Effects of longitudinal short-distance constraints on the hadronic light-by-light contribution to the muon $$\mathbf {g-2}$$
Jan Lüdtke, Massimiliano Procura
Abstract
Abstract We present a model-independent method to estimate the effects of short-distance constraints (SDCs) on the hadronic light-by-light contribution to the muon anomalous magnetic moment $$a_\mu ^\text {HLbL}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>a</mml:mi> <mml:mi>μ</mml:mi> <mml:mtext>HLbL</mml:mtext> </mml:msubsup> </mml:math> . The relevant loop integral is evaluated using multi-parameter families of interpolation functions, which satisfy by construction all constraints derived from general principles and smoothly connect the low-energy region with those where either two or all three independent photon virtualities become large. In agreement with other recent model-based analyses, we find that the SDCs and thus the infinite towers of heavy intermediate states that are responsible for saturating them have a rather small effect on $$a_\mu ^\text {HLbL}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>a</mml:mi> <mml:mi>μ</mml:mi> <mml:mtext>HLbL</mml:mtext> </mml:msubsup> </mml:math> . Taking as input the known ground-state pseudoscalar pole contributions, we obtain that the longitudinal SDCs increase $$a_\mu ^\text {HLbL}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>a</mml:mi> <mml:mi>μ</mml:mi> <mml:mtext>HLbL</mml:mtext> </mml:msubsup> </mml:math> by $$(9.1\pm 5.0) \times 10^{-11}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>9.1</mml:mn> <mml:mo>±</mml:mo> <mml:mn>5.0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>×</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>11</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , where the isovector channel is responsible for $$(2.6\pm 1.5) \times 10^{-11}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2.6</mml:mn> <mml:mo>±</mml:mo> <mml:mn>1.5</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>×</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>11</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . More precise estimates can be obtained with our method as soon as further accurate, model-independent information about important low-energy contributions from hadronic states with masses up to 1–2 GeV become available.