Calculating four-loop tadpoles with one non-zero mass

An efficient method to calculate tadpole diagrams is proposed. Its capability is demonstrated by analytically evaluating two four-loop tadpole diagrams of current interest in the literature, including their $O(\epsilon)$ terms in $D=4-2\epsilon$ space-time dimensions.


Introduction
Calculations of higher-order corrections are very important for precision tests of the Standard Model in present and future high-energy-physics experiments. The complexity of such calculations and the final results strongly increases with the number of quantum loops considered, and one rapidly reaches the limits of the present realm of possibility when one attempts to exactly account for all mass scales of a given problem, already at the two-loop level. To simplify the calculations and also the final expressions, various types of expansions were proposed during the last couple of years (see Ref. [1,2,3] and references cited therein). These approaches usually provide a possibility to reduce the problem of evaluating complicated Feynman integrals to the calculation of tadpoles T 0,m (α 1 , α 2 ) and loops with massless propagators L q (α 1 , α 2 ), defined in Eqs. (2) and (4), respectively, which have simple representations.
As a first step, let us to introduce some definitions to be used below. All the calculations are performed in Euclidean momentum space of dimension D = 4−2ǫ using dimensional regularisation with 't Hooft mass scale µ. For the Euclidean integral measure, we use the short-hand notation Dk ≡ µ 4−D d D k/π D/2 . The dotted and solid lines of any diagram correspond to massless and massive Euclidean propagators, represented graphically as where α and m denote the index and mass of the considered line, respectively. Unless stated otherwise, all solid lines have the same mass m. Lines with index 1 and mass m are not marked.
Let us now discuss the rules used in our calculation. Firstly, massive tadpoles T 0,m (α 1 , α 2 ) are integrated according to the identity where and Γ is Euler's Gamma function. Secondly, massless loops L q (α 1 , α 2 ) with the external momentum q are integrated according to the identity where Recently, two examples of four-loop tadpoles were calculated in Ref. [4]. Certain O(ǫ 0 ) parts of these results, denoted as N 10 and N 20 , were presented there only numerically. 2 Also the O(ǫ) terms beyond N 10 and N 20 are of current interest in the literature [6] and so far only known numerically. In conjunction with the results of Ref. [7], they allow us to derive analytic results for several four-loop master integrals which are indispensable for modern calculations at this level of accuracy [8].
The subject of this letter is to advertise a powerful technique to evaluate such tadpoles analytically. The simplifications encountered in the examples at hand suggest that this technique might also be useful for more complicated diagrams. Therefore, we wish to introduce it to the interested reader.
The content of this letter is as follows. Section 2 explains the calculational technique in general. Sections 3 and 4 describe its application to the two integrals N 10 and N 20 of Ref. [4], respectively, also including their O(ǫ) terms. A summary is given in Section 5.

Technique
The core of the technique is a master formula to represent a loop with two massive propagators as an integral whose integrand contains a new propagator with a mass that depends on the variable of integration. This formula has the following form: It was introduced in Euclidean and Minkowski spaces in Refs. [9] and [10], respectively. Here, we are working in Euclidean space and will, thus, follow Ref. [9]. Some recent applications in Minkowski space may be found in Ref. [11]. Special cases of Eq. (6) were considered in Ref. [12], where a differential-equation method [12,13] was introduced. The latter only requires that quite simple diagrams are calculated directly. The results for more complicated diagrams may be reconstructed by integrating the results for simpler diagrams over external parameters. In many cases, the results for complicated diagrams may be obtained by integrating the one-loop tadpoles of Eq. (2).
The method is now very popular for the calculation of complicated Feynman integrals; for recent articles, see Ref. [14] and references cited therein.
Here, we shall follow a similar strategy. Applying Eq. (6), we shall represent the results of Ref. [4] as integrals over one-loop tadpoles, which in turn contain propagators with masses that depend on the variable of integration. The case of N 10 will be considered in detail.
We would like to note that Eq. (6) can be applied successfully to diagrams containing one-loop self-energy sub-diagrams. Such sub-diagrams are frequently generated by the application of the integration-by-parts technique [15]. Such an application [12,13] leads to a differential equation for the original diagram (in exceptional cases, the equation degenerates to an algebraic relation) with an inhomogeneous term that depends on less complicated diagrams usually containing one-loop self-energy sub-diagrams (see, for example, Refs. [9,10]).
The evaluation of the O(ǫ) term M 10 is rather tedious and cannot be described in this brief communication. Here, we merely list the result, which reads where We note in passing that also the other master integrals presented in Ref. [5] can be written in a more compact form if the combinations b 4 and b 5 of Eqs. (20) and (22), respectively, are introduced.
Lack of space prevents us from going into details with our derivation of the O(ǫ) term M 20 , and we merely list our result, where b 4 is defined in Eq. (20).

Conclusion
We demonstrated the usefulness of a simple technique for the analytic evaluation of four-loop tadpoles, which allows one to obtain the results by integrating one-loop tadpoles with masses that depend on integration variables. By means of this method, we calculated two Feynman diagrams which were presented in Ref. [4] in numerical form. Our results are in full agreement with the analytic results obtained just recently in Ref. [5].
We note that the integral N 10 was found in Ref. [5] with the aid of the PSLQ program [18], which is able to reconstruct the rational-number coefficients multiplying a given set of transcendental numbers from a high-precision numerical result, i.e. the algebraic structure of the result must be known a priori. Such structures are often more complicated, and they are in general not known from other considerations (see, for example, Ref. [19]), so that the PSLQ program cannot be applied. The method presented here does not suffer from such a limitation and should be applicable also for more complicated tadpoles depending on one [5] or two [20] nonzero masses. Moreover, it can be applied in combination with the PSLQ program. In fact, by evaluating some part of a complicated Feynman diagram by means of this method, the underlying set of transcendental numbers may emerge and can then be injected into the reconstruction of the full analytic result with the help of the PSLQ program.