1 0 D ec 2 01 9 Extended Projection Method for Massive Fermion *

Tensor reduction is important for multi-loop amplitude calculation. And the projection method is one of the most popular approaches for tensor reduction. However, projection method could be problematic for amplitude with massive fermions due to the inconsistency between helicity and chirality. We propose an approach to extend the projection method to reduce the loop amplitude containing fermion chain with two massive spinors. The extension is achieved by decomposing one of the massive spinors into two specific massless spinors, ”null spinor” and ”reference spinor”. Then the extended projection method can be safely implemented for all the processes including the production of massive fermions. Finally we present the tensor reduction for the virtual Z boson decaying to top-quark pair to demonstrate our approach.


Introduction
After the discovery of the Higgs boson, the continuous improvement of experiment accuracy at the CERN Large Hadron Collider (LHC) demands the precise theoretical predictions to high order corrections. However, the high order corrections could still be seriously challenging due to the complicacy of multi-loop Feynman diagrams. One of the challenging tasks is to reduce the loop amplitude into linear combination of master integrals.
At the multi-loop level, the achievement of reduction procedure is much harder than the one-loop case. To improve the efficiency, the reduction for multi-loop amplitude can be conventionally separated into two steps, i.e. the tensor reduction and the scalar integrals reduction using integration by part (IBP) identities. For the IBP reduction, many algorithms and codes have been developed after decades of effort [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. On the other hand, tensor reduction is also important. During past decades, many algorithms for tensor reduction have been proposed [39][40][41][42][43][44][45][46]. For some complicated processes, such as full next-to-next-to-leading order QCD correction to single-top production [47], the increasing number of form factors makes the coefficients hard to obtain. And for two-loop five-gluon or six-gluon amplitude, the corre-sponding system of equations for the coefficients can be very complicated to be solved [48,49]. Moreover, some complicated processes, e.g. e + e − → Z ⋆ → tt, can confront serious problem during tensor reduction. This indicates that the further investigation on tensor reduction is still needed.
The projection method [39,40] is one of the most popular approaches for tensor reduction. During past decades, many important researches have been done by using projection method, such as high order QCD corrections to the Higgs production [50][51][52][53] and vector boson production [54,55]. However, for some processes containing fermion chain with two massive spinors, the projection method could be sabotaged by the inconsistency between helicity and chirality, which will be explicitly shown in next section.
In this paper, based on the massive spinor decomposition [56][57][58], we propose to extend the projection method to reduce the loop amplitude for any process including production of massive fermions. The massive spinor can be decomposed by defining "null spinor" and "reference spinor", which have completely different formulas of equation of motion and polarization summation compared to the regular spinor. Then for the processes containing massive spinors the projection method can be safely used.
This paper is organized as follows. In section 2 we briefly review the standard projection method and demonstrate the problem due to the lack of massless spinor. In section 3 based on the massive spinor de-composition we introduce "null spinor" and "reference spinor", which can be used to extend the projection method for all processes. In section 4 and 5, we take oneloop and two-loop diagrams for virtual Z boson decaying to top-quark pair to demonstrate the effectiveness of our approach, respectively. The conclusion is presented in the last section.

Standard Projection method
In projection method, the loop amplitude can be expressed as the linear combination of several monomials. In each monomial the Lorentz structure is composed of the spinors and polarization vectors associated with the contracted momenta, while the remnant factors including coupling constants and scalar products of momenta will not affect the tensor reduction. Then by trimming off the chirality for spinors from the Lorentz structure we can obtain the primitive amplitude. Therefore, the loop amplitude can be decomposed as where X is chirality index and p indicates the different primitive amplitudes. M p,X is the Lorentz structure for certain chirality X, and C p,X is the relevant coefficient. For convenience we can define a map for each primitive amplitude from helicity state H to chirality state X f p : H → X, s.t. P H M p,X = 0, where P H is the helicity projection operator. For instance, the explicit map for the primitive amplitudē where k 2 1 = k 2 2 = 0, can written as Furthermore, for the primitive amplitude with one massive spinor, for examplē where k 2 1 = 0 and k 2 2 = 0, the map still can be constructed as Obviously, the map can be established based on at least one massless spinor, which has equivalence relation between helicity and chirality and further can be used to fix the chirality for the relevant fermion chain by using anti-commute γ 5 scheme. However, the map does not exist for the primitive amplitude containing two massive spinors in the same fermion chain, which is just the case that projection method fails.
Therefore we can obtain where H indicates certain helicity state for the spinors. f −1 p (X) is not the inverse of f p but only represents the set of helicity states that can be mapped to certain chirality state X.
Consequently the amplitude A can be expressed as linear combinations of helicity primitive amplitudes, Now one needs only the tensor reduction on the primitive amplitude M p , and the loop amplitude A can be reconstructed by implementing the helicity projection P H and summing up all primitive amplitude choices p and helicity states H. Meanwhile the above derivation also presents the formula for helicity amplitude In order to make tensor reduction on the primitive amplitude M p , one needs to find a complete set of linear independent form factors {F i }. Then M p can be projected to form factors To obtain the explicit expression of coefficient d i,p , both sides of Eq.(10) can be multiplied by the conjugate form factor F † j . Then the form factor matrix can be defined as The coefficient d i,p can be obtained from the inversion of matrix M Finally the amplitude A can be expressed by the linear where Here P H F i is independent of loop momenta, and it can be further expressed in spinor representation. And its coefficient c i,H contains the scalar integrals, which can be further reduced by IBP method.

Projection method for massive fermion chain
In the above section it can be seen that the problem in projection method is due to the lack of massless spinor. One of the convenient approaches is to decompose the massive momentum k by introducing reference momentum k r [56][57][58], where k 2 0 = k 2 r = 0 and k 2 = m 2 k . And the massive spinor can be decomposed as Here we found in fact that on the right-hand side of each above equation the two terms can be defined as two special spinors so that Explicitly Now we define u r and v r as "reference spinors", although they are not orthogonal to the "null spinors" u 0 and v 0 . The polarization summation formula can be written as Besides, a set of non-trivial Dirac equations for null spinors and reference spinors can be found as Then since one of the fermion spinors becomes massless, the map from helicity state to chirality state for primitive amplitude can be constructed. Finally the projection method can be directly implemented on the decomposed primitive amplitudes.

Conclusion
In this paper, based on the massive spinor decomposition, we proposed an extended projection method to reduce the loop amplitude containing fermion chain with two massive spinors. By decomposing the massive spinor to null spinor and reference spinor, this approach can overcome the difficulty of inconsistency between helicity and chirality. To demonstrate the effectiveness of the extended projection method in high order correction, we present the tensor reduction on both one-loop and two-loop amplitudes for virtual Z boson decaying to top-quark pair. In the future, this approach can be implemented in more complicated processes including the production of multiple massive fermions.