Higher-spin particles at high-energy colliders
Juan Carlos Criado, Abdelhak Djouadi, Niko Koivunen, Martti Raidal, Hardi Veermäe
PPrepared for submission to JHEP
Higher-spin particles at high-energy colliders
Juan C. Criado, a Abdelhak Djouadi, b,c,d
Niko Koivunen, b Martti Raidal, b andHardi Veerm¨ae b a Institute for Particle Physics Phenomenology, Department of Physics, Durham University, DurhamDH1 3LE, United Kingdom b Laboratory of High Energy and Computational Physics, NICPB, R¨avala pst. 10, 10143 Tallinn,Estonia c CAFPE and Departamento de F´ısica Te´orica y del Cosmos, Universidad de Granada, E-18071Granada, Spain d Universit´e Savoie–Mont Blanc, USMB, CNRS, LAPTh, F-74000 Annecy, France
E-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] Abstract:
Using an effective field theory approach for higher-spin fields, we derive theinteractions of colour singlet and electrically neutral particles with a spin higher thanunity, concentrating on the spin-3/2, spin-2, spin-5/2 and spin-3 cases. We compute thedecay rates and production cross sections in the main channels for spin-3/2 and spin-2states at both electron-positron and hadron colliders, and identify the most promisingnovel experimental signatures for discovering such particles at the LHC. The discussion isqualitatively extended to the spin-5/2 and spin-3 cases. Higher-spin particles exhibit a richphenomenology and have signatures that often resemble the ones of supersymmetric andextra-dimensional theories. To enable further studies of higher-spin particles at colliderand beyond, we collect the relevant Feynman rules and other technical details. a r X i v : . [ h e p - ph ] F e b ontents e + e − collisions 82.4 Production at hadron colliders and expectations for the LHC 112.5 Experimental signatures at hadron colliders 15 Testing the standard model (SM) of particle physics and searching for new phenomenabeyond it is the main objective of high-energy colliders such as the Large Hadron Collider(LHC) at CERN. For decades, this search has been focused on new particles predicted bytheories that address the hierarchy problem of the SM and naturally explain the unbearablelightness of the Higgs boson [1, 2]. Among these, supersymmetric theories [3, 4] and modelswith extra space-time dimensions [5, 6] have played a more than considerable role. Thevast majority of these beyond the SM scenarios predict new particles that are similar to theexisting ones and, up to few notable exceptions related to gravity, have a spin smaller thanor equal to unity: new scalar (often Higgs) bosons, additional spin-1 / / .Heavy spin-2 particles have been discussed in the context of extra space-time dimen-sional scenarios [5, 6]. They appear as the massive Kaluza-Klein excitations of the masslessgraviton. However, the interactions of these heavy gravitons are very specific as, for in-stance, they couple universally to all particles [9, 10]. Massive spin-2 particles also appearin extensions of General Relativity, e.g. , in theories of bi-metric gravity [11]. However,these particles also have universal, gravity-strength interactions. These states may thusplay the role of DM [12–14], but their interactions are irrelevant for collider physics.Particles with an even higher spin have been put forward only in very few occurrences.For example, spin-5 / e.g. , Refs. [16, 17] for recent accounts.A reason for the lack of studies of generic higher-spin particles arises from problemsassociated with their nature: the absence of a physically meaningful and mathematicallyconsistent framework for performing computations with interacting higher-spin degrees offreedom, as well as the absence of a consistent ultraviolet completion. For instance, mostof the work on spin-3/2 particles is conducted in the Rarita-Schwinger framework [18]in which the spin-3/2 field is described as a vector spinor with more components thana physical spin-3/2 particle. Due to certain local symmetries, the unphysical degrees offreedom are eliminated by constraints built into the free Lagrangian. The couplings of theRarita-Schwinger field must respect these symmetries. Otherwise, the counting of physicaldegrees of freedom will be inconsistent.This fact has been often ignored in nuclear physics when computing the pion-nucleoninteraction mediated by a spin-3/2 ∆ resonance [19–24], for instance, and in collider studies In fact, a longitudinal gravitino is simply the Goldstino that signals the spontaneous breaking of globalsupersymmetry [4, 8], whose coupling is inversely proportional to the supersymmetry breaking scale, givenby the square-root of the gravitino mass times the Planck mass. In general, this leads to very light gravitinos,which we will not consider here. – 2 –f generic spin-3/2 particles. There have been attempts to cure the problem by rewritingthe interactions of the spin-3/2 fields [25–37]. Difficulties in formulating ∆-resonance in-teractions in a Lagrangian description have been discussed in Ref. [38]. In general, Rarita-Schwinger particles with minimal gauge interaction with photons, massive vector bosonsand gluons, run into inconsistencies [39] such as causality violation as well as uncontrollableunitarity violating processes at energies not far from the mass scale of the new particles.Most of the studies performed in the past were affected by such problems. Essentially, onlysupersymmetric theories with specifically fixed couplings and masses have been known toavoid those problems, suggesting that physical spin-3/2 particles should be identified withthe gravitino [40, 41].Recently, an effective field theory of a generic massive particle of any spin has beendeveloped [42], following an idea originally proposed by Weinberg [43]. Although it doesnot admit a Lagrangian description, this effective theory contains only physical higher-spin degrees of freedom and allows for a consistent computation of physical observablesfor general-spin particles. It avoids the inconsistencies that often appear in other field-theoretical descriptions of higher spin and reproduces the existing results for low-spinstates. As an illustration, this method has been applied successfully to study higher-spinDM particles in terms of a general-spin singlet with symmetric couplings to the Higgsbosons [42], a setup which automatically arises for higher-spin states .In this paper, we use the framework of Ref. [42] to study the collider phenomenology ofhigher-spin particles. In detail, we study higher-spin particles that are singlets under theSM gauge interactions and consider their simplest linear interaction Hamiltonians with SMquarks, leptons and gauge bosons. In this setup, the higher-spin particles are Majorana.Throughout the paper, we will use effective operators to describe the interactions of thehigher-spin fields, that is, j >
1, that we will denote by ψ j . The lowest order operatorslinear in ψ j have dimension 1 + 3 j for bosons and 5 / j for fermions, while the lowestorder operator quadratic in ψ j is ψ ( a ) ψ ( a ) | φ | + h.c., where φ is the SM Higgs doublet, andhas dimension 4 + 2 j . The quadratic operators are thus dominant for spins j ≥ j ≥ / ψ -linear interactions naturally dom-inate, that is, on spin 3 /
2, 2 and 3. We will also present some results for spin-5 / ψ j with the SM fields for all the cases under con-sideration, j = 3 / ,
2, 5 / ψ / and ψ decay ratesand production cross sections both at hadron and lepton colliders. After that, we discussthe most striking experimental signatures and compare those with the ones of supersym-metric and extra-dimensional theories. We shall also discuss the existing constraints onthose particles and outline potential future research directions. We extend this discussion Generic higher-spin DM states have also been studied recently in Ref. [44] using on-shell amplitudemethods for massive particles [45]. Non-relativistic scattering of generic higher-spin DM has recently beenconsidered in Refs. [46, 47]. – 3 –o the higher spin-5/2 and spin-3 cases. One of the objectives of this exploratory work is toopen the possibility of more extensive studies of higher-spin physics. In order to facilitatethis aim, we collect the required technical details in the appendix.The rest of the paper is organized as follows. In Section 2 we focus on the phenomenol-ogy of spin-3/2 particles and, in Section 3, on the one of spin-2 particles. After derivingthe effective interactions of the particles, we discuss their decay modes and productioncross sections and summarise their main signatures at the LHC. The interactions and ex-perimental signatures for spin-5/2 fermions and spin-3 bosons are presented in Section 4and Section 5, respectively. Finally, in Section 6, we discuss other implications of thesehigher-spin particles, outline future research directions and present our conclusions. Theformalism enabling consistent studies of higher-spin particles is outlined in Appendix A:we list the relevant Feynman rules, discuss the narrow width approximation, and give adetailed example of a computation of higher-spin processes. Throughout the paper we usenatural units (cid:126) = c = 1 and the metric signature (+ , − , − , − ). In high-energy particle physics, both spin-3/2 leptons [48–57] and spin-3/2 quarks [58–62]have been considered in the past. For instance, the production of spin-3/2 particles athadron colliders has been discussed in Refs. [59–62], while production at lepton collidershas been considered in Refs. [54, 55]. Indirect effects of spin-3/2 particles, through theirvirtual exchange in high energy processes, has also been discussed with some examplesbeing the t -channel exchange of a spin-3/2 lepton in processes such as e + e − → γ and eγ → eγ [63], and the exchange of spin-3/2 quark in top pair production at hadron colliders, gg, q ¯ q → t ¯ t [64]. However, it is assumed in most of these analyses that the spin-3/2 particleshave colour or electric charge, allowing their pair production in proton-proton or electron-positron collisions. This will not be the case here as such interactions would lead to anunmanageable violation of causality and unitarity [39].For example, at hadron colliders, spin-3/2 quarks that couple to gluons could be pairproduced in gluon fusion and quark-antiquark annihilation, gg, q ¯ q → ψ / ¯ ψ / . The par-tonic cross sections, which depend only on the known gauge coupling constant α s andthe particle mass m / , grows with the third power of the partonic centre-of-mass energy,ˆ σ ∝ ˆ s [59–61, 65]. Such a steep rise leads to unitarity violation at tree-level for ener-gies of the order of ≈ m [65]. The interaction needs, therefore, to be damped by someform-factors in order to remain viable at these energies.To overcome these problems, we consider a generic Majorana spin-3/2 field ψ / thatis a SM singlet. Therefore, the issues related to gauging the higher-spin fields do notappear here, while the interactions of ψ / to gauge bosons are still included. We use theeffective field theory approach of Ref. [42], so we avoid problems related to the existenceof unphysical degrees of freedom that appear in other formulations of higher spin. A briefoverview of the multispinor formalism can be found in Appendix A. Since a spin j fieldcarries an effective dimension of ∆ ψ = j + 1, the lowest dimension of the operators linear– 4 –n a SM-singlet spin-3/2 field is 7. Operators of dimension 5 are allowed when the fieldhas some non-vanishing SM charge. They are always of the form ψ j F f , where F is a SMfield-strength tensor, and f is a SM spin-1 / ψ / is a colourtriplet or sextet, ψ / gq contact interactions are possible at this level.For a singlet spin-3/2 field ψ / , there are 6 independent dimension-7 operators, −H linear = 1Λ ψ abc / (cid:104) c ijkq (cid:15) IJK u i ∗ RIa d j ∗ RJb d k ∗ RKc + c ijkl ( L iTLa (cid:15)L jLb ) e k ∗ Rc + c ijklq ( Q iTLIa (cid:15)L jLb ) d kI ∗ Rc + c φi σ µνab ( D µ ˜ φ ) † D ν L iLc + c Bi ˜ φ † σ µνab B µν L iLc + c Wi ˜ φ † σ µνab σ n W nµν L iLc (cid:105) + h.c. , (2.1)where a, b, c are two-spinor indices, i, j, k are flavour indices, I and J the colour indices and n is the SU(2)-triplet index. The coefficient c ijkl is symmetric in ij , while c ijkq is symmetricin jk . L ia and Q ia are the left-handed lepton and quark doublets L iLa ≡ (cid:32) ν iLa e iLa (cid:33) , Q iLIa ≡ (cid:32) u iLIa d iLIa (cid:33) , (2.2)while e iR , u iR and d iR are the right-handed lepton and quark singlets. B µν and W µν denotethe U(1) Y and SU(2) L field strengths and φ is the SM Higgs doublet. In the unitary gauge,one has φ = 1 √ (cid:32) H + v (cid:33) , (2.3)where v is the vacuum expectation value v = 246 GeV and H the physical Higgs bo-son produced at the LHC [1, 2]. We define D a ˙ a = σ µa ˙ a D µ , with D µ being the usual 4-vector covariant derivative, and in terms of the identity matrix σ and Pauli σ , , matrices( σ µν ) ab ≡ i (cid:104) σ µa ˙ b (¯ σ ν ) ˙ bb − σ νa ˙ b (¯ σ µ ) ˙ bb (cid:105) .The Feynman rules for the various interactions are listed in Appendix A, where we haverestricted ourselves to those operators that lead to the dominant processes, namely, to thequartic point-like interaction of the spin-3/2 particle with three fermions and to the triplevertices involving the spin-3/2 particle, a charged lepton or a neutrino and a gauge or Higgsboson; these vertices will give the dominant effects which will be discussed in this paper.One can add a Higgs or a gauge boson line to turn the three-particle vertices involvingbosons into four-particle vertices, but, in this case, this interaction will be suppressed by apower of the vacuum expectation value v or by an additional weak gauge coupling. Hence,it will lead to subleading processes, which we will not consider here.Another simplification that we will adopt in the following is the absence of flavourviolation and, hence, the spin-3/2 field will couple only to fermions of the same generation.For simplicity, we will assume that the lightest new spin-3/2 particle is the one that couplesto the first generation quarks and leptons, for which one can safely neglect the masses andthe mixing. We restrict our analysis to this case. We further assume that there are no newsources of CP-violation, implying that all couplings of the spin-3/2 field are real.– 5 –n summary, the general Hamiltonian of eq. (2.1) will take a much simpler form in termsof the lepton and quark doublets of the first generation l T = ( ν, e ) L and q T = ( u, d ) L , −H linear = 1Λ ψ abc / (cid:104) c q (cid:15) IJK u ∗ Ia d ∗ Jb d ∗ Kc + c l ( l Ta (cid:15)l b ) e ∗ c + c lq ( q TIa (cid:15)l b ) d ∗ Ic + c B ˜ φ † σ µνab B µν l c + c W ˜ φ † σ µνab σ i W iµν l c + c φ σ µνab ( D µ ˜ φ ) † D ν l c (cid:105) +h.c. . (2.4)In this Hamiltonian, the strength of the various interactions is governed by the couplings c X , which are arbitrary and which, taken one-by-one, are only constrained by the fact thatthey should be small enough for perturbation theory to hold. So, in addition to the scaleΛ, there are six parameters c X that describe the interactions of a generic singlet field ψ / with a mass m / .A few critical comments are in order. Firstly, the higher-spin particles considered inthis work are Majorana and thus, for any of their decay modes, also the conjugate modesmust be present. Secondly, each interaction term in eq. (2.4) involves the SM leptonsand quarks in different ways. Therefore, the interactions can be classified according tothe baryon and lepton number created in each interaction, namely by ∆ B and ∆ L . Indetail, the c q term creates three quarks and no anti-quarks, implying ∆ B = 1, while allother terms in (2.4) have ∆ L = 1. However, since ψ / is Majorana, no B or L quantumnumbers can be assigned to it because the conjugate operators create the configurationswith opposite quantum numbers. This is analogous to, but more general than, the caseof massive Majorana neutrinos. Thirdly, if only one of the couplings in eq. (2.4) is non-vanishing at a time, there are no severe constraints on their strength. On the other hand,if both the lepton and baryon number violating couplings are present, dangerous processeslike proton decay may occur, constraining such combinations. For collider phenomenologypurposes, we will keep only one coupling non-vanishing at a time, unless stated otherwise.We are now in a position to discuss the collider phenomenology of the spin-3/2 particle,its relevant decay modes, present constraints on its mass and couplings and the productioncross sections at hadron colliders, as well as the main signatures to which it leads. The linear Hamiltonian (2.4) permits the following fermionic decay modes of the spin-3/2particle ψ / , adopting, as stated above, the notation of the first family ψ / → udd , ¯ u ¯ d ¯ d,ψ / → e + e − ν e , e + e − ¯ ν e , d ¯ dν e , d ¯ d ¯ ν e , u ¯ de − , ¯ ude + . (2.5)We have disentangled interactions with ∆ B = 1 , ∆ L = 0 (first row) and ∆ B = 0 , ∆ L = 1(second row) as they should be treated separately. If the masses of the final state fermionscan be neglected, which is the case for the first generation, the partial decay widths canbe summarized as Γ( ψ / → f f f ) = κ f f f π m / Λ , (2.6)– 6 –here the overall factor κ f f f depends on the number of quarks in the final state, e.g. , κ e + e − ν e = κ e + e − ¯ ν e = | c l | , κ udd = κ ¯ u ¯ d ¯ d = 3 | c q | , κ d ¯ dν e = κ d ¯ d ¯ ν e = κ u ¯ de − = κ ¯ ude + = 34 | c lq | . (2.7)In the ∆ B = 0 , ∆ L = 1 case, there are also two-body decays into a lepton and amassive gauge or Higgs boson, ψ / → W + e − , W − e + , Zν e , Z ¯ ν e , γν e , γ ¯ ν e , Hν e , H ¯ ν e , . when ψ / is sufficiently heavy. The partial decay widths involving final state gauge bosonsare Γ( ψ / → W + e − ) = ( m / − M W ) πm / Λ (cid:40) c φ ( m / + 6 M W )( m / − M W ) + 16 c φ c W g M W ( m / + 2 m / M W − M W )+ 32 c W g M W ( m / + 2 m / M W + 3 M W ) (cid:41) , Γ( ψ / → Zν e ) = ( m / − M Z ) πm / Λ (cid:40) c φ m / + 6 M Z )( m / − M Z ) + 8 c φ c Z cos θ W g M Z ( m / + 2 m / M Z − M Z )+ 16 c Z cos θ W g M Z ( m / + 2 m / M Z + 3 M Z ) (cid:41) , Γ( ψ / → γν e ) = c γ v π Λ m / , (2.8)where we introduced the couplings c γ ≡ − c B cos θ W + c W sin θ W , c Z ≡ c B sin θ W + c W cos θ W , (2.9)and the partial decay width into a Higgs boson and a neutrino isΓ( ψ / → Hν e ) = c φ π m / Λ (cid:32) − M H m / (cid:33) . (2.10)Due to the Majorana nature of ψ / , decays to final states containing the correspondingantiparticles are also possible and have an equal partial decay width.The ψ / branching ratios are presented in Fig. 1 as a function of the mass m / for thesimple case in which all c X coefficients are equal. As can be seen, the decays into W ± e ± and Zν + Z ¯ ν final states are by far dominant and have branching ratios that approach thelevel of 50% each. The branching ratios for the decays into Hν and γν final states are atthe level of a few percent and are comparable for masses around 300 GeV, but they canreach the level of 10% if the ψ / mass is much larger or smaller (a factor ≈ - - m ( GeV ) B R W + e - + W - e + Z ν + Z νγν + γν H ν + H ν e + e - ν + e + e - ν udd + udddd ν + dd ν + ude - + ude + Figure 1 . The decay branching ratios of the ψ / states into the various final states as functionsof m / for Λ = 1 TeV and c q = c l = c lq = c B = c W = c φ = 1. When the mass of ψ / exceeds the electroweak scale, it roughly holds that Γ ψ / ∝ m / / Λ , thus the total width can grow extremely rapidly with the mass. For instance,taking Λ = 1 TeV and all coefficients equal to unity, c X = 1, we find a total width of about0 . m / = 200 GeV. However, when increasing the mass to m / = 800 GeV ,the total width will grow by several orders of magnitude to about 0 . m / < ∼ Λ, the total width remainsextremely small and cannot be resolved experimentally. e + e − collisions The most stringent constraints on the new physics scale Λ arise from the experimentallimits on proton lifetime. Since the spin-3/2 fermion couples to the SM leptons and quarksaccording to eq. (2.4), if only one interaction term in this Hamiltonian is present, theconstraints on ψ / interactions appear only from collider physics and are not stringent,as will be seen below. However, if for instance c q and c l are both non-vanishing, ψ / mediated processes like ud → ¯ de + e − ν may give rise to the proton decay to a 4-body finalstate. Assuming the ψ / mass to be close to the cut-off scale, m ψ / ∼ Λ , and taking c q = c l = 1 , our order-of-magnitude estimate for the proton lifetime provides a constraintΛ > ∼ O (10) TeV. This scale is so high that there is little chance that it has been probed sofar and, in principle, it should be directly accessible only at future colliders, such as the100 TeV FCC-hh machine [66]. Thus, future colliders may be able to probe the parameterspace of these higher-spin particles which is currently not constrained by any data.Nevertheless, the previous strong constraints from proton decay can be simply evadedby requiring that the operators that lead to the ∆ B = 0 , ∆ L = 1 and ∆ B = 1 , ∆ L = 0possibilities do not occur at the same time and, hence, either c q or c l , c ql should be zero if oneconsiders the four-fermion operators. In this case, the only constraints on the new statescome from collider searches. We will discuss in the following some of these experimentalconstraints which should have been obtained before the start of the LHC.– 8 –he most immediate constraints on the ∆ B = 0 , ∆ L = 1 spin-3/2 interactions wouldcome from W, Z as well as Higgs boson decays. Indeed, for a mass m / (cid:46) M W , M Z , M H ,these particles could decay into a ψ / and a lepton, W ± → ψ / e ± , Z → ψ / ν, ψ / ¯ ν , H → ψ / ¯ ν, ψ / ν. The partial widths of these decay modes areΓ( W ± → ψ / e ± ) = ( M W − m / ) πM W Λ (cid:40) c φ ( m / + 6 M W )( M W − m / ) + 16 c φ c W g M W (3 M W − m / M W − m / )+ 32 c W g M W (3 M W + 2 m / M W + m / ) (cid:41) , Γ( Z → ψ / ν e ) = ( M Z − m / ) πM Z Λ (cid:40) c φ m / + 6 M Z )( M Z − m / ) + 8 c φ c Z cos θ W g M Z (3 M Z − m / M Z − m / )+ 16 c Z cos θ W g M Z (3 M Z + 2 m / M Z + m / ) (cid:41) , Γ( H → ψ / ν ) = | c φ | π Λ M H m / (cid:32) − m / M H (cid:33) . (2.11)Normalized to the total experimentally measured decay widths, Γ tot Z = 2 . tot W =2 .
085 GeV [67] and Γ tot H = 4 .
07 MeV in the SM [68], and, as before, assuming that c φ = c W = c B = 1 and Λ = 1 TeV, the branching ratios for Z, W decays areBR( Z → ψ / ν e ) (cid:39) × − , BR( W ± → ψ / e ± ) (cid:39) × − , (2.12)in the favorable case where phase space effects are ignored, i.e. , m / (cid:28) M W,Z . Theserates are extremely small and cannot be measured despite of the precise measurement ofthe massive gauge boson total widths ∆Γ tot Z / = Γ tot Z (cid:39) .
1% and ∆Γ tot W / Γ tot W (cid:39) H → ψ / ¯ ν e ) → m / (cid:28) M H as it is proportional to m / .Nevertheless, for instance, the process Z → ψ / ν e with subsequent decays ψ / → e + e − ν e , d ¯ dν e or u ¯ de − , should have been observed in Z decays at LEP1 due to its very highstatistics, especially since the signature is rather clean, for O (1) values of the c X coefficientsand not too large scale Λ. For the example given above in which BR( Z → ψ / ν ) ≈ × − ,one obtains about 30 clean events for the 10 Z bosons produced at LEP1. Thus, one ψ / → W ∗ e ± → f ¯ fe ± and ψ / → Z ∗ ν e → f ¯ fν e arealso possible. In this case f is an almost massless SM fermion which, for some values of the coefficients c X could have comparable rates to the direct decays above. The mode ψ / → H ∗ ν e , in turn, should besuppressed by the extremely small total decay width of the H boson. – 9 –resumably already has the lower bound m / < ∼ M Z for parameter values that allow forthe production of the ψ / particles at the LHC.The ψ / particle could have been produced at LEP2 for even larger masses thanabove, as the centre-of-mass energy of the collider slightly exceeded √ s = 200 GeV. Indeed,taking advantage of the four-point interaction ψ / e + e − ¯ ν e , the new state can be producedat electron-positron colliders in the process e + e − → ψ / ¯ ν e , ψ / ν e , with a differential cross section dσ ( e + e − → ψ / ¯ ν e ) d cos θ ≡ dσ ( e + R e − L → ψ / ¯ ν iL ) d cos θ + dσ ( e + L e − R → ψ / ν i ) d cos θ = c l πs s Λ F (cid:48) ( s, m / ) , (2.13)where s, t, u denote the Mandelstam variables, θ is the scattering angle and we defined thefunction F (cid:48) ( s, m ) ≡ (cid:18) − m s (cid:19) stu − m ( st + su + tu ) s (2.14)for future convenience. The corresponding total cross section is σ ( e + e − → ψ / ¯ ν e ) = c l πs s Λ F ( s, m / ) , (2.15)where F ( s, m ) ≡ − m s + m s . (2.16)The cross section of a single ψ / produced in association with a neutrino or an an-tineutrino through the ψ / νe + e − contact interaction is depicted Fig. 2 as a function ofthe mass m / . We have chosen √ s = 200 GeV which typically corresponds to the LEP2centre-of-mass energy, and Λ = 1 TeV and c l = 1. As can be seen, the cross sections arerather small for such parameters, being of the order of a few fb when m / is close to 100GeV. Bearing in mind that the total luminosity at LEP2 was of the order of (cid:82) L ≈ − , this means that such a parameter set could not be probed. The main reason is thatthe rate is suppressed by a factor s / Λ . in our example we used Λ = 1 TeV and a c.m.energy √ s = 200 GeV, corresponding to a suppression factor of 0 . ≈ × − .Hence, for masses m / <
200 GeV, only smaller values of the scale Λ and largercoefficients c l , can be excluded at LEP2 via this channel. At future electron-positroncolliders, these processes should have more chances to be observed. At the planed FCC-eeoption with a c.m. energy of 250 GeV to 350 GeV [69], the still present large suppressionfactor will be compensated by the extremely high integrated luminosity, expected to beof the order of a few ab − , i.e. , four to five orders of magnitude higher than at LEP2.At planned higher energy electron-positron colliders such as the CLIC machine at CERNwith an expected c.m. energy in the TeV range and above, the factor s / Λ ceases to bepenalizing if the scale Λ is not too high. – 10 – - - - m ( GeV ) σ ( p b ) Figure 2 . The production cross section of spin-3/2 particles in the point-like processes e + e − → ψ / ¯ ν e + ψ / ν e at LEP2 energies, √ s = 200 GeV, as a function of the mass m / for the choice ofthe effective Hamiltonian parameters Λ = 1 TeV and c l = 1. The mass of the ψ / particle can be fully reconstructed in the considered process bylooking at the decays ψ / → W e ± → q ¯ qe ± and eventually also ψ / → u ¯ de − as thereis no missing energy involved. Moreover, consider the decay ψ / → Zν e → e + e − ν e andeventually the direct and more rare decays ψ / → e + e − ν e . They generate the sametopology as the pair production of selectrons, the spin–zero superpartners of the electronin the minimal supersymmetric extension of the SM (MSSM), with the selectrons decayinginto an electron and the lightest spin-1/2 neutralino χ , which is supposed to be stableand escapes detection. The process is thus e + e − → ˜ e + ˜ e − → e + e − χ χ for which no eventhas been observed at LEP2 and the limit m ˜ e >
107 GeV has been set [67]. In our case,as we are dealing with single production of the new state in association with a masslessneutrino, this should translate into a limit m / > ∼
200 GeV for optimistic values of theeffective Hamiltonian parameters.Finally, there is another process for producing the ψ / particle at lepton colliders,namely e + e − → γ ∗ , Z ∗ → ψ / ν e through the operator with coefficient c φ . Its cross sectionis expected to be smaller than the one of contact interaction processes and we will discussthis process only in the LHC context to which we turn next. The ψ / state can be produced at hadron colliders in a leading order process through the∆ B = 1, ∆ L = 0 interaction which couples it to three quarks. In the first generation,several sub-processes involving right-handed up and down type quarks are contributing atthe partonic level, u R d R → ψ / ¯ d R , ¯ u R ¯ d R → ψ / d R , d R d R → ψ / ¯ u R , ¯ d R ¯ d R → ψ / u R . – 11 –hese subprocesses have equal partonic cross sections which, in term of the scattering angle θ , take the differential form d ˆ σ ( u R d R → ψ / ¯ d R ) d cos θ = d ˆ σ (¯ u R ¯ d R → ψ / d R ) d cos θ = d ˆ σ ( d R d R → ψ / ¯ u R ) d cos θ = d ˆ σ ( ¯ d R ¯ d R → ψ / u R ) d cos θ = c q π ˆ s ˆ s Λ F (cid:48) (ˆ s, m / ) , (2.17)where √ ˆ s is the partonic centre-of-mass energy and F (cid:48) was given in eq. (2.14). Integratingover the scattering angle, the total cross section for each partonic process simply readsˆ σ i ( q q → ψ / q ) = c q π ˆ s ˆ s Λ F (ˆ s, m / ) , (2.18)with F given in eq. (2.16). To obtain the total hadronic cross section, one should convolutethe four partonic cross sections ˆ σ i , with i = 1 − , over the parton structure functions of thecorresponding quarks in the initial state, and sum over the four possibilities of the partonicprocess. In our numerical analysis, the parton structure functions are chosen to be thoseof the MSTW2008 fit [70].In the case of ∆ B = 0 , ∆ L = 1 interactions, there is a similar process as the oneabove by which ψ / can be produced at hadron colliders at leading order. This is enabledby operator with the c lq coefficient that couples ψ / to two quarks and a lepton. Similarlyto the previous case, there are four possible partonic processes d L ¯ d R → ψ / ¯ ν L , ¯ d L d R → ψ / ν L , u L ¯ d R → ψ / e + L , ¯ u L d R → ψ / e − L . with the differential partonic cross sections d ˆ σ ( d L ¯ d R → ψ / ¯ ν L ) d cos θ = d ˆ σ ( ¯ d L d R → ψ / ν L ) d cos θ = d ˆ σ ( u L ¯ d R → ψ / e + L ) d cos θ + d ˆ σ (¯ u L d R → ψ / e − L ) d cos θ = c lq π ˆ s ˆ s Λ F (cid:48) , (2.19)The total cross sections of the individual partonic processes areˆ σ i ( q ¯ q → ψ / (cid:96) ) = c lq π ˆ s ˆ s Λ F . (2.20)One then should convolute these partonic rates over the corresponding parton structurefunctions and sum over the four possibilities for the individual channels to get the totalhadronic cross section of the process.The resulting cross sections for the production of a single ψ / at the LHC due tocontact interactions are presented in Fig. 3 as a function of the mass m / for a colliderc.m. energy √ s = 14 TeV and effective parameters Λ = 1 TeV and c q = c lq = 1. Onecan easily rescale results for other values of these parameters as the cross sections areproportional to c / Λ .The production cross sections are now fairly large, well above the picobarn level, sincethe chosen scale of new physics is now of the same order as the partonic c.m. energy such– 12 – p → ψ / q + Xpp → ψ / l + X
200 400 600 800 10000.110 m ( GeV ) σ ( p b ) Figure 3 . The single production cross sections of ψ / as a function of m / for the two processes pp → ψ / q + X and pp → ψ / ν + X corresponding to the point-like partonic processes qq → ψ / q and q ¯ q → ψ / (cid:96) at the LHC with √ s = 14 TeV. The effective Hamiltonian parameters have beenset to Λ = 1 TeV and c q = c lq = 1. that the suppression ˆ s / Λ is not effective anymore. One should also keep in mind thatthe integrated luminosity which has been collected at the present Run 2 of the LHC is ofthe order of (cid:82) L ≈
140 fb − and, hence, is four orders of magnitude higher than the totalluminosity obtained at LEP2. Hence, one could collect already a million of ψ / events fora production cross section of the order of 10 pb as expected in the process pp → ψ / l . Theintegrated luminosity is expected to significantly increase at the next Run 3 of the colliderand even more at the high-luminosity option of the LHC (HL-LHC), where 3 ab − of datacould ultimately be collected. Thus, not only parameters that have never been probedbefore can be reached at the LHC but also, the sensitivity will benefit from the increasein luminosity and HL-LHC could explore a completely uncovered territory compared toRun 2.Notice that the two cross sections in Fig. 3 differ by more than an order of magnitude,although c q = c lq = 1, that is, they are both induced by contact interactions with equalcouplings. The first reason is due to the colour multiplicity producing a factor of 8 in therate of the process resulting from the coupling c q compared to the one with the coupling c lq , as can bee seen from eqs. (2.18–2.20). In addition, the cross section in the former caseinvolves only quarks, while the process with the operator of coupling c lq involves a quarkand an anti-quark. In proton-proton collisions, the latter cross section is suppressed bythe sea-quark parton structure functions. These two features explain the factor of 30 to 40difference in the two production rates.For the ψ / particle that interacts via operators involving gauge and Higgs bosons,– 13 –here is another process that allows its production at hadron machines: the one occurringthrough the s -channel exchange of a virtual gauge boson in quark-antiquark annihilation.There is also a process with a Higgs boson exchange in the s -channel but, because the Higgsboson couples extremely weakly to the first generation quarks, this mode gives negligiblecross sections.First, there are the neutral current processes involving photon and Z –boson exchange q ¯ q → γ ∗ , Z ∗ → ψ / ν , ψ / ¯ ν , with a neutrino or an antineutrino in the final state. There are also the charged currentprocesses with W –exchange, leading to a charged lepton in the final state, q ¯ q (cid:48) → W ±∗ → ψ / e ± . In the neutral current channel, the differential production cross section of ψ / , whensumming over the L, R helicities of the initial quarks, reads dσ ( q ¯ q → γ ∗ , Z ∗ → ψ / ν e ) d cos θ = dσ ( q ¯ q → γ ∗ , Z ∗ → ψ / ¯ ν e,L ) d cos θ + dσ ( q ¯ q → γ ∗ , Z ∗ → ψ / ν e,L ) d cos θ = 132 π ˆ s (cid:32) − m / ˆ s (cid:33) (cid:88) α = L,R |M qαα | , (2.21)where the amplitude squared of the process is |M αα | = 4 πv e π (cid:40) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c γ e q ˆ s + c Z g Zqα ˆ s − M Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F + c φ g ( g Zqα ) cos θ W (ˆ s − M Z ) F (cid:41) F . (2.22)where e q denote the electric charges of quarks, e u = 2 / e d = − /
3, the Z couplingsto L/R quarks are given by g Zqα = I q α − e q sin θ W sin θ W cos θ W , (2.23)with the isospin I q = ± / I q = 0 for the right-handedones, and we introduced the three functions F ≡ ˆ s (ˆ u − m / ) , F ≡
116 ˆ u (ˆ s − m / ) , F ≡ m / ˆ t + 3ˆ s ˆ u. (2.24)In the charged current process, the ψ production cross section is driven only by theleft-handed quarks, dσ ( q ¯ q → W ∗ → ψ / e ) d cos θ = dσ ( u L ¯ d L → W + ∗ → ψ / e + L ) d cos θ + dσ ( d L ¯ u L → W −∗ → ψ / e − L ) d cos θ = 132 π ˆ s (cid:32) − m / ˆ s (cid:33) |M ( q L ¯ q L ) | , (2.25)where the amplitude squared is |M ( q L ¯ q L ) | = g v | V ud CKM | (ˆ s − M W ) (cid:104) c W F + c φ F (cid:105) F . (2.26)– 14 – p → ψ / ν + Xpp → ψ / l + X
200 400 600 800 100010 - m ( GeV ) σ ( p b ) Figure 4 . The production cross sections of ψ / particles at the LHC as a function m / forthe processes pp → ψ / ν + X and pp → ψ / e + X , corresponding to the parton level processes q ¯ q → γ ∗ , Z ∗ → ψ / ν and q ¯ q → W ∗ → ψ / e given at eqs. (2.21) and (2.25) respectively. Thechoice of parameters is √ s = 14 TeV, Λ = 1 TeV and c B = c W = c φ = 1. The production cross sections at the LHC for a single ψ / state resulting from s -channel gauge boson exchange are presented in Fig. 4 for √ s = 14 TeV, Λ = 1 TeV and c B = c W = c φ = 1. Again, one can easily rescale the results in Fig. 4 for different values ofthe new physics scale Λ. The cross sections for producing a neutrino or a charged lepton inthe final state are almost identical. A comparison with the cross sections in Fig. 3 showsthat the processes mediated by s -channel gauge bosons are subdominant compared to theones induced by the contact interactions, at least two orders of magnitude smaller. Thisoccurs because, in the effective field theory, the two cross sections scale differently with thepartonic c.m. energy ˆ s . Indeed, the former scales as 1 / ˆ s as is usually the case for s -channelgauge boson exchanges, while the latter scales as ˆ s as a result of the contact interaction.This leads to a very striking difference at the high-energy collisions that occur at the LHC. Since our main goal in this paper is to make a general survey of the phenomenologyof the new particles with a spin higher than unity, a very detailed account of all theexperimental limits that are set by various experiments on their masses and couplings andthe expectations in the search for these particles in the future, is clearly beyond our scope.We will nevertheless briefly describe the various signatures of these particles at hadronmachines and list the various channels that can be exploited efficiently in their search atthe LHC.Starting with the simpler case of spin-3/2 interactions with ∆ B = 1 and ∆ L = 0, the– 15 –nique signature of ψ / production at hadron colliders would be the 4-jet final states qq → ψ / q → q ⇒ pp → j . (2.27)In turn, in the case where ψ / interacts via the ∆ B = 0 and ∆ L = 1 operators, thephenomenology is richer and a plethora of final states and, hence, signatures are possi-ble. Focusing first on the operators that involve only point-like fermionic interactions, thevarious possible final states are qq → ψ / ¯ ν → eeν ¯ ν , qqν ¯ ν , qqeνqq → ψ / e → eeeν , qqeν , qqee ⇒ pp → eeE misT , eeeE misT , qqE misT , qqeE misT , qqee , (2.28)with E misT the transverse missing energy carried by the escaping neutrinos. If, instead, oneconsiders the operators in which ψ / is coupled to a lepton and a gauge or Higgs boson,the various possible topologies become qq → ψ / ¯ ν → W e ¯ ν , Zν ¯ ν , Hν ¯ νqq → ψ / e → W ee , Zeν , Hνe ⇒ pp → ZE misT , HE misT , W eE misT , ZeE misT , HeE misT , W ee . (2.29)In the processes above, as we have the subsequent decays Z → ll, W → lν and Z, W → q ¯ q , the initial and final states are just the same as those that occur in the processes ofeq. (2.28); if one ignores the difference of magnitude in the branching ratios, only theangular distributions are different, so that both should be combined in principle. However,there are also second and third generation leptonic decays of the massive gauge bosons,which can be used to discriminate between the two possibilities.As shown above, the vast majority of these processes involve missing energy in thefinal state. The latter is a typical signature of supersymmetry in models in which thediscrete symmetry called R -parity is conserved [71]. Due to this symmetry, the lightestsupersymmetric particle (LSP), generally the lightest neutralino χ , would be stable andescape detection. Thus, one can use the vast number of supersymmetry searches that havebeen performed at the LHC and adapt them to our specific case.For instance, the signatures pp → eeνν and pp → qqνν are simply those that appear inthe production of right-handed selectrons and first generation quarks that decay directlyinto the LSP and leptons or light quarks, pp → ˜ e + R ˜ e − R → e + e − χ χ and pp → ˜ q ˜ q ∗ → q ¯ qχ χ .This is also the case with signatures like Z/H + E misT which could be due to the productionof the lightest and next-to-lightest neutralino, the latter decaying into the LSP and a gaugeor Higgs boson, q ¯ q → χ χ → Z/Hχ χ . Topologies like Ze/He/W e + E misT , as well asthose with three and two electron final states produced via four-fermion operators, couldbe due to slepton pair production and decays through the chain pp → ˜ e R ˜ ν L → eχ ν e χ → eνχ χ + Z/H → Ze + E misT ( eee + E misT when Z → e + e − ) or pp → ˜ ν ˜ ν → e ∓ χ ± νχ → e ± W ∓ + E misT ( ee + E misT when W → eν ).Signatures involving two quarks and charged leptons or neutrinos in the final states aretypical of those involving leptoquarks [72, 73] produced in pairs, pp → q ¯ q, gg → LQLQ → eqeq, eqqν, qνqν . The latter signature being also similar to what happens in supersymmetricmodels for squark pair production, pp → ˜ q R ˜ q ∗ R → qχ ¯ qχ as discussed above. The signature– 16 –ith W ee final states that appear at the end of eq. (2.29) would be similar to the one inwhich a heavy neutrino N is produced in association with an electron through mixingand decays into an electron and a charged W boson, q ¯ q → W ∗ → eN → eeW ; see forinstance Refs. [74, 75]. Other signatures involving lepton final states can also occur in theproduction of heavy neutral or charged leptons.Finally, for the ∆ B = 1 and ∆ L = 0 spin-3/2 interactions, only the 4-jet final statetopology will be possible, eq. (2.27). This signature is similar to that of squark pairproduction in q ¯ q annihilation or gg fusion, with each squark decaying into two jets in R -parity violating supersymmetric processes [76].In fact, there is a tight connection between our scenario and the one of R -parityviolating supersymmetric models. Indeed, the supersymmetric superpotential describingviolation of R = ( − B + L +2 s parity, with s being the spin-number, is [77] W /R = λ ijk L i L j ¯ E k + λ (cid:48) ijk L i Q j ¯ D k + λ ” ijk ¯ U i ¯ D j ¯ D k , (2.30)where L, E, Q, D, U are doublet and singlet superfields involving SM fermions and theirspin-0 superpartners. One can see that this superpotential is similar to the one that appearsin the first line of the Hamiltonian (2.4) which describes the four–particle interactions of thespin-3/2 particle. Hence, most of the physics of the spin-3/2 particle, as least when the fourpoint-like vertices are concerned, can be described by an R -parity violating phenomenon.For instance, the process discussed above, q ¯ q → ψ / q → qqqq is similar to q ¯ q, gg → ˜ q ˜ q ∗ with the decay ˜ q → q ¯ q occurring through the λ (cid:48)(cid:48) ¯ u ¯ d ¯ d operators.The four lepton signature resembles the one for slepton pair production with subse-quent decays of these into charged leptons or neutrinos via the operator λ ¯ LL ¯ e , leadingto eeeν and eeνν final states. Also, slepton or squark pair production, in which the pairthen decays through the operator λ (cid:48) LQ ¯ d into, respectively, quark pairs and lepton-quarkpairs, which finally lead to the qqee, qqeν and qqνν topologies that also appear in thespin-3/2 case.Hence, many constraints on squarks and sleptons obtained in R -parity violating pro-cesses by the ATLAS and CMS collaborations, e.g. , Refs. [78, 79], can be used to setconstraints on the spin-3/2 particle mass and couplings. As this particle can be producedin association with light SM states, the expectation on the upper limit of the mass m / might range from a few TeV, if the couplings to SM particles are order unity, to the levelof 100 GeV only, if these couplings are extremely small.Note, however, that in all the cases discussed above, the same final states have com-pletely different kinematical distributions. For instance, in the decays of ψ / into threejets or into three electrons, each of these particles carries a comparable amount of energywhich is rather characteristic to a 1 → R -parity conservation as, for instance, thesupersymmetric particles are produced in pairs leading to the presence of two escapingneutralinos in their decays, while only a single ψ / particle is produced in our case, thatis, the process would involve only one escaping particle. Hence, care should be taken in– 17 –dapting the experimental analyses performed in the other scenarios, and it is wiser toconduct some new ones with the specific kinematics of the spin-3/2 particles.In addition, in most of the signatures discussed above, the mass of the ψ / statecannot be directly reconstructed from the four-momenta of the final particles as theyinvolve missing energy due to the escaping neutrinos. However, there are two importantexceptions: the decays ψ / → W e ± → q ¯ qe ± as well as ψ / → ude ± do not involve missingenergy and the momenta of the two very energetic jets and the electron in the final statecombine to form an invariant mass that coincides with the mass of the ψ / state.One spectacular signature of the ψ / particle would be its production in the pp → ψ / ν process and its decay through the ψ / → γν channel leading to a single and veryenergetic photon in the final state and a large amount of missing energy. This might be asignature of supersymmetry in some cases, like when the LSP and next-to-LSP neutralinoare produced in association, pp → χ χ , with a small mass difference that makes the decay χ → χ γ mode rather frequent. This νγ spectacular signature has also been discussed longago in the context of excited neutrinos which can magnetically de-excite into a neutrinoand a photon [80]. There is also the mono-Higgs signature, pp → ψ / ¯ ν → Hν ¯ ν , whichcould be interesting to exploit if the associated rate is not negligible.A more detailed account of all these issues will be postponed to a forthcoming study. The lowest dimension of operators linear in the spin-2 field ψ is also 7. For a SM singletparticle, ψ abcd where a, b, c, d are two-spinor indices, we then have the following effectiveHamiltonian −H linear = 1Λ ψ abcd (cid:104) c B σ µνab σ ρλcd B µν B ρλ + c W σ µνab σ ρλcd W iµν W iρλ + c G σ µνab σ ρλcd G Aµν G Aρλ (cid:105) + h.c. , (3.1)where B µν , W µν and G µν are the U(1) Y , SU(2) L and SU(3) C field strengths, respectively,and A is a colour-octet index. The coefficients c B , c W and c G are in principle arbitrary. Wecould chose them to be equal as was the case in the spin-3/2 discussion. However, mostprobably, if they originate from gauge interactions in the ultraviolet regime, they couldeventually have the same magnitude as, respectively, the electroweak couplings g and g and the strong coupling g such that c G (cid:29) c B , c W . This is the assumption that we willmake here, c B = g ( M Z ), c W = g ( M Z ) and c G = g s ( M Z ).From this Hamiltonian, one can see that the spin-2 particle ψ will couple only togluons and electroweak gauge bosons and that there are no couplings to fermions norcouplings to the Higgs boson at this order. One could immediately ask whether the termsin eq. (3.1) resemble the ones of a massive Kaluza-Klein graviton which was widely discussedin the literature in the context of extra space-time dimensional models, in particular, thosewith large extra dimensions [5] and Randall-Sundrum [6]. There are significant differencesbetween these scenarios and ours. In particular:– 18 – ψ does not couple to fermions and Higgs bosons, while the massive graviton is moredemocratic and couples to all particles;– unlike in extra dimensional models, the ψ couplings to gauge bosons are not universal– the coefficients c B , c W , c G are free parameters and can be different. A practical conse-quence is that the ψ Zγ couplings occur for generic coefficients c W , c B contrary to thecase of extra-dimensional theories;– the structure of the interaction operators are different. As a consequence, angular dis-tributions are different from the case of Kaluza-Klein gravitons.These differences come from the fact that our field ψ is a generic massive spin-2 fieldwith interactions given by eq. (3.1). Thus, it cannot be identified with the massive graviton.Details about the Lorentz group representations of our ψ and the ones of massive gravitoncan be found in Ref. [42]. We now proceed to the phenomenological part and study theproperties of the ψ state, namely its main decay modes and production channels. The spin-2 particle will decay dominantly into the following two-body final states, ψ → γγ, ZZ, Zγ, W W, gg. Due to the non-Abelian nature of the SM gauge bosons, also three- and four-body finalstates are possible, ψ → γW W, ZW W, g,ψ → W, γγW W, ZZW W, γZW W, g. We will ignore these additional modes as they are of higher order and stick to the two-bodyfinal states decays.The partial widths of these dominant decays areΓ( ψ → gg ) = c G π m Λ , (3.2)Γ( ψ → γγ ) = ( c B cos θ W + c W sin θ W ) π m Λ , (3.3)Γ( ψ → ZZ ) = ( c B sin θ W + c W cos θ W ) π ( m − M Z ) / ( m + 2 m M Z + 36 m M Z )Λ , (3.4)Γ( ψ → W + W − ) = c W π ( m − M W ) / ( m + 2 m M W + 36 m M W )Λ , (3.5)Γ( ψ → Zγ ) = ( c B − c W ) sin θ W cos θ W π ( m − m Z ) Λ (cid:18) m + 3 M Z m + 6 M Z m (cid:19) . (3.6)The ψ branching ratios are presented in Fig. 5 as a function of the mass m for thespecific set of coefficients c B = g ( M Z ) = 0 . , c W = g ( M Z ) = 0 .
65 and c G = g s ( M Z ) =1 .
22. As expected, the decays into gluons are by far dominant as they involve the stronginteraction. The decays
W W, ZZ and γγ are at a level of 10% to 2%, respectively, whilethe decay into Zγ final states stays at the permille level.– 19 – m ( GeV ) B R gg γγ ZZWWZ γ Figure 5 . The branching ratios of ψ corresponding to different final states as functions of m .The choice of parameters is Λ = 1 TeV and c B = g Y ( m Z ) , c W = g ( m Z ) , c G = g s ( m Z ). Similarly to the ψ / field, the total decay width of a relatively massive ψ state growsextremely rapidly with the mass since Γ ψ ∝ m / Λ . For instance, taking Λ = 1 TeV andcoupling constants c B = g Y ( m Z ) , c W = g ( m Z ) and c G = g s ( m Z ), one finds a total widthof about 2 MeV when m / = 200 GeV. By increasing the mass to m / = 800 GeV, thetotal width will grow by over four orders of magnitude to about 30 GeV.As expected, our results differ from extra-dimensional theories reported in the litera-ture [9, 10, 81–84], highlighting the point that a generic massive spin-2 field with arbitrarycouplings is not necessarily a massive graviton. Because our spin-2 particle couples directly to two gluons but not to quarks, it will mainlybe produced in gg fusion, gg → ψ , at hadron colliders with the partonic production cross sectionˆ σ ( gg → ψ ) = 16 πc G m ˆ s / Λ δ ( √ ˆ s − m ) . (3.7)As usual, to obtain the total cross section, one has to fold this partonic cross section withthe gluon luminosities, which are extremely high at high-energies. The latter cross sectionis shown in Fig. 6 as a function of the spin-2 particle mass m for the LHC c.m. energy √ s = 14 TeV, and for Λ = 1 TeV and c G = 1. Note that according to Refs. [85, 86] inwhich the higher order corrections to a Kaluza-Klein graviton have been discussed, theremight be a K -factor of order 2 at the LHC for a TeV scale spin-2 state. Such a K -factorhas not been included in the plot. As can be seen in Fig. 6, because we are discussing thesingle production of a resonance in the s -channel, the cross section can be huge. It is atthe level of 10 pb at low masses, m ∼
100 GeV, but increases to a few times 10 pb when– 20 –
00 400 600 800 100010 m ( GeV ) σ ( p b ) Figure 6 . The production cross section of ψ as a function m for the process pp → ψ + X corresponding to the partonic process gg → ψ at the LHC with √ s = 14 TeV for Λ = 1 TeV and c G = 1. m ∼ m being compensated by the lower probability offinding a gluon in the proton at high resonance masses. With the luminosity of 140 fb − already collected at the LHC and the additional data to be collected at Run 3, scales of afew TeV could be probed for the chosen m range and c G value. At HL-LHC, scales up to10 TeV could be probed in this process.Note that spin-2 particles can also be produced at photon-photon colliders as s -channelresonances in the γγ → ψ process. The cross section for this process is similar to that ofthe two gluon process and is given by σ ( γγ → ψ ) = 2 π ( c B cos θ W + c W sin θ W ) m s / Λ δ ( √ s − m ) . (3.8)One should then fold this expression with the relevant luminosity of the two photons: ei-ther Weizs¨cker-Williams photons in the usual e + e − mode when the photons are simplyradiated from the initial beams or Compton backscattered photons from high-power laserbeams in the γγ option of future linear colliders. In addition, one has higher order pro-cesses at electron-positron and hadron machines from vector boson and photon fusion, W W, ZZ, Zγ, γγ → ψ with the former being dominant as usual (because the charged cur-rent couplings are larger than the neutral current ones, in general). If only the transversecomponents of the gauge bosons contribute to the production of the spin-2 object, therates are expected to be tiny as the luminosity for transverse W, Z bosons is small at highenergies, much smaller than the one for the longitudinal component. We shall ignore thesehigher order processes at the moment.The resonant production will, of course, be followed by the decays of ψ into two gaugebosons. Hence, the main topologies to be searched for at proton colliders such as the LHC– 21 –ill be gg → ψ → gg, W W, ZZ, Zγ, γγ. These final states have been searched for at the LHC, in particular in the context of thenotorious 750 GeV two-photon resonance that was thought to be observed at the earlystage of the Run 2 LHC but which turned out to be a statistical fluctuation. The gg final state is expected to be the dominant one, but one should also focus on the V V onesas they are much cleaner. In particular, the γγ signature should be the best as, in themassive bosonic modes, the cleanest final states are those involving charged leptons ( e, µ )or neutrinos (missing energy) which are penalized by the small branching ratios as W, Z bosons dominantly decay into q ¯ q . Hence, the best and most efficient detection signal mightbe gg → γγ . This signature is very clean and has been discussed at length in the literature;see, for instance, Ref. [87].Finally, we should note that in the experimental signatures, there are also some simi-larities with the production of spin-1 new neutral bosons Z (cid:48) and Kaluza-Klein gluons g KK .The production modes should be in both cases due to q ¯ q → Z (cid:48) , g KK . However, gg → g KK can also come from the anti-symmetric part of the triple gluon vertex, which invalidatesFurry theorem that forbids on-shell 3 vector vertices. So, we have the same initial topology.The Z (cid:48) will mainly decay into q ¯ q (the W W state has a low rate and there are no
ZZ, Zγ, γγ final states). This cannot be discriminated from gluons, except if one looks at angular dis-tributions. But new Z (cid:48) bosons have direct decays into lepton pairs [88], and g KK decaysmostly into heavy quark pairs [89, 90]. Therefore, one should easily discriminate betweenthese scenarios, even without studying the angular distributions. Turning to spin-5/2 particles, we will simply list here all their possible effective interac-tions and only briefly describe their collider phenomenology without going into details andperforming a numerical analysis. The lowest dimension of the operators linear in a singletspin-5/2 field ψ / is 10.The effective Hamiltonian at this order is −H int = 1Λ ψ abcde / (cid:104) ( c (1) l ) ijk ( D ˙ aa L TLbi ) (cid:15) ( D c ˙ a L Ldj ) e ∗ Rek + ( c (2) l ) ijk ( D ˙ aa L TLbi ) (cid:15)L Ldj ( D c ˙ a e ∗ Rek )+ ( c (1) q ) ijk (cid:15) IJK ( D ˙ aa d ∗ RbIi )( D c ˙ a d ∗ RdJj ) u ∗ ReKk + ( c (2) q ) ijk (cid:15) IJK ( D ˙ aa d ∗ RbIi ) d ∗ RdJj ( D c ˙ a u ∗ ReKk )+ ( c (1) ql ) ijk ( D ˙ aa Q TLbIi ) (cid:15) ( D c ˙ a L Ldj ) d ∗ ReIk + ( c (2) ql ) ijk ( D ˙ aa Q TLbIi ) (cid:15)L Ldj ( D c ˙ a d ∗ ReIk )+ ( c (3) ql ) ijk Q TLbIi (cid:15) ( D ˙ aa L Ldj )( D c ˙ a d ∗ ReIk )+ ( c (1) qG ) ijk { u ∗ Rai d ∗ Rbj d ∗ Rck } A S σ µνde G Aµν + ( c (2) qG ) ijk { u ∗ Rai d ∗ Rbj d ∗ Rck } A A σ µνde G Aµν + ( c qlG ) ijk Q TLaIi (cid:15)L
Lbj λ AIJ d ∗ RcJk σ µνde G Aµν + ( c qlW ) ijk Q TLaIi (cid:15)σ A L Lbj d ∗ RcIk σ µνde W Aµν – 22 – ( c lW ) ijk L TLai (cid:15)σ A L Lbj e ∗ Rck σ µνde W Aµν + ( c lB ) ijk L TLai (cid:15)L
Lbj e ∗ Rck σ µνde B µν + ( c qB ) ijk (cid:15) IJK d ∗ RbIi d ∗ RdJj u ∗ ReKk σ µνde B µν + ( c qlB ) ijk Q TLbIi (cid:15)L
Ldj d ∗ ReIk σ µνde B µν + ( c φlD ) i (cid:16) D ab (cid:101) φ (cid:17) † ( D cd L Lei )+ ( c (1) φlBD ) i (cid:16) D ab (cid:101) φ (cid:17) † L Lci σ µνde B µν + ( c (2) φlBD ) i (cid:101) φ † (cid:0) D ab L Lci (cid:1) σ µνde B µν + ( c (1) φlW D ) i (cid:16) D ab (cid:101) φ (cid:17) † σ A L Lci σ µνde W Aµν + ( c (2) φlW D ) i (cid:101) φ † σ A (cid:0) D ab L Lci (cid:1) σ µνde W Aµν + ( c φlB ) i (cid:15) ABC (cid:101) φ † L Lci σ µνbc B µν σ ρσde B ρσ + ( c φlBW ) i (cid:15) ABC (cid:101) φ † σ A L Lci σ µνbc B µν σ ρσde W Aρσ + ( c φlW ) i (cid:15) ABC (cid:101) φ † σ A L Lci σ µνbc W Bµν σ ρσde W Cρσ + ( c φlG ) i (cid:15) ABC (cid:101) φ † L Lci σ µνbc G Aµν σ ρσde G Aρσ (cid:105) + h.c. . (4.1)The numbers of independent components in flavour space for each of the Wilson coefficientthat appear are: 18 for c (1) l , c (2) l , c (1) q , c (2) q , c (1) qG , c lB , c q ; 27 for c (1) ql , c (2) ql , c (3) ql , c qlG , c qlW , c qlB ;9 for c (2) qG , c lW ; and 3 for c φlD , c (1) φlBD , c (2) φlBD , c (1) φlW D , c (2) φlW D , c φlB , c φlW B , c φlW , c φlG . Theproduct of 3 SU(2) triplets contains 2 octets, which we denote by {} S and {} A , with {} S being symmetric in the two last triplets and {} A being anti-symmetric in them. λ AIJ arethe Gell-Mann matrices. The rest of the notation is similar to what has been introducedin the spin-3/2 section.To deal with the large number of operators and to make the situation more compre-hensible, we will simply list the allowed field contents assuming, for simplicity, only thefirst generation of fermions similarly to the spin-3/2 case, ψ / l e ∗ D , ψ / u ∗ ( d ∗ ) D , × ( ψ / qd ∗ lD ) ,ψ / u ∗ ( d ∗ ) G, ψ / qd ∗ lG, ψ / qd ∗ lW,ψ / l e ∗ B, ψ / u ∗ ( d ∗ ) B, ψ / qd ∗ lB,ψ / φlD , × ( ψ / φlW D ) , × ( ψ / φlBD ) ψ / φlW , ψ / φlBW, ψ / φlB , ψ / φlG . There are many possible operators even in the simplified case. As already mentioned,we refrain from a detailed discussion of the phenomenology of the spin-5/2 particles andthe computation of the production cross sections and partial decay widths. We consideronly the simplest and the most relevant operators and highlight their potential impacton LHC physics. Obviously, the operators involving the strong interactions give the mostimportant contributions in this context: the dominant production processes should involvethe gluon-gluon fusion channel and the dominant, and most spectacular decays shouldinvolve, respectively, gluons and photons in the final states. From the list of operatorsdisplayed in the previous equation, one such example could be ψ / φlG ⇒ gg → ψ / ¯ ν , ψ / ν, for the production mechanism, and also for the most probable final state, ψ / φlG ⇒ ψ / → gg ¯ ν, ggν, Some terms in the list correspond to 3 independent operators. – 23 –hich means that one gets two jets plus missing energy in the final state. This is the sametopology as in the spin-3/2 case with an interaction of the ψ / field with a quark andlepton where the dominant process was pp → d ¯ d + u ¯ d → j + e + e − , j + eν, j + νν butwhere only the last channel might be present. As noted before, the search resembles theone for leptoquarks.In the case discussed above, one cannot reconstruct the mass of the spin-5 / ψ / u ( d ∗ ) G ⇒ ψ / → gud → j, and the full process would lead to a final state with 4 jets with an invariant mass of m plusmissing energy. This final state signature has been discussed thoroughly in the context ofthe experiment as it is a good signature of supersymmetry and it may also test the spin-5/2particles. Note that as the ψ / interactions are damped by powers of 1 / Λ in our case,all decays and productions rates will be proportional to 1 / Λ . This means that for largevalues of Λ the new physics scale they will be extremely small. Finally, in the case of the spin-3 field, the lowest dimension of the operators linear in ψ is 10 as for ψ / . However, unlike for ψ / , the operators linear in ψ can have a lowerdimension than operators quadratic in ψ . Spin 3 is the highest spin for which this happens.For a singlet ψ abcdef field, the lowest dimensional Hamiltonian is −H linear = 1Λ ψ abcdef σ µνab σ ρσcd σ λ(cid:15)ef (cid:104) c B B µν B ρσ B λ(cid:15) + c BW B µν W iρσ W iλ(cid:15) + c BG B µν G Aρσ G Aλ(cid:15) + c G f ABC G Aµν G Bρσ G Cλ(cid:15) (cid:105) + h.c. , (5.1)where f ABC are the SU(3) structure constants. Thus, as in the spin-2 case, the spin-3 fielddoes not have a coupling to fermions and couples only to gluons and gauge bosons. As thefield strengths involve up to two fields (in the non-Abelian case), there can be interactionsof ψ with three to six vector fields. However, the more fields one has, the higher is theorder in perturbation theory. Thus, the dominant processes will have the minimum numberof fields, namely three, and, for simplicity, we will only discuss this option here.The most spectacular decay modes of the neutral ψ state are thus ψ → γγγ, γγZ, γZZ, ZZZ, W + W − γ, W + W − Z, (5.2)while the most frequent ones would probably involve gluon jets and could be ψ → ggg, ggγ, ggZ. (5.3)For instance, the decay rate of ψ to three photons isΓ( ψ → γγγ ) = 87 c B cos θ W π m Λ . (5.4)– 24 –hus, as in the previous discussion of the spin-2 case, one assumes that the c G factor forstrong interactions is larger than the ones for the weak interactions, c B , c BW , and theinterferences c BG , the main process at hadron colliders will be gg → ψ g, followed by gg → ψ γ, ψ Z, while the main decay modes will be into three gluons, ψ → ggg or two gluons and a photonor Z boson, ψ → ggγ, ggZ .Keeping only the processes that have more than two (or three) powers of g at theamplitude level, one obtains for the various topologies, gg → [4 g ] , [3 gγ, gZ ] , [2 g γ, g Z, gγZ, gW + W − ] , · · · [6 γ, · · · ] , with the first bracket being dominant, the second bracket sub-dominant and the thirdbracket sub-sub-dominant. Of course, the best signal would be gg → γ , but it will have arate that is sub-subleading and very suppressed unless all c X coefficients are comparable.In any case, the dominant process will be either 4 jet events or three jet events witha “monochromatic” photon. Thus, to the first approximation, one can only consider thesetwo options and ignore all the other possibilities. If a discovery is made, one should tryto find the other rare and complicated final states in order to check that it is indeed thesignature of a spin-3 particle and derives the various coefficients in the Hamiltonian.We finally note that at e + e − colliders, one could use the process γγ → ψ + γ/Z → g + γ/Z for production, which might be the dominant one. In the e + e − mode, one shouldrely on vector boson fusion in all channels, V V → V (cid:48) ψ with V, V (cid:48) = γ, Z, W . This alsorepresents the higher order processes that can be probed at hadron colliders too. We have presented an effective field theory for higher-spin particles that are singlets underthe Standard Model gauge group, which involve the lowest order linear operators. Thecomplete set of interactions at that order was explicitly derived for spin-3/2, 2, 5/2 andspin-3 particles and presented in eqs. (2.1), (3.1), (4.1) and (5.1), respectively. We havethen worked out the most important collider phenomenological features of these particles,mostly at hadron colliders and in particular at the LHC, but also in electron-positron colli-sions. The partial widths of the principal decay modes and the production cross sections inthe main channels have been evaluated for spin-3/2 and spin-2 particles using a formalismintroduced earlier, and their implications for constraints on these particles and their searchat the LHC have been summarized. In each case, we have discussed the most relevantfeatures and compared the final state experimental signatures with the ones of supersym-metric models or theories of extra dimensions. In the case of spin-5/2 and spin-3 particles,we simply listed the main decay modes and the production mechanisms and highlightedthe most striking experimental signatures by which they can be searched for at the LHC– 25 –nd beyond. As the general aim of this work was to pave the way for possible future workon higher-spin particle phenomenology, we have therefore collected the relevant and poten-tially useful technical material, including Feynman rules and an example of computation,in several appendices.The higher-spin particles can have a rich phenomenology and, in addition to the collideraspects discussed in this paper, they can play a potentially important role in other areasof high-energy physics. Higher half–integer spin particles that couple to SM leptons andquarks can, for instance, have some impact on proton decay if some effective operatorsare simultaneously present. The Majorana nature of the high–spin particles may also haveimportant cosmological consequences, besides those related to dark matter. For instance, ifone considers more than one higher-spin particle with different masses and different complexcouplings, for example, two different spin-3/2 fermions ψ a / , a = 1 , , interacting accordingto eq. (2.1), the possible interference between the complex amplitudes may give rise todirect baryogenesis in the early Universe, very much the same way as in leptogenesis [91].In this example, the observed baryon asymmetry of the Universe may not be related toneutrino masses but to the Majorana nature of higher-spin particles. This path is worthexploring.Finally, while the main focus of the present work has been on collider phenomenologyof the hypothetical higher-spin particles, higher-spin resonances do exist in low-energyhadronic physics and in nuclear physics. In the introduction, we reviewed some difficultiesrelated to computing the spin-3/2 resonances described by interacting Rarita-Schwingerfields. Our formulation of the massive interacting spin-3/2 fields is free of those problemsand can provide a consistent framework for computing higher-spin nuclear and hadronicphysics observables. This aspect is clearly very important and needs a separate and detaileddiscussion. Acknowledgement.
This work was supported by the Estonian Research Council grants MOBTTP135, PRG803,MOBTT5, MOBJD323 and MOBTT86, and by the EU through the European RegionalDevelopment Fund CoE program TK133 “The Dark Side of the Universe.” J.C.C. is sup-ported by the STFC under grant ST/P001246/1 and A.D. is also supported by the Juntade Andalucia through the Talentia Senior program.
A Appendix
A1 Symmetric multispinor formalism
Our notation, first proposed in Ref. [42], is based on the well-known two-component spinorformalism discussed, e.g. , in Refs. [92, 93]. Undotted indices ( a, b, . . . ) and dotted indices( ˙ a, ˙ b, . . . ) transform in the (1 / ,
0) and (0 , /
2) irrep of the Lorentz group, respectively. Theindices are raised and lowered with antisymmetric tensors (cid:15) ab , (cid:15) ˙ a ˙ b with (cid:15) = − (cid:15) = 1.When possible, we adapt the convention where undotted (dotted) indices are contracted– 26 –n descending (ascending) order, e.g. t a = (cid:15) ab t b , t ˙ a = t ˙ a (cid:15) ˙ a ˙ b so that t a t a = t b (cid:15) ab t a . A paircomprising of a dotted and an undotted spinor index is converted into an vector index µ as p a ˙ a = p µ σ µa ˙ a , p µ = ¯ σ µ ˙ aa p a ˙ a /
2, , where σ is the identity matrix and σ i with i = 1 , , p ˙ aa = p µ ¯ σ µ ˙ aa , where ¯ σ µ ˙ ab is ¯ σ µ = ( σ , − σ i ). It holds that σ µa ˙ a ¯ σ ν ˙ ab + σ νa ˙ a ¯ σ µ ˙ ab = 2 η µν δ ba . (A.1)Objects in the ( j,
0) irrep are denoted by ψ ( a ) ≡ ψ a a ...a j , where ( a ) is a symmet-ric multispinor index. A multispinor object t is converted into a symmetric multispinorby taking the product of 2 j copies of t and symmetrizing the indices. For example themomentum p a ˙ a corresponds to p ( a )( ˙ a ) ≡ j )! (cid:104) p a ˙ a . . . p a j ˙ a j + all permutations of a i and ˙ a j (cid:105) . (A.2)In this way, the (cid:15) ab and (cid:15) ab symbols are also generalized to the (cid:15) ( a )( b ) and (cid:15) ( a )( b ) symbolsthat can be used to raise and lower symmetric multispinor indices. The following identitieshold (cid:15) ( a )( b ) = ( − j (cid:15) ( b )( a ) , (cid:15) ( a )( c ) (cid:15) ( c )( b ) = δ ( b )( a ) , δ ( a )( a ) = 2 j + 1 . (A.3)The object σ µνab ≡ i (cid:0) σ µa ˙ a ¯ σ ν ˙ ab − σ νa ˙ a ¯ σ µ ˙ ab (cid:1) (A.4)makes a frequent appearance in the Feynman rules. It is symmetric in the two-spinorindices ab and antisymmetric in the Lorentz indices µν and thus projects rank 2 tensorsinto their (1 ,
0) subspace.
A2 Feynman rules
Here we present the Feynman rules for higher spin propagators and vertices with up to 3legs that arise from a given effective operator. Below θ W is the Weinberg angle, g Y , g , e = g sin θ W = g Y cos θ W are the U(1) Y , SU(2) gauge couplings and the electric chargerespectively. All vertices are completely symmetric in the spinor indices. Propagators ( ˙ a ) ( a ) p = i p ( a )(˙ a ) p − m , ( a ) ( ˙ a ) p = i p (˙ a )( a ) p − m , (A.5)( b ) ( a ) = i m j δ ( b )( a ) p − m , (˙ b ) ( ˙ a ) = i m j δ (˙ a )(˙ b ) p − m . (A.6)– 27 – xternal lines ( a ) p, σ = u ( a ) ( p, σ ) , ( ˙ a ) p, σ = v ∗ ( ˙ a ) ( p, σ ) , ( ˙ a ) p, σ = u ∗ ( ˙ a ) ( p, σ ) , ( a ) p, σ = v ( a ) ( p, σ ) . (A.7) Completeness relations (cid:88) σ u ( a ) ( p, σ ) u ∗ ( ˙ a ) ( p, σ ) = (cid:88) σ v ( a ) ( p, σ ) v ∗ ( ˙ a ) ( p, σ ) = p ( a )( ˙ a ) , (cid:88) σ u ( a ) ( p, σ ) v ( b ) ( p, σ ) = m j δ ( b )( a ) , (cid:88) σ u ∗ ( ˙ a ) ( p, σ ) v ∗ (˙ b ) ( p, σ ) = m j δ ( ˙ a )(˙ b ) . (A.8) Vertices for spin-3/2 particles
Here, we present only the vertices with 1) the quartic point-like interaction with threefermions and 2) with up to 3 particles for the interactions with Higgs and gauge bosonswhich give the dominant interactions; there are also 4 particle vertices of two types com-pared to the ones below: either one can add a Higgs line and the interaction is suppressedby a power of v or add a gauge boson line and the interaction is suppressed by an additionalgauge coupling. e k ∗ Rf ν jLe ψ abc e iLd = − i c eeνijk + c eeνjik Λ δ da δ eb δ fc ,d k ∗ RKf d j ∗ RJe ψ abc u i ∗ LId = i c uddijk (cid:15) IJK Λ δ da δ eb δ fc ,d k ∗ RJf ν jLe ψ abc d iLId = − i c ddνijk δ IJ Λ δ da δ eb δ fc ,d k ∗ RJf e jLe ψ abc u iLId = i c udeijk δ IJ Λ δ da δ eb δ fc , (A.9)– 28 – iLd hψ abc q q = − i c φi Λ √ σ µνab q µ q ν δ dc , (A.10) ν iLd A ν ψ abc q q = v Λ √ (cid:0) − c Bi cos θ W + c Wi sin θ W (cid:1) σ µνab q µ δ dc ,ν iLd Z ν ψ abc q q = v Λ (cid:16) − i ec φi √ θ W cos θ W q µ + √ c Bi sin θ W + c Wi cos θ W ) q µ (cid:17) σ µνab δ dc ,e iLd W + ν ψ abc q q = v Λ (cid:16) − ic φi g q µ + c Wi q µ (cid:17) σ µνab δ dc . (A.11)– 29 – ertices for spin-2 particles Z ν Z λ ψ abcd q q = − i c B sin θ W + c W cos θ W )Λ σ µνab σ ρλcd q µ q ρ ,A ν Z λ ψ abcd q q = i c B − c W ) sin θ W cos θ W Λ σ µνab σ ρλcd q µ q ρ ,A ν A λ ψ abcd q q = − i c B cos θ W + c W sin θ W )Λ σ µνab σ ρλcd q µ q ρ ,W + ν W − λ ψ abcd q q = − i c W Λ σ µνab σ ρλcd q µ q ρ ,g Aν g Bλ ψ abcd q q = − i c G Λ σ µνab σ ρλcd q µ q ρ δ AB . (A.12) Vertices for spin-3 particles A ν A (cid:15) A σ ψ abcdef q q q = −
12 cos θ W (4 c B cos θ W + c BW sin θ W )Λ σ µνab σ ρσcd σ λ(cid:15)ef q µ q ρ q λ ,Z ν Z (cid:15) Z σ ψ abcdef q q q = 12 sin θ W (4 c B sin θ W + c BW cos θ W )Λ σ µνab σ ρσcd σ λ(cid:15)ef q µ q ρ q λ ,A ν A σ Z (cid:15) ψ abcdef q q q = − θ W cos θ W (6 c B + c BW )Λ σ µνab σ ρσcd σ λ(cid:15)ef q µ q ρ q λ , – 30 – ν Z σ Z (cid:15) ψ abcdef q q q = 8 sin θ W cos θ W (6 c B + c BW )Λ σ µνab σ ρσcd σ λ(cid:15)ef q µ q ρ q λ ,A ν W + σ W − (cid:15) ψ abcdef q q q = − θ W c BW Λ σ µνab σ ρσcd σ λ(cid:15)ef q µ q ρ q λ ,Z ν W + σ W − (cid:15) ψ abcdef q q q = 2 sin θ W c BW Λ σ µνab σ ρσcd σ λ(cid:15)ef q µ q ρ q λ ,A ν G σ G (cid:15) ψ abcdef q q q = −
16 cos θ W c BG Λ σ µνab σ ρσcd σ λ(cid:15)ef q µ q ρ q λ ,A ν G σ G (cid:15) ψ abcdef q q q = 16 sin θ W c BG Λ σ µνab σ ρσcd σ λ(cid:15)ef q µ q ρ q λ ,Z ν G σ G (cid:15) ψ abcdef q q q = −
16 cos θ W c BG Λ σ µνab σ ρσcd σ λ(cid:15)ef q µ q ρ q λ ,G ν G σ G (cid:15) ψ abcdef q q q = − c G f ABC Λ σ µνab σ ρσcd σ λ(cid:15)ef q µ q ρ q λ . (A.13) A3 The narrow width approximation
Consider a process A → B ( ψ → C ) that can be split it into the sub-processes A → Bψ and ψ → C with amplitudes M ( s ) ≡ M ( a )1 v ( a ) ( s ) + M
1( ˙ a ) u ∗ ( ˙ a ) ( s ), M ( s ) ≡ u ( b ) ( s ) M b ) + v ∗ (˙ b ) ( s ) M (˙ b )2 . The symbols A, B, C denote external states that may contain multiple parti-cles and s is the spin of ψ . – 31 –he total amplitude is given by M = M ( a )1 ip ( a )(˙ b ) M (˙ b )2 p − m + im Γ + M
1( ˙ a ) ip ( ˙ a )( b ) M b ) p − m + im Γ+ M
1( ˙ a ) im j δ ( ˙ a )(˙ b ) M (˙ b )2 p − m + im Γ + M ( a )1 im j δ ( a )( b ) M b ) p − m + im Γ , (A.14)where Γ and m are the width and mass of ψ , respectively. The narrow width approximationworks in the limit Γ /M → | p − m − im Γ | − = π/ ( m Γ) δ ( p − m )effectively putting the intermediate ψ on-shell. The spin sums then imply that |M| = π/ ( m Γ) δ ( p − m ) | (cid:80) s M ( s ) M ( s ) | and thus the cross section decomposes as expectedd σ A → B ( ψ → C ) = (cid:88) s ,s d σ s s A → Bψ s dΓ s s ψ s → C / Γ . (A.15)When summing over all final state spins in C and integrating over the C phase space,then Γ s s ψ → C = δ s s Γ ψ → C . The cross section for the complete process is thus obtained byperforming the spin sum in d s A → Bψ and multiplying the latter by the branching ratio of ψ → C , d σ A → B ( ψ → C ) = d σ A → Bψ BR ψ → C . (A.16) A4 Calculation example: spin 3/2 decay
To clarify our formalism, we present here explicit calculation of spin-3/2 particle decay toHiggs and neutrino or anti-neutrino: ψ / ( p ) → H ( q ) ν ( q ) and ψ / ( p ) → H ( q )¯ ν ( q ).The contributing diagrams are ν Ld ( q ) h ( q ) ψ abc ( p ) , ν ∗ L ˙ d ( q ) h ( q ) ψ ˙ a ˙ b ˙ c ( p ) . (A.17)The final states are different and therefore there is no interference between these diagrams.We use two-spinor Feynman rules for the standard model fermions presented in [92]. Byusing the Feynman rules in [92] and in eqs. (A.7) and (A.10) one can write the Feynmanamplitude corresponding to the left diagram: M ψ / → H ¯ ν = − i c φ √ u abc ( p, σ ) σ µνab q µ q ν δ dc y d ( q , s ) , (A.18)where y d ( q , s ) is the wave function corresponding to neutrino. The spin-averaged ampli-tude squared is: |M ψ / → H ¯ ν | = 12 j + 1 | c φ | (cid:32)(cid:88) σ u ∗ ˙ a ˙ b ˙ c ( p, σ ) u def ( p, σ ) (cid:33) (cid:32)(cid:88) s y ∗ ˙ c ( q , s ) y f ( q , s ) (cid:33) ×× ¯ σ µν ˙ b ˙ a σ αβde q µ q α q ν q β . (A.19)– 32 –his can be simplified by using Pauli-matrix identities: σ µνab = i η µν (cid:15) ab + i σ µ ¯ σ ν ) ab and ¯ σ µν ˙ a ˙ b = i η µν (cid:15) ˙ a ˙ b + i σ µ σ ν ) ˙ a ˙ b , (A.20)where the terms proportional to antisymmetric (cid:15) vanish when its indices are contracted withthe symmetric u ∗ ˙ a ˙ b ˙ c ( p, σ ) and u def ( p, σ ). By using the completeness relation in eq. (A.8)and in [92] the amplitude squared becomes: |M ψ / → H ¯ ν | = − j + 1 | c φ | (cid:104) ¯ σ ˙ adµ ¯ σ ˙ beν ¯ σ ˙ cfρ + ¯ σ ˙ adµ ¯ σ ˙ ceν ¯ σ ˙ bfρ + ¯ σ ˙ bdµ ¯ σ ˙ aeν ¯ σ ˙ cfρ ¯ σ ˙ bdµ ¯ σ ˙ ceν ¯ σ ˙ afρ + ¯ σ ˙ cdµ ¯ σ ˙ aeν ¯ σ ˙ bfρ + ¯ σ ˙ cdµ ¯ σ ˙ beν ¯ σ ˙ afρ (cid:105) ×× σ δf ˙ c (¯ σ λ σ γ ) ˙ b ˙ a ( σ α ¯ σ β ) de p µ p ν p ρ q δ q λ q α q γ q β = 12 j + 1 | c φ | (cid:40) − Tr (cid:2) ¯ σ µ σ α ¯ σ β σ ν ¯ σ λ σ γ (cid:3) Tr (cid:2) ¯ σ ρ σ δ (cid:3) − Tr (cid:2) ¯ σ µ σ α ¯ σ β σ ν ¯ σ δ σ ρ ¯ σ λ σ γ (cid:3) +Tr (cid:2) ¯ σ µ σ α ¯ σ β σ ν ¯ σ γ σ λ (cid:3) Tr (cid:2) ¯ σ ρ σ δ (cid:3) + Tr (cid:2) ¯ σ β σ α ¯ σ µ σ λ ¯ σ γ σ ρ ¯ σ δ σ ν (cid:3) +Tr (cid:2) ¯ σ ρ σ δ ¯ σ µ σ α ¯ σ β σ ν ¯ σ γ σ λ (cid:3) − Tr (cid:2) ¯ σ λ σ γ ¯ σ ρ σ δ ¯ σ µ σ α ¯ σ β σ ν (cid:3)(cid:41) ×× p µ p ν p ρ q δ q λ q α q γ q β = 12 j + 1 | c φ | m / ( m / − m H ) . (A.21)The decay rate now becomes:Γ( ψ / → H ¯ ν ) = 12 j + 1 | c φ | π m / Λ (cid:32) − M H m / (cid:33) . (A.22)Finally, when computing the total decay width, one can use the fact that Lorentzsymmetry demands that all spin states must have the same decay width. It is thus sufficientto compute the decay width for the highest spin state only and, the phase space integrationwill yield the spin averaged width. Using the highest spin state is convenient becausethe corresponding multispinor can be constructed from identical two-spinors, and thus,symmetrization is automatic. References [1]
ATLAS
Collaboration, G. Aad et al.,
Observation of a new particle in the search for theStandard Model Higgs boson with the ATLAS detector at the LHC , Phys. Lett. B (2012)1–29, [ arXiv:1207.7214 ].[2]
CMS
Collaboration, S. Chatrchyan et al.,
Observation of a New Boson at a Mass of 125GeV with the CMS Experiment at the LHC , Phys. Lett. B (2012) 30–61,[ arXiv:1207.7235 ].[3] J. Wess and B. Zumino,
Supergauge Transformations in Four-Dimensions , Nucl. Phys. B (1974) 39–50. – 33 –
4] P. Fayet,
Mixing Between Gravitational and Weak Interactions Through the MassiveGravitino , Phys. Lett. B (1977) 461.[5] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, New dimensions at amillimeter to a Fermi and superstrings at a TeV , Phys. Lett. B (1998) 257–263,[ hep-ph/9804398 ].[6] L. Randall and R. Sundrum,
A Large mass hierarchy from a small extra dimension , Phys.Rev. Lett. (1999) 3370–3373, [ hep-ph/9905221 ].[7] M. Dine, A. E. Nelson, and Y. Shirman, Low-energy dynamical supersymmetry breakingsimplified , Phys. Rev. D (1995) 1362–1370, [ hep-ph/9408384 ].[8] R. Casalbuoni, S. De Curtis, D. Dominici, F. Feruglio, and R. Gatto, A gravitino-goldstinohigh-energy equivalence theorem , Phys. Lett. B (1988) 313–316.[9] G. F. Giudice, R. Rattazzi, and J. D. Wells,
Quantum gravity and extra dimensions athigh-energy colliders , Nucl. Phys. B (1999) 3–38, [ hep-ph/9811291 ].[10] T. Han, J. D. Lykken, and R.-J. Zhang,
On Kaluza-Klein states from large extra dimensions , Phys. Rev. D (1999) 105006, [ hep-ph/9811350 ].[11] S. Hassan and R. A. Rosen, Bimetric Gravity from Ghost-free Massive Gravity , JHEP (2012) 126, [ arXiv:1109.3515 ].[12] E. Babichev, L. Marzola, M. Raidal, A. Schmidt-May, F. Urban, H. Veerm¨ae, and M. vonStrauss, Bigravitational origin of dark matter , Phys. Rev. D (2016), no. 8 084055,[ arXiv:1604.08564 ].[13] E. Babichev, L. Marzola, M. Raidal, A. Schmidt-May, F. Urban, H. Veerm¨ae, and M. vonStrauss, Heavy spin-2 Dark Matter , JCAP (2016) 016, [ arXiv:1607.03497 ].[14] L. Marzola, M. Raidal, and F. R. Urban, Oscillating Spin-2 Dark Matter , Phys. Rev. D (2018), no. 2 024010, [ arXiv:1708.04253 ].[15] V. Shklyar, H. Lenske, and U. Mosel, Spin-5/2 fields in hadron physics , Phys. Rev. C (2010) 015203, [ arXiv:0912.3751 ].[16] E. Bergshoeff, D. Grumiller, S. Prohazka, and J. Rosseel, Three-dimensional Spin-3 TheoriesBased on General Kinematical Algebras , JHEP (2017) 114, [ arXiv:1612.02277 ].[17] S. Jafarzade, A. Koenigstein, and F. Giacosa, Phenomenology of j P C = 3 −− tensor mesons ,2021.[18] W. Rarita and J. Schwinger, On a theory of particles with half integral spin , Phys. Rev. (1941) 61.[19] R. D. Peccei, Chiral lagrangian calculation of pion-nucleon scattering lengths , Phys. Rev. (1968) 1812–1821.[20] R. M. Davidson, N. C. Mukhopadhyay, and R. S. Wittman,
Effective Lagrangian approach tothe theory of pion photoproduction in the Delta (1232) region , Phys. Rev. D (1991) 71–94.[21] O. Scholten, A. Y. Korchin, V. Pascalutsa, and D. Van Neck, Pion and photon inducedreactions on the nucleon in a unitary model , Phys. Lett. B (1996) 13–19,[ nucl-th/9604014 ].[22] A. Y. Korchin, O. Scholten, and R. G. E. Timmermans,
Pion and photon couplings of N*resonances from scattering on the proton , Phys. Lett. B (1998) 1–8, [ nucl-th/9811042 ]. – 34 –
23] R. D. Mota, H. Garcilazo, A. Valcarce, and F. Fernandez,
Bound state problem of the NDelta and N Delta Delta systems , Phys. Rev. C (1999) 46–52.[24] A. D. Lahiff and I. R. Afnan, Solution of the Bethe-Salpeter equation for pion nucleonscattering , Phys. Rev. C (1999) 024608, [ nucl-th/9903058 ].[25] V. Pascalutsa, Quantization of an interacting spin - 3 / 2 field and the Delta isobar , Phys.Rev. D (1998) 096002, [ hep-ph/9802288 ].[26] V. Pascalutsa and R. Timmermans, Field theory of nucleon to higher spin baryon transitions , Phys. Rev. C (1999) 042201, [ nucl-th/9905065 ].[27] V. Pascalutsa, Correspondence of consistent and inconsistent spin - 3/2 couplings via theequivalence theorem , Phys. Lett. B (2001) 85–90, [ hep-ph/0008026 ].[28] C. R. Hagen and L. P. S. Singh,
SEARCH FOR CONSISTENT INTERACTIONS OF THERARITA-SCHWINGER FIELD , Phys. Rev. D (1982) 393–398.[29] H. Haberzettl, Propagation of a massive spin 3/2 particle , nucl-th/9812043 .[30] S. Deser, V. Pascalutsa, and A. Waldron, Massive spin 3/2 electrodynamics , Phys. Rev. D (2000) 105031, [ hep-th/0003011 ].[31] T. Pilling, New symmetry current for massive spin-3/2 fields , Mod. Phys. Lett. A (2004)1781, [ hep-ph/0404089 ].[32] T. Pilling, Symmetry of massive Rarita-Schwinger fields , Int. J. Mod. Phys. A (2005)2715–2742, [ hep-th/0404131 ].[33] N. Wies, J. Gegelia, and S. Scherer, Consistency of the pi Delta interaction in chiralperturbation theory , Phys. Rev. D (2006) 094012, [ hep-ph/0602073 ].[34] M. Napsuciale, M. Kirchbach, and S. Rodriguez, Spin 3/2 Beyond the Rarita-SchwingerFramework , Eur. Phys. J. A (2006) 289–306, [ hep-ph/0606308 ].[35] E. G. Delgado-Acosta and M. Kirchbach, Second order theory of ( j, ⊕ (0 , j ) single highspins as Lorentz tensors , arXiv:1312.5811 .[36] E. Delgado Acosta, V. Banda Guzman, and M. Kirchbach, Bosonic and fermionicWeinberg-Joos (j,0) + (0,j) states of arbitrary spins as Lorentz tensors or tensor-spinors andsecond-order theory , Eur. Phys. J. A (2015) 35, [ arXiv:1503.07230 ].[37] T. Mart, J. Kristiano, and S. Clymton, Pure spin-3/2 representation with consistentinteractions , Phys. Rev.
C100 (2019), no. 3 035207, [ arXiv:1909.04282 ].[38] M. Benmerrouche, R. Davidson, and N. Mukhopadhyay,
Problems of Describing Spin 3/2Baryon Resonances in the Effective Lagrangian Theory , Phys. Rev. C (1989) 2339–2348.[39] K. Johnson and E. Sudarshan, Inconsistency of the local field theory of charged spin 3/2particles , Annals Phys. (1961) 126–145.[40] M. T. Grisaru, H. Pendleton, and P. van Nieuwenhuizen, Supergravity and the S Matrix , Phys. Rev. D (1977) 996.[41] M. T. Grisaru and H. Pendleton, Soft Spin 3/2 Fermions Require Gravity andSupersymmetry , Phys. Lett. B (1977) 323–326.[42] J. C. Criado, N. Koivunen, M. Raidal, and H. Veerm¨ae, Dark Matter of Any Spin – anEffective Field Theory and Applications , arXiv:2010.02224 . – 35 –
43] S. Weinberg,
Photons and Gravitons in S -Matrix Theory: Derivation of Charge Conservationand Equality of Gravitational and Inertial Mass , Phys. Rev. (1964) B1049–B1056.[44] A. Falkowski, S. Ganguly, P. Gras, J. M. No, K. Tobioka, N. Vignaroli, and T. You,
Lightquark Yukawas in triboson final states , arXiv:2011.09551 .[45] N. Arkani-Hamed, T.-C. Huang, and Y.-t. Huang, Scattering Amplitudes For All Masses andSpins , arXiv:1709.04891 .[46] P. Gondolo, S. Kang, S. Scopel, and G. Tomar, The effective theory of nuclear scattering fora WIMP of arbitrary spin , arXiv:2008.05120 .[47] P. Gondolo, I. Jeong, S. Kang, S. Scopel, and G. Tomar, The phenomenology of nuclearscattering for a WIMP of arbitrary spin , arXiv:2102.09778 .[48] J. Leite Lopes, J. A. Martins Simoes, and D. Spehler, Production and Decay Properties ofPossible Spin 3/2 Leptons , Phys. Lett. B (1980) 367–372.[49] J. Leite Lopes, D. Spehler, and J. A. Martins Simoes, WEAK INTERACTIONSINVOLVING SPIN 3/2 LEPTONS , Phys. Rev. D (1982) 1854.[50] C. Burges and H. J. Schnitzer, Virtual Effects of Excited Quarks as Probes of a Possible NewHadronic Mass Scale , Nucl. Phys. B (1983) 464–500.[51] D. Spehler, O. J. Eboli, G. Marques, S. Novaes, and A. Natale,
Looking for Spin 3/2 Leptonsin Hadronic Collisions , Phys. Rev. D (1987) 1358.[52] J. Almeida, F.M.L., J. A. Martins Simoes, and A. Ramalho, Spin 3/2 lepton production atHera , Nucl. Phys. B (1993) 502–514.[53] J. Montero and V. Pleitez,
Constraints on spin 3/2 and excited spin 1/2 fermions comingfrom the leptonic Z0 partial width , Phys. Lett. B (1994) 267–270, [ hep-ph/9309210 ].[54] J. Almeida, F.M.L., J. Lopes, J. A. Martins Simoes, and A. Ramalho,
Production and decayof single heavy spin 3/2 leptons in high-energy electron - positron collisions , Phys. Rev. D (1996) 3555–3558, [ hep-ph/9509364 ].[55] O. Cakir and A. Ozansoy, Search for excited spin-3/2 and spin-1/2 leptons at linearcolliders , Phys. Rev. D (2008) 035002, [ arXiv:0709.2134 ].[56] O. J. Eboli, E. Gregores, J. Montero, S. Novaes, and D. Spehler, Excited leptonic states inpolarized e- gamma and e+ e- collisions , Phys. Rev. D (1996) 1253–1263,[ hep-ph/9509257 ].[57] M. Abdullah, K. Bauer, L. Gutierrez, J. Sandy, and D. Whiteson, Searching for spin-3/2leptons , Phys. Rev. D (2017), no. 3 035008, [ arXiv:1609.05251 ].[58] J. Leite Lopes, J. A. Martins Simoes, and D. Spehler, Possible Spin 3/2 Quarks and ScalingViolations in Neutrino Reactions , Phys. Rev. D (1981) 797.[59] B. Moussallam and V. Soni, Production of Heavy Spin 3/2 Fermions in Colliders , Phys. Rev.D (1989) 1883–1891.[60] D. A. Dicus, S. Gibbons, and S. Nandi, Collider production of spin 3/2 quarks , hep-ph/9806312 .[61] D. A. Dicus, D. Karabacak, S. Nandi, and S. K. Rai, Search for spin-3/2 quarks at the LargeHadron Collider , Phys. Rev. D (2013), no. 1 015023, [ arXiv:1208.5811 ]. – 36 –
62] N. D. Christensen, P. de Aquino, N. Deutschmann, C. Duhr, B. Fuks, C. Garcia-Cely,O. Mattelaer, K. Mawatari, B. Oexl, and Y. Takaesu,
Simulating spin- particles at colliders , Eur. Phys. J. C (2013), no. 10 2580, [ arXiv:1308.1668 ].[63] R. Walsh and A. Ramalho, Bounds on the electromagnetic interactions of excited spin 3/2leptons , Phys. Rev. D (1999) 077302, [ hep-ph/9907364 ].[64] W. Stirling and E. Vryonidou, Effect of spin-3/2 top quark excitation on t ¯ t production at theLHC , JHEP (2012) 055, [ arXiv:1110.1565 ].[65] B. Hassanain, J. March-Russell, and J. Rosa, On the possibility of light string resonances atthe LHC and Tevatron from Randall-Sundrum throats , JHEP (2009) 077,[ arXiv:0904.4108 ].[66] FCC
Collaboration, A. Abada et al.,
FCC-hh: The Hadron Collider: Future CircularCollider Conceptual Design Report Volume 3 , Eur. Phys. J. ST (2019), no. 4 755–1107.[67]
Particle Data Group
Collaboration, P. Zyla et al.,
Review of Particle Physics , PTEP (2020), no. 8 083C01.[68] A. Djouadi, J. Kalinowski, M. Muehlleitner, and M. Spira,
HDECAY: Twenty ++ years after , Comput. Phys. Commun. (2019) 214–231, [ arXiv:1801.09506 ].[69]
FCC
Collaboration, A. Abada et al.,
FCC-ee: The Lepton Collider: Future Circular ColliderConceptual Design Report Volume 2 , Eur. Phys. J. ST (2019), no. 2 261–623.[70] A. D. Martin, W. J. Stirling, R. S. Thorne, and G. Watt,
Uncertainties on alpha(S) in globalPDF analyses and implications for predicted hadronic cross sections , Eur. Phys. J. C (2009) 653–680, [ arXiv:0905.3531 ].[71] G. R. Farrar and P. Fayet, Phenomenology of the Production, Decay, and Detection of NewHadronic States Associated with Supersymmetry , Phys. Lett. B (1978) 575–579.[72] A. Djouadi, T. Kohler, M. Spira, and J. Tutas, (eb), (et) type leptoquarks at ep colliders , Z.Phys. C (1990) 679–686.[73] J. L. Hewett and S. Pakvasa, Leptoquark Production in Hadron Colliders , Phys. Rev. D (1988) 3165.[74] F. del Aguila, J. A. Aguilar-Saavedra, and R. Pittau, Heavy neutrino signals at large hadroncolliders , JHEP (2007) 047, [ hep-ph/0703261 ].[75] A. Djouadi, J. Ellis, R. Godbole, and J. Quevillon, Future Collider Signatures of the Possible750 GeV State , JHEP (2016) 205, [ arXiv:1601.03696 ].[76] R. Barbier et al., R-parity violating supersymmetry , Phys. Rept. (2005) 1–202,[ hep-ph/0406039 ].[77] H. K. Dreiner,
An Introduction to explicit R-parity violation , Adv. Ser. Direct. High EnergyPhys. (2010) 565–583, [ hep-ph/9707435 ].[78] ATLAS
Collaboration, G. Aad et al.,
Search for displaced vertices of oppositely chargedleptons from decays of long-lived particles in pp collisions at √ s =13 TeV with the ATLASdetector , Phys. Lett. B (2020) 135114, [ arXiv:1907.10037 ].[79]
CMS
Collaboration, A. M. Sirunyan et al.,
Constraints on models of scalar and vectorleptoquarks decaying to a quark and a neutrino at √ s =
13 TeV , Phys. Rev. D (2018),no. 3 032005, [ arXiv:1805.10228 ]. – 37 –
80] F. Boudjema and A. Djouadi,
Looking for the { LEP } at { LEP } : The Excited NeutrinoScenario , Phys. Lett. B (1990) 485–491.[81] H. M. Lee, M. Park, and V. Sanz,
Gravity-mediated (or Composite) Dark Matter , Eur. Phys.J. C (2014) 2715, [ arXiv:1306.4107 ].[82] A. Falkowski and J. F. Kamenik, Diphoton portal to warped gravity , Phys. Rev. D (2016),no. 1 015008, [ arXiv:1603.06980 ].[83] B. M. Dillon and V. Sanz, Kaluza-Klein gravitons at LHC2 , Phys. Rev. D (2017), no. 3035008, [ arXiv:1603.09550 ].[84] S. B. Giddings and H. Zhang, Kaluza-Klein graviton phenomenology for warpedcompactifications, and the 750 GeV diphoton excess , Phys. Rev. D (2016), no. 11 115002,[ arXiv:1602.02793 ].[85] M. Kumar, P. Mathews, V. Ravindran, and A. Tripathi, Direct photon pair production at theLHC to order α s in TeV scale gravity models , Nucl. Phys. B (2009) 28–51,[ arXiv:0902.4894 ].[86] J. Gao, C. S. Li, B. H. Li, C.-P. Yuan, and H. X. Zhu,
Next-to-leading order QCD correctionsto the heavy resonance production and decay into top quark pair at the LHC , Phys. Rev. D (2010) 014020, [ arXiv:1004.0876 ].[87] A. Carmona, A 750 GeV graviton from holographic composite dark sectors , Phys. Lett. B (2016) 502–508, [ arXiv:1603.08913 ].[88] M. Dittmar, A.-S. Nicollerat, and A. Djouadi,
Z-prime studies at the LHC: An Update , Phys.Lett. B (2004) 111–120, [ hep-ph/0307020 ].[89] A. Djouadi, G. Moreau, and R. K. Singh,
Kaluza-Klein excitations of gauge bosons at theLHC , Nucl. Phys. B (2008) 1–26, [ arXiv:0706.4191 ].[90] B. C. Allanach, F. Mahmoudi, J. P. Skittrall, and K. Sridhar,
Gluon-initiated production of aKaluza-Klein gluon in a Bulk Randall-Sundrum model , JHEP (2010) 014,[ arXiv:0910.1350 ].[91] M. Fukugita and T. Yanagida, Baryogenesis Without Grand Unification , Phys. Lett. B (1986) 45–47.[92] H. K. Dreiner, H. E. Haber, and S. P. Martin,
Two-component spinor techniques andFeynman rules for quantum field theory and supersymmetry , Phys. Rept. (2010) 1–196,[ arXiv:0812.1594 ].[93] V. Borodulin, R. Rogalyov, and S. Slabospitskii,
CORE 3.1 (COmpendium of RElations,Version 3.1) , arXiv:1702.08246 ..