Chiral perturbation theory of the hyperfine splitting in (muonic) hydrogen

Abstract

Abstract The ongoing experimental efforts to measure the hyperfine transition in muonic hydrogen prompt an accurate evaluation of the proton-structure effects. At the leading order in $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> , which is $$O(\alpha ^5)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>α</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math> in the hyperfine splitting (hfs), these effects are usually evaluated in a data-driven fashion, using the empirical information on the proton electromagnetic form factors and spin structure functions. Here we perform a first calculation based on the baryon chiral perturbation theory (B $$\chi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>χ</mml:mi></mml:math> PT). At leading orders it provides a prediction for the proton polarizability effects in hydrogen (H) and muonic hydrogen ( $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>μ</mml:mi></mml:math> H). We find large cancellations among the various contributions leading to, within the uncertainties, a zero polarizability effect at leading order in the B $$\chi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>χ</mml:mi></mml:math> PT expansion. This result is in significant disagreement with the current data-driven evaluations. The small polarizability effect implies a smaller Zemach radius $$R_\textrm{Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mtext>Z</mml:mtext></mml:msub></mml:math> , if one uses the well-known experimental 1 S hfs in H or the 2 S hfs in $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>μ</mml:mi></mml:math> H. We, respectively, obtain $$R_\textrm{Z}(\textrm{H}) = 1.010(9)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mtext>Z</mml:mtext></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mtext>H</mml:mtext><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>1.010</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mn>9</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> fm, $$R_\textrm{Z}(\mu \textrm{H}) = 1.040(33)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mtext>Z</mml:mtext></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>μ</mml:mi><mml:mtext>H</mml:mtext><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>1.040</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mn>33</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> fm. The total proton-structure effect to the hfs at $$O(\alpha ^5)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>α</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math> is then consistent with previous evaluations; the discrepancy in the polarizability is compensated by the smaller Zemach radius. Our recommended value for the 1 S hfs in $$\mu \text {H}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>μ</mml:mi><mml:mtext>H</mml:mtext></mml:mrow></mml:math> is $$182.640(18)\,\textrm{meV}.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>182.640</mml:mn><mml:mo>(</mml:mo><mml:mn>18</mml:mn><mml:mo>)</mml:mo><mml:mspace/><mml:mtext>meV</mml:mtext><mml:mo>.</mml:mo></mml:mrow></mml:math>

Topics & Concepts