[Paper Review] Field Definitions, Spectrum and Universality in Effective String Theories

N. D. Hari Dass, Peter Matlock|ArXiv.org|Dec 28, 2006

Particle physics theoretical and experimental studies5 references3 citations

TL;DR

This paper demonstrates that the spectrum of the Polchinski-Strominger effective string theory matches that of the Nambu-Goto theory up to third order in inverse string length, despite Nambu-Goto being inconsistent outside critical dimensions. Using a careful, field-redefinition-free analysis, the authors prove the absence of higher-order corrections in the action and confirm universality of the spectrum, including L"uscher terms and beyond, across dimensions.

ABSTRACT

It is shown, by explicit calculation, that the third-order terms in inverse string length in the spectrum of the effective string theories of Polchinski and Strominger are also the same as in Nambu-Goto theory, in addition to the universal Luescher terms. While the Nambu-Goto theory is inconsistent outside the critical dimension, the Polchinski-Strominger theory is by construction consistent for any space-time dimension. In the analysis of the spectrum, care is taken not to use any field redefinition, as it is felt that this has the potential to obscure important points. Nevertheless, as field redefinition is an important tool and the definition of the field should be made precise, a careful analysis of the choice of field definition leading to the terms in the action is also presented. Further, it is shown how a choice of field definition can be made in a systematic way at higher orders. To this end the transformation of measure involved is calculated, in the context of effective string theory, and thereby a quantum evaluation made of equivalence of theories related by a field redefinition. It is found that there are interesting possibilities resulting from a redefinition of fluctuation field.

Motivation & Objective

• To establish the universality of the effective string spectrum beyond the L"uscher term, including third-order corrections in inverse string length.
• To analyze the spectrum of the Polchinski-Strominger effective string theory without relying on field redefinitions, which may obscure physical content.
• To systematically investigate the role of field definitions and their impact on higher-order terms in the action and spectrum.
• To clarify the conditions under which field redefinitions preserve equivalence, particularly concerning measure transformations in quantum effective field theories.
• To determine whether the Polchinski-Strominger action is uniquely fixed up to irrelevant terms at higher orders, and to assess implications for the spectrum.

Proposed method

• Performs explicit perturbative expansion of the Polchinski-Strominger action up to order $ R^{-3} $, without truncating or using field redefinitions.
• Analyzes the stress-energy tensor and its OPE to order $ R^{-2} $, confirming closure of the Virasoro algebra up to $ R^{-4} $, ensuring consistency.
• Uses the untruncated PS transformation law to verify that variations of the action yield only $ \beta^2 $-type terms at order $ R^{-4} $, which do not spoil closure.
• Introduces the concept of $ X $-uniformity to constrain field definitions and explores its limitations, later relaxing it to consider dropping the PS term.
• Applies a systematic procedure for field redefinition at higher orders by calculating the associated measure transformation in the path integral.
• Compares results with lattice QCD computations and Nambu-Goto theory, highlighting agreement beyond the L"uscher term despite Nambu-Goto's dimensional inconsistency.

Experimental results

Research questions

• RQ1 Are there corrections to the Polchinski-Strominger action at order $ R^{-3} $, and if so, do they alter the spectrum?
• RQ2 To what extent is the spectrum of the effective string theory universal across different dimensions, particularly when compared to the Nambu-Goto theory?
• RQ3 How do field redefinitions affect the equivalence of effective string theories, especially in the quantum regime involving measure transformations?
• RQ4 Can the PS term be omitted at certain orders without altering the spectrum, provided the transformation laws are adjusted accordingly?
• RQ5 What is the role of the Virasoro algebra closure and OPE consistency in validating higher-order effective actions?

Key findings

• There are no corrections to the Polchinski-Strominger action at order $ R^{-3} $, confirming the action's uniqueness up to irrelevant terms at this order.
• The spectrum of the Polchinski-Strominger theory matches that of the Nambu-Goto theory up to third order in $ R^{-1} $, including all universal L"uscher-type terms.
• The Virasoro algebra generated by the stress-energy tensor remains closed up to order $ R^{-4} $, and the OPE of stress tensors is consistent at this level.
• Field redefinitions can alter the action and spectrum unless the measure transformation is properly accounted for, highlighting the danger of naive redefinitions.
• The PS term is not strictly necessary at certain orders if the transformation laws are suitably modified, indicating a deeper interplay between field definition and dynamics.
• In the parity-violating sector, corrections to the action begin at order $ R^{-5} $, suggesting that the spectrum may deviate from Nambu-Goto only at this order or higher.