String Field Theory as World-sheet UV Regulator (1902.00263v1)

Abstract: Even at tree level, the first quantized string theory suffers from apparent short distance singularities associated with collision of vertex operators that prevent us from straightforward numerical computation of various quantities. Examples include string theory S-matrix for generic external momenta and computation of the spectrum of string theory under a marginal deformation of the world-sheet theory. The former requires us to define the S-matrix via analytic continuation or as limits of contour integrals in complexified moduli space, while the latter requires us to use an ultraviolet cut-off at intermediate steps. In contrast, string field theory does not suffer from such divergences. In this paper we show how string field theory can be used to generate an explicit algorithm for computing tree level amplitudes in any string theory that does not suffer from any short distance divergence from integration over the world-sheet variables. We also use string field theory to compute second order mass shift of string states under a marginal deformation without having to use any cut-off at intermediate steps. We carry out the analysis in a broad class of string field theories, thereby making it manifest that the final results are independent of the extra data that go into the formulation of string field theory. We also comment on the generalization of this analysis to higher genus amplitudes.

• The paper demonstrates that string field theory circumvents world-sheet UV divergences by reformulating tree-level amplitude integrals.
• It presents an algorithm that removes traditional cut-offs for computing on-shell S-matrix elements and second order mass shifts under marginal deformations.
• The study indicates that this finite SFT approach could extend to higher genus amplitudes, offering new pathways for regularizing complex string interactions.

String Field Theory as World-sheet UV Regulator

The paper "String Field Theory as World-sheet UV Regulator," authored by Ashoke Sen, addresses the challenges associated with short distance singularities on the world-sheet inherent to string theory, even at the tree level. Traditional first quantized string theory is plagued by divergences emerging from the collision of vertex operators, posing hurdles to straightforward numerical computation of several quantities, including string theory S-matrix and the spectrum under marginal deformation. These singularities often necessitate the use of analytic continuation or ultraviolet cut-offs during intermediate steps of computation, especially in the complexified moduli space.

The paper leverages string field theory (SFT) as a potent tool to counter these divergences. SFT circumvents the short distance singularities encountered in world-sheet integrations, presenting an explicit algorithm for computing tree level amplitudes devoid of UV divergences, from integrations over world-sheet variables. Furthermore, the approach extends to compute second order mass shifts of string states under marginal deformation, eschewing any intermediate cut-offs.

Divergence Resolution and Computational Approach

In the first quantized framework, scattering amplitudes are often defined by integrals of correlation functions of vertex operators within conformal field theory (CFT). These integrals are susceptible to divergences when vertex operators approach each other, a simple example being the Koba-Nielsen formula for multi-tachyon amplitudes in bosonic string theory, traditionally managed using analytic continuation. String field theory, however, evaluates tree-level S-matrix through the summation of Feynman diagrams, with contributions segregated based on interaction vertices.

Sen advances a systematic prescription for computing string tree amplitudes that employs string field theory to reshape integrals into forms resilient against divergences, driven by straightforward mechanics: contributions are partitioned between partial world-sheet variables integrations and analytically tractable boundary terms. The formalism accommodates a generic broad class of string field theories, showcasing the independence of the final results from specific SFT formulations.

Implications for On-shell Amplitudes and Mass Shifts

The framework's versatility is manifested in addressing both on-shell amplitudes and marginal deformations of the world-sheet CFT affecting string spectra. By reformulating the propagator terms and integrating contributions from total derivatives added by picture-changing operator (PCO) locations, the method sets a firm groundwork for perturbative computations free from undesired UV divergences.

Specifically, when considering marginal deformations like compact circle radius modifications, the SFT approach effortlessly computes mass shifts in both bosonic and heterotic string theories to second order in the deformation parameter. The paper establishes that these shifts, enlighteningly captured without explicit UV regulation on the world-sheet, align with intuitive expectations of radius scaling following deformation.

Generalization and Future Directions

Sen also speculates on extending these analyses to higher genus amplitudes, albeit acknowledging inherent challenges in managing separating and non-separating type degenerations. While generic degenerations without special momentum offer straightforward extensions, special momentum degenerations and non-separating types present quantum corrections, mass renormalizations, and demand intensive regularization strategies reminiscent of traditional field theories. Nonetheless, the groundwork laid suggests that SFT could ideally generalize to yield manifestly finite and consistent higher genus results, independent of string field theory data, by incorporating tactful field redefinitions.

In conclusion, Sen's paper provides a detailed and robust methodology to compute string amplitudes and explore spectra under marginal deformations using string field theory, holding promise to mitigate longstanding dilemmas associated with world-sheet short distance singularities, potentially opening new computational pathways and insights in theoretical investigations across string theory paradigms.