Constructing differential equations

Proof. The demonstration is carried out by induction.

First of all, the formula is verified for the basic case n = 2 (1-loop massless bubble): here it reproduces the analytic result (2.15): For a generic n-propagator massless sunset ((n−1)-loop massless sunset), assuming that the first n−1 internal lines have been reduced to a single one using the method explained in example 5, the equation for the final step is: and this proves the thesis.

2.3 Differential evaluation

2.3.1 Constructing differential equations

As explained in [4, 9], a given FI is a scalar integral depending on a set of not integrated parametersS, as momenta of the external lines, masses of all the lines, Mandelstam invariants.

It can therefore be differentiated with respect to any of the elements of S, obtaining relations among the given FI and other FIs, these last ones all belonging to the subtopology tree of the first.¹

Since scalar equations can be solved and manipulated easily with respect to integration, the focus is set to retrieve scalar differential equations – for that, the variable of differentiation will be only scalar quantities.

S can be divided into two subsets, based on the nature of the variables: a subset of internal variables and a subset of external ones.

1As already said, differentiating can only vary the powers of the elemnts of D and S, without inserting new elements.

Internal variables formed by all the non-zero masses of the internal lines, the construction of the differential equations is straightforward:

∂I(D;S)

∂ m²_i
= C_ijJ_j(D;S) (2.28)

where J_j are FIs that differ from I just for some increased powers in the denominators, therefore belonging to the same topology of I. This is due to the fact that differentiating with respect to m²_i with denominators of the form (K + m²j) does not generate, in the numerator, terms depending on momenta.

External variables formed by quantities containing external momenta or masses. Due to the fact that inside the FIs also scalar products of the form p_e· kⁱ are present, the derivative must be expanded:

∂I(D;S)

∂sα

= ∂p^µ_i

∂sα

∂I(D;S)

∂p^µ_i = ∂s_α

∂p^µ_i

−1

∂I(D;S)

∂p^µ_i =

= Aαj(D;S)J^j(D;S) + Bαk(D;S)Kk(D;S), (2.29) where the term [∂s_α/∂p^µ_i]⁻¹is obtained starting from the expressions ∂I/∂p^µ_i = [∂sα/∂p^µ_i]∂I/∂sα

and inverting these relations. The left-hand side of one of these expressions is a scalar quan-tity, so it is possible to express the right-hand side in terms of denominators and irreducible scalar products, possibly multiplied by mass coefficients. The integrals J_j, as above, will belong to the same topology, but with different powers, while K_k are integrals only of the subtopology tree, specifically of the trench with the same number of loop of the original, otherwise the FIs will be null for dimensional regularization.

The same procedure can be applied to a vector of MIs, all depending on a common set of kinematic variables, obtaining a system of differential relations that can be studied at once.

Most of the integrals J_j and K_k are not MIs for a given topology, therefore, applying again the identities of reduction to MIs, it is possible to express the right-hand side of the relations in terms of MIs only.

To conclude, through differentiation on the kinematic invariants it is possible to write differ-ential equations (DE) for a set I of MIs, with structure:

∂I(D;S)

∂s_α = Mαi(D;S)Iⁱ(D;S) + C^αj(D;S)J^j(D;S) (2.30) with

• I the vector of MIs under study;

• J a vector of MIs belonging to subtopologies of I;

• M and C matrices of rational functions of the kinematic parameters (they can be singular at particular configurations ˜S).

The system for I is not homogeneous, so a bottom-up approach to the MIs becomes natural:

starting with the evaluation of the MIs related to the simplest subtopologies one proceed with the more complex ones, having determined all the K_k for a given “topology complexity” thanks to the “simpler” one(s).

Another way to solve the problem is to start from a basis of MIs wide enough to include all the FIs comparing in the DEs, each one with its own DE, then solving the whole system at once (this is usually the way to proceed, when a MI with a wide number of sub-FIs must be evaluated);

this approach is often used when there are more than one (independent) MI for a given topology, since MIs of the same topology have usually entangled DEs. Matematically speaking, due to the fact that a MIs can be related via DEs only to FIs belonging to his subtopology or topology, it is always possible to put the matrix of the coefficients in a block-triangular form, where the blocks correspond to interactions between MIs with the same topology:

∂I(D;S)

∂sα

= M_αi(D;S)Ii(D;S). (2.31)

Mixing the two methods is also possible, mainly when topologies with more than one MI are involved: the MIs related to the topology under study are treated in a differential way, while all the other MIs are considered known.

Euler’s scaling equation

Another important source of differential relations is the Euler’s scaling equation for homogeneous functions (see [4, 9]).

Proposition 4. An l-loop FI I(D; p₁; . . . ; pg−1; m1; . . . ; ml) depending on dimension D, inde-pendent external momenta p^µ and internal masses m_i, is an homogeneous function of degree lD− 2Pl Differentiating both sides of the relation with respect to λ one obtains:

lD − 2

Now taking the limit λ → 1:

Also the scaling equation is related to the IBP-ids:

Proposition 5. The Euler’s scaling equation for FIs belonging to a certain topology can be written in terms of IBP-ids of the same FIs.

Proof. It is possible to apply the scaling operator (2.32) directly on the integrand function, since all the derivatives commute with the integration on loop momenta:

lD − 2

Now summing and subtractingPl

i=1k_i^µ_∂k^∂fµ

The integrand f is an homogeneous function of degree −2Pl j=1αj:

thanks to the Euler’s theorem for homogeneous functions. Substituting the expression into (2.37):

lD +

This is an IBP-id performed on the integrand of the initial FI.