Scattering Amplitudes from Multivariate Polynomial Division
We show that the evaluation of scattering amplitudes can be formulated as a problem of multivariate polynomial division, with the components of the integration-momenta as indeterminates. We present a recurrence relation which, independently of the number of loops, leads to the multi-particle pole decomposition of the integrands of the scattering amplitudes. The recursive algorithm is based on the Weak Nullstellensatz Theorem and on the division modulo the Gröbner basis associated to all possible multi-particle cuts. We apply it to dimensionally regulated one-loop amplitudes, recovering the well-known integrand-decomposition formula. Finally, we focus on the maximum-cut, defined as a system of on-shell conditions constraining the components of all the integration-momenta. By means of the Finiteness Theorem and of the Shape Lemma, we prove that the residue at the maximum-cut is parametrized by a number of coefficients equal to the number of solutions of the cut itself.
keywords:Scattering amplitudes, Unitarity, Polynomial Division
Scattering amplitudes in quantum field theories are analytic functions of the momenta of the particles involved in the scattering process, and can be determined by their singularity structure. The multi-particle factorization properties of the amplitudes are exposed when propagating particles go on their mass-shell Bern:1994zx (); Britto:2004nc (); Cachazo:2004kj (); Britto:2004ap ().
The investigation of the residues at the singular points has been fundamental for discovering new relations fulfilled by scattering amplitudes. The BCFW recurrence relation Britto:2004ap (), its link to the leading singularity of one-loop amplitudes Britto:2004nc (), and the OPP integrand-decomposition formula for one-loop integrals Ossola:2006us () have shown the underlying simplicity beneath the rich mathematical structure of quantum field theory. Moreover they have become efficient techniques leading to quantitative predictions at the next-to-leading order in perturbation theory Berger:2008sj (); Giele:2008bc (); Badger:2010nx (); Bevilacqua:2011xh (); Hirschi:2011pa (); Cullen:2011ac (); Agrawal:2011tm (); Cascioli:2011va ().
The integrand reduction methods Ossola:2006us () allow to decompose one-loop amplitudes in terms of Master Integrals (MI’s) without performing the loop integration, and are based on the multi-particle pole expansion of the integrand. The expansion is equivalent to the decomposition of the numerator in terms of (a combination of) products of denominators, with polynomial coefficients. In the context of an integrand-reduction, any integration is replaced by polynomial fitting.
The first extension of the integrand reduction method beyond one-loop was proposed in Mastrolia:2011pr (), and it was used to reproduce the results of two-loop 5-point planar and non-planar amplitudes in SYM Bern:2006ew (); Carrasco:2011mn (). A key point of the higher-loop extension is the proper parametrization of the residues at the multi-particle poles. Each residue is a multivariate polynomial in the irreducible scalar products (ISP’s) among the loop momenta and either external momenta or polarization vectors constructed out of them. ISP’s cannot be expressed in terms of denominators, thus any monomial formed by ISP’s is the numerator of a potential MI which may appear in the final result. Hence, a systematic classification of the polynomial structures of the residues is mandatory. In Mastrolia:2011pr (), the residues have been obtained by relating the ISPs to monomials in the components of the loop momenta expressed in a basis chosen according to the topology of the on-shell diagram.
Badger, Frellesvig and Zhang Badger:2012dp () combined on-shell conditions with Gram-identities Gluza:2010ws () to limit the number of monomials appearing in the residues. This technique was applied to the integrand decomposition of two-loop 4-point planar and non-planar diagrams in supersymmetric as well as non-supersymmetric YM theories.
In this work, we show that the shape of the residues is uniquely determined by the on-shell conditions alone, without any additional constraint. We derive a simple integrand recurrence relation that generates the required multi-particle pole decomposition for arbitrary amplitudes, independently of the number of loops.
The algorithm treats the numerator and the denominators of any Feynman integrand, as multivariate polynomials in the components of the loop variables. The properties of multivariate polynomials have been extensively studied in the mathematical literature, see e.g. buch1 (); Cox1 (); Cox2 (); moller (); rouillier (); sturmfels (); verschelde (). The method uses both the weak Nullstellensatz theorem and the multivariate polynomial division modulo appropriate Gröbner basis buch1 (). In the context of the integrand reduction, univariate polynomial division has been already introduced in Mastrolia:2012bu () to improve the decomposition of one-loop scattering amplitudes.
The algorithm, which is described in Section II, relies on general properties of the loop integrand:
When the number of denominators is larger than the total number of the components of the loop momenta, the weak Nullstellensatz theorem yields the trivial reduction of an -denominator integrand in terms of integrands with denominators.
When is equal or less than the total number of components of the loop momenta, we divide the numerator modulo the Gröbner basis of the -ple cut, namely modulo a set of polynomials vanishing on the same on-shell solutions as the cut denominators. The remainder of the division is the residue of the -ple cut. The quotients generate integrands with denominators which should undergo the same decomposition.
By iterating this procedure, we extract the polynomial forms of all residues. The algorithm will stop when all cuts are exhausted, and no denominator is left, leaving us with the integrand reduction formula.
In Section III we apply the algorithm to a generic one-loop integrand, reproducing the -dimensional integrand decomposition formula Ossola:2006us (); Ellis:2007br (); Giele:2008ve (); Ellis:2008ir ().
In Section IV we conclude by proving a theorem on the maximum-cuts, i.e.
the cuts defined by the maximum number of on-shell conditions which can
be simultaneously satisfied by the loop momenta. The on-shell conditions
of a maximum cut lead to a zero-dimensional system.
Finiteness Theorem and the
Shape Lemma ensure that the
residue at the maximum-cut is parametrized
by coefficients, where is the number of
solutions of the multiple cut-conditions.
This guarantees that the
corresponding residue can always be reconstructed by evaluating the numerator
at the solutions of the cut.
2 Multivariate polynomial division
In what follows, we assume 4-dimensional loop-momenta. Extensions to higher-dimensional cases, according to the chosen dimensional regularization scheme, can be treated analogously - as we will show when discussing the one-loop integrand reduction.
The integrand reduction methods Ossola:2006us (); Ossola:2007bb (); Ossola:2007ax (); Ossola:2008xq (); Ellis:2007br (); Giele:2008ve (); Ellis:2008ir (); Mastrolia:2008jb (); Mastrolia:2012bu (); Mastrolia:2011pr (); Badger:2012dp () recast the problem of computing -loop amplitudes with denominators as the reconstruction of integrand functions of the type
where are integration momenta. The generic propagator can be written as follows:
The numerator and any of the denominators are polynomial in the components of the loop momenta, say , i.e.
Let us consider the ideal generated by the denominators in Eq. (3) ,
where is the set of polynomials in . The common zeros of the elements of are exactly the common zeros of the denominators.
The multi-pole decomposition of Eq. (1) is explicitly
achieved by performing multivariate polynomial division,
yielding an expression of
in terms of denominators and residues.
We construct a Gröbner basis buch1 () (see Ch. 2 of Cox1 ()), generating the ideal with respect to a chosen monomial order,
Unless otherwise indicated, we will assume lexicographic order.
In this formalism, the -ple cut-conditions , are equivalent to . The number of elements of the Gröbner basis is the cardinality of the basis. In general, is different from . We then consider the multivariate division of modulo (see Ch. 2, Thm. 3 of Cox1 ()),
is a compact notation for the sum of the products of the quotients
and the divisors .
The polynomial is the remainder of the
division. Since is a Gröbner basis,
the remainder is uniquely determined once the monomial order is fixed.
The term belongs to the ideal , thus it can be expressed in terms of denominators, as
The explicit form of can be found by expressing the elements of the Gröbner basis in terms of the denominators.
2.1 Reducibility criterion.
An integrand is said to be reducible if it can be written in terms of lower-point integrands: that happens when the numerator can be written in terms of denominators. The concept of reducibility can be formalized in algebraic geometry. Indeed a direct consequence of Eqs. (5) and (6) is the following
The integrand is reducible iff the remainder of the division modulo a Gröbner basis vanishes, i.e. iff .
Proposition 1 allows to prove
An integrand is reducible if the cut leads to a system of equations with no solution.
In this case, the system is over-constrained. The propagators cannot vanish simultaneously, i.e.
has no solution. Therefore, according to the weak Nullstellensatz theorem (Thm. 1, Ch. 4 of Cox1 ()),
for some . Irrespective of the monomial order, a (reduced) Gröbner basis is . Eq. (5) becomes
thus is reducible. ∎
2.2 Integrand Recursion Formula
According to Eq. (10), is expressed in terms of integrands, , with denominators. are obtained from by pinching the -th denominator. The numerator of the non-homogeneous term is the remainder of the division (5). By construction, it contains only irreducible monomials with respect to , thus it is identified with the residue at the cut .
The integrands can be decomposed repeating the procedure described in Eqs. (3)-(5). In this case the polynomial division of has to be performed modulo the Gröbner basis of the ideal , generated by the corresponding denominators.
The complete multi-pole decomposition of the integrand is achieved by successive iterations of Eqs. (3)-(5). Like an Erathostene’s sieve, the recursive application of Eqs. (5) and (10) extracts the unique structures of the remainders ’s. The procedure naturally stops when all cuts are exhausted, and no denominator is left, leaving us with the integrand reduction formula.
If all quotients of the last divisions vanish, the integrand is cut-constructible, i.e. it can be determined by sampling the numerator on the solutions of the cuts. If the quotients do not vanish, they give rise to non-cut-constructible terms, i.e. terms vanishing at every multi-pole. They can be reconstructed by sampling the numerator away from the cuts. Non-cut-constructible terms may occur in non-renormalizable theories, where the rank of the numerator is higher than the number of denominators Mastrolia:2012bu ().
3 One-loop integrand decomposition
In this section we decompose an -point integrand of rank- with , using the procedure described in Section 2. The reduction of higher-rank and/or lower-point integrands proceeds along the same lines.
In -dimensions, the generic -point one-loop integrand reads as follows:
For later convenience, for each we define a
If , then , where are four external momenta.
If , then are chosen to fulfill the following relations:
In terms of , the loop momentum can be decomposed as,
Accordingly, each numerator can be treated as a rank- polynomial in ,
Step 1. When , the Proposition 2 guarantees that is reducible, and, by iteration, it can be written as a linear combination of -point integrands .
Step 2. The numerator of each is a rank-5 polynomial in , cfr. Eq. (14). We define the ideal , and compute the Gröbner basis , which is found to have a remarkably simple form:
We observe that each depends linearly on the -th component of .
The division of modulo , see Eq.(5), gives a constant remainder,
The term in Eq. (6) is,
where are the numerators of the 4-point integrands, , obtained by removing the -th denominator.
Step 3. For each , the numerator is a rank- polynomial in . The Gröbner basis of the ideal contains four elements. Dividing modulo , we obtain the remainder. The latter depends on and on the fourth component of the loop momentum in the basis ,
The term ,
contains the numerators of 3-point integrands .
Step 4. The Gröbner basis is formed by three elements, and is used to divide . The remainder is polynomial in and in the third and fourth components of in the basis ,
The term generates the rank- numerators of the -point integrands , , and .
Step 5. The remainder of the division of by the two elements of is:
It is polynomial in and in the last three components of in the basis . The reducible term of the division, , generates the rank- integrands, , and .
Step 6. The numerator of the 1-point integrands is linear in the components of the loop momentum in the basis ,
The only element of the Gröbner basis is , which is quadratic in . Therefore the division modulo , leads to a vanishing quotient, hence
Step 7. Collecting all the remainders computed in the previous steps, we obtain the complete decomposition of in terms of its multi-pole structure
We remark that the basis , defined in Eq.(13) and used for decomposing the integration momentum , depends only on the external momenta of diagram associate to the cut, eventually complemented by orthogonal elements. Therefore, can be used as well to decompose the integration momenta of multi-loop diagrams Mastrolia:2011pr ().
4 The Maximum-cut Theorem
At loops, in four dimensions, we define a maximum-cut as a -ple cut
which constrains completely the components of the loop momenta. In four dimensions this implies the presence of four constraints for each loop momenta. We assume that, in non-exceptional phase-space points, a maximum-cut has a finite number of solutions, each with multiplicity one. Under this assumption we have the following
Theorem 4.1 (Maximum cut).
The residue at the maximum-cut is a polynomial paramatrized by coefficients, which admits a univariate representation of degree .
Let us parametrize the propagators using variables . In this parametrization, the solutions of the maximum-cut read,
Let be the ideal generated by the on-shell denominators,
According to the assumptions, the number of the solutions of (22) is finite, and each of them has multiplicity one, therefore is zero-dimensional moller () and radical111 Given an ideal , the radical of is . is radical iff . , see Cor. 2.6, Ch. 4 of Cox2 (). In this case, the Finiteness Theorem (Prop. 8, Ch. 5 of Cox1 ()) ensures that the remainder of the division of any polynomial modulo can be parametrized exactly by coefficients.
Moreover, up to a linear coordinate change, we can assume that all the solutions of the system have distinct first coordinate , i.e. . We observe that and are in the Shape Lemma position (Prop. 2.3 of sturmfels ()) therefore a Gröbner basis for the lexicographic order is , in the form
The functions are univariate polynomials in . In particular is a rank- square-free polynomial rouillier (),
i.e. it does not exhibits repeated roots. The multivariate division of modulo leaves a remainder which is a univariate polynomial in of degree verschelde (), in accordance with the Finiteness Theorem. ∎
The maximum-cut theorem ensures that the maximum-cut residue, at any loop, is completely determined by the distinct solutions of the cut-conditions. In particular it can be reconstructed by sampling the integrand on the solutions of the maximum cut itself.
At one loop and in -dimensions, the -ple cuts are maximum-cuts. The remarkably simple structure of the Gröbner basis in Eq. (15) is dictated by the maximum-cut theorem. Moreover in this case , thus the residue in Eq. (16) is a constant.
The structures of the residues of the maximum cut, together with the corresponding values of , for a set of one-, two-, and three-loop diagrams are collected in Figure 1.
We presented a new algebraic approach, where the evaluation of scattering amplitudes is addressed by using multivariate polynomial division, with the components of the loop-momenta as indeterminates. We found a recurrence relation to construct the integrand decomposition of arbitrary scattering amplitudes, independently of the number of loops. The recursive algorithm is based on the Weak Nullstellensatz Theorem and on the division modulo the Gröbner basis associated to all possible multi-particle cuts. Using this technique, we rederived the well-known one-loop integrand decomposition formula. Finally, by means of the Finiteness Theorem and of the Shape Lemma, we proved that the residue at the maximum-cuts is parametrised exactly by a number of coefficients equal to the number of solutions of the cut itself.
We are indebted to Simon Badger and Yang Zhang for fruitful discussions, in particular on the properties of the Gröbner basis, and for comments of the manuscript.
E.M. thanks the Center for Theoretical Physics of New York City College of Technology for hospitality during the final stages of this project.
The work of P.M. and T.P. is supported by the Alexander von Humboldt Foundation, in the framework of the Sofja Kovaleskaja Award, endowed by the German Federal Ministry of Education and Research. The work of G.O. is supported in part by the National Science Foundation under Grant PHY-1068550.
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