A symmetry preserving contact interaction treatment of the kaon Zanbin Xingand Lei Changy School of Physics Nankai University Tianjin 300071 China

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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

(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

variable τintegration[23],for example,

ILR

−2α(M2) = lim

N→∞ ZqZ∞

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

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

(q2+M2)α+2

=ZqZ∞

dτ τα+1

Γ(α+ 2)e−τ(q2+M2)

=Z∞

dτ τα−1

Γ(α+ 2)

e−τM2

16π2

→I−2αR(M2) = Zτ2

τ2

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∞

dτqµqν

τα+2

Γ(α+ 3)e−τ(q2+M2)

=ZqZ∞

dτδµν

τα+2

Γ(α+ 3)e−τ(q2+M2)

=δµν Z∞

dτ τα−1

Γ(α+ 3)

e−τM2

32π2

→Iµν

−2αR(M2) = δµν Zτ2

τ2

dτ τα−1

Γ(α+ 3)

e−τM2

32π2,(6)

and

Iµνρσ

−2α(M2) = Zq

qµqνqρqσ

(q2+M2)α+4

=ZqZ∞

dτqµqνqρqσ

τα+3

Γ(α+ 4)e−τ(q2+M2)

=ZqZ∞

dτSµνρσ

τα+3

Γ(α+ 4)e−τ(q2+M2)

=Sµνρσ Z∞

dτ τα−1

Γ(α+ 4)

e−τM2

64π2

→Iµνρσ

−2αR(M2) = Sµνρσ Zτ2

τ2

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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A symmetry preserving contact interaction treatment of the kaon Zanbin Xingand Lei Changy School of Physics Nankai University Tianjin 300071 China

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