2026-10-01 · chapter Yang–Mills and the mass gapPart 4 Quantisationperturbation theory, Feynman diagrams

Perturbation theory and Feynman diagrams — expand around a Gaussian integral, and the terms become diagrams

What can be computed exactly in the Euclidean path integral ((3) of 4-02) is the case where the action is quadratic in the field, that is, a Gaussian integral (2-07 §02). So we split the action into the quadratic part $S_0$ and the interaction $S_{\mathrm{int}}$, and expand $e^{-S_{\mathrm{int}}}$ ($\langle\cdot\rangle_0$ is the expectation with $S_0$ alone).

Exchanging the sum and the integral is formal (§08). Example. In place of a field take one real number $x$, and let $S=\frac12x^2+\frac{u}{4!}x^4$ ($u\ge0$; the same convention as 4-04). It is a "zero-dimensional field theory" whose spacetime is a single point. From $\langle x^{2k}\rangle_0=(2k-1)!!$ (1-08 §06),

(the meaning of $\sim$ is in §08; the coefficients computed as fractions Computed). All the values of the diagrams are numbers, so the bookkeeping can be checked by hand, and in four dimensions the shapes of the diagrams and the bookkeeping of weights are the same. Each term of the expansion can be written only with averages of polynomials under the Gaussian distribution, that is, moments of the normal distribution. Not one hard integral remains; all that remains is counting.

The term of order $n$ in (2) was the number of ways of pairing up $4n$ copies of $x$ two by two (Isserlis' formula, in physics Wick's theorem; (9) of 1-08). Draw each $\frac{u}{4!}x^4$ as a point (vertex) with 4 legs, and join paired legs by lines. Pairings that give the same shape after joining have the same value, so they are grouped by shape. These are Feynman diagrams.

The denominator is the number of permutations of vertices and legs, and usually exactly this many pairings make one shape. If the shape has a symmetry, some of the permutations give back the same pairing and the number drops; that ratio is the symmetry factor $S(\Gamma)$, precisely the number of automorphisms of the diagram (exchanges of vertices and reattachments of legs that do not change how the lines are connected) Known. Example. With one vertex there are 3 pairings, all of them a "figure eight". $3/4!=1/8$ is the coefficient of $u$ in (2), and with the exchange of the loops and the reversal of each loop, $S=2\cdot2\cdot2$.

ComputedThe counting in this figure is done in the browser and has been checked to agree with Table 1. The numbers are the legs of the vertices; x, y and 1–4 are external points (the $x$ put in from outside).

Detour: the 1949 papers and a theorem the following year

Table 1 also includes diagrams containing parts not connected to the external points (vacuum bubbles). The value of a diagram made of separate parts is the product of the values of the parts, and the denominator $Z$ of (1) is the sum of diagrams made only of bubbles, so bubbles vanish by division. Moreover, the sum of all diagrams is the exponential of the sum of connected diagrams. When $k$ copies of the same connected diagram $C$ appear side by side, the exchanges $k!$ are added to $S$, giving $\sum_kC^k/k!=e^C$ Known. In zero dimensions the value of a line is 1, so the value of a diagram is $(-u)^n/S$.

Example. For $u^2$, the $\frac1{16}+\frac1{48}=\frac1{12}$ of the connected vacuum diagrams agrees with the coefficient $\frac{35}{384}-\frac12\cdot\frac1{64}=\frac1{12}$ of $\log Z$, and the $\frac14+\frac14+\frac16=\frac23$ of the connected two-point diagrams is the coefficient of $\langle x^2\rangle$. Read the other way, a single integral is a generating function that counts diagrams, which leads to the counting of diagrams on surfaces (Bessis–Itzykson–Zuber 1980, 7-04).

For a four-dimensional field the procedure is the same. The quadratic part of the $d$-dimensional action

is a Gaussian measure with covariance $G=(-\Delta+m^2)^{-1}$ ($\Delta$ is the Laplacian $\sum_\mu\partial_\mu^2$; not the $\Delta$ of the mass gap) (4-02 §06), and Wick's theorem can be used as it is. Two things change. The value of a line becomes the propagator $G(x-y)$ (the degree to which the fluctuation of the field at the point $x$ reaches the point $y$; it becomes smaller with distance), and vertices can be placed anywhere in spacetime, so their positions $z$ are integrated over.

The second term is the "line with a loop"; the number 1 for $\langle x^2\rangle=1-\frac u2+\cdots$ in zero dimensions has simply been replaced by a propagator. The lowest-order four-point "×" is $-u\int d^dz\prod_iG(x_i-z)$, which can be read as four particles meeting at one point. Precisely, it means integrating the position of that point over all space. Note. $G(0)$ is infinite at $d\ge2$ (1-11), and it diverges from first order. Its treatment is 4-04, and the $T(\Lambda)$ there is this loop.

Putting in $G(x)=\int\frac{d^dp}{(2\pi)^d}\frac{e^{ip\cdot x}}{p^2+m^2}$ (the inverse Fourier transform of the $\hat G$ of (10) in 1-11; the rotation is (5) of 4-02) and integrating over the positions of the vertices gives $\int d^dz\,e^{i(p_1+\cdots+p_4)\cdot z}=(2\pi)^d\delta^{(d)}(p_1+\cdots+p_4)$: momentum is conserved at each vertex. The $\delta$ here is the relation of 1-11 that integrating a plane wave over all space concentrates the value only where the wave number is 0. The integral over positions turns into a condition on adding momenta. There are $V$ conditions for $I$ internal lines, and one of them is overall conservation, involving only the external lines, so the number of momenta left free in a connected diagram is

which is the number of independent cycles (loops) of the diagram Known. The rules are $1/(p^2+m^2)$ for each line, $-u$ for each vertex, $\int d^dk/(2\pi)^d$ for each loop, and $1/S$ overall. Example. The one-loop four-point "bubble" has $I=V=2$, $L=1$, and with momentum $P$ flowing in,

The two-point "joined by three lines" diagram has $I=3$, $V=2$, $L=2$, and is a double integral. (8) behaves like $k^{d-1}dk/k^4$ for large $|k|$, and diverges logarithmically at $d=4$. Adding the three bubbles and changing the sign gives the one-loop term of (4) in 4-04.

Restoring the weight to $e^{-S_E/\hbar}$, propagators are multiplied by $\hbar$ and vertices by $1/\hbar$. For a connected diagram with $I'$ lines including those going to external points, $V$ vertices and $E$ external points, counting external points as points too, $L=I'-V-E+1$, so

Fixing $E$, the power of $\hbar$ is fixed by the number of loops (Srednicki §21) Known. The lowest order is diagrams without loops (trees), and the abstract of Boulware–Brown (1968) writes that the classical field made by a source is 'the generating functional of the tree-graph approximation' Known.

Example. The classical solution of $\frac12x^2+\frac u{4!}x^4-Jx$, zero dimensions with a source added, is $x_{\mathrm{cl}}=J-\frac u6J^3+\frac{u^2}{12}J^5-\frac{u^3}{18}J^7+\cdots$. In general it can be written $x_{\mathrm{cl}}=\sum_k(-1)^kT_k\,(u/6)^kJ^{2k+1}$, where $T_k=1,1,3,12,55,\dots$ is the number of ternary trees. Integrating numerically at $u=1$, $J=0.8$, $(\langle x\rangle-x_{\mathrm{cl}})/\hbar$ approaches $-0.2167,\,-0.2266,\,-0.2277$ at $\hbar=0.1,\,0.01,\,0.001$, the value $-0.2278$ of the one-loop diagram (a line with a loop around $x_{\mathrm{cl}}$) Computed. In Yang–Mills, rescaling as $\hat A=gA$ makes the action $\frac1{2g^2}\int\mathrm{tr}\,\hat F^2$ ((7) of 4-02), which enters the weight only in the combination $g^2\hbar$, so the loop expansion is an expansion in $g^2$ (3-03).

The conventions are the same as in 3-02: $g$ is written split, $F^a_{\mu\nu}=\partial_\mu A^a_\nu-\partial_\nu A^a_\mu+gf^{abc}A^b_\mu A^c_\nu$, $S_E=\frac14\int F^a_{\mu\nu}F^a_{\mu\nu}\,d^4x$. Split into quadratic, cubic and quartic parts ((11) of 3-02 read in Euclidean form as in 4-02 §04; all indices are lower, and repeated indices are summed over 1–4),

The quadratic kernel is $K_{\mu\nu}=p^2\delta_{\mu\nu}-p_\mu p_\nu$ in momentum, and sends $p_\mu$ to 0. This is because $A_\mu=\partial_\mu\lambda$ is the direction of infinitesimal gauge transformations, which do not change the action (4-02 §07). There is no inverse matrix, so no propagator can be made. Adding the gauge-fixing term $\frac1{2\alpha}\int(\partial_\mu A^a_\mu)^2$ (justified in 4-08),

and for $\alpha=1$ (the Feynman gauge), $\delta^{ab}\delta_{\mu\nu}/p^2$ (the same letter as in 4-08). The passage to momentum is $A^a_\mu(x)=\int\frac{d^4p}{(2\pi)^4}e^{ip\cdot x}\tilde A^a_\mu(p)$ (derivatives become $ip_\mu$). A vertex is the term of order $n$ of the action differentiated $n$ times with respect to $\tilde A$, with the sign changed (excluding $(2\pi)^4\delta(\sum p)$); the sign of $e^{-S}$ enters here (all momenta flowing in). Taking the opposite direction of Fourier transform reverses the sign of $i$ in (12).

The three-point vertex is proportional to momentum, the four-point vertex does not depend on momentum, and both contain $f^{abc}$. Example. Putting plane waves into (10), extracting the coefficients directly, and comparing with (11)–(13) for 20 random sets of momenta, polarisations and colours in SU(2) and SU(3), the relative error was at most $8\times10^{-13}$ Computed.

Ghosts. The abstract of Faddeev–Popov (1967) writes 'Feynman and De Witt showed, that the rules must be changed for ... diagrams with closed loops', and proposed a method of obtaining the rules for arbitrary diagrams from the path integral. The result is the rule of adding ghosts with propagator $\delta^{ab}/p^2$ and a vertex with two ghost lines and one gauge field line, and multiplying by $-1$ for each closed ghost loop (Srednicki §71) Known. $-1$ comes from ghosts being anticommuting numbers (4-08). The derivation, and the limits of the local gauge fixing it presupposes (3-04), are in 4-08.

The ratio of the coefficients $a_n$ of (2) is $|a_{n+1}/a_n|=\frac{(4n+1)(4n+3)}{24(n+1)}\approx\frac{2n}{3}$, so $a_n$ grows at a rate of roughly $n!\,(2/3)^n$, and the series does not converge for any $u\gt0$. That the integral (2) diverges for $u\lt0$ shows the same circumstance. The argument of this form in quantum electrodynamics was made by Dyson (1952), whose abstract qualifies the conclusion as 'tentatively' (4-12). Even so, putting $s=\frac{u}{4!}x^4$ into the Taylor remainder $\bigl|e^{-s}-\sum_{n\lt N}(-s)^n/n!\bigr|\le s^N/N!$ ($s\ge0$) and integrating gives

Known. Fixing $N$ and letting $u\to0$, the error vanishes at the rate $u^N$. This is an asymptotic series, and it is the meaning of the $\sim$ in (2). Fixing $u$ and increasing $N$, the error first falls and then rises, and the best is around $N\approx3/(2u)$, where the terms are smallest (the valley of term sizes can be moved in Figure 1 of 4-12).

For $u=1$, adding nothing, $N=0$, is best. (14) also held numerically for $u\le1$, $N\le16$ Computed. For field theory, the problem statement writes that in many known examples ('In most known examples') perturbation series are 'divergent expansions' (Jaffe–Witten §6.1), and that in the two-dimensional models that have been constructed the scattering matrix can be expanded as 'an asymptotic series in the coupling constants' (§6.2). For four-dimensional Yang–Mills there is not yet a theory with which to say what the series is an asymptotic expansion of.

The coefficient at each order is a finite sum of diagram integrals, and that each order is finite after renormalisation is a theorem, gauge theories included (4-04 §08, 4-05) Known. The sum of the series has no meaning in general. There are also things that do not show in perturbation theory. The propagator (11) has a pole at $p=0$ and represents a massless particle. If there is a mass gap, it is thought to come in a form that looks 0 at every order in $g^2$, like $\Lambda\propto e^{-1/(2b_0g^2)}$ (4-07) Physics. §6.6 of the problem statement also writes 'the mass gap is not classically visible'.

Even so, perturbation theory is part of the conditions of the problem statement. §4 requires that the correlation functions of the constructed theory agree at short distances with asymptotic freedom and the predictions of perturbative renormalisation theory (quoted in 0-03 §07). Perturbation theory is not a method of making the theory but a mark that the theory made must meet. Asymptotic freedom is 4-06, and large $N$ is 7-04.

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Next chapter to read: 4-04 Divergences and renormalisation. The $G(0)$ of (6) and the logarithmic divergence of (8) are rewritten in terms of the measured coupling.

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