On-Shell Methods in Perturbative QCD
We review on-shell methods for computing multi-parton scattering amplitudes in perturbative QCD, utilizing their unitarity and factorization properties. We focus on aspects which are useful for the construction of one-loop amplitudes needed for phenomenological studies at the Large Hadron Collider.
Pacs:11.15.Bt, 11.25.Db, 11.55.Bq, 12.38.Bx
UCLA/07/TEP/11 SLAC–PUB–12447 SPhT–T07/039
The Large Hadron Collider (LHC) is poised to begin exploration of the multi-TeV energy frontier within the next year. It is widely anticipated that physics beyond the Standard Model will emerge at this scale, most likely via the production of new, heavy particles which may be associated with the mechanism of electroweak symmetry breaking. At the very least, the scalar Higgs boson of the Standard Model awaits discovery. New, heavy particles typically will decay rapidly into the known leptons, neutrinos, quarks and gluons. When the new physics model includes a dark matter candidate (as in supersymmetric models with conserved parity), this particle can terminate the decay chain. In this case, sharp peaks in invariant-mass distributions may be scarce; they can be replaced by missing-energy signals. Quite often then, the signals for new physics have to be assessed against a significant background of Standard Model physics. The better the background is understood, the better the prospects for discovery. Once new physics is discovered, we will want to measure its properties precisely, via production cross sections, branching ratios and masses. A numerically precise understanding of the backgrounds and of theoretical aspects of luminosity measurement will be essential to this endeavor.
In some cases backgrounds can be understood without much theoretical input. For example, in the decay of a light Higgs boson to a pair of photons, the signal is a very narrow peak in the di-photon invariant mass. The QCD background, in contrast, is a smooth distribution which can be interpolated easily under a candidate peak. However, many signals involve much broader kinematic distributions, with final states including jets and missing energy in addition to charged leptons and photons. A classic example is the production of a Higgs boson in association with a boson at the Tevatron, with the Higgs decaying to a pair, and the decaying to a charged lepton plus a neutrino. For such signals, a much more detailed understanding of the backgrounds is typically required.
Many methods are available for computing Standard Model backgrounds at the leading order (LO) in QCD perturbation theory. For example, MADGRAPH , CompHEP  and AMEGIC++  automatically sum tree-level Feynman graphs for helicity amplitudes. Other programs, such as ALPGEN  and HELAC  are based on ‘off-shell’ recursive algorithms [6, 7]. For these algorithms, the building blocks are quantities in which at least one external leg is off shell (in contrast to the on-shell recursion relations [8, 9] described later in this article). These recursion relations were first constructed in the QCD context by Berends and Giele , and applied early on to matrix elements for backgrounds to top quark production .
For quite a while, simpler processes have been incorporated into parton-shower Monte Carlo programs that provide realistic event simulation at the hadron level. These programs, including PYTHIA  and HERWIG , perform parton showering and hadronization. They implement an approximation to the perturbative expansion that resums leading logarithms. More complex processes are now being included in this framework, using leading-order parton-level matrix elements provided by MADGRAPH or ALPGEN, for example, combined with a matching scheme  that avoids double-counting between the leading-order matrix elements and the parton shower. Yet these leading-order results often have a strong sensitivity to higher-order corrections. Gluon-initiated processes are particularly sensitive. For example, the cross section for production of the Standard Model Higgs boson via gluon-fusion at the LHC is boosted by roughly a factor of two as one goes from leading order to next-to-next-to-leading order (NNLO) in the perturbative QCD expansion .
For this reason, quantitative estimates for most processes require a calculation at next-to-leading order (NLO). Ideally, one would also match such results to a parton-shower Monte Carlo as well [15, 16]. For a growing list of processes, this has been achieved, particularly within the program MC@NLO .
NLO calculations require knowledge of both virtual and real-emission corrections to the basic process. The real-emission corrections are constructed from tree-level amplitudes with one additional parton present, either an additional gluon, or a quark–antiquark pair replacing a gluon in the LO process. They can be computed using the same tree-level techniques used for the basic process. In addition, we need a method for extracting the infrared singularities arising from integration of the real-emission contributions over unresolved regions of phase space. These singularities cancel against those in the virtual corrections or against counterterms from the evolution of parton distributions. Several such methods have been developed for use in generic NLO processes [17, 18, 19, 20]. Subtraction methods based on dipole subtraction [19, 20] are the most widely used today. A related subtraction method, based on so-called antenna factorization , has even been extended to NNLO . In this review, we will focus on techniques for computing the virtual one-loop corrections to processes. Once the virtual corrections are known, their incorporation into a numerical program for the full NLO result is straightforward, because they do not need to be integrated over unresolved regions of phase space, and all of their infrared singularities will be manifest.
The computation of one-loop virtual corrections for processes with multiple partons, plus electroweak particles, is the bottleneck currently limiting availability of NLO results. The state of the art for complete computations is processes with up to three objects — jets, vector bosons, or scalars — in the final state (see e.g. refs. [23, 24, 25, 26, 27, 28, 29]). A number of new approaches aimed at one-loop multi-parton amplitudes are currently under development. Some are already producing results applicable to processes with four final-state objects. These approaches fall into three basic categories: improved traditional (including semi-numerical) [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], purely numerical [43, 29], and on-shell analytic [44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60].
The semi-numerical approach of Ellis, Giele and Zanderighi  has already been used to compute loop corrections to the amplitudes involving a Higgs boson and four external partons , and to the six-gluon amplitude . Nagy and Soper  have proposed a subtraction method for use in a purely numerical evaluation of one-loop amplitudes, and have evaluated the six-photon helicity amplitudes purely numerically . The six-photon result has been reproduced very recently using both on-shell analytical and semi-numerical methods [62, 63]. Binoth et al.  have developed a combined algebraic/numerical algorithm for multi-parton amplitudes; the algebraic part has been used to determine the rational parts of several types of amplitudes, including the six-photon amplitude (for which the rational parts vanish). The full algorithm is also being incorporated into a NLO QCD framework by the GRACE collaboration . A purely numerical approach combining sector decomposition and contour deformations has recently been used to calculate tri-vector boson production . Denner and Dittmaier have developed a numerically stable method for reducing one-loop tensor integrals , which has been used in various electroweak processes, but also in NLO QCD computations, including the production of a top quark pair in association with a jet at hadron colliders .
In this review we describe on-shell analytic methods for one-loop computations. The ‘on-shell’ terminology means that essentially all information is extracted from simpler (lower-loop and lower-point) amplitudes for physical states. In contrast, the conventional Feynman-diagram approach requires building blocks with off-shell states. The on-shell approach effectively restricts the states used in a calculation to physical ones. The restriction falls on some internal as well as external states. The approach relies on three general properties of perturbative amplitudes in any field theory: factorization, unitarity, and the existence of a representation in terms of Feynman integrals. An earlier form of on-shell methods was used to compute the one-loop amplitudes for partons and (by crossing) + 2 jets . The latter have been implemented in the MCFM program . More recently, all six-gluon helicity amplitudes have been computed (primarily) with these methods [44, 45, 48, 50, 51, 57, 55, 56, 40].
On-shell methods provide a means for determining scattering amplitudes directly from their poles and cuts. Perturbative unitarity, applied to a one-loop amplitude, determines its branch cuts in terms of products of tree amplitudes . The unitarity method [44, 45] provides a technique for producing functions with the correct branch cuts in all channels. Its power is enhanced by relying on the decomposition of loop amplitudes into a basis of loop-integral functions. Matching the cuts with the cuts of basis integrals in this decomposition provides a direct and efficient means for producing expressions for amplitudes. It is most convenient to use dimensional regularization to handle both infrared and ultraviolet divergences in massless gauge-theory scattering amplitudes. We take the number of space-time dimensions to be . We can use fully -dimensional states and momenta in the unitarity method to obtain complete amplitudes, at least when all internal particles are massless . (In the older language of dispersion relations , amplitudes can be reconstructed fully from cuts in dimensions because of the convergence of dispersion integrals in dimensional regularization .) In most cases, however, it is much simpler to use four-dimensional states and momenta in the cuts. This procedure correctly yields all terms in the amplitudes with logarithms and polylogarithms , but it generically drops rational terms, which have to be recovered via another method.
The basic framework of the unitarity method was set up in refs. [44, 45]. In ref. , generalized unitarity was introduced as a means for simplifying cut calculations, by limiting the number of integral functions contributing to a cut. These techniques were applied to a variety of calculations in QCD and supersymmetric gauge theories [67, 68]. More recent improvements to the unitarity method, by Britto, Cachazo and Feng , use complex momenta within generalized unitarity, allowing for a simple and purely algebraic determination of all box integral coefficients. Britto, Buchbinder, Cachazo and Feng  have shown how triangle and bubble integral coefficients may be evaluated by extracting residues in contour integrals. This approach has been used to compute various contributions to the six-gluon amplitude [51, 57]. Very recently, Forde has proposed an efficient new approach to computing these coefficients .
The four-dimensional version of the unitarity method leaves undetermined additive rational-function terms in the amplitudes. These rational functions can be characterized by their kinematic poles. An efficient, systematic means for constructing these terms from their poles and residues is to use on-shell recursion relations [52, 53, 54, 55, 56], which were first devised by Britto, Cachazo, Feng and Witten, as a means for constructing tree-level amplitudes [8, 9]. In a related development, Brandhuber et al.  and Anastasiou et al.  have investigated how to determine the rational parts of amplitudes from -dimensional unitarity, extending earlier work [46, 71]. Another approach to computing rational terms uses a clever organization of Feynman diagrams, together with the observation that only a limited set of tensor integrals can contribute to the rational terms [40, 41]. Brandhuber et al.  have argued that rational terms can be obtained using a set of Lorentz-violating counterterms. Finally, Ossola, Papadopoulos and Pittau [42, 63] have developed a new loop-integral decomposition method, which we expect will mesh well with generalized unitarity techniques reviewed here.
Most of the recent development of on-shell techniques at one loop has focused on QCD amplitudes in which all of the external particles are gluons. Amplitudes containing massless external quarks pose no inherent difficulty. The same basis of integrals can be used, and individual amplitudes should have comparable analytic complexity. However, they have fewer discrete symmetries, so the number of amplitudes that need to be computed is larger for the same total number of partons. Amplitudes with external electroweak particles (vector bosons or Higgs bosons) in addition to QCD partons are of even greater phenomenological interest. Such amplitudes can also be built from the same basis of integrals. (For amplitudes with massive internal lines, such as in top-quark production, additional integrals are required.) In this review, in order to simplify the discussion, we focus primarily on amplitudes with external gluons only.
Recent years have also seen the emergence of on-shell methods at tree level, which have provided a basis for some of the recent developments at one loop. We will give some tree-level examples, but refer the reader to refs. [73, 74, 75] for a more extensive exposition.
It is also worth noting that the improved understanding of the structure of scattering amplitudes, stemming from on-shell methods, has also led to advances in more theoretical issues related to the AdS/CFT conjecture and in quantum gravity – see e.g. refs. .
This review is organized as follows. In section On-Shell Methods in Perturbative QCD we describe how the use of complex momenta is intrinsic to on-shell methods. In section On-Shell Methods in Perturbative QCD we review the BCFW on-shell recursion relations for tree amplitudes, giving a few examples. We turn to one-loop amplitudes in section On-Shell Methods in Perturbative QCD, where we describe the unitarity-based method and some recent improvements. In section On-Shell Methods in Perturbative QCD, we show how on-shell recursion is an efficient way to construct rational terms in one-loop amplitudes. We give our conclusions in section On-Shell Methods in Perturbative QCD, and comment on the prospects for evaluating new amplitudes of phenomenological interest at the LHC with these methods.
Multi-parton amplitudes in QCD are functions of a large number of variables, describing polarization states, color quantum numbers, and kinematics. It is important to disentangle the dependence on all these variables to as great an extent as possible. In this section we focus on techniques for separating and simplifying the kinematic behavior. The polarization-state dependence may be handled efficiently by computing helicity amplitudes, using the spinor-helicity formalism for external gluons [77, 78].
The color dependence can be organized using the notion of color-ordered subamplitudes, or primitive amplitudes. Primitive amplitudes are defined for specific cyclic orderings of the external partons, but carry no explicit color indices. They can be computed using ‘color-ordered’ Feynman diagrams. Such diagrams can be drawn on the plane for the specified external ordering of the partons. (There are also restrictions on the direction of fermion flow through the diagrams in loop amplitudes.) The color indices have been stripped from all the vertices and propagators for these diagrams. The full amplitude can be assembled from the primitive amplitudes by dressing them with appropriate color factors.
For example, for -gluon amplitudes, the tree-level color factors have the form of single traces, , where is an generator matrix in the fundamental representation, for gluon [79, 78]. The coefficients of these color structures define the tree-level primitive amplitudes . At one loop, the -gluon color factors can be either single traces — with an additional factor of present — or double traces. The coefficients of the single traces, which would dominate in the large- limit, are given directly by one-loop primitive amplitudes. The coefficients of the double traces are given by sums over permutations of primitive amplitudes [80, 44, 71]. Therefore we can focus on the coefficients of , which we denote here by . (Elsewhere in the literature, they are often denoted , to distinguish them from the double-trace coefficients for .) The one-loop primitive amplitudes have the same symmetry under cyclic permutations of the external legs as the tree amplitudes . Similar results hold when external quarks — fermions in the fundamental representation — are present as well . We refer the reader to previous reviews [78, 71] for a more complete description of helicity and color decompositions.
In this review, we concentrate on methods for computing the building blocks, primitive helicity amplitudes, which are functions only of the kinematic variables. A key property of primitive amplitudes is that their singularities, branch cuts (at loop level) as well as factorization poles, depend only on a restricted set of kinematic variables. This set consists of the squares of sums of cyclically adjacent momenta, , where all indices are taken modulo . Furthermore, the singularities are determined by lower-loop primitive amplitudes in the case of cuts, and by lower-point primitive amplitudes in the case of poles. This reducibility suggests the possibility of developing a recursive computational framework.
The singularity information can be accessed more easily if we define primitive amplitudes for suitable complex, yet on-shell, values of the external momenta. This simple observation came out of Witten’s development of twistor string theory , and many subsequent papers sparked by that article. Witten’s work demonstrated a remarkable simplicity for tree amplitudes, and suitable parts of loop amplitudes, when they were mapped into the twistor space of Penrose  in which twistor strings propagate. This space can be defined as a ‘half Fourier transform’ of the space of left- and right-handed two-component (Weyl) spinors associated with a massless vector, . The left-handed spinors are Fourier transformed, while the right-handed spinors remain as coordinates.
Here we will not rely directly on any concrete properties of twistor space. However, two conceptual ideas from that work underpin our approach:
Use two-component spinor variables as the independent variables for scattering processes, rather than four-component momenta.
Treat opposite-helicity spinors as independent variables. For real Minkowski momenta, there is a complex-conjugation relation between the two. This treatment requires momenta to be generically complex.
Complex momenta are certainly not a new notion. Wick rotation allows the analytic properties of amplitudes for real Minkowski momenta to be deduced by continuation from the Euclidean region, in which the time components of Minkowski momenta are imaginary. Similarly, the complex analyticity of the matrix, expressed as a function of the Lorentz-invariant products , depends implicitly on having complex momenta . On the other hand, arriving at complex momenta by thinking of the spinor variables as fundamental, leads to new ways of organizing the kinematic properties of helicity amplitudes.
The first hint that this approach might be useful comes from investigating three-point helicity amplitudes , displayed in fig. 1(a). For real momenta, an on-shell process with three external massless legs is always singular, because for all three pairs of legs. These conditions force the three momenta to be collinear with each other — if they are real — which in turn makes all kinematic quantities vanish. Using complex momenta to define the amplitude for three massless particles is also not a new idea. For example, Goroff and Sagnotti  used complex momenta to define non-singular three-graviton kinematics, in their on-shell computation of the two-loop divergence in pure Einstein gravity. However, there is a natural way to take the kinematics to be complex using spinor variables, which meshes neatly with the structure of helicity amplitudes.
First we introduce a shorthand notation for the two-component (Weyl) spinors associated with an -parton process :
It is also convenient to describe the spinors with a ‘bra’ and ‘ket’ notation,
One can always reconstruct the momenta from the spinors, using the positive-energy projector for massless spinors, , or
Equation (2.3) shows that a massless momentum vector, written as a bi-spinor, is simply the product of a left-handed spinor with a right-handed one. This result is also valid for complex momenta.
Lorentz-invariant spinor products can be defined using the antisymmetric tensors and for the factors in the Lorentz algebra, :
(The sign of in eq. (2.5) matches that in most of the QCD literature; in much of the ‘twistor’ literature, e.g. refs. [82, 49, 8, 9], the opposite sign is used.) These products are antisymmetric, , . The usual momentum dot products can be constructed from the spinor products using the relation,
We will also use the notation
where the sums run over cyclically ordered labels between and , as well as,
Parity acts on a helicity amplitude by flipping the sign of all helicities. This symmetry may be implemented by exchanging the left- and right-handed spinor products in the amplitude, .
For real momenta, and are complex conjugates of each other. Therefore the spinor products are complex square roots of the Lorentz products,
In this case, if all the vanish, then so do all the spinor products. However, for complex momenta, eq. (2.10) does not hold. In fact, it is possible to choose all three left-handed spinors to be proportional, , , while the right-handed spinors are not proportional, but obey the relation, , which follows from momentum conservation, , and eq. (2.3). Then
while , and are all nonvanishing.
For this kinematical choice, the tree-level primitive amplitude for two negative helicities and one positive helicity, , is nonsingular, even though all momentum invariants vanish according to eq. (2.6). (We assign helicities to particles under the assumption that they are outgoing; if a particle is incoming, its true helicity is opposite to the label.) For three gluons, can be evaluated using the three-gluon-vertex in the color-ordered Feynman rules , as
in terms of null reference momenta , which may be chosen to simplify the computation. In eq. (2.12), if we choose and , then . Upon using a Fierz rearrangement and momentum conservation, the lone surviving term in eq. (2.12) becomes,
In this expression, only gluons and have negative helicity; the remaining gluons have positive helicity. For , these amplitudes are well-defined for real momenta. However, for , formula (2.15) only makes sense for complex momenta of the type (2.11).
There is a class of complex momenta conjugate to eq. (2.11), for which
while , and are all nonvanishing. Such momenta make the parity-conjugate three-point amplitude,
well-defined. When the amplitude appears in the ‘wrong’ kinematics (2.17), it should be set to zero, because more vanishing spinor products appear in the numerator than in the denominator.
For any on-shell complex momenta, the all-positive amplitude vanishes,
This result follows easily by choosing all reference momenta to be the same, which forces to vanish for all pairs . The same result holds for , of course. In a similar fashion, the three-point amplitude for a pair of massless quarks (which must carry opposite helicity) plus one gluon is well-defined and nonzero using the kinematics (2.11) if the gluon helicity is negative, and using the kinematics (2.17) if it is positive. It vanishes for the ‘wrong’ kinematics. These rules will become important later for evaluating generalized unitarity cuts and recursive diagrams containing three-point vertices.
We have seen how suitable complex kinematics can simplify the structure of the three-point amplitude. Much more significantly, however, complex kinematics allow the exploration of generic factorization singularities of on-shell amplitudes, and the use of factorization information to reconstruct the amplitude, as was recognized at tree-level by Britto, Cachazo, Feng and Witten (BCFW) . We will review the BCFW recursion relations in the next section. Here we shall just discuss the simple example of the Parke-Taylor amplitudes. The idea is to embed a tree amplitude into a one-complex-parameter family of on-shell amplitudes . The rationale for introducing the complex parameter is that it allows us to apply the full power of complex variable theory to reconstruct amplitudes from their poles. The simplest way to introduce this parameter is by modifying, or ‘shifting’ the momenta of just two of the partons, in a way that keeps them on shell. Because of eq. (2.3), the two momenta will automatically remain on shell if we shift the spinor variables, and then define the shifted momenta to be the products of the new left- and right-handed spinors.
We define the shift to be ,
where is a complex parameter. The shift leaves untouched , , and the spinors for all the other particles in the process. Under the shift, the corresponding momenta shift as,
which preserves their masslessness, , as well as overall momentum conservation.
Suppose we apply the shift, , , to the MHV amplitude (2.16), for the case (and ). Because the formula contains only right-handed spinors, the only induced dependence arises from terms containing . The spinor product is shifted to The spinor product is unaffected, because , but by antisymmetry. Thus the MHV amplitude becomes
If we divide this shifted amplitude by , we get a function with two poles at finite , one at the origin and one at , as shown in fig. 2. The function behaves like as , so the integral around the circle at infinity vanishes,
Cauchy’s theorem then guarantees that the two poles at finite have equal and opposite residue.
The residue of the pole at the origin is simply the MHV amplitude we want to compute, . The residue of the second pole is determined by the factorization properties of the amplitude, because it occurs at the point that the intermediate momentum goes on shell, . That is, at the value we have,
A hat on a kinematic variable means that a shift of the form (2.20) has been applied to the variable, with then fixed to the value that puts an intermediate momentum on shell. The key point is that, because an intermediate state goes on shell, the amplitude factorizes at this point into the product of two lower-point amplitudes, multiplied by the diverging propagator. In general there is a sum over the helicity of the intermediate state, for intermediate gluons.
Equating the residue at to the negative of the one at the origin, we have
In the second step we evaluated the residue in brackets, and used the vanishing of in eq. (2.19) to reduce the helicity sum to a single term.
Equation (2.25) is the prototype for the tree-level BCFW recursion relation reviewed in section On-Shell Methods in Perturbative QCD. In general, there will be several terms on the right-hand side, corresponding to different nontrivial factorization channels which can be probed for suitable values of . The MHV amplitudes are unique in having no multi-particle poles, which is related to the vanishing of -gluon amplitudes with fewer negative helicities,
For this reason, the recursion relation (2.25) in the MHV case has but a single term.
We can further evaluate eq. (2.25) using the explicit form (2.16) of the MHV amplitudes, as a simple exercise in manipulating the hatted variables that appear, and as an on-shell recursive proof of the formula. (As an historical note, the first proof of eq. (2.16) employed recursion relations of the off-shell variety .) Inserting the forms of the -point and three-point amplitudes, eq. (2.25) becomes,
We need to continue the spinors for into those for . Because a pair of such spinors appears, we pick up a minus sign. (Determining the sign for the case of an intermediate quark line is more subtle.)
Because the shift leaves and alone, we can let and in eq. (2.27). Because the shifted momentum is proportional to , a hatted momentum appearing in a right-handed spinor product with , or in a left-handed spinor product with , can have its hat removed as well. For this reason, it is very convenient to use factors of and to clean up other spinor products containing , inserting them into the numerator and denominator of expressions as needed. In the present case, the necessary factors are already present. Equation (2.27) becomes,
The final form indeed matches the expression (2.16), thus confirming it recursively.
Figure 1(b) shows a second class of configurations for which complex kinematics are useful, namely generalized unitarity conditions, which will be discussed in more detail in section On-Shell Methods in Perturbative QCD. At one loop, conventional unitarity constraints on amplitudes are analyzed by putting two intermediate states on shell. For example, if legs 1, 2, 3 and 4 in fig. 1(b) are outgoing, and legs 5, 6, 7 and 8 are incoming, then the conventional cut in the 1234 channel is computed by imposing . This constraint can be realized with all momenta real in Minkowski space. It can then be interpreted as a particle scattering process, followed by a particle scattering.
Generalized unitarity corresponds to requiring more than two internal particles to be on shell. Often these constraints cannot be realized with real Minkowski momenta. Suppose we try to add the condition to the standard cut constraints in fig. 1(b). The problem is that processes are forbidden for real, non-collinear massless momenta, although processes are allowed. After setting , fig. 1(b) contains a four-point subamplitude in which leg is incoming, and legs 1 and 2 are outgoing with nonzero. We can arrange for this to be a real Minkowski process by taking to be incoming. But then the subamplitude below it in the figure has only leg incoming, and legs , 3 and 4 outgoing. It cannot correspond to a real on-shell process, as long as is nonzero.
Similarly, we cannot impose on top of , in the process , and still have real momenta. Typically then, one needs to allow for complex momenta in order to implement generalized cut conditions. As an even simpler example, if we consider a quadruple cut with fewer than the eight external momenta shown in fig. 1(b), then at least one tree amplitude will have only three external legs, and this fact alone dictates complex momenta.
On the other hand, generalized cut conditions can sometimes be satisfied with only real momenta. In fig. 1(b), suppose now that the scattering process has legs 4, 5, 6 and 7 incoming, and legs 8, 1, 2 and 3 outgoing. Then it is possible to solve all four internal constraints, , with real Minkowski momenta, corresponding to scattering that proceeds from the lower left to the upper right of the figure; that is, , followed by and , followed by .
In this section we describe the construction of tree amplitudes via on-shell recursion. As alluded to in the previous section, the BCFW recursion relation [8, 9] is based on introducing a complex-parameter-dependent shift of two of the external massless spinors, as given in eq. (2.20). The construction of tree amplitudes via on-shell recursion essentially amounts to generalizing and reversing the steps in section On-Shell Methods in Perturbative QCD. Instead of starting with a known amplitude, and verifying its analytic properties under the parameter-dependent shift, we use such properties to systematically construct unknown amplitudes.
Following the same procedure as for the MHV case in eq. (2.22), for generic amplitudes we define an analytically continued amplitude,
which remains on-shell, but depends on the complex parameter . If is a tree amplitude, then is a rational function of . The physical amplitude is given by .
Following the MHV case (2.23) consider the contour integral,
where the contour is taken around the circle at infinity. If as , the contour integral vanishes and we obtain a relationship between the physical amplitude, at , and a sum over residues for the poles of , located at ,
To determine the residues at each pole, we use the general factorization properties that any amplitude must satisfy as an intermediate momentum goes on shell, . In general, the residue is given by a product of lower-point on-shell amplitudes. Only a subset of the possible factorization limits for an amplitude are explored by the -dependent shift. Poles in the plane can develop in any channel that has leg on one side of the pole and leg on the other side, because the intermediate momentum is -dependent. Thus we can solve the on-shell condition,
The solution is
A contribution to the recursion relation from this residue is illustrated diagrammatically in fig. 3. To get the precise form of the contribution, using eq. (3.3), we need to evaluate the residue (as we did in eq. (2.25) for a special case),
Generically we have a double sum, labeled by , over recursive diagrams, with legs and always appearing on opposite sides of the pole. There is also a sum over the helicity of the intermediate state. The squared momentum associated with the pole, , arising from eq. (3.6), is evaluated in the unshifted kinematics. The on-shell tree amplitudes and are evaluated in kinematics that have been shifted by eq. (2.20), with . The shifted momenta for such kinematics are indicated by hats.
Equation (3.7) may alternatively be derived by expressing as a sum over poles multiplied by their residues (under our assumption that as ). This representation, valid for all , is
By setting in this formula, we recover the recursion relation (3.7).
To derive the recursion relation, we assumed that the amplitude vanishes as . If all the external particles are gluons, then the validity of this assumption depends only on the helicity of the two shifted legs. There are four different cases, which we can label by . A shift of type does not generally make vanish at infinity. As an example, the MHV amplitude (2.16) behaves as either or as , because of the factor of in the numerator. The case is the simplest to analyze because each individual Feynman diagram vanishes separately . Shifts of type and also make the amplitude vanish as , but here cancellations between Feynman diagrams are required. The vanishing behavior has been proven using generalizations of the shift (2.20) that affect three or more momenta [88, 75].
The on-shell recursion relation (3.7) contains spinor products involving hatted momenta. For the purposes of numerical evaluation, we can leave the amplitudes in this form, because the complex hatted momenta are built from well-defined spinors, whose inner products can be computed from eqs. (2.4) and (2.5). However, for analytic purposes it is useful to eliminate the hatted momenta in favor of external momenta. We can make use of the following relations,
To simplify the expressions we used, for example,
where the last term vanishes by a Fierz identity. The product is homogeneous in the spinors carrying the intermediate momentum . Because of this, any remaining factors of and can be paired up to give factors of
Recursive diagrams containing three-point amplitudes often vanish because the ‘wrong’ kinematics are present. In general, if a shift is used, and the recursive diagram contains a three-vertex with two positive helicities, one of which is , then the diagram vanishes. The reason is that the spinor is unaffected by the shift, so its product with the spinor for the other external leg in the three-point amplitude, , remains nonvanishing. Therefore , and all of the left-handed spinor products, must vanish, and so the three-vertex with two positive helicities vanishes, as discussed in section On-Shell Methods in Perturbative QCD. Similarly, three-vertices with two negative helicities can also be dropped, when one of the three legs is .
As a first example, consider the amplitude . This amplitude can be constructed recursively from the three-point amplitudes given in eqs. (2.15) and (2.18). As discussed above, for the shift (a shift) the amplitude vanishes for large . Using this shift, there are two potential terms in the recursion relation, corresponding to diagrams (a) and (b) in fig. 4.
The first of these diagrams,
vanishes because of the ‘wrong’ kinematics discussed above.
This form is already satisfactory for the purpose of evaluating the amplitude numerically. It is, however, a useful exercise to eliminate hatted momenta in favor of unhatted external momenta. Applying the substitutions (3.9) and simplifying the expression for diagram (b), we find
in agreement with the MHV formula (2.16).
Interestingly, using the on-shell recursion relations, the four-point amplitude, and indeed all tree amplitudes, can be constructed from the on-shell three-vertices. The four-point vertex in the Yang-Mills Feynman rules is unnecessary. On the other hand, the four-point vertex is related by gauge invariance to the three-point vertex. Gauge invariance is necessary to decouple unphysical states. The recursion relations rely on the fact that such states are not present in factorization limits. In this way, they implicitly incorporate the correct four-point vertex.
Consider now the less trivial example of a six-point next-to-MHV (NMHV) amplitude, . Using a shift (a shift) yields the two recursive diagrams in fig. 5,
The first of these diagrams gives,
Similarly, diagram (b) in fig. 5 is given by
It is interesting that this kind of representation of the amplitude is intimately connected  to the forms in which tree amplitudes appear in the infrared singularities of certain one-loop amplitudes . This feature is related to the appearance of denominator factors such as at one loop, where they arise in the reduction of various loop integrals to a basic set of integrals. They can be thought of as spinor ‘square roots’ of certain Gram determinants.
The expression vanishes on a subspace of phase space. For example, when is any linear combination of and , it vanishes using the massless Dirac equation, . Note that by momentum conservation, and so the latter form also vanishes on the same subspace. This subspace does not correspond to physical factorizations of the amplitude, so and each have spurious singularities on it. However, their sum, the full amplitude , is nonsingular. In principle, a numerical program should check for small values of such spurious denominator factors, in order to avoid round-off errors. (Near a spurious singularity for the above representation (3.16), one could for example make use of a different shift (2.20), whose spurious singularities are located elsewhere.) On the other hand, the singularity is fairly mild, because only one power of appears in the denominator.
Curiously, the introduction of denominators such as , gives a much more compact representation of amplitudes (for ) than forms without such denominators. The compactness is basically due to the more manifest factorization properties of amplitudes constructed via on-shell recursion relations. For example, the representation (3.16) makes manifest the correct behavior as any pair of adjacent momenta become collinear, , because the spinor products are square roots of momentum invariants, as described in eq. (2.10).
Many applications of these techniques have already been carried out at tree level. In the case of -gluon amplitudes, a closed-form formula for the ‘split helicity’ configuration,