Use of equal-time commutators involving the symmetric energy-momentum tensor for the derivation of sum rules

doi:10.1016/0550-3213(71)90338-5

Nuclear Physics B

Volume 34, Issue 2, 15 November 1971, Pages 429-444

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Abstract

We emphasize the central role that second-order Schwinger terms in equal-time commutators involving the energy density T_oo play in determining the scaling behaviour of local operators. Thereby a new derivation is given of the fact (recently derived by Chanowitz) that the time component of a conserved current has dimensions 3 if it has a dimension at all and if a state | A〉 exists such that 〈A|J_o|A〉 ≠ 0. Assuming that the time component of the non-conserved strangeness changing vector current V_μ^K has a dimension we show by saturation of the matrix element $〈π^{+} (q) | [K_{o}, V_{o}^{K} (0) | K^{+} (q)〉$ (with K_μ the conformal charges) in the infinite momentum frame that the root mean square gravitational factor radii (i.e. the slope of the gravitational form factor G₁(t) at t = 0) of π and K are the same. In this case, Regge theory is used to justify the infinite momentum limit. Formal application of this limit also shows that the dimensions of ∂^μV_μ^K and V_k^K (if any) should be 2 and 3, respectively. This result is less reliable since in this case Regge theory implies invalidity of the procedure. Analogous results hold under somewhat stronger assumptions for η too. We discuss the consequences of SU(3) and our results for tensor meson dominance and find that a (numericaly possible small) subtraction that depends on the SU(3) index M must be present in the pseudo-scalr-tensor meson form factor G₁^M(t). Our results are in at least approximate agreement with SU(3) for 2⁺ → 0⁻0⁻ and allow us to predict - correct within the errors - a value for the width $f → K K$ . They also agree with what is known about f → ηη and f′ → ηη. The assumptions include neglect of contributions from the K_A, i.e. the doubtful Q-range 1240–1400 MeV.

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Within the framework of asymptotic scale invariance we analyse the light-cone structure of the hadronic matrix elements 〈p|[θ_μν(x), φ(0)]|0)]〈 and 〈p|[j_μ(x), φ(0)]|0〉 involving the stress tensor θ_μν(x), the electromagnetic current j_μ(x), some scalar field φf(x) and spinless one-particle states |p〉. Provided the off-shell Bjorken limit and the mass shell limit are interchangeable, the asymptotic behaviour of the related form factors is determined by the leading light-cone singularities. Furthermore, within a naive quark model there still exists a connection between the electromagnetic and the mass form factor at high momentum transfer involving the dimension of length l_φ of the meson field φ explicitely. By applying once-substracted dispersion relations a sum rule for l_φ is derived which may be useful in understanding anomalous dimensions.
Energy-momentum tensors and z-diagrams
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The discussion of Z-diagrams is extended to equal-time commutators of the charges Q_D^α(0) = ∫ d³x x^m T_0m^α(x) with space and time components of currents J_μ and their divergences. (We have denoted by T_μν^α any symmetric energy-momentum tensor within the Callan-Coleman-Jackiw class parametrized by α.) The requirement of absence of Z-diagrams in the infinite momentum limit for boson matrix elements of either of the equal time commutators [Q_D^α(0), ∂^μJ_μ(0)], [T_μ^αμ(0)] or [Q_D^α(0), J_k(0) restricts the energy momentum tensor to the new and improved one. This tensor T_μν satisfies the preceeding requirement and, as a consequence, Z-diagrams do not contribute in the infinite momentum limit to boson matrix elements of the commutators [Q_D(0), ∂^μJ_μ(0)], [T_μ^μ(0), Q(0)] and [Q_D(0), J_k(0)] (with Q_D the dilatation charge). For fermion matrix elements of [Q_D(0), ∂^μJ_μ(0)] Z-diagrams do give a contribution in the infinite momentum limit. Furthermore, in this limit no Z-diagrams contribute to fermion matrix elements of [Q_D(0), J_μ(0)] and to boson matrix elements of [Q_D^α(0), J_o(0)] (for any α). Moreover, assuming one particle saturation in the infinite momentum limit of both meson and fermion matrix elements of [Q_D(0), ∂^μ J_μ(0)] and [T_μ^μ(0), Q(0)] one consistently derives that ∂^μJ_μ has dimension two if it has a dimension at all.
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Nuclear Physics B

Use of equal-time commutators involving the symmetric energy-momentum tensor for the derivation of sum rules

Abstract

Nucl. Phys.

Nucl. Phys.

Phys. Rev.

Phys. Rev.

Phys. Rev.

Nucl. Phys.

Phys. Letters

Ann. of Phys.

Phys. Rev.

Phys. Rev.

Z. Phys.

Phys. Rev.

Phys. Rev. Letters

Phys. Rev.

Current algebras

Light-cone and asymptotic behaviour of the electromagnetic and mass form factor of mesons

Energy-momentum tensors and z-diagrams

Role of the Pomeron in tensor-meson dominance

A theoretical estimate for the slope of the scalar K<inf>ℓ3</inf> form factor at the origin

Scale and conformal transformations of currents and tensor-menson dominance