5. Applications and examples 139
5.3. Renormalized subdivergences
5.3.1. Bubble chains
Figure 5.8 defines two families Bn,m and ˆBn,m (n, m ∈ N0) of massless, one-scale ϕ4 vertex graphs in ϕ4-theory. These are logarithmically divergent in D = 4 dimensions,
Bn,m:=n m Bˆn,m :=n m
Figure 5.8.: Two series of one-scale graphs with subdivergences in four dimensions. These graphs arise as vertex graphs inϕ4-theory upon nullification of two external momenta (dashed), incident to the two three-valent vertices.
but they contain a series of bubblesγi ∼= as subdivergences. We denoteγI :=i∈Iγi
for the (edge-disjoint) union of subdivergences indexed by a set I. The coproduct is
∆Bn,m =Bn,m⊗1+
I⊆[n]
J⊆[m]
γIγJ⊗Bn,m/(γIγJ) (5.3.1) where [n] := {1, . . . , n}, we index bubbles in the left row with I and on the right with J. Note that Bn,m/(γIγJ) ∼= Bn−|I|,m−|J|. The same formulas hold for ˆBn,m as well since these two families of graphs differ only by the choice of which of the four external momenta are nullified. One checks that Bn,m and Bn+m,0 = ˆBn+m,0 = ˆB0,n+m define identical Feynman integrals, so it suffices to compute ˆBn,m.
SinceBn,mis not cocommutative forn+m >1, the associated renormalized Feynman rules depend on the renormalization scheme. To point this out we will rather think of Bn,m and ˆBn,m as the same graph, but with different renormalization schemes applied to them. The computation of their periods is elementary in dimensional regularization.
Lemma 5.3.1. The periods of Bˆn,m are given by the exponential generating function
n,m≥0
xnym
n!m!PBˆn,m=
exp−2r≥1 ζ2r+12r+1(x+y)2r+1−x2r+1−y2r+1
1−x−y . (5.3.2)
Proof. In D = 4−2ε dimensions, repeated application of the one-loop master formula (1.2.1) evaluates the unrenormalized Feynman rules to
ΦBˆn,m
= [L(1,1)]n+mΦnǫ mǫ=q−2(n+m+1)ε[L(1,1)]n+mL(1 +nε,1 +mε), in terms of the external momentumq. Now we renormalize by subtraction ats:=q2 →→1, so the counterterm of any bubble is just Φ−(γi) = −Φ|s=1(γi) = −L(1,1) and the multiplicativity of Φ− gives Φ−(γIγJ) = [−L(1,1)]|I|+|J|. With the coproduct (5.3.1), the period (2.3.9) becomes (in D= 4)
PBˆn,m
= lim
ϵ→0
L(1,1)n+m
I⊆[n]
J⊆[m]
(−1)|I|+|J|·ε(1 +|Ic|+|Jc|)L(1 +|Ic|ε,1 +|Jc|ε)
= lim
ε→0ε−n−m
n
i=0
n i
(−1)n+i
m
j=0
m j
(−1)m+j·f(εi, εj, ε), (∗)
where we exploited limε→0[εL(1,1)] = 1 and introduced the power series the only contribution to (∗) left over is
PBˆn,m=an,m,0
The two different renormalization schemes give very different periods indeed: All P(Bn,m) =P(Bn+m,0) =PBˆn+m,0= ∂xn+m(1−x)−1
x=0 = (n+m)! (5.3.3) are integers, while the periods of ˆBn,m involve Riemann zeta values.
Example 5.3.2. P( ˆB1,1) = 2 is still rational, but for all other n, m ≥1 we find zeta values like in P( ˆB1,2) = 6−4ζ3. The values forn+m≤6 are:
P( ˆB1,3) = 24−12ζ3 P( ˆB1,4) = 120−48 (ζ3+ζ5) P( ˆB1,5) = 720−240 (ζ3+ζ5)
P( ˆB2,2) = 24−16ζ3 P( ˆB2,3) = 120−72ζ3−48ζ5 P( ˆB2,4) = 720−384ζ3−288ζ5+ 96ζ32 P( ˆB3,3) = 720−432ζ3−288ζ5+ 144ζ32 We do not want to discuss these particular numbers any further, but only remark thatP( ˆBn,m) only contains products of at most min{n, m} zeta values and has integer coefficients.
Lemma 5.3.3. The periods PBˆn,m∈Z2(2r)!ζ2r+1: r∈N are integer combina-tions of odd zeta values of weight at most ≤n+m.
Proof. Expand the binomial (x+y)2r+1 to rewrite the exponent of (5.3.2) as F(x, y) :=−2 odd zeta values. This property is passed on to the exponential
∂xn∂ymexp(F)
γi=yi yi′ ΓIc = hyperlogarithms7 in the parametric representation. Of course we already know the result and the above calculation in dimensional regularization might seem a lot simpler (in particular to a physicist familiar with dimensional regularization), but the point we want to make is that such a calculation is indeed possible without any regulator, even when many subdivergences are present. For a new result obtained this way, see section 5.3.2.
Lemma 5.3.4. In the parametric representation, the period of Bn,0 can be reduced to a projective integral over n variables x1, . . . , xn∈R+ of the form Note that the summand with I = [n] gives zI = 0, its contribution is understood as (−1)nlimz→0Li1(−z)/z = (−1)n+1. Recall that Ω =δ(1−ni=1λixi)ni=1dxi for arbi-trary λ1, . . . , λn≥0 that do not all vanish.
Proof. Let Γ :=Bn,0 andγi denote the bubble-subgraph consisting of edgesyi andzi as labelled in figure 5.9. The forest formula (2.3.14) for the period delivers
P(Γ) = shorthand for the corresponding cograph. Since a pair of parallel edges yi and yi′ can not be contained in any spanning tree or forest, the graph polynomial
ψΓIc(s, t, y, y′) =ψGIc(s, t, x)
7In fact, our choice of variables allows us to employ only classical polylogarithms of a single variable.
can be expressed in terms of the graph GIc of figure 5.9 where each pair {yi, y′i} is
The dependence of the integrand forP(Γ) above on y and y′ is thus only through the prefactorni=1(yi+yi′)−2 and we can integrate them out using8
such that the integral oversbecomes elementary as well (with integration by parts) and proves the claim. Note that Li1(−zI)/zI =xI/xIc·log(xI/C).
converges absolutely and evaluates tofp(z) =pLip+1(−z).
Proof. Taylor expanding Lip(−x−z) = Lip(−z)+xzLip−1(−z)+Ox2and Lip−x+1z = Lip(−z)−xLip−1(−z) +Ox2 with (3.4.6) reveals the analyticity of the integrand at x → 0. When x → ∞, Lip−x+1z = xz2 +Ox−3 is holomorphic and integrable.
Convergence of (5.3.5) then follows from 1x − x+z1 = xz2 +Ox−3 since Lip(−x−z) diverges at x→ ∞ only logarithmically.
8In this step we choose the constraintδ(1−s) in Ω such that it does not depend ofyandy′.
9The Lip(z)’s in the integrand are well-defined as the analytic continuation of (3.4.3) along the straight path from 0 toz, since this never hits the singularity atz= 1.
Due to absolute convergence we may interchange integration and differentiation to
To finish the inductive proof, we only need to observe limz→0fp(z) = 0 which is clear as we can take this limit on the integrand (which then vanishes).
Lemma 5.3.6. For any 2≤n∈N, p∈N and x1, . . . , xn−1>0, the integral the left-hand side of (5.3.6) becomes
where nowIc:= [n−1]\I. We will now see that the individual integrals are convergent and compute them separately. First we check with (3.4.2) that the last one integrates to and substitute x→→x·xI in the remaining integrals to rewrite them as
∞
The first term is justfp(zI)/zI from (5.3.5), the second evaluates to
To finish the proof we only need to add up all contributions and note that p1 + (−1)n+
corresponds to the term with I = [n−1] on the right-hand side of (5.3.6) in our short-hand convention.
Corollary 5.3.7. For any n, p∈N we compute the following projective integrals (over positive variables x1, . . . , xn), which generalize (5.3.4):
Proof. We perform an induction over n: For n = 1, the integrand is just 1/x1 by our convention and the projective integral tells us to evaluate at x1 = 1. So indeed the left-hand side gives 1 = 1!p−1p−1for allp. Whenn >1, we use (5.3.6) to integrate outxn and obtain, using the statement for smaller values ofn,
Ω
Note that the parametric calculation involves polylogarithms of weight up ton, even though the final result is rational.10 We used (5.3.7) as a test for our implementation HyperInt. Furthermore we used the explicit result (5.3.2) for P( ˆBn,m) to check the program on
Lemma 5.3.8. The parametric representation for the period of Bˆn,m can be reduced to a projective integral over variables x1, . . . , xn and y1, . . . , ym of the form
10We wonder if polylogarithms could be avoided altogether in this case.
Here we set Ic := [n]\I, Jc := [m]\J and ψ := xIcyJc + (xI+yJ)(xIc +yJc). When Ic =∅, the term Li1(−xI2c/ψ)/xIc is understood as zero (its limit when xIc → 0). The same convention applies for Jc=∅.
Proof. The derivation is a straightforward extension of the arguments given in the proof of lemma 5.3.4, so we omit it here.