17 The Renormalization Group 17.1 Quantum field theory as an interpolating theory Probably because they describe point particles, quantum field theories are rife with infinities. Unknown physics at very short distance scales, removes these infinities. Since these infinities really are absent, we can cancel them consistently in renormalizable theories by a procedure called renormal- ization. One starts with an action that contains infinite, unknown constants, such as e0 = e + δe,inwhiche is the charge of the proton. One then uses δe and other infinite parameters to cancel unwanted infinite terms as they appear in perturbation theory. Alternatively, we may consider quantum field theory as a way to interpo- late between the value of a physical quantity as given by experiment at one energy scale and its value at other energies. For instance, in the theory of a scalar field φ described by the action density 1 1 g = @ @⌫φ m2φ2 φ4, (17.1) L −2 ⌫ − 2 − 4! the amplitude for the elastic scattering of two bosons of initial four-momenta p1 and p2 into two of final momenta p10 and p20 is 2 1 g 1 3 2 2 2 A = g k dk dx [k + m sx(1 x)]− − 16⇡2 − − Z0 Z0 (17.2) 2 2 n2 2 2 2 +[k + m tx(1 x)]− +[k + m ux(1 x)]− − − − − o to one-loop order (Weinberg, 1995, section 12.2). In this formula, s, t, and 714 The Renormalization Group u are the Mandelstam variables s = (p + p )2, t = (p p )2, and − 1 2 − 1 − 10 u = (p p )2. − 1 − 20 The amplitude A(s, t, u) diverges logarithmically as k , but its deriva- !1 tives with respect to s, t, and u are convergent. So we can write it as its value A0 = A(s0,t0,u0) at some point (s0,t0,u0) plus convergent integrals of its derivatives with respect to s, t, and u 2 1 s g 1 3 2x(1 x) A(s, t, u) A = k dk dx ds0 − − 0 − 16⇡2 [k2 + m2 s x(1 x)]3 Z0 Z0 " Zs0 − 0 − t 2x(1 x) u 2x(1 x) + dt0 − + du0 − . [k2 + m2 t x(1 x)]3 [k2 + m2 u x(1 x)]3 Zt0 − 0 − Zu0 − 0 − # (17.3) And as long as we finesse the poles in k by keeping the Mandelstam variables o↵-shell and negative, the integrals over k are elementary, as are those over s0,t0, and u0 g2 1 s ds A(s, t, u) A = x(1 x) 0 − 0 − 32⇡2 − m2 s x(1 x) Z0 " Zs0 − 0 − t dt u du + 0 + 0 dx m2 t x(1 x) m2 u x(1 x) Zt0 − 0 − Zu0 − 0 − # g2 1 m2 s x(1 x) = dx ln − 0 − (17.4) − 32⇡2 m2 sx(1 x) Z0 ( − − m2 t x(1 x) m2 u x(1 x) +ln − 0 − +ln − 0 − . m2 tx(1 x) m2 ux(1 x) − − − − ) If we choose as the renormalization point s = t = u = 4µ2/3, then we 0 0 0 − get the usual result (Weinberg, 1995, 1996, sections 12.2, 18.1–2). 17.2 The Renormalization Group in Quantum Field Theory We now use our expression (17.4) to define the running coupling con- stant gµ at energy scale µ as the amplitude A0 = A(s0,t0,u0) at s0 = t0 = u = µ2 0 − g A(s ,t ,u )=A( µ2). (17.5) µ ⌘ 0 0 0 − 17.2 The Renormalization Group in Quantum Field Theory 715 In terms of the sliding scale µ, the scattering amplitude is g2 1 1+(µ/m)2x(1 x) A(s, t, u)=g dx ln − (17.6) µ − 32⇡2 1 s/m2x(1 x) Z0 ( − − 1+(µ/m)2x(1 x) 1+(µ/m)2x(1 x) +ln − +ln − 1 t/m2x(1 x) 1 u/m2x(1 x) − − − − ) plus terms of order g3 and higher in the perturbative expansion. The key observation of Callan, Symanzik, and others is that this scattering ampli- tude, like any physical quantity, should be independent of the sliding scale µ. Thus its derivative with respect to µ must vanish @A(s, t, u) @g 3g2 @ 1 0= = µ dx ln 1+(µ/m)2x(1 x) @µ @µ − 32⇡2 @µ − Z0 (17.7) @g 3g2 1 (2µ/m⇥ 2)x(1 x) ⇤ = µ dx − . @µ − 32⇡2 1+(µ/m)2x(1 x) Z0 − For µ m, the integral is 2/µ. So at high energies, the running coupling constant obeys the di↵erential equation @g 3g2 µ µ β(g )= . (17.8) @µ ⌘ µ 16⇡2 Since the coupling constants g and gµ are equal to lowest order in gµ,we may write the beta function as @g 3g2 µ µ β(g )= µ . (17.9) @µ ⌘ µ 16⇡2 Integrating, we get E E dµ gE dg 16⇡2 gE dg 16⇡2 1 1 ln = = µ = µ = . M µ β(g ) 3 g2 3 g − g ZM ZgM µ ZgM µ ✓ M E ◆ (17.10) So the running coupling constant gµ at energy µ = E is g g = M . (17.11) E 1 3 g ln(E/M)/16⇡2 − M As the energy E = ps rises above M, while staying below the singular value 2 E = M exp(16⇡ /3gM ), the running coupling constant gE slowly increases, as does the scattering amplitude, A g . ⇡ E Example 17.1 (Quantum Electrodynamics). Vacuum polarization makes 716 The Renormalization Group the amplitude for the scattering of two electrons proportional to (Weinberg, 1995, section 11.2) A(q2)=e2 1+⇡(q2) (17.12) rather than to e2. Here e is the renormalized⇥ charge,⇤ q = p p is the 10 − 1 four-momentum transferred to the first electron, and e2 1 q2x(1 x) ⇡(q2)= x(1 x)ln 1+ − dx (17.13) 2⇡2 − m2 Z0 represents the polarization of the vacuum. We define the square of the run- 2 2 2 ning coupling constant eµ to be the amplitude (17.12) at q = µ 2 2 2 2 eµ = A(µ )=e 1+⇡(µ ) . (17.14) For q2 = µ2 m2, the vacuum-polarization⇥ term⇤ ⇡(µ2) is (exercise 17.1) e2 µ 5 ⇡(µ2) ln . (17.15) ⇡ 6⇡2 m − 6 The amplitude (17.12) then is 1+⇡(q2) A(q2)=e2 , (17.16) µ 1+⇡(µ2) and since it must be independent of µ,wehave d A(q2) d e2 d 0= = µ e2 1 ⇡(µ2) . (17.17) dµ 1+⇡(q2) dµ 1+⇡(µ2) ⇡ dµ µ − ⇥ ⇤ So by di↵erentiating eµ and the vacuum-polarization term (17.15), we find de d⇡(µ2) de e2 0=2e µ 1 ⇡(µ2) e2 =2e µ 1 ⇡(µ2) e2 . µ dµ − − µ dµ µ dµ − − µ 6⇡2µ ✓ ◆ ✓ ◆ ⇥ ⇤ ⇥ ⇤ (17.18) But by (17.13) the vacuum-polarization term ⇡(µ2) is of order e2,whichis 2 the same as eµ to lowest order in eµ. Thus we arrive at the Callan-Symanzik equation de e3 µ µ β(e )= µ (17.19) dµ ⌘ µ 12⇡2 which we can integrate E E dµ eE de eE de 1 1 ln = = µ = 12⇡2 µ =6⇡2 M µ β(e ) e3 e2 − e2 ZM ZeM µ ZeM µ ✓ M E ◆ (17.20) 17.2 The Renormalization Group in Quantum Field Theory 717 to e2 e2 = M . (17.21) E 1 e2 ln(E/M)/6⇡2 − M 2 The fine-structure constant eµ/4⇡ slowly rises from ↵ =1/137.036 at me to e2(45.5GeV) ↵ 1 = = (17.22) 4⇡ 1 2↵ ln(45.5/0.00051)/3⇡ 134.6 − at ps = 91 GeV. When all light charged particles are included, one finds that the fine-structure constant rises to ↵ =1/128.87 at E = 91 GeV. Example 17.2 (Quantum Chromodynamics). The beta functions of scalar field theories and of quantum electrodynamics are positive, and so interac- tions in these theories become stronger at higher energy scales. But Yang- Mills theories have beta functions that can be negative because of the cubic interactions of the gauge fields and the ghost fields (16.250). If the gauge group is SU(3), then the beta function is dg 11g3 11g3 µ µ β(g )= = µ (17.23) dµ ⌘ µ − 16⇡2 − 16⇡2 to lowest order in gµ. Integrating, we find E E dµ gE dg 16⇡2 gE dg 8⇡2 1 1 ln = = µ = µ = M µ β(g ) − 11 g3 11 g2 − g2 ZM ZgM µ ZgM µ ✓ M E ◆ (17.24) and 1 11g2 E − g2 = g2 1+ M ln (17.25) E M 8⇡2 M which shows that as the energy E of a scattering process increases, the running coupling slowly decreases, going to zero at infinite energy, an e↵ect called asymptotic freedom. If the gauge group is SU(N), and the theory has nf flavors of quarks with masses below µ, then the beta function is g3 11N n β(g )= µ f (17.26) µ −4⇡2 12 − 6 ✓ ◆ which is negative as long as nf < 11N/2. Using this beta-function with N = 3 and again integrating, we get instead of (17.25) 1 (11 2n /3)g2 E2 − g2 = g2 1+ − f M ln . (17.27) E M 16⇡2 M 2 718 The Renormalization Group Asymptotic Freedom 0.22 0.2 ) E ( s α 0.18 0.16 0.14 Coupling strength 0.12 0.1 20 40 60 80 100 120 140 Energy E (GeV) Figure 17.1 The strong-structure constant ↵s(E) as given by the one-loop formula (17.29) (thin curve) and by a three-loop formula (thick curve) with ⇤= 230 MeV and n = 5 is plotted for m E m .