Polynomial Structures in One-Loop Amplitudes
A general one-loop scattering amplitude may be expanded in terms of master integrals. The coefficients of the master integrals can be obtained from tree-level input in a two-step process. First, use known formulas to write the coefficients of -dimensional master integrals; these formulas depend on an additional variable, , which encodes the dimensional shift. Second, convert the -dependent coefficients of -dimensional master integrals to explicit coefficients of dimensionally shifted master integrals. This procedure requires the initial formulas for coefficients to have polynomial dependence on . Here, we give a proof of this property in the case of massless propagators. The proof is constructive. Thus, as a byproduct, we produce different algebraic expressions for the scalar integral coefficients, in which the polynomial property is apparent. In these formulas, the box and pentagon contributions are separated explicitly.
Detailed calculations of multi-particle scattering events are needed in order to analyze new physics at the experiments of the Large Hadron Collider. Computational complexity increases rapidly with the number of legs, even at the amplitude level. New and improved algorithms are being developed to meet these needs. Recent progress at next-to-leading order has been reviewed in .
Scattering amplitudes at one-loop level can be understood in terms of an expansion in master integrals [2, 3]. The coefficients of the master integrals may be obtained by direct reduction, or alternatively by solving constraint equations derived from singular structures, most notably unitarity cuts [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. In order to obtain complete physical amplitudes from unitarity cuts, we can work in dimensional regularization, where [31, 32, 33, 34]. By now, explicit analytic formulas for these coefficients are available [20, 23, 27, 28]. The input quantities are taken from the complete tree-level amplitudes involved in unitarity cuts. There are other promising algorithms for finding the coefficients in 4 or dimensions [35, 17, 24, 29], or specifically the additional “rational” parts supplementing a pure 4-dimensional expansion [36, 37, 38, 39, 40, 41, 42, 43, 44, 30, 45].
The formulas of , developed in the context of the -dimensional unitarity analysis of [18, 21], are coefficients of -dimensional master integrals; these formulas depend on an additional variable, , which encodes the dimensional shift. To finish the calculation, we convert the -dependent coefficients of -dimensional master integrals to explicit coefficients of dimensionally shifted master integrals.111As an alternative to this last step, complete coefficients of -dimensional master integrals could be obtained with the recursion and reduction formulas of [18, 21].
We are presently concerned with the adaptation of the formulas of  to an efficient numerical algorithm. Two particular issues are addressed in this paper:
Because the coefficients of the -dimensional integrals are polynomials in the variable , a direct numerical implementation is not obvious.
The algebraic expression of boxes includes both box and pentagon contributions. The pentagon contribution is signaled by the factor in the denominator.
Our aim is to solve these two problems. More concretely, in this paper we accomplish the following:
The proof of the polynomial property of : In previous work, some evidence for this assumption was provided. Now, we give a complete proof.
Simplifying our previous expressions: The algebraic expressions for coefficients given in  were the full polynomials in , i.e. a sum of terms of the form . Here, we give expressions for evaluating directly from input quantities.
Separating the coefficients of boxes and pentagons: We give explicit, separate expressions for coefficients of boxes and pentagons.
For simplicity, the results here are specific to amplitudes with massless propagators. Generalization to the massive case is straightforward for the coefficients of master integrals that have nonvanishing cuts. Based on the present paper, the generalization to the massive case has been presented in . We work within the spinor formalism , reviewed in .
The paper is organized as follows. In Section 2, we organize our input quantities from tree amplitudes, define some key vectors and spinors from the input, and briefly discuss the dimensional shift. Then we proceed to the simplifications of the formulas for coefficients, and the proofs that they are polynomials in . These are given in Sections 3 and 4 for triangles and bubbles, respectively. In Section 5, we address box coefficients, and for the first time we present separate formulas for box and pentagon coefficients. Section 6 contains an application of these formulas, within the example of the 5-gluon amplitude. In Section 7, we close with a discussion and comparison to a couple of other recent approaches to the problem of one-loop amplitudes. Appendix A contains our definitions of master integrals and dimensional shift identities. Appendix B contains alternate, more explicit expressions for the triangle coefficients which may be better suited for numerical evaluation, since the derivatives have been taken analytically in every case that arises in a renormalizable theory. Appendix C contains many of the details of the polynomial proof for bubble coefficients. Appendix D contains analytic expressions used in cuts of pentagons.
2 Setup and Definitions
In this section, we set up some key conventions and definitions used in expressing the coefficients of master integrals, and in our proofs of polynomial dependence.
2.1 Unitarity method
The unitarity cut of a one-loop amplitude is its discontinuity across a branch cut in a kinematic region selecting a particular momentum channel. Specifically, we denote the momentum vector by . Then, should be positive, and all other momentum invariants should be negative. The vector will be a sum of momenta of some of the external legs. The discontinuity is given by
where the Lorentz-invariant phase-space (LIPS) of a double cut is defined by inserting two -functions representing the cut conditions:222The delta functions here should properly be denoted by , indicating that they are restricted to the positive light cone. We shall drop the superscript for simplicity.
The master integrals in dimensions with massless propagators are scalar pentagons, scalar boxes, scalar triangles, and scalar bubbles. In the full -dimensional formalism, there are no cut-free terms.
The -point scalar integral with massless propagators is
The coefficients in (3) are, by construction, cut-free rational functions. In the unitarity method, we do not derive the coefficients of master integrals by performing any reduction. Rather, we take the coefficients as unknowns and proceed to constrain them by performing cuts on both sides of (3):
Any realization of the unitarity method must address the problem of isolating the individual coefficients . The unitarity method succeeds because the cuts of master integrals are logarithms of unique functions of the kinematic invariants.
In , it was shown how to obtain scalar box coefficients directly by cutting four propagators rather than two. Similarly explicit analytic formulas for the other coefficients have recently become available [20, 23, 27, 28].
2.2 Input quantities
Having present the general picture, in this subsection we can start with the following most general expression for a unitarity cut integral:
where is -dimensional and is -dimensional. With the integrand in the form of (6), there is a prefactor which depends on the external momenta as well as on . In this discussion we shall be paying careful attention to all dependence on .
From this starting point, the coefficients of master integrals were listed in . Now, we would like to be able produce the complete 4-dimensional expression, by performing the integral over by the recursion and reduction formulas of [18, 21]. To get this complete answer, we need to consider the dependence of the prefactor on , along with the power of in the coefficient formulas of . We consider this dependence in terms of the dimensionless parameter , defined by
With this definition, the cut integral (6) can then be rewritten as
The coefficients listed in , which are summarized below, are the results of the four-dimensional part of the integral (6); they are functions of . The “four-dimensional cut-constructible” part of the amplitude could be obtained by setting in each of these coefficients, inside the expansion of the amplitude in master integrals. The complete -dimensional amplitude requires dealing with this -dependence. Here it is enough to apply the polynomial reduction identities given in [18, 21]. These identities assume polynomial structure of the coefficients , which is proven in the present paper, and which may also be deduced within other approaches . However, if we desire a result only through , it may be more efficient to use the dimensionally-shifted basis discussed in [49, 29]. We shall return to this point in the following subsection.
From the initial expression (6), we extract all the necessary information, as follows. First, notice the triplet of integers
which will play an important role. In particular, the value of constrains the basis of master integrals [4, 5]. If , there are contributions only from boxes and pentagons. If , contributions from triangles will kick in, and finally if , bubble contributions show up as well. This pattern is well known from traditional reduction techniques.
Second, we use the values of , , and from the expression (6) to define the vectors , and related important quantities, as follows: 333These definitions apply specifically to the case with massless propagators. Only a slight modification is necessary for massive propagators [20, 46].
One important observation is that
At this point, we wish to make a few more remarks.
The input quantities are given by . From this we can define and . We make reference to the number of these vectors, encoded in the triple of integers .
To simplify notation when we set , we will write expressions such as , or just .
The coefficients of the master integrals are polynomials in . In this paper, we shall find that the maximum degrees of these polynomials are the following. Pentagon: 0. Box: . Triangle: . Bubble: . Here, denotes the greatest integer less than or equal to .
Knowing the maximum value of the degree of the polynomial in , we can then calculate the coefficient of by the formula
The expression (20) is central in this paper. Since now has an expression where does not appear (as indicated by the right-hand-side of expression (20)), it can be evaluated numerically. 444See  for another approach that is possibly more efficient.
Summary of coefficients of 4-dimensional master
For the box coefficient with momenta ,
For the triangle coefficient with momenta ,
For the bubble coefficient with momentum ,
where we have made the following definitions:
Note that the prefactor has not been included in these formulas for coefficients.
2.3 Some important constructions from input quantities
Given two momenta , we construct two null momenta. If and , are themselves the two null momenta. If at least one of them is not null, for example , then we construct two null momenta as follows.
Then, the following quantities necessarily vanish.
We shall use the following identity:
Any four-dimensional momentum can be expanded in a basis of four other independent momenta by
where the coefficients are given by
At times, we will write these coefficients in the form to emphasize the related quantities.
2.4 The dimensionally shifted basis
Since, as we shall demonstrate, our coefficients are polynomials in , we can translate this information into the dimensionally shifted basis . More explicitly, if we define
then we have
There are two merits of using this basis. First, we can throw away all contributions to make the calculation easier. Second, we improve efficiency. To use the recursion and reduction relations, we first calculated all the contributions by reduction to boxes, triangles and bubbles, and then added them up. With the dimensionally shifted basis, this process of reduction/summation can be done in one step, simplifying calculations. The usefulness of this dimensionally shifted basis has been discussed in [22, 29]. Here, for reference, we discuss this evaluation in Appendix A.
3 Triangle coefficients
Now that we have the necessary background information, it is simplest to start with the coefficients of triangles. Some features of this discussion will apply to bubbles as well.
3.1 Simplifying the formula
We write the formula for triangle coefficients from  in the notation of the previous section, emphasizing -dependence.
where the , as depicted in the indices, are constructed in terms of , as defined in (22), and depend on . In principle we can put (31) into (20) to take derivatives. However, the -dependence everywhere might be an obstacle to taking stable derivatives in terms of in (20). In this subsection, we recast this -dependence in a simpler form.
When we take the square root of , there is a sign ambiguity. It can be shown that the choice of sign does not affect the final result. To be explicit, we choose the minus sign here, i.e.,
where we have defined new scalar quantities, , as follows:
With these results, we can see that
The -dependence has been factored out; here the null momentum does not depend on , since it is constructed from –as indicated in the subscript indices.
To simplify further, apply the identity to derive
If we define two more vectors by
How to use the formula (40): The degree of this polynomial in will be seen to be . Thus we can get the corresponding coefficients by taking derivatives in first (from to , to get coefficients from each term in the polynomial), and then setting .
For example when we can set directly and get
which is suitable for numerical evaluation. For the result will take the form of a linear polynomial, . To get we take one derivative, using
In Appendix B, we have explicit expressions, free of derivatives, for triangle coefficients when .
The formula (40) contains in both numerator and denominator, so it is not so obvious that the total result is simply a polynomial in . The proof of this property is given in the next subsection.
3.2 Proof that the triangle coefficient is a polynomial in
We start by considering two quantities that arise in our expressions, in the course of taking derivatives:
By writing as a linear combination of the ,
and recalling that , we find that
All -dependence has dropped out of this expression.
For , similar manipulations show that
which is a polynomial in .
Now we prove that the full expression (40) for the triangle coefficient is a polynomial in . Throughout this proof, let us abbreviate by and by .
The triangle coefficient is given in terms of derivatives with respect to on an expression where the -dependence appears in the factors (in numerator or denominator). After taking the derivatives, we set . In this process we will produce the following three combinations: ; ; . It is easy to see how the first two combinations arise. The third combination, , appears, for example, in
Consider the -dependent factors in the denominator. With each derivative, we effectively add one overall factor of in the denominator and place one new factor, either or in the numerator. After taking derivatives, there are additional factors in the denominator. Thus, we have exactly factors of