A symmetry preserving contact interaction treatment of the kaon Zanbin Xingand Lei Changy School of Physics Nankai University Tianjin 300071 China
2025-04-30
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A symmetry preserving contact interaction treatment of the kaon
Zanbin Xing∗and Lei Chang†
School of Physics, Nankai University, Tianjin 300071, China
A symmetry-preserving regularization procedure for dealing with the contact interaction model is
proposed in this work. This regularization procedure follows a series of consistency conditions which
are necessary to maintain gauge symmetry. Under this regularization, proofs for the preservation
of the Ward-Takahashi identities are given and the loop integrals in the contact interaction model
are systematically computed. As an application example, the kaon electromagnetic form factor and
Kl3transition form factor are computed and self-consistent results are obtained. Since the proposed
regularization properly handles the divergence, one is freed from the inconsistencies caused by the
regularization and can concentrate more on the physical discussion.
I. INTRODUCTION
Over the years, the Dyson-Schwinger Equations(DSEs)
have proved to be a powerful tool for studying non-
pertubative nature of Quantum Chromodynamics(QCD)
in the continuum[1–3]. Self-consistent treatments of the
quark gap equation and bound state equations, such
as Bethe-Salpeter equation(BSE) and Faddeev equation,
creates a bridge connecting the hadrons to the fundamen-
tal degrees of freedom of QCD, quarks and gluons[4, 5].
Within the framework of DSEs, a vector-vector con-
tact interaction approximation was proposed in Ref. [6].
Despite its simplicity of the contact interaction model
in describing the real world, it has been used to calcu-
late a wide range of hadron properties, including mass
spectrum, various decay processes, electromagnetic form
factors and transition form factors, and the parton dis-
tributions, see Ref. [6–21]. However, due to the non-
renormalizable nature of the contact interaction, the reg-
ularization scheme becomes a crucial part in the practi-
cal calculation and a good regularization scheme should
properly characterise the divergence structure of the the-
ory. It is worth noting that symmetries, and in particular
the Ward-Takahashi identities(WTIs), provide a strong
constraint that must be preserved during the regulariza-
tion process.
The regularization procedures performed in previ-
ous studies of contact model have been unsatisfactory.
Firstly, WTIs do not naturally hold under the previ-
ous regularization procedure. Moreover, there are cases
where inconsistent results occur in the calculation of kaon
electromagnetic form factor, when the principle of charge
conservation is violated [10]. The main reason for these
problems is that previous regularization procedure fails
to properly deal with the quadratic and logarithmic di-
vergent integrals.
Inspired by Ref. [22] we presented a new regularization
procedure in this work. This proper regularization meets
many interesting properties of the dimensional regular-
ization without changing the space-time dimension. One
∗xingzb@mail.nankai.edu.cn
†leichang@nankai.edu.cn
of the most fascinating properties is that gauge symme-
tries are preserved under this regularization.
This paper is organized as follows, Sec. II introduces
the new symmetry preserving regularization that prop-
erly handle the divergent integrals. Sec. III discusses this
new regularization in the contact interaction model, in-
corporating the preserving of (axial-)vector WTIs, and
gives steps for the systematic calculations with contact
model. Sec. IV provides results of the kaon electromag-
netic form factor and Kl3form factor under the new reg-
ularizaiton, and the last section gives a brief summary.
II. SYMMETRY PRESERVING
REGULARIZATION
Before discussing the regularization procedures in de-
tail, it is helpful to introduce the so-called one-fold irre-
ducible loop integrals(ILIs) in Ref. [22].
I−2α(M2) = Zq
1
(q2+M2)α+2 ,
Iµν
−2α(M2) = Zq
qµqν
(q2+M2)α+3 ,
Iµνρσ
−2α(M2) = Zq
qµqνqρqσ
(q2+M2)α+4 (1)
with Rq
.
=Rd4q
(2π)4and α=−1,0,1,···. Here α=−1
represents quadratically divergent integrals and α= 0
represents logarithmically divergent integrals. With the
help of Feynman parametrezation, it is straightforward
to conclude that all one loop integrals can be expressed
in terms of these integrals. Where Mis a function of
Feynman parameters, external momenta and the corre-
sponding mass scales. A regularization procedure can
be implemented after rearranging one loop integrals into
these ILIs.
In Ref. [22], a loop regularization is proposed which
simulates in many interesting features to the momentum
cutoff, Pauli-Villars and dimensional regularization with-
out modifying the original Lagrangian formalism, and is
directly performed in the space-time dimension of orig-
inal theory. The loop regularization is equivalent to in-
troducing a weight function to regularize the proper-time
arXiv:2210.12452v1 [hep-ph] 22 Oct 2022
2
variable τintegration[23],for example,
ILR
−2α(M2) = lim
N→∞ ZqZ∞
0
dτWN(τ, Mc, µs)
×τα+1
Γ(α+ 2)e−τ(q2+M2),(2)
where the superscript LR denotes the loop regularization
and Γ(n) is the gamma function. An explicit form of the
weight function is
WN(τ, Mc, µs) = e−τ µ2
s(1 −e−τM2
R)N,(3)
with M2
R=M2
chw(N)lnN ,hw(N)&1 and hw(N→
∞) = 1. The two energy scale Mcand µsserve as ul-
traviolet(UV) and infrared(IR) cutoff, respectively. It is
worth noting that when N→ ∞ and µs= 0 the weight
function becomes
lim
N→∞ WN(τ, Mc, µs= 0) = θ(τM 2
c−1) .(4)
Thus the weight function regularizes the proper-time in-
tegral just as regularizes it with a hard UV cutoff 1/M 2
c.
We introduce a regularization procedure which is based
on the Schwinger’s proper-time method. The regulariza-
tion procedure for the scalar type ILIs is
I−2α(M2) = Zq
1
(q2+M2)α+2
=ZqZ∞
0
dτ τα+1
Γ(α+ 2)e−τ(q2+M2)
=Z∞
0
dτ τα−1
Γ(α+ 2)
e−τM2
16π2
→I−2αR(M2) = Zτ2
ir
τ2
uv
dτ τα−1
Γ(α+ 2)
e−τM2
16π2.(5)
The label Rin the subscript denotes the regularized inte-
grals. It is already seen that a hard UV cutoff τuv = 1/Mc
is equivalent to the loop regularization with µs= 0. How-
ever, instead of the sliding energy scale µsin the loop reg-
ularization, we introduce an hard IR cutoff τir to imple-
ment confinement, as proposed in Ref. [24]. This way of
dealing with the IR cutoff matches the regulators in pre-
vious contact model studies, which, as we shall see, can
also maintain gauge symmetries if the tensor type ILIs
are properly regularized. Before proceeding, it is noted
that when integer α < −1, the loop integral vanishes un-
der Eq. (5), which happens to be the same property of
dimensional regularization.
Turning now to the regularization of the tensor type
ILIs, the regularization procedure is
Iµν
−2α(M2) = Zq
qµqν
(q2+M2)α+3
=ZqZ∞
0
dτqµqν
τα+2
Γ(α+ 3)e−τ(q2+M2)
=ZqZ∞
0
dτδµν
q2
4
τα+2
Γ(α+ 3)e−τ(q2+M2)
=δµν Z∞
0
dτ τα−1
Γ(α+ 3)
e−τM2
32π2
→Iµν
−2αR(M2) = δµν Zτ2
ir
τ2
uv
dτ τα−1
Γ(α+ 3)
e−τM2
32π2,(6)
and
Iµνρσ
−2α(M2) = Zq
qµqνqρqσ
(q2+M2)α+4
=ZqZ∞
0
dτqµqνqρqσ
τα+3
Γ(α+ 4)e−τ(q2+M2)
=ZqZ∞
0
dτSµνρσ
q4
24
τα+3
Γ(α+ 4)e−τ(q2+M2)
=Sµνρσ Z∞
0
dτ τα−1
Γ(α+ 4)
e−τM2
64π2
→Iµνρσ
−2αR(M2) = Sµνρσ Zτ2
ir
τ2
uv
dτ τα−1
Γ(α+ 4)
e−τM2
64π2,(7)
where Sµνρσ =δµν δρσ +δµρδσν +δµσδνρ is the total sym-
metric tensor. It is obvious that the regularized tensor
type ILIs and scalar type ILIs are related as follows
Iµν
−2αR(M2) = Γ(α+ 2)
2Γ(α+ 3)δµν I−2αR(M2),(8)
Iµνρσ
−2αR(M2) = Γ(α+ 2)
4Γ(α+ 4)Sµνρσ I−2αR(M2).(9)
These relations are precisely the so-called consistency
conditions of gauge symmetry in Ref. [22, 23], which are
independent of regularization and are necessary for pre-
serving the gauge invariance of theories. It is noted that
ILIs under dimensional regularization also satisfy these
conditions. These consistency conditions connect ten-
sor type ILIs and scalar type ILIs and then any gauge
invariant theories can be properly described in terms of
the regularized scalar type ILIs. In fact, there are a series
of consistency conditions for ILIs with even more Lorentz
index which are rarely to encountered and are therefore
not presented here.
We now consider the regularization procedure in previ-
ous contact model studies, such as in Ref. [6]. For exam-
ple, the quadratic divergent tensor type ILI is regularized
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AsymmetrypreservingcontactinteractiontreatmentofthekaonZanbinXingandLeiChangySchoolofPhysics,NankaiUniversity,Tianjin300071,ChinaAsymmetry-preservingregularizationprocedurefordealingwiththecontactinteractionmodelisproposedinthiswork.Thisregularizationprocedurefollowsaseriesofconsistencyconditionswh...
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