The NLO twist-3 contribution to the pion electromagnetic form factors in $k_{T}$ factorization

0.45 MB·

by  Shan Cheng

The NLO twist-3 contribution to the pion electromagnetic form factors in k
T
factorization
Shan Cheng, Ying-Ying Fan, and Zhen-Jun Xiao∗
Department of Physics and Institute of Theoretical Physics,
Nanjing Normal University, Nanjing, Jiangsu 210023, People’s Republic of China,
(Dated: January 22,2014)
Inthispaper, byemployingthek factorization theorem,wecalculate firstlythenext-to-
4 T
1 leading-order (NLO) twist-3 contributions to the pion electromagnetic form factors in the
0 πγ∗ π process. From the analytical and numerical calculations we find the following
2 →
points: (a) For the leading order (LO) twist-2, twist-3 and the NLO twist-2 contributions,

n
our results agree very well with those obtained in previous works; (b) We extract out two
a
J factors F(1)(x ,t,Q2)and F(1)(x ,t,Q2),which describe directly theNLOtwist-3 contri-
T3 i T3 i
0
butionstothepionelectromagneticformfactorsF+(Q2);(c)TheNLOtwist-3contribution
2
isnegativeinsignandcancelpartiallywiththeNLOtwist-2part,thetotalNLOcontribution

]
h canthereforeprovidearoughly 20%correctionstothetotalLOcontributionintheconsid-
±
p eredranges ofQ2;and(d)Thetheoretical predictions forQ2F+(Q2)inthelow-Q2 region
-
p agreewellwithcurrentlyavailabledata,thisagreementcanbeimprovedbytheinclusionof
e
theNLOcontributions.
h
[
PACSnumbers:12.38.Bx,12.38.Cy,12.39.St,13.20.He
1
v
8
1
1
I. INTRODUCTION
5
.
1
0 TheperturbativeQCD(pQCD)factorizationapproach,basedonthek factorizationtheorem[1–
T
4 3], have been wildly used to deal with the inclusiveand exclusiveprocesses [4–7]. In the k fac-
1 T
torization theorem, the end-point singularities are removed by the small but non-zero transverse
:
v
momentum k of the parton propagators. For many years, the application of the k factorization
i T T
X theorem were mainly at the leading order (LO) level. But the situation changed a lot recently. In
r Refs. [8–10], the authors calculated the next-to-leading order (NLO) twist-2 contributions to the
a
π transition form factor, π electromagnetic form factor and B π form factor respectively, ob-
→
tainedtheinfraredfinitek dependentNLOhardkernel,andthereforeconfirmedtheapplicability
T
of the k factorization to these exclusive processes at the NLO and the leading twist (twist-2)
T
level. This fact tell us that the k factorization approach can also be applied to the high order
T
contributionsas mentionedin Ref. [11].
In the framework of the pQCD factorization approach, the contributions to the form factors
includefourparts:
(i) The leading order contribution include the leading order twist-2 (LO-T2) contribution and
theleadingordertwist-3(LO-T3)contribution.
(ii) TheNLOcontributioncontainstheNLOtwist-2(NLO-T2)contributionandtheNLOtwist-
3 (NLO-T3)contribution.
∗ [email protected]
2
At present, the first three parts, namely the LO-T2, LO-T3 and NLO-T2 contributions,have been
evaluatedinRefs. [8–10], buttheNLO-T3contributionis stillabsent now.
At leading order level, the LO-T2 part is smaller than the LO-T3 part, by a ratio of 34%
∼
against 66% as shown in Refs. [9, 12, 13]. The NLO-T2 part is around 20 30% of the total
∼ −
leading order contribution (i.e. LO-T2 plus LO-T3 part ) in the low Q2 region. Since the LO-T3
contribution is large, the remaining unknown fourth part, the NLO twist-3 contribution, maybe
rather important, and should be calculated in order to obtain the pQCD predictions for relevant
form factors at the full NLO level, and to demonstrate that the k factorization theorem is an
T
systematicaltool.
In this paper we concentrate on the calculation for the NLO twist-3 contributionto the π elec-
tromagnetic form factor, which corresponds to the scattering process πγ⋆ π. Our work repre-
→
sents the first calculation for the NLO twist-3 contribution to this quantity in the k factorization
T
theorem.
Weknowthatthecollineardivergenceswouldappearwhenthemasslessgluonisemittedfrom
the light external line as the gluon is paralleled to the initial- or the final-state pion which are
massless assumed. The soft divergences would come from the exchange of the massless gluon
betweentwoon-shellexternallines. Inthisworklightpartonsareconsideredtobeoff-shellbyk2
T
to regulated the infrared divergences in both the QCD quark diagrams and the effective diagrams
for pion wave functions. It’s a nontrivial work to verify that the collinear divergences from the
quark-level diagrams offset those from the pion wave functions and the soft divergences cancel
among quark-level diagrams exactly at the twist-3 level as well as at the leading twist-2 case [9].
AsdemonstratedinRefs. [9,10],boththelargedoublelogarithmsα ln2(k )andα ln2(x ),here
s T s i
x being the parton momentum fraction of the anti-quark in the meson wave functions, could
i
be absorbed through the resummation technology. The double logarithm α ln2(k ) would be
s T
absorbed into the π meson wave functions and then been summed to all orders in the coupling
constantα by thek resummation[3]. Thejet functionwould includedwhen thereexisttheend-
s T
pointsingularityinthehardkernel,andthenthedoublelogarithmα ln2(x )wouldbesummedto
s i
allordersbythethresholdresummation[14,15]. Therenormalizationscaleµandthefactorization
scaleµ areintroducedinthehigh-ordercorrectionstotheQCD quarkdiagramsandtheeffective
f
diagrams, respectively. With the appropriate choice of the scale µ and µ , say setting them as the
f
internalhard scaleas postulatedin[9], theNLOcorrection are undercontrol.
This paper is organized as follows. In section. II, we give a brief review about the evaluations
of the LO diagrams for the process πγ∗ π, for both the twist-2 part and twist-3 part. In
→
section. III, O(α2) QCD quark diagrams for the process will be calculated with the inclusion of
s
the twist-3 contributions. The convolutions of O(α )( NLO) effective diagrams for the meson
s
wave functions and O(α )( LO) hard kernel would also be presented in this section, then the
s
k -dependent NLO hard kernel at twist-3 will be obtained. Section. IV contains the numerical
T
analysis. With appropriate choices for the renormalization scale µ, the factorization scale µ and
f
the input meson wave functions, we make the numerical calculations for all four parts of the LO
and NLO contributions to the pion electromagnetic form factor in the πγ∗ π process. Section
→
V containstheconclusions.
3
II. LOTWIST-2ANDTWIST-3CONTRIBUTIONS
The leading order hard kernels of the π electromagnetic form factor as shown in Fig. 1 are
calculated inthissection. Theπγ⋆ π form factors aredefined viathematrixelement
→
< π(p ) Jµ π(p ) > = f (q2)pµ +f (q2)pµ
2 | | 1 1 1 2 2
= F+(q2)(pµ +pµ), (1)
1 2
wherep (p )referstothemomentumoftheinitial(final)statepion,q = p p isthemomentum
1 2 1 2
−
transferred in the weak vertex. Using the same definitions for the leading case as being used in
Ref. [9], themomentump and p are chosen as
1 2
p = (p+,0,0 ), p = (0,p−,0 ), (2)
1 1 T 2 2 T
with q2 = 2p p = Q2. According to the k factorization, the k = (x p+,0,k ) in the
− 1 · 2 − T 1 1 1 1T
initialpionmesonand k = (0,x p−,k ) inthefinal pionmesonas labeledinFig.1, andx and
2 2 2 2T 1
x beingthemomentumfractions. Thefollowhierarchy ispostulatedinthesmall-xregion:
2
Q2 x Q2 x Q2 x x Q2 k2 ,k2 , (3)
≫ 2 ∼ 1 ≫ 1 2 ≫ 1T 2T
ThefollowingFierz identityis employedto factorizethefermionflow.
1 1 1
I I = I I + (γ ) (γ ) + (γα) (γα)
ij lk ik lj 5 ik 5 lj ik lj
4 4 4
1 1
+ (γ γα) (γ γ ) + (σαβγ ) (σ γ ) . (4)
5 ik α 5 lj 5 ik αβ 5 lj
4 8
The identity matrix I here is a 4 dimension matrix, the structure γ γ in Eq. (4) contribute at
α 5
the leading twist(twist-2), while γ and σ γ contribute at twist-3 level. The identity of SU(3)
5 αβ 5 c
group,
1
I I = I I +2(Tc) (Tc) (5)
ij lk lj ik lj ik
N
c
is also employed to factorize the color flow. In Eq. (5), (i,j,l,k) are color index, N = 3 is the
c
number of the colors, and Tc is the Gel-Mann color matrix of SU(3) . The first term in Eq. (5)
c
corresponds to the color-singlet state of the valence quark and the anti-quark, while the second
term willbeassociated withthecolor-octet state.
Wehere consideronlythesubdiagramFig. 1(a) indetail, wherethequark and anti-quark form
acolor-singletstate. Thehard kernels oftheothersubdiagramscan beobtainedby simplykinetic
replacements. ThewavefunctionΦ (p ,x )fortheinitialandfinalstatepioncanbewrittenasthe
π i i
followingform[17–19]
i
Φ (p ,x ) = γ p/ φA(x )+m φP(x ) (n/ n/ 1)φT(x ) , (6)
π 1 1 √2N 5 1 π 1 0 π 1 − + − − π 1
c
i (cid:8) (cid:2) (cid:3)(cid:9)
Φ (p ,x ) = γ p/ φA(x )+m φP(x ) (n/ n/ 1)φT(x ) , (7)
π 2 2 √2N 5 2 π 2 0 π 2 − − + − π 2
c
(cid:8) (cid:2) (cid:3)(cid:9)
where n = (1,0,0 ) and n = (0,1,0 ) denote the unit vector along with the positive and
+ T − T
negativez-axis direction, m = 1.74 GeV is the chiral mass of pion, N is the number of colors,
0 c
φA(x )aretheleadingtwist-2piondistributionamplitude,whileφP(x )andφT(x )arethetwist-3
π i π i π i
piondistributionamplitudes.
4
P1−k1 P2−k2
k1 k2
(a) (b)
(c) (d)
FIG.1. Leading-order quark diagrams forthe πγ⋆ π form factor with representing thevirtual photon
→ •
vertex.
Combiningthedecompositionsin Eq.(4)and Eq.(5), wethencan sandwichFig. 1(a) withthe
structures
1 1
p/ γ , γ p/ , (8)
1 5 5 2
4N 4N
c c
from theinitialand final staterespectively,in orderto obtain thehard kernel H(0) at twist-2level.
Forthederivationofthetwist-3hardkernel,oneshouldsandwichFig.1(a)withthefollowingtwo
setsofstructures
1 1 1 1
γ , γ ; σαβγ , σ γ . (9)
5 5 5 αβ 5
4N 4N 8N 8N
(cid:18) c c (cid:19) (cid:18) c c (cid:19)
ThentheLO twist-3contributionto thehard kernel from Fig.1(a) can bewrittenas[20]
C
H(0)(x ,k ,x ,k ) = ( 2ie )4πα F m2φP(x )
a 1 1T 2 2T − q s16N 0 π 2
c
4pµ φP(x ) φT(x ) 4x pµ φP(x )+φT(x )
− 2 π 1 − π 1 + 1 1 π 1 π 1 , (10)
· (p k )2(k k )2 (p k )2(k k )2
n 2 (cid:2)− 1 1 − 2 (cid:3) 2 −(cid:2) 1 1 − 2 (cid:3)o
whereα inthestrongcouplingconstant,C = 4/3inscolorfactor, e refers tothechargeofthe
s F q
quark interactingwiththeγ∗ inπγ∗ π process.
→
ThecorrespondingLO twist-2contributiontothehard kerneltakes theformof
C 4x pµ
H(0) (x ,k ,x ,k ) = (ie )4πα F Q2φA(x )φA(x ) 1 1 , (11)
a,T2 1 1T 2 2T q s 16N π 2 π 1 · (p k )2(k k )2
c 1 2 1 2
− −
It is easy to see that all parts of the initial state pion, the twist-2 φA(x ) and twist-3 φP(x )
π 1 π 1
and φT(x ), provide contributions at leading order level, but only the φA(x ) and φP(x ) of
π 1 π 2 π 2
the final state pion contribute at LO level, because the contribution from the φT(x ) component
π 2
become zero when it is contracting with the gluon propagator. For the LO twist-3 hard kernel
H(0)(x ,k ,x ,k ), one can see that it contains two lorentz structures: pµ term and x pµ term,
a 1 1T 2 2T 2 1 1
these two terms all should be included in the numerical calculations. For the LO twist-2 hard
kernel H(0) as given in Eq. (11), it depends on one term x pµ only. From previous studies in
a,T2 1 1
Ref.[9, 12, 13], we know that the LO twist-2 part is only about half of the LO twist-3 part. So
one generally expect that the NLO twist-3 contribution maybe large and essential for considered
transitions,whichis alsooneofthemotivationsofthispaper.
5
III. NLOCORRECTIONS
Under the hierarchy as shown in Eq. (3), only those terms which don’t vanish in the limits
of x 0 and k 0 should be kept, this fact does simplify the expressions of the NLO
iT
→ →
contributionsgreatly.
FromthediscussionsattheendofSec.I,weknowthatbothlorentzstructuresx pµ andpµ will
1 1 2
contribute. From the hard kernel H(0)(x ,k ,x ,k ) as given in Eq. (10), we define those two
a 1 1T 2 2T
parts oftheLO twist-3contribution,H(0)(x pµ)and H(0)(pµ)in theform of
a 1 1 a 2
C 4x pµ[φP(x ) φT(x )]
H(0)(x pµ) ( 2ie )4πα F m2φP(x ) 1 1 π 1 − π 1 , (12)
a 1 1 ≡ − q s16N 0 π 2 (p k )2(k k )2
c 2 1 1 2
− −
C 4pµ[φP(x )+φT(x )]
H(0)(pµ) ( 2ie )4πα F m2φP(x )− 2 π 1 π 1 , (13)
a 2 ≡ − q s16N 0 π 2 (p k )2(k k )2
c 2 1 1 2
− −
H(0) = H(0)(x pµ)+H(0)(pµ). (14)
a a 1 1 a 2
For Figs. 1(b,c,d), one can find the corresponding LO twist-3 contributions by simple replace-
ments. Forthesakeofsimplicity,wewillgenerallyomitthesubscript“a”inH(0) inthefollowing
a
sections,unlessstatedspecifically.
A. NLOtwist-3ContributionsoftheQCDQuarkDiagrams
NowwecalculatetheNLOtwist-3contributionstoFig.1(a),whichcomesfromtheself-energy
diagrams, the vertex diagrams, the box and the pentagon diagrams, as illustrated in Figs. 2,3 and
4 respectively. After completing the calculations for Fig. 1(a), we can obtain the results for other
threefigures: Fig. 1(b,c,d), by simplereplacements.
The ultraviolet(UV) divergences are extracted in the dimensional reduction [21] in order to
avoidtheambiguityfromhandingthematrixγ . Theinfrared(IR)divergencesareidentifiedasthe
5
logarithmslnδ , lnδ and theircombinations,as defined in Ref. [9]
1 2
k2 k2 (k k )2
δ = 1T, δ = 2T, δ = − 1 − 2 . (15)
1 Q2 2 Q2 12 Q2
Theself-energycorrectionsobtainedbyevaluatingtheone-loopFeynmandiagramsinFig.2(a-
f)are oftheform
α C 1 4πµ2
G(1) = s F +ln +2 H(0),
2a − 8π ǫ δ Q2eγE
(cid:20) 1 (cid:21)
α C 1 4πµ2
G(1) = s F +ln +2 H(0), (16)
2b − 8π ǫ δ Q2eγE
(cid:20) 1 (cid:21)
α C 1 4πµ2
G(1) = s F +ln +2 H(0),
2c − 8π ǫ δ Q2eγE
(cid:20) 2 (cid:21)
α C 1 4πµ2
G(1) = s F +ln +2 H(0),
2d − 8π ǫ δ Q2eγE
(cid:20) 2 (cid:21)
α C 1 4πµ2
G(1) = s F +ln +2 H(0),
2e − 4π ǫ x Q2eγE
(cid:20) 1 (cid:21)
α 5 2 1 4πµ2
G(1) = s N N +ln H(0), (17)
2f+2g+2h+2i 4π 3 c − 3 f ǫ δ Q2eγE
(cid:20)(cid:18) (cid:19)(cid:18) 12 (cid:19)(cid:21)
6
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
FIG.2. Self-energy corrections toFig.1(a).
(a) (b) (c)
(d) (e)
FIG.3. Vertexcorrections toFig.1(a).
where1/ǫrepresentstheUVpoleterm,µistherenormalizationscale,γ istheEulerconstant,N
E c
is the number of quark color, N is the number of the active quarks flavors, and H(0) denotes the
f
LOtwist-3hardkerneldescribedinEq.(10). TheFig.2(f,g,h,i)denotestheself-energycorrection
totheexchangedgluon.
It is easy to find that, the NLO self-energy corrections to the LO twist-3 hard kernels as listed
in Eq. (17) are identical in form to those self-energy corrections to the LO twist-2 hard kernels
as given in Eqs. (6-9) in Ref. [9]. The reason is that the self-energy diagrams don’t involve the
externallinesandthereforeareirrelevantwiththetwiststructuresofthewavefunctions. Itshould
be note that an additional symmetry factor 1 appeared from the choice of the gluon endpoint to
2
attach the external line in the self-energy correction Fig. 2(a,b,c,d). The self-energy corrections
to theexternal lines will be canceled by the responding effectivediagrams as shown in Fig. (5,6).
The self-energy correction to the internal quark line as shown in Fig. 2(e) don’t generate any IR
7
divergences.
The vertex corrections obtained by evaluating the one-loop Feynman diagrams in Fig. 3(a-e)
are oftheform
α C 1 4πµ2 1
G(1) = s F +ln + H(0),
3a 4π ǫ Q2eγE 2
(cid:20) (cid:21)
α 1 4πµ2
G(1) = s +ln 1 H(0)
3b −8πN ǫ x Q2eγE −
c (cid:20) 1 (cid:21)
α δ
s 1 ln 2 H(0)(x pµ),
−8πN − x 1 1
c (cid:20) 1(cid:21)
α 1 4πµ2
G(1) = s +ln H(0)
3c −8πN ǫ δ Q2eγE
c (cid:20) 12 (cid:21)
α δ δ δ δ π2
s ln 2 ln 1 +ln 1 +ln 2 + H(0)(pµ),
−8πN δ δ δ δ 3 2
c (cid:20) 12 12 12 12 (cid:21)
α N 3 4πµ2 11
G(1) = s c +3ln + H(0)
3d 8π ǫ δ Q2eγE 2
(cid:20) 12 (cid:21)
α N δ δ
s c ln 1 +ln 2 H(0)(pµ)
− 8π δ δ 2
(cid:20) 12 12(cid:21)
α N 3 4πµ2 11
G(1) = s c +3ln + H(0)
3e 8π ǫ x Q2eγE 2
(cid:20) 1 (cid:21)
α N δ δ
s c ln 2 lnx +ln 2 H(0)(x pµ). (18)
− 8π x 2 x 1 1
(cid:20) 1 1(cid:21)
It is easy to find that the NLO twist-3 corrections to the LO hard kernel H(0) in Eq. (14) have
the UV divergence and they have the same divergence behavior in the self-energy and the vertex
corrections. The summation of these UV divergences leads to the same result as the one for the
NLO twist-2case [9]
α 2 1
s 11 N , (19)
f
4π − 3 ǫ
(cid:18) (cid:19)
whichmeets therequirement oftheuniversalityofthewavefunctions.
TheamplitudeG(1) havenoIRdivergenceduetothefactthatthenumeratorintheamplitudeof
3a
thecollinearregionisdominatedbythetransversecontributionswhicharenegligibleinFig.3(a).
IR divergences in G(1) and G(1) are only relevantwith thehard kernel H(0)(pµ), which is induced
3c 3d 2
by thesingulargluonattaches to the downquark lines. Similarly,IR divergencesin G(1) and G(1)
3b 3e
onlyoccurinthehardkernelH(0)(x pµ)sincethesingulargluonisattachedtotheupquarklines.
1 1
The amplitude G(1) should have collinear divergence because the radiative gluon in Fig. 3(b)
3b
is attached to the light valence quark of the final state pion, and we find that the IR contribution
is regulated by lnδ . Both the collinear and soft divergences are produced in G(1) because the
2 3c
radiative gluon in Fig. 3(c) is attached to the external light valence anti-quarks. The large double
logarithm lnδ lnδ comes from the overlap of the IR divergences, and will be canceled by the
1 2
largedoublelogarithmterm fromFig. 4(f).
TheradiativegluoninFig.3(d)isattachedtothelightvalenceanti-quarksaswellasthevirtual
LOhardgluon,sothesoftdivergenceandthelargedoublelogarithmaren’tgeneratedinG(1). The
3d
8
(a) (b) (c)
(d) (e) (f)
FIG.4. Boxandpentagon corrections toFig.1(a).
radiative gluon in Fig. 3(e) is attached only to the light valence quark as well as the virtual LO
hard gluon, and then G(1) just contains the collinear divergence regulated by lnδ in the l P
3e 2 k 2
region.
The NLO twist-3 contributions from the box and pentagon diagrams as shown in Fig. 4 are
summarizedas
α N 3 1 π2
G(1) = s c (1+lnx )lnδ (1+ lnx )lnx + + H(0)(x pµ),
4a − 8π 1 1 − 2 1 2 8 12 1 1
(cid:20) (cid:21)
α N 3 1 π2
s c (1+lnx )lnδ (1+ lnx )lnx + + H(0)(x pµ),
− 8π 2 2 − 2 2 1 8 12 2 2
(cid:20) (cid:21)
(1)
G 0,
4b ≡
α x x
G(1) = s ln 1 ln 2 +ln2x H(0)(x pµ)
4c −8πN δ δ 2 1 1
c (cid:20) 2 1 (cid:21)
α x x
s ln 2 ln 1 +ln2x H(0)(x pµ),
− 8πN δ δ 1 2 2
c (cid:20) 1 2 (cid:21)
(1)
G 0,
4d ≡
α 5
G(1) = s lnδ lnδ +lnδ + H(0)(x pµ),
4e 8πN 1 2 1 4 1 1
c (cid:20) (cid:21)
α δ δ
G(1) = s ln 1 ln 2 2ln2 1 H(0)(pµ). (20)
4f −8πN δ δ − − 2
c (cid:20) 12 12 (cid:21)
Because of the properties of the propagators in above four- and five-point integrals, there is no
UVdivergencesinaboveamplitudes. Fig.4(b)and4(d)aretwo-particlereduciblediagrams,their
contributionshouldbecanceledbythecorrespondingeffectivediagramsFig.5(c)andFig.6(c)for
the NLO initial and final state meson wave functions due to the requirement of the factorization
theorem,so wecan set themzero safely.
The H(0)(x pµ) terms appeared in G(1) and G(1) are obtained from the evaluation of Fig. 4(a)
2 2 4a 4a
and 4(c) only. The LO hard kernel H(0)(x pµ) has the same form as the H(0)(x pµ) as defined
2 2 1 1
in Eq. (12) but with replacements of x x and pµ pµ. IR regulators only appear to the
hard kernel H(0)(x pµ) of the Fig. 4(a,c1,e→), wh2ich are1de→cide2d by the fact that the left end-point
1 1
of the emission gluon is attached to the up light external line. Similarly, Fig. 4(f) only grow the
IR regulators to thehard kernel H(0)(pµ). Notethat theemissiongluonin Fig. 4(c,e,f) is attached
2
to external light lines, so it’s amplitude would dominated in the collinear regions and soft region,
then the double logarithm would appear. The attaching of the emission gluon in Fig. 4(a) to the
9
initial external line and the LO hard kernel deduce that only IR regulator lnδ is grown in the
1
amplitudeG(1).
4a
Now we just consider the IR parts regulated by lnδ which would not be canceled directly by
i
theircounterparts from theeffectivediagrams of Fig. 5. TheseIR pieces appearin G(1) and
3b,3c,3d,3e
G(1) . We class these amplitudes into two sets according to the hard kernels to which those
4a,4c,4e,4f
IR regulators lnδ give corrections. Then the first set includes G(1) and G(1), while the second
i 3c,3d 4f
set contains G(1) and G(1) terms. These amplitudes are calculated in the leading IR regions
3b,3e 4a,4c,4e
tocheck thek factorizationtheorem.
T
We firstly evaluate the NLO twist-3 corrections to H(0)(pµ). The amplitudes G(1) and G(1)
2 3c,3d 4f
are recalculated byemployingthephasespaceslicingmethod[16],
α δ δ π2
G(1)(l 0) = s ln 1 ln 2 + H(0)(pµ),
3c → 8πN δ δ 3 2
c (cid:20) 12 12 (cid:21)
α δ δ δ
G(1)(l p ) = s ln 1 ln 2 +ln 1 H(0)(pµ),
3c k 1 8πN δ δ δ 2
c (cid:20) 12 12 12(cid:21)
α δ δ δ
G(1)(l p ) = s ln 1 ln 2 +ln 2 H(0)(pµ). (21)
3c k 2 8πN δ δ δ 2
c (cid:20) 12 12 12(cid:21)
α N δ
G(1)(l p ) = s c ln 1 H(0)(pµ),
3d k 1 8π − δ 2
(cid:20) 12(cid:21)
α N δ
G(1)(l p ) = s c ln 2 H(0)(pµ). (22)
3d k 2 8π − δ 2
(cid:20) 12(cid:21)
α δ δ π2
G(1)(l 0) = s ln 1 ln 2 + H(0)(pµ),
4f → −8πN δ δ 3 2
c (cid:20) 12 12 (cid:21)
α δ δ π2
G(1)(l p ) = s ln 1 ln 2 + 1 H(0)(pµ),
4f k 1 −8πN δ δ 6 − 2
c (cid:20) 12 12 (cid:21)
α δ δ π2
G(1)(l p ) = s ln 1 ln 2 + 2ln2 H(0)(pµ). (23)
4f k 2 −8πN δ δ 6 − 2
c (cid:20) 12 12 (cid:21)
By summingupalltermsin Eq.(21,22,23),onefinds thatthesoftcontributionsinthelimitl 0
→
fromFig.3(c)andFig.4(f)arecanceledeachother,whiletheremainingcollinearcontributionsin
theregionsofl p and l p are oftheform,
1 2
k k
α C
G(1) (l p ) = s F [2lnδ ]H(0)(pµ), (24)
3c+3d+4f k 1 − 8π 1 2
α C
G(1) (l p ) = s F [2lnδ ]H(0)(pµ), (25)
3c+3d+4f k 2 − 8π 2 2
10
TheIRcontributionstoNLOtwist-3correctionstoH(0)(x pµ)can beobtainedinsimilarway.
1 1
α δ
G(1)(l p ) = s 1 ln 2 H(0)(x pµ), (26)
3b k 2 −8πN − x 1 1
c (cid:20) 1(cid:21)
α N δ
G(1)(l p ) = s c ln 2(lnx +1) H(0)(x pµ), (27)
3e k 2 − 8π x 2 1 1
(cid:20) 1 (cid:21)
α N 3
(1) s c
G (l p ) = lnδ (lnx +1) lnx ( lnx +1)
4a k 1 − 8π 1 1 − 2 2 1
(cid:20)
π2 1
+ + H(0)(x pµ), (28)
12 8 1 1
(cid:21)
α π2
G(1)(l 0) = s lnδ lnδ + H(0)(x pµ),
4c → −8πN 1 2 3 1 1
c (cid:20) (cid:21)
α δ π2
G(1)(l p ) = s lnδ ln 2 + H(0)(x pµ),
4c k 1 −8πN 1 x 6 1 1
c (cid:20) 1 (cid:21)
α δ π2
G(1)(l p ) = s lnδ ln 1 +lnx (lnx +lnx )+ H(0)(x pµ), (29)
4c k 2 −8πN 2 x 2 2 1 6 1 1
c (cid:20) 2 (cid:21)
α π2
G(1)(l 0) = s lnδ lnδ + H(0)(x pµ),
4e → 8πN 1 2 3 1 1
c (cid:20) (cid:21)
α π2
G(1)(l p ) = s lnδ lnδ +lnδ lnx + +3 H(0)(x pµ),
4e k 1 8πN 1 2 1 − 1 6 1 1
c (cid:20) (cid:21)
α 3 π2 7
G(1)(l p ) = s lnδ lnδ + lnx + H(0)(x pµ). (30)
4e k 2 8πN 1 2 2 1 6 − 4 1 1
c (cid:20) (cid:21)
Again, the soft parts from Fig. 4(c) and Fig. 4(e) are canceled each other, while the remaining
collinear contributions to the LO hard kernel H(0)(x pµ), after summing up the amplitudes as
1 1
givenin Eqs.(26,27,28, 29,30), arethefollowing
α C
G(1) (l p ) = s F [2lnδ (lnx +1)]H(0)(x pµ),
4a+4c+4e k 1 − 8π 1 1 1 1
α C
G(1) (l p ) = s F [2lnδ (lnx +1)]H(0)(x pµ). (31)
3b+3e+4c+4e k 2 − 8π 2 2 1 1
Note that we have dropped the constant terms in Eqs. (25,31) since we here consider the IR parts
only. According to previous studies in Refs.[8–10], we know that these IR divergences could be
absorbed into theNLO wave functionsof the pion mesons. This point will becomeclear after we
complete the calculations for the effective diagrams in Fig. 5 and Fig. 6. This absorption means
thatthek factorizationisvalidat theNLOlevelfortheπγ∗ π process.
T
→
Without the reducible diagrams G(1) , the summation for the NLO twist-3 contribu-
2a,2b,2c,2d,4b,4d
tions from all the irreducible QCD quark diagrams as illustrated by Figs. (2,3,4) leads to the final