Electromagnetic structure and weak decay of meson K in a light-front
QCD-inspired model∗
Fabiano P. Pereira a, J. P. B. C. de Melo b †, T. Frederico c, and Lauro Tomio d
aInstituto de F́ısica, Universidade Federal Fluminense, 24210-900, Niterói, RJ, Brazil
bUniversidade Cruzeiro do Sul, CETEC, 08060-070, São Paulo, SP, Brazil
cInstituto Tecnológico de Aeronáutica, 12228-900, São José dos Campos, SP, Brazil
dInstituto de F́ısica Teórica, UNESP, 01405-900, São Paulo, SP, Brazil
The kaon electromagnetic (e.m.) form factor is reviewed considering a light-front con-
stituent quark model. In this approach, it is discussed the relevance of the quark-antiquark
pair terms for the full covariance of the e.m. current. It is also verified, by considering a
QCD dynamical model, that a good agreement with experimental data can be obtained for
the kaon weak decay constant once a probability of about 80% of the valence component
is taken into account.
1. INTRODUCTION
The kaon, as quark-antiquark bound states, is one appropriate system to study aspects
of QCD at low and intermediate energy regions. By using quantum field theory at the
light-front the subnuclear structure can be more easily studied [ 1, 2, 3]. Within the light-
front framework and an appropriate choice of the frame, it is possible to obtain the pion
electromagnetic form factor at both space- and time-like regimes[ 4]. Using the light-cone
components J+K = J
0+J3 and J−K = J
0−J3 of the kaon electromagnetic current, one can
obtain the corresponding form factors in the light-front formalism, with a pseudoscalar
coupling for the quarks and considering the Breit frame (q+ = 0, ~q⊥ = (qx, 0) 6= 0) [ 5].
In the case of J+K there is no pair term contribution in the Breit frame. However, for the
J−K component of the electromagnetic current, the pair term contribution is different from
zero and necessary to preserve the rotational symmetry of the current.
In the next section, we outline the main equations of the model for the kaon electromag-
netic current, detailed in [ 5], with the corresponding results obtained for the kaon elastic
form factor. In section 3, we briefly review a QCD inspired model, presenting results for
the weak decay pseudoscalar constants compared to data. In section 4 we present our
conclusions.
∗Work partially supported by the Brazilian funding agencies FAPESP and CNPq
†JPBC de Melo thanks Instituto de F́ısica Teórica, UNESP, for supporting facilities
http://arxiv.org/abs/0704.0627v1
2. ELECTROMAGNETIC FORM FACTOR
The initial light-front wave function considered in the present model is given by:
ΦiQ(x, k⊥) =
(1− x)2
−M20 )(m
−M2(mQ, mR))
, (1)
where N is a normalization constant, Q ≡ {q̄, q} is the quark or antiquark index with mQ
is the corresponding quark mass, m2
is the kaon mass, x = k+/P+ is the momentum
fraction, and
M2(mQ, mR) ≡
k2⊥ +m
(P − k)2⊥ +m
− P 2⊥, (2)
with the free quark-mass operator given by M20 = M
2(mq̄, mq). mR is a mass constant
chosen to regularize the triangle diagram. For the corresponding final wave-functions,
q̄ and Φ
q , we just need to exchange P ↔ P
′ in (1) and (2). The relation between the
electromagnetic current Jµ and the space-like kaon electromagnetic form factor FK+(q
is given by 〈P ′|Jµ|P ′〉 = (P ′ + P )µFK+(q
2) . In terms of the initial (Φiq̄) and final (Φ
light-front wave functions, we have
F+q̄ (q
2) = −eq̄
N2g2Nc
4π3P+
d2k⊥dx
N+q̄ θ(x)θ(1− x) Φ
q̄ (x, k⊥)Φ
q̄(x, k⊥) , (3)
F+q (q
2) = [ q ↔ q̄ in F+q̄ (q
2) ] , (4)
where Nc is the color number, g is the coupling constant, eQ is the charge of quark Q, and
N+q̄ = (−1/4)Tr[(/k +mq̄)γ
5(/k − /P ′ +mq)γ
+(/k − /P +mq)γ
. In the light-front
approach, beside the valence contribution, we have also the non-valence contributions
to the currents. In the case of the J+ component, the non-valence component does
not contribute to the corresponding matrix elements [ 5]. The kaon electromagnetic
form factor obtained with J+ is the sum of two contributions from quark and antiquark
currents:
(q2) = F+q (q
2) + F+q̄ (q
2) normalized such that F+
(0) = 1. (5)
In the case that we consider the J− component, to obtain the kaon electromagnetic form
factor, after considering the contribution from the interval 0 < k+ < P+ (interval I), we
need to add a second contribution, which is originated from the pair terms, and non-zero
in the interval P+ < k+ < P ′+ (interval II). The contribution is obtained after a Cauchy
integral in k− is performed in the limit P ′+ → P+ [ 5]. So, instead of (5), we will have:
(q2) =
F−q (q
2) + F−q̄ (q
F−(q2)
, (6)
normalized by the charge conservation to F−
(0) = 1.
The parameters of the model are the constituent quark masses, mq = mu = md = 220
MeV, ms = 419 MeV and the regulator mass mR =946 MeV, adjusted to fit the electro-
magnetic radius of the kaon. The electromagnetic radius is related to the corresponding
form factor, with the mean-square-radius given by
〈r2K+〉 = 6
dFK+(q
. (7)
With the parameters adjusted as given above, we have 〈r2
〉 = 0.354 fm2, which is very
close to the experimental value 〈r2
〉|exp = 0.340 fm
2 [ 6].
Our results for the kaon electromagnetic form factor are presented in Fig. 1, in compar-
ison with available experimental data [ 6]. We observe that the full kaon electromagnetic
form factor is covariant only after the inclusion of the pair terms or non-valence contri-
bution to the J−
component of the electromagnetic current.
0.01 0.10 1.00
 [(GeV/c)
Figure 1. The kaon electromagnetic form factor is obtained with the plus and minus
components of the e.m. current (both cases are shown by the solid-line results) and
compared with experimental data [ 6]. The dashed-line curve shows the form factor
without the pair terms contribution in J−
3. WEAK DECAY CONSTANTS IN A QCD INSPIRED MODEL
Next, we briefly review the calculation of the pseudoscalar constants, in a light-front
QCD-inspired dynamical model. In this case, the constituent quark masses need to be
readjusted in view of the fact that, differently from the approach outlined in section 2,
the wave-function is obtained from an eigenvalue equation, as follows.
The valence wave function is obtained by solving an eigenvalue equation for the effective
square mass operator M2ps [ 7]:
ps ψ(x,
~k⊥) = M
0 (x, k⊥) ψ(x,
~k⊥)−
dx′d~k′⊥θ(x
′)θ(1− x′)
x(1− x)x′(1− x′)
4m1m2
− λpsg(M
0 (x, k⊥))g(M
′, k′⊥))
ψ(x′, ~k′⊥) , (8)
where M20 (x, k⊥) ≡ (
~k2⊥ + m
1)/x + (
~k2⊥ + m
2)/(1 − x) is the free square mass operator
in the meson rest frame, m1,2 are the constituent quark masses, α gives the strength of
the Coulomb-like interaction. g(K) is the model form factor, with λps the strength of the
separable interaction. We consider two expressions for the form factors:
g(a)(K2) =
β(a) +K2
and g(b)(K2) =
, (9)
where the parameters β(a,b) and λps are adjusted to reproduce the experimental val-
ues of the pion electromagnetic radius and mass, mπ. For α = 0.5, we have β
(a) =
−(634.5 MeV)2 and β(b) = (1171 MeV)2. mu = 384 MeV, ms = 508 MeV. In Table 1,
we have the results compared with experimental data [ 8].
Table 1
Results for the kaon and pion weak decay constants, compared with experimental data.
The model is adjusted to reproduce pion radius and mass.
qq f (a)ps (MeV) f
ps (MeV) f
ps (MeV) M
ps (MeV) M
ps (MeV) [ 8]
π+(ud) 110 110 92.4± .07± 0.25 [ 8] 140 140
K+(us) 126 121 113.0± 1.0± 0.31[ 8] 490 494
4. CONCLUSIONS
Considering a light-front model wave-function we have observed a good agreement of
the results for the kaon electromagnetic form factor with experimental data. The electro-
magnetic form factor was obtained using the plus and minus components of the electro-
magnetic current. The inclusion of the non-valence component of the current was shown
to be essential in this approach to obtain covariant results for the calculated matrix ele-
ments. We also show that a good agreement with experimental data is obtained for the
kaon weak decay constants once a probability of the valence component of about 80% is
taken into account.
REFERENCES
1. F. Cardarelli, I. L. Grach, I. M. Narodetsky, E. Pace, G. Salme, S. Simula,
Phys. Rev. D 53 (1996) 6682.
2. J. P. B. C. de Melo, H. W. Naus and T. Frederico, Phy. Rev. C 59 (1999) 2278.
3. B. L. G. Bakker, H.-M. Choi and C.-R. Ji, Phys. Rev. D 63 (2001) 074014.
4. J. P. B. C. de Melo, T. Frederico, E. Pace and G. Salmè, Phy. Rev. D 73 (2006)
074013; J. P. B. C. de Melo, T. Frederico, E. Pace and G. Salmè, Phy. Lett. B 581
(2004) 75.
5. F.P. Pereira, J.P.B.C. de Melo, T. Frederico and L. Tomio, Phys. of Part. and Nucl.
36 (2005) 5217; F.P. Pereira, Fatores de Forma Eletromagnéticos do Ṕıon e do Kaon
na Frente de Luz, Msc Dissertation, IFT, São Paulo, 2005.
6. S. R. Amendolia et al., Phys. Lett. B 178 (1986) 435.
7. T. Frederico and H.-C. Pauli, Phy. Rev. D 64 (2001) 054004; L. A. M. Salcedo, J. P.
B. C. de Melo, D. Hadjmichef and T. Frederico, Eur. Phys. J. A 27 (2006) 213.
8. W.-M. Yao et al., Journal of Physics G 33 (2006) 1.
	INTRODUCTION
	ELECTROMAGNETIC FORM FACTOR
	WEAK DECAY CONSTANTS IN A QCD INSPIRED MODEL
	CONCLUSIONS

  The kaon electromagnetic (e.m.) form factor is reviewed considering a
light-front constituent quark model. In this approach, it is discussed the
relevance of the quark-antiquark pair terms for the full covariance of the e.m.
current. It is also verified, by considering a QCD dynamical model, that a good
agreement with experimental data can be obtained for the kaon weak decay
constant once a probability of about 80% of the valence component is taken into
account.

Electromagnetic structure and weak decay of meson K in a light-front
QCD-inspired model∗
Fabiano P. Pereira a, J. P. B. C. de Melo b †, T. Frederico c, and Lauro Tomio d
aInstituto de F́ısica, Universidade Federal Fluminense, 24210-900, Niterói, RJ, Brazil
bUniversidade Cruzeiro do Sul, CETEC, 08060-070, São Paulo, SP, Brazil
cInstituto Tecnológico de Aeronáutica, 12228-900, São José dos Campos, SP, Brazil
dInstituto de F́ısica Teórica, UNESP, 01405-900, São Paulo, SP, Brazil
The kaon electromagnetic (e.m.) form factor is reviewed considering a light-front con-
stituent quark model. In this approach, it is discussed the relevance of the quark-antiquark
pair terms for the full covariance of the e.m. current. It is also verified, by considering a
QCD dynamical model, that a good agreement with experimental data can be obtained for
the kaon weak decay constant once a probability of about 80% of the valence component
is taken into account.
1. INTRODUCTION
The kaon, as quark-antiquark bound states, is one appropriate system to study aspects
of QCD at low and intermediate energy regions. By using quantum field theory at the
light-front the subnuclear structure can be more easily studied [ 1, 2, 3]. Within the light-
front framework and an appropriate choice of the frame, it is possible to obtain the pion
electromagnetic form factor at both space- and time-like regimes[ 4]. Using the light-cone
components J+K = J
0+J3 and J−K = J
0−J3 of the kaon electromagnetic current, one can
obtain the corresponding form factors in the light-front formalism, with a pseudoscalar
coupling for the quarks and considering the Breit frame (q+ = 0, ~q⊥ = (qx, 0) 6= 0) [ 5].
In the case of J+K there is no pair term contribution in the Breit frame. However, for the
J−K component of the electromagnetic current, the pair term contribution is different from
zero and necessary to preserve the rotational symmetry of the current.
In the next section, we outline the main equations of the model for the kaon electromag-
netic current, detailed in [ 5], with the corresponding results obtained for the kaon elastic
form factor. In section 3, we briefly review a QCD inspired model, presenting results for
the weak decay pseudoscalar constants compared to data. In section 4 we present our
conclusions.
∗Work partially supported by the Brazilian funding agencies FAPESP and CNPq
†JPBC de Melo thanks Instituto de F́ısica Teórica, UNESP, for supporting facilities
http://arxiv.org/abs/0704.0627v1
2. ELECTROMAGNETIC FORM FACTOR
The initial light-front wave function considered in the present model is given by:
ΦiQ(x, k⊥) =
(1− x)2
−M20 )(m
−M2(mQ, mR))
, (1)
where N is a normalization constant, Q ≡ {q̄, q} is the quark or antiquark index with mQ
is the corresponding quark mass, m2
is the kaon mass, x = k+/P+ is the momentum
fraction, and
M2(mQ, mR) ≡
k2⊥ +m
(P − k)2⊥ +m
− P 2⊥, (2)
with the free quark-mass operator given by M20 = M
2(mq̄, mq). mR is a mass constant
chosen to regularize the triangle diagram. For the corresponding final wave-functions,
q̄ and Φ
q , we just need to exchange P ↔ P
′ in (1) and (2). The relation between the
electromagnetic current Jµ and the space-like kaon electromagnetic form factor FK+(q
is given by 〈P ′|Jµ|P ′〉 = (P ′ + P )µFK+(q
2) . In terms of the initial (Φiq̄) and final (Φ
light-front wave functions, we have
F+q̄ (q
2) = −eq̄
N2g2Nc
4π3P+
d2k⊥dx
N+q̄ θ(x)θ(1− x) Φ
q̄ (x, k⊥)Φ
q̄(x, k⊥) , (3)
F+q (q
2) = [ q ↔ q̄ in F+q̄ (q
2) ] , (4)
where Nc is the color number, g is the coupling constant, eQ is the charge of quark Q, and
N+q̄ = (−1/4)Tr[(/k +mq̄)γ
5(/k − /P ′ +mq)γ
+(/k − /P +mq)γ
. In the light-front
approach, beside the valence contribution, we have also the non-valence contributions
to the currents. In the case of the J+ component, the non-valence component does
not contribute to the corresponding matrix elements [ 5]. The kaon electromagnetic
form factor obtained with J+ is the sum of two contributions from quark and antiquark
currents:
(q2) = F+q (q
2) + F+q̄ (q
2) normalized such that F+
(0) = 1. (5)
In the case that we consider the J− component, to obtain the kaon electromagnetic form
factor, after considering the contribution from the interval 0 < k+ < P+ (interval I), we
need to add a second contribution, which is originated from the pair terms, and non-zero
in the interval P+ < k+ < P ′+ (interval II). The contribution is obtained after a Cauchy
integral in k− is performed in the limit P ′+ → P+ [ 5]. So, instead of (5), we will have:
(q2) =
F−q (q
2) + F−q̄ (q
F−(q2)
, (6)
normalized by the charge conservation to F−
(0) = 1.
The parameters of the model are the constituent quark masses, mq = mu = md = 220
MeV, ms = 419 MeV and the regulator mass mR =946 MeV, adjusted to fit the electro-
magnetic radius of the kaon. The electromagnetic radius is related to the corresponding
form factor, with the mean-square-radius given by
〈r2K+〉 = 6
dFK+(q
. (7)
With the parameters adjusted as given above, we have 〈r2
〉 = 0.354 fm2, which is very
close to the experimental value 〈r2
〉|exp = 0.340 fm
2 [ 6].
Our results for the kaon electromagnetic form factor are presented in Fig. 1, in compar-
ison with available experimental data [ 6]. We observe that the full kaon electromagnetic
form factor is covariant only after the inclusion of the pair terms or non-valence contri-
bution to the J−
component of the electromagnetic current.
0.01 0.10 1.00
 [(GeV/c)
Figure 1. The kaon electromagnetic form factor is obtained with the plus and minus
components of the e.m. current (both cases are shown by the solid-line results) and
compared with experimental data [ 6]. The dashed-line curve shows the form factor
without the pair terms contribution in J−
3. WEAK DECAY CONSTANTS IN A QCD INSPIRED MODEL
Next, we briefly review the calculation of the pseudoscalar constants, in a light-front
QCD-inspired dynamical model. In this case, the constituent quark masses need to be
readjusted in view of the fact that, differently from the approach outlined in section 2,
the wave-function is obtained from an eigenvalue equation, as follows.
The valence wave function is obtained by solving an eigenvalue equation for the effective
square mass operator M2ps [ 7]:
ps ψ(x,
~k⊥) = M
0 (x, k⊥) ψ(x,
~k⊥)−
dx′d~k′⊥θ(x
′)θ(1− x′)
x(1− x)x′(1− x′)
4m1m2
− λpsg(M
0 (x, k⊥))g(M
′, k′⊥))
ψ(x′, ~k′⊥) , (8)
where M20 (x, k⊥) ≡ (
~k2⊥ + m
1)/x + (
~k2⊥ + m
2)/(1 − x) is the free square mass operator
in the meson rest frame, m1,2 are the constituent quark masses, α gives the strength of
the Coulomb-like interaction. g(K) is the model form factor, with λps the strength of the
separable interaction. We consider two expressions for the form factors:
g(a)(K2) =
β(a) +K2
and g(b)(K2) =
, (9)
where the parameters β(a,b) and λps are adjusted to reproduce the experimental val-
ues of the pion electromagnetic radius and mass, mπ. For α = 0.5, we have β
(a) =
−(634.5 MeV)2 and β(b) = (1171 MeV)2. mu = 384 MeV, ms = 508 MeV. In Table 1,
we have the results compared with experimental data [ 8].
Table 1
Results for the kaon and pion weak decay constants, compared with experimental data.
The model is adjusted to reproduce pion radius and mass.
qq f (a)ps (MeV) f
ps (MeV) f
ps (MeV) M
ps (MeV) M
ps (MeV) [ 8]
π+(ud) 110 110 92.4± .07± 0.25 [ 8] 140 140
K+(us) 126 121 113.0± 1.0± 0.31[ 8] 490 494
4. CONCLUSIONS
Considering a light-front model wave-function we have observed a good agreement of
the results for the kaon electromagnetic form factor with experimental data. The electro-
magnetic form factor was obtained using the plus and minus components of the electro-
magnetic current. The inclusion of the non-valence component of the current was shown
to be essential in this approach to obtain covariant results for the calculated matrix ele-
ments. We also show that a good agreement with experimental data is obtained for the
kaon weak decay constants once a probability of the valence component of about 80% is
taken into account.
REFERENCES
1. F. Cardarelli, I. L. Grach, I. M. Narodetsky, E. Pace, G. Salme, S. Simula,
Phys. Rev. D 53 (1996) 6682.
2. J. P. B. C. de Melo, H. W. Naus and T. Frederico, Phy. Rev. C 59 (1999) 2278.
3. B. L. G. Bakker, H.-M. Choi and C.-R. Ji, Phys. Rev. D 63 (2001) 074014.
4. J. P. B. C. de Melo, T. Frederico, E. Pace and G. Salmè, Phy. Rev. D 73 (2006)
074013; J. P. B. C. de Melo, T. Frederico, E. Pace and G. Salmè, Phy. Lett. B 581
(2004) 75.
5. F.P. Pereira, J.P.B.C. de Melo, T. Frederico and L. Tomio, Phys. of Part. and Nucl.
36 (2005) 5217; F.P. Pereira, Fatores de Forma Eletromagnéticos do Ṕıon e do Kaon
na Frente de Luz, Msc Dissertation, IFT, São Paulo, 2005.
6. S. R. Amendolia et al., Phys. Lett. B 178 (1986) 435.
7. T. Frederico and H.-C. Pauli, Phy. Rev. D 64 (2001) 054004; L. A. M. Salcedo, J. P.
B. C. de Melo, D. Hadjmichef and T. Frederico, Eur. Phys. J. A 27 (2006) 213.
8. W.-M. Yao et al., Journal of Physics G 33 (2006) 1.
	INTRODUCTION
	ELECTROMAGNETIC FORM FACTOR
	WEAK DECAY CONSTANTS IN A QCD INSPIRED MODEL
	CONCLUSIONS
