# Multi-loop Feynman Integrals and Conformal Quantum Mechanics

###### Abstract

New algebraic approach to analytical calculations of -dimensional integrals for multi-loop Feynman diagrams is proposed. We show that the known analytical methods of evaluation of multi-loop Feynman integrals, such as integration by parts and star-triangle relation methods, can be drastically simplified by using this algebraic approach. To demonstrate the advantages of the algebraic method of analytical evaluation of multi-loop Feynman diagrams, we calculate ladder diagrams for the massless theory. Using our algebraic approach we show that the problem of evaluation of special classes of Feynman diagrams reduces to the calculation of the Green functions for specific quantum mechanical problems. In particular, the integrals for ladder massless diagrams in the scalar field theory are given by the Green function for the conformal quantum mechanics.

To the memory of Sergei Gorishnii 1958 - 1988

## 1 Introduction

It is well known that the evaluation of the multiple integrals associated with the Feynman diagrams is the main source of physical data in the perturbative quantum field theory. Since the number of diagrams grows enormously in higher orders of perturbation theory, the numerical calculations of the integrals for multi-loop Feynman diagrams are not sufficient to obtain results with desirable precision. That is why analytical calculations of the Feynman diagrams (integrals) start to be important.

For last few years considerable progress was achieved in analytical calculations of multi-loop Feynman integrals (see e.g. [1] – [6] and references therein). It is interesting that in many cases analytical results for the Feynman integrals are expressed in terms of the multiple zeta values and polylogarithms. Note that the multiple zeta values and polylogarithms are very interesting and promising subjects for investigations in modern mathematics (see e.g. [7], [8]).

The analytical evaluations of the multi-loop Feynman integrals are usually based on such powerful methods as the integration by parts [9] and star-triangle (uniqueness) relation (see [10], [11] and references therein) methods. These methods have a long history. For example, the star-triangle relations have firstly been considered in the framework of the conformal field theories [12]. Then it was noticed [13] that the star-triangle relation is a kind of the Yang-Baxter equation (see also [14]). In [13], this fact was used to calculate the ”fishing-net” Feynman diagram (of a sufficiently large order) for the four-dimensional () theory (as well as ”triangle-net” and ”honey-comb” diagrams for the () and () theories, respectively).

In this paper a new algebraic approach to analytical calculations of the massless and dimensionally regularized Feynman integrals, e.g., needed for the renormalization group calculations, is developed. In particular, this method is based on using the integration by parts and star-triangle (uniqueness) relation methods. The advantage of our approach is that we change the manipulations with integrals by the manipulations with the algebraic expressions. This drastically simplifies all calculations, as it will be demonstrated by some examples. In particular, we calculate the integrals for ladder Feynman diagrams arising in the field theory for scalar massless particles. The integrals for these diagrams contribute to many important physical quantities and have been extensively used in many applications, e.g. in the calculations of the conformal four-point correlators in the supersymmetric Yang-Mills theory [15], [16]. The remarkable fact which we have observed is that the evaluation of the integrals for special classes of Feynman diagrams reduces to the calculation of the Green functions for specific (integrable) quantum mechanical problems. For example, the integrals for ladder massless diagrams in the scalar field theory are given by the expansion over the coupling constant of the Green function for the D-dimensional conformal quantum mechanics.

The paper is organized as follows. In Section 2, we outline the basic concepts of our operator approach. In Section 3, we explain how the integrals for Feynman diagrams can be represented in the operator form. The analytical calculations of the integrals for multi-loop ladder diagrams are presented in Section 4. In Section 4 we also discuss the relation of multi-loop Feynman integrals to Green functions of specific quantum mechanical problems. In Conclusion we discuss possible generalizations and prospects.

## 2 Operator Formalism

Consider the -dimensional Euclidean space with the coordinates , where . We denote , and in general the parameter is a complex number. Let and be operators of coordinate and momentum, respectively,

(1) |

Introduce the coherent states and which diagonalize the operators of the coordinate and momentum

(2) |

and normalize these states as follows:

(3) |

We also define the inversion operator

(4) |

where .

The well-known formula for the Fourier transformation of the function can be rewritten with the help of eqs. (2) and (3) in the following form:

(5) |

where , is the Euler gamma-function and . Note that obeys the functional equation . Formula (5) can be interpreted as a definition of the infinite dimensional matrix representation for (the matrix ”indices” are the coordinates and ). In this representation the operator is a diagonal matrix

(6) |

We call the function the propagator with the index . Consider the convolution product of two propagators with the indices and :

(7) |

where . This is nothing but the group relation which is written in the matrix form (5).

Using the definition of the inversion operator (4) one can deduce the main formula

(8) |

Then, the group relation

(9) |

which can be represented in the form of the commutativity condition for the operators . Thus, (for all ) generate the commutative set of elements in the algebra of functions of , . Identity (9), represented in the matrix form (5), (6), is the famous star-triangle relation :

(10) |

Consider the dilatation operator which satisfies:

(11) |

and generates the algebra together with the elements and :

(12) |

It is known that the special conformal transformation generators (4), the dilatation operator (11), the elements and generate the -dimensional conformal algebra .

Below the following relations will be important:

(13) | |||

(14) | |||

(15) |

These relations are easily deduced from the Heisenberg algebra (1). We also introduce the notion of degree for the operators which are homogeneous functions of the generators of the Heisenberg algebra. The additive number is called the degree of the operator if . In particular, we find in view of (11).

Using relations (13) and (15) one can deduce the algebraic identity

(16) |

In the matrix form (5), (6) this identity looks like

(17) |

where , , , and we have introduced the concise notation for the ”vertex” integral

(18) |

Identity (17) is called the integration by parts (or “triangle”) rule [9] and plays a very important role in almost all analytical calculations of multi-loop Feynman integrals.

The integration by parts rule (16) is a combination of two more fundamental relations. The first one is

(19) |

and the second one is obtained from (19) by the Hermitian conjugation (for real and ). These relations can also be deduced from (13) – (15).

Consider the sequence of products of the operators ,

(20) |

Then the finite difference equation (19) and its conjugated version can be written in the form of equations where the finite difference operators

generate the algebra with the commutation relations: for and

This - type algebra has the central element 20). which is equal to the degree of the operator (

Another set of equations for the functions (20) follows from the star-triangle identity (9). The action of the operators is

(21) |

The operators (21) define the group which is a direct product of two symmetric groups generated by the sets of even and odd elements, respectively. One can show directly that the whole set of the relations and is closed and combinations of these relations do not lead to new constraints on (20).

It is well known [17] that the dimensional regularization procedure requires the rule: for all . A more precise statement [18] is

(22) |

where is the area of the unit hypersphere in .

In the framework of the dimensional regularization scheme we extend the definition of the integral in the left-hand side of (22) for arbitrary complex numbers as

(23) |

where is the radial delta-function. It is clear that: for analytic in functions .

The important consequence of the definition (23) is that now one can introduce the notion of the trace for the operators (20)

(24) |

It follows from (23) that these traces are proportional to the delta-function:

(25) |

where , is a unit vector in , is a coefficient function which contains the whole information about the correlator . The simple algebraic arguments which lead to the equation (25) are the following. Note that and

which is consistent with (25).

Remark 1. In the case , the operators (20) (which are cut off by the conditions for and ) are expressed as a product of the commutative operators and their inverse:

(26) |

In this case the problem of calculation of the trace (24) reduces to the spectral problem for the commutative operators .

Remark 2. The star-triangle identity (9) can be generalized to

(27) |

where are any tensors being th-order homogeneous polynomials in or () .

## 3 The Diagrams

The Feynman diagrams which will be considered in this paper are graphs with vertices connected by lines labeled by numbers (indices). With each vertex we associate the point in the D-dimensional space while the lines of the graph (with index ) are associated with the propagator

The boldface vertices denote that the corresponding points are integrated over . These diagrams are called the Feynman diagrams in the configuration space.

The figures and operator form of integral expressions for the 3-point, 2-point diagrams and the tetrahedron vacuum diagram are