Physical Sciences

Equation (44) describes the amplitude for the process π-(q)→π+(q)e-e- with on-shell pions q2 = mπ2 and massless electrons with zero four-momentum and not as written in the text above Eq. (44) for π-π-→e-e-. We now consider the more general situation π-(pa)π-(pb)→e-(p1 )e-(p2 ) with on-shell pions pa2 = pb2 = mπ2 and massless electrons p1 2 = p2 2 =0 and introduce the Mandelstam variables s=(pa+pb)2,t=(pa-p1 )2 ,u=(pa-p2 )2. The amplitude Tπ-π-→e-e- can be written as (1) Tπ-π-→e-e-=Tlept2 Fπ2 Sππ+4 GF2 Vud2 mββūL(pe1 )ςμνC ūLT(pe2) Aππμν,where Tlept=4 GF2 Vud2 mββūL(pe1 )C ūLT(pe2 ). The antisymmetric leptonic structure Aππ vanishes if p1 μ= p2 μ. Note that we capture a subset of N2 LO corrections by normalizing the amplitude in terms of the physical pion decay constant Fπ rather than F0. For the amplitude Sππ, we find (2) Sππ=-1 4 1 t + 1 u s-2 mπ2 + 1 (4 πFπ)2 Vππ+ s-2 mπ2 2 5 6 gνππ(μ), where the loop contribution is given by (3) Vππ = 3 s-2 mπ2 2 ln μ2 mπ2 - 2 mπ2 -s 2 ln2 - 1 + 1 - 4 mπ2 s 1 - 1 - 4 mπ2 s 4 s - mπ2 -t ln 1 - t mπ2 mπ4 +6 mπ2 t+t(-s+t) 4 t2 - mπ2 -u ln 1 - u mπ2 mπ4 +6 mπ2 u+u(-s+u) 4 u2 - 6 mπ4 (t+u)+132 mπ2 tu+tu[-45 s+12 (t+u)]24 tu. At threshold, that is, for s=4 mπ2 ,t=- mπ2 ,u=- mπ2 , we obtain (4) Sππ = 1 - mπ2 (4 πFπ)2 3 ln μ2 mπ2 + 7 2 + π2 4 + 5 6 gνππ(μ). At the kinematic point, s=0 ,t= mπ2 ,u= mπ2 ,q2 = mπ2 , which corresponds to the kinematics π-(q)→π+(q)e-(0 )e-(0 ), we recover Eq. (44), (5) Sππ=1 + mπ2 (4 πFπ)2 3 ln μ2 mπ2 +6 + 5 6 gνππ(μ). In this Erratum we specified the correct kinematics for which the amplitude in Eq. (44) of the paper applies. In addition, Eqs. (2) and (3) extend Eq. (44) to a completely general kinematics, which can be useful for the chiral extrapolation of lepton-number-violating amplitudes computed in lattice QCD to the physical point.