Last Latexed: October 24, 2013 at 10:23 1 Lecture 15 Oct. 24, 2013 Fermion Potential, Yukawa and Photon Exchange Copyright c 2007 by Joel A. Shapiro Last week we learned how to calculate cross sections in terms of the S- matrix or the invariant matrix element . Last time we learned, pretty M much, how to evaluate in terms of matrix element of the time ordered product of free fields (i.e.M interaction representation fields) between states given na¨ıvely by creation and annihilation operators for the incoming and outgoing particles. Today we are going to address the first “real” applications. First we will consider the low energy scattering of two different fermions • each coupled to a single scalar field, as proposed by Yukawa to explain the nuclear potential1. Then we will give Feynman rules for the electromagnetic field, that is, • for photons. This will not be rigorously derived until next semester, but we can motivate a set of correct rules, which will get elaborated as we go on. Finally, we will couple the fermions with photons instead of Yukawa • scalar particles, and see that this differs from the Yukawa case not only because the photon is massless, and therefore the range of the Coulomb potential is infinite, but also in being repulsive between two identical particles, and attractive only between particle - antiparticle pairs. We also see that gravity should, like the Yukawa potential, be always attractive. Read Peskin and Schroeder pp. 118–126. Let us consider the scattering of two distinguishable fermions 1This is often mentioned as a great success, with the known range of the nuclear potential predicting the existance of the pion. Of course we now know that the pion is not a scalar, but rather a pseudoscalar, with a coupling igψγ¯ 5ψφ instead of gψψφ¯ . Thenu ¯(~p ′)γ5u(~p) χ†~σ (~p ′ ~p)χ/(2m), which complicates the potential. It becomes spin dependent, but≈ still,· the factor− e−µr/r emerges, modified by a quadratic expression in (1/µr). These extra factors actually play an important role in the nuclear physics potential. 615: Lecture 15: Yukawa and Coulomb Last Latexed: October 24, 2013 at 10:23 2 p' s' k' r' † † 0 a ′ a ′ ψ¯ ψ ψ¯ ψ a a 0 h | Bk Ap B B A A Ap Bk | i A q B p s k r ′ ′ u¯s (p′, m )( ig)us(p, m )¯ur (k′, m )( ig)ur(k, m ) i = i A − A B − B M (p′ p)2 m2 + iǫ − − φ s s √p σξ 0 0 1I 0 Consider at low energy, u (p, m ) = , γ = , p A p · σξs √ ¯ 1I 0 ∼ ′ ′ · m ~p , sou ¯s (p′)us(p) = ξs † p′ σ p σ¯ + p′ σ¯ p σ ξs, √p σ = ≫ | | · · · · · ′ q q ′ q q′ √E√1 ~v ~σ √m,sou ¯s (p′)us(p) 2mξs †ξs =2mδss . − · → ∼ (ig)2 (p′ p)2 in the center of mass is (~p ′ ~p)2. i = i . − − − M [(~p ′ ~p)2 + m2] − − Do 4.123-4.127. Now consider particle-antiparticle ′ ′ † † p k¯ ψψ¯ ψψ¯ pk¯ 0 b ′ a ′ ψ¯ψψ¯ψa b 0 ∼h | k p p k | i D E s′ ′ s r r′ ′ 2 u¯ (p )u (p)¯v (k)v (k ) i = ig − M − (~p ′ ~p)2 m2 − − − The minus sign in the numerator comes as follows: The two lowest shown contractions are both plus, which leaves bψψb¯ †, so the next higher contraction of ψ¯ with b† gives a minus, and then the last gives plus. 0 1I √p′ σ ′ ′ vv¯ = ξr † √p σ, √p σ¯ · ξr 2mδrr · − · 1I 0 √p′ σ¯ → − − · 2 2 g ′ ′ g ′ ′ Again δrr δss (2m)2 δrr δss M→ ~q 2 + m2 −→norm ~q 2 + m2 So the Yukawa potential is attractive for particles and antiparticles in any combination, pp, pp¯,orp ¯p¯. Note that actually the pi meson is actually a pseudoscalar. So the factors are changed ′ ′ 5 ′ 0 5 √p σ u¯s (p′)us(p) u¯s (p′)iγ us(p)= ξs † p′ σ, p′ σ¯ i γ γ ξs p · σ → · · √ ¯ q q 0 1I · |1I{z } 0 − 615: Lecture 15: Yukawa and Coulomb Last Latexed: October 24, 2013 at 10:23 3 ′ = iξs † p′ σ p σ¯ p′ σ¯ p σ ξs, · · − · · q q q q Note to lowest order, p (m, 0, 0, 0) and this vanishes. → To next order, p σ = √E√1 ~v ~σ √E 1 1~v ~σ √m 1 1~v ~σ · − · ∼ − 2 · ∼ − 2 · and p σ¯ √mq1+ 1~v ~σ so · ∼ 2 · q ′ ′ 1 1 1 1 u¯s (p′)iγ5us(p) imξs † ~v ′ ~σ + ~v ~σ ~v ′ ~σ + ~v ~σ ξs ∼ −2 · 2 · − −2 · 2 · ′ ′ = imξs † (~v ′ ~v) σξs imξs †~q σξs. − − · ∼ − · This complicates the form of the effective potential. Back to p 123 After 4.133: µ ′ 0 1I 0 σ √p σ u¯(p′)γµu(p) = ξs † p′ σ, p′ σ¯ · ξs · · 1I 0 σ¯µ 0 √p σ¯ q q ′ · = ξs † p′ σσ¯µ p′ σ + p σσ¯ µ p σ¯ ξs · · · · q′ q q q 2mξs †ξs for µ =0, → ( 0 for µ =0 6 so compared to Yukawa, we have 1 from g00, repulsive,vγ ¯ 0v v†v = ′ ξ†...ξ +2mδrr , so compared to Yukawa,− we have a from g00∼ from → − − − vv¯ , so attractive..