[Paper Review] Once more on the BPS bound for the susy kink

Andrei Litvintsev, P. van Nieuwenhuizen|ArXiv.org|Oct 7, 2000

Quantum Mechanics and Non-Hermitian Physics9 references16 citations

TL;DR

This paper resolves long-standing ambiguities in one-loop quantum corrections to the mass and central charge of N=(1,1) supersymmetric kinks in 1+1 dimensions by introducing two novel regularization schemes. It proposes a momentum cut-off scheme with a smooth function f(k) that modifies zero-point energy sums, and a two-parameter cut-off method (K for Dirac delta functions, Λ for loop momentum) to regulate the central charge. The key result is that M^(1) = Z^(1) = -ħm/(2π), confirming the BPS bound and multiplet shortening in the supersymmetric kink.

ABSTRACT

We consider a new momentum cut-off scheme for sums over zero-point energies, containing an arbitrary function f(k) which interpolates smoothly between the zero-point energies of the modes around the kink and those in flat space. A term proportional to df(k)/dk modifies the result for the one-loop quantum mass M^(1) as obtained from naive momentum cut-off regularization, which now agrees with previous results, both for the nonsusy and susy case. We also introduce a new regularization scheme for the evaluation of the one-loop correction to the central charge Z^(1), with a cut-off K for the Dirac delta function in the canonical commutation relations and a cut-off Λfor the loop momentum. The result for Z^(1) depends only on whether K>Λor K

Motivation & Objective

• To resolve persistent inconsistencies in one-loop quantum corrections to the mass M^(1) and central charge Z^(1) of supersymmetric kinks in (1+1)D field theories.
• To address unresolved regularization and renormalization issues in earlier approaches that led to conflicting results for M^(1) and Z^(1).
• To establish a consistent framework where the BPS bound M^(1) = Z^(1) is saturated, confirming multiplet shortening in N=(1,1) supersymmetric systems.
• To provide new, unambiguous derivations of M^(1) and Z^(1) using novel regularization techniques that resolve ambiguities in prior methods.

Proposed method

• Introduces a momentum cut-off scheme with a smooth function f(k) that interpolates between kink and flat space mode zero-point energies, modifying the density of states via f(k) and f'(k) terms.
• Proposes a two-parameter regularization for Z^(1) using separate cut-offs K (for Dirac delta functions in commutation relations) and Λ (for loop momentum), with results depending on the relative size of K and Λ.
• Applies mode number regularization to a kink-antikink system, showing that 4 bosonic and 3 fermionic discrete states lead to consistent results when combined with twisted boundary conditions.
• Uses twisted boundary conditions for fermions (ψ₁(−L/2) = ±ψ₂(L/2), etc.) to avoid boundary energy contributions and ensure consistency with supersymmetry.
• Demonstrates that the K = Λ case yields the correct Z^(1), while K > Λ or K < Λ reproduce previous inconsistent results, validating the new scheme.
• Establishes that the correct result M^(1) = Z^(1) = -ħm/(2π) is obtained only when the full regularization structure is applied, particularly the f'(k) term and the two-cut-off scheme.

Experimental results

Research questions

• RQ1 Why do previous regularization schemes for the one-loop mass M^(1) of the susy kink yield inconsistent results, and how can this be resolved?
• RQ2 How can the central charge correction Z^(1) be consistently regularized when it classically vanishes due to being a total derivative?
• RQ3 What is the role of the function f(k) in modifying the zero-point energy sum, and why is the term proportional to f'(k) essential for consistency?
• RQ4 Why does the K = Λ case in the two-cut-off scheme for Z^(1) yield the correct result, while other cases do not?
• RQ5 How does mode number regularization applied to a kink-antikink system reproduce the correct M^(1) and Z^(1) values, and why is the number of fermionic modes one less than bosonic modes?

Key findings

• The inclusion of a term proportional to f'(k) in the zero-point energy sum, via a modified quantization condition, corrects the naive momentum cut-off result and yields M^(1) = -ħm(3/(2π) - √3/12) for the nonsusy kink and M^(1) = -ħm/(2π) for the susy kink.
• The two-cut-off scheme for Z^(1), with cut-offs K (for Dirac deltas) and Λ (for loop momentum), yields the correct result only when K = Λ, which saturates the BPS bound.
• The correct value Z^(1) = -ħm/(2π) is obtained only in the K = Λ case, confirming that the anomaly in the central charge is properly regulated and multiplet shortening occurs.
• Mode number regularization applied to a kink-antikink system yields the correct M^(1) and Z^(1) when 4 bosonic and 3 fermionic discrete states are counted, consistent with the anomaly mechanism.
• Twisted boundary conditions for fermions (ψ₁(−L/2) = ±ψ₂(L/2), etc.) are shown to be consistent and free of boundary energy, explaining why they yield correct results.
• The paper confirms that M^(1) = Z^(1) = -ħm/(2π) holds in the N=(1,1) susy kink, validating multiplet shortening and the BPS bound at one-loop order.