Holonomies of gauge fields in twistor space 6: incorporation of massive fermions

Following the previous paper arXiv:1205.4827, we formulate an S-matrix functional for massive fermion ultra-helicity-violating (UHV) amplitudes, i.e., scattering amplitudes of positive-helicity gluons and a pair of massive fermions. The S-matrix functional realizes a massive extension of the Cachazo-Svrcek-Witten (CSW) rules in a functional language. Mass-dimension analysis implies that interactions among gluons and massive fermions should be decomposed into three-point massive fermion subamplitudes. Namely, such interactions are represented by combinations of three-point UHV and next-to-UHV (NUHV) vertices. This feature is qualitatively different from the massive scalar amplitudes where the number of involving gluons can be arbitrary.


Introduction
In a recent paper [19] we construct an S-matrix functional for massive scalar amplitudes, i.e., amplitudes of an arbitrary number of gluons and a pair of massive scalar particles, in the recently proposed holonomy formalism [20,21]. In [19] we show that the on-shell constructibility of massive scalar amplitudes can naturally be implemented by means of offshell continuation of Nair's superamplitude method [22] once a massive holonomy operator is defined. We also show that a careful study of the massive holonomy operator leads to a novel color structure of the massive scalar amplitudes. A main purpose of the present paper is to extend these results to the case of massive fermions in the same framework.
The organization of this paper is as follows. In section 2 we review essential results of the previous paper [19] on the massive scalar amplitudes in the holonomy formalism. We first write down the definitions of the massless and massive holonomy operators. We then present an S-matrix functional for the massive scalar tree amplitudes, focusing on the amplitudes of positive-helicity gluons and a pair of massive scalars, the so-called ultra-helicity-violating (UHV) massive scalar amplitudes.
In section 3 we apply these results to the massive fermion amplitudes. We begin with clarifying our notation of fermions in the spinor-helicity formalism. We then apply off-shell continuation of Nair's superamplitude method such that it lead to tree-level massive fermion UHV amplitudes (scattering amplitudes of an arbitrary number of positive-helicity gluons and a pair of massive fermions) in a form that is consistent with the literature. We also obtain an S-matrix functional for the massive fermion UHV amplitudes.
Mass-dimension analysis on the massive fermion UHV amplitudes implies that the number of gluons involving the amplitudes should be one. This condition means that interactions among gluons and massive fermions should be decomposed into the three-point UHV vertices (with a positive-helicity gluon) and the three-point next-to-UHV (NUHV) vertices (with a negative-helicity gluon). In section 4, for clarification of our arguments, we first construct the massive fermion NUHV amplitudes for arbitrary number of gluons and see how the above constraint is imposed. Lastly, we present a brief conclusion.

S-matrix for massive scalar UHV amplitudes
In this section we review the formulation of an S-matrix functional for the massive scalar UHV amplitudes, i.e., scattering amplitudes of an arbitrary number of positive-helicity gluons and a pair of massive scalar particles, at tree level in the framework of the holonomy formalism, recapitulating the results of the recent study [19].
The massless holonomy operator The original "massless" holonomy operator for gluons is defined as [20] Θ (A) R,γ (u) = Tr R,γ P exp where A is called the comprehensive gluon field. This is expressed as a bialgebraic operator

2)
3) where g is a dimensionless coupling constant and u i (i = 1, 2, · · · , n) denotes the twocomponent spinor momentum for the i-th gluon. (u i u j ) represents a scalar product of the spinor momenta: where the indices A, B take a value of (1, 2) and ǫ AB denotes the rank-2 Levi-Civita tensor. The physical (creation) operators of positive and negative helicity gluons are given by a where Kronecker's deltas show that the non-zero commutators are obtained only when i = j. The remaining commutators, those expressed otherwise, all vanish.
The color degree of freedom can be attached to the physical operators a where t c i 's are the generators of the U(N) gauge group in the R-representation. The symbol P in (2.1) denotes an ordering of the numbering indices. The meaning of the action of P on the exponent of (2.1) can be written explicitly as where h i = ± = ±1 (i = 1, 2, · · · , m) denotes the helicity of the i-th gluon.
In (2.1), γ represents a closed path on the physical configuration space on which the set of gluon operators are defined. S n denotes the rank-n symmetric group. The fundamental homotopy group of C (A) is given by the braid group, Π 1 (C (A) ) = B n . The trace Tr γ in (2.1) is taken over the the generators of the braid group (or the elements of Iwahori-Hecke algebra, see [21] for details on this point) and is called the braid trace. It can be realized by a sum over the permutations of the numbering indices, i.e., where each of the permutations is denoted by σ = 2 3 · · · n σ 2 σ 3 · · · σ n . Explicitly, the braid trace Tr γ over the exponent (2.8) can be expressed as A quintessential point in the holonomy formalism is the mathematical fact that a linear representation of a braid group is equivalent to a monodromy representation of the Knizhnik-Zamolodchikov (KZ) equation. Such a monodromy representation can be given by a holonomy of the so-called KZ connection. The comprehensive gauge one-form (2.2) is a variant of the KZ connection in a sense that it satisfies the infinitesimal braid relations to ensure the integrability of the comprehensive gauge one-form. For details on these fundamental issues, see [20,21]; see also [23,24] for recent studies on KZ connections and two-dimensional holonomies.
The massive holonomy operator Following these basic ideas, we now construct a holonomy operator for a comprehensive gauge one-form which incorporates the helicity-zero (or spin-zero) scalar operators a i . Such a one-form can be defined as where g denotes the dimensionless gauge coupling constant as before. The holonomy operator for B is then defined and calculated as [19] Θ (B) where we use the abbreviated notation (i j) ≡ (u i u j ). In the above expression the massive scalar operators are specified by a pair of the numbering indices (1, r). Accordingly h i = ± = ±1 (i = 2, 3, · · · , r − 1) denotes the helicity of the i-th gluon.
Comments on the color structure of the massive holonomy operator As mentioned in the introduction, the color structure of the massive holonomy operator is qualitatively different from that of the massless holonomy operator in terms of the final realization of the braid trace. In the original massless case, the sum over the permutations of the numbering indices (2.10) is explicit while in the massive case, as shown in (2.14), there appears no explicit sums over permutations. As discussed in [19], this is due to the fact that the calculation of (2.14) is carried out by use of the relation where γ 1r and γ r1 denote open paths on C (B) which compose the closed path, γ = γ 1r γ r1 .
The relation (2.18) is a mathematical equation regarding products of iterated integrals over the logarithmic one-forms ω ij 's. Use of this relation implies that the homogeneous sum (2.17) or the braid trace for Θ (B) R,γ (u) can be absorbed into the expression of (2.14).

Supersymmetrization
In order to relate the massive holonomy operator to massive scalar amplitudes, we need to define the massive holonomy operator in supertwistor space, as in the case of gluon amplitudes. In the massless case the supersymmetrization is carried out by replacing the physical operators a where dµ(p i ) is the ordinary Lorentz-invariant measure for the null momentum p µ i . This is also called the Nair measure when it is written in terms of the spinor momenta [22]. The operators a (ĥ i ) i (x, θ) are physical operators that are defined in a four-dimensional N = 4 chiral superspace (x, θ) where xȦ A = (σ µ )Ȧ A x µ denote coordinates of four-dimensional spacetime and θ α A (A = 1, 2; α = 1, 2, 3, 4) denote their chiral superpartners with N = 4 extended supersymmetry. (Here σ µ is given by σ µ = (1, σ i ), with σ i (i = 1, 2, 3) being the Pauli matrices.) These coordinates emerge from homogeneous coordinates of the supertwistor space CP 3|4 , represented by (u A , vȦ, ξ α ), that satisfy the so-called supertwistor conditions Upon the supersymmetrization, there arise superpartners of the gluon creation operators. The full supermultiplets can be expressed as These operators are characterized by the number of ξ α i = θ α A u A i or the degrees of homogeneities in u A i 's. The number is in one-to-one correspondence with the helicityĥ i = (0, ± 1 2 , ±1) of the superpartner of interest. This can be easily seen from the definition of the helicity operatorĥ (2.22) Notice that we put the hat ofĥ i simply to distinguish it from the non-supersymmetric version h i = ±.
Use of the supermultiplets (2.21) enables us to define gluon operators without introducing the conventional polarization/helicity vectors. This method is known as Nair's superamplitude method [22]. In [19] we show that this prescription is also applicable to the system of gluons and a pair of massive scalars. In the following we review such a massive extension.

Off-shell continuation of Nair's superamplitude method
In carrying out the supersymmetrization of the massive holonomy operator (2.14), we need to consider off-shell version of the scalar operator a (0) i (ξ i ) in (2.21). In the massive spinor-helicity formalism [18], off-shell continuation of the null spinor-momentum u A is defined as where η A is a reference null spinor and m denotes the mass of the massive spinor momentum u A . The complex conjugate of (2.23) is also necessary to define the full massive four-momentum p AȦ = u A ūȦ. We then introduce another set of supertwistor variables (w A , πȦ, ζ α ) such that the supertwistor conditions are satisfied (α = 1, 2, 3, 4).

Using the new Grassmann variable
where we shall specify the numbering index to i = 1, n for massive scalars. Notice that the degrees of homogeneities in u i 's remains the same in the off-shell continuation (2.25). Thus we can naturally interpret a (0) i (ξ i , ζ i ) as massive scalar operators. The chiral superspace representation of the massive operators can be expressed as A supersymmetric version of the massive holonomy operator can be constructed by use of the the expressions (2.19) and (2.26) for physical operators of gluons and massive scalars, respectively. In other words, we can obtain the supersymmetric massive holonomy operator Θ

Choice of reference spinors and functional derivation of the UHV vertex
We now specify the reference null-vectors for the massive scalars as Namely, we choose a specific parametrization of p 1 and p n by Notice that this parametrization is qualitatively different from off-shell prescription for virtual gluons where we set all reference null-vectors identical.
We now introduce a generating functional In terms of this generating functional the massive scalar UHV vertex can be obtained as where a (±)c and a (0) denote sets of source functions of gluons and massive scalars, respectively. .
The integral measures in (2.33) are given by the ordinary Lorentz invariant measures The momenta of the massive scalars are denoted by p 1 and p n , satisfying In deriving the explicit form (2.35), we use the specified reference spinors (2.29) and the Grassmann integral This Grassmann integral guarantees that only the UHV-type vertices survive upon the evaluation of functional derivatives in (2.32). This structure is analogous to the CSW rules, or the MHV rules, in the calculation of gluon amplitudes [1].
The UHV rules and the massive scalar UHV tree amplitudes The UHV rules can be understood as a massive extension of the MHV rules to the massive scalar UHV amplitudes. Namely, the UHV rules state that the massive scalar UHV amplitudes can be obtained by connecting the UHV vertices with massive scalar propagators. This is analogous to the MHV rules in structure but a main difference exists, that is, for the massive scalar UHV amplitudes we have freedom to connect an arbitrary number of the UHV vertices since such procedure, contrary to the MHV rules, does not alter the overall helicity configuration of the UHV amplitudes. In compensation, we no longer take a sum over permutations of gluons for the massive UHV amplitudes and non-UHV amplitudes in general. This peculiar nature of the massive amplitudes is due to the definition of massive holonomy operator (2.14).
Probably the easiest way to understand the UHV rules is to visualize the full expression for the massive scalar UHV amplitudes, see Figure 1. As mentioned above, we have a freedom to arbitrarily add UHV vertices to the UHV amplitudes as long as the total number of the scattering particles is preserved. In this sense, the UHV vertices can be interpreted as 1 UHV irreducible (1UI) subamplitudes. This is why we denote each of the UHV vertices by "1UI" in Figure 1.
Using the UHV rules, we can then obtain an explicit form of the massive scalar UHV amplitudes as follows.
The color-stripped holomorphic massive scalar UHV amplitudes C Figure 1: The UHV rules for the massive scalar UHV amplitudes -"1UI" stands for a 1 UHV irreducible diagram which is equivalent to the massive scalar UHV vertex. The vertices are connected with one another by massive scalar propagators.
In the above expression, (1|n) = (1n) = ǫ AB u A 1 u B n and the uppercase letters label a set of momenta running along each of the propagators. For example, P J µ is defined as where p µ J is the on-shell partner of the off-shell momentum P J , η µ J is the corresponding reference null-vector, and m denotes the mass of the complex scalar particles (φ 1 , φ n ). The spinor momenta (u A J ,ūȦ J ) correspond to the null momentum p µ J and are defined as usual: An S-matrix functional for the massive scalar UHV tree amplitudes It is now straightforward to introduce an S-matrix functional for the massive scalar UHV tree amplitudes (2.39) by use of the supersymmetric massive holonomy operator Θ where in the calculation of W (0) (x) we take the limit y → x, with the time ordering x 0 > y 0 being preserved. An explicit derivation of the x-space UHV massive scalar tree amplitudes is given by where a (±)c and a (0) represent generic source functions for gluons and massive scalars, respectively.

S-matrix for massive fermion UHV amplitudes
In the following sections, we consider the application of the above results to massive fermion amplitudes. In this section we focus on the massive fermion UHV amplitudes, i.e., the scattering amplitudes of an arbitrary number of positive-helicity gluons and a pair of massive fermions.

Notation
We first clarify our definition of fermion operators. In the spinor-helicity formalism fermions are represented by two-component spinors. (For recent reviews of two-component spinor techniques, see, e.g., [25,26].) The ordinary Dirac spinor is defined by where ψ A L (A = 1, 2) and ψȦ R (Ȧ = 1, 2) are the two-component Weyl spinors. Each of these is analogous to the spinor momentum u A . Bearing in mind that its conjugate is denoted as uȦ = (u A ) * in our notation, the conjugate of Ψ can be defined as where we use the chiral representation of the gamma matrix γ 0 .
For both massive and massless fermions, one can naturally assign helicity − 1 2 to ψ L and helicity + 1 2 to ψ R , respectively. (As we shall seen in a moment, one can make an offshell continuation of massless fermions such that the helicity of fermions is not altered; see (3.18)-(3.21). Using such massive deformation, one can and should identify the 'helicity" of massive fermions with the helicity of the corresponding massless fermions.) In the helicityspinor formalism, physical information of any particle is encoded into its helicity and the numbering index. In the framework of the holonomy formalism this is represented by the physical operators a (±) i (x, θ) in (2.19). With the help of the helicity operator (2.22), these physical operators are essentially described by the N = 4 supermultiplets in (2.21).
Therefore, for massless fermions, we can relate the Weyl spinors ψ L and ψ R to the following operators: where i denotes the numbering index. Similarly the conjugates of ψ L and ψ R have the following correspondence: In the chiral-superspace representation, creation operators of these fermions are then defined in the same way as the expression (2.19), with a (±) i being respectively replaced by a Li . In order to incorporate fermions in the holonomy formalism, basically we do not have to use the Weyl spinors but the above creation operators. Notice that these operators do not behave as spinors (which satisfy the ordinary Dirac equation) but as mere Grassmann variables.
We can check the validity of our definitions (3.3)-(3.6) for the massless fermions as follows. From the study of fermionic extension of the CSW rules, we find that the next-to-UHV (NUHV) vertices, i.e., vertices of an arbitrary number of positive-helicity gluon plus one negative-helicity gluon and a pair of fermions, can be calculated as [12,13]: where we include the MHV vertex for clarification of the notation; here the positive-helicity gluons are implicit in the arguments of the left-hand sides, and the color factors are omitted in the right-hand sides. Notation of fermions here is different from the one found in the literature. For example, in [13] light-cone treatment of fermions is used and they are denoted byψ ± , ψ ± ; these are related to our chiral notation by ψ − ⇔ ψ L , ψ + ⇔ ψ R ,ψ − ⇔ψ R and ψ + ⇔ψ L . Carrying out the Grassmann integrals that are analogous to (2.38), we can easily check that the assignments (3.3)-(3.6) indeed lead to the NUHV vertices (3.8) and (3.9) for massless fermions.

Massive fermion UHV vertices and the choice of reference spinors
We continue our discussion to include massive fermions. The massive fermion UHV vertices, the fermionic analogs of (2.35), are given by [13]: where η denotes the identical reference spinor for the pair of massive fermions labeled by the numbering indices (1, n): 14) In (3.10)-(3.13) we follow the notations of (3.8) and (3.9). Notice that V UHV 's are denoted with hats, meaning that the fermions ψ L/R 's andψ L/R 's are massive. From here on, we shall use the notation (ψ L1 , ψ Ln ,ψ R1 , ψ Rn ) to express massive fermions.
The first two vertices (3.10) and (3.11) are explicitly dependent on the identical reference spinor η and would lead to a set of arbitrary expressions, depending on the choice of η's. These η-dependent vertices are consequences of the parametrization (3.14) and (3.15). This parametrization is useful particularly in defining off-shell momentum transfers running along propagators and indeed it has been utilized in the derivation of the CSW rules [1]. In defining exterior (instead of interior) off-shell momenta, however, we can use alternative parametrization as well.
For example, in [5,6] Rodrigo and others parametrize a pair of massive fermions, utilizing two distinct reference spinors. Our choice of the reference spinors for a pair of massive scalars in (2.29) or (2.30) is essentially the same as Rodrigo's. (The difference is that their parametrization imposes an additional conditionp µ 1 +p µ n = p µ 1 + p µ n but ours does not; see appendix of [6] for detail of this point.) The η-dependent vertices (3.10) and (3.11) vanish with suitable choices of η. In order to properly discuss forms of the massive fermion UHV vertices and amplitudes in general, it is therefore indispensable to use η-independent expressions. Otherwise we can not avoid ambiguities arose from different choices of off-shell parametrizations or reference spinors. The vertices (3.12) and (3.13) are the only such η-independent UHV vertices known in the literature and, hence, we shall focus on these vertices in what follows, keeping our choice of reference spinors for the pair of massive fermions in the form of (2.29), rather than (3.14) and (3.15), as in the case of the massive scalars.

Massive fermions in the holonomy formalism
In the massive scalar case, the off-shell continuation of Nair's superamplitude method is carried out by the use of the ξζ-prescription (2.25): where we have introduced the "massive" Grassmann variable Similarly, we can define the off-shell continuation of the massless fermion operators (3.3)-(3.6) such that it leads the η-independent massive UHV vertices (3.12), (3.13). We find that the fermionic off-shell continuation can be carried out by where i = 1, n. Notice that the off-shell continuation by means of the ξζ-prescription is made only for the conjugate fermionsψ Li andψ Ri . The unbar fermions ψ Li and ψ Ri remain onshell. This is consistent with the fact that the nontrivial UHV vertex (3.12) is proportional to m. The rest of the UHV vertices vanish upon the Grassmann integrals over the chiral supervariables θ α A . This can easily be seen by counting the by counting the number of Grassmann variables in each pair. The numbers are given by 8, 4, 6 and 6 for (ψ R1 , ψ Ln ), (ψ L1 , ψ Rn ), (ψ R1 , ψ Rn ) and (ψ L1 , ψ Ln ), respectively. Thus, upon the Grassmann integral over d 8 θ, only the first pair survives.
Our choice (3.18)-(3.21) is also consistent with a recent study on recursion relations for massive fermion currents [9] where a pair of massive fermions are parametrized by an off-shell conjugate fermion and an on-shell fermion.
The chiral superspace representation of the massive fermion operators can then be expressed asā where we specify the numbering indices that are relevant to the UHV vertices of our interest. The off-shell momenta p µ 1 and p µ n are defined by (2.30). As in (2.28), w A 1 is given by where our choice of reference spinors is reflected.

The UHV rules and the massive fermion UHV amplitudes
The basic ingredients of the UHV rules are the UHV vertices and the propagators. In the CSW-type rules, which are originally formulated in twistor space, the vertices correspond to a set of lines in twistor space and they are connected by scalar propagators. Application of the UHV rules to the massive fermion UHV amplitudes is then straightforward because (a) we already know all types of the fermionic UHV vertices and (b) these vertices are connected by massive propagators as in the case of massive scalar amplitudes.
The fermionic UHV vertex V UHV (ψ 1 ψ n ) should be expanded by possible pairs of fermions. Under our choice of reference spinors, only a certain type remains nonzero, i.e., where in the last equation we use (2.35) and (3.12), denoting the color-stripped massive scalar UHV vertices by V UHV (φ 1 φ n ). For the UHV vertices other than V UHV (ψ R1 ψ Ln ), the number of Grassmann variables does not reach the saturating number 8. Thus these vertices vanish upon execution the Grassmann integrals unless the pair of fermions couple to other particles, either massive or massless, such that the total number of Grassmann variables becomes 8. We shall consider such interactions in another paper.
In application of the UHV rules, we can obtain the massive fermion UHV amplitudes from the diagrams in Figure 1 by replacing (φ, φ) with (ψ R , ψ L ). Analytically the massive fermion UHV amplitudes can be written down in the form of (2.42): The full amplitudes in a form holomorphic to the spinor momenta are given, as in (2.41), by We should emphasize again that we do not take a sum over permutations of gluons in the above expression; the sum is already taken care of in the definition of the massive holonomy operator.
To be more precise, the sum is already included in the computation of the braid trace for the massive holonomy operator; see (2.17) and (2.18) for details.
We should comment on the relation between the massive UHV amplitudes of scalars and fermions. In the literature it is often shown that these are proportional to each other as in the case of the UHV vertices (3.27), i.e., C (ψ 1 ψn) . This relation is derived from the supersymmetric Ward identities [13]; see also [10,11,17]. We consider, however, that this relation is contradictory to the UHV rules because in the UHV rules the massive UHV amplitudes are constructed by iterative use of the UHV vertices, see Figure 1 or the analytic expression (3.29). Therefore the above proportional relation holds only for the 1 UHV irreducible (1UI) amplitudes or the UHV vertices in our framework. This interpretation is natural and would improve our understanding of the massive UHV amplitudes since the notion of 1UI amplitudes and its roles in construction of massive UHV amplitudes have been generally ignored so far in the literature.
An S-matrix functional for the massive fermion UHV amplitudes What we have shown so far is that based on the forms of the massive fermion UHV vertices (3.10)-(3.13), one can apply the UHV rules to construct the massive fermion UHV amplitudes in much the same way as the case of massive scalar amplitudes. Motivated by the results in (2.46)-(2.48), we now formulate field theoretic derivation of the massive fermion UHV amplitudes. Since the massive holonomy operator (2.14) does not contain fermion operators (3.18)-(3.21), for this purpose, we need to incorporate the operators of massive fermions, rather than those of massive scalars (a (0) 1 , a (0) n ), into the massive holonomy operator. As in (3.27), this can be done by taking a sum over possible pairs of fermions, i.e., where h i = ± = ±1 (i = 2, 3, · · · , r − 1) denotes the helicity of the i-th gluon. We obtain the above expression based on the UHV rules, which means that we do not need a knowledge of algebra for the fermionic operators. This is related to the fact that in the holonomy formalism the fermionic operators arise upon supersymmetrization of the gluon operators; see (2.21).

Supersymmetrization of Θ
(B)ψ ψ R,γ (u) can be carried out by replacing the gluon and fermion operators with (2.19) and (3.22)-(3.25), respectively. As in the massive scalar case, we denote the supersymmetric holonomy operator by Θ (B)ψ ψ R,γ (u; x, θ). The generating functional for the massive fermion UHV vertices is then expressed as This expression provides a fermionic analog of (2.31).
Similarly, in analogy to (2.46) and (2.47), we can straightforwardly obtain an S-matrix functional for the full massive fermion UHV amplitudes by use of Θ (B)ψ ψ R,γ (u; x, θ): where off-shell momentum transfer P µ J is defined by (2.44). The Lorentz invariant measure dµ( P J ) is given by where m denotes the mass of the fermion. As before, in the calculation of (3.34)-(3.37) we take the limit y → x, while keeping the time ordering x 0 > y 0 .
The massive fermion UHV amplitudes in the x-space representation are then explicitly derived as where a (±)c and (ā (± 1 2 ) , a (∓ 1 2 ) ) represent generic source functions for gluons and massive fermions, respectively. Dimensional analysis and "exclusion rules" for massive fermion amplitudes Lastly, but most importantly, we consider the m dependence of the massive fermion UHV amplitudes A (ψ 1 ψn) UHV (u). In the massive scalar case, the UHV amplitudes are proportional to the squared mass, A The fact that A (ψ 1 ψn) UHV (u) is expanded by m −s+2 (s = 1, 2, · · · , n − 2) is unnatural from a perspective that the fermion-mass dependence is irrelevant to the number of gluons involving the scattering processes. The only way to circumvent this problem is to fix the total number of scattering particles by n = 3 . (3.40) Naively, this constraint leads to the relation where the color factor Tr(t c 2 ) vanishes unless the gluon propagates along an internal line. As in the case of the MHV rules, internal gluons have the U(1) color factor. But, for external gluons, we have the SU(N) color factor and A (ψ 1 g + 2 ψ 3 ) UHV (u) vanishes. To be explicit, Tr(t c 2 ) is given by where SU(N) and U(1) denote the algebras of SU(N) and U(1) groups, respectively. We here use the usual normalization Tr(t c t c ′ ) = δ cc ′ 2 . The three-point interactions (3.41) vanish unless the involving gluon is a virtual gluon which appears only in internal propagators. Intriguingly, this feature is in accord with the pictures of the quark-gluon interactions in QCD.
In the existing literature there have been no considerations on the number of involving particles in the massive UHV amplitudes. This is mainly because the notion of the 1 UHV irreducible (1UI) diagram in Figure 1 is not well-recognized in the literature. In our formulation the 1UI diagram corresponds to the massive UHV vertex but it is often interpreted as the massive UHV amplitude; it is so particularly in the case of massive fermions. Once this point is clarified, one can utilize the relation (3.27) to derive the explicit form of the massive fermion UHV amplitudes (3.28). We can then straightforwardly carry out massdimension analysis of the UHV amplitudes. For scalar amplitudes a pair of massive scalars can couple to an arbitrary number of gluons. This is essentially due to the fact that the massive scalar UHV vertex (2.35) is proportional to m 2 . Since the massive propagator is proportional to m −2 , one can insert as many propagators as possible into the massive scalar UHV amplitudes as far as the mass dimension is concerned. On the other hand, the massive fermion UHV vertex (3.27) is proportional to m. Thus insertion of massive propagators is mass-dimensionally prohibited. This leads to the constraint (3.40) that a pair of massive fermions should couple to a single (internal) gluon.
The constraint n = 3 is crucial when we consider the generalization of the massive fermion UHV amplitudes to the non-UHV amplitudes. For example, it means that all the massive fermion amplitudes are classified by the UHV subamplitude and the next-to-UHV (NUHV) subamplitude, the latter containing a single negative-helicity gluon. In this sense, we can interpret the condition n = 3 for a sort of exclusion rules for the massive fermion amplitudes. This principle is a consequence of (a) the use of massive scalar propagators for the construction of massive fermion amplitudes in the holonomy formalism and (b) the mass-dimension analysis of the massive fermion UHV amplitudes.
In the following section, for clarification of our argument, we first consider the massive fermion NUHV amplitudes for arbitrary n and then study the consequence of the constraint n = 3 afterwards.

Massive fermion NUHV vertices
For the massive scalar amplitudes there are no vertices other than the massive scalar UHV vertices that contribute to the UHV rules. For the massive fermion amplitudes, however, due to our assignments of Grassmann variables in (3.3)-(3.4), there does exists another type of nonzero vertices that contributes to the UHV rules. This is given by the following NUHV vertices (1 a) 2 (a n) 2 (12)(23) · · · (n − 1 n)(n1) (4.1) (2 ≤ a ≤ n − 1) where we choose the reference spinor η 1 = u n as before for the operator (3.20) of the off-shell fermionψ L1 . Notice that one can easily check that there are no other non-vanishing massive fermion vertices, hence, (3.12) and (4.1) are the only vertices that contribute to the massive fermion amplitudes in general.

Massive fermion NUHV amplitudes
The contribution (4.1) is one of the major differences between the UHV rules of massive scalars and fermions. We here study how it reflects in the construction of the massive fermion NUHV amplitudes. Diagrams contributing to the massive fermion NUHV amplitudes A One includes the gluonic MHV amplitudes and the other does not. From the results of A NUHV (u) in [19], we can write down the former contribution A Tr(t σ i · · · t σ i+r t i+r+1 · · · t i−1 ) where we consider the numbering indices in modulo n. The C's are expressed as where the off-shell momentum transfer q µ i i+r is defined in terms of the reference null-vector η µ i i+r and the associated null momentum p µ i i+r : In (4.7) we use the two-component notation of the null vector p µ i i+r . In (4.3) and (4.5) σ denotes the transposition of the numbering indices for gluons: The other contribution to A (ψ 1 g − a ψn) NUHV (u) is essentially given by the massive fermion NUHV vertices: (1 a) 2 (a n) 2 (12)(23) · · · (n − 1 n)(n1) .
This expression is free of the reference spinors.

Imposition of the n = 3 condition
Having written down the massive fermion NUHV amplitudes for arbitrary n, we now impose the condition n = 3 of (3.40) on (4.2). Since the MHV vertex needs more than two legs, the first term in (4.2) vanishes for n = 3, i.e., where we omit the color factor. As discussed in (3.41) and (3.42), the gluon labeled by g − 2 is interpreted as a virtual gluon otherwise the above NUHV amplitude becomes zero. In practice, however, one can connect it with the three-point UHV subamplitude (3.41), in which case the color factor becomes nonzero (regardless the choice of the color for each gluon) due to the normalization Tr(t c t c ′ ) = δ cc ′ 2 . The color-stripped UHV amplitude can similarly be expressed as (23) . (4.11) In general non-UHV amplitudes are labeled by N k UHV amplitudes, specified by the number of negative-helicity gluons, k = 1, 2, · · · n − 3. The massive fermion N k UHV amplitudes then refer to the scattering amplitudes of a pair of massive fermions and k negative-helicity gluons and (n − k − 2) positive-helicity gluons. Therefore, upon the imposition of n = 3, there exist no N k UHV amplitudes for k ≥ 2.
An S-matrix functional for (4.10) and (4.11) To summarize, we find that the interactions among gluons and a pair of massive fermions are given by (4.10) and (4.11). This result is qualitatively different from the massive scalar amplitudes where we can incorporate an arbitrary number of gluons. The difference arises from the the mass dimension analysis which leads to the condition n = 3.
Notice that, apart from the n = 3 constraint, (4.10) and (4.11) are obtained by application of the UHV rules. Since the UHV rules are automatically realized by defining an S-matrix functional, it is still worth deriving the S-matrix functional for (4.10) and (4.11). Since the massive fermion NUHV vertices (4.1) can be obtained from the generating functional for the UHV vertices (3.32) alone, we find that the amplitudes in (4.10) are derived from (3.33) as well. Therefore, together with (3.34)-(3.37), the S-matrix functional for (4.10) and (4.11) is given by (3.33).

Concluding remarks
In this paper we consider incorporation of massive fermions into the holonomy formalism, following the recent study [19] on the massive scalar amplitudes. The upshot of this paper is that the interactions among gluons and a pair of fermions are reduced to the two types of subamplitudes represented by (4.10) and (4.11). This result is simple but there have been several key steps toward it. These steps can be itemized as follows: 1. We clarify our notation of fermions in the spinor-helicity formalism.
2. We carry out off-shell continuation of Nair's superamplitude method for fermions by use of what we call the ξζ-prescription in (3.16)- (3.21). This set of parametrizations leads to the previously known massive fermion UHV vertices [13].
3. We apply the UHV rules, i.e., the massive extension of the CSW rules, to construct massive fermion UHV amplitudes.
4. In the CSW-type rules, vertices correspond to a set of lines in twistor space and they are connected by scalar propagators. In the UHV rules massive fermion amplitudes are therefore obtained by connecting the massive fermion UHV vertices in terms of massive scalar propagators.

5.
We formulate an S-matrix functional for the massive fermion UHV amplitudes that realizes the UHV rules in a functional method.
6. Analysis of mass dimension on the massive fermion UHV amplitudes implies that the number of gluons involving the UHV amplitudes should be one.
7. Interactions among gluons and a pair of massive fermions are then decomposed into the two types of subamplitudes (4.10) and (4.11).
The final result 7 means that interactions among gluons and massive fermions should be decomposed into three-point massive fermions vertices. This is consistent with QCD in terms of the gluon-quark interactions. It is intriguing that this result is derived purely from the mass-dimension analysis. In the framework of the holonomy formalism, the result 7 also illustrates a qualitative difference from the massive scalar amplitudes. In the massive scalar amplitudes there are no restrictions on the number of involving gluons. In this sense, we can interpret the result 6 as a sort of "exclusion rules" for the massive fermion amplitudes.
The results 1-5, on the other hand, elucidate the fact that we can naturally incorporate massive fermions into the holonomy formalism. Motivated by these results, in a forthcoming paper, we shall consider massive deformation of gauge bosons in the same framework.