The other effective fermion compositeness

Journal of High Energy Physics, Nov 2017

Abstract We discuss the only two viable realizations of fermion compositeness described by a calculable relativistic effective field theory consistent with unitarity, crossing symmetry and analyticity: chiral-compositeness vs goldstino-compositeness. We construct the effective theory of \( \mathcal{N} \) Goldstini and show how the Standard Model can emerge from this dynamics. We present new bounds on either type of compositeness, for quarks and leptons, using dilepton searches at LEP, dijets at the LHC, as well as low-energy observables and precision measurements. Remarkably, a scale of compositeness for Goldstino-like electrons in the 2 TeV range is compatible with present data, and so are Goldstino-like first generation quarks with a compositeness scale in the 10 TeV range. Moreover, assuming maximal R-symmetry, goldstino-compositeness of both right- and left-handed quarks predicts exotic spin-1/2 colored sextet particles that are potentially within the reach of the LHC.

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The other effective fermion compositeness

HJE The other e ective fermion compositeness Brando Bellazzini 0 1 2 3 5 7 9 Francesco Riva 0 1 2 4 7 9 Javi Serra 0 1 2 4 7 9 Francesco Sgarlata 0 1 2 7 9 Gif-sur-Yvette 0 1 2 7 9 France 0 1 2 7 9 0 Via Bonomea 265 , 34136, Trieste , Italy 1 CH-1211 Geneve 23 , Switzerland 2 Via Marzolo 8, I-35131 Padova , Italy 3 Dipartimento di Fisica e Astronomia, Universita di Padova 4 CERN, Theory Department 5 Institut de Physique Theorique, Universite Paris Saclay 6 International School for Advanced Studies 7 ories , Space-Time Symmetries 8 Explicitly , @ 9 pulled-back to E We discuss the only two viable realizations of fermion compositeness described by a calculable relativistic e ective eld theory consistent with unitarity, crossing symmetry and analyticity: chiral-compositeness vs goldstino-compositeness. We construct the e ective theory of N Goldstini and show how the Standard Model can emerge from this dynamics. We present new bounds on either type of compositeness, for quarks and leptons, using dilepton searches at LEP, dijets at the LHC, as well as low-energy observables and precision measurements. Remarkably, a scale of compositeness for Goldstino-like electrons in the 2 TeV range is compatible with present data, and so are Goldstino-like rst generation quarks with a compositeness scale in the 10 TeV range. Moreover, assuming maximal R-symmetry, goldstino-compositeness of both right- and left-handed quarks predicts exotic spin-1/2 colored sextet particles that are potentially within the reach of the LHC. E ective Field Theories; Global Symmetries; Supersymmetric E ective The- - 1 2 3 4 5 6 7 3.1.1 3.1.2 3.2.1 3.2.2 3.2.3 3.3.1 3.3.2 3.3.3 LHC LEP Motivation Pseudo-Goldstini The e ective Goldstini theory 3.1 The geometry of N Goldstini 3.2 The Goldstini e ective action 3.3 Embedding quarks and leptons Covariant derivatives and Maurer-Cartan form E ective metric and invariant measure Goldstini self-interactions: Akulov-Volkov Model independent couplings to composite elds Model dependent couplings to composite elds Maximal R-symmetry Embedding leptons Embedding quarks and extra exotics 5.1 5.2 5.3 5.4 Explicit SUSY breaking Collider phenomenology Positivity constraints Outlook | Dibosons Phenomenology of the new colored states Conclusions and outlook A Dileptons at LEP 1 Motivation The LHC de nes the frontiers of our exploration of the universe at microscopic scales. Its primary focus so far has been the search for new physics in the form of narrow resonances, often associated with weakly coupled (calculable) physics beyond the Standard Model (SM). The incredible amount of data that is and will be accumulated at the LHC gives access to unprecedented accuracy in our knowledge of SM processes. A crucial question is then to understand what kind of physics can be tested with all this new information. Amusingly, { 1 { SM particles, can produce sizable deviations in their interactions at high energy and become the principal target of these SM precision tests [1{10]. In this article we present a perspective on how the SM can emerge as the result of such a strong dynamics, a perspective on how high-energy compositeness can appear to a low-energy observer. We are interested in situations where the Compton wavelength of a particle m 1 is much larger than the microscopic scale m 1 of the con ning theory, so that it e ectively appears elementary at energies of order its mass.1 A hierarchy between these di erent scales can be naturally generated by the presence of certain approximate symmetries that characterize the strong dynamics but are explicitly broken by m and by the 4. At low energy only the latter survive and particles look SM-like, while at high-energy the former grow and the new strong interaction is gradually revealed. The more irrelevant the new interactions, the more this scenario would resemble the SM at low energy, simply because the e ect of compositeness would vanish more rapidly as the energy that is used to probe it decreases. It is therefore crucial to understand how well can a framework of compositeness in the multi-TeV region (accessible at present or near-future colliders) fake the SM: how irrelevant can these new interactions be? From a low-energy perspective one would see no obstruction in arbitrarily soft interactions, but dispersion relations can be used to show that the basic requirement of unitarity of the microscopic (UV) theory, imposes certain positivity constraints that guarantee the existence of unsuppressed dimension-8 operators [12{14]. Given the SM eld content and limiting our discussion to lepton and baryon number preserving e ects, this implies that the leading BSM e ects must be characterized either by operators of dimension-6 or by dimension-8, but not higher. While most of the literature focuses on the former case, in this article we discuss instead the latter situation in which the leading BSM interactions saturate the unitarity bound and arise at dimension-8: these are theories that ow maximally fast to the SM as energy is decreased. In fact, systems with very soft behavior have been studied in the context of scalars including the Higgs [7, 15], and vectors [7]. In this article we focus on fermions, and go beyond the traditional paradigm where fermion compositeness relies on chiral symmetry and the associated four-fermion dimension-6 operators. We consider fermions whose composite nature is dominantly captured by dimension-8 operators (four-fermion with two extra derivatives) via a non-linearly realized supersymmetry that forbids operators with D < 8. That is, (some of) the SM fermions are pseudo-Goldstini, in a modern generalization of the 1Note therefore that we are not concerned here with compositeness of the type the proton exhibits for instance. This would give rise to large non-standard e ects for the SM fermions already at energies comparable to their mass, e ects which have been long ruled out experimentally. Instead our focus is on traditional chiral-compositeness of fermions, which gives rise to dimension-6 operators, is incredibly well constrained by LHC data for the light quarks, leaving no hope of ever testing it directly on an earth-based collider (see section 5 or ref. [3]). On the other hand, non-linearly realized supersymmetry, i.e. goldstino-compositeness, provides an alternative explanation for the small fermion masses, while allowing large e ects in the more irrelevant dimension-8 operators, whose collider constraints, as we will show here, are much less severe. Interestingly, the fast decoupling of the Goldstino interactions becomes distinctive as well when considering diboson production, whose leading modi cations come from dimension-8 operators when the Higgs or the transverse gauge bosons are also composite. This paper is organised as follows. In section 2 we present our picture for goldstinocompositeness based on approximate extended supersymmetries. We construct the e ective theory of Goldstini in section 3. In this theoretically oriented section we discuss the geometry of the coset space associated to supersymmetry breaking, the general interactions of the Goldsitini, and the embedding of quarks and leptons. In section 4 we understand the deformations induced by the explicit breaking of supersymmetry. The phenomenology of our scenario is covered in sections 5 and 6, where we discuss, respectively, the 2 ! 2 scattering process that best probe the compositeness of quarks and leptons and the exotic \quixes" that are predicted by maximal R-symmetry. 2 Pseudo-Goldstini We consider a strongly interacting supersymmetric sector. It con nes breaking SUSY spontaneously and leaving only massless Goldstini in the infrared (IR) spectrum, that we identify with (some of) the SM fermions; in addition to kinetic terms, Goldstini have selfinteractions that start at dimension-8. This picture is deformed by non-supersymmetric SM couplings, which we take to be small compared with the new sector's coupling, thus we treat them as perturbative deformations of the exact Goldstino limit.2 Yet, these small e ects are marginal and at small enough energy they dominate, while at high energy Goldstino self-interactions become more important. We will study scenarios where either one, several, or even all of the quarks and leptons are (pseudo-)Goldstini, implying the existence of 1 84 supersymmetries (see section 3.3) | numbers at which the reader might be willing to raise an eyebrow. It is indeed well known that complete massless supermultiplets (i.e. within a linearly realized N 2Another interesting application of this ideology is Goldstino Dark Matter [18]. { 3 { SUSY with massless particles) and N > 8 supercharges imply the existence of massless higher-spin states which are pathological, in at space, on very general grounds [19{21] (see e.g. [22] for a review). Yet, the roots of these arguments are based on the IR properties of the states, i.e. the soft scattering limits of the S-matrix elements or the existence of global charges, which is exactly the regime where one is sensitive to the spontaneous symmetry breaking e ects, i.e. the soft masses. A spontaneously broken extended SUSY does not predict those massless higher spins at all.3 In non-linearly realized SUSY, the would-be higher-spin superpartners are actually multi-particle states obtained by including Goldstino insertions that do in fact raise/lower the spin, but without producing single particle states.4 One can take a step further and argue that the very existence of N > 8 implies the only consistent phase of SUSY in at space is the broken one, which in turn requires the existence of fermions much lighter than the cuto , as we observe in nature. There could then be a fascinating link between the (nearly) vanishing cosmological constant and the existence of (light) spin-1/2 fermions in the spectrum. Moreover, it could be possible that a UV completion in the unbroken phase needs to be formulated in AdS space, given the need to inject a positive contribution to the vacuum energy to move to the broken phase. Notice that massless higher spins in AdS pose no problem [22, 25, 26]. There is in fact another di culty with N 8: the impossibility of interactions of renormalizable dimension D 4, in a Lagrangian description. The only interactions that can be written compatibly with these extended supersymmetries are highly irrelevant. This suggests that, if a UV completion exists, is not of a weakly coupled kind and does not necessarily rely on a Lagrangian or e ective description. It may well be that a UV completion expressed as local Lagrangian is made of a series of higher-dimension operators that become important at energies of order the higher-spin states mass, yet resulting in a well-de ned S-matrix at any nite energy. This is somewhat analogous to what happens in Vasiliev's theories in curved AdS space [25, 26], where the cuto is given by the cosmological constant itself, i.e. the deepest IR observable, such that theory is never weakly coupled in the UV, where the curvature could in principle be neglected. Another example is the highenergy limit of string theory [27], where in nitely many higher-spin conserved currents appear to be restored, corresponding to the tensionless limit. Alternatively, extended SUSY could just be realized as an emergent symmetry which (re)appears in the IR from UV dynamics that do not necessarily exhibits such a symmetry [28{30]. Yet another option is to consider several almost sequestered N 8 super symmetric sectors which are linked by an approximate permutation symmetry, e.g. ZM o [N -SUSY]M (where M is a avor index). In any case, since there are no manifest obstructions to the presence of several supercharges, we assume that a strongly coupled sector that breaks an extended SUSY indeed exists. We call g . 4 its strong coupling at the scale of con nement m . We will consider rst the limit of rigid supersymmetry and no gravity (MP l ! 1), as well as vanishing SM 3Incidentally, in our framework neither the gauge bosons nor the graviton are actually part of the strong sector (more on this below), evading even more the existence of massless supermultiplets. 4This is made manifest for instance in the constrained super eld formalism [23, 24], where the scalar partner of the Goldstino is a pair of Goldstini, X = 2 + + 2F. { 4 { HJEP1(207) couplings gSM with order parameter F fg; g0; gs; Y g ! 0. In this limit we assume SUSY is spontaneously broken m2=g , resulting in N massless Goldstini. A nite MP l corresponds to the gauging of Poincare symmetry, which in the spontaneously broken supersymmetric context implies supergravity with massive Gravitini [31]. This is not how nature looks like: the SM fermions have spin 1=2, excluding N supergravities. Instead, nite MP l with rigid supersymmetry introduces explicit SUSY breaking (the graviton has no superpartner) that can potentially propagate to the matter sector, giving large contributions to relevant operators (e.g. MP2 l contributions to scalar masses) and masses to the Goldstini. However, in theories without elementary scalars [28], or strongly coupled theories where the j j2 operators have D & 4 [29, 30], SUSY is still approximately preserved. Moreover, an exact N = 1 supergravity in addition to N rigid supersymmetries would be enough to protect scalar masses and preserve the necessary amount of SUSY [32]. The strongly interacting supersymmetric sector is also coupled to an elementary sector which includes, possibly, but not necessarily, elementary (non-supersymmetric) scalars, gauge bosons and some fermions. The elementary sector then breaks SUSY explicitly, but in a way that can be kept under control, given that gSM g is a small perturbation of the undeformed theory, in the spirit of ref. [33]. In particular, the SM gauge vectors couple to (part of) the conserved currents in the supersymmetric sector, which are associated to an unbroken R-symmetry under which the Goldstini-like SM fermions transform.5 We further assume that R-symmetry includes (part of) the SM avor group, broken only by the Yukawa couplings to the Higgs boson. This construction reproduces a form of MFV [36] that successfully surpasses avor constraints even for a low SUSY-breaking scale. 3 The e ective Goldstini theory Following the outline in the previous section, we rst construct the Goldstini EFT, in the limit where (rigid) explicit SUSY breaking e ects vanish, that is gSM ! 0 and MP l ! 1. This section is rather formal: the results relevant for the rest of the paper are summarized in tables 1, 2 and 3. 3.1 The geometry of N Goldstini As for every Goldstone eld, the Goldstini i can be chosen to parametrize the coset space associated to spontaneous SUSY breaking [16, 38{41].6 They provide a map from the spacetime to an element of the SUSY transformations, x ! g(x), up to the identi cation g(x) g(x)h(x) where h(x) is an element of the unbroken symmetry group that includes 5There is a priori a di erent starting point to this construction in which R-symmetry is gauged (necessarily together with supergravity) [34]. However, in such theories R-symmetry is broken at very high scales by the VEV of the superpotential, in order to cancel the cosmological constant; see however [35] for a possible caveat. 6Di erent and yet equivalent parametrizations are often adopted for the Goldstini, like the one provided by the constrained super eld formalism, see e.g. [24, 37] and references therein. Constrained super elds are convenient whenever the Goldstini couple to IR elds that ll complete SUSY multiplets, i.e. in linear SUSY representations. This is never the case in our EFT. { 5 { tions, a gravity theory of a sort, although with a non-dynamical composite vielbein E a made of Goldstini. To this end, we choose the following coset representative [P ; Qi ] = [P ; Qy_ i] = fQi ; Qj g = fQy_ i; Qy_j g = 0 ; (3.2) where the latin indexes i; j = 1; : : : ; N label the supercharges, while the greek indexes are spinorial in the (1=2; 0) (undotted) or (0; 1=2) (dotted) representation of the Lorentz group SO(1; 3) SU(2) SU(2). We assume no central charge since, as explained above, we are eventually interested in identifying the unbroken R-symmetry GR U( 1 )R, or some of its subgroups, with the SM avor and gauge groups. U(N ) Under a global transformation g, the representative element U is moved into another group element of the form gU = U 0h which de nes the transformation U (x; (x)) ! U 0(x0; 0(x0)) = gU (x; (x))h 1 ; for the coset representative, hence for the Goldstini. The Goldstini transform linearly under the unbroken symmetries. In particular, they transform under Lorentz as ordinary two-component spinors.8 Similarly, for an unbroken and y carry a linear representation, e.g. the fundamental (3.3) (3.4) HJEP1(207) R-symmetry GR the Goldstini representation of maximal R-symmetry [RSaU(N )R ; Qi] = (T a)ij Q ; j [RSaU(N )R ; Qiy] = T a i Qy ; j j (x(x0)) = (x0 a) as usual. instead a non-linear realization: where T a are the N 2 1 traceless and hermitian N -by-N generators of SU(N )R. Under spacetime translations T = eia P , U 0 = T U where x0 = x + a and 0(x0) = Under the action of the broken SUSY generators g = exp[i j Qj + i jyQjy], we get g U (x; (x)) = U 0(x0; 0(x0)) ! ( 0(x0) = (x(x0)) + x0 (x) = x i y(x) + i y (x) : (3.5) 7Spacetime translations are e ectively considered as broken generators since they are non-linearly realized on spacetime, which is nothing but the coset Poincare/Lorentz, x ! x + c (see e.g. [42]). 8Explicitly, we have LU (x; (x)) = Lei( (x)Q+ y(x)Qy)L 1 Leix P L 1 L = U 0(x0; 0(x))L where one can use LP L 1 = P Goldstini, namely 0 (x0) = , LQi L 1 = e Qi , and LQiy _ L 1 = b _ _ Qi_ to get the Lorentz action on the (x(x0)) e with e the spinorial (1=2; 0) representation of L. { 6 { [RU( 1 )R ; Qi] = Q ; i [RU( 1 )R ; Qiy] = Qy ; i Indeed, under an in nitesimal i it corresponds to the following non-linear action of the SUSY transformations (x) ! 0(x) = (x0(x)) + = (x) + (x) + : : : (3.6) where v ( ; (x)) i y(x) y (x) . For matter elds (scalars and spinors) the non-linear representation is the same as for the Goldstini except that it does not include the Grassmannian shift , namely (x) ! 0(x) = (x0(x)) = (x) (x) + : : : (3.7) HJEP1(207) 3.1.1 Covariant derivatives and Maurer-Cartan form Notice that derivatives transform non-covariantly, that is di erently than the undi erentiated elds.9 In order to deal with proper covariant transformations of derivatives, it is useful to introduce the Maurer-Cartan 1-form in the SUSY algebra (U 1dU )(x) = idx E a Pa + ra Q + ra y Qy ; where for future convenience we have factored out the coe cient E a of the momentum E a = a E aEa = : Since the Maurer-Cartan form is invariant, (U 1dU )0(x0) = (U 1dU )(x), its change at a given point is just the contraction with the Jacobian matrix. This tells us that the E a and Ea transform as vielbeins E a(x) ! E0 a(x) = E a(x0(x)) ; Ea (x) ! Ea0 (x) = Ea (x0(x)) : (3.12) It is therefore clear that ra transforms covariantly, as in eq. (3.7), ra (x) ! (ra )0(x) = (ra )(x0(x)) : Gauge elds behave just as derivatives. The resulting gauge- and SUSY-covariant derivatives read Da ra iA ). Similarly, the covariant eld strength is Fab = Ea E [A ; A ]) : In our EFT we gauge an R-symmetry, as opposed to an ordinary global symmetry, which forces us to break SUSY explicitly. However the gauge elds themselves could in principle be part of the strong sector, that is composite [7]. In this case, one should dress with the vielbeins only the @[ A ] part of the eld strength and not its non-abelian part [A ; A ]. For elementary gauge elds neither term should be dressed with Goldstini. (x)) + : : : and only the rst term on the right-hand side corresponds to the covariant SUSY transformation. { 7 { (3.8) (3.9) (3.10) (3.11) (3.13) (3.14) From the composite vielbeins one can de ne a composite e ective metric g (x) E a(x)E b(x) ab ! g0 (x) = g (x0(x)) (3.15) and use it to build the various invariants that do not involve fermions; the latter requiring the vielbein too. The inverse metric is promptly found to be g = Ea E b ab = + i y (3.16) Note that E a abg the appropriate metrics. The determinant of the Vielbein = Eb , meaning that we can lower and raise the various indexes with d4x det E(x) ! d4x det E(x0(x)) = d4x0 det E(x0) : The coset-ology allows us to work with objects that transform covariantly even with respect to the broken generators, upon vielbein and metric contractions. These objects admit an interesting geometrical interpretation associated to the extended superspace (xa; i where the 1-forms !a = dxa i y ad + id y a , d and d y are invariant under SUSY ; y_ i) transformations (xa; ; y) = ( i y a + i y a ; ; y) [16]. These vector 1-forms are ! (xa(x); i(x); y i(x)). These forms are nothing but the coordinates with respect to the basis Pa, Q, Qy of the Maurer-Cartan form, up to identifying the Goldstino (x) = (x) as a space- lling brane in the extended superspace. The invariant measure (3.18) is nothing but the induced volume element of the Goldstino brane. Every eld of the strong sector that interacts with the induced metric unavoidably interacts with the Goldstini, resulting in model-independent e ects. In addition, our construction allows higher-derivative terms that are model-dependent, following from the direct interactions with r . We explicitly discuss these interactions in the next section. Among the model-dependent contributions, new higher-derivative geometrical objects may appear as well, such as Fbc a(x) and represents a sort of connection [ra; rb] (x) = Fab crc (x). which transforms covariantly, Fbc a(x) ! Fb0c a(x0(x)) (thanks to the anti-symmetry Z 3.2 The Goldstini e ective action We now have all the covariant ingredients to write an e ective action, SSUSY[ ; ] = d4x det E L(ra (x); (x); ra (x); Fbc a(x); : : :) = { 8 { Z d4x p det g L (3.20) which is manifestly invariant under SUSY transformations eq. (3.6) should L be Lorentz symmetric.10 One can also consider nearly secluded sectors, each enjoying its own extended SUSY simply by summing over actions of the type eq. (3.20); see e.g. [43] for the case of N copies of N = 1 SUSY. In the following we focus on a single sector with N supercharges in order to avoid the proliferation of unknown constants of possibly disparate scales.11 Goldstini self-interactions: Akulov-Volkov The most important SUSY-preserving Goldstini interactions are those in eq. (3.20) with the least number of derivatives. There is in fact a unique operator that yields both the leading interactions and the kinetic terms. Not surprisingly, it is the most relevant operator in a gravitational theory, the vacuum energy, injected at the SUSY phase transition, namely LCC = F 2 ; which, dressed with the Goldstini, provides the extended-SUSY generalization [17] of the Akulov-Volkov action [16] SS(0U)SY = Z d4x det h a The scale F is the SUSY-breaking scale. The action SS(0U)SY is accidentally invariant under maximal R-symmetry U(N )R. This will be important when discussing beyond those of the SM. Expanding the Vielbein's determinant avor violations + 1 2 _ a after using the free equations of motion, integrating by parts, and using the Fierz indentity = . The Akulov-Volkov Lagrangian, with a single avor 2=(4F 2) upon integration by parts. This is the only Z d4x 1 i j ) (3.24) as ij U(N )R 10In practice this requires that all the Lorentz indexes a, b, c,. . . must be saturated contracting them in pairs with the Minkowski metric ab, its inverse ab, the fully anti-symmetric abcd symbol, or the sigma matrices . Likewise for the spinorial indexes. To construct an action that is also R-symmetric, the avor indexes i; j; : : : should be contracted among themselves with the relevant invariant tensors, such . Notice that det E alone is automatically, i.e. accidentally, invariant under maximal R-symmetry 11However, one could in principle consider other global symmetries to reduce the unknowns. A trivial example would be semi-direct products of the type ZM o [N -SUSY]M . det E = 1 i jy j + h:c: we extract the kinetic term as well as the leading order interactions. The canonically clear. The leading four-fermion interactions come from a dimension-8 operator that can . In the following we omit the tilde , whenever e normalized Goldstini e be written as d4x N = 1, can be recast as are p2F y a j = e naked terms ; ; F ; 2i iy @ i(x) + h:c: det E 2i iy ara i(x) + h:c: SUSY dressing leading four-body interactions 4F1 2 F A F A scalar particles have these couplings weighted by their degree of compositeness squared j X j2. If the is a composite NGB one should dress the whole kinetic term built with the D-symbol operator with four fermions and two derivatives that can be built with one fermion. With more avors, we note that any dimension-8 operator of this kind that is consistent with the absence of more relevant interactions is necessarily selected by shift and spacetime transformations of the form eq. (3.6), which strongly suggests that they can be linked to a SUSY-breaking pattern, e.g. for two avors, N = 2 or (N = 1)2. 3.2.2 Model independent couplings to composite elds Let us consider now the case where, in addition to the Goldstini, other particles are composite, that is they emerge from the same sector that breaks SUSY spontaneously. These elds have kinetic terms, provided by the SUSY-breaking sector itself, which need to be compensated by Goldstini insertions to make up for the absent kinetic terms of the wouldbe superpartners. Since the kinetic terms can always be rescaled to a canonical form, the resulting Goldstini interactions are model independent, controlled by no free parameter except for the SUSY-breaking scale F (as the Goldstini self-interactions).12 Calling X = , A or any composite fermion, gauge boson or scalar, we construct their SUSY-invariant kinetic terms by making the replacements @ X ! raX, d4x ! d4xp det g, ! g , A ! Aa, F ! Fab. . . , from which we derive the so-called model independent couplings of the composite states to the Goldstini. They are reported in table 1, where we have canonically normalized the Goldstini in the last column, used the free equations of motion and integrated by parts. The curly brackets mean symmetrization, @ . Notice that all these interactions respect accidentally U(N )R. When considering the couplings of the strong sector to external (gauge) sources, an important r^ole is played in our construction by the R-symmetry current, the conserved Noether current associated with in nitesimal R-symmetry transformations i!AT A (T A are the appropriate R-generators). This current can be found by noticing ! that in our geometrical construction Goldstini enter the action only through geometrical 12In fact, this model independency is somewhat of a misnomer: the overall coe cient is controlled by F alone because we assumed full compositeness. Should some of these non-Goldstini states X = , A or be partially composite instead, the would-be model independent couplings would actually get rescaled by the degree of compositeness (squared), j X j2, which measures how large a fraction of the kinetic term arises from the SUSY-breaking sector. Note that X 6= 1 necessarily breaks SUSY. = 1 = 1, NGB = , NGB c det E cij det E (ra iy b a j)( y b ) r dij det E ra iy b a j rb r ra y bT Ara i yT A ! r b leading interactions ) ve-body or / m ; m objects, so that the R-current is directly related to the energy-momentum tensor (associated to the model-independent universal contributions) + h:c: FabFcbE c : T a = = S= E a LEa + det E 2 i iy b From this we nd, for the R-current [44], RA = 1 T F 2 a y aT A = y aT A a + i 2F 2 j y ! (3.26) tensor, namely S 1; : : : ; N . Analogously, the SUSY currents can also be expressed in terms of the energy-momentum j = p2Ta _ jy( a) _ =F and Sy _ = j p2Ta ( a) _ j =F , with j = 3.2.3 Model dependent couplings to composite elds The model dependent couplings may or may not be there, depending on the details of the UV theory. Generically they do not respect maximal R-symmetry, but just some of its subgroups. In table 2 we show a few illustrative examples assuming that the i carry at least a U( 1 )R, i.e. i ! iei R , that the scalar is naturally light because it is a NambuGoldstone boson (NGB), and we restrict to a singlet fermion. In the last row we kept the leading term in the number of NGBs of the E -symbol, EbA = also enforced maximal R-symmetry, GR = U(N )R, for simplicity. This can also be done for the rst and second rows by choosing cij ; dij / i j . Alternatively, choosing e.g. cij ; dij = i yT A ! r b + O( 4 ). We diag(c(N)IN N ; c(M)IM M ) with N U(1)R subgroup of U(N )R under which = (N; 1)x (1; M) xN=M . The size of these interactions depends on the assumptions about the UV theory, e.g. the size of couplings with additional heavy states. The generalization to non-NGB scalars, non-singlet fermions, or other choices of conserved R-symmetries and its representations is straightforward. For example, a scalar without a shift symmetry has a model independent coupling proportional to its mass squared, m2 j j2 det E, which contributes to a 6-body scattering.13 Higher-derivative 13One could naively think that j j2 contributes to the ! elastic Goldstino scattering when gets a VEV, and that there could be a lower bound on how much tachyonic the mass squared can get, from the = N + M corresponds to realize linearly only the non-NGB couplings would be of the form det E (ra y b ! Since the model dependent contributions include extra Goldstini beyond those contained in det E or the metric, the resulting equations of motion and the conserved R-current are di erent than those originating from the model independent contributions. ing R-symmetry multiplets. We still work in the limit where gauge and Yukawa interactions vanish, but ensure that the strong sector itself respects the symmetries of the SM, i.e. gauge and avor symmetries, GSM GGauge GFlav, with GGauge = SU(3)C SU(2)L U( 1 )Y and GFlav = SU(3)5Flav U( 1 )B U( 1 )L.14 Since the only symmetry under which Goldstini transform is by de nition the R-symmetry, GR, the embedding of the SM fermions is possible only if their gauge and avor symmetries can be embedded into the R-symmetry, GSM Maximal R-symmetry N free Weyl fermions, which is what the SM reduces to in the limit gSM 0, enjoys an U(N ) symmetry such that they transform in the fundamental representation. While gauge interactions in the SM break explicitly this U(45) symmetry down to the smaller GSM, the actual representations of quarks and leptons with respect to GSM do not t a fundamental representation of the original U(45), simply because the SU(2)L singlet and doublet quarks come in di erent color representations, namely a 3 and 3 respectively. These facts may be replicated in the strong sector; the lowest-dimensional interactions (the Akulov-Volkov Lagrangian eq. (3.22)) are accidentally U(N )R symmetric, with Goldstini in the fundamental representation. Adding higher-dimensional terms, e.g. cikjlra i a j r rb ykr b ly, generically breaks U(N )R to a subgroup under which the Goldstini transform in various representations that no longer t, generically, into the fundamental of U(N )R. However, if SM matter is to be identi ed as Goldstini, these representations must t at the very least the proper SM representations of GGauge. While it is technically possible that GR = GGauge (or GR = GSM) is strictly smaller than U(N ) and that the Goldstini representations are exactly those of the SM fermions, it would be very surprising, lacking a dynamical reason. With no better option, we assume instead in the following that the SUSY-breaking sector is maximally R-symmetric, i.e. GR = U(N )R. The reduction to GSM and then to GGauge is entirely due to the external SM couplings gSM, rather than the SUSY-preserving parameters of the strong sector. In other words, we extend the paradigm of MFV from GSM to the largest group that can be simultaneously fg; g0; gs; Y g ! preserved by the strong sector and the free theory, i.e. U(N )R. usual positivity of the Goldstino amplitude [12]. However, a SUSY-preserving potential for must have vanishing vacuum energy, e.g. V = (j j of F in eq. (3.21). This ensures that the SUSY-preserving. 2 f 2=2)2, since by construction we include it all in the de nition ! is actually not a ected as long as the potential is 14Whenever including the right-handed neutrinos c we actually consider SU(3)6Flav. Interestingly, in most cases the assumption of maximal R-symmetry does not obstruct a proper embedding of the SM fermions as Goldstini. Maximal R-symmetry is often obtained when the strong sector has the least number of supercharges compatible with the assigned Goldstini content, see table 3. In other words, the representations of the SM fermions often t in the fundamental of U(N )R. Assuming maximal R-symmetry is relevant, for what regards the SM embedding, only when both doublet and singlet quarks are Goldstini, since it requires a number of supercharges larger than the number of Goldstini-like SM fermions, implying extra light fermions in the spectrum. Maximal R-symmetry means, in practice, that we should embed the SM matter repfollowing decompositions turns out to be useful15 resentations that are associated to Goldstini in the fundamental of U(N )R. To do so, the Let us start with the simplest case: the singlet electron, charged only under hypercharge, e c = (1; 1)1 under SU(3)C hypercharge with the U( 1 )R of N = 1, that is = ec. U( 1 )Y . In this case we promptly identify the The case where all three generations of singlet electrons are Goldstini is slightly more interesting as we insist on a SU(3)ec triplet ec = (ec; c ; c) = 31 of SU(3)R avor symmetry. The proper embedding is via a U( 1 )R in N = 3, where the U( 1 )R and SU(3)R factors are identi ed with the hypercharge and the avor group respectively. In this case the R-symmetry index j is nothing but that the avor index, j = ejc. Embedding only one lepton doublet, ` = (1; 2) 1=2 under GGauge, requires N = 2 and again maximal R-symmetry U(2)R electroweak index in this case, `j = ( L; eL)j = j . Similarly, including all lepton doublets requires to consider N (2; 3) 1=2 with respect to SU(2)R = 6 and to decompose the fundamental of U( 6 )R as 6 1=2 = SU(3)R U( 1 )R. One then identi es SU(2)R U( 1 )R with SU(2)L U( 1 )Y and SU(3)R with the avor group. With little extra e ort we can embed all leptons, including the singlet neutrinos c taking N = 12 and the maximal R-symmetry group U(12)R. Such a large R-symmetry contains the proper subgroups SU(12)R U( 1 )R SU( 6 ) SU( 6 ) U( 1 )R U( 1 )A that in turn (SU(3) SU(2)) (SU(3) SU(3) U( 1 )C ) U( 1 )R U( 1 )A, where the three SU(3)'s are identi ed with the avor groups acting on the avor triplets `, ec and c while the SU(2) factor is identi ed with SU(2)L. The last abelian U( 1 )C allows us to give independent hypercharges to doublet and singlet leptons. Explicitly, the fundamental of U(12)R decomposes as 12r = (6; 1)r;a (1; 6)r; a = (3; 2; 1; 1)r;a;0 (1; 1; 3; 1)r; a;c (1; 1; 1; 3)r; a; c = L e c c 15In the rst case, an index I in the fundamental that runs from 1 to N M can be split into a collective pair of indexes I = (i; j) where i = 1; : : : N and j = 1; : : : M . In the second case the collective index I = 1; : : : N + M is split in two separate indexes, i = 1; : : : ; N and j = N + 1; : : : ; N + M . under the chain of subgroups we have mentioned above. The hypercharge is identi ed with Y = A=(2a) + C=(2c), the lepton number is L = A=a, while the U( 1 )R plays no role (we could have demanded just SU(12)R rather than the maximal U(12)R). We summarize these and other cases in table 3. Embedding quarks and extra exotics The embedding of either SU(2)L doublet or singlet quarks works like for leptons, as we show in table 3. Things become more complicated when we embed the quark doublets together with the singlets inside the same fundamental representation of SU(N ). This is due to the di culty in obtaining both a 3 (for dc and uc) and a 3 (for q) of SU(3)C when decomposing the fundamental of SU(N ).16 The solution to this problem is to add extra states to ll a larger multiplet that can give rise to 3 's, since a 3 of SU(3) can be built out of two fundamentals, 3 6. Then, the smallest group that can accommodate all quarks (all avors) is SU(72)R U( 1 )R, with the 72r decomposing as 72r =(2; 3; 3; 1; 1; 1; 1)r;3a;0 (1; 1; 1; 3; 3 ; 1; 1)r; a;c (1; 1; 1; 3; 6; 1; 1)r; a;c (3.29) (1; 1; 1; 1; 1; 3; 3 )r; a; c (1; 1; 1; 1; 1; 3; 6)r; a; c = q u c X 2=3 d c X 1=3 with respect to the subgroup SU(2) [SU(3)]2 [SU(3)]2 [SU(3)]2 U( 1 )R U( 1 )A U( 1 )C : A diagonal SU(3) out of three SU(3)'s is identi ed with the color group while three other SU(3)'s represent the avor group SU(3)qFlav SU(3)dFlav SU(3)uFlav (this matter content is then consistent with two extra global SU(3) factors). The hypercharge reads Y = R=(12r) + A=(12a) C=(2c), whereas the baryon number is B = R=(6r) + A=(6a). The prediction of maximal R-symmetry is then that there are extra (pseudo-)Goldstini X 2=3;1=3 that are colored and charged under Y , transforming as X 2=3;1=3 = (6; 1) 2=3;1=3 (3.30) hypercharge. color sextets. under GGauge. From the decomposition in eq. (3.29) we see that these exotic states (aka quixes) are also triplets of the avor groups, either SU(3)u or SU(3)d, depending on their Similarly, embedding all SM quarks and leptons into Goldstini requires N = 84 supercharges, where the fundamental splits as 84 = 12 72, and we can apply the results derived above. The extra 36 states correspond again to three families of the two exotic 16The 36 is the minimal representation that could in principle accommodate the 18 + 9 + 9 quarks q, uc and dc. However, the decomposition of U(36)R into SU(2) [SU(3)]2 [SU(3)]2 [SU(3)]2 U( 1 )R U( 1 )A U( 1 )C is 36r = (2; 3; 3; 1; 1; 1; 1)r;a (1; 1; 1; 3; 3; 1; 1)r; a;c (1; 1; 1; 1; 1; 3; 3)r; a; c which does not contain a color 3 with the other three SU(3)'s identi ed as the avor groups. HJEP1(207) `e, ec `e, ec, ec dc or uc e c `e e c ` `, ec `; ec; c dc or uc q dc, uc q; dc; uc; X 2=3;1=3 SU(2)L U( 1 )Y U( 1 )Y U( 1 )Y U( 1 )Y U( 1 )Le U( 1 )Le SU(3)C U( 1 )Y U( 1 )Y SU(3)eFlav SU(2)L U( 1 )Y SU(3)`Flav numbers U( 1 )B and U( 1 )L. Boldface characters denote avor triplets. The notation [SU(m)C;L;Y ]n means there are n SU(m) factors of which only one linear combination corresponds to the gauged SU(m)C;L;Y group. The asterisk marks whether SU(N )R, as opposed to U(N )R, is required. The number of supercharges inside parenthesis refers to Nmin for non-maximally R-symmetries (with Goldstini not in the fundamental representation). There are two interesting points related to this extra matter content. First, they give rise to [SU(3)C ]2U( 1 )Y and [U( 1 )Y ]3 anomalies. In order to cancel them there must exist extra light colored states, such as complex conjugate color sextets, Y 2=3; 1=3 = (6 ; 1)2=3; 1=3 ; (3.31) to form, along with X, real representations of GGauge. The Y 's are anti-triplets of avor. We will loosely refer to these Dirac fermions light, relative to the strong coupling scale m 6 = X p Y c as the sextet 6. The sextet is g F , since a mass term m6 XY requires breaking SUSY explicitly (see section 4). Second, the sextet contributes signi cantly to the running of the strong coupling: four extra Weyl fermions (two pseudo-Goldstini X and two extra elds Y ), in the 6 and 6 of SU(3)C , per three generations, imply a contribution b6 = 20 to the 1-loop -function 1 loop = bg3=(16 2). This implies that above m6 the function changes sign and would hit a Landau-pole at roughly 102m6, where we have taken s(m6) 0:09. This sets an upper bound on the scale of ' m6Exp[2 =(b s(m6))] strong coupling m , which should enter before . Equivalently, this sets a lower bound on m6=m & 10 2. 4 Explicit SUSY breaking So far we have discussed the properties that characterize the strongly interacting sector, in di erent cases where one or more SM fermions are Goldstini, but in the limit of vanishing Fermion-Gauge Dipoles ODB = ODW = ODG = L L L H RB aH RW a HT A RGA Fermion-Higgs $ Oy = jHj2 LH R O4 = Four-Fermions OB = D B ( L;R OW = D W a ( L a L;R) L) OL;R = (iHyD H)( L;R L;R) (3) OL = (iHy aD$ H)( L a L) SM interactions gSM ! 0. The leading interactions genuine of goldstino-compositeness are controlled by dimension-8 operators, which are large at high energy, but vanishingly small at small energy. These interactions preserve a maximal R-symmetry, which contains the relevant global groups to be gauged and avor groups. This picture is necessarily distorted at least by the marginal SM interactions, which are small but become leading at su ciently small energy. These interactions break SUSY explicitly, the Goldstino-like SM fermions become pseudo-Goldstini, and new e ects are generated, which we estimate in this section. The SM interactions should be thought as spurions of SUSY breaking, Lbreak = Y ij i j H + : : : + h:c: + gV V ARA + : : : : (4.1) The Yukawas Y also break the avor symmetries and we assume these are the only sources of avor breaking, thus realizing the MFV paradigm [36]. Gauge interactions arise through weakly gauging some of the R-currents RA , eq. (3.26). The dots in eq. (4.1) represent the generalization of the minimal symmetry-breaking interactions (the rst terms in brackets) by operators with the same eld, symmetry, and spurionic content but with extra (covariant) derivatives, suppressed by m . We focus rst on the scenario where the only light composite d.o.f.'s are the pseudoGoldstini, while non-Goldstini SM fermions, the Higgs and the gauge bosons are elementary; we comment below on extensions of this picture. In such minimal scenario no dimension-6 operators are generated directly by the strong dynamics, but some might be generated via loops involving both the strong dynamics and the spurions in eq. (4.1) | a list of the interesting dimension-6 operators is reported in table 4. We estimate these e ects by simple power counting, derived from the leading loop diagrams, some of which are shown in gure 1. Z couplings. The largest e ects involve operators with external fermions and the least number of SUSY-breaking spurions. In particular, loops of pure strong sector with a gauge boson insertion generate OW;B (see e.g. gure 1(a)) with coe cients cW;B g m2 : (4.2) HJEP1(207) (a) g V Y Y (b) H g g (c) blobs denote strong interactions, while little red (blue) ones denote SM gauge (Yukawa) couplings that break SUSY explicitly. HJEP1(207) (3) Through a eld rede nition (which corresponds to the equations of motion) these operators are equivalent to combinations of OL;R, OL and O4 , with coe cients cL;R c(3) L g2=m2 and c4 g2=m2. The former modify the couplings of the Z boson to fermions, which are constrained at the permille level from measurements at LEP-1. Consequently, goldstino-compositeness of doublet leptons (or L-handed, `L, in Dirac notation), is constrained as [45, 46] ` Goldstini: m & 2:5 TeV ; while goldstino-compositeness of singlet leptons (or R-handed, eR), even if probed by the same data, gets a milder constraint due to the (g0=g)2 suppression of the corresponding operators ec Goldstini: m & 2 TeV : We will see in section 5.2 that, because of the low energies accessible at LEP, the constraints on pseudo-Goldstini leptons from their de ning dimension-8 operators are weaker than the indirect ones derived here. Given the constraints on modi cations of the L-handed quark couplings to gauge bosons, for instance from LEP measurements of the Z decay to hadrons, their goldstinocompositeness is indirectly constrained at the same level as that of L-handed leptons q Goldstini: m & 2:5 TeV : On the contrary, goldstino-compositeness of R-handed quarks is not very constrained via the e ects of ORu; d, given that the sensitivity of LEP to non-standard R-handed quark couplings is signi cantly reduced by their small SM coupling to the Z. The only relevant Yukawa-induced operators are those associated to up quarks as pseudo-Goldstini, because of the large top Yukawa. In particular, if both L- and R-handed tops are pseudo-Goldstini, we nd (from e.g. gure 1(b)) an additional contribution to This could be the dominant e ect, since it modi es the Zbb coupling, implying c(3) L cL;R Y 2 m2 t : q3 and tc Goldstini: m & 5 TeV : (4.3) (4.4) (4.5) (4.6) (4.7) like gure 1(b) become weak and thus suppressed by g2=16 2, implying c(3) However, if q3 is elementary this bound disappears, while if tc is elementary, loop diagrams Yt2g2=16 2m2, which might alleviate the bound if the strong coupling is not maximal, g < 4 . Other constraints associated with eq. (4.6) come from top physics only and c L L therefore are mild. Higgs couplings. The operators Oy are generated with coe cients cy Y 3=m2 (or Y g2g2=16 2m2) given that the discrete symmetry H ! H is broken only by the Yukawas. The experimental constraints from measurements of the Higgs couplings to fermions are not competitive enough to make these e ects relevant. Furthermore, this estimate is reduced by g2=16 2 if only the L- or R-handed fermions are pseudo-Goldstini. cients c4 section 5. Four-fermion contact interactions. The operators O4 can be generated with coe g2=m2 from the eld rede nition described below eq. (4.2), or directly from diagrams like gure 1(c), carrying an additional g2=16 2 suppression due to the elementary vector boson in the loop. These operators contribute to the same observables as the bona de Goldstini operators of dimension-8, eq. (3.24). We leave the analysis of these for Dipole moments. When both L- and R-handed SM fermions are pseudo-Goldstini, a Higgs and a gauge insertions are enough to saturate the necessary selection rules to generate dipole operators with V = B; W; G (see e.g. gure 2(a)), Instead, if only L or R are composite, dipole operators can only be generated upon insertion of two extra Yukawa couplings and an elementary Higgs loop similar to the BarrZee one [47], L and R Goldstini: cDV L or R Goldstini: cDV Ymg2V : Y 3gV ( 4 )2m2 : The precise measurement of the anomalous magnetic moment of the muon [48] sets a constraint on goldstino-compositeness of leptons ` and ec Goldstini: m & 3:2 TeV while if only ` or ec are pseudo-Goldstini the bound is negligible. We note that our assumption on R-symmetry, which provides a concrete realization of the MFV paradigm, implies no new sources of CP violation beyond the SM. Therefore we obtain no relevant constraints from electric dipole moments. Flavor transitions. In MFV any avor-violating operator is proportional to the SM Yukawa couplings. This means that in the lepton sector any avor transition is negligible due to the smallness of neutrino masses. In the quark sector instead, MFV implies that any (4.8) (4.9) (4.10) avor-violating process proceeds via the top, and the scale associated to certain operators needs to be above 1 TeV [49]. Some of the potentially problematic operators, e.g. e ( 4 )2m2 dRYdyYuYuy HyqLF ; g 2 ( 4 )4m2 (qLYuYuy qL)2 ; are generated at one or two weak (Higgs) loops respectively, so that the sensitivity of observables such as B ! Xs or K and mBd is suppressed. The strongest avor constraints on goldstino-compositeness of quarks arise instead from the operator (see e.g. gure 1(b)) 1 m2 (iHyD$ H)(qLYuYuy qL) q and uc Goldstini: m & 2:3 TeV ; that contributes to B ! Xs`+` , Bs ! + , giving rise to the bound similar in size to the one from electroweak precision tests eq. (4.5). Goldstini. In such a case diagrams like gure 2(b) generate the operators Comparable constraints arise if both L-handed quarks and leptons are pseudog 2 g 2 ( 4 )2m2 (qLYuYuy qL)(`L `L) ; ( 4 )2m2 (qLYuYuy qL)(eR eR) ; which also contribute to B-decays, implying q and (` or ec) Goldstini: m & (1:7 or 2:7 TeV) g 4 : Other couplings. Operators without external fermions can also be generated, simply because of the existence of a sector with a mass scale m to which all SM elds couple. Yet, such operators do not bene t from any strong coupling enhancement and provide, therefore, less relevant sensitivity to scenarios where (some) fermions are pseudo-Goldstini but the other species are elementary. Other composites. It is certainly plausible that, if SM fermions are indeed pseudoGoldstini, the strong dynamics also involves in some way the Higgs boson (perhaps in a solution to the hierarchy problem), the transverse polarizations of vectors, or perhaps it includes some of the fermions as non-Goldstini composites. These scenarios have been studied extensively already, in e.g. refs. [1, 3, 7], and their predictions in terms of EFTs (4.11) (4.12) (4.13) (4.14) (4.15) are well known: composite fermions imply large O4 , as already discussed in this work, while composite Higgs models imply large OH = (@ jHj2)2 and models of composite gauge vectors are characterized by large O2W = (D W a )2 and O3W = abcW a W b W c . In this context, if the Higgs is composite the size of some of the Yukawa-induced operators discussed above is enhanced. In particular, operators that were generated via an elementary, thus weak, Higgs loop, see e.g. gure 2(b), are no longer suppressed by g2=16 2, since for a composite Higgs the loop becomes strong.17 Such an enhancement applies to eqs. (4.9) and (4.11), (4.14). Similarly, operators that were suppressed by g2=16 2 because either L or R were not pseudo-Goldstini, are enhanced as well when the Higgs is composite. An interesting case is that in which the low-energy EFT includes non-SM light composite states, for which the selection rules di er. We have discussed in section 3.3 that embedding all of the quarks as Goldstini requires the existence of new color sextets X and Y . Masses for these states require an extra source of explicit SUSY breaking, which we simply write as m62=3 X 2=3Y2=3 + m61=3 X1=3Y 1=3 : (4.16) Being this a small deformation of the SUSY-preserving dynamics, we expect the sextets to be naturally light, thus present in the low-energy spectrum. Furthermore, the pseudoGoldstini X have SUSY-preserving interactions which, because of maximal R-symmetry, are invariant under X ! X. Therefore such interactions do not contribute to the sextet's decay to the SM: symmetry breaking e ects can play a major ro^le here. Speci cally, a new SUSY-breaking spurion (with di erent quantum numbers than gSM or m6) can be introduced in association with a linear coupling of the sextet, the lowest-dimensional one being, schematically gs m G ; (4.17) where = Y2=3; 1=3 and = uc; dc, where recall Y is a non-Goldstini state. In fact, Y could well be external to the strong dynamics, in which case its degree of compositeness should be factored in. Besides, 1 is expected since (4.17) is an irrelevant operator. This operator is not only important for the sextet decay but also for its single production at the LHC, as we discuss in section 6. 5 Collider phenomenology In section 3 we have built the Goldstini EFT in the symmetric limit: Goldstini are characterized by interactions of dimension-8 that grow maximally fast with energy. We have further established in section 4 that the less irrelevant dimension-6 operators can only arise from (suppressed) SUSY-breaking e ects. Constrains on the latter imply m & 2 TeV for the compositeness scale in most of the interesting cases (with the exception of R-handed quarks, where the bounds are weaker). 17This can also be understood by identifying, when the Higgs is composite, the Yukawa coupling in eq. (4.1) with a linear mixing a la partial compositeness, see e.g. ref. [7]. p p In this section we attempt instead direct access to the strongly coupled dynamics by considering 2 ! 2 pseudo-Goldstini scattering at the highest possible energies. For comparison, we study both our scenario based on non-linearly realized SUSY and dimension-8 operators, as well as the standard one based on dimension-6 operators, which updates the analysis of ref. [3]. Our main focus is LHC phenomenology, which gives us access to scattering of quarks in the multi-TeV region, but we also brush upon LEP-2 to investigate the extend to which leptons could arise as pseudo-Goldstini. The results on composite quarks are summarized in table 5: we show the bounds on the SUSY-breaking scale F for goldstino-compositeness, as well as those on the scale f for chiral-compositeness, in di erent composite quark scenarios. A direct comparison between the two types of compositeness is also presented in terms of bounds on the strong coupling scale, identi ed as m g F for Goldstini and m g f for chiral composites. The results on composite leptons, speci cally eR and R, are given in eqs. (5.10) and (5.11). We focus on the process that gives, at present, access to the highest energies: dijet events pp ! jj initiated by valence quarks at 13 TeV. The experimental analysis can be found in ref. [50], corresponding to an integrated luminosity of 15:7 fb 1 collected with the ATLAS detector. The relevant pseudo-Goldstini interactions are parametrized by the operators Oqu = Oqd = uR) + h:c: dR) + h:c: ; where in each particular composite quark scenario the Wilson coe cients are cij = 1=(2F 2). These operators contribute to the di erential dijet cross section at the partonic level,18 (5.1) (5.2) (5.3) d dt^ d dt^(qiqi ! qiqi)BSM = 4 s Abq1i + 1 48 (ud ! ud)BSM = u^2Bb3 ; 9 1 16 Bb1qi (2u^2 + 2t^2 + s^2) + 6Bb2qi (u^2 + t^2) ; yj = log q (Ej + pjz)=(Ej pjz), where s^, t^, u^ are the Mandelstam variables (s^+t^+u^ = 0 in the massless approximation) and Au;d = cqq + cuu;dd ; b1 Bu;d = cq2q + c2uu;dd ; b1 Bu;d = cq2u;qd ; b2 Bb3 = cq2q + c2ud + cq2u + cq2d : The BSM contribution is then characterized by a strong energy growth and by being more central ( t s) than the SM one, which instead peaks in the forward region because of t-channel gluon exchange. The analog of the CM scattering angle is conveniently represented by the boost-invariant di erence between the two jet rapidities ejyj1 yj2 j = (1 + cos )=(1 cos ) : 18The leading contributions at high energies come from two initial rst-family quarks (as opposed to quark-antiquark, two antiquarks or second- and third-family quarks), due to the enhanced PDFs in pp collisions. 0.10 40 20 experimental data with its systematic plus statistical uncertainties (black points), the SM prediction with its theoretical error (blue band), and the BSM predictions for the dimension-8 operator Odd (solid orange) or the dimension-6 operator Od(1d) [3] (dashed red). The corresponding (positive) coe cients have been chosen to saturate the bounds eqs. (5.4) and (5.5). Right: bounds on the scale m for a composite dR with a dimension-8 operator jcddj = ( 4 )2=2m4 (orange) or a dimension6 operator jc(d1d)j = (4 =m )2 (red) and for all quarks composite with dimension-8 operators jcijj = ( 4 )2=2m4 (black), for di erent bins (dots) and for all bins combined (lines). The round dots and solid lines correspond to c > 0 while square dots and dashed lines to c < 0. The central region corresponds to small , where the BSM di erential cross section peaks, while the SM is approximately at. Such a behavior can be recognized in gure 3 left panel, where we show the SM distribution and two di erent BSM contributions both corresponding to a composite dR, one from goldstino-compositeness (cdd 6= 0) and the other from chiral-compositeness (c(d1d) 6= 0, operators and cross section formulae can be found in ref. [3]). We compute the BSM particle-level cross section in two di erent ways to double-check: via the analytic di erential cross section eq. (5.2), integrated over the PDFs (CT10 [51]) and binned according to the experimental analysis, and via a MadGraph [52] simulation of our model implemented with FeynRules [53]. This is compared with data from ref. [50], limited to highest-energy bin, with an invariant mass Mjj > 5:4 TeV; we check, a posteriori, that the bounds obtained on the compositeness scale m are compatible with the EFT hypothesis: m Mjj . The analysis includes additional cuts on the transverse momentum of the leading jet pT (j1) > 450 GeV, on the average rapidity of the two jets (yj1 + yj2 )=2 < 1:1, and on the rapidity di erence jyj1 yj2 j=2 < 1:7. For the SM prediction we use the NLO di erential distribution reported in ref. [50], but we normalize it to the total number of SM events, compatible with the above cuts, that we compute using POWHEG [54] and PYTHIA8 [55] with PDF4LHC15 [56], obtaining SM cuts = 50:8 9:1 fb (corresponding to 800 events for the integrated luminosity of ref. [50]). NLO e ects on the new physics distribution are instead neglected, although these could be relatively important compared to SM NLO e ects, in the region of parameters that saturates the bounds (in this region, the SM and BSM contributions are similar in size, see gure 3). Concerning the errors in our analysis, in the small region the dominant uncertainty is statistical, since the SM cross section is dominated by the forward region and therefore in composites dR uR qL uR; dR qL; dR qL; uR qL; uR; dR p goldstino F [m ] (TeV) chiral f [m ] (TeV) F for di erent quark as pseudo-Goldstini scenarios (second column) and on the scale f for di erent cases of chiral-compositeness of quarks (third column). The respective Wilson coe cients are given by cij = 1=(2F 2) for the operators in eq. (5.1) and by ci(j1) = 1=f 2 for the operators Oij ( 1 ) = ( i compositeness scale in each case is given by m p reported in brackets. i)( j j) given in ref. [3]. The g F or m g f with g = 4 , their bounds the central region the statistics is small. Systematic uncertainties (dominantly from the jet energy scale) are relatively small. Theoretical uncertainties are sizable, yet smaller than the statistical at small , becoming the dominant uncertainty at large . Besides, even though the errors of the SM cross section are not gaussian, we treat them as such to simplify the analysis, symmetrizing the distributions such that the 1 band remains unaltered. We extract bounds on the Wilson coe cients in each bin and combine them to obtain the nal constraint. In gure 3 we report these bounds for the case of a composite dR as a function of m , de ned as cdd g2=2m4 for goldstino-compositeness and c(d1d) = (g =m )2 for chiral-compositeness, with strong coupling g = 4 . The nal bounds for positive (m+) or negative coe cients (m ) at the 95% CL are19 dR chiral composite: dR Goldstini: m+ & 36 (g =4 ) TeV ; m+ & 9:4 pg =4 TeV ; m m & 40 (g =4 ) TeV : & 9:0 pg =4 TeV : (5.4) (5.5) Recall these bounds are derived from the bin with Mjj > 5:4 TeV. This means that for our analysis to be safely consistent with the EFT assumption Mjj m , we must require g & 2 in the case of chiral-compositeness and g & 4:5 for goldstino-compositeness, i.e. the new physics must be strongly coupled. Besides, we discussed in section 4 the existence of dimension-6 four-fermion operators O4 generated from explicit SUSY-breaking. The question arises then whether experimental searches, in the case at hand 2 ! 2 quark scattering at the LHC, are more sensitive to these more relevant (but gSM-suppressed) e ects or to the dimension-8 (but g -enhanced) e ects. eqs. (5.4) and (5.5) show that, for g within the validity of the EFT, the constraints on pseudo-Goldstini from the latter are clearly superior. As a matter of fact, bounds on 19Goldstino operators are in fact subject to positivity constraints that require strictly positive coe cients (see section 5.3). Our bounds for c < 0 simply illustrate the relevance of the interference with the SM. goldstino-compositeness from the LHC are stronger than any indirect ones from SUSYbreaking e ects. This is clearly due to the large energies accesible at this collider (roughly E=m gSM=g ). Finally, the scenario of a composite dR is constrained the weakest of all the cases, while when all quarks are composite, the LHC reach is maximal. In gure 3 we show the constraints in this scenario, the nal result being qL; uR; dR Goldstini: m+ & 16:2 pg =4 TeV ; & 14:2 pg =4 TeV : (5.6) The bounds for the rest of composite scenarios are given in table 5. During its second phase, LEP collided electrons and positrons at energies signi cantly higher than the Z-pole, measuring angular distributions with percent precision. Here we focus on how the di erential cross sections for e+e ! e+e ; + are a ected by the goldstino-compositeness of the R-handed electron eR and muon R, and compare them with LEP-2 data [57] to extract bounds on their SUSY-breaking scale F or compositeness Let us rst recall that limits on standard chiral-compositeness of eR, parametrized by the dimension-6 operator c( 6 )(eR eR), were extracted by the LEP collaborations, eR chiral composite: m+ & 43 (g =4 ) TeV ; m & 40 (g =4 ) TeV ; (5.7) where we normalized the corresponding Wilson coe cient as jc( 6 )j = (g =m )2. Bounds were obtained as well on (eR & 41 (g =4 ) TeV and m 33 (g =4 ) TeV. Our interest here is in deriving rst-time bounds on the goldstino-compositeness of eR, which is parametrized by the dimension-8 operator and on the scenario where both eR and R are pseudo-Goldstini, in which besides Oee also eR)(eR eR)( R R) + h:c: (5.8) (5.9) is generated, with the same coe cient.20 These operators induce non-standard contributions to the di erential cross sections for Bhabha scattering and dimuon production | their analytic expressions are reported in the appendix. They share some similarities with the dijet case, in particular the t-channel photon exchange also gives rise to a forward singularity, such that the SM contribution and the interference with the BSM are enhanced for small angles. The experimental sensitivity in e+e production at small angles is approximately 4% (95% CL) of the SM contribution. This means that, contrary to the LHC, LEP is really testing small departures from the SM, and it is sensitive then to the SM-BSM interference 20We do not study R compositeness since the sensitivity of LEP in e+e ! + is weaker. We take the SM prediction provided in ref. [57] and compute the new physics e ects analytically, see the expressions in the appendix. The theoretical uncertainties on the total SM cross section amount to 2% for (e+e ) and 0:4% for ( + ), resulting in an uncertainty for d =d cos jSM of 4% and 1% respectively (assuming the error to be uniformly distributed with the scattering angle ). For the purpose of this analysis, we use samples of events with integrated luminosity of 3 fb 1 and increasing e ective CM energy from 189 to 207 GeV.21 We combine the limits from di erent energy and angular bins and obtain eR; R Goldstini: m+ & 1:8 pg =4 TeV ; m+ & 1:9 pg =4 TeV ; m & 1:4 pg =4 TeV : & 1:5 pg =4 TeV : (5.10) (5.11) The bounds on the scenario where both electron and muon are pseudo-Goldstini is driven by Bhabha scattering, with a SM cross section signi cantly larger than dimuons | the constrains on only Oe are milder m+ & 1:6 pg =4 TeV and m & 1:5 pg =4 TeV. It is amusing to nd that goldstino-compositeness of leptons is allowed at incredibly small scales from direct searches, even for light leptons and maximally strong coupling g = 4 . In fact, precision is relatively less important, compared to the collider energy, when searching for this type of compositeness, as it becomes clear when comparing the reach of LHC vs LEP. In contrast, dimension-6 operators are typically better constrained by LEP (or barely so by the LHC [10]). Indeed, the relative size of our e ects scales as in the linear regime, and as (E=m )8 for large deviations from the SM. In order to increase (E=m ) 4 the bound on the scale m by a factor of 2 we would need to increase the precision at a given energy by at least a factor of 16; the same goal can be achieved by a factor of 2 increase in energy. The LHC high-energy reach makes it the best machine to test for fermions with enhanced soft behavior. Another consequences of this same fact is that the limits on the scale m for chiral-compositeness are much higher, their e ects scaling as (E=m )2. This also means that, for a proper extraction of the bounds, SUSY-breaking e ects which generate four-fermion operators should be included in the analysis [58], even if suppressed by (gSM=g )2. 5.3 Positivity constraints An interesting aspect of the dimension-8 interactions studied in this work is that they are subject to positivity constraints. Indeed, the basic requirement of unitarity in the underlying theory, together with analyticity of the 2 ! 2 scattering amplitudes, implies that the Wilson coe cients of the operators in eqs. (5.1) and (5.8), (5.9), be strictly positive [12, 14]. From a phenomenological perspective, this represents an important prior, from rst principles, to our statistical analysis, that reduces the parameter space by half. Without any prior, our analysis above leads to 95% CL intervals of the form [ c (g ; m ); c+(g ; m+)], while our theory prior implies ]0; c^(g ; m )]. Taking it into ac21Initial-state photon radiation may reduce the CM energy of the dilepton production. In the LEP analyses only events with soft initial-state radiation are retained [57]. count in our statistical analysis, we nd, m & 9:3 pg =4 TeV ; & 1:7 pg =4 TeV : Note that these limits do not improve our knowledge on goldstino-compositeness (compared to eqs. (5.5) and (5.10)). On the contrary, these more conservative bounds highlight the importance of keeping the prior, not to overexclude the physically consistent region of parameter space. Besides, whether c^(g ; m ) < c+(g ; m+) depends on the likelyhood L being symmetric HJEP1(207) or not under re ection of the Wilson coe cients c ! c. Indeed, if it is symmetric then 0:95 = R c1 dc L R c+ dc L 1 R c+ dc L 0 = R01 dc L ; where the last expression de nes the bound with our prior, thus ^c = c+. Under our assumption of symmetric errors, a signi cant asymmetric likelyhood can arise if two conditions are met: an under or upper uctuation in the data is present and the size of the constraints is such that the cross section has a sizable interference term (linear in the Wilson coe cient). In our LHC analysis, departures from the SM are not large (apart from the third bin in gure 3). Most importantly, the experimental resolution, limited by statistical errors at present, is not enough to resolve the SM-BSM interference term in the cross section, which is instead dominated by the quadratic new physics contribution.22 For this reason, positivity constraints do not improve sizably our constraints on m and g . Outlook | Dibosons In the previous sections we have shown that goldstino-compositeness is described by dimension-8 operators, and that currently the LHC is more sensitive to such strongly coupled e ects than to SUSY-breaking e ects, even though the latter give rise to lowestdimensional interactions. If other species in the SM are involved in the strong dynamics, there can also be large e ects in other LHC processes, beyond 2 ! 2 fermion scattering, that are however unique to pseudo-Goldstini. Interestingly, these e ect are characterized by dimension-8 operators as well, as shown in tables 1 and 2. Indeed, if the Higgs is composite in addition to the light quarks, the operator (5.12) (5.13) (5.14) (5.15) (5.16) 1 2F 2 i 1 4F 2 F A F A i + h:c: : modi es the amplitudes for h pair production, as well as WL+WL and ZLZL (L = longitudinal). Similarly, if the transverse (T) polarizations of vectors are strongly interacting, along the lines of ref. [7], the amplitudes for WT+WT , ZT ZT , ZT and are modi ed by 22Notice that this fact is safely compatible with the validity of our EFT expansion, due to the underlying strong coupling, see the discussion below eq. (5.5) and ref. [8]. This is very interesting because, even in a completely model-independent approach, the amplitudes for processes with neutral gauge boson nal states are not modi ed at the level of dimension-6 operators; experimental constraints from ZZ nal states are at present already derived in terms of dimension-8 operators of the form iHyD HD B B . This kind of operators are typically subleading in theories without symmetries and therefore these type of searches have received so far little attention. Pseudo-Goldstini o er instead a context where all dimension-6 operators are naturally suppressed by symmetries and dimension-8 e ects are naturally leading, so that these searches could play the most important ro^le. $ Another reason why the operators we propose in this article are interesting is the following. Currently the entire new physics parametrization of neutral diboson nal states, see ref. [59], only induces nal states with one longitudinal and one transverse vector.23 The corresponding amplitudes decrease at high energy as 1=E compared to the amplitudes for T T or LL nal states, and is therefore typically subdominant. Instead, the e ects we advocate here modify all T T and LL amplitudes (including the one with (+; ) helicity that dominates in the SM) which, beside being more generic, will also be easier to nd. In summary, SM fermions as pseudo-Goldstini provide the rst structurally motivated scenario where processes with neutral gauge boson pair production can be used as valuable BSM search tools. Importantly their parametrization departs from that traditionally adopted in these searches, and involves a richer variety of phenomena with enhanced highenergy behavior and larger SM-BSM interference. We leave this for future work. 6 Phenomenology of the new colored states In section 3, we have found that models with all quarks as pseudo-Goldstini and maximal R-symmetry require the existence of new exotic colored particles and their extra (non-SM fermions and not pseudo-Goldstini) partners. Their quantum numbers under the gauge and avor symmetries U( 1 )Y X 2 = (1; 6; 3; 1) 2 ; Y 2 = (1; 6 ; 3 ; 1) 2 ; 3 3 They have the same quantum numbers as uc and dc except they transform as a 6 of SU(3)C . Their Dirac masses m62=3;1=3 are naturally small, as they break SUSY explicitly, see eq. (4.16). In the following we will always omit the hypercharge of X and Y , being always understood its value from the coupling to the SM quarks. Decays. These sextets can couple to gluons and the SM u or d quarks through the model-dependent SUSY-breaking couplings in eq. (4.17) (recalling that 6 3 = 8 10 and 3 8 = 3 6 15) gs Kib AY i + h:c: ; q = uc; dc (6.3) 23This is due to ref. [59] providing a parametrization for anomalous neutral triple gauge couplings that limit the process to take place through the s-wave, which forbids in this case identical bosons in the nal state; in our case higher waves are allowed and di erent nal states open. where Kib A are the Clebsch-Gordan coe cients linking the anti-sextet to the anti-triplet and the adjoint, normalized such that Kib A j K A b = i ; j Kib AKiA b = 6 : is a SUSY-breaking parameter that depends on assumptions like the degree of compositeness of Y . Despite the interactions (6.3) being suppressed, they represent the main decay mode for the sextet. In particular, they open the following decay channels: with decays widths approximately given by, X; Y ! jj; bj; tj (Y ! qg) s 2 m36 : 4m2 We assume, consistently with being a small symmetry-breaking parameter, that such that the narrow-width approximation applies to the searches described below. Production and direct searches. The sextet can either be singly produced via gq ! Y (see e.g. ref. [60]) or pair produced gg; qq ! XX; Y Y . We assume m62=3 = m61=3 but focus on direct searches of a single sextet avor, coupled to rst generation quarks (for a fully degenerate family spectrum, our analysis can be appropriately rescaled). The relative sensitivity of single production is larger for heavy sextets, although it requires large couplings =m , while double production presents a poorer mass reach, but it is more model-independent given the cross section is xed by QCD. The LO partonic cross section for single production and decay is given by gq!Y !jj = gq!Y BR(Y ! jj) ; gq!Y = 2 s 2 m26): (6.7) where q = u; d. This cross section su ers from signi cant NLO corrections (e.g. from gg ! Y j) that we neglect given the exploratory nature of our analysis. We compare our LO prediction, appropriately convoluted with the PDFs, with experimental searches of singly produced exotic quark-like resonances decaying into dijets [50, 61]. Pair production gives rise to a four-jet signal. We compute the associated cross section at LO with MadGraph [52], requiring four leading jets with pT > 80 GeV and pseudorapidity j j < 1:4. We compare our results with the cross section bounds provided by ref. [62]. We present in gure 4 left panel single- and pair-production cross sections along with their corresponding experimental limits. For single production we take two di erent values of the (inverse) coupling m = = 10; 20 TeV in order to illustrate the variation of the cross section. In the right panel we show the nal bounds in the (m = ; m6) plane. As anticipated, single production dominates the constraints at high masses and large couplings (i.e. small m = ). Note that given the sextet is predicted upon all quarks being pseudoGoldstini (and maximal R-symmetry), for which LHC searches in section 5.1 set a bound (6.4) (6.6) m6, (6.5) single productin 95%CL uper limt pair productin 6 5 4 3 2 1 0 0 10 20 30 40 HJEP1(207) the theoretical cross sections for single-production (blue) with m = = 20 (solid) or m = = 10 (dashed) and pair-production (dot-dashed red). Right: exclusion regions at 95% CL from singleproduction (dark grey; [50] solid line and [61] dashed line) and double-production (light grey; dotdashed line). Also shown are lower bounds (red) on the (inverse) coupling of the sextet, m = 19 TeV (dashed; corresponding to a scale m = 5:4 TeV, the highest energy of our dijet analysis, and a small parameter = 0:3), m = 27 TeV (dot-dashed; same m , = 0:2) and m = 33 TeV (solid; coupling below which pair production dominates the constraints). on the compositeness scale m be too large while keeping & 16:2pg =4 TeV (see table 5), the coupling =m cannot 1. This is exempli ed by the vertical (red) lines in gure 4, which should be understood as upper bounds on the sextet linear coupling, restricting the regime where limits from single production apply. In contrast, pair production sets the robust lower bound m6 & 1:2 TeV. 7 Conclusions and outlook The quest for an answer to the question \what is matter made of ?" has always been at the heart of particle physics research. To keep pursuing this endeavor, the priority of the future high-energy experimental program, both at the High-Luminosity LHC and future colliders, is to understand to what extent those that we call SM particles are indeed elementary entities, or to nd signs of their substructure. Unitarity of the underlying theory implies there are only two relativistic e ective eld theories for fermion compositeness, which can be broadly di erentiated from the point of view of a long-distance observer. The rst such scenario is characterized by a series of e ective four-fermion operators of dimension-6, whose phenomenological implications are well known and studied in the literature. These operators preserve chiral symmetry, providing a structural reason why the new strong interaction associated with compositeness does not percolate to the SM fermion masses and Yukawas. The second scenario has instead a well-de ned limit where the dimension-6 e ects vanish and compositeness is manifested through dimension-8 operators. One certain realization is associated with non-linearly realized supersymmetry; that is, the fermions are composite pseudo-Goldstini. In this article we built the EFT of N Goldstini and discussed how to embed the SM fermions in such a framework. In particular, the SM gauge and avor groups are subgroups of the R-symmetry U(N ). The SM gauge and Yukawa couplings explicitly break supersymmetry, these breakings propagating to other observables | in particular they generate the dimension-6 operators that were forbidden in the exact supersymmetric limit | but can be treated as perturbative and we estimated their sizes. We compared this goldstino-compositeness with the standard chiral-compositeness, confronting them both with experimental data in the form of measurements of dijet distributions at the LHC or e+e scattering at LEP (in addition to other low-energy observables). Wilson coe cients of the standard four-fermion operators scale as g( 6 )=m( 6 ) 2, while the dimension-8 as (g(8))2=(m(8))4, in terms of the physical scales and couplings of the new dynamics. Then, an experiment performing at energy Eexp provides constraints that naively relate as, Eexp (7.1) an intuition that we con rmed with dedicated analyses. Since Eexp=m( 6 ) 1 in a sensible EFT, for g(8) g( 6 ) bounds on goldstino-compositeness are always poorer than those on chiral-compositeness: by a factor of 4 for light quarks (tested at LHC) and by a factor of 20 for light leptons (tested at LEP-2). For pseudo-Goldstini, the onset of the new physics interactions is so dramatic that energy, rather than accuracy, plays the crucial ro^le in experimental searches. As a result, electrons could be pseudo-Goldstini already at scales & 2 TeV, while light quarks require scales & 10 TeV | from a phenomenological point of view goldstino-compositeness is therefore on a better footing than chiral-compositeness, which requires scales & 40 TeV. The scenarios with SM fermions as pseudo-Goldstini bring new interesting questions for future research. From an experimental point of view, these scenarios provide novel and alternative benchmarks to compare the performance of future colliders (such as FCCee, ILC or CLIC in e+e scattering), in which the importance of energy over accuracy is additionally emphasized. Moreover, if other SM species are also composite, novel e ects can be expected in neutral diboson pair production, which we discussed in section 5.4. These probes are very clean experimentally but they have always su ered from the lack of concrete BSM scenarios that would make them interesting: goldstino-compositeness provides a plethora of new e ects that can be searched for in this type of processes (e ects that, even if of the same EFT order as the ones studied so far, are complementary, better measurable and better motivated). From a theoretical point of view goldstini-compositeness relies on the existence of approximate and emergent supersymmetries. While proof-of-principle examples of this possibly exist, it would be interesting to set this on a rmer ground. In this roadmap, the incorporation of a solution to the hierarchy problem through extended suspersymmetry would be an important additional target. HJEP1(207) Acknowledgments We are happy to acknowledge important conversations with Zohar Komargodski, Alexander Monin, Alex Pomarol, Riccardo Rattazzi and Riccardo Torre. We thank Alexander Monin for collaboration in the early stages of this work and Simone Alioli for discussions and for providing us help with POWHEG. B.B. thanks Marco Cirelli and the LPTHE for the kind hospitality during the completion of this work. B.B. is supported in part by the MIURFIRB grant RBFR12H1MW \A New Strong Force, the origin of masses and the LHC". A Dileptons at LEP The set of dimension-8 operators from goldstino-compositeness of leptons relevant at LEP is eL)( L R) + h:c: L) + h:c: ; O`ee O`e O` e eR) + h:c: R) + h:c: eR) + h:c: (A.1) (A.2) (geZR )2cee +(geZL )2c`e`e O`e`e O`e` where c` e = c`e = c` e. d d cos d d cos 16 s u 2 32 s ! e+e )BSM t 2 s e2(cee +c`e`e )+ t s s 2 1 m2Z m2Z + t geZR geZL c`ee 1 m2Z u2(ce2e + c`2e`e ) + 2(t2 + s2)c`2ee ; 1 e2(ce + c`e` ) + s 1 m2Z (geZR )2ce + (geZL )2c`e` 1 m2Z geZR geZL (c`e + c` e) 32 s (ce2 + c`2e` + c`2e + c`2 e) ; = ; and in each particular scenario cij = 1=2F 2. 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Brando Bellazzini, Francesco Riva, Javi Serra, Francesco Sgarlata. The other effective fermion compositeness, Journal of High Energy Physics, 2017, 20, DOI: 10.1007/JHEP11(2017)020