- Email: [email protected]
New QCD sum rules based on canonical commutation relations
Progress in Particle and Nuclear Physics 67 (2012) 136–139
Contents lists available at SciVerse ScienceDirect
Progress in Particle and Nuclear Physics journal homepage: www.elsevier.com/locate/ppnp
Review
New QCD sum rules based on canonical commutation relations Tomoya Hayata Department of Physics, The University of Tokyo, Tokyo 113-0031, Japan
article
abstract
info
Keywords: QCD sum rules Caonical commutation relations Kugo–Ojima operator formalism and Weinberg’s sum rules
New derivation of QCD sum rules by canonical commutators is developed. It is the simple and straightforward generalization of Thomas–Reiche–Kuhn sum rule on the basis of Kugo–Ojima operator formalism of a non-abelian gauge theory and a suitable subtraction of UV divergences. By applying the method to the vector and axial vector current in QCD, the exact Weinberg’s sum rules are examined. Vector current sum rules and new fractional power sum rules are also discussed. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The understanding of the physics of spontaneous symmetry breaking (SSB) is one of the most important goal in a quantum many particle system and a system described by a quantum field theory. There are many theoretical calculations to figure out the characteristic behavior of SSB e.g., Ginzburg–Landau type’s method based on symmetry and its breaking, mean-field type’s self-consistent approach to determine the nonperturbative vacuum, and effective field theories like chiral perturbation theory to reproduce the low energy physics of the system. In quantum many-body systems, sum rule is known to be a powerful tool to study the spectral structure. Here is the well-known example of the dipole sum rule in nuclear physics,
σtot = = =
4π 2 e2 h¯ c 4π 2 e2 h¯ c
ν
(Eν − E0 )|⟨ν|D|0⟩|2
⟨0|[D, [H , D]]|0⟩
2π 2 e2 h¯ NZ mc
A
(1 + K ),
(1)
where σtot is a dipole induced photo-nuclear total cross section, and D is the dipole operator. Such a cross section can be written as an energy weighted sum of the dipole transition probability. It can be also written as a double commutator between the Hamiltonian and D. Furthermore, in this particular example, the commutator can be evaluated exactly. The result is written only by fundamental constants and does not depend on the details of the dynamics. Now what about the sum rules in quantum field theories such as QCD? Of course, we know QCD sum rules based on the operator product expansion (OPE) and dispersion relation. They are originally obtained by Shifman et al. [1]. (SVZ sum rules). Here is an example of the SVZ sum rule, ∞
0
ds 2π
s ρ(s) − ρ con (s) = Condensates.
E-mail address: [email protected]. 0146-6410/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ppnp.2011.12.007
(2)
T. Hayata / Progress in Particle and Nuclear Physics 67 (2012) 136–139
137
Fig. 1. Vector current spectral function and ρ meson resonance. Source: Figure adapted from Ref. [2].
The quantity ρ(s) is the spectral function for the current correlation function. Experimentally it is obtained from the R-ratio in the electron–positron annihilation process. The quantity ρ con (s) means the perturbative part of the spectral function as indicated by the blue region in Fig. 1. The energy weighted sum of ρ(s) − ρ con (s) is dominated by the low-lying resonances and is related to the chiral and gluon condensates through the sum rule. However, there is a large conceptual gap between the previous dipole sum rule and the SVZ sum rule here. A natural question is whether we can derive or generalize the QCD sum rules by using the QCD commutation relations alone. My answer is YES. To show this, we first need to review the canonical operator formalism of QCD. 2. Canonical quantization of QCD There is the elegant operator formalism for the non-abelian gauge theory based on Becchi–Rouet–Stora–Tyutin (BRST) invariant Lagrangian given by Kugo and Ojima [3]. In canonical quantization, a quantum field theory is described by canonical commutation relations of elementary fields and its Heisenberg’s equations i.e., commutators between Hamiltonian. QCD Hamiltonian obtained from covariant gauge fixed BRST Lagrangian becomes
Heff = −gAa0 q¯ f γ 0 t a qf + q¯ f (−iγ k Dk + mf )qf +
+ E⃗ a · (∇ Aa0 − gfabc A⃗ b Ac0 ) + ∂k Ba Aak −
α 2
1 2
⃗ a )2 ) ((E⃗ a )2 + (H
a (Ba )2 + iΠca Πc¯a + gfabc Πca Ab0 c c − i∂ k c¯ a Dad k c ,
(3)
where first line is quark and gluon kinetic energy and interaction energy, second line is Gauss’s law constraint term and covariant gauge fixing term and last line represents the ghost contributions. qf , q¯ f , Aaµ , Ba , Eka , c a , c¯ a , Πca , Πc¯a are Heisenberg’s operators of quark, gluon and ghost fields. They satisfy the canonical commutation relations {q, q¯ }, [Aa0 , Bb ], [Aai , E bj ], {c a , Πcb }, {¯c a , Πc¯b }. Furthermore, in gauge theory, due to the indefinite metric of Asymptotic states, we need the condition to extract the true physical degrees of freedom and the physical vacuum that we are interested in. However, it is well known that a physical state is BRST invariant i.e., QB |state⟩ = 0. In sum up, effective Hamiltonian based on BRST symmetry, canonical commutation relations of elementary fields and BRST chargeless condition for physical states are all of canonical quantization of gauge theory. 3. Sum rules for QCD current correlator We consider the current–current correlator Π (q2 ) in QCD,
Πµν (q) = i
d4 x eiqx ⟨0|T [jµ (x), jν (0)]|0⟩
= (qµ qν − q2 gµν )Π (q2 ).
(4)
The imaginary part of the correlation is nothing but the spectral function ρ(q2 ). It satisfies the dispersion relation,
Π (q2 ) =
ρ(s) . 2π s − q2 − iϵ ds
(5)
138
T. Hayata / Progress in Particle and Nuclear Physics 67 (2012) 136–139
By the Lehman representation of Πµν (q), we obtain the spectral decomposition of the spectral function,
ρ(q2 ) = −
1 3q2
(2π )4 δ (4) (q − p)⟨0|jµ (0)|p⟩⟨p|jµ (0)|0⟩,
(6)
p
where |p⟩ is a energy–momentum eigenstate i.e., P µ |p⟩ = pµ |p⟩. Then, we define the generalized energy-weighted sum of spectral function, ∞
ds 2π
0
sn ρ(s) = −
1
3
d3 x ⟨0|[· · · [jµ (0, ⃗ x), H] · · · , H]2n−1 , jµ (0)|0⟩.
(7)
Without loss of generality, we can take a frame where the three momentum q is zero. It is now straightforward to obtain the right hand side from the spectral function. In the right hand side, we have 2n − 1 commutations. They can be calculated by using the QCD Hamiltonian and the canonical commutation relations in the previous section. There is, however, one caveat. The matrix element in the right hand side is UV divergent in quantum field theory and must be renormalized. Then, we need to subtract out the divergent part from both sides of the sum rule to obtain the finite results. We perform the renormalization by subtracting the high energy perturbative contributions, ∞
ds 2π
0
s (ρ(s) − ρ n
con
(s)) = −
1 3
d3 x ⟨0|[· · · [jµ (0, ⃗ x), H] · · · , H]2n−1 , jµ (0)|0⟩NP ,
(8)
where NP stands for the nonperturbative contribution of vacuum expectation value. The energy-weighted sum of spectral function is dominated by a low energy part. Thus, the sum rule relates a low energy resonance of composite particle (e.g., mesons and baryons) and QCD vacuum structure characterized by condensates. (e.g., chiral condensate and gluon condensate). 4. Weinberg’s sum rules To confirm our method works well, we derive the Weinberg’s sum rules from the QCD commutators. Weinberg’s sum 1 rules are related to the difference between the vector correlator j(ρ) uγµ u − d¯ γµ d) and the axial-vector correlator µ = 2 (¯ (A )
jµ 1 = 12 (¯uγµ γ 5 u − d¯ γµ γ 5 d). By definition, such difference is non-zero only when chiral symmetry is broken explicitly or dynamically. Here is the sum rule for the 1st moment, ∞
0
4mu 4md ¯ s(ρA (s) − ρV (s)) = 0 u¯ u + dd 0 . 2π 3 3 ds
(9)
Since it has only a single commutator between the current and the Hamiltonian, we have only bilinear quark operator in the right hand side. This result is identical to the Weinberg’s sum rule derived from OPE. The sum rule for the 2nd moment requires double commutator. Then, we have following 4-quark condensates in the right hand side, ∞
0
16 3 ← → s (ρA (s) − ρV (s)) = 02im2q q¯ q− mq q¯ q 2π 3 ds
2
π
a µ a µ a ¯ ¯ + αs (¯uL γµ t uL − dL γµ t dL )(¯uR γ t uR − dR γ t dR )0 . 2 a
(10)
The coefficient of the 4-quark operator coincides with the result of OPE. Moreover, we have bilinear quark condensates which are usually neglected in the OPE approach since it is higher orders of the light-quark mass mq . In our approach, we do not need to assume that mq is small, so that we can get exact result in the right hand side. This is an advantage of our commutator approach. Note that these higher order terms in mq may not be negligible for the strange quark. Here is an example of the actual calculation of the commutators. Basic commutator [J , H ] is written like this,
[¯uγ µ u, H ] = −iu¯ γ µ γ 0 γ k Dk u − iD∗k u¯ γ k γ 0 γ µ u + mu u¯ [γ µ , γ 0 ]u.
(11)
If you take double commutator [[J , H ], H ], the result becomes more complicated,
[[¯uγ µ u, H ], H ] = u¯ γ µ γ k Dk γ k Dk′ u − D∗k u¯ γ k γ 0 γ µ γ 0 γ k Dk u + 2imu u¯ γ 0 γ µ γ 0 γ k Dk u + ig u¯ γ µ γ 0 γ k Eka t a u + m2u u¯ (γ µ − γ µĎ )u + h.c. . ′
(12)
The calculation is tedious but straightforward, and we do not need to recourse to the operator product expansion at all.
T. Hayata / Progress in Particle and Nuclear Physics 67 (2012) 136–139
139
5. Sum rules for vector meson Now, let us come back to the vector current correlation and focus on the first moment. Bare sum rule simply obtained by the commutator becomes ∞
0
ds
i ← → sρV (s) = 0 u¯ Dk γ k u − mu u¯ u + (u ⇔ d) 0 . 2π 3
(13)
Since both sides of this sum rule are not finite, let us subtract the UV divergent part by introducing a certain regulator. As pointed out by Fujikawa [4] in the context of the derivation of the trace anomaly, the equation of motion with a gaugeinvariant UV regulator f has the following form,
⟨0|[¯u(i
− m)u]|0⟩ = Tr
d4 k
(2π )
e−ikx f 4
2
M2
eikx .
(14)
Expanding the right hand side by M −2 , we find that the equation of motion has a quartic divergent part which can be absorbed as the continuum part of spectral function ρ con in the left hand side and the finite part proportional to G2 . Using this, the ρ meson resonance in vector current correlator can be related to the chiral and the gluon condensates. Important lesson here is that we should be careful about treating divergent matrix elements. Some non-trivial finite part such as G2 may arise after the subtraction of the divergent part with gauge invariant regulator in manifestly 4 dimensional calculation. We are currently studying the regularization scheme dependence and the consistency between the OPE’s vector sum rules. Before closing, let us mention new types of sum rules obtained from our approach. They are the sum rules with fractional powers like ∞
ds 2π
0
n
s 2 ρ(s) = −
1 0 [· · · [Qµ , H] · · · , H]n−1 , Q µ 0 , 3V
(15)
where Q µ = d3 x jµ (0, ⃗ x). In our approach, they are related to n − 1 commutators, while no such sum rules can be obtained from OPE. Here is an example of 21 -power,
∞
0
2 ds √ sρ(s) = − ⟨0|Qµ Q µ |0⟩ 3V
2π
=−
8 3V
⟨0|Q02 |0⟩.
(16)
It is related to the charge fluctuation of the system. We are currently studying the physical implication of these sum rules. 6. Conclusion New derivation of QCD sum rules by canonical commutators is obtained. It is the simple and straightforward generalization of dipole sum rule on the basis of Kugo–Ojima operator formalism and a suitable subtraction of UV divergences. It gives the quantitative relations between the low energy physics and the vacuum structure characterized by the spontaneous symmetry breaking (the condensates). By applying the method to the vector and axial vector current in QCD, we obtain the exact Weinberg’s sum rules. The coefficients up to linear order of light quark mass mq is consistent with OPE’s results. Moreover, in 2nd moment sum rule, we have bilinear quark condensates which are usually neglected in the OPE approach since it is higher orders of the light-quark mass mq . In our approach, we do not need to assume that mq is small, so that we can get exact result in the right hand side. Note that these higher orders terms in mq may not be negligible for the strange quark sum rules. Vector current sum rules and new fractional power sum rules are also discussed. However, these need further study due to the regularization dependence. Thus, regularization scheme dependence of the sum rules, applications to other mesonic and baryonic currents and in-medium QCD sum rules from the commutators will be examined in future investigation. Acknowledgments This is a work in collaboration with Tetsuo Hatsuda and Schoichi Sasaki in University of Tokyo. We thank Kazuo Fujikawa for valuable comments. References [1] [2] [3] [4]
M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B 147 (1979) 385; 147 (1979) 448. Y. Kwon, M. Procura, W. Weise, Phys. Rev. C 78 (2008) 055203. T. Kugo, I. Ojima, Phys. Lett. B 73 (1978) 459. K. Fujikawa, Phys. Rev. Lett. 44 (1980) 1733; Phys. Rev. D 23 (1981) 2262.
Contents lists available at SciVerse ScienceDirect
Progress in Particle and Nuclear Physics journal homepage: www.elsevier.com/locate/ppnp
Review
New QCD sum rules based on canonical commutation relations Tomoya Hayata Department of Physics, The University of Tokyo, Tokyo 113-0031, Japan
article
abstract
info
Keywords: QCD sum rules Caonical commutation relations Kugo–Ojima operator formalism and Weinberg’s sum rules
New derivation of QCD sum rules by canonical commutators is developed. It is the simple and straightforward generalization of Thomas–Reiche–Kuhn sum rule on the basis of Kugo–Ojima operator formalism of a non-abelian gauge theory and a suitable subtraction of UV divergences. By applying the method to the vector and axial vector current in QCD, the exact Weinberg’s sum rules are examined. Vector current sum rules and new fractional power sum rules are also discussed. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The understanding of the physics of spontaneous symmetry breaking (SSB) is one of the most important goal in a quantum many particle system and a system described by a quantum field theory. There are many theoretical calculations to figure out the characteristic behavior of SSB e.g., Ginzburg–Landau type’s method based on symmetry and its breaking, mean-field type’s self-consistent approach to determine the nonperturbative vacuum, and effective field theories like chiral perturbation theory to reproduce the low energy physics of the system. In quantum many-body systems, sum rule is known to be a powerful tool to study the spectral structure. Here is the well-known example of the dipole sum rule in nuclear physics,
σtot = = =
4π 2 e2 h¯ c 4π 2 e2 h¯ c
ν
(Eν − E0 )|⟨ν|D|0⟩|2
⟨0|[D, [H , D]]|0⟩
2π 2 e2 h¯ NZ mc
A
(1 + K ),
(1)
where σtot is a dipole induced photo-nuclear total cross section, and D is the dipole operator. Such a cross section can be written as an energy weighted sum of the dipole transition probability. It can be also written as a double commutator between the Hamiltonian and D. Furthermore, in this particular example, the commutator can be evaluated exactly. The result is written only by fundamental constants and does not depend on the details of the dynamics. Now what about the sum rules in quantum field theories such as QCD? Of course, we know QCD sum rules based on the operator product expansion (OPE) and dispersion relation. They are originally obtained by Shifman et al. [1]. (SVZ sum rules). Here is an example of the SVZ sum rule, ∞
0
ds 2π
s ρ(s) − ρ con (s) = Condensates.
E-mail address: [email protected]. 0146-6410/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ppnp.2011.12.007
(2)
T. Hayata / Progress in Particle and Nuclear Physics 67 (2012) 136–139
137
Fig. 1. Vector current spectral function and ρ meson resonance. Source: Figure adapted from Ref. [2].
The quantity ρ(s) is the spectral function for the current correlation function. Experimentally it is obtained from the R-ratio in the electron–positron annihilation process. The quantity ρ con (s) means the perturbative part of the spectral function as indicated by the blue region in Fig. 1. The energy weighted sum of ρ(s) − ρ con (s) is dominated by the low-lying resonances and is related to the chiral and gluon condensates through the sum rule. However, there is a large conceptual gap between the previous dipole sum rule and the SVZ sum rule here. A natural question is whether we can derive or generalize the QCD sum rules by using the QCD commutation relations alone. My answer is YES. To show this, we first need to review the canonical operator formalism of QCD. 2. Canonical quantization of QCD There is the elegant operator formalism for the non-abelian gauge theory based on Becchi–Rouet–Stora–Tyutin (BRST) invariant Lagrangian given by Kugo and Ojima [3]. In canonical quantization, a quantum field theory is described by canonical commutation relations of elementary fields and its Heisenberg’s equations i.e., commutators between Hamiltonian. QCD Hamiltonian obtained from covariant gauge fixed BRST Lagrangian becomes
Heff = −gAa0 q¯ f γ 0 t a qf + q¯ f (−iγ k Dk + mf )qf +
+ E⃗ a · (∇ Aa0 − gfabc A⃗ b Ac0 ) + ∂k Ba Aak −
α 2
1 2
⃗ a )2 ) ((E⃗ a )2 + (H
a (Ba )2 + iΠca Πc¯a + gfabc Πca Ab0 c c − i∂ k c¯ a Dad k c ,
(3)
where first line is quark and gluon kinetic energy and interaction energy, second line is Gauss’s law constraint term and covariant gauge fixing term and last line represents the ghost contributions. qf , q¯ f , Aaµ , Ba , Eka , c a , c¯ a , Πca , Πc¯a are Heisenberg’s operators of quark, gluon and ghost fields. They satisfy the canonical commutation relations {q, q¯ }, [Aa0 , Bb ], [Aai , E bj ], {c a , Πcb }, {¯c a , Πc¯b }. Furthermore, in gauge theory, due to the indefinite metric of Asymptotic states, we need the condition to extract the true physical degrees of freedom and the physical vacuum that we are interested in. However, it is well known that a physical state is BRST invariant i.e., QB |state⟩ = 0. In sum up, effective Hamiltonian based on BRST symmetry, canonical commutation relations of elementary fields and BRST chargeless condition for physical states are all of canonical quantization of gauge theory. 3. Sum rules for QCD current correlator We consider the current–current correlator Π (q2 ) in QCD,
Πµν (q) = i
d4 x eiqx ⟨0|T [jµ (x), jν (0)]|0⟩
= (qµ qν − q2 gµν )Π (q2 ).
(4)
The imaginary part of the correlation is nothing but the spectral function ρ(q2 ). It satisfies the dispersion relation,
Π (q2 ) =
ρ(s) . 2π s − q2 − iϵ ds
(5)
138
T. Hayata / Progress in Particle and Nuclear Physics 67 (2012) 136–139
By the Lehman representation of Πµν (q), we obtain the spectral decomposition of the spectral function,
ρ(q2 ) = −
1 3q2
(2π )4 δ (4) (q − p)⟨0|jµ (0)|p⟩⟨p|jµ (0)|0⟩,
(6)
p
where |p⟩ is a energy–momentum eigenstate i.e., P µ |p⟩ = pµ |p⟩. Then, we define the generalized energy-weighted sum of spectral function, ∞
ds 2π
0
sn ρ(s) = −
1
3
d3 x ⟨0|[· · · [jµ (0, ⃗ x), H] · · · , H]2n−1 , jµ (0)|0⟩.
(7)
Without loss of generality, we can take a frame where the three momentum q is zero. It is now straightforward to obtain the right hand side from the spectral function. In the right hand side, we have 2n − 1 commutations. They can be calculated by using the QCD Hamiltonian and the canonical commutation relations in the previous section. There is, however, one caveat. The matrix element in the right hand side is UV divergent in quantum field theory and must be renormalized. Then, we need to subtract out the divergent part from both sides of the sum rule to obtain the finite results. We perform the renormalization by subtracting the high energy perturbative contributions, ∞
ds 2π
0
s (ρ(s) − ρ n
con
(s)) = −
1 3
d3 x ⟨0|[· · · [jµ (0, ⃗ x), H] · · · , H]2n−1 , jµ (0)|0⟩NP ,
(8)
where NP stands for the nonperturbative contribution of vacuum expectation value. The energy-weighted sum of spectral function is dominated by a low energy part. Thus, the sum rule relates a low energy resonance of composite particle (e.g., mesons and baryons) and QCD vacuum structure characterized by condensates. (e.g., chiral condensate and gluon condensate). 4. Weinberg’s sum rules To confirm our method works well, we derive the Weinberg’s sum rules from the QCD commutators. Weinberg’s sum 1 rules are related to the difference between the vector correlator j(ρ) uγµ u − d¯ γµ d) and the axial-vector correlator µ = 2 (¯ (A )
jµ 1 = 12 (¯uγµ γ 5 u − d¯ γµ γ 5 d). By definition, such difference is non-zero only when chiral symmetry is broken explicitly or dynamically. Here is the sum rule for the 1st moment, ∞
0
4mu 4md ¯ s(ρA (s) − ρV (s)) = 0 u¯ u + dd 0 . 2π 3 3 ds
(9)
Since it has only a single commutator between the current and the Hamiltonian, we have only bilinear quark operator in the right hand side. This result is identical to the Weinberg’s sum rule derived from OPE. The sum rule for the 2nd moment requires double commutator. Then, we have following 4-quark condensates in the right hand side, ∞
0
16 3 ← → s (ρA (s) − ρV (s)) = 02im2q q¯ q− mq q¯ q 2π 3 ds
2
π
a µ a µ a ¯ ¯ + αs (¯uL γµ t uL − dL γµ t dL )(¯uR γ t uR − dR γ t dR )0 . 2 a
(10)
The coefficient of the 4-quark operator coincides with the result of OPE. Moreover, we have bilinear quark condensates which are usually neglected in the OPE approach since it is higher orders of the light-quark mass mq . In our approach, we do not need to assume that mq is small, so that we can get exact result in the right hand side. This is an advantage of our commutator approach. Note that these higher order terms in mq may not be negligible for the strange quark. Here is an example of the actual calculation of the commutators. Basic commutator [J , H ] is written like this,
[¯uγ µ u, H ] = −iu¯ γ µ γ 0 γ k Dk u − iD∗k u¯ γ k γ 0 γ µ u + mu u¯ [γ µ , γ 0 ]u.
(11)
If you take double commutator [[J , H ], H ], the result becomes more complicated,
[[¯uγ µ u, H ], H ] = u¯ γ µ γ k Dk γ k Dk′ u − D∗k u¯ γ k γ 0 γ µ γ 0 γ k Dk u + 2imu u¯ γ 0 γ µ γ 0 γ k Dk u + ig u¯ γ µ γ 0 γ k Eka t a u + m2u u¯ (γ µ − γ µĎ )u + h.c. . ′
(12)
The calculation is tedious but straightforward, and we do not need to recourse to the operator product expansion at all.
T. Hayata / Progress in Particle and Nuclear Physics 67 (2012) 136–139
139
5. Sum rules for vector meson Now, let us come back to the vector current correlation and focus on the first moment. Bare sum rule simply obtained by the commutator becomes ∞
0
ds
i ← → sρV (s) = 0 u¯ Dk γ k u − mu u¯ u + (u ⇔ d) 0 . 2π 3
(13)
Since both sides of this sum rule are not finite, let us subtract the UV divergent part by introducing a certain regulator. As pointed out by Fujikawa [4] in the context of the derivation of the trace anomaly, the equation of motion with a gaugeinvariant UV regulator f has the following form,
⟨0|[¯u(i
− m)u]|0⟩ = Tr
d4 k
(2π )
e−ikx f 4
2
M2
eikx .
(14)
Expanding the right hand side by M −2 , we find that the equation of motion has a quartic divergent part which can be absorbed as the continuum part of spectral function ρ con in the left hand side and the finite part proportional to G2 . Using this, the ρ meson resonance in vector current correlator can be related to the chiral and the gluon condensates. Important lesson here is that we should be careful about treating divergent matrix elements. Some non-trivial finite part such as G2 may arise after the subtraction of the divergent part with gauge invariant regulator in manifestly 4 dimensional calculation. We are currently studying the regularization scheme dependence and the consistency between the OPE’s vector sum rules. Before closing, let us mention new types of sum rules obtained from our approach. They are the sum rules with fractional powers like ∞
ds 2π
0
n
s 2 ρ(s) = −
1 0 [· · · [Qµ , H] · · · , H]n−1 , Q µ 0 , 3V
(15)
where Q µ = d3 x jµ (0, ⃗ x). In our approach, they are related to n − 1 commutators, while no such sum rules can be obtained from OPE. Here is an example of 21 -power,
∞
0
2 ds √ sρ(s) = − ⟨0|Qµ Q µ |0⟩ 3V
2π
=−
8 3V
⟨0|Q02 |0⟩.
(16)
It is related to the charge fluctuation of the system. We are currently studying the physical implication of these sum rules. 6. Conclusion New derivation of QCD sum rules by canonical commutators is obtained. It is the simple and straightforward generalization of dipole sum rule on the basis of Kugo–Ojima operator formalism and a suitable subtraction of UV divergences. It gives the quantitative relations between the low energy physics and the vacuum structure characterized by the spontaneous symmetry breaking (the condensates). By applying the method to the vector and axial vector current in QCD, we obtain the exact Weinberg’s sum rules. The coefficients up to linear order of light quark mass mq is consistent with OPE’s results. Moreover, in 2nd moment sum rule, we have bilinear quark condensates which are usually neglected in the OPE approach since it is higher orders of the light-quark mass mq . In our approach, we do not need to assume that mq is small, so that we can get exact result in the right hand side. Note that these higher orders terms in mq may not be negligible for the strange quark sum rules. Vector current sum rules and new fractional power sum rules are also discussed. However, these need further study due to the regularization dependence. Thus, regularization scheme dependence of the sum rules, applications to other mesonic and baryonic currents and in-medium QCD sum rules from the commutators will be examined in future investigation. Acknowledgments This is a work in collaboration with Tetsuo Hatsuda and Schoichi Sasaki in University of Tokyo. We thank Kazuo Fujikawa for valuable comments. References [1] [2] [3] [4]
M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B 147 (1979) 385; 147 (1979) 448. Y. Kwon, M. Procura, W. Weise, Phys. Rev. C 78 (2008) 055203. T. Kugo, I. Ojima, Phys. Lett. B 73 (1978) 459. K. Fujikawa, Phys. Rev. Lett. 44 (1980) 1733; Phys. Rev. D 23 (1981) 2262.