An Introduction to the Nonperturbative Renormalization Group

Abstract

We give in these notes a short presentation of both the main ideas underlying Wilson’s renormalization group (RG) and their concrete implementation under the form of what is now called the non-perturbative renormalization group (NPRG) or sometimes the functional renormalization group (which can be perturbative). Prior knowledge of perturbative field theory is not required for the understanding of the heart of the article.

Appendix

“Nobody ever promised you a rose garden.”

J. Polchinski

2.1.1 Definitions, Conventions

$\bullet$ Integrals in $x$ and $q$ spaces

In real and Fourier spaces we define

$$\begin{aligned} \int _x=\int \mathrm{ d} ^dx,\quad \int _q =\int \frac{\mathrm{ d} ^dq}{(2\pi )^d} \end{aligned}$$

(2.204)

$\bullet$ Fourier transform

$$\begin{aligned} f(x)=\int _q \tilde{f}(q)\, e^{i qx}\!,\quad \tilde{f}(q)=\int _x f(x)\, e^{-i qx}. \end{aligned}$$

(2.205)

Depending on the context, we omit or not the tilde on the Fourier transform.

$\bullet$Definition of$v_d$

$$\begin{aligned} \int _q f(q^2)= 2 v_d\int _0^\infty \mathrm{ d} x\, x^{d/2-1} f(x). \end{aligned}$$

(2.206)

with

$$\begin{aligned} v_d=\frac{1}{2^{d+1}\pi ^{d/2}\varGamma \big (\frac{d}{2}\big )}. \end{aligned}$$

(2.207)

$\bullet$Functional derivatives

$$\begin{aligned} \frac{\delta }{\delta \tilde{\phi }_q}=\int _x\, \frac{\delta \phi (x)}{\delta \tilde{\phi }(q)}\,\frac{\delta }{\delta {\phi }(x)}= \int _x \, \frac{e^{i qx}}{(2\pi )^d} \,\frac{\delta }{\delta \phi _x} \end{aligned}$$

(2.208)

$\bullet$Correlation functions

$\varGamma [M]$ is a functional of $M(x)$. We define the 1PI correlation functions by

$$\begin{aligned} \varGamma ^{(n)}[M(x);x_1,\dots ,x_n]=\frac{\delta ^n\varGamma [M]}{\delta M_{x_1}\dots \delta M_{x_n}} \end{aligned}$$

(2.209)

We also define the Fourier transform of $\varGamma ^{(n)}$ by

$$\begin{aligned} \begin{array}{l} \tilde{\varGamma }^{(n)}[M(x);q_1,\dots ,q_n]= \int _{x_1\dots x_n} e^{-i \sum _iq_ix_i}\,\varGamma ^{(n)}[M(x);x_1,\dots ,x_n]. \end{array} \end{aligned}$$

(2.210)

and also

$$\begin{aligned} \bar{\varGamma }^{(n)}[M(x);q_1,\dots ,q_n]=\frac{\delta ^n\varGamma }{\delta \tilde{M}_{q_1}\dots \delta \tilde{M}_{q_n}}. \end{aligned}$$

(2.211)

The relation between $\tilde{\varGamma }^{(n)}$ and $\bar{\varGamma }^{(n)}$ follows from Eq. (2.208):

$$\begin{aligned} \bar{\varGamma }^{(n)}[M(x);q_1,\dots ,q_n]=(2\pi )^{-nd}\,\tilde{\varGamma }^{(n)}[M(x);q_1,\dots ,q_n]. \end{aligned}$$

(2.212)

$\bullet$Cut-off function in$x$and$q$spaces

$$\begin{aligned} \varDelta H_k[\phi ]= -\frac{1}{2}\int _q \, \tilde{R}_k(q^2)\,\tilde{\phi }_q \tilde{\phi }_{-q} = -\frac{1}{2}\int _{x,y} \, \phi _x\, R_k(x-y)\, \phi _y \end{aligned}$$

(2.213)

One should be careful about the fact that $R_k$ is sometimes considered as a function of $q$ and sometimes as a function of $q^2$. It can be convenient to define a cut-off function with two entries by

$$\begin{aligned} \mathcal{ R}_k(x,y)=R_k(x-y). \end{aligned}$$

(2.214)

Then

$$\begin{aligned} \tilde {\mathcal{R}}_k(q,q^{\prime })=(2\pi )^{d} \delta ^d(q+q^{\prime }) \tilde{ R}_k(q) \end{aligned}$$

(2.215)

$\bullet$$k$-dependent anomalous dimension

By definition:

$$\begin{aligned} k\, {\partial _k}Z_k= -\eta _k Z_k. \end{aligned}$$

(2.216)

$\bullet$ Threshold functions $l_n^{d}$

$$\begin{aligned} l_n^d(w, \eta )= \frac{n+\delta _{n,0}}{2}\, \int _0^\infty \mathrm{ d} y\, y^{d/2-1}\,\frac{s(y)}{\big (y(1+r(y)) +w\big )^{n+1}} \end{aligned}$$

(2.217)

where

$$\begin{aligned} R_k(q^2)= Z_k q^2 r(y)\quad \text{ with}\quad y=\frac{q^2}{k^2} \end{aligned}$$

(2.218)

and, by definition of $s(y)$

$$\begin{aligned} k\, {\partial _k}R_k(q^2) = k\,{\partial _k}\left( Z_k q^2r\left(\frac{q^2}{k^2}\right) \right) = Z_k k^2 \big (-\eta _k\,y\, r(y) -2 y^2 r^{\prime }(y) \big ) {=} Z_k k^2 s(y). \end{aligned}$$

(2.219)

$\bullet$ Threshold functions $m_{n_1,n_2}^{d}$

$$\begin{aligned} m_{n_1,n_2}^d(w)=-\frac{1}{2}Z_k^{-1} k^{d-6}\int _0^\infty \mathrm{ d} x\, x^{d/2}\tilde{\partial }_t\,\frac{(\partial _x P)^2(x,0)}{P^{n_1}(x,0)P^{n_2}(x,w)} \end{aligned}$$

(2.220)

with

$$\begin{aligned} P(x,w)= Z_k\, x + R_k(x) +w \end{aligned}$$

(2.221)

$\bullet$ Universal value of $l_n^{2n}(0,0)$ for $n>0$

For $n>0$ and independently of the choice of cut-off function $R_k$:

$$\begin{aligned} l_n^{2n}(0,0)= \frac{n}{2}\, \int _0^\infty \mathrm{ d} y\, (-2)\,\frac{r^{\prime }(y)}{\left(1+r(y) \right)^{n+1}}=1 \end{aligned}$$

(2.222)

$\bullet$ Derivative of $l_n^{d}$

$$\begin{aligned} \partial _w l_n^{d}(w,\eta )= -(n+\delta _{n,0})\, l_{n+1}^{d}(w,\eta ) \end{aligned}$$

(2.223)

$\bullet$ $\theta$-cut-off

A convenient cut-off function $R_k$ that allows to compute analytically some threshold functions is

$$\begin{aligned} R_k(q)= Z_k\big (k^2-q^2 \big )\theta \left(1-\frac{q^2}{k^2} \right) . \end{aligned}$$

(2.224)

With this cut-off we find

$$\begin{aligned} r(y)= \frac{1-y}{y}\,\theta (1-y) \end{aligned}$$

(2.225)

$\bullet$ Threshold functions $l_n^{d}$ and $m_{2,2}^d$ with the $\theta$-cut-off

With the cut-off function, Eq. (2.224), the $l_n^d$ threshold functions can be computed analytically

$$\begin{aligned} l_n^d(w, \eta )= \frac{2}{d}(n+\delta _{n,0})\left(1-\frac{\eta _k}{d+2} \right)\frac{1}{(1+w)^{n+1}} \end{aligned}$$

(2.226)

$$\begin{aligned} m_{2,2}^d(w)=\frac{1}{(1+w)^2} \end{aligned}$$

(2.227)

2.1.2 Proof of Eq. (2.106)

We define:

$$\begin{aligned} J=\int _{y,z} e^{-y^2/2\alpha -z^2/2\beta } \end{aligned}$$

(2.228)

and we rewrite the exponent:

$$\begin{aligned} -\frac{y^2}{2\alpha }-\frac{z^2}{2\beta }&=-\frac{1}{2}\left(\frac{1}{\alpha }+\frac{1}{\beta }\right) y^2+\frac{xy}{\beta }-\frac{x^2}{2\beta }\nonumber \\&=-\frac{1}{2}\frac{\gamma }{\alpha \beta }\left(y-\frac{\alpha }{\gamma }x\right)^2+\frac{\alpha }{2\beta \gamma } x^2-\frac{x^2}{2\beta } . \end{aligned}$$

(2.229)

We now define

$$\begin{aligned} u=y-\frac{\alpha }{\gamma }x \end{aligned}$$

(2.230)

and change variables: $(y,z)\rightarrow (u,x)$. The jacobian is 1 and thus:

$$\begin{aligned} J=\int _{u,x} e^{-\gamma u^2/2\alpha \beta -x^2/2\gamma } = \sqrt{\frac{2\pi \alpha \beta }{\gamma }}\,I . \end{aligned}$$

(2.231)

2.1.3 The Exact RG Equations

For the sake of simplicity, we consider a scalar theory (e.g. Ising). We have by definition

$$\begin{aligned}&{\displaystyle Z_k[B]=\int \mathcal{ \mathcal D} \phi \, \exp \left(-H[\phi ] - \varDelta H_k[\phi ]+\int B\phi \right)}\nonumber \\ \nonumber \\&\text{ with}\ \ \varDelta H_k[\phi ]=\frac{1}{2}\int _q R_k(q)\, \phi _q\,\phi _{-q}\nonumber \\&\nonumber \\&{\displaystyle W_k[B]= \log Z_k[B]}\nonumber \\&\nonumber \\&{\displaystyle \varGamma _k[M]+ W_k[B]= \int _x B M -\frac{1}{2}\int _{x,y} M_x\, R_{k,x-y}\, M_y}\nonumber \end{aligned}$$

with, by definition of $M(x)$:

$$\begin{aligned} \frac{\delta W_k}{\delta B(x)}=M(x)=\langle \phi (x)\rangle . \end{aligned}$$

(2.232)

When $B(x)$ is taken $k$-independent (as in $Z_k[B]$) then $M(x)$ computed from $W_k$ is $k$-dependent. Reciprocally, if $M(x)$ is taken fixed (as in $\varGamma _k[M]$), then $B(x)$ computed from Eq. (2.236) becomes $k$-dependent.

2.1.3.1 RG Equation for $W_k[B]$

$$\begin{aligned} \partial _k e^{W_k}&=-\frac{1}{2}\int \mathcal{ D} \phi \, \Big (\int _{x,y}\phi _x \,\partial _kR_k(x-y)\,\phi _y\Big )\nonumber \\&\;\,\cdot \exp \Big ( -H[\phi ]-\,\frac{1}{2}\int _q R_k(q) \phi _q\phi _{-q} +\, \int B\phi \Big )\nonumber \\&=\left(-\frac{1}{2}\int _{x,y}{\partial _k}R_k(x-y)\frac{\delta }{\delta B_x}\,\frac{\delta }{\delta B_y}\right)\,e^{ W_k[B]}. \end{aligned}$$

(2.233)

We therefore obtain for $W_k$:

$$\begin{aligned} {\partial _k}W_k[B]= -\frac{1}{2}\int _{x,y}{\partial _k}R_k(x-y)\left(\frac{\delta ^2 W_k}{\delta B_x\,\delta B_y}+\frac{\delta W_k}{\delta B_x}\,\frac{\delta W_k}{\delta B_y} \right) \end{aligned}$$

(2.234)

which is equivalent to the Polchinski equation.

2.1.3.2 RG Equation for $\varGamma _k[M]$

We first derive the reciprocal relation of Eq. (2.232). The Legendre transform is symmetric with respect to the two functions that are transformed. Here the Legendre transform of $W_k$ is $\varGamma _k+1/2 \int R_k M M$. Thus

$$\begin{aligned} \frac{\delta }{\delta M_x} \left(\varGamma _k+ \frac{1}{2}\int _{x,y}M_x\, R_k(x-y)\,M_y\right)=B_x \end{aligned}$$

(2.235)

and then

$$\begin{aligned} \frac{\delta \varGamma _k}{\delta M_x} =B_x- \int _{y} R_k(x-y)M_y . \end{aligned}$$

(2.236)

In the Polchinski equation (2.234), the $k$-derivative is taken at fixed $B_x$. We must convert it to a derivative at fixed $M$:

$$\begin{aligned} {{\partial _k}_{\vert }}_{B}= {{\partial _k}_{\vert }}_{M} + \int _x {\partial _k}{{M_x}_{\vert }}_{B}\frac{\delta }{\delta M_x} \end{aligned}$$

(2.237)

Acting on Eq. () with ${{\partial _k}_{\vert }}_{B}$, we obtain:

$$\begin{aligned} &\quad{{\partial _k}\varGamma _k[M]_{\vert }}_{B}+ {{\partial _k}{W_k[B]}_{\vert }}_{B}=\int _x B\, {\partial _k}{{M}_{\vert }}_{B} \nonumber \\& -\frac{1}{2}\int _{x,y} {\partial _k}R_{k,x-y}\, M_x\, M_y-\int _{x,y} R_{k,x-y} M_x{\partial _k}{{M_y}_{\vert }}_{B} \end{aligned}$$

(2.238)

Substituting Eqs. (2.236, 2.234, 2.237) into this equation we finally obtain

$$\begin{aligned} {\partial _k}\varGamma _k[M]=\frac{1}{2}\int _{x,y}{\partial _k}R_k(x-y) \, \frac{\delta ^2 W_k}{\delta B_x\,\delta B_y} \end{aligned}$$

(2.239)

The last step consists in rewriting the right hand side of this equation in terms of $\varGamma _k$ only. We start from (2.232) and act on it with $\delta /\delta M_z$:

$$\begin{aligned} \delta (x-z)= \frac{\delta ^2 W_k}{\delta B_x\,\delta M_z}=\int _y \frac{\delta ^2 W_k}{\delta B_x\,\delta B_y}\,\frac{\delta B_y}{\delta M_z} . \end{aligned}$$

(2.240)

Now, using (2.236), we obtain

$$\begin{aligned} \delta (x-z)=\int _y \frac{\delta ^2 W_k}{\delta B_x\,\delta B_y}\,\left( \frac{\delta ^2 \varGamma _k}{\delta M_y\,\delta M_z}+R_k(y-z)\right) . \end{aligned}$$

(2.241)

We define

$$\begin{aligned} W^{(2)}_k(x,y)= \frac{\delta ^2 W_k}{\delta B_x\,\delta B_y} \end{aligned}$$

(2.242)

and thus

$$\begin{aligned} \delta (x-z)=\int _y W^{(2)}_k(x,y)\left(\varGamma ^{(2)}_k+\mathcal{ R}_k\right)(y,z) . \end{aligned}$$

(2.243)

$\varGamma ^{(2)}_k+\mathcal{ R}_k$ is therefore the inverse of $W^{(2)}_k$ in the operator sense. Note that although we did not specify it, $W^{(2)}_k$ is a functional of $B(x)$ and $\varGamma ^{(2)}_k$ a functional of $M(x)$. Relation (2.243) is valid for arbitrary $M$. The RG equation (2.239) can now be written in terms of $\varGamma _k$ only:

$$\begin{aligned} {\partial _k}\varGamma _k[M]=\frac{1}{2}\int _{x,y}{\partial _k}R_k(x-y) \,\left(\varGamma ^{(2)}_k+\mathcal{ R}_k\right)^{-1}(x,y) . \end{aligned}$$

(2.244)

In Fourier space this equation becomes:

$$\begin{aligned} {\partial _k}\varGamma _k[M]=\frac{1}{2}\int _{q}{\partial_k}{\tilde{R}}_k(q)\left(\tilde{\varGamma }^{(2)}_k+\tilde{\mathcal{R}}_k\right)^{-1}_{q,-q} . \end{aligned}$$

(2.245)

2.1.3.3 RG Equation for the Effective Potential

The derivative expansion consists in expanding $\varGamma _k$ as

$$\begin{aligned} \varGamma _k[M(x)]= \int _x \left(U_k(M^2) +\frac{1}{2}Z_k(M^2)\, \left(\nabla M\right)^2 +\dots \right) \end{aligned}$$

(2.246)

where we have supposed that the theory is $\mathbb{ Z} _2$ symmetric so that $U_k, Z_k, \dots$ are functions of $M^2$ only. To compute the flow of these functions it is necessary to define them from $\varGamma _k$. The effective potential $U_k$ coincides with $\varGamma _k$ when it is evaluated for uniform field configurations $M_{\text{ unif.}}$:

$$\begin{aligned} \varGamma _k[M_{\text{ unif.}}]= \varOmega \, U_k(M^2_{\text{ unif.}}) \end{aligned}$$

(2.247)

where $\varOmega$ is the volume of the system. It is easy to derive an RG equation from this definition of $U_k$ if we use the local potential approximation (LPA) that consists in truncating $\varGamma _k$ as in (2.246) with $Z_k(M)=1$:

$$\begin{aligned} \varGamma _k^{\text{ LPA}} [M(x)]= \int _x \left(U_k(M^2(x)) +\frac{1}{2}\, \left(\nabla M\right)^2 \right) . \end{aligned}$$

(2.248)

By acting on Eq. (2.247) with ${\partial _k}$ we obtain:

$$\begin{aligned} {\partial _k}U_k(M) = \frac{1}{2\,\varOmega } \int _q {\partial _k}R_k(q){{ \left( \varGamma ^{(2)}_k\,_{\vert _{M_{\text{ unif.}}}} +R_k\right)^{-1}_{q,-q}}} . \end{aligned}$$

(2.249)

Thus, we have to invert $\varGamma ^{(2)}_k+R_k$ for a uniform field configuration and within the LPA. From now on, we omit the superscript LPA on $\varGamma _k$. An elementary calculation leads to

$$\begin{aligned} {\bar{\varGamma }^{(2)}_{k,q,q^{\prime }}\,}_{\vert _{M_{\text{ unif.}}}}=\left(\frac{\partial ^2U_k}{\partial M^2} + q^2 \right)(2\pi )^{-d} \delta (q+q^{\prime }) . \end{aligned}$$

(2.250)

Using $\delta (q=0)=\varOmega (2\pi )^{-d}$ we find

$$\begin{aligned} {\partial _k}U_k=\frac{1}{2}\int _q \frac{{\partial _k}R_k(q)}{ q^2+R_k(q)+\displaystyle {\frac{\partial ^2U_k}{\partial M^2}} } . \end{aligned}$$

(2.251)

It is convenient to re-express this equation in terms of

$$\begin{aligned} \rho =\frac{1}{2}M^2 \end{aligned}$$

(2.252)

which is the $\mathbb{ Z} _2$-invariant.

$$\begin{aligned} {\partial _k}U_k(\rho )=\frac{1}{2}\int _q \frac{{\partial _k}R_k(q)}{ q^2+R_k(q)+ U_k^{\prime }(\rho )+2\rho \, U_k^{\prime \prime }(\rho ) } \end{aligned}$$

(2.253)

where $U_k^{\prime }(\rho )$ and $U_k^{\prime \prime }(\rho )$ are derivatives of $U_k$ with respect to $\rho$.

To obtain the RG equation for the dimensionless potential we have to perform the change of variables of Eq. 2.154. We find

$$\begin{aligned} {\partial _t}_{\vert _{\rho }}={\partial _t}_{\vert _{\tilde{\rho }}} + (2-d-\eta )\tilde{\rho }\,\frac{\partial }{\partial \tilde{\rho }} \end{aligned}$$

(2.254)

and

$$\begin{aligned} {\partial _t}_{\vert _{q^2}}={\partial _t}_{\vert _{y}}-2 y\partial _y . \end{aligned}$$

(2.255)

Inserting these relations together with Eq. (2.154) and with

$$\begin{aligned} \partial _t R_k(q^2)=-y\big (\eta _k\, r(y)+2 y\, r^{\prime }(y)\big ) Z_k\, k^2 \end{aligned}$$

(2.256)

in Eq. (2.253) leads to the RG equation on $\tilde{U}_t$, Eq. (2.155).