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Renormalization April 6, 2020 version 2.0 Quantum Field Theory II Lectures Notes Part II: Renormalization Prof. Dr. Gino Isidori ETH & UZH, Spring Semester 2020 i ii Contents 2 Renormalization 1 2.1 Systematics of Renormalization . .1 2.1.1 General aspects of ultraviolet divergences . .2 2.1.2 Renormalization of the λφ4 theory at the one-loop level . .5 2.1.3 Renormalised perturbation theory and counterterms . 10 2.2 The Renormalization Group . 12 2.2.1 Wilson's approach to renormalization . 12 2.2.2 The renormalization group (Wilsonian approach) . 17 2.2.3 Renormalization scale and the Callan-Symanzik equation . 19 2.2.4 Solutions of the Callan-Symanzik equation . 22 2.2.5 Running coupling . 25 2.2.6 Callan-Symanzik equation with local operators and mass terms . 27 iii iv Chapter 2 Renormalization 2.1 Systematics of Renormalization Within QFT, when computing correlation functions in perturbation theory beyond the tree- level, one often encounters divergent integrals. As you have already seen in QFT-I, the type of divergences one can encounter are of two types: infrared (IR) and ultraviolet (UV) ones. The divergences appear because the system we are considering has an infinite number of degrees of freedom. In particular, the infrared divergences are due to the infinite volume limit, that implies we have to consider field excitations with arbitrary long wave length (unsuppressed in the case of massless fields). The ultraviolet divergences are related to the continuum limit, that implies we have to consider field excitations with arbitrary short wave length. However, both IR and UV divergences are an artefact of perturbation theory: they do not show up in well-defined physical observables. The problem arises because of the non-trivial relation between physical observables (and asymptotic states of the theory) in terms of free fields and Lagrangian parameters. Dealing with these artefacts requires, in general, a three- step procedure: i. regularization of the divergences by means of an appropriate IR and UV cut-off (such as a finite volume for the IR divergences, and a finite lattice spacing or a maximal momentum for the UV divergences, or the more abstract dimensional regularization which can cure both type of divergences); ii. correct identification of physical observables; iii. renormalization of the Lagrangian parameters. In the theories we are interested in, the first two steps are sufficient to deal with IR divergences. In orther words, IR divergneces (or better the dependence of correlation functions from the IR cut-off) cancels automatically when considering realistic observables (in particular when taking into account that we cannot distinguish processes which differ by the emission of quanta with infinitesimal small energy). On the other hand, taming UV divergences usually requires also the last step, i.e. the renormalization of the Lagrangian parameters. In this section we discuss how to deal with this problem in a systematic way. 1 2.1.1 General aspects of ultraviolet divergences As we will see explicitly in the case of the λφ4 theory and in the case of QED (! exercise class), the ultraviolet divergences generated at the one-loop level can be \cured" by a re- definition of the so-called \bare" parameters, i.e via a re-definition of the parameters appearing in the Lagrangian. More explicitly, in the λφ4 theory we need to renormalize the coupling (λ), the mass (m), and the field (i.e. we define a renormalized field, in terms of the \bare" field appearing in the Lagrangian). Two questions naturally arise: • Is it procedure possible in any QFT? • What happens when going to higher order loops? To address these questions in general terms, it is convenient to introduce the concept of super- ficial degree of divergence (D) of a Feynman diagram in d dimension, which is defined as D = d · L − 2P = d · L − 2Pφ − P − 2Pγ (2.1) where d is the space-time dimensions, L is the number of loops and P is the number of propa- gators of a given type (for scalar, fermion, and gauge fields). This relation follows from the fact that each loop brings an integral over ddk and each propagator brings k (1 power for fermions, 2 for bosons) in the denominator. Let us consider an example: a scalar theory with Lagrangian, 1 1 λ L = (@ φ)2 − m2φ2 − n φn : (2.2) 2 µ 2 n! Generally for a fully connected diagram with N external lines and V vertices, each with n lines, we have L = P − Vn + 1 (2.3) nVn = N + 2P (2.4) The first equation tells us that the number of independent momenta over which integration takes place (L) is equal to the number of propagators (P ) minus the number of relations between momenta, which is equal to Vn (momentum conservation at each vertex) minus one (otherwise overall momentum conservation is counted twice). The second equation tells us that the total number of lines attached to the vertices (n for each) is the sum of the number of external lines and twice the number of propagators. If we take the above two relations into (2.1), we get d d D = d − N − 1 + V n − 1 − d : (2.5) 2 n 2 • For n = 4 and d = 4, i.e. for the φ4 interaction in ordinary space-time, the above result implies D = 4 − N (2.6) which is independent from the number of vertices. Hence independently of the number of vertices (or loops) we consider, the superficial degree of divergence of the Feynman 2 diagrams is 8 = 2 for N = 2 quadratic divergence < D = = 0 for N = 4 log divergence : < 0 for N > 4 \na¨ıve convergence" • For n = 6; d = 4 D = 4 − N + 2V6 ; (2.7) hence each insertion of a vertex, with a φ6 interaction, enhances the degree of divergence. • For n = 3; d = 4 D = 4 − N − V3 ; (2.8) hence each insertion of a vertex, with a φ3 interaction, reduces the degree of divergence. Canonical dimension The coefficient multiplying V in D is the canonical dimension of the coupling λ . To define Z n d this concept, note that in natural units (~; c = 1) the action S = d x L is dimensionless. Since the mass/energy dimension of x is d[x] = −1, this imply that the mass/energy dimension of the Lagrangian density is d[L] = d (2.9) Using this property, starting from the kinetic terms (i.e. the quadratic derivative terms in the Lagrangian) of free scalar, fermion and gauge fields, we can derive the canonical mass/energy dimension of each field as (d = 4) d − 2 @ φ∂µφ ! 2d + 2 = d ) d = = 1 µ φ φ 2 d − 1 3 ¯@= ! 2d + 1 = d ) d = = 2 2 d − 2 F F µν ! 2d + 2 = d ) d = = 1 µν A A 2 Using these results, we can then derive the canonical dimension of each coupling in any inter- action term (note that the mass itself can be considered an interaction): (d = 4) d m2 φ2 ! 2d + 2d = d ) d = − d = 1 φ φ mφ mφ 2 φ ¯ m ! 2d + dm = d ) dm = d − 2d = 1 4 λ4φ ! 4dφ + 2dλ4 = d ) dλ4 = d − 4dφ = 0 ¯ µ e γ Aµ ! 2d + dA + de = d ) de = d − 2d − dA = 0 n d λnφ ! ndφ + dλn = d ) dλn = d − n 2 − 1 = 4 − n It is now clear that the result in (2.5), generalised to the case where we consider different n interactions of the type λnφ , can be re-written as d X D = d − N − 1 + V · d (2.10) 2 n Vn n 3 where dVn denotes the canonical dimension of the coupling λn. Proceeding in a similar manner, in QED one finds d d − 1 D = d − − 1 N − N + V · d (2.11) 2 γ 2 e e with de = (d − 2)=2 and where Nγ( ) denotes the number of external photon (fermion) lines. A general classification of QFTs can thus be inferred by looking at the canonical dimension of the interaction terms: • dV > 0: super-renormalizable theories (finite number of divergent diagrams) • dV = 0: renormalizable theories (finite number of divergent basic amplitudes) • dV < 0: \non-renormalizable" theories (infinite number of divergent basic amplitudes) The λφ4 theory and QED, in d = 4, belongs to the class of renormalizable theories. As can be easily understood: only vey few interaction terms belong two the first to cases. A generic interaction terms with high powers of fields or derivatives naturally leads to dV < 0. The so-called non-renormalizable theories (or better theories which are not renormalizable in the classical sense) are not necessarily problematic. However, they are theories that cannot be extrapolated to arbitrary high energies. We will discuss this in more detail in the next section, but we can have a first understanding of the what happens by the following argument. Consider the n-point correlation function Z iS G(x1; : : : ; xn) = h0jT fφ(x1) ··· φ(xn)gj0i = N Dφ φ(x1) ··· φ(xn)e (2.12) expanding the interaction term in perturbation theory brings down powers of λn. Assuming the correlation function corresponds to a process characterised by an energy scale E, then dimensional analysis implies X p −p·dV G ∼ κn cp × (λn) × E (2.13) p where κn is a (dimensional) proportionality factor, which depends on the number of external fields but is independent from the perturbative expansion, cp are dimensionless coefficients, and p is the order of the perturbative expansion. If we now look at the evaluation of Feynman diagrams, in addition to the physical energy (E) we have an extra (unphysical) energy scale associated to the maximal momentum/energy in the loops (i.e.
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