2. Parametric Feynman integrals 11
2.3. 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 [111]. 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 [22, 59].
In particular we recall the convergent integral representation for renormalized Feyn-man integrals, which is based on theforest 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 algebraHof 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 andω(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 everye graphGby
∆,∆:e H −→ H ⊗ H, ∆(G) := X
γ⊆G:ω(γ)=0
γ⊗G/γ=1⊗G+G⊗1+∆(G)e (2.3.2)
18Note that this implies thatG∈ G isone-particle irreducible (1PI), that is, it can not be disconnected by deletion of a single edge.
to extract all subdivergencesγ and the remaining quotientsG/γ (where each connected component ofγ has been shrunken to a single vertex). SinceHis graded by the number of loops, we can compute theantipode S recursively by
S:H −→ H, S(1) =1 and S(G) =− X
γ(G:ω(γ)=0
S(γ)G/γ forG6=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 ofG such that any pair of subgraphs is either (edge-) disjoint or nested:
F ∈ F(G)⇔For anyγ1, γ2 ∈F, eitherγ1∩γ2=∅,γ1 ⊆γ2 orγ2⊆γ1. (2.3.4) Mind that the empty forest∅ ∈ F(G) is always included. If we setγ/F :=γ/Sδ∈F,δ(γδ to the contraction of all proper subgraphsδ of γ that are contained in the forest F, we can state the forest formula as
S(G) =− X
F∈F(G)
(−1)|F|G/F Y
γ∈F
γ/F. (2.3.5)
The Feynman rules are a character on H, that means Φ(G1G2) = Φ(G1)Φ(G2), but in general ill-defined. They depend on the kinematic invariants Θ =m2i ∪{pi·pj} includ-ing the masses of particles in the theory and products of external momenta. We choose a renormalization point Θ and write Φe |
Θe for the Feynman rules with these reference kinematics. The associated counterterms Φ− are given by
Φ−(G) = Φ|?−1
Θe (G) = Φ|
Θe
S(G) =
(2.3.3)− X
γ(G:ω(γ)=0
Φ−(γ)Φ|
Θe(G/γ) (2.3.6) and the renormalized Feynman rules Φ+are determined via the Birkhoff decomposition
Φ+ = Φ−?Φ, meaning Φ+(G) =X
γ(G:ω(γ)=0
Φ−(γ)hΦ(G/γ)−Φ|
Θe(G/γ)i. (2.3.7) Example 2.3.1. If∆(G) = 0 (soe Ghas no subdivergence), we callGprimitive and find S(G) =−G, Φ−(G) =−Φ|
Θe(G) and Φ+(G) = Φ(G)−Φ|
Θe(G) is a simple subtraction.
When G has a single subdivergence∆(G) =e γ⊗G/γ, we findS(G) =−G+γ·G/γ, the counterterm Φ−(G) = Φ|
eΘ(G) + Φ|
Θe(γ)Φ|
Θe(G/γ) and the renormalized Φ+(G) = Φ(G)−Φ|
Θe(G)−Φ|
Θe(γ)hΦ(G/γ)−Φ|
Θe(G/γ)i. In particular, evaluation at the renormalization point always gives Φ+|
eΘ(G) = 0, unless G=1.
Renormalization group
Suppose we choose another renormalization pointΘe0, then we get different renormalized Feynman rules Φ0+. They are related to Φ+ through the renormalization group equation
Φ0+= Φ|?−1
Θe0 ?Φ = Φ|?−1 eΘ0 ?Φ|
eΘ? Φ|?−1
Θe ?Φ = Φ0+
eΘ?Φ+ = Φ+|?−1
eΘ0 ?Φ+. (2.3.8) Equivalently, we can think of this as keeping the scheme (renormalization point) fixed, but varying the actual kinematics instead. Theβ-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` := m2i e` ∪(pi ·pj)e` are simultaneously scaled by a factore`. Then the period mapP:H −→R,
P :=−h∂`Φ+|
Θe`
i
`=0 =−
X
θ∈Θ
(θ∂θ)Φ+
Θ=Θe
(2.3.9) measures the scaling dependence of Φ+ at the renormalization point.
These numbers govern the full scaling dependence, because one can prove [110]
−∂`Φ+|Θ
`=P ?Φ+|Θ
`, such that Φ+|Θ
` = exp?(−P`)?Φ+ (2.3.10) reveals Φ+(G) as a polynomial in `. If Φ+(G) depends only on a single kinematic invariant θ, we call G to be one-scale and conclude that it is a polynomial in log(θ/θ)e and completely determined by the period map alone.
In general, periods depend on the chosen renormalization point Θ. From (2.3.8) onee infers that the periodsP0 for the pointΘe0 are related by the conjugation
P0= Φ+|?−1
eΘ0 ?P ?Φ+|
Θe0. (2.3.11)
This implies thatP(G) =P0(G) is independent of the renormalization point whenG 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 lete ϕeG := ϕG|
Θe 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 [22]:
Φ+(G) =
"
Y
e∈E
Z ∞ 0
αaee−1 dαe
Γ(ae)
# X
F∈F(G)
(−1)|F|e−
ϕG/F ψG/F −e−
ϕG/Fe
ψG/F
ψFD/2
Y
γ∈F
e− eϕγ/F
ψγ/F. (2.3.12)
Here we abbreviateψF :=ψG/F Qγ∈F ψγ/F. This integral is absolutely convergent. As in section 2.1.3 we rescale all Schwinger parameters byλsuch that each forest contributes an integral of the form R0∞dλλ e−λA−e−λB=−lnAB, so
Φ+(G) = 1 Q
e∈EΓ(ae)
Z ΩY
e∈E
αaee−1 X
F∈F(G)
(−1)1+|F| ψFD/2 ln
ϕG/F
ψG/F +Pγ∈F ϕeγ/F
ψγ/F ϕeG/F
ψG/F +Pγ∈F ϕeγ/F
ψγ/F
. (2.3.13) 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
Q
e∈EΓ(ae)
Z Ω X
F∈F(G)
(−1)|F| ψFD/2
ϕeG/F
ψG/F
ϕeG/F
ψG/F +Pγ∈F ϕψeγ/F
γ/F
, (2.3.14)
because ϕ|Θ
`=ϕe` in contrast to ϕe which is independent of Θ and `.
Example 2.3.3 (Primitive divergence). Consider a logarithmically divergent graph G without subdivergences and all indicesae= 1. The renormalized Feynman rule and the period are
Φ+(G) =− Z Ω
ψD/2lnϕ
ϕe and P(G) =Z Ω
ψD/2. (2.3.15) So indeed, P(G) is independent of the renormalization point (the integrand does not contain ϕ) and we see thate P(G) = −∂`|`=0Φ+(G)|
eΘ` holds indeed. If G is one-scale, then Φ+(G) =−`· P(G) is just a logarithm `= ln(ϕ/ϕ) = ln(θ/e θ) of the ratio of thee 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 inD0dimensions. The unrenormalized Feynman rules converge forε >0 and give the Laurent series
Φε(G) = Γ(ω)Z Ω ψD/2
ψ ϕ
ω
= Γ(εh1(G))X
n≥0
(−ε)n n!
Z Ω
ψD0/2 lnn ϕh1(G)
ψ1+h1(G) (2.3.16)
= P(G)
εh1(G) +Oε0. (2.3.17)
So the period appears as the residue of the regularized Feynman rules atε→0. Epsilon-expansions like (2.3.16) can often be computed with hyperlogarithms, see the examples in chapter 5.