Derivation of functional equations for Feynman integrals from algebraic relations

New methods for obtaining functional equations for Feynman integrals are presented. The application of these methods to finding functional equations for various one- and two-loop integrals is described in detail. It is shown that with the aid of the functional equations Feynman integrals in general kinematics can be expressed in terms of simpler integrals.


Introduction
Recently it has been discovered that Feynman integrals obey functional equations [1,2]. In these papers, a systematic method for deriving the functional equations was proposed. Different examples of the functional equations were presented in refs. [1][2][3]. In these articles, only one-loop integrals were considered.
A particular relationship between N -point one-loop integrals with nonzero masses connecting integrals with different sets of kinematical arguments was derived in ref. [4]. A detailed consideration of this relationship for the one-loop propagator, vertex-and box-type integrals was presented in [5].
In the present paper, we propose essentially new systematic methods for deriving functional equations. These methods are based on algebraic relations between propagators and they are suitable for deriving functional equations for multiloop integrals. Also, these methods can be used to derive functional equations for integrals with some propagators raised to non-integer powers.
Our paper is organized as follows. Section 2 gives a short review of the method proposed in ref. [1].

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In section 3, a method for finding algebraic relations between products of propagators is formulated. We describe in detail a derivation of explicit relations for products of two, three and four propagators. Also, the algebraic relation for products of an arbitrary number of proparators is given. These relations are used in section 4. to obtain functional equations for one-as well as two-loop integrals. In particular, the functional equation for the massless one-loop vertex type integral is presented. Also, the functional equation for the two-loop vertex type integral with arbitrary masses is given.
Section 5 proposes another method for obtaining functional equations. The method is based on finding algebraic relations for 'deformed propagators' and a further conversion of the integrals with 'deformed propagators' to usual Feynman integrals by imposing conditions on the deformation parameters. To perform such conversion, the α-parametric representation for both types of integrals is exploited. The method was used to derive a functional equation for the two-loop vacuum type integral with arbitrary masses. From this functional equation we obtained a new hypergeometric representation for the one-loop massless vertex integral.
In conclusion, we formulate our vision of the future applications and developments of the proposed methods.

Deriving functional equations from recurrence relations
The method for deriving functional equations proposed in ref. [1] is based on the use of a different kind recurrence relations. In particular, in refs. [1][2][3] the generalized recurrence relations [6] were utilized to obtain functional equations for one-loop Feynman integrals. In general, such recurrence relations connect a combination of some number of integrals I 1,n , . . . , I k,n corresponding to diagrams, say, with n lines and integrals corresponding to diagrams with fewer lines. Diagrams with fewer lines can be obtained by contracting some lines in integrals with n lines. The integrals corresponding to such diagrams depend on fewer kinematical variables and masses compared to integrals with n lines. Such recurrence relations can be written in the following form: where Q j and R k are polynomials depending on masses m i , scalar products s r of external momenta, powers of propagators ν l and parameter of the space time dimension d. In the left-hand side of eq. (2.1) we combined integrals with n lines and in the right-hand side integrals with fewer lines. In accordance with the method of ref. [1], to obtain functional equation from eq. (2.1), one should eliminate terms on the left-hand side by defining some kinematical variables from the set of equations: If there are nontrivial solutions of this system and for these solutions some R k,r ({m i }, {s m }, ν l , d) are different from zero, then the right-hand side of eq. (2.1) will represent the functional equation. Figure 1. One-loop diagram with n external legs.

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As an illustrative example, we consider one-loop integrals with n propagators Here as usual an imaginary term iη with infinitesimal η > 0 implies a prescription on how to treat the poles of the propagators. The diagram corresponding to this integral is given in figure 1. Different types of recurrence relations for I were given in refs. [6,7]. In particular, in refs. [6,7] the following relation was derived: where the operators k − shift the index of propagators by one unit ν k → ν k − 1, Here p i , p j are external momenta going through lines i, j, respectively, and m j is the mass attributed to the j-th line. The Gram determinant G n−1 and modified Cayley determinant ∆ n are polynomials depending on scalar products of external momenta and masses. In the case where all ν j = 1 the recurrence relation (2.5) for the first time was derived in refs. [8,9].

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It is assumed that these scalar products are made of d dimensional vectors and G n−1 and ∆ n are not subject to any restriction or condition specific to some integer values of d. Equation (2.5) is written in the form corresponding to eq. (2.1). To eliminate integrals with n lines in the left-hand side of eq. (2.5), the following conditions are to be held: Equation (2.5) is valid for arbitrary kinematical variables and masses. The solution of eqs. (2.8) can be easily done with respect to two kinematical variables or masses. Starting from n = 3, substitution of such solutions into eq. (2.5) gives nontrivial functional equations [1]. The method for obtaining the functional equations by eliminating complicated integrals from the recurrence relations is quite general. However, for multiloop integrals depending on several kinematical variables, derivation of equations like eq. (2.5) is computationally challenging. In the next sections, we will describe easier and more powerful methods that can be used for deriving the functional equations for multiloop integrals.

Algebraic relations between products of propagators
Setting ν j = 1 in eq. (2.5) and imposing conditions (2.8) lead to the following equation: n are products of n − 1 propagators depending on different external momenta, i.e. each term in this relation corresponds to the same function but with different arguments. In fact, the functional equations considered in refs. [1][2][3] are of the same form as eq. (3.1). The question naturally arises: does this relationship hold for integrals or it can be obtained as a consequence of a relationship between integrands?
By inspecting eq. (3.1), one can suggest the following form of the algebraic relation between the products of propagators of integrands: where In what follows we will omit the iη term assuming that all masses have such a correction. Now let us consider in detail a derivation of the algebraic relations for products of 2,3 and 4 propagators.

Explicit relations for products of two, three and four propagators
At n = 2 relation (3.2) reads:

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where Multiplying both sides of eq. (3.4) by P 1 P 2 P 3 one gets: In what follows it will be assumed that k 1 is an integration momentum and three vectors p 1 ,p 2 ,p 3 do not depend on it. Differentiating eq. (3.6) with respect to k 1µ yields: This equation is equivalent to a system of two equations: The last equation in (3.8) means that the Gram determinant for the set of three vectors p 1 ,p 2 ,p 3 is equal to zero. Taking into account eqs. (3.8) another equation follows from eq. (3.6): Equation (3.9) and the first equation in (3.8) define the system of equations for x 1 ,x 2 with the solution where λ 2 is the root of the equation This solution can be rewritten in an explicit form: (3.14) Now let us find an algebraic relation for the products of three propagators. At n = 3 eq. (3.2) reads: 1

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where P 1 , P 2 , P 3 are defined in eq. (3.5) and In complete analogy with the previous case we multiply eq. (3.15) by the product of P 1 P 2 P 3 P 4 and obtain Assuming that p j ,x r do not depend on k 1 , and differentiating eq. (3.17) with respect to k 1µ one gets the equation linear in k 1 . This equation is equivalent to the system of two equations: Taking into account eqs. (3.18), (3.19), we obtain from eq. (3.17): where λ 3 is a solution of the equation Here Let us now turn to the derivation of an algebraic relation for the product of four propagators. At n = 4 eq. (3.2) reads: where P 1 , P 2 , P 3 ,P 4 are defined in eqs. (3.5), (3.16), Multiplying eq. (3.24) by the product P 1 P 2 P 3 P 4 P 5 we obtain: As it was in two previous cases, we assume that p j ,x r do not depend on k 1 . Differentiating eq. (3.26) with respect to k 1µ , one gets an equation linear in k 1 from which two equations follow (3.28)

Algebraic relationship for products of any number of propagators
The relations between the products of five and more propagators can be easily derived in the same way as it was done for the products of two-, three-and four-propagators. From eq. (3.2) one can derive a system of equations and find its solution for arbitrary n.
Multiplying both sides of eq. (3.2) by the product n+1 j=1 P j yields Again we assume that k 1 will be our integration momentum and p j , x r do not depend on it. Differentiating eq. (3.34) with respect to k 1µ one gets a linear equation in k 1 from which two equations follow x j x l s lj = 0. It is interesting to note that for any n, the functional equations for integrals with all masses being equal to zero coinside with the functional equations for integrals with all equal masses and different from zero. In the case of equal masses, two mass dependent terms in eq. (3.37) cancel each other due to eq. (3.35). In both cases systems of equations for x i are the same and therefore arguments of integrals are the same.

Prototypes of functional equations
Multiplying algebraic relations (3.4), (3.15), (3.24) by products of any number of propagators raised to arbitrary powers ν j N j=n 0 and integrating with respect to k 1 we get a functional equation for one-loop integrals. Eqs. (3.4), (3.15), (3.24) can also be used to derive functional equations for integrals with any number of loops. Multiplying algebraic relations for propagators by the function corresponding to Feynman integral depending on momentum k 1 and any number of external momenta and then integrating with respect to k 1 will produce functional equations. Just for demonstrational purposes in figure 2 we present graphically a functional equation based on n propagator relation. The blob on this picture corresponds to either a product of the propagators raised to arbitrary powers or an integral with any number of loops and external legs. Also, the blob can be a product of the propagators multiplied by an integral. One of the external momenta of this multiloop integral should be k 1 .

JHEP11(2017)038 4 Some examples of functional equations
In this section several particular examples of the functional equations resulting from algebraic relations for the products of propagators will be considered.

Functional equation for the one-loop propagator type integral
First, we consider the simplest case, namely, a functional equation for the integral I (d) 2 : . (4.1) Integrating both sides of eq. (3.4) with respect to k 1 , we get: The arguments s 13 , s 23 of the integrals on the right-hand side depend on x 1 , x 2 Substituting solution for x j from eq. (3.10) into eq. (4.3) yields:

(4.4)
In these equations m 2 3 is an arbitrary parameter and it can be taken at will. The functional equation can be obtained from eq. (3.4) as well as from eq. (3.15). Multiplying both sides of eq. (3.4) by the factor 1/P 4 where and integrating over k 1 leads to the equation In terms of integrals I There is an essential difference between the functional equation eq. where λ 3 is a root of the quadratic equation If one argument of I This is not surprising because the coefficients of the eq. (4.17) are mass independent and in the integrand m 2 and iη appear in the covariant combination m 2 − iη. For this reason, the similarity of functional equations for massless integrals and integrals with all masses equal take place for integrals with more external legs and more loops.     Figure 3. Diagram corresponding to R(m 2 1 , m 2 2 , m 2 3 , m 2 4 ; q 2 1 , q 2 2 , q 2 3 ).

Functional equations for one-loop box type integrals
Here λ 4 is defined in eq. (3.31) and m 5 ,x 2 , x 3 are arbitrary parameters and In this functional equation the arbitrary parameters can be chosen from the requirement of simplicity of evaluation of integrals on the right-hand side of eq. (4.20) or from some other requirements. For example, one can choose these parameters by transforming arguments to a certain kinematical region needed for the analytic continuation of the original integral.

Functional equation for two-loop vertex type integral
The method described in the previous section can be applied to multiloop integrals. Consider, for example, an integral corresponding to the diagram given in figure 3. If we multiply eq. (3.4) by the one-loop integral depending on k 1 and integrate with respect to momentum k 1 then we obtain the functional equation , (4.24) (4.26) Integrals of this type arise, for example, in calculations of two-loop radiative corrections in the electroweak theory. Instead of the integral R(m 2 1 , m 2 2 , m 2 3 , m 2 4 ; q 2 1 , q 2 2 , q 2 3 ), one can consider a derivative of R with respect to m 2 3 which is UV finite: . (4.28) Integral R 3 satisfies the following functional equations: In fact the integral on the right-hand side is a propagator type integral with one massless line. Applying the recurrence relations given in ref. [11], this integral can be reduced to a simpler integral: . At q 2 = m 2 , the result for J (d) 111 (q 2 ) is known [12]: and it can be used for the ε expansion of R and R 3 . As was already mentioned, at q 2 1 = q 2 2 = m 2 , q 2 3 = 4m 2 integrals on the right-hand side of eq. (4.23) correspond to the propagator type integrals. Analytic result for R reads We checked that the first few terms in the ε = (4 − d)/2 expansion of R and R 3 are in agreement with results of [13]. The main profit from the functional equations for R and R 3 comes from the fact that vertex integrals were expressed in terms of simpler, propagator type integrals.

Deriving functional equation by deforming propagators
The method described in the previous section does not work for deriving functional equations for all kinds of Feynman integrals. For example, we did not find a functional equation for the two-loop vacuum type integral given in figure 4. In this section we shall describe another method that extends the class of integrals for which we can obtain functional equations. The method is based on transformation of functional equations for some auxiliary integrals depending on arbitrary parameters into functional equations for integrals of interest. Such functional equations will be derived from algebraic relations for 'deformed propagators' which will be defined in the next section. These auxiliary integrals will be transformed into α parametric representation. In general characteristic polynomials of these integrals in α parametric representation differ from those for the investigated integral. The functional equation for the integral of interest can be obtained in case when it is possible to map characteristic polynomials of auxiliary integrals with 'deformed propagators' to characteristic polynomials of this integral. Such a mapping will be performed by rescaling α parameters and appropriate choice of arbitrary 'deforming parameters'.

Algebraic relations for products of deformed propagators
In the previous section, in order to derive the functional equation, we added to our consideration a propagator with combination of external momenta taken with arbitrary scalar coefficients. Now we consider a generalization of this method. To find functional equation for L-loop Feynman integral depending on E-external momenta, we start from the relation of the form where D j is defined as and a (j) jl for the time being are arbitrary scalar parameters. Some of these parameters as well as x r will be fixed from eq. (5.1). Another part of these parameters will be fixed from the requirement that the product of propagators in eq. (5.1) should correspond to the integrand of the integral with the considered topology. We would like to remark that instead of the deformation of the propagators proposed in eqs. (5.2), (5.3), one can use other deformations. For example, all terms in denominators of propagators can be taken with arbitrary scalar coefficients: To establish algebraic relation (5.1), we put all terms over a common denominator and then equate coefficients in front of scalar products depending on integration momenta. Solving the obtained system of equations gives some restrictions on the scalar parameters.
In general, the integrals obtained by integrating products of 'deformed propagators' will not correspond to usual Feynman integrals. Further restrictions on parameters should be imposed in order to obtain relations between the integrals corresponding to Feynman integrals coming from a realistic quantum field theory models.

Functional equation for two-loop vacuum type integral with arbitrary masses
As an example, let us consider a derivation of the functional equation for the two-loop vacuum type integral given in figure 4: .

(5.5)
An analytic expression for this integral was presented in ref. [14]. Instead of this integral we will first consider an auxiliary integral with the integrand made from the 'deformed propagators' defined in eqs. (5.2), (5.3): where For the product of three deformed propagators one can try to find an algebraic relation of the form 1 where D 1 , D 2 ,D 3 are defined in eq. (5.7) and Here m k are arbitrary masses, x k , a j , b i , h s , r l are undetermined parameters and k 1 , k 2 will be integration momenta. Putting in eq. (5.8) all over a common denominator and equating to zero coefficients in front of different products of k 2 1 , k 2 2 and k 1 k 2 leads to the system of equations: (5.10) Solving this system for r 1 , x 1 , x 2 , x 3 we have: where λ is a root of the quadratic equation and

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In order to obtain functional equation for the integral J (d) 0 , we integrate first both sides of the eq. (5.8) with respect to k 1 , k 2 and then convert these integrals into the α-parametric representation. Transforming all propagators into a parametric form and using the d-dimensional Gaussian integration formula d d k exp i(ak 2 + 2(pk)) = i π ia with modified arguments: systematic investigation and classification of the proposed functional equations requires application of the methods of algebraic geometry and group theory. At the moment, it is not quite clear whether functional equations derivable from recurrence relations can be reproduced by the methods of algebraic relations between products of propagators described in section 3 and section 5.
A detailed consideration of our functional equations and their application to the oneloop integrals with four, five and six external legs as well as to some two-and three-loop Feynman integrals will be presented in future publications.
Finally, we would like to make a short comparison of our functional equations and the relationship presented in ref. [4]. The relationship given in ref. [4] corresponds to a particular choice of arbitrary parameters appearing in our functional equations. It has no free parameters. In general, our functional equations depend on several arbitrary parameters. This arbitrariness allows one to analytically continue the integrals to the required regions of kinematical variables.
In our approach, by choosing the arbitrary parameters, one can get also relationships between the integrals depending only on the arguments that are free of square roots of kinematical determinants. This was explicitely demonstrated in ref. [2].