Efficient computation of the N -th rank QED polarization tensor: Universal worldline structure of form factors

We derived in [, ] a compact expression for the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>N</a:mi> </a:math> th rank QED polarization tensor <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi mathvariant="normal">Π</c:mi> <c:mrow> <c:msub> <c:mi>μ</c:mi> <c:mn>1</c:mn> </c:msub> <c:mo>⋯</c:mo> <c:msub> <c:mi>μ</c:mi> <c:mi>N</c:mi> </c:msub> </c:mrow> </c:msub> <c:mo stretchy="false">(</c:mo> <c:msub> <c:mi>k</c:mi> <c:mn>1</c:mn> </c:msub> <c:mo>,</c:mo> <c:mo>…</c:mo> <c:mo>,</c:mo> <c:msub> <c:mi>k</c:mi> <c:mi>N</c:mi> </c:msub> <c:mo stretchy="false">)</c:mo> </c:math> in a ( <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mrow> <h:mn>0</h:mn> <h:mo>+</h:mo> <h:mn>1</h:mn> </h:mrow> </h:math> )-dimensional worldline framework. This fully off-shell object, a function of <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:mi>N</j:mi> </j:math> external photon 4-momenta, is a key ingredient in high-order computations of cusp anomalous dimensions and lepton anomalous magnetic moments. We demonstrate here that <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:msub> <l:mi mathvariant="normal">Π</l:mi> <l:mrow> <l:msub> <l:mi>μ</l:mi> <l:mn>1</l:mn> </l:msub> <l:mo>⋯</l:mo> <l:msub> <l:mi>μ</l:mi> <l:mi>N</l:mi> </l:msub> </l:mrow> </l:msub> </l:math> can further be expressed simply in terms of a small number of independent “head” form factors [each representing <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mo stretchy="false">(</o:mo> <o:mi>N</o:mi> <o:mo>−</o:mo> <o:mn>1</o:mn> <o:mo stretchy="false">)</o:mo> <o:mo>!</o:mo> <o:mo>/</o:mo> <o:mn>2</o:mn> </o:math> Feynman diagrams] which have a universal structure in terms of sums over fermion Green’s functions and (proper time derivative of) their boson worldline superpartners. This worldline representation bypasses explicit Wick contractions and avoids tensor reductions to scalar loop integrals Passarino and Veltman, order by order in perturbation theory. We give explicit expressions for the fourth and sixth rank head form factors and provide a computer script generalizing these results to arbitrary <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>N</s:mi> </s:math> external photons. The multiplicity of heads, and their growth with <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi>N</u:mi> </u:math> , can be understood in terms of orbits of the permutation group. We employ the Burnside-Cauchy-Frobenius lemma to show that it scales as <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:msup> <w:mi>e</w:mi> <w:mrow> <w:mi>N</w:mi> <w:mo>−</w:mo> <w:mn>1</w:mn> </w:mrow> </w:msup> <w:mo>/</w:mo> <w:msqrt> <w:mi>N</w:mi> </w:msqrt> </w:math> terms as opposed to the <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:msup> <y:mi>e</y:mi> <y:mrow> <y:mi>N</y:mi> <y:mo>−</y:mo> <y:mn>1</y:mn> </y:mrow> </y:msup> <y:mi>N</y:mi> <y:mo>!</y:mo> <y:mo>/</y:mo> <y:msqrt> <y:mi>N</y:mi> </y:msqrt> </y:math> terms in conventional perturbation theory. We reexpress worldline parameter integrals that define the fourth rank heads as Feynman parameter integrals to reproduce the seminal results by Karplus and Neuman for the on-shell light-by-light amplitude and extend these to the fully off-shell case in massless QED employing a tailored integration-by-parts procedure. In a follow-up paper, we will discuss the direct computation of worldline integrals, potentially providing a further <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:mi>N</ab:mi> <ab:mo>!</ab:mo> </ab:math> advantage relative to Feynman diagram computations at high orders in perturbation theory.