[Paper Review] Another Length Scale in String Theory?

Stephen H. Shenker|ArXiv.org|Sep 24, 1995

Computational Physics and Python ApplicationsComputer Science62 citations

TL;DR

This paper proposes the existence of a new dynamical length scale in string theory, $ m_s/g_s $, which emerges at weak coupling and is much shorter than the string scale $ m_s $. The evidence comes from indirect field-theoretic analogies and string compactifications, particularly near conifold singularities and in $ \mathcal{N}=2 $ supersymmetric theories, where logarithmic divergences suggest a UV cutoff at $ m_{\text{uv}} \sim m_s/g_s $, implying a deeper short-distance structure beyond the standard string scale.

ABSTRACT

We suggest that some of the remarkable results on stringy dynamics which have been found recently indicate the existence of another dynamical length scale in string theory that, at weak coupling, is much shorter than the string scale. This additional scale corresponds to a mass $\sim m_{ m s}/g_{ m s}$ where $m_{ m s}$ is the square root of the string tension and $g_{ m s}$ is the string coupling constant. In four dimensions this coincides with the Planck mass.

Motivation & Objective

• To investigate whether string theory contains a new dynamical length scale beyond the standard string scale $ m_s $, particularly at weak coupling.
• To understand the origin and implications of logarithmic divergences in low-energy effective actions of $ \mathcal{N}=2 $ supersymmetric string compactifications.
• To explore whether the scale $ m_s/g_s $, which coincides with the Planck mass in four dimensions, plays a role in regulating short-distance physics and black hole dynamics.
• To assess whether this new scale resolves puzzles in dual descriptions of conifolds and Ramond-Ramond couplings, or if alternative explanations exist.

Proposed method

• Analyzes the Seiberg-Witten solution for $ \mathcal{N}=2 $ $ SU(2) $ gauge theory, where logarithmic behavior in the dual coupling $ \tau_D \sim \log Z $ signals a UV cutoff at $ m_{\text{uv}} \sim m_W \sim \Lambda $, interpreted as the monopole size.
• Extends this logic to $ SU(N) $ theories to examine hierarchical mass scales and the breakdown of effective field theory near massless monopole points.
• Applies field-theoretic intuition to string compactifications, particularly Strominger’s resolution of conifold singularities in type IIB on Calabi-Yau manifolds.
• Examines the behavior of Ramond-Ramond couplings and their independence from compactification radii, suggesting a universal short-distance scale.
• Considers the role of black hole states and their quantum effects, especially the $ \exp(-1/g_s) $ suppression of non-perturbative processes.
• Uses dual heterotic and mirror symmetry descriptions to probe the structure of moduli spaces and the behavior of solitons near singularities.

Experimental results

Research questions

• RQ1 Does the logarithmic divergence in the effective coupling $ \tau_D \sim \log Z $ near the conifold point imply a new UV cutoff at $ m_s/g_s $, rather than at $ m_s $?
• RQ2 Why does the Ramond-Ramond coupling $ \tau_{RR} $ depend on $ \log Z $ and not $ \log(Z/g_s) $, suggesting a scale independent of coupling?
• RQ3 How can the conifold singularity be mapped to the Seiberg-Witten monopole point if the UV cutoff is not at the string scale?
• RQ4 What is the physical interpretation of the scale $ m_s/g_s $, especially since it coincides with the Planck mass in four dimensions?
• RQ5 Is the finiteness of string perturbation theory at the conifold point sufficient to guarantee UV finiteness, or does it require a new short-distance scale?

Key findings

• The logarithmic behavior in the Seiberg-Witten solution for $ \mathcal{N}=2 $ $ SU(2) $ gauge theory suggests a UV cutoff at $ m_{\text{uv}} \sim m_W \sim \Lambda $, consistent with the semiclassical size of magnetic monopoles.
• In $ SU(N) $ generalizations, the hierarchy of $ W $-boson masses from $ \Lambda $ down to $ \Lambda/N^2 $ implies multiple scales, with the cutoff for each $ U(1) $ factor at $ m_{\text{uv},a} \sim m_{W,a} $.
• Near the conifold singularity in type IIB compactifications, the effective coupling $ \tau_{RR} \sim \log Z $ suggests a UV cutoff at $ m_{\text{uv}} \sim m_s/g_s $, not at $ m_s $.
• The independence of $ \tau_{RR} $ from the compactification radius $ R $ after dimensional reduction implies a universal short-distance scale $ m_s/g_s $.
• The scale $ m_s/g_s $, which equals the Planck mass in four dimensions, may regulate black hole production and non-perturbative effects, though its role in UV finiteness remains unclear.
• The existence of this scale raises deep questions about the correct formulation of string theory, particularly whether it signals a new fundamental structure or an artifact of current dualities.