# Light-front dynamic analysis of the longitudinal charge density using the
solvable scalar field model in (1+1) dimensions

Yongwoo Choi  Department of Physics, Kyungpook National University, Daegu
41566, Korea  Ho-Meoyng Choi<EMAIL_ADDRESS>Department of Physics,
Teachers College, Kyungpook National University, Daegu 41566, Korea  Chueng-
Ryong Ji<EMAIL_ADDRESS>Department of Physics, North Carolina State
University, Raleigh, NC 27695-8202  Yongseok Oh<EMAIL_ADDRESS>Department
of Physics, Kyungpook National University, Daegu 41566, Korea  Asia Pacific
Center for Theoretical Physics, Pohang, Gyeongbuk 37673, Korea

###### Abstract

We investigate the electromagnetic form factor $F(q^{2})$ of the meson by
using the solvable $\phi^{3}$ scalar field model in $(1+1)$ dimensions. As the
transverse rotations are absent in $(1+1)$ dimensions, the advantage of the
light-front dynamics (LFD) with the light-front time $x^{+}=x^{0}+x^{3}$ as
the evolution parameter is maximized in contrast to the usual instant form
dynamics (IFD) with the ordinary time $x^{0}$ as the evolution parameter. In
LFD, the individual $x^{+}$-ordered amplitudes contributing to $F(q^{2})$ are
invariant under the boost, i.e., frame-independent, while the individual
$x^{0}$-ordered amplitudes in IFD are not invariant under the boost but
dependent on the reference frame. The LFD allows to get the analytic result
for the one-loop triangle diagram which covers not only the spacelike
($q^{2}<0$) but also timelike region ($q^{2}>0$). Using the analytic results,
we verify that the real and imaginary parts of the form factor satisfy the
dispersion relations in the entire $q^{2}$ space. Comparing with the results
in $(3+1)$ dimensions, we discuss the transverse momentum effects on
$F(q^{2})$ . We also discuss the longitudinal charge density in terms of the
boost invariant variable $\tilde{z}=p^{+}x^{-}$ in LFD.

##  I Introduction

The formulation of light-front dynamics (LFD) based on the equal light-front
time $x^{+}=x^{0}+x^{3}$ quantization has shown remarkable advantages for
calculations in elementary particle physics, nuclear physics, and hadron
physics. In particular, the light-front (LF) formulation is an essential
theoretical tool for the 3-dimensional imaging and femtography efforts in the
12 GeV upgraded Thomas Jefferson National Accelerator Facility (JLab) and in
the future Electron-Ion Collider project, with the investigation of the form
factors, the generalized parton distributions (GPDs), the transverse momentum
distributions of hadrons, etc. Taking advantage of the LFD, one of the new
experiments planned at JLab is to measure the transverse charge densities of
hadrons CHM14 , which are defined by the 2-dimensional Fourier transforms of
the electromagnetic (EM) form factors describing the distribution of charge
and magnetization in the plane perpendicular to the direction of a fast moving
hadron Soper77 . Due to the Lorentz invariance of the transverse distance and
momentum under the longitudinal boost, the relativistically invariant analysis
of the transverse charge density can be straightforwardly attained in the
$(3+1)$ dimensional LFD. The transverse charge densities are also related to
the GPDs Burkardt00 ; Burkardt02 ; Diehl02  and their properties have been
explored in a number of works Miller07 ; Miller09 ; Miller09a ; CV07 ; SW10 ;
MSW10 ; VAMZ10 ; Miller18 ; Cedric . In particular, it was demonstrated that
the transverse charge density defined by the two-dimensional Fourier transform
can be obtained from the so-called “Drell-Yan-West (DYW)” frame ($q^{+}=0$ and
$q^{2}=-{\bf q}^{2}_{\perp}=-Q^{2}<0$) in LFD using the scalar $\phi^{3}$
model in $(3+1)$ dimensions Miller09 . Although its utility is limited only to
the spacelike region ($q^{2}<0$) due to the intrinsic kinematic constraint
$q^{+}=0$, the DYW formulation BRS73 ; Weinberg66a ; CM69 ; Sawicki91 ; Sa92
may be regarded as the most rigorous and well-established framework to compute
the exclusive processes since it involves typically the particle number
conserving valence contribution. Various studies of two-body bound-states in
the $(3+1)$ dimensional LFD can also be found in the framework of scalar FFT73
; Karmanov80 ; Karmanov81 ; Mueller83 ; BJS85 ; Sawicki85 ; Sawicki86 ; CJS87
; BJ00  and fermion field BJ00 ; BCJ00 ; DNF97 ; DSFS98  models.

On the other hand, the LFD analysis of the longitudinal charge density is not
as straightforward as in the analysis of the transverse charge density due to
the nontrivial spacetime mixture of the LF spatial distance
$x^{-}=x^{0}-x^{3}$ as well as its conjugate momentum $p^{+}=p^{0}+p^{3}$. It
is noteworthy that the new variable $\tilde{z}=p^{+}x^{-}$ was recently
introduced MB19  for the boost invariant analysis in the longitudinal
direction. Although the same level of significant progresses as in the case of
the transverse charge density is yet to be expected in the analysis of the
longitudinal charge density, it may be worthwhile to facilitate the scalar
$\phi^{3}$ model in the $(1+1)$ dimensional LFD extending the previous LFD
analyses in $(1+1)$ dimensions SM88 ; MS89 ; GS90  restricted only for the
spacelike momentum transfer region now to the entire kinematic regions
including the timelike momentum transfers as well. We note that the advantage
of LFD is indeed maximized in $(1+1)$ dimensions due to the absence of the
transverse rotations which are not kinematical but dynamical in LFD. As
evidenced in solving the $(1+1)$ dimensional QCD with large $N_{c}$ limit
tHooft74b , the solution was provided even analytically in LFD first tHooft74b
; Einhorn76  well before its canonical formulation BG78  was presented in the
instant form dynamics (IFD) based on the equal time $x^{0}$ quantization and
later numerically solved in IFD LWB87 ; JLLX17 ; JLXY18 . While in IFD the
individual $x^{0}$-ordered amplitudes contributing to the form factor
$F(Q^{2})$ are not invariant under the boost, i.e., dependent on the reference
frame, the advantage of the LFD with the LF time $x^{+}$ as the evolution
parameter is maximized due to the frame-independence or the boost invariance
of the individual $x^{+}$-ordered amplitudes contributing to $F(Q^{2})$.

The dramatic difference of LFD analysis of form factors in $(1+1)$ dimensions
compared to the case of $(3+1)$ dimensions may also be attributed to the fact
that the DYW frame cannot be taken as it is restricted to $q^{2}=0$ in $(1+1)$
dimensions. As the $q^{+}\neq 0$ frame must be used in $(1+1)$ dimensions for
$Q^{2}\neq 0$, it is inevitable to encounter the non-valence diagram arising
from the particle-antiparticle pair creation (the so-called “Z-graph”). That
is, both valence and non-valence contributions should be included
simultaneously for the form factor analysis in $(1+1)$ dimensions. As
mentioned earlier, the LFD analyses of scalar $\phi^{3}$ model in $(1+1)$
dimensions were reported in Refs. SM88 ; MS89 ; GS90  which were though
restricted only for the spacelike momentum transfer region. Once the
$q^{+}\neq 0$ frame is chosen, however, one does not need to restrict the
analysis only for the spacelike region.

We may also compare the $(1+1)$ dimensional results with the previous $(3+1)$
dimensional results CJ99  within the same solvable scalar $\phi^{3}$ model
since it was shown numerically that the $(3+1)$ dimensional results
analytically continued from the spacelike region $(q^{2}<0)$ coincide exactly
with the results directly obtained in the timelike region $(q^{2}>0)$.
Stemming from the detailed analysis of the solvable and manifestly covariant
model within the framework of the $(3+1)$ dimensional LF calculations, we have
also developed a new method to explore the timelike region directly in the
$q^{+}\neq 0$ frame for the transition form factor
$F_{\mathcal{M}\gamma}(q^{2})$ in the meson-photon transition process,
$\mathcal{M}(p)\to\gamma^{*}(q)+\gamma(p^{\prime})$ CRJ17 . Our direct
calculation in the timelike region showed the complete agreement not only with
the analytic continuation result from the spacelike region but also with the
result from the dispersion relation (DR) between the real and imaginary parts
of the form factor CRJ17 . This direct method of analyzing the timelike region
appears to advance our previous analysis of a solvable model in $(3+1)$
dimensions for the phenomenologically more realistic LF quark model (LFQM).

In this work, we present the $(1+1)$-dimensional analysis of the form factor
in the solvable model both for the spacelike region and the timelike region
obtaining the analytic results both for the valence and non-valence
contributions. Our model is essentially the (1 + 1)-dimensional quantum field
theory model of Mankiewicz and Sawicki SM88 ; MS89 , which was reinvestigated
by several others. (See, for example, Refs. Sawicki91 ; Sa92 ; GS90 ; BH98 ;
CJZero ; SB1 ; SB2 .) In this model, the wave function is obtained as the
solution of the covariant Bethe–Salpeter (BS) equation in the ladder
approximation with a relativistic version of the contact interactions
Sawicki91 ; Sa92 . The covariant model wave function is a product of two free
single-particle propagators, the Dirac delta function for the overall momentum
conservation, and a constant vertex function. Consequently, all our form
factor calculations show various ways of evaluating the Feynman triangle
diagrams in scalar field theory. Previous results reported in Ref. GS90  in
the spacelike region were confirmed, but now the results are extended to the
timelike region. In particular, the anomalous threshold is observed in the
timelike region as in the case of $(3+1)$-dimensional analysis. Apparent
satisfaction of DR is shown explicitly and analytically. Longitudinal charge
density is clearly identified with respect to “intrinsic” versus “apparent”
charge densities. We also discuss the LF longitudinal charge density in terms
of the newly introduced boost invariant variable $\tilde{z}$ of Ref. MB19 .

This paper is organized as follows. In Sec. II, we derive the analytic forms
for both spacelike and timelike EM form factors using the scalar $\phi^{3}$
model in $(1+1)$ dimensions. We obtain the explicit form of the imaginary part
of the form factor in the timelike region so that the DR relation between the
real and imaginary parts of the form factor can be tested. In Sec. III, we
discuss the difference between the intrinsic longitudinal charge density
obtained from the Fourier transform of the form factor and the apparent charge
density including the relativistic corrections such as the Lorentz contraction
in the so-called “Breit” frame. The explicit form of mean-square charge radius
in the longitudinal direction is also derived from the slope of the charge
form factor. We then discuss the LF longitudinal charge density in terms of
the boost invariant variable $\tilde{z}$. Section IV presents our numerical
results for the intrinsic longitudinal charge densities for scalar $(\pi,K,D)$
mesons and their EM form factors in both spacelike and timelike regions
comparing them with the previous $(3+1)$ dimensional results of Ref. CJ99 . We
summarize and conclude in Sec. V. The explicit analytic forms of the valence
and non-valence contributions to the form factor are presented in Appendix.

##  II Form Factor for scalar $\phi^{3}$ model in (1+1) dimensions

###  II.1 Form factor in spacelike region

Figure 1:  One-loop Feynman diagrams that contribute to
$\mathcal{M}(p)+\gamma^{*}(q)\to\mathcal{M}(p^{\prime})$. The covariant
diagram (a) is the same as the sum of the two LF time-ordered diagrams (b) and
(c).

The EM form factor $F_{\mathcal{M}}^{S}(q^{2})$ of a scalar particle
$\mathcal{M}$, a bosonic $q{\bar{Q}}$ bound state, for the process of
$\mathcal{M}(p)+\gamma^{*}(q)\to\mathcal{M}(p^{\prime})$ in spacelike momentum
transfer ($q^{2}<0$) region is defined by the local current $J^{\mu}_{S}(0)$
through

$J^{\mu}_{S}(0)=(p+p^{\prime})^{\mu}F^{S}_{\cal M}(q^{2}),$ (1)

where $p$ $(p^{\prime})$ is the four momentum of the initial (final) state
scalar particle $\mathcal{M}$ and $q=p^{\prime}-p$ is the four-momentum
transfer of the virtual photon ($Q^{2}\equiv-q^{2}>0$). The EM local current
$J^{\mu}_{S}(0)$ in ($1+1$) dimensions obtained from the covariant diagram of
Fig. 1(a) is represented by
$J^{\mu}_{S}(0)=e_{q}I^{\mu}(m_{q},m_{\bar{Q}})+e_{\bar{Q}}I^{\mu}(m_{\bar{Q}},m_{q})$,
where

$I^{\mu}(m_{q},m_{\bar{Q}})=ig^{2}\int\frac{d^{2}k}{(2\pi)^{2}}\frac{2k^{\mu}+q^{\mu}}{N_{k}N_{k+q}\bar{N}_{k-p}},$
(2)

with $N_{k}=k^{2}-m_{q}^{2}+i\epsilon$ and
$\bar{N}_{k}=k^{2}-m_{\bar{Q}}^{2}+i\epsilon$ coming from the bosonic quark
and antiquark propagators of mass $m_{q}$ and $m_{\bar{Q}}$, respectively,
which carry the internal momentum $k$. The normalization constant $g$ is fixed
by the condition that $F^{S}_{\cal M}(q^{2}=0)=e_{q}+e_{\bar{Q}}$. Exchanging
$m_{q}$ and $m_{\bar{Q}}$ in $I^{\mu}(m_{q},m_{\bar{Q}})$ gives
$I^{\mu}(m_{\bar{Q}},m_{q})$.

In LF calculations, we use the metric convention that $k\cdot
q=\frac{1}{2}(k^{+}q^{-}+k^{-}q^{+})$. Using this metric and choosing the plus
component of the currents, $J^{+}_{S}$, the Cauchy integration over $k^{-}$ in
Eq. (2) gives the two time-ordered contributions to the residue calculations,
i.e., one coming from the region ${\rm S1}$ $(0<k^{+}<p^{+})$ [Fig. 1(b)] and
the other from the region ${\rm S2}$ $(-q^{+}<k^{+}<0)$ [Fig. 1(c)]. In the
region of ${\rm S1}$ (S2), the residue is at the pole of $k^{-}=k^{-}_{b}$
($k^{-}=k^{-}_{r}$), which is placed in the upper (lower) half of complex
$k^{-}$ plane. Therefore, the Cauchy integration of $I^{+}(m_{q},m_{\bar{Q}})$
in Eq. (2) over $k^{-}$ in ${\rm S1}$ and ${\rm S2}$ leads to

$\displaystyle I^{+}_{\rm S1}$ $\displaystyle=$
$\displaystyle-\frac{g^{2}}{4\pi}\int^{p^{+}}_{0}dk^{+}\frac{2k^{+}+q^{+}}{C_{k}(k^{-}_{b}-k^{-}_{l})(k^{-}_{b}-k^{-}_{r})},$
$\displaystyle I^{+}_{\rm S2}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{4\pi}\int^{0}_{-q^{+}}dk^{+}\frac{2k^{+}+q^{+}}{C_{k}(k^{-}_{r}-k^{-}_{l})(k^{-}_{r}-k^{-}_{b})},$
(3)

where $C_{k}=k^{+}(k^{+}+q^{+})(k^{+}-p^{+})$ and

$\displaystyle k^{-}_{l}$ $\displaystyle=$
$\displaystyle\frac{m_{q}^{2}}{k^{+}}-i\frac{\epsilon}{k^{+}},$ $\displaystyle
k^{-}_{b}$ $\displaystyle=$ $\displaystyle
p^{-}+\frac{m_{\bar{Q}}^{2}}{k^{+}-p^{+}}-i\frac{\epsilon}{k^{+}-p^{+}},$
$\displaystyle k^{-}_{r}$ $\displaystyle=$
$\displaystyle-q^{-}+\frac{m_{q}^{2}}{k^{+}+q^{+}}-i\frac{\epsilon}{k^{+}+q^{+}}.$
(4)

We note that Eq. (II.1) is obtained from the condition that $q^{+}>0$, which
means that the virtual photon is moving to the positive $z$-direction.

In the spacelike region ($q^{2}=q^{+}q^{-}=-Q^{2}<0$) of $(1+1)$ dimensions,
the momentum transfer $Q^{2}$ is defined as

$Q^{2}=M^{2}{\bar{\beta}}^{2}/\beta,$ (5)

where $\beta=p^{\prime+}/p^{+}=1+q^{+}/p^{+}$, ${\bar{\beta}}=\beta-1$, and
$M^{2}=p^{2}=p^{\prime 2}$. This allows $\beta$ to have two different
solutions:

$\beta_{\pm}=\left(1+\frac{Q^{2}}{2M^{2}}\right)\pm\sqrt{\left(1+\frac{Q^{2}}{2M^{2}}\right)^{2}-1},$
(6)

which leads to $\beta_{\pm}=1$ (or ${\bar{\beta}}_{\pm}=0$) when $Q^{2}=0$.
Using the longitudinal momentum fraction $k^{+}=xp^{+}$ for the struck quark
and $\beta$ for an external momentum transfer, we obtain

$\displaystyle I^{+}_{\rm S1}$ $\displaystyle=$
$\displaystyle\frac{p^{+}g^{2}}{4\pi}\int^{1}_{0}\frac{[dx]}{(1-x^{\prime})}\frac{(2x+\bar{\beta})}{(M^{2}-M^{2}_{0})(M^{2}-M^{\prime
2}_{0})},$ $\displaystyle I^{+}_{\rm S2}$ $\displaystyle=$
$\displaystyle\frac{p^{+}g^{2}}{4\pi}\int^{1}_{0}\frac{[dy]}{(1+\bar{\beta}y)}\frac{(2y-1)\beta\bar{\beta}}{\left(Q^{2}+\frac{m^{2}_{q}}{y(1-y)}\right)\left(M^{2}-M^{2}_{\bar{\beta}y^{\prime}}\right)},$
(7)

where $[dZ]=\frac{dZ}{Z(1-Z)}$ and
$M^{2}_{0}=\frac{m^{2}_{\bar{Q}}}{1-x}+\frac{m^{2}_{q}}{x}$, $M^{\prime
2}_{0}=\frac{m^{2}_{\bar{Q}}}{x^{\prime}}+\frac{m^{2}_{q}}{1-x^{\prime}}$,
$M^{2}_{\bar{\beta}y^{\prime}}=\frac{m^{2}_{\bar{Q}}}{1-\bar{\beta}y^{\prime}}+\frac{m^{2}_{q}}{\bar{\beta}y^{\prime}}$
with $x^{\prime}=(1-x)/\beta$ and $y^{\prime}=(1-y)/\beta$. The change of
variable, $x=-{\bar{\beta}}y$, is made to derive $I^{+}_{\rm S2}$ as given in
Eq. (II.1). While each contribution, $I^{+}_{\rm S1}$ and $I^{+}_{\rm S2}$, is
independent of the choice on $\beta$, we take $\beta=\beta_{+}$ in Eq. (II.1)
since they are obtained using the constraint $q^{+}>0$. Although we do not
explicitly show the results with $q^{+}<0$, where the virtual photon is moving
to the negative $z$-direction, we confirmed that $I^{+}_{\rm S1}$ and
$I^{+}_{\rm S2}$ are indeed independent of the choice of $\beta$, so that
$\beta=\beta_{+}$ and $\beta=\beta_{-}$ lead to the same results.

We then further evaluate the integration over the variables $(x,y)$ in Eq.
(II.1) and combine both contributions as $I_{\rm S}=(I^{+}_{\rm S1}+I^{+}_{\rm
S2})/(p+p^{\prime})^{+}$, to obtain the fully analytic form of $F^{S}_{\cal
M}(Q^{2})$ in $(1+1)$ dimensions as

$F^{S}_{\mathcal{M}}(Q^{2})=e_{q}\,I^{q}_{\rm
S}(Q^{2})+e_{\bar{Q}}\,I^{Q}_{\rm S}(Q^{2}),$ (8)

with

$\displaystyle I^{q}_{\rm S}(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{8\pi
m^{2}_{q}m^{2}_{\bar{Q}}\left(1-\omega^{2}+\gamma_{Q}\right)}$ (9)
$\displaystyle\mbox{}\times\left\\{C_{\omega}+\frac{\sqrt{1+\gamma_{Q}}}{\sqrt{\gamma_{Q}}}\tanh^{-1}\left(\frac{\sqrt{\gamma_{Q}}}{\sqrt{1+\gamma_{Q}}}\right)\right\\},$

where

$\displaystyle
C_{\omega}=\frac{\omega}{\sqrt{1-\omega^{2}}}\left[\tan^{-1}(\frac{\omega_{q}}{\sqrt{1-\omega^{2}}})+\tan^{-1}(\frac{\omega_{Q}}{\sqrt{1-\omega^{2}}})\right],$

and $\omega_{q}=\frac{M^{2}+m^{2}_{q}-m^{2}_{Q}}{2m_{q}m_{\bar{Q}}}$,
$\omega_{Q}=\frac{M^{2}-m^{2}_{q}+m^{2}_{Q}}{2m_{q}m_{\bar{Q}}}$,
$\omega=\frac{M^{2}-m^{2}_{q}-m^{2}_{Q}}{2m_{q}m_{\bar{Q}}}$ are kinematic
factors. We note that the momentum transfer $Q^{2}$ in Eq. (9) comes in only
through the factor $\gamma_{Q}=Q^{2}/4m^{2}_{q}$. The second part $I^{Q}_{\rm
S}(Q^{2})$ of Eq. (8) can be obtained from Eq. (9) by replacing $\gamma_{Q}$
with $Q^{2}/4m^{2}_{\bar{Q}}$. It should be noted that our LF result in Eq.
(8) is identical to the one obtained from the manifestly covariant
calculations. The analytic forms of the valence and non-valence contributions,
i.e., $I^{q}_{\rm S1}(Q^{2})$ and $I^{q}_{\rm S2}(Q^{2})$, are given in
Appendix. As one can see from Eqs. (44) and (45), the second term containing
the $\tanh^{-1}$ function in Eq. (9) comes from the non-valence contribution.
From the form factor normalization $F^{S}_{\cal M}(Q^{2})=e_{q}+e_{\bar{Q}}$,
which means $I^{q}_{\rm S}(Q^{2}=0)=I^{Q}_{\rm S}(Q^{2}=0)=1$, we can obtain
the normalization constant $g$ as

$g=\sqrt{\frac{8\pi m^{2}_{q}m^{2}_{\bar{Q}}(1-\omega^{2})}{C_{\omega}+1}}.$
(11)

In the limit of very high $Q^{2}$, the leading contribution in Eq. (9) comes
from the
$\tanh^{-1}\left(\sqrt{\frac{\gamma_{Q}}{1+\gamma_{Q}}}\right)/\gamma_{Q}$
term in the non-valence diagram. Using the fact that
$\tanh^{-1}\left(\sqrt{\frac{x}{1+x}}\right)\to\ln x$ as $x\to\infty$, the
leading asymptotic behavior of the form factor reads

$\lim_{Q^{2}\to\infty}F^{S}_{\cal M}(Q^{2})\sim\frac{\ln Q^{2}}{Q^{2}},$ (12)

which shows the power-law falloff modified by the presence of a logarithmic
function as was obtained in Ref. GS90 . In Ref. Miller09 , the leading
asymptotic behavior of the form factor using the same $\phi^{3}$ model in
$(3+1)$ dimensions was obtained as $F^{S}_{\cal
M}(Q^{2}\to\infty)\sim\ln^{2}Q^{2}/Q^{2}$, which has one more logarithmic
power than the result in $(1+1)$ dimensions, and this power difference appears
due to the effect of the transverse momentum. Apart from the presence of a
logarithmic function ($\sim\ln Q^{2}$), both asymptotic results in $(1+1)$-and
$(3+1)$-dimensions satisfy the quark counting rules ($\sim 1/Q^{2}$) for the
momentum transfer dependence of the quark and antiquark bound state form
factors. While one may avoid the nonvalence contribution using the $q^{+}=0$
frame in the $(3+1)$ dimensions, the nonvalence contribution cannot be avoided
in the $(1+1)$ dimensions since $q^{+}\neq 0$ for $Q^{2}\neq 0$. It turns out
that the nonvalence contribution dominates in the large $Q^{2}$ region due to
its substantial $\ln Q^{2}$ behavior. This appears the characteristic of the
form factor in the $(1+1)$ dimensional scalar field model that we discuss in
the present work.

Figure 2:  Scalar pion form factor (solid line) and its valence (dotted line)
and non-valence (dashed line) contributions for $0\leq Q^{2}\leq 2$ GeV2
region.

Shown in Fig. 2 is the EM form factor for a ‘scalar pion’ in the spacelike
region of $0\leq Q^{2}\leq 2$ GeV2. In this model calculation, we use $M=0.14$
GeV and $m_{q}=m_{\bar{Q}}=0.25$ GeV. The dotted, dashed, and solid lines
represent the valence contribution $I^{+}_{\rm S1}$, non-valence contribution
$I^{+}_{\rm S2}$, and the total result of the form factor, respectively. We
find that, while the valence contribution dominates for small $Q^{2}$ region,
the non-valence contribution takes over the valence one for $Q^{2}\geq 0.1$
GeV2 and most of the contribution to the form factor for high $Q^{2}$ comes
from the non-valence diagram, indicating significant contributions from the
higher-Fock components.

###  II.2 Form factor in the timelike region

Figure 3:  One-loop Feynman diagrams that contribute to
$\gamma^{*}(q)\to\bar{\mathcal{M}}(p)+{\cal M}(p^{\prime})$. The covariant
diagram (a) is the same as the sum of the two LF time-ordered diagrams (b) and
(c).

The process of $\mathcal{M}(p)+\gamma^{*}(q)\to\mathcal{M}(p^{\prime})$ in
spacelike momentum transfer ($q^{2}<0$) region can be made to the timelike
$(q^{2}>0)$ process such as
$\gamma^{*}(q)\to\bar{\mathcal{M}}(p)+\mathcal{M}(p^{\prime})$, i.e., the
reaction for a virtual photon decaying into a $q{\bar{Q}}$ bound state scalar
particle $\mathcal{M}$ and its antiparticle $\bar{\mathcal{M}}$ from the
principle of crossing symmetry. Therefore, the timelike EM form factor
$F^{T}_{\cal M}(q^{2})$ for
$\gamma^{*}(q)\to\bar{\mathcal{M}}(p)+{\mathcal{M}}(p^{\prime})$ is defined by
the local current $J^{\mu}_{T}(0)$ as

$J^{\mu}_{T}(0)=(p-p^{\prime})^{\mu}~{}F^{T}_{\cal M}(q^{2}),$ (13)

where $q=p+p^{\prime}$ is the four momentum transfer of the virtual photon
satisfying $q^{2}=q^{+}q^{-}>0$. The covariant diagram describing
$\gamma^{*}(q)\to\bar{\cal M}(p)+{\cal M}(p^{\prime})$ process is shown in
Fig. 3(a) and the local current is obtained by
$J^{\mu}_{T}(0)=e_{q}I^{\mu}(m_{q},m_{\bar{Q}})+e_{\bar{Q}}I^{\mu}(m_{\bar{Q}},m_{q})$
as in the case of $\mathcal{M}(p)+\gamma^{*}(q)\to\mathcal{M}(p^{\prime})$
process. Essentially, $I^{\mu}(m_{q},m_{\bar{Q}})$ in this timelike process
has the same form given by Eq. (2) but with the overall sign changed.

In LF calculations, using the plus component of the currents $J^{+}_{T}$, the
Cauchy integration over $k^{-}$ in Eq. (2) gives two time-ordered
contributions to the residue calculations: one coming from the region ${\rm
T1}$ $(0<k^{+}<p^{+})$ [Fig. 3(b)] and the other from the region ${\rm T2}$
$(p^{+}<k^{+}<q^{+})$ [Fig. 3(c)]. In the region of T1 (T2), the residue is at
the pole of $k^{-}=k^{-}_{l}$ ($k^{-}=k^{\prime-}_{r}$), where
$k^{\prime-}_{r}$ is the same as $k^{-}_{r}$ but with replacing $q$ by $-q$.
The poles $k_{l}^{-}$ and $k^{\prime-}_{r}$ are in the lower and upper half of
the complex-$k^{-}$ plane, respectively. This allows one to obtain the Cauchy
integration of $I^{+}(m_{q},m_{\bar{Q}})$ in Eq. (2) over $k^{-}$ in the
regions T1 and T2 as

$\displaystyle I^{+}_{\rm T1}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{4\pi}\int^{p^{+}}_{0}dk^{+}\frac{2k^{+}-q^{+}}{C^{\prime}_{k}(k^{-}_{l}-k^{-}_{b})(k^{-}_{l}-k^{\prime-}_{r})},$
$\displaystyle I^{+}_{\rm T2}$ $\displaystyle=$
$\displaystyle-\frac{g^{2}}{4\pi}\int^{q^{+}}_{p^{+}}dk^{+}\frac{2k^{+}-q^{+}}{C^{\prime}_{k}(k^{\prime-}_{r}-k^{-}_{l})(k^{\prime-}_{r}-k^{-}_{b})},$
(14)

where $C^{\prime}_{k}=C_{k}(q\rightarrow-q)$.

In the timelike region of $(1+1)$ dimensions, the momentum transfer $q^{2}$ is
defined by

$q^{2}=M^{2}(1+\alpha)^{2}/\alpha,$ (15)

where $\alpha=p^{\prime+}/p^{+}=q^{+}/p^{+}-1$ and the two solutions for
$\alpha$ are given by

$\alpha_{\pm}=\left(\frac{q^{2}}{2M^{2}}-1\right)\pm\sqrt{\left(\frac{q^{2}}{2M^{2}}-1\right)^{2}-1}.$
(16)

This shows that both $\alpha_{\pm}=1$ correspond to the threshold
$q^{2}=4M^{2}$. However, the EM form factor $F^{T}_{\mathcal{M}}(q^{2})$ is
independent of the subscript sign of $\alpha$ as in the case of
$F^{S}_{\mathcal{M}}(Q^{2})$ given by Eqs. (8) and (9), which can be seen
below.

With $\alpha$ and $k^{+}=xp^{+}$, we rewrite Eq. (II.2) as

$\displaystyle I^{+}_{\rm T1}$ $\displaystyle=$
$\displaystyle-\frac{p^{+}g^{2}}{4\pi}\int^{1}_{0}\frac{[dx]}{1+\alpha-x}\frac{\alpha(1+\alpha-2x)/(1+\alpha)}{(M^{2}-M^{2}_{0x})\left[M^{2}-\frac{\alpha
m^{2}_{q}}{x(1+\alpha-x)}\right]},$ $\displaystyle I^{+}_{\rm T2}$
$\displaystyle=$
$\displaystyle\frac{p^{+}g^{2}}{4\pi}\int^{1}_{0}\frac{[dx^{\prime}]}{1+\alpha
x^{\prime}}\frac{\alpha(2\alpha
x^{\prime}-\alpha+1)/(1+\alpha)}{(M^{2}-M^{\prime
2}_{0x})\left[M^{2}-\frac{m^{2}_{q}}{(1-x^{\prime})(1+\alpha
x^{\prime})}\right]},$

where $M^{2}_{0x}=\frac{m^{2}_{q}}{x}+\frac{m^{2}_{\bar{Q}}}{1-x}$ and
$M^{\prime 2}_{0x}=M^{2}_{0x}(m_{q}\leftrightarrow m_{Q})$. The change of
variable, $x\equiv\alpha x^{\prime}+1$, is made in the calculation of
$I^{+}_{\rm T2}$.

Then we further integrate over the variables $(x,x^{\prime})$ to combine both
contributions, $I_{\rm T}=(I^{+}_{\rm T1}+I^{+}_{\rm T2})/(p-p^{\prime})^{+}$,
which leads to the fully analytic form of $F^{T}_{\cal M}(q^{2})$ in $(1+1)$
dimensions as

$F^{T}_{\cal M}(q^{2})=e_{q}I^{q}_{\rm T}(q^{2})+e_{\bar{Q}}I^{Q}_{\rm
T}(q^{2}),$ (18)

where

$\displaystyle I^{q}_{\rm T}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{8\pi
m^{2}_{q}m^{2}_{\bar{Q}}\left(1-\omega^{2}-\gamma_{q}\right)}$ (19)
$\displaystyle\mbox{}\times\left\\{C_{\omega}+\sqrt{\frac{1-\gamma_{q}}{\gamma_{q}}}\tan^{-1}\left(\frac{1}{\sqrt{\frac{1-\gamma_{q}}{\gamma_{q}}}}\right)\right\\},$

with $\gamma_{q}=q^{2}/4m^{2}_{q}$. The second part $I^{Q}_{\rm T}(q^{2})$ of
Eq. (18) can be obtained from Eq. (19) by replacing $\gamma_{q}$ with
$q^{2}/4m^{2}_{\bar{Q}}$. It can be easily seen that Eq. (19) and Eq. (9) are
essentially the same through the analytic continuation from spacelike $Q^{2}$
region to timelike $q^{2}(=-Q^{2})$ region. This shows that the timelike EM
form factor can be analytically continued to spacelike region by changing
$q^{2}\to-q^{2}$ in the form factor and vice versa.

It should be noted that the threshold points of the timelike form factor
depend on the bound state condition. That is, the denominator factor
$(1-\omega^{2}-\gamma_{q})$ in Eq. (19) is always nonzero for strong bound
state satisfying both $M<m_{q}+m_{\bar{Q}}$ and
$M^{2}<m^{2}_{q}+m^{2}_{\bar{Q}}$. In this strong bound state case, the
imaginary part of $I^{q(Q)}_{T}(q^{2})$ starts to develop at $q^{2}\geq
4m^{2}_{q(Q)}$. In other words, the thresholds for strong bound state are
given by the “normal” threshold points, i.e., $q^{2}_{\rm min}=4m^{2}_{q}$ and
$4m^{2}_{\bar{Q}}$ for $\gamma^{*}q{\bar{q}}$ and $\gamma^{*}Q{\bar{Q}}$
vertices, respectively. Furthermore, one can easily extract the analytic form
for the imaginary part of the timelike form factor from Eq. (19) as

$\mbox{Im}[I^{q}_{\rm
T}(q^{2})]=-\frac{g^{2}\theta(\gamma_{q}-1)}{16m^{2}_{q}m^{2}_{\bar{Q}}\left(1-\omega^{2}-\gamma_{q}\right)}\sqrt{\frac{\gamma_{q}-1}{\gamma_{q}}},$
(20)

where $\theta(\gamma_{q}-1)$ is the Heaviside step function, i.e.,
$\theta(\gamma_{q}-1)=1$ for $\gamma_{q}>1$ and vanishes otherwise.

On the other hand, for the weakly bound state satisfying $M<m_{q}+m_{\bar{Q}}$
but $M^{2}>m^{2}_{q}+m^{2}_{\bar{Q}}$, the singular $q^{2}$ point, satisfying
$1-\omega^{2}-\gamma_{q}=0$, exists. Because of this, the singularity for the
weakly bound state starts at the “anomalous” threshold points,

$q^{2}_{\rm
min}=\frac{1}{m^{2}_{{\bar{Q}}(q)}}[m^{2}_{q({\bar{Q}})}-(M-m_{{\bar{Q}}(q)})^{2}][(M+m_{{\bar{Q}}(q)})^{2}-m^{2}_{q({\bar{Q}})}],$
(21)

for $\gamma^{*}q{\bar{q}}\;(\gamma^{*}Q{\bar{Q}})$ vertex prior to the normal
thresholds. Our result for the anomalous thresholds given by Eq. (21) is
exactly the same as the one in Ref. Gasiorowicz  for the analysis of the one-
particle matrix elements of a scalar current. Especially, in this weakly bound
state in $(1+1)$ dimensions, the real part of the timelike form factor
diverges at the anomalous threshold.

As a consistency check, we compare our direct result for the form factor with
the dispersion relations (DRs) given by

$\displaystyle{\rm
Re}[F(q^{2})]=\frac{1}{\pi}\mathcal{P}\int_{-\infty}^{\infty}dq^{\prime
2}~{}\frac{{\rm Im}[F(q^{\prime 2})]}{q^{\prime 2}-q^{2}},$ $\displaystyle{\rm
Im}[F(q^{2})]=-\frac{1}{\pi}\mathcal{P}\int_{-\infty}^{\infty}dq^{\prime
2}~{}\frac{{\rm Re}[F(q^{\prime 2})]}{q^{\prime 2}-q^{2}},$ (22)

where ${\mathcal{P}}$ indicates the Cauchy principal value. For the strong
bound state case, we confirm that the real part of the form factor obtained
from the direct calculation in Eq. (19) is exactly the same as the one
obtained from DR using the analytic form of the imaginary part given by Eq.
(20). However, for the weakly bound state case, we note the importance of
taking into account the infinitesimal width $\Gamma$ as
$(1-\omega^{2}-\gamma_{q})\to(1-\omega^{2}-\gamma_{q}-i\Gamma)$ in Eq. (19)
for the timelike form factor in order to remedy the singularity at the
anomalous threshold point. With this care, we explicitly obtain both real and
imaginary part of the timelike form factor for the weakly bound state as

$\displaystyle{\rm Re}[I^{q}_{\rm T}]$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{8\pi
m^{2}_{q}m^{2}_{\bar{Q}}\left((1-\omega^{2}-\gamma_{q})^{2}+\Gamma^{2}\right)}$
$\displaystyle\times\left\\{(1-\omega^{2}-\gamma_{q}){\tilde{C}}^{q}_{\omega}-\frac{\pi}{2}\Gamma\sqrt{\frac{\gamma_{q}-1}{\gamma_{q}}}~{}\theta(\gamma_{q}-1)\right\\},$
$\displaystyle{\rm Im}[I^{q}_{\rm T}]$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{8\pi
m^{2}_{q}m^{2}_{\bar{Q}}\left((1-\omega^{2}-\gamma_{q})^{2}+\Gamma^{2}\right)}$
$\displaystyle\times\left\\{\Gamma{\tilde{C}}^{q}_{\omega}-(1-\omega^{2}-\gamma_{q})~{}\frac{\pi}{2}\sqrt{\frac{\gamma_{q}-1}{\gamma_{q}}}\theta(\gamma_{q}-1)\right\\},$

where

${\tilde{C}}^{q}_{\omega}=C_{\omega}+{\rm
Re}\left[\sqrt{\frac{\gamma_{q}-1}{\gamma_{q}}}\tanh^{-1}\Bigg{(}\frac{1}{\sqrt{\frac{\gamma_{q}-1}{\gamma_{q}}}}\Bigg{)}\right].$
(24)

For an explicit demonstration, we show in Fig. 4 the timelike form factor for
the weakly bound state pion obtained with $M=0.14$ GeV and
$m_{q}=m_{\bar{Q}}=0.08$ GeV together with an infinitesimal width
$\Gamma=0.0001$ GeV. In this case, the normal threshold ($q^{2}=4m^{2}_{q}$)
appears at $q^{2}=0.0256$ GeV while the anomalous threshold defined in Eq.
(21) appears at $q^{2}\simeq 0.0184$ GeV. The solid and dashed lines are the
results for the real and imaginary parts of the form factor obtained from Eq.
(LABEL:eq19-1). The circle and square data represent the results for the real
and imaginary parts of the form factor obtained from the DRs given by Eq.
(II.2). This confirms that our direct results presented in Eq. (LABEL:eq19-1)
are in complete agreement with the DRs.

Figure 4:  Timelike form factor for the weakly bound state pion obtained with
$M=0.14$ GeV and $m_{q}=m_{\bar{Q}}=0.08$ GeV. The direct results of real
(solid line) and imaginary (dashed line) parts of the form factor are compared
with the real (circle) and imaginary (square) parts obtained from the
dispersion relations.

##  III Longitudinal charge density and charge radius

The intrinsic size of the hadron is defined through the slope of the form
factor, i.e., the mean-square charge radius in three dimensional space is
given by $\braket{r^{2}}=-6dF(Q^{2})/dQ^{2}|_{Q^{2}=0}\equiv-6F^{\prime}(0)$.
The size of the hadron may also be computed from the intrinsic charge density
$\rho({\bf r})$ in three dimensional space within a nonrelativistic theory,
defined by the Fourier transform of the form factor,

$\rho({\bf r})=\frac{1}{(2\pi)^{3}}\int d^{3}Q\;e^{-i{\bf Q}\cdot{\bf
r}}F(Q^{2}).$ (25)

At relativistic energies this interpretation becomes obscured because of its
dependence on the reference frame Hofstadter56 ; GR ; Burkardt00 ; Miller09 ;
Miller09a . With this caveat, the mean-square charge radius $\braket{r^{2}}$
may be defined as the second moment of the intrinsic charge density $\rho({\bf
r})$, which gives $\braket{r^{2}}=\int r^{2}\rho({\bf r})d^{3}r$.

However, since the intrinsic charge density is inherently nonrelativistic and
requires relativistic corrections, the transverse charge density $\rho(b)$ was
proposed in Refs. Burkardt02 ; Diehl02 ; Miller09 ; Miller09a  as the true
charge density without the need for relativistic corrections, which is
obtained by the following two-dimensional Fourier transform,

$\rho(b)=\frac{1}{(2\pi)^{2}}\int d^{2}{\bf q}_{\perp}\;e^{-i{\bf
q_{\perp}}\cdot{\bf b}}F(Q^{2}={\bf q}^{2}_{\perp}),$ (26)

where ${\bf b}$ is the two-dimensional transverse variable and $F(Q^{2}={\bf
q}^{2}_{\perp})$ is obtained from the DYW frame (i.e., $q^{+}=0$ and
$q^{2}=-{\bf q}^{2}_{\perp}=-Q^{2}$). This transverse density is also the
integral of the three dimensional infinite-momentum frame (IMF) density
$\rho(x^{-},b)$ over all values of the longitudinal position coordinate
Miller09a . The central charge density of the hadron is determined by
$\rho(b=0)$ because of the Lorentz-contraction of the longitudinal dimension
in the IMF. The mean-square transverse radius $\braket{b^{2}}$ is then given
in terms of $\rho(b)$ as $\braket{b^{2}}=\int
d^{2}b\,b^{2}\rho(b)=-4F^{\prime}(0)$.

While the frame dependence of the intrinsic charge density in the longitudinal
direction due to relativistic corrections, such as the Lorentz contraction,
has been widely discussed in Refs. Burkardt00 ; Miller09 ; Miller09a , the
explicit estimation of the relativistic effect is yet to be fully discussed.
The purpose of this section is to apply our results for the EM form factor in
$(1+1)$ dimensions to obtain the longitudinal charge density $\rho(r_{z})$ and
to understand the difference between the intrinsic charge density obtained
from the one-dimensional Fourier transform of the charge form factor
$F(Q^{2})$ and the relativistic version of the true static charge density
obtained from the $(1+1)$-dimensional Fourier transform of
$J^{0}(Q^{2})=(p+p^{\prime})^{0}F(Q^{2})$ in the so-called “Breit frame (BF)”
where no energy is transferred to the hadron, i.e., $q=(q^{0},q_{z})=(0,Q)$,
$p=(E,p_{z})$, and $p^{\prime}=(E,-p_{z})$. In LFD, the longitudinal charge
density is discussed in terms of the boost invariant variable
$\tilde{z}=p^{+}x^{-}$.

###  III.1 Intrinsic longitudinal charge density

The intrinsic longitudinal charge density (ILD) $\rho_{\rm ILD}(r_{z})$ may be
defined by the one-dimensional Fourier transform of the spacelike form factor
$F^{S}_{\mathcal{M}}(Q^{2})\equiv F(Q^{2})$ as

$\rho_{\rm ILD}(r_{z})=\frac{1}{2\pi}\int dQ\;e^{-iQr_{z}}F(Q^{2}),$ (27)

where $Q$ corresponds to the longitudinal component (i.e., $Q=q_{z}$) of
momentum transfer. The intrinsic longitudinal density $\rho_{\rm ILD}(r_{z})$
represents the probability that electric charge is located at a longitudinal
distance $r_{z}$ from the longitudinal center of momentum with the
normalization condition $\int dr_{z}\,\rho_{\rm ILD}(r_{z})=F(0)$.

Using the inverse Fourier transform and the normalization of $\rho_{\rm
ILD}(r_{z})$, we obtain

$\displaystyle F(Q^{2})$ $\displaystyle=$ $\displaystyle\int
dr_{z}\;e^{iQr_{z}}\rho_{\rm ILD}(r_{z})$ (28) $\displaystyle=$ $\displaystyle
1-\frac{1}{2}Q^{2}\braket{r^{2}_{z}}_{\rm ILD}+\cdots,$

where $\braket{r^{2}_{z}}_{\rm ILD}=\int dr_{z}\;r^{2}_{z}\rho_{\rm
ILD}(r_{z})$. From Eqs. (8) and (9), we can explicitly obtain the analytic
form of the mean square charge radius in the longitudinal direction $r_{z}$,
i.e., $\braket{r^{2}_{z}}_{\rm ILD}=-2~{}\partial F/\partial
Q^{2}|_{Q^{2}=0}$, as

$\braket{r^{2}_{z}}_{\rm
ILD}=\frac{1}{6}\left[\frac{3}{1-\omega^{2}}-\frac{1}{1+C_{\omega}}\right]\left(\frac{e_{q}}{m^{2}_{q}}+\frac{e_{\bar{Q}}}{m^{2}_{\bar{Q}}}\right).$
(29)

It is interesting to compare our result with that obtained from the simple
analysis of nonrelativistic quark model Green , where the mean-square charge
radius, $\braket{r^{2}_{\rm em}}=-6F^{\prime}(0)$, defined in three spatial
dimensions is obtained as $\braket{r^{2}_{\rm
em}}=\frac{(e_{q}m^{2}_{\bar{Q}}+e_{\bar{Q}}m^{2}_{q})\braket{\delta^{2}}}{(m_{q}+m_{\bar{Q}})^{2}}$
with $\bm{\delta}={\bf r}_{q}-{\bf r}_{\bar{Q}}$ being relative coordinate.
This result was derived from $\braket{r^{2}_{\rm
em}}=\braket{\sum^{2}_{i}e^{2}_{i}(r_{i}-R)^{2}}$, i.e., the deviation from
the center-of-mass position $R$ squared weighted by the charge of the quark
and antiquark constituents. While $\braket{r^{2}}_{\rm ILD}$ and
$\braket{r^{2}_{\rm em}}$ were derived from different spacetime dimensions and
different methods, the both have the common factor
$e_{q}m^{2}_{\bar{Q}}+e_{\bar{Q}}m^{2}_{q}$. From this common factor in charge
radius, it is easy to find that the neutral meson such as $K^{0}(d{\bar{s}})$
has a negative square charge radius.111A negative value for
$\braket{r^{2}_{\rm em}}$ happens when the lighter negatively charged $d$
quark is orbiting around the heavier ${\bar{s}}$ quark Green .

Of particular interest, we also obtain $\braket{r^{2}_{z}}_{\rm ILD}$ in terms
of binding energy $B$ defined by $M=2m_{q}-B$ for equal constituent mass case
($m_{q}=m_{Q}$), which leads to

$\braket{r^{2}_{z}}_{\rm
ILD}=\frac{1}{6m^{2}_{q}}\biggl{[}\frac{12}{\varepsilon(4-\varepsilon)(2-\varepsilon)^{2}}-\frac{\sqrt{\varepsilon(4-\varepsilon)}(2-\varepsilon)}{\sqrt{\varepsilon(4-\varepsilon)}(2-\varepsilon)+2(\varepsilon^{2}-4\varepsilon+2)\tan^{-1}\left[\frac{2-\varepsilon}{\sqrt{\varepsilon(4-\varepsilon)}}\right]}\biggr{]},$
(30)

where $\varepsilon=B/m_{q}$ is the dimensionless parameter ranging from zero
binding ($B=0$) to maximal binding (i.e., $B=2m_{q}$) limits. While
$\braket{r^{2}_{z}}_{\rm ILD}\to\infty$ in the zero binding limit, it
decreases monotonically to the minimum value $\braket{r^{2}_{z}}_{\rm ILD}\to
1/(5m^{2}_{q})$ as $B\to 2m_{q}$, which is consistent with the observation
made in Ref. GS90 . As one can see, the charge radius is getting smaller as
the constituent mass $m_{q}$ is getting larger.

###  III.2 Relativistic longitudinal charge density in BF

In $(1+1)$ dimensions, the Fourier transform of the current $J^{\mu}$ is given
by

${\tilde{J}}^{\mu}(t,r_{z})=\frac{1}{(2\pi)^{2}}\int
d^{2}q\;J^{\mu}(q^{2})e^{i(q^{0}t-q^{3}r_{z})}.$ (31)

If we take the BF, where $q^{0}=0$ and $q^{2}=-(q^{3})^{2}=-Q^{2}$, the
momentum of incoming meson, $p$, and that of the outgoing meson,
$p^{\prime}=p+q$, are given by

$\displaystyle(p^{0},p^{3})_{\rm BF}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\sqrt{4M^{2}+Q^{2}},-\sqrt{Q^{2}}\right),$
$\displaystyle(p^{\prime 0},p^{\prime 3})_{\rm BF}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\sqrt{4M^{2}+Q^{2}},\sqrt{Q^{2}}\right).$ (32)

In the BF, only the time component of the currents $J^{\mu}$ in Eq. (1)
survives and the space component is zero so that

$\displaystyle J^{0}_{\rm BF}(q^{2})$ $\displaystyle=$
$\displaystyle\sqrt{4M^{2}+Q^{2}}\;F(q^{2}),$ $\displaystyle J^{3}_{\rm
BF}(q^{2})$ $\displaystyle=$ $\displaystyle 0.$ (33)

Then the Fourier transform of the current $J^{0}$ in the BF results in

$\rho(r_{z})=\frac{1}{2\pi}\int dQ\;{\cal F}(Q^{2})e^{-iQr_{z}},$ (34)

where $\rho(r_{z})=\int dt\,{\tilde{J}}^{0}(t,r_{z})$ corresponds to the
longitudinal charge density and ${\cal
F}(Q^{2})\equiv\sqrt{4M^{2}+Q^{2}}\;F(Q^{2})$. It should be noted that the
prefactor $\sqrt{4M^{2}+Q^{2}}$ in ${\cal F}(Q^{2})$ depends on the reference
frame while the form factor $F(Q^{2})$ is Lorentz-invariant. Since Eq. (34)
leads to the normalization of $\rho(r_{z})$ as $\int dr_{z}\;\rho(r_{z})={\cal
F}(0)=2M$, we redefine Eq. (34) as

$\rho_{\rm BF}(r_{z})=\frac{1}{2\pi}\int dQ\;{\cal F}_{\rm
BF}(Q^{2})e^{-iQr_{z}},$ (35)

where $\rho_{\rm BF}(z)=\rho(z)/2M$ and ${\cal F}_{\rm BF}(Q^{2})={\cal
F}(Q^{2})/2M$. Equation (35) now satisfies $\int dr_{z}\;\rho_{\rm
BF}(r_{z})=1$ and ${\cal F}_{\rm BF}(0)=1$.

Using the inverse Fourier transform and the normalization of $\rho_{\rm
BF}(r_{z})$, we obtain

$\displaystyle{\cal F}_{\rm BF}(Q^{2})$ $\displaystyle=$ $\displaystyle\int
dr_{z}\;e^{iQr_{z}}\rho_{\rm BF}(r_{z})$ (36) $\displaystyle=$ $\displaystyle
1-\frac{1}{2}Q^{2}\braket{r^{2}_{z}}_{\rm BF}+\cdots,$

where $\braket{r^{2}_{z}}_{\rm BF}=\int dr_{z}\;r^{2}_{z}\rho_{\rm
BF}(r_{z})$. Comparing to $\braket{r^{2}_{z}}_{\rm ILD}$, we find

$\braket{r^{2}_{z}}_{\rm BF}=\braket{r^{2}_{z}}_{\rm ILD}-\frac{1}{4M^{2}}.$
(37)

We note that $\braket{r^{2}_{z}}_{\rm BF}$ is independent of time $t$ and is
smaller than $\braket{r^{2}_{z}}_{\rm ILD}$ due to the Lorentz contraction.
For instance, if the mass of the bound state is $M\simeq 1~{}\mbox{GeV}$, then
$\braket{r^{2}_{z}}_{\rm BF}=\braket{r^{2}_{z}}_{\rm
ILD}-0.01~{}\mbox{fm}^{2}$ and the relativistic correction is getting larger
(smaller) as $M$ gets smaller (larger) as expected.

Figure 5:  Comparison of $\rho_{\rm ILD}(r_{z})$ and $\rho_{BF}(r_{z})$ for
the strongly bound state $\pi^{+}$ in (a) and the weakly bound state $D^{+}$
meson in (b).

In Fig. 5, we compare the two longitudinal charge densities, $\rho_{\rm
ILD}(r_{z})$ (solid line) and $\rho_{\rm BF}(r_{z})$ (dashed line), for a
strongly bound state $\pi^{+}$ [Fig. 5(a)] and a weakly bound state $D^{+}$
meson [Fig. 5(b)] in the range of $-1\leq r_{z}\leq 1$ fm. These numerical
results are estimated with $m_{u}=m_{d}=0.25~{}\mbox{GeV}$ and
$m_{c}=1.8~{}\mbox{GeV}$ together with the physical meson masses, i.e.,
$M_{\pi^{+}}=0.14~{}\mbox{GeV}$ and $M_{D^{+}}=1.870~{}\mbox{GeV}$. One can
clearly see that $\rho_{\rm BF}(r_{z})$ are more narrowly peaked near the
longitudinal center of the momentum than $\rho_{\rm ILD}(r_{z})$ due to the
Lorentz contraction, which is more significant for $\pi^{+}$ than for $D^{+}$.

Other reference frame may be obtained by the Lorentz transformation from the
BF. For instance, the target rest frame (TRF), where $(p^{0},p^{3})_{\rm
TRF}=(M,0)$, can be obtained from the BF by the following Lorentz
transformation:

$\left(\begin{array}[]{cc}\gamma&\gamma\beta\\\ \gamma\beta&\gamma\\\
\end{array}\right)\left(\begin{array}[]{c}p^{(\prime)0}_{\rm BF}\\\
p^{(\prime)3}_{\rm BF}\\\ \end{array}\right)=\left(\begin{array}[]{c}p_{\rm
TRF}^{(\prime)0}\\\ p_{\rm TRF}^{(\prime)3}\\\ \end{array}\right),$ (38)

where the Lorentz factors are given by $\gamma=\frac{\sqrt{4M^{2}+Q^{2}}}{2M}$
and $\gamma\beta=\frac{\sqrt{Q^{2}}}{2M}$, which leads to

$(p^{\prime 0},p^{\prime 3})_{\rm
TRF}=\frac{1}{2M}\left(2M^{2}+Q^{2},\sqrt{Q^{2}(4M^{2}+Q^{2})}\right).$ (39)

In the TRF, both time and space components of the current $J^{\mu}$ in Eq. (1)
are non-vanishing and explicitly they are given by

$\displaystyle J_{\rm TRF}^{0}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{4M^{2}+Q^{2}}{2M}F(q^{2}),$ $\displaystyle J_{\rm
TRF}^{3}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{Q^{2}}\sqrt{4M^{2}+Q^{2}}}{2M}F(q^{2}),$ (40)

where $q^{2}=(q^{0})^{2}-(q^{3})^{2}=-Q^{2}$. One should note, however, that
the longitudinal charge density in TRF is not a static quantity but depends on
time because of the fact that $q^{0}=\gamma\beta Q\neq 0$.

###  III.3 Longitudinal charge density in LF coordinate space

Recently, a general procedure was introduced to obtain frame-independent
three-dimensional LF coordinate-space wave functions MB19 . In addition to the
two dimensional transverse spatial variable ${\bf b}$ given by Eq. (26), the
longitudinal boost-invariant dimensionless spatial variable
$\tilde{z}=p^{+}x^{-}$ was also introduced.

In the present $(1+1)$-dimensional model calculations, the longitudinal charge
density in LF coordinate space evaluated at $x^{+}=0$, as a Fourier transform
of the form factor, can be defined by

$\rho_{\rm LF}(x^{-})=\frac{1}{4\pi}\int
dq^{+}~{}F(q^{2})~{}e^{\frac{i}{2}q^{+}x^{-}},$ (41)

where $\rho_{\rm LF}(x^{-})$ satisfies $\int\rho_{\rm LF}(x^{-})~{}dx^{-}=1$.
Using $q^{+}=\bar{\beta}p^{+}$ and Eq. (5), $\rho_{\rm LF}$ can be rewritten
in terms of $({\tilde{z}},{\bar{\beta}})$ modulo the $p^{+}$ factor as

$\rho_{\rm LF}(\tilde{z})=\frac{1}{4\pi}\int
d\bar{\beta}~{}F(\bar{\beta})~{}e^{\frac{i}{2}\bar{\beta}\tilde{z}},$ (42)

where $0\leq\bar{\beta}_{+}\leq\infty$ for ${\bar{\beta}}={\bar{\beta}}_{+}$
and $-1\leq\bar{\beta}_{-}\leq 0$ for ${\bar{\beta}}={\bar{\beta}}_{-}$,
respectively. Since $\bar{\beta}_{-}$ has a closed range, which takes
advantage over $\bar{\beta}_{+}$, we obtain $\rho_{\rm LF}(\tilde{z})$
integrating over $\bar{\beta}_{-}$ in Eq. (42).

Figure 6:  The longitudinal charge density for $\pi^{+}$ in LF coordinate
space ${\tilde{z}}$.

Figure 6 shows the longitudinal charge density $\rho_{\rm LF}(\tilde{z})$ for
$\pi^{+}$ in a wide range of $-100\leq{\tilde{z}}\leq 100$. This shows that
$\rho_{\rm LF}(\tilde{z})$ has a very long and oscillating tail behavior of
$\tilde{z}$, which appears consistent with the result shown in Ref. MB19  for
the case of two-constituents of a Fock-space component. From the inverse
Fourier transform of Eq. (42),

$\displaystyle F(\bar{\beta})$ $\displaystyle=$ $\displaystyle\int
d\tilde{z}~{}e^{-\frac{i}{2}\bar{\beta}\tilde{z}}\rho_{\rm LF}(\tilde{z})$
(43) $\displaystyle=$ $\displaystyle
1-\frac{\bar{\beta}^{2}}{8}\langle\tilde{z}^{2}\rangle+\cdots,$

one may obtain the mean square charge radius in ${\tilde{z}}$ as
$\langle\tilde{z}^{2}\rangle=-8~{}\partial
F/\partial\bar{\beta}^{2}|_{\bar{\beta}^{2}=0}$ where
$\langle\tilde{z}^{2}\rangle=\int d\tilde{z}~{}\tilde{z}^{2}\rho_{\rm
LF}(\tilde{z})$.

##  IV Numerical results

In our numerical calculations, we analyze scalar $\pi$, $K$, and $D$ meson
form factors. For these analyses, we use the constituent quark and antiquark
masses as $m_{u}=m_{d}=0.25~{}\mbox{GeV}$, $m_{s}=0.5~{}\mbox{GeV}$, and
$m_{c}=1.8~{}\mbox{GeV}$ as in Ref. CJ99 . The used physical meson masses are
$M_{\pi^{\pm}}=0.14~{}\mbox{GeV}$, $M_{K^{\pm}}=0.494~{}\mbox{GeV}$,
$M_{K^{0}}=0.497~{}\mbox{GeV}$, $M_{D^{\pm}}=1.870~{}\mbox{GeV}$, and
$M_{D^{0}}=1.865~{}\mbox{GeV}$, respectively. It should be noted from our
constituent masses that $M^{2}<m^{2}_{q}+m^{2}_{\bar{Q}}$ for $\pi$ and $K$
but $M^{2}>m^{2}_{q}+m^{2}_{\bar{Q}}$ for $D$ meson case, while all the mesons
satisfy the bound state condition, $M<m_{q}+m_{\bar{Q}}$. This means that
$\pi$ and $K$ are strongly bound states but $D$ is weakly bound state of which
properties will be discussed in the following numerical calculations.

In the previous work for the $\phi^{3}$ model in $(3+1)$ dimensions CJ99 , two
of us analyzed the form factors in three different reference frames, namely,
(i) purely longitudinal ($q^{+}\neq 0$ and ${\bf q}_{\perp}=0$) frame defined
in the timelike $q^{2}>0$ region, (ii) purely longitudinal ($q^{+}\neq 0$ and
${\bf q}_{\perp}=0$) frame defined in the spacelike $q^{2}<0$ region, and
(iii) the ($q^{+}=0$ with $q^{2}=-{\bf q}^{2}_{\perp}$) frame,222The details
can be found in Eqs. (10), (19), and (23) of Ref. CJ99 . to confirm that all
of three reference frames give exactly the same numerical results for the form
factor in the entire $q^{2}$ region. So, when we refer the “direct results”
from $(3+1)$ dimensions, we mean the results obtained from any of those three
results in Ref. CJ99 . Likewise, the direct results from $(1+1)$ dimensions
indicate those obtained by using Eq. (9) or (19) in the present work. On the
other hand, “DR results” refer to those obtained from the dispersion relations
given by Eq. (II.2). Comparing the two results obtained from both $(1+1)$\-
and $(3+1)$-dimensions, we shall also estimate the effects of the transverse
momenta ${\bf k}_{\perp}$ of the quark and anitquark on the EM form factors.

Shown in Fig. 7 is the profile of the intrinsic longitudinal charge density
$\rho_{\pi}(Q,r_{z})$ for $\pi^{+}$ and its contour plot in phase space
$(Q,r_{z})$ of $-10\leq Q\leq 10~{}\mbox{GeV}$ and $-1\leq r_{z}\leq
1~{}\mbox{fm}$. The momentum-dependent $\rho(Q,r_{z})$ is defined as
$\rho(r_{z})=\frac{1}{2\pi}\int
F(Q^{2})e^{-iQr_{z}}\,dQ\equiv\int\rho(Q,r_{z})\,dQ$. We also show in Fig. 8
the profiles of the intrinsic longitudinal charge densities for
$(K^{+},K^{0})$ and $(D^{+},D^{0})$ in phase space of $-10\leq Q\leq
10~{}\mbox{GeV}$ and $-1\leq r_{z}\leq 1~{}\mbox{fm}$. These figures confirm
that $\rho(Q,r)$ is symmetric under $Q\to-Q$ and $r_{z}\to-r_{z}$ as expected.
The generic structures of $\rho(Q,r_{z})$ for charged particles
($\pi^{+},K^{+},D^{+}$) look similar to each other. Likewise, the generic
structures of neutral particles ($K^{0},D^{0}$) are similar to each other. On
the other hand, the density profiles of charged particles are quite different
from those of neutral ones.

Figure 7:  Profile of the intrinsic longitudinal charge density $\rho_{\rm
ILD}(Q,r_{z})$ for $\pi^{+}$ and its contour plot in phase space $(Q,r_{z})$
of $-10\leq Q\leq 10$ GeV and $-1\leq r_{z}\leq 1$ fm.

Figure 8:  Profiles of the intrinsic longitudinal charge densities $\rho_{\rm
ILD}(Q,r_{z})$ for $(K^{+},K^{0})$ and $(D^{+},D^{0})$ in phase space of
$-10\leq Q\leq 10$ GeV, $-1\leq r_{z}\leq 1$ fm, respectively.

We present the intrinsic longitudinal charge densities $\rho(r_{z})$ for
charged $(\pi^{+},K^{+},D^{+})$ mesons in Fig. 9(a) and for neutral
$(K^{0},D^{0})$ mesons in Fig. 9(b), respectively. The gap between
$\rho_{\pi^{+}}(r_{z})$ (solid line) and $\rho_{K^{+}}(r_{z})$ (dotted line)
is very small and it is hard to distinguish them in Fig. 9(a). Compared to the
light $(\pi,K)$ mesons, the charge density of heavy $D$ meson is narrowly
peaked around $r_{z}=0$. Figure 9(b) also shows the charge density behavior of
neutral $(K^{0},D^{0})$ mesons which satisfy $\int\rho(r_{z})dr_{z}=0$.

Figure 9:  The intrinsic longitudinal charge densities $\rho_{\rm ILD}(r_{z})$
of (a) $(\pi^{+},K^{+},D^{+})$ and (b) $(K^{0},D^{0})$ for $-1\leq r_{z}\leq
1$ fm.

Figure 10:  EM form factor of the pion in $(1+1)$\- and $(3+1)$-dimensions:
(a) Re[$F_{\pi}(q^{2})$] and (b) Im[$F_{\pi}(q^{2})$] and $|F_{\pi}(q^{2})|$
for $-2\leq q^{2}\leq 4$ GeV2 compared with the results obtained from the
dispersion relations.

In Fig. 10, we show the EM form factor of the pion obtained in $(1+1)$\- and
$(3+1)$-dimensions for $-2\leq q^{2}\leq 4~{}\mbox{GeV}^{2}$. The black and
blue lines represent the direct results obtained from the form factors in
$(3+1)$\- and $(1+1)$-dimensions, respectively. The corresponding $(3+1)$\-
and $(1+1)$-dimensional results obtained from the dispersion relations are
denoted by black circles and blue squares. Figure 10(a) represents
Re[$F_{\pi}(q^{2})$], and Fig. 10(b) includes both Im[$F_{\pi}(q^{2})$] and
$|F_{\pi}(q^{2})|$. Close inspection of the figures leads to the following
comments. Firstly, our “direct results” for both Re[$F_{\pi}(q^{2})$] and
Im[$F_{\pi}(q^{2})$] in $(1+1)$\- and $(3+1)$-dimensions show complete
agreement with the “DR results” in corresponding dimensions, respectively.
Secondly, the imaginary parts (dashed lines) of the form factors in Fig. 10(b)
obtained from both $(1+1)$\- and $(3+1)$-dimensions start at the normal
threshold $q^{2}_{\rm min}=4m^{2}_{u(d)}=0.25~{}\mbox{GeV}^{2}$, which is
consistent with the condition for $M^{2}<m^{2}_{q}+m^{2}_{\bar{Q}}$ case. For
high $q^{2}$ region, the imaginary parts of the form factors are shown to
dominate over the real part. Thirdly, the total form factors
$|F_{\pi}(q^{2})|$ (solid lines) in both $(1+1)$\- and $(3+1)$-dimensions
produce a $\rho$ meson-type peaks consistent with the vector meson dominance
(VMD). However, we do not claim that this model indeed reproduces all the
features of the VMD phenomena since more realistic phenomenological models may
have to incorporate more complex mechanisms such as the initial- and final-
state interactions. Finally, in that the difference between the two results in
$(1+1)$\- and $(3+1)$-dimensions measures the effects of transverse momenta of
the constituents, one can see that the effects of ${\bf k}_{\perp}$ reduce the
charge radii and broaden the widths of the peaks. Therefore, while the
qualitative behaviors of the form factors in both dimensions are not much
different from each other, their quantitative behaviors are quite sizable due
to the effects of the transverse momenta of the constituents.

Figure 11:  EM form factors of $K^{+}$ and $K^{0}$ mesons in $(1+1)$\- and
$(3+1)$-dimensions: (a) Re[$F_{K^{+}}(q^{2})$], (b) Im[$F_{K^{+}}(q^{2})$] and
$|F_{K^{+}}(q^{2})|$, (c) Re[$F_{K^{0}}(q^{2})$], and (d)
Im[$F_{K^{0}}(q^{2})$] and $|F_{K^{0}}(q^{2})|$ for $-2\leq q^{2}\leq 4$ GeV2.

Figure 11 shows the EM form factors of $K^{+}(u\bar{s})$ and $K^{0}(d\bar{s})$
mesons obtained from $(1+1)$\- and $(3+1)$-dimensions for $-2\leq q^{2}\leq
4~{}\mbox{GeV}^{2}$. The same line codes are used as in Fig. 10. The direct
results and the DR results for the unequal quark mass cases such as $K$ and
$D$ mesons coincide and we do not explicitly display the DR results in Fig.
11. As in the case of the pion, both $K^{+}$ and $K^{0}$ have the normal
singularities. However, $K$ mesons have two thresholds, namely, one at
$q^{2}_{\rm min}=4m^{2}_{u}$ (or $4m^{2}_{d}$) and the other at $q^{2}_{\rm
min}=4m^{2}_{s}$. While we have in principle two vector-meson-type peak (i.e.
$\rho$ and $\phi$), one can see in Fig. 11 that only the $\phi$ meson-type
peak would be observable for the timelike kaon EM form factors above the
physical threshold at $q^{2}_{\rm min}=4M^{2}_{K^{+(0)}}$ as the $\rho$ meson-
type peak is kinematically below the physical threshold, i.e.
$4m^{2}_{u(d)}<4M^{2}_{K^{+(0)}}$. Again, the differences between the
$(1+1)$\- and $(3+1)$-dimensional results reside in the effects of transverse
momenta of the constituents which play the role of broadening the widths and
flatten the heights of the form factors. The sign flips for both
Re[$F_{K^{0}}(q^{2})$] and Im[$F_{K^{0}}(q^{2})$] between the two peaks come
from the different sign of electric charges of $d$ and $\bar{s}$ quarks. One
can find that the $K^{0}$ meson has the primary and secondary peaks near the
heavy $s$ and light $d$ quark threshold $q^{2}_{\rm min}=4m^{2}_{s(d)}$,
respectively, while it is the opposite for the $K^{+}$ meson. The spacelike
$q^{2}$ region of $|F_{K^{+,0}}(q^{2})|$ in Figs. 11(b,d) shows that $K^{+}$
has a positive mean-square charge radius while $K^{0}$ has a negative one.

Figure 12:  EM form factors of $D^{+}$ and $D^{0}$ mesons in $(1+1)$\- and
$(3+1)$-dimensions: (a) Re[$F_{D^{+}}(q^{2})$], (b) Im[$F_{D^{+}}(q^{2})$] and
$|F_{D^{+}}(q^{2})|$, (c) Re[$F_{D^{0}}(q^{2})$], and (d)
Im[$F_{D^{0}}(q^{2})$] and $|F_{D^{0}}(q^{2})|$ for $-5\leq q^{2}\leq 20$
GeV2.

We present the EM form factors of $D^{+}(c\bar{d})$ and $D^{0}(c\bar{u})$
mesons obtained in $(1+1)$\- and $(3+1)$-dimensions for $-5\leq q^{2}\leq
20~{}\mbox{GeV}^{2}$ in Fig. 12. The same line codes are used as in Fig. 10.
Although the generic features of $D$ meson form factors are similar to the
case of $K$ mesons, several comments are in order. As explained before, the
weakly bound state such as the $D$ meson in the present model calculation has
anomalous thresholds. Both $(1+1)$-and $(3+1)$-dimensional results of
$|F_{D^{+(0)}}(q^{2})|$ in Figs. 12(b,d) indeed show the presence of anomalous
thresholds as given in Eq. (21), i.e., $q^{2}_{\rm min}\sim 0.24$ GeV2
(compared to $4m_{d(u)}=0.25$ GeV2 for normal case) and $q^{2}_{\rm min}\sim
12.4$ GeV2 (compared to $4m_{c}=12.96$ GeV2 for normal case) for the
$\gamma^{*}$-${\bar{d}}(\bar{u})$ and $\gamma^{*}$-$c$ vertices, respectively.
Both anomalous thresholds, however, appear just before the normal thresholds
although the normal thresholds for the light quark sector are hard to be seen
in Fig. 12. Similar to the $K$ meson case, the EM form factors of both charged
and neutral $D$ mesons have two unphysical peaks, i.e., $\rho$ and
$J/\psi(1S)$ meson type peaks due to ${\bar{d}}$ (or ${\bar{u}}$) and $c$
quarks, respectively. However, the timelike form factors of $D$ mesons have no
pole structures for the physical $q^{2}\geq 4M_{D}^{2}$ region. Finally,
unlike the kaon case, the primary peaks for both $D^{+}$ and $D^{0}$ appear
near the thresholds due to the heavy $c$ quark. Figures 12(b,d) also shows
that $D^{+}$ has a positive mean-square charge radius while $D^{0}$ has a
negative one.

##  V Summary and Conclusion

In the present work, we presented $(1+1)$-dimensional analysis of EM form
factors of scalar mesons both for spacelike and timelike region in the
solvable model, obtaining the analytic results of the one-loop triangle
diagram both for the valence and non-valence contributions. Since the
$q^{+}\neq 0$ frame should be used in $(1+1)$ dimensions, it is inevitable to
encounter the non-valence diagram arising from the particle-antiparticle pair
creation (the so-called “Z-graph”). While the valence contribution dominates
for small $Q^{2}$ region, its role is taken over by the non-valence
contribution as $Q^{2}$ gets larger indicating significant contributions from
the higher-Fock components. The leading asymptotic behavior of the form factor
at high $Q^{2}$ in $(3+1)$ dimensions Miller09  has one more logarithmic power
than our result $F^{S}_{\cal M}(Q^{2}\to\infty)\sim\ln Q^{2}/Q^{2}$ in $(1+1)$
dimensions, which is ascribed to the effects of the transverse momentum. Our
analytic results both in the spacelike region and the timelike region
confirmed the analytic continuation from $Q^{2}$ in the spacelike region to
$-Q^{2}$ in the timelike region. In the timelike form factor given by Eq.
(18), the imaginary part of $I^{q(Q)}_{T}(q^{2})$ starts to develop at the
anomalousl threshold given by Eq. (21) for the weakly bound state with
$M<m_{q}+m_{\bar{Q}}$ but $M^{2}>m^{2}_{q}+m^{2}_{\bar{Q}}$, while it starts
at the normal threshold $q^{2}\geq 4m^{2}_{q(Q)}$ for the strongly bound state
with $M<m_{q}+m_{\bar{Q}}$ and $M^{2}<m^{2}_{q}+m^{2}_{\bar{Q}}$. We confirm
that the DRs given by Eq. (II.2) are satisfied by the strongly bound state as
well as by the weakly bound state. In particular, we note the importance of
taking into account the infinitesimal width to remedy the singularity at the
anomalous threshold for the weakly bound state as given by Eq. (LABEL:eq19-1)
in order to satisfy the DRs as shown in Fig. 4.

Defining the intrinsic charge density $\rho({\bf r})$ in three dimensional
space, the transverse charge density $\rho(b)$ in two dimensional space and
the longitudinal charge density $\rho_{\rm ILD}(r_{z})$ in one dimensional
space, respectively, in Eqs. (25), (26) and (27), one may convince that the
mean-square charge radius $\braket{r^{2}}_{\rm em}$ is given by the sum of the
mean-square transverse radius $\braket{b^{2}}$ and the mean-square
longitudinal distance $\braket{r^{2}_{z}}_{\rm ILD}$, i.e.,
$\braket{r^{2}}_{\rm em}=\braket{b^{2}}+\braket{r^{2}_{z}}_{\rm ILD}$. While
$\braket{r^{2}}_{\rm ILD}$ and $\braket{r^{2}}_{\rm em}$ are derived from
different spacetime dimensions and different methods, both have the common
factor $e_{q}m^{2}_{\bar{Q}}+e_{\bar{Q}}m^{2}_{q}$. This observation allows to
understand the negative value of mean-square charge radius of neutral mesons
such as $K^{0}(d{\bar{s}})$ which have negatively charged light quark orbiting
around the heavier ${\bar{s}}$ quark Green . The generic structures of
longitudinal charge densities for charged particles ($\pi^{+}$, $K^{+}$,
$D^{+}$) are similar to each other although the density profiles of charged
particles are quite different from those of neutral ones ($K^{0},D^{0}$). For
the case of equal constituent mass ($m_{q}=m_{Q}$), $\braket{r^{2}_{z}}_{\rm
ILD}$ given by Eq. (30) decreases monotonically to the minimum value
$\braket{r^{2}_{z}}_{\rm ILD}\to 1/(5m^{2}_{q})$ in the maximal binding limit
while $\braket{r^{2}_{z}}_{\rm ILD}\to\infty$ in the zero binding limit, which
is consistent with the observation made in Ref. GS90 . In contrast to the
intrinsic charge density, the apparent charge density defined by the time
component of the current depends on the reference frame and we note that
$\braket{r^{2}_{z}}_{\rm BF}$ is smaller than $\braket{r^{2}_{z}}_{\rm ILD}$
due to the Lorentz contraction.

In terms of the newly introduced boost invariant variable $\tilde{z}$, we also
define the longitudinal charged density in LFD as given by Eq. (41) noting
that the LF longitudinal momentum fraction ${\bar{\beta}}=q^{+}/p^{+}$ is the
variable conjugate to $\tilde{z}$. From Fig. 6, we found that $\rho_{\rm
LF}(\tilde{z})$ has a very long and oscillating tail behavior of $\tilde{z}$,
consistent with the result shown in Ref. MB19  for the case of two-
constituents of a Fock-space component.

Comparing the results in $(1+1)$\- and $(3+1)$-dimensions, we note that the
effects of transverse momentum ${\bf k}_{\perp}$ reduce the charge radii and
broaden the widths of the peaks in charge densities. While the qualitative
behaviors of the form factors in both dimensions are not much different each
other, their quantitative behaviors are quite sizable due to the effects of
the transverse momenta of the quark and antiquark. We thus conclude that the
transverse momentum plays the role of broadening the width of the resonance
and significantly flattens the height of the corresponding form factor. Our
analysis of the solvable scalar field model can be extended to the
phenomenologically more realistic LFQM as we have shown for the transition
form factor $F_{\mathcal{M}\gamma}(q^{2})$ in the meson-photon transition
process, $\mathcal{M}(p)\to\gamma^{*}(q)+\gamma(p^{\prime})$ CRJ17 .

*

## Appendix A

In this appendix we present the analytic forms of the valence and non-valence
contributions to the form factor in the spacelike region. The explicit forms
of $I^{q}_{\rm S1,S2}$ are

$\displaystyle I^{q}_{\rm S1}$ $\displaystyle=$
$\displaystyle\frac{g^{2}C_{S}}{4\pi}\biggl{[}c_{1}{\rm
tan}^{-1}\left(\frac{\omega_{Q}}{\sqrt{1-\omega^{2}}}\right)+c_{2}{\rm
tan}^{-1}\left(\frac{\omega_{q}}{\sqrt{1-\omega^{2}}}\right)$ (44)
$\displaystyle\hskip 28.45274pt+c_{3}{\rm
tan}^{-1}\left(\frac{M^{2}(\beta-2)-(m^{2}_{q}-m^{2}_{Q})\beta}{2m_{q}m_{Q}\beta\sqrt{1-\omega^{2}}}\right)$
$\displaystyle\hskip
28.45274pt+c_{4}\ln\left(\frac{m^{2}_{q}\beta^{2}}{\beta(m^{2}_{q}+{\bar{\beta}}m^{2}_{Q})-M^{2}{\bar{\beta}}}\right)\biggr{]},$
$\displaystyle I^{q}_{\rm S2}$ $\displaystyle=$
$\displaystyle\frac{g^{2}C_{S}}{4\pi}\biggl{[}d_{1}{\rm
tanh}^{-1}\left(\frac{\sqrt{\gamma_{Q}}}{\sqrt{1+\gamma_{Q}}}\right)+d_{2}{\rm
tan}^{-1}\left(\frac{\omega_{q}}{\sqrt{1-\omega^{2}}}\right)$ (45)
$\displaystyle\hskip 28.45274pt+d_{3}{\rm
tan}^{-1}\left(\frac{M^{2}(\beta-2)-(m^{2}_{q}-m^{2}_{Q})\beta}{2m_{q}m_{Q}\beta\sqrt{1-\omega^{2}}}\right)$
$\displaystyle\hskip
28.45274pt+d_{4}\ln\left(\frac{m^{2}_{q}\beta^{2}}{\beta(m^{2}_{q}+{\bar{\beta}}m^{2}_{Q})-M^{2}{\bar{\beta}}}\right)\biggr{]},$

where

$C_{S}=\frac{1-\omega^{2}}{8M^{2}m^{3}_{q}m^{3}_{Q}(1-\omega^{2}+\gamma_{Q})(1-\omega^{2})^{3/2}(\beta^{2}-1)},$
(46)

and

$\displaystyle c_{1}$ $\displaystyle=$ $\displaystyle
4m_{q}m_{Q}\omega\left(2M^{2}{\bar{\beta}}+\beta Q^{2}\right),$ $\displaystyle
c_{2}$ $\displaystyle=$ $\displaystyle
8m^{2}_{q}m^{2}_{Q}(1-\omega^{2})+2{\bar{\beta}}[M^{2}(m^{2}_{q}+m^{2}_{Q})-(m^{2}_{q}-m^{2}_{Q})^{2}]$
$\displaystyle+(M^{2}-m^{2}_{q}+m^{2}_{Q})Q^{2}\beta,$ $\displaystyle c_{3}$
$\displaystyle=$ $\displaystyle-4m_{q}m_{Q}M^{2}\omega{\bar{\beta}}$
$\displaystyle-2m_{Q}\left[4m^{2}_{q}m_{Q}\left(\omega^{2}-1\right)+m_{q}\omega
Q^{2}-m_{Q}Q^{2}\right]\beta,$ $\displaystyle c_{4}$ $\displaystyle=$
$\displaystyle
m_{q}m_{Q}\sqrt{1-\omega^{2}}\left(2M^{2}+2m^{2}_{q}-2m^{2}_{Q}+Q^{2}\right)\beta,$
(47)

and

$\displaystyle d_{1}$ $\displaystyle=$ $\displaystyle
4M^{2}m_{q}m_{Q}\left(\beta^{2}-1\right)\sqrt{1-\omega^{2}}\frac{\sqrt{1+\gamma_{Q}}}{\sqrt{\gamma_{Q}}},$
$\displaystyle d_{2}$ $\displaystyle=$ $\displaystyle
2m_{Q}[\left(\beta^{2}-1\right)M^{2}m_{q}\omega-{\bar{\beta}}^{2}M^{2}m_{Q}-4\beta
m^{2}_{q}m_{Q}(1-\omega^{2})],$ $\displaystyle d_{3}$ $\displaystyle=$
$\displaystyle-c_{3},$ $\displaystyle d_{4}$ $\displaystyle=$ $\displaystyle-
c_{4}.$ (48)

The corresponding results in the timelike region are readily obtained by
changing $Q^{2}\to-Q^{2}$ in Eqs. (44) and (45).

###### Acknowledgements.

H.-M.C. was supported by the National Research Foundation of Korea (NRF) under
Grant No. NRF- 2020R1F1A1067990. The work of Y.C. and Y.O. was supported by
NRF under Grants No. NRF-2020R1A2C1007597 and No. NRF-2018R1A6A1A06024970
(Basic Science Research Program). C.-R.J. was supported in part by the US
Department of Energy (Grant No. DE-FG02-03ER41260).

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11institutetext: Monash University

# Think Physics, Think Man: Barriers to Women’s Participation in Physics
Education (OPINION)

Eliot Walton

##  1 Framework

This text focuses on women because the research that is available tends to
fall into binary gender categories. This work is not generalisable to other
genders without further research. Furthermore, while the discussion draws on
my experience as a trans-woman, I have omitted a discussion of trans-specific
issues (such as transition leave and transphobia) for generality. In my
estimation, the most common situation for a female physics student is that she
is cis-gender and one of a small number among a class primarily composed of
cis-men. I seek to explore two possible barriers to her learning, the notion
of competency and gender dynamics in groups.

##  2 Competency

Having transitioned from male-presenting to female-presenting over the course
of second and third year, I can attest to the degree to which being perceived
as female in physics is synonymous with being perceived as incompetent. My
particular situation was extremely bizarre, where the more I studied (hence
the more I knew) the less I was assumed to know by male peers. Rather than
focusing on mansplaning [1] which is a particular type of condescending
explanation given by a man to a woman where an unreasonable degree of
incompetence is ascribed to the woman, I will focus on why women are assumed
to be incompetent in physics and how the way that physics is taught is likely
disadvantage women.

Educational experiments also link a student’s belief in themselves about their
ability to learn material with success in learning the material [2, 3]. So
self-belief has a tangible effect on actual competence as well as perceived
confidence. Therefore, an individual’s idea of what skills and types of
intelligence are necessary for a discipline have an effect on their success in
the discipline.

Leslie et al., argue that these perceptions can help account for the
distribution of women across academic disciplines [4, 5, 6]. In particular,
they argue that raw intellectual talent, which is associated more with men
than women [4], helps to explain the lack of women in physics, mathematics and
philosophy since these disciplines are assumed to require such skills. If one
must be ’innately’ talented to succeed, and women are taught that innate
talent tends not to reside in them, then it is not surprising that women feel
alienated from the field. It is likely that this alienation encourages women
to experience ”imposter syndrome”, which is most commonly understood to be a
feeling of ”intellectual fraudulence” [7], accompanied by the conviction that
success is unearned and future endeavours are likely to fail [8]. A 2022 study
highlights that minorities (such as women) in ’brilliance’ oriented
disciplines experience higher levels of imposterism, which has negative
consequences on their future career [8].

These results must be carefully interpreted. This is an additional barrier
which women face compared to men, but it does not negate all other barriers
(such as the lack of role models and being a numerical minority in classes).
However, this is an effect of which teachers should be mindful when presenting
class material; emphasis on individual brilliance (regardless of whether it is
by men or by women) produces a set of expectations about the requirements for
success which alienate women from the field.

The other care that must be taken is to avoid adopting the how-to-ask-for-a-
promotion solution where the problem is treated as being with the women rather
than with the way that physics is taught. It is very easy to conclude that
women need to be be more like men in order to be successful in physics and
that we should teach them accordingly. However, this is incorrect because it
places the burden of addressing the inequality on women, and argues they must
be less like themselves to succeed. Rather, we should meet women where they
are and acknowledge that, by the time a student is at university, she has
already internalised such beliefs. To un-learn them is a slow process which
requires that we re-think the way we teach physics.

##  3 Group Dynamics

Gender dynamics operate in all interpersonal relationships. If they are
unnoticed by some that is likely because they have become familiar and have
therefore receded from view but they are often obvious to individuals whose
gender is conspicuous. Since men are the universal default [9] what is
unmarked is assumed to be male. Men make up the majority of people seen or
heard in films, on television or over radio [9]. They make up the majority of
world leaders, CEOs and professors [10]. Therefore man-to-man dynamics are the
most commonly represented gender dynamics and this conspires with the male
default to make the interactions between men appear as the standard rather
than one possible combination of genders. For example, we have the Australian
Football League (AFL), and the Australian Football League Women (AFLW); women
are an afterthought if they are a thought at all. The dynamics in a physics
classroom between members of a group are often male, focusing on competition
and individualism.

This is exclusive of women, but that is not the main point of concern. The
main point of concern is how these dynamics interact with women when women are
introduced [11]. Barbara and Gene Eakings found that, in a recording of seven
university faculty meetings, that men spoke more than women, with one
exception and that the longest comment by a woman was shorter than the
shortest comment by a man [12]. Analysing online discussions about linguistics
between men and women revealed that men’s messages tended to be twice as long
as women’s messages [13].

While I know of no data which gives these numbers for physics classrooms, it
seems reasonable to hypothesise that a similar pattern will hold. While
speaking is not the only form of engagement with class material, in
interactive and flipped classrooms it is a major one. Flipped classrooms are
those where traditional lectures and homework are reversed. Speaking is the
verbal equivalent of taking up space, and women are already taught to minimise
and be conscious of the space they occupy. It is therefore expected that these
formats disadvantage women. Furthermore, while men measurably speak more in
these studies, women are stereotyped as speaking more [14] and this effects
the perception of who speaks more.

My father recently remarked to me about the number of women economists
consulted on the news and pointed out that I would be annoyed if they were all
men economists. The implication being that these situations are equally
inequitable. Despite the perceived abundance women, when measured by the
Global Media Monitoring Project, which has evaluated the world’s media for the
representation of women every five years since 1995, their 2021 report found
that women only make up 25% of persons heard, read about or seen in radio,
newspapers or television [15]. This represents a single point improvement
since 2015 and the first improvement since 2010. Despite this incremental
progress, when Caroline Criado-Perez campaigned in 2013 for a female
historical figure on the back of English banknotes many men were upset with
one man writing ”but women are everywhere now!” [9].

While wrong about the numbers, this man’s experience is enlightening because
he (and those who think like him) experience even a minor intrusion of women
as an inequity. This is important because it highlights that men’s experience
of how many women there are, how often women speak and for how long is biased
against women. Therefore, even there are numerically fewer women and they
measurably speak less, men may still experience this as them ’being
everywhere’. This conclusion is empirically supported by research from the
Global Institute for Women’s Leadership (GIWL), which found that medical
professionals tended to over-estimate women’s true representation in several
different areas of medicine and in different roles [16].

The GIWL research indicates that a man’s degree of overestimation predicted
lower support for gender equality initiatives (with a p-value less than
0.001). Women’s (over)estimations were unrelated to their level of support
(with a p value of 0.92) [16]. This study establishes an empirical
relationship between medical practitioners’ perception of women’s
representation and their willingness to support gender equality initiative but
makes no attempt to explain why such a relationship exists. It is reasonable
to assume that such men see these initiatives as inequitable or unnecessary
and that this is a consequence of their biased perception against women.

While these data have not been collected for physics it is reasonable to
hypothesise that they would show a similar pattern. A recent study of 27 self-
identified progressive white-male physics faculty and graduate students found
that these men despite their self-image as equity champions acted in ways that
maintained the unequal status quo [17]. If this is the case for progressive
men, then it is reasonable to assume those who do not think of themselves as
such will be even more inhibitive to women’s progress and initiatives. In
physics, the degree to which women are perceived as intruding may be greater
than other fields because of the degree of male-domination. An example of this
thinking was provided by Prof. Alessandro Strumia in 2018 where he claimed
that men were discriminated against in the hiring practices of Center for
European Nuclear Research (CERN) [18]. These claims were made despite women
representing 21% of CERN staff in 2019 [19].

He subsequently published a heavily criticised paper attempting to demonstrate
this perceived inequality. This paper did ”not provide a convincing
understanding of the literature or the methods [of the field of gender
studies]” [20] according to Cassidy Sugimots, a researcher on gender
disparities in science [21].

##  4 The Glass Cliff

When all other solutions fail, try women. An educational class in honours
electrodynamics operated via this principle; when a male colleague and I
failed to solve a problem we turned to the quiet observer, the other woman in
the class, and asked directly for her input. In transitioning I have found
particular patterns of thinking which originate in my socialisation including
many of the patterns of thinking critiqued here.

This honours class shows an example of the glass-cliff phenomenon which is the
tendency to appoint women to precarious leadership positions and was first
identified in 2004 [10]. It is reasonable to think this contributed to the
2022 appointment of Elizabeth Truss as Prime Minister of the United Kingdom,
and may also have affected the election of Giorgia Meloni in Italy since this
effect has been measured repeatably in women’s leadership [10]. Perhaps this
effect appears also in physics classrooms, where more space and credence tends
to be made for and lent to women’s input when problems are more difficult.

While this seems to directly contradict the notion that women are less
competent there is no contradiction. Women are sought out of desperation
rather than deference. Discussions of the glass cliff especially in the case
of male-dominated environments often include a discussion of conscious or
unconscious malice; that the woman is set-up to fail or expected to fail. I do
not suggest that men in physics ask their female colleagues in bad faith
(although this is possible). However, it is important to be wary of the
possibility of bias confirmation; if a woman is unable to do the harder
problems when she is turned to this may depress her confidence and reinforce
her peers’ beliefs (conscious or otherwise) about women’s inability to do
physics.

Collaboration is an important aspect of research and teaching attempts to
reflect this in its assessment. While it is reasonably well understood that
group projects will have uneven workloads between members and that this is
reflective of real world practices the way that this type of assessment effect
women verses men should be considered when setting such tasks. In workplace
teams, women tend to do a disproportionate amount of the labour [22] and in
some fields, such as economics, they tend to be given less credit for their
collaborative work [23]. It is not clear if these results are applicable to a
physics classroom; however, I suspect that the amount of work women do will be
disproportionate in group tasks. This suspicion comes from discussions with
undergraduates and is an extension of glass cliff thinking. A student may not
be incorporated if the work is trivial, and she may be given more when it is
difficult solely due to glass-cliff thinking. Combining this with the
disproportionate load women carry in workplace teams, I expect that, when her
contributions are integrated, she will do the majority of the work, or she
will do none of it.

##  5 Conclusion and Further Avenues

###  5.1 Conclusion

A woman in a physics classroom faces obvious and non-obvious barriers to her
learning. The story of physics research is told in a way that emphasises the
innate ability of the individual which biases physics teaching and success
towards men. It is likely that this has a measurable affect on women’s mastery
of the class material. Gender dynamics affect physics classrooms and it is
expected that these dynamics disadvantage women compared to men especially in
the case of flipped class rooms. However, even though women speak less and are
fewer, they still may be perceived as too numerous leading to a reduction in
men’s support of gender equality, which may correlate with a reduction in
their awareness of the ways in which their behaviour excludes women. When
women are included in groups they are treated differently to men by men, who
either expect them to do all of the work or none of it, generally undervalue
their contributions and only turn to them as an emergency resource. Therefore,
it is expected that women will have differential outcomes to men when passing
through a physics curriculum.

###  5.2 Further Avenues

This paper does not explore:

  * •

Intersectionality

  * •

Differential behaviour and expectations

  * •

Gendered violence

  * •

The de-emphasis of women in science history

  * •

Teaching methodology and the built environment

  * •

Role models

  * •

Mentoring programs

  * •

Health (especially access to feminine hygiene products)

  * •

The effect of flexible learning (or lack thereof)

  * •

The effect and distribution of unpaid domestic work

all of which are worthy of exploration. I acknowledge that many expectations
here are based on inference and speculation and that further research is
required to support or refute many claims before action may be taken on them.
This document’s primary purpose is educational and discursive.

## Acknowledgements

I respectfully acknowledge the Boonwurrung people of the Kulin nation as the
traditional owners and custodians of the land on which I live and work. I pay
my respects to Elders past, present and emerging and support the establishment
of an Aboriginal and Torres Strait Islander Voice. I wish to thank Monash’s
Women and Non-Binary People in Physics and Astronomy (WiNPA) for their ongoing
support and companionship. I also wish to thank the Monash Equity, Diversity
and Inclusion (EDI) committee for their support. In particular, I wish to
thank Rowina Nathan, Maddy Howell, Javira Altmann, Sabrina Rock and Dr Mark
Walton for their insightful discussion, comments and support. I also wish to
thank Prof. Ulrik Egede for his willingness to support this work and its
publication.

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