$π^0\toγ^*γ$ transition form factor within Light Front Quark Model

0.18 MB·English

by  Chong-Chung Lih

π0 γ γ transition form factor within Light Front
∗
→
Quark Model
Chong-Chung Lih1,2a and Chao-Qiang Geng3,2b
1Department of Optometry, Shu-Zen College of Medicine
and Management, Kaohsiung Hsien,Taiwan 452
2
1 2Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan 300
0
2 3Department of Physics, National Tsing Hua University, Hsinchu, Taiwan 300

n
a (Dated: January 12, 2012)
J

1 Abstract
1
We study the transition form factor of π0 γ∗γ as a function of the momentum transfer Q2
]
h →
p within the light-front quark model (LFQM). We compare our result with the experimental data
-
p
by BaBar as well as other calculations based on the LFQM in the literature. We show that our
e
h
[ predictedformfactor fitswellwiththeexperimentaldata, particularly thoseatthelargeQ2 region.

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E-mail address: [email protected]
b
E-mail address: [email protected]
1
I. INTRODUCTION
The BaBar collaboration [1] has reported a new data of the π0 γ∗γ transition form
→
factor F (Q2) for the high momentum transfer Q2 up to 40 GeV2. To describe the data
πγ
with the Q2 dependence, the form factor is fitted to satisfy the formula
Q2 β
Q2 F (Q2) = A (1)
| πγ | 10GeV2
(cid:18) (cid:19)
with A = 0.182 0.002 GeV and β = 0.25 0.02. Before the new data, most theoretical
± ±
models predicted that the form factor approaches the QCD asymptotic limit [2], depending
on the pion distribution amplitude (DA) with the Q2 dependence under 10 GeV2 [3–5].
Obviously, the experimental values for Q2 > 10 GeV2 by BaBar are surprisingly much
higher than the QCD asymptotic expectations and thus, cannot be explained by the lowest
perturbative results [2]. Even the high order corrections are considered [6, 7], the large Q2
behavior is still hard to be understood. Recently, many proposals [8–31] have been given
in the literature to understand the transition form factor, particularly the BaBar data for
Q2 > 10 GeV2.
In this note, we will use the phenomenological light front (LF) pion wave function to
evaluate Q2 F (Q2) in the light front quark model (LFQM) [32–36]. We will concentrate
πγ
| |
on the space-like region for the transition form factor. The LF wave function is manifestly
boostinvariantasitisexpressed intermsofthelongitudinalmomentumfractionandrelative
transverse momentum variables. The parameter inthe hadronic wave function is determined
from other information and the meson state of the definite spins can be constructed by the
Melosh transformation. We emphasize that our derivation of the form factor can be applied
to all allowed kinematic region. In Ref. [37], the study on the transition pion form factor
based on the LFQM has been done but the calculation for Q2 is only up to 8 GeV2. With
the same set of parameters in Ref. [37], the high Q2 BaBar data cannot be fitted. The use
of the LFQM to understand the BaBar data has been explored in Ref. [38]. However, the
conclusion in Ref. [38] has failed to explain the data. In this work, we would like to revisit
the LFQM to see if it is indeed the case.
This paper is organized as follows. In Sec. II, we present the relevant formulas for the
matrix element and form factor for the π0 γ∗γ transition. In Sec. III, we show our
→
numerical analysis. We give our conclusions in Sec. IV.
2
II. THE FORM FACTOR
The transition form factor of Fπ0→γ∗γ∗(q12,q22), which describes the vertex of π0γ∗γ∗, is
defined by:
A(π0(P) → γ∗(q1,ǫ1) γ∗(q2,ǫ2)) = ie2Fπ0→γ∗γ∗(q12,q22) εµνρσ ǫµ1 ǫν2 q1ρ q2σ, (2)
where Fπ0→γ∗γ∗(q12,q22) is a symmetric function under the interchange of q12 and q22. From the
quark-meson diagram depicted in Fig. 1, the amplitude in Eq. (2) is found to be
γ∗(q1) γ∗(q1)
p1,u p2,u p3,u¯ p2,u
π0(P) p3,u¯ γ∗(q2) π0(P) p1,u γ∗(q2)
(a) (b)
FIG. 1. Loop diagrams that contribute to π0 γ∗γ∗.
→
A(QQ¯ γ∗(q ) γ∗(q )) = e e N d4p3 Λ Tr γ i(− 6p3 +mQ¯) ǫ i(6p2 +mQ)
→ 1 2 Q Q¯ c (2π)4 P 5 p2 m2 +iǫ 6 2p2 m2 +iǫ
Z ( " 3 − Q¯ 2 − Q
i(p +m )
1 Q
ǫ 6 +(ǫ ǫ , q q )
× 6 1p2 m2 +iǫ 1 ↔ 2 1 ↔ 2
1 − Q # )
+(p p , m m ), (3)
1(3) 3(1) Q Q¯
↔ ↔
whereN isthenumber ofcolors, e isthequarkelectricchargeandΛ isthevertex function
c Q P
related to the π0 meson bound state. To calculate the π0 γ∗γ∗ transition from factor
→
within the LFQM, we have to decompose the π0 meson into QQ¯ Fock states, described as
(uu¯ dd¯)/√2. In the LF approach, the LF meson wave function can be expressed by an
−
anti-quark Q¯ and a quark Q with the total momentum P as:
M(P,S,S ) = [dp ][dp ]2(2π)3δ3(P p p )
z 1 2 1 2
| i − −
λX1λ2Z
×ΦSMSz(z,k⊥)b+Q¯(p1,λ1)d+Q(p2,λ2)|0i, (4)
where
dp+d2p⊥
[dp] = , (5)
2(2π)3
3
Φλ1λ2 is the amplitude of the corresponding q¯(q) andp is the on-massshell LF momentum
M 1(2)
of the internal quark. In the momentum space, the wave function ΦSSz is given by
M
ΦSMSz(k1,k2,λ1,λ2) = RλS1Sλz2(z,k⊥) φ(z,k⊥), (6)
where φ(z,k⊥) represents the momentum distribution amplitude of the constituents in the
bound state and RSSz constructs a spin state (S,S ) out of light front helicity eigenstates
λ1λ2 z
(λ1λ2) [39]. The LF relative momentum variables (z,k⊥) are defined by
p+ = zP+, p+ = (1 z)P+,
1 2 −
p1⊥ = zP⊥ k⊥, p2⊥ = (1 z)P⊥ +k⊥. (7)
− −
The normalization condition of the meson state is given by
hM(P′,S′,Sz′)|M(P,S,Sz)i = 2(2π)3P+δ3(P′ −P)δS′SδSz′Sz , (8)
which leads the momentum distribution amplitude φ(z,k⊥) to
dzd2k⊥
Nc φ(z,k⊥) 2 = 1. (9)
2(2π)3 | |
Z
We note that Eq. (6) can, in fact, be expressed as a covariant form [32, 33, 40]
1
p+p+ 2
ΦSMSz(z,k⊥) = 2[M02 − m1 Q2−mQ¯ 2]! u(p1,λ1)γ5v(p2,λ2)φ(z,k⊥),
m2 +k2 m2 +k2
M2 = Q¯ ⊥ +(cid:0) Q ⊥(cid:1). (10)
0 z 1 z
−
In principle, the momentum distribution amplitude φ(z,k⊥) can be obtained by solving
the LF QCD bound state equation [33]. However, before such first-principle solutions are
available, we would have to be contented with phenomenological amplitudes. One example
that has been used is the Gaussian type wave function [34–36]:
1 dk ~k2
z
φ(z,k⊥) = N exp , (11)
N dz −2ω2
r c M!
where N = 4(π/ωM2 )34, ~k = (k⊥,kz), and kz is defined through
z = EQ +kz , 1 z = EQ¯ −kz , E = m2 +~k2 (12)
E +E − E +E i i
Q Q¯ Q Q¯
q
by
k = z 1 M + m2Q¯ −m2Q , M = E +E . (13)
z 0 0 Q Q¯
− 2 2M
(cid:18) (cid:19) 0
4
−
and dk /dz = E E /z(1 z)M . After integrating over p in Eq. (3), we obtain
z Q Q¯ − 0 3
A(QQ¯ → γ∗(q1) γ∗(q2)) = eQeQ¯Nc q2+dp+3 2(2π)d32p3⊥3 p+ P− pΛ−P p− (I|p−3=p−3on)
Z0 Z i=1 i (cid:20) − 1on − 3on
1
+(ǫ ǫQ, q q ) +(p p ), (14)
− − − 1 2 1 2 1(3) 3(1)
q p p ↔ ↔ ↔
2 − 2on − 3on (cid:21)
and
m2 +p2
I = Tr[γ ( p +m ) ǫ (p +m ) ǫ (p +m )], p− = i i⊥ (15)
5 − 6 3 Q¯ 6 2 6 2 Q 6 1 6 1 Q ion p+
i
where the subscript on stands for the on-shell particles. One can extract the vertex
{ }
function Λ from Eqs. (3), (10) and (14), given by [32, 40, 41]:
P
Λ p+p+
P− −p−1oPn −p−3on = 2[M02 −p m1 Q3−mQ¯ 2] φ(z,k⊥) , (16)
q
To calculate the trace I, we use the definitions of t(cid:0)he LF mom(cid:1)entum variables (z(x),k⊥(k⊥′ ))
and take the frame with the transverse momentum (P q2)⊥ = 0 for the QQ¯ state (P) and
−
photon (q ) in Fig. 1a. Hence, the relevant quark variables are:
2
p+1 = zP+, p+3 = (1−z)P+, p1⊥ = zP⊥ −k⊥, p3⊥ = (1−z)P⊥ +k⊥.
p+2 = xq2+, p+3 = (1−x)q2+, p2⊥ = xq2⊥ −k⊥′ , p3⊥ = (1−x)q2⊥ +k⊥′ . (17)
At the quark loop, it requires that
′
k⊥ = (z −x)q2⊥ +k⊥. (18)
Take the trace I into Eq. (14) and consider the π0 meson Fock states, the form factor
Fπ0→γ∗γ∗(q12,q22) in Eq. (2) can be found to be:
Fπ0→γ∗γ∗(q12,q22) = −34 N6c d2x(2dπ2k)⊥3 Φ z,k⊥2 z(m1Q +z)(q12−z()mm2Q+k⊥2kΘ2)
r Z (cid:26) − 2 − Q ⊥
(cid:0) (cid:1)
¯
+(q q ) +(Q Q), (19)
2 1
↔ ↔
(cid:27)
with
z(1 z) dk ~k2
Φ(z,k2) = N − z exp ,
⊥ s 2M02 r dz −2ωM2 !
1 dΦ(z,k2)
~ ~ ~ ⊥
k = (k⊥,kz), x = zr, Θ = Φ(z,k2) dk2 ,
⊥ ⊥
q+ (m2 +q2 q2)+ (m2 +q2 q2)2 4q2m2
r = 2 = π 2 − 1 π 2 − 1 − 2 π . (20)
P+ 2m2
p P
5
III. NUMERICAL RESULT
To numerically evaluate the transition form factor of π0 γ∗γ, we need to specify the
→
parameters inEq. (19). To constrainthe quarkmasses ofm andthepionscale parameter
u,d,s
of ω , we use the meson decay constant f and the decay branching ratio of π0 2γ, given
π π0
→
by the PDG [42]
f = 130MeV, (π0 2γ) = (98.832 0.034)%, (21)
π0
B → ±
where the explicit expressions of f [43] and (π0 2γ) are
π0
B →
√Nc dxd2k⊥ m
fπ0 = 4 φ(x,k⊥) , (22)
√2 2(2π)3 m2 +k2
Z ⊥
and p
(4πα)2
(π0 2γ) = m3 F(0,0) 2, (23)
B → 64πΓ π| π0→2γ|
π
respectively. As an illustration, we extracte F(0,0) = 0.274 in GeV−1, m = m =
π0→2γ u
| |
m = 0.24 and ω = 0.31 in GeV, which will be used in our following numerical calculations.
d π
We now consider the case with one of the photons on the mass shell. From Eq. (19), the
transition pion form factor becomes
Fπγ(Q2) ≡ Fπ0→γ∗γ(Q2,0) = 4√32 N3c d2x(2dπ2k)⊥3 Φ z,k⊥2 z(1m+z)(Q12−z)(mmk2⊥2+Θk2)
r (cid:26)Z − − ⊥
dxd2k⊥Φ z,k2 m+((cid:0)1−z)(cid:1)mk⊥2Θ . (24)
− 2(2π)3 ⊥ (m2 +k2)
Z ⊥ (cid:27)
(cid:0) (cid:1)
In Fig. 2, we show the form factor in Eq. (24). We note that the first term in Eq. (24)
dominates for the lower region of Q2 and thus, it can be use to describe the experimental
data of BaBar [1], CLEO [44] and CELLO [45] with Q2 10 GeV2. The second one in
≤
Eq. (24), related to the non-valence quark contributions, is small for a small Q2, but it may
enhance the form factor with a high value of Q2. As a result, we will include this term in
our our numerical calculations. To easily examine the Q2 dependence of the form factor, we
have fitted our result in terms of the double-pole form:
F (Q2) = Fπ→γγ(0,0) . (25)
πγ
M +(βQ)2 (αQ)4
−
Explicitly, we find that the dimension parameters of α = 0.325, β = 1.15 and F (0,0) =
π0→γγ
0.274 in GeV−1, and the dimensionless parameter of M = 3.6. In Fig. 3, we concentrate on
6
0.35
Our fit
BaBar fit
0.30 CELLO
CLEO
V) 0.25 BaBar
e
G
(
) 0.20
2
Q
(

F 0.15
2
Q
0.10
0.05
0.00
0 5 10 15 20 25 30 35 40 45
2 2
Q (GeV )
FIG. 2. (Color online) Q2 dependence of F (Q2) in the LFQM.
πγ
the behavior of the form factor in the region with Q2 < 10 GeV2. It is easy to see that our
results fit the data well in this region similar to other theoretical calculations as expected.
In Fig. 4, we show the DA, φ(z), as the function of the momentum fraction of the internal
quark and meson longitudinal momenta, z, obtained by the integration of k⊥ in Eq. (11).
As we mentioned in the introduction, the BaBar result cannot be fitted by extending the
study in Ref. [37] to a high Q2. The main reason is due to the choices of the free parameters,
such as the quark masses and ω , leading to a different sharp of the pion wave function.
π
Similarly, the main difference between our results and those in Ref. [38] comes from DAs. In
particular, our DAshown inFig.4appearstobemuch broader. WenotethatinRefs. [8–12],
a more broader DA of the pion is utilized to fit the BaBar data, particularly in the high Q2
region. The results in these models differ slightly from ours only at large values of Q2. It is
interesting to point out that our result is almost identical with that in the Regge model [21]
and the double logarithmic behavior from the chiral anomaly effects [28]. Finally, we remark
that the single data point at Q2 = 27.31 GeV2 by BaBar, which is apparently consistent
with the QCD asymptotic limit, cannot be explained by this work within the framework of
the LFQM.
7
0.22
0.20
0.18
V) 0.16
e
G 0.14
(
2 ) 0.12
Q
(
0.10
F Our fit
2 0.08
Q BaBar fit
0.06 CELLO
CLEO
0.04
BaBar
0.02
0.00
0 2 4 6 8 10
2 2
Q (GeV )
FIG. 3. (Color online) F (Q2) for Q2 < 10 GeV2 in the LFQM.
πγ
0.25
0.20
0.15
z
0.10
0.05
0.00
0.0 0.2 0.4 0.6 0.8 1.0
z
FIG. 4. φ(z) as a function of z in the LFQM.
IV. CONCLUSIONS
We have studied the form factors of π0 γ∗γ within the LFQM. In our calculation,
→
we have adopted the Gaussian-type wave function and evaluated the form factors for the
momentum dependences in the all allowed Q2 region. We have also parametrized the form
8
factor in terms of the double-pole form. Our numerical values are close to the experimental
results by BaBar. In particular, our results of the transition form factor fit well with the
experimental data inthe highQ2 region, which cannot beexplained inthe previous attempts
based on the framework of the LFQM. Finally, we remark that due to the large uncertainty
in the high Q2 region for the BaBar data, further theoretical studies as well as more precise
experimental data are clearly needed. If some future experiment could not confirm the
BaBar data but be rather in agreement with the QCD asymptotic limit, the parameters of
the LFQM in this study should be either modified or ruled out.
V. ACKNOWLEDGMENTS
This work was partially supported by National Center of Theoretical Science and Na-
tional Science Council (NSC-97-2112-M-471-002-MY3 and NSC-98-2112-M-007-008-MY3)
of R.O.C.
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