We can tie this to something we measure in the lab by choosing some specificmomenta and spins and evaluating the amplitude.At one loop, the relevant corrections are=+++(b) If the momenta areφ(p)ψ(q1)→ψ(q2), the scalar vertex graph is(iλ)3Zd4k(2π)4u2i(6k+6p+m)i(6k+m)iu1((k+p)2-m2)(k2-m2)((k-q1)2-m2h)=-i(iλ)3Zx,y,zZd4k(2π)4u2(6k+6p+m)(6k+m)u1(k2+ 2xk·p+xp2-2zk·q1+zq21-(x+y)m2-zm2h)3=-i(iλ)3Zx,y,zZd4‘(2π)4u2(6‘+ (1-x)6p+z6q1+m)(6‘-x6p+z6q1+m)u1(‘2+x(1-x)p2+z(1-z)q21+ 2xzp·q1-(x+y)m2-zm2h)3where‘=k+xp-zq1. This has two pieces: the term in the numerator that’squadratic in‘gives a logarithmic divergence proportional tou2u1, while every-thing else is some finite dimensionless rational function of the momenta. The log
Phys 253a6includes some energy scale likep2, so after adding counterterms, this graph willcontribute roughlyaλ316π2logp2p20u2u1+ (dimensionless fn ofp2)×(mom./spin-dependent fn ofu2andu1)The first term has the same form as the tree level amplitude, and also dominatesat high energies. So we can think of it as a contribution to an effective Yukawacoupling at momentump2.Thus, we can ignore finite parts of our Feynmangraphs and just focus on the logarithms.Using dim-reg, our graph gives-i(iλ)3u2u1Zx,y,zi(4π)d/2d2Γ(2-d2)Γ(3)1Δ2-d21+ finite rational fn.=(iλ)316π2u2u1Zx,y,z2-log Δ1+ finite rational fn.where Δ1= (x+y)m2+zm2h-x(1-x)p2-z(1-z)q21-2xzp·q1.(c) Our vertex correction, together with the countertermδλ(which we’ll determinebased on our renormalization condition), combine to give a 1PI three point func-tioniλ+λ316π2Zx,y,zlog Δ1-2+δλuu+rational function of momentaIn anMS scheme, we would just set the countertermδλequal to the divergentpart, and stick aμ2in the logs. In a physical renormalization scheme, we needto use our renormalization condition:λ1PI(p2)=λR+λ316π2Zx,y,zlogΔ1(-p2)Δ1(-p20)=λR+λ3R16π2Zx,y,zlogΔ1(-p2)Δ1(-p20)+ higher order≈λR+λ3R16π2logp2p20,pp0We see that this particular contribution makes our coupling grow with energy.
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- Quantum Field Theory, correlation function, momenta, charge conjugation invariance, counterterm