One sum to rule them all: a second order master rate sum rule for charm decays

Abstract

A bstract We show that within the Standard Model any system of hadronic weak charm decays related by U -spin satisfies the following rate sum rule: $$ \frac{\left(\mathrm{sum}\ \mathrm{of}\ \mathrm{CF}\ \mathrm{and}\ \mathrm{DCS}\ \mathrm{CKM}-\mathrm{free}\ \mathrm{rates}\right)}{\left(\mathrm{sum}\ \mathrm{of}\ \mathrm{SCS}\ \mathrm{CKM}-\mathrm{free}\ \mathrm{rates}\right)}=1, $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mfenced> <mml:mrow> <mml:mi>sum</mml:mi> <mml:mspace/> <mml:mtext>of</mml:mtext> <mml:mspace/> <mml:mi>CF</mml:mi> <mml:mspace/> <mml:mtext>and</mml:mtext> <mml:mspace/> <mml:mi>DCS</mml:mi> <mml:mspace/> <mml:mi>CKM</mml:mi> <mml:mo>−</mml:mo> <mml:mtext>free rates</mml:mtext> </mml:mrow> </mml:mfenced> <mml:mfenced> <mml:mrow> <mml:mi>sum</mml:mi> <mml:mspace/> <mml:mtext>of</mml:mtext> <mml:mspace/> <mml:mi>SCS</mml:mi> <mml:mspace/> <mml:mi>CKM</mml:mi> <mml:mo>−</mml:mo> <mml:mtext>free rates</mml:mtext> </mml:mrow> </mml:mfenced> </mml:mfrac> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> </mml:math> which holds up to second order in U -spin breaking. We test this sum rule against available data and find that it is well satisfied in all cases. For systems in which some decay rates have not yet been measured, we use this sum rule to predict the missing rates.