2026-10-01 · chapter Yang–Mills and the mass gapPart 4 Quantisationthe algebra of divergences

The algebra of divergences — nested subtraction becomes a decomposition in a group, yet still does not leave formal power series

In 4-04 we removed one-loop divergences by subtracting the value at a reference scale $\mu$. What was subtracted was a number independent of the external momenta (a counterterm). From two loops on, a part of a diagram (a subdiagram) diverges on its own (a subdivergence). Two subdivergences are either nested, one containing the other, overlapping, sharing a part without either containing the other, or disjoint.

We use a small model with nesting only (Kreimer's toy model; the definition in §3.2 of Panzer 2012). "One loop" is $\int_0^\infty dy/(y+s)$, where $s\gt0$ is the outer scale and $y$ the loop variable. Like the four-dimensional bubble it diverges logarithmically, so we regularise by multiplying by $y^{-z}$ ($z$ is a small positive number, corresponding to the $\varepsilon$ of §04), and build nesting by putting the inner value into the outer integrand.

Writing the part with negative powers of $z$ (the pole part) as $\operatorname{pole}$, with $L=\ln(s/\mu)$ and $\mu=1$,

Computed The pole of the double loop contains $L$. Subtracting only the pole of the whole, the subtracted quantity contains $\ln s$ and is not a local term that can be added to the action. Example: at $z=0.1$, $L=1$, $\varphi(\ell_2)=44.49$, and its pole part is $50-10=40$.

The cure is to subtract from the inside. In Bogoliubov's prescription, one first adds the term in which the subdivergence is replaced by its counterterm $S(\bullet)=-1/z$, and then subtracts the whole.

Computed $L/z$ disappears, and the overall counterterm $S(\ell_2)=1/(2z^2)$ does not depend on the outer scale ($\bar R$: Bogoliubov's map). For general diagrams the sum is over forests — sets of divergent subdiagrams any two of which are nested or disjoint (including the empty set and the whole diagram). Zimmermann's forest formula is

where $T_\gamma$ is the operation extracting the divergent part of $\gamma$ (in momentum space, the low orders of the Taylor expansion in the external momenta), and the product acts from the inside (a form with details omitted). Example: the forests of $\ell_2$ are $\varnothing,\{\bullet\},\{\ell_2\},\{\bullet,\ell_2\}$, and (5) is the expansion $(1-T_{\ell_2})(1-T_\bullet)\,\ell_2$. The $(1-T_\bullet)$ part corresponds to the $\bar R$ of (4), and the remaining $-T_{\ell_2}$ to adding the overall counterterm $S(\ell_2)=1/(2z^2)$, leaving a value without poles (the $\varphi_+$ of §05). It is called BPHZ after the initials of Bogoliubov–Parasiuk (1957), Hepp (1966) and Zimmermann (1969), and the gaps in the first proof are said to have been filled by the work of the latter two (§1 of Ebrahimi-Fard–Kreimer 2005). The forest formula with overlaps also follows from the Hopf algebra of rooted trees (§05) (abstract of Kreimer 1999). Known

This section uses the distributions of 1-11 §04 and §08. We give only the essentials. A test function $g$ is a function that can be differentiated any number of times and goes to 0 quickly far away, and a distribution $t$ is a rule assigning linearly a number $\langle t,g\rangle$ to each test function. $\langle\delta,g\rangle=g(0)$, and its derivative is defined by $\langle\partial^\alpha\delta,g\rangle=(-1)^{|\alpha|}\partial^\alpha g(0)$. A sequence of distributions "converges as distributions" if $\langle t_n,g\rangle$ converges for every test function $g$.

In position space, a divergence is the impossibility of extending a distribution to a singular point. We take examples in Euclidean $\mathbb R^4$. The square of the four-dimensional massless propagator $G=1/(4\pi^2|x|^2)$ (the bubble diagram) behaves as $|x|^{-4}$ near the origin, and its integral diverges logarithmically. The method of Epstein–Glaser (1973) was set up in Minkowski space (spacetime with time); it builds time-ordered products (whose expectations are the correlation functions) inductively in the number of vertices (§4 of Brunetti–Fredenhagen 2000), and extends values fixed off the diagonal, where points coincide, onto the diagonal. Renormalisation is exactly the extension of expectations to the diagonal (abstract of the same paper). Since it is formulated as extension, there is no need to write divergent integrals along the way. The extension problem itself has the same form in Euclidean $\mathbb R^d$, as below. The freedom is measured by the scaling degree ((33) of the same paper; numbering of the arXiv version; convergence as distributions).

For $t_0$ on $\mathbb R^d\setminus\{0\}$ (Theorems 5.2 and 5.3 of the same paper) Known: if $\operatorname{sd}(t_0)\lt d$, the extension preserving the scaling degree is unique. If $d\le\operatorname{sd}(t_0)\lt\infty$, extensions exist, and the difference of two extensions is supported at the origin, of the form

$\rho$ corresponds to the superficial degree of divergence of 4-04, and the constants $c_\alpha$ are the freedom of local counterterms. Example: for $t_0=|x|^{-4}$ ($d=4$, $\rho=0$), the extension that subtracts $g(0)$ only inside the ball of radius $1/\mu$,

agrees with $t_0$ for $x\ne0$, and converges because the numerator is $O(|x|)$. The difference on changing $\mu$ is $2\pi^2\ln(\mu'/\mu)\,g(0)$ ($2\pi^2$ is the area of $S^3$), a constant multiple of $\delta$.

Example: for $g_s=e^{-|x|^2/s^2}$, $\langle u_\mu,g_s\rangle=2\pi^2(\ln(\mu s)-\gamma/2)$ ($\gamma$ is Euler's constant). At $\mu=1$, for $s=0.1,1,10$ it is −51.148, −5.697, 39.754, and the difference from $\mu=2$ is $2\pi^2\ln2=13.682$ in each case. Removing $|x|\lt\eta$ without extending gives 85.21, 357.91 at $\eta=10^{-2},10^{-8}$ ($s=1$), growing. Computed

We regard momentum integrals as functions of the dimension $d$. The bubble of 4-04 (mass $m$) converges for $d\lt4$, giving

($1/(8\pi m)$ at $d=3$). The right-hand side extends meromorphically to complex $d$ (as a function complex-differentiable except at isolated points called poles), and has a pole at $d=4,6,\dots$. Using this is dimensional regularisation. Here we write $d=4-2\varepsilon$ (half the $\epsilon$ of the $d=4-\epsilon$ of 4-04). Then

and the divergence becomes the pole $1/\varepsilon$. At $\varepsilon=0.1,0.01,0.001$, $16\pi^2I$ ($m=1$) is 12.2535, 101.9814, 1001.9565, and the differences from the approximation (10) are 0.2997, 0.0276, 0.0027. The role of $\ln k_{\max}^2$ in the $\ln(k_{\max}^2/m^2)-1$ cut at $|k|\lt k_{\max}$ in momentum is played by $1/\varepsilon$. Computed The MS scheme subtracts only $1/\varepsilon$, and the $\overline{\mathrm{MS}}$ scheme subtracts $1/\varepsilon-\gamma+\ln4\pi$. Since a scale $\mu$ is introduced (the coupling is multiplied by $\mu^{2\varepsilon}$), $-\ln(m^2/\mu^2)$ remains at $\overline{\mathrm{MS}}$. The $\Lambda_{\overline{\mathrm{MS}}}$ of 4-06 is the $\Lambda$ of this scheme.

In 1972 there were papers by Bollini–Giambiagi and by 't Hooft–Veltman (titled "Regularization and renormalization of gauge fields"), and it is said to be used as standard in Yang–Mills perturbation theory as a regularisation that does not break gauge symmetry. The price is that (9) is an analytic continuation diagram by diagram, and gives no measure for non-integer $d$. The problem statement too writes that there are few ways to recover the reflection positivity lost by dimensional regularisation and the like (p.11; quoted in 0-03 §03; 4-10).

Subtraction is the repetition of "extracting a subdiagram $\gamma$" and "the remainder $\Gamma/\gamma$ with $\gamma$ collapsed to a point". Before making this an operation, we prepare three words.

• One-particle irreducible: a diagram that does not split in two when any one internal line is cut.
• $\otimes$ (tensor product): $\gamma\otimes\Gamma/\gamma$ is a symbol that formally sets a pair of diagrams side by side. It is linear in each of left and right (bilinear), and does not multiply left and right. When values are put in later, it is multiplied as $\varphi(\gamma)\,\varphi(\Gamma/\gamma)$.
• Hopf algebra: an algebra with a product, a coproduct (a map splitting one element into a sum of pairs $\otimes$), and auxiliary maps called the counit and the antipode. This chapter uses only the coproduct.

Making subtraction an operation is the Hopf algebra of Kreimer (1998) and Connes–Kreimer (1998, 2000): the generators are divergent one-particle irreducible diagrams, the product is setting diagrams side by side, and extraction is the coproduct ((15) of Ebrahimi-Fard–Kreimer 2005). The sum in (11) is over divergent one-particle irreducible proper subdiagrams and their disjoint sets (set side by side as a product). For a set, $\Gamma/\gamma$ is the remainder with all of them collapsed to points. Writing the coproduct as $\Delta$ is a convention of the literature, unrelated to the $\Delta$ of the mass gap.

With nesting only, rooted trees suffice in place of diagrams. Vertices are divergent subdiagrams, and children are inner subdivergences. The sum in (11) becomes a sum over admissible cuts (choices of edges that cut every path from the root to a leaf at most once), with the cut-off side going on the left and the root side on the right (Figure 1).

$b$ is the tree with two children at the root. Regularised values $\varphi$ (maps taking values in Laurent series in $z$ and carrying products to products; characters) form a group under the convolution $\varphi_1\star\varphi_2=m\circ(\varphi_1\otimes\varphi_2)\circ\Delta$ ($m$ is the product of values). The unit is the character that gives 1 to the empty diagram and 0 to other diagrams. Since there are inverses, the problem can be posed as multiplying $\varphi$ by a factor $S_R$ that cancels its divergences to make $S_R\star\varphi=\varphi_+$ finite. The counterterm $S_R$ plays the role of the inverse of $\varphi$ (§06). Fixing a scheme $R$ (for MS the pole part, for MOM the value at $s=\mu$), the counterterm $S_R$ and the renormalised value $\varphi_+$ are fixed in order of tree size ((21)(25)(26) of Ebrahimi-Fard–Kreimer 2005).

In the toy model, writing the subtree rooted at vertex $v$ as $t_v$ and the number of vertices as $|t|$, the values take closed form (Proposition 3.2.2 of Panzer 2012).

Computing (13) for the 8 trees with up to 4 vertices, (i) none of the counterterms contains $L$ (locality), (ii) the renormalised values are polynomials of degree $|t|$ in $L$ with top-degree term $(-L)^{|t|}/t!$ ($t!=\prod_v|t_v|$ is the tree factorial), and (iii) the MOM values agreed with Panzer's (3.4.2)–(3.4.5). The difference between MS and MOM is finite. Computed

ComputedTable 1 Renormalisation in the toy model (computed exactly in rationals). The right-hand column is the value of $z\to0$. Subtracting by MS changes the constants ($\ell_2$ becomes $\frac{L^2}{2}+\frac{\pi^2}{4}$, $\ell_3$ becomes $-\frac{L^3}{6}-\frac{5\pi^2L}{12}$).

The main theorem of Connes–Kreimer (2000) (abstract) is that the dimensionally regularised bare theory gives a loop $\gamma(z)$ in the group of characters on a small circle around the integer spacetime dimension $D$, and 'the renormalized theory is just the evaluation at z = D of the holomorphic part γ₊ of the Birkhoff decomposition of γ'. $z=D$ is the integer dimension of spacetime, $\gamma$ is the loop given by the regularised theory (the family of characters as the dimension $z$ moves on the circle), and $\gamma_+$ is the holomorphic factor of its decomposition. Known Below we write $z$ as the deviation from $D$ (the same role as the $z$ of the toy model, with $z=0$ the integer dimension).

$\varphi_-$ is a polynomial in $1/z$ (constant term the unit) and is the MS counterterm; $\varphi_+$ is holomorphic at $z=0$, and $\varphi_+(0)$ is the renormalised value. As an analogy, it is the multiplicative version of splitting a Laurent series into its pole part and its holomorphic part. Precisely, since the group is not commutative, they are fixed in order of tree size by the induction of (13). That $S_R$ and $\varphi_+$ are again characters is because $R$ satisfies $R^2=R$ and the Rota–Baxter identity

((20) of Ebrahimi-Fard–Kreimer 2005). Known Example: for the pole part $R$ and $x=1/z+a$, $y=1/z+b$, both sides are $2/z^2+(a+b)/z$.

Figure 2 is the MOM of the ladder $\ell_n$ ($n$ vertices in a row). The sum in (13) becomes $\bar R(\ell_n)=\varphi(\ell_n)+\sum_{j\lt n}S_R(\ell_j)\,\varphi(\ell_{n-j})$; as $z\to0$ the bare value runs away at the rate $z^{-n}$, while the subtracted value settles to a polynomial. Computed

ComputedThe numbers in this figure were computed on this machine. The closed forms of (13)(14) are evaluated in the figure and checked against python3.

If JavaScript is disabled: at $L=1$, the subtracted values at $z=0.02,\ 0.005$ are, for $n=1$: −0.991, −0.998 (limit −1); $n=2$: 0.394, 0.473 (0.5); $n=3$: −1.696, −1.776 (−1.812); $n=4$: 0.655, 1.419 (1.687). The bare values at $z=0.02$ are 49, 1205, 19802, 245195. The difference from the limit shrinks roughly in proportion to $z$. Computed

Detour: rooted trees and Runge–Kutta methods

Another route is to regard the family of actions, one for each scale, as the theory (Wilson's renormalisation group; 6-05). Costello (a 2007 paper, a 2011 book) restricts the $t$ (the square of a length) of the propagator $1/k^2=\int_0^\infty e^{-tk^2}dt$ from $\varepsilon$ to $T$.

Precisely, the paper is written with the heat kernel on a compact manifold, which on flat space becomes $e^{-tk^2}$. Example: at $\varepsilon=0.01$, $T=100$, $k^2P$ is 0.990, 0.368, $1.2\times10^{-4}$ at $k=1,10,30$. Computed Actions at different scales are linked by the renormalisation group equation ($\Gamma(P,S)$ is the sum of diagrams joining vertices of $S$ by $P$).

Theorem B of the paper is that, once a scheme is fixed (a choice of which functions of $\varepsilon$ to call the singular part), there is a unique counterterm $S^{\mathrm{CT}}(\varepsilon)$ such that $\lim_{\varepsilon\to0}\Gamma(P(\varepsilon,T),S-S^{\mathrm{CT}})$ exists, and the counterterms are local and independent of $T$. The counterterms are power series in $\hbar$ (the letter counting loops). The effective action at scale $T$ is the limit as $\varepsilon\to0$ after subtracting these counterterms,

and building diagrams with $P(0,T)$ as it is diverges. The effective action too is a formal power series in $\hbar$ (§08). Theorem C is that, once a scheme is fixed, local actions correspond one-to-one to families of effective actions satisfying (19) and locality as $T\to0$. Changing the scheme amounts to putting different coordinates on the same family. Known

Gauge symmetry is expressed by the quantum master equation of the Batalin–Vilkovisky (BV) formalism (here only the name; it gathers the condition of keeping gauge symmetry into one equation with auxiliary fields added). The 2007 paper distinguishes renormalisation (constructing counterterms) from quantisation (constructing solutions of the quantum master equation), and gives pure Yang–Mills on compact four-dimensional manifolds as an example of the former, and Chern–Simons theory as a non-trivial example of the latter. The 2011 book is described as containing a proof of the perturbative renormalisability of Yang–Mills by cohomology (an algebraic tool that looks at $\ker Q/\operatorname{im}Q$ for maps $Q$ with $Q^2=0$; here only the name). Renormalisability itself is said to have been established by 't Hooft (1971), 't Hooft–Veltman (1972), Lee–Zinn-Justin (1972), Becchi–Rouet–Stora (1976) and others (6-12 §06). Known

All the tools so far work on formal power series in the coupling constant, make each order finite, and confine the freedom to finitely many local terms. Known Three things lie out of reach. (i) The sum of the series: the perturbation series of four-dimensional Yang–Mills is thought to diverge (4-12). Physics (ii) Quantities that are 0 at every order: the one-loop scale (4-07; for pure $SU(N)$, $\beta_0=11N/3$; conversion between four ways of writing the coefficient in Table 1 of 4-06)

(with $b_0=\beta_0/(16\pi^2)$, the same formula as (3) of 4-07) has all its Taylor coefficients in $g$ equal to 0. The mass gap is expected to be proportional to $\Lambda$ Physics, and such quantities cannot be distinguished from 0 within formal power series (6-12 §07). (iii) The measure and positivity: what the problem statement asks for is not a sequence of coefficients but a theory satisfying the axioms (0-03), whose correlation functions should agree at short distances with asymptotic freedom and the predictions of perturbative renormalisation theory (p.6; quoted in 0-03 §07). The algebra of this chapter is on the side of the answers that a construction must reproduce.

An example of the same algebra being used inside a tool outside formal power series is the renormalisation of Hairer's regularity structures. Regularity structures are a framework for solving the equations of stochastic quantisation (6-09; here only the name; a method that solves a time-evolution equation shaken by noise and tries to obtain the measure as its distribution after a long time), and their renormalisation is written with a pair of Hopf algebras and a variant of the Birkhoff decomposition (abstract of Bruned–Hairer–Zambotti 2019). Regularity structures have been used to construct local-in-time solutions (solutions only for a short time) of the stochastic quantisation equation of three-dimensional Yang–Mills–Higgs (abstract of Chandra–Chevyrev–Hairer–Shen 2024). Known What they can handle is subcritical equations (the non-linear term weakens at shorter scales), and four-dimensional Yang–Mills is critical (the strength of the non-linear term does not depend on scale), so it lies outside.

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