Evaluation by Differential Equations
In contrast to the method of alpha parametric representation, the method of MB representation and many other methods of evaluating individual Feynman integrals, the two methods presented in this and subsequent chapter are oriented at the evaluation of mas
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Evaluation by Differential Equations
In contrast to the method of alpha parametric representation, the method of MB representation and many other methods of evaluating individual Feynman integrals, the two methods presented in this and subsequent chapter are oriented at the evaluation of master integrals. This means that we have a solution of the IBP relations [21] for a given family of Feynman integrals, using some technique described in the previous chapter. The method of differential equations (DE) was suggested in [31–35] and developed in [46] and later papers (see references below). The idea of the method is to take some derivatives of a given master integral with respect to kinematical invariants and masses. Then the result of this differentiation is written in terms of Feynman integrals of the given family and, according to the known reduction, in terms of the master integrals. Therefore, one obtains a system of differential equations for the master integrals which can be solved with appropriate boundary conditions. We will consider typical one-loop examples in Sect. 7.1, a two-loop example in Sect. 7.2 and a three-loop example in Sect. 7.3. The status of the method, i.e. its perspectives and open problems will be discussed in Sect. 7.4, together with a brief review of its applications.
7.1 One-Loop Examples Of course, we start with our favourite example. Example 7.1 One-loop propagator diagram corresponding to Fig. 1.1. After solving the corresponding reduction problem in Chap. 6, we know that there are two master integrals, F(1, 1) = I1 and F(1, 0) = I2 . The second one is a simple one-scale integral given by the right-hand side of (6.6). We have started to evaluate I1 in Chap. 1, by differentiating in m 2 and arrived at the Eq. (1.23) for f (m 2 ) = F(1, 1). To be very pedantic, let us rewrite it in terms of our true master integrals,
V. A. Smirnov, Analytic Tools for Feynman Integrals, Springer Tracts in Modern Physics 250, DOI: 10.1007/978-3-642-34886-0_7, © Springer-Verlag Berlin Heidelberg 2012
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7 Evaluation by Differential Equations
∂ 1 2 2 1−ε (1 − 2ε) f (m f (m ) = ) I 2 , ∂m 2 m2 − q2 m2
(7.1)
although this does not make an essential difference here. Let us turn to the new function by f (m 2 ) = iπ d/2 (m 2 )−ε y(m 2 ). We obtain the following differential equation for it: y −
Γ (ε) m 2 (1 − ε) − εq 2 y=− 2 . 2 2 2 m (m − q ) m − q2
(7.2)
It can be solved by the method of the variation of the constant. The general solution to the corresponding homogeneous equation, with a zero on the right-hand side of (7.2), is y(m 2 ) = C(m 2 − q 2 )1−2ε (m 2 )−ε . (7.3) Then we make C = C(m 2 ) dependent on m 2 , solve this equation and obtain f (m ) = iπ 2
d/2
(m − q ) 2
2 1−2ε
m2
−Γ (ε) 0
dx x −ε + C1 , (x − q 2 )2−2ε
(7.4)
where the constant C1 can be determined from the boundary value f (0) which is a massless one-loop diagram evaluated by means of (10.7). This gives f (m 2 ) = −iπ d/2 (m 2 − q 2 )1−2ε Γ (ε) 2 m dx x −ε Γ (1 − ε)2 − × . (x − q 2 )2−2ε Γ (2 − 2ε)(
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