HELAS and MadGraph with spin-3/2 particles

Fortran subroutines to calculate helicity amplitudes with massive spin-3/2 particles, such as massive gravitinos, which couple to the standard model and supersymmetric particles via the supercurrent, are added to the HELAS (HELicity Amplitude Subroutines) library. They are coded in such a way that arbitrary amplitudes with external gravitinos can be generated automatically by MadGraph, after slight modifications. All the codes have been tested carefully by making use of the gauge invariance of the helicity amplitudes.


Introduction
Gravitinos are spin-3/2 superpartners of gravitons in local supersymmetric extensions to the Standard Model (SM). If supersymmetry (SUSY) breaks spontaneously, gravitinos absorb massless spin-1/2 goldstinos and become massive by the super-Higgs mechanism. Therefore, the gravitino mass is related to the scale of SUSY breaking as well as the Planck scale like This implies that the gravitino can take a wide range of mass, depending on the SUSY breaking scale, from eV up to scales beyond TeV, and provide rich phenomenology in particle physics as well as in cosmology [1]. Although the gravitino can play an important role even in collider signatures when it is the lightest supersymmetric particle (LSP), there is few Monte Carlo event generators which can treat them. 1 In this paper, we present new HELAS subroutines [3] for the massive gravitinos and their interactions based on the effective Lagrangian below, and implement them into MadGraph/MadEvent (MG/ME) v4 [4,5,6] so that arbitrary amplitudes with external gravitinos can be generated automatically. 2 a e-mail: kentarou.mawatari@vub.ac.be b e-mail: takaesu@post.kek.jp 1 The SM with gravitino and photino is supported by WHIZARD [2]. 2 The Fortran code for simulations of the massive gravitinos is available at the KEK HELAS/MadGraph/MadEvent Home Page, http://madgraph.kek.jp/KEK/.
The effective interaction Lagrangian relevant to the gravitino phenomenology is [7,8,9] where ψ µ is the spin-3/2 gravitino field, f i and φ i are spinor and scalar fields in the same chiral supermultiplet, P R/L = 1 2 (1 ± γ 5 ) is the chiral-projection operator, and M Pl ≡ M Pl / √ 8π ∼ 2.4 × 10 18 GeV is the reduced Planck mass. The covariant derivative is where g s , g and g ′ are the SU (3) C , SU (2) L and U (1) Y gauge couplings, respectively, and T a 3 , T a 2 and Y are the generators of the SU (3) C (a = 1, · · · , 8), SU (2) L (a = 1, 2, 3) and U (1) Y groups. The field-strength tensors for each gauge group are and the corresponding gauginos λ (α=3,2,1)a are gluinos (g a ), winos (W a ) and bino (B), respectively. The paper is organized as follows: In Sect. 2 we give sample numerical results. Sect. 3 presents our brief summary. In App. A we give the new HELAS subroutines for spin-3/2 particles, and in App. B we describe how to implement the amplitudes into MG.

Sample results
In this section, we present some sample numerical results, using the new HELAS subroutines, which are presented in Appendix A, and the modified MG, which is described in Appendix B.
In the gauge mediated SUSY breaking scenarios, the gravitino is often the LSP, and its phenomenology depends on what is the next-to-lightest supersymmetric particle (NLSP). Here we consider the stau NLSP scenario as well as the neutralino NLSP one.

Stau NLSP
As a sample result for the stau NLSP scenario, we consider radiativeτ decays,τ Here we regard the stau as a purely right-handed stau for simplicity. Feynman diagrams shown in Fig. 1 and the corresponding helicity amplitudes are generated automatically by the modified MG. To study the spin-3/2 nature of the gravitino, we compare theG LSP case (7a) with thẽ where only two decay diagrams contribute; see Fig. 2. We evaluate the amplitudes for the both cases, (7a) and (7b), in theτ rest frame as pτ = (mτ , 0, 0, 0), where the z-axis is taken along the photon momentum direction, and the y-axis is along − → p γ × − → p τ , the normal of the decay plane. Using the generated helicity amplitudes and the above kinematical variables, we investigate photon polarizations by means of Stokes parameters, P 1 , P 2 , and P 3 , which are related with the photon density matrix as with the Pauli sigma matrices σ i . dΓ sum = dρ ++ +dρ −− is the usual spin-summed differential decay rate. The density matrix is calculated as where M λ is the helicity amplitude with the photon helicity λ, and dΦ 3 is the three-body phase space factor. The summation symbol implies the summation over the tau and gravitino/neutralino helicities. By definition, Stokes Feynman diagrams for the radiative stau decay process for theG LSP case,τ → τGγ, generated by MadGraph. ta1, ta, G∼, A, and Ni denote a stau, a tau-lepton, a gravitino, a photon, and neutralinos, respectively.  Fig. 3. Angular dependence of the Stokes parameters of the radiated photon for theτ decay process,τR → τGγ (a) and τR → τχ 0 1 γ (b), where θ is the decay angle between the photon and the tau-lepton. We set mτ = 150 GeV, mLSP = 75 GeV and Eγ = 40 GeV.
parameters take real values from −1 to 1, and P 3 shows the right-left asymmetry of circular polarizations, while P 1 and P 2 present linear polarizations, which reflect the interference between the amplitudes for the right-and lefthanded photons.
In the cos θ > 0 region, the photon bremsstrahlung amplitude (graph 2 in Figs. 1 and 2) is dominant and thẽ G-LSP andχ 0 1 -LSP cases are very similar since only ±1/2helicity states of the gravitino are allowed. In the cos θ < 0 region, on the other hand, the neutralino propagating amplitudes and the four-point interaction amplitude, graph 3 to 7 in Fig. 1, become important, which allow the gravitino to take ±3/2 helicities as well. Note that the amplitude corresponding to the graph 1 in Figs. 1 and 2 always vanishes. Since the gravitino has the large mass in this example, spin-3/2 components dominate spin-1/2 ones, and P 3 for theG LSP shows distinct behavior from those for theχ 0 1 LSP. Especially, for cos θ ∼ −1, the difference is significant; P 3 = −0.8 (almost left-handed photon) for theG LSP, while P 3 = +1 (right-handed photon) for the  Since the photon helicity measurements require a polarized detector, we also examine linear polarizations P 1 and P 2 . In both scenarios, the linear polarization perpendicular to the decay plane vanishes (P 2 = 0), and P 1 tends to behave similarly, but slightly larger |P 1 | is expected in the backward direction (cos θ < 0) for the gravitino LSP case (a).

Neutralino NLSP
As a sample result for the neutralino NLSP scenario, we consider the process Figure 4 shows the distributions of the missing invariant mass at √ s = 190 GeV for the neutralino mass m χ = 75 and 90 GeV with the normalized cross section after kinematical cuts. The gravitino mass is fixed at an eV order so thatχ 0 1 decays instantly without leaving the production point. Here we use the same cuts as in Ref. [11]; with E beam = √ s/2, and our results agree well with Fig. 16 in [11].

Summary
In this paper, we have added new HELAS subroutines to calculate helicity amplitudes with massive spin-3/2 particles (massive gravitinos) to the HELAS library. They are coded in such a way that arbitrary amplitudes with external gravitinos can be generated automatically by MG, after slight modifications. All the codes have been tested carefully by making use of the gauge invariance of the helicity amplitudes.
Acknowledgements We wish to thank Qiang Li for helping us modify MadGraph and Junichi Kanzaki for putting our code on the web. K.H. and Y.T. would like to thank Tilman Plehn and the members of the ITP, Uni. Heidelberg for their warm hospitality, where part of this work has been done. The work presented here has been in part supported by the Concerted Research action "Supersymmetric Models and their Signatures at the Large Hadron Collider" of the Vrije Universiteit Brussel, by the IISN "MadGraph" convention 4.4511.10, by the Belgian Federal Science Policy Office through the Interuniversity Attraction Pole IAP VI/11, and by the Grant-in-Aid for Scientific Research (No. 20340064) from the Japan Society for the Promotion of Science. Y.T. was also supported in part by Institutional Program for Young Researcher Overseas Visits.

A HELAS subroutines for spin-3/2 particles
In this appendix, we list the contents of all the new HELAS subroutines that are needed to evaluate processes based on the effective Lagrangian of (2) with external spin-3/2 gravitinos.
To begin with, in App. A.1 the subroutines to compute external lines for a massive spin-3/2 particle are presented. Next, in Apps. A.2 to A.5, we explain vertex subroutines listed in Table 1, which compute interactions of a gravitino with SM and SUSY particles. Finally, we briefly mention how we test our new subroutines in App. A.6.
The input P(0:3) is a real four-dimensional array which contains the four-momentum p µ of the spin-3/2 particle, RMASS is its mass, NHEL (= ±3, ±1) specifies its helicity λ in unit of 1/2, and NSR specifies whether the fermion is particle or anti-particle. If NSR = 1 the fermion is particle and the subroutine computes the wavefunction with the u-spinor. If NSR = -1 the fermion is anti-particle and the

Vertex Inputs
Output subroutine computes the wavefunction with the v-spinor. 3 The output RI(18) is a complex 18-dimensional array, among which the first 16 components contain the wavefunction as namely Here, i = 1, 2, 3, 4 denotes each u-or v-spinor component. The last two of RI(18) contain the four-momentum along the fermion number flow, (2)).
When the four-momentum of the R-S fermion is given by its helicity states can be expressed as by using the vector boson wavefunctions ǫ µ (p, λ) and the spinor wavefunctions u(p, λ) that obey the relations where J − = J x − iJ y is the J z lowering operator. The vector and spinor wavefunctions in the HELAS convention [3] satisfy above relations. Similarly, ψ µ v (p, λ) is given by the v-spinors and the conjugated vector wavefunctions. The above helicity states satisfy the irreducibility conditions and the Dirac equation, and the completeness relation is where

A.2 FRS vertex
The FRS vertices are obtained from the interaction Lagrangian among a fermion, a R-S fermion and a scalar boson: with the notation R µ = ψ µ u/v and the chiral-projection operator P R/L = 1 2 (1 ± γ 5 ). GR(1) and GR(2) are the relevant left and right coupling constants. For instance, in the case of the quark-gravitino-squark interaction, q-G-q α , those couplings are where  (2)). (2) is the complex coupling constant, such as in (31)  where we use the notations  (5)).

The input GR
The output VERTEX is a complex number: where q µ is the momentum of the scalar boson and we use the notations  (6) gives the off-shell fermion wavefunction multiplied by the fermion propagator and its four-momentum, which is expressed as a complex six-dimensional array: and p is the momentum of the off-shell fermion given in (41) and (42) as p µ = (ℜeFSOR(5), ℜeFSOR (6), ℑmFSOR(6), ℑmFSOR(5)).

A.2.4 FSIRXX
The subroutine computes an off-shell Fermion wavefunction made from the interaction of a Scalar boson and a flowing-In R-S fermion by the FRS vertex, and should be called as CALL FSIRXX(RI,SC,GR,FMASS,FWIDTH , FSIR). The output FSIR(6) is a complex six-dimensional array: and Here we use the notation and the momentum p is p µ = (ℜeFSIR(5), ℜeFSIR(6), ℑmFSIR(6), ℑmFSIR(5)).

A.2.5 HIORXX
This subroutine computes an off-shell scalar current H made from the interaction of a flowing-In fermion and a flowing-Out R-S fermion by the FRS vertex, and should be called as CALL HIORXX(FI,RO,GR,SMASS,SWIDTH , HIOR), where SMASS and SWIDTH are the mass and the width of the scalar boson, m S and Γ S . The output HIOR(3) gives the off-shell scalar current multiplied by the scalar boson propagator and its four-momentum, which is expressed as a complex three-dimensional array: (2) The output HIRO (3) is a complex three-dimensional array: and The momentum q is q µ = (ℜeHIRO(2), ℜeHIRO(3), ℑmHIRO(3), ℑmHIRO (2)).
Before turning to the FRV 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 (40) above, according to the HELAS convention. The HELAS amplitude, obtained by the vertices, such as (34), gives the contribution to the T matrix element without the factor of i. See more details in the HELAS manual [3].

A.3 FRV vertex
The FRV vertices are obtained from the interaction Lagrangian among a fermion, a R-S fermion and a vector boson: (54) We note that, although both a gravitino and a gaugino are Majorana in most cases, the Hermitian conjugate term is necessary for MG; practically, either the first or second term is used in calculations of amplitudes. The corresponding coupling constant to the effective Lagrangian of (2) is The input VC(6) is a complex six-dimensional array which contains the Vector boson wavefunction and its momentum as q µ = (ℜeVC(5), ℜeVC (6), ℑmVC(6), ℑmVC (5)).

A.3.4 FVIRXX
This subroutine computes an off-shell Fermion wavefunction made from the interaction of a Vector boson and a flowing-In R-S fermion by the FRV vertex, and should be called as CALL FVIRXX(RI,VC,GR,FMASS,FWIDTH , FVIR). What we compute here is and where we use the notation and the momentum p is p µ = (ℜeFVIR(5), ℜeFVIR(6), ℑmFVIR(6), ℑmFVIR(5)).

A.3.5 JIORXX
This subroutine computes an off-shell vector current J made from the interaction of a flowing-In fermion and a flowing-Out R-S fermion by the FRV vertex, and should be called as CALL JIORXX(FI,RO,GR,VMASS,VWIDTH , JIOR). The input VMASS and VWIDTH are the mass and the width of the vector boson, m V and Γ V . The output JIOR (6) gives the off-shell vector current multiplied by the vector boson propagator and its four-momentum, which is expressed as a complex six-dimensional array: for the massive vector boson, or Here, q is the momentum of the off-shell vector boson, q µ =(ℜeJIOR(5), ℜeJIOR(6), ℑmJIOR(6), ℑmJIOR (5)).
Note that we use the unitary gauge for the massive vector boson propagator and the Feynman gauge for the massless one, according to the HELAS convention [3].

A.3.6 JIROXX
This subroutine computes an off-shell vector current J made from the interaction of a flowing-In R-S fermion and a flowing-Out fermion by the FRV vertex, and should be called as CALL JIROXX(RI,FO,GR,VMASS,VWIDTH , JIRO). The output JIRO (6) is for the massive vector boson, or for the massless vector boson, and Here the momentum q is q µ = (ℜeJIRO(5), ℜeJIRO(6), ℑmJIRO(6), ℑmJIRO(5)).

A.4 FRVS vertex
The FRVS vertices are obtained from the interaction Lagrangian among a fermion, a R-S fermion, a vector boson and a scalar boson: The coupling constant GR is the product of the FRS coupling constant and the gauge coupling constant of the involving gauge boson. For instance, in the case of the quark-gravitino-gluon-squark interaction, q-G-g-q L , those couplings are where GFRSL is defined in (31) and GG is the strong coupling constant The sign of the coupling constant is fixed by the HELAS convention [3]. The output VERTEX gives a complex number: The output FVSOR is a complex six-dimensional array: for the first four components of FVSOR(6), and The output FVSIR is a complex six-dimensional array: for the first four components of FVSIR(6), and for the momentum p.

A.4.5 JSIORX
This subroutine computes an off-shell vector current J made from the interaction of a Scalar boson, a flowing-In fermion and a flowing-Out R-S fermion by the FRVS vertex, and should be called as CALL JSIORX(FI,RO,SC,GR,VMASS,VWIDTH , JSIOR).
What we compute here is for the massive vector boson, or for the massless vector boson, and for the momentum q. What we compute here is for the massive vector boson, or for the momentum q.

A.4.7 HVIORX
This subroutine computes an off-shell scalar current H made from the interaction of a Vector boson, a flowing-In fermion and a flowing-Out R-S fermion by the FRVS vertex, and should be called as CALL HVIORX(FI, RO, VC, GR, SMASS, SWIDTH , HVIOR).
What we compute here is (2) for the momentum q. What we compute here is for the momentum q.

A.5 FRVV vertex
The FRVV vertices are obtained from the interaction Lagrangian among a fermion, a R-S fermion and two vector bosons: with the structure constant f abc , which can be handled by the MG automatically. The coupling constant GR is the product of the FRV coupling constant and the gauge coupling constant of the involving gauge boson as in the FRVS coupling; see (76).

A.5.1 IORVVX
This subroutine computes an amplitude of the FRVV vertex from a flowing-In fermion, a flowing-Out R-S fermion and two Vector bosons, and should be called as CALL IORVVX(FI,RO,VA,VB,GR , VERTEX).
What we compute here is where we use the notations What we compute here is What we compute here is What we compute here is What we compute here is for the massive vector boson, or (2) What we compute here is A.6 Checking for the new HELAS subroutines The new HELAS subroutines are tested by using the gauge invariance of the helicity amplitudes. In particular, we use the following processes; More explicitly, we express the helicity amplitudes of the above processes as M λGλg =ψ µ (pG, λG) T µν ǫ ν (p g , λ g ) or M λGλg = T µν ψ µ (pG, λG) ǫ ν (p g , λ g ) with an external spin-3/2 and a gluon wavefunction. The identity for the SU (3) gauge invariance p g ν T µν = 0 (121) tests all the above subroutines thoroughly. We also test the agreement of the helicity-summed squared amplitudes at arbitrary Lorentz frames.

B Implementation of spin-3/2 gravitinos into MadGraph
In this appendix, we describe how we implement spin-3/2 gravitinos and their interactions into MG. First, using the default mssm model in MG/ME v4 [6], we make our new model directory, mssm gravitino, including a massive gravitino (particles.dat) and its interactions with SM and SUSY particles (interactions.dat

3-point couplings
GR FRS q gro ql GFRSL q gro qr GFRSR FRV go gro g GFRV 4-point couplings GR FRVS q gro g ql GFRGSL = GFRSL*GG q gro g qr GFRGSR = GFRSR*GG FRVV go gro g g GGORGG = GFRV*G Table 2. List of the coupling constants for each gravitino vertex involving SUSY QCD particles. All the particles and the coupling constants are written in the MG notation. gro stands for a massive gravitino, q represents a light quark, and ql/qr is a left/right-handed squark. g and go are a gluon and a gluino, respectively. GR is a non-renormalizable coupling constant defined in each subroutine in App. A.
and couplings.f); we show the coupling constants for each gravitino vertex involving SUSY QCD particles in Table 2 as examples. Then we add all the new HELAS subroutines for spin-3/2 gravitinos to the HELAS library in MG.
Since the present MG does not handle spin-3/2 particles, we further modify the codes in MG to tell it how to generate the FRS, FRV, FRVS and FRVV type of vertices and helicity amplitudes, and how to deal with the helicity of external spin-3/2 particles.