The setting-sun diagram

(c)

(a) (b)

1 1 1

3 3 3

2 2 2

1

(a’) (b’) (c’)

2 3

Figure 5.7: The setting sun diagram (of which we have drawn 3 copies to indicate the sub-divergences) inφ⁴ theory as an example for overlapping sub-divergences: The three divergent sub-diagrams indicated by the bold lines in the diagrams (a)-(c) have one propagator line in common. Additionally the diagram has also an overall divergence indicated by the box in diagram which we have not depicted here. As comes out from our calculation in the next section by adding the counter terms (a’)-(c’) for the sub-divergences one ends with a local overall divergence which can be absorbed into the bare parameters of the theory. The dot in the diagrams (a’), (b’) and (c’) stands for the one-loop counter term of the “dinosaur subdiagram”. The rule for such diagrams is to set in the local counter term instead of the sub-diagram, keeping all the rest as is in the full diagram. In our case this rule shows that each of the depicted counterterm (a’)-(c’) stand for −iδλi×R

G(l)/3. As is shown below this cancels the leading divergence of1/² in dimensional regularisation.

ically shows that these conjectures are correct. Although it is a rather hard thing to calculate this diagram we shall treat it as the next example for the power of dimensional regularisation to have a fully calculated example with overlapping divergences at hand when switching to a detailed description of the BPHZ renormalisation.

5.6 The setting-sun diagram

Let us now calculate the above introduced setting sun diagram. This calculation is introduced here to present an example for the non-trivial problems concerned with hiding infinities into the Feynman parameterisation which forces us to use some tricks during the calculation. It is also important to realise how important it was to introduce the Wick rotation for the calculation of the standard integrals listed again for reference in appendix C and to use the form with the negative propagators mentioned already above in the very beginning of the calculation because otherwise there we would obtain serious problems in the analytic continuation in the dimension due to factors∝(−1) which are only well defined as long as ∈^Z8. But as in our one-loop example an important step at thevery end of the calculation is an Laurent expansion of the result with respect to around = 0 in order to extract the infinite part for renormalisation as well as the finite physical result of the 1PI vertex function (in our case of course it is a self-energy contribution).

First we write down the dimensionally regularised Feynman integral we have to calculate (the careful reader is invited to understand the symmetry factor 1/6 from our carefully introduced

“counting contractions algorithm”) including the description with the “negative propagators”, where the factors −1 cause no trouble because there is an integer number of propagators (in

8In this section always= (d−4)/2 is understood wheredis the dimension of space time!

our case 3) to take into account: where we have included the−iη-regulator inm²for convenience, andµis the scaling parameter of dimensional regularisation.

Looking back to our previous calculation of the one-loop vertex diagram we would introduce immediately Feynman parameters but this would cause a new sort or trouble, namely that we hide divergences in the Feynman parameter integral. On the other hand in our case we know that the overlapping divergence is cancelled by the three tadpole diagrams with a counter-term vertex from the one-loop vertex diagram (cf. fig. 5.7 diagrams (a’)-(c’)) which are independent of the momentum. Thus if we can split our diagram from the very beginning to a diagram which is logarithmitically and one which is quadratically divergent we can extract the double pole 1/² explicitly. The last part is free from overlapping divergencies and can be calculated in a straight forward way while the other one contains these overlapping divergencies and we have to hunt for the overlapping divergence proportional to 1/².

This can be done by introducing

∂_µ^(k)k^µ+∂_µ^(l)l^µ= 2d= 4ω (5.101) into the integral and then integrating by parts:

Σ(p) =−λ²µ⁴ Differentiating explicitly and algebraic manipulations with some linear shifts of the integration variable, which is allowed because the integrals are dimensionally regularised, gives after combining the original integral on the right hand side of the equation with the left hand side:

Σ(p) =− λ²µ⁴ This enables us to split the integral in the desired to parts:

Σ(p) =− λ² 6(2ω−3)

h

3m²Σ˜⁽¹⁾(p) +p^µΣ˜⁽²⁾_µ (p)i

. (5.104)

Let us start with the less complicated diagram p^µΣ˜⁽²⁾_µ =− we introduce Feynman parameters for the integration over lwhich is the most divergent part

5.6 · The setting-sun diagram

of the integral. This is mainly the same calculation as we have done when we calculated the vertex diagram in section 6.4. After a shift of the integration variable this integral reads

p^µΣ˜⁽²⁾_µ (p) = Now we introduce another Feynman parameter for combining the denominator to the potence of a propagator function. Thereby it is advantageous for the further calculation to take out a factor [x(1−x)]⁻: So we end with the parameter integral

p_µΣ˜⁽²⁾_µ (p) = p²

Now we take the Laurent expansion of this expression around = 0 (and at latest at this place it was important to use the quasi-Euclidean sign convention for the propagators):

p^µΣ˜⁽²⁾_µ = p²

It is quite easy to see that the Feynman parameter integrals over x and y are finite. But because these are rather complicated dilogarithmic functions (in the literature also known as Spence’s function) we leave this for numerical integration where a trial shows that an adaptive integrator of low order is the right method¹⁰.

Now let us turn to the first contribution. With exactly the same steps we did before we obtain Σ˜⁽¹⁾(p) =

This is a good example for the dangers of the Feynman parameterisation. The y-integral is divergent for ≤ 0 at the limit y = 0 of the integral. This makes it impossible to Laurent

9the last one only to show that the integral with pk⁰ in the numerator vanishes because there we have the caseq= 0 in (C.9).

10I used successfully an adaptive Simpson integrator

expand around= 0. But this is only a logarithmic divergence¹¹and thus is cured easily by a partial integration giving an explicit factor 1/as expected from the overlapping divergences.

Now we can Laurent expand around = 0 with the result Σ˜⁽¹⁾(p) =− 1

The finite part is again left for numerical integration.

Now we have only to calculate the two tadpoles c.f. figure 5.6, diagrams (a’) and (b’), which give both the same contribution. From the Feynman rules and (C.8) we immediately read off

δΣ_ov=−iδλ_MS where we have used (5.94). This shows that indeed the divergences ∝ 1/² are cancelled completely by the diagrams (a’)-(c’) in figure 5.6 as expected. The other counterterms nec-essary to cancel the infinities of the diagram for →0 within the MS scheme are due to the subtraction of the overall divergence of the diagram and give rise to both mass (constant in s) and wave function (∝ s) renormalisation counter terms. This was also predicted by the superficial degree of divergence.

11which was the reason for introducing the “trick” (5.101 in (5.102)