Colorphilic Spin-2 Resonances in the LHC Dijet Channel
CColorphilic Spin-2 Resonances in the LHC
Dijet Channel
R. Sekhar Chivukula, Dennis Foren, and Elizabeth H. Simmons
Department of Physics and AstronomyMichigan State University,East Lansing U.S.A.
Abstract
Experiments at the LHC may yet discover a dijet resonance indicative of Beyond the Stan-dard Model (BSM) physics. In this case, the question becomes: what BSM theories areconsistent with the unexpected resonance? One possibility would be a spin-2 object calledthe “colorphilic graviton”–a spin-2 color-singlet particle which couples exclusively to thequark and gluon stress-energy tensors. We assess the possibility of this state’s discovery inthe dijet channel as an s-channel resonance, and report the regions of parameter space wherecolorphilic gravitons have not yet been excluded by LHC-13 data but still may be discoveredin the dijet channel at LHC-14 for integrated luminosities of 0.3, 1, and 3 ab − . We thendelineate which of those regions remain accessible to future collider searches, once one ac-counts for applicability of the narrow-width approximation, mass resolution of the detector,and self-consistency according to tree-level partial-wave unitarity. We discover that–despitethe strong constraints unitarity imposes on collider searches–the colorphilic graviton remainspotentially discoverable in the LHC dijet channel. A means of investigation would be to ap-ply the color discriminant variable, a dimensionless combination of quantities (productioncross-section, total decay width, and invariant mass) that can be quickly measured afterthe discovery of a dijet resonance. Previous publications have demonstrated the color dis-criminant variable’s utility when applied to theories containing vector bosons (colorons, Z (cid:48) ),excited quarks, and diquarks. We extend this analysis to the case of the colorphilic gravitonby applying the color discriminant variable to the appropriate region of parameter space. Weconclude that resolvable, discoverable dijet resonances consistent with colorphilic gravitonsspan a narrower range of masses than those consistent with leptophobic Z (cid:48) models, and canbe distinguished from those originating from coloron, excited quark, and diquark models. a r X i v : . [ h e p - ph ] F e b Introduction
New heavy particles with sizable couplings to quarks and gluons also have relatively largeproduction cross-sections at the Large Hadron Collider (LHC). Such particles could appearas resonances in the dijet channel above the otherwise rapidly-falling QCD background.In this vein, CMS and ATLAS are continually searching the dijet channel for evidence ofnew physics. Because a resonance has yet to be discovered, they report 95% CL exclusionlimits on various benchmark models. As more data is acquired and higher collision energiesexplored, the exclusion limits strengthen, eliminating larger classes of models. Our workutilizes CMS exclusion limits from an analysis of ∼
36 fb − of LHC-13 dijet channel data[1]. These limits are comparable to ATLAS exclusion limits based on ∼
37 fb − of LHC-13dijet channel data [2].Discovery of a new dijet resonance R would indicate a deviation from the Standard Model(SM) of particle physics, and signal the presence of physics beyond the Standard Model(BSM). In the event of a discovery, experiments will immediately measure the dijet cross-section of the resonance ( σ Rjj ), its mass ( m R ), and its total decay width (Γ R ) if possible.We would subsequently wonder what new physics is consistent with these quantities orcombinations of these quantities. In principle, a new dijet resonance could have one of manydifferent color and spin structures. In this work we will focus on spin-2 resonances.Experimental searches typically choose Randall-Sundrum (RS) gravitons as their spin-2benchmark model [3]. RS gravitons appear as Kaluza-Klein excitations of extra-dimensionalgravitational theories. Because RS gravitons couple to the full Standard Model stress-energytensor, their dijet and leptonic couplings are of comparable size, and RS gravitons are morelikely to be discovered through leptonic channels than dijet channels at hadron colliders. Lep-tonic channels are cleaner than dijet channels, largely because they do not have to competeagainst the large QCD background.However, the literature also includes other BSM spin-2 models, such as the various spin-2 resonances described in Refs [4]-[6]. A generic graviton may couple to any number ofstress-energy tensors. We use the term “colorphilic graviton” (labeled X ) to describe aphenomenological massive spin-2 object that couples exclusively to the quark and gluonstress-energy tensors with strengths proportional to parameters κ q and κ g respectively. Byconstruction, a colorphilic graviton is likely to be first observed in the dijet channel as an s -channel resonance. This parallels other particles to which the color discriminant variable hasbeen applied. Explicit specification of a UV completion for the colorphilic graviton modelswe consider is beyond the scope of this article.Phenomenological spin-2 particles warrant caution, because their interactions genericallyviolate unitarity at high enough collision energies [4]. The colorphilic graviton X couplesto qq and gg states through dimension-5 operators, necessitating couplings κ q and κ g withunits of inverse energy. For small partonic center-of-momentum collision energy √ ˆ s betweenpairs of incoming partons, this effective theory is valid order-by-order in powers of κ √ ˆ s . Atsufficiently high partonic scattering energy, κ √ ˆ s grows large enough to destroy perturbativity1f the effective theory, as manifested in unitarity violation.Our work here on production and decay of colorphilic gravitons focuses on two-into-two scat-tering processes. To ensure the validity of our cross-section calculations, we use a tree-levelpartial-wave unitarity analysis to enforce unitarity. Similar analyses have been applied toother spin-2 objects in the literature [7]-[9]. We demonstrate how this further constrains theparameter space accessible to collider searches and find that there exist regions of parameterspace within which a colorphilic graviton is discoverable in the 14 TeV LHC dijet channel.Suppose a new dijet resonance R is discovered within the range of parameter space wherea colorphilic graviton is accessible. How could we immediately distinguish whether R isa spin-2 state? The dimensionless color discriminant variable provides a means of quicklydiagnosing what classes of BSM models might describe a newly discovered resonance: D col,R ≡ m R σ Rjj Γ R (1)This is constructed as to be independent of coupling strength when applied to s -channelresonances with narrow widths. The color discriminant variable is so-named because it isproportional to the color and spin structure of the resonance. Color-singlet objects andcolor-octet objects tend to be well separated in the color discriminant variable vs resonancemass plane [10].Previous analyses have demonstrated the color discriminant variable’s ability to discriminatebetween color-singlet, color-octet, and excited quark dijet resonances [10]-[11]. The methodhas also been broadened to models with flavor-dependent couplings [12]-[13] and applied toscalar diquarks [14]. The present article applies the color discriminant variable to a spin-2particle.The remainder of the paper is organized in the following manner. Section 2 describes thecolorphilic graviton, X . Section 3 details the constraints (including unitarity) that arerelevant to identifying an LHC-discoverable dijet resonance originating from a colorphilicgraviton and plots the surviving parameter space. Section 4 reviews the uncertainties relevantto a color discriminant variable analysis, plots the color discriminant variable (includinguncertainties) vs the mass of X , and then compares the X analysis to an equivalent Z (cid:48) analysis. Section 5 summarizes our conclusions. This section presents the colorphilic graviton, its dijet cross-section, total decay width, andcolor discriminant variable. 2 .1 Lagrangian and Parameterization
A colorphilic graviton is defined to be a massive spin-2 object X that is a SM singlet andcouples exclusively to particles charged under the SM SU (3) C . We write the Lagrangiandensity interaction terms describing how X couples to the Standard Model as follows: L int = − κ g X µν T gµν − (cid:88) q i κ q i X µν T q i µν (2)where X µν labels the X field, T qµν and T gµν are the quark and gluon stress-energy tensors, T gµν = η µν F ρσ F ρσ − F µρ F νρ − ξ η µν (cid:20) ∂ ρ ∂ σ A σ A ρ + 12 ( ∂ ρ A ρ ) (cid:21) + ξ ( ∂ µ ∂ ρ A ρ A ν + ∂ ν ∂ ρ A ρ A µ ) (3) T q i µν = − η µν q i ( iγ ρ D ρ − m ψ ) q i + i q i ( γ µ D ν + γ ν D µ ) q i + i η µν ∂ ρ ( q i γ ρ q i ) − i ∂ µ ( q i γ ν q i ) − i ( q i γ µ q i ) (4)and A and q i are the gluon and quark fields respectively (color indices have been suppressed)[15]. We normalize the spin-2 polarization tensors (cid:15) sµν according to the published version ofRef [15], such that, (cid:15) s,µν (cid:15) s (cid:48) ∗ µν = δ ss (cid:48) (5)where s ∈ {− , − , , , } is a helicity index and µ, ν are Lorentz indices. Eq. (2) impliestree-level couplings for the following interactions: q i q i X , ggX , q i q i gX , q i q i V X , gggX ,and ggggX , where V denotes the electroweak bosons γ, W ± , Z .For the present analysis, we reduce the number of free variables by assuming X has flavoruniversal couplings to quarks ( κ q i = κ q for every quark q i ). This assumption allows us tosimplify the present analysis and avoid flavor constraints. Consequently, there are only twocouplings in the theory: the quark stress-energy coupling κ q and the gluon stress-energycoupling κ g . These couplings have units of inverse energy. By using the mass of X (denoted m X ) as an energy scale, we exchange the dimensionful couplings κ q , κ g ∈ [0 , + ∞ ) TeV forthe dimensionless parameters α ∈ [0 , + ∞ ) and β ∈ [0 , π ], which are defined according to, κ q ≡ αm X cos β κ g ≡ αm X sin β (6)The cases where X couples exclusively to T qµν , where X couples equally to T qµν and T gµν , andwhere X couples exclusively to T gµν correspond to β = 0, β = π , and β = π respectively. Let c β and s β denote cos β and sin β respectively. Eq. (2) yields the following tree-level decaywidths: Γ X → gg = α m X π s β Γ X → q i q i = 3 α m X π c β (cid:18) m q i m X (cid:19) (cid:20) − m q i m X (cid:21) (7)3he model described by Eq. (2) additionally permits three-body decays X → q i q i g , X → q i q i V , and X → ggg where V denotes the electroweak bosons ( γ , W ± , Z ), as well as thefour-body decay X → gggg . After accounting for infrared divergences in processes involvingmultiple massless bosons, these decay channels are numerically negligible relative to the two-body decay widths due to additional coupling and phase space factors. Analysis of theseadditional decay channels lies outside of our tree-level dijet channel analysis, and we ignorethem for the remainder of this article. The total decay width is well described as the sum of the gg and qq channels:Γ X = α m X π (cid:0) s β + 3 n q c β (cid:1) (8)where n q acts as an effective number of massless quarks, n q ≡ (cid:88) q i (cid:18) m q i m X (cid:19) (cid:20) − m q i m X (cid:21) (9)In the limit where m X (cid:29) m q i for every quark flavor, we find n q →
6. Only the top quarkprovides significant deviations from the massless case when m X >
500 GeV, which is themass range relevant to our analysis. As a result, n q is well approximated by the followingexpression: n q = 5 + (cid:18) m t m X (cid:19) (cid:20) − m t m X (cid:21) (10)where m t is the mass of the top quark. As m X increases, n q quickly approaches 6. Forexample, n q (1 TeV) = 5 .
89 and n q (2 TeV) = 5 .
97. We may write the relevant branchingratios as, Br ( X → gg ) = 16 s β
16 + (3 n q − c β (11) Br ( X → q i q i ) = 3 c β
16 + (3 n q − c β · (cid:40) q i = u, d, s, c, bn q − q i = t (12)after some simplification. We are interested in decays of X to pairs of light jets j , which we define as a QCD jetoriginating from a partonic gluon or one of the five lightest quarks. At tree-level, the decaywidth and branching ratio to light dijets equal,Γ X → jj = α m X π (16 − c β ) Br ( X → jj ) = 16 − c β
16 + (3 n q − c β (13) With sufficient luminosity, the three- and four-body decay channels might provide a means to distinguishthe colorphilic graviton from other models, such as the leptophobic Z (cid:48) . ∼ .
15 [16], which is also the range of valueswherein the narrow width approximation (NWA) applies [17]. The dijet cross-section of aresonance R with mass m R equals, in this approximation, σ Rjj = 16 π Γ R m R Br ( R → jj ) (cid:40)(cid:88) i,k (1 + δ ik ) N ik Br ( R → ik ) (cid:20) s dL ik dτ (cid:21) sτ = m R (cid:41) (14)The indices i, k in Eq. (14) label partons in each of the incoming protons. At tree level, X couples to gluons, quarks, and antiquarks, so that in principle i, k ∈ { g , u , d , s , c , b , t , u , d , s , c , b , t } . Because the proton has negligible top quark content and X couplesdiagonally to quark-antiquark pairs, we restrict k = i ∈ { g , u , d , s , c , b } . The contributionof a partonic combination i, k to proton-proton collisions is described by its correspondingparton luminosity function, (cid:20) d L ik dτ (cid:21) ≡
11 + δ ik (cid:90) τ dxx (cid:104) f i ( x, µ F ) f k (cid:16) τx , µ F (cid:17) + f k ( x, µ F ) f i (cid:16) τx , µ F (cid:17)(cid:105) (15)where µ F is the factorization scale and f i ( x, µ F ) is the parton distribution function forparton i [17]. We set the factorization scale to the resonance mass such that µ F = m X and evaluate the parton luminosity functions at τ = s/m X , where s is the proton-protoncenter-of-momentum energy squared. We use the parton distribution functions from theCTEQ6L1 PDF set [18] for our calculations, and take care when extracting data from othersources to use each source’s choice of PDF set.The factor N ik in Eq. (14) counts color and spin degrees of freedom for the partonic combi-nation i, k relative to the resonance R : N ik ≡ N S R N S i N S k · C R C i C k (16) N S and C are a given particle’s number of spin and color states respectively. ( N S , C )equals (5 , , ,
8) for the colorphilic graviton, quarks and antiquarks, and gluonsrespectively, yielding, N q i q i = 536 N gg = 5256 (17)These considerations allow us to simplify Eq. (14) to, σ X jj = 16 π Γ X → jj m X (cid:40) N gg Br ( X → gg ) (cid:20) τ dL gg dτ (cid:21) sτ = m X + N q i q i Br ( X → uu ) b (cid:88) q k = u (cid:20) τ dL q k q k dτ (cid:21) sτ = m X (cid:41) (18)5ubstituting Eqs. (11-13) and Eq. (17) into Eq. (18) yields an explicit expression for thedijet cross-section in terms of m X , α , β , and s . σ X jj = (16 − c β ) πα n q − c β ] m X (cid:40) s β (cid:20) τ dL gg dτ (cid:21) + 2 c β b (cid:88) q i = u (cid:20) τ dL q i q i dτ (cid:21)(cid:41) sτ = m X (19)The color discriminant variable D col,X is, therefore, D col,X ≡ m X σ X jj Γ X (20)= 10 π (16 − c β )3[16 + (3 n q − c β ] (cid:40) s β (cid:20) τ dL gg dτ (cid:21) + 2 c β b (cid:88) q i = u (cid:20) τ dL q i q i dτ (cid:21)(cid:41) sτ = m X (21)Eqs. (19-21) are valid for the process pp → X → jj , where j is a light jet originating froma partonic g , u , d , s , c , or b . In this section, we describe the region of parameter space where the colorphilic gravitonmight be detected by the LHC with 0 .
3, 1, and 3 ab − of LHC-14 integrated luminosity for β = 0, π , and π , and describe the subregion where a color discriminant variable analysis ofsuch a discovery would apply.Note the β = 0 case corresponds to X coupling exclusively to the quark stress-energytensor, whereas the β = π case corresponds to X coupling exclusively to the gluon stress-energy tensor. When plotting the parameter space of the colorphilic graviton, we fix β andplot α vs m x . As discussed in Section 3.3, enforcing tree-level unitarity of a X modelestablishes a scale Λ EFT max up to which our effective field theory respects tree-level unitarity.This scale can be used as an additional parameter to constrain an effective theory. Therefore,we concern ourselves with four parameters: the colorphilic graviton mass m X ; a unitlesscoupling strength α ; an angle β measuring the relative coupling strength of X to quarksvs gluons; and an upper limit Λ EFT max on partonic center-of-momentum energies for which thetheory respects tree-level partial wave unitarity.We eliminate regions of parameter space that are already experimentally 95% CL excludedand only consider regions of parameter space where 5 σ dijet resonance discovery may eventu-ally be observed at LHC-14. The theory must respect unitarity to be self-consistent, leadingus to consider unitarity constraints obtained from tree-level partial-wave amplitudes. Narrowdijet resonance searches by CMS and ATLAS have limited sensitivity for dijet resonanceswith Γ R /m R (cid:38) .
15, and so we eliminate these regions of parameter space as well [16].Because the color discriminant variable analysis is appropriate in regions of parameter space6here the width Γ X of X is wide enough to be measured by the detector, we also considerhow Γ X compares to the detector’s mass resolution M res . As mentioned in the introduction, CMS and ATLAS are searching for resonances in the dijetchannel, and, in the absence of a resonance–establishing 95% CL exclusion limits on dijetresonances [1]-[2]. Both experiments report these limits as upper bounds on acceptance A R times dijet cross-section σ Rjj as a function of resonance mass m R . They also plot A R × σ Rjj for several benchmark models. The exclusion limits presented by ATLAS and CMS arecomparable; we choose to utilize the CMS exclusion limits, which are obtained from 36 fb − of LHC-13 dijet channel data.The acceptance A [ β ] of the colorphilic graviton is calculated for each value of β in Mad-Graph 5 [19] according to the cuts described in Ref [1]. We then demand, A [ β ] · σ X jj ≤ ( A × σ jj ) (22)The region of parameter space that fails to satisfy this constraint, i.e. where a colorphilicgraviton is excluded by current LHC data at 95% CL, is located in the upper left area ofeach plot in Fig. 1, bounded by a thick black curve, and colored with a translucent darkred. All other points of each plot have not been excluded by the limits of Ref [1]. σ Discovery Reach
We also establish regions of parameter space where a dijet resonance might someday bediscovered. The CMS experiment has published how sensitive their detector is towards 5 σ discoveries in the dijet channel for L int = 100 pb − , 1 fb − , and 10 fb − [20]. This is reportedas an upper limit on dijet cross-section times acceptance. We calculate the relevant spin-2acceptance via MadGraph 5 [19] for each value of β according to the cuts described in Ref[20].We assume the systematic uncertainties scale proportionally to the square root of integratedluminosity √L int to extend the 10 fb − discovery prospects of Ref [20] to L int = 0 .
3, 1,and 3 ab − . Any areas of parameter space that require more than 3 ab − worth of LHC-14data according to this scaling are designated as inaccessible and we exclude that region ofparameter space. Naive scaling provides a qualitative idea of LHC-14 discovery prospects;however, we expect the regions we plot ultimately to be conservative because this scalingignores any improved experimental sensitivities in the high luminosity LHC-14 dijet channel.In each pane of Fig. 1, the boundary of every 5 σ discovery region is denoted with a blackcurve, while the regions themselves are denoted in white. Curves corresponding to largervalues of L int are further rightward, with L int = 0 .
3, 1 and 3 ab − appearing from left to7ight respectively. The region above and to the left of a given L int curve corresponds tothe region of parameter space accessible with L int worth of LHC-14 dijet channel data. Thegray region to the bottom-right of each plot is the previously-described inaccessible regionof parameter space, which requires more than 3 ab − worth of LHC-14 dijet channel data todiscover a colorphilic graviton. Because X couples to quarks and gluons via dimension-5 operators, its couplings are dimen-sionful and the colorphilic graviton generically violates unitarity once the energy scales ofthe process exceed some energy scale Λ EFT max [4]. This is indicative of the breakdown of the ef-fective field theory (EFT) approximation implicit in our analysis. Once unitarity is violated,the effective field theory becomes invalid and the method of analysis must be changed. Ifan EFT L EFT arises as an approximation of a more fundamental perturbative theory L fund ,then its breakdown must be circumvented with additional new physics effects relevant to L fund . These new physics effects become relevant at some energy scale Λ NP . In order that L fund respect unitarity at energies higher than L EFT , the new physics must enter before theEFT’s breakdown, such that Λ NP < Λ EFT max . We may demand a model containing a colorphilicgraviton respect tree-level unitarity up to some value of Λ
EFT max and in doing so we ensure thatour analysis including only a colorphilic graviton is self-consistent at energies relevant tocollider searches, and the underlying new physics could be at higher energies than probedby the LHC.Unitarity demands certain relationships between matrix elements, including constraints on2 → s -channel spin-2 particles such as thecolorphilic graviton. In particular, each eigenvalue a of the partial-wave amplitude matrixcorresponding to this process must satisfy | R [ a ] | ≤ /
2. We calculate the tree-level 2 a → X → b amplitudes where 2 a , b ∈ { q i q i , gg } and resulting constraint in Appendix B. Oneconsequence of this constraint is that Γ X /m X must be smaller than .We fix Λ EFT max and seek regions of parameter space where a colorphilic graviton model satisfiesEq. (39) for values of √ ˆ s up to at least Λ EFT max . Combining Eq. (38) and Eq. (36) with Eq.(39), for each value of Λ
EFT max we demand m X , α , and β satisfy:Λ EFT max < m X (cid:34) π (8 + c β ) α (cid:32) (cid:114) − (8 + c β ) α π (cid:33)(cid:35) / (23)In each pane of Fig. 1, we plot the boundary of this constraint for Λ EFT max = 10, 33, and100 TeV. Each boundary is linear at low resonance mass m X and denoted by a red line,with shallower slopes corresponding to larger values of Λ EFT max . The region above a givenΛ
EFT max boundary would encounter unitarity violation for some value of √ ˆ s below Λ EFT max . Thelowest value of Λ
EFT max we might consider setting is Λ
EFT max = m X , so as to ensure consistencyof the theory up to production of the X particle. Beyond this consideration, the choiceof Λ EFT max is arbitrary. Based on the ranges of masses admitted by the other constraints in8he analysis, the largest m X discoverable with 3 ab − of LHC-14 data and consistent withΓ X /m X ≤ is m X ∼ EFT max . We choose EFT consistency up to Λ
EFT max = 10 TeV forthe remainder of the analysis, as to allow room for new physics above m X , and we excludefrom our analysis points in the translucent red region above the Λ EFT max = 10 TeV boundaryline.
As described in Section 2.2, we utilize the narrow width approximation, which provides agood approximation when Γ X /m X ≤ .
15 [16]-[17]. This inequality implies an upper limiton α : α ≤ (cid:115) (0 . π
16 + (3 n q − c β (24)However, as discovered in Appendix B, unitarity demands Γ X /m X ≤ /
8, which is astronger constraint. Therefore, enforcement of tree-level partial wave unitarity ensures col-orphilic graviton models remain in the regime of the narrow width approximation. The simpleform of Eq. 24 motivates plotting X parameter space as ( m X , α ) instead of ( m X , κ ).The boundary at which Γ X /m X = 0 .
15 is marked by a horizontal black line near thetop of each pane of Fig. 1. The grayed region directly above this line corresponds toΓ X /m X > .
15 and is excluded from our analysis.
The dijet mass resolution M res of a detector determines whether experiments can measurethe total decay width Γ R of a particular dijet resonance. Specifically, the resonance widthis measurable if it exceeds the mass resolution; else, the resonance width is irresolvable.Because the color discriminant variable is explicitly constructed from the resonance widthof a dijet resonance, it is only applicable to identifying the nature of the resonance whenΓ R ≥ M res . Dijet resonances for which Γ R < M res may still be detected, but would requirea different kind of identification analysis.The mass resolution of the LHC is an approximately linear function of resonance mass m R . We obtain the LHC dijet mass resolution function by interpolating resonance massbin widths [1], and subsequently demand Γ X ≥ M res . The approximately-horizontal dashedcurve passing through the middle of each plot of Fig. 1 denotes points for which Γ X = M res .Above this curve, colorphilic gravitons satisfy Γ X > M res and possess resolvable widths atthe LHC; below this curve, colorphilic gravitons satisfy Γ X < M res and have irresolvablewidths at the LHC. 9igure 1: Colorphilic graviton ( X ) parameter space for various β values. The translucentdark red region bounded by a thick black curve to the upper left is experimentally excluded at95% CL. The black curves denote points accessible with integrated luminosities of L int = 0 . , , and 3 ab − respectively; the gray dark region to the bottom-right is inaccessible even with3 ab − of LHC-14 data. Points below the rising red diagonal lines respect tree-level unitarityup to Λ EFT max = 10 TeV, 33 TeV, and 100 TeV from steeper to shallower lines respectively.The translucent red region violates tree-level unitarity below 10 TeV and is excluded fromour analysis. The grayed region above the horizontal black line along the upper edge violatesΓ X /m X > .
15. Points below the approximately-horizontal dashed Γ X = M res curve haveirresolvable widths and are excluded from D col,X analysis. For each β -value, this leaves theapproximately triangular purple regions near ( m X , α ) = (3 . , .
5) for D col,X analysis.10 .6 Summary of Available Parameter Space The regions of parameter space in Fig. 1 are, in summary, • Experimentally Excluded: points within the translucent dark red region boundedby a thick black curve and located in the upper left of each plot are experimentallyexcluded at 95% CL. • Experimentally Inaccessible: points within the gray region to the bottom-right ofeach plot are deemed inaccessible at LHC-14, because they require more than 3 ab − worth of LHC-14 dijet channel data for discovery in the dijet channel. • Unitarity: points within the translucent pale red region bounded by a red curve areexcluded from our analysis because they violate unitarity below Λ
EFT max = 10 TeV. • Narrow Width Approximation: points within the grayed region above the hori-zontal black line at each plot’s upper edge satisfy Γ X /m X > .
15 and are excludedbecause the CMS and ATLAS narrow dijet resonance searches have poor sensitivity inthis region [16]. • Mass Resolution: points below the approximately-horizontal dashed curve passingthrough the middle of each plot are excluded from the D col,X analysis because theypossess unresolvable widths: Γ X ≤ M res . Colorphilic gravitons with properties corre-sponding to points below this curve are still potentially discoverable at the LHC, butwould have to be identified by means other than D col,X . • Color Discriminant Variable: points in the approximately triangular purple regionsof each plot are relevant to a color discriminant variable analysis.The black curves correspond to the experimental reach for discovery of a colorphilic gravitonat LHC-14 with luminosities L int = 0 .
3, 1, and 3 ab − from left-to-right, with parameterspace below a given curve being inaccessible with only L int worth of LHC-14 dijet channeldata. The remaining white and purple regions in each Fig. 1 subplot denote areas ofparameter space accessible with L int for the indicated β value. As illustrated, LHC-14has discovery reach for colorphilic gravitons with any β value, even after taking unitarityconstraints into account. Therefore, the colorphilic graviton X is a new physics object thatis generally relevant to searches in the LHC-14 dijet channel.Several conclusions can be reached by comparing the various pieces of information overlap-ping within the panes of Fig. 1:There is a reasonably large range of masses where a graviton can be discovered but a colordiscriminant variable analysis is inapplicable. This is because the discoverable parameter11pace corresponds to gravitons with relatively weak couplings and thus relatively small de-cay widths. There is a much smaller window of masses where the theory is consistent upto Λ EFT max = 10 TeV yet the coupling is also large enough for the color discriminant analysisto work. Note additionally that this region quickly runs towards larger coupling parame-ter α with increasing graviton mass and consequently runs into conflict with the unitarityconstraints. Among the different values of β , the color discriminant analysis applies for aslightly smaller graviton mass in the β = π case than the other cases.Given a fixed value of β , each value of Λ EFT max generates a contour through parameter space.The Λ
EFT max contours corresponding to 10, 33, and 100 TeV are illustrated as red curves inFig. 1. Note additionally the white and purple regions above the L int = 3 ab − curve: thisis the region of parameter space where colorphilic gravitons are discoverable with 3 ab − ofLHC-14 dijet data according to our criteria. As Λ EFT max is continuously increased from 10 TeV,the corresponding contour moves continuously downward, so that less of this discoverableregion lies below the curve. There exists a value of Λ
EFT max for which no parameter space pointsare simultaneously discoverable with L int = 3 ab − LHC-14 data and consistent with ourunitarity bounds up to Λ
EFT max . This value of Λ
EFT max is 34, 38, and 36 TeV for β = 0, π , and π respectively. Any colorphilic graviton arising from a theory consistent with larger values ofΛ EFT max would not be discoverable at LHC-14. Therefore, if a colorphilic graviton is discoveredat the LHC, it necessarily implies additional new physics below about 30 −
40 TeV.The LHC-14 discoverable region relevant to a color discriminant variable analysis (eachpurple region) also corresponds to an upper limit on possible Λ
EFT max values. When L int = 3ab − , this uppermost value of Λ EFT max is 15, 14, and 13 TeV for β = 0, π , and π respectively.While theories of colorphilic gravitons with larger values of Λ EFT max can be constructed, a colordiscriminant variable analysis would not be valid for such objects, because their widths wouldnot be resolvable by the LHC detectors.For ease of discussion during the color discriminant variable analysis in the next section, wedenote by P β [ L int ] the region in the ( m X , α ) plane of X parameter space that is allowed bythe above constraints (Λ EFT max fixed at 10 TeV) for a given luminosity L int and β value, andalso generates dijet resonances with resolvable widths. These are exactly the purple regionsof Figure 1. P β [ L int ] is therefore a proper subset of the parameter space region where acolorphilic graviton is LHC-14 discoverable. X from other Dijet Resonances Suppose a dijet resonance is observed at a mass m R and possesses a measurable color dis-criminant variable D col,R . After accounting for experimental uncertainties, that measurementmay or may not be consistent with the value of D col predicted by a model. In this way, exper-iments can immediately eliminate any class of models inconsistent with the measured D col,R as an explanation of that observed dijet resonance. The utility of the color discriminantvariable analysis hinges on this capability. 12he measurable D col,R values at a fixed m R = m X that are consistent with the colorphilicgraviton model depend on the available parameter space (described in Section 3) as well asthe uncertainties of the measurements involved. In what follows, we review the statisticaland systematic uncertainties relevant to the color discriminant variable; plot the color dis-criminant variable (with uncertainties) for the colorphilic graviton; and show in detail howthe color discriminant variable can be used to distinguish colorphilic gravitons from othermodels.Error propagation of the color discriminant variable is detailed in Ref [10] and summarizedhere. All uncertainties are modeled as Gaussian. The uncertainty of log D col,R is relatedto the relative uncertainties of the dijet cross-section, mass, and decay width according to,∆ [log D col,R ]0 .
434 = ∆ D col,R D col,R = (cid:18) ∆ σ Rjj σ Rjj (cid:19) ⊕ (cid:18) m R m R (cid:19) ⊕ (cid:18) ∆Γ R Γ R (cid:19) (25)The symbol ⊕ denotes addition in quadrature. The relative error of the dijet cross-sectionis, ∆ σ Rjj σ Rjj = 1 √ N σ ⊕ (cid:15) σ sys (26)where N σ denotes the number of events necessary to make a 5 σ discovery and (cid:15) σSY S is thedijet cross-section’s systematic uncertainty. The relative uncertainty of the dijet mass is,∆ m R m R = (cid:20) √ N σ (cid:18) σ Γ m R ⊕ M res m R (cid:19)(cid:21) ⊕ (cid:18) ∆ MM (cid:19) JES (27)where σ Γ (cid:39) Γ R / .
35 is the (Gaussian) standard deviation of the resonance’s intrinsic width. M res denotes the previously-mentioned dijet mass resolution of the experiment and (∆ M JES /M )is mass measurement uncertainty due to jet energy scale uncertainties. The relative uncer-tainty of the dijet resonance total decay width is,∆Γ R Γ R = (cid:118)(cid:117)(cid:117)(cid:116) N σ − (cid:34) (cid:18) M res σ Γ (cid:19) (cid:35) + (cid:18) M res σ Γ (cid:19) (cid:18) ∆ M res M res (cid:19) (28)where ∆ M res /M res is the relative uncertainty of the dijet mass resolution. The variousuncertainties entering this calculation have been estimated using experimental data, and aresummarized as follows: • (cid:15) σ sys is an approximately linear function of m X , extracted from [20]. (cid:15) σ sys (1 TeV) = 0 . (cid:15) σ sys (6 TeV) = 0 . • M res is an approximately linear function of m X obtained as described in Section 3.5,by interpolating bin widths [1]. M res (1 TeV) = 0 .
057 TeV M res (6 TeV) = 0 .
23 TeV13 (∆ M/M ) JES = 0 .
013 [21]. • ∆ M res /M res = 0 . L int = 3 ab − parameter spaces P β [3 ab − ] The results of the color discrim-inant variable analysis are summarized in Fig. 2; the plots within the figure correspond to β = 0, π , and π from top to bottom respectively.Each plot within Fig. 2 contains a dashed red curve that traces out the theoretical colordiscriminant variable D col,X as a function of m X with s = (14 TeV) . By construction, D col,X only depends on m X ; at tree level, any direct α dependence cancels out. However,secondary α dependence lingers in the experimental uncertainty of a D col,R measurement.The various bands we have plotted in ( m R , D col,R ) space correspond to values for which ameasurement of a dijet resonance with mass m R and color discriminant variable D col,R wouldbe consistent with the theoretical D col,X , with different colors signifying different conditions: • Darker Gray:
Values of ( m R , D col,R ) within 1 σ of the theoretical D col,X , where theuncertainty has been obtained by ignoring all constraints and fixing Γ X /m X = 0 . • Lighter Gray:
Values of ( m R , D col,R ) within 1 σ of the theoretical D col,X , where theuncertainty has been obtained by ignoring all constraints and fixing Γ X = M res , theboundary determined by the mass resolution of the LHC. • Solid (Faded) Red:
Values of ( m R , D col,R ) within 1 σ of the theoretical D col,X , wherethe uncertainty is set to the minimum (maximum) uncertainty available in P β [3 ab − ],as plotted in the purple region of Fig. 1. At any given m X , this corresponds to thelargest (smallest) width available for that given mass in P β [3 ab − ]. A black borderoutlines the faded red region.The usefulness of the color discriminant variable emerges from comparing the different re-gions of ( m R , D col,R ) space that various dijet resonances R occupy. For example, considerthe leptophobic Z (cid:48) particle from Ref [10]. Like X , the leptophobic Z (cid:48) possesses a param-eter space restricted by the NWA, mass resolution of the detector, exclusion limits, and L int discovery prospects, yielding an available parameter space P Z (cid:48) [ L int ]. An updated colordiscriminant variable analysis on the leptophobic Z (cid:48) is summarized in Appendix A. Previ-ous works demonstrated that this Z (cid:48) is well-separated from colorons, excited quarks, anddiquarks in ( m R , D col,R ) space [10],[13]-[14]. As a result, a measured D col,R consistent with aleptophobic Z (cid:48) is necessarily inconsistent with coloron, excited quark, and diquark models.Fig. 3 shows how the colorphilic graviton X compares to the leptophobic Z (cid:48) in ( m R , D col,R )space for β = 0, π , and π from top to bottom respectively. As in Fig. 2, L int is fixed at 3ab − and the colorphilic graviton is illustrated in red. The leptophobic Z (cid:48) is illustrated inblue. The dashed blue line plots D col,Z (cid:48) and the regions are colored according to14igure 2: The color discriminant variable and its uncertainties for the colorphilic graviton X . D col,X is plotted as a dashed red curve for each value of β considered. In the longshaded regions: dark gray denotes D col,R values within 1 σ of D col,X when uncertainties arecalculated with Γ X /M X = 0 .
15. Light gray denotes D col,R values within 1 σ of D col,X when uncertainties are calculated with fixed Γ X = M res , the mass resolution of the LHC.In the truncated shaded regions near 3.8 TeV: solid (faded) red denotes D col,R values within1 σ of D col,X when uncertainties are calculated with the minimum (maximum) uncertaintyavailable in the purple L int = 3 ab − parameter space from Fig. 1. A black border outlinesthe faded red region. 15igure 3: Comparison of the color discriminant variable and its uncertainties for the col-orphilic graviton X versus the leptophobic Z (cid:48) . D col,X is plotted as a dashed red curvefor each value of β considered. Solid (faded) red denotes D col,R values within 1 σ of D col,X when uncertainties are calculated with the minimum (maximum) uncertainty available inthe L int = 3 ab − parameter space from Fig. 1; this reproduces the red regions of Figure2. A black border outlines the faded red region. The color discriminant variable D col,Z (cid:48) of the leptophobic Z (cid:48) is plotted as a dashed blue curve. Solid (faded) blue denotes D col,R values within 1 σ of D col,Z (cid:48) when uncertainties are calculated with the minimum (maximum)uncertainty available in the L int = 3 ab − parameter space for Z (cid:48) .16 Solid (Faded) Blue:
Values of ( m R , D col,R ) within 1 σ of the theoretical D col,R , wherethe uncertainty is set to the minimum (maximum) uncertainty available in P Z (cid:48) (3 ab − ).At any given m R , this corresponds to the largest (smallest) width available for thatgiven mass in P Z (cid:48) (3 ab − ).We use the same procedures and sources for the Z (cid:48) data as the X data, and utilize proceduresidentical to those described in Section 3.We immediately see from Fig. 3 that most regions of ( m R , D col,R ) space occupied by thecolorphilic graviton are already occupied by the leptophobic Z (cid:48) . The β = π models arepotentially distinguishable from Z (cid:48) models at resonant masses m R ∼ . σ of the Z (cid:48) models. Meanwhile, effectively all resolvabledijet resonances consistent with β = 0 colorphilic graviton models are also consistent withleptophobic Z (cid:48) models.There is, however, a significant amount of ( m R , D col,R ) space where a resolvable dijet reso-nance would be consistent with a leptophobic Z (cid:48) model but is inconsistent with X models.This is because X is more restricted in its range of available masses by unitarity constraints.For instance, for all values of β , colorphilic gravitons with masses below m X (cid:46) m R (cid:46) X , but may possiblybe described by a leptophobic Z (cid:48) . Similarly, a resolvable dijet resonance with m R (cid:38) Z (cid:48) and X models means many re-solvable dijet resonances could be equally well described by either model, it also meanscolorphilic gravitons can be distinguished from coloron, excited quark, and diquark modelsjust like the leptophobic Z (cid:48) . If a D col,R measurement falls in the regime where the colorphilicgraviton and Z (cid:48) overlap, then a detailed angular distribution analysis of the decay productswill be required to determine the spin of the resonance and thereby select the appropriatemodel. The LHC dijet channel is one of the most powerful tools of modern physics, capable of ex-ploring previously-unseen physical regimes. If a dijet resonance R is discovered, determiningits origin will be of intense interest. Our work here has demonstrated that the LHC mightyet discover a dijet resonance consistent with a “colorphilic graviton” X , a massive spin-2particle that couples to the quark and gluon stress-energy tensors.Analyzing the phenomenology of the X state requires careful consideration of unitarity con-straints: because X couples to qq and gg states via dimension-5 operators, the colorphilicgraviton generically violates tree-level unitarity. Colorphilic graviton models are parameter-17zed by the particle’s mass m X , the overall coupling strength α , a measure of the relativecoupling strength of X to the quark vs gluon stress-energy tensors β , and the scale Λ EFT max up to which the X model respects tree-level unitarity.The parameter space of X relevant to collider searches is constrained by unitarity consider-ations, application of the narrow width approximation, experimental exclusions, and abilityto be discovered with integrated luminosity L int . The region of α vs m X parameter spacethat survives these constraints is illustrated in white and purple in Fig. 1 for L int = 0 .
3, 1,3 ab − , and β = 0, π , π . For every value of β , we found that there does exist a region ofparameter space in which the LHC could discover a X -originating dijet resonance with 3ab − of data. Moreover, across the entire range of β values, we see that there is an upperlimit of about 5 TeV on the mass of a discoverable X resonance. For self-consistency, thevalue of Λ EFT max must be at least this large. Accordingly, we set Λ
EFT max = 10 TeV for this anal-ysis. Due to the combined effect of discovery prospects and unitarity constraints, we alsofound that the discovery of a colorphilic graviton would necessarily imply the presence ofadditional new physics below about 30-40 TeV.While a dijet resonance R consistent with a colorphilic graviton could be discovered at theLHC, such a resonance might also initially be consistent with other models. The colordiscriminant variable D col,R provides a quick means of eliminating potential dijet resonancemodels [10]. D col,R is constructed from quantities immediately measurable after the discoveryof a resolvable resonance: the mass of the resonance m R , the total dijet cross section at thepeak of the resonance σ Rjj , and the width of the resonance Γ R . The color discriminantvariable has proven useful when applied to leptophobic Z (cid:48) , coloron, excited quarks, anddiquark models [10],[13]-[14].In applying the color discriminant variable analysis to X , we fixed L int = 3 ab − to get anidea of which resolvable colorphilic gravitons might be observed in the LHC dijet channel.Our subsequent analysis revealed a limited range of color discriminant variable D col,R valuesconsistent with a colorphilic graviton X to within 1 σ of experimental uncertainties, assummarized in Fig. 2. Furthermore, the region of ( m R , D col,R ) space consistent with X islargely shared with the leptophobic Z (cid:48) , so that resolvable dijet resonances consistent with X are also consistent with Z (cid:48) . This is demonstrated in Fig. 3. Therefore, just as D col,R is able to discriminate Z (cid:48) resonances from coloron, excited quark, and diquark resonances,it can likewise tell an X apart from those other states. Should a D col,R measurement fallin the values applicable to both the colorphilic graviton and Z (cid:48) , additional analyses wouldbe required to distinguish the models. For example, it might be possible to recognize thethree- and four-body decay modes of the colorphilic graviton with sufficient luminosity. Moregenerally, the spin of the new resonance could be determined via a detailed angular analysisof its decay products.We look forward to future opportunities to apply the results of our analysis to newly discov-ered dijet resonances. 18 cknowledgments We thank Kirtimaan Mohan for useful conversations. The work of. R.S.C., D.F., and E.H.S.was supported by the National Science Foundation under Grant PHY-1519045.
A. Parameter Space of Leptophobic Z (cid:48) The (flavor-universal) leptophobic Z (cid:48) is a massive spin-1 particle with the following interac-tion Lagrangian: L Z (cid:48) ,eff = igZ (cid:48) µ (cid:88) i q i γ µ ( g L P L + g R P R ) q i (29)where i sums over the SM quarks, g is the weak coupling, and P L ( R ) is the left (right)projection operator [10]. We calculate quantities in the large Z (cid:48) mass limit, so the couplingsonly occur in the | g L | + | g R | combination.As with the colorphilic graviton, the parameter space of the leptophobic Z (cid:48) is constrained byexperimental exclusions, application of the narrow width approximation, discovery prospects,and the requirement of a resolvable decay width. We utilize the CMS 95% exclusion datafrom [1] and the projected 5 σ discovery prospects from [20], and apply them via the methodsdescribed in Section 3. Fig. 4 summarizes the parameter space that survives this procedure.The color discriminant variable analysis proceeds identically to the analysis of X , utiliz-ing the same experimental uncertainties and application methods. Fig. 3 illustrates thecollection of ( m R , log D col,R ) measurements consistent with the leptophobic Z (cid:48) model for L int = 3 ab − worth of LHC- dijet channel data. Specifically, the solid (faded) blue denotes D col,R values within 1 σ of the theoretical D col,Z (cid:48) when uncertainties are calculated with theminimum (maximum) uncertainty available in the L int = 3 ab − parameter space for theleptophobic Z (cid:48) . B. Tree-Level Partial-Wave Amplitude Details
We now outline the construction of the full scattering matrix in the limit m t (cid:28) √ ˆ s . For s -channel 2 → X → f f to massless fermions f f , M f f → f f = − c β α ˆ s m X (ˆ s − m X ) (cid:18) d , +1 d , − d − , +1 d − , − (cid:19) (30)19igure 4: Leptophobic Z (cid:48) parameter space. The lightly-grayed region above the horizontaldashed line along the upper edge violates Γ Z (cid:48) /m Z (cid:48) > .
15. Points below the approximately-horizontal dotted Γ Z (cid:48) = M res curve have irresolvable widths and are excluded from D col,Z (cid:48) analysis. The translucent dark gray region bounded by a dashed line to the upper left isexperimentally excluded to 95%. The black curves denote points accessible with integratedluminosities of L int = 0 . , , and 3 ab − respectively; the cross-hatched dark region to thebottom-right is inaccessible even with 3 ab − of LHC-14 data. This leaves the white regionsabove the Γ Z (cid:48) = M res curve for D col,Z (cid:48) analysis.20here d Jm ,m (cos θ ) denotes the Wigner d -functions [22] (where we set φ = 0 without loss ofgenerality). Similarly, for massless fermions to massless gauge bosons and vice-versa, M f i f i → γγ = − s β c β α ˆ s m X (ˆ s − m X ) (cid:18) d , +2 d , − d − , +2 d − , − (cid:19) (31) M γγ → f i f i = − s β c β α ˆ s m X (ˆ s − m X ) (cid:18) d , +1 d , − d − , +1 d − , − (cid:19) (32)Finally, for massless gauge bosons to massless gauge bosons, M γγ → γγ = − s β α ˆ s m X (ˆ s − m X ) (cid:18) d , +2 d , − d − , +2 d − , − (cid:19) (33)The rows (columns) label initial (final) state helicity combinations and are organized fromtop-to-bottom (left-to-right) as follows: (+ h, − h ), ( − h, + h ), where h = for the fermion-antifermion states and h = 1 for the massless vector boson states. The (+ h, + h ) and( − h, − h ) combinations yield vanishing matrix elements in the massless limit. To obtain thefull scattering matrix, we piece together matrices of these sort for every flavor and colorof particles that couple to the colorphilic graviton. Columns with outgoing gluons shouldbe divided by two to avoid double-counting identical particles. This final transition matrixpossesses (2 · · · = 52 elements, each of which is converted to a partial-waveamplitude via the following formula: M ( a,b ) → X → ( c,d ) = 80 π A ( a,b ) → X → ( c,d ) d b − a,d − c (34)The partial-wave unitarity constraint is then that every eigenvalue a of the partial waveamplitude matrix A satisfy | R [ a ] | ≤ . The resulting matrix A of partial-wave amplitudesis highly redundant, and possesses only one nonzero eigenvalue a , such that, a = − α ˆ s πm X (ˆ s − m X ) (cid:20) c β · · · · s β · · (cid:21) = − (8 + c β ) α ˆ s πm X (ˆ s − m X ) (35)where s β has been eliminated via the fundamental trigonometric identity. Note the poleat ˆ s = m X , which we generally expect. For convenience, we define the dimensionlesscombinations, ˆ s ≡ ˆ sm X g ≡ (8 + c β ) α π (36)We have defined g intentionally such that g = Γ X /m X . Partial-wave unitarity subsequentlydemands, ≥ | R [ a ] | = ⇒ | ˆ s − | ≥ g ˆ s (37)There is a maximum value of ˆ s that respects this constraint given a specific colorphilicgraviton model. Specifically, for a given g , unitarity requires,ˆ s < ˆ s max ≡ g (cid:104) (cid:112) − g (cid:105) (38)21or g > , no value of ˆ s solves Eq. (38), putting an upper limit on width of the colorphilicgraviton that can be consistent with unitarity. Section 3.4 will show this constraint is evenstronger than the constraint due to the narrow width approximation.We define Λ EFT max such that ˆ s max ≡ (Λ EFT max /m X ) . Given a specific instance of a colorphilicgraviton model with parameters m X ∗ , α ∗ , and β ∗ , tree-level unitarity is respected only forpartonic center-of-momentum energies √ ˆ s that satisfy, √ ˆ s < Λ EFT max where Λ
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