Physical running of the electric charge in a dimensionless theory of gravity

We revisit the renormalization of the gauge coupling in massless QED coupled to a scaleless quadratic theory of gravity. We compare two alternative prescriptions for the running of the electric charge: (i) the conventional <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>μ</a:mi> </a:math> running in minimal subtraction, and (ii) a extracted from the logarithmic dependence of amplitudes on a hard scale <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msup> <c:mi>Q</c:mi> <c:mn>2</c:mn> </c:msup> </c:math> (e.g., <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msup> <e:mi>p</e:mi> <e:mn>2</e:mn> </e:msup> </e:math> or a Mandelstam invariant) after removing IR effects. At one loop, using dimensional regularization with an IR mass regulator <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>m</g:mi> </g:math> , we compute the photon vacuum polarization. We find a clean separation between UV and soft logarithms: the former is gauge and process independent and fixes the beta function, whereas the latter encodes nonlocal, IR-dominated contributions that may depend on gauge parameters and must not be interpreted as UV running. In the quadratic-gravity sector, the photon self-energy is UV finite—the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mrow> <i:mi>ln</i:mi> <i:msup> <i:mrow> <i:mi>μ</i:mi> </i:mrow> <i:mrow> <i:mn>2</i:mn> </i:mrow> </i:msup> </i:mrow> </i:math> pieces cancel—leaving only <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>ln</k:mi> <k:mo stretchy="false">(</k:mo> <k:msup> <k:mi>Q</k:mi> <k:mn>2</k:mn> </k:msup> <k:mo>/</k:mo> <k:msup> <k:mi>m</k:mi> <k:mn>2</k:mn> </k:msup> <k:mo stretchy="false">)</k:mo> </k:math> soft (IR) logs, which do not contribute to the running of the coupling once IR effects are consistently removed. Consequently, quadratic gravity does not modify the one-loop UV coefficient and thus does not alter <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>β</o:mi> <o:mo stretchy="false">(</o:mo> <o:mi>e</o:mi> <o:mo stretchy="false">)</o:mo> </o:math> . Therefore, the physical running coincides with the <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>μ</s:mi> </s:math> running in QED at one loop. Our analysis clarifies how to extract a gauge and process independent running in the presence of gravitational interactions and why soft logs from quadratic gravity should not contribute to <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi>β</u:mi> <u:mo stretchy="false">(</u:mo> <u:mi>e</u:mi> <u:mo stretchy="false">)</u:mo> </u:math> .