Derivation of the Effective Chiral Lagrangian for Pseudoscalar Mesons ...

Derivation of the Effective Chiral Lagrangian for Pseudoscalar Mesons from QCD Qing Wanga,b , Yu-Ping Kuangb,a , Xue-Lei Wanga,c , Ming Xiaoa a b

Department of Physics, Tsinghua University, Beijing 100084, China

∗

China Center of Advanced Science and Technology (World Laboratory), P.O.Box 8730, Beijing 100080, China c

Department of Physics, Henan Normal University, Xinxiang 453002, China (TUIMP-TH-99/100, Feb 28,1999)

We formally derive the chiral Lagrangian for low lying pseudoscalar mesons from the first principles of QCD considering the contributions from the normal part of the theory without taking approximation.

arXiv:hep-ph/9903201v4 24 Dec 1999

The derivation is based on the standard generating functional of QCD in the path integral formalism. The gluon-field integration is formally carried out by expressing the result in terms of physical Green’s functions of the gluon. To integrate over the quark-field, we introduce a bilocal auxiliary field Φ(x, y) representing the mesons. We then develop a consistent way of extracting the local pseudoscalar degree of freedom U (x) in Φ(x, y) and integrating out the rest degrees of freedom such that the complete pseudoscalar degree of freedom resides in U (x). With certain techniques, we work out the explicit U (x)-dependnce of the effective action up to the p4 -terms in the momentum expansion, which leads to the desired chiral Lagrangian in which all the coefficients contributed from the normal part of the theory are expressed in terms of certain quark Green’s functions in QCD. Together with the exsisting QCD formulae for the anomaly contributions, the present results leads to the complete effective chiral Lagrangian for pseudoscalar mesons. The final result can be regarded as the fundamental QCD definition of the coefficients in the chiral Lagrangian. The relation between the present QCD definition of the p2 -order coefficient F02 and the well-known appoximate result given by Pagels and Stokar is discussed.

PACS number(s): 12.39.Fe, 11.30.Rd, 12.38.Aw, 12.38.Lg, This kind of approach has also been applied to the electroweak theory [4] for studying the probe of the electroweak symmetry breaking mechanism [5,6]. Since parity and CP

I. INTRODUCTION

are not conserved in the electroweak theory, there are even

The study of low energy hadron physics in QCD is a

more unknown coefficients in the electroweak chiral La-

long standing difficult problem due to its nonperturbative

grangian than in the case of QCD [4]. So far, this kind

nature. For low lying pseudoscalar mesons, a widely used

of study is of the level of finding out suitable processes

approach is the theory of chiral Lagrangian based on the

at future high energy colliders to determine the unknown

momentum expansion and the consideration of the global

coefficients in the electroweak chiral Lagrangian and inves-

symmetry of the system without dealing with the nonper-

tigating to what precision the determination can be. The

turbative dynamics of QCD [1] [2]. In the chiral Lagrangian

relation between the coefficients in the electroweak chiral

approach, the coefficients in the Lagrangian are all un-

Lagrangian and the underlying model of the electroweak

known phenomenological parameters which are determined

symmetry breaking mechanism is yet not known except for

by experimental inputs. The number of the unknown pa-

some very simple models [5].

rameters increases rapidly with the increase of the precision

Further study on understanding the relation between the

in the momentum expansion. For example, the chiral La-

chiral Lagrangian coefficients and the underlying dynam-

grangian for pseudoscaler mesons with three flavors up to

ical theory will be very helpful both in QCD and in the

the p4 -terms given by Gasser and Leutwyler [2] contains

electroweak theory for reducing the number of independent

14 unknown coefficients. When the p6 -terms are taken into

unknown parameters which makes the theory more predic-

account, there are 143 additional unknown coefficients [3].

tive. There are papers studying approximate formulae for 1

the chiral Lagrangian coefficients based on certain dynam-

example, we show that, under certain approximations, our

ical ansatz [7], but the approach is not completely from

p2 -order coefficient F02 reduces to the well-known approxi-

the first principles of the underlying theory. Attempts to

mate formula given by Pagels and Stokar [11]. A systematic

build closer relations between the chiral Lagrangian and

numerical calculation of the coefficients by solving the re-

the long distance piece of the underlying theory of QCD

lated QCD Green’s functions in certain approximation (the

by considering the anomaly contributions with certain ap-

second step) will be presented in a separate paper [?].

proximation also exist [8,9]. However, several aspects of it

This paper is organized as follows. Sec. II is on the fun-

imply that such kind of approach needs improvement, e.g.

damental generating functional in QCD. We start from it

(a) the theory does not include spontaneous chiral sym-

and formally integrate out the heavy-quark and gluon fields

metry breaking, and the chiral symmetry breaking scale is

to obtain a formal generating functional for the light quark

put in by hand; (b) without putting in the chiral symme-

fields. In Sec. III, we introduce a bilocal auxiliary field re-

try breaking scale, the obtained pion decay constant Fπ

flecting the light meson degrees of freedom with which we

is proportional to an imposed very low (∼ 320 MeV) mo-

can integrate out the light quark fields. Then we develop

mentum cut-off on the underlying theory of QCD; (c) the

a technique for extracting the degree of freedom of the de-

positivity of

Fπ2

depends on a careful choice of the regular-

sired local field U (x) for the pseudoscalar mesons from the

ization scheme. The approach in Ref. [10] does not contain

bilocal auxiliary field, and formally integrate out the the

the above problems. But in Ref. [10], the approximation

remaining degrees of freedom of the bilocal auxiliary field

of large-Nc limit is taken from the beginning and the ap-

to obtain a generating functional for the local field U (x).

proximation of picking up only the local scalar and pseu-

In Sec. IV, we develop certain techniques to work out the

doscalar pieces of the color-singlet quark-antiquark bilocal

complete U (x)-dependence of the effective Lagrangian in

operator arising from integrating the gluon-field is taken in

the sense of momentum expansion, and obtain the effec-

the derivation. With the latter approximation, the formula

tive chiral Lagranian which is of the form given by Gasser

2

F02

in Ref. [10] is expressed in

and Leutwyler [2]. In this process we obtain the QCD ex-

terms of an imposed ultraviolet cut-off, and the formula can

pressions for all the coefficients in the effective chiral La-

hardly be related to the well-known Pagels-Stokar formula

grangian. A discussion on the relation between the present

for the p -order coefficient

F02

[11]. Therefore, further improvement of studying the

QCD definition of the O(p2 ) coefficients F02 and the well-

effective chiral Lagrangian from the fundamental principles

known Pagels-Stokar formula (an approximate result) [11]

of QCD is necessary. Actually, the study can be divided

will be given in Sec. V. Sec. VI is a concluding remark.

for

into two steps. The first step is to formally derive the effective chiral Lagrangian from the fundamental principles II. THE GENERATING FUNCTIONAL

of QCD and express the coefficients in terms of certain dynamical quantities in QCD, which gives the QCD meanings

Consider a QCD-type gauge theory with SU (Nc ) local

of the coefficients. The second step is to calculate the re-

gauge symmetry. Let Aiµ (i = 1, 2, · · · , Nc2 −1) be the gauge

lated dynamical quantities in QCD to obtain the values of

¯η field, ψαaη and Ψaα be , respectively, light and heavy fermion

the coefficients. This paper is mainly devoted to the first

fields with color index α (α = 1, 2, · · · , Nc ), Lorentz spinor

step.

index η, light flavor index a (a = 1, 2, · · · , Nf ) and heavy

In this paper, we develop certain techniques with which

flavor index a ¯ (¯ a = 1, 2, · · · , Nf′ ). For convenience, we sim-

we are able to formally derive the effective chiral La-

¯η ply call ψαaη the ”light quark-field”, Ψaα the ”heavy quark-

grangian for pseudoscalar moesons from the first principles

field” and Aiµ the ”gluon-field”. Let us introduce local ex-

of QCD without taking approximation, and all the coeffi-

ternal sources Jσρ for the composite light quark operators ψ¯σ ψ ρ , where σ and ρ are short notations for the spinor and

cients are expressed in terms of certain Green’s functions in QCD. Such expressions can be regarded as the funda-

flavor indices. The external source J can be decomposed

mental QCD definitions of the coefficients. As a simple

into scalar, pseudoscalar, vector and axial-vector parts 2

J(x) = −s(x) + ip(x)γ5 + v/ (x) + a / (x)γ5 ,

sources Iiµ . For simplicity, the gluon field integration in

(1)

this paper is limited to the topologically trivial sector. In-

where s(x), p(x), vµ (x) and aµ (x) are hermitian matrices,

clusion of topologically non-trivial sectors only changes the

and the light quark masses have been absorbed into the

intermediate results but not the final result (45).

definition of s(x). The vector and axial-vector sources v/ (x)

By Fierz reordering, we can further diagonalize the color

and a / (x) are taken to be traceless.

indices of the light-quark operators, and get

Since the contributions from the anomaly term to the

λi1 )α β γ µ1 ψβa11 (x1 )] · · · 2 1 1

effective chiral Lagrangian has already been studied in Ref.

n ¯a1 Gµi11···i ···µn (x1 , · · · , xn )[ψα1 (x1 )(

[10,9], our aim in this paper is to study the complete normal

λi ×[ψ¯αann (xn )( n )αn βn γ µn ψβann (xn )] 2 Z 4 ′ ···σn ¯ ρσ1···ρ (x1 , x′1 , · · · , xn , x′n ) = d x1 · · · d4 x′n g n−2 G 1 n

part contributions. So, in this paper, we simply ignore the standard CP-violating term related to the anomaly by taking the θ-vacuum parameter θ = 0.

×ψ¯ασ11 (x1 )ψαρ11 (x′1 ) · · · ψ¯ασnn (xn )ψαρnn (x′n ),

Following Gasser and Leutwyler [2], we start from con-

¯ σ1 ···σn (x1 , x′ , · · · , xn , x′ ) is a generalized Green’s where G 1 n ρ1 ···ρn

structing the following generating functional Z ¯ ¯ Z[J] = DψDψDΨD ΨDA µ

function containing 2n space-time points. Then (2) can be written as

Z

¯ Ψ, Ψ, ¯ ¯ Aµ ) + ψJψ} × exp i d4 x{L(ψ, ψ,  Z  Z 4 ¯ ¯ = DψDψ exp i d xψ(i∂/ + J)ψ Z ¯ × DΨDΨDA µ ∆F (Aµ ) Z  1 × exp i d4 x LQCD (A) − [F i (Aµ )]2 2ξ  ¯ / − M − gA −gIiµ Aiµ + Ψ(i∂ /)Ψ ,

(4)

Z[J] =

Z

DψDψ¯ exp i

∞ Z X

Z

¯ / + J)ψ + d4 xψ(i∂

d4 x1 · · · d4 xn d4 x′1 · · · d4 x′n

n=2

(−i)n (g 2 )n−1 n!

···σn ¯ ρσ1···ρ (x1 , x′1 , · · · , xn , x′n ) ×G 1 n

 ×ψ¯ασ11 (x1 )ψαρ11 (x′1 ) · · · ψ¯ασnn (xn )ψαρnn (x′n ) . (2)

(5)

III. THE AUXILIARY FIELDS

where LQCD (A) = − 14 Aiµν Aiµν is the gluon kinetic energy term, M is the heavy quark mass matrix, Iiµ ≡ ψ¯ λi γ µ ψ are colored currents composed of light quark fields,

2 1 [F i (Aµ )]2 − 2ξ

1. The bilocal Auxiliary Field

is the gauge-fixing term and ∆F (Aµ ) is the

¯ we inFor integrating out the light quark fields ψ and ψ,

Fadeev-Popov determinant.

troduce a bilocal auxiliary field Φ(aη)(bζ) (x, x′ ) by inserting

¯ Let us first consider the integration over DΨDΨDA µ for µ ¯ i.e. the current I serves a given configuration of ψ and ψ, i ¯ as an external source in the integration over DΨDΨDA µ.

into (5) the following constant   Z DΦ δ Nc Φ(aη)(bζ) (x, x′ ) − ψ¯αaη (x)ψαbζ (x′ ) .

The result of such an integration can be formally written

(6)

We see from (6) that the bilocal auxiliary field

as

Φ(aη)(bζ) (x, x′ ) embodies the bilocal composite operator ψ¯aη (x)ψ bζ (x′ ) which reflects the meson fields. Inserting (6)

Z  4 ¯ DΨDΨDA ∆ (A ) exp i d x LQCD (A) µ F µ  1 ¯ / − M − gA − [F i (Aµ )]2 − gIiµ Aiµ + Ψ(i∂ /)Ψ 2ξ ∞ Z X (−i)n g n = exp i d4 x1 · · · d4 xn n! n=2 Z

µ1 µn n ×Gµi11···i ···µn (x1 , · · · , xn )Ii1 (x1 ) · · · Iin (xn ),

α

α

into (5) we get Z[J] Z ¯ = DψDψDΦ   ×δ Nc Φ(aη)(bζ) (x, x′ ) − ψ¯αaη (x)ψαbζ (x′ ) Z ¯ / + J]ψ × exp i d4 xψ[i∂

(3)

i n where Gµi11···i ···µn is the full n-point Green’s function of the Aµ -

field containing internal heavy quark lines and with given 3

+Nc

∞ Z X

eiΓ0 [J,Φ,Πc ]  Z ≡ DΠ exp iNc − iTr ln[i∂/ + J − Π] Z + d4 xd4 x′ Φσρ (x, x′ )Πσρ (x, x′ )

d4 x1 · · · d4 xn d4 x′1 · · · d4 x′n

n=2

(−i)n (Nc g 2 )n−1 ¯ σ1 ···σn Gρ1 ···ρn (x1 , x′1 , · · · , xn , x′n ) × n! 

×Φσ1 ρ1 (x1 , x′1 ) · · · Φσn ρn (xn , x′n ) .

(7)

+

(−i)n (Nc g 2 )n−1 ¯ σ1 ···σn Gρ1 ···ρn (x1 , x′1 , · · · , xn , x′n ) × n! 

representation   ′ ¯ δ Nc Φ(x, x′ ) − ψ(x)ψ(x ) Z
R 4 4 ′ ′ ′ ′ ¯ ∼ DΠei d xd x Π(x,x )· Nc Φ(x,x )−ψ(x)ψ(x ) .

×Φσ1 ρ1 (x1 , x′1 ) · · · Φσn ρn (xn , x′n ) .

tion over the Π-field in (8), and express the result by Z Z[J] = DΦ exp{iΓ0 [J, Φ, Πc ]}, (12)

Z[J]  Z = DΦDΠ exp i − iNc Tr ln[i∂/ + J − Π] Z + d4 xd4 x′ Nc Φσρ (x, x′ )Πσρ (x, x′ )

2. Localization Since we are aiming at deriving the low energy effective

d4 x1 · · · d4 xn d4 x′1 · · · d4 x′n

chiral Lagrangian in which the light mesons are approx-

n=2

(−i)n (Nc g 2 )n−1 ¯ σ1 ···σn Gρ1 ···ρn (x1 , x′1 , · · · , xn , x′n ) × n! 

×Φσ1 ρ1 (x1 , x′1 ) · · · Φσn ρn (xn , x′n ) ,

imately described by local fields, we need to consistently extract the local field degree of freedom from the bilocal (8)

auxiliary field Φ(aη)(bζ) (x, x′ ). The extraction should be consistent in the sense that the complete degree of freedom

where Tr is the functional trace with respect to the space-

of the mesons resides in the local fields without leaving any

time, spinor and flavor indices. Let us define the classical field Πc R DΠ ΠeiS , Πc ≡ R DΠ eiS

(11)

With these symbols, we can formally carry out the integra-

With this we can integrate out the ψ and ψ¯ fields and get

∞ Z X

d4 x1 · · · d4 xn d4 x′1 · · · d4 x′n

n=2

The δ-function in (7) can be further expressed in the Fourier

+Nc

∞ Z X

in the coefficients in the chiral Lagrangian. Otherwise, it will affect the validity of the momentum expansion [2]. In this paper, we propose the following way of extraction, and

(9)

we shall see in Sec. IV that it is really consistent.

where S is the argument on the exponential in (8). Let

The auxiliary field Φ introduced in (6) has such a prop-

Γ0 [J, Φ, Πc ] be the effective action for Πc with given J and

erty which allows us to define the fields σ and Ω′ related to

Φ. Πc satisfies∗

the scalar and pseudoscalar sectors of Φ as ∂Γ0 [J, Φ, Πc ] = 0. ′ ∂Πσρ c (x, x )

(Ω′ σΩ′ + Ω′† σΩ′† )ab (x) = (1)ζη Φ(bζ)(aη) (x, x)

(10)

(Ω′ σΩ′ − Ω′† σΩ′† )ab (x) = (γ5 )ζη Φ(bζ)(aη) (x, x).

Then Γ0 [J, Φ, Πc ] is explicitly

(13)

Here the σ-field repesented by a hermitian matrix describes the modular degree of freedom, and the Ω′ -field represented ∗

by an unitary matrix describes the phase degree of freedom,

In the conventional approach, one usually introduces an exter-

i.e.

nal source J coupling to the field Π. With this, the right-handside of (10) euqals to −J [13]. Eq.(10) corresponds to taking

σ † (x) = σ(x),

J = 0. Similarly, when taking J = 0, the effective action

Ω′† (x)Ω′ (x) = 1.

(14)

Γ0 [J, Φ, Πc ] euqals to the generating functional W0 [J, Φ, J ]|J =0

As usual, we can define U ′ (x) ≡ Ω′2 (x) which contains a

for the connected Green’s fuctions. This leads to the left-hand-

U (1) factor such that detU ′ (x) = eiϑ(x) , where the deter-

side of (11).

minant is for the flavor matrix. The unitarity property 4

of U ′ (x) implies that ϑ(x) is a real field. We can fur-

in which

ther extract out the U (1) factor and define a field U (x) as

ant integration measure and the function F [O] is defined

′

U (x) ≡ e

i Nf

ϑ(x)

U (x). It is easy to see that detU (x) = 1.

DU δ(U † U − 1)δ(detU − 1) is an effective invari-

as 1 ≡ detO F [O]

Then we can define a new field Ω and decompose U into U (x) = Ω2 (x),

R

(15)

Z

Dσ δ(O† O − σ † σ)δ(σ − σ † ).

(19)

With the special choice of O(x) = e

which is the conventional decomposition in the literature.

−i

ϑ(x) Nf

trl [PR ΦT (x, x)],

(20)

This U (x), as the desired representation of SU (Nf )R ×

which satisfies det O = det O† , eq.(18) serves as the func-

SU (Nf )L , will be the nonlinear realization of the peu-

tional expression reflecting the relation (16). Inserting (18)

doscalar meson fields in the chiral Lagrangian. Note that

and (20) into the functional (12) and taking the Fourier

the way of introducing the U (x)-field is not unique but is up

representation of the δ-function   δ ΩO† Ω − Ω† OΩ† Z
R † † † ∼ DΞe−iNc dx Ξ· ΩO Ω−Ω OΩ ,

to a chiral rotation which does not affect the final effective chiral Lagrangian since the chiral Lagrangian is chirally invariant. The fields σ and ϑ are intermediate fields which will not appear in the final effective chiral Lagrangian. It

we get

is straightforward to subtract σ from the two equations in

−i

ϑ(x) Nf

=e

Ω† (x)trl [PR ΦT (x, x)]Ω† (x)

ϑ(x) i N f

Ω(x)trl [PL ΦT (x, x)]Ω(x),

(16)

where PR and PL are, resprectively, the projection operators onto the right-handed and left-handed states, the superscript T stands for the functional transposition (trans-

with

position of all indices including the space-time coordinates),

e−iΓI [Φ] Y 1 ≡ F [O(x)] x Y 1 = {det[trl PR ΦT (x, x)]det[trl PL ΦT (x, x)]} 2

and we have expressed the result in terms of Ω. Eq.(16) builds up the relation between Φ(x, x) and U (x) [or Ω(x)]. Taking the determinant of (16) we can express ϑ(x) in terms of ΦT (x, x) as

e2iϑ(x)

Z

DΦDU DΞ δ(U † U − 1)δ(detU − 1)  Z × exp iΓ0 [J, Φ, Πc ] + iΓI [Φ] + iNc d4 x   ϑ(x) −i ×trf Ξ(x) e Nf Ω† (x)trl [PR ΦT (x, x)]Ω† (x)  ϑ(x) i , (22) −e Nf Ω(x)trl [PL ΦT (x, x)]Ω(x)

Z[J] =

(13) and using (14) to get e

(21)

x

  det trl [PR ΦT (x, x)]  , = det trl [PL ΦT (x, x)]

  Dσ δ (trl PR ΦT )(trl PL ΦT ) − σ † σ  ×δ(σ − σ † ) .

× (17)

Z

(23)

In (22), the information about the relation (16) has been

where trl is the trace with respect to the spinor index.

integrated in.

Eqs.(13)-(16) describe our idea of localization. To realize this idea in the functional integration formalism, we need

Next, we deal with the functional integration over the

a technique to integrate in this information to the gener-

Φ-field. For this purpose, we define an effective action ˜ J, Ξ, Φc , Πc ] as Γ[Ω,

ating functional (12). For this purpose, we start from the

˜

following functional identity for an operator O satisfying

eiΓ[Ω,J,Ξ,Φc ,Πc ]  Z Z = DΦ exp iΓ0 [J, Φ, Πc ] + iΓI [Φ] + iNc d4 x   ϑ(x) −i ×trf Ξ(x) e Nf Ω† (x)trl [PR ΦT (x, x)]Ω† (x)  ϑ(x) i N T f −e Ω(x)trl [PL Φ (x, x)]Ω(x) , (24)

det O = det O† (cf. Appendix for the proof) Z DU δ(U † U − 1) δ(detU − 1) ×F[O] δ(ΩO† Ω − Ω† OΩ† ) = const,

(18) 5

dSef f [U, J, Ξc , Φc , Πc ] dJ σρ (x) U fix R ˜ DΞ Φσρ (x, x) exp iΓ[Ω, J, Ξ, Φc , Πc ] R c = Nc ˜ DΞ exp iΓ[Ω, J, Ξ, Φc , Πc ]

in which the classical field Φc (x, x′ ) is defined as ˜

DΦ ΦeS , Φc = R DΦ eS˜ R

(25)

≡ Nc Φσρ c (x, x).

where S˜ stands for the argument in the exponential in (24). Φc satisfies ˜ ∂ Γ[Ω, J, Ξ, Φc , Πc ] = 0. ′ ∂Φσρ c (x, x ) With these symbols, we can formally carry out the integration in (22) and obtain Z Z[J] = DU DΞ δ(U † U − 1)δ(detU − 1) ˜ J, Ξ, Φc , Πc ]}. × exp{iΓ[Ω,

(32)

In (32), the symbol Φc denotes the functional average of Φc ˜ over the field Ξ weighted by the action Γ[Ω, J, Ξ, Φc , Πc ].

(26)

Eq.(32) is crucial in the derivation of the effective chiral R

Lagrangian.

DΦ

IV. THE EFFECTIVE LAGRANGIAN (27)

To derive the effective chiral Lagrangian, we need to ob-

Here we have formally integrated out all the degrees of free-

tain the U (x)- and J-dependence of Sef f [U, J, Ξc , Φc , Πc ].

dom in Φ(x, x′ ) besides the extracted local degree of freedom

Note that Sef f [U, J, Ξc , Φc , Πc ] depends on U and J not

U (x). This localization is different from those in the liter-

only explicitly in (28) but also implicitly via Ξc , Φc and

ature [14].

Πc through (29), (25) and (9).

The remaining task of

Similar to the above procedures, we can formally in-

the derivation of the effective chiral Lagrangian is to

tegrate out the Ξ-field by introducing an effective action

work out explicitly the complete U - and J-dependence of

Sef f [U, J, Ξc , Φc , Πc ] as follows Z iSef f [U,J,Ξc ,Φc ,Πc ] ˜ J, Ξ, Φc , Πc ], e = DΞ exp iΓ[Ω,

Sef f [U, J, Ξc , Φc , Πc ]. The procedure is described as what

where the classical field Ξc is defined as R ˜ DΞ Ξ exp iΓ[Ω, J, Ξ, Φc , Πc ] Ξc = R ˜ J, Ξ, Φc , Πc ] DΞ exp iΓ[Ω,

follows. (28)

First we consider a chiral rotation JΩ (x) = [Ω(x)PR + Ω† (x)PL ] [J(x) + i∂ /] ×[Ω(x)PR + Ω† (x)PL ],

(29)

ΦTΩ (x, y) = [Ω† (x)PR + Ω(x)PL ] ΦT (x, y) ×[Ω† (y)PR + Ω(y)PL ],

and satisfies

∂Sef f [U, J, Ξc , Φc , Πc ] = 0. ∂(Ξc )ab (x)

ΠΩ (x, y) = [Ω(x)PR + Ω† (x)PL ] Π(x, y)

(30)

×[Ω(y)PR + Ω† (y)PL ].

(33)

Then the Ξ-integration in (27) can be formally carried out, The present theory is symmetric under this transformation.

and we obtain Z[J] =

Z

Since the Ξ-field is introduced in (21) and the operator DU δ(U † U − 1)δ(detU − 1)

× exp{iSef f [U, J, Ξc , Φc , Πc ]}.

ΩO† Ω − Ω† OΩ† is invariant under the chiral rotation, there is no need to introduce ΞΩ . Furthermore, since det Ω = 1,

(31)

we can easily see from (17) that ϑΩ (x) = ϑ(x). The explicit We see from (31) that Sef f [U, J, Ξc , Φc , Πc ] is just the action

dependence of Sef f [U, J, Ξc , Φc , Πc ] on U (x) comes from the ˜ explicit Ω(x) (Ω† (x))-dependence of Γ[Ω, J, Φc , Πc ] in (24)

for U with a given J † . Using (11), (24), (30), (26), (25) and (10), one can further

[cf. (24) and (28)]. After the chiral rotation, this term

show the following important relation

becomes   ϑ(x) −i +iNc d x trf Ξ(x) e Nf trl [PR ΦTΩ (x, x)]  ϑ(x) i N T f −e trl [PL ΦΩ (x, x)] , Z

†

Note that Ξc , Φc and Πc are all functionals of U and J through

(29), (25) and (9).

6

4

which no longer depends on U (x) explicitly. Therefore, af-

From that we can derive the effective chiral Lagrangian

ter the chiral rotation, there is no explicit U (x)-dependence

and the expressions for its coefficients. There can be two

of Sef f [U, J, Ξc , Φc , Πc ], i.e. the complete U (x)-dependence

ways of figuring out the JΩ -dependence of ΦΩc . One is to

resides implicitly in the rotated variables with the subscript

write down the dynamical equations for the intermediate

Ω. For instance, the effective actions Γ0 [J, Φc , Πc ], ΓI [Φ], ˜ Γ[Ω, J, Ξ, Φc , Πc ] and Sef f [U, J, Ξc , Φc , Πc ] can be written

fields Ξc , ΦΩc and ΠΩc and solve them (usually this can be done only under certain approximations) to get the JΩ -

as

dependence of these intermediate fields. The other one is to Γ0 [J, Φ, Πc ] = Γ0 [JΩ , ΦΩ , ΠΩc ] + anomaly terms,

(34)

ΓI [Φ] = ΓI [ΦΩ ],

(35)

track back to the original QCD expression for the chirally rotated generating functional (2) through (32) and (25) by reverting the procedures in Sec. III and Sec. II, which

˜ J, Ξ, Φc , Πc ] = Γ[1, ˜ JΩ , Ξ, ΦΩc , ΠΩc ] Γ[Ω, +anomaly terms,

can lead to the fundamental QCD definitions of the chiral (36)

Lagrangian coefficients without taking approximation. We

(37)

take the latter approach in this paper. Because  of the δ (bζ) (aη0 function δ Nc Φ(aη)(bζ) (x, x′ ) − ψ¯α (x)ψα (x′ ) in (7),

Sef f [U, J, Ξc , Φc , Πc ] = Sef f [1, JΩ , Ξc , ΦΩc , ΠΩc ] +anomaly terms.

(aη)(bζ)

we can express ΦΩc

From (24) we see that

(x, y) as

(aη)(bζ)

(x, y) = (40) Nc ΦΩc R ¯ ˆ ψ,Ψ, ¯ aη bζ iS[ψ, Ψ,A,Ξ] ¯ ¯ ¯ DψDψDΨDΨDAµ DΞψα (x)ψα (y)e R ˆ ψ,Ψ, ¯ ¯ iS[ψ, Ψ,A,Ξ] ¯ ¯ DψDψDΨD ΨDA µ DΞ e

˜

Ωc ] eiΓ[1,JΩ ,Ξ,ΦΩc ,Π Z = DΦΩ exp iΓ0 [JΩ , ΦΩ , ΠΩc ] + iΓI [ΦΩ ] Z ϑ(x) +iNc d4 xtrlf {Ξ(x)[−i sin Nf  ϑ(x) T ]ΦΩ (x, x)} , +γ5 cos Nf

where

¯ Ψ, Ψ, ˆ ψ, ¯ A, Ξ] S[ψ, Z 1 ¯ ¯ Ψ, Ψ, ¯ Aµ ) ≡ ΓI [ ψψ] + d4 x{L(ψ, ψ, Nc ¯/Ω+a +ψ[v / Ω γ5 − sΩ + ipΩ γ5 ϑ′ ϑ′ + γ5 cos )]ψ}. +Ξ(−i sin Nf Nf

(38)

where trlf denotes the trace with respect to the spinor and flavor indices. The anomaly terms in (34), (36) and (37) are all the same arising from the non-invariance of Trln[i∂/ + J − Π] under the chiral rotation. Note that the

(41)

functional integration measure does not change under the

In (40) and all later equations in this paper, the symbol ψ is

chiral rotation, i.e. DΦDΠ = DΦΩ DΠΩ since the Jaccobians from Φ → ΦΩ and Π → ΠΩ cancel each other. We

used as a short notation for the chirally rotated quark field ¯ and ϑ′ are the quantities defined ψΩ ‡ . In (41), ΓI [ N1c ψψ]

see that the U (x)-dependence is simplified after the chiral

in (23) and (17) expressed in terms of quark fields, i.e.

rotation.

1

The second approach is the use of eq.(32). As we have mentioned in Sec. II that we ignore the irrelevant anomaly terms in this study. Then after the chiral rotation, eq.(32) becomes dSef f [1, JΩ , Ξc , ΦΩc , ΠΩc ] dJ σρ (x) Ω

=

Nc Φσρ Ωc (x, x).

U

¯

e−iΓI [ Nc ψψ] Y  1 = det{ trlc [ψR (x)ψ¯L (x)]} Nc x  12 1 ¯ ×det{ trlc [ψL (x)ψR (x)]} Nc   Z 1 † ¯ ¯ × Dσ δ ψ ) − σ σ ψ )tr (ψ tr (ψ lc L R lc R L Nc2

fix, anomaly ignored (39)

We see from (39) that once the JΩ -dependence of ΦΩc is ex-

‡

plicitly known, one can integrate (39) over JΩ and get the

As an integration variable, with or without the subscript Ω

makes no difference. Once the classical equation of motion is

U (x)-dependence of Sef f [1, JΩ , Ξc , ΦΩc , ΠΩc ] up to an irrel-

concerned, distinguishing the rotated ψΩ from the unrotated ψ

evant integration constant independent of U (x) and J(x).

will be necessary.

7

 ×δ(σ − σ † )

′

e2iϑ (x)

be rotated away and the frozen degree of freedom becomes just the pesudoscalar degree of freedom ψ¯a γ5 ψ b as it should

(42)

be since this degree of freedom is already included in the in-

  det trlc [ψR (x)ψ¯L (x)]  , ≡ ¯ det trlc [ψL (x)ψR (x)]

tegrating in of the U -field. The automatic occurance of this

(43)

frozen degree of freedom in the present approach implies that our way of extracting the U -field degree of freedom is

where trlc is the trace with respect to the spinor and color

really consistent, i.e. nothing of the pseudoscalar degree of

′

indices. (43) implies that the range of ϑ (x) is [0, π).

freedom is left outside U . After the UA (1) rotation, ΓI and

Note that instantons contribute to both (42) and (43)

the Jacobian due to the rotation will give rise to an extra

[15]. The UA (1) violating field-configurations only cause ¯ nonvanishing ϑ′ but do not contribute to ΓI [ 1 ψψ].

factor in the integrand, which is the compensation factor

With (40)-(43), one can integrate (39) over the rotated

the point of view of the auxiliary field Φ, this corresponds

for the extraction of the U -field degree of freedom. From

Nc

souces and obtain

to the contributions from integrating out the degrees of freedom other than U , say the σ and η ′ mesons.

iSef f [1,JΩ ,Ξc ,ΦΩc ,ΠΩc ] e anomaly ignored  Z 1 ¯ ¯ = DψDψDΨD ΨDA DΞ exp iΓI [ ψψ] µ Nc Z ¯ Ψ, Ψ, ¯ Aµ ) +i d4 x{L(ψ, ψ, / Ω γ5 − sΩ + ipΩ γ5 +ψ[v/ Ω + a  ϑ′ ϑ′ +Ξ(−i sin + γ5 cos )]ψ} . Nf Nf

Now we are ready to explicitly work out the effective chiral Lagrangian to the p2 - and p4 -order. As is pointed out in Ref. [2], the vector and axial-vector sources should be regarded as O(p) and the scalar and pseudoscalar sources should be regarded as O(p2 ) in the momentum expansion. (44)

1. The p2 -Terms

For realistic QCD (Nf = 3), cos(ϑ′ /Nf ) does not vanWe can then shift the integration variable Ξ →

We first consider the p2 -order terms. To this order, the

Ξ − ipΩ / cos(ϑ′ /Nf ) to cancel the pΩ -dependence in the

anomaly can be safely ignored. Expanding (45) up to the

pseudoscalar part of (44). After carrying out the integra-

order of p2 , we obtain

ish.

Sef f [1, JΩ , Ξc , ΦΩc , ΠΩc ]

tion over Ξ, we obtain eiSef f [1,JΩ ,Ξc ,ΦΩc ,ΠΩc ] =

=

anomaly ignored

Z

¯ ¯ DψDψDΨD ΨDA µ   ϑ′
b ϑ′ a ¯ + γ5 cos ψ ×δ ψ − i sin Nf Nf  Z 1 ¯ Ψ, Ψ, ¯ Aµ ) × exp iΓI [ ψψ] + i d4 x{L(ψ, ψ, Nc  ϑ′ ¯ ]ψ} . (45) +ψ[v/ Ω + a / Ω γ5 − sΩ − pΩ tan Nf

p2 -order

Z

′ab d4 xtrf [F ab (x)sab (x)pab Ω (x)] Ω (x) + F Z µ,ab ν,cd + d4 xd4 zGabcd µν (x, z)aΩ (x)aΩ (z),

(46)

where   a b F (x) = − [ψ (x)ψ (x)]   ϑ′ (x) a b ′ab F (x) = − [ψ (x)ψ (x)] tan Nf ab

Gabcd µν (x, z)   i a c b d [ψ (x)γµ γ5 ψ (x)][ψ (z)γν γ5 ψ (z)] = 2    a b c d ¯ ¯ − [ψ (x)γµ γ5 ψ (x)] [ψ (z)γν γ5 ψ (z)] ,

In (45), there is no pΩ -dependence in the pseudoscalar channel, and the pΩ -dependence appears in the scalar channel ′

as the combination sΩ + pΩ tan Nϑf . Eq.(45) shows that Sef f [1, JΩ , , Ξc , ΦΩc , ΠΩc ] is the QCD

(47)

′

generating functional for the rotated sources sΩ +pΩ tan Nϑf ,

and the symbol hOi for an operator O appeared in (47) is

µ vΩ and aµΩ with a special parity odd degree of freedoms a

′

a

defined as

′

−iψ ψ b sin Nϑf +ψ γ5 ψ b cos Nϑf frozen. After making a fur′ ther UA (1) rotation of the ψ and ψ¯ fields, the angle ϑ can

  R Dµ O , O ≡ R Dµ

Nf

8

(48)

2 ] in (46). The reason Note that there is no term like trf [vΩ

where

is that there exists a hidden symmetry sΩ → h† sΩ h, pΩ →

¯ ¯ Dµ ≡ DψDψDΨD ΨDA µ   ′ ϑ′
b ϑ ×δ ψ¯a − i sin + γ5 cos ψ Nf Nf R ¯ ¯ µ) Ψ,A iΓI [ N1c ψψ]+i d4 xL(ψ,ψ,Ψ, . ×e

µ h† pΩ h, aµΩ → h† aµΩ h, and vΩ → h† vΩ h + h† i∂ µ h in which

the vector source transforms inhomogeneously. So that the vector source can only appear together with the derivative i∂ µ to form a covariant derivative, and a hidden symme-

ab

For F (x), translational invariance and flavor conserva-

try covariant quadratic form of the covariant derivative can

tion [16] leads to the conclusion that it is simply a space-

only be an antisymmetric tensor [cf. (56)] which does not

time independent constant proportional to δ ab . So that it

contribute when multiplied by a symmetric coefficient of

can be written as

the type of (52).

For Gabcd µν (x, z), translational invariance leads to the con-

With (50), (51) and (52) the effective action (46) is then Sef f [1, JΩ , Ξc , ΦΩc , ΠΩc ] p2 -order Z = F02 d4 xtrf [a2Ω + B0 sΩ ]  Z 1 = F02 d4 xtrf [∇µ U † ][∇µ U ] 4  1 + B0 [U (s − ip) + U † (s + ip)] , (54) 2

clusion that it can only depend on x − z. We can further

where ∇µ is the covariant derivative related to the exter-

expand this dependence in terms of δ(x − z) and its deriva-

nal sources defined in Ref. [2]. The integrand is just the

tives. To p2 -order, the derivative terms do not contribute, R and the only term left is δ(x − z) d4 zGabcd µν (x, z). The coR 4 abcd efficient d zGµν (x, z) is again independent of the space-

p2 -order chiral Lagrangian given by Gasser and Leutwyler

F ab (x) = F02 B0 δ ab ,

(49)

where F02 B0 ≡ −

  1 ¯ ψψ Nf

(50)

For F ′ab (x), parity conservation [17] leads to F ′ab (x) = 0.

(51)

in Ref. [2]. Now the coefficients F02 and B0 are defined in (53) and (50) and are expressed in terms of certain Green’s

time coordinates due to translational invariance. Then R Lorentz and flavor symmetries imply that d4 xGabcd µν is pro-

functions of the quark fields. These can be regarded as the

ν,cd ture δ ab δ cd since this term is to be multiplied by aµ,ab Ω aΩ ,

2. The p4 -Terms

fundamental QCD definitions of F02 and B0 .

ad bc

portional to gµν δ δ . There cannot be terms of the strucand aµΩ is traceless. Gabcd µν (x, z) is

Therefore the only relevant part of The p4 -order terms can be worked out along the same line. The relevant terms for the normal part contributions

ad bc 2 Gabcd µν (x, z) = δ(x − z)gµν δ δ F0

+irrelevant terms,

(ignoring anomaly contributions) are Sef f [1, JΩ , Ξc , ΦΩc , ΠΩc ] p4 -order, normal  Z = d4 xtrf − K1 [dµ aµΩ ]2

(52)

where F02 ≡

1 4(Nf2

Z

′

d4 x[Gµµ′ ,abba (0, x) −

1 µ′ ,aabb G ′ (0, x)] Nf µ

− 1)   Z i a µ b b a 4 ¯ = d x [ψ (0)γ γ5 ψ (0)][ψ (x)γµ γ5 ψ (x)] 8(Nf2 − 1)   1 a b [ψ (0)γ µ γ5 ψ a (0)][ψ (x)γµ γ5 ψ b (x)] − Nf    a b [ψ (x)γµ γ5 ψ a (x)] − [ψ (0)γ µ γ5 ψ b (0)]    1 a b [ψ (0)γ µ γ5 ψ a (0)] [ψ (x)γµ γ5 ψ b (x)] + Nf . (53)

−K2 (dµ aνΩ − dν aµΩ )(dµ aΩ,ν − dν aΩ,µ ) +K3 [a2Ω ]2 + K4 aµΩ aνΩ aΩ,µ aΩ,ν + K5 a2Ω trf [a2Ω ] +K6 aµΩ aνΩ trf [aΩ,µ aΩ,ν ] + K7 s2Ω +K8 sΩ trf [sΩ ] + K9 p2Ω + K10 pΩ trf [pΩ ] +K11 sΩ a2Ω + K12 sΩ trf [a2Ω ] − K13 VΩµν VΩ,µν  +iK14 VΩµν aΩ,µ aΩ,ν + K15 pΩ dµ aΩ,µ ,

9

(55)

 +(g µ1 µ2 g µ3 µ4 + g µ1 µ3 g µ2 µ4 )K6 ]

where the covariant derivative dµ and the antisymmetric tensor VΩµν are defined as dµ aνΩ VΩµν

≡

=

∂ µ aνΩ

ν ∂ µ vΩ

−

−

+irrelavent terms,

µ ν ivΩ aΩ

µ ∂ ν vΩ

+

µ iaνΩ vΩ

µ ν ν µ − ivΩ vΩ + ivΩ vΩ ,

i 2

(56)

  a b c d ¯ ¯ d x [ψ (0)ψ (0)][ψ (x)ψ (x)]

Z

4

C

= K7 δ ad δ bc + K8 δ ab δ cd

and the fifteen coefficients K1 , · · · K15 are determined by the following integrations of the Green’s functions Z ′ ′ i d4 x xµ xν 4   × [ψ¯a (0)γ µ γ5 ψ b (0)][ψ¯c (x)γ ν γ5 ψ d (x)]

 ϑ′ (0) ¯c [ψ (x)ψ d (x)] d4 x [ψ¯a (0)ψ b (0)] tan Nf  ϑ′ (x) × tan = K9 δ ad δ bc + K10 δ ab δ cd Nf C i 2

Z

C

′ ′ ′ ′ 1 = [( K1 − K2 )(g µµ g νν + g µν g νµ ) 2 ′ ′ +2K2 g µν g µ ν ]δ ad δ bc + irrelavent terms

1 8

Z i d4 xd4 yd4 z 24  × [ψ¯a1 (0)γ µ1 γ5 ψ b1 (0)][ψ¯a2 (x)γ µ2 γ5 ψ b2 (x)]  b4 µ4 a4 b3 µ3 a3 ¯ ¯ ×[ψ (y)γ γ5 ψ (y)][ψ (z)γ γ5 ψ (z)] C   1 a1 b2 a2 b3 a3 b4 a4 b1 1 µ1 µ2 µ3 µ4 [ (g δ δ δ δ g = 6 2 +g µ1 µ4 g µ2 µ3 )K3 + g µ1 µ3 g µ2 µ4 K4 ] 1 +δ a2 b4 δ a4 b3 δ a3 b1 [ (g µ1 µ2 g µ3 µ4 2 

C

K13 =

 i 2gµ1 µ4 gµ2 µ3 TAµ1 µ2 µ3 µ4 + 2gµ1 µ2 gµ3 µ4 TAµ1 µ2 µ3 µ4 36 −gµ1 µ3 gµ2 µ4 TAµ1 µ2 µ3 µ4 − 2gµ1 µ4 gµ2 µ3 TBµ1 µ2 µ3 µ4  µ1 µ2 µ3 µ4 µ1 µ2 µ3 µ4 + gµ1 µ3 gµ2 µ4 TB −2gµ1 µ2 gµ3 µ4 TB

K15 =

+(g

δ

δ

g

δ

+g

[g

µ1 µ4 µ2 µ3

g

g

i 4(Nf2 b

− 1)

Z

d4 x xµ a

×ψ (x)γµ γ5 ψ (x) b

×ψ (x)γµ γ5 ψ b (x)





 C

a

ψ (0)ψ b (0) tan

1 − Nf



a

ϑ′ (0) Nf

ψ (0)ψ a (0) tan

ϑ′ (0) Nf (57)

C

+g µ1 µ2 g µ3 µ4 )K3 + g µ1 µ3 g µ2 µ4 K4 ]

µ1 µ3 µ2 µ4

 Z i 4 d x (5gµν gµ′ ν ′ − 2gµµ′ gνν ′ ) 576(Nf2 − 1)   a b µ′ ν ′ µ b ν a ×x x [ ψ (0)γ ψ (0) ψ (x)γ ψ (x)  C  1 a b ψ (0)γ µ ψ a (0) ψ (x)γ ν ψ b (x) ] − Nf C

K14 =

+g µ1 µ2 g µ3 µ4 )K3 + g µ1 µ4 g µ2 µ3 K4 ]  1 a1 b4 δ a4 b2 δ a2 b3 δ a3 b1 [ (g µ1 µ4 g µ2 µ3 +δ 2 µ1 µ2 µ3 µ4 µ1 µ3 µ2 µ4 K4 ] g )K3 + g g +g 1 +δ a4 b3 δ a3 b2 δ a2 b1 [ (g µ1 µ4 g µ2 µ3 2  +δ

4

1 = K11 (δ a1 b2 δ a2 b3 δ a3 b1 + δ a1 b3 δ a3 b2 δ a2 b1 ) 2 +K12 δ a1 b1 δ a2 b3 δ a3 b2 + irrelavent terms,

+g µ1 µ3 g µ2 µ4 )K3 + g µ1 µ4 g µ2 µ3 K4 ]  1 +δ a1 b3 δ a3 b2 δ a2 b4 δ a4 b1 [ (g µ1 µ3 g µ2 µ4 2 µ1 µ2 µ3 µ4 µ1 µ4 µ2 µ3 K4 ] g )K3 + g g +g a3 b4 a4 b2 a2 b1 1 µ1 µ3 µ2 µ4 g [ (g δ δ +δ 2 

µ1 µ2 µ3 µ4

4

 b3 a3 µ b2 a2 ¯ ¯ ×[ψ (x)γ γ5 ψ (x)][ψ (y)γµ γ5 ψ (y)]

−

a1 b2 a2 b1 a3 b4 a4 b3

 d xd y [ψ¯a1 (0)ψ b1 (0)]

Z

with TA , TB defined as  Z 1 a1 − d4 xd4 y xµ4 ψ (0)γ µ1 ψ b1 (0) 2

2K5

)K6 ]

a2

a3



+δ a1 b3 δ a3 b1 δ a2 b4 δ a4 b2 [g µ1 µ3 g µ2 µ4 2K5

×ψ (x)γ µ2 γ5 ψ b2 (x) ψ (y)γ µ3 γ5 ψ b3 (y)

+(g µ1 µ2 g µ3 µ4 + g µ1 µ4 g µ2 µ3 )K6 ]

= δ a1 b2 δ a2 b3 δ a3 b1 TAµ1 µ2 µ3 µ4 + δ a1 b3 δ a3 b2 δ a2 b1 TBµ1 µ2 µ3 µ4

C

+δ a1 b4 δ a4 b1 δ a2 b3 δ a3 b2 [g µ1 µ4 g µ2 µ3 2K5

+irrelavent terms. 10

(58)

In (57) and (58), h· · ·iC denotes the connected part of h· · ·i,

the normal part contributions to the p4 -order terms in the

and the irrelavent terms are those leading to trf [aµΩ ] or

chiral Lagrangian in Ref. [2], and the coefficients are now

µ trf [vΩ ]

defined by

after multiplied by the corresponding sources.

To further evaluate the rotated source parts in (55), we

(norm)

L1

2

make use of the p -order equation of motion dµ aµΩ = −B0 [pΩ −

(norm)

1 trf (pΩ )] Nf

L2

(59)

(norm)

L3

and the following identities dµ aνΩ − dν aµΩ =

1 † µν [Ω FR Ω − ΩFLµν Ω† ] 2

i † µ Ω [∇ U ]Ω† (x) 2 1 sΩ = [Ω(s − ip)Ω + Ω† (s + ip)Ω† ] 2 i pΩ = [Ω(s − ip)Ω − Ω† (s + ip)Ω† ] 2 1 µν VΩ = [Ω† FRµν Ω + ΩFLµν Ω† ] 2 i + Ω† [−(∇µ U )U † (∇ν U ) 4 +(∇ν U )U † (∇µ U )]Ω† ,

(norm)

L4

(norm)

aµΩ =

L5

(norm)

L6

(norm)

L7

(norm)

L8

(norm)

L9

(60)

(norm)

L10

where FRµν and FLµν are, respectively, the field-strength ten-

(norm)

H1

sors of the right-handed and left-handed souces defined in Ref. [2]. With (59) and (60) and taking Nf = 3, eq. (55)

(norm)

H2

becomes Sef f [1, JΩ , Ξc , ΦΩc , ΠΩc ]

=

Z

(norm)

The twelve standard coefficients L1 p4 −order,

normal

(norm) L10 ,

 (norm) d x L1 [trf (∇µ U † ∇µ U )]2 4

(norm)

+L2

2K4 , K4,5 ≡ K4 + 2K5 , K4,6 ≡ K4 + K6 , K7 , K8 , K1,9,15 ≡ K1 − B12 K9 + B10 K15 , K1,10,15 ≡ K1 +

+ χU ]

+ U χ)]

(norm)

Li = Li

2

Hj = †

+ χU χU ]

(anom)

+ Li

(norm) Hj

(anom)

where Li

†

+

(anom)

and Hj

,

(anom) , Hj

i = 1, · · · , 10, j = 1, 2,

(63)

are the anomaly contributions

to the coefficients given in Ref. [10,9].

R L trf [Fµν ∇µ U ∇ν U † + Fµν ∇µ U † ∇ν U ]

So we have formally derived the p4 -order terms of the

(norm) R +L10 trf [U † Fµν U F L,µν ]

chiral Lagrangian from the fundamental principles of QCD

(norm) R R,µν +H1 trf [Fµν F

without taking approximations and have expressed all the

+

L Fµν F L,µν ]

 (norm) † +H2 trf [χ χ] , where χ ≡ 2B0 (s + ip)

§

+ B10 K15 , K11 ,

The total coefficients are then

†

[trf (χ† U − χU † )]2

(norm)

−iL9

†

+ χU )]

(norm) trf [χ† U χ† U +L8

Nf K B02 10

K12 , K13 , and K14 .

(norm) +L5 trf [∇µ U † ∇µ U (χ† U

(norm)

, · · ·,

are expressed in terms of

0

(norm) +L4 trf [∇µ U † ∇µ U ]trf [χ† U

+L7

(norm) H2

L2

twelve independent p -order coefficients, K2 , K3,4 ≡ K3 −

(norm) +L3 trf [(∇µ U † ∇µ U )2 ]

†

(norm) H1 ,

(norm)

,

(62)

4

trf [∇µ U † ∇ν U ]trf [∇µ U † ∇ν U ]

(norm) +L6 [trf (χ† U

1 1 1 1 K4 + K5 + K13 − K14 , 32 16 16 32 1 1 1 = (K4 + K6 ) + K13 − K14 , 16 8 16 1 = (K3 − 2K4 − 6K13 + 3K14 ), 16 K12 , = 16B0 K11 = , 16B0 K8 = , 16B02 K1 K10 K15 =− − − , 16Nf 16B02 16B0 Nf 1 1 1 1 K15 ], [K1 + 2 K7 − 2 K9 + = 16 B0 B0 B0 1 = (4K13 − K14 ), 8 1 = (K2 − K13 ), 2 1 = − (K2 + K13 ), 4 1 1 1 1 K15 ]. = [−K1 + 2 K7 + 2 K9 − 8 B0 B0 B0 =

§

coefficients in terms of the integrations of certain Green’s

(61)

functions in QCD. Eqs.(57), (58) and (62) give the fundamental QCD definitions of the the twelve coefficients

. The integrand in (61) is just

i

In Ref. [2], χ is defined as χ ≡ 2B0 (s + ip)e 3 θ . In this paper

we have taken θ = 0.

11

(norm)

L1

(norm)

· · · L10

(norm)

, H1

(norm)

and H2

. The procedure

trl [(−i sin

can be carried on order by order in the momentum expan-

ϑ(x) T ϑ(x) + γ5 cos )ΦΩc (x, x)] = 0, Nf Nf

(64)

˜ is a short notation for the following quantity where Ξ Z ∂ ϑc (y) ˜ σρ (x) ≡ Ξ d4 y trlf {Ξc (y)[−i sin ∂Φσρ Nf c (x, x) ϑc (y) T ]ΦΩ,c (y, y)} . (65) +γ5 cos Nf Ξc fixed

sion. The expressions (50), (53), (57), (58) and (62) are convenient for lattice QCD calculations of the fifteen coefficients.

V. ON THE COEFFICIENTS F20 AND B0

In (64) and (65), ϑc (x) depends on Φ(x, x) through (17). Note that the effective action ΓI [ΦΩ ] belongs to O(1/Nc ), so

So far, we have given the formal QCD definitions of the 4

fourteen coefficients of the chiral Lagrangian up to the p -

that it does not contribute in the present approximation. In

order. To get the values of the coefficients, we need to solve

(64), the field ΠΩc can be easily eliminated and the resulting

the relevant Green’s functions which is a hard task, and

equation is

we shall present the calculations in a separate paper [12].

[i∂/ + iΦT,−1 + v/ Ω + a / Ω γ5 − sΩ + ipΩ γ5 Ωc Z ∞ X ˜ σρ (x, y) + +Ξ] d4 x1 · · · d4 xn d4 x′1 · · · d4 x′n

To have an idea of how our present formulae are related to other known approximate results, we take the p2 -order

n=1

coefficients F02 and B0 [eqs. (50) and (53)] as examples and

(−i)n+1 (Nc g 2 )n σσ1 ···σn Gρρ1 ···ρn (x, y, x1 , x′1 , · · · , xn , x′n ) × n! σn ρn (xn , x′n ) = 0. (66) ×ΦσΩc1 ρ1 (x1 , x′1 ) · · · ΦΩc

make the following simple discussion. As we have mentioned in Sec. IV, there can be two ways of figuring out the explicit JΩ -dependence of ΦΩc for evaluating Sef f from (39). One of them is to solve the dynamical

In order to compare our results with the usual dynami-

equations for the intermediate fields ΦΩc , ΠΩc and Ξc , and

cal equations in the ladder approximation, we further take

the other is to track back to the original QCD generating

the ladder approximation which, in the present case, corre-

functional without the intermediate fields. For convenience,

sponds to ignoring all the n > 1 terms in (66) and with

we took the latter way in the above derivation of the chi-

σ1 σ2

Gρ1 ρ2 (x1 , x′1 , x2 , x′2 ) 1 = − Gµ1 µ2 (x1 , x2 )(γ µ1 )σ1 ρ2 (γ µ2 )σ2 ρ1 2 1 ×δ(x′1 − x2 )δ(x′2 − x1 ) + O( ) term, Nc

ral Lagrangian. To compare our results with the known approximate results, we are going to take certain approximations, say the large Nc limit, with which the calculation of the intermediate fields becomes even more convenient.

where Gµν (x, y) is the gluon propagator without internal

Thus we take the former way in the following discussion.

light-quark lines. Then, in the ladder approximation, (66)

First we take the large Nc limit. It can be easily checked

becomes

that, in this limit, the functional integrations in (11), (24) and (28) can be simply carried out by the saddle point

[i∂/ + iΦT,−1 + v/ Ω + a / Ω γ5 − sΩ + ipΩ γ5 Ωc 1 ˜ +Ξ](x, y) + g 2 Nc Gµν (x, y)γ µ ΦTΩc (x, y)γ ν = 0 2

approximation (taking the classical orbit in the semiclassical approximation).

The saddle point equations (10),

(67)

(26) and (30) are just the dynamical equations determining ΠΩc , ΦΩc , Ξc as functions of JΩ , which are (aη)(bζ) ΦΩc (x, y)

−1 (bζ)(aη)

= −i[(i∂/ + JΩ − ΠΩc )

]

On the other hand, in the large Nc limit, ΦΩc is just ΦΩc which is the full physical propagator of the quark with the

(y, x),

rotated sources. When the sources are turned off, ΦΩc can

˜ σρ (x)δ(x − y) + Πσρ (x, y) Ξ Ωc ∞ Z X (−i)n+1 (Nc g 2 )n + d4 x1 · · · d4 xn d4 x′1 · · · d4 x′n n! n=1

be expressed in terms of the quark self-energy Σ(−p2 ) and the wave function renormalization Z(−p2 ) by the standard expression

σσ1 ···σn

×Gρρ1 ···ρn (x, y, x1 , x′1 , · · · , xn , x′n )ΦσΩc1 ρ1 (x1 , x′1 ) · · · σn ρn (xn , x′n ) = 0, ×ΦΩc

12

T (aη)(bζ)

Φ0

(x, y)

≡ [ΦTΩc ](aη)(bζ) (x, y) = δ ab

Z

4

In the literature, a further approximation of dropping

(68)

the last term in (73) is usually taken [18] (It can be shown µ sΩ =pΩ =vΩ =aµ =0 Ω

that to leading order in dynamical perturbation [11], this



−i d p −ip(x−y) e (2π)4 Z(−p2 )p / − Σ(−p2 )

ηζ

term can be reasonably ignored [12].). Moreover, to leading

,

order in dynamical perturbation or in the Landau gauge, Z(−p2 ) = 1. Then F02 becomes Z 1 d4 q 1 iNc trl [γ µ γ5 γµ γ5 ] F02 = 4 2 8 (2π) /q − Σ(−q ) /q − Σ(−q 2 )    Z d4 q 1 ∂ qµ = −4iNc (2π)4 8 ∂q µ q 2 − Σ2 (−q 2 )  2 [Σ(−q ) + 21 q 2 Σ′ (−q 2 )]Σ(−q 2 ) . (74) + [q 2 − Σ2 (−q 2 )]2

in which translational invariance and the flavor and parity conservations have been considered. Plugging (68) into (67) we have Z d4 q 1 −(Z(−p2 ) − 1)p / + Σ(−p2 ) + iNc g 2 2 (2π)4 1 ×Gµν (p − q)γ µ γ ν = 0, Z(−q 2 )q/ − Σ(−q 2 )

(69)

The anomaly contribution to F02 calculated in Ref. [10] is of

where Gµν (p) is the gluon propagator in the momentum ˜ sources=0 = 0 is taken into representation, and the fact Ξ|

the same form as the first term in (74) but with an oposite

account. (69) is just the usual Schwinger-Dyson equation

sign, so that it just cancel the first term in in (74). Then

in the laddar approximation.

(74) is just the well-known Pagels-Stokar formula for F02 [11]. Thus the Pagels-Stokar formula is an approximate

With the solution of the Schwinger-Dyson equation, the formula (50) for F02 B0 can be expressed as Z Σ(−p2 ) d4 p 2 . F0 B0 = 4iNc (2π)4 Z 2 (−p2 )p2 − Σ2 (−p2 )

result of our formula by taking the approximations of the large Nc limit, the ladder approximation and dropping the (70)

last term in (73) (or to leading order in dynamical perturbation).

By definition [cf. (46) and (52)], F02 is related to the coefficient of the term linear in aΩ in the expansion of ΦΩc . We denote Z ηζ x + y − z, x − y)aabµ (71) d4 z ΦT1,µ ( Ω (z) 2 ≡ [ΦTΩc ](aη)(bζ) (x, y) linear in aµΩ Z 4 4 d pd q −ip( x+y −z)−iq(x−y) T ηζ 2 = d4 z Φ1,µ (p, q)aabµ e Ω (z) (2π)8 Then F02 is determined by∗∗ Z d4 q T ζη Nc µ ηζ Φ (0, q). (γ γ5 ) F02 = 8 (2π)4 1,µ ˜ To this order, we still have Ξ| linear in reduces to −i[ΦT0 (q + Z 1 + g 2 Nc 2

aµ Ω

VI. CONCLUSIONS In this paper, we have derived the normal part contributions to the chiral Lagrangian for pseudoscalar mesons up to the p4 -terms from the fundamental principles of QCD without taking approximations. Together with the anomaly part contributions given in Ref. [10,9], it leads to the complete QCD theory of the chiral Lagrangian. We started, in Sec. II, from the fundamental generating

(72)

functional (2) in QCD, and formally expressed the integration over the gluon-field in terms of physical gluon Green’s

= 0, and eq.(67)

functions. Then we integrated out the quark-fields by introducing a bilocal auxiliary field Φ(x, y) [cf. (6)]. To extract the degree of freedom of the local pseudoscalar-meson-field

p −1 T p )] Φ1,µ (p, q)[ΦT0 (q − )]−1 + γµ γ5 2 2 ′ d4 k µ′ T Gµ′ ν ′ (k − q)γ Φ1,µ (p, k)γ ν = 0. (2π)4

U (x), we developed, in Sec. III, a technique for extracting it from the bilocal auxiliary field Φ(x, y) [cf. (13) and (15)], and integrating in the extraction constraint into the gener-

(73)

ating functional [cf. (18), (20) and (22)]. This procedure is consistent in the sense that the complete pseudoscalar me∗∗

son degree of freedom is converted into the U (x)-field such

This is an alternative expression for F02 equivalent to (53).

that the pseudoscalar degree of freedom in the quark sector 13

is automatically frozen in the path-integral formulation of the effective action Sef f . We then developed two techniques for working out the APPENDIX:

explicit U (x)-dependence of Sef f in Sec. IV. The first one is to introduce a chiral rotation (33) which simplifies the

Here we give the proof of eqs.(18) and (19) in the text.

U (x)-dependence in such a way that the U (x)-dependence

Consider a matrix operator O satisfying det O = det O† ,

resides only implicitly in the rotated sources and some ro-

we calculate the following functional integration Z I ≡ DU δ(U † U − 1)δ(detU − 1)

tated intermediate-fields, and the second one is to implement eq.(39) to obtain Sef f from the averaged field ΦΩc . To avoid dealing with the intermediate-fields, we tracked

×F[O] δ(ΩO† Ω − Ω† OΩ† ),

back to the original QCD generating functional with which the implicit U (x)-dependence only resides in the rotated

where DU δ(U † U − 1)δ(detU − 1) serves as the invariant

sources. With all these, we expanded Sef f in power series of

integration measure at the present case. The two delta-

the rotated sources and explicitly derived the p2 -terms and

functions δ(U † U −1), δ(det U −1) constrain the integration

4

p -terms of the chiral Lagrangian for pseudoscalar mesons

to the subspace with unitary and unity determinant of the

[2]. In this formulation, all the fifteen coefficients in the

U -field.

chiral Lagrangian are expressed in terms of certain Green’s

We can rewrite δ(detU − 1)δ(U † U − 1) as

functions in QCD [cf. (50), (53), (62) and (57)]. These forδ(detU − 1)δ(U † U − 1) 1 = δ([detU ]2 − 1)θ(detU )δ(U † U − 1) 2 1 = δ(detU [detU − detU † ])θ(detU )δ(U † U − 1) 2 1 θ(detU ) = δ(detU − detU † ) δ(U † U − 1), 2 detU

mulae can be regarded as the fundamental QCD definitions of the fifteen coefficients in the chiral Lagrangian. These expressions are convenient for lattice QCD calculation of the fifteen coefficients. To see the relation between our QCD definition and the well-known approximate results in the literature, we took

so that

the p2 -order coefficients F02 and B0 as examples in Sec. V. With the approximations of large Nc limit, ladder approx-

I=

imation, and dropping the momentum-integration term in the approximate Bethe-Salpeter equation (73), our formula reduces to the well-known Pagels-Stokar formula for the

1 2

Z

×

θ(detU ) F [O] δ(ΩO† Ω − Ω† OΩ† ). detU

DU δ(U † U − 1)δ(detU − detU † )

˜ and write Next, we introduce two auxiliary fields Σ and Σ

pion decay constant Fπ [11]. The complete calculation of

I as

the fifteen coefficients will be presented in a separate pa-

1 I= 2

per [12]. The derivation of the effective chiral Lagrangian ′

further including the η meson or the ρ meson, and the ap-

Z

˜ δ(U † U − 1)δ(detU − detU † ) DU DΣDΣ

θ(detU ) ˜ δ(Σ − Σ) F [O] δ(Σ − Ω† OΩ† ) detU ˜ − ΩO† Ω) ×δ(Σ Z 1 ˜ δ(U † U − 1) = DU D[Ω† ΣΩ]D[Ω† ΣΩ] 2 θ(detU ) F [O] ×δ(detU − detU † ) detU ˜ − Σ]Ω)δ(Ω† ΣΩ − U † O) ×δ(Ω† [Σ ×

plication of the present approach to the electroweak chiral Lagrangian are all in preparation.

ACKNOWLEDGMENTS

This work is supported by the National Natural Science

˜ − O† U ). ×δ(Ω† ΣΩ

Foundation of China, and the Fundamental Research Foundation of Tsinghua University. One of us (Q. Wang) would

˜ into We then change the integration variables Σ and Σ

like to thank C.P. Yuan for the kind hospitality during his visit to Michigan State University.

Σ → Σ′ = Ω† ΣΩ 14

˜ →Σ ˜ ′ = Ω† ΣΩ, ˜ Σ

(A1)

and get I=

1 2

= Z

˜ ′ δ(U † U − 1)δ(detU − detU † ) DU DΣ′ DΣ

1 detO δ(0) F [O] 2Z

×

θ(detU ) ˜ ′ δ(Σ − Σ′ ) F [O] δ(Σ′ − U † O) detU ˜ ′ − O† U ) ×δ(Σ Z 1 ˜ ′ δ(U † U − 1)δ(detU − detU † ) = DU DΣ′ DΣ 2 θ(detU ) ˜ ′ δ(Σ − Σ′ ) F [O] δ(U Σ′ − O) × [detU ]2 ˜ ′ − Σ′† ) ×δ(Σ Z 1 ˜ ′ det(Σ′5 )δ(Σ ˜ ′ [U † U − 1]Σ′ ) = DU DΣ′ DΣ 2 ˜ ′ U ] − det[Σ ˜ ′ U † ]) θ(detU ) δ(Σ ˜ ′ − Σ′ ) ×δ(det[Σ [det(U Σ′ )]2 ˜ ′ − Σ′† ) ×F[O] δ(U Σ′ − O)δ(Σ Z 1 ˜ ′ det(Σ′4 )δ(Σ′† U † U Σ′ − Σ′† Σ′ ) D[U Σ′ ]DΣ′ DΣ = 2 θ(detU ) ×δ(det[U Σ′ ] − det[Σ′† U † ]) [det(U Σ′ )]2 ˜ ′ − Σ′ ) F [O] δ(U Σ′ − O)δ(Σ ˜ ′ − Σ′† ). ×δ(Σ ×

DΣ′ δ(O† O − Σ′† Σ′ )δ(Σ′ − Σ′† ).

(A3)

In the last step, we have used the property detO = detO† . Taking F [O] to be Z 1 ≡ detO DΣ′ δ(O† O − Σ′† Σ′ )δ(Σ′ − Σ′† ), F [O]

(A4)

eq. (A3) becomes Z DU δ(U † U − 1)δ(detU − 1) ×F[O] δ(ΩO† Ω − Ω† OΩ† ) = const,

(A5)

which is of the form of eq.(18) in the text. Next we look at the meaning of the variable Σ′ in (A4). The constraints on Σ′ in (A4) are Σ′† = Σ′ ,

Σ′2 = O† O.

(A6)

On the other hand, eqs.(13), (14) and (20) in the text show We further change the integration variables U into U → U ′ = U Σ′ ,

that the σ-field is constrained as (A2)

and get I=

1 2

that the definition of σ is not unique. It is up to a hidden

˜ ′ δ(U ′† U ′ − Σ′† Σ′ ) DU ′ DΣ′ DΣ

×δ(detU ′ −

symmetry transformation σ → h† σh. Therefore Σ′ can be

detU ′ ′† det(Σ )θ( detΣ′ ) detU ) [det(U ′ )]2

regarded as an equivalent definition of σ, and thus (A4) can be written as

˜ ′ − Σ′ ) F [O] δ(U ′ − O)δ(Σ ˜ ′ − Σ′† ). ×δ(Σ

1 = detO F [O]

˜ ′ integrations can be carried out and we obtain The U and Σ Z 1 † I= δ(detO − detO ) DΣ′ δ(O† O − Σ′† Σ′ ) 2[detO]2 detO ×det(Σ′4 )θ( )F [O]δ(Σ′ − Σ′† ) detΣ′ Z detO† 1 δ(1 − ) DΣ′ δ(O† O − Σ′† Σ′ ) = 2[detO]3 detO detO ×[det(Σ′† Σ′ )]2 θ( p )F [O]δ(Σ′ − Σ′† ) det[Σ′† Σ′ ] Z 1 detO† = δ(1 − ) DΣ′ δ(O† O − Σ′† Σ′ ) 2[detO]2 detO detO )F [O]δ(Σ′ − Σ′† ) ×[det(O† O)]2 θ( p † det[O O] ′

Z

Dσ δ(O† O − σ † σ)δ(σ − σ † ), (A8)

which is just eq.(19) in the text.

[1] S. Weinberg, Physica, 96 A, 327 (1979). [2] J. Gasser and H. Leutwyler, Ann. Phys. 158, 142 (1984); Nucl. Phys. B 250, 465 (1985). [3] H.W. Fearing and S. Scherer, Phys. Rev. D 53, 315 (1996).

detO† 1 detO δ(1 − ) F [O] 2Z detO

×

(A7)

Comparing (A6) with (A7), we find Σ′ ∼ Ω† σΩ. We know Z

′4

=

(Ω† σΩ)2 = O† O.

σ † = σ,

[4] T. Appelquist and G.-H. Wu, Phys. Rev. D 48, 3235 (1993);

DΣ′ δ(O† O − Σ′† Σ′ )δ(Σ′ − Σ′† )

51, 240 (1995) and references therein.

15

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