1.3 Running coupling and renormalization 27 1.3 Running coupling and renormalization In our discussion so far we have bypassed the problem of renormalization entirely. The need for renormalization is related to the behavior of a theory at infinitely large energies or infinitesimally small distances. In practice it becomes visible in the perturbative expansion of Green functions. Take for example the tadpole diagram in '4 theory, d4k i ; (1.76) (2π)4 k2 m2 Z − which diverges for k . As we will see below, renormalizability means that the ! 1 coupling constant of the theory (or the coupling constants, if there are several of them) has zero or positive mass dimension: d 0. This can be intuitively understood as g ≥ follows: if M is the mass scale introduced by the coupling g, then each additional vertex in the perturbation series contributes a factor (M=Λ)dg , where Λ is the intrinsic energy scale of the theory and appears for dimensional reasons. If d 0, these diagrams will g ≥ be suppressed in the UV (Λ ). If it is negative, they will become more and more ! 1 relevant and we will find divergences with higher and higher orders.8 Renormalizability. The renormalizability of a quantum field theory can be deter- mined from dimensional arguments. Consider φp theory in d dimensions: 1 g S = ddx ' + m2 ' + 'p : (1.77) − 2 p! Z The action must be dimensionless, hence the Lagrangian has mass dimension d. From the kinetic term we read off the mass dimension of the field, namely (d 2)=2. The − mass dimension of 'p is thus p (d 2)=2, so that the dimension of the coupling constant − must be d = d + p pd=2. g − On the other hand, a given 1PI n point function Γn(x ; : : : x ) has mass dimension − 1 n d = d + n nd=2. This is so because the full 1PI function must have the same mass Γ − dimension as its tree-level counterpart, and the latter can be read off from the interac- tion terms in the Lagrangian after removing the fields (here: remove 'p). Therefore we have in four dimensions: d = 4 p and d = 4 n. Analogously, Eqs. (1.10) and (1.13) g − Γ − entail for QCD in four dimensions that the quark fields carry mass dimension 3=2 and the gluon fields dimension 1; the coupling g, the quark-gluon and four-gluon vertices are dimensionless, and the three-gluon vertex has dimension 1. Now consider the perturbative expansion of a 1PI function in 'p theory. Its mass dimension is fixed: d = d + n nd=2. At a given order in perturbation theory, d can Γ − Γ also be determined by counting the number of internal loops L (each comes with mass dimension d), the number of internal propagators I (each with dimension 2), and the − number of vertices V (each with dimension dg): d = dL 2I +d V D = d d V: (1.78) Γ − g ) Γ − g =:D 8On the other hand this| also{z means} that, as long as we are only interested in Λ M, non- renormalizable theories are perfectly acceptable low-energy theories. For example, chiral perturbation theory is a non-renormalizable low-energy expansion of QCD; the non-renormalizable Fermi theory of weak interactions is the low-energy limit of the electroweak theory. 28 QCD D is the degree of divergence of the perturbative diagram: at higher and higher orders, Γ will contain more internal loops. If the momentum powers from the loop integration are overwhelmed by those in the denominators of the propagators, the diagram will converge in the ultraviolet (D < 0); if this is not the case, it will diverge (D 0). The ≥ second equation in (1.78) states that depending on the dimension dg of the coupling, the degree of divergence of a certain n-point function grows (or falls) with the number of internal vertices, i.e., with the order in perturbation theory. Renormalizability means that only a finite number of Green functions have D 0. If ≥ this were not the case, we would have to renormalize every 1PI function of the theory, which means that we would need infinitely many renormalization constants and the theory would lose predictability. This is why it is usually said that non-renormalizable theories cannot describe fundamental interactions since they are not applicable at all scales.9 From Eq. (1.78), renormalizability implies d 0, i.e., the coupling must be g ≥ either dimensionless or have a positive mass dimension. For a renormalizable theory, going to higher orders does not increase the degree of divergence of an n-point function. (For a super-renormalizable theory, defined by dg > 0, divergences only appear in the lowest orders of pertubation theory which is even better.) A renormalizable quantum field theory contains only a small number of superficially divergent amplitudes, namely those with a tree-level counterpart in the Lagrangian, and therefore needs only a finite number of renormalization constants. Renormalization in '4. We will use '4 theory again as our generic example; the generalization to QCD will be straightforward. To start with, we reinterpret the fields, coupling and masses in the Lagrangian (1.77) as 'bare', unphysical quantities 'B, gB and mB. They are related to the renormalized quantities ', g and m via renormalization constants: 1=2 2 2 'B = Z' ' ; gB = Zg g ; mB = Zm m : (1.79) If we insert this in the Lagrangian, we obtain 1 g = Z ' ( + Z m2) ' Z Z2 '4 : (1.80) L −2 ' m − g ' 4! Consequently, when we define 1PI Green functions from the derivative of the resulting effective action with respect to the renormalized field ', the tree-level terms will pick up a dependence on the renormalization constants. The inverse tree-level propagator and tree-level vertex are given by (let's ignore awkward factors of i): ∆−1(p) = Z (p2 Z m2) ; Γ = Z Z2 g : (1.81) 0 ' − m 0 g ' In total, we have three potentially divergent renormalization constants; we will fix them by imposing three renormalization conditions. Higher n-point vertices do not introduce any new divergences; it is sufficient to determine Z', Zg and Zm. 9One should however keep in mind that the renormalizability arguments developed here are based on perturbation theory. In principle, a non-renormalizable theory could also 'renormalize itself' in a nonperturbative way by developing nontrivial fixed points, which leads to the concept of asymptotic safety. 1.3 Running coupling and renormalization 29 Now consider the Dyson-Schwinger equation for the renormalized inverse propagator from Eq. (1.52) (or, equivalently, its perturbation series). It has the generic form ∆−1(p) = Z (p2 Z m2) + Σ(p) : (1.82) ' − m Σ(p) is the self-energy; if we work in perturbation theory, its lowest-order contribution is the tadpole graph, followed by the two-loop diagram and so on. These loop diagrams will depend on internal tree-level propagators ∆0(k) as well as tree-level vertices Γ0. Σ(p) has divergences which we want to eliminate. Therefore, we first have to regularize it in a suitable way. If we use dimensional regularization (in d = 4 dimensions), we − will get terms 1/ that diverge as 0. For dimensional reasons the regularization ∼ ! procedure also introduces a scale Λ. Take for example the tadpole loop at 1-loop order: 10 − ddk i m2 1 m2 /2 = B Γ 1 B (2π)d k2 m2 (4π)2 2 − Λ 4πΛ2 Z − B (1.83) 2 2 ! m 2 m 0 B + (γ 1) + ln B + () : −−−−! (4π)2 − − 4πΛ2 O In d = 4 dimensions the integral, after dividing by m2 , is no longer dimensionless 6 B but goes like 1=Λ. The dimension compensates that of the coupling g that would appear in front of the integral and also has dimension 4 d = . However, even for − 0, the scale Λ in the denominator of the logarithm remains. Therefore, Σ(p) ! generally depends not only on the momentum p, but also on the scale Λ and the parameter 0 which produces the divergence.11 The tadpole diagram above is ! actually momentum-independent; higher-order diagrams won't be. Still, the generic features of these higher-loop contributions can be read off already from Eq. (1.83). To ensure that ∆−1(p) is finite, we impose now two renormalization conditions. In combination with the third one in Eq. (1.88), those will fix the renormalization 2 2 constants Z' and Zm. At some arbitrary renormalization scale p = µ , we demand that the propagator reduces to that of a free particle: d ∆−1(p) =! p2 m2 ; ∆−1(p) =! 1 : (1.84) p2=µ2 − dp2 p2=µ2 If we compare this with Eq. (1.82) and denote the self-energy and its derivative at the renormalization scale by Σ(µ) and Σ0(µ), respectively, we get: m2 + Σ(µ) Σ0(µ) µ2 Z = 1 Σ0(µ) ;Z = − : (1.85) ' − m m2 (1 Σ0(µ)) − The renormalization 'constants' depend now on µ, Λ and and are therefore divergent. They are calculable order by order in perturbation theory. On the other hand, if we 10See e.g. Peskin-Schroeder, p. 249 ff for a derivation. Γ(n) is here the Gamma function and diverges for n = −1; γ is the Euler-Mascheroni constant. 11Regularization always introduces a scale: had we used a hard cutoff instead of dimensional regu- larization, we would have arrived at a similar formula, where Λ would be the cutoff scale that has to be taken to infintiy (instead of ! 0). Pauli-Villars regularization would introduce a regulator mass, lattice regularization an inverse lattice spacing 1=a, etc.