Forests of subdivergences

A forest f of a Feynman graph Γ is a collection of superficially divergent 1PI proper subgraphs γi (Γ (including the empty graph) such that any two distinct subgraphs in a forest are either disjoint or nested (i.e. one is a proper subgraph of the other). The set of all forests f of Γ is then denoted FΓ:

3.3. Subdivergences For disjoint graphs we have seen in eqs. (2.8) and (2.21) how to generalise the graph polynomials and that can be extended to nested graphs. Let f = tiγi be a forest of n nested graphs (sometimes called flag) such that γ₁ ( γ₂ (. . .( γ_n. Then

Ψf ..= Ψγ1

n

Y

i=2Ψγi/γi−1, (3.55)

and

Φf ..= Φγn/γn−1Ψγ1

n−1

Y

i=2Ψγi/γi−1. (3.56) The subdivergences in a forest are partially ordered by their nesting. Following this ordering we can parametrise and subtract each subdivergence analogous to the case of the single subdivergence. This yields a general forest formula integrand M_f^Γ which is the obvious but rather unwieldy generalisation of the terms found in the previous sections. Note that for each propagator subgraph a connector has to be chosen for the squashing E_Γ^sq ⊂E_Γ, ¯Γ = Γ//E_Γ^sq, and the corresponding edge parameters are suppressed in all terms similarly toα_e^γ

2 inM_∅^Γin section 3.3.2.

The fully renormalised integral is then φ^R_Γ ^..= (g^µνq²−q^µq^ν)^Z

σΓ¯

X

f∈FΓ

(−1)^|f|M_f^Γ. (3.57)

3. Renormalisation

4

Structure of the integrand I:

Contraction of Dirac matrices

O caos é uma ordem por decifrar.

Chaos is order yet undeciphered.

José Saramago, O hohem duplicado, 2002

4.1 Dirac matrices

Now that we have made the integrals well-defined we can apply ourselves to the study of the structure of their integrands, which was so far conveniently isolated into the J_Γ^(k). The aim of this chapter is to eliminate the Dirac matrices. To do so, we study the combinatorics of the contraction procedure outlined below, which will allow us to completely circumvent this computation and directly write down the resulting integer factors just from certain combinatorial properties of the Feynman graph.

We begin by recapitulating the basics and usual techniques that are used to deal with the Dirac matrices in Feynman integrals. As was already mentioned in the introduction, Dirac gamma matrices are a set of four complex 4×4 matrices that satisfy the anticommutation relations

γ^µγ^ν +γ^νγ^µ= 2g^µν14×4 µ, ν = 0,1,2,3. (4.1) They generate a Clifford algebra, and there are many different possible choices of concrete representations for these matrices that are useful in different circum-stances. However, one of the main points of this chapter is that we focus entirely on the combinatorics, which are independent of the representation, so we will not delve into this topic her.

4. Structure of the integrand I: Contraction of Dirac matrices

On a related note, since we chose to write our integrals in Euclidean rather than Minkowski space we should actually work with yet another set of Euclidean Dirac matrices γ_E^µ and have a Kronecker delta δ^µν, rather than the metric tensor.

But since the combinatorics are the same either way and we never use an explicit representation we refrain from blowing up the notation with yet another subscript.

Contracting Dirac matrices the old-fashioned way

Traditionally the contraction is computed by iteratively applying the Clifford al-gebra relation eq. (4.1), or rather, an identity that can be derived from it:

γµγν1. . . γνnγ^µ =







−2γνn. . . γν1 if n odd

2(γνnγ_ν1. . . γ_νn−1 +γ_νn−1. . . γ_ν1γ_νn) if n even (4.2) It was first proved (independently and with different methods) by Caianello and Fubini [33] and Chisholm [37]. After all duplicate indices within one product of Dirac matrices are contracted one can continue by combining traces with the Chisholm identity¹ [38]

γµtr(γ^µS) = 2(S+ ˜S) (4.3) whereSis a product containing an odd number of Dirac matrices and ˜Sis the same product reversed. When that identity cannot be applied anymore the remaining traces are expressed in terms of metric tensors with the recursion formula

tr(γ^µ1. . . γ^µn) = ^Xn

i=2(−1)ⁱg^µ1µitr(γ^µ2. . .^dγ^µi. . . γ^µn). (4.4) Remark 4.1.1. Note that the even case of the contraction relation can alterna-tively be expressed in the form

γµγν1. . . γνnγ^µ = 2(γνk+1. . . γνnγν1. . . γνk +γνk. . . γν1γνn. . . γνk+1) (4.5) for any oddk < n. This is discussed in more detail in section 4.2. To summarise:

By exploiting the equivalence of these different choices we can reduce the recursive trace formula eq. (4.4) to a much shorter, non-recursive formula from which – among other things – the Chisholm identity eq. (4.3) follows as a trivial special case. This simplification in turn allows for the diagrammatic and combinatorial interpretation of contraction in section 4.3.

1Sometimes the previous eq. (4.2) is also called Chisholm identity, but here we will always use the name to refer to eq. (4.3)

4.1. Dirac matrices Example 4.1.2. We return to one of our examples from chapter 2. Consider the graph Γ1 from fig. 2.3. Its Dirac matrix structure is given in eq. (2.34):

¯

γΓ1 = (−1) tr(γ^ν1γ^µ2γ^ν2γ^µ3γ^ν3γ^µ4γ^ν4γ^µ5) Its contraction with two metric tensors looks as follows:

gν2ν4gµ2µ4γΓ1 =−tr(γ^ν1γ^µ2γ^ν2γ^µ3γ^ν3γ^µ2γ^ν2

| {z }

=−2γ^µ2γ^ν3γ^µ3

γ^µ5)

= 2 tr(γ^ν1γ^µ2γ^µ2

| {z }

=4

γ^ν3γ^µ3γ^µ5)

= 32(gν1ν3g_µ3µ5−g_ν1µ3g_ν3µ5 +g_ν1µ5g_µ3ν3) (4.6) Products of traces can be combined as follows:

tr(γ^µ1γ^µ2γ^ν1γ^ν2) tr(γ^µ1γ^µ2γ^ν3γ^ν4)

= tr(γ^µ1tr(γ^µ1γ^µ2γ^ν1γ^ν2)

| {z }

=2(γ^µ2γ^ν1γ^ν2+γ^ν2γ^ν1γ^µ2)

γ^µ2γ^ν3γ^ν4)

= 2tr( γ^µ2γ^ν1γ^ν2γ^µ2

| {z }

=2(γ^ν2γ^ν1+γ^ν1γ^ν2)

=4g^ν1ν2

γ^ν3γ^ν4) + tr(γ^ν2γ^ν1γ^µ2γ^µ2

| {z }

=4

γ^ν3γ^ν4)

= 8g^ν1ν2tr(γ^ν3γ^ν4) + tr(γ^ν2γ^ν1γ^ν3γ^ν4)

= 322g^ν1ν2g^ν3ν4 −g^ν1ν4g^ν2ν3 +g^ν1ν3g^ν2ν4 (4.7)

Computer algorithms for contraction (e.g. implemented as trace4 in FORM [132]) typically try to successively apply the three equations (4.2), (4.3) and (4.4) until full contraction is achieved. However, as far back as the 1960s there have been attempts to find alternative contraction methods that bear some similarities to our approach [86]. Kahane developed an algorithm which involves instructions on how to first draw a diagram based on a given sequence of Dirac matrices.

Following that the algorithm describes how to parse the diagram, simultaneously multiplying the result with certain factors depending on what one encounters. In our approach we use chord diagrams – a very well understood type of graph – together with a colouring to carry all the necessary information. Moreover, we isolate the relevant combinatorial property of the chord diagrams – the number of cycle subgraphs with a certain colouring – such that our result is a closed formula instead of an algorithm. Finally, Kahane’s proofs are based on using a certain basis for the Clifford algebra generated by the Dirac matrices, while our results are entirely concluded from the contraction relation eq. (4.2). In fact, in section

4. Structure of the integrand I: Contraction of Dirac matrices

4.2 we completely abstract the process of contraction from Dirac matrices to com-binatorial sequences of letters representing the different space-time indices.

Kahane’s algorithm was later generalised to products of traces by Chisholm [39], using his identity eq. (4.3). Working with Kahane’s diagrams, the com-putations with this generalised algorithm become quite cumbersome². Following our approach the general case follows very directly and with only marginally more complicated notation as corollary 4.3.13 from our single trace result theorem 4.3.9.

There are a multitude of modern methods that have been developed to deal with the problem of overly complicated contractions (e.g. spin-helicity, BCFW recursion [24, 57, 61]) and the reader may not yet be convinced that studying the combinatorics of the “traditional” contraction process is a worthwhile enterprise.

However, especially outside of supersymmetric theories, such on-shell methods are not immune to becoming complicated and tedious either, and the standard contraction of Dirac matrices is still very much used today (e.g. in [36,77]). Instead of circumventing the contraction process, like these methods, we completely work it out, in a way that does not depend on any particular choice of representation for the gamma matrices or spinor basis, and give its end result for any QED graph, at any loop-order, in terms of simple chord diagrams. Moreover, our focus here lies of course on Feynman integrals in the parametric context, in which the above methods are plainly not applicable.

2In the words of J.S.R. Chisholm himself [39] : “The proof of our final result is long and tedious, and even the statement of it is fraught with notational difficulties. We therefore explain it by an example, [...]”

Im Dokument Parametric quantum electrodynamics (Seite 72-79)