Renormalization group

Elementary theory

A generic theory is described by a certain function ${\displaystyle Z}$ of the state variables ${\displaystyle \{s_i\}}$ and a certain set of coupling constants ${\displaystyle \{J_k\}}$. This function may be a partition function, an action, a Hamiltonian, so long as it contains the whole description of the physics of the system.

For a certain transformation of the state variables ${\displaystyle \{s_i\}\to \{{\tilde{s}}_i\}}$, the number of ${\displaystyle {\tilde{s}}_i}$is less than the number of ${\displaystyle s_i}$. If the ${\displaystyle Z}$ function can be rewritten only in terms of the ${\displaystyle {\tilde{s}}_i}$ by a certain change in the parameters, ${\displaystyle \{J_k\}\to \{{\tilde{J}}_k\}}$, then the theory is said to be renormalizable. The change in the parameters is implemented by a certain beta function: ${\displaystyle \{{\tilde{J}}_k\}=\beta (\{J_k\})}$, which is said to induce a renormalization group flow (or RG flow) on the ${\displaystyle J}$-space. The values of ${\displaystyle J}$ under the flow are called running couplings.

Most fundamental theories of physics with the exception of gravity are exactly renormalizable. Similarly, most theories in condensed matter physics are approximately renormalizable.

Conformal symmetry and fixed points

Conformal symmetry is associated with the vanishing of the beta function. This can occur naturally if a coupling constant is attracted, by running, toward a fixed point at which β(g) = 0. In QCD, the fixed point occurs at short distances where g → 0 and is called a (trivial) ultraviolet fixed point. For heavy quarks, such as the top quark, the coupling to the Higgs boson runs toward a fixed non-zero (non-trivial) infrared fixed point The top quark Yukawa coupling lies slightly below the infrared fixed point of the Standard Model suggesting the possibility of additional new physics, such as sequential heavy Higgs bosons.

In string theory, conformal invariance of the string world-sheet is a fundamental symmetry: β = 0 is a requirement. Here, β is a function of the geometry of the space-time in which the string moves. This determines the space-time dimensionality of the string theory and enforces Einstein's equations of general relativity on the geometry. The RG is of fundamental importance to string theory and theories of grand unification.

If the fixed points of the system correspond to a free field theory, the theory is said to exhibit quantum triviality and possesses a Landau pole. For a φ⁴ interaction, Michael Aizenman proved that this theory is indeed trivial, for space-time dimension D ≥ 5. For D = 4, the triviality has yet to be proven rigorously, but lattice computations have provided strong evidence for this. This fact is important as quantum triviality can be used to bound or even predict parameters such as the Higgs boson mass in asymptotic safety scenarios. Numerous fixed points appear in the study of lattice Higgs theories, but the nature of the quantum field theories associated with these remains an open question. Since the RG transformations in such systems are lossy (i.e. the number of variables decreases), there need not be an inverse for a given RG transformation.

Relevant and irrelevant operators

For a certain observable A of a physical system undergoing an RG transformation, the magnitude of the observable as the scale of the system goes from small to large determines the importance of the observable for the scaling law:

A relevant observable is needed to describe the macroscopic behavior of the system; irrelevant observables are not needed. Marginal observables may or may not need to be taken into account. Most observables are irrelevant, i.e., the macroscopic physics is dominated by only a few observables in most systems. As an example, in microscopic physics, to describe a system consisting of a mole of carbon-12 atoms we need of the order of 10²³ (the Avogadro number) variables, while to describe it as a macroscopic system (12 grams of carbon-12) we only need a few.

Critical exponents and universality

Before Wilson's RG approach, it was unclear why the critical exponents describing transitions in very disparate phenomena. In general, thermodynamic features of a system near a phase transition depend only on a small number of variables, such as the dimensionality and symmetry, but are insensitive to details of the underlying microscopic properties of the system. This property is known as universality.

Universality can be explained using the renormalization group, by demonstrating that the differences in phenomena among the individual fine-scale components are determined by irrelevant observables, while the relevant observables are shared in common. Therefore, many macroscopic phenomena may be grouped into a small set of universality classes, specified by the shared sets of relevant observables.

Simple model: block spins

One of the simpler forms of RG is the block spin RG, devised by Leo P. Kadanoff in 1966. For a 2D set of atoms in a perfect square array (Figure 1), atoms interact among themselves only with their nearest neighbors at some temperature T and a certain coupling J. The physics of the system is then described by a certain function of these parameters, such as the Hamiltonian H(T, J).

The solid is then divided into blocks of 2×2 blocks, described in terms of block variables that represent the average behavior within each block. The physics of block variables is approximately described by a formula of the same kind, but with different values for T and J: H(T′, J′). The original problem may be computationally intractable due to the large number of atomic variables involved; in the renormalized problem, there are one fourth as many. Another iteration of the same kind leads to H(T", J"), and only one sixteenth of the atoms. This process increases the observation scale with each RG step.

Iterating until there is only one large block is roughly equivalent to finding the long range behavior of the RG transformation which took (T,J) → (T′,J′) and (T′, J′) → (T", J"). Often, when iterated many times, this RG transformation leads to a certain number of fixed points.

For the test case of an Ising model lattice of spins interacting magnetically, some positive coupling J denotes the trend of neighbor spins to be aligned. The behavior of the system is determined by competition between the J term (which tends to "order" the system by aligning spins) and the disordering effect of temperature, which the alignment of spins to vary randomly.

For models of this type, there are three fixed points:

1. T = 0 and J → ∞. At the largest scale, temperature becomes unimportant, i.e., the disordering factor vanishes and the system appears to be ordered. This is a ferromagnetic phase, where the lack of disorder leads to every spin pointing in the same direction.
2. T → ∞ and J → 0. Temperature dominates, and the system is disordered at large scales.
3. A nontrivial point between them, T = T_c and J = J_c. At this point, changing the scale does not change the physics, because the system will look the same at any choice of scales. This corresponds to the critical point of the Curie phase transition, where the two effects above are perfectly balanced.

Given a certain material with given values of T and J, the large-scale behavior of the system is then determined by these fixed points.

Exact renormalization group equations

An exact renormalization group equation (ERGE) is one that takes irrelevant couplings into account. As there are infinitely many choices of field rescalings, there are also infinitely many different interpolating ERGEs, but the most common approaches are presented below.

Wilson ERGE

The Wilson ERGE is the simplest conceptually, but is difficult to implement. The method begins with a Fourier transform into momentum space after Wick rotating into Euclidean space. By setting a hard momentum cutoff, p² ≤ Λ², the only degrees of freedom are those with momenta less than Λ. The partition function is $${\displaystyle Z={\int }_{p^2\le {\mathrm{\Lambda }}^2}\mathcal{D}\phi \exp \left[-S_{\mathrm{\Lambda }}[\phi ]\right].}$$For any positive Λ′ less than Λ, the functional S_Λ′ is defined as$${\displaystyle \exp \left(-S_{{\mathrm{\Lambda }}^{\prime }}[\phi ]\right)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\int }_{{\mathrm{\Lambda }}^{\prime }\le p\le \mathrm{\Lambda }}\mathcal{D}\phi \exp \left[-S_{\mathrm{\Lambda }}[\phi ]\right].}$$If S_Λ depends only on ϕ and not derivatives thereof, this may be rewritten as $${\displaystyle \exp \left(-S_{{\mathrm{\Lambda }}^{\prime }}[\phi ]\right)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\prod _{{\mathrm{\Lambda }}^{\prime }\le p\le \mathrm{\Lambda }}\int d\phi (p)\exp \left[-S_{\mathrm{\Lambda }}[\phi (p)]\right],}$$ Since only functions φ well-defined between Λ' and Λ are integrated over, the left hand side may still depend on ϕ outside that range. So $${\displaystyle Z={\int }_{p^2\le {{\mathrm{\Lambda }}^{\prime }}^2}\mathcal{D}\phi \exp \left[-S_{{\mathrm{\Lambda }}^{\prime }}[\phi ]\right].}$$In fact, this transformation is transitive. If you compute S_Λ′ from S_Λ and then compute S_Λ′′ from S_Λ′, this gives you the same Wilsonian action as computing S_Λ″ directly from S_Λ.

Polchinski ERGE

The Polchinski ERGE involves a smooth UV regulator cutoff,an improvement over the Wilson ERGE. This suppresses contributions from momenta greater than Λ heavily. The smoothness of the cutoff, however, generates a functional differential equation in the cutoff scale Λ. As in Wilson's approach, there is a different action functional for each Λ. Each of these actions are supposed to describe exactly the same model - which means that their partition functionals have to match exactly.

As an example (in condensed deWitt notation), for a real scalar field, $${\displaystyle Z_{\mathrm{\Lambda }}[J]=\int \mathcal{D}\phi \exp \left(-S_{\mathrm{\Lambda }}[\phi ]+J\cdot \phi \right)=\int \mathcal{D}\phi \exp \left(-\frac{1}{2}\phi \cdot R_{\mathrm{\Lambda }}\cdot \phi -S_{\mathrm{int}\mathrm{\Lambda }}[\phi ]+J\cdot \phi \right)}$$ and Z_Λ is independent of Λ. The action is split into a quadratic kinetic part and an interacting part S_{int Λ}. This split is not clean generically; the "interacting" part can also contain quadratic kinetic terms and if there is any wave function renormalization, it must. This can be reduced by introducing field rescalings. R_Λ is a function of the momentum p and the second term in the exponent is $${\displaystyle \frac{1}{2}\int \frac{d^{d}p}{(2\pi {)}^d}{\tilde{\phi }}^{\ast }(p)R_{\mathrm{\Lambda }}(p)\tilde{\phi }(p)}$$ when expanded.

When ${\displaystyle p\ll \mathrm{\Lambda }}$, R_Λ(p)/p² is essentially 1. When ${\displaystyle p\gg \mathrm{\Lambda }}$, R_Λ(p)/p² approaches infinity. R_Λ(p)/p² is always greater than or equal to 1 and is smooth. This leaves the fluctuations with momenta less than the cutoff Λ unaffected but heavily suppresses contributions from those with momenta greater than the cutoff. The condition that the partition functional be independent of the cutoff (i.e. that the derivative with respect to the cutoff is zero) can be satisfied by, for example, $${\displaystyle \frac{d}{d\mathrm{\Lambda }}{S}_{\mathrm{int}\mathrm{\Lambda }}=\frac{1}{2}\frac{\delta S_{\mathrm{int}\mathrm{\Lambda }}}{\delta \phi }\cdot \left(\frac{d}{d\mathrm{\Lambda }}{R}_{\mathrm{\Lambda }}^{-1}\right)\cdot \frac{\delta S_{\mathrm{int}\mathrm{\Lambda }}}{\delta \phi }-\frac{1}{2}\mathrm{Tr}\left[\frac{{\delta }^2{S}_{\mathrm{int}\mathrm{\Lambda }}}{\delta \phi \delta \phi }\cdot \frac{d}{d\mathrm{\Lambda }}{R}_{\mathrm{\Lambda }}^{-1}\right].}$$Jacques Distler claimed without proof that this ERGE is not correct nonperturbatively.

Effective average action ERGE

The effective average action ERGE involves a smooth IR regulator cutoff, which accounts for all fluctuations up to an IR scale k. The effective average action (EAA) will be accurate for fluctuations with momenta larger than k. As k is lowered, the effective average action approaches the effective action which includes all quantum and classical fluctuations. In contrast, for large k the effective average action is close to the "bare action". So, the effective average action interpolates between the "bare action" and the effective action.

For a real scalar field, one adds an IR cutoff $${\displaystyle \frac{1}{2}\int \frac{d^{d}p}{(2\pi {)}^d}{\tilde{\phi }}^{\ast }(p)R_k(p)\tilde{\phi }(p)}$$ to the action S, where R_k is a function of both k and p such that for ${\displaystyle p\gg k}$, R_k(p) is very tiny and approaches 0 and for ${\displaystyle p\ll k}$, ${\displaystyle R_k(p)\gtrsim k^2}$. R_k is both smooth and nonnegative. Its large value for small momenta leads to a suppression of their contribution to the partition function which is effectively the same thing as neglecting large-scale fluctuations.

Defiine the regulator as $${\displaystyle \frac{1}{2}\phi \cdot R_k\cdot \phi }$$So, $${\displaystyle \exp \left(W_k[J]\right)=Z_k[J]=\int \mathcal{D}\phi \exp \left(-S[\phi ]-\frac{1}{2}\phi \cdot R_k\cdot \phi +J\cdot \phi \right)}$$ where J is the source field. The Legendre transform of W_k ordinarily gives the effective action. However, the initial action is in fact S[φ] + 1/2 φ⋅R_k⋅φ and so the second term must be subtracted to get the effective average action. In other words, $${\displaystyle \phi [J;k]=\frac{\delta W_k}{\delta J}[J]}$$ can be inverted to give J_k[φ] . Finally, define the effective average action Γ_k as $${\displaystyle {\mathrm{\Gamma }}_k[\phi ]\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\left(-W\left[J_k[\phi ]\right]+J_k[\phi ]\cdot \phi \right)-\frac{1}{2}\phi \cdot R_k\cdot \phi .}$$Note the effective action Γ_k is related to Polchinski's effective action S_int via a Legendre transform relation. Taking a derivative with respect to the cutoff completes the process, producing an expression known as the Wetterich equation.

Renormalization group improvement of the effective potential

The renormalization group can also be used to compute effective potentials at orders higher than 1-loop, such as when computing corrections to the Coleman–Weinberg mechanism. To do so, write the renormalization group equation in terms of the effective potential. In the case of the ${\displaystyle {\phi }^4}$ model: $${\displaystyle \left(\mu \frac{\mathrm{\partial }}{\mathrm{\partial }\mu }+{\beta }_{\lambda }\frac{\mathrm{\partial }}{\mathrm{\partial }\lambda }+\phi {\gamma }_{\phi }\frac{\mathrm{\partial }}{\mathrm{\partial }\phi }\right)V_{\text{eff}}=0.}$$In order to determine the effective potential, it is useful to write ${\displaystyle V_{\text{eff}}}$ as $${\displaystyle V_{\text{eff}}=\frac{1}{4}{\phi }^4{S}_{\text{eff}}(\lambda ,L(\phi )),}$$ where ${\displaystyle S_{\text{eff}}}$ is a power series in ${\displaystyle L(\phi )=\log \frac{{\phi }^2}{{\mu }^2}}$: $${\displaystyle S_{\text{eff}}=A+BL+C{L}^2+D{L}^3+\cdots .}$$Using the above ansatz, it is possible to solve the renormalization group equation perturbatively and find the effective potential up to desired order.