BLFQ Collaboration

# Pion to photon transition form factors with basis light-front quantization

Chandan Mondal<EMAIL_ADDRESS>Institute of Modern Physics, Chinese
Academy of Sciences, Lanzhou 730000, China  School of Nuclear Science and
Technology, University of Chinese Academy of Sciences, Beijing 100049, China
Sreeraj Nair<EMAIL_ADDRESS>Institute of Modern Physics, Chinese
Academy of Sciences, Lanzhou 730000, China  School of Nuclear Science and
Technology, University of Chinese Academy of Sciences, Beijing 100049, China
Shaoyang Jia<EMAIL_ADDRESS>Department of Physics and Astronomy, Iowa State
University, Ames, IA 50011, U.S.A.  Physics Division, Argonne National
Laboratory, Argonne, IL 60439, U.S.A.  Xingbo Zhao<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China  School of Nuclear Science and Technology, University of Chinese Academy
of Sciences, Beijing 100049, China  James P. Vary<EMAIL_ADDRESS>Department of Physics and Astronomy, Iowa State University, Ames, IA 50011,
U.S.A.

###### Abstract

We obtain the distribution amplitude (DA) of the pion from its light-front
wave functions in the basis light-front quantization framework. This light-
front wave function of the pion is given by the lowest eigenvector of a light-
front effective Hamiltonian consisting a three-dimensional confinement
potential and the color-singlet Nambu–Jona-Lasinion interaction both between
the constituent quark and antiquark. The quantum chromodynamics (QCD)
evolution of the DA is subsequently given by the perturbative Efremov-
Radyushkin-Brodsky-Lepage evolution equation. Based on this DA, we then
evaluate the singly and doubly virtual transition form factors in the space-
like region for $\pi^{0}\rightarrow\gamma^{*}\gamma$ and
$\pi^{0}\rightarrow\gamma^{*}\gamma^{*}$ processes using the hard-scattering
formalism. Our prediction for the pion-photon transition form factor agrees
well with data reported by the Belle Collaboration. However, in the large
$Q^{2}$ region it deviates from the rapid growth reported by the BaBar
Collaboration. Meanwhile, our result on the
$\pi^{0}\rightarrow\gamma^{*}\gamma^{*}$ transition form factor is also
consistent with other theoretical approaches and agrees with the scaling
behavior predicted by perturbative QCD.

##  I Introduction

The parton distribution amplitudes (DAs) that play essential roles in
describing the various hard exclusive processes of quantum chromodynamics
(QCD) bound states Lepage:1980fj ; Chernyak:1983ej ; Brodsky:1989pv  via the
factorization theorem Collins:1996fb  are among the most basic structure
functions. The DAs are therefore complementary to the parton distribution
functions (PDFs) associated with inclusive processes Sutton:1991ay ;
Wijesooriya:2005ir ; Gluck:1999xe ; Aicher:2010cb . Since the DAs are
longitudinal projections of the hadronic wave functions obtained by
integrating out the transverse momenta of the partons Lepage:1980fj ;
Chernyak:1983ej ; Efremov:1979qk , they carry information on QCD bound states
at the amplitude level. Specifically, the lowest moments of the DAs for a
quark and an antiquark inside a meson are closely related to decay constants
and transition form factors Polyakov:2009je ; Noguera:2010fe ;
Terschlusen:2010gtc ; Petrov:1998kg .

The meson-photon transitions with one or two virtual photons are the simplest
decay processes in QCD, reflecting the structure of the meson. The associated
transition form factors (TFFs) are crucial for determining important
observables, for example the hadronic light-by-light contribution to the
Standard Model prediction of the muon anomalous magnetic moment
Jegerlehner:2009ry ; Nyffeler:2016gnb ; Gerardin:2016cqj ; Aoyama:2020ynm ;
Hoferichter:2018dmo ; Gerardin:2019vio ; Hoferichter:2018kwz , and the rates
of rare pseudoscalar ($P$) decays: $P\to l\bar{l}~{}(l\equiv e,\,\mu)$
Husek:2015wta ; Hoferichter:2021lct . The single-virtual TFFs in the space-
like regions up to a large momentum transfer $(Q^{2}\sim 40$ GeV2) have been
measured experimentally by several collaborations Behrend:1990sr ;
Gronberg:1997fj ; Denig:2014mma ; Uehara:2012ag ; Aubert:2009mc ; BABAR:2011ad
. The results from the BaBar Collaboration Aubert:2009mc  demonstrate a rapid
growth of $Q^{2}F_{\pi\gamma}(Q^{2})$ in the large $Q^{2}$ region,
($F_{P\gamma}(Q^{2})$ being the meson to photon TFF). However, the measurement
by the Belle Collaboration Uehara:2012ag  shows that
$Q^{2}F_{\pi\gamma}(Q^{2})\to$ constant for $Q^{2}>15$ GeV2. The data from the
Belle Collaboration are consistent with the prominent features of perturbative
QCD (pQCD) Lepage:1980fj ; Braaten:1982yp , where TFFs are expected to follow
the asymptotic behavior of, $Q^{2}F_{P\gamma}(Q^{2})\to{\rm constant}$ as
$Q^{2}\to\infty$. Unlike their results showing growth in
$Q^{2}F_{\pi\gamma}(Q^{2})$, the BaBar data BABAR:2011ad  for the
$Q^{2}F_{(\eta,\,\eta^{\prime})\gamma}(Q^{2})$ appear consistent with the pQCD
predication. The inconsistency between the Belle and the BaBar data for
$Q^{2}F_{\pi\gamma}(Q^{2})$ and the disparities in the behaviors of
$Q^{2}F_{\pi\gamma}(Q^{2})$ and $Q^{2}F_{(\eta,\eta^{\prime})\gamma}(Q^{2})$
at the high $Q^{2}$ regime as reported by the BaBar Collaboration have led to
various theoretical studies Polyakov:2009je ; Mikhailov:2009kf ;
Radyushkin:2009zg ; Dorokhov:2013xpa ; Wu:2010zc ; Kroll:2010bf ; Agaev:2012tm
; Roberts:2010rn ; Brodsky:2011xx ; Stefanis:2012yw ; Lucha:2011if ;
deMelo:2013zza ; Agaev:2014wna ; Choi:2017zxn ; Choi:2007yu ; Choi:1997iq ;
Stefanis:2020rnd ; Gerardin:2016cqj ; Ahmady:2018muv ; Zhong:2021epq ;
Gao:2021iqq ; Ding:2018xwy ; Raya:2016yuj .

While the measurements of the single-virtual TFFs having been carried out by
several collaborations, information is less available on the double-virtual
TFFs. Recently, the BaBar Collaboration BaBar:2018zpn  has measured for the
first time the double-virtual
$\gamma^{*}(q_{1})\gamma^{*}(q_{2})\to\eta^{\prime}$ TFF,
$F_{\eta^{\prime}\gamma^{*}}(Q^{1}_{1},Q^{2}_{2})$, in the space-like region
where $Q^{2}_{1(2)}=-q^{2}_{1(2)}>0$. Meanwhile, the meson ($M$) TFFs for the
doubly virtual $M\to\gamma^{*}\gamma^{*}$ transitions have been studied within
the Dyson-Schwinger and Bethe-Salpeter framework Weil:2017knt , the chiral
perturbation theory Bickert:2020kbn , a light-front quark model Choi:2019wqx ,
the anti-de Sitter (AdS)/QCD Brodsky:2011yv ; Stoffers:2011xe , and the
lattice QCD Gerardin:2016cqj ; Lin:2013im ; Feng:2012ck ; Dudek:2006ut ;
Chen:2016yau .

Our theoretical framework for meson structures is based on basis light front
quantization (BLFQ), which provides an approach for solving relativistic many-
body bound state structure in quantum field theories Vary:2009gt ; Li:2021jqb
; Zhao:2014xaa ; Wiecki:2014ola ; Li:2015zda ; Li:2017mlw ; Jia:2018ary ;
Lan:2019vui ; Lan:2019rba ; Lekha ; Tang:2018myz ; Tang:2019gvn ; Xu:2019xhk ;
Xu:2021wwj ; Lan:2019img ; Qian:2020utg ; Lan:2021wok . In this work, we
evaluate the DA of the pion using the light-front wave functions (LFWFs) based
on BLFQ Vary:2009gt , within the valence Fock sector of the pion. We then
evaluate the singly and doubly virtual TFFs in the space-like region for
$\pi^{0}\rightarrow\gamma^{*}\gamma$ and
$\pi^{0}\rightarrow\gamma^{*}\gamma^{*}$ transitions following the hard-
scattering formalism. The effective Hamiltonian includes a three dimensional
confining potential consisting of the light-front holography in the transverse
direction Brodsky:2014yha , a longitudinal confinement Li:2015zda ; Li:2017mlw
, and the color-singlet Nambu–Jona-Lasinio (NJL) interactions Klimt:1989pm ;
Shigetani:1993dx . The color-singlet NJL interactions account for the
dynamical chiral symmetry breaking of QCD. The nonperturbative solutions for
the LFWFs given by the recent BLFQ study of light mesons Jia:2018ary  have
been applied successfully to predict the electromagnetic form factors and
associated charge radii, PDFs, structure functions and generalized parton
distributions of the pion Jia:2018ary ; Lan:2019rba ; Lan:2019vui ; Lekha .
Here, we extend our investigations of the pion to compute its singly and
doubly virtual photon TFFs.

##  II BLFQ-NJL model for the light mesons

The structures of the bound states are embedded in the LFWFs obtainable as the
solutions of the eigenvalue equation of the Hamiltonian
$H_{\mathrm{eff}}|\Psi\rangle=M^{2}|\Psi\rangle$, where $H_{\mathrm{eff}}$ is
the effective Hamiltonian, with $M^{2}$ being the mass squared eigenvalue of
the state $|\Psi\rangle$. Within the current modeling of the meson structure
through the framework of BLFQ Vary:2009gt , we consider an effective light-
front Hamiltonian and solve for its mass eigenvalues and eigenstates at the
model scale suitable for low-resolution probes. In the valence Fock sector,
the effective Hamiltonian for the light mesons from Ref. Jia:2018ary  is given
by

$\displaystyle
H_{\mathrm{eff}}=\frac{\vec{k}^{2}_{\perp}+m_{q}^{2}}{x}+\frac{\vec{k}^{2}_{\perp}+m_{\bar{q}}^{2}}{1-x}+\kappa^{4}\vec{\zeta}_{\perp}^{2}-\frac{\kappa^{4}}{(m_{q}+m_{\bar{q}})^{2}}\partial_{x}\big{(}x(1-x)\partial_{x}\big{)}+H^{\rm
eff}_{\rm NJL}\,,$ (1)

where $m_{q}$ ($m_{\bar{q}}$) is the mass of the quark (antiquark), and
$\vec{k}_{\perp}$ is the relative transverse momentum. The parameter $\kappa$
represents the strength of the confinement. The transverse confinement is
adopted from the light-front holography, where the holographic variable is
defined as $\vec{\zeta}_{\perp}\equiv\sqrt{x(1-x)}\vec{r}_{\perp}$
Brodsky:2014yha . The variable $\vec{r}_{\perp}$ is conjugated to
$\vec{k}_{\perp}$ and measures the transverse separation between the quark and
antiquark. The $x$-derivative in the longitudinal confining potential
Li:2015zda  is defined as $\partial_{x}f(x,\vec{\zeta}_{\perp})=\partial
f(x,\vec{\zeta}_{\perp})/\partial x|_{\vec{\zeta}_{\perp}}$. The effective
Hamiltonian also includes $H_{\mathrm{NJL}}^{\mathrm{eff}}$ that corresponds
to the color-singlet NJL interaction to account for the chiral dynamics
Klimt:1989pm . For the positively-charged pion, the NJL interaction is given
by Jia:2018ary

$\displaystyle H_{\mathrm{NJL},\pi}^{\mathrm{eff}}$
$\displaystyle=G_{\pi}\,\big{\\{}\overline{u}_{\mathrm{u}s1^{\prime}}(p_{1}^{\prime})u_{\mathrm{u}s1}(p_{1})\,\overline{v}_{\mathrm{d}s2}(p_{2})v_{\mathrm{d}s2^{\prime}}(p_{2}^{\prime})+\overline{u}_{\mathrm{u}s1^{\prime}}(p_{1}^{\prime})\gamma_{5}u_{\mathrm{u}s1}(p_{1})\,\overline{v}_{\mathrm{d}s2}(p_{2})\gamma_{5}v_{\mathrm{d}s2^{\prime}}(p_{2}^{\prime})$
$\displaystyle\quad+2\,\overline{u}_{\mathrm{u}s1^{\prime}}(p_{1}^{\prime})\gamma_{5}v_{\mathrm{d}s2^{\prime}}(p_{2}^{\prime})\,\overline{v}_{\mathrm{d}s2}(p_{2})\gamma_{5}u_{\mathrm{u}s1}(p_{1})\big{\\}}\,,$
(2)

which is obtained from the NJL Lagrangian in the two flavor NJL model after
the Legendre transform Klimt:1989pm ; Vogl:1989ea ; Vogl:1991qt ;
Klevansky:1992qe . The nonitalic and italic subscripts in the Dirac spinors,
${u_{\mathrm{f}s}(p)}$ and ${v_{\mathrm{f}s}(p)}$, respectively represent the
flavors and the spins. Meanwhile, $p_{1}$ and $p_{2}$ are the momenta of the
valence quark and the valence antiquark, respectively. The coefficient
$G_{\pi}$ is the coupling constant of the NJL interaction. In this
interaction, only the combinations of Dirac bilinears relevant to the valence
Fock sector LFWFs of the pion have been included with the instantaneous terms
due to the NJL interactions neglected. The explicit expressions for the matrix
elements of the NJL interactions within the BLFQ framework can be found in
Ref. Jia:2018ary .

To compute the Hamiltonian matrix we follow BLFQ Vary:2009gt  and adopt the
two-dimensional (2D) harmonic oscillator (HO) basis functions, to describe the
transverse degrees of freedom, which are defined as Li:2015zda :

$\displaystyle\phi_{nm}\left(\vec{q}_{\perp};\kappa\right)$
$\displaystyle=\dfrac{1}{\kappa}\sqrt{\dfrac{4\pi
n!}{(n+|m|)!}}\left(\dfrac{|\vec{q}_{\perp}|}{\kappa}\right)^{|m|}\exp\left(-\dfrac{\vec{q}_{\perp}^{2}}{2\kappa^{2}}\right)L_{n}^{|m|}\left(\dfrac{\vec{q}_{\perp}^{2}}{\kappa^{2}}\right)\,e^{im\varphi}\,,$
(3)

with $\tan(\varphi)=q_{2}/q_{1}$, $L_{n}^{|m|}(z)$ is the associated Laguerre
polynomial, $n$ and $m$ are the radial and the angular quantum numbers,
respectively. On the other hand, the basis functions in the longitudinal
direction are defined as Li:2015zda

$\displaystyle\quad\chi_{l}(x;\alpha,\beta)=\sqrt{4\pi(2l+\alpha+\beta+1)}\sqrt{\dfrac{\Gamma(l+1)\Gamma(l+\alpha+\beta+1)}{\Gamma(l+\alpha+1)\Gamma(l+\beta+1)}}x^{\beta/2}(1-x)^{\alpha/2}\,P_{l}^{(\alpha,\beta)}(2x-1)\,,$
(4)

where $P_{l}^{(\alpha,\beta)}(z)$ is the Jacobi polynomial with the
dimensionless parameters $\alpha$ $=$
$2m_{\overline{q}}(m_{q}+m_{\overline{q}})/\kappa^{2}$, $\beta$ $=$
$2m_{q}(m_{q}+m_{\overline{q}})/\kappa^{2}$, and $l=0,\,1,\,2,...$. Using
basis functions specified in Eqs. (3) and (4), the valence wave function in a
given spin combination is expanded as

$\displaystyle\quad\psi_{rs}(x,\vec{k}_{\perp})=\sum_{n,m,l}\langle
n,m,l,r,s|\psi\rangle~{}\phi_{nm}\left(\dfrac{\vec{k}_{\perp}}{\sqrt{x(1-x)}};\kappa\right)\chi_{l}(x;\alpha,\beta),$
(5)

where the coefficients $\langle n,m,l,r,s|\psi\rangle$ represent the LFWFs in
our BLFQ basis representation obtained from diagonalizing the truncated
Hamiltonian. We truncate the infinite dimensional Hilbert space of the valence
Fock sector to a finite dimension by imposing the following restrictions on
the quantum numbers Jia:2018ary :

$0\leq n\leq N_{\mathrm{max}},\quad-2\leq m\leq 2,\quad 0\leq l\leq
L_{\mathrm{max}}\,,$ (6)

where $L_{\text{max}}$ specifies the basis resolution in the longitudinal
direction, while $N_{\text{max}}$ controls the transverse momentum covered by
2D HO functions. Since the NJL interactions do not couple to $|m|\geq 3$ basis
states, we have a natural truncation for $m$ Jia:2018ary . The LFWFs of the
mesons $\psi_{rs}(x,\vec{k}_{\perp})$ are normalized as

$\displaystyle\sum_{r,s}\int_{0}^{1}\\!\\!\frac{dx}{2x(1-x)}\\!\int\\!\frac{d^{2}\vec{k}_{\perp}}{(2\pi)^{3}}\big{|}\psi_{rs}(x,\vec{k}_{\perp})\big{|}^{2}\\!\\!=\\!\\!1.$
(7)

Parameters in the BLFQ-NJL model have been fixed to generate the ground state
masses of the light pseudoscalar as well as the vector mesons and the charge
radii of the $\pi^{+}$ and the $K^{+}$ Jia:2018ary . The LFWFs in this model
provide a high quality description of the electromagnetic form factors
Jia:2018ary , PDFs for the pion and the kaon and pion-nucleus induced Drell-
Yan cross sections Lan:2019vui ; Lan:2019rba . The coefficient $\langle
n,m,l,r,s|\psi\rangle$ for the neutral pion is assumed to be identical to that
of the charged pion, resulting in the only difference between their wave
functions being flavor.

##  III Parton distribution amplitude

The DAs are defined using the light-like separated gauge invariant vacuum-to-
meson matrix elements. Explicitly in the light-front formalism, the leading-
twist DAs $\phi_{M}(x;\mu)$ in the light-cone gauge for a pseudoscalar are
defined by Lepage:1980fj ; Brodsky:1997de

$\displaystyle\langle
0|\overline{\psi}(z)\gamma^{+}\gamma_{5}\psi(-z)|M(p)\rangle_{\mu}=\,$
$\displaystyle{\mathrm{i}}p^{+}f_{M}\int_{0}^{1}{\mathrm{d}}x\,e^{{\mathrm{i}}p^{+}z^{-}(x-{\frac{1}{2}})}\phi_{M}(x;\mu)\Big{|}_{z^{+},\vec{z}_{\perp}=0}\,,$
(8)

where $f_{M}$ are the decay constants. The non-local matrix elements as well
as the DAs depend on the scale $\mu$, the renormalization scale which we take
to be the UV cutoff. Following these definitions, the DAs are normalized to
unity:

$\int_{0}^{1}{\mathrm{d}}x\,\phi(x;\mu)=1\,.$ (9)

In terms of LFWF, the DAs of pseudoscalar states can be written as
Lepage:1980fj ,

$\phi(x,\mu_{0})=\frac{2\sqrt{2N_{c}}}{f_{M}}\frac{1}{\sqrt{x(1-x)}}\int\limits\frac{{\mathrm{d}}^{2}\vec{k}_{\perp}}{2(2\pi)^{3}}\psi_{\uparrow\downarrow\mathchar
45\relax\downarrow\uparrow}^{\lambda=0}(x,\vec{k}_{\perp})\,,$ (10)

where
$\psi_{\uparrow\downarrow-\downarrow\uparrow}=(\psi_{\uparrow\downarrow}-\psi_{\downarrow\uparrow})/\sqrt{2}$
and $N_{c}$ is the number of colours. In the BLFQ-NJL model, we use $N_{c}=3$.
We compute the DA of the pion at the model scale using the LFWFs given in Eq.
(5). Recall that the flavor wave function of the neutral pion is the only
difference compared to the wave function of the charged pion in the BLFQ-NJL
model.

The QCD evolution of the DA is specified by the Efremov-Radyushkin-Brodsky-
Lepage (ERBL) equations Efremov:1978rn ; Efremov:1979qk ; Lepage:1980fj . In a
Gegenbauer basis, one has RuizArriola:2002bp

$\phi(x,\mu)=6x(1-x)\sum_{n=0}^{\infty}C_{n}^{3/2}(2x-1)a_{n}(\mu)\,,$ (11)

where $C_{n}^{\frac{3}{2}}(2x-1)$ is a Gegenbauer polynomial and

$a_{n}(\mu)=\frac{2}{3}\frac{2n+3}{(n+1)(n+2)}\left(\frac{\alpha_{s}(\mu)}{\alpha_{s}(\mu_{0})}\right)^{\gamma_{n}^{0}/2\beta_{0}}\int_{0}^{1}\mathrm{d}xC_{n}^{3/2}(2x-1)\phi(x,\mu_{0})\;,$
(12)

with

$\gamma_{n}^{(0)}=-2C_{\rm
F}\left[3+\frac{2}{(n+1)(n+2)}-4\sum_{k}^{n+1}\frac{1}{k}\right];\quad\quad\beta_{0}=\frac{11}{3}C_{\rm
A}-\frac{2}{3}n_{\rm f}\;,$ (13)

in the leading order (LO). Here, the color factors are given by $C_{\rm
F}=\frac{4}{3}$ and $C_{\rm A}=3$. We take $n_{\rm f}=3$ to be the number of
active flavors. The strong running coupling is given by

$\alpha_{\rm
s}(\mu)=\frac{4\pi}{\beta_{0}\ln\left(\mu^{2}/\Lambda^{2}_{\mathrm{QCD}}\right)}\,,$
(14)

with $\Lambda_{\mathrm{QCD}}=0.204$ GeV being the QCD scale parameter. It is
also useful to compute the moments in order to quantitatively compare with
other theoretical predictions. The $p$-th moment of the DA is defined as,

$\displaystyle\langle z_{p}\rangle=\int_{0}^{1}dx~{}z^{p}~{}\phi(x,\mu)\,,$
(15)

where $z\equiv(2x-1)$ when $p\geq 1$ and $z\equiv x$ for $p=-1$. Meanwhile,
the moments of the DA are directly related to the Gegenbauer coefficients in
Eq. (11). The second, fourth and sixth moments are expressed as Choi:2007yu

$\langle z_{2}\rangle=\frac{12}{35}a_{2}+\frac{1}{5}\;,$ (16) $\langle
z_{4}\rangle=\frac{3}{35}+\frac{8}{35}a_{2}+\frac{8}{77}a_{4}\;,$ (17)
$\langle
z_{6}\rangle=\frac{1}{21}+\frac{12}{77}a_{2}+\frac{120}{1001}a_{4}+\frac{64}{2145}a_{6}\;.$
(18)

Figure 1: The valence quark DA of the pion in the BLFQ-NJL model. Left panel:
the DAs for $L_{\rm max}=8$ (blue dashed) $L_{\rm max}=32$ (red dash-dotted)
and extrapolated to $L_{\rm max}\to\infty$ that fits to Eq. (19) with
$a=b=0.62$ (black-solid) at the model scale. Right panel: the evolved DA from
the initial scale ($\mu_{0}^{2}=0.120\pm 0.012$ GeV2) using ERBL equations to
the experimental scale of $10$ GeV2. The black dashed line corresponds to the
DA at the initial scale, while the blue solid line represents the evolved DA.
Our result is compared with the FNAL-E-791 data Aitala:2000hb  and the
asymptotic DA (magenta dotted): $6x(1-x)$ RuizArriola:2002bp .

The LFWFs of the valence quarks in the light mesons have been solved in the
BLFQ framework using the NJL interactions as discussed in the Section II. We
employ the wave functions obtained with truncated basis to compute the DA of
the pion using Eq. (10). On the left panel of Fig. 1, we show the valence
quark DA of the pion at the model scale using the basis truncation
$N_{\text{max}}=8$ with $L_{\text{max}}=8,\,32$, and extrapolation to
$\infty$. The oscillations on the DA are numerical artifacts, while with
increasing $L_{\text{max}}$ the DA tends toward a smooth function with
decreasing oscillation amplitude about a single-peaked function. The DA for
$L_{\text{max}}\to\infty$ is therefore fitted to the following functional form
Jia:2018ary :

$\phi(x,\mu_{0})=\dfrac{x^{a}(1-x)^{b}}{B(a+1,b+1)}\,,$ (19)

with $a=b=0.6$, where $B(a+1,b+1)$ is the Euler beta function. We illustrate
the pion valence quark DA after QCD evolution on the right panel of Fig. 1.
Explicitly, we evolve our input DA from the model scale $\mu_{0}^{2}=0.120\pm
0.012$ GeV2 Lan:2019rba  to the experimental scale of the FNAL-E-791
experiment, $10$ GeV2 Aitala:2000hb . The initial scale was determined by
requiring the PDF result after QCD evolution utilizing the LO DGLAP equation
to fit the pion valence quark PDF data from the FNAL-E-615 experiment
Conway:1989fs . As can be seen, our evolved DA is in excellent agreement with
the FNAL-E-791 data. On the other hand, the pion DA in the BLFQ-NJL model is
very close to the asymptotic DA already at $\mu^{2}=10$ GeV2.

Table 1: Our predictions for the first three non-vanishing moments and inverse moment of the pion DA, compared to other theoretical predictions. |  $\mu{\rm[GeV]}$ |  $\langle z_{2}\rangle$ |  $\langle z_{4}\rangle$ |  $\langle z_{6}\rangle$ |  $\langle x^{-1}\rangle$
---|---|---|---|---|---
BLFQ-NJL (this work) | 1, 2 | 0.221, 0.217 | 0.099, 0.097 | 0.057, 0.055 | 3.21, 3.17
Playkurtic Stefanis:2014nla  | 2 | $0.220^{+0.009}_{-0.006}$ | $0.098^{+0.008}_{-0.005}$ | - | $3.13^{+0.14}_{-0.10}$
NLC Sum Rules Bakulev:2001pa  | 2 | $0.248^{+0.016}_{-0.015}$ | $0.108^{+0.05}_{-0.03}$ | - | 3.16(9)
LF Quark Model Choi:2007yu  | $\sim 1$ | 0.24(22) | 0.11(9) | 0.07(5) | -
Sum Rules Ball:2004ye  | 1 | 0.24 | 0.11 | - | -
AdS/QCD Brodsky:2011yv  | $\sim 1$ | 0.25 | 0.125 | 0.078 | 3.98
LF Holographic ($B=0$) Ahmady:2018muv  | 1, 2 | 0.180, 0.185 | 0.067, 0.071 | - | 2.81, 2.85
LF Holographic ($B\gg 1$) Ahmady:2018muv  | 1, 2 | 0.200, 0.200 | 0.085, 0.085 | - | 2.93, 2.95
Renormalon model Agaev:2005rc  | 1 | 0.28 | 0.13 | - | -
Instanton vacuum (MIA 1) Nam:2006au  | 1, 2 | 0.237, 0.218 | 0.112, 0.094 | 0.066, 0.052 | -
Instanton vacuum (MIA 2) Nam:2006au  | 1, 2 | 0.239, 0.220 | 0.113, 0.096 | 0.067, 0.053 | -
Sum Rules Chernyak:1983ej  | 2 | 0.343 | 0.181 | - | 4.25
Dyson-Schwinger [RL, DB] Chang:2013pq  | 2 | 0.280, 0.251 | 0.151, 0.128 | - | 5.5, 4.6
QCD background field theory sum rule Zhong:2021epq  | 1 | 0.271(13) | 0.138(10) | 0.087(6) | 3.95
QCD background field theory sum rule Zhong:2021epq  | 2 | 0.254(10) | 0.125(7) | 0.077(6) | 3.33
Lattice QCD Arthur:2010xf  | 2 | 0.28(1)(2) | - | - | -
Lattice QCD Braun:2015axa  | 2 | 0.2361(41)(39) | - | - | -
Lattice QCD Braun:2006dg  | 2 | 0.27(4) | - | - | -
Lattice QCD Bali:2017ude  | 2 | 0.2077(43) | - | - | -
Lattice QCD Bali:2019dqc  | 2 | 0.234(6)(6) | - | - | -
Lattice QCD Zhang:2020gaj  | 2 | 0.244(30) | - | - | -
Asymptotic QCD | $\infty$ | 0.200 | 0.086 | 0.048 | 3.00

The numerical values of the first three non-vanishing moments and the inverse
moment of the pion DA in the BLFQ-NJL model are presented in Table 1. We
compare our predictions with the results obtained from various theoretical
approaches. Our predictions for $z_{2}$ and/or $z_{4}$ roughly agree with
those in Refs. Stefanis:2014nla ; Bakulev:2001pa ; Choi:2007yu ; Ball:2004ye ;
Brodsky:2007hb ; Ahmady:2018muv ; Nam:2006au ; Braun:2015axa ; Bali:2017ude ;
Bali:2019dqc ; Zhang:2020gaj . For $z_{6}$, the numerical values are compared
with Refs. Choi:2007yu ; Brodsky:2011yv ; Nam:2006au ; Zhong:2021epq , with
our predictions being close to the results evaluated from the QCD instanton
vacuum Nam:2006au . The inverse moment of the pion DA in the BLFQ-NJL model is
in good agreement with Refs. Stefanis:2014nla ; Bakulev:2001pa , while
differing from other predictions summarized in Table 1.

Based on the BLFQ-NJL model, the decay constant for the pion is given in Table
2. In order to gain a clear impression on how basis truncation affects the
decay constant, we include the numerical values with basis cutoffs $N_{\rm
max}=8$ with $L_{\rm max}=8$, $16$, and $32$ to demonstrate a good convergence
trend. With $L_{\rm max}=32$ we predict $f_{\pi}=145.3$ MeV, while the
experimental data is $f_{\pi}^{\rm exp}=130.2$ MeV ParticleDataGroup:2018ovx .
Note that the decay constants computed here differ from the results presented
in Ref. Jia:2018ary , where an overall factor ($\sqrt{2}$) was erroneously
included.

We now use the Alder-Bell-Jackiw (ABJ) anomaly relations Adler:1969gk ;
Bell:1969ts  to compute the pion-photon TFF at zero momentum transfer as
follows Choi:2017zxn ; Ahmady:2018muv :

$F_{\pi\gamma}^{\mathrm{ABJ}}(0)=\frac{1}{2\sqrt{2}\pi^{2}f_{\pi^{0}}}\;,$
(20)

so that we can evaluate the radiative decay width using

$\Gamma_{\pi\to\gamma\gamma}=\frac{\pi}{4}\alpha^{2}_{\mathrm{EM}}M_{\pi}^{3}|F_{\pi\gamma}(0)|^{2}.$
(21)

Our results are presented in Table 2, where we find reasonable agreement with
the experimental value ParticleDataGroup:2018ovx . With the largest basis size
in our current calculation, i.e., $N_{\rm max}=8$ and $L_{\rm max}=32$, we
obtain $\Gamma_{\pi\to\gamma\gamma}=6.98\times 10^{-3}$ keV, whereas the
experimentally measured value is $\Gamma_{\pi\to\gamma\gamma}=7.82\times
10^{-3}$ keV.

With the original model parameters given in Ref. Jia:2018ary , our prediction
for the pion decay constant is somewhat larger than the experimental data,
which effectively leads to a slightly smaller radiative decay width compared
to the experimental value. However, if we relax the constraint from the charge
radius, we can modify the model parameters into $m_{q}=356.77$ MeV,
$\kappa=182.35$ MeV, and $G_{\pi}=2.6573\times 10^{-5}$ MeV for the basis
cutoff $N_{\rm max}=8$ and $L_{\rm max}=32$ in order to reproduce the
experimental decay constant.

Table 2: Our predictions for the decay constant, $f_{\pi}$, of the pion and the radiative decay width, $\Gamma_{\pi\to 2\gamma}$, compared to the measured values from Particle Data Group (PDG) ParticleDataGroup:2018ovx . The BLFQ-NJL model results are quoted using the LFWFs at the basis cutoff $N_{\rm max}=8$ and $L_{\rm max}=8$, $16$ and $32$, respectively. $[N_{\rm max},\,L_{\rm max}]\to$ |  $[8,8]$ |  $[8,16]$ |  $[8,32]$ |  Experimental data ParticleDataGroup:2018ovx
---|---|---|---|---
$f_{\pi}$ (MeV) | 142.8 | 144.8 | 145.3 | 130.2
$\Gamma_{\pi\to 2\gamma}$ (keV) | 7.22 $\times$ 10-3 | 7.03 $\times$ 10-3 | 6.98 $\times$ 10-3 | (7.82 $\pm$ 0.22) $\times$ 10-3

##  IV Pion to photons transition form factors

The meson-photon transition form factor, $F_{M\gamma}(Q^{2})$, of a
pseudoscalar meson ($M$) for the $M\to\gamma^{*}\gamma$ decay is defined
through the matrix element of electromagnetic current as: Lepage:1980fj

$\displaystyle\langle\gamma(P-q)|J^{\mu}|M(P)\rangle=-ie^{2}F_{M\gamma}(Q^{2})\epsilon^{\mu\nu\rho\sigma}P_{\nu}\epsilon_{\rho}q_{\sigma}\,,$
(22)

where $P$ and $q$ are the momenta of the meson and the virtual photon,
respectively. The invariant 4-momentum transfer squared $Q^{2}=-q^{2}>0$ and
$\epsilon_{\rho}$ is the transverse polarization vector of the final (on-
shell) photon. The TFF, $F_{M\gamma}(Q^{2})$, can be evaluated from the
convolution of a hard scattering amplitude (HSA), $T_{\mathrm{H}}(x,Q^{2})$,
calculable in perturbation theory, with a nonperturbative DA Lepage:1980fj ;
Brodsky:1981rp ; Brodsky:2011yv  using

$\displaystyle
Q^{2}F_{\pi\gamma}(Q^{2})=\frac{\sqrt{2}}{3}f_{\pi}\int_{0}^{1}{\rm
d}x\,T_{\mathrm{H}}(x,Q^{2})\,\phi(x,(1-x)Q),$ (23)

where $T_{\mathrm{H}}(x,Q^{2})$ to the next-to-leading order (NLO) is given by
delAguila:1981nk ; Braaten:1982yp ; Kadantseva:1985kb ; Melic:2001wb ;
Melic:2002ij ,

$\displaystyle T_{\mathrm{H}}(x,Q^{2})$
$\displaystyle=\frac{1}{1-x}+\frac{\alpha_{s}(\mu_{\mathrm{R}})}{4\pi}C_{\mathrm{F}}\frac{1}{1-x}\,\Bigg{[}-9-\frac{1-x}{x}{\rm
ln}(1-x)$ $\displaystyle+{\rm ln}^{2}(1-x)+\left\\{3+2\,{\rm
ln}(1-x)\right\\}{\rm
ln}\left(\frac{Q^{2}}{\mu^{2}_{\mathrm{R}}}\right)\Bigg{]}.$ (24)

For simplicity, the regularization scale, $\mu_{\mathrm{R}}^{2}$, is taken as
$\mu_{\mathrm{R}}^{2}=Q^{2}$ to eliminate large logarithm terms. At leading
order, only the first term in Eq. (24) contributes to the TFF. Using the
standard hard scattering approach, the NLO corrections have been studied in
Refs. delAguila:1981nk ; Braaten:1982yp ; Kadantseva:1985kb ; Melic:2001wb ;
Melic:2002ij  under the assumption that ${\phi(x,(1-x)Q)\simeq\phi(x,Q)}$. The
replacement is reasonable for the behavior at the asymptotic limit
$Q^{2}\to\infty$. However, this approximation is not well justified for the
calculation below the asymptotic region, where one needs to take into account
the ERBL evolution effects. A proper treatment of the NLO calculations with
the evolution effects and $\phi(x,(1-x)Q)$ has been illustrated in Ref.
Brodsky:2011yv .

Figure 2: The $\pi^{0}\to\gamma^{*}\gamma$ transition form factor with BLFQ
basis truncation $N_{\rm max}=8$ and $L_{\rm max}=32$. Left panel: the solid
and dashed lines correspond to the results with LO and NLO hard scattering
amplitudes, respectively. Right panel: Comparison of the TFFs evaluated with
the DAs $\phi(x,Q)$ (magenta lines) and $\phi(x,(1-x)Q)$ (black lines) in Eq.
(23). Effect of the scale evolution is considered following Eq. (11). Figure
3: Comparison of the $\pi^{0}\to\gamma^{*}\gamma$ transition form factors
evaluated with the original model parameters (Set-I) given in Ref. Jia:2018ary
(black lines) and the re-adjusted parameters (Set-II) that fit the pion
experimental pion decay constant (magenta lines) both with BLFQ basis
truncation $N_{\rm max}=8$ and $L_{\rm max}=32$.

Figure 4: Sensitivity of the $\pi^{0}\to\gamma^{*}\gamma$ transition form
factor to BLFQ basis truncation. The magenta dot-dashed, red dashed, black
solid lines correspond to the $\\{N_{\rm max},\,L_{\rm
max}\\}\equiv\\{8,\,8\\}$, $\\{8,\,16\\}$, and $\\{8,\,32\\}$, respectively.
Left and right panels represent the results with LO and NLO hard scattering
amplitude, respectively.

Inserting the evolved DA from Eq. (11) into Eq. (23) we evaluate
$Q^{2}F_{\pi\gamma}(Q^{2})$ with the ERBL evolution considered. Figure 2 (left
panel) shows our results for the pion to photon TFF,
$Q^{2}F_{\pi\gamma}(Q^{2})$, in the BLFQ-NJL model and compares with the
available experimental data from the Belle Uehara:2012ag , the BaBar
Aubert:2009mc , the CLEO CLEO:1997fho , and the CELLO CELLO:1990klc
Collaborations. The gray band corresponds to the correction in the TFF due to
the $\alpha_{s}$ order correction in the HSA in Eq. (24). We need to set
$(1-x)Q=\mu_{0}$ for $(1-x)Q<\mu_{0}$ to ensure the convergence of the
integration in Eq. (23). We obtain the TFF results with the BLFQ basis
truncation: $N_{\rm max}=8$ with $L_{\rm max}=32$ and find good agreement of
our calculated pion TFF with the experiments performed by the Belle
Uehara:2012ag , the CLEO CLEO:1997fho , and the CELLO CELLO:1990klc
Collaborations. However, it deviates from the rapid growth in the large
$Q^{2}$ region reported by the BaBar Collaboration Aubert:2009mc . There are
also theoretical studies suggesting that the BaBar data are incompatible with
QCD calculations Mikhailov:2009sa ; Roberts:2010rn ; Bakulev:2011rp ;
Wu:2011gf . It has also been demonstrated in Ref. Brodsky:2011yv  that the
explanation of BaBar data at large $Q^{2}$ with the QCD calculations using the
asymptotic QCD, AdS/QCD, and Chernyak-Zhitnitsky models for the pion DA is not
possible, but can be accommodated by a flat modeling of the pion DA
Polyakov:2009je ; Brodsky:2011yv . However, the calculations with such a DA
underestimate significantly the pion TFF at low $Q^{2}$. Such a DA also shows
a trend that would appear to violate the Brodsky-Lepage limit of
$Q^{2}F_{\pi\gamma}(Q^{2}\to\infty)=2f_{\pi}$ Brodsky:2011yv . Except those
using the flat modeling of the pion DA Dorokhov:2009dg ; Radyushkin:2009zg ;
Polyakov:2009je ; Li:2009pr , there exist phenomenological studies reproducing
the BaBar data for the pion to photon TFF Wu:2010zc ; Kroll:2010bf ;
RuizArriola:2010mrb ; Gorchtein:2011vf ; Pham:2011zi ; Dorokhov:2010zzb ;
Agaev:2010aq ; Kotko:2009ij .

In Fig. 2 (right panel), we compare the results calculated using
$\phi(x,{(1-x)}Q)$ and $\phi(x,Q)$ in Eq. (23). We notice that it is a
reasonable approximation to use $\phi(x,Q)$ in the perturbative regime, where
both $\phi(x,(1-x)Q)$ and $\phi(x,Q)$ lead to almost identical pion TFF.
However, using $\phi(x,Q)$ fails to reproduce the experimental data at low
$Q^{2}$ region, particularly for the $Q^{2}<10$ GeV2, whereas the results
using $\phi(x,(1-x)Q)$ agree well with the available data.

Parameters in the BLFQ-NJL model have been fixed to fit the charge radius of
the pion Jia:2018ary , while they lead to a slightly larger pion decay
constant compared to the experimental data as summarized in Table 2. However,
the experimental the decay constant can be obtained by adjusting the model
parameters into those mentioned by the end of Section III. In order to gain a
clear impression on how the change in parameters affects
$Q^{2}F_{\pi\gamma}(Q^{2})$, in Fig. 3 we present a comparison of the singly
virtual pion to two photon TFF computed using the original model’s parameters
given in Ref. Jia:2018ary  and the modified parameters. We find that the two
sets of parameters provide qualitatively similar results for this TFF at small
$Q^{2}$. However, $F_{\pi\gamma}(Q^{2})$ with new parameters is narrower than
that obtained using the original model’s parameters, which accounts for the
difference in the TFF for large $Q^{2}$.

The sensitivity of the BLFQ-NJL model prediction to basis truncation is shown
in Fig. 4, where we present the results for $\\{N_{\rm max},\,L_{\rm
max}\\}\equiv\\{8,\,8\\}$, $\\{8,\,16\\}$, and $\\{8,\,32\\}$. The results
show a good convergence trend over the range of $Q^{2}$ as evident by finding
that the $L_{\rm max}=16$ and $L_{\rm max}=32$ results nearly coincide with
each other in contrast to the $L_{\rm max}=8$ results presented in Fig. 4.
This observed convergence in the TFFs is reassuring since the DAs are also
reasonably well converged as can be seen in the left panel of Fig. 1. The
difference between the $L_{\rm max}=32$ and $8$ values is presented as our
uncertainty estimate in the pion TFF.

We now turn our attention to the case in which the photons as decay products
are both off mass-shell, i.e., for the TFF
$F_{\pi\gamma^{*}}(Q_{1}^{2},Q_{2}^{2})$. This pion to doubly virtual photons
($\pi^{0}\to\gamma^{*}\gamma^{*}$) TFF can be obtained by replacing the hard-
scattering amplitude $T_{\mathrm{H}}$ in the previous analysis with an
appropriate expression. The TFF, $F_{M\gamma^{*}}(Q_{1}^{2},Q_{2}^{2})$, can
be expressed from the convolution of a HSA,
$T_{\mathrm{H}}(x,Q_{1}^{2},Q_{2}^{2})$ with a nonperturbative DA
Lepage:1980fj ; Brodsky:1981rp ; Brodsky:2011yv ,

$\displaystyle
F_{\pi\gamma^{*}}(Q_{1}^{2},Q_{2}^{2})=\frac{\sqrt{2}}{3}f_{\pi}\int_{0}^{1}{\rm
d}x\,T_{\mathrm{H}}^{\gamma^{*}\gamma^{*}\to\pi^{0}}(x,Q_{1}^{2},Q_{2}^{2})\,\phi(x,\bar{Q}),$
(25)

where we assume $\bar{Q}=(1-x)Q_{1}+xQ_{2}$ and at the leading order
$T_{\mathrm{H}}$ has the form

$\displaystyle
T_{\mathrm{H}}^{\gamma^{*}\gamma^{*}\to\pi^{0}}(x,Q_{1}^{2},Q_{2}^{2})=\frac{1}{(1-x)Q_{1}^{2}+xQ_{2}^{2}}.$
(26)

Note that the singly virtual TFF $F_{\pi\gamma}(Q^{2})$ can be obtained by
setting one of the momentum transfers to zero in Eq. (25). While the
$F_{\pi\gamma}(Q^{2})$ is sensitive to the shape of the pion DA, the doubly
virtual TFF $F_{\pi\gamma^{*}}(Q_{1}^{2},Q_{2}^{2})$ when $Q_{1}^{2}\neq 0$
and $Q_{2}^{2}\neq 0$ is much less sensitive to the end-point behavior of the
pion DA, since the HSA in Eq. (26) is well-behaved at the end-points,
$x=\\{0,1\\}$. It can also be noted that, for the kinematic region satisfying
$Q_{1}^{2}=Q_{2}^{2}$, the HSA becomes independent of $x$ and thus
$Q_{1}^{2}F_{\pi\gamma^{*}}(Q_{1}^{2},Q_{1}^{2})$ only depends on the
normalization of the pion DA, which is scale and model independent. One then
has $Q^{2}F_{\pi\gamma^{*}}(Q^{2},Q^{2})\to\frac{\sqrt{2}}{3}f_{\pi}$. For
this transition to two off-shell photons, the pQCD expression of
$T_{\mathrm{H}}^{\gamma^{*}\gamma^{*}\to\pi^{0}}$ at NLO can be found in Ref.
Braaten:1982yp .

Table 3: The transition form factors $F_{\pi\gamma^{*}}(Q^{2}_{1},Q^{2}_{2})$ (in units of $10^{-3}$ GeV-1) for some ($Q^{2}_{1},Q^{2}_{2}$) values (in units of GeV2) compared with the LFQM Choi:2019wqx , the vector meson dominance (VMD) model Choi:2019wqx , and the pQCD predictions. The first and second values for each row entry in the second and third columns correspond to the basis truncations $L_{\rm max}=8$ and $32$, respectively. The difference between the $L_{\rm max}=8$ and $32$ values are presented as our uncertainty estimate. $(Q^{2}_{1},Q^{2}_{2})$ | $F^{\rm LO}_{\pi\gamma^{*}}$ | $F^{\rm NLO}_{\pi\gamma^{*}}$ | pQCD | pQCD | LFQM Choi:2019wqx  | VMD Choi:2019wqx
---|---|---|---|---|---|---
| (this work) | (this work) | LO | NLO |  |
(6.48, 6.48) | 10.39 - 10.56 | 9.59 - 9.75 | 9.52 | 8.78 | 9.08 | 1.957 $\pm$ 0.022
(16.85, 16.85) | 3.99 - 4.06 | 3.73 - 3.79 | 3.66 | 2.69 | 3.58 | 0.322 $\pm$ 0.004
(14.83, 4.27) | 7.55 - 7.72 | 7.00 - 7.14 | 6.91 | 6.39 | 6.76 | 1.301 $\pm$ 0.014
(38.11, 14.95) | 2.65 - 2.69 | 2.48 - 2.52 | 2.42 | 2.27 | 2.40 | 0.163 $\pm$ 0.002
(45.63, 45.63) | 1.47 - 1.50 | 1.39 - 1.41 | 1.35 | 1.33 | 1.33 | 0.046 $\pm$ 0.001

Figure 5: The three-dimensional plots for the $\pi^{0}\to\gamma^{*}\gamma^{*}$
transition form factor obtained from Eq. (25) with BLFQ basis truncation
$N_{\rm max}=8$ and $L_{\rm max}=32$. Left and right panels represent the
results with LO and NLO hard scattering amplitude, respectively. Note the
difference in the scales on the vertical axes. The effect of the scale
evolution in the DA is considered following Eq. (11).

The numerical results for $F_{\pi\gamma^{*}}(Q_{1}^{2},Q_{2}^{2})$ within the
BLFQ-NJL model calculated at LO and NLO for selected $(Q_{1}^{2},Q_{2}^{2})$
values are given in Table 3. We compare our results with results from the
light-front quark model (LFQM) and with results from LO and NLO pQCD
Choi:2019wqx . We find that our predictions are close to these results as
summarized in Table 3. On the other hand, our predictions are very different
from the VMD model BaBar:2018zpn ; Choi:2019wqx :

$\displaystyle F^{\rm
VMD}_{{\rm\pi}\gamma^{*}}(Q^{2}_{1},Q^{2}_{2})=\frac{F_{{\rm\pi}\gamma}(0,0)}{(1+Q^{2}_{1}/\Lambda^{2}_{\rm\rho})(1+Q^{2}_{2}/\Lambda^{2}_{\rm\rho})}\,,$
(27)

where $\Lambda_{\rm\rho}=775$ MeV corresponding to the $\rho$-pole, and
$F_{\pi\gamma}(0,0)=0.272(3)$ GeV-1 ParticleDataGroup:2018ovx . The different
behavior between our BLFQ-NJL model result and the VMD model prediction can be
attributed to the fact that the TFF in the BLFQ-NJL model exhibits
$F_{\pi\gamma^{*}}(Q^{2}_{1},Q^{2}_{2})\sim 1/(Q^{2}_{1}+Q^{2}_{2})$ when
$(Q^{2}_{1},Q^{2}_{2})\to\infty$, which is consistent with the pQCD prediction
Lepage:1980fj ; Braaten:1982yp , while the TFF in the VMD model behaves as
$F^{\rm VMD}_{\pi\gamma^{*}}(Q^{2}_{1},Q^{2}_{2})\sim 1/(Q^{2}_{1}Q^{2}_{2})$.
The consistency between our BLFQ-NJL result and pQCD is expected as the
factorization formula in Eq. (25) is applied in our method. Interestingly, for
the singly virtual TFF, i.e, for $Q^{2}_{1}=0$ or $Q^{2}_{2}=0$, both the
models show the expected scaling behavior $F_{\pi\gamma^{*}}(Q^{2},0)\sim
1/Q^{2}$ at the large $Q^{2}$ regime.

We show the three-dimensional plots for
$(Q^{2}_{1}+Q^{2}_{2})F_{\pi\gamma^{*}}(Q^{2}_{1},Q^{2}_{2})$ calculated at LO
and NLO in Fig. 5. The qualitative behavior is found to be consistent with the
LFQM result Choi:2019wqx . As can be seen from the figure, our BLFQ-NJL model
results also show the same scaling behavior as predicted by pQCD Lepage:1980fj
; Braaten:1982yp .

##  V Summary

We have evaluated the valence-quark distribution amplitude from the light-
front wave functions of the pion in the framework of the basis light front
quantization. Our result is based on the wave functions as the eigenfunctions
of an effective Hamiltonian which includes the confinement potentials and the
color-singlet Nambu–Jona-Lasinio interactions. The meson DA then evolves
according to the ERBL evolution equation from pQCD. We have analyzed the QCD
evolution of our pion DA and found that it agrees well with the FNAL-E-791
data. At the scale of 10 GeV2, our DA is close to the asymptotic QCD
prediction.

The non-vanishing moments of the DA in BLFQ framework have been found to be
consistent with various theoretical predictions. We have also investigated the
sensitivity of the pion DA, decay constants, and the radiative decay width to
the BLFQ basis size and found good convergence towards the experimental data
with increasing longitudinal basis size.

We have calculated the singly and doubly virtual pion-photon TFFs for
$\pi^{0}\rightarrow\gamma^{*}\gamma$ and
$\pi^{0}\rightarrow\gamma^{*}\gamma^{*}$ transitions using the DA with the
hard-scattering formalism. We have taken into account the LO evolution effects
of the pion DA and NLO corrections to the hard scattering amplitude for the
calculations at finite $Q^{2}$. We have found that our prediction for
$Q^{2}F_{\pi\gamma}(Q^{2})$ agrees reasonably well with the Belle
Collaboration data, but it disagrees with the rapid growth of the large
$Q^{2}$ pion-photon TFF data reported by the BaBar Collaboration. In the
meantime, we have observed that the doubly virtual pion-photon TFF in the
BLFQ-NJL model is consistent with the LFQM and the pQCD predictions. Our BLFQ-
NJL model result for the doubly virtual TFF manifests as
$F_{{\rm\pi}\gamma^{*}}(Q^{2}_{1},Q^{2}_{2})\sim 1/(Q^{2}_{1}+Q^{2}_{2})$ when
$(Q^{2}_{1},Q^{2}_{2})\to\infty$, which agrees with the pQCD prediction.

## Acknowledgements

C. M. is supported by new faculty start up funding by the Institute of Modern
Physics, Chinese Academy of Sciences, Grant No. E129952YR0. C. M. and S. N.
thank the Chinese Academy of Sciences Presidents International Fellowship
Initiative for the support via Grants No. 2021PM0023 and 2021PM0021,
respectively. X. Z. is supported by new faculty startup funding by the
Institute of Modern Physics, Chinese Academy of Sciences, by Key Research
Program of Frontier Sciences, Chinese Academy of Sciences, Grant No. ZDB-
SLY-7020, by the Natural Science Foundation of Gansu Province, China, Grant
No. 20JR10RA067 and by the Strategic Priority Research Program of the Chinese
Academy of Sciences, Grant No. XDB34000000. S. J. is supported by U.S.
Department of Energy, Office of Science, Office of Nuclear Physics, contract
no. DE-AC02-06CH11357. J. P. V. is supported in part by the Department of
Energy under Grants No. DE-FG02-87ER40371, and No. DE-SC0018223
(SciDAC4/NUCLEI).

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# LLM Guided Inductive Inference for Solving Compositional Problems

Abhigya Sodani  Lauren Moos  Matthew Mirman

###### Abstract

While large language models (LLMs) have demonstrated impressive performance in
question-answering tasks, their performance is limited when the questions
require knowledge that is not included in the model’s training data and can
only be acquired through direct observation or interaction with the real
world. Existing methods decompose reasoning tasks through the use of modules
invoked sequentially, limiting their ability to answer deep reasoning tasks.
We introduce a method, Recursion based extensible LLM (REBEL), which handles
open-world, deep reasoning tasks by employing automated reasoning techniques
like dynamic planning and forward-chaining strategies. REBEL allows LLMs to
reason via recursive problem decomposition and utilization of external tools.
The tools that REBEL uses are specified only by natural language description.
We further demonstrate REBEL capabilities on a set of problems that require a
deeply nested use of external tools in a compositional and conversational
setting.

Machine Learning, ICML

##  1 Introduction

Recently, neural models for natural language generation have demonstrated
impressive results (Koroteev, 2021; Devlin et al., 2018; Brown et al., 2020),
opening significant new avenues for solving natural language reasoning tasks
precisely (Huang & Chang, 2022; Qiao et al., 2022). While LLMs have shown a
unique ability to scale in predictable and efficient ways, is unclear whether
they show this scaling behavior on complex reasoning tasks (Huang & Chang,
2022). Moreover, the limitations of large language models in accessing dynamic
external knowledge sources significantly restrict their usefulness. Human
reasoning involves a combination of observations and interactions with the
world, highlighting the action-oriented nature of reasoning.

In this paper, we address this by introducing the Recursion Based Extensible
LLM (REBEL) framework. REBEL allows LLMs to reason through highly complex
problems that require knowledge from disparate external sources. This is
accomplished using an inference engine that, using the provided tools, gathers
the necessary facts to infer the correct answer. Specifically we show three
contributions:

  1. 1.

Designing a system capable of answering questions using any arbitrary external
tool.

  2. 2.

An evaluation showing that REBEL improves upon the state-of-the-art
performance on multi-Hop fact retrieval and compositional question answering
problems.

  3. 3.

Releasing our code and evaluation suite for open-source usage at
rebel.anarchy.ai.

##  2 Related Works

At a high-level, methods for approaching reasoning tasks using LLMs can be
broken down into prompt engineering techniques (Liu et al., 2023; Schlag et
al., 2023) and fine-tuning (Micheli & Fleuret, 2021; Schick et al., 2023)
techniques, or combinations of the above. Here we focus only on prompt
techniques.

Forward chaining (Liebowitz, 1988) is a reasoning strategy historically used
by expert systems. It operates by repeatedly applying logical inference rules
from an initial repository of known axioms to eventually ideally produce the
goal. This strategy has recently been employed to solve natural language
problems with the assistance of LLMs in Chain of Thought (CoT) (Wei et al.,
2022). ReAct (Yao et al., 2023) builds off of CoT by generating task-specific
actions in response to reasoning. Chameleon (Lu et al., 2023) takes this
further, using LLMs to synthesize tool pipelines including off-the-shelf
computer vision models, web-search engines, and calls to generative models. In
contrast to forward-chaining, the technique of backward-chaining (Russell,
2010) attempts to limit the search-space of possible inferences by determining
what must be true for a goal to be shown (Picco et al., 2021).

Press et al. (2022) demonstrates a method to evaluate problem-solving
abilities on a category of non-trivial reasoning tasks with compositional
structure (Lake & Baroni, 2018; Keysers et al., 2019) that is poorly addressed
by prior methods. They express compositional error as the number of questions
in which two subquestions are answered correctly but the top-level question is
not. Prior work has shown how this can be addressed via problem decomposition
(Yang et al., 2022; Zhou et al., 2022; Drozdov et al., 2022; Khot et al.,
2022). In this work, we show how problem decomposition can be augmented with
tool usage.

##  3 Methods

Figure 1: Visual depiction of the pipeline of the REBEL Algorithm from
Algorithm 1 to answer some $Question_{n}$. Blue boxes contain descriptions of
each step of the pipeline, and the red boxes contains the output variable for
each step that will be used in subsequent steps.

Question Splitting 1) Generates Subquestionsn and recursively gets their
answers and appends them to Memoryn. Memoryn Memory Checking 2) Generates a
boolean MemoryCheckn which indicates if the answer to Questionsn can be found
in Memoryn. MemoryCheckn Tool Picking 3) If MemoryChecknis false, we generate
Tooln, the best tool to answer Questionn Tooln Tool Input Generation 4) Takes
Memoryn and Questionn and generates input for Tooln … ”toolnparamk”:
”toolnvaluek”, …

In this section we introduce the REBEL algorithm as shown in Fig. 1, and all
necessary notation and background. At a high level, it works recursively to
solve questions, breaking questions into subquestions until no further
subquestions can be generated. Let us call the $nth$ question/subquestion
$Question_{n}$ and its answer $Answer_{n}$. For example, the user-provided
question would be $Question_{0}$. Let us call the subquestions that are
generated to answer $Question_{n}$ $Subquestions_{n}$. In each recursive step,
we break $Question_{n}$ into $Subquestions_{n}$. Let us call the answer to the
$ith$ member of $Subquestions_{n}$ $subanswers_{n}[i]$. We recursively call
each member of $Subquestions_{n}$, and each $subanswers_{n}[i]$ is returned as
a $fact$ which is the tuple $(Subquestions_{n}[i],subanswers_{n}[i])$. This
fact is appended to a list of $facts$ that is is global to each
$Question_{n}$. This list of $facts$ becomes $Memory_{n}$ which is used to
inform $Answer_{n}$.

In order to stop unbounded recursion, we delete members of $Subquestions_{n}$
whose featurizations have cosine similarities above 0.98 to the featurization
of $Question_{n}$.

The REBEL system contains a $Tool\textunderscore List$, which is a numbered
list of the tools we have available and their descriptions. If required, we
determine a $Tool_{n}$ for each $Question_{n}$, which is the number of the
tool required to answer $Question_{n}$ given $Memory_{n}$.

Below we define the basic steps of this algorithm: question splitting,
checking memory, picking tools, and using tools. Figure 1 depicts this
pipeline.

Algorithm 1 REBEL

function promptf($Question_{n}$,$facts$,$allowsplit=True$)

if $allowsplit$ then

$Subquestions_{n}$ = split($Question_{n}$, $facts$) {split the question into
subquestions to answer}

for $subquestion$ from 1 to $s$ in $Subquestions_{n}$ do

if cos_similarity($Question_{n}$,$subquestion$) $>$ 0.98 then

Delete $subquestion$

$allowsplit$=False

end if

end for

for $subquestion$ from 1 to $s$ in $Subquestions_{n}$ do

_,$newfacts$=promptf

($subquestion,facts,allowsplit$)

$facts$ += $newfact$

end for

end if

if $\textsc{memorycheck}(Question_{n},facts)$ then

$Answer_{n}=\textsc{callGPT}(Question_{n},facts)$

return $Answer_{n},(Question_{n},Answer_{n})$

else

$tool=\textsc{pick tool}(Question_{n},facts)$

$toolinput=\textsc{callGPT}(tool,Question_{n},facts)$ {to determine tool
input}

$Answer_{n}=\textsc{useTool}(toolinput,facts)$

end if

return $Answer_{n}$,$(Question_{n},Answer_{n})$

end function

###  3.1 Question Splitting

The split subroutine divides $Question_{n}$ into $Subquestions_{n}$ with the
size of $Subquestions_{n}$ being the number of subquestions that the LLM
generates. The LLM is prompted with $Tool\textunderscore List$, and 4 shots of
question splitting examples. This step is representing in step 1 of Figure 1.
To see a single shot of context for question splitting see Appendix A.

We answer each subquestion and its results are returned as a $fact$ (see
Algorithm 1). These facts are accumulated and passed to all subsequent
subquestions. The list $Subquestions_{n}$ is ordered such that the $fact$
gained from answering a lower indexed subquestion will aid in the answering of
a higher indexed subquestion.

###  3.2 Memory Check

We check if a question can be answered without any tool use. This can mean
either that the question can be answered using $Memory_{n}$ or the question
can be answered by an LLM without the use of any tools (see step 2 Figure 1).
If this is the case, we directly provide our base LLM $Memory_{n}$ and
$Question_{n}$ to find $Answer_{n}$. To see the complete memory check prompt
see Appendix B.

###  3.3 Tool Picker

Here we evoke the LLM to decide what member of $Tool\textunderscore List$
(described by integer $Tool_{n}$) would be best to decide the answer to a
question. This is a 0-shot prompted system which can be seen in step 3 of
Figure 1.

###  3.4 Tool Input Generation

We use GPT-3 to generate standardized input to our tools. We provide the tools
to the LLM with 2 fields. The description of the tool and the dynamic
parameters of the tool. We store the 3 more fields about each tool that are
hidden from the LLM, these are: if the tool is a GET/POST request, the
endpoint URL for the tool, and the static parameters of the tool. The dynamic
parameters are parameters that will be adjusted based on each call (for
example, and query field). The static parameters are parameters that stay the
same on each API call (for example, an authentication key).

REBEL uses 3 default tools: search, weather, and Google Maps. We configure the
inputs to every tool as a JSON. A tool input JSON maps a given tool’s dynamic
parameters to the values those parameters should be in order to obtain
information to answer a given question:
$\\{"tool_{n}param_{1}":"tool_{n}value_{1}",$ …
$"tool_{n}param_{k}":"tool_{n}value_{k}"\\}$. A standardized JSON format
reduces the load on the LLM to format an entire API call by itself.

REBEL allows for an arbitrary tools to be added to it, however, the k-shot
examples that are provided to the LLM for generating input given $Tool_{n}$
are designed around the 3 base tools. We have found that this prompting does
extrapolate to 0-shot uses of unseen and arbitrary tools. See Appendix C for a
complete single shot of tool input generation context.

###  3.5 Tool Usage

The UseTool function takes the dynamic parameters (from the LLM generated tool
input), the static parameters that we have stored for each tool, and the API
endpoint and makes a single request URL. This URL is requested, and the return
output is stored as a string. If the return output is longer than 15,000
characters, it is truncated to that amount. Then, we use an LLM, provided with
$Memory_{n}$, $Question_{n}$, and the API request output, to generate an
answer to the $Question_{n}$. This answer is returned from the UseTool
function as $Answer_{n}$. Our approach has some consequences. On the positive
side, users do not have the indicate how to parse the output of the tools they
give us, this makes REBEL extremely extendable and flexible to interpret many
tool return types and formats. On the negative side, because of the extremely
unstructured nature of tool returns, errors are caused by UseTool not being
able to answer a question based on a tool return.

##  4 Evaluation

In this section we first introduce the experimental setup, including the
benchmarks used for evaluation, and then present the results.

###  4.1 Experimental Setup

We tested REBEL on 3 datasets: Compositional Celebrities (Press et al., 2022),
FEVER (Thorne et al., 2018), and HotPotQA (Yang et al., 2018).

On these datasets, correctness was determined by a human experimenter based on
the output of each system. ReAct outputs with simply the answer to the
question, while REBEL often outputs the answer wrapped in reasoning behind the
system’s thoughts. For these experiments, two separate sets of rules had to be
determined for fact verification and fact retrieving questions. For fact
retrieving questions, an answer was considered correct if the desired answer
was contained in the system output. For fact verification, if the model output
determination of the truthfulness of a statement was the same as the desired
truthfulness, then the generated answer was considered correct.

On Compositional Celebrities, due to computational limitations, we tested
using 5 of the 17 categories available, using 100 questions per category,
randomly chosen. These categories can be found in Table 1.

We tested on FEVER and HotPotQA with 100 of the same random questions from
each dataset on both ReAct and REBEL. The accuracy results for this experiment
can be found at Table 2. FEVER has 3 types of potential output labels
(SUPPORTS, REFUTES, NOT ENOUGH INFO). In order to make prevent accidental
correct answers from the REBEL system, only questions with the SUPPORTS and
REFUTES labels were considered.

For this experiment REBEL was only allowed to use a search tool to query the
internet, as that is the only tool that the ReAct system has access to.

Our code, which can be found at rebel.anarchy.ai, was implemented in Python
using the OpenAI Completion API to access GPT-3 (text-davinci-003).

###  4.2 Results

We found that REBEL outperformed ReAct on answering questions that require i)
the gathering of many facts to determine an answer ii) very specific search
queries that return large amounts of unstructured data. With our experimental
results we were able to show that REBEL is a state-of-the-art system in terms
of its ability to consistently answer questions from disparate knowledge
bases.

####  4.2.1 Multi-Hop Fact Retrieval

We used 2 datasets to test multi-hop fact retrieval: Compositional Celebrities
and HotPotQA.

Compositional Celebrities is a dataset consisting of 8.6k questions about
Celebrities in different categories. All questions require retrieving two
facts and basic reasoning. These two facts have never co-occurred in any text
that would conceivably be part of the LLM training and the only way that the
conclusion could be reached is for both of them to be evaluated correctly and
composed with one another. We found that the REBEL system largely outperformed
the ReAct system at all of the 5 categories that were experimented on for
Compositional Celebrities. On average, over the 5 categories tested, REBEL
beat ReAct by 27.6 percent. The reason for this is likely the ability of the
REBEL system to work with unstructured tool return data. This allows the REBEL
system to make and interpret very specific tool queries, whereas other systems
that require standardized output can become constricted by the by a smaller
possible set of tool queries. The results of this experiment can be found in
Table 1.

HotpotQA is a challenging question-answering dataset containing 113,000 pairs
of questions and answers derived from Wikipedia articles. The questions in
HotpotQA necessitate synthesis of information from diverse sources and cannot
be found pre-existing training knowledge bases. ReAct outperformed REBEL on
HotPotQA by 13 percent (Table 2).

HotPotQA has questions that are significantly more than 2-hops, and on these
questions REBEL tends to generate a massive recursive tree of subquestions.
This introduces the issue of generating subquestions that lose context of the
original question. Many times this can lead to the LLM not being able to
reason through the large context window generated when processing these layers
of recursive subquestions, resulting in the LLM finding no solution.

####  4.2.2 Fact Verification

To test fact verification abilities, we employed the FEVER dataset. This
benchmark is designed to evaluate the ability of models to extract factual
information from textual sources and verify claims. The fact verification task
involves determining the accuracy of claims made in a given piece of text.

On FEVER, the REBEL system (78 percent accuracy) performed slightly better
(Table 2) than ReAct system (72 percent). The reason for this out-performance
by the REBEL system is because of the significant amount of ”facts” that it
gathers during its recursive solving of a fact verification problem. On
several occasions, the ReAct system cannot find the information it is looking
for to answer a questions, and therefore reports that it cannot make a
determination if a certain fact is true or not.

Table 1: Accuracy (percent of questions answered correctly) of different
algorithms on the categories of Compositional Celebrities.
Category | ReAct | REBEL
---|---|---
birthplace _rounded _lat | 28 | 59
birthplace _currency | 85 | 94
birthplace _currency _symbol | 35 | 47
birthyear _nobelLiterature | 33 | 82
birthdate _uspresident | 53 | 90
Table 2: Accuracy (percent of questions answered correctly) of different
algorithms on HotPotQA and FEVER.
Dataset | ReAct | REBEL
---|---|---
FEVER | 72 | 78
HotPotQA | 63 | 50
Table 3: Accuracy (percent of questions answered correctly) of different
algorithms on the categories of Compositional Celebrities.
Category | GPT3 | REBEL w/o tools | REBEL
---|---|---|---
birthplace _rounded _lat | 16 | 39 | 59
birthplace _currency | 95 | 94 | 94
birthplace _currency _symbol | 28 | 45 | 47
birthyear _nobelLiterature | 95 | 90 | 82
birthdate _uspresident | 44 | 91 | 90
Table 4: Accuracy (percent of questions answered correctly) of different
algorithms on HotPotQA and FEVER.
Dataset | GPT3 | REBEL w/o tools | REBEL
---|---|---|---
FEVER | 77 | 73 | 78
HotPotQA | 43 | 46 | 50

###  4.3 Ablation Study

In order to determine the efficiency of REBEL, we conducted several ablation
tests. In these tests the aim was to isolate the affect of the REBEL system
upon compositional problem solving. We used plain GPT3 (text-davinci-003) as
our baseline. The results of these tests are in (Table 3 and Table 4).

These tables show that GPT3 outperforms REBEL (with or without an external
search tool) when a question can be easily answered with data that GPT3’s
training set. This is seen in Table 3 in the rows pertaining to
$Birthyear\textunderscore NobelLiterature$ and $Birthplace\textunderscore
Currency$.

The REBEL algorithm without the external search tool outperformed the baseline
when information processing is necessary to determine a final answer. Examples
of this include questions that required the returning of a currency symbol or
a rounded latitude. GPT3 succeeded in fetching the currency name or latitude
correctly, but failed to round the latitude or return the symbol associated
with the currency name. Adding external search augmented the REBEL algorithm’s
ability to reason with current facts, and therefore furthered the REBEL
algorithm performance on most categories of Compositional Celebrities.
Occasionally, the inclusion of an external search tool decreased performance
due to the unstructured nature of return data the external tool provided. An
example of this is on the $Birthyear\textunderscore NobelLiterature$ category
of Compositional Celebrities.

On most categories of Compositional Celebrities and on HotPotQA, REBEL without
the use of an external search tool improved performance over baseline GPT3.
This indicates that our recursive approach adds reasoning capability to GPT3
independently of external tool use.

Table 5: Average time taken to answer a question from Compositional
Celebrities
Algorithm | Time (s)
---|---
GPT3 | 0.94
REBEL w/o tools | 5.358
REBEL w/ tools | 9.76

##  5 Cost Analysis

The recursive search nature of the REBEL algorithm means that it employs many
calls to an LLM before determining an answer to a question. The downsides of
this approach manifest themselves in latency (Table 5) and monetary cost of
LLM queries. Any external tools that are provided to the REBEL system will
also be called very frequently, potentially leading to REBEL being a
monetarily expensive system on that front as well.

If a user desires to use REBEL without any tools, a cost in terms of
hallucination has a potential of arising. Due to the lack of any external
knowledge base, a hallucination on one subquestion has the potential to
pollute the entire tree of reasoning.

##  6 Conclusion

We have introduced REBEL, a recursive reasoning algorithm designed to use any
arbitrary API as an external tool. REBEL outperforms the state-of-the-art on
questions that require the collection of many facts and those that benefit
from the ability to make highly specific queries to outside sources of data,
which may be unstructured. REBEL also has a demonstrable improvement over the
GPT3 LLM when answering questions that require multi-step information
processing. However, the REBEL algorithm tends to over-complicate simple
problems, leading to a reduction in accuracy when compared to baseline GPT3 on
questions that require minimal compositionality.

Future work would ideally address fine-tuning LLMs for each step in the REBEL
pipeline and experimenting with limiting recursive depth of subquestion
generation.

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##  Appendix A Appendix A

Question Splitting Prompt

Tools we have access to = tool 1: The tool returns the results of free-form
queries similar to those used for wolfram alpha. This is useful for
complicated math or live data retrieval. Can be used to get the current date.
tool 2: Find the driving distance and time to travel between two cities. tool
3: Find the weather at a location and returns it in celcius. Q=$Question_{n}$
Look at the tools we have access to. Split Q into subquestions to answer Q
that can each be solved with one use of one tool. Make as few subquestions as
possible. Split each subquestion with a comma and have no extra information
other than the subquestions.

##  Appendix B Appendix B

Memory Check Prompt

Q: ”What’s the time?” Is the answer to Q found in the memory or in your
knowledge base already? Answer with a yes or no. no Q: ”How you feeling?” Is
the answer to Q found in the memory or in your knowledge base already? Answer
with a yes or no. yes Q: ”What color is the sky” Is the answer to Q found in
the memory or in your knowledge base already? Answer with a yes or no. yes Q:
”What is the temperature in Portland?” Is the answer to Q found in the memory
or in your knowledge base already? Answer with a yes or no. no Memory:
$Memory_{n}$ Q: $Question_{n}$ Is the answer to Q found in the memory or in
your knowledge base already? Answer with a yes or no.

##  Appendix C Appendix C

Tool Input Prompt

    <TOOL>
    <ID>1</ID>
    <DESC>Find the driving
    distance and time to
    travel between two
    cities.</DESC>
    <PARAMS>{"origins": the origin city,
    "destinations": the destination
    city}</PARAMS>
    </TOOL>

    <CASE>
    <Q>How long would it take
    to get between
    South Africa and Kenya.
    </Q>
    <THOUGHT>
    <P>What
    should the input for
    tool 1 be to
    answer Q?</P>
    <A ty=JSON>
    {"origins": "South Africa",
    "destinations": "Kenya"}
    </A>
    </THOUGHT>
    </CASE>