Renormalization

It is the aim of perturbative quantum field theory to provide results on measurable quan-tities (like cross sections) that can be compared with the observations in an experiment.

Therefore it is crucial to deal with the divergences occurring in Feynman diagrams and to find a way of absorbing these infinities in order to arrive at finite predictions.

This problem of renormalization has been discussed and developed in the literature for more than sixty years. A rather recent addition to its underpinnings is the concept of Hopf algebra, introduced by Dirk Kreimer first in [112]. It stimulated a plethora of fruitful developments (in physics as well as pure mathematics) which we have no chance

to recall here. Introductory texts into this subject are available by now, the reviews [124, 133] are particularly suitable for our needs here.

We merely want to summarize very briefly the renormalization by kinematic subtrac-tion in the case of logarithmic ultraviolet divergences. Our focus lies on its formulasubtrac-tion in the Schwinger parametric representation, which has been studied in great detail long ago [14, 15] and recently from a modern viewpoint of algebraic geometry [24, 59].

In particular we recall the convergent integral representation for renormalized Feyn-man integrals, which is based on the forest formula from the earliest days of renormal-ization theory. The parametric representation was used widely during those times, but the actual evaluation of the integrals in this form was too complicated. After the inven-tion of dimensional regularizainven-tion, huge progress in the evaluainven-tion of Feynman integrals was possible in momentum space. As of today, the standard machinery in perturbative quantum field theory is almost exclusively centered on dimensional regularization.

Our goal is to advertise the idea to directly compute renormalized integrals using the forest formula in the parametric representation, without ever introducing a regulator in the first place. In section 5.3 we carry out this program in a few examples.

2.3.1. Hopf algebra of ultraviolet divergences

We consider the Hopf algebra H of scalar, logarithmically divergent Feynman diagrams.

As an algebra, H = Q[G] is free, commutative and generated by connected, scalar, logarithmically divergent Feynman graphs

G := {G : π₀(G) = {G} , ω(G) = 0 and ω(γ) ≤ 0 for all subgraphs γ ⊂ G} (2.3.1) that have at worst logarithmically divergent subgraphs.¹⁸ We denote the empty graph by 1. The coproduct ∆ and the reduced coproduct ∆ are linear maps defined on every^ graph G by

∆,∆ : H −→ H ⊗ H,^ ∆(G) := ^

γ⊆G : ω(γ)=0

γ ⊗ G/γ =1 ⊗ G + G ⊗ 1 +∆(G)^ (2.3.2)

to extract all subdivergences γ and the remaining quotients G/γ (where each connected component of γ has been shrunken to a single vertex). Since H is graded by the number of loops, we can compute the antipode S recursively by

S : H −→ H, S(1) = 1 and S(G) = − ^

γ⊊G : ω(γ)=0

S(γ)G/γ for G ̸=1. (2.3.3)

An explicit solution to this relation is given by the forest formula. To state it we let F (G) denote the forests of G, which are those subsets F ⊂ {γ : γ ⊊ G} ∩ G of proper subgraphs of G such that any pair of subgraphs is either (edge-) disjoint or nested:

F ∈ F (G) ⇔ For any γ1, γ2∈ F , either γ₁∩ γ₂ = ∅, γ₁⊆ γ₂ or γ₂ ⊆ γ₁. (2.3.4)

18Note that this implies that G ∈ G is one-particle irreducible (1PI), that is, it can not be disconnected by deletion of a single edge.

Mind that the empty forest ∅ ∈ F (G) is always included. If we set γ/F := γ/^_{δ∈F,δ⊊γ}δ to the contraction of all proper subgraphs δ of γ that are contained in the forest F , we can state the forest formula as

S(G) = − ^ includ-ing the masses of particles in the theory and products of external momenta. We choose a renormalization point Θ and write Φ|^

Θ for the Feynman rules with these reference kinematics. The associated counterterms Φ₋ are given by

Φ₋(G) = Φ|^⋆−1 and the renormalized Feynman rules Φ₊are determined via the Birkhoff decomposition

Φ₊ = Φ₋⋆ Φ, meaning Φ₊(G) =^

γ⊊G : ω(γ)=0

Φ₋(γ)^Φ(G/γ) − Φ|

Θ(G/γ)^. (2.3.7) Example 2.3.1. If∆(G) = 0 (so G has no subdivergence), we call G primitive and find^ S(G) = −G, Φ−(G) = − Φ|

Θ(G) and Φ₊(G) = Φ(G) − Φ|

Θ(G) is a simple subtraction.

When G has a single subdivergence ∆(G) = γ ⊗ G/γ, we find S(G) = −G + γ · G/γ, the counterterm Φ−(G) = Φ|

Θ(G) + Φ|

Θ(γ) Φ|

Θ(G/γ) and the renormalized Φ₊(G) = Φ(G) − Φ|

Θ(G) − Φ|

Θ(γ)^Φ(G/γ) − Φ|

Θ(G/γ)^. In particular, evaluation at the renormalization point always gives Φ₊|

Θ(G) = 0, unless G =1.

Renormalization group

Suppose we choose another renormalization pointΘ^′, then we get different renormalized Feynman rules Φ^′₊. They are related to Φ₊through the renormalization group equation

Φ^′₊= Φ|^⋆−1 Equivalently, we can think of this as keeping the scheme (renormalization point) fixed, but varying the actual kinematics instead. The β-function of a theory is determined by a very special such variation: We rescale all kinematic invariants by a common factor.

Definition 2.3.2. Suppose all kinematic invariants Θ_ℓ := ^m²_i e^ℓ∪^(p_i· p_j) e^ℓ are measures the scaling dependence of Φ₊ at the renormalization point.

These numbers govern the full scaling dependence, because one can prove [110] invariant θ, we call G to be one-scale and conclude that it is a polynomial in log(θ/θ) and completely determined by the period map alone.

In general, periods depend on the chosen renormalization point Θ. From (2.3.8) one infers that the periods P^′ for the pointΘ^′ are related by the conjugation

P^′= Φ₊|^⋆−1

Θ^′ ⋆ P ⋆ Φ₊|

Θ^′. (2.3.11)

This implies that P(G) = P^′(G) is independent of the renormalization point when G is primitive. In section 5.1 we return to the computation of these interesting numbers.

We give a detailed account of the algebraic structures and proofs of the results pre-sented above in [110, 133].

2.3.2. Parametric representation

This general formulation of renormalization is now applied to Feynman integrals in the representation (2.1.8). Our subtractions for the renormalization are determined by a choice Θ of reference values for the kinematic invariants, so we let φ_G := φ_G|

Θ denote the second Symanzik polynomial (2.1.11) evaluated at these values of masses and momenta. The following formula for the renormalized Feynman rules Φ₊ follows from (2.3.5), (2.3.7) and (2.1.8) and was discussed in [24]:

Φ₊(G) = section 2.1.3 we rescale all Schwinger parameters by λ such that each forest contributes an integral of the form ^₀^∞dλ_λ ^e^−λA− e^−λB= − ln_B^A, so

This representation has been studied in great detail and extensions to incorporate quadratic divergences are available [59]. By definition 2.3.2, the period becomes

P(G) = 1

Example 2.3.3 (Primitive divergence). Consider a logarithmically divergent graph G without subdivergences and all indices a_e= 1. The renormalized Feynman rule and the period are

Φ₊(G) = −

 Ω ψ^D/2lnφ

φ and P(G) =

 Ω

ψ^D/2. (2.3.15) So indeed, P(G) is independent of the renormalization point (the integrand does not contain φ) and we see that P(G) = − ∂_{ ℓ}|_ℓ=0Φ₊(G)|

Θ_ℓ holds indeed. If G is one-scale, then Φ₊(G) = −ℓ · P(G) is just a logarithm ℓ = ln(φ/φ) = ln(θ/_ θ) of the ratio of the^ scale Θ = {θ} and its value at the renormalization point.

In dimensional regularization, we set D = D₀ − 2ε and find ω = εh₁(G) if G is logarithmically divergent in D₀dimensions. The unrenormalized Feynman rules converge for ε > 0 and give the Laurent series

Φ_ε(G) = Γ(ω)

 Ω ψ^D/2

ψ φ

ω

= Γ(εh₁(G))^

n≥0

(−ε)ⁿ n!

 Ω

ψ^D0/2lnⁿ φ^h1(G)

ψ^1+h1(G) (2.3.16)

= P(G)

εh1(G)+ O^ε^0. (2.3.17)

So the period appears as the residue of the regularized Feynman rules at ε → 0. Epsilon-expansions like (2.3.16) can often be computed with hyperlogarithms, see the examples in chapter 5.