On-shell Ward Identities Research Articles

Calculation of Zero-Norm States and Reduction of Stringy Scattering Amplitudes

We give a simplified method to generate two types of zero-norm states in the old covariant first quantized (OCFQ) spectrum of open bosonic string. Zero-norm states up to the fourth massive level and general formulas of some zero-norm tensor states at arbitrary mass levels are calculated. On-shell Ward identities generated by zero-norm states and the factorization property of stringy vertex operators can then be used to argue that the string-tree scattering amplitudes of the degenerate lower spin propagating states are fixed by those of higher spin propagating states at each fixed mass level. This decoupling phenomenon is, in contrast to Gross’s high-energy symmetries, valid to all energies. As examples, we explicitly demonstrate this stringy phenomenon up to the fourth massive level (spin-five), which justifies the calculation of two other previous approaches based on the massive worldsheet sigma-model and Witten’s string field theory (WSFT).

Massive inter-particle symmetry and Ward identity in fermionic string theory

We show that there exist enlarged stringy (α′→∞) symmetries for (evenG-parity) massive modes in the 10D fermionic string theory. These symmetries are derived from on-shell Ward identities corresponding to the decoupling of massive gauge states in the spectrum. In the generalized massive supersymmetric σ-model formalism, some symmetry transformations relate particles with different spins in the first order weak field approximation.

Heterotic massive Einstein-Yang-Mills-type symmetry and Ward identity

We show that there exist spontaneously broken symmetries for massive modes with transformation parameters θ(ab)[μμ], θ(ab)μ etc. containing both Einstein and E8⊗E8 (orSO(32)) Yang-Mills indices in the 10D Heterotic string. The corresponding on-shell Ward identities are also constructed.

Generalized On-Shell Ward Identities in String Theory

It is demonstrated that an infinite set of string-tree level on-shell Ward identities, which are valid to all sigma-model loop orders, can be systematically constructed without referring to the string field theory. As examples, bosonic massive scattering amplitudes are calculated explicitly up to the second massive excited states. Ward identities satisfied by these amplitudees are derived by using zero-norm states in the spetrum. In particular, the inter-particle Ward identity generated by the D2xD2' zero-norm state at the second massive level is demonstrated. The four physical propagating states of this mass level are then shown to form a large gauge multiplet. This result justifies our previous consideration on higher inter-spin symmetry from the generalized worldsheet sigma-model point of view.

Generalized On-Shell Ward Identities in String Theory

Generalized On-Shell Ward Identities in String Theory

The master equation of string theory

A general method is presented for deriving on-shell Ward identities in string theory. It is shown that all tree-level Ward identities can be summarized in a single quadratic differential equation for the generating function of all amplitudes. This result is extended to loop amplitudes and leads to a master equation à la Batalin-Vilkovisky for the complete partition function.

Gauge dependence in the Yang-MillsSmatrix

There are formal proofs that $S$-matrix elements are gauge invariant in gauge field theories. These proofs break down unless infrared singularities of the $S$ matrix are handled with care. In a similar fashion, formal on-shell Ward identities can break down. Dimensional regularization of infrared singularities seems to preserve all formal relations.