Effective weak chiral Lagrangian to O(p4) in the chiral quark model

Nuclear Physics B 562 Ž1999. 213–236 www.elsevier.nlrlocaternpe

Effective weak chiral Lagrangian to O Ž p 4 . in the chiral quark model Mario Franz a , Hyun-Chul Kim b, Klaus Goeke a

a,c

Institute for Theoretical Physics II, P.O. Box 102148, Ruhr-UniÕersity Bochum, D-44780 Bochum, Germany b Department of Physics, Pusan National UniÕersity, 609-735 Pusan, South Korea c RCNP, UniÕersity of Osaka, Osaka, Japan Received 9 March 1999; accepted 6 August 1999

Abstract We investigate the D S s 1,2 effective weak chiral Lagrangian within the framework of the chiral quark model. Starting from the effective four-quark operators, we derive the effective weak chiral action by integrating out the constituent quark fields. Employing the derivative expansion, we obtain the effective weak chiral Lagrangian to order O Ž p 4 .. We examine the contributions of the order O Ž Nc . to the ratio g 8rg 27, considering e.g. the quark axial-vector constant gA different from unity. The low energy constants of the counterterms are also presented and discussed. q 1999 Elsevier Science B.V. All rights reserved. PACS: 12.40.-y; 13.25.-k; 14.40.Aq Keywords: Chiral quark model; Effective chiral Lagrangians; Non-leptonic decays; Derivative expansion

1. Introduction Processes involving the creation or annihilation of strangeness are described in the Standard Model by W-exchange. While the theoretical formulation is simple at scales around the W-mass the description of non-leptonic decays of light hadrons at low energies is complicated and difficult because of the presence of the strong interaction. The problem is characterized by the DT s 1r2 selection rule, best known as the fact that the isospin T s 0 amplitude of the K pp decay is about 22 times larger than the T s 2 amplitude. In spite of many efforts this enhancement of the DT s 1r2 channel over the DT s 3r2 channel has not been explained in a satisfactory way. A part of the answer comes from perturbative gluons which are created if one evolves the simple W-exchange-vertex from a scale of 80 GeV down to 1 GeV w1–10x. Another part of the answer is supposed to arise from the structure of the light hadrons, whose description at scales around 1 GeV requires a non-perturbative QCD-method.

™

0550-3213r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 5 5 0 - 3 2 1 3 Ž 9 9 . 0 0 4 8 8 - 5

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In the low energy regime one way to deal with non-perturbative effects is to utilize the large Nc expansion with a s Nc fixed Ž Nc being the number of colors and a s the running coupling constant of QCD.. In the limit of large Nc QCD can be treated as a weakly coupled meson field theory and indeed many experimental consequences have been explained in this way w11–13x. The large Nc limit of QCD was also employed w14–16x in order to understand the DT s 1r2 problem in the K pp decay. However, in contrast to the sector of the pure strong interaction, the large Nc limit in its strict form Žonly leading order in Nc . does not seem to be sufficient to describe the weak non-leptonic decays w17x because it enhances the DT s 3r2 channel while suppressing the DT s 1r2 one making the problem even more difficult. Hence for these processes one is bound to go beyond leading order in the large Nc expansion. At low energies chiral perturbation theory Ž x PT. w18x is known as a proper effective field theory of QCD in the mesonic sector. Based on its success in describing strong interactions x PT was also applied to non-leptonic processes of light mesons w19–22x. However, in this case there are not enough experimental data available to determine the many low energy constants ŽLECs. of the the effective weak chiral Lagrangian to order O Ž p 4 .. Hence, in order to proceed without experiments, one is advised to determine the LECs of the weak chiral Lagrangian by using effective QCD-inspired models. In the present paper we are going to investigate how far the chiral quark model Ž x QM. furnishes a reasonable framework to determine the LECs of the weak chiral Lagrangian. We are motivated to this study by success of the x QM to determine the LECs of the strong chiral Lagrangian. The x QM is characterized by the Euclidean partition function w23x

™

Z s Dc Dc † Dp exp

H

Hd

4

a

a

x c f†a Ž i Euq iMe ig 5 l p . f g cga ,

Ž 1.

where a is the color index, a s 1, . . . , Nc and f and g are flavor indices. The M serves as the coupling parameter between the constituent quark fields c and the Goldstone boson field p a and it can be identified with the constituent quark mass. In this work we want to construct systematically the effective weak chiral Lagrangian without external fields to order O Ž p 2 . and O Ž p 4 . for D S s 1 and D S s 2. We will show that the x QM provides the most general structure of the Lagrangian, known from the work of Refs. w19–22x, and unique descriptions of the LECs in leading and subleading order of the large Nc expansion. The calculations start from the effective weak Hamiltonians for D S s 1,2 of Refs. w10,24–27x. The effective weak Lagrangian for D S s 1 to order O Ž p 2 . in leading and subleading order in Nc has been given already by Antonelli et al. w28x. Bertolini et al. w29x extended the former calculation to O Ž p 4 . in the study of e Xre and BˆK with the D S s 1 Lagrangian, which implies that parts of the O Ž p 4 . effective chiral weak Lagrangian that are necessary for the description of the K pp decays are obtained. They used for this the small field expansion in leading and subleading order in the Nc expansion. In the present paper we prefer the derivative expansion, since we want to evaluate the full effective chiral weak Lagrangian to order O Ž p 4 . including the corresponding LECs in a way that it can be used directly in x PT. This means we have to use an expansion of the x QM which is consistent with the chiral expansion in x PT. In the case of the strong interaction it is known that the small field expansion fulfills this criterion only in the

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leading order in the large Nc expansion. However, for the weak chiral Lagrangian the subleading order in Nc is necessary and hence the derivative expansion seems to us more appropriate than the small field expansion.1 The outline of the present paper is as follows: In Section 2 we sketch the characteristics of the chiral quark model and briefly show how to use the derivative expansion. In Section 3 we review the effective weak chiral action at a scale of 1 GeV and discuss some of its properties relevant for the following. Section 4 is devoted to the derivation of the effective weak chiral Lagrangian to order O Ž p 2 .. We examine the dependence of the LECs on the constituent quark mass, the quark condensate and the quark axial-vector constant. The full effective weak chiral Lagrangian for D S s 1 and D S s 2 to order O Ž p 4 . in leading and next-to-leading order in Nc , as it results from the derivative expansion, is presented in Section 5. The conclusions are given in Section 6. 2. Chiral quark model The characteristic of the chiral quark model is represented by the effective chiral action in Euclidean space given by the functional integral over quark fields w23x: N s exp Ž ySeff . s Dc Dc †exp

H

Hd

4

x c f†a Ž i Euq iMUg 5 . f g cga ,

Ž 2.

where a is the color index, a s 1, . . . , Nc and f and g are flavor indices. M is the constituent quark mass, which is in fact momentum-dependent. However, we regard it as a free parameter for convenience and introduce a cut-off parameter to tame the divergence appearing in the quark loop. It is fixed by producing the pion decay constant. Ug 5 denotes the Goldstone field 1 q g5 1 y g5 Ug 5 s exp Ž ip al ag 5 . s U qU † Ž 3. 2 2 with U sexp Ž ip al a . . Ž 4. The p stands for the meson octet fields

°1 '2 p s p al a s

1

p 0q py

'2

¢

Ky

1

'6

pq

h 1 y

'2

p 0q K0

¶

Kq 1

'6

K0

h 1 y

'2

p 0q

. 1

'6

ß

h

Ž 5. 1 The LECs L1 and L2 r2 of the Gasser–Leutwyler Lagrangian w18x in the strong interaction are the same in the large Nc limit, which makes the small field expansion in the chirl quark model yield the same Lagrangian as in the derivative expansion. However, when one considers higher order corrections L1 and L2 r2 have to deviate from each other, as the experimental extraction of those values implies, a feature, which is only brought out by the derivative expansion.

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Integrating over the quark fields in Eq. Ž2., we obtain the following expression for the effective chiral action Seff w U x s yNcTr ln D,

Ž 6.

where Tr designates the functional trace as well as flavor and spin ones. The D denotes the Dirac operator D s i Euq iMUg 5 .

Ž 7.

Since equation Ž6. is non-Hermitian, one can separate the effective action into the real part and the imaginary one. The real part can be written as Re Seff w U x s y 12 Tr ln

D† D

ž / D†0 D 0

,

Ž 8.

where D†D s yE 2 q M 2 y M Ž EuUg 5 . ,

D†0 D 0 s yE 2 q M 2 .

Ž 9.

It is already well known how to treat the effective action in order to obtain the effective chiral Lagrangian w30–34x. The real part of the effective action may be expanded with respect to the derivatives of the meson field w23x: Re Seff s y

sy

Nc 2 Nc 2

ž

Tr ln 1 y

d4 x

M Ž EuUg 5 . D†0 D 0

d4 k

H H Ž 2p .

ž

/

sy

tr ln 1 y 4

Nc 2

ž

tr d 4 x² x ln 1 y

H

M Ž EuUg 5 .

M Ž EuUg 5 . k 2 q M 2 y Ž 2 ik P E q E 2 .

D†0 D 0

/

P 1.

/

x:

Ž 10 .

The non-vanishing leading term in the expansion is just the kinetic Lagrangian of the strong interaction: Ž2. w U x s y d 4 x LDŽ2.Ss0 , Re Seff

Ž 11 .

H

where fp2 Ž2. LD Ss0 s y 4

² Lm Lm :

Ž 12 .

The Lm Ž Rm . are the Noether currents of SUŽ3.L = SUŽ3.R chiral symmetry: Lm s iU †Em U,

Rm s iUEm U † .

Ž 13 .

The symbol ² : stands for the flavor trace. The fp denote the pion decay constant which is related to the following quark loop integral: fp2 s 4 Nc

d4 k

H Ž 2p .

M2 4

Ž k2qM 2 .

2

.

Ž 14 .

Since the quark loop integral is divergent, which is due to the fact that we regard the M as a constant, we need to introduce the cut-off parameter L via regularization. It is fixed by producing the experimental value of fp s 93 MeV.

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Similarly, we can move up to higher orders in the derivative expansion. The real part of the effective chiral action in the next-to-leading order O Ž p 4 . is given by Ž4. Re Seff s y d 4 x LDŽ4. Ss0 ,

Ž 15 .

H

so that the strong effective chiral Lagrangian to order O Ž p 4 . can be written as w23x LDŽ4. Ss0 s

Nc 192p 2

Hd

4

x 2² Em Lm :2q² Lm Ln Lm Ln : .

Ž 16 .

Those effective Lagrangians in higher orders were extensively studied w30–37x. Ref. w36x investigated also the low energy constants of the effective O Ž p 4 . Lagrangian in relation to chiral perturbation theory. The imaginary part of the effective chiral action is pertinent to the Wess–Zumino– Witten ŽWZW. action w38,39x with the correct coefficient, which arises from the derivative expansion of the imaginary part to order O Ž p 5 . Žsee Ref. w23x for details..

3. Effective weak chiral action The effective chiral action in Eq. Ž1. with the weak D S s 1 or D S s 1 effective Hamiltonian can be written as follows: D Ss1,2 exp Ž ySeff . s Dc Dc †exp

H

4

†

D Ss1,2 eff

H Hd x Ž c DcyH

.

.

Ž 17 .

D Ss1 Here the effective weak quark Hamiltonian Heff consists of ten four-quark operators among which only seven operators are independent, D Ss1 Heff sy

GF

'2

Vu d Vu)s Ý c i Ž m . Qi Ž m . q h.c..

Ž 18 .

i

The GF is the well-known Fermi constant and Vi j denote the Cabibbo–Kobayashi– Maskawa ŽCKM. matrix elements. The t is their ratio given by t s yVt d Vt s) rVu d Vu)s . The c i Ž m . consist of the Wilson coefficients c i Ž m . s z i Ž m . q t yi Ž m .. The functions z i Ž m . and yi Ž m . are the scale-dependent Wilson coefficients given at the scale of the m. The z i Ž m . represent the CP-conserving part, while yi Ž m . stand for the CP-violating one. The four-quark operators Qi contain the dynamic information of the weak transitions, being constructed by integrating out the vector bosons W " and Z and heavy quarks t, b and c. The four-quark operators w10x are given by Q1 s 4 Ž sa† gm PL ub .Ž u†b gm PL da . ,

Ž 19 .

Q2 s 4 Ž sa† gm PL ua .Ž u†b gm PL db . ,

Ž 20 .

Q3 s 4 Ž sa† gm PL da .

Ž 21 .

Ý Ž qb† gm PL qb . , qsu , d , s

Q4 s 4 Ž sa† gm PL db .

Ý Ž qb† gm PL qa . , qsu , d , s

Ž 22 .

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218

Q5 s 4 Ž sa† gm PL da .

Ý Ž qb† gm PR qb . ,

Ž 23 .

qsu , d , s

Q6 s 4 Ž sa† gm PL db .

Ý Ž qb† gm PR qa . ,

Ž 24 .

qsu , d , s

Q7 s 6 Ž sa† gm PL da .

Ý qsu , d , s

Q8 s 6 Ž sa† gm PL db .

Ý qsu , d , s

Q9 s 6 Ž sa† gm PL da .

Ý qsu , d , s

Q10 s 6 Ž sa† gm PL db .

žq

ˆ

/,

Ž 25 .

ˆ

/,

Ž 26 .

ˆ

/,

Ž 27 .

ˆ

/,

† b Qgm PR qb

žq

† b Qgm PR qa

žq

† b Qgm PL qb

Ý qsu , d , s

žq

† b Qgm PL qa

Ž 28 .

where PL, R s 12 Ž 1 " g 5 . are the chiral projection operators and Qˆ s 13 diagŽ2,y 1,y 1. denote the quark charge matrix. The Q1 and Q2 come from the current–current diagrams, while Q3 to Q6 w5–8x and Q7 to Q10 w9x are induced by QCD penguin and electroweak penguin diagrams, respectively. Note that only seven operators in Eq. Ž28. are independent. For example, we can express Q4 , Q9 , and Q10 as follows: Q1 q Q2 q Q3 , Q4 s yQ Q9 s 12 Ž 3 Q1 y Q3 . , Q10 s Q2 q 12 Ž Q1 y Q3 . . Ž 29 . Under the chiral transformation SUŽ3.L = SUŽ3.R the four-quark operators Q3,4,5,6 transform like Ž8L ,1R .. The Q1,2,9,10 transform like the combination of Ž8L ,1R . and Ž27L ,1R ., while the Q7,8 transform like Ž8L ,8R .. The D S s 2 effective weak Hamiltonian is expressed as w24–27,40x D Ss2 Heff sy

GF2 MW2 16p 2

F Ž l c , l t ,m2c ,m2t , MW2 . b Ž m . QD Ss2 Ž m . q h.c.

Ž 30 .

with F s l2ch1 S

m 2c

ž / MW2

q l2t h 2 S

m2t

ž / MW2

q 2 lc lth3 S

ž

m 2c

m2t

, MW2 MW2

/

Ž 31 .

and the parameters l q s Vq d Vq)s denote the pertinent relations of the CKM matrix elements with q s u,c,t. The functions Si are the Inami–Lim functions w40–42x, being obtained by integrating over electroweak loops and describing the < D S < s 2 transition amplitude in the absence of strong interactions. The bŽ m . is again the corresponding Wilson coefficient. The coefficients hi represent the short-distance QCD corrections split off from the bŽ m . w27x. The four-quark operator QDSs2 is written as QDSs2 s 4 Ž sa gm PL da .Ž sb gm PL db . . Ž 32 . Since the Fermi constant GF is very small, one can expand Eq. Ž17. in powers of the GF and keep the lowest order only. Then we can obtain the effective weak chiral Lagrangian 1 D Ss1,2 LeffDSs1,2 s y Dc Dc † Heff exp d 4 x c †Dc . Ž 33 . N

H

H

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If you write a generic operator for the four-quark operator Qi for a given i in Euclidean space such as Qi Ž x . s c † Ž x . gm PR , L L1 c Ž x . c † Ž x . gm PR , L L 2 c Ž x . ,

Ž 34 .

where L1,2 denote the flavor spin operators, then we can calculate the vacuum expectation value ŽVEV. of Qi Ž x . as follows: 1 ² Qi : s

†

H Dc Dc Q Ž x . exp Hd

4

i

N

d4 k

s d4 y

H H Ž 2p . =exp

Hd

4

4

e i kŽ xyy.

z c †Dc

d

d

d JmŽ1.

d JmŽ2.

Ž x.

Ž y.

z² z tr ln D˜ Ž J1 Ž z . , J2 Ž z . . z :

Ž2. s LŽ1. i q Li .

Ž 35 .

J 1 sJ 2 s0

Here, D˜ is D˜ Ž J1 Ž z . , J2 Ž z . . s D q JaŽ1. Ž z . ga PR , L L1 q JbŽ2. Ž z . gb PR , L L2 .

Ž 36.

Ž2. The LŽ1. i and L i are given by 2 ² x< LŽ1. i s yNc tr

s yNc2 tr

D

gm PR , L L1 < x :² x <

1 D

gm PR , L L2 < x q O Ž Nc . i

Ž A1 . m Ž A 2 . m i q O Ž Nc . i s 1,4,6,8,10,

2 ² x< LŽ2. i s Nc tr

s Nc2 tr

1

1 D

gm PR , L L1 < x : tr ² x <

1 D

Ž A1 . m tr Ž A 2 . m i q O Ž Nc .

Ž 37 .

gm PR , L L2 < x : q O Ž Nc . i

i s 2,3,5,7,9,

Ž 38.

where L1,2 are the corresponding flavor matrices. The operators Ž A1,2 .m can be written as

Ž A1,2 . m s

i Euq ku y iMUy g 5

d4 k

H Ž 2p .

4

k 2 q M 2 y E 2 q 2 ik P E y M Ž EuUg 5 .

gm PL , R L1,2 .

Ž 39 .

Assuming that the pion field changes adiabatically, we are able to expand the denominator in Eq. Ž39. in powers of E 2 y 2 ik P E q M Ž EuUg 5 . and obtain the following expression:

Ž A1,2 . m s H

d4 k

1

4 2 2 Ž 2p . k q M

`

Ý ns0

E 2 y 2 ik P E q M Ž EuUg 5 .

n

k2qM 2

= Ž i Euq ku y iMUyg 5 . gm PL , R L1,2 .

Ž 40 .

With the expansion given in Eq. Ž40. we can systematically evaluate effective weak chiral Lagrangian to order O Ž p 4 .: LeffDSs1,2 s LeffDSs1,2 Ž O Ž p 2 . . q LeffDSs1,2 Ž O Ž p 4 . . . We first evaluate the effective weak chiral Lagrangian in the lowest order.

Ž 41 .

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4. Lowest order p 2 and low energy constants

4.1. Leading order in the 1 r Nc expansion The derivation of the LeffDSs1,2 Ž O Ž p 2 .. is straightforward. At lowest leading order in the derivative expansion, i.e. O Ž p 2 . order, we obtain the following results with O Ž Nc2 . considered: ² Q1 q Q1† :O Ž p 2 . s fp4 Ž y 25 ² l6 Lm Lm : q 13 t i j; k l ² l i j Lm :² l k l Lm : . ,

Ž 42 .

² Q2 q Q2† :O Ž p 2 . s fp4 Ž 53 ² l6 Lm Lm : q 31 t i j; k l ² l i j Lm :² l k l Lm : . ,

Ž 43 .

² Q3 q Q3† :O Ž p 2 . s 0,

Ž 44 .

² Q4 q Q4† :O Ž p 2 . s fp4 ² l6

Lm Lm : ,

Ž 45 .

² Q5 q Q5† :O Ž p 2 . s 0, ² Q6 q Q6† :O Ž p 2 . s

ž

Ž 46 .

² qq : fp2

y

² qq : Nc M

M

8p 2

/

fp4 ² l6 Lm Lm : ,

² Q7 q Q7† :O Ž p 2 . s 32 fp4 ² Lm l6 :² Rm Qˆ : , ² Q8 q Q8† :O Ž p 2 . s y

ž

Nc² qq : M 16p 2

q

Ž 47 . Ž 48 .

fp2 ² qq : 2M

q² Ž E 2 U . l6 U † Qˆ : y

/

²Ul6 Ž E 2 U † . Qˆ :

Nc² qq : M 8p 2

= ²Ul6 Ž Em U † . Ž Em U . U † Qˆ : q ² Ž Em U . Ž Em U † . Ul6 U † Qˆ : ,

Ž 49 . ² Q9 q Q9† :O Ž p 2 . s fp4 Ž y 35 ² l6 Lm Lm q 12 t i j; k l ² l i j Lm :² l k l Lm : . ,

Ž 50 .

1 † : 4 2 ² Q10 q Q10 O Ž p 2 . s fp Ž 5 ² l6 Lm Lm q 2 t i j; k l ² l i j Lm :² l k l Lm : . ,

Ž 51 .

² QD Ss2 q QD†Ss2 :O Ž p 2 . s fp4 ² l6 Lm l6 Lm : .

Ž 52 .

The eikosiheptaplet projection operators t i j; k l are defined by t i j; k l s

1 5

1r2 t iTs q t iTs3r2 , j; k l j; k l

Ž 53 .

where 1 Ts1r2 1 Ts 1r2 Ts1r2 Ts1r2 Ts1r2 Ts1r2 Ts1r2 Ts1r2 t 13 ;21 s t 31;12 s t 21;13 s t 12;31 s 2 ,t 23;11 s t 32;11 s t 11;23 s t 11;32 s 2 , 3 Ts 1r2 Ts1r2 Ts1r2 Ts1r2 Ts1r2 Ts1r2 Ts1r2 Ts1r2 t 23 ;22 s t 32;22 s t 22;23 s t 22;32 s 1,t 23;33 s t 32;33 s t 33;23 s t 33;32 s y 2 , 1 Ts3r2 1 Ts 3r2 Ts3r2 Ts3r2 Ts3r2 Ts3r2 Ts3r2 Ts3r2 t 13 ;21 s t 31;12 s t 21;13 s t 12;31 s 2 ,t 13;21 s t 31;12 s t 21;13 s t 12;31 s 2 , 1 Ts1r2 Ts 3r2 Ts3r2 Ts3r2 Ts3r2 t 13 s t iTs3r2 s 0, ;21 s t 31;12 s t 21;13 s t 12;31 s y 2 ,t i j; k l j; k l

for the other i , j,k ,l

Ž 54 .

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and

Ž li j . a b s di a di b .

Ž 55 .

The coefficients appearing in front of the integrals consist of the pion decay constant fp Žsee Eq. Ž14.., quark condensate 2 ² qq :, and constituent quark mass M. The quark condensate is related to the following quadratically divergent integral: ² qq : s 8 Nc

d4 k

H Ž 2p .

M 4

2

k qM 2

.

Ž 56 .

Hence, we find that those coefficients have O Ž Nc2 . order in the Nc counting. Note that the VEV of the operators ² Q3 : and ² Q5 : vanish at leading order in Nc , which implies that in the leading order of the large Nc expansion ² Q3 : and ² Q5 : do not contribute to the effective weak chiral Lagrangian in the order O Ž p 2 .. It is interesting to compare our results with those of Ref. w28x. We find some differences in ² Q6 : and ² Q8 : which, however, disappear when we apply for the quark condensate the same regularization scheme as used in Ref. w28x. Since Ref. w28x employs the expansion of the weak meson field in which the U field is expanded in powers of the p field, one is not able to obtain the full effective chiral Lagrangian to order O Ž p 4 . consistently with the chiral expansion, if one goes beyond the leading order in the large Nc expansion. With the derivative expansion, we can derive the effective weak chiral Lagrangian in the next-to-leading order O Ž p 4 .. In fact, the VEV of the operator Q8 has the zeroth order contribution:

ˆ :. ² Q8 q Q8† :O Ž p 0 . s 34 ² qq :2² l6 U † QU

Ž 57 .

It transforms under SUŽ3.L = SUŽ3.R as Ž8L , 8R .. The effective weak chiral Lagrangian describing the D S s 1 non-leptonic decays of kaons was first introduced by Cronin w43x Žpresented in Minkowski space.: 2

LeffDSs1, O Ž p . s y

GF

'2

Vu d Vu)s fp4 g8² l23 Lm L m :q g27

2 3

ž ²l

12

Lm :² l 31 L m :

q² l32 Lm :² l11 L m : q h.c.s L8Ž 1r2 . q 19 L27Ž 1r2 . q 59 L27Ž 3r2 . ,

/

Ž 58 . where L Ž1r2. sy 8

GF

'2

Vu d Vu)s fp4 g8² l23 Lm L m :q h.c.,

2 In our calculation the quark condensate is defined as ² qq : s ² uuq dd :, since it plays the role of a numerical parameter in our calculation we do not distinguish between the quark condensate in Euclidean and Minkowski space.

M. Franz et al.r Nuclear Physics B 562 (1999) 213–236

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L Ž1r2. sy 27

GF

'2

Vu d Vu)s fp4 g27 ² l12 Lm :² l31 L m :

ž

y² l32 Lm :² l11 L m :y 5² l 32 Lm :² l33 L m : q h.c.,

/

Ž3r2. L 27 sy

GF

'2

Vu d Vu)s fp4 g27 ² l12 Lm :² l31 L m :

ž

q2² l32 Lm :² l11 L m :q² l32 Lm :² l 33 L m : q h.c.

/

Ž 59 .

™

The coupling constants g 8 and g 27 can be extracted from the K pp decay rate and the DT s 1r2 enhancement is reflected in these constants. Now we are in a position to evaluate the constants g 8 and g 27 from the results of ² Qi :. Comparison of Eq. Ž33. and Eqs. Ž42. – Ž52. with Eqs. Ž58., Ž59. yields the following results: g Ž1r2. s y 25 c1 q 35 c 2 q c 4 q 8

ž

² qq : Mfp2

y

² qq : Nc M 8p 2 fp4

/

c6 y 35 c9 q 25 c10 ,

g Ž1r2. s 151 c1 q 151 c 2 q 101 c9 q 101 c10 , 27 Ž1r2. g Ž3r2. s 5 g 27 . 27

Ž 60 .

We employed the Wilson coefficients c i obtained by Buchalla et al. w10x as shown in Table 1. There are three different renormalization schemes. The LO denotes the summation of the leading logarithmic terms ; a s lnŽ MW rm .. n , which were mainly done by Vainshtein et al. w4,5x, Gilman and Wise w6,7x and Guberina and Peccei w8x. The NDR and HV represent respectively ‘‘Naive dimensional regularization’’ and ’t Hooft– Veltman scheme w44,45x Žsee Ref. w10x for details.. In the leading order contribution in the large Nc expansion three parameters are involved: the pion decay constant, the quark Table 1 Wilson coefficients at m s1 GeV. c i can be obtained by the relation c i Ž m . s z i Ž m .qt yi Ž m . which are provided by Ref. w10x Ž4.

Ž4.

LMS s 215 MeV

Ž4.

LMS s 325 MeV

LMS s 435 MeV

Scheme

LO

NDR

HV

LO

NDR

HV

LO

NDR

HV

c1 c2 c3 c4 c5 c6 c7 r a c8 r a c9 r a c10 r a

y0.607 1.333 0.003 y0.008 0.003 y0.009 0.004 0 0.006 0

y0.409 1.212 0.008 y0.022 0.006 y0.022 0.003 0.008 0.008 y0.005

y0.494 1.267 0.004 y0.010 0.003 y0.009 y0.003 0.006 0.001 y0.006

y0.748 1.433 0.004 y0.012 0.004 y0.013 0.008 0.001 0.009 y0.002

y0.509 1.278 0.013 y0.035 0.008 y0.035 0.011 0.014 0.019 y0.008

y0.640 1.371 0.007 y0.017 0.004 y0.014 y0.002 0.010 0.006 y0.010

y0.907 1.552 0.006 y0.017 0.005 y0.018 0.011 0.001 0.013 y0.002

y0.625 1.361 0.023 y0.058 0.009 y0.059 0.021 0.027 0.035 y0.015

y0.841 1.525 0.015 y0.029 0.005 y0.025 y0.001 0.017 0.012 y0.018

M. Franz et al.r Nuclear Physics B 562 (1999) 213–236

223

condensate, and the constituent quark mass. The values of the quark condensate and constituent quark mass are the parameters we can play with. However, these two parameters are to some extent theoretically restricted. The value of the quark condensate lies between yŽ300 MeV. 3 ( ² qq :r2 ( yŽ200 MeV. 3. Larger values give slightly better ratio of the constants g 8 and g 27. The constituent quark mass is in fact the free parameter of the x QM. It is known that the value M , 400 MeV describes consistently very well the static properties of the baryon w46x. However, in the mesonic sector lower values are often voted w47x. We also find that lower values of the constituent quark mass provide better ratios of the constants. Fig. 1 shows the dependence of the g 8rg 27 on the M. In the NDR scheme M-dependence is stronger than in the other two schemes. One can easily understand this dependence. The parameter M appears in front of the coefficient c6 in Eq. Ž60.. From Table 1 we find that the Wilson coefficient c6 based on the NDR scheme Žy0.0022. is larger than in the other two schemes Žy0.009. which causes the strong dependence of the ratio g 8rg 27 on the M in the case of the NDR scheme. Because of the same reason, its dependence on the quark condensate looks very similar, see Fig. 2. In Table 2 we find that the ratio g 8rg 27 is almost seven times underestimated, compared to the empirical data Ž g 8rg 27 , 22 with the counterterms..

Fig. 1. Dependence of the g 8r g 27 on the M. The solid curve denotes the LO renormalization scheme in Ref. w10x, while the dashed curve and dot-dashed one stand for the NDR and the HV schemes, respectively. The value of the quark condensate ² qq :r2 syŽ250 MeV. 3 is used.

M. Franz et al.r Nuclear Physics B 562 (1999) 213–236

224

Fig. 2. Dependence of the g 8 r g 27 on the ² qq :. The solid curve denotes the LO renormalization scheme in Ref. w10x, while the dashed curve and dot-dashed one stand for the NDR and the HV schemes, respectively. The value of the constituent quark mass M s 300 MeV is used.

From the calculation of the ² QDSs2 : in Eq. Ž52., we easily write the effective D S s 2 weak chiral Lagrangian to order O Ž p 2 .: 4

LeffD Ss2, O Ž p . s y

GF2 MW2 4p 2

F Ž l c , l t ,m2c ,m2t , MW2 . b Ž m . fp4 ² l6 Lm :² l6 L m : .

Ž 61 .

4.2. O (Nc ) and axial-Õector coupling corrections So far we concentrate on the leading order in the large Nc expansion. We now want to introduce the next-to-leading order corrections in the large Nc expansion. The Q3 and Q5 survive and the additional terms come into existence in the other quark operators. The VEV of the quark operators are obtained in the O Ž Nc . order as follows: ² Q1 q Q1† :OO ŽŽ Np c2.. s ² Q2 q Q2† :OO ŽŽ Np c2.. s

fp4 Nc fp4 Nc

Ž 35 ² l6 Lm Lm : q 13 ti j; k l ² li j Lm :² lk l Lm : . ,

Ž 62 .

Ž y 25 ² l6 Lm Lm : q 13 ti j; k l ² li j Lm :² lk l Lm : . ,

Ž 63 .

M. Franz et al.r Nuclear Physics B 562 (1999) 213–236

225

Table 2 The low energy constants in O Ž Nc2 . order. The Wilson coefficients are from Ref. w10x as shown in Table 1. M s 300 MeV and ² qq : sy2PŽ250 MeV. 3 are used Ž4.

Ž4.

LMS s 215 MeV Scheme g8 g27 g8 r g27

LO

NDR 1.100 0.436 2.524

N1Ž8 . =10 3 0.16 N2Ž8 . =10 3 26.21 N3Ž8 . =10 3 0 N4Ž8 . =10 3 y11.37 . N Ž8 =10 3 11.29 28

Ž4.

LMS s 325 MeV HV

1.029 0.482 2.135

LO 1.013 0.464 2.184

LMS s 435 MeV

NDR 1.241 0.411 3.019

HV

1.190 0.461 2.578

LO 1.163 0.439 2.652

NDR 1.407 0.387 3.636

1.437 0.442 3.254

HV 1.404 0.410 3.421

0.39 22.01 0 y9.22 9.07

0.16 24.01 0 y10.16 10.08

0.23 0.61 29.05 23.69 0 0 y12.94 y 10.28 12.84 10.08

0.24 0.31 26.89 32.35 0 0 y11.77 y14.74 11.67 14.62

1.03 25.54 0 y11.47 11.24

0.44 30.96 0 y13.99 13.86

N1Ž27 . =10 3 0 0 N2Ž27 . =10 3 y11.03 y12.20 N3Ž27 . =10 3 11.03 12.20 N4Ž27 . =10 3 11.03 12.20 N5Ž27 . =10 3 0 0 N6Ž27 . =10 3 0 0 Ž27 . N 20 =10 3 0 0

0 y11.75 11.75 11.75 0 0 0

0 y10.41 10.41 10.41 0 0 0

0 y11.11 11.11 11.11 0 0 0

0 y11.19 11.19 11.19 0 0 0

0 y10.39 10.39 10.39 0 0 0

0 y11.69 11.69 11.69 0 0 0

. N Ž27 =10 3 y11.03 y12.20 y 11.75 y 10.41 y 11.69 y 11.11 21

² Q3 q Q3† :OO ŽŽ Np c2.. s

fp4 Nc

0 y 9.80 9.80 9.80 0 0 0

y 9.80 y 11.19 y 10.39

² l6 Lm Lm : ,

Ž 64 .

² Q4 q Q4† :OO ŽŽ Np c2.. s 0, ² Q5 q Q5† :OO ŽŽ Np c2.. s

ž

Ž 65 .

fp2 ² qq :

² qq : M y

Nc M

8p 2

/

² l6 Lm Lm : ,

Ž 66 .

² Q6 q Q6† :OO ŽŽ Np c2.. s 0, ² Q7 q Q7† :OO ŽŽ Np c2.. s

ž

Ž 67 .

3 fp2 ² qq : 4 Nc M

3² qq : M y

32p 2

/

ˆ : q ² l6 U †QU ˆ Ž Em U † . Ž Em U . : , = ² l6 Ž Em U † . Ž Em U . U † QU

ž

/

Ž 68 . ² Q8 q Q8† :OO ŽŽ Np c2.. s ² Q9 q Q9† :OO ŽŽ Np c2.. s

3 fp4 2 Nc fp4 Nc

† :O Ž Nc . ² Q10 q Q10 OŽ p2. s

² l6 U † Ž Em U . :² Qˆ Ž Em U . U † : ,

Ž 25 ² l6 Lm Lm : q 12 ti j; k l ² li j Lm :² lk l Lm : . ,

fp4 Nc

Ž y 35 ² l6 Lm Lm :q 12 ti j; k l ² li j Lm :² lk l Lm : . .

Ž 69 . Ž 70 . Ž 71 .

M. Franz et al.r Nuclear Physics B 562 (1999) 213–236

226

Taking into account the O Ž Nc . corrections given above, we get the g 8 and g 27: 2

ž

g8 Ž O Ž Nc . q O Ž Nc . . s y 25 q

q

ž

1 Nc

/ ž c1 q

² qq :

ž

ž

g27 Ž O Ž Nc . q O Ž Nc . . s 1q

1 Nc

3 5

1 y Nc

² qq : M

Nc fp2 M

q y 35 q 2

3 5

y

1

2 5

c2 q

/ ž 2 5

1

3 5

1 Nc

c3 q c4

² qq : fp2 M

/

y

Nc² qq : M 8 fp4p 2

/

c6

c10 ,

Ž 72 .

c q 35 c 2q 109 c9q 109 c10 . .

Ž 73 .

Nc

/Ž

/ ž

/

c5 q

8 fp4p 2 c9 q

2 5

y Nc

3 5 1

Because of the sign in the 1rNc corrections in Eqs. Ž72., Ž73., we can easily see that the O Ž Nc . correction suppresses the octet coupling g 8 while enhancing the eikosiheptaplet coupling g 27 . It indicates that the O Ž Nc . corrections make the ratio of these two couplings even worse than that with only the leading contribution. As shown in Table 3, the ratio g 8rg 27 is completely underestimated. It is also interesting to consider the effect of the quark axial-vector coupling constant. To be more consistent in the large Nc expansion, we can take into account subleading order couplings in the large Nc in addition to the leading order coupling given by c Ug 5c . The simplest way of generalizing the x QM is to introduce the quark axial-vec-

Table 3 Low energy constants with O Ž Nc2 . and O Ž Nc . contributions. The Wilson coefficients are from Ref. w10x as shown in Table 1. M s 300 MeV and ² qq : sy2PŽ250 MeV. 3 are used Ž4.

Ž4.

LMS s 215 MeV Scheme g8 g27 g8 r g27 N1Ž8 . =10 3 N2Ž8 . =10 3 N3Ž8 . =10 3 N4Ž8 . =10 3 . N Ž8 =10 3 28

LO

NDR 0.794 0.581 1.368

0.773 0.642 1.204

Ž4.

LMS s 325 MeV HV

LO 0.739 0.618 1.196

NDR 0.892 0.548 1.628

0.902 0.615 1.467

LMS s 435 MeV HV

LO 0.845 0.585 1.445

NDR 1.009 0.516 1.955

1.117 0.589 1.896

HV 1.025 0.547 1.874

0.14 18.66 0 y6.96 6.96

0.35 15.91 0 y5.46 5.50

0.14 17.26 0 y6.12 6.12

0.20 20.46 0 y8.05 8.05

0.57 16.91 0 y6.18 6.27

0.22 19.08 0 y7.23 7.25

0.29 22.56 0 y9.28 9.30

0.98 17.98 0 y6.96 7.23

0.41 21.68 0 y8.72 8.81

N1Ž27 . =10 3 0 N2Ž27 . =10 3 y14.71 N3Ž27 . =10 3 14.71 N4Ž27 . =10 3 14.71 N5Ž27 . =10 3 0 N6Ž27 . =10 3 0 Ž27 . N 20 =10 3 0

0 y16.27 16.27 16.27 0 0 0

0 y15.66 15.66 15.66 0 0 0

0 y13.88 13.88 13.88 0 0 0

0 y15.59 15.59 15.59 0 0 0

0 y14.81 14.81 14.81 0 0 0

0 y13.07 13.07 13.07 0 0 0

0 y14.92 14.92 14.92 0 0 0

0 y13.86 13.86 13.86 0 0 0

. N Ž27 =10 3 y14.71 21

y16.27

y15.66

y13.88

y15.59

y14.81

y13.07

y14.92

y13.86

M. Franz et al.r Nuclear Physics B 562 (1999) 213–236

227

tor coupling g A different from unity w48,49x. In such a case the g A is known to be smaller than 1. The g A enters in the effective action given in Eq. Ž6.: Seff w p x s yNcTr ln Ž i Euq iMUg 5 q i e AUg 5EuUg 5 . ,

Ž 74 .

where e A s Ž1 y g A .r2. The understanding of this coupling depends on the specific dynamical assumptions. There are two different arguments about the large Nc behavior of the 1 y g A2 . For example, Weinberg argued that 1 y g A2 is of order O Ž1rNc . using the Adler–Weisberger sum rule w50,51x. Also Dicus et al. w52x considered it as 1rNc corrections. On the other hand, Broniowski et al. w53x demonstrated that from the Adler-Weisberger sum rule with the reggeized r meson exchange 1 y g A2 is of order O Ž Nc0 .. The new term i e AUg 5EuUg 5 being considerd, the operators Ž A1,2 .m can be rewritten as

Ž A1,2 . m s H

d4 k

`

1

Ý Ž E 2 y 2 ik E q M Ž EuUg . 5

4 2 2 Ž 2p . k q M

ns0

qe A Ž Ž EuUg 5 . Ž EuUg 5 . q Uyg 5 Ž E 2 Ug 5 . q 2Uyg 5 Ž En Ug 5 . En q2 Ž EuUg 5 . Uyg 5Euy 2 iUyg 5 Ž En Ug 5 . kn y 2 i Ž EuUg 5 . Uyg 5 ku . qe A2

g5

Ž EuU . Ž EuU . .

n

1

g5

k2qM 2

Ž i Euy i MUy g

5

q i e A Ž EuUg 5 . Uy g 5

qku . gm PR , L L1,2 .

Ž 75 .

Since the parameter eA is tiny, we can safely neglect the eA2 terms. Thus, we obtain the following results: ² Q1 q Q1† :OO ŽŽ epA2.. s yfp4 e A

ž

² qq :

q3

2 Mfp2

/

= Ž y 25 ² l6 Lm Lm : q 13 t i j; k l ² l i j Lm :² l k l Lm : . , ² Q2 q Q2† :OO ŽŽ epA2.. s yfp4 e A

ž

² qq :

q3

2 Mfp2

/Ž

3 5

² l6 Lm Lm : q 13 t i j; k l ² l i j Lm :² l k l Lm : . ,

Ž 77 .

² Q3 q Q3† :OO ŽŽ epA2.. s 0, ² Q4 q Q4† :OO ŽŽ epA2.. s yfp4 e A

Ž 78 .

ž

² qq : 2 Mfp2

/

q 3 ² l6 Lm Lm : ,

Ž 79 .

² Q5 q Q5† :OO ŽŽ epA2.. s 0, ² Q6 q Q6† :OO ŽŽ epA2.. s yfp4 e A

Ž 76 .

Ž 80 .

ž

4² qq : Mfp2

y

3 Nc² qq : M 4p 2 fp2

/

² Lm Lm l6 : ,

Ž 81 .

228

M. Franz et al.r Nuclear Physics B 562 (1999) 213–236

Fig. 3. Dependence of the g 8r g 27 on the quark axial-vector constant g A . The solid curve denotes the LO renormalization scheme in Ref. w10x, while the dashed curve and dot-dashed one stand for the NDR and the HV schemes, respectively. The value of the constituent quark mass M s 300 MeV is used and the quark condensate ² qq :r2 syŽ250 MeV. 3 is employed.

² Q7 q Q7† :OO ŽŽ epA2.. s ye A ² Q8 q Q8† :OO ŽŽ epA2.. s ye A

ž ž

3 fp2 ² qq : 4M 3 fp2 ² qq :

q 92 fp4 ² l6 U †Em U :² QˆEm UU † : ,

/

y

9Nc² qq : M 16p 2

M

Ž 82 .

/

ˆ : q ² l6 U †QU ˆ Ž Em U † . Ž Em U . : , = ² l6 Ž Em U † . Ž Em U . U † QU

ž

/

Ž 83 . ² Q9 q Q9† :OO ŽŽ epA2.. s yfp4 e A

ž

² qq : 2 Mfp2

q3

/Ž

y 35 ² l6 Lm Lm q 12 t i j; k l ² l i j Lm :² l k l Lm : . ,

Ž 84 . † :O Ž eA . 4 ² Q10 q Q10 O Ž p 2 . s yfp e A

ž

² qq : 2 Mfp2

q3

/Ž

2 5

² l6 Lm Lm q 12 t i j; k l ² l i j Lm :² l k l Lm : . .

Ž 85 .

M. Franz et al.r Nuclear Physics B 562 (1999) 213–236

229

The LECs can be then obtained as follows:

ž ž ž ž ž

g8g A s 1 y e A

² qq : 2 Mfp2

² qq :

q

fp2 M

g27 g A s 1ye A

y

q3

//Ž

Nc² qq : M 8 fp4p 2

² qq : 2 Mfp2

q3

//Ž

y 25 c1 q 35 c 2 q c 4 y 35 c9 q 25 c10 .

y eA

ž

4² qq : Mfp2

y

3 Nc² qq : M 4p 2 fp2

//

c6 ,

3 5 1

c q 35 c 2q 109 c9q 109 c10 . .

Ž 86 .

Fig. 3 shows the dependence of the g 8rg 27 on the quark axial-vector constant gA ranging from 0.75 to 1.25. The dependence on the gA is stronger again in the case of the NDR scheme. To get a reasonable value for the ratio one should choose a large value of gA which, however, deviates from the physical value gA , 0.75, as easily found from Eq. Ž86.. Thus, corrections from the O Ž Nc . and axial-vector coupling constants turn out to be quite useless if one wants to reproduce the empirical data. The effective D S s 2 weak chiral Lagrangian with the 1rNc and gA corrections is given as follows: 2

LeffD Ss2, O Ž p . s y

GF2 MW2 4p 2

= fp4 q

F Ž l c , l t ,m2c ,m2t , MW2 . b Ž m . fp4

Nc

y eA

ž

fp2 ² qq : 2M

q 3 fp4

/

² l6 Lm :² l6 L m : .

Ž 87 .

5. Next-to-leading order O ( p 4 ) Although the derivative expansion to order O Ž p 4 . is straightforward, reducing the number of terms is quite involved. We first can reduce the terms containing higher-order derivatives by using the following identities: Uq Ž Em En U . s y 12  Lm , Ln 4 y 14 i Wmn ,

Ž 88 .

Ž Em En Uq . U s y 21  Lm , Ln 4 q 41 i Wmn ,

Ž 89 .

Wmn s 2 Ž Em LnqEn Lm . .

Ž 90 .

where

We can then compare the reduced set of terms in the O Ž p 4 . order Lagrangian with that in Refs. w19,20x. To this end the number of terms can be reduced further by employing the equation of motion for the meson fields in the chiral limit Em Lm s 0 and the identities 1 8

²Wmn2 L: s ² Lm Ln Ln Lm L: y ² Lm Ln Lm Ln L: ,

Ž 91 .

M. Franz et al.r Nuclear Physics B 562 (1999) 213–236

230 1 4

i ² Lm L:² Wmn , Ln L: s ² Lm Ln L:² Ln Lm L: y ² Lm Ln L:² Lm Ln L: q ² Lm L:² Ln Lm Ln L: y 12 ² Lm L:²  Lm , Ln Ln 4 L: ,

1 2

Ž 92 .

i ² Lm L:²Wmn Ln : s ² Lm Ln L:² Ln Lm L: y ² Lm Ln L:² Lm Ln L: q ² Lm L:² Ln Lm Ln L: y ² Lm L:² Ln Ln Lm L: ,

Ž 93 .

where L denote arbitrary flavor matrices. The identities Ž91. – Ž93. can be easily obtained by integration by parts and some trace identities from the Cayley-Hamilton theorem. Decomposing the octet and eikosiheptaplet contributions, we end up with the following results for the vacuum expectation values at O Ž p 4 . order and leading order in the large Nc expansion: ² Q1 q Q1† :O Ž p 4 . s

Nc fp2

4 ² L L L L l : y 35 ² Lm l6 :² Lm Ln Ln : 24p 2 5 m n n m 6 y 35 ´abgd ² La l6 :² Lb Lg Ld : y 23 t i jk l Ž ² Lm Ln l i j :² Ln Lm l k l :

y² Lm Ln l i j :² Lm Ln l k l : q ² Lm l i j :² Ln Lm Ln l k l : q´abgd ² La l i j :² Lb Lg Ld l k l : . , ² Q2 q Q2† :O Ž p 4 . s

Ž 94 .

Nc fp2

y 65 ² Lm Ln Ln Lm l6 : q 25 ² Lm l6 :² Lm Ln Ln : 24p 2 q 25 ´abgd ² La l6 :² Lb Lg Ld : y 23 t i jk l Ž ² Lm Ln l i j :² Ln Lm l k l : y² Lm Ln l i j :² Lm Ln l k l : q ² Lm l i j :² Ln Lm Ln l k l : q´abgd ² La l i j :² Lb Lg Ld l k l : . ,

² Q3 q Q3† :O Ž p 4 . s y ² Q4 q Q4† :O Ž p 4 . s y

Nc fp2 24p 2 Nc fp2

Ž 95 .

² Lm l6 :² Lm Ln Ln : q ´a bgd ² La l6 :² Lb Lg Ld : , Ž 96 .

² L L L L l :, 12p 2 m n n m 6 Nc fp2 ² Q5 q Q5† :O Ž p 4 . s ² Lm l6 :² Lm Ln Ln : y ´a bgd ² La l6 :² Lb Lg Ld : , 24p 2 Nc2 M 2 Nc fp2 fp4 ² Q6 q Q6† :O Ž p 4 . s ² Lm Lm Ln Ln l6 : , y q 128p 4 8p 2 2M2

ž

/

Ž 97 . Ž 98 . Ž 99 .

3 Nc fp2

2² l6 Em U †En U :² QˆEn UEm U † : 48p 2 y2² l6 Em U †En U :² QˆEm UEn U † :

² Q7 q Q7† :O Ž p 4 . s y

q² l6 U †Em U :² QˆEn UEm U †En UU † : q² l6 U †En UEm U †En U :² QˆEm UU † : y´abgd ² l6 U †Ea U :² QˆEb UEg U †Ed UU † :

ž

q² l6 U †Ea UEb U †Eg U :² QˆEd UU † : ,

/

Ž 100.

M. Franz et al.r Nuclear Physics B 562 (1999) 213–236

2

² Q8 q Q8† :OO ŽŽ Np c4 .. s

ž

3 Nc2 M 2 256p 4

q

y

3 Nc² qq : 192 Mp 2

3 Nc fp2 16p 2

q

3 fp4 4M 2

/

231

ˆ En U †En U : ² l6 Em U †Em UU † QU

ˆ : ² l6 Em U †En UEm U †En UU † QU

ˆ Em U †En UEm U †En U : y ² l6 Em U †En UEn U †Em UU †QU ˆ : q² l6 U † QU ˆ Em U †En UEn U †Em U : q 12 i ² l6  Em U †En U,Wmn 4 U †QU ˆ : y² l6 U † QU ˆ  Em U †En U,Wmn 4 : y 12 i ² l6 U † QU ˆ : q 18 ² l6 U †QUW ˆ mn2 : q 18 ² l6Wmn2 U † QU ˆ : y2 ´abgd ² l6 Ea U †Eb UEg U †Ed UU † QU

ž

ˆ Ea U †Eb UEg U †Ed U : , y² l6 U † QU

/

² Q9 q Q9† :O Ž p 4 . s

Nc fp2 24p 2

6 5

Ž 101.

² Lm Ln Ln Lm l6 : y 25 ² Lm l6 :² Lm Ln Ln :

y 25 ´abgd ² La l6 :² Lb Lg Ld : y t i jk l Ž ² Lm Ln l i j :² Ln Lm l k l : y² Lm Ln l i j :² Lm Ln l k l : q ² Lm l i j :² Ln Lm Ln l k l : q´abgd ² La l i j :² Lb Lg Ld l k l : . , † : ² Q10 q Q10 OŽ p4. s

Nc fp2 24p 2

Ž 102.

y 45 ² Lm Ln Ln Lm l6 : q 35 ² Lm l6 :² Lm Ln Ln :

q 35 ´abgd ² La l6 :² Lb Lg Ld : y t i jk l Ž ² Lm Ln l i j :² Ln Lm l k l : y² Lm Ln l i j :² Lm Ln l k l : q ² Lm l i j :² Ln Lm Ln l k l : q´abgd ² La l i j :² Lb Lg Ld l k l : . .

Ž 103.

Comparing Eqs. Ž94. – Ž103. with the effective D S s 1 weak chiral Lagrangian to order O Ž p 4 . in x PT given by Ecker et al. w22x and Esposito-Farese ` w21x Žpresented in Minkowski space.: 4

LeffD Ss1, O Ž p . s y

GF

'2

VudVus) fp2

Ž N1Ž8 .² l6 Lm L m Ln Ln : q N2Ž8 . P ² l6 Lm Ln Ln L m :

qN3Ž8.² l6 Lm Ln :² L m Ln : q N4Ž8 .² l6 Lm :² L m Ln Ln : qN28Ž 8. i emnrd ² l6 L m :² Ln L r Ld :

/

q 59 t i jk l Ž N1Ž27 . ² l i j Lm L m :² l k l Ln Ln : qN2Ž27.² l i j Lm Ln :² l k l L m Ln : q N3Ž27.² l i j Lm Ln :² l k l Ln L m :

M. Franz et al.r Nuclear Physics B 562 (1999) 213–236

232

qN4Ž27.² l i j Lm :² l k l Ln L m Ln : q N5Ž27.² l i j Lm :² l k l  L m , Ln Ln 4 : qN6Ž27.² Lm L m :² l i j Ln :² l k l Ln : qN20Ž 27. i emnrd ² l i j L m Ln :² l k l L r Ld : qN21Ž 27. i emnrd ² l i j L m :² l k l Ln L r Ld : ,

Ž 104.

/

we derive the LECs in the case of the leading order in the large Nc expansion:

ž

N1Ž8 . s y

N2Ž8 . s

Nc2 M 2

q 2

128p 4 fp

Nc 60p 2

Nc 8p

y 2

fp2 2M2

/

c6 ,

Ž 105.

Ž y2 c1q3c2q5c4y3c9q2 c10 . ,

Ž 106.

N3Ž8. s 0, N4Ž8. s

Ž 107. Nc

60p 2

N28Ž 8. s

Ž 32 c1yc2q 52 c3y 52 c5qc9y 32 c10 . ,

Nc 60p 2

Ž 108.

Ž y 32 c1qc2y 52 c3y 52 c5yc9q 32 c10 . ,

Ž 109.

Ž 27 . N1Ž27. s N5Ž27. s N6Ž27. s N20 s 0,

Ž 110.

Ž 27 . N2Ž27. s yN3Ž27. s yN4Ž27. s N21 s

Nc 60p 2

Ž y3c1 y 3c2 y 92 c9 y 92 c10 . .

Ž 111.

™

As noted by Ecker et al. w22x, the LECs N1Ž 8., N2Ž 8., N3Ž 8. and N4Ž 8. contribute to the Ž 8. process K 3p while N28Ž 8. does to the radiative K-decays. In particular, the N28 is related to the chiral anomaly w59x. The numerical results can be found in Table 2. Note that the LECs in the eikosiheptaplet can assume only two values. Taking into account the 1rNc corrections, the LECs are extracted as follows:

ž

N1Ž8 . s y

Nc2 M 2

q 2

128p 4 fp

Nc 8p

y 2

fp2 2M2

/ ž

c6 q y

Nc M 2

1 q 2

128p 4 fp

8p

y 2

fp2 2 Nc M 2

/

c5 ,

Ž 112. N2Ž8 . s

Nc 60p

ž

2

žž

1

Nc

1

q y3 q N3Ž8. s 0,

Nc

1

/ ž / / ž / /

y2 q

3 c1 q 3 y

2 c9 q 2 y

1

Nc

Nc

2 c2 q

3 c10 ,

1 Nc

5c 3 q 5c 4

Ž 113. Ž 114.

M. Franz et al.r Nuclear Physics B 562 (1999) 213–236

N4Ž8. s

Nc 60p 1

y Nc N28Ž 8. s

žž

2

1

c1 q y1 q

y

Nc

1

5 2 6

c q 1y

Nc

y 32 q

2

60p 1

y Nc

1

/ ž / ž / ž / / žž / ž / ž / ž / / 3 2

Nc 1

Nc

5 2 6

c q y1 q

3 2

3 2

Nc

c 2 q 52 c 3 q

c9 q y 32 q

c1 q 1 y 1

Nc

3 2

c9 q

1

Nc 3 2

1

Nc

3 2

1 Nc

1

Nc

c 4 y 52 c5

c10 ,

c 2 y 52 c 3 y

y

5 2

233

Ž 115. 1 Nc

5 2

c 4 y 52 c5

c10 ,

Ž 27 . N1Ž27. s N5Ž27. s N6Ž27. s N20 s 0,

Ž 116. Ž 117.

Ž 27 . N2Ž27. s yN3Ž27.s y N4Ž27.s N21

s

Nc 60p

2

ž

1q

1 Nc

/Ž

y3c1 y 3c 2 y 92 c9 y 92 c10 . .

Ž 118.

Table 3 shows the effect of the O Ž Nc . corrections. The effective D S s 2 weak chiral Lagrangian is obtained as LeffDSs2, O Ž p sy

4

.

GF2 MW2 4p 2

F Ž l c , l t ,m 2c ,m2t , MW2 . b Ž m .

Nc fp2 12p 2

Ž ² l6 Lm Ln :² l6 Ln L m :

y² l6 Lm Ln :² l6 L m Ln : q ² l6 Lm :² l6 Ln L m Ln : yi emn a b ² l6 L m :² l6 Ln La L b : . .

Ž 119.

The above formulae present the final result for the effective weak chiral Lagrangian for D S s 1 and D S s 2. It is constructed in a way to be used in x PT without further treatment, since all LECs are explicitely given. The result is a strict outcome of the x QM and the expansion of its Lagrangian in powers of the momentum and in the number of colors. It is assumed, however, that no external fields are present.

6. Summary and conclusions The aim of the present work has been to derive the effective D S s 1,2 effective weak chiral Lagrangian with its low energy constants from the chiral quark model using the weak effective action of Buchalla, Buras and Lautenbacher w10x. In leading order in the large Nc expansion the Lagrangian is already known. The final result of our investigations is the effective weak chiral Lagrangian in next-to-leading order in Nc and to fourth order in the momentum. As is already known, the contribution of leading order in the large Nc expansion to the ratio g 8rg 27 is heavily underestimated in this model. As we show in the present paper the inclusion of the next-to-leading order in the large Nc expansion does not help

234

M. Franz et al.r Nuclear Physics B 562 (1999) 213–236

to improve this result. Hence on the present level of formalism and without further improvements the chiral quark model does not provide low energy constants which can directly be used in chiral perturbation theory for weak processes. Of course such a conclusion can finally only be drawn if a few actual observables have been calculated in chiral perturbation theory by using the above effective weak chiral Lagrangian. However, since the chiral quark model fails heavily in reproducing the ratio g 8rg 27 we do not have much hope that this will work. Actually Antonelli et al. w28x have added to the lowest order in Nc certain corrections from the gluon condensate known as of order O Ž a s Nc . w54x in order to change the ratio g 8rg 27. They have shown that the O Ž a s Nc . corrections indeed improve numerically the DT s 1r2 enhancement. However, in our view there is an important caveat. In fact, the O Ž a s Nc . 2 is of the same order as O Ž a s Nc .. The latter corrections were neglected by the authors of Ref. w54x hoping that they might be smaller since they involve condensates of higher dimensions. In view of the large size of those corrections such an argument requires further substantiation, even though the numerical results are improved. In this paper no attempt was done to obtain gluonic corrections to the next-to-leading order in Nc . At the present level of investigation we see the following ways of investigations, which might improve the low energy constants of the chiral quark model. First, the chiral quark model has been derived from QCD by Diakonov and Petrov w55–58x by assuming a gluonic vacuum configuration which consists of a dilute gas of interacting instantons and anti–instantons. As a result of this approach the constituent quark mass in the chiral quark model is momentum-dependent and it is only an approximation to replace this by a regularization prescription with a properly chosen cut–off parameter. Thus it is interesting to investigate how far the present results change if such a momentum-dependent constituent mass is used. Such an investigation is even necessary if one wants to exploit fully the chiral quark model. Second, all the results in the present paper are based on the assumption that the effective weak Hamiltonian of Buchalla, Buras and Lautenbacher can be used in connection with the chiral quark model. This, however, is not that clear. If one considers the derivation of the chiral quark model from QCD by Diakonov and Petrov the renormalization point of the model is around 600 MeV corresponding to the average size of the instantons of 0.3 fm and the average distance of 1 fm. The Wilson coefficients of the effective weak Hamiltonian are evaluated at a scale of 1 GeV and it is not obvious if they can be used without further change at 600 MeV. Suggestions for investigations in this direction have recently been given in Refs. w29,60,61x. Actually in our next investigations we will follow the first suggestion and will incorporate the momentum-dependent quark mass in the chiral quark model. Such a procedure links the Lagrangian of the chiral quark model to QCD and, perhaps, the results will be improved.

Acknowledgements The authors thank M.V. Polyakov for valuable discussions and comments on the present work. The work is supported in part by COSY, DFG, and BMBF. H.C.K. wishes

M. Franz et al.r Nuclear Physics B 562 (1999) 213–236

235

to acknowledge the financial support of the Korea Research Foundation made in the program year of 1998. K.G. thanks H. Toki and the RCNP ŽOsaka. for hospitality.

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