¯ n
contribution, or equivalently, the crossed-graph contri-
Asx 1,theinvariantmassofthefinalhadronicstate
→ butions. Bychargeconjugationinvariance,the matching
tends to zero. The invariant mass of the final hadronic
coefficients and anomalous dimensions are the same for
state is taken to be of order Q2λ, with Q2 Q2λ
Λ2 , and all results are computed in an ex≫pansion≫in quarks and antiquarks. The final result thus has a sum
QCD over both quark and antiquark distributions.
λ. More details about this power counting scheme are
giveninSec.IIIC. The scaleQ2λis aninfraredscalefor
thetheoryatQ,andcanbesettozeroatleadingorderin
III. KINEMATICS
anexpansioninλ. Inthislimit,thefinalhadronicstateis
massless,andcannotbeintegratedout. Instead,thefinal
Thescatteringprocessise +p e +X. Theproton
hadronic state can be treated as a massless light-like jet, − → −
momentum is P, the incoming momentum of the virtual
and is described by a collinear quark in SCET. Thus for
photon is q, the momentum transfer is Q2 = q2 0,
x 1, the QCD current at scale Q is matched onto an − ≫
→ and x is defined by
SCETcurrentatscaleQ. Thecoordinateaxesarechosen
so that the outgoingquark travels in the nµ =(1,0,0,1) q2 Q2
x= = . (5)
direction, and is described by the collinear field ξ of
n −2P q 2P q
SCET. In the target rest frame, the incoming quark has · ·
Thecoordinateaxesarechosensothatthevirtualphoton
momentumoforderthehadrontargetmomentumΛ ,
QCD
is in the z direction. It is useful to introduce the null
and is an ultrasoft quark. In the Breit frame, the struck
vectorsnµ =(1,0,0,1)andn¯µ =(1,0,0, 1)whichpoint
quark is back-scattered, so the incoming quark is a n¯
−
in the ˆz and ˆz direction, respectively. They satisfy the
collinear quark, and is described by the field ξ , where
n¯ relations n2 =−0, n¯2 = 0, n¯ n = 2. Any four-vector aµ
n¯ = (1,0,0, 1). The first step of the calculation is the
·
matching of−the electromagnetic current ψ¯γµψ onto the can be written as
SCET current ξ¯ γµψ or ξ¯ γµξ in the target or Breit 1 1
n n n¯ aµ = a+n¯µ+ a nµ+aµ, (6)
frame,respectively. Thematchingcoefficientisevaluated 2 2 −
⊥
in Sec. IV.
where
The next scale in the problem is the invariant mass
of the hadronic state, p2 Q2λ Q2(1 x). The a+ n a , a n¯ a , (7)
X ∼ ∼ − ≡ · − ≡ ·
3
and a is in the x y plane. The dot product of two so that
four-v⊥ectors is −
l+
1 x = , (14)
1 1 − Q
a b = a+b + a b+ a b , (8)
− −
· 2 2 − ⊥· ⊥
and
and the integrationmeasurecan be written in light-cone
p2 = Ql+. (15)
coordinates as X
1 The components of momentum in the Breit frame
ddk = dk+ dk− dd−2k . (9) aregive±nbymultiplyingthe componentsofmomentum
2 ⊥ ±
in the target rest frame by (Q/xP+)± (Q/P+)± for
∼
x 1.
≈
A. Target Rest Frame
C. Power Counting
In the target rest frame, q = 0, and Q2 = q2 =
q+q . The deepinelasticlim⊥it Q2 withx fi−xedis
−
g−Tihveenn by taking q− → ∞ at fixed q+→, s∞o that q− ≫ q+. Q2T≫heQen2dλp≫oinΛt2QreCgDio.nInistph2Xis ∼regQio2n(,1t−hexfi)n∼alQst2aλt,ewsittihll
involves a sum overmany hadronic states, but has small
q+q q+ invariant mass and is jet-like. The dimensionless power
−
x = , (10)
−P+q +P q+ ≈−P+ countingparameterλ 1isintroducedastheexpansion
− − ≪
parameter, and 1 x λ.
− ∼
where P+ =M is the target mass, so that The power counting is simplest in the Breit frame,
T
where ℓ+ Qλ and ℓ Λ , so that P+ Q,
− QCD
q+ = −Qx2P+ , PTh−er∼e aΛreQCt∼hDr,eep+Xim∼porQtaλn,tp∼s−Xca∼les:Q(,1q)+Q∼2, tQh,eqin−va∼∼riaQnt.
q− = , massofthevirtualphoton,(2)Q2λ,theinvariantmassof
xP+
the final state hadronic jet, and (3) M2 Λ2 , the in-
p+ = P+(1 x) , T ∼ QCD
X − variantmassofthe target. ThescaleQ2λ2 doesnotplay
p−X = P−+q− ∼q− , animportantrole in deep inelastic scattering;it does for
1 x the shape function in B decays [7].
p2 = Q2 − , (11)
X x Particleswith p− Q, p+ Qλ2 and p Qλ travel
inthe n direction,an∼darede∼scribedbyn-c⊥ol∼linearfields
wherepX =P+q isthemomentumofthefinalhadronic ξ+ (x) in SCET. The large components of momen-
state. n,p−,p⊥
tum, p and p are explicit labels on the field, and mo-
−
mentum of ord⊥er Qλ2 is the Fourier transformof the co-
ordinate x. This is analogous to use of label-momentum
B. Breit Frame
fornon-relativisticquarksinNRQCD[8]. Similarly,par-
ticles with momenta p+ Q, p Qλ2 and p Qλ
−
The virtual photon carries only momentum, and no travelinthen¯ direction,a∼ndarede∼scribedbyn¯-c⊥ol∼linear
energy in the Breit frame. The Breit frame is obtained fields ξ+ . Particleswith momentaof orderQλ2 are
n¯,p+,p⊥
from the target rest frame by boosting along the z-axis, describedbyultrasoftfields. Inthe Breitframe,the out-
so that the proton and virtual photon have no com- going quark is described by a n-collinear field, and the
⊥
ponent of momentum in either frame. The momentum incoming quark is described by a n¯-collinear field. The
components in the Breit frame are choice of coordinate axes is such that the components
⊥
of label momentum are zero. The fields will be referred
q+ = Q, to as ξ , ξ for simplicity.
− n n¯
q− = Q, TheBreitframeisthenaturalframetousetodescribe
p+ = Q+l+, deepinelasticscatteringnearx=1. Thepowercounting
automatically implies that 1 x 0, by Eq. (14). Nev-
p− = l−, ertheless,itisinstructivetoa−lsog→iveresultsinthetarget
p+ = l+, rest frame. The target frame is the best frame to com-
X
p−X = Q+l−, (12) paredeepinelasticscatteringwithB →Xsγ. Thetarget
rest frame is also the natural frame to use for generic
where l are fixed by setting P2 Ql =M and values of x, and is a x-independent frame. The boost
± − T
≈ to the Breit frame depends on x, though for x 1, the
≈
Q Q boost factor is approximately constant. In the target
x = , (13)
Q+l+ M /Q ≈ Q+l+ rest frame, the incoming quark has momentum of order
T
−
4
additional simplification — on-shell graphs in the effec-
p tive theory are usually scaleless integrals, which vanish
2
in pure dimensional regularization,and so have no finite
p part [9]. This eliminates the need to compute the effec-
1
tive theory graphs to determine the matching condition,
whichisgivenbythefinitepartofthefulltheorygraphs.
We will compute Eq. (16) keeping the 1/ǫ terms to
FIG. 2: One-loop vertex correction to the electromagnetic
compare with the matching computation using an in-
current in QCD.
frared regulator. The incoming and outgoing quarks in
Fig. 2 have invariant masses that vanish in the limit
Λ , and is described by an ultrasoft field. The outgo- λ 0, so the matching coefficient is obtained by evalu-
pin−XQgC∼qDuaQr2k/hΛaQsCmD,omp⊥Xen∼tumQλc.omTphoeneonuttsgopi+Xng∼qλuaΛrQkCtDu/rQns, tahti→enignttehgeraglrainphd=on4-s−he2llǫwdiitmhepn21sio=nsp22giv=es0. Evaluating
into a jet moving in the n direction, and so the outgoing
particle can be described by a n-collinear field. V = αsC γµ 1 2 2lnQµ22 +4
4π F "ǫUV − ǫ2IR − ǫIR
IV. MATCHING THE CURRENT AT Q2 FROM µ2 µ2 π2
ln2 3ln 8+ , (17)
QCD TO SCET − Q2 − Q2 − 6 #
The electromagnetic current in QCD is matched onto where we have distinguished the infrared and ultraviolet
theSCETcurrentatthescaleQ. TheQCDcurrentisthe divergences by the subscript on ǫ. However, it is impor-
operator ψ¯γµψ, and the SCET current is ξ¯nWnγµWn¯†ξn¯ tant to keep in mind that all ǫ’s are equal. The integral
intheBreitframe,andξ¯ W γµψinthetargetrestframe. has a 1/ǫ2 infrared divergence arising from a combina-
n n IR
Here W are collinear Wilson lines which are required tion of soft and collinear divergences. It is this double
n,n¯
by collinear gauge invariance. The matching condition divergence that leads to the Sudakov double-logarithmic
will be computed in (a) pure dimensional regularization, behavior in the endpoint region. In pure dimensional
i.e.usingdimensionalregularizationtoregulateboththe regularization,the wavefunction graphs are scaleless,
ultravioletandinfrareddivergences,and(b)byusingdi-
α 1 1
mensional regularization for the ultraviolet divergences I = sC i/p , (18)
w F
and off-shellness for the infrared divergences. 4π (cid:20)ǫUV − ǫIR(cid:21)
Theone-loopvertexgraphfortheelectromagneticcur-
and vanish. The net on-shell matrix element of the elec-
rentinQCDisshowninFig.2, wherep isthe incoming
1 tromagneticcurrentin the full theoryis the difference of
quarkmomentum, andp =p +q is the outgoingquark
2 1 Eqs. (17,18) plus the counterterms, which gives (includ-
momentum. The QCD one-loop graph gives
ing the tree-graph)
ddk /p k/ /p k/ 1
V = −ig2CFµ2ǫZ (2π)dγα(k2−−p2)2γµ(k1−−p1)2γαk(126,) hp2|jµ|p1i = γµ"1+ 4απsCF −ǫ22IR − 2lnǫQµI2R2 +3
µ2 µ2 π2
where CF = 4/3 is the Casimir of the fundamental rep- ln2 3ln 8+ +c.t. .
resentation. − Q2 − Q2 − 6 !#
We firstconsiderthe computationinpuredimensional (19)
regularization, which greatly simplifies the computation
ofmatching conditions in effective field theories. In pure The1/ǫ termscancel,sothereisnocounterterm. This
UV
dimensional regularization, the matching coefficient is cancellationisrequired,sincetheelectromagneticcurrent
obtained by computing the finite parts of on-shell di- isaconservedcurrentandhasnoanomalousdimensionin
agrams and dropping all the 1/ǫ terms, regardless of QCD.Thegraphsinthe effectivetheoryareallscaleless,
whether they arise from ultraviolet or infrared diver- and vanish in dimensional regularization. The matching
gences [9]. The reason this procedure works is that the coefficient of the current in the effective theory is the
ultraviolet divergences in the full and effective theories finite part of Eq. (19),
are canceled by the counterterms in the respective theo-
ries. The remaining 1/ǫ terms are infrared divergences, C(µ) = 1+ αs(µ)C ln2 µ2 3ln µ2 8+ π2 .
which must agree between the full and effective theory. 4π F − Q2 − Q2 − 6
(cid:20) (cid:21)
The1/ǫtermscancelinthematchingcondition,whichis (20)
thedifferencebetweenthefullandeffectivetheory. Thus
thematchingconditionisthedifferenceofthefiniteparts The1/ǫ termsinEq.(19),whicharethenegativeofthe
IR
ofthefullandeffectivetheorycomputation. Thereisone 1/ǫ terms in the effective theory, give the anomalous
UV
5
dimension of the current in the effective theory, as we V. ANOMALOUS DIMENSION OF THE SCET
will see in the next section. CURRENT
The logarithms in the matching coefficient C(µ) can
be minimized by choosing the matching scale µ = Q, at TheelectromagneticcurrentinQCDismatchedatthe
which scale Q onto the SCET current. In the Breit frame, the
current is
α (Q) π2
C(Q) = 1+ s4π CF −8+ 6 . (21) jµ = C(Q) ξ¯n,QWnγµWn¯†ξn¯,Q, (25)
(cid:20) (cid:21)
where the n-collinear quark has label momentum n¯ p=
·
The matching computation can be repeated by reg- Q, p = 0, and the n¯-collinear quark has label momen-
ulating the infrared divergence by using off-shell initial tum⊥n p=Q, p =0. C(Q) is the matching coefficient
and final states, with p2 = p2 = 0. The graph in Fig. 2 comput·ed in Eq.⊥(21).
1 2 6
gives In the target rest frame, the current is
α 1 Q2 p2 p2 jµ = C(Q) ξ¯n,q−Wnγµψu, (26)
V = sC γµ ln 2ln 1 ln 2
4π F "ǫUV − µ2 − Q2 Q2 where ψu is an ultrasoft quark, Wn is a collinear Wilson
line, and ξn,q− is a n-collinear quark field with labels
p2 p2 2π2 n¯ p=q , p =0.
2ln 1 2ln 2 , (22) · − ⊥
− Q2 − Q2 − 3 #
A. Breit Frame
where the 1/ǫ term is purely an ultraviolet divergence,
sincetheinfrareddivergenceshavebeenregulatedbythe
The one-loop anomalous dimension of the SCET cur-
off-shellness. The evaluation of Eq. (22) is considerably
rent in the Breit frame is given by the graphs in Figs. 3,
more complicated than that of Eq. (17). The wavefunc-
as well as wavefunction renormalization graphs.
tion graph is
The ultrasoft graph, Fig. 3(a), gives
αs 1 p2 ddk 1
Iw = 4πCF i/p(cid:20)ǫUV +1−ln−µ2 (cid:21), (23) Is = −ig2CFµ2ǫZ (2π)d nαn·(p2−k)+i0+γµ
1 1
so that the matrix element in the full theory including n¯ . (27)
×n¯ (p k)+i0+ αk2+i0+
the tree-graph is · 1−
Doing the k+ integral by contours and using the substi-
p jµ p = γµ 1+C αs lnQ2 2ln p21 ln p22 tution k− =xp−1 gives
h 2| | 1i " p2 F4π −p2 µ12 − p2Q2 Q2 Is = −g2CFµ2ǫ ∞ d2πx (d2dπ−)2dk⊥2
2ln 1 2ln 2 + ln− 1 Z0 Z −
− Q2 − Q2 2 µ2 γµ
+1ln−p22 1 2π2 +c.t. . (24) ×[1−x+i0+] p+2p−1x−k2⊥+i0+
2 µ2 − − 3 !# = g2 C γµµ2ǫ (cid:2)∞dxΓ(ǫ) −p+2p−1x(cid:3)−ǫ
8π2 F [1 x+i0+]
Z0 −(cid:2) (cid:3)
The ultraviolet counterterm vanishes as before, as it
must, since it does not depend on the choice of infrared = −8gπ22CFγµµ2ǫΓ(ǫ) p+2p−1 −ǫπcscǫπ
regulator. The matching condition is given by subtract-
ingfromEq.(24)thematrixelementintheeffectivethe- = g2 C 1 1ln(cid:2)p+2p−1(cid:3)+ 1ln2 p+2p−1 + π2 .
ory. Theeffectivetheoryintegralsarenolongerscaleless, −8π2 F ǫ2 − ǫ µ2 2 µ2 4
(cid:20) (cid:21)
sincetheydependonp2,andmustbeevaluatedtoobtain (28)
i
thematchingconditionifanoff-shellnessisusedtoregu-
The n-collinear gluon graph Fig. 3(b) gives
latedthe infrareddivergence. The matrixelement inthe
effective theory with an off-shellness is given in the next ddk /n¯nα /nn¯ (p k)
I = ig2C µ2ǫ · 2− γµ
section, where the anomalous dimension of the SCET n − F (2π)d 2 2(p k)2+i0+
current is computed. Taking the difference of the result, Z 2−
1 1
Eq. (34), and Eq. (24) gives the same matching condi- n¯
× n¯ k+i0+ αk2+i0+
tion as before, Eq. (20). The computation of Eq. (20) is − ·
clearly simpler using dimensional regularizationto regu- = 2ig2C µ2ǫ ddk n¯·(p2−k) γµ
late the infrareddivergence,since it does not requirethe − F (2π)d(p k)2+i0+
Z 2−
effective theory computation at this stage, leaving it to 1 1
. (29)
the next section where it properly belongs. × n¯ k+i0+k2+i0+
− ·
6
p p p
2 2 2
p p p
1 1 1
(a) (b) (c)
FIG.3: OneloopcorrectiontotheelectromagneticvertexintheBreitframefrom(a)ultrasoft,(b)n-collinearand(c)n¯-collinear
gluons.
Evaluatingthek+ integralbycontours,andlettingk− = +2ln−p21 lnQ2 +2ln−p22 lnQ2 2ln−p21 ln−p22
zp−2 gives µ2 µ2 µ2 µ2 − µ2 µ2
3 p2 3 p2 5π2
1 dz dd 2k (1 z) ln− 1 ln− 2 +7 . (33)
In = g2CFµ2ǫγµ 2π (2π−)d ⊥2z[z(1 z−)p2 k2] −2 µ2 − 2 µ2 − 6 i
Z0 Z − − 2− ⊥ Theultravioletdivergenceofthisresultagreewiththeul-
g2 1 (1 z) p2z(1 z) −ǫ traviolet divergence in the effective theory inferred from
= µ2ǫγµ dzΓ(ǫ) − − 2 − Eq. (19). Exactly on-shell, the effective theory integrals
−8π2 z
Z0 (cid:2) (cid:3) arescalelessandvanishbecausethe1/ǫ ultravioletdi-
UV
= −8gπ22CFµ2ǫγµΓ(ǫ)ΓΓ((−2ǫ)Γ2(ǫ2)−ǫ) −p22 −ǫ vtrearvgieonlceetsdcivaenrcgeelntcheein1/tǫhIRe einfffercatrievdedthiveeorrgyenshceosu.ldTthheeurel--
−
= −8gπ22CFγµ −ǫ12 − 1ǫ + 1ǫ ln−µp222 (cid:2) (cid:3) afogrreeebsewthitehntehgeatdiviveeorfgethnece1/inǫIREqt.er(m33s)i.nTEhqe.(S1C9E),Twhcuicrh-
h rent operator is multiplicatively renormalized,and there
1 p2 p2 π2
ln2 − 2 +ln− 2 2+ . (30) is no operator mixing. This allows one to compute the
−2 µ2 µ2 − 12 anomalous dimension of the SCET currentdirectly from
i
the 1/ǫ terms in the full theory matrix element. In
The n¯-collinear gluon graph Fig. 3(c) gives IR
cases with operator mixing, the 1/ǫ terms in the full
IR
theory matrix element gives the value of the anomalous
ddk 1
I = ig2C µ2ǫ nα γµ dimension matrix times the operators coefficients evalu-
n¯ − F (2π)d n k+i0+
ated at the matching scale.
Z − ·
/n¯n (p1 k) /nn¯α 1 The infinite part of the matrix element is canceled by
· − , (31)
×2(p k)2+i0+ 2 k2+i0+ thevertexandwavefunctioncountertermsintheeffective
1
−
theory so that the renormalized matrix element is
which is Eq. (30) with p2 p2,
2 → 1 p jµ p
h 2| | 1iren
g2 1 1 1 p2 α Q2 p2 Q2 p2 Q2
I = C γµ + ln− 1 = sC ln2 +2ln− 1 ln +2ln− 2 ln
n¯ −8π2 F −ǫ2 − ǫ ǫ µ2 4π F − µ2 µ2 µ2 µ2 µ2
h h
1 p2 p2 π2 p2 p2 3 p2 3 p2 5π2
ln2 − 1 +ln− 1 2+ . (32) 2ln− 1 ln− 2 ln− 1 ln− 2 +7 .
−2 µ2 µ2 − 12 − µ2 µ2 − 2 µ2 − 2 µ2 − 6
i i
(34)
The remaining graphs are the wavefunction graphs.
The ultrasoft gluon contribution to wavefunction renor- The infrared divergence as p2 0 in the full theory cal-
malization vanishes in Feynman gauge, since n2 = n¯2 = culation, Eq. (24) agrees witih→the infrared divergence of
0. The collinear wavefunction renormalization graph is the effective theory calculation, Eq. (34), and the differ-
the same as the massless quark wavefunction renormal- encegivesthematchingcondition,Eq.(20),whichisfree
ization in QCD, Eq. (23), since the interaction of n- of infrared divergences, and depends only on Q2, not on
collinear quarks with n-collinear gluons is the same as p2.
i
the interaction of quarks with gluons in full QCD. The ultraviolet counterterm for the ultrasoft graph
The matrix element ofthe currentin the effective the- Fig. 3(a), n-collinear graph Fig. 3(b) and n¯-collinear
oryisgivenbythesumofEqs.(28,30,32)andsubtracting graph Fig. 3(c) and wavefunction graph are from
half the wavefunction renormalization Eq. (23) for each Eqs. (28,30,32,23)
external quark. The net result is
ultrasoft: αsC 2 2lnp+2p−1 ,
p jµ p 4π F ǫ2 − ǫ µ2
h 2| | 1ibare (cid:20) (cid:21)
= 4απsCF ǫ22 + 3ǫ − 2ǫ lnQµ22 −ln2 Qµ22 n-collinear: 4απsCF −ǫ22 − 2ǫ + 2ǫ ln−µp222 ,
(cid:20) (cid:21)
h
7
α 2 2 2 p2
n¯-collinear: sC + ln− 1 ,
4π F −ǫ2 − ǫ ǫ µ2 p p
(cid:20) (cid:21) 2 2
α 1
s
wavefunction: 4πCF ǫ , (35) p1 p1
(cid:20) (cid:21)
respectively. The individual counterterms are sensitive (a) (b)
to the infrared through their dependence on the small
components of momentum, p−1 and p+2. The total coun- FthIeG.ta4r:geOtnreeslotofpracmorerefcrotimon(tao)tuhletrealseocfttroamndag(nbe)ticn-vceorltlienxeainr
terterm is the sum of the four terms above,
gluons.
c.t. = αsC 2 3 2lnp+2p−1µ2 ,
4π F −ǫ2 − ǫ − ǫ p2p2 B. Target Rest Frame
(cid:20) 1 2 (cid:21)
α 2 3 2 µ2
s
= C ln ,
4π F (cid:20)−ǫ2 − ǫ − ǫ p−2p+1 (cid:21) Inthetargetrestframe,theelectromagneticcurrentin
the effective theory contains a n-collinear quark and an
α 2 3 2 µ2
= sC ln , (36) ultrasoft quark. The diagrams in the effective theory in
4π F −ǫ2 − ǫ − ǫ Q2
(cid:20) (cid:21) thetargetrestframearethoseinFig.4andwavefunction
graphs. Notethatthereisnon¯-collineargraph,sincethe
anddependsonlyonthelabelmomentap−2 andp+1 which current Eq. (25) does not contain Wn¯ in the target rest
are both Q. The counterterms Eq. (36) give the anoma- frame.
lous dimension for the coefficient of the current in the The n-collinear graph Fig. 4(b) is the same as in the
effective theory, Breitframe,andisgivenbyEq.(30). Theresultisframe-
independent, since the final expression only depends on
dC(µ) p22, not on the individual components p±2 or the momen-
µ dµ = γ1(µ) C(µ) , tum p1 of the incoming quark. The non-zero wavefunc-
tion renormalization graphs also agree between the two
α (µ) µ2
γ (µ) = C s 4ln +6 . (37) theories. Then-collinearwavefunctionrenormalizationis
1 − F 4π Q2 the same as the corresponding graph in the Breit frame,
(cid:20) (cid:21)
andtheultrasoftwavefunctionrenormalizationoftheul-
TheSCETcurrentanomalousdimensiondependsonlnµ, trasoft quark is the same as the n¯-collinear wavefunc-
because the one-loop diagrams have 1/ǫ2 terms from tionrenormalizationofthen¯ quark,sincebothareequal
combined collinear and soft divergences. Consistency of to wavefunction renormalization of a massless quark in
theeffectivetheoryimpliesthattoallorders,theanoma- QCD. Each wavefunction graphs depends only on p2 for
lous dimensionis at most linear in lnµ, as will be shown a single particle.
in Sec. IX. The remaining graph is the ultrasoft graph Fig. 4(a).
Thecancellationof1/ǫlnp−1 and1/ǫlnp+2 betweenthe Theultrasoftgraphinthetargetrestframemustcontain
soft and collinear graphs might suggest that the coef- the contributions of both the ultrasoft and n¯-collinear
ficients in the two sectors must be the same, i.e. that graphs in the Breit frame. The ultrasoft graph in the
the two contributions must have the same value of α . target rest frame is
s
This suggests that SCET should use two-stage running,
in which all coupling constants are evaluated at a single I = ig2C µ2ǫ ddk nα 1 γµ
scale µ, unlike NRQCD, which requires one-stage run- − F (2π)d n (p k)+i0+
Z · 2−
ning using the velocity renormalization group [8, 10]. In /p k/ 1
NRQCD, there is a cancellation of infrared divergences ×(p 1k)−2+i0+γαk2+i0+
1
between the soft and ultrasoft sectors that naively sug- −
= ig2C µ2ǫ
gests that both should have the same value of α . How- F
s −
ever, this is not the case, and a proper treatment of ddk γµ(/p k/)/n 1
1− .
NRQCD has the soft coupling constant evaluated at the (2π)d[n (p k)+i0+](p k)2+i0+k2+i0+
scale mν and ultrasoft coupling constant evaluated at Z · 2− 1−
(38)
the scale mν2 [8, 10]. It has been pointed out that one-
stageandtwo-stagerunninggivethesameresultinSCET Doing the k+ integral, and using k+ =zp+ gives
for quantities which have been computed so far [11]. In 1
NRQCD, the difference betweenone- andtwo-stagerun- 1 dz dd 2k
ningfirstoccursatorderv2inthepowercounting. SCET I = −g2CFµ2ǫγµ 2π (2π−)d ⊥2
anomalousdimensionshavesofarbeencomputedonlyto Z0 Z −
leading order in λ, and do not distinguish between one- p+(1 z)
1 −
stage and two-stage running. ×(p+ p+z)(z(1 z)p2 k2)
2 − 1 − 1−
⊥
8
= 8gπ22CFµ2ǫγµΓ(ǫ) 1dz(pp+1+(1−p+zz)) −p21z(1−z) −ǫ . q n q
Z0 2 − 1
(cid:2) (cid:3)(39)
p p
The ratio p+/p+ is
2 1
p+ p2 xp2 p2 FIG. 5: Tree graph for the product of two currents in the
2 =1 x= 2 = 2 2 = (λ) . (40) target rest frame.
p+1 − p+1p−2 Q2 ≈ Q2 O
InSCET,termsoforder1−xareofordertheexpansion VI. MATCHING AT Q2(1−x) ONTO THE
parameter λ in the power counting. Equation (39) can
PARTON DISTRIBUTION FUNCTION
be evaluated in the limit p+/p+ 0, which simplifies
2 1 →
the computation, and gives
At the scale Q2(1 x), the invariant mass of the final
−
hadronic state p2 can be treated as large, and the final
g2 1 p2 p+ X
I = C γµ ln− 1 1+ln− 2 state can be integrated out. This is done by integrat-
8π2 F ((cid:20)ǫ − µ2 (cid:21)(cid:18) p+1 (cid:19) ing out the n-collinear modes from the effective theory.
SincethecurrentEq.(25,26)containsξ fieldswhichare
π2 1 p+ n
+ 2 ln2 − 2 . (41) integratedout, onematches the productoftwocurrents,
(cid:18) − 3 − 2 p+1 (cid:19)) ratherthanasinglecurrent,ontotheeffectivetheorybe-
lowQ2(1 x). The matchingcoefficients aredetermined
This resultis identicaltothe sumofthe ultrasoftandn¯- −
by computing the matrix elements of
collineargraphsintheBreitframe,giveninEqs.(28,32).1
The ultrasoft graph in the target rest frame has no 1/ǫ2 1
Wµν = d4xeiqxjµ(x)jν(0), (44)
divergence. Since it is the sum of the ultrasoft and n¯- 2π ·
Z
collinear graphs in the Breit frame, these graphs must
have 1/ǫ2 terms of opposite sign, which agrees with the at fixed x and q2. Note that we have a product of cur-
explicit one-loop computation in Eqs. (28,32). This can- rents,ratherthanatime-orderedproduct. Thematrixel-
cellation is expected to persist at higher orders. ementoftheproductofcurrentsisgivenbythe disconti-
The 1/ǫ term in Eq. (41) depends on the infrared reg- nuityinthematrixelementofthetime-orderedproduct.2
ulator through p2, using Eq. (40). This infrared depen- We will use this procedure, since time-ordered products
2
denceiscanceledbythen-collineargraph,whichcanonly can be computed using conventional Feynman diagram
depend on p2. The analog of Eq. (35) is perturbation theory.
2
α 2 2 p+
ultrasoft: 4πsCF −ǫ − ǫ ln−p+2 , A. Target Rest Frame
(cid:20) 1 (cid:21)
α 2 2 2 p2
n-collinear: sC + ln− 2 , The tree graph for the product of two currents with
4π F (cid:20)−ǫ2 − ǫ ǫ µ2 (cid:21) ξn integrated out is shown in Fig. 5. The spin averaged
α 1 matrix element of the tree level graph is
s
wavefunction: C , (42)
F
4π ǫ
(cid:20) (cid:21) 1 1 i/n n¯ (p+q)
Disc Tr /pγν · γµ
so that the total counterterm is 2π2 2 (p+q)2+i0+
i p+(p +q )
c.t. = 4απsCF −ǫ22 − 3ǫ − 2ǫ lnpp+2+µp22 , = Disc 2πTµν(p+q−)2+i−0+
(cid:20) 1 2(cid:21) Tµνδ 1+q+/p+ , (45)
αs 2 3 2 µ2 ≈
= C ln , (43)
4π F (cid:20)−ǫ2 − ǫ − ǫ p+1p−2 (cid:21) where (cid:0) (cid:1)
pµnν +pνnµ
which is the same as Eq. (36), and leads to the same Tµν = g + . (46)
µν
− n p
anomalous dimension, Eq. (37).
·
The on-shell matrix element in the target rest frame Since p is at rest, pµ vµ =(1,0,0,0)
has the same value as in the Breit frame, and leads to ∝
the same matching condition Eq. (20). Tµν = g +vµnν +vνnµ, (47)
µν
−
1 The components of pi have different values in the two frames, 2 The discontinuity of a diagram containing i0+ terms is defined
but p2i and p+2/p+1 are equal, since the Breit frame is obtained by taking the difference of the diagram, and the diagram with
fromthetargetframebyaboostalongthe z axis. i0+→i0−.
9
Note that y and w in this equation are defined with re-
r n r specttothe partonmomentumpratherthanthehadron
momentum P.
p p
The analysis has been restricted to spin-independent
structure functions for simplicity. It is straightforward
to generalize this to spin-dependent structure functions,
FIG. 6: Quark matrix element of the quark distribution op-
which involve spin-dependent quark distribution opera-
erator.
tors [12].
Theone-loopgraphsforthematrixelementofthecur-
independent of the momentum of the target. rent product are shown in Figs. 7. Graphs 7(a–d) have
Define the quark distribution operator by [6] thesamevalueinthetheoriesaboveandbelowQ2(1 x),
−
sincetheinteractionofanultrasoftgluonwithacollinear
O (r+) = 1 ∞ dξe iξr+ψ¯ (nξ)/nY (nξ,0)ψ (0), quark is the same as the vertex generated by the Wil-
q − u u
4π
Z−∞ sonline Y in the operator Oq, and the external ultrasoft
(48) quark fields are unchanged at the matching scale. The
matching correction is given by 7(e–g); these graphs are
whereY isaneikonalWilsonlinefrom0tonξcontaining
absentbelow Q2(1 x) since the n-collinearmodes have
ultrasoft gluons Au, −
been integrated out.
ξ The collinear graph Fig. 7(e) gives for the spin-
Y (nξ,0) = P exp ig n A (nz) dz (49)
"− Z0 · u # averagedmatrix element
andψ areultrasoftquarkfields. TheFeynmanrulesare g2 ddk 1 /nn¯ (p+q)
u I = Disc Tr /pγν ·
given by taking the discontinuity of the diagram, since c,1 2π (2π)d 2 2(p+q)2
the operator is a product, not a time-ordered product. Z
/n¯nα/nn¯ (p+q k) 1 1
The quark distribution in a target T with momentum P · − γµ n¯
× 2 2(p+q k)2 n¯ k αk2
is defined by [6] − − ·
g2 ddk n¯ (p+q)
f (x) = T,P O xP+ T,P . (50) = Disc Tµνp+ ·
q/T h | q | i 2π (2π)d (p+q)2
Z
The onlydifferencebetweenthe(cid:0)quark(cid:1)distributionoper- n¯ (p+q k) 1 1
ator Eq. (48) and the conventional Collins-Soper defini- · − . (55)
× (p+q k)2 n¯ kk2
tion is the replacement of the full theory quark field by − − ·
the ultrasoft quark field in SCET.
Comparing with Eq. (29) gives
The spin-averaged tree-level matrix element of the
quark distribution operator is (see Fig. 6) i g2 n¯ (p+q)
I = Disc − p+Tµν ·
1 i 1 c,1 2π 8π2 (p+q)2
Disc Tr /n/p (cid:18) (cid:19)
Disc 41πn·(p2−ipr+)2 ×"−ǫ12 − 1ǫ + 1ǫ ln−(pµ+2 q)2 − 12ln2 −(pµ+2 q)2
4πp+ r++i0+
= δ 1 r+/p+− , (51) +ln−(p+q)2 2+ π2
− µ2 − 12#
so the tree level rel(cid:0)ation is (cid:1)
i g2 1
Wµν = TµνO ( q+) . (52) Disc − Tµν
q − ≈ 2π 8π2 (1 y)+i0+
(cid:18) (cid:19) −
This is an operator relation independent of the matrix 1 1 1 Q2(y 1) i0+
element since Tµν in Eq. (47) can be written in a form + ln − −
× −ǫ2 − ǫ ǫ yµ2
independent of the target state. The minus sign in the
h
1 Q2(y 1) i0+ Q2(y 1) i0+
argument of Oq arises because q is an incoming momen- ln2 − − +ln − −
tumandr isanoutgoingmomentum. Equation(52)can −2 yµ2 yµ2
also be written as a convolution π2
Wµν = Tµν dr+δ 1 −q+ O (r+) . (53) −2+ 12#θ(y), (56)
r+ − r+ q
Z (cid:18) (cid:19)
It is convenient to write q+ = yp+, and r+ = wp+ so where y 0 since q+ < 0, and q− = Q2/q+ > 0. The
that Eq. (53) becomes − kinemati≥cregionwhereq− <0isinfini−telyfarawayinthe
effective theory, and is described by the effective theory
dw y
Wµν(q+ = yp+) = Tµν δ 1 O (wp+) . withadifferentvalueofthelabelmomentum. Thegraph
q
− w − w with gluon attached to the other vertex gives the same
Z (cid:16) (cid:17)
(54) contribution, Eq. (56).
10
q n q q n q q n q
p p
(a) (b) (c)
q n q q n q q n q
p p p p p p
(d) (e) (f)
q n q
p p
(g)
FIG.7: Oneloop correction totheelectromagnetic currentproductinthetarget rest frame. Graphs(a),(b)and(e)also have
mirror image graphs where the gluon attaches to theother side.
The collinear graph Fig. 7(f) is given by the tree di- The discontinuity of the remaining terms can be writ-
agram, times the negative of the wavefunction diagram ten in terms of + distributions. The distribution 1/(1
−
Eq. (23) evaluated with p2 (p+q)2, and gives y) is defined by
+
→
Ic,2 = Disc 2iπ −8gπ22 Tµν(pp++(pq−)2++qi−0)+ 1dyf(y)(1 1y) ≡ 1dyf((y1)−yf)(1), (59)
(cid:18) (cid:19) Z0 − + Z0 − +
1 1 1 (p+q)2
+ ln− so that
×"2ǫ 2 − 2 µ2
i g2 i1 1 1
Disc Tµν dy = 0. (60)
≈ 2π (cid:18)−8π2(cid:19) (1−y)+i0+ Z0 (1−y)+
1 1 1 Q2(y 1) i0+ The discontinuity of
+ ln − − θ(y).(57)
× 2ǫ 2 − 2 yµ2 #
h i ln(y 1 iη)
− − θ(y) (61)
The graphFig.7(g) vanishes,since the collineargluon 2π 1 y+iη
−
emission vertex is proportional to n¯α, and n¯2 =0.
The total collinear contribution is the sum of twice is given by the difference of the expression for η 0+
→
Eqs. (55) and Eq. (57), and η 0−,
→
i g2 1 i ln(y 1 iη) 1
I = Disc − Tµν Disc − − θ(y)= θ(0 y <1). (62)
c 2π 8π2 (1 y)+i0+ 2π 1 y+iη 1 y ≤
(cid:18) (cid:19) − − −
2 3 2 Q2(y 1) i0+ Thesingulartermsaty =1canbeobtainedbyintegrat-
+ ln − −
×"−ǫ2 − 2ǫ ǫ yµ2 ing Eq. (59) over 0 y Λ, where Λ>1. Then
≤ ≤
Q2(y 1) i0+ 3 Q2(y 1) i0+
ln2 − − + ln − − i Λ ln(y 1 iη) i
− yµ2 2 yµ2 dy − − = ln2Λ ( iπ)2 ,
2π 1 y+iη −4π − −
7 π2 Z0 − (cid:2) (cid:3)(63)
+ . (58)
−2 6 #
irrespective of the sign of η. This has no discontinuous
The collinear counterterms cancel the 1/ǫ terms, so part, so the integral of the discontinuity is zero. This
that the remaining matching contribution is finite. gives
The collinear counterterm contribution (ln p2)/ǫ =
[lnQ2(y 1)]/ǫ is no longer infrared sensitive−, since the i ln(y 1 iη) 1
scale Q2(−1 y) is now considered a large scale. Disc 2π 1 −y+−iη θ(y)= (1 y) . (64)
− − − +