We show how Stokes’ Theorem, in the fashion of the Generalised Cauchy Formula, can be applied for computing doublecut integrals of oneloop amplitudes analytically. It implies the evaluation of phasespace integrals of rational functions in two complexconjugated variables, which are simply computed by an indefinite integration in a single variable, followed by Cauchy’s Residue integration in the conjugated one. The method is suitable for the cutconstruction of the coefficients of 2point functions entering the decomposition of oneloop amplitudes in terms of scalar master integrals.
Unitarity and analyticity are wellknown properties of scattering amplitudes OldUnitarity (). Analyticity grants that amplitudes are determined by their own singularitystructure, while unitarity grants that the residues at the singular points factorize into products of simpler amplitudes. Unitarity and analyticity become tools for the quantitative determinaton of oneloop amplitudes Bern:1994zx () when merged with the existence of an underlying representation of amplitudes as a combination of basic scalar oneloop functions Passarino:1978jh (). These functions, known as Master Integrals (MI’s), are point oneloop integrals, (), with trivial numerator, equal to 1, characterised by external momenta and internal masses present in the denominator. Important improvements of unitaritybased numerical algorithms also make use of the general structure of oneloop integrands Ossola:2006us (); Ellis:2007br (); Giele:2008ve (); Berger:2008sj (). In the context of unitaritybased algorithms, the issue of computing oneloop amplitudes can be addressed in two stages: the computation of the coefficients; and the actual evaluation of the MI’s themeselves. The principle of a unitaritybased method is the extraction of the coefficients multiplying each MI by matching the multiparticle cuts of the amplitude onto the corresponding cuts of the MI’s.
Cutting a propagating particle in an amplitude amounts to applying the onshell condition and replacing its Feynman propagator by the corresponding function, As a result, the original function is substituted by a simpler one, easier to compute, which, nevertheless, still carries nontrivial information. In fact, the particle cut of appears in the 0trascendentality term (rational or irrational) of the corresponding cutamplitude, multiplied by the same coefficient of in the decomposition of the complete amplitude. Highertranscendentality terms, such as logarithms, are associated to the cuts of higherpoint MI’s.
In general, the fulfillment of multiplecut conditions requires loop momenta with complex components. Since the loop momentum has four components, the effect of the cutconditions is to fix some of them according to the number of the cuts. Any quadruplecut Britto:2004nc () fixes the loopmomentum completly, yielding the algebraic determination of the coefficients of ; the coefficient of 3point functions, , are extracted from triplecut MastroliaTriple (); FordeTriBub (); BjerrumBohr:2007vu (); Kilgore:2007qr (); Badger:2008cm (); the evaluation of doublecut Britto:2005ha (); Britto:2006sj (); ABFKM (); FordeTriBub (); Britto:2007tt (); Kilgore:2007qr (); Britto:2008vq (); Britto:2008sw (); Badger:2008cm () is necessary for extracting the coefficient of 2point functions, ; and finally, in processes involving massive particles, the coefficients of 1point functions, , are detected by singlecut Kilgore:2007qr (); NigelGlover:2008ur (); Britto:2009wz (). In cases where fewer than four denominators are cut, the loop momentum is not frozen: the freecomponents are left over as phasespace integration variables.
In this letter, we show a novel efficient method for the
analytic evaluation of the coefficients of oneloop 2point
functions via doublecuts.
Spunoff from the spinorintegration technique
Britto:2005ha (); Britto:2006sj (); ABFKM (); Britto:2007tt (); Britto:2008vq (); Britto:2008sw (),
the method hereby presented is an
application of Stokes’ Theorem.
We analyze the doublecut of massless particles in
fourdimensions, which also is the essential ingredient
for the phasespace integration in the general case
of oneloop massive amplitudes in dimensionalregularization
ABFKM (); Britto:2007tt (); Britto:2008vq (); Britto:2008sw (); MastroliaTriple ().
Due to a special decomposition of the loopmomentum,
the doublecut phasespace integral is written
as parametric integration of rational function
in two complexconjugated variables.
By applying Stokes’ Theorem, the integration is carried on in
two simple steps:
an indefinite integration in one variable, followed by Cauchy’s
Residue Theorem in the conjugated one.
The coefficients of the 2point scalar functions, being proportional to the rational term of the doublecut, can be directly extracted from the indefinite integration by Hermite Polynomial Reduction.
In a framework where factorization properties of scattering amplitudes are accessed via complex momenta, the doublecut integration presented here can be considered as the natural extension of the technique used to prove BCFWrecurrence relation for treelevel amplitudes BCFWproof (). In the latter case, scattering treeamplitudes are holomorphic functions, depending only on one complex variable, and Cauchy’s Residue Theorem is sufficient for their complete determination. In the case of the doublecut of oneloop amplitudes, where the integrand depends on two complexcojugated variables, Stokes’ Theorem in the fashion of Generalised Cauchy Formula, becomes the driving principle.
1 DoubleCut
– PhaseSpace Parametrization. The starting point of our derivation is the spinorial parametrization of the Lorentz invariant phasespace (LIPS) in the channel Cachazo:2004kj (); Britto:2005ha (); Britto:2006sj (),
(1)  
obtained by rescaling the original loopvariable as,
(2) 
with . In terms of spinor variables, the rescaling reads,
(3) 
where , the rescaling parameter, is frozen as a consequence of the (second of the) onshell conditions, and becomes the new loop integration variable.
– Change of Variables. We take two massless momenta, say and fulfilling the conditions,
(4) 
and decompose in a basis of four massless momenta constructed out of them,
(5) 
Notice that the vectors and are trivially orthogonal to both and . The above decomposition can be realized starting from the definition of in terms of spinor variables, and performing the following spinor decomposition,
(6) 
By changing variables as in (6), and using (4), one can write,
(7)  
(8) 
Hence, the LIPS in (1) reduces to the novel form,
(9) 
where is a positive quantity as assured by the argument of the function.
– DoubleCut Integration. The doublecut of a generic point amplitude in the channel is defined as
(10) 
where are the treelevel amplitudes sitting at the two sides of the cut, see Fig.1. After rescaling as in (3), and using expression (9) for the LIPS, one has,
(11)  
whera parametrizes the scaling behaviour of . The integration can be performed trivially, because of the presence of the function. Then, by using the decomposition (5, 6), the doublecut becomes a doubleintegral,
(12) 
where is a rational function of and . As such, it can be expressed as a ratio of two polynomials, say and ,
(13) 
with the following relations between their degrees,
(14) 
We remark that the double integration
in  and variables
appearing in Eq.(12) will be properly
justified in Sec.2.
For the moment, with abuse of notation, we simlpy denote it as
a convolution of an indefinite integral
and a contour integral, which are the actual operations
we are going to carry out.
To begin with the integration,
we find a primitive of with respect
to , say , by keeping as independent variable,
(15) 
so that becomes,
(16) 
where is a shorthand notation for . Before proceeding with the final integration on the variable, let us analyse the structure of . Since is the primitive of a rational function, its general form can only contain two types of terms: a rational term and a logarithimc one,
(17) 
It is important to notice that the presence of the term depends on the powers of in Eq.(11): can be generated, after integrating in , only if . The integration will be performed by applying Cauchy’s Residue Theorem, therefore the final structure of the doublecut is determined by the nature of . Namely, the integration of [] is responsible of the rational [logarithmic] term of .
– 2point Function. We also know apriori that the doublecut of a 2point scalar function in 4dimension is a rational, or better (to account for the massive case as well) a nonlogarithmic term; while the doublecut of higherpoint scalar functions might contain logarithms (with dependent argument). Hence, the coefficient of a 2point function in the channel will appear in , the integration of in ,
(18) 
where the integration is performed via Cauchy’s Residue Theorem. The integrand is rational in , and contains poles whose location in the complex plane is a unique signature of the Feynman integral they come from Britto:2007tt (); Britto:2008vq (); ArkaniHamed:2008gz (). The choice of and specified in Eqs.(4) grants that there exists a pole at associated to the 2point function in the channel, ; while the reduction of higherpoint functions that have as subdiagram can generate poles at finite values. Because of the presence of , through the term , is nonanalytic. The Residue Theorem has to be applied by reading the residues in , and substituting the corresponding complexconjugate values where appears. Therefore, the result of can be implicitly written as,
– Doublecut of the Scalar Function . Let us evaluate the doublecut of the 2point scalar function , which also is a prototype example:
(20)  
where we notice that the primitive in , called in (15), has only a rational term (the logarithmic contribution is absent),
(21) 
For the last integration in , by applying the Residue Theorem, we take the residue of the unique simple pole at , since the term never vanishes, being always positive. The final result of the doublecut of the scalar 2point function reads,
(22) 
– Coefficient of the 2point Function. The expression of the 2point coefficient can be finally obtained by taking the ratio of in (LABEL:eq:DeltaRatfinal) and the doublecut of in (22),
(23)  
– Hermite Polynomial Reduction. To optimize the integration algorithm, one can use the so called Hermite Polynomial Reduction (HPR), a technique enabling the direct extraction of the rational term of the primitive of a rational function, without computing the integral as a whole. Based on the squarefree factorization of the integrand, HPR can be used to write the result of any integral of a rational function as a pure rational term plus another integral that, if explicitly computed, would generate the logarithmic remainder.
As written in Eq.(23), the coefficient of the 2point function comes only from the term , and not from , see Eqs.(15, 17); is the rational term in the result of the integration of , which is rational in , see Eq.(13). Therefore HPR is suitable for extracting out of the integration.
The integration algorithm of Sec.1 can be implemented with S@M Maitre:2007jq () together with the routine HPR () for Hermite Polynomial Reduction.
2 Stokes’ Theorem
In this section we give a formal definition of the  integration used in Sec.1, as an application of Stokes’ Theorem for differential forms. In what follows, we use the notation: and .
Let us recall that the complex 1form
(24) 
which is defined for all except , is a closed form,
(25) 
We consider any complex smooth function and differentiate the 1form ,
(26) 
obtaining the 2form,
(27) 
Now we take a domain in the complex plane and apply Stokes’ Theorem to . Due to the singularity of at , we remove a tiny disk , centered at with radius , from . Then has no singularity in the regulated domain , and we may apply Stokes’ Theorem:
(28) 
Here is a circle around the point , which is described by the parametric equation . Since converges to as the radius shrinks to 0, the last integral in Eq.(28),
(29) 
converges to as goes to 0. Letting in Eq.(28), the disk disappears and fills up . Consequently Stokes’ Theorem can be reformulated as,
(30) 
By using the explicit expression of and , in Eqs.(26, 27), and rearranging terms, we obtain the so called Generalised Cauchy Formula or CauchyPompeiu Formula,
(31) 
Let us discuss two special cases.
First, when is analytic, , hence
we obtain,
(32) 
which is the wellknown Cauchy Formula,
where is any closed curve surrounding .
Secondly, when vanishes on the boundary of ,
that is , Eq.(31) becomes,
(33) 
where we used .
The expression (33) is what needed
to define properly the doublecut given
in Eqs.(12, 16),
which we rewrite here as,
(34) 
by identifying ,
and , where
the functions and were defined in
Eqs.(13, 15, 17).
The integration domain, , is the whole complex plane.
The vanishing of on the boundary is granted
by the structure of the rational integrand and relations
(14)
among the degrees of numerator and denominator.
To deal with the general case, where more than one pole might appear,
the calcualtion of trivially generalises,
by the superimposition principle, to the sum of the residues
at all the poles in ,
(35)  
due to the subtraction of a disk around each of the poles
from the domain .
Finally, Eq.(35)
validates Eq.(LABEL:eq:DeltaRatfinal), hence
the expression for the coefficient in Eq.(23).
Notice that the role of and in the application
of Stokes’ Theorem can be interchanged, reflecting the symmetry
of under the exchange in (4).
– Acknowledgements. I am indebted to Ed Witten for inviting me to consider Stokes’ Theorem as the proper formal framework for the twofold complex integration hereby presented. I also thank Mario Argeri, Simon Badger, Michele Caffo, Bryan Lynn, Bob McElrath, Stefano Pozzorini, and Ciaran Williams for stimulating discussions.
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S. Blake,
http://demonstrations.wolfram.com/
IntegrationUsingHermiteReduction/