8.324 Relativistic Quantum Field Theory II

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1 8.3 Relativistic Quantum Field Theory II MIT OpenCourseWare Lecture Notes Hong Liu, Fall 010 Lecture Firstly, we will summarize our previous results. We start with a bare Lagrangian, L [ 0, ϕ] = g (0) i O i. (1) The path integral Z = k< 0 i Dϕ e d d xl [ 0 ] describes all the physics below the cutoff 0. Most often we are interested in physics at some energy scale E 0. L [ 0 ] is not convenient to use, as it contains the degrees of freedom ϕ(k) : E < k < 0 which are not directly related to the physics at the scale E. However, we cannot simply discard them, as they have indirect effects, which can be taken into account by integrating them out. We write ϕ(k) = ϕ (k < ) + ϕ ( < k < 0 ), and ϕ, Z 0 = Dϕ (k) Dϕ (k) e S[ϕ+ 0] = Dϕ (k) e S[ϕ,], (3) k< <k< 0 k< ({ } ) where S [ϕ, ] = d d x i g (0) i()o i, g i = g i g i, 0 ;. By varying, we obtain a continuous family of S [ϕ, ] or {g i ()}. This is the renormalization group flow. () Infrared E 0 Figure 1: The renormalization flow from the cutoff 0 in the ultraviolet to a cutoff to study processes at an energy scale E in the infrared region. Infinitesimally, we have ds dλ i = F (S ), = β i ({λ j ()}), () d d where λ i = g i δi. The process means that all S should describe the same low-energy physics. By dimensional analysis, we expect that for 0 1, ( ) (d i) λ i () λ i ( 0 ), (5) and so we have three cases: 0 i < d relevant, λ i : i = d marginal, (6) i > d irrelevant. We expect the initial values of irrelevant couplings should not be important for 0. This rough argument can 0 be substantiated by analyzing the flow equation in detail. It turns out that the flow equation can be written in a closed form, as we showed in the last lecture: S [ϕ, ] = S 0 [ϕ, ] + S int [ϕ, ], (7) 1

2 κ (k) k Figure : The propagator G (k) = 1 κ (k) has a cutoff around k. k where S 0 = 1 d k ϕ(k)ϕ( k)g 1 1 (k). If G = k, we have S 0 = 1 d x ( ϕ). We considered the case G = (π) 1 k k κ (k), where κ provides a cut-off at k. An example is a sharp cutoff κ (k) = Θ(1 ). This means ϕ(k) with k > do not propagate, and there is no need to impose k < explicitly in the path integral. Requiring the partition function to be independent of the choice of led to the equation d 1 d k dg (k) [ δs I δs I δ ] S I S I = d (π) d δϕ(k) δϕ( k) δϕ(k)δϕ( k). (8) Remarks: 1. This equation is exact, and fully non-perturbative.. dg is only supported near a thin shell of momentum around. In fact, for G = Θ(1 k d ), we have dg d δ(k ). Physically, this reflects that the flow equation is obtained by integrating out the degrees of freedom around. 3. Expanding S I [ϕ, ] in momentum space as ( ) n 1 d d i S I [ϕ, ] = n! (π) (πk δ (d) (k 1 + k k n ) g(k 1,..., k n ; )ϕ(k 1 )... ϕ(k n ). (9) ) (8) requires that n= i=1 d dg (p) 1 d q dg g(k 1,... k n ; ) = g( p, I1 ; ) g(p, I ; ) d d (π) d g(q, q, k 1,..., k n ; ) {I 1,I } (10) where p = k i I 1 k i, and I 1, I are disjoint subsets of momenta such that I 1 I = {k 1,..., k n }. {I 1,I } is a sum over all possible ways to separate {k 1,..., k n } into groups. Diagramatically, this is shown in figure 3. This corresponds to integrating out a tree-level diagram and a one-loop momentum diagram respectively. 3. We can expand S I [ϕ, ] in coordinate space: S I [ϕ, ] = d x g i ()O i (x), (11) where the O i form a complete set of local operators. From (8), we obtain i dg i = β ij g j + β ijk g j g k, (1) d with β i i = 0. Using dimensionless couplings, λ i () = g i () (d i), we find β i dλ i = β i j λ j + β i jk λ j λ k, (13) d

3 dg Figure 3: Diagramatic representation of (10), where the crossed vertex represents dg (p). d where β j = ( i d) δ j + β ij, and the β j i i i has no diagonal term. We see how this corresponds to the cases i > d damping, i = 0 marginal, (1) i < d growth.. The flow equation, (8), and thus the resulting β-functions, are not unique. It can be written in many other, equivalent forms by using field redefinitions: ϕ(x) ϕ(x) + aϕ (x) + bϕ 3 (x) + c ( ϕ) (15) Such field redefinitions lead to redefinitions of the couplings, but they should not change the physical observables. There is also a scheme dependence on the choice of cutoff functions, κ (k). The equations (1) and (13) are an infinite number of coupled first-order differential equations. It is quite complicated to analyze them, and often requires the development of approximation methods. Let us consider some general features of the flow: 1. Separate the couplings, as before, as {λ i } = {ρ a } + {κ α }, where {ρ a } are the relevant and marginal couplings, and {κ α } are the irrelevant couplings. Then, for a generic theory with all λ (0) i O(1) at 0, we have for 0 1, assuming the {λ i } do not become too large, that the {κ α } only depend on the {ρ a }. For example, if we consider two couplings, λ and λ 6, for terms of the form ϕ and ϕ 6 respectively, we have dλ = λ , d dλ 6 = λ 6 λ +... d The first term in the flow equation for λ 6 provides a damping when going into the infrared regime. λ 6 Infrared λ Figure : Damping and renormalization flow into the point λ = λ 6 = 0 for the irrelevant coupling λ 6 and the marginally irrelevant coupling λ. λ is given by β = dλ = λ 0 0, λ 6 λ 6 = λ 6 (λ ) The flow of λ d as. marginally irrelevant. + O(λ 3 ), and so λ is 3

4 We can now make the connection to the standard renormalization procedure. We consider the ϕ -Lagrangian, L = 1 ( ϕ) 1 m physϕ 1 λ(phys) ϕ + L ct, (16) where m and λ are renormalized physical quantities which can be measure experimentally. They should be interpreted as being defined at a specific infrared scale. For example, λ (phys) = λ ( = 0). (17) (phys) We now keep λ and m phys fixed and take the limit of the cutoff 0. All physical observables only depend on the renormalized quantities. In particular, λ (0) ( 0 ) = 0, λ (0) 6 = 0. That is, we start along the horizontal axis. The Wilsonian approach tells us that such an initial condition is not important. Also note that λ 6 is not zero in the infrared, it is just determined by λ. λ 6 is related to six-particle scatterings, which of course have non-zero amplitude.

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