Bounding anisotropic Lorentz invariance violation from measurements of the effective energy scale of quantum gravity

This is the authors' abstract. We don't add key points for this paper.

Observations of energy-dependent photon time delays from distant flaring sources provide significant constraints on Lorentz invariance violation (LIV). Such effects originate from modified vacuum dispersion relations causing differences in propagation times for photons emitted simultaneously from gamma-ray bursts, active galactic nuclei, or pulsars. These modifications are often parametrized within a general framework by an effective quantum gravity energy scale <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>E</a:mi> <a:mrow> <a:mi>Q</a:mi> <a:mi>G</a:mi> <a:mo>,</a:mo> <a:mi>n</a:mi> </a:mrow> </a:msub> </a:math> . While such general constraints are well established in the LIV literature, their translation into specific coefficients of alternative theoretical frameworks, such as the Standard-Model extension (SME), is rarely carried out. In particular, existing bounds on the quadratic case ( <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>n</c:mi> <c:mo>=</c:mo> <c:mn>2</c:mn> </c:math> ) of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>E</e:mi> <e:mrow> <e:mi>Q</e:mi> <e:mi>G</e:mi> <e:mo>,</e:mo> <e:mi>n</e:mi> </e:mrow> </e:msub> </e:math> can be systematically converted into constraints on the nonbirefringent, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mrow> <g:mi>C</g:mi> <g:mi>P</g:mi> <g:mi>T</g:mi> </g:mrow> </g:math> -conserving SME coefficients <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msubsup> <i:mi>c</i:mi> <i:mrow> <i:mo stretchy="false">(</i:mo> <i:mi>I</i:mi> <i:mo stretchy="false">)</i:mo> <i:mi>j</i:mi> <i:mi>m</i:mi> </i:mrow> <i:mrow> <i:mo stretchy="false">(</i:mo> <i:mn>6</i:mn> <i:mo stretchy="false">)</i:mo> </i:mrow> </i:msubsup> </i:math> . This work provides a concise overview of the relevant SME formalism and introduces a transparent conversion method from <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:msub> <o:mi>E</o:mi> <o:mrow> <o:mi>Q</o:mi> <o:mi>G</o:mi> <o:mo>,</o:mo> <o:mn>2</o:mn> </o:mrow> </o:msub> </o:math> to SME parameters. We review the most stringent time-of-flight-based bounds on <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:msub> <q:mi>E</q:mi> <q:mrow> <q:mi>Q</q:mi> <q:mi>G</q:mi> <q:mo>,</q:mo> <q:mi>n</q:mi> </q:mrow> </q:msub> </q:math> and standardize them by accounting for systematics, applying missing prefactors, and transforming results into two-sided Gaussian uncertainties where needed. We then use these standardized constraints, along with additional bounds from the literature, to improve bounds on the individual SME coefficients of the photon sector by about an order of magnitude. A consistent methodology is developed to perform this conversion from the general LIV framework to the SME formalism.