Next-to-leading-order time-like pion form factors in $k_T$ factorization

0.23 MB·English

by  Hao-Chung Hu

Next-to-leading-order time-like pion form factors in k factorization
T
Hao-Chung Hu1,∗ and Hsiang-nan Li1,2,3,†
1Institute of Physics, Academia Sinica, Taipei, Taiwan 115, ROC
2Department of Physics, National Cheng-Kung University, Tainan, Taiwan 701, ROC
3Department of Physics, National Tsing-Hua University, Hsinchu, Taiwan 300, ROC
We calculate the time-likepion-photon transition form factor and the pion electromagnetic form
factor up to next-to-leading order (NLO) of the strong coupling constant in the leading-twist kT
factorization formalism. It is found that the NLO corrections to the magnitude (phase) are lower
than 30% (30◦) for the former, and lower than 25% (10◦) for the latter at large invariant mass
squaredQ2>30GeV2ofthevirtualphotons. TheincreaseofthestrongphaseswithQ2isobtained,
consistent with the tendency indicated by experimental data. This behavior is attributed to the
3 inclusion of parton transverse momenta kT, implying that the kT factorization is an appropriate
framework for analyzing complex time-like form factors. Potential extensions of our formalism to
1
two-body and three-body hadronic B meson decays are pointed out.
0
2
PACSnumbers: 12.38.Bx,12.38.Cy,12.39.St
n
a
J

0
1 I. INTRODUCTION

]
h The k factorizationformalism[1–6]has beenapplied to next-to-leading-order(NLO)analysisofseveralspace-like
T
p formfactors,suchasthe pion-photontransitionformfactor[7,8],the pionelectromagnetic(EM)formfactor[9],and
- the B π transitionformfactors[10]. Thecalculationsarenontrivial,becausepartonsoff-shellbyk2 areconsidered
p → T
e in bothQCD quarkdiagramsand effective diagramsfor mesonwavefunctions. The gaugeinvarianceof hardkernels,
h derivedfromthedifferenceoftheabovetwosetsofdiagrams,needstobeverified. Theregularizationofthelight-cone
[ singularity in the effective diagrams generates double logarithms, which should be summed to all orders. It has been

found that the NLO corrections, after the above treatments, are negligible in the pion transition form factor, but
3
v reach30%in the latter two cases. In this Letter we shall extend the NLO kT factorizationformalismto the time-like
8 pion transition and EM form factors.
0 One of the widely adopted theoreticalframeworksfor two-bodyhadronic B mesondecaysis the perturbative QCD
7 (PQCD)approach[11]basedonthek factorization. Ithasbeenshownthatfactorizablecontributionstothesedecays
T
6
canbe computedinPQCDwithoutthe ambiguityfromthe end-pointsingularity. Thesecomputationsindicatedthat
.
4 sizablestrongphasesareproducedfrompenguinannihilationamplitudes,withwhichthedirectCPasymmetryinthe
0 B± K±π∓ decayswassuccessfullypredicted. ItisthenaconcernwhetherPQCDpredictionsforstrongphasesare
2 stab→le against radiative corrections. The factorizable penguin annihilation amplitudes involve time-like scalar form
1
factors. BeforecompletingNLOcalculationsfortwo-bodyhadronicB mesondecays,itispossibletoacquireananswer
:
v to the above concern by studying the time-like pion EM form factor. Besides, the PQCD formalism for three-body
i B meson decays [12] has demanded the introduction of two-meson wave functions [13], whose parametrization also
X
involves time-like form factors associated with various currents. If PQCD results for complex time-like form factors
r
a are reliable, a theoretical framework for three-body B meson decays can be constructed.
NLO corrections to time-like form factors are derived easily from those to space-like ones by suitable analytic
continuationfrom Q2toQ2,withQ2denotingthemomentumtransfersquared. Weshallpresentthek factorization
T
−
formulasforthetime-likepiontransitionandEMformfactorsuptoNLOatleadingtwist. Followingtheprescription
proposed in [5, 11], both the renormalization and factorization scales are set to the virtuality of internal particles.
With this scale choice, it will be demonstrated that the NLO corrections to the time-like pion transition and EM
form factors are under control at leading twist. It implies that PQCD predictions for strong phases of factorizable
annihilationamplitudesintwo-bodyhadronicB mesondecaysmaybestableagainstradiativecorrections. Moreover,
weobservetheincreaseofthestrongphasesoftheaboveformfactorswithQ2,consistentwiththetendencyindicated
by experimental data. It will be explained that this behavior is attributed to the inclusion of the parton transverse
momenta k . This consistencysupports the k factorizationas apotential frameworkfor studying complextime-like
T T
form factors and three-body B meson decays.
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
2
P2,ǫ
q=P1+P2 q=P2−P1
xQ2 k2
− T
P1−k
(xQ2+k2)
− T
k P2,ǫ
P1−k k
(a) (b)
FIG. 1: LO quark diagrams for time-like and space-like pion-photon transition form factors with ⊗ representing the virtual
photon vertex. The virtuality of theinternal quark is labeled explicitly.
II. PION-PHOTON TRANSITION FORM FACTOR
Inthissectionwepresenttheleading-twistNLOfactorizationformulaforthe time-likepion-photontransitionform
factor. The leading-order (LO) QCD quark diagram describing γ∗(q) π(P ) γ(P ) is displayed in Fig. 1(a), where
1 2
→
the momentum P of the pion and the momentum P of the outgoing on-shell photon are chosen as
1 2
P =(P+,0,0 ), P =(0,P−,0 ), P+ =P− =Q/√2, (1)
1 1 T 2 2 T 1 2
with Q2 = q2 = (P +P )2 > 0 being the invariant mass squared of the virtual photon γ∗. Figure 1(a) leads to the
1 2
LO hard kernel
N Tr[ /ǫ(P/ +k/)γ γ5P/ ] N Tr[ /ǫP/ γ P/ γ5]
H(LO)(x,Q2,k )= i c 2 µ 1 = i c 2 µ 1 , (2)
πγ T − √2N (P +k)2+iε − r 2 k2 xQ2 iε
c 2 T − −
where N = 3 is the number of colors, ǫ is the polarization vector of the outgoing photon, k = (xP+,0,k ) is the
c 1 T
momentumcarriedbythevalencequark,γ5P/ /√2N istheleading-twistspinprojectorofthepion,andthesubscript
1 c
µ associated with the virtual photon vertex is implicit on the left-hand side. In the previous works on the space-like
transitionform factor [7, 14–16], the internal quark remains off-shell by (P k)2 = (xQ2+k2)<0 as indicated in
2− − T
Fig. 1(b). For the time-like case, the internal quark may go on mass shell, and an imaginary part is generated in the
hard kernel according to the principal-value prescription
1 1
=Pr +iπδ(k2 xQ2). (3)
k2 xQ2 iε k2 xQ2 T −
T − − T −
Fourier transforming Eq. (2) into the impact-parameter b space, we derive the LO pion transition form factor
√2f 1 ∞
F(LO)(Q2)=iπ π dx bdbφ (x)exp[ S(x,b,Q,µ)]H(1) √xQb , (4)
πγ 6 Z Z π − 0
0 0 (cid:0) (cid:1)
with the pion decay constant f , the renormalization and factorization scale µ, the Hankel function of the first kind
π
H(1), and the twist-2 pion distribution amplitude (DA) φ . Here we shall not consider the potential intrinsic k
0 π T
dependence ofthe pionwavefunction[17], because its inclusionwouldintroduce additionalmodeldependence, which
is not the focus of this work. For example, the intrinsic k dependence has been parameterized into the different
T
Gaussian and power forms in [18]. The Sudakov factor e−S sums the double logarithm α ln2k to all orders, and
s T
takes the same expression for both the space-like and time-like form factors [19], since it is part of the universal
meson wave function. For its explicit expression, refer to [5, 20, 21]. Note that Eq. (4) can be obtained from the LO
(1)
space-likepion transitionformfactor in [15] by substituting iπH /2 for the Bessel function K , as a consequence of
0 0
the analytic continuation q2 = Q2 (Q2+iε) in the hard kernel.
− →
Asstatedinthe Introduction,theNLOhardkernelis derivedbytakingthedifference ofthe O(α )quarkdiagrams
s
and the O(α ) effective diagrams for meson wave functions. The ultraviolet divergences in loops are absorbed into
s
3
the renormalizedstrong coupling constantα (µ), andthe infrareddivergencesare subtracted by the nonperturbative
s
mesonwave functions. The above derivationhas been demonstrated explicitly in [7] for the space-likepion transition
formfactor. We repeata similar calculationfor the time-like pion transitionfactor,andderive the NLO hardkernel1
H(NLO)(x,Q2,k ,µ)=h (x,Q2,k ,µ)H(LO)(x,Q2,k ), (5)
πγ T πγ T πγ T
with the NLO correction function
α (µ)C µ2 k2 xQ2 k2 xQ2
h (x,Q2,k ,µ)= s F 3ln ln2 | T − | +2 1 iπ iπΘ k2 xQ2 ln| T − |
πγ T 4π (cid:26)− Q2 − Q2 − − T − Q2
(cid:2) (cid:0) (cid:1)(cid:3)
2lnx+ 4π2 iπ Θ k2 xQ2 3 i5π , (6)
− − T − − − (cid:27)
(cid:0) (cid:1) (cid:0) (cid:1)
C being the color factor. The imaginary pieces proportional to the step function Θ are generated from the O(α )
F s
quarkdiagrams. FortheevaluationoftheO(α )effectivediagrams,wehavechosenthedirectionnµoftheWilsonlines
s
the same as in [7] in order to respect the universality of the meson wave function. Equation (5) can also be achieved
by substituting (Q2+iε) for the virtuality of the external photon, and (xQ2 k2 +iε) for the internal quark in [7],
andthenemployingtherelationsln( Q2 iε)=lnQ2 iπ andln( xQ2+k2−iεT)=ln xQ2 k2 iπΘ(xQ2 k2).
− − − − T− − T − − T
Fourier transforming Eq. (5) to the b space, we arrive at the NLO kT factorization fo(cid:12)rmula for(cid:12)the time-like pion
(cid:12) (cid:12)
transition factor
√2f 1 ∞
F(NLO)(Q2)=iπ π dx bdbφ (x)exp[ S(x,b,Q,µ)]
πγ 6 Z Z π −
0 0
α (µ)C
s F h (x,Q2,k ,µ)H(1) √xQb +H(1)′′ √xQb , (7)
× 4π h πγ T 0 (cid:0) (cid:1) 0 (cid:0) (cid:1)i
e
with
µ2 1 4x 3π 4x
h (x,Q2,k ,µ)= 3ln ln2 +(1+γ i )ln 2lnx
πγ T − Q2 − 4 Q2b2 E − 2 Q2b2 −
e 17π2
+ +π 3 2γ γ2 i(4 3γ )π, (8)
12 − − E − E − − E
γ being the Euler constant. The function
E
∂2
H(1)′′(ρ) H(1)(ρ) , (9)
0 ≡(cid:20)∂α2 α (cid:21)
α=0
whereαdenotestheorderparameteroftheHankelfunction,comesfromtheFouriertransformationofln2( xQ2+k2
− T−
iε)inEq.(6). Forasmallargumentρ=√xQb 0,itsmagnitudebehavesas H(1)′′(ρ) (1/3)ln2ρ H(1)(ρ),which
→ | 0 |∼ | 0 |
representsadouble-logarithmiccorrectionessentially. Theperturbativeexpansioncouldbeimprovedbysummingthe
double logarithm α ln2[x/(Q2b2)] in Eq. (8), which arises from the Fourier transformation of the term α ln2(k2
s s | T −
xQ2 /Q2) in Eq. (6). Strictly speaking, it differs from the threshold resummation of α ln2x performed in [20], and
s
|
deservesa separatestudy. Besides,there is no end-point enhancementinvolvedinthe presentcalculation,so we shall
not perform the resummation here for simplicity. Equations (4) and (7) will be investigated numerically in Sec. IV.
III. PION ELECTROMAGNETIC FORM FACTOR
We then derive the NLO, i.e., O(α2) contribution to the time-like pion EM form factor at leading twist. An LO
s
quarkdiagramforthecorrespondingscatteringγ∗(q) π+(P )π−(P )isdepictedinFig.2(a). Wechooselight-cone
1 2
coordinates,suchthatthemomentaP andP arepara→meterizedthesameasinEq.(1)withQ2 =q2 =(P +P )2 >0.
1 2 1 2
The valencequarkcarriesthemomentumk =(x P+,0,k )andthe valenceanti-quarkcarriesk =(0,x P−,k ).
1 1 1 1T 2 2 2 2T
The LO hard kernel reads
x Tr[P/ P/ γ P/ ]
H(LO)(x ,k ,x ,k ,Q2)=i4πα C 1 2 1 µ 1 , (10)
II 1 1T 2 2T s F(x Q2 k2 +iε)(x x Q2 k +k 2+iε)
1 − 1T 1 2 −| 1T 2T|
1 Comparedto[7],threeeffectivediagramsfortheself-energycorrectionstotheWilsonlineshavebeenincludedinEq.(5).
4
P2−k2 k2 q=P2−P1
P1−k1
q=P1+P2 x1Q2−k12T −(x1Q2+k12T)
x1x2Q2−|k1T+k2T|2
−(cid:16)x1x2Q2+|k1T−k2T|2(cid:17) P2−k2
k1 k2
P1−k1 k1
(a) (b)
FIG. 2: LO quark diagrams for time-like and space-like pion electromagnetic form factors.
where the denominators (x Q2 k2 ) and (x x Q2 k + k 2) are the virtuality of the internal quark and
1 − 1T 1 2 − | 1T 2T|
gluon, respectively. The subscript II denotes that the k -dependent terms in both the internal quark and gluon
T
propagators are retained. When one of the internal particle propagators goes on mass shell, an imaginary part is
produced according to the principle-value prescription in Eq. (3).
Fourier transforming Eq. (10) from the transverse-momentum space (k , k ) to the impact-parameter space
1T 2T
(b , b ), we obtain a double-b convolution for the LO time-like pion EM form factor [22]
1 2
π3f2C 1 ∞
F(LO)(Q2)= π FQ2 dx dx db db b b α (µ)x φ (x )φ (x )exp[ S (x ,b ,x ,b ,Q,µ)]
EM 2N Z 1 2Z 1 2 1 2 s 1 π 1 π 2 − II 1 1 2 2
c 0 0
H(1)(√x x Qb ) H(1)(√x Qb )J (√x Qb )Θ(b b )+H(1)(√x Qb )J (√x Qb )Θ(b b ) , (11)
× 0 1 2 2 h 0 1 1 0 1 2 1− 2 0 1 2 0 1 1 2− 1 i
with the Bessel function of the first kind J , and the Sudakov exponent S (x ,b ,x ,b ,Q,µ) = S(x ,b ,Q,µ)+
0 II 1 1 2 2 1 1
S(x ,b ,Q,µ). The above expressioncan also be obtained via analytical continuationof the space-like formfactor in
2 2
Fig. 2(b) to the time-like region.
The NLO hard kernel for the space-like pion EM form factor has been computed as the difference between the
one-loopQCDquarkdiagramsandeffective diagramsin[9]. To simplify the calculation,the hierarchyx Q2,x Q2
1 2
≫
x x Q2,k2 has been postulated, since the k factorization applies to processes dominated by small-x contributions.
1 2 T T
Ignoring the transverse momenta of the internal quarks, the LO hard kernel in Eq. (10) reduces to
Tr[P/ P/ γ P/ ]
H(LO)(x ,k ,x ,k ,Q2)=i4πα C 2 1 µ 1 . (12)
I 1 1T 2 2T s FQ2(x x Q2 k +k 2+iε)
1 2 1T 2T
−| |
The Fourier transformation of the above expression leads to a single-b convolution [23]
π2f2C 1 ∞
F(LO)(Q2)=i π F dx dx dbbα (µ)φ (x )φ (x )exp[ S (x ,x ,b,Q,µ)]H(1)(√x x Qb), (13)
I N Z 1 2Z s π 1 π 2 − I 1 2 0 1 2
c 0 0
with the simplified Sudakov exponent S (x ,x ,b,Q,µ) S (x ,b,x ,b,Q,µ). Comparing the outcomes from
I 1 2 II 1 2
≡
Eqs. (11) and (13), we can justify the proposed hierarchicalrelation, and tell which particle propagator,the internal
quark or the internal gluon, provides the major source of the strong phase.
Substituting (Q2+iε) for the virtuality of the external photon, and (x x Q2 k +k 2+iε) for the internal
1 2 1T 2T
−| |
gluon in [9], we have the NLO hard kernel for the time-like pion EM form factor
H(NLO)(x ,k ,x ,k ,Q2,µ)=h (x ,x ,δ ,Q,µ)H(LO)(x ,k ,x ,k ,Q2), (14)
EM 1 1T 2 2T EM 1 2 12 I 1 1T 2 2T
with the NLO correction function
α (µ)C 3 µ2 17 27 13 31
h (x ,x ,δ ,Q,µ)= s F ln ln2x + lnx lnx lnx + lnx
EM 1 2 12 4π (cid:20)− 4 Q2 − 4 1 8 1 2− 8 1 16 2
17 23 π2 1 53 3π
ln2δ + lnx + +i2π lnδ + + ln2+ i , (15)
12 1 12
− (cid:18) 4 8 (cid:19) 12 2 4 − 4 (cid:21)
5
and the notation
k +k 2 x x Q2
1T 2T 1 2
lnδ ln(cid:12)| | − (cid:12) +iπΘ k +k 2 x x Q2 . (16)
12 ≡ (cid:12)(cid:12) Q2 (cid:12)(cid:12) (cid:16)| 1T 2T| − 1 2 (cid:17)
Fourier transforming Eq. (14), we derive the k factorization formula for the NLO contribution at leading twist
T
πf2C2 1 ∞
F(NLO)(Q2)=i π F dx dx dbbα2(µ)φ (x )φ (x )exp[ S (x ,x ,b,Q,µ)]
EM 4N Z 1 2Z s π 1 π 2 − I 1 2
c 0 0
h (x ,x ,b,Q,µ)H(1)(√x x Qb)+H(1)′′(√x x Qb) , (17)
×h EM 1 2 0 1 2 0 1 2 i
e
with the function
3 µ2 1 4x x 17 23 π 4x x
h (x ,x ,b,Q,µ)= ln ln2 1 2 + lnx + +γ +i ln 1 2
EM 1 2 − 4 Q2 − 4 Q2b2 (cid:18) 8 1 16 E 2(cid:19) Q2b2
e 17 27 13 17γ 17π 31
ln2x + lnx lnx + E i lnx + lnx
1 1 2 1 2
− 4 8 −(cid:18) 8 4 − 8 (cid:19) 16
π2 1 53 23 171
+(1 2γ )π+ ln2+ γ γ2 +i +γ π. (18)
− 2 − E 2 4 − 8 E − E (cid:18) 16 E(cid:19)
The perturbative expansion could be improved by organizing the double logarithm α ln2x in Eq. (15) into the
s 1
threshold resummation factor S (x ,Q2) [20]. This double logarithm, the same as analyzed in [20], appears in the
t 1
loop correction to the virtual photon vertex under the hierarchical relation x Q2 k2 [9]. Because there is no
1 ≫ T
end-point enhancement involved at leading twist, we shall not perform the threshold resummation here. However,
the end-point enhancement exists in the two-partontwist-3 contribution, for which S will play a crucialrole, and be
t
implemented in Sec. IV. We shall investigate the NLO effect at leading twist in the time-like pion EM form factor
based on Eqs. (11) and (17).
IV. NUMERICAL ANALYSIS
Thenumericalanalysisisperformedinthissection,forwhichweadoptthestandardtwo-loopQCDrunningcoupling
constant α (µ) with the QCD scale Λ = 0.2 GeV, the pion decay constant f = 0.131 GeV, the nonasymptotic
s QCD π
two-parton twist-2 pion DA
φ (x)=6x(1 x) 1+a C3/2(1 2x) , (19)
π − h 2 2 − i
with the Gegenbauer coefficient a =0.2 being fixed by lattice QCD [24], and the Gegenbauer polynomial C3/2(u)=
2 2
(3/2)(5u2 1).
−
We computethe LOandNLOcontributionstothe time-likepion-photontransitionformfactoratleadingtwistvia
Eqs. (4) and (7), with the renormalization and factorization scale µ being set to the virtuality of the internal quark
µ=max(√xQ,1/b). The behavior of Q2F (Q2) for Q2 <20 GeV2 displayedin Fig. 3 reflects the oscillatorynature
πγ
ofthe LOhardkernelinthe b space. TheLOtime-likepiontransitionformfactorexhibits anasymptoticmagnitude,
Q2 F (Q2) 0.225 GeV at large Q2. Recall that an asymptotic scaling is known as Q2F (Q2) √2f = 0.185
πγ πγ π
≈ →
GeV(cid:12) for the(cid:12)space-like piontransitionform factor atlarge q2 =Q2 [25]. The largerasymptotic value for the former
(cid:12) (cid:12) −
isexpectedinthek factorization,becausetheinternalquarkmaygoonmassshellforatime-likemomentumtransfer
T
q, but it is always off-shell for a space-like q. The ratio between the asymptotic values of the above two transition
form factors is roughly 1.22, comparable to the data 1.14 from the η γ transition form factors for Q2 > 100 GeV2
−
[26]. Note that the time-like and space-like transition form factors would have equal magnitudes in the LO collinear
factorization without including the parton transverse momentum k . The NLO contribution to the time-like pion
T
transitionformfactorisalsodisplayedinFig.3,whichdecreaseswithQ2 asexpectedinPQCD.Comparedtothe LO
result, the NLO correction to the magnitude is about 30% at Q2 =30 GeV2, and less than 20% for Q2 >50 GeV2.
Forthephase,theLOresultariseswithQ2,andapproachesanasymptoticvaluecloseto180◦ asshowninFig.3. It
is obviousthatthe variationwith Q2 is alsoattributedto the inclusionofthe partontransversemomentumk . If k
T T
in Eq.(2) was dropped, the LO hard kernel reduces to the traditional expression in the collinear factorization, which
alwaysleads to a real F . A quantitative understanding can be attained via Eq.(3): the contributions from the two
πγ
6
(cid:11)4(cid:17)(cid:14)
55V
(cid:1)(cid:2)
(cid:1)(cid:2)5(cid:4)(cid:1)(cid:2)
(cid:11)4(cid:17)
5VV
(cid:7)(cid:8)(cid:9). (cid:11)4(cid:16)(cid:14) (cid:9)(cid:10)(cid:8)(cid:8)1 3(cid:17)V
(cid:2)+0(cid:1) (cid:4)(cid:5)(cid:4)(cid:6) (cid:7) (cid:11)(cid:11)4(cid:15)4(cid:16)(cid:14) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)+)(cid:7)(cid:8) 3(cid:16)V
3(cid:15)V
(cid:11)4(cid:15)
(cid:1)(cid:2)
(cid:1)(cid:2)0(cid:4)(cid:1)(cid:2)
(cid:11)4(cid:13)(cid:14) 35V
(cid:11) (cid:13)(cid:11) (cid:15)(cid:11) (cid:16)(cid:11) (cid:17)(cid:11) (cid:14)(cid:11) V 3V 5V (cid:20)V (cid:15)V (cid:21)V
(cid:1)(cid:2)+0(cid:7)(cid:8)(cid:9)(cid:15)+. (cid:6)(cid:7)+)(cid:18)(cid:8)(cid:19)5+1
FIG.3: Magnitudeandphaseofthetime-likepion-photontransitionform factoratLO(dashed)anduptoNLO(solid). The
NLO correction is marked in gray.
terms in Eq. (3) are comparable at low Q2, such that the time-like pion transition form factor acquires a nontrivial
phase. At high Q2 >20 GeV2, the phase is dominated by the first term in Eq. (3), since it is unlikely to have a large
parton k2 = xQ2 demanded by the second term. That is, the tiny deviation (less than 5◦) of the asymptotic phase
T
from180◦ is causedby the power-suppressedk2/Q2 effect. The NLO correctionto the phase is about30◦ atQ2 =30
T
GeV2, and fewer than 20◦ for Q2 > 50 GeV2. The above investigation implies that higher-order corrections to the
complextime-liketransitionformfactorsareunder controlinthe k factorization. As statedbefore,the perturbative
T
expansion could be improved by resumming the double logarithm α ln2x in Eq. (8).
s
For the analysis of the time-like pion EM form factor, we first identify the major source of the strong phase by
comparing the results from Eqs. (11) and (13) in Fig. 4. The renormalization and factorization scale µ is set to
µ = max √x Q,1/b ,1/b [5, 9], associated with the virtuality of the internal particles. The curve from Eq. (11)
1 1 2
implies th(cid:0)at the magnitude(cid:1)of the time-like pion EM form factor has an asymptotic behavior Q2 F (Q2) 0.14
EM
GeV2 as Q . Similar to the case of the transition form factor, this asymptotic value is large|r than tha|t→of the
→ ∞
space-like pion EM form factor [9], because of the inclusion of the parton transverse momenta k . The inclusion of
T
k also leads to the variation of the phase with Q2, which arises from the first quadrant, and then approaches an
T
asymptoticvaluecloseto165◦. ForQ2 >15GeV2,thedifferencebetweenusingthesingle-banddouble-bconvolutions
is insignificant in both magnitude and phase, verifying the hierarchical relation x Q2,x Q2 x x Q2,k2, and the
1 2 ≫ 1 2 T
majorsourceforthestrongphaseastheinternalgluonpropagator. WetheninvestigatetheNLOeffectinthetime-like
pionEMformfactorbasedonEqs.(11)and(17),whichisalsoshowninFig.4. ForQ2 >10GeV2,theobservedNLO
correctionis roughly 25% for the magnitude, and less then 10◦ for the phase. That is, the perturbative evaluation of
the time-like pion EM form factor is stable against radiative corrections at leading twist.
Atlast,weincludeanotherpieceofsubleadingeffects,theLOtwo-partontwist-3contribution[27],forcompleteness.
Following the same derivation of the twist-2 contribution in Sec. III, the k factorization formula for the LO two-
T
parton twist-3 contribution to the time-like pion EM form factor was obtained in [22], where an explicit double-b
convolution expression similar to Eq. (11) can be found. The hard kernel in the impact-parameter space is identical
to the one in Eq. (11). We employ the asymptotic two-parton twist-3 DAs,
φP(x)=1, φT(x)=1 2x, (20)
π π −
with the associated chiral scale µ = 1.3 GeV. The threshold resummation factor S (x,Q) with a shape parameter
π t
c=0.4is included, since the importantdouble logarithmα ln2xatsmall x needs to be summed [20]. The numerical
s
outcomesforthe time-likepionEMformfactorarepresentedinFig.5,wherethe availableexperimentaldata[28,29]
are displayed for comparison. It is known that the pion EM form factor is dominated by the two-parton twist-3
contribution, instead of by the twist-2 one at currently accessible energies, because of the end-point enhancement
developed by the above DAs [22, 30, 31]. This enhancement is understood easily as follows: the virtual quark and
gluon propagators behave like 1/x and 1/(x x ), respectively, as indicated in Eq. (10). The twist-2 pion DA is
1 1 2
proportional to φ (x) O(x), but the twist-3 pion DAs remain constant φP,T(x) O(1) at small x, which then
π ∼ π ∼
enhance the end-point contribution dramatically. This enhancement was also observed in perturbative evaluation of
the B π transition form factors [32], and confirmed by the light-cone sum-rule analysis [33]. The relative phase
betwee→n the twist-2 and two-partontwist-3 pieces is about70◦ as indicated by Figs.4 and5, so the magnitude of the
7
rGd
GN0
(cid:1)(cid:2)(cid:3)(cid:4)(cid:1)+)i(cid:8)(cid:9)bi(cid:4)(cid:3)(cid:11)(cid:12)(cid:13)(cid:14)+(cid:15)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:1)+)i(cid:8)(cid:9)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:1)+)i(cid:8)(cid:9)e(cid:17)(cid:8)(cid:9) r4d
GN)1
)(cid:19)(cid:14)(cid:20)ig (cid:20)(cid:14)(cid:12)(cid:14)(cid:14)g r0d
(cid:2)(cid:1) (cid:4)(cid:5)(cid:4)in(cid:6)(cid:7) GN) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:19)in r)d
GNV1
rdd (cid:1)(cid:2)(cid:3)(cid:4)(cid:1)+)i(cid:8)(cid:9)bi(cid:4)(cid:3)(cid:11)(cid:12)(cid:13)(cid:14)+(cid:15)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:1)+)i(cid:8)(cid:9)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:1)+)i(cid:8)(cid:9)e(cid:17)(cid:8)(cid:9)
GNV Gd
G VG )G 0G .G 1G d rd )d 8d 0d 1d
(cid:1)(cid:2)in(cid:19)(cid:14)(cid:20))ig (cid:6)(cid:7)in(cid:27)(cid:14)(cid:28))ig
FIG. 4: Magnitude and phase of the time-like pion EM form factor at leading twist. Contributions from LO with thesingle-b
convolution (dotted),LO (dashed), and LO+NLO(solid) are shown.
formfactoris hardlyaffectedbythe former. However,the twist-2contributiondoeshaveasizableeffectonthe phase
as illustrated in Fig. 5.
Thepredictionsforthemagnitudeofthetime-likepionEMformfactorfromthek factorizationcanaccommodate
T
the data [28] for Q2 > 4 GeV2, an observation consistent with that from the LO analysis [22]. We point out that
the measuredmagnitude ofthe time-like pionEMformfactor is largerthanthe space-likeone [22], andsimultaneous
accommodation of both data is possible in the k factorization, but not in the collinear factorization. Though the
T
perturbative calculations may not be justified for small Q2 <4 GeV2, it is interesting to see the coincidence between
the increasesofthephasewithQ2 fromthe k factorizationandfromthe dataforQ2 <1.3GeV2. InaBreit-Wigner
T
picture, the observed phase increase could be attributed to a resonant ρ meson propagator [29]. It happens that the
parton transverse momentum k plays the role of the ρ meson mass, such that the two curves in Fig. 5 exhibit the
T
similar tendency, and begin to merge for Q2 > 1 GeV2. Again, this coincidence cannot be achieved in the collinear
factorization, which does not generate a significant phase shift.
Theconsistencybetweenthepresentanalysisandthedatasupportsthek factorizationformalismasanappropriate
T
frameworkforstudyingcomplextime-likeformfactors. Ithasbeenunderstoodthatthecomplexpenguinannihilation
contribution is essential for explaining direct CP asymmetries in two-body hadronic B meson decays [11]. This
contribution involves time-like scalar form factors, which can be calculated in the same k factorization formalism.
T
It has been observed that the phase of the S-wave component in ππ scattering shows a similar Q2 dependence to
that of the P-wave [29]. Therefore, the PQCD predictions for the above direct CP asymmetries are expected to be
reliable. Theformalismforthree-bodyhadronicB mesondecays[12]hasrequiredtheintroductionoftwo-mesonwave
functions, whose parametrizationalso involvestime-like form factors of various currents. Stimulated by our work,we
have the confidence on computing these complex time-like form factors directly in the PQCD approach.
V. CONCLUSIONS
In this Letter we have calculated the time-like pion-photon transition and EM form factors up to NLO in the k
T
factorization formalism. The corresponding NLO hard kernels were derived by analytically continuing the space-like
ones to the time-like region of the momentum transfer squared Q2. We have identified the k -dependent internal
T
gluon propagator as the major source for the strong phase of the time-like pion EM form factor, which increases
with Q2, and approaches an asymptotic value [34]. The magnitudes of the time-like form factors are larger than
those of the space-like ones. It has been realizedthat the above features are attributed to the inclusion of the parton
transversemomenta,andconsistentwiththetendencyimpliedbythedata. ItwasobservedthattheNLOcorrections
in magnitude (phase) change the LO leading-twist results by roughly 30% (30◦) for the pion transition form factor,
and 25% (10◦) for the pion EM form factor as Q2 > 30 GeV2. The stability against radiative corrections justifies
the k factorization formalism for both time-like form factors at leading twist. Therefore, the predictions for strong
T
phases of annihilation contributions to two-body hadronic B meson decays in the PQCD approach may be reliable.
The framework presented here will have other applications, for example, to the construction of the two-meson wave
functions for three-body B meson decays.
8
23N
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:4)(cid:8)(cid:9)(cid:10)(cid:11)p(cid:13)(cid:10)(cid:9)(cid:10) at5
(cid:9)(cid:14)(cid:6)(cid:15)(cid:9)-up(cid:18)(cid:19)
2 (cid:9)(cid:14)(cid:6)(cid:15)(cid:9)-up(cid:18)(cid:19)pwp(cid:9)(cid:14)(cid:6)(cid:15)(cid:9)-2p(cid:18)(cid:19) aa5
(cid:9)(cid:14)(cid:6)(cid:15)(cid:9)-up(cid:18)(cid:19)pwp(cid:9)(cid:14)(cid:6)(cid:15)(cid:9)-2p(cid:22)(cid:3)p(cid:9)(cid:23)p(cid:24)(cid:18)(cid:19)
2(cid:27)p (cid:4)(cid:4)8 aD5
(cid:26)(cid:4) +3N (cid:4)(cid:5)(cid:6) a55 D(cid:11)(cid:19)
(cid:2)pl(cid:1) (cid:4)(cid:5)(cid:4)(cid:6)(cid:7) + (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)d7(cid:3) D(cid:13)5 DDD5a(cid:19)5(cid:19)5
D(cid:12)5 (cid:11)(cid:19)
(cid:19)5
O3N D(cid:11)5 a(cid:19) (cid:20)(cid:21)(cid:22)(cid:21)
5
D(cid:9)5 5Va(cid:19) 5V(cid:19) 5V(cid:11)(cid:19) D DVa(cid:19)
O
O +O 2O uO oO NO 5 D5 a5 t5 .5 (cid:19)5
(cid:1)(cid:2)pl(cid:26)(cid:4)(cid:27)2p (cid:6)(cid:7)d7(cid:16)(cid:4)(cid:17)ad8
FIG.5: Contributionstothetime-likepionEMform factorfromtwo-partontwist-3LO(dotted),two-partontwist-3LOplus
twist-2 LO (dashed),and two-parton twist-3 LO plustwist-2 upto NLO (solid).
WethankB.AnanthanarayanandI.Capriniforusefuldiscussions. TheworkwassupportedinpartbytheNational
ScienceCouncilofR.O.C.underGrantNo. NSC-98-2112-M-001-015-MY3,andbytheNationalCenterforTheoretical
Sciences of R.O.C.
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