<p>We consider the simplest non-trivial local composite operators in the massless Sine-Gordon model, which are <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\partial _\mu \phi \, \partial _\nu \phi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>∂</mi> <mi>μ</mi> </msub> <mi>ϕ</mi> <mspace width="0.166667em" /> <msub> <mi>∂</mi> <mi>ν</mi> </msub> <mi>ϕ</mi> </mrow> </math></EquationSource> </InlineEquation> and the stress tensor <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$T_{\mu \nu }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mrow> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>. We show that even in the finite regime <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\beta ^2 < 4 \pi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>β</mi> <mn>2</mn> </msup> <mo><</mo> <mn>4</mn> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation> of the theory, these operators need additional renormalisation (beyond the free-field normal-ordering) at each order in perturbation theory. We further prove convergence of the renormalised perturbative series for their expectation values, both in the Euclidean signature and in Minkowski spacetime, and for the latter in an arbitrary Hadamard state. Lastly, we show that one must add a quantum correction (proportional to <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\hbar $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ħ</mi> </math></EquationSource> </InlineEquation>) to the renormalised stress tensor to obtain a conserved quantity.</p>

Local Operators in the Sine-Gordon Model: $\partial _\mu \phi \, \partial _\nu \phi $ and the Stress Tensor

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2025年4月16日

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Markus B. Fröb,
Daniela Cadamuro

摘要

We consider the simplest non-trivial local composite operators in the massless Sine-Gordon model, which are $\partial _\mu \phi \, \partial _\nu \phi $ $\partial_{μ} ϕ \partial_{ν} ϕ$ and the stress tensor $T_{\mu \nu }$ $T_{μ ν}$ . We show that even in the finite regime $\beta ^2 < 4 \pi $ $β^{2} < 4 π$ of the theory, these operators need additional renormalisation (beyond the free-field normal-ordering) at each order in perturbation theory. We further prove convergence of the renormalised perturbative series for their expectation values, both in the Euclidean signature and in Minkowski spacetime, and for the latter in an arbitrary Hadamard state. Lastly, we show that one must add a quantum correction (proportional to $\hbar $ $ħ$ ) to the renormalised stress tensor to obtain a conserved quantity.

Local Operators in the Sine-Gordon Model: \(\partial _\mu \phi \, \partial _\nu \phi \) and the Stress Tensor

摘要