Introduction to the physics of highly charged ions. Lecture 12: Self-energy and vertex correction

Transcription

1 Introduction to the physics of highly charged ions Lecture 12: Self-energy and vertex correction Zoltán Harman Universität Heidelberg,

2 Recapitulation from the previous lecture Feynman rules for QED processes S-matrix element for a process N: ) N 2 S fi = i(2π) 4 δ (p (4) 1 + p 2 p N j N N k k M fi 2E k=1 j=1 j V 2E k=1 k V. The transition amplitude M fi = n M(n) fi can be constructed according to the following rules: 1 In order n, draw all Feynman diagrams with n vertices, containing the right number of initial and final particles 2 For external lines, write down the following expressions as factors:

3 3 For internal lines: 4 For all vertices, the following factor has to be included: The index µ is contracted with the index of the internal or external photon line: µ γµɛµ or µ γµdµν F 5 At all vertices, 4-momentum conservation holds: a factor δ (4) ( j p j k p k); and integration has to be performed over the undetermined momentum variable p: d 4 p (2π) 4 6 The amplitudes of all contributing diagrams have to be added coherently, with the phase factors -1 for incoming positrons; -1 for fermion exchange graphs; -1 for all closed e /e + loops.

4 The S matrix at higher orders So far: calculations at the lowest orders in e or α. Since α 1 137, this is often a useful approximation. However, with a good theory, one should be able to corrections of higher order (at least in principle). Also, many experiments are more accurate than the leading-order theory only. At first, the small higher-order corrections turned out to be infinitely large. Renormalization is necessary to arrive to finite results.

5 Vacuum polarization The possibility of creating a virtual electron-positron pair influences the photon propagator id Fµν (q) = id Fµν(q) + id Fµλ (q) iπλσ (q) id Fσν (q), 4π with the vacuum polarization tensor [ ] iπ λσ (q) = e 2 d 4 k 4π (2π) 4 Tr 1 1 γ λ /k m + iɛ γσ /k /q m + iɛ Regularized VP tensor: Π µν(q) = (q 2 g µν q µq ν) ( e2 3π withπ R (q 2 ) = 2e2 π e2 π 1 0 q 2 m 2 ) Λ2 ln m 2 + ΠR (q 2 ), dββ(1 β) ln [1 β(1 β) q2 m 2 ( q 2 ) m ]

6 For small q ("weak" scattering), Π R (q 2 ) 0: M (2) fi = Z 3 M (1) fi, with Z 3 = 1 e2 Λ2 ln 3π m 2. The effective, physical charge of the electron is renormalized to e R = Z 3 e, with e being the bare charge. Only e R is observable, and it only weakly depends on Λ. Effect of VP on an atomic electron Modification of the Coulomb potential A 0 ( x) = Ze/ x in momentum space: A 0 ( q) = A 0( q) + Π R ( q 2 )A 0 ( q), or in real space A 0 ( x) = d 3 q ( ) (2π) 3 ei q x 1 + Π R ( q 2 ) A 0 ( q) Ze x e(zα) 4 15m 2 δ(3) ( x). Non-relativistic energy shift of a bound hydrogenic state ψ nl in first-order perturbation theory: E VP nl Value for 2s electrons: E VP 2s 4 = ψ nl e A 0 ψ nl = α(zα) 15m 2 ψ nl(0) 2 = 4m 15πn 3 α(zα)4 δ l0. = ev = 27.1 MHz contradicts the experiment!

7 The self-energy of the electron Modification of the unperturbed e i propagator ig F (p) = through the Feynman /p m+iɛ diagram describing emission and re-absorption of a virtual photon: with the self-energy function (4 4 matrix) ig F (p) = ig F(p) + ig F (p) ( iσ(p)) ig F (p), iσ(p) = ( ie) 2 d 4 k (2π) 4 4πi i k 2 + iɛ γµ γµ (1) /p /k m + iɛ We make the following ansatz: Σ(p) = A + B(/p m) + Σ R (p)(/p m) 2, Taylor expansion around the mass shell /p = m. Here, A, B and Σ R do not contain γ matrices; A and B are constants.

8 To understand the physical meaning of A and B, we sum up higher-order SE diagrams (loop-after-loop diagrams). They can be summed up as a geometrical series ( k=0 xk = 1/(1 x)): ig F(p) = ig F (p) + ig F (p) ( iσ(p)) ig F (p) + ig F (p) ( iσ(p)) ig F (p) ( iσ(p)) ig F (p) +... = 1 ig F (p) 1 Σ(p)G F (p) = i G 1 F (p) Σ(p) = i /p m Σ(p) + iɛ = i /p m A B(/p m) (/p m) 2 Σ(p) + iɛ i (/p m A)(1 B)(1 (/p m)σ R (p)) + iɛ (negl. small terms with A 2, B 2, AB) (1 + B)i (/p m A)(1 (/p m)σ R (p)) + iɛ

9 For free electrons, for which p 2 = m 2 holds (i.e. "on the mass shell"): ig F (p) (1 + B)i /p m A + iɛ = Z 2iG F (p, m m + δm) with Z B ; δm A. The renormalized (physical) mass of the electron is m R = m + δm = 511 kev; sum of the bare mass and the self-energy (self mass), acquired through interaction with the self-generated electromagnetic field ( friction ). Only m R is measurable, this self-field cannot be switched off in an experiment. Z 2 is another e charge renormalization constant, multiplying every electron/positron propagator (just like Z 3 coming from VP, multiplying every photon propagator): e R = Z 2 e It is not Z like for e R = Z 3 e, because there are always 2 electron lines at 1 vertex. Wave function renormalization: u (p) = Z 2 u(p).

10 Now let us calculate and regularize the SE function of Eq. (1). divergence for low values of k, too (IR divergence; we ll see later why) introduce a finite photon mass µ; at the end, we take the limit lim µ 0 The integral diverges linearly in k as k (UV divergence) Pauli-Villars regularization, just like for VP With the photon mass µ added: Regularized function: Σ(p, µ) = 4πie 2 d 4 k 1 /p /k + m (2π) 4 k 2 µ 2 + iɛ γµ (p k) 2 m 2 + iɛ γµ Σ(p, µ) = Σ(p, µ) + i C i Σ(p, µ, m M i ) =... = Σ(p, µ, Λ) = e2 2π 1 dβ(2m β/p) ln 0 βλ 2 (1 β)m 2 + βµ 2 β(1 β)p 2. Here, one regularizing term was used, C 1 = 1, in which the photon mass µ was substituted by the cut-off momentum Λ.

11 The above integral could be solved. We want to identify the constants A = δm and B = Z 2 1 only. A δm = Σ(p, µ, Λ) p 2 =m 2, /p=m = e2 m 2π This integral converges for µ = 0, so we can calculate: δm = e2 m 2π 1 dβ(2 β) ln 0 1 dβ(2 β) [ln β 2 ln(1 β) + ln Λ2 0 m 2 This expression is weakly (logarithmically) divergent for Λ 2 /m 2. βλ 2 (1 β) 2 m 2 + βµ 2 ] = m 3α ( Λ 2 4π ln m ) 2 B Z 2 m = Σ p = e2 1 dββ 2π 0 e2 2π (I 1 + I 2 ). p 2 =m 2, /p=m [ ln βλ 2 ] (1 β) 2 m 2 + βµ 2 2m2 (2 β)(1 β) m 2 (1 β) 2 + µ 2 β

12 The first integral I 2 is convergent even for µ = 0: 1 ] I 1 (µ = 0) = dββ [ln β 2 ln(1 β) + ln Λ2 0 m 2 = The 2nd integral is: ln Λ2 m 2 ; 1 (2 β)(1 β) 1 µ/m I 2 = 2 dββ 0 (1 β) 2 + βµ 2 /m 2 2 (2 β)(1 β) dββ 0 (1 β) 2 ln µ2 m Thus, the SE charge renormalization constant is ( Z 2 = 1 + B = 1 e2 1 2π 2 Λ2 µ2 ln + ln m2 m ). 4 This is also a weak, logarithmic UV divergence in Λ; IR divergence in µ unpleasant...

13 The vertex correction The last radiative correction: modification of an interaction vertex by a virtual photon The vertex factor changes to: ieγ µ ieλ µ(p, p) = ieγ µ ieγ µ(p, p), with the vertex function Γ µ(p, p) = 4πie 2 ( ) d 4 k 1 (2π) 4 k 2 µ 2 γ ν iɛ /p /k m + iɛ γµ /p /k m + iɛ γν (2) logarithmically divergent UV integral also IR divergence introduction of a finite photon mass, µ, again

14 Instead of the (lengthy) calculation of the vertex function, let us discuss some certain properties. It can be rewritten as the sum of an elastic and inelastic part (or rest term): Γ µ(p, p) = Γ µ(p, p) + ( Γ µ(p, p) Γ µ(p, p) ) Γ µ(p, p) + Γ R µ (p, p) It can be shown that the elastic part can be written in the form Γ µ(p, p) = Lγ µ Only Γ µ(p, p) is logarithmically divergent; the rest term is a well-defined finite expression: 1 /p k m + iɛ = = 1 /p k m + (/p /p) + iɛ 1 /p k m + iɛ 1 /p k m + iɛ ( /p 1 /p) /p k m + iɛ... 1st term: 1/k: log. divergence; higher-order terms: regular As discussed already for VP, the electric charge is measured with scattering experiments at low momentum transfer. So, the vertex in this case is modified es ieγ µ ieγ µ ielγ µ + O(q) further charge renormalization has to be done: e R = Z 1 1 e = (1 + L)e, with Z 1 = (1 + L) 1 1 L.

15 The divergent part Γ µ (p, p) does not have to be calculated, it can be traced back to the result for the self-energy by using the Ward identity: Γ µ (p, p) = p µ Σ(p) So, Γ µ (p, p) = Bγ µ + O(/p m). For electrons on the mass shell (free electrons); thus L = B. Therefore, the renormalization constant of the vertex correction is Z 1 = 1 L = 1 + B = Z 2, i.e. the SE and VC renormalization constants are the same. Thus, the total charge renormalization is: e R = Z 1 1 (VC)Z 2(SE) Z 3 (VP)e = Z 3 e So, the renormalization does not depend on the fictitious photon mass µ; charge renormalization originates from VP only.