[Paper Review] Non-orientable string one-loop corrections in the presence of a B field

L. Bonora, M. Salizzoni|ArXiv.org|May 30, 2001

Noncommutative and Quantum Gravity Theories44 references3 citations

TL;DR

This paper proposes a string-theoretic derivation of one-loop corrections in noncommutative SO(N) gauge theories using the Möbius strip worldsheet in the presence of a constant B field. By constructing a boundary string propagator on the Möbius strip and computing 2-, 3-, and 4-gluon amplitudes, the authors show that the field theory limit reproduces the one-loop corrections of a renormalizable noncommutative SO(N) gauge theory, suggesting consistency between string theory and effective field theory despite apparent nonrenormalizability in the field-theory approach.

ABSTRACT

We discuss the problem of noncommutative SO(N) gauge field theories from the string one-loop point of view. To this end we propose an expression for the string propagator on the boundary of the Moebius strip in the presence of a constant B field. We discuss in detail the problems related to its derivation. Then we use it to compute the one-loop corrections to two-, three- and four-gluon amplitudes in an open string theory with orthogonal Chan-Paton factors. We show that these corrections in the field theory limit in 4D are compatible with the one-loop corrections of a renormalizable noncommutative SO(N) gauge field theory.

Motivation & Objective

• To investigate whether noncommutative SO(N) gauge field theories, which appear nonrenormalizable at one loop in field theory, are consistent with string theory in the presence of a B field.
• To construct a string propagator on the boundary of the Möbius strip in the presence of a constant B field, a nontrivial extension of known string Green functions.
• To compute one-loop amplitudes for two-, three-, and four-gluon processes in an open string theory with orthogonal Chan-Paton factors and a B field.
• To compare the field theory limit of these string amplitudes with the one-loop corrections of a noncommutative SO(N) gauge theory, assessing consistency.

Proposed method

• Derives the string propagator on the Möbius strip in the absence of a B field using conformal field theory techniques and boundary conditions on the non-orientable worldsheet.
• Extends the propagator to include a constant B field by modifying the boundary conditions and holonomy structure, ensuring consistency with the open string Chan-Paton factor structure.
• Uses the boundary propagator to compute one-loop amplitudes for 2-, 3-, and 4-gluon vertex operators inserted on the boundary of the Möbius strip.
• Applies the field theory limit (low-energy, large string tension) to extract effective field theory corrections, comparing them with known results for noncommutative SO(N) gauge theories.
• Employs SO(N) Lie algebra tensors and traces in the fundamental and adjoint representations to compute gauge-invariant amplitudes and renormalization constants.
• Analyzes the consistency of the string-theoretic results with the field-theory expectation, particularly regarding renormalizability and gauge symmetry preservation.

Experimental results

Research questions

• RQ1 Does the one-loop correction in string theory with orthogonal Chan-Paton factors and a B field yield a field theory limit consistent with a renormalizable noncommutative SO(N) gauge theory?
• RQ2 How can a consistent string propagator be defined on the boundary of the Möbius strip in the presence of a B field, given the non-orientable nature of the worldsheet?
• RQ3 Why do noncommutative SO(N) gauge theories appear nonrenormalizable at one loop in field theory, while the corresponding string theory appears well-behaved?
• RQ4 Can the discrepancy between string theory and effective field theory results at one loop be resolved by modifying Feynman rules, or is it fundamental?
• RQ5 Is the apparent nonrenormalizability of noncommutative SO(N) theories a sign of inconsistency, or is it an artifact of the field-theory approximation?

Key findings

• The one-loop corrections computed from the string theory on the Möbius strip with a B field reproduce the same renormalization constants as those of a renormalizable noncommutative SO(N) gauge theory in the field theory limit.
• The field theory limit of the string amplitudes for two-, three-, and four-gluon processes matches the known results for noncommutative SO(N) gauge theories, indicating consistency at the one-loop level.
• The string-theoretic approach yields finite and gauge-invariant corrections, suggesting that the apparent nonrenormalizability of the field-theory version may be an artifact of the perturbative approach.
• The construction of the boundary propagator on the Möbius strip in the presence of a B field is nontrivial and requires careful treatment of the bulk extension, which remains a key technical challenge.
• Despite the apparent nonrenormalizability of noncommutative SO(N) field theories, the string theory framework produces a well-defined and finite one-loop correction, indicating that the field theory may not be the correct effective description at loop level.
• The authors find no decisive obstruction to the consistency of the string-theoretic framework, suggesting that the discrepancy between string and field theory results may be due to an incomplete set of Feynman rules in the field theory approach.