Lecture 13: Running of the Coupling Constants

1 Lecture 13: Running of the coupling constants • Recup on renormalization • Coupling constants as functions of the energy. • Non-renormalizable vs. super-renormalizable theories. • The Callan-Symanzik eqution. • Classification of renormalizable quantum theories: asymptotically free vs. asymptotically trivial theories. 2 Review of regularization Let us return to our intuition about regularization as introducing the cutoff - the energy scale Λ such that all higher energies are neglected. Any amplitude is a function of the “bare” parameters, and the cutoff. For example the self-energy and the interaction strength are given by: 2 Π(k ; m0, g0, Λ) V (k1, k2, k3; m0, g0, Λ). In the dimensional regularization one introduces two parameters , µ, with µ of dimension of mass and playing the role of the energy scale Λ. 3 Review of renormalization In renormalization one reformulates the theory so that the unknowns m0, g0, Λ do not appear. This is done by reformulating the theory in terms of “to be physical” parameters m, g and multiplicative renormalization constants, fixing the later so that m, g are indeed the observed parameters, and then taking the limit Λ → ∞. For renormalizable theories - only a finite number of “types” of divergences arises in this limit. Can all be absorbed in renormalization of a finite number of parameters of the theory. The limit can be taken. For non-renormalizable theories infinite number of types of divergences. Cannot absorb them all into physical parameters. Cannot take the limit. Who told us we have to take the limit? Cutoff is physical! Let’s play a different game. 4 Running coupling constants Let us go back to the original theory in terms of the bare parameters, regularize it using a cutoff Λ, and fix the “bare” parameters as well as the cutoff. We see that all amplitudes depend on the energy at which the process is happening. In particular, the strength of interaction is energy dependent! Physics: polarization of the vacuum effects, particles are “dressed” by clouds of virtual particles, and depending on the energy one probes this cloud to different depth. What we define to be the physical parameter (observed on the experiment) is the interaction strength at zero energy. But can consider other energies as well. Let’s study how the coupling constants “run” with energy. This behavior is (qualitatively) independent of the bare parameters and Λ. 5 First look at the running It is convenient and necessary to make the coupling constants dimensionless. One puts gµ[g] for the coupling constants. Here g is dimensionless coupling and µ is “the energy scale”, e.g. the cutoff. How does g change with energy? No physics should depend on how we choose µ. Therefore g(k)k[g] = g(µ)µ[g] or g(k) = g(µ)(k/µ)−[g]. 6 Non-renormalizable couplings We have seen that couplings with [g] < 0 are non-renormalizable. These couplings run according to g(k) = g(µ)(k/µ)|[g]|. Whatever the coupling g(µ) is, at low energies k << µ the coupling is extremelly small! This explains why we mostly see renormalizable interactions!!! Wilson 70’s. This can be used to “explain” why gravity (non-renormalizable) is so weak! 7 Super-renormalizable couplings Couplings [g] > 0 are super-renormalizable (only a finite number of divergent diagrams in such a theory). Their running is g(k) = g(µ)(µ/k)[g]. The coupling constants become small as k → ∞. Such theories are thus best behaved in the UV region of large energies. For small energies (IR region) their behavior is best demonstrated by introducing a length scale L = 1/k instead of the momentum scale. We get: g(L) ∼ L[g]. The interaction strength grows as a power of the distance between objects. All such objects are either clumped together (attractive force) or are zipping away from each other (repulsive force) with more and more increasing force. No wonder we don’t see them! 8 The Callan-Symanzik eqution What happens for renormalizable theories with [g] = 0? Our equation for running can be re-written as a differential equation: dg = −[g]g. d log (k/µ) The linear term on the right hand-side should be thought of as the first term in the expansion of a function β(g): dg = β(g) = −[g]g + ag2 + bg3 + .... d log (k/µ) This is the Callan-Symanzik eqution, or “renormalization group” equation. 9 Renormalizable theories In renormalizable theories [g] = 0 and has to consider the second, quadratic term in the Callan-Symanzik eqution. dg = ag2. d log (k/µ) Its solution is g(µ) g(k) = . 1 − ag(µ) log (k/µ) Thus, the behavior of the coupling constant as a function of energy is determined by the sign of a. 10 Asymptotically free vs. asymptotically trivial theories Consider the case a < 0. For such a theory for large momenta k >> µ, g(k) ∼ 1/ log (k/µ) goes to zero at high energies. Such theories are called asymptotically free theories. Consider the case a > 0. In this case the coupling constant blows up at: 1 ag(µ) kmax = µe . This value of k is called the Landau pole for Landau who first observed this behavior (in QED). The presence of this pole means that the interacting theory is not defined in the limit of infinite energies (UV region). The only version of the theory that is defined is its free version. Such theories where one cannot take the limit k → ∞ are called asymptotically trivial theories..

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