The Effective Theory of Long Strings (1302.6257v2)

Abstract: We present the low-energy effective theory on long strings in quantum field theory, including a streamlined review of previous literature on the subject. Such long strings can appear in the form of solitonic strings, as in the 4d Abelian Higgs model, or in the form of confining strings, as in Yang-Mills theories. The bottom line is that upon expanding in powers of 1/L the energy levels of long (closed) strings (where L is the length of the string), all the terms up to (and including) order 1/L^5 are universal. We argue that for excited strings in D>3 space-time dimensions there is a universal deviation at order 1/L^5 from the naive formula that is usually used to fit lattice results. For D=3 this naive formula is valid even at order 1/L^5. At order 1/L^7 non-universal terms generically appear in all cases. We explain the physical origin of these results, and illuminate them in three different formulations of the effective action of long strings (the relationships among which we partly clarify). In addition, we corroborate these results by an explicit computation of the effective action on long strings in confining theories which have a gravitational dual. These predictions can be tested by precise simulations of 4d Yang-Mills theory on the lattice.

• The paper presents that leading-order energy levels for long strings are universal, valid up to order 1/L across different models.
• It employs multiple formulations—including the static gauge and Polchinski-Strominger framework—to demonstrate equivalent effective actions.
• The findings have practical implications for lattice QCD, predicting universal deviations in excited states for dimensions greater than three.

The Effective Theory of Long Strings

The paper "The Effective Theory of Long Strings" examines the low-energy effective theory for long strings in quantum field theories, such as the solitonic strings in the Abelian Higgs model and confining strings in Yang-Mills theories. It provides a detailed analysis of the energy levels of these strings, expanding them in powers of 1/L, where L is the string length. One of the key findings is that, up to order 1/L, these energy levels are universal, implying model-independent characteristics, whereas non-universal terms typically emerge at order 1/L.

This universal nature of leading-order terms stems from the presence of massless excitations confined to the string's worldsheet, despite the bulk of the theory having a mass gap. These excitations are typically described by Nambu-Goldstone bosons (NGBs) due to symmetry breaking caused by the presence of a string in space-time.

Interestingly, the paper reveals that for dimensions D > 3, excited string states exhibit a universal deviation at order 1/L from naive formulae often used to fit lattice data. For D = 3, however, the naive formula remains valid at this order. To explain these results, the paper introduces three different formulations of the effective action for long strings, demonstrating their equivalence and exploring their implications.

The formulation comparison highlights that despite different presentations, all effective actions yield the same leading-order terms, which are universal. The paper employs the unitary static gauge, reparameterization-invariant formalism, and the Polchinski-Strominger (PS) framework to articulate the subtle interplay between the symmetry constraints and the resulting energy spectra.

The paper fortifies its theoretical predictions with explicit calculations for confining strings in field theories possessing a gravitational dual. This context allows for a direct computation of the effective string action, further substantiating the theoretical conclusions.

The implications of these findings are manifold, particularly in lattice QCD and related fields where precision measurement of string energies can help verify these theoretical predictions. The universality at leading orders ensures a robust framework for predicting string behaviors within a variety of models, providing a powerful tool for exploring and understanding complex quantum field configurations.

Looking forward, the paper proposes further investigations into higher-order deviations and their model-dependent characteristics. The low-energy effective theory's capability to express energy levels in terms of unknown coefficients from higher-order terms implies a potential for the continued impact of this approach as computational techniques advance.

Ultimately, this paper contributes significantly to the understanding of string dynamics within quantum field theory, framing a set of precise predictions amenable to future verification through lattice simulations and broader applications within theoretical physics.