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A model-independent study of the QCD sum rule for the pion-nucleon coupling constant and a projected correlation function approach
ELSEVIER
Nuclear Physics A721 (2003) 597c-6OOc www.elsevier.com/locate/npe
A model-independent study of the QCD sum rule for the pion-nucleon coupling constant and a projected correlation function approach Yoshihiko “Kokugakuin
Kondo” and Osamu Morimatsub University,
Higashi, Shibuya, Tokyo 150-8440, Japan
bInstitute of Particle and Nuclear Studies, High Energy Accelerator Organization, Tukuba, Ibaragi 305-0801, Japan
Research
The QCD sum rule for the pion-nucleon coupling constant, g, is investigated. The physical content of the correlation function is studied without referring to the effective theory. First, we consider the invariant correlation functions by splitting the correlation function into different Dirac structures. We show that the coefficients of the double-pole terms are proportional to g but that the coefficients of the single-pole terms are not determined by g. In the chiral limit the single-pole terms as well as the continuum terms are not well defined in the dispersion integral. Therefore, naive QCD sum rules obtained from the invariant correlation functions are also not well defined. Secondly, we consider the projected correlation function by taking the matrix element of the correlation function with respect to the nucleon spinors. In the chiral limit the double-pole term survives but the single-pole term and the continuum term are well defined in the dispersion integral. Then, we construct a QCD sum rule for g from the projected correlation function without taking the chiral limit. The new sum rule is expected to incorporate O(m,) corrections. The numerical result is in reasonable agreement with the empirical value. There has already been a history in the study of the pion-nucleon coupling constant within the framework of the QCD sum rule [2-71. In those works, the pion-to-vacuum matrix element of the correlation function of two nucleon interpolating fields, v and @, II(p,k)
= 4 / d4zeip”(OlT[q(z)fj(0)]17r(k)),
(1)
was considered and the physical content of the correlation function was investigated by effective theories [2,4-71. They constructed the sum rules from the invariant correlation functions by splitting the correlation function into four Dirac structures:
where M is the nucleon mass. Kim, Lee and Oka studied the dependence correlation functions on the coupling scheme of the effective lagrangians, scheme and the psedovector scheme [6]. Table 1 sh ows the coefficients of single pole terms of the correlation function in the case of the psedoscalar psedovector scheme. IIT4 seems to cont,ain the double pole term only and 0375-9474/03/$ - see front matter 0 2003 Elsevier Science B.V. All rights reserved. doi:lO.l016/SO375-9474(03)01131-X
of the invariant the psedoscalar the double and scheme and the be independent
Y Kondo, 0. Morimatsu/Nuclear
598c Table 1 The coefficients K(P2,Pk)
of the double
and single pole terms of the correlation
= (p2-&f2)[(;-k)1-M2] + (p-t&-M? pseudoscalar scheme 1 I I ‘4 ua
ni
I
1
II,
,I
-ak2/2
n2
-0’
n3
d-f2
n4
gi\/f
1
al2
1
“b
functions
as
+ p&T. II
P ”
A
II
al2
11
pseudovector scheme I I uR I
-ai -0'
-b
0 0
”
Physics A721 (2003) 597~~600~
0 0
I
0
1
0 -g/2
g/2
SM2
0 0
g/2
0
SM2
”P
of the coupling scheme. In the following, we will show that the physical structure of the correlation function can be identified without referring to the effective lagrangian (8,9]. In order to study the physical structure by general principle, we define the vertex function and the invariant vertex functions as I(p, k) = (f - M)II(p, k)($ - $ - M) and r(p, k) = 65ri + 4 r2lM + %F rs/M + a‘7 5iuCLvp’lkYI’4/M2, respectively. The invariant correlation functions are related to the invariant vertex functions as
II,=2 cri- - + r2
+-i n
=
2
2r3
(2 - $$) r4) + -5 (r, + r2) _ &(p”
(P” - M2)((p - k)2 - M2) crl - rz) - &d(P - k)2 - M2 - 24 p2 - M2
r3
-
_ M2 _ 2&)
(P - k)2 - M2 1 r
+
2M2
4’
r2 + r3 - r4
r4
m-
(p-k)2-M2’
n = M2(-rl + r2 + 2r33
(p” - M2)((p
M2 (-rl rI4=
(2 - &)
- k)2 - M2)
+ r2 +2r3 - (2 - &)
(p” - M2)((p The pion-nucleon
Then the coupling
+
(p
-
k)2
-
(p
-
;)2q_
_
p2
M2
ir
r,)
- k)2 - M2)
coupling constant
rz + r3 _ ;r,
r,)
V4 M2
is defined through
constant is related to the invariant
$4 - M2’
-
+’
the coefficient of the pole as
vertex functions
as
g = -rl + r2 + 2r3 - (2 - m~/2M2)r4),2=Mz,(,-k)2=Mz. where m, is the pion mass. From Eq. (2) and Eq. (4), we find that the coefficients of the double poles are proportional to g, but that the coefficients of the single poles are not determined by g. Furthermore, II4 also contains the single pole terms. Since the chiral limit is taken for simplicity in the previous works, we next study the invariant function in the chiral limit. In the chiral limit with vanishing pion momentum, the invariant functions can be regarded as functions of p2 and the absorptive parts of the invariant correlation functions are given by ImII,
=
d(p2 - M ”)I’,(M”)
- &InCl(p2),
I! Kondo, 0. Morimatsu/Nuclear
Physics A721 (2003) 597c-6OOc
ImIIs
=
7r’s($ - M2)12(M2) - --&fpImr2(p2),
ImIIs
=
n’6/(p2 - M2)M2g - nd(p2 - M2)Re!Z>(M2) + cp2 ~;2j21mrA(p2)7
ImII,
=
-X’S’@ - M2)M2g + &(p2 - M2)ReI’i(M2)
599c
- (p2 JfM2)21ml’B(p2),
where rA = hf2[-rl $ p2 r2/b!f2 $ (1 -t p2/kf2) r3 - 2~’ r4/bf2], rB = M2[-rl+r2 + 2IJs - (1 + p2/M2) Id]. The absorpt ive parts of IIs and II4 contain the derivative of the vertex function as well as the continuum. In order to study the physical structure of the derivative term and the continuum term we define the T-matrix as T(q’, k’, q, k) = (4 - M)[4
1 d%eiq’z ~~(~‘)l~~~(~)d(O)ll~~~~~l~~
- w.
The absorptive part of the vertex function can be written as
Imr(P, q, I;) = -g
/
---UP, ($3
q’, fv(el’ + wT*(q’,
k’, 9, h).
(5)
From Eq. (5), we see that the vertex function behaves in the vicinity of the threshold as Imrb.4 4%k) = RertP, 4, J4 =
0 Glt + o(t3)
(PO < M -t %r) (PO > M t m,) ’
(6)
Go t GIT + O(T~) Go t O(t2)
where t = & &$ - (N t m,)2]bg - (M - m,)2] and r = it. In Eq. (6) the behavior of Rel? is determined from that of ImF by analytic continuation. From Eq, (6) one sees that
00 &Rer(~, 4,b)PQ=4f+ma-r = finite z
constant
m, 3 :mT ’
0.
pQ=Mt%t~
We find that the derivative of the vertex function at the threshold is ill defined. We also see that the absorptive part in the vicinity of the threshold is proportional to p2 - M2 in the chiral limit, and the dispersion integral of the continuum term is divergent. When the absorptive part of the invariant correlation function is split into two parts as
each part is ill defined. A way to avoid this difficulty is to construct a sum rule for (pa-M2)ImII(pa), which d oes not include ill-defined term. However, the sum rule obtained by this procedure does not seem to be numerically stable. In this paper, we will propose a different way to construct a sum rule, The basic object of the new approach is the projected correlation function, n&r,
98, k) = q~~)Yo~(P, ~)w(QTs).
(7)
I: Kondo, 0. Morimatsu/Nuclear
600~
Physics A721 (2003) 597c-600~
In order to study the physical structure, we define the projected vertex function as I+@, qs, Ic) = ~(@)l‘(p, q, Ic)u(q’s). The projected vertex function is related to II+ as r+(P’,
qs, k)
=
(PO - M)(Po
where Ek = da
- EJC)U+(PT, qs, k),
+ da.
(8)
From Eq. (8) we obtain
The coupling constant is defined in Eq. (3), while g(p0, i2) in Eq. (9) refers to different kinematical points. However, the difference of g and g(M,O) is O(mz). From this, we expect the sum rule to incorporate O(m$) corrections. Finally, we study the projected function in the chiral limit. The absorptive part of the projected vertex function can be written as
From this we find that the projected Imr+(Pr, ReL
qs, k) =
(PT qs, k) =
vertex function
is expanded as
0
(PO< M+m,) (PO > M+m,)’ o(t") G+ t O(T~) (PO < M + m,) Gt + O(P)
(Po>M+m,)’
In the chiral limit, III, contains no ill-defined term since &Rer+(p,q, IC)lr,,=~+~,f~ = 0. Therefore, we construct the QCD sum rule from the projected correlation function. In practice, we derive the Bore1 sum rule from the dispersion relation of the projected correlation function. Then, we calculate the Wilson coefficients of the OPE and obtain the Bore1 sum rule up to 0 (MB-“) and 0 (m,) where MB is the Bore1 mass. The calculated result is g = 10 f 3, which is in reasonable agreement with the empirical value, gemp, = 13.4. Furthermore, the O(m,) corrections are found to be about 5%.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B14’7(1979)385; Nucl. Phys. B147 (1979) 448. L. J. Reinders, H. R. Rubinstein and S. Yazaki, Phys. Rep. 12’7 (1985) 1. L. J. Reinders, H. R. Rubinstein and S. Yazaki, Nucl. Phys. B213(1983) 109. H. Shiomi and T. Hatsuda, Nucl. Phys. A594 (1995) 294. M. C. Birse and B. Krippa, Phys. Lett. B373 (1996) 9; Phys. Rev. C54 (1996) 3240. H. Kim, S.H. Lee and M. Oka, Phys. Lett. B453 (1999) 199; Phys. Rev. D60 (1999) 034007. T. Doi, H. Kim and M. Oka, Phys. Rev. C62 (2000) 055202. Y. Kondo and 0. Morimatsu, Phys. Rev. C66 (2002) 028201. Y. Kondo and 0. Morimatsu, Prog. Theor. Phys. 100 (1998) 1.
Nuclear Physics A721 (2003) 597c-6OOc www.elsevier.com/locate/npe
A model-independent study of the QCD sum rule for the pion-nucleon coupling constant and a projected correlation function approach Yoshihiko “Kokugakuin
Kondo” and Osamu Morimatsub University,
Higashi, Shibuya, Tokyo 150-8440, Japan
bInstitute of Particle and Nuclear Studies, High Energy Accelerator Organization, Tukuba, Ibaragi 305-0801, Japan
Research
The QCD sum rule for the pion-nucleon coupling constant, g, is investigated. The physical content of the correlation function is studied without referring to the effective theory. First, we consider the invariant correlation functions by splitting the correlation function into different Dirac structures. We show that the coefficients of the double-pole terms are proportional to g but that the coefficients of the single-pole terms are not determined by g. In the chiral limit the single-pole terms as well as the continuum terms are not well defined in the dispersion integral. Therefore, naive QCD sum rules obtained from the invariant correlation functions are also not well defined. Secondly, we consider the projected correlation function by taking the matrix element of the correlation function with respect to the nucleon spinors. In the chiral limit the double-pole term survives but the single-pole term and the continuum term are well defined in the dispersion integral. Then, we construct a QCD sum rule for g from the projected correlation function without taking the chiral limit. The new sum rule is expected to incorporate O(m,) corrections. The numerical result is in reasonable agreement with the empirical value. There has already been a history in the study of the pion-nucleon coupling constant within the framework of the QCD sum rule [2-71. In those works, the pion-to-vacuum matrix element of the correlation function of two nucleon interpolating fields, v and @, II(p,k)
= 4 / d4zeip”(OlT[q(z)fj(0)]17r(k)),
(1)
was considered and the physical content of the correlation function was investigated by effective theories [2,4-71. They constructed the sum rules from the invariant correlation functions by splitting the correlation function into four Dirac structures:
where M is the nucleon mass. Kim, Lee and Oka studied the dependence correlation functions on the coupling scheme of the effective lagrangians, scheme and the psedovector scheme [6]. Table 1 sh ows the coefficients of single pole terms of the correlation function in the case of the psedoscalar psedovector scheme. IIT4 seems to cont,ain the double pole term only and 0375-9474/03/$ - see front matter 0 2003 Elsevier Science B.V. All rights reserved. doi:lO.l016/SO375-9474(03)01131-X
of the invariant the psedoscalar the double and scheme and the be independent
Y Kondo, 0. Morimatsu/Nuclear
598c Table 1 The coefficients K(P2,Pk)
of the double
and single pole terms of the correlation
= (p2-&f2)[(;-k)1-M2] + (p-t&-M? pseudoscalar scheme 1 I I ‘4 ua
ni
I
1
II,
,I
-ak2/2
n2
-0’
n3
d-f2
n4
gi\/f
1
al2
1
“b
functions
as
+ p&T. II
P ”
A
II
al2
11
pseudovector scheme I I uR I
-ai -0'
-b
0 0
”
Physics A721 (2003) 597~~600~
0 0
I
0
1
0 -g/2
g/2
SM2
0 0
g/2
0
SM2
”P
of the coupling scheme. In the following, we will show that the physical structure of the correlation function can be identified without referring to the effective lagrangian (8,9]. In order to study the physical structure by general principle, we define the vertex function and the invariant vertex functions as I(p, k) = (f - M)II(p, k)($ - $ - M) and r(p, k) = 65ri + 4 r2lM + %F rs/M + a‘7 5iuCLvp’lkYI’4/M2, respectively. The invariant correlation functions are related to the invariant vertex functions as
II,=2 cri- - + r2
+-i n
=
2
2r3
(2 - $$) r4) + -5 (r, + r2) _ &(p”
(P” - M2)((p - k)2 - M2) crl - rz) - &d(P - k)2 - M2 - 24 p2 - M2
r3
-
_ M2 _ 2&)
(P - k)2 - M2 1 r
+
2M2
4’
r2 + r3 - r4
r4
m-
(p-k)2-M2’
n = M2(-rl + r2 + 2r33
(p” - M2)((p
M2 (-rl rI4=
(2 - &)
- k)2 - M2)
+ r2 +2r3 - (2 - &)
(p” - M2)((p The pion-nucleon
Then the coupling
+
(p
-
k)2
-
(p
-
;)2q_
_
p2
M2
ir
r,)
- k)2 - M2)
coupling constant
rz + r3 _ ;r,
r,)
V4 M2
is defined through
constant is related to the invariant
$4 - M2’
-
+’
the coefficient of the pole as
vertex functions
as
g = -rl + r2 + 2r3 - (2 - m~/2M2)r4),2=Mz,(,-k)2=Mz. where m, is the pion mass. From Eq. (2) and Eq. (4), we find that the coefficients of the double poles are proportional to g, but that the coefficients of the single poles are not determined by g. Furthermore, II4 also contains the single pole terms. Since the chiral limit is taken for simplicity in the previous works, we next study the invariant function in the chiral limit. In the chiral limit with vanishing pion momentum, the invariant functions can be regarded as functions of p2 and the absorptive parts of the invariant correlation functions are given by ImII,
=
d(p2 - M ”)I’,(M”)
- &InCl(p2),
I! Kondo, 0. Morimatsu/Nuclear
Physics A721 (2003) 597c-6OOc
ImIIs
=
7r’s($ - M2)12(M2) - --&fpImr2(p2),
ImIIs
=
n’6/(p2 - M2)M2g - nd(p2 - M2)Re!Z>(M2) + cp2 ~;2j21mrA(p2)7
ImII,
=
-X’S’@ - M2)M2g + &(p2 - M2)ReI’i(M2)
599c
- (p2 JfM2)21ml’B(p2),
where rA = hf2[-rl $ p2 r2/b!f2 $ (1 -t p2/kf2) r3 - 2~’ r4/bf2], rB = M2[-rl+r2 + 2IJs - (1 + p2/M2) Id]. The absorpt ive parts of IIs and II4 contain the derivative of the vertex function as well as the continuum. In order to study the physical structure of the derivative term and the continuum term we define the T-matrix as T(q’, k’, q, k) = (4 - M)[4
1 d%eiq’z ~~(~‘)l~~~(~)d(O)ll~~~~~l~~
- w.
The absorptive part of the vertex function can be written as
Imr(P, q, I;) = -g
/
---UP, ($3
q’, fv(el’ + wT*(q’,
k’, 9, h).
(5)
From Eq. (5), we see that the vertex function behaves in the vicinity of the threshold as Imrb.4 4%k) = RertP, 4, J4 =
0 Glt + o(t3)
(PO < M -t %r) (PO > M t m,) ’
(6)
Go t GIT + O(T~) Go t O(t2)
where t = & &$ - (N t m,)2]bg - (M - m,)2] and r = it. In Eq. (6) the behavior of Rel? is determined from that of ImF by analytic continuation. From Eq, (6) one sees that
00 &Rer(~, 4,b)PQ=4f+ma-r = finite z
constant
m, 3 :mT ’
0.
pQ=Mt%t~
We find that the derivative of the vertex function at the threshold is ill defined. We also see that the absorptive part in the vicinity of the threshold is proportional to p2 - M2 in the chiral limit, and the dispersion integral of the continuum term is divergent. When the absorptive part of the invariant correlation function is split into two parts as
each part is ill defined. A way to avoid this difficulty is to construct a sum rule for (pa-M2)ImII(pa), which d oes not include ill-defined term. However, the sum rule obtained by this procedure does not seem to be numerically stable. In this paper, we will propose a different way to construct a sum rule, The basic object of the new approach is the projected correlation function, n&r,
98, k) = q~~)Yo~(P, ~)w(QTs).
(7)
I: Kondo, 0. Morimatsu/Nuclear
600~
Physics A721 (2003) 597c-600~
In order to study the physical structure, we define the projected vertex function as I+@, qs, Ic) = ~(@)l‘(p, q, Ic)u(q’s). The projected vertex function is related to II+ as r+(P’,
qs, k)
=
(PO - M)(Po
where Ek = da
- EJC)U+(PT, qs, k),
+ da.
(8)
From Eq. (8) we obtain
The coupling constant is defined in Eq. (3), while g(p0, i2) in Eq. (9) refers to different kinematical points. However, the difference of g and g(M,O) is O(mz). From this, we expect the sum rule to incorporate O(m$) corrections. Finally, we study the projected function in the chiral limit. The absorptive part of the projected vertex function can be written as
From this we find that the projected Imr+(Pr, ReL
qs, k) =
(PT qs, k) =
vertex function
is expanded as
0
(PO< M+m,) (PO > M+m,)’ o(t") G+ t O(T~) (PO < M + m,) Gt + O(P)
(Po>M+m,)’
In the chiral limit, III, contains no ill-defined term since &Rer+(p,q, IC)lr,,=~+~,f~ = 0. Therefore, we construct the QCD sum rule from the projected correlation function. In practice, we derive the Bore1 sum rule from the dispersion relation of the projected correlation function. Then, we calculate the Wilson coefficients of the OPE and obtain the Bore1 sum rule up to 0 (MB-“) and 0 (m,) where MB is the Bore1 mass. The calculated result is g = 10 f 3, which is in reasonable agreement with the empirical value, gemp, = 13.4. Furthermore, the O(m,) corrections are found to be about 5%.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B14’7(1979)385; Nucl. Phys. B147 (1979) 448. L. J. Reinders, H. R. Rubinstein and S. Yazaki, Phys. Rep. 12’7 (1985) 1. L. J. Reinders, H. R. Rubinstein and S. Yazaki, Nucl. Phys. B213(1983) 109. H. Shiomi and T. Hatsuda, Nucl. Phys. A594 (1995) 294. M. C. Birse and B. Krippa, Phys. Lett. B373 (1996) 9; Phys. Rev. C54 (1996) 3240. H. Kim, S.H. Lee and M. Oka, Phys. Lett. B453 (1999) 199; Phys. Rev. D60 (1999) 034007. T. Doi, H. Kim and M. Oka, Phys. Rev. C62 (2000) 055202. Y. Kondo and 0. Morimatsu, Phys. Rev. C66 (2002) 028201. Y. Kondo and 0. Morimatsu, Prog. Theor. Phys. 100 (1998) 1.