JHEP05(2021)079 sglu- sym- R Springer May 11, 2021 April 12, 2021 April 12, 2021 : : : January 26, 2021 : Revised Published Accepted Received symmetry and two sgluons below -symmetric models, despite stark symmetry breaking quantitatively R R R Published for SISSA by https://doi.org/10.1007/JHEP05(2021)079 -symmetric models can be obtained by restor- symmetry. We then compute the rates of all R R ). We construct such models by augmenting minimal [email protected] , . 3 broken models - R 2012.15771 symmetry The Authors. c Supersymmetry Phenomenology, Phenomenological Models

In this work we study the collider phenomenology of color-octet scalars ( R , [email protected] symmetry ( symmetry. In parallel to this discussion, we explore constraints on these models from R R ) in supersymmetric models with Dirac gaugino masses that feature an explicitly bro- -symmetric models with a fairly general set of supersymmetric and softly supersymmetry- E-mail: Department of Physics, The191 Ohio W. State Woodruff University, Ave., Columbus, OH 43210, U.S.A. Open Access Article funded by SCOAP ArXiv ePrint: tively, however, we find thatdifferences in — phenomenology much — as scenariosthe with for TeV broken minimal scale can be accommodated by existing searches. Keywords: detail how the familiar results froming minimal the Large Hadron Collider.affects We find existing that, limits in on general, (pseudoscalar) color-octet sgluons while scalars, opening perhaps one closing for loopholes a for light light CP-even CP-odd (scalar) particle. Qualita- ken R breaking operators that explicitlysignificant break two-body decays andmetry highlight new breaking, features including that enhancementsdecays, appear to and as extant improved a decay cross rates, result sections of novel of tree- single and sgluon loop-level production. We demonstrate in some Abstract: ons Linda M. Carpenter and Taylor Murphy Color-octet scalars in Diracbroken gaugino models with JHEP05(2021)079 32 21 34 48 17 21 14 -broken models 39 -broken models 23 R 38 R 8 16 24 53 4 11 26 43 26 – 1 – 13 -symmetric limit symmetry symmetry breaking 4 R -symmetric limit R R R symmetry breaking 7 28 R 10 14 symmetry breaking 8 R symmetry 5 R 2 -symmetric limits of various expressions -symmetric supersymmetry 42 R R 8.2 Rates of sgluon decays in the 6.2 Production cross sections 8.1 Couplings to gluinos in the 5.2 Novel decays5.3 induced by A brief review of production cross sections 6.1 Decay widths and branching fractions 4.1 Electroweak constraints 4.2 A simple4.3 measure of Reviewing our assumptions and defining benchmarks 5.1 Existing decays modified by 3.2 Color-octet3.3 scalars again Hybrid gluinos3.4 and the electroweakinos Third-generation squark mixing 2.1 Dirac gluinos2.2 and Color-octet2.3 scalars Electroweak adjoint scalars 3.1 A representative model B Technical details II: Lagrangian andC Feynman rules Form factors for color-octet scalar decays 9 Conclusions A Technical details I: masses and mixing 7 Collider constraints on color-octet8 scalars in Interlude: 6 Numerical results and phenomenology 5 Color-octet scalar decays and production in 4 Model discussion II: exploring parameter space 3 Model discussion I: breaking Contents 1 Introduction 2 Review of JHEP05(2021)079 - L ], ˜ q R L 11 ˜ q → L -symmetric q ]. L R q 17 , ]. In minimal 16 56 ]. adjoint (color-octet) – -symmetric models — 50 ); a natural hierarchy ]. Minimal realizations c charge and are allowed 51 – R 2 , R 25 1 SU(3) symmetry [ R ]. Other promising non-minimal ]. These models have provided a -symmetric models, which led to 15 24 R – supersoftness – 13 22 -symmetric superpotential that forbids R – 2 – ]. Accordingly, the theoretical and experimental literature ] of sgluons in minimal ]; and the appearance of new complex adjoint scalars. The 57 58 , 52 21 , ]. Notable among these are an elegant cancellation of quadratic 20 19 ], and general gauge mediation [ 12 ), have received a great share of that attention [ ) due to vanishing amplitudes of certain processes, such as ]. Many varieties of these models have been constructed, all replete with ]. Exclusion limits for the smoking-gun signatures of the colored sector are 18 10 sgluons – 3 symmetry most notably forbids Majorana gaugino masses. In minimal [ -channel gluino [ The aforementioned adjoint scalars, particularly the R t supersafeness catalog of sgluon dynamicsof has these particles, not and yeta many been others, number written. assumed of an gauge-invariant Ourcolor-octet operators scalars. aforementioned that investigation Notable could among affect thesenovel are the sgluon new masses self-couplings, color-symmetric and sgluon-gluino and couplings, dynamics a of wealth the of sgluon couplings to the electroweak sector, these models undergo copious pairdates production for at LHC the searches TeVon [ scale, these making them particles ideal isthis candi- corpus of with considerable a sizeinteresting survey and phenomenological [ continues discoveries, to mostsgluon grow. can notably be that We long-lived. the recently CP-odd added Despite (pseudoscalar) to all of this attention, however, a truly comprehensive vast new terrain for phenomenological exploration at the LHCscalars [ ( symmetric models, these scalarsto are decay assigned to Standard many Model different pairs of Standard Model particles at loop level. Sgluons in ( via abundant particle content andboth rich markedly phenomenology different offormer from far minimal those less of constrained than the that MSSM of — the latter leave [ the parameter space of the superfields that transformsubgroup in [ the adjointinteresting representation features of [ eachdivergences Standard in Model loop gauge between contributions these masses to and scalar those of masses the gauginos; ( the suppression of squark pair production well motivated. Some SUSYHiggsino frameworks with worlds alternative [ spectra includerealizations split involve the SUSY imposition [ of a global continuous models, gauginos obtain Dirac masses via couplings to fermions supplied by new chiral Minimal Supersymmetric Standard Modelconservatively (MSSM); excluded the into lightest the2 MSSM TeV TeV squarks range, are with now close gluinos to impinging excluded on the belowthat LHC masses discoverability extensions limits about for of these the particles. MSSM It with therefore follows non-standard signatures and spectra are increasingly Supersymmetry (SUSY) remains the leadingphysics, candidate as for beyond-Standard it Model makescandidates, (bSM) it and possible can to accommodate stabilizeof gauge supersymmetry, the coupling however, weak unification are hierarchy, [ Large becoming offers Hadron viable increasingly Collider constrained. dark (LHC) matter have Runs by I now and placed II strict of limits on the the strong sector of the 1 Introduction JHEP05(2021)079 , , R 3 68 , we ] and 5 64 – symmetry symmetry 61 R R ]. In the case ], there has not -symmetric mod- , we demonstrate 71 symmetry feature 60 , , R 8 symmetry breaking R 70 symmetry breaking. 59 ]. R R 66 , symmetry-breaking Majo- broken R 65 , . In section 59 symmetry [ and 7 -symmetric models are recovered, R R -broken models, pointing out novel de- , we review minimal R ], there exist high-energy theories featur- 2 symmetry breaking is required in order to 59 R -symmetric results. Here we also briefly discuss – 3 – symmetry breaking due to a fairly general set of ]. While the most reasonable size of symmetry breaking arises and how it affects the R R 72 R ] mediation that predict 32 symmetry is broken, some measure of supersafeness can symmetry breaking is taken once again to vanish. Finally, , we complete our description of these models, examine R R 4 -symmetric models detailed above can be partially preserved R -symmetry. In particular, it has been argued in many places [ R ] and gravity [ 73 symmetry breaking is moderate or tiny. In particular, while supersoft- , -symmetric models. Gauginos with both Dirac and Majorana masses — R R 27 symmetry, models with Dirac gaugino masses ]. Meanwhile, there is at least one reason of a less phenomenological nature to , we study the phenomenology of the sgluons in our set of benchmark scenarios. R ]. On the former point, by the way, we note that there are several ways to gen- 64 6 67 This paper is organized as follows. In section Some of the benefits of symmetry-breaking couplings have been cataloged in various places [ ] that a self-consistent theory of quantum gravity cannot admit global continuous sym- the production of theseIn particles, section which is moderatelyWe confront affected these by results tofor data consistency from the that LHC the inas results section expected, familiar when from our minimal explicit symmetry, taking care to explainparticle how spectrum. Inelectroweak section constraints on oursumptions, multidimensional and establish parameter a space, fewcatalog benchmarks review all for the our quantitative decays model investigation. ofcays In as- the and section color-octet significant scalars modifications in to the in the presence of smallsuperpotential but and measurable softly supersymmetry-breaking operators involving adjoint superfields. els, discussing the generation ofwe Dirac gaugino depart masses from and the this adjoint well scalars. trodden In section ground and define a family of models with broken remains an open question, withas some measured phenomenological by studies the allowing Majorana-Dirac it to massing be ratio both [ quite gauge large [ rana masses suppressed by the Planckwe scale. investigate the With phenomenology all of of the this adjoint motivation scalar in fields mind, — therefore, particularly the sgluons — study models with broken 69 metries, and this restrictionof would supergravity, it appear has to beengenerate apply pointed a to small out cosmological that constant [ ness is immediately lostbe when maintained if the splittingmasses between [ Majorana gluinos is smallerate enough Dirac compared gaugino to masses their withoutbreaking supersoftness, [ so its loss is not unique to two gluinos and sixand neutralinos to — minimal all Majoranahybrid or — so-called in “mixed” stark gauginosfeature — contrast their can to own be interesting both generated and the in distinct MSSM a phenomenology variety [ ofif ways [ the extent of that softly break supersymmetry whileR preserving gauge invariance. Whileyet these been explicitly an extensive explorationalso of dramatically their affects phenomenological the effects. absence gauginos: of since Majorana masses cannot be forbidden in the including to neutralinos and Higgs bosons. There also exist operators of similar structure JHEP05(2021)079 } ]. ), ). , 3 67 and [ 2.1 2.2 (2.2) (2.3) (2.4) (2.1) W α gauge , couple D 2 0 g 3 ψ W 6= ˜ , U(1) . 1 c D 0 , ˜ g (color octet) D 3 {W a κ H.c. O 1 Λ a 3 + t 2  mass, which we write 1 a = √ O α = O a 3 ], 3 W symmetry. m 18 3 term of a hidden ) , yields Dirac gaugino masses. κ R 0 and D + D -symmetric models was the obser- H.c. Dirac gluino , A adjoint Majorana fermion R with + c T a k α A 2 λ a D a k ˜ g W λ a D 2 SU(3) ( ¯ ˜ g κ k 3 + M -symmetric models, including Dirac gauginos, m of which are visible in the decompositions ( – 4 – S R , we obtain a 3 =1 } c α X k , which parameterize the coupling of each adjoint a 3 1 (isospin triplet) and the t ≡ − } 1 2 , 3 3 ) W A κ A 2 1 λ , t T κ SU(3) 2 {  A 2 ⊃ − ] as a basis for direct comparison to models with broken H.c. κ t α , 0 58 1 + = soft κ W symmetry a 3 term, which we denote by { L ψ 1 T Λ a 3 R D θ , which by definition is not self-charge-conjugate: λ 2 ( 0 , D 3 d ˜ g m (1) Z ) and throughout this work, summation is implied over repeated -symmetric supersymmetry = , the generators 2.1 c ⊃ − charge, and the adjoint scalars. This discussion serves to establish nota- R -symmetric models, and draw conclusions. R at which supersymmetry is broken by the R Dirac . In ( Dirac 0 L Λ SU(3) , we summarize our findings, highlight the most interesting contrasts between L W 9 and . These indices refer respectively to Weyl spinors and the adjoint representations of L } a (hypercharge singlet) , symmetry. -broken and S A are prohibited. Suchcan terms, be which forbidden cannot at be large by generated imposing by a the global supersoft continuous operator ( In this scenario, the Majorana gluino to form a DiracThe gluino Dirac nature of this gluinoof is the preserved as form long as softly supersymmetry-breaking terms The dimensionless constants superfield to each Standard Modelneed gauge not field, be can unified. bewith In unique, spinor the so indices case the suppressed of three as gaugino masses at the scale superfield { SU(2) Integrating out the U which generate interactions between the Standardand Model a gauge set superfields of chiral Standard Model adjoint superfields, vation that Dirac gaugino masses canwhile be only generated introducing in finite a manner (supersoft)eters. that corrections breaks to This supersymmetry all feat supersymmetry-breaking is param- made possible by the supersoft operators [ tion and to make contactR with [ 2.1 Dirac gluinosOne and impetus for the historical development of minimal R 2 Review of We begin with aassignments brief of review of minimal in section JHEP05(2021)079 , 3 ϕ (2.7) (2.5) (2.6) ) are -Higgs 2.1 R ) unnecessary, . 2.6 3 -symmetric models R symmetry behind. We term ( R is forbidden and µ symmetry throughout the -invariant contraction B d L R charge assignments consistent H symmetry in the Higgs sector symmetry, with the promise of ) R R R subgroup that does not commute SU(2) and H.c. d u ). a + symmetry in section H H ]. In other scenarios, the Higgs fields U(1) o · i d symmetry, including the amelioration of u R 74 H + H · R a u O θ µ H ( complex adjoint (hence color-octet) scalar charge, then the supersoft operators ( should be of similar size; second, despite the ( 2 2 – 5 – d symmetry. In one popular scheme, all Standard 2 c µ 1 µ ) are not. Beyond this basic requirement, there R √ B Z B R (3) 2.4 ≡ ) is permitted, and a softly supersymmetry-breaking a 3 neutral so that the ⊃ − L ⊃ ϕ and 2.5 R symmetry breaking forces us to abandon exactly Dirac µ soft symmetry breaking concurrent with electroweak symmetry that one might want to avoid.) In this work, we circumvent L R R symmetry elsewhere renders the term ( R µ . ]. In previous work, we adopted a set of Introduction 3.4 charge, a 75 -breaking term of the form , ], depend somewhat on one’s choice of scheme. We discuss the latter point R R 71 76 , , symmetry is a symmetry with at least a 60 71 R The scheme we have just reviewed has two possible disadvantages: first, it suggests An and flavor) and the elimination of mixing between left- and right-chiral squarks and symmetric and terms of the form ( We now consider the adjoint scalars,work the we superpartners restricted of ourselves the adjointwhich to fermions. we the decompose In SU previous according to gauginos anyway. But the noveltydistinct of models and with exciting broken phenomenology,begin is to reason construct enough models to with leave explicitly broken 2.2 Color-octet scalars gaugino masses. (Thisenumerated is in to the say nothingthese of issues the by departing features fromsuperpotential. of convention minimal to Breaking explicitly break and lifting all restrictions on that supersymmetry should bein broken the by a rest different offact mechanism the that in this model, the Higgs is since compromises sector often the than ignored, Dirac the nature explicit of breaking the of gauginos by allowing the generation of Majorana with the latter scheme.the Other supersymmetric consequences flavor of problem (viaR exclusion of operators thatsleptons simultaneously [ violate further in section and explicitly is added to avoid spontaneous breaking [ of the up- andfields down-type must chiral be Higgs addedhave superfields nonzero to generate Higgs masses [ with supersymmetric transformations. Ifare the defined Majorana to gauginos and carryR the equal adjoint and fermions opposite are many ways to endowModel a particles model are with defined to be JHEP05(2021)079 (2.8) (2.9) ) and (2.11) (2.12) (2.10) 2.1 ]. operators, 79 ]. This height- ) according to ), , only the scalar , ], or they can be O 2.2 77 A , . B H.c. a pseudoscalar. We 78 T a 64 o o + . Various solutions to a i a 3 2 ) o O and ) ϕ lemon-twist 2 a 3 M O S ϕ H.c. B ( ≤ supersoft operator ( ), we obtain physical sgluon 2 O + − 2 O c 3 B 2 2.7 O B 1 2 ϕ 3 M and the mass of the pseudoscalar ( − . ϕ b 3 1 2 is a scalar and SU(3) a 3 O tr ϕ ϕ a 3 − m O a 2 O † 3 ϕ a B ϕ ab O δ 2 a O similar terms for + ( 1 2 O term insertions in mass-generating diagrams, M ) + 3 = 2 O – 6 – a ϕ − D † 3 3 ) combine to give B O 2 ) ϕ a ϕ a + 3 2.8 † 3 O tr 2 3 ϕ 0 α ϕ 2 O ) includes all of the terms conventionally included in m tr W + M α , which need not satisfy a 3 0 2 + 4 2.11 O h ) and ( ϕ ( W 2 B O 2 3 2 0 3 2.1 M m κ Λ ⊃ − ( 2 2 1 θ the scalar and pseudoscalar masses. In this expression, the Lie- 2 soft are what we call the sgluons. We continue to assume for simplicity d ] ⊃ − L o Z 54 L ⊃ − soft L and L ⊃ + and split O -symmetric models, but even in these models this expression is incomplete. For , so that ). These operators give contributions to the squared mass of each adjoint that Dirac a 3 R L ϕ a 3 2.1 . These masses, which are in general not equal, receive contributions from multiple t o The tree-level expression ( = m 3 incited by the splittingthis term problem have beenSuch proposed, operators most can of eithergenerated which be via, involve postulated e.g., the using power inclusionassuming symmetry expansions of messenger-based arguments of ultraviolet new [ completions operators. of the supersoft operator [ which are also supersoftators and ( cannot be forbiddenare by large any and symmetry positive that forens allows one concerns the component oper- about and a negative for tachyonic the mass other (for [ one component of the adjoint scalar) already minimal example, the scalar and pseudoscalar masses can be split by the As we alluded toeven above, in splitting the presence between of themass a scalar receives symmetry a and that contribution pseudoscalar forbids from the masses the holomorphic is Dirac mass generic: gluino mass. and after decomposing themasses of adjoint the scalars form according [ to ( Accordingly, the operators ( contribute to algebra valued adjoint scalarϕ fields are decomposed in analogy with ( that the adjoint scalars dodenote not the violate physical CP, so massby that of the scalaroperators sgluon even by in simplethe models. unavoidable soft-breaking At terms minimum, the The particles JHEP05(2021)079 ) L ± 2 ϕ 2.7 . The (2.15) (2.16) (2.17) (2.18) (2.19) (2.13) (2.14) 2 SU(2) ϕ . We take t and 1 → ), ). d 2 ϕ T A as the Goldstone T i i 1 + ∓ . A H d 1 A , and  2 H -symmetric models are T H 2 A s , can be obtained by an ( R M } ), ). symmetry. We write the 2 M + | t s 1 → | ST i i d √ H R A v d S + + ( = , H 2 T S 1 ) = ) = A ± u 1, 2, 3, 4 √ 4 4 + adjoint scalars by + 2 H T H 2 A → S T L  ∈ { m v m v -symmetric models in anticipation of , ( ( , I = 3 H 3 R 2 2 2 H EWSB 1 1 2 A −−−−→ ), | H √ √ with I SU(2) m 0 m d Φ , , A have identical quantum numbers to the up- -broken models. By analogy with the color- 2 ( 2 H , we find two electrically charged states , which we can decompose similarly to ( t 2 H 2 A 2 R 3 2   I and ϕ ϕ + m m – 7 – 3 H EWSB EWSB Y , , −−−−→ −−−−→ ) are physical, as we treat T with 1 1 ) ) and 2 2 H 2 A , and t s −T i i √ T m m 6= 1 U(1) H ) ( ( + , + − u Φ adjoint superfield can be representated analogously s I 3 T S A T , 2 H to be real and symmetric, in which case the masses of i ( ( 2 L T S adjoint (isospin-triplet) scalar, needless to say, transforms diag diag 2 2 M 2 A + 1 √ 1 L √ | H √ u ≡ ≡ M   Φ . H 2 A = 2 1 = 2 H SU(2) 3 2 1 m + m A = and SU(2) ϕ ϕ u v A L ⊃ − 2 H ( T 2 and M 1 2 A √ t boson. Details of the mass and mixing matrices of these and other fields = Z T EWSB −−−−→ ) as 0 u ). When we do so for the scalars H adjoint (singlet) scalar, as its name suggests, transforms trivially under the Standard 2.13 Y Because the adjoint scalars 2.14 Only three of theboson pseudoscalars, of ( the are provided in appendix and an analogous termthe for mass the matrices pseudoscalars with the four scalar (pseudoscalar)orthogonal eigenstates diagonalization of the form all of these particlessignificantly more can complex mix. thansituation that Therefore is of the the qualitatively MSSM. scalar unchangedscalar Here sectors masses in we in of models make the with this gauge basis precise, broken as since the and down-type Higgs bosons to ( and an electrically neutraland complex ( scalar adjoint superfield is given by Each component of the where in passing to thethe last singlet term a we VEV. have initiated The in electroweak a symmetry breaking more to complicated give way. A natural representation of the Lie-algebra valued We now discuss the electroweaktheir adjoint interactions with scalars color-octet in scalarsoctet in scalars, we denote theU(1) complex Model gauge group and is decomposed as 2.3 Electroweak adjoint scalars JHEP05(2021)079 k . G are 1 (3.2) (3.3) (3.1) D} , are once U , 3 { t i j D ). and of establishing o j 3.3 2 ) are the genera- label fields in the i Q ¯ ] k t k } ( j V , ¯ k k i t t { [ are respectively the . k b k g ), 2 V O n . Since this representation is 3.2 o b a ] and c exp 3 † i ), on the other hand, . V k SU(3) c 3 g , forms the basis of the calculations D  t [  3.3 of 3 B + SUSY g  j 8 2 These are the Kähler potentials for the L U n j and ), and + o c j Q i A ] W exp o k symmetry a j L † i V – 8 – ] ¯ k + k O t R [ V † K k k θ = SU(3) g t 2 [ 2 d 3 k n g θ G 2 2 L ⊃ L d n exp † i Z U exp = i h † † O θ Q L † 2 θ d 2 θ d 2 θ . d 2 : kinetic and gauge terms. 6 d Z O Z (with, for instance, L 6=2 X k and k 3 =1 + symmetry breaking and masses and mixings of particles that interact with the symmetry is broken by the adjoint sector of the superpotential, the soft-breaking X k 5 + G q R R = L q L = K subgroup running coupling and vectorfundamental superfield. (antifundamental) representations. Lowered (raised) Inagain indices ( the generators ofreal, the we adjoint routinely representation do not respect adjoint index height, though we do in ( These operators generate interactions betweeninteractions quarks, of squarks, sgluons and with gluinosrespectively and squarks the the and gauge gluinos. up- andtors In these down-type of expressions, quark the fundamental superfields. (antifundamental) representations In of ( the Standard Model gauge and for the Standard Model adjoint fields, particularly for the color octets: L up- and down-type (s)quarks, In the interestsummary of of clarity, the we fields included remark below on and one their quantum piece numbers at is provided a in time table in the sections below. A Majorana gaugino masses, andand soft-breaking trilinear Higgs interactions bosons. between adjointscalars, We scalars so remain we particularly restrict interested ourselvesthe as interest in of much the concreteness, as we dynamics possible firstdivide of to define this the a the Lagrangian Lagrangian relevant color-octet into strong representative components of interactions. such according In models. to We 3.1 A representative model The focus of thiswherein work is on extensions of the models surveyed in the preceding section In this section welighting specify the modelscolor-octet scalars. to be In so investigatednotation. doing, in This we section, the along complete restin with the appendices sections of task this started in work, section high- 3 Model discussion I: breaking JHEP05(2021)079 , 1 0 d H 0 u u (3.6) (3.7) (3.4) (3.5) H of any H − − d ·T ) abc , to be real. H d f  + symmetry is u 2 d H ) Y H.c. H ϕ R 2 TH = + ϕ λ d H.c. 2 and 2 H tr ϕ These are the softly + · u 2 1 √ u These are the super- 3 Y ϕ ϕ H λ + 3 u ST (tr λ a 2 H T = 0, · B + (tr d 3 C 3 1 + . T ϕ M 2 SH ). S B operator contributes to the sgluon ϕ a T † 2 SH c 1 3 ) + SU(3) , 2.16 A ϕ λ / R , T tr + ) R ) + H.c. symmetry-breaking parts of the super- Yukawa matrices 2 d T has scalar component SU(3) 3 + R H θ W u M ABC 2 H.c. · 2 × H f λ d i · Q + 3 2 d are the up- and down-type Higgs superfields. d 3 + Z OO λ ) + 2 – 9 – Y d ). The ϕ OO −H tr D = 3 H ) could be included here if a thorough investigation (tr 3 tr 2.1 2 = ϕ H.c. / does not vanish in R − ABC µ , d S d OOO u M R W + ( 2.12 and + H (tr abc L 2 1 · 1 4 H d tr 2 SO O u · u ϕ % ) + O ( B = H H Q % 2 -symmetric and S TT + u + 1 3 2 , but R B . Y H.c. tr 3 1 2 U + 2 ϕ + µ TT † 3 + : supersymmetry-breaking operators. + TTT 1 OOO ϕ 2 tr symmetry as well. We are interested both in terms involving only + λ d SU(2) | tr 1 tr 1 SSS tr soft S H λ R ϕ · S ( 2 | -symmetric models, and that feature is unchanged when L SS O and the elements of the 1 u % 2 : terms generated by the superpotential. 1 S ST R 1 3 M + % µ µ H M / R W M 1 2 2 1 µ + + L + + 2  -invariant contraction = = Dirac + L / R R L ⊃ − R W W W = L SU(2) ), we take ) generates much of the interesting new dynamics in the models we investigate in soft itself vanishes in terms of the supersoft operators (  = L 3.5 3.5 F  The Similar logic holds for and 1 2 SUSY W  ABC the neutral components of which are decomposed according to ( adjoint fields, the masses in minimal broken. The lemon-twist operators ( of these were desired.some Meanwhile, there of are which many break terms that softly break supersymmetry, first because the contractiongauge of group the with totally antisymmetric ad structure symmetric constants product of superfieldsL must vanish, andand second supersoftly because supersymmetry-breaking terms. The Dirac gaugino masses are given by of ( this work. Herehypercharge-singlet scalar. we We omit otherwise only maintain full a generality, noting tadpole that term whose effect is to shift the VEV of the are respectively the relevant potential. In these expressions, In ( The generation indices contracted with these matrices are suppressed. The second part symmetric terms given by where L JHEP05(2021)079 , ); ). a 3 ST and λ a 2.4 (3.9) (3.8) ) are a 3 S t % symme- = 2.11 and R 3 S -symmetric -symmetric λ ). a R R H.c. charge. Recall, H.c. + R + } I a 3 † R ϕ ˜ ) of strengths d , a 3 ) 2  | ϕ ) 3.5 I † explicit. These operators -broken models, since its 0 d R ) } H.c. ˜ R d H | H.c. † I ) and the first and third lines + 0 u 2 d + H.c. a (and, analogously, 3 † H m 3.7 1, 2, 3 d ( + ϕ symmetry in any scheme (such as A 2 a H + a 3 3 † λ · ∈ { SH R 2 ϕ -axions from spontaneous | ϕ u A 2 λ a I I 1 t u † 3 R H R ϕ ( ϕ similar terms for SO H ) and do not affect the rest of the scalar ˜ 1 † u = 1 { 2 | 2 % µ ϕ I ϕ 2 1 2 ( ϕ B 3.5 2 u ( , break · λ 3 u ), which gives many additional contributions ) + d m − + SO 2 µ -broken models than in minimal I a is unnecessary in ) % we could include are of identical form to the 2 L H R | 3.5 ˜ d SO µ d – 10 – 3 ) + − % + -breaking operators in ( u 2 TH B ) ϕ H | H.c. | a 3 R I , and H − d L + ϕ ) ˜ 2 H d − 3 H.c. | a 3 TH symmetry in any scheme where the adjoint scalars are + 2 I m a ϕ ϕ u L + + a 3 , R H.c. ˜ u + tr H a 3 2 ϕ | 0 u · 2 + ϕ I SH A | d † 2 a 3 H L a 3 a u we have written ) where the Higgs fields are given nonzero ( ˜ ϕ ϕ u H ϕ and | H † 1 I a 3 A | 1 ( T † 2 † R 3 ϕ u I 2.1 ϕ ϕ -broken models, the simple tree-level physical masses ( † 1 ˜ u ϕ † 1 2 Q 2 H ϕ ( u ϕ R adjoint fields. Before examining these, however, we pause to discuss 3 ϕ ( m a m -breaking Majorana gaugino masses mentioned previously in ( SH ( -symmetric models is to avoid ϕ 1 Y SO a R + + + R %  SO µ tr % 1 S ST SO U(1) ϕ % % % ⊃ − 1 2 1 2 1 2 − − soft L ⊃ − W ). The operators involving only adjoint scalars, which have strengths L but do not themselves break Some comments on this sizable set of soft-breaking terms are in order. Notice, first, 3.8 neutral. On the other hand, the trilinear operators involving adjoint scalars and Higgs ST models, due chiefly to the alteredactions nature with of the the gluinos andthe novel self-interactions sgluon and inter- masses.significantly In altered by the superpotentialcomplete ( with additional splitting: scalar components of the lastspectrum, two unlike operators the in terms ( we have included for3.2 the electroweak adjoint Color-octet scalars. scalarsThe again sgluons enjoy much richer dynamics in purpose in minimal try breaking. Nevertheless wemodels. maintain it Finally, for we straightforward note comparisoning that to the we color-octet have scalars. omittedproportional soft-breaking This to trilinear is interactions in involv- the interest of simplicity: the gauge-invariant terms % R bosons, which have strengths those discussed in section further, that the term proportional to the inclusion of the Note also the scalar trilinearof operators on ( the last lineare of in ( direct correspondence with the where we have momentarily made generationgenerate indices many scalar masses and all sorts of scalar trilinear interactions. and in the rest of the scalar potential, where correspondingly to JHEP05(2021)079 c O m (3.14) (3.12) (3.13) (3.10) (3.11) break- SU(3) . R d v u v SH adjoint fermion , we are content λ c  ) ), the sum of terms 2.2 SO % a 2 1 , H.c. O 2.15 SU(3)  a + 3 + O λ 2 T a ) generated by the 3 i v ). 3 ψ , a 3 ψ adjoint (hypercharge-singlet) a 2.3 SO -symmetric models, moreover, o 2  ), and ( ψ split % -breaking couplings generically a H.c. ( Y R 3 o M = R ) that are bilinear in sgluon fields + ST i µ ˜ | g 2.13 % + a 1 2 3 U(1) 1 2 Ψ 3.9 ψ S , all but the last of which are 2 split ), ( a 3 v } + 3 ) + ψ M ( µ is taken positive, and the large splitting 2 S 2.7 SH v −  0 3 λ SO H.c. , S κ S % 2 SO v v 2 + % 3 ST a 3 µ % SO √ ) if is required to obtain a TeV-scale scalar sgluon + , – 11 – ) when the λ % H.c. with a 3 2 + S SO , and in this case the pseudoscalar sgluon becomes 2 O λ v SO % 2 3 3 3.5 1 2.12 + ( % B 2 √ 3 , ˜ m g m SO ) are far from complete.  √ O % M Ψ % 1 2 ˜ S g 1 2 + + 4 , % symmetry breaking in models with Dirac gauginos is that  3.10 S 2 M O 2 -broken models) and ( 1 2 O ˜ | g % R ⊃ − , R M M Ψ . This mass contribution is given by + h T h ⊃ − S 1 2 v ), and the hallmark Dirac mass ( S W 1 2 v 2 1 , v L 1 S 3.7 soft − v µ , L 3 SO L ⊃ − µ ]. Between these options and the variety of additional operators al- + % L ⊃ − , -symmetric models, as we briefly discussed in section 1 64 2 W R ) and ( µ 1 L √ { 3.5 + ]. For example, extremely heavy pseudoscalar sgluons can generically be pro- 2 O 80 B obtains its VEV 1 ) (which still exists in = . We note, however, that boosting the values of o ϕ m 2 2.10 split generated by the last operator in ( 3 M The four gluino mass terms can be combined according to derived from ( supersoft operator. Theψ fourth is a novelscalar Majorana mass for the are now generically two. Therethe are model four defined distinct above terms that (ignoringmasses can Hermitian be conjugation) called in gluino masses. The first three are the Majorana to assume that the expressions ( 3.3 Hybrid gluinosAn and immediate the consequence electroweakinos of (some of) the gauginos need no longer be Dirac. We first discuss the gluinos, of which there extremely heavy. These issues can beoperators ameliorated [ by including new sgluonhibited mass-generating by the lemon-twistbetween operators states ( can beparticles’ tamed masses by exotic [ operatorslowed that in minimal contribute with equal sign to both ing, grant us considerableand freedom in varyingincreases the the individual scalar-pseudoscalar physical splitting. sgluona As masses in large minimal imaginary holomorphicgiven a mass multi-TeV Dirac gluino mass where The parameters in ( becomes If we decompose the adjoint scalars according to ( JHEP05(2021)079 + R u ˜ H (3.21) (3.18) (3.20) (3.15) (3.16) (3.17) of the unitary ± 2 breaking λ R two . 2  / 0 d 1 i ˜ H 2 3 , 0 u m . ˜ RP H  + 4 3 2 − d = λ 2 ˜ and (electrically neutral) H ) N H.c. (3.19) 3 2 3 , and the neutral Higgsinos − 2 B ψ 3 2 + M ψ , 1 ψ i − λ ± U − 2 , and the charged Higgsinos ˜ 0 χ ˜ g 3 with λ 1 ± 2 ] , can be obtained by Ψ  and ψ M M ψ } to be Majorana in a fashion entirely ± , must be Majorana fermions. It has (  | 67 1 ˜ = | χ h } 0 N . ψ U 0 ˜ = χ ˜ − M χ = V ˜ | χ 0 1, 2 + ), to be real and symmetric, in which ± 1, 2, 3 ˜ 3 | χ ). If any contribution to either Majorana M ˜ | χ ˜ ) = Ψ χ is the product of an orthogonal matrix | 2 ∆ Ψ ). ˜ g ∈ { , can be obtained by a unitary diagonaliza- M N ∈ { N A.1 } | A.17 I m + Ψ , ), to be real and symmetric, in which case the I 2.14 , , X , 6 1 + I U and , ) = ˜ g ˜ – 12 – χ ˜ g 0 6 = ± I A.9 with ˜ ... χ m Ψ ˜ χ take the form [ ( ± of the Standard Model  ± m 1, ˜ χ | ˜ 2 χ , + 3 2 u 3 ˜ g m λ ˜ M H ∆ ∈ { m diag H.c. with ... − is given by ( , I ˜ + | χ 2 ≤ , given by ( 0 1 + ≡ ˜ g , | ∓ and ˜ ψ χ Ψ ± 1 0 0 I ˜ 3 g ˜ h ˜ g M ˜ χ 1 , given by ( χ ˜ χ m + 2 0 m λ M 1 2 ( m Ψ ˜ M λ χ 0  + ˜ χ M 0 3 = diag M 0 M ] that the Majorana nature of the mass eigenstates is made manifest, + ). ˜ L ⊃ − | χ ≡ || ˜ | χ 67 Ψ 1 2 0 Ψ ˜ χ A.1 1 2 = m of Majorana phases that ensure positive mass eigenvalues. The charginos, I ˜ g P m L ⊃ − . Accordingly, there are now six neutralinos. These fields obtain a variety of bosons, the charged isospin-triplet fermions 0 These fields obtain masses of similar origin to those of the neutral fermions. We d ˜ 3 H given by ( W . 0 3 symmetry breaking also forces the neutralinos − d ˜ M H R bosons, the neutral electroweak adjoint fermions and These charged fields are decomposed following ( 3 3 0 u ˜ We take thecase mass the matrix masses ofdiagonalizations the of three the charginos form with on the other hand, mustcan be make Dirac: them they Majorana. carrygauge electric degrees Just charge, of as so freedom no forcharged amount with the of identical neutralinos, quantum thereand numbers: are the six superpartners electricallywrite charged the charge fermion masses in the gauge basis as tion of the form where much like the gluinoand mixing a matrix matrix gauge basis as We take the mass matrix masses of the six neutralinos analogous to the gluinos.numbers: There the are superpartners six gaugeW degrees of freedom withH identical quantum supersymmetric and soft-breaking masses. We write the neutral fermion masses in the where the mass eigenvalues and mass is nonzero, then the massbeen eigenstates demonstrated [ and the positivity of bothform mass eigenvalues is assured, by a unitary diagonalization of the where the gluino mass matrix JHEP05(2021)079 . . ) 4 3.7 4.2 (3.24) (3.22) (3.23) charge in -symmetric R R ) terms in ( a  symmetry is vari- -breaking trilinear 2 / R 1 R i 2 ) . If, for instance, we hold 2 LR µ ), to be of the same order . m  ) we take as the ), it need not vanish, and the 3.8 , R 3 ˜ t 8 u O ˜ 2 t + 4( L m ˜ t 2 should strictly be considered unaf- M  ) | ˜ 2 t = O 2 RR and M ˜ | t 3 m , then the enhancement of stop mixing Φ 3 Q symmetry breaking at large, in section to remain complex and perform a chiral ) = u − 2 m ˜ 2 t ˜ 2 t R terms relative to m ), to be real and symmetric. (Alternatively, 2 LL m M a , m 1 with ˜ ( 2 t – 13 – and ). h A.24 m ˜ t 3 ( take the form ± Q Φ , on the third line of ( ) and to the soft-breaking trilinear ( 2 ˜ 2 t A.25 ˜ t , can be obtained from the stop mass matrix by an u ] — which (viz. section m symmetry breaking is the lifting of restrictions on mix- 2 RR } a M diag m 3.5 58 ˜ † t m R ≡ Φ . This restriction is reasonable because the magnitude of ≤ 1, 2 + terms), and on R ˜ 2 t 1 ˜ t / . In that case, while stop mixing can be ignored for simplicity , given by ( a m L 2 LL ∈ { µ ˜ 2 t m b ˜ given by ( m I M L ⊃ − term in (  , I µ and 1 2 2 RR ˜ t m R symmetry scheme and choice of model parameters. In the minimal = to the off-diagonal elements of / . This does happen in the models we are constructing, but some dis- L I ˜ µ 2 t ˜ t R R (to be further discussed below). But there is some subtlety to this de- -symmetric limit is substantial. Again, we elaborate on our choices of ˜ q ]. Both, if they exist, contribute to the off-diagonal elements of the stop m R ˜ 2 t , and 24 M , 2 LR symmetry breaking. Nevertheless, in these scenarios the and m 21 L R , ˜ q )[ 2 LL 3.8 m With all of this in mind, we turn to the mass and mixing matrices. We write the stop The phenomenon of enhanced stop mixing in models without below the TeV scale and choose -symmetric models discussed in [ where the mass eigenvalues with we could allow therotation off-diagonal upon elements the of stopstop fields mass to eigenstates absorborthogonal these diagonalization elements’ of the phases.) form The masses of the two masses in the chirality basis as We take the mass matrix this happens then depends onµ the size of the as the soft-breaking squarkrelative masses to the model parameters (including order to allow non-vanishing (and is small enough forcontribution TeV-scale from soft masses fected by operators cannot be forbidden, and stop mixing is generally enhanced. The extent to which ously attributed to the and ( mass matrix pending on one’s R limit of the models in the present work — the Higgs fields are assigned nonzero mixing for each squark flavordiately depends see on that the first- mass and oftake second-generation the a squark superpartner step mixing quark, further is so always andspace we negligible. we discuss imme- explore In only in fact, stop the we mixing latter half here, of in this view work. of We elaborate the upon regions this of point parameter in section A frequently cited consequence of ing between the superpartnerssquarks) of left- and right-chiralcussion quarks is (the in “left- and order.sbottom) right-chiral” For squarks simplicity, we restrict ourselves to the third-generation (stop and 3.4 Third-generation squark mixing JHEP05(2021)079 . 3 | | and ) ) | 0 u ) − d 2 L ˜ , to avoid ˜ H 3 2 H d 3 2 1 1 ) † R † R 1 λ ψ λ λ d u ψ ψ + 0 − u d L ˜ ˜ u H H Fermions ( ( ( (10 O symmetry is broken by is the electric charge. | | | ) ) ) R . L.C. and R.C. stand for 0 u − Q d L µ ˜ H 3 2 d 1 H † R † R B g ˜ B . ˜ ϕ d ϕ ϕ u W L + 0 u d 3 Bosons u H H (˜ ( ( symmetry whose phenomenology , consistent with sections are wildly unconstrained. Several a R 3 g a 3 t 3 2 1 u d . In this table, gauge indices are implicit S U = D Q T O H H W D W W g Superfield in which the Gell-Mann-Nishijima relation is and Y U ) – 14 – 1 ) ) ) ) ) 2 3 1 2 G 1 6 1 3 1 2 , − − , 0) , 0) , 0) , 0) , 0) , 0) , , , 2 , , 1 3 1 1 3 1 2 1 2 G 1 2 , , , , , , , , , , , , 8 1 1 8 1 1 3 ¯ 3 1 3 ( ( ( ( ( ( ( ¯ 3 ( ( 1 ( ( G ( . 7 is the third component of weak isospin and and 3 I 6 doublets are explicit) but generation indices are ignored. In particular, . All displayed superfields are either vector or left-chiral, despite the name L 1 adjoint adjoint adjoint , where c L Y Y Gauge group representations of selected fields in models where SU(2) boson boson + 3 Gluon W B SU(3) SU(2) U(1) L.C. quark R.C. up-type quark R.C. down-type quark Up-type Higgs Down-type Higgs I The fields relevant to our investigation and their gauge group representations are dis- = 4.1 Electroweak constraints Not all of the myriadare parameters introduced related in to section eachconflict other, with and experiment. somethe Here of magnitude we these of consider some must the of be electroweak the taken physics couplings small, introduced required in of to section estimate (though bold fields are Lie algebra-valued:We use for a example, convention for weakQ hypercharge then choose and justifyanalysis a in few sections interesting and distinct benchmark pointsplayed for in table quantitative “right-chiral” for the left-chiral superfields 4 Model discussion II:In exploring this parameter section we space finish definingwe the models study with in broken the restdefine of our this parameter work. space We and first discuss summarize existing the constraints field from content electroweak of physics. the We theory, then Table 1. the superpotential, the Majorana gauginoleft- masses, and and right-chiral, (unnecessarily) respectively. JHEP05(2021)079 , } L T . and B 2.3 (4.5) (4.1) (4.2) (4.3) (4.4) is its , ]. We i 31 0.01 T SU(2) v H 55 M of which, , | ≤ 0 1 2 )[ GeV. Arbi- ˜ χ are 41 S m , H 1 | 2.5 2 H µ . { , and , and T ]: GeV — must behave d | v 81 31 H 5) . 4 , H ± | u − content of H 10 T × , = (125 2 i 1 v ] ] ] and H 2.0) 2 i 84 . Per our discussion in section 81 , m Y S ± T is its weak hypercharge, and parameter, defined at tree level in the 2 i 2 v − 0.07, i  ρ 2 i Y T the weak mixing angle (3.8 ± v Y , and v i i w . + 1)  θ S P i ρ I faces no similarly simple constraint, but we do 2 ( ∆ masses by [ i = 1.06 – 15 – S I [ v i Z to take less than one percent of its gauge-eigenstate with = 1 + 4 P 1 / R with H and parameter in models with additional complex ρ w average = θ µ ρ 2 (isospin-triplet) adjoint scalar is severely limited by elec- ρ W can be obtained by taking one or more of bSM L sec ρ T ] v 2 Z W 2 m 59 m SU(2) , = 1 + ∆ — which we take to have mass ρ 55 = 1 H -broken models with two Higgs doublets and a complex isospin-triplet SM of the ρ R T v ]. In ) becomes [ 83 is the total weak isospin of multiplet , . The hypercharge-singlet VEV 4.3 i 2 I 82 m Finally, while the search for particle dark matter is not the chief focus of this work, it Some care is required regarding the content of the scalar Higgs mass eigenstates, since The VEV . We therefore ensure, for all of our benchmark choices, that 41 the Higgs mixing matrixH elements that measure the is reasonable to ask thatbeing the the electroweakino spectrum LSP, — may the be lightest member a dark matter candidate — not run afoul of the measured dark and the branchingimportant ratios due of to thesatisfy the tree-level these constraints prospect two-body by allowing of decayscontent tree-level (the from mixing latter the between electroweak being adjoint obviously scalars similarly enough to thebeen Standard demonstrated Model in models Higgs with toconstraints very not similar come field be from content ruled to measurements ours outrecently of that by relative the the experiment. to most Higgs stringent the It global Standard has Model signal prediction strength, as measured [ most trarily low upper bounds on sufficiently large. Thismass goal is particularly easyhave to to ensure achieve that with its a value remains multi-TeV Dirac compatible with wino the the lightest of restrictions these, we discuss next. scalar, ( which immediately implies a strict constraint on the triplet VEV of where VEV [ is measured to be quite close to its Standard Model value of unity [ The tree-level prediction formultiplets the can be written as troweak precision data. In particular,Standard the Model Veltman in terms of the JHEP05(2021)079 , / R and ). In (4.6) (4.7) (4.8) L ]. ), that 3.7 73 -breaking , 4.3 R . SU(2) 32 , it is natural , 1 GeV 27 to grow with / R ]. We accept this 3 symmetry breaking ST 10 87 , a . ]) and the interesting R symmetry breaking on } × ] but larger by an order 86 89 R , 59 and symmetry breaking smaller = 5 TeV in order to reproduce : 88 1, 2, 3 S R d a / R a (1) ∈ { O = : k symmetry, because we assume for symmetry breaking, and the great | = 40 ST R R ]) that the trilinear operators should u % for a | abundant. 90 = / R | over = SO : or smaller. % 2 | ) – 16 – / 1 1 ]. Since, as we discussed in section = − ) 2 | k masses a bit higher, while keeping them all of the GeV and 85 [ O c (10 / M R % | 3 O + symmetry breaking = 2 k | 0.120 µ SU(3) = 10 S R ( : % ≈ | k , discussed below, correspond to 1 m TH / R a ≡ 2 Planck = -symmetric models, it is necessary to quantify in some fashion the k h δ R Ω symmetry is broken in any benchmark we may adopt. This task is SH to control the magnitudes in GeV of the trilinear couplings in ( a R / R = -symmetric models, we adopt a unified measure of . This object measures not only the aforementioned dimensionless ST R / ) for each set of Dirac and Majorana gaugino masses, unifying the R a = 4.7 , with lighter Higgsinos suffering from underabundance [ ], as a reasonable choice. It has been demonstrated in other models, notably the S masses but setting the 58 a Y We also use In order to respect all of these concerns, and to avoid obscuring a clear comparison 2 Planck -breaking couplings should be of h Again some notesdespite are the in fact order. thatsimplicity their (as We is operators allow sometimes dobe the done not in couplings for one-to-one break the correspondence MSSM with their [ supersymmetric analogs, which in the case of than that considered inof some magnitude phenomenological than investigations suggested [ by some supersymmetry-breaking scenarios [ particular, we set We use ( U(1) same order. Our choice of Dirac gaugino mass for each gauge interaction. In other words, we set and, simultaneously, R to minimal denoted by parameters but also the ratio of the Majorana gaugino masses, added in quadrature, to the minimal (erstwhile) extent to which complicated by the existencenumber of of multiple free sources parameters, of vaguely in concerned about the perturbativity: models we we imagine, have in constructed. particular, that We the also dimensionless remain at least (though perhaps naturalnessphenomenology problems of are lighter overstated electroweakinos. herethe LSPs We [ in do our verify, benchmark however scenarios (viz. are section not 4.2 A simpleSince measure of the aim of this work is to explore the effects of explicit for the Diracwith gaugino [ masses toMSSM, be that large, a weΩ Higgsino-like view LSP a should Higgsino-likeproblem have (N)LSP, in mass exchange in for of keeping a somewhat more natural Higgsino mass closer to the weak scale matter relic density JHEP05(2021)079 ]. R R to u GeV does 24 a , ) 0 1 2 ˜ χ 23 , charge so , which is (10 t ) breaks y 21 symmetry), O R for the same 2.6 R of . We maintain u a d , is moderate so a / R -symmetric models 0.120 symmetry breaking. R ≈ R . We are interested in , that / R -symmetric models, which symmetry breaking leaves 3.4 R 2 Planck R h Ω symmetry-breaking operators to R TeV. We ensure that the LSP -breaking couplings. The top quark is a R (10) breaking, as measured by O symmetry breaking as possible by introducing R – 17 – R ) is permitted, and only the operator ( symmetry. We assign larger values to 2.5 R ] under which all Standard Model particles, Higgs bosons, symmetry-breaking measurement 90 [ R break do parity } R mixing, so in fact only stop mixing is affected by ]. These models feature a Higgsino-like (N)LSP at the weak scale, stop ST % R 58 , b ˜ - S term of the form ( L % b ˜ µ } ↔ { ST not produce a relic densitythe hierarchy higher of than stops the and observed gluinoshas established not in yet minimal been ruled out in models featuring significant supersafenesscomplete, [ and highly dependent on one’s choice of ultraviolet completion, we take a without too manyfew decoupled light scalars. electroweakinos, but Webosons we prefer to allow be TeV-scale the sgluons decoupled heaviest with and scalar masses at of and least pseudoscalar a Higgs ear couplings among electroweak adjointbetween electroweak scalars adjoints (which and do Higgs bosons notHiggs (which break bosons do), and (which between do squarkscorrespondence and as between trilinear well). couplings and For their maximal supersymmetric simplicity, analogs. we assume a one-to-one a dimensionless global scenarios where the totalthat amount e.g. of the splittingsnegligible. of gluinos and neutralinos about their Dirac masses are not that a symmetry in the Higgsexplore sector. deviations from Accordingly, we these dointact models. an not We exact vary assume either that and parameter color-octet as scalars we are even and everything else is odd. the superpotential and thestudied soft-breaking in sector [ of thesquarks minimal at the TeV scale, and multi-TeV gluinos. They notably assign a In view of the fact that our collection of sgluon mass-generating operators is far from We are interested in scenarios with reasonably natural scalar spectra; i.e., spectra In the interest of generating realistic scalar spectra, we allow for non-vanishing trilin- We adopt as simple an approach to We consider a family of models created by adding , 5. 4. 3. 2. 1. S . But we note, in a continuation of our discussion in section d a 4.3 Reviewing ourIn assumptions the and interest defining of benchmarks convenience,tioned, we others pause heretofore to tacit collect —symmetry. the that The assumptions characterize our — following approach many arechoices to already of the models men- benchmark most with to broken important the ideas phenomenology that we study inform later everything in from this our work. reason: we take this couplingroughly to an correspond order to of thebit magnitude top-quark more Yukawa larger coupling than than forty our timesa more massive than thehardly affects bottom; hence our chosen ratio of { JHEP05(2021)079 ) 7 3 (1) . O and ], that    6 75 , TeV. The TeV GeV 2 70 using the dark 0 1 ˜ = 400 χ = 1.05 o O m m    . ] and built an input suitable o m 95 – ], which could be raised, as we -symmetric models [ 92 and [ R 100 ]. We used the Universal FeynRules [ O ]. relative to the measured relic density. , the symmetric part of which poses technical m β O 97 0 1 99 , % [ ˜ χ 96 freely while fixing tan [ , which needed to be fairly large — of SARAH    SH o λ O – 18 – outputs for m m , but the sbottom spectrum is static. The gluinos, deviates by less than five percent from that of the    / R SPheno 1 H ] package SPheno MicrOMEGAs 91 both for the collider analysis detailed in sections , vary [    © o O    . These three benchmarks, which are consistent with the assump- SPheno 3 , which we identify as the Standard Model-like Higgs boson. We have 1 ] of and H 98 2 Mathematica GeV scalar. We found, similarly to some when analyzing symmetry is broken. In order to obtain these benchmarks, we implemented our in the 125 R 4 sgluons should be considered not simultaneously, but rather in parallel. not in conflict with experiment: It is critical to keep in mind that our discussions of the scalar and pseudoscalar independently of each other and ofsgluons the depend gluino masses. in Since partstrongly, the in rates on fact) of both on decay our of masses, both treatmentand of our since the picture we other’s mass. generally of wantare In each to the at sgluon interest explore the of depends the weak simplicity viability (quite scale — of — scenarios we where adopt both the sgluons following approach, which we have checked is phenomenological approach to the sgluon masses and assume that they can be varied A final set of remarks is in order regarding the mass spectra in these benchmarks. Before we present our analytic expressions, we conclude this section by describing Our implementation omits the operator proportional to 4 — for a dimensionless coupling.stops Finally, and note gluinos that grows — with as growing expected — the splitting between difficulties but does notsgluon strongly masses affect means the we electroweak do spectrum. not use Our the phenomenological approach to the significantly decoupled andonly even exception the is lightestensured physical that pseudoscalar the around predictedknown mass of accomplishing this required some tuning of heaviest gluino. The Higgsino-likelikely LSP the and chief NLSP, reason in for(Another particular, the might underabundance are of fairly be light, ourhave which found choice is reasonable of spectra abetween two fairly with closely separated low higher neutralinos values.)is in fairly all cases. heavy Notice from Next, also top note that to that the bottom, each Higgs with chargino spectrum the sits heaviest Higgs bosons (not shown in table Output (UFO) [ and to compute the relicmatter density phenomenology associated program with the lightest neutralino None of the electroweakinos are decoupled in the sense of being much heavier than the arately in tables tions and constraints we havewhich just discussed, are principally distinguishedmodel by the extent to for the spectrum-generating program the three benchmark scenariosphenomenological analysis. in We which display we the model will parameters compute and numerical the physical results masses and sep- perform JHEP05(2021)079 . / R 500 800 831 0.50 1.60 0.03 1.00 10.0 1.50 = 0.50 2500 1700 1000 1500 2350 3500 1240 0.722 / R -broken models. This table R Benchmark 3 250 800 415 618 0.25 1.53 0.03 1.00 10.0 1.50 = 0.25 1250 1700 1000 1500 2350 3500 0.729 / R } Benchmark 2 TH , – 19 – SH , ST 100 500 800 116 247 , 0.10 1.50 0.03 1.00 10.0 1.50 = 0.10 1700 1000 1500 2350 3500 0.754 S for physical masses and the relic density. Parameters typeset / R 3 ∈ { Benchmark 1 Y , 3 ∗ } T u 1,2 V B 3 m % M ST -breaking operators, and parameters shaded in red are controlled by , = M R = = ˜ t = β S θ 3 = S SO Q 1,2 3 , % B µ

1,2 3 SH TH Y u S T

O a − a µ λ B λ v tan v − m cos m m µ µ

iesofl( Dimensionful fl aaees(GeV) parameters ’ful) d ls parameters ’less d ∈ { V ∗ Three benchmark scenarios for quantitative investigation of Table 2. displays model parameters; see table in red control the sizeParameters shaded of in gray are constant between benchmarks. JHEP05(2021)079 = 0.50 130.50 1166.3 832.77 5334.5 1495.2 856.61 5527.4 1334.4 1554.7 1886.4 1592.4 2220.8 0.0751 2396.2 3221.8 5530.6 4515.4 3245.3 850.87 1591.8 3222.0 / R Benchmark 3 = 0.25 124.90 1318.2 841.36 5272.2 1475.6 851.10 5531.3 1394.7 1964.3 1927.8 2004.8 2064.4 0.0788 2962.2 2812.2 5533.6 4004.0 2815.9 848.21 2004.5 2816.0 / R Benchmark 2 – 20 – = 0.10 125.00 844.29 1383.3 5231.3 1446.0 848.23 5534.7 1411.2 2262.4 1950.4 1939.1 2298.1 0.0779 3286.3 2507.0 5536.7 3695.1 2523.8 847.06 2298.0 2523.8 / R Benchmark 1 for model parameters. 2 2 1 2 3 2 3 0 1 0 2 0 3 0 4 0 5 0 6 ± 1 ± 2 ± 3 h 1 2 L R 1 2 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ H χ t H t χ H b χ A b χ g χ A g χ χ χ χ ˜ ˜ ˜ χ

m m m m m m m m m m m m Ω m m m m m m m m

Electroweakinos QCD ig bosons Higgs . See table 0 1 ˜ χ This table displays relevant physical masses in GeV and (in the last row) the predicted Table 3. relic density for JHEP05(2021)079 , ), R † J ˜ t I ˜ t (5.1) C.1 , and (b) } . When -symmetric -symmetric D 1, 2 ¯ ˜ g R R D . We first ex- ˜ g 7 , the quantitative } ∈ { ], where the final- J 6 , b I c J 90 and ˜ g ˜ g I { , , and both scalar and ). 6 2 † , | ˜ q ) † J ˜ q ˜ t † J C.4 I ˜ t ˜ t I a -broken models ˜ t o R a -function [ → or O -symmetric models. β , the chirality-basis squarks — at O R 3 c ( ) | |F is given by ( b symmetry breaking removes us further ˜ t (
2 -symmetric models, which formally are SU(3) p | | R ) R 2 J ]. ˜ t  I ˜ t 3 O 101 [ m symmetry breaking – 21 – . These and subsequent diagrams were generated m → . -broken models in sections 1 J  R ˜ g O 3 R I † J I ( ˜ t ˜ t α ˜ g |F ) = is renormalization-group (RG) evolved to the sgluon mass † J ˜ t 2 3 I g ˜ t is applied to the squarks using the generic expression ( = | symmetry is broken (as we show in section → O ˜ t 3 Tikz-Feynman
p R O | πα using the MSSM one-loop Γ( 4 ) a ( 0.12 , and recall our assumption that left-right chiral mixing is negligible for all ≈ X package } E ) T Diagrams for (a) scalar sgluon decays to stop squarks 2 Z A 1, 2 m ( 3 α } ∈ { For illustrative purposes, we restrict ourselves to sgluon decays to stop squarks J , I where as an estimate from state three-momentum and where the squared form factor { other squark flavors. The rate of the altered decay to stops is given by pseudoscalar sgluons can decay atsymmetry is the broken, same as level we to discussedleast pairs at the length of stops in Dirac — section gluinos gluinos. can mix, The and the resultingneed Dirac modifications not gluino match to — splits are decays into displayed tousing in two figure non-degenerate the pairs L Majorana of squarks and gluinos — which We begin withaltered the markedly decays when alreadydifference present depends in smoothly onmodels, the scalar extent sgluons of can the decay symmetry at breaking). tree level In to squark pairs associated partial decay rates.models: These in differ addition significantly toWe from modifications then the to review results known in theproduction decay cross rates, of both we sections particles, note of which several single go novel unchanged production decays. from 5.1 of the scalar Existing sgluon decays and modified of by pair 5 Color-octet scalar decaysIn and this production section in we presentinvestigation the of analytic color-octet results that scalarsamine form in the the basis significant of decay our channels phenomenological of scalar and pseudoscalar sgluons and compute the scalar or pseudoscalar decays to gluinos in particular, show mostfrom clearly familiar how scenarios increasing with a single Dirac gluino. Figure 1. JHEP05(2021)079 . ); o R c b ν µ g g → (5.2) (5.3) C.2 , and O -broken -broken 2 1 gγ k k and with R R , (hence any symmetry, o e d gg R m O O R , depends on ˜ q } → ). The rates of a 1, 2 O and , O 2 C.5 m | L vanish. The rates of 2 ) ˜ q | 2 ) + ). ˜ g I } ∈ { abc 1 c b ν µ ˜ g , J g g ˜ f g I 2 , C.8 1 2 ˜ pseudoscalar sgluons. The g | ) with f k k ). The novel contributions I ) → O { → 5.2 gg , O and J ( C.16 O ˜ g e d → ( I c O b O ν µ ˜ g g g are given by ( | |F O |F ˜ g ( given by ( ˜ g 2
p | a β | 2 ) 1 2 |F | are mediated by squarks, in O k k J ) O 2 O ) 3 O ˜ g J 1 1 I ˜ g m m m I I ˜ , with colors and momenta labeled to facil- gg g + I ˜ ˜ q q 2 3 2 ˜ g c b ν µ is given by ( 2 π structure constants g π π g → → 2 2 1 7 4 3 | → k k ) f K ) ) ) a ˜ g O – 22 – o 1 π 1 1 O π π O ( ( gg Γ( (4 (4 (4 SU(3) |F |F 2 1 d e I J + → ˜ ˜ g g c b ν µ ) = ) = g a g O o 2 I 2 1 ( ) = . The rate of decay of the scalar to a gluon pair is given by ˜ ˜ g g k k I a ˜ q 1 I or gg |F O ˜ C ˜ g g → → → + I I ˜ ˜ q q O O O symmetry breaking does not affect the lowest-order decays to a Γ( Γ( Γ( R -symmetric counterparts. Whereas, in the presence of ) can run in the squark loops. Additional sgluon loops exist with a 2 R O ˜ t and ⊃ ) proportional to the and c — and not just due to kinematics — since if this condition is met, the b ν µ g g 1 ˜ t J B.1 ]. While 1 2 Representative diagrams for sgluon decays to gluons. Both k k = 103 , the speed of either gluino in a matching pair, is given generically by ( I , ˜ g β a 102 o Scalar sgluons remain capable of loop decays to pairs of gauge bosons ( The rate at which a sgluon decays to gluino pairs a )[ or O itate the discussion in appendix where the squared formfrom factor gluinos and sgluons significantly enhance the decay rate when combined with the gluon and an electroweak boson, themodels decay than to a in gluon their pairthe is markedly only different nonvanishing in contributions to models this decay can also berepresentative mediated diagrams by gluinos are and displayed scalar in figure and where the squaredpseudoscalar sgluon form decays to factors gluinoslightly pairs different are squared analogous form to factors ( gZ where whether operators in ( scalar sgluon decays to like and mixed gluino pairs are given by Figure 2. combination of Not displayed are trianglesymmetry diagrams is with reinstated. exchanged final-state gluons. Red diagrams vanish if JHEP05(2021)079 ¯ t t 2 1 (5.4) (5.5) (5.6) p p K . The ˜ t ). The 3 ). Both 0 I b J C.22 symmetry is ˜ ˜ g χ C.44 a R o a or O . We preempt our are non-negligible retain a quadratic decay to gluons, a ¯ q ¯ t ) q t ¯ + q q ¯ t t 1 2 , p p K , → 2 ˜ t 2 | , | ) o 2 ) | ). ¯ t symmetry breaking. None of ) t gg Γ( ¯ t b 0 J t I ˜ R g ˜ ¯ t t χ → pseudoscalar → C.20 2 1 a → + 1 o p p and o ˜ O χ ( o ( a ) ( or ¯ q O |F q |F |F 3 o R R 3 t t / / + β L L β m → b b ˜ ˜ ¯ t t 2 o O π 2 1 O p p K a symmetry. The rate of this decay is given by 7 is given by the squared norm of ( ˜ t is given by ( πm is given by the squared norm of ( ) O Γ( πm 2 2 1 π 6 | R 2 | 6 – 23 – ) | ) ) ) ¯ ) t (4 9 π + ¯ b t t 9 I π c J ˜ t g ˜ 1 2 g gg (4 symmetry breaking (4 0 K 3 2 a → ˜ χ o → → R ¯ t t a O o or o ) = or ) = ( ( O ( ¯ t 1 2 ) = ¯ t b K t p p t ˜ g |F |F |F gg + → → → o I J O o ˜ t ˜ t Γ( Γ( Γ( a is a representative gluino-mediated O 2 ⊃ ¯ t t 1 2 p p Representative diagrams for scalar or pseudoscalar sgluon decays to top-antitop pairs. a o We come finally to the decays of sgluons to quark-antiquark pairs a or O -symmetric models, means that only decays to top-antitop pairs of these decay rates can be naturally5.2 extended to decays to Novel lighter decays quark-antiquark inducedBefore pairs. by we move onpause to to discuss note four sgluon decay production channels and that open (later) as collider a phenomenology, result we of where the squared formcorresponding factor rate for the pseudoscalar sgluon is given by where the squared form factor compared to the gluon decays.rate This at is which reflected a in scalar the diagrams sgluon we decays display to in a figure top-antitop pair is given by where the squared form factor own results by notingdependence that the on decay the rates R mass of the final-state quarks. This feature, familiar from minimal traditional squark contributions,displayed which in still figure vanishremarkable if feature all of models squarks with are broken degenerate. Also Figure 3. Not all diagramsreinstated. contribute to the pseudoscalar decay. Red diagrams vanish if JHEP05(2021)079 0 J a I (5.8) (5.9) (5.7) . The ˜ ˜ g χ B symmetry R a o a or O . The rates of these , , 2 4 ) | , 2 | ) c 2 ) | ( ) 0 J symmetry breaking. For oo ). Finally, the rate of the o ˜ χ I I R → ˜ g A -symmetric models here. The ). The rate of the scalar decay C.10 ); and the rate of the similar O R → → ( C.9 O O |F . All diagrams vanish if ( ( C.11 } o I a β o A 1, 2 | |F | |F O ˜ o g 1

p and with slightly different squared form p m | | o 2 is given by ( } ∈ { ) 2 O 2 O 1 1 3 m J 2 | , m m is given by ( given by ( µ ) I a O 2 o { → 2 | – 24 – | I O % 2 SO 2 SO ) ) ( O % % A 0 J oo π ˜ m χ π π 1 1 1 I ). → 8 8 64 ) ˜ → g b O ( 9 O 32 ( ( → ) = ) = C.11 . As usual, o 0 J |F I 0 J ˜ |F χ O ) = ˜ χ I ( A ˜ g oo |F → → → O O given by ( b c O o o Γ( 2 and reproduce the result for minimal Γ( | Γ( ) 5 0 J ˜ symmetry breaking on the production of scalar and pseudoscalar sgluons χ I ˜ g R and a neutralino a → I ˜ g Diagrams for (a) scalar sgluon decay to pseudoscalar sgluons, (b) scalar decay to a light O o ( |F ) a ( are not nearly aspair striking production as — they whichonly is are the on on dominant their the production couplings particles’convenience, mode to decays. we for gluons, display both This the which particles isis relevant are — foremost helpful) diagrams depends not in (with because affected figure colors by labeled where disambiguation factor 5.3 A brief review ofThe production effects cross of sections with squared form factor pseudoscalar decay is analogous with where the squared formscalar factor decay to a gluino and a neutralino is given by where the squared form factor to a pseudoscalar Higgs boson and a pseudoscalar sgluon is given by in the future. Thedecays diagrams can be for straightforwardly these read decays offrate are of of the displayed the Feynman in rules scalar figure provided decay in to appendix pseudoscalars is given by pseudoscalar Higgs boson andto a a pseudoscalar gluino sgluon, andis (c) reinstated. heavy scalar or pseudoscalar decays these novel decays are to Standard Model particles, but some of them might be interesting Figure 4. JHEP05(2021)079 ] 2 ), . In 51 # s (5.11) (5.10) c d O O OO O O β β ) e and → − O — and the -symmetric ˆ s OO 1 + 1 ) R qq (¯ ln → gg ˆ σ a b µ µ g g O  1 gg ) β → ( 2 O ) σ 2 O ˆ s + O m ) ˆ 2 O m 2 O m , 4 Γ( c 2 ! d m , 4 x O O 2 O 2 ( ˆ s , 4 e is the speed of either sgluon ¯ x 2 f ] m O ( 2 ) x / f ( 58 2 O 1 − ) ] g 1 ) 1 m 2 O − a b µ µ 2 O

m g g , 4 s 1 m 2 O 5 , 4 x 24 1 ( , 4 m 1 x + f 4 − ( x ¯ ( f − + 2 O g  ˆ s O O 2 2 m x x = [1 5 d d 34 1 1 O – 25 – β /sx /sx 1 + g O 2 O 2 , and where the parton-level" cross sections are [ symmetry breaking does affect single sgluon produc- q 1 1 m m O 4 4 R 3 O β Z Z β 1 1 ˆ s 1 x x ˆ s 1 2 3 d d 2 3 O O α g g α /s /s π 2 O 2 O are the quark and antiquark distribution functions with momen- π 16 9 1 1 m m 2 15 ) — due to + 4 4 2 ) Z Z q , g ) = ) = gg O O x ( ) = ) = ¯ f → OO OO o relating the parton- and hadron-level center-of-mass energies OO OO at factorization scale ]. The diagrams for single sgluon production are simply given by figure → → and Γ( s 2 x → → ) ¯ q q g g 58 qq 2 x gg Diagrams for scalar sgluon pair production due to (a) gluon fusion and (b) quark- , (¯ ( 1 q qq gg , ˆ σ ˆ x σ (¯ ( x 52 σ σ ( , ) = everywhere. f a 51 ˆ s (b) ( o On the other hand, it is worth noting that the alteration of and → and section of single sgluonmodels insofar production as differs the corrresponding frombecause decay its rate single differs. counterpart production These in differencession of minimal deserve [ the emphasis scalarwith at momenta the and LHC the is flow due of almost time entirely reversed, to so gluon the fu- amplitudes for these processes are in the pair, and wemass. take the We renormalization emphasize and that factorization thesethe scales expressions pseudoscalar. to are be written twice for the the sgluon scalar but applyappearance equally of to tion. No longer is single production the sole privilege of the scalar sgluon, and the cross with these expressions, the kinematic function where tum fraction O hadron-level cross sections of pair production are given by [ Figure 5. antiquark annihilation. The diagrams for pseudoscalar pair production are given by replacing JHEP05(2021)079 - ]. li- R (a) , in 58 6 5 (5.12) in each 7 Collier ) depends in ) 2 O C m , /sx Numerical evaluation 2 O symmetry. , which includes modes 5 m ˜ q ( † scalars. The former effect R ˜ g q , in the three benchmarks ) 6 2 O channels are combined into a light m , -broken models are given by the x R ( gZ g 1 x and x is the gluon distribution function. For d ) 2 gγ /s . We first study the total decay rates and q 2 O , 1 4 x m ) by ( Z g – 26 – s 1 5.4 ) linked for rapid performance to the ] was carried out here and subsequently using the of the sgluons in gg ) 104 o symmetry breaking that are inversely proportional to → ) and ( Γ( R — that, whenever we vary the mass of one sgluon, the mass of the O 5.3 ). These are plotted in figure Γ( and 4.3 5.9 O ) 2 , as functions of the sgluons’ masses. Package-X π symmetry, using the analytic results presented in section O m 3 )–( Γ( R , and, notably, the channel denoted by 5.1 -symmetric models and is driven by tree-level decays to squarks [ ) = and R gB O 2 ]. There are several interesting features for both particles. In figure package → © 109 gg – ( we merge these two discussions to revisit constraints on these particles from σ , available to the scalar not far beneath the threshold for two-body decays to 0 1 7 105 , ˜ χ ¯ t 91 1 ˜ t The branching fractions for the decays of the scalar are plotted in figure We stress again — viz. section → 5 channel denoted by with both stops andO sbottoms, subsumes an estimate of the three-bodyother decays, is fixed. such as widths narrower than thesymmetric scalar, models. is This nevertheless is significantly achannels broader straightforward and than consequence larger of in partial both minimal widths a in larger number channels of not decay forbiddenof by our three benchmarks. In these plots, the and, more dramatically, the width faris exceeds visible this in mark minimal for The latter effect ispart due on to form the factors loopthe allowed sgluon decay by mass. to gluons, Meanwhile, we which see (viz. that appendix the pseudoscalar, while generally featuring total appropriate sums of ( displayed in tables of the Passarino-Veltman functions [ Mathematica brary [ we see that heavy scalar sgluons attain widths approaching ten percent of their masses — In section searches conducted at the LHC. 6.1 Decay widths andThe branching total decay fractions widths In this sectionin and models the with next broken the we benchmark describe scenarios described thebranching in phenomenology fractions section of for the theof scalar color-octet these and results scalars for pseudoscalar detection sgluons at and the discuss LHC. We the then implications consider the production cross sections. simplicity we take the renormalizationcide and at factorization the scales mass for of these the processes daughter to sgluon. coin- 6 Numerical results and phenomenology the corresponding decay rates ( and similarly for the pseudoscalar, where identical. The hadron-level cross sections of these production modes are given in terms of JHEP05(2021)079 symmetry ], with only R 58 6000 6000 5000 5000 4000 4000 (a) (b) – 27 – 3000 , dominates the loop decay to gluons in the latter 3000 respectively. 2000 = 0.10 2000 / R = 0.50 / 1000 R 1000 , and 0 1 4 6 1 10 - - 100 0.01 0.10 0.01 10 10 = 0.25 / R , Decay widths of the (a) scalar and (b) pseudoscalar sgluons in Benchmarks 1, 2, and 3 = 0.10 / R These plots have a variety of interesting features. As we alluded to above, decays far beneath the squark decay threshold in all of our benchmarks because of the hierarchy -symmetric models. These are eventually overtaken by gluino decay modes, but this ¯ t breaking. The most strikingpseudoscalar example sgluons. is the This increasingthreshold strength in decay, Benchmark of which 1, the is with noveltwo benchmarks. already decay to (For considerable completeness, two we near notet the that the squark gluon decay decay dwarfs the loop decay to to squarks dominate asR soon as they areeffect accessible is in not all visible benchmarks,happens beneath just in the as on-shell our in squark plots, decay minimal threshold which depends on depict the scenarios extent of with multi-TeV gluinos. What with on-shell stops. These three-body decaysthe are appropriate estimated vertex in modifications. close Thisindices: analogy joint with we squark [ simply channel, display like several the others, sum has of no all decays of a certain kind. Figure 6. JHEP05(2021)079 ). to in — , we 8 5.12 7 5.3 ], up to ]. By far , but the 58 112 , -breaking con- = 0.10 ManeParse R 111 / R symmetry breaking. R ], it amounts to roughly package ], which are the targets of © 110 57 ), are provided in [ threshold in order to show the ¯ t 5.11 -broken models is given by ( t in the three benchmarks displayed R 9 -symmetric models.) This effect will R -symmetric models — quickly achieves R Mathematica ]. Both kinds of search should be effective 58 -symmetric models, whose pseudoscalars can . We make clear in that section wherever we R wherever both are accessible. In particular, we color-octet scalars [ 7 color-octet scalars [ ) – 28 – ¯ t t real -broken models because the couplings of sgluons to → channel includes decays to a scalar sgluon and any -symmetric models. But we first dispense with pair R o threshold [ R ¯ t complex Γ( OA t TeV is plotted in figure → and for comparison to collider studies for color-octet scalars. It is ) o = 13 gg -symmetric models by three orders of magnitude or more. For light 2 / R → 1 = 2.0 s o that this behavior leads to complementarity between collider searches K Γ( 7 . This and subsequent plots, which required numerical integration of parton 3 factor, and we do not reproduce them here for want of space. In section and K 2 It is instead single production of both particles that is affected by The branching fractions for the decays of the pseudoscalar are plotted in figure distribution functions, were generated usingread the the CT10 next-to-leading order (NLO)the parton most distribution striking functions [ feature isupon the those in size minimal of thescalars, cross this section increase is in driven all by benchmarks, the inverse which sgluon-mass improves dependence of the several LHC searches we discussrescale in theoretical cross section sections to fit the models weThe study cross in section this of single work. This scalar cross sgluon section production at in in tables choice of choose a generous important to note thatleading-order while (NLO) this cross choice section yieldshalf for a of good the NLO approximation cross of section the for full next-to- We now turn to sgluonis production, a which bit — differentproduction, as than we which in discussed is at minimal unchangedgluons the in are end determined of by section gaugeand invariance. The pseudoscalar leading-order pair (LO) production, cross sections which of are scalar given by ( until novel decays to adecays gluino to and gluinos a neutralino dominateextend become once to available. and pseudoscalar As for sgluon for masses all the large scalar once enough sgluon, they6.2 to are permit allowed, these but decays. Production our cross plots sections do not parity with the latterwill see in in Benchmark section 1,for and color-octet far scalars that exceedsquarks. assume it This decays in is to in Benchmarks gluons starkhardly 2 and contrast be to and those constrained minimal that 3. below assume the decays We to top of the three physicalchannels, pseudoscalar there Higgs are bosons. stillscale, Though some which there interesting we are features have fewerinterplay — between extended available decay particularly just at asee the bit that light beneath the end latter the of decayformer the dominates decay — once which allowed in does Benchmark not 1, exist with in minimal attenuate the effectiveness of futureassume collider decays searches to for multi-TeV gluons. color-octet scalars that each of our threescalar, benchmarks. so The conventions for in instance these the plots are similar to those for the between the gluinos and stops, just as in minimal JHEP05(2021)079 ; (b) = 0.10 / R . 6000 6000 6000 = 0.50 / R 5000 5000 5000 (c) (a) (b) 4000 4000 4000 – 29 – 3000 3000 3000 2000 2000 2000 , and (c) Benchmark 3, with 1000 1000 1000 = 0.25 / R 5 5 5 6 4 6 4 6 4 1 1 1 ------10 10 10 10 10 10 10 10 10 0.001 0.001 0.001 0.100 0.010 0.100 0.010 0.100 0.010 Branching fractions for the scalar sgluon in (a) Benchmark 1, with Figure 7. Benchmark 2, with JHEP05(2021)079 ; (b) = 0.10 / R . 6000 6000 6000 = 0.50 / R 5000 5000 5000 4000 4000 4000 (c) (a) (b) – 30 – 3000 3000 3000 2000 2000 2000 , and (c) Benchmark 3, with 1000 1000 1000 = 0.25 / R 0 0 0 4 4 4 1 1 1 - - - 10 10 10 0.001 0.001 0.001 0.100 0.010 0.100 0.010 0.100 0.010 Branching fractions for the pseudoscalar sgluon in (a) Benchmark 1, with Figure 8. Benchmark 2, with JHEP05(2021)079 . ), ≈ 3 . R O ˜ t 5.12 m and -broken 2 R ) — which take C.16 production cross sections in 5000 symmetry breaking. It therefore , single pseudoscalar production, The valleys, which are not present R scalar 7 6 , which is predominantly composed of 2 ˜ t 4000 , this remarkable effect makes it possible 7 3000 – 31 – — are resonant enhancements at (twice) the on- symmetry, the cross section rises with the value of -symmetric sgluons. The other features worth dis- R R in the three benchmarks displayed in tables 9 = 0.10 2000 ]. Such is the power of / R 58 -symmetric models, are at the left-chiral squark thresholds. R 1000 . As we will see in section ) TeV for TeV in figure gg 3.8 5 7 → = 13 - - ≈ 10 2 10 O 10 1000 / O 0.001 0.100 1 Γ( s m Cross sections of single scalar sgluon production by gluon fusion in Benchmarks 1–3. -symmetric models [ R Finally, the cross section of single pseudoscalar sgluon production, also given by ( This statement is imprecise, since in the presence of stop mixing neither mass eigenstate is purely TeV and 6 . It is worth pointing out — despite the fact that the scalar and pseudoscalar discussions Figure 9. / produced color-octet scalars. right-chiral. We refer at this point to the heavier eigenstate cannot be directly comparedin — our that three the benchmarks crossminimal sections are of roughly single asis pseudoscalar large no production as surprise the that,while possible, as is we too will small in demonstrate these in models section to be constrained by collider searches for singly is plotted at These are considerably more straightforward than thethis plots production for the mode scalar: is as forbiddenR expected, by since in similar plots inThe minimal fact that theat plot these shows points destructive is interference, duemediated rather to parts than intricate of resonant interference this enhancement, between decay.opposite the It sign squark- is, for and in the gluino left- particular, and and the right-chiral sgluon- last mediating two squarks lines — of that generate ( these valleys. for collider searches for singly producedmodels, color-octet in scalars stark to constrain contrast sgluonscussing to in are minimal the peaks2.9 and valleys in theshell plots. right-chiral stop The and peaks sbottom squark — thresholds. for example, around tributions to JHEP05(2021)079 ] → 113 (7.1) O ( ) 2 O — earlier plotted m , the cross sections , 4 -symmetric models, 11 R /sx 2 O m confidence level (CL) [ ( g -broken models ) ]. CMS obtained these limits 5000 R 2 O 95% m 114 , x ( g 1 x 4000 x d at the end of this section. /s 2 O 4 — have appreciable production cross sections. 1 m 3000 – 32 – Z s 1 1 TeV 2 eff k ≤ 3 o α 2000 TeV for dijet resonances [ 2 m π 5 3 or = 13 O 2 ) = m / 1000 1 O s → ]. We reinterpret this search simply by computing the products of the 5 gg - 10 53 ( [ 10 eff 0.001 0.100 2 σ / Cross sections of single pseudoscalar sgluon production by gluon fusion in Benchmarks — are compared to the observed upper limits at = 1 9 . This benchmark model explicitly assumes single color-octet scalar production via 2 eff k We turn first to constraints on single scalar production. In figure ) = 1 with cross sections of singlebenchmark scalar of our production models. andwhich are the In not branching stark produced fractions contrast enough to to for gluons scalars this in in minimal search each to have any constraining power, sgluons by interpreting their datagg for a benchmarkgluon color-octet fusion with scalar a model cross assuming section given BF by results of this analysis are displayed in table for single scalar production in thein three figure benchmarks discussed in section from a CMS search at target the relevant searches, wesgluons — recall particles from with theIn previous scenarios section with that light only sgluons,Model the relatively gauge decays light relevant bosons to phenomenology and— are to our to discussions third-generation two Standard of quark-antiquarkmentary, the pairs. we scalar discuss and Since each pseudoscalar particle — in sgluons once turn, are more beginning in with parallel, the but scalar. not The comple- most important With the cross sectionsfront in searches hand, conducted along at with thebeyond-Standard LHC the Model for sgluons’ color-octet physics branching are scalars. fractions, mostmodels, As we often usual, can it interpreted because con- for is searches the for worth MSSM seeing or for how simplified tightly our unique scenarios are constrained. In order to Figure 10. 1–3. 7 Collider constraints on color-octet scalars in JHEP05(2021)079 . ] or cuts 95% 95% with ) = 1 T ) ¯ t 118 p t GeV for oo } symmetry ]. ATLAS → GeV. This → R O 116 (a), the cross ( O 500 . Unsurprisingly, Γ( 12 ¯ t t . and ) ) and BF ¯ t 1010, 1010, 1010 t TeV) dijet resonance search ) = 1 { . The fact that constraints , the scalar sgluon does not 6.2 13 gg } ( → & 7 1 → O O − O m fb ( Γ( GeV. We reinterpret this search for 36 1400 350 and BF ]; viz. section ] of searches for jets and leptons [ ) 0.10, 0.25, 0.50 57 { O . We note that the due to minimum 1200 = 117 → , TeV for pair-produced resonances in flavorless – 33 – / R 54 gg ) = 1 TeV for four-top quark final states [ ( eff . But here the effect is relatively weak. gg and to increasing = 13 σ 7 = 8 1000 ) 2 → / 2 O 1 ) = / GeV for ( 1 s O O symmetry breaking has almost no effect, since it does not ( } s σ ( symmetry breaking is due both to the effects of σ R R 800 ]. ATLAS obtained these limits by interpreting their results for ) and BF . Here 115 } 600 6.2 1050, 1030, 810 4 6 1 { 10 10 100 & Cross sections in Benchmark 1–3 of single scalar sgluon production with subsequent O , which is visible in figure / m R (b), meanwhile, we offer a similar comparison to the observed upper limits at 0.10, 0.25, 0.50 12 { ]; viz. section -broken models are excluded, roughly speaking, below the TeV scale. More precisely, We now consider constraints on scalar pair production. In figure R = 110 / our real color-octet scalars bysignal rescaling efficiencies, the and cross by section, accounting assuming forbased negligible different on branching differences the fractions in scalar to branchingdecay fractions to we quarks displayed in often figure This enough in particle any also of easily our evades benchmarks recasts to [ be constrained by this search. influence the cross section andfigure does little to theCL branching from fractions in an this ATLAS massobtained search these range. at limits In by interpreting theirassuming results a for a cross model section ofThis complex about search color-octet applies double to scalars resonances ours with ([ masses above ([ on jet energy, thissearch search generally only places appliesthe slightly to CMS weaker dijet resonances TeV-scale resonance with constraintsR search. masses on In above the particular, scalar we have sgluon than sections for scalarCL pair from production an are ATLASfour-jet compared search final to at states the [ a observed model of upper real limits color-octet at scalars assuming a pair production cross section close to ours in we have mildly relax with increasing breaking on the crossgrowing section Figure 11. decay to gluons comparedfor to color-octet exclusion scalars bound assuming from CMS JHEP05(2021)079 ) ¯ t t ] for GeV. → , could 120 o ] implies Γ( -symmetric = 150 ) = 1 116 . The limits R )) restored in O gg GeV where the and (c) we confront GeV as for the m 2.6 ) → , but provides no 12 } 290 gg = 0.50 O ( / & ] that provide robust R → term ( o o µ (290, 500) — this assumption does 119 m -symmetric (Dirac) limits B Γ( 6 ∈ 0.25, 0.50 ) and the Majorana masses R { O 3.5 (a) that the search for four fla- m = ] to constrain the pseudoscalar. threshold, at rates proportional ¯ t / R t 14 114 -symmetric models, the pseudoscalar R threshold. Now, if the search is valid, we (b) that the four-top search [ ¯ t GeV for TeV for pair-produced colored resonances t } 14 but does not constrain = 7 – 34 – } ], the ability of this particle to decay to gluons in 2 (c) the same bound of / 58 1 s 770, 1018 14 { -symmetric models. Finally, in figure R & 0.10, 0.25 o . Finally, as we return to the lower-energy search [ { m = 6.1 ]. These exclusion limits, which were obtained for a complex / R 120 . We also see in figure that, while the exciting prospect of single pseudoscalar production is -symmetric limits of various expressions GeV, leaving at minimum a window for R GeV for 13 = 0.10 290 ] implies that } symmetry is (almost, with the exception of the / R symmetry breaking, the production cross section remains far too small in all & R 115 R O m 820, 590 { We conclude our analysis by considering how all the same searches constrain the pseu- & symmetry breaking closes a loophole in the parameter space of the minimal o -broken models generically keeps it short-lived enough to decay promptly. In this sense, 8 Interlude: In this sectionof we several provide, expressions as inobservables. a this set work, of boththe validity models at checks, we the the consider Lagrangian if level the and dimensionless in couplings terms in of ( some assuming prompt decay to constrainR it [ R pseudoscalar sgluon. scalar in all benchmarks, butthe we pseudoscalar take sgluon this limit remainsing more small upon seriously in because before this the wesgluon mass decay conclude: — width range. able whereas of to in This decayto minimal point only to the is light final-state worth quarks quark elaborat- beneath the masses — can in certain scenarios be too long-lived for searches constraint for m from both searches reflectthat the we discussed complicated in interplay section four between flavorless jets, we find in figure of our benchmarks for theOn CMS the dijet resonance other search hand,strained [ unlike by for all the threedepending scalar, ATLAS on studies, the benchmark. and pseudoscalar the In canvorless particular, two be jets we higher-energy [ see at searches in are least figure competitive partially con- scalar renders this last search ineffective, that window extendsdoscalar to sgluon. at The least situation forfirst this see particle in is quite figure dissimilarenabled to by that for the scalar. We mentioned higher-energy searches. Inall, practice, because however, the these reconstruction limitsthe may of particle not the has apply negligible decaying after width,not sgluon appear and was to — done hold as explicitly for wefind scalar assuming noted that sgluons in that below section the scalar sgluon is unconstrained by any search we have considered. If the large width of the constraints on sgluons in minimal a somewhat older ATLASin search four-jet at final statescolor-octet [ scalar model assuming abe useful cross because section they double apply ours in and principle BF to scalars too light to be constrained by the afore- measurement of the four-top quark production cross section [ JHEP05(2021)079 350 1200 1400 300 1000 1200 (c) (a) (b) – 35 – 250 800 1000 800 600 200 600 400 150 5 5 4 4 7 1 - 10 10 10 10 10 10 5000 × × 100 × × 10 0.001 1000 1 5 1 5 Cross sections in Benchmarks 1–3 of scalar sgluon pair production, with subsequent Figure 12. decays to gluons orfor top-antitop color-octet pairs, scalars. compared to exclusion bounds from three ATLAS searches JHEP05(2021)079 (8.1) X X X ATLAS 13 . j 4 TeV) dijet resonance symmetry. Low-mass 13 R ) = 1 ( 1 gg − . fb X → ATLAS 8   i ¯ t o i 36 ( ¯ tt − t 1 1 5000 , characterized non-uniquely by   D ˜ g 2 and BF 1 CMS X X √ 13 ) Limiting high-mass search j o 2 = 4000 →   i gg ( 1 0 0 eff   σ – 36 –   3000 symmetry breaking. Right-hand side indicates which 820 770 1 1010 1030 1050 1018 High ) = − o R ( 1 1 1 σ   2 1 2000 √ 290 290 290 290 290 290 Lower bounds (GeV) Low = D U 1000 ] 67 = 0.50 = 0.25 = 0.10 = 0.10 = 0.25 = 0.50 5 4 - 10 ) are taken to vanish. In this limit, the Majorana gluinos become degenerate / / / / / / R R R R R R 10 0.01 10

Benchmark

Cross sections in Benchmarks 1–3 of single pseudoscalar sgluon production with 3.7

Pseudo o Scalar O Summary of LHC limits on sgluon masses in models with broken ) and ( 3.5 Figure 13. subsequent decay to gluons compared tosearch exclusion bound for from color-octet CMS scalars assuming the mixing matrix [ search imposes the high-masshigh-mass limit. limits for There the scalar, may but be likely an not for unconstrained the gap pseudoscalar. between low-massin and ( and can be once again viewed as a single Dirac gluino Table 4. limits shaded in gray are unaffected by JHEP05(2021)079 900 1200 350 1100 800 300 1000 700 (c) (a) (b) – 37 – 250 900 600 800 200 500 700 400 150 600 5 5 4 4 4 1 5 10 10 10 10 10 50 10 5000 10 100 500 100 × × × × 0.01 0.10 1000 1000 1 5 1 5 Cross sections in Benchmarks 1–3 of pseudoscalar sgluon pair production, with sub- Figure 14. sequent decays to gluonssearches or for top-antitop color-octet pairs, scalars. compared to exclusion bounds from three ATLAS JHEP05(2021)079 (8.7) (8.4) (8.5) (8.6) (8.2) (8.3) ] and 58 -breaking R ) we do not . ). If we take   ˙ α 3.4 α B.3 † 3 3 , . λ ψ i     . This is consistent c 2 ˙ and α ˜ g D † 3 α 5 3 g λ γ λ ) = 2.1 b 2 2 )) as + 6= ˜ ¯ ˜ g , ˜ g − , and the color-symmetric ˙ i i α c c D D 0 c † 2 3 α ˜ ˜ g g -symmetric limit where they A.8 3 ˜ g + ) + ψ 5 b 2 c 1 ψ R 1 → γ ¯ ˜ g − ). ˜ g . )–( g c b D 5 ). (˜ O c D   R ¯ ˜ c g 2 γ ˜ t ˜ 2 % i g a ˜ b g 1 ) + b 1 A.6 and its charge conjugate: b o ¯ ˜ g c 2 2 √ / g = / g 1 ˜ g a in the color-antisymmetric parts, we D a 2 √ b 1 2 − breaking, these should not strictly be c D ˜ abc g = ˜ ¯ ˜ t g ¯ ˜ g ˜ c g 1 D f = ˜ R g c 3 D + − U + 5 ˜ g g 2 c 1 c γ 1 ˜ g and c as D ˜ b g 2 ˜ g + → ˜ b g ¯ ˜ b g 2 L to obtain b / g ( ¯ ˜ c g ˜ D t i / g ST a ( 1 U ˜ g i D and ¯ ˜ a g a D = b D ˜ – 38 – + g ( ¯ ˜ and g ¯ ˜ g + ( 1 c 1 in terms of -symmetric limit a ˜ t -symmetric limit since in this case the ˜ c g 1 abc 2 -symmetric limit, 5   ˜ g and R O R f ˜ abc g ˙ b 1 γ α   α R 3 f b ¯ ˜ 3 † 3 1 g S The sgluon-gluino couplings are given by the appropri- ˙ α α g 3 The gluon-gluino coupling is given by ( h ¯ ˜ abc g i a † 3 3 λ ψ g h f a i λ λ must vanish. Since (viz. sections and 3   a ]. O g o 1 i + + L ⊃ ˜ in terms of g 58 ˙ TH α α L ⊃ abc ) = † 3 abc 3 2 a 2 f ˜ f g ). In the ψ ψ ˜ 3 g 3 i g g   i L ⊃ − and its charge conjugate can be written as -symmetric limit. However, in order to make contact with [ − and B.4 1 2 2 1 2 and D 1 1 R √ ˜ g g 1 + (˜ ˜ SH g ) to write — are dramatically affected by the Majorana nature of these fermions. 2 a = 1 5 1 √ 8.3 ) and ( L ⊃ − ˜ g = and B.1 D 3 ˜ g everywhere, this operator becomes or the trilinear couplings µ D U can be taken to vanish. In this case, → ˜ t We can again write which matches the results of [ parts of these couplingsobtain vanish. If we take But we can use ( Coupling of sgluon to gluinos. ate sums of ( can be viewed as a single DiracCoupling fermion. of gluon toU gluinos. 8.1 Couplings toWe gluinos begin in with the the interactionsout of sections gluinos, which — asWe we show have now discussed that in these detail particles through- behave as expected in the trilinear couplings regard taken to vanish in the the wider literature, we ignoreθ these parameters in this limit so that the stop mixing angle These states are onwith inspection the Dirac not nature equivalent of to theright-chiral single each squarks fermion. other: is At reduced the in same time, the mixing between the left- and and the Dirac gluino The gluino mass eigenstates can then be written (viz. ( JHEP05(2021)079 - 5 R ] in (8.9) (8.8) , and 58 (8.11) (8.10) 0 → -symmetric -symmetric ˜ t θ R R ) . In this case, 0 -broken models gg R -symmetric limit; and → → R 0 IJ O ( O ]. → R ) is consistent with the . In this limit, the single , 58 F 3 O I i ˜ % t 8.8 and m ν 2 β ). But, as we noted above, k 1 symmetry is restored. At the . . µ 1 2 D =1 D 3 k ˜ g X I R ˜ g → C.2 2 β β 2 3 O O o II − m m -symmetric limit of this decay is to O m m 3 µν R 3 3 η α α α 2 O 2 1 3 2 3 2 , so the sum ( m h R ˜ t so that ) ) = -symmetric limit, is given by ( 2 ) = ) = Scalar sgluons decay to stop squarks at a rate † J 0 2 2 k ˜ t R = ˜ -symmetric limit ( g ˜ g I A – 39 – 1 ˜ 1 t ∗ µ as 2 → Sgluons decay to gluon pairs in R ˜ g ˜ g ε Scalar sgluons decay to gluino pairs at a rate given ˜ t ˜ ) t 0 are forbidden when → 1 θ → → k I ( → O ˜ g o and ∗ ν O breaking is taken to vanish. I ε / ˜ Γ( R g Γ( L Γ( ). In the 2 ˜ t R 0 ) 0 =1 → 2 → 1 π ) vanish, and the total rate of decay to any stop pair can be J → 5.4 = , X lim / † 2 R lim / I R O (4 ˜ t 1 0 1 ˜ t ˜ t − → and then set lim / R } ) = ) and ( of a decay product gg 1, 2 5.3 A β → . We find once more that the analytic results familiar from minimal -symmetric limit } ∈ { R O C ( J ). One way to think clearly about the , vanishes. The amplitude for the scalar decay does not vanish, but reduces to M I -symmetric models. 5.1 { 0 ) R -symmetric limit → R gg lim ), depending on whether the pair is like or mixed. We can consider the / R → 5.2 o Rates of sgluon decaysat to rates gluons. given byΓ( ( Both of these decay rates are consistent with the known results [ The rate of pseudoscalar sgluonlimit decays in to a gluino similar pairs way can as be written in the above that the color-symmetric sgluon-gluinothis coupling implies vanishes that in like decays the same time, the gluinos combinenonvanishing to decay form can a be Dirac written gluino of as mass combined rates of scalarminimal sgluon decays to left- and right-chiralRates squarks of given sgluon decays by to [ by gluinos. ( limit of these decays in a manner similar to our treatment of decays to squarks. We noted where the speed in the given by ( sum over the mixed decays (e.g., to written in the We now perform aand similar appendix analysis onsymmetric the models analytic appear rates when of decay presented inRate section of scalar sgluon decay to stops. 8.2 Rates of sgluon decays in the JHEP05(2021)079 - R and 0 (8.17) (8.14) (8.15) (8.16) (8.12) (8.13) . In the → -breaking ) ¯ t . R O t ) , % L  ˜ o 2 q → ) } 2 m ¯ tt ). Note that we ˜ q o ), , ( ¯ t L t ˜ 2 q → → (4) m C.15 1 → O , , F ˜ q ( L 2 ˜ | 2 q O ) ( (4) R − { m R F gg ) F , and L ˜ q 2 ) )  → ¯ t 2 ]. -symmetric limit, t } m , 0, 0; σ ) + 9 2 , O R , 58 2 O ˜ q ( ¯ L tt 2 ˜ q 2 → p m R ( ( ) proportional to those → m → 0 |I v O , 1 ) ( 3 C L ˜ 2 3 q 1 O O ˜ q 2 L ( σ ˜ 2 q (4) m m , m 1 (1) − { R m F Sgluons decay to top-antitop pairs in 3 3 ). In the p  F α , ( ) ¯ n u 5.6 2 ˜ q ¯ t , 0, 0; t X a 2 t π for the stops, which no longer mix. Finally, we t O 2 5 2 – 40 – m 2 π r → m 4 -symmetric limit as i ) 192 ( 3 π 1 R 0 − 16 ) and ( O C ( (4  and 2 O ) ) = L ˜ ) 2 q 5.5 − (1) ¯ t l m gg gg abc t m t F d  1 m → ) = → → 2 O ¯ t ˜ q 1 1 9 t X O O m O ( ( 2 × 3 → Γ( π F F 2 i 0 m 0 0 O π 3 3 i -symmetric limit. This simplification leaves us with the diagrams ( → 16 → g → lim / lim R / R  R / R M 0 = 2 ≡ = 16 ) → ) = lim lim ¯ t / ) = 2 R t gg gg → → → O , and the diagrams (colored red in figure O ( 0 ( O R R ( -symmetric limit of the squark-generated partial form factor ( F R F → R I -symmetric limit, the amplitude for the scalar decay reduces to SO where % couplings vanish. Not all diagramsof with comparison neutralinos to and the charginos literature,them vanish, where but as for these well the diagrams sake in arethat universally the generate neglected, the we ignore formR factors These expressions are consistent with the known results [ Rates of sgluonbroken decays models to at rates top given quarks. by ( where can write the scalar decay rate in the the have restored the chirality labels with JHEP05(2021)079 - R  (8.20) (8.21) (8.22) (8.23) (8.18) (8.19) ) ) L L ˜ 2 t ˜ 2 t m m , , ] that appears 2 3 2 3 58 ) m m  2 3 , ), , ) o 2 3 ¯ t ) 2 3 ) 2 3 In the latter ex- m t ), ¯ t L , , m ¯ t ˜ 2 t t m m t ; 2 L 7 -symmetric limit to ; , ˜ 2 t

2 t → m 2 t ) ). L → R , ˜ 2 t m → o m 2 3 ¯ tt m ( , , m o , O L 2 t m ( , ( ˜ 2 t 2 t (4) ) → ; C.31 R L R m 2 t ˜ 2 t L m m (4) ˜ 2 t R , F , ; O m I m 2 O ( 2 t ) m 2 O ( × F for the stops. Finally, we can ; 2 R , 0 2 t m m 2 o 2 3 σ m I r ( , B

( 1 , m 0 ) and ( 2 t 0 m m 2 ) + 9 , 3 t ¯ t ; C − p t 2 t m C β t 2 t ( ) 1 ) , and ) 2 m m v L L ) ) m L ˜ 2 t 2 O C.23 ˜ 5 , 2 t ˜ → 2 t t l ( L γ ˜ 2 t × 2 O 0 m m m ) m m ( O 2 2 , 1 B m 3 2 m ( 0 L , σ π ( 2 ˜ 2 t i − C 2 3 , 0 m − (1) R ( 1 ) − m C 2 O m I p L 2 O O ; ) ) ; ˜ 2 t ( present in the corresponding result in [ n m L 2 O 2 t L ) ¯ u m ) m ˜ 2 t ˜ 2 t m 2 t L 3 3 – 41 – a 3 3 ) = 16 ˜ 2 t m m − m ¯ t m − t ( ( m m α − i t , 2 3 0 0 , m 2 3 4 . 3 3 L 2 3 , 2 − 2 ˜ 2 t B B } } m 2 3 1 ) } } m π → m 2 3 − 2 m 9 R R  3 π ; m R R -symmetric limit as ˜ ˜ m t t o  ; ˜ ˜ t t m 2 t 2 O 64 2 O ; ( + + 2 (4 R + 2 2 t 2 t 2 O m m m 2 t 2 t + → → F ( 2 t → → 3 m m m 0 0 1 9 2 t m m L L 3 ) = ( ( m L L B ˜ ˜ t t ) = m ¯ t 0 0 → ˜ ˜ t t m ¯ t t m × 3 3 / t R B B 3 g 3 2 + 2( − { + (2 − { g → + ( − { + (2 − { π → i ) = lim O o ¯ t ) = 2 ) = 2 ( . This term, being independent of the squark masses, disappears after including t ) = ) = 2 ) Γ( ¯ ¯ tt tt ¯ t ¯ ¯ tt tt t 0 M → ) = 16 0 → → → ¯ t → o → → lim t / ( R → lim O O O / R R O O ( ( ( → ( ( F (4) R (4) (1) R R (4) (1) O R R F I I ( F F R I -symmetric limits of the partial form factors ( and R and There is a term proportional to 7 missing from the opposite-sign contributionssymmetric from expressions right-chiral is squarks. not concerning. So the fact that we do not see it in our The amplitude for the pseudoscalar decay, meanwhile, reduces in the where with where the pressions we have again restoredwrite the the chirality scalar labels decay rate in the with JHEP05(2021)079 (8.24) (8.25) (8.26) } 2 ˜ t → . , 1 } 2 ˜ t 2 | ˜ t ) , we briefly discussed ¯ t t − { 3 → symmetry breaking and ].  1 symmetry predict a wide → ˜ ) t 58 L o R ˜ 2 t R ( − { m (4) R , ) 2 3 L ˜ 2 t |I t m ). Finally, we can write the pseu- m , β , 2 3 2 2 3 ) m t m , C.43 , 2 t m 2 3 3 m m , m , -symmetric models whereby a pseudoscalar 2 t ( symmetry-breaking operators in both the su- 2 t o symmetry breaking, to collider constraints on R m R m , m R , 2 o – 42 – 3 3 2 t α m m ( 2 , 0 2 o π 9 C m  64 ( 2 0 ) t C ) = -symmetric models). In an attempt to further quantify the m ¯ t t o R -symmetric limit as ) = m ¯ t R t ( → 3 o symmetry is broken. Otherwise, we have found that exclusion lim- → m Γ( R 3 3 o 0 ( g -symmetric models with → (4) threshold may be too long-lived to be detected by prompt-decay searches lim R / R R I ¯ t ) = t ¯ t t symmetry breaking, but remain around the TeV scale for the scalar and R → symmetry breaking on these models, we have compared our results in a set of symmetry, which we hope we have amply demonstrated in this work, merits o R ( R (4) R -symmetric limit of the partial form factor ( F We think that the distinctiveness of color-octet scalar phenomenology in models with R broken further investigation, theoretical and experimental, both ofat these large. particles As and we of bring the our models inquiry, discussion all to of a which close, are we simply wouldaccounting ideas like that of to exceeded suggest the the some scope routes sgluon of of masses future this work. is First, certainly a warranted. thorough In section its from available searches foraffected color-octet by scalars ataround the half Large that Hadron for Collider thesider pseudoscalar, are taking the mildly the other. mass of Altogether,variety each we of sgluon find interesting to phenomenology be that fixed and models as can with we broadly con- broken be accommodated by existing data. color-octet scalars. In somay doing produce we a have resonance identified broaddemonstrated scenarios enough that in to the which escape loophole a detection,lighter in and light than the — scalar minimal conversely sgluon —is we generically have closed if production modes of single sgluonsin and the pairs former of modes either forsingly sgluon, both noting produced particles strong in (particularly enhancements minimal foreffects the of pseudoscalar, which cannotviable be benchmarks, with different levels of menting minimal perpotential and the softly supersymmetry-breaking sector.the existing We have catalog significantly of extended color-octetcarefully scalar decays demonstrating in models how withidentifying familiar Dirac novel gaugino decays decays masses, and are both calculating altered their rates. by We have also reviewed the significant All of these expressions are consistent with the known9 results [ Conclusions In this work we have studied the color-octet scalars (sgluons) in models constructed by aug- where the doscalar decay rate in the with JHEP05(2021)079 R by o m suggests and 14 O m and 12 would be of some interest. 1 H . Finally, we note that the scalar 11 – 43 – ]. symmetry breaking and understand how extensive we 121 displayed in figure [ R symmetry breaking can occur in multiple sectors and O symmetry breaking — there are interactions between e.g. m R R Diag . It stands to reason that the effects would be similar at least for — that (10) package factor impacts some of our pair-production bounds on O © 4.2 K symmetry breaking on a roughly equal footing. We imagine that a wide R and . We think at the very least that a computation of the sgluon contributions B 3.1 Mathematica matrices — being ofmixing dimension two matrices — we are straightforward thereforemass to matrices provide diagonalize for explicitly. by electroweakinos hand.for and We us Those Higgs then to bosons, turn diagonalizethe which numerically. to become the We complicated perform larger-dimension enough numerical diagonalization with the aid of A Technical details I: masses andIn mixing this appendixmost we relevant provide to explicit our analysis. representations of We first mass discuss matrices the gluinos for and the stop particles squarks, whose mass Acknowledgments This research was supportedgrant in DE-SC0011726. part by the United States Department of Energy under large, but — specifically due to the physical Higgs bosons andin the appendix color-octet scalars, theto rules the for which self-energy we of have the included Standard Model-like Higgs boson factors as large as of the single-production bounds on sectors of the models we havetheir studied are own complicated investigations. and We interesting are and particularlydue entirely interested to worthy in of the adjoint modifications scalars: to the not only, Higgs as sector we have seen, is the number of physical particles quite identify the high-energy origin(s) of should expect it to beof in the NLO the cross infrared. sections Third,in — a particularly order of careful to single (and make sgluon doubtless precisethat production lengthy) our — predictions. calculation would choice Even be of casual necessary inspection of figures have multiple origins. Since ourcomputing goal new was analytic really results) (inwe to addition treated provide to all some introducing quantitative thevariety results model of in and interesting simple scenarios scenarios, couldOf course, be both constructed of these by goalslow-energy abandoning could models, our be which attained simple we as would approach. part find of very an interesting. ultraviolet completion It of would these be particularly useful to to vary the masses independently anda arbitrarily. definitive It set would of be operators, interestingthe but not also two only sgluon to to see choose masses what are happenssymmetry not in breaking independent. the is Second, seemingly warranted. likely a eventin Such much that more a sections nuanced discussion discussion ought of to respect the fact — discussed the plethora of sgluon mass-generating operators and decided, in view of this wide selection, JHEP05(2021)079 . 0 3 )– M (A.1) (A.2) (A.3) (A.5) (A.6) (A.7) (A.8) (A.4) A.6 : R and I ˜ g 3 M 1. − . . ,  = 1, = ) is given by 2  R ˜ ˜ g g / L 2 . 1 2 ˜ g S M M ˜ g 3.14 v  R tr tr L ) , where C is the charge- 1 1 2 I ˜ SO g 0 ˜ 3 g , g %   sgn sgn ), which makes manifest the M 2   = = ˜ 1 2 = ) to recoup the chirality-basis 0 √ − η | R , A.8 | I | L 1, i 2 , 3 ¯ ˜ ˜ g 1 + ˜ G 0 A.7 G for any nonvanishing η    ), C 3   M 3 I (   α µ ˙ ˜ = L g α L † ≡ I M introduced in ( I   2 ˜ = g m ˜ ˜ g g η c I sgn ˜ g − 0 with 3 ˜ θ g ˜ g with |   0 ) and ( 3 θ 0 M 3 M = sin M R L M ( ˜ cos A.6 g ˜ and ˜ 3 G   ε G − ∗ α m – 44 – U ˜ R g 3 ˙ L U α I θ I with ˜ ˜ g g ) can be written as 1, = ˜ M g | = θ sgn ˜ g −   sin ˜ 3 † g now satisfy ˜ g 1 Ψ =   3.15 = 1, = Ψ cos ∆ = ε 3 3 I ˜ g ˜ ˜ g g ˜ g I − ε ) supplies us with the right-chiral components m M ± ˜ g of the gluino mass eigenstates are related to the left-chiral M M   0 1 3 3 ⇐⇒  ⇐⇒ A.6 L tr tr = ] m I M 1 2 ˜ ˜ ˜ g g g ˜ g ˜ † g   67 U  Ψ according to M M Ψ † | = 3 = U The gluino mass matrix ˜ g U ψ ˜ g det det ˜ g θ = θ M = L sgn sgn R and ˜ sin cos G ). The gluinos ˜ G 3 , λ 1, i 3.14    ) from the doublets of two left- or right-chiral Weyl spinors displayed in ( = A.8 1 η ). It is straightforward to check the consistency of ( A.7 doublet ( ( Majorana nature ofLagrangian the ( gluinos, byconjugation using operator. ( Finally, the physical four-component gluinos are given by where for clarity we have momentarily replaced Weyl spinor indices to distinguish the spinor The Hermitian conjugate of ( This scheme guarantees positiveThe mass left-chiral eigenvalues components Weyl fermions and the Majorana phases [ the sign parameter which depends on the mixing angle The unitary gluino mixing matrix ( Majorana gluinos. JHEP05(2021)079 (A.9) : (A.12) (A.13) (A.14) (A.15) (A.16) (A.10) (A.11) R 0 I ˜ χ . . ) is given by   R . L 0 6 of the neutralinos 0 6 ˜ χ 3.17 ˜               L χ = 0. d d 0 I d d d d d ˜ ˜ ˜ ˜ ˜ ˜ ˜ χ H H H H H H ··· ˜ H u d 1 2 1 2 ··· d d ˜ ˜ λ λ ψ ψ H H R v ˜ L H 0 1 u 0 1 m m m m m m S v ˜ χ . Finally, the physical four- ˜ m v χ TH d T  u u u u u u  v v λ ˜ ˜ A SH ˜ ˜ d ˜ ˜ = H H 1 H H H H ST = v u d 2 λ 1 2 1 2 g = % , u 2 ˜ ˜ 1 TH λ λ ψ ψ , H H 0 | ˜ 2 R √ 1 2 g | H 2 L ˜ λ χ 1 ˜ u m m m m 1 introduced in ( m m ˜   N 1 2 √ − µ − ˜ √ 2 N H ˙ α α 1 0 2 2 2 2 2 2 L † L ˜ √ χ λ λ = = = = = λ λ λ λ m I 0 + 0 I u d 1 2 2 1 2 2 ˜ ˜ d d d d d χ 2 ˜ ˜ χ M λ λ ψ ψ − H H = ˜ ˜ ˜ ˜ ˜ H H H H H µ M m   2 2 1 1 S with m m u m m 2 m m with λ λ ˜ v ψ ψ λ H = = = 1 2 2 = 2 2 2 2 ). m m λ m m 2 2 2 ψ ψ ψ ψ ψ ψ m R SH L u d λ λ 1 2 1 2 ψ   ˜ m 2 2 ˜ ˜ ˜ 2 λ λ λ ψ ψ N H H α N λ ψ ψ R α – 45 – ∗ d L 2 T I 0 ˙ A.16 m m m m = N m m 0 I v 1 m m N v m ˜ χ u ˜ √ 2 χ 1 1 1 1 1 1 v = ψ λ λ SH λ λ u λ λ = ST 2   1 u d 1 + 1 2 1 2 0 1 v λ % g λ 1 0 ˜ ˜ ˜ λ λ ψ ψ χ 1 H H ˜ µ χ † = µ M m 2 1 2 1 2 g m Ψ m m ) supplies us with the right-chiral components m m 1 m m 0 I − Ψ √ 1 2 − − = = = ˜ χ = 1 1 1 1 1 1 = 1 1 1 ψ ψ ψ ψ ψ ψ 2 = = = = λ λ 3.17 u d ψ 2 1 2 1 0 λ 1 1 1 ˜ ˜ λ λ u u u u ψ ψ ˜ 1 ⇐⇒ H H χ λ ψ ⇐⇒ ψ ˜ ˜ ˜ ˜ ψ H H H H µ m m m m m m 1 1 2 2 0 m m m 0 m ˜ λ λ ψ ψ χ ˜               χ † Ψ m m m m = Ψ † = ) is made clear by ( | 2 0 N 0 ψ N ˜ χ 1 χ ψ = = M = ˜ The neutral fermion mass matrix L m c R ˜ N 0 I ˜ N ˜ χ With this choice of mixing matrix, the left-chiral components where again weneutralinos have ( made Weyl spinor indices explicit. The Majorana nature of the The Hermitian conjugate of ( Details of the neutralinocomponent mass neutralinos matrix are are given given by in appendix are related to the left-chiral Weyl fermions according to with and and We take this matrix to be real and symmetric, with upper triangular elements Neutralinos. JHEP05(2021)079 ) is (A.24) (A.20) (A.21) (A.22) (A.23) (A.17) (A.18) (A.19) 3.22 . u   v of the charginos , d L L 2 + 3 − 3 v ) is given by ± I g L ˜ ˜ χ ˜ χ ± I χ ± 2 ˜ ˜ χ χ TH ± I 1 L L √ 3.19 ˜ + 2 λ µ − 2 χ ˜ ˜ introduced in ( χ χ ± I = = = ˜ χ ˜ 2 t L L + 1 − 1 + + + u u u m M ˜ ˜ ˜ ˜ ˜ χ . χ H H H . . In the mass basis, we can −   − 2 − − 2 d T      λ ˜ ψ v = H A + = u + = + u u m ˜ m ˜ ˜ H m | H | H TH − d − + 2 − 2 − L L λ ˜ λ H.c. ψ . ˜ H ˜ C S C 2 introduced in ( v m m 1 m +   , √ ˜ ± χ i ST + 2 + 2 + 2 2 LR 2 RR   − % T − ψ M ψ ˙ ψ α α ˜ v m − m C d − 2 − 2 2 with L S u with L 2 ˜ 1 λ ± I ψ ∓† I ± v v H g √ ˜ ˜ | χ ˜ χ χ 2 LL m 2 RL m m − + − L m SH L TH   + m m | ˜ ˜ 2 λ + 2 λ C – 46 – + 2 + 2 C 2 + λ λ = λ   2 m µ − V X ˜ . − d − 2 − 2 1 C ± I ˜ √ λ ψ + = H = = = = = ˜ χ + χ ˜ 2 t m m m − + + + 2 2 2 + − + ˜ ˜ ψ ψ ψ χ M 6= ˜ χ      µ ˜ − 2 − − 2 C d Ψ Ψ − I The stop squark mass matrix λ ˜ ψ = ± H = χ ˜ χ m m ± m ± ˜ m χ ˜ = ˜ χ | ⇐⇒ µ ⇐⇒ c M − T + I ˜ v C ˜ + d − χ 2 h ˜ ˜ χ v χ g 2 1 2 Ψ Ψ g + † † 2 2 2 V X 1 √ M m = = L ⊃ − = = = The charged fermion mass matrix + − L L ˜ ˜ + 2 + 2 + 2 C C λ λ λ ) as − 2 − − 2 d λ ˜ ψ H m m m 3.19 and With these choices of mixing matrices, the left-chiral components Third-generation squarks. given by where once more we haveown made charge Weyl conjugates: spinor indices explicit. These fermions are not their where the physical four-component charginos are given by Details of the charginowrite mass ( matrix are given in appendix are related to the left-chiral Weyl fermions according to with We take this matrix to be real, but not symmetric, with elements Charginos. JHEP05(2021)079 1 A ), (A.28) (A.29) (A.30) (A.25) (A.26) (A.27) 2 d v mixing is . We take − introduced t R 2 u b ˜ 2 H v - , ( → L , M i b ˜  µ  T 2 2 4 2 g H √ symmetry breaking, . − 3 . − 2 1 1 R ˜  2 t , and 2 RR H g T 2 s ˜ t v m 2 1 3 m . 1  H ). − → ˜ t − TH 2 d         1 8  1 2 v λ ˜ 2 t T T S 2 LL u d H = + , 2 ST 2 TT m − −  2 H 2 H m | 2 A u T S m m m m ˜ v v , = v T = ( 2 | S S → ˜ t 2   1 u d m ˜ t SH ˜ t θ g 2 SS 2 TS H 2 θ θ 2 H 2 H λ H g 1 6 h m m m m t with + sin cos − cos 2 d d m d d ˜ t S S H H ˜ t θ v The scalar Higgs mass matrix  ˜ u d v with T θ 1 2 SH 2 TH 1 d 2 H 2 H v O v m sin m m m – 47 – m m and 1 cos 1 ) can be written as  H regardless of the extent of = g − g u u 2 u ˜ t u H 1 3 with µ   H H 1 4 3 1 Φ ˜ u d 2 t √ ˜ t 3.23 2 SH = 2 TH + θ = 2 H 2 H − m + 2 t m m LR 2 2 t H and m m O u − m Φ m m         a ⇐⇒ β 2 2 2 ˜ 2 t u 2 boson). ˜ t v  =  m u Φ tan 2 u v Z v 2 H v | 1 ⇐⇒ v = √  O M  ˜ t H θ = + = + Φ according to 3 3 | ˜ T 2 RL 2 u 2 Q R sin 2 H ˜ t m m m = = = = and H 2 LL 2 LR 2 RR L ˜ t m m m ) is given by and With these choices of mixing matrices, the Higgs mass eigenstates are related to the 2.17 gauge eigenstates according to and similarly for thebeing pseudoscalar the states Goldstone (though of only the three of the latter are physical, The pseudoscalar mass matrixboth is matrices analogous, to with bematrices, as real each and is symmetric. quite lengthy We and refrain they from offer no writing physical the intuition. elements of these negligible for our choices of so we omit those details. Scalar and pseudoscalar Higgs bosons. in ( With this choice ofeigenstates mixing matrix, the stop mass eigenstates are related to the chirality An analogous situation exists for the sbottom squarks, but we find that which depends on the mixing angle The orthogonal stop mixing matrix ( We take this matrix to be real and symmetric, with elements JHEP05(2021)079 (B.4) (B.1) (B.2) (B.3) . c J g ] ˜ R ) that are relevant . P c J ) c J , as g g J 3.1 )˜ )˜ ∗ 2 c J 3 , L Jj R , g U c J ˜ q c )˜ P I P g j 2 J )˜ i R J O ] 1 QCD is significantly richer ∗ 1 L ) U P a 3 a µ b U I t P U ) [ ∗ 1 I g + I J i o 1 ∗ 1 † 1 c I U J µ q U ∗ 1 ab J U U )˜ ∗ 2 ∇ ): I f J U ( − 2 3 + 2 U I † a -symmetric models and the MSSM. g 1 U ) R L O B.2 R o + I U P P + − µ ∗ 2 L J J c ∗ 1 a 1 O R ∇ P + ( δ ( U J ) generates interactions among the adjoint U P µ L − 1 2 1 I I I ∗ 1 ∂ 1 ∗ 1 P J U 3.5 ) ) generates the gauge interactions of the sglu- + 1 U I U U ( J 2 ( a J = ( O 2 b I b I ) ∗ 2 I – 48 – U 3.3 ¯ ˜ g ¯ ˜ c g in the first line acts on sgluons according to U ∗ 1 ( a U O a b I I ( O o µ O ∗ 2 ¯ ˜ g c b I O ∇ ( a a ¯ ˜ g ∇ U ] a ). Here and elsewhere we keep color indices explicit ( abc a µ O abc † a O o d + 1 ) 3 d ∇ O J abc O O 1 m abc % f µ i 3 % f = [ 3 U g 3 g 2 ∇ I 2 a 2 i g ( ∗ 1 ) 1 1 √ √ 1 2 U − − + O 4 4 µ [( symmetry than in minimal ⊃ + b µ ∇ R g ( O enhances the sgluon-gluino couplings: ⊃ − µ L γ / R / R W a I W ¯ ˜ g L -covariant derivative c adjoint Kähler potential ( abc c f 3 g i SU(3) SU(3) is a gluon field (viz. table L ⊃ g The adjoint sector of the superpotential ( The to be the identity for all flavors except the stops, and that in the latter case the mixing -breaking operators in the superpotential that generate several kinds of gauge-invariant in the first line of fields, but again we set those aside for a moment. Here we point out that the last operator where and imply summation over repeatedO indices. Recall that we takematrix the is squark mixing real matrix valued.interact with Before gluons due we to move the on, covariant we derivative ( note that the gluinos are also allowed to where the particles, in terms of the various mixing matrices defined in section section we focus on theadjoint standard scalars strong until interactions, the setting section aside the below. novel interactions of ons. It also enables theto scalar the sgluon sgluon-gluino to couplings. couple at The tree relevant level terms to can squark pairs be and written contributes in the mass basis of all Feynman rules for supersymmetricin chromodynamics. models with broken The inclusion of multipleR soft-breaking sgluon masses,trilinear two scalar Majorana adjoint interactions gluino altogether masses, result in and a reasonably complex model. In this In this appendix weto display parts our of investigation the of representativerequired Lagrangian color-octet to ( obtain scalars. thethat analytic We differ expressions also from in the provide MSSM. this most work, of with the emphasis Feynman given rules to vertices B Technical details II: Lagrangian and Feynman rules JHEP05(2021)079 ), aed (B.7) (B.8) (B.5) (B.6) f bec f ) and the ), to allow + label a par- . 3.2 , 3.5 } bed j . These Kähler f H.c. , c i . L { aec c J + P f ˜ g . b ( i L j } Jj J P q SU(3) ˜ ∗ 1 µν R q j  a η I i P O ] 2 J 3 g I a 3 1 )˜ J g 5 t ) couples sgluons to gluons t ) of i ∗ 1 H.c. L U ¯ 3 )[ N o I O P ( t = + 1 R R I I B.1 y t ∗ 3 2 3 d P c ν P µ U g g I U X +  † L ∗ 2 b J b ˜ R J abc y 2 I U ∗ 1 d P ∗ 3 J O − O i O ∗ 2 V I J % t L − 4 i O ∗ 1 y P R N 2 a b O ] − o o L − I 1 P 3 J t √ ∗ 5 L I ∗ 2 I or or L 4 ∗ 2 b 2 P a N O P t I g O U O I − 2 y i 3 2 J † R 1 J U – 49 – + V 1 b √ ˜ J t O I ∗ 1 J U ( y 3 + ∗ 2 j I O and i 2 X ( − ] J O b ∗ 1 a a 3 I I U -symmetric part of the superpotential, ( y J t ¯ ˜ ∗ 2 g ∗ 1 [ O µ i R i ) † I J − N abc ¯ and appropriate mixing matrix replacements O q ˜ 2 2 q R f I 3 † L 3 p t b 1 3 N g b ˜ g Y g L I 2 ), and elsewhere, lowered (raised) indices 2 + 1 t V 1  → 2 g 1 Y √ b √ I g X t 1 p 2 [ ¯ ˜ g B.6 2 ( g are the denoted fields’ weak isospin and weak hypercharge, and − { a g √ 2 − I h O ⊃ − abc Y i nh ˜ √ χ f + h + I † J 3 R W ˜ t ˜  g χ L ) and ( 0 I and − ˜ = χ 3 L ⊃ − 3.2 † J ⊃ − a b terms with ˜ t T o o + R W or or b a L ⊃ − O O R W 1 2 L p p We now provide the Feynman rules that correspond to these parts of the Lagrangian. Finally, the Kähler potentials for the Standard Model chiral superfields (( is the top-quark Yukawa coupling; and c µ t g where for simplicity we havestants. not Next respected are index the height interactions on of the scalar totally sgluons antisymmetric with con- squarks, which can be summarized Each field is taken to flowas into follows: each vertex. The first line of ( where (recall) y and chargino terms. Up-to and various down-type components left- of andcharges. right-chiral the In (s)quarks physical the couple neutralinos interest differently of accordingour simplicity, we to assumption restrict that their ourselves to only isospins the stop and third squarks generation hyper- mix. and recall We respectively have Recall that in ( ticle in the fundamentalpotentials (antifundamental) representation also conspire withquarks the and squarks to interact with neutralinosgluino and interactions, charginos. there Since, are like the multiple quark-squark- contributions, we further split these into neutralino like) generate interactions of gauge-coupling strength between quarks, squarks, and gluinos: tions with gluino pairs.pairs For are example, given the by left-chiral couplings of a scalar sgluon to gluino It is necessary to consider both terms when writing the Feynman rules for sgluon interac- JHEP05(2021)079 and . L I R P ˜ g R P I L  P ˜ g as  P J ]. We note  1 J O }  ∗ 1 J U ∗ 1 J I U , 1 122 1 I U j j 1, 2 i i ∗ 1 I U ] ] U ∗ 1 I a 3 a 3 U 1 t t U abc U as , } ∈ { )[ )[ abc d j L abc L J } i d R ] P abc O , d P P a 3 ) O d % I I I t i O J % 1, 2 2 { ∗ 2 i 2 O % 2 )[ i U % U 2 U i J 1 J √ J 2 I 2 1 2 ∗ 2 √ ∗ 2 2 2 ]. 1 } ∈ { √ O 2 O 1 O U √ R I 2 J ∗ 2 2 , P − − ) + − + ) I O I ) J − R L { J 1 ) + J ∗ 2 1 ) − ∗ 1 J P P U I ∗ 1 I I U U J J U 1 ∗ 2 2 I I 1 2 I U 2 ∗ 1 U ∗ 2 I U U O U ∗ 2 J U U J J I U 2 1 ∗ 1 ∗ 1 U [( − + U − O O µ O J ( ( − J ( I 1 γ ∗ 1 − 3 3 ∗ 1 I 3 U g g ∗ 1 J U U I 2 2 – 50 – abc m 1 I U 2 J 3 1 f ∗ 2 J U √ √ g 3 U i i ∗ 2 i I U g ( U 2 2 vertex) or to the identical nature of like gluinos (for − − ( 2 U ( U − − 2 + ( abc ( = = ˜ g abc f 1 = = abc 3 f i a a j I I ˜ g ¯ abc q 3 f q g ˜ ˜ g g i a g c I J 3 f g † I Jj ˜  ˜ g g ˜ q 3 ˜ g q i  g  i  − + = = as b b I I c c J J i ˜ ˜ g g ˜ ˜ g g } † J Jj ˜ q ˜ q a b µ g O 1, 2 and following fermion chains opposite this chosen direction [ } ∈ { and a J a , O o I vertices). { I ˜ g I ) vanish, but the color-symmetric parts do not. We conclude with the interactions of ˜ g I We come now to the sgluon-gluino interactions, which — as we have seen — are g ˜ g and fermion flow I ˜ g o quarks, squarks, and gluinos, which can be summarized for fixed Notice that the color-antisymmetric parts of the vertices with like gluinos ( relatively complex. These can be summarized for fixed We evaluate diagrams containing Majoranaof fermions by assigning a(nthe arbitrary) appearance direction of various combinatoricto factors color (such algebra as (for the thethe factor mixed of two here) due either and the gluon-gluino interactions, which can be similarly summarized as for fixed JHEP05(2021)079 L P . (B.9) a  (B.10) J abc O ∗ 1 a d o 3 O a . I I µ a 5 O A O o a N % a t i  O o y ) 2 I J J 1 + 3 √ H H H 2 J I I d ∗ 1  v − H H O ] ] R I a + J J = 4 P 4 4 a ). I O  b c c N o 2 a H H o o J a I I O ∗ 1  H O o 4 4 , as b u 1 2 Now we confront the adjoint I I O o v H H 2 I a (  H H ∗ 5 o 2 ST ST N   g SH % % I t + I ) ) 3 2 3 y λ I I c + + 1 1 a 1 2 H H √ O + . O J J H H  b  3 3 d d } + + J S S O ∗ 2 v v v R H H I v J a 4 I I ∗ 1 O P + + 3 3 O I A SO SO I O ( ∗ 2 I I H H % ∗ 3 T % I 2 2 ) ) 2 v N V abc − H H − and t N d  u u SO SO – 51 – ST 3 y 3 v v 3 2 3 % %  % µ ( ( µ µ − 1 6 2 2 − + O L  + 2 + 2  √ SH SH √ % I P 1 1 λ λ 3 S S abc 2 − g g I + 2 1 2 1 % % A d 1 1 1 2 2 1 √  3 µ 4 X + + µ 1 ) generates all sorts of interesting interactions between the √ √ µ + ( + ( 2 2 µ 2 I I  g O 1 4 4 1 J J 2 {  √ 3.5 % 1 1 √ 1 + i i i H H ⊃ − √ H H T T + 2 − − + I I v v / ). The relevant terms can be written in the mass basis of all R W 1 1 3 + S √ S = = L v 2 v H H S ST ST S 3.5 S v 0 I % % + I t t − % ˜ % ˜ χ S χ SH SH % + + λ λ adjoints. We first have the trilinear sgluon interactions, given by =  [ [  b c  Y O O SO SO SO SO % % % % SO ): 4 1 2 1 4 1 % 1 2 U(1) † L † J + − − + ˜ t B.9 b ˜ ⊃ − and a c / R W O We now provide the Feynman rules that correspond to these parts of the Lagrangian. L and given by ( Each field is again taken to flow into each vertex. The sgluons now have self-interactions Also of interest areseveral the operators interactions in ofparticles, ( sgluons in and terms of Higgs the bosons, mixing which matrices defined are in enabled section by Feynman rules forscalars. adjoint The scalar superpotential self-interactions. ( SU(3) volving top quarks) of which are and the interactions of quarks, squarks, and neutralinos and charginos, two examples (in- JHEP05(2021)079 (B.12) (B.11) ). The , , , ]  B.10   I J ) ) ) 3 4 ]. We conclude I I I 1 J I H H 1 1 4 3 I H  H H 4 H H d S d d I v H v a  K v v 4 a K g S g + . )˜ H ST v )˜ + + SO 0 J L 0 J I % as % R I I 2 ST χ P χ 2 2 SO P )˜ } )˜ + % − H % K H H L K R u 1 J 3 u u ∗ 1 + P v 3 − P v v µ U ( J ( ( U J J 3 2 H J 1 3 ∗ 1 I J I 1 µ 3 ∗ 1 √ SH 3 N 2 H SH SH N U 1, 2, 3, 4 I A λ I I U λ λ H 1 + 3 √ ∗ 1  1 2 ) − 1 2 1 2 1 1 U + H U ∈ { − J ) µ µ R + SO L + + 1 − 1 I + 2 2 % P I P I I J 1 1 µ 4 SO H R 4 4 L 1 √ √ K I K % 2 ∗ 1 H P 1 A 1 P 1 H + 2 H J √ T + + J U T I T H U ∗ 1 S 1 v 1 v J + 2 v S S J % ∗ 1 + N 1 N v v H S SH I ST I ST U ST S S S % ∗ 1 U 1 λ % ( % + ( % v ( [ % % SH a J U U a J S – 52 – + ( λ ¯ ˜ g ( + + ( ¯ ˜ + g [ %   a I I a SO I I ¯ ˜ g ¯ ˜ g % A i a  a H SO I SO SO o I 3 % − % % O 3 i i i A SO H SO % − − = = i SO % i SO % a a a a SO % = = = o o 2 i 2 O O % 1 1 a a a a a a 2 √ 2 √ o o o O O O 1 1 √ √ + 2 2 ⊃ − + W ⊃ − L I J I J I W I H I H H H H A L H and and and the latter by gluino and sgluon-gluino-neutralino interactions. The former are given by Feynman rules forwith interactions the of supersymmetrizations sgluons of and the adjoint adjoint fermions. scalar self-interactions, namely the Higgs- and the four-point interactions can be similarly summarized as three-point interactions can be summarized for fixed They also interact with scalar and pseudoscalar Higgs bosons as a result of ( JHEP05(2021)079 } . fre- ] ] 1, 2 2 1 (C.1) (C.2) (C.3) 3 − 104 ) M 2 − } ∈ { 2 d/ ) π K 2 i , ( ), p ), ` J R − L d , ` P 2 , I d P / { K 2 ][( 1 ), K ∗ 1 ). / ] 2 2 1 o R 1 L 2 ] 2 U # 2 P U M P J 2 ) 1 J J M ∗ 1 J ∗ 1 1 −  B 1 − 2 U U N ) 2 A X N m I ) 1 I + ∗ 1 m p p − 1 m − (for neutralinos) as L U − + U  R A 1 P ` ` } 4 P + − K m ][( ][( , 6 1 K ( L − 2 2 R ∗ 1 1 1 U P 1 produced by the decay of a particle P − U ... J J M M " J 1 1 J ∗ 1 2 X ∗ 1 B − − 1, U N = 2 2 N ( U m I ` ` I ( [ I [ 1 or A ][ ∗ 1 ∈ { 3 I 2 d 4 β 3 U differs from the measure ) ) ) U ` ` A H ( A ( d 4 d π π B d d − . Where practical, we express mixing matrix SO SO m (2 (2 ) SO SO % % π 5 i % i % – 53 – + Z Z with 2 2 2 2 (2 A 1 1 1 1 ` )= )= √ √ √ √ d m A 2 2 2 3 ( d − − − − β . M M } X , , − = = = = 2 2 2 3 1 2 m 2 X 0 J 0 J a a I I a a J J a a K K 1 2 M M M ˜ ˜ ˜ ˜ g g ˜ ˜ g g χ χ ˜ ˜ g g ; , m [ 2 2 1 = p ↔ n ( | M 2 (for Higgs bosons), or 0 1 ; X A 2 2
B p 1 } p M m | , , 2 2 A 2 ) ) a I I 2 m a 2 = O o H p A p → | lim + B A 1, 2, 3, 4 + 1
p m p | 1 p ,( ∈ { ( 2 1 p and ( ↔ 0 is the speed of either particle in a degenerate daughter pair. We express all loop 2 1 C A p β { -dimensional integral measure d and . This expression has the limiting value Our quently used elsewhere, including in thethe original symmetry reference. of In the some three-point placesunder below, function we under exploit certain interchanges of its arguments, e.g. where integrals in terms of the scalar two- and three-point Passarino-Veltman functions [ for the three-momentum of oneX of the particles Here we provide somethe calculation analytic details partial and decay explicitelements rates expressions explicitly in for section in theexpression terms form of factors in constants and mixing angles. On occasion, we use the C Form factors for color-octet scalar decays (for gluinos), The Feynman rules for these interactions can be summarized for fixed JHEP05(2021)079 J ˜ g I ˜ g (C.8) (C.6) (C.7) (C.4) (C.5) O , . )]  2 )] ] ˜  2 g 2 ∗ ] ˜ 2 g ) , ∗ m ) J J m 12 R 1 ∗ 1 + 12 R   ] U + U ] ] (Σ 1 ∗ I I ∗ ˜ ∗ 2 g )] ] (Σ 1 )] ) 1 ∗ 1 ) ) ] 2 12 L ∗ ˜ 2 g ] 2 ˜ 2 g ) ) parts of the I m 12 L ˜ 2 g U II R I U 12 R ˜ II R 2 g ( s m ˜ 2 g R 12 m ( O m O m (Σ + Σ − m (Ω (Σ % % 2 i + Σ + i − (Ω 2 + II L . 12 L 2 O II L 12 R 1 2 2 2 1 L 12 − 2 O ˜ 2 g 12 R − ) ˜ 2 g Σ m 1 1 √ √ ∗ J Σ 2 O m m ][ 2 O + Σ 2 m ) 2 2 N + Ω ∗ + Σ ( 2 ][ ( ) | + Ω m 2 m O 12 L − − II R | 12 R = ][ II R − I 12 L ][ 12 R − 2 R 12 ∗ 2 Σ 2 a Ω 12 R | Σ = = | 2 O Σ ∗ [(Σ 2 O ∗ ∗ Ω | O ) C Σ ) [(Σ II R ) ∗ 2 | II R m IJ L m ˜ ) g IJ R 2 + II L − Σ 12 L ][ II L Σ ˜ g ][ | + | Σ 2 L 12 2 m 2 J | | | 1 2 m 1 ) and symmetric ( + [(Σ | ˜ g [(Ω + [(Σ 1 12 R 12 L and 12 R a ˜ g O I 2 [(Ω 2 I 2 12 L | ˜ 2 g m Ω I ˜ | Σ g ˜ 2 g Ω 2 | | ∗ 1 m | ˜ Σ g II L and | m II L m m 4) O + 1) 1 Σ m + 4) 2 Σ 4) [ | ˜ g 1 | 2 [ − 2 [ 4) [ ˜ – 54 – | g − + 2 | − − ),), Σ The rate of decay of a scalar sgluon to a stop pair m = (  2 − a  2 I 12 L m J 2 − 12 L 2 a a 2 2 a 1 C ∗ 1 | 2 4) Ω a 2 N a 4) Ω C C ) | C U | ( U [ C [ 2 2 ( † J C J I − ˜ − t 2 a 2 2 ∗ 2 a I 2 N a 1 − + ( − 2 a ˜ C t U U C 2 + ( + 2 + 2 ( C C  (  ( − − → f f = f f I J f C C , and where C C ∗ 1 1 . The rates of decays of a pseudoscalar sgluon to gluino pairs O ) C ( U U B N I J = 2 = 2 = 2 = 2 |F 2 ∗ 2 2 2 2 2 | | U U | | ) ) ( ( ) ) SU( 2 I 2 I 3 3 ˜ ˜ g g ˜ ˜ g g g g 1 I 1 I ˜ ˜ g g ˜ ˜ g g = = → → → → IJ L IJ R o o Ω Ω ( ( O O ( ( |F |F |F |F and and are the coefficients ofvertices the given color-antisymmetric in ( appendix depend on the closely related form factors (a) representations of where for generality are the eigenvalues of the quadratic Casimir operators in the fundamental (f) and adjoint depends on the form factor The rates of decays of a scalar sgluon to gluino pairs depend on the form factors Form factors for tree-level decays. JHEP05(2021)079 (C.9) (C.13) (C.14) (C.15) (C.10) (C.11) (C.12) ) , 1 . ) ˜ 2 g 1 2  ˜

2 q )] ) m ) ) . , 2 o I m 2 O 1 0 J 0 J 1 gg , ˜ 2 g ˜ ˜ m χ χ 1 m H ˜ , 2 q , ), m 2 o d → m m , 2 O m I I v 1 gg ˜ ˜ g g m , ˜ 2 g O m , 1 + ( ˜ , 2 q 2 o i m m , m → ˜ g I O ] ] 2 O m 2 m θ 2 2 , F ) )  2 O m H ( J J } u ∗ 1 ∗ 1 ˜ g ,0,0;  v F sin ) + . In these expressions, sums θ } ( ,0,0; N N ,0,0; 2 O 2 i 2 2 ) ) R ,0,0; 2 O 2 O ν ˜ η 2 q gg q 0 J 0 J m 2 sin SH k 2 O ˜ ˜ ( 2 χ 2 χ + ( + ( m ˜ g m 0 µ λ 1 ( = ˜ ( → 2 2 → 4). m m m 0 k → 0 1 2 m ) ) C ( 1 2 2 I I ˜ ˜ ) g 0 q ˜ C q C − O ∗ 1 ∗ 1 − − + 1 θ 1 + ( 2 o , C ˜ 2 g ˜ 2 2 q − ˜ g a I I ˜ g U U L I ˜ ˜ 2 g 2 g θ −{ 2 O m 4 C q m m F 2 ( [ µν 4  m m A ,cos m + f + ( + ( η ˜ = ˜ q 2  T − 2 2 C θ − − 2 O η cos 2 v ebf 1 ) ) 2 ˜ 2 O 1 f q π J J 2 O 2 O m i η → 1 1 ST h m = 2 1 m m % 1 ) ( ˜ g cos2 16 N N dae The amplitude for this decay can be written as ( ( 2 η 2 ˜ g | and d – 55 – , 2 2 + m k ˜ − ) θ q | | 2 h ( X + ( + ( 2 I 1 ˜ J J g ∗ µ 2 3 oo 2 1 1 fcd 2 2  ε = 0 ) ) f A  π ) N N → cos i I I | | 1 2 O ˜ q →  1 1 2 1 1 1 2 O 2 2 k θ 1 | | 1 ˜ 16 g η m ( U U ) + I I µ O m 1 { ∗ ν  1 1 3 ( ˜ g [( 2 ε gg O µ 1 + U U 2 O is over squark flavors, keeping in mind our assumption that m 2 | | √ % |F + [( − 1 O )  2 3 ) m % → 2 1 π g = = + 2 3 3 π gg 2 g i 2 2 O (4 S | | m ( 2 v ) ) √ ˜ 3 q 3 − → 4 S 0 J 0 J √ g F ˜ ˜ % χ χ − +16 O  I I

( ) = ˜ ˜ g g abc ˜ q ) = 2 ) = 3 ) = d = F gg → → 2 gg gg gg | o ) → O ) = ( o ( → → → I : scalar decay to gluons. O gg |F ) ( O O O |F A ( ( ( ˜ ˜ q g → gg O M → F F F O and ( O → ( F O |F ( and where the sum in all squarks other than stops satisfy with, in turn, F where we split the form factor by loop content according to The rates of decayform of factors sgluons to a gluino and a neutralino depend on the closely related factor The rate of decaysgluon depends of on a the scalar form sgluon factor to a pseudoscalar Higgs boson and a pseudoscalar The rate of decay of a scalar sgluon to a pair of pseudoscalar sgluons depends on the form JHEP05(2021)079 (C.22) (C.19) (C.20) (C.21) (C.16) (C.17) (C.18) . .  ) ) i 1 )  gg ˜ 2 g ¯ t } t ˜ g )] m ), → θ , → 1 gg gg O ˜ 2 g ), ( ¯ t sin O ˜ q t → m ( . → , F 2 1 → | O o i → (3) ˜ 2 g ) , ( ( ) ˜ g O θ m  2 O ) F gg | gg i ( F ) β ) − F ) gg ¯ t 2 F ¯ t → ) → t t k gg ¯ ) t , cos )+ t o 2 α , 0, 0; 2 → O ( 1 2 o η → σ ( → → gg ˜ g k o , ∗ → O ( m 2 O O ( O ˜ g |F → → p ) as ( F ( ( 0 The amplitude for this decay has a O ( F 1 O ( O 4) µναβ v C η (6) (4) (  ) S F )+ ˜ , g ˜ g 1 (2) ) − ebf F 2 θ C.14 F 2 t 2 σ f ˜ F g gg 2 F 2 a , k )+ h m 1 ( )[ C ) + 4 → p ∗ µ ( → dae gg , which refer only to the terms preceding the ¯ t cos ( ) + 5 ε gg t d 2 1 } a 1 ¯ t 2 ) 1 ¯ O − u 2 2 t ˜ g ( | 1 → η C ˜ – 56 – q a ) + 6 ) → { ∗ ˜ 1 2 O g k → f ¯ t fcd t ˜ g ( O t The amplitude for this decay can be written as → f F C ( O ∗ ν m gg + 2 → O h ( m ˜ g ) ε t ( 1 2 ∗ O ˜  1 q → 3 π 1 + 2 → ( ˜ 2 q |F m ) = ) F (5) { = π (4 O 1 9 1 O π i (4)  A ( 4) F 2 gg ( − | ˜ h q (4 F 16 ) × − 4) (1) → 2  2 a − |F are external quark spinors. We split the form factor by loop − F gg π ) = + 9 + 3 o C O 2 i a ¯ t )  4) ( ( % t 2 → C 2 2 a 3 ) = F − σ × ( g o C , 2 a a → ( gg f 2 2 C C p C ) = 16 ( f O √ ( |F ¯ t f → ( 8 1 2 v t C C o − ( + − M → = 2 and M O ) = 2 ) ( | : pseudoscalar decay to gluons. 1 ) is the four-dimensional totally antisymmetric symbol, and where gg ) F : scalar decay to tops. σ ) , gg ¯ t 1 gg → t p → ( o µναβ ( ¯ u  ˜ O g → → ( F O o |F ( ( where content according to F For convenience, we write the squared norm of the total form factor as with different Lorentz structure than that of the scalar decay. Namely, we have where F denotation, also apply towrite arguments of the squared Passarino-Veltman functions. norm of For convenience, the we total form factor ( with replacements denoted e.g. by JHEP05(2021)079   ) ) 0 I  1 ˜ 2 χ ) ˜ 2 g 1 (C.27) (C.25) (C.26) (C.23) (C.24) ˜ 2 g m m , , , m 1 1 ˜ 2 t ˜  2 t , ) 2 ¯ t ˜ 2 t m m t  , , ) , m 1 1 ˜ 2 t 0 I ˜ 2 t , i ˜ → 2 χ 2 ) ˜ 2 t m m ¯ t ; ; m t O 2 t m 2 t , ( ; 1 ˜ 2 t m 2 t m → (1) , 2 , 2 t m m 2 t F O , , ( ˜ t 1 m 2 t m ˜ 2 t 4 , , s (2) m 2 2 O m 2 O ) , , ; F ) 2 t m } 2 t 2 O m 1 ˜ t ( 2 ) ( ˜ 2 g 2 m 0 ˜ t 1 m m 0 s ˜ 2 t 4 ( , m C C 0 2 t , ) m ↔ − ) 1 1 } , C ˜ 2 t 1 , with the first diagram counting as ˜ m 2 t ) + 1 2 1 ˜ 2 t ¯ t 2 O 1 2 , ˜ t ˜ ˜ 2 t t t 3 m m { 2 O m , m m ( 1 → − ; ˜ 2 t + m → − ) + 1 4 1 1 ( 2 O 1 ˜ t ˜ 2 g 0 I 0 m  ˜ 2 g O ˜ 2 χ ; . The partial form factor for the diagrams − m } ( C m ˜ t 2 t ( ˜ m g ) −{ ∗ m θ 0 c , in figure ¯ t + (2) ) m 1 2 t  1 B ˜ 2 t 2 t j , + F } I R 2 t → ˜ g − ˜ 2 t t – 57 – m m → cos 2 2 c ˜ g ( ,  m (Ξ c m ) 1 s , 1 ˜ 2 t O ( 2 − , I L 2 O → | ( 2 O 2 ) 1 m Ξ ˜ g − 1 η m I R m ; ˜ 2 t s (1)  1 h ) ( 2 t , Ξ 0 I 0 I 0 | ) m 2 F ˜ → χ ˜ 2 O 2 χ , m 1 η ˜ t C 1 ˜ 2 t 1 , 1 2 + m ˜ m 2 g m 2 t η c 1 2 m 2 ) , t → t , | , m 2 t 1 m  2 1 1 ˜ 1 2 t ; m , m ˜ ˜ 2 t g I L ˜ 2 g η 2 t m }  3 , s 2 O m Ξ 2 4 m t 2 2 | 1 ; ˜ t m ; m → ( ˜ g m 2 t ( η t m − ( 3 0 2 O 1 1 3 0 ˜ g m → g is a shorthand for ˜ g m → 2 O B ( m { C ˜ t m 1  0 ( 1 m  ˜ t 2 corresponds to diagram m 0 ˜ 3 g ˜ 2 g 3 c  B − + ) = ) s { g 6 B =1 ¯ t ¯ t X t I t + + ( − { −  + . (With this bookkeeping, each displayed diagram must be treated as a sum ) = ¯ t → → t ) = 2 ) = ) = ¯ t ¯ t ¯ t = 2 t O O t t ( ( → j ) j → → → (2) ( O ( F O F O O and ( ( ( (1) 2 (1) (1) F 1 (2) 1 = 1 F F F with where for example with two stops and a neutralino is given by and with j of diagrams where appropriate.) Thesein partial terms form of factors Passarino-Veltman have functions. fairlyclarity involved We expressions provide with these concision. results The below, partial tryinga form to gluino balance factor is for given the by diagrams with two stop squarks and where JHEP05(2021)079   .) I 2 R ∗ 3 I R / Ξ IJ L  V ∗ ) 2 (C.30) (C.29) (C.28) Ξ  ) I R . (Note L 1 1 2 b ˜ I R Ξ I R B ∗ m Ξ ) ∗ 1 ) − I R 2 +(Ξ I R J + I 1 ∗ 1 ˜ I L 2 χ O +(Ξ Ξ m I ∗ 1 +(Ξ 5 ) I L 2 2 + I L N Ξ  I L t 2 t ) ∗ Ξ y ) R ∗ ) (Ξ m 2 2 b ˜ ) 2 ( I L + ˜ 2 t 1  t m I L ) J , (Ξ m . 1 ∗ 1 m ,  ˜ 2 t I + t (Ξ 1 )  ˜ 2 χ ˜ O 2 t y ) 2 t m I  4 + I m m ) + m  − ; ˜ ; 2 χ 2  1 N 2 ˜ 2 t 0 I 2 t I ) 2 O I L J ˜ 1 2 χ m L ∗ 1  − m m 2 b , ˜ Ξ m X ( m 1 2 ∗ ( 2 O R O − 0 ) m + I 2 b ˜ 0 I couplings given in appendix  0 I + ˜ 2 , χ ∗ 5 m B ˜ I R B 2 χ 2 t m 2 + 1 J  N , ) m ˜ g t ˜ 2 χ t − m m 2 +( I 0 I ) R ) 2 ( I R y ∗ 1 2 b ) ˜  2 t ˜ m χ + + 1 +(Ξ 2 2 O Ξ t ; R √ ˜ X 2 t 2 χ + m I R 2 b 2 ˜ m ∗ 2 t 2 ), I L ; } ) m 4 Ξ g J m + m 0 I 2 t 2 1 m m ∗ Ξ ∗ 2 ( ˜ , ˜ t 2 χ I R , ( ) J ∗ − L − m )+ – 58 – 2 O 0 ∗ 1 1 ) O R 2 b ˜ m , 2 2 b ˜ I R 1 I ↔ I 2 O ˜ , 2 t B 2 t m O I R ∗ 2 3 2 1 m I m ˜ 2 t +(Ξ ˜ t m m , m 2 N ; − V 2 ( { (Ξ L t , )+ m I L 2 − +(Ξ ) 2 b ˜ 2 O  N  2 , y 2 O + g ˜ 2 t 1 1 L I 0 Ξ ( 1 ˜ 2 t 2 3 2 b ˜ I L m ˜ 2 t ˜ m   ∗ χ t  m ( ) ; m ) ) Ξ 1 6 m ( do apply to the stop mixing matrix elements inside − m 0 I 2 t m 1 m 0 I ∗ m ( 0 ; ∗ 1 , − I L t )   ˜ ) } 2 t 2 χ B 2 t m 1 L t 2 2 y C 2 X ˜ 2 t 1 1  2 b ˜ , I L ˜ t (Ξ m m 2 2 2 g g m  2 t m , | |  , m g 2 m 2 2 1 4 I I 2 t ( (Ξ ; ˜  2 t 2 O m → 3 3 3 2 t =1 t  − , √ √ − 2 O X I m 2 t 1 m m X X , I 2 O m t m ˜ | | ; t t O 2 m 3 = = 2 2 t m 2 O t t { ( − m m m 2 m 0 V ( − 3 m m ( m m t  IJ L IJ R 0  −  ( ( 2 2 b b B y O t t 2 0 m 0 Ξ Ξ C y y   6 =1 3 X C m B I m m g 6 + ( × − − × =1 × + × + − + X I and ) = )= ¯ t ¯ t t t → → O O ( ( (2) 2 (3) F F The partial form factor forby the diagrams with two sbottom squarks and a chargino is given are the coefficients of thethat left- replacements like and right-chiral where and JHEP05(2021)079 )  1 ) ˜ 2 t 1 ) ˜ 2 t ) 1 m ˜ 2 t 1 , (C.36) (C.33) (C.34) (C.35) (C.31) (C.32) ˜ 2 t m 2 ˜ 2 g m m , + , ; m 2 , 2 1 2 ˜ 2 g } ˜ 1 2 g η 2 ˜ , 2 g m 2 I R 1 m i , η Ξ m , ) 1 ) ; 2 ) ¯ t 1 ˜ 2 g ˜ g 2 2 t 1 t ˜ 2 g ˜ 2 g ˜ 2 t ) m m m m 1 ; , m 1 m ↔ − ˜ 2 t → ; , , 2 t ˜ 2 t g 2 t 1 1 ∗ m ˜ ˜ 2 g 2 g ), ) O m m m , ¯ t , m ( 2 , t 1 m m , I L 2 t 2 O ; ˜ , + 2 g 2 2 t (4) 1 S 2 O 2 t m ˜ 2 g m m → m , ( F , (Ξ ; , m m 0 m ˜ t 1 2 t 2 O ( ( ; O 2 I R 2 O C 0 ( 2 t s 2 O m m ) Ξ ( m B ( 1 m (5) ˜ 2 t ( m 0 ) ` 0 , 0 2 2 2 2 t ↔ ) + F B C m η ¯ t C ∗ ) 2 m 2 t  − , ) ˜ 6 1  g =1 ˜ t ˜ 2 t ) − 1 ) ` X 2 O 2 I L 2 1 m → c 2 O . g ˜ 2 t ˜ 2 g m m 2˜ } g 2 + ( m s 2 m 2˜ O m 2 1 , (Ξ 0 ˜ t ( s ) ˜ g ˜ t η − − ) 1 C + ε c + ) 1 2 1 ˜ 2 t 1 ) ˜ (4) ˜ g 2 2 g ˜ 2 t 2 t S 2 O → 1 2 2 1 , ˜ η 2 g m ˜ 2 t η F ↔ 1 m m m m 2 m } } , 2 , ˜ t 1 t ˜ g 2 m ˜ t 2 2 m 1 + η ˜ 2( t ˜ ( ˜ t 2 g s – 59 – ˜ − 2 g 2 1 2 m m SO { ˜ 2 g 2 O ˜ g − 2 2 m − { − % m → ) ˜ 2 g → h ; , − 3 m m 1 1 m 2 O 2 t 1  g ˜ g ˜ 2 g t 1 1 m 2 1 ˜ 2 g 2 2˜ ˜ t + } ˜ g { m η m 1 2 2 t 2 s m 1 m ( + , ) + + η t m − ˜ g 0 ; 1 with 1 m 1 1 − { ˜ 2 g  2 t ˜ 2 g ˜ ˜ 2 g 2 g m B m } ) = ↔ ) (  m + 2 m 2 m ¯ t )(2 ) m m O ; 2 2 t } , ˜ 1 g ¯ t η 8 2 2 1 % η 2 t 2 t 2 O ε η ˜ 2 g η 2 3 + 2 ˜ g , +2 2 − t 2 ˜ ↔ g g → m m 2 t 2 ˜ m 2 t g 1 → ( , ( ˜ g 2 2 O m m η 0 ↔ 2 t m m m O , 2 O 3 3 ( O √ − 2 m 1 1 B + g m ↔ (2 ( (2 ( 2 1 ˜ m g ˜ 2 g 1 2 1 4 2 1 2 t ( (5) 1 2 η 16 1 { 2 1 η η  0 η (5) 1 1 ˜ g ↔ 1 ˜ m g F 1 ˜ g 1 2 − ˜ C { g 1 + + ˜ g F ˜ g m m ) =  m + × + { ( m ¯ t 4 ( t ) = − − +  1 2 ) = ¯ t ¯ t t ) = t + → ¯ t )= t ¯ t → O t → ( → O → O ( (4) ( S O O ( F (4) ( A (6) 2 1 F (4) S F (4) S F F and, conveniently, Finally, the partial form factorsby for the diagrams with a gluino and a neutralino are given and with and the partial form factor for the color-symmetric part of the same diagrams is given by is given by The partial form factor for the color-antisymmetric part of the diagrams with two gluinos JHEP05(2021)079 ) 0 I } ˜ χ 2 2 η  m (C.40) (C.38) (C.39) (C.37) ∗ , ) → 1 1 ˜ t 2 1 I R η  , m ˜ t 2 (Ξ ; s ˜ t 2 g  ∗ ˜ t ) c m 2 → 2 1 ( I R 1 0 } η ˜ g 1 2 B ˜ g η +(Ξ −{  m ˜ t ˜ t )  ) → c ˜ c t 0 I 2 t ∗ ∗ c ˜ } 2 χ 1 ) ) ˜ t ∗ m η 1 1 ) s m , 2 I R R I 2 , 2 ∗ I L 1 ˜ g ) ˜ 2 g − → (Ξ (Ξ 1  t 2 (Ξ ˜ t I L m 2 O 2 1 c ; → − m η , ˜ t 2 O m (Ξ ˜ t 1 1 2 s  c ˜ ˜ g g ˜ t ∗ m { × ) ( s + ( m 1 0 2 1 t I ˜ t  I L 1 ∗ ˜ η t s B → − m 1 s ˜ N 2 O g (Ξ ∗ ˜ t + )  s  m m 2 2 with + 1 , ˜ t t t  I R 2 η ˜ t ˜ t c ˜ t 1 c s ∗ m m ˜ g ) }  ∗ ) ∗  ˜ t ˜ ¯ t t ) ) 2 t → c +(Ξ s m 2 1 I L ˜ t + + 2 ∗ I R L I 1 c + – 60 – ) ˜ t ˜ t ˜ t (Ξ ∗ → 1  → c s { (Ξ ˜ ) t I R ∗ ˜ t − )(Ξ 1 2 O s ) ˜ t O − c t 2 I R ∗ ˜ t ( 1 (Ξ s , ) ) m I L ˜ t s ) + ∗ m 2 2 0 I 2  (Ξ c ∗ ) 1 ˜ (5) I L 2 t η χ 2 ˜ 2 t )  1 (Ξ 2 1 I L 0 I ˜ g m m 0 I m − I R ˜ χ ˜ , χ )  −F (Ξ m +(Ξ → − 1 2 O 2 t I m (Ξ ˜ t  ˜ 2 g m 1 ˜ t  c 0 I  − m 2 t m s ) ˜ ∗ χ m I ( N ) = 2 , 2 2 t ) 1 1 ; I ¯ t m 2 1  η m 1 2 t t ˜ t 2 I L m t N − 1 t ˜ g 2 N  m 6 =1 m 2 O ( I (Ξ → X I →  m − m ∗ 1 0   ( m t 1 t  2 O I 2 O O N B ˜ t 6 =1 ∗ 1 t ( 2 O m X I { m m 6 =1 m + ( + × N X I m (5) m 3 × + +2 + +( − F ) = ¯ t ) = )= t ¯ ¯ t t t t → → → O O ( ( O ( (5) 2 (5) 1 (5) 4 F F F and and and with JHEP05(2021)079 ∗ ) 1 I L ˜ t s (C.45) (C.42) (C.43) (C.44) (C.41) ∗ (Ξ )  ˜ t I . 1 L s  ˜ t } i c , (Ξ ) ) } } ¯ t  1 ˜ t 2 ˜ 2 t ) t )  ˜ t c 1 1 )  ˜ 2 t ˜ 2 t ) . m 2 o ˜ ∗ t ) 1 → } ) ˜ 2 t c → 1 m m 2 ˜ 2 t m − 1 o } 1 η I R m ( ) ˜ − ). t + m 2 O + 2 ¯ t 1 , ˜ 2 t t 2 t (Ξ 2 O 2 (6) → 1 m  ˜ 2 g − ˜ 2 g − { ˜ m t m 1 F m → ( − s m 2 O η m  ), according to + , 2 O ; 2 t } − o } m 2 2 t 1 ( − 0 I ) + 2 ˜ 0 I 2 t ˜ m g m ˜ ¯ t 2 χ ˜ ˜ 1 g 2 χ F m − t ˜ 2 g m ( (3 ) − C.22 m 0 I m 0 → 2 O 2 m ↔ ˜ 2 O ) 2 χ → ( σ 1 B 2 t − + 1 m 2 2 ˜ , m g m o ) 2 t ˜ g 2 η 2 t m ( { 2 2 2 p 2 + η m ˜ g ( m ) + ( 2 (5) ) + 1 − { v ˜ 1 g − 2 t ˜ m 2 g ˜ 2 t 2 O (3 5 F )  m γ h 1 2 O 2 O m ∗ m m m ˜ 2 t ) ) ) 2 with 2 1 ) + m m + − 1 1 + m − ˜ 2 t ( 2 o σ I R The amplitude for this decay can be written ) − 2 , 1 − 1 , 2 t ) ¯ t ) } 2 ˜ 2 g η m m 1 0 I t ˜ t 0 I ¯ t ˜ 2 g (Ξ 2 O ) + 1 , ˜ m p t 2 χ ˜ s 2 χ ˜  g 0 I m 0 I ( − ˜ t m ˜ m ˜ 2 χ – 61 – 2 χ (2 → ¯ m u s m , ( m 2 + → → 1 1 ˜ ( 2 g 2 O ) m a o m ˜ ˜ 2 g 2 g 0 I − ˜ 2 t t − ( t , ˜ i − m O 2 χ m c 1 1 − 1 m m t ( m , ˜ 2 g ˜ 2 g ˜ 2 2 g (4)  ; S ˜ t 1 m 4 − ) ˜ t 2 t ˜ m 2 g c 2 t + m (5) m F 5 m 2 3 π 1  ( ; − 0 I ˜ c 2 g t m m m 2 t 2 t ˜ 2 χ ( , − (4 g 2 2 O m m 2 t 2 t 2˜ −F m m  2 t ( → − m s  , ) + 5 0 I 2 2 m 1 2 t m m ˜ t ¯ t ˜ I 2 t ˜ m g χ ( { 2 o η , 2 t ) = s 1 ε 2 1 m 1 2 o 1 ¯ 2 t ) = 1 { , m (2 m ˜ g t ¯ η t m 2 t 2 N 2 η t t , 2 t t 1 m ˜ → t { t 1 ˜  g m ( m 2 O ˜ g m 1 m 2 m 1 → o  0 m 3 3  → ( { m η  → m m 6 1 2 =1 0 I t C g o ) as 1 I X I ˜ ( ˜ χ 2 O g 2 O 1 × ∗ 1 0 (4) O A 1 2 ˜ t m + × ( 2 2 O m C { m m m N  F M 1 π C.21 i (5) m  6 − × + × + − + + ) = F ¯ t × t ) = ¯ t : pseudoscalar decay to tops. → t ) = 16 ) ¯ t o ¯ t t ( t → → (4) A O → ( o F ( o (5) 5 ( F F The partial form factor foris the given color-antisymmetric by part of the diagrams with two gluinos We split this form factor by loop content, in analogy with ( F in analogy with ( and and JHEP05(2021)079 Phys. (C.48) (C.49) (C.50) (C.46) (C.47) ) , 1 ˜ t 2  } m . ) , 2 , } 2 o ˜ t 2 i (2012) 035 2 ˜ g 2 ) I R m } ¯ t → t Ξ I m 09 − 1 1 , ˜ t 1 1 N ˜ → ˜ 2 g g 2 { o ↔ − m m + ) ( ( ; ∗ 1 2 2  t 2 JHEP ˜ 2 t 2 ) → − (4) η } S 2 , m 2 m 2 I I L F ˜ g 1 , , η ˜ t t 2 1 2 N ˜ m 2 g , (Ξ , s ↔ m 1 2 o o m , 1 I R , o 2 )+ η m 1 Ξ 2 o , ˜ ) 2 g → m 2 1 ( m collision data with the ATLAS ˜ ˜ 2 t g m 0 ↔ O . ) + ; + ¯ m t { } C ∗ 2 t pp ↔ t , 2 ) 2 ) ˜ 2 g 1 ˜ t 1 1 ) m 1 ˜ 2 g I L ˜ t 2 ˜ 1 g , → ˜ 2 t m { 2 t ( m → m TeV o , 2 1 m 2 } ( , (Ξ m 1 1 , with η ˜ t 2 ˜ ˜ t 2 g , )+ , CERN, Geneva, Switzerland (2017). 1 1 1 ˜ c t ˜ 2 o 1 2 g − Natural SUSY endures ˜ g (4) S ˜ 2 t m = 13 ) 2 2 ; m m F m ¯ − { t → ↔ m Supersymmetry and the scale of unification η 2 t s ( ; t t , 1 2 1 0 2 t ˜ t − – 62 –  2 ˜ t √ m η s ˜ m 2 g ) C } { , m 2 → { 2 2 2 2 2 t ˜ ) g ( of m h η η + 1 0 , ˜ 2 t O 2 1 m m g 1 ˜ ( g B  , 2 2˜ 1 ˜ − 2 g m ˜ ↔ 2 o g ) } ]. s m fb 2 2 2 (5) t m 1 m m − η η with ; η − ( F 2 m 2 t ]. 36 1 ), which permits any use, distribution and reproduction in , 0 ˜ g 2 1 ˜ 2 g 2 t 2 ↔ O m η + 2 ) C ˜ g SPIRE m , 1 ¯ t 1 % m m 2 1 2 t ˜ t g 2 t 2 η 3 IN 4 η − , Search for squarks and gluinos in final states with jets and missing g + ↔ [ m 1 m SPIRE 1 m 2 2 1 ˜ g ( → − , 2 t ˜ 2 g 1 η IN (2 1 2 o ˜ g ATLAS-CONF-2017-022 1 2 o o m ˜ 2 g √ m ˜ 1 2 1 g { ( ][ ↔ 2 o ( m m m ( 1 m 16 CC-BY 4.0  − +  0 (5) m ( ˜ g 1 9 1 2  C F − This article is distributed under the terms of the Creative Commons ) = ¯ ) = t  × −{ + + ¯ t t (1981) 1681 t ) = ) = ¯ t collaboration, ¯ t t ) = → t , Tech. Rep. 24 ¯ t → t o → o ( → ( → o o (5) 2 ( o ( (4) S ( arXiv:1110.6926 F [ ATLAS transverse momentum using detector Rev. D (4) F S 1 (6) M. Papucci, J.T. Ruderman and A. Weiler, S. Dimopoulos, S. Raby and F. Wilczek, (4) S F F F [2] [3] [1] any medium, provided the original author(s) and source areReferences credited. and Open Access. Attribution License ( Finally, the partial formneutralinos are factors very for conveniently the given by diagrams with a gluino and one of the lightest and with and the partial form factor for the color-symmetric part of the same diagrams is given by JHEP05(2021)079 03 TeV 42 variable 12 ] JHEP = 13 T2 , TeV in final (1991) 289 s ]. M √ 13 JHEP 352 , , ]. ]. SPIRE IN ]. ][ ]. Aspects of split (2017) 032003 SPIRE SPIRE , CERN, Geneva, IN collisions at ]. IN arXiv:1107.5581 96 SPIRE Acta Phys. Polon. B ][ [ ][ , , CERN, Geneva, Switzerland (2019) 244 IN TeV proton-proton collisions SPIRE pp Nucl. Phys. 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