Schouten identities for Feynman graph amplitudes; the Master Integrals for the two-loop massive sunrise graph

A new class of identities for Feynman graph amplitudes, dubbed Schouten identities, valid at fixed integer value of the dimension d is proposed. The identities are then used in the case of the two loop sunrise graph with arbitrary masses for recovering the second order differential equation for the scalar amplitude in d=2 dimensions, as well as a chained sets of equations for all the coefficients of the expansions in (d-2). The shift from $d\approx2$ to $d\approx4$ dimensions is then discussed.


Introduction
The Feynman integrals associated to the two-loop loop self-mass Feynman graph of Fig.(1), usually referred to as sunrise, have been widely studied in the literature within the framework of the integration by parts identities [1,2], and it is by now well known that they can be expressed in terms of four Master Integrals (M.I.s), [3], which satisfy a system of four first-order coupled differential equations, [4] (equivalent to a single fourth-order differential equation for any of the Master Integrals). Several numerical approaches to the numerical solution of the equations with satisfactory degree of precision have been worked out, (see for instance [5]), but a complete treatment of the general case with three different masses in d = 4 dimensions is still missing. In the equal mass case the number of independent Master Integrals reduces to two, so that the two by two first-order system of differential equations can be rewritten as a single second-order differential equation for one of the Master Integrals, say the full scalar amplitude (see below). In Ref. [6] it is shown how to build the analytic solution of that equation in terms of elliptic integrals, both for d = 2 and d = 4; the two cases, related by the Tarasov's shifting relations [7], are very similar, with the d = 2 case just marginally simpler than the d = 4 case. The analytic solution provides with the necessary information for writing out very precise and fast converging expansions for the accurate numerical evaluation of the two M.I.s [8].
More recently, an interesting paper [9] has shown, by using algebraic geometry arguments, that in d = 2 dimensions the full scalar amplitude satisfies a second-order differential equation also in the different mass case. The equation was then solved in [10] by suitably extending the method of [6]; let us observe here that the analytic solution of the second-order differential equation is equivalent to the analytic knowledge of two (of the four) Master Integrals of the sunrise with different masses.
The problem of extending the approach to d = 4, which is the physically relevant case, remains, as the straightforward use of the Tarasov's dimension-shifting relations is unfortunately not sufficient. Indeed, as will be shown in this paper, by explicitly working out the shifting relations one finds that any of the four Master Integrals at d ≈ 4 dimensions can be expressed as a combination of all the four Master Integrals at d ≈ 2 dimensions and of the first terms of their expansion in (d − 2), while the results of [9] give only two of the four Master Integrals at exactly d = 2 dimensions, but no other information on the remaining Master Integrals and their expansion in (d − 2).
In this paper we introduce a family of particular polynomials in the scalar products of the vectors occurring in the Feynman integrals, dubbed Schouten polynomials, which have the property of vanishing at some fixed integer value of the dimension d. By using those polynomials one can introduce an ad hoc set of amplitudes, from which one can at least in principle extract an independent set of new amplitudes which vanish in a non trivial way (see below) at that value of d (say at d = N for definiteness). If those new amplitudes are expressed in terms of the previously chosen set of Master Integrals, their vanishing gives a set of relations between the Master Integrals, valid at d = N , which we call Schouten identities. Alternatively, one can introduce a new set of Master Integrals including as new Master Integrals some of the independent amplitudes vanishing at d = N , write the system of differential equations satisfied by the new set of Master Integrals and expand them recursively in powers of (d − N ) around d = N . As some of the new Master Integrals vanish at d = N , the system of equations takes a simpler block structure.
The pattern is very general, and applies in principle to the integrals of any Feynman graph. We work out explicitly the case of the sunrise amplitudes at d = 2 with different masses, finding the existence of two independent Schouten identities, i.e. of two independent relations between the usual Master Integrals at d = 2, or, which is the same, we can introduce a new set of Master Integrals, consisting of two "conventional" Master Integrals (say the full scalar amplitude and another M.I.) and two new Master Integrals vanishing at d = 2. The system of differential equations satisfied by the new set of Master Integrals can then be expanded in powers of (d − 2). At zeroth-order we find a two by two system for the two "conventional" M.I.s (the other two Master Integrals vanish), equivalent to the second-order equation found in [9], while at first-order in (d − 2) we find in particular two relatively simple equations for the first terms of the expansion of the two new M.I.s, in which the zeroth-orders of the two "conventional" M.I.s appear as non homogeneous known terms.
One can move from d ≈ 2 to the physically more interesting d ≈ 4 case by means of the Tarasov's shifting relations; it is found that for obtaining the zeroth-order term in (d − 4) of all the four M.I.s (of the old or of the new set) at d ≈ 4 one needs, besides the zeroth-order term in (d − 2) of the two "old" M.I.s at d ≈ 2, also the first term in (d − 2) of the new M.I.s.
The plan of the paper is as follows: in sec. 2 we introduce the Schouten polynomials for an arbitrary number of dimensions, while their applications to Feynman Amplitudes is discussed in sec. 3. In sec. 4 we show how, by using the Schouten Identities, a new set of Master Integrals can be found, whose differential equations in d = 2 take an easier block form and can be therefore re-casted (see sec. 5) as a second-order differential equation for one of the Masters. In sec. 6 we show how the results at d ≈ 4 can be recovered from those at d ≈ 2 through Tarasov's shifting relations. Finally, in sec. 7, which is somewhat pedagogical, we present a thorough treatment of the imaginary parts of the Master Integrals in d = 2 and d = 4 dimensions. Many lengthy formulas and some explicit derivations can be found in the Appendices at the end of the paper.

The Schouten Polynomials
As an introduction, let us recall that in d = 4 dimensions one cannot have more than 4 linearly independent vectors; indeed, given five vectors v α , a µ , b ν , c ρ , d σ in four dimensions they are found to satisfy the following where ǫ µνρσ is the usual Levi-Civita tensor with four indices, with ǫ 1234 = 1, etc., and following the convention introduced in the program SCHOONSCHIP [11] we use Eq.(2.1) is known as the Schouten identity [12]; by squaring it, one gets a huge polynomial, of fifth-order in the scalar products of all the vectors. Due to Eq.(2.1), that polynomial vanishes in d = 4 dimensions (and, a fortiori for any integer dimension d ≤ 4); note however that the polynomial does not vanish identically for any arbitrary value of the dimension; as Eq.(2.1) is valid only when d ≤ 4, for d > 4 the polynomial is not bound to take a vanishing value. As an extension (or rather a simplification) of Eq.(2.1), consider now the quantity where ǫ µν is the Levi-Civita tensor with two indices (defined of course by ǫ 12 = −ǫ 21 = 1, ǫ 11 = ǫ 22 = 0), and a µ , b ν are a couple of two-dimensional vectors. By squaring it, Eq.(2.3) gives at once as can be also verified by an absolutely trivial explicit calculation. Following the elementary procedure leading to Eq.(2.5), given in d = 3 dimensions any triplet of vectors where ǫ µνρ is the Levi-Civita tensor with three indices (defined as usual by ǫ 123 = 1 etc.) and then we evaluate its square We then define the Schouten Polynomial P 3 (d; a, b, c) as where again the r.h.s. is formally the same as in Eq.(2.8), but the three vectors a µ , b ν , c ρ are assumed to be d-dimensional vectors, with continuous d. By construction, P 3 (d; a, b, c) vanishes at d = 1 and at d = 2, Needless to say, the procedure can be immediately iterated to any higher dimension, generating Schouten polynomials involving four vectors and vanishing in d = 1, 2, 3 dimensions, or involving five vectors and vanishing in d = 1, 2, 3, 4 dimensions, corresponding, up to a constant numerical factor, to the square of Eq.(2.1), etc.. As it is apparent from the previous discussion, the Schouten polynomials generated by a given set of vectors are nothing but the Gram determinants of the corresponding vectors; we prefer to refer to them as Schouten polynomials to emphasize that they vanish in any integer dimension d less than the number of the vectors.
In the actual physical applications, as one is interested mainly in the d → 4 limit of Feynman graph amplitudes, one can reach d = 4 starting from a different value of d and then moving to d = 4 by means of the Tarasov's shifting relations [7]. As the shift relates values of d differing by two units, the d = 1 Schouten identities, easily established for any amplitude in which at least two vectors occur, are of no use. The next simplest identities are at d = 2 and occur with any amplitude involving at least three vectors. That is the case of the two-loop self-mass graph (the sunrise), which we will study in this paper in the arbitrary mass case.

The Schouten Identities for the Sunrise graph
We discuss in this Section the use of the Schouten polynomial P 3 (d; a, b, c) in the case of the sunrise, the two-loop self-mass graph of Fig.(1).
The external momentum is p and the internal masses are m 1 , m 2 , m 3 . We use the Euclidean metric, so that p 2 is positive when spacelike; sometimes we will use also s = W 2 = −p 2 , so that the sunrise amplitudes develop an imaginary part when √ s = W > (m 1 + m 2 + m 3 ), the threshold of the Feynman graph. We write the propagators as and define the loop integration measure, in agreement with previous works, as: so that C(2) = 4π and C(4) = 1 . With that definition the Tadpole T (d, m) reads .
In this paper we will use the "double" tadpoles together with the similarly defined T (d; m 1 , m 3 ), T (d; m 2 , m 3 ), and the four amplitudes (3.7) All those amplitudes depend on the three masses m 1 , m 2 , m 3 , even if the masses are not written explicitly in the arguments for simplicity. The four amplitudes are equal, when multiplied by an overall constant factor (2π) 4 , to the four M.I.s used in [4]. S(d; p 2 ), in particular, is the full scalar amplitude already referred to previously. Those amplitudes were chosen in [4] as M.I.s for the sunrise problem, and in the following they will be sometimes referred to as the "conventional" M.I.s . We can now introduce the Schouten amplitudes defined, for arbitrary d, as where the n i are positive integer numbers and P 3 (d; p, q 1 , q 2 ) is the Schouten polynomial defined in Eq.(2.9). The convergence of the integrals, for a given value of d, depends of course on the powers n i , as the Schouten polynomial in the numerator contributes always with four powers of the loop momenta q 1 and q 2 . We are interested here in the d = 2 case. If the Schouten amplitude is convergent at d = 2, due to the second of Eq.s(2.10), it is also vanishing at d = 2, i.e. Z(2; n 1 , n 2 , n 3 , p 2 ) = 0. Note that in the massive case all the integrals we are considering are i.r. finite, therefore the divergences can only be of u.v. nature.
As one can express any sunrise Feynman amplitude in terms of a valid set of M.I.s, we will write in the following a few Schouten amplitudes in terms of the "conventional" M.I.s given in Eq.s(3.7). A few explicit results are now listed: Some comments are in order. Elementary power counting arguments give N = 2(n 1 +n 2 +n 3 ) powers of the integration momenta in the denominator (independently of d) and, in d = 2 dimensions, all together eight powers in the numerator (see Eq. (3.8) for the definition of the integrals), so that the minimum value of N necessary to guarantee the convergence is N = 10. In the case of Z(d; 2, 2, 2, p 2 ) of Eq.(3.12) N = 12, more than the minimum required value N = 10; therefore the integrals in the loop momenta q 1 , q 2 do converge, so that the vanishing of P 3 (d; p, q 1 , q 2 ) in the numerator at d = 2 (and, as a matter of fact at d = 1 as well) does imply the vanishing of the whole amplitude. The explicit result, Eq.(3.12), shows indeed that the amplitude vanishes at d = 1 and d = 2, but that is due to an overall factor (d − 1)(d − 2), so that Eq.(3.12) does not give any useful information. This pattern -the vanishing at d = 2 of the amplitudes with P 3 (d; p, q 1 , q 2 ) in the numerator and N > 10 due to the appearance of an overall factor (d − 2) -is of general nature, and showed up in all the cases which we were able to check (needless to say, the algebraic complications increase quickly with the powers of the denominators). The Z i (d; p 2 ), Eq.s(3.9,3.10,3.11), are more interesting; in this case, N = 10, which is the minimum value needed to guarantee convergence in d = 2 dimensions, so that those amplitudes are expected to vanish at d = 2 (and therefore also at d = 1) as a consequence of the vanishing of P 3 (d; p, q 1 , q 2 ) at d = 1, d = 2, see Eq.s(2.10). The vanishing at d = 1 is trivially given by the overall factor (d − 1) (in d = 1 the minimum value of N to guarantee convergence is N = 8, while in the integrals we are now considering N = 10), but the vanishing at d = 2 is totally non trivial, providing new (and so far not known) relations between the four conventional M.I.s S(d; Any of the three amplitudes Z i (d; p 2 ) can obviously be obtained from the others by a suitable permutation of the three masses, as immediately seen from their explicit expression. When summing the three relations, one obtains 13) showing that at d = 2 they are not independent from each other; indeed, if one takes as input Z 2 (2; p 2 ) = 0 and Z 3 (2; p 2 ) = 0, the previous equation gives Z 1 (2; p 2 ) = 0, showing that the condition Z 1 (2; p 2 ) = 0 depends on the other two. When written explicitly, the vanishing of Z 2 (2; p 2 ) = 0 and Z 3 (2; p 2 ) = 0 reads (3.14) The validity of identities Eq.s(3.14,3.15) in d = 2 has been checked with SecDec [13]. By using the above relations, which hold identically in p 2 , m 2 1 , m 2 2 , m

A New Set of Master Integrals
We have seen in the previous Section that the "conventional" M.I.s in d = 2 dimensions satisfy two independent conditions, written explicitly in Eq.s(3.14,3.15), so that two of them can be expressed as a combination of the other two, which can be taken as independent. On the other hand, it is known that in the equal mass limit the Sunrise has two independent M.I.s (in any dimension, including d = 2) so that no other independent conditions can exist. It can therefore be convenient to introduce a new set of M.I.s, formed by two "conventional" M.I.s , say S(d; p 2 ), S 1 (d; p 2 ) of Eq.(3.7), and two Schouten amplitudes, say Z 2 (d; p 2 ), Z 3 (d; p 2 ) of Eq.s(3.10,3.11). The advantage of the choice is that two conditions at d = 2 take the simple form Z 2 (2; p 2 ) = 0, Z 3 (2; p 2 ) = 0. The actual choice of the new M.I.s satisfying the above criteria is of course not unique (a fully equivalent set could be for instance S(d; In the new basis of M.I.s, the two discarded conventional M.I.s are expressed as Note that P (p 2 , m 1 , m 2 , m 3 ), which is symmetric in the last two arguments, occurs with different arguments in different places.
By substituting the above expressions in the differential equations for the conventional M.I.s as given, for instance, in Ref. [4], one obtains the new equations In the above equations, is the product of all the threshold and pseudo-threshold factors already present in [4], ij (p 2 , m 1 , m 2 , m 3 ) are also polynomials depending on p 2 and the masses; their explicit (and sometimes lengthy expression) is given in Appendix A. Note that a same polynomial can occur in different equations with a different permutation of the masses within its arguments.
We want to stress here an important aspect of the last two equations, Eq.(4.7,4.8), namely the presence of an overall factor (d − 2) in the r.h.s., which plays an important role in the expansion in powers of (d − 2) discussed in the next Subsection.

The expansion of the Equations around d = 2
Let us start off by expanding all M.I.s in powers of (d − 2) around d = 2, Due to the overall factor (d − 2) in the r.h.s., at 0th order in (d − 2) the differential equations Eq.s(4.7,4.8) become d dp 2 Z 2 (2; showing that Z 2 (2; p 2 ), Z 3 (2; p 2 ) must be constants. But we know from Eq.s(3.14,3.15) the actual value of that constant (the two functions vanish identically, Z 2 (2; p 2 ) = 0, Z 3 (2; p 2 ) = 0 ), so that at 0th order in (d − 2) the differential equations Eq.(4.5), Eq.(4.6) for S(2; p 2 ), S 1 (2; p 2 ) become completely decoupled, obviously, from the (trivial) equations for Z 2 (2; p 2 ), Z 3 (2; p 2 ). (See the previous Section and Appendix A for the explicit expression of the polynomials.) Going now one order higher in the expansion, one finds that the first-order terms in (d − 2) of the Z i (d; p 2 ) satisfy the equations It is to be noted that Z 3 (2; p 2 ) do not appear in the r.h.s. of Eq.s (4.15,4.16), which contains only S(2; p 2 ) and S 1 (2; p 2 ), to be considered known once Eq.s(4.13,4.14) for the 0th orders in (d − 2) have been solved. Eq.s (4.15,4.16), indeed, are absolutely trivial when considered as differential equations, as they contain only the derivatives of Z 3 (2; p 2 ), one can move to the differential equations for S (1) (2; p 2 ), S 1 (2; p 2 ) (which we don't write here for the sake of brevity); they involve Z 3 (2; p 2 ) as known inhomogeneous terms, and form again a closed set of two differential equations, decoupled from the equations for the other two M.I.s, as at 0th order in (d − 2).
Thanks to the overall factor (d − 2) in the r.h.s. of Eq.s(4.7,4.8), the pattern -a quadrature for Z (k) 2 (2; p 2 ), Z (k) 3 (2; p 2 ) and a closed set of two differential equations for S (k) (2; p 2 ), S  We go back now to the system of differential equations Eq.s(4.5,4.6), for obtaining a second-order differential equation for S(d; p 2 ). We can use Eq.(4.5) in order to express S 1 (d; p 2 ) in function of S(d; p 2 ) and of its derivative, dS(d; p 2 )/dp 2 . By substituting this expression into Eq.(4.6) we can then derive a secondorder differential equation for S(d; p 2 ) only, which however still contains Z 2 (d; p 2 ) and Z 3 (d; p 2 ) in the inhomogeneous part: The equation above is exact in d but contains, besides S(d; p 2 ) and its derivatives, also Z 2 (d; p 2 ) and Z 3 (d; p 2 ) as inhomogeneous terms. Nevertheless, recalling once more that Z 2 (2; p 2 ) = Z 3 (2; p 2 ) = 0, we can expand Eq.(5.1) in powers of (d − 2) and obtain at leading order in (d − 2) a second-order differential equation for S(2; p 2 ) only: where we made use of the relation Eq.(B.8) of Appendix B. We compared Eq.(5.2) with the second-order differential equation derived in [9], finding perfect agreement. Eq.(5.2) has been solved in reference [10] in terms of one-dimensional integrals over elliptic integrals. Upon inserting the result in Eq.(4.13) one can obtain S 1 (2; p 2 ) in terms of S(2; p 2 ) and dS(2; p 2 )/dp 2 . Inserting then S(2; p 2 ) and S 1 (2; p 2 ) in Eq.s (4.15, 4.16), one obtains by quadrature the first-order terms, Z 2 (2; p 2 ) and Z (1) 3 (2; p 2 ), of the expansion in (d − 2) of Z 2 (d; p 2 ) and Z 3 (d; p 2 ). Having these results on hand, we can now consider the first-order in (d − 2) of the Eq.(5.1), which is now a second-order differential equation for S (1) (2; p 2 ) only, with known inhomogeneous terms (namely S(2; p 2 ), Z 2 (2; p 2 ) and Z (1) 3 (2; p 2 )). Proceeding in this way, at least in principle, the whole procedure can be iterated up to any required order in (d − 2).

Shifting relations from d = 2 to d = 4 dimensions
In the previous Sections we have shown how to use the Schouten identities for writing the differential equations for the M.I.s of the massive sunrise at d = 2 dimensions in block form, and outlined the procedure for obtaining iteratively all the coefficients of the expansion in (d − 2) of the four M.I.s starting from a second-order differential equation for S(2; p 2 ), the leading term of the expansion.
The physically interesting case corresponds however to the expansion of the M.I.s for d ≈ 4; we have therefore to convert the information given by the expansion at d ≈ 2 in useful information at d ≈ 4.
As it is well known, quite in general one can relate any Feynman integral evaluated in d dimensions to the very same integral evaluated in (d − 2) dimensions by means of the Tarasov's shifting relation [7]. This dimensional shift is achieved by acting on the Feynman integral with a suitable combination of derivatives with respect to the internal masses. In the case of the "conventional " M.I.s of the sunrise graph, as defined in Eq.(3.7), the shifting relations read: where the differential operator ∆ takes the form: It is however straightforward to invert the system and, in this way, to obtain the inverse shifting relations, expressing the four M.I.s in d + 2 ≈ 4 dimensions in function of those in d ≈ 2 dimensions. In addition, we can also use Eq.s(4.1,4.2) for expressing S 2 (d; p 2 ) and S 3 (d; p 2 ), in terms of S(d; p 2 ), S 1 (d; p 2 ) and Z 2 (d; p 2 ), Z 3 (d; p 2 ). As a result one arrives at expressing any of the four "conventional" M.I.s S(d+2, p 2 ), S i (d+2; p 2 ), i = 1, 2, 3, as a linear combination (whose coefficients depend -and in a non trivial way -on d and the kinematical variables of the problem) of the "new" M.I.s S(d; p 2 ), S 1 (d; p 2 ), Z 2 (d; p 2 ) and Z 3 (d; p 2 ) (and the tadpoles). Indicating for simplicity the four "conventional" M.I.s with M i (d) and with N j (d) the four "new" M.I.s and the tadpoles, and ignoring for ease of notation all the kinematical variables, the inverse shifting relations can be written as Given a relation of the form by expanding around d = 2 one has, quite in general where r, the first value of the summation index, can be negative (as it is the case in a Laurent expansion), so that F (n) (4) = G (n) (2) .
In the case of the inverse shift Eq. 3) requires also the expansion of the coefficients C i,j (d), but that is not a problem once the inverse shift has been written down explicitly). The explicit formulas of the direct or inverse shifting relations are easily obtained but very lengthy and we decided not to include them entirely here for the sake of brevity. For what follows, it is sufficient to discuss only the general features of one of the inverse shifting relations, namely the relation expressing S(d + 2; p 2 ) in terms of S(d; p 2 ), S 1 (d; p 2 ) and Z 2 (d; p 2 ), Z 3 (d; p 2 ). Keeping for simplicity only the leading term of the expansion in (d − 2) of the coefficients we find: , which will also be used in the following, can be found in Appendix C, Eq.s(C.2-C.5). Note anyway that: By writing the expansion of S(d + 2; p 2 ) at d ≈ 2 as  2) only in the coefficients of Z 2 (d; p 2 ) and Z 3 (d; p 2 ); recalling once more that at d = 2 both Z 2 (2; p 2 ) and Z 3 (2; p 2 ) are identically zero, see Eq.s(3.14,3.15), it is clear that these poles will not generate any singularity of S(d; p 2 ) as d → 2. On the other hand, the tadpoles in the r.h.s. of Eq.(6.4) do generate polar singularities of S(d + 2; p 2 ); recalling Eq.s(3.5,3.6) and by using the lengthy explicit form of the coefficients (which we did not write for brevity) one finds immediately formulas already known for a long time in the literature [4]. As a second observation, let us look at the zeroth-order term S (0) (4; p 2 ) of S(d; p 2 ) in (d − 4), i.e. the zeroth-order term in (d − 2) of Eq.(6.4). We have already commented the apparent polar singularity 1/(d − 2) in the coefficients of Z 2 (d; p 2 ) and Z 3 (d; p 2 ), actually absent because Z 2 (2; p 2 ) and Z 3 (2; p 2 ) are both vanishing. But due to the presence of the 1/(d − 2) polar factor, in order to recover the zeroth-order term S (0) (4; p 2 ), one needs, besides S(2; p 2 ), S 1 (2; p 2 ), also the first-order of the corresponding expansion of Z 2 (d; p 2 ) and Z 3 (d; p 2 ), namely Z (1) 2 (2; p 2 ) and Z (1) 3 (2; p 2 ) -obtained, in our approach, from the systematic expansion of the differential equations, see Eq.s (4.15,4.16) or Section 5.
The complete expression of S (0) (4; p 2 ), which is rather cumbersome, is given by Eq.(C.1) of Appendix C. The corresponding formulas for the other three M.I.s, i.e. the S i (d + 2; p 2 ), can be obtained directly from the authors. 7 The imaginary parts of the Master Integrals.
In this Section, which is somewhat pedagogical, we discuss the relationship between the imaginary parts of the M.I.s at d = 2 and d = 4 dimensions, as a simple but explicit example of functions exhibiting the properties described in the previous sections.
At d = 2 the Cutkosky-Veltman rule [14,15] gives for S(d; p 2 ), as defined by the first of Eq.s(3.7), where the following notations were introduced: We have the relation is the familiar invariant form appearing in the two-body phase space, showing that the system of the three particles, whose masses enter in the definition of R 4 (b; b 1 , b 2 , b 3 , b 4 ), can be considered as the merging of a two-body system of total energy √ b and masses m 2 , m 3 with a two-body system of total energy W and masses √ b, m 1 . According to Eq.s(3.7), for i=1,2,3 the integral representation Eq.(7.1), however, is of no use for obtaining ImS i (2; −W 2 ) through a direct differentiation (due to the appearance of end point singularities). It is more convenient to use Eq.(D.13) of the Appendix D, so that Eq.(7.1) becomes where K(w 2 ) is the complete elliptic integral of the first kind, Eq.(D.6), and Let us observe, in passing, that, even if ImS(2, −W 2 ) (and, more generally S(d; p 2 ) as well) is obviously symmetric in the three masses m 1 , m 2 , m 3 , the symmetry is not explicitly shown by the integral representation Eq.(7.1), while the manifest symmetry is restored in Eq.s(7.6,7.7).
One can now use Eq.(7.6) and Eq.(D.11) to carry out the derivative with respect to the masses m 2 i in Eq.(7.5); the result reads where w 2 is given by the first of Eq.s(7.7), P (p 2 , m 1 , m 2 , m 3 ) is the polynomial already introduced in Eq.(4.3), symmetric in the last two arguments, and E(w 2 ) is the complete elliptic integral of the second kind, see Eq.(D.7). Eq.s(7.6,7.8,7.9,7.10) express the four quantities ImS(2; −W 2 ), ImS i (2, −W 2 ), i = 1, 2, 3 in terms of just two functions, the elliptic integrals K(w 2 ), E(w 2 ); therefore, the four imaginary parts cannot be all linearly independent. It is indeed easy to check that they satisfy the two equations which are nothing but the imaginary parts of Z 2 (2; −W 2 ), Z 3 (2; −W 2 ), Eq.s(3.14,3.15). As a further comment on the imaginary parts at d = 2, let us observe that they take a finite value at threshold, i.e. in the W → (m 1 + m 2 + m 3 ) limit. In that limit, indeed, b 3 → b 2 = (m 2 + m 3 ) 2 , and one finds b3 b2 db The extension to the value at threshold of the ImS i (2, −W 2 ) is similar, even if requiring one more term in the expansion due to the presence of the denominator 1/(b 3 − b 2 ) in their definitions, Eq.s(7.8,7.9,7.10).
The threshold values are At d = 4 the imaginary part of S(d; p 2 ), by using the same notation as in Eq.(7.1), is given by (7.14) At variance with the d = 2 case, the ImS i (4, −W 2 ) can be obtained at once by differentiating with respect to the masses the previous integral representation for ImS ( An explicit calculation gives Again at variance with the d = 2 case, the four imaginary parts are now combinations of four independent elliptic integrals, and therefore all independent of each other. Conversely, one can look at the inverse formulas, giving the imaginary parts at d + 2 → 4 in terms of the imaginary parts at d → 2. For ImS(4, −W 2 ), taking only the imaginary part at d = 2 of Eq.(6.4), one obtains: 3 (2; −W 2 ) are the imaginary parts of the first term of the expansion in (d − 2) of the corresponding functions, see Eq.s(4.11) (let us recall once more that according to Eq.s(3.14,3.15) Z 2 (2; p 2 ), Z 3 (2; p 2 ) vanish identically). An equation similar to Eq.(7.22) holds for ImS 1 (4; −W 2 ); we do not write it explicitly for the sake of brevity. The functions ImS(4; −W 2 ), ImS 1 (4; −W 2 ) and ImS(2; −W 2 ), ImS 1 (2; −W 2 ) are known, see Eq.s (7.16,7.17) and Eq.s(7.6,7.8); by combining Eq.(7.22) and the similar (not written) equation for ImS 1 (4; −W 2 ), one can obtain the explicit values of ImZ From the previous equations and the same procedure giving Eq.s (7.11,7.12) we obtain in particular the values at threshold 1 π ImZ 2 (2; −W 2 ) can also be evaluated solving, by quadrature, the imaginary part of the differential equation Eq.(4.15), i.e. by evaluating where C is an integration constant and dZ (1) 2 (2; p 2 )/dp 2 is obtained from Eq.(4.15) itself. The constant C can be fixed, a posteriori, by requiring that the imaginary parts of the "conventional" M.I. vanish at threshold in d = 4 dimensions, a condition which leads again to Eq.s(7.25).
After many algebraic simplifications, one obtains for ImZ , .
To carry out the integration, we use the integral representations Eq.(D.4) for the elliptic integrals I(n, s) and exchange the order of integration according to where Eq.(7.3) was used. The s integration is then elementary, giving only logarithms of suitable arguments and new square roots quadratic in b ; a subsequent integration by parts in b removes those logarithms with some of the accompanying square roots, and the result is Eq.(7.23), as expected. The same applies also for ImZ 3 (2; −W 2 ), whose value is obtained by simply exchanging m 2 and m 3 in Eq.(7.23).

Conclusions
In this paper we introduced a new class of identities, dubbed Schouten identities, valid at fixed integer value of the dimensions d. We applied the identities valid at d = 2 to the case of the massive two-loop sunrise graph with different masses, finding that in d = 2 dimensions only two of the four Master Integrals (M.I.s) are actually independent, so that the other two can be expressed as suitable linear combinations of the latter.
In the general case of arbitrary dimension d and different masses, the four M.I.s are known to fulfil a system of four first-order coupled differential equations in the external momentum transfer. The system can equivalently be re-phrased as a fourth-order differential equation for one of the M.I.s only.
Using these relations we introduced a new set of four independent M.I.s, valid for any number of dimensions d, whose property is that two of the newly defined integrals vanish identically in d = 2. The new system of differential equations for this set of M.I.s takes then a simpler block form when expanded in (d − 2). Starting from this system, one can derive a second-order differential equation, exact in d, for the full scalar amplitude, which still contains the two integrals, whose value is zero at d = 2, as inhomogeneous terms. We verified that the zeroth-order of our equation corresponds to the equation derived in [9]. Our equations, once expanded in powers of (d − 2), can be used, together with the linear equations for the remaining three M.I.s, for evaluating recursively, at least in principle, all four M.I.s, up to any order in (d − 2).
We then worked out explicitly the Tarasov's shifting relations needed to recover the physically more relevant value of the four M.I.s expanded in (d − 4) at d ≈ 4 starting from the expansion in (d − 2) at d ≈ 2 worked out in our approach.
As an example of this procedure we discussed the relationship between the imaginary parts of the four M.I.s in d = 2 and d = 4. The latter can be computed using the Cutkosky-Veltman rule. We showed how in d = 2 the imaginary parts of the four M.I.s can be written in terms of two independent functions only, namely the complete elliptic integrals of the first and of the second kind. The same is not true in d = 4 dimensions, where four independent elliptic integrals are needed in order to represent the four imaginary parts. We then showed how the Tarasov's shift formulas relate the imaginary parts in d = 2 and d = 4 dimensions. Finally, we gave an explicit example of how the differential equations for the imaginary parts of the master integrals can be integrated by quadrature.
A The polynomials of the first-order differential equations In this appendix we give the explicit expressions for the polynomials appearing in the first-order differential equations in section 4. All polynomials are functions of p 2 and of the three masses m 1 , m 2 , m 3 , while they do not depend on the dimensions d. B The polynomials of the second-order differential equation.
In this second appendix we give the explicit expressions of the polynomials that appear in the second-order differential equation derived in section 5. Also in this case, they are functions of p 2 and of the three masses m 1 , m 2 and m 3 , but they do not depend on the dimensions d.

D Imaginary parts
We work out here in some details the formulas used in Section 7. To start with, let us recall the definitions Eq.s(7.2) In the same way, starting for instance from b3 b2 db d db (D.5) The above four integrals I(n, W ), with n = −1, 0, 1, 2, defined in Eq.s(D.4), are easily expressed in terms of the usual complete elliptic integrals K(w 2 ), E(w 2 ), Π(a; w 2 ) of first, second and third kind, namely: Indeed, the standard change of variable gives With the integral representation of Eq.s(D.6,D.7) it is easy to obtain the formula which is useful for the evaluation of the imaginary parts of the Master Integrals S i (d, p 2 ) in d = 2 dimensions, Eq.(7.5).
One can easily express also K(w 2 ), E(w 2 ) etc. in terms of the I(n, W ) by inverting Eq.s(D.10) or by using the change of variables Eq.(D.9). Indeed, the second of Eq.s(D.10) can also be written as which on account of Eq.(D.5) can also be written as E) . (D.14) Let us discuss shortly also the limit of equal masses m 1 = m 2 = m 3 = m, which gives b 1 = 0, b 2 = 4m 2 , b 3 = (W − m) 2 , b 4 = (W + m) 2 . In that limit, thanks to b 1 = 0 we can read Eq.(D.5) as an identity expressing I(−1, W ) in terms of I(1, W ), I(2, W ). Further, one more identity appears [16] from where R 2 (a, b, c) is defined in Eq.(7.4), giving (in the equal mass limit), I(1, W ) = 1 3 (W 2 + 3m 2 ) I(0, W ) , (D. 15) showing once more that in the equal mass limit the imaginary parts can be expressed in terms of two independent functions only.