HELAS and MadGraph/MadEvent with spin-2 particles

Fortran subroutines to calculate helicity amplitudes with massive spin-2 particles (massive gravitons), which couple to the standard model particles via the energy momentum tensor, are added to the {\tt HELAS} ({\tt HEL}icity {\tt A}mplitude {\tt S}ubroutines) library. They are coded in such a way that arbitrary scattering amplitudes with one graviton production and its decays can be generated automatically by {\tt MadGraph} and {\tt MadEvent}, after slight modifications. All the codes have been tested carefully by making use of the invariance of the helicity amplitudes under the gauge and general coordinate transformations.


Introduction
The idea of extra space dimensions has attracted much attention in recent years, since it can give us a novel solution to the hierarchy problem, or an alternative explanation of the hierarchical difference between the Planck scale (M Pl ∼ 10 19 GeV) and the electroweak scale (m W ∼ 10 2 GeV).
So far, there have been various extra dimension models, which can be divided into two major classes according to the geometry of the background space-time manifold. The first one includes the ADD (Arkani-Hamed, Dimopoulos, and Dvali) model [1][2][3] and its variants, which extend the dimension of the total space-time to D = 4 + δ, with a factorizable metric and large size of the compact extra dimensions ( 1/M Pl ). The second one includes the 5-dimensional RS (Randall and Sundrum) model [4,5] and its variants, in which a warped metric is introduced along the 5-th dimension and the size of the extra dimension needs not to be much larger than the Planck length. In both classes of extra dimension models, there appear Kaluza-Klein (KK) towers of massive spin-2 gravitons, which can interact with the standard model (SM) fields. The effective interaction Lagrangian is given by [6,7] where T ( n)μν is the n-th graviton KK modes, and Λ is the relevant coupling scale. In the ADD model we have where M Pl is the 4-dimensional reduced Planck scale, and in the RS model is at the electroweak scale, where k is a scale of order of the Planck scale and r c is the compactification radius.
with e = g W sin θ W = g Z sin θ W cos θ W and the projection operator P L = 1 2 (1 − γ 5 ). Here, the covariant derivative is Note that the derivative couplings of the W bosons are written explicitly in (5b). Each field-strength tensor for the gauge bosons is Notice that as in the standard HELAS package [8], we use the unitary gauge for the massive vector-boson propagators and the Feynman gauge for the massless ones.
In this paper, we present new HELAS subroutines [8] for the massive gravitons and their interactions based on the effective Lagrangian of (1), and implement them into Mad-Graph/MadEvent (MG/ME) [9][10][11]. 1 The paper is organized as follows: In Sect. 2 we give the new HELAS subroutines. In Sect. 3 we describe how to implement amplitudes with a massive spin-2 graviton into MG/ME. In Sect. 4 we give sample numerical results. Section 5 contains a brief conclusion.

HELAS subroutines for spin-particles
In this section, we list the contents of all the new HELAS subroutines that are needed to evaluate massive spin-2 graviton production at hadron colliders in association with quark and gluon jets, and its decays into a pair of all the SM particles, or into arbitrary numbers of quarks and gluons.
To begin with, in Sect. 2.1 the subroutine to compute the external lines for a spin-2 tensor particle is presented. Next, in Sects. 2.2 to 2.7, the subroutines to compute the interactions of the tensor boson with the SM particles are explained. The new vertex subroutines are listed in Table 1. Those subroutines which we do not present in this paper are FFST, SSST and SSSST type vertices for the interactions with scalars; VVVT and VVVVT for the interactions with the electroweak gauge bosons. These contribute e.g. to the graviton decays into three or more weakly interacting particles. We present the effective Lagrangian of (5a) to (5f) for completeness sake. Finally, we briefly mention how we test our new subroutines in Sect. 2.8.
The helicity states of the tensor boson can be expressed as by using the vector boson wavefunctions μ (p, λ) that obey the relation where J − = J x − iJ y is the J z lowering operator. The spin-1 vector wavefunction in the HELAS convention [8] satisfies this relation, and hence we simply use the HELAS code to obtain the tensor wavefunction. These tensor wavefunctions are traceless, transverse, orthogonal, and symmetric, and the completeness relation is with B μν,αβ (p)

SST vertex
The SST vertices are obtained from the interaction Lagrangian among the tensor and two scalar bosons: for the complex scalar field, or for the real scalar field. Here, is the coupling constant.

SSTXXX
This subroutine computes the amplitude of the SST vertex from two Scalar boson wavefunctions and a Tensor boson wavefunction, and should be called as CALL SSTXXX(S1, S2, TC, GT, SMASS , VERTEX).
The input TC(18) is a complex 18-dimensional array which contains the wavefunction of the Tensor boson, and its four-momentum; see the TXXXXX subroutine in Sect. 2.1.1, GT is the coupling constant in (17) where we used the notation in (8) and (9), and

HSTXXX
This subroutine computes an off-shell scalar current H made from the interactions of a Scalar boson and a Tensor boson by the SST vertex, and should be called as CALL HSTXXX(TC, SC, GT, SMASS, SWIDTH , HST).
The inputs TC(18) and SC(3) are the wavefunctions and momenta of the Tensor and Scalar bosons, respectively, and SWIDTH is the scalar boson width Γ S . The output HST (3) gives the off-shell scalar current multiplied by the scalar boson propagator, which is expressed as a complex threedimensional array: and Here the momenta p and q are (2)).

USSXXX
This subroutine computes an off-shell tensor current U made from two flowing-out Scalar bosons by the SST vertex, and should be called as CALL USSXXX(S1, S2, GT, SMASS, TMASS, TWIDTH , The inputs TMASS and TWIDTH are the tensor boson mass and width, m T and Γ T . The output USS(18) gives the offshell tensor current multiplied by the tensor boson propagator, which is expressed as a complex 18-dimensional array: where we used the notation whose components are assigned into the first 16 component of USS as in (8), and Here, p 1 and p 2 are the momenta of the outgoing scalars, and q is that of the off-shell tensor boson given in (26) and (27) as mUSS (17)).
Although the effective Lagrangian of (1) does not dictates the off-shell behavior of the gravitons, we allow gravitons to propagate just once in the total amplitude where there are no external gravitons in the initial or final states. This is convenient when studying the correlated decays of the graviton production and its subsequent decays.
We may also note that the order of the GT couplings should be restricted to 1 when there is an external graviton, and 2 when a graviton is exchanged among the SM particles.
Before turning to the FFT vertex, it should be noticed here that the conventional factors of i in the vertices and those in the propagators are both included in the off-shell wavefunctions, such as (21) and (24) above, according to the HELAS convention. The HELAS amplitude, obtained by the vertices, such as (18), gives the contribution to the T matrix element without the factor of i. See more details in the HELAS manual [8].

FFT vertex
The FFT vertices are obtained from the interaction Lagrangian among the tensor boson and two fermions: where the coupling constant is

IOTXXX
This subroutine computes the amplitude of the FFT vertex from a flowing-In fermion spinor, a flowing-Out fermion spinor and a Tensor boson wavefunction, and should be called as CALL IOTXXX(FI, FO, TC, GT, FMASS , VERTEX).
The input GT is the coupling constant in (29), and FMASS is the fermion mass m F . What we compute here is where we use the notation (4)). (32)

UIOXXX
This subroutine computes the bi-spinor tensor current U made from flowing-In and flowing-Out fermions by the FFT vertex, and should be called as The output UIO(18) is a complex 18-dimensional array: Here, p 1 and p 2 are the momenta of the flowing-in and flowing-out fermions, respectively, and q is that of the tensor particle.

VVT vertex
The VVT vertices are obtained from the interaction Lagrangian among the tensor and two vector bosons: for the complex vector bosons, or W bosons, and for the real ones, or gluons, photons, and Z bosons. Here the coupling constant is The ξ terms are the gauge-fixing terms, which vanish for massive vector bosons in the unitary gauge. For massless vector bosons we take ξ = 1 in the Feynman gauge.

VVTXXX
This subroutine computes the amplitude of the VVT vertex from two Vector boson polarization vectors and a Tensor wavefunction, and should be called as CALL VVTXXX(V1, V2, TC, GT, VMASS , VERTEX).
The input GT is the coupling constant in (45), and VMASS is the vector boson mass m V . What we compute here is where we use the notation and

JVTXXX
This subroutine computes an off-shell vector current J made from the interactions of a Vector boson and a Tensor boson by the VVT vertex, and should be called as CALL JVTXXX(VC, TC, GT, VMASS, VWIDTH , JVT).

UVVXXX
This subroutine computes an off-shell tensor current U made from two flowing-out Vector bosons by the VVT vertex, and should be called as The output UVV(18) is a complex 18-dimensional array: for the first 16 components of UVV, and Here, p 1 and p 2 are the momenta of the outgoing vector bosons, and q is that of the tensor boson.

FFVT vertex
The FFVT vertices are obtained from the interaction Lagrangian among the tensor boson, vector boson and two fermions: with the chiral-projection operator P R,L = 1 2 (1 ± γ 5 ). The coupling constant GT is List of the coupling constants for each vertex. All the particles and the coupling constants are written in the MG notation. y stands for a massive graviton, f represents all possible fermions, and v is the SM gauge bosons (g, a, z, w). GC is a SM coupling constant, while GT is a non-renormalizable coupling constant defined in each subroutine in Sect. 2

3-point couplings GT
and GC(1) and GC(2) are the relevant FFV left and right coupling constants. The list of the coupling constants is shown in Table 2. For instance, in the case of the interaction with a gluon, the FFV couplings are where the sign of the coupling is fixed by the HELAS convention [8].

IOVTXX
This subroutine computes the amplitude of the FFVT vertex from a flowing-In fermion spinor, a flowing-Out fermion spinor, a Vector boson polarization vector and a Tensor boson wavefunction, and should be called as CALL IOVTXX(FI, FO, VC, TC, GC, GT , VERTEX).
What we compute here is
What we compute here is for the massive vector boson, or Here q is the momentum of the vector boson.

UIOVXX
This subroutine computes an off-shell tensor current by the FFVT vertex, and should be called as CALL UIOVXX(FI, FO, VC, GC, GT, TMASS, TWIDTH ,

UIOV).
What we compute here is Here q is the momentum of the tensor boson.

VVVT vertex
The VVVT vertices are obtained from the interaction Lagrangian among the tensor and three vector bosons: with the structure constant f abc and the coupling constant, as in (45), In this paper we concentrate on the interactions with gluons for the VVVT vertex, so in this case GC is the strong coupling constant, and f abc is the structure constants of the group SU (3), which can be handled by the MG automatically. As in the original subroutines for the VVV vertex [8], the following subroutines, VVVTXX, JVVTXX, and UVVVXX, can be used for the electroweak gauge bosons without any modifications. What we compute here is VERTEX = −GT GCT μν C μν,ρσ (p aλ − p bλ ) with F μν,ρσ λ (p a , p b , p c ) Here, the vector bosons (gluons in this paper) VA, VB, and VC have the momentum p a , p b , and p c , and the color a, b, and c, respectively.
Note that the off-shell gluon JVVT has the color c. What we compute here is Here p a , p b and p c are the momenta of the outgoing vector bosons, and q is that of the tensor boson.

VVVVT vertex
The VVVVT vertices are obtained from the interaction Lagrangian among the tensor and four vector bosons: with the coupling constants GT = GTV in (45) and GC = g s for the interactions with gluons. We should note that the 5-point vertex cannot be generated by MG, and thus we must add the following subroutines by hand, GGGGTX, JGGGTX, or UGGGGX, to the amplitudes which have the corresponding color structures.

GGGGTX
This subroutine computes the portion of the amplitude of the VVVVT vertex from four Gluon polarization vectors and a Tensor boson wavefunction corresponding to the color structure f abe f cde , and should be called as CALL GGGGTX(VA, VB, VC, VD, TC, GC, GT , VERTEX).
To obtain the complete amplitude, this subroutine must be called three times (once for each color structure) with the following permutations: CALL GGGGTX (VA, VB, VC, VD, TC, GC,

JGGGTX
This subroutine computes the portion of the off-shell gluon current by the VVVVT vertex, corresponding to the color structure f abe f cde , and should be called as (couplings.f and interactions.dat); see also Table 2. Then we insert all the new HELAS subroutines for spin-2 tensor bosons into the HELAS library in MG. Since the present MG does not handle external spin-2 particles, we further modify the codes in MG to tell it how to generate the SST, FFT and FFVT type of vertices and helicity amplitudes (for VVT and VVVT type, it has already been done for the Higgs effective field theory (HEFT) model), and how to deal with the helicity of the spin-2 tensor bosons when they are external. Moreover, since MG can only generate Feynman diagrams with up to 4-point vertices, the amplitudes and their HELAS codes with the 5-point vertex, GGGGTX, JGGGTX, or UGGGGX, have been added by hand; see more details in Sect. 2.7.
Finally, we note that since in the ADD model the gravitons are densely populated and we should sum over their contributions by modifying the phase space integration in ME. In the ADD model, the spectrum of KK graviton modes can be treated as continuous for δ ≤ 6 [1][2][3], and the mass density function is given by [6] where M s is the ADD model effective scale. Thus, we modify the phase space generating codes in ME to add one more random number for graviton mass generating and implement the above graviton mass integration.

Sample results
In this section, we present some sample numerical results for the graviton plus mono-jet and di-jet productions at the LHC, using the new HELAS subroutines and the modified MG/ME. We use the following jet definition criteria P j T > 20 GeV, |η j | < 5, where η is the pseudorapidity of the jets and φ is the azimuthal angle around the beam direction, and further require η j 1 · η j 2 < 0, |η j 1 − η j 2 | > 4.2.
The CTEQ6L1 parton distribution functions [14] are employed with the factorization scale chosen as μ f = min(P j T ) of the jets which satisfy the above cuts. For the QCD coupling, we set it as the geometric mean value, α s = α s (P j 1 T ) · α s (P j 2 T ). Figure 1 shows the P miss T distributions for the graviton productions with the mono-jet (via the gg, qg and qq channels) and with the di-jet (via the gg, qg and qQ channels) in the ADD model at the LHC, with Λ = 5 TeV and δ = 4. Note that the unitarity criterion M T n < Λ is used. See more details in Ref. [13]. Figure 2 shows the φ jj distributions for the first KK graviton productions with two jets in the RS model at the LHC, via the gg, qg and qQ channels. Here we set Λ = 4 TeV and M T 1 = 1 TeV. The total cross sections are 0.74, 3.74 and 4.33 pb for the qQ, gg and qg channels, respectively.

Summary
In this paper, we have added new HELAS subroutines to calculate helicity amplitudes with massive spin-2 particles (massive gravitons) to the HELAS library. They are coded in such a way that arbitrary scattering amplitudes with one graviton production and its decays can be generated automatically by MG and ME, after slight modifications. All the codes have been tested carefully by making use of the invariance of the helicity amplitudes under the gauge and general coordinate transformations.