Neutrino force at all length scales

The Standard Model predicts a long-range force mediated by a pair of neutrinos, known as “the neutrino force.” It scales as <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msubsup> <a:mi>G</a:mi> <a:mi>F</a:mi> <a:mn>2</a:mn> </a:msubsup> <a:mo>/</a:mo> <a:msup> <a:mi>r</a:mi> <a:mn>5</a:mn> </a:msup> </a:math> , where <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>G</c:mi> <c:mi>F</c:mi> </c:msub> </c:math> is the Fermi constant. However, as <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>r</e:mi> <e:mo>≲</e:mo> <e:msqrt> <e:msub> <e:mi>G</e:mi> <e:mi>F</e:mi> </e:msub> </e:msqrt> </e:math> , the four-Fermi theory breaks down and the neutrino force no longer has the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mn>1</g:mn> <g:mo>/</g:mo> <g:msup> <g:mi>r</g:mi> <g:mn>5</g:mn> </g:msup> </g:math> scaling. For the first time, we derive a complete expression for the neutrino force that is valid at all distances. For <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>r</i:mi> <i:mo>≫</i:mo> <i:msqrt> <i:msub> <i:mi>G</i:mi> <i:mi>F</i:mi> </i:msub> </i:msqrt> </i:math> , the result reduces to the known <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msubsup> <k:mi>G</k:mi> <k:mi>F</k:mi> <k:mn>2</k:mn> </k:msubsup> <k:mo>/</k:mo> <k:msup> <k:mi>r</k:mi> <k:mn>5</k:mn> </k:msup> </k:math> ; for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>r</m:mi> <m:mo>≪</m:mo> <m:msqrt> <m:msub> <m:mi>G</m:mi> <m:mi>F</m:mi> </m:msub> </m:msqrt> </m:math> , it scales as <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mn>1</o:mn> <o:mo>/</o:mo> <o:mi>r</o:mi> </o:math> . We explore the implications of this result for atomic parity violation (APV) experiments. A key feature of the neutrino force is that it is a long-range effect compared to the atomic length scale. Thus, in general, it cannot be simply treated as a correction to the tree-level <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>Z</q:mi> </q:math> -exchange diagram without considering the atomic wave functions. We calculate the effects in muonium and positronium, finding that the neutrino force contributes about 4% and 16%, respectively, compared to the leading <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>Z</s:mi> </s:math> exchange. This indicates a significant impact on APV, with important implications for detecting the neutrino force and measuring the weak mixing angle in APV experiments.