16 The Perturbative Renormalization Group In this chapter we will reexamine the renormalized perturbation theory dis- cussed in Chapters 12 and 13 from the perspective of the Renormalization Group. We will discuss φ4 theory, the O N non-linear sigma model, and Yang-Mills gauge theories. More detailed presentations of some of this ma- terial can be found in Amit’s book (Amit,( ) 1980), in Zinn-Justin’s book (Zinn-Justin, 2002), and in Peskin and Schroeder (Michael E.Peskinand Daniel V. Schroeder, 1995). 16.1 The perturbative renormalization group We begin by setting up the perturbative renormalization group to φ4 theory. In Sec.13.2 we used renormalized perturbation theory to showthat,totwo- loop order, the theory can be made finite by defining a set of renormalization constants such that the renormalized one-particle irreducible N-point vertex functions are related to the bare functions as N = N 2 N ΓR pi ; mR,gR,κ Zφ Γ pi ,m0,λ, Λ (16.1) ( ) / where Λis the UV momentum regulator and( κ) is the renormalization scale. ({ } ) ({ } ) The relation between the bare and the renormalized theory is encoded in the renormalization constants (the wave function renormalization Zφ,the renormalized mass mR and the renormalized coupling constant gR)such that as the UV cutoffis removed, Λ → ∞,therenormalizedvertexfunctions have a finite limit. Here we will focus our attention on the massless theory, defined by the condition that the renormalized mass vanishes, mR = 0. Thus, we will ex- press the wave function renormalization Zφ and the bare coupling constant λ as functions of the renormalized coupling constant and of therenormal- ization scale. Similarly the bare mass will be tuned to a value mc such that 538 The Perturbative Renormalization Group mR = 0. Thus, we will write Zφ =Zφ gR κ ,κ,Λ (16.2) λ =λ gR κ ,κ,Λ (16.3) 2 2( ( ) ) mc =mc gR κ ,κ,Λ (16.4) ( ( ) ) where we made explicit the fact that( the( renormalized) ) coupling constant gR is actually not actually a constant but depends on the value wechosefor the renormalization scale κ. Since the value of the renormalization scale κ is arbitrary, the vertex functions defined at two different scales κ1 and κ2,mustberelatedtoeach other since they correspond to the same bare theory, N = N 2 N ΓR pi ; gR κ1 ,κ1 Zφ gR κ1 ,κ1 Γ pi ,mc,λ, Λ (N) N/2 ( N) Γ p ; g κ ,κ =Z ; g κ ,κ Γ p ,m ,λ, Λ (16.5) R ({ i} R( 2) 2) φ ( R( )2 )2 ({ }i c ) ( ) / ( ) We see that({ } the renormalized( ) ) vertex( functions( ) ) at the({ two} scales are) related by an expression of the form N = N 2 N ΓR pi ; gR κ1 ,κ1 Z gR κ2 ,κ2; gR κ1 ,κ1 ΓR pi ; gR κ2 ,κ2 ( ) / ( ) (16.6) where({ we} defined( ) ) ( ( ) ( ) ) ({ } ( ) ) Zφ gR κ1 ,κ1, Λ Z gR κ2 ,κ2; gR κ1 ,κ1 = (16.7) Zφ gR κ2 ,κ2, Λ ∂ ( 2( ) ) ( ( ) ( ) ) = Γ p, g κ ,κ (16.8) 2 R R 1 1 2= 2 ∂p( ( ) ) p κ2 ( ) ! which has a finite limit as the UV cutoffis( removed,( ) Λ →)! ∞. ! Equation (16.6) is a relation between finite quantities at different scales and as such it is a finite quantity. It implies that a change in the renormal- 1 2 ization scale κ is equivalent to a rescaling of the fields by Zφ and a change of the renormalized coupling constant g1 = gR κ1 ↦ g2 = gR/ κ2 ,with ≡ = −2 4 g2 F κ2,κ1,g1 Zφ ΓR pi( g1),κ1 ( ) (16.9) SP κ2 ( ) ! where SP κ is the symmetric( ) point of the({ four} momenta)! ( ) pi with each 2 2 ! momentum being at the scale κ,i.e.pi = κ .Themappingg2 = F κ2,κ1,g1 , such that (g =) F κ,κ, g ,definesaflow in the space of coupling{ } constants. i.e. a renormalization group flow. ( ) These relations( apply) to the full generating functional of renormalized 16.1 The perturbative renormalization group 539 vertex functions which then obeys ¯ = 1 2 ¯ ΓR φ, g1,κ1 ΓR Zφ φ,F κ2,κ1,g1 ,κ2 (16.10) / Since the bare theory is independent of our choice (and changes) of a { } { ( ) } renormalization scale, it is kept constant under these transformations. This can be expressed by stating that ∂ N κ Γ p ,λ, m2, Λ = 0, (16.11) ∂κ i c λ,Λ ( ) as the UV regulator Λ → ∞.Consequently,wefind ({ } )∣ ∂ − N 2 N = κ Zφ ΓR pi ,gR κ ,κ 0(16.12) ∂κ λ,Λ / ( ) [ ({ } ( ) )] 16.1.1 The renormalization group equations Therefore, we find that the renormalized N-point vertex functions satisfy the partial differential equation ∂ ∂ N ¯ N = κ + β gR,κ − γφ gR,κ ΓR pi ,gR κ ,κ 0(16.13) ∂κ ∂gR 2 λ,Λ ( ) where[ we used( the) definitions ( )] ({ } ( ) ) ∂gR β¯ gR κ ,κ =κ (16.14) ∂κ λ,Λ ∂ ln Z! φ γφ(gR(κ),κ) =κ ! (16.15) ∂κ! λ,Λ also with Λ → ∞. ! ( ( ) ) ! In general the coupling constant has dimensions.! Let us defineadimen- sionless bare coupling constant u0 ϵ λ = u0κ (16.16) and a dimensionless renormalized coupling constant u such that ϵ gR = uκ (16.17) where ϵ = 4−D = D−∆4,with∆4 = 4 D−2 2beingthescalingdimension of the operator φ4 at the massless free field fixed point. In terms of the dimensionless renormalized( )/ coupling constant u,Eq.(16.13) becomes the Callan-Symanzik Equation ∂ ∂ N N = κ + β u − γφ u ΓR pi ,u κ ,κ 0(16.18) ∂κ ∂u 2 λ,Λ ( ) [ ( ) ( )] ({ } ( ) ) 540 The Perturbative Renormalization Group (again, with Λ → ∞)whereβ u is the renormalization group beta function which is defined by ( ) ∂u β u = κ (16.19) ∂κ λ,Λ ! and ! ( ) ! ∂ ln Z!φ γφ u = (16.20) ∂ ln κ λ,Λ ! Notice that here we defined the( ) sign of the! beta function opposite to the sign we used in Chapter 15. Hence, a positive! beta function means that the coupling constant increases as the momentum scale increases, and viceversa. The formulas of Eqs. (16.19) and Eq.(16.20) as they stand are somewhat awkward to use since they involve he bare dimensionless coupling constant u0 in terms of the dimensionless renormalized coupling constant u instead of the other way around. For this reason we use the chain rule towrite ∂λ ∂u ∂κ κ = −κ u (16.21) ∂κ ∂λ λ ( ) ∂u ( ) κ where, by dimensional analysis, ( ) ϵ λ = κ u0 u, κ Λ (16.22) Using that ( / ) ∂λ κ = ϵλ (16.23) ∂κ u we find that the beta function is( ) ∂u ∂ ln u −1 β u = κ = −ϵ 0 (16.24) ∂κ λ ∂u In this form the beta function( ) β( u )can be expressed( ) as a power series expan- sion in the dimensionless renormalized coupling constant u.Eachcoefficient of this series is a function of ϵ =( 4)− D. Similarly, we can rewrite the anomalous dimension γφ u as ∂ ln Zφ ∂u ∂ ln Zφ γφ u = = κ ( ) (16.25) ∂ ln κ λ ∂κ λ ∂u ! We find ! ( ) ! ( ) ! ∂ ln Zφ γ u = β u (16.26) φ ∂u ( ) ( ) 16.1 The perturbative renormalization group 541 which also can be written as a power series expansion in the dimensionless renormalized coupling constant u. 16.1.2 General solution of the Callan-Symanzik equations We will now solve the Callan-Symanzik equation, Eq.(16.18).Letx = ln κ and write the renormalized vertex function as follows u ′ N N ′ γφ u N Γ p , u, κ = exp du Φ p , uκ (16.27) R i 2 ∫ ′ i u1 β u ( ) ( ) ( ) By requiring({ that} this) expression( satisfies the) Callan-Syma({ } nzik) equation, ( ) Eq.(16.18), we find that the function Φ N must satisfy the simpler equation ( ) ∂ ∂ N + β u Φ p , u, κ = 0(16.28) ∂x ∂u i ( ) It is straightforward[ to see( that) the] solutions({ } to) this equation have the general form u ′ N N du Φ p , u, κ = F p ,x− (16.29) i i ∫ ′ u2 β u ( ) ( ) N ({ } ) ({ } ) where F is a, so far, arbitrary (differentiable) function.( ) We conclude( ) that Callan-Symanzik equation requires the renormalized vertex functions to have the following form u ′ u ′ N N ′ γφ u N du Γ p , uκ = exp du F p ,x− i 2 ∫ ′ i ∫ ′ u1 β u u2 β u ( ) ( ) ( ) (16.30) ({ } ) ( ) ({ } ) where u1 and u2 are two integration constants.( ) It should be apparent( that) the full form scaling function F N cannot be obtained in perturbation theory, only its behavior in special asymptotic( ) limits can be determined by these lim- ited means. In other words, the scaling function contains non-perturbative information about the theory. Let us now rescale all the momenta pi by the same scale factor ρ, pi → ρpi.Dimensionalanalysisimpliesthatthevertexfunctionsshould be rescaled by a prefactor of the form { } N = N+D−ND 2 N ΓR ρpi , u, κ ρ ΓR pi , u, κ (16.31) ( ) / ( ) Using the form of the({ general} ) solution, Eq.(16.30),({ of} the Ca) llan-Symanzik 542 The Perturbative Renormalization Group equation, we find u ′ N N+D−ND N ′ γφ u Γ ρp , u, κ = ρ 2 exp du R i 2 ∫ ′ u1 β u ( ) / ( ) u ′ ({ } ) N( ) du × F p ,x− ln ρ − (16.32) i ( ) ∫ ′ u2 β u ( ) where we used that, since x = ln κ,therescaling({ } κ → κ ρ is equivalent) to ( ) the shift x → x − ln ρ. We will now make explicit the notion that the renormalizatio/ ngroup induces a flow in the space of coupling constants by introducing a running coupling constant u ρ ,whichisafunctionofthescalechangeρ.Tothis effect, we define ( ) u ρ du′ = = s ln ρ ′ (16.33) ∫u ( ) β u such that the running coupling constant u s obeys the differential equation ( ) ∂u s = β u s (16.34) ∂s ( ) with the initial condition for the( flow) u s = 0 = u.