b

-like

depicted
ure of our
s, through
ent with
Physics Letters B 581 (2004) 75–81

www.elsevier.com/locate/physlet

Electromagnetic form factor of the pion in the space- and time
regions within the front-form dynamics

J.P.B.C. de Meloa, T. Fredericob, E. Pacec, G. Salmèd

a Instituto de Física Teórica, Universidade Estadual Paulista, 01405-900 São Paulo, SP, Brazil
b Departamento de Física, Instituto Tecnológico da Aeronáutica, Centro Técnico Aeroespacial,

12.228-900 São José dos Campos, São Paulo, Brazil
c Dipartimento di Fisica, Università di Roma “Tor Vergata” and Istituto Nazionale di Fisica Nucleare, Sezione Tor Vergata,

Via della Ricerca Scientifica 1, I-00133 Roma, Italy
d Istituto Nazionale di Fisica Nucleare, Sezione Roma I, P.le A. Moro 2, I-00185 Roma, Italy

Received 10 November 2003; accepted 28 November 2003

Editor: P.V. Landshoff

Abstract

The pion electromagnetic form factor is calculated in the space- and time-like regions from−10 (GeV/c)2 up to
10(GeV/c)2, within a front-form model. The dressed photon vertex where a photon decays in a quark–antiquark pair is
generalizing the vector meson dominance ansatz, by means of the vector meson vertex functions. An important feat
model is the description of the on-mass-shell vertex functions in the valence sector, for the pion and the vector meson
the front-form wave functions obtained within a realistic quark model. The theoretical results show an excellent agreem
the data in the space-like region, while in the time-like region the description is quite encouraging.
 2003 Elsevier B.V. All rights reserved.

PACS:12.39.Ki; 12.40.Vv; 13.40.Gp; 14.40.Aq

Keywords:Relativistic quark model; Vector-meson dominance; Electromagnetic form factors; Pion
]
ber
lec-
g
l-
e-
a

c-
in
art-
le
ate
er-
ce
rtex
In the framework of the front-form dynamics [1
(see, e.g., [2,3] for extensive reviews) a large num
of papers has been devoted to the study of the e
tromagnetic form factor of the pion, mostly dealin
with the space-like (SL) region [4–10]. To our know
edge, in the front-form quantization the elastic tim
like (TL) form factor has only been calculated in

E-mail address:salmeg@roma1.infn.it (G. Salmè).
0370-2693/$ – see front matter 2003 Elsevier B.V. All rights reserved
doi:10.1016/j.physletb.2003.11.072
scalar field theory model forqQ̄ mesons with point-
like vertices [11].

In this Letter, the pion electromagnetic form fa
tor is evaluated within the front-form dynamics,
both the SL and the TL regions, by using as a st
ing point the Mandelstam formula [12] for the triang
diagrams of Figs. 1 and 2. Our aim is to investig
the possibility of describing the photon–hadron int
action by applying the vector meson (VM) dominan
ansatz (see, e.g., [13]) at the level of the photon ve
function.
.

http://www.elsevier.com/locate/physletb


76 J.P.B.C. de Melo et al. / Physics Letters B 581 (2004) 75–81

n
arks on the

nt
Fig. 1. Diagrammatic representation of the pion elastic form factor forq+ > 0 vs. the globalx+-time flow. Diagram (a) (0� −k+ � P+
π ) is

the contribution of the valence component in the initial pion wave function. Diagram (b) (P+
π � −k+ � P ′+

π ) is the non-valence contributio
to the pion form factor. Both processes contain the contribution from the dressed photon vertex. The crosses correspond to the qu
k− shell (see text).

Fig. 2. Diagrammatic representation of the photon decay (γ ∗ → ππ̄ ) vs. the globalx+-time flow. Diagrams (a) and (b) correspond to differe
x+-time orderings. The crosses correspond to the quarks on thek− shell (see text).
r ra-
re-
(b),
ses
r

y a
ul-
g-
pli-
by

te
e.,
een

nt
ses

ion

k;

to
ta;
2

the
ude
ere
ht
it
The virtual processes where a quark absorbs o
diates a pion are present in both the SL and the TL
gions (see the square blobs in Figs. 1(b) and 2(a),
respectively) [10]. In a recent study of decay proces
within the front-form dynamics [14] the amplitude fo
the pion emission from a quark was described b
pseudoscalar coupling of quark and pion fields, m
tiplied by a constant. Here, we just follow this su
gestion, and use a constant to parametrize the am
tudes for the radiative pion absorption or emission
a quark.

In order to simplify our calculations, we evalua
the pion form factor for a vanishing pion mass, i.
at the chiral limit. As a consequence, the gap betw
the SL region and the TL one (i.e., betweenq2 � 0 and
q2 � 4m2

π ) disappears.
Our starting point is the Mandelstam covaria

expression [12] of the amplitudes for the proces
πγ ∗ → π ′, or γ ∗ → ππ ′, where the mesonπ ′ is a
pion in the elastic case or an antipion in the product
process. For the TL region one has (see Fig. 2)

jµ = −ıe2m
2

f 2
π

Nc

×
∫

d4k

(2π)4
Λ̄π ′(k− Pπ ,Pπ ′)Λ̄π (k,Pπ)

(1)

× Tr
[
S(k −Pπ)γ 5S(k − q)Γ µ(k, q)S(k)γ 5],

where Nc = 3 is the number of colors;S(p) =
1

/p−m+ıε , with m the mass of the constituent quar

qµ is the virtual photon momentum;̄Λπ(k,Pπ ) the
vertex function for the pion, which will be assumed
be a symmetric function of the two quark momen
P
µ
π and Pµ

π ′ are the pion momenta. The factor
stems from isospin algebra. The “bar” notation on
vertex function means that the associated amplit
is the solution of the Bethe–Salpeter equation wh
the two-body irreducible kernel is placed on the rig
of the amplitude, while in the conventional case



J.P.B.C. de Melo et al. / Physics Letters B 581 (2004) 75–81 77

gation.
Fig. 3. Dressed photon vertex. The double-wiggly lines represent the front-form Green function describing the vector-meson propa
de
y
en

i-
er-
ith
ual
ate
by

rm

is
it

7].
tor

o-
-

c-
t of

f

nt
n-
he
n

-
rm
ith
e,
-
x

i-
is placed on the left of the Bethe–Salpeter amplitu
[15]. For the SL regionPµπ should be replaced b
−Pµπ , and then the initial pion vertex should be writt
asΛπ(−k,Pπ)= Λ̄π (k,−Pπ).

The central assumption of the Letter is our m
croscopical description of the dressed photon v
tex, Γ µ(k, q), in the processes where a photon w
q+ > 0 decays in a quark–antiquark pair at eq
light-front times. In these processes we approxim
the plus component of the photon vertex, dressed
the interaction between theqq̄ pair, as follows (see
Fig. 3)

Γ +(k, q)= √
2
∑
n,λ

[
ελ · V̂n(k, k− q)]Λn(k,Pn)

(2)× [ε+λ ]∗fV n
[q2 −M2

n + ıMnΓn(q2)] ,

wherefV n is the decay constant of thenth vector me-
son in a virtual photon (see below),Mn the corre-
sponding mass,Pn the VM total momentum (Pµn ≡
{P−
n = (|q⊥|2 +M2

n)/q
+,Pn⊥ = q⊥,P+

n = q+}; note
that at the production vertex, see Fig. 3, the front-fo
three-momentum is conserved), andελ the VM polar-
ization. The total decay width in the denominator
assumed to be vanishing in the SL region, while
is equal toΓn(q2) = Γnq

2/M2
n in the TL one [16].

For a detailed discussion of Eq. (2) see Ref. [1
In Eq. (2) the sum runs over all the possible vec
mesons, and the quantity[ελ · V̂n(k, k − q)]Λn(k, q)
is the VM vertex function. The momentum comp
nent,Λn(k,Pn), of the VM vertex function, evalu
ated on the quark mass shell (i.e., fork− = k−on =
(|k⊥|2 +m2)/k+), will be related for 0< k+ <P+

n to
the momentum part of the front-form VM wave fun
tion [18], which describes the valence componen
the meson state|n〉,

ψn
(
k+,k⊥;P+

n ,Pn⊥
)

= P+
n

[M2
n −M2

0(k
+,k⊥;P+

n ,Pn⊥)]
(3)× [

Λn(k,Pn)
]
[k−=k−on].

In Eq. (3),M0(k
+,k⊥;P+,P⊥) is the free mass o

a qq̄ system with total momentumP , and individual
kinematical momenta(k+,k⊥) and(P+ − k+,P⊥ −
k⊥). Since in our model we consider for the mome
only the 3S1 vector mesons, we adopt, for the o
the-mass-shell spinorial part of the VM vertex, t
form given in Ref. [18] (that generates the well-know
Melosh rotations for3S1 states)

(4)

V̂ µn (k, k − q)= γ µ − k
µ
on − (q − k)µon

M0(k+,k⊥;q+,q⊥)+ 2m
.

The coupling constant,fV n, of thenth vector meson is
defined by the covariant expression [19]εµλ

√
2fV n =

〈0|q̄γ µq|φn,λ〉, with |φn,λ〉 the VM state. The cou
pling constant can be obtained from the front-fo
VM wave function by evaluating this expression w
µ = + andλ = z, in the rest frame of the resonanc
wherePµn = (Mn, �0), P+

n =Mn (see Fig. 3). Assum
ing thatΛn(k,Pn) does not diverge in the comple
planek− for |k−| → ∞, and neglecting the contr
butions of its singularities in thek− integration, one
obtains

fV n = −ı Nc

4(2π)4

∫
dk− dk+ dk⊥

× Tr[γ+Vnz(k, k − Pn)]Λn(k,Pn)
[k2 −m2 + ıε][(Pn − k)2 −m2 + ıε]



78 J.P.B.C. de Melo et al. / Physics Letters B 581 (2004) 75–81

ell
-

we

ex-

s,
ere

ce
as

0]
n
a
can
ee
ere
are

e

on
n
x

f
of

n

to

al
e

e

e
t

t in
re
ds
ed,

t
r-

nd-
in
of
e,

at

-
ur-

e
po-

ly
the
x-
ton
en
er
ctor
= − Nc

4(2π)3

P+
n∫

0

dk+ dk⊥
Tr[γ+Vnz(k, k −Pn)]

k+(P+
n − k+)

(5)×ψn
(
k+,k⊥;Mn, �0⊥

)
,

where Vnz(k, k − Pn) = (/k − /P n + m)V̂nz(k, k −
Pn)(/k + m). Since both the quarks are on sh
in the front-form wave function and there is sym
metry with respect to the two quark momenta,
do not distinguish between[Λn(k,Pn)][k−=k−on] and
[Λn(k,Pn)][k−=P−

n −(Pn−k)−on] in the range 0< k+ <
P+
n . With our assumptions, one obtains the same

pression forfV n, either if thek− integration is per-
formed in the upper or in the lower complexk− semi-
plane.

For a unified description of TL and SL form factor
it is necessary to choose a reference frame wh
the plus component of the momentum transfer,q+,
is different from zero (otherwise,q2 = q+q− − q2⊥
cannot be positive). It is well known that the choi
of the reference frame has a fundamental role,
shown in previous works in the SL region [7–9,2
and in the TL one [11]. In Ref. [21] it was show
that, within the front-form Hamiltonian dynamics,
Poincaré covariant and conserved current operator
be obtained from the matrix elements of the fr
current, evaluated in the Breit reference frame, wh
the initial and the final total momenta of the system
directed along the spin quantization axis,z. Following
Ref. [21], we calculate Eq. (1) in a reference fram
whereq⊥ = 0 andq+ > 0.

Both in the SL and the TL regions, the integrati
on k− in Eq. (1) is performed with the assumptio
that: (i) Λπ(k,Pπ ) does not diverge in the comple
plane k− for |k−| → ∞, and (ii) the contributions
of the possible singularities ofΛπ(k,Pπ) can be
neglected. Also the contributions of the poles ink−
of the photon vertex function,Γ +(k, q), are supposed
to be negligible.

Then, in the SL case, wherePµ
π ′ = P

µ
π + qµ,

the current matrix elementjµ becomes the sum o
two contributions, corresponding to the diagrams
Fig. 1(a) and (b), respectively,jµSL = j

(I)µ
SL + j

(II)µ
SL

[9]. The contributionj (I)µSL has the integration o

k+ constrained by−P+
π � k+ � 0, and j (II)µSL has

the integration onk+ in the interval 0< k+ < q+.
The valence component of the pion contributes
j
(I)µ
SL only, while j (II)µSL is the contribution of the

pair-production mechanism from an incoming virtu
photon withq+ > 0 [7–9,20,22,23]. In the SL cas
we adopt a frame wherePπ⊥ = Pπ ′⊥ = 0, and we
obtain P+

π = q+(−1 + √
1− 4m2

π/q
2 )/2. Then, in

the limit mπ → 0 the longitudinal momenta of th
pions areP+

π = 0 andP+
π ′ = q+. Therefore only the

contribution of the pair-production mechanism,j (II)µSL ,
survives (Fig. 1(b)). As shown in Ref. [9], in a fram
whereq+ > 0, j (II)µSL (q2) dominates the form factor a
high momentum transfer. Moreover, it turns out tha
the model of Ref. [9] the momentum region, whe
j
(II)µ
SL (q2) starts to dominate the form factor, ten

toward zero if the pion mass is artificially decreas
in agreement with our present discussion.

In the TL case, one hasPµ
π ′ ≡ Pµπ̄ , qµ = Pµπ +Pµπ̄ .

The integration range onk+ for the matrix elemen
of the current,jµ, can be decomposed in two inte
vals, 0< k+ < P+

π andP+
π < k

+ < q+, and thenjµTL
becomes the sum of two contributions, correspo
ing to differentx+-time orderings (see diagrams
Fig. 2(a) and (b), respectively). In the final state
theππ̄ pair we make the purely longitudinal choic
Pπ̄⊥ = −Pπ⊥ = 0. Then, one obtainsP+

π /q
+ = xπ =

(1 ± √
1− 4m2

π/q
2 )2. In the limitmπ → 0, one has

xπ = 1 or 0. Analogously to the SL case, in wh
follows we adopt the choicexπ = 0, which implies
P+
π = 0 andP+

π̄ = q+. Therefore only the contribu
tion corresponding to the diagram of Fig. 2(b) s
vives.

The form factor of the pion in the TL and in th
SL regions can be obtained from the plus com
nent of the proper current matrix elements:jµTL =
〈ππ̄ |q̄γ µq|0〉 = (P

µ
π − P

µ
π̄ )Fπ (q

2), and j
µ
SL =

〈π |q̄γ µq|π ′〉 = (Pµπ + Pµ
π ′ )Fπ(q2) Since in the limit

mπ → 0 the form factor receives contributions on
from the diagrams of Figs. 1(b) and 2(b), where
photon decays in aqq̄ pair, one can apply our appro
imation for the plus component of the dressed pho
vertex (2), both in the SL and in the TL regions. Th
the matrix elementj+ can be written as a sum ov
the vector mesons and consequently the form fa
becomes

(6)Fπ
(
q2) =

∑
n

fV n

q2 −M2
n + ıMnΓn(q2)

g+
V n

(
q2),



J.P.B.C. de Melo et al. / Physics Letters B 581 (2004) 75–81 79

nt
it

ding

rent

rm
e

ce

o-
rm
n

t

m
n-

)),
stant
me
orp-

in
-

n
nt
n

s
.

f a
thin
el
nic

nel.
er-

lo–
e-
ates
l
en
in-

tate
st
nt).
be
-
is

s
ent

or
tal
whereg+
V n(q

2), for q2 > 0, is the form factor for the
VM decay in a pair of pions.

Each VM contribution to the sum (6) is invaria
under kinematical front-form boosts and therefore
can be evaluated in the rest frame of the correspon
resonance. In this frame one hasq+ =Mn andq− =
q2/Mn for the photon andP+

n = P−
n = Mn for the

vector meson. This means that we choose a diffe
frame for each resonance (always withq⊥ = 0), but
all the frames are related by kinematical front-fo
boosts along thez axis to each other, and to the fram
where q+ = −q− = √−q2 (qz = √−q2 ), adopted
in previous analyses of the SL region [9,21]. Sin
in our reference frame one has

∑
λ[ε+λ (Pn)]∗ελ(Pn) ·

Γ̂n = [ε+z (Pn)]∗εz(Pn) · Γ̂n = −Γ̂nz, we obtain from
Eqs. (1)–(3)

g+
V n

(
q2)

= Nc

8π3

√
2

P+
π̄

m

fπ

q+∫

0

dk+

(k+)2(q+ − k+)

×
∫
dk⊥ Tr

[[
ΘzΛ̄π(k;Pπ)

]
(k−=q−+(k−q)−on)

]

×ψ ∗̄
π

(
k+,k⊥;P+

π̄ ,Pπ̄⊥
)

× [M2
n −M2

0(k
+,k⊥;q+,q⊥)]

[q2 −M2
0(k

+,k⊥;q+,q⊥)+ iε]
(7)×ψn

(
k+,k⊥;q+,q⊥

)
,

where Θz = Vnz(k, k − q)γ 5[/k − /Pπ + m]γ 5. To
obtain Eq. (7) we have first performed thek− in-
tegration, and then we have related[Λ̄π̄ (k − Pπ,

Pπ̄ )](k−=q−+(k−q)−on) in the valence sector to the m
mentum component of the corresponding front-fo
pion wave function through the following equatio
(see Eq. (3))

ψπ
(
k+,k⊥;P+

π ,Pπ⊥
)

(8)= m

fπ

P+
π [Λπ(k,Pπ)][k−=k−on]

[m2
π −M2

0(k
+,k⊥;P+

π ,Pπ⊥)]
.

As for the VM vertex function, we do no
distinguish between[Λπ(k,Pπ )][k−=k−on] and [Λπ(k,
Pπ )][k−=P−

π −(Pπ−k)−on], in the range 0< k+ <P+
π .

Following Ref. [14], in our model the momentu
part of the quark-pion emission vertex in the no
valence sector which appears in Eq. (7),m

f
[Λ̄π (k;
π

Pπ)](k−=q−+(k−q)−on) (see the square blob in Fig. 2(b
is assumed to be a constant. The value of the con
is fixed by the pion charge normalization. The sa
constant value is assumed for the quark-pion abs
tion vertex (see the square blob in Fig. 1(b)).

Let us stress that, within our assumptions,g+
V n(q

2)

is given by the same expression both in the TL and
the SL (Pπ → −Pπ , see below Eq. (1)) regions. In
deed in the limitmπ → 0, one hasP+

π = 0, Pπ⊥ = 0,
and±P−

π = q2/Mn with the positive (negative) sig
in the TL (SL) region, and therefore the sign in fro
of /P π in the quantityΘz of Eq. (7) has no effect. The
the form factor is continuous in this limit, atq2 = 0. It
turns out that formπ = 0 only the instantaneous term
(in x+-time, see, e.g., [9]) contribute to Eq. (7) [17]

In order to describe the pion and the interactingqq̄
pairs in the 1− channel, we use the eigenfunctions o
square mass operator proposed in Refs. [24,25], wi
a relativistic constituent quark model. This mod
takes into account confinement through a harmo
oscillator potential and theπ–ρ splitting through a
Dirac-delta interaction in the pseudoscalar chan
It achieves a satisfactory description of the exp
imental masses for both singlet and tripletS-wave
mesons, with a natural explanation of the “Iachel
Anisovitch law” [26,27], namely the almost-linear r
lation between the square mass of the excited st
and the radial quantum numbern. Since the mode
of Refs. [24,25] does not include the mixing betwe
isoscalar and isovector mesons, in this Letter we
clude only the contributions of the isovectorρ-like
vector mesons.

The eigenfunctionψn(k+,k⊥;q+,q⊥), which de-
scribes the valence component of the meson s
|n〉, is normalized to the probability of the lowe
(qq̄) Fock state (i.e., of the valence compone
The qq̄ probability can be roughly estimated to
∼ 1/

√
2n+ 3/2 in a simple model [17] that repro

duces the “Iachello–Anisovitch law” [26,27], and
based on an expansion of the VM state|n〉, in terms of
properly weighted Fock states|i〉0, with i > 0 quark–
antiquark pairs.

Our calculation of the pion form factor contain
a very small set of parameters: (i) the constitu
quark mass, (ii) the oscillator strength,ω, and
(iii) the VM widths, Γn, for Mn > 2.150 GeV. The
up-down quark mass is fixed at 0.265 GeV [25]. F
the first four vector mesons the known experimen



80 J.P.B.C. de Melo et al. / Physics Letters B 581 (2004) 75–81

28].

M
us-

or

ne,
ss
re

gle
e-
ta
al-

ns
ay

be

he

e,
ve

uare
nd

lue
h
ns-
ized

4
om

ntly
he
nc-
nly
l

tum
the
ely.

de-

TL
ons
at
tent

ure

atz
l
ive

m
st
for

the
ta,
r
ate

n
ella
masses and widths are used in the calculations [
However, the non-trivialq2 dependence ofg+

V n(q
2) in

our microscopical model implies a shift of the V
masses, with respect to the values obtained by
ing Breit–Wigner functions with constant values f
g+
V n. As a consequence, the value of theρ meson

mass is moved in our model from the usual o
0.775 to 0.750 GeV. For the other VM, the ma
values corresponding to the model of Ref. [25] a
used, while for the unknown widths we use a sin
valueΓn = 0.15 GeV, which presents the best agre
ment with the compilation of the experimental da
of Ref. [29]. We consider 20 resonances in our c
culations to obtain stability of the results up toq2 =
10 (GeV/c)2.

The oscillator strength is fixed atω= 1.39 GeV2

[27]. The values of the coupling constants,fV n,
are evaluated from the model VM wave functio
through Eq. (5). The corresponding partial dec
width [19] Γe+e− = 8πα2f 2

V n/(3M
3
n), where α is

the fine structure constant, can be considered to
in agreement with the experimental data for theρ
meson (Γ th

e+e− = 6.37 keV,Γ exp
e+e− = 6.77± 0.32 keV),

and for ρ′ and ρ′′ (Γ th
e+e− = 1.61 keV andΓ th

e+e− =
1.23 keV, respectively) to be consistent with t
experimental lower bounds (Γ exp

e+e− > 2.30± 0.5 keV
andΓ exp

e+e− > 0.18± 0.1 keV, respectively) [28].
We perform two sets of calculations. In the first on

we use the asymptotic form of the pion valence wa
function, obtained withΛπ(k,Pπ ) = 1 in Eq. (8); in
the second one, we use the eigenstate of the sq
mass operator of Refs. [24,25]. The pion radius fou
for the asymptotic wave function israsymp

π = 0.65 fm
and for the full model wave function isrmodel

π =
0.67 fm, to be compared with the experimental va
r

exp
π = 0.67± 0.02 fm [30]. The good agreement wit

the experimental form factor at low momentum tra
fers is expected, since we have built-in the general
ρ-meson dominance.

The calculated pion form factor is shown in Fig.
in a wide region of square momentum transfers, fr
−10 (GeV/c)2 up to 10(GeV/c)2. A general quali-
tative agreement with the data is seen, independe
of the detailed form of the pion wave function. T
results obtained with the asymptotic pion wave fu
tion and the full model, present some differences o
above 3(GeV/c)2. The SL form factor is notably wel
described, except near−10 (GeV/c)2. It has to be
Fig. 4. Pion electromagnetic form factor vs. the square momen
transfer q2. Dashed and solid lines are the results with
asymptotic (see text) and the full pion wave function, respectiv
Experimental data are from Ref. [29].

stressed that the heights of the TL bumps directly
pend on the calculated values offV n andg+

V n.
The introduction ofω-like [31] andφ-like mesons

could improve the description of the data in the
region. For instance, the introduction of these mes
could smooth out the oscillations of the form factor
high momentum transfer values. However, a consis
dynamical description ofω-like andφ-like states is far
beyond the present work, and we leave it for fut
developments of the model.

Our results show that a VM dominance ans
for the (dressed photon)-(qq̄) vertex, within a mode
consistent with the meson spectrum, is able to g
a unified description of the SL and TL pion for
factor. Using the experimental widths for the fir
four vector mesons and a single free parameter
the unknown widths of the other vector mesons,
model gives a qualitative agreement with the TL da
while in the SL region it works surprisingly well. Ou
VM dominance model can be also applied to evalu
other observables, as theγ ∗ → πγ form factor or the
nucleon TL form factor.

Acknowledgements

This work was partially supported by the Brazilia
agencies CNPq and FAPESP and by Ministero d



J.P.B.C. de Melo et al. / Physics Letters B 581 (2004) 75–81 81

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ucl.

;
50;
04;

2)

02)

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44

v.
Ricerca Scientifica e Tecnologica. J.P.B.C.M. and T
acknowledge the hospitality of the Dipartimento
Fisica, Università di Roma “Tor Vergata” and
Istituto Nazionale di Fisica Nucleare, Sezione T
Vergata and Sezione Roma I.

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	Electromagnetic form factor of the pion in the space- and time-like regions within the front-form dynamics
	Acknowledgements
	References