Prof. Dr. Stefan Floerchinger

Researcher and professor for quantum field theory.

Properties of the flow equation

  1. The flow equation is exact, we have made no approximations. All non-perturbative effects are included, such as topological defects, etc.

  2. The particular form of the matrix \(R_{k\alpha\beta}\) is not important. It is only important that \(\Delta S_k[\phi]\) is a quadratic form in the fields. This allows generalisations to a wide range of situations where \(R_k\) is not necessarily interpreted as a momentum cutoff.

  3. Finite momentum integrals. For certain choices of the regulator \(R_k\) one obtains finite momentum integrals. This is best discussed in examples later on.

  4. Flow equations for correlation functions. One may take functional derivatives with respect to the field \(\Phi^\alpha\) and obtain further flow equations for correlation functions.

Derivative expansion

The flow equation is a functional differential equation. Except for a few particular cases (leading order large \(N\) expansion, few non-relativistic particles), it cannot be solved exactly.

Approximate solutions are constructed by truncation. A truncation is an ansatz for the general form of the effective average action in terms of a few free parameters or free functions. One computes the flow of these parameters or functions by inserting the ansatz on the right hand side of the flow equation, specifically for the computation of \(\Gamma^{(2)}[\Phi]\).

For the derivative expansion, one expands \(\Gamma_k[\Phi]\) in terms of powers of derivatives of the fields. For example, for a theory with \(\text{O}(N)\) symmetry, this leads to \[\Gamma_k[\Phi] = \int_x \left\{ U_k(\rho) + \frac{g^{\mu\nu}}{2}Z_k(\rho)\partial_\mu\Phi_n(x)\partial_\nu\Phi_n(x) +\frac{g^{\mu\nu}}{4}Y_k(\rho)\partial_\mu\rho(x)\partial_\nu\rho(x) + \ldots \right\},\] where \(\rho(x)= \frac{1}{2} \Phi_n(x)\Phi_n(x)\). The derivative expansion to the given order neglects terms with four or more derivatives. In this order, one has three functions, \(U_k(\rho)\), \(Z_k(\rho)\) and \(Y_k(\rho)\) which parametrize the flowing action. If we simplify further: \(Y_k=0\), \(Z_k\) independent of \(\rho\), then this is called “leading (order) potential approximation”.

Flow of effective potential

We want to compute the flow equation for the effective potential \(U_k(\rho)\). For this purpose, we evaluate \(\partial_t \Gamma_k[\Phi]\) for homogeneous fields. One needs to evaluate \(\Gamma_k^{(2)}\) for constant \(\Phi\). In momentum space, it reads \[(\Gamma_k^{(2)})_{nm}(p,q)[\Phi] = \left[Z_k p^2\delta_{nm} + \frac{\partial^2 U_k(\rho) }{\partial \Phi_n\partial\Phi_m}\right] (2\pi)^{(d)}\delta^{(d)}(p-q).\] We also specify the regulator term to be of the form \[\Delta S_k[\Phi] = \int_x \left\{ \frac{1}{2} \Phi_n(x) R_k(-\Box_x) \Phi_n(x) \right\} = \int_p \left\{ \frac{1}{2}\Phi_n^*(p) R_k(p^2) \Phi_n(p) \right\},\] with some function \(R_k(p^2)\). The functional trace for the flow equation is then easily performed in momentum space, and one obtains the flow equation for the effective potential, \[\begin{split} \partial_t U_k(\rho) = \frac{1}{2}\int_q \frac{\partial_t R_k(q^2)}{Z_k q^2+R_k(q^2)+U_k'(\rho)+2\rho U_k''(\rho)} + \frac{N-1}{2}\int_q \frac{\partial_t R_k(q^2)}{Z_k q^2 + R_k(q^2) + U_k'(\rho)}. \end{split}\] This can be compared to the one-loop approximation \[\begin{split} U_\text{1-loop} = \frac{1}{2}\int_q \ln(Zq^2+V'(\rho)+2\rho V''(\rho)) + \frac{N-1}{2}\int_q \ln (Zq^2+V'(\rho)). \end{split}\] Replace \(V(\rho)\to U(\rho)\), add \(R_k(q^2)\) by \(Zq^2 \to Z_k q^2+R_k(q^2)\) and take the \(\tilde\partial_t\)-derivative. This leads from the one-loop approximation to the flow equation.

Ultraviolet finite flow

There are choices of regulator functions such that no ultraviolet divergencies arise in \(\partial_t U_k(\rho)\). As an example, consider the function \[R_k(q^2) = Z_k (k^2-q^2)\theta(k^2-q^2).\] The anomalous dimension is defined by \[\begin{split} \eta = -\partial_t \ln Z_k. \end{split}\] It is typically very small and we can neglect the term proportional to \(\eta\) in \(\partial_t R_k\). The result is a simple expression for the flow of the effective potential, \[\partial_t U_k(\rho) = \frac{1}{2} \int_{q^2<k^2} \frac{2Z_kk^2}{Z_k k^2 + U_k'(\rho) + 2\rho U_k''(\rho)} + \frac{N-1}{2} \int_{q^2<k^2} \frac{2Z_kk^2}{Z_kk^2 + U_k'(\rho)}.\] We define the renormalized dimensionless mass terms \(w_1\) for the radial mode and \(w_2\) for the Goldstone modes. \[\begin{split} w_1 = \frac{U_k'(\rho)+2\rho U_k''(\rho)}{Z_k k^2},\qquad w_2 = \frac{U_k'(\rho)}{Z_k k^2}. \end{split}\] The momentum integrals are trivial, \[\int_{q^2<k^2} = \alpha_d k^d,\] Examples for the coefficients that arise here are \(\alpha_2 = 1/(4\pi)\), \(\alpha_3 = 1/(6\pi^2)\) and \(\alpha_4 = 1/(32\pi^2)\). We arrive at a very simple flow equation for the effective potential, \[\partial_t U_k(\rho) = \alpha_d k^d \left[ \frac{1}{1+w_1}+ \frac{N-1}{1+w_2} \right].\] Since \(w_1\) and \(w_2\) involve \(\rho\)-derivatives of \(U_k(\rho)\), this is a differential equation for a single function \(U\) of the two variables \(k\) and \(\rho\). For a given \(\eta\) or \(\eta=0\), it is closed.

Scale dependent minimum

We now specialize to \(d=4\) dimensions and investigate the implications of the flow equation in more detail. If the effective average potential \(U_k(\rho)\) has a minimum at \(\rho_0(k)\), the condition for the minimum is for all \(k\), \[U'_k(\rho_0(k)) = 0.\] The flow equation for \(U'(\rho)\) at fixed \(\rho\) is obtained by taking a \(\rho\)-derivative of the flow equation for the potential, \[\begin{split} \partial_t U_k'(\rho) &= \frac{k^4}{32\pi^2} \frac{\partial}{\partial \rho} { \frac{1}{1+w_1}+ \frac{N-1}{1+w_2}}\\ &= - \frac{k^4}{32\pi^2} { -\frac{1}{(1+w_1)^2} \frac{\partial w_1}{\partial \rho} + \frac{N-1}{(1+w_2)^2} \frac{\partial w_2}{\partial \rho} }. \end{split}\] For simplicity, we take \(Z=1\) (equivlent to \(\eta=0\)), and use \[\begin{split} w_1 &= \frac{U_k'(\rho)+2\rho U_k''(\rho)}{k^2},\quad\quad\quad w_2 = \frac{U_k'(\rho)}{k^2},\\ \frac{\partial w_1}{\partial \rho} &= \frac{3U_k''(\rho)+2\rho U_k'''(\rho)}{k^2},\quad\quad\quad \frac{\partial w_2}{\partial \rho} = \frac{U_k''(\rho)}{k^2}. \end{split}\] One infers \[\begin{split} \partial_t U_k'(\rho) &= - \frac{k^2}{32\pi^2} \left[ -\frac{3U''(\rho)+2\rho U'''(\rho)}{(1+w_1)^2} + \frac{(N-1)U''(\rho)}{(1+w_2)^2} \right]. \end{split}\] For \(\rho=\rho_0\), one has \(U_k'(\rho_0)=0\), \(w_2=0\), and \(w_1=2\rho_0 U_k''(\rho_0)\). We define \(\lambda=U_k''(\rho_0)\) and \(\nu=U_k'''(\rho_0)\) such that \[\begin{split} \partial_t U_k'(\rho_0) &= - \frac{k^2}{32\pi^2} \left[ \frac{3\lambda+2\rho_0\nu}{(1+2\rho_0 \lambda)^2} + (N-1)\lambda \right]. \end{split}\] For a fixed location \(\rho_0\), the derivation \(U_k'(\rho_0)\) does not remain zero. The location of the minimum therefore depends on \(k\), according to \[\partial_t U_k'(\rho_0)+ \partial_t U_k''(\rho_0) \frac{\partial \rho}{\partial t} = 0,\] which implies \[\frac{\partial \rho_0}{\partial t} = - \frac{1}{\lambda}\partial_t U_k'(\rho_0).\] The location of the minimum moves according to \[\begin{split} \frac{\partial \rho_0}{\partial t} &= \frac{k^2}{32\pi^2} \left[ \frac{3+2\rho_0\nu/\lambda}{(1+2\rho_0 \lambda)^2} + (N-1) \right]. \end{split}\] As \(k\) is lowered, \(\rho_0\) becomes smaller. Depending on the initial value at \(k=\Lambda\), it may reach zero at some \(k>0\) or not.

Leading order at small coupling

For small \(\lambda\), we will see that \(\nu\sim \lambda^3\). To lowest order in \(\lambda\), the flow equation for \(\rho_0\) simplifies, \[\frac{\partial \rho_0}{\partial t} = \frac{k^2}{32\pi^2}(N+2).\] This has the simple solution \[\begin{split} \rho_0(k) = \frac{k^2}{64\pi^2}(N+2) + c_\Lambda, \end{split}\] with integration constant \(c_{\Lambda}\), or, with \(\rho_\Lambda = \rho_0(k=\Lambda)\), \[\rho_0(k) = \rho_\Lambda - \frac{\Lambda^2-k^2}{64\pi^2}(N+2).\] Different initial values \(\rho_\Lambda\) label different “flow trajectories”.

Phase transition

There is a critical value \(\rho_{\Lambda,\text{critilcal}}\) for which \(\rho_0(k=0)=0\), namely \[\rho_{\Lambda,\text{critical}} = \frac{\Lambda^2}{64\pi^2}(N+2).\] For \(\rho_\Lambda > \rho_{\Lambda,\text{critical}}\), one has \(\rho_0(k=0)>0\). This corresponds to the phase with spontaneous symmetry breaking (SSB).

On the other hand, for \(\rho_\Lambda<\rho_{\Lambda,\text{critical}}\), one finds \(\rho_0(k_t)=0\) for \(k_t>0\). For \(k<k_t\), the minimum is located at \(\rho=0\). The flow of \(U\) is then better described by the flow of \(m_0^2 = U'(0)\). It increases with decreasing \(k\). At \(k=0\), one finds \(m_0^2>0\). Then the model is in the symmetric phase (SYM).

Quadratic divergence

For a given macroscopic or “remormalized” \(\rho_{0,\text{R}}=\rho_0(k=0)\), one finds for the microscopic or “bare” \(\rho_\Lambda\), \[\begin{split} \rho_{\Lambda} = \rho_{0,\text{R}}+\frac{\Lambda^2}{64\pi^2}(N+2). \end{split}\] For \(\Lambda\to\infty\), this diverges quadratically. The divergence arises from the relation between bare and renormalized parameters which in turn arises due to the flow that is generated by fluctuations.

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